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\begin{enumerate}gin{document}
\title{Analytic classification of a class of cuspidal foliations in
$({\mathbb C}^3,0)$}
\author{Percy Fern\'{a}ndez-S\'{a}nchez}
\address[Percy Fern\'{a}ndez]{Dpto. Ciencias - Secci\'{o}n Matem\'{a}ticas, Pontificia Universidad Cat\'{o}lica del Per\'{u}, Av. Universitaria 1801,
San Miguel, Lima 32, Peru}
\epsilonmail{[email protected]}
\author{Jorge Mozo-Fern\'{a}ndez}
\address[Jorge Mozo Fern\'{a}ndez]{Dpto. \'{A}lgebra, An\'{a}lisis Matem\'{a}tico, Geometr\'{\i}a y Topolog\'{\i}a \\
Facultad de Ciencias, Universidad de Valladolid \\
Campus Miguel Delibes\\
Paseo de Bel\'{e}n, 7\\
47011 Valladolid - Spain}
\epsilonmail{[email protected]}
\author{Hern\'{a}n Neciosup}
\address[Hern\'{a}n Neciosup]{Dpto. Ciencias - Secci\'{o}n Matem\'{a}ticas, Pontificia Universidad Cat\'{o}lica del Per\'{u}, Av. Universitaria 1801,
San Miguel, Lima 32, Peru}
\epsilonmail{[email protected]}
\thanks{First and third authors partially supported by the Pontificia Universidad Cat\'{o}lica del Per\'{u} project VRI-DGI 2014-0025. \\ Second author partially supported by the Ministerio de Econom\'{\i}a y Competitividad from Spain, under the Project ``\'{A}lgebra y Geometr\'{\i}a en Din\'{a}mica Real y Compleja III" (Ref.: MTM2013-46337-C2-1-P)}
\date{\today}
\begin{enumerate}gin{abstract}
In this article we study the analytic classification of certain types of quasi-homogeneous cuspidal holomorphic foliations in $({\mathbb C}^3,{\bf 0})$ via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation.
\epsilonnd{abstract}
\maketitle
\section{Introduction} \label{introduccion}
The objective of this paper is to give a contribution to the analytical classification of germs of codimension one holomorphic foliations defined on an ambient space of dimension three. More generally, assume that the dimension of the ambient space is arbitrary, call it $n$.
Let ${\mathcal F}_1, {\mathcal F}_2$ be two such germs, generated by integrable 1-forms $\omegaega_1, \omegaega_2$, respectively, holomorphic in a neighbourhood of ${\mathbf 0}\in {\mathbb C}^n$. We will say that ${\mathcal F}_1, {\mathcal F}_2$ are analytically equivalent if there exists a germ of diffeomorphism $\phi:({\mathbb C}^n, {\bf 0})\to ({\mathbb C}^n, {\bf0})$ such that $\phi^*\omegaega_1\wedge\omegaega_2=0$.
Let us mention here some of the previous results concerning this subject. When $n=2$, J. Martinet and J.-P. Ramis \cite{MR1,MR2} consider simple (reduced) singularities, both in the saddle-node case and in the resonant case. Concerning non-simple singularities, consider the nilpotent case (i.e. foliations defined by a 1-form $\omegaega$ such that its dual vector field has nilpotent, non-zero, linear part), in which these foliations admit a formal normal form $d(y^2+x^n)+x^p.u(x)dy$, where $n\geq3, p\geq2$ \cite{Takens}, and $u(0)\neq0$, and in fact an analytic pre-normal form $\omegaega=d(y^2+x^n)+A(x,y)(nydx-2xdy)$ (see \cite{Cerveau-Moussu}). These foliations have been the object of study of R. Moussu \cite{R.Moussu2}, D. Cerveau \cite{Cerveau-Moussu}, R. Meziani \cite{Meziani}, M. Berthier, P. Sad \cite{BMS} and E. Str\'{o}\.{z}yna \cite{Strozyna}. It is worth to mention here other previous works that contribute to the analytic classification of holomorphic foliations, as the ones by J.-F. Mattei \cite{Mattei-modules,Mattei} and Y. Genzmer \cite{Genzmer}. Other authors focus in the topological classification, that we will not treat in this paper. Among them, let us mention, without trying to be exhaustive, works of D. Mar\'{\i}n and J.-F. Mattei \cite{MarinMattei}, and of R. Rosas \cite{Rosas}.
In dimension $n=3$, simple singularities have been studied by D. Cerveau and J. Mozo in \cite{Cerveau-Mozo}. Concerning non-simple singularities, P. Fern\'{a}ndez and J. Mozo studied in \cite{FM} the case of quasi-ordinary, cuspidal singularities. In these works, the main tool used is, either the holonomy of one of the separatrices of the foliation or, more frequently, the projective holonomy of a certain component of the exceptional divisor that appears after the reduction of the singularities. In order to apply this technique, specially in dimension higher than two, it is of great importance to study the topology of the divisor, as the holonomy group of each component is a representation of its fundamental group.
In this paper, we will apply this technique to a kind of cuspidal foliations more general than the quasi-ordinary ones studied in \cite{FM}, more precisely, admissible quasi-homogeneous foliation (see Definition \ref{def_quasi_homogeneous} in Section \ref{general}) of generalized surface type. Recall, from \cite{FM2} and \cite{FMNeciosup1}, that a germ ${\mathcal F}$ of non dicritical foliation in $({\mathbb C}^3,{\bf 0})$, generated by an integrable 1-form $\omegaega$, is a generalized surface if every generically transverse plane section $\varphi:({\mathbb C}^2,{\bf 0})\to ({\mathbb C}^3,{\bf 0})$ (i.e., $\varphi^{*}\omegaega\not\epsilonquiv0 $) defines a generalized curve in the sense of \cite{CLS}. In particular, for generalized surfaces, once its set of separatrices has been reduced, the foliation has only simple singularities, as shown in \cite{FM2} and generalized in \cite{FMNeciosup1} for higher dimensions.
In Section \ref{general}, a cuspidal foliation in $({\mathbb C}^3,{\bf 0})$ will be defined as a generalized surface ${\mathcal F}$ having a separatrix written as $z^2+\varphi(x,y)=0$, in appropriate coordinates. We will consider in this paper the situation in which the separatrix is quasi-homogeneous. The first point would be to find an analytic normal form for these type of foliations, in the style of the ones considered by D. Cerveau and R. Moussu in dimension two. This is the main problem studied in \cite{FMNeciosup2}. In that paper, it is shown that a cuspidal, quasi-homogeneous foliation ${\mathcal F}$ with a separatrix $z^2+\varphi(x,y)=0$, can be generated by an integrable 1-form
$$\omegaega=d(z^2+\varphi)+G(\Psi,z)\cdot z\cdot\Psi\cdot\left(2{dz\over z}-{d\varphi\over\varphi}\right),$$
where $\Psi^r=\varphi, \Psi$ is not a power, and $G$ is a germ of holomorphic function in two variables. This is the starting point of the present paper.
Also in Section \ref{general}, the main definitions of the paper are presented. We will work with a concrete reduction of singularities of cuspidal quasi-homogeneous foliations, obtained via the Weierstrass-Jung method. For the sake of completeness, this reduction will be sketched in Section \ref{section3}. In Section \ref{section_toplogy_of_the_divisor}, the topology of the components of the exceptional divisor is studied. As we said before, this is important in order to identify the component of the exceptional divisor where the holonomy is computed. In particular there is one component such that, once removed the singular locus, it is homeomorphic to $({\mathbb C}^*\times{\mathbb C} )\smallsetminus\mathcal{C}$, where $\mathcal{\mathcal{C}}$ is a certain algebraic plane curve, whose topology will be interesting for us. The fundamental group will be computed in this section.
In order to lift the holonomy of a component of the divisor, a fibration transverse to the foliation outside the separatrices, is constructed in Section \ref{section_holonomy_of_the_essential_component}. Finally, Section \ref{section_cladification}, is devoted to study the main problem of the paper: to classify analytically quasi-homogeneous generalized surfaces following this method. Unfortunately, the answer is not complete, and we must impose some conditions (properties $\wp_1, \wp_2$ of Definition \ref{defini_propiedades_p_1andp_2}), that guarantee the linearization of the holonomy over some ``end component'' of the divisor, that we will call \textit{special components}. Under these conditions, Theorem \ref{thm_clasificacion_analitica} of this section is the main result of the paper, and gives the analytic classification.
As a future work, we would like to study a wider class of foliations. The existing works of Mattei and Genzmer \cite{Mattei-modules, Mattei,Genzmer} in dimension two lead us to think that different techniques should be developed, as for instance, techniques concerning deformations of foliations. We will not follow this method here.
\section{Generalities} \label{general}
In dimension $n\geq2$, a germ of codimension one holomorphic foliation ${\mathcal F}$ is called \textit{cuspidal} if, besides being generalized hypersurface (see Section \ref{introduccion} and \cite{FM2}), its separatrix is defined in appropriate coordinates $(\mathbf{x}, z)$, where $\textbf{x}:=(x_1,x_2,\cdots,x_{n-1})$, by a reduced equation $z^2+\varphi(\textbf{x})=0$, with $\nu_0 (\varphi (\textbf{x}))\geq 2$. In \cite[Thm. 1.4]{FMNeciosup2} we constructed a pre-normal form for germs of holomorphic foliation of cuspidal type. In particular, if $z^2+\varphi(\textbf{x})=0$ is a quasi-homogeneous hypersurface, we know that there exist analytic coordinates such that a generator of ${\mathcal F}$ is the 1-form
$$
\omega = d(z^2+\varphi) + G(\Psi, z) (z\cdot \Psi)\cdot \left(
2\frac{dz}{z}-\frac{d\varphi}{\varphi}\right) ,
$$
where $\varphi= \Psi^r$ for some $r\in{\mathbb N},\;\Psi$ is not a power, and $G$ is a germ of holomorphic function in two variables (\cite[Cor. 4.2]{FMNeciosup2}). \\
In dimension three, there exist analytic coordinates $(x,y,z)$ such that
$$\Psi=x^{n_1}y^{n_2}\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i},$$
where $n_i,p,q\in\mathbb{N}$, and $a_i\in {\mathbb C}^*$ are pairwise distinct. After \cite[Thm. 2.4]{FMNeciosup2} and \cite[Thm. 11]{FM2}, the reduction of the singularities of this type of foliations follows the same scheme that the desingularization of their separatrix $S$,
$$S:=z^2+x^{n_1}y^{n_2}\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i}=0.$$
A key to prove the main result of this paper (Theorem \ref{thm_clasificacion_analitica}) is the existence of the first integral in the intersections of the components of the exceptional divisor.
Let us mention in dimension two the work \cite{D.Marin-Thesis}, where it is said that the classification technique from R. Moussu \cite{R.Moussu2} and F. Loray \cite{Loray-Thesis} can be used in the quasi-homogeneous framework, \textit{provided that the coordinate axis are not part of this set of separatrices}. i.e., after reduction of singularities, all the separatrices lie on the same component of the exceptional divisor. Because of this, we will assume an analogous hypothesis in dimension three, i.e., $n_1=n_2=0$,
and Theorem \ref{thm_clasificacion_analitica} will be proved based on this property. This means that coordinate hyperplanes are not part of its set of separatrices. Let us mention that on the other hand, in \cite{Genzmer} this hypothesis is not considered, but other techniques of a rather different nature have been used, which will not be addressed here. According to this, let us state the following definition:
\begin{enumerate}gin{defin}\label{def_quasi_homogeneous}
A cuspidal holomorphic foliation, ${\mathcal F}$, in $({\mathbb C}^3,{\bf 0})$ is called {\bf quasi-homogeneous of admissible type}, if its separatrix, in appropriate coordinates, is defined by the equation
$$S=z^2+\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i}=0,$$
where $p,q\geq2,\;a_i\in{\mathbb C}^*$, and $a_i\neq a_j,$ if $i\neq j$.
\epsilonnd{defin}
So, a quasi-homogeneous foliations of admissible type is generated by a 1-form
\begin{enumerate}gin{equation}\label{forma_quasi_homogeneous}
\Omegaega=d(z^2+\varphi)+G(\Psi,z)\cdot z\cdot\Psi\left({d\varphi\over\varphi}-2{dz\over z}\right),
\epsilonnd{equation}
where $\varphi=\Psi^r=\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i},\;p,q\geq2,\; r=\gcd (d_1,\cdots,d_l)$.
In general, let us denote by ${\mathcal F}_{\omegaega}$ a holomorphic foliation of codimension one generated by an integrable 1-form $\omegaega\in\Omegaega^1({\mathbb C}^n,{\bf 0})$.
Not every such 1-form defines a quasi-homogeneous foliation of admissible type, as we have imposed to the definition the condition of being a generalized surface. In general, such a foliation may have more separatrices, or be dicritical. In \cite[Thm. 5.2]{FMNeciosup2} a sufficient condition is stated for being a generalized surface. More precisely, if we write $G(\Psi,z)=\displaystyleplaystyle\sum_{\alpha,\begin{enumerate}ta}G_{\alpha,\begin{enumerate}ta}\Psi^{\alpha}z^{\begin{enumerate}ta}$, and
$$\nu_{2,r}(G):=\min\left\{{2\alpha+r\begin{enumerate}ta\over \gcd(2,r)}; G_{\alpha,\begin{enumerate}ta}\neq0\right\},$$
then, if $\nu_{2,r}(G)\geq{r-2\over \gcd(2,r)}$, the foliation is a generalized surface.
On the other hand, let us observe that if $d_i=1$ for all $i$, $\varphi$ is reduced and the singular locus of ${\mathcal F}_{\Omegaega}$ is the origin of coordinates. By Frobenius singular Theorem \cite{Malgrange}, ${\mathcal F}_{\Omegaega}$ admits holomorphic first integral and the study of these foliations is then equivalent to that of the surfaces. We will suppose, at any time, that $d_i>1$ for some $i$, and we will denote $\Sigma_{p,q}^{(d_1,\cdots,d_l)}$ the set of integrable 1-forms analytically equivalent to a 1-form as in (\ref{forma_quasi_homogeneous}), with $d_i>1$ for some $i$.
\section{Desingularization of quasi homogeneous foliations}\label{section3}\label{Desingularization}
Let us consider a quasi-homogeneous foliation of admissible type, ${\mathcal F}_{\Omegaega}$, generated by an integrable 1-form $\Omegaega\in\displaystyleplaystyle \Sigma_{p,q}^{d_1,\cdots,d_l}$, according with the notations of Section \ref{general}. The reduction of the singularities of ${\mathcal F}_{\Omegaega}$ is achieved after the reduction of its separatrices. We use in this paper a precise reduction, namely the one obtained by Weierstrass-Jung method, that we shall sketch here for the sake of completeness. This precise reduction will be useful in the sequel in order to extend the conjugation of the holonomy to a whole neighbourhood of the exceptional divisor.
This reduction will be described in three steps:
{\bf Step I}: The separatrix $S$ being defined by $z^2+\varphi(x,y)=z^2+\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i}=0$, we shall proceed to desingularize first the curve $\varphi(x,y)=0$. This is done algorithmically taking into account its characteristic exponents, more precisely, the continuous fraction expansion of $\displaystyleplaystyle{p\over q}=[c_0;c_1,\cdots,c_N]$. It is necessary to do $k=\displaystyleplaystyle\sum_{\nu
=0}^{N}c_{\nu}$ quadratic transformations (point blow-ups), whose composition will be denoted $\pi_I:$
$$\pi_I:(M_I,E_I)\to({\mathbb C}^3,{\bf 0}).$$
Denote $\widetilde{\Omegaega}$ the strict transform of $\Omegaega$. Locally, in appropriate coordinates, it is given as
$$\widetilde{\Omegaega}=\displaystyleplaystyle(z^2+x^ay^bU_{\alpha})\omegaega_{\alpha}+xy\epsilonta_{\alpha},$$ where $a,b$ are natural numbers, depending on $p,q$ and the chosen chart, $\omegaega_{\alpha}=mydx+nxdy$ is a linear form, with $m$, $n\in {\mathbb N}$, $U_{\alpha}$ is a holomorphic function, that in the ``most interesting'' chart obtained after the last blow-up is
$\displaystyleplaystyle U_{\alpha}=h^r=\left(\prod_{i=1}^l(y^{\delta}-a_i)^{d_i'}\right)^r$, where $r=\gcd(d_1,\cdots,d_l)$, $d_i'=d_i/r$, $\delta=\gcd (p,q)$ and $\displaystyleplaystyle\epsilonta_{\alpha}=d\left(z^2+x^ay^bU_{\alpha}\right)+\Delta_{\alpha}\cdot \left(a{dx\over x}+b{dy\over y}+{dU_{\alpha}\over U_{\alpha}}-2{dz\over z}\right)$. Here $\Delta_{\alpha}$ is a certain germ of holomorphic function that we will not precise here.
{\bf Step II:} Consider the foliation defined by $\widetilde{\Omegaega}$. We will blow-up certain curves biholomorphic to $\mathbb{P}_{{\mathbb C}}^1$. This process will depend of the nature of integers $a$, $b$, and follows the scheme of the quasi-ordinary case studied in \cite{FM}. More precisely:
{\bf II.a) If $d$ is even,} $a, b$ are also even. Blow-up $(z=y=0)\:\; {a\over 2}$ times and $(z=x=0)\:\; {b\over 2}$ times. The final result is schematized in Figure \ref{figura2}.
\begin{enumerate}gin{center}
\includegraphics[width=11.5cm]{Caso-i-final-segunda-etapa}\\
\figcaption{$\widetilde{S}$, strict transform by $\pi_{_{II}}$ of $S$. }\label{figura2}
\epsilonnd{center}
{\bf II.b) If $d$ is odd, $p$ even and $q$ odd}, a certain number of suitable monoidal transformations, depending on the chart, is necessary to perform. The final result is schematized in Figure \ref{figuraCasoii_1}.
\begin{enumerate}gin{center}
\includegraphics[width=10.5cm]{Caso-ii1-final-segunda-etapa}\\
\figcaption{$\widetilde{S}$, strict transform by $\pi_{II}$ of $S$ (Case II.b)}\label{figuraCasoii_1}
\epsilonnd{center}
{\bf II.c) If $d, a$ and $b$ are odd}, after the sequence of monoidal transformations, a final quadratic transformation will be necessary to do, as in the quasi-ordinary case of \cite{FM}. Figure \ref{figure_trans_monoidal_caso_impar} represents the final result in an appropriate chart.
\begin{enumerate}gin{center}
\includegraphics[width=11cm,height=3.5cm]{impar-ultima-componente}\\
\figcaption{$\widetilde{S}$, strict transform by $\pi_{II}$ of $S$ (Case II.c)}\label{figure_trans_monoidal_caso_impar}
\epsilonnd{center}
We shall denote $\pi_{II}$ the composition of the transformations done in this step.
{\bf Step III}. At the end of Step II, there exist analytic coordinates in which the local equation of the strict transform of the foliation is
$$\Omegaega_{PQ}=(t^2+h^r)\omegaega_{PQ}+xy\epsilonta_{PQ},$$
where
$$\begin{enumerate}gin{array}{rcl}
\omegaega_{_{PQ}} & = & ({pq\over\delta}d)ydx+(nqd)xdy\\
\epsilonta_{_{PQ}} & = & d(t^2+h^r)+x^{{pq\over \delta}d'(1-{r\over2})}y^{nqd'(1-{r\over 2})}th.G_{1PQ}\Bigg(r{dh\over h}-2{dt\over t}\Bigg)\\
G_{1PQ} & = & G\Big(x^{{pq\over\delta}d'}y^{nqd'}h,x^{{pq\over
\delta}{d\over2}}y^{nq{d\over2}}t\Big).\\
P & = & {pq\over\delta}d-2\left({p+q\over\delta}-1\right).\\
Q & = & nqd-(m+n-1).
\epsilonnd{array}$$
It is necessary to blow-up the lines $z=0, y=a_i^{1\over \delta}$, according to the nature of each $d_i$. The components of the divisor are, topologically, $\mathcal{A}_j\times \mathbb{D}$, where $\mathcal{A}_j\approx\mathbb{P}_{{\mathbb C}}^1$ and $\mathbb{D}$ a disc around the origin.
Denote $\widetilde{D}$ the last component generate in Step II, i.e., the main component where the separatrix cuts the exceptional divisor. This component will be called {\it essential component}.
Note that the modifications done in this step do not alter the topology of this component. This will be precised in Section \ref{section_toplogy_of_the_divisor}.
Analogously as in the previous steps, $\pi_{III}$ will denote the composition of the transformations done here.
\section{Topology of the divisor}\label{section_toplogy_of_the_divisor}
Let $\widetilde{{\mathcal F}}$ be the strict transform of a quasi-homogeneous holomorphic foliation of admissible type via the morphism $\pi:(M,E)\to({\mathbb C}^3,{\bf0})$ ($\pi=\pi_{_{I}}\circ\pi_{_{II}}\circ\pi_{_{III}}$), as described in Section \ref{section3}.
With the conditions imposed, the singular locus $\mathcal{S}:=\Sing(\widetilde{{\mathcal F}})$, is an analytic space of dimension $1$ with normal crossings. $\mathcal{S}$ is given by the intersection of the components of the divisor $E$, along with the intersection of the strict transform of $S$ with the divisor. These components are denoted as $D_{\alpha},\; D_{\alpha j},\; \mathcal{A}_j\times \mathbb{D}$ (see Section \ref{section3} for notations).
It will be useful, in order to study the projective holonomy of the foliation, to know their homotopy type, once we have removed the singular locus. Let us describe it briefly in this section.
In the first step of the reduction we have produced several components $D_{\alpha}$. After removing the singular locus they are homotopically equivalent to:
$$D_{\alpha}\smallsetminus\mathcal{S}\approx
\left\{
\begin{enumerate}gin{array}{ll}
{\mathbb C}\times{\mathbb C}, & \hbox{if}\; \alpha=1; \\\\
{\mathbb C}\times{\mathbb C}^*, & \hbox{if}\;1<\alpha\leq c_0+1; \\\\
{\mathbb C}^*\times{\mathbb C}^*, & \hbox{ in other cases.}
\epsilonnd{array}
\right.
$$
The components $D_{\alpha j}$ appearing in the second step, removing the singular locus, are homotopically equivalent to ${\mathbb C}\times{\mathbb C}^*$, to ${\mathbb C}^*\times{\mathbb C}^*,$ to ${\mathbb C}\times({\mathbb C}\smallsetminus\{\text{2 points}\})$, or, in the most interesting case, to $({\mathbb C}^*\times{\mathbb C})\smallsetminus\mathcal{C}$, where, in coordinates $(y,t), \mathcal{C}$ is the affine curve defined by $t^2-y^{a}.v=0$, where $v$ is a unit, or
$$t^2-\displaystyleplaystyle\left(\prod_{i=1}^{l}(y^{\delta}-a_i)^{d'_i}\right)^{r}=0,$$
with previous notations.
Finally, Step III produces new components homotopically equivalent to ${\mathbb C}\times{\mathbb C}$, ${\mathbb C}^*\times{\mathbb C}$ or $({\mathbb C}\smallsetminus\{\text{2 points}\} )\times{\mathbb C}$, that are not relevant for our study. However, we highlight the component $D_{\alpha j}$ (generated in the second step) which is modified now to $\widetilde{D}$ (essential component defined in Section \ref{Desingularization}), but whose topology does not change in this step and consequently $\widetilde{D}\smallsetminus\Sing(\widetilde{{\mathcal F}}_{\Omegaega})\thickapprox ({\mathbb C}^*\times{\mathbb C} )\smallsetminus\mathcal{C}$.
The fundamental group of $({\mathbb C}^*\times{\mathbb C})\smallsetminus\mathcal{C}$ can be computed using the Zariski-Van Kampen method, as described, for instance, in \cite{ACT} (see also \cite{VK} as a classical reference). Even if it is standard enough, for the sake of completeness we shall briefly review here this method. Consider the projection $\rho:{\mathbb C}^2\to{\mathbb C},\;\rho(x,t)=x$.
Let us denote by $\Delta:=\{a_1,a_2,\cdots,a_l\}$ and $\mathcal{L}:=\displaystyleplaystyle\begin{itemize}gcup_j\rho^{-1}(a_j)$. The restriction \begin{enumerate}gin{equation}\label{fibracion_localmente_trivial}
\rho:{\mathbb C}^2\smallsetminus\mathcal{C}\cup\mathcal{L}\to{\mathbb C}\smallsetminus\Delta,
\epsilonnd{equation}
is a locally trivial fibration, with fibres isomorphic to ${\mathbb C}\smallsetminus\{\text{2 points}\}$. So, for every point in ${\mathbb C}\smallsetminus\Delta$, we obtain the exact sequence
$$
\xymatrix{1\ar[r] & \pi_1\left({\mathbb C}\smallsetminus\{\text{2 points}\}\right)\ar[r]^-{i_{*}} & \pi_1({\mathbb C}^2\smallsetminus \mathcal{C}\cup\mathcal{L}) \ar[r]^-{\rho_{*}}& \ar@/_2pc/[l]^-{s_{*}}\pi_1({\mathbb C}\smallsetminus\Delta)\ar[r]& 1},
$$
where $i^*$ and $s^*$ are map induced in homotopy from inclusion and section of $\rho$ respectively.
Let $\mathbb{D}\subset {\mathbb C}$ be a sufficiently big closed disk, such that $\Delta$ is contained in its interior. Choose a point $\star$ on $\partial \mathbb{D}$. Take a small disk $\mathbb{D}_j$ centered at ${a_j}\in\Delta$ containing no other elements of $\Delta$ and choose a point $R\in\partial\mathbb{D}_j$. Consider a path $\alpha\in{\mathbb C}\smallsetminus\Delta$ joining $R$ and $\star$, and denote by $\epsilonta_{R,\mathbb{D}_j}$ the closed path based at $R$ that runs counterclockwise along $\partial\mathbb{D}_j$. The homotopy class of the loop $\gamma_j:=\alpha^{-1}.\epsilonta_{R,\mathbb{D}_j}.\alpha$ is called a \textit{meridian} of $a_j$ in ${\mathbb C}\smallsetminus\Delta$. Then, the collections of meridians in ${\mathbb C}\smallsetminus\Delta$ (one for each point of $\Delta$) define bases of $\pi_1({\mathbb C}\smallsetminus\Delta,\star).$ Similarly we can choose meridians $g_1,g_2$ in a fiber of the restriction (\ref{fibracion_localmente_trivial}) and $\gamma$ a meridian around the straight line $x=0$. It follows that
$$
\pi_1( ({\mathbb C}^*\times {\mathbb C}) \smallsetminus \mathcal{C})=\Big\langle g_1,g_2,\gamma;\ g_i^{\sigma^r}=g_i\;\; \wedge \;\; g_i^{\sigma^b}=\gamma^{-1}g_i\gamma\Big\rangle,
$$
where $g_i^{\sigma^r}$ is the action on $g_i$ of $\gamma_j$ (known as the factorization of the braid monodromy, see \cite{ACT}).
The expression of the group can be simplified obtaining:
$$\pi_1(({\mathbb C}^*\times {\mathbb C} ) \smallsetminus \mathcal{C})=
\left\{
\begin{enumerate}gin{array}{ll}
\Big\langle \alpha, \begin{enumerate}ta,\gamma: \begin{enumerate}ta\alpha^r=\alpha^r\begin{enumerate}ta\; \wedge\; \gamma\alpha=\alpha\gamma \Big\rangle, & \hbox{if $r=2m+1$ is odd;} \\\\
\Big\langle \alpha, \begin{enumerate}ta,\gamma: \alpha^r=\begin{enumerate}ta^2\; \wedge\; \gamma\alpha=\alpha\gamma \Big\rangle, & \hbox{if $r=2m$ is even.}
\epsilonnd{array}
\right.
$$
where $\alpha:=g_2g_1$ and $\begin{enumerate}ta:=(g_2g_1)^mg_2$.
\section{Holonomy of the essential component}\label{section_holonomy_of_the_essential_component}
The dynamical behavior of one foliation may be studied in a neighbourhood of a leaf by the representation of its fundamental group, as introduced by C. Ehresmann in 1950. In this study we will mainly follow J.-F. Mattei and R. Moussu \cite{Mattei-Moussu}: let us choose a point ${\bf p}$ of a leaf $L$ and a germ of a transversal section $\Sigma$ in ${\bf p}$. The lifting of a closed path $\gamma$ starting in ${\bf p}$, following the leaves of the foliation, induces germs of diffeomorphisms
$$h_{\gamma}:(\Sigma,{\bf p})\to (\Sigma,{\bf p}),$$
that only depend on the homotopy class of the path. The map $h_{\gamma}$ is called {\it holonomy } of the leaf $L$. The representation of the holonomy of $\pi_1(L,{\bf p})$ is the morphism defined by
$$\begin{enumerate}gin{array}{cccc}
Hol(L)&:\pi_1(L,{\bf p})&\to & Diff(\Sigma,{\bf p})\\
& \gamma &\longmapsto & h_{\gamma},
\epsilonnd{array}$$
and the {\it holonomy group} of the foliation along $L$ is the image of this application $Hol(L)$. Different points in the leaf and different transversal sections define conjugated representations.
In order to lift the path $\gamma$ it is necessary to have a fibration, that is transverse to the foliation. Let us describe it. In general let $\Omegaega$ be an integrable 1-form in $({\mathbb C}^3,{\bf 0})$, $\pi:(M,E)\to ({\mathbb C}^3,{\bf 0})$ a minimal reduction of singularities of the foliation ${\mathcal F}_{\Omegaega}$. Let $\widetilde{{\mathcal F}}_{\Omegaega}$ be the strict transform of ${\mathcal F}_{\Omegaega}$ under $\pi$ and $D$ a component of the exceptional divisor $E$.
\begin{enumerate}gin{defin}\label{Def_Hopf_fibration}
A {\bf Hopf fibration adapted to ${\mathcal F}_{\Omegaega}$}, $\mathcal{H}_{{\mathcal F}_{\Omegaega}}$, is a holomorphic fibration $f:M\to D$ transverse to the foliation ${\mathcal F}_{\Omegaega}$, i.e.,
\begin{enumerate}gin{enumerate}
\item $f$ is a submersion and $f|_{D}=Id_{D}.$
\item The fibres $f^{-1}(p)$ of $\mathcal{H}_{{\mathcal F}_{\Omegaega}}$ are contained in the separatrices of $\widetilde{{\mathcal F}}_{\Omegaega}$, for all $p\in D\cap\Sing(\widetilde{{\mathcal F}}_{\Omegaega})$.
\item The fibres $f^{-1}(p)$ of $\mathcal{H}_{{\mathcal F}_{\Omegaega}}$ are transverse to the foliation ${\mathcal F}_{\Omegaega}$, for all $p\in D\smallsetminus\Sing(\widetilde{{\mathcal F}}_{\Omegaega})$.
\epsilonnd{enumerate}
\epsilonnd{defin}
Consider now the particular case $\Omegaega\in\Sigma_{p,q}^{(d_1,\cdots,d_l)}$. The vector field
$$\mathcal{X}=px{\partial\over\partial x}+qy{\partial\over \partial y} + {pqd\over 2}z{\partial\over\partial z}. $$
verifies $\Omegaega(X)=pqd(z^2+\varphi)$, so, it is transverse to ${\mathcal F}_{\Omegaega}$ outside the separatrix $S$, and $S$ is a union of trajectories of $\mathcal{X}$ (equivalently, $S$ is invariant by $\mathcal{X}$). The trajectories $\mathcal{X}$ give the fibres of the Hopf fibration adapted to $\Omegaega$.
Let $\widetilde{D}$ be the essential component of the reduction of singularities of ${\mathcal F}_{\Omegaega}$, as defined in Section \ref{Desingularization}.
\begin{enumerate}gin{defin}
The \textbf{exceptional holonomy group} is the group of holonomy of $\widetilde{D}$. Let us denote it $H_{\Omegaega,\widetilde{D}}$.
\epsilonnd{defin}
As a consequence of the results of Section \ref{section_toplogy_of_the_divisor}, $H_{\Omegaega,\widetilde{D}}$, is generated by elements $h_{\alpha}, h_{\begin{enumerate}ta}, h_{\gamma}$, which are the holonomy diffeomorphisms of, respectively, paths $\alpha, \begin{enumerate}ta, \gamma$.
Consider, now, two such foliations ${\mathcal F}_{\Omegaega_1}, {\mathcal F}_{\Omegaega_2}$, where $\Omegaega_1,\Omegaega_2\in\Sigma_{p,q}^{(d_1,\cdots,d_l)}$, with respective exceptional holonomy groups
$$H_{\Omegaega_1,\widetilde{D}}=\langle h_{\alpha}^{1},h_{\begin{enumerate}ta}^{1},h_{\gamma}^{1}\rangle, H_{\Omegaega_2,\widetilde{D}}=\langle h_{\alpha}^{2},h_{\begin{enumerate}ta}^{2},h_{\gamma}^{2}\rangle$$
represented on a the transverse section. Suppose they are analytically conjugated by an element $\psi\in Diff({\mathbb C},0)$ such that
$$\psi^*(h_{\alpha_i}^1):=\psi^{-1}h_{\alpha_i}^1\psi=h^2_{\alpha_i};\; \alpha_i\in\{\alpha,\begin{enumerate}ta, \gamma\}.$$
Then we have the following result.
\begin{enumerate}gin{lema}\label{lema-extension-de-holonomia--}
There exist a fibered diffeomorphism $\phi, \phi(x,{\bf p})=(\varphi(x,{\bf p}),{\bf p})$ between two open neighbourhoods $V_j$ of $\widetilde{D}$ in the space $(M,E)$ such that
\begin{enumerate}gin{enumerate}
\item $\phi$ sends the leaves of the foliation ${\widetilde{{\mathcal F}}_{\Omegaega_1}}{|_{V_1}}$ to the leaves of the foliation ${\widetilde{{\mathcal F}}_{\Omegaega_2}}{|_{V_2}}$.
\item The restriction of $\phi$ to the transverse section $\Sigma$ is $\psi$.
\epsilonnd{enumerate}
\epsilonnd{lema}
\begin{enumerate}gin{proof}
Let ${\bf p}\in \mathcal{L}=\widetilde{D}\smallsetminus \mathcal{S}$ and $\gamma_{\bf p}\subset \mathcal{L}$ be a path from ${\bf p}$ to ${\bf p}_j$. We have that the foliation ${\mathcal F}_{\Omegaega_j}$ is generically transverse to the Hopf fibration relative to $\widetilde{D}$, $\mathcal{H}_{{\mathcal F}_{\Omegaega_j}}$, outside the strict transform of the separatrix $\widetilde{S}$ and of the components $D_{\alpha j}, \mathcal{A}_j$, such that $\widetilde{D}\cap D_{\alpha j}\neq\epsilonmptyset$, $\widetilde{D}\cap \mathcal{A}_j\neq\epsilonmptyset$.
The projection $({\mathbb C},0)\times\mathcal{L}\to \mathcal{L}$, is locally a covering map. Then, for each point $(x,{\bf p})\in{\mathbb C}\times\mathcal{L}$, with $|x|$ small enough, we can consider the lifting $\widetilde{\gamma}^1_{\bf p}$, following a leaf of the foliation $\widetilde{{\mathcal F}}_{\Omegaega_1}$, of the path $\gamma_{\bf p}$, such that $\widetilde{\gamma}_{\bf p}^1(0)=(x,{\bf p}).$\\
If $(x_1,{\bf p}_1)=\widetilde{\gamma}^1_{\bf p}(1)$ is the end point of $\widetilde{\gamma}^1_{\bf p}$, we lift the path $\gamma^{-1}_{\bf p}$ to $\widetilde{\gamma}^2_{\bf p}$ in the foliation $\widetilde{{\mathcal F}}_{\Omegaega_2}$ such that $\widetilde{\gamma}^{2}_{\bf p}(0)=(\psi(x_1),{\bf p}_2)$.
Let $(x_2,{\bf p})=\widetilde{\gamma}^{2}_{\bf p}(1)$. Define
$$\phi(x,{\bf p})=(x_2,{\bf p})$$
as $\phi_{|\Sigma_1}=\psi,\; \phi(x,{\bf p})$ does not depend on the chosen path.
Let us observe that we can define $\phi$ as close as we want to the points ${\bf q}\in\mathcal{S}$, in $\widetilde{D}$. In fact, $\widetilde{D}\smallsetminus \mathcal{S}\cong {\mathbb C}^*\times{\mathbb C}\smallsetminus \mathcal{C}$, so ${\bf q}\in \mathcal{C}$ or ${\bf q}\in{\mathbb C}^*$. We can consider
a meridian relative to ${\bf q}$ with base point ${\bf p}$, (see Section \ref{section_toplogy_of_the_divisor}), and define the path $\begin{enumerate}ta_{\bf p}=\alpha_{\bf q}\gamma_{\bf p}$ with starting point ${\bf p}$ and end point ${\bf p}_j$. The path $\gamma^{-1}_{\bf p}\begin{enumerate}ta_{\bf p}$ is a meridian relative to ${\bf q}$ with base point ${\bf p}_j$, so its homotopy class define an element of $\pi_1(\mathcal{L},{\bf p}_j)$
and
$$h^1_{\gamma_{\bf p}^{-1}\begin{enumerate}ta_{\bf p}}=\psi^*(\gamma_{\bf p}^{-1}\begin{enumerate}ta_{\bf p}).$$
Therefore $\phi$ extends to a neighbourhood of $\widetilde{D}\smallsetminus \mathcal{S}$.
On the other hand, the singular points ${\bf q}\in\mathcal{S}$ are points of intersection of $\widetilde{D}$ with the other components of the divisor, and with the separatrix. These points are of dimensional type two or three and in a neighbourhood of all them, $\widetilde{D}$ is a strong separatrix\footnote{In dimension two, \textit{strong} separatrices around simple singular points are the ones corresponding to non-zero eigenvalues of the linear part. In higher dimension, the separatrix is called \textit{strong} if it corresponds to a strong separatrix in dimension two for a generic plane transversal section.}
From \cite{Cerveau-Mozo}, it follows that the conjugation of the holonomy of $\widetilde{D}$ implies the conjugation of the reduced foliations in a neighbourhood of these points. This conjugation coincides, outside of the separatrix, with the diffeomorphism $\phi$, and in consequence, $\phi$ is extended to a diffeomorphism around $\widetilde{D}$.
\epsilonnd{proof}
\section{Analytic classification}\label{section_cladification}
In this section we will study the analytic classification of quasi-homogeneous cuspidal foliations ${\mathcal F}_{\Omegaega}$ in $({\mathbb C}^3,{\bf 0})$, using as the main tool the lifting of the projective holonomy, as stated in previous sections.
As described in Section \ref{section_toplogy_of_the_divisor}, after the reduction of singularities of a quasi-homogeneous cuspidal holomorphic foliation in $({\mathbb C}^3,{\bf 0})$, the first component, $D_1$, of the exceptional divisor $E$, is either
\begin{enumerate}gin{itemize}
\item $D_1\smallsetminus \mathcal{S}\simeq {\mathbb C}^*\times{\mathbb C}^*$, if the separatrix of ${\mathcal F}_{\Omegaega}$ is defined by the equation
$$z^2+x^{n_1}y^{n_2}\prod_{i=1}^{l}\begin{itemize}g(y^{p}-a_ix^{q}\begin{itemize}g)^{d_i}=0,\ \text{with } (n_1,n_2)\neq (0,0),$$
\item or $D_1\smallsetminus \mathcal{S}\simeq {\mathbb C}^2$, if the separatrix of ${\mathcal F}_{\Omegaega}$ is defined by the equation
$$z^2+\prod_{i=1}^{l}\begin{itemize}g(y^{p}-a_ix^{q}\begin{itemize}g)^{d_i}=0.$$
\epsilonnd{itemize}
In the admissible case that we are studying, $D_1\smallsetminus \mathcal{S}$ is simply connected, so, the existence of a first integral around it is guaranteed due to the results of Mattei and Moussu \cite{Mattei-Moussu}. In order to be able to extend this first integral, and consequently, to extend the conjugation diffeomorphism, it would be necessary to impose additional technical conditions on one, or possibly several components of the exceptional divisor, that we will be call in the sequel \textit{special components}. These special components will be
those that arise at the end of the monoidal transformations with center the projective lines $D_1\cap S_1$, $D_{c_0+1}\cap S_{c_0+1}$ that we will denote by $\widetilde{D}', \;\widetilde{D}''$ respectively. Note that
$$\widetilde{D}'\smallsetminus \mathcal{S}\simeq{\mathbb C}\times({\mathbb C}\smallsetminus\{2pts\})\approx \widetilde{D}''\smallsetminus \mathcal{S}.$$
This will motivate the following definition:
\begin{enumerate}gin{defin}\label{defini_propiedades_p_1andp_2}
Let $\Omegaega\in\Sigma_{p,q}^{{(d_1,\cdots,d_l)}}$ be, we will say that the foliation ${\mathcal F}_{\Omegaega}$:
\begin{enumerate}gin{enumerate}
\item For $d-$even ({\bf Case i}), satisfies the property $\wp_1$: if the holonomy of the leaves $\widetilde{D}'\smallsetminus \mathcal{S},\; \widetilde{D}''\smallsetminus \mathcal{S}$, is linearizable.
\item For $d-$odd ({\bf Case ii.1}), satisfies the property $\wp_2$: if the holonomy of the leaf $\widetilde{D}''\smallsetminus \mathcal{S}$, is linearizable.
\epsilonnd{enumerate}
\epsilonnd{defin}
The following theorem is the main result of this paper.
\begin{enumerate}gin{teorema}\label{thm_clasificacion_analitica}
Let $\Omegaega_1, \Omegaega_2$ be elements of $\Sigma_{p,q}^{(d_1,\cdots,d_l)}$. Consider the foliations ${\mathcal F}_{\Omegaega_1}$ and ${\mathcal F}_{\Omegaega_2}$ that satisfy one of the properties $\wp_1, \wp_2$ if we are in one of the cases described above, with exceptional holonomy groups $H_{\Omegaega_i, \tilde{D}}=\langle h_{g_1}^i,h_{g_2}^i,h_{\alpha}^i\rangle, \; i=1,2$. Then, the foliations are analytically conjugated if and only if the triples $(h_{g_1}^i,h_{g_2}^i,h_{\alpha}^i)$ are also analytically conjugated.
\epsilonnd{teorema}
\begin{enumerate}gin{proof}
If the foliations are conjugated, then their essential holonomy groups are conjugated. The arguments are exactly the same that those described in \cite{Cerveau-Mozo}.
Assume that the holonomies are conjugated via $\Psi$, and let $\widetilde{{\mathcal F}}_{{\Omegaega}_i}$ be the respective strict transform of the foliations ${\mathcal F}_{\Omegaega_i}$, after its reduction of singularities. From the existence of the Hopf fibration relative to $\widetilde{D}$ (see Figure \ref{secuencia-explosiones-segun-di}) and Lemma \ref{lema-extension-de-holonomia--}, $\Psi$ may be extended to a neighbourhood $V_i\subset (M,E)$ of $\widetilde{D}$ and as a consequence, we have that $\widetilde{{\mathcal F}}_{{\Omegaega}_i}$ are conjugated in $V_i$. Now, we need to conjugate the foliations in a neighbourhood of all the other components of the divisor. We must first check the existence of the first integral around of the singular points outside $\widetilde{D}$. In fact, note that $D_1\smallsetminus \mathcal{S}$ is simply connected, so its holonomy group is trivial. As a consequence, the holonomy of the leaf $D_{2}\smallsetminus \mathcal{S}\approx {\mathbb C}\times{\mathbb C}^{*}$, generated by a loop around $L_1:=D_1\cap D_{2}$, is periodic (see \cite{Mattei-Moussu}). The same argument proves that $D_{\alpha}\smallsetminus \mathcal{S}$ has periodic holonomy, for all $1<\alpha\leq c_0+1$. So, $\widetilde{{\mathcal F}}_{{\Omegaega}_i}$ has first integral around
$$L_{\alpha}:=D_{\alpha}\cap D_{\alpha+1},\;1\leq\alpha \leq c_0+1.$$
On the other hand, the leaves $D_{\alpha}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}^*\times{\mathbb C}^*$ have periodic holonomy, generated by two loops around the lines
$$L_{\alpha}=D_{\alpha}\cap D_{\alpha+1};$$
$$L_{s_\nu}^{\alpha}=D_{\alpha}\cap D_{s_\nu}.$$
The above arguments prove the existence of the first integral around the lines
$$L_{\alpha j}:=D_{\alpha j}\cap D_{\alpha(j-1)}$$
where $\alpha, j$ are such that $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox{\mathbb C}\times{\mathbb C}^*$ or $\thickapprox {\mathbb C}^*\times{\mathbb C}^*$.
Finally, we need to guarantee the existence of the first integral around the points that represent the intersection of the divisor with the strict transform of the separatrix. These points are in the components $D_{\alpha j}$, and:
\begin{enumerate}gin{itemize}
\item Either $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}\times({\mathbb C}\smallsetminus\{{\rm 2\;points}\})$,
\item or $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}^*\times({\mathbb C}\smallsetminus\{{\rm 2\;points}\})$,
\item or $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox ({\mathbb C}^*\times{\mathbb C} )\smallsetminus \mathcal{C}$.
\epsilonnd{itemize}
Let us denote by $\mathcal{D}:=D_{\alpha j}$ such that $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}\times({\mathbb C}\smallsetminus\{{\rm 2\;points}\})$.
Note that
$\mathcal{D}\cap\widetilde{S}=\mathcal{L}_1\cup \mathcal{L}_2$ or $\mathcal{D}\cap\widetilde{S}=\mathcal{L}$, where $\widetilde{S}$ represents the strict transform of the separatrix, and $\mathcal{L},\mathcal{L}_1,\mathcal{L}_2$ are projective lines.
Let us suppose now that $d$ is even, then there exist exactly two components of the exceptional divisor such that $\mathcal{D}\cap\widetilde{S}=\mathcal{L}_1\cup \mathcal{L}_2$. The points of $\mathcal{L}_1, \mathcal{L}_2$ are singular points of dimensional type two.
Assuming that $\Omegaega_j\in\Sigma_{p,q}^{(d_1,\cdots,d_l)}$, ${\mathcal F}_{\Omegaega_j}$ satisfies the property $\wp_1$, i.e., the holonomy of the leaves $\mathcal{D}\smallsetminus \mathcal{S}$ is linearizable. This fact guarantees the existence of the first integral around $\mathcal{L}_1,\; \mathcal{L}_2$.
\begin{enumerate}gin{center}
\includegraphics[width=10cm]{TorreSuperfiesCaso-Par-existencia-integral-primera}
\figcaption{Existence of the first integral around the points $\rho_i$ and $\mu_i$}\label{sin nombre}
\epsilonnd{center}
Now let us prove the existence of the first integral around the points (see Figure \ref{sin nombre}) $$\rho_i:=D_{\alpha j}\cap\widetilde{S}\cap\mathcal{D}.$$
Let $h_{\alpha_i}$ be the holonomy associated to $\alpha_i$, loop around $\mathcal{L}_i$; $h_{\alpha_i}$ is linearizable. Now let us consider $\begin{enumerate}ta\subset\mathcal{D}\smallsetminus\mathcal{S}\simeq {\mathbb C}\times({\mathbb C}\smallsetminus\{2\;{\rm points}\})$, a loop around $D_{\alpha j}\cap\mathcal{D}$. Note that $\begin{enumerate}ta\subset {\mathbb C}\times\{1\;{\rm point}\}\simeq{\mathbb C}$; so, $\begin{enumerate}ta$ is homotopically trivial and the associated holonomy to $\begin{enumerate}ta$ is $1_{{\mathbb C}}$, the identity map, $h_{\begin{enumerate}ta}=1_{{\mathbb C}}$. Then, the holonomy group of $D_{\alpha j}\smallsetminus \mathcal{S}$ is linearizable and therefore, around $\rho_i$, $\widetilde{\Omegaega}_i$ is linearizable. So, there exists a first integral around $\rho_i$. Consider now the points $\mu_i$ (see Figure \ref{sin nombre}), and $h_\gamma$, the holonomy associated to $\gamma$, loop around the projective line $D_{(\alpha+1)j}\cap D_{\alpha j}$. Note that $\gamma$ is such that $\gamma^{-1}$ is a loop around $D_{\alpha-1}\cap D_{\alpha j}$, so $h_{\gamma^{-1}}$ is the holonomy of the leaf $D_{\alpha-1}\smallsetminus \mathcal{S}$, which is periodic. Then around $\mu_i$ there exists a first integral. In a similar way, a first integral can be found around $E\cap \widetilde{S}$.
Let us suppose that $d,q$ are odd and $p$ is even. There exist exactly a component $\mathcal{D}$ for which $\mathcal{D}\cap\widetilde{S}=\mathcal{L}_1\cup \mathcal{L}_2$ and a finite number of components for which we have $\mathcal{D}\cap\widetilde{S}=\mathcal{L}$. It is easy to see that around $\mathcal{L}$ there exist a first integral, and the existence of first integral around the lines $\mathcal{L}_1, \mathcal{L}_2$ follows, from property $\wp_2$, as in the previous case.
Let us see now the extension of $\Psi$ around all the exceptional divisor. The idea is, first, to extend $\Psi$ to a neighborhood of all the components of the divisor in which the separatrix intersects, and finally to extend it to the rest of the components. In both situations we should respect the fibration and the first integral that we have constructed previously. We have that $\widetilde{{\mathcal F}}_{{\Omegaega}_j}$ are analytically conjugated, via $\Psi$, in a neighbourhood of $\widetilde{D}$. We want to extend $\Psi$ to a neighbourhood of the exceptional divisor.
$$E=E_I\cup E_{II}\cup E_{III}.$$
Note that $E_{III}=\begin{itemize}gcup\mathcal{A}_j\times{\mathbb C}$, and these components are topologically equivalent to ${\mathbb P}_{{\mathbb C}}^1\times\mathbb{D}$. As $\widetilde{{\mathcal F}}_{{\Omegaega}_j}$ are analytically conjugate in a neighbourhood of $\widetilde{D}$, it follows that $\Psi$ is extended to a neighbourhood of $\widetilde{D}\cup E_{III}$
\begin{enumerate}gin{figure}[h]
\begin{enumerate}gin{center}
\includegraphics[width=11.5cm]{extension-analitica-1}\\
\figcaption{Extension of $\Psi=\Psi_2\Theta\Psi_1^{-1}$ to a neighbourhood of $D_{(\alpha-1)\begin{enumerate}ta}\cap D_{(\alpha-1)(\begin{enumerate}ta-1)}$.}\label{extension-analitica}
\epsilonnd{center}
\epsilonnd{figure}
On the other hand, for the case that $d$ is even, around ${\bf p}:=D_{\alpha-2}\cap D_{(\alpha-1)\begin{enumerate}ta}\cap D_{(\alpha-1)(\begin{enumerate}ta-1)}$, with $\alpha=s_{_N}$ and $\begin{enumerate}ta={\tilde{Q}_2\over2}$ (where $\tilde{Q}= \left( \frac{pq}{\delta} -nq\right) d-2\left( \frac{p+q}{\delta} -(m+n+1)\right)$, see Figure \ref{extension-analitica}) a first integral is given by the equation
$$F_j:= x^{m_{\nu}}s^{n_{\nu}+2(\varepsilonilon-1)}t^{n_{\nu}-2\varepsilonilon}U_j(x,s,t);\ U_j({\bf 0})\neq0$$
where for simplicity we denote $x:=x_{\alpha-1};\; \varepsilonilon=\begin{enumerate}ta-1,\; \nu=N\;; j_{\nu}=c_{\nu}-1$.
Let $$\mathcal{B}_c=\Big\{x\in{\mathbb C}: |x|<c\Big\}$$ be, for $c\in\mathbb{R}^{+}$ large enough, and
$$D_{\varepsilon}:=\{s\in{\mathbb C}:|s|<\varepsilon_1\}\times\{t\in {\mathbb C}: |t|<\varepsilon_2\}.$$
Similar arguments as in \cite{Meziani} and \cite{FM} are now used. First define a diffeomorphism $\Psi_j=(\Psi_{j1},\Psi_{j2},\Psi_{j3})$, on an open set $\mathcal{B}_c\times D_{\varepsilon}$, that transforms the first integral in $x^{m_{\nu}}s^{n_{\nu}+2(\varepsilonilon-1)}t^{n_{\nu}-2\varepsilonilon}$ and respects the fibration, i.e.:
\begin{enumerate}gin{equation}\label{trasforma-Int.Primera}
\Psi_{j1}^{m_{\nu}}\Psi_{j2}^{n_{\nu}+2(\varepsilonilon-1)}\Psi_{j3}^{n_{\nu}-2\varepsilonilon}=x^{m_{\nu}}s^{n_{\nu}+2(\varepsilonilon-1)}t^{n_{\nu}-2\varepsilonilon}U_j,
\epsilonnd{equation}
and
\begin{enumerate}gin{equation}\label{respeta-fibracion}
\begin{enumerate}gin{array}{ll}
\Psi_{j3}\Psi_{j1}^{{P\over2}-\widetilde{Q}_2+2}& =tx^{{P\over2}-\widetilde{Q}_2+2}\\\\
\Psi_{j2}\Psi_{j1}^{\widetilde{Q}_2-{P\over2}} & =sx^{\widetilde{Q}_2-{P\over2}}.
\epsilonnd{array}
\epsilonnd{equation}
Consider now $\Theta:=\Psi_2\circ\Psi\circ\Psi_1^{-1}$, defined in the open set $U_{c,\epsilonta,\varepsilon}$. $\Theta$ sends $\Psi_1(\widetilde{{\mathcal F}}_{{\Omegaega}_1})$ on $\Psi_2(\widetilde{{\mathcal F}}_{{\Omegaega}_2})$, and respects the first integral $$x^{m_{\nu}}s^{n_\nu+2(\varepsilonilon-1)}t^{n_\nu-2\varepsilonilon}.$$
It is defined on a set of the type
$${|x^{m_{\nu}}s^{n_\nu+2(\varepsilonilon-1)}t^{n_\nu-2\varepsilonilon}|<\varepsilon},$$ in the considered charts, and this set intersects the domain of definition of $\Psi$. So $\Psi=\Psi_2\circ\Theta\circ\Psi_1^{-1}$ can be extended to a neighbourhood of $D_{(\alpha-1)\begin{enumerate}ta}\cap D_{(\alpha-1)(\begin{enumerate}ta-1)}$.
Repeat the same argument to extend $\Psi$ to a neighbourhood of
$$\begin{itemize}g(D_{(\alpha-1)\begin{enumerate}ta}\cap D_{\alpha_2}\begin{itemize}g)\cup\Big(D_{(\alpha-1)\begin{enumerate}ta}\cap(\cup D_{(\alpha-2)j})\Big)\cup (D_{(\alpha-1)\begin{enumerate}ta}\cap \widetilde{S}),$$
and in consequence, $\Psi$ can be extended to a neighbourhood of $D_{(\alpha-1)\begin{enumerate}ta}$. This process can be repeated and extend $\Psi$ to a neighbourhood of the exceptional divisor $E$. So ${\mathcal F}_{\Omegaega_i}$ are analytically conjugated, outside of the singular locus of codimension two. Finally, using Hartogs theorem we can obtain the extension of the conjugation around of the origin.
In the case $d, q$ odd and $p$ even, we proceed analogously.
\epsilonnd{proof}
\section*{Acknowledgments}
The authors want to thank the Pontificia Universidad Cat\'{o}lica del Per\'{u} and the Universidad de Valladolid for their hospitality during the visits while preparing this paper.
The authors would like to thank the anonymous referee for many valuable and constructive suggestions, that have helped to improve the paper.
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\epsilonnd{thebibliography}
\epsilonnd{document} | math | 54,680 |
\begin{document}
\title[The quadratic covariation for a weighted-fBm]{The quadratic covariation for a weighted fractional Brownian motion${}^{*}$}
\footnote[0]{${}^{*}$The Project-sponsored by NSFC (No. 11571071, 11426036), Innovation Program of Shanghai Municipal Education
Commission (No. 12ZZ063) and Natural Science Foundation of Anhui
Province (No.1408085QA10)}
\author[X. Sun, L. Yan and Q. Zhang]{Xichao Sun${}^{\dag,\natural}$, Litan Yan${}^{\ddag,\S}$ and Qinghua Zhang${}^{\ddag}$}
\footnote[0]{${}^{\natural}[email protected], ${}^{\S}[email protected] (Corresponding Author)}
\date{}
\keywords{Weighted fractional Brownian motion, local time, Malliavin calculus, quadratic covariation, It\^{o} formula}
\subjclass[2000]{60G15, 60H05, 60G17}
\maketitle
\begin{center}
{\footnotesize {\it ${}^{\dag}$Department of Mathematics and Physics, Bengbu University\\
1866 Caoshan Rd., Bengbu 233030, P.R. China\\
${}^{\ddag}$Department of Mathematics, College of Science, Donghua University\\
2999 North Renmin Rd. Songjiang, Shanghai 201620, P.R. China}}
\end{center}
\maketitle
\begin{abstract}
Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic property
$$
E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim \varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0)
$$
for all $s>0$, we consider the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ defined by
$$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow
0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon}
\left\{f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds,
$$
provided the limit exists uniformly in probability. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(\Omega)$ and the generalized Bouleau-Yor identity
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
$$
holds for all $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)=\int_0^t\delta(B^{a,b}_s-x)ds^{1+a+b}$ is the weighted local time of $B^{a,b}$ and ${\mathbb B}(\cdot,\cdot)$ is the Beta function.
\end{abstract}
\section{Introduction}
Long/short range dependence (or long/short memory) stochastic processes with self-similarity have been intensively used as models for different physical phenomena. These properties appeared in empirical studies in areas like hydrology and geophysics; and they appeared to play an important role in network traffic analysis, economics and telecommunications. As a consequence, some efficient mathematical models based on long/short range dependence processes with self-similarity have been proposed in these directions. We refer to the monographs of self-similar processes by Embrechts-Maejima~\cite{Embrechts-Maejima}, Sheluhin et al~\cite{Sheluhin et al.}, Samorodnitsky~\cite{Samorodnitsky}, Samorodnitsky-Taqqu~\cite{Samorodnitsky-Taqqu}, Taqqu~\cite{Taqqu3} and Tudor~\cite{Tudor}.
The fractional Brownian motion is a simple stochastic process with long/short range dependence and self-similarity which is a suitable generalization of standard Brownian motion. Some surveys and complete literatures on fractional Brownian motion could be found in Biagini {\it et al}~\cite{BHOZ}, Gradinaru et al.~\cite{Grad1}, Hu~\cite{Hu2}, Mishura~\cite{Mishura2}, Nualart~\cite{Nualart1}. On the other hand, many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about self-similar Gaussian
processes and fields in general. Therefore, some generalizations of
the fBm has been introduced. However, contrast to the extensive
studies on fractional Brownian motion, there has been little
systematic investigation on other self-similar Gaussian processes.
The main reason for this is the complexity of dependence structures
for self-similar Gaussian processes which do not have stationary
increments. Thus, it seems interesting to study some extensions of fractional Brownian motion such as bi-fractional Brownian motion and the weighted fractional Brownian motion.
In this paper we consider the weighted fractional Brownian motion (weighted-fBm). Recall that the so-called weighted-fBm $B^{a,b}$ with parameters $a$ and $b$ is a zero mean Gaussian process with long/short-range dependence and self-similarity. It admits the relatively simple covariance as follows
$$
E\left[B^{a,b}_tB^{a,b}_s\right]=\frac{1}{2{\mathbb
B}(a+1,b+1)}\int_0^{s\wedge t}u^a((t-u)^b+(s-u)^b)du
$$
where ${\mathbb B}(\cdot,\cdot)$ is the beta function and
$a>-1,|b|<1,|b|<a+1$. Clearly, if $a=0$, the process coincides with
the standard fractional Brownian motion with Hurst parameter
$H=\frac{b+1}2$, and it admits the explicit significance. We have
(see, Lemma~\ref{lem3.0} in Section~\ref{sec3}, see also Bojdecki
{\em et al}~\cite{Bojdecki1})
\begin{equation}\label{sec1-eq1-1}
c_{a,b}(t\vee s)^a|t-s|^{b+1}\leq
E\left[\left(B^{a,b}_t-B^{a,b}_s\right)^2\right]\leq C_{a,b} (t\vee
s)^a|t-s|^{b+1}
\end{equation}
for $s,t\geq 0$. Thus, Kolmogorov's continuity criterion implies
that weighted-fractional Brownian motion is $\gamma$-H\"{o}lder
continuous for any $\gamma<\frac{1+b}2$, where $\frac{1+b}2$ is
called the H\"older continuous index. The process $B^{a,b}$ is $\frac12(a+b+1)$-self similar and its increments are not stationary. It is important to note that the following fact:
\begin{itemize}
\item The H\"older continuous index $\frac12(1+b)$ is not equal either to the its self-similar index nor the order of the infinitesimal $\sqrt{E[(X_t)^2]}\to 0$ as $t\downarrow 0$, provided $a\neq 0$.
\end{itemize}
However, the three indexes are coincident for many famous self-similar Gaussian processes such as fractional Brownian motion,
sub-fractional Brownian motion and bi-fractional Brownian motion.
That is causing trouble for the research, and it is also our a motivation to study the weighted-fBm. Before making the decision to study the weighted-fBm we first try to investigate in Yan et al.~\cite{Yan3} some path properties including strong local nondeterminism, Chung's law of the iterated logarithm and the smoothness of the collision local time. In particular, we showed that it is strongly locally $\phi$-nondeterministic with $\phi(r)=r^{1+b}$. In general, the function $\phi$ depends on the self-similar index of the process, but the fact is, for the weighted-fBm, $\phi(r)=r^{1+b}$ is independent of parameter $a$, which enhances further our interesting to study the weighted-fBm.
The weighted-fBm appeared in Bojdecki {\em et al}~\cite{Bojdecki1} in a limit of occupation time fluctuations of a system of independent particles moving in ${\mathbb R}^d$ according a symmetric $\alpha$-stable L\'evy process, $0<\alpha\leq 2$, started from an inhomogeneous
Poisson configuration with intensity measure
$$
\frac{dx}{1+|x|^\gamma}
$$
and $0<\gamma\leq d=1<\alpha$, $a=-\gamma/\alpha$, $b=1-1/\alpha$,
the ranges of values of $a$ and $b$ being $-1< a < 0$ and $0 < b\leq
1+a$. The process also appears in Bojdecki {\em et
al}~\cite{Bojdecki2} in a high-density limit of occupation time
fluctuations of the above mentioned particle system, where the
initial Poisson configuration has finite intensity measure, with
$d=1<\alpha$, $a=-1/\alpha$, $b=1-1/\alpha$. Moreover, the
definition of the weighted-fBm $B^{a,b}$ was first introduced by
Bojdecki {\em et al}~\cite{Bojdecki0}, and it is neither a
semimartingale nor a Markov process if $b\neq 0$, so many of the
powerful techniques from stochastic analysis are not available when
dealing with $B^{a,b}$. There has been little systematic investigation on weighted-fBm since it it has been introduced by Bojdecki {\em et al}~\cite{Bojdecki0}.
In this paper, we consider the the {\it generalized quadratic covariation} when $b<0$, and it is important to note that a large class of Gaussian processes with similar characteristics as weighted-fBm could be handled in uniform approach used here. Clearly, by the estimates~\eqref{sec1-eq1-1}, we have
$$
E\bigl[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2\bigr]=O( s^a\varepsilon^{1+b})\qquad (\varepsilon\to 0)
$$
for all $s>0$, which implies that
$$
\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon} \int_{t_0}^{t}(B_{s+\varepsilon}-B_s)^2ds=
\begin{cases}
0, & \text{ {if $0<b<1$}}\\
+\infty,& \text{ {if $-1<b<0$}}
\end{cases}
$$
for all $t\geq t_0>0$, where the limit is uniformly in probability. Additional results on the quadratic variation can be found in Russo-Vallois~\cite{Russo2}. Thus, we need a substitute tool of the quadratic variation for $b\neq 0$. Inspired by~\eqref{sec1-eq1-1}, the fact
$$
E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim \varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0)
$$
for all $s>0$ and Cauchy's principal value, one can naturally give the following definition.
\begin{definition}
Let $a>-1,|b|<1,|b|<a+1$ and let the integral
$$
J_\varepsilon(f,t):=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}\left\{ f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds
$$
exists for all $\varepsilon>0$ and all Borel functions $f$. The limit
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t:=\lim_{\varepsilon\downarrow 0}J_\varepsilon(f,t)
$$
is called the {\it generalized quadratic covariation} of $f(B^{a,b})$ and $B^{a,b}$, provided the limit exists uniformly in probability.
\end{definition}
\begin{remark}\label{rem}
{\rm
It is important to note that the above definition is available for a large class of Gaussian processes with similar characteristics as weighted-fBm. Let now $X$ be a self-similar Gaussian process with H\"older continuous paths of order $\alpha\in (0,1)$. We then can define the generalized quadratic covariation $[f(X),X]^{(a,b)}$ as follows
$$
[f(X),X]^{(a,b)}_t=\lim_{\varepsilon\downarrow
0}\frac{2\alpha}{\varepsilon^{2\alpha}}\int_\varepsilon^{t+\varepsilon}
\left\{f(X_{s+\varepsilon})
-f(X_s)\right\}(X_{s+\varepsilon}-X_s)s^{2\alpha-1}ds
$$
for any Borel functions $f$, provided the limit exists uniformly in probability. When $0<\alpha<\frac12$, we can get some similar results for the process $X$ to weighted-fBm with $-1<b<0$.
}
\end{remark}
We shall see in Section~\ref{sec6} that
$$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t =\kappa_{a,b}\int_0^tf'(B^{a,b}_s)ds^{1+a+b},\qquad t\geq 0
$$
for all $f\in C^1({\mathbb R})$ and all $b\in (-1,1)$, where
$$
\kappa_{a,b}=\frac1{(1+b){\mathbb B}(a+1,b+1)}.
$$
In the present paper we prove the existence of the generalized quadratic covariation for $-1<b<0$, our start point is to consider the decomposition
\begin{equation}\label{sec1-eq1-2}
\begin{split}
\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} &\left\{f(B^{a,b}_{
s+\varepsilon})-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)s^{b}ds\\
&=\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_{
s+\varepsilon})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds\\ &\qquad\qquad-\frac{1}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_s)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds
\end{split}
\end{equation}
for all $-1<b<0$. It is important to note that the above decomposition is unavailable for $0<b<1$. For example, we have
$$
\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} E\left(B^{a,b}_{s} (B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right)s^bds\longrightarrow \infty
$$
for all $b>0$ and $t>0$, as $\varepsilon$ tends to zero, because
\begin{equation}\label{sec1-eq1-3}
E\left(B^{a,b}_{s} (B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right) \sim s^{a+b}\varepsilon\qquad (\varepsilon\to 0).
\end{equation}
The above asymptotic property follows from
\begin{align*}
E[B^{a,b}_s(B^{a,b}_{s+\varepsilon}&-B^{a,b}_s)]=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a[({s+\varepsilon}-u)^b+(s-u)^b]du-s^{1+a+b}\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a\left[({s+\varepsilon}-u)^b-(s-u)^b\right]du\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a({s+\varepsilon}-u)^b\left[1- (\frac{s-u}{{s+\varepsilon}-u})^b\right]du\\
&\sim s^{a+b}\varepsilon
\end{align*}
for all $b>0$ by the fact $1-x^b\sim 1-x$ as $x\to 1$. Thus, the method used here is different from that need to handle the case $0<b<1$.
This paper is organized as follows. In Section~\ref{sec2} we present some preliminaries for weighted-fBm and Malliavin calculus. In Section~\ref{sec3}, we establish some technical estimates associated with weighted-fBm with $-1<b<0$. In Section~\ref{sec6}, as an example, when $f\in C^1({\mathbb R})$ we show that the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ exists in $L^2$ for all $a,b$ and
$$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\kappa_{a,b}\int_0^t f'(B^{a,b}_s)ds^{1+a+b}.
$$
In particular, we have
$$
\bigl[B^{a,b},B^{a,b}\bigr]^{(a,b)}_t=\kappa_{a,b}t^{1+a+b}.
$$
In Section~\ref{sec5}, in more general cases we consider the existence of generalized quadratic covariation for $-1<b<0$. By estimating the two terms of the right hand side in the decomposition~\eqref{sec1-eq1-2}, respectively, we construct a Banach space ${\mathscr H}$ of measurable functions $f$ on
${\mathbb R}$ such that $\|f\|_{{\mathscr H}}<\infty$, where
\begin{align*}
(\|f\|_{{\mathscr H}})^2:=\int_0^{T+1}\int_{\mathbb
R}|f(x)|^2e^{-\frac{x^2}{2s^{1+a+b}
}}\frac{dxds}{\sqrt{2\pi}s^{(1-a-b)/2}}.
\end{align*}
We show that the {\it generalized quadratic covariation} $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ exists for all $f\in {\mathscr H}$ and
$$
E\left|\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\right|^2\leq C_{a,b,T}\|f\|_{\mathscr H}^2,\qquad 0\leq t\leq T.
$$
In Section~\ref{sec7}, for $-1<b<0$ we consider the integral
\begin{equation}\label{sec1-eq1-4}
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
\end{equation}
for $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)$ denotes the weighted local time defined by
$$
{\mathscr L}^{a,b}(x,t)=(1+a+b)\int_0^t\delta(B^{a,b}_s-x)s^{a+b}ds.
$$
In order to study the integral we obtain the following It\^o formula :
$$
F(B^H)=F(0)+\int_0^tf(B^{a,b}_s)dB^{a,b}_s+\frac12(\kappa_{a,b})^{-1}
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t,
$$
where $F$ is an absolutely continuous function with $F'=f\in {\mathscr H}$, and show that the {\em generalized Bouleau-Yor identity}
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\kappa_{a,b}\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
$$
holds for all $f\in {\mathscr H}$. As a corollary we get the Tanaka formula
$$
|B^{a,b}_t-x|=|-x|+\int_0^t{\rm sign}(B^{a,b}_s-x)dB^{a,b}_s+{\mathscr L}^{a,b}(x,t)
$$
for $-1<b<0$.
\section{The weighted fractional Brownian motion}\label{sec2}
Let $B^{a,b}$ be a weighted-fBm with parameters $a,b$
($a>-1,|b|<1,|b|<a+1$), defined on the complete probability space
$(\Omega,\mathcal{F},P)$. As we pointed out before, the weighted-fBm
$B^{a,b}=\left\{B^{a,b}_t,0\leq t\leq T \right\}$ with indices $a$
and $b$ is a mean zero Gaussian processes such that $B_0^{a,b}=0$ and
\begin{equation}\label{sec2-eq2.1}
E\left[B^{a,b}_tB^{a,b}_s\right]=\frac{1}{2{\mathbb
B}(a+1,b+1)}\int_0^{s\wedge t}u^a((t-u)^b+(s-u)^b)du
\end{equation}
for $s,t\geq 0$. It is known that the function $(t,s)\mapsto
R^{a,b}(t,s)$ is positive-definite if and only if $a$ and $b$
satisfy the conditions
\begin{equation}\label{sec2-eq2.2}
a>-1,\;|b|<1,\;|b|<a+1,
\end{equation}
and the following statements hold (see Bojdecki~\cite{Bojdecki0}):
\begin{itemize}
\item $B^{a,b}$ is $\frac12(a+b+1)$-self similar;
\item $B^{a,b}$ has independent increments for $b=0$;
\item $B^{a,b}$ is neither a semimartingale nor a Markov process if $b\neq 0$;
\item If $b>0$, then $B^{a,b}$ is long-range dependence;
\item If $b<0$, then $B^{a,b}$ is short-range dependence.
\end{itemize}
Thus, throughout this paper we let $b\neq 0$ for simplicity.
As a Gaussian process, it is possible to construct a stochastic calculus of variations with respect to $B^{a,b}$. We refer to Al\'os {\it et al}~\cite{Nua1} and Nualart~\cite{Nualart1} for the complete descriptions of stochastic calculus with respect to Gaussian processes. Here we recall only the basic elements of this theory. Throughout this paper we assume that~\eqref{sec2-eq2.2} holds.
Let ${\mathcal H}_{a,b}$ be the completion of the linear space ${\mathcal
E}$ generated by the indicator functions $1_{[0,t]}, t\in [0,T]$
with respect to the inner product
$$
\langle 1_{[0,s]},1_{[0,t]} \rangle_{{\mathcal H}_{a,b}}=R^{a,b}(s,t)=\frac{1}{2{\mathbb
B}(a+1,b+1)}\int_0^{s\wedge t}u^a((t-u)^b+(s-u)^b)du.
$$
The application $\varphi\in {\mathcal E}\to B(\varphi)$ is an
isometry from ${\mathcal E}$ to the Gaussian space generated by $B^{a,b}$ and it can be extended to ${\mathcal H}_{a,b}$.
\begin{remark}
{\rm
For $b>0$ we can characterize ${\mathcal H}_{a,b}$ as
$$
{\mathcal H}_{a,b}=\{f:[0,T]\to {\mathbb R}\;|\;\|f\|_{a,b}<\infty\},
$$
where
$$
\|f\|^2_{a,b}=\int_0^T\int_0^Tf(t)f(s)\frac{\partial^2}{\partial s\partial t}R^{a,b}(s,t)dsdt.
$$
Clearly, we can write its covariance as
\begin{align*}
\phi_{a,b}(s,t):=\frac{\partial^2}{\partial t\partial s}R^{a,b}(t,s) &=\frac{b}{2{\mathbb
B}(a+1,b+1)}(t\wedge s)^a|t-s|^{b-1}
\end{align*}
for $b>0$. Thus, $R^{a,b}$ is the distribution function of an absolutely continuous positive measure with density $\frac{b}{2{\mathbb
B}(a+1,b+1)}(t\wedge s)^a|t-s|^{b-1}$ which belongs of course to
$L^1([0,T ]^2)$.
}
\end{remark}
Let us denote by ${\mathcal S}^{a,b}$ the set of smooth functionals of the form
$$
F=f(B(\varphi_1),B(\varphi_2),\ldots,B(\varphi_n)),
$$
where $f\in C^{\infty}_b({\mathbb R}^n)$ ($f$ and all its derivatives are bounded) and $\varphi_i\in {\mathcal H}_{a,b}$. The {\em Malliavin derivative} $D^{a,b}$ of a functional $F$ as above is given by
$$
D^{a,b}F=\sum_{j=1}^n\frac{\partial f}{\partial
x_j}(B(\varphi_1),B(\varphi_2), \ldots,B(\varphi_n))\varphi_j.
$$
The derivative operator $D^{a,b}$ is then a closable operator from $L^2(\Omega)$ into $L^2(\Omega;{\mathcal H}_{a,b})$. We denote by ${\mathbb D}^{1,2}$ the closure of ${\mathcal S}_{a,b}$ with respect to the norm
$$
\|F\|_{1,2}:=\sqrt{E|F|^2+E\|D^{a,b}F\|^2_{a,b}}.
$$
The {\it divergence integral} $\delta^{a,b}$ is the adjoint of derivative operator $D^{a,b}$. That is, we say that a random variable $u$ in $L^2(\Omega;{\mathcal H}_{a,b})$ belongs to the domain of the divergence operator $\delta^{a,b}$, denoted by ${\rm {Dom}}(\delta^{a,b})$, if
$$
E\left|\langle D^{a,b}F,u\rangle_{{\mathcal H}_{a,b}}\right|\leq
c\|F\|_{L^2(\Omega)}
$$
for every $F\in {\mathbb D}^{1,2}$, where $c$ is a constant depending only on $u$. In this case $\delta^{a,b}(u)$ is defined by the duality relationship
\begin{equation}
E\left[F\delta^{a,b}(u)\right]=E\langle D^{a,b}F,u\rangle_{{\mathcal H}_{a,b}}
\end{equation}
for any $F\in {\mathbb D}^{1,2}$, we have ${\mathbb D}^{1,2}\subset
{\rm {Dom}}(\delta^{a,b})$. We will denote
$$
\delta^{a,b}(u)=\int_0^Tu_sdB^{a,b}_s
$$
for an adapted process $u$, and it is called Skorohod integral. We have the following It\^o formula.
\begin{theorem}[Al\'os {\it et al}~\cite{Nua1}]\label{theorem-Ito}
Let $f\in C^{2}({\mathbb R})$ such that
\begin{equation}\label{sec2-Ito-con1}
\max\left\{|f(x)|,|f'(x)|,|f''(x)|\right\}\leq \kappa e^{\beta x^2},
\end{equation}
where $\kappa$ and $\beta$ are positive constants with
$\beta<\frac14T^{-(1+a+b)}$. Then we have
\begin{align*}
f(B^{a,b}_t)=f(0)&+\int_0^t\frac{d}{dx}f(B^{a,b}_s)dB^{a,b}_s
+\frac12(1+a+b)\int_0^t\frac{d^2}{dx^2}f(B^{a,b}_s)s^{a+b}ds
\end{align*}
for all $t\in [0,T]$.
\end{theorem}
\section{Some basic estimates}\label{sec3}
For simplicity throughout this paper we let $C$ stand for a positive constant depending only on the subscripts and its value may be different in different appearance, and this assumption is also adaptable to $c$. Moreover, the notation $F\asymp G$ means that there are positive constants $c_1$ and $c_2$
so that
$$
c_1G(x)\leq F(x)\leq c_2G(x)
$$
in the common domain of definition for $F$ and $G$. For $x,y\in
\mathbb{R}$, $x\wedge y:=\min\{x,y\}$ and $x\vee y:=\max\{x,y\}$.
\begin{lemma}\label{lem3.0}
Let $a>-1,\;|b|<1,\;|b|<a+1$. We then have
\begin{equation}\label{sec3-eq3.1}
Q(t,s):=E\left[\left(B^{a,b}_t-B^{a,b}_s\right)^2\right]\asymp (t\vee
s)^a|t-s|^{1+b}
\end{equation}
for $s,t\geq 0$. In particular, we have
\begin{equation}\label{sec3-eq3.2}
E\left[\left(B^{a,b}_t-B^{a,b}_s\right)^2\right]\leq
C_{a,b}|t-s|^{1+a+b}
\end{equation}
for $a\leq 0$.
\end{lemma}
The estimates~\eqref{sec3-eq3.1} are first considered by
Bojdecki~\cite{Bojdecki0}. The present form is a slight
modification given by Yan et al.~\cite{Yan3}. Thus, Kolmogorov's continuity criterion and the Gaussian property of the process imply that
weighted-fractional Brownian motion is $\gamma$-H\"{o}lder
continuous for any $\gamma<\frac{1+b}2$, where $\frac{1+b}2$ is
called the {\em H\"older continuity index}. It is important to note
that the H\"older continuity index $\frac{1+b}2$ is not equal to the
order $\frac{1+a+b}2$ of the infinitesimal $\sqrt{E[(X_t)^2]}\to 0$
as $t\downarrow 0$ unless $a=0$. However, the H\"{o}lder continuity
index of many popular self-similar Gaussian processes equals to the
order of the infinitesimal such as fractional Brownian motion,
sub-fractional Brownian motion and bi-fractional Brownian motion.
\begin{lemma}\label{lem3.1}
Let $a>-1,\;|b|<1,\;|b|<a+1$. We then have
\begin{equation}\label{sec3-eq3.3}
t^{1+a+b}s^{1+a+b}-\mu^2\asymp (ts)^{a}(t\wedge s)^{1+b}|t-s|^{1+b}
\end{equation}
for all $s,t>0$, where $\mu=E(B^{a,b}_tB^{a,b}_s)$.
\end{lemma}
\begin{proof}
Without loss of generality we may assume that $t>s>0$. Then
\begin{align*}
t^{1+a+b}&s^{1+a+b}-\mu^2\\
&=t^{1+a+b}s^{1+a+b}-\frac1{4{\mathbb B}^2(1+a,1+b)}\left(\int_0^su^a\left((t-u)^b+(s-u)^b\right)du\right)^2\\
&\equiv t^{2(1+a+b)}G(x)
\end{align*}
with $x=\frac{s}{t}$, where
$$
G(x)=x^{1+a+b}-\frac1{4{\mathbb B}^2(1+a,1+b)}\left(\int_0^xu^a(1-u)^bdu+{\mathbb B}(1+a,1+b)x^{1+a+b}\right)^2
$$
with $x\in [0,1]$. Noting that
$$
t^{1+a+b}s^{1+a+b}-\mu^2\geq 0
$$
and
$$
t^{1+a+b}s^{1+a+b}-\mu^2=0\qquad\Longleftrightarrow\qquad s=t\quad {\text {or }}\quad s=0
$$
for all $t\geq s>0$, we see that
$G(x)\geq 0$ and
$$
G(x)=0\qquad \Longleftrightarrow \qquad x=0\quad{\text {or }}\quad x=1.
$$
Decompose $G(x)$ as follows
\begin{align*}
G(x)&=\frac1{4(K_{a,b})^2}\left(2K_{a,b} x^{\frac{1+a+b}2}-\int_0^xu^a(1-u)^bdu-K_{a,b}x^{1+a+b}\right)\\
&\qquad\quad\cdot\left(2K_{a,b} x^{\frac{1+a+b}2}+\int_0^xu^a(1-u)^bdu+K_{a,b}x^{1+a+b}\right)\\
&\equiv \frac1{4(K_{a,b})^2}G_1(x)G_2(x)
\end{align*}
for all $x\in [0,1]$, where $K_{a,b}={\mathbb B}(1+a,1+b)$.
Obviously, we have
$$
K_{a,b} x^{\frac{1+a+b}2}\leq G_2(x)\leq C_{a,b}x^{\frac{1+a+b}2}
$$
for all $x\in [0,1]$. In fact, the left inequality is clear, and the right inequality follows from
the fact
$$
\int_0^xv^a(1-v)^bdv\leq \int_0^xv^a\left((x-v)^b+1\right)dv\leq K_{a,b}x^{1+a+b}+\frac1{1+a}x^{1+a}
$$
for all $x\in [0,1]$. On the other hand, we also have
\begin{align*}
\lim_{x\uparrow 1}\frac{G_1(x)}{x^{\frac{1+a+b}2}(1-x)^{1+b}}=\frac1{1+b},\qquad \lim_{x\downarrow 0}\frac{G_1(x)}{x^{\frac{1+a+b}2}(1-x)^{1+b}}=2{\mathbb B}(1+a,1+b),
\end{align*}
which deduces
$$
G_1(x)\asymp x^{\frac{1+a+b}2}(1-x)^{1+b}
$$
by the continuity. Thus, we have showed that
$$
G(x)=G_1(x)G_2(x)\asymp x^{1+a+b}(1-x)^{1+b}
$$
and the lemma follows.
\end{proof}
\begin{lemma}\label{lem3.3}
Let $t>s>t'>s'>0$ and let $-1<b<1$, $a>-1$, $|b|<1+a$. We then have
\begin{equation}\label{sec3-eq3.6}
\begin{split}
|E(B^{a,b}_{t}-&B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})|
\\
&\leq C_{a,b,\alpha}\left((s')^a\vee s^a\right)^\alpha(tt')^{\frac12a(1-\alpha)} \frac{[(t-s)(t'-s')]^{\alpha+\frac12(1-\alpha)(1+b)} }{(t-t')^{(1-b)\alpha}}
\end{split}
\end{equation}
for all $\alpha\in [0,1]$.
\end{lemma}
\begin{proof}
Let $b<0$. Denote
\begin{align*}
\mu(t,s,t',s'):&=E(B^{a,b}_{t}-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_{s'}^{t'}u^a\left[(s-u)^b-(t-u)^b\right]du.
\end{align*}
It follows from the fact
\begin{equation}\label{sec3-eq3.7=0}
\begin{split}
y^\gamma-x^\gamma&=y^\gamma\left(1-(\frac{x}{y})^\gamma\right)\leq C_{\gamma}y^\gamma\left(1-\frac{x}{y}\right)\\
&\leq C_{\gamma,\beta}y^\gamma\left(1-\frac{x}{y}\right)^\beta
\leq C_{\gamma,\beta}y^{\gamma-\beta}(y-x)^\beta
\end{split}
\end{equation}
for $y>x>0$, $\gamma\geq 0$, $0\leq \beta\leq 1$ and the inequality
\begin{align*}
t-u=(t-t')+(t'-u)\geq (t-t')^{1-\nu}(t'-u)^\nu\qquad (0<u<t'<t)
\end{align*}
for all $0\leq \nu\leq 1$ that
\begin{equation}\label{sec3-eq3.8==}
\begin{split}
|\mu(t,s,&t',s')|=\frac1{2{\mathbb B}(1+a,1+b)}\int_{s'}^{t'}u^a\left[(s-u)^b -(t-u)^b\right]du\\
&\leq C_{a,b}\int_{s'}^{t'}u^a\frac{(t-u)^{-b} -(s-u)^{-b}}{
(s-u)^{-b}(t-u)^{-b}}du\leq C_{a,b,\beta}\int_{s'}^{t'}u^a\frac{(t-s)^\beta}{
(s-u)^{-b}(t-u)^\beta}du\\
&\leq C_{a,b}\frac{(t-s)^\beta}{(t-t')^{\beta(1-\nu)}} \int_{s'}^{t'}\frac{u^a}{(t'-u)^{-b+\beta\nu}}du\\
&\leq C_{a,b}\left[(s')^a\vee s^a\right]\frac{(t-s)^\beta(t'-s')^{1+b-\beta\nu}}{(t-t')^{\beta(1-\nu)}}
\end{split}
\end{equation}
for all $0\leq \beta\leq 1$ and $0\leq \nu\beta\leq 1+b$.
On the other hand, noting that
\begin{align*}
|E[(B^{a,b}_t-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|^2&\leq E\left[(B^{a,b}_t-B^{a,b}_s)^2\right]E\left[(B^{a,b}_{t'} -B^{a,b}_{s'})^2\right]\\
&\leq C_{a,b}(tt')^a(t-s)^{1+b}(t'-s')^{1+b},
\end{align*}
we see that
\begin{align*}
&\frac{|E[(B^{a,b}_t-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|}{ {\sqrt{C_{a,b}(tt')^a(t-s)^{1+b}(t'-s')^{1+b}}}}\leq
\left(\frac{|E[(B^{a,b}_t-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|}{ {\sqrt{C_{a,b}(tt')^a(t-s)^{1+b}(t'-s')^{1+b}}}}\right)^\alpha
\end{align*}
for all $\alpha\in [0,1]$. Combining this with~\eqref{sec3-eq3.8==} (taking $\beta=1+b$ and $\nu=0$), we get
\begin{align*}
|E[(B^{a,b}_t-&B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|\\
&\leq
C_{a,b,\alpha}\left((s')^a\vee s^a\right)^\alpha(tt')^{\frac12a(1-\alpha)} \frac{[(t-s)(t'-s')]^{\alpha+\frac12(1-\alpha)(1+b)} }{(t-t')^{(1+b)\alpha}}
\end{align*}
and the lemma follows for all $\alpha\in [0,1]$. Similarly, we can show that the lemma holds for $b>0$.
\end{proof}
\begin{lemma}\label{lem3.4}
For $a>-1$, $-1<b<0$ and $|b|<1+a$ we have
\begin{align}\label{lem3.4-eq1}
&E[B^{a,b}_t(B^{a,b}_s-B^{a,b}_r)]\leq C_{a,b}(s-r)^{1+b}s^a\\ \label{lem3.4-eq2}
&E[B^{a,b}_s(B^{a,b}_t-B^{a,b}_s)]\leq C_{a,b}(t-s)^{1+b}s^a\\ \label{lem3.4-eq3}
&E[B^{a,b}_s(B^{a,b}_s-B^{a,b}_r)]\leq C_{a,b}(s-r)^{1+b}s^a\\ \label{lem3.4-eq4}
&E[B^{a,b}_{s}(B^{a,b}_t-B^{a,b}_r)]\leq C_{a,b}(t-r)^{1+b}s^a\\ \label{lem3.4-eq5}
&E[B^{a,b}_{r}(B^{a,b}_t-B^{a,b}_s)]\leq C_{a,b}(t-s)^{1+b}r^a
\end{align}
for all $t>s>r>0$.
\end{lemma}
\begin{proof}
Let $t>s>r>0$. By~\eqref{sec3-eq3.7=0} we have
\begin{equation}\label{lem3.4-eq6}
s^{1+a+b}-r^{1+a+b}\leq C_{a,b}s^{a+b}(s-r)\leq C_{a,b}(s-r)^{1+b}s^a
\end{equation}
for all $a>-1$, $-1<b<0$ and $0\leq \beta\leq 1$. Notice that
\begin{equation}\label{lem3.1-eq3}
\int_{x}^1r^a(1-r)^bdr \asymp (1-x)^{1+b}
\end{equation}
for all $x\in [0,1]$ by the continuity and the convergence
\begin{align*}
\lim_{x\to 1}\frac{f(x)}{(1-x)^{1+b}}=\frac1{1+b}
\end{align*}
for all $a,b>-1$. We get
\begin{align*}
\int_r^su^a(t-u)^bdu&\leq \int_r^su^a(s-u)^bdu=s^{1+a+b}\int_{r/s}^1v^a(1-v)^bdv\\
&\leq C_{a,b}s^{1+a+b}(1-\frac{r}s)^{1+b} =C_{a,b}(s-r)^{1+b}s^a
\end{align*}
for $-1<b<0,a>-1$. It follows that
\begin{align*}
E[B^{a,b}_t&(B^{a,b}_s-B^{a,b}_r)]=\frac1{2{\mathbb B}(1+a,1+b)}\left(\int_0^su^a[(t-u)^b+(s-u)^b]du\right.\\
&\hspace{3cm}\left.-
\int_0^ru^a[(t-u)^b+(r-u)^b]du\right)\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_r^su^a(t-u)^bdu +\frac12\left(s^{1+a+b}-r^{1+a+b}\right)\\
&\leq C_{a,b}(s-r)^{1+b}s^a.
\end{align*}
This establishes the estimate~\eqref{lem3.4-eq1}. In order to prove the estimate~\eqref{lem3.4-eq2} we have
\begin{equation}\label{lem3.4-eq7}
\int_0^{x}v^a(1-v)^bdv\asymp x^{1+a}
\end{equation}
for all $x\in [0,1]$ by the continuity and the convergence
$$
\lim_{x\downarrow 0}\frac{1}{x^{1+a}}\int_0^xv^a(1-v)^bdv=\frac1{1+a}
$$
for all $a,b>-1$. It follows that
\begin{align*}
|E[B^{a,b}_s(B^{a,b}_t&-B^{a,b}_s)]|=\left|\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a[(t-u)^b+(s-u)^b]du-s^{1+a+b}\right|\\
&=\left|\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a(t-u)^bdu-\frac12s^{1+a+b}\right|\\
&=\left|\frac{t^{1+a+b}}{2{\mathbb B}(1+a,1+b)}\int_0^{s/t}v^a(1-v)^bdv-\frac12s^{1+a+b}\right|\\
&\leq \frac{1}{2{\mathbb B}(1+a,1+b)}\int_0^{s/t}v^a(1-v)^bdv\left(t^{1+a+b}-s^{1+a+b}\right)\\
&\qquad +s^{1+a+b}\frac{1}{2{\mathbb B}(1+a,1+b)}\left({\mathbb B}(1+a,1+b)-\int_0^{s/t}v^a(1-v)^bdv\right)\\
&\leq C_{a,b} (t-s)^{1+b}s^a,
\end{align*}
which gives~\eqref{lem3.4-eq2}. Similarly, we can prove the estimate~\eqref{lem3.4-eq3}. The estimate~\eqref{lem3.4-eq4} follows from~\eqref{lem3.4-eq2} and~\eqref{lem3.4-eq3}.
Finally, in order to prove~\eqref{lem3.4-eq5} we have
\begin{align*}
|E[B^{a,b}_{r}&(B^{a,b}_t-B^{a,b}_s)]|=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^ru^a[(s-u)^b-(t-u)^b]du\\
&\leq \frac{r^a}{2{\mathbb B}(1+a,1+b)}\int_0^r[(s-u)^b-(t-u)^b]du\\
&\leq \frac{r^a}{2{\mathbb B}(1+a,1+b)}\int_0^s[(s-u)^b-(t-u)^b]du\\
&=C_{a,b}r^a\left(s^{1+b}-t^{1+b}+(t-s)^{1+b}\right)\leq C_{a,b}(t-s)^{1+b}r^a
\end{align*}
for all $a\geq 0$ and $-1<b<0$. Moreover, we also have
\begin{align*}
s^{1+a+b}\Bigl(\int_0^{r/s}& v^a(1-v)^bdv -\int_0^{r/t}v^a(1-v)^bdv\Bigr) =s^{1+a+b}\int_{r/t}^{r/s}v^a(1-v)^bdv\\
&\leq \frac{s^{1+a+b}}{1+b}(\frac{r}t)^a \left(\frac{r}s-\frac{r}{t}\right)^{1+b}\\
&=\frac{s^{1+a+b}}{1+b}(\frac{r}t)^a \frac{(t-s)^{1+b}r^{1+b}}{(ts)^{1+b}}\leq \frac{1}{1+b}r^a (t-s)^{1+b}
\end{align*}
for $-1<a<0$. It follows from~\eqref{lem3.4-eq7} that
\begin{align*}
|E[B^{a,b}_{r}&(B^{a,b}_t-B^{a,b}_s)]|=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^ru^a[(s-u)^b-(t-u)^b]du\\
&=C_{a,b}\left(s^{1+a+b}
\int_0^{r/s}v^a(1-v)^bdv-t^{1+a+b}\int_0^{r/t}v^a(1-v)^bdv\right)\\
&\leq C_{a,b}
s^{1+a+b}
\left(\int_0^{r/s}v^a(1-v)^bdv-\int_0^{r/t}v^a(1-v)^bdv\right)\\
&\qquad +C_{a,b}\int_0^{r/t}v^a(1-v)^bdv
\left(t^{1+a+b}-s^{1+a+b}\right)\\
&\leq C_{a,b} (t-s)^{1+b}r^a+C_{a,b}(\frac{r}{t})^{1+a} \left(t^{1+a+b}-s^{1+a+b}\right)\\
&\leq C_{a,b}(t-s)^{1+b}r^a,
\end{align*}
and the lemma follows.
\end{proof}
Let $\varphi_{t,s}(x,y)$ denote the density function of $(B^{a,b}_t,B^{a,b}_s)$ ($t>s>0$). That is
\begin{equation}\label{sec4-eq4.2}
\varphi_{t,s}(x,y)=\frac1{2\pi\rho_{t,s}}\exp\left\{
-\frac{1}{2\rho^2_{t,s}}\left(s^{1+a+b}x^2-2\mu_{t,s}
xy+t^{1+a+b}y^2\right)\right\},
\end{equation}
where $\mu_{t,s}=E(B^{a,b}_tB^{a,b}_s)$, $E\left[(B^{a,b}_t)^2\right]=t^{1+a+b}$ and
$\rho^2_{t,s}=(ts)^{1+a+b}-\mu_{t,s}^2$.
\begin{lemma}\label{lem3.5}
Let $a>-1,\;|b|<1,\;|b|<a+1$. If $f\in C^1({\mathbb R})$ admits compact support, we then have
\begin{align}\label{lemma3.5-1}
|Ef'(B^{a,b}_{s})f'(B^{a,b}_{r})|&\leq
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(E[f^2(B^{a,b}_s)]
+E[f^2(B^{a,b}_r)]\right)\\ \label{lemma3.5-2}
|Ef''(B^{a,b}_{s})f(B^{a,b}_{r})|&\leq
\frac{r^{1+a+b}}{\rho^2_{s,r}}\left(E[f^2(B^{a,b}_s)]
+E[f^2(B^{a,b}_r)]\right)
\end{align}
for all $s,r>0$.
\end{lemma}
\begin{proof}
Elementary computation shows that
\begin{align*}
\int_{\mathbb{R}^2}f^2(y)&(x-\frac{\mu_{t,s}}{r^{1+a+b}}
y)^2\varphi_{s,r}(x,y)dxdy\\
&=\frac{\rho^2_{s,r}}{r^{1+a+b}}\int_{\mathbb{R}}f^2(y)
\frac{1}{\sqrt{2\pi}r^{(1+a+b)/2}}
e^{-\frac{y^2}{2r^{1+a+b}}}dy=\frac{\rho^2_{s,r}}{r^{1+a+b}}
E|f(B^{a,b}_r)|^2,
\end{align*}
which implies that
\begin{align*}
\frac1{\rho^4_{s,r}}\int_{\mathbb{R}^2}|f(x)f(y)
(&s^{1+a+b}y-\mu_{s,r}x)(r^{1+a+b}x-\mu_{s,r}y)|\varphi_{s,r}(x,y)dxdy\\
&\leq \frac{r^{(1+a+b)/2}s^{(1+a+b)/2}}{\rho^2_{s,r}}\left(
E|f(B^{a,b}_s)|^2E|f(B^{a,b}_r)|^2\right)^{1/2}
\end{align*}
for all $s,r>0$. It follows that
\begin{align*}
|E&[f'(B^{a,b}_{s})f'(B^{a,b}_{r})]|=|\int_{\mathbb{R}^2}
f(x)f(y)\frac{\partial^{2}}{\partial x\partial
y}\varphi_{s,r}(x,y)dxdy|\\
&=|\int_{\mathbb{R}^2} f(x)f(y)\left\{\frac1{\rho^4_{s,r}}(s^{1+a+b}y-\mu_{s,r}x)(r^{1+a+b}x -\mu_{s,r}y)+\frac{\mu_{s,r}}{\rho^2_{s,r}}\right\}\varphi_{s,r}(x,y)dxdy|\\
&\leq \left(\frac{(rs)^{(1+a+b)/2}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(
E|f(B^{a,b}_s)|^2E|f(B^{a,b}_r)|^2\right)^{1/2}.
\end{align*}
Similarly, one can show that the estimate~\eqref{lemma3.5-2} holds.
\end{proof}
\section{The generalized quadratic covariation, an example} \label{sec6}
In this section, for $|b|<1$ we consider an example of Borel functions such that the generalized quadratic covariation exists. Recall that
$$
J_\varepsilon(f,t)=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}\left\{ f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds
$$
for all $\varepsilon>0$ and all Borel functions $f$, and the generalized quadratic covariation defined as follows
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=\lim_{\varepsilon\downarrow 0}J_\varepsilon(f,t),
$$
provided the limit exists uniformly in probability (in short, ucp). For ucp-convergence we have the next result due to Russo {\em et al}~\cite{Russo2}.
\begin{lemma}[Russo {\em et al}~\cite{Russo2}]\label{lemm3.1-1}
Let $\{X^\varepsilon,\;\varepsilon>0\}$ be a family of continuous processes. We suppose
\begin{itemize}
\item For any $\varepsilon>0$, the process $t\mapsto X^\varepsilon_t$ is increasing;
\item There is a continuous process $X=(X_t,t\geq 0)$ such that $X^\varepsilon_t\to X_t$ in probability as $\varepsilon$ goes to zero.
\end{itemize}
Then $Z^\varepsilon$ converges to $X$ ucp.
\end{lemma}
\begin{proposition}\label{prop4.1}
Let $a>-1$, $|b|<1$ and $|b|<1+a$. Then we have
\begin{equation}\label{sec4-eq4.111111}
[B^{a,b},B^{a,b}]^{(a,b)}_t=\kappa_{a,b}t^{1+a+b},
\end{equation}
where
$$
\kappa_{a,b}=\frac1{(1+b){\mathbb B}(a+1,b+1)}.
$$
\end{proposition}
Let now $(X,Y)$ be a $2$-dimensional normal random variable with the density
$$
\varphi(x,y)=\frac1{2\pi}e^{-\frac1{2\rho^2} (\sigma_2^2x^2-2\mu xy+\sigma_1^2y^2)},
$$
where $E[X]=E[Y]=0,\sigma_1^2=E[X^2],\sigma_2^2=E[Y^2],\mu= E[XY]$ and $\rho^2=\sigma_1^2\sigma_2^2-\mu^2$. Then, an elementary calculus can show that
\begin{equation}\label{sec4-eq4.1-11}
E[X^2Y^2]=E[X^2]E[Y^2]+2(E[XY])^{2}.
\end{equation}
Denote by
\begin{align*}
h_s(\varepsilon):=E(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2 -\kappa_{a,b}\varepsilon^{1+b}s^a
\end{align*}
for $\varepsilon\in (0,1)$ and $s>0$. By making substitution $u-s=+v\varepsilon$ we have
\begin{align*}
h_s(\varepsilon)&=\frac{1}{{\mathbb B}(a+1,b+1)}\int_s^{s+\varepsilon}u^a(s+\varepsilon-u)^bdu -\kappa_{a,b}\varepsilon^{1+b}s^a\\
&=\frac{\varepsilon^{1+b}}{{\mathbb B}(a+1,b+1)}\int_0^{1}(s+\varepsilon v)^a(1-v)^bdu-\kappa_{a,b}\varepsilon^{1+b}s^a\\
&=\varepsilon^{1+b}\kappa_{a,b}\left((1+b)\int_0^{1}(s+\varepsilon v)^a(1-v)^bdu-s^a\right)\\
&=\frac{\varepsilon^{1+b}}{{\mathbb B}(a+1,b+1)}\int_0^{1}\left\{(s+\varepsilon v)^a-s^a\right\}(1-v)^bdu.
\end{align*}
Notice that
$$
\left|(s+\varepsilon v)^a-s^a\right|=(s+\varepsilon v)^a-s^a\leq
(\varepsilon v)^a
$$
for all $0<a\leq 1,\;\varepsilon>0,\;0\leq v\leq 1$, and
$$
\left|(s+\varepsilon v)^a-s^a\right|=(s+\varepsilon v)^a-s^a\leq
C_a\varepsilon v(s+\varepsilon v)^{a-1}\leq C_a\varepsilon v(s+\varepsilon)^{a-1}
$$
for all $a>1,\;\varepsilon>0,\;0\leq v\leq 1$, and
\begin{align*}
\left|(s+\varepsilon v)^a-s^a\right|&=s^a-(s+\varepsilon v)^a
=s^{a}\left(1-(\frac{s}{s+\varepsilon v})^{-a}\right)\\
&\leq s^{a}\left(1-\frac{s}{s+\varepsilon v}\right)=s^{a}\frac{\varepsilon v}{s+\varepsilon v}
\leq s^a\frac{\varepsilon v}{s^\nu(\varepsilon v)^{1-\nu}}
=(\varepsilon v)^{\nu}s^{a-\nu}
\end{align*}
for all $-1<a<0,\;\varepsilon>0,\;0\leq v\leq 1$ and all $0<\nu<1+a$ by Young's inequality. We get
\begin{equation}\label{sec6-eq6.6}
h_s(\varepsilon)\leq
\begin{cases}
C_{a,b}(s+\varepsilon)^{a-1}\varepsilon^{2+b},& {\text { if $a>1$}},\\
C_{a,b}\varepsilon^{1+b+a},& {\text { if $0<a\leq 1$}},\\
C_{a,b}s^{a-\nu}\varepsilon^{1+b+\nu},& {\text { if $-1<a<0$ \;($0<\nu<1+a$)}}
\end{cases}
\end{equation}
for all $s>0$ and $\varepsilon>0$.
\begin{proof}[Proof of Proposition~\ref{prop4.1}]
Denote
$$
X_\varepsilon(t)=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds
$$
for all $0<\varepsilon<1$. Then, it is sufficient to show that
$$
E\left(X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b}\right)^2\longrightarrow 0,
$$
as $\varepsilon$ tends to zero. We have
\begin{align*}
X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b} &=\frac{1+a+b}{\varepsilon^{1+b}}\left(\int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds -\kappa_{a,b}\frac{\varepsilon^{1+b}t^{1+a+b}}{1+a+b}\right)\\
&=\frac{1+a+b}{\varepsilon^{1+b}}\left(\int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds -\kappa_{a,b}\varepsilon^{1+b} \int_\varepsilon^{t+\varepsilon}s^{a+b}ds\right)\\
&\qquad+\kappa_{a,b}\left( (1+a+b) \int_\varepsilon^{t+\varepsilon}s^{a+b}ds- t^{1+a+b}\right)\\
&\equiv \Xi_1(a,b,\varepsilon,t)+\Xi_2(a,b,\varepsilon,t).
\end{align*}
We deduce from above
\begin{equation}\label{sec6-eq6.7}
E\left|X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b}\right|^2
\leq 2E|\Xi_1(a,b,\varepsilon,t)|^2+2|\Xi_2(a,b,\varepsilon,t)|^2
\end{equation}
for $t\geq 0$ and $\varepsilon>0$. Clearly, we have
\begin{align}\notag
|\Xi_2(a,b,\varepsilon,t)|&=\kappa_{a,b}\left| (1+a+b)\int_\varepsilon^{t+\varepsilon}s^{a+b}ds-t^{1+a+b}\right|\\
\label{sec6-eq6.8}
&=\kappa_{a,b}\left|(t+\varepsilon)^{1+a+b}-\varepsilon^{1+a+b} -t^{1+a+b}\right|=O(\varepsilon^{1\wedge (1+a+b)})
\end{align}
for each $t\geq 0$.
Now, let us estimate $E|\Xi_1(a,b,\varepsilon,t)|^2$. We have
\begin{equation}\label{sec6-eq6.9}
\begin{split}
E|\Xi_1(a,b,\varepsilon,t)|^2&=\frac{(1+a+b)^2}{\varepsilon^{2+2b}} E\left(\int_\varepsilon^{t+\varepsilon}
\left\{(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2-\kappa_{a,b} \varepsilon^{1+b}s^{a}\right\}s^{b}ds\right)^2\\
&=\frac{(1+a+b)^2}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{t+\varepsilon}A_\varepsilon(s,r)(sr)^{b}dsdr
\end{split}
\end{equation}
for each $t\geq 0$, where
\begin{align*}
A_\varepsilon(s,r):&=E\left(
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2-\kappa_{a,b} \varepsilon^{1+b}s^a\right)\left(
(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2-\kappa_{a,b} \varepsilon^{1+b}r^a\right)\\
&=E(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2(B^{a,b}_{r+\varepsilon} -B^{a,b}_r)^2+(\kappa_{a,b})^2\varepsilon^{2+2b}(sr)^a\\
&\qquad-\kappa_{a,b}\varepsilon^{1+b}
E\left((B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2s^a +(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2r^a\right).
\end{align*}
For all $r,s\geq 0$ and $\varepsilon>0$ by decomposing
\begin{align*}
E(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2 =h_s(\varepsilon)+\kappa_{a,b}\varepsilon^{1+b}s^a
\end{align*}
we have
\begin{align*}
E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2(B^{a,b}_{r+\varepsilon} -&B^{a,b}_r)^2]=
E\left[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2\right]
E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2\right]\\
&\hspace{2cm}+2\left(E\left[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)
(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]\right)^{2}\\
&=\left[h_s(\varepsilon)+\kappa_{a,b}\varepsilon^{1+b}s^a\right]
\left[h_r(\varepsilon)+\kappa_{a,b}\varepsilon^{1+b}s^a \right]+2(\mu_{s,r})^2
\end{align*}
by~\eqref{sec4-eq4.1-11}, where $\mu_{s,r}:=E\left[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)
(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]$, which yields
\begin{equation}\label{sec6-eq6.10}
A_\varepsilon(s,r)=h_s(\varepsilon)h_r(\varepsilon)+2(\mu_{s,r})^2.
\end{equation}
On the other hand, for $0<s-r<\varepsilon\leq 1$ we have
\begin{align*}
(\mu_{s,r})^2&\leq E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2]E[(B^{a,b}_{r+\varepsilon} -B^{a,b}_{r})^2]\\
&\leq (s+\varepsilon)^a(r+\varepsilon)^a\varepsilon^{2+2b}
\leq (s+\varepsilon)^a(r+\varepsilon)^a\frac{\varepsilon^{2+2b+\gamma} }{(s-r)^\gamma}
\end{align*}
for all $0\leq \gamma\leq 1$. It follows from Lemma~\ref{lem3.3} that
\begin{equation}\label{sec6-eq6.11}
\begin{split}
(\mu_{s,r})^2&\leq C_{a,b}\left(r^a\vee s^a\right)^{2\alpha} [(s+\varepsilon)(r+\varepsilon)]^{a(1-\alpha)} \frac{\varepsilon^{2+2b+2\alpha(1-b)} }{(s-r)^{2(1+b)\alpha}}1_{\{s-r>\varepsilon\}}\\
&\qquad+(s+\varepsilon)^a(r+\varepsilon)^a \frac{\varepsilon^{2+2b+\alpha} }{(s-r)^\alpha}1_{\{s-r\leq \varepsilon\}}
\end{split}
\end{equation}
for all $0<\alpha<\frac1{2(1+b)}\wedge 1$. Combining this with~\eqref{sec6-eq6.6},~\eqref{sec6-eq6.7},~\eqref{sec6-eq6.8}, ~\eqref{sec6-eq6.9} and~\eqref{sec6-eq6.10} we see that
\begin{align*}
E|\Xi_1(a,b,\varepsilon,t)|^2&=\frac{(1+a+b)^2}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{t+\varepsilon}A_\varepsilon(s,r)(sr)^{b}drds \longrightarrow 0
\end{align*}
for all $t\geq 0$, as $\varepsilon$ tends to zero, and the proposition follows.
\end{proof}
Recall that the local H\"{o}lder index $\gamma_0$
of a continuous paths process $\{X_t: t\geq 0\}$ is the supremum of
the exponents $\gamma$ verifying, for any $T>0$:
$$
P(\{\omega: \exists L(\omega)>0, \forall s,t \in[0,T],
|X_t(\omega)-X_s(\omega)|\leq L(\omega)|t-s|^\gamma\})=1.
$$
The next lemma is considered by Gradinaru-Nourdin~\cite{Grad3}.
\begin{lemma}\label{Grad-Nourdin}
Let $g:{\mathbb R}\to {\mathbb R}$ be a continuous function satisfying
\begin{equation}\label{eq4.2-Gradinaru--Nourdin}
|g(x)-g(y)|\leq C|x-y|^a(1+x^2+y^2)^b,\quad (C>0,0<a\leq 1,b>0),
\end{equation}
for all $x,y\in {\mathbb R}$ and let $X$ be a locally H\"older
continuous paths process with index $\gamma\in (0,1)$. Assume that
$V$ is a bounded variation continuous paths process. Set
$$
X^{g}_\varepsilon(t)=\int_\varepsilon^{t+\varepsilon} g(\frac{X_{s+\varepsilon}-X_s
}{\varepsilon^\gamma})ds
$$
for $t\geq 0$, $\varepsilon>0$. If for each $t\geq 0$, as
$\varepsilon\to 0$,
\begin{equation}\label{condition}
\|X^{g}_\varepsilon(t)-V_t\|_{L^2}^2=O(\varepsilon^\alpha)
\end{equation}
with $\alpha>0$, then, for any $t\geq 0$, $\lim_{\varepsilon\to
0}X^{g}_\varepsilon(t)=V_t$ almost surely, and if $g$ is
non-negative, for any continuous stochastic process $\{Y_t:\;t\geq
0\}$,
\begin{equation}
\lim_{\varepsilon\to 0}
\int_\varepsilon^{t+\varepsilon} Y_sg(\frac{X_{s+\varepsilon}-X_s}{\varepsilon^\gamma})ds
\longrightarrow \int_0^tY_sdV_s,
\end{equation}
almost surely, uniformly in $t$ on each compact interval.
\end{lemma}
Recall that
$$
X_\varepsilon(t)=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds
$$
for all $0<\varepsilon<1$. The proof of Proposition~\ref{prop4.1} points out that there exists $\beta\in (0,1)$ such that
\begin{align*}
\|X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b}\|_{L^2}=O(\varepsilon^{\beta}), \qquad \varepsilon\to 0
\end{align*}
for all $t\geq 0$. Notice that $g(x)=x^2$ satisfies the condition~\eqref{eq4.2-Gradinaru--Nourdin}. We obtain the convergence
\begin{align*}
\lim_{\varepsilon\downarrow 0}\frac{1+a+b}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_s)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds
\longrightarrow \kappa_{a,b}\int_0^tf(B^{a,b}_s)ds^{1+a+b},
\end{align*}
almost surely, uniformly in $t$ on each compact interval by taking $Y_s=f(B^{a,b}_s)s^{b}$ for $s\geq 0$. Clearly, by the H\"older continuity of weighted-fBm $B^{a,b}$, we get
\begin{align*}
\frac{1+a+b}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon}o(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds\longrightarrow 0
\end{align*}
in $L^1(\Omega)$, as $\varepsilon \to 0$. Thus, we have obtained the next result.
\begin{corollary}
Let $a>-1$, $|b|<1$, $|b|<1+a$ and let $f\in
C^1({\mathbb R})$. Then the generalized quadratic covariation $[f(B^{a,b}),B^{a,b}]^{(a,b)}$ exists and
\begin{align}\label{sec6-eq6.2}
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t&=\kappa_{a,b} \int_0^tf'(B^{a,b}_s)ds^{1+a+b}.
\end{align}
\end{corollary}
\section{The generalized quadratic covariation, existence}~\label{sec5}
In this section, for $-1<b<0$ we shall study the existence of the
{\it generalized quadratic covariation} more general class of
functions than the one of class $C^1$. Recall that
$$
J_\varepsilon(f,t):=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}\left\{ f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds
$$
for all $\varepsilon>0$ and all Borel functions $f$. Consider the decomposition
\begin{equation}\label{sec4-eq4.000000}
\begin{split}
\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} &\left\{f(B^{a,b}_{
s+\varepsilon})-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)s^{b}ds\\
&=\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_{
s+\varepsilon})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds\\
&\qquad\qquad\qquad-\frac{1}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_s)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds\\
&\equiv I_\varepsilon^{+}(f,t)-I_\varepsilon^{-}(f,t),
\end{split}
\end{equation}
and define the set ${\mathscr H}$ of measurable functions $f$ on
${\mathbb R}$ such that $\|f\|_{{\mathscr H}}<\infty$, where
\begin{align*}
\|f\|_{{\mathscr H}}:=&\sqrt{\int_0^{T+1}\int_{\mathbb
R}|f(x)|^2e^{-\frac{x^2}{2s^{1+a+b}
}}\frac{dxds}{\sqrt{2\pi}s^{(1-a-b)/2}}}.
\end{align*}
For the set ${\mathscr H}$ we have
\begin{itemize}
\item [(1)] ${\mathscr H}$ is a Banach space.
\item [(2)] ${\mathscr H}\supset C^\infty_0({\mathbb R})$, the set of infinitely differentiable functions with compact support.
\item [(3)] the set ${\mathscr E}$ of elementary functions of the form
$$
f_\triangle(x)=\sum_{i}f_{i}1_{(x_{i-1},x_{i}]}(x)
$$
is dense in ${\mathscr H}$, where $\{x_i,0\leq i\leq l\}$ is an
finite sequence of real numbers such that $x_i<x_{i+1}$.
\item [(4)] the space ${\mathscr H}$ contains all Borel functions $f$ satisfying
$$
|f(x)|\leq Ce^{\beta x^2},\qquad x\in {\mathbb R}
$$
with $\beta<\frac14T^{-(1+a+b)}$.
\end{itemize}
For simplicity we let $T=1$ in the following discussions. The main result of this section is to explain and prove the
following theorem.
\begin{theorem}\label{th6.2}
Let $a>-1$, $-1<b<0$, $|b|<1+a$ and $f\in {\mathscr H}$. Then the generalized quadratic covariation $[f(B^{a,b}),B^{a,b}]^{(a,b)}$ exists, and we have
\begin{align}\label{th4.1-eq}
E\left|[f(B^{a,b}),B^{a,b}]^{(a,b)}_t\right|^2\leq C_{a,b,T}\|f\|_{{\mathscr H}}^2
\end{align}
for all $a\geq 0,t\in [0,1]$.
\end{theorem}
For simplicity we let $T=1$ in the rest of this section.
\begin{lemma}\label{lem6.2}
Let $a>-1,\;-1<b<0,\;|b|<1+a$ and let $f\in {\mathscr H}$. We then have
\begin{align}
&E\left|I_\varepsilon^{-}(f,t)\right|^2\leq C_{a,b}\|f\|_{{\mathscr H}}^2,\\
&E\left|I_\varepsilon^{+}(f,t)\right|^2\leq C_{a,b}\|f\|_{{\mathscr H}}^2
\end{align}
for all $0<\varepsilon<1$ and $t\in [0,1]$.
\end{lemma}
\begin{proof}
We prove only the first estimate, and similarly one can prove the second estimate. By the denseness of ${\mathscr E}$ in ${\mathscr H}$ we only need to show that the lemma holds for all $f\in {\mathscr E}$. Consider the function $\zeta$ on ${\mathbb R}$ by
\begin{equation}
\zeta(x):=
\begin{cases}
ce^{\frac1{(x-1)^2-1}}, &{\text { $x\in (0,2)$}},\\
0, &{\text { otherwise}},
\end{cases}
\end{equation}
where $c$ is a normalizing constant such that $\int_{\mathbb
R}\zeta(x)dx=1$, and define the mollifiers
\begin{equation}\label{sec7-eq7.4}
\zeta_n(x):=n\zeta(nx),\qquad n=1,2,\ldots.
\end{equation}
For $f_\triangle\in {\mathscr E}$ consider the sequence of smooth functions
$$
f_{\triangle,n}(x):=\int_{\mathbb R}f_{\triangle}(x-{y})\zeta_n(y)dy=\int_0^2 f_{\triangle}(x-\frac{y}n)\zeta(y)dy,\quad x\in {\mathbb R}.
$$
Then $f_{\triangle,n}\in C_0^\infty({\mathbb R})$ for all $n\geq 1$ and $f_{\triangle,n}\to f_{\triangle}$ in ${\mathscr H}$, as $n$ tends to infinity. Thus, by approximating we may assume that $f$ is an infinitely differentiable function with compact support. It follows from the duality relationship that
\begin{equation}\label{sec4-eq4.800}
\begin{split}
E[f(&B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]\\
&=E\left[f(B^{a,b}_{s})f(B^{a,b}_{
r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) \int_r^{r+\varepsilon}dB^{a,b}_l\right]\\
&=E\left\langle D^{a,b}\left(
f(B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right), 1_{[r,r+\varepsilon]}\right\rangle_{{\mathcal H}_{a,b}}\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]E\left[
f'(B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) \right]\\
&\qquad +E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]E\left[
f(B^{a,b}_{s})f'(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) \right]\\
&\qquad
+E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s) \right]E\left[
f(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]\\
&\qquad
+E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)\right]E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&\equiv \sum_{i=1}^5 \Psi_i(s,r,\varepsilon).
\end{split}
\end{equation}
In order to end the proof we claim now that
\begin{equation}\label{eq4.4}
\frac{1}{\varepsilon^{2+2b}}\left| \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon} \Psi_i(s,r,\varepsilon)(sr)^{b}dsdr \right|\leq C_{a,b}\|f\|^2_{{\mathscr H}},\qquad i=1,2,\ldots,5
\end{equation}
for all $\varepsilon>0$ small enough.
{\bf For $i=5$} we have
\begin{align*}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|&\leq \sqrt{E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2E(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)^2]}\\
&\leq C_{a,b}[(r+\varepsilon)(s+\varepsilon)]^{a/2} \varepsilon^{1+b}\\
&\leq C_{a,b}[(r+\varepsilon)(s+\varepsilon)]^{a/2}\frac{ \varepsilon^{2+2b}}{(s-r)^{1+b}}
\end{align*}
for $0<s-r<\varepsilon\leq 1$, which gives
\begin{align*}
I_\varepsilon:&= \frac{1}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_{s-\varepsilon}^s
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|E[f^2(B^{a,b}_{s})](sr)^{b}
drds\\
&\leq C_{a,b}\int_\varepsilon^{t+\varepsilon}\int_{s-\varepsilon}^s \frac{(r^a\vee s^a)(sr)^{b}}{(s-r)^{1+b}} E[f^2(B^{a,b}_{s})]drds\\
&\leq C_{a,b}\int_\varepsilon^{t+\varepsilon}
s^bEf^2(B^{a,b}_{s})ds
\int_0^s \frac{r^{b}(r^a\vee s^a)}{(s-r)^{1+b}} dr\\
&=C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{b+a}Ef^2(B^{a,b}_{s})ds \leq C_{a,b}\int_0^{2}s^{a+b}Ef^2(B^{a,b}_{s})ds
\end{align*}
for $0<s-r<\varepsilon\leq 1$. Moreover, by~\eqref{sec3-eq3.8==} with $\beta=1$ and $\nu=-b$ we have
\begin{align*}
II_\varepsilon:&=\frac{1}{\varepsilon^{2+2b}} \int_{\varepsilon}^{t+\varepsilon} \int_\varepsilon^{s-\varepsilon}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|E[f^2(B^{a,b}_{s})] (sr)^{b}drds \\
&\leq C_{a,b}\int_{\varepsilon}^{t+\varepsilon}
s^bE[f^2(B^{a,b}_{s})]ds\int_\varepsilon^{s-\varepsilon}(r^a\vee s^a)\frac{r^bdr}{(s-r)^{1+b}} \\
&\leq C_{a,b} \int_{\varepsilon}^{t+\varepsilon}s^b Ef^2(B^{a,b}_{s})ds \int_0^s
\frac{r^{b}(r^a\vee s^a)}{(s-r)^{1+b}}dr\\
&\leq C_{a,b}\int_{\varepsilon}^{t+\varepsilon}s^{a+b} Ef^2(B^{a,b}_{s})ds\leq C_{a,b}\int_{0}^{2}s^{a+b} Ef^2(B^{a,b}_{s})ds
\end{align*}
for all $0<\varepsilon\leq 1$. It follows that
\begin{align*}
& \frac{1}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon} |\Psi_5(s,r,\varepsilon)|(sr)^{b} drds \\
&\leq \frac{1 }{2\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|[Ef^2(B^{a,b}_{s})+Ef^2(B^{a,b}_{r})] (sr)^{b}drds\\
&\leq \frac{1}{\varepsilon^{2+2b}}\int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{t+\varepsilon}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|Ef^2(B^{a,b}_{s})(sr)^{b}drds\\
&\leq \frac{2}{\varepsilon^{2+2b}}\int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{s}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|Ef^2(B^{a,b}_{s})(sr)^{b}drds\\
&\leq 2(I_\varepsilon+II_\varepsilon)\leq C_{a,b}
\int_{0}^{2}s^{a+b} E[f^2(B^{a,b}_{s})]ds
\end{align*}
for all $0<\varepsilon\leq 1$.
{\bf For $i=1$}, by Lemma~\ref{lem3.4} and Lemma~\ref{lem3.5} we have
\begin{align*}
\frac{1}{\varepsilon^{2+2b}}& \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon} |\Psi_1(s,r,\varepsilon)|(sr)^{b}drds\\
&\leq \frac{C_{a,b}}{2\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon}
|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]
E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(Ef^2(B^{a,b}_s)
+Ef^2(B^{a,b}_r)\right)(sr)^{b}dsdr\\
&= \frac{C_{a,b}}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon}
|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]
E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right) Ef^2(B^{a,b}_s)(sr)^{b}dsdr\\
&\leq C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{a+b}ds\int_\varepsilon^{s}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)Ef^2(B^{a,b}_s)r^{a+b}dr\\
&\qquad +C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{a+b}ds \int_s^{s+\varepsilon}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)Ef^2(B^{a,b}_s)r^{a+b}dr\\
&\qquad +C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{2a+b}ds \int_{s+\varepsilon}^{t+\varepsilon}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)Ef^2(B^{a,b}_s)r^{b}dr\\
&\leq C_{a,b}
\int_{0}^{2}s^{a+b} E[f^2(B^{a,b}_{s})]ds
\end{align*}
for all $0<\varepsilon\leq 1$. Similarly, we can obtain the estimate~\eqref{eq4.4} for $i=2,3,4$, and
the lemma follows.
\end{proof}
\begin{lemma}\label{lem6.3}
Let $a>-1,\;-1<b<0,\;|b|<1+a$ and let $f$ be an infinitely differentiable function with compact support. We then have
\begin{align}
\Pi_1(t,\varepsilon_i,\varepsilon_j):= \frac1{\varepsilon_i^{2+2b}} E\Bigl|\int_{t+\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_i}-B^{a,b}_s)s^bds\Bigr|^2\longrightarrow 0
\end{align}
for all $0<\varepsilon_2<\varepsilon_1<1$ and $t\in [0,1]$, as $\varepsilon_1\to 0$, where $i,j=1,2$ and $i\neq j$.
\end{lemma}
\begin{proof}
From the proof of Lemma~\ref{lem6.2} one can easily prove the result.
\end{proof}
Now we can show our main result.
\begin{proof}[Proof of Theorem~\ref{th6.2}]
From Lemma~\ref{lem6.2}, it is enough to show that $I_{\varepsilon_1}^{-}(f,t)$ and $I_{\varepsilon_1}^{+}(f,t)$ are two Cauchy's sequences in $L^2(\Omega)$ for all $t\in [0,1]$. That is,
\begin{equation}\label{sec40-eq3-1}
E\left|I_{\varepsilon_1}^{-}(f,t)-I_{\varepsilon_2}^{-}(f,t)\right|^2
\longrightarrow 0,
\end{equation}
and
\begin{equation}\label{sec40-eq3-2}
E\left|I_{\varepsilon_1}^{+}(f,t)-I_{\varepsilon_2}^{+}(f,t)\right|^2
\longrightarrow 0
\end{equation}
for all $t\in [0,1]$, as $\varepsilon_1,\varepsilon_2\downarrow 0$. We prove only the convergence~\eqref{sec40-eq3-1}, and similarly one can prove~\eqref{sec40-eq3-2}. Without loss of generality one may assume that $\varepsilon_1>\varepsilon_2$. It follows that
\begin{align*}
&E\bigl|I_{\varepsilon_1}^{-}(f,t)-I_{\varepsilon_2}^{-}(f,t)\bigr|^2
\\
&\leq 3E\Bigl|\frac1{\varepsilon_1^{1+b}} \int_{t+\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)s^bds\Bigr|^2\\
&\qquad+3E\Bigl|\frac1{\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_2}-B^{a,b}_s)s^bds\Bigr|^2\\
&+3E\Bigl|\frac1{\varepsilon_1^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)s^bds- \frac1{\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_2}-B^{a,b}_s)s^bds\Bigr|^2\\
&\equiv \Pi_1(t,\varepsilon_1,\varepsilon_2) +\Pi_1(0,\varepsilon_2,\varepsilon_1) +\Pi_2(t,\varepsilon_1,\varepsilon_2).
\end{align*}
In order to end the proof, it is enough to check $\Pi_2(t,\varepsilon_1,\varepsilon_2)\to 0$, as $\varepsilon_1,\varepsilon_2\to 0$. We have
\begin{align*}
\Pi_2&(t,\varepsilon_1,\varepsilon_2)\\
&=\frac1{\varepsilon_1^{2+2b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} Ef(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s) (B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)(sr)^bdrds\\
&\quad-
\frac2{\varepsilon_1^{1+b}\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} Ef(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)(sr)^bdrds\\
&\quad+\frac1{\varepsilon_2^{2+2b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} Ef(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_2}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)(sr)^bdrds\\
&\equiv \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}\left\{\varepsilon_2^{1+b} \Phi_{s,r}(1,\varepsilon_1)-\varepsilon_1^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\right\}(sr)^bdrds\\
&\quad+
\frac1{\varepsilon_1^{1+b}\varepsilon_2^{2+2b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} \left\{\varepsilon_1^{1+b}\Phi_{s,r}(1,\varepsilon_2)
-\varepsilon_2^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\right\}(sr)^bdrds,
\end{align*}
where
$$
\Phi_{s,r}(1,\varepsilon) =E\left[f(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right],
$$
and
$$
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2) =E\left[f(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right].
$$
In order to estimate $\Phi_{s,r}(1,\varepsilon)$ and $\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)$, by approximating we may assume that $f$ is an infinitely differentiable function with compact support. We then have by~\eqref{sec4-eq4.800},
\begin{align*}
\Phi_{s,r}(1,\varepsilon)&=\sum_{i=1}^5\Psi_i(s,r,{\varepsilon})\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&+E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right] E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]\\
&\qquad\qquad+E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]E\left[
f(B^{a,b}_{s})f(B^{a,b}_{r})\right]
\end{align*}
and
\begin{align*}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)&=
E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]E\left[
f'(B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) \right]\\
&\qquad +E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]E\left[
f(B^{a,b}_{s})f'(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) \right]\\
&\qquad\qquad +E\left[(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)
\right]E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]
E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&+E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad +E\left[B_{r}(B^{a,b}_{r+\varepsilon_2}-B_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad +E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]
E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]\\
&\qquad\qquad +E\left[(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)
\right]E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right].
\end{align*}
Denote
\begin{align*}
A_1(s,r,\varepsilon,j):&=\varepsilon_j^{1+b} E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b} E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_{11}(s,r,\varepsilon,j):&=\varepsilon_j^{1+b}
E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right] E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b} E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_{12}(s,r,\varepsilon,j):&=\varepsilon_j^{1+b}
E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b}
E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_3(s,r,\varepsilon,j):&=\varepsilon_j^{1+b} E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b} E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_4(s,r,\varepsilon,j):&=\varepsilon_j^{1+b}
E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r) (B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b}
E\left[(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
\end{align*}
with $j=1,2$. It follows that
\begin{align*}
\varepsilon_j^{1+b}&\Phi_{s,r}(1,\varepsilon_i) -\varepsilon_i^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\\
&=A_1(s,r,\varepsilon_i,j)E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&\qquad
+(A_{11}(s,r,\varepsilon_i,j)+A_{12}(s,r,\varepsilon_i,j)) E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+A_3(s,r,\varepsilon_i,j)E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]
+A_4(s,r,\varepsilon_i,j)E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right]
\end{align*}
with $i,j\in\{1,2\}$ and $i\neq j$. In order to end the proof we claim
that the following convergence
\begin{align}\label{sec4-Con-eq1}
\frac1{\varepsilon_i^{2+2b}\varepsilon_j^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} \left\{\varepsilon_j^{1+b}\Phi_{s,r}(1,\varepsilon_i) -\varepsilon_i^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\right\}(sr)^bdrds \longrightarrow 0
\end{align}
with $i,j\in\{1,2\}$ and $i\neq j$, as $\varepsilon_1,\varepsilon_2\to 0$. By symmetry, we only need to show that this holds for $i=1,j=2$. This will be done in three parts.
{\bf Part I}. The following convergence holds:
\begin{align}\label{step1-eq1}
\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\frac{A_4(s,r,\varepsilon_1,2)} {\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} E[f(B^{a,b}_{s})f(B^{a,b}_{r})](sr)^bdrds \longrightarrow 0,
\end{align}
as $\varepsilon_1,\varepsilon_2\to 0$. Clearly, we have
\begin{align*}
|E[(B^{a,b}_{r+\varepsilon_i}-B^{a,b}_r)&(B^{a,b}_{s+\varepsilon_j} -B^{a,b}_s)]|\leq \sqrt{E[(B^{a,b}_{r+\varepsilon_i}-B^{a,b}_r)^2 E(B^{a,b}_{s+\varepsilon_j} -B^{a,b}_s)^2]}\\
&\leq C_{a,b}[(r+\varepsilon_i)(s+\varepsilon_j)]^{a/2} \varepsilon_i^{(1+b)/2}\varepsilon_j^{(1+b)/2}\\
&\leq C_{a,b}[(r+\varepsilon_i)(s+\varepsilon_j)]^{a/2} \frac{\varepsilon_i^{1+b+\gamma} \varepsilon_j^{1+b}
}{(s-r)^{1+b+\gamma}}
\end{align*}
for $0<s-r<\varepsilon_i\wedge \varepsilon_j\leq 1$ and $0<\gamma<-b$, where $i,j\in \{1,2\}$. Combining this with~\eqref{sec3-eq3.8==} (taking $\beta=1$), we get
\begin{align*}
|E[(&B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)]|\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right) \left(\frac{\varepsilon_1^{2+b-\nu}}{(s-r)^{1-\nu}}
1_{\{s-r>\varepsilon_1\}}+ \frac{\varepsilon_1^{2+2b+\gamma}}{(s-r)^{1+b+\gamma}}
1_{\{0<s-r\leq \varepsilon_1\}}\right)\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right) \frac{\varepsilon_1^{2+b-\nu}}{(s-r)^{1-\nu}}
\end{align*}
and
\begin{align*}
|E[(&B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right)\left( \frac{\varepsilon_1^{1+b-\nu}\varepsilon_2}{(s-r)^{1-\nu}}
1_{\{s-r>\varepsilon_2\}}+ \frac{\varepsilon_1^{1+b+\gamma}\varepsilon_2^{1+b}}{(s-r)^{1+b+\gamma}}
1_{\{0<s-r\leq \varepsilon_2\}}\right)\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right) \frac{\varepsilon_1^{1-\nu}\varepsilon_2^{1+b}}{(s-r)^{1-\nu}}
\end{align*}
for all $s>r>0$ and $0<\nu<(1+b)\wedge (-b)$ by taking $b+\gamma+\nu\geq 0$. It follows that
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |A_4(s,r,\varepsilon_1,2)|&\leq \frac{C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right)}{(s-r)^{1-\nu}} \varepsilon_1^{-b-\nu}\longrightarrow 0
\end{align*}
for all $s>r>0$ and $\nu<-b$, as $\varepsilon_1,\varepsilon_2\to 0$.
On the other hand, from the above proof we have also
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |A_4(s,r,\varepsilon_1,2)|&\leq \frac1{\varepsilon_1^{2+2b}} |E[(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)]|\\
&\qquad+\frac1{\varepsilon_1^{1+b}\varepsilon_2^{1+b}}
|E[(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\\
&\leq C_{a,b}\frac{r^a\vee (s+\varepsilon_1)^a}{(s-r)^{1+b}}
\end{align*}
for all $s>r>0$ and $\varepsilon_1,\varepsilon_2>0$, and
$$
\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} \frac{r^a}{(s-r)^{1+b}} |E[f(B^{a,b}_{s})f(B^{a,b}_{r})]|(sr)^bdrds\leq C_{a,b}\|f\|^2_{\mathscr H}
$$
for any $0<\varepsilon_1,\varepsilon_2<1$. Using Lebesgue's dominated convergence theorem we can deduce the convergence~\eqref{step1-eq1}.
{\bf Part II}. The following convergence holds:
\begin{align}\label{step2-eq1}
\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\frac{A_{11}(s,r,\varepsilon_1,2)+A_{12}(s,r,\varepsilon_1,2)}{ \varepsilon_1^{2+2b}\varepsilon_2^{1+b}} E[f'(B^{a,b}_{s})f'(B^{a,b}_{r})](sr)^bdrds\to 0,
\end{align}
as $\varepsilon_1,\varepsilon_2\to 0$. By Lemma~\ref{lem3.4} we have
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}}& \left(|A_{11} (s,r,\varepsilon_1,2)|+|A_{12}(s,r,\varepsilon_1,2)|\right)\\
&\leq |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot\left(\varepsilon_2^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]| +\varepsilon_1^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\quad +|E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot\left(
\varepsilon_2^{1+b}|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]| +\varepsilon_1^{1+b}
|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\leq C_{a,b}s^ar^a
\end{align*}
for all $s>r>0$, and moreover, by Lemma~\ref{lem3.1} and Lemma~\ref{lem3.5} we have
\begin{align*}
&\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
s^ar^a|E[f'(B^{a,b}_{s})f'(B^{a,b}_{r})]|(sr)^bdrds\\
&\;\;\leq C_{a,b}\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(E|f(B^{a,b}_s)|^2
+E|f(B^{a,b}_r)|^2\right)(sr)^{a+b}drds\\
&\;\;=C_{a,b}\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)E|f(B^{a,b}_s)|^2(sr)^{a+b}drds \leq C_{a,b}\|f\|_{\mathscr H}^2
\end{align*}
for all $\varepsilon_1,\varepsilon_2>0$
On the other hand, we have
\begin{align*}
E[B^{a,b}_{r}&(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]\\
&=\frac1{2{\mathbb B}(1+a,1+b)}
\left(\int_0^ru^a(r+\varepsilon-u)^bdu-{\mathbb B}(1+a,1+b)r^{1+a+b}\right)\\
&=\frac1{2{\mathbb B}(1+a,1+b)} \int_0^{\frac{r}{r+\varepsilon}}x^a(1-x)^bdx \left\{(r+\varepsilon)^{1+a+b}- r^{1+a+b}\right\}\\
&\qquad-\frac{r^{1+a+b}}{2{\mathbb B}(1+a,1+b)} \int_{\frac{r}{r+\varepsilon}}^1x^a(1-x)^bdx\\
&\equiv \frac1{2{\mathbb B}(1+a,1+b)} \left\{\Lambda_1(r,\varepsilon)-\Lambda_2(r,\varepsilon)\right\}
\end{align*}
for $r>0$ and $\varepsilon>0$, and
\begin{align*}
\Bigl|\varepsilon_2^{1+b}
E[&B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]
-\varepsilon_1^{1+b}E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2} -B^{a,b}_r)]\Bigr|\\
&\leq \varepsilon_2^{1+b}|\Lambda_1(r,\varepsilon_1)| +\varepsilon_1^{1+b}|\Lambda_1(r,\varepsilon_2)| +\left|\varepsilon_2^{1+b}\Lambda_2(r,\varepsilon_1) -\varepsilon_1^{1+b}\Lambda_2(r,\varepsilon_2)\right|
\end{align*}
for $r>0$ and $\varepsilon_1>\varepsilon_2>0$. Notice that for $r>0$ and $\varepsilon>0$,
\begin{align*}
|\Lambda_1(r,\varepsilon)|&= \int_0^{\frac{r}{r+\varepsilon}}x^a(1-x)^bdx \left|(r+\varepsilon)^{1+a+b}- r^{1+a+b}\right|\\
&\leq C_{a,b}\left(\frac{r}{r+\varepsilon}\right)^{1+a} \left|(r+\varepsilon)^{1+a+b}- r^{1+a+b}\right|\\
&\leq C_{a,b}\left(\frac{r}{r+\varepsilon}\right)^{1+a}
(r+\varepsilon)^{a+b}\varepsilon\leq C_{a,b}r^{a+b}\varepsilon
\end{align*}
by~\eqref{lem3.4-eq7} and the fact
\begin{equation}\label{step2-eq2-1}
y^\alpha-x^\alpha\leq C_{\alpha}y^{\alpha-1}(y-x),\quad y>x>0,\quad \alpha>0,
\end{equation}
and
\begin{align*}
|\varepsilon_2^{1+b}&\Lambda_2(r,\varepsilon_1) -\varepsilon_1^{1+b}\Lambda_2(r,\varepsilon_2)|\\
&=C_{a,b}r^{1+a+b}\left|\varepsilon_2^{1+b} \int_{\frac{r}{r+\varepsilon_1}}^1x^a(1-x)^bdx-
\varepsilon_1^{1+b} \int_{\frac{r}{r+\varepsilon_2}}^1x^a(1-x)^bdx\right|\\
&=C_{a,b}r^{1+a+b}(\varepsilon_1\varepsilon_2)^{1+b} \left|\frac1{\varepsilon_1^{1+b}} \int_{\frac{r}{r+\varepsilon_1}}^1x^a(1-x)^bdx-
\frac1{\varepsilon_2^{1+b}} \int_{\frac{r}{r+\varepsilon_2}}^1x^a(1-x)^bdx\right|\\
&\equiv C_{a,b}r^{1+a+b}(\varepsilon_1\varepsilon_2)^{1+b}
\Lambda_3(r,\varepsilon_1,\varepsilon_2)
\end{align*}
for $r>0$ and $\varepsilon_1>\varepsilon_2>0$. We get
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}}& |A_{11}(s,r,\varepsilon_1,2)|\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\cdot\left(\varepsilon_2^{1+b}
E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)] -\varepsilon_1^{1+b} E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]\right)\\
&\leq C_{a,b}s^ar^{a+b}\left(\varepsilon_1^{-b}+\varepsilon_2^{-b}+
r\Lambda_3(r,\varepsilon_1,\varepsilon_2)\right)\longrightarrow 0
\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
for all $s,r>0$ and $-1<b<0$ because
$$
\Lambda_3(r,\varepsilon_1,\varepsilon_2)\longrightarrow 0
$$
as $0<\varepsilon_2<\varepsilon_1\to 0$ by the convergence
$$
\lim_{x\uparrow 1}\frac1{(1-x)^{1+b}}\int_x^1u^a(1-u)^bdu=\frac1{1+b},
$$
and similarly we also have
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}}&|A_{12} (s,r,\varepsilon_1,2)| \longrightarrow 0\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
for all $s,r>0$, and the convergence~\eqref{step2-eq1} follow from Lebesgue's dominated convergence theorem again.
{\bf Part III}. The following convergence holds:
\begin{equation}\label{step3-eq1}
\begin{split}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} &\int_{\varepsilon_2}^{t+\varepsilon_1} \Bigl(|A_1(s,r,\varepsilon_1,2)E[f''(B^{a,b}_{s})f(B^{a,b}_{r})]|\\
&\qquad +|A_3(s,r,\varepsilon_1,2)E[f(B^{a,b}_{s})f''(B^{a,b}_{r})]|
\Bigr)(sr)^bdrds \longrightarrow 0,
\end{split}
\end{equation}
as $\varepsilon_1,\varepsilon_2\to 0$. Clearly, for all $s>r>0$ and all $\varepsilon_1,\varepsilon_2>0$ we have
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} (|&A_1(s,r,\varepsilon_1,2)|+|A_3(s,r,\varepsilon_1,2)|)
\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\cdot\left(\varepsilon_2^{1+b}| E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|+ \varepsilon_1^{1+b}|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)] |\right)\\
&+\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad \cdot\left(
\varepsilon_2^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|
+\varepsilon_1^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\leq C_{a,b}s^ar^a
\end{align*}
by Lemma~\ref{lem3.4}, and
\begin{align*}
\int_{\varepsilon_2}^{t+\varepsilon_1} &\int_{\varepsilon_2}^{t+\varepsilon_1} s^ar^a\Bigl(|E[f''(B^{a,b}_{s})f(B^{a,b}_{r})]| +|E[f(B^{a,b}_{s})f''(B^{a,b}_{r})]|
\Bigr)(sr)^bdrds\\
&\leq C_{a,b}\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} s^ar^a\frac{r^{1+a+b}+s^{1+a+b}}{\rho^2_{s,r}}( E|f(B^{a,b}_s)|^2+E|f(B^{a,b}_r)|^2)(sr)^bdrds\\
&\leq C_{a,b} \int_{\varepsilon_2}^{t+\varepsilon_1} s^{a+b}E|f(B^{a,b}_s)|^2ds\leq C_{a,b}\|f\|_{{\mathscr H}}^2
\end{align*}
by Lemma~\ref{lem3.5} and Lemma~\ref{lem3.1}.
On the other hand, by the fact~\eqref{step2-eq2-1} we have
\begin{equation}\label{sec4-eq4.2200-2}
\begin{split}
\int_r^{r+\varepsilon}u^a(s-u)^bdu&\leq \frac1{1+b}\left(r^a\vee (r+\varepsilon)^a\right) \left((s-r)^{1+b}-(s-r-\varepsilon)^{1+b}\right)\\
&\leq \frac1{1+b}\left(r^a\vee s^a\right)(s-r)^{1+b-\beta}\varepsilon^\beta
\end{split}
\end{equation}
for $r+\varepsilon<s$ and $1+b<\beta<1$, which implies that
\begin{equation}\label{sec4-eq4.2200}
\begin{split}
|E[B^{a,b}_s&(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]| 1_{\{s-r>\varepsilon\}}=C_{a,b}\left(\int_0^{r+\varepsilon}u^a[(s-u)^b +({r+\varepsilon}-u)^b]du\right.\\
&\hspace{3cm}\left.-
\int_0^ru^a[(s-u)^b+(r-u)^b]du\right)1_{\{s-r>\varepsilon\}}\\
&=C_{a,b}\left(\int_r^{r+\varepsilon}u^a(s-u)^bdu +(r+\varepsilon)^{1+a+b}-r^{1+a+b} \right)1_{\{s-r>\varepsilon\}}\\
&\leq C_{a,b}\varepsilon^\beta \left((r^a\vee s^a)(s-r)^{1+b-\beta} +(r+\varepsilon)^{1+a+b-\beta}\right) 1_{\{s-r>\varepsilon\}}
\end{split}
\end{equation}
for all $1+b<\beta<1$. Moreover, by Lemma~\ref{lem3.4} we have
\begin{align*}
|E[B^{a,b}_s(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]| 1_{\{s-r<\varepsilon\}}&\leq C_{a,b}s^a\varepsilon^{1+b}1_{\{0<s-r<\varepsilon\}}\\
&\leq C_{a,b}s^a \varepsilon^{\beta}(s-r)^{1+b-\beta}1_{\{0<s-r<\varepsilon\}}
\end{align*}
for all $1+b<\beta<1$. It follows from Lemma~\ref{lem3.4} that
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} &|A_1(s,r,\varepsilon_1,2)|
\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad \cdot\left(\varepsilon_2^{1+b}| E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|+ \varepsilon_1^{1+b}|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)] |\right)\\
&\leq C_{a,b}s^a
\left((r^a\vee s^a)(s-r)^{1+b-\beta} +(r+\varepsilon_1)^{1+a+b-\beta}\right) \varepsilon_1^{\beta-1-b}
1_{\{s-r>\varepsilon_1\}}\\
&\quad+C_{a,b}s^a
\left((r^a\vee s^a)(s-r)^{1+b-\beta} +(r+\varepsilon_2)^{1+a+b-\beta}\right) \varepsilon_2^{\beta-1-b}
1_{\{s-r>\varepsilon_2\}}\\
&\quad+C_{a,b}s^{2a}(s-r)^{1+b-\beta}\varepsilon^{\beta}_1
1_{\{0<s-r<\varepsilon_1\}} +C_{a,b}s^{2a}(s-r)^{1+b-\beta}\varepsilon^{\beta}_2
1_{\{0<s-r<\varepsilon_2\}}\\
&\longrightarrow 0\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
for all $s>r>0$. Similarly, for all $s>r>0$ and $1+b<\beta<1$ we have also
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} &|A_3(s,r,\varepsilon_1,2)|\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad \cdot\left(
\varepsilon_2^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|
+\varepsilon_1^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\leq C_{a,b}(s-r)^{b}s^{a-\gamma+1}r^a\varepsilon_1^{\gamma-1-b} \longrightarrow 0\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
by Lemma~\ref{lem3.4} and using the estimate
\begin{align*} |E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]&=
\frac1{2{\mathbb B}(1+a,1+b)} \int_0^ru^a[(s-u)^b -({s+\varepsilon_1}-u)^b]du \\
&=\frac1{2{\mathbb B}(1+a,1+b)} \int_0^ru^a\frac{({s+\varepsilon_1}-u)^{-b}-(s-u)^{-b}}
{({s+\varepsilon_1}-u)^{-b}(s-u)^{-b}}du\\
&\leq \frac1{2{\mathbb B}(1+a,1+b)}\varepsilon_1^\gamma
\int_0^r\frac{u^a}
{({s+\varepsilon_1}-u)^{\gamma}(s-u)^{-b}}du\\
&\leq \frac1{2{\mathbb B}(1+a,1+b)}\frac{\varepsilon_1^\gamma}{(s-r)^{-b}}
\int_0^s\frac{u^a}{(s-u)^{\gamma}}du\\ &=C_{a,b}(s-r)^{b}s^{a-\gamma+1}\varepsilon_1^\gamma
\end{align*}
for all $s>r>0,\;(-b)\vee (1+b)<\gamma<1$. Consequently, Lebesgue's dominated convergence theorem implies that the convergence~\eqref{step3-eq1} holds again.
Thus, we have established the convergence~\eqref{sec4-Con-eq1} for $i=1,j=2$ and the theorem follows.
\end{proof}
\begin{corollary}\label{cor5.2}
Let $f,f_1,f_2,\ldots\in {\mathscr H}$. If $f_n\to f$ in ${\mathscr
H}$, as $n$ tends to infinity, then we have
$$
[f_n(B^{a,b}),B^{a,b}]^{(a,b)}_t\longrightarrow [f(B^{a,b}),B^{a,b}]^{(a,b)}_t
$$
in $L^2$ as $n\to \infty$.
\end{corollary}
\begin{proof}
The corollary follows from
$$
E\left|[f_n(B^{a,b}),B^{a,b}]^{(a,b)}_t -[f(B^{a,b}),B^{a,b}]^{(a,b)}_t\right|^2\leq
C_{a,b}\|f_n-f\|_{\mathscr H}^2\to 0,
$$
as $n$ tends to infinity.
\end{proof}
\section{The generalized Bouleau-Yor identity with $b<0$}\label{sec7}
In this section, we study the Bouleau-Yor identity. It is known that the quadratic covariation $[f(B),B]$ of Brownian motion $B$ can be characterized
as
$$
[f(B),B]_t=-\int_{\mathbb R}f(x){\mathscr L}^{B}(dx,t),
$$
where $f$ is locally square integrable, ${\mathscr L}^{B}(x,t)$
is the local time of Brownian motion and the quadratic covariation $\bigl[f(B),B\bigr]$ (see Russo-Vallois~\cite{Russo2}) is defined by
$$
\bigl[f(B),B\bigr]:=\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon} \int_0^{t}\left\{ f(B_{s+\varepsilon})
-f(B_s)\right\}(B_{s+\varepsilon}-B_s)ds,
$$
provided the limit exists uniformly in probability. This is called the Bouleau-Yor identity. More works for this can be found in
Bouleau-Yor~\cite{Bouleau}, Eisenbaum~\cite{Eisen1}, F\"ollmer {\it
et al}~\cite{Follmer}, Feng--Zhao~\cite{Feng,Feng3},
Peskir~\cite{Peskir1}, Rogers--Walsh~\cite{Rogers2},
Yan et al~\cite{Yan4,Yan1,Yan2} and the references therein. However, the Bouleau-Yor identity is not true for weighted-fBm with $b<0$ because
$$
[B^{a,b},B^{a,b}]_t=+\infty
$$
for all $t>0$ and $-1<b<0$. In this section, we shall obtain a generalized Bouleau-Yor identity based on the generalized quadratic covariation defined in Section~\ref{sec6}
Recall that for any closed interval $I\subset {\mathbb R}_{+}$ and for any $x\in {\mathbb R}$, the local time $L(x,I)$ of $B^{a,b}$ is defined as
the density of the occupation measure $\mu_I$ defined by
$$
\mu_I(A)=\int_I1_A(B^{a,b}_s)ds
$$
It can be shown (see Geman and Horowitz~\cite{Geman}, Theorem 6.4) that the following occupation density formula holds:
$$
\int_Ig(B^{a,b}_s,s)ds=\int_{\mathbb R}dx\int_Ig(x,s) L(x,ds)
$$
for every Borel function $g(x,t)\geq 0$ on $I\times {\mathbb R}$. Thus, Lemma~\ref{lem3.0} in Section~\ref{sec3} and Theorem 21.9 in Geman-Horowitz~\cite{Geman} together imply that the following result holds.
\begin{corollary}
Let $a>-1,|b|<1,|b|<1+a$ and let $L(x,t):=L(x, [0,t])$ be the local time of $B^{a,b}$ at $x$. Then $L\in L^2(\lambda\times P)$ for all $t\geq 0$ and $(x,t)\mapsto L(x,t)$ is jointly continuous if and only if $a+b<3$, where $\lambda$ denotes Lebesgue measure. Moreover, the occupation formula
\begin{equation}\label{sec2-1-eq1}
\int_0^t\psi(B^{a,b}_s,s)ds=\int_{\mathbb R}dx\int_0^t\psi(x,s) L(x,ds)
\end{equation}
holds for every continuous and bounded function $\psi(x,t):{\mathbb
R}\times {\mathbb R}_{+}\rightarrow {\mathbb R}$ and any $t\geqslant
0$.
\end{corollary}
Define the weighted local time ${\mathscr L}^{a,b}$ by
\begin{align*}
{\mathscr L}^{a,b}(x,t)&=(1+a+b)\int_0^ts^{a+b}d_sL(s,x)\\
&\equiv (1+a+b)\int_0^t\delta(B^{a,b}_s-x)s^{a+b}ds
\end{align*}
for $t\geq 0$ and $x\in {\mathbb R}$, where $\delta$ is the Dirac delta function. In this section, we define the integral
\begin{equation}\label{sec7-eq7.1}
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t),
\end{equation}
for $b<0$, where $f$ is a Borel function. We shall use the generalized quadratic covariation to study it and obtain the following generalized Bouleau-Yor identity:
\begin{equation}\label{Bouleau-Yor}
[f(B^{a,b}),B^{a,b}]_t^{(a,b)}=-\kappa_{a,b}\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
\end{equation}
for all $f\in {\mathscr H}$, $-1<b<0$, $a>-1$, $-1<a+b<3$ and $t\geq 0$. We first give an extension of It\^{o} formula stated as follows.
\begin{theorem}\label{th7.1}
Let $a>-1$, $-1<b<0$, $|b|<1+a$ and let $f\in {\mathscr H}$ be a left continuous function with right limits. If $F$ is an absolutely continuous function with $F'=f$ and
\begin{equation}\label{sec7-Ito-con1}
\max\left\{|F(x)|,|f(x)|\right\}\leq C e^{\beta x^2},
\end{equation}
where $C$ and $\beta$ are some positive constants with
$\beta<\frac14T^{-(1+a+b)}$, then the It\^o type formula
\begin{equation}\label{sec7-eq7.3}
F(B^{a,b}_t)=F(0)+\int_0^tf(B^{a,b}_s)dB^{a,b}_s+\frac12(\kappa_{a,b})^{-1}
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t
\end{equation}
holds for all $t\in [0,T]$.
\end{theorem}
Clearly, this is an analogue of F\"ollmer-Protter-Shiryayev's
formula F\"ollmer (see, for example, F\"ollmer {\it et al}~\cite{Follmer}). It is an improvement in terms of the
hypothesis on $f$ and it is also quite interesting itself.
Based on the localization argument and smooth approximation one can prove Theorem~\ref{th7.1}. The localization is that one can localize the domain ${\rm Dom}(\delta^{a,b})$ of the operator $\delta^{a,b}$ (see Nualart~\cite{Nualart1}). Suppose that
$\{(\Omega_n, u^{(n)}), n\geq 1\}\subset {\mathscr F}\times {\rm
Dom}(\delta^{a,b})$ is a localizing sequence for $u$, i.e., the
sequence $\{(\Omega_n, u^{(n)}), n\geq 1\}$ satisfies
\begin{itemize}
\item [(i)] $\Omega_n\uparrow \Omega$, a.s.;
\item [(ii)] $u=u^{(n)}$ a.s. on $\Omega_n$.
\end{itemize}
If $\delta^{a,b}(u^{(n)})= \delta^{a,b}(u^{(m)})$ a.s. on $\Omega_n$ for
all $m\geq n$, then, the divergence $\delta^{a,b}$ is the random
variable determined by the conditions
$$
\delta^{a,b}(u)|_{\Omega_n}=\delta^{a,b} (u^{(n)})|_{\Omega_n}\qquad
{\rm { for\;\; all\;\;}} n\geq 1,
$$
which may depend on the localizing sequence.
\begin{lemma}[Nualart~\cite{Nualart1}]\label{complete2.1} Let
$\{v^{(n)}\}$ be a sequence such that $v^{(n)}\to v$ in $L^2$, as $n\to \infty$ and let
$$
\delta^{a,b}(v^{(n)})=\int_0^Tv_s^{(n)}dB^{a,b}_s,\qquad n\geq 1
$$
exist in $L^2$. If $\delta^{a,b}(v^{(n)})\to G$ in $L^2$, then
$\delta^{a,b}(v)=\int_0^Tv_sdB^{a,b}_s$ exists in $L^2$ and equals to $G$.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{th7.1}]
If $F\in C^2({\mathbb R})$, then this is It\^o's formula since
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=(1+a+b) \kappa_{a,b} \int_0^tf'(B^{a,b}_s)s^{a+b}ds.
$$
by~\eqref{sec6-eq6.2}. The assumption $F\in C^2({\mathbb R})$ is not correct. By a localization argument we may
assume that the function $f$ is uniformly bounded. In fact, for any
$k\geq 0$ we may consider the set
$$
\Omega_k=\left\{\sup_{0\leq t\leq T}|B^{a,b}_t|<k\right\}
$$
and let $f^{[k]}$ be a measurable function such that $f^{[k]}=f$ on
$[-k,k]$ and such that $f^{[k]}$ vanishes outside. Then $f^{[k]}\in
{\mathscr H}$ and uniformly bounded. Set
$\frac{d}{dx}F^{[k]}=f^{[k]}$ and $F^{[k]}=F$ on $[-k,k]$. If the
theorem is true for all uniformly bounded functions on ${\mathscr
H}$, then we get the desired formula
$$
F^{[k]}(B^{a,b}_t)=F^{[k]}(0)+\int_0^t
f^{[k]}(B^{a,b}_s)dB^{a,b}_s+\frac12(\kappa_{a,b})^{-1}
\bigl[f^{[k]}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
$$
on the set $\Omega_k$. Letting $k$ tend to infinity we deduce the
It\^o formula~\eqref{sec7-eq7.3} for all $f\in {\mathscr H}$ being
left continuous and locally bounded.
Let now $F'=f\in {\mathscr H}$ be uniformly bounded such that the conditions in the theorem hold. Define the sequence of smooth functions
$$
F_n(x):=\int_{\mathbb R}F(x-{y})\zeta_n(y)dy,\quad x\in {\mathbb R},
$$
where the mollifiers $\zeta_n,n\geq 1$ are given by~\eqref{sec7-eq7.4}. Then $F_n\in C^\infty({\mathbb R})$ and $F_n$ satisfies the condition~\eqref{sec2-Ito-con1}, and the It\^{o} formula
\begin{equation}\label{sec7-eq7.5}
F_n(B^{a,b}_t)=F_n(0)+\int_0^tf_n(B^{a,b}_s)dB^{a,b}_s+
\frac12(1+a+b)\int_0^tf_n'(B^{a,b}_s)s^{a+b}ds
\end{equation}
holds for all $n\geq 1$, where $f_n=F_n'$. We also have
$$
F_n \longrightarrow F,\quad f_n \longrightarrow f,
$$
uniformly in $\mathbb R$, as $n$ tends to infinity, and moreover, $\{f_n\}\subset {\mathscr H}$, $f_n\to f$ in ${\mathscr H}$, as $n$ tends to infinity. It follows that
\begin{align*}
(1+a+b)\int_0^tf_n'(B^{a,b}_s)s^{a+b}ds &=(\kappa_{a,b})^{-1}\bigl[f_n(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\\
&\longrightarrow (\kappa_{a,b})^{-1}\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
\end{align*}
in $L^2(\Omega)$ by Corollary~\ref{cor5.2}, as $n$ tends to infinity. It follows that
\begin{align*}
\int_0^tf_n(B^{a,b}_s)dB^{a,b}_s&=F_n(B^{a,b}_t)-F_n(0)-
\frac12(\kappa_{a,b})^{-1}\bigl[f_n(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\\
&\longrightarrow F(B^{a,b}_t)-F(0)-\frac12(\kappa_{a,b})^{-1} \bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
\end{align*}
in $L^2(\Omega)$, as $n$ tends to infinity. This completes the proof since
the integral is closed in $L^2(\Omega)$.
\end{proof}
Now, let us study the integral~\eqref{sec7-eq7.1} for $-1<b<0$.
\begin{lemma}
Let $-1<b<0$, $a>-1$ and $-1<a+b<3$. For any $f_\triangle=\sum_jf_j1_{(a_{j-1},a_j]}\in {\mathscr H}$, we
define
$$
\int_{\mathbb R}f_\triangle(x){\mathscr
L}^{a,b}(dx,t):=\sum_jf_j\left[{\mathscr L}^{a,b}(a_j,t)-{\mathscr
L}^{a,b}(a_{j-1},t)\right].
$$
Then the integral is well-defined and
\begin{equation}\label{sec6-eq6.4}
\kappa_{a,b}\int_{\mathbb R}f_{\Delta}(x)\mathscr{L}^{a,b}(dx,t)=
-\bigl[f_\triangle(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
\end{equation}
almost surely, for all $t\in [0,T]$.
\end{lemma}
\begin{proof}
For the function $f_\triangle(x)=1_{(a,b]}(x)$ we define the sequence of smooth functions $f_n,\;n=1,2,\ldots$ by
\begin{align}
f_n(x)&=\int_{\mathbb
R}f_\triangle(x-y)\zeta_n(y)dy=\int_a^b\zeta_n(x-u)du
\end{align}
for all $x\in \mathbb R$, where $\zeta_n,n\geq 1$ are the so-called
mollifiers given in~\eqref{sec7-eq7.4}. Then $\{f_n\}\subset
C^{\infty}({\mathbb R})\cap {\mathscr H}$ and $f_n$ converges to $f_\triangle$ in ${\mathscr H}$, as $n$ tends to infinity. It follows from the occupation formula that
\begin{align*}
\bigl[f_n(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t& =\kappa_{a,b}\int_0^tf'_n(B^{a,b}_s)ds^{1+a+b}\\
&=\kappa_{a,b}\int_{\mathbb R}f_n'(x){\mathscr L}^H(x,t)dx=\int_{\mathbb
R}\left(\int_a^b\zeta_n'(x-u)du\right){\mathscr L}^{a,b}(x,t)dx\\
&=-\kappa_{a,b}\int_{\mathbb R}{\mathscr
L}^{a,b}(x,t)\left(\zeta_n(x-b)-\zeta_n(x-a)\right)dx\\
&=\kappa_{a,b}\int_{\mathbb R}{\mathscr L}^{a,b}(x,t)\zeta_n(x-a)dx
-\int_{\mathbb R}{\mathscr L}^{a,b}(x,t)\zeta_n(x-b)dx\\
&\longrightarrow \kappa_{a,b}\left({\mathscr L}^{a,b}(a,t)-{\mathscr L}^{a,b}(b,t)\right)
\end{align*}
almost surely, as $n\to \infty$, by the continuity of $x\mapsto
{\mathscr L}^{a,b}(x,t)$. On the other hand, Corollary~\ref{cor5.2}
implies that there exists a subsequence $\{f_{n_k}\}$ such that
$$
\bigl[f_{n_k}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\longrightarrow \bigl[1_{(a,b]}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
$$
for all $t\in [0,T]$, almost surely, as $k\to \infty$. Then we have
$$
\bigl[1_{(a,b]}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t =\kappa_{a,b}\left({\mathscr L}^{a,b}(a,t)-{\mathscr L}^{a,b}(b,t)
\right)
$$
for all $t\in [0,T]$, almost surely. Thus, the identity
$$
\kappa_{a,b}\sum_jf_j\bigl[{\mathscr L}^{a,b}(a_j,t)-{\mathscr
L}^{a,b}(a_{j-1},t)\bigr]=
-\bigl[f_\triangle(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
$$
follows from the linearity property, and the lemma follows.
\end{proof}
As a direct consequence of the above lemma we can show
that
\begin{equation} \lim_{n\to \infty}\int_{\mathbb
R}f_{\triangle,n}(x){\mathscr L}^{a,b}(x,t)dx =\lim_{n\to
\infty}\int_{\mathbb R}g_{\triangle,n}(x){\mathscr L}^{a,b}(x,t)dx
\end{equation}
in $L^2(\Omega)$ if
$$
\lim_{n\to \infty}f_{\triangle,n}(x)=\lim_{n\to
\infty}g_{\triangle,n}(x)=f(x)
$$
in ${\mathscr H}$, where $\{f_{\triangle,n}\},\{g_{\triangle,n}\}\subset
{\mathscr E}$. Thus, by the density of ${\mathscr
E}$ in ${\mathscr H}$ we can define
$$
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t):=\lim_{n\to
\infty}\int_{\mathbb R}f_{\triangle,n}(x){\mathscr L}^{a,b}(dx,t)
$$
for any $f\in {\mathscr H}$, where $\{f_{\triangle,n}\}\subset
{\mathscr E}$ and
$$
\lim_{n\to \infty}f_{\triangle,n}(x)=f(x)
$$
in ${\mathscr H}$. These considerations are enough to prove the following theorem.
\begin{theorem}\label{th7.2}
Let $-1<b<0$, $a>-1$ and $-1<a+b<3$. For any $f\in {\mathscr H}$, the integral
$$
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
$$
is well-defined in $L^2(\Omega)$ and the Bouleau-Yor type formula
\begin{equation}\label{sec6-eq6.3}
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=-\kappa_{a,b}\int_{\mathbb R}f(x)\mathscr{L}^{a,b}(dx,t)
\end{equation}
holds, almost surely, for all $t\in [0,T]$.
\end{theorem}
\begin{corollary}[Tanaka formula]
Let $-1<b<0$, $a>-1$ and $-1<a+b<3$. For any $x\in {\mathbb R}$ we have
\begin{align*}
(B^{a,b}_t-x)^{+}=(-x)^{+}+\int_0^t{1}_{\{B^{a,b}_s>x\}} dB^{a,b}_s
+\frac12{\mathscr L}^{a,b}(x,t),\\
(B^{a,b}_t-x)^{-}=(-x)^{-}-\int_0^t{1}_{\{B^{a,b}_s<x\}}
dB^{a,b}_s+\frac12{\mathscr L}^{a,b}(x,t),\\
|B^{a,b}_t-x|=|x|+\int_0^t{\rm sign}(B^{a,b}_s-x)dB^{a,b}_s+{\mathscr L}^{a,b}(x,t).
\end{align*}
\end{corollary}
\begin{proof}
Take $F(y)=(y-x)^{+}$. Then $F$ is absolutely continuous and
$$
F(x)=\int_{-\infty}^y1_{(x,\infty)}(y)dy.
$$
It follows from It\^o's formula~\eqref{sec7-eq7.3} and the
identity~\eqref{sec6-eq6.4} that
\begin{align*}
{\mathscr
L}^{a,b}(x,t)&=(\kappa_{a,b})^{-1} \bigl[1_{(x,+\infty)}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\\
&=2(B^{a,b}_t-a)^{+}-2(-x)^{+}-2 \int_0^t{1}_{\{B^{a,b}_s>x\}}dB^{a,b}_s
\end{align*}
for all $t\in [0,T]$, which gives the first identity. In the same
way one can obtain the second identity. By subtracting the last
identity from the previous one, we get the third identity.
\end{proof}
\begin{corollary}
Let $-1<b<0$, $a>-1$, $-1<a+b<3$ and let $f,f_1,f_2,\ldots\in {\mathscr H}$. If $f_n\to f$ in ${\mathscr H}$, as $n$ tends to infinity, we then have
\begin{align*}
\int_{\mathbb R}f_n(x){\mathscr L}^{a,b}(dx,t)\longrightarrow
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
\end{align*}
in $L^2$, as $n$ tends to infinity.
\end{corollary}
According to Theorem~\ref{th7.1}, we get an analogue of the It\^o
formula (Bouleau-Yor type formula).
\begin{corollary}\label{cor7.1}
Let $-1<b<0$, $a>-1$, $-1<a+b<3$ and let $f\in {\mathscr H}$ be a left continuous function with right limits. If $F$ is an absolutely continuous function with $F'=f$ and the condition~\eqref{sec7-Ito-con1} is satisfied, then the following It\^o type formula holds:
\begin{equation}\label{sec7-eq7.11}
F(B^{a,b}_t)=F(0)+\int_0^tf(B^{a,b}_s)dB^{a,b}_s -\frac12\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t).
\end{equation}
\end{corollary}
Recall that if $F$ is the difference of two convex functions, then
$F$ is an absolutely continuous function with derivative of bounded
variation. Thus, the It\^o-Tanaka formula
\begin{align*}
F(B^{a,b}_t)&=F(0)+\int_0^tF^{'}(B^{a,b}_s)dB^{a,b}_s +\frac12\int_{\mathbb R}{\mathscr L}^{a,b}(x,t)F''(dx)\\
&\equiv F(0)+\int_0^tF^{'}(B^{a,b}_s)dB^{a,b}_s -\frac12\int_{\mathbb R}F'(x){\mathscr L}^{a,b}(dx,t)
\end{align*}
holds for all $-1<b<0$, $a>-1$ and $-1<a+b<3$.
\end{document} | math | 92,694 |
\begin{document}
\thetaitle[Navier-Stokes-Boltzmann equations]{Existence results for compressible radiation hydrodynamic equations with vacuum }
\alphauthor{yachun li }
\alphaddress[Y. C. Li]{Department of Mathematics and Key Lab of Scientific and Engineering Computing (MOE), Shanghai Jiao Tong University,
Shanghai 200240, P.R.China} \email{\thetat [email protected]}
\alphauthor{Shengguo Zhu}
\alphaddress[S. G. Zhu]{Department of Mathematics, Shanghai Jiao Tong University,
Shanghai 200240, P.R.China; School of Mathematics, Georgia Tech,
Atlanta 30332, U.S.A.}
\email{\thetat [email protected]}
\begin{abstract}In this paper, we consider the three-dimensional compressible isentropic radiation hydrodynamic (RHD) equations. The existence of unique local strong solutions is firstly proved when the initial data are arbitrarily large, contain vacuum and satisfy some initial layer compatibility condition. The initial mass density does not need to be bounded away from zero and may vanish in some open set. We also prove that if the initial vacuum is not so irregular, then the initial layer compatibility condition is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we establish a blow-up criterion for the strong solution that we obtained. The similar results also hold for the barotropic flow with general pressure law $p_m=p_m(\rhoho)\in C^1(\mathbb{\overline{R}}^+)$.
\end{abstract}
\date{Dec. 05, 2013}
\keywords{Radiation, Navier-Stokes-Boltzmann equations, Strong solutions, Vacuum, Blow-up criterion.\\
}
\maketitle
\sigmaection{Introduction}
The system of radiation hydrodynamic equations appears in high-temperature plasma physics \cite{sjx} and in various astrophysical contexts \cite{kr}. The couplings between fluid field and radiation field involve momentum source and energy source depending on the specific radiation intensity driven by the so-called radiation transfer equation \cite{gp}.
Suppose that the matter is in local thermodynamical equilibrium (LTE), the coupled system of Navier-Stokes-Boltzmann (RHD) equations for the mass density $\rhoho(t,x)$, the velocity $u(t,x)=(u^{(1)},u^{(2)},u^{(3)})$ of the fluid and the specific radiation intensity $I(v,\mathbb{R}^3mega,t,x)$ in three-dimensional space reads as \cite{gp}
\begin{equation}
\label{eq:1.1}
\begin{cases}
\displaystyle
\frac{1}{c}I_t+\mathbb{R}^3mega\cdot\nabla I=A_r,\\[10pt]
\displaystyle
\rhoho_t+\thetaext{div}(\rhoho u)=0,\\[10pt]
\displaystyle
\left(\rhoho u+\frac{1}{c^{2}}F_r\rhoight)_t+\thetaext{div}(\rhoho u\otimes u+P_r)
+\nabla p_m =\thetaext{div}\mathbb{T},
\end{cases}
\end{equation}
where $t\geq 0$ and $x\in \mathbb{R}^3$ are the time and space variables, respectively. $p_m$ is the material pressure satisfying the equation of state:
\begin{equation}
\label{eq:1.3}
p_m=A\rhoho^{\gamma},
\end{equation}
where $A>0$ and $\gamma>1 $ are both constants, $\gamma $ is the adiabatic exponent.
$\mathbb{T}$ is the viscosity stress tensor given by
\begin{equation}
\label{eq:1.4}
\mathbb{T}=\mu(\nabla u+(\nabla u)^\thetaop)+\lambda \thetaext{div}u\,\mathbb{I}_3,
\end{equation}
where $\mathbb{I}_3$ is the $3\thetaimes 3$ unit matrix,
$\mu$ is the shear viscosity coefficient, $\lambda+\frac{2}{3}\mu$ is the bulk viscosity coefficient, $\mu$ and $\lambda$ are both real constants satisfying
\begin{equation}
\label{eq:1.5}
\mu > 0, \quad \lambda+\frac{2}{3}\mu \geq 0
\end{equation}
which ensure the ellipticity of the Lam$\alphacute{ \thetaext{e} }$ operator defined by
\begin{equation}\label{lame}
\displaystyle
Lu=-\thetaext{div}\mathbb{T}=-\mu\thetariangle u-(\lambda+\mu)\nabla \thetaext{div} u.
\end{equation}
$v$ and $\mathbb{R}^3mega$ are radiation variables. $v\in \mathbb{R}^+$ is the frequency of photon, and $\mathbb{R}^3mega\in S^2 $ is the travel direction of photon. The radiation flux $F_r$ and the radiation pressure tensor $P_r$ are defined by
\begin{equation*}
\displaystyle
F_r=\int_0^\infty \int_{S^{2}} I(v,\mathbb{R}^3mega,t,x)\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v,\
P_r=\frac{1}{c}\int_0^\infty \int_{S^{2}} I(v,\mathbb{R}^3mega,t,x)\mathbb{R}^3mega\otimes\mathbb{R}^3mega\thetaext{d}\mathbb{R}^3mega \thetaext{d}v,
\end{equation*}
where $S^2$ is the unit sphere in $\mathbb{R}^3$. The collision term on the right-hand side of the radiation transfer equation is
$$
A_r=S-\sigmaigma_aI
+\int_0^\infty \int_{S^{2}} \Big(\frac{v}{v'}\sigmaigma_sI'
-\sigmaigma'_sI\Big) \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',
$$
where $I=I(v,\mathbb{R}^3mega,t,x),\,\,I'=I(v',\mathbb{R}^3mega',t,x)$; $S=S(v,\mathbb{R}^3mega,t,x)\geq 0$ is the rate of energy emission due to spontaneous process; $\sigmaigma_a=\sigmaigma_a(v,\mathbb{R}^3mega,t,x,\rhoho)\geq 0$ denotes the absorption coefficient that may also depend on the mass density $\rhoho$; $\sigmaigma_s$ is the ``differential scattering coefficient'' such that the probability of a photon being
scattered from $v'$ to $v$ contained in $\thetaext{d}v$, from $\mathbb{R}^3mega'$ to
$\mathbb{R}^3mega$ contained in $\thetaext{d}\mathbb{R}^3mega$, and travelling a distance $\thetaext{d}s$ is
given by $\sigmaigma_s(v' \rhoightarrow v,\mathbb{R}^3mega'\cdot\mathbb{R}^3mega)\thetaext{d}v \thetaext{d}\mathbb{R}^3mega
\thetaext{d}s$, and
\begin{equation*}
\begin{split}
\sigmaigma_s\equiv \sigmaigma_s(v' \rhoightarrow v, \mathbb{R}^3mega'\cdot\mathbb{R}^3mega,\rhoho)=O(\rhoho),\ \sigmaigma'_s\equiv \sigmaigma_s(v \rhoightarrow v', \mathbb{R}^3mega\cdot\mathbb{R}^3mega',\rhoho)=O(\rhoho).
\end{split}
\end{equation*}
When there is no radiation effect, the local existence of strong solutions with vacuum has been solved by many authors, we refer the reader to \cite{CK2}\cite{CK3}\cite{CK}. Huang-Li-Xin \cite{HX1} obtained the well-posedness of classical solutions with large oscillations and vacuum for Cauchy problem \cite{HX1} to the isentropic flow.
In general, the study of radiation hydrodynamics equations is challenging due to the high complexity and mathematical difficulty of the equations themselves. For the Euler-Boltzmann equations of the inviscid compressible radiation fluid, Jiang-Zhong \cite{sjx} obtained the local existence of $C^1$ solutions for the Cauchy problem away from vacuum. Jiang-Wang \cite{pjd} showed that some $ C^1$ solutions will blow up in finite time, regardless of the size of the initial disturbance. Li-Zhu \cite{sz1} established the local existence of Makino-Ukai-Kawashima type (see \cite{tms1}) regular solutions with vacuum, and also proved that the regular solutions will blow up if the initial mass density vanishes in some local domain.
For the Navier-Stokes-Boltzmann equations of the viscous compressible radiation fluid, under some physical assumptions, Chen-Wang \cite{zcy} studied the classical solutions of the Cauchy problem with the mass density away from vacuum. Ducomet and Ne$\check{\thetaext{c}}$asov$\alphacute{\thetaext{a}}$ \cite{BD}\cite{BS} obtained the global weak solutions and their large time behavior for the one-dimensional case. Li-Zhu \cite{sz2} considered the formation of singularities on classical solutions in multi-dimensional space ($d\geq 2$), when the initial mass density is compactly supported and the initial specific radiation intensity satisfies some directional condtions. Some special phenomenon has been observed, for example, it is known in contrast with the second law of thermodynamics, the associated entropy equation may contain a negative production term for RHD system, which has already been observed in Buet and Despr$\alphacute{\thetaext{e}}$s \cite{add1}. Moreover, from Ducomet, Feireisl and Ne$\check{\thetaext{c}}$asov$\alphacute{\thetaext{a}}$ \cite{add2}, in which they obtained the existence of global weak solution for some RHD model, we know that the velocity field $u$ may develop uncontrolled time oscillations on the hypothetical vacuum zones.
The purpose of this paper is to provide a local theory of strong solutions (see Definition \rhoef{strong1}) to the RHD equations in the framework of Sobolev spaces.
Via the radiation transfer equation $(\rhoef{eq:1.1})_1$ and the definitions of $F_r$ and $P_r$, system (\rhoef{eq:1.1}) can be rewritten as
\begin{equation}
\label{eq:1.2}
\begin{cases}
\displaystyle
\frac{1}{c}I_t+\mathbb{R}^3mega\cdot\nabla I=A_r,\\[10pt]
\displaystyle
\rhoho_t+\thetaext{div}(\rhoho u)=0,\\[10pt]
\displaystyle
(\rhoho u)_t+\thetaext{div}(\rhoho u\otimes u)
+\nabla p_m +Lu=-\frac{1}{c}\int_0^\infty \int_{S^2}A_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v,
\end{cases}
\end{equation}
where $L$ is the Lam$\alphacute{ \thetaext{e} }$ operator defined by \eqref{lame}.
We consider the Cauchy problem of \eqref{eq:1.2} with the following initial data
\begin{equation} \label{eq:2.2hh}
I|_{t=0}=I_0(v,\mathbb{R}^3mega,x),\quad (\rhoho, u)|_{t=0}=(\rhoho_0(x), u_0(x)),\ (v,\mathbb{R}^3mega,x)\in \mathbb{R}^+\thetaimes S^2\thetaimes \mathbb{R}^3.
\end{equation}
For (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}), inspired by the argument used in \cite{CK2}\cite{CK}, we introduce a similar initial layer compatibility condition (\rhoef{kkkkk}), which will be used to compensate the loss of positive lower bound of the initial mass density when vacuum appears. The key point is to get a priori estimates independent of the lower bound of the initial mass density by this compatibility condition. Then the existence of the local strong solutions can be obtained by the approximation process from non-vacuum to vacuum. We also prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient for the existence of a unique strong solution. Finally, we give a blow-up criterion for the local strong solution: if $\overline{T}< +\infty$ is the maximal existence time of the local strong solution $(I,\rhoho,u)$, then
\begin{equation*}
\begin{split}
\lim \sigmaup_{t\mapsto \overline{T}} \thetaotog(\|I(t)\|_{L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3))}+\|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+|u(t)|_{\mathbb{D}^1}\thetaotog)=+\infty,
\end{split}
\end{equation*}
where $3< q\leq 6$ and $\overline{\rhoho}\geq 0$ are both constants. The similar results also hold for the barotropic flow with general pressure law $p_m=p_m(\rhoho)\in C^1(\mathbb{\overline{R}}^+)$.
Throughout this paper, we use the following simplified notations for standard homogenous and inhomogeneous Sobolev spaces:
\begin{equation*}\begin{split}
&\|f(v,\mathbb{R}^3mega,t,x,\rhoho(t,x))\|_{X_1(\mathbb{R}^+\thetaimes S^2;X_2(\mathbb{R}^3))}=\thetaotog\|\|f(v,\mathbb{R}^3mega,t,\cdot,\rhoho(t,\cdot))\|_{X_2(\mathbb{R}^3)}\thetaotog\|_{X_1(\mathbb{R}^+\thetaimes S^2)},\\
&\|f(v,\mathbb{R}^3mega,t,x,\rhoho(t,x))\|_{X_1(\mathbb{R}^+\thetaimes S^2;X_2([0,T]\thetaimes \mathbb{R}^3))}
=\thetaotog\|\|f(v,\mathbb{R}^3mega,\cdot,\cdot,\rhoho(\cdot,\cdot))\|_{X_2([0,T]\thetaimes\mathbb{R}^3)}\thetaotog\|_{X_1(\mathbb{R}^+\thetaimes S^2)},\\
& \|(f,g)\|_X=\|f\|_X+\|g\|_X,\quad |\|g\||_{X,T}=|\|g(t,x)\||_{X,T}=\sigmaup_{t\in [0,T]}\|g(t,\cdot)\|_{X},\\
& \|f\|_{W^{m,p}}=\|f\|_{W^{m,p}(\mathbb{R}^3)},\quad \|f\|_s=\|f\|_{H^s(\mathbb{R}^3)},\quad |f|_p=\|f\|_{L^p(\mathbb{R}^3)},\\
&D^{k,r}=\{f\in L^1_{loc}(\mathbb{R}^3): |\nabla^kf|_{r}<+\infty\},\,\, D^k=D^{k,2},\,\, \mathbb{D}^1=\{f\in L^6(\mathbb{R}^3): |\nabla f|_{2}<\infty\},\\
& |f|_{D^{k,r}}:=\|f\|_{D^{k,r}(\mathbb{R}^3)}=|\nabla^kf|_{r},\ |f|_{D^{k}}:=\|f\|_{D^{k}(\mathbb{R}^3)}=|\nabla^kf|_{2}, \,\,|f|_{\mathbb{D}^1}:=\|f\|_{\mathbb{D}^1(\mathbb{R}^3)}=|\nabla f|_{2},
\end{split}
\end{equation*}
where $0< T<\infty$ and $1\leq p \leq\infty$ are both constants, $X$, $X_1$, and $X_2$ are some Sobolev spaces.
The following inequalities will be used in our paper:
$$
|u|_6\leq C|u|_{\mathbb{D}^1},\quad |u|_{\infty}\leq C\|u\|_{\mathbb{D}^1\cap D^2}, \quad |u|_{\infty}\leq C\|u\|_{W^{1,q}},
$$
where $3< q\leq 6$ and $\|u\|_{X_1\cap X_2}=\|u\|_{X_1}+\|u\|_{X_2}$. A detailed study on homogeneous Sobolev spaces may be found in \cite{gandi}.
Now we make some assumptions on the physical coefficients $\sigmaigma_a$ and $\sigmaigma_s$. First, let
$$ \sigmaigma_s=\overline{\sigmaigma}_s(v' \rhoightarrow v, \mathbb{R}^3mega'\cdot\mathbb{R}^3mega)\rhoho,\quad \sigmaigma'_s=\overline{\sigmaigma}'_s(v \rhoightarrow v', \mathbb{R}^3mega\cdot\mathbb{R}^3mega')\rhoho,$$
where the functions $\overline{\sigmaigma}_s\geq 0$ and $\overline{\sigmaigma}'_s\geq 0$ satisfy
\begin{equation}\label{zhen1}
\begin{cases}
\displaystyle
\int_0^\infty \int_{S^{2}}\Big(\int_{0}^{\infty}\int_{S^{2}} \Big|\frac{v}{v'}\Big|^2\overline{\sigmaigma}^2_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{\lambda_1}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\leq C,\\[12pt]
\displaystyle
\int_0^\infty \int_{S^{2}}\Big(\int_{0}^{\infty}\int_{S^{2}} \overline{\sigmaigma}'_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{\lambda_2}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v+\int_{0}^{\infty}\int_{S^{2}} \overline{\sigmaigma}'_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\leq C,
\end{cases}
\end{equation}
where $\lambda_1=1$ or $\frac{1}{2}$, and $\lambda_2=1$ or $2$. Hereinafter we denote by $C$ a generic positive constant depending only on the fixed constants $\mu$, $\lambda$, $\gamma$, $q$, $T$ and the norms of $S$.
Second, let
$$\sigmaigma_a=\sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho)\rhoho,$$
then for $\rhoho^{i}(t)\,(i=1,2)$ satisfying
\begin{equation}\label{zhen3}
\begin{split}
\|\rhoho^i(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}(\mathbb{R}^3)}+\|\rhoho^i_{t}(t)\|_{L^2\cap L^q(\mathbb{R}^3)}<& +\infty,\
\end{split}
\end{equation}
we assume that
\begin{equation}\label{jia345}
\begin{cases}
\|\sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho^i)\|_{L^2\cap L^\infty(\mathbb{R}^+\thetaimes S^2; L^\infty(\mathbb{R}^3))}\leq M( |\rhoho^i(t)|_\infty),\\[6pt]
\|\nabla \sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho^i)\|_{L^2\cap L^\infty(\mathbb{R}^+\thetaimes S^2; L^r(\mathbb{R}^3))}\leq M( |\rhoho^i(t)|_\infty) (|\nabla \rhoho^i(t)|_r+1),\\[6pt]
\|\sigmaigma_t(v,\mathbb{R}^3mega,t,x,\rhoho^i)\|_{L^2 (\mathbb{R}^+\thetaimes S^2; L^2(\mathbb{R}^3))}
\leq M(|\rhoho^i(t)|_\infty)(|\rhoho^i_t(t)|_2+1),\\[6pt]
|\sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho_1)-\sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho_2)|\leq \overline{\sigmaigma}(v,\mathbb{R}^3mega,t,x,\rhoho_1,\rhoho_2)|\rhoho^1(t)-\rhoho^2(t)|, \\[6pt]
\|\overline{\sigmaigma}(v,\mathbb{R}^3mega,t,x,\rhoho_1,\rhoho_2)\|_{L^\infty\cap L^2(\mathbb{R}^+\thetaimes S^2;L^\infty( \mathbb{R}^3))}\leq M(|(\rhoho^1,\rhoho^2)(t)|_\infty),
\end{cases}
\end{equation}
for $t\in [0,T]$ and $r \in [2,q]$, where $M=M(\cdot):\,[0,+\infty) \rhoightarrow[1,+\infty) $ denotes a strictly increasing continuous function, and $\sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho^i)\in C([0,T]; L^2(\mathbb{R}^+\thetaimes S^2; L^{\infty}(\mathbb{R}^3)))$.
\begin{remark}\label{con}
These assumptions are similar to those in \cite{sjx} for the local existence of classical solutions to the Euler-Boltzmann equations with initial mass density away form vacuum and the assumptions in \cite{sz1} for the local existence of regular solutions with vacuum. The evaluation of these radiation quantities is a difficult problem of quantum mechanics, and their general forms are usually not known. The expressions of $\sigmaigma_a$ and $\sigmaigma_s$ used for describing Compton Scattering process in \cite{gp} are given by
\begin{equation}\label{kk}
\begin{split}
&\sigmaigma_a(v,t,x,\rhoho,\thetaheta)=D_1\rhoho \thetaheta^{-\frac{1}{2}}\exp\Big(-\frac{D_2}{\thetaheta^{\frac{1}{2}}}\Big(\frac{v-v_0}{v_0}\Big)^2\Big),\ \sigmaigma_s=\overline{\sigmaigma}_s(v \rhoightarrow v', \mathbb{R}^3mega\cdot\mathbb{R}^3mega')\rhoho,
\end{split}
\end{equation}
where $v_0$ is the fixed frequency, $D_i(i=1,2)$ are positive constants and $\thetaheta$ is the temperature.
\end{remark}
The rest of this paper is organized as follows. In Section $2$, we give our main results including the local existence of strong solutions with vacuum, the necessity and sufficiency of the initial layer compatibility condition and the corresponding blow-up criterion for the local strong solution that we obtained.
In Section $3$, we prove the existence and uniqueness of local strong solutions via establishing a priori estimates independent of the lower bound of $\rhoho_0$. In Section $4$, we show that the initial layer compatibility condition is necessary and sufficient for the existence of a unique local strong solution. Finally in Section $5$, we prove the blow-up criterion that we claimed in Section $2$.
\sigmaection{Main results}
We state our main results in this section.
First, we give the definition of strong solutions to Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}).
\begin{definition}[\thetaextrm{Strong solutions}]\label{strong1}
$(I,\rhoho,u)$ is a strong solution on $\mathbb{R}^+\thetaimes S^2\thetaimes [0,T]\thetaimes\mathbb{R}^3 $ to Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}) if the following holds:
\begin{enumerate}
\item $(I,\rhoho,u)$ solves (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}) in the following sense of distribution:
\begin{equation}\label{weak}\begin{split}
&\int_0^\infty \int_{S^{2}}\int_{0}^{T}\int_{\mathbb{R}^3}\Big(\frac{1}{c}I{\bf x}i_t+I\mathbb{R}^3mega\cdot\nabla{\bf x}i\Big)\thetaext{d}x \thetaext{d}t\thetaext{d}\mathbb{R}^3mega \thetaext{d}v+\int_0^\infty \int_{S^{2}}\int_{\mathbb{R}^3}\frac{1}{c}I_0{\bf x}i(0)\thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
=&-\int_0^\infty \int_{S^{2}}\int_{0}^{T}\int_{\mathbb{R}^3}A_r{\bf x}i\thetaext{d}x \thetaext{d}t\thetaext{d}\mathbb{R}^3mega \thetaext{d}v;\\[6pt]
&\int_{0}^{T}\int_{\mathbb{R}^3}\Big(\rhoho\zeta_t+\rhoho u \cdot\nabla\zeta\Big)\thetaext{d}x \thetaext{d}t+\int_{\mathbb{R}^3}\rhoho_0\zeta(0)\thetaext{d}x=0;\\[6pt]
&\int_{0}^{T}\int_{\mathbb{R}^3}\Big(\rhoho u\varphi_t+\rhoho u\otimes u:\nabla\varphi+p_m\thetaext{div}\varphi-\mu \nabla u:\nabla \varphi-(\lambda+\mu)\thetaext{div}u\thetaext{div}\varphi \Big)\thetaext{d}x \thetaext{d}t\\
\displaystyle
=&-\int_{\mathbb{R}^3}\rhoho_0u_0\varphi(0)\thetaext{d}x+\frac{1}{c}\int_{0}^{T}\int_{\mathbb{R}^3}A_r\mathbb{R}^3mega\cdot\varphi\thetaext{d}x \thetaext{d}t;
\end{split}
\end{equation}
for any test functions ${\bf x}i={\bf x}i(v,\mathbb{R}^3mega,t,x)\in C^\infty_c(\mathbb{R}^+\thetaimes S^2\thetaimes [0,T)\thetaimes\mathbb{R}^3 )$, $\zeta=\zeta(t,x)\in C^\infty_c( [0,T)\thetaimes\mathbb{R}^3 )$ and $\varphi \in \mathbb{R}^3$ with $\varphi \in C^\infty_c( [0,T)\thetaimes\mathbb{R}^3 )$.
\item $(I,\rhoho,u)$ satisfies the following regularities:
\begin{equation*}\begin{split}
&I\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];H^1\cap W^{1,q}(\mathbb{R}^3))),\ I_t\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];L^{2}\cap L^{q}(\mathbb{R}^3))),\\
&\rhoho\geq 0,\ \rhoho-\overline{\rhoho}\in C([0,T];H^1\cap W^{1,q}),\quad \rhoho_t\in C([0,T];L^{2}\cap L^{q}),\\
&u\in C([0,T];\mathbb{D}^1\cap D^2)\cap L^2([0,T];D^{2,q}),\ u_t\in L^2([0,T];\mathbb{D}^1), \ \sigmaqrt{\rhoho}u_t\in L^\infty([0,T];L^2).
\end{split}
\end{equation*}
\end{enumerate}
\end{definition}
As has been observed in $3$-D compressible isentropic Navier-Stokes equations \cite{CK}, in order to make sure that the Cauchy problem with initial mass density containing vacuum is well-posed, the lack of a positive lower bound of the initial mass density $\rhoho_0$ should be compensated by some initial layer compatibility condition on the initial data $(\rhoho_0,u_0)$. Now considering the $3$-D compressible isentropic radiation hydrodynamic equations (\rhoef{eq:1.2}), if we denote $p^0_m=A\rhoho^\gamma_0$, $S_0=S(v,\mathbb{R}^3mega,t=0,x)$, and
\begin{equation*}
\begin{split} A^0_r=&S_0-\sigmaigma_a(v,\mathbb{R}^3mega,t=0,x,\rhoho_0)I_0
+\int_0^\infty \int_{S^{2}} \Big(\frac{v}{v'}\sigmaigma_s(\rhoho_0)I'_0
-\sigmaigma'_s(\rhoho_0)I_0\Big) \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',\\
\sigmaigma_s(\rhoho_0)=& \sigmaigma_s(v' \rhoightarrow v, \mathbb{R}^3mega'\cdot\mathbb{R}^3mega,\rhoho_0),\ \sigmaigma'_s(\rhoho_0)= \sigmaigma_s(v \rhoightarrow v', \mathbb{R}^3mega\cdot\mathbb{R}^3mega',\rhoho_0),
\end{split}
\end{equation*}
then the main result of this paper on the existence of the unique local strong solutions can be shown as
\begin{theorem} [{\thetaextbf{Local existence of strong solutions}}]\ \\[4pt]
\label{th1}
Let the assumptions (\rhoef{zhen1})-(\rhoef{jia345}) hold, and assume that
$$\|S(v,\mathbb{R}^3mega,t,x)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;C^1([0,\infty);H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C^1([0,\infty); L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2(\mathbb{R}^3)))}< +\infty.$$
If the initial data $(I_0,\rhoho_0, u_0)$ satisfy the regularities
\begin{equation}\label{gogo}
\begin{split}
& I_0(v,\mathbb{R}^3mega,x)\in L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3)),\\
&\rhoho_0\geq 0, \quad \rhoho_0-\overline{\rhoho}\in H^1\cap W^{1,q},\quad u_0\in \mathbb{D}^1\cap D^2,\\
&(I_0, \rhoho_0, u_0)\rhoightarrow (0,\overline{\rhoho},0), \ \thetaext{as}\ |x|\mapsto\infty,\ \forall \ (v,\mathbb{R}^3mega)\in\mathbb{R}^+\thetaimes S^2,
\end{split}
\end{equation}
and the initial layer compatibility condition
\begin{equation}\label{kkkkk}
\begin{split}
Lu_0+\nabla p^0_{m}+\frac{1}{c}\int_0^\infty \int_{S^2}A^0_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v=&\rhoho^{\frac{1}{2}}_0 g_1
\end{split}
\end{equation}
for some $g_1 \in L^2$, then there exists a time $T_*>0$ and a unique strong solution $(I,\rhoho,u)$ on $\mathbb{R}^+\thetaimes S^2\thetaimes [0,T_*]\thetaimes\mathbb{R}^3 $ to Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}).
\end{theorem}
\begin{remark}\label{zhen99}
For the case that the rate of energy emission $S$ depends on the mass density $\rhoho$, that is, $S=S(v,\mathbb{R}^3mega, t,x,\rhoho)$, similar results can be obtained via the same argument as the case $S=S(v,\mathbb{R}^3mega,t,x)$, if we assume, for $\rhoho^{i}(t) \ (i=1,2)$ satisfying (\rhoef{zhen3}), that,
\begin{equation}\label{jia666}
\begin{cases}
\|S(v,\mathbb{R}^3mega,t,x,\rhoho^i)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;L^\infty( \mathbb{R}^3))\cap L^1(\mathbb{R}^+\thetaimes S^2;L^r( \mathbb{R}^3))} \leq M(|\rhoho^i(t)|_\infty),\\[6pt]
\|\nabla S(v,\mathbb{R}^3mega,t,x,\rhoho^i)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;L^r( \mathbb{R}^3))}\leq M(|\rhoho^i(t)|_\infty)(|\nabla \rhoho(t)|_r+1),\\[6pt]
\|S_t(v,\mathbb{R}^3mega,t,x,\rhoho^i)\|_{ L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2( \mathbb{R}^3))} \leq M(|\rhoho^i(t)|_\infty)(|\rhoho_t(t)|_2+1),\\[6pt]
|S(v,\mathbb{R}^3mega,t,x,\rhoho_1)-S(v,\mathbb{R}^3mega, t,x,\rhoho_2)|\leq \overline{S}(v,\mathbb{R}^3mega,t,x,\rhoho_1,\rhoho_2)|\rhoho^1(t)-\rhoho^2(t)|, \\[6pt]
\|\overline{S}(v,t,x,\rhoho_1,\rhoho_2)\|_{ L^1(\mathbb{R}^+\thetaimes S^2;L^3( \mathbb{R}^3))\cap L^2 (\mathbb{R}^+\thetaimes S^2;L^2( \mathbb{R}^3))}\leq M(|(\rhoho^1,\rhoho^2)(t)|_\infty)
\end{cases}
\end{equation}
for $t\in [0,T]$ and $r \in [2,q]$.
\end{remark}
Our second result can be regarded as an explanation for the compatibility between (\rhoef{gogo}) and (\rhoef{kkkkk}) when the initial vacuum is not so irregular. To be more precise, we denote by $V$ the initial vacuum set, i.e, the interior of the zero-set of the initial density in $\mathbb{R}^3$, and define the Sobolev space $D^{1}_0(V)$ as
$$ D^{1}_0(V)=\{f\in L^6(V): |f|_{D^{1}_0}=|\nabla f|_{L^2}<\infty,\ f|_{\partial V}=0\}.$$
Then we have
\begin{theorem} [{\thetaextbf{Necessity and sufficiency of the compatibility condition}}]\ \\[4pt]
\label{th2}
Let conditions in Theorem \rhoef{th1} hold. We assume that either the initial vacuum set $V$ is empty or the elliptic system
\begin{equation}\label{zhen101}
L\phi=-\mu\thetariangle \phi-(\lambda+\mu)\nabla \emph{div}\phi=0
\end{equation}
has only zero solution in $\mathbb{D}^1_0(V)\cap D^2(V)$. Then there exists a unique local strong solution $(I,\rhoho,u)$ satisfying
\begin{equation}\label{coco}
\begin{split}
&\|I(t)-I_0\|_{H^1\cap W^{1,q}(\mathbb{R}^3)}\rhoightarrow 0, \quad \thetaext{as}\ t\rhoightarrow 0, \ \forall \ (v,\mathbb{R}^3mega) \in \mathbb{R}^+\thetaimes S^2,\\
&\|\rhoho(t)-\rhoho_0\|_{H^1\cap W^{1,q}(\mathbb{R}^3)}+|u(t)-u_0|_{\mathbb{D}^1\cap D^2(\mathbb{R}^3)}\rhoightarrow 0, \quad \thetaext{as}\ t\rhoightarrow 0,
\end{split}
\end{equation}
if and only if the initial data satisfy the initial layer compatibility condition (\rhoef{kkkkk}).
\end{theorem}
\begin{remark}\label{initial}
From the regularities of the strong solution $(I,\rhoho,u)$ in the Definition \rhoef{strong1}, we know that
\begin{equation*}
\begin{split}
&I(v,\mathbb{R}^3mega,t,x)\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];H^1\cap W^{1,q}(\mathbb{R}^3))), \\
&\rhoho(t,x)-\overline{\rhoho}\in C([0,T];H^1\cap W^{1,q}(\mathbb{R}^3)), \\
&u(t,x)\in C([0,T];\mathbb{D}^1\cap D^2(\mathbb{R}^3)).
\end{split}
\end{equation*}
But since the strong solution $(I,\rhoho,u)$ satisfies the Cauchy problem only in the sense of distribution, we only have $I(v,\mathbb{R}^3mega,t=0,x)=I_0$, $\rhoho(t=0,x)=\rhoho_0$ and $\rhoho u(t=0,x)=\rhoho_0 u_0$. In the vacuum domain, the relation $u(t=0,x)=u_0$ maybe not hold.
Theorem \rhoef{th2} tells us that if the initial vacuum set $V$ has a sufficiently simple geometry, for instance, it is a domain with Lipschitz boundary, we then have $u(t=0,x)=u_0$.
\end{remark}
Finally, we give a blow-up criterion for strong solutions obtained in Theorem \rhoef{th1}.
\begin{theorem} [{\thetaextbf{Blow-up criterion for the local strong solution}}]\ \\[4pt]
\label{th3}
Let conditions in Theorem \rhoef{th1} hold.
If $\overline{T}< +\infty $ is the maximal existence time of the local strong solution $(I,\rhoho, u)$ obtained in Theorem \rhoef{th1}, then we have
\begin{equation}\label{blowcr}
\begin{split}
\lim \sigmaup_{t\mapsto \overline{T}} \thetaotog(\|I(t)\|_{L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3))}+\|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+|u(t)|_{\mathbb{D}^1}\thetaotog)=+\infty.
\end{split}
\end{equation}
\end{theorem}
\begin{remark}[\thetaextbf{General barotropic flow}]
\label{co1}
Similar results also hold for general barotropic flow.
Let (\rhoef{zhen1}) and (\rhoef{jia345}) hold, $p_m=p_m(\rhoho)\in C^1(\mathbb{\overline{R}}^+)$ and
$$\|S(v,\mathbb{R}^3mega,t,x)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;C^1([0,\infty);H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C^1([0,\infty); L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2(\mathbb{R}^3)))}< +\infty.$$
Assume that the initial data $(I_0, \rhoho_0,u_0)$ satisfy the regularity conditions
\begin{equation}\label{gogggo}
\begin{split}
& I_0(v,\mathbb{R}^3mega,x)\in L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3)), \\
&\rhoho_0\geq 0,\quad \rhoho_0-\overline{\rhoho}\in H^1\cap W^{1,q},\quad u_0\in \mathbb{D}^1\cap D^2,\\
&(I_0, \rhoho_0, u_0)\rhoightarrow (0,\overline{\rhoho},0), \ \thetaext{as}\ |x|\mapsto\infty,\ \forall (v,\mathbb{R}^3mega)\in\mathbb{R}^+\thetaimes S^2,
\end{split}
\end{equation}
and the compatibility condition
\begin{equation}\label{kkcv}
\begin{split}
Lu_0+\nabla p^0_{m}+\frac{1}{c}\int_0^\infty \int_{S^2}A^0_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v=&\rhoho^{\frac{1}{2}}_0 g_2
\end{split}
\end{equation}
for some $g_2 \in L^2$, where $p^0_m=p_m(\rhoho_0)$, $ A^0_r$ is defined as before.
Then the conclusions obtained in Theorems \rhoef{th1}-\rhoef{th3} also hold for (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}).
\end{remark}
\sigmaection{The existence and uniqueness of local strong solutions}
We prove Theorem \rhoef{th1} in this section, i.e., the existence and uniqueness of local strong solutions. For the rate of energy emission $S$, we always assume that
$$\|S(v,\mathbb{R}^3mega, t,x)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;C^1([0,\infty);H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C^1([0,\infty); L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2(\mathbb{R}^3)))}< +\infty.$$
In order to prove
the local existence of strong solutions to the nonlinear problem, we need to consider the linearized system
\begin{equation}
\label{eq:1.weeer}
\begin{cases}
\displaystyle
\rhoho_t+\thetaext{div}(\rhoho w)=0,\\[8pt]
\displaystyle
\frac{1}{c}I_t+\mathbb{R}^3mega\cdot\nabla I=\overline{A}_r,\\[8pt]
\displaystyle
(\rhoho u)_t+\thetaext{div}(\rhoho w\otimes u)
+\nabla p_m +Lu=-\frac{1}{c}\int_0^\infty \int_{S^2}A_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v,
\end{cases}
\end{equation}
with the initial data (\rhoef{eq:2.2hh}), where
$w=w(t,x)\in \mathbb{R}^3$ is a known vector, the terms $\overline{A}_{r}$ and $A_{r}$ are defined by
\begin{equation*}
\begin{split}
\overline{A}_r=&S-\sigmaigma_a(\rhoho)I+\int_0^\infty \int_{S^2} \Big(\frac{v}{v'}\sigmaigma_s(\rhoho)\psi-\sigmaigma'_s(\rhoho)I \Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',\\
A_r=&S-\sigmaigma_a(\rhoho)I+\int_0^\infty \int_{S^2} \Big(\frac{v}{v'}\sigmaigma_s(\rhoho)I'-\sigmaigma'_s(\rhoho)I \Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',
\end{split}
\end{equation*}
where $\psi=\psi(v',\mathbb{R}^3mega',t,x)$ is a known function. We assume that
\begin{equation}\label{tyuu}
\begin{split}
& I_0(v,\mathbb{R}^3mega,x)\in L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}),\ \rhoho_0\geq 0,\ \rhoho_0-\overline{\rhoho}\in H^1\cap W^{1,q}, \ u_0\in \mathbb{D}^1\cap D^2,\\
& w\in C([0,T];\mathbb{D}^1\cap D^2)\cap L^2([0,T];D^{2,q}), \ w_t\in L^2([0,T];\mathbb{D}^1),\\
&\psi\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];H^1\cap W^{1,q})),\ \psi_t\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];L^{2}\cap L^{q})),\\
& (w,\psi)|_{t=0}=(u_0,I_0).
\end{split}
\end{equation}
\sigmaubsection{A priori estimates to the linearized problem away from vacuum}\ \\
We immediately have the global existence of a unique strong solution $(I,\rhoho,u)$ to (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}) by the standard methods at least for the case that the initial mass density is away from vacuum.
\begin{lemma}\label{lem1}
Assume in addition to (\rhoef{tyuu}) that $\rhoho_0\geq \delta$ for some constant $\delta>0$ and the compatibility condition (\rhoef{kkkkk}) holds.
Then there exists a unique strong solution $(I,\rhoho,u)$ to Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}) such that
\begin{equation*}\begin{split}
&I\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C([0,T]; L^2(\mathbb{R}^+\thetaimes S^2; L^{2}\cap L^{q}(\mathbb{R}^3))),\\
&I_t\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];L^{2}\cap L^{q}(\mathbb{R}^3))),\\
&\rhoho-\overline{\rhoho}\in C([0,T];H^1\cap W^{1,q}),\ \rhoho_t\in C([0,T];L^{2}\cap L^{q}),\ \rhoho \geq {\bf u}nderline{\delta},\\
&u\in C([0,T];H^2)\cap L^2([0,T];D^{2,q}),\ u_t\in C([0,T];L^2)\cap L^2([0,T];H^1),\ u_{tt}\in L^2([0,T];H^{-1}),
\end{split}
\end{equation*}
for some constants $3<q \leq 6$ and ${\bf u}nderline{\delta}>0$.
\end{lemma}
\begin{proof}
First, the existence and regularity of the unique solution $\rhoho$ to $(\rhoef{eq:1.weeer})_1$ can be obtained essentially according to the same argument in \cite{CK} for Navier-Stokes equations, and $\rhoho$ can be expressed by
\begin{equation}
\label{eq:bb1}
\rhoho(t,x)=\rhoho_0(U(0;t,x))\exp\Big(-\int_{0}^{t}\thetaextrm{div}\, w(s,U(s,t,x))\thetaext{d}s\Big),
\end{equation}
where $U\in C([0,T]\thetaimes[0,T]\thetaimes \mathbb{R}^3)$ is the solution to the initial value problem
\begin{equation}
\label{eq:bb1}
\begin{cases}
\displaystyle\frac{\thetaext{d}}{\thetaext{d}s}U(s;t,x)=w(s,U(s;t,x)),\quad 0\leq s\leq T,\\
U(t;t,x)=x, \quad \ \qquad \quad 0\leq t\leq T,\ x\in \mathbb{R}^3,
\end{cases}
\end{equation}
so we can easily get the positive lower bound of $\rhoho$.
Second, $(\rhoef{eq:1.weeer})_2$ can be rewritten into
\begin{equation}\label{rst}
\frac{1}{c}I_t+\mathbb{R}^3mega\cdot\nabla I+\Big(\sigmaigma_a+\int_0^\infty \int_{S^2} \sigmaigma'_s(\rhoho)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)I=F(v,\mathbb{R}^3mega,t,x),
\end{equation}
where
$$F=S+\int_0^\infty \int_{S^2} \frac{v}{v'}\sigmaigma_s(\rhoho)\psi\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];H^1\cap W^{1,q}(\mathbb{R}^3))),$$
then we easily get the existence and regularity of a unique solution $I$ to (\rhoef{rst}) such that
$$
I\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];H^1\cap W^{1,q}(\mathbb{R}^3))),\ I_t\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T];L^{2}\cap L^{q}(\mathbb{R}^3))),
$$
and according to the classical imbedding theory for Sobolev spaces, it is easy to show that
$$ I\in C([0,T]; L^2(\mathbb{R}^+\thetaimes S^2; L^{2}\cap L^{q}(\mathbb{R}^3))).$$
Finally, the momentum equations $(\rhoef{eq:1.weeer})_3$ can be written into
\begin{equation}\label{rst1}
\displaystyle
u_t+ w\cdot\nabla u
+\rhoho^{-1}Lu= -\rhoho^{-1}\nabla p_m -\frac{1}{c}\rhoho^{-1}\int_0^\infty \int_{S^2}A_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v,
\end{equation}
then the existence and regularity of the unique solution $u$ to the corresponding linear parabolic problem can be obtained by standard methods as in \cite{CK2}\cite{CK3}.
\end{proof}
In order to pass to the limit as $\delta\rhoightarrow 0$, we need to establish a priori estimates independent of $\delta$ for the solution $(I,\rhoho,u)$ to Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}) obtained in Lemma \rhoef{lem1}.
We fix a positive constant $c_0$ sufficiently large such that
\begin{equation*}\begin{split}
&2+\|\rhoho_0-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+|u_0|_{\mathbb{D}^1\cap D^2}+\|I_0\|_{L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3))}+|g_1|_2\\
&+\|S(v,\mathbb{R}^3mega, t,x)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;C^1([0,\infty);H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C^1([0,\infty); L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2(\mathbb{R}^3)))}\leq c_0,
\end{split}
\end{equation*}
and
\begin{equation*}\begin{split}
\displaystyle
\|\psi\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3)))}\leq& c_1,\\
\displaystyle
\|\psi_t\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];L^{2}\cap L^{q}(\mathbb{R}^3)))}\leq& c_2,\\
\displaystyle
\sigmaup_{0\leq t \leq T^*}|w(t)|^2_{\mathbb{D}^1}+\int_{0}^{T^*}|w(s)|^2_{D^{2}}\thetaext{d}s\leq& c^2_3,\\
\displaystyle
\sigmaup_{0\leq t \leq T^*}|w(t)|^2_{D^2}+\int_{0}^{T^*}\Big(|w(s)|^2_{D^{2,q}}+|w_t(s)|^2_{\mathbb{D}^1}\Big)\thetaext{d}s\leq& c^2_4,
\end{split}
\end{equation*}
for some time $T^*\in (0,T)$ and constants $c_i$ ($i=1,2,3,4$) such that
$$1< c_0\leq c_1 \leq c_2 \leq c_3 \leq c_4. $$ The constants $c_i$ ($i=1,2,3,4$) and $T^*$ will be determined later and depend only on $c_0$ and the fixed constants $\overline{\rhoho}$, q, A, $\mu$, $\lambda$, $\gamma$, $c$ and $T$.
As defined in assumption (\rhoef{jia345}), $M=M(\cdot): [0,+\infty) \rhoightarrow[1,+\infty)$ still denotes a strictly increasing continuous function depending only on fixed constants $\overline{\rhoho}$, q, A, $\mu$, $\lambda$, $\gamma$, $c$ and $T$.\\
We first give the a priori estimates for density $\rhoho$. Hereinafter, we use $C\geq 1$ to denote a generic positive constant depending only on fixed constants $\overline{\rhoho}$, q, A, $\mu$, $\lambda$, $\gamma$, $c$ and $T$.
\begin{lemma}[\thetaextbf{Estimates for the mass density $\rhoho$}]\label{lem:2}
For the strong solution $(I,\rhoho,u)$ to the Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}), there exists a time $T_1>0$ such that
\begin{equation*}\begin{split}
\|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+\|p_m(t)-\overline{p}\|_{H^1\cap W^{1,q}}\leq& M(c_0),\\
|\rhoho_t(t)|_{2}+ |(p_m)_t(t)|_{2}\leq M(c_0)c_3,\quad |\rhoho_t(t)|_{q}+ |(p_m)_t(t)|_{q}\leq& M(c_0)c_4,
\end{split}
\end{equation*}
for $0\leq t \leq T_1=\min(T^*,(1+c^2_4)^{-1})$ and $\overline{p}=A\overline{\rhoho}^\gamma$.
\end{lemma}
\begin{proof}
From the continuity equation and the standard energy estimates as shown in \cite{CK}, for $2\leq r\leq q$, we have
\begin{equation*}\begin{split}
\|\rhoho(t)-\overline{\rhoho}\|_{W^{1,r}}\leq \Big(\|\rhoho_0-\overline{\rhoho}\|_{W^{1,r}}+\int_0^t \|\nabla w(s)\|_{W^{1,r}}\thetaext{d}s\Big) \exp\Big(C\int_0^t \|\nabla w(s)\|_{W^{1,q}}\thetaext{d}s\Big).
\end{split}
\end{equation*}
Therefore, the desired estimate for $\rhoho$ follows by observing that
$$
\int_0^t \|\nabla w(s)\|_{W^{1,r}}\thetaext{d}s\leq t^{\frac{1}{2}}\Big(\int_0^t \|\nabla w(s)\|^2_{ W^{1,r}}\thetaext{d}s\Big)^{\frac{1}{2}}\leq C(c_1t+(c_1t)^{\frac{1}{2}}),
$$
for $0\leq t \leq T_1=\min(T^*,(1+c^2_4)^{-1})$. The estimate for $\rhoho_t$ is clear from $\rhoho_t=-\thetaext{div}(\rhoho w)$.
Due to $p_m=A\rhoho^\gamma$ ($\gamma>1$),
then the estimate for $p_m$ follows immediately from above.
\end{proof}
Now we give the a priori estimates for $I$.
\begin{lemma}[\thetaextbf{Estimates for specific radiation intensity $I$}]\label{lem:3}
For the strong solution $(I,\rhoho,u)$ to the Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}), there exists a time $T_2>0$ such that
\begin{equation}\label{2I}
\begin{split}
\|I\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T_2];H^1\cap W^{1,q}(\mathbb{R}^3)))}\leq &Cc_0,\\
\|I_t\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T_2];L^{2}\cap L^{q}(\mathbb{R}^3)))}\leq& M(c_0)c_0,
\end{split}
\end{equation}
for $T_2= \min(T^*,(1+M(c_0)c^2_4)^{-1})$.
\end{lemma}
\begin{proof}Let $2\leq r\leq q$. First, multiplying $ (\rhoef{eq:1.weeer})_2$ by $r|I|^{r-2}I$ and integrating over $\mathbb{R}^3$ with respect to $x$, we have
\begin{equation}\label{thyt}
\begin{split}
\frac{\thetaext{d}}{\thetaext{d}t}|I|_{r}
\leq C|S|_{r}
+C|\rhoho|_\infty\int_0^\infty \int_{S^2} \frac{v}{v'}|\psi|_{r} \overline{\sigmaigma}_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',
\end{split}
\end{equation}
where we used the fact that $\sigmaigma_a\geq 0$ and $\sigmaigma'_s\geq 0$.
According to the assumptions (\rhoef{zhen1})-(\rhoef{jia345}) and H\"older's inequality, it is not hard to deduce that
\begin{equation}\label{thyt1}
\begin{split}
\frac{\thetaext{d}}{\thetaext{d}t}|I|^2_{r}
\leq& C\Big(|I|^2_{r}+|S|^2_{r}+ |\psi|^2_{L^2(\mathbb{R}^+\thetaimes S^2; L^r)}|\rhoho|^2_\infty \int_0^\infty \int_{S^2}\Big|\frac{v}{v'}\Big|^2 \overline{\sigmaigma}^2_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)\\
\leq& C\Big(|I|^2_{r}+|S|^2_{r}
+M(c_0)c^2_1\int_0^\infty \int_{S^2}\Big|\frac{v}{v'}\Big|^2 \overline{\sigmaigma}^2_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big),
\end{split}
\end{equation}
for $0\leq t\leq T_1$.
Second, differentiating $ (\rhoef{eq:1.weeer})_2$ $\beta$-times ($|\beta|=1$) with respect to $x$, then multiplying the resulting equation by $r|\partial^\beta_xI|^{r-2}\partial^\beta_xI$ and integrating over $\mathbb{R}^3$ with respect to $x$, we get
\begin{equation}\label{thytffg}
\begin{split}
\frac{\thetaext{d}}{\thetaext{d}t}|\partial^\beta_x I|_{r}
\leq& C\thetaotog(|\partial^\beta_x S|_{r}+|\sigmaigma_a|_{D^{1,q}}|I|_{\frac{qr}{q-r}}\thetaotog)+C|\nabla\rhoho|_{q}|I|_{\frac{qr}{q-r}}\int_0^\infty \int_{S^2} \overline{\sigmaigma}'_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\\
&+C \int_0^\infty \int_{S^2} \frac{v}{v'}\Big(|\rhoho|_{\infty}|\psi|_{D^{1,r}} \overline{\sigmaigma}_s+
|\rhoho|_{D^{1,r}} \|\psi\|_{W^{1,q}} \overline{\sigmaigma}_s\Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',
\end{split}
\end{equation}
where we also used the fact that $\sigmaigma_a\geq 0$ and $\sigmaigma'_s\geq 0$.
Since $r<\frac{qr}{q-r}\leq \frac{3r}{3-r} $ if $2\leq r \leq 3$ and $r<\frac{qr}{q-r}\leq +\infty $ if $3<r \leq q$, it follows from Sobolev's imbedding theorem that $|I|_{\frac{qr}{q-r}}\leq C \|I\|_{W^{1,r}}$. From assumption (\rhoef{jia345}) and Lemma \rhoef{lem:2}, we easily have
\begin{equation}\label{da1}
\begin{split}
|\sigmaigma_a|_{D^{1,q}}\leq |\nabla \sigmaigma|_q |\rhoho|_\infty+|\nabla\rhoho|_q |\sigmaigma|_\infty\leq M( |\rhoho|_\infty)\thetaotog(|\rhoho|_\infty|\nabla\rhoho|_q+|\rhoho|_\infty+|\nabla\rhoho|_q\thetaotog).
\end{split}
\end{equation}
According to assumptions (\rhoef{zhen1})-(\rhoef{jia345}), estimates (\rhoef{thytffg})-(\rhoef{da1}), and the H\"older's inequality, we obtain
\begin{equation}\label{thytq}
\begin{split}\frac{\thetaext{d}}{\thetaext{d}t}|\partial^\beta_x I|^2_{r}
\leq& C\thetaotog(1+|\sigmaigma_a|_{D^{1,q}}+|\nabla\rhoho|_{q}\int_0^\infty \int_{S^2} \overline{\sigmaigma}'_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaotog)\|I\|^2_{W^{1,r}}+C|\partial^\beta_x S|^2_{r}\\
&+M(c_0)\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2 \overline{\sigmaigma}^2_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\cdot\int_0^\infty \int_{S^2} |\psi|^2_{D^{1,r}} \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\\
&+M(c_0)\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2\overline{\sigmaigma}^2_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\cdot\int_0^\infty \int_{S^2} \|\psi\|^2_{W^{1,q}} \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\\
\leq&M(c_0)\|I\|^2_{W^{1,r}}+C|\partial^\beta_x S|^2_{r}+M(c_0)c^2_1\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2 \overline{\sigmaigma}^2 _s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'
\end{split}
\end{equation}
for $0\leq t\leq T_1$.
Then combining (\rhoef{thyt1}) and (\rhoef{thytq}), it turns out that
\begin{equation}\label{thytgb}
\begin{split}
&\frac{\thetaext{d}}{\thetaext{d}t}\|I\|^2_{W^{1,r}}
\leq M(c_0)\|I\|^2_{W^{1,r}}+C\| S\|^2_{W^{1,r}}+M(c_0)c^2_1\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2 \overline{\sigmaigma}^2_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'.
\end{split}
\end{equation}
From Gronwall's inequality, we get
\begin{equation}\label{coti}
\begin{split}
&\|I(v,\mathbb{R}^3mega,\cdot,\cdot)\|^2_{C([0,T_2];H^1\cap W^{1,q}(\mathbb{R}^3))}
\leq \exp (M(c_0)T_2)\|I_0\|^2_{W^{1,r}}\\
&+\exp (M(c_0)T_2)\Big(\int_0^{T_2}\|S\|^2_{W^{1,r}}\thetaext{d}s+M(c_0)c^2_1T_2\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2 \overline{\sigmaigma}^2_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big),
\end{split}
\end{equation}
where $ T_2= \thetaext{min}(T^*,(1+M(c_0)c^2_4)^{-1})$.
Then integrating the above inequality in $\mathbb{R}^+\thetaimes S^2$ with respect to $(v,\mathbb{R}^3mega)$ and using assumptions (\rhoef{zhen1})-(\rhoef{jia345}), we arrive at
$$
\|I(v,\mathbb{R}^3mega,t,x)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T_2];H^1\cap W^{1,q}(\mathbb{R}^3)))}\leq Cc^2_0.
$$
Finally, due to $ I_t=-c\mathbb{R}^3mega\cdot\nabla I+c\overline{A}_r$, the desired estimates for $I_t$ are obvious.
\end{proof}
Now we give the a priori estimates for $u$.
\begin{lemma}[\thetaextbf{Estimates for velocity $u$}]\label{lem:4}
For the strong solution $(I,\rhoho,u)$ to the Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}), there exists a time $T_3$ such that
\begin{equation}\label{gai}
\begin{split}
|u(t)|^2_{\mathbb{D}^1}+\int_{0}^{t}\Big(|u(s)|^2_{D^2}+|\sigmaqrt{\rhoho}u_t(s)|^2_{2}\Big)\thetaext{d}s\leq& M(c_0)c^{2}_0,\\
|u(t)|^2_{\mathbb{D}^1\cap D^2}+|\sigmaqrt{\rhoho}u_t(t)|^2_{2}+\int_{0}^{t}\Big(|u(s)|^2_{D^{2,q}}+|u_t(s)|^2_{\mathbb{D}^1}\Big)\thetaext{d}s\leq& M(c_0)c^{10}_3,
\end{split}
\end{equation}
for $0\leq t \leq T_3= \min(T^*,(1+M(c_0)c^{8}_4)^{-1})$.
\end{lemma}
\begin{proof}
$\thetaext{{\bf u}nderline{Step 1}}$. The estimate of $|u|_{\mathbb{D}^1}$. Multiplying $ (\rhoef{eq:1.weeer})_3$ by $u_t$ and integrating over $\mathbb{R}^3$, we have
\begin{equation}\label{f1}
\begin{split}
&\int_{\mathbb{R}^3}\rhoho |u_t|^2 \thetaext{d}x+\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}\Big(\mu|\nabla u|^2+\thetaotog(\lambda+\mu\thetaotog)(\thetaext{div}u)^2\Big) \thetaext{d}x\\
=& \int_{\mathbb{R}^3} \Big(-\nabla p_m-\rhoho w\cdot \nabla u\Big)\cdot u_t\thetaext{d}x-\frac{1}{c}\int_{\mathbb{R}^3} \int_0^\infty \int_{S^2} A_r u_t\cdot \mathbb{R}^3mega\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x\\
=&\frac{d}{dt}\Lambda_1(t)- \Lambda_2(t)+E_I,
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
\Lambda_1(t)=\int_{\mathbb{R}^3} ( p_m-\overline{p})\thetaext{div}u\thetaext{d}x,\quad \Lambda_2(t)=\int_{\mathbb{R}^3} \Big((p_m)_t\thetaext{div}u+\rhoho (w\cdot \nabla u)\cdot u_t \Big)\thetaext{d}x.
\end{split}
\end{equation*}
According to Lemma \rhoef{lem:2}, H\"older's inequality, Gagliardo-Nirenberg inequality and Young's inequality, we have
\begin{equation*}
\begin{split}
\Lambda_1(t)\leq& C|\nabla u|_2 |p_m-\overline{p}|_2
\leq M(c_0)|\nabla u|_2,\\
\Lambda_2(t)\leq& C\thetaotog(|\nabla u|_2 |(p_m)_t|_2+|\rhoho|^{\frac{1}{2}}_{\infty} |\sigmaqrt{\rhoho}u_t|_2 |w|_{\infty} |\nabla u|_2\thetaotog)\\
\leq& M(c_0)c_3|\nabla u|_2+M(c_0)c^2_4|\nabla u|^2_2+\frac{1}{10}|\sigmaqrt{\rhoho} u_t|^2_2,
\end{split}
\end{equation*}
for $0< t\leq T_2$.
Now we estimate the radiation term $E_{I}$, where
\begin{equation*}
\begin{split}
E_{I}=&-\frac{1}{c}\int_{\mathbb{R}^3} \int_0^\infty \int_{S^2} A_r u_t\cdot \mathbb{R}^3mega\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x \\
=&-\frac{1}{c}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} \Big(S-\sigmaigma_a I+ \int_0^\infty \int_{S^2} \frac{v}{v'}\sigmaigma_s I'_t\thetaext{d}\mathbb{R}^3mega'\thetaext{d}v'\Big)u_t \cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&+\frac{1}{c}\int_0^\infty \int_{S^2}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} \sigmaigma'_s I u_t\cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega'\thetaext{d}v' \thetaext{d}\mathbb{R}^3mega \thetaext{d}v
=:\sigmaum_{j=1}^{4} J_j.
\end{split}
\end{equation*}
We estimate $J_{j}$ term by term. From Lemmas \rhoef{lem:2}-\rhoef{lem:3}, Gagliardo-Nirenberg inequality, H\"older's inequality, Young's inequality and (\rhoef{zhen1})-(\rhoef{jia345}), for $0\leq t\leq T_2$, we have
\begin{equation*}
\begin{split}
J_1= &-\frac{1}{c} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S u_t\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
=&-\frac{1}{c} \frac{d}{dt}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S u\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v+\frac{1}{c} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S_t u\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& -\frac{1}{c} \frac{d}{dt}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S u\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v+C|u|_{\mathbb{D}^1}\int_0^\infty \int_{S^2} |S_t |_{\frac{6}{5}}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& -\frac{1}{c} \frac{d}{dt}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S u\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v+Cc_0| \nabla u|_2,\\
J_{2}=&\frac{1}{c} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} \sigmaigma_a I u_t\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v
\leq C|\rhoho|^{\frac{1}{2}}_{\infty}\int_0^\infty \int_{S^2}|\sigmaqrt{\rhoho}u_t|_2 |\sigmaigma|_{\infty}|I|_2\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& \frac{1}{20}| \sigmaqrt{\rhoho} u_t|^2_{2}+C|\rhoho|_\infty\int_0^\infty \int_{S^2} |I |^2_{2}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v \int_0^\infty \int_{S^2} |\sigmaigma|^2_{\infty}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v \\
\leq& \frac{1}{20}| \sigmaqrt{\rhoho} u_t|^2_{2} +M(c_0)c^{2}_0,\\
J_{3}= &-\frac{1}{c}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3}\frac{v}{v'}\sigmaigma_s I'
u_t\cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq &C|\sigmaqrt{\rhoho}u_t|_{2} |\rhoho|^{\frac{1}{2}}_\infty \int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \frac{v}{v'}
\overline{\sigmaigma}_s|I'|_2\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq &M(c_0)\int_0^\infty \int_{S^2} |I'|^2_2\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big(\int_0^\infty \int_{S^2} \Big(\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2
\overline{\sigmaigma}^2_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{\frac{1}{2}}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\Big)^2 \\
&+\frac{1}{20}| \sigmaqrt{\rhoho} u_t|^2_{2}\leq \frac{1}{20}| \sigmaqrt{\rhoho} u_t|^2_{2}+M(c_0)c^{2}_0,\\
J_{4}= &\frac{1}{c}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3}\sigmaigma'_s I
u_t\cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq &C|\sigmaqrt{\rhoho}u_t|_{2} |\rhoho|^{\frac{1}{2}}_\infty \int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}
\overline{\sigmaigma}'_s |I|_2\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq &M(c_0)\int_0^\infty \int_{S^2} |I|^2_2\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\int_0^\infty \int_{S^2} \Big(\int_0^\infty \int_{S^2}
\overline{\sigmaigma}'_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{2}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v+\frac{1}{20}| \sigmaqrt{\rhoho} u_t|^2_{2} \\
\leq& \frac{1}{20}| \sigmaqrt{\rhoho} u_t|^2_{2}+M(c_0)c^{2}_0.
\end{split}
\end{equation*}
Combining the above estimates for $\Lambda_i$ and $J_j$, it turns out that
\begin{equation}\label{fsd1}
\begin{split}
&\frac{1}{2}\int_{\mathbb{R}^3}\rhoho |u_t|^2 \thetaext{d}x+\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}\Big(\mu|\nabla u|^2+\thetaotog(\lambda+\mu\thetaotog)(\thetaext{div}u)^2\Big) \thetaext{d}x\\
\leq &M(c_0)c^2_4|\nabla u|^2_2+M(c_0)c^2_4 -\frac{1}{c} \frac{d}{dt}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S u\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v.
\end{split}
\end{equation}
Then integrating (\rhoef{fsd1}) over $(0,t)$, we have
\begin{equation*}
\begin{split}
&\int_{0}^{t}|\sigmaqrt{\rhoho} u_t(s)|^2_2\thetaext{d}s+|\nabla u(t)|^2_2\\
\leq & M(c_0)(1+c^2_4t)|\nabla u(t)|^2_2+M(c_0)c^2_4t+Cc^2_0,
\end{split}
\end{equation*}
for $0 \leq t \leq T_2$, where we have used the fact that
$$
\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S u\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\leq C|u|_{\mathbb{D}^1}\int_0^\infty \int_{S^2} |S |_{\frac{6}{5}}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\leq Cc_0| \nabla u|_2.
$$
From Gronwall's inequality, we have
\begin{equation}\label{plh}
\begin{split}
&\int_{0}^{t}|\sigmaqrt{\rhoho} u_t(s)|^2_2\thetaext{d}s+|\nabla u(t)|^2_2\\
\leq& \thetaotog(M(c_0)c^2_4t+Cc^2_0\thetaotog)\exp \thetaotog(M(c_0)(1+c^2_4t)t\thetaotog)\leq Cc^{2}_0, \quad \thetaext{for}\quad 0\leq t \leq T_2.
\end{split}
\end{equation}
According to Lemma \rhoef{lem:2}, (\rhoef{plh}) and the standard elliptic regularity estimate, we have
\begin{equation}\label{nv980}
\begin{split}
| u(t)|_{D^2}\leq & C\Big(|\rhoho u_t|_2+|\rhoho w\cdot\nabla u|_2+|\nabla p_m|_2+\int_0^\infty \int_{S^2}|A_r|_2 \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\Big)(t)\\
\leq & (|\rhoho|^{\frac{1}{2}}_\infty|\sigmaqrt{\rhoho}u_t(t)|_{2}+|\rhoho|_\infty|w|_6 |\nabla u(t)|_{3}+M(c_0)c_0)\\
\leq & M(c_0)\Big(|\sigmaqrt{\rhoho}u_t(t)|_{2}+c_3|\nabla u|^{\frac{1}{2}}_2|\nabla^2 u|^{\frac{1}{2}}_2+c_0\Big),
\end{split}
\end{equation}
which means that
\begin{equation}\label{nv9800}
\begin{split}
| u(t)|_{D^2}\leq & M(c_0)\thetaotog(|\sigmaqrt{\rhoho}u_t(t)|_{2}+c^2_3c_0\thetaotog).
\end{split}
\end{equation}
From (\rhoef{plh}) and (\rhoef{nv9800}), we know that
\begin{equation}\label{plmn}
\begin{split}
\int_{0}^{t} |u|^2_{D^2}\thetaext{d}s\leq&C \int_{0}^{t}M(c_0)\thetaotog(|\sigmaqrt{\rhoho}u_t(t)|_{2}+c^2_3c_0\thetaotog)^2\thetaext{d}s\leq M(c_0)c^{2}_0.
\end{split}
\end{equation}
$\thetaext{{\bf u}nderline{Step 2}}$. The estimate of $|u|_{D^2}$.
Differentiating $ (\rhoef{eq:1.weeer})_3$ with respect to $t$, we have
\begin{equation}\label{da2}
\rhoho u_{tt}+Lu_t=-\rhoho_tu_t-(\rhoho w\cdot\nabla u)_t-(\nabla p_m)_t-\frac{1}{c}\int_0^\infty \int_{S^2}(A_r)_t\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v,
\end{equation}
multiplying (\rhoef{da2}) by $u_t$ and integrating the resulting equations over $\mathbb{R}^3$, we obtain
\begin{equation*}
\begin{split}
&\frac{1}{2}\frac{\thetaext{d}}{\thetaext{d}t}\int_{\mathbb{R}^3}\rhoho |u_t|^2 \thetaext{d}x+\int_{\mathbb{R}^3}(\mu|\nabla u_t|^2+(\lambda+\mu)(\thetaext{div}u_t)^2) \thetaext{d}x\\
\leq& C \int_{\mathbb{R}^3} \Big(|\rhoho_t w \cdot \nabla u\cdot u_t|+|\rhoho w_t \cdot \nabla u \cdot u_t| +|\rhoho w\cdot \nabla u_t \cdot u_t|
+|\ (p_m)_t||\nabla u_t|\Big)\thetaext{d}x\\
-&\frac{1}{c}\int_{\mathbb{R}^3} \int_0^\infty \int_{S^2} (A_r)_t u_t\cdot \mathbb{R}^3mega\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x=:\sigmaum_{i=1}^{4}I_i+E_{II}.
\end{split}
\end{equation*}
First, we estimate the fluid terms $\sigmaum_{i=1}^{4}I_i$. According to Lemmas \rhoef{lem:2}-\rhoef{lem:3}, Gagliardo-Nirenberg inequality, H\"older's inequality and Young's inequality, we easily have
\begin{equation}\label{gu1}
\begin{split}
I_1=& \int_{\mathbb{R}^3} |\rhoho_t w \cdot \nabla u\cdot u_t|\thetaext{d}x \leq C|\rhoho_t|_{3} |w|_{\infty} |\nabla u|_{2} | u_t|_{6}\leq M(c_0)c^6_4+\frac{\mu}{20}| \nabla u_t|^2_{2},\\
\end{split}
\end{equation}
\begin{equation}\label{gu1gh}
\begin{split}
I_2=& \int_{\mathbb{R}^3} |\rhoho w_t \cdot \nabla u \cdot u_t| \thetaext{d}x
\leq C|\rhoho|^{\frac{1}{2}}_{\infty} | w_t|_{6} |\nabla u|_{2} |\sigmaqrt{\rhoho}u_t|_{3}\\
\leq &\frac{1}{c^2_4} |\nabla w_t|^2_2+M(c_0)c^2_0c^2_4|\sigmaqrt{\rhoho}u_t|_2|\sigmaqrt{\rhoho}u_t|_6 \\
\leq& \frac{1}{c^2_4} |\nabla w_t|^2_2+M(c_0)c^8_4|\sigmaqrt{\rhoho}u_t|^2_2+\frac{\mu}{20}|\nabla u_t|^2_2,\\
I_3=& \int_{\mathbb{R}^3} |\rhoho w\cdot \nabla u_t \cdot u_t| \thetaext{d}x \\
\leq& C|\rhoho|^{\frac{1}{2}}_{\infty} |w|_{\infty} |\nabla u_t|_{2} |\sigmaqrt{\rhoho}u_t|_{2}\leq M(c_0)c^2_4|\sigmaqrt{\rhoho}u_t|^2_{2}+\frac{\mu}{20}| \nabla u_t|^2_{2},\\
I_{4}=& \int_{\mathbb{R}^3} |\ (p_m)_t||\nabla u_t| \thetaext{d}x \leq C|(p_m)_t|_{2} |\nabla u_t|_{2}\leq M(c_0)c^{2}_4+\frac{\mu}{20}| \nabla u_t|^2_{2}.
\end{split}
\end{equation}
Now we estimate the radiation term $E_{II}$.
\begin{equation*}
\begin{split}
E_{II}=&-\frac{1}{c}\int_{\mathbb{R}^3} \int_0^\infty \int_{S^2} (A_r)_t u_t\cdot \mathbb{R}^3mega\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x \\
=&-\frac{1}{c}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} \Big(S_t-(\sigmaigma_a)_t I-\sigmaigma_a I_t+ \int_0^\infty \int_{S^2} \frac{v}{v'}(\sigmaigma_s I')_t\thetaext{d}\mathbb{R}^3mega'\thetaext{d}v'\Big)u_t \cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&+\frac{1}{c}\int_0^\infty \int_{S^2}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} (\sigmaigma'_s I)_t u_t\cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega'\thetaext{d}v' \thetaext{d}\mathbb{R}^3mega \thetaext{d}v
=:\sigmaum_{j=5}^{9} J_j.
\end{split}
\end{equation*}
We estimate $J_{j}$ term by term. From Lemmas \rhoef{lem:2}-\rhoef{lem:3}, Gagliardo-Nirenberg inequality, H\"older's inequality, Young's inequality and (\rhoef{zhen1})-(\rhoef{jia345}), for $0\leq t\leq T_2$ we have
\begin{equation*}
\begin{split}
J_5= &-\frac{1}{c} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} S_t u_t\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& C|u_t|_{\mathbb{D}^1}\int_0^\infty \int_{S^2} |S_t |_{\frac{6}{5}}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v
\leq \frac{\mu}{20}| \nabla u_t|^2_{2}+Cc^2_0,\\
J_{6}=&\frac{1}{c} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} (\sigmaigma_a)_t I u_t\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v
\leq C|u_t|_{6}\int_0^\infty \int_{S^2} |(\sigmaigma_a)_t |_{2}\|I\|_1\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& \frac{\mu}{20}| \nabla u_t|^2_{2}+C\int_0^\infty \int_{S^2} \|I \|^2_{1}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v \int_0^\infty \int_{S^2} \thetaotog(|\sigmaigma_t|^2_{2}|\rhoho|^2_\infty+|\sigmaigma|^2_{\infty}|\rhoho_t|^2_2\thetaotog)\thetaext{d}\mathbb{R}^3mega \thetaext{d}v \\
\leq& \frac{\mu}{20}| \nabla u_t|^2_{2} +M(c_0)c^{4}_4 \int_0^\infty \int_{S^2} \thetaotog(|\sigmaigma_t|^2_{2}+|\sigmaigma|^2_{\infty}\thetaotog)\thetaext{d}\mathbb{R}^3mega \thetaext{d}v \\
\leq& \frac{\mu}{20}| \nabla u_t|^2_{2} + M(c_0)c^{4}_4 (|\rhoho_t |^2_{2}|\rhoho|^2_\infty+|\rhoho|^2_\infty) \leq \frac{\mu}{20}| \nabla u_t|^2_{2} +M(c_0)c^{6}_4,\\
\end{split}
\end{equation*}
where we used the fact that
\begin{equation*}
\begin{split}
&\int_0^\infty \int_{S^2} |(\sigmaigma_a)_t |^2_{2}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\leq \int_0^\infty \int_{S^2}\thetaotog(|\sigmaigma_t|^2_{2}|\rhoho|^2_\infty+|\sigmaigma|^2_{\infty}|\rhoho_t|^2_2\thetaotog)\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\leq M(|\rhoho|_\infty)|\rhoho|^2_\infty (|\rhoho_t |^2_{2}+1).
\end{split}
\end{equation*}
And similarly
\begin{equation*}
\begin{split}
J_{7}=& \frac{1}{c}\int_0^\infty \int_{S^2} \int_{\mathbb{R}^3} \sigmaigma_a I_t u_t\cdot \mathbb{R}^3mega \thetaext{d}x \thetaext{d}\mathbb{R}^3mega \thetaext{d}v
\leq C|\rhoho|^{\frac{1}{2}}_\infty|\sigmaqrt{\rhoho}u_t|_{2}\int_0^\infty \int_{S^2} |\sigmaigma |_{\infty}|I_t|_2\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& C|\sigmaqrt{\rhoho}u_t|^2_{2}+M(c_0) \int_0^\infty \int_{S^2} |I_t|^2_2\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\int_0^\infty \int_{S^2} |\sigmaigma |^2_{\infty}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& C|\sigmaqrt{\rhoho}u_t|^2_{2}+M(c_0)c^{2}_4,\\
J_{8}= &-\frac{1}{c}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3}\frac{v}{v'}(\sigmaigma_s I')_t
u_t\cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\qquad\qquad\qquad\quad\\
\leq &C|u_t|_{\mathbb{D}^1} |\rhoho_t|_2 \int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \frac{v}{v'}
\overline{\sigmaigma}_s\|I'\|_1\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\qquad\qquad\qquad\quad\\
&+C|\rhoho|^{\frac{1}{2}}_\infty|\sigmaqrt{\rhoho}u_t|_{2}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \frac{v}{v'}
\overline{\sigmaigma}_s |I'_t|_2\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\leq &M(c_0)c^2_4\int_0^\infty \int_{S^2} \|I'\|^2_1\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big(\int_0^\infty \int_{S^2} \Big(\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2
\overline{\sigmaigma}^2_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{\frac{1}{2}}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\Big)^2 \\
& + M(c_0)\int_0^\infty \int_{S^2} |I'_t|^2_2\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big(\int_0^\infty \int_{S^2} \Big(\int_0^\infty \int_{S^2} \Big|\frac{v}{v'}\Big|^2
\overline{\sigmaigma}^2_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{\frac{1}{2}}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\Big)^2 \\
&+\frac{\mu}{20}|u_t|^2_{\mathbb{D}^1}+C|\sigmaqrt{\rhoho}u_t|^2_{2}\leq \frac{\mu}{20}|u_t|^2_{\mathbb{D}^1}+C|\sigmaqrt{\rhoho}u_t|^2_{2}+M(c_0)c^{4}_4,\\
J_{9}= &\frac{1}{c}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2} \int_{\mathbb{R}^3}(\sigmaigma'_s I)_t
u_t\cdot \mathbb{R}^3mega\thetaext{d}x \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq &C|u_t|_{\mathbb{D}^1}|\rhoho_t|_2 \int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}
\overline{\sigmaigma}'_s \|I\|_1\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&+C|\rhoho|^{\frac{1}{2}}_\infty|\sigmaqrt{\rhoho}u_t|_{2}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}
\overline{\sigmaigma}'_s |I_t|_2\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq &M(c_0)c^2_4\int_0^\infty \int_{S^2} \|I\|^2_1\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\int_0^\infty \int_{S^2} \Big(\int_0^\infty \int_{S^2}
\overline{\sigmaigma}'_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{2}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v+\frac{\mu}{20}|u_t|^2_{\mathbb{D}^1} \\
&+M(c_0)\int_0^\infty \int_{S^2} |I_t|^2_2\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\int_0^\infty \int_{S^2}\Big( \int_0^\infty \int_{S^2}
\overline{\sigmaigma}'_s \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)^{2}\thetaext{d}\mathbb{R}^3mega \thetaext{d}v+C|\sigmaqrt{\rhoho}u_t|^2_{2} \\
\leq& \frac{\mu}{20}|u_t|^2_{\mathbb{D}^1}+C|\sigmaqrt{\rhoho}u_t|^2_{2}+M(c_0)c^{4}_4.
\end{split}
\end{equation*}
Combining the above estimates for $I_i$ and $J_j$, it turns out that
\begin{equation}\label{kaka}
\begin{split}
&\frac{1}{2}\frac{\thetaext{d}}{\thetaext{d}t}\int_{\mathbb{R}^3}\rhoho |u_t|^2 \thetaext{d}x+\frac{1}{3}\int_{\mathbb{R}^3}(\mu|\nabla u_t|^2+(\lambda+\mu)(\thetaext{div}u_t)^2) \thetaext{d}x\\
\leq& M(c_0)c^{6}_4+\frac{1}{c^2_4} |\nabla w_t|^2_2+M(c_0)c^8_4|\sigmaqrt{\rhoho}u_t|^2_{2}+M(c_0)c_0| u|_{D^2}.
\end{split}
\end{equation}
Integrating (\rhoef{kaka}) over $(\thetaau,t)$ for $\thetaau\in (0,t)$, we easily get
\begin{equation}\label{nv4}
\begin{split}
&|\sigmaqrt{\rhoho}u_t(t)|^2_{2}+\int_{\thetaau}^{t}|u_t(s)|^2_{\mathbb{D}^1}\thetaext{d}s\\
\leq& |\sigmaqrt{\rhoho}u_t(\thetaau)|^2_{2}
+\int_{\thetaau}^{t}\Big(M(c_0)c^8_4|\sigmaqrt{\rhoho}u_t|^2_{2}+M(c_0)c_0| u|_{D^2}\Big)\thetaext{d}s+M(c_0)c^{6}_4t+C.
\end{split}
\end{equation}
From the momentum equations $(\rhoef{eq:1.weeer})_3$, we have
\begin{equation}\label{nv5}
\begin{split}
&|\sigmaqrt{\rhoho}u_t(\thetaau)|^2_{2}\leq C|\rhoho(\thetaau)|_{\infty} \|\nabla w(\thetaau) \|^2_1 |\nabla u(\thetaau)|^2_2+C\int_{\mathbb{R}^3}\frac{|\Phi(\thetaau)|^2}{\rhoho(\thetaau)}\thetaext{d}x,
\end{split}
\end{equation}
where
$$
\Phi(\thetaau)=\nabla p_m(\thetaau) +Lu(\thetaau)+\frac{1}{c}\int_0^\infty \int_{S^2}A_r(\thetaau)\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v.
$$
From the assumptions (\rhoef{zhen1})-(\rhoef{jia345}), Lemma \rhoef{lem1}, the regularity of $S(v,\mathbb{R}^3mega, t,x)$ and Minkowski inequality, we easily have
\begin{equation*}
\begin{split}
&\lim_{\thetaau \mapsto 0}\int_{\mathbb{R}^3}\Big(\frac{|\Phi(\thetaau)|^2}{\rhoho(\thetaau)}-\frac{|\Phi(0)|^2}{\rhoho_0}\Big)\thetaext{d}x\\
\leq&
\lim_{\thetaau \mapsto 0}\Big(\frac{1}{{\bf u}nderline{\delta}}\int_{\mathbb{R}^3}|\Phi(\thetaau)-\Phi(0)|^2\thetaext{d}x+\frac{1}{\delta{\bf u}nderline{\delta}}|\rhoho(\thetaau)-\rhoho_0|_\infty\int_{\mathbb{R}^3}|\Phi(0)|^2\thetaext{d}x\Big)
=0.
\end{split}
\end{equation*}
According to the compatibility condition (\rhoef{kkkkk}), it is easy to show that
\begin{equation}\label{nv6}
\begin{split}
\limsup_{\thetaau \rhoightarrow 0}|\sigmaqrt{\rhoho}u_t(\thetaau)|^2_{2}\leq C|\rhoho_0|_{\infty} \|\nabla u_0 \|^2_1 |\nabla u_0|^2_2+C|g_1|^2_2\leq Cc^5_0.
\end{split}
\end{equation}
Therefore, by letting $\thetaau\rhoightarrow 0$ in (\rhoef{nv4}) and (\rhoef{plmn}), we have
\begin{equation}\label{ghj}
\begin{split}
&|\sigmaqrt{\rhoho}u_t(t)|^2_{2}+\int_{0}^{t}|u_t(s)|^2_{\mathbb{D}^1}\thetaext{d}s\\
\leq&\int_{0}^{t}M(c_0)c^8_4|\sigmaqrt{\rhoho}u_t|^2_{2}\thetaext{d}s+M(c_0)c^{6}_4t+M(c_0)c^5_0,
\end{split}
\end{equation}
for $0\leq t\leq T_2$.
From Gronwall's inequality, we get
\begin{equation}\label{nv98nn}
\begin{split}
&|\sigmaqrt{\rhoho}u_t(t)|^2_{2}+\mu\int_{0}^{t}| u_t(s)|^2_{\mathbb{D}^1}\thetaext{d}s\\
\leq& (M(c_0)c^{6}_4t+M(c_0)c^5_0) \exp\thetaotog(M(c_0)c^8_4t\thetaotog)\leq M(c_0)c^{5}_0
\end{split}
\end{equation}
for $0\leq t \leq T_3= \min(T^*,(1+M(c_0)c^{8}_4)^{-1})$. Combining (\rhoef{nv9800}) and (\rhoef{nv98nn}) yields
\begin{equation*}
\begin{split}
| u(t)|_{D^2}
\leq M(c_0)\thetaotog(|\sigmaqrt{\rhoho}u_t(t)|_{2}+c^2_3c_0\thetaotog)\leq M(c_0)c^{3}_3.
\end{split}
\end{equation*}
Finally, from the standard elliptic regularity estimate (see \cite{CK2}) and Minkowski inequality, we conclude that
\begin{equation*}
\begin{split}
&\int_{0}^{t}|u(s)|^2_{D^{2,q}}\thetaext{d}s
\leq \int_{0}^{t}\Big( |\rhoho u_t|^2_q+|\rhoho w\cdot\nabla u|^2_q+|\nabla p_m|^2_q\Big)(s)\thetaext{d}s\\
&\qquad \qquad \qquad+\int_{0}^{t} \Big(\int_0^\infty \int_{S^2}|A_r|_q\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\Big)^2(s)\thetaext{d}s\leq M(c_0)c^{10}_3
\end{split}
\end{equation*}
for $0\leq t \leq T_3$.
\end{proof}
Based on Lemmas \rhoef{lem:2}-\rhoef{lem:4}, we obtain the following local (in time) a priori estimate independent of the lower bound $\delta$ of the initial mass density $\rhoho_0$:
\begin{equation}\label{priaa}\begin{split}
\|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+\|p_m(t)-\overline{p}\|_{H^1\cap W^{1,q}}\leq& M(c_0),\\
|\rhoho_t(t)|_{2}+ |(p_m)_t(t)|_{2}\leq M(c_0)c_3,\quad |\rhoho_t(t)|_{q}+ |(p_m)_t(t)|_{q}\leq& M(c_0)c_4,\\
\displaystyle
\|I\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T_*];H^1\cap W^{1,q}(\mathbb{R}^3)))}\leq& M(c_0)c_0\\
\|I_t\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T_*];L^{2}\cap L^{q}(\mathbb{R}^3)))}\leq& M(c_0)c_0,\\
\displaystyle
|u(t)|^2_{\mathbb{D}^1}+\int_{0}^{T_*}\Big(|u|^2_{D^2}+|\sigmaqrt{\rhoho}u_t|^2_{2}\Big)(t)\thetaext{d}t\leq& M(c_0)c^{2}_0,\\
\displaystyle
\thetaotog(|u(t)|^2_{\mathbb{D}^1\cap D^2}+|\sigmaqrt{\rhoho}u_t(t)|^2_{2}\thetaotog)+\int_{0}^{T_*}\Big(|u|^2_{D^{2,q}}+|u_t|^2_{\mathbb{D}^1}\Big)(t)\thetaext{d}t\leq& M(c_0)c^{10}_3,
\end{split}
\end{equation}
for $0\leq t\leq T_3$. Therefore, if we define the constants $c_i$ ($i=1,2,3,4$) and $T^*$ by
\begin{equation}\label{dingyi}
\begin{split}
&c_1=c_2=c_3=M(c_0)c_0,\\
& c_4= M(c_0)c^5_3=M^6(c_0)c^5_0, \quad \thetaext{and} \quad T^*=(T, (1+M(c_0)c^8_4)^{-1}),
\end{split}
\end{equation}
then we deduce that
\begin{equation}\label{pri}\begin{split}
\displaystyle
\|I\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3)))}\leq& c_1,\\
\|I_t\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];L^{2}\cap L^{q}(\mathbb{R}^3)))}\leq& c_2,\\
\displaystyle
\sigmaup_{0\leq t\leq T^*}|u(t)|^2_{\mathbb{D}^1}+\int_{0}^{T^*}\Big(|u|^2_{D^2}+|\sigmaqrt{\rhoho}u_t|^2_{2}\Big)(t)\thetaext{d}t\leq& c^2_3,\\
\displaystyle
\thetaext{ess}\sigmaup_{0\leq t\leq T^*}\thetaotog(|u(t)|^2_{\mathbb{D}^1\cap D^2}+|\sigmaqrt{\rhoho}u_t(t)|^2_{2}\thetaotog)+\int_{0}^{T^*}\Big(|u|^2_{D^{2,q}}+|u_t|^2_{\mathbb{D}^1}\Big)(t)\thetaext{d}t\leq& c^2_4,\\
\sigmaup_{0\leq t\leq T^*}(\|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+\|p_m(t)-\overline{p}\|_{H^1\cap W^{1,q}})\leq& c_1,\\
\sigmaup_{0\leq t\leq T^*} (|\rhoho_t(t)|_{2}+ |(p_m)_t(t)|_{2}+|\rhoho_t(t)|_{q}+ |(p_m)_t(t)|_{q})\leq& c^2_4.
\end{split}
\end{equation}
\sigmaubsection{The unique solvability of the linearized problem with vacuum}\ \\
First we give the following key lemma for the proof of our main result - Theorem \rhoef{th1}.
\begin{lemma}\label{lemk1}
Let (\rhoef{tyuu}) hold.
If $(I_0,\rhoho_0, u_0)$ satisfies the compatibility condition
\begin{equation}\label{kk2k}
\begin{split}
Lu_0+\nabla p^0_{m}+\frac{1}{c}\int_0^\infty \int_{S^2}A^0_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v=&\rhoho^{\frac{1}{2}}_0 g_1
\end{split}
\end{equation}
for some $g_1\in L^2$, where $p^0_m=A\rhoho^\gamma_0,\ A^0_r=A_r(v,\mathbb{R}^3mega,t=0,x,\rhoho_0,I_0,I'_0),$ then there exists a unique strong solution $(I,\rhoho,u)$ to the Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}) such that
\begin{equation*}\begin{split}
&I\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3))), \ I_t\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];L^{2}\cap L^{q}(\mathbb{R}^3))),\\
&\rhoho\geq 0,\ \ \rhoho-\overline{\rhoho}\in C([0,T^*];H^1\cap W^{1,q}),\quad \rhoho_t\in C([0,T^*];L^{2}\cap L^{q}),\\
&u\in C([0,T^*];\mathbb{D}^1\cap D^2)\cap L^2([0,T^*];D^{2,q}),\ u_{t}\in L^2([0,T^*];\mathbb{D}^1),\ \sigmaqrt{\rhoho}u_{t}\in L^\infty([0,T^*];L^2).
\end{split}
\end{equation*}
Moreover, $(I,\rhoho,u)$ satisfies the local estimate (\rhoef{pri}).
\end{lemma}
\begin{proof}We divide the proof into three steps.\\
{\bf u}nderline{Step 1}: Existence.
Let $\delta> 0$ be a constant, and for each $\delta \in (0,1)$, define
\begin{equation*}
\begin{split}
&\rhoho_{\delta0}=\rhoho_0+\delta,\,\, (p_m)_{\delta0}=A(\rhoho_0+\delta)^\gamma,\\
&A^{\delta0}_r=A_r(v,\mathbb{R}^3mega,t=0,x,\rhoho_{\delta0},I_0,I'_0).
\end{split}
\end{equation*}
Then from the compatibility condition (\rhoef{kkkkk}) we have
\begin{equation*}
\begin{split}
&Lu_0+A\nabla \rhoho^\gamma_{\delta0}+\frac{1}{c}\int_0^\infty \int_{S^2}A^{\delta0}_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v=(\rhoho_{\delta0})^{\frac{1}{2}} g^\delta_1,
\end{split}
\end{equation*}
where
\begin{equation*}\begin{split}
g^\delta_1=&\Big(\frac{\rhoho_0}{\rhoho_{\delta0}}\Big)^{\frac{1}{2}}g_1+A\frac{ \nabla(\rhoho^\gamma_{\delta0}-\rhoho^\gamma_0)}{(\rhoho_{\delta0})^{\frac{1}{2}}}-\frac{1}{c}\int_0^\infty \int_{S^{2}}\frac{(A^{0}_r-A^{\delta0}_r)}{(\rhoho_{\delta0})^{\frac{1}{2}}}\mathbb{R}^3mega\thetaext{d}\mathbb{R}^3mega \thetaext{d}v.
\end{split}
\end{equation*}
It is easy to know from the assumptions (\rhoef{zhen1})-(\rhoef{jia345}), that for all small $\delta> 0$,
\begin{equation*}\begin{split}
&1+\|\rhoho_{\delta0}-\overline{\rhoho}-\delta\|_{H^1\cap W^{1,q}}+|u_0|_{\mathbb{D}^1\cap D^2}+\|I_0\|_{L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3))}+|g^\delta_1|_2\\
&+\|S(v,\mathbb{R}^3mega, t,x)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;C^1([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C^1([0,T^*]; L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2(\mathbb{R}^3)))}\leq c_0.
\end{split}
\end{equation*}
Therefore, corresponding to initial data $(I_0,\rhoho_{\delta0},u_0)$, there exists a unique strong solution $(I^\delta,\rhoho^\delta, u^\delta)$ satisfying the local estimate (\rhoef{pri}). Thus we can choose a subsequence of solutions (still denoted by $(I^\delta,\rhoho^\delta,u^\delta)$) converging to a limit $(I,\rhoho,u)$ in weak or weak* sense. Furthermore, for any $R> 0$, thanks to the compact property \cite{jm}, there exists a subsequence (still denoted by $(I^\delta,\rhoho^\delta,u^\delta)$) satisfying
\begin{equation}\label{ert}\begin{split}
& I^\delta\rhoightarrow I\ \thetaext{weakly}\ \thetaext{in}\ L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3))),\\
&\rhoho^\delta\rhoightarrow \rhoho\ \thetaext{in }\ C([0,T^*];L^2(B_R)),\\
&u^\delta\rhoightarrow u \ \thetaext{in } \ C([0,T^*];H^1(B_R)),
\end{split}
\end{equation}
where $B_{R}=\{x \in \mathbb{R}^3: |x|<R\}$. By the lower semi-continuity of norms (see \cite{gandi}), it follows from (\rhoef{ert}) that $(I,\rhoho,u)$ also satisfies the estimate (\rhoef{pri}).
For any $\varphi\in C^\infty_c(\mathbb{R}^+\thetaimes S^2\thetaimes [0,T^*]\thetaimes \mathbb{R}^3)$, from (\rhoef{pri}), (\rhoef{ert}) and assumptions (\rhoef{zhen1})-(\rhoef{jia345}) we easily have
\begin{equation*}\begin{split}
\int_0^\infty \int_{S^2}\int_0^{T^*} \int_{\mathbb{R}^3} (\Lambda(v,\mathbb{R}^3mega,t,x,\rhoho^\delta)I^\delta -\Lambda(v,\mathbb{R}^3mega,t,x,\rhoho)I)\varphi \thetaext{d}x \thetaext{d}t\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\rhoightarrow 0,
\ \thetaext{as} \ \delta\rhoightarrow 0,
\end{split}
\end{equation*}
where $\displaystyle \Lambda=\sigmaigma_a+\int_{0}^{\infty}\int_{S^{2}} \sigmaigma'_s\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'$,
and
\begin{equation*}\begin{split}
\int_0^\infty \int_{S^{2}}\int_{0}^{T^*}\int_{\mathbb{R}^3}\int_0^\infty \int_{S^{2}}\frac{v}{v'}\Big(\sigmaigma_s(\rhoho^\delta)I'^\delta-\sigmaigma_s(\rhoho)I'\Big){\bf x}i
\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}x \thetaext{d}t\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\rhoightarrow 0,
\ \thetaext{as} \ \delta\rhoightarrow 0.
\end{split}
\end{equation*}
Then it is easy to show that $(I,\rhoho,u)$ is a weak solution in the sense of distribution and satisfies the following regularities:
\begin{equation*}\begin{split}
&I\in L^2(\mathbb{R}^+\thetaimes S^2;L^\infty([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3))),\ I_t\in L^2(\mathbb{R}^+\thetaimes S^2;L^\infty([0,T^*];L^{2}\cap L^{q}(\mathbb{R}^3))),\\
&\rhoho\geq 0,\quad \rhoho-\overline{\rhoho}\in L^\infty([0,T^*];H^1\cap W^{1,q})),\quad \rhoho_t\in L^\infty([0,T^*];L^{2}\cap L^{q}),\\
&u\in L^\infty([0,T^*];\mathbb{D}^1\cap D^2)\cap L^2([0,T^*];D^{2,q}),\ u_{t}\in L^2([0,T^*];\mathbb{D}^1),\ \sigmaqrt{\rhoho}u_{t}\in L^\infty([0,T^*];L^2).
\end{split}
\end{equation*}
{\bf u}nderline{Step 2}: Uniqueness. Let $(I_1,\rhoho_1,u_1)$ and $(I_2,\rhoho_2,u_2)$ be two solutions obtained in step 1 with the same initial data. Then by the same method as in \cite{CK} we can get$\rhoho_1=\rhoho_2$ and $u_{1}=u_{2}$. Here we omit the details. It is easy to show that $I_1-I_2$ satisfies the following Cauchy problem:
$$\frac{1}{c}(I_1-I_2)_t+\mathbb{R}^3mega\cdot\nabla (I_1-I_2)+\Lambda (I_1-I_2)=0, \quad (I_1-I_2)|_{t=0}=0.$$
It follows immediately that $I_{1}=I_{2}$. \\
{\bf u}nderline{Step 3}: The time-continuity. The continuity of $\rhoho$ can be obtained analogously to \cite{CK}. As for $I$,
by Lemma \rhoef{lem:2}, for $\forall$ $(v,\mathbb{R}^3mega)\in R^+\thetaimes S^2$ we have
$$
I(v,\mathbb{R}^3mega,\cdot,\cdot)\in C([0,T^*]; L^2\cap L^q(\mathbb{R}^3)) \cap C([0,T^*]; W^{1,2}\cap W^{1,q}(\mathbb{R}^3)-\thetaext{weak}).
$$
According to ({\rhoef{coti}}), we have
\begin{equation}\label{thymm}
\begin{split}
\limsup_{t\rhoightarrow 0}\|I(v,\mathbb{R}^3mega,\cdot,\cdot)\|^2_{W^{1,r}}\leq \|I_0\|^2_{W^{1,r}},
\end{split}
\end{equation}
which implies that $I(v,\mathbb{R}^3mega,t,x)$ is right-continuous at $t=0$ (see \cite{teman}). Similarly, form Lemmas \rhoef{lem:3}-\rhoef{lem:4} we have
$$
u\in C([0,T^*]; \mathbb{D}^1) \cap C([0,T^*]; D^2-\thetaext{weak}).
$$
From equations $(\rhoef{eq:1.weeer})_3$ and Lemmas \rhoef{lem:2}-\rhoef{lem:4}, we also know that
$$
\rhoho u_t\in L^2([0,T^*]; L^2), \ \thetaext{and}\ (\rhoho u_t)_t\in L^2([0,T^*]; H^{-1}).
$$
From Aubin-Lions lemma we then have $ \rhoho u_t \in C([0,T^*]; L^2)$.
From
\begin{equation}\label{thghj}
\displaystyle
Lu=-\rhoho u_t-\rhoho w\cdot \nabla u
-\nabla p_m +\frac{1}{c}\int_0^\infty \int_{S^2}A_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v,
\end{equation}
and the standard elliptic regularity estimates (see \cite{CK2}), we get $u\in C([0,T^*]; D^2)$.
\end{proof}
\sigmaubsection{Proof of Theorem \rhoef{th1}}\ \\
Our proof is based on the classical iteration scheme and the existence results for the linearized problem obtained in Section $3.2$. Like in Section 3.2, we define constants $c_{0}$ and $c_{1}$, $c_2$, $c_3$, $c_4$ and assume that
\begin{equation*}\begin{split}
&2+\|\rhoho_0-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+|u_0|_{\mathbb{D}^1\cap D^2}+\|I_0\|_{L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3))}+|g_1|_2\\
&+\|S(v,\mathbb{R}^3mega, t,x)\|_{ L^2(\mathbb{R}^+\thetaimes S^2;C^1([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3)))\cap C^1([0,T^*]; L^1(\mathbb{R}^+\thetaimes S^2;L^1\cap L^2(\mathbb{R}^3)))}\leq c_0.
\end{split}
\end{equation*}
Let $u^0\in C([0,T^*];\mathbb{D}^1\cap D^2)\cap L^2([0,T^*];D^{2,q}) $ be the solution to the linear parabolic problem
$$
h_t-\thetariangle h=0 \quad \thetaext{in} \quad (0,+\infty)\thetaimes \mathbb{R}^3 \quad \thetaext{and} \quad h|_{t=0}=u_0 \quad \thetaext{in} \quad \mathbb{R}^3.
$$
Let $I^0\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T^*];H^1\cap W^{1,q}(\mathbb{R}^3)))$ be the solution to the linear parabolic problem
$$
f_t+c\mathbb{R}^3mega\cdot\nabla f=0, \ \thetaext{in} \ \mathbb{R}^+\thetaimes S^2\thetaimes (0,+\infty)\thetaimes \mathbb{R}^3 \quad \thetaext{and} \quad f|_{t=0}=I_0 \quad \thetaext{in} \quad \mathbb{R}^+\thetaimes S^2\thetaimes \mathbb{R}^3.
$$
Taking a small time $\overline{T}_1\in (0,T^*)$, we then have
\begin{equation*}\begin{split}
\|I^0\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,\overline{T}_1];H^1\cap W^{1,q}(\mathbb{R}^3)))}\leq& c_1,\\
\|I^0_t\|_{L^2(\mathbb{R}^+\thetaimes S^2;C([0,\overline{T}_1];L^{2}\cap L^{q}(\mathbb{R}^3)))}\leq& c_2,\\
\sigmaup_{0\leq t\leq \overline{T}_1 }|u^0(t)|^2_{\mathbb{D}^1}+\int_{0}^{\overline{T}_1}|u^0(t)|^2_{D^2}\thetaext{d}t\leq& c^2_3,\\
\sigmaup_{0\leq t\leq \overline{T}_1 }|u^0(t)|^2_{ D^2}+\int_{0}^{\overline{T}_1}\thetaotog(|u^0(t)|^2_{D^{2,q}}+|u^0_t(t)|^2_{\mathbb{D}^1}\thetaotog)\thetaext{d}t\leq& c^2_4.
\end{split}
\end{equation*}
We divided the proof of Theorem \rhoef{th1} into two cases: $\overline{\rhoho}>0$ and $\overline{\rhoho}=0$.
\sigmaubsubsection{\bf{ Case $\overline{\rhoho}>0$.}}\ \\
\begin{proof} The proof of this case is divided into three steps.\\[2mm]
{\bf u}nderline{Step 1}. The existence of strong solutions. Let $(I^1, \rhoho^1, u^1)$ be the strong solution to Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}) and $(w,\psi)=(u^0,I'^0)$. Then we construct approximate solutions $(I^{k+1}, \rhoho^{k+1}, u^{k+1})$ inductively as follows. Assume that $(I^{k},\rhoho^k, u^{k})$ was defined for $k\geq 1$, let $(I^{k+1}, \rhoho^{k+1}, u^{k+1})$ be the unique solution to the Cauchy problem (\rhoef{eq:1.weeer}) with (\rhoef{eq:2.2hh}) with $(w,\psi)$=$( u^{k},I'^{k})$:
\begin{equation}
\label{eq:1.wgo}
\begin{cases}
\displaystyle
\rhoho^{k+1}_t+\thetaext{div}(\rhoho^{k+1} u^k)=0,\\[5pt]
\displaystyle
\frac{1}{c}I^{k+1}_t+\mathbb{R}^3mega\cdot\nabla I^{k+1}=\overline{A}^{k}_r,\\[5pt]
\displaystyle
\rhoho^{k+1} u^{k+1}_t+\rhoho^{k+1} u^{k}\cdot \nabla u^{k+1}
+\nabla p^{k+1}_m +Lu^{k+1}=-\frac{1}{c}\int_0^\infty \int_{S^2}A^{k}_r\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v
\end{cases}
\end{equation}
with initial data
$$
(I^{k+1}, \rhoho^{k+1}, u^{k+1})|_{t=0}=(I_0, \rhoho_0, u_0),
$$
where
\begin{equation*}
\begin{split}
&\overline{A}^{k}_r=S-\sigmaigma^{k+1}_aI^{k+1}+\int_0^\infty \int_{S^2}\Big( \frac{v}{v'}\sigmaigma^{k+1}_sI'^{k} -(\sigmaigma'_s)^{k+1}I^{k+1} \Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',\\
&A^{k}_r=S-\sigmaigma^{k+1}_aI^{k+1}+\int_0^\infty \int_{S^2}\Big( \frac{v}{v'}\sigmaigma^{k+1}_sI'^{k+1} -(\sigmaigma'_s)^{k+1}I^{k+1} \Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',\\
&p^{k+1}_m=A(\rhoho^{k+1})^\gamma,\quad
\sigmaigma^{k+1}_a=\sigmaigma_a(v,\mathbb{R}^3mega,t,x,\rhoho^{k+1}),\\
& \sigmaigma^{k+1}_s=\sigmaigma_s(v'\rhoightarrow v,\mathbb{R}^3mega'\cdot\mathbb{R}^3mega,\rhoho^{k+1}),\ (\sigmaigma'_s)^{k+1}=\sigmaigma_s(v\rhoightarrow v',\mathbb{R}^3mega\cdot\mathbb{R}^3mega',\rhoho^{k+1}).
\end{split}
\end{equation*}
According to the arguments in Sections $3.1$-$3.2$, we know that the solution sequence $(I^k,\rhoho^k,p^k_m,u^k)$ still satisfies the priori estimates (\rhoef{pri}).
Now we show that $(I^k, \rhoho^k, u^k)$ converges to a limit in a strong sense. Let
$$
\overline{I}^{k+1}=I^{k+1}-I^k,\ \overline{\rhoho}^{k+1}=\rhoho^{k+1}-\rhoho^k,\ \overline{p}^{k+1}_m=p^{k+1}_m-p^k_m,\ \overline{u}^{k+1}=u^{k+1}-u^k,
$$
then we have
\begin{equation}
\label{eq:1.2w}
\begin{cases}
\displaystyle
\overline{\rhoho}^{k+1}_t+\thetaext{div}(\overline{\rhoho}^{k+1} u^k)+\thetaext{div}(\rhoho^{k} \overline{u}^k)=0,\\[8pt]
\displaystyle
\frac{1}{c}\overline{I}^{k+1}_t+\mathbb{R}^3mega\cdot\nabla \overline{I}^{k+1}+\Big(\sigmaigma^{k+1}_a+\int_0^\infty \int_{S^2}(\sigmaigma'_s)^{k+1}\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\Big)\overline{I}^{k+1}
=L_1,\\[8pt]
\displaystyle
\rhoho^{k+1} \overline{u}^{k+1}_t+\rhoho^{k+1} u^k\cdot\nabla \overline{u}^{k+1}+L\overline{u}^{k+1}\\[8pt]
=\overline{\rhoho}^{k+1}(-u^k_t-u^{k-1}\cdot\nabla u^k)
-\rhoho^{k+1}\overline{u}^{k}\cdot\nabla u^k-\nabla \overline{p}^{k+1}_m+L_2,
\end{cases}
\end{equation}
where $L_1$ and $L_2$ are given by
\begin{equation*}\begin{split}
L_1=&-I^k(\sigmaigma^{k+1}_a-\sigmaigma^{k}_a)-\int_0^\infty \int_{S^2}
\Big((\sigmaigma'_s)^{k+1}-(\sigmaigma'_s)^{k}\Big)I^{k}\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\\
&+\int_0^\infty \int_{S^2} \Big(\frac{v}{v'}
\thetaotog(\sigmaigma^{k}_s\overline{I}'^{k}+I'^k(\sigmaigma^{k+1}_s-\sigmaigma^{k}_s)\thetaotog)\Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v',\\[6pt]
L_2=&-\frac{1}{c}\int_0^\infty \int_{S^2}\mathbb{R}^3mega\Big(-\sigmaigma^{k+1}_a\overline{I}^{k+1}-I^k(\sigmaigma^{k+1}_a-\sigmaigma^{k}_a)\Big) \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&-\frac{1}{c}\int_0^\infty \int_{S^2}\int_0^\infty \int_{S^2}\mathbb{R}^3mega \frac{v}{v'}\Big(\sigmaigma^{k+1}_s\overline{I}'^{k+1}+I'^k(\sigmaigma^{k+1}_s-\sigmaigma^{k}_s)\Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&-\frac{1}{c}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}-\mathbb{R}^3mega\Big(I^{k}\thetaotog((\sigmaigma'_s)^{k+1}-(\sigmaigma'_s)^{k}\thetaotog)+(\sigmaigma'_s)^{k+1}\overline{I}^{k+1}\Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v.
\end{split}
\end{equation*}
First, we estimate sequence $\overline{\rhoho}^{k+1}$. Multiplying $ (\rhoef{eq:1.2w})_1$ by $\overline{\rhoho}^{k+1}$ and integrating over $\mathbb{R}^3$, we have
\begin{equation}\label{go64}\begin{cases}
\frac{\thetaext{d}}{\thetaext{d}t}|\overline{\rhoho}^{k+1}|^2_2\leq A^k_\eta(t)|\overline{\rhoho}^{k+1}|^2_2+\eta |\nabla\overline{u}^k|^2_2,\\[8pt]
A^k_\eta(t)=C(|\nabla u^k|^2_{W^{1,q}}+\frac{1}{\eta}|\nabla\rhoho^{k}|^2_{3} +\frac{1}{\eta}|\rhoho^{k}|^2_{\infty}),\ \thetaext{and} \ \int_0^t A^k_\eta(s)\thetaext{d}s\leq \widehat{C}+\widehat{C}_{\eta}t
\end{cases}
\end{equation}
for $t\in[0,\overline{T}_1]$, where $0<\eta \leq \frac{1}{10}$ is a constant and $\widehat{C}_{\eta}$ is a positive constant depending on $\frac{1}{\eta}$ and constant $\widehat{C}$.
Second, we estimate sequence $\overline{I}^{k+1}$.
Multiplying $(\rhoef{eq:1.2w})_2$ by $\overline{I}^{k+1}$ and integrating over $\mathbb{R}^+\thetaimes S^2\thetaimes\mathbb{R}^3$, from (\rhoef{zhen1})-(\rhoef{jia345}) and the similar arguments used in Lemma \rhoef{lem:3} we have
\begin{equation}\label{go66}\begin{split}
&\frac{\thetaext{d}}{\thetaext{d}t}\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}\\
\leq & \int_0^\infty \int_{S^2} \Big(|\sigmaigma^{k+1}|_{\infty}|I^k|_{\infty}| \overline{\rhoho}^{k+1}|_2|\overline{I}^{k+1}|_2+ |\overline{\sigmaigma}^{k+1,k}|_{\infty}| \rhoho^{k}|_\infty|I^k|_{\infty}| | \overline{\rhoho}^{k+1}|_2|\overline{I}^{k+1}|_2\Big)\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&+\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}\frac{v}{v'}\overline{\sigmaigma}_s\Big(|\rhoho|_\infty|\overline{I}'^k|_2|\overline{I}^{k+1}|_2+\|I'^k\|_{W^{1,q}}|\overline{I}^{k+1}|_2|\overline{\rhoho}^{k+1}|_2\Big) \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
&+\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}\overline{\sigmaigma}'_s\|I^k\|_{W^{1,q}}|\overline{I}^{k+1}|_2|\overline{\rhoho}^{k+1}|_2 \thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\\
\leq& D^k_\eta(t)\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}+|\overline{\rhoho}^{k+1}|^2_2+\eta\|\overline{I}^{k}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))},
\end{split}
\end{equation}
where we have used the facts $\sigmaigma_a\geq 0$ and $\sigmaigma'_s\geq 0$. $\sigmaigma^{k+1}$,
$\overline{\sigmaigma}^{k+1,k}$
and $D^k_\eta(t)$ are defined by
\begin{equation*}
\begin{split}
&\sigmaigma^{k+1}=\sigmaigma(v,\mathbb{R}^3mega,t,x,\rhoho^{k+1}),\quad \overline{\sigmaigma}^{k+1,k}=\overline{\sigmaigma}(v,\mathbb{R}^3mega,t,x,\rhoho^{k+1},\rhoho^k),\\
&D^k_\eta(t)=C\Big(\thetaotog(1+|\sigmaigma^{k+1}|^2_\infty+|\overline{\sigmaigma}^{k+1,k}|^2_\infty\thetaotog)\|I^{k}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2; H^1\cap W^{1,q}(\mathbb{R}^3))}+\frac{1}{\eta}
|\rhoho|^2_{\infty}\Big).
\end{split}
\end{equation*}
From the estimate (\rhoef{pri}), we also have $\int_0^t D^k_\eta(s)\thetaext{d}s\leq \widehat{C}+\widehat{C}_{\eta}t$, for $t\in[0,\overline{T}_1]$.
Finally, multiplying $ (\rhoef{eq:1.2w})_3$ by $\overline{u}^{k+1}$ and integrating over $\mathbb{R}^3$, we have
\begin{equation*}\begin{split}
&\frac{1}{2}\frac{\thetaext{d}}{\thetaext{d}t}|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|^2_2+\mu|\nabla\overline{u}^{k+1} |^2_2+(\lambda+\mu)|\thetaext{div}\overline{u}^{k+1} |^2_2\\
=& \int_{\mathbb{R}^3}\Big(-\overline{\rhoho}^{k+1}u^k_t\cdot\overline{u}^{k+1}-\overline{\rhoho}^{k+1}(u^{k-1}\cdot\nabla u^k)\cdot\overline{u}^{k+1}
-\rhoho^{k+1}(\overline{u}^{k}\cdot\nabla u^k)\cdot\overline{u}^{k+1}\Big)\thetaext{d}x\\
&+\int_{\mathbb{R}^3}\Big(-\nabla \overline{p}^{k+1}_m\cdot\overline{u}^{k+1}+L_2\overline{u}^{k+1}\Big)\thetaext{d}x
=:\sigmaum_{i=5}^{14} I_i.
\end{split}
\end{equation*}
For the fluid terms $I_{5}-I_{8}$, according to the Gagliardo-Nirenberg inequality, Minkowski inequality and H\"older's inequality, it is not hard to show that
\begin{equation*}\begin{split}
& I_{5}= -\int_{\mathbb{R}^3}\overline{\rhoho}^{k+1}u^k_t\cdot\overline{u}^{k+1}\thetaext{d}x \leq C\int_{\mathbb{R}^3}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x, \\
& I_{6}= -\int_{\mathbb{R}^3}\overline{\rhoho}^{k+1}(u^{k-1}\cdot\nabla u^k)\cdot\overline{u}^{k+1}
\thetaext{d}x\leq C|\overline{\rhoho}^{k+1}|_{2}|\nabla\overline{u}^{k+1}|_2\|\nabla u^k\|_1\|\nabla u^{k-1}\|_1,\\
&I_{7}= -\int_{\mathbb{R}^3}\rhoho^{k+1}(\overline{u}^{k}\cdot\nabla u^k)\cdot\overline{u}^{k+1}\thetaext{d}x\leq C|\overline{\rhoho}^{k+1}|^{\frac{1}{2}}_{\infty}|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|_2\|\nabla u^k\|_1|\nabla \overline{u}^k|_2,\\
& I_{8}=-\int_{\mathbb{R}^3}\nabla \overline{p}^{k+1}_m\cdot\overline{u}^{k+1}\thetaext{d}x=\int_{\mathbb{R}^3}
\overline{p}^{k+1}_m\thetaext{div}\overline{u}^{k+1}\thetaext{d}x
\leq C|\overline{p}^{k+1}_m|_{2}|\nabla\overline{u}^{k+1}|_2.
\end{split}
\end{equation*}
For the radiation related terms $I_{9}-I_{14}$,
\begin{equation*}\begin{split}
I_{9}=&-\frac{1}{c}\int_{\mathbb{R}^3}\int_0^\infty\int_{S^2}\mathbb{R}^3mega\cdot\overline{u}^{k+1}\Big(-\sigmaigma^{k+1}_a\overline{I}^{k+1}\Big)\thetaext{d}\mathbb{R}^3mega\thetaext{d}v\thetaext{d}x \quad\\
\leq & C|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|_2|\rhoho^{k+1}|^{\frac{1}{2}}_\infty\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}\|\sigmaigma^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^\infty(\mathbb{R}^3))},\\
I_{10}=&-\frac{1}{c}\int_{\mathbb{R}^3}\int_0^\infty \int_{S^2}\mathbb{R}^3mega\cdot\overline{u}^{k+1}\Big(-I^k(\sigmaigma^{k+1}_a-\sigmaigma^{k}_a)\Big) \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x\\
\leq &C|\overline{\rhoho}^{k+1}|_{2}|\nabla\overline{u}^{k+1}|_2\|I^{k}\|_{L^2(\mathbb{R}^+\thetaimes S^2;H^1(\mathbb{R}^3))}\|\sigmaigma^{k+1}\|_{L^2(\mathbb{R}^+\thetaimes S^2;L^\infty(\mathbb{R}^3))}\\
&+C|\rhoho^k|_\infty|\overline{\rhoho}^{k+1}|_2|\nabla\overline{u}^{k+1}|_2\|I^{k}\|_{L^2(\mathbb{R}^+\thetaimes S^2;H^1(\mathbb{R}^3))}\|\overline{\sigmaigma}^{k+1,k}\|_{L^2(\mathbb{R}^+\thetaimes S^2;L^\infty(\mathbb{R}^3))},\\
I_{11}=&-\frac{1}{c}\int_{\mathbb{R}^3}\int_0^\infty \int_{S^2}\int_0^\infty \int_{S^2}\frac{v}{v'}\mathbb{R}^3mega\cdot\overline{u}^{k+1}\sigmaigma^{k+1}_s\overline{I}'^{k+1}\thetaext{d}\mathbb{R}^3mega'\thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x \qquad\qquad\quad\\
\leq & C|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|_2|\rhoho^{k+1}|^{\frac{1}{2}}_\infty\|\overline{I}^{k+1}\|_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))},\\
\end{split}
\end{equation*}
\begin{equation*}\begin{split}
I_{12}=&-\frac{1}{c}\int_{\mathbb{R}^3}\int_0^\infty \int_{S^2}\int_0^\infty \int_{S^2}\frac{v}{v'}\mathbb{R}^3mega\cdot\overline{u}^{k+1}I'^k(\sigmaigma^{k+1}_s-\sigmaigma^{k}_s)\thetaext{d}\mathbb{R}^3mega'\thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x\\
\leq &C|\nabla\overline{u}^{k+1}|_2|\overline{\rhoho}^{k+1}|_{2}\|I^k\|_{L^2(\mathbb{R}^+\thetaimes S^2;H^1(\mathbb{R}^3))},\\
I_{13}=&\frac{1}{c}\int_{\mathbb{R}^3}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}\mathbb{R}^3mega \cdot\overline{u}^{k+1}I^{k}\Big((\sigmaigma'_s)^{k+1}-(\sigmaigma'_s)^{k}\Big)\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x\\
\leq& C|\nabla{u}^{k+1}|_2|\overline{\rhoho}^{k+1}|_{2}\|I^{k}\|_{L^2(\mathbb{R}^+\thetaimes S^2;H^1(\mathbb{R}^3))},\\
I_{14}=&\frac{1}{c}\int_{\mathbb{R}^3}\int_0^\infty \int_{S^2} \int_0^\infty \int_{S^2}\mathbb{R}^3mega \cdot\overline{u}^{k+1} (\sigmaigma'_s)^{k+1}\overline{I}^{k+1}\thetaext{d}\mathbb{R}^3mega' \thetaext{d}v'\thetaext{d}\mathbb{R}^3mega \thetaext{d}v\thetaext{d}x\\
\leq& C|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|_2|\rhoho^{k+1}|^{\frac{1}{2}}_\infty\|\overline{I}^{k+1}\|_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}.
\end{split}
\end{equation*}
Then combining the above estimates, we have
\begin{equation}\label{gogo1}\begin{split}
&\frac{\thetaext{d}}{\thetaext{d}t}|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|^2_2+\mu|\nabla\overline{u}^{k+1} |^2\\
\leq& F^k_\eta(t)|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|^2_2+F^k_2(t)|\overline{\rhoho}^{k+1}|^2_{2 }+M(C(c_0,c_1))|\overline{\rhoho}^{k+1}|^2_{2 }\\
&+F^k_3(t)\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}+\eta|\nabla\overline{u}^{k}|^2_2+C\int_{\mathbb{R}^3}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x,
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
F^k_\eta(t)=&C\Big(1+\frac{1}{\eta}|\rhoho^{k+1}|_{\infty}\|\nabla u^{k}\|^2_1\Big),\\
F^k_2(t)=&C\thetaotog(\|\nabla u^{k-1}\|^2_1\|\nabla u^{k}\|^2_1+
\|I^{k}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;H^1(\mathbb{R}^3))}(\|\sigmaigma^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^\infty(\mathbb{R}^3))}+1)\thetaotog)\\
&+C|\rhoho^k|^2_\infty\|I^{k}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;H^1(\mathbb{R}^3))}\|\overline{\sigmaigma}^{k+1,k}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^\infty(\mathbb{R}^3))},\\
F^k_3(t)=&C|\rhoho^{k+1}|_{\infty}\thetaotog(\|\sigmaigma^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^\infty(\mathbb{R}^3))}+1\thetaotog),\\
\end{split}
\end{equation*}
and we also have $\int_0^t \thetaotog(F^k_\eta(s)+F^k_2(s)+F^k_3(s)\thetaotog)\thetaext{d}s\leq \widehat{C}+\widehat{C}_\eta t$, for $t\in [0,\overline{T}_1]$.
To estimate $\int_{\mathbb{R}^3}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x$,
we need the following Lemma.
\begin{lemma}[\thetaextbf{The lower bound of the mass density at far field}]\label{zhengding}\ \\
There exists a sufficiently large $R>1$ and a time $\overline{T}_2\in (0,\overline{T}_1)$ small enough such that
$$
\frac{3}{8}\leq \rhoho^{k+1}(t,x)\leq \frac{5}{2}, \quad \forall \ (t,x)\in [0,\overline{T}_2]\thetaimes B^C_R,
$$
where the constants $R> 0$ and $\overline{T}_2$ is independent of $k$, and $B^C_R=\mathbb{R}^3\sigmaetminus B_R$.
\end{lemma}
\begin{proof}
From $\rhoho_0-\overline{\rhoho}\in H^1\cap W^{1,q}$ and the embedding $W^{1,q}\hookrightarrow C_0$, where $C_0$ is the set of all continuous functions on
$\mathbb{R}^3$ vanishing at infinity, we can choose a sufficiently large $R> 1$ such that
\begin{equation}\label{zhenglem}
\frac{3}{8}\overline{\rhoho}\leq \rhoho_0\leq \frac{5}{2}\overline{\rhoho} \quad \thetaext{for} \quad x \in B^C_R.
\end{equation}
From the proof of Lemma \rhoef{lem1} we know that
\begin{equation}\label{zhengleme}
\rhoho^{k+1}(t,x)=\rhoho_0(U^{k+1}(0;t,x))\exp\Big(-\int_{0}^{t}\thetaextrm{div} u^k(s;U^{k+1}(s,t,x))\thetaext{d}s\Big),
\end{equation}
where $U^{k+1}\in C([0,\overline{T}_1]\thetaimes[0,\overline{T}_1]\thetaimes \mathbb{R}^3)$ is the solution to the initial value problem
\begin{equation}\label{gongshi}
\begin{cases}
\frac{d}{ds}U^{k+1}(s;t,x)=u^k(s,U^{k+1}(s;t,x)),\quad 0\leq s\leq \overline{T}_1,\\[8pt]
U^{k+1}(t,t,x)=x, \qquad \qquad \qquad 0\leq t\leq \overline{T}_1,\ x\in \mathbb{R}^3.
\end{cases}
\end{equation}
The local estimate (\rhoef{pri}) leads to
\begin{equation}\label{zhengle}
\begin{split}
&\int_0^t|\thetaextrm{div} u^k(s,U^{k+1}(s,t,x))|\thetaext{d}s\leq \int_0^t|\nabla u^k|_\infty\thetaext{d}s\leq \overline{C}t^{1/2}\leq \ln 2.
\end{split}
\end{equation}
From the ODE problem (\rhoef{gongshi}), we get
\begin{equation}\label{zhengle1}
\begin{split}
&|U^{k+1}(0;t,x)-x|=|U^{k+1}(0;t,x)-U^{k+1}(t;t,x)|\\
\leq& \int_0^t| u^k(\thetaau,U^{k+1}(\thetaau;t,x))|\thetaext{d}\thetaau\leq \overline{C}t\leq R/2,
\end{split}
\end{equation}
for all $(t,x)\in [0,\overline{T}_2]\thetaimes \mathbb{R}^3$, where $\overline{C}$ is a positive constant independent of $k$, and $\overline{T}_2$ is a small positive time depending only on $\overline{C}$ and $\overline{T}_1$. That means,
$$ U^{k+1}(0,t,x)\in B^C_{R/2}, \quad \thetaext{for}\quad (t,x)\in [0,\overline{T}_2]\thetaimes B^C_{R}.$$
Then combining (\rhoef{zhenglem}), (\rhoef{zhengleme}), (\rhoef{zhengle}) and (\rhoef{zhengle1}), the desired conclusion is obtained.
\end{proof}
Based on Lemma \rhoef{zhengding}, for $t\in [0,\overline{T}_2]$, we have
\begin{equation}\label{zhengchu}
\begin{split}
&\int_{B_R}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x
\leq C|\nabla u^k_t |^2_2|\overline{\rhoho}^{k+1} |^2_2+\frac{1}{8}\mu|\nabla\overline{u}^{k+1} |^2_2,
\end{split}
\end{equation}
and
\begin{equation}\label{zhengchu1}
\begin{split}
&\int_{B^C_R}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x
\leq\frac{C}{\sigmaqrt{\rhoho}^{k+1}} |\overline{\rhoho}^{k+1} |_2|\nabla u^k_t |_2|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|_3\\
&\quad \quad \leq C|\nabla u^k_t |^2_2|\overline{\rhoho}^{k+1} |^2_2+\frac{1}{8}\mu|\nabla\overline{u}^{k+1} |^2_2+C|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|^2_2.
\end{split}
\end{equation}
Define
\begin{equation*}\begin{split}
\Gamma^{k+1}=&\sigmaup_{0\leq t \leq \overline{T}_2}\Big(\|\overline{I}^{k+1}(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}+|\overline{\rhoho}^{k+1}(t)|^2_{
2}+|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}(t)|^2_2\Big),
\end{split}
\end{equation*}
then from (\rhoef{go64})-(\rhoef{gogo1}), (\rhoef{zhengchu}) and (\rhoef{zhengchu1}) we have
\begin{equation*}\begin{split}
&\Gamma^{k+1}(\overline{T}_2)+\mu\int_0^{\overline{T}_2} |\nabla\overline{u}^{k+1} |^2_2\thetaext{d}t\\
\leq& \int_0^{\overline{T}_2} G^k_{\eta} \Gamma^{k+1}(t)\thetaext{d}t+3\eta\int_0^{\overline{T}_2}|\nabla\overline{u}^{k}(t) |^2_2\thetaext{d}t+ \eta \overline{T}_2 \sigmaup_{0\leq t \leq \overline{T}_2}\|\overline{I}^{k}(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}
\end{split}
\end{equation*}
for some $G^k_{\eta}$ such that $\int_{0}^{t}G^k_\eta(s)\thetaext{d}s\leq \widehat{C}+\widehat{C}_{\eta} t$ for $0\leq t \leq \overline{T}_2$.
By using Gronwall's inequality, we have
\begin{equation*}\begin{split}
&\Gamma^{k+1}(\overline{T}_2)+ \mu\int_{0}^{\overline{T}_2}|\nabla\overline{u}^{k+1}|^2_2\thetaext{d}s\\
\leq & \exp{(\widehat{C}+\widehat{C}_{\eta}t)}\Big(3\eta\int_0^{\overline{T}_2}|\nabla\overline{u}^{k}(t) |^2_2\thetaext{d}t+ \eta \overline{T}_2 \sigmaup_{0\leq t \leq \overline{T}_2}\|\overline{I}^{k}(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}\Big).
\end{split}
\end{equation*}
Since $0< \overline{T}_2\leq 1$, we first choose $\eta=\eta_0$ small enough such that
$$
3\eta_0\exp(\widehat{C})\leq \min\Big\{\frac{1}{8},\ \frac{\mu}{8}\Big\},
$$
then we choose $\overline{T}_2=T_*$ small enough such that
$$ 3\eta_0\exp(\widehat{C}_{\eta_0} T_*)\leq 4.$$
So, when $\Gamma^{k+1}=\Gamma^{k+1}(T_*)$, we have
\begin{equation*}\begin{split}
\sigmaum_{k=1}^{\infty}\Big( \Gamma^{k+1}(T_*)+\mu\int_{0}^{T_*} |\nabla\overline{u}^{k+1}|^2_2\thetaext{d}s\Big)\leq \widehat{C}<+\infty.
\end{split}
\end{equation*}
Therefore, the Cauchy sequence $(I^k,\rhoho^k,u^k)$ converges to a limit $(I,\rhoho,u)$ in the following strong sense:
\begin{equation}\label{strong}
\begin{split}
&I^k\rhoightarrow I \ \thetaext{in}\ L^\infty([0,T_*];L^2(R^+\thetaimes S^2;L^2(\mathbb{R}^3))),\\
&\rhoho^k\rhoightarrow \rhoho \ \thetaext{in}\ L^\infty([0,T_*];L^2(\mathbb{R}^3)),\\
&u^k\rhoightarrow u\ \thetaext{in}\ L^2([0,T_*];\mathbb{D}^1(\mathbb{R}^3)).
\end{split}
\end{equation}
Thanks to the local uniform estimate (\rhoef{pri}), the strong convergence in (\rhoef{strong}) and the lower semi-continuity of norms, we also have $(I,\rhoho, u)$ still satisfies the a priori estimates (\rhoef{pri}).
Then it is easy to show that $(I,\rhoho,u)$ is a weak solution in the sense of distribution satisfying the a priori estimates (\rhoef{pri}).\\[2mm]
{\bf u}nderline{Step 2}. The uniqueness of strong solutions. Let $(I_1,\rhoho_1,u_1)$ and $(I_2,\rhoho_2,u_2)$ be two strong solutions to Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}) satisfying the a priori estimates (\rhoef{pri}). Denote
$$
\overline{I}=I_1-I_2,\quad \overline{\rhoho}=\rhoho_1-\rhoho_2,\quad \overline{p}_m=p_m(\rhoho_1)-p_m(\rhoho_2),\quad \overline{u}=u_1-u_2.
$$
Let
$$
\Gamma(t)=\|\overline{I}(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}+|\overline{\rhoho}(t)|^2_{ 2}+|\sigmaqrt{\rhoho}_1\overline{u}|^2_2.
$$
By the same method for deriving (\rhoef{go64})-(\rhoef{gogo1}), we similarly have
\begin{equation}\label{gonm}\begin{split}
\frac{\thetaext{d}}{\thetaext{d}t}\Gamma(t)+\mu|\nabla \overline{u}|^2_2\leq H(t)\Gamma (t),
\end{split}
\end{equation}
where $ \int_{0}^{t}H(s)ds\leq \widehat{C}$, for $t\in [0,T_*]$. Then from the Gronwall's inequality, we immediately conclude that
$$\overline{I}=\overline{\rhoho}=\nabla \overline{u}=0,$$
Since $\overline{u}(t,x)\rhoightarrow 0$ as $|x|\rhoightarrow \infty$, the uniqueness follows.\\[2mm]
{\bf u}nderline{Step 3}. The time-continuity of the strong solution. It can be obtained by the same method used in the proof of Lemma \rhoef{lem1}.
\end{proof}
\sigmaubsubsection{\bf{ Case $\overline{\rhoho}=0$.}}\ \\
\begin{proof}Multiplying the first equation in (\rhoef{eq:1.2w}) by $\thetaext{sign}(\overline{\rhoho}^{k+1})|\overline{\rhoho}^{k+1}|^{\frac{1}{2}}$ and integrating over $\mathbb{R}^3$, we have
\begin{equation}\label{go65e}\begin{cases}
\displaystyle \frac{\thetaext{d}}{\thetaext{d}t}|\overline{\rhoho}^{k+1}|^2_{3/2}\leq B^k_\eta(t)|\overline{\rhoho}^{k+1}|^2_{3/2}+\eta |\nabla\overline{u}^k|^2_2,\\[8pt]
\displaystyle
B^k_\eta(t)=C\left(|\nabla u^k|_{W^{1,q}}+\frac{1}{\eta}|\rhoho^{k}|^2_{H^1}\rhoight),\ \thetaext{and} \ \int_0^t B^k_\eta(s)\thetaext{d}s\leq \widehat{C}+\widehat{C}_{\eta}t,
\end{cases}
\end{equation}
for $t\in [0,\overline{T}_1]$.
Similarly, we can establish the estimates
\begin{equation}\label{go64e}\begin{cases}
\displaystyle \frac{\thetaext{d}}{\thetaext{d}t}|\overline{\rhoho}^{k+1}|^2_2\leq A^k_\eta(t)|\overline{\rhoho}^{k+1}|^2_2+\eta |\nabla\overline{u}^k|^2_2,\\[8pt]
\displaystyle \frac{\thetaext{d}}{\thetaext{d}t}\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}\\[8pt]
\leq D^k_\eta(t)\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}+|\overline{\rhoho}^{k+1}|^2_2+\eta\|\overline{I}^{k}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))},\\[8pt]
\displaystyle \frac{1}{2}\frac{\thetaext{d}}{\thetaext{d}t}|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|^2_2+\mu\|\nabla\overline{u}^{k+1} \|^2\\[8pt]
\leq F^k_\eta(t)|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}|^2_2+F^k_2(t)|\overline{\rhoho}^{k+1}|_{L^2\cap L^{\frac{3}{2}}}+M(C(c_0,c_1))|\overline{p}^{k+1}_m|^2_2\\[8pt]
\displaystyle
+F^k_3(t)\|\overline{I}^{k+1}\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}
+\eta|\nabla\overline{u}^{k}|^2_2+C\int_{\mathbb{R}^3}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x,
\end{cases}
\end{equation}
for $t\in [0,\overline{T}_1]$, and from the local a priori estimate (\rhoef{pri}) we have
\begin{equation*}\begin{split}
\int_0^t \thetaotog(A^k_\eta(s)+D^k_\eta(s)+F^k_\eta(s)+F^k_2(s)+F^k_3(s)\thetaotog)\thetaext{d}s\leq \widehat{C}+\widehat{C}_\eta t.
\end{split}
\end{equation*}
According to (\rhoef{go65e}), the key term can be estimated by
\begin{equation}\label{go6ge}\begin{split}
&\int_{\mathbb{R}^3}|\overline{\rhoho}^{k+1}||u^k_t||\overline{u}^{k+1}|\thetaext{d}x
\leq C|\nabla u^k_t |^2_2|\overline{\rhoho}^{k+1} |^2_{3/2}+\frac{1}{8}\mu|\overline{u}^{k+1}|^2_2.
\end{split}
\end{equation}
We can also define the energy function by
\begin{equation*}\begin{split}
\Gamma^{k+1}=&\sigmaup_{0\leq t \leq \overline{T}_1}\Big(\|\overline{I}^{k+1}(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}+|\overline{\rhoho}^{k+1}(t)|^2_{
2}+|\sigmaqrt{\rhoho}^{k+1}\overline{u}^{k+1}(t)|^2_2\Big).
\end{split}
\end{equation*}
Then from (\rhoef{go65e})-(\rhoef{go6ge}) and Gronwall's inequality, we easily obtain
\begin{equation*}\begin{split}
&\Gamma^{k+1}(\overline{T}_1)+\mu\int_0^{\overline{T}_1} |\nabla\overline{u}^{k+1} |^2_2\thetaext{d}t\\
\leq& \int_0^{\overline{T}_1} G^k_{\eta} \Gamma^{k+1}(t)\thetaext{d}t+3\eta\int_0^{\overline{T}_1}|\nabla\overline{u}^{k}(t) |^2_2\thetaext{d}t+ \eta \overline{T}_1 \sigmaup_{0\leq t \leq \overline{T}_1}\|\overline{I}^{k}(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2;L^2(\mathbb{R}^3))}
\end{split}
\end{equation*}
for some $G^k_{\eta}$ such that $\int_{0}^{t}G^k_\eta(s)\thetaext{d}s\leq \widehat{C}+\widehat{C}_{\eta} t$ for $0\leq t \leq \overline{T}_1$.
The rest of the proof are analogous to the proof for the case $\overline{\rhoho}>0$. We omit the details here.
\end{proof}
Thus the proof of Theorem \rhoef{th1} is finished.
\sigmaection{Necessity and sufficiency of the initial layer compatibility condition}
We prove Theorem \rhoef{th2} in this section, that is, the initial layer compatibility condition is not only sufficient but also necessary if the initial vacuum set is not very irregular. Since the strong solution $(I,\rhoho,u)$ only satisfies the Cauchy problem in the sense of distribution, we only have
$$I(v,\mathbb{R}^3mega,0,x)=I_0,\ \rhoho(0,x)=\rhoho_0,\ \rhoho u(0,x)=\rhoho_0 u_0,\ x\in \mathbb{R}^3.$$
So, the key point of the proof is to make sure that the relation $u(0,x)=u_0$ holds in the vacuum domain. Now we give the proof of Theorem \rhoef{th2}.
\begin{proof} We prove the necessity and sufficiency, respectively.\\
{\bf u}nderline{\bf{Step 1: to prove the necessity.}} Let $(I,\rhoho,u)$ be a strong solution of the Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}) with the regularity as shown in Definition \rhoef{strong1}. Then from the momentum equations in (\rhoef{eq:1.2}) we have
\begin{equation}\label{da9}\begin{split}
Lu(t)+\nabla p_{m}(t)+\frac{1}{c}\int_0^\infty \int_{S^2}A_r(t)\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v=\sigmaqrt{\rhoho}(t) G(t)
\end{split}
\end{equation}
for $0\leq t \leq T_*$, where $G(t)=\rhoho^{\frac{1}{2}}(t) (-u_t-u\cdot\nabla u)$. Since
$$
\sigmaqrt{\rhoho}u_t\in L^\infty([0,T_*];L^2),\quad \sigmaqrt{\rhoho}u\cdot\nabla u \in L^\infty([0,T_*];L^2),
$$
we have $G(t)\in L^\infty([0,T_*];L^2)$. So there exists a sequence $\{t_k\}$, $t_k\rhoightarrow 0$, such that
$$
G(t_k)\rhoightarrow g \quad \thetaext{in} \quad L^2 \quad \thetaext{for some} \quad g\in L^2.
$$
Taking $t=t_k\rhoightarrow 0$ in (\rhoef{da9}), we obtain
\begin{equation}\label{jojo}\begin{split}
Lu(0)+\nabla p_{m}(\rhoho(0))+\frac{1}{c}\int_0^\infty \int_{S^2}A_r(0)\mathbb{R}^3mega \thetaext{d}\mathbb{R}^3mega \thetaext{d}v=\sigmaqrt{\rhoho}(0) g.
\end{split}
\end{equation}
Together with the strong convergence (\rhoef{coco}) and the construction of our strong solutions, initial layer compatibility condition \eqref{kkkkk} holds with $g_1=g$.\\
{\bf u}nderline{\bf{Step 2: to prove the sufficiency.}} Let $(I_0,\rhoho_0,u_0)$ be the initial data satisfying (\rhoef{gogo})-(\rhoef{kkkkk}). Then there exists a unique strong solution $(I,\rhoho,u)$ to the Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}) with the regularity
\begin{equation*}
\begin{split}
&I(v,\mathbb{R}^3mega,t,x)\in L^2(\mathbb{R}^+\thetaimes S^2;C([0,T_*];H^1\cap W^{1,q}(\mathbb{R}^3))), \\
&\rhoho(t,x)-\overline{\rhoho}\in C([0,T_*];H^1\cap W^{1,q}), \ u(t,x)\in C([0,T_*];\mathbb{D}^1\cap D^2(\mathbb{R}^3)).
\end{split}
\end{equation*}
So we only need to verify the initial conditions
$$
I(v,\mathbb{R}^3mega,0,x)=I_0,\ \rhoho(0,x)=\rhoho_0,\ u(0,x)=u_0(x), \ x\in \mathbb{R}^3.
$$
From the weak formulation of the strong solution, it is easy to know that
$$
I(v,\mathbb{R}^3mega,0,x)=I_0,\ \rhoho(0,x)=\rhoho_0,\ \rhoho(0,x)u(0,x)=\rhoho_0u_0, \ x\in \mathbb{R}^3.
$$
So it remains to prove that $u(0,x)=u_0(x), \ x\in V$. Let $\overline{u}_0(x)=u_0(x)-u(0,x)$. According to the proof of the necessity, we know that $(I(v,\mathbb{R}^3mega,0,x),\rhoho(0,x),u(0,x))$ also satisfies the relation (\rhoef{kkkkk}) for $g_1\in L^2$. Then $\overline{u}_0\in \mathbb{D}^1_0(V)\cap D^2(V)$ is the unique solution of the elliptic problem (\rhoef{zhen101}) in $V$ and thus $\overline{u}_0=0$ in $V$, which implies that
$u(0,x)=u_0(x), \ x\in V$.
\end{proof}
Finally we remark that, for a special case that the mass density $\rhoho(t,x)= 0$ only holds in some single point or only decay in the far field, $u(0,x)=u_0$ obviously hold according to our proof of the sufficiency.
\sigmaection{Blow-up criterion of strong solutions}
In this section, we prove Theorem \rhoef{th3} in which we establish a blow-up criterion for strong solutions. Firstly we define the following two auxiliary quantities:
\begin{equation*}\begin{split}
\Phi(t)=&1+\|I\|_{L^2(\mathbb{R}^+\thetaimes S^2; C([0,t];H^1\cap W^{1,q}(\mathbb{R}^3)))}+\sigmaup_{0\leq s\leq t}\|\rhoho(s)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+\sigmaup_{0\leq s\leq t}|u(s)|_{\mathbb{D}^1},\\[6pt]
\Theta(t)=&1+\|I\|_{L^2(\mathbb{R}^+\thetaimes S^2; C([0,t];H^1\cap W^{1,q}(\mathbb{R}^3)))}+\|I_t\|_{L^2(\mathbb{R}^+\thetaimes S^2; C([0,t];L^2\cap L^q(\mathbb{R}^3)))}\\[6pt]
&+\|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}+|\rhoho_t(t)|_{L^2\cap L^q}
\\
&+|u(t)|_{\mathbb{D}^1\cap D^2}+|\sigmaqrt{\rhoho}u_t(t)|_2+\int_{0}^{t}\Big(|u(s)|^2_{D^{2,q}}+|u_t(s)|^2_{\mathbb{D}^1}\Big)\thetaext{d}s.
\end{split}
\end{equation*}
Then according to the definition of the maximal existence time $\overline{T}$ of the local strong solution in Definition (\rhoef{strong1}), we know that
\begin{equation}\label{up}\begin{split}
\Theta(t)\rhoightarrow +\infty, \quad \thetaext{as} \quad t\rhoightarrow \overline{T}.
\end{split}
\end{equation}
Based on (\rhoef{up}), our purpose in the following proof is to show that
\begin{equation}\label{ga1}\begin{split}
\Phi(t)\rhoightarrow+\infty, \quad \thetaext{as} \quad t\rhoightarrow \overline{T}.
\end{split}
\end{equation}
\thetaextbf{Now we give the proof for Theorem {\rhoef{th3}}.}
\begin{proof}
Firstly, let $(I,\rhoho,u)$ be the unique strong solution of the Cauchy problem (\rhoef{eq:1.2})-(\rhoef{eq:2.2hh}) with the regularities shown in Definition \rhoef{strong1}. Then from similar arguments as shown in Lemmas \rhoef{lem:2} and \rhoef{lem:3}, for $0< t < \overline{T}$, we easily get
\begin{equation}\label{vb1}
\qquad \qquad \|\rhoho(t)-\overline{\rhoho}\|_{H^1\cap W^{1,q}}\leq C\Big(1+\int_0^t \|\nabla u\|_{H^1\cap W^{1,q}}\thetaext{d}s\Big) \exp\Big(\int_0^t \|\nabla u\|_{W^{1,q}}\thetaext{d}s\Big),
\end{equation}
and
\begin{equation}\label{vb1qq}
\quad \|I\|^2_{L^2(\mathbb{R}^+\thetaimes S^2; C([0,t]; H^1\cap W^{1,q}(\mathbb{R}^3)))}
\leq \exp\thetaotog(t M(\Phi(t)) \thetaotog)\thetaotog(1+tM(\Phi(t))\thetaotog).
\end{equation}
Directly from the continuity equation and the radiation transfer equation in (\rhoef{eq:1.2}), for $0< t < \overline{T}$, we deduce that
\begin{equation}\label{vb2}\begin{cases}
|(\rhoho_t,(p_m)_t)(t)|_{2}\leq M(|\rhoho(t)|_\infty)\thetaotog(1+|\rhoho(t)|_\infty |\nabla u(t)|_{2} +|u(t)|_6 |\nabla\rhoho(t)|_{3}\thetaotog),\\[6pt]
|(\rhoho_t,(p_m)_t)(t)|_{q}\leq M(|\rhoho(t)|_\infty)\thetaotog(1+|\rhoho(t)|_\infty |\nabla u(t)|_{q} +|u(t)|_\infty |\nabla\rhoho(t)|_q\thetaotog),\\[6pt]
\|I_t(t)\|_{L^2(\mathbb{R}^+\thetaimes S^2; C([0,t];L^2\cap L^q(\mathbb{R}^3)))}\leq M(\Phi(t)).
\end{cases}
\end{equation}
Secondly, for the velocity vector $u$ of the fluid, standard energy estimates as shown in Lemma \rhoef{lem:4} lead to
\begin{equation}\label{vb4}\begin{split}
&\frac{1}{2}\frac{\thetaext{d}}{\thetaext{d}t}\int_{\mathbb{R}^3}\rhoho |u_t|^2 \thetaext{d}x+\int_{\mathbb{R}^3}\thetaotog(\mu|\nabla u_t|^2+(\lambda+\mu)|\thetaext{div}u_t|^2\thetaotog)\thetaext{d}x\\
\leq& C\thetaotog(1+|\rhoho|^3_{\infty}|\nabla u|^4_2 \thetaotog)|\sigmaqrt{\rhoho}u_t|^2_{2}+C\thetaotog(1+|\rhoho_t|^2_{2}|\nabla u|^4_2 +|(p_m)_t|^2_2\thetaotog)\\
&+ M(|\rhoho|_\infty)\thetaotog(1+|\rhoho|_\infty)\thetaotog(1+|\rhoho_t|^2_2\|I(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2; L^2(\mathbb{R}^3))}+\|I_t(t)\|^2_{L^2(\mathbb{R}^+\thetaimes S^2; L^2(\mathbb{R}^3))}\thetaotog).
\end{split}
\end{equation}
Integrating (\rhoef{vb4}) over $(\thetaau,t)$ for $\thetaau\in (0,t)$, and according to (\rhoef{vb1})-(\rhoef{vb2}), we easily have
\begin{equation}\label{vb7}
\begin{split}
|\sigmaqrt{\rhoho}u_t(t)|^2_{2}+\int_{\thetaau}^{t}|\nabla u_t|^2_{2}\thetaext{d}s
\leq&C+|\sigmaqrt{\rhoho}u_t(\thetaau)|^2_{2}
+C\int_{\thetaau}^{t}(1+|\sigmaqrt{\rhoho}u_t|^2_{2})M(\Phi)\thetaext{d}s.
\end{split}
\end{equation}
According to Gronwall's inequality, we have
\begin{equation}\label{vb11}
\begin{split}
|\sigmaqrt{\rhoho}u_t(t)|^2_{2}+\int_{\thetaau}^{t}|\nabla u_t(s)|^2_{2}\thetaext{d}s
\leq&C\Theta (\thetaau)\exp\thetaotog(\overline{T}M(\Phi(t))\thetaotog).
\end{split}
\end{equation}
Thirdly, we consider the higher order terms $|u|_{D^2}$ and $|\nabla u|_{W^{1,q}}$.
From the standard elliptic regularity estimates and Minkowski inequality, we have
\begin{equation*}
\begin{split}
| u|_{D^2}\leq & C\Big(|\rhoho u_t|_2+|\rhoho u\cdot\nabla u|_2+|\nabla p_m|_2+\int_0^\infty \int_{S^2}|A_r|_2 \thetaext{d}\mathbb{R}^3mega \thetaext{d}v\Big)\\
\leq & C\thetaotog(1+|\rhoho|^{\frac{1}{2}}_\infty |\sigmaqrt{\rhoho}u_t|_{2}+|\rhoho|_\infty | u|^{\frac{3}{2}}_{\mathbb{D}^1}| u|^{\frac{1}{2}}_{D^2}+M(|\rhoho|_\infty)(|\rhoho|_2+\|I\|_{L^2(\mathbb{R}^+\thetaimes S^2; L^2(\mathbb{R}^3))})\thetaotog).
\end{split}
\end{equation*}
This implies, by Young's inequality, that,
\begin{equation}\label{vb5}
\begin{split}
| u(t)|_{\mathbb{D}^1\cap D^2}\leq & C\thetaotog(1+|\sigmaqrt{\rhoho} u_t(t)|_2\thetaotog)M(\Phi(t))
\end{split}
\end{equation}
for $t\in (\thetaau,\overline{T})$.
Similarly, due to (\rhoef{vb5}), for $t\in (\thetaau,\overline{T})$, we have
\begin{equation}\label{vb6}
\begin{split}
|u(t)|_{D^{2,q}}
\leq C\thetaotog( (1+|\sigmaqrt{\rhoho} u_t(t)|^2_2)M(\Phi(t))+|\nabla u_t(t)|_2\thetaotog).
\end{split}
\end{equation}
Finally, we combine (\rhoef{vb1})-(\rhoef{vb6}) and conclude that for each $t\in (\thetaau,\overline{T})$,
\begin{equation}\label{vb12}
\begin{split}
\Theta(t)
\leq&C(1+\overline{T})(1+\Theta (\thetaau))^2 M(\Phi(t))\exp\thetaotog(\overline{T}M(\Phi(t))\thetaotog).
\end{split}
\end{equation}
From (\rhoef{up}), the blow-up criterion \eqref{blowcr} as shown in Theorem \rhoef{th3} follows immediately by letting $t\rhoightarrow \overline{T}$ in (\rhoef{vb12}).
\end{proof}
\thetaotogskip
{\bf Acknowledgement:} The research of Y. Li and S. Zhu were supported in part
by National Natural Science Foundation of China under grant 11231006 and Natural Science Foundation of Shanghai under grant 14ZR1423100. S. Zhu was also supported by China Scholarship Council.
\thetaotogskip
\end{document} | math | 111,012 |
\begin{document}
\title{Additive decompositions of large multiplicative subgroups in finite fields}
\begin{abstract}
We show that a large multiplicative subgroup of a finite field $\mathbb{F}_q$ cannot be decomposed into $A+A$ or $A+B+C$ nontrivially. We also find new families of multiplicative subgroups that cannot be decomposed as the sum of two sets nontrivially. In particular, our results extensively generalize the results of S\'{a}rk\"{o}zy and Shkredov on the additive decomposition of the set of quadratic residues modulo a prime.
\end{abstract}
\section{Introduction}
Throughout the paper, let $p$ be a prime and $q$ a power of $p$. Let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $d \mid (q-1)$ such that $d>1$, we denote $S_d=\{x^d: x \in \mathbb{F}_q^*\}$ to be the subgroup of $\mathbb{F}_q^*$ with order $\frac{q-1}{d}$.
A celebrated conjecture of S\'{a}rk\"{o}zy \cite{S12} asserts that $S_2$ cannot be written as $S_2=A+B$, where $A,B \subset \mathbb{F}_q$ and $|A|, |B| \geq 2$, provided that $q$ is a sufficiently large prime. Recall that for two sets $A,B \subset \mathbb{F}_q$, their {\em sum} is $A+B=\{a+b: a \in A, b \in B\}$. It is necessary to assume that $|A|,|B| \geq 2$ in the conjecture, otherwise there are many {\em trivial} additive decompositions of $S_2$ into the sum of two sets. The motivation of the conjecture is clear: sumsets have rich additive structure, while multiplicative subgroups do not possess too much additive structure. It is natural to consider the analogue of this conjecture over all finite fields and all multiplicative subgroups, namely, given a proper multiplicative subgroup $G$ of $\mathbb{F}_q$, can we find a nontrivial additive decomposition of $G$ as the sum of two sets $A, B \subset \mathbb{F}_q$? This natural extension was first studied by Shparlinski \cite{S13}, shortly after S\'{a}rk\"{o}zy's seminal paper \cite{S12} was published. It is likely that the answer remains negative provided that the index of the subgroup $G$ is fixed and $q$ is sufficiently large, and this can be regarded as \emph{the generalized S\'{a}rk\"{o}zy's conjecture}. S\'{a}rk\"{o}zy's conjecture, as well as the generalized S\'{a}rk\"{o}zy's conjecture, have attracted lots of attention in the last decade; see for example \cite{S13, S14, BKS15, S16, S20, HP, CY21, CX22, WS23} (listed in chronological order).
Recently, Hanson and Petridis \cite{HP} showed that if $q$ is a prime and $S_d=A+B$ with $A, B \subset \mathbb{F}_q$ and $|A|,|B| \geq 2$, then $|A||B|=|S_d|$, that is, all sums $a+b$ must be distinct. This is a breakthrough on S\'{a}rk\"{o}zy's conjecture, and in particular, it implies that S\'{a}rk\"{o}zy's conjecture holds for almost all primes \cite[Corollary 1.4]{HP}. We also refer to the recent work by Chen and Xi \cite{CX22} on the best-known lower and upper bounds on $|A|$ and $|B|$ provided that $A+B=S_2$ and $A, B \subset \mathbb{F}_p$.
In this paper, we make progress on the generalized S\'{a}rk\"{o}zy's conjecture for large multiplicative subgroups in finite fields. In particular, we show that large multiplicative subgroups cannot be written as $A+A$ or $A+B+C$ nontrivially, extending and refining existing results in the literature significantly. We also obtain new families of $(d,q)$ such that $S_d \subset \mathbb{F}_q$ admits no nontrivial additive decomposition, as well as new structural information on $A$ and $B$ if $A+B=S_d$ (Proposition~\ref{prop:structure}).
Our first result extends \cite[Theorem 1.2]{HP}, which is the main result in \cite{HP} by Hanson and Petridis, to all finite fields, with an extra assumption on the non-vanishing of a binomial coefficient.
\begin{thm}\label{thm:main}
Let $d \mid (q-1)$ such that $d>1$. If $A,B \subset \mathbb{F}_q$ such that $A+B\subset S_d \cup \{0\}$ and $
\binom{|A|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\not \equiv 0 \pmod p,$
then
$$|A||B|\leq \frac{q-1}{d}+|A \cap (-B)|.$$
\end{thm}
We remark that in general the condition on the binomial coefficient cannot be dropped; see Example~\ref{ex:ex1}. Note that when $q$ is a prime, it is easy to see that the condition on the binomial coefficient is automatically satisfied and thus Theorem~\ref{thm:main} recovers \cite[Theorem 1.2]{HP}.
With a bit extra work, Theorem~\ref{thm:main} implies the following corollary (see also Proposition~\ref{prop:structure}), which generalizes \cite[Corollary 1.3]{HP} and makes new progress towards the generalized S\'{a}rk\"{o}zy's conjecture.
\begin{cor}\label{cor:a+b}
Let $d \mid (q-1)$ such that $\frac{q-1}{d}\leq \frac{2p}{3}$. If $A,B \subset \mathbb{F}_q$ such that $A+B=S_d$, then $|A||B|\leq \frac{q-1}{d}$; moreover, if $A+B=S_d$, then $|A||B|=\frac{q-1}{d}$ and all sums $a+b$ are distinct. In particular, if $\frac{q-1}{d}$ is a prime, then there is no nontrivial additive decomposition of $S_d$ into the sum of two sets.
\end{cor}
We remark that Corollary~\ref{cor:a+b} does not extend to all pairs $(d,q)$ with $d \geq 2$ and $d \mid (q-1)$; see Example~\ref{ex:ex1}. Nevertheless, if $A+B=S_d$ and $|A||B|>|S_d|$, we can still say something about the structure of $A$ and $B$; see Proposition~\ref{prop:structure} and Remark~\ref{rem:structure}.
Next we proceed with the discussion on the possibility of $S_d=A+A$. Before stating our contributions, we introduce the following two definitions for convenience. Let $q=p^n$ and let $d \mid (q-1)$ with $2 \leq d<q-1$. Write the base-$p$ expansion of $\frac{q-1}{d}$ as $$\frac{q-1}{d}=\sum_{j=0}^{n} e_jp^j,$$ where $0 \leq e_j \leq p-1$. For a positive real number $\delta$, we say a pair $(d,q)$ is \emph{$\delta$-good} if $$e_j \leq \lfloor (1-\delta)(p-1) \rfloor$$ for each $0 \leq j<(n+1)/2$. Furthermore, we say a pair $(d,q)$ is $\emph{good}$ if one of the following conditions holds:
\begin{itemize}
\item $\frac{q-1}{d}\leq \frac{2p}{3}$;
\item $e_j \leq \frac{p-1}{2}$ for each $0 \leq j<n/2$;
\item $n=2r+1$ with $r \geq 1$, $d\leq 2p-2$, and $e_r\leq p-1-\lceil \sqrt{p} \rceil/2$;
\item $n=2r$, $d \leq 2p^2$, and $e_{r-1} \leq \frac{p-3}{2}$.
\end{itemize}
Note that if a pair $(d,q)$ is $\frac{1}{2}$-good, then it is good.
Shkredov \cite{S14} showed that $S_2$ cannot be written as $A+A$ when $q=p$ is a prime such that $p>3$. However, as remarked by Shparlinski \cite{S13} (see also Remark~\ref{rmk:A+A}), his method does not seem to extend to other subgroups. Instead, Theorem~\ref{thm:main}, together with other tools and observations, allows us to extend Shkredov's result to other subgroups.
\begin{thm}\label{thm:A+A}
Let $d \mid (q-1)$ with $2 \leq d<q-1$. If the pair $(d,q)$ is good, then there is no additive decomposition $S_d=A+A$ for any $A \subset \mathbb{F}_q$.
\end{thm}
We remark that when $q$ is a proper prime power, we do need some extra assumptions on the pair $(d,q)$ for Theorem~\ref{thm:A+A} to hold; see Example~\ref{ex:ex1} for a counterexample. In certain cases, it is easy to verify that a pair $(d,q)$ is good without actually expanding $\frac{q-1}{d}$ in base-$p$. The following corollary describes a few such cases and is thus an immediate consequence of Theorem~\ref{thm:A+A}.
\begin{cor}\label{cor:A+A}
Let $q=p^n$ and $d \mid (q-1)$ with $2 \leq d<q-1$. Let $k$ be the order of $p$ modulo $d$. Then there is no additive decomposition $S_d=A+A$ for any $A \subset \mathbb{F}_q$, provided that one of the following conditions holds:
\begin{itemize}
\item $d \mid (p-1)$ (in particular, if $d=2$ or $q$ is a prime);
\item $\frac{p^k-1}{d}\leq \frac{p-1}{2}$;
\item $2k \mid n$ and $d \leq 2p^2$.
\end{itemize}
\end{cor}
Finally, we turn to the discussion of the {\em ternary decomposition}, that is, decomposing $S_d$ into the sum of three sets. In his paper, S\'{a}rk\"{o}zy \cite{S12} confirmed the ternary version of his conjecture. Recently, Chen and Yan \cite{CY21} showed that for any prime $p$, there is no additive decomposition $A+B+C=S_2$ with $|A|,|B|,|C| \geq 2$, refining S\'{a}rk\"{o}zy's result \cite{S12}. Very recently, Wu and She \cite{WS23} showed that for a prime $p> 184291$, there is no additive decomposition $A+B+C=S_3$ with $|A|,|B|,|C| \geq 2$. We refer to \cite{GS15, GS17} for a general discussion on (ternary)-irreducible subsets of $\mathbb{F}_p$. On the other hand, Shkredov \cite{S20} showed that any small multiplicative subgroup is not a sumset in the sense that if $G$ is a multiplicative subgroup of $\mathbb{F}_p$ with $1 \ll_{\epsilon} |G| \leq p^{2/3-\epsilon}$, then there is no additive decomposition $G=A+B$ with $A, B \subset \mathbb{F}_p$ and $|A|,|B| \geq 2$.
Our next result shows that as long as $G$ is a proper multiplicative subgroup of some prime field $\mathbb{F}_p$ with $|G| \gg 1$, then $G$ cannot be written as $A+B+C$ nontrivially. In particular, this confirms the ternary version of the generalized S\'{a}rk\"{o}zy's conjecture over prime fields in a stronger form.
\begin{thm}\label{thm:prime3}
There exists a constant $M>0$, such that whenever $G$ is a proper multiplicative subgroup of $\mathbb{F}_p$ with $|G|>M$ and $p$ a prime, there is no additive decomposition $G=A+B+C$ with $A,B,C \subset \mathbb{F}_p$ and $|A|,|B|,|C| \geq 2$.
\end{thm}
Theorem~\ref{thm:prime3} fails to extend to general finite fields $\mathbb{F}_q$; a simple counterexample can be found in Example~\ref{ex:ex2}. Instead, we show that a large multiplicative subgroup $S_d$ cannot be written as $A+B+C$ nontrivially.
\begin{thm}\label{thm:dq}
Let $\epsilon>0$. There is a constant $Q=Q(\epsilon)$, such that for each prime power $q>Q$ and a divisor $d$ of $q-1$ with $2 \leq d \leq q^{1/10-\epsilon}$, there is no additive decomposition $S_d=A+B+C$ with $A,B,C \subset \mathbb{F}_q$ and $|A|,|B|,|C| \geq 2$.
\end{thm}
In particular, Theorem~\ref{thm:dq} implies the following corollary immediately, which confirms the ternary version of the generalized S\'{a}rk\"{o}zy's conjecture.
\begin{cor}\label{cor:dfixed}
There is an absolute constant $M$, such that for each pair $(d,q)$ such that $q \equiv 1 \pmod d$ and $q>Md^{11}$, there is no additive decomposition $S_d=A+B+C$ with $A,B,C \subset \mathbb{F}_q$ and $|A|,|B|,|C| \geq 2$. In particular, if $d$ is fixed, and $q \equiv 1 \pmod d$ is sufficiently large, then there is no nontrivial ternary decomposition of $S_d$.
\end{cor}
Assuming that the base-$p$ representation of $\frac{q-1}{d}$ behaves ``nicely", we can further improve the range of $d$ from $q^{1/10}$ in Theorem~\ref{thm:dq} to roughly $q^{1/4}$.
\begin{thm}\label{thm:deltagood}
Let $\epsilon>0$. There are constants $Q=Q(\epsilon)$ and $P=P(\epsilon)$, such that for each prime power $q>Q$ with $p>P$, and a divisor $d$ of $q-1$ such that the pair $(d,q)$ is $\delta$-good with $\delta>0$ and $2 \leq d \leq q^{1/4-\epsilon} \delta^{3/2}$, there is no nontrivial ternary decomposition of $S_d$.
\end{thm}
\textbf{Notation.} We follow the Vinogradov notation $\ll$. We write $X \ll Y$ if there is an absolute constant $C>0$ so that $|X| \leq CY$.
\textbf{Structure of the paper.}
In Section~\ref{sec:prelim}, we provide additional background and prove some preliminary results. In Section~\ref{sec:Stepanov}, we prove Theorem~\ref{thm:main} and Corollary~\ref{cor:a+b}. In Section~\ref{sec:A+A}, we prove Theorem~\ref{thm:A+A}. In Section~\ref{sec:A+B+C}, we prove Theorem~\ref{thm:prime3}, Theorem~\ref{thm:dq}, and Theorem~\ref{thm:deltagood} in Section~\ref{sec:A+B+C}.
\section{Preliminaries}\label{sec:prelim}
\subsection{Estimates of the size of $A$ and $B$ for $A+B=S_d$} In this section, we collect a few known results on the size of $A$ and $B$ for a given additive decomposition $S_d=A+B$.
\begin{lem}\label{lem:ub}
Let $A,B \subset \mathbb{F}_q$. If $A+B\subset S_d$, then $|A||B|<q$.
\end{lem}
\begin{proof}
Let $\chi$ be a multiplicative character of $\mathbb{F}_q$ with order $d$. The following double character sum estimate is well-known (see for example \cite[Theorem 2.6]{Y22}):
\begin{equation}\label{eq:double}
\bigg|\sum_{a\in A,\, b\in B}\chi(a+b)\bigg| \leq \sqrt{q|A||B|}\bigg(1-\frac{|A|}{q}\bigg)^{1/2}\bigg(1-\frac{|B|}{q}\bigg)^{1/2}.
\end{equation}
Since $A+B\subset S_d $, we have $\chi(a+b)=1$ for all $a \in A$, $b \in B$. It follows that
$$
|A||B| \leq \sqrt{q|A||B|}\bigg(1-\frac{|A|}{q}\bigg)^{1/2}\bigg(1-\frac{|B|}{q}\bigg)^{1/2}<\sqrt{q|A||B|}
$$
and thus $|A||B|<q$.
\end{proof}
Note that the right-hand side of inequality~\eqref{eq:double} is symmetric in $|A|$ and $|B|$. Based on the application of a bound of Karatsuba~\cite{K91} (more precisely, see \cite[Lemma 2.2]{S13}) on double character sums, that is, an ``asymmetric version" of inequality~\eqref{eq:double}, Shparlinski \cite{S13} proved the following remarkable theorem.
\begin{thm}[Shparlinski {\cite[Theorem 7.1]{S13}}]\label{thm:lball}
If $G$ is a proper multiplicative subgroup of $\mathbb{F}_p$ such that $G=A+B$ for some $A,B \subset \mathbb{F}_p$ with $|A|,|B| \geq 2$, then
$$
|G|^{1/2 + o(1)}= \min\{|A|, |B|\} \leq \max \{|A|, |B|\}=|G|^{1/2 + o(1)}
$$
as $|G| \to \infty$.
\end{thm}
As remarked by Shparlinski~\cite{S13}, one of the main ingredients to prove Theorem~\ref{thm:lball}, namely \cite[Lemma 3.1]{S13} on the size of the intersection of shifted subgroups, has been established only for prime fields. Indeed, Theorem~\ref{thm:lball} fails to extend to finite fields in general; see Example~\ref{ex:ex1} and Example~\ref{ex:ex2}. Nevertheless, for a large multiplicative subgroup of a general finite field, we have the following weaker estimate, which was implicitly discussed in \cite[Section 6]{S13}:
\begin{lem}[Shparlinski \cite{S13}]\label{lem:maxmin}
Let $\epsilon>0$. Let $d \mid (q-1)$ such that $d \leq q^{1/4-\epsilon}$. If $A+B=S_d$ for some $A,B \subset \mathbb{F}_q$ with $|A|, |B| \geq 2$, then
$$
\frac{\sqrt{q}}{d} \ll \min \{|A|,|B|\} \leq \max \{|A|,|B|\} \ll q^{1/2}.
$$
\end{lem}
\begin{proof}
Since $d \leq q^{1/4-\epsilon}$, we have $|S_d| \gg q^{3/4+\epsilon}$. Thus, \cite[Theorem 6.1]{S13} implies that
\begin{equation}\label{eq:max}
\max \{|A|,|B|\} \ll q^{1/2}.
\end{equation}
On the other hand, note that $A+B=S_d$ implies $|A||B|\geq |S_d| \gg \frac{q}{d}$. Thus, it follows that
$$
\min \{|A|,|B|\}=\frac{|A||B|}{ \max \{|A|,|B|\}} \gg \frac{\sqrt{q}}{d}.
$$
\end{proof}
\subsection{Hyper-derivatives}
The proof of Theorem~\ref{thm:main} replies on computing derivatives of a polynomial over $\mathbb{F}_q$. Note that the $p$-th derivative of a polynomial over $\mathbb{F}_q$ is trivially $0$ due to its characteristic. To overcome the characteristic issue, we need to work on hyper-derivatives (also known as Hasse derivatives) instead. We recall a few basic properties of hyper-derivative; a general discussion can be found in \cite[Section 6.4]{LN97}.
\begin{defn}
Let $c_0,c_1, \ldots c_d \in \mathbb{F}_q$. If $n$ is a non-negative integer, then the $n$-th hyper-derivative of $f(x)=\sum_{j=0}^d c_j x^j$ is
$$
E^{(n)}(f) =\sum_{j=0}^d \binom{j}{n} c_j x^{j-n},
$$
where we follow the standard convention that $\binom{j}{n}=0$ for $j<n$, so that the $n$-th hyper-derivative is a polynomial.
\end{defn}
\begin{lem}[{\cite[Corollary 6.48]{LN97}}]\label{lem:differentiate}
Let $n,d$ be positive integers. If $c \in \mathbb{F}_q$, then
$$E^{(n)}\big((x+c)^d\big)=\binom{d}{n} (x+c)^{d-n}.$$
\end{lem}
\begin{lem}[{\cite[Lemma 6.51]{LN97}}]\label{lem:multiplicity}
Let $f$ be a nonzero polynomial in $\mathbb{F}_q[x]$. If $c$ is a root of $E^{(k)}(f)$ for $k=0,1,\ldots, m-1$, then $c$ is a root of multiplicity at least $m$.
\end{lem}
\subsection{Tools from additive combinatorics}
In this section, we list two useful results from additive combinatorics.
The following theorem is a generalization of the classical Cauchy-Davenport theorem.
\begin{thm}[K\'{a}rolyi \cite{K05}]\label{thm:CD}
Let $G$ be a finite group with group operation $+$, and let $A, B \subset G$ be non-empty subsets. Further, let $\rho(G)$ denote the minimum of the orders of nontrivial subgroups of $G$. If $\rho(G)\geq |A|+|B|-1$, then $|A+B|\geq |A|+|B|-1$.
\end{thm}
\begin{cor}\label{cor:CD}
Let $A, B \subset \mathbb{F}_q$ be non-empty subsets. Then $|A+B|\geq \min \{p, |A|+|B|-1\}$.
\end{cor}
\begin{proof}
The minimum size of a nontrivial (additive) subgroup of $\mathbb{F}_q$ is $p$. Thus, the corollary follows from Theorem~\ref{thm:CD}.
\end{proof}
The following lemma turns out to be useful for studying ternary decompositions.
\begin{lem}[Gyarmati, Matolcsi, and Ruzsa \cite{GMR10}]\label{lem:addcomb}
Let $A, B, C$ be nonempty subsets of $\mathbb{F}_q$. Then $|A+B+C|^2 \leq |A+B||B+C||C+A|$.
\end{lem}
\begin{proof}
This is a special case of \cite[Theorem 1.2]{GMR10} due to Gyarmati, Matolcsi, and Ruzsa.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}\label{sec:Stepanov}
The proof of Theorem~\ref{thm:main} is based on Stepanov's method. The key idea is to construct a low degree \emph{nonzero} polynomial that vanishes on each element of $B$ with high multiplicity.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
If $|A|=1$, the result is immediate. Next we assume that $|A|\geq 2$. Let $r=|A \cap (-B)|$. Let $A=\{a_1,a_2,\ldots, a_n\}$ and $B=\{b_1,b_2, \ldots, b_m\}$ such that $b_{r+1}, \ldots, b_m \notin (-A)$. Since $A+B \subset S_d \cup \{0\}$, we have $$(a_i+b_j)^{\frac{q-1}{d}+1}=a_i+b_j$$ for each $1 \leq i \leq n$ and $1 \leq j \leq m$. This simple observation will be used repeatedly in the following computation.
Let $c_1,c_2,...,c_n$ be the unique solution of the following system of equations:
\begin{equation} \label{system}
\left\{
\TABbinary\tabbedCenterstack[l]{
\sum_{i=1}^n c_i a_i^j=0, \quad 0 \leq j \leq n-2\\\\
\sum_{i=1}^n c_i a_i^{n-1}=1
}\right.
\end{equation}
This is justified by the invertibility of the coefficient matrix of the system (a Vandermonde matrix).
Consider the following auxiliary polynomial
$$
f(x)=-1+\sum_{i=1}^n c_i (x+a_i)^{n-1+\frac{q-1}{d}}\in \mathbb{F}_q[x].
$$
We claim that the degree of $f$ is $\frac{q-1}{d}$. Indeed, for each $0 \leq j \leq n-1$, the coefficient of $x^{n-1+\frac{q-1}{d}-j}$ in $f(x)$ is
$$
\binom{n-1+\frac{q-1}{d}}{j} \cdot \sum_{i=1}^n c_i a_i^{j}.
$$
Thus, system~\eqref{system} implies that the coefficient of $x^{n-1+\frac{q-1}{d}-j}$
is $0$ for $j=0,1, \ldots, n-2$, and the coefficient of $x^{\frac{q-1}{d}}$
is $$\binom{n-1+\frac{q-1}{d}}{n-1}= \binom{n-1+\frac{q-1}{d}}{\frac{q-1}{d}}\neq 0$$ by the assumption.
Next, we compute the hyper-derivatives of $f$ on $B$. For each $1\leq j \leq m$, system~\eqref{system} implies that
\begin{align*}
E^{(0)} f (b_j)
&= f (b_j)\\
&= -1+\sum_{i=1}^n c_i (b_j+a_i)^{n-1+\frac{q-1}{d}} \\
&= -1+\sum_{i=1}^n c_i (b_j+a_i)^{n-1} \\
&= -1+\sum_{\ell=0}^{n-1} \binom{n-1}{\ell} \bigg(\sum_{i=1}^n c_i a_i^\ell\bigg)b_j^{n-1-\ell} \\
&= -1+\binom{n-1}{n-1} \cdot \sum_{i=1}^n c_i a_i^{n-1}
=0.
\end{align*}
For each $1\leq j \leq m$ and $1 \leq k \leq n-2$, Lemma~\ref{lem:differentiate} implies that
\begin{align*}
E^{(k)} f (b_j)
&= \binom{n-1+\frac{q-1}{d}}{k} \sum_{i=1}^n c_i (b_j+a_i)^{n-1+\frac{q-1}{d}-k} \\
&= \binom{n-1+\frac{q-1}{d}}{k} \sum_{i=1}^n c_i (b_j+a_i)^{n-1-k} \\
&= \binom{n-1+\frac{q-1}{d}}{k} \sum_{\ell=0}^{n-1-k} \binom{n-1-k}{\ell} \bigg(\sum_{i=1}^n c_i a_i^\ell\bigg)b_j^{n-1-k-\ell}
=0,
\end{align*}
where we again use the assumptions in system~\eqref{system}.
For each $r+1\leq j \leq m$, by the assumption, $b_j \notin (-A)$, that is, $b_j+a_i \neq 0$ for each $1 \leq i \leq n$. Thus, by Lemma~\ref{lem:differentiate}, for each $r+1\leq j \leq m$, we additionally have
\begin{align*}
E^{(n-1)} f (b_j)
&= \binom{n-1+\frac{q-1}{d}}{n-1} \sum_{i=1}^n c_i (b_j+a_i)^{\frac{q-1}{d}} \\
&= \binom{n-1+\frac{q-1}{d}}{n-1} \sum_{i=1}^n c_i =0.
\end{align*}
Given Lemma \ref{lem:multiplicity}, we conclude that each of $b_1,b_2, \ldots b_r$ is a root of $f$ with multiplicity at least $n-1$, and each of $b_{r+1},b_{r+2}, \ldots b_m$ is a root of $f$ with multiplicity at least $n$.
Therefore
$$
r(n-1)+(m-r)n= mn-r \leq \operatorname{deg}f =\frac{q-1}{d}.
$$
\end{proof}
\begin{rem}\label{rem:Paley}
The proof of Theorem~\ref{thm:main} is inspired by the arguments in \cite{HP}, as well as the arguments in \cite{Yip1, Y21} related to the clique number of (generalized) Paley graphs. Recall that if $q \equiv 1 \pmod {2d}$, then a subset $C \subset \mathbb{F}_q$ in the $d$-Paley graph over $\mathbb{F}_q$ is a clique if and only if $C-C \subset S_d \cup \{0\}$. Let $C$ be a maximum clique in the $d$-Paley graph over $\mathbb{F}_q$; by taking $A$ to be a subset of $C$ with the desired condition on the binomial coefficient and $B=-C$ in the Theorem~\ref{thm:main}, it recovers \cite[Theorem 1.6]{Yip1} and \cite[Theorem 5.8]{Y21} immediately, which are key ingredients in establishing the best-known upper bound on the clique number of Paley graphs \cite{Yip1} and generalized Paley graphs \cite{Y21} over finite fields. In particular, the estimate on the clique number of cubic Paley graphs by the author~\cite[Theorem 1.8]{Y21} has already improved the upper bound on the size of $A \subset \mathbb{F}_p$ such that $A-A=S_3 \cup \{0\}$ proved in the recent work of Wu and She \cite[Theorem 1.4]{WS23} (in fact the result by the author works for a general finite field $\mathbb{F}_q$), perhaps because this connection has been sometimes overlooked. On the other hand, although the work of Hanson and Petridis~\cite{HP} has received lots of attention in the study of Paley graphs, surprisingly, their work did not receive much attention in the study of additive decompositions.
\end{rem}
The following corollary is a simple consequence of Theorem~\ref{thm:main}.
\begin{cor}\label{cor:cora+b}
If $A,B \subset \mathbb{F}_q$ such that $A+B\subset S_d$ and $
\binom{|A|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\not \equiv 0 \pmod p,$
then
$|A||B|\leq \frac{q-1}{d}$.
\end{cor}
\begin{proof}
Since $0 \notin A+B$, it follows that $A \cap (-B)$ is empty. The corollary follows from Theorem~\ref{thm:main} immediately.
\end{proof}
As a special case, we deduce Corollary~\ref{cor:a+b} as a partial progress towards the generalized S\'{a}rk\"{o}zy's conjecture.
\begin{proof}[Proof of Corollary~\ref{cor:a+b}]
Let $A,B \subset \mathbb{F}_q$ such that $A+B \subset S_d$. Without loss of generality, we may assume that $|A| \leq |B|$. Since $$|A+B|\leq |S_d|=\frac{q-1}{d}<p,$$ Corollary~\ref{cor:CD} implies that
$$
2(|A|-1) \leq |A|+|B|-2 \leq |A+B|-1=\frac{q-1}{d}-1.
$$
It follows that
$$
\frac{q-1}{d}\leq |A|-1+\frac{q-1}{d}<\frac{3(q-1)}{2d}\leq p,
$$
and thus $
\binom{|A|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\not \equiv 0 \pmod p.$
Corollary~\ref{cor:cora+b} then implies that $|A||B| \leq \frac{q-1}{d}$. If we further assume that $A+B=S_d$, then $$|A||B| \geq |A+B|=|S_d|=\frac{q-1}{d},$$ which forces that $|A||B|=\frac{q-1}{d}$ and thus all sums $a+b$ are distinct.
\end{proof}
Next we take a closer look at the proof of Theorem~\ref{thm:main} and make new observations.
\begin{rem}
Suppose $A,B \subset \mathbb{F}_q$ such that $A+B=S_d$ and $
\binom{|A|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\equiv 0 \pmod p$. Then Theorem~\ref{thm:main} does not apply. However, we can still say something nontrivial from the proof of Theorem~\ref{thm:main}. In the proof of Theorem~\ref{thm:main}, the assumption on the binomial coefficient guarantees that the polynomial $f$ is nonzero, so that we can apply Lemma~\ref{lem:multiplicity} to obtain an upper bound on $|A||B|$ based on the degree of $f$. It is clear that without the condition on the binomial coefficient, we can still conclude the same upper bound if the polynomial $f$ is nonzero.
Indeed, we can say something stronger. We follow the same notations as in the proof of Theorem~\ref{thm:main}. Since $A+B=S_d$, we have $r=0$. The same computations would show that each element in $B$ is a root of $f$ with multiplicity at least $n=|A|$. However, since $
\binom{|A|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\equiv 0 \pmod p$, the degree of $f$ is strictly less than $\frac{q-1}{d}$. If $f$ is a nonzero polynomial, then Lemma~\ref{lem:multiplicity} implies that
$$
\frac{q-1}{d}=|S_d|=|A+B| \leq |A||B|\leq \deg f <\frac{q-1}{d},
$$
a contradiction. Therefore, the polynomial $f$ must be zero.
\end{rem}
We summarize the above discussions into the following proposition. Roughly speaking, it predicts that any supposed additive decompositions of $S_d$ must have a very rigid structure, providing some evidence for the generalized S\'{a}rk\"{o}zy's conjecture.
\begin{prop}\label{prop:structure}
Let $d \mid (q-1)$ such that $d>1$. If $A,B \subset \mathbb{F}_q$ such that $A+B=S_d$ with $|A|, |B| \geq 2$, then one of the following two situations happens:
\begin{itemize}
\item $|A||B|=|S_d|$, that is, all sums $a+b$ are distinct;
\item $|A||B|>|S_d|$. In this case, $|A|$ and $|B|$ must satisfy
$$
\binom{|A|-1+\frac{q-1}{d}}{\frac{q-1}{d}} \equiv 0 \pmod p, \quad \binom{|B|-1+\frac{q-1}{d}}{\frac{q-1}{d}} \equiv 0 \pmod p.
$$
Moreover, if we write $A=\{a_1,a_2, \ldots, a_n\}$ and $B=\{b_1, b_2, \ldots, b_m\}$, then
$$
\binom{n-1+\frac{q-1}{d}}{j} \cdot \sum_{i=1}^n c_i a_i^{j}=0, \quad \binom{m-1+\frac{q-1}{d}}{\ell} \cdot \sum_{k=1}^m d_k b_k^{\ell}=0
$$
for all $0 \leq j<n-1+\frac{q-1}{d}$ and $0 \leq \ell<m-1+\frac{q-1}{d}$, and
$$
\sum_{i=1}^n c_i a_i^{n-1+\frac{q-1}{d}}=1, \quad \sum_{k=1}^m d_k b_k^{m-1+\frac{q-1}{d}}=1,
$$
where $c_i$'s and $d_k$'s are uniquely determined by the following systems of linear equations:
\begin{equation} \label{system111}
\left\{
\TABbinary\tabbedCenterstack[l]{
\sum_{i=1}^n c_i a_i^j=0, \quad 0 \leq j \leq n-2\\\\
\sum_{i=1}^n c_i a_i^{n-1}=1\\\\
\sum_{k=1}^m d_k b_k^\ell=0, \quad 0 \leq \ell \leq m-2\\\\
\sum_{k=1}^m d_k b_k^{m-1}=1
}\right.
\end{equation}
\end{itemize}
\end{prop}
\begin{rem}\label{rem:structure}
Following the notations used in Proposition~\ref{prop:structure}, suppose that $A+B=S_d$ with $|A||B|>|S_d|$. Proposition~\ref{prop:structure} implies that there are lots of generalized Vandermonde matrices associated with $A$ that are singular. In particular, for each $n \leq j_0<n-1+\frac{q-1}{d}$ such that $\binom{n-1+\frac{q-1}{d}}{j_0} \not \equiv 0 \pmod p$, we must have $\sum_{i=1}^n c_i a_i^{j_0}=0$, and thus the generalized Vandermonde matrix
$$(a_i^j)_{1 \leq i \leq n, j \in \{0,1, \ldots, n-2\} \cup \{j_0\}}$$ must be singular (for otherwise, all $c_i$ must be 0). For such a $j_0$, we can use Jacobi's bialternant formula to compute the determinant of the above generalized Vandermonde matrix and it follows that
$$0=s_{(j_0-(n-1),0, 0, \ldots, 0)}(a_1,a_2, \ldots, a_n),
$$
where $s$ is the Schur polynomial, and thus
$$
h_{j_0-(n-1)}(a_1,a_2, \ldots, a_n)=\sum_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_{j_0-(n-1)}\leq n} a_{i_1}a_{i_2} \cdots a_{i_{j_0-(n-1)}}=0
$$
where $h_{j_0-(n-1)}$ is the complete homogeneous symmetric polynomial of degree $j_0-(n-1)$ (see for example \cite[Chapter 4]{S03}). This observation, together with some basic properties of symmetric polynomials, allows us to predict the algebraic structure of $A$, as well as eliminate certain sizes of $A$.
\end{rem}
\begin{ex}
Let $q=p^2$, where $p$ is a sufficiently large prime. Assume that $A, B \subset \mathbb{F}_q$ such that $A+B=S_2$ with $|A|, |B| \geq 2$ and $|A||B|>|S_2|$. Then we can use Proposition~\ref{prop:structure} to obtain some nontrivial information of $|A|$ and $|B|$. Without loss of generality, assume that $|A| \leq |B|$. Note that the $p$-adic expression of $\frac{q-1}{2}$ is simply $(\frac{p-1}{2}, \frac{p-1}{2})_p$. Since $\binom{|A|-1+\frac{q-1}{2}}{\frac{q-1}{2}} \equiv 0 \pmod p$, we must have $|A|\geq \frac{p+3}{2}$. On the other hand, Lemma~\ref{lem:ub} implies that $|A||B|< q=p^2$ and thus $|B| \leq 2(p-3)$. Since $\binom{|B|-1+\frac{q-1}{2}}{\frac{q-1}{2}} \equiv 0 \pmod p$, we must have $|B|=ap+b$, where $b \in \{0, \frac{p+3}{2}, \frac{p+5}{2}, \ldots, p-1\}$. Thus, there are two possibilities:
\begin{itemize}
\item $\frac{3p+3}{2}\leq |B| \leq 2p-6$, and $\frac{p+3}{2}\leq |A|<\frac{2p}{3}$.
\item $\sqrt{\frac{p^2-1}{2}}<|B|\leq p$, and $\frac{p+3}{2} \leq |A| \leq |B|$.
\end{itemize}
In particular, these estimates improves the best-known estimate that $\min \{|A|, |B|\} \geq (2+o(1)) \frac{p \log 2}{8 \log p}$ \cite[Lemma 5.1]{S13} under the assumption that $|A||B|>|S_2|$. One can also compare our bounds with the best-known estimates on $|A|$ and $|B|$ over prime fields by Chen and Xi \cite{CX22}.
\end{ex}
\section{Proof of Theorem~\ref{thm:A+A}}
In this section, we study additive decompositions of the special form $A+A$.
\begin{proof}[Proof of Theorem~\ref{thm:A+A}]\label{sec:A+A}
Let $q=p^n$, and let $(d,q)$ be a good pair. For the sake of contradiction, assume that $A+A=S_d$ for some $A \subset \mathbb{F}_q$. Then Lemma~\ref{lem:ub} implies that $|A|<\sqrt{q}=p^{n/2}$. On the other hand, since $a+a'=a'+a$ for each $a,a' \in A$, it follows that
\begin{equation}\label{eq/2}
\frac{q-1}{d}=|S_d|=|A+A|\leq \frac{|A|^2+|A|}{2}.
\end{equation}
If $\frac{q-1}{d} \leq \frac{2p}{3}$ (in particular, if $q=p$ is a prime), then by Corollary~\ref{cor:a+b}, $|A|^2=\frac{q-1}{d}$. Thus, inequality~\eqref{eq/2} implies that $|A|=1$. It follows that $|S_d|=|A+A|=1$ and thus $d=q-1$, violating our assumption.
Next, we assume that $q$ is a proper prime power. Let $k$ be the unique integer such that $p^k \leq |A|-1<p^{k+1}$. Since $|A|<p^{n/2}$, it follows that $k<n/2$. Thus we can write $$|A|-1=(c_k, c_{k-1}, \ldots, c_1,c_0)_p$$ in base-$p$, that is, $|A|-1=\sum_{i=0}^k c_ip^i$ with $0 \leq c_i \leq p-1$ for each $0 \leq i \leq k$ and $c_k \geq 1$. Also write $$\frac{q-1}{d}=(e_n, e_{n-1}, \ldots, e_1,e_0)_p$$ in base-$p$; note that it is possible that $e_n=0$.
The key idea is to apply Corollary~\ref{cor:cora+b}. To do so, we take a subset $A'$ of $A$ such that
\begin{equation}\label{binomial}
\binom{|A'|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\not \equiv 0 \pmod p,
\end{equation}
then Corollary~\ref{cor:cora+b} implies that $|A||A'|\leq \frac{q-1}{d}$. To derive the contradiction, in view of inequality~\eqref{eq/2}, it suffices to show that we can pick $A'$ so that the inequality $|A'|> \frac{|A|+1}{2}$ and equation~\eqref{binomial} both hold. Also note that if $|A'|=\frac{|A|+1}{2}$, then we must have $|A||A'|=\frac{q-1}{d}$.
Next, we consider the other three sufficient conditions for a good pair $(d,q)$ separately:
(1) Assume that $e_j \leq \frac{p-1}{2}$ for each $j<n/2$. Since $k<n/2$, we have $e_j \leq \frac{p-1}{2}$ for each $0 \leq j \leq k$. We construct a desired $A'$ according to the following two cases:
\begin{itemize}
\item
If $c_k\leq \frac{p-1}{2}$, then we can pick $A'$ such that $|A'|-1=c_kp^k$ so that $$c_k+e_k\leq \frac{p-1}{2}+\frac{p-1}{2} \leq p-1.$$
Thus, Lucas' theorem implies that
$$
\binom{|A'|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\equiv \prod_{j=k+1}^n \binom{e_j}{e_j} \cdot \binom{c_k+\frac{p-1}{d}}{\frac{p-1}{d}} \cdot \prod_{j=0}^{k-1} \binom{e_j}{e_j}\not \equiv 0 \pmod p,
$$
In this case, $|A|\leq (c_k+1)p^k$ and thus
$$
|A'|=c_kp^k+1= \frac{2c_kp^k+2}{2}>\frac{(c_k+1)p^k+1}{2}\geq \frac{|A|+1}{2}
$$
by the assumption that $c_k \geq 1$.
\item
If $c_k>\frac{p-1}{2}$, then we can pick $A'$ such that $|A'|-1=\frac{p^{k+1}-1}{2}$ so that
$$
\binom{|A'|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\equiv \prod_{j=k+1}^n \binom{e_j}{e_j} \cdot \prod_{j=0}^{k} \binom{\frac{p-1}{2}+e_j}{e_j}\not \equiv 0 \pmod p,
$$
where we used Lucas' theorem. Note that in this case, by the assumption, $|A|-1<p^{k+1}$ so $|A|\leq p^{k+1}$. It follows that $|A'|=\frac{p^{k+1}+1}{2} \geq \frac{|A|+1}{2}$. Thus, it suffices to rule out the case of equality. However, if $|A'|=\frac{|A|+1}{2}$, then we must have $|A|=p^{k+1}$ and $|A||A'|=\frac{q-1}{2}$, implying that $q-1=p^{2k+2}+p^{k+1}$, which is clearly impossible by considering modulo $p$ on both sides of the equation.
\end{itemize}
(2) Assume that $n=2r+1$ with $r \geq 1$, $d\leq 2p-2$, and $e_r\leq p-1-\lceil \sqrt{p} \rceil/2$. Since $d\leq 2p-2$, inequality~\eqref{eq/2} implies that
$$
p^{2r}+p^r<\sum_{i=0}^{2r} p^i=\frac{p^{2r+1}-1}{p-1} \leq \frac{2(q-1)}{d}\leq |A|^2+|A| \implies p^r<|A|.
$$
Thus, $p^r \leq |A|-1<\sqrt{q}-1=\sqrt{p} \cdot p^r-1$. It follows that $k=r$ and $1\leq c_r\leq \lfloor \sqrt{p} \rfloor$.
\begin{itemize}
\item If $c_r+e_r\leq p-1$, then we can pick $A'$ such that $|A'|-1=c_rp^r$. Similar to the analysis in case (1), we can show that the inequality $|A'|> \frac{|A|+1}{2}$ and equation~\eqref{binomial} both hold.
\item If $c_r+e_r \geq p$, then we can pick $A'$ such that $|A'|-1=(p-1-e_r)p^r$. We can verify that equation~\eqref{binomial} holds in a similar way using Lucas' theorem. By the assumption, $e_r\leq p-1-\lceil \sqrt{p} \rceil/2$. It follows that $p-1-e_r \geq \lceil \sqrt{p} \rceil/2$ and thus
$$|A'| \geq (\lceil \sqrt{p} \rceil/2)p^r+1>\frac{p^{r+1/2}}{2}+1=\frac{\sqrt{q}+2}{2}>\frac{|A|+1}{2}.
$$
\end{itemize}
(3) Assume that $n=2r$, $d \leq 2p^2$, and $e_{r-1} \leq (p-3)/2$.
Similar to the analysis in case (2), we can show that $p^{r-1} \leq |A|-1<\sqrt{q}=p \cdot p^{r-1}$ so that $k=r-1$. Let $c_{r-1}'=\min \{c_{r-1}, \frac{p+1}{2}\}$ so that $e_{r-1}+c_{r-1}'\leq p-1$. Take a subset $A'$ of $A$ with $|A'|-1=c_{r-1}' \cdot p^{r-1}$. Similar to the analysis in cases (1) and (2), we can verify that the inequality $|A'|> \frac{|A|+1}{2}$ and equation~\eqref{binomial} both hold.
\end{proof}
The following corollary is a special case of Corollary~\ref{cor:A+A}.
\begin{cor}
Let $q=p^n$ with $q \equiv 1 \pmod d$ and $d \geq 2$. Assume that $p \equiv 1 \pmod d$ holds, or $p \equiv -1 \pmod d$ and $4 \mid n$ both hold. Then there is no additive decomposition $S_d=A+A$ for any $A \subset \mathbb{F}_q$.
\end{cor}
The next example shows that it is necessary to assume that $d \geq 2$ in Theorem~\ref{thm:A+A}, and have some additional assumptions when $q$ is a proper prime power.
\begin{ex}\label{ex:ex1}
Let $p\geq 7$ be an odd prime, and $n$ be a positive integer. We claim that $\mathbb{F}_{p^n}^*$ can be decomposed as $A+A$. Indeed, since $\mathbb{F}_{p^n}$ is isomorphic to the $n$-dimensional space over $\mathbb{F}_p$, it suffices to show that $\mathbb{F}_{p}^n \setminus \{\mathbf{0}\}$ can be decomposed as $A+A$, where $A \subset \mathbb{F}_{p}^n$. We can take
$$
A=\bigg(\bigg\{0,1,2, \ldots, \frac{p-3}{2}\bigg\} \cup \bigg\{\frac{p+1}{2}\bigg\}\bigg)^n \setminus \{\mathbf{0}\}
$$
to achieve this purpose. Also note that $|A|=(\frac{p+1}{2})^n-1$ and thus $|A|^2/p^n$ could be arbitrarily large.
Let $q=p^n$, with $p \geq 7$ and $n \geq 2$. Let $k$ be a proper divisor of $n$ and let $d=\frac{q-1}{p^k-1}$. Then $S_d=\mathbb{F}_{p^k}^*$ can be decomposed as $A+A$ by the above argument. This example shows that when $q$ is a proper prime power, one needs to impose some conditions on the pair $(d,q)$ so that Theorem~\ref{thm:A+A} holds. It also shows that the condition on the binomial coefficient in Theorem~\ref{thm:main} cannot be dropped, and shows that Corollary~\ref{cor:a+b} does not extend to all pairs $(d,q)$.
\end{ex}
We end the section by comparing our new results with existing results.
\begin{rem}\label{rmk:A+A}
Shkredov's proof \cite{S14} that $A+A \neq S_2$ for any $A \subset \mathbb{F}_p$ with $p\geq 5$ is elegant and remarkable. In fact, the identical proof shows that $A+A \neq S_2$ for any $A \subset \mathbb{F}_q$, where $q \geq 5$ is an odd prime power. Here we sketch his proof. From inequality~\eqref{eq:double}, we can deduce that $|A|\leq \sqrt{q} (1-\frac{|A|}{q})$, equivalently, $q+\frac{|A|^2}{q} \geq |A|^2+2|A|$. On the other hand, $|A+A|=|S_2|$ implies that $|A|^2+|A| \geq q-1$ (see inequality~\eqref{eq/2}). It is easy to deduce a contradiction by comparing the above two inequalities. Unfortunately, it seems his proof does not extend to $S_d$ for $d \geq 3$. Nevertheless, Shkredov also showed a similar result for the restricted sumset $A \hat{+} A = \{ a + b: a,b \in A, \, a \neq b \}$, for which our techniques do not seem to apply.
\end{rem}
\begin{rem}
In fact, in his paper, Shkredov~\cite{S14} established a much stronger result, namely, $A+A$ is impossible to be too close to $S_2$. We remark that under the same assumption on the pair $(d,q)$ in Corollary~\ref{cor:a+b}, Corollary~\ref{cor:a+b} implies that: if $A+A \subset S_d$, then $|A+A|\leq (\frac{1}{2}+o(1))|S_d|$. In particular, this improves and generalizes Shkredov's result \cite[Theorem 3.2]{S14}. The proof of Theorem~\ref{thm:A+A} could be also adapted to estimate the maximum size of $A+A$ (or the maximum size of $A$) provided that $A+A \subset S_d$.
\end{rem}
\begin{rem}
Wu and She \cite{WS23} recently studied the additive decomposition of $S_3$. Let $p \equiv 1 \pmod 3$. They showed that if $A+B=S_3$ holds for some $A, B \subset \mathbb{F}_p$ with $|A|=|B|$, then $\sqrt{\frac{p-1}{3}} \leq |A| \leq \sqrt{p}$ \cite[Theorem 1.3]{WS23}. Note that Corollary~\ref{cor:a+b} shows that we must have $|A|=|B|=\sqrt{\frac{p-1}{3}}$. They also had an improved lower bound on $|A|$ if $A=B$, however, this case has been ruled out by Corollary~\ref{cor:A+A}.
\end{rem}
\begin{rem}
The possibility of writing $S_d \cup \{0\}$ as the difference set $A-A$ have also been studied extensively; see for example \cite{S16, LS17, S18, HP, Y21,Y22, Yip1}. As mentioned in Remark~\ref{rem:Paley}, this problem is also closely related to the clique number of generalized Paley graphs. More generally, this problem is related to the Paley graph conjecture (see for example \cite[Section 2.2]{Y22}), which is widely open.
Murphy, Petridis, Roche-Newton, Rudnev, and Shkredov \cite[Theorem 16]{MPRRS19} showed that multiplicative subgroups of size less than $p^{6/7-o(1)}$ cannot be represented in the form $(A-A) \setminus \{0\}$ for any $A \subset \mathbb{F}_p$. Unfortunately, it seems the proof techniques in Theorem~\ref{thm:A+A} fail to extend to $A-A$. Nevertheless, the proof of Corollary~\ref{cor:a+b} can be modified slightly to show that under the same condition on the pair $(d,q)$, if $A \subset \mathbb{F}_q$ such that $A-A=S_d \cup \{0\}$, then $|A|^2-|A|=|S_d|$. In particular, this leads to new families of pairs $(d,q)$ such that $A-A \neq S_d \cup \{0\}$ for any $A \subset \mathbb{F}_q$, which is beyond \cite{S16} and \cite[Corollary 1.6]{HP}.
\end{rem}
\section{No nontrivial ternary decomposition}\label{sec:A+B+C}
In this section, we establish various sufficient conditions on the pair $(d,q)$ so that $S_d \subset \mathbb{F}_q$ admits no nontrivial ternary decomposition.
\begin{proof}[Proof of Theorem~\ref{thm:prime3}]
Assume that there exists a prime $p$ and a multiplicative subgroup $G$ of $\mathbb{F}_p$, such that $G$ admits a nontrivial additive decomposition: $G=A+B+C$ with $|A|,|B|, |C| \geq 2$. Then we can write $G$ in three different ways: $$A+(B+C), B+(C+A), C+(A+B).$$ Applying Theorem~\ref{thm:lball} to each sumset, we have $$|A|, |B|, |C| \gg |G|^{1/2+o(1)}.$$ We can also apply Corollary~\ref{cor:a+b} to obtain that
$$
|A+B||C|,|B+C||A|,|C+A||B|\ll |G|.
$$
Therefore, from Lemma~\ref{lem:addcomb} and the fact $|A+B+C|=|G|$, we have
$$
|G|^2 |A||B||C|\ll |A+B+C|^2 |A||B||C| \leq (|A+B||C|)(|B+C||A|)(|C+A||B|) \ll |G|^3.
$$
It follows that
$$
|G|^{3/2+o(1)} \ll |A||B||C| \ll |G|,
$$
that is, $|G|\ll 1$, where the implicit constant is absolute. This completes the proof of the theorem.
\end{proof}
\begin{rem}
When $d$ is fixed, and $p \equiv 1 \pmod d$ is sufficiently large, there is an alternative way to show that there is no nontrivial ternary decomposition of $S_d$. Suppose $A+B+C=S_d$, where $A, B, C \subset \mathbb{F}_p$ and $|A|, |B|, |C| \geq 2$. Then Lemma~\ref{lem:maxmin} implies that $|A|, |B|, |C| \gg \sqrt{p}$. However, when $p$ is sufficiently large, this would violate a ternary character sum estimate proved by Hanson \cite[Theorem 1]{H17} that
$$
\sum_{a \in A, b \in B, c \in C} \chi(a+b+c)=o(|A||B||C|),
$$
where $\chi$ is a multiplicative character with order $d$.
\end{rem}
Next, we modify the proof of Theorem~\ref{thm:prime3} to establish Theorem~\ref{thm:dq}.
\begin{proof}[Proof of Theorem~\ref{thm:dq}]
The proof is similar to that of Theorem~\ref{thm:prime3}. Let $\epsilon$ be fixed. Assume that $(d,q)$ is a pair with $d \leq q^{1/10-\epsilon}$, such that $S_d$ can be decomposed into $S_d=A+B+C$ for some $A,B,C \subset \mathbb{F}_q$ with $|A|,|B|, |C| \geq 2$. Applying Lemma~\ref{lem:maxmin} and Lemma~\ref{lem:ub} to each of the sumsets $A+(B+C), B+(C+A), C+(A+B)$, we have
$$
|A|, |B|, |C| \gg \frac{\sqrt{q}}{d}, \quad |A||B+C|,|B+C||A|, |C+A||B| <q.
$$
Therefore, the same argument as in the proof of Theorem~\ref{thm:prime3} gives the estimate
$$
\frac{q^{3.5}}{d^5} \ll |A+B+C|^2 |A||B||C| \leq (|A+B||C|)(|B+C||A|)(|C+A||B|) \ll q^3.
$$
It follows that $d \gg q^{1/10}$. Together with the assumption $d\leq q^{1/10-\epsilon}$, we conclude that $q\ll 1$, where the implicit constant depends only on $\epsilon$. This completes the proof of the theorem.
\end{proof}
The next example provides a counterexample when the conditions in Theorem~\ref{thm:prime3} and Theorem~\ref{thm:dq} are weakened.
\begin{ex}\label{ex:ex2}
Let $p\geq 5$ be a prime, and $n$ be a positive integer. We claim that $\mathbb{F}_{p^n}^*$ can be decomposed as $A+B+C$ nontrivially. Indeed, since $\mathbb{F}_{p^n}$ is isomorphic to the $n$-dimensional space over $\mathbb{F}_p$, it suffices to show that $\mathbb{F}_{p}^n \setminus \{\mathbf{0}\}$ can be decomposed as $A+B+C$, where $A,B,C \subset \mathbb{F}_{p}^n$. We can take
$$
A=\{0,1\}^n, \quad B=\{0,1\}^n, \quad \text{and } C=\{0,1,2, r+3, r+6, \ldots, p-3\}^n \setminus \{\mathbf{0}\}
$$
to achieve this purpose, where $p \equiv r \pmod 3$ with $r \in \{1,2\}$.
Let $q=p^n$, with $p \geq 5$ and $n \geq 2$. Let $k$ be a proper divisor of $n$ and let $d=\frac{q-1}{p^k-1}$. Then $S_d=\mathbb{F}_{p^k}^*$ can be decomposed as $A+B+C$ nontrivially by the above argument. Note that $|S_d|$ could be arbitrarily large, which shows that Theorem~\ref{thm:prime3} fails to extend to all finite fields. It also shows that in the statement of Theorem~\ref{thm:dq}, it is necessary to impose the condition that $q$ is sufficiently large compared to $d$; in particular, the exponent $\frac{1}{10}$ cannot be replaced by any constant which is greater than $\frac{1}{2}$. It is interesting to explore if the exponent $\frac{1}{10}$ could be improved. Under extra assumptions, we improve the exponent to roughly $\frac{1}{4}$ in Theorem~\ref{thm:deltagood}.
\end{ex}
Next, we combine the ideas used in the previous discussions to prove Theorem~\ref{thm:deltagood}.
\begin{proof}[Proof of Theorem~\ref{thm:deltagood}]
Let $\epsilon$ be fixed. Let $M$ be the implicit constant from inequality~\eqref{eq:max}; note $M$ only depends on $\epsilon$. Let $P=M^2+1$.
Let $q=p^n$ with $p>P$. Assume that $(d,q)$ is a $\delta$-good pair with $\delta \in (0,1)$ and $d \leq q^{1/4-\epsilon}\delta^{3/2}$, such that $S_d$ can be decomposed into $S_d=A+B+C$ for some $A,B,C \subset \mathbb{F}_q$ with $|A|,|B|, |C| \geq 2$. Then we can write $S_d$ in three different ways: $A+(B+C), B+(C+A), C+(A+B)$. Applying Lemma~\ref{lem:maxmin} to each of these sumsets, we have
\begin{equation}\label{eq:lb}
|A|, |B|, |C| \gg \frac{\sqrt{q}}{d},
\end{equation}
and
$$
|A|, |B|, |C| \leq M\sqrt{q}<p^{(n+1)/2}.
$$
Let $k$ be the unique integer such that $p^k \leq |A|-1<p^{k+1}$. By the assumption, $k<(n+1)/2$. Then we can write $$|A|-1=(c_k, c_{k-1}, \ldots, c_1,c_0)_p$$ in base-$p$, where $c_k\geq 1$. Also write $$\frac{q-1}{d}=(e_n, e_{n-1}, \ldots, e_1,e_0)_p$$ in base-$p$, where $e_n$ is possibly $0$.
Since $(d,q)$ is a $\delta$-good pair, we have $e_k\leq \lfloor(1-\delta)(p-1)\rfloor$. Let $A'$ be a subset of $A$ such that $|A'|-1=c_k' p^k$, where $c_k'=\min \{c_k, \lceil \delta (p-1) \rceil\}$ so that
$$c_k'+e_k\leq \lceil \delta (p-1) \rceil +\lfloor(1-\delta)(p-1)\rfloor= p-1.$$
Then Lucas' theorem implies that
$$
\binom{|A'|-1+\frac{q-1}{d}}{\frac{q-1}{d}}\equiv \prod_{j=k+1}^n \binom{e_j}{e_j} \cdot \binom{c_k'+e_k}{e_k} \cdot \prod_{j=0}^{k-1} \binom{e_j}{e_j}\not \equiv 0 \pmod p.
$$
Similar to the proof of Theorem~\ref{thm:A+A}, it is easy to verify that $$|A'| \gg \min \{|A|/2,\delta|A|\} \gg \delta |A|$$ from our construction. Since $A' +(B+C) \subset A+B+C=S_d$, Corollary~\ref{cor:a+b} implies that
$$
|A||B+C| \ll \frac{|A'||B+C|}{\delta}\ll \frac{q}{d\delta}.
$$
Using a similar argument, we can also show that
$$
|A+B||C|\ll \frac{q}{d\delta}, \quad |C+A||B|\ll \frac{q}{d\delta}.
$$
Therefore, from Lemma~\ref{lem:addcomb} and the fact $|A+B+C|=|S_d|$, we have
$$
\frac{q^2}{d^2} |A||B||C|\ll |A+B+C|^2 |A||B||C| \leq (|A+B||C|)(|B+C||A|)(|C+A||B|) \ll \frac{q^3}{d^3\delta^3}.
$$
Combining inequality~\eqref{eq:lb}, we have
$$
\frac{q^{3/2}}{d^3} \ll |A||B||C| \ll \frac{q}{d \delta^3},
$$
that is, $d \gg q^{1/4} \delta^{3/2}$. Together with the assumption $d\leq q^{1/4-\epsilon} \delta^{3/2}$, we conclude that $q\ll 1$, where the implicit constant depends only on $\epsilon$. This completes the proof of the theorem.
\end{proof}
Finally, we deduce a simple corollary of Theorem~\ref{thm:deltagood}.
\begin{cor}\label{cor:order1}
There is an absolute constant $P$, such that for pair of prime power $q=p^n$ and $d$ such that $n \geq 5$, $p>P$, and $d \mid (p-1)$, we have $S_d \neq A+B+C$ for any $A,B,C \subset \mathbb{F}_q$ with $|A|,|B|,|C| \geq 2$.
\end{cor}
\begin{proof}
Take $\epsilon=\frac{1}{20}$ so that $d \leq p-1<p\leq q^{1/5}=q^{1/4-\epsilon}$. Note that all digits in the base-$p$ representation of $\frac{q-1}{d}$ are equal to $\frac{p-1}{d}\leq \frac{p-1}{2}$. So the pair $(d,q)$ is $\delta$-good with $\delta \geq \frac{1}{2}$. The corollary follows from Theorem~\ref{thm:deltagood}.
\end{proof}
\textbf{Note added.} A slightly different version of the manuscript
has been submitted to a journal for publication on March 2023. Very recently, a related paper by Wu, Wei, and Li \cite{WWL23} appeared in arXiv. We remark that Theorem~\ref{thm:prime3} is better than Theorem 1.2 in their preprint.
\end{document} | math | 46,343 |
\begin{document}
\title[Models for Tetrablock Contractions]{Dilation theory and functional models for tetrablock contractions}
\author[J. A. Ball]{Joseph A. Ball}
\address{Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA} \email{[email protected]}
\author[H. Sau]{Haripada Sau}
\address{Department of Mathematics, Indian Institute of Science Education and Research Pune, Maharashtra 411008}
\email{[email protected]; [email protected]}
\subjclass{Primary: 47A13. Secondary: 47A20, 47A25, 47A56, 47A68, 30H10}
\keywords{commutative contractive operator-tuples; functional model; unitary dilation; isometric lift; spectral set; pseudo-commutative
contractive lift}
\thanks{The research of the second named author was supported by DST-INSPIRE Faculty Fellowship DST/INSPIRE/04/2018/002458.}
\begin{abstract}
A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary ${\mathcal U}$, i.e.,
$T^n = P_{\mathcal H} {\mathcal U}^n|{\mathcal H}$ for all $n =0,1,2,\dots$ . A more general multivariable setting
for these ideas is the setup where (i) the unit disk is replaced by a domain $\Omega$ contained in ${\mathbb C}^d$, (ii) the contraction
operator $T$ is replaced by an $\Omega$-contraction, i.e., a commutative operator $d$-tuple ${\mathbf T} = (T_1, \dots, T_d)$ on a Hilbert space ${\mathcal H}$
such that $\| r(T_1, \dots, T_d) \|_{{\mathcal L}({\mathcal H})} \le \sup_{\langlem \in \Omega} | r(\langlem) |$ for all rational functions with no singularities in
$\overline{\Omega}$ and the unitary operator ${\mathcal U}$ is replaced by an $\Omega$-unitary operator tuple, i.e., a commutative operator $d$-tuple
${\mathbf U} = (U_1, \dots, U_d)$ of commuting normal operators with joint spectrum contained in the distinguished boundary $b\Omega$
of $\Omega$. For a given domain $\Omega \subset {\mathbb C}^d$, the {\em rational dilation question} asks: given an $\Omega$-contraction ${\mathbf T}$ on ${\mathcal H}$,
is it always possible to find an $\Omega$-unitary
${\mathbf U}$ on a larger Hilbert space ${\mathcal K} \supset {\mathcal H}$ so that, for any $d$-variable rational function without singularities in $\overline{\Omega}$, one can recover
$r(T)$ as $r(T) = P_{\mathcal H} r({\mathbf U})|_{\mathcal H}$.
We focus here on the case where $\Omega = {\mathbb E}$, a domain in ${\mathbb C}^3$ called the {\em tetrablock}.
(i) We identify a complete set of unitary invariants for a ${\mathbb E}$-contraction $(A,B,T)$ which can then be used to write down a functional model for $(A,B,T)$,
thereby extending earlier results only done for a special case,
(ii) we identify the class of {\em pseudo-commutative ${\mathbb E}$-isometries} (a priori slightly larger than the class of ${\mathbb E}$-isometries) to which any
${\mathbb E}$-contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a ${\mathbb E}$-isometric lift
$(V_1, V_2, V_3)$ of a special type for a ${\mathbb E}$-contraction $(A,B,T)$.
\end{abstract}
\dedicatory{\textit{Dedicated to the memory of J\"org Eschmeier: a fine mentor, researcher, and colleague}}
\maketitle
\section{Introduction}
Suppose that we are given a commutative tuple ${\mathbf T} = (T_1, \dots, T_d)$ of operators on a Hilbert space ${\mathcal H}$ together with a bounded domain $\Omega$ contained
in $d$-dimensional Euclidean space ${\mathbb C}^d$. We now recall the notion of $\Omega$ being a {\em spectral set} and $\Omega$ being a {\em complete spectral
set} for the commutative $d$-tuple ${\mathbf T}$ and refer to the original paper of Arveson \cite{Arveson} for additional details on this and the related matters which follow.
We say that $\Omega$ is a {\em spectral set} for ${\mathbf T}$ if it is the case that
\begin{equation} \langlebel{sp-ineq}
\| r({\mathbf T}) \|_{_{{\mathcal B}({\mathcal H})}} \le \sup_{\langlem \in \Omega} |r(\langlem)| \text{ for } r \in \operatorname{Rat}(\Omega)
\end{equation}
where we set $R(\Omega)$ equal to the space of all $d$-variable scalar-valued rational functions $r$ having no singularities in $\overline{\Omega}$
and ${\mathcal B}({\mathcal H})$ equal to the Banach algebra of all bounded linear operators on ${\mathcal H}$. Here $r({\mathbf T})$ can be defined via the functional calculus given by
$$
r({\mathbf T}) = p(T_1, \dots, T_d) q(T_1, \dots, T_d)^{-1}
$$
where $(p,q)$ is a coprime pair of $d$-variable polynomials such that $r = p/q$.
In analogy with what happens for the case $\Omega$ equal to
the unit disk ${\mathbb D}$ (see the discussion below), we say simply that ${\mathbf T}$ is an $\Omega$-contraction if it is the case that $\Omega$ is a spectral set
for ${\mathbf T}$.
We say that $\Omega$ is a {\em complete spectral set} for ${\mathbf T}$ if \eqref{sp-ineq} continues to hold when one substitutes matrix rational functions $R(\langlem)
= [ r_{ij}(\langlem)]_{i,j=1, \dots, n} = [ p_{ij}(\langlem) q_{ij}(\langlem)^{-1}]$ having no singularities in $\overline{\Omega}$:
$$
\| R({\mathbf T}) \|_{_{{\mathcal B}({\mathcal H}^n)}} \le \sup_{\langlem \in \Omega} \| R(\langlem) \|_{_{{\mathbb C}^{n \times n}}}.
$$
The seminal result of Arveson (see \cite{Arveson}) is that $\Omega$ is a complete spectral set for ${\mathbf T}$ if and only if there is a commutative $d$-tuple of normal
operators ${\mathbf N} = (N_1, \dots, N_d)$ on a larger Hilbert space $\widetilde {\mathcal K} \supset {\mathcal H}$ with joint spectrum contained in the distinguished boundary
$b\Omega$ of $\Omega$ (in which case we say that ${\mathbf N}$ is a {\em $\Omega$}-unitary for short)
so that, for any rational function $r$ with no singularities in $\overline\Omega$ as above,
it is the case that $r({\mathbf T})$ on ${\mathcal H}$ can be represented as the compression of $r({\mathbf N})$ to ${\mathcal H}$, i.e.,
$$
r({\mathbf T}) = P_{\mathcal H} r({\mathbf N}) |_{\mathcal H}
$$
where $P_{\mathcal H}$ is the orthogonal projection of ${\mathcal K}$ onto ${\mathcal H}$. It is easy to see that a necessary condition for $\Omega$ to be a complete spectral set
for a given operator $d$-tuple ${\mathbf T}$ is that $\Omega$ be a spectral set for ${\mathbf T}$. The {\em rational dilation problem} for a given domain $\Omega$ is to determine
if the converse holds: {\em given $\Omega$, is it always the case that an operator tuple ${\mathbf T}$ having $\Omega$ as a spectral set in fact has $\Omega$
as a complete spectral set} (and hence then any ${\mathbf T}$ having $\Omega$ as a spectral set has a $d\Omega$-normal dilation ${\mathbf N}$)?
Let us mention that it is often convenient to reformulate the problem of existence of an {\em $\Omega$-unitary dilation} instead as the problem of
existence of a {\em $\Omega$-isometric lift} (see e.g.~the introduction of \cite{BS-failure}). Here we say that the operator tuple ${\mathbf V} = (V_1, \dots, V_d)$
on a Hilbert space ${\mathcal K}$ is a {\em $\Omega$-isometry} if ${\mathbf V}$ extends to a $\Omega$-unitary operator tuple ${\mathbf U} = (U_1, \dots, U_d)$
on a Hilbert spaces $\widetilde {\mathcal K} \supset {\mathcal K}$. We say that ${\mathbf V} = (V_1, \dots, V_d)$ on ${\mathcal K}$ is a {\em lift} of ${\mathbf T} = (T_1, \dots, T_d)$ on ${\mathcal H}$ if
${\mathcal H} \subset {\mathcal K}$ and $r({\mathbf V})^*|_{\mathcal H} =r({\mathbf T})^*$ for $r \in \operatorname{Rat}(\Omega)$, or equivalently, ${\mathbf V}$ is a {\em coextension} of ${\mathbf T}$ in the sense that
$$
P_{\mathcal H} r({\mathbf V})|_{\mathcal H} = r({\mathbf T}) \text{ and } r({\mathbf V}){\mathcal H}^\perp \subset {\mathcal H}^\perp \text{ for } r \in \operatorname{Rat}(\Omega).
$$
It suffices to consider only {\em minimal} $\Omega$-unitary dilations and {\em minimal} $\Omega$-isometric lifts. It is always the case that the restriction
of a $\Omega$-unitary dilation to the subspace $\bigvee_{r \in \operatorname{Rat}(\Omega)} r({\mathbf U}) {\mathcal H}$ gives rise to a minimal
$\Omega$-isometric lift, and conversely, the minimal $\Omega$-unitary extension of a minimal $\Omega$-isometric lift gives rise to a minimal $\Omega$-unitary
dilation for ${\mathbf T}$. Finally we point out that it is often convenient to be more flexible in the definition of an $\Omega$-isometric lift and of an $\Omega$-unitary dilation
by not insisting that ${\mathcal H}$ is a subspace of ${\mathcal K}$ or $\widetilde {\mathcal K}$ but rather allow an isometric identification map $\Pi \colon {\mathcal H} \to {\mathcal K}$
and $\widetilde \Pi \colon {\mathcal H} \to \widetilde {\mathcal K}$. Thus we say that the pair $(\Pi, {\mathbf V})$ is an an $\Omega$-isometric lift for ${\mathbf T}$ on ${\mathcal H}$ if
$\Pi \colon {\mathcal H} \to {\mathcal K}$ is an isometric embedding, ${\mathbf V}$ is $\Omega$-isometric on ${\mathcal K}$ and $r({\mathbf V})^* \Pi = \Pi r({\mathbf T})^*$ for $r \in \operatorname{Rat}(\Omega)$,
while $(\widetilde \Pi, {\mathbf U})$ is a $\Omega$-unitary dilation of ${\mathbf T}$ if $\widetilde \Pi \colon {\mathcal H} \to \widetilde {\mathcal K}$ is an isometric embedding,
${\mathbf U}$ is $\Omega$-unitary on $\widetilde {\mathcal K}$, and $\Pi^* r({\mathbf U}) \Pi = r({\mathbf T})$ for $ r \in \operatorname{Rat}(\Omega)$.
The motivating classical example for this setup is the case where $\Omega$ is the unit disk ${\mathbb D} \subset {\mathbb C}$. In this case, the distinguished
boundary $b{\mathbb D}$ of ${\mathbb D}$ is the same as the boundary $\partial {\mathbb D}$ which is the unit circle ${\mathbb T}$
and a $b {\mathbb D}$-normal operator is just a unitary operator. Since ${\mathbb D}$ is polynomially convex, it suffices to work with polynomials
rather than rational functions with no poles in $\overline{\mathbb D}$. By choosing the polynomial $p$ to be $p = \chi$ and
$\chi(\langlem) = \langlem$, we see that $\| T \| \le 1$ (i.e., that $T$ be a contraction) is necessary for ${\mathbb D}$ to be a spectral set for $T$. The fact that
this condition is also sufficient, i.e., that the inequality
$$
\| p(T) \| \le \sup_{\langlem \in {\mathbb D}} |p(\langlem)|
$$
holds for any contraction operator $T$ and polynomial $p$,
is a classical inequality known as von Neumann's inequality going back to \cite{vonN-Wold}. to show that ${\mathbb D}$ is a complete spectral set for any
contraction operator $T$, we may use the easier side of Arveson's theorem and show instead that any contraction operator $T$ has a ${\mathbb D}$-unitary dilation.
But for the case $\Omega = {\mathbb D}$, according to our conventions, a ${\mathbb D}$-unitary operator is just a unitary operator
$U$ (i.e., $U^*U = U U^* = I_{\widetilde {\mathcal K}}$). But any contraction operator $T$ on ${\mathcal H}$ dilating to a unitary operator $U$ on $\widetilde {\mathcal K} \supset {\mathcal H}$
is exactly the content of the Sz.-Nagy dilation theorem (see \cite[Chapter II]{Nagy-Foias}).
Over the ensuing decades there have been sporadic attempts to find other domains (both contained in ${\mathbb C}$ or more generally contained in ${\mathbb C}^d$)
for which one can settle the rational dilation question one way or the other (i.e., positively or negatively). Among single-variable domains (as observed in the
introduction of \cite{BS-failure} where precise references are given), it is known that rational dilation holds if $\Omega \subset {\mathbb C}$ is a simply connected domain
(simply use a conformal map to reduce to the disk case) or is doubly-connected, but fails if $\Omega$ has two or more holes (see \cite{AHR_Memoir, DM_JAMS}).
As for multivariable domains,
perhaps the first class to be understood are the polydisks ${\mathbb D}^d$ with $d \ge 2$: for $d = 2$ rational dilation holds due to the And\^o dilation theorem
\cite{ando} while for $d \ge 3$ rational dilation fails (see \cite{Parrott, Varopoulos}).
More recently the rational dilation problem has been investigated for other concrete multivariable domains originally discovered due to connections
with the $\mu$-synthesis problem in Robust Control Theory (see the original Doyle-Packard paper \cite{DoylePackard} as well as the book \cite{DP} for
a more expository treatment). We mention in particular the symmetrized bidisk
\begin{equation} \langlebel{symdisk}
\Gamma = \{ (s, p) \in {\mathbb C}^2 \colon s = (\langlem_1 + \langlem_2), \, p = \langlem_1 \langlem_2 \text{ for some } (\langlem_1, \langlem_2) \in {\mathbb D}^2 \}
\end{equation}
and a domain in ${\mathbb C}^3$ called the tetrablock and denoted by ${\mathbb E}$:
\begin{equation} \langlebel{tetrablock}
\mathbb E:=\left\{(a,b,\text{det}X): X=\begin{bmatrix} a & a' \\ b' & b \end{bmatrix}\text{ with }\lVert X \rVert <1\right\}.
\end{equation}
As might be expected, the domain $\Gamma$ behaves like ${\mathbb D}^2$ with respect to the rational dilation problem as both domains are contained in
${\mathbb C}^2$: specifically, rational dilation holds for the domain $\Gamma$ (see \cite{AglerYoung00, AglerYoung03, B-P-SR}) and there is a functional model
analogous to the Sz.-Nagy-Foias model for the disk case (see \cite{AglerYoung03, BPJOT}), at least for the pure case.
The situation of the rational dilation problem for the tetrablock ${\mathbb E}$ is less clear: there is a sufficient and a necessary condition for the existence of a
${\mathbb E}$-isometric lift of a certain form \cite{Tirtha14, BS-failure} but a definitive resolution of the problem in full generality remains elusive
(see \cite{BS-failure, PalFailure}).
However it is shown in \cite{SauNYJM} that, at least in the pure case, it is still possible to construct a functional representation of a pure
$\Gamma$-contraction as the compression to ${\mathcal H}$ of a certain lift triple $(A_\ell, B_\ell, T_\ell)$ which formally looks like an tetrablock isometry but is not
guaranteed to satisfy all of the required commutativity conditions.
A similar phenomenon holds for the case where $\Omega = {\mathbb D}^d$ with $d \ge 2$ (see \cite{BallSauDougVol}): for this case, as pointed out above,
there are indeed counterexamples to show that rational dilation fails, but there is nevertheless a weaker type of lift (called {\em pseudo-commutative
${\mathbb D}^d$-isometric lift)}) which generates a functional model for the given ${\mathbb D}^d$-contractive $d$-tuple ${\mathbf T} = (T_1, \dots, T_d)$ even when
rational dilation fails.
In this paper we focus on the case $\Omega = {\mathbb E}$. As was the case in \cite{Tirtha14}, the most definitive results are for the case of what we shall call
a \textbf{special tetrablock contraction}, i.e., a tetrablock contraction $(A,B,T)$ which has a tetrablock isometric lift $(V_1, V_2, V_3)$ such that $V_3 = V$
is a Sz.-Nagy-Foias minimal isometric lift for the single contraction operator $T$. As in \cite{Tirtha14}, we identify the additional commutativity conditions
\eqref{com=} which must be imposed on the Fundamental Operator pair $(G_1, G_2)$ of $(A^*,B^*,T^*)$ which characterizes when $(A,B,T)$ is special.
There results a Douglas-type functional model (as in \cite{Doug-Dilation} for the single contraction operator setting) for the tetrablock contraction
which also exhibits the tetrablock isometric lift $(V_1, V_2, V_3)$, all in a functional-model form rather than via block-matrix constructions as in \cite{Tirtha14}.
This Douglas-type model can in turn be converted to a Sz.-Nagy-Foias-type model; the Sz.-Nagy-Foias characteristic function $\Theta_T$ for the
contraction operator $T$, together with the the fundamental operators $(G_1, G_2)$ for the adjoint tetrablock contraction $(A^*, B^*, T^*)$, along
with some additional information needed to handle the case where $T$ is not a pure contraction, form what we call a {\em characteristic tetrablock data set}
for $(A,B,T)$ in terms of which one can write down the functional model. Conversely, we identify a collection of objects which we call a
{\em special tetrablock data set}:
specifically, (i) a pure contractive operator function $({\mathcal D}, {\mathcal D}_*, \Theta)$, (ii) a pair of operators $(G_1, G_2)$ on the coefficient space ${\mathcal D}_*$,
(iii) a tetrablock unitary $(R,S,W)$ acting on $\overline{D_\Theta \cdot L^2({\mathcal D})}$, such that (iv) all these together satisfy a natural invariant-subspace compatibility
condition. From such a characteristic tetrablock data set we construct a functional model such that the embedded functional-model operator triple is the most
general special tetrablock contraction up to unitary equivalence, with its special tetrablock isometric lift also embedded in the functional model.
We also are careful to push the theory as far as we can without the assumption that the original tetrablock contraction is special. In this case
we identify a class of operator triples $(V_1, V_2, V_3)$ with $V_3$ equal to a minimal isometric lift for $T$ to which $(A,B,T)$
can be lifted: here $V_!$ and $V_2$ commute with $V_3$ but not necessarily with each other and it appears that $V_1$, $V_2$ need not be contractions.
In this case there is no converse direction: there is no guarantee that the compression of a general pseudo-commutative tetrablock isometry $(V_1, V_2, V_3)$
on ${\mathcal K}$ back to ${\mathcal H}$ will yield a tetrablock contraction.
Let us note that the recent paper of Bisai and Pal \cite{BisaiPal21} contains closely related results. These authors basically compute the $Z$-transform of
the Sch\"affer-type construction of the unique special tetrablock isometric lift $(V_1, V_2, V_3)$ (where $V_3$ is equal to the minimal Sz.-Nagy isometric lift
of $T$) to arrive at a functional model for this lift. Our approach on the other hand
uses the Douglas lifting approach to construct the functional model directly with the existence and uniqueness of the special tetrablock isometric lift
falling out as part of the construction. When the tetrablock contraction is not special and no such lift is possible, the same construction still leads
to a functional model but $(V_1, V_2, V_3)$ is only a pseudo-commutative tetrablock isometry and there is no tetrablock isometric lift constructed in this way.
The results for the special case arise as a special case (the case where the Fundamental Operator pair $(G_1, G_2)$ for the tetrablock contraction $(A^*,B^*,T^*)$
satisfy the additional commutativity conditions \eqref{com=}) of the general functional-model construction. The paper \cite{BisaiPal21} also obtains a
noncommtative functional model for a non-special case, based on the work of Durszt \cite{Durszt} (a variation of the approach of Douglas for the construction
of the minimal isometric lift for the case of a single contraction operator $T$), but with the additional hypothesis that $A$ and $B$ commute not only with $T$
but also with $T^*$. It is clear that the complete unitary invariant for a pure tetrablock contraction $(A,B,T)$ consists of the characteristic function $\Theta_T$ of $T$
together with the Fundamental Operator pair $(G_1, G_2)$ of $(A^*, B^*, T^*)$; for the non-pure case (where $\Theta_T$ is no longer inner) we add
a certain tetrablock unitary $(R,S,W)$ acting on $\overline{\Delta_T H^2({\mathcal D}_T) }$ which is part of our model (see Theorem \ref{Thm:CompUniInv} below),
while Bisai-Pal add the Fundamental Operator pair $(F_1, F_2)$ for $(A,B,T)$ and argue that $(\Theta_T, (F_1, F_2), G_1, G_2))$ is a complete unitary
invariant. It remains to be seen which is the more relevant and useful in the future.
It is now becoming clear that the domains ${\mathbb D}^d$ (polydisk), $\Gamma$ (symmetrized bidisk), ${\mathbb E}$ (tetrablock) as well as ${\mathbb D}^d_s$
(symmetrized polydisk) all have common features with respect to the associated operator theory and applications to the rational dilation problem
for each of these domains. The paper \cite{BallSauDougVol} shows how a program completely parallel to that done here for the tetrablock case
can be worked out equally well for the polydisk case $\Omega = {\mathbb D}^d$ (where rational dilation is known to fail when $d \ge 3$).
In all these settings, there appear a pair of unitary invariants called Fundamental Operators which play a key role as part of a set (including the Sz.-Nagy--Foias
characteristic function of an appropriate contraction operator determined by the operator tuple) of unitary invariants for the operator tuple of whatever class.
The notion of Fundamental Operators as a fundamental object of interest seems to have appeared first in connection with the symmetrized bidisk
$\Gamma$ \cite{BPJOT}, then in connection with the un-symmetrized polydisk \cite{sauAndo, BallSauDougVol}, and now also in connection with the
symmetrized polydisk (see \cite{Pal21}).
Often the proper notion of Fundamental Operators for one setting is found by making a
correspondence of the less understood setting with some other better understood setting, and then adapting definitions for the first to become
definitions for the second. In particular, many of the results for the tetrablock case were originally found by adapting from results for the
symmetrized bidisk case (see e.g.~\cite{Tirtha14}), and it has been shown how one can deduce the bidisk functional model from the tetrablock
functional model (see \cite{sauAndo}). In this spirit in a future publication we plan to show how the results
from \cite{BallSauDougVol} for the polydisk case (most of which are just statements parallel to what is done here for the tetrablock case)
can alternatively be derived as a corollary of the corresponding results for the tetrablock case via the simple observation:
if ${\mathbf T} = (T_1, \dots, T_d)$ is a commutative, contractive operator $d$-tuple, then for $1 \le i \le d$, if we set $T_{(i)} = \Pi_{1 \le j \le d \colon j \ne i} T_j$, then for each
$i = 1, \dots, d$ the $d$-tuple $(A_i, B_i, P) = (T_i, T_{(i)}, \Pi_{1 \le j \le d} T_j)$ is a tetrablock contraction; the $d=2$ case can be found in \cite[Section 3, Version 3]{sauAndo}.
Finally, let us point out that
it is possible to reformulate the rational dilation problem for a given domain $\Omega$ as a problem about unital representations of a unital function algebra:
given a contractive representation
$\pi \colon f \in {\mathcal A} \mapsto \pi(f) \in {\mathcal B}({\mathcal H})$ which is contractive ($\| \pi(f) \|_{{\mathcal B}({\mathcal H})} \le \| f \|_{{\mathcal B}({\mathcal H})}$ where the unital representation property is that
$\pi(1_{\mathcal A}) = I_{\mathcal H}$ and $\pi(f_1 \cdot f_2) = \pi(f_1) \pi(f_2)$, is it automatically the case that the representation is {\em completely contractive}, i.e.,
still contractive after tensoring with ${\mathbb C}^{n \times n}$ for any $n \in {\mathbb N}$? To recover the original formulation as a special case,
one can take ${\mathcal A} = \overline{\operatorname{Rat}(\Omega)}$ where the closure is in the $C^*$-algebra $C(b\Omega)$
(continuous functions on the distinguished boundary $b \Omega$). However with this more general formulation one can consider function algebras
which go beyond $\overline{\operatorname{Rat}({\Omega})}$, e.g., the
constrained subalgebra ${\mathbb C} \cdot 1 + z^2 {\mathbb A}({\mathbb D})$ of the disk algebra ${\mathbb A}({\mathbb D}) =
\overline{\operatorname{Rat}({\mathbb D})}$. Alternatively, it is often possible to represent the algebra ${\mathcal A}$ as conformally equivalent to the algebra of all
functions analytic on some algebraic curve $\Omega = {\mathbf C}$ embedded in some higher-dimensional closed complex manifold (the Neil
parabola intersected with the bidisk for the case of ${\mathbb C} \cdot 1 + z^2 {\mathbb A}({\mathbb D})$).
For the state of knowledge (up to 2018) on this direction of dilation theory
including much discussion and references on earlier work, we refer to the paper of Dritschel and Undrakh \cite{DU}. We shall not pursue this direction here.
The paper is organized as follows. After the present Introduction, in Section 2 we collect assorted definitions and illustrative results concerning
tetrablock contractions, tetrablock isometries, and tetrablock unitaries, including a direct proof of the existence of the Fundamental Operator pair
for a given tetrablock contraction, which will be needed in the sequel. Here we also show how to associate a tetrablock unitary $(R,S,W)$ with a
tetrablock contraction $(A,B,T)$ in a canonical way; this is the key ingredient needed to eliminate the {\em purity} assumption on the contraction operator
$T$ required in earlier work on this problem (see \cite{SauNYJM}).
Section 3 shows how a lifting framework for the tetrablock-contraction setting can be constructed as an embellishment of the Douglas-model lifting framework
\cite{Doug-Dilation} originally formulated as an approach to the Sz.-Nagy dilation theorem for a single contraction operator $T$, with the
pseudo-commutative tetrablock-isometric lift $(V_1, V_2, V_3)$ having $V_3 = T$ and $V_1$ and $V_2$ constructed by making use of the
Fundamental Operator pair for the tetrablock contraction $(A^*, B^*, T^*)$. The final Section 4 identifies the invariants required to write down a functional
model equipped with a model operator triple $(A,B,T)$ which is concrete functional-model version of a general tetrablock contraction.
\section{The fundamentals of tetrablock contractions}
This section gives a brief introduction to the operator theory associated with the tetrablock.
\subsection{Tetrablock contractions}
The {\bf tetrablock}, denoted by ${\mathbb E}$, is the non-convex but polynomially convex domain in $\mathbb C^3$ given by \eqref{tetrablock}. From this formula
for ${\mathbb E}$ is is easy to read off the following symmetry properties.
\begin{proposition} \langlebel{P:symmetries}
The tetrablock ${\mathbb E}$ has the following symmetry properties:
\begin{enumerate}
\item ${\mathbb E}$ is invariant under complex conjugation:
$$
(a,b,t) \in {\mathbb E} \Leftrightarrow (\overline{a}, \overline{b}, \overline{t}) \in {\mathbb E}.
$$
\item ${\mathbb E}$ is invariant under interchange of the first two coordinates:
$$
(a,b,t) \in {\mathbb E} \Leftrightarrow (b,a,t) \in {\mathbb E}.
$$
\end{enumerate}
\end{proposition}
The distinguished boundary of ${\mathbb E}$, i.e., the \v Silov boundary with respect to the algebra of functions that are analytic in $\bE$ and
continuous on $\overline{\bE}$, is given by
$$
b\bE:=\left\{(a,b,\text{det}X): X=\begin{bmatrix} a & a' \\ b' & b \end{bmatrix}\text{ is a unitary }\right\}
$$
(see \cite[Theorem 7.1]{awy07}). From this characterization it is easy to see that $b{\mathbb E}$ is also invariant under the two involutions
$(a,b,t) \mapsto (\overline{a}, \overline{b}, \overline{t})$ and $(a,b,t) \mapsto (b,a, t)$.
Several tractable characterizations of the tetrablock can be found in \cite[Theorem 2.2]{awy07};
we pick two of these that will be used in what follows.
\begin{thm}\langlebel{T:CharacTetra} For a point $(a,b,t)\in\bC^3$, the following are equivalent:
\begin{itemize}
\item[(i)] $(a,b,t)\in\bE;$
\item[(ii)] with the rational function $\Psi:\overline{{\mathbb D}}\times\bC^3\to\bC$ defined as
\begin{align}\langlebel{Psi}
\Psi(z,(a,b,t))=\frac{a-z t}{1-z b},
\end{align}$\sup_{z\in\overline{{\mathbb D}}}|\Psi(z,(a,b,t))|<1$; and if $ab=t$ then, in addition, $|b|<1$;
\item[(iii)] with $\Psi$ as in \eqref{Psi}, $\sup_{z\in\overline{{\mathbb D}}}|\Psi(z,(b,a,t))|<1$; and if $ab=t$ then, in addition, $|a|<1$.
\end{itemize}Moreover, when item (i) is replaced by $(a,b,t)\in\overline{\bE}$, then all the strict inequalities in items (ii) and (iii) are replaced by non-strict inequalities.
\end{thm}
\begin{remark} \langlebel{R:symmetry} Note that in Theorem \ref{T:CharacTetra}, the equivalence of (i) $\Leftrightarrow$ (iii) is an immediate consequence of the equivalence of
(i) $\Leftrightarrow$ (ii) in view of the
invariance of ${\mathbb E}$ under the involution $(a,b,t) \mapsto (b,a,t)$.
\end{remark}
Recall the notions of \textbf{tetrablock unitary}, \textbf{tetrablock
isometry} and \textbf{tetrablock contraction} given in the Introduction.
Several algebraic characterizations of tetrablock isometries and tetrablock unitaries are known; see Theorems 5.4 and 5.7 in \cite{Tirtha14}.
We recall the ones that are useful for our purposes here. Here we use the notation
$r(X)$ for the \textbf{spectral radius} of a Hilbert-space operator $X$. It is then not difficult to see that the ${\mathbb E}$-symmetries noted in
Proposition \ref{P:symmetries} imply the same symmetries on the respective operator classes (with respect to the class of ${\mathbb E}$-isometries
which requires a little extra care), as noted in the next result. We leave the
easy verification as an exercise for the reader.
\begin{proposition} \langlebel{P:operatorsym}
Suppose that $(A,B,T)$ is a triple of bounded operators on a Hilbert space ${\mathcal H}$.
Then:
\begin{enumerate}
\item $(A,B,T)$ is a ${\mathbb E}$-contraction $\Leftrightarrow$ $(A^*, B^*, T^*)$ is a ${\mathbb E}$-contraction $\Leftrightarrow$
$(B,A,T)$ is a ${\mathbb E}$-contraction.
\item $(A,B,T)$ is a ${\mathbb E}$-isometry $\Leftrightarrow$ $(B,A,T)$ is a ${\mathbb E}$-isometry.
\item $(A,B,T)$ is a ${\mathbb E}$-unitary $\Leftrightarrow$ $(A^*, B^*, T^*)$ is a ${\mathbb E}$-unitary $\Leftrightarrow$ $(B,A,T)$ is a ${\mathbb E}$-unitary.
\end{enumerate}
\end{proposition}
\begin{thm}\langlebel{T:IsoChar}
Let $(A,B,T)$ be a commutative triple of bounded Hilbert space operators. Then the following are equivalent:
\begin{itemize}
\item[(i)] $(A,B,T)$ is a tetrablock isometry (respectively unitary);
\item[(ii)] $(A,B,T)$ is a tetrablock contraction and $T$ is an isometry (respectively unitary);
\item[(iii)] $A=B^*T$, $B$ is a contraction and $T$ is an isometry (respectively unitary); and
\item [(iv)] $B=A^*T$, $A$ is a contraction and $T$ is an isometry (respectively unitary).
\item[(v)] $B = A^* T$, $r(A) \le 1$ and $r(B) \le 1$, and $T$ is an isometry (respectively unitary).
\end{itemize}
\end{thm}
\subsection{Pseudo-commutative tetrablock isometries and unitaries} \langlebel{SS:pc-tetra-isom/unit}
We propose to introduce the notions of \textbf{pseudo-commutative tetrablock unitary} and \textbf{pseudo-commutative tetrablock isometry}
for an operator triple $(A,B,T)$ by using criterion (iii) or equivalently (iv) in Theorem \ref{T:IsoChar} but with the weakening the commutativity hypothesis imposed on the
whole triple $(A,B,T)$ to just the condition that $A$ and $B$ commute with $T$ (but not necessarily with each other). As we are also dropping the condition
that $A$ or $B$ be a contraction, a more proper term would be {\em noncontractive pseudo-commutative tetrablock isometry}, but, as this term will be
consistent throughout, we settle on the shorter term for brevity. The resulting definition is as
follows. We leave it to the reader to verify that the two formulations are equivalent.
\begin{definition} \langlebel{D:pc}
Let $(A,B,T)$ be a triple of bounded Hilbert-space operators. We say that the triple
$(A,B,T)$ is a \textbf{pseudo-commutative tetrablock isometry} (respectively, \textbf{unitary}) if any of the following equivalent conditions holds:
\begin{enumerate}
\item $T$ is an isometry (respectively, unitary) and
\begin{equation} \langlebel{PseudoIdentity1}
AT=TA,\quad BT=TB, \quad A = B^*T.
\end{equation}
\item $T$ is an isometry (respectively, unitary), and
\begin{equation} \langlebel{PseudoIdentity2}
AT = TA, \quad BT = TB, \quad B = A^*T.
\end{equation}
\end{enumerate}
\end{definition}
\begin{remark} \langlebel{R:pctetra-vs-tetra}
From Definition \ref{D:pc} and Theorem \ref{T:IsoChar}, we see that any tetrablock isometry/unitary is also a pseudo-commutative tetrablock isometry/unitary
but not conversely. If we wish to emphasize that we are referring to the logically more special {\em tetrablock isometry/ unitary} rather than the more general
{\em pseudo-commutative tetrablock isometry/unitary}, we often will say
{\em strict tetrablock isometry/unitary} for emphasis.
\end{remark}
\begin{remark} \langlebel{R:pseudocom} We now present a couple of elementary observations on pseudo-commutative versus strict tetrablock unitaries
unitaries which we hope give the reader some additional insight.
\noindent
(1) We remark that {\em if $(A,B,T)$ is a pseudo-commutative tetrablock unitary, then $A$ and $B$ are not necessarily normal operators} as would happen
in the strict case. For example, pick a non-normal
contraction $G_1$ acting on a Hilbert space ${\mathcal E}$ and consider the triple $(M_{G_1^*},M_{\zeta G_1},M_\zeta)$ on $L^2({\mathcal E})$. It is easy to see that this triple is a
pseudo-commutative tetrablock unitary. However, neither $A$ nor $B$ is normal unless $G_1$ is so. Also note that if $(A,B,T)$ is a pseudo-commutative tetrablock
unitary, then so is the adjoint triple $(A^*,B^*,T^*)$.
This can be seen by observing that the adjoint of the identities in \eqref{PseudoIdentity1} with $T$ unitary can be converted to the identities
\eqref{PseudoIdentity2} for $(A^*,B^*,T^*)$ with $T^*$ still unitary. Note next that if $(A,B,T)$ is a pseudo-commutative tetrablock unitary, then
\begin{align*}
&A^*A = T^* B B^* T= B T^* T B^* = B B^*, \\
& B^*B = T^* A A^* T = A T^* T A^* = A A^*.
\end{align*}
Thus we always have
\begin{equation} \langlebel{pc1}
A^*A = BB^*, \quad AA^* = B^* B
\end{equation}
for a pseudo-commutative tetrablock unitary $(A,B,T)$.
As a first consequence of \eqref{pc1}, we see that
if $A$ is normal, then
$$
B^*B = A A^* = A^* A = B B^*
$$
and $B$ is also normal. Similarly if $B$ is normal, then $A$ is also normal. In conclusion,
{\em if $(A,B,T)$ is a pseudo-commutative unitary such that one of $A$ or $B$ is normal, then so is the other.}
\noindent
(2)
We note as a consequence of (iii) $\Rightarrow$ (i) in Theorem \ref{T:IsoChar} that in particular {\em if $(A, B, T)$ is a
strict tetrablock unitary
(so we also have $AB = BA$), then the operators $A$ and $B$ are normal. } One can see this directly from the considerations here as follows.
As a strict tetrablock unitary in particular meets all the requirements for membership in the {\em pseudo-commutative tetrablock unitary} class,
we know that \eqref{pc1} holds.
Combining this with the commutativity relation $AB = BA$ then gives us
$$
A^* A = A^* B^* T = B^* A^* T = B^* B = A A^*
$$
showing that $A$ is normal. The same computation with the roles of $A$ and $B$ interchanged then shows that $B$ is also normal.
The full strength of (iii) $\Rightarrow$ (i) in Theorem \ref{T:IsoChar} is that in addition the commutative normal triple $(A,B,T)$ has joint spectrum
in the boundary of the tetrablock ${\mathbb E}$; for this somewhat deeper fact we refer to \cite{Tirtha14}.
\end{remark}
The next result gives a feel for how close pseudo-commutative tetrablock isometries come to being strict tetrablock isometries.
\begin{thm} \langlebel{T:pc-vs-strict-Eisom} Let $(A,B,T)$ be a pseudo-commutative tetrablock isometry on a Hilbert space ${\mathcal H}$.
\begin{enumerate}
\item Then the spectral radius $r(AB)$ of the product operator $AB$ is given by
\begin{equation} \langlebel{norm/rad}
r(AB) = \operatorname{max} \{ \| A \|^2, \, \| B \|^2 \|.
\end{equation}
\item Suppose in addition that $AB = BA$ and $r(A) \le 1$, $r(B) \le 1$. Then both $A$ and $B$ are contraction operators
($\max \{ \| A \|, \| B \| \} \le 1$) and $(A,B, T)$ is a strict tetrablock isometry.
\end{enumerate}
\end{thm}
\begin{proof} The proof follows the ideas of Bhattacharyya \cite[pp.1619-1620]{Tirtha14}.
We first consider statement (1). Form two operators $X_1 = \sbm{ 0 & A \\ B & 0 }$ and $X_2 = \sbm{ T & 0 \\ 0 & T}$ on $\sbm{ {\mathcal H} \\ {\mathcal H}}$.
From the two relations $B^*T = A$ and $A^*T = B$ we deduce that $X_1 = X_1^* X_2$ where $X_2 X_2^* = \sbm{ T T^* & 0 \\ 0 & T T^*} \preceq
\sbm{ I & 0 \\ 0 & I}$ since $T$ is an isometry. Hence
$$
X_1 X_1^* = X_1^* X_2 X_2^* X_1 \preceq X_1^* X_1,
$$
i.e., $X_1$ is a {\em hyponormal operator}. By a theorem of Stampfli (see \cite[Proposition 4.6]{Conway}, it follows that $r(X_1) = \| X_1 \|$.
We compute the operator norm of $X_1$ as follows:
$$
\| X_1\|^2 = \bigg\| \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \begin{bmatrix} 0 & B^* \\ A^* & 0 \end{bmatrix} \bigg\|
= \bigg\| \begin{bmatrix} A A^* & 0 \\ 0 & B B^* \end{bmatrix} \bigg\| = \operatorname{max} \{ \| A \|^2, \| B \\^2 \}.
$$
and hence $\| X_1 \| = \max \{ \| A \|, \| B \| \}$. To compute $r(X_1)$, note first that $X_1^*X_1 = \sbm{ A B & 0 \\ 0 & BA }$ and hence
$$
X_1^{2n} = \begin{bmatrix} (AB)^n & 0 \\ 0 & (BA)^n \end{bmatrix}.
$$
Consequently,
$$
r(X_1) = \lim_{n \to \infty} \max \| \| (AB)^n \|^{\frac{1}{2n}}, \| (BA)^n \|^{\frac{1}{2n}} \} = \max \{ r(AB)^{\frac{1}{2}}, r(BA)^{\frac{1}{2}} \}.
$$
However a general fact is that the nonzero spectrum of $AB$ is the same as the nonzero spectrum of $BA$, and hence $r(AB) = r(BA)$.
Thus $r(X_1) = \| X_1\|$ gives us \eqref{norm/rad} and the proof of statement (1) is complete.
As for statement (2), a known fact is that if $A$ and $B$ commute, then the spectrum of the product operator $AB$ is given by
$$
\sigma(AB) = \{ \langlem \cdot \mu \colon \langlem \in \sigma(A), \, \mu \in \sigma(B) \}.
$$
Hence the hypothesis that $r(A) \le 1$ and $r(B) \le 1$ implies that $r(AB) \le 1$ as well. But then from the conclusion of statement (1)
already proved, we conclude that both $A$ and $B$ are contraction operators, and the proof of statement (2) is now complete.
(Note that this also proves (v) $\Rightarrow$ (iii) or (iv) in Theorem \ref{T:IsoChar}.)
\end{proof}
\begin{example} \langlebel{E:model}
(1) \textbf{A pseudo-commutative/strict tetrablock isometry.}
Let ${\mathcal E}$ be a coefficient Hilbert space and $H^2({\mathcal E}) = H^2 \otimes {\mathcal E}$ be the associated Hardy space of ${\mathcal E}$-valued functions.
Let $G_1$ and $G_2$ be operators on ${\mathcal E}$ and set
\begin{equation} \langlebel{pcform}
A = M_{G_1^* + z G_2}, \quad B = M_{G_2^* + z G_1}, \quad T = M_z \text{ on } H^2({\mathcal E}).
\end{equation}
Then it is immediate that $T$ is an isometry and that $A$ and $B$ commute with $T$. The special coupled form of the pencils defining $A$ and $B$
enables us to show that $A = B^*T$:
\begin{align*}
B^* T& = (I_{H^2} \otimes G_2^* + M_z \otimes G_1)^* \cdot (M_z \otimes I_{\mathcal G}) \\
& = (I_{H^2} \otimes G_2 + M_z^* \otimes G_1^*) \cdot (M_z \otimes I_{\mathcal G}) \\
& = (I_{H^2} \otimes G_1^*) + (M_z \otimes G_2)
= M_{G_1^* + z G_2} = A
\end{align*}
and similarly $B = A^* T$.
For $(A,B,T)$ to be a strict tetrablock isometry, we need in addition that $AB = BA$ and that $\|A \| \le 1$ (in which case also
$\| B \| = \| A^* T \| \le 1$ as well). To ensure that $\| A \| \le 1$ requires that $G_1$ and $G_2$ are not too large in the precise sense that
\begin{equation} \langlebel{PencilContr}
\sup_{z \in {\mathbb T}} \| G_1^* + z G_2 \| \le 1.
\end{equation}
To check the condition $AB = BA$, we compute
\begin{align*}
A B & = M_{G_1^* + z G_2} M_{G_2^* + z G_1} \\
& = (I_{H^2} \otimes G_1^* + M_z \otimes G_2) \cdot (I_{H^2} \otimes G_2^* + M_z \otimes G_1) \\
& = I_{H^2} \otimes G_1^* G_2^* + M_z \otimes (G_1^* G_1 + G_2 G_2^*) + M_z^2 \otimes G_1 G_2
\end{align*}
while a similar computation gives us
$$
BA = I_{H^2} \otimes G_2^* G_1^* + M_z \otimes (G_1 G_1^* + G_2^* G_2) + M_z^2 \otimes G_2 G_1.
$$
We conclude that in this example, $(A, B, T)$ is a strict tetrablock isometry exactly when \eqref{PencilContr} together with the following commutativity conditions hold:
\begin{equation} \langlebel{com=}
G_1 G_2 = G_2 G_1, \quad G_1^* G_1 + G_2 G_2^* = G_1 G_1^* + G_2^* G_2,
\end{equation}
sometimes also written more compactly in terms of commutators as
$$
[ G_1, G_2] = 0, \quad [G_1^*, G_1] = [G_2^*, G_2]
$$
where in general $[X,Y]$ is the commutator:
$$
[ X, Y] = X Y - Y X.
$$
(2) \textbf{A pseudo-commutative/strict tetrablock unitary.}
It is easy to use the spectral theory for unitary operators (a particular case of the spectral theory of normal operators) to write down a model for the general
pseudo-commutative/strict tetrablock unitary $(R,S,W)$. as follows. By Definition \ref{D:pc} we see in particular that $W$ is unitary.
By the spectral theory for general normal operators (see e,g,~any of \cite{Dix, Arv} or \cite[Chapter 2]{Arveson02}), after a unitary change of
coordinates, we can represent $W$ as the operator $M_\zeta$ of multiplication by the coordinate function ($M_\zeta \colon h(\zeta) \mapsto \zeta h(\zeta)$) on a
direct-integral space $\bigoplus\int_{\mathbb T} {\mathcal H}_\zeta\, \nu({\tt d}\zeta)$ determined by a scalar spectral measure $\nu$ supported on ${\mathbb T}$ and a
measurable multiplicity function
$\zeta \mapsto \dim {\mathcal H}_\zeta$.. Since the operators $R$ and $S$ commute with $W = M_\zeta$, it follows that $R$ and $S$
are represented as {\em decomposable operators} on $\bigoplus \int_{\mathbb T} {\mathcal H}_\zeta\, \tt{d} \nu(\zeta)$, i.e.,
$R = M_\phi \colon h(\zeta) \mapsto \phi(\zeta) h(\zeta)$ and $S = M_\psi \colon h(\zeta) \mapsto \psi(\zeta) h(\zeta)$
for measurable functions such that $\phi(\zeta) \in {\mathcal B}({\mathcal H}_\zeta)$, $\psi(\zeta) \in {\mathcal B}({\mathcal H}_\zeta)$ for a.e.~$\zeta$.
The fact that in addition $R = S^*W$ then forces $\phi(\zeta) = \psi(\zeta)^* \cdot \zeta $ for a.e.~$\zeta$, Thus any pseudo-commutative tetrablock unitary has the form
\begin{equation} \langlebel{gen-tetra-unitary}
(R, S, W) = (M_{\psi^* \cdot \zeta}, M_\psi, M_\zeta) \text{ acting on } \bigoplus \int_{\mathbb T} {\mathcal H}_\zeta \, \nu(\tt{d} \zeta)
\end{equation}
If $(R,S,W)$ is a strict tetrablock unitary, then in addition we must have that $\psi(\zeta)$ is a {\em contractive normal operator} on ${\mathcal H}_\zeta$ for a.e.~$\zeta$
in order to guarantee in addition that $RS = SR$ and that $\| R \| \le 1$, $\| S \| \le 1$. By this analysis we conclude that \eqref{gen-tetra-unitary}
(with $\psi(\zeta)$ constrained to be contractive normal for a.e.~$\zeta$ for the strict case)
is the general form for a pseudo-commutative/strict tetrablock unitary. In a less-functional form, to write a pseudo-contractive tetrablock triple $(R,S,W)$, the free parameters
are: (i) a unitary operator $W$, and (ii) an operator $S$ commuting with $W$; then the associated pseudo-commutative tetrablock contraction is
$(W^*S, S, W)$; for this to be strict, one must require in addition that the operator $S$ in the commutant of $W$ be a normal contraction.
\end{example}
To deduce the von Neumann-Wold decomposition for a tetrablock isometry, the next lemma is useful. Only the special case where the operator $S$
in the statement is a shift will be needed for our application, in which case the result is well-known (see e.g. \cite[page 22]{NFintertwine}).
For completeness we present here a proof of the general result.
\begin{lemma} \langlebel{L:zero}
Let $W$ be a unitary operator on ${\mathcal H}_2$, $S$ an operator on ${\mathcal H}_1$ such that $S^{*n}\to 0$ in the strong operator topology as $n\to\infty$.
If $X $ is a bounded operator from ${\mathcal H}_2$ to ${\mathcal H}_1$ such that $XW=SX$, then $X=0$.
\end{lemma}
\begin{proof} From $XW = SX$ we get by iteration that $XW^n = S^n X$ for $n = 1,2,\dots$. Taking adjoints gives then $W^{*n} X^* = X^* S^{*n}$.
Apply this identity to an arbitrary fixed vector $x \in {\mathcal H}_2$ to get
$ W^{*n} X^* x = X^* S^{*n} x$ for all $n \ge 1$. Apply $W^n$ to both sides of this equation to get $W^n W^{*n} X^*x = W^n X^* S^{*n} x$ for $n=1,2, \dots$.
As $W$ is unitary, this becomes $X^*x = W^n X^* S^{*n} x$. Taking norms then gives
$$
\| X^* x \| = \| W^n X^* S^{*n} x \| = \| X^* S^{*n} x \| \le \| X^*\| \| S^{*n} x \| \to 0 \text{ as } n \to \infty
$$
by the assumed strong convergence of powers of $S^*$ to zero, forcing $X^*$ (and hence also $X$) to be the zero operator.
\end{proof}
The von Neumann-Wold decomposition (see \cite{Wold, vonN-Wold, Nagy-Foias}) ensures that if $T$ is an isometry acting on a Hilbert space ${\mathcal H}$, then
$T$ can be represented as an operator as the external direct sum $M_z \oplus U$ of a shift operator $M_z$ acting on a Hardy space $H^2({\mathcal E})$
and a unitary operator $W$ on ${\mathcal F}$ for some coefficient Hilbert spaces ${\mathcal E}$ and ${\mathcal F}$.
The following result not only gives a model for an arbitrary pseudo-commutative/strict tetrablock isometry $(A,B,T)$, but also can be seen
as a pseudo-commutative/strict tetrablock-isometry analogue of the classical von Neumann--Wold decomposition for a
single isometric Hilbert space operator $T$.
\begin{thm}\langlebel{T:WoldPC} Let $(A,B,T)$ be an operator-triple on the Hilbert space ${\mathcal H}$.
\noindent
{\rm(1)}
Then $(A,B,T)$ is a pseudo-commutative tetrablock isometry on ${\mathcal H}$ if and only if there exist Hilbert spaces ${\mathcal E}$, ${\mathcal F}$, operators $G_1,G_2$ acting on ${\mathcal E}$
subject to \eqref{PencilContr}, along with a
pseudo-commutative tetrablock unitary $(R,S,W)$ acting on ${\mathcal F}$, such that ${\mathcal H}$ is isomorphic to $\sbm{H^2({\mathcal E}) \\ {\mathcal F}}$ and under the same isomorphism
$(A,B,T)$ is unitarily equivalent to
\begin{align}\langlebel{Pencils}
\left(\begin{bmatrix}M_{G_1^*+z G_2}&0\\0&R \end{bmatrix},\begin{bmatrix}M_{G_2^*+z G_1}&0\\0&S \end{bmatrix},\begin{bmatrix}M_z &0\\0&W \end{bmatrix}\right).
\end{align}
\noindent
{\rm (2)} Then $(A,B,T)$ is a strict tetrablock isometry on ${\mathcal H}$ if and only if $(A,B,T)$ is unitarily equivalent to the operator triple as in \eqref{Pencils}
(with $G_1$, $G_2$ subject to \eqref{PencilContr})
acting on a space $\sbm{H^2({\mathcal E}) \\ {\mathcal F}}$, where in addition the operator-pencil coefficients $(G_1, G_2)$ satisfy the system of operator identities
\eqref{com=}, and the triple $(R,S, W)$ is a strict tetrablock unitary (i.e., we also have the relation $RS = SR$ with $R$ and $S$ contraction operators).
\end{thm}
\begin{remark} \langlebel{R:FuncModel} We shall think of a triple of operators on $\sbm{ H^2({\mathcal E}) \\ {\mathcal F}}$ as in \eqref{Pencils} as a \textbf{functional model} for a pseudo-commutative/strict
tetrablock isometry/unitary. The $H^2({\mathcal E})$-component clearly has a functional form while the second component can be brought to a measure-theoretic
functional form as in item (2) in Example \ref{E:model}.
\end{remark}
\begin{proof}
The sufficiency (for both the pseudo-commutative and the strict case) follows from Example \ref{E:model}.
We now suppose that $(A,B,T)$ is a strict tetrablock isometry.
Let us apply the Wold decomposition to the isometry $T$: there exist Hilbert spaces
${\mathcal E}$, ${\mathcal F}$, and a unitary $\tau:{\mathcal H}\to\sbm{H^2({\mathcal E})\\ {\mathcal F}}$ such that
$$
\tau T\tau^*=
\begin{bmatrix}M_z&0\\0&W \end{bmatrix}:\begin{bmatrix} H^2({\mathcal E})\\ {\mathcal F}\end{bmatrix} \to \begin{bmatrix} H^2({\mathcal E})\\ {\mathcal F}\end{bmatrix}
$$
for some unitary $W$ on ${\mathcal F}$. Next assume that
$$
\tau(A,B)\tau^*=\left(\begin{bmatrix}
A_{11}&A_{12}\\A_{21}&R
\end{bmatrix},\begin{bmatrix}
B_{11}&B_{12}\\B_{21}&S
\end{bmatrix}\right):\begin{bmatrix}
H^2({\mathcal E})\\ {\mathcal F}
\end{bmatrix}\to
\begin{bmatrix}
H^2({\mathcal E})\\ {\mathcal F}
\end{bmatrix}.
$$
Now use these matrix representations and equate the (12)-entries of the relation $AT=TA$ to get $A_{12}W=M_zA_{12}$. Therefore $A_{12}=0$ by Lemma \ref{L:zero}.
Similarly, from the relation $BT=TB$ we have $B_{12}=0$. Compare the (21)-entries of the relation $A=B^*T$ to get $A_{21}=0$. The same treatment for the relation
$B=A^*T$ gives $B_{21}=0$. Therefore we are left with the following relations
\begin{align*}
&A_{11}M_z=M_zA_{11}, \, B_{11}M_z=M_zB_{11},\, A_{11}=B_{11}^*M_z \text{ (and $B_{11} = A_{11}^* M_z$);} \\
& RW=WR, \, SW=WS, \, R=S^*W \text{ (and $S = W R^*$).}
\end{align*}
The second set of the above relations together with the fact that $W$ is a unitary implies that $(R,S,W)$ is a pseudo-commutative tetrablock unitary. The first two
intertwining relations in the first set implies that
there exist bounded analytic functions $\Phi,\Psi:\mathbb D\to {\mathcal B}({\mathcal E})$ such that $A_{11} = M_\Phi$, $B_{11} = M_\Psi$. The remaining relations in the first set
then give us
$$
M_\Phi = M_\Psi^* M_z, \quad M_\Psi = M_\Phi^* M_z.
$$
There now only remains a tedious computation with the power series expansions of $\Phi$ and $\Psi$ to see that the remaining relations in the second set forces
$\Phi$ and $\Psi$ to have the coupled linear forms
$$
\Phi(z)=G_1^*+zG_2 \quad\mbox{and}\quad \Psi(z)=G_2^*+zG_1
$$
for some operators $G_1,\, G_2 \in {\mathcal B}({\mathcal E})$. Again the relation \eqref{PencilContr} is equivalent to $M_\Phi$ being a contraction operator.
From Definition \ref{D:pc} it follows that $(M_\Phi, M_\Psi, M_z)$ is a pseudo-commutative tetrablock isometry.
The completes the proof for the pseudo-commutative setting.
Suppose now that $(A, B, T)$ is a tetrablock isometry. Then in particular $(A,B,T)$ satisfies all the requirements to be a pseudo-commutative tetrablock isometry,
so all the preceding analysis applies. We then see that $(A,B,T)$ is unitarily equivalent to the triple in \eqref{Pencils}. As $(A,B,T)$ now is
actually a tetrablock isometry, we have that $AB = BA$. The unitary equivalence then forces
$$
\begin{bmatrix} M_{G_1^* + z G_2} & 0 \\ 0 & R \end{bmatrix} \begin{bmatrix} M_{G_2^* + z G_1} & 0 \\ 0 & S \end{bmatrix}
= \begin{bmatrix} M_{G_2^* + z G_1} & 0 \\ 0 & S \end{bmatrix} \begin{bmatrix} M_{G_1^* + z G_2} & 0 \\ 0 & R \end{bmatrix}
$$
which can be split up as two commutativity conditions
\begin{align}
& M_{G_1^* + z G_2} M_{G_2^* + z G_1} = M_{G_2^* + z G_1} M_{G_1^* + z G_2} \langlebel{com=1} \\
& RS = SR. \langlebel{com=2}
\end{align}
By reversing the computations done in item (1) of Example \ref{E:model}, we see that the intertwining \eqref{com=1} forces the set of conditions \eqref{com=}
Moreover, the condition \eqref{com=2} is exactly the missing ingredient needed to promote $(R,S,W)$ from a pseudo-commutative tetrablock unitary
to a strict tetrablock unitary. This completes the proof.
\end{proof}
\begin{remark} \langlebel{R:semi-strict} It will be useful to have a terminology for an intermediate class of operator triples $(A,B,T)$ which sits between
strict tetrablock isometries and general pseudo-commutative tetrablock isometries. Let us say that the triple $(A,B,T)$ is a
\textbf{semi-strict tetrablock isometry} if $(A,B,T)$ is a pseudo-commutative isometry with Wold decomposition as in \eqref{Pencils} is
such that the pseudo-commutative tetrablock unitary component $(R,S,W)$ is actually a strict tetrablock unitary, i.e., $R$ and $S$ are contractions
which commute with each other as well as with $W$ which is unitary.
\end{remark}
We next present an analogue of the single-variable operator theory fact that any isometry can always be extended to a unitary.
\begin{corollary} \langlebel{C:pc} Pseudo-commutative/strict tetrablock isometries can be extended to pseudo-commutative/strict tetrablock unitaries. More precisely:
\noindent
{\rm (1)} A triple $(A,B,T)$ is a pseudo-commutative tetrablock isometry if and only if it extends to a pseudo-commutative tetrablock unitary. Moreover there exists an extension that acts on the space of minimal unitary extension of the isometry $T$.
\noindent
{\rm (2)} A triple $(A,B,T)$ is a strict tetrablock isometry if and only if it extends to a strict tetrablock unitary acting on the space of the minimal unitary extension
of the isometry $T$.
\end{corollary}
\begin{proof}
If a triple extends to a pseudo-commutative tetrablock unitary, then from Definition \ref{D:pc} we can read off that it is also a pseudo-commutative tetrablock isometry.
Similarly, if a triple extends to a strict tetrablock unitary, we can read off from criterion (iii) or (iv) in Theorem \ref{T:IsoChar} that the triple itself must be
a strict tetrablock isometry.
We now address the converse.
In view of Theorem \ref{T:WoldPC}, we can assume without loss of generality that a pseudo-commutative tetrablock isometry $(A,B,T)$ is given in the form:
$$
\left(\begin{bmatrix}M_{G_1^*+z G_2}&0\\0&R \end{bmatrix},\begin{bmatrix}M_{G_2^*+z G_1}&0\\0&S \end{bmatrix},\begin{bmatrix}M_z &0\\0&W \end{bmatrix}\right): \begin{bmatrix}H^2({\mathcal E})\\{\mathcal F}\end{bmatrix}\to \begin{bmatrix}H^2({\mathcal E})\\{\mathcal F}\end{bmatrix}
$$
for some operators $G_1$, $G_2$ on ${\mathcal E}$ and for some pseudo-commutative tetrablock unitary $(R,S,W)$ acting on ${\mathcal F}$. Consider $H^2({\mathcal E})\oplus{\mathcal F}$
as a subspace of $L^2({\mathcal E})\oplus{\mathcal F}$ in the natural way. Then the triple
$$
\left(\begin{bmatrix}M_{G_1^*+\zeta G_2}&0\\0&R \end{bmatrix}, \begin{bmatrix}M_{G_2^*+\zeta G_1}&0\\0&S \end{bmatrix},
\begin{bmatrix}M_\zeta &0\\0&W \end{bmatrix}\right)\colon
\begin{bmatrix}L^2({\mathcal E})\\ {\mathcal F}\end{bmatrix} \to \begin{bmatrix}L^2({\mathcal E})\\ {\mathcal F}\end{bmatrix}
$$
is an extension of $(A,B,T)$. The unitary $M_\zeta\oplus W$ is clearly a minimal unitary extension of the isometry $M_z\oplus W$. And since $M_{G_1^*+\zeta G_2}=M_{G_2^*+\zeta G_1}^*M_\zeta$ and $(R,S,W)$ is a pseudo-commutative tetrablock unitary, the above triple is a pseudo-commutative tetrablock unitary by Definition \ref{D:pc}.
If we start with a strict tetrablock isometry, then we shall also have that
$\sbm{ M_{G_1^*+ z G_2} & 0 \\ 0 & R }$ commutes with $\sbm{M_{G_2^* + z G_1} & 0 \\ 0 & S}$ on $\sbm{ H^2({\mathcal E}) \\ {\mathcal F}}$, or equivalently,
$RS = SR$ and the Toeplitz operator symbols equal to the pencils $G_1^* + z G_2$ and $G_2^* + z G_1$ commute:
$$
(G_1^* + z G_2)(G_2^* + z G_1 ) = (G_2^* + z G_1 ) (G_1^* + z G_2).
$$
But then it is straightforward to see that this implies the commutativity of the associated Laurent operators acting on $L^2({\mathcal E})$:
$$
M_{ G_1^* + \zeta G_2} M_{G_2^* + \zeta G_1} = M_{G_2^* + \zeta G_1} M_{ G_1^* + \zeta G_2} .
$$
and hence also
$$
\begin{bmatrix} M_{ G_1^* + \zeta G_2} & 0 \\ 0 & R \end{bmatrix} \begin{bmatrix} M_{G_2^* + \zeta G_1} & 0 \\ 0 & S \end{bmatrix} =
\begin{bmatrix} M_{G_2^* + \zeta G_1} & 0 \\ 0 & S \end{bmatrix} \begin{bmatrix} M_{ G_1^* + \zeta G_2} & 0 \\ 0 & R \end{bmatrix}.
$$
on $\sbm{ L^2({\mathcal E}) \\ {\mathcal F}}$.
Moreover the extension of $M_{ G_1^* + z G_2}$ on $H^2({\mathcal E})$ to $M_{ G_1^* + \zeta G_2}$ on $L^2({\mathcal E})$ is norm-preserving, and hence the latter
is contractive whenever the former is contractive, and similarly for $M_{G_2^* + z G_1}$ and $M_{G_2^* + \zeta G_1}$.
We now have enough observations to conclude by criterion (iii) or (iv) in Theorem \ref{T:IsoChar} that
$\big( \left[ \begin{smallmatrix} M_{ G_1^* + \zeta G_2} & 0 \\ 0 & R \end{smallmatrix} \right],
\left[ \begin{smallmatrix} M_{G_2^* + \zeta G_1} & 0 \\ 0 & S \end{smallmatrix} \right] , \left[ \begin{smallmatrix} M_\zeta & 0 \\ 0 & W \end{smallmatrix} \right] \big)$
on $\sbm{ L^2({\mathcal E}) \\ {\mathcal F}}$ is a tetrablock unitary as required.
\end{proof}
Another one-variable fact is the result due to Sz.-Nagy-Foias (see \cite[Theorem I.3.2]{Nagy-Foias}): {\em any contraction operator $T$ on a Hilbert space ${\mathcal H}$
can be decomposed as $T = \sbm { T_{cnu} & 0 \\ 0 & U}$ where $T_{cnu}$ is a {\em completely nonunitary} (c.n.u.) contraction operator (meaning there is no
reducing subspace of ${\mathcal H}$ such that $T|_{\mathcal H}$ is unitary) and where $U$ is unitary}. There is a analogous result for the setting of tetrablock unitaries
and tetrablock contractions. We say that the tetrablock contraction $(A,B,T)$ on ${\mathcal H}$ is a \textbf{c.n.u.~tetrablock contraction} if there is no nontrivial
jointly reducing subspace ${\mathcal H}_u \subset {\mathcal H}$ for $(A,B,T)$ such that $(A,B,T)|_{{\mathcal H}_u}$ is a tetrablock unitary. The following result appears in
\cite{PalJMAA16}
\begin{thm} \langlebel{T:tetra-can-decom} Let $(A,B,T)$ be a tetrablock contraction on ${\mathcal H}$. Then ${\mathcal H}$ has an internal orthogonal direct-sum decomposition
${\mathcal H} = {\mathcal H}_{c.n.u.} \oplus {\mathcal H}_u$ with ${\mathcal H}_{c.n.u.}$ and ${\mathcal H}_u$ jointly reducing for $(A,B,T)|_{{\mathcal H}_{c.n.u.}}$ equal to a c.n.u.\ tetrablock contraction
and $(A,B,T)|_{{\mathcal H}_u}$ equal to a tetrablock unitary.
\end{thm}
The at first surprising fact is that the same decomposition ${\mathcal H} = {\mathcal H}_{c.n.u.} \oplus {\mathcal H}_u$ inducing the canonical decomposition of the contraction operator
$T$ into is c.n.u.~part $T_{c.n.u.}$ and its unitary part $T_u$ turns out to also be jointly reducing for the whole operator triple $(A,B,T)$ and induces
the canonical tetrablock decomposition for the tetrablock contraction $(A,B,T)$, i.e.,
$(A,B,T)|_{{\mathcal H}_{c.n.u.}}$ is a c.n.u.\ tetrablock contraction, and $(A,B,T)|_{{\mathcal H}_u}$ is a tetrablock unitary.
\begin{remark} \langlebel{R:tetra-can-decom} The import of Theorem \ref{T:tetra-can-decom} for model theory is the same as is the case for the classical case:
since the model theory for a (strict) tetrablock unitary is already well understood (see item (2) in Remark \ref{R:pseudocom}, it follows that it is perfectly
satisfactory to focus on the case where $(A,B,T)$ is a c.n.u.~tetrablock contraction for the purposes of model theory.
\end{remark}
\subsection{A canonical construction of a tetrablock unitary from a tetrablock contraction}\langlebel{SS:canonical}
In this section we start with a tetrablock contraction $(A,B,T)$ and construct a tetrablock unitary which is uniquely associated with $(A,B,T)$ in a sense that will be made precise later in this subsection. This will be used later to construct a concrete functional model for a pseudo-commutative tetrablock-isometric lift for $(A,B,T)$
which can be viewed as a functional model for $(A,B,T)$ itself.
We start by using the fact that the last entry $T$ in our tetrablock contraction $(A, B, T)$ is a contraction operator. Hence there exist a positive semidefinite operator $Q_{T^*}$ such that
\begin{align}\langlebel{Q}
Q_{T^*}^2:=\operatorname{SOT-}\lim T^nT^{* n}.
\end{align}
Define the operator $X_{T^*}^*:\overline{\operatorname{Ran}}\;Q_{T^*}\to\overline{\operatorname{Ran}}\;Q_{T^*}$ densely by
\begin{align}\langlebel{theX}
X_{T^*}^*Q_{T^*}h=Q_{T^*}T^*h.
\end{align}
This is an isometry because for all $h\in{\mathcal H}$,
\begin{align}
& \| X_{T^*}^* Q_{T^*}h\|^2 = \langlengle Q_{T^*}^2 T^*h, T^*h\ranglengle=\lim_{n\to\infty}\langlengle T^{n}T^{* n}T^*h,T^*h\ranglengle \notag \\
& \quad = \lim_{n \to \infty} \langlengle T^{n+1} T^{*(n+1)} h, h \ranglengle =\langlengle Q_{T^*}^2 h, h\ranglengle = \| Q_{T^*} h \|^2.
\langlebel{Xisoprf}
\end{align}
Since $A$ is a contraction, we have for all $h\in{\mathcal H}$
\begin{align*}
\langlengle AQ_{T^*}^2A^*h,h\ranglengle=\langlengle \lim_n T^nAA^*T^{*n} h,h\ranglengle \leq \lim_n\langlengle T^nT^{*n}h,h\ranglengle=\langlengle Q_{T^*}^2h,h \ranglengle.
\end{align*}
The same computation for the contraction $B$ will yield the same inequality involving $B$ in place of $A$. Consequently, the operators
$A_{T^*}.B_{T^*}:\overline{\operatorname{Ran}}\;Q_{T^*}\to\overline{\operatorname{Ran}}\;Q_{T^*}$ defined densely by
\begin{align}\langlebel{AB}
A_{T^*}^*Q_{T^*}h=Q_{T^*}A^*h \quad\text{and}\quad B_{T^*}^*Q_{T^*}h=Q_{T^*}B^*h
\end{align}
are contractions, and extend contractively to all of $\overline{\operatorname{Ran}}\, Q_{T^*}$ by a limiting process.
Furthermore, from the definitions it is easy to see that $(A_{T^*}, B_{T^*}, X_{T^*})$ is a commutative triple since
by assumption we know that $(A^*, B^*, T^*)$ is commutative.
This and \eqref{theX} imply that if $f$ is a three-variable polynomial, then for all $h\in{\mathcal H}$,
\begin{align*}
& \| f(A_{T^*}^*,B_{T^*}^*,X_{T^*}^*) Q_{T^*} h\| = \| Q_{T^*} f(A^*,B^*,T^*) h\| \\
& \quad \le \| f(A^*, B^*, T^*) \| \| h || \leq \| f(A^*,B^*,T^*)\| \| h \| \leq \big( \sup_{\mathbb E} |f| \big) \| h \|
\end{align*}
where in the last inequality we used the fact that $(A,B,T)$ is a tetrablock contraction. This inequality together with the fact that $X_{T^*}^*$ is an isometry implies that
$(A_{T^*}^*,B_{T^*}^*,X_{T^*}^*)$ is a tetrablock isometry. By Corollary \ref{C:pc}, $(A_{T^*}^*,B_{T^*}^*,X_{T^*}^*)$ has a tetrablock unitary extension
$(R_D^*,S_D^*,W_D^*)$ acting on a space which we shall call ${\mathcal Q}_{T^*} \supseteq \overline{\operatorname{Ran }}\;Q_{T^*}$,
where $W_D^*$ acting on ${\mathcal Q}_{T^*}$ is the minimal unitary extension of $X_{T^*}^*$.
\begin{definition}\langlebel{D:canonical}
Let $(A,B,T)$ be a tetrablock contraction and let $(R_D,S_D,W_D)$ be the tetrablock unitary constructed from $(A,B,T)$ as above. We say that $(R_D,S_D,W_D)$
is the {\bf canonical tetrablock unitary} associated with the tetrablock contraction $(A,B,T)$.
\end{definition}
The next result assures us that canonical tetrablock unitaries associated with the same tetrablock contraction $(A,B,T)$ are the same up to unitary equivalence.
\begin{thm}\langlebel{T:UniCanon}
Let $(A,B,T)$ on ${\mathcal H}$ and $(A',B',T')$ on ${\mathcal H}'$ be two tetrablock contractions with $(R_D,S_D,W_D)$ and $(R_D',S_D',W_D')$ equal to the respective canonical
tetrablock unitaries. If $(A,B,T)$ and $(A',B',T')$ are unitarily equivalent via $\tau$, then $(R_D,S_D,W_D)$ and $(R_D',S_D',W_D)$ are unitarily equivalent via
$\omega_\tau:{\mathcal Q}_{T^*}\to{\mathcal Q}_{T^{'*}}$
\begin{align}\langlebel{omega-tau}
\omega_\tau: W_D^n Q_{T^*}h\mapsto W_D^{'n} Q_{T'^*}\tau h
\end{align}for all $n\geq0$ and $h\in{\mathcal H}$.
\end{thm}
\begin{proof}
Let the spaces ${\mathcal Q}_{T^*}, {\mathcal Q}_{T^{'*}}$ and the operators $\{A_{T^*},B_{T^*},Q_{T^*}\}$, $\{A_{T'^*},B_{T'^*},Q_{T'^*}\}$ be obtained as above from $(A,B,T)$ and
$(A',B',T')$, respectively. Since $\tau$ is a unitary intertwining $T$ and $T'$, it intertwines $T^*$ and $T'^*$ and thus $\tau Q_{T^*}=Q_{T'^*} \tau$.
Therefore by definition \eqref{AB} it follows that $\tau(A_{T^*},B_{T^*},Q_{T^*})=(A_{T'^*},B_{T'^*},Q_{T'^*})\tau$.
By definition of $\omega_\tau$ it is clear that $\omega_\tau W_D=W_D'\omega_\tau$. Therefore for every $h\in{\mathcal H}$ and $n\geq 0$,
\begin{align*}
\omega_\tau R_DW_D^n Q_{T^*}h&=\omega_\tau W_D^{n+1}W_D^*R_DQ_{T^*}h\\
&=\omega_\tau W_D^{n+1}S_D^*Q_{T^*}h \quad \mbox{[using Theorem \ref{T:IsoChar}, part (iv)]}\\
&=W_D'^{(n+1)}\tau B_{T^*}^*Q_{T^*}h=W_D'^{(n+1)} B_{T'^*}^* Q_{T'^*}\tau h\\
&=W_D'^{(n+1)} S_D'^* Q_{T'^*}\tau h\\
&=S_D'^*W_D'^{(n+1)}Q_{T^*} \quad \mbox{[since $S_D'^*W_D'=W_D'S_D'^*$]}\\
&=R_D' W_D'^nQ_{T'^*}\tau h\quad \mbox{[using Theorem \ref{T:IsoChar}, part (iii)]}\\
&=R_D'\omega_\tau W_D^nQ_{T^*}h.
\end{align*}
A similar computation shows that $\omega_\tau S_D=S_D'\omega_\tau$.
\end{proof}
\subsection{The fundamental operators}\langlebel{SS:FundOps}
Much of the theory of tetrablock contractions is heavily based on a pair of operators that is uniquely associated with a tetrablock contraction. These are called the
fundamental operators, the existence of which was proved in \cite{Tirtha14} with appeal to connections between tetrablock contractions and symmetrized-bidisk
contractions. We state the result and sketch a more self-contained proof with the appeal to symmetrized-bidisk theory eliminated.
In the sequel we shall use the notation $\nu(X)$ to denote the numerical radius of the operator $X$ on the Hilbert space ${\mathcal H}$:
$$
\nu(X):= \sup_{x \in {\mathcal H}\colon \| x \| =1} | \langlengle X x, x \ranglengle_{\mathcal H} |.
$$
\begin{thm}\langlebel{T:FundTetra}
Let $(A,B,T)$ be a tetrablock contraction on a Hilbert space $\mathcal H$.
\begin{enumerate}
\item[(i)]{\rm (See \cite[Theorem 3.4]{Tirtha14})} There exist two unique operators $F_1$ and $F_2$ acting on ${\mathcal D}_T$ with the numerical radii at most one such that
\begin{align}\langlebel{FundEqns}
A-B^*T=D_TF_1D_T \quad\text{and}\quad B-A^*T=D_TF_2D_T.
\end{align}
Moreover, the operators $F_1,F_2$ are such that $\nu(F_1+zF_2)\leq1$ for all $z\in \overline{\mathbb{D}}$.
\item[(ii)]{\rm (See \cite[Corollary 4.2]{Tirtha14})} The operators $F_1,F_2$ are alternatively characterized as the unique bounded operators on
${\mathcal D}_T$ such that $(X_1, X_2) = (F_1, F_2)$ satisfies the system of operator equations
\begin{align}\langlebel{Determining}
D_TA=X_1D_T+X_2^*D_TT \quad\text{and}\quad D_TB=X_2D_T+X_1^*D_TT.
\end{align}
\end{enumerate}
\end{thm}
\begin{proof}
Let $(A,B,T)$ be a tetrablock contraction on ${\mathcal H}$. Since for every $z\in{\mathbb D}$, $\Psi(z,\cdot)$ as in item (ii) of Theorem \ref{T:CharacTetra} is analytic in an open set
containing $\overline{\bE}$, and $\overline{\bE}$ is polynomially convex, a limiting argument implies that $\Psi(\zeta,(A,B,T))$ is a contraction for every $\zeta\in{\mathbb{T}}$,
or equivalently, on simplifying $I-\Psi(\zeta,(A,B,T))^*\Psi(\zeta,(A,B,T))\succeq 0$ we get
\begin{align*}
(I-T^*T)+(B^*B-A^*A) - \zeta(B-A^*T) - \overline{\zeta}(B-A^*T)^*\succeq 0.
\end{align*}Similarly, applying item (iii) of Theorem \ref{T:CharacTetra} we have for every $\alpha\in{\mathbb{T}}$,
\begin{align*}
(I-T^*T)+(A^*A-B^*B) - \alpha(A-B^*T) - \overline{\alpha}
(A-B^*T)^*\succeq 0.
\end{align*}
On adding the above two positive operators and then simplifying we get
\begin{align}\langlebel{Posit3}
D_T^2 \succeq \operatorname{Re}\alpha \left[(A-B^*T)+\beta(B-A^*T)\right]
\end{align}for every $\alpha,\beta\in{\mathbb{T}}$.
We now make use the following lemma of independent interest:
\begin{lemma}{\rm (See \cite[Lemma 4.1]{B-P-SR})} \langlebel{L:Tirtha14}
Let $\Sigma$ and $D$ be two operators such that
$$
DD^*\succeq\operatorname{Re}\alpha\Sigma \quad\mbox{for all }\alpha\in{\mathbb{T}}.
$$
Then there exists an operator $F$ acting on $\overline{\operatorname{Ran}} \, D^*$ with numerical radius at most one such that $\Sigma=DFD^*$.
\end{lemma}
\begin{proof}[Sketch of proof.] Apply the Fejer-Riesz factorization theorem of Dritschel-Rovnyak \cite[Theorem 2.1]{DM-FRThm} to the Laurent operator-valued polynomial
$P(e^{i\theta}) = 2DD^*-e^{i\theta}\Sigma-e^{-i\theta}\Sigma^*$. Along the way one makes use of the standard Douglas lemma ($\exists$ $X \in {\mathcal B}({\mathcal H})$ with
$AX = B$ $\Leftrightarrow$ $B B^* \preceq A A^*$) and a criterion for a
Hilbert space operator to have numerical radius at most $1$: $X \in {\mathcal B}({\mathcal H})$ has $\nu(X) \le 1$ $\Leftrightarrow$ $\operatorname{Re} \, ( \beta X)
\preceq I_{\mathcal H}$ for all $\beta \in {\mathbb T}$. Note that Lemma \ref{L:Tirtha14} can itself be viewed as a quadratic, numerical-radius version of the Douglas lemma.
\end{proof}
We apply Lemma \ref{L:Tirtha14} to the case \eqref{Posit3} for each $\beta$ to get a numerical contraction $F(\beta)$ such that
\begin{align}\langlebel{FundEqnsBeta}
(A-B^*T)+\beta(B-A^*T)=D_TF(\beta)D_T.
\end{align}
On adding equations \eqref{FundEqnsBeta} for the cases $\beta=1$ and $-1$, we get
\begin{align}\langlebel{beta=1,-1}
A-B^*T=D_TF_1D_T\quad\mbox{where}\quad F_1:=\frac{F(1)+F(-1)}{2}.
\end{align}
Thus putting $\beta=1$ in \eqref{FundEqnsBeta} and combining with \eqref{FundEqnsBeta} gives us
\begin{equation} \langlebel{beta=1}
B-A^*T=D_T(F(1)-F_1)D_T=D_TF_2D_T\quad\mbox{where}\quad F_2:=\frac{F(1)-F(-1)}{2}.
\end{equation}
We conclude that $F_1$ and $F_2$ so constructed satisfy equations \eqref{FundEqns}. It is easy to see that in general
\begin{equation} \langlebel{easy}
X \in {\mathcal B}({\mathcal D}_T) \text{ with } D_T X D_T = 0 \Rightarrow X = 0.
\end{equation}
Applying this observation to the homogeneous version of equations \eqref{FundEqns} implies that the solutions $(F_1, F_2)$
of \eqref{FundEqns} must be unique whenever they exist.
On the other hand, if we combine \eqref{beta=1,-1} with \eqref{beta=1} we see that
$\widetilde F(\beta) := F_1 + \beta F_2$ gives us a second solution of \eqref{FundEqnsBeta}.
Hence
$$
D_T (\widetilde F(\beta) - F(\beta)) D_T = 0 \text{ where } \widetilde F(\beta) - F(\beta) \in {\mathcal B}({\mathcal D}_T) \text{ for all } \beta \in {\mathbb T}.
$$
Again by \eqref{easy}, we see that
$\widetilde F(\beta) = F(\beta)$ for all $\beta \in {\mathbb T}$. But we saw above (as a consequence of Lemma \ref{L:Tirtha14}) that
$F(\beta)$ is a numerical contraction for all $\beta \in {\mathbb T}$. As we now know that $\widetilde F(\beta) = F(\beta)$, we conclude that
the pencil $\widetilde F(\beta) = F_1 + \beta F_2$ is a numerical contraction for all $\beta \in {\mathbb T}$. By applying the Maximum Modulus Theorem to the
holomorphic function $\langlengle (F_1 + \beta F_2)h, h \ranglengle$ for each fixed $h \in {\mathcal H}$, we see that $F_1 + z F_2$ is a numerical contraction
for all $z \in \overline{\mathbb D}$. This completes the proof of item (i) in Theorem \ref{T:FundTetra}.
To see that $F_1$ and $F_2$ satisfy equations \eqref{Determining}, simply multiply $D_T$ on the left of each equation and use the identities \eqref{FundEqns}
to simplify. To show that $F_1,F_2$ are the unique operators on ${\mathcal D}_T$ satisfying these two equations, it is enough to show that $X= 0$ and $Y=0$
are the only operators in ${\mathcal B}({\mathcal D}_T)$ satisfying
$$
XD_T+Y^*D_TT=0, \quad YD_T+X^*D_TT=0.
$$
To show that $X=0$, compute
\begin{align*}
D_TXD_T=-D_TY^*D_TT=T^*D_TXD_TT&=-T^*D_TY^*D_TT^2\\
&=T^{*2}D_TXD_TT^2.
\end{align*}
Thus by iteration of the above process $D_TXD_T=T^{*n}D_TXD_TT^n$. This shows that $X=0$ because for every $h\in{\mathcal H}$
$$
\lim_n\|D_TT^nh\|^2=\lim_n\|T^nh\|^2-\lim_n\|T^{n+1}h\|^2=0.
$$
A similar argument gives $Y=0$. This is the idea of the proof due by Bhattacharyya \cite{Tirtha14}.
\end{proof}
The unique operators $F_1,F_2$ in item (i) of Theorem \ref{T:FundTetra} will be referred to as the {\bf fundamental operators} for the tetrablock contraction $(A,B,T)$, as in \cite{Tirtha14}.
As we have seen in Proposition \ref{P:symmetries}, if $(A,B,T)$ is a ${\mathbb E}$-contraction, so also is $(A^*, B^*, T^*)$.
For the construction of the functional model for a tetrablock contraction $(A,B,T)$,
it turns out to be more convenient to work with the fundamental operators for the
adjoint tetrablock contraction $(A^*, B^*, T^*)$ which we denote as $(G_1, G_2)$. Thus there exists exactly one solution
$(X_1, X_2) = (G_1, G_2)$ of the system of equations
\begin{equation} \langlebel{FundEquns*}
A^* - B T^* = D_{T^*} X_1 D_{T^*}, \quad B^* - A T^* = D_{T^*} X_2 D_{T^*}
\end{equation}
with equivalent characterization as the unique solution $(X_1, X_2) = (G_1, G_2)$ of the second system of equations
\begin{equation} \langlebel{Determining*}
D_{T^*} A^* = X_1 D_{T^*} + X_2^* D_{T^*} T^*, \quad D_{T^*} B^* = X_2 D_{T^*} + X_1^* D_{T^*} T^*
\end{equation}
Then from Example \ref{E:model} we see immediately that the operator pair $(M_{G_1^* + z G_2}, M_{G_2^* + z G_1}, M_z)$ acting on the Hardy space
$H^2({\mathcal D}_{T^*})$ is of the correct form to be
a pseudo-commutative tetrablock isometry. We would like to establish conditions under which this a priori only pseudo-commutative ${\mathbb E}$-isometry
is actually a strict ${\mathbb E}$-isometry. This follows from the following result.
\begin{thm} \langlebel{T:pseudo-strict} {\rm (See \cite{Tirtha14}.)} Suppose that $(G_1, G_2)$ is the Fundamental Operator pair for the ${\mathbb E}$-contraction
$(A^*, B^*, T^*)$. Let $(V_1, V_2, V_3)$ be the operator triple
\begin{equation} \langlebel{Eisom-candidate}
(V_1, V_2, V_2) = (M_{G_1^* + z G_2}, M_{G_2^* + z G_1}, M_z) \text{ on } H^2({\mathcal D}_{T^*}).
\end{equation}
Then:
\begin{enumerate}
\item $(V_1, V_2, V_3)$ is a pseudo-commutative tetrablock isometry having the additional property that
$$
r(V_1) \le 1, \quad r(V_2) \le 1.
$$
\item Suppose in addition that the Fundamental Operator pair $(G_1, G_2)$ satisfy the commutativity conditions. Then
$(V_1, V_2, V_3)$ is a strict tetrablock isometry.
\end{enumerate}
\end{thm}
\begin{corollary} \langlebel{C:strict} Let $(G_1, G_2)$ be the Fundamental Operator pair for the tetrablock contraction $(A^*, B^*, T^*)$ and set $(V_1, V_2, V_3)$
as in \eqref{Eisom-candidate}. Then $(V_1, V_2, V_3)$ is a (strict) tetrablock isometry if and only if the commutativity conditions \eqref{com=} hold.
\end{corollary}
\begin{proof}[Proof of Corollary \ref{C:strict}] If the commutativity conditions \eqref{com=} are satisfied, then statement (2) of Theorem \ref{T:pseudo-strict}
says that $(V_1, V_2, V_3)$ is a tetrablock isometry. Conversely, if $(V_1, V_2, V_3)$ is a tetrablock isometry, in particular $(V_1, V_2, V_3)$ is
commutative so the commutativity conditions \eqref{com=} are satisfied.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:pseudo-strict}]
We first consider statement (1). That $(V_1, V_2, V_3)$ is a pseudo-commutative tetrablock isometry follows from the fact that it has the required form \eqref{pcform}
as presented in Example \ref{E:model}. It remains to use the fact that $(G_1, G_2)$ is a Fundamental Operator pair for the tetrablock contraction $(A^*, B^*, T^*)$
(to see that we also have $r(V_1) \le 1$ and $r(V_2) \le 1$.
By Theorem \ref{T:FundTetra} (applied to $(A^*, B^*, T^*)$ in place of $(A,B,T)$), we know that
$\nu(G_1 + z G_2) \le 1$ for all $z \in \overline{\mathbb D}$. By the And\^o criterion for the numerical radius of an operator to be no more than 1, this means that
$$
\beta (G_1 + \alpha G_2) + \overline{\beta} (G_1^* + \overline{\alpha} G_2^*) \preceq 2 I_{{\mathcal D}_{T^*}} \forall \alpha, \beta \in \overline{\mathbb D}.
$$
Rearrange this inequality as
\begin{align*}
& (\overline{\beta} G_1^* + \beta \alpha G_2) + ( \beta G_1 + \overline{\beta} \overline{\alpha} G_2^*) \\
& \quad = \overline{\beta} (G_1^* + \beta^2 \alpha G_2) + \beta (G_1 + \overline{\beta}^2 \overline{\alpha} G_2^*) \preceq 2I_{{\mathcal D}_{T^*}}.
\end{align*}
By the And\^o criterion applied in the reverse direction, this tells us that
$$
\nu(G_1^* + z G_2) \le 1 \text{ for all } z \in \overline{\mathbb D}.
$$
But in general the numerical radius dominates the spectral radius; thus
$$
r(G_1^* + z G_2) \le \nu(G_1^* + z G_2) \le 1 \text{ for all } z \in \overline{\mathbb D}.
$$
If we choose $\langlem \in {\mathbb C}$ with $|\langlem| > 1$, then $\langlem I_{{\mathcal D}_{T^*}} - (G_2^* + z G_1)$ is invertible, and in fact
the operator-valued function $z \mapsto (\langlem I_{{\mathcal D}_{T^*}} - (G_2^* + z G_1))^{-1}$ is in $H^\infty({\mathcal B}({\mathcal D}_{T^*})$. We thus conclude that in fact
$$
r(M_{G_2^* + z G_1}) \le 1.
$$
All the above analysis applies to the pair $(G_2, G_1)$ in place of $(G_1, G_2)$, as $(G_2, G_1)$ is the Fundamental Operator pair for the ${\mathbb E}$-contraction
$(B^*, A^*, T^*)$; hence we also have also
$$
r(M_{G_2^* + z G_1}) \le 1.
$$
This completes the proof of statement (1).
We now consider statement (2). As we are now assuming that $(G_1, G_2)$ satisfy the commutativity conditions \eqref{com=}, it follows that
$M_{G_1^* + z G_2}$ commutes with $M_{G_2^* + z G_1}$.
We are now in a position to apply statement (3) in Remark \ref{R:pseudocom} (or, what is the same, statement (2) in Theorem \ref{T:pc-vs-strict-Eisom}
to conclude that the a priori only pseudo-commutative ${\mathbb E}$-contraction
$ (M_{G_1^* + z G_2}, M_{G_2^* + z G_1},$ $ M_z)$ is in fact a strict ${\mathbb E}$-contraction.
\end{proof}
\section{Functional models for tetrablock contractions}\langlebel{S:fmTetr}
In this section we produce two functional models for tetrablock contractions, the first inspired by model theory of Douglas
\cite{Doug-Dilation}, and the second by the model theory of Sz.-Nagy and Foias \cite{Nagy-Foias}. We note that so far only a functional model is
known for the special case when the last entry is a pure contraction; see \cite[Theorem 4.2]{SauNYJM}.
\subsection{A Douglas-type functional model}\langlebel{SS:fmTet}
Let $T$ be any contraction on ${\mathcal H}$. Define the operators ${\mathcal O}_{D_{T^*}, T^*}:{\mathcal H}\to H^2({\mathcal D}_{T^*})$ as
\begin{align}\langlebel{observ}
{\mathcal O}_{D_{T^*}, T^*}(z) h = \sum_{n=0}^\infty z^nD_{T^*} T^{*n} h, \text{ for every }h\in{\mathcal H},
\end{align}
and $\Pi_D:{\mathcal H}\to\sbm{H^2({\mathcal D}_{T^*})\\ {\mathcal Q}_{T^*}}$ by
\begin{align}\langlebel{Pi-D}
\Pi_Dh=
\begin{bmatrix}
{\mathcal O}_{D_{T^*}, T^*}(z) h \\ Q_{T^*}h\end{bmatrix} \quad\text{for all } h\in{\mathcal H},
\end{align}
where $Q_{T^*}$ is as in \eqref{Q}. Then the computation
\begin{align*}
\|\Pi_D h\|^2=&\| {\mathcal O}_{D_{T^*}, T^*}(z) h\|_{H^2({\mathcal D}_{T^*})}^2+ \|Q_{T^*}h\|^2\\
&=\sum_{n=0}^\infty \|D_{T^*} T^{*n} h\|^2+\lim_{n\to\infty}\|T^{*n}h\|^2\\
&=(\|h\|^2-\|T^*h\|^2)+(\|T^{*}h\|^2-\|T^{*2}h\|^2)+\cdots) +\lim_{n\to\infty}\|T^{*n}h\|^2\\
&=\|h\|^2
\end{align*}
shows that $\Pi_D$ is an isometry. Note the following intertwining property of ${\mathcal O}_{D_{T^*}, T^*}$:
\begin{align}
{\mathcal O}_{D_{T^*}, T^*}(z)T^*h & =\sum_{n=0}^\infty z^nD_{T^*} T^{*n+1} h=M_z^*\sum_{n=0}^\infty z^nD_{T^*} T^{*n} h \notag \\
& =M_z^* {\mathcal O}_{D_{T^*}, T^*}(z)h.
\end{align}
This together with intertwining \eqref{theX} of $Q_{T^*}$ implies
\begin{align}\langlebel{lift}
\Pi_DT^*=
\begin{bmatrix}
M_z&0\\0&W_D
\end{bmatrix}^*\Pi_D.
\end{align}
This shows that the the pair
$$
V_D:=\begin{bmatrix} M_z&0\\0&W_D \end{bmatrix}: \begin{bmatrix} H^2({\mathcal D}_{T^*})\\ {\mathcal Q}_{T^*} \end{bmatrix}
\to \begin{bmatrix} H^2({\mathcal D}_{T^*})\\ {\mathcal Q}_{T^*} \end{bmatrix}
$$
is an isometric lift of $T$. This construction is by Douglas, where he also showed that this lift is minimal (see \cite{Doug-Dilation}).
Now let $(A,B,T)$ be a tetrablock contraction acting on ${\mathcal H}$ and $G_1,G_2$ be the fundamental operators of $(A^*,B^*,T^*)$.
Let $(R_D,S_D,W_D)$ acting on ${\mathcal Q}_{T^*}$ be the canonical tetrablock unitary associated with $(A,B,T)$. Consider the operators
\begin{align}
\left(\begin{bmatrix}
M_{G_1^*+zG_2}&0\\0&R_D
\end{bmatrix},\begin{bmatrix}
M_{G_2^*+zG_1}&0\\0&S_D
\end{bmatrix},\begin{bmatrix}
M_z&0\\0& W_D
\end{bmatrix}\right)
\mbox{ on }\begin{bmatrix}
H^2({\mathcal D}_{T^*})\\ {\mathcal Q}_{T^*}
\end{bmatrix}.
\end{align}
We claim that
\begin{align}\langlebel{claim}
\notag &\Pi_D(A^*,B^*,T^*)\\
&=
\left(\begin{bmatrix}
M_{G_1^*+zG_2}&0\\0&R_D
\end{bmatrix}^*,\begin{bmatrix}
M_{G_2^*+zG_1}&0\\0&S_D
\end{bmatrix}^*,\begin{bmatrix}
M_z&0\\0& W_D
\end{bmatrix}^*\right)\Pi_D,
\end{align}
where $\Pi_D:{\mathcal H}\to \sbm{H^2({\mathcal D}_{T^*})\\ {\mathcal Q}_{T^*}}$ is the isometry as in \eqref{Pi-D}.
Recalling the definition $\Pi_D = \sbm{ {\mathcal O}_{D_{T^*}, T^*} \\ {\mathcal Q}_{T^*}} \colon {\mathcal H} \to \sbm{H^2({\mathcal D}_{T^*}) \\ {\mathcal Q}_{T^*}}$, we see that the three-fold
intertwining condition \eqref{claim} splits into two three-fold intertwining conditions
\begin{align}
& {\mathcal O}_{D_{T^*}, T^*} (A^*, B^*, T^*) = (M^*_{G_1^* + z G_2}, M^*_{G_2^* + z G_1}, M^*_z) {\mathcal O}_{D_{T^*}, T^*}, \langlebel{claim1} \\
& Q_{T^*} (A^*, B^*, T^*) = (R^*_D, S^*_D, W^*_D) Q_{T^*}. \langlebel{claim2}
\end{align}
The last equation in \eqref{claim1} combined with the last equation in \eqref{claim2} we have already seen as the condition that $\Pi_D$ is the isometric
identification map implementing $\sbm{ M_z & 0 \\ 0 & W_D}$ as the Douglas minimal isometric lift of $T$ (see \eqref{lift}). We shall next check only the first equation
in \eqref{claim1} and the first equation in \eqref{claim2} as the verification of the respective second equations is completely analogous. Thus it remains to check
\begin{align}
& {\mathcal O}_{D_{T^*}, T^*} A^* = M^*_{G_1^* + z G_2} {\mathcal O}_{D_{T^*}, T^*}, \langlebel{claim1'} \\
& Q_{T^*} A^* = R^*_D Q_{T^*}. \langlebel{claim2'}
\end{align}
Note that \eqref{claim2'} is part of the construction of the canonical tetrablock unitary associated with the original tetrablock contraction $(A,B,T)$ (see \eqref{AB}).
To check \eqref{claim1'}, let us rewrite the condition in function form:
$$
D_{T^*} (I - z T^*)^{-1} A^* = G_1 D_{T^*} (I - z T^*)^{-1} + G_2^* D_{T^*}(I - z T^*)^{-1} T^*
$$
As $A$ commutes with $T$, we can rewrite this as
$$
D_{T^*} A^* (I - z T^*)^{-1} = (G_1 D_{T^*} + G_2^* D_{T^*} T^*) (I - z T^*)^{-1}.
$$
We may now cancel off the resolvent term $(I - z T^*)^{-1}$ to get a pure operator equation
\begin{equation} \langlebel{need}
D_{T^*} A^* = G_1 D_{T^*} + G_2^* D_{T^*} T^*.
\end{equation}
Let us now recall that the operators $(G_1, G_2)$ on ${\mathcal D}_{T^*}$ were chosen to be the Fundamental Operators for the tetrablock contraction
$(A^*, B^*, T^*)$. Thus by our earlier discussion of Fundamental Operators for tetrablock contractions (see Theorem \ref{T:FundTetra}), we know that
$(X_1, X_2) = (G_1, G_2)$ satisfies the identities \eqref{Determining*}, the first of which gives the same condition on $(G_1, G_2)$ as \eqref{need}.
Thus this choice of $(G_1, G_2)$ indeed leads to a solution of \eqref{claim1'}, and the proof of \eqref{claim} is complete. (Of course the second equation
in \eqref{Determining*} amounts to the verification of the second equation in \eqref{claim1}.)
This then means that the operator triple with isometric embedding operator $\Pi_D$
\begin{align}
& (V_1, V_2, V_3) := \notag \\
& \quad\bigg(\Pi_D, \bigg( \begin{bmatrix} M_{G_1^* + z G_2} & 0 \\ 0 & R_D \end{bmatrix},
\begin{bmatrix} M_{G_2^* + z G_1} & 0 \\ 0 & S_D \end{bmatrix}, \begin{bmatrix} M_z & 0 \\ 0 & W_D \end{bmatrix} \bigg)\bigg) \langlebel{candidate}
\end{align}
is a lift of the tetrablock contraction $(A,B,T)$. As $(R_D, S_D, W_D)$ is ${\mathbb E}$-unitary as part of the canonical construction in Section
\ref{SS:canonical}, we see from the form of the top components of $(V_1, V_2, V_3)$ in \eqref{candidate} that $(V_1, V_2, V_3)$ is a semi-strict
${\mathbb E}$-isometry and is a strict ${\mathbb E}$-isometry exactly when the top-component triple
$( M_{G_1^* + z G_2}, M_{G_2^* + z G_1}, M_z)$ is a strict ${\mathbb E}$-isometry. By Theorem \ref{T:pseudo-strict}, this in turn happens exactly
when the Fundamental Operator pair $(G_1, G_2)$ satisfies the commutativity conditions \eqref{com=}.
In summary we have verified most of the following result. We note that item (2) recovers a result of Bhattacharyya-Sau \cite{BS-CAOT}
via functional-model methods rather than by block-matrix-construction methods.
\begin{thm}\langlebel{T:Dmodel}
{\rm (1)}
Let $(A,B,T)$ be a tetrablock contraction on ${\mathcal H}$ and let $V_3$ on ${\mathcal K} \supset {\mathcal H}$ be the (essentially unique) minimal isometric lift for the
contraction operator $T$.
Then there is a unique choice of operators $(V_1, V_2)$ on ${\mathcal K}$ so that the triple ${\mathbf V} = (V_1, V_2, V_3)$ is a semi-strict tetrablock isometric lift for $(A,B,T)$.
\noindent
{\rm (2)} A necessary and sufficient condition that there be a strict tetrablock isometric lift
${\mathbf V} = (V_1, V_2, V_3)$ for $(A,B,T)$ with the isometry $V_3$ equal to a minimal isometric lift for $T$ is that the fundamental operators $(G_1, G_2)$ for the adjoint tetrablock contraction $(A^*, B^*, T^*)$ satisfy the system of operator equations \eqref{com=}. In this case the operator pair $(V_1, V_2)$ on ${\mathcal K}$
is uniquely determined once one fixes a choice (essentially unique) for a minimal isometric lift $V_3$ for $T$.
\noindent
{\rm (3)}
The lift $(V_1, V_2, V_3)$ can be given in functional form in the coordinates of the Douglas model as follows.
For $(A,B,T)$ equal to a tetrablock contraction on a Hilbert space $\mathcal{H}$ let $(G_1,G_2)$ be the fundamental operators of $(A^*,B^*,T^*)$,
let $(R_D, S_D, W_D)$ be the tetrablock unitary canonically associated with $(A,B,T)$ as in Definition \ref{D:canonical},
and let
$\Pi_D = \sbm{ {\mathcal O}_{D_{T^*}, T^*} \\ Q_{T^*}}$ be the Douglas isometric embedding map from ${\mathcal H}$ into $\sbm{ H^2({\mathcal D}_{T*}) \\ {\mathcal Q}_{T^*} }$.
Then
\begin{equation} \langlebel{unique-lift}
\bigg( \Pi_D, \bigg( \begin{bmatrix} M_{G_1^* + z G_2} & 0 \\ 0 & R_D \end{bmatrix},
\begin{bmatrix} M_{G_2^* + z G_1} & 0 \\ 0 & S_D \end{bmatrix}, \begin{bmatrix} M_z & 0 \\ 0 & W_D \end{bmatrix} \bigg) \bigg)
\end{equation}
is a semi-strict (strict exactly when $(G_1, G_2)$ satisfies \eqref{com=}) tetrablock isometric lift for $(A,B,T)$. In particular $(A,B,T)$ is jointly unitarily equivalent to
\begin{align}\langlebel{SemiModel}
P_{{\mathcal H}_D}\left(\begin{bmatrix} M_{G_1^*+zG_2}&0\\0&R_D \end{bmatrix},\begin{bmatrix} M_{G_2^*+zG_1}&0\\0&S_D\end{bmatrix},
\begin{bmatrix} M_z&0\\0& W_D \end{bmatrix}\right) \bigg|_{{\mathcal H}_D},
\end{align}
where ${\mathcal H}_D$ is the functional model space given by
\begin{align}\langlebel{HD}
{\mathcal H}_D:=\operatorname{Ran}\Pi_D\subset \begin{bmatrix} H^2({\mathcal D}_{T^*}) \\ {\mathcal Q}_{T^*} \end{bmatrix}.
\end{align}
and any other semi-strict ${\mathbb E}$-isometric lift $(V'_1, V'_2, V'_3)$ with $V'_3 = \sbm{ M_z & 0 \\ 0 & W_D}$ on $\sbm{ H^2({\mathcal D}_{T^*}) \\ {\mathcal Q}_{T^*}}$
is equal to \eqref{unique-lift}.
\end{thm}
\begin{proof} The discussion preceding the statement of the theorem amounts to a proof of statement (3) in Theorem \ref{T:Dmodel}. Statements (1) and (2)
apart from the uniqueness assertion amounts to a coordinate-free (abstract, model-free) interpretation of the results of statement (3). It remains only
to discuss the uniqueness assertion in statements (1) and (2). This can also be formulated in terms of the model as follows:
{\em Given a tetrablock contraction $(A,B,T)$, let $\Pi_D = \sbm{ {\mathcal O}_{D_{T^*}, T^*} \\ Q_{T^*}}$ be the Douglas embedding map and let ${\mathcal K}_D= \sbm{ H^2({\mathcal D}_{T^*})
\\ {\mathcal Q}_{T^*}}$ be the Douglas minimal isometric lift space for $T$ with $V_D = \sbm{ M_z & 0 \\ 0 & W_D}$ on ${\mathcal K}_D$ equal to the Douglas minimal
isometric lift for $T$. Suppose that
$$
\widetilde A = \begin{bmatrix} \widetilde A_{11} & \widetilde A_{12} \\ \widetilde A_{21} & \widetilde A_{22} \end{bmatrix}, \quad
\widetilde B = \begin{bmatrix} \widetilde B_{11} & \widetilde B_{12} \\ \widetilde B_{21} & \widetilde B_{22} \end{bmatrix}
$$
are two operators on ${\mathcal K}_D = \sbm{ H^2({\mathcal D}_{T^*} \\ {\mathcal Q}_{T^*} }$ such that
$ \big(\widetilde A, \widetilde B, \sbm{ M_z & 0 \\ 0 & W_D}\big)$ is a pseudo-commutative tetrablock isometric lift for $T$. Then necessarily}
$$
\widetilde A = \begin{bmatrix} M_{G_1^* + z G_2} & 0 \\ 0 & R_D \end{bmatrix} , \, \widetilde B = \begin{bmatrix} M_{G_2^* + z G_1} & 0 \\ 0 & S_D \end{bmatrix}.
$$
{\em where the pair $(G_1, G_2)$ is equal to the pair of Fundamental Operators for the tetrablock contraction $(A^*, B^*, T^*)$, and where
$(R_D, S_D, W_D)$ is the tetrablock unitary canonically associated with the tetrablock contraction $(A, B, T)$ as in Definition \ref{D:canonical}.}
To prove this model-theoretic reformulation of the uniqueness problem, we proceed as follows. We are given first of all that the triple
\begin{align}\langlebel{pcLift}
\bigg( \widetilde A = \begin{bmatrix} \widetilde A_{11} & \widetilde A_{12} \\ \widetilde A_{21} & \widetilde A_{22} \end{bmatrix}, \quad
\widetilde B = \begin{bmatrix} \widetilde B_{11} & \widetilde B_{12} \\ \widetilde B_{21} & \widetilde B_{22} \end{bmatrix}, \quad
\begin{bmatrix} M_z & 0 \\ 0 & W_D \end{bmatrix} \bigg)
\end{align}
is a pseudo-commutative tetrablock isometry. According to Definition \ref{D:pc} we have the following:
\begin{itemize}
\item{(i)} $ \widetilde A \sbm{ M_z & 0 \\ 0 & W_D} = \sbm{ M_z & 0 \\ 0 & W_D} \widetilde A$ and $ \widetilde B \sbm{ M_z & 0 \\ 0 & W_D} = \sbm{ M_z & 0 \\ 0 & W_D} \widetilde B$;
\item{(ii)} $\widetilde B = \widetilde A^* \sbm{ M_z & 0 \\ 0 & W_D },$ and $\widetilde B = \widetilde A^* \sbm{ M_z & 0 \\ 0 & W_D }$; and
\item{(iii)} $\| \widetilde A \| \le 1$.
\end{itemize}
As in the proof of Theorem \ref{T:WoldPC}, conditions (i) ad (ii) force
$\widetilde A$ and $\widetilde B$ to have the block-diagonal form
$$(\widetilde A,\widetilde B) = \left(\begin{bmatrix}
M_{\widetilde G_1^*+z\widetilde G_2} & 0 \\ 0 & \widetilde A_{22}
\end{bmatrix} , \begin{bmatrix} M_{\widetilde G_2^*+z\widetilde G_1}& 0 \\ 0 & \widetilde B_{22}\end{bmatrix}\right).$$
for some pseudo-commutative tetrablock unitary $(\widetilde A_{22},\widetilde B_{22},W_D)$, and operators $\widetilde G_1,\widetilde G_2\in{\mathcal B}({\mathcal D}_{T^*})$ so that the linear pencils $\widetilde G_1^*+z\widetilde G_2$ and $\widetilde G_1^*+z\widetilde G_2$ are contraction-valued for all $z\in{\mathbb D}$. We now use the fact that the triple \eqref{pcLift} is a pseudo-commutative lift of $(A,B,T)$, i.e., the operators $\widetilde A,\widetilde B$ satisfy the conditions
$$
\begin{bmatrix}
M_{\widetilde G_1^*+z\widetilde G_2}^* & 0 \\ 0 & \widetilde A_{22}^*
\end{bmatrix} \begin{bmatrix}
{\mathcal O}_{D_{T^*},T^*}\\ Q_{T^*}
\end{bmatrix}=\begin{bmatrix}
{\mathcal O}_{D_{T^*},T^*}\\ Q_{T^*}
\end{bmatrix}A^*
$$
and
$$
\begin{bmatrix} M_{\widetilde G_2^*+z\widetilde G_1}^*& 0 \\ 0 & \widetilde B_{22}^*\end{bmatrix}
\begin{bmatrix}
{\mathcal O}_{D_{T^*},T^*}\\ Q_{T^*}
\end{bmatrix}=\begin{bmatrix}
{\mathcal O}_{D_{T^*},T^*}\\ Q_{T^*}
\end{bmatrix}B^*.
$$Equivalently,
\begin{align}\langlebel{uniquness1}
&M_{\widetilde G_1^*+z\widetilde G_2}^* {\mathcal O}_{D_{T^*},T^*}={\mathcal O}_{D_{T^*},T^*}A^*,\; M_{\widetilde G_1^*+z\widetilde G_2}^* {\mathcal O}_{D_{T^*},T^*}={\mathcal O}_{D_{T^*},T^*}A^*\\
&\mbox{and}\quad(\widetilde A_{22}^*,\widetilde B_{22}^*,W_D^*)Q_{T^*}=Q_{T^*}(A^*,B^*,T^*).\langlebel{uniquness2}
\end{align}
We first show that $(\widetilde A_{22},\widetilde B_{22})=(R_D,S_D)$. Note that since $\widetilde A_{22}$ commutes with $W_D$ and $W_D$ is a unitary, $\widetilde A_{22}$ commutes with $W_D^*$
as well, and so we use \eqref{uniquness2} to compute
\begin{align*}
\widetilde A_{22}^*(W_D^nQ_{T^*}h)=W_D^n\widetilde A_{22}^*Q_{T^*}h&=W_D^nQ_{T^*}A^*h\\
&=W_D^nR_D^*Q_{T^*}h=R_D^*(W_D^nQ_{T^*}h).
\end{align*}Since $\{W_D^nQ_{T^*}h:n\geq0\mbox{ and }h\in{\mathcal H}\}$ is dense in ${\mathcal Q}_{T^*}$, we have $\widetilde A_{22}=R_D$. Similarly, $\widetilde B_{22}=S_D$. Next we show that
$(\widetilde G_1,\widetilde G_2)$ are the fundamental operators of $(A^*,B^*,T^*)$. To this end, we can use \eqref{uniquness1} and the power series expansion of
${\mathcal O}_{D^*,T^*}(z)=\sum_{n\geq0}D_{T^*}T^{*n}h$ to arrive at the equations
$$
\widetilde G_1D_{T^*}+\widetilde G_2^*D_{T^*}T^*=D_{T^*}A^*\quad\text{and}\quad
\widetilde G_2D_{T^*}+\widetilde G_1^*D_{T^*}T^*=D_{T^*}B^*.
$$By part (ii) of Theorem \ref{T:FundTetra} applied to the tetrablock contraction $(A^*,B^*,T^*)$, $(\widetilde G_1,\widetilde G_2)$ must be equal to the Fundamental Operator
pair for $(A^*, B^*,T^*)$.
\end{proof}
\subsection{A Sz.-Nagy--Foias type functional model}
Sz.-Nagy and Foias gave a function-space realization of ${\mathcal Q}_{T^*}$ and thereby produced a concrete functional model for a contraction $T$.
In their analysis a crucial role is played by what they called the {\bf characteristic function} associated with $T$:
\begin{align}\langlebel{CharcFunc}
\Theta_T(z): = -T+z{\mathcal O}_{D_{T^*}, T^*}T|_{{\mathcal D}_T}:{\mathcal D}_T\mapsto {\mathcal D}_{T^*}.
\end{align}
The name suggests the fact the characteristic function $\Theta_T$ enables one to write down an explicit functional model on which there is a
compressed multiplication operator ${\mathbf T}$ which recovers the original operator $T$ up to unitary equivalence in case $T$ is a
c.n.u.\ contraction (see Chapter VI of \cite{Nagy-Foias}). Let $\Theta_T(\zeta)$ be the radial limit of the characteristic function defined almost everywhere on ${\mathbb{T}}$.
Consider
\begin{equation}\langlebel{DefectCharc}
\Delta_{T}(\zeta) := (I - \Theta_T(\zeta)^* \Theta_T(\zeta))^{1/2}.
\end{equation}
Sz.-Nagy and Foias showed in \cite{Nagy-Foias} that
$$
V_{\rm NF}:=\begin{bmatrix}
M_z&0\\0&M_{\zeta}|_{\overline{\Delta_T L^2({\mathcal D}_T)}}
\end{bmatrix}:\begin{bmatrix}
H^2({\mathcal D}_{T^*})\\ \overline{\Delta_T L^2({\mathcal D}_T)}
\end{bmatrix}\to \begin{bmatrix}
H^2({\mathcal D}_{T^*})\\ \overline{\Delta_T L^2({\mathcal D}_T)}
\end{bmatrix}
$$
is a minimal isometric lift of $T$ via some isometric embedding
$$
\Pi_{\rm NF}:{\mathcal H}\to \begin{bmatrix}
H^2({\mathcal D}_{T^*})\\ \overline{\Delta_T L^2({\mathcal D}_T)}\end{bmatrix}=:{\mathcal K}_{\rm NF}
$$ such that
\begin{align}\langlebel{RanPinf}
{\mathcal H}_{\rm NF}:=\operatorname{Ran}\Pi_{\rm NF}=\begin{bmatrix} H^2({\mathcal D}_{T^*}) \\ \overline{ \Delta_{T}L^2({\mathcal D}_T)} \end{bmatrix}
\ominus \begin{bmatrix} \Theta_T \\ \Delta_{T} \end{bmatrix} \cdot H^2({\mathcal D}_T).
\end{align}
Any two minimal isometric lifts of a given contraction $T$ are unitarily equivalent; see Chapter I of \cite{Nagy-Foias}. In \cite{BS-Memoir} an explicit unitary $u_{\text{min}}:{\mathcal Q}_{T^*}\to\overline{\Delta_T L^2({\mathcal D}_T)}$ is found that intertwines $W_D$ and $M_{\zeta}|_{\overline{\Delta_T L^2({\mathcal D}_T)}}$ and
\begin{equation}\langlebel{Pinf}
\Pi_{\rm NF}=\begin{bmatrix} I_{H^2({\mathcal D}_{T^*})} & 0 \\ 0 & u_{\text{min}} \end{bmatrix} \Pi_D.
\end{equation}
It is possible to give a concrete Sz.-Nagy--Foias type functional model using the transition map $u_{\rm{min}}:{\mathcal Q}_{T^*}\to \overline{\Delta_TL^2({\mathcal D}_T)}$ as appeared (see \eqref{Pinf}) in the case of a single contractive operator above. We must replace the canonical tetrablock unitary $(R_D,S_D,W_D)$ by its avatar on the function space $\overline{\Delta_TL^2({\mathcal D}_T)}$:
\begin{align}\langlebel{CanonTetraUniNF}
(R_{\rm{NF}}, S_{\rm{NF}},W_{\rm{NF}})=u_{\rm{min}}^*(R_D,S_D,W_D)u_{\rm{min}}.
\end{align}Then the following functional model is a straightforward consequence of Theorem \ref{T:Dmodel} and \eqref{Pinf}.
\begin{thm}\langlebel{T:NFmodel}
Let $(A,B,T)$ be a tetrablock contraction on a Hilbert space $\mathcal{H}$ such that $T$ is c.n.u., and $(G_1,G_2)$ be the fundamental operators of
$(A^*,B^*,T^*)$. Then $(A,B,T)$ is jointly unitarily equivalent to
\begin{align}\langlebel{FunctModel}
P_{{\mathcal H}_{\rm{NF}}}\left(\begin{bmatrix}
M_{G_1^*+zG_2}&0\\0&R_{\rm{NF}}
\end{bmatrix},
\begin{bmatrix} M_{G_2^*+zG_1}&0\\0&S_{\rm{NF}} \end{bmatrix},
\begin{bmatrix} M_z&0\\0& W_{\rm{NF}} \end{bmatrix}\right)\bigg|_{{\mathcal H}_{\rm{NF}}},
\end{align}
where ${\mathcal H}_{\rm{NF}}$ is the functional model space given by (see \eqref{RanPinf})
$$
{\mathcal H}_{\rm{NF}}=\operatorname{Ran}\Pi_{\rm{NF}}=\begin{bmatrix} H^2({\mathcal D}_{T^*}) \\ \overline{ \Delta_{T}L^2({\mathcal D}_T)} \end{bmatrix}
\ominus \begin{bmatrix} \Theta_T \\ \Delta_{T} \end{bmatrix} \cdot H^2({\mathcal D}_T).
$$
\end{thm}
Note that in the special case when $T^{*n}\to 0$ strongly as $n\to\infty$, the space ${\mathcal Q}_{T^*}=0$ and hence also $\overline{ \Delta_{T}L^2({\mathcal D}_T)} = 0$,
i.e., $\Theta_T$ is inner. Therefore in this special case the models above simply boil down to the following which was obtained in \cite{SauNYJM}.
\begin{thm}{\rm (See \cite[Theorem 4.2]{SauNYJM})}\langlebel{T:fm}
Let $(A,B,T)$ be a pure tetrablock contraction on a Hilbert space $\mathcal{H}$ and $(G_1,G_2)$ be the fundamental operators of $(A^*,B^*,T^*)$. Then $(A,B,T)$ is jointly unitarily equivalent to
$$P_{\operatorname{Ran}{\mathcal O}_{D_{T^*}, T^*}}(M_{G_1^*+zG_2}, M_{G_2^*+zG_1}, M_z))|_{\operatorname{Ran}{\mathcal O}_{D_{T^*}, T^*}}.$$
\end{thm}
\section{Tetrablock data sets: characteristic and special} \langlebel{S:DataSets}
In this section we provide some preliminary results towards a Sz.-Nagy-Foias-type model theory for tetrablock-contraction operator tuples $(A,B,T)$. Note that if $T$ is unitary, then the characteristic function $\Theta_T$, as in \eqref{CharcFunc}, is trivial (i.e., equal to the zero operator between the zero spaces). As the most general contraction operator $T$
is the direct sum of a unitary $T_u$ with a completely nonunitary (c.n.u.) part $T_{\rm cnu}$ and the model theory for unitary operators is easily handled by spectral theory,
it is natural for model theory purposes to restrict to the case where $T$ is c.n.u.
One then associates a functional model spaces
$$
{\boldsymbol{\mathcal K}}_T = \begin{bmatrix} H^2({\mathcal D}_{T^*} \\ \overline{\Delta_T L^2({\mathcal D}_T) } \end{bmatrix}
\quad {\boldsymbol{\mathcal H}}_T ={\boldsymbol{\mathcal K}}_T \ominus \begin{bmatrix} \Theta_T \\ \Delta_T \end{bmatrix} H^2({\mathcal D}_T) \subset {\boldsymbol{\mathcal K}}_T
$$
together with functional-model operators ${\mathbf T}_T$ and ${\mathbf V}_T$ by
$$
{\mathbf T}_T = P_{{\boldsymbol{\mathcal H}}_T} \begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \bigg|_{{\boldsymbol{\mathcal H}}_T} \text{ on } {\boldsymbol{\mathcal H}}_T, \quad
{\mathbf V}_T = \begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \text{ on } {\boldsymbol{\mathcal K}}_{\mathbb{T}}.
$$
Then it is immediate that:
\begin{enumerate}
\item [(NF1)] ${\mathbf V}_T$ is an isometry on ${\boldsymbol{\mathcal K}}_T$.
\item[(NF2)] ${\boldsymbol{\mathcal H}}_{\mathbf T}^\perp = \sbm{ \Theta_T \\ \Delta_T} H^2({\mathcal D}_T)$ is invariant for ${\mathbf V}_T$, and hence ${\mathbf V}_T$ is an isometric lift for ${\mathbf T}_T$.
\end{enumerate}
Less immediately obvious are other features of the model:
\begin{enumerate}
\item[(NF3)] (See \cite[Theorem VI.2.3]{Nagy-Foias}) If $T$ is c.n.u., then ${\mathbf T}_T$ is unitarily equivalent to $T$.
\item[(NF4)] (See \cite[Theorem VI.3.4]{Nagy-Foias}.)
If $T$ on ${\mathcal H}$ and $T'$ on ${\mathcal H}'$ are two c.n.u.~contraction operators, then $T$ is unitarily equivalent to $T'$
if and only if $\Theta_T$ coincides with $\Theta_{T'}$ in the following sense: {\em there exist unitary change-of-coordinate maps
$\phi \colon {\mathcal D}_T \to {\mathcal D}_{T'}$ and $\phi_* \colon {\mathcal D}_{T^*} \to {\mathcal D}_{T^{\prime *}}$ so that
$\phi_* \Theta_T(\langlem) = \Theta_{T'}(\langlem) \phi$ for all $\langlem \in {\mathbb D}$.} Often in the literature this property is described simply as:
{\em the characteristic function $\Theta_T$ is a complete unitary invariant for c.n.u.~contraction operator $T$.}
\end{enumerate}
The Sz.-Nagy-Foias theory goes still further by identifying the {\em coincidence-envelop} of the characteristic functions \eqref{ThetaT},
i.e., the set of all contractive operator functions $({\mathcal D}, {\mathcal D}_*, \Theta)$ coinciding with the characteristic function $\Theta_T$ for some c.n.u.~contraction operator
$T$,
as simply any contractive operator function $({\mathcal D}, {\mathcal D}_*, \Theta)$ which is \textbf{pure} in the sense that
$$
\| \Theta(0) u \| < \| u \| \text{ for any } u \in {\mathcal D} \text{ such that } u \ne 0.
$$
Then we can start with any pure COF $({\mathcal D}, {\mathcal D}_*, \Theta)$, form ${\boldsymbol{\mathcal K}}(\Theta)$ and ${\boldsymbol{\mathcal H}}(\Theta)$ according to
$$
{\boldsymbol{\mathcal K}}(\Theta) = \begin{bmatrix} H^2({\mathcal D}_*) \\ \overline{ D_\Theta \cdot L^2({\mathcal D})} \end{bmatrix}, \quad
{\boldsymbol{\mathcal H}}(\Theta) = {\boldsymbol{\mathcal K}}(\Theta) \ominus \begin{bmatrix} \Theta \\ D_\Theta \end{bmatrix} H^2({\mathcal D}) \subset {\boldsymbol{\mathcal K}}(\Theta)
$$
where $D_\Theta$ is the $\Theta$-defect operator function $D_\Theta(\zeta) = ( I_{\mathcal D} - \Theta(\zeta)^* \Theta(\zeta))^{\frac{1}{2}}$.
Then we can form the model operators
$$
{\mathbf V}(\Theta) = \begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \text{ on } {\boldsymbol{\mathcal K}}(\Theta), \quad
{\mathbf T}(\Theta) = P_{{\boldsymbol{\mathcal H}}(\Theta)} {\mathbf V}(\Theta) \bigg|_{{\boldsymbol{\mathcal H}}(\Theta)}.
$$
Then we have the additional results:
\begin{enumerate}
\item[(NF5)] (See \cite[Theorem VI.3.1]{Nagy-Foias}.) Given any pure COF $\Theta$, ${\mathbf T}(\Theta)$ is a c.n.u.~contraction operator on ${\boldsymbol{\mathcal H}}(\Theta)$ with characteristic operator function
$\Theta_{{\mathbf T}(\Theta)}$ coinciding with $\Theta$.
\end{enumerate}
In this way the loop is closed: the study of c.n.u.~contraction operators is the same as the study of pure COFs.
To explain generalizations to the setting of tetrablock contractions, we first introduce some useful terminology.
For this discussion, just as in the classical Sz.-Nagy-Foias settings, it makes sense to restrict to c.n.u.~tetrablock contractions
(see Theorem \ref{T:tetra-can-decom} and Remark \ref{R:tetra-can-decom}).
\begin{definition} \langlebel{D:Edata} Let us say that any collection of objects
$$
\Xi = (\Theta, (G_1, G_2), \psi \}
$$
consisting of
\begin{enumerate}
\item[(i)] a pure COF function $({\mathcal D}, {\mathcal D}_*, \Theta)$,
\item[(ii)] a pair of operators $(G_1, G_2)$ on the coefficient space ${\mathcal D}_*$, and
\item[(iii)] a measurable function $\psi$ on ${\mathbb T}$ such that, for a.e.~$\zeta \in {\mathbb T}$,
$\psi(\zeta)$ is a contractive normal operator on ${\mathcal D}_\Theta(\zeta) = \overline{\operatorname{Ran} \, D_\Theta(\zeta)}$.
\end{enumerate}
is a \textbf{tetrablock data set}.
\end{definition}
\begin{remark} \langlebel{R:Edata}
Canonically associated with any such $\psi$ as in item (iii) in Definition \ref{D:Edata} is the tetrablock unitary triple $(R,S,W)$ on the direct integral space
$\oplus \int_{\mathbb T} {\mathcal D}_{\Theta(\zeta)} \frac{ |{\tt d} \zeta|}{2 \pi}$ given by
$$
R = M_{\psi^* \cdot \zeta}, \quad S = M_\psi, \quad W = M_\zeta
$$
and (as one sweeps over all possible such $\psi$), this is the general tetrablock unitary operator triple $(R,S,W)$ on the space
$\oplus \int_{\mathbb T} {\mathcal D}_{\Theta(\zeta)} \frac{ |{\tt d} \zeta|}{2 \pi}$ with the unitary operator $W$ equal to $W = M_\zeta$
(multiplication by the coordinate function) (see Example \ref{E:model} (2)). Thus item (iii) in the definition of tetrablock data set can be equivalently rephrased as:
\begin{enumerate}
\item[(iii')] a tetrablock unitary operator-triple $(R,S,W)$ on the direct-integral space $\int_{\mathbb T} {\mathcal D}_{\Theta(\zeta)} \frac{|{\tt d} \zeta|}{2 \pi}$ such
that the last unitary component $W$ is equal to multiplication by the coordinate function $W = M_\zeta$.
\end{enumerate}
However, for convenience of notation, we shall continue to use the notation $(R,S,W)$ for the third component of a tetrablock data set $\Xi = (\Theta, (G_1, G_2),
(R,S,W))$ with the convention (iii') also part of the definition.
\end{remark}
\begin{definition} \langlebel{D:charEdata}
Given a c.n.u.~tetrablock contraction $(A,B,T)$ we say that $\Xi_{A,B,T} = (\Theta, (G_1, G_2), \psi)$ is the
\textbf{characteristic tetrablock data set} for $(A,B,T)$ if
\begin{enumerate}
\item[(i)] $({\mathcal D}, {\mathcal D}_*, \Theta)$ is equal to the Sz.-Nagy-Foias characteristic function $({\mathcal D}_T, {\mathcal D}_{T^*}, \Theta_T)$ for the c.n.u.~contraction operator $T$,
\item[(ii)] $(G_1, G_2)$ is equal to the Fundamental Operator pair for the adjoint tetrablock contraction $(A^*, B^*, T^*)$, and
\item[(iii)] $(R,S,D)$ is given by
$$
((R,S,W))= (R_{\rm NF}, S_{\rm NF}, W_{\rm NF}):= u_{\rm min}^* (R_D, S_D, W_D) u_{\rm min}
$$
where $(R_D, S_D, W_D)$ is the tetrablock unitary on ${\mathcal Q}_{T^*}$ determined by tetrablock contraction
$(A,B,T)$ according to Definition \ref{D:canonical}, and where $u_{\rm min} \colon \overline{\Delta_T L^2({\mathcal D}_T)} \to {\mathcal Q}_{T^*}$
is the unitary identification map identifying the Sz.-Nagy-Foias lifting residual space $\overline{\Delta_T L^2({\mathcal D}_T)}$ with the
Douglas lifting residual space ${\mathcal Q}_{T^*}$.
\end{enumerate}
\end{definition}
Then it is clear that the characteristic tetrablock data set $\Xi_{A,B,T}$ for a c.n.u.~tetrablock contraction $(A,B,T)$ is a tetrablock data set.
The natural notion of equivalence for tetrablock-data sets is the following.
\begin{definition} \langlebel{D:coincide}
Let $(\mathcal{D},\mathcal{D}_*,\Theta)$, $(\mathcal{D'},\mathcal{D'_*},\Theta')$ be two purely contractive
analytic functions. Let $G_1,G_2\in{\mathcal B}({\mathcal D}_*)$, $G_1',G_2'\in{\mathcal B}(\mathcal{D'_*})$, and $(R,S,W)$ on $\overline{\Delta_\Theta L^2(\mathcal{D})}$ and $(R',S',W')$ on
$\overline{\Delta_{\Theta'} L^2(\mathcal{D'})}$ be two tetrablock unitaries (with $W$ and $W'$ equal to $M_\zeta$ on their respective spaces).
We say that the two triples $(\Theta, (G_1,G_2),(R,S,W))$
and $(\Theta', (G_1',G_2'),(R',S',W'))$ {\bf coincide} if:
\begin{enumerate}
\item[(i)] $(\mathcal{D},\mathcal{D}_*,\Theta)$ and $(\mathcal{D'},\mathcal{D'_*},\Theta')$ coincide,
\item[(ii)] the unitary operators $\phi$, $\phi_*$ involved in the coincidence of $(\mathcal{D},\mathcal{D}_*,\Theta)$ and $(\mathcal{D'},\mathcal{D'_*},\Theta')$ satisfy
the additional intertwining conditions:
\begin{align*}
\phi_*(G_1,G_2)=(G_1',G_2')\phi_*\quad\mbox{and}\quad \omega_{\phi}(R,S,W)=(R',S',W')\omega_{\phi},
\end{align*}
where $\omega_{\phi}:\overline{\Delta_{\Theta} L^2(\mathcal{D})}\to\overline{\Delta_{\Theta'} L^2(\mathcal{D'})}$
is the unitary map induced by $\phi$ according to the formula
\begin{equation} \langlebel{omega-u}
\omega_{\phi}:=(I_{L^2}\otimes \phi)|_{\overline{\Delta_{\Theta} L^2(\mathcal{D})}}.
\end{equation}
\end{enumerate}
\end{definition}
Given a characteristic tetrablock data set $\Xi_{(A,B,T)} = (\Theta, (G_1, G_2), (R,S,W))$ for a tetrablock contraction $(A,B,T)$ we can write down a functional model:
$$
{\boldsymbol{\mathcal K}}(\Xi) = \begin{bmatrix} H^2({\mathcal D}_*) \\ \overline{D_\Theta L^2({\mathcal D})} \end{bmatrix}, \quad
{\boldsymbol{\mathcal H}}(\Xi) = {\boldsymbol{\mathcal K}} \ominus \begin{bmatrix} \Theta \\ D_\Theta \end{bmatrix} H^2({\mathcal D})
$$
with functional-model operators
\begin{align*}
& {\mathbf V}(\Xi) = \bigg( \begin{bmatrix} M_{G_1^* + z G_2} & 0 \\ 0 & R \end{bmatrix}, \begin{bmatrix} M_{G_2^* + z G_1} & 0 \\ 0 & S \end{bmatrix},
\begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \bigg) \text{ on } {\boldsymbol{\mathcal K}}(\Xi), \\
& {\mathbf T}(\Xi) = P_{{\boldsymbol{\mathcal H}}(\Xi)} {\mathbf V}(\Xi) |_{{\boldsymbol{\mathcal H}}(\Xi)}.
\end{align*}
The tetrablock analogue of items (NF1)-(NF2) in our discussion of the Sz.-Nagy-Foias model above fails without the additional assumptions:
namely, it is not the case that ${\mathbf V}$ is a tetrablock isometry as well as that ${\mathbf V}$ is a lift for ${\mathbf T}$
unless we also impose the condition \eqref{com=} on $(G_1, G_2)$. Nevertheless, the analogue of (NF3) does hold: {\em given that $\Xi$ is the
characteristic tetrablock triple for $(A,B,T)$, it is the case that $(A,B,T)$ is unitarily equivalent to ${\mathbf T}(\Xi)$}: this is the content of the last part of
item (3) in Theorem \ref{T:Dmodel} (after a conversion to Sz.-Nagy-Foias rather than Douglas coordinates): see \eqref{SemiModel} and \eqref{HD}.
The next theorem amounts to the analogue of (NF4) in our list of features for the Sz.-Nagy-Foias model above.
\begin{thm}\langlebel{Thm:CompUniInv}
Let $(A,B,T)$ and $(A',B',T')$ be two tetrablock contractions acting on ${\mathcal H}$ and ${\mathcal H}'$, respectively. Let
$$
(\Theta_T, (G_1,G_2), (R_{\rm NF}, S_{\rm NF},W_{\rm NF}), \quad (\Theta_{T'}, (G_1',G_2'), (R_{\rm NF}', S_{\rm NF}',W_{\rm NF}'))
$$
be the characteristic tetrablock data sets for $(A,B,T)$ and $(A',B',T')$, respectively.
\noindent
{\rm (1)} If $(A,B,T)$ and $(A',B',T')$ are unitarily equivalent, then their characteristic tetrablock data sets coincide.
\noindent
{\rm (2)}
Conversely, if $T$ and $T'$ are c.n.u.\ contractions and the characteristic tetrablock data sets of $(A,B,T)$ and $(A',B',T')$ coincide, then $(A,B,T)$ and $(A',B',T')$
are unitarily equivalent.
\end{thm}
\begin{proof}
First suppose that $(A,B,T)$ and $(A',B',T')$ are unitarily equivalent via a unitary $\tau:{\mathcal H}\to{\mathcal H}'$. The fact that $\Theta_T$ and $\Theta_{T'}$ coincide is a part of the Sz.-Nagy--Foias theory \cite{Nagy-Foias}. Indeed, note that
\begin{align*}
\tau(I-T^*T)=(I-T'^*T')\tau \text{ and }\tau(I-TT^*)=(I-T'T'^*)\tau
\end{align*}
and therefore by the functional calculus for positive operators,
\begin{align}\langlebel{IntwinDefects}
\tau D_T=D_{T'}\tau\quad\mbox{and}\quad\tau D_{T^*}=D_{T'^*}\tau
\end{align} and thereby inducing two unitary operators
\begin{align}\langlebel{u&u*}
\phi:=\tau|_{{\mathcal D}_T}:{\mathcal D}_T\to{\mathcal D}_{T'} \text{ and }\phi_*:=\tau|_{{\mathcal D}_{T^*}}:{\mathcal D}_{T^*}\to{\mathcal D}_{T'^*}.
\end{align}
Consequently $\phi_*\Theta_{T}=\Theta_{T'}\phi$. Next, since the fundamental operators are the unique operators that satisfy the equations for $(X_1,X_2)=(G_1,G_2)$:
\begin{align*}
A^*-BT^*=D_{T^*}X_1D_{T^*} \quad\mbox{and}\quad B^*-AT^*=D_{T^*}X_2D_{T^*},
\end{align*}
one can easily obtain using (\ref{IntwinDefects}) that
\begin{align}\langlebel{IntwinFunds}
\phi_*(G_1,G_2)=(G_1',G_2') \phi_*.
\end{align}
Finally the proof of the forward direction will be complete if we establish that
\begin{align}\langlebel{ForwardLast}
(R_{\rm NF},S_{\rm NF},W_{\rm NF})=\omega_\phi^*(R'_{\rm NF},S'_{\rm NF},W'_{\rm NF})\omega_\phi
\end{align}
where $\omega_\phi=(I_{L^2}\otimes \phi)|_{\overline{\Delta_{T} L^2(\mathcal{D}_T)}}:
\overline{\Delta_{T} L^2(\mathcal{D}_T)}\to \overline{\Delta_{T'} L^2(\mathcal{D}_{T'})}$.
For this we first note that
\begin{align*}
\begin{bmatrix}
I&0\\0& u_{\rm min}
\end{bmatrix}\begin{bmatrix}
{\mathcal O}_{D_{T^*},T^*}\\ Q_{T^*}
\end{bmatrix}=\Pi_{\rm NF}&=\begin{bmatrix}
I_{H^2}\otimes \phi_*^*&0\\0& \omega_\phi^*
\end{bmatrix}\Pi_{\rm NF}'\tau \\
&=\begin{bmatrix}
I_{H^2}\otimes \phi_*^*&0\\0& \omega_\phi^*
\end{bmatrix}\begin{bmatrix}
I&0\\0& u_{\rm min}'
\end{bmatrix}\begin{bmatrix}
{\mathcal O}_{D_{T'^*},T'^*}\\ Q_{T'^*}
\end{bmatrix}\tau,
\end{align*}
from which we read off that
\begin{align}\langlebel{u-minQ}
u_{\rm min}Q_{T^*}=\omega_\phi^* u_{\rm min}'Q_{T'^*}\tau.
\end{align}
Now since ${\mathcal Q}_{T'^*}=\overline{\operatorname{span}}\{W_D'^nQ_{T'^*}\tau h:h\in{\mathcal H},n\geq 0\}$ and $u_{\rm min}$ has the intertwining property $u_{\rm min} W_D=M_\zeta u_{\rm min}$, we use \eqref{u-minQ} to compute
\begin{align*}
\omega_{\phi}\cdot u_{\rm min}\cdot \omega_\tau^* (W_D'^nQ_{T'^*}\tau h)
&=\omega_{\phi}\cdot u_{\rm min} (W_D^nQ_{T^*}h)\\
&=\omega_{\phi} M_\zeta ^n u_{\rm min} Q_{T^*}h\\
&=M_\zeta^n\omega_\phi \cdot u_{\rm min} Q_{T^*}h\\
&=M_\zeta^n u_{\rm min}' Q_{T'^*}\tau h=u_{\rm min}' (W_D'^nQ_{T'^*}\tau h).
\end{align*}Consequently
\begin{align}\langlebel{u-min}
\omega_\phi\cdot u_{\rm min}\cdot \omega_\tau^*= u_{\rm min}'.
\end{align}Using this identity and the intertwining properties of the unitaries involved, it is now easy to establish \eqref{ForwardLast}.
Conversely, suppose $T$ and $T'$ are c.n.u.\ contractions, $\phi:{\mathcal D}_T\to{\mathcal D}_{T'}$ and $\phi_*:{\mathcal D}_{T^*}\to{\mathcal D}_{T'^*}$ be the unitary operators involved
in the coincidence of the characteristic tetrablock data sets
$$
((G_1,G_2), (R_{\rm NF}, S_{\rm NF},W_{\rm NF}),\Theta_T), \quad
((G_1',G_2'), (R_{\rm NF}', S_{\rm NF}',W_{\rm NF}'),\Theta_{T'}).
$$
By Definition \ref{D:coincide}, it follows that the unitary
$$
U=\begin{bmatrix} I_{H^2}\otimes\phi_* & 0 \\ 0& \omega_\phi \end{bmatrix} \colon
\begin{bmatrix} H^2({\mathcal D}_{T^*}) \\ \overline{ \Delta_{T}L^2({\mathcal D}_T)} \end{bmatrix} \to
\begin{bmatrix} H^2({\mathcal D}_{T'^*}) \\ \overline{ \Delta_{T'}L^2({\mathcal D}_{T'})} \end{bmatrix}
$$
identifies the model spaces ${\mathcal H}_{\rm NF}$ and ${\mathcal H}_{\rm NF}'$ and intertwines the model operators as in \eqref{FunctModel} associated with
$(A,B,T)$ and $(A',B',T')$, respectively. This completes the proof of Theorem \ref{Thm:CompUniInv}.
\end{proof}
The question remains
as to what additional coupling conditions must be imposed on a tetrablock data set $\Xi$ to assure that $\Xi$ coincides with the characteristic tetrablock data set
for a c.n.u.~ tetrablock contraction $(A,B,T)$. In the Sz.-Nagy-Foias theory, the data set (or invariant) consists of a single COF, and the only
additional requirement is that it must be pure.
From the results of Section \ref{SS:FundOps} we see that any characteristic tetrablock data
$$
\Xi_{A,B,T} = (\Theta, (G_1, G_2), (R,S,W))
$$
for a c.n.u.~tetrablock contraction operator-triple $(A,B,T)$ satisfies the additional conditions
(expressed directly in terms of the components of $\Xi_{A,B,T}$ rather than in terms of $(A,B,T)$):
\begin{enumerate}
\item[(i)] $\Theta$ is a pure COF (see \cite[Theorem VI.3.1]{Nagy-Foias}),
\item[(ii)] the numerical radius conditions
$$
\nu(G_1^* + z G_2) \le 1, \quad \nu(G_2^* + z G_1) \le 1 \text{ for all } z \in \overline{\mathbb D}
$$
hold, implying that also the spectral radius conditions
$$
r(M_{G_1^* + z G_2}) \le 1, \quad r(M_{G_2^* + z G_1}) \le 1
$$
(see Theorems \ref{T:pseudo-strict} (1)).
\item[(iii)] It is almost the case that the spectral radius and norm agree for $M_{G_1^* + z G_2}$ and $M_{G_2^* + z G_1}$ in the following sense
(see Theorem \ref{T:pc-vs-strict-Eisom} (1)):
$$
r(M_{G_1^* + z G_2} \cdot M_{G_2^* + z G_1} ) = \max\{ \| M_{G_1^* + z G_2} \|^2, \, \|M_{G_2^* + z G_1} \|^2 \}.
$$
\end{enumerate}
However we do not expect that just imposing these conditions is sufficient to guarantee that such a tetrablock data set $\Xi$ will coincide with
the characteristic tetrablock data set for some tetrablock contraction, so we do not expect to have an analogue of (NF5) at this level of generality.
Let us now specialize our class of tetrablock contractions to what we call \textbf{special tetrablock contractions}, i.e., any tetrablock contraction
$(A,B,T)$ with the special property that the Fundamental Operator pair $(G_1, G_2)$ for $(A^*, B^*, T^*)$ satisfies the additional pair of operator equations
\eqref{com=}. By Theorem \ref{T:Dmodel}, this is equivalent to $(A,B,T)$ having a minimal tetrablock isometric lift $(V_1, V_2, V_3)$ acting on a
minimal Sz.-Nagy isometric-lift space for $T$ with $V_3$ equal to a minimal isometric lift for the single contraction operator $T$.
This suggests that we define a \textbf{special tetrablock data set} as follows. For convenience in later discussion we shall now write the last component
$(R,S,W)$ simply as $\psi$ for a measurable contractive-normal operator-valued function $\zeta \mapsto \psi(\zeta) \in {\mathcal B}({\mathcal D}_{\Theta(\zeta)})$
according to the convention explained in Remark \ref{R:Edata}.
\begin{definition}\langlebel{D:Adm}
We say that the tetrablock data set
\begin{equation} \langlebel{Edata}
\Xi = (({\mathcal D}, {\mathcal D}_*, \Theta), (G_1, G_2), \psi)
\end{equation}
is a \textbf{special tetrablock data set} if the following conditions hold:
\begin{enumerate}
\item[(i)] The operators $G_1, G_2 \in {\mathcal B}({\mathcal D}_*)$ satisfy the commutativity conditions \eqref{com=}, i.e.,
$$
[G_1,G_2]=0,\quad [G_1^*,G_1]=[G_2^*,G_2]
$$
as well as the pencil-contractivity condition
$$
\|G_1^*+zG_2\|\leq 1 \text{ for all } z\in\overline{{\mathbb D}}
$$
and then also
$$
\|G_2^*+zG_1 \|\leq 1 \text{ for all } z \in \overline{{\mathbb D}}.
$$
\item[(ii)] the space $\left\{\sbm{\Theta\\ D_\Theta}f:f\in H^2({\mathcal D})\right\}$ is jointly invariant under the operator triple
\begin{align}\langlebel{AdmLifts}
\bigg( \begin{bmatrix} M_{G_1^* + zG_2} & 0 \\ 0 & M_{\psi(\zeta)^*\cdot \zeta} \end{bmatrix}, \begin{bmatrix} M_{G_2^* + zG_1} & 0 \\ 0 & M_{\psi(\zeta)}
\end{bmatrix},
\begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \bigg).
\end{align}
\end{enumerate}
\end{definition}
Given a special tetrablock data set $(\Theta, (G_1,G_2),\psi)$, we say that the space
\begin{align}\langlebel{AdmSpace}
{\boldsymbol {\mathcal H}} = \begin{bmatrix} H^2 ({\mathcal D}_*) \\ \overline{ D_\Theta L^2({\mathcal D})} \end{bmatrix} \ominus
\begin{bmatrix} \Theta \\ D_\Theta \end{bmatrix} H^2({\mathcal D})
\end{align}
is the {\bf functional model space} and the (commutative) operator triple $({\mathbf A}, {\mathbf B}, {\mathbf T})$ given by
\begin{align}\langlebel{AdmOps}
P_{{\boldsymbol{\mathcal H}}}\bigg( \begin{bmatrix} M_{G_1^* + zG_2} & 0 \\ 0 & M_{\psi(\zeta)^* \cdot \zeta} \end{bmatrix}, \begin{bmatrix} M_{G_2^* + zG_1} & 0 \\ 0 & M_{\psi(\zeta)} \end{bmatrix},
\begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \bigg) \bigg|_{\boldsymbol{\mathcal H}}
\end{align}
the {\bf functional-model operator triple} associated with the data set. The following theorem is our one analogue of item (NF5) in our list of features
of the Sz.-Nagy-Foias model.
\begin{thm}
If $\Xi = (\Theta, (G_1,G_2),\psi)$ is a special tetrablock data set, then the associated model operator triple $(\bf A,\bf B, \bf T)$ as in \eqref{AdmOps},
is a tetrablock contraction that lifts to the tetrablock isometry as in \eqref{AdmLifts}. Moreover, the tetrablock data set $\Xi$ coincides with the characteristic
triple of $(\bf A, \bf B, \bf T)$.
\end{thm}
\begin{proof}
By part (i) of Definition \ref{D:Adm}, the triple as in \eqref{AdmLifts} is a strict tetrablock isometry, and by part (ii), it lifts the model triple $(\bf A, \bf B, \bf T)$ as in \eqref{AdmOps}. Thus in particular, $(\bf A, \bf B, \bf T)$ is a tetrablock contraction and the first part of the theorem follows. For the second part, we use the Sz.-Nagy--Foias model theory for single contractions and Theorem \ref{T:Dmodel} as follows. Apply Theorem VI.3.1 in \cite{Nagy-Foias} to the purely contractive analytic function $\Theta$ to conclude that the characteristic function $\Theta_{\bf T}$ of $\bf T$ coincides with $\Theta$, i.e., there exists unitary operators $u:{\mathcal D}\to{\mathcal D}_{\bf T}$ and $u_*:{\mathcal D}_*\to{\mathcal D}_{\bf T^*}$ such that $u_*\cdot\Theta(z)=\Theta_{\bf T}(z)\cdot u$ for all $z\in{\mathbb D}$. Let us set $(G_1',G_2'):=u_*(G_1,G_2)u_*^*$ and $(R',S',W'):=\omega_u(M_{\psi(\zeta)^*\cdot \zeta},M_{\psi(\zeta)},M_\zeta)\omega_u^*$. Since $G_1,G_2$ satisfy the commutativity conditions, $G_1',G_2'$ also satisfy the same conditions, and consequently the triple
\begin{align}\langlebel{ConjLift}
\bigg( \begin{bmatrix} M_{G_1'^* + zG_2'} & 0 \\ 0 & \omega_u M_{\psi(\zeta)^*\cdot \zeta}\omega_u^* \end{bmatrix}, \begin{bmatrix} M_{G_2'^* + zG_1'} & 0 \\ 0 & \omega_u M_\psi\omega_u^* \end{bmatrix},
\begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \bigg)
\end{align}is a strict tetrablock isometry. Note that since
\begin{align*}
\begin{bmatrix}
u_*&0\\0&\omega_u
\end{bmatrix}\begin{bmatrix}
\Theta\\ \Delta_\Theta
\end{bmatrix}=\begin{bmatrix}
\Theta_{\bf T}\\ \Delta_{\Theta_{\bf T}}
\end{bmatrix}\begin{bmatrix}
u&0\\0&\omega_u\end{bmatrix},
\end{align*} the unitary operator
$$\tau:=\begin{bmatrix}u_*&0\\o&\omega_u\end{bmatrix}:\begin{bmatrix}H^2({\mathcal D}_*)\\ \overline{ \Delta_\Theta L^2({\mathcal D})} \end{bmatrix}
\to
\begin{bmatrix}H^2({\mathcal D}_{\bf T^*})\\ \overline{ \Delta_{\Theta_{\bf T}} L^2({\mathcal D}_{\bf T})}
\end{bmatrix}$$takes the functional model space ${\boldsymbol{\mathcal H}}$ as in \eqref{AdmSpace} onto
\begin{align*}
\begin{bmatrix} H^2 ({\mathcal D}_{\bf T^*}) \\ \overline{ \Delta_{\Theta_{\bf T}} L^2({\mathcal D}_{\bf T})} \end{bmatrix} \ominus
\begin{bmatrix} \Theta_{\bf T} \\ \Delta_{\Theta_{\bf T}} \end{bmatrix} H^2({\mathcal D}_{\bf T}).
\end{align*}Therefore by part (ii) of Definition \ref{D:Adm}, the tetrablock isometry as in \eqref{ConjLift} is a lift of $(\bf A, \bf B, \bf T)$ via the embedding $\iota\cdot \tau|_{\boldsymbol{\mathcal H}}$, where
$$\iota:\begin{bmatrix}H^2 ({\mathcal D}_{\bf T^*}) \\ \overline{ \Delta_{\Theta_{\bf T}} L^2({\mathcal D}_{\bf T})} \end{bmatrix} \ominus
\begin{bmatrix} \Theta_{\bf T} \\ \Delta_{\Theta_{\bf T}} \end{bmatrix} \to
\begin{bmatrix}H^2({\mathcal D}_{\bf T}) \\ \overline{ \Delta_{\Theta_{\bf T}} L^2({\mathcal D}_{\bf T})} \end{bmatrix}$$is the inclusion map. Since
$$
\begin{bmatrix}
M_z&0\\0&M_\zeta
\end{bmatrix}:\begin{bmatrix}H^2(D_{\bf T}) \\ \overline{ \Delta_{\Theta_{\bf T}} L^2({\mathcal D}_{\bf T})} \end{bmatrix}\to
\begin{bmatrix}H^2(D_{\bf T}) \\ \overline{ \Delta_{\Theta_{\bf T}} L^2({\mathcal D}_{\bf T})} \end{bmatrix}
$$is a minimal isometric lift of $\bf T$, by part (2) of Theorem \ref{T:Dmodel}, there is a unique such tetrablock isometric lift with the last entry of the lift fixed.
If $\bf G_1,\bf G_2$ are the Fundamental Operators of $(\bf A^*,\bf B^*, \bf T^*)$, then by Theorem \ref{T:NFmodel},
\begin{align*}
\left(\begin{bmatrix}
M_{\bf G_1^*+z\bf G_2}&0\\0&R_{\rm{NF}}
\end{bmatrix},
\begin{bmatrix} M_{\bf G_2^*+z\bf G_1}&0\\0&S_{\rm{NF}} \end{bmatrix},
\begin{bmatrix} M_z&0\\0& W_{\rm{NF}} \end{bmatrix}\right)
\end{align*}
is another tetrablock isometric lift of $(\bf A,\bf B, \bf T)$, where $(R_{\rm NF}, S_{\rm NF},W_{\rm NF})$ is the canonical tetrablock unitary associated
with $(\bf A,\bf B, \bf T)$. Therefore we must have
$({\bf G}_1,{\bf G}_2)=(G_1',G_2')=u_*(G_1,G_2)u_*^*$ and
$$
(R_{\rm NF}, S_{\rm NF},W_{\rm NF})=\omega_u(M_{\psi(\zeta)^*\cdot \zeta},M_{\psi(\zeta)},M_\zeta)\omega_u^*.
$$
This is what was needed to be shown.
\end{proof}
\noindent
\textbf{Epilogue.} It is easy to write down tetrablock data sets as in \eqref{Edata}. Given such a tetrablock data set, it may not be very tractable to determine
if in addition it satisfies conditions (i) and (ii) in Definition \ref{D:Adm}.
However it is not so difficult to cook up viable examples. For example,
we note that the commutativity conditions in (i) are automatic if we choose $G_1$ and $G_2$ to be scalar operators on ${\mathcal D}_*$.
We can arrange the pencil contractivity condition to hold just by choosing $G_1$ and $G_2$ to be sufficiently small. If we choose the operator
$\psi(\zeta)$ to be a scalar for each $\zeta$, then we are forced to choose $\psi(\zeta) = G_2^* \zeta + G_1$. Since we have already chosen $G_2$
and $G_1$ so that the pencil contractivity condition holds, then $\psi(\zeta)$ is contractive and of course a scalar operator is normal. Then all conditions
are satisfied. In this way we get a whole class of tractable examples of special tetrablock contractions for any pure COF $\Theta$.
If ${\mathcal D}_*$ and ${\mathcal D}$ are both at most one-dimensional, all the examples are of this form.
\noindent
\textbf{Conflict of Interest/Dataset Statement.} The authors state that there are no conflicts of interest. No data sets were generated or analyzed during the
current study.
\end{document} | math | 123,642 |
\begin{document}
\author{Yamen Hamdouni}
\ead{[email protected]}
\address{Department of Physics, Faculty of Exact Sciences, Mentouri University, Constantine, Algeria}
\title{Random spin distributions and the diffusion equation}
\begin{abstract}
We show that the probability distribution corresponding to a fully random tracial state of a system of spin-S particles satisfies a diffusion-like equation.
The diffusion coefficient turns out to be equal
to $S(S+1)/6$, where $S$ is the magnitude of the spin of each particle. We also present a bosonization scheme for the lowering and raising total spin operators.
\end{abstract}
\begin{keyword}
probability distribution, spin, bosonization
\end{keyword}
\maketitle
\section{Introduction}
Many-spin systems exhibit interesting properties~\cite{kittel}, which explains the reason for which they are among the most studied systems in current research. These studies are motivated
by several potential applications in mesoscopic and atomic physics, among which we mention the promising quantum technologies that are slowly emerging. The theoretical
investigation
of the dynamics of spin systems requires, very often, the use of statistical and probabilistic techniques. In particular, the probability distribution associated with the
addition of the spin degrees of freedom turns out to be of great importance and usefulness.
For example, we have employed random distributions of spin angular momentum to deal with the description of the decoherence and the entanglement evolution of qubits interacting
with spin environments~\cite{ham1,ham2}. In this letter we address the relation between these Gaussian distributions and the diffusion equation.
To be more explicit in our discussion, we denote by $\vec{J} $ the sum of the individual spins of a set of $N$ spin-$S$ particles, i.e., $\vec{J}=\sum_{i=1}^N \vec{S}_i$, and by
$\lambda_j=j(j+1)$ and $m$ the eigenvalues of the operators $\hat J^2$ and $\hat J_z$ respectively. This means that, given a complete set $\{|j,m\rangle\}$ of common eigenvectors of the above operators, we may write
$\hat J^2|j,m\rangle=j(j+1)|j,m\rangle$, and $\hat J_z|j,m\rangle=m |j,m\rangle$. For a given $j$, the multiplicity of $m$ is simple to calculate and is equal to $2j+1$. On the other hand,
the degeneracy $\nu(N,j;S)$ corresponding to the quantum number $j$ is shown to satisfy the relation \cite{ham2}(see the appendix for the proof):
\begin{equation}
\nu(j,N+1;S)=\sum\limits_{j'=|j-S|}^{j+S}\nu(j',N;S).\label{main}
\end{equation}
Furthermore, a fully
random tracial state $\omega_N$ of $N$ independent spin-S particles is described by the fully mixed density operator
\begin{equation}
\hat\omega_N= \bigotimes_{i=1}^N \mathbb I_{2S+1}/{ \rm tr} \mathbb I_{2S+1}=\mathbb I_{2S+1}^{\otimes N}/(2S+1)^N
\end{equation}
where $\mathbb I_{2S+1}$ refers to the $2S+1$-dimensional unit matrix. The action of $ \omega_N$ on any operator $\hat A$ defined on $\mathbb C^{(2S+1)\otimes N}$ (here $\mathbb C$ is
the field of complex numbers) is
prescribed as \cite{vals}
\begin{equation}
\omega_N(\hat A)={\rm tr}(\hat\omega_N \hat A).
\end{equation}
With respect to this state, the quantum number $j$ is assigned a probability, which reads:
\begin{equation}
P(j,N)=\frac{2j+1}{(2S+1)^N} \nu(j,N;S)\label{rand}.
\end{equation}
The latter is the probability that the eigenvalue of the operator $\hat J^2$ is equal to $j(j+1)$, where, once more, $\vec{J}$ is the sum of the spin operators of $N$
independent spin-S particles. Knowing $P(j,N)$, one can deduce the probability that the eigenvalue of $J_z$ is equal to the quantum number $m$, namely,
\begin{equation}
P(m,N)=\sum_{j=|m|}^{SN} \frac{P(j,N)}{2j+1}=\sum_{j=|m|}^{SN} \frac{\nu(j,N;S)}{(2S+1)^N},
\end{equation}
which stems from the fact that, for fixed $m$, all the values of $j$ such that $j\geq |m|$ yield the eigenvalue $m$ of the operator $\hat J_z$, and that
for a given $j$, all the values of $m$ such that $-j\leq m \leq j$ are equiprobable.
In what follows, we will be interested in the case where $N$ is sufficiently large. We will show that, under the above condition, equation~(\ref{main})
allows us to prove that the probability distribution $P(j,N)$
verifies a diffusion-like equation, that is of great usefulness in deriving its explicit asymptotic Gaussian form.
\section{Explicit form of the probability distribution}
To begin
we note that it is more convenient to deal with each of the components of the vector operator $\vec{J}$ separately; the reasoning could equally be carried out in terms of vector
quantities but the notation will be a little cumbersome.
This being said, let $P(m,N)$ be the probability associated with the eigenvalue $m$ of the operator $\hat J_z$, as defined above. By noting that the multiplicity of $S_z$ is $2S+1$, we infer
from Eq.(\ref{main})
that the following equality holds:
\begin{equation}
(2S+1)P(m,N+1)=\sum\limits_{\tau=-S}^SP(m-\tau,N).\label{probsum}
\end{equation}
Indeed, we have:
\begin{eqnarray}
P(m,N+1)&=&\sum_{j=|m|}^{S(N+1)} \frac{\nu(j,N+1;S)}{(2S+1)^{N+1}}\nonumber \\
&=&\sum_{j=|m|}^{S(N+1)} \frac{1}{(2S+1)^{N+1}} \sum\limits_{j'=|j-S|}^{j+S}\nu(j',N;S).
\end{eqnarray}
It follows that
\begin{equation}
(2S+1)P(m,N+1)=\sum_{j=|m|}^{S(N+1)} \sum\limits_{j'=|j-S|}^{j+S} \frac{\nu(j',N;S)}{(2S+1)^{N}}.
\end{equation}
Since $P(m,N)=P(-m,N)$, we assume without loss of generality that $m>0$; then by introducing the variable $\alpha=j-S$, we see that
\begin{equation}
(2S+1)P(m,N+1)=\sum_{\alpha=m-S}^{m+S} \sum\limits_{j'=|\alpha|}^{NS} \frac{\nu(j',N;S)}{(2S+1)^{N}}=\sum_{\alpha=m-S}^{m+S} P(\alpha,N),
\end{equation}
from which equation~(\ref{probsum}) readily follows.
Let us now focus on the case of $N$ large; in this case, one usually uses Stirling's formula to obtain a Gaussian approximation for $P(m,N)$, which is
fairly good only for values of $m$ that are close to the most probable value of $m$
(typically within an interval of the order of the standard deviation of the distribution), which is clearly $m=0$ in our case because of the symmetry.
As we move farther away from this value, the Gaussian approximation becomes less and less precise, and we should use the exact form of the degeneracy. However,
for very large values of $N$, it is plausible to deal with $N$ and $m$ as continuous random variables; for instance, $N$ may directly be linked to the density
of particles, while $m$ corresponds to the $z$-projection of the total classical spin vector of the whole system, which is clearly a continuous quantity.
Then one is rather dealing with a probability density function, for which
the quantity of interest is
the probability that $m$ falls within a certain interval, say $[m,m+dm]$. With that in mind, we may expand in Taylor series,
up to second order, both $P(m,N+1)$ and $P(m-\tau,N)$ to obtain
\begin{equation}
(2S+1)\biggl[P(m,N)+\frac{\partial P }{\partial N}+\frac{1}{2}\frac{\partial^2 P }{\partial N^2}\biggr]=\sum\limits_{\tau=-S}^S P(m,N)-\frac{\partial P }{\partial m}
\sum\limits_{\tau=-S}^S\tau + \frac{1}{2}\frac{\partial^2 P }{\partial m^2}\sum\limits_{\tau=-S}^S \tau^2.
\end{equation}
On account of the fact that
\begin{equation}
\sum\limits_{\tau=-S}^S \tau^2=\frac{1}{3} S(2S+1) (S+1),
\end{equation}
we end up with the equation
\begin{equation}
\frac{\partial P }{\partial N}+\frac{1}{2}\frac{\partial^2 P }{\partial N^2}=\frac{1}{6}S(S+1)\frac{\partial^2 P }{\partial m^2}.\label{daya}
\end{equation}
From a mathematical point of view, the density $P(m,N)$ is a slowly varying function of $N$. This property is physically explained by the fact that, when $N$ is very large
(e.g. $N\sim 10^4$), adding or removing a small number of spins does not affect much the distribution of the system. This means that
\begin{equation}
\biggl|\frac{\partial P }{\partial N}\biggr|>> \biggl|\frac{\partial^2 P }{\partial N^2}\biggr| ,
\end{equation}
because $\partial P /\partial N\sim 1/N$ while $\partial^2 P /\partial N^2 \sim 1/N^2$, which allows us to safely neglect the second term in the left-hand side of Eq.(\ref{daya}) when $N$ is large. Whence:
\begin{equation}
\frac{\partial P }{\partial N}=\frac{1}{6}S(S+1)\frac{\partial^2 P }{\partial m^2}\label{diffspin}.
\end{equation}
It is worth mentioning that the same equation could be derived using a somewhat different method which makes use of the discrete Laplacian. Indeed, we can write
Eq.(\ref{probsum}) as:
\begin{eqnarray}
(2S+1)(P(m,N+1)-P(m,N))&=&\sum_{\tau=-S}^{S}(P(m-\tau,N)-P(m,N))\nonumber \\
&\approx&\sum_{\tau=1}^{S} \tau^2 \Delta_\tau P
\end{eqnarray}
where the discrete laplacian is defined by
\begin{equation}
\Delta_\tau P=\frac{P(m+\tau,N)+P(m-\tau,N)-2P(m,N)}{\tau^2}.
\end{equation}
Now, using the mean value theorem for $\Delta_\tau P$, we arrive at equation (\ref{diffspin}). The latter reminds us the diffusion equation in one dimension without drift, namely,~\cite{zwanz}
\begin{equation}
\frac{\partial \rho(x,t)}{\partial t}=D\frac{\partial^2 \rho(x,t) }{\partial x^2}, \label{diff}
\end{equation}
where $D$ is the diffusion coefficient. Thus it is tempting to interpret Eq.(\ref{diffspin}) as being a diffusion equation where the number of spins plays the role of time,
and the role of the position $x$ is played by the magnitude of the spin $m$. The corresponding diffusion coefficient is
\begin{equation}
D\equiv \frac{1}{6}S(S+1).
\end{equation}
The solution of Eq.(\ref{diffspin}) should be subject to the boundary condition
\begin{equation}
P(m,0)=0,
\end{equation}
which is physically obvious. The corresponding explicit form can be found by noting that the solution of Eq.(\ref{diff}) is given by
\begin{equation}
\rho(x,t)=\frac{1}{\sqrt{4\pi D t}}e^{-x^2/(4 D t)}.
\end{equation}
Hence, by comparison, we obtain:
\begin{equation}
P(m,N)=\sqrt{\frac{3}{2\pi N S(S+1)}}\exp\Bigl\{-\dfrac{3 m^2}{2 N S(S+1)}\Bigr\},\label{dist}
\end{equation}
which is the desired result.
Some comments would be very useful at this stage. First, in the context of the Brownian motion, we may also write:
\begin{equation}
P(m',N')=\int_{-\infty}^{+\infty} P(m'-m,N'-N) P(m,N) dN,
\end{equation}
meaning that $P(m'-m,N'-N)$ can be thought of as a transition density, linking the probability densities of the system for different values of the spin magnitudes, and particles
numbers. In the previous expression, however, the semi-group condition $N<N'$ is implied. The latter assertions are equivalent to the well established notion of Greens's function, met in different problems in physics.
Furthermore, notice that the expectation value of $m^2$ turns out to be
\begin{equation}
\langle m^2\rangle =\frac{NS(S+1)}{3},
\end{equation}
that is
\begin{equation}
\langle m^2\rangle =2 D N
\end{equation}
This result is analogous to the Brownian motion mean-square displacement.
Another remark worth mentioning is that the probability distribution of the random variables associated with the spin operators $\hat J_x/\sqrt{N}$, $\hat J_y/\sqrt{N}$,
and $\hat J_z/\sqrt{N}$ (which are the components of the operator $\vec{J}=\sum_{i=1}^N\vec{ S}_i$ scaled by $\sqrt{N}$) when $N\to\infty$ has already been derived
using the trace properties of the angular
momentum for $S=1/2$~\cite{ham1}.
Again, the quantity $N$ refers to the total number of the spin-$\frac{1}{2}$
particles, meaning that for finite $N$, the dimension of the total spin space is $2^N$. For an arbitrary value of the spin $S$, the same method may be employed to obtain the above
results as follows:
The trace of even powers of the components of the total spin operator is given, according to the multinomial theorem, by
\begin{equation}
{\rm tr}\bigl( \hat J_z\bigr)^{2\ell}=\sum_{r_1 r_2\cdots r_N}\frac{2\ell!}{r_1! r_2!\cdots r_N!}{\rm tr} \hat S_{1z }^{r_1}\otimes \hat S_{2z}^{\rm r_2}
\otimes\cdots\otimes \hat S_{Nz}^{r_N}.
\end{equation}
where the natural numbers $r_k$ satisfy $\sum_k r_k=2\ell$. It can be see that the main contribution to the sum comes from the term where all the $r_k$'s are equal to two; then, by
taking into account all the possible partitions of the spins in sets of $\ell$ elements (the remaining terms come from the unit matrix), we find
\begin{eqnarray}
{\rm tr}\bigl(\hat J_z\bigr)^{2\ell}&=&\frac{2\ell!}{\underbrace{2! \times 2!\cdots 2!}_{\ell \ \rm terms }}\sum_{\Pi_\ell} \Biggl(\prod_{k=1}^\ell{\rm tr}\hat S^2_{z k}
\prod_{k=\ell+1}^N{\rm tr}{\mathbb I}_{2S+1}\Biggr)_{\Pi_\ell[1,2,\cdots N]}\nonumber \\&+&Q_{\ell-1}(N),
\end{eqnarray}
where as indicated, the products should be evaluated for all the possible partitions $\Pi_\ell[1,2,\cdots, N]$ of
$N$ elements into subsets of $\ell$ elements; the quantity $Q_{\ell-1}(N)$ is a polynomial in $N$ whose degree is at most equal to $\ell-1$. The latter equation means that
\begin{equation}
{\rm tr}\bigl(\hat J_z\bigr)^{2\ell}= N^\ell \Biggl[\frac{(2\ell)!}{2^\ell \ell!} (2S+1)^{N-\ell} [\tfrac{1}{3} S(S+1)(2S+1)]^\ell +O\Biggr(\frac{1}{N}\Biggr)\Biggr].
\end{equation}
By rescaling the spin operator and taking the limit $N\to\infty$, we obtain
\begin{equation}
\lim_{N\to\infty}(2S+1)^{-N}{\rm tr}(\hat J_z/\sqrt{N})^{2\ell}=\xi^{2\ell}:=\frac{(2\ell)!}{2^\ell \ell !} [\tfrac{1}{3} S(S+1)]^\ell.\label{mom}
\end{equation}
The characteristic function, with moments given by the right-hand side of Eq.(\ref{mom}), is
\begin{eqnarray}
\Phi(t)&=&\sum_{n=0}^\infty (it)^{2n}\xi^{2n}/(2n)!=\sum_{n=0}^\infty \frac{ (it)^{2n}}{ n! } [\tfrac{1}{6} S(S+1)]^{n}\nonumber \\ &=&\exp\{-\tfrac{1}{6} S(S+1)t^2\},
\end{eqnarray}
where we used the fact that the odd moments vanish. Afterwards, by Fourier transforming the characteristic function
\begin{equation}
P(m)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\Phi(t) e^{-it m}dt
\end{equation}
and making the substitution $m\to m/\sqrt{N}$ we obtain exactly Eq.(\ref{dist}).
Notice that the joint distribution of three such identical independent random variables, say $m_1,m_2,m_3$,
turns out to be
\begin{eqnarray}
&& P(m_1,m_2,m_3)= P(m_1)P(m_1)P(m_3)\nonumber \\ &&=\Biggl[\frac{3}{2\pi N S(S+1)}\Biggr]^{3/2}\exp\Bigl\{-\dfrac{3 (m_1^2+m_2^2+m_3^2)}{2 N S(S+1)}\Bigr\}.
\end{eqnarray}
Furthermore if we require that $\sqrt{m_1^2+m_2^2+m_3^2}=j$, then the probability distribution becomes~\cite{ham2}
\begin{equation}
\tilde P(m_1,m_2,m_3)\equiv P(j)=4\pi j^2 P(m_1,m_2,m_3).
\end{equation}
\section{Bosonization}
We may now proceed to the bozonization of the scaled lowering and raising operators $\hat J_\pm/\sqrt{N}$. To this end, we stress that given a function
$f(\hat J_x/\sqrt{N}, \hat J_y/\sqrt{N})$ of the spin
operators $\hat J_x/\sqrt{N}$ and $\hat J_y/\sqrt{N}$, we can write:
\begin{equation}
\lim_{N\to \infty} \omega_N(f(\hat J_x/\sqrt{N}, \hat J_y/\sqrt{N}))=\iint\limits_{-\infty}^{\infty}\bar P(m_1)\bar P(m_2) f(m_1,m_2) dm_1 dm_2,
\end{equation}
where the probability distribution $\bar P$ is given (after a proper rescaling) by
\begin{equation}
\bar P(m)=\sqrt{\frac{3}{2\pi S(S+1)}}\exp\Bigl\{-\dfrac{3 m^2}{2 S(S+1)}\Bigr\}.
\end{equation}
In fact we can express the above result in terms of the lowering and raising operators, by introducing the polar coordinates $z$ and $z^*$, with $dz dz^*=dm_1dm_2$,
such that:
\begin{eqnarray}
\lim_{N\to \infty} \omega_N(f(\hat J_+/\sqrt{N}, \hat J_-/\sqrt{N}))&=&\frac{3}{2\pi S(S+1)}\iint_{\mathbb C}f(z^*,z)
\exp\Bigl\{-\dfrac{3 |z|^2}{2 S(S+1)}\Bigr\} dz^* dz \nonumber \\
&=&\iint_{\mathbb C}\mathcal P(z^*,z) {\mathcal N}f(z^*,z) dz^* dz.
\end{eqnarray}
We identify the latter equality as being the coherent state representation of the mean value of $ f $ written in normal order, which is expressed in terms of the so-called
P-representation~\cite{louis}
\begin{equation}
\mathcal P(z^*,z)=\sum_n\langle n|\hat\rho\delta(z^*-a^\dag)\delta(z-a)|n\rangle.
\end{equation}
In the above, $a^\dag$ and $a$ are bosonic creation and annihilation operators, and $\hat \rho$ is a density matrix to be determined. Explicitly we have
\begin{equation}
\hat\rho=\iint_{\mathbb C} \mathcal P(z^*,z)|z\rangle \langle z|dz^* dz,
\end{equation}
where $|z\rangle=e^{-|z|^2/2}\sum_{k=0}^{\infty}\frac{z^k}{k!}|k\rangle$ is the usual bosonic coherent state.
Using the polar coordinates $(r,\phi)$, the elements of the density matrix can be calculated as
\begin{eqnarray}
\langle n|\hat\rho|k\rangle&=&\frac{3}{2\pi S(S+1)} \int_{z^*} dz^*\int_{z} dz \ \exp\Bigl\{-\dfrac{3 |z|^2}{2 S(S+1)}-|z|^2\Bigr\}
\frac{z^n{z^{*}}^k}{\sqrt{n!k!}}\nonumber \\
&=& \delta_{kn}\frac{3}{ S(S+1)} \int_0^\infty \frac{r^{n+k}}{\sqrt{n!k!}} \exp\Bigl\{-\dfrac{3 r^2}{2 S(S+1)}-r^2\Bigr\} r dr \nonumber \\
&=&\delta_{kn}\frac{\Bigl(\frac{2S(S+1)}{3}\Bigl)^n}{\Bigl(1+{\frac{2S(S+1)}{3}}\Bigl)^{n+1}}.
\end{eqnarray}
Thus $\hat \rho$ is the density matrix of a harmonic oscillator which is in thermal equilibrium, at temperature $T$, the value of which
may be obtained by nothing that the expectation value of the occupation
number $\hat n$ is
\begin{equation}
\langle \hat n \rangle=\frac{2S(S+1)}{3}.
\end{equation}
The latter equation immediately yields
\begin{equation}
\frac{\hbar\omega}{k_B T}=\ln\Bigl(\frac{3+2S(S+1)}{2S(S+1)}\Bigl)
\end{equation}
where $k_B$ is Boltzmann's constant, and $\omega$ is the frequency of the harmonic oscillator.
From the above discussion we conclude that for any well-behaved function of two variables $f$, we can write
\begin{eqnarray}
\lim_{N\to\infty}(2S+1)^{-N}& {\rm tr} & \Biggl\{f\Bigl(\frac{\hat J_+}{\sqrt{N}},\frac{\hat J_-}
{\sqrt{N}}\Bigr)\Biggr\}=\frac{3}{2S(S+1)}\sqrt{\frac{2S(S+1)}{3+2S(S+1)}}\nonumber \\ &\times &{\rm tr}\Bigl\{e^{-\ln (\frac{3+2S(S+1)}{2S(S+1)})(a^\dag a+\frac{1}{2})}{\mathcal N}f(a^\dag,a)\Bigr\}
\end{eqnarray}
where $\mathcal N$ denotes the normal ordering. We thus have constructed a scheme that anables us to map the spin opetarors to their bosonic couterparts. In fact we can also
conjecture the following result:
\begin{eqnarray}
\lim_{N\to\infty}(2S+1)^{-N}& {\rm tr} & \Biggl\{ {\mathcal N}f\Bigl(\frac{\hat J_+}{\sqrt{N}},\frac{\hat J_-}
{\sqrt{N}}\Bigr)\Biggr\}=\frac{3}{2S(S+1)}\sqrt{\frac{2S(S+1)}{3+2S(S+1)}}\nonumber \\ &\times &{\rm tr}\Bigl\{e^{-\ln (\frac{3+2S(S+1)}{2S(S+1)})(a^\dag
a+\frac{1}{2})}f(a^\dag,a)\Bigr\},
\end{eqnarray}
which can be checked by numerical calculation.
\section{Discussion and concluding remarks}
To summarize, we have shown that the probability distribution corresponding to a random tracial state of a set of $N$ spin-$S$ particles satisfies a
diffusion-like equation when $N$ is large. We have used this fact to derive its explicit form, and extended the use of the method developed in Ref.~\cite{ham1} to arbitrary
values of the spin. We also were able to come out with a bosonization scheme for the lowering and raising operators. We found that the mean value of the square of the spin is
proportional to the number of particles, the coefficient of proportionality being equal to $2D$. Hence by knowing the value of $ \langle m^2\rangle$ and $S$
it is possible to deduce the exact value
of $N$, a fact that may exploited experimentally as an indirect method of the measurement of the number of constituents of random spin systems.
Equivalently, if we measurement$ \langle m^2\rangle$ and $N$
it is possible to determine the value of the spin $S$. Moreover,
the diffusion-like equation would be of great usefulness in studying random spin systems in which transport phenomena are present (e.g by absorptions or adsorption),
when the number of constituents change with time. This would be complementary to ordinary transport equations rendering thus the full investigation of the evolution of
such systems more complete. Another possible application could be in studying high-temperature plasma systems~\cite{hora} where diffusion in space and time is present.
For instance, to verify the validity of the diffusion-like equation, it would be interesting to study a gas of electrons at high temperature~\cite{port}, and to identify the change of
the spin distribution of the total system as electrons are removed or added to the gas. Finally we note that the use of the explicit form of the distribution makes it possible
to investigate the decoherence of qubits in large spin systems, which is undoubtedly of great importance.
\section{}
The aim of this appendix is to derive Eq.(\ref{main})~\cite{ham2}. To this end we note that the spin space of a system of $N$ spin-S particles is given by $\mathbb C^{(2S+1)\otimes N}$.
This space can be decomposed as the direct sum
\begin{equation}
\mathbb C^{(2S+1)\otimes N}=\bigoplus_{j}^{NS} \nu(j,N;S) \mathbb C^{2j+1}.
\end{equation}
It follows that
\begin{eqnarray}
\mathbb C^{(2S+1)\otimes (N+1)}&=&\bigoplus_{j}^{(N+1)S} \nu(j,N+1;S) \mathbb C^{2j+1}\nonumber \\
&=&\bigoplus_{j}^{NS} \nu(j,N;S) \mathbb C^{2j+1}\otimes \mathbb C^{2S+1} \nonumber \\
&=&\bigoplus_{j}^{NS} \nu(j,N;S) \bigoplus_{j'=|j-S|}^{j+S} \mathbb C^{2j'+1}
\end{eqnarray}
By a suitable change of dummy summation indices, and respecting the triangle rule, we find by comparison that
\begin{equation}
\nu(j,N+1;S)=\sum\limits_{j'=|j-S|}^{j+S}\nu(j',N;S).
\end{equation}
\end{document} | math | 21,268 |
\begin{document}
\title{\textbf{\sc A Study on Set-Graphs}
\begin{abstract}
A \textit{primitive hole} of a graph $G$ is a cycle of length $3$ in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of that graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$. In this paper, we introduce the notion of set-graphs and study the properties and characteristics of set-graphs. We also check the primitive hole number and primitive degree of set-graphs. Interesting introductory results on the nature of order of set-graphs, degree of the vertices corresponding to subsets of equal cardinality, the number of largest complete subgraphs in a set-graph etc. are discussed in this study. A recursive formula to determine the primitive hole number of a set-graph is also derived in this paper.
\end{abstract}
\noindent \textbf{Key Words:} set-graphs, primitive hole, primitive degree.
\noindent \textbf{MS Classification}: 05C07, 05C38, 05C78.
\section{Introduction}
For general notations and concepts in graph theory, we refer to \cite{BM}, \cite{FH} and \cite{DBW}. All graphs mentioned in this paper are simple, connected undirected and finite, unless mentioned otherwise.
A \textit{hole} of a simple connected graph $G$ is a chordless cycle $C_n$ , where $n \in N$, in $G$. The \textit{girth} of a simple connected graph $G$, denoted by $g(G)$, is the order of the smallest cycle in $G$. The following notions are introduced in \cite{KS1}.
\begin{defn}{\rm
\cite{KS1} A \textit{primitive hole} of a graph $G$ is a cycle of length $3$ in $G$. The number of primitive holes in a given graph $G$ is called the \textit{primitive hole number} of that graph $G$. The primitive hole number of a graph $G$ is denoted by $h(G)$. }
\end{defn}
\begin{defn}{\rm
\cite{KS1} The \textit{primitive degree} of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$ and the primitive degree of the vertex $v$ in the graph $G$ is denoted by $d^p_G(v)$.}
\end{defn}
Some studies on primitive holes of certain graphs have been made in \cite{KS1}. The number of primitive holes in certain standard graph classes, their line graphs and total graphs were determined in this study. Some of the major results proved in \cite{KS1} are the following.
\begin{thm}
{\rm \cite{KS1}} The number of primitive holes in a complete graph $K_n$ is $h(K_n)=\binom{n}{3}$.
\end{thm}
\begin{thm}
{\rm \cite{KS1}} For any subgraph $H$ of a graph $G$, we have $h(H)\le h(G)$. Moreover, if $G$ is a graph on $n$ vertices, then $0\le h(G)\le \binom{n}{3}$.
\end{thm}
\section{Set-Graphs}
In this paper, we introduce the notion of set-graphs and study certain characteristics of set-graphs and also present a number of interesting results related to graph properties and invariants. A set-graph is defined as follows.
\begin{defn}\label{D-SG}{\rm
Let $A^{(n)} = \{a_1, a_2, a_3, \ldots, a_n\}, n\in \mathbb{N}$ be a non-empty set and the $i$-th $s$-element subset of $A^{(n)}$ be denoted by $A_{s,i}^{(n)}$. Now consider $\mathcal{S} = \{ A_{s,i}^{(n)}: A_{s,i}^{(n)} \subseteq A^{(n)}, A_{s,i}^{(n)} \ne \emptyset\}$. The \textit{set-graph} corresponding to set $A^{(n)}$, denoted $G_{A^{(n)}}$, is defined to be the graph with $V(G_{A^{(n)}}) = \{v_{s,i}: A_{s,i}^{(n)} \in \mathcal{S}\}$ and $E(G_{A^{(n)}}) = \{v_{s,i}v_{t,j}:~ A_{s,i}^{(n)}\cap A_{t,j}^{(n)}\ne \emptyset\}$, where $s\ne t~ \text{or}~ i\ne j$. }
\end{defn}
It can be noted from the definition of set-graphs that $A^{(n)}\ne\emptyset$ and if $|A^{(n)}|$ is a singleton, then $G_{A^{(n)}}$ to be the trivial graph. Hence, all sets we consider here are non-empty, non-singleton sets.
Let us now write the vertex set of a set-graph $G_{A^{(n)}}$ as $V(G_{A^{(n)}})=\{v_{s,r}: 1\le r\le \binom{n}{s}\}$, where $s$ is the cardinality of the subset $A^{(n)}_{s,r}$ of $A^{(n)}$ corresponding to the vertex $v_{s,r}$.
\noindent The following result is perhaps obvious, but an important property of set-graphs.
\begin{prop}
If $G$ is a set-graph, then $G$ has odd number of vertices.
\end{prop}
\begin{proof}
Let $G$ be a set-graph with respect to the set $A^{(n)}$. It is to be noted the number of non-empty subsets of $A^{(n)}$ is $2^n-1$. Since every vertex of $G$ corresponds to a non-empty subset of $A^{(n)}$, the number of vertices in $G$ must be $2^n-1$, an odd integer.
\end{proof}
\begin{ill}{\rm
Consider the set-graph with respect to the set $A^{(3)} = \{a_1, a_2, a_3\}$. Here we have the subsets of $A^{(3)}$ which are $A^{(3)}_{1,1}=\{a_1\}, A^{(3)}_{1,2}=\{a_2\}, A^{(3)}_{1,3}=\{a_3\}, A^{(3)}_{2,1}=\{a_1, a_2\}, A^{(3)}_{2,2}=\{a_1, a_3\},A^{(3)}_{2,3} =\{a_2, a_3\}, A^{(3)}_{3,1}=\{a_1, a_2, a_3\}$. Then, the vertices of $G_{A^{(3)}}$ have the labeling as follows. $v_{1,1} =\{a_1\}, v_{1,2} =\{a_2\}, v_{1,3} =\{a_3\}, v_{2,1}=\{a_1, a_2\}, v_{2,2} = \{a_1, a_3\}, v_{2,3} =\{a_2, a_3\}, v_{3,1} = \{a_1, a_2, a_3\}$.}
\end{ill}
\noindent Figure \ref{fig-1} depicts the above mentioned labeling procedure of the set-graph $G_{A^{(3)}}$.
\begin{figure}\label{fig-1}
\end{figure}
\begin{thm}\label{T-SGDV}
Let $G_{A^{(n)}}$ be a set-graph. Then, the vertices $v_{s,i}, v_{s,j}$ of $G_{A^{(n)}}$, corresponding to subsets $A^{(n)}_{s,i}$ and $A^{(n)}_{s,j}$ in $\mathcal{S}$ of equal cardinality, have the same degree in $G_{A^{(n)}}$. That is, $d_{G_{A^{(n)}}}(v_{s,i}) = d_{G_{A^{(n)}}}(v_{s,j})$.
\end{thm}
\begin{proof}
Consider the set-graph $G=G_{A^{(n)}}, n \in \mathbb{N}$. We begin by considering the vertices of $G$ corresponding to the $n$ singleton subsets of $A^{(n)}$. Let these vertices be denoted by $v_{1,i}$, where $1\le i\le n$. Clearly, for all ${j\ne i}$, we have $\{a_i\} \cap \{a_k\}=\emptyset$. Hence, by the definition of set-graphs, it follows that no edges are induced amongst the vertices $v_{1,1}, v_{1,2}, v_{1,3},\ldots, v_{1,n}$.
Now, construct all the two element subsets of $A^{(n)}$. Now choose two arbitrary vertices $v_{2,i}$ and $v_{2,j}$, where $i\ne j$. Then, here we have the subsets of $A^{(n)}$ of the form $\{a_i, a_j\}$, for $1\le i\ne j \le n$. It can be observed that the subsets of the form $\{a_i, a_j\}$ and $\{a_j, a_k\}$ are the elements of $\mathcal{S}$, where $1 \le i\ne j \ne k \le n$. Moreover, $\{a_i\} \cap \{a_i, a_j\}\ne \emptyset$ for all $1 \le i \ne j \le n$. In a similar way, we can extend this argument for the sets $\{a_i\}$ and an arbitrary subset of $A^{(n)}$ containing the element $a_i$. That is, the vertex $v_{1,i}$ is adjacent to those vertices of $G$ whose corresponding sets have $m$ elements including the common elements $a_i$, for $m\ge 2$. Therefore, $d_G(v_{1,i})=2^{n-1}-1$. Since the choice of $i$ is arbitrary, we have $d_G(v_{1,i})=d_G(v_{1,j})=2^{n-1}-1$ for all $1\le i,j \le n$. Therefore, the result holds for $s=1$.
Now, assume that the result holds for $s=k$, where $k$ is a positive integer. That is, we have $d_G(v_{k,i})=d_G(v_{k,j})$ for all $1\le i,j \le \binom{n}{k}$.
Next, consider the vertices of $G$ corresponding to the $(k+1)$-element subsets of $A^{(n)}$. Let $A_{(k+1),i}^{(n)}$ be a $(k+1)$-element subset of $A^{(n)}$ and let $v_{(k+1),i}$ be the vertex of $G$ corresponding to the set $A_{(k+1),i}^{(n)}$. Let $a_l$ be an arbitrary element of the set $A_{(k+1),i}^{(n)}$ and let $A_{(k+1),i}^{(n)'}=A_{(k+1),i}^{(n)}-\{a_l\}$. Then, the vertex $v_{(k+1),i}$ is adjacent to the vertices of $G$ corresponding to the sets containing the element $a_l$ in addition to the vertices of $G$ corresponding to the proper subsets of $A_{(k+1),i}^{(n)}$ and $A_{(k+1),i}^{(n)'}$. Hence, the difference between the number of edges incident on $v_{(k+1),i}$ and the number of edges incident on the vertex $v'_{(k+1),i}$ corresponding to the set $A_{(k+1),i}^{(n)'}$ is equal to the number of subsets of $A^{(n)}$ containing the element $a_l$, other than $A_{(k+1),i}^{(n)}$. This number is a constant for any set of $(k+1)$-element sets. Therefore, $d_G(v_{(k+1),i})=d_G(v_{(k+1),j})$ for all $1\le i,j \le \binom{n}{k+1}$. That is, the result is true for $s=k+1$ if it is true for $s=k$. Therefore, the theorem follows by induction.
\end{proof}
A question that arouses much interest in this context is what the degree of an arbitrary vertex of a set-graph $G_{A^{(n)}}$. The following result provides a solution to this problem.
\begin{thm}
Let $G$ be a set-graph with respect to a non-empty set $A^{(n)}=\{a_1,a_2,a_3,\ldots,a_n\}$ and let $v_{k,i}$ be an arbitrary vertex of $G$ corresponding to an $k$-element subset of $A^{(n)}$. Then, $d_G(v_{k,i})= (\sum\limits_{J}(-1)^{|J|-1}|\bigcap\limits_{j\in J}\mathcal{S}_j|)-1$, where $J$ is an indexing set such that $\emptyset\ne J\subseteq \{0,1,2,\ldots,k\}$ and $\mathcal{S}_j$ is the collection of subsets of $A^{(n)}$ containing the element $a_j$.
\end{thm}
\begin{proof}
Let $G$ be a set-graph with respect to a non-empty set $A^{(n)}$. Without loss of generality, let $A_{k,i}^{(n)}$ be a $k$-element subset of $A^{(n)}$, say $\{a_1,a_2,a_3,\ldots,a_k\}$ and let $v_{k,i}$ be the vertex of $G$ corresponding to the set $A_{k,i}^{(n)}$. Therefore, the vertex $v_{k,i}$ is adjacent to the vertices of $G$ which correspond to the subsets of $A^{(n)}$, containing the at least one element of $A_{k,i}^{(n)}$. That is $d_G(v_{k,i})= |\bigcup\limits_{j\in J}\mathcal{S}_j|-1$. But, by principle of inclusion and exclusion of sets, we have $|\bigcup\limits_{j\in J}\mathcal{S}_j|=\sum\limits_{J}(-1)^{|J|-1}|\bigcap\limits_{j\in J}\mathcal{S}_j|$, where $\emptyset\ne J\subseteq \{0,1,2,\ldots,k\}$. Therefore, $d_G(v_{k,i})= (\sum\limits_{J}(-1)^{|J|-1}|\bigcap\limits_{j\in J}\mathcal{S}_j|)-1$, where $\emptyset\ne J\subseteq \{0,1,2,\ldots,k\}$.
\end{proof}
Determining the degree of vertices of a set-graph is an important and interesting problem at this time. The following result determines a lower and upper limits for the degree of vertices of a given set-graph.
\begin{thm}\label{T-SGDV1}
For any vertex $v_{s,i}$ of a set-graph $G=G_{A^{(n)}}$, we have $2^{n-1}-1\le d_G(v_{s,i})\le 2(2^{n-1}-1)$.
\end{thm}
\begin{proof} Let $G=G_{A^{(n)}}$ be a set-graph with respect to a non-empty set $A^{(n)}$. Here, we need to consider the following two cases.
\noindent {\em Case-1:} It is to be noted that the vertices of $G$ corresponding to singleton subsets of $G$ have the minimum degree in $G$. Without loss of generality, let the vertex $v_{s,i}$ of $G$ corresponds to the set $\{a_i\}$. Then, $v_{s,i}$ should be adjacent to the vertices of $G$ corresponding to the subsets of $A^{(n)}$, other than itself, containing the element $a_i$. Therefore, degree of the vertex $v_{s,i}$ is equal to the the number of $m$-element subsets of $A^{(n)}$ containing the element $a_i$ for $m\ge2$. By binomial theorem, the total number of subsets of an $n$-element set, containing a particular element is $2^{n-1}$. Therefore, the minimum degree of a vertex in $G$ is $2^{n-1}-1$.
\noindent {\em Case-2:} Note that we need to consider $2^n -1$ of the subsets of $A^{(n)}$ only excluding $\emptyset$. Hence, the final vertex $v_{n,1}$ of the graph $G=G_{A^{(n)}}$ corresponding to the set $A^{(n)}$ in $\mathcal{S}$ will be adjacent to all its preceding vertices. Since $G$ has $2^n-1$ vertices, $d_G(v_{n,1})=2n-2$. No other vertices in $G$ can be adjacent to all other vertices of $G$, the vertex $v_{n,1}$ has the maximum possible degree in $G$. That is, the maximum degree of a vertex in $G$ is $2^n-2=2(2^{n-1}-1)$.
\end{proof}
\noindent The following results are immediate consequences of the above theorem.
\begin{cor}
For any set-graph $G=G_{A^{(n)}}$, $\Delta(G)=2\,\delta(G)$.
\end{cor}
\begin{proof}
From the proof the above theorem, we have $\delta (G)= 2^{n-1}-1$ and $\Delta (G) = 2^n-2= 2(2^{n-1}-1)$. This completes the proof.
\end{proof}
\begin{cor}\label{C-UDG}
There exists a unique vertex $v_{n,1}$ in a set-graph $G_{A^{(n)}}$ having degree $\Delta(G_{A^{(n)}})$.
\end{cor}
\begin{proof}
The proof follows from Case-2 of Theorem \ref{T-SGDV1}.
\end{proof}
The following result indicates the nature of the minimal and maximal degrees of the vertices of a set-graph.
\begin{cor}
The maximal degree of vertex in a set-graph $G$ is always an even number and the minimal degree of a vertex in $G$ is always an odd number.
\end{cor}
\begin{proof}
Let $G$ be a set-graph with respect to a non-empty set $A^{(n)}$. Then, by Theorem \ref{T-SGDV1}, the maximum degree of a vertex in a set-graph $G$ is $\Delta (G)=2(2^{n-1}-1)$, which is always an even number and the minimal degree of a vertex $G$ is $\delta(G)=2^{n-1}-1$, which is always an odd number.
\end{proof}
We have already proved that the vertex of the set-graph $G$ corresponding to the set $A^{(n)}$ itself has the maximum degree $2^n-2$ in $G$. Analogous to this result, we propose the following result on the primitive degree of this vertex $v_{n,1}$.
\begin{prop}
For set-graph $G=G_{A^{(n)}}$, the primitive degree of the vertex corresponding the set $A^{(n)}$ is $d^p_{G}(v_{n,1}) = |E(G)|-\Delta(G)$.
\end{prop}
\begin{proof}
Let $G=G_{A^{(n)}}$ be a set-graph with respect to the set $A^{(n)}$. Consider the subgraph $G'= G-v_{n,1}$. By Theorem \ref{T-SGDV}, we have $d(v_{n,1})=2^n-2$ and hence $|E(G')| = |E(G)|-(2^n - 2)$. Furthermore, both ends ends of every edge $vu \in E(G')$ are adjacent to vertex $v_{n,1}$ in $G_{A^{(n)}}$. Hence, each such edge $uv$ in $G'$ corresponds to a primitive hole $C_3$ in $G_{A^{(n)}}$ on the vertices $u,v, v_{n,1}$. Hence, $d^p_G(v_{n,1})=|E(G)|-(2^n - 2)= |E(G)|-\Delta(G)$.
\end{proof}
In the result given below, we describe a recursive formula to determine the number of edges of a set-graph.
\begin{thm}\label{T-SGRD}
For a set-graph $G_{A^{(n+1)}}$ we have
\begin{enumerate}\itemsep0mm
\item[(i)] $|E(G_{A^{(n+1)}})| = 3|E(G_{A^{(n)}})| + |V(G_{A^{(n)}})| + |E(K_{|V(G_{A^{(n)}})| + 1})|$
\item[(ii)] $|V(G_{A^{(n+1)}})| = 2|V(G_{A^{(n)}})| +1$.
\end{enumerate}
\end{thm}
\begin{proof}
Consider the set-graph $G_{A^{(n)}}$. To extend it to $G_{A^{(n+1)}}$, we proceed in five steps as explained below.
\begin{enumerate}\itemsep0.25cm
\item[(i)] Replicate the vertices of $G_{A^{(n)}}$ as an \textit{edgeless} graph and add the new element $a_{n+1}$ as an element to all subsets corresponding to all $v_{s,i} \in V(G_{A^{(n)}})$ and label each \textit{replica vertex}, $v^{\ast}_{s,i}$. Also add the new vertex $v_{1,(n+1)}$ corresponding to the single element subset $\{a_{n+1}\}$.,
\item[(ii)] Apply the definition of a set-graph to these new vertices. Clearly, we obtain the complete graph $K_{|V(G_{A^{(n)}})| + 1}$.
\item[(iii)] Each vertex $v_{s,i} \in V(G_{A^{(n)}})$ corresponding to the subset $A_{s,i}^{(n)}$ can be linked with its replica vertex corresponding to the new subset $A_{s,i}^{(n)} \cup \{a_{n+1}\}$. We refer to \textit{parallel linkage} and exactly $|V(G_{A^{(n)}})|$ such edges are added.
\item[(iv)] For the ends of each edge in $E(G_{A^{(n)}})$ say $v_{s,l}$ and $v_{t,l'}$ with corresponding subsets say, $A_{s,k}^{(n)}$ and $A_{t,m}^{(n)}$ we have $A_{s,k}^{(n)} \cap (A_{t,m}^{(n)}\cup \{a_{n+1}\}) \ne \emptyset$ and $A_{t,m}^{(n)}\cap (A_{s,k}^{(n)} \cup \{a_{n+1}\}) \ne \emptyset$. So the edges $v_{s,l} v_{s,l}^{\ast}$ and $v_{t,l'}v_{t,l'}^{\ast}$, with $v_{s,l}^{\ast}, v_{t,l'}^{\ast}$ corresponding to subsets $A_{s,k}^{(n)} \cup \{a_{n+1}\}$ and $A_{t,m}^{(n)} \cup \{a_{n+1}\}$ respectively, exist. Hence $2|E(G_{A^{(n)}})|$ additional edges are linked.
\item[(v)] Relabel the vertices according to the Definition \ref{D-SG} to obtain the set-graph $G_{A^{(n+1)}}$.
\end{enumerate}
The summation of the edges added through steps (i) to (v) plus the existing edges of $G_{A^{(n)}}$ provides the result: $|E(G_{A^{(n+1)}})| = 3|E(G_{A^{(n)}})| + |V(G_{A^{(n)}})| + |E(K_{|V(G_{A^{(n)}})| + 1})|$.
The second part of the Theorem is an immediate consequence of the above proof of first part. Then, the proof is complete.
\end{proof}
\noindent The following is a result related to the largest complete graphs found in a set-graph.
\begin{prop}\label{P-SGKN}
The set-graph $G_{A^{(n)}}, n \ge 2$ has exactly two largest complete graphs, $K_{2^{n-1}}$.
\end{prop}
\begin{proof}
Consider the set-graph $G_{A^{(n-1)}}$ and extend to $G_{A^{(n)}}$. From step (ii) in the proof of Theorem \ref{T-SGRD}, we construct a largest complete graph amongst the replica vertices and the new vertex $v_{1,n}$ because no vertex of $G_{A^{(n-1)}}$ is linked to $v_n$. However, the erstwhile vertex $v_{(n-1),1}$ of $G_{A^{(n-1)}}$ is also linked to all the \textit{replica} vertices hence, inducing the complete graph $K_{2^{n-1}}$. Clearly, another largest complete graph does not exist.
\end{proof}
The primitive hole number of a set-graph is determined recursively in the following theorem.
\begin{thm}
For a set-graph $G_{A^{(n)}}, n \geq 3$ we find the number of primitive holes through the recursive formula $h(G_{A^{(n+1)}}) = h(G_{A^{(n)}}) + \binom{2^n}{3} + 4|E(G_{A^{(n)}})|$.
\end{thm}
\begin{proof}
Consider the set-graph $G_{A^{(n)}}$ whose primitive hole number is denoted by $h(G_{A^{(n)}})$. Extending $G_{A^{(n)}}$ to $G_{A^{(n+1)}}$ will only increase the number of primitive holes. What we need here to determine the number of additional primitive holes formed on extending $G_{A^{(n)}}$ to $G_{A^{(n+1)}}$. This calculation is done as follows.
The set of replica vertices together with the vertex $v_{(n+1),1}$ induce a complete subgraph $K_{|V(G_{A^{(n)}}| +1}$ and hence an additional $\binom{2^n}{3}$ primitive holes are added to the extended graph. Finally, for each edge $v_{s,i}v_{t,j} \in E(G_{A^{(n)}})$ the vertices $v_{s,i}, v_{t,j}, v^{\ast}_{s,i}, v^{\ast}_{t,j}$ induce a $K_4$ subgraph and the number of primitive holes thus formed is $\binom{4}{3} = 4$. Hence, a further $4|E(G_{A^{(n)}}|$ primitive holes are added to the extended graph. This completes the proof.
\end{proof}
\noindent Next, we introduce the following notions for a set-graph as follows.
\begin{defn}{\rm
Let $G=G_{A^{(n)}}$ be a set-graph on a non-empty set $A^{(n)}$ and let $A_{s,i}^{(n)}$ be an arbitrary subset of the set $A^{(n)}$. The \textit{characteristic function} of a subset $A_{t,j}^{(n)}$ of $A^{(n)}$ with respect to $A_{s,i}^{(n)}$, denoted by $\xi_{A_{s,i}^{(n)}}(A_{t,j}^{(n)})$, is defined as
\begin{equation*}
\xi_{A_{s,i}^{(n)}}(A_{t,j}^{(n)})=
\begin{cases}
1 & \text{if} \quad A_{s,i}^{(n)}\cap A_{t,j}^{(n)}\ne \emptyset\\
0 & \text{if} \quad A_{s,i}^{(n)}\cap A_{t,j}^{(n)}=\emptyset.
\end{cases}
\end{equation*} }
\end{defn}
\begin{defn}{\rm
The \textit{tightness number} of a subset $A^{(n)}_{s,k}$, denoted $\varsigma(A_{s,k}^{(n)})$ is the number of subsets distinct from $A_{s,k}^{(n)}$ for which the intersection with $A_{s,k}^{(n)}$ is non-empty. Hence, $\varsigma(A_{s,k}^{(n)}) = \sum\limits_{j\ne k}\xi_{A_{s,i}^{(n)}}(A_{t,j}^{(n)})$.}
\end{defn}
We note that in terms of the definition of a set-graph and for the vertex $v_{s,i}$ corresponding to the subset $A_{s,k}^{(n)}$ we have, $d_{G_{A^{(n)}}}(v_{s,i}) = \varsigma(A_{s,k}^{(n)})$. Also, we have that $|E(G_{A^{(n)}})| = \frac{1}{2}\sum\limits_{1 \le k \le 2^n-1}\varsigma(A_{s,k}^{(n)})$.
The next theorem enables us to employ a step-wise recursive formula to determine the tightness number of all non-empty subsets of $A^{(n+1)}$ if the the tightness number of all non-empty subsets of $A^{(n)}$ are known.
\begin{thm}
Consider a set-graph $G_{A^{(n)}}, n \ge 1$ and its extended set-graph $G_{A^{(n+1)}}$. Then we have
\begin{enumerate}\itemsep0mm
\item[(i)] $\varsigma (\{a_{n+1}\}) = 2^n - 1$,
\item[(ii)] For each erstwhile subset $A^{(n)}_{s,i}$ with $\varsigma (A^{(n)}_{s,i}) = k$ in $G_{A^{(n)}}$, we have $\varsigma (A_{s,i}^{(n+1)}) = 2k+1$ in $G_{A^{(n+1)}}$,
\item[(iii)] For a replica vertex say, $v^{\ast}_{s,i}$ representing the new subset $A_{s,i}^{(n)} \cup \{a_{n+1}\}$ we have $\varsigma (A_{s,i}^{(n)}\cup \{a_{n+1}\}) = 2^n+k$.
\end{enumerate}
\end{thm}
\begin{proof} Let $G_{A^{(n)}}$ be a set-graph and $G_{A^{(n+1)}}$ be its extended graph obtained by introducing a new element, say $a_{n+1}$, to the set $A^{(n)}$. Then,
\begin{enumerate}\itemsep0.25cm
\item[(i)] To generate the set-graph $G_{A^{(n+1)}}$ by extending the set-graph $G_{A^{(n)}}$, we initially add the subsets $\{a_{n+1}\}$ and $A_{s,i}^{(n)} \cup \{a_{n+1}\}$, for all applicable values of $s$ and $i$. Clearly, $\{a_{n+1}\} \cap (A_{s,i}^{(n)}\cup \{a_{n+1}\}) \ne \emptyset$. So, $\varsigma (\{a_{n+1}\}) \ge 2^n -1$. Also, $\{a_{n+1}\} \cap A_{s,i}^{(n)}=\emptyset$. Therefore, we have $\varsigma (\{a_{n+1}\}) = 2^n -1$.
\item[(ii)] If $\varsigma (A_{s,i}^{(n)}) = k$ in $G_{A^{(n)}}$, the subset $A_{s,i}^{(n)}$ has non-zero intersections with exactly $k$ distinct subsets of $A^{(n)}$. Since in the replication, we have $A_{s,i}^{(n)}\cup \{a_{n+1}\}$ together with the subsets $A_k^{(n)}\cup \{a_{n+1}\}$, for all $k$, the result follows.
\item[(iii)] The replica vertices are $2^n-1$ in number and induce a complete graph together with vertex $v_{1,(n+1)}$. This partially represents $2^n -1$ non-zero intersections in respect of any subset say, $A_{s,i}^{(n)}\cup \{a_{n+1}\}$ corresponding to any replica vertex say, $v^{\ast}_{s,i}$. Clearly, $(A_{s,i}^{(n)} \cup \{a_{n+1}\}) \cap A_{s,i}^{(n)} \ne \emptyset$ and $(A_{s,i}^{(n)}\cup \{a_{n+1}\}) \cap A_{t,j}^{(n)}\ne \emptyset$, for all $A_{s,i}^{(n)}\cap A_{t,j}^{(n)}\ne \emptyset$ . It implies that an additional $(k+1)$ non-zero intersections exist in respect of $A_{s,i}^{(n)}\cup \{a_{n+1}\}$. Hence, $\varsigma (A_{s,i}^{(n)}\cup \{a_{n+1}\}) = 2^n-1+(k+1)= 2^n+k$.
\end{enumerate}
\noindent This completes the proof.
\end{proof}
\section{Certain Parameters of Set-Graphs}
Let $G$ be a given non-trivial finite graph. The \textit{chromatic number}, denoted by $\chi(G)$, of $G$ is the minimum $k$ for which $G$ is $k$-colourable.
\begin{thm}
The chromatic number of a set-graph $G_{A^{(n)}}$ is $\chi(G_{A^{(n)}}) = 2^{n-1}$.
\end{thm}
\begin{proof}
It is easy to see that $\chi(G_{A^{(1)}}) = 1 = 2^{1-1}$, $\chi(G_{A^{(2)}})=2=2^{2-1}$, $\chi(G_{A^{(3)}})=4= 2^{3-1}$ (See figure \ref{fig-1}). Assume the result holds for the set-graph $G_{A^{(k)}}$. Therefore, we have $\chi(G_{A^{(k)}}) = 2^{k-1}$.
Now consider the set-graph $G_{A^{(k+1)}}$. From the steps to be followed to extend from $G_{A^{(k)}}$ to $G_{A^{(k+1)}}$ (See proof of Theorem \ref{T-SGRD}), we have the erstwhile vertices of $G_{A^{(k)}}$, and in addition, the replica vertices corresponding to the vertices of $G_{A^{(k)}}$ and one more vertex $v_{1,(k+1)}$. From the proof of Proposition \ref{P-SGKN} we can notice that the replica vertices and vertex $v_{1,(k+1)}$ induce a largest complete subgraph, $K_{2^k}$ in the extended graph of $G_{A^{(n)}}$. We also note that the replica vertices and vertex $v_{1,k}$ form a second largest complete subgraph, $K_{2^k}$.
Since the vertices $v_{1,k}$ and $v_{1,{k+1}}$ are not adjacent in $G_{A^{(n)}}$, both of them have the same colour, say $c_1$ and colour the replica vertices by the colours $c_2, c_3, c_4, \ldots, c_{2^k}$. Since, no other largest complete graph exists it is always possible to find at least one pair of erstwhile-replica vertices which are non-adjacent. Hence the erstwhile vertex can carry the colour of such a replica vertex. This can be done in such a way that two adjacent erstwhile vertices do not carry the same colour by using the colours $c_2, c_3, c_4, ..., c_{2^k}$ accept for the colour of $v_{n,1}$, exhaustively. So the result $\chi(G_{A^{(k+1)}}) = 2^k = 2^{(k+1)-1}$ follows. Hence, the main result follows by induction.
\end{proof}
An \textit{independent set} of graph $G$ is a set of mutually non-adjacent vertices of $G$. The \textit{independence number}, denoted by $\alpha(G)$, of $G$ is the cardinality of a maximal independent set of $G$. The independence number of a set-graph is determined in the following theorem.
\begin{thm}
The independence number of a set-graph $G_{A^{(n)}}$ is $\alpha(G_{A^{(n)}}) = n$.
\end{thm}
\begin{proof}
Let $G= G_{A^{(n)}}$ be a given set-graph. Then, as explained in the proof of Theorem \ref{T-SGDV}, the vertices $v_{1,1}, v_{1,2}, v_{1,3}, ..., v_{1,n}$, corresponding to the singleton subsets of $A^{(n)}$, are pairwise non-adjacent. Hence, the set $I= \{v_{1,1}, v_{1,2}, v_{1,3}, ..., v_{1,n}\}$ is an independent set. By the Definition \ref{D-SG}, we note that any vertex in $V(G)-I$ is adjacent to at least one vertex in $I$. Therefore, $I$ is the maximal set of mutually non-adjacent vertices and hence is the maximal independent set in $G$. Hence, $\alpha(G_{A^{(n)}}) = n$.
\end{proof}
A \textit{domianting set} of a graph is a set of vertices $D$ such that every vertex of $G$ is either in $D$ or is adjacent to at least one vertex in $D$. The \textit{domination number}, denoted by $\gamma(G)$, of a graph $G$ is the cardinality of the minimal dominating set of $G$. The following discusses the domination number of a set-graph.
\begin{thm}
The domination number of a set-graph$G_{A^{(n)}}$ is $\gamma(G_{A^{(n)}}) = 1$.
\end{thm}
\begin{proof}
Let $G=G_{A^{(n)}}$ be a given set-graph. By Corollary \ref{C-UDG}, the vertex $v_{n,1}$, corresponding to the $n$-element set $A^{(n)}$, is the unique vertex in the set-graph $G$ that is adjacent to all other vertices of the set-graph $G$. Therefore, the singleton set $\{v_{n,1}\}$ is the minimal set such that every vertex of $G$ is adjacent to the unique element in the set $D$. Therefore, $\gamma(G_{A^{(n)}})=1$.
\end{proof}
Another parameter we consider here is the bondage number of a graph $G$, which is denoted by $b(G)$ defined as the minimum number of edges to be removed to increase the domination number $\gamma(G)$ by $1$.
\begin{thm}
The bondage number of a set-graph $G_{A^{(n)}}$ is $b(G_{A^{(n)}}) =1$.
\end{thm}
\begin{proof}
Since $\{v_{n,1}\}$ is the minimal dominating set of the set-graph $G=G_{A^{(n)}}$, the removal of any edge $v_{s,i}v_{n,1}$ will increase the domination number by $1$ in the reduced graph $G_{A^{(n)}} - v_{s,i}v_{n,1}$.
\end{proof}
Another parameter we are going to discuss here is the McPhersion Number of undirected graphs. For this, let us now recall the definition McPhersion Number, as given in \cite{JS1}.
\begin{defn}{\rm
\cite{JS1} The \textit{McPherson recursion} is a series of \textit{vertex explosions} such that on the first iteration a vertex $v \in V(G)$ explodes to arc (directed edges) to all vertices $u \in V(G)$ for which the edge $vu \notin E(G)$, to obtain the mixed graph $G'_1$. Now $G'_1$ is considered on the second iteration and a vertex $w \in V(G'_1) = V(G)$ may explode to arc to all vertices $z \in V(G'_1)$ if edge $wz \notin E(G)$ and arc $(w,z)$ or $(z,w) \notin E(G'_1)$.
The \textit{McPherson number}, denoted by $\Upsilon(G)$, of a simple connected graph $G$ is the minimum number of iterative vertex explosions say $l$, to obtain the mixed graph $G'_l$ such that the underlying graph $G^{\ast}_l \cong K_n$.}
\end{defn}
The McPherson number of a set-graph $G_{A^{(n)}}$ is determined in the following theorem.
\begin{thm}
For a set-graph $G_{A^{(n)}}$ we have $\Upsilon(G_{A^{(n)}}) = 2^{n-1}-1$.
\end{thm}
\begin{proof}
Consider the set-graph $G_{A^{(n-1)}}$ which has $2^{n-1}-1$ vertices. On extending the set-graph $G_{A^{(n)}}$, the replica vertices, as explained in Theorem \ref{T-SGRD}, together with vertex $v_{n,1}$ induce a complete subgraph and hence no further vertex explosions are required to ensure complete induced by these vertices. However, at least all the erstwhile vertices require vertex explosions to ensure complete connectivity amongst themselves and the replica vertices. Hence, $\Upsilon(G_{A^{(n)}}) = 2^{n-1} -1$.
\end{proof}
\section{Conclusion}
We have discussed particular types of graphs called set-graphs and studied certain characteristics and structural properties of these graphs. The study seems to be promising as it can be extended to certain standard graph classes and certain graphs that are associated with the given graphs. More problems in this area are still open and hence there is a wide scope for further studies. Now, we have the notion of {M\`ela numbers} as follows.
\begin{defn}{\rm
The set of \textit{M\`ela numbers} is defined to be the set $\mathbb{M} =\{m_i: m_1 = 1, m_i = 2m_{i-1} + 1, i\in \mathbb{N}, i\ge 2\}$.}
\end{defn}
\noindent Invoking the above definition, it can immediately be noted that $|V(G_{A^n})| = m_n$.
\noindent Some open problems\footnote{The first author wishes to dedicate these open problems to Ms. M\`ela Odendaal, who is expected to grow up as a great mathematician.} we wish to mention in this context are the following.
\begin{prob}
Show that $m_i + m_j \notin \mathbb{M},$ $m_im_j \notin \mathbb{M}$ and if $m_i > m_j$ then $m_i - m_j \notin \mathbb{M}$.
\end{prob}
\begin{prob}
Show that $m_{ki}$ is divisible by $m_i$ but, $\frac{m_{ki}}{m_i} \notin \mathbb{M}$.
\end{prob}
Finding other number theoretical results for \textit{M\`ela numbers} are also challenging problems which seems to be promising. All these facts indicate that there is a wide scope for further research in this area.
\end{document} | math | 29,600 |
\begin{document}
\selectlanguage{english}
\title{Strongly anisotropic diffusion problems; asymptotic analysis}
\author{Mihai Bostan
\thanks{Laboratoire d'Analyse, Topologie, Probabilit\'es LATP, Centre de Math\'ematiques et Informatique CMI, UMR CNRS 7353, 39 rue Fr\'ed\'eric Joliot Curie, 13453 Marseille Cedex 13
France. E-mail : {\tt [email protected]}}
}
\date{ (\today)}
\maketitle
\begin{abstract}
The subject matter of this paper concerns anisotropic diffusion equations: we consider heat equations whose diffusion matrix have disparate eigenvalues. We determine first and second order approximations, we study the well-posedness of them and establish convergence results. The analysis relies on averaging techniques, which have been used previously for studying transport equations whose advection fields have disparate components.
\end{abstract}
\paragraph{Keywords:}
Anisotropic diffusion, Variational methods, Multiple scales, Average operator.
\paragraph{AMS classification:} 35Q75, 78A35.
\section{Introduction}
\label{Intro}
\indent
Many real life applications lead to highly anisotropic diffusion equations: flows in porous media, quasi-neutral plasmas, microscopic transport in magnetized plasmas \cite{Bra65}, plasma thrusters, image processing \cite{PerMal90}, \cite{Wei98}, thermal properties of crystals \cite{DiaShaYun91}. In this paper we investigate the behavior of the solutions for heat equations whose diffusion becomes very high along some direction. We consider the problem
\begin{equation}
\label{Equ1} \partial _t \ue - \divy ( D(y) \nabla _y \ue ) - \frac{1}{\eps} \divy ( b(y) \otimes b(y) \nabla _y \ue ) = 0, \;\;(t,y) \in \R_+ \times \R ^m
\end{equation}
\begin{equation}
\label{Equ2}
\ue (0,y) = \uein (y), \;\;y \in \R^m
\end{equation}
where $D(y) \in {\cal M}_m (\R)$ and $b(y) \in \R^m$ are smooth given matrix field and vector field on $\R^m$, respectively. For any two vectors $\xi, \eta$, the notation $\xi \otimes \eta$ stands for the matrix whose entry $(i,j)$ is $\xi _i \eta _j$, and for any two matrix $A, B$ the notation $A:B$ stands for $\mathrm{trace}(^t AB) = A_{ij} B_{ij}$ (using Einstein summation convention).
We assume that at any $y \in \R^m$ the matrix $D(y)$ is symmetric and $D(y) + b (y) \otimes b(y)$ is positive definite
\begin{equation}
\label{Equ3}
^t D (y) = D(y),\;\;\exists \;d >0 \;\;\mbox{such that}\;\;D(y)\xi\cdot\xi + (b(y) \cdot \xi)^2 \geq d \;|\xi|^2,\;\;\xi \in \R ^m,\;\;y \in \R ^m.
\end{equation}
The vector field $b(y)$, to which the anisotropy is aligned, is supposed divergence free {\it i.e.,} $\divy b = 0$. We intend to analyse the behavior of \eqref{Equ1}, \eqref{Equ2} for small $\eps$, let us say $0 < \eps \leq 1$. In that cases $D(y) + \frac{1}{\eps} b(y) \otimes b(y)$ remains positive definite and if $(\uein)_\eps$ remain in a bounded set of $\lty$, then $(\ue)_\eps$ remain in a bounded set of $\litlty{}$ since, for any $t \in \R_+$ we have
\begin{align*}
\frac{1}{2}\inty{(\ue (t,y))^2} & + d \intsy{|\nabla _y \ue (s,y)|^2} \leq \frac{1}{2}\inty{(\ue (t,y))^2} \\
& + \intsy{\left \{D(y) + \frac{1}{\eps} b(y) \otimes b(y) \right \} : \nabla _y \ue (s,y) \otimes \nabla _y \ue (s,y)} \\
& = \frac{1}{2}\inty{(\uein (y))^2}.
\end{align*}
In particular, when $\eps \searrow 0$, $(\ue)_\eps$ converges, at least weakly $\star$ in $\litlty{}$ towards some limit $u \in \litlty{}$. Notice that the explicit methods are not well adapted for the numerical approximation of \eqref{Equ1}, \eqref{Equ2} when $\eps \searrow 0$, since the CFL condition leads to severe time step constraints like
\[
\frac{d}{\eps} \frac{\Delta t}{|\Delta y |^2} \leq \frac{1}{2}
\]
where $\Delta t$ is the time step and $\Delta y $ is the grid spacing. In such cases implicit methods are desirable \cite{BalTilHow08}, \cite{ShaHam10}.
Rather than solving \eqref{Equ1}, \eqref{Equ2} for small $\eps >0$, we concentrate on the limit model satisfied by the limit solution $u = \lime \ue$. We will see that the limit model is still a parabolic problem, decreasing the $\lty$ norm and satisfying the maximum principle. At least formally, the limit solution $u$ is the dominant term of the expansion
\begin{equation}
\label{Equ6} \ue = u + \eps u ^1 + \eps ^2 u ^2 + ...
\end{equation}
Plugging the Ansatz \eqref{Equ6} into \eqref{Equ1} leads to
\begin{equation}
\label{Equ7} \divy (b \otimes b \nabla _y u ) = 0,\;\;(t,y) \in \R_+ \times \R ^m
\end{equation}
\begin{equation}
\label{Equ8} \partial _t u - \divy (D \nabla _y u ) - \divy ( b \otimes b \nabla _y u^1) = 0,\;\;(t,y) \in \R_+ \times \R ^m
\end{equation}
\[
\vdots
\]
Clearly, the constraint \eqref{Equ7} says that at any time $t \in \R_+$, $b \cdot \nabla _y u = 0$, or equivalently $u(t,\cdot)$ remains constant along the flow of $b$, see \eqref{EquFlow}
\[
u(t, Y(s;y)) = u(t,y),\;\;s \in \R,\;\;y \in \R^m.
\]
The closure for $u$ comes by eliminating $u^1$ in \eqref{Equ8}, combined with the fact that \eqref{Equ7} holds true at any time $t \in \R_+$. The symmetry of the operator $\divy (b \otimes b \nabla _y)$ implies that $\partial _t u - \divy (D \nabla _y u)$ belongs to $(\ker (b \cdot \nabla _y ))^\perp$ and therefore we obtain the weak formulation
\begin{equation}
\label{Equ9}
\frac{\md}{\md t}\inty{u(t,y) \varphi (y)} + \inty{D \nabla _y u (t,y) \cdot \nabla _y \varphi (y) } = 0,\;\;\varphi \in \hoy \cap \kerbg{}.
\end{equation}
The above formulation is not satisfactory, since the choice of test functions is constrained by \eqref{Equ7}; \eqref{Equ9} is useless for numerical simulation. A more convenient situation is to reduce \eqref{Equ9} to another problem, by removing the constraint \eqref{Equ7}. The method we employ here is related to the averaging technique which has been used to handle transport equations with diparate advection fields \cite{BosAsyAna}, \cite{BosTraSin}, \cite{BosGuidCent3D}, \cite{Bos12}
\begin{equation}
\label{Equ10} \partial _t \ue + a(t,y) \cdot \nabla _y \ue + \frac{1}{\eps} b (y) \cdot \nabla _y \ue = 0,\;\;(t,y) \in \R_+ \times \R^m
\end{equation}
\begin{equation}
\label{Equ11} \ue (0,y) = \uein (y),\;\;y \in \R^m.
\end{equation}
Using the same Ansatz \eqref{Equ6} we obtain as before that $b \cdot \nabla _y u (t,\cdot) = 0, t \in \R_+$ and the closure for $u$ writes
\begin{equation}
\label{Equ12}
\mathrm{Proj}_{\kerbg} \{ \partial _t u + a\cdot \nabla _y u \} = 0
\end{equation}
or equivalently
\begin{equation}
\label{Equ14} \frac{\md }{\md t} \inty{u(t,y) \varphi (y) } - \inty{u(t,y) \;a \cdot \nabla _y \varphi } = 0
\end{equation}
for any smooth function satisfying the constraint $b \cdot \nabla _y \varphi = 0$. The method relies on averaging since the projection on $\kerbg$ coincides with the average along the flow of $b$, cf. Proposition \ref{AverageOperator}. As $u$ satisfies the constraint $b \cdot \nabla _y u = 0$, it is easily seen that $\mathrm{Proj}_{\kerbg} \partial _t u = \partial _t u$. A simple case to start with is when the transport operator $a \cdot \nabla _y$ and $b \cdot \nabla _y$ commute {\it i.e.,} $[b \cdot \nabla _y, a \cdot \nabla _y ] = 0$. In this case $a \cdot \nabla _y$ leaves invariant the subspace of the constraints, implying that $\mathrm{Proj}_{\kerbg} \{a \cdot \nabla _y u \} = a \cdot \nabla _y u$. Therefore \eqref{Equ12} reduces to a transport equation and it is easily seen that this equation propagates the constraint, which allows us to remove it. Things happen similarly when the transport operators $a \cdot \nabla _y, b \cdot \nabla _y$ do not commute, but the transport operator of the limit model may change. In \cite{BosTraSin} we prove that there is a transport operator $A \cdot \nabla _y$, commuting with $b \cdot \nabla _y$, such that for any $u \in \kerbg$ we have
\[
\mathrm{Proj}_{\kerbg} \{a \cdot \nabla _y u \} = A \cdot \nabla _y u.
\]
Once we have determined the field $A$, \eqref{Equ12} can be replaced by $\partial _t u + A \cdot \nabla _y u = 0$, which propagates the constraint $b \cdot \nabla _y u (t) = 0$ as well.
Comming back to the formulation \eqref{Equ9}, we are looking for a matrix field $\tilde{D}(y)$ such that $\divy (\tilde{D} \nabla _y)$ commutes with $b \cdot \nabla _y$ and
\[
\mathrm{Proj}_{\kerbg} \{\divy (D(y) \nabla _y u ) \}= \divy (\tilde{D}(y)\nabla _y u),\;\;u \in \kerbg{}.
\]
We will see that, under suitable hypotheses, it is possible to find such a matrix field $\tilde{D}$, and therefore \eqref{Equ9} reduces to the parabolic model
\begin{equation}
\label{Equ15} \partial _t u - \divy (\tilde{D}(y) \nabla _y u ) = 0,\;\;(t,y) \in \R_+ \times \R^m.
\end{equation}
The matrix field $\tilde{D}$ will appear as the orthogonal projection of the matrix field $D$ (with respect to some scalar product to be determined) on the subspace of matrix fields $A$ satisfying $[b\cdot \nabla _y, \divy(A \nabla _y)] = 0$. The field $\tilde{D}$ inherits the properties of $D$, like symmetry, positivity, etc.
Our paper is organized as follows. The main results are presented in Section \ref{ModMainRes}. Section \ref{AveOpe} is devoted to the interplay between the average operator and first and second order linear differential operators. In particular we justify the existence of the {\it averaged} matrix field $\tilde{D}$ associated to any field $D$ of symmetric, positive matrix. The first order approximation is justified in Section \ref{FirstOrdApp} and the second order approximation is discussed in Section \ref{SecOrdApp}. Several technical proofs are gathered in Appendix \ref{A}.
\section{Presentation of the models and main results}
\label{ModMainRes}
\noindent
We assume that the vector field $b :\R^m \to \R^m$ is smooth and divergence free
\begin{equation}
\label{Equ21} b \in W^{1,\infty}_{\mathrm{loc}} (\R^m),\;\;\divy b = 0
\end{equation}
with linear growth
\begin{equation}
\label{Equ22}
\exists \;C > 0\;\;\mbox{such that}\;\; |b(y)| \leq C (1 + |y|),\;\;y \in \R^m.
\end{equation}
We denote by $Y(s;y)$ the characteristic flow associated to $b$
\begin{equation}
\label{EquFlow} \frac{\md Y}{\md s} = b(Y(s;y)),\;\;Y(s;0) = y,\;\;s \in \R,\;\;y \in \R^m.
\end{equation}
Under the above hypotheses, this flow has the regularity $Y \in W^{1,\infty} _{\mathrm{loc}} (\R \times \R^m)$ and is measure preserving.
We concentrate on matrix fields $A(y) \in \loloc{}$ such that $[b(y) \cdot \nabla _y, \divy ( A(y) \nabla _y)] = 0$, let us say in $\dpri$. We check that the commutator between $b \cdot \nabla _y $ and $\divy (A \nabla _y)$ writes cf. Proposition \ref{ComSecOrd}
\[
[b(y) \cdot \nabla _y, \divy ( A(y) \nabla _y)] = \divy ( [b,A]\nabla _y)\;\;\mbox{in}\;\;\dpri
\]
where the bracket between $b$ and $A$ is given by
\[
[b,A] := (b \cdot \nabla _y) A - \partial _y b A (y) - A(y) \;^t \partial _y b,\;\;y \in \R^m.
\]
Several characterizations for the solutions of $[b,A] = 0$ in $\dpri$ are indicated in the Propositions \ref{MFI}, \ref{WMFI}, among which
\begin{equation}
\label{Equ16} A(Y(s;y)) = \partial _y Y (s;y) A(y) \;{^t \partial _y Y} (s;y),\;\;s\in \R,\;\;y \in \R ^m.
\end{equation}
We assume that there is a matrix field $P(y)$ such that
\begin{equation}
\label{Equ56} ^t P = P,\;\;P(y) \xi \cdot \xi >0,\;\;\xi \in \R^m,\;\; y \in \R^m,\;\;P^{-1}, P \in \ltloc{},\;\;[b,P]= 0 \;\mbox{in}\;\dpri.
\end{equation}
We introduce the set
\[
H_Q = \{ A = A(y)\;:\; \inty{Q(y) A(y) : A(y) Q(y) } < +\infty\}
\]
where $Q = P ^{-1}$, and the scalar product
\[
(A,B)_Q = \inty{QA:BQ},\;\;A, B \in H_Q.
\]
The equality \eqref{Equ16} suggests to introduce the family of applications $G(s): H_Q \to H_Q$, $s \in \R$, $G(s)A = (\partial _y Y )^{-1}(s; \cdot) A(Y(s;\cdot)) \;^t (\partial _y Y )^{-1}(s;\cdot)$ which is a $C^0$-group of unitary operators on $H_Q$ cf. Proposition \ref{Groupe}. This allows us to introduce $L$, the infinitesimal generator of $(G(s))_{s\in \R}$. The operator $L$ is skew-adjoint on $H_Q$ and its kernel coincides with $\{A \in H_Q\subset \loloc{} : [b,A] = 0\;\mbox{in} \; \dpri\}$ cf. Proposition \ref{PropOpeL}. The averaged matrix field denoted $\ave{D}_Q$, associated to any $D \in H_Q$ appears as the long time limit of the solution of
\begin{equation}
\label{Equ67}
\partial _t A - L(L(A)) = 0,\;\;t \in \R_+
\end{equation}
\begin{equation}
\label{Equ68} A(0) = D.
\end{equation}
The notation $\ave{\cdot}$ stands for the orthogonal projection (in $\lty{}$) on $\kerbg{}$.
\begin{thm}
\label{AveMatDif} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ56} hold true. Then for any $D \in H_Q \cap \liy{}$ the solution of \eqref{Equ67}, \eqref{Equ68} converges weakly in $H_Q$ as $t \to +\infty$ towards the orthogonal projection of $D$ on $\ker L$
\[
\lim _{t \to +\infty} A(t) = \ave{D}_Q\;\mbox{ weakly in }\;H_Q,\;\;\ave{D}_Q := \mathrm{Proj} _{\ker L } D.
\]
If $D$ is symmetric and positive, then so is the limit $\ave{D}_Q = \lim _{t \to +\infty} A(t)$, and satisfies
\begin{equation}
\label{Equ72} L (\ave{D}_Q) = 0,\;\;\nabla _y u \cdot \ave{D}_Q \nabla _y v = \ave{\nabla _y u \cdot D\nabla _y v},\;\;u, v \in H^1(\R^m) \cap \kerbg{}
\end{equation}
\begin{equation}
\label{Equ72Bis}\ave{\nabla _y u \cdot \ave{D}_Q \nabla _y (b \cdot \nabla _y \psi )} = 0,\;\;u \in H^1(\R^m) \cap \kerbg{},\;\;\psi \in C^2_c (\R^m).
\end{equation}
\end{thm}
The first order approximation (for initial data not necessarily well prepared) is justified by
\begin{thm}
\label{MainResult1} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ56}, \eqref{Equ26} hold true and that $D$ is a field of symmetric positive matrix, which belongs to $H_Q$. Consider a family of initial conditions $(\uein)_{\eps } \subset \lty$ such that $(\ave{\uein})_\eps$ converges weakly in $\lty{}$, as $\eps \searrow 0$, towards some function $\uin$. We denote by $\ue$ the solution of \eqref{Equ1}, \eqref{Equ2} and by $u$ the solution of
\begin{equation}
\label{Equ75}
\partial _t u - \divy ( \ave{D}_Q \nabla _y u ) = 0,\;\;t \in \R_+,\;\;y \in \R^m
\end{equation}
\begin{equation}
\label{Equ76}
u(0,y) = \uin (y),\;\;y \in \R^m
\end{equation}
where $\ave{D}_Q$ is associated to $D$, cf. Theorem \ref{AveMatDif}. Then we have the convergences
\[
\lime \ue = u\;\;\mbox{weakly} \star \mbox{ in } \litlty{}
\]
\[
\lime \nabla _y \ue = \nabla _y u\;\;\mbox{weakly} \mbox{ in } \lttlty{}.
\]
\end{thm}
The derivation of the second order approximation is more complicated and requires the computation of some other matrix fields. For simplicity, we content ourselves to formal results. The crucial point is to introduce the decomposition given by
\begin{thm}
\label{Decomposition} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ56}, \eqref{Equ26} hold true and that $L$ has closed range. Then, for any field of symmetric matrix $D \in H_Q$, there is a unique field of symmetric matrix $F \in \dom (L^2) \cap (\ker L )^\perp$ such that
\[
- \divy ( D \nabla _y) = - \divy ( \ave{D}_Q \nabla _y ) + \divy (L^2 (F)\nabla _y )
\]
that is
\begin{align*}
& \inty{D \nabla _y u \cdot \nabla _y v } - \inty{\ave{D}_Q\nabla _y u \cdot \nabla _y v } \\
& = \inty{L(F) \nabla _y u \cdot \nabla _y (b \cdot \nabla _y v)} + \inty{L(F) \nabla _y ( b \cdot \nabla _y u ) \cdot \nabla _y v} \\
& = - \inty{F \nabla _y ( b \cdot \nabla _y ( b \cdot \nabla _y u)) \cdot \nabla _y v} - 2 \inty{F \nabla _y ( b \cdot \nabla _y u) \cdot \nabla _y ( b \cdot \nabla _y v)}\\
& - \inty{F \nabla _y u \cdot \nabla _y ( b \cdot \nabla _y ( b \cdot \nabla _y v))}
\end{align*}
for any $u, v \in C^3_c(\R^m)$.
\end{thm}
After some computations we obtain, at least formally, the following model, replacing the hypothesis \eqref{Equ56} by the stronger one: there is a matrix field $R(y)$ such that
\begin{equation}
\label{Equ90}
\det R(y)\neq 0,\;y \in \R^m,\;Q = {^t R} R \mbox{ and }P = Q^{-1} \in \ltloc{},\;b \cdot \nabla _y R + R \partial _y b = 0 \mbox{ in } \dpri.
\end{equation}
\begin{thm}
\label{MainResult2}
Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ23}, \eqref{Equ26}, \eqref{Equ90} hold true and that $D$ is a field of symmetric positive matrix which belongs to $H_Q \cap \liy{}$. Consider a family of initial conditions $(\uein)_\eps \subset \lty{}$ such that $(\frac{\ave{\uein} - \uin }{\eps} ) _{\eps >0}$ converges weakly in $\lty{}$, as $\eps \searrow 0$, towards a function $\vin{}$, for some function $\uin \in \kerbg{}$. Then, a second order approximation for \eqref{Equ1} is provided by
\begin{equation}
\label{IntroEqu87}\partial _t \tue - \divy ( \ave{D}_Q \nabla _y \tue) + \eps [ \divy ( \ave{D}_Q \nabla _y ), \divy (F \nabla _y ) ]\tue - \eps S(\tue) = 0,\;\;(t,y) \in \R_+ \times \R^m
\end{equation}
\begin{equation}
\label{NewIC} \tue (0,y) = \uin (y) + \eps ( \vin (y) + \win (y)),\;\;\win = \divy ( F \nabla _y \uin),\;\;y \in \R ^m
\end{equation}
for some fourth order linear differential operator $S$, see Proposition \ref{DifOpe}, and the matrix field $F$ given by Theorem \ref{Decomposition}.
\end{thm}
\section{The average operator}
\label{AveOpe}
\noindent
We assume that the vector field $b : \R^m \to \R^m$ satisfies \eqref{Equ21}, \eqref{Equ22}. We consider the linear operator $u \to b \cdot \nabla _y u = \divy(ub)$ in $\lty{}$, whose domain is defined by
\[
\dom (b \cdot \nabla _y ) = \{ u \in \lty{} \;:\; \divy(ub) \in \lty\}.
\]
It is well known that
\[
\kerbg = \{ u \in \lty{}\;:\; u (Y(s;\cdot)) = u (\cdot), \;s \in \R\}.
\]
The orthogonal projection on $\kerbg{}$ (with respect to the scalar product of $\lty{}$), denoted by $\ave{\cdot}$, reduces to average along the characteristic flow $Y$ cf. \cite{BosTraSin} Propositions 2.2, 2.3.
\begin{pro}
\label{AverageOperator} For any function $u \in \lty{}$ the family $\ave{u}_T : = \frac{1}{T} \int _0 ^T u (Y(s;\cdot))\md s, T>0$ converges strongly in $\lty{}$, when $T \to + \infty$, towards the orthogonal projection of $u$ on $\kerbg{}$
\[
\lim _{T \to +\infty} \ave{u}_T = \ave{u},\;\;\ave{u} \in \kerbg{} \;\mbox{and} \; \inty{(u - \ave{u}) \varphi } = 0,\;\forall\; \varphi \in \kerbg{}.
\]
\end{pro}
Since $b \cdot \nabla _y$ is antisymmetric, one gets easily
\begin{equation}
\label{Equ24} \overline{\ran (b \cdot \nabla _y ) } = (\kerbg{} ) ^\perp = \ker ( \mathrm{Proj}_{\kerbg{}} ) = \ker \ave{\cdot}.
\end{equation}
\begin{remark}
\label{DetFun} If $u \in \lty{}$ satisfies $\inty{u(y) b \cdot \nabla _y \psi } = 0, \forall \;\psi \in C^1 _c (\R^m)$ and $\inty{u\varphi }= 0, \forall \; \varphi \in \kerbg{}$, then $u = 0$. Indeed, as $u \in \lty{} \subset \loloc{}$, the first condition says that $b \cdot \nabla _y u = 0$ in $\dpri{}$ and thus $u \in \kerbg{}$. Using now the second condition with $\varphi = u$ one gets $\inty{u^2} = 0$ and thus $u = 0$.
\end{remark}
In the particular case when $\ran (b \cdot \nabla _y)$ is closed, which is equivalent to the Poincar\'e inequality (cf. \cite{Brezis} pp. 29)
\begin{equation}
\label{Equ23} \exists\;C_P >0\;:\; \left ( \inty{(u - \ave{u})^2}\right ) ^{1/2} \leq C_P \left ( \inty{(b \cdot \nabla _y u ) ^2} \right ) ^{1/2},\;\;u \in \dom (b \cdot \nabla _y)
\end{equation}
\eqref{Equ24} implies the solvability condition
\[
\exists \; u \in \dom ( b \cdot \nabla _y ) \;\mbox{ such that }\; b \cdot \nabla _y u = v\;\mbox{ iff } \ave{v} = 0.
\]
If $\|\cdot \|$ stands for the $\lty{}$ norm we have
\begin{pro}
\label{Inverse}
Under the hypothesis \eqref{Equ23}, $b \cdot \nabla _y $ restricted to $\ker \ave{\cdot}$ is one to one map onto $\ker \ave{\cdot}$. Its inverse, denoted $(b \cdot \nabla _y )^{-1}$, belongs to ${\cal L}(\ker \ave{\cdot}, \ker \ave{\cdot})$ and
\[
\|(b \cdot \nabla _y ) ^{-1} \|_{{\cal L}(\ker \ave{\cdot}, \ker \ave{\cdot})} \leq C_P.
\]
\end{pro}
Another operator which will play a crucial role is ${\cal T } = - \divy (b \otimes b \nabla _y)$ whose domain is
\[
\dom ({\cal T}) = \{ u \in \dom (b \cdot \nabla _y)\;:\; b \cdot \nabla _y u \in \dom ( b \cdot \nabla _y )\}.
\]
The operator ${\cal T}$ is self-adjoint and under the previous hypotheses, has the same kernel and range as $b\cdot \nabla _y$.
\begin{pro}
\label{KerRanTau} Under the hypotheses \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ23} the operator ${\cal T}$ satisfies
\[
\ker {\cal T} = \kerbg,\;\;\ran {\cal T} = \ran (b \cdot \nabla _y ) = \ker \ave{\cdot}
\]
and $\| u - \ave{u}\| \leq C_P ^2 \|{\cal T} u \|,u \in \dom ({\cal T})$.
\end{pro}
\begin{proof}
Obviously $\kerbg{} \subset \ker {\cal T}$. Conversely, for any $u \in \ker {\cal T}$ we have $\inty{\;(\bg u )^2} = \inty{\;u {\cal T}u} = 0$ and therefore $ u \in \kerbg{}$.
Clearly $\ran {\cal T} \subset \ran ( \bg{}) = \ker \ave{\cdot}$. Consider now $w \in \ker \ave{\cdot} = \ran ( \bg{})$. By Proposition \ref{Inverse} there is $v \in \ker \ave{\cdot} \cap \dom (\bg)$ such that $\bg v = w$. Applying one more time Proposition \ref{Inverse}, there is $ u \in \ker \ave{\cdot} \cap \dom (\bg)$ such that $\bg u = v$. We deduce that $u \in \dom {\cal T}, w = {\cal T}(-u)$. Finally, for any $u \in \dom {\cal T}$ we apply twice the Poincar\'e inequality, taking into account that $\ave{\bg u } = 0$
\[
\| u - \ave{u}\| \leq C_P \|\bg u \| \leq C_P ^2 \|{\cal T} u \|.
\]
\end{proof}
\begin{remark}
\label{AveLone}
The average along the flow of $b$ can be defined in any Lebesgue space $L^q (\R^m)$, $q \in [1,+\infty]$. We refer to \cite{BosTraSin} for a complete presentation of these results.
\end{remark}
\subsection{Average and first order differential operators}
\label{FirstOrdDiffOpe}
\noindent
We are looking for first order derivations commuting with the average operator. Recall that the commutator $[\xi \cdot \nabla _y, \eta \cdot \nabla _y]$ between two first order differential operators is still a first order differential operator, whose vector field, denoted by $[\xi, \eta]$, is given by the Poisson bracket between $\xi$ and $\eta$
\[
[\xi \cdot \nabla _y, \eta \cdot \nabla _y]:= \xi \cdot \nabla _y ( \eta \cdot \nabla _y ) - \eta \cdot \nabla _y ( \xi \cdot \nabla _y ) = [\xi, \eta] \cdot \nabla _y
\]
where $[\xi, \eta] = (\xi \cdot \nabla _y ) \eta - ( \eta \cdot \nabla _y ) \xi$. The two vector fields $\xi$ and $\eta$ are said in involution iff their Poisson bracket vanishes.
Assume that $c(y)$ is a smooth vector field, satisfying $c(Y(s;y)) = \partial _y Y (s;y) c(y), s\in \R, y \in \R^m$, where $Y$ is the flow of $b$ (not necessarily divergence free here). Taking the derivative with respect to $s$ at $s = 0$ yields $(b \cdot \nabla _y ) c = \partial _y b \;c(y)$, saying that $[b,c] = 0$. Actually the converse implication holds true and we obtain the following characterization for vector fields in involution, which is valid in distributions as well (see Appendix \ref{A} for proof details).
\begin{pro}
\label{VFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$ (not necessarily divergence free), with linear growth and $c \in \loloc{}$. Then $(b \cdny) c - \partial _y b \;c = 0$ in $\dpri$ iff
\begin{equation}
\label{Equ34} c (Y(s;y)) = \partial _y Y(s;y) c(y),\;\;s\in \R,\;\;y \in \R^m.
\end{equation}
\end{pro}
We establish also weak formulations characterizing the involution between two fields, in distribution sense (see Appendix \ref{A} for the proof). The notation $w_s$ stands for $w \circ Y(s;\cdot)$.
\begin{pro}
\label{WVFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$, with linear growth and zero divergence and $c \in \loloc{}$. Then the following statements are equivalent\\
1.
\[[b,c] = 0 \;\mbox{in}\; \dpri{}
\]
2.
\begin{equation}
\label{Equ41}
\inty{(c \cdny u )v_{-s} } = \inty{(c\cdny u_s) v },\;\;\forall \;u, v \in C^1_c(\R^m)
\end{equation}
3.
\begin{equation}
\label{Equ42} \inty{c \cdny u \;b \cdny v } + \inty{c \cdny (b \cdny u ) v } = 0,\;\;\forall\; u \in C^2 _c (\R^m),\;\;v \in C^1 _c (\R^m).
\end{equation}
\end{pro}
\begin{remark}
\label{VecDiv}
If $[b,c]=0$ in $\dpri{}$, applying \eqref{Equ41} with $v = 1$ on the support of $u_s$ (and therefore $v_{-s} = 1$ on the support of $u$) yields
\[
\inty{c \cdny u} = \inty{c\cdny u_s},\;\;u \in C^1_c(\R^m)
\]
saying that $\divy c$ is constant along the flow of $b$ (in $\dpri{}$).
\end{remark}
We claim that for vector fields $c$ in involution with $b$, the derivation $c \cdny $ commutes with the average operator.
\begin{pro}
\label{AveComFirstOrder} Consider a vector field $c \in \loloc{}$ with bounded divergence, in involution with $b$, that is $[b,c] = 0$ in $\dpri{}$. Then the operators $u \to c \cdny u$, $u \to \divy(uc)$ commute with the average operator {\it i.e.,} for any $u \in \dom ( c\cdny )= \dom (\divy (\cdot \;c))$ we have $\ave{u} \in \dom ( c\cdny )= \dom (\divy (\cdot \;c))$ and
\[
\ave{c \cdny u} = c\cdny \ave{u},\;\;\ave{\divy(uc) } = \divy ( \ave{u}c).
\]
\end{pro}
\begin{proof}
Consider $u \in \dom (c \cdny ), s \in \R$ and $\varphi \in C^1 _c (\R^m)$. We have
\begin{align}
\label{Equ43} \inty{u_s c \cdny \varphi } & = \inty{u (c \cdny \varphi)_{-s}} \\
& = \inty{u (c \cdny ) \varphi _{-s} } \nonumber \\
& = - \inty{\divy (uc) \varphi _{-s}} \nonumber \\
& = - \inty{(\divy (uc))_s \varphi (y)} \nonumber
\end{align}
saying that $u _s \in \dom ( c \cdny ) = \dom ( \divy ( \cdot \;c ))$ and $ \divy (u_s c ) = ( \divy (uc))_s$. We deduce $c \cdny u_s = (c \cdny u )_s$ cf. Remark \ref{VecDiv}. Integrating \eqref{Equ43} with respect to $s$ between $0$ and $T>0$ one gets
\begin{align*}
\inty{\frac{1}{T} \int _0 ^T u_s \md s \;c \cdny \varphi } & = \frac{1}{T} \int _0 ^T \inty{u_s c \cdny \varphi }\md s \\
& = - \frac{1}{T} \int _0 ^T \inty{(\divy (uc ))_s \varphi (y) }\md s \\
& = - \inty{\frac{1}{T}\int _0 ^T ( \divy (uc))_s \md s \;\varphi (y) }.
\end{align*}
By Proposition \ref{AverageOperator} we know that $\frac{1}{T} \int _0 ^T u_s \md s \to \ave{u}$ and $\frac{1}{T}\int _0 ^T (\divy (uc))_s \md s \to \ave{\divy (uc)}$ strongly in $\lty{}$, when $T \to +\infty$, and thus we obtain
\[
\inty{\ave{u} c \cdny \varphi } = - \inty{\ave{\divy(uc)} \varphi (y) }
\]
saying that $\ave{u} \in \dom ( c \cdny )$ and $\divy (\ave{u}c) = \ave{\divy (uc)}$, $c \cdny \ave{u} = \ave{c \cdny u }$.
\end{proof}
\subsection{Average and second order differential operators}
\label{SecondOrdDiffOpe}
\noindent
We investigate the second order differential operators $- \divy (A(y) \nabla _y)$ commuting with the average operator along the flow of $b$, where $A(y)$ is a smooth field of symmetric matrix. Such second order operators leave invariant $\kerbg{}$. Indeed, for any $u \in \dom (- \divy (A(y) \nabla _y)) \cap \kerbg{}$ we have
\[
- \divy (A(y) \nabla _y u ) = - \divy (A(y) \ave{u}) = \ave{ - \divy (A(y) \nabla _y u )} \in \kerbg{}.
\]
For this reason it is worth considering the operators $- \divy (A(y) \nabla _y )$ commuting with $b \cdny$. A straightforward computation shows that
\begin{pro}
\label{ComSecOrd} Consider a divergence free vector field $b \in W^{2,\infty} (\R^m)$ and a matrix field $A \in W^{2,\infty} (\R^m)$. The commutator between $b \cdny $ and $- \divy (A(y) \nabla _y)$ is still a second order differential operator
\[
[b\cdny, - \divy(A\nabla _y )] = - \divy ([b,A] \nabla _y )
\]
whose matrix field, denoted by $[b,A]$, is given by
\[
[b,A] = (b \cdny )A - \dyb A(y) - A(y)\; {^t \dyb},\;\;y \in \R^m.
\]
\end{pro}
\begin{remark}
We have the formula ${^t [b,A]} = [b, {^t A}]$. In particular if $A(y)$ is a field of symmetric (resp. anti-symmetric) matrix, the field $[b,A]$ has also symmetric (resp. anti-symmetric) matrix.
\end{remark}
As for vector fields in involution, we have the following characterization (see Appendix \ref{A} for proof details).
\begin{pro}
\label{MFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$ (not necessarily divergence free) with linear growth and $A(y) \in \loloc{}$. Then $[b,A] = 0$ in $\dpri{}$ iff
\begin{equation}
\label{Equ35} A(\ysy) = \dyy A(y) \;{^t \dyy},\;\;s\in \R,\;\;y \in \R^m.
\end{equation}
\end{pro}
For fields of symmetric matrix we have the weak characterization (see Appendix \ref{A} for the proof).
\begin{pro}
\label{WMFI} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$ with linear growth, zero divergence and $A \in \loloc{}$ a field of symmetric matrix. Then the following statements are equivalent\\
1.
\[
[b,A] = 0 \;\mbox{ in } \; \dpri{}.
\]
2.
\[
\inty{A(y) \nabla _y u_s \cdot \nabla _y v_s } = \inty{A(y) \nabla _y u \cdot \nabla _y v }
\]
for any $s \in \R$, $u, v \in C^1 _c ( \R^m)$.\\
3.
\[
\inty{A(y) \nabla _y ( b \cdny u ) \cdot \nabla _y v } + \inty{A(y) \nabla _y u \cdot \nabla _y ( b \cdny v ) } = 0
\]
for any $u, v \in C^2 _c (\R^m)$.
\end{pro}
We consider the (formal) adjoint of the linear operator $A \to [b,A]$, with respect to the scalar product $(U,V) = \inty{U(y) : V(y)}$, given by
\[
Q \to - (b \cdny ) Q - {^t \dyb} Q(y) - Q(y) \dyb
\]
when $\divy b = 0$. The following characterization comes easily and the proof is left to the reader.
\begin{pro}
\label{AdyMatFieInv} Consider $b \in W^{1,\infty}_{\mathrm{loc}} (\R^m)$, with linear growth and $Q \in L^1 _{\mathrm{loc}} (\R^m)$. Then $- (b \cdny ) Q - {^t \dyb} Q(y) - Q(y) \dyb = 0$ in $\dpri {}$ iff
\begin{equation}
\label{Equ36} Q(\ysy) = {^t \partial _y Y }^{-1}(s;y) Q(y) \partial _y Y ^{-1}(s;y),\;\;s\in \R,\;\;y \in \R^m.
\end{equation}
\end{pro}
\begin{remark}
\label{InverseQ}
If $Q(y)$ satisfies \eqref{Equ36} and is invertible for any $y \in \R^m$ with $Q^{-1} \in L^1 _{\mathrm{loc}}(\R^m)$, then $Q^{-1} (\ysy) = \dyy Q^{-1} (y) {^t \dyy}$, $s \in \R, y \in \R^m$ and therefore $[b,Q^{-1}] = 0$ in $\dpri{}$. If $P(y)$ satisfies \eqref{Equ35} and is invertible for any $y \in \R^m$, then
\[
P^{-1} (\ysy) = {^t \partial _y Y} ^{-1} (s;y) P ^{-1} (y) \partial _y Y ^{-1} (s;y),\;\;s \in \R,\;\; y \in \R^m
\]
and therefore $- (b \cdny ) P - {^t \dyb} P(y) - P(y) \dyb = 0$ in $\dpri{}$.
\end{remark}
As for vector fields in involution, the matrix fields in involution with $b$ generate second order differential operators commuting with the average operator.
\begin{pro}
\label{AveComSecondOrder} Consider a matrix field $A \in \loloc{}$ such that $\divy A \in \loloc{}$ and $[b,A] = 0$ in $\dpri{}$. Therefore the operator $u \to - \divy (A \nabla _y u )$ commutes with the average operator {\it i.e.,} for any $u \in \dom ( - \divy (A \nabla _y ))$ we have $\ave{u} \in \dom ( - \divy (A \nabla _y ))$ and
\[
-\ave{\divy (A \nabla _y u )} = - \divy (A \nabla _y \ave{u}).
\]
\end{pro}
\begin{proof}
Consider $u \in \dom( - \divy (A\nabla _y )) = \{w \in \lty{}: -\divy (A\nabla _y w ) \in \lty{}\}$. For any $s \in \R, \varphi \in C^2 _c (\R^m)$ we have
\begin{equation}
\label{Equ51} - \inty{u_s \;\divy ( \;{^t A} \nabla _y \varphi ) } = - \inty{u \;( \divy ( \;{^t A} \nabla _\varphi ))_{-s}}.
\end{equation}
By the implication $1.\implies 2.$ of Proposition \ref{WMFI} (which does not require the symmetry of $A(y)$) we know that
\[
\inty{{^t A } \nabla _y \varphi \cdot \nabla _y \psi _s } = \inty{{^t A} \nabla _y \varphi _{-s} \cdot \nabla _y \psi }
\]
for any $\psi \in C^2 _c (\R^m )$. We deduce that
\[
- \inty{\divy ( {^t A} \nabla _y \varphi ) \psi _s } = - \inty{\divy ( {^t A } \nabla _y \varphi _{-s} ) \psi }
\]
and thus $(\divy ( {^t A} \nabla _y \varphi ))_{-s} = \divy ( {^t A} \nabla _y \varphi _{-s})$. Combining with $\eqref{Equ51}$ yields
\begin{eqnarray}
\label{Equ52} - \inty{\;u_s \divy ( {^t A} \nabla _y \varphi )} & = - \inty{\;u\; \divy ( {^t A} \nabla _y \varphi _{-s})} \\
& = - \inty{\;\divy (A \nabla _y u ) \varphi _{-s}} \nonumber \\
& = - \inty{\;(\divy (A \nabla _y u ))_s \varphi (y)} \nonumber
\end{eqnarray}
saying that $u_s \in \dom ( - \divy (A \nabla _y ))$ and
\[
- \divy ( A \nabla _y u_s) = ( - \divy (A \nabla _y u ))_s.
\]
Integrating \eqref{Equ52} with respect to $s$ between $0$ and $T$ we obtain
\[
\inty{\frac{1}{T} \int _0 ^T u_s \;\md s\; \divy ( {^t A } \nabla _y \varphi )} = \inty{\frac{1}{T} \int _0 ^T (\divy (A \nabla _y u))_s \;\md s \;\varphi (y)}.
\]
Letting $T \to +\infty$ yields
\[
\inty{\ave{u} \divy ( {^t A }\nabla _y \varphi ) } = \inty{\ave{\divy (A \nabla _y u )} \varphi (y)}
\]
and therefore $\ave{u} \in \dom ( \divy (A \nabla _y ))$, $\divy (A \nabla _y \ave{u}) = \ave{\divy (A \nabla _y u )}$.
\end{proof}
\subsection{The averaged diffusion matrix field}
\label{AveDifMatFie}
\noindent
We are looking for the limit, when $\eps \to 0$, of \eqref{Equ1}, \eqref{Equ2}. We expect that the limit $u = \lime \ue $ satisfies \eqref{Equ7}, \eqref{Equ8}. By \eqref{Equ7} we deduce that at any time $t \in \R_+$, $u(t,\cdot) \in \kerbg{}$. Observe also that $\divy(b \otimes b \nabla _y u^1) = b \cdny (b \cdny u^1) \in \ran ( b \cdny ) \subset \ker \ave{\cdot}$ and therefore the closure for $u$ comes by applying the average operator to \eqref{Equ8} and by noticing that $\ave{\partial _t u } = \partial _t \ave{u} = \partial _t u $
\begin{equation}
\label{Equ54} \partial _t u - \ave{\divy ( D \nabla _y u )} = 0,\;\;t\in \R_+,\;\;y \in \R^m.
\end{equation}
At least when $[b,D] = 0$, we know by Proposition \ref{AveComSecondOrder} that
\[
\ave{\divy (D \nabla _y u)} = \divy (D \nabla _y \ave{u}) = \divy (D \nabla _y u)
\]
and \eqref{Equ54} reduces to the diffusion equation associated to the matrix field $D(y)$. Nevertheless, even if $[b,D] \neq 0$, \eqref{Equ54} behaves like a diffusion equation. More exactly the $\lty{}$ norm of the solution decreases with a rate proportional to the $\lty{}$ norm of its gradient under the hypothesis \eqref{Equ3}
\begin{align*}
\frac{1}{2}\frac{\md }{\md t} \inty{(u(t,y))^2} & = \inty{\ave{\divy ( D \nabla _y u )} u (t,y) } \\
& = \inty{\divy ( D \nabla _y u ) u }\\
& = - \inty{D \nabla _y u \cdot \nabla _y u }\\
& = - \inty{(D + b\otimes b ) : \nabla _y u \otimes \nabla _y u } \\
& \leq - d \inty{|\nabla _y u (t,y) |^2 }.
\end{align*}
We expect that, under appropriate hypotheses, \eqref{Equ54} coincides with a diffusion equation, corresponding to some {\it averaged} matrix field ${\cal D}$, that is
\begin{equation}
\label{Equ55}
\exists \; {\cal D} (y)\;:\; [b, {\cal D}] = 0\;\mbox{ and } \; \ave{- \divy ( D \nabla _y u )} = - \divy ( {\cal D} \nabla _y u ),\;\;\forall \;u \in \kerbg{}.
\end{equation}
It is easily seen that in this case the limit model \eqref{Equ54} reduces to
\[
\partial _t u - \divy ( {\cal D}\nabla _y u ) = 0,\;\;t \in \R_+,\;\;y \in \R^m.
\]
In this section we identify sufficient conditions which guarantee the existence of the matrix field ${\cal D}$. We will see that it appears as the long time limit of the solution of another parabolic type problem, whose initial data is $D$, and thus as the orthogonal projection of the field $D(y)$ (with respect to some scalar product to be defined) on a subset of $\{A \in \loloc{}:[b,A] = 0\mbox{ in } \dpri{}\}$. We assume that \eqref{Equ56} holds true. We introduce the set
\[
H_Q = \{A = A(y)\;:\; \inty{Q(y)A(y) : A(y) Q(y)} < +\infty\}
\]
where $Q = P^{-1}$ and the bilinear application
\[
(\cdot, \cdot)_Q : H_Q \times H_Q \to \R,\;\;(A,B)_Q = \inty{Q(y)A(y):B(y)Q(y)}
\]
which is symmetric and positive definite. Indeed, for any $A \in H_Q$ we have
\[
(A,A)_Q = \inty{Q^{1/2}AQ^{1/2} : Q^{1/2}AQ^{1/2}} \geq 0
\]
with equality iff $Q^{1/2}AQ^{1/2}= 0$ and thus iff $A = 0$. The set $H_Q$ endowed with the scalar product $(\cdot, \cdot)_Q$ becomes a Hilbert space, whose norm is denoted by $|A|_Q = (A, A)_Q ^{1/2}, A \in H_Q$. Observe that $H_Q \subset \{A(y):A \in \loloc{}\}$. Indeed, if for any matrix $M$ the notation $|M|$ stands for the norm subordonated to the euclidian norm of $\R^m$
\[
|M| = \sup _{\xi \in \R^m \setminus \{0\}} \frac{|M\xi|}{|\xi|} \leq ( M : M ) ^{1/2}
\]
we have for a.a. $y \in \R^m$
\begin{eqnarray}
\label{Equ57} |A(y)| & = & \sup _{\xi, \eta \neq 0} \displaystyle \frac{A(y) \xi \cdot \eta}{|\xi|\;|\eta|} \\
& = & \sup _{\xi, \eta \neq 0} \displaystyle \frac{Q^{1/2}AQ^{1/2} P ^{1/2}\xi \cdot P^{1/2} \eta}{|P^{1/2} \xi|\;|P^{1/2} \eta|}\;\frac{|P^{1/2}\xi|}{|\xi|}\;\frac{|P^{1/2}\eta|}{|\eta|} \nonumber \\
& \leq & |Q^{1/2}AQ^{1/2} |\;|P^{1/2}|^2 \nonumber \\
& \leq & ( Q^{1/2}AQ^{1/2}:Q^{1/2}AQ^{1/2}) ^{1/2} \;|P|.\nonumber
\end{eqnarray}
We deduce that for any $R>0$
\[
\int_{B_R} |A(y)|\;\md y \leq \int _{B_R} ( Q^{1/2}AQ^{1/2}:Q^{1/2}AQ^{1/2}) ^{1/2} \;|P|\;\md y \leq (A,A)_Q ^{1/2} \left ( \int _{B_R} |P(y)|^2 \;\md y \right ) ^{1/2}.
\]
\begin{remark}
\label{Ortho}
We know by Remark \ref{InverseQ} that $Q_s = {^t \partial _y Y ^{-1} }(s;y) Q(y) \partial _y Y ^{-1}(s;y) $ which writes ${^t {\cal O}}(s;y) {\cal O}(s;y) = I$ where ${\cal O}(s;y) = Q_s ^{1/2} \dyy Q^{-1/2}$. Therefore the matrix ${\cal O}(s;y)$ are orthogonal and we have
\begin{equation}
\label{Equ58}
Q_s ^{1/2} \dyy Q^{-1/2} = {\cal O}(s;y) = {^t {\cal O}}^{-1} (s;y) = Q_s ^{-1/2} \;{^t \partial _y Y }^{-1} Q^{1/2}
\end{equation}
\begin{equation}
\label{Equ59}
Q ^{-1/2} \;{^t \dyy} Q_s ^{1/2} = {^t {\cal O}}(s;y) = { {\cal O}}^{-1} (s;y) = Q ^{1/2} { \partial _y Y }^{-1} Q_s^{-1/2}.
\end{equation}
\end{remark}
\begin{pro}
\label{Groupe} The family of applications $A \to G(s)A : = \partial _y Y ^{-1} (s; \cdot) A_s \; {^t \partial _y Y } ^{-1} (s; \cdot)$ is a $C^0$- group of unitary operators on $H_Q$.
\end{pro}
\begin{proof}
For any $A\in H_Q$ observe, thanks to \eqref{Equ59}, that
\begin{align*}
\left | \partial _y Y ^{-1}(s; \cdot) A_s {^t\partial _y Y ^{-1}(s; \cdot) }\right | ^2 _Q &
= \!\!\inty{Q^{1/2}\partial _y Y ^{-1} A_s {^t \partial _y Y ^{-1}}Q^{1/2}:Q^{1/2}\partial _y Y ^{-1} A_s {^t \partial _y Y ^{-1}}Q^{1/2}}\\
& = \!\!\inty{\!\!\!\!{^t {\cal O}} (s;y) Q_s ^{1/2} A_s Q_s ^{1/2} {\cal O}(s;y) \!:\! {^t {\cal O}} (s;y) Q_s ^{1/2} A_s Q_s ^{1/2} {\cal O}(s;y)}\\
& = \inty{Q_s ^{1/2} A_s Q_s ^{1/2} : Q_s ^ {1/2} A_s Q_s ^{1/2}}\\
& = \inty{Q^{1/2}AQ^{1/2} : Q^{1/2}AQ^{1/2}} \\
& = |A|^2 _Q.
\end{align*}
Clearly $G(0)A = A, A\in H_Q$ and for any $s, t \in \R$ we have
\begin{align*}
G(s) G(t) A & = \partial _y Y ^{-1} (s;\cdot) (G(t)A)_s {^t \partial _y Y ^{-1} (s;\cdot)}\\
& = \partial _y Y ^{-1} (s;\cdot) (\partial _y Y )^{-1} (t; Y(s;\cdot))(A_t)_s {^t (\partial _y Y )^{-1} (t; Y(s;\cdot))}{^t \partial _y Y ^{-1} (s;\cdot)} \\
& = \partial _y Y ^{-1} (t + s;\cdot)A_{t+s} {^t \partial _y Y ^{-1} (t + s;\cdot)} = G(t+s) A,\;\;A \in H_Q.
\end{align*}
It remains to check the continuity of the group, {i.e.,} $\lim _{s \to 0 } G(s)A = A$ strongly in $H_Q$ for any $A \in H_Q$. For any $s \in \R$ we have
\begin{align*}
|G(s) A - A|^2 _Q = |G(s)A|^2 _Q + |A|^2 _Q - 2 ( G(s)A, A)_Q = 2|A|^2 _Q - 2 (G(s)A, A)_Q
\end{align*}
and thus it is enough to prove that $\lim _{s \to 0 } G(s)A = A$ weakly in $H_Q$. As $|G(s)| = 1$ for any $s \in \R$, we are done if we prove that $\lim _{s \to 0} (G(s)A, U)_Q = (A, U)_Q$ for any $U \in C^0 _c (\R^m) \subset H_Q$. But it is easily seen that $\lim _{s\to 0} G(-s)U = U$ strongly in $H_Q$, for $U \in C^0 _c (\R^m) $ and thus
\[
\lim _{s \to 0} ( G(s)A, U)_Q = \lim _{s \to 0} (A, G(-s)U)_Q = (A,U)_Q,\;\;U \in C^0 _c (\R^m).
\]
\end{proof}
We denote by $L$ the infinitesimal generator of the group $G$
\[
L:\dom(L) \subset H_Q \to H_Q,\;\;\dom L = \{ A\in H_Q\;:\; \exists \;\lim _{s \to 0} \frac{G(s)A-A}{s}\;\mbox{ in } \;H_Q\}
\]
and $L(A) = \lim _{s \to 0} \frac{G(s)A-A}{s}$ for any $A \in \dom(L)$.
Notice that $C^1 _c (\R^m) \subset \dom(L)$ and $L(A) = b \cdny A - \dyb A - A \;{^t \dyb}$, $A \in C^1 _c (\R^m)$ (use the hypothesis $Q \in \ltloc{}$ and the dominated convergence theorem). Observe also that the group $G$ commutes with transposition {\it i.e.} $G(s) \;{^t A} = {^t G(s)}A$, $s \in \R, A \in H_Q$ and for any $A \in \dom (L)$ we have
$^t A \in \dom (L)$, $L({^t A}) = {^t L(A)}$.
The main properties of the operator $L$ are summarized below (when $b$ is divergence free).
\begin{pro}
\label{PropOpeL} $\;$\\
1. The domain of $L$ is dense in $H_Q$ and $L$ is closed.\\
2. The matrix field $A \in H_Q$ belongs to $\dom (L)$ iff there is a constant $C >0$ such that
\begin{equation}
\label{Equ61} |G(s)A - A |_Q \leq C |s|,\;\;s \in \R.
\end{equation}
3. The operator $L$ is skew-adjoint.\\
4. For any $A \in \dom (L)$ we have
\[
- \divy (L(A) \nabla _y ) = b\cdny ( - \divy (A \nabla _y)) + \divy (A \nabla _y ( b \cdny ))\;\mbox{ in } \; \dpri{}
\]
that is
\[
\inty{L(A) \nabla _y u \cdot \nabla _y v } = - \inty{A \nabla _y u \cdot \nabla _y ( b \cdny v)} - \inty{A\nabla _y ( b \cdny u ) \cdot \nabla _y v }
\]
for any $u, v \in C^2 _c (\R^m)$.
\end{pro}
\begin{proof}
1. The operator $L$ is the infinitesimal generator of a $C^0$-group, and therefore $\dom(L)$ is dense and $L$ is closed. \\
2.
Assume that $A \in \dom(L)$. We know that $\frac{\md }{\md s} G(s)A = L(G(s)A) = G(s)L(A)$ and thus
\[
|G(s)A - A|_Q = \left | \int _0 ^t G(\tau) L(A)\;\md \tau\right |_Q \leq \left | \int _0 ^s |G(\tau)L(A)|_Q \;\md \tau \right | = |s| \;|L(A)|_Q,\;\;s \in \R.
\]
Conversely, assume that \eqref{Equ61} holds true. Therefore we can extract a sequence $(s_k)_k$ converging to $0$ such that
\[
\limk \frac{G(s_k) A - A}{s_k} = V \;\mbox{ weakly in } \;H_Q.
\]
For any $U \in \dom (L)$ we obtain
\[
\left ( \frac{G(s_k) A - A}{s_k}, U \right ) _Q = \left ( A, \frac{G(-s_k)U - U}{s_k} \right ) _Q
\]
and thus, letting $k \to +\infty$ yields
\begin{equation}
\label{Equ62} (V, U)_Q = - (A, L(U))_Q.
\end{equation}
But since $U \in \dom (L)$, all the trajectory $\{G(\tau)U:\tau \in \R\}$ is contained in $\dom(L)$ and $G(-s_k)U = U + \int _0 ^{-s_k}L(G(\tau)U)\md \tau$. We deduce
\begin{align*}
(G(s_k)A - A, U)_Q & = \left ( A, \int _0 ^{-s_k} L(G(\tau)U ) \;\md \tau \right ) \\
& = \int _0 ^{-s_k}( A, L(G(\tau) U))_Q \;\md \tau \\
& = - \int _0 ^{-s_k}( V, G(\tau) U )_Q \;\md \tau \\
& = - \left ( V, \int _0 ^{-s_k} G(\tau)U\;\md \tau \right ) _Q.
\end{align*}
Taking into account that $\left | \int _0 ^{-s_k} G(\tau ) U \md \tau \right |_Q \leq |s_k| \;|U|_Q$ we obtain
\[
\left | \left ( \frac{G(s_k)A - A}{s_k}, U \right ) _Q \right | \leq |V|_Q|U|_Q,\;\;U \in \dom (L)
\]
and thus, by the density of $\dom (L)$ in $H_Q$ one gets
\[
\left | \frac{G(s_k)A - A}{s_k} \right |_Q \leq |V|_Q,\;\;k \in \N.
\]
Since $V$ is the weak limit in $H_Q$ of $\left ( \frac{G(s_k)A - A}{s_k} \right )_k$, we deduce that $\limk \frac{G(s_k)A - A}{s_k} = V$ strongly in $H_Q$. As the limit $V$ is uniquely determined by \eqref{Equ62}, all the family $\left ( \frac{G(s)A - A}{s} \right )_s$ converges strongly , when $s \to 0$, towards $V$ in $H_Q$ and thus $A \in \dom (L)$.\\
3. For any $U, V \in \dom (L)$ we can write
\[
(G(s)U - U, V)_Q + (U, V - G(-s)V)_Q = 0,\;\;s\in \R.
\]
Taking into account that
\[
\lim _{s \to 0} \frac{G(s)U - U}{s} = L(U),\;\;\lim _{s \to 0} \frac{V - G(-s)V}{s} = L(V)
\]
we obtain $(L(U), V)_Q + (U, L(V))_Q = 0$ saying that $V\in \dom (L^\star)$ and $L^\star (V) = - L(V)$. Therefore $L \subset (-L^\star)$. It remains to establish the converse inclusion. Let $V \in \dom (L^\star)$, {\it i.e.,} $\exists C >0$ such that
\[
|(L(U), V)_Q|\leq C|U|_Q,\;\;U \in \dom (L).
\]
For any $s \in \R$, $U \in \dom (L)$ we have
\[
(G(s)V - V , U)_Q = (V, G(-s)U - U)_Q = (V, \int _0 ^{-s}LG(\tau)U \;\md \tau )_Q = \int _0 ^{-s} (V, LG(\tau)U)_Q \;\md \tau
\]
implying
\[
|(G(s) V - V , U )_Q|\leq C |s| \;|U|_Q,\;\;s\in \R.
\]
Therefore $|G(s)V - V|_Q \leq C |s|, s \in \R$ and by the previous statement $V \in \dom (L)$. Finally $\dom (L) = \dom (L^\star)$ and $L^\star (V) = - L(V), V \in \dom (L) = \dom (L^\star)$.\\
4. As $L$ is skew-adjoint, we obtain
\[
- \inty{L(A)\nabla _y u \cdot \nabla _y v } = - ( L(A), Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}\;)_Q = ( A, L ( Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1})\;)_Q.
\]
Recall that $P = Q^{-1}$ satisfies $L(P) = 0$, that is, $G(s)P = P, s \in \R$ and thus
\begin{align*}
L(Q^{-1} \nabla _y v \otimes & \nabla _y u Q^{-1}) = \lim _{ s \to 0} \frac{G(s)P\nabla _y v \otimes \nabla _y u P - P \nabla _y v \otimes \nabla _y u P}{s} \\
& = \lim _{ s \to 0} \frac{\partial _y Y ^{-1} (s;\cdot) P_s (\nabla _y v )_s \otimes (\nabla _y u )_s P_s {^t \partial _y Y ^{-1}}(s;\cdot) - P \nabla _y v \otimes \nabla _y u P}{s}\\
& = \lim _{ s \to 0} \frac{P{^t \partial _y Y} (s;\cdot) (\nabla _y v )_s \otimes (\nabla _y u )_s \partial _y Y (s;\cdot)P - P \nabla _y v \otimes \nabla _y u P}{s} \\
& = \lim _{ s \to 0} \frac{P \nabla _y v_s \otimes \nabla _y u_s P -P \nabla _y v \otimes \nabla _y u P }{s} \\
& = P \nabla _y ( b \cdny v ) \otimes \nabla _y u P + P \nabla _y v \otimes \nabla _y ( b \cdny u ) P.
\end{align*}
Finally one gets
\begin{align*}
- \inty{L(A) \nabla _y u \cdot \nabla _y v } & = ( A, P \nabla _y ( b \cdny v ) \otimes \nabla _y u P) + P \nabla _y v \otimes \nabla _y ( b \cdny u )P)_Q \\
& = \inty{A\nabla _y u \cdot \nabla _y ( b \cdny v)} + \inty{A\nabla _y ( b \cdny u ) \cdot \nabla _y v }.
\end{align*}
\end{proof}
We claim that $\dom (L)$ is left invariant by some special (weighted with respect to the matrix field $Q$) positive/negative part functions. The notations $A^\pm$ stand for the usual positive/negative parts of a symmetric matrix $A$
\[
A^\pm = S \Lambda ^\pm \;{^t S},\;\;A = S\Lambda \;{^t S}
\]
where $\Lambda, \Lambda ^\pm $ are the diagonal matrix containing the eigenvalues of $A$ and the positive/negative parts of these eigenvalues respectively, and $S$ is the orthogonal matrix whose columns contain a orthonormal basis of eigenvectors for $A$. Notice that
\[
A^+ : A^- = 0,\;\;A^+ - A^- = A,\;\;A^+ : A^+ + A^- : A^- = A: A.
\]
We introduce also the positive/negative part functions which associate to any field of symmetric matrix $A(y)$ the fields of symmetric matrix $A^{Q\pm}(y)$ given by
\[
Q^{1/2} A^{Q\pm} \;Q^{1/2} = (Q^{1/2} AQ^{1/2})^\pm.
\]
Observe that $A^{Q+} - A^{Q-} = A$.
\begin{pro}
\label{InvPosNeg}$\;$\\
1. The applications $A \to A^{Q\pm}$ leave invariant the subset $\{A\in \dom (L): {^t A} = A\}$.\\
2. For any $A \in \dom (L), {^t A } = A$ we have
\[
(A^{Q+}, A^{Q-})_Q = 0,\;\;( L(A^{Q+}), L(A^{Q-}))_Q \leq 0.
\]
\end{pro}
\begin{proof}
1. Consider $A \in \dom (L), {^t A} = A$. It is easily seean that ${^t A^{Q\pm}} = A^{Q\pm}$ and
\begin{align*}
|A^{Q+}|^2 _Q + |A^{Q-}|^2 _Q & = \inty{(Q^{1/2}A Q^{1/2} ) ^ + : ( Q^{1/2} A Q^{1/2})^+} \\
& + \inty{(Q^{1/2}A Q^{1/2} ) ^ - : ( Q^{1/2} A Q^{1/2})^-}\\
& = \inty{Q^{1/2}A Q^{1/2} : Q^{1/2} A Q^{1/2}} = |A|^2 _Q < +\infty
\end{align*}
and therefore $A ^{Q\pm} \in H_Q$. The positive/negative parts $A^{Q\pm}$ are orthogonal in $H_Q$
\[
( A^{Q+} , A^{Q-})_Q = \inty{(Q^{1/2}AQ^{1/2})^+ : (Q^{1/2}AQ^{1/2})^-} = 0.
\]
We claim that $A^{Q\pm}$ satisfies \eqref{Equ61}. Indeed, thanks to \eqref{Equ59} we can write, using the notation $X^{:2} = X : X$
\begin{align}
\label{Equ63} |G(s)A^{Q\pm}- A^{Q\pm}|^2 _Q & =
\inty{\{ Q^{1/2} ( \partial _y Y ^{-1} (A ^{Q\pm})_s {^t \partial _y Y ^{-1}} - A^{Q\pm})Q^{1/2} \} ^{:2}}\\
& = \inty{\{{^t {\cal O}} (s;y) Q_s ^{1/2} (A ^{Q\pm})_sQ_s ^{1/2} {\cal O}(s;y) - Q^{1/2} A^{Q\pm}Q^{1/2} \}^{:2}} \nonumber \\
& = \inty{\{ {^t {\cal O}} (s;y) ( Q_s ^{1/2} A_s Q_s ^{1/2})^{\pm} {\cal O}(s;y) - (Q^{1/2}A Q^{1/2} ) ^{\pm} \} ^{:2}}. \nonumber
\end{align}
Similarly we obtain
\begin{equation}
\label{Equ64} |G(s)A - A|^2 _Q = \inty{\{{^t {\cal O}}(s;y) Q^{1/2}_s A_s Q^{1/2}_s {\cal O}(s;y) - Q^{1/2}AQ^{1/2} \} ^{:2}}.
\end{equation}
We are done if we prove that for any symmetric matrix $U, V$ and any orthogonal matrix $R$ we have the inequality
\begin{equation}
\label{Equ65}
( \;{^t R } U ^{\pm} R - V ^\pm \;) : ( \;{^t R } U ^{\pm} R - V ^\pm \;)\leq ( \;{^t R } U R - V \;):( \;{^t R } U R - V \;).
\end{equation}
For the sake of the presentation, we consider the case of positive parts $U^+, V^+$. The other one comes in a similar way. The above inequality reduces to
\[
2 \;{^t R } U R : V - 2 \;{^t R } U ^+ R : V ^+ \leq {^t R } U ^- R : {^t R } U ^- R + V^- : V^-
\]
or equivalently, replacing $U$ by $ U^+ - U^-$ and $V$ by $V^+ - V^-$, to
\[
- 2 \;{^t R } U ^+ R : V^- - 2 \;{^t R } U ^- R : V^+ + 2 \;{^t R } U ^- R : V ^- \leq {^t R } U ^- R : {^t R } U ^- R + V ^- : V^-.
\]
It is easily seen that the previous inequality holds true, since ${^t R } U ^+ R : V^- \geq 0$, ${^t R } U ^- R : V^+ \geq 0$ and
\[
2 \;{^t R } U ^- R : V^- \leq 2 ( {^t R} U ^- R : {^t R} U ^- R) ^{1/2} ( V^- : V^- ) ^{1/2} \leq {^t R} U ^- R : {^t R} U ^- R + V^- : V^-.
\]
Combining \eqref{Equ63}, \eqref{Equ64} and \eqref{Equ65} with
\[
U = Q^{1/2} _s A_s Q^{1/2}_s, \;\;V = Q^{1/2}AQ^{1/2},\;\;R = {\cal O}
\]
yields
\[
\sup _{s \neq 0} \frac{|G(s)A^{Q\pm} - A^{Q\pm} |_Q}{|s|} \leq \sup _{s \neq 0} \frac{|G(s)A - A|_Q}{|s|} \leq |L(A)|_Q
\]
saying that $A^{Q\pm} \in \dom (L)$. \\
2. For any $A \in \dom (L)$, $^t A = A$ we can write
\begin{align*}
(A^{Q+}, A^{Q-})_Q & = \inty{Q^{1/2}A^{Q+}Q^{1/2}: Q^{1/2}A^{Q-}Q^{1/2}} \\
& = \inty{(Q^{1/2}AQ^{1/2})^+ : (Q^{1/2}AQ^{1/2})^-} = 0.
\end{align*}
Since $A^{Q\pm} \in \dom (L)$ we have
\[
L(A^{Q\pm}) = \lim _{s \to 0} \frac{G(s/2) A^{Q\pm} - G(-s/2) A^{Q\pm}}{s}
\]
and therefore, thanks to \eqref{Equ59}, we obtain
\begin{align*}
& ( L(A^{Q+}), L(A^{Q-}))_Q = \lim _{s \to 0} \left (\frac{G(\frac{s}{2}) A^{Q+} - G(-\frac{s}{2}) A^{Q+}}{s}, \frac{G(\frac{s}{2}) A^{Q-} - G(-\frac{s}{2}) A^{Q-}}{s} \right ) _Q\\
& = \lim _{s \to 0} \inty{\frac{Q^{1/2} (\;G(\frac{s}{2}) A^{Q+} - G(-\frac{s}{2}) A^{Q+} \;) Q^{1/2} }{s} : \frac{Q^{1/2} (\;G(\frac{s}{2}) A^{Q-} - G(-\frac{s}{2}) A^{Q-} \;) Q^{1/2}}{s} }\\
& = \lim _{s \to 0} \inty{ \frac{{^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^+ {\cal O}(\frac{s}{2};y) - {^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^+ {\cal O}(-\frac{s}{2};y)}{s} \\
& : \frac{{^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^- {\cal O}(\frac{s}{2};y) - {^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^- {\cal O}(-\frac{s}{2};y)}{s}}\\
& = - \lim _{s \to 0} \inty{\frac{{^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^+ {\cal O}(\frac{s}{2};y) : {^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^- {\cal O}(-\frac{s}{2};y)}{s^2}} \\
& - \lim _{ s \to 0}\inty{\frac{{^t {\cal O}(-\frac{s}{2};y)}( Q^{1/2}_{-\frac{s}{2}} A_{-\frac{s}{2}} Q^{1/2}_{-\frac{s}{2}})^+ {\cal O}(-\frac{s}{2};y) : {^t {\cal O}(\frac{s}{2};y)}( Q^{1/2}_{\frac{s}{2}} A_{\frac{s}{2}} Q^{1/2}_{\frac{s}{2}})^- {\cal O}(\frac{s}{2};y)}{s^2}} \\
& \leq 0
\end{align*}
since
\[
{^t {\cal O}}(\pm s/2;\cdot) ( Q^{1/2} A Q^{1/2}) _{\pm s/2} ^\pm {\cal O}(\pm s/2;\cdot) \geq 0,\;\;{^t {\cal O}}(\mp s/2;\cdot) ( Q^{1/2} A Q^{1/2}) _{\mp s/2} ^\pm {\cal O}(\mp s/2;\cdot) \geq 0.
\]
\end{proof}
We intend to solve the problem \eqref{Equ67}, \eqref{Equ68} by using variational methods. We introduce the space $V_Q = \dom (L) \subset H_Q$ endowed with the scalar product
\[
((A, B))_Q = (A, B)_Q + (L(A), L(B))_Q,\;\;A, B \in V_Q.
\]
Clearly $(V_Q, ((\cdot, \cdot))_Q)$ is a Hilbert space (use the fact that $L$ is closed) and the inclusion $V_Q \subset H_Q$ is continuous, with dense image. The notation $\|\cdot \|_Q$ stands for the norm associated to the scalar product $((\cdot, \cdot))_Q$
\[
\|A\|^2 _Q = ((A, A))_Q = (A, A)_Q + (L(A), L(A))_Q = |A|^2 _Q + |L(A)|^2_Q,\;\;A\in V_Q.
\]
We introduce the bilinear form $\sigma : V_Q \times V_Q \to \R$
\[
\sigma (A, B) = (L(A), L(B))_Q,\;\;A, B \in V_Q.
\]
Notice that $\sigma$ is coercive on $V_Q$ with respect to $H_Q$
\[
\sigma (A, A) + |A|^2_Q = \|A\|^2_Q,\;\;A \in V_Q.
\]
By Theorems 1,2 pp. 620 \cite{DauLions88} we deduce that for any $D \in H_Q$ there is a unique variational solution for \eqref{Equ67}, \eqref{Equ68} that is $A \in C_b (\R_+; H_Q) \cap L^2 (\R_+; V_Q)$, $\partial _t A \in L^2 (\R_+; V_Q ^\prime)$
\[
A(0) = D,\;\;\frac{\md }{\md t } (A(t), U)_Q + \sigma (A(t), U) = 0,\;\;\mbox{in}\;\;\dpri{},\;\;\forall \;U \in V_Q.
\]
The long time limit of the solution of \eqref{Equ67}, \eqref{Equ68} provides the averaged matrix field in \eqref{Equ55}.
\begin{proof}
(of Theorem \ref{AveMatDif}) The identity
\[
\frac{1}{2}\frac{\md }{\md t} |A(t) |^2 _Q + |L(A(t))|^2 _Q = 0,\;\;t \in \R_+
\]
gives the estimates
\[
|A(t)|_Q \leq |D|_Q,\;\;t \in \R_+,\;\;\int _0 ^{+\infty} |L(A(t))|^2 _Q \;\md t\leq \frac{1}{2}|D|^2 _Q.
\]
Consider $(t_k)_k$ such that $t_k \to +\infty$ as $k \to +\infty$ and $(A(t_k))_k$ converges weakly towards some matrix field $X$ in $H_Q$. For any $U \in \ker L$ we have
\[
\frac{\md }{\md t} (A(t), U)_Q = 0,\;\;t \in \R_+
\]
and therefore
\begin{equation}
\label{Equ70} (\mathrm{Proj}_{\ker L} D, U)_Q = (D,U)_Q = (A(0), U)_Q = (A(t_k), U)_Q = ( X, U)_Q,\;\;U \in \ker L.
\end{equation}
Since $L(A) \in L^2 (R_+;H_Q)$ we deduce that $\limk L(A(t_k)) = 0$ strongly in $H_Q$. For any $V \in V_Q$ we have
\[
(X, L(V))_Q = \limk (A(t_k), L(V))_Q = - \limk (L(A(t_k)), V)_Q = 0.
\]
We deduce that $X \in \dom (L^\star) = \dom (L)$ and $L(X) = 0$, which combined with \eqref{Equ70} says that $X = \mathrm{Proj}_{\ker L} D$, or $X = \ave{D}_Q$. By the uniqueness of the limit we obtain $\lim _{t \to +\infty} A(t) = \mathrm{Proj}_{\ker L} D$ weakly in $H_Q$. Assume now that ${^t D } = D$. As $L$ commutes with transposition, we have $\partial _t {^t A} - L (L({^t A})) = 0$, ${^t A }(0) = D$. By the uniqueness we obtain ${^t A } = A$ and thus
\[
^t \ave{D}_Q = \;^t ( \mbox{w}-\lim _{t \to +\infty} A(t) ) = \mbox{w}-\lim _{t \to +\infty} {^t A(t)} = \mbox{w}-\lim _{t \to +\infty} A(t)= \ave{D}_Q.
\]
Suppose that $D\geq 0$ and let us check that $\ave{D}_Q \geq 0$. By Proposition \ref{InvPosNeg} we know that $A^{Q\pm}(t) \in V_Q$, $t \in \R_+$ and
\[
(A^{Q+}(t), A^{Q-}(t))_Q = 0,\;\;(L(A^{Q+}(t)), L(A^{Q-}(t)))_Q \leq 0,\;\;t \in \R_+.
\]
It is sufficient to consider the case of smooth solutions. Multiplying \eqref{Equ67} by $-A^{Q-}(t)$ one gets
\begin{align}
\label{Equ71} \frac{1}{2}\frac{\md }{\md t} |A^{Q-}(t) |^2 _Q + |L(A^{Q-}(t)|^2 _Q & = ( \partial _t A^{Q+}, A^{Q-}(t))_Q + (L(A^{Q+}(t)),L(A^{Q-}(t)) )_Q \\
& \leq ( \partial _t A^{Q+}, A^{Q-}(t))_Q. \nonumber
\end{align}
But for any $0 < h < t $ we have
\begin{align*}
(A^{Q+}(t) - A^{Q+}(t-h), A^{Q-}(t))_Q = - (A^{Q+}(t-h), A^{Q-}(t))_Q \leq 0
\end{align*}
and therefore $(\partial _t A^{Q+}(t), A^{Q-}(t))_Q \leq 0 $. Observe that $Q^{1/2}A^{Q-}(0)Q^{1/2} = (Q^{1/2} D Q^{1/2})^- = 0$, since $Q^{1/2} D Q^{1/2}$ is symmetric and positive. Thus $A^{Q-}(0) = 0$, and from \eqref{Equ71} we obtain
\[
\frac{1}{2} |A^{Q-}(t) |^2 _Q \leq \frac{1}{2}|A^{Q-}(0)|^2 _Q = 0
\]
implying that $Q^{1/2} A(t) Q^{1/2} \geq 0$ and $A(t) \geq 0$, $t \in \R_+$.
Take now any $U \in H_Q$, ${^t U } = U$, $U \geq 0$. By weak convergence we have
\[
( \ave{D}_Q, U)_Q = \lim _{t \to +\infty} (A(t), U)_Q = \lim _{t \to +\infty} \inty{Q^{1/2} A(t)Q^{1/2} :Q^{1/2} UQ^{1/2} }\geq 0
\]
and thus $\ave{D}_Q \geq 0$. By construction $\ave{D}_Q = \mathrm{Proj}_{\ker L} D \in \ker L$. It remains to justify the second statement in \eqref{Equ72}, and \eqref{Equ72Bis}. Take a bounded function $\varphi \in \liy{}$ which remains constant along the flow of $b$, that is $\varphi _s = \varphi, s \in \R$, and a smooth function $u \in C^1 (\R^m)$ such that $u_s = u, s \in \R$ and
\[
\inty{(\nabla _y u \cdot Q^{-1} \nabla _y u )^2 } < +\infty.
\]
We introduce the matrix field $U$ given by
\[
U(y) = \varphi (y) Q^{-1} (y) \;\nabla _y u \otimes \nabla _y u \; Q^{-1}(y),\;\;y \in \R^m.
\]
By one hand notice that $U \in H_Q$
\begin{align*}
|U|^2_Q & = \inty{Q^{1/2}UQ^{1/2}:Q^{1/2}UQ^{1/2}} = \inty{\varphi ^2|Q^{-1/2} \nabla _y u |^4}\\
& \leq \|\varphi \|_{L^\infty} ^2\inty{(\nabla _y u \cdot Q^{-1} \nabla _y u )^2 }.
\end{align*}
By the other hand, we claim that $U \in \ker L$. Indeed, for any $s \in \R$ we have
\[
\nabla _y u = \nabla _y u_s = {^t \dyy}(\nabla _y u )_s
\]
and thus
\begin{align*}
Q_s U_s Q_s & = \varphi _s (\nabla _y u )_s \otimes ( \nabla _y u )_s \\
& = \varphi \;( {^t \partial _y Y ^{-1}}\nabla _y u ) \otimes ( {^t \partial _y Y ^{-1}}\nabla _y u ) \\
& = \varphi \;{^t \partial _y Y ^{-1}}\;\nabla _y u \otimes \nabla _y u \; \partial _y Y ^{-1}\\
& = {^t \partial _y Y ^{-1}} QUQ \partial _y Y ^{-1}.
\end{align*}
Taking into account that $Q_s = {^t \partial _y Y ^{-1}}Q { \partial _y Y ^{-1}}$ we obtain
\[
{^t \partial _y Y ^{-1}} Q { \partial _y Y ^{-1}}U_s {^t \partial _y Y ^{-1}} Q { \partial _y Y ^{-1}} = {^t \partial _y Y ^{-1}} Q U Q { \partial _y Y ^{-1}}
\]
saying that $U_s (y)= \dyy U(y) {^t \dyy}$. As $\ave{D}_Q = \mathrm{Proj}_{\ker L }D$ one gets
\begin{align*}
0 = (D - \ave{D}_Q, U ) _Q & = \inty{(D - \ave{D}_Q) : QUQ} \\
& = \inty{\varphi (y) (D - \ave{D}_Q) :\nabla _y u \otimes \nabla _y u }\\
& = \inty{\varphi (y) \{ \nabla _y u \cdot D \nabla _y u - \nabla _y u \cdot \ave{D}_Q \nabla _y u \}}.
\end{align*}
In particular, taking $\varphi = 1$ we deduce that $\nabla _y u \cdot \ave{D}_Q \nabla _y u \in \loy{}$ and
\[
\inty{\nabla _y u \cdot \ave{D}_Q \nabla _y u } = \inty{\nabla _y u \cdot D \nabla _y u } = (D, Q^{-1}\;\nabla _y u \otimes \nabla _y u \;Q^{-1} )_Q < +\infty
\]
since $D \in H_Q$, $Q^{-1}\nabla _y u \otimes \nabla _y u Q^{-1} \in H_Q$. Since $\ave{D}_Q \in \ker L$, the function $\nabla _y u \cdot \ave{D}_Q \nabla _y u $ remains constant along the flow of $b$
\[
(\nabla _y u )_s \cdot (\ave{D}_Q)_s (\nabla _y u )_s = (\nabla _y u )_s \cdot \dyy \ave{D}_Q \;{^t \dyy} (\nabla _y u )_s = \nabla _y u \cdot \ave{D}_Q\nabla _y u.
\]
Therefore the function $\nabla _y u \cdot \ave{D}_Q \nabla _y u $ verifies the variational formulation
\begin{equation}
\label{Equ73} \nabla _y u \cdot \ave{D}_Q \nabla _y u \in \loy{},\;\;(\nabla _y u \cdot \ave{D}_Q \nabla _y u)_s = \nabla _y u \cdot \ave{D}_Q \nabla _y u,\;\;s \in \R
\end{equation}
and
\begin{equation}
\label{Equ74}
\inty{\nabla _y u \cdot D \nabla _y u \;\varphi } = \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y u\;\varphi },\;\;\forall \;\varphi \in \liy{},\;\varphi _s = \varphi,\; s \in \R.
\end{equation}
It is easily seen, thanks to the hypothesis $D \in \liy{}$, that \eqref{Equ73}, \eqref{Equ74} also make sense for functions $u \in \hoy{}$ such that $u _s = u$, $s \in \R$. We obtain
\[
\nabla _y u \cdot \ave{D}_Q \nabla _y u = \ave{\nabla _y u \cdot D \nabla _y u },\;\;u \in \hoy{},\;\;u_s = u,\;\;s\in \R
\]
where the average operator in the right hand side should be understood in the $\loy{}$ setting cf. Remark \ref{AveLone}. Moreover, if $u, v \in \hoy{} \cap \kerbg{}$ then $\ave{D}_Q ^{1/2} \nabla _y u, \ave{D}_Q ^{1/2} \nabla _y v$ belong to $\lty{}$ implying that $\nabla _y u \cdot \ave{D}_Q \nabla _y v \in \loy{}$. As before we check that $\nabla _y u \cdot \ave{D}_Q \nabla _y v$ remains constant along the flow of $b$ and for any $\varphi \in \liy{}$, $\varphi _s = \varphi, s \in \R$ we can write
\begin{align*}
2 \inty{\nabla _y u \cdot D \nabla _y v \;\varphi } & = \inty{\nabla _y (u + v) \cdot D \nabla _y (u + v) \;\varphi}\\
& - \inty{\nabla _y u \cdot D \nabla _y u \;\varphi} - \inty{\nabla _y v \cdot D \nabla _y v \;\varphi}\\
& = \inty{\nabla _y (u + v) \cdot \ave{D}_Q \nabla _y (u + v) \;\varphi}\\
& - \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y u \;\varphi} - \inty{\nabla _y v \cdot \ave{D}_Q \nabla _y v \;\varphi}\\
& = 2 \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y v \;\varphi }.
\end{align*}
Finally one gets
\[
\nabla _y u \cdot \ave{D}_Q \nabla _y v = \ave{\nabla _y u \cdot D \nabla _y v},\;\;u, v \in \hoy{} \cap \kerbg{}.
\]
Consider now $u \in \hoy{} \cap \kerbg{}$ and $\psi \in C^2 _c (\R^m)$. In order to prove that $\ave{\nabla _y u \cdot \ave{D}_Q \nabla _y ( b \cdny \psi )} = 0$, where the average is understood in the $\loy{}$ setting, we need to check that
\[
\inty{\varphi (y) \;\nabla _y u \cdot \ave{D}_Q \nabla _y ( b \cdny \psi ) } = 0
\]
for any $\varphi \in \liy{}$, $\varphi _s = \varphi, s \in \R$. Clearly $B(y) := \varphi (y) \ave{D}_Q (y) \in \ker L$ and therefore it is enough to prove that
\[
\inty{\nabla _y u \cdot B \nabla _y ( b \cdny \psi ) }= 0
\]
for any $B \in \ker L$, which comes by the third statement of Proposition \ref{WMFI}.
\end{proof}
\begin{remark}
\label{Parametrization} Assume that there is $u_0$ satisfying $u_0 (\ysy) = u_0 (y) + s$, $s \in \R, y \in \R^m$. Notice that $u_0$ could be multi-valued function (think to angular coordinates) but its gradient satisfies for a.a. $y \in \R^m$ and $ s \in \R$
\[
\nabla _y u_0 = {^t \dyy } (\nabla _y u_0 )_s
\]
exactly as any function $u$ which remains constant along the flow of $b$. For this reason, the last equality in \eqref{Equ72} holds true for any $u, v \in \hoy{} \cap \kerbg{} \cup \{u_0\}$. In the case when $m-1$ independent prime integrals of $b$ are known {\it i.e.,} $\exists u_1, ..., u_{m-1} \in \hoy{}\cap \kerbg{}$, the average of the matrix field $D$ comes by imposing
\[
\nabla _y u_i \cdot \ave{D}_Q \nabla _y u_j = \ave{\nabla _y u_i \cdot D \nabla _y u_j},\;\;i, j \in \{0,...,m-1\}.
\]
\end{remark}
\section{First order approximation}
\label{FirstOrdApp}
\noindent
We assume that the fields $D(y), b(y)$ are bounded on $\R^m$
\begin{equation}
\label{Equ26} D \in \liy{},\;\;b \in \liy{}.
\end{equation}
We solve \eqref{Equ1}, \eqref{Equ2} by using variational methods. We consider the Hilbert spaces $V:= \hoy{} \subset H := \lty{}$ (the injection $V \subset H$ being continuous, with dense image) and the bilinear forms $\aeps : V \times V \to \R$ given by
\[
\aeps (u,v) = \inty{D(y) \nabla _y u \cdot \nabla _y v } + \frac{1}{\eps} \inty{(b \cdny u ) \;(b \cdny v)},\;\;u, v \in V.
\]
Notice that for any $0 < \eps \leq 1$ and $v \in V$ we have
\begin{align*}
\aeps (v, v) + d |v|_H ^2 & \geq \inty{D(y) \nabla _y v \cdot \nabla _y v + (b \cdny v ) \;(b \cdny v) } + d \inty{(v(y))^2} \\
& \geq d \inty{|\nabla _y v |^2} + d \inty{(v(y))^2} \\
& = d |v|_V ^2
\end{align*}
saying that $\aeps$ is coercive on $V$ with respect to $H$. By Theorems 1,2 pp. 620 \cite{DauLions88} we deduce that for any $\uein \in H$, there is a unique variational solution for \eqref{Equ1}, \eqref{Equ2}, that is
$\ue \in C_b (\R_+; H) \cap L^2(\R_+;V)$ and
\[
\ue (0) = \uein,\;\;\frac{\md}{\md t } \inty{\ue (t,y) v(y) } + \aeps (\ue (t), v) = 0,\;\;\mbox{in}\;\dpri{},\;\;\forall \; v \in V.
\]
By standard arguments one gets
\begin{pro}
\label{UnifEstim}
The solutions $(\ue)_\eps$ satisfy the estimates
\[
\|\ue \|_{C_b (\R_+; H)} \leq |\uein|_H,\;\;\int _0 ^{+\infty} \!\!\!\!\inty{|\nabla _y \ue |^2}\md t \leq \frac{|\uein |^2 _H}{2d}
\]
and
\[
\|b \cdny \ue \|_{L^2(\R_+; H)} \leq \left ( \frac{\eps}{2(1 - \eps)}\right ) ^{1/2} |\uein |_H,\;\;\eps \in (0,1).
\]
\end{pro}
We are ready to prove the convergence of the family $(\ue )_\eps$, when $\eps \searrow 0$, towards the solution of the heat equation associated to the averaged diffusion matrix field $\ave{D}_Q$.
\begin{proof} (of Theorem \ref{MainResult1}) Based on the uniform estimates in Proposition \ref{UnifEstim}, there is a sequence $(\eps _k)_k$, converging to $0$, such that
\[
\uek \rightharpoonup u \;\mbox{ weakly } \star \mbox{ in } L^\infty(\R_+; H),\;\;\nabla _y \uek \rightharpoonup \nabla _y u \;\mbox{ weakly in }\;L^2(\R_+;H).
\]
Using the weak formulation of \eqref{Equ1} with test functions $\eta (t) \varphi (y)$, $\eta \in C^1 _c (\R_+), \varphi \in C^1 _c (\R^m)$ yields
\begin{align}
\label{Equ77} - \intty{\eta ^\prime (t) \varphi (y) \uek (t,y) } & - \eta (0) \inty{\varphi \uekin } + \intty{\eta \nabla _y \uek \cdot D \nabla _y \varphi } \nonumber \\
& = - \frac{1}{\eps _k} \intty{\eta (t) ( b \cdny \uek) ( b \cdny \varphi )}.
\end{align}
Multiplying by $\eps _k$ and letting $k \to +\infty$, it is easily seen that
\[
\intty{\eta (b \cdny u ) \;(b \cdny \varphi ) } = 0.
\]
Therefore $u(t,\cdot) \in \ker {\cal T} = \kerbg$, $t \in \R_+$, cf. Proposition \ref{KerRanTau}. Clearly \eqref{Equ77} holds true for any $\varphi \in V$. In particular, for any $\varphi \in V \cap \kerbg{}$ one gets
\begin{align}
\label{Equ78} - \intty{\eta ^\prime \uek \varphi } - \eta (0) \inty{\uekin \varphi } + \intty{\eta \nabla _y \uek \cdot D \nabla _y \varphi } = 0.
\end{align}
Thanks to the average properties we have
\[
\inty{\uekin \varphi } = \inty{\ave{\uekin} \varphi } \to \inty{\uin \varphi}
\]
and thus, letting $k \to +\infty$ in \eqref{Equ78}, leads to
\begin{align}
\label{Equ79} - \intty{\eta ^\prime u \varphi } - \eta (0) \inty{\uin \varphi } + \intty{\eta \nabla _y u \cdot D \nabla _y \varphi } = 0.
\end{align}
Since $u(t, \cdot), \varphi \in V \cap \kerbg{}$ we have cf. Theorem \ref{AveMatDif}
\[
\inty{\nabla _y u \cdot D \nabla _y \varphi } = \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y \varphi}
\]
and \eqref{Equ79} becomes
\begin{align}
\label{Equ80} - \intty{\eta ^\prime u \varphi } - \eta (0) \inty{\uin \varphi } + \intty{\eta \nabla _y u \cdot \ave{D}_Q \nabla _y \varphi } = 0.
\end{align}
But \eqref{Equ80} is still valid for test functions $\varphi = b \cdny \psi$, $\psi \in C^2 _c (\R^m)$ since $u(t,\cdot) \in \kerbg$, $\uin = \mbox{w}-\lime \ave{\uein} \in \kerbg$ and $\ave{D}_Q \in \ker L$
\[
\inty{u(t,y) b \cdny \psi } = 0,\;\;\inty{\uin b \cdny \psi } = 0,\;\;\inty{\nabla _y u \cdot \ave{D}_Q \nabla _y ( b \cdny \psi ) }= 0
\]
cf. Theorem \ref{AveMatDif}. Therefore, for any $v \in V$ one gets
\[
\frac{\md}{\md t} \inty{u (t,y) v(y) } + \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y v } = 0\;\mbox{ in } \dpri{}
\]
with $u(0) = \uin$. By the uniqueness of the solution of \eqref{Equ75}, \eqref{Equ76} we deduce that all the family $(\ue)_\eps$ converges weakly to $u$.
\end{proof}
\begin{remark}
\label{Propagation} Notice that \eqref{Equ75} propagates the constraint $b \cdny u = 0$, if satisfied initially. Indeed, for any $v \in C^1 _c (\R^m)$ we have
\begin{equation}
\label{Equ81}\frac{\md }{\md t } \inty{u (t,y) v (y) } + \inty{ \nabla _y u \cdot \ave{D}_Q \nabla _y v } = 0\;\mbox{ in } \dpri{}.
\end{equation}
Since $\ave{D}_Q \in \ker L$, we know by the second statement of Proposition \ref{WMFI} that
\begin{equation*}
\inty{\nabla _y u_s \cdot \ave{D}_Q \nabla _y v } = \inty{\nabla _y u \cdot \ave{D}_Q \nabla _y v_{-s}}.
\end{equation*}
Replacing $v$ by $v_{-s}$ in \eqref{Equ81} we obtain
\[
\frac{\md }{\md t } \inty{u_s v } + \inty{\nabla _y u_s \cdot \ave{D}_Q \nabla _y v } = 0\;\mbox{ in } \dpri{}
\]
and therefore $u_s$ solves
\[
\partial _t u_s - \divy ( \ave{D}_Q \nabla _y u_s) = 0,\;\;(t, y) \in \R_+ \times \R^m
\]
and $u_s (0,y) = \uin (\ysy) = \uin (y), y \in \R^m$. By the uniqueness of the solution of \eqref{Equ75}, \eqref{Equ76} one gets $u_s = u$ and thus, at any time $t \in \R_+$, $b \cdny u (t,\cdot) = 0$.
\end{remark}
\section{Second order approximation}
\label{SecOrdApp}
\noindent
For the moment we have determined the model satisfied by the dominant term in the expansion \eqref{Equ6}. We focus now on second order approximation, that is, a model which takes into account the first order correction term $\eps u ^1$. Up to now we have used the equations \eqref{Equ7}, \eqref{Equ8}. Finding a closure for $u + \eps u ^1$ will require one more equation
\begin{equation}
\label{Equ83} \partial _t u^1 - \divy ( D \nabla _y u^1 ) - \divy ( b \otimes b \nabla _y u^2) = 0,\;\;(t, y) \in \R_+ \times \R ^m.
\end{equation}
Let us see, at least formally, how to get a second order approximation for $(\ue )_\eps$, when $\eps $ becomes small. The first order approximation {\it i.e.}, the closure for $u$, has been obtained by averaging \eqref{Equ8} and by taking into account that $u \in \kerbg{}$
\[
\partial _t u = \ave{\divy( D \nabla _y u ) } = \divy ( \ave{D}_Q \nabla _y u ).
\]
Thus $u^1$ satisfies
\begin{equation}
\label{Equ84} \divy ( \ave{D}_Q \nabla _y u ) - \divy ( D \nabla _y u ) - \divy ( b \otimes b \nabla _y u^1) = 0
\end{equation}
from which we expect to express $u^1$, up to a function in $\kerbg{}$, in terms of $u$.
\begin{proof} (of Theorem \ref{Decomposition})
We claim that $\ran L^2 = \ran L $ and thus $\ran L^2 $ is closed as well. Clearly $\ran L^2 \subset \ran L$. Consider now $Z = L(Y)$ for some $Y \in \dom (L)$. But $Y - \mathrm{Proj}_{\ker L} Y \in \ker L ^\perp = (\ker L^\star ) ^\perp = \overline{\ran L} = \ran L$ and there is $X \in \dom (L)$ such that $Y - \mathrm{Proj}_{\ker L} Y = L(X)$. Finally $X \in \dom (L^2)$ and
\[
Z = L(Y) = L(Y - \mathrm{Proj} _{\ker L} Y ) = L(L(X)).
\]
By construction we have $D - \ave{D}_Q \in ( \ker L)^\perp = ( \ker L^\star ) ^\perp = \overline{\ran L} = \ran L = \ran L^2$ and thus there is a unique $F \in \dom (L^2) \cap ( \ker L )^\perp $ such that $D = \ave{D}_Q - L(L(F))$. As $F \in ( \ker L )^\perp$, there is $C \in \dom (L)$ such that $F = L(C)$ implying that ${^t F} = {^t L(C)} = L ({^t C})$. Therefore ${^t F } \in \dom (L^2) \cap ( \ker L )^\perp$ and satisfies the same equation as $F$
\[
L(L({^t F})) = {^t L}(L(F)) = \ave{D}_Q - D.
\]
By the uniqueness we deduce that $F$ is a field of symmetric matrix.
By Proposition \ref{PropOpeL} we know that
\[
- \divy(L(F) \nabla _y ) = [b \cdot \nabla _y, - \divy ( F \nabla _y )]\;\mbox{ in }\; \dpri{}
\]
{\it i.e.,}
\[
\inty{L(F) \nabla _y u \cdot \nabla _y v } = - \inty{F \nabla _y u \cdot \nabla _y ( b \cdny v ) } - \inty{F \nabla _y ( b \cdny u ) \cdny v }
\]
for any $u, v \in C^2 _c (\R^m)$. Similarly, $E := L(F)$ satisfies
\[
- \divy ( L^2 (F) \nabla _y ) = - \divy (L(E) \nabla _y ) = [b\cdny, - \divy ( E \nabla _y )]\;\mbox{ in }\;\dpri{}
\]
and thus, for any $u, v \in C^3_c (\R^m)$ one gets
\begin{align*}
& \inty{(\ave{D}_Q - D) \nabla _y u \cdny v } = \inty{L^2(F)\nabla _y u \cdny v } \\
& = - \inty{L(F) \nabla _y u \cdny ( b \cdny v ) }- \inty{L(F) \nabla _y ( b \cdny u ) \cdny v } \\
& = \inty{F \nabla _y u \cdny ( b \cdny ( b \cdny v ))} + \inty{F \nabla _y ( b \cdny u ) \cdny ( b \cdny v)} \\
& + \inty{F \nabla _y ( b \cdny u ) \cdny ( b \cdny v)} + \inty{F \nabla _y ( b \cdny ( b \cdny u )) \cdny v}.
\end{align*}
\end{proof}
\noindent
The matrix fields $F \in \dom (L^2)$ and $E = L(F) \in \dom (L)$ have the following properties.
\begin{pro}
\label{PropOpeF} For any $u, v \in C^1 (\R^m)$ which are constant along the flow of $b$ we have in $\dpri{}$
\[
D \nabla _y u \cdny v - \ave{D}_Q \nabla _y u \cdny v = - b \cdny ( E \nabla _y u \cdny v ) = - \divy ( b \otimes b \nabla _y ( F \nabla _y u \cdny v ))
\]
and
\[
\ave{E \nabla _y u \cdny v } = \ave{ F \nabla _y u \cdny v } = 0.
\]
In particular
\[
\inty{E \nabla _y u \cdny v } = \inty{\ave{E \nabla _y u \cdny v}}= 0
\]
\[
\inty{F \nabla _y u \cdny v } = \inty{\ave{F \nabla _y u \cdny v}}= 0
\]
saying that $\ave{\divy ( E \nabla _y u ) } = \ave{\divy ( F \nabla _y u )} = 0$ in $\dpri{}$.
\end{pro}
\begin{proof}
Consider $\varphi \in C^1 _c (\R^m)$, $u, v \in C^1 (\R^m)$ such that $u_s = u, v_s = v$, $s \in \R$ and the matrix field $U = \varphi Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} \in H_Q$. Actually $U \in \dom (L)$ and, as in the proof of the last statement in Proposition \ref{PropOpeL}, one gets
\begin{align*}
L(U) & = (b \cdny \varphi ) Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} + \varphi \;L ( Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}) \\
& = (b \cdny \varphi) Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}
\end{align*}
since $ Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} \in \ker (L)$. Multiplying by $U$ the equality $D - \ave{D}_Q = - L(E)$, $E = L(F)$, one gets
\[
\inty{\varphi ( D - \ave{D}_Q)\nabla _y u \cdny v } = - (L(E), U)_Q = (E, L(U))_Q = \inty{(b \cdny \varphi) ( E \nabla _y u \cdny v )}
\]
implying that $D \nabla _y u \cdny v = \ave{D}_Q \nabla _y u \cdny v - b \cdny ( E \nabla _y u \cdny v)$ in $\dpri{}$. Multiplying by $U$ the equality $E = L(F)$ yields
\[
\inty{\varphi E \nabla _y u \cdny v } = (E, U)_Q = (L(F), U)_Q = - (F, L(U))_Q = - \inty{(b \cdny \varphi) F \nabla _y u \cdny v}.
\]
We obtain
\[
E \nabla _y u \cdny v = b \cdny ( F \nabla _y u \cdny v) \;\mbox{ in }\; \dpri{}
\]
and thus
\[
D \nabla _y u \cdny v - \ave{D}_Q \nabla _y u \cdny v = - b \cdny (E \nabla _y u \cdny v ) = - b \cdny ( b \cdny ( F \nabla _y u \cdny v ))
\]
in $\dpri{}$. Consider now $U = \varphi Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1}$ with $\varphi \in \kerbg{}$. We know that $L(U) = 0$ and since, by construction $F \in (\ker L )^\perp$, we deduce
\[
\inty{\varphi F \nabla _y u \cdny v } = (F, U)_Q = 0
\]
saying that $\ave{F \nabla _y u \cdny v} = 0$. Similarly $E = L(F) \in (\ker L )^\perp$ and $\ave{E \nabla _y u \cdny v } = 0$.
\end{proof}
\begin{remark}
\label{ParametrizationBis}
Assume that there is $u_0$ (eventually multi-valued) satisfying $u_0 (\ysy{}) = u_0 (y) + s$, $s \in \R, y \in \R^m$. Its gradient changes along the flow of $b$ exactly as the gradient of any function which is constant along this flow cf. Remark \ref{Parametrization}. We deduce that $Q^{-1} \nabla _y v \otimes \nabla _y u Q^{-1} \in \ker L$ for any $u, v \in \kerbg \cup \{u_0\}$ and therefore the arguments in the proof of Proposition \ref{PropOpeF} still apply when $u, v \in \kerbg{} \cup \{u_0\}$. In the case when $m-1$ independent prime integrals $\{u_1, ..., u_{m-1}\}$ of $b$ are known, the matrix fields $E, F$ come, by imposing for any $i, j \in \{0,1,...,m-1\}$
\[
- b \cdny (E \nabla _y u_i \cdny u _j) = D \nabla _y u_i \cdny u_j - \ave{D \nabla _y u_i \cdny u_j},\;\;\ave{E \nabla _y u_i \cdny u_j} = 0
\]
and
\[
b \cdny (F \nabla _y u_i \cdny u _j) = E \nabla _y u_i \cdny u _j,\;\;\ave{F \nabla _y u_i \cdny u _j } = 0.
\]
\end{remark}
We indicate now sufficient conditions which guarantee that the range of $L$ is closed.
\begin{pro}
\label{CompleteIntegr} Assume that \eqref{Equ21}, \eqref{Equ22}, \eqref{Equ23} hold true and that there is a matrix field $R(y)$ such that \eqref{Equ90} holds true. Then the range of $L$ is closed.
\end{pro}
\begin{proof}
Observe that \eqref{Equ90} implies \eqref{Equ56}. Indeed, it is easily seen that $b \cdny R + R \partial _y b = 0$ in $\dpri{}$ is equivalent to $R = R_s \partial _y Y ( s; \cdot)$, $s \in \R$. We deduce that $P = R ^{-1} \;{^t R} ^{-1}$ satisfies
\[
G(s)P = \partial _y Y ^{-1} (s; \cdot) P_s {^t \partial _y Y ^{-1} (s; \cdot)} = \partial _y Y ^{-1} (s; \cdot)R_s ^{-1} \;{^t R_s} ^{-1} \;{^t \partial _y Y ^{-1} (s; \cdot)} = R^{-1} \;{^t R}^{-1} = P
\]
saying that $[b,P] = 0$ in $\dpri{}$. Therefore we can define $L$ as before, on $H_Q$, which coincides in this case with $\{A:RA\;{^t R} \in \lty{}\}$. We claim that $i \circ L = ( b \cdny ) \circ i$ where $i : H_Q \to \lty{}$, $i(A) = R A\; {^t R}$, $A \in H_Q$, which comes immediately from the equalities
\[
(i\circ G(s))A = RG(s)A {^t R} = R \partial _y Y ^{-1}( s; \cdot ) A_s {^t \partial _y Y }^{-1} {^t R} = R_s A_s {^t R_s} = (i(A))_s,\;s\in \R, A\in H_Q.
\]
In particular we have
\[
\ker L = \{A \in H_Q\;:\; i(A) \in \kerbg\}
\]
and
\begin{align*}
(\ker L )^\perp & = \{A \in H_Q\;:\; \inty{i(A) : U }
= 0\;\forall\;U \in \kerbg{}\} \\
& = \{A \in H_Q\;:\; i(A) \in ( \kerbg)^\perp \}.
\end{align*}
For any $A \in (\ker L)^\perp$ we can apply the Poincar\'e inequality \eqref{Equ23} to $i(A) \in (\kerbg)^\perp$ and we obtain
\[
|A|_Q = |i(A)|_{L^2} \leq C_P |b \cdny (i(A))|_{L^2} = C_P |i (L(A))|_{L^2} = C_P |L(A)|_Q.
\]
Therefore $L$ satisfies a Poincar\'e inequality as well, and thus the range of $L$ is closed.
\end{proof}
\begin{remark}
\label{ClosedRanL}
The hypothesis $b \cdny R + R \dyb = 0$ in $\dpri{}$ says that the columns of $R^{-1}$ form a family of $m$ independent vector fields in involution with respect to $b$, cf. Proposition \ref{VFI}
\[
R_s ^{-1} (y) = \dyy R ^{-1} (y),\;\;s\in \R,\;\;y \in \R^m.
\]
\end{remark}
\begin{remark}
\label{ExplicitAve} For any $U \in \ker L$, that is $i(U) \in \kerbg{}$, we have
\[
\inty{R ( D - \ave{D}_Q) {^t R } : i(U)} = 0.
\]
As $\ave{D}_Q \in \ker L $, we know that $i(\ave{D}_Q) = R \ave{D}_Q {^t R } \in \kerbg{}$ and thus the matrix field $R \ave{D}_Q {^t R }$ is the average (along the flow of $b$) of the matrix field $RD\;{^t R}$, which allows us to express $\ave{D}_Q$ in terms of $R$ and $D$
\[
R \ave{D}_Q {^t R } = \ave{R D \;{^t R }}.
\]
\end{remark}
From now on we assume that \eqref{Equ90} holds true. Applying the decomposition of Theorem \ref{Decomposition} with the dominant term $u \in \kerbg$ in the expansion \eqref{Equ6} and any $v \in C^3 _c (\R^m)$ yields
\[
\inty{(D - \ave{D}_Q) \nabla _y u \cdny v } = - \inty{F \nabla _y u \cdny ( b \cdny ( b \cdny v ))}.
\]
From \eqref{Equ84} one gets
\[
\inty{(D - \ave{D}_Q ) \nabla _y u \cdny v } - \inty{u^1 b \cdny ( b \cdny v )} = 0
\]
and thus
\begin{equation}
\label{CorrSplit}
u^1 = \divy ( F \nabla _y u ) + v^1,\;\;v^1 \in \ker ( b \cdny ( b \cdny )) = \kerbg.
\end{equation}
Notice that $\ave{u^1} = v^1$, since $\ave{\divy ( F \nabla _y u )} = 0$, cf. Proposition \ref{PropOpeF}. The time evolution for $v^1 = \ave{u^1}$ comes by averaging \eqref{Equ83}
\[
\partial _t v ^1 - \ave{\divy ( D \nabla _y v^1)} - \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = 0.
\]
As $v^1 \in \kerbg$ we have
\[
- \ave{\divy ( D \nabla _y v^1)} = - \divy ( \ave{D}_Q \nabla _y v^1)
\]
and we can write, with the notation $w^1 = \divy (F \nabla _y u)$
\begin{align}
\label{Equ86} \partial _t \{u + \eps u^1\} - \divy ( \ave{D}_Q \nabla _y \{u + \eps u^1\}) = \eps \partial _t w^1 - \eps \divy ( \ave{D}_Q \nabla _y w^1 ) + \eps \ave{\divy ( D \nabla _y w^1)}.
\end{align}
But the time derivative of $w^1$ is given by
\[
\partial _t w^1 = \divy ( F \nabla _y \partial _t u ) = \divy ( F \nabla _y ( \divy ( \ave{D}_Q \nabla _y u )))
\]
which implies
\begin{align*}
\partial _t w^1 - \divy ( \ave{D}_Q\nabla _y w^1) & = \divy ( F \nabla _y ( \divy ( \ave{D}_Q \nabla _y u )))- \divy ( \ave{D}_Q \nabla _y ( \divy ( F \nabla _y u ))) \\
& = - [\divy(\ave{D}_Q \nabla _y ), \divy ( F \nabla _y )]u.
\end{align*}
Up to a second order term, the equation \eqref{Equ86} writes
\begin{align}
\label{Equ102} \partial _t \{u + \eps u^1\} - \divy ( \ave{D}_Q \nabla _y \{u + \eps u ^1\}) & + \eps [\divy(\ave{D}_Q \nabla _y ), \divy ( F \nabla _y )]\{u + \eps u^1\} \nonumber \\
& - \eps \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = {\cal O}(\eps ^2).
\end{align}
We claim that for any $u \in \kerbg$ we have
\begin{equation}
\label{Equ87} \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u)))} = \ave{\divy ( E \nabla _y ( \divy ( E \nabla _y u )))}.
\end{equation}
By Proposition \ref{PropOpeF} we know that $\ave{\divy ( F \nabla _y u )} = 0$. As $L(\ave{D}_Q) = 0$ we have
\[
[b \cdny, - \divy ( \ave{D}_Q \nabla _y )] = - \divy ( L ( \ave{D}_Q) \nabla _y ) = 0
\]
and thus $\divy ( \ave{D}_Q\nabla _y)$ leaves invariant the subspace of functions which are constant along the flow of $b$. By the symmetry of the operator $\divy ( \ave{D}_Q \nabla _y )$, we deduce that the subspace of zero average functions is also left invariant by $\divy ( \ave{D}_Q \nabla _y )$. Therefore $\ave{\divy ( \ave{D}_Q \nabla _y ( \divy ( F \nabla _y u )))} = 0$ and
\[
\ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = \ave{\divy ((D - \ave{D}_Q) \nabla _y ( \divy ( F \nabla _y u )))}.
\]
Thanks to Theorem \ref{Decomposition} we have
\begin{align*}
\divy((D - \ave{D}_Q)\nabla _y ) & = [b \cdny, [b \cdny, - \divy ( F \nabla _y )]\;] \\
& = [b \cdny, - \divy (L(F)\nabla _y )]\\
& = [b \cdny, - \divy (E\nabla _y )]
\end{align*}
which implies that
\begin{align*}
& \ave{\divy ( D \nabla _y ( \divy ( F \nabla _y u )))} = \ave{\divy ( (D - \ave{D}_Q) \nabla _y ( \divy ( F \nabla _y u )))} \\
& = \ave{\divy ( E \nabla _y ( b \cdny ( \divy ( F \nabla _y u )))) - b \cdny ( \divy ( E \nabla _y ( \divy ( F \nabla _y u ))))}\\
& = \ave{\divy ( E \nabla _y ( b \cdny ( \divy ( F \nabla _y u ))))}.
\end{align*}
Finally notice that
\[
- \divy ( E \nabla _y u ) = - \divy ( L(F)\nabla _y u ) = [b \cdny, - \divy ( F \nabla _y u )] = - b \cdny ( \divy ( F \nabla _y u ))
\]
and \eqref{Equ87} follows.
We need to average the differential operator $\divy ( E \nabla _y ( \divy ( E \nabla _y )))$ on functions $u \in \kerbg$. For simplicity we perform these computations at a formal level, assuming that all fields are smooth enough. The idea is to express the above differential operator in terms of the derivations ${^t R }^{-1} \nabla _y $ which commute with the average operator (see Proposition \ref{AveComFirstOrder}), since the columns of $R^{-1}$ contain vector fields in involution with $b(y)$.
\begin{lemma}
\label{ChangeOfCoord} Under the hypothesis \eqref{Equ90}, for any smooth function $u(y)$ and matrix field $E(y)$ we have
\begin{equation}
\label{Equ100} \divy ( E \nabla _y u) = \divy ( R \;{^t E}) \cdot ( {^t R}^{-1} \nabla _y u ) + R E \;{^t R} : ( {^t R } ^{-1} \nabla _y \otimes {^t R }^{-1} \nabla _y ) u.
\end{equation}
\end{lemma}
\begin{proof}
Applying the formula $\divy (A\xi) = \divy {^t A} \cdot \xi + {^t A } : \partial _y \xi$, where $A(y)$ is a matrix field and $\xi (y)$ is a vector field, one gets
\[
\divy ( E \nabla _y u ) = \divy ( E \;{^t R } \;{^t R ^{-1}} \nabla _y u ) = \divy ( R \;{^t E}) \cdot ( {^t R }^{-1} \nabla _y u ) + R \;{^t E} : \partial _y ( {^t R }^{-1} \nabla _y u ).
\]
The last term in the above formula writes
\begin{align*}
R \;{^t E } : \partial _y ( {^t R }^{-1} \nabla _y u ) & = R \;{^t E} \;{^t R}\; {^t R } ^{-1} : \partial _y ( {^t R } ^{-1} \nabla _y u ) \\
& = R \;{^t E } \;{^t R } : \partial _y ( {^t R} ^{-1} \nabla _y u ) R ^{-1} \\
& = R E \;{^t R} : {^t R }^{-1} \;{^t \partial _y } ( {^t R } ^{-1} \nabla _y u ) \\
& = R E \;{^t R} : ( {^t R} ^{-1} \nabla _y \otimes {^t R} ^{-1} \nabla _y ) u
\end{align*}
and \eqref{Equ100} follows.
\end{proof}
Next we claim that the term $\ave{\divy ( E \nabla _y ( \divy ( E \nabla _y u )))}$ reduces to a differential operator, if $u \in \kerbg{}$.
\begin{pro}
\label{DifOpe} Under the hypothesis \eqref{Equ90}, for any smooth matrix field $E$ there is a linear differential operator $S(u)$ of order four, such that, for any smooth $u \in \kerbg{}$
\begin{equation}
\label{Equ101} \ave{\divy ( E \nabla _y ( \divy ( E \nabla _y u )))} = S(u).
\end{equation}
\end{pro}
\begin{proof}
For any smooth functions $u, \varphi \in \kerbg{}$ we have, cf. Lemma \ref{ChangeOfCoord}
\begin{align*}
& \inty{\ave{\divy(E \nabla _y ( \divy ( E \nabla _y u )))}\varphi } = \inty{\divy(E \nabla _y ( \divy ( E \nabla _y u ))) \varphi }\\
& = \inty{\divy ( E \nabla _y u ) \;\divy ( E \nabla _y \varphi )} \\
& = \inty{\{\divy ( R \; ^t E) \cdot ( ^t R ^{-1} \nabla _y u ) + R E \; ^t R : ( ^t R ^{-1} \nabla _y \otimes {^t R } ^{-1} \nabla _y )u \}\\
& \times \{\divy ( R \; ^t E) \cdot ( ^t R ^{-1} \nabla _y \varphi ) + R E \; ^t R : ( ^t R ^{-1} \nabla _y \otimes {^t R } ^{-1} \nabla _y )\varphi \}}\\
& = \inty{[\divy (R \;\;^t E) \otimes \divy ( R \;\;^t E)] : [^t R ^{-1} \nabla _y u \otimes {^t R} ^{-1} \nabla _y \varphi] }\\
& + \inty{[R E \;\;^t R \otimes \divy ( R \;\;^t E)] : [( ^t R ^{-1} \nabla _y \otimes {^t R }^{-1}\nabla _y )u \otimes {^t R } ^{-1} \nabla _y \varphi] }\\
& + \inty{[\divy( R \;\;^t E) \otimes R E \;\;^t R] : [(^t R ^{-1} \nabla _y u ) \otimes ( ^t R ^{-1} \nabla _y \otimes {^t R}^{-1} \nabla _y ) \varphi]}\\
& + \inty{[R E \;\;^t R \otimes R E \;\;^t R]:[ ( ^t R ^{-1} \nabla _y \otimes {^t R }^{-1}\nabla _y )u \otimes ( ^t R ^{-1} \nabla _y \otimes {^t R }^{-1}\nabla _y )\varphi ]}
\end{align*}
Recall that $^tR ^{-1} \nabla _y $ leaves invariant $\kerbg$ and therefore
\[
{^t R }^{-1} \nabla _y u \otimes {^t R } ^{-1} \nabla _y \varphi \in \kerbg{}
\]
implying that
\begin{align*}
& \inty{[\divy (R \;\;^t E) \otimes \divy ( R \;\;^t E)] : [^t R ^{-1} \nabla _y u \otimes {^t R} ^{-1} \nabla _y \varphi] }\\
= &
\inty{\ave{\divy (R \;\;^t E) \otimes \divy ( R \;\;^t E)} : [^t R ^{-1} \nabla _y u \otimes {^t R} ^{-1} \nabla _y \varphi] }.
\end{align*}
Similar transformations apply to the other three integrals above, and finally one gets
\begin{align*}
\inty{\ave{\divy(E \nabla _y ( \divy ( E \nabla _y u )))}\varphi } & = \inty{X : [\nablar u \otimes \nablar \varphi ]} \\
& + \inty{Y : [( \nablar \otimes \nablar )u \otimes \nablar \varphi ]}\\
& + \inty{Z : [\nablar u \otimes ( \nablar \otimes \nablar ) \varphi] } \\
& + \inty{T : [( \nablar \otimes \nablar )u \otimes ( \nablar \otimes \nablar ) \varphi]}\\
& = I_1 (u, \varphi) + I_2 (u, \varphi) + I_3 (u, \varphi) + I_4 (u, \varphi)
\end{align*}
where $\nablar := {^t R} ^{-1} \nabla _y $ and $X, Y, Z, T$ are tensors of order two, three, three and four respectively
\[
X_{ij} = \ave{\divy ( R \;\;^t E) _i \;\divy(R \;\;^t E)_j},\;\;i,j\in \{1,...,m\}
\]
\[
Y_{ijk} = \ave{(R E \;\;^t R) _{ij} \;\divy (R \;\;^t E)_k},\;\;Z_{ijk} = \ave{\divy ( R \;\;^t E)_i \;\;(RE \;\;^t R)_{jk}} ,\;\;i,j, k\in \{1,...,m\}
\]
\[
T_{ijkl} = \ave{(RE \;\;^t R)_{ij} \;\;(RE \;\;^tR)_{kl}},\;\;i,j, k, l\in \{1,...,m\}.
\]
Integrating by parts one gets
\[
I_1 (u, \varphi) = \inty{X \nablar u \cdot \nablar \varphi } = \inty{R^{-1} X \nablar u \cdot \nabla _y \varphi } = \inty{S_1 (u) \varphi}
\]
where $S_1 (u) = - \divy ( R^{-1} X \nablar u)$. Notice that the differential operator
\[
\xi \to \divy ( R^{-1} \xi) = \divy (\;^t R ^{-1}) \cdot \xi + {^t R}^{-1} : \partial _y \xi
\]
maps $(\kerbg{})^m$ to $\kerbg{}$, since the columns of $R^{-1}$ contain fields in involution with $b$, and therefore $S_1$ leaves invariant $\kerbg{}$, that is, for any $u \in \kerbg{}$, $\xi = X \nablar u \in (\kerbg{})^m$ and $S_1 (u) = - \divy ( R^{-1} X \nablar u ) = - \divy ( R^{-1} \xi ) \in \kerbg{}$. Similarly we obtain
\[
I_2 (u, \varphi)
= \inty{S_2 (u) \varphi },\;\;
I_3 (u, \varphi) = \inty{S_3 (u) \varphi },\;\;I_4 (u, \varphi) = \inty{S_4 (u) \varphi }
\]
where $S_2, S_3, S_4$ are differential operators of order three, three and four respectively, which leave invariant $\kerbg{}$. We deduce that
\[
\inty{\ave{\divy (E \nabla _y ( \divy ( E \nabla_y u )))} \varphi } = \inty{S(u) \varphi}
\]
for any $u, \varphi \in \kerbg{}$, with $S = S_1 + S_2 + S_3 + S_4$, saying that
\[
\ave{\divy (E \nabla _y ( \divy ( E \nabla_y u )))} - S(u) \perp \kerbg{}.
\]
But we also know that
\[
\ave{\divy (E \nabla _y ( \divy ( E \nabla_y u )))} - S(u) \in \kerbg{}
\]
and thus \eqref{Equ101} holds true.
\end{proof}
Combining \eqref{Equ102}, \eqref{Equ87}, \eqref{Equ101} we obtain
\begin{align*}
\partial _t \{u + \eps u^1\} - \divy ( \ave{D}_Q \nabla _y \{u + \eps u ^1\}) & + \eps [\divy(\ave{D}_Q \nabla _y ), \divy ( F \nabla _y )]\{u + \eps u^1\} \\
& - \eps S(u + \eps u^1) = {\cal O}(\eps ^2)
\end{align*}
which justifies the equation introduced in \eqref{IntroEqu87}. The initial condition comes formally by averaging the Ansatz \eqref{Equ6}
\[
\ave{\ue} = u + \eps v^1 + {\cal O}(\eps ^2).
\]
One gets
\[
v ^1 (0, \cdot) = \mbox{w-} \lime \frac{\ave{\uein} - \uin }{\eps} = \vin
\]
implying that $u ^1 (0, \cdot) = \vin + \divy(F \nabla _y \uin )$, cf. \eqref{CorrSplit}, which justifies \eqref{NewIC}.
\section{An example}
Let us consider the vector field $b(y) = {^\perp y} := (y_2, - y_1)$, for any $y = (y_1, y_2) \in \R^2$ and the matrix field
\[
D (y) = \left(
\begin{array}{cc}
\lambda _ 1 (y) & 0 \\
0 & \lambda _2 (y)
\end{array}
\right),\;\;y \in \R^2
\]
where $\lambda _1, \lambda _2 $ are given functions, satisfying $\min _{y\in \R^2} \{\lambda _1 (y), \lambda _2 (y)\} \geq d>0$. We intend to determine the first order approximation, when $\eps \searrow 0$, for the heat equation
\begin{equation}
\label{Equ91} \partial _t \ue - \divy ( D(y) \nabla _y \ue ) - \frac{1}{\eps} \divy ( b(y) \otimes b(y) \nabla _y \ue ) = 0,\;\;(t, y ) \in \R_+ \times \R ^2
\end{equation}
with the initial condition
\[
\ue (0, y) = \uin (y),\;\;y \in \R^2.
\]
The flow of $b$ is given by $Y(s;y) = {\cal R}(-s)y$, $s \in \R, y \in \R^2$ where ${\cal R}(\alpha)$ stands for the rotation of angle $\alpha \in \R$. The functions in $\kerbg{}$ are those depending only on $|y|$. Notice that the matrix field
\[
R(y) = \frac{1}{|y|}\left(
\begin{array}{rr}
y_2 & -y_1 \\
y_1 & y_2
\end{array}
\right)
\]
satisfies $b \cdot \nabla _y R + R \partial _y b = 0$ and $Q = {^t R } R = I_2$. The averaged matrix field $\ave{D}_Q$ comes, thanks to Remark \ref{ExplicitAve}, by the formula $R \ave{D}_Q {^t R} = \ave{R D \;{^t R}}$ and thus
\[
\ave{D}_Q = {^t R} \ave{RD\; {^t R}} R,\;\;\ave{RD\; {^t R}} = \left(
\begin{array}{rr}
\ave{\frac{\lambda _1 y _2 ^2 + \lambda _2 y_1 ^2 }{|y|^2}} & \ave{\frac{(\lambda _1 - \lambda _2)y_1 y _2 }{|y|^2}} \\
\ave{\frac{(\lambda _1 - \lambda _2)y_1 y _2 }{|y|^2}} & \ave{\frac{\lambda _1 y _1 ^2 + \lambda _2 y_2 ^2 }{|y|^2}}
\end{array}
\right).
\]
In the case when $\lambda _1, \lambda _2$ are left invariant by the flow of $b$, that is $\lambda _1, \lambda _2$ depend only on $|y|$, it is easily seen that
\[
\ave{\frac{y_1 ^2}{|y|^2}} = \ave{\frac{y_2 ^2}{|y|^2}} = \frac{1}{2},\;\;\ave{\frac{y_1 y_2}{|y|^2}} = 0
\]
and thus
\[
\ave{D}_Q = {^t R } \frac{\lambda _1 + \lambda _2}{2} I_2 R = \frac{\lambda _1 + \lambda _2}{2} I_2.
\]
The first order approximation of \eqref{Equ91} is given by
\[
\left\{
\begin{array}{ll}
\partial _t u - \divy \left ( \frac{\lambda _1 (y) + \lambda _2 (y)}{2} \nabla _y u \right ) = 0,& \;\;(t, y ) \in \R_+ \times \R ^2 \\
u(0,y) = \uin (y),& \;\;y \in \R^2.
\end{array}
\right.
\]
We consider the multi-valued function $u_0 (y) = - \theta (y)$, where $y = |y| ( \cos \theta (y), \sin \theta (y))$, which satisfies $b \cdot \nabla _y u_0 = 1$, or $u_0 (Y(s;y)) = u_0 (y) + s$. Notice that the averaged matrix field $\ave{D}_Q$ satisfies (with $u_1 (y) = |y|^2 /2 \in \kerbg{}$\;)
\[
\nabla _y u_i \cdot \ave{D}_Q \nabla _y u _j = \ave{\nabla _y u _i \cdot D \nabla _y u_j },\;\;i, j \in \{0,1\}
\]
as predicted by Remark \ref{Parametrization}.
\appendix
\section{Proofs of Propositions \ref{VFI}, \ref{WVFI}, \ref{MFI}, \ref{WMFI}}
\label{A}
\begin{proof} (of Proposition \ref{VFI}) For simplicity we assume that $b$ is divergence free. The general case follows similarly. Let $c(y)$ be a vector field satisfying \eqref{Equ34}. For any vector field $\phi \in C^1 _c (\R^m)$ we have, with the notation $u _\tau = u (Y(\tau;\cdot))$
\begin{align*}
\inty{c \cdot ( \phi _{-h} - \phi )} = \inty{(c_h - c) \cdot \phi } = \inty{(\partial _y Y (h;y) - I) c \cdot \phi }.
\end{align*}
Multiplying by $h^{-1}$ and passing to the limit when $h \to 0$ imply
\[
- \inty{c ( b \cdny \phi ) } = \inty{\partial _y b c\cdot \phi }
\]
and therefore $(b \cdny ) c - \partial _y b c = 0$ in $\dpri{}$.
Conversely, assume that $[b,c] = 0$ in $\dpri{}$. We introduce $e(s,y) = c(Y(s;y)) - \partial _y Y(s;y) c(y)$. Notice that $e(s,\cdot) \in \loloc{}, s \in \R$ and $e(0, \cdot) = 0$. For any vector field $\phi \in C^1 _c (\R^m)$ we have
\[
E_\phi (s) : = \inty{e(s,y) \cdot \phi (y)} = \inty{c(y) \cdot \phi _{-s}} - \inty{\partial _y Y(s;y) c(y) \cdot \phi (y) }
\]
and thus
\begin{align*}
\frac{\md }{\md s} E_\phi (s) & = - \inty{c(y) \cdot ((b \cdny )\phi )_{-s}} - \inty{\partial _y (b(\ysy)) \;c(y) \cdot \phi (y) } \\
& = - \inty{c \cdot (b \cdny ) \phi _{-s}} - \inty{\partial _y b (Y(s;y)) \dyy c(y) \cdot \phi (y) }\\
& = \inty{\dyb \;c(y) \cdot \phi _{-s}} - \inty{\dyb (\ysy) \dyy c(y) \cdot \phi (y)}\\
& = \inty{\dyb (\ysy) ( c(\ysy) - \dyy c(y)) \cdot \phi (y)} \\
& = \inty{e(s,y)\cdot {^t \dyb} (\ysy) \phi (y)}.
\end{align*}
In the previous computation we have used the fact that the derivation and tranlation along $b$ commute
\[
((b \cdny ) \phi )_{-s} = (b\cdny )\phi _{-s}.
\]
After integration with respect to $s$ one gets
\[
E_\phi (s) = \int _0 ^s \inty{e(\tau,y) \cdot {^t \dyb} (Y(\tau;y)) \phi (y) } \;\md \tau.
\]
Clearly, the above equality still holds true for any $\phi \in C_c (\R^m)$. Consider $R>0, T>0$ and let $K = \|{^t \dyb} \circ Y\|_{L^\infty([-T,T] \times B_R)}$. Therefore, for any $s \in [-T, T]$ we obtain
\begin{align*}
\|e(s,\cdot)\|_{L^\infty(B_R)} & = \sup \{ |E_\phi (s)|\;:\;\phi \in C_c (B_R),\;\;\|\phi \|_{\loy{}} \leq 1\}\\
& \leq K \left |\int _0 ^s \|e (\tau, \cdot) \|_{L^\infty (B_R)}\md \tau \right |.
\end{align*}
By Gronwall lemma we deduce that $\|e(s,\cdot)\|_{L^\infty(B_R)} = 0$ for $-T \leq s \leq T$ saying that $c(\ysy) - \dyy c(y) = 0, s \in \R, y \in \R^m$.
\end{proof}
\begin{proof} (of Proposition \ref{WVFI})\\
1.$\implies$ 2. By Proposition \ref{VFI} we deduce that $c(\ysy) = \dyy c(y)$ and therefore
\begin{align*}
\inty{(c\cdny u) v_{-s}} & = \inty{c(\ysy) \cdot (\nabla _y u ) (\ysy) v(y)}\\
& = \inty{c(y) \cdot {^t \dyy} (\nabla _y u )(\ysy) v(y) } = \inty{(c(y) \cdot \nabla _y u_s) v(y)}.
\end{align*}
2.$\implies$ 3. Taking the derivative with respect to $s$ of \eqref{Equ41} at $s = 0$, we obtain \eqref{Equ42}.
3.$\implies$ 1. Applying \eqref{Equ42} with $v \in C^1 _c (\R^m)$ and $u _i = y_i \varphi (y)$, $\varphi \in C^2 _c (\R^m)$, $\varphi = 1$ on the support of $v$, yields
\[
\inty{c_i \;b \cdny v } + \inty{c \cdny b_i \;v (y)}= 0
\]
saying that $b \cdny c_i = (\dyb \;c) _i$ in $\dpri{}$, $i \in \{1,...,m\}$ and thus $[b,c] = b \cdny c - \dyb c = 0$ in $\dpri{}$.
\end{proof}
\begin{proof} (of Proposition \ref{MFI}) The arguments are very similar to those in the proof of Proposition \ref{VFI}. Let us give the main lines. We assume that $b$ is divergence free, for simplicity. Let $A(y)$ be a matrix field satisfying \eqref{Equ35}. For any matrix field $U \in C^1 _c (\R^m)$ we have
\begin{align*}
\inty{A(y) & : ( U(Y(-h;y)) - U(y) )} = \inty{(A(Y(h;y)) - A(y)) : U(y) } \\
& = \inty{( \partial _y Y (h;y) A(y) {^t \partial _y Y (h;y)} - A(y)) : U(y)} \\
& = \inty{\{( \partial _y Y (h;y) - I ) A(y) {^t \partial _y Y (h;y)} : U(y) + A(y) {^t (\partial _y Y (h;y) - I)} : U(y)\}}.
\end{align*}
Multiplying by $\frac{1}{h}$ and passing $h \to 0$ we obtain
\[
- \inty{A(y) : ( b \cdny U ) } = \inty{(\dyb A(y) + A(y) {^t \dyb }):U(y) }
\]
saying that $[b,A] = 0$ in $\dpri{}$.
For the converse implication define, as before
\[
f(s,y) = A(\ysy ) - \dyy A(y) {^t \dyy},\;\;s \in \R,\;\;y \in \R^m.
\]
For any $U \in C^1 _c (\R^m)$ we have
\begin{align*}
F_U (s)& := \inty{f(s,y) : U(y)} \\
&= \inty{A(y) : U(Y(-s;y))} - \inty{\dyy A(y) {^t \dyy } : U (y)}
\end{align*}
and thus
\begin{align*}
\frac{\md }{\md s} F_U (s) & = - \inty{A(y) : (\;(b \cdny )U\;)_{-s}} - \inty{\partial _y ( b (\ysy)) A(y) {^t \dyy } : U(y)} \\
& - \inty{\dyy A(y) {^t \partial _y ( b (\ysy ))} : U(y)} \\
& = - \inty{A(y) : (b \cdny ) U_{-s} } - \inty{\dyb (\ysy) \dyy A(y) {^t \dyy} : U}\\
& - \inty{\dyy A(y) {^t \dyy} {^t \dyb (\ysy) } : U(y)}\\
& = \inty{ \{ \dyb (\ysy) f(s,y) + f(s,y) {^t \dyb (\ysy)}\} : U(y)} \\
& = \inty{f(s,y) : \{ {^t \dyb (\ysy)} U(y) + U(y) \dyb (\ysy) \}}.
\end{align*}
The previous equality still holds true for $U \in C_c (\R^m)$, and our conclusion follows as in the proof of Proposition \ref{VFI}, by Gronwall lemma.
\end{proof}
\begin{proof} (of Proposition \ref{WMFI})\\
$1.\implies 2.$ By Proposition \ref{MFI} we deduce that $A(\ysy) = \dyy A(y) {^t \dyy}$. Using the change of variable $y \to \ysy$ one gets
\begin{align*}
\inty{A(y) \nabla _y u \cdot \nabla _y v } & = \inty{A(\ysy) (\nabla _y u )(\ysy) \cdot (\nabla _y v ) (\ysy)} \\
& = \inty{A(y) {^t \dyy }(\nabla _y u ) (\ysy) \cdot {^t \dyy } (\nabla _y v ) (\ysy) } \\
& = \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s }.
\end{align*}
$2.\implies 3.$ Taking the derivative with respect to $s$ at $s = 0$ of the constant function $s \to \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s}$ yields
\[
\inty{A(y) \nabla _y ( b \cdny u ) \cdot \nabla _y v } + \inty{A(y) \nabla _y u \cdot \nabla _y ( b \cdny v) } = 0.
\]
$3.\implies 2.$ For any $u, v \in C^2 _c (\R^m)$ we can write, thanks to 3. applied with the functions $u_s, v_s$
\begin{align*}
\frac{\md }{\md s} \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s} & = \inty{A(y) \nabla _y ( \;( b \cdny u)_s )\cdot \nabla _y v_s} \\
& + \inty{A(y) \nabla _y u_s \cdot \nabla _y ( \; (b \cdny v )_s)} \\
& = \inty{A(y) \nabla _y ( b \cdny u_s) \cdot \nabla _y v_s } \\
& + \inty{A(y) \nabla _y u_s \cdot \nabla _y ( b \cdny v_s ) } = 0.
\end{align*}
Therefore the function $s \to \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s}$ is constant on $\R$ and thus
\[
\inty{A(y) \nabla _y u_s \cdot \nabla _y v_s} = \inty{A(y) \nabla _y u \cdot \nabla _y v},\;\;s\in \R.
\]
Up to now, the symmetry of the matrix $A(y)$ did not play any role. We only need it for the implication $2.\implies 1.$\\
$2.\implies 1.$ We have
\begin{align*}
\inty{A(y) \nabla _y u \cdot \nabla _y v } & = \inty{A(y) \nabla _y u_s \cdot \nabla _y v_s } \\
& = \inty{A(y) {^t \dyy} ( \nabla _y u)_s \cdot {^t \dyy } (\nabla _y v )_s } \\
& = \inty{\dyy A(y) {^t \dyy } (\nabla _y u )_s \cdot ( \nabla _y v )_s } \\
& = \inty{(\partial _y Y A \;{^t \partial _y Y})_{-s} \nabla _y u \cdot \nabla _y v }
\end{align*}
where $(\partial _y Y A {^t \partial _y Y})_{-s} = \partial _y Y (s; Y(-s;y)) A(Y(-s;y)) {^t \partial _y Y (s; Y(-s;y))}$. We deduce that
\[
\inty{(A(y) - (\partial _y Y A \;{^t \partial _y Y })_{-s}) \nabla _y u \cdot \nabla _y v} = 0,\;\;u, v \in C^1 _c (\R^m).
\]
Since $A(y) - (\partial _y Y A \;{^t \partial _y Y })_{-s}$ is symmetric, it is easily seen, cf. Lemma \ref{Divergence} below, that $A(y) - (\partial _y Y A \;{^t \partial _y Y })_{-s}= 0$. Therefore we have $A(\ysy) = \dyy A(y) {^t \dyy}$, $s \in \R,y \in \R^m$ and by Proposition \ref{MFI} we deduce that $[b,A] = 0$ in $\dpri{}$.
\end{proof}
\begin{lemma}
\label{Divergence}
Consider a field $A(y) \in \loloc{}$ of symmetric matrix satisfying
\begin{equation}
\label{Equ38} \inty{A(y) \nabla _y u \cdot \nabla _y v} = 0,\;\;u, v \in C^1 _c (\R^m).
\end{equation}
Therefore $A(y) = 0$ a.a. $y \in \R^m$.
\end{lemma}
\begin{proof}
Applying \eqref{Equ38} with $v_j = y_j v$, $v \in C^1 _c (\R^m)$, $u_i = y_i \varphi (y)$ where $\varphi \in C^1 _c (\R^m)$ and $\varphi = 1$ on the support of $v$, yields
\begin{equation}
\label{Equ39}
\inty{A(y) e_i \cdot ( y_j \nabla _y v + v e_j )} = 0.
\end{equation}
Applying \eqref{Equ38} with $v$ and $u_{ij} = y_i y_j \varphi (y)$ one gets
\begin{equation}
\label{Equ40}
\inty{A(y) ( y_j e_i + y_i e _j ) \cdot \nabla _y v } = 0.
\end{equation}
Combining \eqref{Equ39}, \eqref{Equ40} we obtin for any $i, j \in \{1,...,m\}$
\[
2 \inty{(A(y)e_i \cdot e_j) \;v(y)} = \inty{( A(y) e_i \cdot e_j + A(y) e_j \cdot e_i)v(y)} = 0
\]
saying that $A(y) = 0$, a.a. $y \in \R^m$.
\end{proof}
\end{document} | math | 98,374 |
\begin{document}
\begin{center}
{\Large Some remarks about Fibonacci elements in an arbitrary
algebra}
\begin{equation*}
\end{equation*}
Cristina FLAUT and Vitalii SHPAKIVSKYI
\begin{equation*}
\end{equation*}
\end{center}
\textbf{Abstract. }{\small In this paper, we prove some relations between
Fibonacci elements in an arbitrary algebra. Moreover, we define imaginary
Fibonacci quaternions and imaginary Fibonacci octonions and we prove that
always three arbitrary imaginary Fibonacci quaternions are linear
independents and the mixed product of three arbitrary imaginary Fibonacci
octonions is zero.}
\begin{equation*}
\end{equation*}
Keywords: Fibonacci quaternions, Fibonacci octonions, Fibonacci elements.
2000 AMS Subject Classification: 11B83, 11B99.
\begin{equation*}
\end{equation*}
\textbf{1. Introduction}
\begin{equation*}
\end{equation*}
Fibonacci elements over some special algebras were intensively studied in
the last time in various papers, as for example: [Akk; ], [Fl, Sa; 15], [Fl,
Sh; 13(1)], [Fl, Sh; 13(2)], [Ha; ],[Ha1; ],[Ho; 61],[Ho; 63],[Ke; ]. All
these papers studied properties of Fibonacci quaternions, Fibonacci
octonions in Quaternion or Octonion algebras or in generalized Quaternion or
Octonion algebras or studied dual vectors or dual Fibonacci quaternions (
see [Gu;], [Nu; ]).
In this paper, we will prove that some of the obtained identities can be
obtained over an arbitrary algebras. We introduce the notions of imaginary
Fibonacci quaternions and imaginary Fibonacci octonions and we prove, using
the structure of the quaternion algebras \ and octonion algebras, that
always arbitrary three of such elements are linear dependents. For other
details, properties and applications regarding quaternion algebras \ and
octonion algebras, the reader is referred, for example, to [Sc; 54], [Sc;
66], [Fl, St; 09], [Sa, Fl, Ci; 09].
\textbf{2. Fibonacci elements in an arbitrary algebra}
\begin{equation*}
\end{equation*}
Let $A$ be a unitary algebra over $K$ ($K=\mathbb{R},\mathbb{C}$)
with a basis $\{e_{0}=1,e_{1},e_{2},...,e_{n}\}.$ Let $\{f_{n}\}_{n\in
\mathbb{N}}$ be the Fibonacci sequence
\begin{equation*}
f_{n}=f_{n-1}+f_{n-2},n\geq 2,f_{0}=0,f_{1}=1.
\end{equation*}
In algebra $A,$ we define the Fibonacci element as follows:
\begin{equation*}
F_{m}=\overset{n}{\underset{k=0}{\sum }}f_{m+k}e_{k}.
\end{equation*}
\textbf{Proposition 2.1.} \textit{With the above notations, the following
relations hold:}
\textit{1)} $F_{m+2}=F_{m+1}+F_{m};$
\textit{2)} $\overset{p}{\underset{i=0}{\sum }}F_{i}=F_{p+2}-F_{i}.$
\textbf{Proof.} 1) $F_{m+1}+F_{m}=\overset{n}{\underset{k=0}{\sum }}
f_{m+k+1}e_{k}+\overset{n}{\underset{k=0}{\sum }}f_{m+k}e_{k}=\overset{n}{
\underset{k=0}{\sum }}(f_{m+k+1}+f_{m+k})e_{k}=\overset{n}{\underset{k=0}{
\sum }}$ $f_{m+k+2}e_{k}=F_{m+2}.$
2) $\overset{p}{\underset{i=0}{\sum }}F_{i}=F_{1}+F_{2}+...+F_{p}=$\newline
$=\overset{n}{\underset{k=0}{\sum }}f_{k+1}e_{k}+\overset{n}{\underset{k=0}{
\sum }}f_{k+2}e_{k}+...+\overset{n}{\underset{k=0}{\sum }}f_{k+p}e_{k}=$
\newline
$=e_{0}\left( f_{1}+...+f_{p}\right) +e_{1}\left( f_{2}+...+f_{p+1}\right) +$
\newline
$+e_{2}\left( f_{3}+...+f_{p+2}\right) +...+e_{n}\left(
f_{k+n}+...+f_{p+n}\right) =$\newline
$=$ $e_{0}\left( f_{p+2}-1\right) +e_{1}\left( f_{p+3}-1-f_{1}\right)
+e_{2}\left( f_{p+4}-1-f_{1}-f_{2}\right) +$\newline
$+e_{3}\left( f_{p+5}-1-f_{1}-f_{2}-f_{3}\right) +...+e_{n}\left(
f_{p+n+2}-1-f_{1}-f_{2}-...-f_{n}\right) =$\newline
$=F_{p+2}-F_{2}.$
We used the identity $\overset{p}{\underset{i=1}{\sum }}f_{i}=f_{p+2}-1$
(for usual Fibonacci numbers) and $1+f_{1}+f_{2}+...+f_{n}=f_{n+2}.\Box
$
\textbf{Remark 2.2}. The equalities 1, 2 from the above proposition
generalize the corresponding formulae from [Ke; ] [Ha; ] [Nu; ] [Ha1;
].
\textbf{Proposition 2.3.} \textit{We have the following formula (Binet's
fomula):}
\begin{equation*}
F_{m}=\frac{\alpha ^{\ast }\alpha ^{m}-\beta ^{\ast }\beta ^{m}}{\alpha
-\beta },
\end{equation*}
\textit{where} $\alpha =\frac{1+\sqrt{5}}{2},\beta =\frac{1-\sqrt{5}}{2}
,\alpha ^{\ast }=\underset{k=0}{\overset{n}{\sum }}\alpha ^{k}e_{k},~\beta
^{\ast }=\underset{k=0}{\overset{n}{\sum }}\beta ^{k}e_{k}.$
\textbf{Proof.} \ Using the formula for the real quaternions, $f_{m}=\frac{
\alpha ^{m}-\beta ^{m}}{\alpha -\beta },$ we obtain\newline
$F_{m}=\overset{n}{\underset{k=0}{\sum }}f_{m+k}e_{k}=\frac{\alpha
^{m}-\beta ^{m}}{\alpha -\beta }e_{0}+\frac{\alpha ^{m+1}-\beta ^{m+1}}{
\alpha -\beta }e_{1}+\frac{\alpha ^{m+2}-\beta ^{m+2}}{\alpha -\beta }
e_{2}+...+$\newline
$+\frac{\alpha ^{m+n}-\beta ^{m+n}}{\alpha -\beta }e_{n}=\frac{a^{m}}{\alpha
-\beta }\left( e_{0}+\alpha e_{1}+\alpha ^{2}e_{2}+...+\alpha
^{n}e_{n}\right) +$\newline
$+\frac{\beta ^{m}}{\alpha -\beta }\left( e_{0}+\beta e_{1}+\beta
^{2}e_{2}+...+\beta ^{n}e_{n}\right) =\frac{\alpha ^{\ast }\alpha ^{m}-\beta
^{\ast }\beta ^{m}}{\alpha -\beta }.
\Box
$
\textbf{Remark 2.4.} The above result generalizes the Binet formulae from
the papers [Gu;] [Akk; ] [Ke; ] [Ha; ] [Nu; ] [Ha1; ].
\textbf{Theorem 2.5.} \textit{The generating function for the Fibonacci
number over an algebra is of the form}
\begin{equation*}
G\left( t\right) =\frac{F_{0}+\left( F_{1}-F_{0}\right) t}{1-t-t^{2}}.
\end{equation*}
\textbf{Proof.} We consider the generating function of the form
\begin{equation*}
G\left( t\right) =\overset{\infty }{\underset{m=0}{\sum }}F_{m}t^{m}.
\end{equation*}
We consider the product\newline
$G\left( t\right) \left( 1-t-t^{2}\right) =\overset{\infty }{\underset{m=0}{
\sum }}F_{m}t^{m}=$ $\overset{\infty }{\underset{m=0}{\sum }}F_{m}t^{m}-
\overset{\infty }{\underset{m=0}{\sum }}F_{m}t^{m+1}-\overset{\infty }{
\underset{m=0}{\sum }}F_{m}t^{m+2}=$\newline
$=F_{0}+F_{1}t+F_{2}t^{2}+F_{3}t^{3}+...-F_{0}t-F_{1}t^{2}-F_{2}t^{3}-...-$
\newline
$-F_{0}t^{2}-F_{1}t^{3}-F_{2}t^{4}-...=F_{0}+\left( F_{1}-F_{0}\right)
t.\Box
$
\textbf{Remark 2.6.} The above Theorem generalizes results from the papers
[Gu;], [Akk; ], \ [Ke; ], [Ha; ],[Nu; ].
\begin{equation*}
\end{equation*}
\textbf{The Cassini identity}
\begin{equation*}
\end{equation*}
First, we obtain the following identity.
\textbf{Proposition 2.7.}
\begin{equation}
F_{-m}=\left( -1\right) ^{m+1}f_{m}F_{1}+\left( -1\right) ^{m}f_{m+1}F_{0}.
\tag{2.1.}
\end{equation}
\textbf{Proof.} We use induction. \ For $m=1,$ we obtain $
F_{-1}=f_{1}F_{1}-f_{2}F_{0},$ which is true. Now, we assume that it is true
for an arbitrary integer $k$
\begin{equation*}
F_{-k}=\left( -1\right) ^{k+1}f_{k}F_{1}+\left( -1\right) ^{k}f_{k+1}F_{0}
\end{equation*}
For $k+1,$ we obtain\newline
$F_{-(k+1)}=\left( -1\right) ^{k+2}f_{k+1}F_{1}+\left( -1\right)
^{k+1}f_{k+2}F_{0}=$\newline
$=\left( -1\right) ^{k}f_{k}F_{1}+\left( -1\right) ^{k}f_{k-1}F_{1}+\left(
-1\right) ^{k-1}f_{k+1}F_{0}+$\newline
$+\left( -1\right) ^{k-1}f_{k}F_{0}=F_{-\left( n-1\right) }-F_{-n}.$
Therefore, this statement is true.$\Box
$
\textbf{Theorem 2.8.} (Cassini's identity) \textit{With the above notations,
we have the following formula}
\begin{equation*}
F_{m-1}F_{m+1}-F_{m}^{2}=\left( -1\right) ^{m}(F_{-1}F_{1}-F_{0}^{2}).
\end{equation*}
\textbf{Proof.}
We consider\newline
$
F_{m-1}=f_{m-1}e_{0}+f_{m}e_{1}+f_{m+1}e_{2}+f_{m+2}e_{3}+...+f_{m+n-1}e_{n},
$\newline
$
F_{m+1}=f_{m+1}e_{0}+f_{m+2}e_{1}+f_{m+3}e_{2}+f_{m+4}e_{3}+...+f_{m+n+1}e_{n},
$\newline
$F_{m}=f_{m}e_{0}+f_{m+1}e_{1}+f_{m+2}e_{2}+f_{m+2}e_{3}+...+f_{m+n}e_{n}.$
We compute\newline
$F_{m-1}F_{m+1}=$\newline
$=\left[
f_{m-1}f_{m+1}e_{0}^{2}+f_{m-1}f_{m+2}e_{0}e_{1}+f_{m-1}f_{m+3}e_{0}e_{2}+f_{m-1}f_{m+4}e_{0}e_{3}...+f_{m-1}f_{m+n+1}e_{0}e_{n}
\right] +$\newline
$
+[f_{m}f_{m+1}e_{1}e_{0}+f_{m}f_{m+2}e_{1}^{2}+f_{m}f_{m+3}e_{1}e_{2}+f_{m}f_{m+4}e_{1}e_{3}...+f_{m}f_{m+n+1}e_{1}e_{n}]+
$\newline
$
+[f_{m+1}^{2}e_{2}e_{0}+f_{m+1}f_{m+2}e_{2}e_{1}+f_{m+1}f_{m+3}e_{2}^{2}+f_{m+1}f_{m+4}e_{1}e_{3}...+f_{m+1}f_{m+n+1}e_{2}e_{n}]+
$\newline
$
+[f_{m+2}f_{m+1}e_{3}e_{0}+f_{m+2}^{2}e_{3}e_{1}+f_{m+2}f_{m+3}e_{3}e_{2}+f_{m+2}f_{m+4}e_{3}^{2}...+f_{m+2}f_{m+n+1}e_{3}e_{n}]+...+
$\newline
$
+[f_{m+n-1}f_{m+1}e_{n}e_{0}+f_{m+n-1}f_{m+2}e_{n}e_{1}+f_{m+n-1}f_{m+3}e_{n}e_{2}+f_{m+n-1}f_{m+4}e_{n}e_{3}...+f_{m+n-1}f_{m+n+1}e_{n}^{2}].
$
Now, we compute \newline
$F_{m}^{2}=\left[
f_{m}^{2}e_{0}^{2}+f_{m}f_{m+1}e_{0}e_{1}+f_{m}f_{m+2}e_{0}e_{2}+f_{m}f_{m+3}e_{0}e_{3}+...+f_{m}f_{m+n}e_{0}e_{n}
\right] +$\newline
$+$ $\left[
f_{m+1}f_{m}e_{1}e_{0}+f_{m+1}^{2}e_{1}^{2}+f_{m+1}f_{m+2}e_{1}e_{2}+f_{m+1}f_{m+3}e_{1}e_{3}+...+f_{m+1}f_{m+n}e_{1}e_{n}
\right] +$\newline
$+\left[
f_{m+2}f_{m}e_{2}e_{0}+f_{m+2}f_{m+1}e_{2}e_{1}+f_{m+2}^{2}e_{2}^{2}+f_{m+2}f_{m+3}e_{2}e_{3}+...+f_{m+2}f_{m+n}e_{2}e_{n}
\right] +$\newline
$+\left[
f_{m+2}f_{m}e_{2}e_{0}+f_{m+2}f_{m+1}e_{2}e_{1}+f_{m+2}^{2}e_{2}^{2}+f_{m+2}f_{m+3}e_{2}e_{3}+...+f_{m+2}f_{m+n}e_{2}e_{n}
\right] +$\newline
$+\left[
f_{m+3}f_{m}e_{3}e_{0}+f_{m+3}f_{m+1}e_{3}e_{1}+f_{m+3}f_{m+2}e_{3}e_{2}+f_{m+3}^{2}e_{3}^{2}+...+f_{m+3}f_{m+n}e_{3}e_{n}
\right] +...+$\newline
$+\left[
f_{m+n}f_{m}e_{n}e_{0}+f_{m+n}f_{m+1}e_{n}e_{1}+f_{m+n}f_{m+2}e_{n}e_{2}+f_{m+n}f_{m+3}e_{n}e_{3}+...+f_{m+n}^{2}e_{n}^{2}
\right] .$
Consider the difference\newline
$F_{m-1}F_{m+1}-F_{m}^{2}=$\newline
$=e_{0}\left[ e_{0}\left( f_{m-1}f_{m+1}-f_{m}^{2}\right) +e_{1}\left(
f_{m-1}f_{m+2}-f_{m}f_{m+1}\right) +...+e_{n}\left(
f_{m-1}f_{m+n+1}-f_{m}f_{m+n}\right) \right] +$\newline
$+e_{1}\left[ e_{0}\left( f_{m}f_{m+1}-f_{m+1}f_{m}\right) +e_{1}\left(
f_{m}f_{m+2}-f_{m+1}^{2}\right) +...+e_{n}\left(
f_{m}f_{m+n+1}-f_{m+1}f_{m+n}\right) \right] +$\newline
$+e_{2}\left[ e_{0}\left( f_{m+1}^{2}-f_{m+2}f_{m}\right) +e_{1}\left(
f_{m+1}f_{m+2}-f_{m+2}f_{m+1}\right) +...+e_{n}\left(
f_{m+1}f_{m+n+1}-f_{m+2}f_{m+n}\right) \right] +$\newline
$+e_{3}\left[ e_{0}\left( f_{m+2}f_{m+1}-f_{m+3}f_{m}\right) +e_{1}\left(
f_{m+2}^{2}-f_{m+3}f_{m+1}\right) +...+e_{n}\left(
f_{m+2}f_{m+n+1}-f_{m+3}f_{m+n}\right) \right] +...+$\newline
$+e_{n}\left[ e_{0}\left( f_{m+n-1}f_{m+1}-f_{m+n}f_{m}\right) +e_{1}\left(
f_{m+n-1}f_{m+2}-f_{m+n}f_{m+1}\right) +...+e_{n}\left(
f_{m+n-1}f_{m+n+1}-f_{m+n}^{2}\right) \right] .$
Using the formula $f_{i}f_{j}-f_{i+k}f_{j-k}=\left( -1\right)
^{j-k}f_{i+k-j}f_{k}$ (see Koshy, p. 87, formula 2) and the identities $
f_{1}=1,f_{-m}=\left( -1\right) ^{m+1}f_{m}$ (see Koshy, p. 84), we obtain
\newline
$F_{m-1}F_{m+1}-F_{m}^{2}=e_{0}\left( -1\right) ^{m+1}\left[
e_{0}f_{1}+e_{1}f_{2}+e_{2}f_{3}+...+e_{n}f_{n+1}\right] +$\newline
$+e_{1}\left( -1\right) ^{m+1}\left[
e_{0}f_{0}+e_{1}f_{1}+e_{2}f_{2}+...+e_{n}f_{n}\right] +$\newline
$+e_{2}\left( -1\right) ^{m}\left[
e_{0}f_{-1}+e_{1}f_{0}+e_{2}f_{1}+...+e_{n}f_{n-1}\right] +$\newline
$+e_{3}\left( -1\right) ^{m}\left[
e_{0}f_{-2}+e_{1}f_{-1}+e_{2}f_{0}+...+e_{n}f_{n-2}\right] +...+$\newline
$+\left( -1\right) ^{m\ast n}e_{n}\left[
e_{0}f_{-n+1}+e_{1}f_{-n+2}+e_{2}f_{-n+3}+...+e_{n}f_{1}\right] =$\newline
$=\left( -1\right) ^{m}\left(
e_{0}F_{1}-e_{1}F_{0}+e_{2}F_{-1}-e_{3}F_{-2}+...+\left( -1\right)
^{n}e_{n}F_{-n+1}\right) .$
Using Proposition 2.7, we have\newline
$F_{m-1}F_{m+1}-F_{m}^{2}=\left( -1\right)
^{m}[e_{0}F_{1}-e_{1}F_{0}+e_{2}\left( F_{1}-F_{0}\right) -$\newline
$-e_{3}\left( 2F_{0}-F_{1}\right) +e_{4}\left( 2F_{1}-3F_{0}\right)
-e_{5}\left( -3F_{1}+5F_{0}\right) +...+$\newline
$+e_{n}\left( -1\right) ^{n}\left( \left( -1\right) ^{n}f_{n-1}F_{1}+\left(
-1\right) ^{n-1}f_{n}F_{0}\right) ]=$\newline
$=\left( -1\right)
^{m}[(e_{0}f_{-1}+e_{1}f_{0}+e_{2}f_{1}+...+e_{n}f_{n-1})F_{1}-$\newline
$-\left( f_{0}e_{0}+f_{1}e_{1}+f_{2}e_{2}+...+f_{n}e_{n}\right) F_{0}]=$
\newline
$=\left( -1\right) ^{m}\left[ F_{-1}F_{1}-F_{0}^{2}\right] .$ The theorem is
now proved.
\textbf{Remark 2.9. }
i) Similarly, we can prove an analogue of Cassini's formula:
\begin{equation*}
F_{m+1}F_{m-1}-F_{m}^{2}=\left( -1\right) ^{m}\left[ F_{1}F_{-1}-F_{0}^{2}
\right] .
\end{equation*}
ii) Theorem 2.8 generalizes Cassini's formula for all real algebras.
iii) If the algebra $A$ is algebra of the real numbers $\mathbb{R},$ in this
case, we have $F_{m}=f_{m}.$ From the above theorem, it \ results that
\begin{equation*}
f_{m+1}f_{m-1}-f_{m}^{2}=\left( -1\right) ^{m}\left[ f_{1}f_{-1}-f_{0}^{2}
\right] =\left( -1\right) ^{m},
\end{equation*}
which it is the classical Cassini's identity.
\begin{equation*}
\end{equation*}
\textbf{3. Imaginary Fibonacci quaternions and imaginary Fibonacci
octonions }
\begin{equation*}
\end{equation*}
In the following, we will consider a field $K$ with $charK\neq 2,3,$ $V$ a
finite dimensional vector space and $A$ a finite dimensional unitary algebra
over a field $\ K$, associative or nonassociative.
Let $\mathbb{H}\left( \alpha ,\beta \right) $ be the generalized real\
quaternion algebra, the algebra of the elements of the form $a=a_{1}\cdot
1+a_{2}\mathbf{i}+a_{3}\mathit{j}+a_{4}\mathbf{k},$ where $a_{i}\in \mathbb{R
},\mathbf{i}^{2}=-\alpha ,\mathbf{j}^{2}=-\beta ,$ $\mathbf{k}=\mathbf{ij}=-
\mathbf{ji}.$ We denote by $\mathbf{t}\left( a\right) $ and $\mathbf{n}
\left( a\right) $ the trace and the norm of a real quaternion $a.$ The norm
of a generalized quaternion has the following expression $\mathbf{n}\left(
a\right) =a_{1}^{2}+\alpha a_{2}^{2}+\beta a_{3}^{2}+\alpha \beta a_{4}^{2}$
and $\mathbf{t}\left( a\right) =2a_{1}.$ It is known that for $a\in $ $
\mathbb{H}\left( \alpha ,\beta \right) ,$ we have $a^{2}-\mathbf{t}\left(
a\right) a+\mathbf{n}\left( a\right) =0.$ The quaternion algebra $\mathbb{H}
\left( \alpha ,\beta \right) $ is a \textit{division algebra} if for all $
a\in \mathbb{H}\left( \alpha ,\beta \right) ,$ $a\neq 0$ we have $\mathbf{n}
\left( a\right) \neq 0,$ otherwise $\mathbb{H}\left( \alpha ,\beta \right) $
is called a \textit{split algebra}.
Let $\mathbb{O}(\alpha ,\beta ,\gamma )$ be a generalized octonion algebra
over $\mathbb{R},$ with basis $\{1,e_{1},...,e_{7}\},$ the algebra of the
elements of the form $\
a=a_{0}+a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}+a_{4}e_{4}+a_{5}e_{5}+a_{6}e_{6}+a_{7}e_{7}\,
$and the multiplication given in the following table:
\begin{center}
{\footnotesize $
\begin{tabular}{c||c|c|c|c|c|c|c|c|}
$\cdot $ & $1$ & $\,\,\,e_{1}$ & $\,\,\,\,\,e_{2}$ & $\,\,\,\,e_{3}$ & $
\,\,\,\,e_{4}$ & $\,\,\,\,\,\,e_{5}$ & $\,\,\,\,\,\,e_{6}$ & $
\,\,\,\,\,\,\,e_{7}$ \\ \hline\hline
$\,1$ & $1$ & $\,\,\,e_{1}$ & $\,\,\,\,e_{2}$ & $\,\,\,\,e_{3}$ & $
\,\,\,\,e_{4}$ & $\,\,\,\,\,\,e_{5}$ & $\,\,\,\,\,e_{6}$ & $
\,\,\,\,\,\,\,e_{7}$ \\ \hline
$\,e_{1}$ & $\,\,e_{1}$ & $-\alpha $ & $\,\,\,\,e_{3}$ & $-\alpha e_{2}$ & $
\,\,\,\,e_{5}$ & $-\alpha e_{4}$ & $-\,\,e_{7}$ & $\,\,\,\alpha e_{6}$ \\
\hline
$\,e_{2}$ & $\,e_{2}$ & $-e_{3}$ & $-\,\beta $ & $\,\,\beta e_{1}$ & $
\,\,\,\,e_{6}$ & $\,\,\,\,\,e_{7}$ & $-\beta e_{4}$ & $-\beta e_{5}$ \\
\hline
$e_{3}$ & $e_{3}$ & $\alpha e_{2}$ & $-\beta e_{1}$ & $-\alpha \beta $ & $
\,\,\,\,e_{7}$ & $-\alpha e_{6}$ & $\,\,\,\beta e_{5}$ & $-\alpha \beta
e_{4} $ \\ \hline
$e_{4}$ & $e_{4}$ & $-e_{5}$ & $-\,e_{6}$ & $-\,\,e_{7}$ & $-\,\gamma $ & $
\,\,\,\gamma e_{1}$ & $\,\,\gamma e_{2}$ & $\,\,\,\,\,\gamma e_{3}$ \\ \hline
$\,e_{5}$ & $\,e_{5}$ & $\alpha e_{4}$ & $-\,e_{7}$ & $\,\alpha e_{6}$ & $
-\gamma e_{1}$ & $-\,\alpha \gamma $ & $-\gamma e_{3}$ & $\,\alpha \gamma
e_{2}$ \\ \hline
$\,\,e_{6}$ & $\,\,e_{6}$ & $\,\,\,\,e_{7}$ & $\,\,\beta e_{4}$ & $-\,\beta
e_{5}$ & $-\gamma e_{2}$ & $\,\,\,\gamma e_{3}$ & $-\beta \gamma $ & $-\beta
\gamma e_{1}$ \\ \hline
$\,\,e_{7}$ & $\,\,e_{7}$ & $-\alpha e_{6}$ & $\,\beta e_{5}$ & $\alpha
\beta e_{4}$ & $-\gamma e_{3}$ & $-\alpha \gamma e_{2}$ & $\beta \gamma
e_{1} $ & $-\alpha \beta \gamma $ \\ \hline
\end{tabular}
\
$ }
Table 1
\end{center}
The algebra $\mathbb{O}(\alpha ,\beta ,\gamma )$ is non-commutative and
non-associative.
If $a\in \mathbb{O}(\alpha ,\beta ,\gamma ),$ $
a=a_{0}+a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}+a_{4}e_{4}+a_{5}e_{5}+a_{6}e_{6}+a_{7}e_{7}
$ then $\bar{a}
=a_{0}-a_{1}e_{1}-a_{2}e_{2}-a_{3}e_{3}-a_{4}e_{4}-a_{5}e_{5}-a_{6}e_{6}-a_{7}e_{7}
$ is called the \textit{conjugate} of the element $a.$ The scalars $\mathbf{t
}\left( a\right) =a+\overline{a}\in \mathbb{R}$ and
\begin{equation}
\,\mathbf{n}\left( a\right) =a\overline{a}=a_{0}^{2}+\alpha a_{1}^{2}+\beta
a_{2}^{2}+\alpha \beta a_{3}^{2}+\gamma a_{4}^{2}+\alpha \gamma
a_{5}^{2}+\beta \gamma a_{6}^{2}+\alpha \beta \gamma a_{7}^{2}\in \mathbb{R},
\tag{3.1.}
\end{equation}
are called the \textit{trace}, respectively, the \textit{norm} of the
element $a\in $ $A.$ \thinspace It\thinspace \thinspace \thinspace
follows\thinspace \thinspace \thinspace that$\,$\newline
\thinspace \thinspace $a^{2}-\mathbf{t}\left( a\right) a+\mathbf{n}\left(
a\right) =0,\forall a\in A.$The octonion algebra $\mathbb{O}\left( \alpha
,\beta ,\gamma \right) $ is a \textit{division algebra} if for all $a\in
\mathbb{O}\left( \alpha ,\beta ,\gamma \right) ,$ $a\neq 0$ we have $\mathbf{
n}\left( a\right) \neq 0,$ otherwise $\mathbb{O}\left( \alpha ,\beta ,\gamma
\right) $ is called a \textit{split algebra}.
Let \ $V$ be a real vector space of dimension $n$ and $<,>$ be the inner
product. The\textit{\ cross product} on $V$ is a continuos map
\begin{equation*}
X:V^{s}\rightarrow V,s\in \{1,2,...,n\}
\end{equation*}
with the following properties:
1) $<X\left( x_{1},...x_{s}\right) ,x_{i}>=0,i\in \{1,2,...,s\};$
2) $<X\left( x_{1},...x_{s}\right) ,X\left( x_{1},...x_{s}\right) >=\det
\left( <x_{i},x_{j}>\right) .($see [Br; ]$)$
In [Ro; 96], was proved that if $d=\dim _{\mathbb{R}}V,$ therefore $d\in
\{0,1,3,7\}.$(see [Ro; 96], Proposition 3)
The values $0,1,3$ and $7$ for dimensions are obtained from Hurwitz's
theorem, since the real Hurwitz division algebras $\mathcal{H}$ exist only
for dimensions $1,2,4$ and $8$ dimensions. In this situations, the cross
product is obtained from the product of the normed division algebra,
restricting it to imaginary subspace of the algebra $\mathcal{H},$ which can
be of \ dimension $0,1,3$ or $7$.(see [Ja; 74]) It is known that the real
Hurwitz division algebras are only: the real numbers, the complex numbers,
the quaternions and the octonions.
In $\mathbb{R}^{3}$ with the canonical basis $\{i_{1},i_{2},i_{3}\},$ the
cross product of two linearly independent vectors $
x=x_{1}i_{1}+x_{2}i_{2}+x_{3}i_{3}$ and $y=y_{1}i_{1}+y_{2}i_{2}+y_{3}i_{3}$
is a vector, denoted by $x\times y$ and can be expressed computing the
following formal determinant
\begin{equation}
x\times y=\left\vert
\begin{array}{ccc}
i_{1} & i_{2} & i_{3} \\
x_{1} & x_{2} & x_{3} \\
y_{1} & y_{2} & y_{3}
\end{array}
\right\vert . \tag{3.2.}
\end{equation}
The cross product can also be described using the quaternions and the basis $
\{i_{1},i_{2},i_{3}\}$ as a standard basis for $\mathbb{R}^{3}.$ If a vector
$x\in \mathbb{R}^{3}$ has the form $x=x_{1}i_{1}+x_{2}i_{2}+x_{3}i_{3}$ and
is represented as the quaternion $x=x_{1}\mathbf{i}+x_{2}\mathbf{j}+x_{3}
\mathbf{k}$, therefore the cross product of two vectors has the form $
x\times y=xy+<x,y>,$ where $<x,y>=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$ is the
inner product.
A cross product for 7-dimensional vectors can be obtained in the same way by
using the octonions instead of the quaternions. If $x=\underset{i=0}{\overset
{7}{\sum }}x_{i}e_{i}$ and $y=\underset{i=0}{\overset{7}{\sum }}y_{i}e_{i}$
are two imaginary octonions, therefore
\begin{eqnarray*}
x\times y
&=&(x_{2}y_{4}-x_{4}y_{2}+x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})
\,e_{1}+ \\
&&+(x_{3}y_{5}-x_{5}y_{3}+x_{4}y_{1}-x_{1}y_{4}+x_{6}y_{7}-x_{7}y_{6})
\,e_{2}+ \\
&&+(x_{4}y_{6}-x_{6}y_{4}+x_{5}y_{2}-x_{2}y_{5}+x_{7}y_{1}-x_{1}y_{7})
\,e_{3}+ \\
&&+(x_{5}y_{7}-x_{7}y_{5}+x_{6}y_{3}-x_{3}y_{6}+x_{1}y_{2}-x_{2}y_{1})
\,e_{4}+ \\
&&+(x_{6}y_{1}-x_{1}y_{6}+x_{7}y_{4}-x_{4}y_{7}+x_{2}y_{3}-x_{3}y_{2})
\,e_{5}+ \\
&&+(x_{7}y_{2}-x_{2}y_{7}+x_{1}y_{5}-x_{5}y_{1}+x_{3}y_{4}-x_{4}y_{3})
\,e_{6}+ \\
&&+(x_{1}y_{3}-x_{3}y_{1}+x_{2}y_{6}-x_{6}y_{2}+x_{4}y_{5}-x_{5}y_{4})
\,e_{7},
\end{eqnarray*}
\begin{equation}
\tag{3.3.}
\end{equation}
see [Si; 02].
Let $\mathbb{H}$ be the real division quaternion algebra and $\mathbb{H}
_{0}=\{x\in \mathbb{H}$ $\mid $ $\mathbf{t}\left( x\right) =0\}.$ An element
$F_{n}\in \mathbb{H}_{0}$ is called an\textit{\ imaginary Fibonacci
quaternion element} if it is of the form $F_{n}=f_{n}\mathbf{i}+f_{n+1}
\mathbf{j}+f_{n+2}\mathbf{k,}$ where $\left( f_{n}\right) _{n\in \mathbb{N}}$
is the Fibonacci numbers sequence$.$ Let $F_{k},F_{m},F_{n}$ be three
imaginary Fibonacci quaternions. Therefore, we have the following result.
In the proof of the following results, we will use some relations between
Fibonacci numbers, namely:
\textit{D'Ocagne's identity}
\begin{equation}
f_{m}f_{n+1}-f_{n}f_{m+1}=\left( -1\right) ^{n}f_{m-n} \tag{3.4.}
\end{equation}
see relation (33) from [Wo], and
\textit{Johnson's identity}
\begin{equation}
f_{a}f_{b}-f_{c}f_{d}=\left( -1\right) ^{r}\left(
f_{a-r}f_{b-r}-f_{c-r}f_{d-r}\right) , \tag{3.5.}
\end{equation}
for arbitrary integers $a,b,c,d,$ and $r$ with $a+b=c+d,$ see relation (36)
from [Wo].
\textbf{Proposition 3.1.} \textit{With the above notations, for three
arbitrary Fibonacci imaginary quaternions, } \textit{we have}
\begin{equation*}
<F_{k}\times F_{m},F_{n}>=0.
\end{equation*}
\textit{Therefore, the vectors} $F_{k},F_{m},F_{n}$ \textit{are linear
dependents}.
The above result is similar with the result for dual Fibonacci vectors
obtained in [Gu;], Theorem 11.
Let $\mathbb{O}$ be the real division octonion algebra and $\mathbb{O}
_{0}=\{x\in \mathbb{H}$ $\mid $ $\mathbf{t}\left( x\right) =0\}.$ An element
$F_{n}\in \mathbb{O}_{0}$ is called an \textit{imaginary Fibonacci octonion
element} if it is of the form $
F_{n}=f_{n}e_{1}+f_{n+1}e_{1}+f_{n+2}e_{1}+f_{n+3}e_{1}+f_{n+4}e_{1}+f_{n+5}e_{1}+f_{n+6}e_{1}
\mathbf{,}$ where $\left( f_{n}\right) _{n\in \mathbb{N}}$ is the Fibonacci
numbers sequence$.$ Let $F_{k},F_{m},F_{n}$ be three imaginary Fibonacci
octonions.
\
\textbf{Proposition 3.2.} \textit{With the above notations, for three
arbitrary Fibonacci imaginary octonions, we have}
\begin{equation*}
<F_{k}\times F_{m},F_{n}>=0.
\end{equation*}
\qquad
\textbf{Proof.} Using formulae ($3.3$), $\left( 3.4\right) $ and $\left(
3.5\right) $, we will compute
$F_{k}\times F_{m}.$ \newline
The coefficient of \ $e_{1}$ is\newline
$f_{m+2}f_{k+4}-f_{k+2}f_{m+4}+f_{m+3}f_{k+7}-f_{k+3}f_{m+7}+f_{m+5}f_{k+6}-$
$f_{k+5}f_{m+6}=$\newline
$=f_{m}f_{k+2}-f_{k}f_{m+2}-f_{m}f_{k+4}+f_{k}f_{m+4}-f_{m}f_{k+1}+$ $
f_{k}f_{m+1}=$\newline
$=f_{m}\left( f_{k+2}-f_{k+4}-f_{k+1}\right) +f_{k}\left(
-f_{m+2}+f_{m+4}+f_{m+1}\right) =$\newline
$=f_{m}\left( f_{k}-f_{k+4}\right) +f_{k}\left( f_{m+4}-f_{m}\right) =$
\newline
$=-f_{m}\left( 3f_{k+1}+f_{k}\right) +f_{k}\left( 3f_{m+1}+f_{m}\right) =$
\newline
$=-3\left( f_{m}f_{k+1}-f_{k}f_{m+1}\right) =-3\left( -1\right) ^{k}f_{m-k}.$
\newline
The coefficient of $e_{2}$ is\newline
$f_{m+3}f_{k+5}-f_{k+3}f_{m+5}+f_{m+4}f_{k+1}-f_{k+4}f_{m+1}+f_{m+6}f_{k+7}-$
$f_{k+6}f_{m+7}=$\newline
$=$ $-f_{m}f_{k+2}+f_{k}f_{m+2}-f_{m+3}f_{k}+f_{k+3}f_{m}+f_{m}f_{k+1}-$ $
f_{k}f_{m+1}=$\newline
$=f_{m}\left( -f_{k+2}+f_{k+3}+f_{k+1}\right) +f_{k}\left(
f_{m+2}-f_{m+3}-f_{m+1}\right) =$\newline
$=2\left( f_{m}f_{k+1}-f_{k}f_{m+1}\right) =2\left( -1\right) ^{k}f_{m-k}.$
\newline
The coefficient of $e_{3}$ is \newline
$f_{m+4}f_{k+6}-f_{m+3}f_{k+5}+f_{m+5}f_{k+2}-f_{m+2}f_{k+5}+f_{m+7}f_{k+1}-$
$f_{k+7}f_{m+1}=$\newline
$
=f_{m}f_{k+2}-f_{m+2}f_{k}+f_{m+3}f_{k}-f_{m}f_{k+3}-f_{m+6}f_{k}+f_{m}f_{k+6}=
$\newline
$=f_{m}\left( f_{k+2}-f_{k+3}+f_{k+6}\right) +f_{k}\left(
-f_{m+2}+f_{m+3}-f_{m+6}\right) =$\newline
$=7\left( f_{m}f_{k+1}-f_{k}f_{m+1}\right) =7\left( -1\right) ^{k}f_{m-k}.$
\newline
The coefficient of $e_{4}$ is\newline
$f_{m+5}f_{k+7}-f_{k+5}f_{m+7}+f_{m+6}f_{k+3}-f_{k+6}f_{m+3}+f_{m+1}f_{k+2}-$
$f_{m+2}f_{k+1}=$\newline
$
=-f_{m}f_{k+2}+f_{k}f_{m+2}-f_{m+3}f_{k}+f_{k+3}f_{m}-f_{m}f_{k+1}+f_{k}f_{m+1}=
$\newline
$=f_{m}\left( -f_{k+2}+f_{k+3}-f_{k+1}\right) =0.$\newline
The coefficient of $e_{5}$ is\newline
$f_{m+6}f_{k+1}-f_{k+6}f_{m+1}+f_{m+7}f_{k+4}-f_{k+7}f_{m+4}+f_{m+2}f_{k+3}-$
$f_{k+2}f_{m+3}=$\newline
$
=-f_{m+5}f_{k}+f_{k+5}f_{m}+f_{m+3}f_{k}-f_{k+3}f_{m}+f_{m}f_{k+1}-f_{k}f_{m+1}=
$\newline
$=f_{m}\left( f_{k+5}-f_{k+3}+f_{k+1}\right) +f_{k}\left(
-f_{m+5}+f_{m+3}-f_{m+1}\right) =$\newline
$=4\left( f_{m}f_{k+1}-f_{k}f_{m+1}\right) =4\left( -1\right) ^{k}f_{m-k}.$
\newline
The coefficient of $e_{6}$ is\newline
$f_{m+7}f_{k+2}-f_{k+7}f_{m+2}+f_{m+1}f_{k+5}-f_{k+1}f_{m+5}+f_{m+3}f_{k+4}-$
$f_{k+3}f_{m+4}=$\newline
$=f_{m+5}f_{k}-f_{k+5}f_{m}-f_{m}f_{k+4}+f_{k}f_{m+4}-f_{m}f_{k+1}+$ $
f_{k}f_{m+1}=$\newline
$=f_{m}\left( -f_{k+5}-f_{k+4}-f_{k-1}\right) +f_{k}\left(
f_{k+5}+f_{k+4}+f_{k-1}\right) =$\newline
$=-9\left( f_{m}f_{k+1}-f_{k}f_{m+1}\right) =-9\left( -1\right) ^{k}f_{m-k}.$
\newline
The coefficient of $e_{7}$ is\newline
$f_{m+1}f_{k+3}-f_{k+1}f_{m+3}+f_{m+2}f_{k+6}-f_{k+2}f_{m+6}+f_{m+4}f_{k+5}-$
$f_{k+4}f_{m+5}=$\newline
$=f_{m}\left( -f_{k+2}+f_{k+4}+f_{k+1}\right) +f_{k}\left(
f_{m+2}-f_{m+4}-f_{m+1}\right) =$\newline
$=3\left( f_{m}f_{k+1}-f_{k}f_{m+1}\right) =3\left( -1\right) ^{k}f_{m-k}.$
\newline
We obtain that\newline
$F_{k}\times F_{m}=\left( -1\right) ^{k}f_{m-k}\left(
-3e_{1}+2e_{2}+7e_{3}+4e_{5}-9e_{6}+3e_{7}\right) .$ \newline
Therefore\newline
$<F_{k}\times F_{m},F_{n}>=\left( -1\right) ^{k}f_{m-k}\left(
-3f_{n+1}+2f_{n+2}+7f_{n+3}+4f_{n+5}-9f_{n+6}+3f_{n+7}\right) =$\newline
$=-2f_{n+2}+2f_{n+1}+2f_{n}=0.$
\begin{equation*}
\end{equation*}
\textbf{References}
\begin{equation*}
\end{equation*}
\newline
\newline
[Akk; ] I Akkus, O Kecilioglu, \textit{Split Fibonacci and Lucas Octonions},
accepted in Adv. Appl. Clifford Algebras.\newline
[Br; ] R. Brown, A. Gray (1967), \textit{Vector cross products}, Commentarii
Mathematici Helvetici \qquad 42 (1)(1967), 222--236.\newline
[Fl, Sa; 15] C. Flaut, D. Savin, \textit{Quaternion Algebras and Generalized
Fibonacci-Lucas Quaternions}, accepted in Adv. Appl. Clifford Algebras
\newline
[Fl, Sh; 13(1)] C. Flaut, V. Shpakivskyi,\textit{\ Real matrix
representations for the complex quaternions}, Adv. Appl. Clifford Algebras,
23(3)(2013), 657-671.\newline
[Fl, Sh; 13(2)] Cristina Flaut and Vitalii Shpakivskyi, \textit{On
Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions}, Adv.
Appl. Clifford Algebras, 23(3)(2013), 673-688.\newline
[Fl, St; 09] C. Flaut, M. \c{S}tef\~{a}nescu, \textit{Some equations over
generalized quaternion and octonion division algebras}, Bull. Math. Soc.
Sci. Math. Roumanie, 52(100), no. 4 (2009), 427-439.\newline
[Gu;] I. A. Guren, S.K. Nurkan, \textit{A new approach to Fibonacci, Lucas
numbers and dual vectors}, accepted in Adv. Appl. Clifford Algebras.\newline
[Ha; ] S. Halici, On Fibonacci Quaternions, Adv. in Appl. Clifford Algebras,
22(2)(2012), 321-327.\newline
[Ha1; ] S Halici, \textit{On dual Fibonaci quaternions}, Selcuk J. Appl
Math, accepted.\newline
[Ho; 61] A. F. Horadam, \textit{A Generalized Fibonacci Sequence}, Amer.
Math. Monthly, 68(1961), 455-459.\newline
[Ho; 63] A. F. Horadam, \textit{Complex Fibonacci Numbers and Fibonacci
Quaternions}, Amer. Math. Monthly, 70(1963), 289-291.\newline
[Ja; 74] Nathan Jacobson (2009). \textit{Basic algebra I}, Freeman 1974 2nd
ed., 1974, p. 417--427.\newline
[Ke; ] O Kecilioglu, I Akkus, \textit{The Fibonacci Octonions,} accepted in
Adv. Appl. Clifford Algebras.\newline
[Ro; 96] M. Rost, \textit{On the dimension of a composition algebra}, Doc.
Math. J.,1(1996), 209-214.\newline
[Nu; ] S.K. Nurkan, I.A. Guren, \textit{Dual Fibonacci quaternions},
accepted in Adv. Appl. Clifford Algebras.\newline
[Sa, Fl, Ci; 09] D. Savin, C. Flaut, C. Ciobanu, \textit{Some properties of
the symbol algebras, Carpathian Journal of Mathematics}, 25(2)(2009), p.
239-245.\newline
[Si; 02] Z. K. Silagadze, \textit{Multi-dimensional vector product}, arxiv
\newline
[Sc; 66] R. D. Schafer, \textit{An Introduction to Nonassociative Algebras},
Academic Press, New-York, 1966.\newline
[Sc; 54] R. D. Schafer, \textit{On the algebras formed by the Cayley-Dickson
process}, Amer. J. Math. 76 (1954), 435-446.\newline
[Sm; 04] W.D.Smith, \textit{Quaternions, octonions, and now, 16-ons, and
2n-ons; New kinds of numbers,\newline
}[Sw; 73] M. N. S. Swamy, \textit{On generalized Fibonacci Quaternions}, The
Fibonacci Quaterly, 11(5)(1973), 547-549.\newline
[Wo] http://mathworld.wolfram.com/FibonacciNumber.html
\begin{equation*}
\end{equation*}
Cristina FLAUT
Faculty of Mathematics and Computer Science,
Ovidius University,
Bd. Mamaia 124, 900527, CONSTANTA,
ROMANIA
http://cristinaflaut.wikispaces.com/
http://www.univ-ovidius.ro/math/
e-mail:
[email protected]
cristina [email protected]
Vitalii SHPAKIVSKYI
Department of Complex Analysis and Potential Theory
Institute of Mathematics of the National Academy of Sciences of Ukraine,
3, Tereshchenkivs'ka st.
01601 Kiev-4
UKRAINE
http://www.imath.kiev.ua/
e-mail: [email protected]
\begin{equation*}
\end{equation*}
\end{document} | math | 29,070 |
\begin{document}
\title{Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller--Segel type models}
\author{M. Di Francesco and D. Donatelli}
\maketitle
\begin{abstract}
In this paper we deal with diffusive relaxation limits of
nonlinear systems of Euler type modeling chemotactic movement of
cells toward Keller--Segel type systems. The approximating systems
are either hyperbolic--parabolic or hyperbolic--elliptic. They all
feature a nonlinear pressure term arising from a \varepsilonmph{volume
filling effect} which takes into account the fact that cells do
not interpenetrate. The main convergence result relies on
compensated compactness tools and is obtained for large initial
data under suitable assumptions on the approximating solutions. In
order to justify such assumptions, we also prove an existence
result for initial data which are small perturbation of a constant
state. Such result is proven via classical Friedrichs's
symmetrization and linearization. In order to simplify the
coverage, we restrict to the two--dimensional case with periodical
boundary conditions.
\varepsilonnd{abstract}
\section{Introduction}
This paper deals with diffusive relaxation limits for the
nonlinear hyperbolic model describing chemotactic movement of
cells, also known as the \varepsilonmph{persistence and chemotaxis} model,
\begin{equation}\label{eq:beforescaling_intro}
\begin{cases}
\partial_{\tau}\rho +\nabla\cdot (\rho v)=0 & \\
\partial_{\tau} v + v\cdot \nabla v + \nabla
g(\rho)=\nabla c - d v & \\
\sigma\partial_{\tau} c = \Delta c + \alpha \rho -\beta c,
\varepsilonnd{cases}
\varepsilonnd{equation}
with $\alpha, \beta, d,\sigma$ positive constants. The function
$g(\rho)$ is taken of the form $g(\rho)=\rho^\gamma$ with
$\gamma>0$, we shall discuss this choice later on in this
introduction. The model (\ref{eq:beforescaling_intro}) has been
introduced and motivated very precisely in
\cite{ambrosi_gamba_serini}, whereas similar models have been also
discussed in
\cite{preziosi_et_al,kowalczyk_gamba_preziosi,dolak_hillen,filbet_laurencot_perthame}.
We shall briefly summarize the biological motivations behind
(\ref{eq:beforescaling_intro}) by framing them in the general
context of PDE systems describing chemotactical phenomena.
The analysis of partial differential equations modeling chemotaxis
goes back to the work of Keller and Segel \cite{KS70}, who
proposed a macroscopic model for aggregation of cellular slime
molds, and to the earlier related work of Patlak \cite{Pat53}, who
derived similar models with applications to the study of
long-chain polymers. In the successive decades, the term
\varepsilonmph{chemotaxis} has been used to represent the dynamics of
several biological systems (such as the bacteria Escherichia Coli,
or the amoebae Dyctiostelium Discoideum, or endothelial cells of
the human body responding to angiogenic factors secreted by a
tumor) in which the motion of a species is biased by the gradient
of a certain chemical substance. Typically these models consist of
a system of drift--diffusion equations of the form
\begin{equation}\label{eq:limit_intro}
\begin{cases}
\rho_t = \Delta \rho -\nabla\cdot \left(\rho\chi(\rho,c)\nabla
c\right) & \\
c_t =\Delta c +r(\rho,c), &
\varepsilonnd{cases}
\varepsilonnd{equation}
with diffusion terms modeling random motion for the density $\rho$
of the individuals (cells, bacteria and so on) and for the
concentration $c$ of the \varepsilonmph{chemoattractant} (the chemical
substance responsible of the chemotactical movement), first order
drift terms modeling chemotactical aggregation and zero order
reaction terms in the equation for the chemoattractant. The
coefficient $\chi(\rho,c)$ is called \varepsilonmph{chemotactical
sensitivity}. The simplest model combining diffusion and
chemotaxis is the well known parabolic--elliptic
Patlak--Keller--Segel system (or simply Keller--Segel system)
\begin{equation}\label{KeSeintro}
\begin{cases}
\rho_t = \Delta \rho -\nabla\cdot \left(\chi \rho\nabla
c\right) & \\
0=\Delta c +\rho, &
\varepsilonnd{cases}
\varepsilonnd{equation}
which has the interesting mathematical feature of producing smooth
solutions for small initial norms (in the appropriate space) and
blow-up (in the form of concentration to deltas) for large initial
norms. The rigorous analysis of such mathematical issue (also
extended to fully parabolic systems and to more complex models)
has attracted the attention of many mathematicians in the last
decades. We mention the pioneering works of J\"{a}ger--Luckaus
\cite{JL92}, Nagai \cite{Nag95}, Herrero--Velazquez \cite{HV96}
among others. The existence vs. blow--up problem in two space
dimensions for the classical Keller--Segel model (\ref{KeSeintro})
has been completely solved in \cite{DP04}, where the authors
proved that if the initial mass is less than a threshold value
$m^*$ (depending on the coefficient $\chi$) then the solution
exists globally in time in $L^1$, whereas if the initial mass is
larger than $m^*$ then the solution blows up in a finite time. We
refer to the surveys \cite{HorstmannI,HorstmannII} for a complete
and detailed description of the literature of this topic.
In the last years, some authors
\cite{kowalczyk,carrillo_calvez,LS06} have proposed variants of
Keller--Segel type models featuring global existence of solutions
no matter how large the initial mass is, obtained by replacing the
linear diffusion term in the equation for the population density
by a \varepsilonmph{degenerate nonlinear diffusion} term with super--linear
growth for large densities. This choice can be motivated by taking
into account the fact that cells do not interpenetrate (that is,
they are full bodies with nonzero volume) and therefore diffusion
is supposed to inhibit singular aggregation effects when the
density is very high. We mention here that other authors proposed
the use of a nonlinear chemotaxis coefficient $\rho\chi(\rho)$
which attains the value zero when the population density $\rho$
reaches a fixed maximal value -- see for instance
\cite{PaHil,BO04,BDiFD06} -- being this choice motivated by the
fact that individuals stop aggregating when the density is too
high. In both cases, in the resultant model overcrowding of cells
(concentration to deltas for the cells density $\rho$) is
prevented independently on any initial parameter.
In the last years, several authors have started to describe
biological systems with chemotaxis from a hydrodynamical point of
view, i. e. via nonlinear hyperbolic systems of Euler type, see in
particular
\cite{ambrosi_gamba_serini,preziosi_et_al,kowalczyk_gamba_preziosi,dolak_hillen,filbet_laurencot_perthame}.
The models obtained are of the form of system
(\ref{eq:beforescaling_intro}), where the chemotactical force
$\nabla c$ in (\ref{eq:beforescaling_intro}) and the pressure
contribute to balance the rate of change of the momentum.
Moreover, our model (\ref{eq:beforescaling_intro}) features a
friction term modeling the drag between cells and the substrate
material (some authors also considered models with a linear
viscous term). In this framework, the nonlinear pressure term
$g(\rho)$ in (\ref{eq:beforescaling_intro}) plays the role of the
diffusive one in the drift--diffusion equation. Therefore, one can
interpret the overcrowding--preventing effect described before
(sometimes referred to as \varepsilonmph{volume filling effect}) by
thinking of the cellular matter as a medium with \varepsilonmph{limited
compressibility}, i. e. closely packed cells exhibit a limited
amount of resistance to compression. In this sense, a reasonable
choice of a pressure $g(\rho)$ is a function of the form
$g(\rho)=\rho^\gamma$, $\gamma>0$. Such an expression also has the
advantage of modeling absence of stresses for low densities (see
\cite{ambrosi_gamba_serini} for a more detailed description).
In this paper we want to contribute to the problem of establishing
a rigorous mathematical link between the system
(\ref{eq:beforescaling_intro}) and several Keller Segel type
models of the form (\ref{eq:limit_intro}) in terms of diffusive
relaxation limits. A typical example of diffusive scaling on the
system (\ref{eq:beforescaling_intro}) that we shall consider (see
subsection \ref{sec:firstscaling}) is the following
\begin{equation*}
d =\frac{1}{\varepsilonpsilon},\qquad \tau=\frac{t}{\varepsilon},\quad
v^\varepsilon(x,t)=\frac{1}{\varepsilon}v\left(x,\frac{t}{\varepsilon}\right)
\varepsilonnd{equation*}
which transforms (\ref{eq:beforescaling_intro}) into the following
rescaled system
\begin{equation}\label{eq:afterscalingintro}
\begin{cases}
\partial_{t}\rho\varepsilone +\nabla\cdot (\rho\varepsilone v\varepsilone)=0 & \\
\varepsilon^2\left[\partial_{t} v\varepsilone + v\varepsilone\cdot \nabla v\varepsilone\right] + \nabla
g(\rho\varepsilone)=\nabla c\varepsilone - v\varepsilone & \\
\varepsilon\partial_{t} c\varepsilone = \Delta c\varepsilone + \alpha \rho\varepsilone -\beta
c\varepsilone.
\varepsilonnd{cases}
\varepsilonnd{equation}
Formally, as $\varepsilon\rightarrow 0$, we expect the solution
$(\rho^\varepsilon,v^\varepsilon,c^\varepsilon)$ to system (\ref{eq:afterscalingintro}) to
behave like the solution $(\rho^{0},u^{0},c^{0})$ to the
drift--diffusion system of Keller--Segel type
\begin{equation}\label{eq:reducedintro}
\begin{cases}
\partial_{t}\rho^{0}+\nabla\cdot (\rho^{0} \nabla (c^{0}-g(\rho^{0}) )=0 & \\
\Delta c^{0} +\alpha \rho^{0} -\beta c^{0}=0, &
\varepsilonnd{cases}
\varepsilonnd{equation}
where the loss of the persistence term in the equation for the
momentum yields a constitutive law for the velocity $v^{0}=\nabla
c^{0}- \nabla g(\rho^{0})$ (which can be considered as an equivalent
of the Darcy law in \cite{MM90}).
A way to understand the meaning of this phenomena is to consider
it as the large time behaviour of dissipative nonlinear hyperbolic
systems and to look at the asymptotic profile as the relaxed
equilibrium. This is the case for many relevant situations in
mathematical physics and applied mathematics. Singular limits with
a structure similar to \varepsilonqref{eq:afterscalingintro} have been
analyzed by Marcati and Milani \cite{MM90}. In that paper they
investigate the porous media flow as the limit of the Euler
equation in $1-D$, later generalized by Marcati and Rubino
\cite{MR00} to the $2-D$ case. Relaxation phenomena of the same
nature appear also in the zero relaxation limits for the
Euler-Poisson model for semiconductors devices and they were
investigated by Marcati and Natalini \cite{MN95a, MN95b} in the
$1-D$ case and by Lattanzio and Marcati \cite{lattanzio_marcati}
in the multi-D case. For a general overview of the theory of the
singular limits see the survey \cite{DM02} and the paper
\cite{DM04}, where the theory is completely set up.
To perform the relaxation limit we follow the same techniques
developed in \cite{MM90,MR00,DM04} (among others), which are
crucially based on the method of compensated compactness of Tartar
and Murat (see \cite{Tar79,Tar83,Mur78}) combined with the Young
measures associated to the relaxing sequence $\rho\varepsilone$ (see
\cite{Tar79, DiP83a, DiP83b, DiP85b, DiP85a}). Throughout the
whole paper, we shall restrict ourselves to the case of two space
dimensions, which is also the most treated case in the literature
concerning Keller--Segel type systems. Moreover, for the sake of
simplicity we shall work on the $2$-dimensional torus.
We shall prove that this singular limit can be rigorously justified as far as the new time
variable $\tau$ stays in a bounded interval $[0,T]$ for an
arbitrary $T$ and provided that certain a priori assumptions holds
for the solution to (\ref{eq:afterscalingintro}) (see assumption
\ref{basicassumption} below). These a priori assumptions are usual
in the framework of relaxation limits for nonlinear hyperbolic
systems (see also \cite{lattanzio_marcati}, \cite{MR00}) and they don't include
any smallness assumption on the initial conditions. The rigorous statements of these results are contained in Theorem \ref{tconv}.
In order to produce a nontrivial class of solutions to the
nonlinear hyperbolic system (\ref{eq:beforescaling_intro}) which
relax toward a Keller--Segel type model after a proper rescaling,
we shall also provide an existence theorem for the approximating
system (\ref{eq:afterscalingintro}) and prove the uniform
estimates needed to justify the assumptions
(\ref{basicassumption}) in case of initial densities $\rho_0$
which are small perturbation of an arbitrary non zero constant
state (see Theorem \ref{tpert}). This result is achieved by means
of the classical FRiedrichs' symmetrization technique and by a
linearization argument, see \cite{Fri54,KM81}. We remark that, in
many of the estimates performed here, the pressure term need not
to be of the form $g(\rho)=\rho^\gamma$. Indeed, some of the
estimates proven are still valid if one considers a logarithmic
pressure $g(\rho)=\log \rho$, which corresponds to a linear
diffusion term in the limit problem (\ref{eq:limit_intro}) (this
fact is not in contradiction with the blow--up of the density in
the limit problem with linear diffusion, see the Remark
\ref{remblowup}). However, while considering the alternative
scaling introduced in subsection \ref{sec:scalingpoisson} (where
the limit is the classical Keller--Segel system
(\ref{KeSeintro})), such an expression for the pressure seems to
be essential in order to achieve the needed estimates no matter
how large the initial mass is, in a very similar fashion to what
happens in \cite{carrillo_calvez}. We remark that our convergence
results hold on an arbitrary time interval. Therefore, at least in
the case of the second scaling treated in \ref{sec:scalingpoisson}
(where the expression $g(\rho)=\rho^\gamma$ is crucial in order to
achieve the desired estimates), our result can be seen as a new
interpretation of the overcrowding--preventing effect due to the
power--like expression of the pressure. More precisely, the global
smoothness of the limit density $\rho^0$ (and the absence of
concentration to deltas for all times of $\rho^0$ as a byproduct)
can be obtained as a consequence of our relaxation result,
alternatively to the more direct proof developed in
\cite{kowalczyk,carrillo_calvez}.
The paper is organized as follows. In chapter \ref{chap:prelim} we
state the three different scalings we shall deal with. In chapter
\ref{chapest} we perform the uniform estimate we need in order to
prove the main convergence theorem. In chapter \ref{chapconv} we
prove the main convergence theorem for large data under the a
priori assumption \ref{basicassumption} by means of compensated
compactness and Minty's argument. In chapter \ref{chappert} we
prove an existence theorem for the approximating rescaled system
in order to provide a class of solutions satisfying the basic
assumptions (\ref{basicassumption}).
\section{Preliminaries and rescalings}\label{chap:prelim}
We consider the following \varepsilonmph{persistence and chemotaxis} model
\begin{equation}\label{eq:beforescaling}
\begin{cases}
\partial_{\tau}\rho +\nabla\cdot (\rho v)=0 & \\
\partial_{\tau} v + v\cdot \nabla v + \nabla
g(\rho)=\nabla c - d v & \\
\sigma\partial_{\tau} c = \Delta c + \alpha \rho -\beta c
\varepsilonnd{cases}
\varepsilonnd{equation}
where $\alpha, \beta, d,\sigma$ are nonnegative constants. The
system \varepsilonqref{eq:beforescaling} is endowed with the following
$1$--periodic initial data
\begin{equation*}
\rho(0,x)=\rho_{0}(x), \qquad v(0,x)=v_{0}(x), \qquad c(0,x)=c_{0}(x)
\varepsilonnd{equation*}
The nonlinear function $g(\rho)$ has the form
$$g(\rho)=\rho^{\gamma},\ \hbox{for some}\
\gamma>0.$$
\begin{rem}
\varepsilonmph{The nonlinear function $g(\rho)$ grows faster than
$\kappa^{\ast}\log \rho$, for large $\rho$, where
$\kappa^{\ast}=M/4\pi$ and $M$ is the total mass of $\rho$. More
precisely, there exists $\mathcal{U}>0$, such that
$$\text{for any $\rho\geq \mathcal{U}$ and $\kappa>\kappa^{\ast}$}\quad g(\rho)\geq \kappa\log\rho,$$
(see \cite{carrillo_calvez}). \label{r1}}
\varepsilonnd{rem}
Some of the results contained in the present paper hold in any
space dimension $n$, whereas some of them are true only in the
case $n=2$. In order to simplify the coverage, we shall always
restrict ourselves to the latter case. In the sections
\ref{chap:prelim}, \ref{chapest} and \ref{chapconv} we shall not
deal with the existence theory of (\ref{eq:beforescaling}),
whereas we shall work under the following basic assumption.
\begin{assumption}\label{basicassumption}
There exists a global solution $(\rho,v,c)$ to
(\ref{eq:beforescaling}), smooth enough in order to justify the
estimates contained in section \ref{chapest} and such that
\begin{itemize}
\item [(A1)] the total mass $M=\int \rho dx$ is conserved,
\item [(A2)] $\rho(x,t)\geq k>0$
\item [(A3)] $(\rho, \rho v)\in L^\infty(\mathbb{T}^2 \times [0,+\infty))$.
\varepsilonnd{itemize}
\varepsilonnd{assumption}
Let us now explain in detail the relaxation limits we want to
perform. We shall deal with three different asymptotic regimes for
(\ref{eq:beforescaling}), corresponding to small parameter limits
of three different types of scaling.
\subsection{First scaling: large times and large
damping}\label{sec:firstscaling}
For a fixed constant $\varepsilon>0$ we consider the large damping rate
$d =\frac{1}{\varepsilonpsilon}$ in (\ref{eq:beforescaling}). Then, we
introduce the fast time variable $$\tau=\frac{t}{\varepsilon},$$ and the
new independent variables
\begin{equation}
v^{\varepsilon}(x,t)=\frac{1}{\varepsilon}v(x,\tau),\quad
\rho^{\varepsilon}(x,t)=\rho(x,\tau),\quad c^{\varepsilon}(x,t)=c(x,\tau).
\label{s1}
\varepsilonnd{equation}
Moreover, we fix $\sigma=1$ in the third equation. Then, system
(\ref{eq:beforescaling}) in the new variables reads
\begin{equation}\label{eq:afterscaling}
\begin{cases}
\partial_{t}\rho\varepsilone +\nabla\cdot (\rho\varepsilone v\varepsilone)=0 & \\
\varepsilon^2\left[\partial_{t} v\varepsilone + v\varepsilone\cdot \nabla v\varepsilone\right] + \nabla
g(\rho\varepsilone)=\nabla c\varepsilone - v\varepsilone & \\
\varepsilon\partial_{t} c\varepsilone = \Delta c\varepsilone + \alpha \rho\varepsilone -\beta
c\varepsilone.
\varepsilonnd{cases}
\varepsilonnd{equation}
The formal limit as $\varepsilon\rightarrow 0$ is given by the
parabolic--elliptic system
\begin{equation}\label{eq:reduced}
\begin{cases}
\rho^{0}_t+\nabla\cdot (\rho^{0} \nabla (c^{0}-g(\rho^{0}) )=0 & \\
\Delta c^{0} +\alpha \rho^{0} -\beta c^{0}=0. &
\varepsilonnd{cases}
\varepsilonnd{equation}
\subsection{Second scaling: large time and large damping with Poisson
coupling}\label{sec:scalingpoisson}
A simplified version of (\ref{eq:beforescaling}), namely with
$\beta=0$ and $\sigma=0$, is given by the following system
\begin{equation}\label{eq:beforescaling2}
\begin{cases}
\partial_{\tau}\rho +\nabla\cdot (\rho v)=0 & \\
\partial_{\tau} v + v\cdot \nabla v + \nabla
g(\rho)=\nabla c - \gamma v & \\
0= \Delta c + \alpha \rho.
\varepsilonnd{cases}
\varepsilonnd{equation}
By performing the same scaling as before, namely
\begin{equation}
\tau=\frac{t}{\varepsilon},\quad v^{\varepsilon}(x,t)=\frac{1}{\varepsilon}v(x,\tau),\quad
\rho^{\varepsilon}(x,t)=\rho(x,\tau),\quad c^{\varepsilon}(x,t)=c(x,\tau),
\label{s2}
\varepsilonnd{equation}
and by
putting $d=\frac{1}{\varepsilonpsilon}$, we obtain
\begin{equation}\label{eq:afterscaling2}
\begin{cases}
\partial_{t}\rho\varepsilone +\nabla\cdot (\rho\varepsilone v\varepsilone)=0 & \\
\varepsilon^2\left[\partial_{t} v\varepsilone + v\varepsilone\cdot \nabla v\varepsilone\right] + \nabla
g(\rho\varepsilone)=\nabla c\varepsilone - v\varepsilone & \\
0= \Delta c\varepsilone + \alpha \rho\varepsilone.
\varepsilonnd{cases}
\varepsilonnd{equation}
The formal limit as $\varepsilon\rightarrow 0$ leads to the usual
Keller--Segel model with nonlinear diffusion (see
\cite{carrillo_calvez})
\begin{equation}\label{eq:reduced2}
\begin{cases}
\rho^{0}_t+\nabla\cdot (\rho^{0} \nabla (c^{0}-g(\rho^{0}) )=0 & \\
\Delta c^{0} +\alpha \rho^{0} =0. &
\varepsilonnd{cases}
\varepsilonnd{equation}
\subsection{Third scaling: diffusive scaling with small reaction
rates}\label{sec:thirdscaling}
Starting once again by (\ref{eq:beforescaling}), we consider the
case $\sigma=d=1$. We consider $\varepsilonpsilon$--depending reaction
coefficients $\alpha$ and $\beta$, namely we require
$\alpha=\varepsilonpsilon\widetilde{\alpha}$ and
$\beta=\varepsilonpsilon\widetilde{\beta}$ for fixed $\widetilde{\alpha},
\widetilde{\beta}>0$. We then perform the diffusive scaling
\begin{equation}
\tau=\frac{t}{\varepsilonpsilon^2},\quad y=\frac{x}{\varepsilonpsilon},\quad v^{\varepsilon}(x,t)=\frac{1}{\varepsilon}v(x,\tau),\quad
\rho^{\varepsilon}(x,t)=\rho(x,\tau),\quad c^{\varepsilon}(x,t)=c(x,\tau).
\label{s3}
\varepsilonnd{equation}
This
leads to the rescaled system
\begin{equation}\label{eq:afterscaling3}
\begin{cases}
\partial_{t}\rho\varepsilone +\nabla\cdot (\rho\varepsilone v\varepsilone)=0 & \\
\varepsilon^2\left[\partial_{t} v\varepsilone + v\varepsilone\cdot \nabla v\varepsilone\right] + \nabla
g(\rho\varepsilone)=\nabla c\varepsilone - v\varepsilone & \\
\partial_t c\varepsilone= \Delta c\varepsilone + \widetilde{\alpha} \rho\varepsilone
-\widetilde{\beta}c.
\varepsilonnd{cases}
\varepsilonnd{equation}
Therefore, the formal limit as $\varepsilonpsilon\rightarrow 0$ is given in
this case by the following fully parabolic model (we drop the \
$\widetilde{}$\, symbol for simplicity)
\begin{equation}\label{eq:reduced3}
\begin{cases}
\rho^{0}_t+\nabla\cdot (\rho^{0} \nabla (c^{0}-g(\rho^{0}) )=0 & \\
c^{0} _t =\Delta c^{0} +\alpha \rho^{0} -\beta c^{0}.&
\varepsilonnd{cases}
\varepsilonnd{equation}
\begin{rem}
\label{r2} \varepsilonmph{From the hypotheses (A3) and the scalings
\varepsilonqref{s1}, \varepsilonqref{s2}, \varepsilonqref{s3}, it follows that the sequences
$\{\rho\varepsilone\}$, $\{\varepsilon\rho\varepsilone v\varepsilone\}$ are uniformly bounded in
$L^{\infty}(\mathbb{T}^{2}\times [0,+\infty))$ with respect to $\varepsilon$.}
\varepsilonnd{rem}
\section{Estimates}\label{chapest}
In this section we provide suitable estimates on the solutions of
the three rescaled models (\ref{eq:afterscaling}),
(\ref{eq:afterscaling2}) and (\ref{eq:afterscaling3}). For future
use we define
\begin{equation}\label{generalizedpressure}
P(\rho):=\int_{0}^{\rho}g(n)dn=\frac{1}{\gamma+1}\rho^{\gamma+1}.
\varepsilonnd{equation}
\subsection{First scaling}
We have the following (standard) energy estimate for the rescaled
system (\ref{eq:afterscaling}).
\begin{prop}\label{p1} The following identity is
satisfied for any $t\in[0,T]$, by any solution $(\rho\varepsilone,v\varepsilone,c\varepsilone)$ to
(\ref{eq:afterscaling}):
\begin{align}
& \int_{\mathbb{T}^{2}} \left[\frac{\varepsilon^2}{2}\rho\varepsilone(x,t)|v\varepsilone(x,t)|^2
+P(\rho\varepsilone(x,t))\right]dx+\frac{1}{2}\int_{0}^{t}\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)|v\varepsilone(x,s)|^2dxds\nonumber
\\
& \ =\int_{\mathbb{T}^{2}} \left[\frac{\varepsilon^2}{2}\rho_{0}\varepsilone(x)|v_{0}\varepsilone(x)|^2
+P(\rho_{0}\varepsilone(x))\right]dx+\left(\tilde{K}t+\varepsilon\int_{\mathbb{T}^{2}}\frac{c_{0}\varepsilone(x)}{2}dx\right).
\label{estimate1}
\varepsilonnd{align}
\varepsilonnd{prop}
\proof By multiplying second equation in
(\ref{eq:afterscaling}) by $\rho\varepsilone v\varepsilone$ by using the first equation in (\ref{eq:afterscaling}) and by integration by parts it follows that
\begin{align}
&\frac{d}{dt}\int_{\mathbb{T}^{2}}\left[\frac{\varepsilon^2}{2}\rho\varepsilone(x,t)|v\varepsilone(x,t)|^2
+P(\rho\varepsilone(x,t))\right]dx+\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)|v\varepsilone(x,s)|^2 dx=\notag\\
&\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)v\varepsilone(x,s)\nabla c\varepsilone(x,t)dx\leq \frac{1}{2}\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)|v\varepsilone(x,s)|^2 dx+\frac{1}{2}\|\rho\varepsilone\|_{\infty}\int_{\mathbb{T}^{2}}|\nabla c\varepsilone(x,t)|^{2}dx.
\label{estimate2}
\varepsilonnd{align}
Now, by multiplying the third equation of \varepsilonqref{eq:afterscaling} by $c\varepsilone$ we get for any $\delta>0$
\begin{align}
\frac{d}{dt}\int_{\mathbb{T}^{2}}\frac{\varepsilon}{2}|c\varepsilone(x,t)|^{2}dx&=-\int_{\mathbb{T}^{2}}|\nabla c\varepsilone(x,t)|^{2}dx+\alpha\|\rho\varepsilone\|_{\infty}\left(\frac{|\mathbb{T}^{2}|^{2}}{4\delta}+\delta\int_{\mathbb{T}^{2}}|c\varepsilone(x,t)|^{2}dx\right)\notag\\
&-\beta\int_{\mathbb{T}^{2}}|c\varepsilone(x,t)|^{2}dx
\label{estimate3}
\varepsilonnd{align}
By choosing $\delta<\frac{\beta}{2}$, by integrating in time we obtain, for fixed constant $\tilde{K}$, independent on $\varepsilon$, that $c\varepsilone$ satisfies the following inequality
\begin{equation}
\int_{\mathbb{T}^{2}}\frac{\varepsilon}{2}|c\varepsilone(x,t)|^{2}dx+\frac{\beta}{2}\int_{0}^{t}\int_{\mathbb{T}^{2}}|c\varepsilone(x,t)|^{2}dxds+\int_{0}^{t}
\int_{\mathbb{T}^{2}}|\nabla c\varepsilone(x,t)|^{2}dxds\leq \tilde{K} t+\varepsilon\int_{\mathbb{T}^{2}}\frac{|c_{0}\varepsilone(x)|^{2}}{2}dx.
\label{estimate4}
\varepsilonnd{equation}
The estimate \varepsilonqref{estimate1} follows now by using together \varepsilonqref{estimate2} with \varepsilonqref{estimate4} and by taking into account the hypothesis (A3).
\varepsilonndproof
\begin{cor}
\label{c1}
Let $(\rho\varepsilone,v\varepsilone,c\varepsilone)$ be a solution to (\ref{eq:afterscaling})
satisfying assumption \ref{basicassumption} and with initial datum
$(\rho\varepsilone_0,v\varepsilone_0,c\varepsilone_0)$ satisfying
\begin{equation}\label{i1}
\int_{\mathbb{T}^{2}}\left[\frac{\varepsilon^2}{2}\rho\varepsilone_0|v\varepsilone_0|^2dx + (\rho\varepsilone_0)^{\gamma+1}+\varepsilon|c\varepsilone_0|^2\right]dx\quad\hbox{uniformly bounded w.r.t.}\
\varepsilon\ll1.
\varepsilonnd{equation}
Then, for all $T>0$,
\begin{align}
&\varepsilon\sqrt{\rho\varepsilone}v\varepsilone &\quad&\text{is uniformly bounded in
$L^{\infty}([0,T],L^p(\mathbb{T}^{2}))$, for all $p\geq 1$,}\label{c1.1}\\
&\rho\varepsilone &\quad& \text{is uniformly bounded in
$L^{\infty}([0,T],L^{p}(\mathbb{T}^{2}))$, for all $p\geq 1$,}\label{c1.2}\\
&\sqrt{\rho\varepsilone}v\varepsilone &\quad& \text{is uniformly bounded in
$L^{2}([0,T]\times\mathbb{T}^{2})$,}\label{c1.3}\\
&\sqrt{\varepsilon} c\varepsilone &\quad&\text{ is uniformly bounded in
$L^{\infty}([0,T],L^{2}(\mathbb{T}^{2}))$,}\label{c1.5}\\
& c\varepsilone &\quad&\text{ is uniformly bounded in
$L^{2}([0,T],H^{1}(\mathbb{T}^{2}))$.}\label{c1.6}
\varepsilonnd{align}
\varepsilonnd{cor}
\proof \varepsilonqref{c1.1} and \varepsilonqref{c1.2} are a consequence of the
assumption \ref{basicassumption}, while \varepsilonqref{c1.3} follows from
the inequality \varepsilonqref{estimate1}. Finally
\varepsilonqref{c1.5}, \varepsilonqref{c1.6} follow from \varepsilonqref{estimate4}.
\varepsilonndproof
\subsection{Second scaling}
We consider the following energy for the solution to
(\ref{eq:afterscaling2})
\begin{equation}\label{energypoisson}
E_\varepsilon(t)=\int_{\mathbb{R}^2}\left[\frac{\varepsilon^{2}}{2}\rho\varepsilone |v\varepsilone|^2 + P(\rho\varepsilone)
-\frac{1}{2}\rho\varepsilone c\varepsilone\right]dx,
\varepsilonnd{equation}
where $P$ is given by (\ref{generalizedpressure}). For semplicity we will take $\alpha=1$. We have the following estimate.
\begin{prop}
\label{p2}
The following inequality is valid for a solution
$(\rho\varepsilone,v\varepsilone,c\varepsilone)$ to (\ref{eq:afterscaling2}):
\begin{equation}
E_{\varepsilon}(t)+\int_0^t\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)|v\varepsilone(x,s)|^2 dxds= E_{\varepsilon}(0)
\label{e1}
\varepsilonnd{equation}
\varepsilonnd{prop}
\proof By using the Poisson equation of the system \varepsilonqref{eq:afterscaling2} we
easily have
\begin{align*}
\frac{d}{dt} E_\varepsilon(t) &= -\int_{\mathbb{T}^{2}} \rho\varepsilone |v\varepsilone|^2 dx
+\int_{\mathbb{T}^{2}}\rho\varepsilone v\varepsilone\cdot\nabla c\varepsilone dx +\int_{\mathbb{T}^{2}}c\varepsilone
\Delta c_{t}\varepsilone dx\\
&=-\int_{\mathbb{T}^{2}} \rho\varepsilone |v\varepsilone|^2 dx+\int_{\mathbb{T}^{2}}\rho\varepsilone v\varepsilone\cdot\nabla c\varepsilone dx+\int_{\mathbb{T}^{2}}(c\varepsilone\nabla\cdot(\rho\varepsilone v\varepsilone))dx,
\varepsilonnd{align*}
and this implies
$$ \frac{d}{dt} E_\varepsilon(t)=-\int_{\mathbb{T}^{2}}\rho\varepsilone|v\varepsilone|^{2}dx.$$
Integration with respect to time yields \varepsilonqref{e1}.
\varepsilonndproof
In order to recover an estimate for $\nabla c\varepsilone$, let us introduce the following convex functional
$$J[\rho\varepsilone]=\int_{\mathbb{T}^{2}}(P(\rho\varepsilone(t))-\rho\varepsilone c\varepsilone(t))dx.$$
Now we proceed by estimating the functional $J[\rho\varepsilone]$ from
below, using the same strategy of \cite{carrillo_calvez}. Let us
recall that if $c\varepsilone\in W^{1,1}(\mathbb{T}^{2})$, then the convex
functional $J[\rho\varepsilone]$ has a critical point $\rho^{\ast}$ which
is a solution of
\begin{equation}
g(\rho^{\ast})-c\varepsilone=\lambda
\label{l1}
\varepsilonnd{equation}
whenever $\rho^{\ast}>0$ and null otherwise. Here $\lambda$ is the Lagrange multiplier associated to the constraint given by the mass conservation $\int \rho^{\ast}=M$ and fixed by this condition. We refer to \cite{carrillo_calvez} and (\cite{CJM01}, Proposition 5) for details. Therefore, we have
$$J[\rho\varepsilone]\geq\int_{\mathbb{T}^{2}}(P(\rho^{\ast})-\rho^{\ast}c\varepsilone)dx=\int_{\{\rho^{\ast}>0\}}(P(\rho^{\ast})-\rho^{\ast}g(\rho^{\ast})+\lambda \rho^{\ast})dx.$$
By taking into account the Remark \ref{r1} we can introduce the corrective term $R$ such that $g(\rho^{\ast})=\kappa\log\rho^{\ast}+R(\rho^{\ast})$, then we have
\begin{equation}
\label{l2}
J[\rho\varepsilone]\geq\int_{\mathbb{T}^{2}}(P(\rho^{\ast})-\kappa\rho^{\ast}\log\rho^{\ast})dx-\int_{\{\rho^{\ast}>0\}}\rho^{\ast}R(\rho^{\ast})dx+\lambda M.
\varepsilonnd{equation}
Now, \varepsilonqref{l1} implies $\kappa\log \rho^{\ast}+R(\rho^{\ast})=\lambda+c\varepsilone$, whenever $\rho^{\ast}>0$ and thus
$$\int_{\{\rho^{\ast}>0\}}exp\left(\frac{R(\rho^{\ast})}{\kappa}\right)\rho^{\ast}dx=e^{\lambda/\kappa}\int_{\{\rho^{\ast}>0\}}exp\left(\frac{c\varepsilone}{\kappa}\right)dx,$$
so we have
\begin{equation}
\label{l3}
\lambda=\kappa\log\left(\int_{\{\rho^{\ast}>0\}}e^{R/\kappa}\rho^{\ast}dx\right)-\kappa\log\left(\int_{\{\rho^{\ast}>0\}}e^{c\varepsilone/\kappa}dx\right).
\varepsilonnd{equation}
If we replace $\lambda$ by its expression in the inequality \varepsilonqref{l2}, we conclude that
\begin{align}
J[\rho\varepsilone]&\geq\int_{\mathbb{T}^{2}}(P(\rho^{\ast})-\kappa\rho^{\ast}\log\rho^{\ast})dx-\int_{\{\rho^{\ast}>0\}}\rho^{\ast}R(\rho^{\ast})dx\notag\\
&+\kappa M\log\left(\int_{\{\rho^{\ast}>0\}}e^{R/\kappa}\rho^{\ast}dx\right)-\kappa M\log\left(\int_{\{\rho^{\ast}>0\}}e^{c\varepsilone/\kappa}dx\right).
\label{l4}
\varepsilonnd{align}
By taking into account the Remark \ref{r1} we have that
$$\int_{\{\rho^{\ast}\geq \mathcal{U}\}}(P(\rho^{\ast})-\kappa\rho^{\ast}\log\rho^{\ast})dx\geq C.$$
On the other hand, we have
$$\int_{\{\rho^{\ast}< \mathcal{U}\}}(P(\rho^{\ast})-\kappa\rho^{\ast}\log\rho^{\ast})dx\geq -\left(\sup_{[0,\mathcal{U})}(P-\kappa\rho\log\rho)^{-}\right)|\mathbb{T}^{2}|.$$
Therefore,
$$\int_{\mathbb{T}^{2}}(P(\rho^{\ast})-\rho^{\ast}\log\rho^{\ast})dx$$
is uniformly bounded form below. Now, the Jensen inequality for the probability density $\rho^{\ast}/M$ over the set where $\rho^{\ast}>0$, gives us that
$$exp\int_{\{\rho^{\ast}>0\}}\left(\frac{R(\rho^{\ast})}{\kappa}\frac{\rho^{\ast}}{M}dx \right)\leq \int_{\{\rho^{\ast}>0\}} e^{R/\kappa}\frac{\rho^{\ast}}{M}dx,$$
ans thus
$$\kappa M\log\left(\int_{\{\rho^{\ast}>0\}} e^{R/\kappa}\frac{\rho^{\ast}}{M}dx\right)-
\int_{\{\rho^{\ast}>0\}}\rho^{\ast}R(\rho^{\ast})dx\geq 0. $$
Finally, we recall and use the Trudinger - Moser inequality \cite{Mos71, CY88, GZ98}.
\begin{thm}
Assume that $\Omega\subset \mathbb{R}^{2}$ is a $C^{2}$, bounded, connected domain. It exists a constant $C_{\Omega}$, such that for all $h\in H^{1}$ with $\int_{\Omega}h=0$ we have
$$\int_{\Omega}exp(|h|)dx\leq C_{\Omega}exp\left(\frac{1}{8\pi}\int_{\Omega}|\nabla h |^{2}dx\right).$$
\varepsilonnd{thm}
By applying the previous theorem to $c\varepsilone/\kappa$ we obtain
$$\int_{\{\rho^{\ast}>0\}}e^{c\varepsilone/\kappa}dx\leq \int_{\mathbb{T}^{2}}e^{c\varepsilone/\kappa}dx\leq exp\left(\frac{1}{8\pi\kappa^{2}}\int_{\mathbb{T}^{2}}|\nabla c\varepsilone|^{2}dx\right)$$
and thus
$$-\kappa M\log\left(\int_{\{\rho^{\ast}>0\}}e^{c\varepsilone/\kappa}dx\right)\geq -\frac{M}{8\pi\kappa^{2}}\int_{\mathbb{T}^{2}}|\nabla c\varepsilone|^{2}dx.$$
So by \varepsilonqref{l4} we have that
\begin{equation}
\label{l5}
J[\rho\varepsilone]\geq C-\frac{M}{8\pi\kappa^{2}}\int_{\mathbb{T}^{2}}|\nabla c\varepsilone|^{2}dx
\varepsilonnd{equation}
\begin{prop}
\label{p3}
Assume $(\rho\varepsilone,v\varepsilone,c\varepsilone)$ be a solution to (\ref{eq:afterscaling2})
satisfying assumption \ref{basicassumption} then
\begin{equation}\label{l6}
\int_{\mathbb{T}^{2}}|\nabla c\varepsilone|^{2}dx \qquad \text{is uniformly bounded}.
\varepsilonnd{equation}
\varepsilonnd{prop}
\proof
We can rewrite \varepsilonqref{e1} as
\begin{equation}
E_{\varepsilon}(0)=\int_{\mathbb{T}^{2}}\frac{\varepsilon^{2}}{2}\rho\varepsilone |v\varepsilone|^2dx+\int_{\mathbb{T}^{2}}J[\rho\varepsilone]+\frac{1}{2}\int_{\mathbb{T}^{2}}|\nabla c\varepsilone(t)|^{2} dx+\int_0^t\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)|v\varepsilone(x,s)|^2 dxds.
\label{l7}
\varepsilonnd{equation}
Combining \varepsilonqref{l7} with \varepsilonqref{l5} we get that
\begin{align}
E_{\varepsilon}(0)&\geq\int_{\mathbb{T}^{2}}\frac{\varepsilon^{2}}{2}\rho\varepsilone |v\varepsilone|^2dx+\int_0^t\int_{\mathbb{T}^{2}}\rho\varepsilone(x,s)|v\varepsilone(x,s)|^2 dxds\notag\\
&+C|\mathbb{T}^{2}|+\frac{1}{2}\left(1-\frac{M}{4\pi\kappa}\right)\int_{\mathbb{T}^{2}}|\nabla c\varepsilone(t)|^{2} dx.
\label{l8}
\varepsilonnd{align}
Finally, Remark \ref{r1} implies $ \kappa>\kappa^{\ast}$, i.e. $\left(1-\frac{M}{4\pi\kappa}\right)>0$ and thus
$$\int_{\mathbb{T}^{2}}|\nabla c\varepsilone|^{2}dx$$
is uniformly bounded.
\varepsilonndproof
\begin{cor}
\label{c2}
Let $(\rho\varepsilone,v\varepsilone,c\varepsilone)$ be a solution to (\ref{eq:afterscaling2})
satisfying assumption \ref{basicassumption} and with initial datum
$(\rho\varepsilone_0,v\varepsilone_0,c\varepsilone_0)$ satisfying
\begin{equation}\label{i2}
\int_{\mathbb{T}^{2}}\left[\frac{\varepsilon^2}{2}\rho\varepsilone_0|v\varepsilone_0|^2dx + (\rho\varepsilone_0)^{\gamma+1}-\frac{1}{2}\rho\varepsilone_0 c\varepsilone_0\right]dx\quad\hbox{uniformly bounded w.r.t.}\
\varepsilon\ll1.
\varepsilonnd{equation}
Then, for all $T>0$,
\begin{align}
&\varepsilon\sqrt{\rho\varepsilone}v\varepsilone &\quad&\text{is uniformly bounded in
$L^{\infty}([0,T],L^p(\mathbb{T}^{2}))$, for all $p\geq 1$,}\label{c2.1}\\
&\rho\varepsilone &\quad& \text{is uniformly bounded in
$L^{\infty}([0,T],L^{p}(\mathbb{T}^{2}))$, for all $p\geq 1$,}\label{c2.2}\\
&\sqrt{\rho\varepsilone}v\varepsilone &\quad& \text{is uniformly bounded in
$L^{2}([0,T]\times\mathbb{T}^{2})$,}\label{c2.3}\\
& c\varepsilone &\quad&\text{ is uniformly bounded in
$L^{\infty}([0,T], H^{1}(\mathbb{T}^{2}))$.}\label{c2.4}
\varepsilonnd{align}
\varepsilonnd{cor}
\proof \varepsilonqref{c2.4} follows from Proposition \ref{p3} and by
taking into account that we are in a periodic domain. \varepsilonqref{c2.1}
and \varepsilonqref{c2.2} are a consequence of the assumption
\ref{basicassumption}, while \varepsilonqref{c2.3} is a consequence of
\varepsilonqref{e1} and (\ref{l6}).
\varepsilonndproof
\subsection{Third scaling}
With the same procedure as in the Proposition \ref{p1} we are able
to prove that
\begin{prop}
\label{p33}
Let $(\rho\varepsilone,v\varepsilone,c\varepsilone)$ be a solution to (\ref{eq:afterscaling3})
satisfying assumption \ref{basicassumption} and with initial datum
$(\rho\varepsilone_0,v\varepsilone_0,c\varepsilone_0)$ satisfying
\begin{equation}\label{i3}
\int_{\mathbb{T}^{2}}\left[\frac{\varepsilon^2}{2}\rho\varepsilone_0|v\varepsilone_0|^2dx + (\rho\varepsilone_0)^{\gamma+1}+|c\varepsilone_0|^2\right]dx\quad\hbox{uniformly bounded w.r.t.}\
\varepsilon\ll1.
\varepsilonnd{equation}
Then, for all $T>0$,
\begin{align}
&\varepsilon\sqrt{\rho\varepsilone}v\varepsilone &\quad&\text{is uniformly bounded in
$L^{\infty}([0,T],L^p(\mathbb{T}^{2}))$, for all $p\geq 1$,}\label{c3.1}\\
&\rho\varepsilone &\quad& \text{is uniformly bounded in
$L^{\infty}([0,T],L^{p}(\mathbb{T}^{2}))$, for all $p\geq 1$,}\label{c3.2}\\
&\sqrt{\rho\varepsilone}v\varepsilone &\quad& \text{is uniformly bounded in
$L^{2}([0,T]\times\mathbb{T}^{2})$,}\label{c3.3}\\
&c\varepsilone &\quad&\text{ is uniformly bounded in
$L^{\infty}([0,T],L^{2}(\mathbb{T}^{2}))\cap L^{2}([0,T],H^{1}(\mathbb{T}^{2}))$.}\label{c3.5}
\varepsilonnd{align}
\varepsilonnd{prop}
\section{Strong convergence}\label{chapconv}
This section is devoted to the study of the relaxation of the systems
\varepsilonqref{eq:afterscaling}, \varepsilonqref{eq:afterscaling2}, \varepsilonqref{eq:afterscaling3} towards their formal limit \varepsilonqref{eq:reduced}, \varepsilonqref{eq:reduced2}, \varepsilonqref{eq:reduced3}, respectively. As a consequence of the Corollary \ref{c1} and the Propositions \ref{p3}, \ref{p33} we have that, extracting if necessary a subsequence,
$$\nabla c\varepsilone\rightharpoonup \nabla c^{0}\quad \text{as $\varepsilon\downarrow 0$ weakly in $L^{2}([0,T]\times \mathbb{T}^{2})$ }.$$
This convergence for $c\varepsilone$ is enough to pass into the limit in
\varepsilonqref{eq:afterscaling}, \varepsilonqref{eq:afterscaling2},
\varepsilonqref{eq:afterscaling3} to get in the sense of distribution
\varepsilonqref{eq:reduced}, \varepsilonqref{eq:reduced2}, \varepsilonqref{eq:reduced3},
respectively, p rovided that $\rho\varepsilone$ converges in a strong
topology. In fact by the Remark \ref{r2} we know that
$\rho\varepsilone\rightarrow \rho^{0}$ in $L^{\infty}$ $\ast-$weakly, while
by \varepsilonqref{c1.2}, \varepsilonqref{c2.2}, \varepsilonqref{c3.2} we have
$\rho\varepsilone\rightharpoonup \rho^{0}$ weakly in $L^{p}$, for any
$p>1$. These convergence are clearly too weak to pass into the
limit in the nonlinear terms of \varepsilonqref{eq:afterscaling},
\varepsilonqref{eq:afterscaling2}, \varepsilonqref{eq:afterscaling3}. So, in this
section we will investigate the strong convergence of the
approximating sequence $\rho\varepsilone$. The analysis of this convergence
reduces to the analysis of the convergence of quadratic forms with
constant coefficients via the classical compensated compactness
technique due to Tartar \cite{Tar79, Tar83} and Murat \cite{Mur78}
(see Dacorogna \cite{Dac82}). As we will see later on, these
techniques will apply in the same way to the three scalings
\varepsilonqref{s1}, \varepsilonqref{s2}, \varepsilonqref{s3}, so we will discuss them
together. Let us recall the following theorem
\begin{thm}\label{cc}
{\bf (Tartar's Compensated compactness)}\\
Let us consider
\begin{enumerate}
\item
a bounded open set $\Omega\subset\mathbb{R}^{n}$;
\item
a sequence $\{l^{\nu}\}_{\nu = 1}^{\infty}$,
$l^{\nu}:~\Omega\subset \mathbb{R}^{n}
\longrightarrow\mathbb{R}^{m} $;
\item
a symmetric matrix
$\mathbb{T}heta:~\mathbb{R}^{m}\longrightarrow\mathbb{R}^{m} $;
\item
constants $a_{jk}^{i}\in \mathbb{R}$, $i = 1,\ldots,q$,
$j = 1,\ldots,m$, $k = 1,\ldots,n$.
\varepsilonnd{enumerate}
Let us define
\begin{align*}
f(\alpha) & = \left\langle \mathbb{T}heta \alpha,\alpha \right\rangle,
\quad\text{for all $\alpha \in \mathbb{R}^{m}$;} \\
\Lambda & = \left\{ \lambda \in \mathbb{R}^{m}:~
\varepsilonxists \varepsilonta \in \mathbb{R}^{n}\setminus
\{0\}, ~\sum\limits _{j,k}a_{jk}^{i} \lambda_{j}
\varepsilonta_{k} = 0, i = 1,\ldots,q\right\}.
\varepsilonnd{align*}
Assume that
\begin{itemize}
\item[\bf(a)]
there exists $\widetilde{l}\in L^{2}\left(\Omega\right)$
such that $l^{\nu}\rightharpoonup \widetilde{l}$ in
$L^{2}\left(\Omega\right)$ as $\nu\uparrow\infty$;
\item[\bf(b)]
$\mathcal{A}^{i}l^{\nu} = \sum\limits _{j,k}a^{i}_{jk}\frac{\partial
l_{j}^{\nu}}{\partial x_{k}}$, $i=1,\ldots q$ are relatively compact in
$H^{-1}_{loc}\left(\Omega\right)$;
\item[\bf(c)]
$f_{|\Lambda}\varepsilonquiv 0$;
\item[\bf(d)] there exists
$\widetilde{f}\in \mathbb{R}$ such that $f(l)\rightharpoonup
\widetilde{f}$ in the sense of measures $\mathcal{M}(\Omega)$.
\varepsilonnd{itemize}
Then we have $\widetilde{f} = f(\widetilde{l})$.
\varepsilonnd{thm}
\subsection{Weak convergence of $\rho\varepsilone P(\rho\varepsilone)$}
First of all we start by studying the weak convergence
of $\rho\varepsilone P(\rho\varepsilone)$. Our goal will be to prove that
$$\rho\varepsilone P(\rho\varepsilone)\rightharpoonup \rho^{0}P(\rho^{0}),$$
where $\rho^{0}$ is the weak limit of $\rho\varepsilone$. To this end we are going to apply the Theorem \ref{cc} in the same spirit of \cite{MR00}. In order to fit the into the hypotheses of the Theorem \ref{cc} we rewrite the first two equations of the systems
\varepsilonqref{eq:afterscaling}, \varepsilonqref{eq:afterscaling2}, \varepsilonqref{eq:afterscaling3}, as
\begin{equation}
\begin{cases}
\rho\varepsilone_{t} +m\varepsilone_{x}+n\varepsilone_{y} = 0 \\
\displaystyle
{\varepsilon^{2}m\varepsilone_{t}+\left(\varepsilon^{2}\frac{(m\varepsilone)^{2}}{\rho\varepsilone}+\gamma P(\rho\varepsilone)\right)_{x}+\left(\varepsilon^{2}\frac{m\varepsilone n\varepsilone}{\rho\varepsilone}\right)_{y} = \rho\varepsilone c\varepsilone_{x}-m\varepsilone}\\
\displaystyle{
\varepsilon^{2}n\varepsilone_{t}+\left(\varepsilon^{2}\frac{m\varepsilone n\varepsilone}{\rho^{\varepsilone}}\right)_{x}+\left(\varepsilon^{2}\frac{(n\varepsilone)^{2}}{\rho\varepsilone}+\gamma P(\rho\varepsilone)\right)_{y} = \rho\varepsilone c\varepsilone_{y}-n\varepsilone.}
\varepsilonnd{cases}
\label{4.1.2}
\varepsilonnd{equation}
where
\begin{equation}
v\varepsilone=(v_{1}\varepsilone, v_{2}\varepsilone)\qquad \rho\varepsilone v\varepsilone=(\rho\varepsilone v_{1}, \rho\varepsilone v_{2})=(m\varepsilone, n\varepsilone).
\label{4.1.1}
\varepsilonnd{equation}
It will be usefull rewrite \varepsilonqref{4.1.2} in the following way
\begin{align}
\rho\varepsilone_{t} +m\varepsilone_{x}+n\varepsilone_{y} &= 0 \notag\\
\gamma P(\rho\varepsilone)_{x}&=-\varepsilon^{2}m\varepsilone_{t}-\varepsilon^{2}\left(\frac{(m\varepsilone)^{2}}{\rho\varepsilone}\right)_{x}-\varepsilon^{2}\left(\frac{m\varepsilone n\varepsilone}{\rho\varepsilone}\right)_{y} + \rho\varepsilone c\varepsilone_{x}-m\varepsilone\label{4.1.11}\\
\gamma P(\rho\varepsilone)_{y}&=-\varepsilon^{2}n\varepsilone_{t}-\varepsilon^{2}\left(\frac{m\varepsilone n\varepsilone}{\rho\varepsilone}\right)_{x}-\varepsilon^{2}\left(\frac{(n\varepsilone)^{2}}{\rho\varepsilone}\right)_{y}+\rho\varepsilone c\varepsilone_{y}-n\varepsilone.\notag
\varepsilonnd{align}
By using \varepsilonqref{c1.3}, \varepsilonqref{c2.3}, \varepsilonqref{c3.3}, \varepsilonqref{l6}, and the assumption (A3) we get that $\rho\varepsilone v\varepsilone, \ \rho\varepsilone\nabla c\varepsilone\in L^{2}([0,T]\times \mathbb{T}^{2})$. In fact
\begin{align}
&\|\rho\varepsilone v\varepsilone\|_{L^{2}([0,T]\times \mathbb{T}^{2})}\leq \|\sqrt{\rho\varepsilone}\|_{\infty}\|\sqrt{\rho\varepsilone}v\varepsilone\|_{L^{2}([0,T]\times \mathbb{T}^{2})}\label{cc1}\\
&\|\rho\varepsilone\nabla c\varepsilone\|_{L^{2}([0,T]\times \mathbb{T}^{2})}\leq \|\rho\varepsilone\|_{\infty}\|\nabla c\varepsilone\|_{L^{2}([0,T]\times \mathbb{T}^{2})}\label{cc2}
\varepsilonnd{align}
Moreover, by taking into account the assumptions (A2) and (A3) and \varepsilonqref{c1.3}, \varepsilonqref{c2.3}, \varepsilonqref{c3.3} we have that $\varepsilon^{2}\left(\frac{(m\varepsilone)^{2}}{\rho\varepsilone}\right)_{x}$, is relatively compact in $H^{-1}([0,T]\times \mathbb{T}^{2})$. In fact let us consider $\omega$ relatively compact in $[0,T]\times \mathbb{T}^{2}$, then by taking into account (A2), (A3) and the Remark \ref{r2} we have,
\begin{equation}
\left\|\varepsilon^{2}\left(\frac{(m\varepsilone)^{2}}{\rho\varepsilone}\right)_{x}\right\|_{H^{-1}(\omega)}\leq
\sup _{\|\phi\|_{H^{1}_{0}(\omega)}=1}\left|\left\langle
\varepsilon^{2}\left(\frac{(m\varepsilone)^{2}}{\rho\varepsilone}\right)_{x} ,
\phi\right\rangle\right|\leq \varepsilon\|\rho\varepsilone
v\varepsilone\|_{\infty}\frac{1}{\sqrt{k}}\|\sqrt{\rho\varepsilone}v\varepsilone\|_{L^{2}(\omega)}
\label{cc2bis}
\varepsilonnd{equation}
In a similar way it can be proved that $\varepsilon^{2}\left(\frac{m\varepsilone
n\varepsilone}{\rho\varepsilone}\right)_{y}$, $\varepsilon^{2}\left(\frac{m\varepsilone
n\varepsilone}{\rho\varepsilone}\right)_{x}$,
$\varepsilon^{2}\left(\frac{(n\varepsilone)^{2}}{\rho\varepsilone}\right)_{y}$,
$\varepsilon^{2}(\rho\varepsilone v\varepsilone)_{t}$ are relatively compact in
$H^{-1}([0,T]\times \mathbb{T}^{2})$. Now, \varepsilonqref{cc1}--\varepsilonqref{cc2bis},
imply that
\begin{equation}
\begin{pmatrix}
\rho\varepsilone_{t}+m\varepsilone_{x}+n\varepsilone_{y} \\
P(\rho\varepsilone)_{x} \\
P(\rho\varepsilone)_{y}
\varepsilonnd{pmatrix}
\qquad
\text{is relatively compact in $\left(H^{-1}_{loc}\right)^{3}$.} \label{4.1.3}
\varepsilonnd{equation}
In order to fit into the framework of the Theorem \ref{cc} we set $x_{1}=x$, $x_{2}=y$, $x_{3}=t$, $l\varepsilone=(m\varepsilone, n\varepsilone, \rho\varepsilone, P(\rho\varepsilone))$, hence $m=4$. In our case the differential constraints are $q=3$. So we can define the matrices $\mathcal{A}^{1}, \mathcal{A}^{2}, \mathcal{A}^{3}\in \mathcal{M}_{4\times 3}$, where $\mathcal{A}^{i}=\{a^{i}_{jk}\}$, $i=1,2,3$, $j=1, \ldots,4$, $k=1,2,3$ as follows:
\begin{equation*}
\mathcal {A}^{1}=\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0\\
0 & 0 & 0
\varepsilonnd{pmatrix} \quad
\mathcal{A}^{2}=\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
1 & 0 & 0
\varepsilonnd{pmatrix}
\quad
\mathcal{A}^{3}=\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 1 & 0
\varepsilonnd{pmatrix}.
\varepsilonnd{equation*}
The characteristic manifold $\Lambda$ is then given by
\begin{equation*}
\Lambda =\left\{\lambda \in \mathbb{R}^{4} \mid \varepsilonxists \xi \in
\mathbb{R}^{3}\setminus \{0\}, B(\xi,\lambda)=0\right\}
\varepsilonnd{equation*}
where
\begin{equation*}
B(\xi,\lambda)=\begin{pmatrix}
\lambda_{1}\xi_{1}+\lambda_{2}\xi_{2}+\lambda_{3}\xi_{3}\\
\lambda_{4}\xi_{1} \cr \lambda_{4}\xi_{2}.
\varepsilonnd{pmatrix}
\varepsilonnd{equation*}
Therefore
\begin{equation*}
\Lambda =\left\{\lambda\in \mathbb{R} ^{4} \mid
det\begin{pmatrix}\lambda_{1} &
\lambda_{2} & \lambda_{3} \\
\lambda_{4} & 0 & 0 \\
0 & \lambda_{4} & 0
\varepsilonnd{pmatrix} \right\}=\left\{\lambda \in \mathbb{R}^{4} \mid
\lambda_{3}\lambda_{4}=0\right\}.
\varepsilonnd{equation*}
If we define
$$M=\frac{1} {2}\begin{pmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0
\varepsilonnd{pmatrix} \in \mathcal {M}_{4\times 4}, $$
then
$f(\lambda)={\lambda}^{T}M \lambda =\lambda_{3}\lambda_{4}$ and, of course
$f_{|\Lambda}\varepsilonquiv 0$.
Now, by applying the Theorem \ref{cc} we have
$f(l\varepsilone)\rightharpoonup f(\tilde{l})$, and in our case this means
\begin{equation*}
\rho\varepsilone P(\rho\varepsilone)\rightharpoonup \rho^{0} P^{0},
\varepsilonnd{equation*}
where $P^{0} =w-\lim P (\rho\varepsilone)$.
\subsection{Strong convergence of $\rho\varepsilone$}
In the previous section we proved that
\begin{equation*}
\rho\varepsilone P(\rho\varepsilone)\rightharpoonup \rho^{0} P^{0}.
\varepsilonnd{equation*}
Here we will be able to prove that
$$\rho\varepsilone\rightarrow \rho^{0}\quad \text{strongly in $L^{p}_{loc}$, $p<+\infty$}.$$
At this point we can follow the methods of \cite{MM90}, \cite{MR00}.
First of all let as use Minty's argument (\cite{L-JL84}, \cite{MM90}) to prove that
$P^0=P(\rho^0)$. Since the function $P$ is monotone, for any $w\in
L^\infty$ e $\varphi$ test function, $\varphi >0$, we have that
\begin{equation}
H(\varepsilon)\varepsilonquiv \int\!\!\int (P(\rho\varepsilone)-P(w))(\rho\varepsilone-w)\varphi dxdt\geq 0;
\label{5.1}
\varepsilonnd{equation}
But for $\varepsilon\downarrow 0$, we have that
$$\int\!\!\int P(\rho\varepsilone)\rho\varepsilone\varphi dxdt\rightarrow \int\!\!\int P^0
\rho^0\varphi dxdt,$$
So from \varepsilonqref{5.1} we get that for $\varepsilon \downarrow 0$
$$H(\varepsilon)\rightarrow H\varepsilonquiv \int\!\!\int(P^0-p(w))(\rho^{0}-w) \varphi dxdt \geq
0$$
If we choose $w=\rho^{0} +\lambda z$, with $\lambda \leq 0$ and arbitrary $z \in L^\infty$, we have
\begin{align*}
G(\lambda,z) &\varepsilonquiv \int\!\!\int (P^0-P(\rho^{0} +\lambda z))z\varphi dxdt\\
&
=\frac{1}{\lambda}\int\!\!\int (P^0-P(\rho^{0} +\lambda z))\lambda z\varphi dxdt
\leq 0
\varepsilonnd{align*}
and for $\lambda \uparrow 0$, $G(0,z)\leq 0$ for any $z \in L^\infty$,
then
$$G(0,z)=\int\!\!\int (P^0-P(\rho^{0}))z\varphi dxdt =0,$$
and finally $P^0=P(\rho^0)$.\\
Our next step now, is to prove the strong convergence for
$\rho\varepsilone \rightarrow \rho^0$ in $L^{p}_{loc}$. To this end we characterize the weak convergence by means of Young's probability measures
(see \cite{Tar79},\cite{DiP83a}, \cite{DiP83b}, \cite{DiP85b},
\cite{DiP85a}). Let us recall that if $\{u\varepsilone\}$ is sequence converging to
$U$ in $L^\infty$ $\ast$-weakly , we can associate to
$\{u\varepsilone\}$ a family $\{\nu_{(x,t)}(\lambda)\}$ of probability measures such that for any continuos function
$F(\cdot)$
$$\ast-\lim_{\varepsilon \rightarrow 0} F(u\varepsilone)=\int
F(\lambda)\nu_{(x,t)}(d\lambda)\qquad a.e$$
If $\nu_{(x,t)}=\delta _{U(x,t)}$ then $u\varepsilone \rightarrow U$ strongly in
$L^{p}_{loc}$ for any $p\in(1, +\infty)$ (see \cite{Dac82}, Corollary 6.2).
Let $\{\nu_{(x,t)}\}$ be the family of Young's probability measures
associated to the sequence $\{\rho\varepsilone\}$: since
$\rho\varepsilone \longrightarrow \rho^{0}$ in $L^{\infty}$ $\ast$-weakly, we can find a closed interval $\left[a,b\right]$, $0\leq a\leq b$,
such that $supp \ \nu_{(x,t)}\subseteq \left[a,b\right]$. Since
$P(r) = r^{\alpha}$, $\alpha> 1$, we have three possibilities:
\begin{enumerate}
\item $P\in C^{2}\left(\mathbb{R}\setminus \{0\}\right)$ e $P''(r)\uparrow
+\infty$ for $r\downarrow 0$, if $1< \alpha <2$;
\item $P\in C^{2}\left(\mathbb{R}\right)$ e $P''(0) = 1$, if $\alpha = 2$;
\item $P\in C^{2}\left(\mathbb{R}\right)$ and $P''(0) = 0$, if $\alpha >2$.
\varepsilonnd{enumerate}
Let us assume that $1< \alpha \leq 2$. Then we can write for any $\lambda, \lambda_{0}$
$$P(\lambda) - P(\lambda_{0}) = P'(\lambda_{0})
\left(\lambda - \lambda_{0}\right) + \frac {1}{2}
P''\left(\lambda^{*}\right) \left(\lambda - \lambda_{0}\right)^{2},$$
where $\lambda^{*}$ belongs to the segment between $\lambda$ and $\lambda_{0}$.
If we choose
$$\lambda_{0} = \int_{a}^{b} \lambda \nu_{(x,t)}\left(d\lambda\right)=\rho_{0},$$
since $P^0=P(\rho^0)$
$$P\left(\lambda_{0}\right) = \int_{a}^{b}
P\left(\lambda\right)\nu_{(x,t)}\left(d\lambda\right)$$
so that
$$\int_{a}^{b} \{ P(\lambda) - P(\lambda_{0}) \} \nu_{(x,t)}\left(d\lambda\right) = 0.$$
On the other hand we also have
\begin{align*}
\int_{a}^{b}P'(\lambda_{0})\left(\lambda - \lambda_{0}\right)
\nu_{(x,t)} \left(d\lambda\right)
= P'(\lambda_{0})\left\{\int_{a}^{b} \lambda \nu_{(x,t)}\left(d\lambda\right)-\lambda_{0}\int_{a}^{b}\nu_{(x,t)}\left(d\lambda\right)\right\}=0,
\varepsilonnd{align*}
so we can conclude that
$$\int_{a}^{b} P''\left(\lambda^{*}\right) \left(\lambda - \lambda_{0}
\right)^{2} \nu_{(x,t)}\left(d\lambda\right) = 0.$$
Taking
$E=\displaystyle{\min_{ \lambda \in \left[a,b\right]}} P''\left(\lambda\right) > 0$, we get
$$E\int_{a}^{b} \left(\lambda - \lambda_{0}\right)^{2} \nu_{(x,t)}\left(d\lambda\right) \leq 0,$$
namely
$$\int_{a}^{b} \left(\lambda - \lambda_{0}\right)^{2} \nu_{(x,t)}\left(d\lambda\right)= 0,$$
and it follows $a = b$ and $\nu_{(x,t)}=\delta$, a point mass and so we finally get
$$\rho\varepsilone\rightarrow \rho_{0}\qquad \text{strongly in $L^{p}_{loc}$}.$$
To conclude we remark that in the case $\alpha>2$ this result can be obtained in the same way by using the function $-P^{-1}$.
\begin{rem}
\varepsilonmph{The strong convergence result for $\rho\varepsilone$ obtained in this
section is still valid in the case of linear diffusion, namely if
we consider $g(\rho)=\log \rho$ and consequently
$P(\rho)=\rho\log\rho -\rho$. }\varepsilonnd{rem} By using the estimates
and the strong convergence of the sequence $\{\rho\varepsilone\}$ obtained
in the previous section we are able to prove the following main
theorem.
\begin{thm}
\label{tconv} Let $T>0$ be arbitrary and let $(\rho\varepsilone, v\varepsilone,
c\varepsilone)$ be a family of solutions to the system
\varepsilonqref{eq:afterscaling} (\varepsilonqref{eq:afterscaling2} and
\varepsilonqref{eq:afterscaling3} respectively) with initial data
satisfying (\ref{i1}) ((\ref{i2}) and (\ref{i3}) respectively).
Assume that the assumption \ref{basicassumption} holds, then,
there exist $\rho^{0}\in L^{\infty}([0,T]\times \mathbb{T}^{2})$ and
$c^{0}\in L^{2}([0,T],H^{1}(\mathbb{T}^{2}))$, such that, extracting if
necessary a subsequence,
\begin{align*}
\rho\varepsilone&\longrightarrow\rho^{0} \qquad \text{strongly in $L^{p}_{loc}([0,T]\times \mathbb{T}^{2})$ for any $p<\infty$}\\
\nabla c\varepsilone&\rightharpoonup \nabla c^{0} \qquad \text{weakly in $L^{2}([0,T]\times \mathbb{T}^{2})$.}
\varepsilonnd{align*}
Moreover the couple $(\rho^{0}, c^{0})$, satisfies the system
\varepsilonqref{eq:reduced} (\varepsilonqref{eq:reduced2} and \varepsilonqref{eq:reduced3}
respectively) in the sense of distributions.
\varepsilonnd{thm}
\section{Perturbation of constant states in the approximating
system} \label{chappert}
In this section we deal with the rescaled system
\begin{equation}\label{eq:pert1}
\begin{cases}
\partial_{t}\rho\varepsilone +\nabla\cdot (\rho\varepsilone v\varepsilone)=0 & \\
\varepsilon^2\left[\partial_{t} v\varepsilone + v\varepsilone\cdot \nabla v\varepsilone\right] + \nabla
g(\rho\varepsilone)=\nabla c\varepsilone - v\varepsilone & \\
\varepsilon\partial_{t} c\varepsilone = \Delta c\varepsilone + \alpha \rho\varepsilone -\beta
c\varepsilone.
\varepsilonnd{cases}
\varepsilonnd{equation}
with $x\in \mathbb{T}^2$, $t\geq 0$, where $\mathbb{T}^2$ is the flat normalized
two--dimensional torus. The system (\ref{eq:pert1}) is
complemented with the $1$-periodical initial data
\begin{equation*}
\rho\varepsilone(x,0)=\rho\varepsilone_0(x),\quad v\varepsilone(x,0)=v\varepsilone_0(x),\quad
c\varepsilone(x,0)=c\varepsilone_0(x).
\varepsilonnd{equation*}
We shall consider small perturbations of the stationary state
\begin{equation}\label{constantstates}
(\rho,v,c)=(\widetilde{\rho},\widetilde{v},\widetilde{c}),\quad \widetilde{\rho}>0, \quad \widetilde{v}=0,\quad
\widetilde{c}=\frac{\alpha}{\beta}\widetilde{\rho}
\varepsilonnd{equation}
and prove the existence of solutions $(\rho\varepsilone,v\varepsilone,c\varepsilone)$ such
that the density $\rho\varepsilone$ stays away from zero, uniformly in
$\varepsilon$, on a small enough time interval $[0,T]$ with $T$
independent on $\varepsilon$ (see similar results in \cite{KM81} and \cite{DiFM}). In order to perform this task, we
shall use an iterative method, namely we define recursively the
sequence $(\rho^n,v^n,c^n)$ as follows:
$(\rho^0(x,t),v^0(x,t),c^0(x,t))=(\rho\varepsilone_0(x),v\varepsilone_0(x),c\varepsilone_0(x))$
and, for $n\geq 1$, $(\rho^n,v^n,c^n)$ solves the linear system
\begin{equation}\label{eq:sequence}
\begin{cases}
\displaystyle{\partial_{t}\rho^n +v^{n-1}\cdot \nabla \rho^n + \rho^{n-1}\nabla\cdot u^n=0} & \\
\displaystyle{\partial_{t} v^n + v^{n-1}\cdot \nabla v^n + \frac{g'(\rho^{n-1})}{\varepsilon^2}\nabla
\rho^n=\frac{1}{\varepsilon^2}\nabla c^n - \frac{1}{\varepsilon^2}v^n}
& \\
\displaystyle{\partial_{t} c^n = \frac{1}{\varepsilon}\Delta c^n + \frac{\alpha}{\varepsilon} \rho^n
-\frac{\beta}{\varepsilon}c^n.}
\varepsilonnd{cases}
\varepsilonnd{equation}
From now on we shall drop the dependency on $\varepsilon$ on the solutions
$(\rho,v,c)$ to simplify the notation. Moreover, we shall use the
following notation: the variables taken at the step $n-1$ will be
denoted e. g. by $\rho^{n-1}=\widehat{\rho}$; the variables taken at the
step $n$ will be denoted without any further symbol, e. g. $\rho^n
= \rho$; the deviation from the aforementioned constant stationary
states will be denoted e. g. by $\bar \rho=\rho^n - \widetilde{\rho}$ and
$\underline{\rho}=\rho^{n-1}-\widetilde{\rho}$.
The first two equations in system (\ref{eq:sequence}) can be
easily viewed as a hyperbolic system in vectorial form. More
precisely, let us define the $3$--dimensional variable $U$ as
\begin{equation*}
U:=(\rho,v^1,v^2),
\varepsilonnd{equation*}
where $v=(v^1,v^2)$. Let us denote
\begin{equation*}
A_1(\widehat{U}):=\left[\begin{matrix} \widehat{v}^1 & \widehat{\rho} & 0 \\
\frac{g'(\widehat{\rho})}{\varepsilon^2} & \widehat{v}^1 & 0 \\ 0 & 0 & \widehat{v}^1
\varepsilonnd{matrix}\right],\qquad A_2(\widehat{U}):=\left[\begin{matrix} \widehat{v}^2 & 0 & \widehat{\rho} \\
0 & \widehat{v}^2 & 0 \\ \frac{g'(\widehat{\rho})}{\varepsilon^2} & 0 & \widehat{v}^2
\varepsilonnd{matrix}\right],\qquad B(U):=\frac{1}{\varepsilon^2}\left(\begin{matrix} 0 \\
\partial_{x_1}c - v^1 \\ \partial_{x_2}c -
v^2\varepsilonnd{matrix}\right).
\varepsilonnd{equation*}
Then, with all these notations, the system (\ref{eq:sequence}) can
be rephrased as
\begin{equation}\label{eq:pert2}
\begin{cases}
\partial_t U + + A_1(\widehat{U}) \partial_{x_1}U + A_2(\widehat{U})
\partial_{x_2}U= B(U) & \\
\displaystyle{\partial_t c = \frac{1}{\varepsilon}\Delta c
+\frac{\alpha}{\varepsilon}\rho - \frac{\beta}{\varepsilon}c}. &
\varepsilonnd{cases}
\varepsilonnd{equation}
The first line in (\ref{eq:pert2}) corresponds to a linear
hyperbolic system which can be \varepsilonmph{symmetrized} by means of the
matrix
\begin{equation}\label{symmetrizer}
S(\widehat{U}):=\mathrm{diag}\left(\frac{g'(\widehat{\rho})}{\varepsilon^2}, \widehat{\rho}, \widehat{\rho}
\right).
\varepsilonnd{equation}
The matrix $S(\widehat{U})$ is uniformly positive definite provided the
variable $\widehat{\rho}$ satisfies a condition of the form $0<c\leq
\widehat{\rho}\leq C$ (we recall that $g'(\rho)=\gamma \rho^{\gamma-1}$
exhibits a singularity at zero in case of $\gamma<1$). The two
matrices $S(\widehat{U})A_1(\widehat{U})$ and $S(\widehat{U})A_2(\widehat{U})$ can be easily
proven to be symmetric. We now rewrite system (\ref{eq:pert2}) in
terms of the deviations $\bar U$ and $\bar c$:
\begin{equation}\label{eq:pert3}
\begin{cases}
\partial_t \bar U + A_1(\widetilde{U}+\underline{U}) \partial_{x_1}\bar U + A_2(\widetilde{U}+\underline{U})
\partial_{x_2}\bar U= B(\widetilde{U}+\bar U)=B(\bar U) & \\
\displaystyle{\partial_t \bar c = \frac{1}{\varepsilon}\Delta \bar c
+\frac{\alpha}{\varepsilon}\bar \rho - \frac{\beta}{\varepsilon}\bar c}. &
\varepsilonnd{cases}
\varepsilonnd{equation}
We introduce the energy functional
\begin{equation*}
\mathcal{E}(U,c):=\frac{1}{2}\int_{0}^{t}o \left[ U^T S(\widehat{U}) U + \lambda c^2\right]dx =
\frac{1}{2}\int_{0}^{t}o \left[ \frac{g'(\widehat{\rho})}{\varepsilon^2}\rho^2 + \widehat{\rho}
|v|^2 + \lambda c^2\right] dx,
\varepsilonnd{equation*}
where $\lambda>0$ is a constant to be chosen later on. We have the
following
\begin{prop} Let $T>0$. There exist constants $\varepsilon_0,\delta\in (0,1)$, $K\in(0,\widetilde{\rho} /2)$ such that, if
\begin{align}
& \|\rho_0^\varepsilon-\widetilde{\rho}\|_{H^4(\mathbb{T}^2)} +
\varepsilon\|v_0^\varepsilon\|_{H^4(\mathbb{T}^2)}+\sqrt{\varepsilon}\|c_0^\varepsilon-\widetilde{c}\|_{H^4(\mathbb{T}^2)}
\leq \delta \quad \hbox{and}\nonumber\\
& \quad \sup_{0\leq t\leq T}\left(\|\underline{\rho}(t)\|_{H^4(\mathbb{T}^2)}+\varepsilon\|\underline{v}(t)\|_{H^4(\mathbb{T}^2)} +\sqrt{\varepsilon}\|\underline{c}(t)-\widetilde{c}\|_{H^4(\mathbb{T}^2)}
\right) \leq
K,\label{ipotesi_induttiva}
\varepsilonnd{align}
for all $\varepsilon\in(0,\varepsilon_0)$, then,
\begin{equation}\label{final1}
\sup_{0\leq t\leq
T}\left(\|\bar \rho(t)\|_{H^4(\mathbb{T}^2)}+\varepsilon
\|\bar v(t)\|_{H^4(\mathbb{T}^2)}+\sqrt{\varepsilon}\|\bar c(t)-\widetilde{c}\|_{H^4(\mathbb{T}^2)}\right)
\leq K
\varepsilonnd{equation}
for all $\varepsilon\in(0,\varepsilon_0)$.
\varepsilonnd{prop}
\proof During the proof of this proposition we shall often make
use of the Sobolev inequality $\|f\|_{L^\infty(\mathbb{T}^2)}\leq
C\|f\|_{H^2(\mathbb{T}^2)}$.
\textsc{Step 1}. Due to the symmetry of the two matrices $SA_1$
and $SA_2$, we can use integration by parts in the evolution of
$\mathcal{E}(\bar U,\bar c)$ as follows:
\begin{align}
\frac{d}{dt} \mathcal{E}(\bar U,\bar c) & = \frac{1}{2}\int_{0}^{t}o\bar U^T\left[S(\widehat{U})A_1(\widehat{U})\right]_{x_1}\bar U dx +
\frac{1}{2}\int_{0}^{t}o\bar U^T\left[S(\widehat{U})A_2(\widehat{U})\right]_{x_2}\bar U dx\nonumber\\
& \ + \int_{0}^{t}o\bar U^T S(\widehat{U})B(\bar U)dx -\frac{\lambda}{\varepsilon}\int_{0}^{t}o |\nabla
\bar c|^2 dx + \frac{\lambda\alpha}{\varepsilon}\int_{0}^{t}o\bar \rho \bar c dx
-\frac{\lambda\beta}{\varepsilon}\int_{0}^{t}o \bar c^2 dx.\label{energyest1}
\varepsilonnd{align}
Due to the assumption (\ref{ipotesi_induttiva}) we have
\begin{align}
\frac{d}{dt} \mathcal{E}(\bar U,\bar c) & \leq
C(K) \left(\|\nabla \underline{\rho}\|_{L^\infty} + \|\nabla \underline{v}\|_{L^\infty}\right)
\frac{1}{2}\int_{0}^{t}o\left(\frac{\bar \rho^2}{\varepsilon^2} + \frac{|\bar v|^2}{\varepsilon^2}\right)
dx+\frac{\|\widehat{\rho}\|_{L^\infty}}{\varepsilon^2}\int_{0}^{t}o \bar v\cdot \nabla \bar c dx\nonumber\\
& \
-\frac{(\widetilde{\rho} - K)}{\varepsilon^2}\int_{0}^{t}o |\bar v|^2 dx-\frac{\lambda}{\varepsilon}\int_{0}^{t}o |\nabla
\bar c|^2 dx + \frac{\lambda\alpha}{\varepsilon}\int_{0}^{t}o\bar \rho \bar c dx
-\frac{\lambda\beta}{\varepsilon}\int_{0}^{t}o \bar c^2 dx\label{energyest2}
\varepsilonnd{align}
for a function $C(K)>0$ of the constant $K$ such that $C$ is
continuous on $K\in [0,\widetilde{\rho} /2]$. By choosing
\begin{equation*}
\lambda=\frac{(\widetilde{\rho} +K)^2}{\varepsilon(\widetilde{\rho} - K)}
\varepsilonnd{equation*}
and $\varepsilon_0< 1$, we can use once again (\ref{ipotesi_induttiva}) and
find a constant $C_1>0$ such that
\begin{align*}
\frac{d}{dt}\mathcal{E}(\bar U,\bar c)& \leq K C(K)\frac{1}{2}\int_{0}^{t}o\left(\frac{\bar \rho^2}{\varepsilon^2} + \frac{|\bar v|^2}{\varepsilon^2}\right)
dx -\frac{(\widetilde{\rho}-K)}{2\varepsilon^2}\int_{0}^{t}o |\bar v|^2 dx -\frac{(\widetilde{\rho}
+K)^2}{2(\widetilde{\rho} -K)
\varepsilon^2}\int_{0}^{t}o|\nabla \bar c|^2 dx\\
& \ -\frac{(\widetilde{\rho} +K)^2\beta}{2(\widetilde{\rho}-K)\varepsilon}\int_{0}^{t}o
\bar c^2 dx + \frac{(\widetilde{\rho}+K)^2 \alpha^2}{2(\widetilde{\rho}-K)\varepsilon \beta}\int_{0}^{t}o \bar \rho^2
dx.
\varepsilonnd{align*}
We now choose the constant $K$ such that $K C(K)<\frac{1}{2}(\widetilde{\rho}
-K)$. By using the coercivity property
\begin{equation}
\mathcal{E}(U,c)\geq c(K)\int_{0}^{t}o\left[\frac{\rho^2}{\varepsilon^2} +|v|^2 +
\frac{c^2}{\varepsilon}\right]dx,\label{coercivity}
\varepsilonnd{equation}
which holds for a certain $c(K)>0$, due to Gronwall inequality we
easily obtain
\begin{equation*}
\mathcal{E}(\bar U(t),\bar c(t)) +\frac{1}{\varepsilon^2}\int_0^t\int_{0}^{t}o
\left[|\bar v(\tau)|^2 dx + |\nabla \bar c(\tau)|^2\right]dx d\tau +
\frac{1}{\varepsilon}\int_{0}^{t}\int_{0}^{t}o \bar c(\tau)^2 dxd\tau \leq
A\mathcal{E}(\bar U(0),\bar c(0))e^{Bt}
\varepsilonnd{equation*}
for certain constants $A,B>0$ depending only on $K$ and $\varepsilon_0$.
The above implies in particular
\begin{equation*}
\sup_{0\leq t\leq T}\int_{0}^{t}o\left[\bar \rho(t)^2 +\varepsilon^2 |\bar v(t)|^2 +\varepsilon
\bar c(t)^2\right] dxdt \leq C(K) \delta e^{BT},
\varepsilonnd{equation*}
for a certain $C(K)$ depending on $K$. Therefore, by choosing
$\delta$ small enough such that $C(K)\delta e^{BT}\leq K^2$ we
obtain
\begin{equation*}
\sup_{0\leq t\leq
T}\left(\|\bar \rho(t)\|_{L^2(\mathbb{T}^2)}+\varepsilon\|\bar v(t)\|_{L^2(\mathbb{T}^2)}
+\sqrt{\varepsilon}\|\bar c(t)\|_{L^2(\mathbb{T}^2)}
\right) \leq
K.
\varepsilonnd{equation*}
\textsc{Step 2}. We now perform the energy estimate of the space
derivatives of $(\bar U,\bar c)$. For $j=1,2$ we denote the derivatives
with respect to $x_j$ by the subscript
$\rho_j=\partial_{x_j}\rho$. The system satisfied by the first
derivatives of $(\bar U,\bar c)$ is
\begin{equation}\label{eq:pert4}
\begin{cases}
\partial_t \bar U_j + A_1(\widetilde{U}+\underline{U}) \partial_{x_1}\bar U_j + A_2(\widetilde{U}+\underline{U})
\partial_{x_2}\bar U_j= B(\bar U)_j- A_1(\widetilde{U}+\underline{U})_j \bar U_1 - A_2(\widetilde{U}+\underline{U})_j \bar U_2 & \\
\displaystyle{\partial_t \bar c_j = \frac{1}{\varepsilon}\Delta \bar c_j
+\frac{\alpha}{\varepsilon}\bar \rho_j - \frac{\beta}{\varepsilon}\bar c_j}. &
\varepsilonnd{cases}
\varepsilonnd{equation}
The evaluation of the energy
\begin{equation*}
\mathcal{E}(\bar U_j,\bar c_j)=\frac{1}{2}\int_{0}^{t}o \left[ \bar U_j^T S(\widehat{U}) \bar U_j + \lambda \bar c_j^2\right]dx
\varepsilonnd{equation*}
in a similar way as in (\ref{energyest1}) yields
\begin{align*}
\frac{d}{dt} \mathcal{E}(\bar U_j,\bar c_j) & = \frac{1}{2}\int_{0}^{t}o\bar U_j^T\left[S(\widehat{U})A_1(\widehat{U})\right]_{x_1}\bar U_j dx +
\frac{1}{2}\int_{0}^{t}o\bar U_j^T\left[S(\widehat{U})A_2(\widehat{U})\right]_{x_2}\bar U_j dx\\
& \ + \int_{0}^{t}o\bar U_j^T S(\widehat{U})B(\bar U)_j dx -\frac{\lambda}{\varepsilon}\int_{0}^{t}o |\nabla
\bar c_j|^2 dx + \frac{\lambda\alpha}{\varepsilon}\int_{0}^{t}o\bar \rho_j \bar c_j dx
-\frac{\lambda\beta}{\varepsilon}\int_{0}^{t}o \bar c_j^2 dx\\
& \ -\int_{0}^{t}o \bar U_j^T S(\widehat{U})A_1(\widehat{U})_j \bar U_1 dx -\int_{0}^{t}o \bar U_j^T S(\widehat{U})A_2(\widehat{U})_j \bar U_2 dx.
\varepsilonnd{align*}
Assumption (\ref{ipotesi_induttiva}) allows for the estimate of
the first two terms above as in (\ref{energyest1}), as well as for
the estimate of the last two terms in a similar fashion. The
result is the following estimate
\begin{align*}
\frac{d}{dt} \mathcal{E}(\bar U_j,\bar c_j) & \leq
\widetilde{C}(K) \left(\|\nabla \underline{\rho}\|_{L^\infty} + \|\nabla \underline{v}\|_{L^\infty}\right)
\frac{1}{2}\int_{0}^{t}o\left(\frac{\bar \rho_j^2}{\varepsilon^2} + \frac{|\bar v_j|^2}{\varepsilon^2}\right)
dx+\frac{\|\widehat{\rho}\|_{L^\infty}}{\varepsilon^2}\int_{0}^{t}o \bar v_j\cdot \nabla \bar c_j dx\\
& \
-\frac{(\widetilde{\rho} - K)}{\varepsilon^2}\int_{0}^{t}o |\bar v_j|^2 dx-\frac{\lambda}{\varepsilon}\int_{0}^{t}o |\nabla
\bar c_j|^2 dx + \frac{\lambda\alpha}{\varepsilon}\int_{0}^{t}o\bar \rho_j \bar c_j dx
-\frac{\lambda\beta}{\varepsilon}\int_{0}^{t}o \bar c_j^2 dx,
\varepsilonnd{align*}
which is the equivalent of the estimate (\ref{energyest2}) where
$(\bar U,\bar c)$ are replaced by their first derivatives. Therefore, we
can easily conclude as before
\begin{equation*}
\sup_{0\leq t\leq
T}\left(\|\nabla\bar \rho(t)\|_{L^2(\mathbb{T}^2)}+\varepsilon\|\nabla\bar v(t)\|_{L^2(\mathbb{T}^2)}
+\sqrt{\varepsilon}\|\nabla \bar c(t)\|_{L^2(\mathbb{T}^2)}
\right) \leq
K.
\varepsilonnd{equation*}
\textsc{Step 3}. The second space derivatives of $(\bar U,\bar c)$
satisfy the system
\begin{equation}\label{eq:pert5}
\begin{cases}
\partial_t \bar U_{ij} + A_1(\widetilde{U}+\underline{U}) \partial_{x_1}\bar U_{ij} + A_2(\widetilde{U}+\underline{U})
\partial_{x_2}\bar U_{ij}&\hspace{-3mm}= B(\bar U)_{ij} -A_1(\widetilde{U}+\underline{U})_i\bar U_{1j} - A_2(\widetilde{U}+\underline{U})_i\bar U_{2j} \\
& \ - A_1(\widetilde{U}+\underline{U})_{ij} \bar U_1 - A_2(\widetilde{U}+\underline{U})_{ij} \bar U_2 \\
& \ -A_1(\widetilde{U}+\underline{U})_j \bar U_{1j}-A_2(\widetilde{U}+\underline{U})_j \bar U_{2j} \\
\displaystyle{\partial_t \bar c_{ij} = \frac{1}{\varepsilon}\Delta \bar c_{ij}
+\frac{\alpha}{\varepsilon}\bar \rho_{ij} - \frac{\beta}{\varepsilon}\bar c_{ij}}, &
\varepsilonnd{cases}
\varepsilonnd{equation}
for $i,j=1,2$. The structure of system (\ref{eq:pert5}) is similar
to (\ref{eq:pert4}) and therefore the estimate of the energy
\begin{equation*}
\mathcal{E}(\bar U_{ij},\bar c_{ij})=\frac{1}{2}\int_{0}^{t}o \left[ \bar U_{ij}^T S(\widehat{U}) \bar U_{ij} + \lambda \bar c_{ij}^2\right]dx
\varepsilonnd{equation*}
can be performed as in step 2. The only extra terms which needs to
be analyzed are the following, for $i,j,k=1,2$ ($C(K)$ denotes a
generic constant depending on $K$):
\begin{align*}
& \int_{0}^{t}o \bar U_{ij}^T S(\widehat{U})A_k(\widehat{U})_{ij}\bar U_k \leq \frac{C(K)}{\varepsilon^2} \int_{0}^{t}o |\bar U_{ij}|\left[|\underline{U}_i\underline{U}_j| +|\underline{U}_{ij}|\right]|\bar U_k|dx \\
& \ \leq \frac{K^2 C(K)}{\varepsilon^2}\int_{0}^{t}o |\bar U_{ij}||\bar U_k|dx + \frac{K C(K)}{\varepsilon^2}\int_{0}^{t}o
|\bar U_{ij}||\underline{U}_{ij}|dx\\
& \ \leq \frac{(K+K^2) C(K)}{\varepsilon^2}\left[\int_{0}^{t}o |\bar U_{ij}|^2 dx
+ K^2\right],
\varepsilonnd{align*}
where we have used once again (\ref{ipotesi_induttiva}) and the
result in step 2. Notice that so far we have used $L^\infty$
estimates only up to the firs order derivatives of $\bar U$ and
$\underline{U}$. In the last inequality above, the second derivatives are
only estimated in $L^2$. We have therefore obtained, for $0<K<1$,
\begin{align*}
\frac{d}{dt} \mathcal{E}(\bar U_{ij},\bar c_{ij}) & \leq
\frac{K^3 C(K)}{\varepsilon^2}+ C(K)K
\frac{1}{2}\int_{0}^{t}o\left(\frac{\bar \rho_{ij}^2}{\varepsilon^2} + \frac{|\bar v_{ij}|^2}{\varepsilon^2}\right)
dx+\frac{\|\widehat{\rho}\|_{L^\infty}}{\varepsilon^2}\int_{0}^{t}o \bar v_{ij}\cdot \nabla \bar c_{ij} dx\\
& \
-\frac{(\widetilde{\rho} - K)}{\varepsilon^2}\int_{0}^{t}o |\bar v_{ij}|^2 dx-\frac{\lambda}{\varepsilon}\int_{0}^{t}o |\nabla
\bar c_{ij}|^2 dx + \frac{\lambda\alpha}{\varepsilon}\int_{0}^{t}o\bar \rho_{ij} \bar c_{ij} dx
-\frac{\lambda\beta}{\varepsilon}\int_{0}^{t}o \bar c_{ij}^2 dx
\varepsilonnd{align*}
and, by using the same choice of $\lambda$ and $K$ as in step 1,
after using Gronwall Lemma we obtain
\begin{equation*}
\mathcal{E}(\bar U(t),\bar c(t)) \leq C(K)\left[\mathcal{E}(\bar U(0),\bar c(0))
+\frac{K^3}{\varepsilon^2}\right] e^{Bt}.
\varepsilonnd{equation*}
Then, the coercivity property (\ref{coercivity}) and the
assumptions (\ref{ipotesi_induttiva}) imply
\begin{equation*}
\sup_{0\leq t\leq
T}\left(\|D^2\bar \rho(t)\|_{L^2(\mathbb{T}^2)}+\varepsilon\|D^2\bar v(t)\|_{L^2(\mathbb{T}^2)}
+\sqrt{\varepsilon}\|D^2 \bar c(t)\|_{L^2(\mathbb{T}^2)}
\right) \leq C(K)(\delta + K^3)
\varepsilonnd{equation*}
and clearly, a choice of $\delta$ and $K$ small enough implies
$C(K)(\delta + K^3)<K^2$, which concludes the estimate of the
second derivatives.
\textsc{Step 4}. In order to conclude the proof of the
proposition, one needs to perform the same energy estimate also on
the space derivatives of order $3$ and $4$. All the estimates on
the nonlinear terms on the right--hand side are analogous to those
in Step 3. The integrals with over-quadratic terms always contains
not more than two terms involving more than two derivatives.
Therefore, all the extra terms can be estimated in $L^\infty$ by
using assumption (\ref{ipotesi_induttiva}) and the results in the
previous steps. We shall skip the details of these computations.
The proof is complete.
\varepsilonndproof
We are now ready to state the main theorem of this section.
\begin{thm}
\label{tpert}
Let $T>0$ and let $0<s<4$. Let $(\widetilde{\rho},\widetilde{v},\widetilde{c})$ be the constant state in (\ref{constantstates}). There exists constants $\delta,\varepsilon_0\in
(0,1)$ such that, if the initial data $\rho_0,v_0,c_0$ satisfy
\begin{equation*}
\|\rho_0^\varepsilon-\widetilde{\rho}\|_{H^4(\mathbb{T}^2)} +
\varepsilon\|v_0^\varepsilon\|_{H^4(\mathbb{T}^2)}+\sqrt{\varepsilon}\|c_0^\varepsilon-\widetilde{c}\|_{H^4(\mathbb{T}^2)}
\leq \delta,
\varepsilonnd{equation*}
for all $\varepsilon\in(0,\varepsilon_0)$, then there exists a classical solution
$(\rho^\varepsilon,v^\varepsilon, c^\varepsilon)$ to (\ref{eq:pert1}) such that the quantity
\begin{equation*}
\sup_{0\leq t\leq
T}\left(\|\rho^\varepsilon(t)\|_{H^s(\mathbb{T}^2)}+\varepsilon
\|v^\varepsilon(t)\|_{H^s(\mathbb{T}^2)}+\sqrt{\varepsilon}\|c^\varepsilon(t)\|_{H^s(\mathbb{T}^2)}\right)
\varepsilonnd{equation*}
is uniformly bounded with respect to $\varepsilon\in(0,\varepsilon_0)$ and such that
the density $\rho^\varepsilon$ satisfies
\begin{equation*}
\rho^\varepsilon (x,t)>\widetilde{\rho}/2>0
\varepsilonnd{equation*}
for all $\varepsilon\in(0,\varepsilon_0)$.
\varepsilonnd{thm}
\proof For any fixed $\varepsilon\in(0,\varepsilon_0)$, the sequence
$(\rho^n,v^n,c^n)$ has all space derivatives up to order $4$ in
$L^2$ and all time derivatives up to order $3$ in $L^2$.
Therefore, $(\rho^n,v^n,c^n)$ is relatively strongly compact in
$W^{1,\infty}$ and it converge (up to a subsequence) to a solution
to the original problem (\ref{eq:pert1}). Moreover, the estimate
\begin{equation*}
\sup_{0\leq t\leq
T}\left(\|\rho^\varepsilon(t)\|_{H^s(\mathbb{T}^2)}+\varepsilon
\|v^\varepsilon(t)\|_{H^s(\mathbb{T}^2)}+\sqrt{\varepsilon}\|c^\varepsilon(t)\|_{H^s(\mathbb{T}^2)}\right)\leq
K
\varepsilonnd{equation*}
can be passed to the limit by weak lower semicontinuity and the
proof is complete.
\varepsilonndproof
\begin{rem}
\varepsilonmph{The whole procedure developed in the proof of the above
theorem can be easily generalized to the case of the third scaling
introduced in section \ref{sec:thirdscaling}.}
\varepsilonnd{rem}
\begin{rem}\label{remblowup}\varepsilonmph{We observe here that the power like expression for the
pressure $g(\rho)=\rho^\gamma$ can be replaced by a more general
one in order to achieve the same existence result as in the above
theorem. In particular one can use $g(\rho)=\log \rho$, thus
obtaining a system which relaxes toward a Keller--Segel type
system with linear diffusion. Therefore, some of the relaxation
results contained in chapter \ref{chapconv} would include
Keller--Segel type system with linear diffusion as possible
limits. This fact is not in contradiction with the finite time
blow up phenomena occurring in the latter, because the class of
initial data for which the above theorem holds is not significant
enough in order to see the appearance of blow--up in the limit
system.}
\varepsilonnd{rem}
\varepsilonnd{document} | math | 75,419 |
\begin{document}
\title{Decentralized Quantum Federated
Learning for Metaverse: Analysis, Design and Implementation
}
\author{Dev Gurung, Shiva Raj Pokhrel and Gang Li
\thanks{The authors are with the School of Information Technology, Deakin University, Geelong, VIC 3220, Australia}}
\maketitle
\begin{abstract}
With the emerging developments of the Metaverse, a virtual world where people can interact, socialize, play, and conduct their business, it has become critical to ensure that the underlying systems are transparent, secure, and trustworthy.
To this end, we develop a decentralized and trustworthy quantum federated learning (QFL) framework.
The proposed QFL leverages the power of blockchain to create a secure and transparent
system that is robust against cyberattacks and fraud.
In addition, the decentralized QFL system addresses the risks associated with a centralized server-based approach.
With extensive experiments and analysis, we evaluate classical federated learning (CFL) and QFL in a distributed setting and demonstrate the practicality and benefits of the proposed design.
Our theoretical analysis and discussions develop a genuinely decentralized financial system essential for the Metaverse.
Furthermore, we present the application of blockchain-based QFL in a hybrid metaverse powered by a metaverse observer and world model.
Our \href{https://github.com/s222416822/BQFL} {implementation details and code} are publicly available \footnote{https://github.com/s222416822/BQFL}.
\end{abstract}
\begin{IEEEkeywords}
Quantum Federated Learning, Metaverse, Blockchain
\end{IEEEkeywords}
\section{Introduction}\label{sec-intro}
Quantum Machine Learning (QML)~\cite{biamonte2017quantum} has emerged as a promising paradigm in a number of computationally demanding fields, thanks to the proliferation of quantum computers and the ensuing surge in linear/algebraic computation and operational capabilities.
The underlying physics of QML, such as entanglements, teleportation, and superposition \cite{kwakQuantumDistributedDeep2022} are the key enablers of such high computational efficiency.
When such enablers can be developed under the Federated Learning (FL) framework, such as by employing Quantum Neural Networks
(QNNs) \cite{abbasPowerQuantumNeural2021}, the training,
learning, federation, prediction, and optimization capabilities can leapfrog simultaneously, leading to the development of Quantum FL (QFL).
\begin{figure}
\caption{Peer-to-Peer Blockchain-based Quantum Federated Learning (BQFL) integration into Metaverse.
Such a P2P model with blockchain improves reliability
and trustworthiness considerably.}
\label{fig:bqfl}
\end{figure}
On the other hand, the blockchain has successfully guaranteed the immutability and trustworthiness of FL~\cite{pokhrelFederatedLearningBlockchain2020, pokhrelBlockchainBringsTrust2021} while
facilitating decentralized financial transactions (e.g., cryptocurrency).
It is essential to investigate the potential of blockchain for decentralizing
QFL functionalities. However, the greenfield development of QFL and decentralizing its capabilities are nontrivially challenging tasks.
Despite these challenges, we must overcome them so that the developed QFL not only becomes a part of our future but molds it, propelling us toward a more environmentally and technologically sophisticated civilization. The emerging Metaverse can be taken as an example of such a civilization, which aims to bring together multiple sectors into one
ecosystem and create a virtual environment that mirrors its natural equivalent and maintains persistence.
Taking the metaverse as a motivating example, we
aim to build QFL capabilities to support an interactive
a platform that blends social networking, gaming, and simulation to
create a virtual space replicating the real world.
Furthermore, such QFL can facilitate collaborative learning, which is very important in the metaverse.
\subsection{Motivation and Background}
In
this work, our primary focus is on designing a robust blockchain-based QFL (BQFL) framework that is specifically tailored to support
Metaverse.
In addition to that, we also shed light on the hybrid metaverse.
As illustrated in Figure~\ref{fig:bqfl},
our objective is to develop a peer-to-peer blockchain-based QFL framework suitable for the Metaverse.
One main motivation for this work is the increasing interest in
the growing demand for secure, decentralized, and
trustworthy ML algorithms in the Metaverse.
A QFL based on a centralized global server is prone
to single-point failure issues.
To address this issue, in this work, we propose a
decentralized QFL protocol that takes advantage of the
immutable and distributed nature of blockchain technology to create a more trustworthy
QFL framework.
Eventually, the design of decentralized QFL
ensures the security and trustworthiness of machine learning
algorithms and cryptocurrency transactions in the Metaverse.
\subsection{Related Works}
QFL is an emerging area with several studies in recent years \cite{larasatiQuantumFederatedLearning2022, kaewpuangAdaptiveResourceAllocation2022a,yangDecentralizingFeatureExtraction2021a,yunSlimmableQuantumFederated,xiaQuantumFedFederatedLearning2021a, xiaDefendingByzantineAttacks2021, zhangFederatedLearningQuantum2022}.
Most QFL works focus on optimization
\cite{kaewpuangAdaptiveResourceAllocation2022a,yangDecentralizingFeatureExtraction2021a,yunSlimmableQuantumFederated,xiaQuantumFedFederatedLearning2021a}.
Some works are for security aspects \cite{xiaDefendingByzantineAttacks2021, zhangFederatedLearningQuantum2022, gurungSECURECOMMUNICATIONMODEL2023} and some in terms of implementation \cite{qiFederatedQuantumNatural2022, yamanyOQFLOptimizedQuantumBased2021,abbasPowerQuantumNeural2021, huangQuantumFederatedLearning2022, chehimiQuantumFederatedLearning2022}.
BCFL is well studied in the literature; see \cite{pokhrelFederatedLearningBlockchain2020, pokhrelFederatedLearningMeets2020, chenRobustBlockchainedFederated2021, pokhrelDecentralizedFederatedLearning2020} etc. Pokhrel \textit{et
al.}~\cite{pokhrelFederatedLearningBlockchain2020} introduced the BCFL design to
address privacy concerns and
communication cost in vehicular networks.
Whereas, Chen \textit{et al.}~\cite{chenRobustBlockchainedFederated2021}
addressed the single point of failure and malicious device detection through the distributed validation mechanism.
These works~\cite{zhaoExactDecompositionQuantum2022a,
huangQuantumFederatedLearning2022, chehimiQuantumFederatedLearning2022,
pokhrelFederatedLearningBlockchain2020, chenRobustBlockchainedFederated2021}
have built enough ground to investigate BQFL, to harness the benefits of both blockchain and QFL, which is the main research problem considered in this paper.
Duan \textit{et. al} \cite{duanMetaverseSocialGood2021b} proposed a three-layer
metaverse architecture consists of infrastructure, interaction, and
ecosystem.
Metaverse is based on blockchain and has been studied in a few types of
literature \cite{yangFusingBlockchainAI2022a}, but the use and integration of machine learning, especially QFL, in Metaverse, has not yet been explored.
Bhattacharya \textit{et. al}
\cite{metaverseP2PGaming} proposed FL integration metaverse for the gaming environment.
Chang \textit{et. al}
\cite{chang6GEnabledEdgeAI2022} provided a survey on AI-enabled by 6G for Metaverse.
Zeng \textit{et. al}
\cite{zengHFedMSHeterogeneousFederated2022} proposed a high-performance and efficient FL system for Industrial Metaverse.
The authors partly incorporated a new concept of Sequential-to-parallel (STP) training mode with a fast Inter-Cluster Grouping (ICG) grouping algorithm and claim that it effectively addresses the heterogeneity issues of streaming industrial data for better learning.
\subsection{Contributions}
In summary, the main contributions of this paper are as follows.
\begin{enumerate}
\item We analyze and develop a novel trustworthy blockchain-based quantum federated learning (BQFL), i.e., a decentralized approach to QFL, by presenting and combining the principles of blockchain, quantum computing, and federated learning.
\item We implement BQFL and develop new insights into the feasibility and practicality of BQFL. In addition, we develop substantial reasoning with a thorough theoretical analysis in terms of employing BQFL under Metaverse, considering both privacy and security concerns.
\end{enumerate}
\section{Identified Research Challenges}
We have identified several bottleneck challenges in
the field of classic and quantum FL and their potential
applications for orchestrating Metaverse, all of which are explained in the following.
\begin{enumerate}
\item \textit{Limitations of CFL}:
One of the key limitations of conventional CFL, which is used to
train models across heterogeneous clients and aggregate them at a
the central server is its restricted computational power
\cite{kwakQuantumDistributedDeep2022}. This can undermine the advantages of decentralized learning, especially when clients are a mix of pioneers and stragglers. To address this limitation, the use of QNNs is required, which is known as QFL \cite{abbasPowerQuantumNeural2021}. However, the literature currently lacks a complete analysis and understanding of QNNs over FL, and more research is needed in the field of BQFL to investigate how QNNs work in various FL scenarios, such as blockchain integration and data availability.
\item \textit{Central-Serverless QFL}:
In QFL, only a central server typically aggregates the model parameters, leading to single points of failure and a lack of incentive mechanisms that can limit the performance of QNNs.
To address these issues, blockchain technology can be highly effective in introducing trust into QNNs due to its immutable nature. However, the architecture of blockchain-based QNNs is poorly understood in the literature, and we aim to investigate this by developing a BQFL framework that can resolve these issues.
\item \textit{Challenges of P2P Blockchain QFL}:
Implementing the P2P blockchain FL is a complex and resource-intensive task requiring significant network bandwidth and computational power.
As a result, extensive research work and studies are required in this direction.
\item \textit{External use of Blockchain for QFL}:
This approach will make the implementation simpler and more scalable than the P2P approach. However, it will be less decentralized and trustless than the P2P approach, as it might need a central server.
\item \textit{Challenges in the development of Metaverse}:
Progress in technologies such as Virtual Reality and Augmented Reality (VR/AR) has led to the possibility of the existence and development of Metaverse.
However, problems such as the digital economy's transparency, stability, and sustainability cannot be solved with these technologies alone \cite{duanMetaverseSocialGood2021b}.
Also, one of the main challenges or shortcomings in the development of Metaverse is the lack of a clear and distinctive architectural definition that could be used as a standard blueprint approach.
Some challenges in the development of Metaverse can be:
\begin{enumerate}
\item \textit{How to cope with the computational requirement of Metaverse?}
\item \textit{How to achieve efficient resources allocation and
solve large-scale data complexities?}
\item \textit{Can QFL and Blockchain together solve the
issue?}
\end{enumerate}
\item \textit{Blockchain QFL for Metaverse}:
BQFL is a decentralized approach to federated learning that
combines blockchain technology, quantum computing, and machine learning.
On the other hand, Metaverse is a blockchain-based virtual world that allows users to interact with each other along with their digital assets and experiences, such as Virtual land where one can buy, purchase, and build businesses on the land.
Blockchain technology can facilitate trusted digital ownership, interoperability, and decentralization to improve user experience and create new business models.
\item \textit{Problem with today's quantum computers}:
Current quantum computers are a sort of proof-of-concept that the technology can be built \cite{preskillQuantumComputingNISQ2018}.
But the problem is their infeasibility in real-life applications and the frequent error that occurs in them, referred to as 'noise' in terms of computation.
Thus, this leads to the need for a thorough study and investigation of the implementation of how the blockchain works along with QFL networks.
\item \textit{Multi-Model AI for Metaverse}:
Metaverse requires different types of model learning, not just limited to text recognition.
Thus, the model training needs to learn different forms of data images, speeches, videos, etc.
One of the current works being done towards it is by Meta AI, which refers to as self-supervised learning \cite{baevskiData2vecGeneralFramework}.
Also, a world model is needed specifically for Metaverse because it needs to be able to interpret and understand different forms of data such as text, video, image, etc.
\item \textit{Hybrid Metaverse}:
It won't be an overstatement to say that Metaverse will be our future in some way or another for sure.
However, the metaverse, especially as proposed by Meta (former Facebook), has come under a number of criticisms that doubt its future.
There are many limitations to the purely virtual metaverse.
First, not everyone would love to be stuck in a virtual world 24/7.
Thus, architecture design in the form of a hybrid metaverse that can incorporate both AR and VR simultaneously needs to be redesigned.
Thus, the question is, can we create a metaverse that is a replica of the real world where an actor in that environment can quickly switch between the virtual (VR) and semi-real (AR) world simultaneously?
For example, suppose that a replica of an existing shopping center is created as a shopping center metaverse.
An actor/actress within VR equipment can enter the metaverse replica.
However, while doing so, is it possible for her/him to be
teleported to the actual store that s/he wants to visit and see items, in real interaction with the people in the shopping center as if s/he visits there?
\item \textit{Data Utilization}:
Most metaverse today focuses on avatar creation, with avatar actions limited to interaction.
Thus, with so many actors and devices working together,
the data thus generated must be utilized for an end-to-end solution \cite{metaverseP2PGaming}.
The metaverse involves sensor data, such as the user's physical movement and motion capture, and personal data, like biometric data, etc. which are highly sensitive and personal.
\item \textit{Full potential of Metaverse}:
Due to limited resources and computing power, Metaverse is still far from reaching its full potential of total immersion, materialization, and interoperability \cite{chang6GEnabledEdgeAI2022}.
\end{enumerate}
\section{Preliminaries, Theories, and Ideas}
In this section,
we cover fundamental concepts of quantum machine learning and present
the theoretical concept behind the implementation of the system framework proposed \cite{zhaoExactDecompositionQuantum2022a}.
\subsection{Terms and Terminologies}
Qubit is the fundamental unit of data storage in quantum computing \cite{QuantumDataBaidu}.
Unlike classical computers, which use bits with only two values (0 and 1),
qubits can also take on a range of values due to mechanical superposition.
The number of qubits needed depends on the computational problems that need
to be solved.
The quantum channel refers to the medium used for transferring quantum
information (qubits) \cite{zhaoExactDecompositionQuantum2022a}. Quantum Federated Averaging aims to find a quantum channel that takes an input state and transforms it into the desired output.
Quantum Classifiers are devices that solve classification problems.
The quantum circuit takes the quantum state as an input.
Tensor Circuit \cite{zhangTensorCircuitQuantumSoftware2022a} is an
open-source quantum circuit simulator that supports different features such as automatic
differentiation,
hardware acceleration, etc.
It is especially useful for simulating complex quantum
circuits used in variational algorithms that rely on parameterized quantum
circuits.
Noisy Intermediate-Scale Quantum (NISQ) computers \cite{preskillQuantumComputingNISQ2018} with a limited
number of error-prone qubits are currently the most advanced quantum
computers available \cite{yunSlimmableQuantumFederated2022}.
Quantum computers that are fully fault-tolerant and capable of
running large-scale quantum algorithms are not available at the moment.
Since real quantum computers are not easily accessible, quantum circuit
simulation on classical computers is necessary. The tensor circuit library is
commonly used for this purpose.
The quantum neural network is a variational quantum circuit used in quantum
computing.
Variational quantum circuits (VQC) \cite{arthurHybridQuantumClassicalNeural2022, cerezoVariationalQuantumAlgorithms2021} are a technique that mimics
classical neural networks in quantum computing. It involves training a
dataset, encoding the quantum states as input, producing output quantum
states, and then converting the output back into classical data.
Data Encoding \cite{jerbiQuantumMachineLearning2023} is the process of transforming classical information into
quantum states that can be manipulated by a quantum computer.
Amplitude encoding stores data in the amplitudes of quantum states, while
binary encoding stores information in the state of a qubit. Binary encoding
is preferable for arithmetic computations, while analogue encoding is
suitable for mapping data into the Hilbert space of quantum devices.
Quantum Convolutional Neural Network (QCNN) \cite{yangDecentralizingFeatureExtraction2021a} is a type of neural network
used in quantum computing.
Quantum perceptron is the smallest building block in Quantum Neural Networks (QNNs) \cite{gargAdvancesQuantumDeep2020}.
Blockchain is a decentralized ledger system that
relies on distributed nodes to keep a record of transactions that cannot be
altered once committed.
Smart contracts are programs that automatically execute and follow the terms
of a contract or agreement. Blockchain consensus is a crucial task that
ensures the system's overall reliability and addresses unexpected
behavior from clients or malicious nodes on the network.
Decentralized Applications run autonomously on decentralized computing systems such as the blockchain. In peer-to-peer networks, each
the participating device has equal privileges and the distributed network
architecture is embraced.
\subsection{Design Ideas and Fundamentals}
\subsubsection{Gradient Descent}
To minimize a function $f(\overrightarrow{w})$ and its gradient $\delta
f(\overrightarrow{w})$ starting from an initial point, the optimal approach
is to update the parameters in the direction of the steepest descent, given
by $\overrightarrow{w}_{n+1} = \overrightarrow{w}_n - \eta \Delta
f(\overrightarrow{w})$. This process can be repeated until the function reaches a
local minimum $f(\overrightarrow{w}^*)$. Here, $\eta$ is the step size or
learning rate, which handles the magnitude of the update at each iteration.
\subsubsection{Local Training}
Initially, the first trainer accesses the global parameters $w_g$.
Then, the Adam optimizer, a popular optimization algorithm used in machine
learning, is utilized through the \textit{optax} library for gradient-based optimization.
The learning rate is set to a commonly used value of $1e-2$.
Next, the optimizer state $opt\_state$ is initialized with the current
parameters $w_d$. This state represents the optimizer's internal variables
and state, which are used to update the model parameters during training.
The optimizer state is updated during the training process, which involves
iterating over the training data for a certain number of epochs, $epochs$.
In each iteration, the loss value $loss\_val$ and the gradient value $grad\_val$
are calculated for the current batch using the model parameters $w_d$, input
data $x$, output data $y$ and variable $k$.
The optimizer state $opt\_state$ is updated using the calculated gradients
$grad\_val$ with the current parameters $params$, and the updated values
$updates$ are stored. These updates are then applied to the current model
parameters.
Finally, the mean loss $loss\_mean$ for the current batch is calculated using
$loss\_val$, which is a list of individual loss values for each example in the
batch.
\subsubsection{Class filter [filter function]}
To remove certain labels from a given dataset, one approach is to iterate
over the dataset and filter out samples with undesired labels.
This can be accomplished using a conditional statement to check if each
sample's label is in the list of labels to be removed.
If a sample's label is found in the list, it can be skipped or removed from
the dataset.
\subsubsection{Quantum Circuit [clf function]}
To create a quantum circuit, we define a function that applies $k$ layers of
quantum gates to a given circuit $c$. Each layer consists of gates applied to
each qubit in the circuit. To do this, we iterate over each layer first and then over each qubit in the circuit.
If the 2D numpy array $params$ represents the circuit parameters,
the rotation angles for each qubit are determined based on the corresponding
parameters at a particular index. To create an entangled state, a Controlled
NOT (CNOT) gate is applied to each neighboring pair of qubits in the circuit.
Next, each qubit is rotated about the x-axis with a rotation angle determined by
the corresponding parameter at index [3 * j, i].
Then, a z-axis rotation is applied to each qubit with a rotation angle determined by the
corresponding parameter at index [3 * j + 1, i].
Finally, another x-axis rotation is applied to each qubit with an angle of rotation determined by
the parameters at index [3 * j + 2, i].
Here, $j$ represents the layer number, while $i$ refers to the qubit number.
By applying this series of gates to the circuit, we can create a quantum state with desired
properties.
It is important to note that the specific combination of gates used can have a significant
impact on the resulting quantum state i.e. a careful selection of gates and parameters is critical
for achieving desired outcomes.
\subsubsection{Parameterized Quantum Circuits (PQC) }
When training a parameterized quantum circuit model, the objective is to
learn an arbitrary function from the data. This is done by minimizing a cost or
loss function, denoted as $f(\overrightarrow{w})$, with respect to the
parameter vector $\overrightarrow{w}$. The process involves minimizing the
expectation value, $\bra{\psi(\overrightarrow{w})}\hat{H}\ket{\psi(\overrightarrow{w})}$, where
$\hat{H}$ is the Hamiltonian of the system.
To achieve this, the trainers first send the parameters $w_n$ to the server.
Then, the expectation value is computed as $\bra{\psi(w_n)}\hat{H}\ket{\psi(w_n)}$.
Parameters are updated to $w_{n+1}$, and the process is repeated until
convergence.
Gradient-based algorithms are commonly used to optimize the parameters of a
variational circuit, denoted $\mathbb{U_w}$.
For a PQC, its output is a quantum state $\ket{\psi(w_n)}$, where $w_n$ is a vector of parameters that can be tuned
\cite{zhangTensorCircuitQuantumSoftware2022a}.
\subsubsection{Prediction probabilities [readout function]}
The purpose of this function is to extract probabilities from a given quantum circuit.
The function takes a quantum circuit $c$ as input and generates probabilities using one of two modes, namely
softmax and sampling method.
In "softmax" mode, the function first computes the logits for each node in the neural network,
which are the outputs of the last layer before applying the activation function.
These logits are then used to compute the softmax probabilities.
On the other hand, if the "sample" mode is selected, the function computes the wave-function
probabilities directly and then normalize them to obtain the output probabilities.
\subsubsection{Loss Function}
A neural network loss function is optimized in a quantum circuit taking four input arguments:
network parameters $params$, input data $x$, target data $y$, and the number of quantum circuit layers $k$.
First, a quantum circuit with $n$ qubits is constructed using the input data $x$.
The circuit is then modified by a classifier function that takes input arguments $params$, $c$, and $k$, thus transforming the circuit into a quantum neural network.
The modified circuit is then passed to a read-out function that produces predicted
probabilities for each input. Finally, the loss is computed as the negative logarithmic likelihood
of the predicted probabilities and is averaged over all samples in the batch.
\subsubsection{Accuracy Calculation}
To evaluate the accuracy of the quantum classifier model with given parameters $params$,
input data $x$, target labels $y$ and a number of layers $k$, the following steps are taken:
First, a quantum circuit $c$ is created using the input data $x$.
Then, the function $clf$ is applied to the circuit $c$ with the parameters $params$ to update the
circuit.
The updated circuit is then passed to the $readout$ function to obtain the predicted probabilities
for each label.
Then, the highest probability index is obtained for each input in $x$
and then compared with the true class label index for each input in $y$.
Finally, the precision is calculated by dividing the number of correct predictions
by the total number of inputs in $x$.
\begin{figure}
\caption{Blockchained QFL}
\label{fig:BQFL}
\end{figure}
\section{Proposed BQFL Framework}
\label{sec:proposed}
\begin{figure*}
\caption{Metaverse Powered by BQFL. 1) BQFL training to create a world model for Metaverse Observer, 2) World Model used for different purposes, 3) Actors in Metaverse using AR/VR technology to access Metaverse.}
\label{fig:metaverse}
\end{figure*}
We have proposed a trustworthy Blockchain-based QFL (BQFL) framework
that integrates blockchain with QFL to address various issues related
to privacy, security, and decentralization in machine learning.
The framework consists of two approaches, one where the blockchain is separate from QFL,
and the other where the blockchain is within QFL in a completely peer-to-peer network.
\subsection{Motivating example: Metaverse with BQFL}
Consider a Metaverse space for a city center replicated in a digital twin creating a hybrid space for visiting shops,
shopping, looking at items closely, and meeting people.
Unlike a complete virtual immersive experience,
this metaverse could be a hybrid one that has options for complete immersiveness into the virtual world as well as in augmented reality in the actual store.
In this way, we can respond to people with AR by pushing a button or some specific method.
For this purpose, the whole system network must perform extremely fast with the least delay issues.
Thus, in this regard, as we have found out from experimental and theoretical aspects, BQFL indeed performs better.
\subsection{The BQFL Framework Design}
As shown in Figure \ref{fig:metaverse},
the proposed framework consists of an actual physical space and a virtual replica of the real-world space.
The three main building blocks are the QFL framework, Blockchain, and Metaverse.
For QFL, we have nodes or devices with QNN for training with the local data.
Once all local devices do the local training, then the final averaging of the local models for final global model generation can be done by Metaverse Observer or done in a pure decentralized fashion.
The Metaverse observer is the main entity in the framework which looks after overall activities in the metaverse.
This entity could be designed for each type of specific part in the metaverse to perform specific tasks.
For example, with the proposed framework, the main task for the metaverse observer is to provide knowledge inference, prediction, recommendation, etc. to the actors.
The actors or people participating in the metaverse use AR/VR technology to enter into the virtual (purely virtual world with avatars) or augmented reality version of real physical space.
The overall workflow for the framework is presented in Algorithm \ref{alg:qfl_metaverse}, whereas, the overall local training in the higher level representation is presented in Algorithm \ref{alg:qfl_blockchain}.
\begin{algorithm} [h!]
\caption{Hybrid Metaverse with BQFL powered Observer}
\label{alg:qfl_metaverse}
\begin{algorithmic}[1]
{\cal P}rocedure{\textcolor{blue}{Initialization}}{}
\State Define $N$ number of nodes for QFL.
\State Hybrid Metaverse environment with both AR and VR.
\State $m$ actors and
Metaverse Observer, $\mathbb{O}$
\State Blockchain network.
\State VR/AR replica of the real world.
\State Actors in Metaverse, \{$\mathbb{A}_i\}$ use AR/VR technology.
{\cal E}ndProcedure
{\cal P}rocedure{\textcolor{blue}{Actor Activity}}{}
\State Access replica metaverse using AR or VR headset.
\State Meet people with avatars and interact.
{\cal I}f{AR is chosen}
\State Go to the actual shopping center (teleportation)
{\cal E}lse
\State Stay in VR world.
{\cal E}ndIf
{\cal E}ndProcedure
{\cal P}rocedure{\textcolor{blue}{Device Training}}{}
{\cal F}or{device in devices}
\State Train Local parameters.
\State Send learned parameters for FedAvg.
\State Retrieve global model after FedAvg.
\State
{\cal E}ndFor
{\cal E}ndProcedure
{\cal P}rocedure{\textcolor{blue}{Metaverse Observor}}{}
{\cal F}or{device in devices}
\State Observer performs various tasks.
\State Recommendation System.
\State Actor activity detection.
{\cal I}f{Actor activity legal}
\State Pass
{\cal E}lse
\State Stop Actor.
{\cal E}ndIf
{\cal E}ndFor
{\cal E}ndProcedure
\end{algorithmic}
\end{algorithm}
\subsection{Different Approaches}
\subsubsection{Blockchain separate from QFL - Blockchain externally}
Blockchain can be integrated into QFL in various ways depending on the need for decentralization.
One approach is to use blockchain only for storing transactions such as rewards, model weights,
etc., while the clients that train the model do not have copies of the blockchain.
In this approach, the blockchain nodes can act as miners, and the QFL nodes can be different from the miners.
This approach of using the blockchain externally is a secure and transparent way to
store model updates and other metadata.
The blockchain is not used for coordination or
communication between clients.
The steps involved in integrating blockchain externally in QFL are as follows:
\begin{enumerate}[a.]
\item Set up a blockchain network with nodes to store and validate transactions.
\item Deploy a smart contract.
\item Submit model updates to the smart contract after each round of training by the clients.
\item The smart contract aggregates the updates and generates a global model for the next round of training.
\end{enumerate}
\subsubsection{Blockchain within QFL- Peer to Peer BQFL}
In another approach, we consider decentralized QML in a completely peer-to-peer network
where each client has a copy of the blockchain.
We present a decentralized peer-to-peer network QFL for demonstration and experimental purposes.
The steps involved in integrating blockchain within QFL are as follows.
\begin{enumerate}[a.]
\item Clients communicate with each other through the blockchain network.
\item Blockchain is used to facilitate communication and coordination between clients, along with the storage of updates and other metadata.
\item Each client in the network validates transactions and contributes to the consensus mechanism, ensuring the integrity and security of the system.
\end{enumerate}
This QFL P2P blockchain will allow for a fully decentralized and trustworthy system where clients
communicate and collaborate directly without any central authority or third-party service.
Some of the obvious advantages of this approach are greater privacy, security, and transparency,
with reduced risk of a single point of failure or attack.
\subsubsection{Foundations of BQFL for Metaverse}
The architecture of the proposed BQFL consists of a quantum computing infrastructure, the QFL Algorithm, and Metaverse.
\begin{enumerate}[a.]
\item Quantum Computing Infrastructure:
To train machine learning models using QFL, we require quantum computing resources that are capable of running QFL algorithms.
\item QFL Algorithm:
The QFL algorithm should work in a distributed and privacy-preserving manner among multiple users contributing their local data for the training process.
\item Metaverse:
Metaverse is a virtual world where clients can collaborate and communicate with each other.
\end{enumerate}
\subsubsection{Assumption for the framework design}
We have made the following assumptions.
\begin{enumerate} [a.]
\item FedAvg performs $E$ steps of SGDs in parallel on a set of devices. Opposite to QFL with a central server,
model averaging occurs without a central server architecture.
\item In the presence of stragglers, devices that can become inactive are inevitable. Thus, all devices are assumed to be active throughout the process.
\item Data sets are non-IID.
\end{enumerate}
\begin{algorithm}
\caption{Blockchain QFL}
\label{alg:qfl_blockchain}
\begin{algorithmic}[1]
{\cal P}rocedure{\textcolor{blue}{Initialization}}{}
\State Total $n$ devices, $\{d_1, d_2, d_3, ... , d_n\}$.
{\cal E}ndProcedure
{\cal P}rocedure{\textcolor{blue}{deviceTraining}}{{$pk, sk$}}
{\cal F}or{device $d_i$ in $\{d_i\}$}
\State Encode data.
\State Resize data to fit the quantum circuit.
\State Remove data classes.
\State Train local params $w_i$ with QNN.
\State Send \{$w_i$\} to FedAvg.
{\cal E}ndFor
{\cal E}ndProcedure
{\cal P}rocedure{\textcolor{blue}{FedAvg}}{}
\State Recieve trained model from the nodes.
\State Create a global model.
{\cal E}ndProcedure
\end{algorithmic}
\end{algorithm}
\section{Theoretical Analysis}
In this section, we present theoretical analysis in terms of convergence for blockchain-based quantum federated learning.
\subsection{Convergence Study}
In this section, we examine the convergence properties of the proposed
algorithms.
Here, we analyze the convergence and conditions for convergence under different
assumptions about the data distribution and communication patterns.
From \cite{liConvergenceFedAvgNonIID2020}, we follow these assumptions.
\begin{assumption}
Objective functions ${\cal P}hi_1, \ldots, {\cal P}hi_N$ are all $L$-smooth, which is a standard assumption in federated learning.
\begin{equation}
{\cal P}hi_k(\beta ) \leq {\cal P}hi_k(\theta) + (\beta - \theta)^T \nabla {\cal P}hi_k(\theta) + \frac{L}{2} ||\beta - \theta||^2_2.
\end{equation}
\end{assumption}
where, for any vectors $\beta $ and $\theta$, the function value at $\beta $ is upper-bounded by the function
value at $\theta$, plus a term that depends on the gradient of $A_k$ at $\theta$ and the distance between $\beta $ and $\theta$.
\begin{assumption}
The objective functions ${\cal P}hi_1, \dots, {\cal P}hi_N$ are all $\mu$-strongly convex.
This means that for all $\beta $ and $\theta$, the following inequality holds:
\begin{equation}
{\cal P}hi_k(\beta ) \geq {\cal P}hi_k(\theta) + (\beta -\theta)^T \nabla A_k(\theta) + \frac{\mu}{2}||\beta -\theta||^2_2.
\end{equation}
\end{assumption}
where, $k \in \{1,\dots,N\}$, $\beta$ and $\theta$ are vectors in the same space as the gradients,
and $\mu$ is a positive constant that controls the strength of the convexity of the functions.
\begin{assumption}
Let suppose $\xi_k^t$ be sampled uniformly at random from the local data of the $k^{th}$ device.
The variance of the stochastic gradients in each device is bounded by a constant $\sigma_k^2$ i.e.
\begin{equation}
E ||\nabla {\cal P}hi_{k}\left(\theta_{t}^k, \xi_t^k\right)-\nabla {\cal P}hi_{k}\left(\theta_{t}^k\right)||^{2} \leq \sigma{k}^{2}, \text { for } k=1, \ldots, N.
\end{equation}
\end{assumption}
Here, $\nabla {\cal P}hi_k(\theta_{t}^k, \xi_t^k)$ represents the stochastic gradient of the $k^{th}$ device's
objective function with respect to its local model parameter at iteration $t$, evaluated at the
random data sample $\xi_k^t$.
$\nabla {\cal P}hi_k(\theta_{t}^k)$ represents the average stochastic gradient of
the $k^{th}$ device's objective function with respect to its local model parameter at iteration $t$
evaluated on all the local data samples of the device.
\begin{assumption}
For stochastic gradients, its expected squared norm is uniformly bounded, i.e.,
\begin{equation}
\mathbb{E}||\nabla {\cal P}hi_k(\theta_{t}^k, \xi_t^k)||^2 \leq G^2 \text{ for all } k=1,\dots,N \text{ and } t=1,\dots,T-1.
\end{equation}
\end{assumption}
where, $A_k$ is the objective function of the $k^{th}$ client,
$\theta^k_t$ is the local model parameter of the $k^{th}$ client at iteration $t$,
$\xi^k_t$ is the random data sample from the $k^{th}$ client's local data at iteration $t$,
$\Delta {\cal P}hi_k(\theta^k_t, \xi^k_t)$ is the stochastic gradient of the $k^{th}$ client's objective function with
respect to its local model parameter at iteration $t$, evaluated at the random data sample $\xi^k_t$.
$G$ is the bound on the expected squared norm of stochastic gradients.
Then, from \cite{liConvergenceFedAvgNonIID2020} if assumptions 1-4 hold, then
fedAvg satisfies,
\begin{equation} \label{eq:convergence_fedAvg}
\mathbb{E}[{\cal P}hi(\theta_T)] - {\cal P}hi^* \leq \frac{\kappa}{\gamma + (T-1)}\left(\frac{2B}{\mu} + \frac{\mu \gamma}{2} E||\theta_1 - \theta^*||^2\right),
\end{equation}
where,
\begin{align*}
B = \sum_{k=1}^N p_k^2 \sigma_k^2 + 6L\Gamma + 8(E-1)^2G^2.
\end{align*}
In equation \ref{eq:convergence_fedAvg},
$\mathbb{E}[{\cal P}hi(\theta_{T})] - {\cal P}hi^*$ represents the expected excess risk, $B$ is a constant term,
and $p_k, \sigma_k, L, \Gamma$, and $G$ are parameters that depend on the problem and the algorithm.
Here, $\kappa$ and $\gamma$ are constants defined as $\kappa = \frac{L}{\mu}$ and $\gamma = \max \{ 8\kappa,
E \}$. Whereas, $\eta_t = \frac{2}{\mu(\gamma + t)}$.
\subsection{Considering encoding and decoding time for QFL}
Data encodings for quantum computing is the process of data representation for the quantum state of a system \cite{weigoldEncodingPatternsQuantum2021}.
This process of encoding classical data into a quantum state can be a
significant bottleneck, as it can introduce significant time delays.
The choice of encoding scheme
used will depend on the specific task and the available hardware.
Here, we consider a vanilla data encoding method which is known as "amplitude encoding".
It involves mapping classical data to the amplitudes of a quantum state.
Given an input data vector of length $L$, the quantum state can be represented as:
$$|\psi\rangle = \sum_{i=1}^{L} a_i |i\rangle.$$
Here, $a_i$ is the $i$th element of the input data vector, and $|i\rangle$ is the basic state
corresponding to the binary representation of $i$.
A sequence of gates is applied to encode the data vector into a quantum circuit.
This sets the amplitudes of the quantum state to the values in the input data vector.
In order to do that, a set
of rotation, gates can be used that adjust the phase of each basis state, followed by a set of controlled-NOT (CNOT) gates for the amplitudes.
Even though, the time required to apply a single gate in a quantum circuit is typically on the order of
nanoseconds or less,
but for the whole data vector, it will depend on the number
of qubits and the complexity of the encoding process.
Assuming that we have a quantum circuit with $n$ qubits, the time required to encode a single
element of the input data vector can be approximated as:
$$t_{\text{i}} \approx n \cdot t_{\text{gate}}$$
Here, $t_{\text{gate}}$ is the time required to apply a single gate in the quantum circuit.
Therefore, for the entire input data vector, the total time required to encode the can be approximated as:
\begin{equation} \label{eqn:encoding_time}
t_{\text{T}} \approx L \cdot n \cdot t_{\text{gate}}
\end{equation}
The above equation follows the assumption that we encode each element of the input data vector one at a time.
\subsection{Blockchain Time Delay}
Blockchain time delay can include both communication delay and consensus delay.
Also, the time required to share the new copy of the blockchain ledger with each other can be added to the total delay time.
We also need to consider the time it takes for a block to be appended to the blockchain after it is proposed by a node as well as the block propagation delay.
Suppose we have a blockchain network with $n$ nodes, and each node has a stake $Stake_i$ and a
probability $prob_i$ of being selected as the next validator to create a block.
The probability of a
the node $i$ being selected as the validator can be represented as,
$$prob_i = \frac{Stake_i}{Stake}$$
Let's assume that each node takes $t$ seconds to create a block and the network latency is $L$
seconds.
The time it takes for a block to be created and validated is:
$$T = \max(t, L) + t$$
Now, to calculate the expected time it takes for a block to be created and
validated by the network, given the stake and probability of each node, we can write,
\begin{equation}\label{eqn:blockchain_time}
E[T] = \frac{1}{\sum_{i=1}^n prob_i} \sum_{i=1}^n prob_i T
\end{equation}
\begin{theorem} \label{theorem:main_convergence}
The proposed blockchain-based quantum federated learning algorithm satisfies:
\begin{align}
E[total\_time] \leq \frac{\kappa}{\gamma + (T-1)}\left(\frac{2B}{\mu} + \frac{\mu \gamma}{2} E||\theta_1 - \theta^*||^2\right) + \nonumber\\
\frac{1}{\sum_{i=1}^n prob_i} \sum_{i=1}^n prob_i T + L \cdot n \cdot t_{\text{gate}}
\end{align}
\end{theorem}
\begin{proof}
Using \eqref{eq:convergence_fedAvg}, \eqref{eqn:encoding_time} and \eqref{eqn:blockchain_time},
we can prove that the total time convergence is satisfied as in Theorem \ref{theorem:main_convergence}.
\end{proof}
\subsection{Metaverse Factors}
With important insights form \cite{sicknessReductionTechnology2020}, we can express the meta immersion experience $E_{meta}$ for any user $k$ as,
\begin{align}
E_{meta}^k = D_{rate}^k(1 - Ulink_{errorRate}^k) * VR_e^k
\end{align}
where, $D_{rate}$ is the download data link that impacts lossless virtual experience, $Ulink_{errorRate}$ is uplink tracking bit error rate and $VR_e$ is a virtual experience that is subjective to the user.
For quantifying virtual experience, we can say,
\begin{equation}
VR_e \propto \{activity, onlineTime, ...\}
\label{eqn:vr_experience}
\end{equation}
From Equation \ref{theorem:main_convergence} and \ref{eqn:vr_experience},
\begin{equation}
VR_e \propto E[total\_time]
\label{eqn:vr_experience_prop}
\end{equation}
From our experimental results \ref{fig:performance}, QFL performs faster than CFL.
Thus, this directly implies that with QFL we will have better $VR_e$ than CFL and, in general, better service indicators and technical indicators.
\begin{theorem}
With BQFL, the experience of Metaverse $VR_e$ is always greater or more satisfactory than CFL.
\end{theorem}
\begin{proof}
Different technical and service indicators such as $D_{rate}, Uplink_{errorRate}$, etc. are impacted proportionally to $VR_e$, $E[total\_time]$. Using \eqref{eqn:vr_experience_prop} and Theorem \ref{theorem:main_convergence}, we can prove \textit{Theorem 2}. We will discuss later how our experimental analysis supports the conclusion of \textit{Theorem 2}.
\end{proof}
\subsection{Metaverse Ecosystem}
Different aspects of Metaverse include user behavior prediction, content recommendation, object
recognition, training data, etc.
The metaverse can be considered as a network
of interconnected nodes, where each node is a user,
object, or virtual space, and edges are the relationships or
connections between them.
Graph theory can be used for mathematical
analysis and modeling of such networks.
In the metaverse, various events,
interactions, and behaviors occur probabilistically.
Thus, probability
theory can be used to model the likelihood of events happening
like the probability of encountering a particular object or
meeting a specific user.
With statistical analysis,
understanding patterns, trends, and distributions within the
metaverse can be achieved.
Also, user behavior can be analyzed within the metaverse in a probabilistic manner.
Finally, machine learning algorithms could
be used to predict and simulate user actions based on historical
data, contextual information, and user preferences.
To this end, we have considered three key aspects of metaverse ecosystems that can be orchestrated with BQFL. They are:
\subsubsection{PQ security}
\cite{duanMetaverseSocialGood2021b} For a fair and transparent ecosystem, security is crucial.
This demonstrates an impeding need for the post-quantum secure BQFL.
\subsubsection{Autonomous Governance}
Autonomous Governance is the key to the success of the whole system.
This prevents the system from being controlled by a certain group of people.
\subsubsection{AI-Driven Metaverse Observer}
Duan \textit{et. al} \cite{duanMetaverseSocialGood2021b} presented the idea of an AI-Driven Metaverse Observer, who can track real-time operation data from the Metaverse and analyze it.
This observer can make a recommendation on ongoing events to users.
For this approval rating system can be implemented.
This would provide global information that can assist users to capture timely events in a better way.
\section{Experiments and Results}
To study the integration of blockchain in QFL, we have inherited implementation approaches in
\cite{pokhrelFederatedLearningBlockchain2020},
\cite{chenRobustBlockchainedFederated2021} for BCFL and \cite{zhaoExactDecompositionQuantum2022a} for QFL.
The experiments were run in Google Colab Pro as well as on a local computer.
\subsection{Preprocessing}
Image data are preprocessed for training and testing purposes as follows.
The first pixel values of the input images are scaled in the range of [0,1].
After that, encoding is applied to the images depending on the type of encoding.
With the "vanilla" encoding, the mean is set to 0.
With "mean" encoding, the mean of training images is subtracted from all images.
While with the "half" encoding approach, the images are shifted by 0.5.
Another important step in preprocessing is resizing the image to the size of
$[int(2^{n/2}), int(2^{n/2})]$, where $n$ is the number of qubits used in the quantum circuit.
Eventually, the resized images are flattened to a 1D array of size $2^n$.
Finally, the pixel values are normalized by dividing each image by the square root of the sum of its squared pixel values so that we have a unit length.
The labels for the input images are hot encoded to match the output format of the model.
\subsection{Dataset Preparation}
Quantum computers cannot directly process the classical representations of datasets. The data preparation consists of the following steps:
\begin{enumerate}
\item Data Loading: Libraries like TensorFlow can be used for the loading and splitting of data sets into a training and test set.
With the initial processing of normalization and encoding, the downscaling of the images needs to be performed afterward.
\item Downscaling images: An image size of 28 × 28 is too large for existing quantum computers.
Thus, they need to be resized to a size of 16 × 16 (for an 8-qubit quantum circuit) or \num{4} X \num{4}.
\item Data Encoding is an essential step in QML.
We perform encoding of classical data into states of qubits.
\end{enumerate}
We use the MNIST dataset for experimental purposes.
MNIST dataset consists of 70,000, 28 × 28 images of handwritten digits.
The digits consist of 10 classes (0 to 9).
For this work, we have performed data sharding similar to \cite{zhaoExactDecompositionQuantum2022a}.
In doing so, we remove the samples with labels equal to '8' and '9' from both the training and the testing sets.
We follow a cycle-m structure, where each client has access to the data set to only
$m$ classes at a time.
We also consider $n = 9$ clients as a whole with $7$ assigned as workers and $2$ miners.
Both quantum FedAvg and quantum FedInference \cite{zhaoExactDecompositionQuantum2022a} are experimented with.
For classical learning, normal FedAveraging is used as in \cite{chenRobustBlockchainedFederated2021}.
The batch size is 128 and the learning rate is 0.01.
In terms of evaluation, top 1 accuracy and loss are used as performance metrics.
\begin{figure}
\caption{Cycle-m Sharding}
\label{fig:cycle}
\caption{Dataset Sharding}
\label{fig:data_sharding}
\end{figure}
Figures
\ref{fig:bfql-inf},
\ref{fig:bqfl-avg}, and \ref{fig:bcfl} display the individual test
accuracy plots for
BQFL-inf,
BQFL-avg, and BCFL-avg, respectively.
BQFL-inf performs exceptionally well with non-IID data, as evident from Figure \ref{fig:bfql-inf}.
However, BQFL-avg struggles more with non-IID data, as depicted in Figure \ref{fig:bqfl-avg},
with highly fluctuating accuracy between the Top 1 and the lowest accuracy, as shown in Figure \ref{fig:obervations_avg}.
On the other hand, BCFL-avg performs well with the non-IID dataset, as illustrated in Figure \ref{fig:bcfl}.
\begin{figure}
\caption{Accuracy observations}
\label{fig:obervations_inf}
\caption{Against Classes}
\label{fig:against_classes_inf}
\caption{BQFL-inference Test Accuracy}
\label{fig:bfql-inf}
\end{figure}
\begin{figure}
\caption{Accuracy observations}
\label{fig:obervations_avg}
\caption{Against Classes}
\label{fig:against_classes_avg}
\caption{BQFL-avg Test Accuracy}
\label{fig:bqfl-avg}
\end{figure}
\begin{figure}
\caption{Accuracy observations}
\label{fig:obervations_bfl}
\caption{Against Classes}
\label{fig:against_classes_bfl}
\caption{BCFL-avg Test Accuracy}
\label{fig:bcfl}
\end{figure}
\subsection{Test Performance}
Figure \ref{fig:test_performance} shows the test accuracy plots for BQFL-avg,
BQFL-inf,
and BCFL-avg. Among these, BCFL-avg outperforms
both BQFL-inf and
BQFL-avg in terms of test accuracy.
As the degree of non-IID decreases, the test accuracy of BCFL-avg increases.
However, for BQFL-inf, the test accuracy decreases slightly with an increase in the degree of non-IID.
On the other hand, BQFL-avg suffers greatly with a higher degree of non-IID, especially when training workers with each having only two classes, resulting in a low final test accuracy.
\begin{figure}
\caption{Accuracy observations}
\label{fig:observations_all}
\caption{Against Classes}
\label{fig:against_classes_all}
\caption{Test Accuracy}
\label{fig:test_performance}
\end{figure}
\subsection{Training Performance}
In addition to evaluating the test accuracy, we also analyzed the training performance of the
different FL frameworks.
As shown in Figure \ref{fig:training_performance}, both BQFL-avg
and BQFL-inf
converge faster than BCFL-avg.
BQFL-avg and BQFL-inf share similar training performance, indicating that the additional
communication overhead incurred by BQFL-inf does not result in significant performance degradation.
However, as shown in Figure \ref{fig:test_performance}, BQFL-inf performance is declining when the degree of non-IID increases.
This is the opposite of BQFL-avg in terms of
test accuracy.
In contrast, as shown in Figure \ref{fig:training_performance}, BCFL-avg has a slower convergence rate compared to the BQFL frameworks,
especially when the degree of non-IID is high in terms of training.
Overall, these results suggest that BQFL-avg
and BQFL-inf
can achieve better training performance
than BCFL-avg, while BQFL-inf may not be as robust as BQFL-avg in handling non-IID data.
\begin{figure}
\caption{Training Accuracy}
\label{fig:against_classes}
\caption{Training Loss}
\label{fig:obervations}
\caption{Training Performance}
\label{fig:training_performance}
\end{figure}
\subsection{Impact of Degree of Non-IID}
The impact of the degree of non-IID on the test accuracy of BQFL-Avg,
BQFL-inf and BCFL-
avg is shown in Figure \ref{fig:against_classes_all}. The results reveal that the degree of non-IID
has varying effects on the performance of the different federated learning algorithms.
First, we observe that BQFL-Avg is the most impacted by a higher degree of non-IID, as evidenced by
its decreasing test accuracy with increasing non-IID.
This indicates that BQFL-Avg may not be the
best choice for federated learning scenarios with highly non-IID data distributions.
In contrast, BQFL-inf is not that significantly impacted by the degree of non-IID.
This suggests that
BQFL-inf may be a suitable algorithm for federated learning with non-IID data distributions.
BQFL-avg, on the other hand, shows a more obvious decrease in test accuracy with increasing non-
IID, particularly at higher levels.
This indicates that BQFL-avg may not perform well in
federated learning scenarios with highly non-IID data distributions.
Finally, we observe that BCFL-avg performs consistently well across all levels of non-IID.
In fact,
BCFL-avg outperforms BQFL-avg at all levels of non-IID, indicating that BCFL-avg may be a more robust
algorithm for federated learning with non-IID data distributions.
\subsection{Quantifying Stake Accumulation}
The plot in Figure \ref{fig:stake} indicates that there is a similar trend in stake accumulation for
all three cases of BQFL-avg,
BQFL-inf,
and BCFL-avg.
However, it is worth noting that there are some variations in the initial stages of stake
accumulation, especially for BCFL-avg.
As shown in the plot, stake accumulation starts at a relatively lower level for BCFL-avg,
but it catches up with the other two methods as the stake accumulation progresses.
It is important to consider stake accumulation as it directly affects the selection of
representatives for the consensus protocol.
In scenarios where a selection mechanism is implemented,
higher stake accumulation indicates a higher
probability of a node being selected as a representative, which in turn,
increases its influence in the consensus process.
Therefore, having a steady and predictable stake accumulation rate is crucial for the stability and
security of the consensus protocol.
However, its actual implementation is limited in this work.
\begin{figure}
\caption{BQFL-avg}
\label{fig:stake_avg}
\caption{BQFL-inf}
\label{fig:stake_inf}
\caption{BCFL-avg}
\label{fig:stake_bcfl}
\caption{Stake Accumulation}
\label{fig:stake}
\end{figure}
\subsection{Delay Performance}
In terms of communication time as shown in Figure \ref{fig:comm_time},
BQFL-inf takes the longest time compared to the other algorithms.
BQFL-avg is faster than BCFL-avg in this aspect, which indicates that BQFL-avg can achieve faster
convergence compared to BCFL-avg. However, it's important to note that there is a slight difference
in the way test accuracy and test loss are computed between BQFL and BCFL.
Regarding block generation time as shown in Figure \ref{fig:block_generation}, BCFL-avg takes the longest time of all algorithms.
BQFL-avg and BQFL-inf
have similar performance in terms of block generation time. It is worth mentioning that block
generation time can have a significant impact on the overall performance of the federated learning
algorithm, especially in scenarios where the network has limited resources.
Therefore, a trade-off must be made between communication time and block generation time to achieve optimal performance for a given system.
\begin{figure}
\caption{Block Generation Time}
\label{fig:block_generation}
\caption{Communication Time}
\label{fig:comm_time}
\caption{Performance}
\label{fig:performance}
\end{figure}
\subsection{Accounting Metaverse measures}
For a fully immersive experience in the metaverse, a few metrics are known as the service and technical indicators of user experience and feelings in the metaverse
For example, for a lossless visual experience, a download data link is required to be high enough, i.e., 20–40 Mbps \cite{sicknessReductionTechnology2020}, which impacts the resolution, frame rate, motion blur, etc., that determines the feeling of presence and are service indicators.
In this instance, BQFL can assist in achieving its goal because of its high-performance training and testing, which can be used to create digital twins or any other tasks faster.
From Figures \ref{fig:performance} and \ref{fig:comparison_average_min_max}, it is clear that BQFL-avg performs better than BCFL-avg implying that it is better suited for applications such as metaverse and can fulfill today's increasing data and computational needs.
\begin{figure}
\caption{Max Min Time}
\label{fig:block_generation_minmax}
\caption{Average Time}
\label{fig:comm_time_average}
\caption{Communication Delay Comparison}
\label{fig:comparison_average_min_max}
\end{figure}
\section{Concluding Remarks}
In this work, we have developed a rigorous analysis and design of a BQFL framework, considering
its practicality and implementation in Metaverse.
Extensive theoretical and experimental analysis is done to design and understand the behavior of integration between QFL with the blockchain.
We have developed new insights and explained significant results with new findings. Our experimental results demonstrated the practicality of BQFL.
However, extensive further research is required to fully understand the practicality of BQFL and its application for the metaverse application.
which requires further investigation.
\end{document} | math | 58,098 |
\begin{document}
\title{Solution of Hypergraph Turan problem }
\date{July 5, 2015}
\author{Vladimir Blinovsky\footnote{The author was supported by the Sao Paulo Research Foundation (FAPESP), Project no 2014/23368-6 and
NUMEC/USP (Project MaCLinC/USP).
}}
\date{\small
Instituto de Matematica e Estatistica, USP,\\
Rua do Matao 1010, 05508- 090, Sao Paulo, Brazil\\
Institute for Information Transmission Problems, \\
B. Karetnyi 19, Moscow, Russia,\\
[email protected]}
\maketitle
\begin{center}
{\bf Abstract}
Using original {\it Symmetrical Smoothing Method} we solve hypergraph $(3,k)$-Turan problem \end{center}
Let $X$ be a finite set $|X|=n$. Define ${X\choose k}$ to be the family of $k$-element subsets of $X$. We say that family (hypergraph) ${\cal A}\subset {X\choose 3}$ satisfies $(3,k)$-Turan property if for the arbitrary
$A\in{[n]\choose k}$ it follows that
$$
|B\in{\cal A}: B\in A |<{k\choose 3}.
$$
There is a number of sites and conferences devoted to this problem (see~\cite{9},~\cite{10},~\cite{11}).
We investigate the following
{\bf Problem}. For given $n>k>3$ find the maximal (at least one) family which satisfies $(3,k)$- Turan property.
This is famous Turan problem. Actually Turan problem in general case (not only $3$- hypergraph but $m$-hypergraph) is the key problem in Erdos's extremal combinatorics. And the case $m=3$ is the first nontrivial
case of great importance. For the surveys and references see~\cite{1},~\cite{2}.
Next we assume that $(k-1)|n$. For other cases it is necessary to find proper explicit symmetrical constructions
for the optimal Turan hypergraph. I am lasy to do this and it is not needed for asymptotic Turan problem (see~\cite{3},~\cite{4}).
To solve this problem we make some preliminary preparations.
We use the natural bijection between $2^{[n]}$ and $\{ 0,1\}^n$ and don't make difference between these two sets.
Define
$$
\varphi (x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\xi^2 /2}d\xi ,
$$
\begin{eqnarray*}
N(\{\sigma\} )&=&\sum_{x\in {[n]\choose 3}}\varphi\left(\left( \left(\sum_{i=1}^M \varphi\left(((x,\beta_i )-3)/\sigma\right)\right) -1/2 )\right)/\sigma\right),\\
R(\{ \sigma\} ) &=& {n\choose k} -\sum_K \sum_{x\in K}\varphi\left(\left(\left(\varphi\left(
\left(\sum_{i=1}^{M}\varphi\left(((x,\beta_i )-3)/\sigma\right)\right) -1/2 )\right)/\sigma\right) -(1-\epsilon )\right)/\sigma\right) .
\end{eqnarray*}
Here $K's$ are complete $3$-hypergraphs on vertices $K\subset [n],\ |K|=k$. It can be easily seen
that $N$ is convex and $R$ is concave function of $\{\beta_{i}\}$. We assume that only three coordinates
are nonzero and these nonzero coordinates are equal to $1$.
We have for $N\to M/2$ as $\sigma\to 0$ and if hypergraph has $(3,k)$- Turan property, then $R=o(1)$ as $\sigma\to 0$.
Necessary and sufficient condition for minimization of $M$ is the Kuhn- Tucker conditions:
\begin{eqnarray}
\label{e1}
&&N^\prime_{\beta_{i,j}}=\lambda R^\prime_{\beta_{i,j}} ,\\
&&R =o(1), \ \sigma\to\infty .\nonumber
\end{eqnarray}
Consider the following well known construction of $(3,k)$-Turan hypergraph. We describe the complement hypergraph $\bar{T}={[n]\choose 3}\setminus T$. We divide set of vertices $[n]$ into $k-1$ equal parts $B_1 ,
\ldots , B_{k-1}$ of size $n/(k-1)$.
Hypergraph $\bar{T}$ consists of edges in each part $B_i$ and all edges such that each of them has two vertices in $B_i$ and one vertex in $B_{i+1 |\hbox{mod} k-1}$.
It is easy to see, that $T$ satisfies $(3,k)$-Turan property.
Note that we find solution of~(\ref{e1}) not among all vertices $\beta_i$ but with specific condition that only three coordinates $\beta_{i,1},\beta_{i,2},\beta_{i,3}$ are nonzero and $\beta_{i,1}+\beta_{i,2}+\beta_{i,3}=3$.
Assume that $\beta_{i,1}=\beta_{i,2}=\beta_{i,3}=1$.
It is easy to see that
$$
N^\prime_{\beta_i,1}=R^\prime_{\beta_i ,1}=0
$$
if $(\beta_{i,1},\beta_{i,3})$ are on the positions of $x$ which belongs the same cell $\bar{T}_r$ and
\begin{eqnarray*}
N^\prime_{\beta_{i_1 ,1}}&=&N^\prime_{\beta_{i_2 ,2}} ,\\
R^\prime_{\beta_{i_1 ,1}}&=&R^\prime_{\beta_{i_2 ,2}}
\end{eqnarray*}
if $(\beta_{i_1 ,1},\beta_{i_2 ,2})$ are components of consecutive (cyclically) sells. From here it follows that there exists unique $\lambda$ such that equations (\ref{e1}) are satisfied and hence construction of $T$ is optimal.
Easy consequence of optimality of $T$ is the validity of the following famous Turan
\begin{Co}
$$
\lim_{n\to\infty}\frac{|T|}{{n\choose 3}}=1-\left(\frac{2}{k-1}\right)^2 .
$$
\end{Co}
At last note that original Symmetrical Smoothing Method allows to prove optimality of the solution of extremal
problems for (binary and not only) sequences in many cases when sufficiently symmetric constructions which are optimal are suggested. We are going to show how to solve many such problems in forthcoming papers.
\end{document} | math | 4,888 |
\begin{equation}gin{document}
\title{\vspace*{-2cm}
Lattice paths and branched continued fractions \\[5mm]
II.~Multivariate Lah polynomials \\ and Lah symmetric functions
}
\author{ \\
\hspace*{-1cm}
{\large Mathias P\'etr\'eolle${}^1$ and Alan D.~Sokal${}^{1,2}$}
\\[5mm]
\hspace*{-1.3cm}
{\mathbf{n}}ormalsize
${}^1$Department of Mathematics, University College London,
London WC1E 6BT, UK \\[1mm]
\hspace*{-2.9cm}
{\mathbf{n}}ormalsize
${}^2$Department of Physics, New York University,
New York, NY 10003, USA
\\[5mm]
\hspace*{-0.5cm}
{\tt [email protected]}, {\tt [email protected]} \\[1cm]
}
\maketitle
\thispagestyle{empty}
\begin{equation}gin{abstract}
We introduce the generic Lah polynomials $L_{n,k}(\bm{\phi})$,
which enumerate unordered forests of increasing ordered trees
with a weight $\phi_i$ for each vertex with $i$ children.
We show that, if the weight sequence $\bm{\phi}$ is Toeplitz-totally positive,
then the triangular array of generic Lah polynomials is totally positive
and the sequence of row-generating polynomials $L_n(\bm{\phi},y)$
is coefficientwise Hankel-totally positive.
Upon specialization we obtain results for the
Lah symmetric functions and multivariate Lah polynomials
of positive and negative type.
The multivariate Lah polynomials of positive type
are also given by a branched continued fraction.
Our proofs use mainly the method of production matrices;
the production matrix is obtained by a bijection
from ordered forests of increasing ordered trees
to labeled partial \L{}ukasiewicz paths.
We also give a second proof of the continued fraction
using the Euler--Gauss recurrence method.
\end{abstract}
{\mathbf{n}}oindent
{\bf Key Words:}
Lah polynomial, Bell polynomial, Eulerian polynomial, symmetric function,
increasing tree, forest, \L{}ukasiewicz path, Stirling permutation,
continued fraction, branched continued fraction,
production matrix, Toeplitz matrix, Hankel matrix,
total positivity, Toeplitz-total positivity, Hankel-total positivity.
{\mathbf{n}}oindent
{\bf Mathematics Subject Classification (MSC 2010) codes:}
05A15 (Primary);
05A05, 05A18, 05A19, 05A20, 05C30, 05E05, 11B37, 11B73,
15B48, 30B70 (Secondary).
\vspace*{1cm}
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\tableofcontents
\section{Introduction and statement of main results}
In a seminal 1980 paper, Flajolet \cite{Flajolet_80}
showed that the coefficients in the Taylor expansion
of the generic Stieltjes-type (resp.\ Jacobi-type) continued fraction
--- which he called the {\em Stieltjes--Rogers}\/
(resp.\ {\em Jacobi--Rogers}\/) {\em polynomials}\/ ---
can be interpreted as the generating polynomials
for Dyck (resp.\ Motzkin) paths with specified height-dependent weights.
Very recently it was independently discovered by several authors
\cite{Fusy_15,Oste_15,Josuat-Verges_18,Sokal_totalpos}
that Thron-type continued fractions also have an interpretation of this kind:
namely, their Taylor coefficients
--- which we call, by analogy, the {\em Thron--Rogers polynomials}\/ ---
can be interpreted as the generating polynomials
for Schr\"oder paths with specified height-dependent weights.
In a recent paper \cite{latpath_SRTR}
we presented an infinite sequence
of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials,
which are parametrized by an integer $m \ge 1$
and reduce to the classical Stieltjes--Rogers and Thron--Rogers polynomials
when $m=1$;
they are the generating polynomials of $m$-Dyck and $m$-Schr\"oder paths,
respectively, with height-dependent weights,
and are also the Taylor coefficients of certain branched continued fractions.
We proved that these generalizations all possess the fundamental property of
coefficientwise Hankel-total positivity \cite{Sokal_flajolet,Sokal_totalpos},
jointly in all the (infinitely many) indeterminates.
These facts were known when $m = 1$ \cite{Sokal_flajolet,Sokal_totalpos}
but were new when $m > 1$.
By specializing the indeterminates we were able to give many examples
of Hankel-totally positive sequences
whose generating functions do not possess nice classical continued fractions.
(The concept of Hankel-total positivity \cite{Sokal_flajolet,Sokal_totalpos}
will be explained in more detail later in this Introduction.)
In particular, in \cite[section~12]{latpath_SRTR}
we introduced the multivariate Eulerian polynomials
and Eulerian symmetric functions:
these are generating polynomials for increasing trees and forests
of various types (see below for precise definitions),
which vastly extend the classical univariate Eulerian
and $r$th-order Eulerian polynomials;
we proved their coefficientwise Hankel-total positivity.
Here we would like to refine this analysis by considering
(among other things) the row-generating polynomials:
this leads to defining multivariate Lah polynomials
and Lah symmetric functions,
which extend the classical univariate Lah polynomials.
So let us begin by reviewing briefly some well-known univariate
combinatorial polynomials;
then we define our multivariate and symmetric-function extensions.
Recall first that the {\em Bell number}\/ $B_n$
is the number of partitions of an $n$-element set into nonempty blocks;
by convention $B_0 = 1$.
Refining this, the {\em Stirling subset number}\/
(also called {\em Stirling number of the second kind}\/)
$\stirlingsubset{n}{k}$ is
the number of partitions of an $n$-element set into $k$ nonempty blocks;
by convention $\stirlingsubset{0}{k} = \delta_{k0}$.
The {\em Bell polynomials}\/ $B_n(x)$ are then defined as
$B_n(x) = \sum_{k=0}^n \stirlingsubset{n}{k} x^k$.\footnote{
See \cite[A008277/A048993]{OEIS} for further information
on the Stirling subset numbers and Bell polynomials.
}
Similarly, the {\em Lah number}\/ $L_n$
is the number of partitions of an $n$-element set
into nonempty linearly ordered blocks (also called {\em lists}\/);
we set $L_0 = 1$.
Refining this, the {\em Lah number}\/ $L(n,k)$ is
the number of partitions of an $n$-element set
into $k$ nonempty linearly ordered blocks;
we set $L(0,k) = \delta_{k0}$.
The Lah numbers also have the explicit expression
\begin{equation}
L(n,k) \;=\; {n! \over k!} \binom{n-1}{n-k}
\;=\; \begin{equation}gin{cases}
\delta_{k0} & \textrm{if $n=0$} \\[2mm]
{n! \over k!} \binom{n-1}{k-1}
& \textrm{if $n \ge 1$}
\end{cases}
\end{equation}
The {\em Lah polynomials}\/ $L_n(x)$ are then defined as
$L_n(x) = \sum_{k=0}^n L(n,k) \: x^k$.\footnote{
See \cite[A008297/A105278/A271703/A111596/A066667]{OEIS}
for further information on the Lah numbers and Lah polynomials.
}
More generally, let $x$ and ${\mathbf{w}} = \{w_m\}_{m \ge 1}$ be indeterminates,
and let $P_n(x,{\mathbf{w}})$ be the generating polynomial
for partitions of an $n$-element set into nonempty blocks
in~which each block of cardinality $m$ gets a weight $x w_m$:
\begin{equation}
P_n(x,{\mathbf{w}})
\;\eqdef\;
\sum_{\pi \in \Pi_n} x^{|\pi|} \prod_{B \in \pi} w_{|B|}
\;.
\label{def.Pnxw}
\end{equation}
(In particular, the empty set has a unique partition into nonempty blocks
--- namely, the partition with zero blocks --- so that $P_0(x,{\mathbf{w}}) = 1$.)
Then the Bell polynomials correspond to $w_m = 1$,
while the Lah polynomials correspond to $w_m = m!$.
It is not difficult to show that the polynomials $P_n(x,{\mathbf{w}})$
have the exponential generating function
\begin{equation}
\sum_{n=0}^\infty P_n(x,{\mathbf{w}}) \, {t^n \over n!}
\;=\;
\exp \biggl( x \sum_{m=1}^\infty w_m \, {t^m \over m!} \biggr)
\;.
\end{equation}
The polynomials $P_n(x,{\mathbf{w}})$ are also known
\cite[pp.~133--134]{Comtet_74}
as the
{\em complete Bell polynomials}\/ ${\bf Y}_n(xw_1,\ldots,xw_n)$.
Let us now express the Bell and Lah polynomials
in terms of a different combinatorial object,
namely, unordered forests of increasing ordered trees.
Recall first \cite[pp.~294--295]{Stanley_86}
that an {\em ordered tree}\/ (also called {\em plane tree}\/)
is a rooted tree in which the children of each vertex are linearly ordered.
An {\em unordered forest of ordered trees}\/
is an unordered collection of ordered trees.
An {\em increasing ordered tree}\/ is an ordered tree
in which the vertices carry distinct labels from a linearly ordered set
(usually some set of integers) in such a way that
the label of each child is greater than the label of its parent;
otherwise put, the labels increase along every path downwards from the root.
An {\em unordered forest of increasing ordered trees}\/
is an unordered forest of ordered trees with the same type of labeling.
Now let ${\bm{p}}hi = (\phi_i)_{i \ge 0}$ be indeterminates,
and let $L_{n,k}({\bm{p}}hi)$ be the generating polynomial for
unordered forests of increasing ordered trees on the vertex set $[n]$,
having $k$ components (i.e.\ $k$ trees),
in which each vertex with $i$ children gets a weight $\phi_i$.
Clearly $L_{n,k}({\bm{p}}hi)$ is a homogeneous polynomial of degree $n$
with nonnegative integer coefficients;
it is also quasi-homogeneous of degree $n-k$
when $\phi_i$ is assigned weight~$i$.
The first few polynomials $L_{n,k}({\bm{p}}hi)$
[specialized for simplicity to $\phi_0 = 1$] are
\vspace*{2mm}
\begin{equation}gin{table}[H]
\centering
\footnotesize
\begin{equation}gin{tabular}{c|cccccc}
$n \setminus k$ & 0 & 1 & 2 & 3 & 4 & 5 \\
{\rm hl}ine
0 & 1 & & & & & \\
1 & 0 & 1 & & & & \\
2 & 0 & $\phi_1$ & 1 & & & \\
3 & 0 & $\phi_1^2 + 2 \phi_2$ & $3 \phi_1$ & 1 & & \\
4 & 0 & $\phi_1^3 + 8 \phi_1 \phi_2 + 6 \phi_3$ & $7 \phi_1^2 + 8 \phi_2$ & $6 \phi_1$ & 1 & \\
5 & 0 & $\phi_1^4 + 22 \phi_1^2 \phi_2 + 16 \phi_2^2 + 42 \phi_1 \phi_3 + 24 \phi_4$ & $15 \phi_1^3 + 60 \phi_1 \phi_2 + 30 \phi_3$ & $25 \phi_1^2 + 20 \phi_2$ & $10 \phi_1$ & 1 \\
\end{tabular}
\end{table}
{\mathbf{n}}oindent
(see also the Appendix for $n \le 7$).
Now let $y$ be an additional indeterminate,
and define the row-generating polynomials
$L_n({\bm{p}}hi,y) = \sum_{k=0}^n L_{n,k}({\bm{p}}hi) \: y^k$.
Then $L_n({\bm{p}}hi,y)$ is quasi-homogeneous of degree $n$
when $\phi_i$ is assigned weight~$i$ and $y$ is assigned weight~1.
We call $L_{n,k}({\bm{p}}hi)$ and $L_n({\bm{p}}hi,y)$
the \textbfit{generic Lah polynomials},
and we call the lower-triangular matrix ${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$
the \textbfit{generic Lah triangle}.
Here ${\bm{p}}hi = (\phi_i)_{i \ge 0}$ are in the first instance indeterminates,
so that $L_{n,k}({\bm{p}}hi) \in {\mathbb Z}[{\bm{p}}hi]$ and $L_n({\bm{p}}hi,y) \in {\mathbb Z}[{\bm{p}}hi,y]$;
but we can then, if we wish, substitute specific values for ${\bm{p}}hi$
in any commutative ring $R$,
leading to values $L_{n,k}({\bm{p}}hi) \in R$ and $L_n({\bm{p}}hi,y) \in R[y]$.
When doing this we use the same notation $L_{n,k}({\bm{p}}hi)$ and $L_n({\bm{p}}hi,y)$,
as the desired interpretation for ${\bm{p}}hi$ should be clear from the context.
Note, finally, that an
unordered forest of increasing ordered trees on the vertex set $[n]$,
with $k$~components, can be obtained by first choosing a partition of $[n]$
into $k$~nonempty blocks, and then constructing an increasing ordered tree
on each block. It follows that the generic Lah polynomial $L_n({\bm{p}}hi,y)$
equals the set-partition polynomial $P_n(y,{\mathbf{w}})$
evaluated at $w_m = L_{m,1}({\bm{p}}hi)$.
Now let $X = (x_i)_{i \ge 1}$ be indeterminates,
and let $\begin{equation}e = (e_n(X))_{n \ge 0}$ and ${\bm{h}} = (h_n(X))_{n \ge 0}$
be the elementary symmetric functions
and complete homogeneous symmetric functions, respectively;
they are elements of the ring ${\mathbb Z}[[X]]_{\rm sym}$ of symmetric functions,
considered as a subring of the formal-power-series ring ${\mathbb Z}[[X]]$.
We then define the \textbfit{Lah symmetric functions of positive type} by
$L_{n,k}^{(\infty)+}(X) = L_{n,k}(\begin{equation}e)$ and
$L_n^{(\infty)+}(X,y) = L_n(\begin{equation}e,y)$,
and the \textbfit{Lah symmetric functions of negative type} by
$L_{n,k}^{(\infty)-}(X) = L_{n,k}({\bm{h}})$ and
$L_n^{(\infty)-}(X,y) = L_n({\bm{h}},y)$.
Also, for any integer $r \ge 1$ we can imagine specializing $X$
by setting $x_i = 0$ for $i > r$;
we then define the
\textbfit{multivariate Lah polynomials of positive type} by
$L_{n,k}^{(r)+}(x_1,\ldots,x_r) = L_{n,k}(\begin{equation}e(x_1,\ldots,x_r))$ and
$L_n^{(r)+}(x_1,\ldots,x_r;y) = L_n(\begin{equation}e(x_1,\ldots,x_r),y)$,
and the \textbfit{multivariate Lah polynomials of negative type} by
$L_{n,k}^{(r)-}(x_1,\ldots,x_r) = L_{n,k}({\bm{h}}(x_1,\ldots,x_r))$ and
$L_n^{(r)-}(x_1,\ldots,x_r;y) = L_n({\bm{h}}(x_1,\ldots,x_r),y)$.\footnote{
In \cite[section~12]{latpath_SRTR} we considered these quantities
only for $n,k \ge 1$, and we used the notations
${\mathcal{Q}}_{n,k}^{(\infty)}$, ${\mathcal{Q}}_{n,k}^{(r)}$,
${\mathcal{Q}}_{n,k}^{(\infty)-}$, ${\mathcal{Q}}_{n,k}^{(r)-}$
(with $n,k \ge 0$)
for what we are now calling
$L_{n+1,k+1}^{(\infty)+}$, $L_{n+1,k+1}^{(r)+}$,
$L_{n+1,k+1}^{(\infty)-}$, $L_{n+1,k+1}^{(r)-}$,
respectively;
we called these the {\em Eulerian symmetric functions}\/
and {\em multivariate Eulerian polynomials}\/.
We now think that it might be preferable to reserve the term ``Eulerian''
for quantities associated to trees, and to use instead the term ``Lah''
for quantities associated to forests.
}
In the Appendix we report the Lah symmetric functions
$L_{n,k}^{(\infty)+}$ and $L_{n,k}^{(\infty)-}$ for $n \le 7$
in terms of the monomial symmetric functions $m_\lambda$.
These multivariate Lah polynomials and symmetric functions
can also be interpreted as generating polynomials for increasing
$r$-ary and multi-$r$-ary trees and forests ($1 \le r \le \infty$).
Let us recall first \cite[p.~295]{Stanley_86}
the recursive definition of an {\em $r$-ary tree}\/ ($1 \le r < \infty$):
it is either empty or else consists of a root together with
an ordered list of $r$ subtrees, each of which is an $r$-ary tree
(which may be empty). We draw an edge from each vertex to the root
of each of its nonempty subtrees; an edge from a vertex to the root
of its $i$th subtree will be called an {\em $i$-edge}\/.
Similarly, we can define recursively an {\em $\infty$-ary tree}\/:
it is either empty or else consists of a root together with
an ordered list of subtrees indexed by the positive integers ${\mathbb P}$,
each of which is an $\infty$-ary tree (which may be empty)
{\em and only finitely many of which are nonempty}\/;
we define $i$-edges as before.\footnote{
Please note that such a graph is necessarily finite
(as always, the recursion is carried out only finitely many times).
}
But we can now view $r$-ary trees from a slightly different point of view:
an $r$-ary (resp.\ $\infty$-ary) tree is simply an ordered tree
in which each edge carries a label $i \in [r]$ (resp.\ $i \in {\mathbb P}$)
and the edges emanating outwards from each vertex consist, in order,
of zero or one edges labeled 1, then zero or one edges labeled 2,
and so forth; an edge with label $i$ will be called an $i$-edge.
Let us now consider the generating polynomial for
unordered forests of increasing $\infty$-ary trees on the vertex set $[n]$,
having $k$~components, in which each $i$-edge gets a weight $x_i$.
Since the choice of labels on the edges emanating outwards from a vertex $v$
can be made independently for each $v$,
this is equivalent to evaluating the generating polynomial $L_{n,k}({\bm{p}}hi)$
at $\phi_i = e_i(X)$;
in other words, it is the Lah symmetric function of positive type
$L_{n,k}^{(\infty)+}(X) = L_{n,k}(\begin{equation}e)$.
Similarly, $L_n^{(\infty)+}(X,y) = L_n(\begin{equation}e,y)$
is the generating polynomial for
unordered forests of increasing $\infty$-ary trees on the vertex set $[n]$,
in which each $i$-edge gets a weight $x_i$
and each tree (or equivalently, each root) gets a weight $y$.
And if we set $x_i = 0$ for $i > r$ so as to obtain $r$-ary trees or forests,
we get the multivariate Lah polynomials
$L_{n,k}^{(r)+}(x_1,\ldots,x_r) = L_{n,k}(\begin{equation}e(x_1,\ldots,x_r))$
and $L_n^{(r)+}(x_1,\ldots,x_r;y) = L_n(\begin{equation}e(x_1,\ldots,x_r),y)$.
The multivariate Lah polynomials and Lah symmetric functions of negative type
can be interpreted in a similar way.
We begin by adopting the reinterpretation of $r$-ary and $\infty$-ary trees
as ordered trees with labeled edges,
and then consider \cite[section~10.3.2]{latpath_SRTR}
the variant in which the number of edges
of each label emanating from a given vertex,
instead of being ``zero or one'', is ``zero or more'':
we call this a \emph{multi-$r$-ary}
(resp.\ \emph{multi-$\infty$-ary}) \emph{tree}.
We now consider the generating polynomial for
unordered forests of increasing multi-$\infty$-ary trees
on the vertex set $[n]$,
having $k$~components, in which each $i$-edge gets a weight $x_i$.
This is equivalent to evaluating the generating polynomial $L_{n,k}({\bm{p}}hi)$
at $\phi_i = h_i(X)$;
in other words, it is the Lah symmetric function of negative type
$L_{n,k}^{(\infty)-}(X) = L_{n,k}({\bm{h}})$.
Similarly, $L_n^{(\infty)-}(X,y) = L_n({\bm{h}},y)$
is the generating polynomial for
unordered forests of increasing multi-$\infty$-ary trees
on the vertex set $[n]$,
in which each $i$-edge gets a weight $x_i$
and each tree (or equivalently, each root) gets a weight $y$.
And if we set $x_i = 0$ for $i > r$ so as to obtain
multi-$r$-ary trees or forests,
we get the multivariate Lah polynomials
$L_{n,k}^{(r)-}(x_1,\ldots,x_r) = L_{n,k}({\bm{h}}(x_1,\ldots,x_r))$
and $L_n^{(r)-}(x_1,\ldots,x_r;y) = L_n({\bm{h}}(x_1,\ldots,x_r),y)$.
Let us now consider the further specialization
of the multivariate Lah polynomials of positive type
to $x_1 = \ldots = x_r = 1$, corresponding to $\phi_i = \binom{r}{i}$.
It is well known \cite[p.~24]{Stanley_86}
that the number of increasing binary trees on $n$ vertices is $n!$,
and more generally that the number of increasing $r$-ary trees
on $n$ vertices is the multifactorial $F_n^{(r-1)}$
\cite[p.~30, Example~1]{Bergeron_92},
where
\begin{equation}
F_n^{(r)} \;\eqdef\; \prod_{j=0}^{n-1} (1+jr)
\;.
\label{def.multifactorial}
\end{equation}
Therefore, the univariate $r$th-order Lah polynomials of positive type,
$L_n^{(r)+}({\bm{1}};y)$, coincide with the set-partition polynomials $P_n(y,{\mathbf{w}})$
defined in \reff{def.Pnxw} when we set $w_m = F_m^{(r-1)}$.
In particular, for $r=1$ we have $w_m = 1$
and obtain the Bell polynomials $B_n(y)$;
for $r=2$ we have $w_m = m!$
and obtain the univariate Lah polynomials $L_n(y)$;
for $r=3,4,5$ we have $w_m = (2m-1)!!, (3m-2)!!!, (4m-3)!!!!$
and obtain the row-generating polynomials
of \cite[A035342, A035469, A049029]{OEIS}.
In a similar way, we can specialize
the multivariate Lah polynomials of negative type
to $x_1 = \ldots = x_r = 1$, corresponding to $\phi_i = \binom{r+i-1}{i}$.
It is known \cite[p.~30, Corollary~1(iv)]{Bergeron_92}
that the number of increasing multi-unary trees on $n$ vertices
is $(2n-3)!!$;
more generally,
it was observed in \cite[section~12.3]{latpath_SRTR}
that the number of increasing multi-$r$-ary trees
on $n$ vertices is the shifted multifactorial ${\mathbf{w}}idetilde{F}_{n-1}^{(r+1)}$,
where
\begin{equation}
{\mathbf{w}}idetilde{F}_n^{(r)} \;\eqdef\; \prod\limits_{j=0}^{n-1} [(r-1)+jr]
\;.
\end{equation}
Therefore, the univariate $r$th-order Lah polynomials of negative type,
$L_n^{(r)-}({\bm{1}};y)$, coincide with the set-partition polynomials $P_n(y,{\mathbf{w}})$
defined in \reff{def.Pnxw} when we set $w_m = {\mathbf{w}}idetilde{F}_{m-1}^{(r+1)}$.
In particular, for $r=1$ we have $w_m = (2m-3)!!$
and obtain a variant of the Bessel polynomials \cite[A001497]{OEIS}
(see also Example~\ref{exam.phi=1} below);
for $r=2,3$ we have $w_m = (3m-4)!!!, (4m-5)!!!!$
and obtain the row-generating polynomials of \cite[A004747, A000369]{OEIS}.
Let us now explain how all this relates to total positivity.
Recall first that a finite or infinite matrix of real numbers is called
{\em totally positive}\/ (TP) if all its minors are nonnegative,
and {\em totally positive of order~$r$} (TP${}_r$)
if all its minors of size $\le r$ are nonnegative.
Background information on totally positive matrices can be found
in \cite{Karlin_68,Gantmacher_02,Pinkus_10,Fallat_11};
they have application to many fields of pure and applied mathematics.
In particular, it is known
\cite[Th\'eor\`eme~9]{Gantmakher_37} \cite[section~4.6]{Pinkus_10}
that a Hankel matrix $(a_{i+j})_{i,j \ge 0}$
of real numbers is totally positive if and only if the underlying sequence
$(a_n)_{n \ge 0}$ is a Stieltjes moment sequence
(i.e.\ the moments of a positive measure on $[0,\infty)$).
And a Toeplitz matrix $(a_{i-j})_{i,j \ge 0}$ of real numbers
(where $a_n = 0$ for $n < 0$)
is totally positive if and only if its ordinary generating function
can be written as
\begin{equation}
\sum_{n=0}^\infty a_n t^n
\;=\;
C e^{\gamma t} t^m \prod_{i=1}^\infty {1 + \alpha_i t \over 1 - \begin{equation}ta_i t}
\label{eq.thm.aissen}
\end{equation}
with $m \in {\mathbb N}$, $C,\gamma,\alpha_i,\begin{equation}ta_i \ge 0$,
$\sum \alpha_i < \infty$ and $\sum \begin{equation}ta_i < \infty$
\cite[Theorem~5.3, p.~412]{Karlin_68}.
But this is only the beginning of the story,
because we are here principally concerned,
not with sequences and matrices of real numbers,
but with sequences and matrices of polynomials
(with integer or real coefficients) in one or more indeterminates~${\mathbf{x}}$:
they will typically be generating polynomials that enumerate
some combinatorial objects with respect to one or more statistics.
We equip the polynomial ring ${\mathbb R}[{\mathbf{x}}]$ with the coefficientwise
partial order: that is, we say that $P$ is nonnegative
(and write $P \succeq 0$)
in case $P$ is a polynomial with nonnegative coefficients.
We then say that a matrix with entries in ${\mathbb R}[{\mathbf{x}}]$ is
\textbfit{coefficientwise totally positive}
if all its minors are polynomials with nonnegative coefficients;
and analogously for coefficientwise total positivity of order~$r$.
We say that a sequence ${\bm{a}} = (a_n)_{n \ge 0}$ with entries in ${\mathbb R}[{\mathbf{x}}]$
is \textbfit{coefficientwise Hankel-totally positive}
(resp.\ \textbfit{coefficientwise Toeplitz-totally positive})
if its associated infinite Hankel (resp.\ Toeplitz) matrix
is coefficientwise totally positive;
and likewise for the versions of order $r$.
Similar definitions apply to the formal-power-series ring ${\mathbb R}[[{\mathbf{x}}]]$.
Most generally, we can consider sequences and matrices
with values in an arbitrary partially ordered commutative ring
(a precise definition will be given in Section~\ref{subsec.totalpos.prelim});
total positivity, Hankel-total positivity and Toeplitz-total positivity
are then defined in the obvious way.
Let us also explain some partial orders on the ring of symmetric functions.
Let~$R$ be a commutative ring
and let ${\mathbf{X}} = (x_i)_{i \ge 1}$ be a countably infinite collection
of indeterminates.
Then let $R[[{\mathbf{X}}]]_{\rm sym}$ be the ring of symmetric functions
with coefficients in $R$
\cite[Chapter~1]{Macdonald_95} \cite[Chapter~7]{Stanley_99};
it is a subring of the formal-power-series ring $R[[{\mathbf{X}}]]$.
(It goes without saying that ``function'' is a misnomer;
these are formal power series.)
Now let $R$ carry a partial order ${\mathcal{P}}$.
When the coefficientwise order on $R[[{\mathbf{X}}]]$
is restricted to $R[[{\mathbf{X}}]]_{\rm sym}$,
it becomes the \textbfit{monomial order}:
an element $f \in R[[{\mathbf{X}}]]_{\rm sym}$ is monomial-nonnegative
if and only~if it can be written as a (finite) nonnegative linear combination
of monomial symmetric functions $m_\lambda({\mathbf{X}})$.
However, the ring of symmetric functions can also be equipped
with a stronger order, namely the \textbfit{Schur order}:
an element $f \in R[[{\mathbf{X}}]]_{\rm sym}$ is Schur-nonnegative
if and only~if it can be written as a (finite) nonnegative linear combination
of Schur functions $s_\lambda({\mathbf{X}})$.
This indeed defines a partial order compatible with the ring structure,
since any product of Schur functions
is a nonnegative linear combination of Schur functions
(Littlewood--Richardson coefficients \cite[Section~7.A1.3]{Stanley_99}).
And it is strictly stronger than the monomial order,
because every Schur function
is a nonnegative linear combination of monomial symmetric functions
(Kostka numbers \cite[eq.~(7.35), p.~311]{Stanley_99})
but not conversely (e.g.\ $m_2 = s_2 - s_{11}$).
We can now state our main result:
\begin{equation}gin{theorem}[Total positivity of Lah matrices and Lah polynomials]
\label{thm1.1}
{\bm{r}}eak{\mathbf{n}}oindent
Fix $1 \le r \le \infty$.
Let $R$ be a partially ordered commutative ring,
and let ${\bm{p}}hi = (\phi_i)_{i \ge 0}$ be a sequence in $R$
that is Toeplitz-totally positive of order $r$.
Then:
\begin{equation}gin{itemize}
\item[(a)] The lower-triangular matrix
${\sf L}({\bm{p}}hi) = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$
is totally positive of order $r$ in the ring $R$.
\item[(b)] The sequence ${\bm{L}}({\bm{p}}hi) = (L_n({\bm{p}}hi,y))_{n \ge 0}$
is Hankel-totally positive of order $r$ in the ring $R[y]$
equipped with the coefficientwise order.
\item[(c)] The sequence ${\bm{L}}^\mathop{\rm tr}{\mathbf{n}}olimitsiangle({\bm{p}}hi) = (L_{n+1,1}({\bm{p}}hi))_{n \ge 0}$
is Hankel-totally positive of order $r$ in the ring $R$.
\end{itemize}
\end{theorem}
Specializing this to ${\bm{p}}hi = \begin{equation}e(X)$ or ${\bm{h}}(X)$
and using the Jacobi--Trudi identity
\cite[Theorem~7.16.1 and Corollary~7.16.2]{Stanley_99},
we obtain:
\begin{equation}gin{corollary}[Total positivity of Lah symmetric functions]
\label{cor1.2}
{\bm{r}}eak{\mathbf{n}}oindent
\vspace*{-6mm}
\begin{equation}gin{itemize}
\item[(a)] The unit-lower-triangular matrices
${\sf L}^{(\infty)+} = (L_{n,k}(\begin{equation}e(X)))_{n,k \ge 0}$ and
${\sf L}^{(\infty)-} =$ \linebreak $(L_{n,k}({\bm{h}}(X)))_{n,k \ge 0}$
are totally positive with respect to the Schur order
on the ring of symmetric functions (with coefficients in ${\mathbb Z}$).
\item[(b)] The sequences
${\bm{L}}^{(\infty)+} = (L_n(\begin{equation}e(X),y))_{n \ge 0}$ and
${\bm{L}}^{(\infty)-} = (L_n({\bm{h}}(X),y))_{n \ge 0}$
are Hankel-totally positive with respect to the Schur order
on the ring of symmetric functions (with coefficients in ${\mathbb Z}$)
and the coefficientwise order on polynomials in $y$.
\item[(c)] The sequences
${\bm{L}}^{(\infty)+\mathop{\rm tr}{\mathbf{n}}olimitsiangle} = (L_{n+1,1}(\begin{equation}e(X)))_{n \ge 0}$ and
${\bm{L}}^{(\infty)-\mathop{\rm tr}{\mathbf{n}}olimitsiangle} = (L_{n+1,1}({\bm{h}}(X)))_{n \ge 0}$
are Hankel-totally positive with respect to the Schur order
on the ring of symmetric functions (with coefficients in ${\mathbb Z}$).
\end{itemize}
\end{corollary}
Weakening this result from the Schur order to the monomial order,
and then further specializing by setting $x_i = 0$ for $i > r$,
we obtain:
\begin{equation}gin{corollary}[Total positivity of multivariate Lah polynomials]
{\mathbf{n}}opagebreak
\label{cor1.3}
{\bm{r}}eak{\mathbf{n}}oindent
{\mathbf{n}}opagebreak
\vspace*{-6mm}
\begin{equation}gin{itemize}
\item[(a)] The unit-lower-triangular matrices
${\sf L}^{(r)+} = (L_{n,k}(\begin{equation}e(x_1,\ldots,x_r)))_{n,k \ge 0}$ and
${\sf L}^{(r)-} = (L_{n,k}({\bm{h}}(x_1,\ldots,x_r)))_{n,k \ge 0}$
are totally positive with respect to the coefficientwise order
on the polynomial ring ${\mathbb Z}[x_1,\ldots,x_r]$.
\item[(b)] The sequences
${\bm{L}}^{(r)+} = (L_n(\begin{equation}e(x_1,\ldots,x_r),y))_{n \ge 0}$ and
${\bm{L}}^{(r)-} =$ \qquad
\linebreak
$(L_n({\bm{h}}(x_1,\ldots,x_r),y))_{n \ge 0}$
are Hankel-totally positive with respect to the coefficientwise order
on the polynomial ring ${\mathbb Z}[x_1,\ldots,x_r,y]$.
\item[(c)] The sequences
${\bm{L}}^{(r)+\mathop{\rm tr}{\mathbf{n}}olimitsiangle} = (L_{n+1,1}(\begin{equation}e(x_1,\ldots,x_r)))_{n \ge 0}$ and
${\bm{L}}^{(r)-\mathop{\rm tr}{\mathbf{n}}olimitsiangle} =$ \qquad
\linebreak
$(L_{n+1,1}({\bm{h}}(x_1,\ldots,x_r)))_{n \ge 0}$
are Hankel-totally positive with respect to the coefficientwise order
on the polynomial ring ${\mathbb Z}[x_1,\ldots,x_r]$.
\end{itemize}
\end{corollary}
{\bf Remarks.}
1. In Theorem~\ref{thm1.1} and its two corollaries,
part~(c) follows trivially from part~(b)
by dividing $L_{n+1}({\bm{p}}hi,y)$ by $y$
and then specializing to $y=0$.
But in Section~\ref{sec.proofs} we will introduce a generalization
where the analogue of~(c) still holds (by a different proof),
but it is unknown whether there is any analogue of~(b).
2. Corollaries~\ref{cor1.2}(a) and \ref{cor1.3}(a)
are essentially already contained in
\cite[Corollaries~12.5 and 12.25 and Remark after the proof of Lemma~12.13]{latpath_SRTR}.
But the extension in Theorem~1.1(a) to the generic Lah polynomials is new.
3. Similarly, Corollaries~\ref{cor1.2}(c) and \ref{cor1.3}(c)
are already contained in \cite[Corollaries~12.3 and 12.24]{latpath_SRTR}.
Indeed, a slightly stronger version of Corollaries~\ref{cor1.2}(c)
and \ref{cor1.3}(c) for ${\sf L}^{(\infty)+}$ and ${\sf L}^{(r)+}$
--- in which the Hankel-totally positive sequence is extended backwards
by prepending one element ---
is given in
\cite[Theorem~12.1(a) and Corollaries~12.3 and 12.7]{latpath_SRTR}.
There is also an analogous result for the case of negative type
\cite[Theorem~12.20(a) and Corollary~12.22]{latpath_SRTR},
but it does not seem to imply (at least in any obvious way)
the negative-type case of Corollaries~\ref{cor1.2}(c) and \ref{cor1.3}(c).
${\rm bl}acksquare$
Our proof of Theorem~\ref{thm1.1} will be based on the method of
{\em production matrices}\/ \cite{Deutsch_05,Deutsch_09}.
We shall review this theory
in Sections~\ref{subsec.production} and \ref{subsec.totalpos.prodmat},
so now we state only the bare-bones definitions.
Let $P = (p_{ij})_{i,j \ge 0}$ be an infinite matrix
with entries in a commutative ring~$R$;
we assume that $P$ is either row-finite
(i.e.\ has only finitely many nonzero entries in each row)
or column-finite.
Now define an infinite matrix $A = (a_{nk})_{n,k \ge 0}$ by
\begin{equation}
a_{nk} \;=\; (P^n)_{0k}
\;.
\end{equation}
We call $P$ the \textbfit{production matrix}
and $A$ the \textbfit{output matrix}, and we write $A = {\mathcal{O}}(P)$.
The two key facts here are the following \cite{Sokal_totalpos}:
if $R$ is a partially ordered commutative ring
and $P$ is totally positive of order $r$,
then ${\mathcal{O}}(P)$ is totally positive of order $r$
and the zeroth column of ${\mathcal{O}}(P)$ is Hankel-totally positive of order $r$.
See Section~\ref{subsec.totalpos.prodmat} for precise statements and proofs.
We shall prove Theorem~\ref{thm1.1} by explicitly constructing
the production matrix that generates the
the generic Lah triangle ${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$,
and then verifying its total positivity.
Let ${\mathbb D}elta = (\delta_{i+1,j})_{i,j \ge 0}$
be the matrix with 1 on the superdiagonal and 0 elsewhere.
We then have:
\begin{equation}gin{proposition}[Production matrix for the generic Lah triangle]
\label{prop.prodmat}
Let ${\bm{p}}hi = (\phi_i)_{i \ge 0}$ and $y$ be indeterminates,
and work in the ring ${\mathbb Z}[{\bm{p}}hi,y]$.
Define the lower-Hessenberg matrix $P = (p_{ij})_{i,j \ge 0}$ by
\begin{equation}
p_{ij}
\;=\;
\begin{equation}gin{cases}
0 & \textrm{if $j=0$ or $j > i+1$} \\[1mm]
{\displaystyle {i! \over (j-1)!}} \, \phi_{i-j+1}
& \textrm{if $1 \le j \le i+1$}
\end{cases}
\label{eq.prop.prodmat}
\end{equation}
and the unit-lower-triangular $y$-binomial matrix $B_y$ by
\begin{equation}
(B_y)_{nk} \;=\; \binom{n}{k} \, y^{n-k}
\;.
\end{equation}
Then:
\begin{equation}gin{itemize}
\item[(a)] $P$ is the production matrix for the generic Lah triangle
${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$.
\item[(b)] $B_y^{-1} P B_y = P (I + y {\mathbb D}elta^{\rm T})$
is the production matrix for ${\sf L} B_y$.
\end{itemize}
\end{proposition}
We will prove Proposition~\ref{prop.prodmat}(a)
by constructing a bijection from
{\em ordered}\/ forests of increasing ordered trees
to labeled partial \L{}ukasiewicz paths,
along the lines of \cite[proofs of Theorems~12.11 and 12.28]{latpath_SRTR}.
Then Proposition~\ref{prop.prodmat}(b) will follow
by a straightforward but slightly nontrivial computation
(Lemma~\ref{lemma.ByinvPBy}).
In fact, we will prove a generalization of Proposition~\ref{prop.prodmat}(a)
[and hence also of Theorem~\ref{thm1.1}(a,c)
and Corollaries~\ref{cor1.2}(a,c) and \ref{cor1.3}(a,c)]
for some polynomials ${\widehat{L}}_{n,k}({\bm{p}}hihat)$,
to be defined in Section~\ref{subsec.proofs.prodmat},
that depend on a refined set of indeterminates
${\bm{p}}hihat = (\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$
and that reduce to $L_{n,k}({\bm{p}}hi)$ when $\phi_i^{[L]} = \phi_i$ for all $L$.
However, no analogue of Proposition~\ref{prop.prodmat}(b)
[and hence also of Theorem~\ref{thm1.1}(b)
and Corollaries~\ref{cor1.2}(b) and \ref{cor1.3}(b)]
appears to exist for these more general polynomials.
{\bf Remark.}
If we were to work in the ring ${\mathbb Q}[{\bm{p}}hi,y]$ instead of ${\mathbb Z}[{\bm{p}}hi,y]$,
we would have $P = D T_\infty({\bm{p}}hi) D^{-1} {\mathbb D}elta$
where $T_\infty({\bm{p}}hi)$ is the infinite lower-triangular Toeplitz matrix
associated to the sequence ${\bm{p}}hi$,
and $D = \mathop{\rm diag}{\mathbf{n}}olimits\bigl( (n!)_{n \ge 0} \bigr)$.
${\rm bl}acksquare$
Now return to the situation of Theorem~\ref{thm1.1}.
If the ring $R$ contains the rationals (with their usual order),
it follows from $P = D T_\infty({\bm{p}}hi) D^{-1} {\mathbb D}elta$
that $P$ is totally positive of order $r$
whenever ${\bm{p}}hi$ is Toeplitz-totally positive of order $r$;
and the same holds for $B_y^{-1} P B_y = P (I + y {\mathbb D}elta^{\rm T})$.
And even if $R$ does not contain the rationals,
it~turns out that the same conclusions are true,
as we can show with a bit more work (Lemma~\ref{lemma.diagmult.TP}).
Theorem~\ref{thm1.1} is then an immediate consequence
of Proposition~\ref{prop.prodmat} and Lemma~\ref{lemma.diagmult.TP}
together with the general theory of production matrices and total positivity
(Section~\ref{subsec.totalpos.prodmat}).
Now fix an integer $r \ge 1$, and let us consider the
multivariate Lah polynomials of positive type
by specializing the production matrix \reff{eq.prop.prodmat}
to $\phi_n = e_n(x_1,\ldots,x_r)$.
Recall that the product of two lower-triangular Toeplitz matrices
corresponds to the convolution of their generating sequences,
or equivalently the product of their ordinary generating functions;
and since
$\sum\limits_{n=0}^\infty e_n(x_1,\ldots,x_r) \, t^n
= \prod\limits_{i=1}^r (1 + x_i t)$,
it follows that the lower-triangular Toeplitz matrix
$T_\infty(\begin{equation}e(x_1,\ldots,x_r))$
has the factorization $T_\infty(\begin{equation}e) = L_1 \cdots L_r$
where $L_i = L(1,1,\ldots;x_i,x_i,\ldots)$
is the lower-bidiagonal Toeplitz matrix with 1 on the diagonal
and $x_i$ on the subdiagonal.
Therefore, $D T_\infty(\begin{equation}e) D^{-1}$
has the factorization $L'_1 \cdots L'_r$
where $L'_i = D L_i D^{-1} = L(1,1,\ldots;x_i,2x_i,3x_i,\ldots)$;
the production matrix
$P(x_1,\ldots,x_r) = D T_\infty(\begin{equation}e) D^{-1} {\mathbb D}elta$
has the factorization $L'_1 \cdots L'_r {\mathbb D}elta$;
and the modified production matrix
$B_y^{-1} P B_y = P (I + y {\mathbb D}elta^{\rm T})$ has the factorization
\begin{equation}
B_y^{-1} \, P(x_1,\ldots,x_r) \, B_y
\;=\;
L'_1 \,\cdots\, L'_r \, ({\mathbb D}elta + yI)
\label{eq.lah.prodmat.y}
\end{equation}
(since ${\mathbb D}elta {\mathbb D}elta^{\rm T} = I$).
On the other hand, \reff{eq.lah.prodmat.y}
is precisely the production matrix for an $r$-branched S-fraction
with coefficients
\begin{equation}
{\bm{a}}lpha \;=\; (\alpha_i)_{i \ge r}
\;=\; y,x_1,\ldots,x_r,y,2x_1,\ldots,2x_r,y,3x_1,\ldots,3x_r,\ldots
\label{eq.lah.alphas}
\end{equation}
(see~\cite[Propositions~7.2 and 8.2(b)]{latpath_SRTR}
and eq.~\reff{eq.prop.contraction} below).
Since the zeroth column of the matrix ${\sf L} B_y$
is given by the Lah polynomials $L_n({\bm{p}}hi,y)$,
it follows that the multivariate Lah polynomials of positive type
$L_n^{(r)+}(x_1,\ldots,x_r;y)$
are given by an $r$-branched S-fraction with coefficients \reff{eq.lah.alphas}:
\begin{equation}gin{theorem}[Branched S-fraction for multivariate Lah polynomials of positive type]
\label{thm.lah.S-fraction}
We have $L_n^{(r)+}(x_1,\ldots,x_r;y) = S_n^{(r)}({\bm{a}}lpha)$
where the coefficients ${\bm{a}}lpha$ are given by \reff{eq.lah.alphas}.
[Here $S_n^{(r)}({\bm{a}}lpha)$ is the $r$-Stieltjes--Rogers polynomial
of order $n$; see Section~\ref{subsec.mSR} for the precise definition.]
\end{theorem}
{\bf Remarks.}
1. For $r=1$ the multivariate Lah polynomials of positive type
are simply the homogenized Bell polynomials
$L_n^{(1)+}(x_1;y) = x_1^n B_n(y/x_1)$,
and this is the well-known classical S-fraction
with coefficients ${\bm{a}}lpha = y,x_1,y,2x_1,y,3x_1,\ldots$.
2. Since $L_n^{(r)+}(x_1,\ldots,x_r;y)$ is invariant under
permutations of $x_1,\ldots,x_r$, it is actually represented by
$r!$ different $r$-branched S-fractions in which the coefficients ${\bm{a}}lpha$
are obtained from \reff{eq.lah.alphas} by permuting $x_1,\ldots,x_r$.
This illustrates the nonuniqueness of $r$-branched S-fractions for $r \ge 2$
\cite{latpath_SRTR}.
3. In Section~\ref{sec.euler-gauss} we will also give a completely independent
proof of Theorem~\ref{thm.lah.S-fraction}, based on the Euler--Gauss
recurrence method.
${\rm bl}acksquare$
For the multivariate Lah polynomials of negative type,
the lower-triangular Toeplitz matrix $T_\infty({\bm{h}}(x_1,\ldots,x_r))$
has the factorization $T_\infty({\bm{h}}) = {\widetilde{L}}_1 \cdots {\widetilde{L}}_r$
where ${\widetilde{L}}_i =$ \linebreak $L(1,1,\ldots;-x_i,-x_i,\ldots)^{-1}$;
so $D T_\infty({\bm{h}}) D^{-1}$
has the factorization ${\widetilde{L}}'_1 \cdots {\widetilde{L}}'_r$
where
${\widetilde{L}}'_i = D {\widetilde{L}}_i D^{-1} = L(1,1,\ldots;-x_i,-2x_i,-3x_i,\ldots)^{-1}$;
and the production matrix $D T_\infty({\bm{h}}) D^{-1} {\mathbb D}elta$
has the factorization ${\widetilde{L}}'_1 \cdots {\widetilde{L}}'_r {\mathbb D}elta$.
But the matrices ${\widetilde{L}}_i$ and ${\widetilde{L}}'_i$
are dense lower-triangular, not lower-bidiagonal,
so we do not see any way of interpreting this as the production matrix
of a branched S-fraction.
Indeed, we have verified
that the multivariate Lah {\em numbers}\/ of negative type
$L_n^{(r)-}(1,\ldots,1;1)$
{\em cannot}\/ be expressed as an $m$-branched S-fraction
of the following types:
\begin{equation}gin{itemize}
\item For $r=1,2,3$, the numbers $L_n^{(r)-}(1,\ldots,1;1)$
cannot be expressed as a 2-branched S-fraction
with {\em nonnegative integer}\/ coefficients:
this was verified by exhaustive computer search using
$n \le 7,8,7$, respectively.
\item For $m > r+1$, the numbers $a_n = L_n^{(r)-}(1,\ldots,1;1)$
cannot be expressed as an $m$-branched S-fraction
with {\em positive integer}\/ coefficients:
this is simply because $a_0 = a_1 = 1$ and $a_2 = r+1$,
while the $m$-Stieltjes--Rogers polynomial
$S_2^{(m)}({\bm{a}}lpha)$ equals
$\alpha_m (\alpha_m + \alpha_{m+1} + \cdots + \alpha_{2m-1})$.
\end{itemize}
For $(r,m) = (1,3)$ and $(2,3)$,
our computations (as far as we were able to go)
were unable to give either a proof of nonexistence
(with nonnegative integer coefficients)
or a comprehensible candidate for ${\bm{a}}lpha$.
Although the present paper is a follow-up to our paper \cite{latpath_SRTR},
we have endeavored, for the convenience of the reader,
to make it as self-contained as possible.
We have therefore begun (Section~\ref{sec.prelim})
with a brief review of the key definitions and results from \cite{latpath_SRTR}
that will be needed in the sequel.
We then proceed as follows:
In Section~\ref{sec.proofs} --- which is the technical heart of the paper ---
we prove Proposition~\ref{prop.prodmat},
from which we deduce Theorem~\ref{thm1.1};
indeed, we state and prove a generalization involving
a refined set of indeterminates.
In Section~\ref{sec.differential}
we give expressions for the multivariate Lah polynomials
of positive and negative type
in terms of the action of certain first-order linear differential operators.
In Section~\ref{sec.euler-gauss} we give a second proof
of Theorem~\ref{thm.lah.S-fraction},
based on the differential operators and the Euler--Gauss recurrence method.
In Section~\ref{sec.decorated} we interpret
the multivariate Lah polynomials of positive type $L_n^{(r)+}({\mathbf{x}},y)$
as generating polynomials for partitions of the set $[n]$
in~which each block is ``decorated'' with an additional structure.
In Section~\ref{sec.exponential} we compute explicit expressions
for the generic Lah polynomials $L_{n,k}({\bm{p}}hi)$
by using exponential generating functions.
In the Note Added (Section~\ref{sec.exp_riordan})
we give an alternate proof of Proposition~\ref{prop.prodmat}(a),
using the theory of exponential Riordan arrays.
In the Appendix we report the generic Lah polynomials
and Lah symmetric functions for $n \le 7$.
\section{Preliminaries} \label{sec.prelim}
Here we review some definitions and results from \cite{latpath_SRTR}
that will be needed in the sequel.
\subsection{Partially ordered commutative rings and total positivity}
\label{subsec.totalpos.prelim}
In this paper all rings will be assumed to have an identity element 1
and to be nontrivial ($1 {\mathbf{n}}e 0$).
A \textbfit{partially ordered commutative ring} is a pair $(R,{\mathcal{P}})$ where
$R$ is a commutative ring and ${\mathcal{P}}$ is a subset of $R$ satisfying
\begin{equation}gin{itemize}
\item[(a)] $0,1 \in {\mathcal{P}}$.
\item[(b)] If $a,b \in {\mathcal{P}}$, then $a+b \in {\mathcal{P}}$ and $ab \in {\mathcal{P}}$.
\item[(c)] ${\mathcal{P}} \cap (-{\mathcal{P}}) = \{0\}$.
\end{itemize}
We call ${\mathcal{P}}$ the {\em nonnegative elements}\/ of $R$,
and we define a partial order on $R$ (compatible with the ring structure)
by writing $a \le b$ as a synonym for $b-a \in {\mathcal{P}}$.
Please note that, unlike the practice in real algebraic geometry
\cite{Brumfiel_79,Lam_84,Prestel_01,Marshall_08},
we do {\em not}\/ assume here that squares are nonnegative;
indeed, this property fails completely for our prototypical example,
the ring of polynomials with the coefficientwise order,
since $(1-x)^2 = 1-2x+x^2 {\mathbf{n}}ot\succeq 0$.
Now let $(R,{\mathcal{P}})$ be a partially ordered commutative ring
and let ${\mathbf{x}} = \{x_i\}_{i \in I}$ be a collection of indeterminates.
In the polynomial ring $R[{\mathbf{x}}]$ and the formal-power-series ring $R[[{\mathbf{x}}]]$,
let ${\mathcal{P}}[{\mathbf{x}}]$ and ${\mathcal{P}}[[{\mathbf{x}}]]$ be the subsets
consisting of polynomials (resp.\ series) with nonnegative coefficients.
Then $(R[{\mathbf{x}}],{\mathcal{P}}[{\mathbf{x}}])$ and $(R[[{\mathbf{x}}]],{\mathcal{P}}[[{\mathbf{x}}]])$
are partially ordered commutative rings;
we refer to this as the \textbfit{coefficientwise order}
on $R[{\mathbf{x}}]$ and $R[[{\mathbf{x}}]]$.
A (finite or infinite) matrix with entries in a
partially ordered commutative ring
is called \textbfit{totally positive} (TP) if all its minors are nonnegative;
it is called \textbfit{totally positive of order~$\bm{r}$} (TP${}_r$)
if all its minors of size $\le r$ are nonnegative.
It follows immediately from the Cauchy--Binet formula that
the product of two TP (resp.\ TP${}_r$) matrices is TP
(resp.\ TP${}_r$).\footnote{
For infinite matrices, we need some condition to ensure that
the product is well-defined.
For instance, the product $AB$ is well-defined whenever
$A$ is row-finite (i.e.\ has only finitely many nonzero entries in each row)
or $B$ is column-finite.
}
This fact is so fundamental to the theory of total positivity
that we shall henceforth use it without comment.
We say that a sequence ${\bm{a}} = (a_n)_{n \ge 0}$
with entries in a partially ordered commutative ring
is \textbfit{Hankel-totally positive}
(resp.\ \textbfit{Hankel-totally positive of order~$\bm{r}$})
if its associated infinite Hankel matrix
$H_\infty({\bm{a}}) = (a_{i+j})_{i,j \ge 0}$
is TP (resp.\ TP${}_r$).
We say that ${\bm{a}}$
is \textbfit{Toeplitz-totally positive}
(resp.\ \textbfit{Toeplitz-totally positive of order~$\bm{r}$})
if its associated infinite Toeplitz matrix
$T_\infty({\bm{a}}) = (a_{i-j})_{i,j \ge 0}$
(where $a_n \eqdef 0$ for $n < 0$)
is TP (resp.\ TP${}_r$).\footnote{
When $R = {\mathbb R}$, Toeplitz-totally positive sequences are traditionally called
{\em P\'olya frequency sequences}\/ (PF),
and Toeplitz-totally positive sequences of order $r$
are called {\em P\'olya frequency sequences of order $r$}\/ (PF${}_r$).
See \cite[chapter~8]{Karlin_68} for a detailed treatment.
}
We will need an easy fact about the total positivity of special matrices:
\begin{equation}gin{lemma}[Bidiagonal matrices]
\label{lemma.bidiagonal}
Let $A$ be a matrix with entries in a partially ordered commutative ring,
with the property that all its nonzero entries belong to two consecutive
diagonals.
Then $A$ is totally positive if and only if all its entries are nonnegative.
\end{lemma}
\par
\noindent{\sc Proof.\ }
The nonnegativity of the entries (i.e.\ TP${}_1$)
is obviously a necessary condition for TP.
Conversely, for a matrix of this type it is easy to see that
every nonzero minor is simply a product of some entries.
$\square$
\subsection{Production matrices} \label{subsec.production}
The method of production matrices \cite{Deutsch_05,Deutsch_09}
has become in recent years an important tool in enumerative combinatorics.
In the special case of a tridiagonal production matrix,
this construction goes back to Stieltjes' \cite{Stieltjes_1889,Stieltjes_1894}
work on continued fractions:
the production matrix of a classical S-fraction or J-fraction is tridiagonal.
In~the present paper, by contrast,
we shall need production matrices that are lower-Hessenberg
(i.e.\ vanish above the first superdiagonal)
but are not in general tridiagonal.
We therefore begin by reviewing briefly
the basic theory of production matrices.
The important connection of production matrices with total positivity
will be treated in the next subsection.
Let $P = (p_{ij})_{i,j \ge 0}$ be an infinite matrix
with entries in a commutative ring $R$.
In~order that powers of $P$ be well-defined,
we shall assume that $P$ is either row-finite
(i.e.\ has only finitely many nonzero entries in each row)
or column-finite.
Let us now define an infinite matrix $A = (a_{nk})_{n,k \ge 0}$ by
\begin{equation}
a_{nk} \;=\; (P^n)_{0k}
\label{def.iteration}
\end{equation}
(in particular, $a_{0k} = \delta_{0k}$).
Writing out the matrix multiplications explicitly, we have
\begin{equation}
a_{nk}
\;=\;
\sum_{i_1,\ldots,i_{n-1}}
p_{0 i_1} \, p_{i_1 i_2} \, p_{i_2 i_3} \,\cdots\,
p_{i_{n-2} i_{n-1}} \, p_{i_{n-1} k}
\;,
\label{def.iteration.walk}
\end{equation}
so that $a_{nk}$ is the total weight for all $n$-step walks in ${\mathbb N}$
from $i_0 = 0$ to $i_n = k$, in~which the weight of a walk is the
product of the weights of its steps, and a step from $i$ to $j$
gets a weight $p_{ij}$.
Yet another equivalent formulation is to define the entries $a_{nk}$
by the recurrence
\begin{equation}
a_{nk} \;=\; \sum_{i=0}^\infty a_{n-1,i} \, p_{ik}
\qquad\hbox{for $n \ge 1$}
\label{def.iteration.bis}
\end{equation}
with the initial condition $a_{0k} = \delta_{0k}$.
We call $P$ the \textbfit{production matrix}
and $A$ the \textbfit{output matrix},
and we write $A = {\mathcal{O}}(P)$.
Note that if $P$ is row-finite, then so is ${\mathcal{O}}(P)$;
if $P$ is lower-Hessenberg, then ${\mathcal{O}}(P)$ is lower-triangular;
if $P$ is lower-Hessenberg with invertible superdiagonal entries,
then ${\mathcal{O}}(P)$ is lower-triangular with invertible diagonal entries;
and if $P$ is unit-lower-Hessenberg
(i.e.\ lower-Hessenberg with entries 1 on the superdiagonal),
then ${\mathcal{O}}(P)$ is unit-lower-triangular.
In all the applications in this paper, $P$ will be lower-Hessenberg.
The matrix $P$ can also be interpreted as the adjacency matrix
for a weighted directed graph on the vertex set ${\mathbb N}$
(where the edge $ij$ is omitted whenever $p_{ij} = 0$).
Then $P$ is row-finite (resp.\ column-finite)
if and only if every vertex has finite out-degree (resp.\ finite in-degree).
This iteration process can be given a compact matrix formulation.
Let ${\mathbb D}elta = (\delta_{i+1,j})_{i,j \ge 0}$
be the matrix with 1 on the superdiagonal and 0 elsewhere.
Then for any matrix $M$ with rows indexed by ${\mathbb N}$,
the product ${\mathbb D}elta M$ is simply $M$ with its zeroth row removed
and all other rows shifted upwards by 1.
(Some authors use the notation $\overline{M} \eqdef {\mathbb D}elta M$.)
The recurrence \reff{def.iteration.bis} can then be written as
\begin{equation}
{\mathbb D}elta \, {\mathcal{O}}(P) \;=\; {\mathcal{O}}(P) \, P
\;.
\label{def.iteration.bis.matrixform}
\end{equation}
It follows that if $A$ is a row-finite matrix
that has a row-finite inverse $A^{-1}$
and has first row $a_{0k} = \delta_{0k}$,
then $P = A^{-1} {\mathbb D}elta A$ is the unique matrix such that $A = {\mathcal{O}}(P)$.
This holds, in particular, if $A$ is lower-triangular with
invertible diagonal entries and $a_{00} = 1$;
then $A^{-1}$ is lower-triangular
and $P = A^{-1} {\mathbb D}elta A$ is lower-Hessenberg.
And if $A$ is unit-lower-triangular,
then $P = A^{-1} {\mathbb D}elta A$ is unit-lower-Hessenberg.
We shall repeatedly use the following easy fact:
\begin{equation}gin{lemma}[Production matrix of a product]
\label{lemma.production.AB}
Let $P = (p_{ij})_{i,j \ge 0}$ be a row-finite matrix
(with entries in a commutative ring $R$),
with output matrix $A = {\mathcal{O}}(P)$;
and let $B = (b_{ij})_{i,j \ge 0}$
be a lower-triangular matrix with invertible (in $R$) diagonal entries.
Then
\begin{equation}
AB \;=\; b_{00} \, {\mathcal{O}}(B^{-1} P B)
\;.
\end{equation}
That is, up to a factor $b_{00}$,
the matrix $AB$ has production matrix $B^{-1} P B$.
\end{lemma}
\par
\noindent{\sc Proof.\ }
Since $P$ is row-finite, so is $A = {\mathcal{O}}(P)$;
then the matrix products $AB$ and $B^{-1} P B$
arising in the lemma are well-defined. Now
\begin{equation}
a_{nk}
\;=\;
\sum_{i_1,\ldots,i_{n-1}}
p_{0 i_1} \, p_{i_1 i_2} \, p_{i_2 i_3} \,\cdots\,
p_{i_{n-2} i_{n-1}} \, p_{i_{n-1} k}
\;,
\end{equation}
while
\begin{equation}
{\mathcal{O}}(B^{-1} P B)_{nk}
\;=\;
\sum_{j,i_1,\ldots,i_{n-1},i_n}
(B^{-1})_{0j} \,
p_{j i_1} \, p_{i_1 i_2} \, p_{i_2 i_3} \,\cdots\,
p_{i_{n-2} i_{n-1}} \, p_{i_{n-1} i_n} \, b_{i_n k}
\;.
\end{equation}
But $B$ is lower-triangular with invertible diagonal entries,
so $(B^{-1})_{0j} = b_{00}^{-1} \delta_{j0}$.
It follows that $AB = b_{00} \, {\mathcal{O}}(B^{-1} P B)$.
$\square$
We will also need the following easy lemma:
\begin{equation}gin{lemma}[Production matrix of a down-shifted matrix]
\label{lemma.down-shifted}
Let $P = (p_{ij})_{i,j \ge 0}$ be a row-finite or column-finite matrix
(with entries in a commutative ring $R$),
with output matrix $A = {\mathcal{O}}(P)$;
and let $c$ be an element of $R$.
Now define
\begin{equation}
Q
\;=\;
\left[
\begin{equation}gin{array}{c|c@{\hspace*{2mm}}c@{\hspace*{2mm}}c}
0 & c & 0 & \cdots \\
{\rm hl}ine
0 & & & \\
0 & & P & \\[-1mm]
\vdots & & & \\
\end{array}
\right]
\;=\;
c {\bf e}_{01} \,+\, {\mathbb D}elta^{\rm T} P {\mathbb D}elta
\end{equation}
and
\begin{equation}
B
\;=\;
\left[
\begin{equation}gin{array}{c|c@{\hspace*{2mm}}c@{\hspace*{2mm}}c}
1 & 0 & 0 & \cdots \\
{\rm hl}ine
0 & & & \\
0 & & cA & \\[-1mm]
\vdots & & & \\
\end{array}
\right]
\;=\;
{\bf e}_{00} \,+\, c {\mathbb D}elta^{\rm T} A {\mathbb D}elta
\;.
\end{equation}
Then $B = {\mathcal{O}}(Q)$.
\end{lemma}
\par
\noindent{\sc Proof.\ }
We use \reff{def.iteration.walk} and its analogue for $Q$:
\begin{equation}
{\mathcal{O}}(Q)_{nk}
\;=\;
\sum_{i_1,\ldots,i_{n-1}}
q_{0 i_1} \, q_{i_1 i_2} \, q_{i_2 i_3} \,\cdots\,
q_{i_{n-2} i_{n-1}} \, q_{i_{n-1} k}
\;.
\label{def.iteration.walk.BQ}
\end{equation}
In \reff{def.iteration.walk.BQ}, the only nonzero contributions come from
$i_1 = 1$, with $q_{01} = c$;
and then we must also have $i_2,i_3,\ldots \ge 1$ and $k \ge 1$,
with $q_{ij} = p_{i-1,j-1}$.
Hence ${\mathcal{O}}(Q)_{nk} = c a_{n-1,k-1}$ for $n \ge 1$.
$\square$
\subsection{Production matrices and total positivity}
\label{subsec.totalpos.prodmat}
Let $P = (p_{ij})_{i,j \ge 0}$ be a matrix with entries in a
partially ordered commutative ring $R$.
We will use $P$ as a production matrix;
let $A = {\mathcal{O}}(P)$ be the corresponding output matrix.
As before, we assume that $P$ is either row-finite or column-finite.
When $P$ is totally positive, it turns out \cite{Sokal_totalpos}
that the output matrix ${\mathcal{O}}(P)$ has {\em two}\/ total-positivity properties:
firstly, it is totally positive;
and secondly, its zeroth column is Hankel-totally positive.
Since \cite{Sokal_totalpos} is not yet publicly available,
we shall present briefly here (with proof) the main results
that will be needed in the sequel.
The fundamental fact that drives the whole theory is the following:
\begin{equation}gin{proposition}[Minors of the output matrix]
\label{prop.iteration.homo}
Every $k \times k$ minor of the output matrix $A = {\mathcal{O}}(P)$
can be written as a sum of products of minors of size $\le k$
of the production matrix $P$.
\end{proposition}
In this proposition the matrix elements ${\mathbf{p}} = \{p_{ij}\}_{i,j \ge 0}$
should be interpreted in the first instance as indeterminates:
for instance, we can fix a row-finite or column-finite set
$S \subseteq {\mathbb N} \times {\mathbb N}$
and define the matrix $P^S = (p^S_{ij})_{i,j \in {\mathbb N}}$ with entries
\begin{equation}
p^S_{ij}
\;=\;
\begin{equation}gin{cases}
p_{ij} & \textrm{if $(i,j) \in S$} \\[1mm]
0 & \textrm{if $(i,j) {\mathbf{n}}otin S$}
\end{cases}
\end{equation}
Then the entries (and hence also the minors) of both $P$ and $A$
belong to the polynomial ring ${\mathbb Z}[{\mathbf{p}}]$,
and the assertion of Proposition~\ref{prop.iteration.homo} makes sense.
Of course, we can subsequently specialize the indeterminates ${\mathbf{p}}$
to values in any commutative ring $R$.
\par
\noindent{\sc Proof.\ }of{Proposition~\ref{prop.iteration.homo}}
Consider any minor of $A$ involving only the rows 0 through $N$.
We will prove the assertion of the Proposition by induction on $N$.
The statement is obvious for $N=0$.
For $N \ge 1$, let $A_N$ be the matrix consisting of rows
0 through $N-1$ of $A$, and let $A'_N$ be the matrix consisting of rows
1 through $N$ of $A$. Then we have
\begin{equation}
A'_N \;=\; A_N P
\;.
\label{eq.proof.prop.iteration}
\end{equation}
If the minor in question does not involve row 0,
then obviously it involves only rows 1 through $N$.
If the minor in question does involve row 0,
then it is either zero (in case it does not involve column~0)
or else equal to a minor of $A$ (of one size smaller)
that involves only rows 1 through $N$
(since $a_{0k} = \delta_{0k}$).
Either way it is a minor of $A'_N$;
but by \reff{eq.proof.prop.iteration} and the Cauchy--Binet formula,
every minor of $A'_N$ is a sum of products of minors (of the same size)
of $A_N$ and $P$.
This completes the inductive step.
$\square$
If we now specialize the indeterminates ${\mathbf{p}}$
to values in some partially ordered commutative ring $R$,
we can immediately conclude:
\begin{equation}gin{theorem}[Total positivity of the output matrix]
\label{thm.iteration.homo}
Let $P$ be an infinite matrix that is either row-finite or column-finite,
with entries in a partially ordered commutative ring $R$.
If $P$ is totally positive of order~$r$, then so is $A = {\mathcal{O}}(P)$.
\end{theorem}
{\bf Remarks.}
1. In the case $R = {\mathbb R}$, Theorem~\ref{thm.iteration.homo}
is due to Karlin \cite[pp.~132--134]{Karlin_68};
see also \cite[Theorem~1.11]{Pinkus_10}.
Karlin's proof is different from ours.
2. Our quick inductive proof of Proposition~\ref{prop.iteration.homo}
follows an idea of Zhu \cite[proof of Theorem~2.1]{Zhu_13},
which was in turn inspired in part by Aigner \cite[pp.~45--46]{Aigner_99}.
The same idea recurs in recent work of several authors
\cite[Theorem~2.1]{Zhu_14}
\cite[Theorem~2.1(i)]{Chen_15a}
\cite[Theorem~2.3(i)]{Chen_15b}
\cite[Theorem~2.1]{Liang_16}
\cite[Theorems~2.1 and 2.3]{Chen_19}
\cite{Gao_non-triangular_transforms}.
However, all of these results concerned only special cases:
\cite{Aigner_99,Zhu_13,Chen_15b,Liang_16}
treated the case in which the production matrix $P$ is tridiagonal;
\cite{Zhu_14} treated a (special) case in which $P$ is upper bidiagonal;
\cite{Chen_15a} treated the case in which
$P$ is the production matrix of a Riordan array;
\cite{Chen_19,Gao_non-triangular_transforms}
treated (implicitly) the case in which $P$ is upper-triangular and Toeplitz.
But the argument is in fact completely general, as we have just seen;
there is no need to assume any special form for the matrix $P$.
${\rm bl}acksquare$
Now define
${\mathcal{O}}o_0(P)$ to be the zeroth-column sequence of ${\mathcal{O}}(P)$, i.e.
\begin{equation}
{\mathcal{O}}o_0(P)_n \;\eqdef\; {\mathcal{O}}(P)_{n0} \;\eqdef\; (P^n)_{00}
\;.
\label{def.scroo0}
\end{equation}
Then the Hankel matrix of ${\mathcal{O}}o_0(P)$ has matrix elements
\begin{equation}gin{eqnarray}
& &
\!\!\!\!\!\!\!
H_\infty({\mathcal{O}}o_0(P))_{nn'}
\;=\;
{\mathcal{O}}o_0(P)_{n+n'}
\;=\;
(P^{n+n'})_{00}
\;=\;
\sum_{k=0}^\infty (P^n)_{0k} \, (P^{n'})_{k0}
\;=\;
{\mathbf{n}}onumber \\
& &
\sum_{k=0}^\infty (P^n)_{0k} \, ((P^{\rm T})^{n'})_{0k}
\;=\;
\sum_{k=0}^\infty {\mathcal{O}}(P)_{nk} \, {\mathcal{O}}(P^{\rm T})_{n'k}
\;=\;
\big[ {\mathcal{O}}(P) \, {{\mathcal{O}}(P^{\rm T})}^{\rm T} \big]_{nn'}
\;.
\qquad
\end{eqnarray}
(Note that the sum over $k$ has only finitely many nonzero terms:
if $P$ is row-finite, then there are finitely many nonzero $(P^n)_{0k}$,
while if $P$ is column-finite,
there are finitely many nonzero $(P^{n'})_{k0}$.)
We have therefore proven:
\begin{equation}gin{lemma}[Identity for Hankel matrix of the zeroth column]
\label{lemma.hankel.karlin}
Let $P$ be a row-finite or column-finite matrix
with entries in a commutative ring $R$.
Then
\begin{equation}
H_\infty({\mathcal{O}}o_0(P))
\;=\;
{\mathcal{O}}(P) \, {{\mathcal{O}}(P^{\rm T})}^{\rm T}
\;.
\end{equation}
\end{lemma}
{\bf Remark.}
If $P$ is row-finite, then ${\mathcal{O}}(P)$ is row-finite;
${\mathcal{O}}(P^{\rm T})$ need not be row- or column-finite,
but the product ${\mathcal{O}}(P) \, {{\mathcal{O}}(P^{\rm T})}^{\rm T}$
is anyway well-defined.
${\rm bl}acksquare$
Combining Proposition~\ref{prop.iteration.homo}
with Lemma~\ref{lemma.hankel.karlin} and the Cauchy--Binet formula,
we obtain:
\begin{equation}gin{corollary}[Hankel minors of the zeroth column]
\label{cor.iteration2}
Every $k \times k$ minor of the infinite Hankel matrix
$H_\infty({\mathcal{O}}o_0(P)) = ((P^{n+n'})_{00})_{n,n' \ge 0}$
can be written as a sum of products
of the minors of size $\le k$ of the production matrix $P$.
\end{corollary}
And specializing the indeterminates ${\mathbf{p}}$
to nonnegative elements in a partially ordered commutative ring,
in such a way that $P$ is row-finite or column-finite,
we deduce:
\begin{equation}gin{theorem}[Hankel-total positivity of the zeroth column]
\label{thm.iteration2bis}
Let $P = (p_{ij})_{i,j \ge 0}$ be an infinite row-finite or column-finite
matrix with entries in a partially ordered commutative ring $R$,
and define the infinite Hankel matrix
$H_\infty({\mathcal{O}}o_0(P)) = ((P^{n+n'})_{00})_{n,n' \ge 0}$.
If $P$ is totally positive of order~$r$, then so is $H_\infty({\mathcal{O}}o_0(P))$.
\end{theorem}
\subsection[$m$-Stieltjes--Rogers polynomials]{$\bm{m}$-Stieltjes--Rogers
polynomials} \label{subsec.mSR}
Throughout this subsection we fix an integer $m \ge 1$.
We recall \cite{Aval_08,Cameron_16,Prodinger_16,latpath_SRTR}
that an \textbfit{$\bm{m}$-Dyck path}
is a path in the upper half-plane ${\mathbb Z} \times {\mathbb N}$,
starting and ending on the horizontal axis,
using steps $(1,1)$ [``rise'' or ``up step'']
and $(1,-m)$ [``$m$-fall'' or ``down step''].
More generally, an \textbfit{$\bm{m}$-Dyck path at level $\bm{k}$}
is a path in ${\mathbb Z} \times {\mathbb N}_{\ge k}$,
starting and ending at height $k$,
using steps $(1,1)$ and $(1,-m)$.
Since the number of up steps must equal $m$ times the number of down steps,
the length of an $m$-Dyck path must be a multiple of $m+1$.
Now let ${\bm{a}}lpha = (\alpha_i)_{i \ge m}$ be an infinite set of indeterminates.
Then \cite{latpath_SRTR}
the \textbfit{$\bm{m}$-Stieltjes--Rogers polynomial} of order~$n$,
denoted $S^{(m)}_n({\bm{a}}lpha)$, is the generating polynomial
for $m$-Dyck paths of length~$(m+1)n$ in which each rise gets weight~1
and each $m$-fall from height~$i$ gets weight $\alpha_i$.
Clearly $S_n^{(m)}({\bm{a}}lpha)$ is a homogeneous polynomial
of degree~$n$ with nonnegative integer coefficients.
Let $f_0(t) = \sum_{n=0}^\infty S^{(m)}_n({\bm{a}}lpha) \, t^n$
be the ordinary generating function for $m$-Dyck paths with these weights;
and more generally, let $f_k(t)$ be the ordinary generating function
for $m$-Dyck paths at level $k$ with these same weights.
(Obviously $f_k$ is just $f_0$ with each $\alpha_i$ replaced by $\alpha_{i+k}$;
but we shall not explicitly use this fact.)
Then straightforward combinatorial arguments \cite{latpath_SRTR}
lead to the functional equation
\begin{equation}
f_k(t) \;=\; 1 \:+\: \alpha_{k+m} t \, f_k(t) \, f_{k+1}(t) \,\cdots\, f_{k+m}(t)
\label{eq.mSRfk.1}
\end{equation}
or equivalently
\begin{equation}
f_k(t) \;=\; {1 \over 1 \:-\: \alpha_{k+m} t \, f_{k+1}(t) \,\cdots\, f_{k+m}(t)}
\;.
\label{eq.mSRfk.2}
\end{equation}
Iterating \reff{eq.mSRfk.2}, we see immediately that $f_k$
is given by the branched continued fraction
\begin{equation}gin{subeqnarray}
f_k(t)
& = &
\cfrac{1}
{1 \,-\, \alpha_{k+m} t
\prod\limits_{i_1=1}^{m}
\cfrac{1}
{1 \,-\, \alpha_{k+m+i_1} t
\prod\limits_{i_2=1}^{m}
\cfrac{1}
{1 \,-\, \alpha_{k+m+i_1+i_2} t
\prod\limits_{i_3=1}^{m}
\cfrac{1}{1 - \cdots}
}
}
}
\slabel{eq.fk.mSfrac.a} \\[2mm]
& = &
\Scale[0.6]{
\cfrac{1}{1 - \cfrac{\alpha_{k+m} t}{
{\sf B}iggl( 1 - \cfrac{\alpha_{k+m+1} t}{
{\sf B}igl( 1 - \cfrac{\alpha_{k+m+2} t}{(\cdots) \,\cdots\, (\cdots)} {\sf B}igr)
\,\cdots\,
{\sf B}igl( 1 - \cfrac{\alpha_{k+2m+1} t}{(\cdots) \,\cdots\, (\cdots)} {\sf B}igr)
}
{\sf B}iggr)
\,\cdots\,
{\sf B}iggl( 1 - \cfrac{\alpha_{k+2m} t}{
{\sf B}igl( 1 - \cfrac{\alpha_{k+2m+1} t}{(\cdots) \,\cdots\, (\cdots)} {\sf B}igr)
\,\cdots\,
{\sf B}igl( 1 - \cfrac{\alpha_{k+3m} t}{(\cdots) \,\cdots\, (\cdots)} {\sf B}igr)
}
{\sf B}iggr)
}
}
}
{\mathbf{n}}onumber \\
\slabel{eq.fk.mSfrac.b}
\label{eq.fk.mSfrac}
\end{subeqnarray}
and in particular that $f_0$ is given by
the specialization of \reff{eq.fk.mSfrac} to $k=0$.
We shall call the right-hand side of \reff{eq.fk.mSfrac}
an \textbfit{$\bm{m}$-branched Stieltjes-type continued fraction},
or \textbfit{$\bm{m}$-S-fraction} for short.
{\bf Remark.}
In truth, we hardly ever use the branched continued fraction
\reff{eq.fk.mSfrac};
instead, we work directly with the $m$-Dyck paths
and/or with the recurrence \reff{eq.mSRfk.1}/\reff{eq.mSRfk.2}
that their generating functions satisfy.
${\rm bl}acksquare$
We now generalize these definitions as follows.
A \textbfit{partial $\bm{m}$-Dyck path}
is a path in the upper half-plane ${\mathbb Z} \times {\mathbb N}$,
starting on the horizontal axis but ending anywhere,
using steps $(1,1)$ [``rise'']
and $(1,-m)$ [``$m$-fall''].
A partial $m$-Dyck path starting at $(0,0)$
must stay always within the set
$V_m = \{ (x,y) \in {\mathbb Z} \times {\mathbb N} \colon\: x=y \bmod m+1 \}$.
Now let ${\bm{a}}lpha = (\alpha_i)_{i \ge m}$ be an infinite set of indeterminates,
and let $S^{(m)}_{n,k}({\bm{a}}lpha)$ be the generating polynomial
for partial $m$-Dyck paths from $(0,0)$ to ${((m+1)n,(m+1)k)}$
in~which each rise gets weight~1
and each $m$-fall from height~$i$ gets weight $\alpha_i$.
We call the $S^{(m)}_{n,k}$ the
\textbfit{generalized $\bm{m}$-Stieltjes--Rogers polynomials}.
Obviously $S^{(m)}_{n,k}$ is nonvanishing only for $0 \le k \le n$,
and $S^{(m)}_{n,n} = 1$.
We therefore have an infinite unit-lower-triangular array
${\sf S}^{(m)} = \big( S^{(m)}_{n,k}({\bm{a}}lpha) \big)_{n,k \ge 0}$
in which the first ($k=0$) column displays
the ordinary $m$-Stieltjes--Rogers polynomials $S^{(m)}_{n,0} = S^{(m)}_n$.
The production matrix for the triangle ${\sf S}^{(m)}$
was found in \cite[sections~7.1 and 8.2]{latpath_SRTR}.
We begin by defining some special matrices $M = (m_{ij})_{i,j \ge 0}$:
\begin{equation}gin{itemize}
\item $L(s_1,s_2,\ldots)$ is the lower-bidiagonal matrix
with 1 on the diagonal and $s_1,s_2,\ldots$ on the subdiagonal:
\begin{equation}
L(s_1,s_2,\ldots)
\;=\;
\begin{equation}gin{bmatrix}
1 & & & & \\
s_1 & 1 & & & \\
& s_2 & 1 & & \\
& & s_3 & 1 & \\
& & & \ddots & \ddots
\end{bmatrix}
\;.
\label{def.L}
\end{equation}
\item $U^\star(s_1,s_2,\ldots)$ is the upper-bidiagonal matrix
with 1 on the superdiagonal and $s_1,s_2,\ldots$ on the diagonal:
\begin{equation}
U^\star(s_1,s_2,\ldots)
\;=\;
\begin{equation}gin{bmatrix}
s_1 & 1 & & & & \\
& s_2 & 1 & & & \\
& & s_3 & 1 & & \\
& & & s_4 & 1 & \\
& & & & \ddots & \ddots
\end{bmatrix}
\;.
\label{def.Ustar}
\end{equation}
\end{itemize}
Then the production matrix for the triangle ${\sf S}^{(m)}$ is
\begin{equation}gin{eqnarray}
P^{(m)\mathrm{S}}({\bm{a}}lpha)
& \eqdef &
L(\alpha_{m+1}, \alpha_{2m+2}, \alpha_{3m+3}, \ldots)
\:
L(\alpha_{m+2}, \alpha_{2m+3}, \alpha_{3m+4}, \ldots)
\:\cdots\:
\hspace*{1cm}
{\mathbf{n}}onumber \\
& & \qquad
L(\alpha_{2m}, \alpha_{3m+1}, \alpha_{4m+2}, \ldots)
\:
U^\star(\alpha_m, \alpha_{2m+1}, \alpha_{3m+2}, \ldots)
\;,
\hspace*{1cm}
\label{eq.prop.contraction}
\end{eqnarray}
that is, the product of $m$ factors $L$ and one factor $U^\star$
\cite[Proposition~8.2]{latpath_SRTR}.
Finally, we proved the following fundamental results on total positivity
\cite[Theorems~9.8 and 9.10]{latpath_SRTR}:\footnote{
In fact, we gave two independent proofs of these results:
a graphical proof, based on the Lindstr\"om--Gessel--Viennot lemma;
and an algebraic proof, based on the production matrix
\reff{eq.prop.contraction}.
}
\begin{equation}gin{itemize}
\item[(a)] For each integer $m \ge 1$, the lower-triangular matrix
${\sf S}^{(m)} = \big( S^{(m)}_{n,k}({\bm{a}}lpha) \big)_{n,k \ge 0}$
of generalized $m$-Stieltjes--Rogers polynomials
is totally positive in the polynomial ring ${\mathbb Z}[{\bm{a}}lpha]$
equipped with the coefficientwise partial order.
\item[(b)] For each integer $m \ge 1$,
the sequence ${\bm{S}}^{(m)} = ( S^{(m)}_n({\bm{a}}lpha) )_{n \ge 0}$
of $m$-Stieltjes--Rogers polynomials
is a Hankel-totally positive sequence in the polynomial ring ${\mathbb Z}[{\bm{a}}lpha]$
equipped with the coefficientwise partial order.
\end{itemize}
Of course, we can then substitute for ${\bm{a}}lpha$
any sequence of nonnegative elements
of any partially ordered commutative ring $R$,
and the resulting matrix ${\sf S}^{(m)}$ (resp.\ sequence ${\bm{S}}^{(m)}$)
will be totally positive (resp.\ Hankel-totally positive) in $R$.
\section{Proofs of main results}
\label{sec.proofs}
In this section we prove Theorem~\ref{thm1.1} and its corollaries,
by the following steps:
First we prove Proposition~\ref{prop.prodmat}(a),
which asserts that the matrix $P$ defined in \reff{eq.prop.prodmat}
is the production matrix for the generic Lah triangle
${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$.
Next we prove the matrix identity $B_y^{-1} P B_y = P (I + y {\mathbb D}elta^{\rm T})$.
Once this is done, Lemma~\ref{lemma.production.AB} implies
that $P (I + y {\mathbb D}elta^{\rm T})$ is the production matrix for ${\sf L} B_y$,
which completes the proof of Proposition~\ref{prop.prodmat}(b).
Finally, we show that if the Toeplitz matrix $T_\infty({\bm{p}}hi)$ is TP${}_r$,
then so is $P$
(Lemma~\ref{lemma.diagmult.TP} and Corollary~\ref{cor.diagmult.TP}).
Then Theorem~\ref{thm1.1} follows from
the general theory of production matrices and total positivity
(Theorems~\ref{thm.iteration.homo} and \ref{thm.iteration2bis}).
In fact, we will prove a generalization of Proposition~\ref{prop.prodmat}(a)
[and hence also of Theorem~\ref{thm1.1}(a,c)
and Corollaries~\ref{cor1.2}(a,c) and \ref{cor1.3}(a,c)]
for some polynomials ${\widehat{L}}_{n,k}({\bm{p}}hihat)$
that depend on a refined set of indeterminates
${\bm{p}}hihat = (\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$
and that reduce to $L_{n,k}({\bm{p}}hi)$ when $\phi_i^{[L]} = \phi_i$ for all $L$.
However, no analogue of Proposition~\ref{prop.prodmat}(b)
[and hence also of Theorem~\ref{thm1.1}(b)
and Corollaries~\ref{cor1.2}(b) and \ref{cor1.3}(b)]
appears to exist for these more general polynomials.
\subsection{A generalization of Proposition~\ref{prop.prodmat}(a)}
\label{subsec.proofs.prodmat}
In this subsection, we shall state and prove a generalization
of Proposition~\ref{prop.prodmat}(a).
We begin by introducing the notion of {\em level}\/
of a vertex in a forest, as was done in \cite{latpath_SRTR}:
\begin{equation}gin{definition}[Level of a vertex]
\label{def.level}
Let $F$ be a forest of increasing trees on a totally ordered vertex set,
with $k$ trees.\footnote{
Here the forest can be either ordered or unordered;
and the trees in the forest can be either ordered or unordered
(in the sense that the children at each vertex
can be either ordered or unordered).
Neither of these orderings, if present,
will play any role in the definition of ``level''.
}
For each vertex $j$ in $F$, let $r_j$ be the number of trees in $F$
that contain at least one vertex $\le j$.
Then the \textbfit{level} of the vertex~$j$ in the forest~$F$,
denoted ${\rm lev}_F(j)$,
is the number of children of vertices $< j$ whose labels are $>j$,
plus $k+1-r_j$.
\end{definition}
{\mathbf{n}}oindent
Note that $1 \le r_j \le k$, and hence ${\rm lev}_F(j) \ge 1$.
{\bf Remark.}
This definition of ``level'' is slightly different
from the one given in \cite{latpath_SRTR},
since our forests here have $k$ trees rather than $k+1$
as in \cite{latpath_SRTR},
and our levels here are $\ge 1$ rather than $\ge 0$.
${\rm bl}acksquare$
We can now define a generalization of our generic Lah triangle, as follows:
Let ${\bm{p}}hihat = (\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$ be indeterminates,
and let ${\widehat{L}}_{n,k}({\bm{p}}hihat)$ be the generating polynomial
for unordered forests of increasing ordered trees on the vertex set $[n]$,
having $k$ trees, in~which each vertex with $i$ children and level $L$
gets a weight $\phi_i^{[L]}$.
We shall refer to the lower-triangular matrix
${\sf L}hat = ({\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \ge 0}$
as the \textbfit{refined generic Lah triangle}.
Of course, when $\phi_i^{[L]} = \phi_i$ for all $L$,
it reduces to the original generic Lah triangle
${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$.
We shall see later that
the polynomial ${\widehat{L}}_{n,k}({\bm{p}}hihat)$ has a factor
$\phi_0^{[1]} \phi_0^{[2]} \cdots \phi_0^{[k]}$.
So we can, if we wish, pull this factor out,
and consider also the lower-triangular array
${\sf L}tilde = \big( {\widehat{L}}_{n,k}({\bm{p}}hihat) /
(\phi_0^{[1]} \phi_0^{[2]} \cdots \phi_0^{[k]})
\big)_{\! n,k \ge 0}$.
Finally, it turns out that, in proving the formula for the production matrix,
it~is most convenient to work with {\em ordered}\/ forests
of increasing ordered trees, not unordered ones.
Since the trees of our forests are labeled and hence distinguishable,
the generating polynomial for ordered forests
on the vertex set $[n]$ with $k$ components
is simply $k!$ times the generating polynomial for unordered forests.
So we will begin by finding the production matrix $P^{\rm ord}$
for the triangle
${\sf L}hat^{\rm ord} = (k! \, {\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \geq 0}$;
then we will deduce from it the production matrix $P$
for the triangle ${\sf L}hat = ({\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \geq 0}$,
and the production matrix ${\widetilde{P}}$ for the triangle
${\sf L}tilde = \big( {\widehat{L}}_{n,k}({\bm{p}}hihat) /
(\phi_0^{[1]} \phi_0^{[2]} \cdots \phi_0^{[k]})
\big)_{\! n,k \ge 0}$.
We now claim that the following generalization of
Proposition~\ref{prop.prodmat}(a) holds:
\begin{equation}gin{proposition}[Production matrix for the refined generic Lah triangle]
\label{prop.prodmat.phiL}
{\bm{r}}eak{\mathbf{n}}oindent
Let ${\bm{p}}hihat = (\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$ be indeterminates,
and work in the ring $\mathbb{Z}[{\bm{p}}hihat]$.
Define the lower-Hessenberg matrices
$P^{\rm ord} = (p^{\rm ord}_{ij})_{i,j \ge 0}$,
$P = (p_{ij})_{i,j \ge 0}$ and ${\widetilde{P}} = ({\widetilde{p}}_{ij})_{i,j \ge 0}$ by
\begin{equation}gin{eqnarray}
p^{\rm ord}_{ij}
& = &
\begin{equation}gin{cases}
0 & \textrm{if $j=0$ or $j > i+1$} \\[1mm]
j \: \phi_{i-j+1}^{[j]}
& \textrm{if $1 \le j \le i+1$}
\end{cases}
\label{eq.prop.prodmat.phiL.0}
\\[4mm]
p_{ij}
& = &
\begin{equation}gin{cases}
0 & \textrm{if $j=0$ or $j > i+1$} \\[1mm]
{\displaystyle {i! \over (j-1)!}} \: \phi_{i-j+1}^{[j]}
& \textrm{if $1 \le j \le i+1$}
\end{cases}
\label{eq.prop.prodmat.phiL.1}
\\[4mm]
{\widetilde{p}}_{ij}
& = &
\begin{equation}gin{cases}
0 & \textrm{if $j=0$ or $j > i+1$} \\[1mm]
{\displaystyle {i! \over (j-1)!}}
\; \phi_0^{[j+1]} \cdots \phi_0^{[i]}
\; \phi_{i-j+1}^{[j]}
& \textrm{if $1 \le j \le i$} \\[4mm]
1 & \textrm{if $j=i+1$}
\end{cases}
\label{eq.prop.prodmat.phiL.2}
\end{eqnarray}
Then:
\begin{equation}gin{itemize}
\item[(a)] $P^{\rm ord}$ is the production matrix for the triangle
${\sf L}hat^{\rm ord} = (k! \, {\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \geq 0}$.
\item[(b)] $P$ is the production matrix for the
refined generic Lah triangle
${\sf L}hat = ({\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \geq 0}$.
\item[(c)] ${\widetilde{P}}$ is the production matrix for the triangle
${\sf L}tilde = \big( {\widehat{L}}_{n,k}({\bm{p}}hihat) /
(\phi_0^{[1]} \phi_0^{[2]} \cdots \phi_0^{[k]})
\big)_{\! n,k \ge 0}$.
\end{itemize}
\end{proposition}
As preparation for the proof of Proposition~\ref{prop.prodmat.phiL},
we recall the definition of the \textbfit{depth-first-search labeling}
of an ordered forest of ordered trees.
(The more precise name is {\em preorder traversal}\/,
i.e.\ parent first, then children in order from left to right,
carried out recursively starting at the root.)
The recursive definition can be found in \cite[pp.~33--34]{Stanley_99},
but there is a simple informal description:
for each tree, we walk clockwise around the tree,
starting at the root, and label the vertices in the order
in which they are first seen;
this is done successively for the trees of the forest, in the given order.
Note that, in the depth-first-search labeling,
all the children of a vertex~$j$ will have labels $> j$;
that is, the depth-first-search labeling is a (very special)
increasing labeling.
Note also that, in the depth-first-search labeling, if $r < r'$,
then all the vertices of the $r$th tree will have labels smaller than
all the vertices of the $r'$th tree;
of course, this property need {\em not}\/ hold
in a general increasing labeling.
Finally, we recall that a \textbfit{partial \L{}ukasiewicz path}
(in our definition)
is a path in the upper half-plane ${\mathbb Z} \times {\mathbb N}$
using steps $(1,s)$ with $-\infty < s \le 1$,
while a \textbfit{reversed partial \L{}ukasiewicz path}
is a path in the upper half-plane ${\mathbb Z} \times {\mathbb N}$
using steps $(1,s)$ with $-1 \le s < \infty$.
\par
\noindent{\sc Proof.\ }of{Proposition~\ref{prop.prodmat.phiL}}
We will construct a bijection from the set of {\em ordered}\/ forests
of increasing ordered trees on the vertex set $[n]$, with $k$ components,
to a set of ${\mathbf{L}}$-labeled reversed partial \L{}ukasiewicz paths
from $(0,k)$ to $(n,0)$,
where the label sets ${\mathbf{L}}$ will be defined below.
The case $n=k=0$ is trivial, so we assume $n,k \ge 1$.
Given an ordered forest $F$ of increasing ordered trees
on the vertex set $[n]$, with $k$ components,
we define a labeled reversed partial \L{}ukasiewicz path $(\omega,{\mathbf{x}}i)$
of length~$n$ as follows (see Figure~\ref{fig.bijection} for an example):
{\bf Definition of the path ${\bm{\omega}}$.}
The path $\omega$ starts at height $h_0 = k$
and takes steps $s_1,\ldots,s_n$ with $s_i = \deg(i) - 1$,
where $\deg(i)$ is the number of children of vertex $i$.
Therefore, the heights $h_0,\ldots,h_n$ are
\begin{equation}
h_j \;=\; k \,+\, \sum_{i=1}^j [\deg(i) - 1]
\;.
\end{equation}
Since $\sum_{i=1}^n \deg(i) = n-k$, we have $h_n = 0$.
We will show later that $h_1,\ldots,h_{n-1} \ge 1$.
{\bf Definition of the labels ${\bm{x}}i$.}
The label ${\mathbf{x}}i_j$ is, by definition,
1 plus the number of vertices $> j$
that are either children of $\{1,\ldots,j-1\}$ or roots
and that precede $j$ in the depth-first-search order.\footnote{
Here the depth-first-search order could be replaced by any chosen order
on the vertices of $F$ that commutes with truncation.
The key property we need is that
the order on the truncated forest $F_{j-1}$ to be defined below
is the restriction of the order on the full forest $F$.
}
Obviously ${\mathbf{x}}i_j$ is an integer $\ge 1$;
we will show later that ${\mathbf{x}}i_j \le h_{j-1}$ (Corollary~\ref{cor.xij}).
\begin{equation}gin{figure}[t]
\begin{equation}gin{center}
\includegraphics[scale=1]{bijectiongenerictree.pdf}
\hspace*{45pt}
\includegraphics[scale=1]{bijectiongenericpath.pdf}
\end{center}
\caption{An ordered forest of two increasing ordered trees
on the vertex set $[10]$,
and its image under the bijection.
We put the label ${\mathbf{x}}i_j$ above the step $s_j$.
}
\label{fig.bijection}
\end{figure}
{\bf Interpretation of the heights $\bm{h_j}$.}
Recall that $r_j$ is the number of trees in $F$ that contain
at least one of the vertices $\{1,\ldots,j\}$.
We then claim:
\begin{equation}gin{lemma}
\label{lemma.hj}
For $1 \le j \le n$,
the height $h_j = k+ \sum_{i=1}^j s_i$ has the following interpretations:
\begin{equation}gin{itemize}
\item[(a)] $h_j$ is the number of children of the vertices $\{1,\ldots,j\}$
whose labels are $> j$, plus $k-r_j$.
\item[(b)] $h_{j-1}$ is the number of children of the vertices
$\{1,\ldots,j-1\}$ whose labels are $> j$, plus $k+1-r_j$.
That is, $h_{j-1}$ is the level of the vertex $j$
as given in Definition~\ref{def.level}.
\end{itemize}
In particular, $h_j > k-r_j$ whenever $j$
is not the highest-numbered vertex of its tree,
and $h_j \ge k-r_j$ always.
\end{lemma}
\par
\noindent{\sc Proof.\ }
By induction on $j$.
For the base case $j=1$, the claims are clear
since $r_1 = 1$, $h_0 = k$ and $h_1 = k + \deg(1) - 1$.
For $j>1$, the vertex $j$ is either the child of another node,
or the root of a tree. We consider these two cases separately:
(i) Suppose that $j$ is the child of another node
(obviously numbered ${\le j-1}$).
By the inductive hypothesis~(a),
$h_{j-1}$ is the number of children of the vertices ${\{1,\ldots,j-1\}}$
whose labels are $\ge j$, plus $k-r_{j-1}$;
and since one of these children is $j$,
it follows that
$h_{j-1} - 1$ is the number of children of the vertices $\{1,\ldots,j-1\}$
whose labels are $> j$, plus $k-r_{j-1}$.
Now vertex $j$ has $\deg(j)$ children, all of which have labels $> j$;
so $h_j = h_{j-1} + s_j = h_{j-1} - 1 + \deg(j)$
is the number of children of the vertices $\{1,\ldots,j\}$
whose labels are $> j$, plus $k-r_{j-1}$.
Since $r_j = r_{j-1}$, the preceding two sentences prove
claims (b) and (a), respectively.
(ii) Suppose that $j$ is a root.
By the inductive hypothesis~(a),
$h_{j-1}$ is the number of children of the vertices $\{1,\ldots,j-1\}$
whose labels are $\ge j$, plus $k-r_{j-1}$;
and since $j$ is not one of these children,
it follows that
$h_{j-1}$ is also the number of children of the vertices $\{1,\ldots,j-1\}$
whose labels are $> j$, plus $k-r_{j-1}$.
Now vertex $j$ has $\deg(j)$ children, all of which have labels $> j$;
so $h_j = h_{j-1} + \deg(j) - 1$
is the number of children of the vertices $\{1,\ldots,j\}$
whose labels are $> j$, plus $k-r_{j-1}-1$.
Since $r_j = r_{j-1} +1$, the preceding two sentences prove
claims (b) and (a), respectively.
$\square$
It follows from Lemma~\ref{lemma.hj}(b)
that $h_0,\ldots,h_{n-1} \ge 1$ and $h_n = 0$.
So the path $\omega$ is indeed a reversed partial \L{}ukasiewicz path
from $(0,k)$ to $(n,0)$,
which reaches level 0 only at the last step.
\begin{equation}gin{corollary}
\label{cor.xij}
${\mathbf{x}}i_j \le h_{j-1}$.
\end{corollary}
\par
\noindent{\sc Proof.\ }
By Lemma~\ref{lemma.hj}(b), the number of vertices $> j$
that are children of ${ \{1,\ldots,j-1\} }$
is $h_{j-1} - (k+1 - r_j)$.
The number of vertices $> j$ that are roots is at most $k-r_j$
(since any tree containing a vertex $\le j$ necessarily has its root $\le j$).
So ${\mathbf{x}}i_j \le 1 + h_{j-1} - (k+1 - r_j) + (k-r_j) = h_{j-1}$.
$\square$
{\bf The inverse bijection.}
We claim that this mapping $F \mapsto (\omega,{\mathbf{x}}i)$ is a bijection
from the set of ordered forests of increasing ordered trees
on the vertex set $[n]$ with $k$ components
to the set of labeled reversed partial \L{}ukasiewicz paths
from $(0,k)$ to $(n,0)$
that reach level 0 only at the last step,
with integer labels satisfying $1 \le {\mathbf{x}}i_j \le h_{j-1}$.
To prove this, we explain the inverse mapping.
Given a labeled reversed partial \L{}ukasiewicz path $(\omega,{\mathbf{x}}i)$,
where $\omega$ reaches level 0 only at the last step,
and $1 \le {\mathbf{x}}i_j \le h_{j-1}$ for all $j$,
we build up the ordered forest $F$ vertex-by-vertex:
after stage $j$ we will have an ordered forest $F_j$
in~which some of the vertices are labeled $1,\ldots,j$
and some others are unnumbered ``vacant slots''.
The starting forest $F_0$ has $k$ singleton components,
each of which is a vacant slot (these components are of course ordered).
We now ``read'' the path step-by-step, from $j=1$ through $j=n$.
When we read a step $s_j$ with label ${\mathbf{x}}i_j$,
we insert a new vertex $j$ into one of the vacant slots of $F_{j-1}$:
namely, the ${\mathbf{x}}i_j$th vacant slot
in the depth-first-search order of $F_{j-1}$.
We also create $s_j + 1$ new vacant slots that are children of~$j$.
This defines $F_j$.
Since $F_0$ has $k = h_0$ vacant slots,
and at stage $j$ we remove one vacant slot and add $s_j + 1$ new ones,
it follows by induction that $F_j$ has $h_j$ vacant slots.
(In particular, the placement of the vertex $j$
into the ${\mathbf{x}}i_j$th vacant slot of $F_{j-1}$ is well-defined,
since $1 \le {\mathbf{x}}i_j \le h_{j-1}$ by hypothesis.)
Since by hypothesis the path $\omega$ satisfies
$h_0,\ldots,h_{n-1} \ge 1$ and $h_n = 0$,
it follows that each forest $F_0,\ldots,F_{n-1}$ has at least one vacant slot,
while the forest $F_n$ has no vacant slot.
We define $F = F_n$.
It is fairly clear that this insertion algorithm
defines a map $(\omega,{\mathbf{x}}i) \mapsto F$
that is indeed the inverse of the mapping $F \mapsto (\omega,{\mathbf{x}}i)$
defined previously:
this follows from the proof of Lemma~\ref{lemma.hj}
and the definition of the insertion algorithm.
{\bf Computation of the weights.}
We want to enumerate ordered forests of increasing ordered trees
on the vertex set $[n]$ with $k$ components,
in~which each vertex at level $L$ with $i$ children
gets a weight $\phi_i^{[L]}$.
We use the bijection to push these weights from the forests
to the labeled reversed partial \L{}ukasiewicz paths.
Given a forest $F$, each vertex $j \in [n]$
contributes a weight $\phi_{\deg(j)}^{[{\rm lev}(j)]}$.
Under the bijection, this vertex
is mapped to a step $s_j = \deg(j) - 1$
from height $h_{j-1} = {\rm lev}(j)$ to height $h_j = h_{j-1} + s_j$.
Therefore, the weight in the labeled path $(\omega,{\mathbf{x}}i)$
corresponding to this vertex is $\phi_{s_j+1}^{[h_{j-1}]}$,
and the weight of the labeled path $(\omega,{\mathbf{x}}i)$
is the product of these weights over $1 \le j \le n$.
Now we sum over the labels ${\mathbf{x}}i$ to get the total weight
for each path $\omega$:
summing over ${\mathbf{x}}i_j$ gives a factor $h_{j-1}$.
Therefore, the weight in the reversed partial \L{}ukasiewicz path
for a step~$s$ ($-1 \le s < \infty$) starting from height~$h$ will be
\begin{equation}
W(s,h) \;=\; h \, \phi_{s+1}^{[h]}
\;.
\label{eq.Wsh}
\end{equation}
Note that $W(s,0) = 0$;
this implements automatically the constraint that
the reversed partial \L{}ukasiewicz path is not allowed to reach level 0
before the last step.
We now want to read the path $\omega$ backwards,
so that it becomes an ordinary partial \L{}ukasiewicz path ${\widehat{\omega}}$
from $(0,0)$ to $(n,k)$.
A step~$s$ starting at height~$h$ in $\omega$
becomes a step~$s' = -s$ starting at height~$h' = h+s$ in ${\widehat{\omega}}$.
Therefore, in the ordinary partial \L{}ukasiewicz path ${\widehat{\omega}}$,
the weight will be
\begin{equation}
W'(s',h') \;=\; (h'+s') \, \phi_{1-s'}^{[h'+s']}
\;.
\label{eq.Wsh.prime}
\end{equation}
That is, a step from height $i$ to height $j$ gets a weight
\begin{equation}
p^{\rm ord}_{ij} \;=\; W(j-i,i) \;=\; j \, \phi_{i-j+1}^{[j]}
\;.
\end{equation}
(Note that $p^{\rm ord}_{i0} = 0$, i.e.\ steps to level 0 are forbidden.)
Then $P^{\rm ord} = (p^{\rm ord}_{ij})_{i,j \ge 0}$
is the production matrix for the triangle
$(k! \, {\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \geq 0}$
that enumerates ordered forests.
This proves Proposition~\ref{prop.prodmat.phiL}(a).
We then apply Lemma~\ref{lemma.production.AB}
with $B = \mathop{\rm diag}{\mathbf{n}}olimits\big( (1/k!)_{k \ge 0} \big)$,
working temporarily in the ring ${\mathbb Q}[{\bm{p}}hihat]$.
It follows that the production matrix $P$
for the triangle $({\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \geq 0}$
is given by $P = B^{-1} P^{\rm ord} B$,
which is precisely \reff{eq.prop.prodmat.phiL.1}.
This proves Proposition~\ref{prop.prodmat.phiL}(b).
It also follows that the polynomial ${\widehat{L}}_{n,k}({\bm{p}}hihat)$ has a factor
$\phi_0^{[1]} \phi_0^{[2]} \cdots \phi_0^{[k]}$,
since every partial \L{}ukasiewicz path from $(0,0)$ to $(n,k)$
must have rises $0 \to 1$, $1 \to 2$, \ldots, $k-1 \to k$.
To prove Proposition~\ref{prop.prodmat.phiL}(c),
we apply Lemma~\ref{lemma.production.AB} once again,
this time with
${\sf B}tilde = \mathop{\rm diag}{\mathbf{n}}olimits\big( (1/\phi_0^{[1]} \cdots \phi_0^{[k]})_{k \ge 0} \big)$,
working temporarily in the ring ${\mathbb Z}[{\bm{p}}hihat,{\bm{p}}hihat^{-1}]$.
Of course the matrix elements of ${\widetilde{P}} = {\sf B}tilde^{-1} P {\sf B}tilde$
lie in the subring ${\mathbb Z}[{\bm{p}}hihat] \subseteq {\mathbb Z}[{\bm{p}}hihat,{\bm{p}}hihat^{-1}]$.
This completes the proof of Proposition~\ref{prop.prodmat.phiL}.
$\square$
{\bf Remark.} The reasoning here using Lemma~\ref{lemma.production.AB}
corresponds, at the level of \L{}ukasiewicz paths,
to pairing each $\ell$-fall $i \to i-\ell$ ($\ell \ge 1$)
with the corresponding rises $i-\ell \to i-\ell+1 \to \ldots \to i$
and then transferring the weights (or part of the weights)
from those rises to the $\ell$-fall
(as was done in \cite{latpath_SRTR}).
${\rm bl}acksquare$
\par
\noindent{\sc Proof.\ }of{Proposition~\ref{prop.prodmat}{\rm (a)}}
Specialize Proposition~\ref{prop.prodmat.phiL}(b) to the case
$\phi_i^{[L]} = \phi_i$ for all $L$.
$\square$
Note now that the triangular arrays
${\sf L}hat^{\rm ord}$, ${\sf L}hat$, ${\sf L}tilde$ are each of the form
$ \left[
\begin{equation}gin{array}{c|c@{\hspace*{2mm}}c@{\hspace*{2mm}}c}
1 & 0 & 0 & \cdots \\
{\rm hl}ine
0 & & & \\
0 & & L^{\mathbf{w}}edge & \\[-1mm]
\vdots & & & \\
\end{array}
\right]
$.
So it is of some interest to find the production matrix
for the corresponding submatrices $L^{\mathbf{w}}edge$.
Let us use the following notation:
For any matrix $M = (m_{ij})_{i,j \ge 0}$,
write $M^{\mathbf{w}}edge \eqdef {\mathbb D}elta M {\mathbb D}elta^{\rm T}$
for $M$ with its zeroth row and column removed.
We then have:
\begin{equation}gin{corollary}
\label{cor.prodmat.phiL}
Let ${\bm{p}}hihat = (\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$ be indeterminates,
and work in the ring $\mathbb{Z}[{\bm{p}}hihat]$.
Define the lower-Hessenberg matrices $P^{\rm ord}$, $P$, ${\widetilde{P}}$
and the lower-triangular matrices
${\sf L}hat^{\rm ord}$, ${\sf L}hat$, ${\sf L}tilde$
as in Proposition~\ref{prop.prodmat.phiL}.
Then:
\begin{equation}gin{itemize}
\item[(a)] $({\sf L}hat^{\rm ord})^{\mathbf{w}}edge =
\phi_0^{[1]} \, {\mathcal{O}}((P^{\rm ord})^{\mathbf{w}}edge)$.
\item[(b)] ${\sf L}hat^{\mathbf{w}}edge = \phi_0^{[1]} \, {\mathcal{O}}(P^{\mathbf{w}}edge)$.
\item[(c)] ${\sf L}tilde^{\mathbf{w}}edge = {\mathcal{O}}({\widetilde{P}}^{\mathbf{w}}edge)$.
\end{itemize}
\end{corollary}
\par
\noindent{\sc Proof.\ }
(a) By Proposition~\ref{prop.prodmat.phiL}(a) we have
${\sf L}hat^{\rm ord} = {\mathcal{O}}(P^{\rm ord})$.
Now use Lemma~\ref{lemma.down-shifted}
with $P,Q,c,A,B$ replaced by
$(P^{\rm ord})^{\mathbf{w}}edge, P^{\rm ord}, \phi_0^{[1]}, ({\sf L}hat^{\rm ord})^{\mathbf{w}}edge,
{\sf L}hat^{\rm ord}$, respectively.
(b) and (c) are analogous.
$\square$
{\rm sm}allskip
{\bf Remark.}
In \cite[sections~12.2 and 12.3]{latpath_SRTR}
we considered the output matrix ${\sf L}hat^{\mathbf{w}}edge$ rather than ${\sf L}hat$,
and therefore obtained the production matrix $P^{\mathbf{w}}edge$ rather than $P$.
Also, in that paper we considered only the cases ${\bm{p}}hi = \begin{equation}e$ and ${\bm{h}}$;
but the proof for the generic case ${\bm{p}}hi$, given here,
is completely analogous and indeed slightly simpler.
$\square$
\subsection[Identity for $B_y^{-1} P B_y$ and proof of
Proposition~\ref{prop.prodmat}(b)]{Identity for $\bm{B_y^{-1} P B_y}$
and proof of Proposition~\ref{prop.prodmat}(b)}
We now wish to prove the following identity:
\begin{equation}gin{lemma}[Identity for $B_y^{-1} P B_y$]
\label{lemma.ByinvPBy}
Let ${\bm{p}}hi = (\phi_i)_{i \ge 0}$ and $y$ be indeterminates,
and work in the ring ${\mathbb Z}[{\bm{p}}hi,y]$.
Define the lower-Hessenberg matrix $P = (p_{ij})_{i,j \ge 0}$ by
\begin{equation}
p_{ij}
\;=\;
\begin{equation}gin{cases}
0 & \textrm{if $j=0$ or $j > i+1$} \\[1mm]
{\displaystyle {i! \over (j-1)!}} \, \phi_{i-j+1}
& \textrm{if $1 \le j \le i+1$}
\end{cases}
\label{eq.lemma.ByinvPBy.1}
\end{equation}
and the unit-lower-triangular $y$-binomial matrix $B_y$ by
\begin{equation}
(B_y)_{nk} \;=\; \binom{n}{k} \, y^{n-k}
\;.
\end{equation}
Let ${\mathbb D}elta = (\delta_{i+1,j})_{i,j \ge 0}$
be the matrix with 1 on the superdiagonal and 0 elsewhere.
Then
\begin{equation}
B_y^{-1} P B_y \;=\; P (I + y {\mathbb D}elta^{\rm T})
\;.
\label{eq.lemma.ByinvPBy}
\end{equation}
\end{lemma}
\par
\noindent{\sc Proof.\ }
It is easy to see, using the Chu--Vandermonde identity,
that $B_y B_z = B_{y+z}$ and hence that $B_y^{-1} = B_{-y}$.
Therefore
\begin{equation}
(B_y^{-1} P B_y)_{ij}
\;=\;
\sum_{k,\ell} (-1)^{i+k} \, {i! \over k! \, (i-k)!} \, y^{i-k}
\: {k! \over (\ell-1)!} \, \phi_{k-\ell+1}
\: {\ell! \over j! \, (\ell-j)!} \, y^{\ell-j}
\;.
\label{eq.proof.lemma.ByinvPBy.1}
\end{equation}
Then the coefficient of $\phi_m$ in this is (setting $k=\ell+m-1$)
\begin{equation}gin{subeqnarray}
& & \!\!\!\!
[\phi_m] \, (B_y^{-1} P B_y)_{ij}
\;=\;
(-1)^{i+m-1} \, {i! \over j!} \, y^{i-j+1-m}
\sum_\ell (-1)^\ell \, {\ell \over (i-\ell-m+1)! \, (\ell-j)!}
\qquad {\mathbf{n}}onumber \\[-1mm] \\[1mm]
& & \qquad =\;
(-1)^{i+m-1} \, {i! \over j! \, (i-j+1-m)!} \, y^{i-j+1-m}
\sum_\ell (-1)^\ell \, \ell \, \binom{i-j+1-m}{\ell-j}
\;.
{\mathbf{n}}onumber \\
\slabel{eq.proof.lemma.ByinvPBy.2.b}
\label{eq.proof.lemma.ByinvPBy.2}
\end{subeqnarray}
Now
\begin{equation}
\sum_\ell (-1)^\ell \, \binom{i-j+1-m}{\ell-j} \, x^\ell
\;=\;
(-x)^j \, (1-x)^{i-j+1-m}
\;,
\end{equation}
so that
\begin{equation}gin{subeqnarray}
\sum_\ell (-1)^\ell \, \ell \, \binom{i-j+1-m}{\ell-j}
& = &
{d \over dx}
\bigl[ (-x)^j \, (1-x)^{i-j+1-m} \bigr]
\biggr|_{x=1}
\\[2mm]
& = &
(-1)^j \, \bigl[ j \delta_{m,i-j+1} \,-\, \delta_{m,i-j} \bigr]
\;.
\end{subeqnarray}
Substituting this into \reff{eq.proof.lemma.ByinvPBy.2.b} gives
\begin{equation}gin{subeqnarray}
(B_y^{-1} P B_y)_{ij}
& = &
{i! \over j!} \, \bigl[ j \phi_{i-j+1} \,+\, y \phi_{i-j} \bigr]
\\[2mm]
& = &
p_{ij} \,+\, y p_{i,j+1}
\;,
\end{subeqnarray}
which is precisely \reff{eq.lemma.ByinvPBy}.
$\square$
We can now prove Proposition~\ref{prop.prodmat}(b):
\par
\noindent{\sc Proof.\ }of{Proposition~\ref{prop.prodmat}{\rm (b)}}
By Proposition~\ref{prop.prodmat}(a),
the matrix $P$ defined in \reff{eq.prop.prodmat}/\reff{eq.lemma.ByinvPBy.1}
is the production matrix for the generic Lah triangle ${\sf L}$.
By Lemma~\ref{lemma.production.AB},
the production matrix for ${\sf L} B_y$ is then $B_y^{-1} P B_y$;
and by Lemma~\ref{lemma.ByinvPBy} this equals $P (I + y {\mathbb D}elta^{\rm T})$.
$\square$
\subsection{Total positivity of the production matrix}
We shall use the following general lemma:
\begin{equation}gin{lemma}
\label{lemma.diagmult.TP}
Let $A = (a_{ij})_{i,j \ge 0}$ be a lower-triangular matrix
with entries in a partially ordered commutative ring $R$,
and let ${\bm{d}} = (d_i)_{i \ge 1}$.
Define the lower-triangular matrix $B = (b_{ij})_{i,j \ge 0}$ by
\begin{equation}
b_{ij} \;=\; d_{j+1} d_{j+2} \cdots d_i \, a_{ij}
\;.
\end{equation}
Then:
\begin{equation}gin{itemize}
\item[(a)] If $A$ is TP${}_r$ and ${\bm{d}}$ are indeterminates,
then $B$ is TP${}_r$ in the ring $R[{\bm{d}}]$ equipped with
the coefficientwise order.
\item[(b)] If $A$ is TP${}_r$ and ${\bm{d}}$ are nonnegative elements of $R$,
then $B$ is TP${}_r$ in the ring $R$.
\end{itemize}
\end{lemma}
\par
\noindent{\sc Proof.\ }
(a) Let ${\bm{d}} = (d_i)_{i \ge 1}$ be commuting indeterminates,
and let us work in the ring $R[{\bm{d}},{\bm{d}}^{-1}]$
equipped with the coefficientwise order.
Let $D = \mathop{\rm diag}{\mathbf{n}}olimits(1,\, d_1,\, d_1 d_2,\, \ldots)$.
Then $D$ is invertible, and both $D$ and
$D^{-1} = \mathop{\rm diag}{\mathbf{n}}olimits(1,\, d_1^{-1},\, d_1^{-1} d_2^{-1},\, \ldots)$
have nonnegative elements.
It follows that $B = D A D^{-1}$ is TP${}_r$ in the ring $R[{\bm{d}},{\bm{d}}^{-1}]$
equipped with the coefficientwise order.
But the matrix elements $b_{ij}$
actually belong to the subring $R[{\bm{d}}] \subseteq R[{\bm{d}},{\bm{d}}^{-1}]$.
So $B$ is TP${}_r$ in the ring $R[{\bm{d}}]$
equipped with the coefficientwise order.
(b) follows from (a) by specializing indeterminates.
$\square$
\begin{equation}gin{corollary}
\label{cor.diagmult.TP}
Let $(\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$ be
elements of a partially ordered commutative ring~$R$,
with $\phi_i^{[L]} \eqdef 0$ for $i < 0$.
Suppose that the lower-triangular matrix
$\Phi = (\phi_{i-j}^{[j+1]})_{i,j \ge 0}$
is TP${}_r$.
Then the matrices $P^{\rm ord}$ and $P$
defined by \reff{eq.prop.prodmat.phiL.0}/\reff{eq.prop.prodmat.phiL.1}
are also TP${}_r$.
\end{corollary}
\par
\noindent{\sc Proof.\ }
(a) We have $P^{\rm ord} = \Phi {\mathbb D}elta D$
where $D = \mathop{\rm diag}{\mathbf{n}}olimits\big( (j)_{j \ge 0} \big)$.
(b) Applying Lemma~\ref{lemma.diagmult.TP}(b) with $A = \Phi$
and $d_i = i!$,
we see that the lower-triangular matrix $P' = (p'_{ij})_{i,j \ge 0}$
with entries $p'_{ij} = (i!/j!) \, \phi_{i-j}^{[j+1]}$ is TP${}_r$.
But then $P = P' {\mathbb D}elta$ is also TP${}_r$.
$\square$
The doubly-indexed sequence
$(\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$ is very general,
but precisely because of its generality it is somewhat difficult to work with:
indeed, the corresponding matrix $\Phi$ is a
{\em completely arbitrary}\/ lower-triangular matrix,
for which it may or may not be feasible to determine its total positivity.
It is therefore of interest to consider specializations
for which the total positivity may be proven more easily.
One such specialization is the following:
Let ${\bm{p}}hi= (\phi_i)_{i \geq 0}$ and ${\mathbf{c}}= (c_L)_{L \ge 1}$
be two sequences of indeterminates,
and set $\phi_i^{[L]} = \phi_i c_L$;
we denote this specialization by the shorthand ${\bm{p}}hihat = {\bm{p}}hi {\mathbf{c}}$.
We then have the following easy fact:
\begin{equation}gin{lemma}
\label{lemma.phic}
Let ${\bm{p}}hi= (\phi_i)_{i \geq 0}$ be a sequence
in a partially ordered commutative ring~$R$,
with $\phi_i \eqdef 0$ for $i < 0$,
and let ${\bm{c}} = (c_L)_{L \ge 1}$;
and define the lower-triangular matrix
$\Phi = (\phi_{i-j} c_{j+1})_{i,j \ge 0}$
Then:
\begin{equation}gin{itemize}
\item[(a)] If ${\bm{p}}hi$ is Toeplitz-TP${}_r$
and ${\bm{c}}$ are indeterminates,
then $\Phi$ is TP${}_r$ in the ring $R[{\mathbf{c}}]$ equipped with
the coefficientwise order.
\item[(b)] If ${\bm{p}}hi$ is Toeplitz-TP${}_r$
and ${\bm{c}}$ are nonnegative elements of $R$,
then $\Phi$ is TP${}_r$ in the ring $R$.
\end{itemize}
\end{lemma}
\par
\noindent{\sc Proof.\ }
We have $\Phi = T_\infty({\bm{p}}hi) \, \mathop{\rm diag}{\mathbf{n}}olimits\big( (c_{j+1})_{j \ge 0} \bigr)$.
$\square$
By combining Lemma~\ref{lemma.phic} and Corollary~\ref{cor.diagmult.TP},
we deduce, under the same hypotheses, the TP${}_r$ property
for the production matrices $P^{\rm ord}$ and $P$
evaluated at ${\bm{p}}hihat = {\bm{p}}hi {\mathbf{c}}$.
\subsection{Generalization of Theorem~1.1(a,c)}
We can now state and prove a generalization of Theorem~1.1(a,c):
\begin{equation}gin{theorem}[Total positivity of the refined generic Lah polynomials]
\label{thm1.1.generalized}
{\bm{r}}eak{\mathbf{n}}oindent
Let ${\bm{p}}hihat = (\phi_i^{[L]})_{i \ge 0,\, L \ge 1}$ be
elements of a partially ordered commutative ring~$R$,
with $\phi_i^{[L]} \eqdef 0$ for $i < 0$,
such that the lower-triangular matrix $\Phi = (\phi_{i-j}^{[j+1]})_{i,j \ge 0}$
is TP${}_r$.
Then:
\begin{equation}gin{itemize}
\item[(a)] The lower-triangular matrix
${\sf L}hat({\bm{p}}hihat) = ({\widehat{L}}_{n,k}({\bm{p}}hihat))_{n,k \ge 0}$
is TP${}_r$.
\item[(c)] The sequence ${\bm{L}}hat^\mathop{\rm tr}{\mathbf{n}}olimitsiangle({\bm{p}}hihat) =
({\widehat{L}}_{n+1,1}({\bm{p}}hihat))_{n \ge 0}$
is Hankel-TP${}_r$.
\end{itemize}
\end{theorem}
\par
\noindent{\sc Proof.\ }
(a) By Corollary~\ref{cor.diagmult.TP}, the production matrix $P$
defined in \reff{eq.prop.prodmat.phiL.1} is TP${}_r$.
By Proposition~\ref{prop.prodmat.phiL}(b),
the corresponding output matrix is ${\sf L}hat = {\mathcal{O}}(P)$.
So Theorem~\ref{thm.iteration.homo} implies that ${\sf L}hat$ is TP${}_r$.
(c) By Corollary~\ref{cor.diagmult.TP}, the matrix $P$ is TP${}_r$;
hence so is $P^{\mathbf{w}}edge = {\mathbb D}elta P {\mathbb D}elta^{\rm T}$.
By Corollary~\ref{cor.prodmat.phiL}(b),
we have ${\sf L}hat^{\mathbf{w}}edge = \phi_0^{[1]} \, {\mathcal{O}}(P^{\mathbf{w}}edge)$.
So Theorem~\ref{thm.iteration2bis} implies that
the zeroth column of ${\sf L}hat^{\mathbf{w}}edge$ is Hankel-TP${}_r$.
But that is precisely $({\widehat{L}}_{n+1,1}({\bm{p}}hihat))_{n \ge 0}$.
$\square$
{\bf Remark.}
Since an {\em arbitrary}\/ lower-triangular matrix $\Phi$
can be written in the form $\Phi = (\phi_{i-j}^{[j+1]})_{i,j \ge 0}$,
it follows that $\Phi \mapsto {\sf L}hat({\bm{p}}hihat)$
is a well-defined polynomial mapping
of the lower-triangular matrices into themselves,
which preserves total positivity of each order $r$.
However, this mapping seems rather complicated,
even when restricted to Toeplitz matrices $\Phi$
(see the comments in Section~\ref{sec.exponential} below).
It would be interesting to better understand this mapping
from an algebraic point of view.
${\rm bl}acksquare$
As an immediate consequence of Lemma~\ref{lemma.phic}
and Theorem~\ref{thm1.1.generalized}, we have:
\begin{equation}gin{corollary}
\label{cor.thm1.1.generalized}
Let ${\bm{p}}hi= (\phi_i)_{i \geq 0}$ be a sequence
in a partially ordered commutative ring~$R$,
and let ${\mathbf{c}} = (c_L)_{L \ge 1}$ be indeterminates.
If ${\bm{p}}hi$ is Toeplitz-TP${}_r$, then:
\begin{equation}gin{itemize}
\item[(a)] The lower-triangular matrix
${\sf L}hat({\bm{p}}hi,{\mathbf{c}}) = ({\widehat{L}}_{n,k}({\bm{p}}hi{\mathbf{c}}))_{n,k \ge 0}$
is TP${}_r$ in the ring $R[{\mathbf{c}}]$ equipped with
the coefficientwise order.
\item[(c)] The sequence ${\bm{L}}hat^\mathop{\rm tr}{\mathbf{n}}olimitsiangle({\bm{p}}hi,{\mathbf{c}}) =
({\widehat{L}}_{n+1,1}({\bm{p}}hi{\mathbf{c}}))_{n \ge 0}$
is Hankel-TP${}_r$ in the ring $R[{\mathbf{c}}]$ equipped with
the coefficientwise order.
\end{itemize}
\end{corollary}
\par
\noindent{\sc Proof.\ }of{Theorem~\ref{thm1.1}{\rm (a,c)}}
Specialize Corollary~\ref{cor.thm1.1.generalized}(a,c) to ${\mathbf{c}} = {\bm{1}}$.
$\square$
\subsection{Completion of the proofs}
\par
\noindent{\sc Proof.\ }of{Theorem~\ref{thm1.1}{\rm (b)}}
Since the zeroth column of the matrix ${\sf L} B_y$
is given by the Lah polynomials $L_n({\bm{p}}hi,y)$,
Theorem~\ref{thm1.1}(b) is an immediate consequence
of Proposition~\ref{prop.prodmat}(b), Corollary~\ref{cor.diagmult.TP}
and Theorem~\ref{thm.iteration2bis}.
$\square$
\par
\noindent{\sc Proof.\ }of{Corollary~\ref{cor1.2}}
The Jacobi--Trudi identity
\cite[Theorem~7.16.1 and Corollary~7.16.2]{Stanley_99}
expresses all the Toeplitz minors of $\begin{equation}e$ or ${\bm{h}}$
as skew Schur functions.
Furthermore, every skew Schur function
is a nonnegative linear combination of Schur functions
(Littlewood--Richardson coefficients \cite[Section~7.A1.3]{Stanley_99}).
So the sequences $\begin{equation}e$ and ${\bm{h}}$ are Toeplitz-totally positive
with respect to the Schur order.
Corollary~\ref{cor1.2} is then an immediate consequence
of Theorem~\ref{thm1.1}.
$\square$
Furthermore, by using Corollary~\ref{cor.thm1.1.generalized} here
in place of Theorem~\ref{thm1.1}(a,c),
we obtain a generalization of Corollary~\ref{cor1.2}(a,c),
whose precise statement we leave to the reader;
and likewise for Corollary~\ref{cor1.3}.
\section{Differential operators for the multivariate Lah polynomials}
\label{sec.differential}
\subsection{Differential operator for positive type}
In \cite[Proposition~12.6]{latpath_SRTR} we gave expressions
for the multivariate Eulerian polynomials of positive type
in terms of the action of certain first-order linear differential operators.
Translated to our current notation, we proved the following:\footnote{
Strictly speaking, what we proved in \cite[Proposition~12.6]{latpath_SRTR},
when translated to our current notation,
puts $\delta_{n1} \delta_{k1}$ instead of $\delta_{n0} \delta_{k0}$
in \reff{eq.scrqnk.differential}, and holds only for $n,k \ge 1$.
But it is then easy to see that also \reff{eq.scrqnk.differential}
holds as written for $n,k \ge 0$.
Also, the statement in \cite[Proposition~12.6]{latpath_SRTR}
applied only to $r \ge 2$.
But for $r=1$ we have
$L_{n,k}^{(1)+}(x_1) = \stirlingsubset{n}{k} x_1^{n-k}$,
so that \reff{eq.scrqnk.differential} is the well-known recurrence
for the Stirling subset numbers (note that ${\mathcal{D}}_1 = 0$).
}
\begin{equation}gin{proposition}
{$\!\!\!$ \bf \protect\cite[Proposition~12.6]{latpath_SRTR}\ }
\label{prop.MVeulerian.differential}
For every integer ${r \ge 1}$, we have
\begin{equation}gin{eqnarray}
L_{n,1}^{(r)+}({\mathbf{x}}) & = &
{\sf B}igl( {\mathcal{D}}_r \,+\, \sum_{i=1}^r x_i {\sf B}igr)^{\! n-1} \: 1
\qquad\hbox{for $n \ge 1$}
\label{eq.scrq.differential} \\
L_{n,k}^{(r)+}({\mathbf{x}}) & = &
{\sf B}igl( {\mathcal{D}}_r \,+\, k \sum_{i=1}^r x_i {\sf B}igr)
L_{n-1,k}^{(r)+}({\mathbf{x}}) \:+\: L_{n-1,k-1}^{(r)+}({\mathbf{x}})
\:+\: \delta_{n0} \delta_{k0}
\quad\hbox{for $n,k \ge 0$}
\qquad
\label{eq.scrqnk.differential}
\end{eqnarray}
where
\begin{equation}
{\mathcal{D}}_r
\;=\;
\sum_{i=1}^r \biggl( \! x_i
\!\!\!\!
\sum_{\begin{equation}gin{scarray}
1 \le j \le r \\
j {\mathbf{n}}e i
\end{scarray}}
\!\!\!\!\!\!
x_j \!
\biggr) \, {\partial \over \partial x_i}
\;.
\label{def.dm}
\end{equation}
\end{proposition}
Now we would like to extend this to give a
differential expression for the row-generating polynomials
$L_n^{(r)+}({\bf x},y)$:
\begin{equation}gin{proposition}
\label{prop.diff.op}
For every integer $r\geq 1$, we have:
\begin{equation}gin{equation}
L_n^{(r)+}({\bf x},y) \;=\; ({\mathcal{D}}tilde_r +y)^n \: 1
\;,
\label{eq.prop.diff.op}
\end{equation}
where
\begin{equation}
{\mathcal{D}}tilde_r
\;=\;
{\mathcal{D}}_r \:+\: \sum_{i=1}^r x_i y \, {\partial \over \partial y}
\;.
\label{def.diff.opp}
\end{equation}
and ${\mathcal{D}}_r$ is defined in \reff{def.dm}.
\end{proposition}
\par
\noindent{\sc Proof.\ }
Multiply \reff{eq.scrqnk.differential} by $y^k$ and sum over $k$:
the factor $k$ becomes $y \, \partial/\partial y$, and we have
\begin{equation}
L_n^{(r)+}({\mathbf{x}},y)
\;=\;
{\sf B}igl( {\mathcal{D}}_r \,+\, \sum_{i=1}^r x_i y \, {\partial \over \partial y}
\,+\, y {\sf B}igr)
L_{n-1}^{(r)+}({\mathbf{x}},y)
\:+\: \delta_{n0} \delta_{k0}
\;.
\end{equation}
Iterating this yields \reff{eq.prop.diff.op}.
$\square$
\subsection{Differential operator for negative type}
Similarly, in \cite[Proposition~12.26]{latpath_SRTR} we gave expressions
for the multivariate Eulerian polynomials of negative type
in terms of the action of certain first-order linear differential operators.
Translated to our current notation, we proved the following:\footnote{
Strictly speaking, what we proved in \cite[Proposition~12.26]{latpath_SRTR},
when translated to our current notation,
puts $\delta_{n1} \delta_{k1}$ instead of $\delta_{n0} \delta_{k0}$
in \reff{eq.scrqnk.differential.multi}, and holds only for $n,k \ge 1$.
But it is then easy to see that also \reff{eq.scrqnk.differential.multi}
holds as written for $n,k \ge 0$.
}
\begin{equation}gin{proposition}
{$\!\!\!$ \bf \protect\cite[Proposition~12.26]{latpath_SRTR}\ }
\label{prop.MVeulerian.differential.multi}
For every integer ${r \ge 1}$, we have
\begin{equation}gin{eqnarray}
L_{n,1}^{(r)-}({\mathbf{x}}) & = &
{\sf B}igl( {\mathcal{D}}^-_r \,+\, \sum_{i=1}^r x_i {\sf B}igr)^{\! n-1} \: 1
\qquad\hbox{for $n \ge 1$}
\label{eq.scrq.differential.multi} \\
L_{n,k}^{(r)-}({\mathbf{x}}) & = &
{\sf B}igl( {\mathcal{D}}^-_r \,+\, k \sum_{i=1}^r x_i {\sf B}igr)
L_{n-1,k}^{(r)-}({\mathbf{x}}) \:+\: L_{n-1,k-1}^{(r)-}({\mathbf{x}})
\:+\: \delta_{n0} \delta_{k0}
\quad\hbox{for $n,k \ge 0$}
\qquad
\label{eq.scrqnk.differential.multi}
\end{eqnarray}
where
\begin{equation}
{\mathcal{D}}^-_r
\;=\;
\sum_{i=1}^r \biggl( \! x_i^2 \:+\:
x_i \sum_{j=1}^r x_j \!
\biggr) \, {\partial \over \partial x_i}
\;.
\label{def.dm-}
\end{equation}
\end{proposition}
Now we would like to extend this to give a
differential expression for the row-generating polynomials
$L_n^{(r)-}({\bf x},y)$,
analogously to what we did for the positive type.
The result is:
\begin{equation}gin{proposition}
\label{prop.diff.op.minus}
For every integer $r\geq 1$, we have:
\begin{equation}gin{equation}
L_n^{(r)-}({\bf x},y) \;=\; ({\mathcal{D}}tilde_r^- +y)^n \: 1
\;,
\label{eq.prop.diff.op.minus}
\end{equation}
where
\begin{equation}
{\mathcal{D}}tilde_r^-
\;=\;
{\mathcal{D}}^-_r \:+\: \sum_{i=1}^r x_i y \, {\partial \over \partial y}
\;.
\label{def.diff.opp.minus}
\end{equation}
and ${\mathcal{D}}^-_r$ is defined in \reff{def.dm-}.
\end{proposition}
\par
\noindent{\sc Proof.\ }
Identical to the proof of Proposition~\ref{prop.diff.op},
but with ${\mathcal{D}}_r$ replaced by ${\mathcal{D}}^-_r$.
$\square$
\section{Proof of Theorem~\ref{thm.lah.S-fraction} by the Euler--Gauss recurrence method}
\label{sec.euler-gauss}
In the Introduction we explained how,
for the multivariate Lah polynomials of {\em positive}\/ type,
the matrix $B_y^{-1} P B_y = P (I + y {\mathbb D}elta^{\rm T})$,
which is the production matrix for ${\sf L} B_y$,
has the bidiagonal factorization \reff{eq.lah.prodmat.y};
and we explained how this in turn implies,
by virtue of \reff{eq.prop.contraction},
that the multivariate Lah polynomials of positive type $L_n^{(r)+}({\mathbf{x}},y)$
are given by an $r$-branched S-fraction with coefficients
\begin{equation}
{\bm{a}}lpha \;=\; (\alpha_i)_{i \ge r}
\;=\;
y,x_1,\ldots,x_r,y,2x_1,\ldots,2x_r,y,3x_1,\ldots,3x_r, \,\ldots
\;,
\label{eq.lah.alphas.bis}
\end{equation}
as stated in Theorem~\ref{thm.lah.S-fraction}.
In this section we would like to give a second (and completely independent)
proof of Theorem~\ref{thm.lah.S-fraction},
based on the Euler--Gauss recurrence method for proving continued fractions,
generalized to $m$-S-fractions as in \cite[Proposition~2.3]{latpath_SRTR}.
Let us recall briefly the method:
if $(g_k(t))_{k \ge -1}$ are formal power series with constant term 1
(with coefficients in some commutative ring $R$)
satisfying a recurrence
\begin{equation}
g_k(t) - g_{k-1}(t) \;=\; \alpha_{k+m} t \, g_{k+m}(t)
\qquad\hbox{for } k \ge 0
\label{eq.recurrence.gkm.0.bis}
\end{equation}
for some coefficients ${\bm{a}}lpha = (\alpha_i)_{i \ge m}$ in $R$,
then $g_0(t)/g_{-1}(t) = \sum_{n=0}^\infty S^{(m)}_n({\bm{a}}lpha) \, t^n$,
where $S^{(m)}_n({\bm{a}}lpha)$ is the $m$-Stieltjes--Rogers polynomial
evaluated at the specified values ${\bm{a}}lpha$.
As in \cite[sections~12.2.4 and 12.3.4]{latpath_SRTR},
we will apply this method with the choice $g_{-1}(t) = 1$.
We need to find series $(g_k(t))_{k \ge 0}$ with constant term 1
satisfying \reff{eq.recurrence.gkm.0.bis},
where here $m=r$.
Let us write $g_k(t) = \sum_{n=0}^\infty g_{k,n} \, t^n$
and define $g_{k,n} = 0$ for $n < 0$.
Then \reff{eq.recurrence.gkm.0.bis} can be written as
\begin{equation}
g_{k,n} \,-\, g_{k-1,n} \;=\; \alpha_{k+r} \, g_{k+r,n-1}
\qquad\hbox{for } k,n \ge 0
\;.
\label{eqrec.gkn}
\end{equation}
Here are the required $g_{k,n}$:
\begin{equation}gin{proposition}[Euler--Gauss recurrence for multivariate Lah polynomials
of positive type]
\label{prop.euler-gauss.quasi-affine}
Let ${\mathbf{x}} = (x_1,\ldots,x_r)$ be indeterminates;
we work in the ring $R = {\mathbb Z}[{\mathbf{x}}]$.
Set $g_{k,n} = \delta_{n0}$ for $k < 0$,
and then define $g_{k,n}$ for $k,n \ge 0$ by the recurrence
\begin{equation}
g_{k,n}
\;=\;
{\sf B}igl( {\mathcal{D}}tilde_r \,+\, \sum_{i=1}^r \alpha_{k+i} {\sf B}igr) g_{k,n-1}
\:+\: g_{k-r,n}
\label{def.gkn}
\end{equation}
where ${\mathcal{D}}tilde_r$ is given by \reff{def.diff.opp}
and ${\bm{a}}lpha$ are given by
\begin{equation}
{\bm{a}}lpha \;=\; (\alpha_i)_{i \ge 0}
\;=\;
\underbrace{0,\ldots,0}_\text{$r$ times},
y,x_1,\ldots,x_r,y,2x_1,\ldots,2x_r,y,3x_1,\ldots,3x_r, \,\ldots
\label{eq.lah.alphas.bis2}
\end{equation}
or in detail
\begin{equation}
\alpha_i
\;=\;
\begin{equation}gin{cases}
{\sf B}ig\lfloor\displaystyle\frac{i}{r+1} {\sf B}ig\rfloor \, x_{j+1}
& \textrm{if $i \equiv j \bmod r\!+\!1$
with $0 \le j \le r\!-\!1$}
\\[3mm]
y & \textrm{if $i \equiv r \bmod r\!+\!1$}
\end{cases}
\label{eq.lah.alphas.bis3}
\end{equation}
Then:
\begin{equation}gin{itemize}
\item[(a)] $g_{k,0} = 1$ for all $k \in {\mathbb Z}$.
\\[-5mm]
\item[(b)] $(g_{k,n})$ satisfies the recurrence
\reff{eqrec.gkn} for all $k \ge -r$ and $n \ge 0$.
\\[-5mm]
\item[(c)] $S^{(r)}_n({\bm{a}}lpha) = g_{0,n} = ({\mathcal{D}}tilde_r + y)^n \, 1$.
\end{itemize}
\end{proposition}
\par
\noindent{\sc Proof.\ }
(a) We see trivially using \reff{def.gkn} that $g_{k,0} = 1$
for all $k \in {\mathbb Z}$, i.e.\ $g_k(t)$ has constant term 1.
(b) We will now prove that the recurrence \reff{eqrec.gkn} holds.
The proof will be by an outer induction on $k$ (in steps of $r$)
in which we encapsulate an inner induction on~$n$.
The base cases $k = -r,\ldots,-1$ for the outer induction
hold trivially because $g_{k,n} = g_{k-1,n} = \delta_{n0}$ for $k < 0$
and $\alpha_0,\ldots,\alpha_{r-1} = 0$.
We now assume that \reff{eqrec.gkn} holds for a given $k$ and all $n \ge 0$;
we want to prove that it still holds when we replace $k$ by $k+r$, i.e.\ that
\begin{equation}gin{equation}
g_{k+r,n} \:-\: g_{k+r-1,n} \:-\: \alpha_{k+2r} \, g_{k+2r,n-1}
\;=\;
0
\quad \hbox{for all $n \ge 0$}
\;.
\label{eqrec.gkn.inn}
\end{equation}
We will prove \reff{eqrec.gkn.inn} by induction on $n$.
Clearly \reff{eqrec.gkn.inn} holds for $n=0$
because $g_{k+r,0} = g_{k+r-1,0} = 1$ [from part~(a)]
and $g_{k+2r,-1} = 0$ (by definition of the $g$'s).
When $n > 0$, we use \reff{def.gkn}
on each of the three $g$'s on the left-hand side of
\reff{eqrec.gkn.inn}, giving
\begin{equation}gin{subeqnarray}
g_{k+r,n}
& = &
{\sf B}igl( {\mathcal{D}}tilde_r \,+\, \sum_{i=1}^r \alpha_{k+r+i} {\sf B}igr) g_{k+r,n-1}
\:+\: g_{k,n}
\\
g_{k+r-1,n}
& = &
{\sf B}igl( {\mathcal{D}}tilde_r \,+\, \sum_{i=1}^r \alpha_{k+r-1+i} {\sf B}igr) g_{k+r-1,n-1}
\:+\: g_{k-1,n}
\\
\alpha_{k+2r} \, g_{k+2r,n-1}
& = &
\alpha_{k+2r} \,
{\sf B}igl( {\mathcal{D}}tilde_r \,+\, \sum_{i=1}^r \alpha_{k+2r+i} {\sf B}igr) g_{k+2r,n-2}
\:+\: \alpha_{k+2r} \, g_{k+r,n-1}
\qquad
\end{subeqnarray}
We can then rewrite the left-hand-side of \reff{eqrec.gkn.inn} as
\begin{equation}gin{eqnarray}
\text{LHS of \reff{eqrec.gkn.inn}}
& = &
g_{k,n}-g_{k-1,n}-\alpha_{k+2r}g_{k+r,n-1}
{\mathbf{n}}onumber \\[1mm]
& &
\quad+\,
\sum_{i=1}^r \alpha_{k+r+i} \,
(g_{k+r,n-1}-g_{k+r-1,n-1}-\alpha_{k+2r}g_{k+2r,n-2})
{\mathbf{n}}onumber \\[1mm]
& &
\quad+\,
{\mathcal{D}}tilde_r (g_{k+r,n-1}-g_{k+r-1,n-1})
\:-\:
\alpha_{k+2r} \, {\mathcal{D}}tilde_r g_{k+2r,n-2}
{\mathbf{n}}onumber \\[1mm]
& &
\quad-\,
\sum_{i=1}^r (\alpha_{k+r-1+i}-\alpha_{k+r+i}) \, g_{k+r-1,n-1}
{\mathbf{n}}onumber \\[1mm]
& &
\quad-\,
\sum_{i=1}^r (\alpha_{k+2r+i}-\alpha_{k+r+i})
\, \alpha_{k+2r} \, g_{k+2r,n-2}
\;.
\label{eq.LHS.eqrec.gkn.inn}
\end{eqnarray}
On the right-hand side of \reff{eq.LHS.eqrec.gkn.inn}, the first line is
\begin{equation}gin{eqnarray}
& &
g_{k,n}-g_{k-1,n}-\alpha_{k+2r}g_{k+r,n-1}
{\mathbf{n}}onumber \\[1mm]
& &
\qquad =\;
g_{k,n}-g_{k-1,n}-\alpha_{k+r}g_{k+r,n-1} +
(\alpha_{k+r}-\alpha_{k+2r}) g_{k+r,n-1}
{\mathbf{n}}onumber \\[1mm]
& &
\qquad =\;
(\alpha_{k+r}-\alpha_{k+2r})g_{k+r,n-1}
\;,
\end{eqnarray}
where the first equality is trivial
and the second one comes from the induction hypothesis on~$k$.
The second line of \reff{eq.LHS.eqrec.gkn.inn}
is zero by the induction hypothesis on~$n$.
The fourth line of \reff{eq.LHS.eqrec.gkn.inn} is a telescoping sum over $i$,
yielding simply $-(\alpha_{k+r}-\alpha_{k+2r}) \, g_{k+r-1,n-1}$.
For the third line of \reff{eq.LHS.eqrec.gkn.inn}, we use the fact that
${\mathcal{D}}tilde_r$ is a pure first-order differential operator;
then the Leibniz rule implies that the third line equals
\begin{equation}gin{eqnarray}
& &
{\mathcal{D}}tilde_r (g_{k+r,n-1}-g_{k+r-1,n-1})
\:-\: \alpha_{k+2r}{\mathcal{D}}tilde_r g_{k+2r,n-2}
{\mathbf{n}}onumber \\[1mm]
& &
\qquad =\;
{\mathcal{D}}tilde_r (g_{k+r,n-1}-g_{k+r-1,n-1}-\alpha_{k+2r}g_{k+2r,n-2})
\:+\: g_{k+2r,n-2} {\mathcal{D}}tilde_r \alpha_{k+2r}
{\mathbf{n}}onumber \\[1mm]
& &
\qquad =\;
g_{k+2r,n-2} {\mathcal{D}}tilde_r \alpha_{k+2r}
\;,
\end{eqnarray}
where the last equality comes from the induction hypothesis on $n$.
We now need to do a distinction of cases to compute
${\mathcal{D}}tilde_r \alpha_{k+2r}$.
If $k +2r\equiv r \bmod r+1$, we have $\alpha_{k+2r}=y$, and so
\begin{equation}
{\mathcal{D}}tilde_r \alpha_{k+2r}
\;=\;
{\mathcal{D}}tilde_r y
\;=\;
y \sum_{i=1}^r x_i
\;=\;
\alpha_{k+2r} \sum_{i=1}^r x_i
\;.
\end{equation}
On the other hand, if $k +2r\equiv j \bmod r+1$ with $j {\mathbf{n}}eq r$,
we then have $\alpha_{k+2r}=\lfloor \frac{ k+2r}{r+1}\rfloor x_{j+1}$,
and so
\begin{equation}
{\mathcal{D}}tilde_r \alpha_{k+2r}
\;=\; {\mathcal{D}}tilde_r {\sf B}igl( {\sf B}igl\lfloor \frac{ k+2r}{r+1} {\sf B}igr\rfloor
x_{j+1} {\sf B}igr)
\;=\; {\sf B}igl\lfloor \frac{ k+2r}{r+1} {\sf B}igr\rfloor x_{j+1}
\!\!\!\!
\sum_{\begin{equation}gin{scarray}
1 \le i \le r \\
i {\mathbf{n}}eq j+1
\end{scarray}}
\!\!\!\!
x_i
\;=\; \alpha_{k+2r}
\!\!\!\!
\sum_{\begin{equation}gin{scarray}
1 \le i \le r \\
i {\mathbf{n}}eq j+1
\end{scarray}}
\!\!\!\!
x_i
\;.
\end{equation}
Finally, the fifth line of of \reff{eq.LHS.eqrec.gkn.inn} is
\begin{equation}gin{eqnarray}
& &
-\sum_{i=1}^r (\alpha_{k+2r+i}-\alpha_{k+r+i}) \, \alpha_{k+2r} \, g_{k+2r,n-2}
{\mathbf{n}}onumber \\[1mm]
& &
\qquad =\;
- {\sf B}igl( \alpha_{k+r}-\alpha_{k+2r} +
\sum_{i=1}^r (\alpha_{k+2r+i}-\alpha_{k+r+i-1})
{\sf B}igr)\alpha_{k+2r} \, g_{k+2r,n-2}
\qquad
\end{eqnarray}
by a change of index $i \to i-1$ in the second sum.
Again, we need to do a distinction of cases to compute the sum.
If $k +2r\equiv r \bmod r+1$, we then have
\begin{equation}
\sum_{i=1}^r \left(\alpha_{k+2r+i}-\alpha_{k+r+i-1}\right)
\;=\;
\sum_{i=1}^r x_i
\end{equation}
by definition of the $\alpha$'s;
whereas when $k +2r\equiv j \bmod r+1$ with $j {\mathbf{n}}eq r$,
we have
\begin{equation}
\sum_{i=1}^r \left(\alpha_{k+2r+i}-\alpha_{k+r+i-1}\right)
\;=\;
\!\!\!\!
\sum_{\begin{equation}gin{scarray}
1 \le i \le r \\
i {\mathbf{n}}eq j+1
\end{scarray}}
\!\!\!\!
x_i
\;.
\end{equation}
And we still have the term
$-(\alpha_{k+r}-\alpha_{k+2r}) \alpha_{k+2r}g_{k+2r,n-2}$.
In both cases, the sum involving the $x_i$'s
cancels between the third and fifth lines;
therefore, all that remains of the third and fifth lines is
$-(\alpha_{k+r}-\alpha_{k+2r}) \alpha_{k+2r}g_{k+2r,n-2}$.
Now, adding all the lines gives
\begin{equation}
(\alpha_{k+r}-\alpha_{k+2r}) \,
(g_{k+r,n-1}-g_{k+r-1,n-1}-\alpha_{k+2r}g_{k+2r,n-2})
\;,
\end{equation}
which vanishes by the induction hypothesis on $n$.
This concludes the inductive step in $n$ to prove \reff{eqrec.gkn.inn},
which in turn concludes the induction on $k$ and finishes the proof
of part~(b).
(c) Putting $k=0$ in \reff{def.gkn} gives
$g_{0,n} = ({\mathcal{D}}tilde_r + y) g_{0,n-1} + \delta_{n0}$,
which proves $g_{0,n} = ({\mathcal{D}}tilde_r + y)^n \, 1$;
and this equals $L_n^{(r)+}({\mathbf{x}},y)$ by Proposition~\ref{prop.diff.op}.
On the other hand, starting from the recurrence \reff{eqrec.gkn}
and applying the Euler--Gauss recurrence method
\cite[Proposition~2.3]{latpath_SRTR},
we conclude that $g_{0,n} = S_n^{(r)}({\bm{a}}lpha)$.
$\square$
\section{Multivariate Lah polynomials in terms of decorated set partitions}
\label{sec.decorated}
In this section we would like to interpret the
multivariate Lah polynomials of positive type $L_n^{(r)+}({\mathbf{x}},y)$
as generating polynomials for partitions of the set $[n]$
in~which each block is ``decorated'' with an additional structure,
where the nature of this structure depends on the value of $r$.
For $r=1,2$ we observed already in the Introduction how this goes:
for $r=1$ the additional structure is empty,
while for $r=2$ it is a linear ordering on the block.
More precisely:
\begin{equation}gin{proposition}
\label{prop.bell}
The polynomial $L_n^{(1)+}(1,y)$ is the Bell polynomial $B_n(y)$,
that is, the generating polynomial of set partitions of $n$ elements
with a weight $y$ for each block.
More generally, $L_n^{(1)+}(x_1,y)$
is the homogenized Bell polynomial $x_1^n B_n(y/x_1)$.
\end{proposition}
\par
\noindent{\sc Proof.\ }
By definition, the polynomial $L_n^{(1)+}(1,y)$ is the generating
polynomial of unordered forests of increasing unary trees.
Since, given a set of integer labels,
there is only one way to increasingly label a unary tree,
there is a natural bijection between increasing unary trees and sets of labels.
An unordered forest of increasing unary trees is then
an unordered collection of disjoint sets of integers, whose union is $[n]$.
But this is nothing other than a set partition of $[n]$.
The final statement follows from the fact that $L_n^{(1)+}(x_1,y)$
is homogeneous of degree $n$.
$\square$
\begin{equation}gin{proposition}
\label{prop.lah}
The polynomial $L_n^{(2)+}({\bm{1}};y)$ is the Lah polynomial $L_n(y)$,
that is, the generating polynomial of set partitions of $n$ elements
into any number of nonempty lists (= linearly ordered subsets),
with a weight $y$ for each list.
More generally, the polynomial $L_n^{(2)+}(x_1,x_2;y)$
is the generating polynomial of partitions of the set $[n]$
into any number of nonempty lists, with a weight $y$ for each list,
$x_1$ for each descent in a list, and $x_2$ for each ascent in a list.
\end{proposition}
\par
\noindent{\sc Proof.\ }
By definition, the polynomial $L_n^{(2)+}(x_1,x_2;y)$
is the generating polynomial of unordered forests
of increasing binary trees,
with a weight $y$ for each root (or equivalently, each tree)
and a weight $x_1$ (resp.\ $x_2$) for each left (resp.\ right) child.
Now, a classical bijection \cite[pp.~23--25]{Stanley_86}
sends increasing binary trees to permutations.
Since a permutation is the same thing as a list,
applying this classical bijection to each tree of the forest
maps bijectively an unordered forest of increasing binary trees
to a set of lists whose union is $[n]$,
where the number of trees in the forest equals the number of lists.
The second claim comes out naturally, as this classical bijection maps
a left (resp.\ right) child of the tree to a descent (resp.\ ascent)
in the resulting permutation.
$\square$
This can be generalized to any positive integer $r$ by using the concept
of {\em Stirling permutation}\/ \cite{Gessel_78,Gessel_78a,Park_94a,Park_94b}
as discussed in \cite[section~12.5]{latpath_SRTR}.
Recall that a word ${\mathbf{w}} = w_1 \cdots w_L$
on a totally ordered alphabet $\mathbb{A}$
is called a \textbfit{Stirling word}
if $i < j < k$ and $w_i = w_k$ imply $w_j \ge w_i$:
that is, between any two occurrences of any letter $a$,
only letters that are larger than or equal to $a$ are allowed.
(Equivalently, between any two successive occurrences of the letter $a$,
only letters that are larger than $a$ are allowed.)
Now let $\mathbb{A}$ be a totally ordered alphabet
of finite cardinality $\ell$, and let $r$ be a nonnegative integer;
we denote by $r \mathbb{A}$ the multiset
consisting of $r$ copies of each letter $a \in \mathbb{A}$.
A~\textbfit{permutation} of $r \mathbb{A}$ is a word $w_1 \cdots w_{r\ell}$
containing exactly $r$ copies of each letter $a \in \mathbb{A}$;
it is called a \textbfit{Stirling permutation} of $r \mathbb{A}$
if it is also a Stirling word.
Now let $n$ be a nonnegative integer.
We define an \textbfit{$\bm{r}$-Stirling set partition} of $[n]$
to be a set partition of $[n]$ in which each block $B$
is decorated by a Stirling permutation of $rB$,
where the total order on $B$ is of course the one inherited from
the usual total order on the integers.
In particular, when $r=0$, the decoration is empty,
and we get back to classical set partitions;
and when $r=1$, we get a partition of the set $[n]$
in which each block is decorated by a permutation of the letters of that block,
or in other words, a partition of the set $[n]$ into nonempty lists.
\begin{equation}gin{proposition}
\label{prop.stirling}
The polynomial $L_n^{(r)+}({\bm{1}}, y)$ is the generating polynomial
of $(r-1)$-Stirling set partitions of $[n]$, with a weight $y$ for each block.
More generally, the polynomial $L_n^{(r)+}({\mathbf{x}}, y)$
is the generating polynomial of $(r-1)$-Stirling set partitions of $[n]$,
with a weight $y$ for each block,
a weight $x_i$ ($1 \le i \le r-1$)
for each time the $i$th occurrence of a letter is the end of a descent,
and a weight $x_r$ for each time the last occurrence of a letter
is the beginning of an ascent.
\end{proposition}
\par
\noindent{\sc Proof.\ }
By definition, the polynomial $L_n^{(r)+}({\mathbf{x}}, y)$ is the
generating polynomial for unordered forests of increasing $r$-ary trees,
with a weight $y$ for each root and a weight $x_i$ for each $i$-child.
Now a classical bijection \cite{Gessel_78a,Janson_11,Kuba_09}
(see also \cite[section~12.5]{latpath_SRTR})
sends increasing $r$-ary trees on the vertex set $[n]$
to Stirling permutations of the multiset $(r-1) [n]$,
such that\footnote{
See \cite[Lemma~12.34]{latpath_SRTR}
after some slight translation of notation.
}:
\begin{equation}gin{itemize}
\item[1)] A vertex $j$ has a 1-child if and only if
the first occurrence of the letter $j$ in the word ${\mathbf{w}}$
is the end of a descent.
\item[2)] A vertex $j$ has an $r$-child if and only if
the last occurrence of the letter $j$ in the word ${\mathbf{w}}$
is the beginning of an ascent.
\item[3)] A vertex $j$ has an $i$-child ($2 \le i \le r-1$) if and only if
in the word ${\mathbf{w}}$, between the $(i-1)$st and $i$th occurrences
of the letter $j$ there is a nonempty subword;
or equivalently, the $(i-1)$st occurrence of the letter $j$
is the beginning of an ascent;
or equivalently, the $i$th occurrence of the letter $j$
is the end of a descent.
\end{itemize}
Applying this bijection to each tree of the forest maps bijectively
unordered forests of increasing $r$-ary trees on the vertex set $[n]$
to a collection of Stirling permutations of multisets $(r-1)B_i$,
where the $B_i$ taken together form a partition of the set $[n]$.
This collection is nothing other than
an $(r-1)$-Stirling set partition of $[n]$.
$\square$
\section{Exponential generating functions} \label{sec.exponential}
By using exponential generating functions together with
the Lagrange inversion formula, we can obtain explicit expressions
for the generic Lah polynomials $L_{n,k}({\bm{p}}hi)$.
The method is due to Bergeron, Flajolet and Salvy \cite{Bergeron_92};
see also \cite[Chapter~5, especially pp.~364--365]{Bergeron_98}
and \cite[section~12.2.1]{latpath_SRTR}.
We will use Lagrange inversion in the following form \cite{Gessel_16}:
If $A(u)$ is a formal power series
with coefficients in a commutative ring $R$ containing the rationals,
then there exists a unique formal power series $f(t)$
with zero constant term satisfying
\begin{equation}
f(t) \;=\; t \, A(f(t))
\;,
\end{equation}
and it is given by
\begin{equation}
[t^n] \, f(t) \;=\; {1 \over n} \, [u^{n-1}] \, A(u)^n
\quad\hbox{for $n \ge 1$}
\;;
\end{equation}
and more generally, if $H(u)$ is any formal power series, then
\begin{equation}
[t^n] \, H(f(t)) \;=\; {1 \over n} \, [u^{n-1}] \, H'(u) \, A(u)^n
\quad\hbox{for $n \ge 1$}
\;.
\label{eq.lagrange.H}
\end{equation}
Let ${\bm{p}}hi = (\phi_i)_{i \ge 0}$ and $y$ be indeterminates;
we will employ formal power series with coefficients in
${\mathbb Q}[{\bm{p}}hi]$ or ${\mathbb Q}[{\bm{p}}hi,y]$.
Recall that $L_{n,k}({\bm{p}}hi)$ is the generating polynomial
for unordered forests of increasing ordered trees on $n$ total vertices
with $k$ components,
in which each vertex with $i$ children gets a weight $\phi_i$;
in particular, $L_{n,1}({\bm{p}}hi)$ is the generating polynomial
for increasing ordered trees.
And $L_n({\bm{p}}hi,y) = \sum_{k=0}^n L_{n,k}({\bm{p}}hi) \: y^k$
are the row-generating polynomials.
Define now the exponential generating function for trees:
\begin{equation}
U(t) \;=\; \sum_{n=1}^\infty L_{n,1}({\bm{p}}hi) \, {t^n \over n!}
\;.
\label{def.U}
\end{equation}
It is easy to see that the exponential generating function
for $k$-component unordered forests is
\begin{equation}
{U(t)^k \over k!} \;=\; \sum_{n=0}^\infty L_{n,k}({\bm{p}}hi) \, {t^n \over n!}
\;.
\label{eq.Uk}
\end{equation}
Multiplying this by $y^k$ and summing over $k$ then gives
the exponential generating function for the row-generating polynomials:
\begin{equation}
e^{y U(t)} \;=\; \sum_{n=0}^\infty L_n({\bm{p}}hi,y) \, {t^n \over n!}
\;.
\end{equation}
Here is the key step:
standard enumerative arguments \cite[Theorem~1]{Bergeron_92}
show that $U(t)$ satisfies the ordinary differential equation
\begin{equation}
U'(t) \;=\; \Phi(U(t))
\;,
\label{eq.bergeron.ODE}
\end{equation}
where $\Phi(w) \eqdef \sum_{k=0}^\infty \phi_k w^k$
is the ordinary generating function for ${\bm{p}}hi$.
At this point it is convenient to specialize to $\phi_0 = 1$;
at the end we can restore the missing factors of $\phi_0$
by recalling that $L_{n,k}({\bm{p}}hi)$ is homogeneous of degree $n$ in ${\bm{p}}hi$.
(Alternatively, we could keep $\phi_0$ and work over the ring
${\mathbb Q}[{\bm{p}}hi,\phi_0^{-1}]$ instead of ${\mathbb Q}[{\bm{p}}hi]$.)
We can now rewrite the differential equation \reff{eq.bergeron.ODE}
as the implicit equation
\begin{equation}
t \;=\; \int\limits_0^{U(t)} {dw \over \Phi(w)}
\;.
\label{eq.bergeron}
\end{equation}
Introducing $\Psi(w) \eqdef 1/\Phi(w) \eqdef 1 + \sum_{i=1}^\infty \psi_i w^i$,
we then have
\begin{equation}
t \;=\; U(t) \, {\mathbf{w}}idehat{\Psi}(U(t))
\qquad\hbox{where}\qquad
{\mathbf{w}}idehat{\Psi}(z) \;=\; 1 + \sum_{i=1}^\infty {\psi_i \over i+1} \, z^i
\;.
\label{eq.bergeron.fZ}
\end{equation}
Solving $U(t) = t/{\mathbf{w}}idehat{\Psi}(U(t))$
by Lagrange inversion \reff{eq.lagrange.H}
with $A(u) = 1/{\mathbf{w}}idehat{\Psi}(u)$ and $H(u) = u^k/k!$ gives
\begin{equation}gin{subeqnarray}
& &
\hspace*{-1.45cm}
L_{n,k}({\bm{p}}hi) {\sf B}igr|_{\phi_0 = 1}
\;=\; n! \: [t^n] \, {U(t)^k \over k!}
\;=\; {(n-1)! \over (k-1)!} \: [z^{n-k}] \, {\mathbf{w}}idehat{\Psi}(z)^{-n}
\\[2mm]
& & \;
=\;
{(n-1)! \over (k-1)!}
\!\! \sum_{\begin{equation}gin{scarray}
l_1, l_2, \ldots \ge 0 \\
\sum i l_i = n-k
\end{scarray}}
\!\!\!\!
\binom{-n}{-n-\sum l_i,\, l_1,\, l_2,\, \ldots}
\prod_{i=1}^\infty {\sf B}igl( {\psi_i \over i+1} {\sf B}igr) ^{\! l_i}
\\[2mm]
& & \;
=\;
{(n-1)! \over (k-1)!}
\!\! \sum_{\begin{equation}gin{scarray}
l_1, l_2, \ldots \ge 0 \\
\sum i l_i = n-k
\end{scarray}}
\!\!
(-1)^{\sum l_i} \,
\binom{n+\sum l_i - 1}{n-1,\, l_1,\, l_2,\, \ldots}
\prod_{i=1}^\infty {\sf B}igl( {\psi_i \over i+1} {\sf B}igr) ^{\! l_i}
\,.
\;
\slabel{eq.bergeron.Zn.c}
\label{eq.bergeron.Zn}
\end{subeqnarray}
\begin{equation}gin{example}[Forests of increasing multi-unary trees]
\label{exam.phi=1}
\rm
The multivariate Lah polynomials of negative type
$L_{n,k}^{(r)-}(x_1,\ldots,x_r) = L_{n,k}({\bm{h}}(x_1,\ldots,x_r))$
specialized to $r=1$ and $x_1 = 1$
--- which count unordered forests of increasing multi-unary trees ---
correspond to $\phi = {\bm{1}}$,
hence $\psi_1 = -1$ and $\psi_i = 0$ for $i \ge 2$.
It follows that in \reff{eq.bergeron.Zn.c} we have
$l_1 = n-k$ and $l_i = 0$ for $i \ge 2$, hence
\begin{equation}
L_{n,k}({\bm{1}})
\;=\;
{(2n-k-1)! \over 2^{n-k} \, (n-k)! \, (k-1)!}
\;.
\end{equation}
These are a shifted version of the coefficients
of the (reversed) Bessel polynomials \cite[A001497]{OEIS}.
${\rm bl}acksquare$
\end{example}
\begin{equation}gin{example}[$r=\infty$]
\label{exam.r=infty}
\rm
If we consider $r$-ary or multi-$r$-ary trees and take $r \to\infty$,
then after an appropriate rescaling we get $\phi_i = 1/i!$ and $\Phi(w) = e^w$.
Solving \reff{eq.bergeron.ODE} gives $U(t) = -\log(1-t)$ and hence
\begin{equation}
L_{n,k}({\bm{p}}hi)
\;=\;
{n! \over k!} \; [t^n] \, (-\log(1-t))^k
\;=\;
\stirlingcycle{n}{k}
\;,
\end{equation}
the Stirling cycle numbers \cite[A132393]{OEIS}.
${\rm bl}acksquare$
\end{example}
{\bf Final remark.} Here \reff{eq.bergeron.Zn.c} gives a nice explicit
expression for $L_{n,k}({\bm{p}}hi)$,
but it~is in~terms of the coefficients $\psi_i$ in $\Psi(w) = 1/\Phi(w)$,
not directly in terms of the ${\bm{p}}hi$.
Indeed, if one computes from \reff{eq.bergeron.Zn.c}
the polynomials $L_{n,k}({\bm{p}}hi)$,
one finds some coefficients that have modestly (but not hugely) large
prime factors:
for instance, one of the terms in $L_{11,1}({\bm{p}}hi)$ is
$24950808\, \phi_1^2 \phi_2 \phi_3^2$,
where $24950808 = 2^3 \cdot 3^3 \cdot 115513$;
and one of the terms in $L_{13,1}({\bm{p}}hi)$ is
$2318149824\, \phi_1^3 \phi_2^3 \phi_3$,
where $2318149824 = 2^6 \cdot 3 \cdot 12073697$.
This suggests that the polynomials $L_{n,k}({\bm{p}}hi)$
{\em might}\/ not have any simple explicit expression.
Or alternatively, they might have a simple explicit expression,
but with coefficients that are given by sums and not just by products.
We leave it as an open problem to find such an expression.
${\rm bl}acksquare$
\section{Note Added: Exponential Riordan arrays}
\label{sec.exp_riordan}
After completing this paper we realized that the key
Proposition~\ref{prop.prodmat}(a),
which expresses the production matrix of the generic Lah triangle
and which we proved combinatorially in Section~\ref{subsec.proofs.prodmat}
by bijection onto labeled partial \L{}ukasiewicz paths,
can also be proven algebraically by using the theory of
exponential Riordan arrays \cite{Deutsch_04,Deutsch_09,Barry_16}.
Here we would like to present briefly this alternate proof.
Let $R$ be a commutative ring containing the rationals,
and let $F(t) = \sum_{n=0}^\infty f_n t^n/n!$
and $G(t) = \sum_{n=1}^\infty g_n t^n/n!$ be formal power series
with coefficients in $R$; we set $g_0 = 0$.
Then the \textbfit{exponential Riordan array} associated to the pair $(F,G)$
--- or equivalently to the pair of sequences
${\bm{f}} = (f_n)_{n \ge 0}$ and ${\bm{g}} = (g_n)_{n \ge 1}$ ---
is the infinite lower-triangular matrix
${\mathcal{R}}[F,G] = ({\mathcal{R}}[F,G]_{nk})_{n,k \ge 0}$ defined by
\begin{equation}
{\mathcal{R}}[F,G]_{nk}
\;=\;
{n! \over k!} \:
[t^n] \, F(t) G(t)^k
\;.
\end{equation}
That is, the $k$th column of ${\mathcal{R}}[F,G]$
has exponential generating function $F(t) G(t)^k/k!$.
Please note that the diagonal elements of ${\mathcal{R}}[F,G]$
are ${\mathcal{R}}[F,G]_{nn} = f_0 g_1^n$,
so the matrix ${\mathcal{R}}[F,G]$ is invertible
in the ring $R^{{\mathbb N} \times {\mathbb N}}_{\rm lt}$ of lower-triangular matrices
if and only if $f_0$ and $g_1$ are invertible in $R$.
We shall use an easy but important result that is sometimes called
the \emph{fundamental theorem of exponential Riordan arrays} (FTERA):
\begin{equation}gin{lemma}[Fundamental theorem of exponential Riordan arrays]
\label{lemma.FETRA}
Let ${\bm{b}} = (b_n)_{n \ge 0}$ be a sequence with
exponential generating function $B(t) = \sum_{n=0}^\infty b_n t^n/n!$.
Considering ${\bm{b}}$ as a column vector and letting ${\mathcal{R}}[F,G]$
act on it by matrix multiplication, we obtain a sequence ${\mathcal{R}}[F,G] {\bm{b}}$
whose exponential generating function is $F(t) \, B(G(t))$.
\end{lemma}
\par
\noindent{\sc Proof.\ }
We compute
\begin{equation}gin{subeqnarray}
\sum_{k=0}^n {\mathcal{R}}[F,G]_{nk} \, b_k
& = &
\sum_{k=0}^\infty {n! \over k!} \, [t^n] \, F(t) G(t)^k \, b_k
\\[2mm]
& = &
n! \: [t^n] \: F(t) \sum_{k=0}^\infty b_k \, {G(t)^k \over k!}
\\[2mm]
& = &
n! \: [t^n] \: F(t) \, B(G(t))
\;.
\end{subeqnarray}
$\square$
We can now determine the production matrix of an exponential
Riordan array ${\mathcal{R}}[F,G]$:
\begin{equation}gin{theorem}[Production matrices of exponential Riordan arrays]
\label{thm.riordan.exponential.production}
Let $L$ be a lower-triangular matrix
(with entries in a commutative ring $R$ containing the rationals)
with invertible diagonal entries and $L_{00} = 1$,
and let $P = L^{-1} {\mathbb D}elta L$ be its production matrix.
Then $L$ is an exponential Riordan array
if and only~if $P = (p_{nk})_{n,k \ge 0}$ has the form
\begin{equation}
p_{nk}
\;=\;
{n! \over k!} \: (z_{n-k} \,+\, k \, a_{n-k+1})
\label{eq.thm.riordan.exponential.production}
\end{equation}
for some sequences ${\bm{a}} = (a_n)_{n \ge 0}$ and ${\bm{z}} = (z_n)_{n \ge 0}$
in $R$.
More precisely, $L = {\mathcal{R}}[F,G]$ if and only~if $P$
is of the form \reff{eq.thm.riordan.exponential.production}
where the ordinary generating functions
$A(t) = \sum_{n=0}^\infty a_n t^n$ and $Z(t) = \sum_{n=0}^\infty z_n t^n$
are connected to $F(t)$ and $G(t)$ by
\begin{equation}
G'(t) \;=\; A(G(t)) \;,\qquad
{F'(t) \over F(t)} \;=\; Z(G(t))
\label{eq.prop.riordan.exponential.production.1}
\end{equation}
or equivalently
\begin{equation}
A(t) \;=\; G'({\bm{a}}r{G}(t)) \;,\qquad
Z(t) \;=\; {F'({\bm{a}}r{G}(t)) \over F({\bm{a}}r{G}(t))}
\label{eq.prop.riordan.exponential.production.2}
\end{equation}
where ${\bm{a}}r{G}(t)$ is the compositional inverse of $G(t)$.
\end{theorem}
\par
{\mathbf{n}}oindent{\sc Proof}
(mostly contained in \cite[pp.~217--218]{Barry_16}).
Suppose that $L = {\mathcal{R}}[F,G]$.
The hypotheses on $L$ imply that $f_0 = 1$
and that $g_1$ is invertible in $R$;
so $G(t)$ has a compositional inverse.
Now let $P = (p_{nk})_{n,k \ge 0}$ be a matrix;
its column exponential generating functions are, by definition,
$P_k(t) = \sum_{n=0}^\infty p_{nk} \, t^n/n!$.
Applying the FTERA to each column of $P$,
we see that ${\mathcal{R}}[F,G] P$ is a matrix
whose column exponential generating functions
are $\big( F(t) \, P_k(G(t)) \big)_{k \ge 0}$.
On~the other hand, ${\mathbb D}elta \, {\mathcal{R}}[F,G]$
is the matrix ${\mathcal{R}}[F,G]$ with its zeroth row removed
and all other rows shifted upwards,
so it has column exponential generating functions
\begin{equation}
{d \over dt} \, \big( F(t) \, G(t)^k/k! \big)
\;=\;
{1 \over k!} \: {\sf B}ig[ F'(t) \, G(t)^k
\:+\: k \, F(t) \, G(t)^{k-1} \, G'(t) {\sf B}ig]
\;.
\end{equation}
Comparing these two results, we see that
${\mathbb D}elta \, {\mathcal{R}}[F,G] = {\mathcal{R}}[F,G] \, P$
if and only~if
\begin{equation}
P_k(G(t))
\;=\;
{1 \over k!} \:
{F'(t) \, G(t)^k \:+\: k \, F(t) \, G(t)^{k-1} \, G'(t)
\over
F(t)}
\;,
\end{equation}
or in other words
\begin{equation}
P_k(t)
\;=\;
{1 \over k!} \:
\biggl[ {F'({\bm{a}}r{G}(t)) \over F({\bm{a}}r{G}(t))} \, t^k
\:+\: k \, t^{k-1} \, G'({\bm{a}}r{G}(t))
\biggr]
\;.
\end{equation}
Therefore
\begin{equation}gin{subeqnarray}
p_{nk}
& = &
{n! \over k!} \: [t^n] \,
\biggl[ {F'({\bm{a}}r{G}(t)) \over F({\bm{a}}r{G}(t))} \, t^k
\:+\: k \, t^{k-1} \, G'({\bm{a}}r{G}(t))
\biggr]
\\[2mm]
& = &
{n! \over k!} \:
\biggl[ [t^{n-k}] \: {F'({\bm{a}}r{G}(t)) \over F({\bm{a}}r{G}(t))}
\:+\: k \, [t^{n-k+1}] \: G'({\bm{a}}r{G}(t))
\biggr]
\\[2mm]
& = &
{n! \over k!} \: (z_{n-k} \,+\, k \, a_{n-k+1})
\end{subeqnarray}
where ${\bm{a}} = (a_n)_{n \ge 0}$ and ${\bm{z}} = (z_n)_{n \ge 0}$
are given by \reff{eq.prop.riordan.exponential.production.2}.
Conversely, suppose that $P = (p_{nk})_{n,k \ge 0}$ has the form
\reff{eq.thm.riordan.exponential.production}.
Define $F(t)$ and $G(t)$
as the unique solutions (in the formal-power-series ring $R[[t]]$)
of the differential equations \reff{eq.prop.riordan.exponential.production.1}
with initial conditions $F(0) = 1$ and $G(0) = 0$.
Then running the foregoing computation backwards
shows that ${\mathbb D}elta \, {\mathcal{R}}[F,G] = {\mathcal{R}}[F,G] \, P$.
$\square$
\par
{\mathbf{n}}oindent{\sc Alternate Proof of Proposition~\ref{prop.prodmat}}(a).
We use the expressions for the exponential generating functions
of the generic Lah polynomials, which were determined in
Section~\ref{sec.exponential}.
{}From \reff{def.U}/\reff{eq.Uk} we see that the
generic Lah triangle ${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$
is the exponential Riordan array ${\mathcal{R}}[F,G]$
with $F(t) = 1$ and $G(t) = U(t)$.
Comparing \reff{eq.prop.riordan.exponential.production.1}
with \reff{eq.bergeron.ODE},
we see that $A(t) = \Phi(t)$ and $Z(t) = 0$.
The production matrix \reff{eq.thm.riordan.exponential.production}
then becomes \reff{eq.prop.prodmat},
which proves Proposition~\ref{prop.prodmat}(a).
$\square$
{\bf Remark.}
This proof shows that the generic Lah triangle
${\sf L} = (L_{n,k}({\bm{p}}hi))_{n,k \ge 0}$
is in fact the {\em general}\/ exponential Riordan array ${\mathcal{R}}[F,G]$
of the ``associated subgroup'' $F=1$,
expressed in terms of its $A$-sequence ${\bm{a}} = {\bm{p}}hi$.
In this way, the theory of the generic Lah triangle
is {\em equivalent}\/ to the theory of
exponential Riordan arrays of the ``associated subgroup'' ${\mathcal{R}}[1,G]$,
expressed in the combinatorial language of
unordered forests of increasing ordered trees.
It would be interesting to work out the combinatorial interpretation
of exponential Riordan arrays ${\mathcal{R}}[F,G]$ with $F {\mathbf{n}}eq 1$.
${\rm bl}acksquare$
This algebraic proof of Proposition~\ref{prop.prodmat}(a)
is arguably much simpler than the combinatorial proof
presented in Section~\ref{subsec.proofs.prodmat}.
On the other hand, the combinatorial method seems to be more powerful:
we do not see (at least at present) how to extend the algebraic proof
to obtain the more general Proposition~\ref{prop.prodmat.phiL},
which expresses the production matrix of the refined generic Lah triangle.
\section*{Acknowledgments}
One of us (A.D.S.)\ wishes to thank Xi Chen for helpful conversations
concerning exponential Riordan arrays.
This research was supported in part by
the U.K.~Engineering and Physical Sciences Research Council grant EP/N025636/1.
\appendix
\section{Lah polynomials for $\bm{n \le 7}$}
In this appendix we report the generic Lah polynomials $L_{n,k}({\bm{p}}hi)$
for $n \le 7$ (specialized for simplicity to $\phi_0 = 1$).
We also report the Lah symmetric functions
$L_{n,k}^{(\infty)+}$ and $L_{n,k}^{(\infty)-}$ for $n \le 7$
in terms of the monomial symmetric functions $m_\lambda$;
from these the reader can easily reconstruct explicit expressions
for the multivariate Lah polynomials
$L_{n,k}^{(r)+}(x_1,\ldots,x_r)$ and $L_{n,k}^{(r)-}(x_1,\ldots,x_r)$
for any chosen value of $r$.
The conversions from $e_\lambda$ or $h_\lambda$ to $m_\lambda$
were performed using the {\tt SymFun} {\sc Mathematica} package
(version~3.1), developed by Curtis Greene and collaborators
\cite{Greene_symfun}.
\subsection{Generic Lah polynomials}
\vspace*{-5mm}
\begin{equation}gin{eqnarray*}
L_{1,1}({\bm{p}}hi) & = & 1 \\[3mm]
L_{2,1}({\bm{p}}hi) & = & \phi_{1} \\[0.5mm]
L_{2,2}({\bm{p}}hi) & = & 1 \\[3mm]
L_{3,1}({\bm{p}}hi) & = & \phi_{1}^2+2 \phi_{2} \\[0.5mm]
L_{3,2}({\bm{p}}hi) & = & 3 \phi_{1} \\[0.5mm]
L_{3,3}({\bm{p}}hi) & = & 1 \\[3mm]
L_{4,1}({\bm{p}}hi) & = & \phi_{1}^3+8 \phi_{1} \phi_{2}+6 \phi_{3} \\[0.5mm]
L_{4,2}({\bm{p}}hi) & = & 7 \phi_{1}^2+8 \phi_{2} \\[0.5mm]
L_{4,3}({\bm{p}}hi) & = & 6 \phi_{1} \\[0.5mm]
L_{4,4}({\bm{p}}hi) & = & 1 \\[3mm]
L_{5,1}({\bm{p}}hi) & = & \phi_{1}^4+22 \phi_{1}^2 \phi_{2}+16 \phi_{2}^2+42 \phi_{1} \phi_{3}+24 \phi_{4} \\[0.5mm]
L_{5,2}({\bm{p}}hi) & = & 15 \phi_{1}^3+60 \phi_{1} \phi_{2}+30 \phi_{3} \\[0.5mm]
L_{5,3}({\bm{p}}hi) & = & 25 \phi_{1}^2+20 \phi_{2} \\[0.5mm]
L_{5,4}({\bm{p}}hi) & = & 10 \phi_{1} \\[0.5mm]
L_{5,5}({\bm{p}}hi) & = & 1 \\[3mm]
L_{6,1}({\bm{p}}hi) & = & \phi_{1}^5+52 \phi_{1}^3 \phi_{2}+136 \phi_{1} \phi_{2}^2+192 \phi_{1}^2 \phi_{3}+180 \phi_{2} \phi_{3}+264 \phi_{1} \phi_{4}+120 \phi_{5} \\[0.5mm]
L_{6,2}({\bm{p}}hi) & = & 31 \phi_{1}^4+292 \phi_{1}^2 \phi_{2}+136 \phi_{2}^2+342 \phi_{1} \phi_{3}+144 \phi_{4} \\[0.5mm]
L_{6,3}({\bm{p}}hi) & = & 90 \phi_{1}^3+240 \phi_{1} \phi_{2}+90 \phi_{3} \\[0.5mm]
L_{6,4}({\bm{p}}hi) & = & 65 \phi_{1}^2+40 \phi_{2} \\[0.5mm]
L_{6,5}({\bm{p}}hi) & = & 15 \phi_{1} \\[0.5mm]
L_{6,6}({\bm{p}}hi) & = & 1 \\[3mm]
L_{7,1}({\bm{p}}hi) & = & \phi_{1}^6+114 \phi_{1}^4 \phi_{2}+720 \phi_{1}^2 \phi_{2}^2+272 \phi_{2}^3+732 \phi_{1}^3 \phi_{3}+2304 \phi_{1} \phi_{2} \phi_{3}+540 \phi_{3}^2 \\
& & \quad +\, 1824 \phi_{1}^2 \phi_{4}+1248 \phi_{2} \phi_{4}+1920 \phi_{1} \phi_{5}+720 \phi_{6} \\[0.5mm]
L_{7,2}({\bm{p}}hi) & = & 63 \phi_{1}^5+1176 \phi_{1}^3 \phi_{2}+1848 \phi_{1} \phi_{2}^2+2436 \phi_{1}^2 \phi_{3}+1680 \phi_{2} \phi_{3}+2352 \phi_{1} \phi_{4}+840 \phi_{5} \\[0.5mm]
L_{7,3}({\bm{p}}hi) & = & 301 \phi_{1}^4+1792 \phi_{1}^2 \phi_{2}+616 \phi_{2}^2+1512 \phi_{1} \phi_{3}+504 \phi_{4} \\[0.5mm]
L_{7,4}({\bm{p}}hi) & = & 350 \phi_{1}^3+700 \phi_{1} \phi_{2}+210 \phi_{3} \\[0.5mm]
L_{7,5}({\bm{p}}hi) & = & 140 \phi_{1}^2+70 \phi_{2} \\[0.5mm]
L_{7,6}({\bm{p}}hi) & = & 21 \phi_{1} \\[0.5mm]
L_{7,7}({\bm{p}}hi) & = & 1
\end{eqnarray*}
\subsection{Lah symmetric functions of positive type}
\vspace*{-5mm}
\begin{equation}gin{eqnarray*}
L_{1,1}^{(\infty)+} & = & 1 \\[3mm]
L_{2,1}^{(\infty)+} & = & m_{1} \\[0.5mm]
L_{2,2}^{(\infty)+} & = & 1 \\[3mm]
L_{3,1}^{(\infty)+} & = & m_{2}+4 m_{11} \\[0.5mm]
L_{3,2}^{(\infty)+} & = & 3 m_{1} \\[0.5mm]
L_{3,3}^{(\infty)+} & = & 1 \\[3mm]
L_{4,1}^{(\infty)+} & = & m_{3}+11 m_{21}+36 m_{111} \\[0.5mm]
L_{4,2}^{(\infty)+} & = & 7 m_{2}+22 m_{11} \\[0.5mm]
L_{4,3}^{(\infty)+} & = & 6 m_{1} \\[0.5mm]
L_{4,4}^{(\infty)+} & = & 1 \\[3mm]
L_{5,1}^{(\infty)+} & = & m_{4}+66 m_{22}+26 m_{31}+196 m_{211}+576 m_{1111} \\[0.5mm]
L_{5,2}^{(\infty)+} & = & 15 m_{3}+105 m_{21}+300 m_{111} \\[0.5mm]
L_{5,3}^{(\infty)+} & = & 25 m_{2}+70 m_{11} \\[0.5mm]
L_{5,4}^{(\infty)+} & = & 10 m_{1} \\[0.5mm]
L_{5,5}^{(\infty)+} & = & 1 \\[3mm]
L_{6,1}^{(\infty)+} & = & m_{5}+302 m_{32}+57 m_{41}+1898 m_{221}+848 m_{311}+5244 m_{2111}+14400 m_{11111} \\[0.5mm]
L_{6,2}^{(\infty)+} & = & 31 m_{4}+906 m_{22}+416 m_{31}+2446 m_{211}+6576 m_{1111} \\[0.5mm]
L_{6,3}^{(\infty)+} & = & 90 m_{3}+510 m_{21}+1350 m_{111} \\[0.5mm]
L_{6,4}^{(\infty)+} & = & 65 m_{2}+170 m_{11} \\[0.5mm]
L_{6,5}^{(\infty)+} & = & 15 m_{1} \\[0.5mm]
L_{6,6}^{(\infty)+} & = & 1 \\[3mm]
L_{7,1}^{(\infty)+} & = & m_{6}+2416 m_{33}+1191 m_{42}+120 m_{51}+28470 m_{222}+13644 m_{321}+3228 m_{411} \\
& & \quad +\, 75216 m_{2211}+36240 m_{3111}+197856 m_{21111}+518400 m_{111111} \\[0.5mm]
L_{7,2}^{(\infty)+} & = & 63 m_{5}+6006 m_{32}+1491 m_{41}+31794 m_{221}+15624 m_{311}+82152 m_{2111}+211680 m_{11111} \\[0.5mm]
L_{7,3}^{(\infty)+} & = & 301 m_{4}+6006 m_{22}+2996 m_{31}+15316 m_{211}+38976 m_{1111} \\[0.5mm]
L_{7,4}^{(\infty)+} & = & 350 m_{3}+1750 m_{21}+4410 m_{111} \\[0.5mm]
L_{7,5}^{(\infty)+} & = & 140 m_{2}+350 m_{11} \\[0.5mm]
L_{7,6}^{(\infty)+} & = & 21 m_{1} \\[0.5mm]
L_{7,7}^{(\infty)+} & = & 1
\end{eqnarray*}
\subsection{Lah symmetric functions of negative type}
\vspace*{-5mm}
\begin{equation}gin{eqnarray*}
L_{1,1}^{(\infty)-} & = & 1 \\[3mm]
L_{2,1}^{(\infty)-} & = & m_{1} \\[0.5mm]
L_{2,2}^{(\infty)-} & = & 1 \\[3mm]
L_{3,1}^{(\infty)-} & = & 3 m_{2}+4 m_{11} \\[0.5mm]
L_{3,2}^{(\infty)-} & = & 3 m_{1} \\[0.5mm]
L_{3,3}^{(\infty)-} & = & 1 \\[3mm]
L_{4,1}^{(\infty)-} & = & 15 m_{3}+25 m_{21}+36 m_{111} \\[0.5mm]
L_{4,2}^{(\infty)-} & = & 15 m_{2}+22 m_{11} \\[0.5mm]
L_{4,3}^{(\infty)-} & = & 6 m_{1} \\[0.5mm]
L_{4,4}^{(\infty)-} & = & 1 \\[3mm]
L_{5,1}^{(\infty)-} & = & 105 m_{4}+250 m_{22}+210 m_{31}+380 m_{211}+576 m_{1111} \\[0.5mm]
L_{5,2}^{(\infty)-} & = & 105 m_{3}+195 m_{21}+300 m_{111} \\[0.5mm]
L_{5,3}^{(\infty)-} & = & 45 m_{2}+70 m_{11} \\[0.5mm]
L_{5,4}^{(\infty)-} & = & 10 m_{1} \\[0.5mm]
L_{5,5}^{(\infty)-} & = & 1 \\[3mm]
L_{6,1}^{(\infty)-} & = & 945 m_{5}+3010 m_{32}+2205 m_{41}+5810 m_{221}+4760 m_{311}+9156 m_{2111}+14400 m_{11111} \\[0.5mm]
L_{6,2}^{(\infty)-} & = & 945 m_{4}+2590 m_{22}+2100 m_{31}+4130 m_{211}+6576 m_{1111} \\[0.5mm]
L_{6,3}^{(\infty)-} & = & 420 m_{3}+840 m_{21}+1350 m_{111} \\[0.5mm]
L_{6,4}^{(\infty)-} & = & 105 m_{2}+170 m_{11} \\[0.5mm]
L_{6,5}^{(\infty)-} & = & 15 m_{1} \\[0.5mm]
L_{6,6}^{(\infty)-} & = & 1 \\[3mm]
L_{7,1}^{(\infty)-} & = & 10395 m_{6}+48160 m_{33}+42525 m_{42}+27720 m_{51}+122010 m_{222}+97860 m_{321} \\
& & \quad +\, 69300 m_{411}+197904 m_{2211}+158928 m_{3111}+320544 m_{21111}+518400 m_{111111} \\[0.5mm]
L_{7,2}^{(\infty)-} & = & 10395 m_{5}+38430 m_{32}+26775 m_{41}+79170 m_{221}+63000 m_{311}+129528 m_{2111} \\
& & \quad +\, 211680 m_{11111} \\[0.5mm]
L_{7,3}^{(\infty)-} & = & 4725 m_{4}+14350 m_{22}+11340 m_{31}+23660 m_{211}+38976 m_{1111} \\[0.5mm]
L_{7,4}^{(\infty)-} & = & 1260 m_{3}+2660 m_{21}+4410 m_{111} \\[0.5mm]
L_{7,5}^{(\infty)-} & = & 210 m_{2}+350 m_{11} \\[0.5mm]
L_{7,6}^{(\infty)-} & = & 21 m_{1} \\[0.5mm]
L_{7,7}^{(\infty)-} & = & 1
\end{eqnarray*}
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\end{document} | math | 185,006 |
\betaegin{document}
\title[The M\"obius disjointness conjecture for distal flows]
{The M\"obius disjointness conjecture \\ for distal flows}
\alphauthor{Jianya Liu \& Peter Sarnak}
\alphaddress{School of Mathematics
\\
Shandong University
\\
Jinan
\\
Shandong 250100
\\
China}
\email{[email protected]}
\alphaddress{Department of Mathematics
\\
Princeton University \& Institute for Advanced Study
\\
Princeton, NJ 08544-1000
\\
USA}
\email{[email protected]}
\deltaate{\today}
\sigmaubjclass[2000]{11L03, 37A45, 11N37}
\keywords{The M\"obius function, distal flow, affine linear map,
skew product, nilmanifold}
\maketitle
\alphaddtocounter{footnote}{1}
\sigmaection{The M\"obius disjointness conjecture}
\sigmaetcounter{equation}{0}
Let $\mathscr{X}=(T, X)$ be a flow, namely $X$ is a compact topological space
and $T: X\to X$ a continuous map. The sequence $\xi(n)$
is observed in $\mathscr{X}$ if there is an $f\in C(X)$ and an $x\in X$,
such that $\xi(n)=f(T^n x)$. Let $\mu(n)$ be the M\"obius function, that
is $\mu(n)$ is $0$ if $n$ is not square-free, and is $(-1)^t$ if $n$ is a
product of $t$ distinct primes. We say that $\mu$ is linearly
disjoint from $\mathscr{X}$ if
\betaegin{eqnarray}\lambdaabel{def/DISJO}
\frac{1}{N}\sigmaum_{n\lambdaeq N} \mu(n)\xi(n) \to 0, \quad \mbox{as } N\to\infty,
\end{eqnarray}
for every observable $\xi$ of $\mathscr{X}$.
The M\"obius Disjointness Conjecture of
the second author (\cite{Sar}, \cite{Sar1}) states the following.
\betaegin{conjecture} [The M\"obius Disjointness Conjecture]
The M\"obius function $\mu$ is linearly disjoint from
every $\mathscr{X}$ whose entropy is $0$.
\end{conjecture}
This Conjecture has been established for many flows
$\mathscr{X}$ (see \cite{Dav}, \cite{MauRiv}, \cite{GreTao2}, \cite{BouSarZie}, \cite{Bou})
however all of these flows are quasi-regular (or rigid) in the sense that
the Birkhoff averages
\betaegin{eqnarray}\lambdaabel{def/Birkh}
\frac{1}{N}\sigmaum_{n\lambdaeq N} \xi(n)
\end{eqnarray}
exist for every $\xi$ observed in $\mathscr{X}$.
In \cite{LiuSar} we establish some new cases of the Disjointness Conjecture and
in particular for irregular flows $\mathscr{X}$, that is ones for which (\ref{def/Birkh})
fails. These flows are complicated in terms of the behavior of their individual
orbits but they are distal and of zero entropy, so that the
disjointness is still expected to hold.
\sigmaection{Results}
\sigmaetcounter{equation}{0}
In this section we summarize the results we have established
in \cite{LiuSar}. The first result in \cite{LiuSar}
is concerned with certain regular flows, namely affine linear
maps of a compact abelian group $X$. Such a flow $(T, X)$
is given by
\betaegin{eqnarray}\lambdaabel{def/AFF}
T(x)=Ax+b
\end{eqnarray}
where $A$ is an automorphism of $X$ and $b\in X$ (see \cite{Hah}, \cite{HoaPar}).
\betaegin{theorem}\lambdaabel{thm1}
Let $\mathscr{X}=(T, X)$ be an affine linear flow on a compact
abelian group which is of zero entropy. Then $\mu$
is linearly disjoint from $\mathscr{X}$.
\end{theorem}
Theorem~\ref{thm1} actually holds with a rate of convergence.
We first reduce to the torus case and then handle
the torus case by Fourier analysis and classical results of Davenport \cite{Dav}
and Hua \cite{Hua} on exponential sums concerning the
M\"{o}bius function.
The flows in Theorem~\ref{thm1} are distal,
and our main
result in \cite{LiuSar} is concerned with nonlinear distal flows on such spaces.
We restrict to $X={\Bbb T}^2$ the two dimensional torus
${\Bbb R}^2/{\Bbb Z}^2$ and consider nonlinear smooth (or even analytic)
skew products as discussed in Furstenberg \cite{Fur61}. $T: {\Bbb T}^2\to {\Bbb T}^2$
is given by
\betaegin{eqnarray}\lambdaabel{def/SKEW}
T(x, y)= (ax+\alpha, cx+dy+ h(x))
\end{eqnarray}
where $a, c, d\in \Bbb Z, ad=\pm 1, \alpha\in \Bbb R$
and $h$ is a smooth periodic function of period $1$.
The affine
linear part is in the form
$$
\lambdaeft[
\betaegin{array}{ccc}
a & 0\\
c & d
\end{array}
\rightght] \in GL_2 (\Bbb Z),
$$
ensuring that $T$ has zero entropy (and it can always be brought into
this form). The flow $(T, {\Bbb T}^2)$ is distal and this skew product
is a basic building block (with $e(h(x))$ continuous) in Furstenberg's
classification theory of minimal distal flows \cite{Fur63}. If
$\alpha$ is diophantine, that is
$$
\betaigg|\alpha-\frac{a}{q}\betaigg|\gammaeq \frac{c}{q^B}
$$
for some $c>0, B<\infty$ and all $a/q$ rational, then $T$ can be conjugated
by a smooth map of ${\Bbb T}^2$ to its affine linear part
\betaegin{eqnarray}\lambdaabel{DIO/conjugate}
(x, y)\mapsto (ax+\alpha, cx+dy+ \beta)
\end{eqnarray}
where
$$
\beta=\int_0^1 h(x)dx
$$
(see \cite{SanUrz}).
Hence the disjointness of $\mu$ from
$\mathscr{X}=(T, {\Bbb T}^2)$ for a $T$ with a diophantine
$\alpha$, follows from Theorem~\ref{thm1}.
However if $\alpha$ is not diophantine the dynamics of the
flow $(T, {\Bbb T}^2)$ can be very different from an affine
linear flow. For example, as Furstenberg shows it may be
irregular (i.e. the limits in (\ref{def/Birkh}) fail to exist for certain observables).
Our main result is that these nonlinear skew products
are linearly disjoint from $\mu$, at least if $h$ satisfies some further
technical hypothesis. Firstly we assume that $h$ is analytic, namely that
if
\betaegin{eqnarray}\lambdaabel{h=/FOUR}
h(x)=\sigmaum_{m\in \Bbb Z} \hat{h}(m) e(mx)
\end{eqnarray}
then
\betaegin{eqnarray}\lambdaabel{hhat/UPP}
\hat{h}(m)\lambdal e^{-\tau |m|}
\end{eqnarray}
for some $\tau>0$. Secondly we assume that there is
$\tau_2<\infty$ such that
\betaegin{eqnarray}\lambdaabel{hhat/LOW}
|\hat{h}(m)|\gammag e^{-\tau_2 |m|}.
\end{eqnarray}
This is not a very natural condition being an artifact of our proof.
However it is not too restrictive and the following applies rather
generally (and most importantly there is no condition on $\alpha$).
\betaegin{theorem}\lambdaabel{thm2}
Let $\mathscr{X}=(T, {\Bbb T}^2)$ be of the form (\ref{def/SKEW}),
with $h$ satisfying (\ref{hhat/UPP}) and (\ref{hhat/LOW}).
Then $\mu$ is linearly disjoint from $\mathscr{X}$.
\end{theorem}
The assertion of
Thereom~\ref{thm2} holds for all $\alpha$, and so we have to consider all
diophantine possibilities of $\alpha$. The tool is the Bourgain-Sarnak-Ziegler
\cite{BouSarZie}
finite version of the Vinogradov method, incorporated with various analytic methods
such as Poisson's summation and stationary phase.
Furstenberg \cite{Fur61} gives examples of skew product transformations
of the form (\ref{def/SKEW}) which are not regular in the sense of (\ref{def/Birkh}).
Many of the flows ${\mathscr X}$
in Theorem~\ref{thm2} have this property and we show in \cite{LiuSar}
that Furstenberg's
examples are smoothly conjugate to such ${\mathscr X}$'s. In particular his examples are linearly
disjoint from $\mu$.
Theorem~\ref{thm1} deals with the affine
linear distal flows on the $n$-torus. A different source
of homogeneous distal flows are the affine linear flows on nilmanifold
$X=G/\Gamma$ where $G$ is a nilpotent Lie group and
$\Gamma$ a lattice in $G$. For
$\mathscr{X}=(T, G/\Gamma)$ where $T(x)=\alpha x\Gamma$
with $\alpha\in G$, i.e. translation on $G/\Gamma$, the
linear disjointness of $\mu$ and $\mathscr{X}$ is proven in
\cite{GreTao1} and \cite{GreTao2}. Using the classification of zero entropy (equivalently distal)
affine linear flows on nilmanifolds \cite{Dan}, and
Green and Tao's results we establish
in \cite{LiuSar} the following.
\betaegin{theorem}\lambdaabel{thm3}
Let $\mathscr{X}=(T, G/\Gamma)$ where $T$ is an affine
linear map of the nilmanifold $G/\Gamma$ of zero entropy.
Then $\mu$ is linearly disjoint from $\mathscr{X}$.
\end{theorem}
The above results for $\mu(n)$ can be
proved in the same way for similar multiplicative functions such as
$\lambda(n)=(-1)^{\tau(n)}$ where $\tau(n)$ is the number of
prime factors of $n$.
\sigmaection{Disjointness of $\mu$ from Furstenberg's systems}
\sigmaetcounter{equation}{0}
As a consequence of Theorem~\ref{thm2},
it is proved in \cite{LiuSar} that $\mu$ is linearly
disjoint from Furstenberg's systems.
But no rate of convergence is obtained there
since Theorem~\ref{thm2} in general offers no rate.
In this
section we show that, for Furstenberg's systems, rate of convergence is actually
available if we work on these systems directly rather than appeal to
Theorem~\ref{thm2}.
\sigmaubsection{The continued fraction expansion of $\alpha$.}
We assume that $\alpha$ is irrational, and our argument will
depend on the continued fraction expansion of $\alpha$.
Every real number $\alpha$ has its continued fraction
representation
\betaegin{eqnarray}\lambdaabel{a=a0a1+}
\alpha=a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots}}
\end{eqnarray}
where $a_0=[\alpha]$ is the integral part of $\alpha$,
and $a_1, a_2, \lambdadots$ are positive integers.
The expression (\ref{a=a0a1+}) is infinite since $\alpha\not\in \Bbb Q$.
We write $[a_0; a_1, a_2, \lambdadots]$ for the
expression on the right-hand side of (\ref{a=a0a1+}), which is the limit of
the finite continued expressions
\betaegin{eqnarray}\lambdaabel{a=a0a1F}
[a_0; a_1, a_2, \lambdadots, a_k]=a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots+\frac{1}{a_k}}}
\end{eqnarray}
as $k\to \infty$.
Writing
\betaegin{eqnarray*}
\frac{l_k}{q_k}=[a_0; a_1, a_2, \lambdadots, a_k],
\end{eqnarray*}
we have $l_0=a_0, l_1=a_0a_1+1, q_0=1, q_1=a_1, $
and for $k\gammaeq 2$,
\betaegin{eqnarray*}
l_k=a_k l_{k-1}+l_{k-2}, \quad
q_k=a_k q_{k-1}+q_{k-2}.
\end{eqnarray*}
Since $\alpha$ is irrational we have $q_{k+1}\gammaeq q_k+1$ for all $k\gammaeq 1$.
An induction argument gives the stronger
assertion that
$q_k\gammaeq 2^{(k-1)/2}$
for all $k\gammaeq 2$, and thus $q_k$ increases at least like an exponential function of $k$.
The irrationality of $\alpha$ also implies that, for all $k\gammaeq 2$,
\betaegin{eqnarray}\lambdaabel{ratAPP}
\frac{1}{2 q_k q_{k+1}}<\betaigg|\alpha-\frac{l_k}{q_k}\betaigg| <\frac{1}{q_kq_{k+1}}.
\end{eqnarray}
\sigmaubsection{Furstenberg's examples.}
Furstenberg \cite{Fur61}
gave examples of smooth transformation $T: {\Bbb T}^2\to {\Bbb T}^2$
such that the ergodic averages do not all exist.
Let $\alpha$ be as above such that
\betaegin{eqnarray}\lambdaabel{qk+1>eqk}
q_{k+1}\gammaeq e^{q_k}
\end{eqnarray}
for all positive $k$.
Define $q_{-k}= - q_k$ and set
\betaegin{eqnarray}
h(x)=\sigmaum_{k\not=0}\frac{e(q_k\alpha)-1}{|k|} e(q_k x).
\end{eqnarray}
It follows from (\ref{ratAPP}) and (\ref{qk+1>eqk}) that
$h(x)$ is a smooth function. We also have
$h(x)=g(x+\alpha)-g(x)$
where
\betaegin{eqnarray}
g(x)=\sigmaum_{k\not=0}\frac{1}{|k|}e(q_k x)
\end{eqnarray}
so that $g(x)\in L^2(0,1)$ and in particular
defines and measurable function. But $g(x)$ cannot
correspond to a continuous function, as shown in Furstenberg \cite{Fur61}.
\sigmaubsection{Disjointness of $\mu$ from Furstenberg's systems.}
In the following we consider slightly more general $h$'s with
\betaegin{eqnarray}\lambdaabel{def/hx=sum}
h(x)=\sigmaum_{k\not=0} c_k (1-e(q_k\alpha))e(q_k x)
\end{eqnarray}
where $\alpha$ satisfies (\ref{qk+1>eqk}) and, for all positive $k$, the coefficients
$c_k$ satisfy
\betaegin{eqnarray}
c_k= c_{-k}, \quad |c_k|\lambdaeq C
\end{eqnarray}
for some positive constant $C$ which we may
assume to be greater than $1$.
We note that if $c_k$ is not bounded then $g(x)$ will not be $L^2$.
Now what we want to estimate is essentially
\betaegin{eqnarray}\lambdaabel{defSN}
S(N):=\sigmaum_{n\lambdaeq N} \mu(n) e\betaigg(\sigmaum_{j=0}^{n-1}h(j\alpha)\betaigg),
\end{eqnarray}
and our result is the following.
\betaegin{proposition}\lambdaabel{thm:2-2}
Let $S(N)$ be as in (\ref{defSN}). Then
\betaegin{eqnarray}\lambdaabel{S2est}
S(N)\lambdal_{} N\lambdaog^{-A}N
\end{eqnarray}
where the implied constant depends on $A$,
but is independent of $\alpha$.
\end{proposition}
\betaegin{proof}
By (\ref{def/hx=sum}) we have
\betaegin{eqnarray*}
\sigmaum_{j=0}^{n-1}h(j\alpha)
=\sigmaum_{k\not=0} c_k (1-e(q_k\alpha))\sigmaum_{j=0}^{n-1}e(q_k j\alpha)
=\sigmaum_{k\not=0} c_k (1-e(n q_k\alpha)).
\end{eqnarray*}
We should cut the last sum at some point. Let $K$ be such that
$q_{K-1}< 2\lambdaog N \lambdaeq q_{K}$.
Then for $k\gammaeq K$ we have
\betaegin{eqnarray}\lambdaabel{qk-lk}
|q_k\alpha-l_k|
\lambdaeq \frac{1}{q_{k+1}}
\lambdaeq e^{-q_k},
\end{eqnarray}
so that
\betaegin{eqnarray*}
\sigmaum_{|k|\gammaeq K} |c_k (1-e(n q_k\alpha))|
\lambdal C\sigmaum_{|k|\gammaeq K} n e^{-q_{k}}
\lambdal CN e^{-q_{K}}
\lambdal CN^{-1}
\end{eqnarray*}
where we the implied constant does not depend on $C$.
It follows that
\betaegin{eqnarray*}
S(N)=\sigmaum_{n\lambdaeq N} \mu(n)
e\betaigg\{\sigmaum_{1\lambdaeq |k|\lambdaeq K-1} c_k (1-e(n q_k\alpha))+O\betaigg(\frac{C}{N}\betaigg)\betaigg\}.
\end{eqnarray*}
The $O$-term as well as $\sigmaum\lambdaimits_{1\lambdaeq |k|\lambdaeq K-1} c_k $
above are harmless,
and so from now on we concentrate on
\betaegin{eqnarray}\lambdaabel{TilS=}
\widetilde{S}(N)
&:=&\sigmaum_{n\lambdaeq N} \mu(n)
e\betaigg\{\sigmaum_{1\lambdaeq |k|\lambdaeq K-1} c_k e(n q_k\alpha)\betaigg\} \nonumber\\
&=&\sigmaum_{n\lambdaeq N} \mu(n)
e\betaigg\{\sigmaum_{1\lambdaeq |k|\lambdaeq K-1} c_k e(n \theta_k)\betaigg\}
\end{eqnarray}
on writing $\theta_k=\|q_k\alpha\|$. Note that by
(\ref{qk-lk}) we have $\theta_k\lambdaeq e^{-q_k}$.
We have a sequence
$$
q_1<\exp(q_1) \lambdaeq q_2 < \exp(q_2) \lambdaeq q_3< \lambdadots,
$$
and therefore
$q_k> \exp \cdots\exp (q_2)$
with $k-2$ repeated $\exp$'s.
Taking $k=K-1$ and noting that $q_2\gammaeq 2$
give
\betaegin{eqnarray}\lambdaabel{repEXP}
\exp \cdots\exp (2) \lambdaeq \exp \cdots\exp (q_2) < q_{K-1}\lambdaeq 2\lambdaog N
\end{eqnarray}
where on the left-hand side there are $K-3$ $\exp$'s.
Now let
\betaegin{eqnarray}\lambdaabel{DEFfi}
\phi(x)
= 2c_1\cos (2\pi x), \quad x\in [\theta_1,\theta_1 N].
\end{eqnarray}
Then $e(\phi(x))$ is a smooth periodic function
and hence can expanded into
Fourier series
\betaegin{eqnarray}
e(\phi(x))=\sigmaum_{l\in \Bbb Z} a_l(c_1)e(lx),
\end{eqnarray}
where
\betaegin{eqnarray}
a_l(c_1)=\int_0^1 e(\phi(x))e(-lx)dx.
\end{eqnarray}
We must compute the dependence of $a_l(c_1)$ on $c_1$.
By partial integration for $l\not=0$ we have
\betaegin{eqnarray*}
a_l(c_1)
&=& -\frac{1}{2\pi i l}\int_0^1 e(\phi(x))d e(-lx) \\
&=& -\frac{1}{l}\int_0^1 e(\phi(x))\phi'(x) e(-lx) dx\\
&=& \frac{1}{2\pi i l^2}\int_0^1 \frac{d[e(\phi(x))\phi'(x)]}{dx} e(-lx) dx.
\end{eqnarray*}
Since
\betaegin{eqnarray*}
\phi'(x)= - 4\pi c_1\sigmain (2\pi x), \quad
\phi''(x)= - 8\pi^2 c_1\cos (2\pi x),
\end{eqnarray*}
we have
\betaegin{eqnarray*}
\betaigg|\frac{d[e(\phi(x))\phi'(x)]}{dx}\betaigg|
=|e(\phi(x)) [\phi''(x)+2\pi i\phi'(x)\phi'(x)]|
\lambdal |c_1|+|c_1|^2\lambdal C^2
\end{eqnarray*}
and consequently
\betaegin{eqnarray}\lambdaabel{al<<}
a_l(c_1)
\lambdal \frac{C^2}{l^2}
\end{eqnarray}
for all $l\not=0$.
The above $\lambdal$-constant is absolute. Of course the above $l^2$
can be improved to $l^A$ for arbitrary $A>0$, but we are not going to
use this.
The sum $\widetilde{S}(N)$ in (\ref{TilS=}) is of the form
\betaegin{eqnarray*}
\widetilde{S}(N)
&=&\sigmaum_{n\lambdaeq N} \mu(n)e(\phi(n\theta_1)+F(n))\\
&=& \sigmaum_{n\lambdaeq N} \mu(n) e(F(n))\sigmaum_{l\in \Bbb Z} a_{l}(c_1) e(ln\theta_1)\\
&=& \sigmaum_{l\in \Bbb Z} a_{l}(c_1)
\sigmaum_{n\lambdaeq N} \mu(n) e(ln\theta_1+F(n)),
\end{eqnarray*}
where $F(n)$ stands for the remaining terms in the $e(\cdot)$ in (\ref{TilS=}).
It follows from this and (\ref{al<<}) that
\betaegin{eqnarray*}
|\widetilde{S}(N)|
&\lambdaeq & \sigmaum_{l\in \Bbb Z} |a_{l}(c_1) |
\betaigg|\sigmaum_{n\lambdaeq N} \mu(n) e(ln\theta_1+F(n))\betaigg|\\
&\lambdal & C^2 \sigmaum_{l\in \Bbb Z}\frac{1}{l^2+1}
\betaigg|\sigmaum_{n\lambdaeq N} \mu(n) e(ln\theta_1+F(n))\betaigg|\\
&\lambdal & C^2
\sigmaup_{l_1} \betaigg|\sigmaum_{n\lambdaeq N} \mu(n) e(n l_1\theta_1 +F(n))\betaigg|.
\end{eqnarray*}
Repeating this procedure in (\ref{TilS=}), we get
\betaegin{eqnarray*}
|\widetilde{S}(N)|
\lambdal C^{2(K-1)}
\sigmaup_{l_1, \lambdadots, l_{K-1}} \betaigg|\sigmaum_{n\lambdaeq N}
\mu(n) e(n l_1\theta_1+ \lambdadots+ n l_{K-1}\theta_{K-1}) \betaigg|.
\end{eqnarray*}
The inner sum involving $\mu$ can be estimated by classical results of
Davenport \cite{Dav} and Hua \cite{Hua} as
$$
\lambdal N\lambdaog^{-A}N
$$
where $A>0$ is arbitrary, and the implied constant
depends at most on $A$, i.e. it does not depend on the coefficients
$l_1, l_2, \lambdadots, l_{K-1}$ or $\theta_1, \theta_2, \lambdadots, \theta_{K-1}$.
By (\ref{repEXP}) we have $C^{2(K-1)}\lambdaeq \lambdaog N$. The proposition is proved.
\end{proof}
\noindent
{\betaf Acknowledgements.}
The first author is supported by the 973 Program,
NSFC grant 11031004, and IRT1264 from the Ministry of Education.
The second author is supported by an NSF grant.
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\end{document} | math | 19,909 |
\begin{document}
\title[Connection coefficients and orthogonalizing measures]{On affinity
relating two positive measures and the connection coefficients between
polynomials orthogonalized by these measures}
\author{Pawe\l\ J. Szab\l owski}
\address{Department of Mathematics and Information Sciences,\\
Warsaw University of Technology\\
pl. Politechniki 1, 00-661 Warsaw, Poland}
\email{[email protected]}
\date{January 30, 2012}
\subjclass[2010]{Primary 42C05, 26C05; Secondary 42C10, 12E05}
\keywords{Orthogonal polynomials, positive measures, Kesten--McKay measure,
Jacobi polynomials, Charlier polynomials}
\thanks{The author is grateful to the unknown referee for drawing his
attention to valuable works of P. Maroni.}
\begin{abstract}
We consider two positive, normalized measures $dA\left( x\right) $ and $
dB\left( x\right) $ related by the relationship $dA\left( x\right) =\frac{C}{
x+D}dB\left( x\right) $ or by $dA\left( x\right) \allowbreak =\allowbreak
\frac{C}{x^{2}+E}dB\left( x\right) $ and $dB\left( x\right) $ is symmetric.
We show that then the polynomial sequences $\left\{ a_{n}\left( x\right)
\right\} ,$ $\left\{ b_{n}\left( x\right) \right\} $ orthogonal with respect
to these measures are related by the relationship $a_{n}\left( x\right)
=b_{n}\left( x\right) +\kappa _{n}b_{n-1}\left( x\right) $ or by $
a_{n}\left( x\right) \allowbreak =\allowbreak b_{n}\left( x\right)
\allowbreak +\allowbreak \lambda _{n}b_{n-2}\left( x\right) $ for some
sequences $\left\{ \kappa _{n}\right\} $ and $\left\{ \lambda _{n}\right\} .$
We present several examples illustrating this fact and also present some
attempts for extensions and generalizations. We also give some universal
identities involving polynomials $\left\{ b_{n}\left( x\right) \right\} $
and the sequence $\left\{ \kappa _{n}\right\} $ that have a form of Fourier
series expansion of the Radon--Nikodym derivative of one measure with
respect to the other.
\end{abstract}
\maketitle
\section{Introduction}
We study relationship between the pair of orthogonal polynomials and the
pair of measures that make these polynomials orthogonal. This problem has
practical importance. If solved in full generality would enable quick and
easy way of finding sets of orthogonal polynomials for a given measure
simplifying the usual path of the Gram-Smith orthogonalization. Besides it
would provide quick and easy way of finding 'connection coefficients'
between the two analyzed sets of orthogonal polynomials. On its side
'connection coefficients', as it is well known supply many useful
informations about the properties of the involved sets of polynomials. So
far in the literature devoted to connection coefficients like \cite{Szwarc92}
, \cite{Dimitrov01}, \cite{Area 04} the authors studied the properties of
these coefficients and their relationship to zeros of orthogonal polynomials
in question without referring to the properties of orthogonalizing measures.
We are solving the problem of affinity between connection coefficients and
measures that make polynomials orthogonal only partially. There are still
many challenging questions that we pose in Section \ref{ext} and which are
unsolved to our knowledge.
To be more precise we will assume throughout the paper the following setting:
We consider two sequences of monic, orthogonal polynomials $\left\{
a_{n}\right\} $ and $\left\{ b_{n}\right\} $ such that their 3-term
recurrences are as given below:
\begin{eqnarray}
a_{n+1}\left( x\right) \allowbreak &=&\allowbreak (x-\alpha _{n})a_{n}\left(
x\right) -\hat{\alpha}_{n-1}a_{n-1}\left( x\right) , \label{aa} \\
b_{n+1}\left( x\right) &=&\left( x-\beta _{n}\right) b_{n}\left( x\right) -
\hat{\beta}_{n-1}b_{n-1}\left( x\right) \label{bb}
\end{eqnarray}
with $a_{-1}\left( x\right) \allowbreak =\allowbreak b_{-1}\left( x\right)
\allowbreak =\allowbreak 0,$ $a_{0}\left( x\right) \allowbreak =\allowbreak
b_{0}\left( x\right) \allowbreak =\allowbreak 1.$
In \cite{Szab2010}, Proposition 1 it was shown that if these measures are
such that $\limfunc{supp}A=\allowbreak \limfunc{supp}B$ and $dA\left(
x\right) \allowbreak =\allowbreak \frac{1}{P_{r}\left( x\right) }dB\left(
x\right) $, where $P_{r}$ is a polynomial of order $r,$ then there exist $r$
sequences $\left\{ c_{n}^{\left( j\right) }\right\} _{n\geq 1,1\leq j\leq r}$
such that
\begin{equation}
a_{n}\left( x\right) \allowbreak =\allowbreak b_{n}\left( x\right)
+\sum_{j=1}^{r}c_{n}^{\left( j\right) }b_{n-j}\left( x\right) . \label{qo}
\end{equation}
In the cited result it was not presented how to relate 3-term recurrence
satisfied by say the set $\left\{ b_{n}\right\} $ and the form of the
polynomial $P_{r}$ to the form of the coefficients $\left\{ c_{n}^{\left(
j\right) }\right\} _{n\geq 1,1\leq j\leq r}.$
\begin{remark}
Notice that our assumptions mean in fact that $dA<<dB$ and $\frac{dA(x)}{
dB(s)}\allowbreak =\allowbreak \frac{1}{P_{r}\left( x\right) },$ where $
\frac{dA(x)}{dB(s)}$ denotes Radon--Nikodym derivative of $dA$ with respect
to $dB.$
\end{remark}
Relationship like (\ref{qo}) between sets of orthogonal polynomials is
called quasi-orthogonality as defined in \cite{Chih57} and \cite{Dic61}.
More precisely polynomials $\left\{ a_{n}\right\} $ that are related to
polynomials $\left\{ b_{n}\right\} $ by (\ref{qo}) are called
quasi-orthogonal provided $b_{n}^{\prime }s$ are orthogonal. Thus our
problem can be expressed in the following way: If the measures $dA$ and $dB$
are related by $dA\left( x\right) \allowbreak =\allowbreak \frac{1}{
P_{r}\left( x\right) }dB\left( x\right) ,$ then there exists $r$ sequences
of numbers $\left\{ c_{n}^{\left( j\right) }\right\} _{n\geq 1,1\leq j\leq
r} $ such that quasi-orthogonal polynomials defined by (\ref{qo}) are
orthogonal (with respect to $dA).$
There exits also another, similar in a way, path of research followed by
Pascal Maroni and his associates. The results of their research were
presented in the series of papers \cite{M1}-\cite{M6}. The problem
considered by Maroni concerns general linear regular (i.e. possessing sets
of orthogonal polynomials) functionals $u$ and $v$ (not necessarily in the
form of measure) defined on the linear space of polynomials and related to
one another by the relationship
\begin{equation*}
x^{m}u=\lambda v,
\end{equation*}
where $m$ is a fixed integer (in Maroni's papers $m\leq 4)$ and $\lambda $ a
fixed complex number. In majority of cases he assumes that regular $v$ has a
form $<v,p>\allowbreak =\allowbreak \int V(x)p(x)dx$ where $p$ is a
polynomial while $V$ is a locally integrable function rapidly decaying at
infinity. Maroni is interested in conditions for the existence of regular $u$
and also in the relationship between the sets of polynomials orthogonal with
respect to $v$ and $u.$ In his studies he showed that orthogonal polynomials
of $u$ must be linear combinations of the last $m+1$ (i.e. are
quasi-orthogonal). He also obtained some (mostly in the case of $
m\allowbreak =\allowbreak 1)$ recursive relations relating sets of
orthogonal polynomials of $u$ and $v$ and coefficients of the
quasi-orthogonality.
Notice that even if $v$ is a measure $u$ may be not. Moreover it is
expressed (as it follows from Maroni's papers) by derivatives of Dirac's
delta and Cauchy's principal value. The existence conditions are not simple
and constitute major part of these papers.
That is why although some of the results obtained below were first
discovered by Maroni we will repeat them for the sake of uniformity of
treatment. Of course we will point out which of them were mentioned in
Maroni's papers.
In the present paper we continue the research started in \cite{Szab2010} and
relate the 3-term recurrence coefficients satisfied by $\left\{
b_{n}\right\} $ and the exact form of the polynomial $P_{r}$ for $r=1,2$ to
coefficients $c_{n}^{\left( 1\right) }$ for $r\allowbreak =\allowbreak 1$
and $c_{n}^{\left( 1\right) },$ $c_{n}^{\left( 2\right) }$ for $r\allowbreak
=\allowbreak 2$. We also give the 3-term recurrence coefficients of the
polynomials $\left\{ a_{n}\right\} $. Besides we also provide certain
universal identities satisfied by the polynomials $\left\{ b_{n}\right\} ,$
coefficients $\left\{ \hat{\beta}_{n}\right\} ,$ $\left\{ c_{n}^{\left(
1\right) }\right\} $ and the parameters of the polynomials $P_{r}.$
The paper is organized as follows: In the next Section \ref{gl} we present
our main result concerning the case $r\allowbreak =\allowbreak 1$ and
illustrate it by $3$ examples concerning well known families of polynomials
like Jacobi or Charlier. Examples are presented in Section \ref{przyklady}.
Less complete or less simple and nice results are presented in Section \ref
{ext}. Here also we will illustrate the developed ideas by a few examples.
Finally Section \ref{dowody} contains less interesting or lengthy proofs of
our results.
\section{Main results\label{gl}}
The simplest but also the most important case is when $r\allowbreak
=\allowbreak 1.$ This case is treated by the theorem below:
\begin{theorem}
\label{glowny}Let the sequence of monic, orthogonal polynomials $\left\{
b_{n}\right\} $ be defined by the 3-term recurrence (\ref{bb}). Suppose that
$dB\left( x\right) $ is the positive measure that makes these polynomials
orthogonal. Let us consider another normalized measure $dA\left( x\right) $
related to $dB\left( x\right) $ by the relationship:
\begin{equation}
dA\left( x\right) \allowbreak =\allowbreak \frac{C}{x+D}dB\left( x\right) ,
\label{jednomian}
\end{equation}
so that $\frac{C}{\left( x+D\right) }\geq 0$ on the support of $dB$ (and of $
dA$).
Then there exists a number sequence $\left\{ \kappa _{n}\right\} $ defined
by the relationship:
\begin{equation}
\kappa _{n}=\beta _{n-1}-\frac{\hat{\beta}_{n-2}}{\kappa _{n-1}}+D,
\label{_1}
\end{equation}
$n\geq 2$ with $\kappa _{1}\allowbreak =\allowbreak \beta _{0}+D-C,$ such
that the sequence of monic polynomials defined by:
\begin{equation}
a_{n}\left( x\right) =b_{n}\left( x\right) +\kappa _{n}b_{n-1}\left(
x\right) . \label{zaleznosc}
\end{equation}
satisfies the 3-term recurrence (\ref{aa}) with:
\begin{eqnarray}
\alpha _{n}\allowbreak &=&\allowbreak \beta _{n}+\kappa _{n}-\kappa _{n+1},
\label{new1} \\
\hat{\alpha}_{n-1} &=&\kappa _{n}\frac{\hat{\beta}_{n-2}}{\kappa _{n-1}},
\label{new2}
\end{eqnarray}
and is orthogonal with respect to the measure $dA\left( x\right) .$
\end{theorem}
\begin{remark}
Recursive equations (\ref{_1}) and (\ref{new1}) were obtained by P. Maroni
in \cite{M1} as sidelines of his results obtained in a slightly different
context. (\ref{new2}) was obtained in a different but equivalent form.
\end{remark}
\begin{proof}
Is shifted to section \ref{dowody}.
\end{proof}
We have immediate remarks, observations and corollaries
\begin{remark}
All coefficients $\kappa _{n}$ have the same sign i.e. are either all
positive or all negative. This follows the fact that since $dA\left(
x\right) $ and $dB\left( x\right) $ are positive measures we must have
nonnegative both $\hat{\alpha}_{n}$ and $\hat{\beta}_{n}.$ Then we use (\ref
{new2}).
\end{remark}
\begin{remark}
Notice that following relationship $\kappa _{1}\allowbreak =\allowbreak
\beta _{0}+D-C$ and (\ref{zaleznosc}), $\frac{C}{x+D}$ can be written as $
\frac{1}{a_{1}\left( x\right) /C+1}$ which fits assumptions of Proposition 1
of \cite{Szab2010}.
\end{remark}
\begin{remark}
Following (\ref{new2}) and the fact that $\hat{\alpha}_{n-1}\geq 0$ we
deduce that either $\beta _{n}\allowbreak +\allowbreak D\allowbreak \geq
\allowbreak 0$ or $\beta _{n}\allowbreak +\allowbreak D\allowbreak \leq
\allowbreak 0$ for all $n\geq 1.$ Consequently either we have for all $n\geq
0$
\begin{equation*}
\beta _{n-1}+D\leq \kappa _{n}\leq 0~~\text{or~~}0\leq \kappa _{n}\leq \beta
_{n-1}+D.
\end{equation*}
\end{remark}
\begin{corollary}
\label{rozw}Under assumptions of Theorem \ref{glowny} we have
\begin{equation}
b_{n}\left( x\right) =a_{n}\left( x\right) +\sum_{j=1}^{n}\left( -1\right)
^{j}\left( \prod_{k=n-j+1}^{n}\kappa _{k}\right) a_{n-j}\left( x\right)
\label{odwr}
\end{equation}
for $n\allowbreak =\allowbreak 0,1,2,\ldots $ . Further under additional
assumption that $\int_{\limfunc{supp}B}\frac{1}{(x+D)^{2}}dB\left( x\right)
<\infty $ we have:
\begin{equation*}
1+\sum_{n\geq 1}\left( \prod_{k=1}^{n}\frac{\kappa _{k}}{\hat{\beta}_{k-1}}
\right) ^{2}=C^{2}\int_{\limfunc{supp}B}\frac{1}{(x+D)^{2}}dB\left( x\right)
\end{equation*}
and
\begin{equation}
\frac{C}{x+D}=1+\sum_{n\geq 1}\left( -1\right) ^{n}\left( \prod_{k=1}^{n}
\frac{\kappa _{k}}{\hat{\beta}_{k-1}}\right) b_{n}\left( x\right) ,
\label{szereg}
\end{equation}
on $\limfunc{supp}B$ in $L_{2}\left( \limfunc{supp}B,\mathcal{B},dB\left(
x\right) \right) .$ If additionally $\sum_{n\geq 1}\left( \prod_{k=1}^{n}
\frac{\kappa _{k}}{\hat{\beta}_{k-1}}\right) ^{2}\log ^{2}n<\infty ,$ then
convergence in (\ref{szereg}) is almost ($dB\left( x\right) )$ pointwise on $
\limfunc{supp}B.$
\end{corollary}
\begin{proof}
Using (\ref{odwr}) we have $b_{n}\left( x\right) \allowbreak +\allowbreak
\kappa _{n}b_{n-1}\left( x\right) \allowbreak =\allowbreak
a_{n}+\sum_{j=1}^{n}\left( -1\right) ^{j}\left( \prod_{k=n-j+1}^{n}\kappa
_{k}\right) a_{n-j}\left( x\right) \allowbreak +\allowbreak \kappa
_{n}a_{n-1}\allowbreak +\allowbreak \sum_{j=1}^{n-1}\left( -1\right)
^{j}\left( \prod_{k=n-j}^{n}\kappa _{k}\right) a_{n-1-j}\left( x\right)
\allowbreak =\allowbreak a_{n}.$ Now we apply idea of ratio of density
expansion presented in \cite{Szab2010} and use (\ref{odwr}). On the way we
notice that the requirement that both measures (i.e. $dA$ and $dB)$ have
densities with respect to Lebesgue measure can be dropped. We also utilize
the fact that $\int_{\limfunc{supp}B}b_{n}^{2}\left( x\right) dB\left(
x\right) \allowbreak =\allowbreak \prod_{k=0}^{n-1}\hat{\beta}_{k}$ which
follows Favard's Theorem. The we also use Rademacher--Menshov theorem
concerning almost sure convergence of $L_{2}$ converging Fourier series.
\end{proof}
\section{Examples\label{przyklady}}
To illustrate how simple and easy is to utilize the presented in the
previous section observations and rules let us consider the following few
examples:
It should be remarked that although examples concerning Jacobi and Legendre
polynomials were considered by Maroni in \cite{M1} they were illustrating
different phenomena discovered by Maroni in this paper. In particular
formulae (\ref{Jkappa}), (\ref{Jexp}) and (\ref{expJ}) do not appear in
Maroni's paper. We present them in order to illustrate the use of equations (
\ref{_1}), (\ref{new1}), (\ref{new2}) and expansion (\ref{szereg}).
\begin{example}[Jacobi polynomials]
Recall (e.g. basing on \cite{Andrews1999} or \cite{IA}) that monic Jacobi
polynomials $J_{n}^{\left( \alpha ,\beta \right) }\left( x\right) $ satisfy
the following 3-term recurrence:
\begin{subequations}
\label{Jacobi}
\begin{gather}
J_{n+1}^{(\alpha ,\gamma )}(z)=\left( x+\frac{\alpha ^{2}-\gamma ^{2}}{
(2n+\alpha +\gamma +2)(\alpha +\gamma +2n)}\right) J_{n}^{\left( \alpha
,\gamma \right) }\left( x\right) \label{Jacobi1} \\
-\frac{4n\left( \alpha +\gamma +n\right) (n+\alpha )(n+\gamma )}{\left(
\alpha +\gamma +2n-1\right) (2n+\alpha +\gamma )^{2}(\alpha +\gamma +2n+1)}
J_{n-1}^{(\alpha ,\gamma )}(z). \label{Jacobi2}
\end{gather}
Besides one knows also that the normalized measure that makes these
polynomials orthogonal is the following:
\end{subequations}
\begin{equation*}
f\left( x;\alpha ,\gamma \right) =\frac{\Gamma \left( \alpha +\gamma
+2\right) }{2^{\alpha +\gamma +1}\Gamma \left( \gamma +1\right) \Gamma
\left( \alpha +1\right) }\left( 1-x\right) ^{\alpha }(1+x)^{\gamma },
\end{equation*}
where $\Gamma \left( \eta \right) $ denotes value of the Gamma function at $
\eta ,$ for $\left\vert x\right\vert <1$ and $\alpha ,\gamma >-1.$
Now let us take $\alpha >0,\gamma >-1,$ $dB\left( x\right) \allowbreak
=\allowbreak f\left( x;\alpha ,\gamma \right) dx$ and $dA\left( x\right)
\allowbreak =\allowbreak f\left( \alpha -1,\gamma \right) dx,$ $b_{n}\left(
x\right) \allowbreak =\allowbreak J_{n}^{\left( \alpha ,\gamma \right)
}\left( x\right) $ and $a_{n}\left( x\right) \allowbreak =\allowbreak
J_{n}^{\left( \alpha -1,\gamma \right) }\left( x\right) .$ One can easily
notice that $dA\left( x\right) \allowbreak =\allowbreak \frac{C}{(-1+x)}
dB\left( x\right) ,$ where $C=\allowbreak -\frac{2\alpha }{\alpha +\gamma +1}
,$ hence $D\allowbreak =\allowbreak -1.$ From (\ref{Jacobi}) it follows also
that
\begin{eqnarray}
\beta _{n}\allowbreak &=&\allowbreak -\frac{\alpha ^{2}-\gamma ^{2}}{
(2n+\alpha +\gamma +2)(\alpha +\gamma +2n)}, \label{_bJ} \\
\hat{\beta}_{n-1} &=&\frac{4n\left( \alpha +\gamma +n\right) (n+\alpha
)(n+\gamma )}{\left( \alpha +\gamma +2n-1\right) (2n+\alpha +\gamma
)^{2}(\alpha +\gamma +2n+1)}. \label{_bbJ}
\end{eqnarray}
Thus $\kappa _{1}\allowbreak =\allowbreak \beta _{0}+D-C\allowbreak =-\frac{
\alpha ^{2}-\gamma ^{2}}{(\alpha +\gamma +2)(\alpha +\gamma )}\allowbreak +
\frac{2\alpha }{\alpha +\gamma +1}-1\allowbreak \allowbreak =\allowbreak -2
\frac{\gamma +1}{\left( \alpha +\gamma +1\right) \left( \alpha +\gamma
+2\right) }$ and consequently coefficients $\kappa _{n}$ satisfy recursive
equation:
\begin{equation*}
\kappa _{n}=\beta _{n-1}-1-\frac{\hat{\beta}_{n-2}}{\kappa _{n-1}},
\end{equation*}
for $n\geq 2.$ One can also easily notice that
\begin{equation}
\kappa _{n}=-\frac{2n(n+\gamma )}{\left( \alpha +\gamma +2n\right) \left(
\alpha +\gamma +2n-1\right) } \label{Jkappa}
\end{equation}
satisfies above mentioned recursive equation. Hence
\begin{equation}
J_{n}^{\left( \alpha -1,\gamma \right) }\left( x\allowbreak \right)
=J_{n}^{\left( \alpha ,\gamma \right) }\left( x\right) +\kappa
_{n}J_{n-1}^{\left( \alpha ,\gamma \right) }(x). \label{Jexp}
\end{equation}
As far as application of Corollary \ref{rozw} is concerned we have the
following identity true for $\alpha >1,$ $\gamma >-1,$ and almost all $
\left\vert x\right\vert <1:$
\begin{equation}
1=\frac{\alpha +\gamma +1}{2\alpha }(1-x)(1+\sum_{n\geq 1}^{\infty }\frac{
(\alpha +\gamma +1)_{2n}}{2^{n}(\alpha +\gamma +1)\left( \alpha +1\right)
_{n}\left( \alpha +\gamma +1\right) _{n}}J_{n}^{\left( \alpha ,\gamma
\right) }\left( x\right) ), \label{expJ}
\end{equation}
where we use the so called Pochhammer symbol $\left( a\right)
_{n}\allowbreak =\allowbreak a\left( a+1\right) \ldots (a+n-1).$ This is so
since $\frac{\kappa _{n}}{\gamma _{n-1}}\allowbreak =\allowbreak -\frac{
\left( \alpha +\gamma +2n+1\right) (\alpha +\gamma +2n)}{2\left( \alpha
+n\right) \left( \alpha +\gamma +n\right) },$ by (\ref{_bbJ}) and (\ref
{Jkappa}) and because $\int_{-1}^{1}\frac{1}{(1-x)^{2}}dB\left( x\right)
<\infty $ for $\gamma >-1,$ $\alpha -2>-1.$
\end{example}
\begin{example}[Charlier polynomials]
Basing on \cite{Koek} let us recall that monic Charlier polynomials $\left\{
c_{n}\left( x;\lambda \right) \right\} _{n\geq -1}$ are polynomials given by
the following 3-term recurrence
\begin{equation*}
c_{n+1}\left( x;\lambda \right) =(x-n-\lambda )c_{n}\left( x;\lambda \right)
-n\lambda c_{n-1}\left( x;\lambda \right) ,
\end{equation*}
with $c_{-1}\left( x;\lambda \right) \allowbreak =\allowbreak 0,$ $
c_{0}\left( x;\lambda \right) \allowbreak =\allowbreak 1.$ For $\lambda >0$
they are orthogonal with respect to discrete measure concentrated at
nonnegative integers with mass at $n$ equal to $\exp \left( -\lambda \right)
\frac{\lambda ^{n}}{n!},$ $n\geq 0.$ Another words this measure is the
Poisson normalized measure.
In order not to complicate too much let us take $dB\left( n\right)
\allowbreak =\allowbreak $ $\exp \left( -\lambda \right) \frac{\lambda ^{n}}{
n!}$ and $dA\left( n\right) \allowbreak =\allowbreak \frac{C}{n+1}dB\left(
n\right) $ for $n\allowbreak =\allowbreak 0,1,\ldots $ . Since $\sum_{n\geq
0}^{\infty }\frac{\lambda ^{n}}{\left( n+1\right) !}\allowbreak =\allowbreak
\frac{\left( \exp \left( \lambda \right) -1\right) }{\lambda }$ we see that $
C\allowbreak =\allowbreak \frac{\lambda \exp \left( \lambda \right) }{\exp
\left( \lambda \right) -1}.$ Naturally we have also $D=1$ and $\beta
_{n}\allowbreak =\allowbreak n+\lambda $ and $\hat{\beta}_{n-1}\allowbreak
=\allowbreak n\lambda ,$ hence $\kappa _{1}\allowbreak =\allowbreak \lambda
+1-\frac{\lambda \exp \left( \lambda \right) }{-1+\exp \left( \lambda
\right) }\allowbreak =\allowbreak $ $\frac{\exp \left( \lambda \right)
-1-\lambda }{\exp (\lambda )-1}.$ Thus recursive equation satisfied by
coefficients $\kappa _{n}$ is the following:
\begin{equation*}
\kappa _{n}=n+\lambda -\frac{\left( n-1\right) \lambda }{\kappa _{n-1}},
\end{equation*}
$n\geq 2.$ In particular we have $\kappa _{2}\allowbreak =\allowbreak 2\frac{
\exp \left( \lambda \right) -1-\lambda -\lambda ^{2}/2}{\exp \left( \lambda
\right) -1-\lambda },$ $\kappa _{3}\allowbreak =\allowbreak 3\frac{\exp
\left( \lambda \right) -1-\lambda -\lambda ^{2}/2-\lambda ^{3}/3!}{\exp
\left( \lambda \right) -1-\lambda -\lambda ^{2}/2}$ and in general it is
easy to see that
\begin{equation*}
\kappa _{n}\allowbreak =\allowbreak n\frac{\exp \left( \lambda \right)
-\sum_{j=0}^{n}\frac{\lambda ^{j}}{j!}}{\exp \left( \lambda \right)
-\sum_{j=0}^{n-1}\frac{\lambda ^{j}}{j!}}=n\frac{\sum_{j\geq n+1}\frac{
\lambda ^{j}}{j!}}{\sum_{j\geq n}\frac{\lambda ^{j}}{j!}}.
\end{equation*}
Thus we have in particular
\begin{eqnarray*}
a_{n}\left( x\right) \allowbreak &=&\allowbreak c_{n}\left( x\right)
+\kappa _{n}c_{n-1}\left( x\right) , \\
\alpha _{n} &=&n+\lambda +\kappa _{n}-\kappa _{n+1}, \\
\hat{\alpha}_{n-1} &=&\frac{\lambda n(\sum_{j\geq n+1}\frac{\lambda ^{j}}{j!}
)\left( \sum_{j\geq n-1}\frac{\lambda ^{j}}{j!}\right) }{(\sum_{j\geq n}
\frac{\lambda ^{j}}{j!})^{2}}.
\end{eqnarray*}
As the application of Corollary \ref{rozw} we have the following identity
true for $\lambda >0$ and $x\allowbreak =\allowbreak 0,1,\ldots $
\begin{equation}
e^{\lambda }=(1+x)(\frac{e^{\lambda }-1}{\lambda }+\sum_{n\geq 1}\left(
-1\right) ^{n}c_{n}\left( x;\lambda \right) \sum_{k\geq n+1}\frac{\lambda
^{k-n-1}}{k!}). \label{CA}
\end{equation}
This is so since $\frac{\kappa _{n}}{\hat{\beta}_{n-1}}=\frac{\exp \left(
\lambda \right) -\sum_{j=0}^{n}\frac{\lambda ^{j}}{j!}}{\lambda (\exp \left(
\lambda \right) -\sum_{j=0}^{n-1}\frac{\lambda ^{j}}{j!})}$ and consequently
$\prod_{k=1}^{n}\frac{\kappa _{k}}{\hat{\beta}_{k-1}}\allowbreak
=\allowbreak \frac{\exp \left( \lambda \right) -\sum_{k=0}^{n}\frac{\lambda
^{k}}{k!}}{\lambda ^{n}}\allowbreak =\allowbreak \sum_{k\geq n+1}\frac{
\lambda ^{k-n}}{k!}.$ Let us observe that (\ref{CA}) is not satisfied for
non-positive integer $x.$
\end{example}
\begin{example}[Legendre polynomials]
As it is known Legendre polynomials are the Jacobi polynomials with $\alpha
,\gamma \allowbreak =\allowbreak 0$. As $dB\left( x\right) $ let us consider
measure with the density $f(x;0,0)\allowbreak =\allowbreak 1/2$ on $[-1,1].$
As $dA\left( x\right) $ let us consider measure with the density $\frac{C}{
2(3-x)}$ on $[-1,1].$ Parameter $C$ we get by direct integration, namely $
C\allowbreak =\allowbreak \frac{-2}{\ln 2}$ while $D\allowbreak =\allowbreak
\allowbreak -3.$ Further using (\ref{_bJ}) and (\ref{_bbJ}) we get:
\begin{equation*}
\beta _{n}\allowbreak =\allowbreak 0,\hat{\beta}_{n-1}\allowbreak =\frac{
n^{2}}{(2n-1)(2n+1)}.
\end{equation*}
Hence $\kappa _{1}\allowbreak =\allowbreak 0-3+\frac{2}{\ln 2}$ and
consequently coefficients $\kappa _{n}$ are given by the following recursive
equation:
\begin{equation*}
\kappa _{n+1}\allowbreak =\allowbreak -3-\frac{n^{2}}{(2n-1)(2n+1)\kappa _{n}
},
\end{equation*}
for $n\geq 1.$ In particular we get $\kappa _{2}\allowbreak =\allowbreak -3-
\frac{1}{3(-3+2/\ln 2)}\allowbreak =\allowbreak $ $-\frac{\left( 26\ln
2-18\right) }{9\ln 2-6}.$ Finally we deduce that polynomials defined by
\begin{equation*}
a_{n}\left( x\right) =J_{n}^{\left( 0,0\right) }+\kappa _{n}J_{n-1}^{\left(
0,0\right) }(x),
\end{equation*}
are orthogonal with respect to the measure with the density: $\frac{2}{
(3-x)\ln 2}$ on $[-1,1].$
\end{example}
\section{Extensions and open problems\label{ext}}
In this section we are going to present some generalizations of the results
of Section \ref{gl}. The results are not as nice and compact as the ones
presented above that is why we present them here. We will also pose some
open problems that appeared immediately when writing the article.
Let us return to the setting that was presented in the Introduction and
consider the case $r\allowbreak =\allowbreak 2.$ Let us assume that measures
$dA$ and $dB$ are related to one another by the relationship
\begin{equation}
dA\left( x\right) =\frac{C}{x^{2}+Dx+E}dB\left( x\right) \label{dwumian}
\end{equation}
and that constants $C,D,E$ are such that $\frac{C}{x^{2}+Cx+E}\geq 0$ on $
\limfunc{supp}B$ and that measure $dA$ is normalized. Following cited
already \cite{Szab2010}, Proposition 1 we deduce that then polynomials $
\left\{ a_{n}\right\} $ and $\left\{ b_{n}\right\} $ orthogonal with respect
to these measures are related by the relationship
\begin{equation}
a_{n}\left( x\right) \allowbreak =\allowbreak b_{n}\left( x\right) +\kappa
_{n}b_{n-1}\left( x\right) +\lambda _{n}b_{n-2}\left( x\right) , \label{zal}
\end{equation}
for some number sequences $\left\{ \kappa _{n}\right\} $ and $\left\{
\lambda _{n}\right\} .$ given in the Proposition below:
\begin{proposition}
\label{uog}Suppose normalized, positive measures $dA$ and $dB$ are related
to one another by (\ref{dwumian}). Let further polynomial sequences $\left\{
a_{n}\right\} $ and $\left\{ b_{n}\right\} $ orthogonal with respect to
these measures satisfy respectively 3-term recurrence (\ref{aa}) and (\ref
{bb}). Then there exist two number sequences $\left\{ \kappa _{n}\right\} $
and $\left\{ \lambda _{n}\right\} $ such that (\ref{zal}) is satisfied.
Moreover number sequences $\left\{ \kappa _{n}\right\} ,$ $\left\{ \lambda
_{n}\right\} $, $\left\{ \alpha _{n}\right\} $ $\left\{ \hat{\alpha}
_{n}\right\} ,$ $\left\{ \beta _{n}\right\} ,$ $\left\{ \hat{\beta}
_{n}\right\} $ are related to one another by the system of equations:
\begin{eqnarray}
\kappa _{n+1}+\alpha _{n} &=&\beta _{n}+\kappa _{n}, \label{s1} \\
\lambda _{n+1}+\alpha _{n}\kappa _{n}+\hat{\alpha}_{n-1} &=&\hat{\beta}
_{n-1}+\kappa _{n}\beta _{n-1}+\lambda _{n}, \label{s2} \\
\alpha _{n}\lambda _{n}+\hat{\alpha}_{n-1}\kappa _{n-1} &=&\kappa _{n}\hat{
\beta}_{n-2}+\lambda _{n}\beta _{n-2}, \label{s3} \\
\hat{\alpha}_{n-1}\lambda _{n-1} &=&\lambda _{n}\hat{\beta}_{n-3}.
\label{s4}
\end{eqnarray}
with $\lambda _{1}\allowbreak =\allowbreak 0$ and $\kappa _{1},$ $\kappa
_{2} $ and $\lambda _{2}$ defined as solutions of the system of $7$
equations $0\allowbreak =\allowbreak \int_{\limfunc{supp}B}\left(
b_{1}\left( x\right) +\kappa _{1}\right) dA\left( x\right) \allowbreak
=\allowbreak \int_{\limfunc{supp}B}\left( b_{2}\left( x\right) +\kappa
_{2}b_{1}\left( x\right) +\lambda _{2}\right) dA\left( x\right) \allowbreak
=\allowbreak \int_{\limfunc{supp}B}\left( b_{2}\left( x\right) +\kappa
_{2}b_{1}\left( x\right) +\lambda _{2}\right) (b_{1}\left( x\right) +\kappa
_{1})dA\left( x\right) \allowbreak =\allowbreak \int_{\limfunc{supp}
B}b_{1}\left( x\right) \left( x^{2}+Dx+E\right) dA\left( x\right)
\allowbreak =\allowbreak \int_{\limfunc{supp}B}b_{2}\left( x\right) \left(
x^{2}+Dx+E\right) dA\left( x\right) ,$ $\int_{\limfunc{supp}B}\left(
x^{2}+Dx+E\right) dA\left( x\right) \allowbreak =\allowbreak C,$ \newline
$\int_{\limfunc{supp}B}b_{1}^{2}\left( x\right) \left( x^{2}+Dx+E\right)
dA\left( x\right) \allowbreak =\allowbreak \hat{\beta}_{0}$ with $4$
additional unknowns \newline
$\int_{\limfunc{supp}B}b_{1}\left( x\right) dA\left( x\right) ,$ $\int_{
\limfunc{supp}B}b_{2}\left( x\right) dA\left( x\right) ,$ $\int_{\limfunc{
supp}B}b_{1}\left( x\right) b_{2}\left( x\right) dA\left( x\right) $, $\int_{
\limfunc{supp}B}b_{2}^{2}\left( x\right) dA\left( x\right) .$
\end{proposition}
\begin{proof}
Uninteresting proof is shifted to Section \ref{dowody}.
\end{proof}
Visibly it is hard to solve system of equations (\ref{s1})-(\ref{s4}) for $
\{\kappa _{n},\lambda _{n},\alpha _{n},\hat{\alpha}_{n}\}$ in general. Below
we will present one example where it is simple.
\begin{example}[Kesten--McKay distribution]
This example concerns measure that is called Kesten--McKay. It appeared in
probability in the context of random matrices. One of the particular
examples of its density is the density of the form:
\begin{equation*}
f\left( x;y,\rho \right) =\frac{(1-\rho ^{2})\sqrt{4-x^{2}}}{2\pi ((1-\rho
^{2})^{2}-\rho (1+\rho ^{2})xy+\rho ^{2}(x^{2}+y^{2}))},
\end{equation*}
for $\left\vert x\right\vert ,\left\vert y\right\vert \leq 2,$ $\rho ^{2}<1.$
To see that $\int_{-2}^{2}f\left( x;y,\rho \right) dx\allowbreak
=\allowbreak 1$ for all $\left\vert y\right\vert \leq 2$ $and$ $\rho ^{2}<1$
is difficult hence to obtain sequence of polynomials orthogonal with respect
to it by Gram-Schmidt procedure is quite hard. As one can easily see this
density is a particular example of the relationship (\ref{dwumian}) with $
dB(x)\allowbreak =\allowbreak \frac{1}{2\pi }\sqrt{4-x^{2}}dx,$ $
b_{n}(x)\allowbreak =\allowbreak U_{n}(x/2),$ where $U_{n}(x)$ are the
Chebyshev polynomials of the second kind (for details see e.g. \cite
{Andrews1999})). One can notice that polynomials $b_{n}$ satisfy the
following 3-term recurrence:
\begin{equation*}
b_{n+1}(x)=xb_{n}(x)-b_{n-1}(x),
\end{equation*}
with $b_{-1}(x)\allowbreak =\allowbreak 0,$ $b_{0}(x)\allowbreak
=\allowbreak 1.$ First of all notice that if $\rho \allowbreak =\allowbreak 0
$ then we deal with trivial case. Hence let us assume that $0<\left\vert
\rho \right\vert <1.$ We have $\beta _{n}\allowbreak =\allowbreak 0$ and $
\hat{\beta}_{n}\allowbreak =\allowbreak 1$ and further $C\allowbreak
=\allowbreak \frac{(1-\rho ^{2})}{\rho ^{2}},$ $D\allowbreak =\allowbreak -
\frac{(1+\rho ^{2})y}{\rho },$ $E\allowbreak =\allowbreak (\frac{1-\rho ^{2}
}{\rho })^{2}+y^{2}.$ By direct computation we check that $\kappa
_{1}\allowbreak =\allowbreak \kappa _{2}\allowbreak =\allowbreak -\rho y$
and $\lambda _{2}\allowbreak =\allowbreak \rho ^{2}.$ Now inserting all of
the ingredients to equations (\ref{s1})-(\ref{s4}) we see that $\kappa
_{n}\allowbreak =\allowbreak -\rho y$ for $n\geq 1,$ $\lambda
_{n}\allowbreak =\allowbreak \rho ^{2}$ for all $n\geq 2$ , $\alpha
_{n}\allowbreak =\allowbreak 0,$ and $\hat{\alpha}_{n}\allowbreak
=\allowbreak 1$ for all $n\geq 1.$ Thus $a_{n}(x)\allowbreak =\allowbreak
b_{n}(x)-\rho yb_{n-1}(x)+\rho ^{2}b_{n-2}(x)$ for all $n\geq 2.$
\end{example}
Now let us simplify calculations by assuming that the measure $dB$ and the
polynomial $\left( x^{2}+Dx+E\right) $ are symmetric which implies that
polynomials orthogonal with respect to $dB$ (i.e. $b_{n})$ must contain only
either even or odd powers of $x$. Hence coefficients $\beta _{n}$ are equal
to zero for $n\geq 0$ and also that $D\allowbreak =\allowbreak 0.$
Consequently measure $dA$ must also be symmetric and by similar argument we
deduce that coefficients $\alpha _{n}=0$ for $n\geq 0.$ This results in the
fact that coefficients $\kappa _{n}$ are also zero for all $n\geq 0.$
As a result we have the following Lemma which is in fact a corollary of the
Proposition \ref{uog}.
\begin{lemma}
Suppose normalized, positive measures $dA$ and $dB$ are related to one
another by the relationship:
\begin{equation*}
dA\left( x\right) \allowbreak =\allowbreak \frac{C}{x^{2}+E}dB\left(
x\right) .
\end{equation*}
Let further respectively polynomial sequences $\left\{ a_{n}\right\} $ and $
\left\{ b_{n}\right\} $ orthogonal with respect to these measures satisfy
3-term recurrence (\ref{aa}) and (\ref{bb}). With $\beta _{n}\allowbreak
=\allowbreak 0$ for $n\geq 0.$ Then there exist a number sequences $\left\{
\lambda _{n}\right\} $ such that
\begin{equation}
a_{n}\left( x\right) \allowbreak =\allowbreak b_{n}\left( x\right) +\lambda
_{n}b_{n-2}\left( x\right) , \label{_2}
\end{equation}
and $\alpha _{n}\allowbreak =\allowbreak 0$ for $n\geq 0$. Moreover number
sequence $\left\{ \lambda _{n}\right\} $, satisfies the following second
order recursive equation for $n\geq 3:$
\begin{equation}
\lambda _{n+1}=\lambda _{n}+\hat{\beta}_{n-1}-\frac{\lambda _{n}}{\lambda
_{n-1}}\hat{\beta}_{n-3}, \label{eqn2}
\end{equation}
with $\lambda _{1}\allowbreak =\allowbreak 0,$ $\lambda _{2}\allowbreak
=\allowbreak \hat{\beta}_{0}+E-C,$ $\lambda _{3}=\hat{\beta}_{1}+E-\frac{E}{
C-E}\hat{\beta}_{0}.$
Coefficients $\hat{\alpha}_{n}$ are given by relationship:
\begin{equation*}
\hat{\alpha}_{n}=\frac{\lambda _{n+1}}{\lambda _{n}}\hat{\beta}_{n-2}.
\end{equation*}
\end{lemma}
\begin{proof}
We apply assumptions to the system of equations (\ref{s1}-\ref{s4}) getting:
\begin{eqnarray*}
\lambda _{n+1}+\hat{\alpha}_{n-1} &=&\lambda _{n}+\hat{\beta}_{n-1}, \\
\hat{\alpha}_{n-1}\lambda _{n-1} &=&\lambda _{n}\hat{\beta}_{n-3},
\end{eqnarray*}
from which we get (\ref{eqn2}). To get initial conditions we notice that $
a_{2}\left( x\right) \allowbreak =\allowbreak x^{2}-\hat{\beta}_{0}+\lambda
_{2}\allowbreak =\allowbreak x^{2}\allowbreak +\allowbreak E+(\lambda
_{2}-\beta _{0}-E),$ so from the relationships $\int_{\limfunc{supp}
B}a_{2}\left( x\right) dA\left( x\right) \allowbreak =\allowbreak 0$ and $
\int_{\limfunc{supp}B}(x^{2}+E)dA\left( x\right) \allowbreak \allowbreak
=\allowbreak C$ we get $C\allowbreak +\allowbreak (\lambda _{2}-\beta
_{0}-E)\allowbreak =\allowbreak 0.$ Now to get $\lambda _{3}$ we use
relationship: $\int_{\limfunc{supp}B}a_{1}\left( x\right) a_{3}\left(
x\right) dA\left( x\right) \allowbreak =\allowbreak 0,$ using on the way the
fact that $a_{3}\left( x\right) \allowbreak =\allowbreak b_{3}\left(
x\right) +\lambda _{3}x\allowbreak =\allowbreak x(x^{2}-\hat{\beta}
_{0})\allowbreak -\allowbreak \hat{\beta}_{1}x\allowbreak +\allowbreak
\lambda _{3}x\allowbreak =\allowbreak x^{3}\allowbreak +\allowbreak
x(\lambda _{3}-\hat{\beta}_{0}-\hat{\beta}_{1})\allowbreak $ and that $\int_{
\limfunc{supp}B}(x^{2}-\hat{\beta}_{0})(x^{2}+E)dA\left( x\right)
\allowbreak =\allowbreak 0.$ We have: $0\allowbreak =\allowbreak \int_{
\limfunc{supp}B}((x^{2}-\hat{\beta}_{0})(x^{2}+E)+(\lambda _{3}-\hat{\beta}
_{1}-E)x^{2}+\hat{\beta}_{0}E)dA\left( x\right) \allowbreak =\allowbreak
\int_{\limfunc{supp}B}((\lambda _{3}-\hat{\beta}_{1}-E)(x^{2}+E)+\hat{\beta}
_{0}E-E(\lambda _{3}-\hat{\beta}_{1}-E))dA\left( x\right) \allowbreak
=\allowbreak C(\lambda _{3}-\hat{\beta}_{1}-E)-E(\lambda _{3}-\hat{\beta}
_{1}-E-\hat{\beta}_{0})=0$.
\end{proof}
We will briefly illustrate this Lemma$\allowbreak $ by the following example.
\begin{example}[Jacobi polynomials revisited]
Let us consider the symmetric case i.e. assuming that parameters $\alpha $
and $\gamma $ are equal say to $a$. Then $\beta _{n}\allowbreak =\allowbreak
0$ and $\hat{\beta}_{n-1}\allowbreak =\allowbreak \frac{n(n+2a)}{
(2a+2n-1)(2a+2n+1)}.$ Parameters $C$ and $E$ are now equal to $-\frac{2a}{
2a+1}$ and $-1$ respectively. Hence
\begin{eqnarray*}
\lambda _{2} &=&\hat{\beta}_{0}+E-C=-\frac{2}{(2a+1)(2a+3)} \\
\lambda _{3} &=&\hat{\beta}_{1}+E-\frac{E}{C-E}\hat{\beta}_{0}=-\frac{6}{
(2a+3)(2a+5)}.
\end{eqnarray*}
Besides one can easily check that the sequence $\left\{ -\frac{n(n-1)}{
(2a+2n-1)(2a+2n-3)}\right\} $ satisfies (\ref{eqn2}). So $\lambda
_{n}\allowbreak =\allowbreak -\frac{n(n-1)}{(2a+2n-1)(2a+2n-3)},$ $n\geq 2.$
Hence we have:
\begin{equation*}
J_{n}^{\left( \alpha -1,\alpha -1\right) }\left( x\right) =J_{n}^{\left(
\alpha ,\alpha \right) }-\frac{n(n-1)}{(2a+2n-1)(2a+2n-3)}J_{n-2}^{\left(
\alpha ,\alpha \right) }.
\end{equation*}
In particular notice that for $\alpha \allowbreak =\allowbreak 1/2$ we have $
J_{n}^{(1/2,1/2)}\left( x\right) \allowbreak =\allowbreak U_{n}\left(
x\right) /2^{n}$ where $U_{n}$ are the Chebyshev polynomials of the second
kind, while $J_{n}^{(-1/2,-1/2)}\left( x\right) \allowbreak =\allowbreak
T_{n}\left( x\right) /2^{n-1}$ for $n\geq 2$ and $J_{n}^{(-1/2,-1/2)}\left(
x\right) \allowbreak =\allowbreak T_{n}\left( x\right) $ for $n\allowbreak
=\allowbreak 0,1,$ where $\allowbreak T_{n}\left( x\right) ,$ are the
Chebyshev polynomials of the first kind. Besides $\left[ -\frac{n(n-1)}{
(2a+2n-1)(2a+2n-3)}\right] _{\alpha =1/2}\allowbreak =\allowbreak -\frac{1}{4
}$ and we end up with well known relationship between Chebyshev polynomials
of the first and second kind:
\begin{equation*}
T_{n}\left( x\right) \allowbreak =\allowbreak (U_{n}(x)-U_{n-2}\left(
x\right) )/2.
\end{equation*}
\end{example}
\begin{remark}
Notice that we could have reached the result in the above mentioned example
by applying the procedure described in Section \ref{gl} twice once for
monomial $1-x$ and then for $1+x.$
\end{remark}
\begin{remark}
Notice also that one could invert relationship (\ref{_2}) and find
connection coefficients of polynomials $\left\{ b_{n}\right\} $ expressed in
terms of polynomials $\left\{ a_{n}\right\} .$ Like in the setting of
Corollary \ref{rozw} they would be expressed in terms of products of
coefficients $\left\{ \lambda _{n}\right\} $ (in fact either only with odd
or even numbers) and consequently obtain expansion similar (\ref{szereg})
(in fact involving polynomials $\left\{ b_{n}\right\} $ with even numbers).
\end{remark}
\subsection{Open problems}
\begin{itemize}
\item Is it possible to simplify equation (\ref{eqn2}) and reduce it to the
first order recursive equation?
\item Is it possible to simplify system of equations (\ref{s1}-\ref{s4}) and
reduce it to the problem of solving system of the first order recursive
equations?
\item More generally is it possible to solve general system of equations
presented in Proposition 1 of \cite{Szab2010} or at least deduce more
properties of coefficients $\left\{ c_{n}^{\left( j\right) }\right\} _{n\geq
1,1\leq j\leq r}$ not only that for $j>r$ they are zeros.
\item Is it possible give some properties of connection coefficients between
the two sets of orthogonal polynomials given the fact that orthogonalizing
measures are related by the known relationship $dA\left( x\right)
\allowbreak =\allowbreak F\left( x\right) dB\left( x\right) $ for functions $
F$ different from the reciprocal of a polynomial. It seems possible to
consider rational functions to start generalization.
\end{itemize}
\section{Proofs\label{dowody}}
\begin{proof}[Proof of Theorem \protect\ref{glowny}]
Noticing that the proof of Proposition 1, iii) does not require the measures
$dA\left( x\right) $ and $dB\left( x\right) $ to have densities we can apply
its assertion and deduce that if positive, normalized measures are related
by the relationship (\ref{jednomian}) then the polynomial sequences $\left\{
a_{n}\right\} $ and $\left\{ b_{n}\right\} $ orthogonal respectively with
respect to these measures are related by (\ref{zaleznosc}). Hence sequence $
\left\{ \kappa _{n}\right\} $ exists and consequently we have $a_{n}\left(
x\right) \allowbreak =\allowbreak b_{n}\left( x\right) +\kappa
_{n}b_{n-1}\left( x\right) .$ Remembering that sequences of polynomials $
\left\{ a_{n}\right\} $ and $\left\{ b_{n}\right\} $ are orthogonal and
satisfy the following 3-term recurrences:
\begin{eqnarray*}
a_{n+1}\left( x\right) \allowbreak &=&\allowbreak (x-\alpha _{n})a_{n}\left(
x\right) -\hat{\alpha}_{n-1}a_{n-1}\left( x\right) , \\
b_{n+1}\left( x\right) &=&\left( x-\beta _{n}\right) b_{n}\left( x\right) -
\hat{\beta}_{n-1}b_{n-1}\left( x\right) .
\end{eqnarray*}
So on one hand we have
\begin{eqnarray*}
xa_{n}\left( x\right) \allowbreak &=&\allowbreak a_{n+1}\left( x\right)
+\alpha _{n}a_{n}\left( x\right) +\hat{\alpha}_{n-1}a_{n-1}\allowbreak \\
&=&\allowbreak b_{n+1}\left( x\right) \allowbreak +\allowbreak (\kappa
_{n+1}+\alpha _{n}\kappa _{n})b_{n}\allowbreak +\allowbreak (\alpha
_{n}\kappa _{n}+\hat{\alpha}_{n-1})b_{n-1}\left( x\right) \allowbreak
+\allowbreak \hat{\alpha}_{n-1}\kappa _{n-1}b_{n-2}\left( x\right) .
\end{eqnarray*}
On the other we have:
\begin{eqnarray*}
xa_{n}\left( x\right) \allowbreak &=&\allowbreak xb_{n}\left( x\right)
+x\kappa _{n}b_{n-1}\left( x\right) \allowbreak \\
&=&\allowbreak b_{n+1}\left( x\right) \allowbreak +\allowbreak \left( \beta
_{n}+\kappa _{n}\right) b_{n}\left( x\right) \allowbreak +\allowbreak (\hat{
\beta}_{n-1}+\kappa _{n}\beta _{n-1})b_{n-1}\allowbreak +\allowbreak \kappa
_{n}\hat{\beta}_{n-2}b_{n-2}\left( x\right) .
\end{eqnarray*}
Hence we must have:
\begin{eqnarray*}
\kappa _{n+1}+\alpha _{n} &=&\beta _{n}+\kappa _{n}, \\
\alpha _{n}\kappa _{n}+\hat{\alpha}_{n-1} &=&\hat{\beta}_{n-1}+\kappa
_{n}\beta _{n-1}, \\
\hat{\alpha}_{n-1}\kappa _{n-1} &=&\kappa _{n}\hat{\beta}_{n-2}
\end{eqnarray*}
Now let us get $\alpha _{n}$ from the first of the equations
\begin{equation*}
\alpha _{n}\allowbreak =\allowbreak \beta _{n}+\kappa _{n}-\kappa _{n+1}.
\end{equation*}
We get further
\begin{equation*}
\hat{\alpha}_{n-1}\allowbreak =\allowbreak -\beta _{n}\kappa _{n}-\kappa
_{n}^{2}+\kappa _{n}\kappa _{n+1}+\hat{\beta}_{n-1}+\kappa _{n}\beta _{n-1}.
\end{equation*}
So finally we have:
\begin{equation*}
\kappa _{n-1}(-\beta _{n}\kappa _{n}-\kappa _{n}^{2}+\kappa _{n}\kappa
_{n+1}+\hat{\beta}_{n-1}+\kappa _{n}\beta _{n-1})\allowbreak =\allowbreak
\kappa _{n}\hat{\beta}_{n-2}
\end{equation*}
dividing both sides by $\kappa _{n}\kappa _{n-1}$ we get:
\begin{equation*}
\kappa _{n+1}=\kappa _{n}+\frac{\hat{\beta}_{n-2}}{\kappa _{n-1}}-\frac{\hat{
\beta}_{n-1}}{\kappa _{n}}+\beta _{n}-\beta _{n-1}
\end{equation*}
Now notice that we can rearrange terms on both sides of this equation in the
following way:
\begin{equation*}
\kappa _{n+1}+\frac{\hat{\beta}_{n-1}}{\kappa _{n}}-\beta _{n}=\kappa _{n}+
\frac{\hat{\beta}_{n-2}}{\kappa _{n-1}}-\beta _{n-1},
\end{equation*}
proving that quantity $\kappa _{n}+\frac{\hat{\beta}_{n-2}}{\kappa _{n-1}}
-\beta _{n-1}$ does not depend on $n$ and is equal to $\kappa
_{2}\allowbreak +\allowbreak \frac{\hat{\beta}_{0}}{\kappa _{1}}\allowbreak
-\allowbreak \beta _{1}.$
We can easily find this quantity by finding directly quantities $\kappa _{1}$
and $\kappa _{2}.$
Naturally we have $\kappa _{0}=1$. Remembering that $dB\left( x\right)
=\allowbreak (d+cx)dA\left( x\right) ,$ that
\begin{equation*}
b_{n+1}\left( x\right) =\left( x-\beta _{n}\right) b_{n}\left( x\right) -
\hat{\beta}_{n}b_{n-1}\left( x\right)
\end{equation*}
and since $\int_{\limfunc{supp}A}a_{1}\left( x\right) dA\left( x\right)
\allowbreak =\allowbreak 0$ we must have
\begin{equation*}
1\allowbreak =\int_{\limfunc{supp}A}dA\left( x\right) \allowbreak
=\allowbreak \int_{\limfunc{supp}A}\frac{C}{\left( D+x\right) }dB\left(
x\right) .
\end{equation*}
Now since $a_{1}\left( x\right) \allowbreak =\allowbreak b_{1}\left(
x\right) \allowbreak +\allowbreak \kappa _{1}$ we have\
\begin{eqnarray*}
0\allowbreak &=&\allowbreak \int_{\limfunc{supp}A}\frac{C\left( b_{1}\left(
x\right) +\kappa _{1}\right) }{\left( D+x\right) }dB\left( x\right)
\allowbreak \\
&=&\allowbreak C+(\kappa _{1}-\beta _{0}-D)\int \frac{C}{x+D}dB\left(
x\right) \allowbreak =\allowbreak C+\kappa _{1}-\beta _{0}-D,
\end{eqnarray*}
So
\begin{equation*}
\kappa _{1}\allowbreak =\allowbreak \beta _{0}+D-C.
\end{equation*}
To find $\kappa _{2}$ we use the fact that $a_{2}\left( x\right) \allowbreak
=\allowbreak b_{2}\left( x\right) +\kappa _{2}b_{1}\left( x\right) .$ Hence
we have:
\begin{eqnarray*}
0\allowbreak &=&\allowbreak C\int_{\limfunc{supp}A}\frac{\left( b_{2}\left(
x\right) +\kappa _{2}b_{1}\left( x\right) \right) }{\left( D+x\right) }
dB\left( x\right) \allowbreak \\
&=&\allowbreak \allowbreak C\int_{\limfunc{supp}A}\frac{\left( (-D-\beta
_{1}+\kappa _{2})(b_{1}\left( x\right) +\kappa _{1}-\kappa _{1})-\hat{\beta}
_{0}\right) }{\left( D+x\right) }dB\left( x\right) \allowbreak \\
&=&\allowbreak C\int_{\limfunc{supp}A}\frac{\left( (D+\beta _{1}-\kappa
_{2})\kappa _{1}-\hat{\beta}_{0}\right) }{\left( D+x\right) }dB\left(
x\right) \allowbreak =\allowbreak \left( (D+\beta _{1}-\kappa _{2})\kappa
_{1}-\hat{\beta}_{0}\right) .
\end{eqnarray*}
Hence we see that $\kappa _{2}\allowbreak +\allowbreak \frac{\hat{\beta}_{0}
}{\kappa _{1}}\allowbreak -\allowbreak \beta _{1}\allowbreak =\allowbreak D.$
\end{proof}
\begin{proof}[Proof of Proposition \protect\ref{uog}]
Assuming that both sequences of polynomials i.e. $\left\{ a_{n}\right\} $
and $\left\{ b_{n}\right\} $ are orthogonal we have on one hand:
\begin{eqnarray*}
xa_{n}\left( x\right) \allowbreak &=&\allowbreak a_{n+1}+\alpha
_{n}a_{n}\left( x\right) +\hat{\alpha}_{n-1}a_{n-1}\left( x\right)
\allowbreak =\allowbreak \\
&=&\allowbreak b_{n+1}\left( x\right) \allowbreak +\allowbreak \left( \kappa
_{n+1}+\alpha _{n}\right) b_{n}\left( x\right) \allowbreak +\allowbreak
\left( \lambda _{n+1}+\alpha _{n}\kappa _{n}+\hat{\alpha}_{n-1}\right)
b_{n-1}\left( x\right) \allowbreak \\
&&+\allowbreak \left( \alpha _{n}\lambda _{n}+\hat{\alpha}_{n-1}\kappa
_{n-1}\right) b_{n-2}\left( x\right) \allowbreak +\allowbreak \hat{\alpha}
_{n-1}\lambda _{n-1}b_{n-3}\left( x\right) .
\end{eqnarray*}
and on the other:
\begin{eqnarray*}
xa_{n}\left( x\right) \allowbreak &=&\allowbreak x\left( b_{n}\left(
x\right) +\kappa _{n}b_{n-1}\left( x\right) +\lambda _{n}b_{n-2}\left(
x\right) \right) \allowbreak \allowbreak \\
&=&\allowbreak b_{n+1}\allowbreak +\allowbreak \left( \beta _{n}+\kappa
_{n}\right) b_{n}\left( x\right) \allowbreak +\allowbreak \left( \hat{\beta}
_{n-1}+\kappa _{n}\beta _{n-1}+\NEG{\lambda}_{n}\right) b_{n-1}\left(
x\right) \allowbreak +\allowbreak \\
&&\left( \kappa _{n}\hat{\beta}_{n-2}+\lambda _{n}\beta _{n-2}\right)
b_{n-2}\left( x\right) +\allowbreak \lambda _{n}\hat{\beta}
_{n-3}b_{n-3}\left( x\right) .
\end{eqnarray*}
Comparing expressions by $b_{n},$ $b_{n-1},$ $b_{n-2}$ and $b_{n-3}$ we get
equations (\ref{s1}-\ref{s4}).
\end{proof}
\end{document} | math | 45,906 |
\begin{document}
\begin{abstract}
We investigate a new family of locally harmonic Maass forms which correspond to periods of modular forms. They transform like negative weight modular forms and are harmonic apart from jump singularities along infinite geodesics. Our main result is an explicit splitting of the new locally harmonic Maass forms into a harmonic part and a locally polynomial part
that captures the jump singularities. As an application, we obtain finite rational formulas for suitable linear combinations of periods of meromorphic modular forms associated to positive definite binary quadratic forms.
\end{abstract}
\title{Locally harmonic Maass forms and periods of meromorphic modular forms}
\section{Introduction and statement of results}
\subsection{Locally harmonic Maass forms and cycle integrals} In the early 2000s, Zwegers \cite{zwegers} made the groundbreaking discovery that Ramanujan's mock theta functions, whose precise automorphic nature had been a long-standing conundrum, could be viewed as the holomorphic parts of \emph{harmonic weak Maass forms} of weight $1/2$ whose shadows are unary theta functions of weight $3/2$. The theory of harmonic weak Maass forms was developed systematically around the same time by Bruinier and Funke \cite{bruinierfunke}. Since then, it has become a vital area of research in number theory and has found many fascinating applications, for example to the singular theta correspondence and Borcherds products \cite{bruinierhabil, bruinierfunke}, the partition function and its variants \cite{ahlgrenandersen, bruinieronoalgebraic}, special values of $L$-functions of elliptic curves \cite{bruinieronoheegner}, and CM values of higher Green functions \cite{bringmannkanevonpippich, bruinierehlenyang, li}.
More recently, Bringmann, Kane, and Kohnen \cite{bringmannkanekohnen} constructed a new type of harmonic weak Maass forms which have jump singularities along certain geodesics in the upper half-plane $\mathbb{H}$, and hence are called \emph{locally harmonic Maass forms} (see also \cite{brikavia, crawford, crawfordfunke, hoevel}). Specifically, for $k \in \mathbb{Z}$ with $k \geq 2$ the authors of \cite{bringmannkanekohnen} associated to each non-square discriminant $D > 0$ the function ($\tau = u+iv \in \mathbb{H}$)
\[
\mathcal{F}_{1-k,D}(\tau) := \frac{(-1)^k D^{\frac{1}{2}-k}}{\binom{2k-2}{k-1}\pi}\sum_{Q = [a,b,c]\in \mathcal{Q}_{D}}\sgn\left(a|\tau|^{2}+bu+c\right)Q(\tau,1)^{k-1}\psi\left(\frac{D^{2}v}{|Q(\tau,1)|^{2}} \right),
\]
where $\mathcal{Q}_{D}$ denotes the set of all (positive definite if $D < 0$) integral binary quadratic forms $Q(x,y) = ax^{2}+bxy + cy^{2}$ of discriminant $D = b^{2}-4ac$, and $\psi(v) := \frac{1}{2}\int_{0}^{v}t^{k-\frac{3}{2}} (1-t)^{-\frac{1}{2}}dt$ is a special value of the incomplete $\beta$-function. The function $\mathcal{F}_{1-k,D}(\tau)$ transforms like a modular form of negative even weight $2-2k$ for $\Gamma := \mathrm{SL}_{2}(\mathbb{Z})$, it is bounded at the cusp, and it is harmonic on $\mathbb{H}$ up to jump singularities along the exceptional set
\[
E_{D} :=\{\tau=u+iv \in \mathbb{H} \,:\, a|\tau|^{2}+bu+c = 0 \, , \, [a,b,c] \in \mathcal{Q}_{D}\}.
\]
Note that $E_{D}$ is a union of semi-circles centered at the real line if $D > 0$ is not a square. A special feature of the locally harmonic Maass form $\mathcal{F}_{1-k,D}(\tau)$ is the fact that its images under the two differential operators
\[
\xi_{2-2k} := 2iv^{2-2k}\overline{\frac{\partial}{\partial \overline{\tau}}}, \qquad \mathcal{D}^{2k-1} := \left(\frac{1}{2\pi i}\frac{\partial}{\partial \tau} \right)^{2k-1},
\]
are non-zero multiples of the weight $2k$ cusp form
\begin{align}\label{eq fkD}
f_{k,D}(\tau) := \frac{D^{k-\frac{1}{2}}}{\pi}\sum_{Q \in \mathcal{Q}_{D}}Q(\tau,1)^{-k}.
\end{align}
In contrast, it is impossible for a harmonic weak Maass form to map to a non-zero multiple of the same cusp form under both operators $\xi_{2-2k}$ and $\mathcal{D}^{2k-1}$. The cusp form $f_{k,D}(\tau)$ can be characterized by the fact that the Petersson inner product $\langle f,f_{k,D}\rangle$ of a cusp form $f(\tau)$ of weight $2k$ with $f_{k,D}(\tau)$ is a certain multiple of the $D$-th trace of cycle integrals
\[
\tr_{D}(f) := \sum_{Q \in \mathcal{Q}_{D}/\Gamma}\int_{\Gamma_{Q}\backslash S_{Q}}f(z)Q(z,1)^{k-1}dz
\]
of $f(\tau)$. Here $\Gamma_{Q}$ denotes the stabilizer of $Q$ in $\Gamma$ and $S_{Q}$ for $Q = [a,b,c]$ is the semi-circle consisting of all $\tau = u+iv \in \mathbb{H}$ with $a|\tau|^{2}+bu+c = 0$. In this sense, the locally harmonic Maass forms $\mathcal{F}_{1-k,D}(\tau)$ correspond to the (traces of) cycle integrals of cusp forms.
\subsection{Locally harmonic Maass forms and periods} In the present article we are concerned with the construction of a new family of locally harmonic Maass forms $\mathcal{H}_{1-k,n}(\tau)$, for $k \in \mathbb{Z}$ with $k \geq 2$ and integral $0 \leq n \leq 2k-2$, which correspond to the \emph{periods} of cusp forms, in a sense which will become apparent in a moment. In order to give the definition of $\mathcal{H}_{1-k,n}(\tau)$, for $k,\ell \in \mathbb{Z}$ with $k\geq 2$ we consider Petersson's Poincar\'e series ($z, \tau = u+iv \in \mathbb{H}$)
\begin{align}\label{eq Peterssons Poincare series}
H_{k,\ell}(z,\tau) := \sum_{M \in \Gamma}v^{k+\ell}(z-\tau)^{\ell-k}(z-\overline{\tau})^{-\ell-k}\Bigl|_{-2\ell,\tau}M,
\end{align}
with the usual slash operator applied in the $\tau$-variable. It transforms like a modular form of weight $2k$ in $z$ and of weight $-2\ell$ in $\tau$. We will be particularly interested in the case $\ell = k-1$. Then the function $H_{k,k-1}(z,\tau)$ is a meromorphic modular form of weight $2k$ in $z$ which has poles precisely at the $\Gamma$-translates of $\tau$ and decays like a cusp form towards $i\infty$. Furthermore, it transforms like a modular form of weight $2-2k$ in $\tau$ and is harmonic on $\mathbb{H} \setminus \Gamma z$. We can now make the following definition.
\begin{definition}
For $k \in \mathbb{Z}$ with $k \geq 2$ and integral $0 \leq n \leq 2k-2$ we define the function
\begin{align}\label{definition curlyH}
\mathcal{H}_{1-k,n}(\tau):= \frac{(2i)^{2k-2}}{2\pi}\int_{0}^{\infty}H_{k,k-1}(iy,\tau)y^{n}dy.
\end{align}
If $\tau \in \bigcup_{M \in \Gamma}M(i \mathbb{R}_{+})$ then the integral is defined using the Cauchy principal value (see \cite{ls}, Section~3.5).
\end{definition}
In other words, the function $\mathcal{H}_{1-k,n}(\tau)$ is essentially the $n$-th period of the meromorphic modular form $z \mapsto H_{k,k-1}(z,\tau)$. In \cite{ls} we showed that the locally harmonic Maass form $\mathcal{F}_{1-k,D}(\tau)$ from \cite{bringmannkanekohnen} can be viewed as the $D$-th trace of cycle integrals of the function $z \mapsto H_{k,k-1}(z,\tau)$, which inspired the above definition of $\mathcal{H}_{1-k,n}(\tau)$.
It is clear from the construction that $\mathcal{H}_{1-k,n}(\tau)$ transforms like a modular form of weight $2-2k$ for $\Gamma$ and is harmonic on $\mathbb{H} \setminus\bigcup_{M \in \Gamma}M(i \mathbb{R}_{+})$. Further, it follows from Theorem~\ref{theorem splitting} below that $\mathcal{H}_{1-k,n}(\tau)$ is of moderate growth at the cusp and has jump singularities along the $\Gamma$-translates of the positive imaginary axis $i\mathbb{R}_{+}$. In particular, the function $\mathcal{H}_{1-k,n}(\tau)$ defines a locally harmonic Maass form of weight $2-2k$ for $\Gamma$ with exceptional set $E_{1} = \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$ in the sense of \cite{bringmannkanekohnen}.
\begin{figure}
\caption{The set $E_1$ of singularities of $\mathcal{H}
\label{figure}
\end{figure}
We first determine the images of $\mathcal{H}_{1-k,n}(\tau)$ under the differential operators $\xi_{2-2k}$ and $\mathcal{D}^{2k-1}$. To state the result, we let $E_{2k}(\tau)$ be the usual normalized Eisenstein series of weight $2k$, and for $0 \leq n \leq 2k-2$ we let $R_{n}(\tau)$ be the weight $2k$ cusp form which is characterized by the fact that its Petersson inner product $\langle f,R_{n} \rangle$ with a cusp form $f(\tau)$ of weight $2k$ equals the $n$-th period
\begin{align}\label{definition periods}
r_{n}(f) := \int_{0}^{\infty}f(iy)y^{n}dy
\end{align}
of $f(\tau)$. Note that the periods satisfy the symmetry $r_n(f) = (-1)^k r_{2k-2-n}(f)$. The cusp form $R_n(\tau)$ is related to $\mathcal{H}_{1-k,n}(\tau)$ in the following way.
\begin{proposition}\label{proposition diffops} For $\tau \in \mathbb{H} \setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$ we have
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,n}(\tau) &= -R_{n}(\tau), \\
\mathcal{D}^{2k-1}\mathcal{H}_{1-k,n}(\tau) &= (-1)^{n+1}\frac{(2k-2)!}{(4\pi)^{2k-1}}R_{n}(\tau) -\left((-1)^k\delta_{n = 0}+\delta_{n=2k-2}\right)\frac{(2k-2)!}{(2\pi)^{2k-1}}E_{2k}(\tau).
\end{align*}
\end{proposition}
We refer the reader to Section~\ref{section proof proposition diffops} for the proof of Proposition~\ref{proposition diffops}. The above proposition implies that $\mathcal{H}_{1-k,n}(\tau)$ can be written as a sum of the holomorphic and non-holomorphic Eichler integrals of $R_{n}(\tau)$ (and of $E_{2k}(\tau)$ if $n = 0$ or $n = 2k-2$) and a locally polynomial part, which captures the singularities of $\mathcal{H}_{1-k,n}(\tau)$. Recall that the holomorphic and non-holomorphic Eichler integrals of a cusp form $f(\tau) = \sum_{n=1}^{\infty}c_{f}(n)e(n\tau)$ of weight $2k$ (with $e(x):= e^{2\pi i x}$ for $x \in \mathbb{C}$) are defined by
\begin{align}\label{eq eichler integrals}
\begin{split}
\mathcal{E}_{f}(\tau) &:= \frac{(-2\pi i)^{2k-1}}{(2k-2)!}\int_{\tau}^{i\infty}f(z)(z-\tau)^{2k-2}dz= \sum_{n\geq 1}\frac{c_{f}(n)}{n^{2k-1}}e(n\tau), \\
f^{*}(\tau) &:= (-2i)^{1-2k}\int_{-\overline{\tau}}^{i\infty}\overline{f(-\overline{z})}(z+\tau)^{2k-2}dz = -\sum_{n\geq 1}\frac{\overline{c_{f}(n)}}{(4\pi n)^{2k-1}}\Gamma(2k-1, 4\pi nv)e(-n\tau),
\end{split}
\end{align}
where $\Gamma(s,x):=\int_{x}^\infty e^{-t} t^{s-1}dt$ is the incomplete Gamma function. They satisfy
\begin{align}\label{eq diffops eichler}
\xi_{2-2k}f^{*}(\tau) = f(\tau), \qquad \mathcal{D}^{2k-1}f^{*}(\tau) = 0, \qquad \xi_{2-2k}\mathcal{E}_{f}(\tau) = 0, \qquad \mathcal{D}^{2k-1}\mathcal{E}_{f}(\tau) = f(\tau).
\end{align}
By using the series expansions on the right-hand sides of \eqref{eq eichler integrals}, we can extend the definitions of the Eichler integrals to the Eisenstein series $E_{2k}(\tau)$. The main result of this work is the following splitting of $\mathcal{H}_{1-k,n}(\tau)$.
\begin{theorem}\label{theorem splitting}
We have the decomposition
\begin{align}\begin{split}\label{split}
\mathcal{H}_{1-k,n}(\tau) &= \mathcal{P}_{1-k,n}(\tau) + (-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau) \\
&\quad-((-1)^k\delta_{n=0}+\delta_{n=2k-2})\frac{(2k-2)!}{(2\pi)^{2k-1}}\mathcal{E}_{E_{2k}}(\tau),
\end{split}
\end{align}
where $\mathcal{P}_{1-k,n}(\tau)$ is locally a polynomial on the connected components of $\mathbb{H} \setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$. It is explicitly given for $\tau\in\mathbb{H} \setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$ by
\begin{align*}
\mathcal{P}_{1-k,n}(\tau) &= r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}\mathbb{B}_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}\mathbb{B}_{2k-1-n}(\tau)
\\
&\quad +\frac{i^{1-n}}{2}\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac>0, \mathbb{R}e(M\tau)< 0}}(\tau^n -(-1)^n \tau^{2k-2-n})\Bigl |_{2-2k}M.
\end{align*}
Here $\mathbb{B}_{m}(\tau)$ is the $1$-periodic function on $\mathbb{H}$ which agrees with the Bernoulli polynomial $B_{m}(\tau)$ for $0 < u < 1$ and
$r_{n}(E_{2k})$ is the $n$-th period of the Eisenstein series $E_{2k}(\tau)$ given in \eqref{periodEisenstein}. \\
If there is a matrix $M\in\Gamma$ with $\mathbb{R}e(M\tau)=0$, then $\mathcal{P}_{1-k,n}(\tau)$ is given by the average value
\[
\lim_{\varepsilon\rightarrow 0+}\frac{1}{2}\big(\mathcal{P}_{1-k,n}(M^{-1}(M\tau - \varepsilon))+\mathcal{P}_{1-k,n}(M^{-1}(M\tau + \varepsilon))\big).
\]
\end{theorem}
\begin{remark}Note that the sum in the second line of $\mathcal{P}_{1-k,n}(\tau)$ is finite for every $\tau\in\mathbb{H}$ and vanishes if $\mathrm{Im}(\tau) > \frac{1}{2}$. Indeed, the condition $ac > 0$ for $M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma$ implies that $M^{-1}(i\mathbb{R}_+)$ is a semi-circle of radius $\frac{1}{2ac}$ centered at the real line, and $\mathbb{R}e(M \tau) < 0$ means that $\tau$ lies in the bounded component of $\mathbb{H}\setminus M^{-1}(i\mathbb{R}_+)$. See also Figure 1.
\end{remark}
Our proof of the above theorem is quite different from the proof of the analogous splitting of the locally harmonic Maass form $\mathcal{F}_{1-k,D}(\tau)$ given in \cite{bringmannkanekohnen}. As the main step, we will employ an explicit formula for the period polynomial of $R_{n}(\tau)$ due to Kohnen and Zagier \cite{kohnenzagierrationalperiods} in order to directly show the modularity of the expression on the right-hand side of \eqref{split}. We refer the reader to Section~\ref{section proof theorem splitting} for the details of the proof.
\subsection{Periods of meromorphic modular forms} As an application of the above results, we study the periods of certain meromorphic modular forms. Namely, for a fixed positive definite quadratic form $P \in \mathcal{Q}_{d}$ of discriminant $d < 0$ we consider the function
\begin{align}\label{definition fkP}
f_{k,P}(z):= \frac{|d|^{k-\frac{1}{2}}}{\pi}\sum_{Q \in [P]}Q(z,1)^{-k},
\end{align}
where $[P]$ denotes the class of $P$ in $\mathcal{Q}_{d}/\Gamma$. We let
\[
f_{k,d}(z) := \sum_{P \in \mathcal{Q}_{d}/\Gamma}f_{k,P}(z) = \frac{|d|^{k-\frac{1}{2}}}{\pi}\sum_{Q \in \mathcal{Q}_{d}}Q(z,1)^{-k}.
\]
Notice the analogy with the definition of the cusp form $f_{k,D}(\tau)$ in \eqref{eq fkD} for $D > 0$. The function $f_{k,P}(z)$ transforms like a modular form of weight $2k$ for $\Gamma$ and decays like a cusp form towards $i\infty$. Moreover, it is meromorphic on $\mathbb{H}$ and has poles of order $k$ precisely at the $\Gamma$-translates of the CM point $\tau_{P}$ defined by $P(\tau_{P},1) = 0$.
It was shown in \cite{anbs, anbms, ls} that certain linear combinations of geodesic cycle integrals of $f_{k,P}(z)$ are rational. Motivated by these results, in the present work we investigate the rationality of the periods
\[
r_{n}(f_{k,P}) = \int_{0}^{\infty}f_{k,P}(iy)y^{n}dy
\]
for $0 \leq n \leq 2k-2$, where the integral is defined using the Cauchy principal value as in \cite{ls}, Section~3.5, if a pole of $f_{k,P}$ lies on the positive imaginary axis. To get a first idea how these periods look, we consider the quadratic form $P = [1,1,1]$ of discriminant $d=-3$. In this case, there is only one class in $\mathcal{Q}_{-3}/\Gamma$, so we have $f_{k,[1,1,1]} = f_{k,-3}$. Numerical integration yields the following values of the periods of $f_{k,-3}$ for small values of $k$.
\begin{align*}
\renewcommand{1.2}{1.2}
\begin{array}{|c||c|c|c|c|c|c|c|}
\hline
n & 0 & 1 & 2 & 3 & 4 & 5 & 6\\
\hline \hline
r_n(f_{2,-3}) & -2.05670 & -2 & -2.05670 & - & - & - & -\\
\hline
r_n(f_{3,-3}) & -3.25653 & -1.5 & 0 & 1.5 & 3.25652 & - & -\\
\hline
r_n(f_{4,-3}) & -6.76949 & -2.22222 & 0 & 0.66666 & 0 & -2.22222 & -6.76949\\
\hline
r_n(f_{5,-3}) & -15.65457 & -4.08333 &0 & 0.66666 & 0 & -0.66666 & 0 \\
\hline
r_n(f_{6,-3}) & -38.31573 & -8.36729 &0 & 0.89204 & 0 & -0.48637 & 0\\
\hline
r_n(f_{7,-3}) & -97.17273 & -18.33333 &0 & 1.4 & 0 & -0.5 & 0\\
\hline
\end{array}
\end{align*}
We notice that the even periods of $f_{k,-3}(z)$ for $0 < n < 2k-2$ seem to vanish and the odd periods for $k \neq 6$ appear to be rational numbers, but the outer periods $r_{0}(f_{k,-3})$ and $r_{2k-2}(f_{k,-3})$ and the odd periods of $f_{6,-3}(z)$ do not seem to be particularly nice. In order to state the result explaining these observations, following \cite{kohnenzagierrationalperiods} we introduce the functions
\[
f_{k,P}^{+}(z) := \frac{1}{2}\left(f_{k,P}(z)+f_{k,P'}(z)\right), \qquad f_{k,P}^{-}(z):= \frac{i}{2}\left(f_{k,P}(z)-f_{k,P'}(z) \right),
\]
where we put $P' := [a,-b,c]$ for $P = [a,b,c]$.
\begin{theorem}\label{theorem rationality}
\begin{enumerate}
\item For $0<n<2k-2$ we have
\[
r_n\left(f_{k,P}^{\varepsilon}\right)=0,
\]
where $\varepsilon$ is the sign of $(-1)^{n}$.
\item The outer periods of $f_{k,P}^{+}(z)$ are given by
\[
r_0(f_{k,P}^{+}) = (-1)^{k}r_{2k-2}(f_{k,P}^{+}) = -\frac{|d|^{k-\frac12}\zeta_P(k)}{2(2k-1)|\overline{\Gamma}_P|\zeta(2k)},
\]
where $\zeta_{P}(s) := \sum_{(x,y) \in \mathbb{Z}^2 \setminus \{(0,0)\}}P(x,y)^{-s}$ is the Epstein zeta function associated to $P$, and $\overline{\Gamma}_P$ denotes the stabilizer of $P$ in $\overline{\Gamma} := \Gamma/\{\pm 1\}$.
\item Let $a_{n} \in \mathbb{Q}$ for odd $0 < n < 2k-2$ be coefficients such that
\[
\sum_{\substack{0 < n < 2k-2 \\ n \text{ odd}}}a_{n}R_{n}(\tau) = 0
\]
in $S_{2k}$. Then the linear combination
\[
\sum_{\substack{0 < n < 2k-2 \\ n \text{ odd}}}a_{n}r_{n}(f_{k,P})
\]
of odd periods of $f_{k,P}(z)$ is rational. Similarly, if a rational linear combination of the cusp forms $R_n$ with even $0 < n < 2k-2$ vanishes in $S_{2k}$, then the corresponding linear combination of even periods of $f_{k,P}(z)$ is in $i\mathbb{Q}$.
\end{enumerate}
\end{theorem}
The proof of Theorem~\ref{theorem rationality} will be given in Section~\ref{section proof rationality}. A nice feature of the proof is that it also yields an exact rational formula for the linear combinations of periods of $f_{k,P}(z)$ considered in item (3) of Theorem~\ref{theorem rationality}, compare Theorem~\ref{theorem rn formula} below. In contrast, the algebraic nature of the special values $\zeta_P(k)$ appearing in the outer periods of $f_{k,P}^{+}(z)$ is more mysterious. For example, for $k = 3$ and $P = [1,1,1]$ we have $\zeta_P(3) = 6\zeta(3)L(\chi_{-3},3) = \frac{8\pi^3}{27\sqrt{3}}\zeta(3)$ (compare \cite{zagierzetafunctions}, Proposition 3), which is expected to be transcendental.
We will now give an example to illustrate item (3) of Theorem~\ref{theorem rationality}.
\begin{example}
We first notice that for $k = 2,3,4,5,$ and $7$, the space of cusp forms of weight $2k$ is trivial. Hence, Theorem~\ref{theorem rationality} shows that in these cases the odd periods of $f_{k,P}(z)$ are rational and the even periods for $0 < n < 2k-2$ are in $i\mathbb{Q}$ for all positive definite forms $P$, in accordance with the numerical values given in the above table. More generally, Cohen \cite{cohen} has found relations between the cusp forms $R_{n}(\tau)$ in any weight, for example
\[
\sum_{\substack{0 < n \leq j-1 \\ n \text{ odd}}}\binom{j}{n}(-1)^{\frac{n-1}{2}}R_{n}(\tau)+\sum_{\substack{j \leq n < 2k-2 \\ n \text{ odd}}}\binom{2k-2-j}{n-j}(-1)^{\frac{n-1}{2}}R_{n}(\tau) = 0
\]
for all $0 \leq j \leq 2k-2$. Relations of this kind can be proved using the Eichler-Shimura isomorphism. From Theorem~\ref{theorem rationality} we obtain that the linear combinations
\[
\sum_{\substack{0 < n \leq j-1 \\ n \text{ odd}}}\binom{j}{n}(-1)^{\frac{n-1}{2}}r_{n}(f_{k,P})+\sum_{\substack{j \leq n < 2k-2 \\ n \text{ odd}}}\binom{2k-2-j}{n-j}(-1)^{\frac{n-1}{2}}r_{n}(f_{k,P})
\]
of odd periods of $f_{k,P}(z)$ are rational for all $0 \leq j \leq 2k-2$ and all $P$. For instance, for $k = 6, d = -3$, and $j = 4$, we find that the linear combination
\[
10r_{1}(f_{6,-3}) - 24r_{3}(f_{6,-3}) + 6r_{5}(f_{6,-3})
\]
is rational. Indeed, plugging in the numerical values given in the above table, we find that this linear combination of periods is numerically close to the integer $-108$. Using our exact formula for the periods of $f_{k,P}(z)$ given in Theorem~\ref{theorem rn formula} below one can show that this really is the correct value.
\end{example}
We will explain the results from Theorem~\ref{theorem rationality} by using the following connection between the periods of $f_{k,P}(z)$ and special values of (derivatives of) the locally harmonic Maass forms $\mathcal{H}_{1-k,n}(\tau)$.
\begin{proposition}\label{proposition special value}
We have
\[
r_{n}(f_{k,P}) = \frac{(-1)^{k-1}|d|^\frac{k-1}{2}}{2^{k-2}(k-1)!|\overline{\Gamma}_P|}R_{2-2k}^{k-1}\mathcal{H}_{1-k,n}(\tau_{P}),
\]
where $\tau_{P} \in \mathbb{H}$ denotes the CM point characterized by $P(\tau_{P},1) = 0$, and $R_{2-2k}^{k-1} := R_{-2}\circ \dots \circ R_{2-2k}$ is an iterated version of the Maass raising operator $R_{\kappa} := 2i\frac{\partial}{\partial \tau}+\kappa v^{-1}$.
\end{proposition}
The proof of the above proposition can be found in Section~\ref{section proof rationality}. In Theorem~\ref{theorem rn formula} below we will apply the iterated raising operator $R_{2-2k}^{k-1}$ to the splitting of $\mathcal{H}_{1-k,n}(\tau)$ from Theorem~\ref{theorem splitting}. Together with Proposition~\ref{proposition special value} we obtain an explicit formula for the periods $r_{n}(f_{k,P})$, which we will then use to prove Theorem~\ref{theorem rationality}.
Eventually, we remark that the methods of this work can be used to study the rationality of periods of certain linear combinations of the meromorphic modular forms $f_{k,P}(z)$, similar to \cite{ls}. For example, in analogy to Theorem~2.4 from \cite{ls}, one can show that certain linear combinations of Hecke-translates of $f_{k,P}(z)$ have rational periods.
We start with a section on the necessary preliminaries. In the remaining sections, we give the proofs of the above results.
\section{Preliminaries}\label{section preliminaries}
\subsection{Derivatives of Eichler integrals}
The holomorphic and the non-holomorphic Eicher integrals defined in \eqref{eq eichler integrals} are related by the iterated raising operator $R_{2-2k}^{k-1} = R_{-2}\circ \dots \circ R_{2-2k}$, with $R_{\kappa} = 2i\frac{\partial}{\partial \tau} + \kappa v^{-1}$, as follows.
\begin{proposition}\label{proposition eichler integral relations}
For any holomorphic modular form $f \in M_{2k}$ we have the relation
\[
R_{2-2k}^{k-1}f^*(\tau) = -\frac{(2k-2)!}{(4\pi)^{2k-1}}\overline{R_{2-2k}^{k-1}\mathcal{E}_f(\tau)}.
\]
\end{proposition}
\begin{proof}
Using the Fourier expansions of the Eichler integrals given in \eqref{eq eichler integrals} we see that the claim is equivalent to
\begin{align}\label{eq eichler integrals simplified}
R_{2-2k}^{k-1}\left(\Gamma(2k-1,4\pi n v)e(-n\tau)\right) = (2k-2)! \overline{R_{2-2k}^{k-1}e(n\tau)}
\end{align}
for all $n \geq 1$. Following \cite{bruinierhabil}, Section 1.3, we consider the function
\[
\mathcal{W}_{\kappa,s}(y) := |y|^{-\kappa/2}W_{\frac{\kappa}{2}\sgn(y),s-\frac{1}{2}}(|y|), \qquad (\kappa \in \mathbb{R}, \ s \in \mathbb{C}, \ y \in \mathbb{R} \setminus \{0\}),
\]
where $W_{\nu,\mu}(y)$ is the usual $W$-Whittaker function. At $s = 1-\frac{\kappa}{2}$ it simplifies to
\[
\mathcal{W}_{\kappa,1-\frac{\kappa}{2}}(y) = \begin{cases}
e^{-y/2}, & \text{if $y > 0$}, \\
e^{-y/2}\Gamma(1-\kappa,|y|), & \text{if $y < 0$}.
\end{cases}
\]
In particular, \eqref{eq eichler integrals simplified} is equivalent to
\begin{align}\label{eq eichler integrals simplified 2}
R_{2-2k}^{k-1}\left(\mathcal{W}_{2-2k,k}(-4\pi n v)e(-nu) \right) = (2k-2)! \overline{R_{2-2k}^{k-1}\left(\mathcal{W}_{2-2k,k}(4\pi n v)e(nu)\right)}.
\end{align}
On the other hand, using (13.4.33) and (13.4.31) in \cite{abramowitz}, we obtain the formula
\begin{align*}
&R_{\kappa}\left(\mathcal{W}_{\kappa,s}(4\pi m v)e(mu)\right) \\
&\quad = \begin{cases}
-4\pi |m|\left(s+\frac{\kappa}{2}\right)\left(s-\frac{\kappa}{2}-1\right)\mathcal{W}_{\kappa+2,s}(4\pi m v)e(mu), & \text{if $m < 0$}, \\
-4\pi |m|\mathcal{W}_{\kappa+2,s}(4\pi m v)e(mu), & \text{if $m > 0$},
\end{cases}
\end{align*}
for $u+iv \in \mathbb{H}$ and $m \in \mathbb{R} \setminus \{0\}$. This easily implies \eqref{eq eichler integrals simplified 2} and finishes the proof.
\end{proof}
\subsection{Periods of cusp forms and non-cusp forms}\label{section prelims periods}
We recall from \cite{kohnenzagierrationalperiods} that the period polynomial of a cusp form $f \in S_{2k}$ is defined by
\[
r_f(\tau) := \int_0^{i\infty} f(z)(z - \tau)^{2k-2}dz = \sum_{n = 0}^{2k-2}i^{-n+1}\binom{2k-2}{n}r_n(f)\tau^{2k-2-n}.
\]
The even period polynomial $r_f^+(\tau)$ of $f$ is defined as $-i$ times the even part of $r_f(\tau)$, and the odd period polynomial $r_f^-(\tau)$ of $f$ is the odd part of $r_f(\tau)$, such that $r_f(\tau) = r_f^-(\tau)+ir_f^+(\tau)$. It is well-known that the errors of modularity of the Eichler integrals of $f$ can be expressed in terms of the period function of $f$ as
\begin{align}\label{holomorphiceichlertrafo}
\mathcal{E}_{f}(\tau)\Bigl|_{2-2k}(I-S) &= \frac{(-2\pi i )^{2k-1}}{(2k-2)!}r_{f}(\tau)
\end{align}
and
\begin{align}\label{nonholomorphiceichlertrafo}
f^*(\tau)\Bigl|_{2-2k}(I-S) &= (-2i)^{1-2k}r_{f}^c(\tau),
\end{align}
where $r_f^c(\tau) := \overline{r_f(\overline{\tau})}$ is the polynomial whose coefficients are the complex conjugates of the coefficients of $r_f(\tau)$.
For $0 \leq n \leq 2k-2$ the periods of a (not necessarily cuspidal) modular form $f \in M_{2k}$ are defined by
\[
r_n(f) := \frac{n!}{(2\pi)^{n+1}}L_f(n+1),
\]
where $L_f(s)$ denotes the usual $L$-function associated to $f$. For a cusp form $f \in S_{2k}$ this agrees with the definition~\eqref{definition periods}. Note that the functional equation of the $L$-function of $f$ implies the symmetry $r_n(f) = (-1)^k r_{2k-2-n}(f)$.
The periods of the normalized Eisenstein series $E_{2k}$ are given by
\begin{align}\label{periodEisenstein}
r_{n}(E_{2k}) = \begin{dcases}
-\frac{\pi \zeta(2k-1) }{(2k-1)\zeta(2k)}, & \text{if $n=0$,}\\
0, & \text{if $0<n<2k-2$ is even,}\\
(-1)^{\frac{n-1}{2}}\frac{2k B_{n+1}B_{2k-1-n}}{B_{2k}(n+1)(2k-1-n)}, & \text{if $0<n<2k-2$ is odd,}\\
(-1)^{k-1}\frac{\pi \zeta(2k-1) }{(2k-1)\zeta(2k)}, & \text{if $n =2k-2$,}
\end{dcases}
\end{align}
compare p.240 of \cite{kohnenzagierrationalperiods}. Following \cite{zagierperiods}, we define the corresponding period function by
\begin{equation}\label{EisenSplit}
r_{E_{2k}}(\tau) :=\frac{1}{2k-1}\left(\tau^{2k-1}+\frac{1}{\tau}\right) + \sum_{n=0}^{2k-2}i^{1-n}\binom{2k-2}{n}r_{n}(E_{2k})\tau^{2k-2-n} = r^-_{E_{2k}}(\tau)+ir^+_{E_{2k}}(\tau)
\end{equation}
with
\begin{align}\label{EisenEven}
r^+_{E_{2k}}(\tau) := r_{0}(E_{2k})(\tau^{2k-2}-1)
\end{align}
and
\begin{align}\label{EisenPeriodOddPoly}
r^-_{E_{2k}}(\tau)
:=\frac{2k(2k-2)!}{B_{2k}}\sum_{\substack{-1\leq n\leq 2k-1, \\ \text{$n$ odd}}}\frac{B_{n+1}B_{2k-1-n}}{(n+1)!(2k-1-n)!}\tau^{2k-2-n}.
\end{align}
Then we also have
\begin{align}\label{EisenEichlerTrafo}
\mathcal{E}_{E_{2k}}(\tau)\Bigl|_{2-2k}(I-S) = \frac{(-2\pi i )^{2k-1}}{(2k-2)!}r_{E_{2k}}(\tau).
\end{align}
\subsection{The cusp forms $R_{n}(\tau)$}\label{section period polynomial Rn}
Recall that $R_n(\tau)$ denotes the unique cusp form of weight $2k$ for $\Gamma$ which satisfies the inner product formula $\langle f,R_n \rangle = r_n(f)$ for every cusp form $f(\tau) \in S_{2k}$. We will need the following well-known series representation of $R_n(\tau)$.
\begin{proposition}\label{proposition Rn}
For $0 < n < 2k-2$ the cusp form $R_n(\tau) \in S_{2k}$ is given by
\[
R_{n}(\tau) = \frac{2^{2k-3}}{i^{2k-1-n}\binom{2k-2}{n}\pi}\sum_{M \in \Gamma}\tau^{-n-1}\Bigl|_{2k}M.
\]
Moreover, for $n = 0$ or $n = 2k-2$ it can be constructed as
\[
R_{2k-2}(\tau) = (-1)^k R_{0}(\tau) = 2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau),
\]
where $P_{2k,m}(\tau) := \sum_{M \in \Gamma_\infty \backslash \Gamma}e(m\tau)|_{2k}M$ with $\Gamma_\infty := \{\pm\left(\begin{smallmatrix}1 & n \\ 0 & 1 \end{smallmatrix} \right): n\in \mathbb{Z}\}$ is the usual cuspidal Poincar\'e series of weight $2k$.
\end{proposition}
\begin{proof}
The above series representation of $R_n(\tau)$ for $0 < n < 2k-2$ was first given by Cohen in \cite{cohen}. We also refer the reader to the Lemma in Section 1.2 of \cite{kohnenzagierrationalperiods} for a proof.
For $n = 2k-2$ and $f(\tau) = \sum_{m=1}^\infty c_f(m)e(m\tau) \in S_{2k}$ we have
\[
\left\langle f,2^{2k-1}\sum_{m=1}^\infty P_{2k,m}\right\rangle = \frac{(2k-2)!}{(2\pi)^{2k-1}}\sum_{m=1}^{\infty}\frac{c_f(m)}{m^{2k-1}} = \frac{(2k-2)!}{(2\pi)^{2k-1}}L_f(2k-1) = r_{2k-2}(f) = \langle f,R_{2k-2} \rangle
\]
by the usual Petersson inner product formula for $P_{2k,m}(\tau)$. In other words, this means that $2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau)$ converges weakly to $R_{2k-2}(\tau)$, which implies $2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau) = R_{2k-2}(\tau)$ since $S_{2k}$ is finite-dimensional.
\end{proof}
We state the explicit formulas of Kohnen and Zagier \cite{kohnenzagierrationalperiods} for the even and odd period polynomials of the cusp forms $R_n(\tau)$, in a form that is convenient for our purposes.
\begin{theorem}[\cite{kohnenzagierrationalperiods}, Theorem 1']\label{Periods R_n}
For $0\leq n\leq 2k-2$ even, the odd period polynomial of $R_n(\tau)$ is given by
\begin{multline*}
\left(\frac{i}{2}\right)^{2k-2}ir^-_{R_n}(\tau)
=i^{n+1}\left(\frac{B_{n+1}(\tau)}{n+1}- \frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S)\\
+i^{n+1}(\tau^n-\tau^{2k-2-n})+i(\delta_{n=0}+(-1)^k\delta_{n=2k-2})r^-_{E_{2k}}(\tau),
\end{multline*}
and for $0< n< 2k-2$ odd, the even period polynomial of $R_n(\tau)$ is given by
\begin{multline*}
\left(\frac{i}{2}\right)^{2k-2}r^+_{R_n}(\tau)
=-i^{n+1}\left(\frac{B_{n+1}(\tau)}{n+1}+ \frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S) \\
-i^{n+1}(\tau^{n} + \tau^{2k-2-n})+r_n(E_{2k})(\tau^{2k-2}-1).
\end{multline*}
\end{theorem}
\section{The proof of Proposition~\ref{proposition diffops}}\label{section proof proposition diffops}
First note that the Poincar\'e series defined in \eqref{eq Peterssons Poincare series} satisfy the differential equations
\begin{align*}
R_{-2\ell,\tau}H_{k,\ell}(z,\tau) &= (k-\ell)H_{k,\ell-1}(z,\tau), \\
L_{-2\ell,\tau}H_{k,\ell}(z,\tau) &= (k+\ell)H_{k,\ell+1}(z,\tau),
\end{align*}
which can be checked by a direct computation. Furthermore, by Bol's identity the iterated derivative $\mathcal{D}^{2k-1}$ can be expressed in terms of the iterated raising operator by
\[
\mathcal{D}^{2k-1} = -\frac{1}{(4\pi)^{2k-1}}R_{2-2k}^{2k-1},
\]
compare equation (56) in Zagier's part of \cite{zagier123}. In particular, we find
\begin{align*}
\xi_{2-2k,\tau}H_{k,k-1}(z,\tau) &= v^{-2k}\overline{L_{2-2k,\tau}H_{k,k-1}(z,\tau)} = (2k-1)v^{-2k}\overline{H_{k,k}(z,\tau)}, \\
\mathcal{D}_\tau^{2k-1}H_{k,k-1}(z,\tau) &= -\frac{1}{(4\pi)^{2k-1}}R_{2-2k, \tau}^{2k-1}H_{k,k-1}(z,\tau)=-\frac{(2k-1)!}{(4\pi)^{2k-1}}H_{k,-k}(z,\tau).
\end{align*}
For $\tau \in \mathbb{C}\setminus i\mathbb{R}_{\geq 0}$ we will need the evaluation
\begin{align}\label{yintegraleval}
\int_{0}^{\infty} \frac{y^ndy}{(iy-\tau)^{2k}} = \frac{i^{n+1}}{2k-1}\binom{2k-2}{n}^{-1}\tau^{n+1-2k},
\end{align}
which can be shown directly for $\tau = iv$ with $v < 0$ and then follows by analytic continuation for all $\tau \in \mathbb{C} \setminus i\mathbb{R}_{\geq 0}$.
For $0 < n < 2k-2$ we can now compute
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,n}(\tau) &=(2k-1) \frac{(2i)^{2k-2}}{2\pi}v^{-2k}\overline{\sum_{M\in\Gamma}\left(v^{2k}\int_{0}^{\infty} \frac{y^ndy}{(iy-\overline{\tau})^{2k}}\right)\Bigl|_{-2k,\tau}M}\\
&=(-i)^{n+1}\frac{(2i)^{2k-2}}{2\pi}\binom{2k-2}{n}^{-1}v^{-2k}\overline{\sum_{M\in\Gamma}v^{2k}\overline{\tau}^{n+1-2k}\Bigl|_{-2k,\tau}M} \\
&=(-i)^{n+1}\frac{(2i)^{2k-2}}{2\pi}\binom{2k-2}{n}^{-1}\sum_{M\in\Gamma}\tau^{n+1-2k}\Bigl|_{2k,\tau}M\\
&= (-1)^{k-1}R_{2k-2-n}(\tau)= - R_n(\tau),
\end{align*}
and
\begin{align*}
\mathcal{D}^{2k-1}\mathcal{H}_{1-k,n}(\tau) &=(-1)^k\frac{(2k-1)!}{2(2\pi)^{2k}}\sum_{M\in\Gamma}\left(\int_{0}^{\infty} \frac{y^ndy}{(iy-\tau)^{2k}}\right)\Bigl|_{2k,\tau}M\\
&=i^{n+1}(-1)^k\frac{(2k-2)!}{2(2\pi)^{2k}}\binom{2k-2}{n}^{-1}\sum_{M\in\Gamma}\tau^{n+1-2k}\Bigl|_{2k,\tau}M\\
& = (-1)^{n+1} \frac{(2k-2)!}{(4\pi)^{2k-1}}(-1)^k R_{2k-2-n}(\tau) = (-1)^{n+1}\frac{(2k-2)!}{(4\pi)^{2k-1}}R_{n}(\tau),
\end{align*}
where we used the series representation of $R_n(\tau)$ from Proposition~\ref{proposition Rn}.
However, for $n = 0$ or $n = 2k-2$ the series representation of $R_n(\tau)$ used above does not converge, so we need to proceed differently. By the symmetries $\mathcal{H}_{1-k,n}(\tau) = (-1)^k \mathcal{H}_{1-k,2k-2-n}(\tau)$ and $R_{n}(\tau) = (-1)^k R_{2k-2-n}(\tau)$ it suffices to treat the case $n = 2k-2$. First, we write
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,2k-2}(\tau) &= 2(2k-1) \frac{(2i)^{2k-2}}{2\pi}v^{-2k}\overline{\sum_{M\in\Gamma_\infty\backslash\Gamma}\left(v^{2k}\int_{0}^{\infty} \sum_{m \in \mathbb{Z}} \frac{y^{2k-2}dy}{(iy-\overline{\tau}+m)^{2k}}\right)\Bigl|_{-2k,\tau}M}.
\end{align*}
Using the Lipschitz formula
\begin{align}\label{eq Lipschitz formula}
\sum_{m \in \mathbb{Z}}(z + m)^{-2k} = \frac{(2\pi i )^{2k}}{(2k-1)!}\sum_{m = 1}^\infty m^{2k-1}e(mz),
\end{align}
which is valid for $z \in \mathbb{H}$, we can rewrite the integral as
\begin{align*}
\int_{0}^{\infty} \sum_{m \in \mathbb{Z}} \frac{y^{2k-2}dy}{(iy-\overline{\tau}+m)^{2k}} &= \frac{(2\pi i)^{2k}}{(2k-1)!}\int_{0}^{\infty}y^{2k-2} \sum_{m = 1}^{\infty}m^{2k-1}e^{-2\pi m y}e(-m\overline{\tau})dy \\
&= \frac{2\pi (-1)^k}{2k-1}\sum_{m=1}^\infty e(-m \overline{\tau}).
\end{align*}
Putting everything together, we obtain
\begin{align*}
\xi_{2-2k}\mathcal{H}_{1-k,2k-2}(\tau) &= -2^{2k-1}v^{-2k}\overline{\sum_{M \in \Gamma_\infty\backslash \Gamma}\left(v^{2k}\sum_{m=1}^\infty e(-m \overline{\tau})\right)\Bigl|_{-2k,\tau}M} \\
&= -2^{2k-1}\sum_{m=1}^\infty \sum_{M \in \Gamma_\infty \backslash \Gamma}e(m \tau)\Bigl|_{2k,\tau}M \\
&= -2^{2k-1}\sum_{m=1}^\infty P_{2k,m}(\tau) = - R_{2k-2}(\tau),
\end{align*}
where we used Proposition~\ref{proposition Rn} in the last equality.
To compute $\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau)$, we write as before
\begin{align*}
\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau) &= -\frac{(2i)^{2k-2}(2k-1)!}{\pi (4\pi)^{2k-1}}\sum_{M \in \Gamma_\infty\backslash\Gamma}\left(\int_0^\infty \sum_{m \in \mathbb{Z}}\frac{y^{2k-2}}{(iy-\tau+m)^{2k}} dy\right)\Bigl|_{2k,\tau}M.
\end{align*}
It now suffices to prove the identity
\begin{align}\label{eq D identity}
\int_0^\infty \sum_{m \in \mathbb{Z}}\frac{y^{2k-2}}{(iy-\tau+m)^{2k}}dy = \frac{2^{2k-1}\pi}{(2i)^{2k-2}(2k-1)}\cdot\frac{1}{1-e(\tau)}
\end{align}
for $\tau \in \mathbb{H}$ with $0 < u < 1$, since we can write $\frac{1}{1-e(\tau)} = 1+\sum_{m=1}^\infty e(m\tau)$ and then finish the computation of $\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau)$ in the same way as for the $\xi_{2-2k}$-image, using Proposition~\ref{proposition Rn}. Note that we cannot directly apply the Lipschitz formula \eqref{eq Lipschitz formula} to prove the identity \eqref{eq D identity} since $iy-\tau$ has negative imaginary part for $0 < y < v$. However, since both sides of \eqref{eq D identity} are holomorphic functions on the vertical strip consisting of all $\tau \in \mathbb{C}$ with $0 < u < 1$, it is enough to prove \eqref{eq D identity} for all $\tau$ in this strip with $v < 0$ and then use analytic continuation. Under the assumption $v < 0$ we can apply the Lipschitz formula~\eqref{eq Lipschitz formula} and obtain similarly as before
\begin{align*}
\int_0^\infty \sum_{m \in \mathbb{Z}}\frac{y^{2k-2}}{(iy-\tau+m)^{2k}}dy
&= \frac{(2\pi i)^{2k}}{(2k-1)!} \int_0^\infty y^{2k-2}\sum_{m=1}^\infty m^{2k-2}e^{-2\pi m y}e(-m\tau)dy \\
&= \frac{(2\pi i)^{2k}}{(2k-1)!}\cdot \frac{(2k-2)!}{(2\pi)^{2k-1}}\sum_{m=1}^\infty e(-m\tau) \\
&= -\frac{2^{2k-1}\pi}{(2i)^{2k-2}(2k-1)}\left(\frac{1}{1-e(-\tau)}-1 \right) \\
&= \frac{2^{2k-1} \pi}{(2i)^{2k-2}(2k-1)}\cdot\frac{1}{1-e(\tau)},
\end{align*}
which yields \eqref{eq D identity} and concludes the computation of $\mathcal{D}^{2k-1}\mathcal{H}_{1-k,2k-2}(\tau)$. This finishes the proof of Proposition~\ref{proposition diffops}.
\section{The proof of Theorem~\ref{theorem splitting}}\label{section proof theorem splitting}
In this section we prove Theorem~\ref{theorem splitting}, i.e., the decomposition of the locally harmonic Maass form $\mathcal{H}_{1-k,n}(\tau)$ into a sum of a locally polynomial part and Eichler integrals of the cusp forms $R_n(\tau)$. For brevity, we let
\begin{align*}
\widetilde{\mathcal{H}}_{1-k,n}(\tau) &:= \mathcal{P}_{1-k,n}(\tau) + (-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau) \\
&\quad-((-1)^k\delta_{n=0}+\delta_{n=2k-2})\frac{(2k-2)!}{(2\pi)^{2k-1}}\mathcal{E}_{E_{2k}}(\tau)
\end{align*}
be the expression on the right-hand side of Theorem~\ref{theorem splitting}. Then the theorem is equivalent to the identity $\mathcal{H}_{1-k,n}(\tau) = \widetilde{\mathcal{H}}_{1-k,n}(\tau)$. In order to prove this identity, we show that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is a locally harmonic Maass form of weight $2-2k$ with the same singularities as $\mathcal{H}_{1-k,n}(\tau)$ and the same images under the differential operators $\xi_{2-2k}$ and $\mathcal{D}^{1-2k}$. This implies that the difference $\mathcal{H}_{1-k,n}(\tau)-\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is a polynomial on $\mathbb{H}$ which is also modular of negative weight $2-2k$, and hence vanishes identically.
We first show that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is modular. To this end, we study the modularity properties of the local polynomial $\mathcal{P}_{1-k,n}(\tau)$. We start by rewriting $\mathcal{P}_{1-k,n}(\tau)$ in a more convenient form.
\begin{lemma}\label{lemma alternative polynomial}
For $\tau \in \mathbb{H}$ we have
\begin{align*}
\mathcal{P}_{1-k,n}(\tau) &= r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}B_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}B_{2k-1-n}(\tau)
\\
&\quad +\frac{(-i)^{n+1}}{2}\left(\tau^n-(-1)^n\tau^{2k-2-n}\right)+\frac{i^{1-n}}{4}\sum_{M\in\Gamma}((\sgn(u)-\sgn(M))\tau^n)\Bigl |_{2-2k}M,
\end{align*}
where
\[
\sgn\left(\begin{pmatrix} a & b\\ c& d \end{pmatrix}\right):=\begin{cases}
\sgn(ac), & \text{if $ac\neq 0$,}\\
\sgn(bd), & \text{otherwise.}
\end{cases}
\]
\end{lemma}
\begin{proof}
For simplicity, we sketch the computations in the case that $\tau\notin\bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$ and leave the general case to the reader. We first rewrite
\begin{multline*}
\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac>0, \mathbb{R}e(M\tau)> 0}}(\tau^n -(-1)^n \tau^{2k-2-n})\Bigl |_{2-2k}M = \sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac>0, \mathbb{R}e(M\tau)> 0}}\tau^n \Bigl |_{2-2k}M - \tau^n|_{2-2k}SM \\
=\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac \mathbb{R}e(M\tau) < 0 }}\sgn(M)\tau^n \Bigl |_{2-2k}M
=-\frac12\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac \neq 0 }}\left((\sgn(u)-\sgn(M))\tau^n\right) \Bigl |_{2-2k}M.
\end{multline*}
Here we used that $ac>0$ if and only if $SM = \left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta \end{smallmatrix}\right)$ has $\alpha\gamma <0$ and $\sgn(\mathbb{R}e(SM\tau))\neq \sgn(\mathbb{R}e(M\tau))$.
Using $\frac{B_m(\tau+1)}{m} = \frac{B_m(\tau)}{m} + \tau^{m-1}$ we write
\begin{align*}
\frac{\mathbb{B}_m(\tau)}{m} -\frac{B_m(\tau)}{m} &=
\begin{cases}-\sum_{j=1}^{\lfloor u \rfloor}(\tau-j)^{m-1},& \text{if $u> 0$,} \\
\sum_{j=0}^{\lfloor -u\rfloor } (\tau+j)^{m-1}, & \text{if $u<0$,}
\end{cases} \\
&= \frac12\tau^{m-1}+\frac12\sum_{j\in\mathbb{Z}}(\sgn(j)-\sgn(u+j))(\tau+j)^{m-1} \\
&=\frac12\tau^{m-1}+\frac14\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ c = 0 }}(\sgn(M)-\sgn(u))\tau^{m-1}\Bigl |_{2-2k}M.
\end{align*}
Hence, we find after a short computation
\begin{align*}
\Biggl(\frac{\mathbb{B}_{n+1}(\tau)}{n+1}&-(-1)^n\frac{\mathbb{B}_{2k-1-n}(\tau)}{2k-1-n}\Biggr)-\Biggl(\frac{B_{n+1}(\tau)}{n+1}-(-1)^n\frac{B_{2k-1-n}(\tau)}{2k-1-n}\Biggr) \\
&=\frac12\left(\tau^n -(-1)^n\tau^{2k-2-n}\right)- \frac14\sum_{\substack{M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\Gamma \\ ac = 0 }}(\sgn(u)-\sgn(M))\tau^{n}\Bigl |_{2-2k}M.
\end{align*}
Gathering everything together, we obtain the formula given in the lemma.
\end{proof}
We can now state the transformation law for the locally polynomial part $\mathcal{P}_{1-k,n}(\tau)$.
\begin{lemma}\label{lemma trafo P}
For $\tau \in \mathbb{H}$ we have
\begin{align*}
\mathcal{P}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-T) = 0
\end{align*}
and
\begin{align*}
\mathcal{P}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-S) = \begin{dcases}
-\left(\frac{i}{2}\right)^{2k-2}r^+_{R_n}(\tau),&\text{if $n$ is odd,} \\
\begin{aligned} &-\left(\frac{i}{2}\right)^{2k-2}ir^-_{R_n}(\tau) \\
&\quad+i(\delta_{n=0} +(-1)^k \delta_{n=2k-2})r_{E_{2k}}(\tau),
\end{aligned}&\text{if $n$ is even.}
\end{dcases}
\end{align*}
\end{lemma}
\begin{proof}
For the $T$-transformation, we write
$$
\left(\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(M))\tau^n\right)\Bigl |_{2-2k}M\right)\Bigl|_{2-2k}T =
\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(MT^{-1}))\tau^n\right)\Bigl |_{2-2k}M.
$$
Now one can show that the only matrices with $\sgn(MT^{-1})\neq \sgn(M)$ are $M\in\pm \{I, S, T, ST\}$. Therefore we get
\begin{align*}
\Biggl(&\sum_{M\in\Gamma}((\sgn(u)-\sgn(M))\tau^n)\Bigl |_{2-2k}M\Biggr)\Bigl|_{2-2k}(I-T)\\
&= 2\sum_{M\in\{I, S, T, ST\}}((\sgn(MT^{-1})-\sgn(M))\tau^n)\Bigl |_{2-2k}M\\
&= 2(-\tau^n + (-1)^n\tau^{2k-2-n} - (\tau+1)^n + (-1)^n(\tau+1)^{2k-2-n}).
\end{align*}
The Bernoulli polynomials satisfy $
\frac{B_m(\tau+1)}{m} = \frac{B_m(\tau)}{m} + \tau^{m-1}$,
so all in all we obtain
\begin{align*}
&\mathcal{P}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-T) = \frac12 (-i)^{n+1}\Bigl(-2\tau^n +2(-1)^n\tau^{2k-2-n} + \tau^n -(\tau+1)^n -(-1)^n\tau^{2k-2-n} \\
&\qquad+ (-1)^n(\tau+1)^{2k-2-n}+\tau^n -(-1)^n\tau^{2k-2-n}+(\tau+1)^n -(-1)^n(\tau+1)^{2k-2-n} \Bigr) = 0.
\end{align*}
For the $S$-transformation, note that
\begin{align*}
\left(\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(M))\tau^n\right)\Bigl |_{2-2k}M\right)\Bigl|_{2-2k}S
&=\sum_{M\in\Gamma}\left((\sgn(u)-\sgn(MS^{-1}))\tau^n\right)\Bigl |_{2-2k}M \\
&= \sum_{M\in\Gamma}\left((\sgn(u)-\sgn(M))\tau^n\right)\Bigl |_{2-2k}M,
\end{align*}
since we have $\sgn(MS^{-1}) = \sgn(M)$ for all $M\in\Gamma$. Furthermore, if $n$ is odd, then $(-i)^{n+1} = i^{n+1}$ and Theorem \ref{Periods R_n} gives
\begin{align*}
&\Bigl(r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}B_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}B_{2k-1-n}(\tau) \\
&\qquad+\frac{(-i)^{n+1}}{2}(\tau^n-(-1)^n \tau^{2k-2-n}\Bigr)\Bigl|_{2-2k}(I-S) \\
&= -r_n(E_{2k})(\tau^{2k-2}-1) +i^{n+1}\left(\frac{B_{n+1}(\tau)}{n+1}+\frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S) \\
&\qquad +i^{n+1}(\tau^n+\tau^{2k-2-n}) \\
&=-\left(\frac{i}{2}\right)^{2k-2}r^+_{R_n}(\tau).
\end{align*}
If $n$ is even, then $(-i)^{n+1} = -i^{n+1}$ and Theorem~\ref{Periods R_n} yields
\begin{align*}
&\Bigl( r_n(E_{2k}) +\frac{(-i)^{n+1}}{n+1}B_{n+1}(\tau)+\frac{i^{n+1}}{2k-1-n}B_{2k-1-n}(\tau) \\
&\qquad +\frac{(-i)^{n+1}}{2}(\tau^n-(-1)^n\tau^{2k-2-n})\Bigr)\Bigl|_{2-2k}(I-S) \\
&= -r_n(E_{2k})(\tau^{2k-2}-1) +i^{n+1}\left(-\frac{B_{n+1}(\tau)}{n+1}+\frac{B_{2k-1-n}(\tau)}{2k-1-n}\right)\Bigl|_{2-2k}(I-S) \\
&\qquad -i^{n+1}(\tau^n-\tau^{2k-2-n}) \\
&=-i\left(\frac{i}{2}\right)^{2k-2}r^-_{R_n}(\tau)+i(\delta_{n=0} +(-1)^k \delta_{n=2k-2})r_{E_{2k}}(\tau).
\end{align*}
In the last step we used that $r_n(E_{2k}) =0$ for even $n$ unless $n=0$ or $n=2k-2$, as well as \eqref{EisenSplit} and \eqref{EisenEven}. Pulling everything together, we obtain the stated result.
\end{proof}
Combining the above results, we obtain the modularity of the function $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ defined in the beginning of this section.
\begin{proposition}\label{proposition modularity Htilde}
The function $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ transforms like a modular form of weight $2-2k$ for $\Gamma$.
\end{proposition}
\begin{proof}
Since the Eichler integrals are one-periodic by definition and the local polynomial $\mathcal{P}_{1-k,n}(\tau)$ is one-periodic by Lemma~\ref{lemma trafo P}, the same is true for $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$.
Next, we show the invariance of $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ under $S$. Using \eqref{holomorphiceichlertrafo} and \eqref{nonholomorphiceichlertrafo}, and the fact that $R_n(\tau)$ has real Fourier coefficients, we compute
\begin{align*}
\left((-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau)\right)\Bigl|_{2-2k}(I-S)
&=\left(\frac{i}{2}\right)^{2k-2}
\begin{cases}
r_{R_n}^+(\tau), & \text{if $n$ is odd,}\\
ir_{R_n}^-(\tau),& \text{if $n$ is even.}
\end{cases}
\end{align*}
Combining this with Lemma~\ref{lemma trafo P} (and \eqref{EisenEichlerTrafo} if $n = 0$ or $n = 2k-2$), we see that
\[
\widetilde{\mathcal{H}}_{1-k,n}(\tau)\Bigl|_{2-2k}(I-S) = 0.
\]
This finishes the proof.
\end{proof}
Next, we determine the singularities of $\mathcal{H}_{1-k,n}(\tau)$ and $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$. Here, we say that a function $f$ has a \emph{singularity of type $g$} at a point $\tau_{0}$ if there exists a neighbourhood $U$ of $\tau_{0}$ on which $f-g$ is harmonic.
\begin{lemma}\label{jumps}
The functions $\mathcal{H}_{1-k,n}(\tau)$ and $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ are harmonic on $\mathbb{H}\setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$. At a point $\tau_{0} \in \mathbb{H}$ they have a singularity of type
\[
\frac{i^{1-n}}{4}\sum_{\substack{M\in\Gamma \\ \mathbb{R}e(M\tau_0)=0}}\left(\sgn(u)\tau^{n}\right)\Bigl|_{2-2k}M.
\]
\end{lemma}
\begin{proof}
We start with the function $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$. Note that the weight $2-2k$ invariant Laplace operator can be written as $\Delta_{2-2k} = -\xi_{2k}\circ \xi_{2-2k}$ and that $\xi_{2k}$ annihilates holomorphic functions. Hence, the action \eqref{eq diffops eichler} of $\xi_{2-2k}$ on Eichler integrals and the fact that $\mathcal{P}_{1-k,n}(\tau)$ is a polynomial on each connected component of $\mathbb{H}\setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$ imply that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is harmonic on this set. The singularities of $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ come from the sum in the locally polynomial part $\mathcal{P}_{1-k,n}(\tau)$ and are given by the stated formula by Lemma~\ref{lemma alternative polynomial}.
Now we consider the function $\mathcal{H}_{1-k,n}(\tau)$. Since the function $\tau \mapsto H_{k,k-1}(z,\tau)$ is harmonic on $\mathbb{H} \setminus \Gamma z$, the function $\mathcal{H}_{1-k,n}(\tau)$ is harmonic on $\mathbb{H} \setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$. To determine the singularities, we keep $\tau_{0}\in\mathbb{H}$ fixed and consider the function
\[
G_{\tau_0}(z, \tau) := \sum_{\substack{M\in\Gamma \\ \mathbb{R}e(M\tau_0) = 0}} \left(\frac{v^{2k-1}}{(z-\tau)(z-\overline{\tau})^{2k-1}}\right)\Big|_{2-2k, \tau} M.
\]
Note that $\mathbb{R}e(M\tau_0)=0$ means that $\tau_0$ lies on the geodesic $M^{-1}(i\mathbb{R}_+)$, hence the above sum is finite (see also Figure 1 in the introduction). Furthermore, the function $G_{\tau_0}(z,\tau)$ is meromorphic in $z$ and harmonic in $\tau$ on $\mathbb{H}\setminus\Gamma z$. We split $\mathcal{H}_{1-k,n}(\tau)$ into
\[
\mathcal{H}_{1-k,n}(\tau) =\frac{(2i)^{2k-2}}{2\pi}\left(\int_0 ^\infty (H_{k,k-1}(iy,\tau) - G_{\tau_0}(iy,\tau))y^ndy + \int_0 ^\infty G_{\tau_0}(iy,\tau)y^ndy\right).
\]
The function
\[
z\mapsto H_{k,k-1}(z,\tau) - G_{\tau_0}(z,\tau)
\]
is meromorphic and has no singularities near $\tau_0$ and the function
\[
\tau \mapsto \int_0 ^\infty (H_{k,k-1}(iy,\tau) - G_{\tau_0}(iy,\tau))y^ndy
\]
is harmonic in a neighborhood of $\tau_0$. For the second summand we compute for any $\tau\notin E_1$
\[
\int_0 ^\infty G_{\tau_0}(iy,\tau)y^ndy =(-i)^{n+1}\sum_{\substack{M\in\Gamma \\ \mathbb{R}e(M\tau_0)=0}}\left(v^{2k-1}\int_0 ^{i\infty} \frac{z^ndz}{(z-\tau)(z-\overline{\tau})^{2k-1}}\right)\Big|_{2-2k, \tau} M.
\]
We now evaluate the inner integral for fixed $\tau$. If we shift the path of integration to the left towards $-\infty$, we pick up a residue $2\pi i\frac{\tau^n}{(2iv)^{2k-1}}$ if $\mathbb{R}e(\tau) <0$. If we shift towards $\infty$, we pick up a residue of $-2\pi i\frac{\tau^n}{(2iv)^{2k-1}}$ if $\mathbb{R}e(\tau) >0$. Thus we get
\begin{multline*}
\int_0 ^\infty G_{\tau_0}(iy,\tau)y^ndy =(-i)^{n+1}\sum_{\substack{M\in\Gamma \\ \mathbb{R}e(M\tau_0)=0}}\Bigl(\frac{v^{2k-1}}{2}\left(\int_{-\infty}^0-\int_0 ^\infty\right)\frac{z^ndz}{(z-\tau)(z-\overline{\tau})^{2k-1}} \\
-\sgn(u)\pi i\frac{\tau^n}{(2i)^{2k-1}}\Bigr)\Big|_{2-2k, \tau} M.
\end{multline*}
The function
\[
\tau\mapsto \left(\int_{-\infty}^0-\int_0 ^\infty\right)\frac{z^ndz}{(z-\tau)(z-\overline{\tau})^{2k-1}}
\]
is harmonic on $\mathbb{H}$, so it does not contribute to the singularity. The sum over the signed terms yields the claimed singularity.
\end{proof}
We can now finish the proof of Theorem~\ref{theorem splitting}. It follows from Proposition~\ref{proposition modularity Htilde} and Lemma~\ref{jumps} that $\widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is a locally harmonic Maass form of weight $2-2k$ with the same singularities as $\mathcal{H}_{1-k,n}(\tau)$, i.e., the difference $\mathcal{H}_{1-k,n}(\tau) - \widetilde{\mathcal{H}}_{1-k,n}(\tau)$ transforms like a modular form of weight $2-2k$ and is harmonic on all of $\mathbb{H}$. Furthermore, it follows from \eqref{eq diffops eichler} and Proposition~\ref{proposition diffops} that $\mathcal{H}_{1-k,n}(\tau) - \widetilde{\mathcal{H}}_{1-k,n}(\tau)$ is annihilated by $\xi_{2-2k}$ and $\mathcal{D}^{2k-1}$, which implies that it is a polynomial on $\mathbb{H}$. But the only one-periodic polynomials on $\mathbb{H}$ are the constant functions, and the only constant function which transforms like a modular form of non-zero weight is the constant 0 function. This shows $\mathcal{H}_{1-k,n}(\tau) = \widetilde{\mathcal{H}}_{1-k,n}(\tau)$ and concludes the proof of Theorem~\ref{theorem splitting}.
\section{The proof of Theorem~\ref{theorem rationality} and Proposition~\ref{proposition special value}}\label{section proof rationality}
Let $P \in \mathcal{Q}_d$ be a positive definite binary quadratic form and let $\tau_P \in \mathbb{H}$ be the associated CM point defined by $P(\tau_P,1) = 0$. For simplicity, we assume throughout this section that $\tau_P$ does not lie on any $\Gamma$-translate of the imaginary axis $i\mathbb{R}_+$ and leave the necessary adjustments in the general case to the reader.
We start with the proof of Proposition~\ref{proposition special value}. We can write
\[
P(z,1) = \frac{\sqrt{|d|}}{2\mathrm{Im}(\tau_P)}(z-\tau_P)(z-\overline{\tau}_P),
\]
which implies
\begin{align*}
H_{k,0}(z,\tau_P) &= \sum_{M \in \Gamma}j(M,z)^{-2k}\left(\frac{(Mz - \tau_P)(Mz - \overline{\tau}_P)}{\mathrm{Im}(\tau_P)} \right)^{-k} \\
&= 2^{-k}|d|^{\frac{k}{2}}\sum_{M \in \Gamma}j(M,z)^{-2k}P(Mz,1)^{-k} = 2^{1-k}|d|^\frac{1-k}{2}|\overline{\Gamma}_P|\pi f_{k,P}(z).
\end{align*}
Using $(k-1)!H_{k,0}(z,\tau) = R_{2-2k,\tau}^{k-1}H_{k,k-1}(z,\tau)$ we obtain
\begin{align*}
r_n(f_{k,P}) &= \int_0^\infty f_{k,P}(iy)y^n dy \\
&= \frac{2^{k-1}|d|^\frac{k-1}{2}}{|\overline{\Gamma}_P| \pi }\int_0^\infty H_{k,0}(iy,\tau_P)y^n dy \\
&= \frac{2^{k-1}|d|^\frac{k-1}{2}}{(k-1)!|\overline{\Gamma}_P| \pi }\int_0^\infty R_{2-2k,\tau}^{k-1}H_{k,k-1}(iy,\tau)\big|_{\tau = \tau_P}y^n dy \\
&=\frac{(-1)^{k-1}|d|^\frac{k-1}{2}}{2^{k-2}(k-1)!|\overline{\Gamma}_P|}R_{2-2k}^{k-1}\mathcal{H}_{1-k,n}(\tau_P).
\end{align*}
This finishes the proof of Proposition~\ref{proposition special value}.
Before we come to the proof of Theorem~\ref{theorem rationality}, we give a general formula for the periods $r_n(f_{k,P})$ of the meromorphic modular forms $f_{k,P}(z)$.
\begin{theorem}\label{theorem rn formula}
Assume that $\tau_P \in \mathbb{H} \setminus \bigcup_{M \in \Gamma}M(i\mathbb{R}_{+})$ and $0 < \mathbb{R}e(\tau_P) < 1$. Then we have the formula
\begin{align*}
r_n(f_{k,P}) &= \frac{|d|^{\frac{k-1}{2}}}{|\overline{\Gamma}_P|}\bigg(\frac{(-1)^{k-1}}{2^{k-2}(k-1)!}R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P) \\
&\qquad \qquad \quad +\frac{(-1)^{k}}{2^{k-2}(k-1)!}\left((-1)^{n+1}\overline{R_{2-2k}^{k-1}R_n^*(\tau_P)}+ R_{2-2k}^{k-1}R_n^*(\tau_P)\right) \\
&\qquad \qquad \quad +(\delta_{n=0}+(-1)^k \delta_{n=2k-2})\frac{2^{k+1}(2k-2)!}{(4\pi)^{2k-1}(k-1)!}R_{2-2k}^{k-1}\mathcal{E}_{E_{2k}}(\tau_P)\bigg),
\end{align*}
where $R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau)$ is explicitly given by
\begin{align*}
R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau) &= (-v)^{1-k}\frac{(2k-2)!}{(k-1)!}r_n(E_{2k}) \\
&\quad + \frac{(-i)^{n+1}}{n+1}R_{2-2k}^{k-1}B_{n+1}(\tau) + \frac{i^{n+1}}{2k-1-n}R_{2-2k}^{k-1}B_{2k-1-n}(\tau) \\
&\quad + \frac{i^{1-n}}{2}\sum_{\substack{M \in \Gamma \\ ac > 0, \mathbb{R}e(M\tau) \leq 0}}\left(R_{2-2k}^{k-1}\tau^{n}-(-1)^n R_{2-2k}^{k-1}\tau^{2k-2-n} \right)\bigg|_0 M.
\end{align*}
\end{theorem}
\begin{proof}
First, by Proposition~\ref{proposition special value} we have
\[
r_n(f_{k,P}) = \frac{(-1)^{k-1}|d|^\frac{k-1}{2}}{2^{k-2}(k-1)!|\overline{\Gamma}_P|}R_{2-2k}^{k-1}\mathcal{H}_{1-k,n}(\tau_P).
\]
Furthermore, by Theorem~\ref{theorem splitting} the locally harmonic Maass form $\mathcal{H}_{1-k,n}(\tau)$ has the splitting
\begin{align*}
\mathcal{H}_{1-k,n}(\tau) &= \mathcal{P}_{1-k,n}(\tau) + (-1)^{n+1}\frac{(2k-2)!}{(4\pi )^{2k-1}}\mathcal{E}_{R_n}(\tau) - R_{n}^*(\tau) \\
&\quad-((-1)^{k}\delta_{n=0}+\delta_{n=2k-2})\frac{(2k-2)!}{(2\pi)^{2k-1}}\mathcal{E}_{E_{2k}}(\tau).
\end{align*}
By Proposition~\ref{proposition eichler integral relations}, we can rewrite
\[
\frac{(2k-2)!}{(4\pi)^{2k-1}}R_{2-2k}^{k-1}\mathcal{E}_{R_n}(\tau) = -\overline{R_{2-2k}^{k-1}R_n^*(\tau)}.
\]
Finally, using the formula $R_{\kappa}v^j = (j+\kappa)v^{j-1}$ we can compute the action of the iterated raising operator on the constant $r_n(E_{2k})$ in the locally polynomial part $\mathcal{P}_{1-k,n}(\tau)$. This finishes the proof.
\end{proof}
Note that the Fourier expansion of the raised Eichler integrals appearing in Theorem~\ref{theorem rn formula} can be computed using Proposition~\ref{proposition eichler integral relations} and \eqref{raising and derivative}. In order to understand the algebraic nature of the expressions appearing in $R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)$, the following formula will be useful.
\begin{lemma}\label{lemma raising taupower}
For $k \in \mathbb{N}$, $0 \leq \ell \leq 2k-1$, and $\tau = u+iv \in \mathbb{H}$ we have the formula
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell &= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2)^j\sum_{\substack{\alpha = 0 \\ \ell-\alpha \text{ even }}}^{\ell-j}\binom{\ell-j}{\alpha}u^{\alpha}(iv)^{\ell-\alpha} \\
&\quad + i\delta_{\ell=2k-1}(-1)^{k-1}2^{2k-2}(k-1)!v^k.
\end{align*}
\end{lemma}
\begin{proof}
We first apply the formula
\begin{align}\label{raising and derivative}
R_{2-2k}^{k-1}f(\tau) = (-v)^{1-k}(k-1)!\sum_{j=0}^{k-1}\frac{(-2iv)^j}{j!}\binom{2k-2-j}{k-1}\frac{\partial^j}{\partial \tau^j}f(\tau),
\end{align}
which holds for any smooth function $f: \mathbb{H} \to \mathbb{C}$ (see equation (56) in Zagier's part of \cite{zagier123}). This yields
\begin{align}\label{eq raising taupower}
R_{2-2k}^{k-1}\tau^\ell = (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2iv)^j\tau^{\ell-j}.
\end{align}
On the other hand, using $R_{\kappa}v^j = (j+\kappa)v^{j-1}$ we compute
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell &= R_{2-2k}^{k-1}(\overline{\tau}+2iv)^\ell = R_{2-2k}^{k-1} \sum_{j=0}^{\ell}\binom{\ell}{j}(2iv)^j\overline{\tau}^{\ell-j} \\
&= (-v)^{1-k}\sum_{j=0}^{\ell}\binom{\ell}{j}(2iv)^j\overline{\tau}^{\ell-j} (2k-2-j)(2k-3-j)\cdots (k-j).
\end{align*}
Note that $(2k-2-j)(2k-3-j)\cdots (k-j)$ equals $\frac{(2k-2-j)!}{(k-1-j)!}$ for $0 \leq j \leq k-1$, it vanishes for $k \leq j \leq 2k-2$, and it equals $(-1)^{k-1}(k-1)!$ if $j = 2k-1$, which can occur only if $\ell = 2k-1$. Hence we obtain
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell &= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(2iv)^j\overline{\tau}^{\ell-j}+\delta_{\ell=2k-1}(2i)^{2k-1}(k-1)!v^{k}.
\end{align*}
Comparing this with \eqref{eq raising taupower}, we see that
\[
R_{2-2k}^{k-1}\tau^\ell - \frac{1}{2}\delta_{\ell=2k-1}(2i)^{2k-1}(k-1)!v^k
\]
is real. Thus, expanding $\tau^{\ell-j}$ in \eqref{eq raising taupower} using the binomial theorem, we obtain
\begin{align*}
R_{2-2k}^{k-1}\tau^\ell
&= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2)^j\sum_{\alpha = 0}^{\ell-j}\binom{\ell-j}{\alpha}u^{\alpha}(iv)^{\ell-\alpha} \\
&= (-v)^{1-k}\sum_{j=0}^{\min\{k-1,\ell\}}\binom{\ell}{j}\frac{(2k-2-j)!}{(k-1-j)!}(-2)^j\sum_{\substack{\alpha = 0 \\ \ell-\alpha \text{ even }}}^{\ell-j}\binom{\ell-j}{\alpha}u^{\alpha}(iv)^{\ell-\alpha} \\
&\quad + i\delta_{\ell=2k-1}(-1)^{k-1}2^{2k-2}(k-1)!v^k.
\end{align*}
Here we used that the terms with odd $\ell-\alpha$ would be purely imaginary and hence cannot occur, apart from possibly the summand for $\alpha = 0$ in the case $\ell = 2k-1$, which is a purely imaginary multiple of $v^k$ and hence has to be equal to $i(-1)^{k-1}2^{2k-2}(k-1)!v^k$. This finishes the proof.
\end{proof}
We obtain the following rationality result for the special values $R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)$.
\begin{lemma}\label{lemma P rational}
We have
\begin{align*}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P) \in \begin{cases}\mathbb{Q}, & \text{if $0 < n < 2k-2$ is odd,} \\
i\mathbb{Q}, & \text{if $0 < n < 2k-2$ is even.}
\end{cases}
\end{align*}
\end{lemma}
\begin{proof}
If we write $P=[a,b,c]$ then the corresponding CM point is given by
\[
\tau_P = -\frac{b}{2a}+i\frac{\sqrt{|d|}}{2a}.
\]
Noting that $B_m(\tau) = \sum_{\ell=0}^m \binom{m}{\ell}B_{m-\ell}\tau^\ell$, we see from Lemma~\ref{lemma raising taupower} that
\begin{align}\label{eq rational 1}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}B_m(\tau_P) \in \mathbb{Q}
\end{align}
for every $0 \leq m \leq 2k-2$, and
\begin{align}\label{eq rational 2}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}B_{2k-1}(\tau_P) = C + i|d|^{\frac{k-1}{2}}(-1)^{k-1}2^{2k-2}(k-1)!v_P^k
\end{align}
for some constant $C \in \mathbb{Q}$. By the same lemma we see that
\begin{align}\label{eq rational 3}
|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}\tau^\ell \in \mathbb{Q}
\end{align} for every $0 \leq \ell \leq 2k-2$ and every CM point $\tau$ of discriminant $d$. Note that $M\tau_P$ is a CM point of discriminant $d$ for every $M \in \Gamma$. Recall from Section~\ref{section prelims periods} that $r_n(E_{2k})$ vanishes for even $0 < n < 2k-2$ and is rational for odd $n$. Summarizing, we obtain the stated rationality result.
\end{proof}
Now we come to the proof of Theorem~\ref{theorem rationality}. First, it follows from the definition of $f_{k,P}(z)$ that
\[
f_{k,P'}(iy) = \overline{f_{k,P}(iy)},
\]
where we put $P' = [a,-b,c]$ for $P = [a,b,c]$. This implies
\begin{align}\label{eq real imag}
r_n(f_{k,P}^{+}) = \mathbb{R}e(r_n(f_{k,P})), \qquad r_n(f_{k,P}^{-}) = -\mathrm{Im}(r_n(f_{k,P})).
\end{align}
It is now easy to see from Theorem~\ref{theorem rn formula} and Lemma~\ref{lemma P rational} that $r_n(f_{k,P})$ for $0 < n < 2k-2$ is a real number if $n$ is odd, and a purely imaginary number if $n$ is even. Therefore, \eqref{eq real imag} implies $r_n(f_{k,P}^\varepsilon) = 0$ for $0 < n < 2k-2$, where $\varepsilon$ is the sign of $(-1)^n$. This concludes the proof of the first part of Theorem~\ref{theorem rationality}.
Concerning the period $r_0(f_{k,P}^+) = \mathbb{R}e(r_0(f_{k,P}))$, we obtain from Theorem~\ref{theorem rn formula} and equations \eqref{eq rational 1}--\eqref{eq rational 3} the formula
\begin{align*}
&r_0(f_{k,P}^+) = -\frac{2^{k-1}|d|^{\frac{k-1}{2}}}{(2k-1)|\overline{\Gamma}_P|\zeta(2k)} \\
&\times\bigg(2v_P^k \zeta(2k) + 2^{3-2k}v_P^{1-k}\pi \zeta(2k-1)\binom{2k-2}{k-1}-\frac{4(2k-1)!\zeta(2k)}{(4\pi)^{2k-1}(k-1)!} \mathbb{R}e(R_{2-2k}^{k-1}\mathcal{E}_{2k} (\tau_P))\bigg).
\end{align*}
Here we used the evalution of $r_0(E_{2k})$ given in Section~\ref{section prelims periods}. Writing out the Fourier expansion of $R_{2-2k}^{k-1}\mathcal{E}_{2k}(\tau_P)$ explicitly using the formula \eqref{raising and derivative} and the well-known expansion
\[
E_{2k}(\tau) = 1+\frac{(2\pi i)^{2k}}{(2k-1)!\zeta(2k)}\sum_{n=1}^\infty \sigma_{2k-1}(n)e(n\tau),
\]
and then using the formula for the Epstein zeta function given in equation (2.4) in \cite{smartepstein} (note that the quadratic form $Q$ in \cite{smartepstein} is normalized to have to discriminant $-4$), we see that the expression in the second line in the above formula for $r_0(f_{k,P}^+)$ equals $2^{-k}|d|^{\frac{k}{2}}\zeta_P(k)$. This proves the second part of Theorem~\ref{theorem rationality}.
In order to prove the rationality statement in the third item in Theorem~\ref{theorem rationality}, we note that the assumption $\sum_n a_n R_n(\tau) = 0$ implies that the raised Eichler integrals of the cusp forms $R_n(\tau)$ in Theorem~\ref{theorem rn formula} cancel out in the linear combination $\sum_n a_n r_n(f_{k,P})$. In particular, $\sum_n a_n r_n(f_{k,P})$ is given by a rational linear combination of the values $|d|^{\frac{k-1}{2}}R_{2-2k}^{k-1}\mathcal{P}_{1-k,n}(\tau_P)$. Now Lemma~\ref{lemma P rational} implies the third item in Theorem~\ref{theorem rationality}. This finishes the proof.
\end{document} | math | 61,654 |
\begin{document}
\title{On weak-strong uniqueness property for the full compressible magnetohydrodynamics flows
}
\author{Weiping Yan
}
\institute{College of of Mathematics, Jilin University, Changchun
130012, P.R. China.\\
Beijing International Center for Mathematical Research, Peking University, Beijing 100871, P.R. China.\\
\email{[email protected]}}
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract}
This paper is devoted to the study of the weak-strong uniqueness property for the full compressible magnetohydrodynamics flows.
The governing equations for magnetohydrodynamic flows are expressed by the
full Navier-Stokes system for compressible fluids enhanced by
forces due to the presence of the magnetic field as well as
the gravity and with an additional equation which describes the
evolution of the magnetic field. Using the relative entropy inequality, we prove that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists.
\keywords{magnetohydrodynamic flows \and weak solution \and strong solution \and entropy}
\subclass{ 76W05 \and 35D30 \and 35D35 \ and 54C70}
\end{abstract}
\section{Introduction and Main results}
\label{Sec:1} \indent
This paper studies the weak-strong uniqueness property of the viscous compressible magnetohydrodynamic flows
\begin{eqnarray}\label{E1-1}
&&\partial_t\rho+\textbf{div}_x(\rho\textbf{u})=0,\\
\label{E1-2}
&&\partial_t(\rho\textbf{u})+\textbf{div}_x(\rho\textbf{u}\otimes\textbf{u})+\nabla_xP(\rho,\theta)=\textbf{div}_x\textbf{S}+\textbf{J}\times\textbf{H},\\
\label{E1-1'}
&&\partial_t(\rho s(\rho,\theta))+\textbf{div}_x(\rho s(\rho,\theta)\textbf{u})+\textbf{div}_x(\frac{\textbf{q}}{\theta})=\sigma,\\
\label{E1-1R}
&&\partial_t\textbf{H}-\nabla\times(\textbf{u}\times\textbf{H})+\nabla\times(\nu\nabla\times\textbf{H})=0.
\end{eqnarray}
where $\textbf{u}$ is the vector field, $\rho$ is the density, $\theta$ is the temperature, $\textbf{J}$ is the electronic current, $e(\rho,\theta)$ is the (specific) internal energy and $\textbf{H}$ is
the magnetic field. The electronic
current satisfies Amp\`{e}re's law
\begin{eqnarray}\label{E1-3}
\textbf{J}=\nabla\times\textbf{H},
\end{eqnarray}
whereas the Lorentz force is given by
\begin{eqnarray}\label{E1-4}
\textbf{J}\times\textbf{H}=\textbf{div}_x(\frac{1}{\mu}\textbf{H}\otimes\textbf{H}-\frac{1}{2\mu}|\textbf{H}|^2\textbf{I}),
\end{eqnarray}
with $\mu$ being a permeability constant of free space, which here is assumed to be $\mu=1$ for simplicity
of the presentation. The electronic current $\textbf{J}$, the electric field $\textbf{E}$ and the magnetic field $\textbf{H}$ are related
through Ohm's law
\begin{eqnarray}\label{E1-5}
\textbf{J}=\sigma(\textbf{E}+\textbf{u}\times\textbf{H}).
\end{eqnarray}
The interaction described by the theory of magnetohydrodynamics, ``collective effects,'' is governed by
the Faraday's law,
\begin{eqnarray}\label{E1-6}
\partial_t\textbf{H}+\nabla\times\textbf{E}=0,~~\textbf{div}_x\textbf{H}=0.
\end{eqnarray}
Taking into consideration (\ref{E1-5}) we are able to write (\ref{E1-6}) in the following form
\begin{eqnarray}\label{E1-7}
\partial_t\textbf{H}+\nabla\times(\textbf{H}\times\textbf{u})+\nabla\times(\nu\nabla\times\textbf{H})=0,
\end{eqnarray}
where $\nu=\frac{1}{\sigma}$.
Motivated by several recent studies devoted to the scale analysis as well as numerical experiments
related to the proposed model (see Klein et al. \cite{Klein}), we suppose that the viscous stress $\textbf{S}$ is a linear function of the velocity gradient, therefore described by Newton's law
\begin{eqnarray}\label{E1-8}
\textbf{S}(\theta,\nabla_x\textbf{u})=\mu(\theta)(\nabla_x\textbf{u}+\nabla_x^{\perp}\textbf{u}-\frac{2}{3}\textbf{div}_x\textbf{u}\textbf{I})+\eta(\theta)\textbf{div}_x\textbf{u}\textbf{I},
\end{eqnarray}
while $\textbf{q}$ is the heat flux satisfying Fourier's law
\begin{eqnarray}\label{E1-9}
\textbf{q}=-\kappa(\theta)\nabla_x\theta,
\end{eqnarray}
and $\sigma$ stands for the entropy production rate which is non-negative measure given by
\begin{eqnarray}\label{E1-10}
\sigma\geq\frac{1}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta}).
\end{eqnarray}
We supplement compressible magnetohydrodynamic flows (\ref{E1-1})-(\ref{E1-1R}) with conservation boundary condition:
\begin{eqnarray}\label{E1-11}
\textbf{u}|_{\partial\Omega}=\textbf{q}\cdot\textbf{n}|_{\partial\Omega}=0,
\end{eqnarray}
and
\begin{eqnarray}\label{E1-12}
\textbf{H}|_{\partial\Omega}=0.
\end{eqnarray}
The concept of weak solution in fluid dynamics was introduced by Leray \cite{Ler} in the context of incompressible, linearly viscous fluids.
The original ideas of Leray have been put into the elegant framework of generalized derivatives
(distributions) and the associated abstract function spaces of Sobolev type (For example, see Ladyzhenskaya \cite{La} and Temam \cite{T}).
Lions \cite{Lions} extended the theory to the class of barotropic flows (see also \cite{Fei0}). One of meaningful compressible flow models is the compressible magnetohydrodynamics (MHD). It is a combination of the compressible Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.
Ducomet and Feireisl \cite{Du} proved that the existence of global in time weak solutions to
a multi-dimensional nonisentropic MHD system for gaseous stars coupled with the Poisson equation with
all the viscosity coefficients and the pressure depending on temperature and density asymptotically, respectively.
Hu and Wang \cite{Hu1} studied the global variational weak
solution to the three-dimensional full magnetohydrodynamic equations
with large data by an approximation scheme and a weak convergence
method. Jiang, et all. \cite{Jiang} obtained that the convergence towards the strong solution of the
ideal incompressible MHD system in the periodic domains. Recently, Kwon, et all \cite{Kwon} established the incompressible limits of weak solutions to the compressible magnetohydrodynamics flows (\ref{E1-1})-(\ref{E1-1R}) on both
bounded and unbounded domains.
The physical properties of the magnetohydrodynamics flows are reflected through various constitutive
relations which are expressed as typically non-linear functions relating the pressure $P=P(\rho,\theta)$,
the internal energy $e(\rho,\theta)$, the specific entropy $s=s(\rho,\theta)$ to the macroscopic variables $\rho$, $\textbf{u}$,
and $\theta$. According to the fundamental principles of thermodynamics, the specific internal energy $e$ is
related to the pressure $P$, and the specific entropy s through Gibbs' relation
\begin{eqnarray}\label{E1-13}
\theta Ds(\rho,\theta)=De(\rho,\theta)+P(\rho,\theta)D(\frac{1}{\rho}),
\end{eqnarray}
where $D$ denotes the differential with respect to the state variables $\rho$ and $\theta$.
Since the lack of information resulting from the inequality sign in (\ref{E1-10}), we need supplement the resulting system with the energy inequality,
\begin{eqnarray}\label{E1-14}
\frac{d}{dt}\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}|^2+\rho e(\rho,\theta)+\frac{1}{2}|\textbf{H}|^2)dx+\int_{\Omega}(|\nabla\textbf{u}|^2+\nu|\nabla\times\textbf{H}|^2)\leq0.
\end{eqnarray}
Thus the total energy $\mathcal{E}$ is given by
\begin{eqnarray}\label{E1-14R}
\mathcal{E}=\frac{1}{2}\rho|\textbf{u}|^2+\rho e(\rho,\theta)+\frac{1}{2}|\textbf{H}|^2.
\end{eqnarray}
Under these circumstances, it
can be shown (see \cite{Fei2}, Chapter 2) that any weak solution of (\ref{E1-1}) that is sufficiently
smooth satisfies, instead of (\ref{E1-10}), the standard relation
\begin{eqnarray}\label{E1-10R1}
\sigma=\frac{1}{\theta}\left(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta}\right).
\end{eqnarray}
The pressure $P=P(\rho,\theta)$ is here expressed as
\begin{eqnarray}\label{E1-15}
P=P_F+P_R,~~P_R=\frac{a}{3}\theta^4,~a>0,
\end{eqnarray}
where $P_R$ denotes the radiation pressure. Moreover, we shall assume that $P_F=P_M+P_E$ , where $P_M$ is the classical molecular pressure
obeying Boyle's law, while $P_E$ is the pressure of electron gas constituent behaving like a Fermi
gas in the degenerate regime of high densities and/or low temperatures (see Chapters 1, 15 in
Eliezer et al. \cite{E}). Thus necessarily $P_F$ takes the form
\begin{eqnarray}\label{E1-16}
P_F=\theta^{\frac{5}{2}}p(\frac{\rho}{\theta^{\frac{3}{2}}}),
\end{eqnarray}
where $p\in\textbf{C}^1[0,\infty)$ satisfies
\begin{eqnarray}\label{E1-17}
p(0)=0,~~p'(Z)>0~for~all~z\geq0.
\end{eqnarray}
In agreement with Gibbs' relation (\ref{E1-13}), the internal energy can be taken as
\begin{eqnarray}\label{E1-18}
e=e_F+e_R,~~with~e_R=a\frac{\theta^4}{\rho},
\end{eqnarray}
where $e_F=e_F(\rho,\theta)$, $P_F(\rho,\theta)$ are interrelated through the following equation of state
\begin{eqnarray}\label{E1-18RR}
P_F(\rho,\theta)=\frac{2}{3}\rho e_F(\rho,\theta).
\end{eqnarray}
We need the thermodynamic stability hypothesis:
\begin{eqnarray}\label{E1-18R}
\frac{\partial P(\rho,\theta)}{\partial\rho}>0,~~\frac{\partial e(\rho,\theta)}{\partial\theta}>0~~for~all~~\rho,\theta>0.
\end{eqnarray}
The second inequality in thermodynamic stability hypothesis (\ref{E1-18R}) gives that
\begin{eqnarray}\label{E1-19}
0<\frac{\frac{5}{3}p(Z)-p'(Z)Z}{Z}<c~~for~all~~Z>0,
\end{eqnarray}
which implies that the function $Z\mapsto\frac{p(Z)}{Z^{\frac{5}{3}}}$ is decreasing and we suppose that
\begin{eqnarray}\label{E1-20}
\lim_{Z\longrightarrow\infty}\frac{p(Z)}{Z^{\frac{5}{3}}}=p_{\infty}>0.
\end{eqnarray}
In accordance with (\ref{E1-13}) and (\ref{E1-18}) we set the entropy as
\begin{eqnarray}\label{E1-21}
s=s_F+s_R,~~wtih~s_F=S(\frac{\rho}{\theta^{\frac{3}{2}}}),~~s_R=\frac{4a}{3\rho}\theta^3.
\end{eqnarray}
Furthermore, by the Third law of thermodynamics,
\begin{eqnarray}\label{E1-22}
S'(Z)=-\frac{3}{2}\frac{\frac{5}{3}p(Z)-p'(Z)Z}{Z^2}<0,~~\lim_{Z\longrightarrow\infty}S(Z)=0.
\end{eqnarray}
We choose the transport coefficients in the form
\begin{eqnarray}\label{E1-23}
&&\mu(\theta)=\mu_0+\mu_1\theta,~~\mu_0,\mu_1>0,~~\eta\equiv0,\\
\label{E1-24}
&&\kappa(\theta)=\kappa_0+\kappa_2\theta^2+\kappa_3\theta^3,~~\kappa_i>0,~~i=0,2,3.
\end{eqnarray}
A fundamental test of admissibility of a class of weak solutions to a given evolutionary
problem is the property of weak-strong uniqueness. More specifically, the
weak solution must coincide with a (hypothetical) strong solution emanating from the same initial data as long as the latter exists. This problem has been intensively
studied for the incompressible Navier-Stokes system, for example, see \cite{Che,Pi1,Ser}. It is a bit more delicate in the case of compressible cases. The weak-
strong uniqueness of compressible barotropic Navier-Stokes system and isentropic compressible Navier-Stokes system were established in \cite{Fei11,Fei3} and \cite{Pi2}, respectively. P. Germain \cite{Pi2} provides only a partial
and conditional answer to the weak-strong uniqueness problem for the compressible Navier-Stokes equations. This question is definitely solved in \cite{Fei3}.
More recently, Feireisl and Novotn\'{y} \cite{Fei1} extended the problem to compressible Navier-Stokes-Fourier system by the relative entropy inequality.
The relative entropy in \cite{Fei1} is reminiscent to C.M. Dafermos \cite{Daf} (who introduced
the relatives entropies via the entropy flux pairs for the conservation laws),
but is different from the C.M. Dafermos concept (in contrast to \cite{Daf}, it is based on
the thermodynamic stability conditions).
Inspired by the work of Feireisl and Novotn\'{y} \cite{Fei1},
we prove that the weak-
strong uniqueness of compressible three-dimensional magnetohydrodynamic equations. Our contribution is to
construct suitable relative entropy inequality to (\ref{E1-1})-(\ref{E1-1'}). Then we overcome the presence of the magnetic field and its interaction with the
hydrodynamic motion in the MHD flow of large oscillation.
We organize the rest of this paper as follows. In section 2, we recall the definition of
the weak solutions and strong solutions to the magnetohydrodynamic
flows on bounded domains. Meanwhile, the relative entropy inequality of (\ref{E1-1})-(\ref{E1-1R}) is derived.
In the last section, we give the rigorous proof of the weak-strong uniqueness property for the compressible magnetohydrodynamic
flows on bounded domains in the spirit of Feireisl and Novotn\'{y} \cite{Fei1}.
\section{Relative entropy and Main result}
Let $\Omega\subset\textbf{R}^3$ be a bounded Lipschitz domain. We recall the definition of weak solution for (\ref{E1-1})-(\ref{E1-1R}).
\begin{definition}
We say that a quantity $(\rho,\textbf{u},\theta,\textbf{H})$ is a weak solution of the
full magnetohydrodynamic flows (MHD) (\ref{E1-1})-(\ref{E1-1R}) supplemented with the initial data $(\rho_0,\textbf{u}_0,s(\rho_0,\theta_0),\textbf{H}_0)$, and $\rho_0\geq0,\theta_0>0$ provided
that the following holds.
\textbf{i)} The density $\rho$ is a non-negative function, $\rho\in\textbf{C}_{weak}([0,T];\textbf{L}^{\frac{5}{3}}(\Omega))$, the velocity field $\textbf{u}\in\textbf{L}^2(0,T;\textbf{W}_0^{1,2}(\Omega;\textbf{R}^3))$, $\rho\textbf{u}\in\textbf{C}_{weak}([0,T];\textbf{L}^{\frac{5}{4}}(\Omega;\textbf{R}^3))$. Equation (\ref{E1-1}) is replaced by a family of integral identities
\begin{eqnarray}\label{E2-1}
\int_{\Omega}\rho(\tau,\cdot)\varphi(\tau,\cdot)dx-\int_{\Omega}\rho_0\varphi(0,\cdot)dx=\int_0^{\tau}\int_{\Omega}(\rho\partial_t\varphi+\rho\textbf{u}\cdot\nabla_x\varphi)dxdt
\end{eqnarray}
for any $\varphi\in\textbf{C}^1([0,T]\times\bar{\Omega})$, and any $\tau\in[0,T]$.
\textbf{ii)} The balance of momentum holds in distributional sense, namely
\begin{eqnarray}\label{E2-2}
&&\int_{\Omega}\rho\textbf{u}(\tau,\cdot)\cdot\varphi(\tau,\cdot)dx-\int_{\Omega}\rho_0\textbf{u}_0\cdot\varphi(0,\cdot)dx\nonumber\\
&=&\int_0^{\tau}\int_{\Omega}(\rho\textbf{u}\cdot\partial_t\varphi+\rho\textbf{u}\otimes\textbf{u}:\nabla_x\varphi+P\textbf{div}_x\varphi-\textbf{S}:\nabla_x\varphi\nonumber\\
&&+[(\nabla\times\textbf{H})\times\textbf{H}]\cdot\varphi)dxdt
\end{eqnarray}
for any $\varphi\in\textbf{C}^1([0,T]\times\bar{\Omega};\textbf{R}^3)$, $\varphi|_{\partial\Omega}=0$ and any $\tau\in[0,T]$.
\textbf{iii)} The entropy balance (\ref{E1-1'}) and (\ref{E1-10}) are replaced by a family of integral inequalities
\begin{eqnarray}\label{E2-3}
&&\int_{\Omega}\rho s(\rho_0,\theta_0)\varphi(0,\cdot)dx-\int_{\Omega}\rho s(\rho,\theta)(\tau,\cdot)\varphi(\tau,\cdot)dx\nonumber\\
&+&\int_0^{\tau}\int_{\Omega}\left(\frac{\varphi}{\theta}(\textbf{S}:\nabla_x\textbf{u}-\frac{\textbf{q}\cdot\nabla_x\theta}{\theta}\right)dxdt\nonumber\\
&\leq&-\int_0^{\tau}\int_{\Omega}\left(\rho s(\rho,\theta)\partial_t\varphi+\rho s(\rho,\theta)\textbf{u}\cdot\nabla_x\varphi+\frac{\textbf{q}\cdot\nabla_x\varphi}{\theta}\right)dxdt
\end{eqnarray}
for any $\varphi\in\textbf{C}^1([0,T]\times\bar{\Omega})$, $\varphi\geq0$ and almost all $\tau\in[0,T]$. Here the quantities $\textbf{S}$ and $\textbf{q}$ are given through the constitutive equations (\ref{E1-8}) and (\ref{E1-9}). Moreover,
similarly to the above, all quantities must be at least integrable on $(0,T)\times\Omega$. In particular,
$\theta$ belongs to $\textbf{L}^{\infty}(0,T;\textbf{L}^4(\Omega))\cap\textbf{L}^2(0,T;\textbf{W}^{1,2}(\Omega))$. In addition, we require $\theta$ to be positive for almost all $(t,x)\in(0,T)\times\Omega$.
\textbf{iv)} The total energy of the system satisfies the following inequality
\begin{eqnarray}\label{E2-4}
\int_{\Omega}\left(\frac{1}{2}\rho|\textbf{u}|^2+\rho e(\rho,\theta)+\frac{1}{2}|\textbf{H}|^2\right)dx
&+&\int_0^{\tau}\int_{\Omega}(|\nabla\textbf{u}|^2+\nu|\nabla\times\textbf{H}|^2)dxdt\nonumber\\
&\leq&\int_{\Omega}\left(\frac{1}{2}\rho_0|\textbf{u}_0|^2+\rho_0e(\rho_0,\theta_0)+\frac{1}{2}|\textbf{H}_0|^2\right)dx~~~~~~~~
\end{eqnarray}
for almost all $\tau\in[0,T]$.
\textbf{v)} The magnetic field $\textbf{H}\in\textbf{L}^2(0,T;\textbf{W}^{1,2}(\Omega;\textbf{R}^3))$. The Maxwell equation (\ref{E1-1R}) verifies
\begin{eqnarray}\label{E2-5}
\int_{\Omega}\textbf{H}(\tau,\cdot)\varphi(\tau,\cdot)dx&-&\int_{\Omega}\textbf{H}_0\varphi_0dx\nonumber\\
&=&\int_0^{\tau}\int_{\Omega}\left(\textbf{H}\cdot\partial_t\varphi-(\textbf{H}\times\textbf{u}+\nu\nabla\times\textbf{H})\cdot(\nabla\times\varphi)\right)dxdt,~~~~~~~~
\end{eqnarray}
where $\varphi\in\textbf{C}^1([0,T]\times\bar{\Omega};\textbf{R}^3)$, $\varphi|_{\partial\Omega}=0$ and any $\tau\in[0,T]$.
\end{definition}
The definition of strong solution is
\begin{definition}
We say that $(\rho',\textbf{u}',\theta',\textbf{H}')$ is a classical (strong) solution to the full magnetohydrodynamic system (\ref{E1-1})-(\ref{E1-1R}) in $(0,T)\times\Omega$ if
\begin{eqnarray}\label{E2-x1}
&&\rho'\in\textbf{C}^1([0,T]\times\bar{\Omega}),~~\theta',\partial_t\theta',\nabla^2\theta'\in\textbf{C}([0,T]\times\Omega),\nonumber\\
&&\textbf{u}',\partial_t\textbf{u}',\nabla^2\textbf{u}'\in\textbf{C}([0,T]\times\Omega;\textbf{R}^3),~~
\textbf{H}',\partial_t\textbf{H}',\nabla^2\textbf{H}'\in\textbf{C}([0,T]\times\Omega;\textbf{R}^3),~~~~~\nonumber\\
&&\rho'(t,x)\geq\rho>0,~~\theta'(t,x)\geq\theta'_0>0,~~for~all~(t,x),
\end{eqnarray}
and $\rho',\textbf{u}',\theta',\textbf{H}'$ satisfy equations (\ref{E1-1})-(\ref{E1-1R}), (\ref{E1-10R1}), together with the boundary conditions
(\ref{E1-11})-(\ref{E1-12}). Observe that hypothesis (\ref{E2-x1}) implies the following regularity properties of
the initial data:
\begin{eqnarray}\label{E2-x2}
&&\rho(0)=\rho_0\in\textbf{C}^1(\bar{\Omega}),~~\rho_0\geq\rho_0'>0,\nonumber\\
&&\textbf{u}(0)=\textbf{u}_0\in\textbf{C}^2(\bar{\Omega}),\nonumber\\
&&\theta(0)=\theta_0\in\textbf{C}^2(\bar{\Omega}),~~\theta_0\geq\theta_0'>0,\nonumber\\
&&\textbf{H}(0)=\textbf{H}_0\in\textbf{C}^2(\bar{\Omega}).
\end{eqnarray}
\end{definition}
Before giving the main result, we deduce a relative entropy inequality which is satisfied by any weak solution to the full magnetohydrodynamic system (\ref{E1-1})-(\ref{E1-1R}).
Let $\{A,B,C,D\}$ be a quantity of smooth function, $A$ and $C$ bounded below away from zero in $[0,T]\times\Omega$, and $B|_{\partial\Omega}=D|_{\partial\Omega}=0$. Moreover, we assume that smooth functions $B$ and $D$ satisfy that
\begin{eqnarray}\label{E2-5R1}
\partial_tD-\nabla\times(B\times D)+\nabla\times(\nu\nabla\times D)=0.
\end{eqnarray}
Taking $\varphi=\frac{1}{2}|B|^2$, $\varphi=B$ and $\varphi=C>0$ as a test function in (\ref{E2-1}), (\ref{E2-2}) and the entropy inequality (\ref{E2-3}), respectively, we get
\begin{eqnarray}\label{E2-x3}
\int_{\Omega}\frac{1}{2}\rho|B|^2(\tau,\cdot)dx-\int_{\Omega}\frac{1}{2}\rho_0|B|^2(0,\cdot)dx=\int_{0}^{\tau}\int_{\Omega}(\rho B\cdot\partial_t B+\rho\textbf{u}\cdot\nabla_xB\cdot B)dxdt,~~~~
\end{eqnarray}
\begin{eqnarray}\label{E2-x4}
\int_{\Omega}\rho\textbf{u}\cdot B(\tau,\cdot)dx-\int_{\Omega}\rho_0\textbf{u}_0\cdot B(0,\cdot)dx&=&\int_{0}^{\tau}\int_{\Omega}(\rho\textbf{u}\cdot\partial_tB
+\rho\textbf{u}\otimes\textbf{u}:\nabla_x B+P(\rho,\theta)\textbf{div}_xB\nonumber\\
&&-\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB+((\nabla\times\textbf{H})\times\textbf{H})\cdot B)dxdt~~~~~~
\end{eqnarray}
and
\begin{eqnarray}\label{E2-x5}
\int_{\Omega}\rho_0 s(\rho_0,\theta_0)C(0,\cdot)dx&-&\int_{\Omega}\rho s(\rho,\theta)C(\tau,\cdot)dx+\int_{0}^{\tau}\int_{\Omega}\frac{C}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&-\int_{0}^{\tau}\int_{\Omega}(\rho s(\rho,\theta)\partial_tC+\rho s(\rho,\theta)\textbf{u}\cdot\nabla_x C+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x C)dxdt.~~~~~
\end{eqnarray}
It follows from (\ref{E2-x3}), (\ref{E2-x4}) and the energy inequality (\ref{E2-4}) that
\begin{eqnarray}\label{E2-x6}
\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-B|^2&+&\rho e(\rho,\theta)+\frac{1}{2}|\textbf{H}|^2)(\tau,\cdot)dx
+\int_0^{\tau}\int_{\Omega}(|\nabla\textbf{u}|^2+\nu|\nabla\times\textbf{H}|^2)dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-B(0,\cdot)|^2+\rho_0 e(\rho_0,\theta_0)+\frac{1}{2}|\textbf{H}_0|^2)dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\rho\partial_tB+\rho\textbf{u}\cdot\nabla_xB)\cdot(B-\textbf{u})-P(\rho,\theta)\textbf{div}_xB\nonumber\\
&&+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB-((\nabla\times\textbf{H})\times\textbf{H})\cdot B)dxdt.
\end{eqnarray}
Then summing up (\ref{E2-x5}) and (\ref{E2-x6}), we deduce that
\begin{eqnarray}\label{E2-x6'}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-B|^2+\rho e(\rho,\theta)+\frac{1}{2}|\textbf{H}|^2-C\rho s(\rho,\theta))(\tau,\cdot)dx\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}(|\nabla\textbf{u}|^2+\nu|\nabla\times\textbf{H}|^2)dxdt+\int_{0}^{\tau}\int_{\Omega}\frac{C}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-B(0,\cdot)|^2+\rho_0 e(\rho_0,\theta_0)+\frac{1}{2}|\textbf{H}_0|^2+C(0,\cdot)\rho_0 s(\rho_0,\theta_0))dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\rho\partial_tB+\rho\textbf{u}\cdot\nabla_xB)\cdot(B-\textbf{u})-P(\rho,\theta)\textbf{div}_xB+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB-((\nabla\times\textbf{H})\times\textbf{H})\cdot B)dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\rho s(\rho,\theta)\partial_tC+\rho s(\rho,\theta)\textbf{u}\cdot\nabla_x C+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x C)dxdt.
\end{eqnarray}
Taking a test function $\varphi=D$ in (\ref{E2-5}) and $\varphi=\partial_{\rho}H_{C}(A,C)$ in (\ref{E2-1}), we have
\begin{eqnarray}\label{E2-5R}
\int_{\Omega}\textbf{H}(\tau,\cdot)D(\tau,\cdot)dx-\int_{\Omega}\textbf{H}_0D_0dx
=\int_0^{\tau}\int_{\Omega}\left(\textbf{H}\cdot\partial_tD-(\textbf{H}\times\textbf{u}+\nu\nabla\times\textbf{H})\cdot(\nabla\times D)\right)dxdt,~~~~
\end{eqnarray}
\begin{eqnarray}\label{E2-x7}
\int_{\Omega}\rho\partial_{\rho}H_{C}(A,C)(\tau,\cdot)dx&-&\int_{\Omega}\rho_0\partial_{\rho}H_{C(0,\cdot)}(A(0,\cdot),C(0,\cdot))dx\nonumber\\
&=&\int_0^{\tau}\int_{\Omega}(\rho\partial_t(\partial_{\rho}H_{C}(A,C)))+\rho\textbf{u}\cdot\nabla_x(\partial_{\rho}H_{C}(A,C))dxdt,~~~~~~
\end{eqnarray}
Multiplying (\ref{E2-5R1}) by $D$ and integrate over $(0,\tau)\times\Omega$, we find
\begin{eqnarray}\label{E2-5R2}
\int_{\Omega}\frac{1}{2}|D|^2(\tau,\cdot)dx-\int_{\Omega}\frac{1}{2}|D_0|^2dx
=-\int_0^{\tau}\int_{\Omega}\left((D\times B+\nu\nabla\times D)\cdot(\nabla\times D)\right)dxdt.~~~~
\end{eqnarray}
So by (\ref{E2-x6'})-(\ref{E2-5R2}), we have
\begin{eqnarray}\label{E2-x8}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-B|^2+\frac{1}{2}|\textbf{H}-D|^2+H_{C}(\rho,\theta)-\partial_{\rho}(H_{C})(A,C)(\rho-A)-H_{C}(A,C))(\tau,\cdot)dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left(|\nabla\textbf{u}|^2+\nu(|\nabla\times D|^2-|\nabla\times D||\nabla\times\textbf{H}|+|\nabla\times\textbf{H}|^2)\right)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\frac{C}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-B(0,\cdot)|^2+\frac{1}{2}|\textbf{H}_0-D_0|^2+(H_{C(0,\cdot)}(\rho(0,\cdot),\theta(0,\cdot))\nonumber\\
&&-\partial_{\rho}(H_{C(0,\cdot)})(A(0,\cdot),C(0,\cdot))(\rho_0-A(0,\cdot))-H_{C(0,\cdot)}(A(0,\cdot),C(0,\cdot))))dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left((\rho\partial_tB+\rho\textbf{u}\cdot\nabla_xB)\cdot(B-\textbf{u})-P(\rho,\theta)\textbf{div}_xB
+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB\right)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(-\textbf{H}\cdot\partial_tD-((\nabla\times\textbf{H})\times\textbf{H})\cdot B-(D\times B)\cdot(\nabla\times D)+(\textbf{H}\times\textbf{u})\cdot(\nabla\times D)) dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\rho s(\rho,\theta)\partial_tC+\rho s(\rho,\theta)\textbf{u}\cdot\nabla_x C+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x C)dxdt\nonumber\\
&&-\int_0^{\tau}\int_{\Omega}(\rho\partial_t(\partial_{\rho}(H_{C})(A,C)))+\rho\textbf{u}\cdot\nabla_x(\partial_{\rho}H_{C}(A,C))dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}\partial_t(A\partial_{\rho}(H_{C})(A,C)-H_{C}(A,C))dxdt.
\end{eqnarray}
Replacing $\partial_tD$ by (\ref{E2-5R1}) in (\ref{E2-x8}) to find
\begin{eqnarray}\label{E2-R5}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-B|^2+\frac{1}{2}|\textbf{H}-D|^2+H_{C}(\rho,\theta)-\partial_{\rho}(H_{C})(A,C)(\rho-A)-H_{C}(A,C))(\tau,\cdot)dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left(|\nabla\textbf{u}|^2+\nu|\nabla\times D-\nabla\times\textbf{H}|^2\right)dxdt+\int_{0}^{\tau}\int_{\Omega}\frac{C}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-B(0,\cdot)|^2+\frac{1}{2}|\textbf{H}_0-D_0|^2+\frac{1}{2}\rho_0|B_0|^2+(H_{C(0,\cdot)}(\rho(0,\cdot),\theta(0,\cdot))\nonumber\\
&&-\partial_{\rho}(H_{C(0,\cdot)})(A(0,\cdot),C(0,\cdot))(\rho_0-A(0,\cdot))-H_{C(0,\cdot)}(A(0,\cdot),C(0,\cdot))))dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left((\rho\partial_tB+\rho\textbf{u}\cdot\nabla_xB)\cdot (B-\textbf{u}))-P(\rho,\theta)\textbf{div}_xB
+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB\right)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left(-((\nabla\times\textbf{H})\times\textbf{H})\cdot B-(D\times B)\cdot(\nabla\times D)+(\textbf{H}\times\textbf{u})\cdot(\nabla\times D)\right) dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\nabla\times(B\times D))\cdot\textbf{H}dxdt-\int_{0}^{\tau}\int_{\Omega}(\rho s(\rho,\theta)\partial_tC+\rho s(\rho,\theta)\textbf{u}\cdot\nabla_x C+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x C)dxdt\nonumber\\
&&-\int_0^{\tau}\int_{\Omega}(\rho\partial_t(\partial_{\rho}(H_{C})(A,C)))+\rho\textbf{u}\cdot\nabla_x(\partial_{\rho}H_{C}(A,C))dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}\partial_t(A\partial_{\rho}(H_{C})(A,C)-H_{C}(A,C))dxdt.
\end{eqnarray}
Note that
\begin{eqnarray}\label{E9-5}
\int_{\Omega}\left((\nabla\times\textbf{H})\times\textbf{H}\right)\cdot Bdx=-\int_{\Omega}\left(\textbf{H}^{\top}B\textbf{H}+\frac{1}{2}\nabla(|\textbf{H}|^2)\cdot B\right)dx,
\end{eqnarray}
\begin{eqnarray*}
\int_{\Omega}\left(\nabla\times(B\times\textbf{H})\right)\cdot Bdx=\int_{\Omega}\left(\textbf{H}^{\top}B\textbf{H}+\frac{1}{2}\nabla(|\textbf{H}|^2)\cdot B\right)dx.
\end{eqnarray*}
So direct calculation shows that
\begin{eqnarray}\label{E2-8}
&&\int_{0}^{\tau}\int_{\Omega}\left(-((\nabla\times\textbf{H})\times\textbf{H})\cdot B-(D\times B)\cdot(\nabla\times D)+(\textbf{H}\times\textbf{u})\cdot(\nabla\times D)\right) dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}\nabla\times(B\times D)\cdot\textbf{H}dxdt\nonumber\\
&=&\int_{0}^{\tau}\int_{\Omega}\left(-((\nabla\times\textbf{H})\times\textbf{H})\cdot B-\nabla\times(B\times D)\cdot\textbf{H}\right) dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left(-(D\times B)\cdot(\nabla\times D)+(\textbf{H}\times\textbf{u})\cdot(\nabla\times D)\right) dxdt\nonumber\\
&=&\int_{0}^{\tau}\int_{\Omega}((\textbf{H}-D)^{\top}\nabla B(\textbf{H}-D)+\frac{1}{2}\nabla(|\textbf{H}-D|^2)\cdot B)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(D-\textbf{H})^{\top}\nabla(\textbf{u}-B)D+\frac{1}{2}\nabla(D(D-\textbf{H}))(\textbf{u}-B)dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(D^{\top}\nabla(\textbf{u}-B)D+\frac{1}{2}\nabla(|D|^2)(\textbf{u}-B))dxdt.
\end{eqnarray}
Note that
\begin{eqnarray*}
\partial_y(\partial_{\rho}H_{C}(A,C))&=&-s(A,C)\partial_yC-A\partial_{\rho}s(A,C)\partial_yC+\partial^2_{\rho,\rho}H_C(A,C)\partial_y\rho\nonumber\\
&&+\partial^2_{\rho,\theta}H_C(A,C)\partial_yC,~~for~y=t,x.
\end{eqnarray*}
Thus it follows from (\ref{E2-R5}) and (\ref{E2-8}) that
\begin{eqnarray}\label{E2-6}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-B|^2+\frac{1}{2}|\textbf{H}-D|^2+H_{C}(\rho,\theta)-\partial_{\rho}(H_{C})(A,C)(\rho-A)-H_{C}(A,C))(\tau,\cdot)dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left(|\nabla\textbf{u}|^2+\nu|\nabla\times D-\nabla\times\textbf{H}|^2\right)dxdt+\int_{0}^{\tau}\int_{\Omega}\frac{C}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-B(0,\cdot)|^2+\frac{1}{2}|\textbf{H}_0-D_0|^2+(H_{C(0,\cdot)}(\rho(0,\cdot),\theta(0,\cdot))\nonumber\\
&&-\partial_{\rho}(H_{C(0,\cdot)})(A(0,\cdot),C(0,\cdot))(\rho_0-A(0,\cdot))-H_{C(0,\cdot)}(A(0,\cdot),C(0,\cdot))))dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\rho\partial_tB+\rho\textbf{u}\cdot\nabla_xB)\cdot(B-\textbf{u})-P(\rho,\theta)\textbf{div}_xB+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\textbf{H}-D)^{\top}\nabla B(\textbf{H}-D)+\frac{1}{2}\nabla(|\textbf{H}-D|^2)\cdot B)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(D-\textbf{H})^{\top}\nabla(\textbf{u}-B)D+\frac{1}{2}\nabla(D(D-\textbf{H}))(\textbf{u}-B)dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(D^{\top}\nabla(\textbf{u}-B)D+\frac{1}{2}\nabla(|D|^2)(\textbf{u}-B))dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\rho(s(\rho,\theta)-s(A,C))\partial_tC+\rho(s(\rho,\theta)-s(A,C))\textbf{u}\cdot\nabla_x C+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x C)dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}\rho(A\partial_{\rho}s(A,C)\partial_tC+r\partial_{\rho}s(A,C)\textbf{u}\cdot\nabla_xC)dxdt\nonumber\\
&&-\int_0^{\tau}\int_{\Omega}\rho(\partial^2_{\rho,\rho}(H_C)(A,C)\partial_tA+\partial^2_{\rho,\theta}(H_{C})(A,C)\partial_tC)dxdt\nonumber\\
&&-\int_0^{\tau}\int_{\Omega}\rho\textbf{u}(\partial^2_{\rho,\rho}(H_C)(A,C)\nabla_xA+\partial^2_{\rho,\theta}(H_{C})(A,C)\nabla_xC)dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}\partial_t(A\partial_{\rho}(H_{C})(A,C)-H_{C}(A,C))dxdt.
\end{eqnarray}
Following \cite{Car,Fei1,Sa}, introducing the quantity as
\begin{eqnarray*}
\Gamma(\rho,\theta|C,C)=H_{C}(\rho,\theta)-\partial_{\rho}H_{C}(A,C)(\rho-A)-H_{C}(A,C),
\end{eqnarray*}
where
\begin{eqnarray*}
H_C(\rho,\theta)=\rho e(\rho,\theta)-C\rho s(\rho,\theta).
\end{eqnarray*}
Note that
\begin{eqnarray}\label{E2-12}
&&\partial^2_{\rho,\rho}H_C(A,C)=\frac{1}{A}\partial_{\rho}P(A,C),~~A\partial_{\rho}s(A,C)=-\frac{1}{C}\partial_{\theta}P(A,C),\nonumber\\
&&\partial^2_{\rho,\theta}H_C(A,C)=\partial_{\rho}(\rho(\theta-C)\partial_{\theta}s)(A,C)=(\theta-C)\partial_{\rho}(\rho\partial_{\theta}s(\rho,\theta))(A,C)=0,~~~~~~\\
&&A\partial_{\rho}(H_{C})(A,C)-H_{C}(A,C)=P(A,C).\nonumber
\end{eqnarray}
Therefore, we can obtain a kind of relative entropy inequality by simplifying (\ref{E2-6}) as
\begin{eqnarray}\label{E2-7}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-B|^2+\frac{1}{2}|\textbf{H}-D|^2+\Gamma(\rho,\theta|A,C))(\tau,\cdot)dx+\nu\int_{0}^{\tau}\int_{\Omega}|\nabla\times D-\nabla\times\textbf{H}|^2dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\frac{C}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-B(0,\cdot)|^2+\frac{1}{2}|\textbf{H}_0-D_0(0,\cdot)|^2+\Gamma(\rho_0,\theta_0|A(0,\cdot),C(0,\cdot)))\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}\rho(\textbf{u}-B)\cdot\nabla_xB\cdot(B-\textbf{u})dxdt+\int_{0}^{\tau}\int_{\Omega}\rho(s(\rho,\theta)-s(A,C))(B-\textbf{u})\cdot\nabla_xCdxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\rho\partial_tB+\rho\textbf{u}\cdot\nabla_xB)\cdot(B-\textbf{u})-P(\rho,\theta)\textbf{div}_xB+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_xB)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\textbf{H}-D)^{\top}\nabla B(\textbf{H}-D)+\frac{1}{2}\nabla(|\textbf{H}-D|^2)\cdot B)dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(D-\textbf{H})^{\top}\nabla(\textbf{u}-B)D+\frac{1}{2}\nabla(D(D-\textbf{H}))(\textbf{u}-B)dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(D^{\top}\nabla(\textbf{u}-B)D+\frac{1}{2}\nabla(|D|^2)(\textbf{u}-B))dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\rho(s(\rho,\theta)-s(A,C))\partial_tC+\rho(s(\rho,\theta)-s(A,C))\textbf{u}\cdot\nabla_x C+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x C)dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}((1-\frac{\rho}{A})\partial_tP(A,C)-\frac{\rho}{A}\textbf{u}\cdot\nabla_xP(A,C))dxdt.
\end{eqnarray}
Now we state the weak-strong uniqueness property to the full magnetohydrodynamic system (\ref{E1-1})-(\ref{E1-1R})
on a bounded Lipschitz domains with Dirichlet
boundary conditions.
\begin{theorem}
Let $\Omega\subset\textbf{R}^3$ be a bounded Lipschitz domain and $(\rho,\textbf{u},\theta,\textbf{H})$ be a weak solution of the full magnetohydrodynamic system (\ref{E1-1})-(\ref{E1-1R}) in $(0,T)\times\Omega$ and $(\rho',\textbf{u}',\theta',\textbf{H}')$ be a strong solution emanating from the same initial data (\ref{E2-x2}).
Assume that the thermodynamic
functions $P,$ $e$, $s$ satisfy hypotheses (\ref{E1-15})-(\ref{E1-22}), and that the transport
coefficients $\mu$, $\eta$ and $\kappa$ satisfy (\ref{E1-23})-(\ref{E1-24}). Then
\begin{eqnarray*}
\rho\equiv\rho',~~\textbf{u}=\textbf{u}',~~\theta=\theta',~~\textbf{H}=\textbf{H}'.
\end{eqnarray*}
\end{theorem}
\section{Proof of Theorem 1}
In this section, we apply the relative entropy inequality to finish the proof of Theorem 1. Assume that $(\rho',\textbf{u}',\theta',\textbf{H}')$ is a classical (strong) solution to the full magnetohydrodynamic system in $(0,T)\times\Omega$, it satisfies that
\begin{eqnarray*}
\rho'(0,\cdot)=\rho_0,~~\textbf{u}'(0,\cdot)=\textbf{u}_0,~~\theta'(0,\cdot)=\theta_0,~~\textbf{H}'(0,\cdot)=\textbf{H}_0.
\end{eqnarray*}
Following \cite{Fei2,Fei1}, we introduce essential and residual component of
each quantity appearing in (\ref{E2-12}).
Thermodynamic stability hypothesis (\ref{E1-18R}) implies that
$\rho\mapsto H_{\theta'}(\rho,\theta')$ is strictly convex,
while $\theta\mapsto H_{\theta'}(\rho,\theta')$ attains its global minimum at $\theta=\theta'$.
Thus it has
\begin{eqnarray}\label{E3-2}
\Gamma(\rho,\theta|\rho',\theta')\geq c\left\{
\begin{array}{lll}
&&|\rho-\rho'|^2+|\theta-\theta'|^2~~if~~(\rho,\theta)\in[\rho_0',\rho_1']\times[\theta_0',\theta_1']\\
&&1+|\rho s(\rho,\theta)|+\rho e(\rho,\theta)~~otherwise,
\end{array}
\right.
\end{eqnarray}
where $[\rho',\theta']\in[\rho_0',\rho_1']\times[\theta_0',\theta_1']$, the constant $c$ depends on positive constants $\rho_0',\rho_1',\theta_0',\theta_1'$ and the structural properties of the thermodynamic function $e$, $s$. More precisely, the restriction of positive constants $\rho_0',\rho_1',\theta_0',\theta_1'$ can be found in \cite{Fei1}.
Thus we can write each measurable function $h=h_{ess}+h_{res}$,
where
\begin{eqnarray*}
h_{ess}=\left\{
\begin{array}{lll}
&&h(t,x)~~if~~(\rho,\theta)\in[\rho_0',\rho_1']\times[\theta_0',\theta_1']\\
&&0~~otherwise.
\end{array}
\right.
\end{eqnarray*}
Taking $(A,B,C,D)=(\rho',\textbf{u}',\theta',\textbf{H}')$ in (\ref{E2-7}). By the fact that the initial data coincide, we have
\begin{eqnarray}\label{E3-1}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2+\frac{1}{2}|\textbf{H}-\textbf{H}'|^2+\Gamma(\rho,\theta|\rho',\theta))(\tau,\cdot)dx+\nu\int_{0}^{\tau}\int_{\Omega}|\nabla\times\textbf{H}'-\nabla\times\textbf{H}|^2dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\frac{\theta'}{\theta}(\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}-\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta})dxdt\nonumber\\
&\leq&\int_{\Omega}(\frac{1}{2}\rho_0|\textbf{u}_0-\textbf{u}'(0,\cdot)|^2+\frac{1}{2}|\textbf{H}_0-\textbf{H}'_0(0,\cdot)|^2+\Gamma(\rho_0,\theta_0|\rho'(0,\cdot),\theta'(0,\cdot)))\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\rho|\textbf{u}-\textbf{u}'|^2|\nabla_x\textbf{u}'|dxdt+\int_{0}^{\tau}\int_{\Omega}\rho(s(\rho,\theta)-s(\rho,\theta))(\textbf{u}'-\textbf{u})\cdot\nabla_x\theta'dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\rho\partial_t\textbf{u}'+\rho\textbf{u}\cdot\nabla_x\textbf{u}')\cdot(\textbf{u}'-\textbf{u})-P(\rho,\theta)\textbf{div}_x\textbf{u}'+\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}')dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}((\textbf{H}-\textbf{H}')^{\top}\nabla\textbf{u}'(\textbf{H}-\textbf{H}')+\frac{1}{2}\nabla(|\textbf{H}-\textbf{H}'|^2)\cdot \textbf{u}')dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(\textbf{H}'-\textbf{H})^{\top}\nabla(\textbf{u}-\textbf{u}')\textbf{H}'+\frac{1}{2}\nabla(\textbf{H}'(\textbf{H}'-\textbf{H}))(\textbf{u}-\textbf{u}')dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\textbf{H}'^{\top}\nabla(\textbf{u}-\textbf{u}')\textbf{H}'+\frac{1}{2}\nabla(|\textbf{H}'|^2)(\textbf{u}-\textbf{u}'))dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\rho(s(\rho,\theta)-s(\rho',\theta'))\partial_t\theta'+\rho(s(\rho,\theta)-s(\rho',\theta'))\textbf{u}'\cdot\nabla_x\theta'+\frac{\textbf{q}(\theta,\nabla_x\theta)}{\theta}\cdot\nabla_x\theta')dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}((1-\frac{\rho}{\rho'})\partial_tP(\rho',\theta')-\frac{\rho}{\rho'}\textbf{u}\cdot\nabla_xP(\rho',\theta'))dxdt.
\end{eqnarray}
In what follows, we estimate the right-hand side of (\ref{E3-1}). It is easy to see that
\begin{eqnarray}\label{E3-3}
\int_{\Omega}\rho|\textbf{u}-\textbf{u}'|^2|\nabla_x\textbf{u}'|dx\leq\|\nabla_x\textbf{u}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}\int_{\Omega}\rho|\textbf{u}-\textbf{u}'|^2dx.
\end{eqnarray}
By virtue of (\ref{E3-2}), using interpolation inequality, for any $\epsilon>0$, we derive
\begin{eqnarray}\label{E3-4}
&&\int_{\Omega}\rho(s(\rho,\theta)-s(\rho,\theta))(\textbf{u}'-\textbf{u})\cdot\nabla_x\theta'dx\nonumber\\
&\leq&2\rho_1'\|\nabla_x\theta'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}(\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{L}^2(\Omega;\textbf{R}^3)}+c(\epsilon)\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx)\nonumber\\
&&+\|\nabla_x\theta'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}(\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{L}^6(\Omega;\textbf{R}^3)}+c(\epsilon)\|[\rho(s(\rho,\theta)-s(\rho',\theta'))]_{res}\|^2_{\textbf{L}^{\frac{6}{5}}(\Omega)}).~~~~~
\end{eqnarray}
It follows from (\ref{E1-20})-(\ref{E1-21}) that
\begin{eqnarray}\label{E3-5}
|[\rho(s(\rho,\theta)-s(\rho',\theta'))]_{res}|
\leq c(\rho+\rho[\log\theta]^++\rho|\log\rho|+\theta^3).
\end{eqnarray}
Using (\ref{E1-18}), (\ref{E1-19})-(\ref{E1-20}),
\begin{eqnarray}\label{E3-6}
\rho e(\rho,\theta)\geq c(\rho^{\frac{5}{3}}+\theta^4),
\end{eqnarray}
and (\ref{E2-3})-(\ref{E2-x1}) imply
\begin{eqnarray}\label{E3-7}
t\mapsto\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx\in\textbf{L}^{\infty}(0,T).
\end{eqnarray}
By (\ref{E3-2}), (\ref{E3-5})-(\ref{E3-7}) and H\"{o}lder inequality,
\begin{eqnarray*}
\|[\rho(s(\rho,\theta)-s(\rho',\theta'))]_{res}\|^2_{\textbf{L}^{\frac{6}{5}}(\Omega)}\leq c(\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta,))^{\frac{5}{3}}.
\end{eqnarray*}
So by (\ref{E3-4}), for any $\epsilon>0$, we obtain
\begin{eqnarray}\label{E3-8}
\int_{\Omega}\rho(s(\rho,\theta)-s(\rho,\theta))(\textbf{u}'-\textbf{u})\cdot\nabla_x\theta'dx\leq\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{W}_0^{1,2}(\Omega;\textbf{R}^3)}+c'(\epsilon,\cdot)\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx,~~~
\end{eqnarray}
where $c'(\epsilon,\cdot)$ is a generic constant depending on $\epsilon$, $\rho'$, $\textbf{u}'$ and $\theta'$ through the norms induced by (\ref{E2-x1})-(\ref{E2-x2}), while $c'(\cdot)$ is independent of $\epsilon$ but depends on $\rho'$, $\textbf{u}'$, $\theta'$, $\rho_0'$ and $\theta_0,$ through the norms induced by (\ref{E2-x1})-(\ref{E2-x2}).
Similar with estimating (\ref{E3-8}), we get
\begin{eqnarray}\label{E3-9}
&&\int_{\Omega}\frac{1}{\rho'}(\rho-\rho')(\textbf{u}'-\textbf{u}')\cdot(\textbf{div}_x\textbf{S}(\theta',\nabla_x\textbf{u})-\nabla_xP(\rho',\theta')+(\nabla\times\textbf{H}')\times\textbf{H}'))dx\nonumber\\
&=&\int_{\Omega}[\rho'^{-1}(\rho-\rho')(\textbf{u}'-\textbf{u}')\cdot[\textbf{div}_x\textbf{S}(\theta',\nabla_x\textbf{u})-\nabla_xP(\rho',\theta')+(\nabla\times\textbf{H}')\times\textbf{H}')]_{ess}dx\nonumber\\
&&+\int_{\Omega}[\rho'^{-1}(\rho-\rho')(\textbf{u}'-\textbf{u}')\cdot[\textbf{div}_x\textbf{S}(\theta',\nabla_x\textbf{u})-\nabla_xP(\rho',\theta')+(\nabla\times\textbf{H}')\times\textbf{H}')]_{ess}dx\nonumber\\
&\leq&c'(\epsilon,\cdot)\|[\rho-\rho']_{ess}\|^2_{\textbf{L}^2(\Omega)}+\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{L}^2(\Omega;\textbf{R}^3)}\nonumber\\
&&+c'(\epsilon,\cdot)(\|[\rho]_{ess}\|^2_{\textbf{L}^{\frac{6}{5}}(\Omega)}+\|[1]_{res}\|^2_{\textbf{L}^{\frac{6}{5}}(\Omega)})
+\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{L}^6(\Omega;\textbf{R}^3)}.
\end{eqnarray}
So using integrating by parts, then virtue of (\ref{E3-2}), (\ref{E3-7}), (\ref{E3-9}) and $\textbf{W}^{1,2}(\Omega)\hookrightarrow\textbf{L}^6(\Omega)$, we derive
\begin{eqnarray}\label{E3-10}
&&|\int_{\Omega}\rho(\partial_t\textbf{u}'+\textbf{u}'\cdot\nabla_x\textbf{u}')\cdot(\textbf{u}'-\textbf{u})dx|\nonumber\\
&=&|\int_{\Omega}\frac{\rho}{\rho'}(\textbf{u}'-\textbf{u})\cdot(\textbf{div}_x\textbf{S}(\theta',\nabla_x\textbf{u}')-\nabla_xP(\rho',\theta')+(\nabla\times\textbf{H}')\times\textbf{H}')dx|\nonumber\\
&\leq&\int_{\Omega}|\frac{\rho-\rho'}{\rho'}(\textbf{u}'-\textbf{u})\cdot(\textbf{div}_x\textbf{S}(\theta',\nabla_x\textbf{u}')-\nabla_xP(\rho',\theta')+(\nabla\times\textbf{H}')\times\textbf{H}')|dx\nonumber\\
&&+|\int_{\Omega}(\textbf{u}'-\textbf{u})\cdot(\textbf{div}_x\textbf{S}(\theta',\nabla_x\textbf{u}')-\nabla_xP(\rho',\theta')+(\nabla\times\textbf{H}')\times\textbf{H}')dx|\nonumber\\
&\leq&c'(\epsilon,\cdot)\|[\rho-\rho']_{ess}\|^2_{\textbf{L}^2(\Omega)}+\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{L}^2(\Omega;\textbf{R}^3)}+c'(\epsilon,\cdot)(\|[\rho]_{ess}\|^2_{\textbf{L}^{\frac{6}{5}}(\Omega)}+\|[1]_{res}\|^2_{\textbf{L}^{\frac{6}{5}}(\Omega)})\nonumber\\
&&+\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{L}^6(\Omega;\textbf{R}^3)}\nonumber\\
&&+|\int_{\Omega}\left(\textbf{S}(\theta',\nabla_x\textbf{u}'):\nabla_x(\textbf{u}'-\textbf{u})+P(\rho',\theta')\textbf{div}_x(\textbf{u}'-\textbf{u})+((\nabla\times\textbf{H}')\times\textbf{H}')\cdot(\textbf{u}'-\textbf{u})\right)dx|\nonumber\\
&\leq&|\int_{\Omega}\left(\textbf{S}(\theta',\nabla_x\textbf{u}'):\nabla_x(\textbf{u}'-\textbf{u})+P(\rho',\theta')\textbf{div}_x(\textbf{u}'-\textbf{u})+((\nabla\times\textbf{H}')\times\textbf{H}')\cdot(\textbf{u}'-\textbf{u})\right)dx|\nonumber\\
&&+\left(\epsilon\|\textbf{u}-\textbf{u}'\|_{\textbf{W}_0^{1,2}(\Omega;\textbf{R}^3)}+c(\epsilon)\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx\right).
\end{eqnarray}
By H\"{o}lder inequality and (\ref{E1-11})-(\ref{E1-12}), we derive
\begin{eqnarray}\label{E9-1}
\int_{0}^{\tau}\int_{\Omega}((\textbf{H}-\textbf{H}')^{\top}\nabla\textbf{u}'(\textbf{H}-\textbf{H}')dxdt
\leq\int_{0}^{\tau}(\|\nabla\textbf{u}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}\int_{\Omega}|\textbf{H}-\textbf{H}'|^2dx)dt,~~~~~~
\end{eqnarray}
\begin{eqnarray}\label{E9-2}
\frac{1}{2}\int_{0}^{\tau}\int_{\Omega}\nabla(|\textbf{H}-\textbf{H}'|^2)\cdot\textbf{u}'dxdxdt
\leq\frac{1}{2}\int_{0}^{\tau}(\|\nabla\textbf{u}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}\int_{\Omega}|\textbf{H}-\textbf{H}'|^2dx)dt,~~~~~
\end{eqnarray}
\begin{eqnarray}\label{E9-3}
\int_{0}^{\tau}\int_{\Omega}(\textbf{H}'-\textbf{H})^{\top}\nabla(\textbf{u}-\textbf{u}')\textbf{H}'dxdt
&\leq&\int_{0}^{\tau}c_{\epsilon}\|\textbf{H}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}
\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dxdt\nonumber\\
&&+\int_0^{\tau}\epsilon\|\textbf{H}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}dt,~~~~~~~
\end{eqnarray}
\begin{eqnarray}\label{E9-4}
\frac{1}{2}\int_{0}^{\tau}\int_{\Omega}\nabla(\textbf{H}'(\textbf{H}'-\textbf{H}))(\textbf{u}-\textbf{u}')dxdt
&\leq&\frac{1}{4}\int_{0}^{\tau}c_{\epsilon}\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dxdt\nonumber\\
&&+\frac{1}{4}\int_0^{\tau}\epsilon\|\textbf{H}'\|^2_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)}\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}dt,~~~~~
\end{eqnarray}
Thus combing with (\ref{E9-1})-(\ref{E9-4}) and (\ref{E9-5}), we have
\begin{eqnarray}\label{E3-11}
&&\int_{0}^{\tau}\int_{\Omega}((\textbf{H}-\textbf{H}')^{\top}\nabla\textbf{u}'(\textbf{H}-\textbf{H}')+\frac{1}{2}\nabla(|\textbf{H}-\textbf{H}'|^2)\cdot \textbf{u}')dx\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(\textbf{H}'-\textbf{H})^{\top}\nabla(\textbf{u}-\textbf{u}')\textbf{H}'+\frac{1}{2}\nabla(\textbf{H}'(\textbf{H}'-\textbf{H}))(\textbf{u}-\textbf{u}')dxdt\nonumber\\
&&-\int_{0}^{\tau}\int_{\Omega}(\textbf{H}'^{\top}\nabla(\textbf{u}-\textbf{u}')\textbf{H}'+\frac{1}{2}\nabla(|\textbf{H}'|^2)(\textbf{u}-\textbf{u}'))dxdt\nonumber\\
&\leq&\int_{0}^{\tau}\left(\epsilon c''\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+c_{\epsilon}\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dx\right)dt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}((\nabla\times\textbf{H}')\times\textbf{H}')(\textbf{u}-\textbf{u}')dxdt,
\end{eqnarray}
where $c''=c''(\|\textbf{H}'\|^2_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)})$ and
$c_{\epsilon}=c_{\epsilon}(\|\nabla\textbf{u}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)},\|\textbf{H}'\|_{\textbf{L}^{\infty}(\Omega;\textbf{R}^3)})$ denote constants, $\epsilon>0$ sufficient small.
In what follows, we estimate the next term. Using Taylor-Lagrange formula, we derive
\begin{eqnarray}\label{E3-12}
&&\int_{\Omega}\rho(s(\rho,\theta)-s(\rho',\theta'))\partial_t\theta'dx\nonumber\\
&\leq&\int_{\Omega}\rho'[\partial_{\rho}s(\rho',\theta')(\rho-\rho')+\partial_{\theta}s(\rho',\theta')(\theta-\theta')]\partial_t\theta'dx+4c(\cdot)\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx.~~~~~~
\end{eqnarray}
Similar with getting (\ref{E3-12}) and (\ref{E3-4}), respectively, we have
\begin{eqnarray}\label{E3-13}
&&-\int_{\Omega}\rho(s(\rho,\theta)-s(\rho',\theta'))\textbf{u}'\cdot\nabla_x\theta'dx\nonumber\\
&\leq&-\int_{\Omega}\rho'[\partial_{\rho}s(\rho',\theta')(\rho-\rho')+\partial_{\theta}s(\rho',\theta')(\theta-\theta')]\textbf{u}'\cdot\nabla_x\theta'dx+c(\cdot)\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx~~~~~~
\end{eqnarray}
and
\begin{eqnarray}\label{E3-14}
&&|\int_{\Omega}\frac{\rho-\rho'}{\rho'}\nabla_xP(\rho',\theta')\cdot(\textbf{u}-\textbf{u}')dx|\nonumber\\
&\leq&c(|\nabla_x\rho',|\nabla_x\theta'|)\left(\epsilon\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta,)dx\right).~~~~~
\end{eqnarray}
Now we estimate the last term of right-hand side of (\ref{E3-1}). By (\ref{E3-14}),
\begin{eqnarray}\label{E3-15}
&&|\int_{\Omega}\left(\frac{\rho-\rho'}{\rho'}\partial_tP(\rho',\theta')-\frac{\rho}{\rho'}\textbf{u}\cdot\nabla_xP(\rho',\theta')\right)dx\nonumber\\
&\leq&\int_{\Omega}\frac{\rho-\rho'}{\rho'}(\partial_tP(\rho',\theta')+\textbf{u}\cdot\nabla_xP(\rho',\theta'))dx
+\int_{\Omega}P(\rho',\theta')\textbf{div}_x\textbf{u}dx\nonumber\\
&&+c(|\nabla_x\rho',|\nabla_x\theta'|)\left(\epsilon\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta,)dx\right).~~
\end{eqnarray}
Thus by (\ref{E3-1})-(\ref{E3-3}), (\ref{E3-10})-(\ref{E3-12}) and (\ref{E3-15}), we obtain the following relative entropy inequality
\begin{eqnarray}\label{E3-16}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2+\frac{1}{2}|\textbf{H}-\textbf{H}'|^2+\Gamma(\rho,\theta|\rho',\theta))(\tau,\cdot)dx+\nu\int_{0}^{\tau}\int_{\Omega}|\nabla\times\textbf{H}'-\nabla\times\textbf{H}|^2dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(\frac{\theta'}{\theta}\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}
-\textbf{S}(\theta',\nabla_x\textbf{u}'):(\nabla_x\textbf{u}-\nabla_x\textbf{u}')-\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}'dxdt)\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}\left(\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta'}{\theta}-\frac{\theta'}{\theta}\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta}dxdt \right)\nonumber\\
&\leq&\int_{0}^{\tau}\left(\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{W}_0^{1,2}(\Omega;\textbf{R}^3)}+c'(\epsilon,\cdot)\int_{\Omega}(\Gamma(\rho,\theta|\rho',\theta')+\frac{\rho}{2}|\textbf{u}-\textbf{u}'|^2)dx\right)dt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}(P(\rho',\theta')-P(\rho,\theta))\textbf{div}_x\textbf{u}'+((\nabla\times\textbf{H}')\times\textbf{H}')\cdot(\textbf{u}'-\textbf{u})dxdt\nonumber\\
&&+\int_{0}^{\tau}\left(\epsilon c''\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+c_{\epsilon}\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dx\right)dt\nonumber\\
&&-\int_0^{\tau}\int_{\Omega}\rho'[\partial_{\rho}s(\rho',\theta')(\rho-\rho')+\partial_{\theta}s(\rho',\theta')(\theta-\theta')](\partial_t\theta'+\textbf{u}'\cdot\nabla_x\theta')dxdt\nonumber\\
&&+\int_0^{\tau}\int_{\Omega}\frac{\rho-\rho'}{\rho'}(\partial_tP(\rho',\theta')+\textbf{u}\cdot\nabla_xP(\rho',\theta'))dxdt.
\end{eqnarray}
Next we simplify the relative entropy inequality (\ref{E3-16}). Using (\ref{E2-12}) and the fact that $\rho'$ and $\textbf{u}'$ satisfy the equation of continuity (\ref{E1-1}), we derive
\begin{eqnarray}\label{E3-17}
&&-\int_{\Omega}\rho'[\partial_{\rho}s(\rho',\theta')(\rho-\rho')+\partial_{\theta}s(\rho',\theta')(\theta-\theta')](\partial_t\theta'+\textbf{u}'\cdot\nabla_x\theta')dx+\int_{\Omega}\frac{\rho-\rho'}{\rho'}(\partial_tP(\rho',\theta')+\textbf{u}\cdot\nabla_xP(\rho',\theta'))dx\nonumber\\
&=&-\int_{\Omega}\rho'(\theta-\theta')\left(\frac{1}{\theta}(\textbf{S}(\theta',\nabla_x\textbf{u}'):\nabla_x\textbf{u}'-\frac{\textbf{q}(\theta',\nabla_x\theta')\cdot\nabla_x\theta'}{\theta'})-\textbf{div}_x(\frac{\textbf{q}(\theta',\nabla_x\theta')}{\theta})\right)dx\nonumber\\
&&+\int_{\Omega}\left((\theta-\theta')\partial_{\theta}P(\rho',\theta')+(\rho-\rho')\partial_{\rho}P(\rho',\theta')\right)\textbf{div}_x\textbf{u}'dx.
\end{eqnarray}
Note that
\begin{eqnarray}\label{E3-18R}
&&|\int_{\Omega}\left(P(\rho',\theta')-\partial_{\rho}P(\rho',\theta')(\rho'-\rho)-\partial_{\theta}P(\rho',\theta')(\theta'-\theta)-P(\rho,\theta)\textbf{div}_x\textbf{u}'\right)dx|\nonumber\\
&\leq&c\|\textbf{div}_x\textbf{u}\|_{\textbf{L}^{\infty}(\Omega)}\int_{\Omega}\Gamma(\rho,\theta|\rho',\theta')dx.
\end{eqnarray}
Thus by (\ref{E3-16})-(\ref{E3-17}) and (\ref{E3-18R}), for any $\epsilon>0$, we obtain
\begin{eqnarray}\label{E3-18}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2+\frac{1}{2}|\textbf{H}-\textbf{H}'|^2+\Gamma(\rho,\theta|\rho',\theta))(\tau,\cdot)dx+\nu\int_{0}^{\tau}\int_{\Omega}|\nabla\times\textbf{H}'-\nabla\times\textbf{H}|^2dxdt\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(\frac{\theta'}{\theta}\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}
-\textbf{S}(\theta',\nabla_x\textbf{u}'):(\nabla_x\textbf{u}-\nabla_x\textbf{u}')-\textbf{S}(\theta,\nabla_x\textbf{u}):\nabla_x\textbf{u}'+\frac{\theta-\theta'}{\theta'}\textbf{S}(\theta',\nabla_x\textbf{u}':\nabla_x\textbf{u}')dxdt)\nonumber\\
&&+\int_{0}^{\tau}\int_{\Omega}(\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta'}{\theta}-\frac{\theta'}{\theta}\frac{\textbf{q}(\theta,\nabla_x\theta)\cdot\nabla_x\theta}{\theta}+(\theta'-\theta)\frac{\textbf{q}(\theta',\nabla_x\theta')}{\theta'^2}+\frac{\textbf{q}(\theta',\nabla_x\theta')}{\theta'}\cdot\nabla_x(\theta-\theta'))dxdt\nonumber\\
&\leq&\int_{0}^{\tau}\left(\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{W}_0^{1,2}(\Omega;\textbf{R}^3)}+c'(\epsilon,\cdot)\int_{\Omega}(\Gamma(\rho,\theta|\rho',\theta')+\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2)dx\right)dt\nonumber\\
&&+\int_{0}^{\tau}\left(\epsilon c''\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+c_{\epsilon}\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dx\right)dt\nonumber\\
&&+\int_{\Omega}\left(P(\rho',\theta')-\partial_{\rho}P(\rho',\theta')(\rho'-\rho)-\partial_{\theta}P(\rho',\theta')(\theta'-\theta)-P(\rho,\theta)\textbf{div}_x\textbf{u}'\right)dx\nonumber\\
&\leq&\int_{0}^{\tau}\left(\epsilon\|\textbf{u}'-\textbf{u}\|^2_{\textbf{W}_0^{1,2}(\Omega;\textbf{R}^3)}+c'(\epsilon,\cdot)\int_{\Omega}(\Gamma(\rho,\theta|\rho',\theta')+\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2)dx\right)dt\nonumber\\
&&+\int_{0}^{\tau}\left(\epsilon c''\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+c_{\epsilon}\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dx\right)dt.
\end{eqnarray}
The estimate of the terms on $\textbf{S}$ and $\textbf{q}$ has been founded in \cite{Fei1}. So we omit it.
To conclude that we get
\begin{eqnarray*}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2+\frac{1}{2}|\textbf{H}-\textbf{H}'|^2+\Gamma(\rho,\theta|\rho',\theta))(\tau,\cdot)dx+\nu\int_{0}^{\tau}\int_{\Omega}|\nabla\times\textbf{H}'-\nabla\times\textbf{H}|^2dxdt\nonumber\\
&&+c_1\int_{0}^{\tau}\int_{\Omega}\left(|\nabla_x\textbf{u}'-\nabla_x\textbf{u}|^2+|\nabla_x\theta'-\nabla_x\theta|^2+|\nabla_x\log\theta'-\nabla_x\log\theta|^2\right)dxdt\nonumber\\
&\leq&c_2\int_{0}^{\tau}\int_{\Omega}\left(\Gamma(\rho,\theta|\rho',\theta')+\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2\right)dxdt\nonumber\\
&&+\int_{0}^{\tau}\left(\epsilon c''\|\textbf{u}-\textbf{u}'\|^2_{\textbf{W}^{1,2}(\Omega;\textbf{R}^3)}+c_{\epsilon}\int_{\Omega}|\textbf{H}'-\textbf{H}|^2dx\right)dt,~~for~almost~all~\tau\in(0,T).
\end{eqnarray*}
Furthermore, for sufficient small $\epsilon>0$, we obtain
\begin{eqnarray*}
&&\int_{\Omega}(\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2+\frac{1}{2}|\textbf{H}-\textbf{H}'|^2+\Gamma(\rho,\theta|\rho',\theta))(\tau,\cdot)dx+\nu\int_{0}^{\tau}\int_{\Omega}|\nabla\times\textbf{H}'-\nabla\times\textbf{H}|^2dxdt\nonumber\\
&&+(c_1-c''\epsilon)\int_{0}^{\tau}\int_{\Omega}\left(|\nabla_x\textbf{u}'-\nabla_x\textbf{u}|^2+|\nabla_x\theta'-\nabla_x\theta|^2+|\nabla_x\log\theta'-\nabla_x\log\theta|^2\right)dxdt\nonumber\\
&\leq&c_2\int_{0}^{\tau}\left(\int_{\Omega}(\Gamma(\rho,\theta|\rho',\theta')+\frac{1}{2}\rho|\textbf{u}-\textbf{u}'|^2+\frac{1}{2}|\textbf{H}'-\textbf{H}|^2)\right)dx
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\rho\equiv\rho',~~\textbf{u}\equiv\textbf{u}',~~\theta\equiv\theta',~~\textbf{H}\equiv\textbf{H}'.
\end{eqnarray*}
This completes the proof.
\end{document} | math | 54,471 |
\begin{document}
\title{Precise Propagation of Upper and Lower Probability
Bounds in System P}
\author{Angelo Gilio \\ Dipartimento di Matematica e Informatica, Citt\`a Universitaria \\
Viale A. Doria, 6 - 95125 Catania (Italy).\\
e-mail: [email protected]}
\maketitle
\begin{abstract}
\begin{quote}
In this paper we consider the inference rules of System P in the framework of
coherent imprecise probabilistic assessments. Exploiting our algorithms, we
propagate the lower and upper probability bounds associated with the
conditional assertions of a given knowledge base, automatically obtaining the
precise probability bounds for the derived conclusions of the inference rules.
This allows a more flexible and realistic use of System P in
default reasoning and provides an exact illustration of the degradation of
the inference rules when interpreted in probabilistic terms. We also examine
the {\em disjunctive Weak Rational Monotony} of System P$^+$ proposed by Adams
in his extended probability logic.
\end{quote}
\end{abstract}
{\bf Keywords:} Nonmonotonic reasoning, System P, Conditional probability
bounds, Precise propagation, Coherence. \\ \ \\
\section{Introduction}
In the applications of intelligent systems to automated uncertain
reasoning the explicit knowledge of the agent is represented by a
knowledge base $\cal K$, constituted by a set of {\em conditional
assertions} (i.e. {\em defaults}). The
nonmonotonic inferential process is developed using a suitable
set of rules. Among the many formalisms which have been proposed for
default reasoning, the so-called System P developed in
(Kraus, Lehmann, and Magidor 1990) is
widely accepted and has a probabilistic semantics based on infinitesimal
probabilities, see (Adams 1975, Pearl 1988). An extended
probability logic has been proposed in (Adams 1986) by allowing
{\em disjunction of conditionals}. The corresponding system P$^+$ has been
studied in (Schurz 1998) where the perspectives of the infinitesimal probability
semantics and that of a noninfinitesimal one, based on probabilistic
inequality relations, have been unified.
In (Hawthorne 1996) many logics of nonmonotonic conditionals, that behave like
conditional probabilities at various levels of probabilistic support, have
been examined. In the quoted paper the author shows that, for each
given conditional $\rightarrow$, there is a probabilistic support level $r$
and a conditional probability $P$ such that, for all sentences $B, A$,
it is $B \rightarrow A$ only if $P(A|B) \geq r$.
We recall that an early examination of Adams rules by means of imprecise
probabilities has been given in (Dubois, and Prade 1991). In the quoted
paper a semantic interpretation in terms of intervals has been given for the
relations of negligibility, closeness and comparability examined in the
system of qualitative reasoning proposed in (Raiman 1986). Moreover,
an application to the inference rules of Adams has been given interpreting
"$P(A|B)$ is large" by means of the relation of closeness to 1
(making the infinitesimal parameter $\epsilon$ explicit).
In this way a quantification of
the degradation of the validity of Adams' rules when reasoning with
noninfinitesimal probabilities has been obtained.
While in practical applications the semantics of infinitesimal probabilities
may involve some difficulties, the approach based
on lower (and possibly upper) probability bounds, proposed also in
(Bourne, and Parsons 1998), is clearly more flexible and realistic.
In this paper the propagation of probability bounds to the conditional
assertions associated with the rules of System P is examined
in the framework of the de Finetti's probabilistic
methodology, based on the coherence principle. Notice that
a coherent set of conditional probability assessments satisfies all
the usual properties of conditional probabilities.
A short examination of the logic of conditionals of Adams
from the point of view of coherence has been given in (Gilio 1997),
where the propagation of probabilistic bounds has not been considered.
We point out that the peculiarity of our approach is given by the
possibility of looking at the conditional probability $P(A|B)$
as a primitive concept,with no need of assuming that the probability of
the {\em conditioning} event $B$ be positive.
On the contrary, in the probabilistic approaches usually adopted
in the literature, see, e.g., (Adams 1975, Hawthorne 1996, Schurz 1998),
by definition the quantity $P(A|B)$ is the {\em ratio} of $P(AB)$ and $P(B)$
if $P(B) \neq 0$, with $P(A|B) = 1$ if $P(B) = 0$. Notice that a clear
{\em rationale} of this latter assumption does not seem to exist.
Indeed, in the framework of coherence, this
assumption is not made and the case of {\em conditioning} events of zero
probability is managed without any problem: as an example, the
condition $P(A|B) + P(A^c|B) = 1$, where $A^c$ denotes the negation of $A$,
is satisfied also when $P(B) = 0$. We think that the opportunity, offered
by the probabilistic approach based on coherence, of developing a completely
general treatment of probabilistic default reasoning is important specially in
the field of nonmonotonic reasoning where infinitesimal probabilities
play a significant role. Moreover, exploiting the algorithms developed in our framework,
the lower and upper probability bounds associated with the conditional assertions
of a given knowledge base can be propagated to further conditional assertions,
obtaining in all cases the tightest probability bounds.
Beside allowing a more flexible and realistic approach to probabilistic default reasoning,
this provides an exact illustration of the degradation of System P rules
when interpreted in probabilistic terms.
The probabilistic approach based on coherence has been adopted in many
recent papers, see, e. g., (Biazzo, and Gilio 1999), (Capotorti, and
Vantaggi 1999), (Coletti 1994), (Coletti, and Scozzafava 1996),
(Coletti, and Scozzafava 1999), (Gilio 1995a), (Gilio 1995b), (Gilio 1999),
(Gilio, and Ingrassia 1998), (Gilio, and Scozzafava 1994), (Lad 1999),
(Lad, Dickey, and Rahman 1990), (Scozzafava 1994).
The algorithms described in (Biazzo, and Gilio 1999), based on the linear programming technique,
have been implemented with Maple V.
The paper is organized as follows. In Section 2 we give some preliminary concepts on
coherence and probability logic. In Section 3 we consider the definitions
of probabilistic consistency and entailment given by Adams; then we recall the main
inference rules of his probability logic. In Section 4 we consider the propagation of lower
and upper probability bounds in System P. We also examine the disjunctive Weak Rational
Monotony of System P$^+$. In Section 5 we examine the propagation of lower bounds with
real $\epsilon-$values. Then, in Section 6 we apply the results to (a slightly modified
version of) an example given in (Bourne, and Parsons 1998). Finally, in Section 7 we give some
conclusions.
\section{Some preliminaries}
We recall some preliminary concepts on coherence of imprecise probability assessments and
on probability logic. Given a family ${\cal F}_n = \{E_1 | H_1, \ldots, E_n|H_n \}$
and a vector ${\cal A}_n = (\alpha_1, \ldots, \alpha_n)$ of lower bounds
$P(E_i|H_i) \geq \alpha_i$, with $i \in J_n = \{1, \ldots, n\}$,
we consider the following definition of generalized coherence (g-coherence),
given in (Biazzo, and Gilio 1999), which essentially coincides with a previous one
introduced in (Gilio 1995a).
\begin{Def} \label{IMPR}{\rm
The vector of lower bounds ${\cal A}_n$ on ${\cal F}_n$ is said
g-coherent if and only if there exists a precise coherent assessment
${\cal P}_n = (p_1, \ldots, p_n)$
on ${\cal F}_n$, with $p_i = P(E_i|H_i)$, which is consistent with ${\cal A}_n$, that is
such that $p_i \geq \alpha_i$ for each $i \in J_n$.
}\end{Def}
The Definition ~\ref{IMPR} can be also applied to
imprecise assessments like
\[
\alpha_i \leq P(E_i|H_i) \leq \beta_i \; , \; \; i \in J_n,
\]
since each inequality $P(E_i|H_i) \leq \beta_i$ amounts to the inequality
$P(E_i^c|H_i) \geq 1-\beta_i$. \\
Then, given an imprecise assessment ${\cal A}_n= (\alpha_1, \ldots, \alpha_n)$
on ${\cal F}_n$, a suitable procedure, given in (Gilio 1995b),
can be used to check the g-coherence of ${\cal A}_n$.
The g-coherent extension of ${\cal A}_n$ to a further conditional event
$E_{n+1}|H_{n+1}$ has been studied in (Biazzo, and Gilio 1999) where, defining a
suitable interval $[p_{\circ},p^{\circ}] \subseteq [0,1]$, the following
result has been obtained.
\begin{Thm}\label{GENdF}{\rm
Given a g-coherent imprecise assessment
${\cal A}_n = ([\alpha_i, \beta_i], i \in J_n)$ on the family
${\cal F}_n = \{E_i|H_i, \; i \in J_n\}$, the extension
$[\alpha_{n+1}, \beta_{n+1}]$ of ${\cal A}_n$ to a further conditional event
$E_{n+1}|H_{n+1}$ is g-coherent if and only if the following condition is
satisfied
\begin{equation}\label{INTERS}
[\alpha_{n+1}, \beta_{n+1}] \cap [p_{\circ},p^{\circ}] \neq \emptyset \; .
\end{equation}
} \end{Thm}
In the quoted paper an algorithm has been proposed to determine
$[p_{\circ},p^{\circ}]$. Moreover, starting with a g-coherent assessment ${\cal A}_n$, by the same
algorithm it is possible to determine the corresponding assessment ${\cal A}^*_n$
coherent wrt definition given in (Walley 1991). \\
We can frame our approach to the problem of propagating imprecise
conditional probability assessments ({\em probabilistic deduction})
from the {\em probability logic} point of view, see, e. g.,
(Frisch, and Haddawy 1994), (Lukasiewicz 1998), (Nilsson 1986). \\
We associate to each conditional assertion $H \; | \hspace{-1.8 mm} \sim E$ in the
knowledge base ${\cal K}$ a probability interval $[\alpha,\beta]$.
In particular, given a family ${\cal F}_n$ of $n$ conditional assertions, we
consider an interval-valued probability
assessment ${\cal A}_n = ([\alpha_i,\beta_i], i = 1, \ldots, n)$.
Then, we can look at the pair $({\cal F}_n, {\cal A}_n)$ as
a probabilistic knowledge base, where each imprecise assessment
$\alpha_i \leq P(E_i|H_i) \leq \beta_i$ is a probabilistic formula
denoted by $(E_i|H_i)[\alpha_i,\beta_i]$. In our approach a probabilistic
interpretation is just a coherent precise conditional probability assessment ${\cal P}_n$ on
${\cal F}_n$. A probabilistic interpretation ${\cal P}_n = (p_1, \ldots, p_n)$ is a {\em
model} of a probabilistic formula $(E_i|H_i)[\alpha_i, \beta_i]$ iff ${{\cal P}}_n
\models (E_i|H_i)[\alpha_i, \beta_i]$, that is $\alpha_i \leq p_i \leq
\beta_i$. ${\cal P}_n$ is a model of the probabilistic knowledge base ${\cal K} = ({\cal F}_n,
{\cal A}_n)$, denoted ${\cal P}_n \models {\cal K}$, iff ${\cal P}_n \models (E|H)[\alpha, \beta]$ for
every $(E|H)[\alpha, \beta] \in {\cal K}$. Therefore, ${\cal P}_n$ is a model of ${\cal K} =
({\cal F}_n, {\cal A}_n)$ iff ${\cal P}_n$ is consistent with ${\cal A}_n$. A set of probabilistic
formulas ${\cal K}$ is {\em satisfiable} iff a model of ${\cal K}$ exists, therefore the
concept of satisfiability of ${\cal K} = ({\cal F}_n, {\cal A}_n)$ coincides with that of
g-coherence of ${\cal A}_n$ on ${\cal F}_n$. A probabilistic formula
$(E_{n+1}|H_{n+1})[\alpha_{n+1}, \beta_{n+1}]$ is a {\em logical consequence}
of ${\cal K} = ({\cal F}_n, {\cal A}_n)$, denoted ${\cal K} \models (E_{n+1}|H_{n+1})[\alpha_{n+1},
\beta_{n+1}]$, iff
\[
\alpha_{n+1} \leq inf \; {\cal I} \; , \; \; \; \beta_{n+1} \geq sup \; {\cal I} \; ,
\]
where ${\cal I}$ is the set of the real values $p$ such that there exists a model of
${\cal K} \cup \{(E_{n+1}|H_{n+1})[p, p]\}$. As shown by the condition
(\hspace{-1.5 mm}~\ref{INTERS}), in our approach this amounts to
\[
[p_{\circ},p^{\circ}] \subseteq [\alpha_{n+1}, \beta_{n+1}].
\]
A probabilistic formula
$(E_{n+1}|H_{n+1})[\alpha_{n+1}, \beta_{n+1}]$ is a
{\em tight logical consequence} of ${\cal K} = ({\cal F}_n, {\cal A}_n)$, denoted
${\cal K} \models_{tight} (E_{n+1}|H_{n+1})[\alpha_{n+1}, \beta_{n+1}]$,
iff
\[
\alpha_{n+1} = inf \; {\cal I} \; , \; \; \; \beta_{n+1} = sup \; {\cal I} \; ,
\]
that is
\[
\alpha_{n+1} = p_{\circ} \; , \; \; \; \beta_{n+1} = p^{\circ} \; .
\]
Considering a {\em probabilistic }{\em query} $(E_{n+1}|H_{n+1})[\alpha,
\beta]$, where $\alpha$ and $\beta$ are two different variables, to a
probabilistic knowledge base ${\cal K} = ({\cal F}_n, {\cal A}_n)$ a {\em correct answer} is any
$[\alpha, \beta] = [\alpha_{n+1}, \beta_{n+1}] \supseteq
[p_{\circ},p^{\circ}]$, that is such that ${\cal K} \models
(E_{n+1}|H_{n+1})[\alpha_{n+1}, \beta_{n+1}]$. The {\em tight answer} is
$[\alpha, \beta] = [p_{\circ},p^{\circ}]$.
\section{Probabilistic consistency and entailment}
We recall that in (Adams 1975) the
conditional assertion {\em "if} $A$ {\em then} $B$ {\em "} is looked at as
$P(B|A) \geq 1 - \epsilon \; \; \; (\forall \epsilon > 0) $.
Adopting a more realistic point of view we may look at the same conditional assertion as the
probabilistic formula $(B|A)[\alpha, \beta]$, with $0 \leq \alpha \leq \beta \leq 1$,
where usually $\beta = 1$.
Then, a (probabilistic) knowledge base might be defined as a family of
probabilistic formulas ${\cal K} = \{(E|H)[\alpha, \beta]\}$.
In (Adams 1975) the following definition has been given.
\begin{Def}{\rm
The knowledge base $\cal K$ is {\em probabilistically
consistent} ({\em p-consistent}) if, for every $\epsilon > 0$,
there exists a probability $P$ on {\cal A}, proper for $\cal K$,
such that $P(E|H) \geq 1-\epsilon$ for every $E|H \in \cal K$ .
} \end{Def}
In our framework the concept of probabilistic consistency can be defined
in the following way.
\begin{Def}{\rm
The knowledge base $\cal K$ is {\em probabilistically
consistent} ({\em p-consistent}) if, for every set of lower bounds
${\cal A} = \{\alpha_{E|H}, E|H \in {\cal K}\}$ on ${\cal K}$, there exists a precise coherent
probability assessment $P = \{p_{E|H}, E|H \in {\cal K}\}$ on ${\cal K}$, with $p_{E|H} = P(E|H)$,
such that, for each $E|H \in {\cal K}$, $p_{E|H} \geq \alpha_{E|H}$.
} \end{Def}
We recall the concept of probabilistic entailment as defined in
(Adams 1975).
\begin{Def}{\rm
A p-consistent knowledge base $\cal K$ {\em probabilistically entails}
({\em p-entails}) the conditional $B|A$ if, for every $\epsilon > 0$,
there exists $\delta > 0$ such that for all probabilities $P$,
proper for ${\cal K} \cup \{B|A\}$, if $P(E|H) \geq 1 - \delta$
for each $E|H \in \cal K$, then $P(B|A) \geq 1 - \epsilon$.
} \end{Def}
In our framework the concept of probabilistic entailment can be defined in
the following way.
\begin{Def}\label{PE}{\rm
A p-consistent knowledge base $\cal K$ {\em probabilistically entails}
the conditional $B|A$ if there exists a subfamily ${\cal F} \subseteq {\cal K}$ such
that, for every $\alpha_{B|A}$, there exists a set of lower bounds
${\cal A} = \{\alpha_{E|H}, E|H \in {\cal F}\}$ on ${\cal F}$ such that for all precise coherent
probability assessment $P = \{p_{B|A}, p_{E|H}, E|H \in {\cal F}\}$ on ${\cal F} \cup \{B|A\}$,
with $p_{B|A} = P(B|A), p_{E|H} = P(E|H)$, if $p_{E|H} \geq \alpha_{E|H}$
for each $E|H \in {\cal F}$, then $p_{B|A} \geq \alpha_{B|A}$.
} \end{Def}
The probabilistic entailment of $B|A$ by $\cal K$ is denoted by the symbol
${\cal K} \cal Rightarrow B|A$.
In (Adams 1975) a suitable set ${\cal R}$ of seven inference rules
has been introduced, by means of which an inferential system
can be developed to deduce in an automatic way all the
plausible conclusions of a knowledge base $\cal K$.See also (Pearl 1988).
The fundamental rules of the set $\cal R$ are the following ones:
\[
\begin{array}{ll}
R1. & A \; | \hspace{-1.8 mm} \sim C \; , \; A \; | \hspace{-1.8 mm} \sim B \; \cal Rightarrow \;
AB \; | \hspace{-1.8 mm} \sim C \\ & (Triangularity) \\ \ \\
R2. & AB \; | \hspace{-1.8 mm} \sim C \; , \; A \; | \hspace{-1.8 mm} \sim B \; \cal Rightarrow \;
A \; | \hspace{-1.8 mm} \sim C \\ & (Bayes) \\ \ \\
R3 & A \; | \hspace{-1.8 mm} \sim C \; , \; B \; | \hspace{-1.8 mm} \sim C \; \cal Rightarrow \;
A \vee B \; | \hspace{-1.8 mm} \sim C \\ & (Disjunction)
\end{array}
\]
The previous rules, among others, have been used (with different
names and in the framework of symbolic approaches too) by many
authors, with the aim of developing some nonmonotonic logics to
formalize the plausible reasoning (see e.g. (Kraus, Lehmann, and Magidor 1990),
(Lehmann, and Magidor 1992), (Dubois, and Prade 1994)).
See also the survey given in (Benferhat, Dubois, and Prade 1997). \\
We recall that in (Gilio 1997) the following concept of
{\em strict probabilistic consistency} has been introduced.
\begin{Def}{\rm
The knowledge base $\cal K$ is strictly p-consistent if
the probability assessment $P$ on $\cal K$, such that $P(E|H) = 1$
for each $E|H \in \cal K$, is coherent.
}
\end{Def}
Then the following result, which has some relations with the Theorem 3
given in (Schurz 1998), has been given (without proof).
\begin{Thm} \label{STRICT} {\rm
$\cal K$ is p-consistent if and only if $\cal K$
is strictly p-consistent.
}
\end{Thm}
The proof of Theorem ~\ref{STRICT} is based on the following
three assertions:
\begin{description}
\item[a.]If $\cal K$ is strictly p-consistent, then $\cal K$ is
p-consistent;
\item[b.] If $\cal K$ is p-consistent, then $\cal K$ is consistent;
\item[c.] If $\cal K$ is consistent, then $\cal K$ is strictly
p-consistent.
\end{description}
Hence, the property of p-consistency can be simply defined on
the basis of the property of strict p-consistency. Moreover, the
following well known result
\begin{Thm}{\rm
If $\cal K$ is consistent, then ${\cal K} \cal Rightarrow B|A$
if and only if ${\cal K} \cup \{B^c|A\}$ is inconsistent.
}
\end{Thm}
can be reformulated in the following way:
\begin{Thm}{\rm
Given a consistent knowledge base $\cal K$ and a
conditional $B|A$, $\cal K$ p-entails $B|A$ if and only if the
probability assessment $P$ on ${\cal K} \cup \{B^c|A\}$, with
$P(B^c|A) = P(E|H) = 1$ for each $E|H \in \cal K$, is not coherent.
}
\end{Thm}
\section{Exact propagation of probability bounds in System P}
We recall that the inference rules in System P are the following ones:
\[
\begin{array}{ll}
1. & A \; | \hspace{-1.8 mm} \sim A \\
& (Reflexivity) \\ \ \\
2. & \models A \leftrightarrow B, \; A \; | \hspace{-1.8 mm} \sim C
\; \Longrightarrow \; B \; | \hspace{-1.8 mm} \sim C \\
& (Left \; Logical \; Equivalence) \\ \ \\
3. & \models B \; \rightarrow C, \; A \; | \hspace{-1.8 mm} \sim B \;
\Longrightarrow \; A \; | \hspace{-1.8 mm} \sim C \\
& (Right \; Weakening) \\ \ \\
4. & A \; | \hspace{-1.8 mm} \sim B, \; A \; | \hspace{-1.8 mm} \sim C \;
\Longrightarrow \; A \; | \hspace{-1.8 mm} \sim B C \\
& (And) \\ \ \\
5. & A \; | \hspace{-1.8 mm} \sim C, \; A \; | \hspace{-1.8 mm} \sim B \; \Longrightarrow \;
AB \; | \hspace{-1.8 mm} \sim C \\
& (Cautious \; Monotonicity) \\ \ \\
6. &
A \; | \hspace{-1.8 mm} \sim C, \; B \; | \hspace{-1.8 mm} \sim C \; \Longrightarrow \;
A \vee B \; | \hspace{-1.8 mm} \sim C \\
& (Or) \\
\end{array}
\]
Two derived rules in System P are
\[
\begin{array}{lll}
7. \hspace{0.5 cm}
A B \; | \hspace{-1.8 mm} \sim C, \; \; A \; | \hspace{-1.8 mm} \sim B \;
\Longrightarrow \; A \; | \hspace{-1.8 mm} \sim C & & ({\em Cut}) \\ \ \\
8. \hspace{0.5 cm}
A B \; | \hspace{-1.8 mm} \sim C \; \Longrightarrow \; A \; | \hspace{-1.8 mm} \sim B \rightarrow C
& & ({\em S})
\end{array}
\]
As we can see, the rules $R1, R2, R3$ coincide respectively with the
rules {\em Cautious Monotonicity, Cut, Or} in System P.
In (Adam86) an extended probability logic (System P$^+$) was developed
by allowing the disjunction of conditionals in the conclusion of inferences
and by adding the following dWRM rule.
\[
\begin{array}{ll}
9. & A \; | \hspace{-1.8 mm} \sim C \; \Longrightarrow
\; A \; | \hspace{-1.8 mm} \sim B^c \; \; \vee \; \; AB \; | \hspace{-1.8 mm} \sim C \\
& (disjunctive \; Weak \; Rational \; Monotony)
\end{array}
\]
Now we will show how the probability
intervals associated with the antecedents in each rule of System P propagate
in a precise way to the consequent of the given rule. We will also examine the
rules Cut, S and dWRM. These bounds will provide
an exact illustration of the degradation of System P rules when interpreted in
probabilistic terms. Perhaps, as a by-product, also a better
appreciation of the results given in (Hawthorne 1996) on the interpretation of
nonmonotonic conditionals in terms of probabilistic support levels could be obtained.
We assume that the basic events $A, B, C$ are logically independent.
\begin{enumerate}
\item {\em Reflexivity rule}. As for every assertion $A$ the
assessment $P(A|A) = p$ is coherent if
and only if $p = 1$, to the conditional assertion $A \; | \hspace{-1.8 mm} \sim A$ we associate
the interval $[\alpha, \beta] = [1,1]$.
\item {\em Left Logical Equivalence rule}. If two assertions $A, B$ are
equivalent, then for every assertion $C$ the assessment $(x,y)$ on
the pair of conditional events $C|A, C|B$ is coherent if and only $x=y$.
Therefore we associate the same probability interval to the conditional
assertions $C|A, C|B$. In other words, the assessment $[\alpha, \beta]$ on
$C|A$ propagates to the same interval on $C|B$.
\item {\em Right Weakening rule}. If $B \subseteq C$ then, defining the inclusion
among conditional events as in (Goodman, and Nguyen 1988),
it is $B|A \subseteq C|A$ and then the assessment $(x,y)$ on
the pair of conditional events $B|A, C|A$ is coherent if and only
$x \leq y$, see (Gilio 1993). Therefore, the assessment $[\alpha, \beta]$
on $B|A$ propagates to the interval $[\alpha, 1]$ on $C|A$.
\item {\em And rule}. Given the assessment
$(x,y)$ on the pair of conditional events $B|A, C|A$, as well known
the extension $P(BC|A) = z$ is coherent if and only if
\[
Max \; \{0, x + y - 1\} = z' \leq z \leq z'' = Min \; \{x,y\}.
\]
Therefore, the probability intervals
$[\alpha_1, \beta_1], [\alpha_2, \beta_2]$ on the antecedents $B|A, C|A$
of the rule propagate to the exact interval $[\alpha_3, \beta_3]$, with
\begin{equation}\label{AND}
\begin{array}{l}
\alpha_3 = Min \; z' = Max \; \{0, \alpha_1 + \alpha_2 - 1\} \; , \\ \ \\
\beta_3 = Max \; z'' = Min \; \{\beta_1, \beta_2\} \; ,
\end{array}
\end{equation}
on the consequent $BC|A$.
\item {\em Cautious Monotonicity rule}. Given the assessment
$(x,y)$ on the pair of conditional events $C|A, B|A$, as proved in (Gilio 1995b),
the extension $P(C|AB) = z$ is coherent if and only if $z \in [z', z'']$, with
\[
\begin{array}{l}
z' \; = \; \left\{\begin{array}{lcl}
\frac{x+y-1}{y} \; , & \mbox{if} & x+y > 1 \\
0 \; , & \mbox{if} & x+y \leq 1
\end{array} \right. \; , \\ \ \\
z'' \; = \; \left\{\begin{array}{lcl}
\frac{x}{y} \; , & \mbox{if} & x < y \\
1 \; , & \mbox{if} & x \geq y
\end{array} \right. \; .
\end{array}
\]
We observe that, for $(x,y) \in [\alpha_1, \beta_1] \times [\alpha_2, \beta_2]$
the function $f(x,y)$ attains its minimum value at the point $(\alpha_1, \alpha_2)$.
Then, it follows
\begin{equation}
\begin{array}{ll}
\alpha_3 \: = \:
\left\{\begin{array}{lll}
\frac{\alpha_1+\alpha_2-1}{\alpha_2}, & \mbox{if } &
\alpha_1 + \alpha_2 > 1 \\
0, & \mbox{if } & \alpha_1 + \alpha_2 \leq 1
\end{array} \right. \label{ACM}
\end{array}
\end{equation}
Moreover, the function $g(x,y)$ attains its maximum
value at the point $(\beta_1, \alpha_2)$. Then it follows
\begin{equation}
\beta_3 \: = \: \left\{\begin{array}{lll}
\frac{\beta_1}{\alpha_2}, & \mbox{if } & \beta_1 < \alpha_2 \\
1, & \mbox{if } & \beta_1 \geq \alpha_2
\end{array} \right. \label{zeta^*}
\end{equation}
Then, in order to determine the interval $[\alpha_3, \beta_3]$ we have to
consider the position of the vertices $(\alpha_1, \alpha_2), (\beta_1, \alpha_2)$ wrt
diagonals of the unitary square $[0,1]^2$.
\item {\em Or rule}. Given the assessment
$(x,y)$ on the pair of conditional events $C|A, C|B$, it can be proved that
the extension $P(C|(A \vee B)) = z$ is
coherent if and only if $z \in [z', z'']$, with
\[
\begin{array}{l}
z' \; = \; \left\{\begin{array}{lll}
\frac{xy}{x+y-xy}, & \mbox{if} & (x,y) \neq (0,0) \\
0 , & \mbox{if} & (x,y) = (0,0)
\end{array}
\right. \; , \\ \ \\
z'' \; = \; \left\{\begin{array}{lll}
\frac{x+y-2xy}{1- xy}, & \mbox{if} & (x,y) \neq (1,1) \\
1 , & \mbox{if} & (x,y) = (1,1)
\end{array}
\right. \; .
\end{array}
\]
Moreover, we observe that both $z'$ and $z''$ increase as either $x$ or $y$
increase. Therefore, the probability intervals
$[\alpha_1, \beta_1], [\alpha_2, \beta_2]$ on the antecedents $C|A, C|B$
of the rule propagate, under the condition
$(\alpha_1,\alpha_2) \neq (0,0), \; (\beta_1,\beta_2) \neq (1,1)$,
to $[\alpha_3, \beta_3]$, with
\begin{equation}\label{OR}
\alpha_3 \; = \; \frac{\alpha_1\alpha_2}{\alpha_1+\alpha_2-\alpha_1\alpha_2} \; ,
\end{equation}
\begin{equation}\label{ORR}
\beta_3 \; = \; \frac{\beta_1+\beta_2-2\beta_1\beta_2}{1- \beta_1\beta_2} \; ,
\end{equation}
on the consequent $C|(A \vee B)$.
\end{enumerate}
Concerning the rules Cut and S we have the following results. \\
(e) {\em Cut rule}. Given the assessment
$(x,y)$ on the pair of conditional events $C|AB, B|A$, it can be proved that
the extension $P(C|A) = z$ is coherent if and only if
\[
xy \leq z \leq xy + 1 - y \; .
\]
Therefore, the probability intervals
$[\alpha_1, \beta_1], [\alpha_2, \beta_2]$ on the antecedents $C|AB, B|A$
of the rule propagate to $[\alpha_3, \beta_3]$, with
\begin{equation}\label{CUT}
\alpha_3 =
\alpha_1\alpha_2 \; , \; \; \;
\beta_3 = \beta_1\alpha_2 + 1 - \alpha_2 \; ,
\end{equation}
on the consequent $C|A$. \\
(f) {\em S rule}. As $C|AB \subseteq (B^c \vee C)|A$, then the assessment
$(x,y)$ on the conditional events $C|AB, (B^c \vee C)|A$ is coherent if and
only if $x \leq y$.
Therefore, the probability interval $[\alpha_1, \beta_1]$ on the antecedent
$C|AB$ of the rule propagates to $[\alpha_2, \beta_2]$, with
\[
\alpha_2 = \alpha_1, \; \; \; \beta_2 = 1 \; ,
\]
on the consequent $(B^c \vee C)|A$. \\
(g) {\em dWRM rule}. Let ${\cal P} = (x,y,z)$ a probability assessment on
the family $\{C|A, B^c|A, C|AB\}$. For this family the constituents
(possible worlds) are
\[
\begin{array}{l}
C_0 = A^c \; , \; \; C_1 = ABC \; , \; \; C_2 = ABC^c \; , \\ \ \\
C_3 = AB^cC \; , \; \; C_4 = AB^cC^c \; .
\end{array}
\]
To the constituents $C_1, \ldots, C_4$ we associate the points
\[
\begin{array}{l}
Q_1 = (1,0,1) \; , \; \; Q_2 = (0,0,0) \; , \\ \ \\
Q_3 = (1,1,z) \; , \; \; Q_4 = (0,1,z) \; .
\end{array}
\]
Then, based on the method given in (Gilio 1995b) and denoting by ${\cal I}$
the convex hull of the points $Q_1, \ldots, Q_4$, it can be proved that
the coherence of ${\cal P}$ amounts to the condition ${\cal P} \in {\cal I}$. Notice that
in general this condition is necessary but not sufficient for the coherence
of an assessment ${\cal P}_n = (p_1, \ldots, p_n)$ on a family
${\cal F}_n = \{E_1|H_1, \ldots, E_n|H_n\}$. The study of the condition
${\cal P} \in {\cal I}$ requires considering the equations of the four planes determined
respectively by the terns of points
\[
\begin{array}{l}
\{Q_1, Q_2, Q_3\} \; , \; \; \; \{Q_2, Q_3, Q_4\} \; , \\ \ \\
\{Q_1, Q_2, Q_4\} \; , \; \; \; \{Q_1, Q_3, Q_4\} \; .
\end{array}
\]
Denoting by $X,Y,Z$ the axes' coordinates, the equations are given
respectively by
\[
\begin{array}{l}
Z = X + (z-1)Y \; , \; \; \; \; Z = zY \; , \\ \ \\
Z = X + zY \; , \; \; \; \; Z = (z-1)Y + 1 \; .
\end{array}
\]
Then, given the values $x,y$, it is
\[
{\cal P} \in {\cal I} \; \Longleftrightarrow \; z' \leq z \leq z'' \; ,
\]
where
\[
\begin{array}{l}
z' \; = \; \left\{\begin{array}{lcl}
\frac{x-y}{1-y} \; , & \mbox{if} & x > y \\
0 \; , & \mbox{if} & x \leq y
\end{array} \right. \; , \\ \ \\
z'' \; = \; \left\{\begin{array}{lcl}
\frac{x}{1-y} \; , & \mbox{if} & x+y < 1 \\
1 \; , & \mbox{if} & x+y \geq 1
\end{array} \right. \; .
\end{array}
\]
In order to examine the probabilistic interpretation of the rule
we introduce a partition $\{{\cal R}_1, {\cal R}_2, {\cal R}_3, {\cal R}_4\}$ of the unitary
square $[0,1]^2$, with
\[
\begin{array}{lll}
{\cal R}_1 & = & \{(x,y): x+y < 1, x \geq y \} \; , \\ \\
{\cal R}_2 & = & \{(x,y): x+y < 1, x < y \} \; , \\ \ \\
{\cal R}_3 & = & \{(x,y): x+y \geq 1, x < y \} \; , \\ \ \\
{\cal R}_4 & = & \{(x,y): x+y \geq 1, x \geq y \} \; .
\end{array}
\]
We have to examine the case in which $x$ is "high", therefore ${\cal R}_2$
is not of interest. In ${\cal R}_3$, since $x < y$, if $x$ is "high" then $y$ is
"high" too. In ${\cal R}_1$ and ${\cal R}_4$ it is $z \geq z' = \frac{x-y}{1-y}$
so that, if $x$ is "high" and $y$ is "not high", then $z$ is "high". \\
Concerning propagation of probability intervals, if we consider
the assessments $[\alpha_1, \beta_1], [\alpha_2, \beta_2]$ on the
conditional events $C|A, B^c|A$, then for the interval $[\alpha_3, \beta_3]$
associated with $C|AB$ we first observe that the quantity $\frac{x}{1-y}$
attains its maximum value at the point $(\beta_1, \alpha_2)$, while the
quantity $\frac{x-y}{1-y}$ attains its minimum value at the point
$(\alpha_1, \beta_2)$. Then, we have:
\begin{equation}
\alpha_3 \; = \;
\left\{\begin{array}{lll}
\frac{\alpha_1-\beta_2}{1-\beta_2}, & \mbox{if} & \alpha_1 \geq \beta_2 \\
0 , & \mbox{if} & \alpha_1 < \beta_2
\end{array} \right.
\end{equation}
\begin{equation}
\beta_3 \; = \;
\left\{\begin{array}{lll}
\frac{\beta_1}{1-\alpha_2}, & \mbox{if} & \beta_1 + \alpha_2 < 1 \\
1 , & \mbox{if} & \beta_1 + \alpha_2 \geq 1
\end{array} \right.
\end{equation}
\begin{Rem}{\rm
Using the lower bounds computed previously, we can verify the probabilistic entailment
in each inference rule on the basis of Definition ~\ref{PE}. We have
\begin{itemize}
\item {\em And rule.} For each given value $\alpha_3$, from (\hspace{-1.5 mm}~\ref{AND}) we have
that, for every $(\alpha_1, \alpha_2) \in [\alpha_3,1] \times [\alpha_3,1]$ such that
$\alpha_1 + \alpha_2 = 1 + \alpha_3$, if $P(B|A) \geq \alpha_1, P(C|A) \geq \alpha_2$
then $P(BC|A) \geq \alpha_3$.
\item {\em Cautious Monotonicity rule.}
For each given value $\alpha_3$, from (\hspace{-1.5 mm}~\ref{ACM}) we have
that, for every $(\alpha_1, \alpha_2) \in [\alpha_3,1] \times (0,1]$ such that
$\alpha_1 + (1 - \alpha_3)\alpha_2 = 1$, if $P(C|A) \geq \alpha_1, P(B|A) \geq \alpha_2$
then $P(C|AB) \geq \alpha_3$.
\item {\em Or rule.} For each given value $\alpha_3$, from (\hspace{-1.5 mm}~\ref{OR}) we have
that, for every $(\alpha_1, \alpha_2) \in [\alpha_3,1] \times [\alpha_3,1]$ such that
$\alpha_2 = \frac{\alpha_1\alpha_3}{\alpha_1(1+\alpha_3)-\alpha_3}$,
if $P(C|A) \geq \alpha_1, P(C|B) \geq \alpha_2$ then $P(C|A \vee B) \geq \alpha_3$.
\item {\em Cut rule.} For each given value $\alpha_3$, from (\hspace{-1.5 mm}~\ref{CUT}) we have
that for every $(\alpha_1, \alpha_2) \in [\alpha_3,1] \times [\alpha_3,1]$ such that
$\alpha_2 = \frac{\alpha_3}{\alpha_1}$, if $P(C|AB) \geq \alpha_1, P(B|A) \geq \alpha_2$
then $P(C|A) \geq \alpha_3$.
\end{itemize}
}\end{Rem}
\section{Propagation with $\epsilon-$values}
The results of the previous section can be examined in the particular case
in which for $i = 1,2$ it is $[\alpha_i, \beta_i] = [1 - \epsilon_i, 1]$.
As it can be verified, the $\epsilon-$values propagate in the following way.
\begin{itemize}
\item {\em And rule}. From (\hspace{-1.5 mm}~\ref{AND}), the probability bounds \linebreak
$[1 - \epsilon_1, 1], [1 - \epsilon_2, 1]$ on the antecedents $B|A, C|A$
of the rule propagate, on the consequent $BC|A$, to the exact bounds
$[1 - \epsilon_3, 1]$, with
\begin{equation}\label{AND2}
\epsilon_3 = \epsilon_1 + \epsilon_2
\end{equation}
\item {\em Cautious Monotonicity rule}.
From (\hspace{-1.5 mm}~\ref{ACM}), the probability intervals $[1 - \epsilon_1, 1]$,
$[1 - \epsilon_2, 1]$ on the antecedents $C|A, B|A$
of the rule propagate, on the consequent $C|AB$,
to $[1 - \epsilon_3, 1]$, with
\begin{equation}\label{CM}
\epsilon_3 = \frac{\epsilon_1}{1 - \epsilon_2}
\end{equation}
\item {\em Or rule}.
From (\hspace{-1.5 mm}~\ref{OR}), the probability intervals \linebreak
$[1 - \epsilon_1, 1], [1 - \epsilon_2, 1]$ on the antecedents $C|A, C|B$
of the rule propagate, on the consequent $C|(A \vee B)$,
to \linebreak $[1 - \epsilon_3, 1]$, with
\begin{equation}\label{OR2}
\epsilon_3 = \frac{\epsilon_1 + \epsilon_2 - 2\epsilon_1\epsilon_2}
{1 - \epsilon_1\epsilon_2}
\end{equation}
\item {\em Cut rule}.
From (\hspace{-1.5 mm}~\ref{CUT}), the probability intervals \linebreak
$[1 - \epsilon_1, 1], [1 - \epsilon_2, 1]$ on the antecedents $C|AB, B|A$
of the rule propagate, on the consequent $C|A$,
to \linebreak $[1 - \epsilon_3, 1]$, with
\begin{equation}\label{CUT2}
\epsilon_3 = \epsilon_1 + \epsilon_2 - \epsilon_1\epsilon_2
\end{equation}\end{itemize}
\begin{Rem}{\rm
Our results concerning the value of $\epsilon_3$ coincide with that
ones obtained in (Bourne, and Parsons 1998) for the rules {\em And} and
{\em Cautious Monotonicity} and are better for the rules {\em Or} and
{\em Cut}, as from (\hspace{-1.5 mm}~\ref{OR2}) and \linebreak (\hspace{-1.5 mm}~\ref{CUT2}) one has respectively
\[
\begin{array}{cc}
\epsilon_3 \; = \; \frac{\epsilon_1 + \epsilon_2 - 2\epsilon_1\epsilon_2}
{1 - \epsilon_1\epsilon_2} \; < \; \epsilon_1 + \epsilon_2 &
(\epsilon_1 < 1, \epsilon_2 < 1) , \\ \ \\
\epsilon_3 \; = \; \epsilon_1 + \epsilon_2 - \epsilon_1\epsilon_2 \; < \;
\epsilon_1 + \epsilon_2 & (\epsilon_1 > 0, \epsilon_2 > 0) .
\end{array}
\]
The use of the precise bounds may have some relevance when the inference rules
are applied with real $\epsilon-$values.
}
\end{Rem}
\section{An application}
We will now examine an example to give an idea, on one hand, of how much the
conclusions may be sensible
to the use of methods of exact propagation of probability bounds and, on another
hand, of the related phenomenon of degradation
of inference rules when interpreted in probabilistic terms.
The example is a modified version of an application
considered in (Bourne, and Parsons 1998) which was inspired by examples given in (Kraus, Lehmann, and Magidor 1990).
We consider a probabilistic knowledge base consisting of some conditional assertions,
which concern the fact that a given party has various attributes (the party is {\em great,
noisy}, {\em Linda} and {\em Steve} are {\em present}, and so on).
By the symbol
$A \; | \hspace{-1.8 mm} \sim_{\epsilon}B$ we denote the assessment $P(B|A) \geq 1 - \epsilon$.
We start with a knowledge
base which has the following rules and $\epsilon-$values:
\[
\begin{array}{ccc}
1. & & Linda \; | \hspace{-1.8 mm} \sim_{0.05} great \\ \ \\
2. & & Linda \; | \hspace{-1.8 mm} \sim_{0.2} Steve \\ \ \\
3. & & Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.1} \neg noisy \\ \ \\
4. & & Steve \; | \hspace{-1.8 mm} \sim_{0.05} Linda \\ \ \\
5. & & \neg noisy \; | \hspace{-1.8 mm} \sim_{0.2} \neg great
\end{array}
\]
Notice that the conditional $"Linda \; | \hspace{-1.8 mm} \sim_{0.05} great"$ means that the
probability of the conditional event
\begin{center}
"(The party will be great $|$ Linda goes to the party)"
\end{center}
is greater than or equal to $1 - 0.05$, and so on.
We are interested in propagating the previous bounds to find the
$\epsilon-$values of the following conditionals:
\[
\begin{array}{ccc}\label{SET}
(a) & & Linda \; | \hspace{-1.8 mm} \sim_{\epsilon} \neg noisy \\ \ \\
(b) & & \top \; | \hspace{-1.8 mm} \sim_{\epsilon}\neg Linda \\ \ \\
(c) & & Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{\epsilon} great \wedge \neg noisy \\ \ \\
(d) & & Steve \; | \hspace{-1.8 mm} \sim_{\epsilon} \neg noisy \\ \ \\
(e) & & Linda \vee Steve \; | \hspace{-1.8 mm} \sim_{\epsilon} \neg noisy
\end{array}
\]
By the symbol $\top$ we denote (any tautology representing) the certain
event. \\ Applying the Cut rule to the conditionals
\[
Linda \; | \hspace{-1.8 mm} \sim_{0.2} Steve \; , \; \; \;
Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.1} \neg noisy \; ,
\]
we obtain the conditional
\[
Linda \; | \hspace{-1.8 mm} \sim_{0.28} \neg noisy \; .
\]
Then, applying the And rule to the conditionals
\[
Linda \; | \hspace{-1.8 mm} \sim_{0.28} \neg noisy \; , \; \; \; Linda \; | \hspace{-1.8 mm} \sim_{0.05} great \; ,
\]
we obtain the conditional
\[
Linda \; | \hspace{-1.8 mm} \sim_{0.33} great \wedge \neg noisy \; .
\]
Applying the S rule to
\[
Linda \; | \hspace{-1.8 mm} \sim_{0.33} great \wedge \neg noisy
\]
we obtain
\[
\top \; | \hspace{-1.8 mm} \sim_{0.33} \neg Linda \vee great \wedge \neg noisy \; .
\]
Applying the S rule to
\[
\neg noisy \; | \hspace{-1.8 mm} \sim_{0.2} \neg great
\]
we obtain
\[
\top \; | \hspace{-1.8 mm} \sim_{0.2} noisy \vee \neg great \; .
\]
Finally, applying the And rule to the conditionals
\[
\top \; | \hspace{-1.8 mm} \sim_{0.33} \neg Linda \vee great \wedge \neg noisy , \;
\top \; | \hspace{-1.8 mm} \sim_{0.2} noisy \vee \neg great ,
\]
we obtain the conditional
\[
\top \; | \hspace{-1.8 mm} \sim_{0.53} \neg Linda \wedge (noisy \vee \neg great).
\]
Then, by the Right Weakening rule we have
\[
\top \; | \hspace{-1.8 mm} \sim_{0.53} \neg Linda \wedge (noisy \vee \neg great)
\; \Longrightarrow \;
\top \; | \hspace{-1.8 mm} \sim_{0.53} \neg Linda.
\]
Concerning the conditional $(c)$, applying the Cautious Monotonicity rule to
\[
Linda \; | \hspace{-1.8 mm} \sim_{0.05} great \; , \; \; \;
Linda \; | \hspace{-1.8 mm} \sim_{0.2} Steve \; ,
\]
we obtain
\[
Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.0625} great \; .
\]
Then, applying the And rule to the conditionals
\[
Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.0625} great, \;
Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.1} \neg noisy ,
\]
we obtain the conditional
\[
Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.0725} great \wedge \neg noisy \; .
\]
Concerning the conditional $(d)$, applying the Cut rule to the conditionals
\[
Linda \wedge Steve \; | \hspace{-1.8 mm} \sim_{0.1} \neg noisy \; , \; \;
Steve \; | \hspace{-1.8 mm} \sim_{0.05} Linda \; ,
\]
we obtain the conditional
\begin{equation}\label{15}
Steve \; | \hspace{-1.8 mm} \sim_{0.145} \neg noisy
\end{equation}
Then, applying the Or rule to the conditionals
\[
Steve | \hspace{-1.8 mm} \sim_{0.145} \neg noisy \; , \; \;
Linda | \hspace{-1.8 mm} \sim_{0.28} \neg noisy \; ,
\]
we obtain the conditional
\begin{equation}\label{16}
Linda \vee Steve | \hspace{-1.8 mm} \sim_{0.358} \neg noisy
\end{equation}
We observe that, propagating the bounds with
$\epsilon_3 = \epsilon_1 + \epsilon_2$, instead of the
conditionals (\hspace{-1.5 mm}~\ref{15}) and (\hspace{-1.5 mm}~\ref{16}) we would obtain
respectively
\[
Steve \; | \hspace{-1.8 mm} \sim_{0.15} \neg noisy \; ,
\]
and
\[
Linda \vee Steve \; | \hspace{-1.8 mm} \sim_{0.425} \neg noisy \; .
\]
\section{Conclusions}
In this paper the inference rules of System P have been considered in the
framework of coherence.
We have also examined the {\em disjunctive Weak Rational Monotony}
proposed by Adams in his extended probability logic, corresponding to
System P$^+$. Differently from the probabilistic approaches generally given
in the literature, see, in particular, (Hawthorne 1996) and (Schurz 1998),
within our framework we can directly manage conditional probability
assessments, even if some (or possibly all the) conditioning events have
zero probability. We think that this opportunity is important specially in
the field of nonmonotonic reasoning where infinitesimal probabilities
play a significant role. Moreover, exploiting our algorithms, the lower
and upper probability bounds
associated with the conditional assertions of a given knowledge base
can be propagated to further conditional assertions,
obtaining in all cases the precise probability bounds. In particular,
beside allowing a more flexible
and realistic approach to probabilistic default reasoning,
this provides an exact illustration of the degradation of System P rules
when interpreted in probabilistic terms. \\ \ \\
\subsection{Acknowledgments}
The author is grateful to the referees for the very helpful criticisms and suggestions.
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\end{document} | math | 45,367 |
\begin{document}
\title{Simulation of Zitterbewegung by modelling the Dirac equation in Metamaterials}
\author{Sven Ahrens}
\email[]{[email protected]}
\affiliation{Beijing Computational Science Research Center, Beijing 100094, China}
\author{Jun Jiang}
\affiliation{Tongji University, Shanghai 200092, China}
\author{Yong Sun}
\affiliation{Tongji University, Shanghai 200092, China}
\author{Shi-Yao Zhu}
\affiliation{Beijing Computational Science Research Center, Beijing 100094, China}
\date{\today}
\begin{abstract}
Strong field processes, which occur in intense laser fields are a test area for relativistic quantum field theory, but difficult to study experimentally and theoretically. Thus, modelling relativistic quantum dynamics and therewith the Dirac equation can help to understand quantum field theory. We develop a dynamic description of an effective Dirac theory in metamaterials, in which the wavefunction is modeled by the corresponding electric and magnetic field in the metamaterial. This electro-magnetic field can be probed in the experimental setup, which means that the wavefunction of the effective theory is directly accessible by measurement. Our model is based on a plane wave expansion, which ravels the identification of Dirac spinors with single-frequency excitations of the electro-magnetic field in the metamaterial. We proof the validity of our relativistic quantum dynamics simulation by demonstrating the emergence of Zitterbewegung and verifying it with an analytic solution.
\end{abstract}
\pacs{42.70.Qs, 78.67.Pt, 81.05.Xj}
\keywords{Metamaterial}
\maketitle
\section{Introduction}
Fundamental properties in quantum electro-dynamics can be studied in strong field physics, in which for example intense laser beams can create electron positron pairs in the unstable vacuum\cite{schwinger_1951_pair_creation}. The study of these processes analytically\cite{huayu_carsten_2010_pair_creation} or numerically \cite{hebenstreit_2008_pair_production,matthias_carsten_2009_pair_creation,steinacher_ahrens_grobe_2014_charge_density} demands extended efforts. Therefore modelling systems are discussed in the literature, which allow to re-engineer physics in extreme conditions\cite{szpak_2011_qed_simulator,szpak_2012_qed_simulator}. The capability of simulating intrinsic relativistic quantum processes like pair-creation or Zitterbewegung can therefore improve and test the understanding of quantum field theory.
The latter mentioned Zitterbewegung, first considered by Schr\"odinger \cite{Schroedinger_1930_Zitterbewegung}, is a small quivering motion of particles in relativistic quantum mechanics. The theoretical existence of this quivering motion has been evidenced by numerical simulations of the Dirac equation and of quantum-field-theory\cite{Grobe_1999_Numerical_Dirac_equation_Zitterbewegung,Grobe_2004_Lack_of_Zitterbewegung,Grobe_2008_Zitterbewegung_distinguishable_fermions}. However, the motion itself is very small and very fast. It can be estimated to be at the time scale of electron-positron pair creation ($10^{-20}$ seconds) and has an amplitude on the order of the Compton wavelength of an electron ($10^{-12}$ meters). Therefore, the experimental observation of this counter-intuitive property of nature is challenging.
Nevertheless, the Zitterbewegung can be modeled in various other effective systems and it's appearance has been subject of diverse investigations. It has been discussed for superconductors\cite{Lurie_1970_Zitterbewegung_Superconducting_id_25} in 1970, for one-dimensional chains\cite{Cannata_1990_tight_binding_Zitterbewegung_id_23} in 1990 and many considerations set in on semiconductors\cite{Ferrari_Russo_1990_nonrelativistic_zitterbewegung_in_two-band_systems_id_8,Cannata_Ferrari_1991_Zitterbewegung_two-band_systems_id_9,Zawadzki_2005_zitterbewegung_in_semiductors_id_10,Schliemann_Westervelt_2005_Zitterbewegung_semiconductor_quantum_wells_id_3,Shung-Qing_Shen_2005_zitterbewegung_spin_transverse_force_id_12,Schliemann_Westervelt_2006_Zitterbewegung_semiconductor_quantum_wells_id_11,Rusin_Zawadazki_2007_Zitterbewegung_semiconductors_id_24}. In 2005 renewed discussions on Zitterbewegung in Spintronics\cite{Jiang_2005_zitterbewegung_luttinger_hamiltonian_id_13,Brusheim_2006_electron_waveguide_id_14}, Graphene\cite{Katsnelson_2006_Zitterbewegung_Graphene_id_26,Peres_Novoselov_2009_zitterbewegung_the_electronic_properties_of_graphene_id_18} and carbon nanotubes\cite{Zawadzki_2006_zitterbewegung_carbon_nanotubes_id_15,Tomasz_Wlodek_Zitterbewegung_charge_carriers_graphene_id_5} raised up, (see also\cite{Cserti_2006_zitterbewegung_spintronic_graphene_superconductin_systems_id_1}). Zitterbewegung has also been investigated for trapped ions\cite{Bermudez_2007_zitterbewegung_Jaynes-Cummings_model_id_16,Lamata_2007_zitterbewegung_single_trapped_ion_id_6,Xiong_2008_zitterbewegung_by_quantum_field-theory_considerations_id_17,Wunderlich_2009_cold_trapped_ion_zitterbewegung_id_28}, ultracold atoms\cite{Vaishnav_2008_Zitterbewegung_Ultracold_Atoms_id_4} and Bose-Einstein condensates\cite{Engels_2013_zitterbewegung_bose-einstein_condensate_id_32} with an experimental demonstration of the effect in 2010\cite{Blatt_2010_Zitterbewegung_First_Experiment_id_31}. More considerations on Zitterbewegung aroused for photonic crystals\cite{Zhang_2008_Zitterbewegung_Dirac_Point_id_29}, negative-zero-positive index materials\cite{Zhu_2009_NZPI_Zitterbewegung_id_33} and binary waveguide arrays\cite{Longhi_2010_Zitterbewegung_Binary_Waveguide_Array_id_30}.
However, it is difficult to access the wavefunction itself in the listed experiments, where in particular higher order processes in quantum field theory are at least based on the knowledge of the wavefunction\cite{fradkin_1991_qed_unstable_vacuum}. Recent experimental investigations demonstrated, that the wavefunction of the Dirac equation can be imitated by the electro-magnetic field of a waveguide structure with designable electro-dynamic properties \cite{topological_excitations_tan_2014}. Since this waveguide operates in the microwave regime, the simulated wavefunction can be directly probed in the experiment. But only quasi-stationary field configurations have been investigated and the question arises, whether the
time-independent description in terms of frequency-eigensolutions is capable of forming dynamics, which correspond to the Dirac equation.
Here, we extend the quasi-static theory to a dynamic description of an effective Dirac equation (section \ref{sec:theory_description}) and use it for the demonstration of the Zitterbewegung in theory. In the subsections \ref{sec:maxwell_equations} and \ref{sec:effectice_dirac_equation} we repeat the already established description \cite{topological_excitations_tan_2014} of the Maxwell equations in metamaterials and show how solutions of the Dirac equation can be deduced. Next, we formulate a formal solution of the Maxwell equations by an expansion in plane-wave solutions in subsection \ref{sec:time-evolution_maxwell-equations}. This solution can be identified with the unitary time-evolution of the Dirac equation, which is discussed in subsection \ref{sec:time-evolution_dirac-equation}, in which we also give an explicit mapping between the electro-magnetic field and the Dirac wave-function in frequency- and momentum space. The effective parameters for the mass and the speed of light of the emulated Dirac equation depend on the metamaterial an are derived in subsection \ref{sec:scaled_dirac_equation}. In section \ref{sec:boundary_conditions} we refer back to the Maxwell equations for deriving boundary conditions, with which electro-magnetic input pulses can be injected at the metamaterial interfaces in our simulation.
After the introduction of the theory foundations, we describe the numerical implementation in section \ref{sec:numercial_implementation}, in which we also consider the properties of periodic boundary conditions, which are implied by our simulation method.
In the Results section \ref{sec:results}, we present the metamaterial simulations in three kinds of dynamical scenarios, in which we first consider the easiest possible setup for a Zitterbewegung with a Gaussian wavepacket excitation \ref{sec:gaussian_wavepacket}. This setup is modified into a counterpropagating excitation, such that an experimental realization might be more feasible \ref{sec:counterpropagating_wavepacket} and finally we also account for the influence of the boundaries of the metamaterial \ref{sec:boundary_wavepacket}.
The appendix contains a derivation of a formula of the expectation value of the position expectation operator, with which the Zitterbewegung can be computed semi-analytically\ref{sec:zitterbewegung_of_real_electron}. Another section discusses the absence of imaginary values in a real experiment, while the corresponding Dirac wavefunction consists of complex numbers\ref{sec:imaginary_part}.
\section{Theory description\label{sec:theory_description}}
\subsection{The Maxwell equations\label{sec:maxwell_equations}}
The description of the one-dimensional Maxwell equations
\begin{subequations}
\begin{alignat}{3}
- \partial_x E_z &= &i &\omega \mu_0 \mu_r(\omega) H_y \label{eq:maxwell_equations_Ez}\\
\partial_x H_y &= &-i &\omega \epsilon_0 \epsilon_r(\omega) E_z\,.\label{eq:maxwell_equations_Hy}
\end{alignat}\label{eq:maxwell_equations}
\end{subequations}
in the photonic crystal is adapted from an effective model for the electro-magnetic field in a metamaterial waveguide, which has been developed recently \cite{topological_excitations_tan_2014}. Here, the electric and magnetic field is propagating in the $x$-direction and $\epsilon_0$ and $\mu_0$ are the vacuum permittivity and vacuum permeability, respectively. The frequency dependent relative permittivity and relative permeability is given by
\begin{subequations}
\begin{align}
\epsilon_r(\omega) &= \frac{1}{p \epsilon_0} \left( C_0 - \frac{1}{\omega^2 L d} \right) \quad \textrm{and}\label{eq:effective_permittivity} \\
\mu_r(\omega) &= \frac{p}{\mu_0} \left( L_0 - \frac{1}{\omega^2 C d} \right)\,, \label{eq:effective_permeability}
\end{align}\label{eq:permittivity_and_permeability}
\end{subequations}
where we assume, that damping of the electro-magnetic field is negligible and the metamaterial is homogeneous along the $x$-direction in space. Damping terms may lead to a decay of electro-magnetic excitations, which might be subject to future studies. We assume specific material parameters for our model, which are listed in table \ref{tab:metamaterial_properties}.
\begin{table}[!ht]
\caption{
\bf{Metamaterial parameters}}
\begin{tabular}{ll}
$d=8\,\textrm{mm}$ & element length\\
$p=4$ & geometric factor \\
$C=2.82\,\textrm{pF}$ & series capacitance of the loading elements \\
$C_0=58.8\,\textrm{pF/m}$ & per-unit-length capacitance of the trans-\\
& mission line segment \\
$L=19.5\,\textrm{nH}$ & shunt inductance of the loading elements \\
$L_0=314\,\textrm{nH/m}$ & per-unit-length inductance of the trans-\\
& mission line segment
\end{tabular}
\begin{flushleft}
This table contains a list of the parameter properties of the permittivity and permeability in equations \eqref{eq:permittivity_and_permeability}.
\end{flushleft}
\label{tab:metamaterial_properties}
\end{table}
As a result, the electro-magnetic metamaterial properties assume the specific relations
\begin{subequations}
\begin{align}
\epsilon_r(\omega) &= 1.66 - 188 \left(\frac{\textrm{GHz}}{\omega}\right)^2 \quad \textrm{and}\\
\mu_r(\omega) &= 1.00 - 141 \left(\frac{\textrm{GHz}}{\omega}\right)^2\,.
\end{align}
\end{subequations}
\subsection{Effective Dirac equation\label{sec:effectice_dirac_equation}}
As discussed in \cite{topological_excitations_tan_2014}, the Maxwell equations can be transformed to a one-dimensional, two-component Dirac equation by introducing
\begin{subequations}
\begin{align}
\varphi_1 &= \sqrt{\epsilon_0} E_z\,, \\
\varphi_2 &= \sqrt{\mu_0} H_y\,.
\end{align}\label{eq:wavefunction_em-field_relations}
\end{subequations}
If one identifies the effective mass
\begin{equation}
m(\omega) = \frac{\omega}{2 c}\left[ \epsilon_r(\omega) - \mu_r(\omega) \right]\label{eq:Dirac_mass}
\end{equation}
and the effective energy
\begin{equation}
\mathcal{E}(\omega) = - \frac{\omega}{2 c} \left[ \epsilon_r(\omega) + \mu_r(\omega) \right]\,.\label{eq:Dirac_energy}
\end{equation}
one can rewrite the Maxwell equations, such that they are of the same structural form as the one-dimensional Dirac equation
\begin{equation}
- i \sigma_x \partial_x \varphi + m(\omega) \sigma_z \varphi = \mathcal{E}(\omega) \varphi \,. \label{eq:metamaterial_dirac_equation}
\end{equation}
Here, $\varphi$ is the two-component wavefunction
\begin{equation}
\varphi =
\begin{pmatrix}
\varphi_1 \\ \varphi_2
\end{pmatrix}\,,\label{eq:em-wavefunction_transformation}
\end{equation}
$c=1/\sqrt{\epsilon_0 \mu_0}$ is the vacuum speed light and $\sigma_x$ and $\sigma_z$ are the first and third Pauli matrices
\begin{equation}
\sigma_x =
\begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}
\,,\quad
\sigma_z =
\begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}\,.
\end{equation}
\subsection{Time evolution of Maxwell equations\label{sec:time-evolution_maxwell-equations}}
In order to establish a dynamic Dirac theory, we first consider the time-evolution of the Maxwell equations \eqref{eq:maxwell_equations}. Since the Maxwell equations are specified in frequency space, we evolve the electro-magnetic field in the metamaterial by the time-evolution of an expansion of plane wave momentum eigenfunctions $e^{i k x}$. Applying these plane waves yields the Maxwell equations in frequency- and momentum-space
\begin{subequations}
\begin{align}
k E_{z,k}(\omega) &= - \omega \mu_0 \mu_r(\omega) H_{y,k}(\omega) \label{eq:maxwell_equations_frequency_space_E_z_k}\,,\\
k H_{y,k}(\omega) &= - \omega \epsilon_0 \epsilon_r(\omega) E_{z,k}(\omega) \label{eq:maxwell_equations_frequency_space_H_y_k}\,,
\end{align} \label{eq:maxwell_equations_frequency_space_k}
\end{subequations}
where the $E_{z,k}(\omega)$ and $H_{y,k}(\omega)$ are the electric and magnetic field amplitudes for a certain field frequency $\omega$ and a certain momentum $k$. From these Maxwell equations one can derive the dispersion relation
\begin{equation}
k^2 = \frac{\omega^2}{c^2} \epsilon_r(\omega) \mu_r(\omega)\,. \label{eq:dispersion-relation}
\end{equation}
The dispersion relation implies, that there exists a positive and negative momentum $k$ for each frequency $\omega$. Furthermore, if one inserts the permittivity \eqref{eq:effective_permittivity} and permeability \eqref{eq:effective_permeability}, one obtains a polynomial of second order in $\omega^2$. This means, that there exist two frequencies (upper band $\omega_+$ and lower band $\omega_-$) and the identical negative values $-\omega_+$ and $-\omega_-$ for each momentum $k$. This is plotted in figure \ref{fig:dispersion_relation} (a) for the two positive frequencies $\omega_+$ and $\omega_-$.
\begin{figure}
\caption{(a) Dispersion relation of the metamaterial \eqref{eq:dispersion-relation}
\label{fig:dispersion_relation}
\end{figure}
Upper and lower band are divided by the band gap in which the product of $\epsilon_r(\omega) \mu_r(\omega)$ turns negative. Outside of the band gap, the product $\epsilon_r(\omega) \mu_r(\omega)$ is always positive, and if also $k^2$ is positive (ie. $k\in\mathbb{R}$), it follows, that $\omega^2$ is positive too and therefore $\omega \in \mathbb{R}$. The upper and lower boundary of the band gap in frequency space is given by the frequencies, at which $\epsilon_r$ and $\mu_r$ are zero. This happens at the frequency
\begin{equation}
\omega_1 = 10.4\,\textrm{GHz}\,,
\end{equation}
at which $\epsilon_r$ is zero and at the frequency
\begin{equation}
\omega_2 = 11.8\,\textrm{GHz}\,,
\end{equation}
at which $\mu_r$ is zero, for the parameters of table \ref{tab:metamaterial_properties}. The comparison of the dispersion relation of the metamaterial in Fig. \ref{fig:dispersion_relation} (a) with the relativistic energy-momentum relation of the scaled, exact Dirac theory in Fig. \ref{fig:dispersion_relation} (b) illustrates a possible dynamical similarity of both descriptions around the momentum $k=0\,\textrm{m}^{-1}$.
Since the Maxwell equations are linear differential equations, the superposition of solutions is a solution again. Therefore, the time-evolution of the electro-magnetic field can be written down by the expansion
\begin{subequations}
\begin{align}
E_z(x,t) &= \sum_{k \atop{\{\omega_+,\omega_-,\atop{-\omega_+,-\omega_-\}}}} E_{z,k}(\omega) e^{i (k x - \omega t)} \ \Delta k\,, \label{eq:E-field_expansion}\\
H_y(x,t) &= \sum_{k \atop{\{\omega_+,\omega_-,\atop{-\omega_+,-\omega_-\}}}} H_{y,k}(\omega) e^{i (k x - \omega t)} \ \Delta k\,, \label{eq:H-field_expansion}
\end{align}\label{eq:field_expansion}
\end{subequations}
in which the sum runs over all discrete momenta $k$ and all frequencies $\{\omega_+(k),\omega_-(k),-\omega_+(k),-\omega_-(k)\}$ for each momentum. The factor $\Delta k$ is a measure for the summation in momentum space and is considered to be analogous to the measure $dk$ of an integral. Since momentum space is spaced equidistantly in our approach, we write down a sum in \eqref{eq:field_expansion} with measure $\Delta k$ and focus on the plane wave solution in this more fundamental theory part. The spacing of $\Delta k$ and it's relation to position space is discussed in the section on numerical implementation \ref{sec:numercial_implementation}.
In the expansion \eqref{eq:field_expansion}, the magnetic field can be expressed by the electric field by using one of the two Maxwell equations \eqref{eq:maxwell_equations_frequency_space_k}, yielding
\begin{subequations}
\begin{align}
H_{y,k}(\omega) &= - \frac{k}{\omega \mu_0 \mu_r(\omega)} E_{z,k}(\omega)\textrm{, or}\label{eq:E-H-relation_mu} \\
H_{y,k}(\omega) &= - \frac{\omega \epsilon_0 \epsilon_r(\omega)}{k} E_{z,k}(\omega)\,.\label{eq:E-H-relation_epsilon}
\end{align}\label{eq:E-H-relation}
\end{subequations}
Therefore, the sum \eqref{eq:E-field_expansion} is already sufficient for a unique specification of the electro-magnetic field in the metamaterial.
\subsection{Time evolution of the effective Dirac equation\label{sec:time-evolution_dirac-equation}}
In order to write down the time-evolution in Dirac theory, the plane wave solutions of the Dirac equation \eqref{eq:metamaterial_dirac_equation} are to be considered. These are usually obtained by applying the plane waves $e^{i k x}$ at the Hamiltonian and solving the eigenvector and eigenvalue problem of the Hamiltonian matrix. The left-hand side of the Dirac equation \eqref{eq:metamaterial_dirac_equation} turns into the matrix
\begin{equation}
\begin{pmatrix}
m(\omega) & k \\
k & - m(\omega)
\end{pmatrix}\,,\label{eq:dirac_matrix}
\end{equation}
when applying the plane wave $e^{i k x}$. The definition of the mass \eqref{eq:Dirac_mass} and energy \eqref{eq:Dirac_energy} together with the dispersion relation \eqref{eq:dispersion-relation} ensure the identity
\begin{equation}
\mathcal{E}(\omega)^2 = k^2 + m(\omega)^2\,,\label{eq:energy_momentum_relation}
\end{equation}
which can be seen as relativistic energy-momentum relation of an electro-magnetic excitation in the metamaterial with wave number $k$ and corresponding frequency $\omega$. Making use of \eqref{eq:energy_momentum_relation} one obtains the eigenvector
\begin{subequations}
\begin{equation}
u_{k}^+(\omega) =
\frac{1}{\sqrt{2 |\mathcal{E}(\omega)|[|\mathcal{E}(\omega)| + m(\omega)]}}
\begin{pmatrix}
|\mathcal{E}(\omega)| + m(\omega) \\ k
\end{pmatrix}\label{eq:positive_spinor}
\end{equation}
with eigenvalue $|\mathcal{E}(\omega)|$ and the eigenvector
\begin{equation}
u_{k}^-(\omega) =
\frac{1}{\sqrt{2 |\mathcal{E}(\omega)|[|\mathcal{E}(\omega)| + m(\omega)]}}
\begin{pmatrix}
- k \\ |\mathcal{E}(\omega)| + m(\omega)
\end{pmatrix}\label{eq:negative_spinor}
\end{equation}\label{eq:spinors}
\end{subequations}
with eigenvalue $-|\mathcal{E}(\omega)|$, for the matrix \eqref{eq:dirac_matrix}. The solutions \eqref{eq:spinors} are commonly known as bi-spinors of the Dirac equation in one dimension \cite{parthasarathy_2010_rel_quantum_mech}. We have normalized the bi-spinors in our notion, such that they form an an orthonormal base. Note, that the spinors \eqref{eq:spinors} have the identical analytic expressions
\begin{subequations}
\begin{equation}
u_{k}^+(\omega) =
\frac{\textrm{sign}(k)}{\sqrt{2 |\mathcal{E}(\omega)|[|\mathcal{E}(\omega)| - m(\omega)]}}
\begin{pmatrix}
k \\ |\mathcal{E}(\omega)| - m(\omega)
\end{pmatrix}\label{eq:positive_spinor_alternative}
\end{equation}
and
\begin{equation}
u_{k}^-(\omega) =
\frac{\textrm{sign}(k)}{\sqrt{2 |\mathcal{E}(\omega)|[|\mathcal{E}(\omega)| - m(\omega)]}}
\begin{pmatrix}
- |\mathcal{E}(\omega)| + m(\omega) \\ k
\end{pmatrix}\label{eq:negative_spinor_alternative}
\end{equation}\label{eq:spinor_alternative}
\end{subequations}
respectively, with the signum of $k$
\begin{equation}
\textrm{sign}(k) :=
\begin{cases}
\phantom{-}1 & \textrm{, if } 0 \le k \\
-1 & \textrm{\, else.}
\end{cases}
\end{equation}
Even though the eigensolutions of the matrix \eqref{eq:dirac_matrix} are determined by (\ref{eq:spinors},\ref{eq:spinor_alternative}), one has to keep in mind, that this matrix is an integrated part of equation \eqref{eq:metamaterial_dirac_equation}. As a consequence the plane wave eigensolutions are formed by the electro-dynamics in the metamaterial. Since $\mathcal{E}(\omega)$ is negative in the upper band and positive in the lower band, only the negative eigensolution $u_k^- e^{i k x}$ appears in the upper band and only the positive eigensolution $u_k^+ e^{i k x}$ appears in the lower band. In fact, the spinors \eqref{eq:spinors} can be expressed in terms of the electro-magnetic field. Keeping in mind, that $\mathcal{E}(\omega)$ is positive at the lower band, the spinor $u_k^+$ can be written as
\begin{subequations}
\begin{align}
u_{k}^+(\omega_-) &=
\frac{1}{\sqrt{2 \mathcal{E}(\omega_-)[\mathcal{E}(\omega_-) + m(\omega_-)]}}
\begin{pmatrix}
\mathcal{E}(\omega_-) + m(\omega_-) \\ k
\end{pmatrix}\nonumber\\
&=\frac{1}{\sqrt{[\epsilon_r(\omega_-) + \mu_r(\omega_-)] \mu_r(\omega_-)}}
\begin{pmatrix}
- \mu_r(\omega_-) \\ \frac{c k}{\omega_-}
\end{pmatrix} \,.\label{eq:positive_spinor_em}
\end{align}
Similarly, $\mathcal{E}(\omega)$ is negative at the upper band and by using the solution \eqref{eq:negative_spinor_alternative} for $u_k^-$, one can write
\begin{align}
u_{k}^-(\omega_+) &=
\frac{\textrm{sign}(k)}{\sqrt{2 \mathcal{E}(\omega_+)[\mathcal{E}(\omega_+) + m(\omega_+)]}}
\begin{pmatrix}
\mathcal{E}(\omega_+) + m(\omega_+) \\ k
\end{pmatrix}\nonumber\\
&=\frac{\textrm{sign}(k)}{\sqrt{[\epsilon_r(\omega_+) + \mu_r(\omega_+)] \mu_r(\omega_+)}}
\begin{pmatrix}
- \mu_r(\omega_+) \\ \frac{c k}{\omega_+}
\end{pmatrix} \,,\label{eq:negative_spinor_em}
\end{align}\label{eq:spinor_em}
\end{subequations}
which is identical to \eqref{eq:positive_spinor_em}, except the sign of $k$. The $\textrm{sign}(k)$ factor in \eqref{eq:negative_spinor_em} is necessary, because we want to keep the convention of the spinor solutions \eqref{eq:spinors}, as close to common relativistic quantum dynamics as possible (see also \eqref{eq:spinors_vacuum}). Note, that the fraction of upper and lower component of the spinors \eqref{eq:spinor_em} corresponds to the factor \eqref{eq:E-H-relation_mu} between electric and magnetic field, if one accounts for the transformation law \eqref{eq:wavefunction_em-field_relations} of the electro-magnetic field of the Dirac wave function. Therefore, the free eigensolutions of the Dirac equation are linearly dependent to the plane wave solutions of the Maxwell equations and it just remains to determine the proportionality constant, to make a complete, unique link between the Maxwell equations of the electro-magnetic field in the metamaterial and the corresponding dynamic Dirac equation. To do so, we write down the time evolution of the wave function by expanding it with respect to the obtained eigenfunctions $u_k^+ e^{i k x}$ and $u_k^- e^{i k x}$ of the Dirac equation \eqref{eq:metamaterial_dirac_equation}.
\begin{equation}
\varphi(x,t) = \frac{1}{\sqrt{N}}\sum_{k} \Bigg[\phi_k^+(\omega_-) \, u_k^+(\omega_-) \, e^{i (k x - \omega_- t)}
+ \phi_k^-(\omega_+) \, u_k^-(\omega_+) \, e^{i (k x - \omega_+ t)}\Bigg] \Delta k\,,\label{eq:wavefunction_expansion}
\end{equation}
Here, $\phi_k^+(\omega)$ and $\phi_k^-(\omega)$ are expansion coefficients,
\begin{equation}
N=\int_{-\infty}^\infty |\varphi(x,t)|^2 dx \label{eq:position_space_normalization}
\end{equation}
is a normalization constant and $\Delta k$ is the measure for the momentum space summation, which is already mentioned in subsection \ref{sec:time-evolution_maxwell-equations} and further discussed in section \ref{sec:numercial_implementation}. Since \eqref{eq:wavefunction_expansion} is a unitary transformation of the initial state of the wave function, the normalization constant will keep constant in time and $\varphi(x,t)$ will always stay normalized at one. Note, that according to the considerations above, the positive eigensolutions which are proportional to $u_k^+$ will evolve with the lower band frequency $\omega_-$ and the negative eigensolutions which are proportional to $u_k^-$ will evolve with the upper band frequency $\omega_+$.
If one compares the expansion of the electric field \eqref{eq:E-field_expansion} with the first component of the wave function \eqref{eq:wavefunction_expansion} and requires equality for the prefactors of the exponential $e^{i k x - \omega_+}$ one obtains
\begin{subequations}
\begin{equation}
E_{z,k}(\omega_+) = - \phi_k^-(\omega_+) \frac{\textrm{sign}(k) c \sqrt{\mu_0 |\mu_r(\omega_+)|}}{\sqrt{|\epsilon_r(\omega_+) + \mu_r(\omega_+)|}}\,. \label{eq:expansion_coefficient_equality_upper_band}
\end{equation}
Similarly, one obtains
\begin{equation}
E_{z,k}(\omega_-) = \phi_k^+(\omega_-) \frac{c \sqrt{\mu_0 |\mu_r(\omega_-)|}}{\sqrt{|\epsilon_r(\omega_-) + \mu_r(\omega_-)|}} \label{eq:expansion_coefficient_equality_lower_band}
\end{equation}\label{eq:expansion_coefficient_equality}
\end{subequations}
if one requires equality for the prefactors of $e^{i k x - \omega_-}$. One may also equalize the expansion of the magnetic field \eqref{eq:H-field_expansion} with the second component of the wave function \eqref{eq:wavefunction_expansion} for the exponentials $e^{i k x - \omega_+}$ and $e^{i k x - \omega_-}$. The result will be equivalent to equations \eqref{eq:expansion_coefficient_equality}, because the spinors \eqref{eq:spinor_em} are linearly dependent on the electro-magnetic excitations in the expansions \eqref{eq:field_expansion}.
\subsection{The metamaterial's scaled Dirac equation\label{sec:scaled_dirac_equation}}
According to the considerations in the sections \ref{sec:time-evolution_maxwell-equations} and \ref{sec:time-evolution_dirac-equation} the metamaterial's emulated quantum dynamics is expected to evolve as in Dirac theory, but natural constants like the electron mass and the effective speed of light depend on the metamaterial properties.
For deducing these properties, we first consider the time-evolution equation
\begin{equation}
i \hbar \partial_t \varphi(x,t) = H \varphi(x,t)\,,\label{eq:quantum_mechanical_time_evolution}
\end{equation}
with the Dirac Hamiltonian
\begin{equation}
H = c p_x \sigma_x + m_0 c^2 \sigma_z\,,\label{eq:vacuum_dirac_hamiltonian}
\end{equation}
in relativistic quantum mechanics. Here, $m_0$ is the electron rest mass and
\begin{equation}
p_x=- i \hbar \partial_x \label{eq:momentum_operator}
\end{equation}
is the momentum operator. After a Fourier transform of this equation into frequency space, the time-derivative $i \partial_t$ is substituted by $\omega$, yielding
\begin{equation}
H \varphi(x,\omega) = \hbar \omega \varphi(x,\omega)\,.\label{eq:desired-dirac_form}
\end{equation}
If equation \eqref{eq:metamaterial_dirac_equation} could be cast in a similar shape with a linear $\omega$ at the right-hand side, it's generalization to the time-domain would be straight forward and the mass and speed of light parameters of the emulated Dirac equation could be read of from the corresponding Hamiltonian on the left-hand side.
This can be achieved by a Taylor expansion of the effective Energy $\mathcal{E}(\omega)$ and effective mass $m(\omega)$ of the metamaterial Dirac equation \eqref{eq:metamaterial_dirac_equation}, at the frequency $\omega_0$, at which $\mathcal{E}(\omega)$ is zero. Thus, $\omega_0$ is determined by setting the definition \eqref{eq:Dirac_energy} of $\mathcal{E}(\omega)$ to zero and solving for $\omega$, resulting in
\begin{equation}
\omega_0 = \sqrt{\frac{1}{CLd}\frac{\mu_0 C + p^2 \epsilon_0 L}{\mu_0 C_0 + p^2 \epsilon_0 L_0}}\,.
\end{equation}
For the parameters chosen in table \ref{tab:metamaterial_properties} $\omega_0$ has the value $11.0\,\textrm{GHz}$. The first relevant order of the Taylor expansion is sufficient for our consideration. For the mass $m(\omega)$, the first relevant order is the zeroth order, whereas for the effective Energy $\mathcal{E}(\omega)$, the first relevant order is the first order. The reason is, that $\omega_0$ is chosen such that the zeroth order of $\mathcal{E}(\omega)$ vanishes. Accordingly, one obtains
\begin{equation}
- i \sigma_x \partial_x \varphi + m(\omega_0) \sigma_z \varphi = \left. \frac{\partial \mathcal{E}}{\partial \omega}\right|_{\omega_0} (\omega - \omega_0) \,\varphi \label{eq:taylored_dirac_equation}
\end{equation}
for equation \eqref{eq:metamaterial_dirac_equation}. If one defines the shifted frequency
\begin{equation}
\Delta \omega := \omega - \omega_0\,,
\end{equation}
the effective speed of light
\begin{equation}
c_D := - \left(\left.\frac{\mathcal{E}(\omega)}{\partial \omega}\right|_{\omega_0}\right)^{-1},
\end{equation}
the new effective mass
\begin{equation}
m' = m(\omega_0)\frac{\hbar}{c_D}
\end{equation}
and multiplies equation \eqref{eq:taylored_dirac_equation} with $\hbar\, c_D$, then equation \eqref{eq:taylored_dirac_equation} can be transformed into
\begin{equation}
p_x c_D \sigma_x \varphi + m' c_D^2 \sigma_z \varphi = - \hbar \, \Delta \omega \,\varphi\,.\label{eq:scaled_dirac_equation}
\end{equation}
This is consistent with the Hamiltonian in equation \eqref{eq:desired-dirac_form} and a negative time evolution, according to equation \eqref{eq:quantum_mechanical_time_evolution}. Comparing the corresponding Hamiltonian
\begin{equation}
H = p_x c_D \sigma_x + m' c_D^2 \sigma_z
\end{equation}
with the Hamiltonian of Dirac theory \eqref{eq:vacuum_dirac_hamiltonian}, one deduces the substitution
\begin{subequations}
\begin{align}
c &\rightarrow c_D\,, \\
m_0 &\rightarrow m'\,,
\end{align}\label{eq:scaling_replacements}
\end{subequations}
which scales the Dirac theory of electrons such that it is similar to the emulated Dirac dynamics of the metamaterial.
We point out, that equation \eqref{eq:scaled_dirac_equation} implies a negative time-evolution, because it's right-hand side contains a minus sign in front of the $\hbar \Delta \omega \varphi$ expression. In a Dirac equation with positive time-evolution, this minus sign is a plus sign (see equation \eqref{eq:desired-dirac_form}). The minus sign originates from our chosen convention (\ref{eq:wavefunction_em-field_relations},\ref{eq:Dirac_mass},\ref{eq:Dirac_energy}). A different convention might require an inversion of the $x$-axis. Therefore, if we want to compare the metamaterial dynamics with results from exact Dirac theory, we have to revert time. In particular we have to add a minus sign in the time argument of the semi-analytic expression \eqref{eq:zitterbewegung_expectation_complex} and \eqref{eq:zitterbewegung_expectation} of the Zitterbewegung, which is derived in appendix \ref{sec:zitterbewegung_of_real_electron}.
\section{Boundary conditions\label{sec:boundary_conditions}}
In this section, we consider electro-dynamic boundary conditions, for accounting for the finite space extension of the metamaterial. We assume, that the 'left' end of the metamaterial is at location $x_a$ and the 'right' end of the metamaterial is at location $x_b$, where the notion 'left' and 'right' implies that $x_a < x_b$ on the $x$-coordinate. As a consequence, the positions $x_a$ and $x_b$ are the limiters of three different regions, which are denoted by the indices $(1)$, $(2)$ and $(3)$, in which the physical space is divided into
\begin{align}
x < x_a &:\quad \textrm{region (1), the left input wave guide,} \nonumber \\
x_a < x < x_b &:\quad \textrm{region (2), the metamaterial,} \\
x_b < x \phantom{< x_a,}&:\quad \textrm{region (3), the right input wave guide.} \nonumber
\end{align}
The boundary conditions are obtained by integration of equation \eqref{eq:maxwell_equations} over the infinitesimal interval $[x_a - \epsilon,x_a + \epsilon]$ and $[x_b - \epsilon,x_b + \epsilon]$.
We make the reasonable assumption, that the right-hand side of equation \eqref{eq:maxwell_equations} is bound and has a finite value. In this case the integrals over the intervals with infinitely small $\epsilon$ imply, that the electric and magnetic fields must be smooth at the boundaries $x_a$ and $x_b$, which means that
\begin{subequations}
\begin{align}
E_z^{(1)}(x_a,\omega) &= E_z^{(2)}(x_a,\omega)\,,\\
E_z^{(2)}(x_b,\omega) &= E_z^{(3)}(x_b,\omega)\,,\\
H_y^{(1)}(x_a,\omega) &= H_y^{(2)}(x_a,\omega)\,,\\
H_y^{(2)}(x_b,\omega) &= H_y^{(3)}(x_b,\omega)
\end{align}\label{eq:E-and-H_continuity_omega}
\end{subequations}
holds.
We assume that the relative permittivity and permeability are just $1$ for the propagation along the input wave guides of the regions (1) and (3), like it is for electro-magnetic waves in vacuum. Then, the dispersion relation \eqref{eq:dispersion-relation} simplifies to
\begin{equation}
|\omega| = c |k|.
\end{equation}
For left propagating waves, in which $E_{z,k}(\omega) = H_{y,k}(\omega) = 0$ and $E_{z,-k}(-\omega) = H_{y,-k}(-\omega) = 0$, the field expansion \eqref{eq:field_expansion} can be simplified to
\begin{subequations}
\begin{align}
E_{z,\textrm{left}}(x,t) &= \sum_{k , \{\omega,-\omega\}} E_{z,k}(\omega) e^{i k (x - c t)}\,,\\
H_{y,\textrm{left}}(x,t) &= \sum_{k ,\{\omega,-\omega\}} H_{y,k}(\omega) e^{i k (x - c t)}\,,
\end{align}\label{eq:left-moving_field_expansion}
\end{subequations}
for the regions (1) and (3). Similarly, for right propagating waves, in which $E_{z,-k}(\omega) = H_{y,-k}(\omega) = 0$ and $E_{z,k}(-\omega) = H_{y,k}(-\omega) = 0$, the field expansion \eqref{eq:field_expansion} can be simplified to
\begin{subequations}
\begin{align}
E_{z,\textrm{right}}(x,t) &= \sum_{k ,\{\omega,-\omega\}} E_{z,k}(\omega) e^{i k (x + c t)}\,,\\
H_{y,\textrm{right}}(x,t) &= \sum_{k ,\{\omega,-\omega\}} H_{y,k}(\omega) e^{i k (x + c t)}\,,
\end{align}\label{eq:right-moving_field_expansion}
\end{subequations}
for the regions (1) and (3). The simplified expansions imply a dispersionless translation of the signal
\begin{subequations}
\begin{align}
E_{z,\textrm{left}}(x,t) &= E_{z,\textrm{left}}(x + c \Delta t,t + \Delta t) \,,\\
E_{z,\textrm{right}}(x,t) &= E_{z,\textrm{right}}(x - c \Delta t,t + \Delta t) \,,\\
H_{z,\textrm{left}}(x,t) &= H_{z,\textrm{left}}(x + c \Delta t,t + \Delta t) \,,\\
H_{z,\textrm{right}}(x,t) &= H_{z,\textrm{right}}(x - c \Delta t,t + \Delta t)
\end{align}\label{eq:signal_displacement}
\end{subequations}
in the regions (1) and (3). In other words: Once the left and right propagating input and output signals are known in the regions (1) and (3) at any position $x$, they can be trivially deduced from the above equations. It is most convenient to know the input signal at the metamaterial boundaries $x_a$ and $x_b$. Therefore, we introduce the new coordinates
\begin{subequations}
\begin{align}
x' &:= x - x_a \quad \textrm{for region (1) and}\\
x^{\prime\prime} &:= x - x_b \quad\, \textrm{for region (3)}\,.
\end{align}
\end{subequations}
The conditions \eqref{eq:E-and-H_continuity_omega} change into
\begin{subequations}
\begin{align}
E_z^{(1)}(0,\omega) &= E_z^{(2)}(x_a,\omega)\,,\\
E_z^{(2)}(x_b,\omega) &= E_z^{(3)}(0,\omega)\,,\\
H_y^{(1)}(0,\omega) &= H_y^{(2)}(x_a,\omega)\,,\\
H_y^{(2)}(x_b,\omega) &= H_y^{(3)}(0,\omega)
\end{align}
\end{subequations}
in terms of these new coordinates. If one inserts the field expansion \eqref{eq:field_expansion} in these continuity conditions and keeps in mind, that the exponentials $e^{-i \omega t}$ are linearly independent on the time interval $]-\infty,\infty[$ for each frequency $\omega$, one obtains the boundary conditions in frequency space
\begin{subequations}
\begin{equation}
E_{z,k}^{(1)}(\omega) + E_{z,-k}^{(1)}(\omega) = E_{z,k}^{(2)}(\omega) e^{i k x_a} + E_{z,-k}^{(2)}(\omega) e^{- i k x_a}\label{eq:explicit_boundary_conditions_left_E}
\end{equation}
\begin{equation}
E_{z,k}^{(2)}(\omega) e^{i k x_b} + E_{z,-k}^{(2)}(\omega) e^{- i k x_b} = E_{z,k}^{(3)}(\omega) + E_{z,-k}^{(3)}(\omega) \label{eq:explicit_boundary_conditions_right_E}\\
\end{equation}
\begin{equation}
H_{y,k}^{(1)}(\omega) + H_{y,-k}^{(1)}(\omega) = H_{y,k}^{(2)}(\omega) e^{i k x_a} + H_{y,-k}^{(2)}(\omega) e^{- i k x_a}\label{eq:explicit_boundary_conditions_H_left}
\end{equation}
\begin{equation}
H_{y,k}^{(2)}(\omega) e^{i k x_b} + H_{y,-k}^{(2)}(\omega) e^{- i k x_b} = H_{y,k}^{(3)}(\omega) + H_{y,-k}^{(3)}(\omega) \,.\label{eq:explicit_boundary_conditions_H_right}
\end{equation}\label{eq:explicit_boundary_conditions}
\end{subequations}
The relations \eqref{eq:E-H-relation} allow for the substitution of the magnetic field expansion coefficients in \eqref{eq:explicit_boundary_conditions_H_left} and \eqref{eq:explicit_boundary_conditions_H_right} with the electric field expansion coefficients, resulting in
\begin{subequations}
\begin{equation}
-E_{z,k}^{(1)}(\omega) \frac{1}{c \mu_0} + E_{z,-k}^{(1)}(\omega) \frac{1}{c \mu_0} = E_{z,k}^{(2)}(\omega) F^{(2)}_k(\omega) e^{i k x_a} + E_{z,-k}^{(2)}(\omega) F^{(2)}_{-k}(\omega) e^{- i k x_a}\,,
\end{equation}
\begin{equation}
E_{z,k}^{(2)}(\omega) F^{(2)}_k(\omega) e^{i k x_b} + E_{z,-k}^{(2)}(\omega) F^{(2)}_{-k}(\omega) e^{- i k x_b} = -E_{z,k}^{(3)}(\omega) \frac{1}{c \mu_0} + E_{z,-k}^{(3)}(\omega) \frac{1}{c \mu_0}\,,
\end{equation}\label{eq:explicit_boundary_conditions_II}
\end{subequations}
where
\begin{equation}
F^{(2)}_k(\omega) = - \frac{k}{\omega \mu_0 \mu_r^{(2)}(\omega)} = - \frac{\omega \epsilon_0 \epsilon_r^{(2)}(\omega)}{k}
\end{equation}
is an abbreviation for convenience.
\section{Numerical implementation\label{sec:numercial_implementation}}
The numerical implementation makes use of an equidistant grid of states in momentum space, which by Fourier transform also implies an equidistant grid in position space. Assume, there are $n$ grid points, each of them spaced by the extension $\Delta x=8\,\textrm{mm}$ of the metamaterial's unit element size (see table \ref{tab:metamaterial_properties}). This implies a length $l$ of the metamaterial, which is $l=n \,\Delta x$. Since the plane waves $e^{i k x}$ should be $2\pi$ periodic after this extension $l$, the spacing in momentum space must be $\Delta k = 2 \pi/l$. Thus, in momentum space, the momenta reach from $-\Delta k (n-1)/2$ till $\Delta k (n-1)/2$, where $n$ must be an odd number. The advantage of an equidistant momentum spacing is, that the plane wave spinors $u_k^{\pm} e^{i k x}$ in the expansion \eqref{eq:wavefunction_expansion} are complete and orthonormal basis functions on the spacial interval $[-l/2 , l/2]$. Furthermore, the simulation has periodic boundary conditions at the interval limits, because all exponentials are periodically continuing. Therefore, the simulations in the subsections \ref{sec:gaussian_wavepacket} and \ref{sec:counterpropagating_wavepacket} are valid, as long as the wavefunction's probability density stays within the boundaries of the simulation area.
In subsection \ref{sec:boundary_wavepacket} the initial condition of the simulation is not specified at an initial time, but imposed by additional boundary conditions instead. These boundary conditions, which are discussed in section \ref{sec:boundary_conditions}, are modeling the injection of input pulses at the physical boundaries of the metamaterials. However, the positions $x_a$ and $x_b$, at which the injection boundary conditions are imposed are within the simulation area (see illustrative figure \ref{fig:periodic_boundary_conditions}), such that they don't interfere with each other.
\begin{figure}
\caption{
Implementation of boundary conditions within the periodic simulation area in subsection \ref{sec:boundary_wavepacket}
\label{fig:periodic_boundary_conditions}
\end{figure}
The time-evolution is computed according to equation \eqref{eq:wavefunction_expansion}, which is implemented numerically by first multiplying each expansion coefficient $\phi_k$ with it's time-evolving phase factor $e^{- i \omega t}$ at the certain time $t$ and then summing over $k$. Since the sum over $k$ runs over the exponentials $e^{i k x}$ and the discrete set of momenta $k$ is equidistantly spaced, the sum is executed by performing a Fourier transformation of the $\phi_k \, e^{- i \omega(k) t}$ array.
\section{Results from simulation\label{sec:results}}
In this results section, we apply the theory considerations from above, for demonstrating a Zitterbewegung of the emulated Dirac dynamics in the metamaterial. In a series of three different simulations, we first consider a Gaussian wavepacket excitation as simplest possible setup in subsection \ref{sec:gaussian_wavepacket}. For easier experimental implementation we also consider a moving Gaussian wavepacket in subsection \ref{sec:counterpropagating_wavepacket}. In the last subsection \ref{sec:boundary_wavepacket}, we implement boundary conditions and account for the influence of these boundaries in the simulation.
\subsection{Gaussian wavepacket excitation\label{sec:gaussian_wavepacket}}
\subsubsection{Initial condition}
It is known\cite{Grobe_1999_Numerical_Dirac_equation_Zitterbewegung}, that the Zitterbewegung only shows up for a simultaneous excitation of the positive eigen energy spectrum (above the mass gap of $m_0 c^2$) and the negative eigen energy spectrum (below the mass gap of $m_0 c^2$). Therefore, the simplest initial condition for a wavepacket, which exhibits Zitterbewegung dynamics is two Gaussian wavepackets in momentum space: One Gaussian wavepacket for the positive energy eigenstates and one Gaussian wavepacket for the negative energy eigenstates. Both Gaussians should be centered at momentum $k=0\,\textrm{m}^{-1}$. This description corresponds to the initial condition of the wavepacket
\begin{subequations}
\begin{align}
\phi_k^+ &= e^{- \left(\frac{k}{\sigma_k}\right)^2}\,,\\
\phi_k^- &= e^{- \left(\frac{k}{\sigma_k}\right)^2}\,,
\end{align}\label{eq:gaussian_wavepacket}
\end{subequations}
at time $t=0\,\textrm{ns}$, where the positive- and negative energy eigenstates are excited equally. The width of the wavepacket should be of the order of the Compton wavelength $\hbar/(m_0 c)$, which is the typical scale of the Zitterbewegung. By applying the metamaterial scaling \eqref{eq:scaling_replacements}, this turns into $\hbar/(m' c_D)$. Thus we choose the width in frequency space to be $\sigma_k=m' c_D \sqrt{2}/\hbar$. Correspondingly, the wavefunction \eqref{eq:wavefunction_expansion_vacuum} takes the form
\begin{equation}
\varphi(x) = \frac{1}{\sqrt{N}} \int_{-\infty}^{\infty} dk \left(u_k^+ e^{i k x} + u_k^- e^{i k x}\right) e^{- \left(\frac{k}{\sigma_k}\right)^2}\,, \label{eq:initial_wavepacket_nonmoving}
\end{equation}
with norm
\begin{equation}
N = \sqrt{2 \pi} \sigma_k\,. \label{eq:wavepacket_norm}
\end{equation}
\subsubsection{Simulation}
The simulation is carried with a resolution of 401 grid points in momentum space. Therefore the simulation area has an extension of $3.2\,\textrm{m}$ from the $1^\textrm{st}$ till the $401^\textrm{th}$ index, according to the numerical considerations section.
We have plotted the probability density of the metamaterial simulation with initial condition \eqref{eq:gaussian_wavepacket} in figure \ref{fig:simple_zitterbewegung_probability_density}, in which we also compute the position expectation value
\begin{equation}
\Braket{\varphi|x(t)|\varphi} = \int_{-\infty}^\infty dx \, x \, |\varphi(x,t)|^2 \label{eq:position_expectation_simple}
\end{equation}
of the simulated probability density $|\varphi(x,t)|^2$ (red line). Equation \eqref{eq:position_expectation_simple} thereby shows the general formula for the position expectation, in which the limits of the infinite integration interval $[-\infty,\infty]$ are constrained down to the interval $[-1.6\,\textrm{m},1.6\,\textrm{m}]$ in the case of the numerical simulation. For further investigation, we plot the red line of figure \ref{fig:simple_zitterbewegung_probability_density} again in figure \ref{fig:simple_zitterbewegung_probability_amplitude} as solid black line. We compare this function with the position expectation value which is computed by using the simulated probability density of the exact Dirac equation (dotted line). The term `exact Dirac equation' means, that the spinors $u_k^+$ and $u_k^-$ are replaced by the spinors $\tilde u_k^+$ and $\tilde u_k^-$ of equation \eqref{eq:spinors_vacuum} and the frequencies $\omega_+$ and $\omega_-$ are replaced by $\pm \tilde{\mathcal{E}}(k)/\hbar$ of equation \eqref{eq:relativistic_energy-momentum-relation} in the time evolution equation \eqref{eq:wavefunction_expansion} of the Dirac equation. Another graph (plus-marked line) in figure \ref{fig:simple_zitterbewegung_probability_amplitude} is the position expectation value, which is derived analytically in appendix \ref{sec:zitterbewegung_of_real_electron}. The final equation \eqref{eq:zitterbewegung_expectation} can be adapted to our problem by inserting the initial condition \eqref{eq:gaussian_wavepacket} and applying the scaling rule \eqref{eq:scaling_replacements}, which yields the semi-analytic equation
\begin{equation}
\Braket{\varphi|x(t)|\varphi} = \frac{1}{N} \int_{-\infty}^{\infty} dk \frac{m' \hbar c_D^3 }{{\mathcal{E}(k)}^2} e^{-2 \left(\frac{k}{\sigma_k}\right)^2} \sin\left(- \frac{2 \,\mathcal{E}(k) t}{\hbar}\right)\,.\label{eq:gaussian_zitterbewegung_expectation}
\end{equation}
Note, that we have inserted a minus sign in the argument of the sine function by hand, according to the negative time-evolution of the metamaterial simulation, which we have discussed at the end of subsection \ref{sec:scaled_dirac_equation}.
\subsubsection{Discussion}
The line `\emph{scaled Dirac}' and the line `\emph{analytic}' in figure \ref{fig:simple_zitterbewegung_probability_amplitude} are both based on exact Dirac theory. For this reason, they are on top of each other and therewith identical.
Furthermore, the comparison of the metamaterial simulation of the effective Dirac equation with the exact Dirac simulation in figure \ref{fig:simple_zitterbewegung_probability_amplitude} yields good agreement.
The match of the `\emph{simulation}' line with the `\emph{scaled Dirac}' line and the `\emph{analytic}' line is one of our main results. It implies, that our effective dynamic Dirac theory appears in metamaterial simulations such that the well-known Zitterbewegung can emerge. We conclude that metamaterials are capable of emulating the time-dependent Dirac equation in the case of well-suited parameters as for this setup.
\begin{figure}
\caption{
Time evolution of a Gaussian wavepacket. The figure shows the probability density of the metamaterial simulation of the effective Dirac theory according to the dynamical evolution equation \eqref{eq:wavefunction_expansion}
\label{fig:simple_zitterbewegung_probability_density}
\end{figure}
\begin{figure}
\caption{
Zitterbewegung of a metamaterial simulation. We show the position expectation value \eqref{eq:position_expectation_simple}
\label{fig:simple_zitterbewegung_probability_amplitude}
\end{figure}
\subsection{Counterpropagating excitation\label{sec:counterpropagating_wavepacket}}
\subsubsection{Initial condition}
The Gaussian wavepacket excitation in the above subsection is entering and exiting the simulation region very slowly, which means that the required electro-magnetic input pulse at the metamaterial interfaces will have an infinitely long head and tail. This is no useful property for an experimental realization and therefore, we demand a wavepacket which performs a Zitterbewegung but evolves more suitable in a prospective experiment. This property can be achieved by shifting the Gaussian wave packet \eqref{eq:gaussian_wavepacket} to the right in momentum space by the value $k_0 = 20\,\textrm{m}^{-1}$, implying the new initial condition
\begin{subequations}
\begin{align}
\phi_k^+ &= e^{- \left(\frac{k - k_0}{\sigma_k}\right)^2}\,, \\
\phi_k^- &= e^{- \left(\frac{k - k_0}{\sigma_k}\right)^2}\,,
\end{align}\label{eq:initial_condition_colliding}
\end{subequations}
at time $t=0\,\textrm{ns}$. Here, the width of the wavepacket in frequency space is chosen to be $\sigma_k=m' c_D/\hbar$. Consequently, the wavefunction \eqref{eq:initial_wavepacket_nonmoving} changes into
\begin{equation}
\varphi(x) = \frac{1}{\sqrt{N}} \sum_k \left(u_k^+ e^{i k x} + u_k^- e^{i k x} \right) e^{- \left(\frac{k - k_0}{\sigma_k}\right)^2}\,, \label{eq:initial_wavepacket_colliding}
\end{equation}
where the norm \eqref{eq:wavepacket_norm} remains unchanged, except of course, the change of $\sigma_k$.
\subsubsection{Simulation}
For the new setup, we increase the number of spacial grid points to 625.
As a consequence, the simulation region is now extended over the interval $[-2.496\,\textrm{m},2.496\,\textrm{m}]$.
Like in the above subsection \ref{sec:gaussian_wavepacket} we have plotted the time-evolution of the probability density of the metamaterial simulation for the initial condition \eqref{eq:initial_condition_colliding} in figure \ref{fig:collision_zitterbewegung_probability_density}. One can see, that the positive and negative energy eigenstates are counterpropagating from $x=-\infty$ and $x=\infty$ at time $t=-\infty$, collide at $x=0$ at time $t=0$ and move apart from each other to $x=-\infty$ and $x=\infty$ at time $t=+\infty$. An oscillatory pattern is appearing at the colliding point (see also figure \ref{fig:collision_zitterbewegung_probability_amplitude}), which we attribute as Zitterbewegung dynamics. The reader might expect, that positive and negative states move in the same direction, because we have shifted the excitation by momentum $k_0$ to the right in momentum space. But since the upper and lower band of the dispersion relation have the opposite slope at $k_0$ the shift in momentum space results in two counterpropagating wave packets.
The metamaterial simulation of the effective Dirac equation and the simulation of the exact Dirac equation differ significantly from each other. Therefore, we plot them in two different plots (a) and (b) in figure \ref{fig:collision_zitterbewegung_probability_density}, respectively.
In the case of the effective Dirac theory of the metamaterial, there is an asymmetry which we explain with the asymmetry of the dispersion relation (see figure \ref{fig:dispersion_relation} (a)). The positive energy eigenstates are moving with group velocity $(\partial \omega_- / \partial k) |_{k_0}$ at $k_0$ of the lower band, while the negative energy eigenstates are moving with group velocity $\partial (\omega_+ / \partial k) |_{k_0}$ at $k_0$ of the upper band. Since the upper and lower band are differently shaped, the positive and negative energy-eigenstates are propagating at different speed through the medium. On the other hand, in the case of the exact Dirac equation in figure \ref{fig:collision_zitterbewegung_probability_density} (b), the dynamics appears symmetric, which we explain with the symmetric upper and lower band of the relativistic energy-momentum relation in figure \ref{fig:dispersion_relation} (b). As a result of the differences of the dispersion relations, there is an additional effective motion superimposed to the Zitterbewegung (see red line in figure \ref{fig:collision_zitterbewegung_probability_density} (a) and solid black line in figure \ref{fig:collision_zitterbewegung_probability_amplitude} (a)), as compared to the wavepacket in the exact Dirac theory (see red line in figure \ref{fig:collision_zitterbewegung_probability_density} (b) and dotted black line in figure \ref{fig:collision_zitterbewegung_probability_amplitude} (b)).
Similarly as in the above subsection \ref{sec:gaussian_wavepacket}, we want to verify the exact Dirac theory by comparison with the semi-analytic solution in appendix \ref{sec:zitterbewegung_of_real_electron}. The new initial condition \eqref{eq:initial_condition_colliding}, substituted in equation \eqref{eq:zitterbewegung_expectation} yields
\begin{equation}
\Braket{\varphi|x(t)|\varphi} = \frac{1}{N} \int_{-\infty}^{\infty} dk \frac{m' \hbar c_D^3}{{\mathcal{E}}(k)^2} e^{- 2 \left(\frac{k - k_0}{\sigma_k}\right)^2} \sin\left(- \frac{2 \,\mathcal{E}(k) t}{\hbar}\right)\,.\label{eq:colliding_analytic_Zitterbewegung}
\end{equation}
Note, that we inserted in a minus sign by hand in the argument of the sine function, as it has been done for equation \eqref{eq:gaussian_zitterbewegung_expectation} already. We plot the value of the solution \eqref{eq:colliding_analytic_Zitterbewegung} as plus-marked line in figure \ref{fig:collision_zitterbewegung_probability_amplitude} (b).
\subsubsection{Discussion}
The lines `\emph{exact Dirac}' of the exact Dirac simulation and the `\emph{analytic}' solution in figure \ref{fig:collision_zitterbewegung_probability_amplitude} (b) are coinciding. This is to expect, because the analytic solution is based on exact Dirac theory and is a validation of both descriptions.
For a comparison of the metamaterial simulation with the exact Dirac theory, we subtract the value $t/(49.7\,\textrm{ns})$ from the position expectation value in figure \ref{fig:collision_zitterbewegung_probability_amplitude} (a), and plot it as the solid black line `\emph{modified simulation}' in figure \ref{fig:collision_zitterbewegung_probability_amplitude} (b). We obtain very good agreement of both descriptions and take this match as further confirmation for the reliability of the simulation of the effective Dirac dynamics in the metamaterial.
\begin{figure*}
\caption{
Time evolution of a colliding wavepacket. (a) The metamaterial time-evolution according to equation \eqref{eq:wavefunction_expansion}
\label{fig:collision_zitterbewegung_probability_density}
\end{figure*}
\begin{figure}
\caption{
Zitterbewegung of colliding wavepackets. The solid black line in (a) and the dotted black line in (b), are the red lines in figure \ref{fig:collision_zitterbewegung_probability_density}
\label{fig:collision_zitterbewegung_probability_amplitude}
\end{figure}
\subsection{Excitation input at the boundaries\label{sec:boundary_wavepacket}}
\subsubsection{Initial condition}
In the case of a metamaterial with physical boundaries, the initial wavefunction \eqref{eq:initial_wavepacket_colliding} has to be replaced by an electric and magnetic field, which is propagating from the regions (1) and (3) into region (2) of the metamaterial. This can be done by specifying the incoming electric field of the regions (1) and (3) at the boundaries by
\begin{subequations}
\begin{align}
E_z^{(1)}(x_a,t) &= \hat E \, e^{-i \omega_a t} e^{-\left(\frac{\sigma_\omega t}{2}\right)^2} \,\label{eq:initial_wavepacket_boundaries_left} \,,\\
E_z^{(3)}(x_b,t) &= \hat E \, e^{-i \omega_b t} e^{-\left(\frac{\sigma_\omega t}{2}\right)^2} \,.\label{eq:initial_wavepacket_boundaries_right}
\end{align}\label{eq:initial_wavepacket_boundaries}
\end{subequations}
with an arbitrary electric field amplitude, which is chosen to be $\hat E = 1\,\textrm{V/m}$, the frequency width $\sigma_\omega = 0.52\,\textrm{GHz}$ and the two frequencies $\omega_a = 13.81\,\textrm{GHz}$ and $\omega_b = 8.95\,\textrm{GHz}$. Here, the frequency $\omega_a$ is above the band gap and $\omega_b$ is below the band gap, corresponding to the positive and negative excitation \eqref{eq:initial_condition_colliding}.
The expansion coefficients of the right propagating input pulse \eqref{eq:initial_wavepacket_boundaries_left} in region (1) are determined by the inverse Fourier transform
\begin{equation}
E_{z,k}^{\prime(1)}(\omega) = \frac{1}{2 \pi} \int_{- \infty}^\infty dt \, E_{z}^{(1)}(x_a,t) e^{i(k x_a + \omega t)}\label{eq:time_fourier_transform}\,,
\end{equation}
in time, with corresponding Fourier transform
\begin{equation}
E_{z}^{(1)}(x_a,t) = \int_{- \infty}^\infty d\omega \, E_{z,k}^{\prime(1)}(\omega) e^{i(k x_a - \omega t)}\label{eq:frequency_fourier_transform}
\end{equation}
in frequency space. However, the equidistant grid of the simulated time evolution \eqref{eq:wavefunction_expansion} in momentum space implies, that the sum of the Fourier transform has to be expressed in terms of an integral over the variable $k$ instead of $\omega$. Such a momentum space integral of \eqref{eq:frequency_fourier_transform} in the fashion of \eqref{eq:field_expansion} would read as
\begin{equation}
E_z^{(1)}(x,t)=\int_0^\infty dk \Bigg[ E_{z,k}^{(1)}(\omega) e^{i(- k x - \omega t)} + E_{z,k}^{(1)}(-\omega) e^{i(- k x + \omega t)} \Bigg]\,, \label{eq:momentum_fourier_transform}
\end{equation}
where the sum with momentum space measure $\Delta k$ is replaced by an integral over $dk$. We consider a right propagating input pulse, therefore only right propagating modes with $0 \le k$ are accounted for. Since we want to adapt the same density of states in region (1) as in region (2) in the numerical implementation, we relate frequency space to momentum space by the dispersion relation \eqref{eq:dispersion-relation} in the Fourier transform. Accordingly, the function $\omega(k)$ implies an inner derivative in the integral
\begin{equation}
E_z^{(1)}(x,t)=\int_0^\infty d k \left| \frac{\partial \omega}{\partial k} \right| \Bigg[ E_{z,k}^{\prime(1)}(\omega) e^{i(- k x - \omega t)}
+ E_{z,k}^{\prime(1)}(-\omega) e^{i(- k x + \omega t)} \Bigg]\,, \label{eq:momentum_fourier_transform_iner_derivative}
\end{equation}
and by comparison with \eqref{eq:momentum_fourier_transform}, one obtains the relation
\begin{equation}
E_{z,k}^{(1)}(\omega) = \left| \frac{\partial \omega}{\partial k} \right| E_{z,k}^{\prime(1)}(\omega) \ \Leftrightarrow \ E_{z,k}^{\prime(1)}(\omega) = \left| \frac{\partial k}{\partial \omega} \right| E_{z,k}^{(1)}(\omega)\,.
\end{equation}
By these relations, the expansion coefficients of the Fourier transform in time and frequency space, which is required for obtaining expansion coefficients of the input pulse \eqref{eq:initial_wavepacket_boundaries_left} can be related to the Fourier transform between momentum and position space \eqref{eq:field_expansion}, which is required for the metamaterial simulation and accordingly the Dirac time-evolution \eqref{eq:wavefunction_expansion}.
The same considerations also hold in an analogous way for the left propagating input pulse \eqref{eq:initial_wavepacket_boundaries_right} with expansion coefficients $E_{z,-k}^{(3)}(\omega)$ in region (3), as well as for the right propagating reflected pulse with expansion coefficients $E_{z,k}^{(3)}(\omega)$ in region (3) and the left propagating reflected pulse $E_{z,-k}^{(1)}(\omega)$ in region (1). Note, that in the expressions \eqref{eq:momentum_fourier_transform} and \eqref{eq:momentum_fourier_transform_iner_derivative} the integral over $k$ is replaced by a sum over an equidistant momentum grid with momentum space spacing length $\Delta k$ in the numerical implementation.
Within the framework of the herein considered integral measures, the two reflected `output' pulses and the electric field inside of the metamaterial are determined for each momentum, ie. each frequency according to the system of equations of the boundary conditions \eqref{eq:explicit_boundary_conditions_left_E}, \eqref{eq:explicit_boundary_conditions_right_E} and \eqref{eq:explicit_boundary_conditions_II} of section \ref{sec:boundary_conditions}. The fully determined electric, and therewith magnetic field in region (2) in form of the expansion coefficients $E_{z,k}^{(2)}$ and $H_{y,k}^{(2)}$ can be used to compute the wavefunction's expansion coefficients $\phi_k$ by making use of the relations \eqref{eq:expansion_coefficient_equality}. Once the wavefunction is determined by this procedure, we normalize it by
\begin{equation}
N=\int_{-\infty}^\infty \left[ |\phi_k^+(\omega_-)|^2 + |\phi_k^-(\omega_+)|^2 \right] \Delta k\,,\label{eq:momentum_space_normalization}
\end{equation}
which according to Parseval's theorem, is equivalent to normalization in position space \eqref{eq:position_space_normalization}. In contrast to the initial condition at time $t=0$ along the whole $x$-axis in the above two subsections, this most realistic simulation specifies the `initial condition' at the boundary positions $x_a=-1.0\,\textrm{m}$ for \eqref{eq:initial_wavepacket_boundaries_left} and $x_b=1.0\,\textrm{m}$ for \eqref{eq:initial_wavepacket_boundaries_right} along the time-axis. However, through the relations (\ref{eq:left-moving_field_expansion},\ref{eq:right-moving_field_expansion},\ref{eq:signal_displacement}), the alignment of the `initial condition' can be recast from the time-axis to the $x$-axis regions (1) and (3) at a certain time $t \ll 0\,\textrm{ns}$.
\subsubsection{Simulation}
For the setup of this simulation we use $1001$ grid points in position and momentum space, which implies that the simulation is extended on the interval $[-4\,\textrm{m},4 \textrm{m}]$. For this setup, the situation of figure \ref{fig:periodic_boundary_conditions} applies, in which the boundary conditions of the metamaterial are implemented inside of the simulation interval, in which the physical metamaterial (corresponding to region (2)) is extended from $x_a$ to $x_b$ on the interval $[-1\,\textrm{m},1\,\textrm{m}]$. The intervals $[-4\,\textrm{m},-1\,\textrm{m}]$ and $[1\,\textrm{m},4\,\textrm{m}]$ correspond to unphysical regions, which are depicted as dump space in figure \ref{fig:periodic_boundary_conditions}. Therefore, even though the time-evolution has periodic boundary conditions, the wavepacket does not immediately reenter the simulation region (2) from the other side, after it went out before.
\subsubsection{Discussion}
The simulation of a Zitterbewegung with real metamaterial boundaries is shown in figure \ref{fig:boundary_zitterbewegung_probability_density} and the position expectation value of this simulation is overlayed as red line and plotted a second time in figure \ref{fig:boundary_zitterbewegung_probability_amplitude}. A similar simulation for the exact Dirac equation with the same initial condition differs much from the effective Dirac dynamics of the metamaterial (see also figure \ref{fig:collision_zitterbewegung_probability_density}) and avoids the feasibility of a direct comparison. Therefore, there is no comparison with the corresponding dynamics of the exact Dirac equation in this subsection.
In figure \ref{fig:boundary_zitterbewegung_probability_density} one can see, that the electro-magnetic excitation in the setup is propagating from the physical boundaries towards the central region of the simulation area. In the overlap region of both excitations an oscillatory pattern emerges (see figure \ref{fig:boundary_zitterbewegung_probability_amplitude}) which we attribute as Zitterbewegung of the effective Dirac equation in the metamaterial. This demonstrates, that in the case of an ideal metamaterial, given by the equations \eqref{eq:maxwell_equations} and \eqref{eq:permittivity_and_permeability} a Zitterbewegung of a simulated Dirac equation may occur for a real experiment.
\begin{figure}
\caption{
Time-evolution with boundary conditions. This figure shows a similar time-evolution as in figure \ref{fig:collision_zitterbewegung_probability_density}
\label{fig:boundary_zitterbewegung_probability_density}
\end{figure}
\begin{figure}
\caption{
Zitterbewegung with boundary conditions. The plotted graph is the position expectation value, which is computed as red line in the boundary specified simulation in figure \ref{fig:boundary_zitterbewegung_probability_density}
\label{fig:boundary_zitterbewegung_probability_amplitude}
\end{figure}
\section{Conclusion and Outlook}
The article discusses the feasibility on an exactly mappable, time-dependent Dirac simulation, which is actually emulated by electro-dynamics of Maxwell equations with designed electro-magnetic properties of a metamaterial structure. We present a time-evolution equation in a Dirac-like fashion \eqref{eq:wavefunction_expansion} and give arguments of why it evolves very similar to exact Dirac theory. Furthermore, we consider scaling relations in subsection \ref{sec:scaled_dirac_equation}, with which the slower dynamics of the metamaterial can be mapped to the faster dynamics of elementary particles in Dirac theory. Based on the scaling rules and the exact Dirac equation, we have derived an explicit, semi-analytic expression for the position expectation of a general wavefunction in appendix \ref{sec:zitterbewegung_of_real_electron}.
We demonstrate the dynamic description with three simulations: In subsection \ref{sec:gaussian_wavepacket} and \ref{sec:counterpropagating_wavepacket}, we compare our metamaterial simulations with exact Dirac theory and with the analytic solution, which is derived within this exact Dirac theory. We get very good agreement between the emulated Dirac dynamics and the exact Dirac dynamics, which we interpret as validation of our description. It is also a proof for the emergence of Zitterbewegung, which is a characteristic property of the free Dirac equation. Subsequently, we present a more realistic scenario, in which boundary conditions are included, such that the simulation can be realized in experiment in subsection \ref{sec:boundary_wavepacket}. In this most realistic simulation we also observe an oscillatory pattern, which matches the Zitterbewegung.
We conclude that a Dirac wavefunction can be emulated by the electro-magnetic field in a metamaterial, such that the Zitterbewegung occurs. Next steps are an experimental implementation of this theoretical study of the effect and also an investigation of the quantization of the system.
\begin{acknowledgments}
S. A. would like to thank Hong Chen for the support nice hospitality of the collaborative visits at the Tongji University. S. A. also gratefully thanks for the helpful discussions about Zitterbewegung and periodic boundery conditions with Rainer Grobe from the Illinois State University.
\end{acknowledgments}
\appendix
\section{Zitterbewegung of a real electron in Dirac theory\label{sec:zitterbewegung_of_real_electron}}
For verifying exact Dirac theory simulations, we consider the time-dependent position operator $x(t)$ of the free Dirac equation in the Heisenberg picture in this section. This is already discussed in the literature. We simplify the already known derivation\cite{Schroedinger_1930_Zitterbewegung,Milonni_1994_Quantum__Vacuum,Grobe_1999_Numerical_Dirac_equation_Zitterbewegung} of the position operator for the one-dimensional case. Applying a general wavefunction to this position operator, yields a semi-analytic formula for the computation of the wavepacket's position expectation value.
We start out with the velocity operator $c \sigma_x$ in the Heisenberg picture
\begin{equation}
\partial_t x = \frac{i}{\hbar} \left[ H,x \right] = c \sigma_x\,,
\end{equation}
where $H$ is the Hamiltionan of the exact Dirac theory \eqref{eq:vacuum_dirac_hamiltonian}.
A second application of the Heisenberg equation yields the acceleration operator
\begin{equation}
\partial_t c \sigma_x = \frac{i}{\hbar} \left[ H, c \sigma_x \right] = - \frac{2 c \sigma_y m_0 c^2}{\hbar} = \frac{2 i c}{\hbar} \left( c p_x - \sigma_x H \right)\,.
\end{equation}
If one accounts for the time-independent constants $H$ and $p_x$, one finds the formal solution
\begin{equation}
c \sigma_x(t) = \left( c \sigma_x(0) - c^2 p_x H^{-1} \right) e^{-i 2 H t/\hbar} + c^2 p_x H^{-1}\,.
\end{equation}
Integration of the velocity operator with respect to time gives the solution of the time-dependent position operator in the Heisenberg picture
\begin{equation}
x(t) = x_0 + c^2 p_x H^{-1} t
- \frac{i}{2} \hbar c \left( \sigma_x H^{-1} - c p_x H^{-2} \right) \left( 1 - e^{- i 2 H t/\hbar} \right)\,.\label{eq:time-dependent_position_space_operator}
\end{equation}
In the following, we deduce the expectation value of this operator for a general wavefunction
\begin{equation}
\varphi(x,t) = \frac{1}{\sqrt{N}} \int_{-\infty}^{\infty} \left(\phi_k^+ \, \tilde u_k^+ \, e^{i k x} + \phi_k^- \, \tilde u_k^- \, e^{i k x} \right) dk\,.\label{eq:wavefunction_expansion_vacuum}
\end{equation}
Note, that in contrast to \eqref{eq:wavefunction_expansion} the spinors no longer have a frequency dependent mass in the Dirac theory of an elementary particle. Accordingly, due to the equations of motion \eqref{eq:quantum_mechanical_time_evolution} and \eqref{eq:vacuum_dirac_hamiltonian} the energy-momentum relation
\begin{equation}
\tilde{\mathcal{E}}(k) = \sqrt{c^2 \hbar^2 k^2 + m_0^2 c^4}\label{eq:relativistic_energy-momentum-relation}
\end{equation}
and the spinors
\begin{subequations}
\begin{align}
\tilde u_{k}^+ &=
\frac{1}{\sqrt{2 |\tilde{\mathcal{E}}(k)|(|\tilde{\mathcal{E}}(k)| + m_0 c^2)}}
\begin{pmatrix}
|\tilde{\mathcal{E}}(k)| + m_0 c^2 \\ c \hbar k
\end{pmatrix}\,,\label{eq:positive_spinor_vacuum}\\
\tilde u_{k}^- &=
\frac{1}{\sqrt{2 |\tilde{\mathcal{E}}(k)|(|\tilde{\mathcal{E}}(k)| + m_0 c^2)}}
\begin{pmatrix}
- c \hbar k \\ |\tilde{\mathcal{E}}(k)| + m_0 c^2
\end{pmatrix}\label{eq:negative_spinor_vacuum}
\end{align}\label{eq:spinors_vacuum}
\end{subequations}
contain the natural constants $m_0$, $c$ and $\hbar$. The computation of the expectation value of the position operator is as follows. First we define the function
\begin{equation}
f(k) := \phi_k^+ \, \tilde u_k^+ + \phi_k^- \, \tilde u_k^-\,.
\end{equation}
With this abbreviation, the position expectation value of the position operator can be written as
\begin{equation}
\braket{\varphi|x(t)|\varphi} = \int_{- \infty}^\infty dx \int_{- \infty}^\infty dk \, e^{-i k x} f(k)^\dagger x(t)
\int_{- \infty}^\infty dk' e^{i k' x} f(k')\,.\label{eq:position_expectation_value_definition}
\end{equation}
The operator $x(t)$ acts at on the right hand side integral, which is the integral over $k'$. This integral is a continuous sum over the exponential functions $e^{-i k' x}$ and one can commute the operator with the right hand side integration, where it solely acts at these exponentials. Hence, the position operator \eqref{eq:time-dependent_position_space_operator} turns into a function of $k'$, which means, that all momentum operators $p_x$ of the operator $x(t)$ and $H$ in $x(t)$ turn into the number $\hbar k'$. It remains a triple integral over the function
\begin{equation}
e^{-i (k-k') x} f(k)^\dagger x(t,k') f(k')
\end{equation}
with the three integration variables $x$, $k$ and $k'$. If one performs the integration over $x$ first, the exponential $e^{-i (k-k') x}$ turns into the delta function $2 \pi \, \delta( k- k')$. Another integration over $k'$ gives contributions only at $k=k'$, such that equation \eqref{eq:position_expectation_value_definition} turns into
\begin{equation}
\braket{\varphi|x(t)|\varphi} = 2 \pi \int_{- \infty}^\infty dk \,f(k)^\dagger x(t,k) f(k')\,.
\end{equation}
The remaining integrand can be reshaped by making use of the eigenvalue relations
\begin{align}
H(k)\, \tilde u_k^+ &= \phantom{-}\mathcal{E}(k)\, \tilde u_k^+ \,,\\
H(k)\, \tilde u_k^- &= -\mathcal{E}(k)\, \tilde u_k^-\,,
\end{align}
the orthonormal property
\begin{subequations}
\begin{align}
\tilde u_k^{+\dagger} \tilde u_k^+ &= 1\,, \\
\tilde u_k^{+\dagger} \tilde u_k^- &= 0\,, \\
\tilde u_k^{-\dagger} \tilde u_k^+ &= 0\,, \\
\tilde u_k^{-\dagger} \tilde u_k^- &= 1
\end{align}
\end{subequations}
and the basic identities
\begin{subequations}
\begin{align}
\tilde u_k^{+\dagger} \sigma_1 \tilde u_k^+ &= \frac{c \hbar k}{\tilde{\mathcal{E}}(k)}\,, \\
\tilde u_k^{+\dagger} \sigma_1 \tilde u_k^- &= \frac{m_0 c^2}{\tilde{\mathcal{E}}(k)}\,, \\
\tilde u_k^{-\dagger} \sigma_1 \tilde u_k^+ &= \frac{m_0 c^2}{\tilde{\mathcal{E}}(k)}\,, \\
\tilde u_k^{-\dagger} \sigma_1 \tilde u_k^- &= - \frac{c \hbar k}{\tilde{\mathcal{E}}(k)}\,.
\end{align}
\end{subequations}
Applying these identities, canceling some terms and rearranging the remaining terms yields
\begin{widetext}
\begin{multline}
\Braket{\varphi|x(t)|\varphi}
= \frac{1}{N} \int_{-\infty}^{\infty} dk \bigg\{ x_0 + \left[ |\phi^+(k)|^2 - |\phi^-(k)|^2 \right] \frac{c^2 \hbar k t}{\tilde{\mathcal{E}}(k)}\\
+ \frac{i}{2} \frac{m_0 \hbar c^3}{\tilde{\mathcal{E}}(k)^2} \left[ \phi^+(k)^* \phi^-(k) \left( 1 - e^{i 2 \tilde{\mathcal{E}}(k) t/\hbar} \right) - \phi^+(k) \phi^-(k)^* \left( 1 - e^{- i 2 \tilde{\mathcal{E}}(k) t/\hbar} \right) \right] \bigg\}\,.\label{eq:zitterbewegung_expectation_complex}
\end{multline}
With a further usage of Euler's formula this result can be further rewritten in the form
\begin{multline}
\Braket{\varphi|x(t)|\varphi}
= \frac{1}{N} \int_{-\infty}^{\infty} dk \bigg\{ x_0 + \left[ |\phi^+(k)|^2 - |\phi^-(k)|^2 \right] \frac{c^2 \hbar k t}{\tilde{\mathcal{E}}(k)} \\
- \frac{m_0 \hbar c^3}{\tilde{\mathcal{E}}(k)^2} \left[ Im\left[\phi^+(k)^* \phi^-(k) \right]\left[ 1 - \cos\left(\frac{2 \tilde{\mathcal{E}}(k) t}{\hbar}\right) \right]
- Re\left[\phi^+(k)^* \phi^-(k) \right] \sin\left(\frac{2 \tilde{\mathcal{E}}(k) t}{\hbar}\right) \right] \bigg\}\,.\label{eq:zitterbewegung_expectation}
\end{multline}
\end{widetext}
\section{The wavefunction's imaginary part in reality\label{sec:imaginary_part}}
The whole description of the simulation of the electro-magnetic waves in the metamaterial and even the basic equations \eqref{eq:maxwell_equations}, on which our theory is built on, as well as in the description of the Dirac equation makes use of complex numbers. As a result, the electric and magnetic field has an imaginary part. However, in reality nothing is imaginary and there must be an answer of what happens with this kind of `degree of freedom', which is not explicitly showing up in the real experiment.
Consider the expansion
\begin{equation}
f(x) = \sum_k \tilde f(k) e^{i k x}\,.\label{eq:simple_expansion}
\end{equation}
The expansion will be exclusively real, if all expansion coefficients only have a symmetric real part
\begin{equation}
Re\left(\tilde f(k)\right) = Re\left(\tilde f(-k)\right)
\end{equation}
and an anti-symmetric imaginary part
\begin{equation}
Im\left(\tilde f(k)\right) = -Im\left(\tilde f(-k)\right)\,.
\end{equation}
On the other hand, the expansion will be exclusively imaginary, if all expansion coefficients only have an anti-symmetric real part
\begin{equation}
Re\left(\tilde f(k)\right) = -Re\left(\tilde f(-k)\right)
\end{equation}
and a symmetric imaginary part
\begin{equation}
Im\left(\tilde f(k)\right) = Im\left(\tilde f(-k)\right)\,.
\end{equation}
In \eqref{eq:field_expansion} the sum in the expansion runs over terms with negative frequencies and negative momenta at the same time and the discussion of exclusively real or imaginary functions in analogy to \eqref{eq:simple_expansion} gets more involved. Table \ref{tab:real_and_imaginary_wavefunction} lists the different combination of symmetries, which are possible for the expansion coefficients $E_{z,k}$ and $H_{y,k}$. Note, that the real part and the imaginary part can be chosen independently from each other. This means, that the resulting function will be only exclusively real or exclusively imaginary, if the symmetries of the real part and the symmetries of the imaginary part are both chosen either real or imaginary at the same time. According to these considerations, it is possible to choose expansion coefficients, such that the approximated function is exclusively real.
\begin{table}[!ht]
\caption{
\bf{Expansion coefficient symmetries}}
\begin{center}
\begin{tabular}{r|r|r||c}
coefficient & frequency & momentum & function \\ \hline \hline
real part & symmetric & symmetric & real \\
real part & symmetric & anti-symmetric & imaginary \\
real part & anti-symmetric & symmetric & imaginary \\
real part & anti-symmetric & anti-symmetric & real \\ \hline
imaginary part & symmetric & symmetric & imaginary \\
imaginary part & symmetric & anti-symmetric & real \\
imaginary part & anti-symmetric & symmetric & real \\
imaginary part & anti-symmetric & anti-symmetric & imaginary
\end{tabular}
\end{center}
\begin{flushleft}
This table lists the different symmetries of the real part and the imaginary part of the expansion coefficients $E_{z,k}$ and $H_{y,k}$ in the sum \eqref{eq:field_expansion} with respect to frequency $\omega$ and momentum $k$, in analogous extension to the symmetry considerations of the expansion \eqref{eq:simple_expansion}. The first column `\emph{coefficient}' tells if the real part or the imaginary part of the wave function is considered, the second `\emph{frequency}' and third `\emph{momentum}' column tells, whether the real/imaginary part of the expansion coefficients are assumed to be fully symmetric or fully anti-symmetric with respect to frequency or momentum, respectively. The last line `\emph{function}' tells if the resulting function of this expansion will be exclusively real or exclusively imaginary.
\end{flushleft}
\label{tab:real_and_imaginary_wavefunction}
\end{table}
Having said that the electric and magnetic field can be exclusively real in the expansion \eqref{eq:simple_expansion}, we additionally point out, that the Maxwell equations in momentum- and frequency space \eqref{eq:maxwell_equations_frequency_space_k} are flipping the symmetry type from symmetric to anti-symmetric and vice versa with respect to frequency and momentum simultaneously, when they are coupling the electric expansion coefficients $E_{z,k}$ to the magnetic expansion coefficients $H_{y,k}$. By comparing this flipping operation with table \ref{tab:real_and_imaginary_wavefunction}, one realizes, that functions which are real, are only mapped to functions which are real again. As a consequence, it is possible to choose expansion coefficients, such that the electric and magnetic fields are real in space and time and furthermore are compatible with the Maxwell equations \eqref{eq:maxwell_equations_frequency_space_k}. So the theory, which we have developed in the subsections above is compatible with a real electric field and a real magnetic field, even though complex numbers are showing up in the equations.
However, the Dirac equation (\ref{eq:quantum_mechanical_time_evolution},\ref{eq:vacuum_dirac_hamiltonian}) explicitly implies complex wavefunctions. The time evolution \eqref{eq:wavefunction_expansion} of the effective Dirac equation can be directly translated to the electric and magnetic field by the relations \eqref{eq:wavefunction_em-field_relations} in real space and \eqref{eq:expansion_coefficient_equality} in momentum and frequency space. Therefore, the question why the electro-magnetic field does not contain an imaginary part in reality, but in contrast the Dirac wavefunction does contain an imaginary part is still to be answered!
The difference of the description of the Maxwell equations and the Dirac equation can be found in the expansions \eqref{eq:field_expansion} and \eqref{eq:wavefunction_expansion}. While the sum in \eqref{eq:wavefunction_expansion} only runs over the positive frequencies $\omega_+$ and $\omega_-$, it also runs over the negative frequencies $-\omega_+$ and $-\omega_-$ in \eqref{eq:field_expansion}. Therefore, on one hand, the simulated Dirac equation only makes use of the upper two bands of the dispersion relation \eqref{eq:dispersion-relation} and is consistent with the two bands of the positive and negative energy eigenstates of the effective Dirac equation \eqref{eq:scaled_dirac_equation} (see also figure \ref{fig:dispersion_relation}). On the other hand, the dispersion relation \eqref{eq:dispersion-relation} has four bands, of which the two negative ones have the just the negative value of the positive ones. So the expansion of the electro-magnetic field runs additionally over these two negative energy bands. As a conclusion, the time-evolution of the Dirac equation \eqref{eq:wavefunction_expansion} omits the sum over the negative energy bands of the electro-magnetic field, without explicitly mentioning it. This omittance is introduced to be consistent with the two bands of conventional Dirac theory. However, the real- and imaginary part of the expansion coefficients $E_{z,k}(-\omega)$, $H_{y,k}(-\omega)$ of the negative energy band of the Maxwell equations can be chosen according to the symmetry scheme of table \ref{tab:real_and_imaginary_wavefunction}, such that the electric and magnetic field, which is transformed from the wavefunction by the relations \eqref{eq:expansion_coefficient_equality} becomes exclusively real. If one eliminates the imaginary part by the above discussed symmetry procedure and divides by a factor of two, one acted as if one just took the real part of the wavefunction. Therefore, the considerations of this appendix are justifying that the real part of the simulated Dirac wavefunction is proportional to the electric and magnetic field of the underlying Maxwell equations, which can be measured in experiment.
\end{document} | math | 82,324 |
\begin{document}
\title{Seemingly Unrelated Multi-State processes: a Bayesian semiparametric approach}
\author[1]{Andrea Cremaschi}
\author[2,1,3,4]{Raffele Argiento}
\author[2,1,3,4]{Maria De Iorio}
\author[1]{Cai Shirong}
\author[2,1]{Yap Seng Chong}
\author[1,2,5]{Michael J. Meaney}
\author[1]{Michelle Z. L. Kee}
\affil[1]{Singapore Institute for Clinical Sciences, A*STAR, Singapore}
\affil[2]{Yong Loo Lin School of Medicine, National University of Singapore, Singapore}
\affil[3]{Division of Science, Yale-NUS College, Singapore}
\affil[4]{Department of Statistical Science, University College London, UK}
\affil[5]{Department of Psychiatry, McGill University, Montreal, Canada}
\date{}
\maketitle
\textbf{Abstract}: Many applications in medical statistics as well as in other fields can be described by transitions between multiple states (e.g. from health to disease) experienced by individuals over time. In this context, multi-state models are a popular statistical technique, in particular when the exact transition times are not observed. The key quantities of interest are the transition rates, capturing the instantaneous risk of moving from one state to another. The main contribution of this work is to propose a joint semiparametric model for several possibly related multi-state processes (Seemingly Unrelated Multi-State, SUMS, processes), assuming a Markov structure for the transitions over time. The dependence between different processes is captured by specifying a joint random effect distribution on the transition rates of each process. We assume a flexible random effect distribution, which allows for clustering of the individuals, overdispersion and outliers. Moreover, we employ a graph structure to describe the dependence among processes, exploiting tools from the Gaussian Graphical model literature. It is also possible to include covariate effects. We use our approach to model disease progression in mental health. Posterior inference is performed through a specially devised MCMC algorithm.
\textbf{Keywords}: Multi-State Models, Normalized Point Processes, Graphical Models, Mixture Models, Markov Chain Monte Carlo
\section{Introduction}\label{sec:Intro}
Biomedical data are characterized by a high number of different variables, in many cases mostly categorical and recorded on a (nowadays often large) set of subjects. This is mainly due to the practice in clinical settings to record the absence/presence of symptoms and/or to use ordinal scales to represent disease markers. Typically, we only observe clinical variables at fixed time points (usually corresponding to follow up or hospital visits), and as such these variables are interval-censored (i.e., panel data).
The objective of clinical studies is often to model a patient's \textit{disease progression}, as captured by the evolution over time of one or more responses of interest, e.g. representing the disease status, and associated clinical markers. A popular approach to disease progression is to use multi-state models describing the transition of individuals among multiple states in continuous time \citep[see, for instance,][]{Cook_1999, jackson_2011, van_2015, DeIorio_etal_2018}. In this framework, it is straightforward to include time-homogeneous covariates and time varying continuous covariates (leading to a Markov regression model).
In this work, we propose a joint modelling approach for several categorical variables evolving simultaneously through time. More in details, our approach is based on a combination of ideas from multi-state models, seemingly unrelated regression \citep{zellner1963estimators, wang2010sparse}, Gaussian Graphical models and Product Partition Models with Covariates (PPMx) \citep{muller2011product}. In a Bayesian framework, we define a joint model for several multi-state processes, which represent the evolution of, for instance, clinical markers of interest as in the disease progression application of Section \ref{sec:GUSTO_application}. The main idea is that the different markers provide complementary information on the underlying health status and, as such, they are regarded as stochastic processes defined on a finite state-space, evolving in continuous time according to \textit{dependent} Markov processes. We link the different Markov processes through the specification of a flexible prior distribution on the instantaneous transition rates, specifically a mixture distribution with random number of components \citep{argiento2019infinity}. In this way, we build a robust modelling strategy, which leads to covariate-driven clustering of the subjects and enables the inclusion of different types of covariates/responses in a natural and efficient way \citep{barcella2017comparative}. Each multi-state process is then, conditionally on the vector of instantaneous transition rates, independent from the other processes, resembling the seemingly unrelated regressions (SUR) setting of \cite{zellner1963estimators}. Furthermore, we allow the dependence structure between the transition rates to be encoded into a random graph, which is also object of posterior inference, as it is done in sparse SUR models \citep[SSUR,][]{wang2010sparse}. Thus, the nature of the dependence is learnt from the data, spanning from independence to full inter-dependence. As such, we refer to our model as \textit{Seemingly Unrelated Multi-State} (SUMS) processes. Briefly, the proposed model allows for: (i) multiple responses; (ii) processes with more than two states; (iii) patient- and process-specific times of observation; (iv) inclusion of mixed-type covariates; (v) covariate-driven clustering of the subjects; (vi) missing initial state information.
One of the main advantages of our modelling strategy is that the relationship between different multi-state processes is encoded into a
graph structure. Indeed, if there is an edge linking two processes, it means that they are conditionally dependent, while the lack of an edge implies conditional independence. This gives insight into the co-regulatory mechanisms of the different processes. This is relevant in many application as often it is of interest also to identify important factors affecting disease progression, for better prognosis and therapeutic choices. Moreover, the model allows for the inclusion of time-homogeneous covariates (of any type) and time-varying continuous covariates in a regression component, for which standard variable selection techniques (e.g. shrinkage, spike and slab priors) can be employed.
The manuscript is organised as follows: Section \ref{sec:SUMS} introduces the SUMS model, by presenting how its key components - the joint multi-state model, the mixture prior with unknown number of components and the graphical structure describing the dependence among processes - interrelate, as well as the specifically designed MCMC algorithm. Section \ref{sec:GUSTO_application} presents an application of the model to the analysis of mental health indicators obtained from the GUSTO cohort study. Section \ref{sec:Conclusions} concludes the work. In Supplemetary Material we include a detailed description of the algorithm and of the GUSTO dataset, a sensitivity analysis and a simulation study, as well as further results from the analysis of the GUSTO data.
\section{SUMS: Seemingly Unrelated Multi-State Processes}\label{sec:SUMS}
\subsection{Modelling of multi-state processes}\label{sec:modelling-of-multi-state-processes}
Multi-state models can be used to describe how an individual moves between a set of states in continuous time. In this work, we focus on multi-state processes for panel data, where the states of several processes are observed only at certain time points, and their exact transition times are not known. For each $h = 1, \dots, p$, let $\{ Y^{(h)}(t), t \in \mathbb{R}^+ \}$ be a continuous time Markov process, where $Y^{(h)}(t)$ represents the state of the $h-$th process over time, with state-space $\mathcal{S}^{(h)} = \{1, \dots, d^{(h)}\}$ of dimension $d^{(h)}$, i.e. $Y^{(h)}(t) \in \mathcal{S}^{(h)}$; the elements of $\mathcal{S}^{(h)}$ represent the states that the $h$-th process can visit between transitions. The exact times of transition of the processes $Y^{(h)}(t)$ are not known, but in applications the processes are observed on a discrete set of time points, $\bm t_i^{(h)} = \left(t_{i1}^{(h)},\ldots, t_{in^{(h)}_i}^{(h)}\right)$, where $n^{(h)}_i$ denotes the number of observed time points for the $i$-th individual and $h$-th process. Notice that the times of observation and their number are both process- and subject-specific. We indicate with $Y^{(h)}_{ij} = Y^{(h)}(t_{ij}^{(h)})$ the value of the $h$-th process $Y^{(h)}(t)$ at the $j$-th observed time $t_{ij}^{(h)}$ for the $i$-th subject. Hence, for each subject $i = 1, \dots, N$, we observe the random vector $\bm Y^{(h)}_i = \left(Y^{(h)}_{i1}, \dots, Y^{(h)}_{in^{(h)}_i}\right)$, whose joint distribution is the finite-dimensional law of the process $Y^{(h)}(t)$ at the times of observation, for $h = 1, \dots, p$. The aim of this work is to jointly model the processes $Y^{(h)}(t)$, capturing their time evolution and possible dependencies. For each process, we assume that the Markov property holds, i.e. conditionally on current and past events, future transitions only depend on the current state. The probability law of the $h$-th process $Y^{(h)}(t)$ is assigned via the matrix of instantaneous transition rates $\bm Q^{(h)}(t) =[\lambda^{(h)}(r,s;t)]_{r,s} $, which is also time-dependent, and whose elements are the instantaneous transition rates $\lambda^{(h)}(r,s;t)$ with $r, s \in \mathcal{S}^{(h)}$. In what follows, the vector $\bm \lambda^{(h)}(t) = \{\lambda^{(h)}(r,s;t): r, s \in \mathcal{S}^{(h)}, r \neq s\}$, of dimension $D_p = \sum_{h=1}^{p}d^{(h)}(d^{(h)}-1)$, indicates the off-diagonal transition rates of the matrix $\bm Q^{(h)}(t)$ at time $t > 0$, concatenated by row from top to bottom. For simplicity, we indicate a transition of the $h$-th process between different states of $\mathcal{S}^{(h)}$ with the notation $r \rightarrow s$. The instantaneous transition rates can be made covariate-dependent by specifying a Cox proportional hazard model. This allows the inclusion of both time-homogeneous covariates as well as time-varying continuous covariates. Alternatively, a semi-proportional intensity model can be easily specified for the covariates as in \cite{kim2012bayesian}. Note that the decision of including either type of covariates is process-specific. The time-homogeneous covariates are straightforwardly incorporated in the model, and we denote them here by $\bm X^{(h)}_i = \left(X^{(h)}_{i1}, \dots, X^{(h)}_{ig^{(h)}}\right)$, for the $i$-th individual and $h$-th process. On the other hand, the time-varying continuous ones, denoted by $\bm Z^{(h)}_i(t) = \left(Z^{(h)}_{i1}(t), \dots, Z^{(h)}_{iq^{(h)}}(t)\right)$, require additional assumptions. They are usually included by assuming a piece-wise constant effect over each interval of observations \citep{andersen2012statistical}, or by modelling them as longitudinal processes, linking their distribution to the ones of the multi-state processes via the inclusion of suitable random effects \citep{ferrer2016joint}. The first option has a clear computational advantage, while the latter has the potential to yield better inference on the overall disease progression. In the application presented in Section \ref{sec:GUSTO_application} we are not provided with any time-varying continuous covariates. However, the code provided with this manuscript allows for the implementation of the first method. This assumption leads to a piecewise constant model for the instantaneous transition rates $\lambda^{(h)}_{ij}(r,s)$, with $r, s \in \mathcal{S}^{(h)}$, and for the matrix $\bm Q^{(h)}_{ij} := \bm Q^{(h)}_i(t^{(h)}_{ij})$, for $j = 1, \dots, n^{(h)}_i$, $i = 1, \dots, N$ and $h = 1, \dots, p$. The model for the instantaneous $\log$-transition rates for $i = 1, \dots, N$ is then:
\begin{equation}\label{eq:Lambdas_Y}
\log\left(\lambda^{(h)}_{ij}(r,s)\right) = \phi^{(h)}_i(r,s) + \bm X^{(h)}_i \bm \beta^{(h)}_{rs} + \bm Z^{(h)}_{ij} \bm \gamma^{(h)}_{rs}, \quad j = 1, \dots, n^{(h)}_i, h = 1, \dots, p
\end{equation}
where $\phi^{(h)}_i(r,s)$ represents the baseline transition rate (on a log scale) of a transition $r \rightarrow s$. The parameters $\bm \beta^{(h)}_{rs} \in \mathbb{R}^{g^{(h)}}$ and $\bm \gamma^{(h)}_{rs} \in \mathbb{R}^{q^{(h)}}$ are the vectors of regression coefficients for the $h$-th process and the $r\rightarrow s$ transition.
Let $\epsilon^{(h)}_{ij} = t^{(h)}_{ij} - t^{(h)}_{ij-1}$ indicate the length of the $j$-th time interval, for $j = 2, \dots, n^{(h)}_i$, $i = 1, \dots, N$ and $h = 1, \dots, p$. Thanks to the piecewise constant assumption, the Chapman-Kolmogorov equations can be solved to obtain the interval-specific transition probabilities, $\bm p_{ij}^{(h)} (\bm \lambda^{(h)}_{ij} ,\epsilon^{(h)}_{ij}) = \left\{p_{ij}^{(h)} (r, s; \bm \lambda^{(h)}_{ij} ,\epsilon^{(h)}_{ij}): r, s \in \mathcal{S}^{(h)}\right\} $, for the vector of random variables $\bm Y^{(h)}_i$ \citep[see][]{Ross_1996}.
When $d^{(h)} = 2$, closed-form solutions are readily available \citep{cox1977theory}, while problems involving more than three states are usually tackled numerically \citep{moler2003nineteen}.
It can be shown that for each process $h$ a unique stationary distribution exists \citep{grimmet1992probability}, and we denote it by $\bm \pi^{(h)}_{ij}\left(k; \bm \lambda^{(h)}_{ij}\right)$, with $k \in \mathcal{S}^{(h)}$, highlighting the fact that these are functions of the subject-specific instantaneous transition rates \citep[see][for details]{Ross_1996}.
The stationary distribution can be used as marginal distribution for modelling the state of the processes at time $j = 1$, considering the vectors of instantaneous transition rates $\bm \lambda^{(h)}_{i1}$, in contrast to the general practice in multi-state modelling of specifying the model conditionally on the state at the first time of observation. This is important, as it allows Bayesian imputation of missing observations at time one, since they are treated as unknown parameters in the model. This aspect is particularly useful in our application, where the initial time presents a non-negligible missing rate.
The specification of the process- and subject-specific transition probabilities, together with the existence of the stationary distribution, leads to the joint likelihood for the vector of observed states $\bm Y^{(h)}_i$ for $i = 1, \dots, N$ and $h = 1, \dots, p$, as follows:
\begin{equation}\label{eq:likelihood_Y}
p\left(\bm Y \mid \bm \lambda^Y \right) = \prod_{i = 1}^N\prod_{h = 1}^{p}\prod_{j = 2}^{n^{(h)}_i} \left( p^{(h)}_{ij} \left( Y^{(h)}_{ij-1}, Y^{(h)}_{ij}; \bm \lambda^{(h)}_{ij},\epsilon^{(h)}_{ij}\right) \right) \pi^{(h)}_{i1}\left(Y^{(h)}_{i1}; \bm \lambda^{(h)}_{i1}\right)
\end{equation}
where $\bm Y$ and $\bm \lambda^Y$ indicate the multi-dimensional arrays containing the observation vectors $\bm Y^{(h)}_i$ and the instantaneous transition rate vectors $\bm \lambda^{(h)}_i$, while $p_{ij}^{(h)}$ denotes the transition probabilities and $\pi^{(h)}_{i1}$ is the (stationary) distribution at time one.
\subsection{Random effect distribution and relationship with SUR}\label{sec:BNP_Model}
Consider the vector of $\log$-baseline transition rates for the $h$-th process $\bm \phi^{(h)}_i = \{\phi^{(h)}_i(r,s): r \rightarrow s\}$, for $h = 1, \dots, p$ and for each subject $i$, and let $\bm \phi_i = (\bm \phi^{(1)}_i, \dots, \bm \phi^{(p)}_i)$ be the vector containing the $\log$-baseline transition rates of all the $p$ processes. To capture the inter-individual heterogeneity and allow for clustering of the subjects, we choose as random effect distribution for $\bm \phi_1, \dots, \bm \phi_N$ a mixture prior with random number of components, where the distribution of the weights is given by the normalization of a finite point process, as proposed by \cite{argiento2019infinity}. This approach has several advantages, allowing for flexible modelling of the weights in the mixture as well as efficient posterior computations (e.g., as compared to traditional reversible jump algorithms for mixture models). In more details, we assume the following mixture prior:
\begin{align}\label{eq:NIFPP_model}
\bm \phi_i &= \bm \phi^\star_{c_i}, \quad i = 1, \dots, N \nonumber \\
\bm \phi^\star_1, \dots, \bm \phi^\star_M \mid M &\iid P_0(\bm \phi^\star \mid \bm \theta) \nonumber \\
\mathbb{P}(c_i = m)& \propto S_m, \quad i = 1, \dots, N \\
S_1, \dots, S_M & \iid \text{Gamma}(\gamma_S, 1) \nonumber \\
M - 1 & \sim \text{Poi}(\Lambda) \nonumber
\end{align}
where we denote by $\text{Gamma}(a, b)$ the Gamma distribution with mean $a/b$, and by $\text{Poi}(\Lambda)$ the Poisson distribution with mean $\Lambda$. The variables $\bm c = (c_1, \dots, c_N)$ indicate the component allocations of the subjects and their corresponding prior probabilities are proportional to the unnormalized weights $\bm S = \left(S_1, \dots, S_M\right)$. As shown by \cite{argiento2019infinity}, posterior computation is greatly simplified via the introduction of a latent variable, conditionally on which the unnormalized weights of the mixture become independent. This computational trick is borrowed from the Bayesian nonparametric literature \citep{jamesetal09}.
Finally, the vectors $\bm \phi^\star_1, \dots, \bm \phi^\star_M$ are a finite sequence of locations for the mixture distribution and are, conditionally on the number of components $M$, i.i.d. from the base measure $P_0$. The specification of a joint random effect distribution for $\bm \phi_1, \dots, \bm \phi_N$ in model \eqref{eq:NIFPP_model} and the choice of $P_0$ are crucial in our modelling strategy, as it will be shown in Section \ref{sec:GGM}, since this allows inference on the shared dependence structure among the components of the vectors $\bm \phi^\star_m$, for $m = 1, \dots, M$ and, consequently, on
the dependence structure
among the $p$ different processes. As an alternative, a Bayesian nonparametric prior could have been specified as random effect distribution such as the Dirichlet process \citep{DeIorio_etal_2018} and the beta-Dirichlet process prior \citep{kim2012bayesian}, or, taking a complete different approach, flexible modelling of the baseline transition intensities can be achieved using penalised splines \citep{kneib2008bayesian}.
Our modelling approach resembles the one underlying the SUR framework of \cite{zellner1963estimators}, where $p$ different regression models are linked by specifying a joint error distribution, usually multivariate normal. The SUR methodology is one of the main techniques for handling multiple responses and offers a way to share information between models which are \textit{seemingly} unrelated, since they describe different data-generating processes. However, since these are observed for the same set of subjects and measurements are taken on often related processes, the study of their interdependency is of great interest in most applications. For this reason SUR-type models have gained vast popularity in different fields, such as Phenomics \citep{houle2010phenomics, banterle2018sparse}. In our application, for instance, where we deal with several processes associated to different aspects of maternal mental health (e.g., depression, anxiety, sleep quality), it is important to understand the relationships between such processes in order to have a broader view of the phenomenon under study. As in the SUR framework, in our context each process is modelled by its own \textit{seemingly unrelated} multi-state Markov process, but then they are \textit{related} through the joint random effect distribution on $\bm \phi = \left(\bm \phi_1, \dots, \bm \phi_N\right)$. Motivated by this parallelism, we name the proposed model as Seemingly Unrelated Multi-State (SUMS) processes.
\subsection{Gaussian Graphical model}\label{sec:GGM}
We use tools from the Gaussian Graphical models literature to describe the dependence
among the $p$ processes. Referring to model \eqref{eq:NIFPP_model}, we assume that $\bm \phi^\star_1, \dots, \bm \phi^\star_M \mid M \iid P_0 = \text{N}\left(\bm \mu , \bm \Omega_G\right)$, where the key modelling feature is the specification of the prior on the precision matrix $\bm \Omega_G$ conditional on a graph $G$, which captures the conditional dependence structure among the $\log$-baseline transition rates. The novelty of our modelling strategy is that $G$ is modelled conditionally on another random graph $G_0$, which characterizes the dependence structure among processes and is one of the main aspects of our inference.
In details, consider the graph $G_0 = (V_0, E_0)$, defined over the set of nodes $V_0 = \{1, \dots, p\}$, i.e. each node in the graph corresponds to a multi-state process
$Y^{(h)}(t)$. The edge set $E_0$ is formed of the pairs $E_0 \subseteq \{(h,k)\in V_0\times V_0: h <k \}$ such that an edge exists between nodes $h, k \in V_0$. We consider only simple graphs, i.e. undirected graphs, without self-loops nor multiple edges. As mentioned earlier, to introduce dependence among the elements of the vector $\bm \phi^\star_m = (\bm \phi^{\star, (1)}_m, \dots, \bm \phi^{\star, (p)}_m)$, with $\bm \phi^{\star, (h)}_m = \{\phi^{\star, (h)}_m(r,s): r \rightarrow s\}$ for $h = 1, \dots, p$, we define a second graph $G$ whose structure is determined by $G_0$. In particular, we let $G = (V, E)$ be the graph whose nodes are the indices of the vector $\bm \phi^\star_m$, i.e. $V = \{1, \dots, D_p\}$, with $D_p = \sum_{h \in V_0} d^{(h)}(d^{(h)} - 1)$. $G$ is a deterministic function of $G_0$ specified as follows. First, whatever the form of $G_0$, there exists an edge in $G$ between transition rates of the same process. Therefore, an empty graph $G_0$ corresponds to a graph $G$ with $p$ cliques, one for each process. Second, if there is an edge between nodes $h$ and $k$ in $G_0$ (i.e., $(h,k) \in E_0$), then there is and edge between all the possible pairs of elements of $\bm \phi^{\star, (h)}_m$ with those of $\bm \phi^{\star, (k)}_m$. An illustration for the case of three binary processes is given in Figure \ref{fig:tikz_G_G0}. We write $G = f(G_0)$, $f$ being the transformation described above. Note that $f$ is bijective and, as such, the specification of a prior on $G_0$ implies a prior on $G$. This construction is advantageous in terms of dimension reduction, as the dimension of the graph space where $G_0$ is defined can be significantly smaller than the one of $G$, leading to more efficient exploration of the posterior space.
\begin{figure}
\caption{Example of graphical structures underlying the SUMS processes: the graph $G_0$ describes the conditional dependence between the processes in $V_0 = \{1, 2, 3\}
\label{fig:tikz_G_G0}
\end{figure}
Following the literature on GGMs, the conditional independence structure of the multivariate Gaussian vectors $\bm \phi^\star_m \sim \text{N}(\bm \mu, \bm \Omega_G)$, for $m = 1, \dots, M$, is described by constraining the elements of the precision matrix $\bm \Omega_G$ \citep{Dempster}. Namely, two elements of the vector $\bm \phi^\star_m$ are, conditionally on the others, independent if and only if there is a zero in the corresponding entry of the precision matrix $\bm \Omega_G$. Since $G$ is a deterministic function of $G_0$, it is the latter that encodes the conditional independence structure of the vectors $\bm \phi^\star_m$, for $m = 1, \dots, M$ (see Figure \ref{fig:tikz_G_G0}). The standard conjugate prior for the precision matrix $\bm \Omega_G$ is the G-Wishart distribution, specified conditionally on the graph structure $G$ \citep{Roverato2002}. The last component needed to fully specify this part of the model, is the prior distribution for the graph $G$. We do not assign this prior directly, but rather it is inherited by the prior we choose for the graph $G_0$:
$$
\pi(G_0 \mid \eta) \propto \eta^{|E_0|} (1 - \eta)^{\binom{p}{2} - |E_0|}, \quad \eta \in (0,1)
$$
where $|E_0|$ is the number of edges in graph $G_0$ (i.e., the size of $E_0$), while $\binom{p}{2}$ is the number of possible graphs with nodes $V_0 = \{1, \ldots, p\}$. This prior is equivalent to assuming a Bernoulli prior with probability of success (here inclusion) $\eta$ on each edge of the graph $G_0$, independently across edges. Small values of $\eta$ favour sparser graphs \citep{armstrong2009}. Finally, we point out that, while the prior for the graph $G_0$ is defined over all possible graphs, including the non-decomposable ones, the resulting prior distribution on $G$ is defined on a restricted space due to the clique constraints imposed on the transitions of the same process which need to be fully connected.
\subsection{Relationship with PPMx models}\label{sec:relationship-with-ppmx-models}
The SUMS model has wide applicability in biomedical research as some processes can be regarded as responses and some as covariates. Indeed, time-varying categorical covariates are very common in the field of medical research, for instance in association with the monitoring of a patient's disease status over time.
In the application to disease progression in Section \ref{sec:GUSTO_application}, some processes represent mental health outcomes of interest, while others correspond to categorical clinical markers, and the goal of the analysis is to model the joint evolution of outcomes and clinical factors to determine how the symptom variables influence the disease course.
Handling of time-varying multivariate categorical information can be problematic in several applications and the SUMS approach provide a natural framework to deal with this problem. In a Bayesian framework, \cite{DeIorio_etal_2018} discuss possible solutions and propose an approach based on a latent health function borrowing ideas from Item Response Theory \citep{thissen2009item}. The latter approach, although computationally efficient, does not allow for a direct quantification of the covariate effect on the clinical response of interest and it may lead to identifiability problems. A simpler and more common approach to deal with time-varying categorical covariates is to introduce appropriate dummy variables, considerably increasing the number of parameters to be estimated, resulting in slower computations and lower effectiveness in high dimensional problems.
Another computational effective solution is to summarize the covariates into an often arbitrary time-varying score, but at the cost of losing information and interpretability.
When some of the multi-state processes can be seen as covariates, then the SUMS model has interesting connections with Product Partition Models with Covariates (PPMx), very popular in the Bayesian nonparametric literature \cite{muller1996bayesian, muller2011product}. Indeed, our approach provides a flexible and robust modelling strategy, which leads to covariate-driven clustering of the subjects and enables the inclusion of different types of covariates/responses in a natural and efficient way \citep{barcella2017comparative}. We now clarify the relationship between SUMS and PPMx. The main modelling idea behind the PPMx is to include covariate information into the partition model (e.g., into a Dirichlet Process Mixture model framework) by treating each covariate as a random variable \citep{muller1996bayesian, muller2011product, barcella2017comparative}. In the SUMS approach we can consider a set of processes $(Y^{(1)}(t), \dots, Y^{(p_Y)}(t))$ as responses and another set, denoted by $(H^{(1)}(t), \dots, H^{(p_H)}(t))$, as explanatory factors. Then Eq.\eqref{eq:likelihood_Y} specifies a suitable probability model for the joint vector of processes $\{(Y^{(h)}(t), H^{(l)}(t)); h=1, \ldots, p_Y; l=1,\ldots,p_H; t \in \mathbb{R}^+ \}$, with $p_Y+p_H=p$. Let $\bm c$ be the set of allocation variables introduced in \eqref{eq:NIFPP_model}, and let $\rho_N$ be the partition of the indices $\{1, \dots, N\}$ induced by $\bm c$. We indicate by $C_j$ the set of indices belonging to the $j$-th cluster, i.e. $C_j = \{i \in \{1, \dots, N\} \mid c_i = j\}$, and thus a partition with $K_N$ clusters corresponds to $\rho_N = \{C_1, \dots, C_{K_N}\}$. In the PPMx framework, the SUMS model induces a prior on the partition $\rho_N$ which depends on the covariates $\bm H$:
\begin{equation}\label{eq:rho_H_1}
p\left(\rho_N \mid \bm H \right) = V(N, K_N) \prod_{j = 1}^{K_N} \mathcal{C}(C_j) \mathcal{G}(\bm H^\star_j)
\end{equation}
where $\mathcal{C}(C_j)$ is the \textit{cohesion}, i.e. a function of the $j$-th cluster $C_j$, $\mathcal{G}(\bm H^\star_j)$ is the \textit{similarity}, i.e. a function of the array of covariates corresponding to the subjects in cluster $j$, denoted as $\bm H^\star_j := \{H_i : i \in C_j\}$, for $j = 1, \dots, K_N$, and $V(N, K_N)$ is a constant depending only on the sample size $N$ and the number of clusters $K_N$. The cohesion function $\mathcal{C}$ expresses prior information about the partition, such as the average size of a cluster, while the similarity function $\mathcal{G}$ captures the contribution of the covariates to the clustering structure. The presence of $\mathcal{G}$ in \eqref{eq:rho_H_1} allows subjects with similar covariates to be more likely assigned to the same cluster. Under our modelling assumptions it can be shown that \eqref{eq:rho_H_1} is given by:
\begin{gather}
p\left(\rho_N \mid \bm H \right) \propto p\left( \rho_N \right) p\left(\bm H \mid \rho_N \right) = \nonumber \\
V(N, K_N) \prod_{j = 1}^{K_N} \frac{\Gamma(\gamma_S + n_j)}{\Gamma(\gamma_S)} \int \left( \prod_{i \in C_j} p\left(\bm H_i \mid \bm \lambda^H_i \right) \right) P_0(d\bm \phi^{\star, (p_Y + 1)}_j, \dots, d\bm \phi^{\star, (p)}_j) \label{eq:rho_H_2}
\end{gather}
See Supplementary Material Section 1 for a proof. The proposed model induces the similarity function $\mathcal{G}(\bm H^\star_j) = \int \prod_{i \in C_j} p\left(\bm H_i \mid \bm \lambda^H_i \right) P_0(d\bm \phi^{\star, (p_Y + 1)}_j, \dots, d\bm \phi^{\star, (p)}_j)$, for $j = 1, \dots, K_N$. The similarity function $\mathcal{G}$ is not known in closed form, differently from the common PPMx specification, where the similarity function is usually obtained from a conjugate model for the covariates vector via marginalization to simplify computations. In the proposed approach, the evaluation of \eqref{eq:rho_H_2} would require an expensive numerical approximation. For this reason, we resort to a conditional MCMC algorithm analogous to the one proposed by \cite{argiento2019infinity}, not requiring the evaluation of the integral in \eqref{eq:rho_H_2}.
\subsection{MCMC Algorithm}\label{sec:mcmc-algorithm}
Posterior inference is performed through a MCMC algorithm, described in details in Supplementary Material Section 2. The numerous non-conjugate updates required by the proposed model are tackled using adaptive Metropolis-Hastings sampling schemes \citep{haario2001adaptive, atchade2005adaptive}, which need an additional short burn-in period. Additionally, inference under the proposed model is challenging given the presence of the graphs $G$ and $G_0$. We adopt the birth-and-death approach of \cite{mohammadi2015bayesian}, and extend their algorithm to accommodate for MCMC moves on cliques instead of single edges, recalling that each edge in $G_0$ corresponds to a clique in $G$ through the map $f$. Indeed, the original algorithm of \cite{mohammadi2015bayesian} is based on theoretical results from the GGM literature \citep[see][]{wang2012efficient}, which can be extended to our modelling setting. In Supplementary Material Section 4, we also compare the performance of our model with the approach of \cite{DeIorio_etal_2018} and with two alternative versions of the proposed model (i.e., DP and parametric versions). The results of the comparison show that the proposed model outperforms the parametric approach, as well as the nonparametric competitors in terms of clustering, leading to comparable results with respect to the estimation of regression coefficients.
\section{Application to the GUSTO study}\label{sec:GUSTO_application}
The GUSTO study \citep[Growing Up in Singapore Towards healthy Outcomes, ][]{soh2014cohort} is a longitudinal birth cohort study started in 2009 and involving Singaporean mothers and their children. The study is one of the most carefully phenotyped parent-offspring cohorts, focusing on the roles of foetal, developmental and epigenetic factors involved in early body composition as well as neuro-development. In this work we consider data on $N = 301$ mothers, followed during pre- and post-natal periods, starting from three months before childbirth. The main focus of the analysis is understanding the relationship among five psychometric indicators obtained from specific questionnaires: the Beck's Depression Inventory II \citep[BDI II,][]{Beck_etal_1961}; the Edinburgh Postnatal Depression Scale \citep[EPDS,][]{matthey2006variability}; the State-Trait Anxiety Inventory \citep[STAI,][]{spielberger1983manual} that can be decomposed into two different scores describing the anxious states (STAI-s), reflecting characteristics that can vary with time, and the anxiety traits (STAI-t), reflecting more stable characteristics; and the Pittsburgh Sleep Quality Index \citep[PSQI,][]{buysse1989pittsburgh}. The score ranges of these questionnaires are discretized to obtain clinically relevant categories, and are recorded at different time points, as reported in Supplementary Material Table 2. These five processes represent time-varying categorical observations and are modelled jointly via SUMS, to capture significant relationships between them \citep[see also][]{van2018relation}. In our setting, the four mental health indicators (BDI, EPDS, STAI-s, STAI-t) represent the main clinical responses of interest, while the sleep quality indicator (PSQI) is treated as a time-varying categorical covariate. For all processes, we assume missingness at random and impute missing values at the first time of observation from their full conditionals (see Section 2.2 of Supplementary Material).
We are also provided with information regarding socio-demographic and clinical characteristics, as well as scoring obtained from additional questionnaires measuring personality traits. In particular, we have individual scores for the Big Five Inventory \citep[BFI, ][]{john1999big} (including the scores for \textit{Extraversion}, \textit{Agreeableness}, \textit{Conscientiousness}, \textit{Neuroticism}, \textit{Openness}, and \textit{Liking}) and for the Maternal Childhood Adversity \citep[MCA, ][]{bouvette2015maternal}. Many of the remaining covariates are time-homogeneous categorical, while no time-varying continuous covariates are available. The time-homogeneous continuous covariates are centred and scaled so that each column has null mean and unitary standard deviation, thus estimating the corresponding regression coefficients $\bm \beta^{(h)}$ on the same scale across processes. The full set of covariates ($g^{(h)} = 22$, for $h = 1, \dots, 4$, including dummy coding for the categorical ones) is described in more details in Supplementary Material Table 3, and is included in the specification of the four psychometric processes, but not of PSQI.
\paragraph{Full model specification}
We describe the full model used in the application presented in this section, which is the same implemented in the sensitivity analysis on the hyperparameters $\Lambda$ and $\gamma_S$ appearing in Supplementary Section 3. For each $i = 1, \ldots, N$:
\begin{eqnarray}\label{eq:MixtureModel1}
&&\bm Y^{(1)}_i, \dots, \bm Y^{(4)}_i, \bm H_i \mid \{\bm \lambda^{(h)}_i, \bm \beta^{(h)}, h = 1, \ldots, 4\}, \bm \lambda^H_i \ind \prod_{h = 1}^{4} p(\bm Y^{(h)}_i \mid \bm \lambda^{(h)}_i, \bm \beta^{(h)}) p(\bm H_i \mid \bm \lambda^H_i) \nonumber \\
&&\log \left(\lambda^{(h)}_i(r,s) \right) = \phi^{\star,h}_{c_i}(r,s) + \bm X^{(h)}_i \bm \beta^{(h)}_{rs}, \ r \rightarrow s, \quad h = 1, \dots, 4 \nonumber\\
&&\log \left(\lambda^H_i(r,s) \right) = \phi^{\star,H}_{c_i}(r,s), \ r \rightarrow s \nonumber\\
&& \bm \beta^{(h)} \sim \text{MN}_{g^{(h)} \times d^{(h)}(d^{(h)}-1)}(\bm 0, \bm U_{\bm \beta^{(h)}}, \bm V_{\bm \beta^{(h)}}), \quad h = 1, \dots, 4 \nonumber \\
&&\bm \phi^\star_m = (\bm \phi^{\star,1}_m, \dots, \bm \phi^{\star,4}_m, \bm \phi^{\star,H}_m) \mid M, \bm \mu, \bm \Omega_G \sim P_0 =\text{N}_{D_p}(\bm \mu, \bm \Omega_G), \quad m = 1, \dots, M \nonumber \\
&&\bm \mu, \bm \Omega_G \mid G, \bm m_{\mu}, k_0 \sim \text{N}_{D_p}(\bm \mu \mid \bm m_{\bm \mu}, k_0 \Omega_G) G\text{-Wishart}_G(\bm \Omega_G \mid \nu, \bm \Psi) \\
&&k_0 \sim \text{Gamma}(a_{k_0}, b_{k_0}) \nonumber \\
&&\mathbb{P}(c_i = m) \propto S_m, \quad m = 1, \dots, M \nonumber \\
&&S_1, \dots, S_M \mid M, \gamma_S \iid \text{Gamma}(\gamma_S, 1) \nonumber \\
&&M - 1 \mid \Lambda \sim \text{Poi}(\Lambda) \nonumber \\
&&G = f(G_0), \quad p(G_0) \propto \eta^{|E_0|} (1 - \eta)^{\binom{p}{2} - |E_0|} \nonumber
\end{eqnarray}
where we indicate with $\bm \phi_m^\star$ the vectors of unique $\log$-baseline transition rates for the $m$-th component in the model and $M$ is the unknown number of components in the mixture. Here $\bm{c}= \{c_i, i=\ldots, N\}$ represents the allocation vector, i.e. it specifies to which component the $i$-th observation is assigned to, characterised by $\bm \phi_i =\bm \phi_{c_i}^\star$. The probability of $c_i$ being equal to the $m$-th component of the mixture is proportional to the unnormalized weights $S_m$, for $m = 1, \dots, M$. Therefore, due to the discrete property of the mixing measure, the parameters $\bm \phi_i$ are assigned to $K_N$ different clusters, with $K_N \leq M$. We impose a conditionally conjugate hyper-prior on $k_0$, and fix the hyperparameters $\gamma_S, \Lambda$. We refer to \cite{argiento2019infinity} for a thorough discussion on prior specification in mixture models with unknown number of components. However, we point out that the mixture component of the model is specified conditionally to the graph structure $G$. Finally, $\text{MN}_{n \times p}(\bm 0, \bm U, \bm V)$ is the matrix-variate Normal distribution of dimension $n \times p$ centred on the null matrix $\bm 0$ and with covariance matrices $\bm U$ and $\bm V$ of dimensions $n \times n$ and $p \times p$, respectively.
\paragraph{Hyper-Prior elicitation}\label{SMsec:hyper-prior-elicitation}
We need to specify the hyperparameters for the priors in the three components of the model: the transition rates, the mixture model with random number of components and the graphical model.
In order to induce sparsity in the graph structure and identify meaningful relationship between the SUMS processes, we set the a-priori probability of edge inclusion to $\eta = 0.1$. The hyperparameters of the centring measure $P_0$ are such that $\bm m_{\bm \mu} = \bm 0$, $k_0 \sim \text{Gamma}(1,1)$, $\nu = D_p + 2$ and $\bm \Psi = \mathbb{I}_{D_p} / \nu$, where $\mathbb{I}_p$ is the identity matrix of size $p$. In the case of a full graph $G$, the latter corresponds to $\mathbb{E}(\bm \Omega_G\mid G) = \mathbb{I}_{D_p}$. The regression coefficients $\bm \beta^{(h)}$ are a-priori independent and identically distributed, i.e. $\bm U_{\bm \beta^{(h)}} = \bm V_{\bm \beta^{(h)}} = \mathbb{I}_{g^{(h)} d^{(h)} (d^{(h)} - 1)}$, for $h = 1, \ldots,4$. The mixture prior for the $\log$-baseline transition rates $\bm \phi^\star_1, \dots, \bm \phi^\star_M$ is controlled by the hyperparameters $\Lambda$ and $\gamma_S$. These parameters determine the distribution of the number of components and the corresponding allocation of the subjects, and are the object of an extensive sensitivity analysis presented in Supplementary Section 3. In this application, we fix these parameters to $\Lambda = 0.01$ and $\gamma_S = 0.1$.
\paragraph{Posterior inference}
We run the MCMC algorithm described in Section \ref{sec:mcmc-algorithm} for 50000 iterations, after an initial burn-in period of 1000 iterations used to initialise the adaptive Metropolis-Hastings, discarding 40000 iterations as burn-in and thinning every 2, obtaining a final sample of 5000.
We explore the relationship between the multi-state processes by imposing dependencies via the graphical model approach described in Section \ref{sec:GGM}. Inference on the posterior distribution of the graphical structure $G_0$ is obtained by reporting the posterior edge inclusion probability for each pair of nodes. In Figure \ref{fig:Graph_G0} we report the posterior median graph, obtained by including only those edges with posterior edge inclusion probability greater than 0.5 \citep{barbieri2004optimal}. The four clinical mental health indicators BDI, EPDS, STAI-s and STAI-t are strongly associated, presenting a clique in the posterior median graph. Interestingly, the sleep quality index PSQI is only related to the anxiety indices STAI-s and STAI-t, forming a clique as well. Links between probable anxiety and sleeping quality have been reported in previous studies \citep{swanson2011relationships, ibrahim2012sleep}, and it is confirmed by our findings. Moreover, as previously reported, poor sleep quality may feed into poor emotional and mental health states \citep{ruiz2015sleep, osnes2019insomnia}.
\begin{figure}
\caption{Posterior median graph of $G_0$: each edge included in the graph has posterior edge inclusion probability greater than 0.5. Each edge of the median graph is labelled with the corresponding posterior edge inclusion probability.}
\label{fig:Graph_G0}
\end{figure}
Another important aspect of the proposed model is the possibility of including covariates in the specification of the transition rates via \eqref{eq:Lambdas_Y}. Posterior inference on the coefficient $\bm \beta^{(h)}$, for $h = 1, \dots, 4$ is not trivial, due to the high number of parameters involved. The importance of each covariate can be assessed through Bayes Factors (BF), defined as the ratio of the marginal contributions derived from the model with the corresponding regression coefficient set to zero versus the full model \citep{kass1995bayes}. Closed form expressions for the Bayes Factor under the SUMS model are not available, and thus we use the Savage-Dickey density ratio method \citep{wagenmakers2010bayesian,verdinelli1995computing}. The applicability of this method is guaranteed by the component-wise assumption of independence a-priori for the regression coefficients $\bm \beta^{(h)}$, for $h = 1, \dots, 4$ (see the full model specification in \eqref{eq:MixtureModel1}). For each process $h$, the values of $-\log_{10}\left(\text{BF}^{(h)}_{jk}\right)$ are reported in the heatmap of Figure \ref{fig:Beta_BFs}, for $j = 1, \dots, g^{(h)}$ and $k = 1, \dots, d^{(h)} (d^{(h)} - 1)$. The magnitude of $-\log_{10}\left(\text{BF}^{(h)}_{jk}\right)$ measures the evidence in favour of the full model \citep{kass1995bayes}. The majority of the coefficients is characterized by a low value of $-\log_{10}\left(\text{BF}^{(h)}_{jk}\right)$, supporting the hypothesis of no association, particularly in the case of the STAI processes. However, some coefficients are characterized by $-\log_{10}(BF)$ values above 1 or 2, indicating strong evidence in support of the inclusion of the corresponding covariate in the specific process. Of particular interest are the coefficients relative to the BFI and MCA scores, representing different traits of personality, trauma and parental relationship. We present the posterior mean and 95\% credible intervals of the regression coefficients relative to BFI and MCA in details in Figures 5 of Supplementary Material Section 6.
\begin{figure}
\caption{Heatmap of Bayes Factors ($- \log_{10}
\label{fig:Beta_BFs}
\end{figure}
The personality traits of the mothers as described by the BFI scores have been previously associated with increased likelihood for both antenatal and postnatal mood disorder traits \citep{ritter2000stress, leigh2008risk}. Our analysis supports this as BFI traits have a relevant impact on both BDI and EPDS (95\% credible interval does not contain zero). An interesting result appears through the estimates of the BFI's \textit{Neuroticism} dimension, which characterizes transitions 2 $\rightarrow$ 3 (deterioration, positive regression coefficient) and 3 $\rightarrow$ 1 (improvement, negative regression coefficient) in both BDI and EPDS scores, indicating that higher \textit{Neuroticism} scores are associated with higher depressive symptoms during the peripartum period \citep{kitamura1993psychological, o1996rates}. On the other hand, \textit{Openness} and \textit{Conscientiousness} in EPDS (see Supplementary Figure 5) positively influence the transition 3 $\rightarrow$ 1 (improvement). We also notice the effects of BFI's \textit{Extraversion} and \textit{Agreeableness} differ for BDI and EPDS's transitions. This could be explained by the fact that the social behaviors associated with \textit{Extraversion} and \textit{Agreeableness} are distinct \citep{tobin2000personality, jensen2001agreeableness}. Extraverts tend to actively seek out social interactions, whereas people scoring high on \textit{Agreeableness} prefer harmonious relationships. Maternal history of developmental adversity is linked to increased risk for depression \citep{leigh2008risk}, of which childhood abuse is a strong risk factor \citep{seng2014complex}, as highlighted by the importance of the MCA covariate for the transition 2 $\rightarrow$ 3 (deterioration) in BDI (see Figure \ref{fig:Beta_BFs} and Supplementary Figure 5). This result is also confirmed by \cite{mandelli2015role} who found that women who were victims of childhood neglect or abuse were at least twice as likely to suffer from depression. The quality of relationship with the women’s parents may also contribute to maternal developmental adversity. Mothers who received low parental care and high control during childhood are at risk for peripartum anxiety \citep{grant2012parental} and depression \citep{mcmahon2005psychological}.
The choice of the mixture prior \eqref{eq:NIFPP_model} as random effect distribution for the vector of $\log$-transition rates $\bm \phi_1, \dots, \bm \phi_N$ allows for clustering of the subjects. Inference on the random partition is shown in Supplementary Material Figure 6, where the posterior distributions of the number of clusters, components and of the co-clustering probabilities are reported. An estimate of the random partition induced on the subjects under study is obtained by minimizing the Binder's loss function \citep{Binder_1978} with equal costs. We obtain a partition with three clusters, which also corresponds to the posterior mode of the number of clusters. The three clusters contain 140, 126 and 35 subjects, respectively, and are labelled according to their sizes in decreasing order. In Figure \ref{fig:phi_inClusters} we report the posterior distribution of $\phi^{(h)}(r,s)$ conditional on the Binder's partition, for $r \rightarrow s$ and $h = 1, \dots, p$. Cluster-specific estimates of transition rates differ among clusters (see Supplementary Material Section 7 for a discussion). For instance, transition rates corresponding to improvement in the BDI or EPDS scores are higher in Clusters 1 and 2 rather than Cluster 3. The binary processes (STAI-s, STAI-t and PSQI) also seem to present differences between clusters in the same direction, identifying Cluster 3 as the one most prone to a deterioration of the mental health status of its subjects.
\begin{figure}
\caption{Posterior means and 95\% credible intervals of the instantaneous $\log$-transition rates $\phi^{(h)}
\label{fig:phi_inClusters}
\end{figure}
\section{Conclusions}\label{sec:Conclusions}
Observations on time-evolving related processes are very common in biomedical applications and beyond. In this work, we present a Bayesian semiparametric approach for joint modelling of several multi-state Markov processes describing an individual's transitions between different states in continuous time. The proposed model builds on the multi-state Markov models, GGM and PPMx literature. The different multi-state processes are linked by imposing a flexible prior distribution for the instantaneous transition rates, which allows for data-driven clustering of the subjects. The dependence among the processes is captured by a graph and posterior inference is performed through a tailored MCMC algorithm.
The proposed model finds wide applicability, due to its flexibility, interpretability and relative ease of computations.
In this work, we analyse data from the GUSTO cohort study with the aim of understanding the evolution and relationships between mental health indicators over time. Our findings are in agreement with existing medical literature and shed more light on the influence of childhood and parental factors on mental health progression. Potential extensions include higher order Markov dependency and joint modelling of multi-state processes and continuous longitudinal trajectories.
A possible alternative to our approach is to represent the categorical covariates with continuous Gaussian latent variables linked to the categorical outcome by thresholding \citep{albert1993bayesian}, allowing for the inclusion of a time component through auto-regressive terms in the likelihood \citep[e.g.][]{ barcella2018modelling}. To the best of our knowledge, this strategy has not been employed in the context of multi-state models, and it represents an interesting direction for future developments. However, this formulation could suffer from limited interpretability \citep{garcia2007conditional} and could induce further computational challenges \citep{zhang2006sampling}.
\end{document} | math | 50,026 |
\begin{equation}gin{document}
\title
[conformally flat generalized Ricci recurrent manifolds]
{A classification of conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds}
\author{Tee-How Loo \and Avik De}
\address{T. H. Loo\\
Institute of Mathematical Sciences\\
Universiti Malaya\\
50603 Kuala Lumpur\\
Malaysia}
\email{[email protected]}
\address{A. De\\
Department of Mathematical and Actuarial Sciences\\
Universiti Tunku Abdul Rahman\\
Jalan Sungai Long\\
43000 Cheras\\
Malaysia}
\email{[email protected]}
\footnotetext{A.D. and L.T.H. are supported by the grant FRGS/1/2019/STG06/UM/02/6.}
\begin{equation}gin{abstract}
Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be either a de Sitter space or an anti-de Sitter space.
In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.
\end{abstract}
\date{}
\maketitle
\section{\textbf{Introduction}}
The importance of the Generalized Ricci recurrent structure and its interaction with several gravity theories, like the standard theory of gravity \cite{avik}, the modified $f(R)$-theory \cite{avik-2}, the modified Gauss-Bonnet $f(R,G)$-theory \cite{frg} are well established. And of course, mathematically the structure raised curiosity among researchers in several occasions, details can be seen in \cite{avik} and the references therein.
De et al. \cite{de} introduced the notion of $(GR)_n$ as an $n$-dimensional non-flat pseudo-Riemannian manifold whose Ricci tensor $R_{ij}$ satisfies the following:
\begin{equation} \nabla _{i}R_{jl}= A_iR_{jl}+B_ig_{jl},\langlebel{eqn:1}\end{equation}
where $A_i$ and $B_i$ are two non-zero 1-forms. The structure is considered to be a generalization of Patterson introduced Ricci recurrent manifolds $R_n$ \cite{patterson} in which the Ricci tensor satisfies $\nabla_{i}R_{jl}= A_iR_{jl}$, with a non-zero 1-form $A_i$. Obviously, if the one-form $B_i$ vanishes, it reduces to a $R_n$.
A $(GR)_4$ spacetime is a generalized Robertson Walker spacetime with Einstein fibre for a Codazzi type $R_{ij}$ \cite{pinaki}. In \cite{avik}, a conformally flat $(GR)_4$ was shown to be a perfect fluid. Its interaction with general relativistic cosmology was discussed thoroughly in the same paper. Continuing to this study, in \cite{avik-2}, the authors studied a conformally flat $(GR)_4$ with constant $R$ as a solution of $f(R)$-gravity theory. The presently accepted homogeneous and isotropic model of our universe, the Robertson-Walker spacetime is $(GR)_4$ if and only if it is Ricci symmetric. The equation of state (EoS) was shown to have $\omega = -1$. Several energy conditions were also analyzed and validated by current observational dataset. Very recently, the impact of $(GR)_4$ structure is investigated in modified Gauss-Bonnet, $f(R,G)$ theory of gravity \cite{frg}. The obtained results were examined for two particular $f(R,G)$-models and for them both, the weak, null and dominant energy conditions were validated while the strong energy condition was violated, which is a good agreement with the recent observational studies revealing that the current universe is in accelerating phase. Despite having such recognised physical significance, a complete classification of conformally flat Lorentzian manifolds with generalized Ricci recurrent structure is still due.
This motivated us to do a careful scrutiny through all the possible cases.
We shall give a complete classification of this structure in a general setting as below:
\begin{equation}gin{theorem}\langlebel{res1}
A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is an Einstein manifold, in particular,
it is either a de Sitter space or an anti-de Sitter space.
\end{theorem}
We can then deduce a classification of conformally flat generalized Ricci recurrent spacetime.
\begin{equation}gin{corollary}\langlebel{res2}
A conformally flat generalized Ricci recurrent spacetime is either a de Sitter spacetime or an anti-de Sitter spacetime.
\end{corollary}
\section{The Proof}
Let $M$ be an $n$-dimensional pseudo-Riemannian manifold, $n\geq 3$, with index $q$ ($0\leq q\leq n$).
Suppose $M$ is a conformally flat $(GR)_n$, that is, the Ricci tensor satisfies (\ref{eqn:1})
and the Riemannian curvature tensor satisfies \cite[pp.302]{neil}
\begin{equation}gin{align}\langlebel{eqn:2}
R^l{}_{ijk}=\frac{R_{ki}\delta^l_j-R_{ji}\delta^l_k+g_{ki}R^l_j-g_{ji}R^l_k}{n-2}
-\frac{g_{ki}\delta^l_j-g_{ji}\delta^l_k}{n-1}\frac{R}{n-2}.
\end{align}
The divergence of the above equation gives
\begin{equation}gin{align}\langlebel{eqn:3}
\nabla_jR_{ik}-\nabla_kR_{ij}=\frac12\frac{\nabla_jR g_{ki}-\nabla_kR g_{ji}}{n-1}.
\end{align}
Contracting $j$ and $l$ in (\ref{eqn:1}) we obtain
\begin{equation} \nabla_iR=RA_i+nB_i.\langlebel{eqn:6}\end{equation}
Contracting $i$ and $l$ in (\ref{eqn:1}) we obtain
\begin{equation} \frac{1}{2}\nabla_jR=A^lR_{lj}+B_j.\langlebel{eqn:7}\end{equation}
It follows from (\ref{eqn:6})--(\ref{eqn:7}) that
\begin{equation}\langlebel{eqn:8} 2A^lR_{lj}=RA_j+(n-2)B_j.\end{equation}
By using (\ref{eqn:1}), (\ref{eqn:3})--(\ref{eqn:6}) we calculate
\begin{equation} \langlebel{eqn:9}
2(n-1)\{A_iR_{kl}-A_kR_{il}\}=RA_ig_{kl}-RA_kg_{il}-(n-2)B_ig_{kl}+(n-2)B_kg_{il}.
\end{equation}
Next we split into two cases: $A^i$ is not a lightlike vector and $A^i$ is a lightlike vector.
\textbf{(I) $A^i$ is either a timelike or spacelike vector.}
Write
\[
A_i=\varepsilon\alpha U_i; \quad U_lU^l=\varepsilon=\pm 1; \quad \alpha\neq0.
\]
Then (\ref{eqn:8})--(\ref{eqn:9}) become
\begin{equation}gin{align}
2\varepsilon\alpha U^lR_{ij}=&\varepsilon\alpha RU_j+(n-2)B_j, \langlebel{eqn:8b}\\
2(n-1)\varepsilon\alpha\{ U_iR_{kl}-U_kR_{il}\}
=&\varepsilon\alpha\{ RU_ig_{kl}- RU_kg_{il}\}-(n-2)B_ig_{kl}+(n-2)B_kg_{il}. \langlebel{eqn:9b}
\end{align}
Transvecting (\ref{eqn:9b}) with $U^i$ we obtain
\begin{equation}a
2(n-1)\alpha R_{kl}
=\{\alpha R-(n-2)\begin{equation}ta\} g_{kl}+(n-2)\{\varepsilon\alpha RU_kU_l+U_lB_k+(n-1)U_kB_l\},
\langlebel{eqn:10}
\end{equation}a
where $\begin{equation}ta=B_lU^l$.
The skew-symmetry part gives
\[
U_lB_k+(n-1)U_kB_l = U_kB_l+(n-1)U_lB_k,
\]
which implies that
\begin{equation} B_i={\varepsilon}\begin{equation}ta U_i, \quad (\begin{equation}ta= U^lB_l).\langlebel{eqn:11}\end{equation}
Armed with this significant result, using (\ref{eqn:6}), (\ref{eqn:8b}) and (\ref{eqn:10}) we can conclude that the Ricci curvature tensor and the Ricci scalar satisfy
\begin{equation}gin{align}
2(n-1)R_{kl}=&\{R-(n-2)\langlembda\}g_{kl} +(n-2)\{R+n\langlembda\}\varepsilon U_kU_l, \langlebel{eqn:12} \\
\nabla_iR=&\{R+n\langlembda\}\alpha \varepsilon U_i, \langlebel{eqn:13} \\
2U^lR_{lj}=&\{R+(n-2)\langlembda\}U_j, \langlebel{eqn:14}
\end{align}
where
\[
\langlembda=\frac\begin{equation}ta\alpha.
\]
Covariantly differentiating (\ref{eqn:12}) with respect to $i$ we obtain
\begin{equation}gin{align*}
2(n-1)\nabla_iR_{kl}
=& \{\nabla_iR-(n-2)\nabla_i\langlembda\}g_{kl}+(n-2)(\nabla_iR+n\nabla_i\langlembda)\varepsilon U_kU_l \notag\\
& +(n-2)(R+n\langlembda){\varepsilon}\nabla_i(U_kU_l) \notag\\
=&(n-2)\{-\nabla_i\langlembda g_{kl}+n\nabla_i\langlembda\varepsilon U_kU_l+(R+n\langlembda){\varepsilon}\nabla_i(U_kU_l)\} \notag\\
&+\nabla_iR \{g_{kl}+(n-2)
\varepsilon U_kU_l\}.
\end{align*}
On the other hand, (\ref{eqn:1}), (\ref{eqn:11})--(\ref{eqn:12}) give
\begin{equation}gin{align*}
2(n-1)\nabla_iR_{kl}
=&2(n-1)\alpha\varepsilon U_i R_{kl}+2(n-1)\langlembda\alpha\varepsilon U_i g_{kl} \notag\\
=&\{R-(n-2)\langlembda\}\alpha\varepsilon U_ig_{kl} +(n-2)\{R+n\langlembda\}\alpha\varepsilon U_i \varepsilon U_kU_l \notag \\
&+2(n-1)\langlembda\alpha\varepsilon U_i g_{kl} \notag \\
=& (R+n\langlembda)\alpha\varepsilon U_i \{g_{kl}+(n-2)\varepsilon U_kU_l\} \notag\\
=&\nabla_iR \{g_{kl}+(n-2)\varepsilon U_kU_l\}.
\end{align*}
These two equations give
\begin{equation}gin{align*}
-\nabla_i\langlembda g_{kl}+n\nabla_i\langlembda\varepsilon U_kU_l+(R+n\langlembda){\varepsilon}\nabla_i(U_kU_l)=0.
\end{align*}
Transvecting with $U^k$ and $U^l$ we obtain
\begin{equation}gin{align}\langlebel{eqn:22}
\nabla_i\langlembda=0.
\end{align}
It follows from these two equations that
\begin{equation}gin{align*}
(R+n \langlembda)\nabla_i(U_kU_l)=0.
\end{align*}
We consider two subcases: $R+n\langlembda\neq0$ and $R+n\langlembda=0$.
\begin{equation}gin{description}
\item[a] If $R+n\langlembda\neq0$, we have $\nabla_i U_l=0$ and so $U_lR_i^l=0$.
It follows from (\ref{eqn:14}) that
\[
R+(n-2)\langlembda=0.
\]
It follows from this equation and (\ref{eqn:22}) that
\[
\nabla_iR=-(n-2)\nabla_i\langlembda=0.
\]
After applying this to (\ref{eqn:13}) gives
$R+n\langlembda=0$. Hence we have $R=\langlembda=0$ and so $R_{lk}=0$; violating the hypothesis of being $(GR)_n$.
Hence this case is impossible.
\item[b] Suppose that
\[ R+n\langlembda=0.
\]
Then, using (\ref{eqn:12}) we obtain
\[
R_{kl}=-\langlembda g_{kl},
\]
meaning that it is an Einstein space.
\end{description}
\textbf{(II) $A^i$ is a lightlike vector.}
Take another lightlike vector $K^i$ such that
\[
A_lA^l=K_lK^l=0; \quad A_lK^l=-1.
\]
Transvecting (\ref{eqn:9}) with $A^i$, with the help of (\ref{eqn:8}), we obtain
\begin{equation}
-A^jB_j g_{kl}+ RA_kA_l+A_lB_k+(n-1)A_kB_l=0.
\langlebel{eqn:30}
\end{equation}
The skew-symmetry part gives
\[
A_lB_k= A_kB_l,
\]
which implies that $A^kB_k=0$ and
\begin{equation} B_i=-\langlembda A_i, \quad (\langlembda=K^lB_l).\langlebel{eqn:31}\end{equation}
Substituting these into (\ref{eqn:6}) and (\ref{eqn:30}) respectively give
\begin{equation}gin{align}
\nabla_iR=R-n\langlembda=0,
\langlebel{eqn:32}\end{align}
and so
\begin{equation}\nabla_i\langlembda=0.
\langlebel{eqn:32b}\end{equation}
By using (\ref{eqn:31})--(\ref{eqn:32}), we can simply (\ref{eqn:9}) as
\begin{equation}\langlebel{eqn:33}
A_iR_{kl}-A_kR_{il}=\langlembda A_ig_{kl}-\langlembda A_kg_{il}.
\end{equation}
Transvecting with $K^i$ and $K^l$ gives
\[
K^lR_{lk}=-\tau A_k+\langlembda K_k; \quad (\tau=K^iK^jR_{ij}).
\]
Transvecting (\ref{eqn:33}) with $K^i$ and applying the above equation gives
\begin{equation}\langlebel{eqn:34}
R_{lk}=\langlembda g_{lk}+\tau A_lA_k.
\end{equation}
Suppose
\begin{equation}\langlebel{eqn:tau} \tau\neq0. \end{equation}
Applying (\ref{eqn:32}) and (\ref{eqn:34}), we can simplfy (\ref{eqn:2}) as
\begin{equation}gin{align}\langlebel{eqn:35}
R^l{}_{ijk}=\frac\langlembda{n-1}\{g_{ki}\delta^l_j-g_{ji}\delta^l_k\}
+\frac\tau{n-2}\{A_kA_i\delta^l_j-A_jA_i\delta^l_k+g_{ki}A^lA_j-g_{ji}A^lA_k\}.
\end{align}
On the other hand, by substituting (\ref{eqn:31}) and (\ref{eqn:34}) into (\ref{eqn:1}) give
\begin{equation}\langlebel{eqn:36}
\nabla_iR_{jk}=\tau A_iA_jA_k.
\end{equation}
It follows from (\ref{eqn:2}), (\ref{eqn:32})--(\ref{eqn:32b}) and (\ref{eqn:34}) that
\[
\nabla_iR_{hj}=\nabla_hR_{ij}=\nabla_h(\tau A_iA_j).
\]
Covariantly differentiating (\ref{eqn:36}) with respect to $h$ gives
\[
\nabla_h\nabla_iR_{jk}=\nabla_h(\tau A_iA_j)A_k+\tau A_iA_j\nabla_hA_k
=\nabla_hR_{ij}A_k+\tau A_iA_j\nabla_hA_k,
\]
and so
\begin{equation}gin{align}
\tau A_jA_i\nabla_hA_k-\tau A_jA_h\nabla_iA_k
=&
(\nabla_h\nabla_i-\nabla_i\nabla_h)R_{jk}
=-R^l{}_{jhi}R_{lk}-R^l{}_{khi}R_{lj} \notag
\end{align}
Next, by applying (\ref{eqn:34}), the properties $R_{kjhi}+R_{jkhi}=0$ and (\ref{eqn:tau}) we have
\begin{equation}gin{align*}
A_jA_i\nabla_hA_k-A_jA_h\nabla_iA_k=&-R^l{}_{jhi}A_lA_k + R^l{}_{khi}A_lA_j.
\end{align*}
Furthermore, by using (\ref{eqn:35}) we obtain
\begin{equation}gin{align}\langlebel{eqn:37}
A_jA_i\Big\{\nabla_hA_k-\frac\langlembda{n-1}g_{hk}\Big\}-A_jA_h\Big\{\nabla_iA_k-&\frac\langlembda{n-1}g_{ik}\Big\} \notag\\
=&\frac\langlembda{n-1}\{-g_{ij}A_h+g_{hj}A_i\}A_k.
\end{align}
Transvecting with $K^i$ and $K^j$ we obtain
\begin{equation}gin{align*}
\nabla_hA_k-\frac\langlembda{n-1}g_{hk}=-A_h\Omega_k-\frac\langlembda{n-1}K_hA_k; \quad
\left(\Omega_k=K^i\nabla_iA_k-\frac\langlembda{n-1}K_k\right).
\end{align*}
Substituting this into (\ref{eqn:37}) gives
\begin{equation}\notag
\frac\langlembda{n-1}\{-A_jA_iK_h+A_jA_hK_i\}A_k=\frac\langlembda{n-1}\{-g_{ij}A_h+g_{hj}A_i\}A_k.
\end{equation}
Since $\langlembda\neq0$, it is descended to
\[
-A_jA_iK_h+A_jA_hK_i=-g_{ij}A_h+g_{hj}A_i.
\]
Transvecting with $K^h$ and $g^{ij}$ we obtain $1=n-1$. This is impossible. Hence
(\ref{eqn:tau}) is false or we must have $\tau=0$. It follows from
(\ref{eqn:34}) that it is also an Einstein space in this case.
Hence we conclude that $M$ is an Einstein space in both cases.
As a conformaly flat Einstein space is of constant sectional curvature and the manifold is non-flat,
We conclude that a conformally flat generalized Ricci recurrent spacetime is either a de Sitter space or an anti-de Sitter space.
\begin{equation}gin{thebibliography}{99}
\bibitem{avik-2} A. De, T. H. Loo, R. Solanki and P. K. Sahoo,
A conformally flat generalized Ricci recurrent spacetime in $f(R)$ gravity,
{\it Phys. Scripta}, {\bf 96}, 085001 (2021).
\bibitem{frg} A. De, T. H. Loo, R. Solanki, P.K. Sahoo, How a conformally flat (GR)4 impacts Gauss-Bonnet gravity? arXiv:2107.02889.
\bibitem{de}U. C. De, N. Guha and D. Kamilya, On generalized Ricci-recurrent manifolds,
\textit{Tensor (N.S.)}, \textbf{56}, 312--317 (1995).
\bibitem{pinaki} C. Dey, On Generalized Ricci Recurrent Spacetimes,
{\it Kyungpook Math. J.}, {\bf 60}, 571--584 (2020).
\bibitem{avik} S. Mallick, A. De and U. C. De, On Generalized Ricci Recurrent Manifolds with Applications To Relativity, {\it Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.}, {\bf83}, 143--152 (2013).
\bibitem{neil} B. O'Neill, The Geometry of Kerr Black holes, A K Peters., Wellesly, Massasuchetts, (1995).
\bibitem{patterson}E. M. Patterson, Some theorems on Ricci-recurrent spaces,
\textit{J. London Math. Soc.}, \textbf{27}, 287--295 (1952).
\end{thebibliography}
\end{document} | math | 14,157 |
\begin{document}
\address{Huaiyu Jian: Department of Mathematics, Tsinghua University, Beijing 100084, China.}
\address{You Li:Department of Mathematics, Tsinghua University, Beijing 100084, China.}
\address{Xushan Tu:Department of Mathematics, Tsinghua University, Beijing 100084, China.}
\email{[email protected]; [email protected]; [email protected]. }
\maketitle
\baselineskip=15.8pt
\partial arskip=3pt
\centerline {\bf Huaiyu Jian \ \ \ \ You Li \ \ \ \ Xushan Tu}
\centerline {Department of Mathematics, Tsinghua University}
\centerline {Beijing 100084, China}
\vskip20pt
\noindent {\bf Abstract}:
In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of
a class of Monge-Amp\`ere type equations which may be degenerate and singular on the boundary of convex domains.
We will establish a relation of the H$\ddot{o}$lder exponent for the solutions with the convexity for the domains.
\vskip20pt
\noindent{\bf Key Words:} existence, uniqueness, global regularity, degenerate, singular, Monge-Amp\`ere equation
\vskip20pt
\noindent {\bf AMS Mathematics Subject Classification}: 35J60, 35J96, 53A15.
\vskip20pt
\noindent {\bf Running head}:
Degenerate And Singular Monge-Amp\`ere Equations
\vskip20pt
\baselineskip=15.8pt
\partial arskip=3pt
\centerline {\bf On A Class of Degenerate And Singular Monge-Amp\`ere Equations }
\vskip10pt
\centerline { Huaiyu Jian\ \ \ \ You Li\ \ \ \ Xushan Tu}
\maketitle
\baselineskip=15.8pt
\partial arskip=3.0pt
\section {Introduction}
In this paper we study the Monge-Amp\`ere type equation
\begin{equation}
\begin{split}\label{1.1}
\det D^2 u&= F(x, u)\ \ \text{in}\ \Omega,\\
u &=0\ \ \text{on}\ \partial om,
\end{split}
\end{equation}
where $\Omega$ is a bounded convex domain in $R^n$ $(n\geq 2)$, and $F$
satisfies the following (1.2)-(1.3):
\begin{equation}\label{1.2}
\begin{split} F(x, t)\sqrt{-1}n C (\Omega\times(-\sqrt{-1}nfty,0)) \text{ is
non-decreasing in}\ t \text{ for any}\ x\sqrt{-1}n \Omega;
\end{split}
\end{equation}
\begin{equation}\label{1.3}
\begin{split}
&\text{there are constants }\ A>0,\ \alpha\geq0,\ \beta\geq n+1 \ \text{ such that} \\
&0<F(x, t )\leq A d_{x}^{\beta-n-1}|t|^{-\alpha} \ \ \ \forall (x,t) \sqrt{-1}n \Omega\times(-\sqrt{-1}nfty,0),\\
\end{split}
\end{equation}
where $d_{x}=dist(x, \partial om)$. Obviously, this problem is singular and degenerate at the boundary of the domain.
The particular case of problem (1.1) includes a few geometric problems. When $F= |t|^{-(n+2)}$ and $u$ is a solution to problem (1.1),
then the Legendre transform of $u$ is a complete affine hyperbolic sphere \cite {[Ca2], [CY], [CY2], [JL], [JLW]}, and
$(-u)^{-1}\sum u_{x_ix_j} dx_idx_j$ gives the Hilbert metric (Poincare metric) in the convex domain $\Omega$ \cite {[LN]}. When $F =f(x)|t|^{-p},$
problem (1.1) may be obtained from $L_p$-Minkowski problem \cite{[Lut]} and the Minkowski problem in centro-affine geometry\cite {[CW], [JLZ]}. Also see p.440-441 in
\cite{[JW]}. Generally, problem (1.1) can be applied to construct non-homogeneous complete Einstein-K$\ddot{a}$hler metrics on a tubular domain \cite {[CY], [CY1]}.
Cheng and Yau in \cite{[CY]} proved that
if $\Omega$ is a strictly convex $C^2$-domain and $F\sqrt{-1}n C^{k}$ ($k\geq3$) satisfies (1.2)-(1.3),
then problem (1.1) admits an unique convex generalized solution $u\sqrt{-1}n C(\bar \Omega)$. Moreover,
$u\sqrt{-1}n C^{k+1, \varepsilon}(\Omega)\bigcap C^{\gamma}(\bar \Omega)$ for any $\varepsilon \sqrt{-1}n ( 0 , 1)$ and some
$\gamma=C(\beta, \alpha, A, n, \partial artial\Omega)\sqrt{-1}n (0, 1)$. We should emphasize that their methods need the strict convexity and the smoothness of $\Omega$, and the differentiability of $F$.
In this paper we find that the global H$\ddot{o}$lder regularity for problem (1.1) is independent of the smoothness of $\Omega$ and $F$, and the H$\ddot{o}$lder exponent depends only on the convexity of the domain. As a result, we can remove the smoothness of $\Omega$ as well as the differentiability of $F$ in \cite{[CY]}. Moreover, using the concept of $(a, \eta)$ type introduced in \cite{[JL]} to describe the convexity of the domain, we obtain a relation of the H$\ddot{o}$lder exponent for $u$ with the convexity for $\Omegaega$.
We have noticed that there are many papers on global regularity for equations of Monge-Amp\`ere type. See, for example, \cite{[CNS], [F], [GT],[LS], [Sa], [TW], [U1]} and the references therein. But, generally speaking, those results require that the domain $\Omegaega$ should be strictly convex and $\partial artial \Omegaega \sqrt{-1}n C^{1,1} $.
Our first result is stated as the following
\begin{Theorem}\label{1.1}
Supposed that $\Omega$ is a bounded convex domain in $R^n$ and $F(x, t)$ satisfies (1.2)-(1.3). Let
\begin{equation}\label{1.4}
\gamma_1:=\left\{\begin{array}{cc}
\frac{\beta-n+1}{n+\alpha}, & {\rm if } \beta<\alpha+2n-1,\\
\text{any number in} (0, 1), & {\rm if } \beta\geq \alpha+2n-1.
\end{array}
\right.
\end{equation}
Then problem (1.1) admits an unique convex generalized solution $u\sqrt{-1}n C^{\gamma_1}(\overline{\Omega})$.
Furthermore, $u\sqrt{-1}n C^{2,\gamma_1}(\Omega)$ if $F(x, t)\sqrt{-1}n C^{0,1}(\Omega\times(-\sqrt{-1}nfty,0))$.
\end{Theorem}
Here a generalized solution means the well-known Alexandrov solution. See, for example, \cite{[F], [G], [TW1]} for the details.
To improve the regularity for the solution obtained in Theorem 1.1, we use the $(a, \eta)$ type in \cite{[JL]} to describe the convexity
of $\Omega$. From now on, we denote
$$x=(x_1, x_2, \dots , x_n)=(x', x_n), \ \
x'=(x_{1},...,x_{n-1})$$ and
$$ |x'|=\sqrt{x_{1}^{2}+...+x_{n-1}^{2}}.$$
\noindent{\bf Definition 1.1}. {\sl Supposed that $\Omega$ is a bounded convex domain in $R^n$, and $x_0\sqrt{-1}n \partial om$.
$x_{0}$ is called to be $(a, \eta)$ type if
there are numbers $a\sqrt{-1}n[1,+\sqrt{-1}nfty)$ and $\eta>0$, after translation and rotation transforms, we have
$$x_{0}=0 \ \ \text{and} \ \ \Omega\subseteq\{x\sqrt{-1}n R^{n}|x_{n}\geq\eta|x'|^{a}\}.$$
$\Omega$ is called $(a, \eta)$ type domain if every point of $\partial artial \Omega$ is $(a, \eta)$ type.}
\noindent{\bf Remark 1.1}. The convexity requires that the number $a$ should be no less than 1. The less is $a$, the more convex is the domain.
There is no $(a, \eta)$ type domain for $a\sqrt{-1}n [1,2)$, although part of
$\partial artial \Omega$ may be $(a, \eta)$ type point for $a\sqrt{-1}n [1,2)$.
\noindent{\bf Definition 1.2}. {\sl We say that a domain
$\Omega$ in $R^n$ satisfies exterior (or interior) sphere condition with radius $R$ if for each $x_{0}\sqrt{-1}n \partial om$, there
is a $B_R(y_0) \supseteq \Omega $ (or $B_R(y_0) \subseteq \Omega $, respectively) such that $\partial artial B_R(y_0) \bigcap \partial om\ni x_0$. }
In \cite{[JL]}, we have proved that $(2, \eta)$ type domain is equivalent to the domain satisfies exterior sphere condition.
The following two theorems show the relation of the H$\ddot{o}$lder exponent for $u$ on $\bar \Omega$ with the convexity for $\Omegaega$.
\begin{Theorem} \label {1.2} Supposed that $\Omega$ is (a, $\eta$) type domain in $R^n$ with $a\sqrt{-1}n (2,+\sqrt{-1}nfty)$, and $F $ satisfies (1.2)-(1.3). Let Let
\begin{equation}\label{1.5}
\gamma_2:=\left\{\begin{array}{cc}
\frac{\beta-n+1}{n+\alpha}+\frac{2n-2}{a(n+\alpha)}, & {\rm if } \beta<\alpha+2n-1-\frac{2n-2}{a},\\
\text{any number in} (0, 1), & {\rm if } \beta\geq \alpha+2n-1-\frac{2n-2}{a}.
\end{array}
\right.
\end{equation}
Then the convex generalized solution to problem (1.1)
\begin{equation} \label{1.6} u\sqrt{-1}n C^{ \gamma_2 } (\overline{\Omega}).\end{equation}
Furthermore $u\sqrt{-1}n C^{2, \gamma_2 }(\Omega)$ if $F(x, t)\sqrt{-1}n C^{0,1}(\Omega\times(-\sqrt{-1}nfty,0))$.
\end{Theorem}
\begin{Theorem} \label {1.3} Let $\Omega$ be a bounded convex domain in $R^n$ and $u$ be a convex generalized solution to problem (1.1).
(i) Suppose that $\Omega$ satisfies exterior sphere condition and $F$ satisfies (1.2)-(1.3). Let
\begin{equation}\label{1.7}
\gamma_3:=\left\{\begin{array}{cc}
\frac{\beta}{n+\alpha}, & {\rm if } \beta<\alpha+n,\\
\text{any number in} (0, 1), & {\rm if} \alpha+n\leq \beta< \alpha+n+1,\\
1, & {\rm if} \beta\geq \alpha+n+1.
\end{array}
\right.
\end{equation}
Then
\begin{equation} \label{1.8}
u\sqrt{-1}n C^{ \gamma_3 } (\overline{\Omega}).\end{equation}
Furthermore $u\sqrt{-1}n C^{2, \gamma_3}(\Omega)$ if $ F(x, t)\sqrt{-1}n C^{0,1}(\Omega\times(-\sqrt{-1}nfty,0)).$
(ii) If $\Omega$ satisfies interior sphere condition with radius $R$ and $F$ satisfies (1.2) and
\begin{equation} \label{1.9}
A d_{x}^{\beta-n-1}|t|^{-\alpha} \leq F(x, t ) , \ \forall (x,t)\sqrt{-1}n \Omega\times(-\sqrt{-1}nfty,0)
\end{equation}
for some constants $A>0$,
then
\begin{equation} \label {1.10} |u(y)|\geq C (d_y)^{\gamma_4}, \ \ \forall y\sqrt{-1}n \Omega \end{equation}
for some constant $C=C(\beta, \alpha, A, n, R)>0$, where
\begin{equation}\label{1.11} \gamma_4:=\frac{\beta}{n+\alpha}\sqrt{-1}n(0, 1),
\end{equation}
\end{Theorem}
\noindent{\bf Remark 1.2}. The H$\ddot{o}$lder regularity result of Theorem 1.1 can be viewed as the limit case of Theorem 1.2 as $a\to \sqrt{-1}nfty$. Theorem 1.3 (i) shows that Theorem 1.2 is true for $a=2$, since a $(2, \eta)$ type domain
is equivalent to that the domain satisfies exterior sphere condition.
In the following Sections 2, 3, and 4, we will prove Theorems 1.1, 1.2, and 1.3, respectively.
\section {Proof of Theorem 1.1}
We start at a primary result which is useful to proving that a convex function in $\Omega$ is H$\ddot{o}$lder continuous in
$\bar \Omega$.
\begin{Lemma}\label{2.1}
Let $\Omega$ be a bounded convex domain and $u\sqrt{-1}n C(\overline{\Omega})$ be a convex function in $\Omega$ with $u|_{\partial om}=0$. If there are $\gamma\sqrt{-1}n(0,1]$ and $M>0$
such that
\begin{equation} \label {2.1} |u(x)|\leq M{d_{x}}^{\gamma},\ \ \forall x\sqrt{-1}n \Omega ,\end{equation}
then $u\sqrt{-1}n C^{\gamma}(\overline{\Omega})$ and
$$|u|_{C^{\gamma}}(\overline{\Omega})\leq M\{1+[diam(\Omega)]^{\gamma}\} .$$
\end{Lemma}
\begin{proof} This was proved in \cite{[JL]}. Here we copy the arguments for the convenience.
For any two point $x_{1}$, $x_{2}\sqrt{-1}n \Omega$, consider the line determined by $x_{1}$ and $x_{2}$.
The line will intersect $\partial om$ at two points $y_{1}$ and $y_{2}$.
Without loss generality we assume the four points are $y_{1}$, $x_{1}$, $x_{2}$, $y_{2}$ in order.
By restricted onto the line, $u$ is one dimension convex function. By the monotonic proposition of convex functions, we have
$$|u(x_{2})-u(x_{1})|\leq \max \{|u(y_{1}+(x_{2}-x_{1}))-u(y_{1})|, \ |u(y_{2})-u(y_{2}-(x_{2}-x_{1}))|\}.$$
Moreover, since $y_{1}\sqrt{-1}n\partial om$, by the assumption (2.1) we have
\begin{equation*}
\begin{split}
|u(y_{1}+(x_{2}-x_{1}))-u(y_{1})|
=&|u(y_{1}+(x_{2}-x_{1}))|\\
\leq& M \{dist(y_{1}+x_{2}-x_{1}, \partial om)\}^{\gamma}\\
\leq& M |x_{2}-x_{1}|^{\gamma}.\\
\end{split}
\end{equation*}
Similarly,
$$|u(y_{2})-u(y_{2}-(x_{2}-x_{1}))|\leq M |x_{2}-x_{1}|^{\gamma}.$$
The above three inequalities, together with (2.1), implies the desired result.
\end{proof}
To prove Theorem 1.1, we need an
a priori estimate result as follows, which holds without strictly convexity of $\Omega$ or any smoothness of $\Omega$ and of $F$.
\begin{Lemma} \label{2.2}
Supposed that $\Omega$ is a bounded convex domain in $\Bbb R^n$ and $F(x, t)$ satisfies (1.2) and (1.3). If
$u$ is a convex generalized solution to problem (1.1),
then $u\sqrt{-1}n C^{\gamma_1}(\overline{\Omega})$ and
\begin{equation} \label {2.2} |u|_{C^{\gamma_1}(\overline{\Omega})}
\leq C(\alpha,\ \beta,\ A,\ diam(\Omega),\ n),\end{equation}
\end{Lemma}
where $\gamma_1$ is given by (1.4).
\begin{proof} First, we may assume
\begin{equation} \label{2.3} \beta < \alpha+2n-1.\end{equation}
Since for the case $\beta\geq \alpha+2n-1$, we take a $\hat \beta<\alpha+2n-1$ such that
$\frac{\hat \beta-n+1}{n+\alpha}$ can be any number in $(0, 1)$. (Note $n \geq 2$). Obviously,
(1.3) still holds with $\beta$ replaced by $\hat \beta$. Hence, this case is reduced to the case (2.3).
Next, we assume for the time being that
\begin{equation} \label{2.4} 0\sqrt{-1}n \overline{\Omega}\subseteq R_{+}^{n}.\end{equation}
Then we are going to construct a sub-solution to problem (1.1).
For brevity, write $l=diam(\Omega)$. Set
$$W=-Mx_{n}^{\gamma}\cdot\sqrt{N^{2}l^{2}-r^{2}}$$
where $r=\sqrt{x_{1}^{2}+...+x_{n-1}^{2}}$. We will choose positive constants $\gamma$, $M$, $N$ such that $W$ is an sub-solution to problem (1.1)
under the assumptions (2.3) and (2.4).
For $i, j\sqrt{-1}n\{1, 2, ..., n-1\}$, write $W_{i}=\frac{\partial artial W}{\partial artial x_i}, W_{ij}=\frac{\partial artial^2 W}{\partial artial x_i\partial artial x_j}$. Then we have
\begin{equation*}
\begin{split}
W_{i}&=Mx_{n}^{\gamma}\cdot \frac{x_{i}}{\sqrt{N^{2}l^{2}-r^{2}}},\\
W_{ij}&=Mx_{n}^{\gamma}\cdot \frac{1}{\sqrt{N^{2}l^{2}-r^{2}}}(\delta_{ij}+\frac{x_{i}x_{j}}{N^{2}l^{2}-r^{2}}),\\
W_{n}&=-M\gamma x_{n}^{\gamma-1}\cdot\sqrt{N^{2}l^{2}-r^{2}},\\
W_{in}&=M\gamma x_{n}^{\gamma-1}\cdot \frac{x_{i}}{\sqrt{N^{2}l^{2}-r^{2}}},\\
W_{nn}&=M\gamma(1-\gamma)x_{n}^{\gamma-2}\cdot\sqrt{N^{2}l^{2}-r^{2}}.
\end{split}
\end{equation*}
Denote
$$D^{2}W:=\begin{pmatrix} G & \xi \\ \xi^{T} & W_{nn} \end{pmatrix}$$
where $\xi^{T}=(W_{n1}, ..., W_{n (n-1)})$, and $G$ is the $(n-1)$-order matrix. Then
$$det D^{2}W=det G\cdot(W_{nn}-\xi^{T}G^{-1}\xi).$$
Since all the eigenvalues of $G$ are
$$Mx_{n}^{\gamma}\frac{1}{\sqrt{N^{2}l^{2}-r^{2}}},..., \ \ Mx_{n}^{\gamma}\frac{1}{\sqrt{N^{2}l^{2}-r^{2}}},\ \ Mx_{n}^{\gamma}\frac{N^{2}l^{2}}{(N^{2}l^{2}-r^{2})\sqrt{N^{2}l^{2}-r^{2}}},$$
$$det G=M^{n-1}N^{2}l^{2}x_{n}^{(n-1)\gamma}\cdot(\frac{1}{\sqrt{N^{2}l^{2}-r^{2}}})^{n+1}.$$
It is direct to verify that
$$ G\xi= \frac {N^{2}l^{2}Mx_{n}^{\gamma}}{(N^{2}l^{2}-r^{2})^{\frac{3}{2}}}\xi.$$
It follows that
\begin{equation*}
\begin{split}
\xi^{T}G^{-1}\xi &= \frac{(N^{2}l^{2}-r^{2})^{\frac{3}{2}}}{N^{2}l^{2}Mx_{n}^{\gamma}}|\xi|^2\\
&=\frac{M\gamma^{2}}{N^{2}l^{2}} x_{n}^{\gamma-2} r^{2} \sqrt{N^{2}l^{2}-r^{2}} .
\end{split}
\end{equation*}
Hence, we obtain that
\begin{equation} \label {2.5}
\begin{split}
det D^{2}W &=det G (W_{nn}-\xi^{T}G^{-1}\xi) \\
&=M^{n-1}N^{2}l^{2}x_{n}^{(n-1)\gamma} (\frac{1}{\sqrt{N^{2}l^{2}-r^{2}}})^{n+1} M\gamma x_{n}^{\gamma-2}\sqrt{N^{2}l^{2}-r^{2}} \\
& \ \ \cdot [1-(1+\frac{r^{2}}{N^{2}l^{2}})\gamma]\\
&=M^{n}N^{2}l^{2}\gamma x_{n}^{n\gamma-2} (\frac{1}{\sqrt{N^{2}l^{2}-r^{2}}})^{n} [1-(1+\frac{r^{2}}{N^{2}l^{2}})\gamma].
\end{split}
\end{equation}
We want to prove
\begin{equation} \label{2.6} detD^{2}W\geq F(x, W) \ \ \text{in }\ \ \Omega.\end{equation}
Since (1.3) and (2.4) implies that
$$F(x, W)\leq A d_{x}^{\beta-n-1}|W|^{-\alpha}\leq A x_{n}^{\beta-n-1}|W|^{-\alpha},$$
we see that (2.6) can be deduced from
\begin{equation} \label{2.7}
det D^{2} W\geq A x_{n}^{\beta-n-1}|W|^{-\alpha} \ \ \text{in }\ \ \Omega ,\end{equation}
which is equivalent to
\begin{equation} \label {2.8}
det D^{2} W\cdot \frac{1}{A} x_{n}^{n+1-\beta}|W|^{\alpha}\geq1 \ \ \text{in }\ \ \Omega.\end{equation}
By (2.5), (2.8) is nothing but
\begin{equation}\label{2.9}
\frac{1}{A}M^{n+\alpha}N^{2}l^{2}\gamma x_{n}^{(n+\alpha)\gamma-(\beta- n+1)}[1-(1+\frac{r^{2}}{N^{2}l^{2}})\gamma]\cdot(\sqrt{N^{2}l^{2}-r^{2}}\ )^{\alpha-n}\geq 1
\ \ \text{in }\ \ \Omega.\end{equation}
Now we choose $\gamma=\frac{\beta- n+1}{n+\alpha}$ such that
$$(n+\alpha)\gamma-(\beta- n+1)=0.$$
Since $\gamma \sqrt{-1}n(0, 1)$ by (2.3) and
$r=|x'|\leq diam(\Omega)=l$ in $\Omega$, we first take $N=C(\gamma)$ large enough such that
$$1-(1+\frac{r^{2}}{N^{2}l^{2}})\gamma>0.$$
Noting $N^{2}l^{2}-r^{2}\sqrt{-1}n [(N^{2}-1)l^{2},\ N^{2}l^{2}]$, \ we
then take $M=C(A, \alpha, \gamma, N, n, l)$ large enough such that
$$\frac{1}{A}M^{n+\alpha}N^{2}l^{2}\gamma x_{n}^{(n+\alpha)\gamma-(\beta- n+1)}[1-(1+\frac{r^{2}}{N^{2}l^{2}})\gamma]\cdot(\sqrt{N^{2}l^{2}-r^{2}}\ )^{\alpha-n}\geq 1,$$
we obtain (2.9) and thus have proved (2.6).
Finally, for any point $y\sqrt{-1}n \Omega$, letting $z\sqrt{-1}n \partial om$ be the nearest boundary point to $y$,
by some translations and rotations, we assume $z=0$, $\Omega\subseteq R_{+}^{n}$ and the line $yz$ is the $x_{n}-axis$.
This is to say that (2.4) is satisfied. Therefore we have (2.6). Obviously, $W\leq 0$ on $\overline \Omega$.
Hence, $W$ is a sub-solution to problem (1.1).
By comparison principle for generalized solutions (see \cite{[F],[G],[TW1]} for example), we have
$$|u(y)|\leq|W(y)|\leq MNl y_{n}^{\frac{\beta- n+1}{n+\alpha}}=MNld_{y}^{\frac{\beta- n+1}{n+\alpha}},$$
which, together with Lemma 2.1, implies the desired result (2.2).
Note that we have used the fact that problem (1.1) is invariant under translation and rotation transforms, since $det D^2u$ is invariant and $F(x,u)$ is transformed to the one
satisfying the same condition as $F$. This fact will be again used a few times in the following.
\end{proof}
\vskip 0.5cm
{\bf Proof of Theorem 1.1.}\ We prove the theorem by three steps.
{\bf Step 1.}\ Suppose that $\Omega$ is bounded convex but $F(x, t)\sqrt{-1}n C^{k}(\Omega\times(-\sqrt{-1}nfty,0))$ $(k\geq3)$ satisfies (1.2) and (1.3).
We choose a sequence of bounded and strictly convex domains $\{\Omega_{i}\}$ such that
\begin{equation} \label {2.10} \Omega_{i}\sqrt{-1}n C^{2}\ \ \text{and} \ \ \Omega_{i}\subseteq\Omega_{i+1}, i=1, 2, \cdots, \ \ \bigcup_{i=1}^{\sqrt{-1}nfty}\Omega_{i}=\Omega. \end{equation}
Then by Theorem 5 in \cite{[CY]}, there exists a convex generalized solution $u_{i}$ to problem (1.1) in the domain $\Omega_{i}$ for each $i$.
We assume $u_{i}(x)=0$ for all $x\sqrt{-1}n R^{n}\setminus\Omega_{i}$.
By Lemma 2.2, We have the uniform estimations
\begin{equation} \label{2.11}|u_{i}|_{C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega})}=|u_{i}|_{C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega_{i}})}\leq C(\alpha,\ \beta,\ A,\ diam(\Omega),\ n),
\end{equation}
which implies that there is a subsequence, still denoted by itself, convergent to a $u$ in the space $C(\overline{\Omega})$. Moreover, by (2.11) again, we have
$$|u|_{C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega})}\leq C(\alpha,\ \beta,\ A,\ diam(\Omega),\ n).$$
By the well-known convergence result for convex generalized solutions (see Lemma 1.6.1 in \cite{[G]} for example), we see that $u$ is a convex generalized solution to problem (1.1).
{\bf Step 2.}\ Drop the restriction on the smoothness for $F$.
Suppose $F_{j}\sqrt{-1}n C^{k}(\Omega\times(-\sqrt{-1}nfty,0))$ $(k\geq3)$ satisfy the same assumption as $F$ in the Step 1
and $F_{j}$ locally uniform convergence to $F$ in as $j\to \sqrt{-1}nfty$. (For example we can take $F_{j}=F*\eta_{\varepsilon_{j}}$, $\varepsilon_{j}$ convergence to 0 as $j$ tend to $+\sqrt{-1}nfty$.)
Then by the result of Step 1, for each $j$, there exists a convex generalized solution $u_{j}\sqrt{-1}n C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega})$ to problem (1.1) with $F$ replaced by $F_{j}$. Moreover, we
have
\begin{equation} \label {2.12} |u_{j}|_{C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega})}\leq C(\alpha,\ \beta,\ A,\ diam(\Omega),\ n)\end{equation} for all $j$. Using this estimate, Lemma 1.6.1 in \cite{[G]}, and the same argument as in Step 1,
we obtain a solution $u$ to problem (1.1), which is the limit of a subsequence of $u_j$ in the space space $C(\overline{\Omega})$. Furthermore, we have
$u\sqrt{-1}n C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega})$ by (2.12). The uniqueness for (1.1) is directly from the comparison principle (see \cite{[F],[G],[TW1]} for example).
{\bf Step 3.} \ We are going to prove $u\sqrt{-1}n C^{2,\ \frac{\beta-n+1}{n+\alpha}}(\Omega)$ if $F(x, t)\sqrt{-1}n C^{0,1}(\Omega\times(-\sqrt{-1}nfty,0))$.
It is enough to prove
\begin{equation} \label{2.13} u\sqrt{-1}n C^{2,\ \frac{\beta-n+1}{n+\alpha}}(\overline{\Omega_1})\end{equation} for any convex $\Omega_1\subset\subset\Omega$.
Taking a convex $\Omega'$ such that $\Omega_1\subset\subset\Omega'\subset\subset\Omega$, if there exists $z\sqrt{-1}n\overline{\Omega'}\subset\Omega$ such that $u(z)=0$, then $u\equiv0$ in $\Omega$ by convexity and the boundary condition $u|_{\partial om}=0$. Hence
we obtain (2.13). Otherwise, $u(x)<0$ \ for all $x\sqrt{-1}n\overline{\Omega'}$. Then
$F(x, u(x))\sqrt{-1}n C^{\frac{\beta-n+1}{n+\alpha}}(\overline{\Omega'})$ and is positive on $\overline{\Omega'}$. By the Caffarelli's local $C^{2,\alpha}$ regularity in \cite{[Caf]} (also see \cite{[JW1]} for another proof),
we obtain (2.13), too.
\vskip20pt
\section {Proof of Theorem 1.2}
In this section we establish the relation between the H$\ddot{o}$lder exponent and the convexity of the domain $\Omega$ and thus prove Theorem 1.2.
Assume that $\Omega$ is a $(a, \eta)$ type domain with $a\sqrt{-1}n (2, \sqrt{-1}nfty )$, $F$ satisfies (1.2)-(1.3), and $u$ is the unique solution to problem (1.1) as in Theorem 1.1.
To prove Theorem 1.2, it is sufficient to prove (1.6). See the Step 3 in the proof of Theorem 1.1.
As (2.3) we may assume
\begin{equation} \label {3.1} \beta <\alpha+2n-1-\frac{2n-2}{a}.\end{equation}
Hence, in the following we have
$$\gamma_2=\frac{\beta-n+1}{n+\alpha}+\frac{2n-2}{a(n+\alpha)}\sqrt{-1}n (0, 1).$$
By Lemma 2.1, (1.6) can be deduced from
\begin{equation} \label{3.2}
|u(y)|\leq C \ {d_{y}}^{ \gamma_2}, \ \ \forall y\sqrt{-1}n \Omega
\end{equation}
for some positive constant$C=C(a,n, \alpha, \eta, A, diam\Omega)$.
We are going to prove (3.2). For any $y\sqrt{-1}n \Omega$, we can find $z\sqrt{-1}n \partial om$, such that $|y-z|=d_{y}.$
Since the domain $\Omega$ is $(a,\eta)$ type and the problem (1.1) is invariant under translation and rotation transforms,
we may assume $z=0$, and take the line determined by $z$ and $y$
as the $x_{n}-axis$ such that
$$\Omega\subseteq\{x\sqrt{-1}n R^{n}|x_{n}\geq\eta|x'|^{a}\}.$$
We will prove (3.2) by three steps.
{\bf Step 1.} Let
$$W(x_{1}, ..., x_{n})=W(r, x_n)=-[(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-x_{1}^{2}-...-x_{n-1}^{2}]^{\frac{1}{b}},$$
where $r=|x'|=\sqrt{x_1^2+\cdots, x_{n-1}^2}$, $b$ and $\varepsilon$ are positive constants to be determined.
We want to find a sufficient condition for which $W$ is a sub-solution to problem (1.1).
For $i, j\sqrt{-1}n\{1, 2, ..., n-1\}$, by direct computation we have
\begin{equation} \label {3.3}
\begin{split}
W_{i}=&W_{r}\frac{x_{i}}{r},\\
W{ij}=&\frac{W_{r}}{r}\delta_{ij}+(W_{rr}-\frac{W_{r}}{r})\frac{x_{i}}{r}\frac{x_{j}}{r},\\
W_{in}=&W_{rn}\frac{x_{i}}{r}.\\
\end{split}
\end{equation}
Let $$D^{2}W:=\begin{pmatrix} G & \xi \\ \xi^{T} & W_{nn} \end{pmatrix}$$
where $\xi^{T}=(W_{n1}, ..., W_{n (n-1)})$, and $G$ is the matrix of $n-1$ order all of which eigenvalues
are $$\frac{W_{r}}{r}, ..., \frac{W_{r}}{r}, W_{rr},$$ and one of which
eigenvector with respect to the eigenvalue $W_{rr}$
is $\xi$. As obtaining (2.5), we have
$$det D^{2}W=(\frac{W_{r}}{r})^{n-2}W_{rr}(W_{nn}-\frac{|W_{rn}|^{2}}{W_{rr}}).$$
Obviously, $W\leq 0$ on $\partial om$.
Therefore we conclude that {\sl $W$ is a sub-solution to problem (1.1)
if and only if
\begin{equation} \label{3.4} H[W]:=(\frac{W_{r}}{r})^{n-2}(W_{rr}W_{nn}-|W_{rn}|^{2})[F(x,W)]^{-1}\geq 1\ \ \text{in} \ \ \Omega .\end{equation}}
We use the expression of $W$ to compute
\begin{equation*}
\begin{split}
W_{r}&=\frac{2}{b}((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-1}\cdot r,\\
W_{n}&=-\frac{2}{ab}((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-1}\cdot
(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-1}\cdot\frac{1}{\varepsilon},\\
W_{rr}&=\frac{4}{b}(1-\frac{1}{b})((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-2}\cdot r^{2}
+\frac{2}{b}((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-1},\\
W_{nn} &=\frac{4(b-1)}{a^{2}b^{2}}((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-2}\cdot
(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\cdot(\frac{1}{\varepsilon})^{2} \\
& \ \ \ +\frac{2(a-2)}{a^{2}b}((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-1}\cdot
(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-2}\cdot(\frac{1}{\varepsilon})^{2},\\
W_{rn}&=\frac{4(1-b)}{ab^{2}}((\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2})^{\frac{1}{b}-2}\cdot
(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-1}\cdot r\cdot \frac{1}{\varepsilon}.
\end{split}
\end{equation*}
Using the expression of $W$ again we have
\begin{equation} \label{3.5} W_{r}=\frac{2}{b}|W|^{1-b}\cdot r,\end{equation}
\begin{equation*}
\begin{split}
W_{n}&=-\frac{2}{ab}|W|^{1-b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-1}\cdot\frac{1}{\varepsilon},\\
W_{rr}&=\frac{4(b-1)}{b^{2}}|W|^{1-2b}\cdot r^{2}+\frac{2}{b}|W|^{1-b},\\
W_{nn}&=\frac{4(b-1)}{a^{2}b^{2}}|W|^{1-2b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\cdot\frac{1}{\varepsilon^{2}}
+\frac{2(a-2)}{a^{2}b}|W|^{1-b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-2}\cdot\frac{1}{\varepsilon^{2}},\\
W_{rn}&=\frac{4(1-b)}{ab^{2}}|W|^{1-2b}(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-1}\cdot r\cdot\frac{1}{\varepsilon}.
\end{split}
\end{equation*}
Hence,
\begin{equation} \label{3.6}
\begin{split}
W_{rr}\cdot W_{nn}-(W_{rn})^{2}&=\frac{8(a-2)(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-2}\cdot r^{2}\cdot(\frac{1}{\varepsilon})^{2}\\
&+\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\cdot(\frac{1}{\varepsilon})^{2}\\
&+\frac{4(a-2)}{a^{2}b^{2}}|W|^{2-2b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-2}\cdot(\frac{1}{\varepsilon})^{2}\\
& :=I_{1}+I_{2}+I_{3}.
\end{split}
\end{equation}
To estimate $I_{1}$, $I_{2}$ and $I_{3}$, \ we will choose a small $\delta=C(a, \alpha, \beta, n)>0$. Now for this $\delta$, we choose
a small $ \varepsilon =C(\delta, a, \eta)>0$ such that
\begin{equation} \label {3.7}
\varepsilon (\frac{1}{\delta})^{\frac{a}{2}}\leq\eta .
\end{equation}
Then we have
\begin{equation}\label{3.8}
\Omega\subseteq\{x\sqrt{-1}n R^{n}|x_{n}\geq\eta|x'|^{a}\}
\subseteq \{ x\sqrt{-1}n R^{n}| \delta(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}\geq r^{2}\}.
\end{equation}
By (3.8) we have
\begin{equation} \label{3.9}
|W|^{b}=(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}-r^{2}\sqrt{-1}n[(1-\delta)(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}, \ (\frac{x_{n}}{\varepsilon})^{\frac{2}{a}}]\end{equation}
Since $a>2$, we have two case: $a\geq \frac{2\alpha+2}{\beta-n+1}$ and $a< \frac{2\alpha+2}{\beta-n+1}$ if $\frac{2\alpha+2}{\beta-n+1}>2$.\\
{\bf Step 2.} Assume that $\frac{2\alpha+2}{\beta-n+1}>2$ and $2<a< \frac{2\alpha+2}{\beta-n+1}$. We want to find $b>1$ and $ \varepsilon>0$ such that
(3.4) is satisfied, by which we will prove (3.2).
Since $a>2$ and $b>1$, $I_{1}, I_{2}$ and $ I_{3}$ in (3.6) are all positive.
$$W_{rr}\cdot W_{nn}-(W_{rn})^{2}\geq I_{2}=\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\cdot(\frac{1}{\varepsilon})^{2}.$$
Observe that $d_x\leq x_n$ in $\Omega$. Hence, by (1.3), (3.4) and (3.5) we obtain
\begin{equation*}
\begin{split}
H[W]&=(\frac{W_{r}}{r})^{n-2}(W_{rr}W_{nn}-|W_{rn}|^{2}) [F(x,W)]^{-1}\\
&\geq (\frac{2}{b})^{n-2}\cdot|W|^{(1-b)(n-2)}\cdot\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot
(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\cdot(\frac{1}{\varepsilon})^{2}\cdot
\frac{1}{A}d_{x}^{n+1-\beta}|W|^{\alpha}\\
&\geq(\frac{2}{b})^{n-2}\cdot|W|^{(1-b)(n-2)}\cdot\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot
(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\cdot(\frac{1}{\varepsilon})^{2}\cdot\frac{1}{A}x_{n}^{n+1-\beta}|W|^{\alpha}.
\end{split}
\end{equation*}
it follows from (3.9) that
\begin{equation*}
\begin{split}
&x_{n}\leq \varepsilon (\frac{1}{1-\delta})^{\frac{a}{2}}|W|^{\frac{ab}{2}} \ \ \ \text{in}\ \Omega,\\
&(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\geq [\ (\frac{1}{1-\delta})^{\frac{a}{2}}|W|^{\frac{ab}{2}}\ ]^{\frac{4}{a}-2},\\
&x_{n}^{n+1-\beta}\geq [\ \varepsilon (\frac{1}{1-\delta})^{\frac{a}{2}}|W|^{\frac{ab}{2}}\ ]^{n+1-\beta}
\end{split}
\end{equation*}
Therefore, we arrive at
\begin{equation*}
\begin{split}
H[W]\geq&(\frac{2}{b})^{n-2}\cdot|W|^{(1-b)(n-2)}\cdot\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot
(\frac{1}{1-\delta})^{2-a}|W|^{\frac{ab}{2}(\frac{4}{a}-2)}\\
& \cdot(\frac{1}{\varepsilon})^{2}\cdot\frac{1}{A}\varepsilon^{n+1-\beta} (\frac{1}{1-\delta})^{\frac{a}{2}(n+1-\beta)}|W|^{\frac{ab}{2}(n+1-\beta)}|W|^{\alpha}\\
=&(\frac{1}{\varepsilon})^{\beta-n+1}\frac{1}{A}(\frac{2}{b})^{n-2}\cdot\frac{8(b-1)}{a^{2}b^{3}}\cdot
(\frac{1}{1-\delta})^{2-a+\frac{a}{2}(n+1-\beta)}\\
& \cdot |W|^{(1-b)(n-2)+2-3b+\frac{ab}{2}(\frac{4}{a}-2)+\frac{ab}{2}(n+1-\beta)+\alpha}.\\
\end{split}
\end{equation*}
Now, we set
$$(1-b)(n-2)+2-3b+\frac{ab}{2}(\frac{4}{a}-2)+\frac{ab}{2}(n+1-\beta)+\alpha=0$$
which is equivalent to
$$b=\frac{2(n+\alpha)}{a(\beta-n+1)+2n-2}.$$ Since $a\sqrt{-1}n(2,\frac{2\alpha+2}{\beta-n+1}),$ we see that $b>1$ by (3.1).
Observing that $\beta-n+1> 0$, we can
choose $\varepsilon=C(a, \eta, A, \alpha, \beta, n)>0$ small enough again,
such that $H[W]\geq1$. This proves (3.4), which is to say that
$W$ is a sub-solution to problem (1.1). By
comparison principle, we have
$$|u(x)|\leq |W(x)|, \ \ \forall x\sqrt{-1}n \Omega .$$
Restricting this inequality onto the $x_{n}$ axis,
we obtain $$|u(y)|\leq(\frac{y_n}{\varepsilon})^{\frac{2}{ab}}=({\frac{d_{y}}{\varepsilon}})^{\frac{\beta-n+1}{n+\alpha}+\frac{2n-2}{a(n+\alpha)}},$$
which is (3.2) exactly.
{\bf Step 3. } Assume that $a\geq \frac{2\alpha+2}{\beta-n+1}$. Note that $a>2$ by the assumption of the theorem.
We will find $b\sqrt{-1}n (0,1)$ and $\varepsilon>0$ such that the function $W$
is a sub-solution to problem (1.1), and thus prove (3.2).
By (3.9) we have
$$I_{1}\geq\frac{8(a-2)(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-2}\cdot\delta(\frac{x_{n}}
{\varepsilon})^{\frac{2}{a}}\cdot(\frac{1}{\varepsilon})^{2} =\delta(a-2)I_{2}.$$
Since $a>2$, $b\sqrt{-1}n (0, 1)$ and (3.9) yields
$$(\frac{x_{n}}{\varepsilon})^{\frac{4}{a}-2}\leq|W|^{b(2-a)},$$
we obtain
\begin{equation*}
\begin{split}
I_{1}+I_{2}&\geq(1+\delta(a-2))I_{2}\\
&\geq (1+\delta(a-2))\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-3b}\cdot|W|^{b(2-a)}\cdot(\frac{1}{\varepsilon})^{2}\\
&=(1+\delta(a-2))\frac{8(b-1)}{a^{2}b^{3}}|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2}.
\end{split}
\end{equation*}
Again by (3.9), we have
$$(\frac{x_{n}}{\varepsilon})^{\frac{2}{a}-2}\geq(\frac{1}{1-\delta})^{1-a}|W|^{b(1-a)}.$$
Hence, we have
\begin{equation*}
\begin{split}
I_{3}&\geq\frac{4(a-2)}{a^{2}b^{2}}|W|^{2-2b}\cdot(\frac{1}{1-\delta})^{1-a}|W|^{b(1-a)}\cdot(\frac{1}{\varepsilon})^{2}\\
&=\frac{4(a-2)}{a^{2}b^{2}}(\frac{1}{1-\delta})^{1-a}\cdot|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2}.
\end{split}
\end{equation*}
Therefore, we obtain
\begin{equation*}
\begin{split}
W_{rr}\cdot W_{nn}-(W_{rn})^{2}
&=I_{1}+I_{2}+I_{3}\\
&\geq [(1+\delta(a-2))\frac{8(b-1)}{a^{2}b^{3}}+\frac{4(a-2)}{a^{2}b^{2}}(\frac{1}{1-\delta})^{1-a}]|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2}\\
&:=\sigma(a, b, \delta)|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2},
\end{split}
\end{equation*}
where $$\sigma(a, b, \delta)=(1+\delta(a-2))\frac{8(b-1)}{a^{2}b^{3}}+\frac{4(a-2)}{a^{2}b^{2}}(\frac{1}{1-\delta})^{1-a}.$$
Using above estimates, together with (1.3) and (3.9) we have
\begin{equation*}
\begin{split}
H[W]&=(\frac{W_{r}}{r})^{n-2}(W_{rr}W_{nn}-|W_{rn}|^{2})(F(x, W))^{-1}\\
&\geq(\frac{2}{b})^{n-2}\cdot|W|^{(1-b)(n-2)}\cdot \sigma(a, b, \delta)|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2}\cdot (F(x, W))^{-1}\\
&\geq(\frac{2}{b})^{n-2}\cdot|W|^{(1-b)(n-2)}\cdot \sigma(a, b, \delta)|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2}\cdot \frac{1}{A}d_{x}^{n+1-\beta}|W|^{\alpha}\\
&\geq(\frac{2}{b})^{n-2}\cdot|W|^{(1-b)(n-2)}\cdot \sigma(a, b, \delta)|W|^{2-b-ab}\cdot(\frac{1}{\varepsilon})^{2}\cdot \frac{1}{A}x_{n}^{n+1-\beta}|W|^{\alpha}\\
&=(\frac{1}{\varepsilon})^{\beta-n+1}(\frac{2}{b})^{n-2}\frac{1}{A}\cdot\sigma(a, b, \delta)|W|^{2-b-ab}\cdot|W|^{(1-b)(n-2)}\cdot (\frac{x_{n}}{\varepsilon})^{n+1-\beta}|W|^{\alpha}\\
&\geq(\frac{1}{\varepsilon})^{\beta-n+1}(\frac{2}{b})^{n-2}\frac{1}{A}\cdot\sigma(a, b, \delta)|W|^{2-b-ab}\cdot|W|^{(1-b)(n-2)}\\
&\ \ \cdot (\frac{1}{1-\delta})^{\frac{a(n+1-\beta)}{2}}|W|^{\frac{ab(n+1-\beta)}{2}}\cdot |W|^{\alpha}\\
&=(\frac{1}{\varepsilon})^{\beta-n+1}(\frac{2}{b})^{n-2}\frac{1}{A}\cdot(\frac{1}{1-\delta})^{\frac{a(n+1-\beta)}{2}}\sigma(a, b, \delta)|W|^{2-b-ab+(1-b)(n-2)+\frac{ab(n+1-\beta)}{2}+\alpha} .
\end{split}
\end{equation*}
Now, we set
\begin{equation} \label {3.10} 2-b-ab+(1-b)(n-2)+\frac{ab(n+1-\beta)}{2}+\alpha=0,\end{equation}
which ie equivalent to
$$ b=\frac{2(n+\alpha)}{a(\beta-n+1)+2n-2}.$$
Since $a \geq \frac{2\alpha+2}{\beta-n+1} $, we see that $b\sqrt{-1}n (0, 1]$.
Of course, we also need
\begin{equation} \label {3.11} \sigma(a, b, \delta)=(1+\delta(a-2))\frac{8(b-1)}{a^{2}b^{3}}+\frac{4(a-2)}{a^{2}b^{2}}(\frac{1}{1-\delta})^{1-a}>0,\end{equation}
which is equivalent to
\begin{equation} \label{3.12}(a-2)(1-\delta)^{a-1}>(1+\delta(a-2))(\frac{2(1-b)}{b}).\end{equation}
Since $\gamma_2=\frac{\beta-n+1}{n+\alpha}+\frac{2n-2}{a(n+\alpha)}\sqrt{-1}n(0,1)$ by (3.1), we see that
$$a-2>\frac{a(\beta-n+1)+2n-2}{n+\alpha}-2 =(\frac{2(1-b)}{b}).$$ Using this and taking $\delta=C(a, \alpha, \beta, n)>0$ small enough,
we obtain (3.12) and thus (3.11).
Finally, choosing a positive $$\varepsilon=C(a, \eta, A, \alpha, \beta, b(a, \alpha, \beta, n), \delta(a, \alpha, \beta, n))=C(a, \eta, A, \alpha, \beta, n)$$
smaller if necessary, by (3.10) and (3.11) we obtain that $H[W]\geq1$ in $\Omega$, which implies $W$ is an sub-solution to problem (1.1) by (3.4). As in the end of Step 2, we have proved (3.2).
\vskip20pt
\section {Proof of Theorem 1.3}
As the proof of Theorem 1.2, the proof of (i) of Theorem 1.3 follows directly from
\begin{equation} \label{4.1}
|u(y)|\leq C \ {d_{y}}^{\gamma_3}, \ \ \forall y\sqrt{-1}n \Omega
\end{equation}
for some positive constant$C=C(a,n, \alpha, \eta, A, diam\Omega)$.
For any $y\sqrt{-1}n \Omega$, we can find $z\sqrt{-1}n \partial om$, such that $|y-z|=d_{y}.$ Since the domain $\Omega$ satisfies exterior sphere condition with
radius $R$ and the problem (1.1) is invariant under translation and rotation transforms,
we may assume
\begin{equation} \label{4.2} z=\mathbf{0}\sqrt{-1}n \partial artial\Omega\bigcap \partial artial B_R(y_0), \ \ \Omega\subseteq B_R(y_0).\end{equation}
Since $z=\mathbf{0}$\ satisfies $|y-z|=d_{y}$, the tangent plane of $\Omega$ at $z=\mathbf{0}$ is unique. And it is easy to check $y$ is on the line dertermined by $\mathbf{0}$ and $y_{0}$.
Hence $d_{y}=|y|=|y_{0}|-|y_{0}-y|=R-|y_{0}-y|$.\\
Consider the function
\begin{equation} \label{4.3} W(x)=-M (R^2-|x-y_0|^2)^b=-M (R^2-r^2)^b,\end{equation}
where $r=|x-y_0|$, $M$ and $b$ are positive constants to be determined later. As (3.3), we obtain
that
$$det D^2W=(\frac{W_r}{r})^{n-1}W_{rr}.$$
But
$$W_r=2Mbr (R^2-r^2)^{b-1},$$
$$W_{rr}=2Mb (R^2-r^2)^{b-2}[R^2-(2b-1)r^2].$$
Hence
\begin{equation} \label{4.4}det D^2W=(2Mb)^n (R^2-r^2)^{n(b-1)-1}[R^2-(2b-1)r^2].\end{equation}
Observing that $W\leq 0$ on $\partial artial \Omega$, we see that $W$ is a sub-solution
to problem (1.1) if and only if
\begin{equation} \label{4.5} H[W]:=(2Mb)^n (R^2-r^2)^{n(b-1)-1}[R^2-(2b-1)r^2][F(x,W)]^{-1}\geq 1\end{equation}
for all $x\sqrt{-1}n \Omega$ and $r=|x-y_0|$.
First, we consider the case
\begin{equation} \label{4.6}\beta<n+\alpha+1.\end{equation}
As (2.3), we need only to consider the case $\beta<n+\alpha.$
We take \begin{equation} \label{4.7} b=\frac{\beta}{n+\alpha}=\gamma_3.\end{equation} Then in this case
$b=\gamma_3\sqrt{-1}n (0, 1)$
and $|2b-1|<1$. Hence,
\begin{equation} \label{4.8}R^2-(2b-1)r^2\geq(1-|2b-1|)R^{2}.\end{equation}
It follows from (4.2) that
\begin{equation} \label{4.9}d_x\leq R -|x-y_0|=R-r, \ \ \forall x\sqrt{-1}n \Omega .\end{equation}
Therefore, by (1.3), (4.5), (4.8) and (4.9) that
\begin{equation}\label{4.10}
\begin{split}
H[W]& \geq (1-|2b-1|)R^{2}(2Mb)^n (R^2-r^2)^{n(b-1)-1}\frac{1}{A}(d_x)^{n+1-\beta}|W|^{\alpha}\\
&\geq(1-|2b-1|)R^{2} \frac{1}{A}(2Mb)^n (R^2-r^2)^{n(b-1)-1}(R-r)^{n+1-\beta}|W|^{\alpha}\\
&=(1-|2b-1|)R^{2}\frac{1}{A}M^{\alpha}(2Mb)^n (R^2-r^2)^{n(b-1)+b\alpha-1}(R-r)^{n+1-\beta}\\
&=(1-|2b-1|)R^{2}\frac{1}{A}M^{n+\alpha}(2b)^n (R+r)^{n(b-1)+b\alpha-1}(R-r)^{n(b-1)+b\alpha+n-\beta}.
\end{split}
\end{equation}
Note that
\begin{equation} \label{4.11} n(b-1)+b\alpha+n-\beta=0\end{equation}
by (4.7).
Hence, by (4.10) and (4.11)
we can choose a large $M=C(A, b, R, \alpha, n, \beta)$ such that
\begin{equation} \label{4.12} H[W]\geq 1\ \ in \ \ \Omega .\end{equation}
Next, we consider the case
$$\beta\geq n+\alpha+1.$$
In this case, we take
$$ b=1=\gamma_3.$$
Then , by (1.3) and (4.4) we have
\begin{equation*}
\begin{split}
H[W]& =(2M)^n [F(x,W)]^{-1}\\
& \geq \frac{1}{A}2^n M^{n+\alpha} (R+r)^{ \alpha}(R-r)^{\alpha+n+1-\beta} \\
& = \frac{1}{A} 2^n M^{n+\alpha} (R+r)^{ \alpha}.
\end{split}
\end{equation*}
Therefore, (4.12) still holds true.
To sum up, we have obtained (4.5). By comparison principle, we see that
\begin{equation} \label{4.13} W(x)\leq u(x)\leq 0.\end{equation}
In particular, we obtain that
$$|u(y)|\leq |W(y)|=M(R+|y-y_0|)^{\gamma_3} (R-|y-y_0|)^{\gamma_3} \leq M(2R)^{\gamma_3}(d_y)^{\gamma_3}.$$
This is desired (4.1) and hence we have proved the (i) of Theorem 1.3.
\vskip 0.5cm
To prove (ii) of Theorem 1.3, we notice that $u\sqrt{-1}n C(\bar \Omega)$ and $u<0$ in $\Omega$
and $u=0$ on $\partial om$. By comparing the graph of the convex function $ u$ with the cone whose vortex is $(x_0, u(x_0))$ and whose upper bottom is $\bar \Omega$, where $u(x_0)=\min_{\bar\Omega}u$, we
see easily that (1.10) is true for $\gamma_4\geq 1$. Hence, we need only to consider that case $\gamma_4< 1$ in the following, which implies that $\beta< n+1$.
Since (1.10)
holds naturally for all $y\sqrt{-1}n \{x\sqrt{-1}n \Omega: d_x\geq \frac{R}{2}\}$, where $R$ is the radius of the interior sphere for
the $\Omega$. Hence, it is sufficient to prove
\begin{equation} \label{4.14} |u(y)|\geq (d_y)^{\gamma_4}, \ \ \forall y\sqrt{-1}n \{x\sqrt{-1}n \Omega: d_x< \frac{R}{2}\}.\end{equation}
Take such a $y$. We can find $z\sqrt{-1}n \partial om$, such that $|y-z|=d_{y}.$
we may assume
\begin{equation} \label{4.15} z=\mathbf{0}\sqrt{-1}n \partial artial\Omega\bigcap \partial artial B_R(y_0), \ \ B_R(y_0)\subseteq\Omega.\end{equation}
Since the tangent plane of $\Omega$ at $z=\mathbf{0}$ is unique. And it is easy to check $y$ is on the line determined by $\mathbf{0}$ and $y_{0}$.
Hence $d_{y}=|y|=|y_{0}|-|y_{0}-y|=R-|y_{0}-y|$.\\
Observing that in this case, instead of (4.8) we have
\begin{equation} \label{4.16}d_x\geq R -|x-y_0|=R-r, \ \ \forall x\sqrt{-1}n B_R(y_0).\end{equation}
First, we require $b\sqrt{-1}n(0, 1)$, which implies $2b-1\sqrt{-1}n(-1, 1)$. Similarly to the arguments of (i),\ by (4.16) we find that the function $W$, given by (4.3),
satisfies
\begin{equation}\label{4.17}
\begin{split}
H[W]& \leq \frac{1}{A}(2Mb)^n [R^2- r^2)]^{n(b-1)-1} [R^2-(2b-1)r^2)]d_{x}^{n+1-\beta}|W|^{\alpha}\\
&\leq \frac{1}{A}M^{\alpha}(2Mb)^n2R^{2} [R^2- r^2)]^{n(b-1)-1+b\alpha} (R-r)^{n+1-\beta}\\
&\leq \frac{1}{A}M^{\alpha+n}(2b)^n2R^{2}(2R)^{n(b-1)-1+b\alpha} (R- r)^{n(b-1)+b\alpha+n-\beta}.
\end{split}
\end{equation}
Taking $b=\frac{\beta}{n+\alpha}=\gamma_{4}\sqrt{-1}n (0, 1)$ we have
\begin{equation} \label{4.18} n(b-1)+b\alpha+n-\beta = 0.\end{equation}
Using (4.17)-(4.18), we see that $W$ is a super-solution to problem (1.1) in the domain
$B_R(y_0)$ for sufficiently small $M=C(A, b, R, \alpha, n, \beta)>0$. Since $u$ is a solution on $\Omega$ and $u|_{\partial artial B_R(y_0)}\leq0$, thus $u$ is a sub-solution on $B_R(y_0)$. Therefore, we have
\begin{equation*}
\begin{split}
|u(y)|&\geq |W(y)|\\
&=M(R+|y-y_0|)^{\gamma_4}(R-|y-y_0|)^{\gamma_4}\\
&\geq MR^{\gamma_3}(d_y)^{\gamma_4},
\end{split}
\end{equation*}
Which is the desired (4.14) exactly. In this way, the proof of Theorem 1.3 has been completed.
\end{document} | math | 40,641 |
\begin{equation}gin{document}
\title[Pushforwards of klt pairs]{Pushforwards of klt pairs under morphisms to abelian varieties}
\author{Fanjun Meng}
\address{Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA}
\email{[email protected]}
\thanks{2010 \emph{Mathematics Subject Classification}: 14F17, 14E30.\newline
\indent \emph{Keywords}: generic vanishing, global generation, klt pairs.}
\begin{equation}gin{abstract}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. We prove that the sheaf $f_*\mathcal{O}_X(D)$ becomes globally generated after pullback by an isogeny and has the Chen-Jiang decomposition, along with some related results. These are applied to some effective results for $\mathcal{O}_X(D)$ when $X$ is irregular.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
In this paper, we prove several results about pushforwards of klt pairs under morphisms to abelian varieties over ${\mathbb C}$ and give some applications. These results are natural generalizations of \cite{LPS20} in the presence of singularities.
Many facts are known about the positivity properties of pushforwards of pluricanonical bundles under morphisms from smooth projective varieties to abelian varieties. For example, they are GV-sheaves by \cite{GL87, Hac04, PP11a, PS14}. Their cohomological support loci are finite unions of torsion subvarieties by \cite{GL91, Sim93, Lai11, LPS20}. They have the Chen-Jiang decomposition by \cite{CJ18, PPS17, LPS20}. For the definitions of the concepts mentioned above, we refer to Section \ref{2}. It is natural to ask what happens if we allow singularities. Our work treats the case of klt pairs.
The first three theorems stated below are the main results of the paper. They are all equivalent by Proposition \ref{equi}. For the first result, the case of pushforwards of canonical bundles and pluricanonical bundles is obtained in \cite{CJ18, PPS17, LPS20} in increasing generality.
\begin{equation}gin{thm}\label{main1}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. Then there exists an isogeny $\varphi\colon A'\to A$ such that $\varphi^*f_*\mathcal{O}_X(lD)$ is globally generated for every $l\geq 1$.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X' \arrow[r, "\varphi'"] \arrow[d, "f'"] & X \arrow[d, "f"] \\
A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
\end{thm}
We first need to do substantial work in Section \ref{3} to perform some reduction steps which imply that it suffices to prove Theorem \ref{main1} for the Cartier divisor $ND$ for some positive integer $N$, in order to deduce it for $D$ itself. Then we adapt the strategy in \cite[Sections 8 and 9]{LPS20} to our setting to prove Proposition \ref{SNC}. The proof relies in part on analytic results based on the theory of singular Hermitian metrics from \cite{CP17, HPS18}. See Lemma \ref{split} and Proposition \ref{SNC} for details. The invariance of plurigenera for smooth families of smooth varieties is used in the proof of the main theorems in \cite{LPS20}. Since we do not know it in our case, we use a different technique to avoid it. See Remark \ref{genera} for details.
Theorem \ref{main2} is a consequence of Theorem \ref{main1} and Proposition \ref{equi}. The case of pushforwards of pluricanonical bundles is obtained in \cite[Theorem A]{LPS20}.
\begin{equation}gin{thm}\label{main2}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. Then there exists a generically finite surjective morphism $h\colon Z\to X$ from a smooth projective variety $Z$ such that $f_*\mathcal{O}_X(D)$ is a direct summand of $(f\circ h)_*\mathcal{O}_Z(K_Z)$.
\end{thm}
Theorem \ref{main2} implies that all the properties of pushforwards of canonical bundles mentioned above, along with many other properties, carry over to pushforwards of klt pairs. See Section \ref{4} for details.
The third theorem regards the Chen-Jiang decomposition property for the sheaf $f_*\mathcal{O}_X(D)$. The case of pushforwards of canonical bundles and pluricanonical bundles is obtained in \cite{CJ18, PPS17, LPS20} in increasing generality.
\begin{equation}gin{thm}\label{main3}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. Then $f_*\mathcal{O}_X(D)$ admits a finite direct sum decomposition
$$f_*\mathcal{O}_X(D)\cong \bibitemgoplus_{i\in I}(\alpha_i\otimes p_i^*\mathcal{F}_i),$$
where each $A_i$ is an abelian variety, each $p_i\colon A\to A_i$ is a fibration, each $\mathcal{F}_i$ is a nonzero M-regular coherent sheaf on $A_i$, and each $\alpha_i\in\Pic^0(A)$ is line bundle which becomes trivial when pulled back by the isogeny $\varphi$ in Theorem \ref{main1}.
\end{thm}
When attacking these theorems in Section \ref{3}, we in fact first prove Theorem \ref{main3}. This theorem gives a detailed description about the positivity of the sheaf $f_*\mathcal{O}_X(D)$. It essentially says that the sheaf $f_*\mathcal{O}_X(D)$ is not just semipositive, but semiample, since M-regular sheaves are ample by \cite[Corollary 3.2]{Deb06}.
We briefly explain how the main theorems are proved. When we consider pluricanonical bundles on smooth varieties, Viehweg's cyclic covering trick applies in order to perform some reduction steps. However, this trick does not apply in our case when $m>1$. Instead, in Section \ref{3} we need to do substantial work in order to overcome this, using a technique from \cite[Theorem 1.7]{PS14} and some other methods. These statements imply that it suffices to show that $f_*\mathcal{O}_X(ND)$ satisfies the conclusions of Theorems \ref{main1}, \ref{main2} and \ref{main3} for some positive integer $N$ in order to prove that $f_*\mathcal{O}_X(D)$ has the same properties, which gives us extra flexibility. When $N$ is sufficiently big and divisible, we follow the strategy in \cite[Sections 8 and 9]{LPS20}, with some modifications, to prove that $f_*\mathcal{O}_X(ND)$ has the required properties. See Proposition \ref{SNC} for details.
We note that there is an alternative approach to proving the main theorems for $f_*\mathcal{O}_X(ND)$ when $N$ is sufficiently big and divisible, different from the one described above. We believe that this is worth mentioning since the techniques are rather different and interesting in their own right. It is based on the use of the minimal model program but it requires the assumption that the general fiber $(F, \Delta|_F)$ of $f$ has a good minimal model. This approach is not purely algebraic either. See Proposition \ref{MMP} and Remark \ref{not} for details.
With more work, in Section \ref{4} we prove that pushforwards of klt pairs under the Albanese morphism satisfy stronger positivity which is valid only for $m>1$. The crucial point is the next theorem which generalizes \cite[Proposition 2.12]{HP02}, \cite[Lemma 2.2]{Jia11} and \cite[Theorem 11.2]{HPS18} to klt pairs. We consider a smooth model of the Iitaka fibration associated to a Cartier divisor $D$ where $\kappa(X, D)\geq0$.
\begin{equation}gin{thm}\label{main6}
Let $f\colon X\to Y$ be a smooth model of the Iitaka fibration associated to a Cartier divisor $D$ on $X$ where $D\sim_{{\mathbb Q}}m(K_X+\Delta)$, $m>1$ is a rational number, $Y$ is smooth and $(X, \Delta)$ is a klt pair. Let $a_Y\colon Y\to \Alb(Y)$ be the Albanese morphism of $Y$ and $g$ the morphism $a_Y\circ f$. Then:
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] For every torsion point $\alpha\in\Pic^0(X)$, every $\begin{equation}ta\in\Pic^0(Y)$ and every nef divisor $L$ on $\Alb(Y)$, we have
$$\quad h^0(X, \mathcal{O}_X(D+g^*L)\otimes\alpha)=h^0(X, \mathcal{O}_X(D+g^*L)\otimes\alpha\otimes f^*\begin{equation}ta).$$
\item[$\mathrm{(ii)}$] There exist finitely many torsion points $\alpha_i\in\Pic^0(X)$ such that
$$V^0(X, \mathcal{O}_X(D))=\bibitemgcup_{i\in I}(\alpha_i\otimes f^*\Pic^0(Y)).$$
\end{enumerate}
\end{thm}
To prove Theorem \ref{main6}, we follow the strategy used for example in \cite[Lemma 2.2]{Jia11} and \cite[Theorem 11.2]{HPS18} for the case of smooth varieties. However, the techniques involved are somewhat different in our case. Once we have Theorem \ref{main6}, the next theorem below follows from Theorem \ref{main3} and standard arguments. The case when $D=mK_X$ and $m\geq2$ is an integer is obtained in \cite[Theorem D]{LPS20}.
\begin{equation}gin{thm}\label{main4}
Let $f\colon X\to Y$ be a smooth model of the Iitaka fibration associated to a Cartier divisor $D$ on $X$ where $D\sim_{{\mathbb Q}}m(K_X+\Delta)$, $m>1$ is a rational number, $Y$ is smooth and $(X, \Delta)$ is a klt pair. Let $a_X\colon X\to \Alb(X)$ be the Albanese morphism of $X$ and $a_f\colon \Alb(X)\to\Alb(Y)$ the induced morphism between Albanese varieties. Then $(a_X)_*\mathcal{O}_X(lD)$ admits, for every positive integer $l$, a finite direct sum decomposition
$$(a_X)_*\mathcal{O}_X(lD)\cong \bibitemgoplus_{i\in I}(\alpha_i\otimes a_f^*\mathcal{F}_i),$$
where each $\mathcal{F}_i$ is a coherent sheaf on $\Alb(Y)$ satisfying $\IT$ and each $\alpha_i\in\Pic^0(X)$ is a torsion line bundle whose order can be bounded independently of $l$.
\end{thm}
\begin{equation}gin{rem}
After finishing this manuscript, I was informed by Zhi Jiang that he has also proved a result in \cite{Jia20} which is essentially the same as Theorem \ref{main4}, by a different but related method. He uses it to obtain some very interesting geometric applications. I would like to thank him for sharing his draft.
\end{rem}
We apply Theorem \ref{main4} to studying effective freeness and very ampleness for klt pairs on irregular varieties in Section \ref{4}. The following statement is a special case of Theorem \ref{main5}. It deals with the case when $(X, \Delta)$ is of log general type and $X$ is of maximal Albanese dimension, and generalizes some effective results about varieties with at worst canonical singularities from \cite{PP03, PP11b, LPS20} to klt pairs. See \cite[Corollary E]{LPS20} for instance.
\begin{equation}gin{coro}\label{coro5}
Let $(X, \Delta)$ be a klt pair of log general type, $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$, $m>1$ a rational number and $a_X\colon X\to \Alb(X)$ the Albanese morphism of $X$. Assume that $a_X$ is generically finite onto its image and denote the union of the positive dimensional fibers of $a_X$ by $\Exc(a_X)$. Then for every $\alpha\in\Pic^0(X)$:
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] $\mathcal{O}_{X}(2D)\otimes\alpha$ is globally generated away from $\Exc(a_X)$.
\item[$\mathrm{(ii)}$] $\mathcal{O}_{X}(3D)\otimes\alpha$ is very ample away from $\Exc(a_X)$.
\end{enumerate}
\end{coro}
Finally, it is natural to ask what happens if we consider log canonical pairs instead of klt pairs. Although some of the ideas we use here still work, at this point we do not know how to fully generalize our theorems to this case even when $m=1$. Moreover, we do not have the generalization of Proposition \ref{SNC} to log canonical pairs, because the results from the singular Hermitian metrics are not yet known in this setting.
\begin{equation}gin{ac}
{I would like to express my sincere gratitude to my advisor Mihnea Popa for proposing this problem and for helpful discussions and generous support. I would also like to thank Bingyi Chen for helpful discussions, and Zhi Jiang for sharing \cite{Jia20}.}
\end{ac}
\section{Preliminaries}\label{2}
We work over ${\mathbb C}$ and all varieties are projective throughout the paper. A \emph{fibration} is a projective surjective morphism with connected fibers. A \emph{birational contraction} is a birational map whose inverse does not contract any divisor. For the definitions and basic results on the singularities of pairs and the minimal model program (MMP) we refer to \cite{KM98}. We always ask the boundary $\Delta$ in a pair $(X,\Delta)$ to be effective.
First, we give the definition for the general fiber of a morphism which is not necessarily surjective.
\begin{equation}gin{defi}\label{general fiber}
Let $f\colon X \to Y$ be a morphism between two normal projective varieties. The Stein factorization of $f$ gives a decomposition of $f$ as $g\circ h$ where $h$ is a fibration and $g$ is a finite morphism. The \emph{general fiber} of $f$ is defined as the general fiber of $h$. The definition is similar when $X$ is replaced with a log canonical pair $(X, \Delta)$.
\end{defi}
We include the definition of good minimal models.
\begin{equation}gin{defi}
Let $(X, \Delta)$ be a log canonical pair over a normal projective variety $Z$. A birational contraction $\xi \colon (X, \Delta) \dashrightarrow (Y, \Delta_Y)$ over $Z$ to a ${\mathbb Q}$-factorial log canonical pair $ (Y,\Delta_Y) $ is a \emph{good minimal model} over $Z$ of the pair $(X,\Delta)$ if $ \Delta_Y =\xi_*\Delta$, if $K_Y+\Delta_Y$ is semiample over $Z$ and if
$$a(F, X, \Delta) < a(F, Y, \Delta_Y)$$
for any prime divisor $F$ on $X$ which is contracted by $\xi$.
\end{defi}
Next we recall the definition for the irregularity of a projective variety $X$.
\begin{equation}gin{defi}
Let $X$ be a smooth projective variety. The \emph{irregularity} $q(X)$ is defined as $h^1(X,\mathcal{O}_X)$. If $X$ is a projective variety, the \emph{irregularity} $q(X)$ is defined as the irregularity of any resolution of $X$.
\end{defi}
If $X$ is smooth, the irregularity $q(X)$ is equal to the dimension of its Albanese variety $\Alb(X)$.
Let $A$ be an abelian variety and $\hat{A}\cong \Pic^0(A)$ its dual abelian variety. We denote by
$$\bold{R}\hat{\mathcal{S}}\colon \bold{D}(A)\to \bold{D}(\hat{A}),\quad \bold{R}\hat{\mathcal{S}}\mathcal{F}:=\bold{R}p_{2*}(p_1^*\mathcal{F}\otimes P)$$
the Fourier-Mukai functor induced by a normalized Poincar\'e bundle $P$ on $A\times\hat{A}$ where $p_1$ and $p_2$ are projections onto $A$ and $\hat{A}$ respectively.
We recall several definitions.
\begin{equation}gin{defi}
A coherent sheaf $\mathcal{F}$ on an abelian variety $A$
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] is a GV-\emph{sheaf} if $\codim$ $\Supp$ $\bold{R}^i\hat{\mathcal{S}}\mathcal{F}\geq i$ for every $i>0$.
\item[$\mathrm{(ii)}$] is \emph{M-regular} if $\codim$ $\Supp$ $\bold{R}^i\hat{\mathcal{S}}\mathcal{F}> i$ for every $i>0$.
\item[$\mathrm{(iii)}$] \emph{satisfies} $\IT$ if $\bold{R}^i\hat{\mathcal{S}}\mathcal{F}=0$ for every $i>0.$
\end{enumerate}
\end{defi}
We include a proposition about determining when a GV-sheaf satisfies $\IT$ which is \cite[Proposition 2.3]{LPS20}.
\begin{equation}gin{prop}\label{IT}
Let $\mathcal{F}$ be a $\GV$-sheaf on an abelian variety $A$. If $h^0(A, \mathcal{F}\otimes\alpha)$ is independent of $\alpha\in\Pic^0(A)$, then $\mathcal{F}$ satisfies $\IT$ and $\bold{R}\hat{\mathcal{S}}\mathcal{F}$ is locally free.
\end{prop}
\begin{equation}gin{defi}
Let $\mathcal{F}$ be a coherent sheaf on an abelian variety $A$. The \emph{cohomological support loci} $V_l^i(A, \mathcal{F})$ for $i\in{\mathbb N}$ and $l\in{\mathbb N}$ are defined by
$$V_l^i(A, \mathcal{F})=\{\alpha\in\Pic^0(A)\mid\dim H^i(A, \mathcal{F}\otimes\alpha)\geq l\}.$$
We use $V^i(A, \mathcal{F})$ to denote $V_1^i(A, \mathcal{F})$.
\end{defi}
\begin{equation}gin{defi}
Let $A$ be an abelian variety. A \emph{torsion subvariety} of $A$ is a translate of an abelian subvariety of $A$ by a torsion point which is a closed point of finite order in $A$.
\end{defi}
Next we define the useful notion of continuous evaluation morphisms (cf. \cite{PP03, LPS20}) and give a simple lemma about it (cf. \cite[Proposition 3.1]{Deb06}).
\begin{equation}gin{defi}
Let $\mathcal{F}$ be a coherent sheaf on an abelian variety $A$. The \emph{continuous evaluation morphism} associated to the coherent sheaf $\mathcal{F}$ is defined by
$$e_{\mathcal{F}}\colon \bibitemgoplus_{\alpha\in \Pic^0(A)\atop \mathrm{torsion}}H^0(A, \mathcal{F}\otimes\alpha)\otimes\alpha^{-1}\to \mathcal{F}$$
which is induced from the evaluation morphisms.
\end{defi}
\begin{equation}gin{lemma}\label{tor}
Let $\mathcal{F}$ be a coherent sheaf on an abelian variety $A$. Then there exists an isogeny $\varphi\colon A'\to A$ such that $\varphi^*\mathcal{F}$ is globally generated if and only if the continuous evaluation morphism $e_{\mathcal{F}}$ associated to $\mathcal{F}$ is surjective.
\end{lemma}
\begin{equation}gin{proof}
First, we prove the if part. Since $\mathcal{F}$ is coherent, there exist finitely many torsion line bundles $\alpha_1, \dots, \alpha_N$ such that
$$\bibitemgoplus^{N}_{i=1}H^0(A, \mathcal{F}\otimes\alpha_i)\otimes\alpha_i^{-1}\to \mathcal{F}$$
is surjective. Then we can choose an isogeny $\varphi$ on $A$ such that $\varphi^*\alpha_i$ becomes trivial for each $1\leq i\leq N$ and thus $\varphi^*\mathcal{F}$ is globally generated.
We prove the only if part now. Consider the sheaf $\mathcal{G}:=\Image e_{\mathcal{F}}\subseteq \mathcal{F}$. By the same argument, we can choose an isogeny $\psi$ such that $\psi^*\mathcal{G}$ is globally generated. By taking a fiber product of $\psi$ and $\varphi$, we can get a new isogeny $\xi\colon B\to A$ such that both $\xi^*\mathcal{G}$ and $\xi^*\mathcal{F}$ are globally generated. Since $\xi$ is \'{e}tale, we have that $\xi^*\mathcal{G}\subseteq\xi^*\mathcal{F}$. We only need to prove $\xi^*\mathcal{G}=\xi^*\mathcal{F}$ to deduce $\mathcal{G}=\mathcal{F}$ since $\xi$ is faithfully flat. We know that
$$\xi_*\mathcal{O}_B\cong\bibitemgoplus_{\alpha\in\Ker \hat{\xi}}\alpha$$
where $\hat{\xi}$ is the dual isogeny of $\xi$. We have that $H^0(A, \mathcal{G}\otimes\alpha)=H^0(A, \mathcal{F}\otimes\alpha)$ for every torsion line bundle $\alpha\in\Pic^0(A)$ by the definition of $\mathcal{G}$. By the base change theorem, we have
$$H^0(B, \xi^*\mathcal{G})\cong H^0(A, \mathcal{G}\otimes\xi_*\mathcal{O}_B)\cong\bibitemgoplus_{\alpha\in\Ker \hat{\xi}}H^0(A, \mathcal{G}\otimes\alpha)$$
$$\cong\bibitemgoplus_{\alpha\in\Ker \hat{\xi}}H^0(A, \mathcal{F}\otimes\alpha)\cong H^0(A, \mathcal{F}\otimes\xi_*\mathcal{O}_B)\cong H^0(B, \xi^*\mathcal{F}).$$
Thus we have $\xi^*\mathcal{G}=\xi^*\mathcal{F}$ since $\xi^*\mathcal{G}$ and $\xi^*\mathcal{F}$ are globally generated.
\end{proof}
We now give the definition of the Chen-Jiang decomposition which is one of the main properties we will discuss in the following sections. The paper \cite{CJ18} justifies the name of this property.
\begin{equation}gin{defi}
Let $\mathcal{F}$ be a coherent sheaf on an abelian variety $A$. The sheaf $\mathcal{F}$ is said to have the \emph{Chen-Jiang decomposition} if $\mathcal{F}$ admits a finite direct sum decomposition
$$\mathcal{F}\cong \bibitemgoplus_{i\in I}(\alpha_i\otimes p_i^*\mathcal{F}_i),$$
where each $A_i$ is an abelian variety, each $p_i\colon A\to A_i$ is a fibration, each $\mathcal{F}_i$ is a nonzero M-regular coherent sheaf on $A_i$, and each $\alpha_i\in\Pic^0(A)$ is a torsion line bundle.
\end{defi}
We include a useful proposition which is \cite[Proposition 3.6]{LPS20}.
\begin{equation}gin{prop}\label{sum}
Let $\mathcal{F}\cong \mathcal{F'}\oplus\mathcal{F''}$ be a coherent sheaf on an abelian variety $A$. Then $\mathcal{F}$ has the Chen-Jiang decomposition if and only if both $\mathcal{F'}$ and $\mathcal{F''}$ have the Chen-Jiang decomposition.
\end{prop}
Next, we give a lemma about the splitting of morphisms which is based on the minimal extension property of some special singular Hermitian metrics. For clarity, we will use the analytic language for the statement and the proof of it. For details, we refer to \cite{CP17, HPS18}.
\begin{equation}gin{lemma}\label{split}
Let $f$ be a projective surjective morphism from a log smooth klt pair $(X, \Delta)$ on a projective manifold $X$ to an abelian variety $A$. Then for every positive integer $N$ which is sufficiently big and divisible such that $f_*\mathcal{O}_X(N(K_X+\Delta))\neq0$, the sheaf $f_*\mathcal{O}_X(N(K_X+\Delta))$ admits a singular Hermitian metric with semipositive curvature and the minimal extension property. In particular, any nonzero morphism $f_*\mathcal{O}_X(N(K_X+\Delta))\to \mathcal{O}_A$ splits.
\end{lemma}
\begin{equation}gin{proof}
We choose $N$ sufficiently divisible such that $N\Delta$ is an integral divisor. Define a new divisor $L_N:=(N-1)K_{X/A}+N\Delta$. By the discussion at the beginning of \cite[Section 4]{CP17}, if $N$ is sufficiently big, then there exists a singular Hermitian metric $h_N$ on the line bundle $\mathcal{O}_X(L_N)$ with semipositive curvature such that the inclusion
$$f_*(\mathcal{O}_X(K_{X/A}+L_N)\otimes \mathcal{I}(h_N))\hookrightarrow f_*\mathcal{O}_X(K_{X/A}+L_N)$$
is generically an isomorphism where $\mathcal{I}(h_N)\subseteq \mathcal{O}_X$ is the multiplier ideal sheaf associated to the singular Hermitian metric $h_N$. Thus we know the sheaf $f_*\mathcal{O}_X(N(K_X+\Delta))\cong f_*\mathcal{O}_X(K_{X/A}+L_N)$ admits a singular Hermitian metric with semipositive curvature and the minimal extension property by \cite[Theorem 21.1 and Corollary 21.2]{HPS18}. Then the property of splitting follows from \cite[Theorem 26.4]{HPS18}.
\end{proof}
Let $\varphi\colon A'\to A$ be an isogeny in Lemma \ref{split}. We have the following base change diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X' \arrow[r, "\varphi'"] \arrow[d, "f'"] & X \arrow[d, "f"] \\
A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
We consider the log smooth klt pair $(X', \Delta')$ given by $K_{X'}+\Delta'=\varphi'^*(K_X+\Delta)$. Then the same $N$ as in $f_*\mathcal{O}_X(N(K_X+\Delta))$ will make Lemma \ref{split} work for $f'_*\mathcal{O}_{X'}(N(K_{X'}+\Delta'))$ by the construction of the singular Hermitian metric on it. See \cite[Section 4]{CP17} for details.
\section{Positivity of pushforwards of klt pairs}\label{3}
In this section, we prove our main theorems. The next four statements are needed for the reduction steps. We start with a basic lemma which treats the case when $m=1$.
\begin{equation}gin{lemma}\label{m=1}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$ and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}K_X+\Delta$. Then there exists a generically finite surjective morphism $h\colon Z\to X$ from a smooth projective variety $Z$ such that $R^if_*\mathcal{O}_X(D)$ is a direct summand of $R^i(f\circ h)_*\mathcal{O}_Z(K_Z)$ for every $i\in {\mathbb N}$. In particular, the sheaf $R^if_*\mathcal{O}_X(D)$ has the Chen-Jiang decomposition for every $i\in {\mathbb N}$.
\end{lemma}
\begin{equation}gin{proof}
We take a log resolution of $(X, \Delta)$ denoted by $\mu\colon Y\to X$ as in the following diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
Y \arrow[r, "\mu"] \arrow[dr, "g" swap] & X \arrow[d, "f"] \\
& A
\end{tikzcd}
\end{center}
Then we have
$$K_Y+\Delta_Y\sim_\mathbb{Q}\mu^*(K_{X}+\Delta)+E,$$
where the ${\mathbb Q}$-divisors $\Delta_Y$ and $E$ are effective and have no common components and $E$ is $\mu$-exceptional. We also have
$$K_Y+\Delta_Y+\lceil E\rceil-E\sim_\mathbb{Q}\mu^*(K_{X}+\Delta)+\lceil E\rceil.$$
Let $\Delta'_Y$ be $\Delta_Y+\lceil E\rceil-E$, then the pair $(Y, \Delta'_Y)$ is klt and log smooth since $(X, \Delta)$ is klt. We know that $\mu_*\mathcal{O}_Y(\mu^*D+\lceil E\rceil)\cong \mathcal{O}_X(D)$ since $\lceil E\rceil$ is $\mu$-exceptional and effective. By \cite[Corollary 10.15]{Kol95}, $R^i\mu_*\mathcal{O}_Y(\mu^*D+\lceil E\rceil)$ is torsion-free for every $i\in {\mathbb N}$ since $\mu^*D+\lceil E\rceil\sim_{{\mathbb Q}}K_Y+\Delta'_Y$ and the pair $(Y, \Delta'_Y)$ is klt and log smooth. Thus $R^i\mu_*\mathcal{O}_Y(\mu^*D+\lceil E\rceil)=0$ for $i>0$, since it is $0$ on an open subset of $X$ and torsion-free. Then we deduce that
$$R^ig_*\mathcal{O}_Y(\mu^*D+\lceil E\rceil)\cong R^if_*(\mu_*\mathcal{O}_Y(\mu^*D+\lceil E\rceil))\cong R^if_*\mathcal{O}_X(D)$$
for every $i\in {\mathbb N}$ by Grothendieck spectral sequence. Thus we can assume $(X, \Delta)$ is log smooth from the start.
We can choose a positive integer $N$ such that $ND\sim N(K_X+\Delta)$ and $N\Delta$ becomes an integral divisor. Then we have $N(D-K_X)\sim N\Delta$. Thus we can take the associated branched covering along $N\Delta$ and resolve the singularities. This gives us a generically finite surjective morphism $h\colon Z\to X$ of degree $N$. By \cite[Lemma 2.3]{Vie83}, $h_*{O}_Z(K_Z)$ contains as a direct summand the sheaf
$$\mathcal{O}_X(D-K_X+K_X)\otimes \mathcal{O}_X(-\lfloor \frac{1}{N}\cdot N\Delta \rfloor)\cong \mathcal{O}_X(D).$$
By Grauert-Riemenschneider vanishing theorem, we have that $R^ih_*\mathcal{O}_Z(K_Z)=0$ for $i>0$. We deduce that
$$R^i(f\circ h)_*\mathcal{O}_Z(K_Z)\cong R^if_*h_*\mathcal{O}_Z(K_Z)$$
contains $R^if_*\mathcal{O}_X(D)$ as a direct summand for every $i\in {\mathbb N}$ by Grothendieck spectral sequence. Since $R^i(f\circ h)_*\mathcal{O}_Z(K_Z)$ has the Chen-Jiang decomposition for every $i\in {\mathbb N}$ by \cite[Theorem A]{PPS17}, the sheaf $R^if_*\mathcal{O}_X(D)$ has the Chen-Jiang decomposition for every $i\in {\mathbb N}$ by Proposition \ref{sum}.
\end{proof}
Our next proposition establishes an equivalence between four statements about the positivity of pushforwards of klt pairs.
\begin{equation}gin{prop}\label{equi}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. Then the following four statements are equivalent:
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] The sheaf $f_*\mathcal{O}_X(D)$ has the Chen-Jiang decomposition.
\item[$\mathrm{(ii)}$] There exists an isogeny $\varphi\colon A'\to A$ such that $\varphi^*f_*\mathcal{O}_X(D)$ is globally generated.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X' \arrow[r, "\varphi'"] \arrow[d, "f'"] & X \arrow[d, "f"] \\
A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
\item[$\mathrm{(iii)}$] There exists a generically finite surjective morphism $h\colon Z\to X$ from a smooth projective variety $Z$ such that $f_*\mathcal{O}_X(D)$ is a direct summand of $(f\circ h)_*\mathcal{O}_Z(K_Z)$.
\item[$\mathrm{(iv)}$] The continuous evaluation morphism $e_{f_*\mathcal{O}_X(D)}$ associated to $f_*\mathcal{O}_X(D)$ is surjective.
\end{enumerate}
\end{prop}
\begin{equation}gin{proof}
We can assume that $f_*\mathcal{O}_X(D)\neq0$. The fact that statement (i) implies (ii) follows from the proof of \cite[Theorem 5.1]{LPS20}.
We now prove that statement (ii) implies (iii). We have an isogeny $\varphi\colon A'\to A$ such that $\varphi^*f_*\mathcal{O}_X(D)$ is globally generated. We define a ${\mathbb Q}$-divisor $\Delta'$ by $K_{X'}+\Delta'=\varphi'^*(K_X+\Delta)$. Since $\varphi'$ is an \'etale morphism, the new pair $(X', \Delta')$ is klt and $\Delta'$ is effective. Define $D'$ as $\pi^*D$ then we have $D'\sim_{{\mathbb Q}}m(K_{X'}+\Delta')$. By the flat base change theorem, we know that $f'_*\mathcal{O}_{X'}(D')\cong\varphi^*f_*\mathcal{O}_X(D)$ is globally generated.
Next, we consider the following adjoint morphism
$$f'^*f'_*\mathcal{O}_{X'}(D')\to \mathcal{O}_{X'}(D').$$
The image of this morphism is $D'\otimes\mathcal{I}$ and $\mathcal{I}$ is the relative base ideal of $D'$. We can take a log resolution $\mu\colon Y\to X'$ of $\mathcal{I}$ and $(X', \Delta')$ as in the following diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
Y \arrow[r, "\mu"] \arrow[dr, "g" swap] & X' \arrow[r, "\varphi'"] \arrow[d, "f'"] \arrow[dr] & X \arrow[d, "f"] \\
& A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
Then we have
$$K_Y+\Delta_Y\sim_\mathbb{Q}\mu^*(K_{X'}+\Delta')+E,$$
where the ${\mathbb Q}$-divisors $\Delta_Y$ and $E$ are effective and have no common components and $E$ is $\mu$-exceptional. We also have that the image of the new adjoint morphism
$$g^*g_*\mathcal{O}_{Y}(D_Y)\to \mathcal{O}_{Y}(D_Y)$$
is a line bundle $\mathcal{O}_{Y}(D_Y-F)$ where $D_Y=\mu^*D'$ is a Cartier divisor, $F$ is an effective divisor and $\Delta_Y+E+F$ has simple normal crossings support. Since $(X', \Delta')$ is klt, we know $(Y, \Delta_Y':=\Delta_Y+\frac{\lceil mE\rceil}{m}-E)$ is also klt and
$$m(K_Y+\Delta_Y')\sim_\mathbb{Q}\mu^*(m(K_{X'}+\Delta'))+\lceil mE\rceil\sim_{{\mathbb Q}} D_Y+\lceil mE\rceil.$$
Denote $D_Y+\lceil mE\rceil$ by $G$. We have $g_*\mathcal{O}_{Y}(G)\cong f'_*\mathcal{O}_{X'}(D')$ since the effective divisor $\lceil mE\rceil$ is $\mu$-exceptional and thus $g_*\mathcal{O}_{Y}(G)$ is globally generated. The image of the adjoint morphism
$$g^*g_*\mathcal{O}_{Y}(G)\to \mathcal{O}_{Y}(G)$$
is $\mathcal{O}_{Y}(G-F-\lceil mE\rceil)$. Let $G'$ be $F+\lceil mE\rceil$. We claim that $g_*\mathcal{O}_{Y}(G-G'')\cong g_*\mathcal{O}_{Y}(G)$ for any effective divisor $G''\leq G'$. This is because we can factor the adjoint morphism as
$$g^*g_*\mathcal{O}_{Y}(G)\to \mathcal{O}_{Y}(G-G')\hookrightarrow \mathcal{O}_{Y}(G-G'')\hookrightarrow \mathcal{O}_{Y}(G)$$
and the morphism induced from pushforwards
$$g_*\mathcal{O}_{Y}(G)\to g_*\mathcal{O}_{Y}(G-G')\hookrightarrow g_*\mathcal{O}_{Y}(G-G'')\hookrightarrow g_*\mathcal{O}_{Y}(G)$$
is identity.
We claim that we can find an effective divisor $T\leq G'$ and a klt pair $(Y, N)$ such that $G-T\sim_{{\mathbb Q}}K_Y+N$. To do this, we use a technique from the proof of \cite[Theorem 1.7]{PS14}. Since $g_*\mathcal{O}_{Y}(G)$ is globally generated, we have $g^*g_*\mathcal{O}_{Y}(G)$ is globally generated and so is $\mathcal{O}_{Y}(G-G')$. By Bertini's theorem, we can pick a general Cartier divisor $H\sim G-G'$ which is reduced and effective such that $H$ and $\Delta_Y'+G'$ have no common components and $H+\Delta_Y'+G'$ has simple normal crossings support. We define an effective divisor $T$ as
$$T:=\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{m}G' \bibitemgg\rfloor.$$
We have $T\leq G'$ since the coefficient of each irreducible component of $\Delta_Y'$ is smaller than $1$. We deduce
$$G-T\sim_{{\mathbb Q}}m(K_Y+\Delta_Y')-T=K_Y+\Delta_Y'+(m-1)(K_Y+\Delta_Y')-T$$
$$\sim_{{\mathbb Q}}K_Y+\Delta_Y'+\frac{m-1}{m}G-T\sim_{{\mathbb Q}}K_Y+\Delta_Y'+\frac{m-1}{m}(G'+H)-T$$
$$=K_Y+\frac{m-1}{m}H+\Delta_Y'+\frac{m-1}{m}G'-\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{m}G' \bibitemgg\rfloor.$$
We define $N$ as
$$N:=\frac{m-1}{m}H+\Delta_Y'+\frac{m-1}{m}G'-\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{m}G' \bibitemgg\rfloor.$$
The effective ${\mathbb Q}$-divisor $N$ has simple normal crossings support and the coefficient of each irreducible component of itself is smaller than $1$. Thus $(Y, N)$ is a klt pair and we have proved our claim.
We know that $g_*\mathcal{O}_{Y}(G-T)\cong g_*\mathcal{O}_{Y}(G)$ since $0\leq T\leq G'$. Thus we have
$$(\varphi\circ g)_*\mathcal{O}_{Y}(G-T)\cong (\varphi\circ g)_*\mathcal{O}_{Y}(G)\cong \varphi_*f'_*\mathcal{O}_{X'}(D')$$
$$\cong f_*\varphi'_*\mathcal{O}_{X'}(\varphi'^*D)\cong f_*(\mathcal{O}_{X}(D)\otimes\varphi'_*\mathcal{O}_{X'}).$$
We know that $\mathcal{O}_{X}$ is a direct summand of $\varphi'_*\mathcal{O}_{X'}$. Then $f_*(\mathcal{O}_{X}(D))$ is a direct summand of $(\varphi\circ g)_*\mathcal{O}_{Y}(G-T)$. Since $G-T\sim_{{\mathbb Q}}K_Y+N$, there exists a generically finite surjective morphism $h'\colon Z\to Y$ from a smooth projective variety $Z$ as in the following diagram such that $g_*\mathcal{O}_{Y}(G-T)$ is a direct summand of $(g\circ h')_*\mathcal{O}_Z(K_Z)$ by Lemma \ref{m=1}.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
Z \arrow[r, "h'"] &Y \arrow[r, "\mu"] \arrow[dr, "g"] & X' \arrow[r, "\varphi'"] \arrow[d, "f'"] \arrow[dr] & X \arrow[d, "f"] \\
&& A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
Then $(\varphi\circ g)_*\mathcal{O}_{Y}(G-T)$ is a direct summand of $(\varphi\circ g\circ h')_*\mathcal{O}_Z(K_Z)$ and so is $f_*(\mathcal{O}_{X}(D))$. Define the morphism $h\colon Z\to X$ as $\varphi'\circ\mu\circ h'$. We have that $\varphi\circ g\circ h'=f\circ h$ and $h$ is a generically finite surjective morphism such that $f_*(\mathcal{O}_{X}(D))$ is a direct summand of $(f\circ h)_*\mathcal{O}_Z(K_Z)$.
Next, we prove that statement (iii) implies (i). It follows from \cite[Theorem A]{PPS17} and Proposition \ref{sum}.
Lemma \ref{tor} implies that statements (ii) and (iv) are equivalent.
\end{proof}
\begin{equation}gin{coro}\label{red}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. Let $\varphi\colon A'\to A$ be an isogeny. Then $f_*\mathcal{O}_X(D)$ has the Chen-Jiang decomposition if and only if $f'_*\mathcal{O}_{X'}(D')$ has the Chen-Jiang decomposition where $D'=\varphi'^*D$.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X' \arrow[r, "\varphi'"] \arrow[d, "f'"] & X \arrow[d, "f"] \\
A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
\end{coro}
\begin{equation}gin{proof}
We first prove the if part. By Proposition \ref{equi}, there exists an isogeny $\psi: A''\to A'$ such that $\psi^*f'_*\mathcal{O}_{X'}(D')\cong \psi^*\varphi^*f_*\mathcal{O}_X(D)$ is globally generated as in the following diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X''\arrow[r,"\psi'"]\arrow[d, "f''"] & X' \arrow[r, "\varphi'"] \arrow[d, "f'"] & X \arrow[d, "f"] \\
A''\arrow[r, "\psi" ] &A' \arrow[r, "\varphi" ] & A
\end{tikzcd}
\end{center}
By Proposition \ref{equi} again, $f_*\mathcal{O}_X(D)$ has the Chen-Jiang decomposition.
Next, we prove the only if part. By Proposition \ref{equi}, there exists an isogeny $g\colon B\to A$ such that $g^*f_*\mathcal{O}_X(D)$ is globally generated. Consider the following diagram for the fiber product $B':=B\times_{A}A'$.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
B' \arrow[r, "p"] \arrow[d, "q"] & A' \arrow[d, "\varphi"] \\
B \arrow[r, "g" ] & A
\end{tikzcd}
\end{center}
The morphisms $p$ and $q$ are the projections and isogenies between abelian varieties since $g$ and $\varphi$ are isogenies. Since
$$p^*f'_*\mathcal{O}_{X'}(D')\cong p^*\varphi^*f_*\mathcal{O}_X(D)\cong q^*g^*f_*\mathcal{O}_X(D)$$
is globally generated, $f'_*\mathcal{O}_{X'}(D')$ has the Chen-Jiang decomposition by Proposition \ref{equi}.
\end{proof}
The following proposition claims that we only need to find a positive integer $N$ such that $f_*\mathcal{O}_X(ND)$ has the Chen-Jiang decomposition in order to prove that $f_*\mathcal{O}_X(D)$ has the same property.
\begin{equation}gin{prop}\label{N}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. If there exists a positive integer $N$ such that $f_*\mathcal{O}_X(ND)$ has the Chen-Jiang decomposition, then $f_*\mathcal{O}_X(D)$ has the Chen-Jiang decomposition.
\end{prop}
\begin{equation}gin{proof}
If the sheaf $f_*\mathcal{O}_X(ND)$ is $0$, then $f_*\mathcal{O}_X(D)=0$ and the statement is trivial. Thus we can assume $f_*\mathcal{O}_X(ND)\neq0$. By Proposition \ref{equi} and Corollary \ref{red}, we can assume $f_*\mathcal{O}_X(ND)$ is globally generated.
We use a similar method as in the proof of Proposition \ref{equi}. We consider the following two adjoint morphisms
$$f^*f_*\mathcal{O}_{X}(D)\to \mathcal{O}_{X}(D)\quad \text{and} \quad f^*f_*\mathcal{O}_{X}(ND)\to \mathcal{O}_{X}(ND).$$
We can take a log resolution $\mu\colon Y\to X$ of $(X, \Delta)$ and the relative base ideals of $D$ and $ND$.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
Y \arrow[r, "\mu"] \arrow[dr, "g" swap] & X \arrow[d, "f"] \\
& A
\end{tikzcd}
\end{center}
Then we have
$$K_Y+\Delta_Y\sim_\mathbb{Q}\mu^*(K_{X}+\Delta)+E,$$
where the ${\mathbb Q}$-divisors $\Delta_Y$ and $E$ are effective and have no common components and $E$ is $\mu$-exceptional. We also have that the images of the new adjoint morphisms
$$g^*g_*\mathcal{O}_{Y}(D_Y)\to \mathcal{O}_{Y}(D_Y)\quad \text{and} \quad g^*g_*\mathcal{O}_{Y}(ND_Y)\to \mathcal{O}_{Y}(ND_Y)$$
are line bundles $\mathcal{O}_{Y}(D_Y-F)$ and $\mathcal{O}_{Y}(ND_Y-F_N)$ where $D_Y=\mu^*D$ is a Cartier divisor, $F$ and $F_N$ are effective divisors and $\Delta_Y+E+F+F_N$ has simple normal crossings support. Since $(X, \Delta)$ is klt, we know $(Y, \Delta_Y':=\Delta_Y+\frac{\lceil mE\rceil}{m}-E)$ is also klt and
$$m(K_Y+\Delta_Y')\sim_\mathbb{Q}\mu^*(m(K_{X}+\Delta))+\lceil mE\rceil\sim_{{\mathbb Q}} D_Y+\lceil mE\rceil.$$
Denote $D_Y+\lceil mE\rceil$ by $G$. We have $g_*\mathcal{O}_{Y}(G)\cong f_*\mathcal{O}_{X}(D)$ and $g_*\mathcal{O}_{Y}(NG)\cong f_*\mathcal{O}_{X}(ND)$ since $\lceil mE\rceil$ is $\mu$-exceptional and effective. The images of the two adjoint morphisms
$$g^*g_*\mathcal{O}_{Y}(G)\to \mathcal{O}_{Y}(G)\quad \text{and} \quad g^*g_*\mathcal{O}_{Y}(NG)\to \mathcal{O}_{Y}(NG)$$
are $\mathcal{O}_{Y}(G-F-\lceil mE\rceil)$ and $\mathcal{O}_{Y}(NG-F_N-N\lceil mE\rceil)$. Let $G'$ be the divisor $F+\lceil mE\rceil$ and $G'_N$ the divisor $F_N+N\lceil mE\rceil$. We claim that $G'_N\leq NG'$. We have the natural morphism
$$(g_*\mathcal{O}_{Y}(G))^{\otimes N}\to g_*\mathcal{O}_{Y}(NG)$$
and the following commutative diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
(g^*g_*\mathcal{O}_{Y}(G))^{\otimes N}\arrow[r] \arrow[d] & g^*g_*\mathcal{O}_{Y}(NG) \arrow[d] \\
(\mathcal{O}_{Y}(G))^{\otimes N} \arrow[r] & \mathcal{O}_{Y}(NG)
\end{tikzcd}
\end{center}
We deduce from the diagram that $\mathcal{O}_{Y}(N(G-G'))$ is contained in $\mathcal{O}_{Y}(NG-G'_N)$ and thus $\mathcal{O}_{Y}(-NG')$ is contained in $\mathcal{O}_{Y}(-G'_N)$ as subsheaves of $\mathcal{O}_{Y}$ which proves the claim.
We know $g_*\mathcal{O}_{Y}(NG)$ is globally generated and thus $\mathcal{O}_{Y}(NG-G'_N)$ is globally generated. By Bertini's theorem, we can pick a general Cartier divisor $H\sim NG-G'_N$ which is reduced and effective such that $H$ and $\Delta_Y'+G'+G'_N$ have no common components and $H+\Delta_Y'+G'+G'_N$ has simple normal crossings support. We define an effective divisor $T$ as
$$T:=\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{Nm}G'_N \bibitemgg\rfloor.$$
We have
$$T\leq\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{m}G' \bibitemgg\rfloor\leq G'$$
since $G'_N\leq NG'$ and the coefficient of each irreducible component of $\Delta_Y'$ is smaller than $1$. Since $0\leq T\leq G'$, we know $g_*\mathcal{O}_{Y}(G)\cong g_*\mathcal{O}_{Y}(G-T)$ by the same argument of Proposition \ref{equi}. We have
$$G-T\sim_{{\mathbb Q}}m(K_Y+\Delta_Y')-T=K_Y+\Delta_Y'+(m-1)(K_Y+\Delta_Y')-T$$
$$\sim_{{\mathbb Q}}K_Y+\Delta_Y'+\frac{m-1}{m}G-T\sim_{{\mathbb Q}}K_Y+\Delta_Y'+\frac{m-1}{Nm}(G'_N+H)-T$$
$$=K_Y+\frac{m-1}{Nm}H+\Delta_Y'+\frac{m-1}{Nm}G'_N-\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{Nm}G'_N \bibitemgg\rfloor.$$
We define $N$ as
$$N:=\frac{m-1}{Nm}H+\Delta_Y'+\frac{m-1}{Nm}G'_N-\bibitemgg\lfloor \Delta_Y'+\frac{m-1}{Nm}G'_N \bibitemgg\rfloor.$$
The effective ${\mathbb Q}$-divisor $N$ has simple normal crossings support and the coefficient of each irreducible component of itself is smaller than $1$. Thus $(Y, N)$ is a klt pair. Since $G-T\sim_{{\mathbb Q}}K_Y+N$, the sheaf $f_*\mathcal{O}_{X}(D)\cong g_*\mathcal{O}_{Y}(G)\cong g_*\mathcal{O}_{Y}(G-T)$ has the Chen-Jiang decomposition by Lemma \ref{m=1}.
\end{proof}
We will prove our main theorems by induction on $\dim A$. Our next lemma treats the central part of the induction. To prove it, we follow the strategy used in \cite[Sections 8 and 9]{LPS20} together with some additional techniques.
\begin{equation}gin{prop}\label{SNC}
Assume that Theorem \ref{main3} is true when the base abelian variety is of dimension up to $n-1$. Let $f$ be a morphism from a log smooth klt pair $(X, \Delta)$ to an abelian variety $A$ of dimension $n$. Then $f_*\mathcal{O}_X(N(K_X+\Delta))$ has the Chen-Jiang decomposition for every positive integer $N$ which is sufficiently big and divisible.
\end{prop}
\begin{equation}gin{proof}
We choose $N$ sufficiently divisible such that $N(K_X+\Delta)$ is Cartier. We can assume that $f_*\mathcal{O}_X(N(K_X+\Delta))\neq0$ for every positive integer $N$ which is sufficiently big and divisible. We choose $N$ sufficiently big and divisible such that Lemma \ref{split} works for $f_*\mathcal{O}_X(N(K_X+\Delta))$.
We denote $f_*\mathcal{O}_X(N(K_X+\Delta))$ by $\mathcal{F}$ and $N(K_X+\Delta)$ by $D$. The sheaf $\mathcal{F}$ is a GV-sheaf by \cite[Variant 5.5]{PS14}. We consider the continuous evaluation morphism associated to $\mathcal{F}$
$$e_{\mathcal{F}}\colon \bibitemgoplus_{\alpha\in \Pic^0(A)\atop \mathrm{torsion}}H^0(A, \mathcal{F}\otimes\alpha)\otimes\alpha^{-1}\to \mathcal{F}$$
and the sheaf $\mathcal{G}:=\Image e_{\mathcal{F}}\subseteq \mathcal{F}$. Lemma \ref{split} will work for the new pair for the same $N$ if we take an isogeny on $A$ and do the base change by the discussion after it. Thus we can assume that $\mathcal{G}$ is globally generated by Lemma \ref{tor} and Corollary \ref{red}. By Proposition \ref{equi} we only need to prove that $\mathcal{G}=\mathcal{F}$ to deduce our lemma. We consider the adjoint morphism
$$f^*\mathcal{G}\to\mathcal{O}_X(N(K_X+\Delta)).$$
By taking a log resolution, we can assume the image of the adjoint morphism is of the form $\mathcal{O}_X(N(K_X+\Delta)-E)$ where $E$ is an effective divisor and that $E+\Delta$ has simple normal crossings support. Since $\mathcal{G}$ is globally generated, the line bundle $\mathcal{O}_X(N(K_X+\Delta)-E)$ is globally generated and thus there exists an effective and reduced divisor $H\sim N(K_X+\Delta)-E$ such that $H$ and $E+\Delta$ have no common components and $H+E+\Delta$ has simple normal crossings support. By the same argument of Proposition \ref{equi}, there exists an effective divisor $T\leq E$ such that $D-T\sim_{{\mathbb Q}}K_X+\Delta'$ where $(X, \Delta')$ is a log smooth klt pair. We claim that the inclusion
$$\mathcal{G}\hookrightarrow f_*\mathcal{O}_X(D-T)$$
is an identity. Since $D-T\sim_{{\mathbb Q}}K_X+\Delta'$, the continuous evaluation morphism associated to $f_*\mathcal{O}_X(D-T)$ is surjective by Lemma \ref{m=1} and Proposition \ref{equi}. We know that
$$H^0(A, \mathcal{G}\otimes\alpha)\subseteq H^0(A, f_*\mathcal{O}_X(D-T)\otimes\alpha)\subseteq H^0(A, f_*\mathcal{O}_X(D)\otimes\alpha)$$
for every torsion line bundle $\alpha\in\Pic^0(A)$ and the two section spaces on the left and right are equal from the definition of $\mathcal{G}$. Thus we deduce that
$$\mathcal{G}=f_*\mathcal{O}_X(D-T)$$
since the continuous evaluation morphisms associated to $\mathcal{G}$ and $f_*\mathcal{O}_X(D-T)$ are surjective. By Lemma \ref{m=1}, there exists a generically finite surjective morphism $h\colon W\to X$ from a smooth projective variety $W$ such that $\mathcal{G}=f_*\mathcal{O}_X(D-T)$ is a direct summand of $(f\circ h)_*\mathcal{O}_W(K_W)$. Thus $\mathcal{G}$ is a GV-sheaf by \cite{GL87} and \cite{Hac04} and the cohomological support loci $V_l^i(A, \mathcal{G})$ are finite unions of torsion subvarieties of $\Pic^0(A)$ for every $i$ and $l$ by \cite{GL91}, \cite{Sim93} and \cite[Lemma 10.3]{HPS18}. Thus the sheaf $\mathcal{H}:=\mathcal{F}/\mathcal{G}$ is also a GV-sheaf. We only need to prove $\mathcal{H}=0$ or $V^0(A, \mathcal{H})$ is empty to deduce our lemma. By \cite[Theorem 1.3]{Shi16}, the cohomological support loci $V_l^0(A, \mathcal{F})$ are finite unions of torsion subvarieties of $\Pic^0(A)$ for every $l$. We deduce that
$$H^0(A, \mathcal{G}\otimes\alpha)=H^0(A, \mathcal{F}\otimes\alpha)$$
for every $\alpha\in \Pic^0(A)$ since it is true when $\alpha$ is a torsion point and torsion points are dense inside $V_l^0(A, \mathcal{G})$ and $V_l^0(A, \mathcal{F})$ by the structures of these sets. By the exact sequence
$$0\to H^0(A, \mathcal{G}\otimes\alpha)\to H^0(A, \mathcal{F}\otimes\alpha)\to H^0(A, \mathcal{H}\otimes\alpha)\to H^1(A, \mathcal{G}\otimes\alpha),$$
we deduce that $V^0(A, \mathcal{H})\subseteq V^1(A, \mathcal{G})$. Thus $V^0(A, \mathcal{H})$ is contained in a finite union of torsion subvarieties of codimension $\geq1$. We can choose an isogeny and do the base change using this isogeny freely by Corollary \ref{red}. Thus we can assume that all the irreducible components of $V^1(A, \mathcal{G})$ are abelian varieties. If $V^1(A, \mathcal{G})$ is empty, then $\mathcal{H}=0$. Assume that $V^1(A, \mathcal{G})$ is not empty. If $V^1(A, \mathcal{G})$ does not contain any positive-dimensional abelian subvariety, then $V^1(A, \mathcal{G})=\{\mathcal{O}_A\}$. Then $\mathcal{H}=0$ follows from Lemma \ref{split} and the same proof of \cite[Proposition 8.3]{LPS20}.
We now assume that $V^1(A, \mathcal{G})$ contains a positive-dimensional abelian subvariety. If $V^0(A, \mathcal{H})$ does not intersect any of these positive-dimensional abelian subvarieties, then $V^0(A, \mathcal{H})$ is empty and thus $\mathcal{H}=0$. We assume that $V^0(A, \mathcal{H})$ intersects at least one of them which is denoted by $C\subseteq V^1(A, \mathcal{G})$. Then the abelian subvariety $C$ is the image of the pullback morphism $p^*\colon\Pic^0(B)\to\Pic^0(A)$ where $p\colon A\to B$ is a fibration to an abelian variety $B$. Since $\mathcal{G}$ is a GV-sheaf, we have $1\leq\dim B\leq n-1$. After a base change by an isogeny on $B$, we can assume $A=B\times F$ where $B$ and $F$ are abelian varieties, $p$ is the first projection and $q$ is the second projection. Denote $p\circ f$ by $g$.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X \arrow[r, "f"] \arrow[rr, bend right, "g"] & B\times F \arrow[r, "p"] & B
\end{tikzcd}
\end{center}
We know the continuous evaluation morphism $e_{p_*\mathcal{G}}$ associated to $p_*\mathcal{G}$ is surjective since $p_*\mathcal{G}$ is a direct summand of $(g\circ h)_*\mathcal{O}_W(K_W)$. Since we assume that Theorem \ref{main3} is true when the base abelian variety is of dimension up to $n-1$, the continuous evaluation morphism $e_{p_*\mathcal{F}}$ associated to $p_*\mathcal{F}$ is surjective by Proposition \ref{equi} and thus $p_*\mathcal{G}=p_*\mathcal{F}$ since we have
$$H^0(B, p_*\mathcal{G}\otimes\begin{equation}ta)=H^0(B, p_*\mathcal{F}\otimes\begin{equation}ta)$$
for every $\begin{equation}ta\in\Pic^0(B)$. By a similar argument, we can prove that
$$p_*(\mathcal{G}\otimes\alpha)=p_*(\mathcal{F}\otimes\alpha)$$
for every torsion point $\alpha\in\Pic^0(A)$.
We now consider $Z=g(X)$ which is a reduced and irreducible subvariety of $B$ and $\mathcal{G}$, $\mathcal{F}$ and $\mathcal{H}$ as coherent sheaves on $f(X)\subseteq Z\times F$. For any $b\in Z$, let $X_b=g^{-1}(b)$ and $f_b$ the induced morphism as in the following base change diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X_b \arrow[r, hook, "h_b"] \arrow[d, "f_b"] & X \arrow[d, "f"] \arrow[dd, bend left=60, "g"] \\
F \arrow[r, hook, "k_b"] \arrow[d, ""] & Z\times F \arrow[d, "p"] \\
\{b\} \arrow[r, hook, "i_b" ] & Z
\end{tikzcd}
\end{center}
The morphism $k_b$ is $(i_b, \id_F)$. By the generic smoothness theorem, there exists a nonempty open subset $U'\subseteq Z$ such that $g$ is smooth over $U'$ and $(X_b, \Delta_b)$ is a log smooth klt pair for every $b\in U'$ where $\Delta_b=\Delta|_{X_b}$. Denote $(f_{b})_*\mathcal{O}_{X_b}(N(K_{X_b}+\Delta_b))$ by $\mathcal{F}_b$. By \cite[Proposition 4.1]{LPS20}, we can shrink $U'$ such that the base change morphism
$$k_b^*\mathcal{F}\to\mathcal{F}_b$$
is an isomorphism for every $b\in U'$ and thus the base change morphism
$$k_b^*(\mathcal{F}\otimes q^*\gamma)\to\mathcal{F}_b\otimes\gamma$$
is an isomorphism for every $b\in U'$ and every torsion point $\gamma\in\Pic^0(F)$ where $q$ is the second projection onto $F$. Since we assume that Theorem \ref{main3} is true when the base abelian variety is of dimension up to $n-1$ and $1\leq\dim F\leq n-1$, the continuous evaluation morphism associated to $\mathcal{F}_b$
$$e_{\mathcal{F}_b}\colon \bibitemgoplus_{\gamma\in \Pic^0(F)\atop \mathrm{torsion}}H^0(F, \mathcal{F}_b\otimes\gamma)\otimes\gamma^{-1}\to\mathcal{F}_b$$
is surjective for every $b\in U'$. For every torsion point $\gamma\in\Pic^0(F)$, there exists a nonempty open subset $U_{\gamma}\subseteq U'$ such that the base change morphism
$$i_b^*\Big(g_*\bibitemg(\mathcal{O}_X(N(K_X+\Delta))\otimes f^*q^*\gamma\bibitemg)\Big)\to H^0(X_b, \mathcal{O}_{X_b}(N(K_{X_b}+\Delta_b))\otimes f_b^*\gamma)$$
is surjective for every $b\in U_{\gamma}$ by the base change theorem. Thus we know
$$i_b^*(p_*(\mathcal{F}\otimes q^*\gamma))\to H^0(F, \mathcal{F}_b\otimes\gamma)$$
is surjective for every $b\in U_{\gamma}$ by the projection formula. Consider the set
$$U:=\bibitemgcap_{\gamma\in\Pic^0(F)\atop\mathrm{torsion}}U_{\gamma}$$
which is an intersection of countably many nonempty open subsets of $Z$ since $\Pic^0(F)$ only has countably many torsion points. The set $U$ is not empty since an irreducible variety over an uncountable and algebraically closed field cannot be a countable union of proper closed subsets. By the surjective morphisms above, we deduce that the morphism $e$ induced from the adjoint morphisms
$$e\colon\bibitemgoplus_{\gamma\in \Pic^0(F)\atop \mathrm{torsion}}p^*p_*(\mathcal{F}\otimes q^*\gamma)\otimes q^*\gamma^{-1}\to\mathcal{F}$$
is surjective after restricted to the fiber $p^{-1}(b)$ for every $b\in U$ since $e_{\mathcal{F}_b}$ is surjective for every $b\in U$. Then $e$ is surjective at every point of the fiber $p^{-1}(b)$ for every $b\in U$ by Nakayama's lemma since $\mathcal{F}$ is coherent. Since
$$p_*(\mathcal{G}\otimes q^*\gamma)=p_*(\mathcal{F}\otimes q^*\gamma)$$
for every torsion point $\gamma\in\Pic^0(F)$, the morphism $e$ factors through the subsheaf $\mathcal{G}$ of $\mathcal{F}$. We deduce that $\mathcal{G}=\mathcal{F}$ on $p^{-1}(U)$ and $\Supp\mathcal{H}$ does not intersect the set $p^{-1}(U)$. Denote $Z\times F\setminus p^{-1}(U_{\gamma})$ by $V_{\gamma}$ which is a proper closed subset of $Z\times F$. Then we have
$$\Supp\mathcal{H}\subseteq \bibitemgcup_{\gamma\in\Pic^0(F)\atop\mathrm{torsion}}V_{\gamma}.$$
Since $\mathcal{H}$ is coherent, $\Supp\mathcal{H}$ is closed and can be decomposed as a union of irreducible components
$$\Supp\mathcal{H}=\bibitemgcup_{k\in K}Z_k$$
where $K$ is a finite index set. We deduce that
$$Z_k=\bibitemgcup_{\gamma\in\Pic^0(F)\atop\mathrm{torsion}}(Z_k\cap V_{\gamma}).$$
Since an irreducible variety over an uncountable and algebraically closed field cannot be a countable union of proper closed subsets, we deduce that $Z_k\subseteq V_{\gamma_k}$ for some torsion point $\gamma_k\in\Pic^0(F)$. Thus $\Supp\mathcal{H}$ does not intersect the nonempty open set
$$p^{-1}(\bibitemgcap_{k\in K}U_{\gamma_k})$$
and we deduce that $p_*\mathcal{H}$ is a torsion sheaf on $Z$. We have the exact sequence
$$0\to p_*\mathcal{G}\to p_*\mathcal{F}\to p_*\mathcal{H}\to R^1p_*\mathcal{G}.$$
We know $\mathcal{G}$ is a direct summand of $(f\circ h)_*\mathcal{O}_W(K_W)$. By the five-term exact sequence, we deduce that $R^1p_*((f\circ h)_*\mathcal{O}_W(K_W))$ is a subsheaf of $R^1(p\circ f\circ h)_*\mathcal{O}_W(K_W)$ which is a torsion-free sheaf by \cite[Theorem 2.1]{Kol86}. Since $p_*\mathcal{G}=p_*\mathcal{F}$, the torsion sheaf $p_*\mathcal{H}$ is a subsheaf of $R^1p_*\mathcal{G}$ which is a torsion-free sheaf. Thus we deduce that $p_*\mathcal{H}=0$ and
$$H^0(A, \mathcal{H}\otimes p^*\begin{equation}ta)\cong H^0(B, p_*\mathcal{H}\otimes\begin{equation}ta)=0$$
for every $\begin{equation}ta\in\Pic^0(B)$. Thus $V^0(A, \mathcal{H})$ does not intersect $C$ which is a contradiction and we finish our proof.
\end{proof}
\begin{equation}gin{rem}\label{genera}
The invariance of plurigenera for smooth families of smooth varieties is used in the proof of \cite[Lemma 9.2]{LPS20} to conclude that there exists a nonempty open subset $O\subseteq Z$ such that $\Supp\mathcal{H}$ does not intersect $O\times F$. Our argument in Proposition \ref{SNC} shows that we can prove the same result even in the case of klt pairs without appealing to the invariance of plurigenera.
\end{rem}
Next, we prove our main theorems.
\begin{equation}gin{proof}[Proof of Theorem \ref{main3}]
We prove the theorem by induction on $\dim A$. The existence of the Chen-Jiang decomposition follows from Propositions \ref{N} and \ref{SNC} by taking a log resolution. The statement that we can choose $\alpha_i$ which becomes trivial when pulled back by the isogeny $\varphi$ in Theorem \ref{main1} follows from the same proof of \cite[Theorem C]{LPS20}.
\end{proof}
\begin{equation}gin{proof}[Proof of Theorem \ref{main1}]
For an individual $l$, the existence of such an isogeny follows from Theorem \ref{main3} and Proposition \ref{equi}. Next we prove that we can find an isogeny which works for every $l$. By \cite[Theorem 1.2]{BCHM}, for $N\in{\mathbb N}$ sufficiently divisible, the $\mathcal{O}_A$-algebra
$$\mathfrak{R}(f, ND)=\bibitemgoplus_{i\in{\mathbb N}}f_*\mathcal{O}_X(iND)$$
is finitely generated. Thus the $\mathcal{O}_A$-algebra $\mathfrak{R}(f, D)$ is finitely generated by a Theorem of E. Noether on the finiteness of integral closure. Thus we can find an isogeny which works for every $l$.
\end{proof}
\begin{equation}gin{proof}[Proof of Theorem \ref{main2}]
It follows from Theorem \ref{main3} and Proposition \ref{equi}.
\end{proof}
As mentioned in the introduction, for the sake of completeness we also give an alternative approach to the main theorems, which uses the minimal model program as in the following lemma. However, we need an extra assumption which is that the general fiber $(F, \Delta|_F)$ of $f$ has a good minimal model. The good minimal model conjecture is a famous conjecture in birational geometry which states that every klt pair $(X, \Delta)$ has a good minimal model if $K_X+\Delta$ is pseudoeffective. The conjecture is known to hold in dimensions up to $3$ by a lot of work and when the pair $(X, \Delta)$ is of log general type by \cite{BCHM}.
\begin{equation}gin{prop}\label{MMP}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m>0$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. If the general fiber $(F, \Delta|_F)$ of $f$ has a good minimal model, then $f_*\mathcal{O}_X(ND)$ has the Chen-Jiang decomposition for every positive integer $N$ which is sufficiently big and divisible.
\end{prop}
\begin{equation}gin{proof}
The Stein factorization of $f$ gives a decomposition of $f$ as $g\circ h$ where $h\colon X\to Z$ is a fibration to a normal projective variety $Z$ and $g\colon Z\to A$ is a finite morphism. By assumption, the general fiber $(F, \Delta|_F)$ of $h$ has a good minimal model and $(X, \Delta)$ is klt. Thus by \cite[Theorem 1.2]{BCHM} and \cite[Theorem 2.12]{HX13}, $(X, \Delta)$ has a good minimal model $(Y, \Delta_Y)$ over $Z$. Let $\xi\colon X\dasharrow Y$ be the birational contraction and $(p,q)\colon W\to X\times Y$ a smooth resolution of indeterminacies of $\xi$ as in the following diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
& W \arrow[dl, "p" swap] \arrow[dr, "q"] & \\
X \arrow[rr, dashed, "\xi" ] \arrow[dr, "h"]&& Y \arrow[dl, "h'" swap]\arrow[d, "f'"]\\&Z \arrow[r, "g"]&A
\end{tikzcd}
\end{center}
The pair $(Y, \Delta_Y)$ is klt and we have
$$ p^*(K_X+\Delta)\sim_{\mathbb Q} q^*(K_Y+\Delta_Y)+E,$$
where $E$ is an effective $q$-exceptional ${\mathbb Q}$-divisor. We can choose a positive integer $N$ sufficiently divisible such that $Nm(K_X+\Delta)$, $Nm(K_Y+\Delta_Y)$ and $NmE$ are Cartier divisors, $ND\sim Nm(K_X+\Delta)$ and
$$p^*(Nm(K_X+\Delta))\sim q^*(Nm(K_Y+\Delta_Y))+NmE.$$
Note that $m>0$ is a rational number. We can also make $N$ sufficiently big such that $Nm\geq 1$. We deduce that
$$f_*\mathcal{O}_X(ND)\cong f_*\mathcal{O}_X(Nm(K_X+\Delta))\cong f_*p_*\mathcal{O}_W\bibitemg(p^*(Nm(K_X+\Delta))\bibitemg)$$
$$\cong f'_*q_*\mathcal{O}_W\bibitemg(q^*(Nm(K_Y+\Delta_Y))+NmE\bibitemg)\cong f'_*\mathcal{O}_Y(Nm(K_Y+\Delta_Y))$$
since $NmE$ is an effective $q$-exceptional divisor. Since $g$ is a finite morphism and $K_Y+\Delta_Y$ is semiample over $Z$, we deduce that $K_Y+\Delta_Y$ is semiample over $A$. By \cite[Corollary 1.2]{Hu16}, $K_Y+\Delta_Y$ is semiample and thus $Nm(K_Y+\Delta_Y)$ is semiample. Thus we can find a positive integer $M$ which is sufficiently big such that $MNm(K_Y+\Delta_Y)\sim H$ where $H$ is a reduced and effective divisor such that $(Y,\Delta_Y+\frac{1}{M}H)$ is still klt. We have that
$$Nm(K_Y+\Delta_Y)\sim_{{\mathbb Q}}K_Y+\Delta_Y+\frac{Nm-1}{MNm}H.$$
The pair $(Y, \Delta_Y+\frac{Nm-1}{MNm}H)$ is klt since $Nm\geq 1$. Thus $f'_*\mathcal{O}_Y(Nm(K_Y+\Delta_Y))$ has the Chen-Jiang decomposition by Lemma \ref{m=1}. Then $f_*\mathcal{O}_X(ND)$ has the Chen-Jiang decomposition.
\end{proof}
\begin{equation}gin{rem}\label{not}
The proof of Proposition \ref{MMP} is not purely algebraic since \cite[Corollary 1.2]{Hu16} builds on the result by \cite{BC15} which exploits the extension theorem from \cite{DHP13}. The proof of the extension theorem needs applications of the theory of singular Hermitian metrics.
\end{rem}
Next, we give a simple corollary of Theorems \ref{main1}, \ref{main2} and \ref{main3} regarding some special dlt pairs.
\begin{equation}gin{coro}
Let $f$ be a morphism from a ${\mathbb Q}$-factorial dlt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number, $H$ an ample ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$ and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta+H)$. Then the conclusions of Theorems \ref{main1}, \ref{main2} and \ref{main3} are true.
\end{coro}
\begin{equation}gin{proof}
Denote $\lfloor\Delta\rfloor$ by $S$. Since $H$ is ample, we can choose a rational number $\varepsilon>0$ which is sufficiently small such that $H':=H+\varepsilon S$ is an ample ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor. Since $(X, \Delta)$ is dlt, the pair $(X, \Delta-\varepsilon S)$ is klt and thus we can choose an effective ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $E\sim_{{\mathbb Q}}H'$ such that $(X, \Delta':=\Delta-\varepsilon S+E)$ is klt. We have that
$$D\sim_{{\mathbb Q}}m(K_X+\Delta+H)\sim_{{\mathbb Q}}m(K_X+\Delta-\varepsilon S+H')\sim_{{\mathbb Q}}m(K_X+\Delta')$$
and this finishes the proof.
\end{proof}
\section{Applications}\label{4}
We start with a corollary of Theorem \ref{main2}. Note that \cite{PS14} and \cite{Shi16} have some results related to statements (i) and (ii) of Corollary \ref{coro1} under different assumptions.
\begin{equation}gin{coro}\label{coro1}
Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{{\mathbb Q}}m(K_X+\Delta)$. Then:
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] The sheaf $f_*\mathcal{O}_X(D)$ is a $\GV$-sheaf.
\item[$\mathrm{(ii)}$] The cohomological support loci $V_l^i(A, f_*\mathcal{O}_X(D))$ are finite unions of torsion subvarieties of $\Pic^0(A)$ for every $i$ and $l$.
\item[$\mathrm{(iii)}$] The Fourier-Mukai transform $\bold{R}\hat{\mathcal{S}}\bold{R}\mathcal{H}om_{\mathcal{O}_A}(f_*\mathcal{O}_X(D), \mathcal{O}_A)$ is computed locally around each point by a linear complex of trivial vector bundles.
\end{enumerate}
\end{coro}
\begin{equation}gin{proof}
By Theorem \ref{main2}, statement (i) follows from \cite{GL87} and \cite{Hac04}, (ii) follows from \cite{GL91}, \cite{Sim93} and \cite[Lemma 10.3]{HPS18} and (iii) follows from \cite{CH02b} and \cite{LPS11} which are based on \cite{GL91}.
\end{proof}
Next we prove Theorem \ref{main6}. Given a klt pair $(X, \Delta)$ and a Cartier divisor $D\sim_{{\mathbb Q}}m(K_X+\Delta)$ on $X$ such that $\kappa(X, D)\geq0$ where $m>1$ is a rational number, we consider the Iitaka fibration associated to $D$. After a birational modification of $X$, we may assume from the beginning that the Iitaka fibration is a morphism $f\colon X\to Y$ where $Y$ is a smooth projective variety of dimension $\kappa(X, D)$. Note that we do not assume $(X, \Delta)$ is log smooth. Since $(X, \Delta)$ is klt, $X$ is of rational singularities. Thus its Albanese variety coincides with the Albanese variety of any of its log resolution by \cite[Proposition 2.3]{Rei83} and \cite[Lemma 8.1]{Kaw85a}. Thus by the universal property of the Albanese variety, we have the following commutative diagram
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X \arrow[r, "a_X"] \arrow[d, "f"] \arrow[dr, "g"]& \Alb(X) \arrow[d, "a_f"] \\
Y \arrow[r, "a_Y" ] & \Alb(Y)
\end{tikzcd}
\end{center}
where $\Alb(X)$ and $\Alb(Y)$ are the two Albanese varieties and $a_X$ and $a_Y$ are the two Albanese morphisms. Denote $a_Y\circ f$ by $g$. We will give an explicit description of the cohomological support locus
$$V^0(X, \mathcal{O}_X(D))=\{\alpha\in\Pic^0(X)\mid\dim H^0(X, \mathcal{O}_X(D)\otimes\alpha)\geq 1\}$$
using the Iitaka fibration $f$. The morphism $f^*\colon \Pic^0(Y)\to \Pic^0(X)$ is injective since $f$ is a fibration.
\begin{equation}gin{proof}[Proof of Theorem \ref{main6}]
We claim that it suffices to prove the theorem when $(X, \Delta)$ is log smooth. We take a log resolution of $(X, \Delta)$ denoted by $\mu\colon W\to X$. Then we have
$$K_W+\Delta_W\sim_\mathbb{Q}\mu^*(K_{X}+\Delta)+E,$$
where the ${\mathbb Q}$-divisors $\Delta_W$ and $E$ are effective and have no common components, $E$ is $\mu$-exceptional and $\Delta_W+E$ has simple normal crossings support. We know $(W, \Delta_W':=\Delta_W+\frac{\lceil mE\rceil}{m}-E)$ is log smooth and klt since $m>1$ and
$$m(K_W+\Delta_W')\sim_\mathbb{Q}\mu^*(m(K_{X}+\Delta))+\lceil mE\rceil\sim_{{\mathbb Q}} \mu^*D+\lceil mE\rceil.$$
By \cite[Lemma 8.1]{Kaw85a},
$$\mu^*\colon\Pic^0(X)\to \Pic^0(W)$$
is an isomorphism since $X$ is of rational singularities. Thus it suffices to prove the theorem when $(X, \Delta)$ is log smooth by considering $\mu^*D+\lceil mE\rceil$ since $\lceil mE\rceil$ is effective and $\mu$-exceptional.
We prove statement (i) first. We use $\alpha'$ to denote the Cartier divisor corresponding to the line bundle $\alpha$. We fix the torsion point $\alpha$. Given a very ample divisor $H'$ on $\Alb(Y)$, we can find a positive integer $c$ which is sufficiently big such that $cD\sim g^*H'+B'$ where $B'$ is an effective divisor on $X$ since $g$ factors through the Iitaka fibration associated to $D$. We have that $c(D+g^*L)\sim g^*(H'+cL)+B'$ where the divisor $H'+cL$ is ample. Thus we can find a positive integer $d$ which is sufficiently big and divided by $c$ such that
$$d(D+g^*L)\sim g^*H+B$$
where $B$ is an effective divisor on $X$ and $H$ is a very ample divisor on $\Alb(Y)$.
We take a log resolution $\mu\colon W\to X$ of $(X, \Delta)$ and the linear systems $|D+g^*L+\alpha'|$ and $|p(D+g^*L)|$ where $p$ is a positive integer which is sufficiently divisible such that $\alpha^{\otimes p}$ is trivial and $p$ is divided by $d$ as in the following diagram.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
W \arrow[r, "\mu"] & X \arrow[r, "a_X"] \arrow[d, "f"] \arrow[dr, "g"]& \Alb(X) \arrow[d, "a_f"] \\
& Y \arrow[r, "a_Y" ] & \Alb(Y)
\end{tikzcd}
\end{center}
Then we have
$$K_W+\Delta_W\sim_\mathbb{Q}\mu^*(K_{X}+\Delta)+E,$$
where the ${\mathbb Q}$-divisors $\Delta_W$ and $E$ are effective and have no common components and $E$ is $\mu$-exceptional. We also have that
$$\mu^*|D+g^*L+\alpha'|=|H_1|+F_1\quad \text{and} \quad \mu^*|p(D+g^*L)|=|H_p|+F_p,$$ where the linear systems $|H_1|$ and $|H_p|$ are base point free, the effective divisors $F_1$ and $F_p$ are the fixed parts of the corresponding linear systems and $\Delta_W+E+F_1+F_p$ has simple normal crossings support. We observe that $pF_1\geq F_p$ and $\frac{p}{d}\mu^*B\geq F_p$. We define a divisor $T$ as
$$T:=\bibitemgg\lfloor \Delta_W-E+\frac{m-1}{mp}F_p\bibitemgg\rfloor.$$
We have that $T\geq\lfloor-E\rfloor$ and thus $-T\leq\lceil E\rceil$. We also have that
$$T\leq\bibitemgg\lfloor \Delta_W+\frac{m-1}{mp}F_p\bibitemgg\rfloor\leq \bibitemgg\lfloor\Delta_W+\frac{m-1}{m}F_1\bibitemgg\rfloor\leq F_1$$
since $(W, \Delta_W)$ is klt. We denote $D+g^*L+\alpha'$ by $D'$. Thus we have
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
H^0(W, \mathcal{O}_W(\mu^*D'-F_1)) \arrow[r, "a_1"] \arrow[d, "a_2"] & H^0(W, \mathcal{O}_W(\mu^*D'-T))\arrow[d, "a_3"] \\
H^0(W, \mathcal{O}_W(\mu^*D')) \arrow[r, "a_4"] & H^0(W, \mathcal{O}_W(\mu^*D'+\lceil E\rceil))
\end{tikzcd}
\end{center}
where $a_2$ and $a_4$ are isomorphisms since $F_1$ is the fixed part of the linear system $\mu^*|D'|$ and $\lceil E\rceil$ is $\mu$-exceptional and effective. We also know $a_1$ and $a_3$ are injective and thus $a_1$ and $a_3$ are isomorphisms. Thus we have
$$H^0(X, \mathcal{O}_X(D'))\cong H^0(W, \mathcal{O}_W(\mu^*D'-T)).$$
Since $\alpha$ is a torsion line bundle, we have $\alpha'\sim_{{\mathbb Q}}0$. For any $\varepsilon\in{\mathbb Q}$, we have
$$\mu^*(D+\alpha')\sim_{{\mathbb Q}}\frac{1}{m}\mu^*D+(\frac{m-1}{m}-\varepsilon)\mu^*D+\varepsilon\mu^*D$$
$$\sim_{{\mathbb Q}}\mu^*(K_X+\Delta)+\frac{m-1-m\varepsilon}{m}(\frac{1}{p}(H_p+F_p)-\mu^*g^*L)+\varepsilon\mu^*(\frac{1}{d}(g^*H+B)-g^*L)$$
$$\sim_{{\mathbb Q}}\frac{\varepsilon}{d}\mu^*g^*H+K_W+\Delta_W-E+\frac{\varepsilon}{d}\mu^*B-\frac{m-1}{m}\mu^*g^*L+\frac{m-1-m\varepsilon}{mp}(H_p+F_p).$$
Thus we deduce that
$$\mu^*D'-T\sim_{{\mathbb Q}}\mu^*(D+g^*L+\alpha')-T$$
$$\sim_{{\mathbb Q}}\mu^*g^*(\frac{\varepsilon}{d}H+\frac{1}{m}L)+K_W+\Delta_W-E+\frac{\varepsilon}{d}\mu^*B+\frac{m-1-m\varepsilon}{mp}(H_p+F_p)-T.$$
Since $|H_p|$ is base point free and $\Delta_W+E+F_p$ has simple normal crossings support, we can choose a reduced and effective divisor $E_p\sim H_p$ such that $(W, \frac{m-1}{mp}E_p+\Delta_W-E+\frac{m-1}{mp}F_p-T)$ is klt by the definition of $T$. Since $\frac{p}{d}\mu^*B\geq F_p$ and $m>1$, we can choose $\varepsilon\geq0$ sufficiently small and deduce that
$$\Delta'_W(\varepsilon):=\Delta_W-E+\frac{\varepsilon}{d}\mu^*B+\frac{m-1-m\varepsilon}{mp}(E_p+F_p)-T$$
$$\geq\Delta_W-E+\frac{\varepsilon}{p}F_p+\frac{m-1-m\varepsilon}{mp}(E_p+F_p)-\bibitemgg\lfloor \Delta_W-E+\frac{m-1}{mp}F_p\bibitemgg\rfloor$$
$$=\frac{m-1-m\varepsilon}{mp}E_p+\Delta_W-E+\frac{m-1}{mp}F_p-\bibitemgg\lfloor \Delta_W-E+\frac{m-1}{mp}F_p\bibitemgg\rfloor\geq0.$$
Thus we can choose $\varepsilon>0$ sufficiently small such that $(W, \Delta'_W(\varepsilon))$ is klt since $(W, \Delta'_W(0))$ is klt and $\Delta'_W(\varepsilon)\geq0$. We identify $\Pic^0(Y)$ and $\Pic^0(\Alb(Y))$ by $a_Y^*$. We use $\begin{equation}ta'$ to denote the Cartier divisor corresponding to the line bundle $\begin{equation}ta$. We deduce that
$$\mu^*g^*\begin{equation}ta'+\mu^*D'-T\sim_{{\mathbb Q}}\mu^*g^*(\frac{\varepsilon}{d}H+\frac{1}{m}L+\begin{equation}ta')+K_W+\Delta'_W(\varepsilon)$$
where $\frac{\varepsilon}{d}H+\frac{1}{m}L+\begin{equation}ta'$ is an ample ${\mathbb Q}$-divisor on $\Alb(Y)$ since $\varepsilon>0$, $L$ is nef and $\begin{equation}ta'$ is numerically trivial. By \cite[Corollary 10.15]{Kol95}, we have that
$$H^i(\Alb(Y), \begin{equation}ta\otimes R^j(g\circ\mu)_*\mathcal{O}_W(\mu^*D'-T))=0$$
for every $i>0$, every $j\geq0$ and every $\begin{equation}ta\in\Pic^0(Y)$. Denote the sheaf $(g\circ\mu)_*\mathcal{O}_W(\mu^*D'-T)$ by $\mathcal{F}$. We deduce that
$$h^0(\Alb(Y), \mathcal{F})=\chi(\Alb(Y), \mathcal{F})=\chi(\Alb(Y), \begin{equation}ta\otimes\mathcal{F})=h^0(\Alb(Y), \begin{equation}ta\otimes\mathcal{F})$$
for every $\begin{equation}ta\in\Pic^0(Y)$.
Since $-T\leq\lceil E\rceil$, we have that
$$h^0(X, \mathcal{O}_X(D'))=h^0(W, \mathcal{O}_W(\mu^*D'-T))=h^0(\Alb(Y), \mathcal{F})$$
$$=h^0(\Alb(Y), \begin{equation}ta\otimes\mathcal{F})=h^0(W, \mu^*g^*\begin{equation}ta\otimes\mathcal{O}_W(\mu^*D'-T))$$
$$\leq h^0(W, \mu^*g^*\begin{equation}ta\otimes\mathcal{O}_W(\mu^*D'+\lceil E\rceil))=h^0(X, g^*\begin{equation}ta\otimes\mathcal{O}_X(D')).$$
Recall that $D'=D+g^*L+\alpha'$. Since the inequality above is true for every $\begin{equation}ta\in\Pic^0(Y)$ and every nef divisor $L$ on $\Alb(Y)$, we can replace $L$ with $L-\begin{equation}ta'$ and get
$$h^0(X, g^*\begin{equation}ta^{-1}\otimes\mathcal{O}_X(D'))\leq h^0(X, \mathcal{O}_X(D'))\leq h^0(X, g^*\begin{equation}ta\otimes\mathcal{O}_X(D'))$$
for every $\begin{equation}ta\in\Pic^0(Y)$. Thus we conclude that for every $\begin{equation}ta\in\Pic^0(Y)$
$$h^0(X, \mathcal{O}_X(D+g^*L)\otimes\alpha)=h^0(X, \mathcal{O}_X(D+g^*L)\otimes\alpha\otimes f^*\begin{equation}ta).$$
We now prove statement (ii). We choose a positive integer $N$ sufficiently divisible such that $h^0(F, \mathcal{O}_F(ND|_{F}))>0$ where $F$ is the general fiber of $f$ such that $(F, \Delta|_{F})$ is a log smooth klt pair. We have $ND|_{F}\sim_{{\mathbb Q}}Nm(K_F+\Delta|_{F})$. The cohomological support locus
$$V^0(F, \mathcal{O}_F(ND|_{F}))=\{\gamma\in\Pic^0(F)\mid\dim H^0(F, \mathcal{O}_F(ND|_{F})\otimes\gamma)\geq 1\}$$
is a finite union of torsion subvarieties of $\Pic^0(F)$ by Corollary \ref{coro1}. We claim that if $\alpha\in V^0(X, \mathcal{O}_X(D))$, then $(\alpha|_{F})^{\otimes N}\cong \mathcal{O}_F$ if $F$ is sufficiently general. Since $\alpha\in V^0(X, \mathcal{O}_X(D))$, we deduce that $(\alpha|_{F})^{\otimes N}\in V^0(F, \mathcal{O}_F(ND|_{F}))$ if $F$ is sufficiently general. If $(\alpha|_{F})^{\otimes N}$ is not trivial, then $V^0(F, \mathcal{O}_F(ND|_{F}))$ contains two different points since it contains $\mathcal{O}_F$ and thus it contains a torsion point $\gamma$ which is not trivial by the structure of $V^0(F, \mathcal{O}_F(ND|_{F}))$. However, $\gamma$ can only be trivial by $\kappa(F, D|_{F})=0$ and it is a contradiction. We conclude that $(\alpha|_{F})^{\otimes N}\cong \mathcal{O}_F$ and by \cite[Lemma 2.6]{CH04} we know that a nonzero multiple of $\alpha$ belongs to $f^*\Pic^0(Y)$. Thus there exist countably many torsion points $\alpha_j\in\Pic^0(X)$ such that
$$V^0(X, \mathcal{O}_X(D))\subseteq\bibitemgcup_{j\in J}(\alpha_j\otimes f^*\Pic^0(Y))$$
where $J$ is a countable index set. Since $V^0(X, \mathcal{O}_X(D))$ is a closed subset of $\Pic^0(X)$, it can be decomposed as a union of irreducible components
$$V^0(X, \mathcal{O}_X(D))=\bibitemgcup_{k\in K}Z_k$$
where $K$ is a finite index set. We deduce that
$$Z_k=\bibitemgcup_{j\in J}(Z_k\cap(\alpha_j\otimes f^*\Pic^0(Y))).$$
Note that we are only considering closed points of varieties now. Since an irreducible variety over an uncountable and algebraically closed field cannot be a countable union of proper closed subsets, we deduce that $Z_k\subseteq\alpha_{j_k}\otimes f^*\Pic^0(Y)$ for some $j_k\in J$. By statement (i) of our theorem, we have
$$Z_k\subseteq\alpha_{j_k}\otimes f^*\Pic^0(Y)\subseteq V^0(X, \mathcal{O}_X(D))$$
and thus $Z_k=\alpha_{j_k}\otimes f^*\Pic^0(Y)$ since $Z_k$ is an irreducible component of $V^0(X, \mathcal{O}_X(D))$. Then we deduce that
$$V^0(X, \mathcal{O}_X(D))=\bibitemgcup_{k\in K}(\alpha_{j_k}\otimes f^*\Pic^0(Y)).$$
\end{proof}
We are ready to prove Theorem \ref{main4} now.
\begin{equation}gin{proof}[Proof of Theorem \ref{main4}]
By the same argument of Theorem \ref{main6}, we can assume $(X, \Delta)$ is log smooth. By Theorem \ref{main3}, we know $(a_X)_*\mathcal{O}_X(lD)$ admits the Chen-Jiang decomposition for every positive integer $l$
$$(a_X)_*\mathcal{O}_X(lD)\cong \bibitemgoplus_{i\in I}(\alpha_i\otimes p_i^*\mathcal{G}_i)$$
where each $A_i$ is an abelian variety, each $p_i\colon \Alb(X)\to A_i$ is a fibration, each $\mathcal{G}_i$ is a nonzero M-regular coherent sheaf on $A_i$, and each $\alpha_i\in\Pic^0(X)$ is a torsion line bundle whose order is bounded independently of $l$. We identify $\Pic^0(X)$ and $\Pic^0(\Alb(X))$ by $a_X^*$. By \cite[Lemma 3.3]{LPS20}, we have that
$$V^0(X, \mathcal{O}_X(lD))=\bibitemgcup_{i\in I}(\alpha^{-1}_i\otimes p_i^*\Pic^0(A_i)).$$
We know that $V^0(X, \mathcal{O}_X(lD))$ is a finite union of torsion translates of $(a_f)^*\Pic^0(Y)$ by Theorem \ref{main6} and thus for every $i\in I$, we have a factorization
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
\Alb(X) \arrow[r, "a_f"] \arrow[rr, bend right, "p_i"]& \Alb(Y) \arrow[r, "q_i"] & A_i
\end{tikzcd}
\end{center}
where each $q_i$ is a fibration since each $p_i$ is a fibration. We define $\mathcal{F}_i=q_i^*\mathcal{G}_i$ and then we have
$$(a_X)_*\mathcal{O}_X(lD)\cong \bibitemgoplus_{i\in I}(\alpha_i\otimes a_f^*\mathcal{F}_i).$$
We claim that each $\mathcal{F}_i$ satisfies $\IT$. Fix a torsion point $\alpha\in\Pic^0(X)$. By Theorem \ref{main6}, $h^0(X, \mathcal{O}_X(lD)\otimes\alpha\otimes f^*\begin{equation}ta)$ is constant for every $\begin{equation}ta\in\Pic^0(Y)$. We know
$$h^0(X, \mathcal{O}_X(lD)\otimes\alpha\otimes f^*\begin{equation}ta)=\sum_{i\in I}h^0(\Alb(X), \alpha\otimes\alpha_i\otimes a_f^*\mathcal{F}_i\otimes a_f^*\begin{equation}ta)$$
and thus each term on the right is constant for every $\begin{equation}ta\in\Pic^0(Y)$ since those terms are upper semi-continuous functions with respect to $\begin{equation}ta$. In particular, we can take $\alpha$ to be $\alpha_i^{-1}$ and deduce that $h^0(\Alb(X), a_f^*\mathcal{F}_i\otimes a_f^*\begin{equation}ta)$ is constant for every $\begin{equation}ta\in\Pic^0(Y)$. By the base change theorem, we know that
$$h^0(\Alb(X), a_f^*\mathcal{F}_i\otimes a_f^*\begin{equation}ta)=h^0(\Alb(Y), \mathcal{F}_i\otimes\begin{equation}ta)$$
since $a_f$ is a fibration by \cite[Lemma 2.6]{CH04}. Thus $h^0(\Alb(Y), \mathcal{F}_i\otimes\begin{equation}ta)$ is constant for every $\begin{equation}ta\in\Pic^0(Y)$. Next we prove that each $\mathcal{F}_i$ is a GV-sheaf. We can take an isogeny $k\colon B\to\Alb(X)$ such that each $k^*\alpha_i$ is trivial and consider the following commutative diagram for the fiber product $X':=X\times_{\Alb(X)}B$.
\begin{equation}gin{center}
\begin{equation}gin{tikzcd}
X' \arrow[r, "p"] \arrow[d, "q"] & B \arrow[d, "k"] \arrow[dr, "k'"]\\
X \arrow[r, "a_X" ] & \Alb(X) \arrow[r, "a_f"] &\Alb(Y)
\end{tikzcd}
\end{center}
The morphisms $p$ and $q$ are the projections and $q$ is \'{e}tale. Consider the log smooth klt pair $(X', \Delta')$ defined by $K_{X'}+\Delta'=q^*(K_{X}+\Delta)$. By Corollary \ref{coro1}, we deduce that each $\mathcal{F}_i$ is a GV-sheaf. Then each $\mathcal{F}_i$ satisfies $\IT$ by Proposition \ref{IT}.
\end{proof}
The next corollary is a special case of Theorem \ref{main4}, which applies to klt pairs of log general type.
\begin{equation}gin{coro}\label{ITL}
Let $f\colon X\to Y$ be a smooth model of the Iitaka fibration associated to a Cartier divisor $D$ on $X$ where $D\sim_{{\mathbb Q}}m(K_X+\Delta)$, $m>1$ is a rational number, $Y$ is smooth and $(X, \Delta)$ is a klt pair. Let $a_X\colon X\to \Alb(X)$ be the Albanese morphism of $X$. If $q(X)=q(Y)$ and $(a_X)_*\mathcal{O}_X(D)\neq0$, then:
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] $V^0(X, \mathcal{O}_X(D))=\Pic^0(X)$.
\item[$\mathrm{(ii)}$] $(a_X)_*\mathcal{O}_X(D)$ satisfies $\IT$, hence in particular it is ample.
\end{enumerate}
\end{coro}
\begin{equation}gin{proof}
By the same argument of Theorem \ref{main6}, we can assume $(X, \Delta)$ is log smooth. Since $(a_X)_*\mathcal{O}_X(D)\neq0$ is a GV-sheaf by Corollary \ref{coro1}, we deduce $V^0(X, \mathcal{O}_X(D))$ is not empty by \cite[Lemma 7.4]{HPS18}. Since $q(X)=q(Y)$, the fibration $a_f$ is an isomorphism. Then statement (i) follows from Theorem \ref{main6}. By Theorem \ref{main4}, $(a_X)_*\mathcal{O}_X(D)$ satisfies $\IT$. It is ample by \cite[Proposition 2.13]{PP03} and \cite[Corollary 3.2]{Deb06}.
\end{proof}
Next we use Corollary \ref{ITL} to give effective bounds for the generation of klt pairs of log general type on irregular varieties. This extends some results in \cite{PP03, PP11b, LPS20} to klt pairs. Theorem \ref{main5} follows from Corollary \ref{ITL} and standard arguments.
\begin{equation}gin{thm}\label{main5}
Let $f\colon X\to Y$ be a smooth model of the Iitaka fibration associated to a Cartier divisor $D$ on $X$ where $D\sim_{{\mathbb Q}}m(K_X+\Delta)$, $m>1$ is a rational number, $Y$ is smooth and $(X, \Delta)$ is a klt pair. Let $a_X\colon X\to \Alb(X)$ be the Albanese morphism of $X$. Assume that $q(X)=q(Y)$ and there exists a nonempty open subset $W\subseteq\Alb(X)$ consisting of points $a$ such that $\mathcal{O}_{X_a}(D)$ is globally generated and the natural map
$$(a_X)_*\mathcal{O}_{X}(D)\otimes{\mathbb C}(a)\to H^0(X_a, \mathcal{O}_{X_a}(D))$$
is surjective. Then:
\begin{equation}gin{enumerate}
\item[$\mathrm{(i)}$] $\mathcal{O}_{X}(2D)\otimes\alpha$ is globally generated on $a_X^{-1}(W)$ for every $\alpha\in\Pic^0(X)$.
\item[$\mathrm{(ii)}$] Assuming in addition that for every $x\in a_X^{-1}(W)$ and every $a\in W$, $\mathcal{O}_{X_a}(2D)\otimes\mathcal{I}_{x}|_{X_a}$ is globally generated and the natural map
$$\quad (a_X)_*(\mathcal{O}_{X}(2D)\otimes\mathcal{I}_{x})\otimes{\mathbb C}(a)\to H^0(X_a, \mathcal{O}_{X_a}(2D)\otimes\mathcal{I}_{x}|_{X_a})$$
is surjective, then $\mathcal{O}_{X}(3D)\otimes\alpha$ is very ample on $a_X^{-1}(W)$ for every $\alpha\in\Pic^0(X)$.
\end{enumerate}
\end{thm}
\begin{equation}gin{proof}
It follows from the same proof of \cite[Theorem 12.2]{LPS20} and we include some details for the reader's convenience. We can assume that $a_X^{-1}(W)$ is not empty. We can identify $\Pic^0(X)$ and $\Pic^0(\Alb(X))$ using $a_X^*$ by \cite[Lemma 8.1]{Kaw85a}. We prove statement (i) first. We know $(a_X)_*\mathcal{O}_{X}(D)\neq0$ by assumptions and thus it satisfies $\IT$ by Corollary \ref{ITL}. Take a $\begin{equation}ta\in\Pic^0(X)$ such that $\begin{equation}ta^{\otimes2}\cong\alpha$. Then $(a_X)_*(\mathcal{O}_{X}(D)\otimes\begin{equation}ta)$ satisfies $\IT$. By \cite[Proposition 2.13]{PP03}, it is continuously globally generated which means that there exists an integer $N$ such that for general line bundles $\alpha_1, \dots, \alpha_N\in\Pic^0(\Alb(X))$ the sum of the twisted evaluation morphisms
$$\bibitemgoplus_{i=1}^NH^0(\Alb(X), (a_X)_*(\mathcal{O}_{X}(D)\otimes\begin{equation}ta)\otimes\alpha_i)\otimes\alpha_i^{-1}\to (a_X)_*(\mathcal{O}_{X}(D)\otimes\begin{equation}ta)$$
is surjective. We know $a_X^*(a_X)_*\mathcal{O}_{X}(D)\to\mathcal{O}_{X}(D)$ is surjective on $a_X^{-1}(W)$ by assumptions. Thus
$$\bibitemgoplus_{i=1}^NH^0(X, \mathcal{O}_{X}(D)\otimes\begin{equation}ta\otimes a_X^*\alpha_i)\otimes a_X^*\alpha_i^{-1}\to \mathcal{O}_{X}(D)\otimes\begin{equation}ta$$
is surjective on $a_X^{-1}(W)$. We conclude that
$$\mathcal{O}_{X}(2D)\otimes\alpha\cong(\mathcal{O}_{X}(D)\otimes\begin{equation}ta)^{\otimes2}$$
is generated by global sections on $a_X^{-1}(W)$ by the same argument of \cite[Proposition 2.12]{PP03}.
We prove statement (ii) now. By a similar argument as above and the additional assumptions, we only need to prove $(a_X)_*(\mathcal{O}_{X}(2D)\otimes\mathcal{I}_{x})$ satisfies $\IT$ for every $x\in a_X^{-1}(W)$. By Corollary \ref{ITL}, $(a_X)_*(\mathcal{O}_{X}(2D))$ satisfies $\IT$. Since $\mathcal{O}_{X}(2D)$ is generated by global sections on $a_X^{-1}(W)$ by statement (i) of our theorem, we can prove that $(a_X)_*(\mathcal{O}_{X}(2D)\otimes\mathcal{I}_{x})$ satisfies $\IT$ for every $x\in a_X^{-1}(W)$ by a standard argument using exact sequences.
\end{proof}
Corollary \ref{coro5} follows from Theorem \ref{main5} directly.
\begin{equation}gin{proof}[Proof of Corollary \ref{coro5}]
It follows from Corollary \ref{ITL} and Theorem \ref{main5} since the klt pair $(X, \Delta)$ is of log general type and $a_X$ is generically finite onto its image.
\end{proof}
\bibitembliographystyle{amsalpha}
\bibitembliography{biblio}
\end{document} | math | 80,255 |
\begin{document}
\title{ Remarks on the Well-Posedness of
the Nonlinear Cauchy Problem }
\author{Guy M\'etivier}
\address{MAB, Universit\'e de Bordeaux I, 33405 Talence Cedex, France}
\email{[email protected]}
\thanks{Research partially supported by European network HYKE, HPRN-CT-2002-00282 .}
\subjclass{Primary 35 }
\date{January 1, 1994 and, in revised form, June 22, 1994.}
\dedicatory{This paper is dedicated to Fran\c{c}ois Treves.}
\keywords{Cauchy Problem, well posedness}
\begin{abstract}
We show that hyperbolicity is a necessary condition for
the well posedness of the noncharacteristic Cauchy problem
for nonlinear partial differential equations.
We
give conditions on the initial data which are necessary for the existence of solutions
and we analyze Hadamard's instabilities in Sobolev spaces.
We also show that genuinely nonlinear
equations raise new interesting problems.
\end{abstract}
\maketitle
\section{Introduction}
The question of the well-posedness of the Cauchy problem was first raised by Hardamard
(\cite{Ha}, \cite{Ha1}) who proved that it is ill-posed in the case of linear second order elliptic equations. But the introduction in \cite{Ha1} clearly indicates that
Hadamard was interested in nonlinear equations as well.
In modern words, Hadamard's proof is based on the analytic regularity
of linear elliptic boundary problems. This
regularity has been extended to nonlinear elliptic equations by Morrey (\cite{MN})
so that Hadamard's argument
also applies to general nonlinear elliptic equations.
For general {\sl linear} equations,
it is well known that hyperbolicity is a necessary condition
for the well-posedness of the noncharacteristic Cauchy problem in $C^\infty$,
that is for the existence of solutions for general $C^\infty$ data
(see Lax \cite{La}, Mizohata \cite{Mi} and Ivrii-Petkov \cite{IP} for
a simplified proof and further developments; see also \cite{Ho}).
Moreover, for several classes of nonhyperbolic equations, explicit necessary conditions on the initial data for the existence of solutions have been given (see
\cite{Ni1} \cite{Ni2}).
For {\sl nonlinear} equations, Wakabayashi \cite{Wak} has proved
that the existence of a smooth {\it stable} solution implies hyperbolicity,
stability meaning that one can perturb the initial data and
the source terms in the
equations. In a previous paper, Yagdjian obtained this result, with a much weaker definition of stability,
in the sense of continuous dependence on the initial data, for the special case of ``gauge invariant''
equations (\cite{Yag1}).
We also mention \cite{Yag} for a particular case and \cite{Hou} for
first order scalar complex equations.
In this paper, we continue the analysis of Hadamard's instabilities
for nonhyperbolic nonlinear equations in two directions.
First, in the spirit of Hadamard's examples, we give necessary conditions on the
initial data for the existence of smooth solutions of a given equation, without perturbing
the equation or the source terms.
Next, we also want to point out that
the nonlinear theory yields interesting
and difficult new problems.
There are many interesting examples, for instance in multi-phase fluid dynamics,
where the equations are not {\sl everywhere} hyperbolic.
To mention one occurrence of this phenomenon, consider Euler's equations of gas dynamics in Lagrangian coordinates:
\begin{equation}
\langlebel{ex11}
\left\{\begin{aligned}
&\partial_t u + \partial_x v \, =\, 0\, ,
\cr
& \partial_t v + \, \partial_x p(u) \, =\, 0\, .
\end{aligned}\right.
\end{equation}
The system is hyperbolic [resp. elliptic] when
$p'(u) > 0$ [resp. $p'(u) < 0$].
For van der Waals state laws, it happens that $p$ is decreasing on an interval
$[u_*, u^*]$.
A mathematical example is
\begin{equation}
p(u) = u \, (u^2 - 1)\,
\langlebel{ex12}
\end{equation}
Hadamard's argument (see e.g. \cite{Ha}, \cite{Ha1}) shows that the Cauchy problem with data taking values in the elliptic region is ill-posed: if $u_{\vert t = 0} $ is real analytic near $\underline x$
and $u(\underline x)$ belongs to the elliptic interval, then
any local
$C^1$ solution is analytic (see e.g. \cite{MN}); thus the initial data $v_{ \vert t = 0} $ {\sl must} be analytic for
the initial value problem to have a solution.
However, there are other interesting questions about the system \eqref{ex11}.
There are classical solutions
with values in one of the hyperbolic region
$u < u_*$ or $u > u^*$,
but there are also discontinuous solutions,
modeling for instance phase transitions, which take values in both regions.
They have been extensively studied, see e.g. \cite{Sl}, \cite{Be}, \cite{Fr}.
Another remarkable fact is that \eqref{ex11} has a conserved
energy. Let $P(u)$ satisfy $P' = p$.
Then the energy:
\begin{equation}
E(t) \, := \, \int \big({1\over 2}\, v^2(t,x) + P(u (t,x))\big) \, dx \,
\end{equation}
is conserved for solutions of \eqref{ex11}.
For the example \eqref{ex12},
$ P(u) = {1 \over 4} \, u^2 \, (u^2 - 2)$. If one considers the
periodic problem, the $L^2$ norm is dominated by
the $L^4$ norm on $[0, 2 \pi]$, thus the boundedness of $E$
controls the $L^4$ and $L^2$ norm of the solutions.
Of course, this is formal, and the validity of {\sl a priori} bounds does not prove
the {\sl existence} of solutions. However, this indicates that the nonexistence of solutions
is much more subtle than in the linear case. In particular, there is no blow up phenomenon
in $L^p$ norms.
The equations \eqref{ex11} are thought of as approximations or limits of
more complicated models which may include for instance viscosity or capillarity
(see e.g. \cite{Be});
numerical schemes have also been considered (see e.g. \cite{Fan}, \cite{HW}).
In the case of periodic solutions,
spectral methods lead to filter high frequencies and to consider
the ``approximate'' system
\begin{equation}
\langlebel{ex14}
\left\{\begin{aligned}
&\partial_t u^\langlembda + \partial_x v^\langlembda = 0 ,
\\
& \partial_t v^\langlembda + \partial_x S_\langlembda p(u^\langlembda ) = 0,
\end{aligned}\right. \qquad
\left\{\begin{aligned}
& u^\langlembda{}_{\vert t = 0} = S_\langlembda h ,
\\
& v^\langlembda{}_{\vert t = 0} = S_\langlembda k,
\end{aligned}\right.
\end{equation}
where $S_\langlembda$ is the projector on Fourier modes of index
$\vert n \vert \le \langlembda$. For instance, when $p$ is given by \eqref{ex12},
the conservation of energy and the Cauchy-Lipschitz theorem imply that
in the periodic case:
\begin{enumerate}
\item[] {\sl for all $h \in L^4$ and $k \in L^2$, the equations
$\eqref{ex14}$ have global solutions $(u^\langlembda, v^\langlembda)$ which
are uniformly bounded in $C^0([0, \infty[ ; L^4 \times L^2)$.}
\end{enumerate}
Note that there are no conditions on $h$, which can take values in the elliptic region
$u^2 < 1/3$. The question is to analyze the behavior of
$(u^\langlembda, v^\langlembda)$ as $\langlembda \to + \infty$.
Because of the bounds on $(u^\langlembda, v^\langlembda)$ and
$(\partial _t u^\langlembda, \partial _t v^\langlembda)$, there are subsequences
which converge weakly and strongly in $C^0([0, 1], H^{- \varepsilon })$. In particular,
the weak limits $(\underline u, \underline v)$ are bounded with values in $L^2$ and continuous
in time for the weak tolopogy of $L^2$. Thus $\underline u(0) = h$ and
$\underline v (0) = k$. Taking the weak limit
$\underline p$ of $p(u^\langlembda)$, there holds
$$
\partial _t \underline u + \partial _x \underline v = 0 \,, \quad \partial _t \underline v + \partial _x \underline p = 0\,.
$$
The question is to express $\underline p$ in terms of $\underline u$ and $\underline v$.
As mentioned above, the answer,
$\underline p = p(\underline u)$, $(\underline u, \underline v)$ smooth,
{\sl cannot} be true in general when $h$ takes values in the elliptic zone.
The common idea is that the limits $(\underline u, \underline v)$ ``escape'' from the elliptic region,
as suggested by numerical calculations (\cite{Fan}, \cite{HW}), but
no rigorous proof of this fact seems available in the literature.
A detailed answer to the questions above remains a very interesting
open problem. Motivated by this problem, we consider in Section 5
a modified nonlocal system:
\begin{equation}
\langlebel{ex15}
\left\{\begin{aligned}
&\partial_t u = a(t) \partial_x v ,
\cr
& \partial_t v = \vert a(t) \vert \partial_x u ,
\end{aligned}\right. \qquad \mathrm{with}\quad
a(t) = \Vert u(t) \Vert^2_{L^2} -1 .
\end{equation}
This a version of Kirchhoff equations (\cite{K}, \cite{Li}
\cite{AS}), which is non hyperbolic when
$a < 0$. As \eqref{ex11}, this system has a natural (formal) energy:
\begin{equation}
E (t) = \Vert v(t, \cdot) \Vert^2_{L^2} + \Big\vert \Vert u(t, \cdot) \Vert^2_{L^2} - 1 \Big \vert .
\end{equation}
This implies that the equations with filtered initial data $(S_\langlembda h, S_\langlembda k)$
has global solutions $(u^\langlembda, v^\langlembda)$ uniformly bounded in
$L^\infty([0, + \infty[, L^2({\mathbb T}))$.
For large classes of ``nonanalytic'' initial data $(h, k) \in L^2 $,
with $\Vert h \Vert_{L^2} < 1$, we show that
the limits are $\underline u = h$, $\underline v = k$, constant in time, remaining in the elliptic region.
The limit equations, $\partial _t \underline u = \partial _t \underline v = 0$, have little to see with the original ones.
This indicates that the answers to the questions above might be very delicate.
\bigbreak
Now we review the results of Sections
2 to 4.
To fix a framework, we consider first order square systems
\begin{equation}
\partial_t u = F(t, x, u, \partial_x u) \, , \quad u_{\vert t = 0} = h \, .
\langlebel{CP}
\end{equation}
where
$F $ is a smooth function of
$(t, x, u, v) \in {\mathbb R} \times {\mathbb R}^d \times {\mathbb R}^N \times ({\mathbb R}^N)^d$.
The principal symbol
of the equation reads
\begin{equation}
\tau \mathrm{Id} - \xi \cdot \partial_v F (t, x, u, v) \, .
\end{equation}
Hyperbolicity means that all the eigenvalues of
$\partial_v F$ are real.
We consider the local Cauchy problem \eqref{CP} near
$(0, \underline x)$ and a given base point $(\underline u, \underline v)$, assuming that the initial data
satisfy
\begin{equation}
h(\underline x) = \underline u \, , \quad \partial_x h(\underline x ) = \underline v .
\langlebel{D0}
\end{equation}
The results in Sections 2 to 4 illustrate
the idea that
if $\partial _v F(0, \underline x, \underline u, \underline v)$ has a non real eigenvalue, then
the Cauchy problem \eqref{CP} \eqref{D0}
for classical solutions is ill-posed.
Well posedness means first {\sl solvability}.
Hadamard's counterexamples (see the example \eqref{ex11} above)
prove that analyticity type conditions on the data are necessary for the existence of solutions
of the elliptic Cauchy problem.
In the same vein,
consider the equation\footnote{
This explicit elementary example was suggested by
Nicolas Lerner.}:
\begin{equation}
(\partial _x+ i \partial _y ) u = u^2 \,, \quad x > 0, \qquad u_{\vert x= 0} = h\,,
\end{equation}
with $h(0) \ne 0$. Any $C^0$ solution on $\{ x \ge 0 \} $ near the origin is $ C^1$,
does not vanish and $1/u$ satisfies the equation
$
(\partial _x+ i \partial _y )(1/u) = -1/2\,.
$
Therefore, $1/u + \overline z/2 $ is holomorphic and $1/h $ is the trace of an holomorphic function in $\{ x > 0 \}$, implying that
necessarily $1/h$ is microlocally analytic in the direction $+ 1$ at the origin.
In particular, if $h$ is real valued, $h$ must be real analytic near the origin.
In Section 2, we extend this analysis
to first order scalar complex equations.
For such equations, is is proved in \cite{Hou} that the existence of solutions
for all complex data close to a given $h$, implies that the system
must be semilinear and hyperbolic.
We give a more precise result, showing that, in the non hyperbolic case,
{\sl microlocal analyticity conditions on the initial data are necessary for the existence of classical solutions},
hence that the Cauchy problem has no classical solution for most initial data.
The proof is based on the analysis of \cite{Me1} (see also \cite{BGT}), which provides
approximate integral representation of the $C^1$ or $C^2$ solutions.
Taking real and imaginary parts, this provides examples of $2 \times 2$ systems
where the Cauchy problem has no classical solutions.
This analysis does not extend to general systems: the representation
and approximation theorems valid in the scalar case have no analogue;
the local uniqueness of the Cauchy problem may be false (\cite{Me2});
there are no microlocal analytic regularity theorem at elliptic directions for
$C^1$ or $C^2$ solutions.
However, there are results about $H^{s'}$ microlocal regularity for
$H^{s}$ solutions when $s' \le 2s - s_0$ (see Bony \cite{Bo} and Sabl\'e-Tougeron \cite{ST}).
In Section 4, we show that
if $\partial _v F(0, \underline x, \underline u, \underline v )$ has a nonreal eigenvalue, then for
all $H^s $ local solution, with $s > \frac{d}{2} + 1$,
the polarized $H^{s'}$ wave front set of the initial data
is not arbitrary, when $s' < 2s - \frac{d}{2} - 1$.
In particular, for most data in $H^{s'}$, the Cauchy problem has
no local $H^s$ solution.
Note that $s'$ can be taken larger than $s$,
and for any ``loss'' $k $, it applies to $s' = s + k$, if $s$ is large enough.
The restriction $s > \frac{d}{2} + 1$ is natural in order to have a $C^1$ classical solution.
The restriction $s' < 2s - d -1/2$ is forced by the Theorem of
microlocal ellipticity for nonlinear equations (see \cite{Bo} \cite{ST}).
Under additional assumptions on the equations, one can
show that for arbitrary large $s'$ there are $H^{s'} $ initial data such that
the Cauchy problem has no local $H^{s}$ solution.
\medbreak
For linear equations, standard functional analysis results convert
well-posedness into estimates, and necessary conditions are found
by contradicting the estimates. Solvability implies
continuous dependence on the data (see also F.John \cite{Jo} for general
remarks on this notion). For nonlinear equations,
there are no such abstract argument and it is reasonable to include the continuous
dependence in the definition of the well posedness.
In addition, because local uniqueness is not
guaranteed, we also include it in the following definition of H\"older continuous
solvability.
In the next statement,
$B_r$ denotes the ball $\{ \vert x - \underline x \vert < r \}$
and $\Omega_{r, \delta} $ the lens shaped domain
$$
\Omega_{r, \delta} = \big\{ (t, x) \ : \ 0 < t \,, \vert x -\underline x \vert^2 + \delta t < r^2
\big\}
$$
\begin{definition}
\langlebel{def12}
We say that the Cauchy problem $\eqref{CP}$ is H\"older well posed
on $H^\sigma$, if there are constants $r_0 > r_1 > 0$, $\delta> 0$, $c > 0$, $C$ and
$\alpha \in ]0, 1]$, such that for all
$h \in H^\sigma (B_{r_0})$ satisfying
$\Vert h - \underline u - x \cdot \underline v \Vert _{H^\sigma(B_{r_0})} \le c$ and
all $r \in ]0, r_1]$, the Cauchy problem $\eqref{CP}$ has
a unique solution in $C^1(\overline \Omega_{r, \delta})$,
with norm bounded by $C$. Morever, given $h_1$ and $h_2$, the
corresponding solutions satisfy for all $r < r_1$:
\begin{equation}
\Vert u_1 - u_2 \Vert_{L^\infty(\Omega_{r, \delta}) }
\le C \Vert h_1 - h_2 \Vert^\alpha_{H^\sigma(B_{r_0})}
\langlebel{Hest}
\end{equation}
\end{definition}
In Section 3, we show that {\sl if $F$ is real analytic
and if $\partial _v F(0, x_0, u_0, v_0)$ has a nonreal eigenvalue, then
the Cauchy problem $\eqref{CP}$ is not H\"older well posed
on $H^\sigma$, for all $\sigma$}.
Note that the definition above differs strongly from
the notion of {\it stable} solution introduced in \cite{Wak} in the sense that
we do not allow perturbations of the equations, while
the stability used in \cite{Wak} is related to the solvability of
\begin{equation}
\partial _t u + F(t, x, u, \partial _x u ) = f \,, \quad u_{\vert t = 0} = h + h'.
\langlebel{CPc}
\end{equation}
for all $f$ and $h'$ small.
The analysis is based on the construction of
asymptotic solutions using WKB or geometric optics expansions.
But they are not exact solutions, yielding error terms $f$
which are precisely the source terms
chosen in \cite{Wak}. In this analysis, the choice of
$f$ is dictated by the choice of $h$.
It is interesting and much stronger to consider exact solutions of \eqref{CP}
(as in \cite{Yag1}), or to be able to choose
$h$ and $f$ independently.
In Section 3, we construct exact solutions close to the approximate solutions,
by Cauchy-Kowalewski type arguments. This is where we use the analyticity of the equation.
In this respect, the results of this section give a detailed account of the $H^s$ instability
of analytic solutions when hyperbolicity fails.
\section{Necessary conditions for scalar complex equations}
To simplify the discussion, consider a quasilinear scalar equation
\begin{equation}
\partial_t u \, + \, \sum_{j=1}^d a_j(t, x, u) \partial_{x_j} u + b(t, x, u) \, = \, 0\, , \quad
u_{\vert t= 0} = h\, .
\langlebel{eq21}
\end{equation}
where the $a_j $ are holomorphic functions of $(t,x,u)$ on a neighborhood
of $(0, \underline x, \underline u )$. The Cauchy data $h$ is always assumed to satisfy
$h(\underline x) = \underline u$.
The nonhyperbolicity condition reads
\begin{equation}
{\rm Im}\, a(0, \underline x , \underline u ) \ne 0\, .
\langlebel{nH}
\end{equation}
\bigbreak
\begin{theorem}
\langlebel{th21}
If the Cauchy problem $\eqref{eq21}$ has a
$C^1$ solution for $t \ge 0$ on a neighborhood of $(0, \underline x)$, then for all
$\xi \in {\mathbb R}^d $ such that $\xi \cdot {\rm Im}\, a(0, \underline x , \underline u ) > 0$,
$(\underline x, \xi) $
does not belong to the analytic wave front set of $h$.
\end{theorem}
\bigbreak
For the definition of the analytic wave front set, we refer to \cite{Sj} or
\cite{Ho}. In particular, it contains the $C^\infty $ wave front set
and the theorems implies that if the local Cauchy problem
has a $C^1$ solution, then $h$ must be $C^\infty $ at
$(\underline x, \underline xi)$ if $\underline xi \cdot {\rm Im}\, a(0, \underline x , \underline u ) > 0$.
This means that for all
$C^\infty$ cut-off function $\chi$ supported in
a sufficiently small neighborhood of $\underline x$,
the Fourier transform of $\chi h$ is rapidly decreasing
in any small conical neighborhood of $\underline xi$.
For ``most'' functions $h$ in $H^s$, $(\underline x, \underline xi)$ belongs to the
the $C^\infty $ wave front set.
Theorem \ref{th21} implies that for most $h$, the Cauchy problem
\eqref{eq21} has no $C^1$ solution.
\begin{example}
\textup{Taking real and imaginary parts of the unknowns yields nonexistence theorem
for $2 \times 2$ real systems. With
$\alpha_j (u, v) = {\rm Re\, } a_j (u+ i v)$, $\beta_j (u, v) = {\rm Im\, } a_j(u+iv)$,
the equation \eqref{eq21} with $b = 0$ is equivalent to:}
\begin{equation}
\left\{\begin{aligned}
& \partial _t u + \sum \partial _{x_j} \alpha_j(u, v) = 0 \,, \quad u_{\vert t = 0} = h,
\\
& \partial _t v + \sum \partial _{x_j} \beta_j(u, v) = 0 \,, \quad v_{\vert t = 0} = k.
\end{aligned}\right.
\langlebel{ex22}
\end{equation}
\textup{Suppose that $\underline \beta = \beta(h(\underline x), k(\underline x) ) \ne 0$ and
choose $\underline xi$ such that $\underline xi\cdot \underline \beta > 0$. If
$h+ ik$ is not microlocally analytic at $(\underline x, \underline \xi )$, then the Cauchy problem
\eqref{ex22} has no local $C^1$ solution near $(0, \underline x)$. }
\textup{ For instance, this applies to the the system:}
\begin{equation}
\left\{\begin{aligned}
& \partial _t u + u \partial _x u - v \partial _x v + \partial _y u = 0 \,, \quad u_{\vert t = 0} = h,
\\
& \partial _t v + v \partial _x u + u \partial _x v + \partial _y v = 0 \,, \quad v_{\vert t = 0} = k,
\end{aligned}\right.
\langlebel{ex23}
\end{equation}
\textup{when $k(\underline x ) \ne 0$.
For functions independent of $y$, or equivalently dropping
the $\partial _y$, the system is elliptic for $v \ne 0$ and Hadamard's argument applies.
The example \eqref{ex23} shows that Theorem \ref{th21} also applies to nonelliptic systems. }
\end{example}
\bigbreak
\begin{proof}[Proof of Theorem $\ref{th21}$]
{\bf a)}
The complex characteristic curves are integral curves of
the holomorphic vector field:
$$
\mathcal{ L} = \partial_t + \sum a_j(t,x,u) \partial_{x_j} \, - \,
b(t,x,u) \partial_u\, .
$$
They are given by
$Z_j (t, x, u) = c_j , U(t, x, u) = c_0$, where $Z_j$ and $U$ are local
holomorphic solutions of
$$
\left\{ \begin{aligned} \mathcal{L} Z_j = 0\, ,& \quad Z_j{}_{\vert t = 0} = x_j\, ,
\\
\mathcal{L} U = 0\, ,& \quad U{}_{\vert t = 0} = u\, .
\end{aligned}\right.
$$
We also introduce the additional
variables $v = (v_1, \ldots, v_d)$, which are placeholders for $\partial _{x_j} u$,
and the function
$$
J(t,x,u,v) := \det \Big({ \partial Z_j (t,x,u) \over \partial{x_k} } +
v_k {\partial Z_j(t,x,u) \over \partial u} \Big).
$$
Let $G(t, x, u)$ be a holomorphic solution of ${\mathcal L} G = 0$
on a complex neighborhood ${\mathcal O}$ of $(0, \underline x, \underline u)$.
Suppose that $u$ is $C^1$ solution
of \eqref{eq21} on
$[0, T] \times \Omega$ such that $(t, x, u(t, x)) \in {\mathcal O}$
for all $(t, x) \in [0,T] \times \Omega$. Then,
by Lemma 2.2.2 of \cite{Me1}, there holds for all $s \in [0, T]$
and
$\chi \in C^1_0(\Omega)$:
\begin{equation}
\begin{aligned}
\int_\Omega & G(0, x, h(x) ) \chi(x) \, dx =
\int_\Omega G(s, x, u(s,x)) \, \chi(x)\, \widetilde J (t,x) \, dx \,
\\ \,
& - \int_{[0,s] \times \Omega}
\sum_j \partial_{x_j} \chi (x)\, a_j(t,x,u(t,x))\, G(t, x, u(t,x) )
\, \widetilde J (t, x) \, dt dx
\end{aligned}
\langlebel{eq23}
\end{equation}
with $\widetilde J(t, x) := J(t, x, u(t, x), \partial_xu (t, x))$.
\medbreak
{\bf b)}
We use \eqref{eq23} with
\begin{equation}
G_{\langlembda, y}(t,x,u) := \Big({\langlembda \over \pi}\Big)^{d/2}\,
U(t,x,u) \, e^{ - \langlembda q ( Z(t,x,u) - y)}\, .
\langlebel{eq24}
\end{equation}
where $q(y) = \langlengle Q y, y \ranglengle$ is a quadratic form, with real coefficients,
positive definite on ${\mathbb R}^d$.
The $G_{\langlembda, y}$ are defined and holomorphic for
$\vert t \vert \le T$, $ \vert x - \underline x \vert \le r$,
$\vert u- \underline u \vert \le \mathrm{h}o $, for some
$T> 0$, $ r > 0$ and $\mathrm{h}o > 0$.
We can also assume that the given solution $u$ of \eqref{eq21}
is defined and $C^1$ for real $(t, x) \in [0, T] \times \overline \Omega$
where $\Omega$ is the ball $ \{ \vert x - \underline x \vert < r \}$ and that
$\vert u(t, x) - \underline u \vert < \mathrm{h}o $ on this domain.
We fix $\chi \in C^\infty_0(\Omega)$ equal to 1 on a smaller
neighborhood of $\underline x$.
Because
$
Z(t, x, u) = x + O\big(\vert t \vert )\big)
$,
${\rm Re} Z (t, x, u(t, x) ) \ne 0$ for
$t$ small and $x $ in the support of $d\chi$.
Because $Z(0, x , u) - \underline x = x - \underline x \ne 0$ on the support of
$d\chi$,
there are
$\Omega_0 \subset \Omega $, real neighborhood of
$ \underline x$, $\varepsilon > 0$,
$\delta > 0$ and $T_0 > 0$, such that
\begin{equation}
\begin{aligned}
\forall y \in \Omega_0 +i [-\delta, \delta]^d, \ & \forall t \in [0, T_0] ,
\ \forall x \in \mathrm{supp} d \chi \ :
\\ \quad
& {\rm Re}q (Z(t, x, u(t, x)) - y) \ge 2 \varepsilon > 0.
\end{aligned}
\langlebel{eq25}
\end{equation}
Consider
\begin{equation}
T h(y, \langlembda) \, := \, \Big({\langlembda \over \pi}\Big)^{d/2}\,
\int_\Omega e^{ -\langlembda q(x-y) }\, h(x) \, \chi(x) \, dx.
\langlebel{eq26}
\end{equation}
We apply \eqref{eq23} to $G= G_{\langlembda, y}$ given by \eqref{eq24}.
The estimate \eqref{eq25} shows that the second integral
in the right hand side is
$O(e^{- \varepsilon \langlembda})$. Therefore, there is $C$ such that
for all
$y \in \Omega_0 +i [-\delta, \delta]$ and $t \in [0, T_0]$,
\begin{equation}
\begin{aligned}
\Big\vert T h (y, \langlembda) -
\Big({\langlembda \over \pi}\Big)^{d/2}\, \int_\Omega
\widetilde U(t,x) e^{ - \langlembda q (\widetilde Z(t,x) - y) } \chi(x)
\widetilde J (t, x) & dx \Big\vert
\\ & \le C e^{ -\varepsilon \langlembda} \,
\langlebel{eq27}
\end{aligned}
\end{equation}
where $\widetilde Z(t, x) := Z(t,x, u(t,x)) $ and we use similar notations for $\widetilde U$ and $\widetilde J$
(see the estimate (4.3.1) in \cite{Me1}).
\medbreak
{\bf c)} We now make use of Assumption \eqref{nH}. Shrinking
$\Omega$ if necessary, in addition to the previous requirements, we can further assume that
\begin{equation}
\forall x \in \Omega \ : \quad
\vert {\rm Im} (a (0, x, h(x) ) - \underline a \vert \le \mathrm{h}o ,
\end{equation}
where $\underline a = {\rm Im} (a (0, \underline x, h(\underline x) )$ and
$\mathrm{h}o > 0$ to be chosen later on. We use the estimate \eqref{eq27} with
\begin{equation}
y = \underline x - i t {\rm Im} \underline a + y'\,, \quad y' \in {\mathbb C}^d, \quad
\vert y' \vert \le \mathrm{h}o t .
\langlebel{eq29}
\end{equation}
Because
$ Z(t,x, u) = x - t a (0,x,u) + O(t^2) $ and $u \in C^1([0, T] \times \overline \Omega)$,
there holds:
$$
{\rm Im } \big( \widetilde Z(t,x) - y \big) = - { \rm Im} y' - t \big({\rm Im} a(0, x, h(x)) - \underline a \big) + O(t^2)\,.
$$
Thus, there is $T_1 > 0$ such that for $t \in [0, T_1]$:
\begin{equation*}
\vert {\rm Im }\big( \widetilde Z(t,x) - y \big) \vert \le 3 \mathrm{h}o t .
\langlebel{eq210}
\end{equation*}
Hence
\begin{equation*}
q \big( {\rm Im }\big( \widetilde Z(t,x) - y \big) \big) \vert \le 9 \Vert Q \Vert \mathrm{h}o^2 t^2 .
\langlebel{eq210}
\end{equation*}
On the other hand:
$ {\rm Im} y = t \underline a - {\rm Im }y' $ and therefore if $\mathrm{h}o$ is small enough
and $\vert y' \vert \le \mathrm{h}o t$,
\begin{equation*}
q ( {\rm Im} y ) \ge t^2 q (\underline a) / 2 .
\end{equation*}
Hence, if
$\mathrm{h}o$ small enough, for $t \in ]0, T_1]$, $y$ satisfying \eqref{eq29} and $x \in \Omega$ there holds:
\begin{equation}
- {\rm Re} q (\widetilde Z(t, x) - y )
\le q \big( {\rm Im} (\widetilde Z(t, x) - y) \big) \le q \big( {\rm Im} y \big)
- \frac{t^2}{4}q( \underline a )
\langlebel{eq211}
\end{equation}
We now fix $t > 0$, $t \le \min(T_0, T_1)$, such that
$y \in \Omega_0 + i [- \delta, \delta]^d$ for all
$y$ satisfying \eqref{eq29}.
Thus, the estimates \eqref{eq27} and \eqref{eq211} imply that there are
$\varepsilon_1 > 0$ and $C >0$ such that for all
$y$ in the complex ball of radius $\mathrm{h}o t $ centered at
$\underline x - it \underline a$ and for all $\langlembda \ge 1$, there holds
\begin{equation}
\langlebel{n213}
\vert T h(y, \langlembda) \vert \le C e^{ \langlembda( q ( {\rm Im} y ) - \varepsilon_1) } \, .
\end{equation}
Since the quadratic form $q$ is definite positive on ${\mathbb R}^d$,
for $y \in {\mathbb C}^d$, the unique real critical point of
$x \mapsto \mathrm{Re} q(y - x) $ is $x = \mathrm{Re} y$ and at this point
$ - \partial _x q( y - x)$ is equal to $- Q \mathrm{Im} y$.
By Proposition 7.2 of Sj\"ostrand \cite{Sj} (see also \cite{Del}, section I.2),
the estimate \eqref{n213} on a neighborhood of $\underline x - it \underline a$ implies that
$(\underline x, t Q \underline a)$ does not belong to the analytic wave front set of $h$.
\medbreak
{\bf d) } For all $\xi$ such that $ \xi \cdot \underline a > 0$, there is a definite positive real symmetric $Q$ such that $Q \underline a = \xi$. We apply the previous step to
$q(x) = \langlengle Q x, x \ranglengle$ which implies that
there is $t > 0$ such that $(\underline x, t \xi)$ does
not belong to the analytic wave front set of $h$. Since the wave front is conic in
$\xi$, the theorem is proved.
\end{proof}
\section{Hadamard's instabilities in Sobolev spaces}
We consider systems, and for simplicity we state the results for quasi-linear
systems:
\begin{equation}
\partial _t u = \sum_{j=1}^d A_j(t, x, u) \partial _{x_j} u + F(t,x, u) , \quad
u_{\vert t = 0} = h.
\langlebel{eq31}
\end{equation}
We assume that the $A_j$ and $F$ are real valued and real analytic near
$(0, \underline x, \underline u) \in {\mathbb R} \times {\mathbb R}^d \times {\mathbb R}^N$.
We want to compare two solutions of \eqref{eq31} with initial data $h_1$ and
$h_2$. We can choose $h_1$ to be analytic, for instance
$h_1 (x) = \underline u$, and find an analytic local solution
$u_1$ by Cauchy-Kowalewski theorem.
Changing $u$ to $u- u_1$, we get an equation similar
to \eqref{eq31}, with the additional information that $0$ is a solution, that is:
\begin{equation}
F(t, x, 0) = 0 \quad \mathrm{ or } \quad F(t, x, u) = F_1(t,x, u) u .
\end{equation}
We look for solutions of \eqref{eq31} in lens shaped domains
\begin{equation}
\Omega_{r, \delta} = \big\{ (t, x) \ : \ t \ge 0 , \ \vert x - \underline x \vert^2 + \delta t < r^2 \big\},
\end{equation}
assuming that the equation is not hyperbolic at $(0, \underline x, 0)$:
\begin{assumption}
There is $\underline xi \in {\mathbb R}^d $ such that the
matrix $\underline A :=
\sum \underline xi_j A_j(0, \underline x, 0)$ has a nonreal eigenvalue.
\end{assumption}
The next theorem shows that the Cauchy problem is not
H\"older well posed. We denote by $B_r$ the ball of radius $r$
centered at $\underline x$.
\begin{theorem}
\langlebel{theo32}
For all $m$, $\alpha \in ]0, 1]$, $r_0 > 0$ and $\delta > 0$,
there are $r_\varepsilon \to 0$, families of initial data $h_\varepsilon \in H^m(B_{r_0})$ and solutions
$u_\varepsilon $ of $\eqref{eq31}$ on $\Omega_{r_\varepsilon , \delta}$, such that
\begin{equation}
\lim_{\varepsilon \to 0} \Vert u_\varepsilon \Vert_{L^2(\Omega_{r_\varepsilon , \delta})} /
\Vert h_\varepsilon \Vert^\alpha_{H^m(B_{r_0})} = + \infty.
\end{equation}
\end{theorem}
\bigbreak
Let $\langlembda_0$ denote an eigenvalue of $\underline xi \cdot A(0, \underline x, 0)$ such that
$\gamma_0 = \vert {\rm Im } \langlembda_0 \vert > 0$ is maximum.
Let $\underline r $ denote an eigenvector associated to $\langlembda_0$.
We consider initial data
\begin{equation}
\langlebel{eq35}
h_{\varepsilon }(x) := \varepsilon ^M \ {\rm Re } \big( e^{i x \cdot \underline xi / \varepsilon } \underline r \big) \,.
\end{equation}
We look for solutions
\begin{equation}
u_\varepsilon (t, x) = \mathrm{u}(t, x, t/\varepsilon , x \cdot \underline xi/\varepsilon )
\end{equation}
where $\mathrm{u} (t, x, s, \theta)$ is $2 \pi$ periodic in $\theta$. For $u_\varepsilon $ to be
solution of the equation, it is sufficient that $\mathrm{u}$ solves an equation of the form
\begin{equation}
\langlebel{eq37}
\partial _s \mathrm{u} = \mathrm{A} (y, \mathrm{u}) \partial _\theta \mathrm{u} +
\varepsilon \big( \mathrm{B} (y, \mathrm{u}) \partial _y u + F(y, \mathrm{u}) \big) ,
\end{equation}
with $y= (t,x-\underline x)$ and
$\mathrm{A} (y, u) = \sum \underline xi_j A_j (t, x, u)$. In particular
$ \mathrm{A} (0, 0) = \underline A$ and the equation reads
\begin{equation}
(\partial _s - \underline A \partial _\theta) \mathrm{u} = \mathrm{F} (\mathrm{u}) :=
( \mathrm{A} - \underline A) \partial _\theta \mathrm{u} +
\varepsilon \big( \mathrm{B} \partial _y u + F(y, \mathrm{u}) \big) .
\langlebel{eq37b}
\end{equation}
The solution of the Cauchy problem is given by
\begin{equation}
\langlebel{eq39}
\mathrm{u} = e^{ s \underline A \partial _\theta} \mathrm{h} + {\mathcal T}(\mathrm{u}) \,,
\quad {\mathcal T}(\mathrm{u}) (s) := \int_{0}^s e^{ (s - {s'}) \underline A \partial _\theta} \mathrm{F} (\mathrm{u}(s') ) ds'.
\end{equation}
We solve this equation following the method explained in Wagschall \cite{Wa}
(see also the references therein).
\medbreak
\noindent{\sl Function spaces and existence of solutions. }
Given power series
$\mathrm{u} = \sum u_\alpha y^\alpha$ and $\Phi = \sum \Phi_\alpha y^\alpha$, we say that
$\mathrm{u} \ll \Phi$ when $\vert u_\alpha \vert \le \Phi_\alpha $ for all $\alpha$.
Consider the series
$$
\phi (z) = c_0 \sum_{n=0}^{+\infty} \frac{z^n}{n^2 + 1}
$$
where $c_0$ is taken such that $\phi^2 \ll \phi$ (cf \cite{La1}, \cite{Wa}).
For $y \in {\mathbb R}^{1+d}$, we denote by $Y = \sum {y_j}$, and we will consider
power series $\mathrm{u}(y)$ such that there is a constant $C$ such that
$$
\mathrm{u}(y) \ll C \phi (R Y + R_0 ).
$$
$R > 0 $ and $R_0 \in ]0, 1]$ are given parameters. These power series are convergent
for $R \sum \vert y_j \vert + R_0 \le 1$.
Next we introduce the weight function on ${\mathbb Z}$:
$$
\langlengle n \ranglengle = \vert n \vert \quad \mathrm{when} \ n \ne 0,
\qquad \langle 0 \rangle = 2.
$$
Note that for all $p$ and $q$ in ${\mathbb Z}$:
\begin{equation}
\langlebel{eq39}
\langle p+q\rangle \le \langle p \rangle + \langle q \rangle.
\end{equation}
Given positive parameters $\gamma$, $\kappa$, $\varepsilon $, $R$ and $\mathrm{h}o$, we consider
formal Fourier series
\begin{equation}
\mathrm{u} (s, \theta, y) = \sum_{- \infty} ^{+ \infty} \mathrm{u}_n (s, y) e^{i n \theta}
\langlebel{eq310}
\end{equation}
where the $\mathrm{u}_n(s, y)$ are power series in $y$, with coefficient
$C^\infty$ in $s \in [0, \underline s ]$, where
\begin{equation}
\langlebel{eq312}
\underline s := \min\{ \kappa/\gamma , 1/ \varepsilon \mathrm{h}o\}.
\end{equation}
We denote by
$\EE$ the space of $\mathrm{u}$
such that there is a constant $C$ such that for all
$s \in [0, \underline s ]$:
\begin{equation}
\mathrm{u}_n( s, y) \ll C \frac{c_1}{n^2 + 1} e^{( \gamma s - \kappa)\langle n \rangle}
\phi (R Y + \varepsilon \mathrm{h}o s) \,.
\langlebel{eq311}
\end{equation}
The number $c_1$ is chosen such that
$$
\sum_{p+ q = n} \frac{c_1}{p^2+1} \frac{c_1}{q^2+1} \le \frac{c_1}{n^2 + 1} .
$$
Elements $\mathrm{u} \in \EE$ define smooth functions on the domain
\begin{equation}
\langlebel{dom314}
\begin{aligned}
\partial elta= \big\{ (s, \theta, y) \ :
0 \le s < & \underline s , \\
& \theta \in {\mathbb T} , \ R \sum \vert y_j \vert + \varepsilon \mathrm{h}o s < 1 \big \}.
\end{aligned}
\end{equation}
The best constant $C$ in \eqref{eq311} defines a norm $\vert\!\vert\!\vert u \vert\!\vert\!\vert$ on
$\EE$. Equipped with this norm, $\EE$ is a Banach space.
The choice of $c_0$ and $c_1$ and \eqref{eq39} imply that
$\EE$ is a Banach algebra:
\begin{equation}
\langlebel{eq313}
\vert\!\vert\!\vert \mathrm{u} \mathrm{v} \vert\!\vert\!\vert \le \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \ \vert\!\vert\!\vert \mathrm{v} \vert\!\vert\!\vert .
\end{equation}
When $\mathrm{u}$ is valued in ${\mathbb C}^N$, we denote by
$\vert\!\vert\!\vert u \vert\!\vert\!\vert$ the sup of the norms of the components of $\mathrm{u}$.
\begin{lemma}
\langlebel{lem32}
If $F(y, u)$ is holomorphic on a neighborhood
of the origin in ${\mathbb C}^{1+d} \times {\mathbb C}^{N}$, there are constants
$R_0$, $C_0$ and $a_0$ such that for all parameters
$\gamma$, $\kappa$, $\varepsilon $, $R \ge R_0$ and $\mathrm{h}o$,
the mapping $\mathrm{u} \mapsto F(\cdot, \mathrm{u})$ maps the ball of radius
$a_0$ of $\EE$ into the ball of radius $C_0$ in $\EE$.
\end{lemma}
\begin{proof}
There are constants $R_0$, $C$ and $a$ such that
$$
F(y, u) \ll C \phi(R_0 Y) \prod_{j=1}^N \frac{1}{a - u_j} ,
$$
in the sense of power series in $(y,u)$.
Substituting $u= u(y)$ in the expansion, using \eqref{eq313} as well as the
identities $\phi^2 \ll \phi$ and $\phi (R_0 Y) \ll \phi(R Y + \varepsilon \mathrm{h}o s)$,
yields for $\vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert < a$:
$$
\vert\!\vert\!\vert F( \cdot, \mathrm{u}) \vert\!\vert\!\vert \le C \frac{1}{(a - \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert)^N} .
$$
\end{proof}
We further denote by $\vert\!\vert\!\vert \cdot \vert\!\vert\!\vert'$ the norm obtained
when $\phi$ is replaced by its derivative $\phi'$ in \eqref{eq311},
and by $ \vert\!\vert\!\vert \cdot \vert\!\vert\!\vert_1$ the norm obtained when
$c_1/ (n^2+1)$ is replaced by $ c_1 / \sqrt{ n^2+1}$. In particular, there holds:
\begin{eqnarray*}
\vert\!\vert\!\vert \partial _y \mathrm{u} \vert\!\vert\!\vert' & \le & R \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert,
\\
\vert\!\vert\!\vert \partial _\theta \mathrm{u} \vert\!\vert\!\vert_1 & \le & \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert.
\end{eqnarray*}
Moreover, differentiating the estimate $\phi^2 \ll \phi $ implies that
$2 \phi \phi' \ll \phi'$, thus
\begin{equation*}
2 \vert\!\vert\!\vert \mathrm{u} \mathrm{v} \vert\!\vert\!\vert' \le \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \ \vert\!\vert\!\vert \mathrm{v} \vert\!\vert\!\vert' .
\end{equation*}
Similarly, there is $c_2$ independent of all the parameters such that:
\begin{equation*}
\vert\!\vert\!\vert \mathrm{u} \mathrm{v} \vert\!\vert\!\vert_1 \le c_2 \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \ \vert\!\vert\!\vert \mathrm{v} \vert\!\vert\!\vert_1.
\end{equation*}
Factoring out $y$ and $u$ in $\mathrm{A}(y, u) - \underline A$ and $u$ in $F(y, u)$, using that
$y \ll ( 2 / c_0 R) \phi ( R Y ) \ll (2 c_0 / R) \phi (RY + \varepsilon \mathrm{h}o s)$ and
that $\phi' \ll \phi$, we deduce from
Lemma \ref{lem32} and the estimates above that for
$R \ge R_0$ and $\vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \le a_0$:
\begin{eqnarray}
\langlebel{eq316} \vert\!\vert\!\vert (\mathrm{A} (y, \mathrm{u}) - \underline A) \partial _\theta \mathrm{u} \vert\!\vert\!\vert_1 & \le &C
\big( R^{-1} + \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \big) \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert ,
\\
\langlebel{eq317} \vert\!\vert\!\vert \mathrm{B}(y, \mathrm{u}) \partial _y \mathrm{u} + F(y, \mathrm{u}) \vert\!\vert\!\vert' & \le& C
R \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert.
\end{eqnarray}
Next, we investigate the action of the operator
$$
\mathrm{v} (s) = {\mathcal I}(\mathrm{f}) (s) := \int_0^s e^{ (s- {s'}) \underline A \partial _\theta } \mathrm{f} (s') ds' \,.
$$
On each Fourier component, it reads
$$
\mathrm{v}_n(s, y ) = \int_0^s e^{ i n (s- {s'}) \underline A } \mathrm{f}_n (s', y ) ds'.
$$
By the definition of $\gamma_0$, for all
$\gamma > \gamma_0$ there is a constant $K_\gamma$ such that:
\begin{equation}
\langlebel{eq318}
\forall n \in {\mathbb Z}, \ \forall s \in [0, + \infty [:
\quad e^{ i n s \underline A } \le K_\gamma e^{ \vert n \vert \gamma s } .
\end{equation}
By definition of the norm $\vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert_1$, there holds:
$$
\mathrm{f}_n (s', y) \ll \frac{c_1}{\sqrt{n^2 +1}} \vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert_1 e^{ (s' \gamma - \kappa) \langle n \rangle}
\phi( R Y + \varepsilon \mathrm{h}o s') .
$$
Using that for $s' \le s$, $\phi( R Y + \varepsilon \mathrm{h}o s') \ll \phi( R Y + \varepsilon \mathrm{h}o s) $, and integrating
term by term the power series in $y$, implies that
$$
\mathrm{v}_n (s, y) \ll \frac{c_1 K_{\gamma_1} }{\sqrt{n^2 +1}} \vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert_1
\phi( R Y + \varepsilon \mathrm{h}o s)
\int_0^s e^{ \vert n \vert \gamma_1 (s- {s'})} e^{ (s' \gamma - \kappa) \langle n \rangle}
ds'
$$
For $\gamma > \gamma_1 > \gamma_0$, the last integral is estimated by
$$
e^{ ( s \gamma - \kappa) \langle n \rangle}
\int_0^s e^{ {s'} (\vert n \vert \gamma_1 - \langle n \rangle \gamma) } ds'
\le \frac {C}{ (\gamma- \gamma_1) \langle n \rangle } e^{ ( s \gamma - \kappa) \langle n \rangle} \,.
$$
Choosing $\gamma_1 = (\gamma + \gamma_0) /2$, this shows that for
all $\gamma > \gamma_0$, there is a constant $K_\gamma$ such that
$$
\vert\!\vert\!\vert {\mathcal I}( \mathrm{f}) \vert\!\vert\!\vert \le K_\gamma \vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert_1.
$$
Similarly, there holds
$$
\mathrm{v}_n(s, y) \ll \frac{K_ \gamma} {n^2 + 1} \vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert' e^{ (s \gamma - \kappa) \langle n \rangle }
\int_0^s e^{ \gamma (s - s') ( \vert n \vert - \langle n \rangle) }
\phi' (R Y + \varepsilon \mathrm{h}o s') ds' .
$$
Since $\vert n \vert \le \langle n \rangle$, we can ignore the exponential in the integral.
Moreover,
$$
\varepsilon \mathrm{h}o \int_0^s
\phi' (R Y + \varepsilon \mathrm{h}o s') ds' \ll \phi (RY + \varepsilon \mathrm{h}o s) - \phi(RY) \ll
\phi(RY + \varepsilon \mathrm{h}o s) .
$$
Therefore,
$$
\vert\!\vert\!\vert {\mathcal I}( \mathrm{f}) \vert\!\vert\!\vert \le \frac{K_\gamma}{ \varepsilon \mathrm{h}o} \vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert'.
$$
Using \eqref{eq316} \eqref{eq317}, these inequalities yield estimates for the operator ${\mathcal T}(\mathrm{u}) $ defined in \eqref{eq39}.
Similarly, one obtains estimates for increments ${\mathcal T}(\mathrm{u}) - {\mathcal T}(\mathrm{v})$:
\begin{prop}
There are $R_0$ and $a_0$ and for all $\gamma > \gamma_0$ there
is a constant $K_\gamma$, such that for all $R \ge R_0$, all $\kappa >0$, all
$\mathrm{h}o > 0$ and all $\varepsilon \in ]0, 1]$, there holds for
all $\mathrm{u}$ and $\mathrm{v}$ in $ \EE$ such that
$\vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \le a_0$ and $\vert\!\vert\!\vert \mathrm{v} \vert\!\vert\!\vert \le a_0$:
$$
\begin{aligned}
&\vert\!\vert\!\vert {\mathcal T}(\mathrm{u}) \vert\!\vert\!\vert \le K_\gamma \big( R^{-1} + 2 \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert + R \mathrm{h}o^{-1} \big) \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert,
\\
&\vert\!\vert\!\vert {\mathcal T}(\mathrm{u}) - {\mathcal T}(\mathrm{v}) \vert\!\vert\!\vert \le K_\gamma \big( R^{-1} + \vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert
+ \vert\!\vert\!\vert \mathrm{v} \vert\!\vert\!\vert + R \mathrm{h}o^{-1} \big)
\vert\!\vert\!\vert \mathrm{u} - \mathrm{v} \vert\!\vert\!\vert
\end{aligned}
$$
\end{prop}
\begin{cor}
\langlebel{cor35}
With notations as above, if
$$
K_\gamma \big( R^{-1} + 4 a + R \mathrm{h}o^{-1} \big) < \frac{1}{2},
$$
then for all $\mathrm{f} \in \EE$ with $\vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert \le a$, the equation
$$
\mathrm{u} = \mathrm{f} + {\mathcal T}(\mathrm{u})
$$
has a unique solution $\mathrm{u} \in \EE$ such that
$\vert\!\vert\!\vert \mathrm{u} \vert\!\vert\!\vert \le 2 a$. Moreover,
$$
\vert\!\vert\!\vert \mathrm{u} - \mathrm{f} \vert\!\vert\!\vert \le K_\gamma \big( R^{-1} + 4 \vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert + R \mathrm{h}o^{-1} \big)
\vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert.
$$
\end{cor}
\bigbreak
\noindent{\sl Application.} In accordance with \eqref{eq35}, we solve the Cauchy problem
\eqref{eq37} with initial data
\begin{equation}
\langlebel{eq319}
\mathrm{u}_{\vert s = 0} = \mathrm{h} := \varepsilon ^M \mathrm{Re} ( e^{ i \theta} \underline r ) .
\end{equation}
Let
\begin{equation}
\mathrm{f} = e^{s \underline A \partial _\theta} \mathrm{h} = \varepsilon ^M \mathrm{Re} \big( e^{i s \langlembda_0 + i \theta} \underline r \big) .
\langlebel{eq320}
\end{equation}
We consider only small values of the parameter $\varepsilon $, and we use the notation
$\varepsilon ^M = e^{- \kappa_1}$, that is $\kappa_1 = M \vert \ln \varepsilon \vert$.
Consider a small parameter $\beta > 0 $, to be chosen later on, such that
$\beta M < 1/2 $. We fix
\begin{equation}
\langlebel{eq321}
\left\{ \begin{array}{ll}
\gamma = (1 + \beta) \gamma_0, \quad & \kappa = (1- \beta) \kappa_1, \quad
\\
R = e^{ \beta \kappa_1} = \varepsilon ^{ - \beta M} , \quad & \mathrm{h}o = R^2 = \varepsilon ^{- 2 \beta M}.
\end{array}\right.
\end{equation}
Introduce $\sigma = (1 - \beta)/ (1 + \beta) < 1$.
For $\varepsilon $ small enough, $\kappa/ \gamma = \sigma \kappa_1/\gamma_0 \le
(\varepsilon \mathrm{h}o)^{-1} = \varepsilon ^{ -1 + 2 \beta M}$, thus, the end point \eqref{eq312}
is $\underline s = \sigma \kappa_1/ \gamma_0 $.
\begin{prop}
\langlebel{prop36}
There is a constant $c > 0$ such that
for all $M\ge 1 $ and $\beta \in ]0, 1/ 2M[ $, there is $\varepsilon _0$ such that for all $\varepsilon \in ]0, \varepsilon _0]$,
and parameters as in $\eqref{eq321}$, the Cauchy problem
$\eqref{eq37}, \eqref{eq319}$ has a solution
$\mathrm{u} \in \EE$, and
\begin{equation}
\langlebel{est322}
\forall (s, \theta, y) \in \partial elta \, : \quad
\vert \mathrm{u}(s, \theta, y) \vert \ge c e^{ s \gamma_0 - \kappa_1} .
\end{equation}
\end{prop}
\begin{proof}
For $\varepsilon $ small enough, there holds
$$
\vert\!\vert\!\vert \mathrm{f} \vert\!\vert\!\vert = \frac{2}{c_1 c_0} \vert \underline r \vert \max_{s \in [0, \underline s[}
e^{ \kappa - \kappa_1 + s (\gamma_0- \gamma) } \le C e^{ - \beta \kappa_1}
$$
for some constant $C$ independent of $\varepsilon $ and $\beta$.
By Corollary \ref{cor35}, there is $K$, depending only on $\beta$, such that
for $ K \varepsilon ^{ \beta M} < 1$, the problem has a unique solution
$\mathrm{u} \in \EE$ and
$$
\vert\!\vert\!\vert \mathrm{u} - \mathrm{f} \vert\!\vert\!\vert \le K e^{ - 2 \beta \kappa_1}.
$$
For $(s, \theta, y) \in \partial elta$, there holds
$R \sum \vert y_j \vert + \varepsilon \mathrm{h}o s \le 1$. Since the series
$\phi (z)$ converges at $z =1$,
$$
\vert (\mathrm{u} - \mathrm{f}) (s, \theta, y) \vert
\le K e^{ - 2 \beta \kappa_1} \sum_{n \in {\mathbb Z}} \frac{c_1}{n^2 +1} \phi(1)
e^{ ( s \gamma - \kappa)\langle n \rangle} .
$$
Since $s \gamma - \kappa \le 0$ and $\langle n \rangle \ge 1$, this implies that there is
$K'$ such that
$$
\begin{aligned}
\vert (\mathrm{u} - \mathrm{f}) (s, \theta, y) \vert
& \le K'
e^{ - 2 \beta \kappa_1}
e^{ ( s \gamma - \kappa) }
\\
& \le K' e^{ s \gamma_0- \kappa_1}
e^{ - \beta \kappa_1} e^{ \underline s \beta \gamma_0} =
K' e^{ s \gamma_0- \kappa_1}
e^{ - \beta (1- \sigma) \kappa_1 } .
\end{aligned}
$$
Because $\underline r$ and $\overline {\underline r}$ are eigenvectors associated to distinct eigenvalues
$\langlembda_0$ and $\overline{ \langlembda_0}$, they are linearly independent and there is
$c > 0$ such that
$$
\vert f( s, \theta) \vert \ge 2 c e^{ s \gamma_0 - \kappa_1} .
$$
Since $\sigma < 1$, the two estimates above imply that for $\varepsilon $ small enough \eqref{est322}
is satisfied.
\end{proof}
\bigbreak
\begin{proof}[Proof of Theorem $\ref{theo32} $]
The integer $m\ge 1$ and the H\"older exponent $\alpha\in ]0, 1] $ are given, as well as
the parameter $\delta > 0$.
We fix $M$ large enough, such that
\begin{equation}
\langlebel{rel323}
\alpha' := \frac{M - m}{M} \alpha - \frac{ 1 + d}{ 2 M } > 0 .
\end{equation}
Note that $\alpha' < \alpha \le 1$. Next we choose $\beta > 0$ such that
\begin{equation}
\langlebel{rel324}
1 - \alpha' < \sigma := \frac{1- \beta}{1+ \beta} \quad \mathrm{and} \quad
2 M \beta < 1
\end{equation}
and we fix the parameters $\gamma$, $\kappa$, $R$ and $\mathrm{h}o$ as in
\eqref{eq321}.
By Proposition \ref{prop36}, for $\varepsilon $ small enough, we have a solution
$\mathrm{u}$ of \eqref{eq37} \eqref{eq319} on the domain $\partial elta$ defined in \eqref{dom314}.
Thus
$$
u_\varepsilon (t, x) = \mathrm{u}(\frac{t}{\varepsilon }, \frac{x \cdot \underline xi}{\varepsilon }, t, y)
$$
is a solution of \eqref{eq31} \eqref{eq35} on the domain
\begin{equation*}
\tilde \partial elta_\varepsilon = \big\{ (t, x) \ :
0 \le t < \varepsilon \underline s , \
\ \sum \vert x_j \vert + t + t \varepsilon ^{ - \beta M} < \varepsilon ^{ \beta M} \big \}.
\end{equation*}
Since $\underline t_\varepsilon := \varepsilon \underline s = \sigma \gamma_0^{-1} M \varepsilon \vert \ln \varepsilon \vert $ and
$2 \beta M < 1$, for $\varepsilon $ small enough, this domain contains
\begin{equation*}
\partial elta_\varepsilon = \big\{ (t, x) \ :
0 \le t < \underline t_ \varepsilon , \
\ \vert x \vert < c \varepsilon ^{ \beta M} \big \}
\end{equation*}
with $c = 1/(2 \sqrt d)$. For $\varepsilon $ small, it also contains the lens shaped domain
$\Omega_{r_\varepsilon , \delta} $, with
\begin{equation}
\langlebel{def325}
r_\varepsilon = (t_ \varepsilon / \delta)^{ 1/2} .
\end{equation}
Moreover, for $\varepsilon $ small, $\Omega_{r_\varepsilon , \delta} $ contains
the cube
$$
\big\{ \underline t_\varepsilon - \varepsilon \le t \le \underline t_\varepsilon , \ \vert x - \underline x \vert \le \varepsilon \big\} \,.
$$
Thus, Proposition \ref{prop36} implies that there is $c > 0$ such that for all
$\varepsilon $ small enough:
$$
\Vert u_ \varepsilon \Vert_{L^2(\Omega_{r_\varepsilon , \delta})} \ge c e^{ \underline s \gamma_0 - \kappa_1}
\varepsilon ^{1+ d} = c \varepsilon ^{ M ( 1 - \sigma) + (1 + d)/2}.
$$
On the other hand, the Sobolev norm of the initial data
on a fixed ball $B_{r_0}$ centered at $\underline x$ is of order:
\begin{equation*}
\Vert h_{\kappa, \varepsilon } \Vert_{H^m (B_{r_0})} \le C \varepsilon ^{ M - m} \,.
\end{equation*}
Thus, using the notation \eqref{rel323},
\begin{equation}
\Vert u_\varepsilon \Vert_{L^2(\Omega_{r_\varepsilon , \delta})} /
\Vert h_\varepsilon \Vert^\alpha_{H^m(B_{r_0})} \ge \frac{c}{C^\alpha}
\varepsilon ^{ M (1 - \sigma - \alpha')}
\end{equation}
which, by \eqref{rel324}, tends to $+ \infty$ as $\varepsilon $ tends to zero.
\end{proof}
\section{Solvability in Sobolev spaces}
In this section, we consider the fully nonlinear Cauchy problem in ${\mathbb R}^{1+d}$:
\begin{equation}
\langlebel{eq41}
\left\{\begin{aligned}
& \partial _t u = F(t, x, u, \partial _{x_1} u, \ldots, \partial _{x_d} u ) , \quad t \ge 0
\\
& u_{\vert t = 0} = h,
\end{aligned}\right.
\end{equation}
near $(0, \underline x)$. We assume that $F$ is $C^\infty$ in a neighborhood of $\underline p := (0, \underline x, \underline u, \underline v)$
in ${\mathbb R}\times {\mathbb R}^d \times {\mathbb R}^N \times ({\mathbb R}^N)^d$. The initial data $h$ is smooth
and satisfies
\begin{equation}
\langlebel{eq42}
h(\underline x) = \underline u , \quad \partial _x h(\underline x)= \underline v.
\end{equation}
\begin{assumption}
\langlebel{ass41}
There is $\underline xi \in {\mathbb R}^d$ such that the matrix
$\underline xi \cdot \partial_v F ( \underline p )= \sum \underline xi_j \partial _{v_j} F(\underline p) $ has nonreal eigenvalues.
\end{assumption}
Because $\xi \cdot \partial _v F$ is real, this implies that there is at least one eigenvalue
with positive imaginary part. Denote by $\underline \Pi $ the spectral
projector of
$\underline xi \cdot \partial_v F ( \underline p )$
associated to eigenvalues in $ \{ \mathrm{Im} \mu > 0 \}$.
\begin{theorem}
\langlebel{theo42}
Let $s > d/2+1$ and $ s \le s' < 2 s - 1 - d/2 $.
Suppose that the Cauchy data $h$ satisfies $\eqref{eq42}$ and
\begin{equation}
\langlebel{b43}
(\mathrm{Id} - \underline \Pi) h \in H^{s'} \quad near \ \underline x.
\end{equation}
If
the Cauchy problem $\eqref{eq41}$ has a solution in
$C^0([0, T]; H^{ s }(\omega))$, then
\begin{equation}
\langlebel{b44}
(\underline x, \underline xi) \notin WF_{H^{s'}} ( \underline \Pi h).
\end{equation}
\end{theorem}
In this statement, $WF_{H^{s'}}$ denotes the $H^{s'}$ wave front set.
In the spirit of Hadamard's argument and of Theorem \ref{th21}, Theorem \ref{theo42} shows
that smoothness of part of the Cauchy data, here $\underline \Pi h$, implies
smoothness of the other components.
For all $s'' \in [s, s' [$, there are many Cauchy data $h$ such that
\begin{equation}
\langlebel{b45}
h \in H^{s''}, \quad (\mathrm{Id} - \underline \Pi) h \in H^{s'} , \quad
(\underline x, \underline xi) \in WF_{H^{s'}} ( \underline \Pi h).
\end{equation}
For such data, all $T >0$ and all neighborhood $\omega$ of $\underline x$,
Theorem \ref{theo42} implies that
\eqref{eq41} has no local solution
in $C^0([0, T]; H^{ s }(\omega))$.
This implies that the Cauchy problem is not locally well posed from
$H^{s''}$ to $C^0(H^s)$ for all
$s'' = 2 s -1 - d/2 - \varepsilon > s$ when $s > 1 + d/2$.
The proof is an application of the results of
Monique Sabl\'e-Tougeron \cite{ST} about the propagation of microlocal singularities
for nonlinear boundary value problems.
For the convenience of the reader, we sketch a proof
within the class of spaces $C^0(H^s)$ instead of the class $H^{s,s'}({\mathbb R}^{1+d})$
used in \cite{ST}.
\begin{proof}
Decreasing slightly
$s$, we can assume that
$\mathrm{h}o := s -1 - d/2 \notin {\mathbb N}$. Suppose that $h \in H^{s}({\mathbb R}^d)$ satisfies
\eqref{eq42} and that
$u \in C^0([0, T]; H^{s}(\omega))$ is a solution of \eqref{eq41}.
{\bf a)}
The product is continuous from
$H^{\sigma- \alpha} \times H^{\sigma- \beta}$ into $H^{\sigma - \alpha - \beta}$,
when $\sigma > d/2$, $\alpha \ge 0$, $\beta \ge 0$ and
$\alpha + \beta \le 2 \sigma$.
By induction on $k$, \eqref{eq41} implies that
\begin{equation*}
\partial_t^k u \, \in \, C^0 ( [0; T] H^{ s - k }(\omega)) \,, \quad k \in \{0, \ldots, [2 s - 2] \}\,
\end{equation*}
Therefore, for all smooth function
$G $,
\begin{equation*}
\partial_t^k G(\, \cdot \, , \,\cdot \, , u, \partial_x u) \, \in \, C^0 ( H^{ s- 1 - k }) \, , \quad k \in \{0, \ldots,
[2 s - 2]
\}\, .
\end{equation*}
Since, $\mathrm{h}o = 2s - 2 - (s - 1 + d/2) < 2s -2 - d$ this
property is true up to $ k = [\mathrm{h}o ] + 1$.
Denoting by $C^\alpha({\mathbb R}^d)$ the usual H\"older space
for $\alpha \in {\mathbb R} \backslash {\mathbb Z}$,
this implies that $ g = G( \cdot, u(\cdot), \partial _x u (\cdot))$ satisfies
\begin{equation}
\langlebel{eq43}
\partial_t^k g \, \in \, C^0 ([0, T]; C^{ \mathrm{h}o - k }(\omega)) \, ,
\quad k \in \{0, \ldots, [\mathrm{h}o]+ 1 \}\, .
\end{equation}
For $\mathrm{h}o > 0 $, $\mathrm{h}o \notin {\mathbb N}$, we denote by $\widetilde C _\mathrm{h}o$ the set of functions $g$ which satisfy this
property.
\medbreak
{\bf b)} Localizing near $(0, \underline x)$, and using Bony's paralinearization theorem
in $x$ (\cite{Bo}, \cite{ST}),
\eqref{eq41} implies that $\tilde u = \chi_1 u$ satisfies
\begin{equation}
\langlebel{eq44}
\partial_t \tilde u \, - \, T_A (t, x, \partial_x) \, \tilde u \, = \, f ,
\end{equation}
where $\chi_1 \in C_0^\infty(\omega)$ is equal to one
near $\underline x$ and $f \in C^0([0, T]; H^{ s - 1 + \mathrm{h}o}(\omega')) $
for some smaller neighborhood $\omega'$ of $\underline x$.
In this equation, $T_A$ denotes a paradifferential operator
in $x$ of symbol
\begin{equation}
A (t, x, \xi):=
\sum_{j=1}^d \xi \cdot \partial_v F (p(t, x) ) + \partial_u F p(t, x)
\end{equation}
with $ p(t, x) = (t, x, u(t, x), \partial_x u(t,x))$. The coefficients belong to
$\widetilde C^\mathrm{h}o$.
\medbreak
{\bf c)} We perform a microlocal block diagonalization of $A$.
Near $(0, \underline x, \underline xi)$, there are symbols
\begin{equation*}
P \, = \, \sum_{j = 0} ^{ [\mathrm{h}o]} P_{j }\, , \quad
D \, = \, \sum_{j = 0}^{ [\mathrm{h}o]} D_j\,
\end{equation*}
such that
\begin{equation}
\langlebel{eq46}
P \sharp A - \partial_t (P - P_{[\mathrm{h}o]}) \, = \, D \sharp P\,.
\end{equation}
The terms $P_j(t, x, \xi)$
[resp. $D_j$] are $C^\infty$ and homogeneous of degree $ -j$
[resp. $1-j$] in $\xi$ and $\widetilde C^{\mathrm{h}o - j}$ in $(t,x)$.
In \eqref{eq46}, $\sharp$ denotes the composition of symbols (cf \cite{Bo}):
$$
\Big( \sum_{j \le [\mathrm{h}o]} P_j \Big) \sharp
\Big( \sum_{j \le [\mathrm{h}o]} Q_j\Big) \, := \,
\sum_{ l + m +\vert \alpha \vert \le [\mathrm{h}o]}
\ {1\over \alpha!} (\partial_\xi^\alpha P_l) \, ( (- i \partial _x)^\alpha Q_m)
$$
Moreover, $D$ is block diagonal and there is one block $D^I$ associated to
the spectrum of $ A(0, \underline x, \underline xi)$ in
$\{ \mathrm{Im} \mu > 0 \}$. In particular
\begin{equation}
\langlebel{eq47}
\mathrm{Im \ spec} (D^I_0(0, \underline x, \underline xi) ) > 0
\end{equation}
The construction is classical. The principal terms $P_0$ and $D_0$ are chosen such that
$$
P_0 A_0 (P_0)^{-1} = D_0,
$$
where $A_0 = \xi \cdot \partial _v F$ is the principal symbol of $A$.
Next, one proceeds by induction on $j$, choosing $P_j$ and $D_j$ such that
the terms of degree $1-j$ in the two sides of \eqref{eq46} are equal.
In particular, \eqref{eq46} is an identity between symbols of degree $1$,
like $D$, and the degree of the last term, like $D_{[\mathrm{h}o]}$ is
$1 - [\mathrm{h}o]$. This is why, in $\partial_t P$, we can ignore the last term which would be
$\partial_t P_{[\mathrm{h}o]}$, of degree $- [\mathrm{h}o]$.
In this computation, we only use symbols with positive degrees of smoothness
$\mathrm{h}o -j$, with $j \le [\mathrm{h}o]$.
However, the term $\partial_t P_{[\mathrm{h}o]}$
which will appear in the remainders, requires one more derivative.
This is why we took $k \le [\mathrm{h}o]+1$ in \eqref{eq43}.
\medbreak
{\bf d)} Suppose that $\chi (t, x , \xi) $ is a microlocal cut-off
function supported in a conical neighborhood of $(0, \underline x, \underline xi)$ where
\eqref{eq46} is satisfied.
The equation \eqref{eq44} implies that
\begin{equation*}
(\partial_t - T_ {\chi D} ) T_{\chi P} \tilde u \, = \,
T_{ \chi^2 P} (\partial_t - T_{A} \tilde u) \, + \, T_{ \chi \partial_t P_{[\mathrm{h}o]}} \tilde u \, + \,
T_{ Q} \tilde u \, + \, R u
\end{equation*}
where $R$ is a remainder in the $[\mathrm{h}o]$-calculus $x$. If
$\mathrm{h}o < 1$ then $Q = 0$ and if $\mathrm{h}o > 1$, $Q$ is a symbol of degree zero,
$(\mathrm{h}o-1)$ smooth in $x$ and equal to zero near $(0, \underline x , \underline xi)$.
In particular, near $(0, \underline x)$,
$R u \in C^0( H^{s -1 + \mathrm{h}o})$
and $T_ Q u (t, \cdot) \in H^{s -1 + \mathrm{h}o})$ near $(\underline x, \underline xi))$, uniformly in $t$.
Moreover, since $\partial_t P_{[\mathrm{h}o]}$ is of degree $- [\mathrm{h}o]$
with smoothness $C^{\mathrm{h}o - [\mathrm{h}o]-1}$
in $x$, the operator $ T_{ \chi \partial_t P_{[\mathrm{h}o]}}$ is of order
$- [\mathrm{h}o]+ (1 - \mathrm{h}o +[\mathrm{h}o]) = 1 - \mathrm{h}o$.
Therefore, we see that
$$
w := T_{\chi P} \tilde u
$$
satisfies
\begin{equation}
\langlebel{eq48}
\partial_t w - T_{\chi D} w \in C^0( H^{ s-1 + \mathrm{h}o} ) \,, \quad
{\rm near }\ (0, \underline x, \underline xi)\, .
\end{equation}
\medbreak
{\bf e)} $D$ is block diagonal. Denote by $w_I$ the components of
$w$ which correspond to the bloc $D^I$. The equation \eqref{eq48} implies that
\begin{equation}
\langlebel{eq49}
\partial_t w_I - T_{\chi D^I} w \in C^0( H^{ s-1 + \mathrm{h}o} ) \,, \quad
{\rm near }\ (0, \underline x, \underline xi)\, .
\end{equation}
By \eqref{eq47}, this problem is elliptic and the backward Cauchy problem is well posed
(\cite{ST}). This implies that $ w \in C^0( H^{ s + \mathrm{h}o} )$
near $(0, \underline x, \underline xi)$.
By construction,
$w_I = T_\Pi u $ where $ \Pi = \sum_{j \le [\mathrm{h}o]} \Pi_j $, with $\Pi_j$ of
degree $-j$ in $\xi$ and $\widetilde C^{\mathrm{h}o -j}$ in $(t,x)$.
In particular,
\begin{equation}
\langlebel{eq49b}
w_I {}_{\vert t = 0} = T_{\Pi_{\vert t = 0} } h \in H^{ s + \mathrm{h}o} \, ,
\, \quad
{\rm near }\ (\underline x, \underline xi)\, .
\end{equation}
For $(t,x, \xi)$ close to
$(0, \underline x, \underline xi)$, the principal symbol $\Pi_0 (t, x, \xi) $ is the spectral
projector of
$\xi \cdot \partial_v F (t,x, u(t,x), \partial_xu(t,x) )$
corresponding to eigenvalues in $\{ \mathrm{ Im } \mu > 0 \}$.
In particular
\begin{equation}
\langlebel{eq410}
\underline \Pi := \Pi_0(0, \underline x, \underline xi).
\end{equation}
Since the system $(\Pi_0, \mathrm{Id} - \underline \Pi ) $ is elliptic near $(\underline x, \underline xi)$,
there are symbols $U = \sum_{j \le [\mathrm{h}o]} U_j$ and
$V_j = \sum_{j \le [\mathrm{h}o]}$ of degree zero such that
$$
\underline \Pi \, = \, U \, \sharp \, \Pi _{\vert t = 0} \, + \, V \, \sharp \, (\mathrm{Id} - \underline \Pi),
\quad \mathrm{near} \ (\underline x, \underline xi) \, .
$$
This implies that for $\chi_1$ supported in a sufficiently small conical neighborhood
of $(\underline x, \underline xi)$
\begin{equation}
\langlebel{eq411}
\chi_1(x, D_x) \underline \Pi = T_U T_{\Pi_{\vert t = 0}} + T_V (\mathrm{Id} - \underline \Pi) + R
\end{equation}
with $R$ of order $-\mathrm{h}o$.
Suppose that the initial data satisfies
\begin{equation}
\langlebel{eq413}
h \in H^{s}({\mathbb R}^d) \, , \quad (\mathrm{Id} - \underline \Pi ) h \in H^{s + \mathrm{h}o} ({\mathbb R}^d) .
\end{equation}
Then
\eqref{eq411} and \eqref{eq49b} imply that
$ \chi_1 (x, D_x) \underline \Pi h \in H^{s+\mathrm{h}o}$,
that is
$ \underline \Pi h \in H^{s+ \mathrm{h}o}$ near $ (\underline x, \underline xi)$ or
$(\underline x , \underline xi) \notin WF_{H^{s+ \mathrm{h}o}} (\underline \Pi h)$.
\end{proof}
\begin{remark}
a) Note that only the condition $u \in C^0( H^s) $ is used to prove \eqref{eq49b}.
b) The proof only relies on the ellipticity
$\partial_t - T_{\chi D_I}$. Thus, the Sobolev spaces
$H^s$ do not play any particular role and there are analogous results
in the H\"older spaces $C^\mu$.
c) For semilinear equation, the critical index $1 + d/2$
can be decreased to $d/2$ as usual.
This is also the case for system of conservation laws, since we only need
to paralinearize functions of $u$.
\end{remark}
\bigbreak
One can push a little further the analysis when Assumption \ref{ass41} is strengthened.
\begin{assumption}
\langlebel{ass51}
The real eigenvalues of
$\xi \cdot \partial_v F ( p(t,x))$ are semi-simple and have constant multiplicity,
and there are nonreal eigenvalues.
\end{assumption}
In this case, the condition $s' < 2s - 1 - d/2$ in Theorem \ref{theo42} can be relaxed.
\begin{theorem} Under Assumption $\ref{ass51}$,
for all $\sigma > d/2+1$, there are Cauchy data
$h \in H^{\sigma } ({\mathbb R}^d)$, satisfying $\eqref{eq42} $ such that
for all $s > 2+ d/2$, all $T> 0$ and all neighborhood
$ \omega$ of $\underline x$, the Cauchy problem $\eqref{eq41}$ has
no solution
$u \in C^0([0, T]: H^s(\omega))$.
\end{theorem}
\bigbreak
The meaning is that one can take $\sigma$ very large and
$s$ very close to $ 2+d/2+ 2$, so that $u$
will be of class $C^2$, but not much smoother, while the initial data is
as smooth as we want.
\bigbreak
\begin{proof}
{\bf a)} Suppose that $u \in C^0([0, T]; H^s(\omega))$ solves \eqref{eq41}.
We show that
$u \in C^0([0, T']; H^{s'}(\omega'))$ for $T' < T$,
$ \omega' \subset\!\subset \omega$ and $s'\le \sigma $ such that
$s' < 2s -2 - d/2 $.
It is sufficient to prove that $u \in C^0([0, T']; H^{s'}(\omega'))$
with $s' =\min (\sigma, 2s - 2 - d/2)$ when
$\mathrm{h}o := s- 1 - d/2 \notin {\mathbb N}$.
The analogue is proved in \cite{ST}, for $m$-th order scalar equations, when
the real roots of the principal symbol are simple.
As in the proof of Theorem \ref{theo42}, near any $\xi \ne 0$, there is an elliptic symbol
$P = \sum _{j \le [\mathrm{h}o]}$ such that
$ w := T_{ P} u $
satisfies
$$
\partial_t w - T_{ D} w \in C^0( H^{ s'} ) \,, \quad
w_{\vert t = t_0} \in H^{\sigma} \, \quad
{\rm near }\ (0, \underline x, \xi )\, .
$$
The matrix $D$ is block diagonal. By Assumption \ref{ass51}, the blocks of
the principal symbol $D_0$ are either hyperbolic, that is of the form $ i \langlembda \mathrm{Id}$ with
$\langlembda(t, x, \xi)$ real, or elliptic, meaning that the imaginary part of the eigenvalues
is either positive or negative.
Since the Cauchy data is $H^\sigma$, the equation implies that hyperbolic blocks
are microlocally $H^{s'}$. The same result holds for negative elliptic blocks.
For positive elliptic block, we use the backward elliptic
regularity, and decreasing the interval of time,
we see that the elliptic modes are $C^0(H^{s + \mathrm{h}o})$. This shows that
$w\in C^0(H^{s '}) $. Since $P$ is elliptic, $u$ has the
same regularity.
\medbreak
{\bf b)} Repeating the argument in a), we deduce that any solution
$u\in C^0(H^s)$ with initial data in $H^\sigma$ is necessarily in $C^0(H^\sigma)$ on a smaller domain.
Therefore, by Theorem \ref{theo42}, if
$$
h \in H^{\sigma}({\mathbb R}^d) \, , \quad (\mathrm{Id} - \underline \Pi ) h \in H^{\sigma' + \mathrm{h}o'} ({\mathbb R}^d) \,, \quad
\underline \Pi_0 h \notin H^{\sigma'+ \mathrm{h}o'} (\underline x, \underline xi)\, ,
$$
the Cauchy problem has no solution in
$C^0(H^{\sigma'})$, thus no solution $u\in C^0(H^s)$.
\end{proof}
\bigbreak
\section{An example of problems with elliptic zones}
In this section we consider a modified version of \eqref{ex11}.
This is a nonhyperbolic form of Kirchhoff equation.
The advantage is that we can make explicit computations, the drawback is
that the equation is nonlocal\footnote{Thierry Colin
proposed the simpler example
: $\partial_t u \, = - (1 - \Vert \partial_x u \Vert^2_{L^2}) \, \partial_x^2 u $.
The system \eqref{eq56} is first order and fits the general
presentation of this paper. }.
The modified system reads:
\begin{equation}
\langlebel{eq56}
\left\{\begin{aligned}
&\partial_t u = a(t) \partial_x v ,
\cr
& \partial_t v = \vert a(t) \vert \partial_x u ,
\end{aligned}\right. \qquad \mathrm{with}\quad
a(t) = \Vert u(t) \Vert^2_{L^2} -1 .
\end{equation}
As \eqref{ex11}, this system has a natural (formal) energy:
\begin{equation}
E (t) = \Vert v(t, \cdot) \Vert^2_{L^2} + \Big\vert \Vert u(t, \cdot) \Vert^2_{L^2} - 1 \Big \vert .
\end{equation}
If $u$ is $C^1 (L^2)$, the mapping $t \mapsto U(t) := \Vert u(t, \cdot) \Vert^2_{L^2}$ is
$C^1$, thus $\vert U(t) - 1 \vert$ is Lipschitzian and
$$
\frac{d}{dt} \vert U(t) - 1 \vert = \mathrm{sign} \big(U(t)- 1\big) \frac{d U(t)}{dt}
\quad a.e.
$$
If $(u, v)$ is in addition $C^0(H^1)$, then
$$
\frac{d E(t) }{dt} = \vert a \vert \int v \partial _x u dx + a \mathrm{sign}(a)
\int u \partial _x v dx = \vert a \vert \int \partial _x (u v) dx = 0
$$
In the spirit of \eqref{ex14}, we considered the filtered system,
with truncated frequencies. Since the system has constant coefficients in $x$,
it is sufficient to filter the initial data:
\begin{equation}
\langlebel{eq57}
\left\{\begin{aligned}
&\partial_t u^\langlembda = a^\langlembda (t) \partial_x v^\langlembda ,
\cr
& \partial_t v^\langlembda = \vert a^\langlembda (t) \vert \partial_x u^\langlembda ,
\end{aligned}\right. \qquad
\left\{\begin{aligned}
& u^\langlembda{}_{\vert t = 0} = 0 ,
\cr
& v^\langlembda {}_{\vert t = 0} = S_\langlembda h ,
\end{aligned}\right.
\end{equation}
with $a^\langlembda (t) = \Vert u^\langlembda (t) \Vert^2_{L^2} -1$ and $S_\langlembda$
is defined on the Fourier side by:
$$
\widehat{S_\langlembda h } (\xi) = 1_{\{ \vert \xi \vert \le \langlembda\} } \hat h(\xi).
$$
For Fourier transforms, the system reads for
$\vert \xi \vert \le \langlembda$:
\begin{equation}
\langlebel{eq58}
\left\{\begin{aligned}
&\partial_t \hat u^\langlembda = i \xi a^\langlembda (t) \hat v^\langlembda ,
\cr
& \partial_t \hat v^\langlembda = i \xi \vert a^\langlembda (t) \vert \hat u^\langlembda ,
\end{aligned}\right. \qquad
\left\{\begin{aligned}
& \hat u^\langlembda{}_{\vert t = 0} = 0 ,
\cr
& \hat v^\langlembda {}_{\vert t = 0} = \hat h ,
\end{aligned}\right.
\end{equation}
and $\hat u^\langlembda = \hat v^\langlembda = 0$ for $\vert \xi \vert \ge \langlembda$.
This is a system of ordinary differential equations, and it has local
solutions, $C^1$ in time with values in $L^2$. One can use the energy $E$
to prove that the solutions are global in time, but we provide a direct proof.
Suppose that $(u^\langlembda, v^\langlembda)$ is defined
and that $U^\langlembda(t) := \Vert u^\langlembda(t) \Vert^2_{L^2} \le 1$ on
$[0, T]$. This is certainly true for $T$ small. Then,
$\vert a^\langlembda \vert = - a^\langlembda$ on this interval and for $\vert \xi \vert \le \langlembda$:
\begin{equation}
\langlebel{eq59}
\left\{\begin{aligned}
& u^\langlembda (t, \xi ) = i \sinh ( \xi A^\langlembda(t) \big) h(\xi)
\\
& v^\langlembda (t, \xi) = \cosh ( \xi A^\langlembda(t) \big) h(\xi)
\end{aligned}\right. \qquad
A^\langlembda (t) = t - \int_0^t U^\langlembda(s) ds.
\end{equation}
Therefore
$$
U^\langlembda(t) = \frac{1}{2\pi} \int_{\vert \xi\vert \le \langlembda}
\sinh^2 (\xi A^\langlembda(t)) \,
\vert \hat h (\xi)\vert^2 \, d\xi ,
$$
and
\begin{equation}
\frac{ d U^\langlembda }{dt } = (1 - U^\langlembda(t)) I^\langlembda (t) \,
\langlebel{eq510}
\end{equation}
with
$$
I^\langlembda (t) = \frac{1}{2\pi} \int_{\vert \xi\vert \le \langlembda}
\xi \sinh ( 2 \xi A^\langlembda(t)) \,
\vert \hat h (\xi)\vert^2 \, d\xi .
$$
Since $U^\langlembda(0) = 0$, the equation \eqref{eq510} implies that $1 - U^\langlembda$ does not
vanish and remains positive on $[0, T]$. In particular
$U^\langlembda (T) < 1$. By \eqref{eq59}, there holds
$$
\Vert v^\langlembda (t) \Vert^2_{L^2} = U^\langlembda(t) + \Vert S_\langlembda h \Vert^2_{L^2}
\le 1 + \Vert h \Vert^2_{L^2}.
$$
Therefore, by continuation, this implies that \eqref{eq57} has a unique global solution
in $C^0([0, + \infty[ ; L^2({\mathbb R})$ and that
$U^\langlembda(t) < 1$ for all time.
Moreover, $A^\langlembda \ge 0$ and the integral
$I^\langlembda $ is positive, implying that $U^\langlembda$ is strictly increasing.
Since $U^\langlembda$ is increasing, there holds
$A(t) \le t (1 - U^\langlembda(t))$. Therefore
$$
\begin{aligned}
1 \ge U^\langlembda(t) & \ge
\frac{1}{8 \pi} \int_{C_ \langlembda }
\big( e^{2 \vert \xi \vert t(1 - a^\langlembda(t)) } - 2 \Big) \,
\vert h (\xi)\vert^2 \, d\xi
\\
& \ge
e^{ t(1 - U^\langlembda(t)) \langlembda } \int_{C_\langlembda } \frac{1}{8 \pi}
\vert h (\xi)\vert^2 \, d\xi - \frac{1}{2} \Vert h \Vert^2_{L^2}.
\end{aligned}
$$
where $C_\langlembda := \{ \langlembda 2 \le \vert \xi\vert \le \langlembda\}$.
Hence
\begin{equation}
\langlebel{eq510b}
1 - U^\langlembda(t) \, \le \, \frac{1} { t \langlembda } ( \mu(\langlembda ) + K ) \, .
\end{equation}
where $K = \ln ( 8 \pi (1 + \Vert h \Vert^2)) $ and
\begin{equation}
\mu( \langlembda) = - \ln \Big( \int_{C_\langlembda } \vert h (\xi)\vert^2 \, d\xi \Big)
\end{equation}
The condition $\mu (\langlembda) \le C \langlembda$ implies that
$h$ is real analytic. On the other hand, for general non analytic functions,
there holds
\begin{equation}
\langlebel{eq511}
\lim_{ \langlembda \to + \infty} \frac{ \mu (\langlembda) }{\langlembda} = 0.
\end{equation}
Typically, for general $H^s$ functions which are not smoother than $H^s$,
$\mu \le C \ln \langlembda$ with $C$ related to $s$.
\begin{prop}
Suppose that $h \in L^2( {\mathbb R})$ is not analytic in the sense that it satisfies
$\eqref{eq511}$. Then solutions $(u^\langlembda, v^\langlembda)$ of
$\eqref{eq57}$ converge weakly to $(0, h)$.
\end{prop}
This means that the weak limits satisfy
\begin{equation}
\langlebel{eq512}
\left\{\begin{aligned}
&\partial_t \underline u = 0 , \qquad \underline u_{\vert t = 0} = 0,
\cr
& \partial_t \underline v = 0 , \qquad \underline v_{\vert t = 0} = h.
\end{aligned}\right.
\end{equation}
These ``limit'' equations have nothing to see with the original ones \eqref{eq56},
implying that \eqref{eq57} are {\sl not} approximations of \eqref{eq56}.
\begin{proof}
The estimate \eqref{eq510b} and \eqref{eq511} imply that for all
$t > 0$, $U^\langlembda(t) \to 1$ when
$\langlembda \to \infty$. Since $U^\langlembda < 1$ and is increasing,
this implies that $A^\langlembda(t) \to 0 $, uniformly on compacts subsets of $]0, + \infty[$.
By \eqref{eq59}
$$
\widehat u^\langlembda(t, \xi) \ \to \ 0\qquad \hat v^\langlembda(t, \xi) \to \hat (\xi),
$$
uniformly on compacts of $]0, + \infty[\times {\mathbb R}$.
\end{proof}
\bigbreak
\begin{remark}
\textup{The same analysis applies to more general initial data.
One can for instance take $u(0, \cdot) = h \ne 0$, with
$\Vert h \Vert_{L^2} < 1$ and $v(0, \cdot) = 0$. This only amounts to interchange
$\cosh$ and $\sinh $ in \eqref{eq59}.}
\end{remark}
\end{document} | math | 73,507 |
\begin{equation}gin{document}
\title[Distance difference functions]
{Distance difference functions on non-convex boundaries
of Riemannian manifolds}
\author{Sergei Ivanov}
\address{St.~Petersburg Department of Steklov Mathematical Institute of
Russian Academy of Sciences,
Fontanka 27, St.Petersburg 191023, Russia}
\address{Saint Petersburg State University,
7/9 Universitetskaya emb., St. Petersburg 199034, Russia
}
\email{[email protected]}
\thanks{Research is supported by the Russian Science Foundation grant 16-11-10039}
\keywords{Distance functions, inverse problems}
\subjclass[2010]{53C20}
\begin{equation}gin{abstract}
We show that a complete Riemannian manifold with boundary
is uniquely determined, up to an isometry,
by its distance difference representation on the boundary.
Unlike previously known results, we do not impose any
restrictions on the boundary.
\end{abstract}
\maketitle
\section{Introduction}
Let $M=(M,g)$ be a complete, connected, $C^\infty$
Riemannian manifold with boundary $F=\partial M\ne\emptyset$.
We study the following structure introduced by Lassas and Saksala \cite{LS}.
For every $x\in M$ consider the \textit{distance difference function}
$$
D_x\colon F\times F \to\mathbb R
$$
defined by
\begin{equation}\lambdabel{e:ddef}
D_x(y,z) = d_M(x,y)-d_M(x,z), \qquad y,z\in F ,
\end{equation}
where $d_M$ is the arclength distance in~$M$.
Note that for every $x_1,x_2\in M$, the distance $d_M(x_1,x_2)$
is realized by a shortest path
which is a $C^1$ curve, see e.g.~\cite{AA}.
The boundary is not assumed convex, thus
a shortest path can touch the boundary, bend along it, etc.
We regard the collection of functions $D_x$, $x\in M$,
as a map
$$
\mathcal D\colon M\to\mathcal C(F\times F)
$$
given by
\begin{equation}\lambdabel{e:ddef2}
\mathcal D(x)=D_x , \qquad x\in M .
\end{equation}
Here $\mathcal C(F\times F)$ denotes the space of all continuous functions
on $F\times F$.
The map $\mathcal D$ is called the \textit{distance difference representation} of $M$,
and the set $\mathcal D(M)\subset\mathcal C(F\times F)$
is called the \textit{distance difference data} \cite{LS}
or \textit{travel time difference data} \cite{HS}.
In this paper we use the former term.
The main result of this paper tells that the distance difference data $\mathcal D(M)$
determine $M$ uniquely up to a Riemannian isometry.
See Theorem \ref{t:whole} below for the precise statement.
This improves results from \cite{HS} and \cite{I20}
where similar theorems were obtained under additional
assumptions on the boundary.
Moreover in Theorem \ref{t:part} we show that the geometry of an arbitrary open region $U\subset M$
is determined by the partial distance difference data $\mathcal D(U)$.
The intuition behind the problem is the following.
Imagine that $M$ is some material object of interest,
for example the Earth, and $g$ represents the speed of
wave propagation in $M$.
A point $x\in M$ can be a spontaneous spherical wave source
(for example, think of microseismic events in the Earth's crust).
An observer measures arrival times
of the wave at a dense set of points on the surface.
Since the time of the event in the interior is unknown,
the information obtained from the measurement is precisely
the same as that provided by the function $D_x$.
Knowing the set $\mathcal D(M)$ means that such information
is collected from a dense set of points in the interior,
and the goal is to learn the geometry of $M$ from these data.
See \cite{LS,HS} for more detailed discussion of applications and \cite{KKL}
for applications of ordinary distance representations.
i.e., plain distance functions $d_M(x,\cdot)$ rather than
the differences.
\subsection*{Previous results}
Lassas and Saksala \cite{LS} proved the unique determination of $M$
in a different setting where $M$ is compact and has no boundary,
and the ``observation domain'' $F$ is an open subset of $M$
rather than the boundary.
In \cite{I20} this is generalized to the case
of a complete but possibly non-compact manifold $M$
and partial distance difference data $\mathcal D(U)$ where $U\subset M$
is an open set whose geometry is to be determined.
The case when $F=\partial M$ turns out to be more difficult.
It is partly addressed in \cite{HS} and \cite{I20}
where the unique determination of the manifold is proved
under various additional assumptions on the boundary.
The assumption in \cite{I20} is the nowhere concavity
of the boundary and in \cite{HS} it is a less restrictive
``visibility'' condition.
We emphasize that in this paper we do not make additional
assumptions on the boundary.
The similar problem about boundary distance data
(without the ``difference'' part) was solved earlier,
see \cite{KKL,Kur}. See also \cite{KLU} for similar problems
in the Lorentzian setting.
\subsection*{Statement of the results}
We begin with a global determination result:
\begin{equation}gin{theorem}\lambdabel{t:whole}
Let $M=(M,g)$ and $M'=(M',g')$
be complete, connected
Riemannian $n$-manifolds ($n\ge 2$) with a common
boundary $F=\partial M=\partial M'\ne\emptyset$.
Assume that the distance difference data of $M$ and $M'$
coincide as subsets of $\mathcal C(F\times F)$,
i.e.\ $\mathcal D(M)=\mathcal D'(M')$ where $\mathcal D$ and $\mathcal D'$ are
their respective distance difference representations,
see \eqref{e:ddef} and \eqref{e:ddef2}.
Then $M$ and $M'$ are isometric via a Riemannian isometry that fixes $F$.
\end{theorem}
The words ``common boundary'' in the theorem require clarification.
Their precise meaning is the following: $F$ is a topological manifold
and it is identified with $\partial M$ and $\partial M'$ by means of some
homeomorphisms. In other words, we assume that $M$ and $M'$
induce the same topology on $F$ but do not assume that they
induce the same differential structure and metric.
Like in \cite{I20}, we actually prove a more general result on
unique determination of local geometry from the corresponding
partial data:
\begin{equation}gin{theorem}\lambdabel{t:part}
Let $M=(M,g)$ and $M'=(M',g')$ be complete, connected
Riemannian $n$-manifolds ($n\ge 2$)
with a common boundary $F=\partial M=\partial M'\ne\emptyset$.
Let $\mathcal D$ and $\mathcal D'$ denote the distance difference representations
of $M$ and $M'$ in $\mathcal C(F\times F)$, see \eqref{e:ddef} and \eqref{e:ddef2}.
Let $U\subset M$ and $U'\subset M'$ be open sets such that
\begin{equation}\lambdabel{e:sameset}
\mathcal D(U) = \mathcal D'(U') .
\end{equation}
Then there exists a Riemannian isometry
$\phi\colon U\to U'$
(with respect to the metrics $g$ and $g'$)
such that $\phi|_{U\cap F}$ is the identity.
\end{theorem}
Note that Theorem \ref{t:whole} is a special case
of Theorem \ref{t:part} for $U=M$ and ${U'=M'}$.
The proof of Theorem \ref{t:part} occupies the rest of the paper.
It builds upon results from \cite{I20} and \cite{KKL}.
In particular one of the important ingredients of the proof is
Proposition \ref{p:bilip} borrowed from \cite{I20}.
The paper is organized as follows.
In Section \ref{sec:prelim} we fix notation,
collect preliminaries and construct a candidate
for the isometry $\phi$.
In Section \ref{sec:nearest} we find a region on
the boundary where a given distance function is regular,
see Lemma \ref{l:no-cut}.
This step of the proof is essentially borrowed from \cite{KKL}.
In Section \ref{sec:shortest}
we show that certain minimizing
geodesics in $M$ are mapped by $\phi$ to geodesics in $M'$
(but not necessarily preserving the arc length).
Similar proof steps can be found in \cite{LS,HS,I20}
but the details in each case are different.
Finally in Section \ref{sec:final} we prove the key Lemma \ref{l:dphi},
which in a sense reconstructs the metric
tensor at a point,
and deduce the theorems.
The arguments in Section \ref{sec:final}
are similar to those in \cite{I20} with some
modifications.
\subsection*{Remarks on regularity}
The Riemannian manifolds in this paper are $C^\infty$.
This is different from \cite{I20} where the proof of the main result
requires only sectional curvature bounds
and works for Alexandrov spaces as well.
More regularity is needed in Proposition~\ref{p:bilip}
whose proof in \cite{I20} depends
on smooth extension of the metric beyond the boundary,
and in Lemma \ref{l:no-cut} where we rely on properties of cut loci.
It is plausible that the result holds under weaker regularity
assumptions such as uniform bounds on the sectional curvature and
the second fundamental form of the boundary.
Such an improvement would
imply stability of manifold determination with respect
to Gromov-Hausdorff topology, cf.~\cite[Proposition 6.4]{I20}.
\section{Preliminaries and notation}
\lambdabel{sec:prelim}
Let $M,g,\mathcal D,U,M',g',\mathcal D',U'$ be as in Theorem \ref{t:part}.
We denote by $D_x$ and $D'_x$ the distance difference functions
defined by \eqref{e:ddef} for $M$ and $M'$, resp.
In the sequel a number of lemmas are stated
only for $M$ but they apply to both $M$ and $M'$.
\begin{equation}gin{notation}
For $x\in M$, we denote by $T_xM$ the tangent space of $M$ at $x$
and by $S_xM$ the unit sphere of $T_xM$ (with respect to~$g$).
That is,
$$
S_xM = \{v\in T_xM : \|v\|_g = 1\} .
$$
As usual we write $\<u,v\>$ instead of $g(u,v)$ for $u,v\in T_xM$.
For $x\ne y\in M$, we denote by $[xy]$ a shortest path from $x$ to $y$.
In general, a shortest path is not unique;
we assume that some choice of $[xy]$ is fixed for every pair $x,y$.
By $\overrightarrow{xy}$ we denote the unit tangent vector of $[xy]$ at~$x$.
That is, $\overrightarrow{xy}\in S_xM$ is the initial velocity of the unit-speed
parametrization of $[xy]$.
\end{notation}
We need the following standard implication of the Gauss Lemma:
If $x\ne y\in M$ and the distance function
$d_M(\cdot,y)$ is differentiable at $x$ then the Riemannian gradient
of this function at $x$ is given by
\begin{equation}\lambdabel{e:grad0}
\operatorname{grad}_g d_M(\cdot,y)|_x = - \overrightarrow{xy}
\end{equation}
Equivalently, the differential of $d_M(\cdot,y)$ at $x$ is given by
\begin{equation}\lambdabel{e:grad}
d_x d_M(\cdot,y) = -\<\cdot , \overrightarrow{xy} \>_g
\end{equation}
for all $v\in T_xM$.
To see why \eqref{e:grad0} holds
(even in manifolds with boundary),
observe that $d_M(\cdot,y)$ is a 1-Lipschitz function and
it decays with speed 1 along $[xy]$.
Hence $-\overrightarrow{xy}$ is the direction of maximum growth
of this function
and the growth rate is~1, therefore it is the gradient.
We regard the target space $\mathcal C(F\times F)$ of $\mathcal D$
with the sup-norm distance,
$$
\|u-v\| = \sup_{x,y\in F} |u(x,y)-v(x,y)|, \qquad u,v\in \mathcal C(F\times F).
$$
If $F$ is not compact then this distance can
attain infinite values.
However the distance between functions from $\mathcal D(M)$ is always finite.
Indeed, the triangle inequality for $d_M$ implies that
$$
\|\mathcal D(x)-\mathcal D(y)\| \le 2d_M(x,y) < \infty
$$
for all $x,y\in M$.
This inequality also shows that $\mathcal D$ is a 2-Lipschitz map.
We need the following result from \cite{I20}.
\begin{equation}gin{proposition}[{\cite[Proposition 7.1]{I20}}]\lambdabel{p:bilip}
The map $\mathcal D\colon M\to\mathcal C(F\times F)$ is a locally bi-Lipschitz
homeomorphism between $M$ and $\mathcal D(M)$.
\end{proposition}
Now, applying Proposition \ref{p:bilip} to $M$ and $M'$
and using the assumption \eqref{e:sameset}
of Theorem \ref{t:part},
we can define a locally bi-Lipschitz homeomorphism
$
\phi \colon U\to U'
$
by
$$
\phi = (\mathcal D')^{-1}\circ\mathcal D|_{U} .
$$
The definition of $\phi$ implies that
\begin{equation}\lambdabel{e:phi0}
D_x = D'_{\phi(x)}
\end{equation}
for all $x\in U$.
Our ultimate goal is to prove that $\phi$ is a Riemannian isometry.
\section{Nearest and almost nearest boundary points}
\lambdabel{sec:nearest}
Since $M$ is complete, all closed balls of the metric $d_M$
are compact, see e.g.\ \cite[Proposition 2.5.22]{BBI}.
Therefore for every $x\in M$
there exists at least one nearest boundary point,
i.e., a point $y\in F$ realizing the minimum of
the function $d_M(x,\cdot)|_F$.
\begin{equation}gin{lemma}\lambdabel{l:phi-nearest}
Let $x\in U$ and let $y\in F$ be a nearest boundary point to $x$ in $M$.
Then $y$ in a nearest boundary point to $\phi(x)$ in $M'$.
\end{lemma}
\begin{equation}gin{proof}
A point $y\in F$ is a nearest boundary point to $x$ if and only if
$$
\forall z\in F \qquad D_x(y,z) \le 0 ,
$$
see \eqref{e:ddef}.
By \eqref{e:phi0}, this relation implies the same one for $D'_{\phi(x)}$
in place of $D_x$.
Hence $y$ in a nearest boundary point to $\phi(x)$ in $M'$.
\end{proof}
Now we can prove that $\phi$ satisfies the last
requirement of Theorem \ref{t:part}.
\begin{equation}gin{lemma}\lambdabel{l:id-on-boundary}
$U\cap F=U'\cap F$ and $\phi|_{U\cap F}$ is the identity map.
\end{lemma}
\begin{equation}gin{proof}
Since $U$ and $U'$ are open subsets of $M$ and $M'$,
they are topological manifolds, possibly with boundaries
$\partial U= U\cap F$ and $\partial U'=U'\cap F$.
Since $\phi$ is a homeomorphism between $U$ and $U'$,
it sends boundary to boundary. Thus
\begin{equation}\lambdabel{e:id-on-boundary1}
\phi(U\cap F)=U'\cap F .
\end{equation}
Any point $x\in U\cap F$ is a unique nearest boundary
point to itself.
This and Lemma \ref{l:phi-nearest} imply that
$x$ is a unique nearest boundary point to $\phi(x)$.
Since $\phi(x)\in F$ by \eqref{e:id-on-boundary1}, it follows that $\phi(x)=x$.
Thus $U\cap F=U'\cap F$ and $\phi|_{U\cap F}$ is the identity.
\end{proof}
Let $p\in M\setminus F$ and let $q\in F$
be a nearest boundary point to $p$.
Consider a shortest path $[pq]$.
It meets $F$ only at $q$,
therefore it is a Riemannian geodesic.
Furthermore, the first variation formula implies that $[pq]$
meets $F$ orthogonally at~$q$.
These properties imply that $[pq]$ is a unique shortest path
between $p$ and $q$.
As shown in \cite[Lemma 2.13]{KKL}, $p$ and $q$ are not
conjugate along $[pq]$, hence $p$ is not a cut point of $q$
and therefore the distance function $d_M(\cdot,q)$ is smooth at~$p$.
Since the cut-point relation is closed, similar properties hold
for all boundary points sufficiently close to~$q$.
Namely we have the following lemma.
\begin{equation}gin{lemma}\lambdabel{l:no-cut}
Let $p\in M\setminus F$
and let $q\in F$ be a nearest boundary point to $x$.
Then there exists an neighborhood $V\subset F$ of $q$
such that for every $z\in V$ the following holds:
\begin{equation}gin{enumerate}
\item There is a unique shortest paths $[pz]$ in $M$;
\item $[pz]\cap F = \{z\}$ and $[pz]$ meets $F$ transversally;
\item the function $d_M(p,\cdot)$ is differentiable at $z$;
\item the function $d_M(\cdot,z)$ is differentiable at $p$.
\end{enumerate}
\end{lemma}
\begin{equation}gin{proof}
The proof is similar to that of Lemma 2.14 in \cite{KKL} and its essence
is explained above. Here are the formal details.
In order to use standard properties of Riemannian cut loci, we extend
$M$ beyond the boundary to obtain
a complete boundaryless Riemannian $n$-manifold $\widehat M$
such that $F$ is a smooth hypersurface in $\widehat M$
separating $M$ from its complement.
Since $q$ is a nearest to $p$ point of $F$,
$[pq]$ is a unique
shortest path between $p$ and $q$ in both $M$ and $\widehat M$.
By \cite[Lemma 2.13]{KKL}, $p$ and $q$ are not conjugate along $[pq]$.
Therefore $q$ is not a cut point of $p$ in $\widehat M$.
Hence there is a neighborhood $W$ of $q$ in $\widehat M$
such $d_{\widehat M}(p,\cdot)$ is smooth on $W$ and
every point $z\in W$ is connected to $p$
by a unique $\widehat M$-minimizing geodesic whose
direction at $z$ depends smoothly on~$z$.
Since $[pq]$ meets $F$ orthogonally at $q$
and has no other points on $F$,
one can choose a neighborhood
$W_0\subset W$ of $q$ such that, for every $z\in W_0$ the $\widehat M$-minimizing
geodesic $[pz]$ intersects $F$ at most once and transversally.
For $z$ from the ``half-neighborhood''
$W_0\cap M$, this property implies that $[pz]\subset M$
and therefore $[pz]$ is a shortest path in both $\widehat M$ and $M$.
Hence $d_M(p,z)=d_{\widehat M}(p,z)$ for all $z\in W_0\cap M$.
Thus for all $z\in V:=W_0\cap F$ the requirements (1)--(3) of the lemma are satisfied.
To prove (4), recall that the cut-point relation is
symmetric.
Hence, for every $z\in V$, $p$ is not a cut point of $z$
and a similar argument shows that
$d_M(\cdot,z)=d_{\widehat M}(\cdot,z)$ in a neighborhood of $p$.
These properties imply that $d_M(\cdot,z)$ is differentiable at~$p$.
\end{proof}
\begin{equation}gin{remark}\lambdabel{r:twice good}
By Lemma \ref{l:phi-nearest}, a nearest boundary point
$y$ to $x\in U$ is also a nearest boundary
point to $\phi(x)$ in $M'$.
Applying Lemma \ref{l:no-cut} to both manifolds
and taking the intersection of the respective neighborhoods,
we obtain a neighborhood $V\subset F$ satisfying
the requirements (1)--(4) of the lemma
for both $x\in M$ and $\phi(x)\in M'$.
\end{remark}
\section{Shortest paths to boundary points}
\lambdabel{sec:shortest}
The main result of this section is Lemma \ref{l:phi-segment}
about $\phi$-images of certain geodesics
(compare with \cite[Lemma 2.9]{LS} and \cite[Lemma 6.3]{I20}).
The following preparation lemma
characterizes these geodesics in terms of distance difference
functions.
\begin{equation}gin{lemma}\lambdabel{l:segment}
Let $p\in M\setminus F$
and let $z\in F$ be a point satisfying
conditions (1)--(4) from Lemma \ref{l:no-cut}.
Then for every $x\in M$ the following holds:
$x\in[pz]$ if and only if the function
$\Phi\colon F\to\mathbb R$ given by
\begin{equation}\lambdabel{e:Dpx}
\Phi(y) = D_p(y,z)-D_x(y,z), \qquad y\in F,
\end{equation}
where $D_p=\mathcal D(p)$ and $D_x=\mathcal D(x)$
(see \eqref{e:ddef} and \eqref{e:ddef2})
attains its maximum at~$z$.
\end{lemma}
\begin{equation}gin{proof}
Substituting \eqref{e:ddef} into \eqref{e:Dpx} yields that
$\Phi(y)=\Psi(y)+C$ where
$$
\Psi(y) = d_M(p,y) - d_M(x,y)
$$
and $C=d_M(x,z)-d_M(p,z)$ does not depend on~$y$.
We prove the statement of lemma for $\Psi$ instead of $\Phi$.
The statements for $\Phi$ и $\Psi$ are equivalent since
the two functions have the same points of maxima.
To prove the ``only if'' part, consider $x\in[pz]$.
For all $y\in F$ we have
$$
\Psi(y) = d_M(p,y) - d_M(x,y) \le d_M(p,x)
$$
by the triangle inequality.
Since $x\in[pz]$, this inequality turns into equality for $y=z$.
Hence
$$
\Psi(z) = d_M(p,x) = \max_{y\in F} \Psi(y) .
$$
Thus $z$ is a point of maximum of $\Psi$.
To prove the ``if'' part, consider $x\in M$ and assume that
$\Psi$ attains its maximum at~$z$.
First we show that $\overrightarrow{zx}=\overrightarrow{zp}$.
Suppose the contrary.
Let $v\in T_zF$ be the orthogonal projection of
the vector $\overrightarrow{zx}-\overrightarrow{zp}$ to $T_zF$.
Since $\overrightarrow{zx}$ and $\overrightarrow{zp}$ belong to the hemisphere
of $S_zM$ bounded by the hyperplane $T_zF\subset T_zM$,
these two vectors have different projections to $T_zF$.
Hence $v\ne 0$ and moreover
\begin{equation}\lambdabel{e:segment2}
\< v, \overrightarrow{zx}-\overrightarrow{zp} \> > 0 .
\end{equation}
Let $\gamma\colon[0,\varepsilon)\to F$ be a smooth curve with $\gamma(0)=z$
and $\dot\gamma(0)=v$.
By our assumptions the function $d_M(p,\cdot)$ is
differentiable at $z$, hence by \eqref{e:grad},
\begin{equation}\lambdabel{e:segment1}
\frac d{dt} d_M(p,\gamma(t))\big|_{t=0} = - \< v, \overrightarrow{zp} \> .
\end{equation}
Construct a smooth variation of curves $\{\sigma_t\}$, $t\in[0,\varepsilon)$,
where $\sigma_0=[xz]$ and $\sigma_t$ connects $x$ to $\gamma(t)$ for every~$t$.
By the first variation formula,
$$
\frac d{dt}\operatorname{length}(\sigma_t) = - \< v, \overrightarrow{zx} \> .
$$
Hence
$$
d_M(x,\gamma(t)) \le \operatorname{length}(\sigma_t)
\le d_M(x,z) -t \< v, \overrightarrow{zx} \> + o(t), \qquad t\to 0.
$$
This and \eqref{e:segment1} imply that
$$
\Psi(\gamma(t)) = d_M(p,\gamma(t)) - d_M(x,\gamma(t))
\ge \Psi(z) + t \< v, \overrightarrow{zx}-\overrightarrow{zp} \> + o(t), \qquad t\to 0.
$$
By \eqref{e:segment2}, this implies that
$
\Psi(\gamma(t)) > \Psi(z)
$
for a sufficiently small $t>0$.
Hence $\Psi(z)$ is not a maximum of $\Psi$, a contradiction.
This contradiction shows that $\overrightarrow{zx}=\overrightarrow{zp}$,
hence either $x\in [zp]$ or $p\subset[zx]$.
It remains to rule out the latter case.
Suppose that $p\subset[zx]$.
Then we can repeat the argument of the ``only if'' part
with $p$ and $z$ swapped. Namely, for all $y\in F$,
$$
\Psi(y) = d_M(p,y) - d_M(x,y) \ge -d_M(p,x)
$$
by the triangle inequality. This inequality turns into equality
only for $y=z$ since $[pz]$ intersects $F$ only at $z$
and this intersection is transversal.
Thus $\Psi(z)$ is a strict minimum of $\Psi$ rather than
the maximum, a contradiction.
This finishes the proof of the ``if'' part and of the lemma.
\end{proof}
Now we are in a position to prove the main result of this section.
\begin{equation}gin{lemma}\lambdabel{l:phi-segment}
Let $p\in U\setminus F$, $p'=\phi(p)$ and let $V\subset F$ be a neighborhood
constructed in Remark \ref{r:twice good}. Then for every $z\in V$,
$$
\phi([pz]\cap U) = [p'z]\cap U'
$$
where the shortest paths in the left- and right-hand side
are in $M$ and $M'$, resp.
\end{lemma}
\begin{equation}gin{proof}
Since $\phi$ is a bijection between $U$ and $U'$,
we can reformulate the lemma as follows:
a point $x\in U$ belongs to $[pz]$ if and only if $\phi(x)$
belongs to $[p'z]$.
By Lemma \ref{l:segment},
$x\in[pz]$ if and only if $z$ is a point of maximum of the function
\begin{equation}\lambdabel{e:phiseg1}
\Phi(y) = D_p(y,z)-D_x(y,z), \qquad y\in F.
\end{equation}
on $F$.
By the same lemma applied to $M'$, $\phi(x)\in[p'z]$ if and only if
$z$ is a point of maximum of the function
\begin{equation}\lambdabel{e:phiseg2}
\Phi'(y) = D'_{\phi(p)}(y,z)-D'_{\phi(x)}(y,z), \qquad y\in F.
\end{equation}
By \eqref{e:phi0} we have $\Phi=\Phi'$, hence the two
maximality conditions are equivalent.
\end{proof}
\section{Derivative of $\phi$ and proof of the theorems}
\lambdabel{sec:final}
Recall that $\phi\colon U\to U'$ is a locally bi-Lipschitz homeomorphism.
By Rademacher's theorem, every locally Lipschitz is differentiable
almost everywhere.
Applying this to $\phi$ and $\phi^{-1}$ yields that
$\phi$ is differentiable a.e.\ and
its differential $d_x\phi$ at any differentiability point $x\in U$
is a non-degenerate linear map from $T_xM$ to $T_{\phi(x)}M'$.
In the next key lemma we show that this differential is an isometry.
\begin{equation}gin{lemma}\lambdabel{l:dphi}
Let $p\in U$ be a point where $\phi$ is differentiable and $p'=\phi(p)$.
Then the differential $d_p\colon T_pM\to T_{p'}M'$ is a linear isometry
with respect to $g$ and $g'$.
\end{lemma}
\begin{equation}gin{proof}
Let $V\subset F$ be a neighborhood constructed in Remark \ref{r:twice good}.
Then every point $z\in V$ satisfies conditions (1)--(4)
of Lemma \ref{l:no-cut} for both $p$ in $M$ and $p'$ in~$M'$.
These conditions imply that
for every $z\in V$, there is a unique shortest path $[pz]$ and
it initial direction $\overrightarrow{pz}$ depends continuously on~$z$.
Hence the map $z \mapsto \overrightarrow{pz}$ is a homeomorphism
from $V$ onto an open subset $\Sigma$ of the sphere $S_pM$.
Pick two different vectors $v_1,v_2\in\Sigma$
and let $z_1,z_2\in V$ be such that $v_i=\overrightarrow{pz_i}$, $i=1,2$.
Let $\gamma_i$, $i=1,2$, denote the unit-speed parametrization of $[pz_i]$
with $\gamma_i(0)=p$.
By the choice of $V$ (see Lemma \ref{l:no-cut}(4)) the function
$$
t\mapsto D_{\gamma_1(t)}(z_1,z_2) = d_M(\gamma_1(t),z_1)-d_M(\gamma_1(t),z_2)
$$
is differentiable at $t=0$, and by \eqref{e:grad} its derivative is given by
\begin{equation}\lambdabel{e:dphi1}
\frac d{dt} D_{\gamma_1(t)}(z_1,z_2)\big|_{t=0} = -1 + \<v_1,v_2\> .
\end{equation}
Similarly (swapping $v_1$ and $v_2$),
\begin{equation}\lambdabel{e:dphi2}
\frac d{dt} D_{\gamma_2(t)}(z_2,z_1)\big|_{t=0} = -1 + \<v_1,v_2\> .
\end{equation}
The scalar products above are $g$-products in $T_pM$.
Now consider the images of these curves and vectors under $\phi$ and $d_p\phi$.
For $i=1,2$, define
$$
\lambda_i=\|d_p\phi(v_i)\|_{g'}
$$
and
$$
w_i=\frac{d_p\phi(v_i)}{\lambda_i} .
$$
Let $\varepsilon>0$ be such that $\gamma_i([0,\varepsilon))\subset U$ for $i=1,2$.
Then by Lemma \ref{l:phi-segment}, $\phi\circ\gamma_i|_{[0,\varepsilon)}$
parametrizes an initial interval of $[p'z_i]$.
The velocity of $\phi\circ\gamma_i|_{[0,\varepsilon)}$ at 0 equals
$d_p\phi(v_i)=\lambda_i w_i$, hence $w_i = \overrightarrow{p'z_i}$, $i=1,2$.
Now similarly to \eqref{e:dphi1} we calculate the derivative
\begin{equation}\lambdabel{e:dphi3}
\frac d{dt} D'_{\phi(\gamma_1(t))}(z_1,z_2)\big|_{t=0} = \lambda_1(-1 + \<w_1,w_2\>) .
\end{equation}
By \eqref{e:phi0}, the functions differentiated in \eqref{e:dphi1} and \eqref{e:dphi3} are the same, hence
\begin{equation}\lambdabel{e:dphi4}
-1 + \<v_1,v_2\> = \lambda_1(-1 + \<w_1,w_2\>)
\end{equation}
where $\<w_1,w_2\>$ is the scalar product with respect to $g'$.
Similarly from \eqref{e:dphi2} we obtain that
\begin{equation}\lambdabel{e:dphi5}
-1 + \<v_1,v_2\> = \lambda_2(-1 + \<w_1,w_2\>) .
\end{equation}
By \eqref{e:dphi4} and \eqref{e:dphi5},
$$
\lambda_1(-1 + \<w_1,w_2\>) = \lambda_2(-1 + \<w_1,w_2\>) ,
$$
therefore $\lambda_1=\lambda_2$
(note that $\<w_1,w_2\> \ne 1$ since $w_1$ and $w_2$ are different unit vectors).
Substituting the definitions of $\lambda_1$ and $\lambda_2$ we obtain that
$$
\|d_p\phi(v_1)\|_{g'}=\|d_p\phi(v_2)\|_{g'} .
$$
Since $v_1$ and $v_2$ are arbitrary vectors from $\Sigma$,
this identity implies that
the function $v\mapsto \|d_p\phi(v)\|_{g'}$ is constant on $\Sigma$.
We denote this constant by~$\lambda$.
Since $\Sigma$ is an open subset of the sphere $S_pM$,
it follows that $d_p\phi$ is a $\lambda$-homothetic linear map:
$$
\|d_p\phi(v)\|_{g'} = \lambda \|v\|_{g}
$$
for all $v\in T_pM$.
Hence $d_p\phi$ preserves the angles, in particular $\<v_1,v_2\>=\<w_1,w_2\>$.
Now \eqref{e:dphi4} implies that $\lambda=\lambda_1=1$.
Thus $d_p\phi$ is an isometry.
\end{proof}
\subsection*{Proof of the theorems \ref{t:whole} and \ref{t:part}}
As shown in Lemma \ref{l:dphi}, the derivative of our bi-Lipschitz
homeomorphism $\phi$ is an isometry almost everywhere.
Hence $\phi$ is a 1-Lipschitz map, i.e.\
it does not increase arclength distances.
The same holds for $\phi^{-1}$, therefore $\phi$ is a distance isometry.
By the Myers-Steenrod theorem (\cite{MS}, see also \cite[Ch.~5, Theorem 18]{Pe})
every distance isometry between Riemannian manifolds
is a smooth Riemannian isometry.
Thus $\phi$ is a Riemannian isometry.
Lemma \ref{l:id-on-boundary} implies the last claim of Theorem \ref{t:part}
and this finishes the proof of Theorem~\ref{t:part}.
As explained in the introduction, Theorem \ref{t:part} implies
Theorem \ref{t:whole}.
\begin{equation}gin{thebibliography}{99}
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Indiana Univ. Math. J. {\bf 30} (1981), no. 4, 481--488.
\bibitem{BBI}
D. Burago, Yu. Burago, S. Ivanov,
\emph{A course in metric geometry},
Graduate Studies in Mathematics, {\bf 33}, Amer. Math. Soc., 2001.
\bibitem{HS}
M. V. de Hoop, T. Saksala,
{\em Inverse problem of Travel time difference functions on compact Riemannian manifold with boundary},
J. Geom. Anal. 29 (2019), no. 4, 3308--3327.
\bibitem{I20}
S. Ivanov, {\em Distance difference representations of Riemannian manifolds},
Geometriae Dedicata {\bf 207} (2020), 167--192.
\bibitem{KKL}
A. Katchalov, Y. Kurylev, M. Lassas,
Inverse boundary spectral problems.
Chapman \& Hall/CRC, Boca Raton, FL, 2001.
\bibitem{Kur}
Y. Kurylev, {\em Multidimensional Gel'fand inverse problem and boundary distance map},
In ``Inverse Problems Related with Geometry'', Ed. H. Soga (1997), 1--15.
\bibitem{KLU}
Y. Kurylev, M. Lassas, G. Uhlmann.
{\em Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations}, Invent. Math. {\bf 212} (2018), no. 3, 781--857.
\bibitem{LS}
M. Lassas, T. Saksala.
{\em Determination of a Riemannian manifold from the distance difference functions}, Asian J. Math {\bf 23} (2019), no. 2, 173--200.
\bibitem{MS}
S.B. Myers, N.E. Steenrod,
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\bibitem{Pe}
P. Petersen, {\em Riemannian geometry}, 2nd edition, Springer, 2006.
\end{thebibliography}
\end{document} | math | 28,471 |
\begin{document}
\title{Quantum-classical correspondence of the Dirac matrices:\\ The
Dirac Lagrangian as a Total Derivative}
\author{S. Savasta$^1$, O. Di Stefano$^{1\, , 2}$, and O.M. Marag\`{o}$^3$}
\affiliation{$^1$Dipartimento di Fisica della Materia e Ingegneria
Elettronica, Universit\`{a} di Messina Salita Sperone 31, I-98166
Messina, Italy.
\\ $^2$Dipartimento di Matematica, Universit\`{a} di Messina, Salita
Sperone 31, I-98166 Messina, Italy. \\
$^3$CNR-Istituto per i Processi Chimico-Fisici, Salita Sperone c.da
Papardo, I-98158 Faro Superiore Messina, Italy.}
\begin{abstract}
{The Dirac equation provides a description of spin $1/2$ particles,
consistent with both the principles of quantum mechanics and of
special relativity. Often its presentation to students is based on mathematical propositions
that may hide the physical meaning of its contents. Here we show that Dirac
spinors provide the quantum description of two unit classical vectors: one
whose components are the speed of an elementary particle and the
rate of change of its proper time
and a second vector which fixes the velocity direction. In this
context both the spin degree of freedom and antiparticles can be understood
from the rotation symmetry of these unit vectors. Within this approach the Dirac Lagrangian acquires a direct physical meaning as the quantum operator describing the total time-derivative.
}
\end{abstract}
\pacs{11.30.-j,11.30.Cp, 11.10.-z,03.65.-w}
\maketitle
\section{Introduction}
Historically, Paul Dirac found the Klein-Gordon equation
physically unsatisfactory for the appearence of negative
probabilities arising from the second-order time derivative
\cite{Dirac}. Thus he sought for a relativistically invariant
Schr\"{o}dinger-like wave equation of first order in time derivative
of the form ($\hbar =c=1$):
\begin{equation}
i\partial_t \psi = H \psi\, .
\label{Sl}\end{equation} In order to have a more symmetric
relativistic wave equation in the 4-momentum components, he also
looked for an equation linear in space derivatives i.e. momentum
${\bf P}= -i {\bm \nabla}$, so that $H$ takes the form
\cite{Dirac}:
\begin{equation}
H = {\bm \alpha} \cdot {\bf P} + \beta m \, , \label{DH}
\end{equation}
with ${\bm \alpha}$ and $\beta$ being independent on space, time and
4-momentum. The condition that Eq.\ (\ref{DH}) provides the correct
relativistic relationship involving energy, rest mass and momentum,
\begin{equation}
E^2 = {\bf p}^2+m^2\, ,
\label{Ec}\end{equation}
requires that ${\bm \alpha}$ and $\beta \equiv \alpha_4$ obey the
anticommutation rules $\left \{ \alpha_i, \alpha_j \right \}= 2
\delta_{ij}$ ($i=1,4$) \cite{weinberg}. Dirac found that a set of $4
\times 4$ matrices satisfying this relation provides the lowest
order representation of the four $\alpha_i$. They can be expressed
as a tensor product of ($2 \times 2$) Pauli matrices $\rho_i$ and
$\sigma_i$ belonging to two different Hilbert spaces: ${\alpha}_i =
\rho_1 \sigma_i$ ($i=1,3$) and $\beta=\rho_3$ \cite{Dirac}.
As a consequence $\psi$ in Eq.\ (\ref{Sl}) is a 4-component wave
function.
It turns out that ${\bm \alpha}$ is the velocity
operator\cite{Dirac,Breit}. So that $\psi^\dag {\bm \alpha} \psi$ is
the current density, determining the coupling with the
electromagnetic field \cite{Dirac,weinberg,peskin}
This derivation, although straightforward, does not help for a clear physical understanding of the four component wave function and of Dirac matrices.
Indeed Richard Feynman in his Nobel lecture pointed out that {\em Dirac obtained his equation for the
description of the electron by an almost purely mathematical proposition. A
simple physical view by which all the contents of this equation can be seen is
still lacking}.
The fact that relativistic Dirac theory automatically includes spin
leads to the conclusion that spin is a purely quantum relativistic
effect originating from the finite dimensional representations of
the Lorentz group \cite{Fuchs}. Nevertheless this
interpretation is not generally accepted, e.g. following Weinberg
argument (Ref.\ \cite{weinberg} Chapter 1) {\em \dots it is
difficult to agree that there is anything fundamentally wrong with
the relativistic equation for zero spin that forced the development
of the Dirac equation -- the problem simply is that the electron
happens to have spin $\hbar /2$, not zero.} Technically speaking,
the homogeneous Lorentz group (in contrast e.g. to the group of
rotations) is not a compact group, hence the implementation of this
symmetry in quantum mechanics does not need the use of finite
dimensional representations. On the contrary, being non compact, it
has no faithful finite dimensional representation that is unitary
\cite{Fuchs,peskin}. Thus the homogeneous Lorentz group is the only
group of relativistic quantum field theory acting on
multiple-components quantum fields non-unitarily. This rather
surprising fact conflicts with a theorem proved by Wigner in 1931
(see Ref. \cite{weinberg} Chapter 2) which states that any symmetry
operation on quantum states must be induced by a unitary (or
anti-unitary) transformation.
The conflict is usually overcome,
either by regarding the field not as a multicomponent quantum
wavefunction but as a classical field \cite{peskin}, or by pointing
out that the fundamental group is not the (homogeneous) Lorentz
group but the Poincar\'e group \cite{weinberg}. Independently on the
point of view, one consequence is that the Hermitean conjugate
$\psi^\dag$ of the (four-component) spinor field $\psi$ does not
have the inverse transformation property of $\psi$ as requested by
quantum mechanics. The rather ad hoc, though generally accepted,
solution is to define $\bar \psi = \psi^\dag \gamma^0$ called the
{\em Dirac conjugate} of $\psi$, being $\gamma^0\equiv \beta$ the
{\em time} Dirac matrix \cite{weinberg,peskin,maggiore,hey}.
In this paper we address some naturally
arising questions: is there a fundamental compact symmetry group
that requires the occurrence of spin? is there a reason why the
velocity operator ${\bm \alpha}$ of elementary matter particles is a vector made of $4
\times 4$ matrices instead of continuous variable operators? is
there a simple interpretation of Dirac four-component spinors?
Here we discuss how
the spin of elementary particles can be understood as a consequence
of a rotation symmetry displayed by the kinematic variables
describing their space-time motion. Furthermore we show that such rotation
symmetry implies the existence of antiparticles.
Before starting our analysis, it is worth pointing out that here we follows the Dirac {\em old fashioned} point of view, regarding the Dirac equation as a Schroedinger-like quantum mechanical equation providing the quantum description of a relativistic point-like particle. In contrast the modern view regards the Dirac equation as a classical wave equation describing the $1/2$ spin field. The field is then quantized by means of canonical quantization for relativistic fields \cite{weinberg,peskin}.
\section{Rotations in quantum mechanics}
A rotation in classical physics is implemented by a $3 \times 3$
orthogonal matrix $R$ which, acting on a given vector ${\bf v}$,
gives the rotated vector ${\bf v}' = R {\bf v}$. Rotations are a
symmetry group whose generator is the angular momentum
\cite{Goldstein}. In quantum mechanics a symmetry transformation is
a transformation of the state kets describing the physical system
and of the operators $\hat O$ in the ket space. The transformation
operators ${\hat D}(R)$ are unitary and have the same group
properties as $R$ (see also supplementary information). The
expectation values of the angular momentum operators transform under
rotation as classical vectors:
\begin{equation}
\left< J_k \right> \to R_{kl}\left<J_l \right>
\end{equation}
where repeated indices are implicitly summed over \cite{Sakurai}.
The lowest number $N=2j+1$ of dimensions in which
the angular momentum commutation relations are realized is $N=2$ (j
= 1/2). In this case the angular momentum operators can be
represented in terms of the Pauli matrices: $J_k = \sigma_k/2$.
Independently on the physical state in the 2D Hilbert space, they
obey the following relationship:
\begin{equation}
\sum_i \langle \sigma_i \rangle^2=1
\end{equation}
which is not satisfied by higher order angular momentum
operators. Hence Pauli matrices are the {\em best} quantum
correspondents of classical unit vectors ${\bf \hat u}$ (with
$\sum_i {\hat u}^2_i = 1$) \cite{Sakurai}. If there is any classical
physical variable which is described by a unit vector, Pauli
matrices provide its natural quantization: ${\hat u}_j \to
\sigma_j$, which preserves rotation symmetry and ensures expectation
values which maps on the classical values.
\section{From relativistic kinematics to unit vectors}
Let us consider a relativistic point-like particle moving with
velocity $\mathbf{v}(t)$ with respect to an observer's reference
frame $K$ \cite{Jackson}. A possible spacetime worldline trajectory is shown in Fig.\ 1a.
We can consider the Lorentz-invariant quantity
\begin{equation}
d \tau(t)^2 \equiv dt^2 - d{\bf x}(t)^2\, .
\label{propertime}\end{equation}
In the instantaneous rest frame $K'$ of the particle $d{\bf x}=0$ and hence $d \tau(t)^2 = dt^2$ which admit solutions
$d \tau = \pm dt$.
The solution $d \tau = dt$ is the one usually presented in the introductory texts on special relativity (see e.g. \cite{Jackson}).
More recently Costella {\em et al.} \cite{Costella} pointed out that the negative solution provide a classical description of antiparticles
in agreement with the Feynman's interpretation of antiparticles as particles {\it going
back in time} \cite{hey}.
Thus from Eq.\ (\ref{propertime}) it follows that:
\begin{equation}
d\tau = \pm \frac{d t}{\gamma(t)}\, ,
\end{equation}
where $\gamma=(1-v^2)^{-1/2}$ is the usual Lorentz boost parameter yielding time dilation.
The time $\tau$ is the proper time of the particle. Thus the
quantity $d\tau/d t \equiv \dot\tau= \pm 1/\gamma(t)$ expresses the rate
of change of the proper time with respect to the time $t$ of the
reference frame i.e. proper time velocity.
The usual way to describe spacetime trajectories (worldlines) relies on $t-{\bf
x}$ Minkowski diagrams, as shown in fig. \ref{worldlines}a, where
the light cones separates causal events from space-like events.
Additional information can be inferred by drawing
proper time-time trajectories (timelines) as shown in fig. \ref{worldlines}b. The solid line describes the trajectory of a particle
with $\dot \tau >0$. Trajectories with
$\dot \tau >0$ can be identified by classical particle motion states, while
solutions with $\dot \tau <0$ correspond to classical anti-particle
states \cite{Costella}.
\begin{figure}
\caption{Worldlines of relativistic point-like particle and
antiparticles. a) In the usual Minkowski space-time diagram
particles and antiparticles worldlines are not distinguished. b) A
proper time-time diagram manifests the different kinematics of
particles ($\dot \tau >0$) and antiparticles ($\dot \tau <0$) with
respect to an observer's reference frame.}
\label{worldlines}
\end{figure}
When attempting to understand the spin and Dirac spinors on a
physical ground, the following question arises: {\em are there
kinematic variables which can be described as components of unit
vectors?}
The speed of a particle $v(t) = \left| {\bf \dot x}(t) \right|$ is
limited (by the speed of light $c =1$) like the component of a unit
vector. Additionally the quantity $\dot \tau$ can be viewed as the
complementary component of this unit vector. Indeed by definition of
$\gamma$, it follows that:
\begin{equation}
\left| {\bf \dot x} \right|^2 + \dot \tau^2 =1.\\
\end{equation}
Thus we can introduce a convenient unit vector ${\bf \hat r} \equiv(
\left| {\bf \dot x} \right| ,0,{\dot \tau})$ lying on the $r_1r_3$ plane (see Fig.
\ref{unitvectors}a) and an angle $\phi$ so that:
\begin{equation}
\left| {\bf \dot x} \right|=\sin \phi, \ \dot \tau =\cos \phi.\\
\end{equation}
A particle changing its kinematic state will move on the unitary
circle defined by the angle $\phi$.
\begin{figure}
\caption{Geometric
representations of unit vectors defining the kinematic state of a
relativistic point-like particle. (a) The unit vector ${\bf \hat r}
\label{unitvectors}
\end{figure}
Figure \ref{unitvectors}a displays one such kinematic vector with
positive components. If the particle is at rest with respect to the
reference frame, the vector lies on the $r_3$ axis (${\dot \tau}
= 1$). Moreover $\dot \tau$ decreases when the particle-speed increases
as predicted by special relativity (time dilation).
The unit vector ${\bf \hat r}$, besides ${\dot \tau}$, is able only
to describe the modulus of the particle velocity $v = \left| {\bf
\dot x} \right|$. The particle velocity is indeed a 3D vector and
the direction of ${\bf \dot x}$ can be accounted for by one
additional 3D unit vector ${\bf \hat s}$ providing just the
direction of ${\bf \dot x}$. Hence the motion state of a particle
can be described by a specific couple of unit vectors ${\bf \hat
r}$ and ${\bf \hat s}$:
\begin{equation}
{\dot \tau} = {r}_3 \, ,\hspace{1 cm} {\bf \dot x}= { r}_1\, {\bf
\hat s}\, .
\end{equation}
Within this approach changes of the particle velocity respect to an
inertial frame can be accounted for by rotations of ${\bf \hat r}$
in the kinematic plane (changes of the modulus) and rotations of
${\bf \hat s}$ (changes of the direction). The unit vector ${\bf
\hat s}$ can be transformed according to arbitrary 3D rotations
around an arbitrary 3D unit vector ${\bf \hat n}$:
\begin{equation}
s_i \to s_i' =
[{R}_{\bf \hat n}(\theta)]_{ij}\, s_j,
\end{equation}
where $\theta$ labels the angle of
rotation about ${\bf \hat n}$. On the other hand, physical
kinematic states ${\bf \hat r}$ admit only rotations about the
$r_2$-axis:
\begin{equation}
r_i \to r_i' = [R'_{{\bf j}}(\phi)]_{ij}\, r_j\, .
\end{equation}
We observe that this rotation
symmetry suggests that states obtained rotating ${\bf \hat r}$, and
${\bf \hat s}$ should be considered as accessible states. In
particular ${\bf \hat r}$ also describes kinematic states in the
second and third quadrant with ${\dot \tau} < 0$, corresponding to classical antiparticles \cite{Costella}
From the point of view of classical special relativity, the
description in terms of ${\bf \hat s}$ and ${\bf \hat r}$ appears to
be redundant: a given velocity is described by two different states.
For example a particle with given velocity along direction ${\bf
\hat d}= {\bf \dot x}/ \left| {\bf \dot x} \right|$ can be described
by the unit vectors ${\bf \hat s}_\uparrow = {\bf \hat d}$ and ${\bf
\hat r}_\uparrow = (\sin \phi ,0,\cos \phi)$ with $\phi = \arcsin
\left| {\bf \dot x} \right|$, or equivalently by the unit vectors
${\bf s}_\downarrow =-{\bf \hat d}$ and ${\bf \hat r}_\downarrow =
(\sin \phi' ,0,\cos \phi')$ with $\phi' = -\phi$. This
two-valuedness can be described in terms of the helicity variable $h
= ({\bf \hat s} \cdot {\bf p})/p$. Below we show that this classical
twofold degeneracy is indeed the classical correspondent of the
helicity states determined by quantum spin. It is worth pointing out
that, although the present approach describes a spin-like degree of
freedom yet at a classical level, the interaction of a classical
particle with the electromagnetic field is not affected by this
additional degree of freedom, in contrast to what happens after
quantization.
\begin{figure}
\caption{Sections of the {\em velocity space}
\label{tintegrated ky=0}
\end{figure}
Figure\ 3 provides a clear geometric interpretation of the different
kind of kinematic states: the first quadrant contains spin up
particles, the second one spin up antiparticles, the third spin down
antiparticles and the fourth quadrant spin down particles (see
Supplementary Information).
\section{From unit vectors to Dirac matrices}
As discussed earlier, a quantum mechanical description of unit
vectors is obtained replacing the classical vector components with
Pauli matrices: $r_i \to \rho_i$ and $s_i \to \sigma_i$, where
$\rho_i$ and $\sigma_i$ are now Pauli matrices acting on two
different Hilbert spaces $\mathcal{R}$ and $\mathcal{S}$. Hence:
\begin{eqnarray}
{\dot \tau} &=& {r}_3 \to \rho_3 \otimes I = \beta \nonumber\\
{\bf \dot x}&=& {r}_1\, {\bf \hat s} \to \rho_1 \otimes {\bm
\sigma} = {\bm \alpha}\, , \label{corr1}\end{eqnarray} being $I$ the
identity operator in the $s$ space describing the direction of
velocity. The ${\bm \alpha}$ and $\beta$ are indeed the well-known
Dirac matrices in the standard representation \cite{Dirac}. Thus we
find a direct symmetry-driven correspondence showing that ${\bm
\alpha}$ is the velocity vector operator, and $\beta$ the
proper-time velocity operator. In this quantum picture the helicity
operator for a particle of given momentum becomes $h = ({\bf \sigma}
\cdot {\bf p})/p$ in agreement with the Dirac theory \cite{peskin}
and with our classical derivation of helicity states. In summary, at
a classical level, the rotation symmetries we introduced describe
the correct relativistic kinematics. Furthermore they lead to two
possible helicities for a given motion state and the existence of
particles with $\dot \tau < 0$. After quantization, without any
other additional assumption, they give rise to the 4-component Dirac
spinors fully describing quantum spin and antiparticles with the
corresponding expectation value $\langle \beta \rangle < 0$. We
point out that in this framework the Dirac matrices are derived only
on the basis of the fundamental rotation symmetry without any
reference to the Dirac equation or to the energy of a relativistic
free particle.
At this point it is interesting to show that the energy of a relativistic free particle can be written as observed By Breit as $E = {\bf v} \cdot {\bf p} + \dot \tau m$. This equation has the same structure of the Dirac Hamiltonian.
\section{The Dirac Lagrangian as a Total Derivative}
A crucial feature of this symmetyry based derivation of the Dirac matrices
is that it has been obtained by
regarding the position-coordinate and the proper time of the
particle on the same footing as functions of the time-coordinate of
the reference frame, i.e. we use $t$ as the relevant {\it meter} to
which compare the dynamical evolution of all other observables (as
happens in non-relativistic mechanics). Accordingly if a quantum
wavefunction depends on position ${\bf x}$, it should
also depend on $\tau$ i.e. $\psi(t,{\bf x},\tau)$. We now consider a
particle with such wavefunction and require the conservation of its
probability density $\mathcal{D}(t,{\bf x},\tau) \equiv
\psi^*(t,{\bf x},\tau) \psi(t,{\bf x},\tau)$:
\begin{equation} \frac{d }{dt}\, \mathcal{D} =
\frac{d \psi^*}{dt} \psi + \psi^* \frac{d \psi}{dt} = 0\, .
\label{Liouville}\end{equation} Stationarity of $\psi$ (and
$\psi^*$) ensures that a continuity equation holds. In
turn if, as in Lagrangian formulations with complex
fields, $\psi$ and $\psi^*$ are regarded as independent,
Eq.\ (\ref{Liouville}) implies stationarity of $\psi$ (and of
$\psi^*$) i.e. $d \psi/dt=0$. Because the time dependence of the
wavefunction is both explicit and implicit (through the time
dependence of ${\bf x}$ and $\tau$), we have:
\begin{equation}
\frac{d}{dt} \psi = \partial_t\, \psi +
{\bf \dot x} \cdot \partial_{\bf x}\, \psi
+ {\dot \tau} \partial_\tau\, \psi\, =0.
\end{equation}
Thus the request of stationarity and the crucial symmetry-based
quantization Eq.\ (\ref{corr1}), provides a generalized Dirac-like
equation:
\begin{equation}
i {\partial}_t \psi(t,{\bf x},\tau) = H_g \psi(t,{\bf x}, \tau)\, ,
\label{gDe}\end{equation}
with
\begin{equation}
H_g = {\bm \alpha} \cdot (-i \partial_{\bf x}) + \beta (-i
\partial_\tau)\, . \label{Hg}\end{equation}
The corresponding Lagrangian density can be written as
\begin{equation}
\psi^\dag {\cal L} \psi = \psi^\dag i(\partial_t+ {\bm \alpha} \cdot {\partial}_{\bf x} + \beta {\partial}_\tau) \psi \, .
\label{gLag}\end{equation}
This Lagrangian operator ${\cal L}$ has a precise physical meaning, being the Hermitean quantum operator describing the total time derivative: $i d / dt \to {\cal L}$ after symmetry-based
quantization Eq.\ (\ref{corr1}).
Equation (\ref{Hg}) is more
general than the Dirac equation since the mass parameter is now replaced
by the internal time momentum operator. Elementary matter particles
of given mass can be viewed as eigenstates of this operator. In
this case Eq.(\ref{Hg}) reduces to the standard Dirac equation. On
the other hand, this new degree of freedom offers a chance to
shed new light on some fundamental aspects and concepts.
As an example this new degree of freedom could be exploited to discuss charge conservation.
Charge conservation is generally accounted by invariance
under phase change of a field, Eq.\ (\ref{Hg}) allows the interpretation of this phase
change as the consequence of proper time translation applied to an
eigenstate of the proper time operator. Charge conservation could thus be viewed as arising from invariance of the Lagrangian under proper time translations.
It is also interesting to address the proper time reversal
symmetry $\tau \rightarrow -\tau$. Applying this symmetry to a
positive energy solution of Eq. \ref{Hg} which is also an eigenstate
of the proper time momentum (i.e. a particle with fixed mass and
charge), we obtain its antiparticle with positive energy and
opposite charge, so antiparticle solutions with positive energy emerge without the need for second
quantization adjustments.
\section{Conclusions}
Within the approach here presented, changes of the velocity modulus and direction of a relativistic particle can
simply be accounted by rotations of two independent unit vectors. Dirac spinors just provide the quantum description
of these rotations. These transformations are able to describe the helicity of a pointlike particle yet at a classical
relativistic level, rendering spin a less mysterious degree of freedom. This analysis sacrifies explicit covariance by
making explicit rotation symmetry which nevertheless is the fundamental symmetry on which the algebra of the
Lorentz group is based.
We observe that the addressed rotation symmetries form a compact group hence with unitary finite-dimensional
representations as all other symmetry groups in quantum field theory. A feature of this analysis is that it has been
obtained by regarding the position-coordinate and the proper time of the particle on the same footing as functions
of the time-coordinate of the reference frame, i.e. t is used as the relevant meter to which compare the dynamical
evolution of all other observables (as happens in non-relativistic quantum mechanics).
Within this approach we derived the Dirac equation just invoking
total stationarity of the wavefunction with respect to the reference
time. No assumptions about the classical relativistic energy of a
particle or about the quantum operator replacements ($E \to i
\partial_t$ and ${\bf p} \to -i \partial_{\bf x}$) have been
performed.
Finally we observe that the quantum replacement $m \to -i
\partial_\tau$ implies an internal time-energy uncertainty
principle $\Delta \tau\, \Delta m$, e.g. as required by a gedanken
experiment recently proposed by Aharonov and Rezni \cite{Ahranov}.
\end{document} | math | 23,487 |
\begin{document}
\title{2-uniform covers of $2$-semiequivelar toroidal maps}
\author {Dipendu Maity}
\affil{Department of Sciences and Mathematics,
Indian Institute of Information Technology Guwahati, Bongora, Assam-781\,015, India.~~
Email id : [email protected]}
\date{\today}
\maketitle
\begin{abstract}
If every vertex in a map has one out of two face-cycle types, then the map is said to be $2$-semiequivelar. A 2-uniform tiling is an edge-to-edge tiling of regular polygons having $2$ distinct transitivity classes of vertices. Clearly, a $2$-uniform map is $2$-semiequivelar. The converse of this is not true in general. There are 20 distinct 2-uniform tilings (these are of $14$ different types) on the plane. In this article, we prove that a $2$-semiequivelar toroidal map $K$ has a finite $2$-uniform cover if the universal cover of $K$ is $2$-uniform except of two types.
\end{abstract}
\noindent {\small {\em MSC 2010\,:} 52C20, 52B70, 51M20, 57M60.
\noindent {\em Keywords:} Polyhedral map on torus; 2-uniform maps; 2-semiequivelar maps; Symmetric group.}
\section{Introduction}
A map is a connected $2$-dimensional cell complex on a surface. Equivalently, it is a cellular embedding of a connected graph on a surface. In this article, a map will mean a polyhedral map on a surface, that is, non-empty intersection of any two faces is either a vertex or an edge.
For a map $\mathcal{K}$, let $V(\mathcal{K})$ be the vertex set of $\mathcal{K}$ and $u\in V(\mathcal{K})$. The faces containing $u $ form a cycle (called the {\em face-cycle} at $u $) $C_u $ in the dual graph of $\mathcal{K} $. That is, $C_u $ is of the form $(F_{1,1}\mbox{-}\cdots\mbox{-}F_{1,n_1})\mbox{-}\cdots\mbox{-}(F_{k,1}\mbox{-}\cdots \mbox{-}F_{k,n_k})\mbox{-}$ $F_{1,1} $, where $F_{i,\ell} $ is a $p_i $-gon for $1\leq \ell \leq n_i $, $1\leq i \leq k $, $p_r\neq p_{r+1} $ for $1\leq r\leq k-1 $ and {$p_k\neq p_1 $}. In this case, the vertex $u$ is said to be of type $ [p_1^{n_1}, \dots, p_k^{n_k}]$ (addition in the suffix is modulo $k $). A map $\mathcal{K} $ is said to be {\em 2-semiequivelar} of type $[p_1^{n_1}, \dots, p_k^{n_k};q_1^{m_1}, \dots, q_k^{m_s}]$ if $V(\mathcal{K}) = V_1 \sqcup V_2$ such that the vertices of $V_1$ is of type $[p_1^{n_1}, \dots, p_k^{n_k}]$ and the vertices of $V_2$ is of type $[q_1^{m_1}, \dots, q_k^{m_s}]$. So, clearly, if $[p_1^{n_1}, \dots, p_k^{n_k}] = [q_1^{m_1}, \dots, q_k^{m_s}]$ then the map is called $semiequivelar$ of type $[p_1^{n_1}, \dots, p_k^{n_k}]$. A semiequivelar map is said to be an $equivelar~map$ if it consists of same type of faces.
A {\em $2$-uniform tiling} is an edge-to-edge tiling of regular polygons having $2$ distinct transitivity classes of vertices. A vertex-transitive map is a map on a closed surface on which the automorphism group acts transitively on the set of vertices. A $2$-uniform tiling or map will have vertices that we could label $X$, and others that we could label $Y$. Each $X$ vertex can be mapped onto every other $X$ vertex, but cannot be mapped to any $Y$ vertex.
Clearly, an $2$-uniform map is $2$-semiequivelar.
A {\em semiregular} tiling of $\mathbb{R}^2$ is also known as {\em Archimedean}, or {\em homogeneous}, or {\em uniform} tiling. In \cite{GS1977}, Gr\"{u}nbaum and Shephard showed that there are exactly eleven types of Archimedean tilings on the plane. These types are $[3^6]$, $[3^4,6^1]$, $[3^3,4^2]$, $[3^2,4^1,3^1,4^1]$, $[3^1,6^1,3^1,6^1]$, $[3^1,4^1,6^1,4^1]$, $[3^1,12^2]$, $[4^4]$, $[4^1,6^1,12^1]$, $[4^1,8^2]$, $[6^3]$.
Clearly, a {\em semiregular} tiling on $\mathbb{R}^2$ gives a semiequivelar map on $\mathbb{R}^2$. But, there are semiequivelar maps on the plane which are not (not isomorphic to) an Archimedean tiling. In fact, there exists $[p^q]$ equivelar maps on $\mathbb{R}^2$ whenever $1/p+1/q<1/2$ (e.g., \cite{CM1957}, \cite{FT1965}). Thus, we have
\begin{proposition} \label{prop:plane}
There are infinitely many types of equivelar maps on the plane $\mathbb{R}^2$.
\end{proposition}
We know that the plane is the universal cover of the torus. Since there are infinitely many equivelar maps on the plane, it is natural to ask that what are the other types of semiequivelar maps exist on the torus. Here we have the following result.
\begin{proposition} \cite{DM2017, DM2018} \label{theo:GrSh}
Let $X$ be a semiequivelar map on a surface $M$. If $M$ is the torus then the type of $X$ is $[3^6]$, $[6^3]$, $[4^4]$, $[3^4,6^1]$, $[3^3,4^2]$, $[3^2,4^1,3^1,4^1]$, $[3^1,6^1,3^1,6^1]$,
$[3^1,4^1,6^1,4^1]$, $[3^1,12^2]$, $[4^1,8^2]$ or $[4^1,6^1,12^1]$.
\end{proposition}
We know that all the Archimedean tiling are vertex-transitive. But, it not true on the torus. Here, we know the following.
\begin{proposition} \cite{DM2017, DM2018} \label{prop:36&44}
Let $X$ be an equivelar map on the torus. If the type of $X$ is $[3^6]$, $[4^4]$, $[6^3]$ or $[3^3,4^2]$ then $X$ is vertex-transitive. If the type is $[3^2,4^1,3^1,4^1]$, $[3^1,6^1,3^1,6^1]$, $[3^1,4^1,6^1,4^1]$, $[3^1,12^2]$, $[4^1,8^2]$, $[3^4,6^1]$ or $[4^1,6^1,12^1]$ then there exists a semiequivelar toroidal map of which is not vertex-transitive.
\end{proposition}
We know that the $2$-sphere $\mathbb{S}^2$ is simply connected. So, the boundary of the pseudo-rhombicuboctahedron (which is a semiregular spherical map of type $[4^3, 3]$) has no other cover. We also know that this map is not vertex-transitive. Thus, this map has no vertex-transitive cover. We know that the Archimedean tilings are vertex-transitive. So, each semiequivelar toroidal map has vertex-transitive universal cover. Here, we know
\begin{proposition} \cite{BD2020} If $X$ is a semiequivelar toroidal map then there exists a covering $\gammamma \colon Y \to X
$ where $Y$ is a vertex-transitive toroidal map.
\end{proposition}
The {\em $2$-uniform} tilings of the plane $\mathbb{R}^2$ are the generalization of vertex-transitive tilings on the plane. We know from \cite{GS1977, GS1981, Otto1977} that there are 20 2-uniform tiling of types
\begin{align*}
& [3^6;3^3;4^{2}], [3^{6};3^2,4^1,3^1,4^1], [3^4,6^1;3^2,6^{2}], [3^{3}, 4^2;3^1,4^1,6^1,4^1], [3^3, 4^2;3^2,4^1,3^1,4^1],\\
& [3^{6}; 3^2, 4^1,12^1], [3^1, 4^1, 6^1, 4^1; 4^1, 6^1, 12^1], [3^2,4^1,3^1,4^1;3^1, 4^1,6^1,4^1], [3^2,6^2; 3^1, 6^1, 3^1, 6^1],\\
& [3^1, 4^1, 3^1, 12^1; 3^1, 12^2], [3^1,4^2,6^1; 3^1, 4^1, 6^1, 4^1], [3^1,4^2, 6^1; 3^1, 6^1, 3^1, 6^1], [3^3,4^2;4^4], [3^{6};3^4,6^1].
\end{align*}
on the plane (see in Section \ref{2uniform}).
Since the plane is the universal cover of the torus, so, these types of maps also exist on the torus. Here we know the following.
\begin{proposition} \cite{MDD2020}
Let $X$ be a $2$-semiequivelar map on the torus that is the quotient of the plane's $2$-uniform lattice. Let the vertices of $X$ form $m$ Aut$(X)$-orbits. Then, $m \leq k$ for some positive integer $k$.
\end{proposition}
In this article, we prove the following.
\begin{theorem}\label{theo1}
\begin{enumerate}
\item[(a)] If $X$ is a $2$-semiequivelar toroidal map that is the quotient of the plane's $2$-uniform lattice of type other than $[3^3, 4^2;3^2,4^1,3^1,4^1]$ and $[3^1,4^2, 6^1; 3^1, 6^1,$ $3^1, 6^1]$ then there exists a covering $\alphapha : Y \to X$ where $Y$ is a $2$-uniform toroidal map.
\item[(b)] If $X$ is a $2$-semiequivelar toroidal map that is the quotient of the plane's $2$-uniform lattice of type $[3^3, 4^2;3^2,4^1,3^1,4^1]$ or $[3^1,4^2, 6^1; 3^1, 6^1, 3^1, 6^1]$ then there does not exist any covering $\gammamma : Y \to X$ where $Y$ is a $2$-uniform toroidal map.
\item[(c)] If $X$ is a $2$-semiequivelar toroidal map that is not the quotient of the plane's $2$-uniform lattice then there does not exist any $2$-uniform toroidal map $Y$ such that $\beta : Y \to X$ is a covering map.
\end{enumerate}
\end{theorem}
\section{$2$-uniform tilings of the plane}\label{2uniform}
We first present $20$ $2$-uniform tilings on the plane. These are also given in \cite{MDD2020}. We need these for the proofs of our results in Section \ref{sec:proofs-1}.
\section{Proof }\label{sec:proofs-1}
Let $K_i$ (given in Section \ref{2uniform}) be of type $A$, where
\begin{align*}
A \in &\{ [3^{6};3^4,6^1], [3^6;3^3;4^{2}], [3^{6};3^2,4^1,3^1,4^1], [3^{6}; 3^2,4^1,12^1],\\
& [3^4,6^1;3^2,6^{2}], [3^3, 4^2;3^2,4^1,3^1,4^1], [3^{3}, 4^2;3^1,4^1,6^1,4^1], [3^3,4^2;4^4], \\
& [3^2,4^1,3^1,4^1;3^1,4^1,6^1,4^1], [3^2,6^2; 3^1, 6^1, 3^1, 6^1], [3^1, 4^1, 3^1, 12^1; 3^1, 12^2],\\
& [3^1,4^2,6^1; 3^1, 4^1, 6^1, 4^1], [3^1,4^2, 6^1; 3^1, 6^1, 3^1, 6^1], [3^1, 4^1, 6^1, 4^1; 4^1, 6^1, 12^1] \}.
\end{align*}
Gr\"{u}nbaum and G. C. Shephard \cite{GS1977, GS1981} and Kr\"{o}tenheerdt \cite{Otto1977} have discussed the existence and uniqueness of the $2$-uniform tilings $K_i$, $i =1, 2, \dots, 20$ of the plane. Thus, we have the following.
\begin{proposition}\label{prop1}
The $2$-uniform maps $K_{i}$ ($1 \le i \le 20$) are unique up to isomorphism.
\end{proposition}
\begin{proof}[Proof of Theorem \ref{theo1}(a)]
\noindent Case 1. Let $X_1$ be a $2$-semiequivelar map on the torus that is the quotient of the plane's $2$-uniform lattice $K_1$ (see in Section \ref{2uniform}).
Let $V_{1} = V(K_1)$ be the vertex set of $K_1$. Let $H_{1}$ be the group of all the translations of $K_1$. So, $H_1 \leq $Aut$(K_1)$.
By assumption, $X_1$ is a $2$-semiequivelar map on the torus and it is quotient of $K_1$, so, we can assume, there is a polyhedral covering map $\eta_{1} : K_1 \to X_1$ where $X_1 = K_1/\Gammamma_{1}$ for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{1} \le $Aut$(K_1)$. Hence $\Gammamma_{1}$
consists of translations and glide reflections. Since $X_1 =K_1/\Gammamma_{1}$ is orientable, $\Gammamma_{1}$ does not contain any glide reflection. Thus $\Gammamma_{1} \leq H_{1}$.
We take the middle point of the line segment joining vertices $a_{0}$ and $a_3$ as the origin $(0,0)$ of $K_1$. Let $A_1 := a_{9} - a_{0}$, $B_1 := a_{14}- a_{0}$ and $F_1 := a_{20} - a_{0}$ in $K_1$. Clearly, $B_1 = \ell_1 A_1+t_1 F_1$ for some $\ell_1, t_1 \in \mathbb{Z}$. Then $$H_1 := \langle \alphapha_1 \colon x\mapsto x+A_1, \beta_1 \colon x\mapsto x+B_1\rangle.$$ Under the action of $H_1$, vertices of $K_1$ form twelve orbits. The orbits are
\begin{align*}
& O_1 :=\langle a_{0} \rangle, O_2 :=\langle a_{1} \rangle, O_3 :=\langle a_{2} \rangle,
O_4 :=\langle a_{3} \rangle, O_5 :=\langle a_{4} \rangle, O_6 :=\langle a_{5} \rangle,\\
& O_7 :=\langle b_{18} \rangle, O_8 :=\langle b_{20} \rangle, O_9 :=\langle b_{28} \rangle, O_{10} :=\langle b_{39} \rangle, O_{11} :=\langle b_{37} \rangle, O_{12} :=\langle b_{27} \rangle.
\end{align*}
Let $\rho_1$ be the function obtained by $60$ degrees anticlockwise rotation. Then $\rho_1 \in Aut(K_1)$ and $\rho_1(A_1) = B_1, \rho_1(B_1) = F_1$.
Let $G_1 = \langle \alphapha_1, \beta_1, \rho_1\rangle$. Clearly, vertices of $K_1$ form $O_1 :=\langle a_{0} \rangle, O_2 :=\langle b_{18} \rangle$ $G_1$-orbit. So, $G_1$ acts $2$-uniformly on $K_1$.
Since $\Gammamma_{1} \leq H_{1}$, $\Gammamma_{1} = \langle \gammamma_1 \colon x\mapsto x+C_1, \delta_1 \colon x\mapsto x+D_1\rangle$ where $C_1 = a_1A_1 + b_1B_1$ and $D_1 = c_1A_1 + d_1B_1$, for some $a_1, b_1, c_1, d_1 \in \mathbb{Z}$.
\noindent {\bf Claim 1.} $L_1 := \langle \alphapha_1^m, \beta_1^m \rangle \le \Gammamma_{1}$ for some $m \in \mathbb{Z}$.
Since $K_1/\Gammamma_1$ is compact, $C_1$ and $D_1$ are linearly independent. Therefore, there exists $a, b, c, d \in \mathbb{Q}$ such that $A_1 = aC_1 + bD_1$ and $B_1 = cC_1 + dD_1$. Let $m$ be the smallest positive integer such that $ma,mb,mc,md \in \mathbb{Z}$. Then $mA_1 = (ma)C_1 + (mb)D_1, mB_1 = (mc)C_1 + (md)D_1$. Thus, $\alphapha_1^m (z) = z + mA_1 = (\gammamma_1^{ma} \circ \delta_1^{mb})(z), \beta_1^m(z)= z+mB_1 = (\gammamma_1^{mc} \circ \delta_1^{md})(z)$ and hence $\alphapha_1^m, \beta_1^m \in \Gammamma_1$.
This proves Claim 1.
Since $\Gammamma_1$ is abelian, we have $L_1 \unlhd \Gammamma_1 \unlhd H_1 \le G_1 \le Aut(K_1)$.
\noindent {\bf Claim 2.} $L_1 \unlhd G_1 $.
For $u, v \in \mathbb{R}$ and $p \in \mathbb{Z}$, $(\rho_1 \circ \alphapha_1^p \circ \rho_1^{-1})(uA_1+vB_1) = (\rho_1 \circ \alphapha_1^p)(vA_1 - uB_1) = \rho_1((vA_1-uB_1)+pA_1) = \rho_1((p+v)A_1-uB_1)=(p+v)B_1+uA_1 = (uA_1+vB_1)+pB_1 = \alphapha_2^p(uA_1+vB_1).$ Thus, $\rho_1 \circ \alphapha_1^p \circ \rho_1^{-1} = \alphapha_2^p$. Again, for $u, v \in \mathbb{R}$, $\rho_1\circ\alphapha_2^p\circ\rho_1^{-1} (uA_1+vB_1) = (\rho_1 \circ \alphapha_2^p)(vA_1-uB_1) = \rho_1((vA_1-uB_1)+pB_1) = \rho_1(vA_1+(p-u)B_1) = vB_1-(p-u)A_1 = (uA_1+vB_1)-pA_1 = \alphapha_1^{-p}(uA_1+vB_1)$. Thus, $\rho_1 \circ \alphapha_2^p \circ \rho_1^{-1} = \alphapha_1^{-p}.$ In particular, $\rho_1 \circ \alphapha_1^m \circ \rho_1^{-1} = \alphapha_2^m$, $\rho_1 \circ \alphapha_2^m\circ\rho_1^{-1}=\alphapha_1^{-m} \in L_1.$ Since $\alphapha_1, \alphapha_2$ commute, $L_1 \unlhd G_1 $.
By Claim 2, $L_1$ is a normal subgroup of $G_1$. Therefore, $G_1/L_1$ acts on $Y_1= K_1/L_1$ and $u + L_1 \mapsto u+ \Gammamma_1$ gives a covering $\gammamma_1 \colon Y_1 \to X_1$.
Since $\langle a_{0} \rangle, \langle b_{18} \rangle$
are the $G_1$-orbits, it follows that $O_j/L_1$ for $j=1, 2$ are the $(G_1/L_1)$-orbits. Clearly, $G_1/L_1 \le Aut(Y_1)$. It follows that the number of Aut$(Y_1)$-orbits of vertices is $2$. This completes Theorem \ref{theo1} when $X_1$ is associated to $K_1$.
\noindent Case 2. Let $X_2$ be a $2$-semiequivelar map on the torus and $X_2 = K_2/\Gammamma_{2}$ (see $K_2$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{2} \le $Aut$(K_2)$.
Let $V_{2} = V(K_2)$ be the vertex set of $K_2$. Let $H_{2}$ be the group of all the translations of $K_2$. So, $H_2 \leq $Aut$(K_2)$. Since $X_2 = K_2/\Gammamma_{2}$, $\Gammamma_{2}$
consists of translations and glide reflections. Since $X_2 =
K_2/\Gammamma_{2}$ is orientable, $\Gammamma_{2}$ does not contain any glide reflection. Thus $\Gammamma_{2} \leq H_{2}$.
We take the middle point of the line segment joining vertices $a_{0}$ and $a_3$ as the origin $(0,0)$ of $K_2$. Let $A_2 := a_{6} - a_0$, $B_2 := a_{18} - a_0$ and $F_2 := a_{24} - a_{0}$ $\in \mathbb{R}^2$. Similarly as in Case 1, define $H_2, G_2, L_2$. The result follows in this case by similar argument as in Case 1.
\noindent Case 3. Let $X_5$ be a $2$-semiequivelar map on the torus and $X_5 = K_5/\Gammamma_{5}$ (see $K_5$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{5} \le $Aut$(K_5)$.
We take the middle point of the line segment joining vertices $b_{0}$ and $b_{3}$ as the origin $(0,0)$ of $K_5$ (see in Section \ref{2uniform}). Let $A_5 := a_1 - a_0$, $B_5 := a_{7} - a_0$ and $F_5 := a_{6} - a_0$ $\in \mathbb{R}^2$. Similarly as above in Case 1, define $H_5, G_5, L_5$. The result follows in this case by exactly same argument as in in Case 1.
\noindent Case 4. Let $X_6$ be a $2$-semiequivelar map on the torus and $X_6 = K_6/\Gammamma_{6}$ (see $K_6$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{6} \le $Aut$(K_6)$.
We take the middle point of the line segment joining vertices $a_{0}$ and $a_{6}$ as the origin $(0,0)$ of $K_6$. Let $A_6 := a_{27} - a_1$, $B_6 := a_{32} - a_{0}$ and $F_6 := a_{37} - a_{11}$ $\in \mathbb{R}^2$. Similarly as in Case 1, define $H_6, G_6, L_6$. The result follows in this case by similar argument as in Case 1.
\noindent Case 5. Let $X_7$ be a $2$-semiequivelar map on the torus and $X_7 = K_7/\Gammamma_{7}$ (see $K_7$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{7} \le $Aut$(K_7)$.
We take the middle point of the line segment joining vertices $b_{1}$ and $b_{4}$ as the origin $(0,0)$ of $K_7$. Let $A_7 := a_{9} - a_0$, $B_7 := a_{2} - a_{0}$ and $F_7 := a_{1} - a_{0}$ $\in \mathbb{R}^2$. Similarly as above in Case 1, define $H_7, G_7, L_7$. The result follows in this case by similar argument as in Case 1.
\noindent Case 6. Let $X_{11}$ be a $2$-semiequivelar map on the torus and $X_{11} = K_{11}/\Gammamma_{11}$ (see $K_{11}$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{11} \le $Aut$(K_{11})$.
We take origin $(0,0)$ is the middle point of the line segment joining vertices $a_{0}$ and $a_{3}$ of $K_{11}$. Let $A_{11} := a_{37} - a_0$, $B_{11} := a_6 - a_{0}$ and $F_{11} := a_{15} - a_{0} \in \mathbb{R}^2$. Similarly as in Case 1, define $H_{11}, G_{11}, L_{11}$. The result follows in this case by similar argument as in Case 1.
\noindent Case 7. Let $X_{14}$ be a $2$-semiequivelar map on the torus and $X_{14} = K_{14}/\Gammamma_{14}$ (see $K_{14}$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{14} \le $Aut$(K_{14})$.
We take origin $(0,0)$ is the middle point of the line segment joining vertices $a_{0}$ and $a_{3}$ of $K_{14}$. Let $A_{14} := a_{10} - a_4$, $B_{14} := a_{29} - a_{4}$ and $F_{14} := a_{35} - a_{4} \in \mathbb{R}^2$. Similarly as in Case 1, define $H_{14}, G_{14}, L_{14}$. The result follows in this case by similar argument as in Case 1.
\noindent Case 8. Let $X_{16}$ be a $2$-semiequivelar map on the torus and $X_{16} = K_{16}/\Gammamma_{16}$ (see $K_{16}$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{16} \le $Aut$(K_{16})$.
We take origin $(0,0)$ is the middle point of the line segment joining vertices $a_{0}$ and $a_{4}$ of $K_{16}$. Let $A_{16} := a_{5} - a_0$ and $B_{16} := a_{8} - a_1 \in \mathbb{R}^2$. Similarly as above in Case 1, define $H_{16}, G_{16}, L_{16}$. The result follows in this case by similar argument as in Case 1.
\noindent Case 9. Let $X_{17}$ be a $2$-semiequivelar map on the torus and $X_{17} = K_{17}/\Gammamma_{17}$ (see $K_{17}$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{17} \le $Aut$(K_{17})$.
We take origin $(0,0)$ is the middle point of the line segment joining vertices $a_0$ and $a_3$ of $K_{17}$. Let $A_{17} := a_{20} - a_4$, $B_{17} := a_7 -a_4$ and $F_{17} :=a_{17} - a_3 \in \mathbb{R}^2$. Similarly as in Case 1, define $H_{17}, G_{17}, L_{17}$. The result follows in this case by similar argument as in Case 1.
\noindent Case 10. Let $X_{20}$ be a $2$-semiequivelar map on the torus and $X_{20} = K_{20}/\Gammamma_{20}$ (see $K_{20}$ in Section \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{20} \le $Aut$(K_{20})$.
We take origin $(0,0)$ is the middle point of the line segment joining vertices $a_{0}$ and $a_{6}$ of $K_{20}$. Let $A_{20} := a_{19} - a_3$, $B_{20} := a_{14} -a_5$ and $F_{20} := a_{23} - a_8 \in \mathbb{R}^2$. Similarly as in Case 1, define $H_{20}, G_{20}, L_{20}$. The result follows in this case by similar argument as in Case 1.
\noindent Case 11. Let $K = K_3, K_4, K_8, K_{12}, K_{13}$ or $K_{15}$ (see $K_i$ for $i = 3, 4, 8, 12, 13, 15$ in Section \ref{2uniform}). Let $X$ be a $2$-semiequivelar map on the torus and $X = K/\Gammamma_i$ for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{i} \le $Aut$(K_i)$. Let the vertices of $X$ form $m_i$ Aut$(X_i)$-orbits. Then, from \cite{MDD2020}, we know $m_i =2$. So, the identity map $X \to X$ is a covering map.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theo1}(b)] Let $X_9$ be a $2$-semiequivelar map on the torus and $X_9 = K_9/\Gammamma_{9}$ (see $K_9$ in Sec. \ref{2uniform}) for some fixed element (vertex, edge or face) free subgroup $\Gammamma_{9} \le $Aut$(K_9)$. Since $X_9 = K_9/\Gammamma_{9}$ is orientable, $\Gammamma_{9}$ does not contain any glide reflection. Let $H_{9}$ be the group of all the translations of $K_9$. Thus $\Gammamma_{9} \leq H_{9} \leq$ Aut$(K_9)$. In $K_9$, observe that the elements in the orbit $O(b_{34})$ maps to the elements in the orbit $O(b_{33})$ only under some glide reflection symmetry or reflection symmetry about a line. Here, both are not fixed element free. Since $\Gammamma_{9}$ does not contain these two symmetries, the number of vertex Aut$(X_9)$-orbit is at least three. Hence, there does not exist any covering $\gammamma : M \to X_9$ where $M$ is a $2$-uniform toroidal map. Similarly, if $X_{i}$ is a $2$-semiequivelar map on the torus and $X_i = K_i/\Gammamma_{i}$ for $i=10, 18, 19$, the number of vertex Aut$(X_i)$-orbit is at least three. Hence, in these cases also, covering map $\gammamma : M \to X_i$ where $M$ is a $2$-uniform toroidal map does not exist.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theo1}(c)] Let $X$ be a $2$-semiequivelar map on the torus that is not the quotient of the plane's $2$-uniform lattice $K_i$ for $1 \le i \le 20$ (see in Section \ref{2uniform}). We know that the plane is the universal cover of the torus. So, there exists a tiling $Y$ and a covering group $\Gammamma$ such that $X = Y /\Gammamma$. Since $Y \not\cong K_i$ $\forall ~i$, $Y$ is not $2$-uniform and the number of Aut$(Y)$-orbit is at least three. So, for any fixed element free subgroup $H$ of Aut$(Y)$, the number of $H$-orbit is at least three. Hence, there does not exist any covering $\beta : M \to X$ where $M$ is a $2$-uniform toroidal map. This completes the part (c).
\end{proof}
{\small
}
\end{document} | math | 21,620 |
\begin{document}
\begin{abstract}
We adopt the viewpoint that topological And\'e-Quillen
theory for commutative $S$-algebras should provide usable
(co)homology theories for doing calculations in the sense
traditional within Algebraic Topology. Our main emphasis
is on homotopical properties of universal derivations,
especially their behaviour in multiplicative homology
theories. There are algebraic derivation properties, but
also deeper properties arising from the homotopical
structure of the free algebra functor $\mathfrak{m}athbb{P}_R$ and
its relationship with extended powers of spectra. In the
connective case in ordinary $\bmod{\,p}$ homology, this
leads to useful formulae involving Dyer-Lashof operations
in the homology of commutative $S$-algebras. Although
many of our results could no doubt be obtained using
stabilisation, our approach seems more direct. We also
discuss a reduced free algebra functor $\tilde{\mathfrak{m}athbb{P}}_R$.
\end{abstract}
\mathfrak{m}aketitle
\section*{Introduction}
Topological Andr\'e-Quillen homology and cohomology theories
for commutative $S$-algebras were introduced by Maria Basterra,
building on ideas of Igor {K\v{r}\'{i}\v{z}}{} as well as algebraic Andr\'e-Quillen
theory. Subsequent work, both individually and jointly in various
combinations, by Basterra, Gilmour, Goerss, Hopkins, Kuhn, Lazarev,
Mandell, McCarthy, Minasian, Reinhard, Richter, Robinson, Whitehouse
as well as the present author, has laid out the basic structure
and provided key relationships with other areas.
In this work we continue to adopt the viewpoint of~\cite{BGRtaq},
regarding $\TAQ$ as providing usable (co)homology theories for
doing calculations in the sense traditional within Algebraic
Topology.
Our main emphasis is on homotopical properties of universal
derivations, especially their behaviour in multiplicative
homology theories. As the name suggests, there are algebraic
derivation properties, but also deeper properties arising out
of the homotopical structure of the free algebra functor and
its relationship with extended powers of spectra. In the
connective case and in ordinary mod~$p$ homology, this leads
to useful formulae involving Dyer-Lashof operations in the
homology of commutative $S$-algebras. It seems likely that
many of these results are obtainable using stabilisation, but
our approach seems more direct. We remark that work of Mike
Mandell~\cite{MM:TAQ} suggests that it might be more natural
to replace commutative $S$-algebras by algebras in $\mathfrak{m}athcal{M}_S$
over his operad $\mathfrak{m}athcal{G}$ and work with those. Related
results on the homology of the free commutative $S$-algebra
functor also appear in work of Nick Kuhn \& Jason
McCarty~\cites{NJK:Transfers,NJK&JBMcC:HomLoopSpces}.
We also discuss a \emph{reduced free algebra} functor
$\tilde{\mathfrak{m}athbb{P}}_R$ which we learnt of from Tyler Lawson.
This takes as input $R$-modules under a fixed cofibrant
replacement for the $R$-module~$R$ and gives rise to a Quillen
adjunction.
\[
\xymatrix{
{\mathfrak{m}athscr{C}_R} \ar@/_8pt/[rr]_{\tilde{\mathfrak{m}athbb{U}}}
&& {S^0_R/\mathfrak{m}athscr{M}_R} \ar@/_8pt/[ll]_{\tilde{\mathfrak{m}athbb{P}}_R}
}
\]
We will use this in a sequel to study spectral sequences related
to those studied by Maria Basterra~\cite{MBtaq}*{section~5} and
Haynes Miller~\cite{HRM:SS}.
We give some sample calculations, but our main concern is with
laying the groundwork for future applications.
In two brief appendices we supply a proof of a basic result,
an adjunction result, and some formulae for calculating
Dyer-Lashof operations.
\subsection*{Notation, etc}
When working over a fixed commutative ground ring $\Bbbk$ such
as $\mathfrak{m}athbb{F}_p$, we often write $\circtimes$ for $\circtimes_{\Bbbk}$,
$\mathfrak{m}athbb{H}om$ for $\mathfrak{m}athbb{H}om_{\Bbbk}$, etc.
\section{Recollections on topological Andr\'e-Quillen theory}
\leftarrowbel{sec:Recollections}
We will assume the reader is familiar with Basterra's foundational
paper~\cite{MBtaq} and the further development of its ideas
in~\cite{BGRtaq}. All of this is founded on the notions of $S$-modules
and commutative $S$-algebras of~\cite{EKMM}. We briefly spell
out some of the main ingredients.
If $R$ is a commutative $S$-algebra, then its category of (left)
$R$-modules $\mathfrak{m}athscr{M}_R$ is a model category and the category
of commutative $R$-algebras $\mathfrak{m}athscr{C}_R$ consists of the
commutative monoids in $\mathfrak{m}athscr{M}_R$ with monoidal morphisms.
There is a free $R$-algebra functor
$\mathfrak{m}athbb{P}_R\colon\mathfrak{m}athscr{M}_R\to\mathfrak{m}athscr{C}_R$ left adjoint to
the forgetful functor $\mathfrak{m}athbb{U}\colon\mathfrak{m}athscr{C}_R\to\mathfrak{m}athscr{M}_R$,
and this pair gives a Quillen adjunction. We denote the derived
(or homotopy) categories of these model categories by
$\bar{h}\mathfrak{m}athscr{M}_R$ and $\bar{h}\mathfrak{m}athscr{C}_R$.
For every pair of commutative $R$-algebras $A\to B$, their
\emph{cotangent complex} is a $B$-module $\Omega_A(B)$ which
is well defined up to isomorphism in $\bar{h}\mathfrak{m}athscr{M}_B$.
This comes with a canonical morphism in $\bar{h}\mathfrak{m}athscr{M}_A$,
the \emph{universal derivation}
\[
\partialta_{(B,A)}\colonB\to\Omega_A(B),
\]
characterised by a natural isomorphism
\[
\bar{h}\mathfrak{m}athscr{C}_A/B(B, B\vee X) \cong \bar{h}\mathfrak{m}athscr{M}_B(\Omega_A(B),X),
\]
where $X\in\mathfrak{m}athscr{M}_B$ and $B\vee X$ denotes the \emph{square
zero extension} of~$B$ by~$X$ viewed as an $A$-algebra over~$B$.
\emph{Topological Andr\'e-Quillen homology} and \emph{cohomology}
with coefficients in a $B$-module $M$ are defined by
\begin{align*}
\TAQ_*(B,A;M) &= \pi_*(M\wedge_B \Omega_A(B)), \\
\TAQ^*(B,A;M) &= \pi_{-*}(F_B(\Omega_A(B),M))
= \bar{h}\mathfrak{m}athscr{M}_B(\Omega_A(B),M)^*,
\end{align*}
where
\[
\bar{h}\mathfrak{m}athscr{M}_B(\Omega_A(B),X)^n
= \bar{h}\mathfrak{m}athscr{M}_B(\Omega_A(B),\Sigma^n X).
\]
When $E$ is a (unital) $B$ ring spectrum, the composition
\begin{equation}\leftarrowbel{eq:TAQ-Hurewiczhom}
\xymatrix{
\pi_*(B) \ar[r]\ar@/^19pt/[rrrr]^{\theta}
& E_*(B) \ar[rr]_{(\partialta_{(B,A)})_*}
&& E_*(\Omega_A(B))\ar[r]_(.35){\theta'}
& E^B_*(\Omega_A(B)) = \TAQ_*(B,A;E)
}
\end{equation}
is the \emph{$\TAQ$-Hurewicz homomorphism}.
In \cite{BGRtaq} we showed how this could be interpreted
as a cellular theory for cellular commutative $R$-algebras.
A key ingredient was the basic observation that for a
cofibrant $R$-module $X$, cotangent complex of the free
$R$-algebra $\mathfrak{m}athbb{P}_RX$ is
\begin{equation}\leftarrowbel{eq:OmegaPX}
\Omega_R(\mathfrak{m}athbb{P}_RX) \cong \mathfrak{m}athbb{P}_RX\wedge_R X,
\end{equation}
see~\cite{BGRtaq}*{proposition~1.6} for example. For
completeness we give a proof of this in
Appendix~\ref{sec:Missingpf}, and discuss the universal
derivation for $\mathfrak{m}athbb{P}_RX$ in Section~\ref{sec:PX}.
In \cite{BGRtaq} we developed the theory of connective
$p$-local commutative $S$-algebras along the lines
of~\cite{AJB-JPM} for spectra, making crucial use of
$\TAQ$ with coefficients in $H\mathfrak{m}athbb{F}_p$. In both of those
works, one important outcome was the ability to detect
\emph{minimal atomic} objects using the vanishing of
the appropriate Hurewicz homomorphism in positive
degrees.
\section{Homotopical properties of universal derivations}
\leftarrowbel{sec:UnivDer}
Let $A$ be a commutative $S$-algebra. As pointed out
in~\cite{AL:Glasgow}, for a commutative $A$-algebra~$B$,
the universal derivation $\partialta_{(B,A)}\colonB\to\Omega_A(B)$
is a homotopy derivation in the sense of the following
discussion.
Suppose that $R$ is a commutative $S$-algebra, let~$E$
be an~$R$ ring spectrum and let $M$ be a left $E$-module.
We remind the reader this means that there are morphisms
$\eta\colonR\to E$, $\varphi\colonE\wedge_R E\to E$ and
$\mathfrak{m}u\colonE\wedge_R M\to M$ in $\bar{h}\mathfrak{m}athscr{M}_R$ which
satisfy appropriate associativity and unital conditions.
\begin{defn}\leftarrowbel{defn:HtpyDeriv}
A morphism $\partial\colonE\to M$ in $\bar{h}\mathfrak{m}athscr{M}_R$ is
a \emph{homotopy derivation} if the following diagram
in $\bar{h}\mathfrak{m}athscr{M}_R$ commutes.
\begin{equation}\leftarrowbel{eq:hder}
\xymatrix{
&E\wedge_R E\ar[rr]^{\varphi}\ar[dl]_{I\wedge\partial\vee\partial\wedge I}&&E\ar[dr]^{\partial}& \\
E\wedge_R M\vee M\wedge_R E\ar[dr]_{I\vee\mathfrak{m}athrm{switch}} &&&& M \\
&E\wedge_R M\vee E\wedge_R M\ar[rr]^(.6){\mathfrak{m}athrm{fold}} &&E\wedge_R M\ar[ur]_{\mathfrak{m}u}&
}
\end{equation}
\end{defn}
Now let $A$ be a commutative $S$-algebra and let $B$ be
a commutative $A$-algebra. Following the remarks at end
of~\cite{AL:Glasgow}*{section~2}, we recall that the
universal derivation $\partialta_{(B,A)}\colonB\to\Omega_A(B)$
is a morphism in the derived category of $A$-modules
$\bar{h}\mathfrak{m}athscr{M}_{A}$ which is also a homotopy
derivation in the sense that the following diagram
commutes in $\bar{h}\mathfrak{m}athscr{M}_{A}$
\begin{equation}\leftarrowbel{eq:Deriv}
\xymatrix{
B\wedge_AB\ar[rr]^(.57){\mathfrak{m}athrm{prod}}
\ar[dd]_{I\wedge\partialta_{(B,A)}+\mathfrak{m}athrm{switch}\circ(\partialta_{(B,A)}\wedge I)}
&& B\ar[dd]^{\partialta_{(B,A)}} \\
&& \\
B\wedge_A\Omega_A(B)\ar[rr]^(.57){\mathfrak{m}athrm{mult}} && \Omega_A(B)
}
\end{equation}
where elements of $\bar{h}\mathfrak{m}athscr{M}_{A}(X,Y)$ are added
in the usual way.
Now suppose that $E$ is a commutative $B$ ring spectrum;
this implies that $E$ is a $B$-module and there is a unit
morphism of $B$ ring spectra $B\to E$ in $\bar{h}\mathfrak{m}athscr{M}_B$.
Then on smashing with copies of $E$,~\eqref{eq:Deriv}
gives another commutative diagram
\begin{equation}\leftarrowbel{eq:Deriv-E}
\xymatrix{
E\wedge_AB\wedge_AE\wedge_AB\ar[rr]^{\mathfrak{m}athrm{switch}}
\ar[dd]^{\substack{ I\wedge I\wedge I\wedge\partialta_{(B,A)} \ph{abcabcabcabcabc}\\
\ph{123} + \mathfrak{m}athrm{switch}\circ(I\wedge \partialta_{(B,A)}\wedge I\wedge I)} }
&& E\wedge_A E\wedge_A B\wedge_AB\ar[rr]^{\mathfrak{m}athrm{prod}}
\ar[dd]^{\substack{I\wedge I\wedge I\wedge\partialta_{(B,A)} \ph{abcabcabcabcabc} \\
\ph{123} + I\wedge I\wedge\mathfrak{m}athrm{switch}\circ(\partialta_{(B,A)}\wedge I)}}
&& E\wedge_AB\ar[dd]^{I\wedge\partialta_{(B,A)}} \\
&& && \\
E\wedge_AB\wedge_AE\wedge_A\Omega_A(B)\ar[rr]^{\mathfrak{m}athrm{switch}}
&& E\wedge_A E\wedge_AB\wedge_A\Omega_A(B)
\ar[rr]^(.57){\mathfrak{m}athrm{prod}\wedge\mathfrak{m}athrm{mult}}
&& E\wedge_A\Omega_A(B)
}
\end{equation}
which shows that the commutative $E_*$-algebra
$E^A_*B=\pi_*(E\wedge_A B)$ admits the $E_*$-module
homomorphism
\[
(\partialta_{(B,A)})_*\colonE^A_*B\to E^A_*\Omega_A(B).
\]
Of course $E^A_*\Omega_A(B)$ is also a left $E^A_*B$-module
since $\Omega_A(B)$ is a left $B$-module.
Composing $(\partialta_{(B,A)})_*$ with the natural homomorphism
$E^A_*\Omega_A(B)\to E^B_*\Omega_A(B)$, we obtain an $E_*$-module
homomorphism
\[
\mathfrak{m}athbb{D}elta_{(B,A)}\colonE^A_*B \to E^A_*\Omega_A(B)\to E^B_*\Omega_A(B).
\]
We also have an augmentation $\varepsilon\colonE^A_*B\to E_*$ induced
by applying $\pi_*(-)$ to the evident composition
\[
E\wedge_AB \to E\wedge_AE \to E.
\]
Clearly $\varepsilon$ is a morphism of $E_*$-algebras.
\begin{lem}\leftarrowbel{lem:Deriv-E}
$(\partialta_{(B,A)})_*$ and $\mathfrak{m}athbb{D}elta_{(B,A)}$ are $E_*$-derivations,
so for $u,v\in E^A_*B$,
\begin{align*}
(\partialta_{(B,A)})_*(uv) &=
u(\partialta_{(B,A)})_*(v)\pm v(\partialta_{(B,A)})_*(u), \\
\mathfrak{m}athbb{D}elta_{(B,A)}(uv) &=
\varepsilon(u)\mathfrak{m}athbb{D}elta_{(B,A)}(v)\pm\varepsilon(v)\mathfrak{m}athbb{D}elta_{(B,A)}(u),
\end{align*}
where the signs are determined from the degrees of $u,v$
with the usual sign convention. In particular, if
$u,v\in\Bbbker\varepsilon\colonE^A_*B\to E_*$, then
\[
\mathfrak{m}athbb{D}elta_{(B,A)}(uv) = 0,
\]
so $\mathfrak{m}athbb{D}elta_{(B,A)}$ annihilates non-trivial products.
\end{lem}
\begin{proof}
This involves diagram chasing using the definitions.
\end{proof}
We will often write $\partialta$ and $\mathfrak{m}athbb{D}elta$ for $\partialta_{(B,A)}$
and $\mathfrak{m}athbb{D}elta_{(B,A)}$ when $(B,A)$ is clear from the context.
\section{The free commutative algebra functor}\leftarrowbel{sec:PX}
For a $R$-module $X$ there is a free commutative $R$-algebra
\[
\mathfrak{m}athbb{P}_RX = \bigvee_{j\geqslant 0} X^{(j)}/\Sigma_j.
\]
When $R=S$ or a localisation of $S$, we will set $\mathfrak{m}athbb{P}=\mathfrak{m}athbb{P}_S$.
If $X$ is cofibrant as an $R$-module then $\mathfrak{m}athbb{P}_RX$ is
cofibrant as an commutative $R$-algebra. The functor $\mathfrak{m}athbb{P}_R$
is left adjoint to the forgetful functor
$\mathfrak{m}athbb{U}\colon\mathfrak{m}athscr{C}_R\to\mathfrak{m}athscr{M}_R$, so for $A\in\mathfrak{m}athscr{C}_R$,
\[
\mathfrak{m}athscr{C}_R(\mathfrak{m}athbb{P}_R(-),A) \cong \mathfrak{m}athscr{M}_R(-,A),
\]
where $A=\mathfrak{m}athbb{U}A$ is regarded as an $R$-module. In
fact,
\begin{equation}\leftarrowbel{eq:QuillenAdj-C-M}
\xymatrix{
{\mathfrak{m}athscr{C}_R} \ar@/_8pt/[rr]_{\mathfrak{m}athbb{U}}
&& {\mathfrak{m}athscr{M}_R} \ar@/_8pt/[ll]_{\mathfrak{m}athbb{P}_R}
}
\end{equation}
is a Quillen adjunction~\cite{EKMM}.
As it is a left adjoint, $\mathfrak{m}athbb{P}_R$ preserves colimits,
including pushouts. As cell and CW $R$-modules are defined
as iterated pushouts, applying $\mathfrak{m}athbb{P}_R$ to the skeleta
leads to cell or CW skeleta. To make this explicit, suppose
that $X$ is an $R$-module with CW skeleta $X^{[n]}$, and
attaching maps
\[
j_n\colon\bigvee_i S^n_R \to X^{[n]},
\]
where $S^n_R=\mathfrak{m}athbb{F}_RS^n$ is the cofibrant model for the
sphere spectrum in $\mathfrak{m}athscr{M}_R$. Setting $D^n_R=\mathfrak{m}athbb{F}_RD^n$,
the $(n+1)$-skeleton $X^{[n+1]}$ is defined by the pushout
diagram
\[
\xymatrix{
\bigvee_i S^n_R\ar@{}[dr]|{\text{\pigpenfont R}}
\ar[r]\ar[d] & X^{[n]}\ar[d] \\
\bigvee_i D^{n+1}_R\ar[r] & X^{[n+1]}
}
\]
which induces the pushout diagram
\[
\xymatrix{
\mathfrak{m}athbb{P}_R(\bigvee_i S^n_R)
\ar@{}[dr]|{\text{\pigpenfont R}}
\ar[r]\ar[d]
& \mathfrak{m}athbb{P}_RX^{[n]}\ar[d] \\
\mathfrak{m}athbb{P}_R(\bigvee_i D^{n+1}_R)\ar[r] & \mathfrak{m}athbb{P}_RX^{[n+1]}
}
\]
in $\mathfrak{m}athscr{C}_R$. So we obtain a CW filtration
on $\mathfrak{m}athbb{P}_RX$ with $n$-skeleton
\[
\mathfrak{m}athbb{P}_R^{\leftarrowngle n\rightarrowngle}X =
(\mathfrak{m}athbb{P}_RX)^{\leftarrowngle n\rightarrowngle} = \mathfrak{m}athbb{P}_R(X^{[n]}).
\]
Now we discuss the cotangent complex and universal derivation
for free algebras. Recalling~\eqref{eq:OmegaPX}, we know that
in the homotopy category $\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_RX}$,
\[
\Omega_R(\mathfrak{m}athbb{P}_RX) \cong \mathfrak{m}athbb{P}_RX \wedge_R X,
\]
and the universal derivation
\[
\partialta_{(\mathfrak{m}athbb{P}_RX,R)}
\in\bar{h}\mathfrak{m}athscr{M}_R(\mathfrak{m}athbb{P}_RX,\Omega_R(\mathfrak{m}athbb{P}_RX))
=\bar{h}\mathfrak{m}athscr{M}_R(\mathfrak{m}athbb{P}_RX,\mathfrak{m}athbb{P}_RX\wedge_R X)
\]
has the homotopy derivation property shown in the homotopy
commutative diagram~\eqref{eq:hder}. Furthermore,
$\partialta_{(\mathfrak{m}athbb{P}_RX,R)}$ corresponds to the inclusion
$X\to\mathfrak{m}athbb{P}_RX\wedge_R X$ under the sequence of
isomorphisms
\begin{align}
\bar{h}\mathfrak{m}athscr{C}_R/\mathfrak{m}athbb{P}_RX(\mathfrak{m}athbb{P}_RX,\mathfrak{m}athbb{P}_RX\vee\Omega_R(\mathfrak{m}athbb{P}_RX))
&\cong
\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_RX}(\Omega_R(\mathfrak{m}athbb{P}_RX),\Omega_R(\mathfrak{m}athbb{P}_RX))
\notag \\
&\cong
\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_RX}(\mathfrak{m}athbb{P}_RX\wedge_R X,\mathfrak{m}athbb{P}_RX\wedge_R X)
\notag \\
&\cong
\bar{h}\mathfrak{m}athscr{M}_R(X,\mathfrak{m}athbb{P}_RX\wedge_R X).
\leftarrowbel{eq:univder-modulemap}
\end{align}
We will describe $\partialta_{(\mathfrak{m}athbb{P}_RX,R)}$ as a morphism
in the homotopy category $\bar{h}\mathfrak{m}athscr{M}_R$ using this
identification. Nick Kuhn has pointed out that~\cite{NJK:Transfers}
gives a closely related analysis of extended powers, and
proves far more about their coproduct structure induced
by the pinch map.
Suppose that $X'$ is a second copy of $X$. A representative
for the homotopy class of the pinch map
$\pinch\colonX\to X\vee X'$ induces morphisms of
commutative $R$-algebras
\[
\xymatrix@C=0.5cm@R=1.0cm{
&& \mathfrak{m}athbb{P}_R X\ar[dll]_{\mathfrak{m}athbb{P}_R\pinch}\ar[drr] && \\
\mathfrak{m}athbb{P}_R(X\vee X')\ar[dr]_(.45){\cong}
&& && \mathfrak{m}athbb{P}_R X \vee \mathfrak{m}athbb{P}_R X\wedge_R X' \\
&\mathfrak{m}athbb{P}_R X\wedge_R\mathfrak{m}athbb{P}_R X'\ar[rr]
&& \mathfrak{m}athbb{P}_R X\wedge_R (R\vee X')\ar[ur]_{\cong} &
}
\]
where $R\vee X'$ and $\mathfrak{m}athbb{P}_R X\vee\mathfrak{m}athbb{P}_R X\wedge_R X'$
are square zero extensions of $R$ and $\mathfrak{m}athbb{P}_RX$ respectively,
and the horizontal morphism kills the wedge summands $(X')^{(r)}/\Sigma_r$
with $r\geqslant2$. Restricting to the summand $X$ in $\mathfrak{m}athbb{P}_R X$
we obtain the pinch map~$\pinch$, and then on applying the
isomorphism of~\eqref{eq:univder-modulemap} we find that
the resulting composition
\[
\xymatrix{
\mathfrak{m}athbb{P}_R X \ar[r]\ar@/^17pt/[rr]^\partialta
& \mathfrak{m}athbb{P}_R X \vee \mathfrak{m}athbb{P}_R X\wedge_R X' \ar[r]
& \mathfrak{m}athbb{P}_R X\wedge_R X'
}
\]
agrees with the universal derivation $\partialta_{(\mathfrak{m}athbb{P}_RX,R)}$.
In the homotopy category of $S$-modules, $\partialta_{(\mathfrak{m}athbb{P}_RX,R)}$
is equivalent to a coproduct of maps
\[
\partialta_{(\mathfrak{m}athbb{P}_RX,R),n}\colon
E\Sigma_n\ltimes_{\Sigma_n}X^{(n)}
\to (E\Sigma_{n-1}\ltimes_{\Sigma_{n-1}}X^{(n-1)})\wedge X,
\]
where we have of course identified $X'$ with $X$. In fact
these are the transfer maps $\tau_{n-1,1}$
of~\cite{LNM1176}*{definition~II.1.4}, \emph{i.e.},
\begin{equation}\leftarrowbel{eq:delta-cpt}
\partialta_{(\mathfrak{m}athbb{P}_RX,R),n}=\tau_{n-1,1}\colon
E\Sigma_n\ltimes_{\Sigma_n}X^{(n)}
\to (E\Sigma_{n-1}\ltimes_{\Sigma_{n-1}}X^{(n-1)})\wedge X.
\end{equation}
The derivation property of $\partialta_{(\mathfrak{m}athbb{P}X,S)}$ is
just a consequence of the commutativity of the
diagram~\eqref{eq:DoubCoset} below. We will give a brief
explanation of this.
For detailed accounts of the stable homotopy theory involved,
see~\cites{LNM1176,LNM1213,JPM:Silverbook}. We remark that
in~\cite{LNM1176}*{chapter~II, p.~24}, the pinch map is
referred to as the `diagonal' since in the stable category
finite products and coproducts coincide. To ease notation
and exposition, we take $R=S$ and set $\mathfrak{m}athbb{P}=\mathfrak{m}athbb{P}_S$;
however the general case is similar.
Let $E\Sigma_{m+n}$ be a free contractible $\Sigma_{m+n}$-space,
and let~$Y$ be a $\Sigma_{m+n}$-spectrum for $m,n\geqslant1$; we
are interested in the case where $Y=X^{(m+n)}$, the $(m+n)$-th
smash power of~$X$. The equivariant half smash product
$E\Sigma_{m+n}\ltimes Y$ is a free $\Sigma_{m+n}$-spectrum,
and the evident inclusions of subgroups
\[
\xymatrix{
\Sigma_m\times\Sigma_n\ar@{^{(}->}[r]
& \Sigma_{m+n}
& \ar@{_{(}->}[l]\Sigma_{m+n-1}
}
\]
induce morphisms of spectra
\[
E\Sigma_{m+n}\ltimes_{\Sigma_m\times\Sigma_n} Y
\to E\Sigma_{m+n}\ltimes_{\Sigma_{m+n}} Y
\longleftarrow E\Sigma_{m+n}\ltimes_{\Sigma_{m+n-1}} Y
\]
on orbit spectra. There are also transfer maps
\[
\xymatrix{
E\Sigma_{m+n}\ltimes_{\Sigma_{m+n}} Y \ar[rr]^(.45){\tau_{m,n}}
&& E\Sigma_{m+n}\ltimes_{\Sigma_{m}\times\Sigma_{n} } Y \\
E\Sigma_{m+n}\ltimes_{\Sigma_{m+n}} Y \ar[rr]^(.45){\tau_{m+n-1,1}}
&& E\Sigma_{m+n}\ltimes_{\Sigma_{m+n-1} } Y
}
\]
associated with these inclusions of subgroups. We will
use the double coset formula of \cite{LNM1213}*{\S IV.6}.
We are in the situation of~\cite{LNM1213}*{theorem~IV.6.3},
and our first task is to identify representatives for
the double cosets in
\[
\Sigma_m\times\Sigma_n\backslash\Sigma_{m+n}/\Sigma_{m+n-1}.
\]
An elementary exercise with cycle notation shows that the
following are true:
\begin{itemize}
\item
the elements of $\Sigma_{m+n}/\Sigma_{m+n-1}$ are the distinct
left cosets $(r,m+n)\Sigma_{m+n-1}$ where $1\leqslant r\leqslant m+n-1$,
together with $\Sigma_{m+n-1}$;
\item
by definition, the elements of
$\Sigma_m\times\Sigma_n\backslash\Sigma_{m+n}/\Sigma_{m+n-1}$
are the $\Sigma_m\times\Sigma_n$-orbits in $\Sigma_{m+n}/\Sigma_{m+n-1}$
and these are represented by $(m,m+n)$ and $\id$. In fact the
orbit of $(m,m+n)$ contains all the $(r,m+n)$ with $1\leqslant r\leqslant m$,
and the orbit of the identity $I$ contains all of the transpositions
$(m+r,m+n)$ with $1\leqslant r\leqslant n$.
\end{itemize}
It is straightforward to verify the two identities
\begin{align*}
\Sigma_m\times\Sigma_{n-1} &=
\Sigma_m\times\Sigma_n \cap \Sigma_{m+n-1}, \\
\Sigma_{m-1}\times\Sigma_n &=
\Sigma_m\times\Sigma_n \cap (m,m+n)\Sigma_{m+n-1}(m,m+n).
\end{align*}
Now the double coset formula tells us that in the homotopy
category, there is a commutative diagram having the following
form.
\begin{equation}\leftarrowbel{eq:DoubCoset}
\xymatrix@C=0.3cm@R=1.0cm{
& E\Sigma_{m+n}\ltimes_{\Sigma_m\times\Sigma_n} Y\ar[drr]\ar[dl]_{\tau_{n-1,1}\vee\tau_{m-1,1}\;}
&& \\
E\Sigma_{m+n}\ltimes_{\Sigma_{m}\times\Sigma_{n-1}} Y \vee E\Sigma_{m+n}\ltimes_{\Sigma_{m-1}\times\Sigma_n} Y
\ar[d] & && E\Sigma_{m+n}\ltimes_{\Sigma_{m+n}} Y \ar[d]^{\tau_{m+n-1,1}} \\
E\Sigma_{m+n}\ltimes_{\Sigma_{m+n-1}} Y\vee E\Sigma_{m+n}\ltimes_{\Sigma_{m+n-1}} Y
\ar[rrr]^(.54){\mathfrak{m}athrm{fold}} &&
& E\Sigma_{m+n}\ltimes_{\Sigma_{m+n-1}} Y
}
\end{equation}
\section{Power operations and the free functor}\leftarrowbel{sec:PowOps+PX}
We will describe another result on the effect of certain
transfer maps in homology that sheds light on the calculation
of universal derivations. We begin by recalling some standard
facts about the homology of extended powers.
Let $p$ be a prime and let $V=V_*$ be a graded $\mathfrak{m}athbb{F}_p$-vector
space. The inclusion $C_p\leqslant\Sigma_p$ of the subgroup of
cyclic permutations $C_p=\leftarrowngle\gamma\rightarrowngle$ with
$\gamma=(1,2,\ldots,p)$ has index $(p-1)!$, so the associated
transfer homomorphism provides a splitting for the induced
homomorphism in group homology with coefficients in the
$p$-fold tensor power $V^{\circtimes p}$ with the obvious action.
\[
\xymatrix{
\mathfrak{m}athrm{H}_*(C_p;V^{\circtimes p})\ar@{->>}[r]
& \ar@/_19pt/@{ >->}[l]_{\Tr_{C_p}^{\Sigma_p}}
\mathfrak{m}athrm{H}_*(\Sigma_p;V^{\circtimes p})
}
\]
Furthermore, the homology of the subgroup $\Sigma_{p-1}\leqslant\Sigma_p$
is trivial in positive degrees, \emph{i.e.},
\[
\mathfrak{m}athrm{H}_*(\Sigma_{p-1};V^{\circtimes p})
= \mathfrak{m}athrm{H}_0(\Sigma_{p-1};V^{\circtimes p})
= (V^{\circtimes p})_{\Sigma_{p-1}}.
\]
Hence the associated transfer homomorphism is also
zero in positive degrees, \emph{i.e.}, for $k>0$,
\begin{equation}\leftarrowbel{eq:Tr=0}
0=\Tr_{\Sigma_{p-1}}^{\Sigma_p}\colon\mathfrak{m}athrm{H}_k(\Sigma_p;V^{\circtimes p})
\to\mathfrak{m}athrm{H}_k(\Sigma_{p-1};V^{\circtimes p}).
\end{equation}
In fact the diagram of subgroup inclusions
\[
\xymatrix{
1\ar@{^(->}[r]\ar@{^(->}[d] & C_p\ar@{^(->}[d] \\
\Sigma_{p-1}\ar@{^(->}[r] & \Sigma_p
}
\]
induces a commutative diagram of split epimorphisms.
\[
\xymatrix{
V^{\circtimes p}\ar[r]\ar@{->>}[d]
& \mathfrak{m}athrm{H}_*(C_p;V^{\circtimes p})\ar@{->>}[d] \\
{\ph{\Sigma_{p-1}}}(V^{\circtimes p})_{\Sigma_{p-1}}\ar[r]\ar@/^17pt/@{ >->}[u]^{\Tr_{1}^{\Sigma_{p-1}}}
& \mathfrak{m}athrm{H}_*(\Sigma_p;V^{\circtimes p})\ar@/_17pt/@{ >->}[u]_{\Tr_{C_p}^{\Sigma_p}}
}
\]
This can be generalised to $\Sigma_{p^m}$ where $m\geqslant2$.
Then the $p$-order of $|\Sigma_{p^m}|$ is
\[
\circrd_p|\Sigma_{p^m}| = \frac{(p^m-1)}{(p-1)}
= p^{m-1}+p^{m-2}+\cdots+p+1.
\]
Writing
\[
\Sigma_{p^{m-1}}^k =
\circverset{k}{\circverbrace{\Sigma_{p^{m-1}}\times\cdots\times\Sigma_{p^{m-1}}}},
\]
the wreath product
\[
\Sigma_p\wr\Sigma_{p^{m-1}} =
\Sigma_p\ltimes\Sigma_{p^{m-1}}^p \leqslant \Sigma_{p^m}
\]
has $p$-order
\[
\circrd_p|\Sigma_p\wr\Sigma_{p^{m-1}}|
= 1 + p\frac{(p^{m-1}-1)}{(p-1)}
= \frac{(p^m-1)}{(p-1)},
\]
so an argument using transfer shows that the inclusion
induces a split epimorphism.
\[
\xymatrix{
H_*(\Sigma_p\wr\Sigma_{p^{m-1}};\mathfrak{m}athbb{F}_p)\ar@{->>}[rr] &&
\ar@/_19pt/@{ >->}[ll]_{\Tr_{\Sigma_p\wr\Sigma_{p^{m-1}}}^{\Sigma_{p^m}}}
H_*(\Sigma_{p^m};\mathfrak{m}athbb{F}_p)
}
\]
Another calculation shows that
\[
\circrd_p|\Sigma_{p^m-1}| = \frac{(p^m-1)}{(p-1)} - m
= (p^{m-1}+p^{m-2}+\cdots+p+1) - m
\]
and
\begin{align*}
\circrd_p|\Sigma_{p^{m-1}}^{(p-1)}\times\Sigma_{p^{m-1}-1}|
&= (p-1)\frac{(p^{m-1}-1)}{(p-1)} + \frac{(p^{m-1}-1)}{(p-1)} - (m-1) \\
&= (p^{m-1}+p^{m-2}+\cdots+p+1) - m \\
&= \circrd_p|\Sigma_{p^m-1}|.
\end{align*}
Therefore
\[
\Sigma_{p^{m-1}}^{(p-1)}\times\Sigma_{p^{m-1}-1}\leqslant\Sigma_{p^m-1}
\]
and these subgroups of $\Sigma_{p^m}$ have the same $p$-order,
hence the inclusion induces an isomorphism
\[
H_*(\Sigma_{p^{m-1}}^{(p-1)}\times\Sigma_{p^{m-1}-1};\mathfrak{m}athbb{F}_p)
\xrightarrow{\;\cong\;} H_*(\Sigma_{p^m-1};\mathfrak{m}athbb{F}_p).
\]
Consider the commuting diagram of subgroup inclusions
\[
\xymatrix@C=0.05cm@R=1.0cm{
& \Sigma_{p^m} & \\
\Sigma_{p}\wr\Sigma_{p^{m-1}}\ar@{^{(}->}[ur]^{1} && \\
&& \Sigma_{p^m-1}\ar@{_{(}->}[uul]_{p^m} \\
\Sigma_{p-1}\wr\Sigma_{p^{m-1}}\times\Sigma_{p^{m-1}}\ar@{^{(}->}[uu]^{p}
&& \\
&\Sigma_{p^{m-1}}^{(p-1)}\times\Sigma_{p^{m-1}-1}
\ar@{_{(}->}[ul]_{p^{m-1}}\ar@{^{(}->}[uur]^{1}\ar@{^{(}->}[uuuu]^{p^{m}}
& \\
}
\]
in which the arrows are decorated with the $p$-power factors
of the indices, \emph{i.e.}, if $H\leqslant G$ then the number would be
$p^{\circrd_p|G:H|}$. Applying homology with coefficients in
$V^{\circtimes p^m}$ with the evident action of $\Sigma_{p^m}$,
$\mathfrak{m}athrm{H}_*(-;V^{\circtimes p^m})$, we obtain a commutative
diagram of induced homomorphisms (solid arrows) and transfer
homomorphisms (dashed arrows).
\[
\xymatrix@C=0.05cm@R=1.0cm{
& \mathfrak{m}athrm{H}_*(\Sigma_{p^m};V^{\circtimes p^m}) & \\
\mathfrak{m}athrm{H}_*(\Sigma_{p}\wr\Sigma_{p^{m-1}};V^{\circtimes p^m})
\ar@<0.5ex>@{->>}[ur]\ar@<-0.5ex>@{<--<}[ur]
&& \\
&& \mathfrak{m}athrm{H}_*(\Sigma_{p^m-1};V^{\circtimes p^m})
\ar@<-0.5ex>[uul]\ar@<0.5ex>@{<--}[uul]^{\Tr_{\Sigma_{p^m-1}}^{\Sigma_{p^m}}}\ar@<-0.5ex>@{ >-->}[ddl] \\
\mathfrak{m}athrm{H}_*(\Sigma_{p-1}\wr\Sigma_{p^{m-1}}\times\Sigma_{p^{m-1}};V^{\circtimes p^m})
\ar@<0.5ex>[uu]
\ar@<-0.5ex>@{<--}[uu]_{\Tr_{\Sigma_{p-1}\wr\Sigma_{p^{m-1}}\times\Sigma_{p^{m-1}}}^{\Sigma_{p}\wr\Sigma_{p^{m-1}}}}
&& \\
&\mathfrak{m}athrm{H}_*(\Sigma_{p^{m-1}}^{(p-1)}\times\Sigma_{p^{m-1}-1};V^{\circtimes p^m})
\ar@<0.5ex>[ul]\ar@<-0.5ex>@{<--}[ul]\ar@<-0.5ex>@{>>}[uur]
\ar@<-0.5ex>[uuuu] \ar@<0.5ex>@{<--}[uuuu] & \\
}
\]
As the transfer is contravariantly functorial with respect
to homomorphisms induced from inclusions, it is enough to
show that
$\Tr_{\Sigma_{p-1}\wr\Sigma_{p^{m-1}}\times\Sigma_{p^{m-1}}}^{\Sigma_{p}\wr\Sigma_{p^{m-1}}}$
is zero in positive degrees to deduce that the same holds
for $\Tr_{\Sigma_{p^m-1}}^{\Sigma_{p^m}}$. But this follows
since
\[
\mathfrak{m}athrm{H}_*(\Sigma_{p}\wr\Sigma_{p^{m-1}};V^{\circtimes p^m})
\cong
\mathfrak{m}athrm{H}_*(\Sigma_{p};\mathfrak{m}athrm{H}_*(\Sigma_{p^{m-1}};V^{\circtimes p^{m-1}})^{\circtimes p})
\]
and by~\eqref{eq:Tr=0} we already know the result for
all transfer homomorphisms of the form
\[
\Tr_{\Sigma_{p-1}}^{\Sigma_{p}}\colon
\mathfrak{m}athrm{H}_*(\Sigma_{p};W^{\circtimes p})
\to \mathfrak{m}athrm{H}_*(\Sigma_{p-1};W^{\circtimes p})
\]
for some $\mathfrak{m}athbb{F}_p$-vector space~$W$. To summarise, we
have verified
\begin{lem}\leftarrowbel{lem:Tr=0}
For $m\geqslant1$, the transfer
\[
\Tr_{\Sigma_{p^m-1}}^{\Sigma_{p^m}}\colon
\mathfrak{m}athrm{H}_*(\Sigma_{p^m};V^{\circtimes p^m})
\to
\mathfrak{m}athrm{H}_*(\Sigma_{p^m-1};V^{\circtimes p^m})
\]
is zero in positive degrees.
\end{lem}
Recall the standard $2$-periodic projective $\mathfrak{m}athbb{F}_p[C_p]$-resolution
of~$\mathfrak{m}athbb{F}_p$,
\begin{equation}\leftarrowbel{eq:Cp-resolution}
\xymatrix{
\mathfrak{m}athbb{F}_p & \ar@{->>}[l] \mathfrak{m}athbb{F}_p[C_p]e_0 &&\ar[ll]_{1-\gamma}\mathfrak{m}athbb{F}_p[C_p]e_1
&&\ar[ll]_{1+\gamma+\cdots+\gamma^{p-1}}
\mathfrak{m}athbb{F}_p[C_p]e_2 && \ar[ll]_(.4){1-\gamma} \cdots
}
\end{equation}
where $C_p=\leftarrowngle\gamma\rightarrowngle$ generated by the $p$-cycle
$\gamma=(1,2,\ldots,p)$. Tensoring over $\mathfrak{m}athbb{F}_p[C_p]$ gives
a complex
\[
\xymatrix{
0 & \ar@{->}[l] \mathfrak{m}athbb{F}_pe_0\circtimes V^{\circtimes p}
&&\ar[ll]_{1-\gamma}\mathfrak{m}athbb{F}_pe_1\circtimes V^{\circtimes p}
&&\ar[ll]_{1+\gamma+\cdots+\gamma^{p-1}}\mathfrak{m}athbb{F}_pe_2\circtimes V^{\circtimes p}
&&\ar[ll]_(.4){1-\gamma}\cdots
}
\]
whose homology is $\mathfrak{m}athrm{H}_*(C_p;V^{\circtimes p})$.
Let $X$ be a connective cofibrant $S$-module, and let
$x\in H_n(X;\mathfrak{m}athbb{F}_p)$ be a non-zero element. Recall the
algebraic results of~\cite{JPM:Steenrodops}*{lemma~1.4}.
If~$p$ is a prime, then there are elements
\[
e_r\circtimes x^{\circtimes p} =
e_r\circtimes\circverset{p}{\circverbrace{x\circtimes\cdots\circtimes x}}
\]
which survive to non-zero homology classes in
$\mathfrak{m}athrm{H}_*(C_p;H_*(X;\mathfrak{m}athbb{F}_p)^{\circtimes p})$ for $r\geqslant0$,
and where if~$p$ is odd,
\begin{itemize}
\item
$n$ is even and $r=2s(p-1)$ or $r=2(s+1)(p-1)-1$
for $0\leqslant s\in\mathbb{Z}$,
\item
$n$ is odd, $r=(2s+1)(p-1)$ or $r=(2s+1)(p-1)-1$
for $0\leqslant s\in\mathbb{Z}$.
\end{itemize}
These map to non-zero elements
\[
\tilde{\dlQ}_rx,\beta\tilde{\dlQ}_rx
\in\mathfrak{m}athrm{H}_*(\Sigma_p;H_*(X;\mathfrak{m}athbb{F}_p)^{\circtimes p})
\]
depending on the parity of $r$. There is a canonical
isomorphism
\[
\mathfrak{m}athrm{H}_*(\Sigma_p;H_*(X;\mathfrak{m}athbb{F}_p)^{\circtimes p})
\xrightarrow{\;\cong\;}
H_*(E\Sigma_p\ltimes_{\Sigma_p}X^{(p)})
\]
and we also denote the images of $\tilde{\dlQ}_sx,\tilde{\beta\dlQ}_sx$
by the same symbols. The natural weak equivalence
\[
E\Sigma_p\ltimes_{\Sigma_p}X^{(p)}
\xrightarrow{\;\sim\;} X^{(p)}/\Sigma_p
\]
induces an isomorphism
\[
\xymatrix{
\mathfrak{m}athrm{H}_*(\Sigma_p;H_*(X;\mathfrak{m}athbb{F}_p)^{\circtimes p})\ar[r]_{\cong}\ar@/^19pt/[rr]^{\cong}
& H_*(E\Sigma_p\ltimes_{\Sigma_p}X^{(p)};\mathfrak{m}athbb{F}_p)\ar[r]_(.57){\cong}
& H_*(X^{(p)}/\Sigma_p;\mathfrak{m}athbb{F}_p)
}
\]
sending $\tilde{\dlQ}_rx$ to the element which we will denote
by $\bar{\dlQ}_rx\in H_*(X^{(p)}/\Sigma_p;\mathfrak{m}athbb{F}_p)$. When~$p$
is odd, whenever $2r\geqslant n$ we will set
\begin{align*}
\bar{\dlQ}^rx &= (-1)^r\nu(n)\bar{\dlQ}_{(2r-n)(p-1)}x,
& \beta\bar{\dlQ}^rx &= (-1)^r\nu(n)\bar{\dlQ}_{(2r-n)(p-1)-1}x,
\end{align*}
in keeping with upper indexing for Dyer-Lashof operations,
where
\[
\nu(n) = (-1)^{n(n-1)(p-1)/4}\biggl(((p-1)/2)!\biggr)^n,
\]
which does not depend on~$r$. If $p=2$, whenever $r\geqslant n$
we set
\[
\bar{\dlQ}^rx = \bar{\dlQ}_{r-n}x.
\]
The action of the Dyer-Lashof operations $\dlQ^r$ and
$\beta\dlQ^r$ on $H_*(\mathfrak{m}athbb{P}X;\mathfrak{m}athbb{F}_p)$ described by
Steinberger in~\cite{LNM1176}*{chapter~III} is consistent
with this notation; however, we will sometimes write
$\dlQ^r\cdotx$ or $\beta\dlQ^r\cdotx$ when applying such an
operation to an element
$x\in H_*(X;\mathfrak{m}athbb{F}_p)\subseteqeq H_*(\mathfrak{m}athbb{P}X;\mathfrak{m}athbb{F}_p)$ to
avoid potential confusion when $X$ is itself a
commutative $S$-algebra.
\begin{lem}\leftarrowbel{lem:DLops-compare}
For a connective cofibrant $S$-module $X$, and an element
$x\in H_*(X;\mathfrak{m}athbb{F}_p)$, under the natural map
\[
\xymatrix{
E\Sigma_p\ltimes_{\Sigma_p}X^{(p)}
\ar[r]_(.57){\;\sim\;}\ar@/^19pt/[rr]^\rho
& X^{(p)}/\Sigma_p\ar[r]
& \mathfrak{m}athbb{P}X
}
\]
in $H_*(-;\mathfrak{m}athbb{F}_p)$ we have
\[
\dlQ^rx = \rho_*(\bar{\dlQ}^rx),
\quad
\beta\dlQ^rx = \rho_*(\beta\bar{\dlQ}^rx).
\]
\end{lem}
\begin{proof}
The basic observation is that a commutative $S$-algebra
is an algebra over the monad $\mathfrak{m}athbb{P}\circ\mathfrak{m}athbb{U}$
where the two model categories $\mathfrak{m}athscr{M}_S$ and
$\mathfrak{m}athscr{C}_S$ are related by the Quillen adjunction
of~\eqref{eq:QuillenAdj-C-M} with $R=S$.
\[
\xymatrix{
{\mathfrak{m}athscr{C}_S} \ar@/_8pt/[rr]_{\mathfrak{m}athbb{U}}
&& {\mathfrak{m}athscr{M}_S} \ar@/_8pt/[ll]_{\mathfrak{m}athbb{P}}
}
\]
The definition of the Dyer-Lashof operations for a
commutative $S$-algebra~$A$ involves the composition
\[
\xymatrix{
E\Sigma_p\ltimes_{\Sigma_p} (A^c)^{(p)}
\ar[r]\ar@/^21pt/[rrrr]
& (A^c)^{(p)}/\Sigma_p \ar[r]
& A^{(p)}/\Sigma_p\ar[r]
& \mathfrak{m}athbb{P}A\ar[r] & A
}
\]
where $(-)^c$ denotes cofibrant replacement in $\mathfrak{m}athscr{M}_S$.
When $X$ is cofibrant in $\mathfrak{m}athscr{M}_S$ and $A=\mathfrak{m}athbb{P}X$,
we obtain a commutative diagram of the form
\[
\xymatrix{
E\Sigma_p\ltimes_{\Sigma_p} X^{(p)}\ar[d]\ar@/^21pt/[drrrr]^\rho
&&& \\
E\Sigma_p\ltimes_{\Sigma_p} ((\mathfrak{m}athbb{P}X)^c)^{(p)}\ar[r]
& ((\mathfrak{m}athbb{P}X)^c)^{(p)}/\Sigma_p \ar[r]
& (\mathfrak{m}athbb{P}X)^{(p)}/\Sigma_p\ar[r]
& \mathfrak{m}athbb{P}(\mathfrak{m}athbb{P}X)\ar[r] & \mathfrak{m}athbb{P}X
}
\]
and applying $H_*(-;\mathfrak{m}athbb{F}_p)$ to this gives the result.
\end{proof}
Using iterated extended powers associated with the iterated
wreath power subgroups
\[
\Sigma_p^{\wr p} =
\circverset{\ell}{\circverbrace{\Sigma_p\wr\cdots\wr\Sigma_p}}
\leqslant\Sigma_{p^\ell}
\]
we can form elements
\[
\dlQ^Ix =
\beta^{\varepsilon_1}\bar{\dlQ}^{i_1}
\cdots\beta^{\varepsilon_\ell}\bar{\dlQ}^{i_\ell}x
\in H_*(\mathfrak{m}athbb{P}X;\mathfrak{m}athbb{F}_p)
\]
for $x\in H_*(X;\mathfrak{m}athbb{F}_p)$, where
$I = (\varepsilon_1,i_1,\ldots,\varepsilon_\ell,i_\ell)$,
$i_k>0$ and $\varepsilon_k=0,1$ (with $\varepsilon_k=0$
when $p=2$) can be interpreted in terms of the
homology of iterated wreath powers, and we obtain
the compatibility formula
\[
\dlQ^Ix = \rho^\ell_*(\bar{\dlQ}^Ix),
\]
where $\rho^\ell$ is defined in an obvious way from
the following diagram.
\[
\xymatrix{
E\Sigma_p^{\wr p}
\ltimes_{\Sigma_p^{\wr p}} X^{(p^\ell)}
\ar[d]\ar@/^21pt/[ddrrrr]^{\rho^\ell}
&&& \\
E\Sigma_{p^\ell}\ltimes_{\Sigma_{p^\ell}} X^{(p^\ell)} \ar[d]
&&& \\
E\Sigma_{p^\ell}\ltimes_{\Sigma_{p^\ell}}((\mathfrak{m}athbb{P}X)^c)^{({p^\ell})}
\ar[r]
& ((\mathfrak{m}athbb{P}X)^c)^{({p^\ell})}/\Sigma_{p^\ell}\ar[r]
& (\mathfrak{m}athbb{P}X)^{({p^\ell})}/\Sigma_{p^\ell}\ar[r]
& \mathfrak{m}athbb{P}(\mathfrak{m}athbb{P}X)\ar[r] & \mathfrak{m}athbb{P}X
}
\]
We can now state an important result.
\begin{thm}\leftarrowbel{thm:deltaQ^Ix=0}
Let $X$ be a connective cofibrant $S$-module. Then
\[
(\partialta_{(\mathfrak{m}athbb{P}X,S)})_*\colonH_*(\mathfrak{m}athbb{P}X;\mathfrak{m}athbb{F}_p)
\to H_*(\Omega_S(\mathfrak{m}athbb{P}X);\mathfrak{m}athbb{F}_p)
\]
annihilates every element of the form $\dlQ^Ix$,
where $\len(I)>0$ and $x\in H_*(X;\mathfrak{m}athbb{F}_p)$.
\end{thm}
\begin{proof}
This follows from the observation~\eqref{eq:delta-cpt}
together with Lemma~\ref{lem:Tr=0}.
\end{proof}
We will give an explicit description of the algebra
in Theorem~\ref{thm:H*PX-Fp}.
Recall that if $A$ is a connective commutative $S$-algebra
for which $\pi_0(A)$ is augmented over $\mathfrak{m}athbb{F}_p$, there is
an induced augmentation of commutative $S$-algebras
$A\to H\mathfrak{m}athbb{F}_p$, making $H\mathfrak{m}athbb{F}_p$ into an $A$-module.
Theorem~\ref{thm:deltaQ^Ix=0} generalises to give the
following result.
\begin{thm}\leftarrowbel{thm:deltaQ^Ix=0-gen}
Let $A$ be a connective commutative $S$-algebra. Then
\[
(\partialta_{(A,S)})_*\colonH_*(A;\mathfrak{m}athbb{F}_p)\to H_*(\Omega_S(A);\mathfrak{m}athbb{F}_p)
\]
annihilates every element of the form $\bar{\dlQ}^Ia$ with
$\len(I)>0$ and $a\in H_*(A;\mathfrak{m}athbb{F}_p)$. Hence, if
$H\mathfrak{m}athbb{F}_p$ is an $A$-module the homomorphism
\[
\mathfrak{m}athbb{D}elta_{(A,S)}\colonH_*(A;\mathfrak{m}athbb{F}_p)
\to H^A_*(\Omega_S(A);\mathfrak{m}athbb{F}_p) = \TAQ_*(A,S;H\mathfrak{m}athbb{F}_p)
\]
also annihilates all such elements.
\end{thm}
\begin{proof}
Using the observation in the Proof of
Lemma~\ref{lem:DLops-compare}, we know there is a morphism
of commutative $S$-algebras $\mathfrak{m}athbb{P}A\to A$ extending
the multiplication. Choose a cofibrant replacement $A^c\to A$
for the underlying $S$-module of $A$. By naturality there
is a commutative diagram in the homotopy category of $S$-modules
\[
\xymatrix{
\mathfrak{m}athbb{P}A^c\ar[d]_{\partialta_{(\mathfrak{m}athbb{P}A^c,S)}}\ar[rr]
&& \mathfrak{m}athbb{P}A\ar[d]_{\partialta_{(\mathfrak{m}athbb{P}A,S)}}\ar[rr]
&& A\ar[d]_{\partialta_{(A,S)}} \\
\mathfrak{m}athbb{P}A^c\wedge A^c\ar[rr]
&& \Omega_S(\mathfrak{m}athbb{P}A)\ar[rr]
&& \Omega_S(A)
}
\]
and on applying $H_*(-)=H_*(-;\mathfrak{m}athbb{F}_p)$ we obtain an algebraic
commutative diagram.
\[
\xymatrix{
H_*(\mathfrak{m}athbb{P}A^c)\ar[d]_{(\partialta_{(\mathfrak{m}athbb{P}A^c,S)})_*}\ar[rr]
&& H_*(\mathfrak{m}athbb{P}A)\ar[d]_{(\partialta_{(\mathfrak{m}athbb{P}A,S)})_*}\ar[rr]
&& H_*(A)\ar[d]_{(\partialta_{(A,S)})_*} \\
H_*(\mathfrak{m}athbb{P}A^c\wedge A^c)\ar[rr]
&& H_*(\Omega_S(\mathfrak{m}athbb{P}A))\ar[rr]
&& H_*(\Omega_S(A))
}
\]
Since an element of the form $\dlQ^Ia$ lifts back to an
element $\dlQ^Ia'\in H_*(\mathfrak{m}athbb{P}A^c)$ as explained
above, the result follows. The result about $\theta'$ is
immediate from the definition~\eqref{eq:TAQ-Hurewiczhom}.
\end{proof}
\section{The reduced free commutative algebra functor}
\leftarrowbel{sec:redPX}
In this section we describe a modification $\tilde{\mathfrak{m}athbb{P}}_R$
of the usual free functor $\mathfrak{m}athbb{P}_R$ from $R$-modules to
commutative $R$-algebras. Recall that $\mathfrak{m}athbb{P}_R$ is
left adjoint to the forgetful functor and when viewed as
an endofunctor of $\mathfrak{m}athscr{M}_R$, its algebras are precisely
the commutative $R$-algebras. Then $\tilde{\mathfrak{m}athbb{P}}_R$
gives an endofunctor on the comma category $S^0_R/\mathfrak{m}athscr{M}_R$
of $R$-modules under the $R$-sphere $S^0_R$. Roughly speaking,
the difference between $\mathfrak{m}athbb{P}_RX$ and $\tilde{\mathfrak{m}athbb{P}}_RX$
is that in the latter, the morphism $S^0_R\to X$ becomes identified
with the unit. There are two reasons why this is useful to us.
First we will investigate the free algebra for an $R$-module
with such a `unit' morphism corresponding to a bottom cell (at
least when $R=S$). Second, a unital commutative $R$-algebra has
a natural choice of such morphism induced by the unit, and algebras
over $\tilde{\mathfrak{m}athbb{P}}_R$ in $S^0_R/\mathfrak{m}athscr{M}_R$ are the
commutative $R$-algebras; a similar observation was made by
{K\v{r}\'{i}\v{z}}{} \& May~\cite{IK&JPM:Asterisque}, see the discussion
leading up to and including proposition~3.7.
Throughout, we fix a cofibrant commutative $S$-algebra~$R$.
The two model categories $\mathfrak{m}athscr{M}_R$ and $\mathfrak{m}athscr{C}_R$
are related by the Quillen adjunction
\[
\xymatrix{
{\mathfrak{m}athscr{C}_R} \ar@/_8pt/[rr]_{\mathfrak{m}athbb{U}}
&& {\mathfrak{m}athscr{M}_R} \ar@/_8pt/[ll]_{\mathfrak{m}athbb{P}_R}
}
\]
where the right adjoint $\mathfrak{m}athbb{U}$ is the forgetful functor.
For a cofibrant $R$-module~$Z$, inclusion of the basepoint
$*\to X$ induces a cofibration of commutative $R$-algebras
$R=\mathfrak{m}athbb{P}_R*\to\mathfrak{m}athbb{P}_RZ$, so $\mathfrak{m}athbb{P}_RZ$ is
cofibrant in the model category $\mathfrak{m}athscr{C}_R$. More
generally a(n acyclic) cofibration $f\colonX\to Y$ in
$\mathfrak{m}athscr{M}_R$ induces a(n acyclic) cofibration
$\mathfrak{m}athbb{P}_Rf\colon\mathfrak{m}athbb{P}_R X\to\mathfrak{m}athbb{P}_RY$ in
$\mathfrak{m}athscr{C}_R$.
In $\mathfrak{m}athscr{M}_R$, $R$ is not cofibrant and we denote its
functorial cofibrant replacement by $S^0_R=\mathfrak{m}athbb{F}_RS^0$
and a weak equivalence induced by a map of spectra $S_R^0\to R$
which represents the unit.
\[
\xymatrix{
{*}\ar@{ >->}[r]\ar[dr] & S^0_R\ar@{->>}[d]^{\sim}\\
& R
}
\]
There is a unique induced morphism $\mathfrak{m}athbb{P}_R S^0_R\to R$
in $\mathfrak{m}athscr{C}_R$, but this need not be a cofibration.
Using the functorial factorisation in~$\mathfrak{m}athscr{C}_R$
we obtain a commutative diagram in $\mathfrak{m}athscr{C}_R$
\begin{equation}\leftarrowbel{eq:tildeR}
\xymatrix{
& \ar@{ >->}[dl]\mathfrak{m}athbb{P}_R S^0_R\ar[dr] & \\
\widetilde{R}\ar@{->>}[rr]_{\sim} & & R
}
\end{equation}
which we use to fix the left hand arrow.
We will make use of the comma category $S^0_R/\mathfrak{m}athscr{M}_R$
of $R$-modules under $S^0_R$, whose objects are the morphisms
$S^0_R\to X$ and whose morphisms are the commuting diagrams
\[
\xymatrix{
& \ar[dl] S^0_R\ar[dr] & \\
X\ar[rr] & & Y
}
\]
with initial object $\Id_{S^0_R}$ and terminal object $S^0_R\to*$.
This inherits a model structure from $\mathfrak{m}athscr{M}_R$. Given
$i\colonS^0_R\to X$ in $S^0_R/\mathfrak{m}athscr{M}_R$, we obtain the induced
morphism $\mathfrak{m}athbb{P}_Ri\colon\mathfrak{m}athbb{P}_RS^0_R\to\mathfrak{m}athbb{P}_RX$ in
$\mathfrak{m}athscr{C}_R$. If $i$ is a cofibration, then $i$ is cofibrant
in $S^0_R/\mathfrak{m}athscr{M}_R$ and
$\mathfrak{m}athbb{P}_Ri\colon\mathfrak{m}athbb{P}_RS^0_R\to\mathfrak{m}athbb{P}_RX$ is a cofibration
in $\mathfrak{m}athscr{C}_R$; we will then write $X/S^0_R$ for the cofibre
of~$i$.
We obtain a pushout diagram of commutative $R$-algebras
\[
\xymatrix{
\mathfrak{m}athbb{P}_RS^0_R\ar@{}[dr]|{\text{\pigpenfont R}}
\ar[r]^{\mathfrak{m}athbb{P}_Ri}\ar@{ >->}[d]
& \mathfrak{m}athbb{P}_RX\ar@{ >->}[d] \\
\tilde{R}\ar[r] & \tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX
}
\]
and we set
\[
\tilde{\mathfrak{m}athbb{P}}_RX =
\tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX.
\]
If $i^c\colonS^0_R\to X^c$ is the cofibrant replacement of~$i$
in the comma category, then the pushout diagram of solid
arrows in
\[
\xymatrix{
& & \mathfrak{m}athbb{P}_RX\ar@/^19pt/@{ >-->}[ddd] \\
\mathfrak{m}athbb{P}_RS^0_R\ar@{}[dr]|{\text{\pigpenfont R}}
\ar@/^19pt/@{-->}[urr]^{\mathfrak{m}athbb{P}_Ri}\ar@{ >->}[r]^{\mathfrak{m}athbb{P}_Ri^c}\ar@{ >->}[d]
& \mathfrak{m}athbb{P}_RX^c\ar@{ >->}[d]\ar@{-->}[ur] & \\
\tilde{R}\ar@{ >->}[r]\ar@/_19pt/@{-->}[drr]
& \tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX^c\ar@{-->}[dr] \\
& & \tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX
}
\]
defines the homotopy pushout of the first diagram,
\[
\tilde{\mathfrak{m}athbb{P}}^h_RX =
\tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX^c.
\]
Of course $\tilde{\mathfrak{m}athbb{P}}^h_RX$ is well-defined
in the homotopy category $\bar{h}\mathfrak{m}athscr{C}_R$, and
we have in effect defined it by making functorial
choices.
The model categories $S^0_R/\mathfrak{m}athscr{M}_R$ and $\mathfrak{m}athscr{C}_R$
are related by the Quillen adjunction
\[
\xymatrix{
{\mathfrak{m}athscr{C}_R} \ar@/_8pt/[rr]_{\tilde{\mathfrak{m}athbb{U}}}
&& {S^0_R/\mathfrak{m}athscr{M}_R} \ar@/_8pt/[ll]_{\tilde{\mathfrak{m}athbb{P}}_R}
}
\]
where the right adjoint $\tilde{\mathfrak{m}athbb{U}}$ is the
forgetful functor sending $A$ to the composition
\[
\xymatrix{
S^0_R\ar[r]\ar@/^15pt/[rr] & R \ar[r] & A
}
\]
in $\mathfrak{m}athscr{M}_R$. The total left derived functor of
$\tilde{\mathfrak{m}athbb{P}}_R$ is $\tilde{\mathfrak{m}athbb{P}}^h_R$
and
\[
\xymatrix{
{\bar{h}\mathfrak{m}athscr{C}_R} \ar@/_8pt/[rr]_{\tilde{\mathfrak{m}athbb{U}}^h}
&& {\bar{h}(S^0_R/\mathfrak{m}athscr{M}_R)}\ar@/_8pt/[ll]_{\tilde{\mathfrak{m}athbb{P}}^h_R}
}
\]
is a derived adjunction on homotopy categories.
\begin{rem}\leftarrowbel{rem:S^0/pushouts}
In $S_R^0/\mathfrak{m}athscr{M}_R$, pushouts are defined using pushouts
in $\mathfrak{m}athscr{M}_R$, and we use the symbol $\undervee{S^0_R}$
to indicate such a pushout.
\[
\xymatrix{
S^0_R \ar[r] \ar[d]
\ar@{}[dr]|{\text{\pigpenfont R}} & Y\ar[d] \\
X\ar[r] & X\undervee{S^0_R}Y
}
\]
Since the reduced free algebra functor
$\tilde{\mathfrak{m}athbb{P}}_R\colon\mathfrak{m}athscr{M}_R\to\mathfrak{m}athscr{C}_R$ is
a left adjoint it preserves pushouts, so for two $R$-modules
$X,Y$ under $S^0_R$,
\begin{equation}\leftarrowbel{eq:S^0/pushouts}
\tilde{\mathfrak{m}athbb{P}}_R(X\undervee{S^0_R}Y) \cong
\tilde{\mathfrak{m}athbb{P}}_RX \wedge_R \tilde{\mathfrak{m}athbb{P}}_RY.
\end{equation}
In particular, if we have a connective CW $R$-module with
distinguished bottom cell $S^0_R\to X$, then the $n$-skeleton
$X^{[n]}$ gives rise to the $n$-skeleton of the CW commutative
$R$-algebra $\tilde{\mathfrak{m}athbb{P}}_R X$,
\begin{equation}\leftarrowbel{eq:S^0/skeleta}
\tilde{\mathfrak{m}athbb{P}}_R^{\left\leftarrownglen\right\rightarrowngle} X
= (\tilde{\mathfrak{m}athbb{P}}_R X)^{\left\leftarrownglen\right\rightarrowngle}
= \tilde{\mathfrak{m}athbb{P}}_R(X^{[n]}).
\qedhere
\end{equation}
\end{rem}
We already know that for cofibrant $X$, in the homotopy
category $\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_RX}$,
\[
\Omega_R(\mathfrak{m}athbb{P}_RX) \cong \mathfrak{m}athbb{P}_RX\wedge_R X.
\]
\begin{prop}\leftarrowbel{prop:Omega-redP}
Let $X\in S^0_R/\mathfrak{m}athscr{M}_R$ be cofibrant. Then in
the homotopy category $\bar{h}\mathfrak{m}athscr{M}_{\tilde{\mathfrak{m}athbb{P}}_RX}$,
\[
\Omega_R(\tilde{\mathfrak{m}athbb{P}}_RX) \cong \tilde{\mathfrak{m}athbb{P}}_R X\wedge_R X/S^0_R.
\]
\end{prop}
\begin{proof}
First we recall some observations appearing in~\cite{BGRtaq}.
For a pushout diagram of cofibrations of commutative
$R$-algebras
\[
\xymatrix{
A \ar@{ >->}[r] \ar@{ >->}[d]
\ar@{}[dr]|{\text{\pigpenfont R}}
& B\ar@{ >->}[d] \\
C\ar@{ >->}[r] & B\wedge_A C
}
\]
by \cite{MBtaq}*{proposition~4.6} we have
\begin{equation}\leftarrowbel{eq:Omega-redP-1}
\Omega_B(B\wedge_A C) \sim B\wedge_A \Omega_A(C).
\end{equation}
Now assume that $i\colonS^0_R\to X$ is a cofibration, and consider
\[
\tilde{\mathfrak{m}athbb{P}}_RX =
\tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX.
\]
Notice that by \eqref{eq:Omega-redP-1},
\[
\Omega_{\mathfrak{m}athbb{P}_RX}(\tilde{\mathfrak{m}athbb{P}}_RX) \sim
\mathfrak{m}athbb{P}_RX\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\Omega_{\mathfrak{m}athbb{P}_RS^0_R}(\tilde{R}),
\]
and since there is a weak equivalence of $R$-algebras
$\tilde{R}\xrightarrow{\sim}R$, the sequence
\[
R\to\mathfrak{m}athbb{P}_RS^0_R\to\tilde{R}
\]
has an associated cofibre sequence of the form
\[
\xymatrix{
\tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\Omega_R(\mathfrak{m}athbb{P}_RS^0_R)\ar[r] &
\Omega_R(\tilde{R})\ar[r] & \Omega_{\mathfrak{m}athbb{P}_RS^0_R}(\tilde{R})\ar[r] &
\cdots
}
\]
where
\[
\Omega_R(\tilde{R})\sim\Omega_R(R)\sim *.
\]
Therefore
\[
\Omega_{\mathfrak{m}athbb{P}_RS^0_R}(\tilde{R}) \sim
\tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\Sigma\Omega_R(\mathfrak{m}athbb{P}_RS^0_R)
\sim \tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RS^0_R\wedge_R\Sigma S^0_R
\sim \tilde{R}\wedge_R\Sigma S^0_R,
\]
and so
\[
\Omega_{\mathfrak{m}athbb{P}_RX}(\tilde{\mathfrak{m}athbb{P}}_RX) \sim
\tilde{\mathfrak{m}athbb{P}}_RX\wedge_R\Sigma S^0_R.
\]
Similarly, from the sequence
\[
R \to \mathfrak{m}athbb{P}_RX \to \tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RX
\]
we obtain a cofibre sequence of $\tilde{\mathfrak{m}athbb{P}}_RX$-modules
\[
\xymatrix@C=0.7cm@R=1.0cm{
\tilde{\mathfrak{m}athbb{P}}_RX \wedge_{\mathfrak{m}athbb{P}_RX}\Omega_R(\mathfrak{m}athbb{P}_RX)\ar[r]
& \Omega_R(\tilde{\mathfrak{m}athbb{P}}_RX) \ar[r]
& \Omega_{\mathfrak{m}athbb{P}_RX}(\tilde{\mathfrak{m}athbb{P}}_RX)\ar[r]
& \tilde{\mathfrak{m}athbb{P}}_RX \wedge_{\mathfrak{m}athbb{P}_RX}\Sigma\Omega_R(\mathfrak{m}athbb{P}_RX)
\ar[r] & \cdots
}
\]
which is equivalent to
\[
\xymatrix{
\tilde{\mathfrak{m}athbb{P}}_RX \wedge_R X \ar[r]
& \Omega_R(\tilde{\mathfrak{m}athbb{P}}_RX) \ar[r]
& \tilde{\mathfrak{m}athbb{P}}_RX\wedge_R \Sigma S^0_R \ar[r]
& \tilde{\mathfrak{m}athbb{P}}_RX \wedge_R \Sigma X \ar[r]
& \cdots
}
\]
and so
\[
\Omega_R(\tilde{\mathfrak{m}athbb{P}}_RX) \sim
\tilde{\mathfrak{m}athbb{P}}_RX \wedge_R X/S^0_R,
\]
as required.
\end{proof}
Now let $A$ be a commutative $R$-algebra. On composing
the unit $R\to A$ with the weak equivalence $S^0_R\to R$
we obtain the object
\[
S^0_R\xrightarrow{\;\sim\;}R
\xrightarrow{\;\ph{\sim}\;} A
\]
in $S^0_R/\mathfrak{m}athscr{M}_R$. Using functorial factorisation
we obtain a cofibrant replacement
\[
\xymatrix{
& \ar@{ >->}[dl] S^0_R\ar[dr] & \\
A^c\ar@{->>}[rr]^{\sim} && A
}
\]
and so
\[
\tilde{\mathfrak{m}athbb{P}}^h_RA =
\tilde{R}\wedge_{\mathfrak{m}athbb{P}_RS^0_R}\mathfrak{m}athbb{P}_RA^c.
\]
\begin{rem}\leftarrowbel{rem:tildePA->A}
The multiplication on a commutative $R$-algebra~$A$ extends
to a morphism of commutative $R$-algebras
$\tilde{\mathfrak{m}athbb{P}}_RA\to A$. This follows from the evident
commutative diagram of solid arrows
\[
\xymatrix{
\mathfrak{m}athbb{P}_RS^0_R
\ar[r]\ar@{ >->}[d]\ar@{}[dr]|{\text{\pigpenfont R}}
& \mathfrak{m}athbb{P}_RA\ar[d]\ar@/^17pt/[ddr] & \\
\tilde{R} \ar[r]\ar@{->>}[d]^{\sim} & \tilde{\mathfrak{m}athbb{P}}_RA\ar@{-->}[dr] & \\
R\ar@/_17pt/[rr] && A
}
\]
\noindent
where the curved arrows come from the unit and extension
of the product respectively.
\end{rem}
For completeness we recall a useful relationship between
the free functor on a suspension spectrum and the suspension
spectrum of the associated infinite loop space. This can
be found in~\cite{NJK:LocAQG}*{section~4} for example.
Let $Z$ be a connected based space. The infinite loop space
$\Omega^\infty \Sigma^\infty Z$ gives rise to a commutative
$S$-algebra $\Omega^\infty\Sigma^\infty_+Z$, \emph{i.e.}, the
suspension spectrum of the based space
$\Omega^\infty\Sigma^\infty Z$ with a disjoint basepoint.
The natural (based) map $Z\to\Omega^\infty\Sigma^\infty Z$
viewed as an unbased map induces a based map
\[
\Sigma^\infty_+Z\to\Sigma^\infty_+\Omega^\infty\Sigma^\infty Z
\]
which extends uniquely to a morphism of ring spectra
\[
\mathfrak{m}athbb{P}\Sigma^\infty_+Z\to\Sigma^\infty_+\Omega^\infty\Sigma^\infty Z.
\]
The base points in $Z$ and $\Omega^\infty\Sigma^\infty Z$
pick out maps from the sphere $S$ and there is a commutative
diagram of commutative $S$-algebras
\begin{equation}\leftarrowbel{eq:PZ-QZ}
\xymatrix{
\mathfrak{m}athbb{P}S^0\ar[r]\ar@{ >->}[d]\ar@{}[dr]|{\text{\pigpenfont R}}
& \mathfrak{m}athbb{P}\Sigma^\infty_+ Z\ar@/^17pt/[dddr]\ar@{-->}[d] & \\
\tilde{S}\ar@{->>}[d]_{\sim}\ar@/_17pt/[ddrr]\ar@{-->}[r]
& \tilde{\mathfrak{m}athbb{P}}\Sigma^\infty_+ Z \ar@{-->}[ddr]&\\
S\ar@/_17pt/[drr]&& \\
&& \mathfrak{m}athbb{P}\Sigma^\infty_+\Omega^\infty\Sigma^\infty Z
}
\end{equation}
where the acyclic fibration $\tilde{S}\to S$
is that defined in~\eqref{eq:tildeR}.
\begin{prop}\leftarrowbel{prop:PZ-QZ}
For a connected based space $Z$, the morphism
\[
\tilde{\mathfrak{m}athbb{P}}\Sigma^\infty Z
\xrightarrow{\;\sim\;}
\Sigma^\infty_+\Omega^\infty\Sigma^\infty Z
\]
of~\eqref{eq:PZ-QZ} is a weak equivalence.
\end{prop}
\begin{proof}
If $\Bbbk=\mathbb{Q}$ and $\Bbbk=\mathfrak{m}athbb{F}_p$ for $p$ a prime, comparison of the
known answers for $H_*(\tilde{\mathfrak{m}athbb{P}}\Sigma^\infty Z;\Bbbk)$
and $H_*(\Omega^\infty\Sigma^\infty Z;\Bbbk)$ given in
Section~\ref{sec:HomologyPX} shows that this morphism induces
an isomorphism $H_*(-;\Bbbk)$. It follows that it induces an
isomorphism on $H_*(-;\mathbb{Z})$, hence it is a weak equivalence.
\end{proof}
\section{The ordinary homology of free commutative \protect$S$-algebras}
\leftarrowbel{sec:HomologyPX}
In this section we recall some results on ordinary homology
of free commutative $S$-algebras.
For a commutative ring $\Bbbk$, and a graded $\Bbbk$-module~$V_*$,
we will write $\Bbbk\leftarrowngle V_*\rightarrowngle$ for the free commutative
graded $\Bbbk$-algebra on $V_*$. If $V_*$ is connective and
$V_0$ is a cyclic $\Bbbk$-module, we set $\tilde{V}_* = V_*/V_0$.
Let $X$ be a cofibrant connective spectrum.
\begin{thm}\leftarrowbel{thm:H*PX-Q}
The rational homology of\/ $\mathfrak{m}athbb{P}X$ is given by
\[
H_*(\mathfrak{m}athbb{P}X;\mathbb{Q}) = \mathbb{Q}\leftarrowngle H_*(X;\mathbb{Q})\rightarrowngle.
\]
\end{thm}
In positive characteristic, the next result is fundamental.
We use the standard convention for Dyer-Lashof monomials
so that in an indexing sequence
\[
I =
(\varepsilon_1,i_1,\varepsilon_2,i_2,\ldots,\varepsilon_\ell,i_\ell),
\]
each $i_r$ is positive, when $p=2$ all the $\varepsilon_i$
are zero, while for odd~$p$, $\varepsilon_i=0,1$. The length
of $\dlQ^I$ is $\len(\dlQ^I)=\ell$.
\begin{thm}\leftarrowbel{thm:H*PX-Fp}
For~$p$ a prime, $H_*(\mathfrak{m}athbb{P}X;\mathfrak{m}athbb{F}_p)$ is the free
commutative graded $\mathfrak{m}athbb{F}_p$-algebra generated by elements
$\dlQ^Ix_j$, where $x_j$ for $j\in J$ gives a basis
for $H_*(X;\mathfrak{m}athbb{F}_p)$ and
$I=(\varepsilon_1,i_1,\varepsilon_2,\ldots,\varepsilon_\ell,i_\ell)$
is admissible with $\exc(I)+\varepsilon_1>|x_j|$.
\end{thm}
So for $p=2$, this gives the polynomial ring
\[
H_*(\mathfrak{m}athbb{P}X;\mathfrak{m}athbb{F}_2) =
\mathfrak{m}athbb{F}_2[\dlQ^Ix_j:\text{$j\in J$, $\exc(I)+\varepsilon_1>|x_j|$}\;].
\]
Of course these results are very similar to those for the
homology of $\Omega^\infty\Sigma^\infty Z$ for a space~$Z$;
for a convenient overview of the latter, see~\cite{JPM:HomOps}.
The Dyer-Lashof operations are compatible with the notation
used in Section~\ref{sec:PowOps+PX}.
\begin{proof}[Sketch of why \emph{Theorems~\ref{thm:H*PX-Q}}
and~\emph{\ref{thm:H*PX-Fp}} hold]
We learnt some of the following from Mike Mandell, see
also~\cite{MBtaq}*{section~6}.
Let $R$ be a commutative $S$-algebra. The free functor
$\mathfrak{m}athbb{P}_R$ for, sends $R$-modules to commutative
$R$-algebras. As it is a left adjoint it preserves
pushouts, so for $R$-modules $X,Y$,
\[
\mathfrak{m}athbb{P}_R(X\vee Y) \cong
\mathfrak{m}athbb{P}_R(X)\wedge_R \mathfrak{m}athbb{P}_R(Y).
\]
For any commutative $R$-algebra $A$, base change gives
\[
A\wedge_R \mathfrak{m}athbb{P}_R(-) = \mathfrak{m}athbb{P}_A(A\wedge_R(-)).
\]
If $R=S$ and $H=H\Bbbk$ for a field $\Bbbk$, then
\[
H\wedge\mathfrak{m}athbb{P}(-) = \mathfrak{m}athbb{P}_H(H\wedge (-)).
\]
Applying $\pi_*(-)$ gives a functor sending spectra
to commutative graded $\Bbbk$-algebras,
\[
X \mathfrak{m}apsto H_*(\mathfrak{m}athbb{P}(X)) = \pi_*(\mathfrak{m}athbb{P}_H(H\wedge(-)),
\]
which preserves pushouts, in particular it sends wedges
to tensor products. For any spectrum $X$, as an $H$-module,
$H\wedge X$ is equivalent to a wedge of suspensions of $H$,
so the calculation of $H_*(\mathfrak{m}athbb{P}(X))=H_*(\mathfrak{m}athbb{P}(X);\Bbbk)$
reduces to that for spheres. For $\Bbbk=\mathbb{Q}$ this gives the
rational result.
When $\Bbbk=\mathfrak{m}athbb{F}_p$, for a sphere $S^n$ the answer is the free
commutative $\Bbbk$-algebra on admissible Dyer-Lashof monomials
$\dlQ^I$ applied to an element~$s_n$, \emph{i.e.}, elements of the
form $\dlQ^Is_n=\dlQ^I\cdots_n$, where the indexing
sequences~$I$ are admissible and satisfy $\exc(I)>|s_n|=n$.
Although this makes sense for $n\in\mathbb{Z}$ (for the general case
see~\cite{LNM1176}*{chapter~III}), we only require the case
where $n\geqslant0$.
Using the ideas of Section~\ref{sec:PowOps+PX}, we know
that for $x\in H_*(X)$, the element $\dlQ^Ix$ is the image
under the canonical homomorphism
\[
\xymatrix{
H_*(E\Sigma_{p^\ell}\ltimes_{\Sigma_{p^\ell}}X^{(p^\ell)})\ar[rr]^(.55){\cong}\ar[drr]
&& H_*(X^{(p^\ell)}/\Sigma_{p^\ell})\ar@{^{(}->}[d] \\
&& H_*(\mathfrak{m}athbb{P}X)
}
\]
of an element obtained by forming iterated wreath powers
of~$x$ in the homology
\[
H_*(E\Sigma_p\ltimes_{\Sigma_p}(E\Sigma_{p^k}\ltimes_{\Sigma_{p^k}}X^{(p^k)})).
\]
Of course we can view $\bar{\dlQ}^Ix$ as obtained from~$x$
by applying the Dyer-Lashof monomial
\[
\dlQ^I =
\beta^{\varepsilon_1}\dlQ^{i_1}\cdots\beta^{\varepsilon_\ell}\dlQ^{i_\ell}
\]
which exists as an operation on the homology of any commutative
$S$-algebra.
\end{proof}
We will describe the analogous results for $\tilde{\mathfrak{m}athbb{P}}X$
where $X\in S^0/\mathfrak{m}athscr{M}_S$ is a cofibrant $S$-module under
$S^0$, i.e., equipped with a cofibration $S^0\to X$. Of course
our results can also be interpreted in the category of modules
over the $p$-local sphere.
Our main computational tool is the K\"unneth spectral sequence,
and we will use its multiplicative properties and compatibility
with the action of the Dyer-Lashof operations, see~\cite{MB-MM:BP-E4}
for some related results on this.
We begin by stating the rational result whose proof we leave
to the reader.
\begin{thm}\leftarrowbel{thm:H*redPX-Q}
The rational homology of\/ $\tilde{\mathfrak{m}athbb{P}}X$ is given by
\[
H_*(\tilde{\mathfrak{m}athbb{P}}X;\mathbb{Q}) = \mathbb{Q}\leftarrowngle \tilde{H}_*(X;\mathbb{Q})\rightarrowngle.
\]
\end{thm}
The positive characteristic case is of course more interesting.
\begin{thm}\leftarrowbel{thm:H*redPX-Fp}
Let~$p$ be a prime. If $X\in S^0/\mathfrak{m}athscr{M}_S$ is cofibrant,
then $H_*(\tilde{\mathfrak{m}athbb{P}}X;\mathfrak{m}athbb{F}_p)$ is the free commutative
graded $\mathfrak{m}athbb{F}_p$-algebra generated by elements $\dlQ^Ix_j$, where
$x_j$ for $j\in J$ gives a basis for
$\tilde{H}_*(X;\mathfrak{m}athbb{F}_p)=H_*(X/S^0;\mathfrak{m}athbb{F}_p)$ and
$I=(\varepsilon_1,i_1,\varepsilon_2,\ldots,\varepsilon_\ell,i_\ell)$
is admissible with $\exc(I)+\varepsilon_1>|x_j|$.
\end{thm}
\begin{proof}
We set $H_*(-) = H_*(-;\mathfrak{m}athbb{F}_p)$.
Since the pushout agrees with the smash product over $\mathfrak{m}athbb{P}S^0$,
there is a first quadrant K\"unneth spectral sequence with
\[
\mathfrak{m}athrm{E}^2_{s,t} =
\Tor^{H_*(\mathfrak{m}athbb{P}S^0)}_{s,t}(H_*(\mathfrak{m}athbb{P}X),\mathfrak{m}athbb{F}_p)
\Longrightarrow H_{s+t}(\tilde{\mathfrak{m}athbb{P}}X).
\]
Here $i'_*\colonH_*(\mathfrak{m}athbb{P}S^0)\to H_*(\mathfrak{m}athbb{P}X)$ embeds
the domain as a subalgebra since $x_0=i_*(s_0)$ generates
$H_0(X)=\mathfrak{m}athbb{F}_p$ and
\begin{align*}
i'_*(s_0) &= x_0-1, \\
i'_*(\dlQ^Is_0) &= \dlQ^I(x_0)-\dlQ^I(1) = \dlQ^I(x_0)
\quad \text{if $\len(I)>0$}.
\end{align*}
By the freeness of $H_*(\mathfrak{m}athbb{P}X)$, it is a free
$i'_*H_*(\mathfrak{m}athbb{P}S^0)$-module, hence
\begin{align*}
\mathfrak{m}athrm{E}^2_{*,*} &= \Tor^{H_*(\mathfrak{m}athbb{P}S^0)}_{0,*}(H_*(\mathfrak{m}athbb{P}X),\mathfrak{m}athbb{F}_p) \\
&= H_*(\mathfrak{m}athbb{P}X)\circtimes_{i'_*H_*(\mathfrak{m}athbb{P}S^0)}\mathfrak{m}athbb{F}_p \\
&= H_*(\mathfrak{m}athbb{P}X)/(x_0-1,\dlQ^Ix_0:\len(I)>0).
\end{align*}
Thus the spectral sequence collapses at the $\mathfrak{m}athrm{E}^2$-term
and the result follows.
\end{proof}
Armed with Theorem~\ref{thm:deltaQ^Ix=0}, we have
\begin{thm}\leftarrowbel{thm:univder-Q^I}
Let $p$ be a prime and let $X$ be a $p$-local connective
cofibrant $S$-module. Then the universal derivation
$\partialta_{(\tilde{\mathfrak{m}athbb{P}}X,S)}$ induces the derivation
\[
\mathfrak{m}athbb{D}elta_{(\tilde{\mathfrak{m}athbb{P}}X,S)}\colon H_*(\tilde{\mathfrak{m}athbb{P}}X;\mathfrak{m}athbb{F}_p)
\to \TAQ_*(\tilde{\mathfrak{m}athbb{P}}X,S;H\mathfrak{m}athbb{F}_p) \cong H_*(X/S^0;\mathfrak{m}athbb{F}_p)
\]
which acts on the elements $\dlQ^Ix$ with $x\in H_*(X;\mathfrak{m}athbb{F}_p)$
by the rule
\[
\mathfrak{m}athbb{D}elta(\dlQ^Ix) =
\begin{cases}
x & \text{\rm if $\len(I)=0$}, \\
0 & \text{\rm if $\len(I)>0$}.
\end{cases}
\]
\end{thm}
We remark that the universal derivation is not
an $\mathfrak{m}athcal{A}(p)_*$-comodule homomorphism.
The interaction between Dyer-Lashof operations
and the $\mathfrak{m}athcal{A}(p)_*$-coaction is described
in~\cite{Nishida}.
\section{Some calculations}\leftarrowbel{sec:Calculations}
Equipped with our earlier results, we revisit some
of the calculations of~\cite{BGRtaq}*{section~5}.
First we consider the $\TAQ$-Hurewicz homomorphism
for the \mathfrak{m}athrm{E}infty Thom spectrum $MU$. By work of
Basterra and Mandell, in the homotopy category
$\bar{h}\mathfrak{m}athscr{M}_{MU}$,
\[
\Omega_S(MU) \cong MU\wedge \Sigma^2 ku.
\]
At the prime~$p=2$, the Hurewicz homomorphism factors through
$H_*(MU;\mathfrak{m}athbb{F}_2)$
\[
\xymatrix{
\pi_*(MU) \ar[r]\ar@/^19pt/[rrr]^\theta & H_*(MU;\mathfrak{m}athbb{F}_2) \ar[r]_(.45){\theta'}
& \TAQ_*(MU,S;\mathfrak{m}athbb{H}\mathfrak{m}athbb{F}_2) \ar[r]_(.56){\cong}
& H_*(\Sigma^2 ku;\mathfrak{m}athbb{F}_2)
}
\]
where
\[
\xymatrix{
H_*(MU;\mathfrak{m}athbb{F}_2)\ar[r]^{\theta'}\ar@{=}[d] & H_{*-2}(\Sigma^2 ku;\mathfrak{m}athbb{F}_2)\ar@{=}[d] \\
\mathfrak{m}athbb{F}_2[b_r:r\geqslant1] & \Sigma^2\mathfrak{m}athbb{F}_2[\zeta_1^2,\zeta_2^2,\zeta_3,\ldots]
}
\]
is a derivation and
\[
\theta'(b_r) =
\begin{cases}
\Sigma^2\xi_s^2 & \text{if $r = 2^s$}, \\
\;\;0 & \text{otherwise}.
\end{cases}
\]
Here $\zeta_s=\chi(\xi_s)$ is the conjugate of the Milnor generator
in $\mathfrak{m}athbb{F}_2[\zeta_1^2,\zeta_2^2,\zeta_3,\ldots]\subseteqeq\mathfrak{m}athcal{A}(2)_*$.
This tells us that the elements $b_{2^s}$ are not decomposable in
terms of the Dyer-Lashof action, and recovers part of Kochman's
result Theorem~\ref{thm:DL-MU-QH}.
For an odd prime $p$, the $\TAQ$-Hurewicz homomorphism decomposes
into $(p-1)$ pieces corresponding to the Adams splitting of
$p$-local connective $K$-theory, giving
\[
\Sigma^2ku_{(p)} \sim \bigvee_{1\leqslant r\leqslant p-1}\Sigma^{2r}\ell.
\]
This gives an equivalent homomorphism
\[
\theta'\colonH_*(MU;\mathfrak{m}athbb{F}_p) \to
\bigoplus_{1\leqslant r\leqslant p-1}H_*(\Sigma^{2r}\ell;\mathfrak{m}athbb{F}_p).
\]
Here
\[
H_*(\ell;\mathfrak{m}athbb{F}_p) = \mathfrak{m}athbb{F}_p[\zeta_i:i\geqslant1]\circtimes\leftarrowmbda_{\mathfrak{m}athbb{F}_p}(\bar{\tau}_j:j\geqslant2)
\subseteqeq\mathfrak{m}athcal{A}(p)_*,
\]
and the component corresponding to $\Sigma^{2r}\ell$ is
determined in terms of generating functions by
\[
\sum_{i\geqslant0}b_it^i \mathfrak{m}apsto t^r\biggl(\sum_{j\geqslant0}\xi_jt^{p^j-1}\biggr)^r.
\]
It follows that $b_k\mathfrak{m}apsto0$ unless $k\equiv r\bmod{(p-1)}$.
Write $k=np^e$ with $p\nmid n$, so $n\equiv r\bmod{(p-1)}$.
Now set
\[
n-r = (p-1)(s_0+s_1p+\cdots+s_dp^d)
\]
with $0\leqslant s_i\leqslant p-1$ and $s_d\neq0$. Then we obtain
\[
b_{(s(p-1)+r)p^e} \mathfrak{m}apsto
\binom{r}{r-s_0,s_0-s_1,\ldots,s_{d-1}-s_d,s_d}
\xi_e^{r-s_0}\xi_{e+1}^{s_0-s_1} \cdots
\xi_{e+d-1}^{s_{d-2}-s_{d-1}}\xi_{e+d}^{s_{d-1}-s_d}\xi_{e+d+1}^{s_d}.
\]
Notice that this can only give a non-zero answer if the
following inequalities are satisfies:
\[
1\leqslant s_d\leqslant s_{d-2}\leqslant\cdots \leqslant s_1\leqslant s_0\leqslant r.
\]
In these cases $b_{(s(p-1)+r)p^e}$ must be Dyer-Lashof
indecomposable, and so we again recover Kochman's odd
primary result of Theorem~\ref{thm:DL-MU-QH}.
Here is another example, the reader is invited to compare
it with that of $MU_{(2)}$ and $MSp_{(2)}$ in~\cite{BGRtaq}*{proposition~5.1}.
\begin{prop}\leftarrowbel{prop:MSU2}
The $2$-local commutative $S$-algebra $MSU_{(2)}$ is not
minimal atomic.
\end{prop}
\begin{proof}
We recall that $H_*(MSU;\mathfrak{m}athbb{F}_2)$ is a polynomial algebra with
a generator in each even degree greater than~$2$. There are
many explicit generating families known, for example
see~\cites{AB:HomGenBSO+BSU,AB:HWdecomp}. In fact, $H_*(MSU;\mathfrak{m}athbb{F}_2)$
can be identified as a subalgebra of $H_*(MU;\mathfrak{m}athbb{F}_2)$, and then
there are polynomial generators $a_n\in H_{2n}(MU;\mathfrak{m}athbb{F}_2)$ so
that
\[
H_*(MSU;\mathfrak{m}athbb{F}_2) =
\mathfrak{m}athbb{F}_2[a_{2^s}^2:s\geqslant0]\circtimes\mathfrak{m}athbb{F}_2[a_{2^sk}:\text{$s\geqslant0$, $k>1$ odd}]
\subseteqeq H_*(MU;\mathfrak{m}athbb{F}_2).
\]
We will write $a'_n$ the generator in degree $2n$ where
$n\geqslant2$, for our purposes it is not important which
choice we make here.
By \cite{SOK:DLops}*{theorem~19(a)}, the Dyer-Lashof indecomposables
in $H_*(MSU;\mathfrak{m}athbb{F}_2)$ are the algebra generators appearing in degrees
of the form $2^m+2^n$ where $m,n\geqslant0$; this includes the case
$2^s=2^{s-1}+2^{s-1}$ where $s\geqslant1$.
Since there is a weak equivalence of infinite loop spaces
$BSU\sim\Omega^\infty\Sigma^4ku$, by~\cite{MB-MM:TAQ},
\[
\Omega_S(MSU) \cong MSU\wedge \Sigma^4ku.
\]
Therefore the $\TAQ$-Hurewicz homomorphism factors as
\[
\xymatrix{
\pi_*(MSU) \ar[r]\ar@/^19pt/[rrr]^{\theta}
& H_*(MSU;\mathfrak{m}athbb{F}_2)\ar[r]_(.45){\theta'}
& \TAQ_*(MSU,S;H\mathfrak{m}athbb{F}_2)\ar[r]_(.6){\cong}
& H_*(\Sigma^4ku;\mathfrak{m}athbb{F}_2)
}
\]
and in fact using the geometrically defined generators
described in~\cite{AB:HomGenBSO+BSU} it can be shown
that
\[
\im\theta' =\mathfrak{m}athbb{F}_2\{\Sigma^4\xi_m^2\xi_n^2:m,n\geqslant1\}.
\]
Here $\theta'$ has the effect
\[
a'_{2^{n+1}} \mathfrak{m}apsto \Sigma^4\xi_n^4\quad (n\geqslant0),
\qquad
a'_{2^m+2^n} \mathfrak{m}apsto \Sigma^4\xi_m^2\xi_n^2\quad (n>m\geqslant0).
\]
However, this alone does not give us the result. We would
like to use~\cite{BGRtaq}*{theorems~3.2,3.4}, so we must show
that
\[
\theta\colon\pi_n(MSU)\to\TAQ_n(MSU,S;H\mathfrak{m}athbb{F}_2)
\]
is non-trivial for some $n>0$. For this we will use work of
Pengelley~\cite{DJP:MSU} on the Adams spectral sequence for
$\pi_n(MSU_{(2)})$. In~\cite{DJP:MSU}*{theorem~2.6} it is
shown that there are polynomial generators
$y'_{8k}\in H_{8k}(MSU;\mathfrak{m}athbb{F}_2)$ for which the Adams differential
$d_2$ satisfies
\[
d_2y'_{8k} =
\begin{cases}
hq'_{s-1}\neq0 & \text{if $k=2^s$}, \\
0 & \text{if $k$ is not a power of $2$},
\end{cases}
\]
and furthermore all trivial higher differentials in the spectral
sequence are trivial. For our purposes what matters here is that
each generator $y'_{2^{m+3} + 2^{n+3}}$ where $n>m\geqslant0$ is in
the image of the classical Hurewicz homomorphism and under the
$\TAQ$-homomorphism it maps to $\Sigma^4\xi_{m+2}\xi_{n+2}\neq0$.
This means that $MSU_{(2)}$ cannot be minimal atomic.
\end{proof}
If the summand $BoP$ of $MSU_{(2)}$ were to have a commutative
$S$-algebra structure, then Pengelley's results would imply that
the $\bmod{\;2}$ $\TAQ$-Hurewicz homomorphism was trivial, hence
$BoP$ would be minimal atomic. However, this depends on the
observation that the classical $\bmod{\;2}$ Hurewicz homomorphism
is trivial so we already know it is minimal atomic as a
spectrum~\cite{AJB-JPM} and hence it would be as a commutative
$S$-algebra. So the use of $\TAQ$ would not be really necessary.
Here are some more examples.
\begin{examp}\leftarrowbel{examp:HFp}
Let $p$ be a prime and set $H=H\mathfrak{m}athbb{F}_p$, $H_*(-)=H_*(-;\mathfrak{m}athbb{F}_p)$. Then
$\TAQ$-Hurewicz homomorphism
\[
\theta'\colonH_*(H)\to \TAQ_*(H,S;H)
\]
has the following effect on
\[
H_*(H) = \mathfrak{m}athcal{A}(p)_* =
\begin{cases}
\mathfrak{m}athbb{F}_p[\xi_i:i\geqslant1]\circtimes\Lambda(\tau_j:j\geqslant0) & \text{if $p$ is odd}, \\
\mathfrak{m}athbb{F}_2[\xi_i:i\geqslant1] & \text{if $p=2$}.
\end{cases}
\]
When $p$ is odd,
\[
\theta'(\tau_0) \neq 0,
\quad
\theta'(\tau_i) = \theta'(\xi_i) = 0 \quad (i\geqslant1).
\]
When $p=2$,
\[
\theta'(\xi_1) \neq0,
\quad
\theta'(\xi_i) = 0 \quad (i\geqslant2).
\]
The vanishing results follows from Steinberger's calculations
of Dyer-Lashof operations in~\cite{LNM1176}*{chapter~III, theorem~2.3}.
The non-triviality results use the fact that the unit $S\to H$
is $0$-connected, hence by Basterra~\cite{MBtaq}*{lemma~8.2},
$\Omega_S(H)$ is $0$-connected, see also~\cite{BGRtaq}*{corollary~1.3}.
\end{examp}
Next we will consider the case of $MO$. The infinite loop
space $BO$ has Thom spectrum $MO$ which admits the structure
of an \mathfrak{m}athrm{E}infty ring spectrum or equivalently of a commutative
$S$-algebra. By Thom's theorem, this is known to split as a
wedge of suspensions of $H=H\mathfrak{m}athbb{F}_2$ even as a ring spectrum
\[
MO \sim \bigvee_\alpha\Sigma^\alpha H.
\]
But as we will see, no such splitting can happen in
$\bar{h}\mathfrak{m}athscr{C}_S$ because of obstructions lying in
$\TAQ$. Here the underlying infinite loop space is $BO$ and
the associated spectrum is~$ko\left\leftarrowngle1\right\rightarrowngle$, the $0$-connected cover
of~$ko$. In the above splitting, the generalized Eilenberg-Mac~Lane
ring spectrum on the right hand side realises the graded
polynomial ring
\begin{equation}\leftarrowbel{eqn:pi*MO}
MO_* = \pi_*(MO)
= \mathfrak{m}athbb{F}_2[z_n : \text{$n\geqslant 1$ is not of the form $2^s-1$}],
\end{equation}
where $z_n$ has degree $n$. For more on such ring spectra,
see~\cite{JMB:GEM}. Let $\underline{h}\colon\pi_n(MO) \to H_n(MO)$
denote the usual mod~$2$ homology Hurewicz homomorphism.
By Thom's theorem, $\underline{h}$ is a monomorphism and
for the polynomial generators $z_n$ of~\eqref{eqn:pi*MO},
the Hurewicz images $\underline{h}(z_n)$ form part of a
set of polynomial generators for $H_*(MO)$ which has one
generator in each positive degree.
By a result of Basterra and Mandell~\cite{MB-MM:TAQ},
\[
\Omega_S(MO) = MO \wedge ko\left\leftarrowngle1\right\rightarrowngle,
\]
where $ko\left\leftarrowngle1\right\rightarrowngle$ is the $0$-connected cover of $ko$, defined
by the cofibre sequence of $ko$-modules
\[
ko\left\leftarrowngle1\right\rightarrowngle \to ko \to H\mathbb{Z} \to \Sigma ko\left\leftarrowngle1\right\rightarrowngle.
\]
On applying~$\bmod{\;2}$ homology $H_*(-)$ we obtain a short
exact sequence
\[
0\rightarrow H_*(\Sigma^{-1} ko) \to H_*(\Sigma^{-1} H\mathbb{Z}) \to H_*(ko\left\leftarrowngle1\right\rightarrowngle) \rightarrow0
\]
from which we deduce that as an $H_*ko$-module,
\begin{equation}\leftarrowbel{eqn:H*ko<1>}
H_*(ko\left\leftarrowngle1\right\rightarrowngle) =
H_*(ko)\{\Sigma^{-1}\zeta_1^2,\Sigma^{-1}\zeta_2,\Sigma^{-1}\zeta_1^2\zeta_2\},
\end{equation}
\emph{i.e.}, the free $H_*(ko)$-module on the generators
$\Sigma^{-1}\zeta_1^2,\Sigma^{-1}\zeta_2,\Sigma^{-1}\zeta_1^2\zeta_2$
which have degrees~$1,2,4$ respectively.
We will make use of the $\TAQ$-Hurewicz homomorphism
\[
\theta\colon \pi_n(MO) \to \TAQ_n(MO,S;\mathfrak{m}athbb{F}_2) = H_n(ko\left\leftarrowngle1\right\rightarrowngle),
\]
and so we need to understand the mod~$2$ homology $H_*(ko\left\leftarrowngle1\right\rightarrowngle)$.
In the dual Steenrod algebra
\[
\mathfrak{m}athcal{A}(2)_* = H_*(H) = \mathfrak{m}athbb{F}_2[\xi_r:r\geqslant1] = \mathfrak{m}athbb{F}_2[\zeta_r:r\geqslant1],
\]
each generator $\xi_r\in\mathfrak{m}athcal{A}(2)_{2^r-1}$ is in the image
of the natural map
\[
H_{2^r}(\mathbb{R}P^\infty) \to H_{2^r}(\Sigma H) = \mathfrak{m}athcal{A}(2)_{2^r-1},
\]
and $\zeta_r=\chi(\xi_r)$, the Hopf-algebra conjugate of $\xi_r$.
Now since $\pi_1(ko\left\leftarrowngle1\right\rightarrowngle)=\mathfrak{m}athbb{F}_2$, there is a canonical non-trivial
homotopy class $\psi\colonko\left\leftarrowngle1\right\rightarrowngle\to\Sigma H$ inducing an isomorphism
on $\pi_1(-)$. The horizontal composition in the diagram
\[
\xymatrix{
H\mathbb{Z} \ar[r]\ar@{..>}@/_0.8pc/[dr]_{\text{reduction $\bmod{\;2}$\;}}
& \Sigma ko\left\leftarrowngle1\right\rightarrowngle \ar[r]^{\ph{a}\psi} & \Sigma^2H \\
& H\ar@{..>}@/_0.8pc/[ur]_{\mathrm{Sq}^2}&
}
\]
factors as shown. In order to calculate the effect of the
$H_*(ko)$-module homomorphism
\[
\psi_*\colonH_*(ko\left\leftarrowngle1\right\rightarrowngle)\to H_{*-1}(H),
\]
we first note that for $r=1,2$, the composition
\[
ko\to H \xrightarrow{\mathrm{Sq}^r} \Sigma^2H
\]
is trivial, hence it induces the trivial map on $H_*(ko)$.
Using the Cartan formula for $\mathrm{Sq}^2_*$, for any element
$w\in H_*(ko)$ we obtain
\begin{equation}\leftarrowbel{eq:H*ko<1>}
\psi_*(w\Sigma^{-1}\zeta_1^2) = w, \quad
\psi_*(w\Sigma^{-1}\zeta_2) = w\zeta_1 = w\xi_1, \quad
\psi_*(w\Sigma^{-1}\zeta_1^2\zeta_2) = w(\zeta_2+\zeta_1^3) = w\xi_2.
\end{equation}
In particular it follows that $\psi_*\colonH_*(ko\left\leftarrowngle1\right\rightarrowngle)\to H_{*-1}(H)$
is a monomorphism. We also note that the factorisation
of $\eta\colon\Sigma ko\to ko$ through a $ko$-module map
$\tilde\eta\colon\Sigma ko\to ko\left\leftarrowngle1\right\rightarrowngle$ induces
\[
\tilde\eta_*\colonH_*(\Sigma ko) \to H_*(ko\left\leftarrowngle1\right\rightarrowngle);
\quad
\tilde\eta_*(w) = w\Sigma^{-1}\zeta_1^2.
\]
\begin{prop}\leftarrowbel{prop:MO-theta}
For any choice of generators $z_n$ in~\eqref{eqn:pi*MO},
the\/ $\TAQ$-Hurewicz homomorphism
$\theta\colon\pi_*(MO)\to H_*(ko\left\leftarrowngle1\right\rightarrowngle)$ satisfies
\[
\theta(z_n) =
\begin{cases}
\ph{abc}0 & \text{\rm if $n\neq2^s$}, \\
\Sigma^{-1}\zeta_2 & \text{\rm if $n=2$}, \\
\Sigma^{-1}\zeta_1^2\zeta_2 & \text{\rm if $n=4$}, \\
\Sigma^{-1}\zeta_1^2\xi_s & \text{\rm if $n=2^s$ with $s\geqslant3$}. \\
\end{cases}
\]
Hence $MO$ is not a minimal atomic $2$-complete commutative
$S$-algebra.
\end{prop}
\begin{proof}
Choose polynomial generators $a_n\in H_n(MO)$ so that
when~$n+1$ is not a power of~$2$,
\[
\underline{h}(z_n) = a_n.
\]
Note that Kochman's results in~\cite{SOK:DLops} give the
action of the Dyer-Lashof operations on $H_*(BO)$ and the
Dyer-Lashof indecomposables are spanned by the polynomial
generators $a_{2^s}$ for $s\geqslant0$. Thus we should only
expect $\theta(z_n)$ to be non-zero when $n=2^s$ for some
$s\geqslant0$.
The calculation of $\psi_*\circ\theta$ require similar methods
to those used for $MU$ in~\cite{BGRtaq}*{section~3}. The
crucial point is the determination of the homomorphism
\[
H_*(\mathbb{R}Pi) \to H_*(BO) \to H_*(ko\leftarrowngle1\rightarrowngle)
\]
induced by the natural inclusion $\mathbb{R}Pi\to BO$ and the
evaluation
\[
\Sigma^\infty BO =
\Sigma^\infty\Omega^\infty\Sigma^\infty ko\to ko.
\]
Composing with $\psi$ and applying homology we obtain
\[
H_*(\mathbb{R}Pi) \to H_*(ko\leftarrowngle1\rightarrowngle)
\xrightarrow{\;\psi_*\;}
H_*(\Sigma H) = \mathfrak{m}athcal{A}(2)_{*-1},
\]
where $\psi_*$ is monic. Since $H^1(\mathbb{R}Pi)=\mathfrak{m}athbb{F}_2$, our
composition is the standard one which maps the generator
$\gamma_n\in H_n(\mathbb{R}Pi)$ according to the rule
\[
\gamma_n \mathfrak{m}apsto
\begin{cases}
\xi_s & \text{if $n=2^s-1$}, \\
0 &\text{otherwise}.
\end{cases}
\]
Using~\eqref{eq:H*ko<1>} we see that $\theta'(a_{2^s})$
has the form claimed.
The statement about $MO$ not being minimal atomic follows
from Thom's result since by definition, for each $s\geqslant1$,
$a_{2^s}$ is the Hurewicz image of a homotopy element.
\end{proof}
For completeness, we mention the following result which
appeared in the unpublished preprint of {K\v{r}\'{i}\v{z}}~\cite{IK:BP},
unpublished work of Basterra and Mandell, and
Lazarev~\cite{AL:Glasgow}.
\begin{thm}\leftarrowbel{thm:Kriz-HF2}
There is an isomorphism
\[
\TAQ^*(H\mathfrak{m}athbb{F}_2,S;H\mathfrak{m}athbb{F}_2) \cong
\mathfrak{m}athbb{F}_2\{\Sigma\mathrm{SQ}^I:
\text{\rm $I=(i_1,\ldots,i_t)$ admissible, $i_t\geqslant4$}\}.
\]
\end{thm}
Here the symbols $\mathrm{SQ}^I$ behave like the analogous symbols
$\mathrm{Sq}^I$ in the Steenrod algebra $\mathfrak{m}athcal{A}(2)^*$, and we
regard the empty sequence as admissible. However, the right
hand side should not be regarded as a module over the Steenrod
algebra $\mathfrak{m}athcal{A}(2)^*$, and this is merely an isomorphism
of vector spaces. Here the suspension $\Sigma(-)$ indicates
a degree shift of~$+1$. There is a duality between
$\TAQ^*(H\mathfrak{m}athbb{F}_2,S;H\mathfrak{m}athbb{F}_2)$ and $\TAQ_*(H\mathfrak{m}athbb{F}_2,S;H\mathfrak{m}athbb{F}_2)$, i.e.,
\[
\TAQ^n(H\mathfrak{m}athbb{F}_2,S;H\mathfrak{m}athbb{F}_2) \cong \mathfrak{m}athbb{H}om_{\mathfrak{m}athbb{F}_2}(\TAQ_n(H\mathfrak{m}athbb{F}_2,S;H\mathfrak{m}athbb{F}_2),\mathfrak{m}athbb{F}_2).
\]
We end with a result that is essentially a generalisation
of~\cite{Hu-Kriz-May}*{proposition~2.11}; several examples of this
kind were given in Helen Gilmour's thesis~\cite{HG:PhD}.
In the planned part~II of this work, we will describe computations
in the setting of a Miller-type spectral sequence for computing
$\TAQ$ which also lead to such results.
\begin{prop}\leftarrowbel{prop:H->MO}
There is no morphism of commutative $S$-algebras $H\mathfrak{m}athbb{F}_2\to MO$.
\end{prop}
\begin{proof}
Once again we set $H=H\mathfrak{m}athbb{F}_2$.
If such a morphism $H\to MO$ existed, the generator
$\zeta_1\in H_1H=\mathfrak{m}athcal{A}(2)_1$ would map to the
algebra generator $a_1\in H_1MO$. Using the Dyer-Lashof
action calculated by Kochman~\cite{SOK:DLops}, see
Theorem~\ref{thm:DL-MU}, we have
\[
\dlQ^4a_1 \equiv a_5 \mathfrak{m}od{\text{decomposables}}.
\]
As in the proof of~\cite{Hu-Kriz-May}*{proposition~2.11},
this leads to a contradiction since there is no degree $5$
indecomposable in $\mathfrak{m}athcal{A}(2)_*$.
\end{proof}
We will not give the details here, but it seems worth mentioning
that the Thom spectrum $MU/O$ associated to the infinite loop
space $U/O$ which is the fibre in the sequence
\[
U/O \to BO \to BU,
\]
is a core for $MO$. It turns out that $H_*(MU/O;\mathfrak{m}athbb{F}_2)$ embeds
into $H_*(MO;\mathfrak{m}athbb{F}_2)$ as a polynomial subalgebra on odd degree
generators the only Dyer-Lashof indecomposable has degree~$1$.
In fact
\[
\Omega_S(MU/O) \cong MU/O\wedge \Sigma ko
\]
and so
\[
\TAQ_*(MU/O,S;H\mathfrak{m}athbb{F}_2) = H_*(\Sigma ko;\mathfrak{m}athbb{F}_2),
\]
and under the $\TAQ$-Hurewicz homomorphism the Dyer-Lashof
indecomposable generator is sent to $\Sigma1$.
\section{\mathfrak{m}athrm{E}infty orientations for complex line bundles}
\leftarrowbel{sec:UnivCplxOrient}
We end with a discussion involving the suspension spectrum
$\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi=\Sigma^{-2}\Sigma^{\infty}\mathfrak{m}athbb{C}Pi$
discussed in~\cite{AB-BR:Mxi}. A complex orientation
for a ring spectrum $E$ is the homotopy class of a map
$\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to E$ whose restriction
\[
S^0 \sim \Sigma^{-2}\Sigma^{\infty}\mathfrak{m}athbb{C}P^1
= \Sigma^{-2}\Sigma^{\infty}S^2\to E
\]
is homotopic to the unit of~$E$. When $E$ is a commutative
$S$-algebra, such a map $\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to E$ induces
a unique morphism of commutative $S$-algebras
\[
\mathfrak{m}athbb{P}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi \to E.
\]
Because of the condition involving the bottom cell, there
is a commutative diagram of solid arrows
\[
\xymatrix{
\mathfrak{m}athbb{P}S^0
\ar@{}[dr]|{\text{\pigpenfont R}}
\ar[r]\ar@{ >->}[d]
& \mathfrak{m}athbb{P}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\ar@{ >->}[d]\ar@/^21pt/[ddr] & \\
\mathfrak{m}athbb{P}D^1\ar[r]\ar@/_21pt/[drr]
& \tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\ar@{.>}[dr] & \\
& & E
}
\]
\noindent
and hence a unique dotted arrow making the whole diagram
commute. This shows that $\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi$
is universal for maps to $E$ which give complex orientations
for complex line bundles. Of course the inclusion map
$\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi$
itself provides a complex orientation.
\begin{lem}\leftarrowbel{lem:UnivCplxOrient}
The universal complex orientation
\[
\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi = \Sigma^{\infty-2}MU(1) \to MU
\]
induces a rational equivalence of commutative $S$-algebras
$\sigma\colon\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to MU$.
Furthermore, the inclusion
$\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi$
induces a morphism of ring spectra
\[
MU\to\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi
\]
which provides a splitting of $\sigma$ in the homotopy
category $\bar{h}\mathfrak{m}athscr{M}_{MU}$.
\end{lem}
\begin{proof}
The rational result is straightforward since an argument
using the K\"unneth spectral sequence gives
\[
H_*(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi;\mathbb{Q})
= \mathbb{Q}[\tilde{\beta}_r : r\geqslant1],
\]
where $\tilde{\beta}_r$ is the image of the canonical
generator $\beta_r\in H_{2r}(\mathfrak{m}athbb{C}Pi)$. Then the morphism
$\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to MU$ induces
an isomorphism of rings
\[
H_*(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi;\mathbb{Q})
\to H_*(MU;\mathbb{Q}).
\]
It is easy to see that
\[
H_*(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi;\mathbb{Z})
\to H_*(MU;\mathbb{Z})
\]
is epic.
The composition
\[
\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to MU
\to\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi
\xrightarrow{\;\sigma\;} MU
\]
is homotopic to the canonical orientation, so the composition
\[
MU\to\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi
\xrightarrow{\;\sigma\;} MU
\]
is homotopic to the identity by the classical universality of
the commutative ring spectrum~$MU$ described by
Adams~\cite{JFA:Blue}.
\end{proof}
We remark that
$\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi$ is weakly equivalent
to the Thom spectrum associated with the infinite loop map
$\Omega^{\infty}\Sigma^{\infty}\mathfrak{m}athbb{C}Pi\to BU$ extending the natural
map $\mathfrak{m}athbb{C}Pi\to BU$ whose fibre $F$ has torsion homotopy groups;
in fact, a result of Graeme Segal shows that there is an
equivalence of spaces
\[
\Omega^{\infty}\Sigma^{\infty}\mathfrak{m}athbb{C}Pi \sim BU\times F.
\]
The morphism $\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi\to MU$
can be converted into a fibration (in either of the two model
categories $\mathfrak{m}athscr{M}_S$ or $\mathfrak{m}athscr{C}_S$), giving a
commutative diagram
\[
\xymatrix{
\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi \ar[dr]\ar@{ >->}[rr]^{\sim}
& &
(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)'\ar@{->>}[dl] \\
& MU &
}
\]
where $(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)'$ is cofibrant
in $\mathfrak{m}athscr{C}_S$. The map
$(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)'\to MU$ is a morphism
in the subcategory
$\mathfrak{m}athscr{M}_{(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)'}$
of $\mathfrak{m}athscr{M}_S$.
\begin{cor}\leftarrowbel{cor:UnivCplxOrient}
The fibre of\/ $(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)'\to MU$
is rationally trivial.
\end{cor}
A version of the next result appears in \cite{AB-BR:Mxi}.
\begin{prop}\leftarrowbel{prop:MU->PCP}
For a prime $p$, there can be no morphism of commutative
$S_{(p)}$-algebras
$\theta\colonMU\to(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)_{(p)}$
for which $\sigma\circ\theta$ is a weak equivalence. Hence
there can be no morphism of commutative $S$-algebras
$\theta\colonMU\to\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi$ for
which $\sigma\circ\theta$ is a weak equivalence.
\end{prop}
\begin{proof}
It suffices to prove the result for a prime~$p$, and we will
assume all spectra are localised at~$p$. Assume such a morphism
$\theta$ existed. Then by naturality of $\Omega_S$, there are
(derived) morphisms of $MU$-modules and a commutative diagram
\[
\xymatrix{
\Omega_{S}(MU) \ar[r]_(.4){\theta} \ar@/^19pt/[rr]^{\sim}
& \Omega_{S}(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi)\ar[r]_(.6){\sigma}
& \Omega_{S}(MU)
}
\]
which induces a commutative diagram in $\TAQ_*(-;H\mathfrak{m}athbb{F}_p)$
\[
\xymatrix{
H_*(\Sigma^2ku;\mathfrak{m}athbb{F}_p) \ar[r]_(.46){\theta_*} \ar@/^19pt/[rr]^{\cong}
& H_*(\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_2;\mathfrak{m}athbb{F}_p)\ar[r]_(.56){\sigma_*}
& H_*(\Sigma^2ku;\mathfrak{m}athbb{F}_p)
}
\]
where $\mathfrak{m}athbb{C}Pi_2 = \mathfrak{m}athbb{C}Pi/\mathfrak{m}athbb{C}P^1$, and we use a result
due to Basterra \& Mandell~\cite{MB-MM:TAQ} (for
further details see~\cite{BGRtaq}*{sections~4~\&~5})
to identify
$\TAQ_*(MU,S;H\mathfrak{m}athbb{F}_p)$, namely
\[
\Omega_S(MU) \sim MU\wedge\Sigma^2ku.
\]
It is standard that
\[
H_n(\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_2;\mathfrak{m}athbb{F}_p) =
\begin{cases}
\mathfrak{m}athbb{F}_p & \text{if $n\geqslant2$ and is even}, \\
\;0 & \text{otherwise}.
\end{cases}
\]
On the other hand, when $p=2$,
\[
H_*(ku;\mathfrak{m}athbb{F}_2) = \mathfrak{m}athbb{F}_2[\zeta_1^2,\zeta_2^2,\zeta_3,\zeta_4,\ldots]
\subseteqeq\mathfrak{m}athcal{A}(2)_*
\]
with $|\zeta_s|=2^s-1$, while when $p$ is odd,
$\Sigma^2ku\sim \bigvee_{1\leqslant r\leqslant p-1}\Sigma^{2r}\ell$
with
\[
H_*(\ell;\mathfrak{m}athbb{F}_2) =
\mathfrak{m}athbb{F}_p[\zeta_1,\zeta_2,\zeta_3,\ldots]\circtimes\Lambda(\bar{\tau}_r:r\geqslant2)
\]
where $|\zeta_s|=2p^s-2$ and $|\bar{\tau}_s|=2p^s-1$. Clearly
this means that no such $\theta$ can exist.
\end{proof}
At the prime~$2$, $\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi$ is known to be minimal
atomic~\cite{AJB-JPM}*{proposition~5.9}. The next result shows
that the functor $\tilde{\mathfrak{m}athbb{P}}$ need not preserve this
property; see Proposition~\ref{prop:MinAtom} for a converse
to this.
\begin{prop}\leftarrowbel{prop:P-minatom}
The $2$-local commutative $S$-algebra
$\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)}$ is not minimal
atomic.
\end{prop}
\begin{proof}
If $\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)}$ were minimal
atomic then by~\cite{BGRtaq}*{theorem~3.3}, the TAQ Hurewicz
homomorphism (induced from the universal derivation)
\[
\theta\colon\pi_n(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)})
\to \TAQ_n(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)},S;H\mathfrak{m}athbb{F}_2))
\]
would be trivial for $n>0$.
By naturality, there is a commutative diagram
\[
\xymatrix{
\pi_*(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)})
\ar@{->>}[rr]^{\sigma_*}\ar[d]_{\theta} && \pi_*(MU_{(2)})\ar[d]^{\theta} \\
\TAQ_n(\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)},S;H\mathfrak{m}athbb{F}_2)
\ar[rr]^{\sigma_*}\ar[d]_{\cong} & & \TAQ_n(MU_{(2)},S;H\mathfrak{m}athbb{F}_2))\ar[d]^{\cong} \\
H_*(\mathfrak{m}athbb{C}Pi_2;\mathfrak{m}athbb{F}_2)\ar[rr]^{\sigma_*} & & H_*(\Sigma^2ku;\mathfrak{m}athbb{F}_2)
}
\]
in which the surjectivity of the top row follows from
Lemma~\ref{lem:UnivCplxOrient}. The $2$-primary calculations
of~\cite{BGRtaq}*{section~5} show that the right hand Hurewicz
homomorphism~$\theta$ is non-zero in positive degrees, hence
so is the left hand one. Therefore
$\tilde{\mathfrak{m}athbb{P}}\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_{(2)}$ cannot be
minimal atomic.
\end{proof}
We leave the interested reader to formulate and verify analogues
for an odd prime~$p$ based on desuspensions of the $p$-local
summands of $\Sigma^\infty\mathfrak{m}athbb{C}Pi_{(p)}$.
\begin{rem}\leftarrowbel{rem:TAQ-comparison}
We point out that $\sigma_*\colon H_*(\mathfrak{m}athbb{C}Pi_2;\mathfrak{m}athbb{F}_2)\to H_*(\Sigma^2ku;\mathfrak{m}athbb{F}_2)$
is different from the homomorphism induced by any map of spectra
$\Sigma^{\infty-2}\mathfrak{m}athbb{C}Pi_2\to\Sigma^2ku$ which is an equivalence
on the bottom cell. For such a map composed with the natural map
$\Sigma^2ku\to H\mathfrak{m}athbb{F}_2$ induces the homomorphism in homology given
by
\[
\Sigma^{-2}\beta_n \mathfrak{m}apsto
\begin{cases}
\xi_s^4 &\text{if $n=2^s$}, \\
0 &\text{otherwise}.
\end{cases}
\]
But also $\Sigma^{-2}\beta_n \mathfrak{m}apsto b_{n-1}$ in $H_*(MU;\mathfrak{m}athbb{F}_2)$
and under the $\TAQ$-Hurewicz homomorphism,
\[
b_{n-1} \mathfrak{m}apsto
\begin{cases}
\xi_s^2 &\text{if $n=2^s+1$}, \\
0 &\text{otherwise}.
\end{cases}
\]
\end{rem}
As promised above, here is a positive result relating the
additive and multiplicative notions of minimal atomic. We
regard $p$-local spectra as equivalent to $S$-modules.
\begin{prop}\leftarrowbel{prop:MinAtom}
Let $p$ be a prime and let $S$ be the $p$-local sphere
spectrum. Suppose that $X$ is a connective Hurewicz
$S$-module with chosen bottom cell $S^0\to X$. If\/
$\tilde{\mathfrak{m}athbb{P}}X$ is minimal atomic as a commutative
$S$-algebra, then $X$ is minimal atomic as an $S$-module.
\end{prop}
\begin{proof}
Working in $S^0/\mathfrak{m}athscr{M}_S$ we can replace $X$ by a
CW spectrum which is weakly equivalent to it so we will
assume that this has been done. Using observations in
Remark~\ref{rem:S^0/pushouts}, we can relate the two
$n$-skeleta. The $(n+1)$-skeleton $X^{[n+1]}$ is
constructed using a map of $S$-modules
\[
i^n\colon\bigvee_iS^n \to X^{[n]}
\]
for which
\[
\Bbbker[i^n_*\colon\pi_n(\bigvee_iS^n) \to \pi_n(X)^{\left\leftarrownglen\right\rightarrowngle}]
\subseteqeq p\,\pi_n(\bigvee_iS^n).
\]
Similarly, we form the $(n+1)$-skeleton
$\tilde{\mathfrak{m}athbb{P}}^{\left\leftarrownglen+1\right\rightarrowngle}X$ is constructed from
$\tilde{\mathfrak{m}athbb{P}}^{\left\leftarrownglen\right\rightarrowngle}X$ using a morphism of
$S$-modules
\[
j^n\colon\bigvee_iS^n \to \tilde{\mathfrak{m}athbb{P}}^{\left\leftarrownglen\right\rightarrowngle}X
\]
for which
\[
\Bbbker[j^n_*\colon\pi_n(\bigvee_iS^n) \to \pi_n(\tilde{\mathfrak{m}athbb{P}}^{\left\leftarrownglen\right\rightarrowngle}X)]
\subseteqeq p\,\pi_n(\bigvee_iS^n).
\]
In $S^0/\mathfrak{m}athscr{M}_S$ there is a commutative diagram
\[
\xymatrix{
& \ar[dl]_{i^n}\bigvee_iS^n\ar[dr]^{j^n} & \\
X^{[n]}\ar[rr]^{\mathfrak{m}athrm{incl}} & & \tilde{\mathfrak{m}athbb{P}}^{\left\leftarrownglen\right\rightarrowngle}X
}
\]
in which~$i^n$ provides the attaching maps for the $(n+1)$-cells
of~$X$. Clearly
\[
\Bbbker[i^n_*\colon\pi_n(\bigvee_iS^n) \to \pi_n(X^{[n]})]
\subseteqeq
\Bbbker[j^n_*\colon\pi_n(\bigvee_iS^n) \to \pi_n(\tilde{\mathfrak{m}athbb{P}}^{\left\leftarrownglen\right\rightarrowngle}X)]
\subseteqeq p\,\pi_n(\bigvee_iS^n),
\]
and it follows that $X$ is nuclear.
\end{proof}
\appendix
\section{A proof and a Lemma}\leftarrowbel{sec:Missingpf}
For completeness we outline a proof of~\cite{BGRtaq}*{proposition~1.6},
due to Philipp Reinhard; unfortunately this was only produced
after that paper was published. Our approach is similar to that
of McCarthy and Minasian in~\cite{RM&VM}*{theorem~6.1}, however
this appears to be incorrect as stated (at one stage they seem
to assume that~$M$ is an algebra).
\begin{prop}\leftarrowbel{prop:TAQ-prop1.6}
Let $R$ be a commutative $S$-algebra and let $X$ be a cofibrant
$R$-module. Then there is a weak equivalence of
$\mathfrak{m}athbb{P}_RX$-modules
\[
\Omega_R(\mathfrak{m}athbb{P}_R X) \sim \mathfrak{m}athbb{P}_R X \wedge_R X.
\]
\end{prop}
\begin{proof}
For every $M\in\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_R X}$ there is an adjunction
\[
\mathfrak{m}athscr{C}_R / \mathfrak{m}athbb{P}_R X (\mathfrak{m}athbb{P}_R X,\mathfrak{m}athbb{P}_R X\vee M)
\cong \mathfrak{m}athscr{M}_R / \mathfrak{m}athbb{P}_R X(X,\mathfrak{m}athbb{P}_R X\vee M),
\]
where $\mathfrak{m}athscr{M}_R / \mathfrak{m}athbb{P}_R X$ denotes the category of
$R$-modules over $\mathfrak{m}athbb{P}_R X$. Because the forgetful functor
$\mathfrak{m}athscr{C}_R / \mathfrak{m}athbb{P}_R X \to\mathfrak{m}athscr{M}_R/\mathfrak{m}athbb{P}_R X$
respects fibrations and acyclic fibrations, the adjunction passes
to homotopy categories, giving
\[
\bar{h}\mathfrak{m}athscr{C}_R/\mathfrak{m}athbb{P}_R X(\mathfrak{m}athbb{P}_R X,\mathfrak{m}athbb{P}_R X\vee M)
\cong \bar{h}\mathfrak{m}athscr{M}_R /\mathfrak{m}athbb{P}_R X(X,\mathfrak{m}athbb{P}_R X\vee M).
\]
Now we have
\[
\mathfrak{m}athscr{M}_R /\mathfrak{m}athbb{P}_R X (\mathfrak{m}athbb{P}_R X, M)
\cong \mathfrak{m}athscr{M}_R / X(X,X\vee M)
\]
and the adjunction again passes to homotopy categories
and gives
\[
\bar{h}\mathfrak{m}athscr{M}_R /\mathfrak{m}athbb{P}_R X (\mathfrak{m}athbb{P}_R X, M)
\cong \bar{h}\mathfrak{m}athscr{M}_R / X(X,X\vee M).
\]
Since in the homotopy category $X \vee M$ is the product
of~$X$ and~$M$, we have
\[
\bar{h}\mathfrak{m}athscr{M}_R / X(X,X\vee M)\cong\bar{h}\mathfrak{m}athscr{M}_R(X,M).
\]
By using the free functor from $R$-modules to $\mathfrak{m}athbb{P}_R X$-modules,
we obtain
\[
\bar{h}\mathfrak{m}athscr{M}_R / X(X,X\vee M) \cong
\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_R X}(\mathfrak{m}athbb{P}_R X \wedge_R X, M).
\]
Thus we have shown that
\begin{align*}
\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_R X}(\Omega_R(\mathfrak{m}athbb{P}_R X), M)
&\cong \bar{h}\mathfrak{m}athscr{C}_R / \mathfrak{m}athbb{P}_R X(\mathfrak{m}athbb{P}_R X,\mathfrak{m}athbb{P}_R X \vee M) \\
&\cong \bar{h} \mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_R X}(\mathfrak{m}athbb{P}_R X \wedge_R X, M).
\end{align*}
Using Yoneda's lemma, we obtain the desired equivalence
\[
\Omega_R(\mathfrak{m}athbb{P}_R X)\sim \mathfrak{m}athbb{P}_R X \wedge_R X.
\qedhere
\]
\end{proof}
We also give a useful result on the adjunction for a
commutative $R$-algebra. Let~$A$ be a cofibrant
commutative $R$-algebra and let
\[
\xymatrix{
A^c\ar@{->>}[r]^{\;\sim\;} & A
}
\]
be its functorial cofibrant replacement in the model
category of $R$-modules $\mathfrak{m}athscr{M}_R$. Let
\[
\tilde{\mathfrak{m}u}\colon\mathfrak{m}athbb{P}_RA^c\to\mathfrak{m}athbb{P}_RA\to A
\]
be the extension of the multiplication. We have
\[
\Omega_R(\mathfrak{m}athbb{P}_RA^c) \cong \mathfrak{m}athbb{P}_RA^c \wedge_R A^c,
\]
and also the $A$-module $\Omega_R(A)$ becomes a
$\mathfrak{m}athbb{P}_RA^c$-module via pullback along $\tilde{\mathfrak{m}u}$.
Writing $\partialta$ (without decorations) for universal
derivations, we obtain a commutative diagram in
$\bar{h}\mathfrak{m}athscr{M}_{R}$ (with the pentagon commuting
in $\bar{h}\mathfrak{m}athscr{M}_{\mathfrak{m}athbb{P}_RA^c}$).
\begin{equation}\leftarrowbel{eq:Diffadjunct}
\xymatrix{
&R\wedge_R A^c\ar[rd]^{\mathfrak{m}athrm{unit}}&& \\
A^c\ar[r]\ar[rdd]_{\sim}\ar@{<->}[ur]^{\cong} & \mathfrak{m}athbb{P}_RA^c\ar[dd]^{\tilde{\mathfrak{m}u}}\ar[r]^(.4){\partialta}
& \mathfrak{m}athbb{P}_RA^c\wedge_R A^c\ar[dr]^{\tilde{\mathfrak{m}u}\wedge\partialta}\ar[dd]^{\circmega(\tilde{\mathfrak{m}u})} & \\
&& & A\wedge_R\Omega_R(A)\ar[ld]^{\mathfrak{m}athrm{mult}} \\
& A\ar[r]^(.4){\partialta} & \Omega_R(A) &
}
\end{equation}
Here $\circmega(\tilde{\mathfrak{m}u})$ denotes the induced `derivative'
morphism $\Omega_R(\mathfrak{m}athbb{P}_RA^c)\to\Omega_R(A)$.
\begin{lem}\leftarrowbel{lem:Diffadjunct}
Suppose that $M$ is an $A$-module and therefore an
$\mathfrak{m}athbb{P}_RA^c$-module. Then the induced morphism on
$\TAQ_*(-)$, $\tilde{\mathfrak{m}u}_*$, is given by the following
commutative diagram.
\[
\xymatrix{
\TAQ_*(\mathfrak{m}athbb{P}_RA^c,R;M)\ar@{=}[d] \ar[rr]^{\tilde{\mathfrak{m}u}_*}
&& \TAQ_*(A,R;M)\ar@{=}[d] \\
\pi_*(M\wedge_R A^c)\ar[dr]_{(I\wedge\partialta_{(A,R)})_*}
&& \pi_*(M\wedge_A\Omega_R(A)) \\
& \pi_*(M\wedge_R\Omega_R(A))\ar[ur] &
}
\]
\end{lem}
\begin{proof}
This is obtained by applying $\pi_*(M\wedge_R-)$ to~\eqref{eq:Diffadjunct}.
\end{proof}
Since the universal derivation restricts trivially to~$R$,
there is an induced map
\[
\bar{\partialta}_{(A,R)}\colonA^c/S_R^0\to\Omega_R(A).
\]
So for the reduced free algebra there is a similar commutative
diagram.
\[
\xymatrix{
\TAQ_*(\tilde{\mathfrak{m}athbb{P}}_RA^c,R;M)\ar@{=}[d] \ar[rr]^{\tilde{\mathfrak{m}u}_*}
&& \TAQ_*(A,R;M)\ar@{=}[d] \\
\pi_*(M\wedge_R A^c/S_R^0)\ar[dr]_{(I\wedge\bar{\partialta}_{(A,R)})_*}
&& \pi_*(M\wedge_A\Omega_R(A)) \\
& \pi_*(M\wedge_R\Omega_R(A))\ar[ur] &
}
\]
\section{Some formulae}\leftarrowbel{sec:Formulae}
We begin by recalling formula due to Kochman~\cite{SOK:DLops}.
Actually his results are for the infinite loop spaces such
as $BU$, but the Thom isomorphism commutes with the Dyer-Lashof
operations so we will interpret them in the homology of the
Thom spectrum $MU$ with its \mathfrak{m}athrm{E}infty structure inherited
from that of $BU$.
Let $p$ be a prime. We will write $H_*(-)=H_*(-;\mathfrak{m}athbb{F}_p)$. Let
$b_r\in H_{2r}(MU)$ be the generator obtained as the image
of $\beta_{r+1}\in H_{2r+2}(MU(1))\cong H_{2r+2}(\mathfrak{m}athbb{C}Pi)$
under the homomorphism induced by the canonical map
$MU(1)\to \Sigma^2MU$ as in~\cite{JFA:Blue}. We will use
the notation $x\approx y$ as shorthand for
$x\equiv y\bmod{\text{decomposables}}$. We also interchangeably
use the notations
\[
(a,b) = (b,a) = \binom{a+b}{a} = \binom{a+b}{b}
\]
for binomial coefficients, where this is taken to be zero
if $a<0$ or $b<0$. We will use the well-known congruence
\begin{equation}\leftarrowbel{eq:Binom-modp}
\binom{n_0+n_1p+\cdots +n_kp^k}{m_0+m_1p+\cdots +m_kp^k}
\equiv
\binom{n_0}{m_0}\binom{n_0}{m_0}\cdots \binom{n_k}{m_k}
\mathfrak{m}od{p}
\end{equation}
when $0\leqslant m_i,n_i\leqslant p-1$.
\begin{thm}\leftarrowbel{thm:DL-MU}
In $H_*(MU)$ we have
\begin{itemize}
\item
if $p$ is odd,
\[
\dlQ^rb_n \approx (-1)^{r+n+1}(n,r-n-1)b_{n+r(p-1)},
\]
\item
if $p=2$,
\[
\dlQ^{2r}b_n \approx (n,r-n-1)b_{n+r}.
\]
\end{itemize}
\end{thm}
Note that in the $p=2$ case there are analogous results
for $H_*(MO;\mathfrak{m}athbb{F}_2)$.
The Dyer-Lashof operations annihilate $1$ and the Cartan
formula implies that they act on the indecomposable
quotient. In~\cite{SOK:DLops}*{theorem~10}, Kochman
determined the indecomposable generators which are not
in the image of any Dyer-Lashof operations of positive
degree. We set
\[
\mathfrak{m}athrm{Q}_{\mathfrak{m}athrm{DL}}H_*(MU) =
\mathfrak{m}athrm{Q}H_*(MU)/\{\dlQ^sx:s\geqslant1,\;x\in\mathfrak{m}athrm{Q}H_*(MU)\}.
\]
\begin{thm}\leftarrowbel{thm:DL-MU-QH}
The indecomposables $\mathfrak{m}athrm{Q}_{\mathfrak{m}athrm{DL}}H_*(MU)$
have the following elements as a basis:
\begin{itemize}
\item
if $p$ is odd, $b_{np^t}$ where $p\nmid n$ and
$n=(\sum_{i=0}^ks_ip^i)(p-1)+r$ with $r=1,2,\ldots,p-1$
and if $\sum_{i=0}^ks_ip^i\neq0$, $0\leqslant s_i\leqslant(p-1)$
and $1\leqslant s_k\leqslant s_{k-1}\leqslant\cdots\leqslant s_0\leqslant r$,
\item
if $p=2$, $b_{2^t}$ where $t\geqslant0$.
\end{itemize}
\end{thm}
The Dyer-Lashof indecomposability of the stated generators
can be deduced from our results on the $\TAQ$-Hurewicz
homomorphism. As an exercise in computing with binomial
coefficients modulo a prime, we have
\begin{prop}\leftarrowbel{prop:DL-decomp}
Suppose that $p$ is a prime and $n$ has $p$-adic expansion
\[
n = n_sp^s+\cdots n_{s+t}p^{s+t}
\]
where $n_s\neq0\neq n_{s+t}$ and $t>0$. \\
If $p$ is odd then
\[
\dlQ^{n-n_sp^s} b_{n_sp^s} \approx \pm\binom{n-n_sp^s-1}{n_sp^s} b_n
\not\approx 0.
\]
If $p=2$ then
\[
\dlQ^{2n-2^{s+1}} b_{2^s} \approx \binom{n-2^s-1}{2^s} b_n
\not\approx 0.
\]
\end{prop}
\begin{proof}
In each case working modulo~$p$ we have
\begin{align*}
\binom{n-n_sp^s-1}{n_sp^s} &\equiv \binom{(n-n_sp^s-p^k) + (p^k-1)}{n_sp^s} \\
&\equiv \binom{n-n_sp^s-p^k}{0}\binom{p^k-1}{n_sp^s} \\
&\not\equiv 0,
\end{align*}
where $p^k$ is the highest power of $p$ dividing $(n-n_sp^s)$,
and we use the fact that
\[
p^k-1 = (p-1)p^{k-1} +\cdots+(p-1)p^s+\cdots(p-1)p+(p-1)
\]
with $n_s\leqslant p-1$.
\end{proof}
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volume={91},
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S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees,
L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou
and S. Waner},
date={1996},
}
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author={McCarthy, R.},
author={Minasian, V.},
title={HKR theorem for smooth $S$-algebras},
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volume={185},
date={2003},
pages={239\ndash258},
}
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author={Miller, H. R.},
title={A spectral sequence for the homology
of an infinite delooping},
journal={Pacific J. Math.},
volume={79},
date={1978},
number={1},
pages={139\ndash155},
}
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author={Pengelley, D. J.},
title={The homotopy type of $M\mathfrak{m}athrm{SU}$},
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volume={104},
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pages={1101\ndash1123},
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\end{bibdiv}
\end{document} | math | 108,072 |
\begin{document}
\title{An extension of a Theorem of V. \v{S}ver\'ak to variable exponent spaces}
\author[C. Baroncini, J. Fern\'andez Bonder]{Carla Baroncini and Juli\'an Fern\'andez Bonder}
\address{IMAS - CONICET and Departamento de Matem\'atica, FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pabell\'on I (1428) Buenos Aires, Argentina.}
\email[J. Fernandez Bonder]{[email protected]}
\urladdr[J. Fernandez Bonder]{http://mate.dm.uba.ar/~jfbonder}
\email[C. Baroncini]{[email protected]}
\subjclass[2010]{49Q10,49J45}
\keywords{Shape optimization, sensitivity analysis, nonstandard growth}
\begin{abstract}
In 1993, V. \v{S}ver\'ak proved that if a sequence of uniformly bounded domains $\Omega_n\subset {\mathbb {R}}^2$ such that $\Omega_n\to \Omega$ in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source $f\in L^2({\mathbb {R}}^2)$ converges to the solution of the limit domain with same source. In this paper, we extend \v{S}ver\'ak result to variable exponent spaces.
\end{abstract}
\maketitle
\section{Introduction}
One important problem in partial differential equations is the stability of solutions with respect to perturbations on the domain. This problem has fundamental applications in numerical computations of the solutions and is also fundamental in optimal shape design problems. See \cite{Allaire, Henrot, Pironneau} and references therein.
The famous example of Cioranescu and Murat \cite{C-M} shows that this problem presents severe difficulties when treated in full generality. In fact, in \cite{C-M} the authors take $D=[0,1]\times [0,1]\subset {\mathbb {R}}^2$ and define the domains $\Omega_n = D\setminus \cup_{i,j=1}^{n-1} B_{r_n}(x_{i,j}^n)$ where the centers of the balls $x_{i,j}^n = (i/n, j/n)$, $1\le i,j\le n-1$ and the radius $r_n = n^{-2}$. Then these domains $\Omega_n$ converge to the empty set in the Hausdorff complementary topology, but if $u_n\in H^1_0(\Omega_n)$ is the solution to
$$
\begin{cases}
-{\mathcal{D}}elta u_n = f & \text{in }\Omega_n,\\
u_n=0 & \text{on }\partial\Omega_n,
\end{cases}
$$
then $u_n\rightharpoonup u^*$ weakly in $H^1_0(D)$ to the solution of
$$
\begin{cases}
-{\mathcal{D}}elta u^* + \frac{2}{\pi} u^* = f & \text{in }D,\\
u^*=0 & \text{on }\partial D.
\end{cases}
$$
This example can be generalized to other space dimensions, to different bounded sets $D$ and also to different types of {\em holes}. See the original work \cite{C-M} and also \cite{Tartar}.
There are some simple cases where the continuity can be granted. For instance, if $\Omega$ is convex and $\{\Omega_n\}_{n\in{\mathbb {N}}}$ is an increasing sequence of convex polygons such that $\Omega = \cup_{n\in {\mathbb {N}}}\Omega_n$, then the solutions of the approximating domains $\Omega_n$ converges to the one of $\Omega$. This fact can be traced back to the late 50's and the beginning of the 60's, see \cite{Babuska, Hong57, Hong58, Hong59}. Then, this result can be generalized in terms of the capacity of the symmetric differences of $\Omega$ and $\Omega_n$. See the book of Henrot, \cite{Henrot}.
In practical applications, when one does not have control on the sequence of approximating domains, this hypothesis is uncheckable, so a different condition is needed. \v{S}ver\'ak in \cite{Sverak} gave such a condition. In fact, given a bounded domain $D\subset {\mathbb {R}}^2$ and a sequence of domains $\Omega_n\subset D$ such that $\Omega_n\to \Omega$ in the sense of the Hausdorff complementary topology the condition that guaranty the convergence of the solutions in $\Omega_n$ to the one in $\Omega$ is that the number of connected components of $D\setminus\Omega_n$ be bounded. c.f. with the example of Cioranescu-Murat.
The reason why \v{S}ver\'ak's result holds in dimension 2 is because the capacity of curves in dimension 2 is positive, while in higher dimension curves have zero capacity.
\v{S}ver\'ak's result was later generalized to nonlinear elliptic equations of $p-$Laplace type. In fact, in \cite{Bucur}, the authors prove the continuity of the solutions of
$$
\begin{cases}
-{\mathcal{D}}elta_p u_n = f & \text{in }\Omega_n\subset {\mathbb {R}}^N, \\
u_n = 0 & \text{on }\partial\Omega_n,
\end{cases}
$$
when the domains $\Omega_n$ converges to $\Omega$ in the Hausdorff complementary topology under the assumption that the number of connected components of its complements remains bounded. The idea of the proof is similar to the original one of \v{S}ver\'ak and so they end up with the restriction $p>N-1$ that is needed for the curves to have positive $p-$capacity.
Recall that ${\mathcal{D}}elta_p u = \operatorname {\text{div}}(|\nabla u|^{p-2} \nabla u)$ is the so-called $p-$laplace operator.
In recent years a lot of attention have been put in nonlinear elliptic equations with nonstandard growth. One of the most representative of such equations is the so-called $p(x)-$laplacian, that is defined as ${\mathcal{D}}elta_{p(x)} u = \operatorname {\text{div}}(|\nabla u|^{p(x)-2}\nabla u)$. This operator became very popular due to many new interesting applications, for instance in the mathematical modeling of electrorheological fluids (see \cite{Ru}) and also in image processing (see \cite{CLR}). Here, the exponent $p(x)$ is assumed to be measurable and bounded away from 1 and infinity.
So, the purpose of this paper is the extension of the result of \v{S}ver\'ak (and also the results of \cite{Bucur}) to the variable exponent setting.
\subsection*{Organization of the paper} The rest of the paper is organized as follows. In section 2 we collect some preliminaries on variable exponent spaces that are needed in this paper. The standard reference for this is the book \cite{Diening}. Some results are slight variations of the ones found in \cite{Diening} and in these cases we present full proofs of those facts (c.f. Theorem \ref{teocaracterizacion}).
In section 3, we study the Dirichlet problem for the $p(x)-$laplacian, the main result being the continuity of the solution with respect to the source. Although some of the results are well known, we decided to present the proofs of all of the results since we were unable to find a reference for these.
In section 4 we analyze the dependence of the solution of the Dirichlet problem for the $p(x)-$laplacian with respect to variations on the domain. Our two main theorems here are Theorem \ref{teo2dirichlet} where a capacity condition on the sequence of approximating domains is given in order for the continuity of solutions to hold, and Theorem \ref{teoindep} where it is shown that the continuity only depends on the approximating domains and not on the source term.
In section 5 after giving some capacity estimates that are needed in the remaining of the paper, collect all of our results and prove the main result of the paper, namely the extension of \v{S}ver\'ak's result to the variable exponent setting, i.e. Theorem \ref{teosverak}.
\section{Preliminaries}
\subsection{Definitions and well-known results}
Given $\Omega\subset {\mathbb {R}}^N$ a bounded open set, we consider the class of exponents ${\mathcal{P}}(\Omega)$ given by
$$
{\mathcal{P}}(\Omega) := \{p\colon \Omega\to [1,\infty)\colon p \text{ is measurable and bounded}\}.
$$
The variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is defined by
$$
L^{p(x)}(\Omega):= \Big\{f\in L^1_{\text{loc}}(\Omega)\colon \rho_{p(x)}(f)<\infty\Big\},
$$
where the modular $\rho_{p(x)}$ is given by
$$
\rho_{p(x)}(f) := \int_{\Omega} |f|^{p(x)}\, dx.
$$
This space is endowed with the luxemburg norm
$$
\|f\|_{L^{p(x)}(\Omega)} = \|f\|_{p(x),\Omega} = \|f\|_{p(x)} := \sup\Big\{\lambda>0\colon \rho_{p(x)}(\tfrac{f}{\lambda})<1\Big\}.
$$
The infimum and the supremum of the exponent $p$ play an important role in the estimates as the next elementary proposition shows. For further references, the following notation will be imposed
$$
1\le p_-:= \inf_{\Omega}p \le \sup_{\Omega} p =: p_+<\infty.
$$
The proof of the following proposition can be found in \cite[Theorem 1.3, p.p. 427]{FanyZhao}.
\begin{prop}\label{propdesigualdades}
Let $f\in L^{p(x)}(\Omega)$, then
$$
\min\{\|f\|_{p(x)}^{p_-}, \|f\|_{p(x)}^{p_+}\} \leq \rho_{p(x)}(f)\leq \max\{\|f\|_{p(x)}^{p_-}, \|f\|_{p(x)}^{p_{+}}\}.
$$
\end{prop}
\begin{remark}\label{minmax}
Proposition \ref{propdesigualdades}, is equivalent to
$$
\min\{\rho_{p(x)}(f)^{\frac{1}{p_-}}, \rho_{p(x)}(f)^{\frac{1}{p_{+}}} \} \leq \|f\|_{p(x)}\leq \max\{\rho_{p(x)}(f)^{\frac{1}{p_-}}, \rho_{p(x)}(f)^{\frac{1}{p_{+}}} \}.
$$
\end{remark}
We will use the following form of H\"older's inequality for variable exponents. The proof, which is an easy consequence of Young's inequality, can be found in \cite[Lemma 3.2.20]{Diening}.
\begin{prop}[H\"older's inequality]\label{propholder}
Assume $p_->1$. Let $u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$, then
$$
\int_{\Omega} |u v|\, dx\leq 2\|u\|_{p(x)} \|v\|_{p'(x)},
$$
where $p'(x)$ is, as usual, the conjugate exponent, i.e. $p'(x):= p(x)/(p(x)-1)$.
\end{prop}
The variable exponent Sobolev space $W^{1,p(x)}$ is defined by
$$
W^{1,p(x)}(\Omega):=\Big\{u\in W^{1,1}_{\text{loc}}(\Omega)\colon u\in L^{p(x)}(\Omega) \text{ and } \partial_i u\in L^{p(x)}(\Omega)\ i=1,\operatorname{\text{dist}}ots,N\Big\},
$$
where $\partial_i u$ stands fot the $i-$th partial weak derivative of $u$.
This space posses a natural modular given by
$$
\rho_{1,p(x)}(u) := \int_\Omega |u|^{p(x)} + |\nabla u|^{p(x)}\, dx,
$$
so $u\in W^{1,p(x)}(\Omega)$ if and only if $\rho_{1,p(x)}(u)<\infty$.
The corresponding luxenburg norm associated to this modular is
$$
\|u\|_{W^{1,p(x)}(\Omega)} = \|u\|_{1,p(x),\Omega} = \|u\|_{1,p(x)} := \sup\Big\{\lambda>0\colon \rho_{1,p(x)}(\tfrac{u}{\lambda})<1\Big\}.
$$
Observe that this norm turns out to be equivalent to $\|u\|:= \|u\|_{p(x)} + \|\nabla u\|_{p(x)}$.
One important subspace of $W^{1,p(x)}(\Omega)$ is the functions with zero boundary values. This is the content of the next definition.
\begin{defi}
We define $W^{1,p(x)}_{0}(\Omega)$ as the closure in $W^{1,p(x)}(\Omega)$ of functions with compact support.
\end{defi}
In most applications is very helpful to have test functions to be dense in $W^{1,p(x)}_0(\Omega)$. It is well known, see \cite{Diening}, that this property fails in general, even for continuous exponents $p(x)$. In order to have this desired property one need to impose some regularity conditions on the exponent $p(x)$.
\begin{defi}We say that $p \colon \Omega\rightarrow {{\mathbb {R}}}$ is {\em log-H\"older continuous} in $\Omega$ if
\begin{equation}\label{logholder}
\sup_{\substack{x,y\in \Omega\\ x\neq y}}\log(|x-y|^{-1})|p(x)-p(y)| < \infty.
\end{equation}
Set ${\mathcal{P}}^{log}(\Omega)=\{p \in {\mathcal{P}}(\Omega) \colon p \text{ satisfies \eqref{logholder}}\}$.
\end{defi}
Under this condition, the following theorem holds,
\begin{teo}[Theorem 9.1.6 in \cite{Diening}]
Assume that $p \in \mathcal P^{log}(\Omega)$, then $C^{\infty}_{c}(\Omega)$ is dense in $W^{1,p(x)}_0(\Omega)$.
\end{teo}
The proof of the following theorem can be found in \cite[Theorem 8.2.4]{Diening}.
\begin{teo}[Poincar\'e's inequality.]\label{teopoincare}
Let $p \in {\mathcal{P}}^{log}(\Omega)$. Then there exists a constant $c>0$ such that
$$
\|u\|_{p(x)} \leq c \|\nabla u\|_{p(x)},\quad u\in W^{1,p(x)}_0(\Omega).
$$
\end{teo}
\begin{remark}
Thanks to Poincar\'e inequality, as usual, in $W^{1,p(x)}_0(\Omega)$ the following norm will be used,
$$
\|u\|_{W^{1,p(x)}_0(\Omega)} = \|\nabla u\|_{p(x)}.
$$
This norm, is equivalent to the usual norm in $W^{1,p(x)}(\Omega)$ for functions $u\in W^{1,p(x)}_0(\Omega)$.
\end{remark}
\begin{defi}
We denote by $W^{-1, p'(x)}(\Omega)$ the topological dual space of $W^{1,p(x)}_0(\Omega)$.
The duality product between $f\in W^{-1, p'(x)}(\Omega)$ and $u\in W^{1, p(x)}_0(\Omega)$ will be denoted, as usual, by $\langle f, u\rangle$.
The norm in this space will be denoted by
$$
\|f\|_{W^{-1,p'(x)}(\Omega)} = \|f\|_{-1,p'(x)} := \sup\{\langle f, u\rangle \colon u\in W^{1,p(x)}_0(\Omega),\ \|\nabla u\|_{p(x)}\le 1\}.
$$
\end{defi}
We now present a result which we will find most useful later.
\begin{prop} \label{propdensidad}
The space $L^{\infty}(\Omega)$ is dense in $W^{-1,p'(x)}(\Omega)$.
\end{prop}
\begin{proof}
By H\"older's inequality we have that $W^{1,p_{+}}_{0}(\Omega)\subset W^{1,p(x)}_{0}(\Omega) \subset W^{1,p_-}_{0}(\Omega)$ with continuous embeddings. Since $C^\infty_c(\Omega)\subset W^{1,p_{+}}_{0}(\Omega)$ and $p\in {\mathcal{P}}^{log}(\Omega)$ we have the embeddings are dense.
Therefore,
$$
W^{-1,(p_-)'}(\Omega) \subset W^{-1,p'(x)}(\Omega) \subset W^{-1,(p_{+})'}(\Omega),
$$
with dense embeddings. Finally, since $L^{\infty}(\Omega)$ is dense in $W^{-1,(p_{-})'}(\Omega)$, we have that $L^{\infty}(\Omega)$ is dense in $W^{-1,p'(x)}(\Omega)$.
\end{proof}
Analogous to the constant exponent case, we have the following characterization of $W^{-1, p'(x)}(\Omega)$.
\begin{prop}
Let $f\in W^{-1, p'(x)}(\Omega)$. Then, there exists $\{f_i\}_{i=0}^N\subset L^{p'(x)}(\Omega)$ such that
$$
\langle f, u\rangle = \int_\Omega f_0 u\, dx - \sum_{i=1}^N \int_{\Omega} f_i\partial_i u\, dx.
$$
We will then say that $f = f_0 + \sum_{i=1}^N \partial_i f_i$. Moreover,
$$
\|f\|_{*} = \inf\left\{\sum_{i=0}^N \|f_i\|_{p'(x)}\colon f = f_0 + \sum_{i=1}^N \partial_i f_i,\ f_i\in L^{p'(x)}(\Omega), i=0,\operatorname{\text{dist}}ots,N\right\},
$$
defines an equivalent norm in $W^{-1,p'(x)}(\Omega)$.
\end{prop}
\begin{proof}
The characterization of $W^{-1,p'(x)}(\Omega)$ follows exactly as in the constant exponent case. It remains to see the equivalence of the norms $\|\cdot\|_{-1,p'(x)}$ and $\|\cdot\|_*$.
Observe that $\|\cdot\|_{*}$ clearly defines a norm in $W^{-1,p'(x)}(\Omega)$.
Let us now take $f_0, f_{1},\operatorname{\text{dist}}ots,f_{n}\in L^{p'(x)}(\Omega)$ such that $f= f_0 + \sum_{i=1}^N \partial_i f_i$ and consider $v \in W_{0}^{1,p(x)}(\Omega)$ such that $\|\nabla v\|_{p(x)}=1$.
By H\"older's inequality (Proposition \ref{propholder}) and Poincar\'e's inequality (Theorem \ref{teopoincare}), we have
\begin{align*}
\langle f,v \rangle &=\int_{\Omega}\left(f_{0}v+\sum_{i=1}^{N} f_{i}\partial_{i}v\right)\, dx\\
& \leq 2 \|f_0\|_{p'(x)}\|v\|_{p(x)} + 2 \sum_{i=1}^N \|f_i\|_{p'(x)}\|\partial_i v\|_{p(x)}\\
&\leq C \left(\|f_0\|_{p'(x)}+\sum_{i=1}^N \|f_i\|_{p'(x)}\right).
\end{align*}
Therefore,
$$
\|f\|_{-1,p'(x)}=\inf_{\|\nabla v\|_{p(x)}=1}\langle f,v \rangle \leq C \left(\|f_0\|_{p'(x)}+\sum_{i=1}^N \|f_i\|_{p'(x)}\right),
$$
so
$$
\|f\|_{-1,p'(x)}\le C\|f\|_*
$$
Now, the reverse inequality is a direct consequence of the Open Mapping Theorem (cf. \cite{Brezis}).
\end{proof}
\begin{remark}
Let now $D\subset {\mathbb {R}}^N$ be a bounded, open set and let $\Omega\subset D$ be open. Then, we have that $W^{1,p(x)}_0(\Omega)\subset W^{1,p(x)}_0(D)$, the inclusion being canonical, extending by zero. This inclusion induces $W^{-1, p'(x)}(D)\subset W^{-1,p'(x)}(\Omega)$ by restriction. Therefore, when dealing with sets $\Omega$ that are subsets of $D$, if one is considering $f\in W^{-1,p'(x)}(D)$ and $u\in W^{1,p(x)}_0(\Omega)$ there is no ambiguity in the notation $\langle f, u\rangle$.
\end{remark}
\subsection{$p(x)$-capacity and pointwise properties of Sobolev functions} We need the concept of capacity modified to deal with pointwise properties of functions in $W^{1,p(x)}_0(\Omega)$. This is the concept of $p(x)-$capacity. See \cite[Chapter 10]{Diening}.
\begin{defi}
Given $E\subset \mathbb{R}^{N}$, we consider the set
$$
S_{p(x)}(E)=\left\{u\in W^{1,p(x)}(\mathbb{R}^{N})\colon u\geq 0 \text{ and } u\geq 1 \text{ in an open set containing } E \right\}.
$$
If $S_{p(x)}(E)\neq \emptyset$, we define {\em $p(x)-$Sobolev capacity} of E as follows
$$
\text{\rm{cap}}_{p(x)}(E)=\inf_{u\in S_{p(x)}(E)}\int_{\mathbb{R}^N} |u|^{p(x)}+|\nabla u|^{p(x)}dx = \inf_{u\in S_{p(x)}(E)} \rho_{1,p(x)}(u).
$$
If $S_{p(x)}(E)=\emptyset$, we set $\text{\rm{cap}}_{p(x)}(E)=\infty$.
\end{defi}
\begin{defi}
Let $p \in {\mathcal{P}}^{log}(\Omega)$ and $K \subset \Omega$ compact, we define the \textit{$p(x)-$relative capacity} as
$$
\text{\rm{cap}}^{*}_{p(x)}(K,\Omega)=\inf_{u \in R_{p(x)}(K,\Omega)} \rho_{p(x),\Omega}(\nabla u)
$$
where $R_{p(x)}(K,\Omega)=\{u \in W^{1,p(x)}_0(\Omega)\colon u>1 \text{ in } K \text{ and } u\geq 0\}$.
If $U \subset \Omega$ is an open set, we define $\text{\rm{cap}}_{p(x)}(U,\Omega)=\operatorname{\text{dist}}isplaystyle{\sup_{\substack {K \subset U\\ K \text{ compact}}} \text{\rm{cap}}^{*}_{p(x)}(K,\Omega)}$.
Finally, if $E \subset \Omega$ is arbitrary, we define the $p(x)-$ relative capacity of $E$ with respect to $\Omega$ as
$$
\text{\rm{cap}}_{p(x)}(E,\Omega)=\inf_{\substack{E \subset U \subset \Omega\\ U \text{ open}}} \text{\rm{cap}}_{p(x)}(U,\Omega).
$$
\end{defi}
The main advantage of the relativa capacity is the fact that is possible to obtain a {\em capacitary potential}, i.e. a function whose modular gives the capacity of a set.
To this end, let $A\subset D$ and consider the class
$$
{\mathcal{G}}amma_{A}=\overline{\left\{v\in W_{0}^{1,p(x)}(D) \colon v\geq 1 \text{ a.e. in an open set containing } A\right\}},
$$
the closure being taken in $W^{1,p(x)}_0(D)$.
\begin{remark}\label{wclosed}
Observe that since ${\mathcal{G}}amma_{A}\subset W_{0}^{1,p(x)}(D)$ is closed and convex (the closure of a convex set is convex), it follows that is weakly convex. This fact will be used in the next proposition.
\end{remark}
Now we show that the relative capacity of a set is realized by a function in ${\mathcal{G}}amma_A$.
\begin{prop}
If ${\mathcal{G}}amma_{A}\neq \emptyset$, then there exists a unique $u_{A}\in {\mathcal{G}}amma_{A}$ such that
$$
\text{\rm{cap}}_{p(x)}(A,D)=\int_{D}\left|\nabla u_{A}\right|^{p(x)}dx.
$$
\end{prop}
\begin{proof}
Consider $\{v_n\}_{n\in{\mathbb {N}}}\subset W_{0}^{1,p(x)}(D)$ such that $v_{n}\geq 1$ a.e. in an open set containing $A$ and
$$
\int_{D}\left|\nabla v_{n}\right|^{p(x)}\, dx \to \text{\rm{cap}}_{p(x)}(A,D).
$$
By Theorem \ref{teopoincare} and Proposition \ref{propdesigualdades}, we have
$$
\|\nabla v_{n}\|_{p(x)}\leq \max \{\rho_{p(x)}(\nabla v_{n})^{\frac{1}{p_+}}, \rho_{p(x)}(\nabla v_{n})^{\frac{1}{p_-}}\}.
$$
Then, $\{v_n\}_{n\in{\mathbb {N}}}$ is bounded in $W_{0}^{1,p(x)}(D)$, which is a reflexive space. By Alaoglu's Theorem, there is a subsequence $v_{n_{j}}\rightharpoonup v_{\infty}$ en $W_{0}^{1,p(x)}(D)$. By Remark \ref{wclosed}, $v_{\infty}\in {\mathcal{G}}amma_{A}$.
Observe that
$$
\int_{D}\left|\nabla v_{\infty}\right|^{p(x)}dx\leq \liminf \int_{D}\left|\nabla v_{n_{j}}\right|^{p(x)}dx=\text{\rm{cap}}_{p(x)}(A,D).
$$
Since the reverse inequality is obvious, the first part of the Proposition is proved.
The uniqueness is an immediate consequence of the strict convexity of the modular, since $p_->1$. We leave the details to the reader.
\end{proof}
We can now give the definition of capacitary potential.
\begin{defi}
We define the capacitary potential of $A$ such as the only $u_{A}$ that verifies
$$
\int_{D}\left|\nabla u_{A}\right|^{p(x)}dx=\inf_{ v \in {\mathcal{G}}amma_{A}} \int_{D}\left|\nabla v\right|^{p(x)}dx = \text{\rm{cap}}_{p(x)}(A,D).
$$
\end{defi}
It is well known that when dealing with pointwise properties of Sobolev functions, the concept of {\em almost everywhere} needs to be changed to {\em quasi everywhere}. This is the content of the next definition.
\begin{defi}
An statement is valid {\em $p(x)-$quasi everywhere} ($p(x)-$q.e.) if it is valid except in a set of null Sobolev $p(x)-$capacity.
\end{defi}
\begin{defi}
Let $D \subset {{\mathbb {R}}}^{N}$ be an open bounded set, $\Omega \subset D$ is {\em $p(x)-$quasi open} if there is a decreasing sequence $\{W_{n}\}_{n\in{\mathbb {N}}}$ of open sets such that $\text{\rm{cap}}_{p(x)}(W_{n},D)$ converges to $0$ and $\Omega\cup W_{n}$ is an open set for each $n$.
\end{defi}
\begin{defi}
A function $u \colon \Omega \rightarrow {{\mathbb {R}}}$ is {\em $p(x)-$quasi continuous} if for every $\varepsilon>0$, there is an open set $U$ such that $\text{\rm{cap}}_{p(x)}(U)<\varepsilon$ and $u|_{\Omega \setminus U}$ is continuous.
\end{defi}
The proof of the next theorem can be found in \cite[Corollary 11.1.5]{Diening}.
\begin{teo}\label{teolog}
Let $p \in {\mathcal{P}}^{log} (\Omega)$ with $1<p_-\leq p_+ < \infty$. Then for each $u \in W^{1,p(x)}(\Omega)$ there exists a $p(x)-$quasicontinuous function $v \in W^{1,p(x)}(\Omega)$ such that $u=v$ almost everywhere in $\Omega$.
\end{teo}
\begin{remark}
It is easy to see that two $p(x)-$quasi continuous representatives of a given function $u\in W^{1,p(x)}(\Omega)$ can only differ in a set of zero $p(x)-$capacity. Therefore, the unique $p(x)-$quasi continuous representative (defined $p(x)-$q.e.) of $u\in W^{1,p(x)}(\Omega)$ will be denoted by $\tilde u$.
\end{remark}
The proof of the next proposition can be found in \cite[Section 11.1.11]{Diening}.
\begin{prop} \label{propctp}
Let $v_{j}\rightarrow v$ in $W^{1,p(x)}_{0}(D)$. Then, there is a subsequence $\{v_{j_{k}}\}_{k \in {\mathbb {N}}}$ such that $\tilde{v}_{j_{k}}\rightarrow \tilde{v}$ $p(x)-$q.e.
\end{prop}
Now we need a characterization of the space $W^{1,p(x)}_{0}(\Omega)$ as the restriction of quasi continuous functions that vanishes quasi everywhere on ${\mathbb {R}}^N\setminus\Omega$. This theorem is esentialy contained in \cite[Corollary 11.2.5, Theorem 11.2.5]{Diening}. We include here the proof since a minor modification of the above mentioned result is needed and for the reader's convenience.
\begin{teo}[Characterization Theorem]\label{teocaracterizacion}
Let $D\subset {{\mathbb {R}}}^{N}$ be an open set, $\Omega\subset D$ an open subset and $p \in \mathcal P^{log}(\Omega)$. Then,
$$
u\in W^{1,p(x)}_{0}(\Omega) \Leftrightarrow u\in W^{1,p(x)}_{0}(D)\text{ and } \tilde{u}=0\ p(x)-q.e.\text{ in } D \setminus \Omega.
$$
\end{teo}
\begin{proof}
Let $u\in W^{1,p(x)}_0(\Omega)$, then, it is immediate that $u\in W^{1,p(x)}_0(D)$.
Now, let $\{\varphi_{n}\}_{n \in {\mathbb {N}}}\subset C^{\infty}_{c}(\Omega)$ such that $\varphi_{n}\to u$ in $W^{1,p(x)}_{0}(\Omega)$ (and therefore in $W^{1,p(x)}_{0}(D)$).
Let $\{\varphi_{{n_{j}}}\}_{j \in {\mathbb {N}}}\subset \{\varphi_{n}\}_{n \in {\mathbb {N}}}$ be a subsequence such that $\varphi_{n_j}\to \tilde{u}$ $p(x)-$q.e. Then, since $\varphi_{{n_{j}}}=0$ in $D \setminus \Omega$, we have that $\tilde{u}=0$ $p(x)-$q.e. in $D \setminus \Omega$.
To see the converse, let us assume that $D={{\mathbb {R}}}^{N}$ (or else, we extend by zero). Since $u=u^{+}-u^{-}$, we can assume that $u\geq 0$. Moreover, since $\min\{u,n\}\in W^{1,p(x)}({{\mathbb {R}}}^{N})$ converges to $u$ in $W^{1,p(x)}({{\mathbb {R}}}^{N})$, we can assume that $u$ is bounded.
Finally, let us consider $\xi \in C^{\infty}_{c}(B(0,2))$ such that $0\leq \xi \leq 1$ and $\xi\equiv 1$ in $B(0,1)$. Setting $\xi_{n}(x)=\xi(\frac{x}{n})$, we have that $\xi_{n} u$ converges to $u$ in $W^{1,p(x)}({{\mathbb {R}}}^{N})$. Therefore we can assume that $u(x)=0$ for every $x\in (B(0,R))^{c}$ with $R$ large enough.
Therefore, we need to prove the converse for bounded, compactly supported and nonnegative functions $u\in W^{1,p(x)}_0({\mathbb {R}}^N)$ such that $\tilde u = 0$ $p(x)-$q.e. in $\Omega^c$.
Since $\tilde u$ is $p(x)-$quasi continuous, there is a decreasing sequence of open sets $\{W_{n}\}_{n \in {\mathbb {N}}}$ such that $\text{\rm{cap}}_{p(x)}(W_{n},D)\to 0$ and $\tilde{u}|_{{{\mathbb {R}}}^{N} \setminus W_{n}}$ is continuous.
We can assume that $W_{n}$ contains the set of null capacity of ${{\mathbb {R}}}^{N} \setminus \Omega$ where $\tilde{u}\neq 0$. Therefore, $\tilde{u}= 0$ in $(\Omega\cup W_{n})^{c}=\Omega^{c}\cap W_{n}^{c}$.
Given $\operatorname{\text{dist}}elta>0$, set $V_{n}=\{x \colon \tilde{u}(x)<\operatorname{\text{dist}}elta\}\cup W_{n}$. Since $\tilde u$ is continuous in ${\mathbb {R}}^N\setminus W_n$, $V_{n}$ is an open set. Therefore, $V_{n}^{c}$ is a closed set. It is also bounded since $V_{n}^{c}\subset B(0,R)$. Then, $ V_{n}^{c}$ is compact.
Let $u_{W_{n}}$ be the capacitary potential of $W_{n}$, then $(u-\operatorname{\text{dist}}elta)^{+} (1-u_{W_{n}})=0$ a.e. in $\Omega \setminus V_{n}^{c}$.
Consider now a regularizing sequence $\{\phi_{j}\}_{j \in {\mathbb {N}}}$. Therefore, for $j$ sufficiently large we have that
$$
\phi_{j}*\left[(u-\operatorname{\text{dist}}elta)^{+} (1-u_{W_{n}}) \right]\in C_{}^{\infty}(\Omega).
$$
Observe that
$$
\rho_{p(x)}(\nabla u_{W_{n}})=\text{\rm{cap}}_{p(x)}(W_{n},D)\rightarrow 0.
$$
By Proposition \ref{propdesigualdades}, we can conclude that $\|\nabla u_{W_n}\|_{p(x)}\to 0$ and, by Poincar\'e's inequality, $\|u_{W_n}\|_{1,p(x)}\to 0$. Therefore, $1-u_{W_n} \to 1$ in $W^{1,p(x)}(D)$ when $n\to \infty$.
Obviously, $(u-\operatorname{\text{dist}}elta)^{+}\to u^{+}=u$ in $W^{1,p(x)}(D)$ when $\operatorname{\text{dist}}elta\to 0$ and observe that
\begin{align*}
\left\|(u-\operatorname{\text{dist}}elta)^{+}(1-u_{W_{n}})-u \right\|_{1,p(x)} \leq &\left\|1-u_{W_{n}}\right\|_{1,p(x)} \left\|(u-\operatorname{\text{dist}}elta)^{+}-u\right\|_{1,p(x)}\\
&+\left\|u \right\|_{1,p(x)} \left\|u_{W_{n}}\right\|_{1,p(x)}.
\end{align*}
Finally, taking the limit when when $j\to \infty$, $n\to \infty$ and $\operatorname{\text{dist}}elta\to 0$, we have that
$$
\phi_{j}*\left[(u-\operatorname{\text{dist}}elta)^{+} (1-u_{W_{n}}) \right]\to u,
$$
which completes the proof.
\end{proof}
We end this subsection with a lemma that will be much helpful in the sequel.
\begin{lema} \label{lemainf}
Let $v \in W^{1,p(x)}_{0}(\mathbb{R}^{N})$ and $w \in W_{0}^{1,p(x)}(D)$ such that $|v|\leq w$ a.e. in $D$. Then, $v \in W_{0}^{1,p(x)}(D)$.
\end{lema}
\begin{proof}
It is enough to see that $v^{+} \in W_{0}^{1,p(x)}(D)$ (for $v^{-}$ we prodece similarly and haven shown this result for $v^{+}$ and $v^{-}$, we can state that is valid for $v=v^{+}-v^{-}$).
Since $w\geq 0$, by density we can consider $\{w_{n}\}_{n \in {\mathbb {N}}}\subset C^{\infty}_{c}(D)^{+}$ such that $\{w_{n}\}_{n \in {\mathbb {N}}}$ converges to $w$ in $W^{1,p(x)}(D)$.
Therefore, $\inf\{w_{n}, v^{+}\}$, which has compact support in $D$ (for each $w_{n}$ has so) converges to $\inf\{w, v^{+}\}$ which coincides with $v^{+}$ since $|v|\leq w$ a.e. in $D$.
Then, taking an adequate regularizing sequence, we obtain a sequence of $C^{\infty}_{c}(D)$ convergent to $v^{+}$, which completes the proof.
\end{proof}
\section{The Dirichlet problem for the $p(x)-$laplacian.}
We define the $p(x)-$laplacian as
$$
{\mathcal{D}}elta_{p(x)}u:=\operatorname {\text{div}}(\left|\nabla u \right|^{p(x)-2}\nabla u).
$$
Observe that when $p(x)=2$ this operator agrees with the classical Laplace operator, and when $p(x)=p$ is constant is the well-known $p-$laplacian.
The Dirichlet problem for the $p(x)-$laplacian consists of finding $u$ $\in W^{1,p(x)}_{0}(\Omega)$ such that
\begin{equation}\label{eq.dirichlet}
\left\{
\begin{array}{rl}
-{\mathcal{D}}elta_{p(x)}u=f & \text{en } \Omega,\\
u=0 & \text{en } \partial\Omega,
\end{array} \right.
\end{equation}
where $f\in L^{p'(x)}(\Omega)$ or, more generally, $f\in W^{-1,p'(x)}(\Omega)$.
In its weak formulation, this problem consists of finding $u$ $\in W^{1,p(x)}_{0}(\Omega)$ such that
$$
\int_{\Omega}\left|\nabla u \right|^{p(x)-2}\nabla u \nabla v\, dx=\langle f, v\rangle \text{ for every } v \in W^{1,p(x)}_{0}(\Omega).
$$
Setting
$$
I(v):= \int_{\Omega}\frac{1}{p(x)}\left|\nabla v \right|^{p(x)}dx - \langle f, v\rangle,
$$
the problem can be reformulated as finding $u \in W^{1,p(x)}_{0}(\Omega)$ such that $$
I(u)=\min\{I(v) \colon v \in W^{1,p(x)}_{0}(\Omega)\}.
$$
By standard methods, we obtain the following result
\begin{teo}
Assume $p_->1$. Then there exists a unique minimizer of $I(v)$ in $W^{1,p(x)}_0(\Omega)$ and a unique weak solution of \eqref{eq.dirichlet} $u\in W^{1,p(x)}_{0}(\Omega)$.
\end{teo}
\begin{proof}
The proof is standard and uses the direct method of the calculus of variations. We omit the details.
\end{proof}
\begin{remark}
The unique weak solution of \eqref{eq.dirichlet} will be denoted by $u^{f}_{\Omega}$.
\end{remark}
\begin{prop}\label{acotacion.solucion}
Let $f\in W^{-1,p'(x)}(\Omega)$ and let ${\mathcal{A}}>0$ be such that $\|f\|_{-1,p'(x)}\le {\mathcal{A}}$. Then, there exists a constant $C$ depending only on ${\mathcal{A}}$, $p_-$ and $p_+$ such that
$$
\|\nabla u_\Omega^f\|_{p(x)}\le C.
$$
\end{prop}
\begin{proof}
Let us assume that $\|\nabla u_\Omega^f\|_{p(x)}>1$ (otherwise, the result is clear). By Proposition \ref{propdesigualdades},
$$
\int_\Omega |\nabla u_\Omega^f|^{p(x)}=\langle f,u_\Omega^f \rangle \leq \|f\|_{-1,p'(x)}\|u_\Omega^f\|_{p(x)}\leq \|f\|_{-1,p'(x)} (\rho_{p(x)}(u_\Omega^f))^{\frac{1}{p_-}}.
$$
Therefore,
$$
\int_\Omega |\nabla u_\Omega^f|^{p(x)}\leq \|f\|_{-1,p'(x)}^{\frac{p_-}{p_{-}-1}},
$$
which completes the proof.
\end{proof}
In what follows, the monontonicity of the $p(x)-$laplacian is crucial. This fact is a consequence of the following well-known lemma that is proved in \cite[p.p. 210]{Simon}.
\begin{lema}\label{lemadesigualdad}
There is a constant $c_{1}>0$ such that for every $a,b\in {{\mathbb {R}}}^{N}$,
$$
(|b|^{p-2}b-|a|^{p-2}a)\cdot (b-a) \geq \begin{cases}
c_{1}|b-a|^{p} & \text{if } p \geq 2,\\
c_{1}\frac{|b-a|^{2}}{(|b|+|a|)^{2-p}} & \text{if } p \leq 2.
\end{cases}
$$
\end{lema}
\begin{remark}\label{deltap}
Observe that if $u\in W^{1,p(x)}_0(\Omega)$, then $-{\mathcal{D}}elta_{p(x)}u\in W^{-1,p'(x)}(\Omega)$. In fact,
$$
\langle -{\mathcal{D}}elta_{p(x)}u, v\rangle = \int_{\Omega} |\nabla u|^{p(x)-2}\nabla u\nabla v\, dx.
$$
\end{remark}
\begin{defi}
Let $f\in W^{-1,p'(x)}(\Omega)$. We say that $f\ge 0$ if $\langle f, v\rangle\ge 0$ for every $v\in W^{1,p(x)}_0(\Omega)$ such that $v\ge 0$.
Let $f, g\in W^{-1,p'(x)}(\Omega)$. We say that $g\le f$ if $f-g\ge 0$.
\end{defi}
We now prove the comparison principle for \eqref{eq.dirichlet}
\begin{lema}[Comparison Principle] \label{ppiomax}
Let $u,v \in W_{0}^{1,p(x)}(D)$ be such that
$$\begin{cases}
-{\mathcal{D}}elta_{p(x)}u\leq -{\mathcal{D}}elta_{p(x)}v & \text{in } D,\\
u\leq v & \text{on } \partial D.
\end{cases}
$$
Then, $u\leq v$ en $D$.
\end{lema}
\begin{proof}
Let us call $g:=-{\mathcal{D}}elta_{p(x)}u$ and $f:=-{\mathcal{D}}elta_{p(x)}v$. Then, by Remark \ref{deltap}, we obtain that, given $\varphi \in W_{0}^{1,p(x)}(D)$,
$$
\int_{D}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v) \nabla \varphi(x)\, dx=\langle g-f, \varphi \rangle.
$$
In particular, taking $\varphi=(u-v)^{+} \in W_{0}^{1,p(x)}(D)$, since $g \leq f$ we have that
$$
\int_{D}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v) \nabla (u-v)^{+}\, dx
=\langle g-f, (u-v)^{+}\rangle \leq 0.
$$
Taking into account that $\nabla (u-v)^{+}=(\nabla u-\nabla v)\chi_{\{u>v\}}$, we conclude that
$$
\int_{\{u>v\}}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)(\nabla u-\nabla v)\, dx\leq 0.
$$
Now, let us define $\Omega_1':=\{x\in D \colon p(x)\ge 2\}$ and $\Omega_1'':=\{x\in D\colon p(x)< 2\}$. Therefore, $D = \Omega_1' \cup \Omega_1''$ (disjoint union).
Now, by Lemma \ref{lemadesigualdad}, there is a constant $c>0$ such that
\begin{align*}
&\int_{\{u\geq v\}}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v) (\nabla u-\nabla v)\, dx \\
&\geq c \int_{\{u\geq v\}\cap \Omega_{1}'}|\nabla u-\nabla v|^{p(x)}\, dx + c \int_{\{u\geq v\}\cap \Omega_{1}''}\frac{|\nabla u-\nabla v|^{2}}{(|\nabla u|+|\nabla v|)^{2-p(x)}}\, dx.
\end{align*}
Therefore, since $\nabla (u-v)^{+}=(\nabla u-\nabla v)\chi_{u>v}$, we conclude that
$$
0 \geq \int_{\Omega_{1}'} |\nabla (u-v)^{+}|^{p(x)}\, dx, + \int_{\Omega_{1}''}\frac{|\nabla (u-v)^{+}|^{2}}{(|\nabla u|+|\nabla v|)^{2-p(x)}}\, dx.
$$
Then, $\nabla (u-v)^{+}=0$ in $D$. So $(u-v)^{+}$ is constant in $D$. Since $(u-v)^{+} \in W_{0}^{1,p(x)}(D)$, we have that $(u-v)^{+}=0$. Therefore $u-v\leq 0$, which completes the proof.
\end{proof}
\begin{corol}[Weak maximum principle]
Let $f\in W^{-1,p'(x)}(\Omega)$ be such that $f\ge 0$. Then $u_\Omega^f\ge 0$.
\end{corol}
\begin{proof}
Just apply Lemma \ref{ppiomax} with $u=0$ and $v=u_\Omega^f$.
\end{proof}
The following proposition gives the monotonicity property of the solution with respect to the domain. The proof follows the ideas of \cite[Theorem 3.2.5.]{Henrot} where the linear case $p(x)=2$ is treated. Nevertheless, since the $p(x)-$laplacian is nonlinear, the monotonicity property of this operator comes into play replacing linearity in the argument.
\begin{prop}[Property of monotonicity with respect to the domain.]\label{propmonotonia}
Let $\Omega_{1}\subset \Omega_{2}$ and $f\in W^{-1,p'(x)}(\Omega_2)$ be such that $f\geq 0$. Then, $u_{\Omega_1}^{f}\leq u_{\Omega_2}^{f}$.
\end{prop}
\begin{proof}
We will denote $u_{1}=u_{\Omega_1}^{f}$ and $u_{2}=u_{\Omega_2}^{f}$.
Given $v \in W_{0}^{1,p(x)}(\Omega_{1})\subset W^{1,p(x)}_0(\Omega_2)$,
\begin{equation}\label{primeraintegral}
\int_{\Omega_{i}}|\nabla u_{i}|^{p(x)-2}\nabla u_{i} \nabla v\, dx=\langle f, v\rangle,\quad i=1,2.
\end{equation}
Therefore,
\begin{equation}\label{segundaintegral}
\int_{\Omega_{1}}(|\nabla u_{1}|^{p(x)-2}\nabla u_{1}-|\nabla u_{2}|^{p(x)-2}\nabla u_{2}) \nabla v\, dx=0,
\end{equation}
for every $v\in W^{1,p(x)}_0(\Omega_1)$.
Since $f\geq 0$, we have that $u_{2}\geq 0$. Then, $(u_{1}-u_{2})^{+}\leq u_{1}^{+} \in W_{0}^{1,p(x)}(\Omega_{1})$ and hence, by Lemma \ref{lemainf}, $(u_{1}-u_{2})^{+} \in W_{0}^{1,p(x)}(\Omega_{1})$. Therefore
$$
\int_{\Omega_{1}}(|\nabla u_{1}|^{p(x)-2}\nabla u_{1}-|\nabla u_{2}|^{p(x)-2}\nabla u_{2}) \nabla (u_{1}-u_{2})^{+}\, dx=0.
$$
Now, let us define $\Omega_1':=\{x\in \Omega_1\colon p(x)\ge 2\}$ and $\Omega_1'':=\{x\in \Omega_1\colon p(x)< 2\}$. Therefore, $\Omega_1 = \Omega_1' \cup \Omega_1''$ (disjoint union).
Now, by Lemma \ref{lemadesigualdad}, there is a constant $c>0$ such that
\begin{align*}
0&=\int_{\{u_{1}\geq u_{2}\}\cap \Omega_{1}}(|\nabla u_{1}|^{p(x)-2}\nabla u_{1}-|\nabla u_{2}|^{p(x)-2}\nabla u_{2}) (\nabla u_{1}-\nabla u_{2})\, dx \\
&\geq c \int_{\{u_{1}\geq u_{2}\}\cap \Omega_{1}'}|\nabla u-\nabla v|^{p(x)}\, dx + c \int_{\{u_{1}\geq u_{2}\}\cap \Omega_{1}''}\frac{|\nabla u-\nabla v|^{2}}{(|\nabla u|+|\nabla v|)^{2-p(x)}}\, dx.
\end{align*}
Therefore, since $\nabla (u-v)^{+}=(\nabla u-\nabla v)\chi_{u>v}$, we conclude that
$$
0 \geq \int_{\Omega_{1}'} |\nabla (u-v)^{+}|^{p(x)}\, dx, + \int_{\Omega_{1}''}\frac{|\nabla (u-v)^{+}|^{2}}{(|\nabla u|+|\nabla v|)^{2-p(x)}}\, dx.
$$
Then, $\nabla (u_{1}-u_{2})^{+}=0$ in $\Omega_{1}$. Hence, $(u_{1}-u_{2})^{+}$ is constant in $\Omega_{1}$. Since $(u_{1}-u_{2})^{+} \in W_{0}^{1,p(x)}(\Omega_{1})$, we have that $(u_{1}-u_{2})^{+}=0$. Therefore $u_{1}-u_{2}\leq 0$, which completes the proof.
\end{proof}
We now end this section with an stability result for solutions of the Dirichlet problem
\begin{teo}\label{estabilidadDirichlet}
Let $D\subset {\mathbb {R}}^N$ be open, and let $f_i\in W^{-1,p'(x)}(D)$, $i=1,2$. There exists a constant $C>0$ depending only on $p_-$, $p_+$ and $\max\{\|f_i\|_{-1,p'(x)}\}$ such that, if $\Omega\subset D$,
$$
\int_{D} |\nabla u_\Omega^{f_1} - \nabla u_\Omega^{f_2}|^{p(x)}\, dx \le C (\|f_1-f_2\|_{-1,p'(x)} + \|f_1-f_2\|_{-1,p'(x)}^\beta),
$$
where the constant $\beta>0$ depends only on $p_-$ and $p_+$.
\end{teo}
Theorem \ref{estabilidadDirichlet} immediately implies the following Corollary.
\begin{corol}\label{continuidad.dato}
Let $f_n, f\in W^{-1,p'(x)}(\Omega)$ be such that $\|f_n-f\|_{-1,p'(x)}\to 0$. Then
$$
\|\nabla u_\Omega^{f_n} - \nabla u_\Omega^{f}\|_{p(x)}\to 0.
$$
\end{corol}
Now we proceed with the proof of the Theorem.
\begin{proof}[Proof of Theorem \ref{estabilidadDirichlet}]
Let us denote $u_i = u_{\Omega}^{f_i}$.
Given $\varphi \in W_{0}^{1,p(x)}(\Omega)$, we have that
\begin{equation}\label{uno}
\int_{\Omega}|\nabla u_i|^{p(x)-2}\nabla u_i \nabla \varphi \, dx=\langle f_i, \varphi \rangle,\quad i=1,2.
\end{equation}
In particular, considering $\varphi=u_1-u_2\in W_{0}^{1,p(x)}(\Omega)$ and subtracting, we obtain
\begin{align*}
\int_{\Omega}(|\nabla u_1|^{p(x)-2}\nabla u_1 - |\nabla u_2|^{p(x)-2}\nabla u_2)& (\nabla u_1 - \nabla u_2)\, dx \\
&= \langle f_{1}-f_{2}, u_1-u_2\rangle\\
& \leq \|f_{1}-f_{2}\|_{-1,p'(x)} \|\nabla u_1 - \nabla u_2\|_{p(x)}\\
&\leq \|f_{1}-f_{2}\|_{-1,p'(x)} (\|\nabla u_1\|_{p(x)} + \|\nabla u_2\|_{p(x)})\\
&\le C\|f_{1}-f_{2}\|_{-1,p'(x)}
\end{align*}
where we have used Proposition \ref{acotacion.solucion} in the last inequality.
On the other hand, naming $\Omega_{1}=\Omega \cap \{p(x)\geq 2\}$ and $\Omega_{2}=\Omega \cap \{p(x)< 2\}$, we have that
\begin{align*}
\int_{\Omega}&(|\nabla u_1|^{p(x)-2}\nabla u_1-|\nabla u_2|^{p(x)-2}\nabla u_2) (\nabla u_1 - \nabla u_2)\, dx\\
&=\sum_{i=1}^2 \int_{\Omega_{i}}(|\nabla u_1|^{p(x)-2}\nabla u_1 - |\nabla u_2|^{p(x)-2}\nabla u_2) (\nabla u_1 - \nabla u_2)\, dx.
\end{align*}
Let us study each of these integrals.
By Lemma \ref{lemadesigualdad},
$$
\int_{\Omega_{1}}(|\nabla u_1|^{p(x)-2}\nabla u_1 - |\nabla u_2|^{p(x)-2}\nabla u_2) (\nabla u_1-\nabla u_2)\, dx
\geq c \int_{\Omega_{1}} |\nabla (u_1-u_2)|^{p(x)}\, dx.
$$
Let us now analyze the integral over $\Omega_{2}$.
\begin{align*}
\int_{\Omega_{2}} |\nabla (u_1-u_2)|^{p(x)}dx &= \int_{\Omega_{2}} (|\nabla u_1|+|\nabla u_2|)^{\frac{(2-p(x))p(x)}{2}} \left(\frac{|\nabla (u_1-u_2)|}{(|\nabla u_1|+|\nabla u_2|)^{\frac{2-p(x)}{2}}}\right)^{p(x)}\, dx\\
&\leq 2 \|(|\nabla u_1|+|\nabla u_2|)^{\frac{(2-p(x))p(x)}{2}}\|_{\frac{2}{2-p(x)}} \left\| \left(\frac{|\nabla (u_1-u_2)|}{(|\nabla u_1|+|\nabla u_2|)^{\frac{2-p(x)}{2}}}\right)^{p(x)}\right\|_{\frac{2}{p(x)}}\\
&\le 2 \left(\int_{\Omega_2}(|\nabla u_1|+|\nabla u_2|)^{p(x)}dx \right)^{\alpha}
\left(\int_{\Omega_2}\frac{|\nabla (u_1-u_2)|^2}{(|\nabla u_1|+|\nabla u_2|)^{2-p(x)}}\, dx\right)^{\beta}.
\end{align*}
for some constants $\alpha$ and $\beta$ depending only on $p_-$ and $p_+$. Let us observe that for the first inequality we took into account H\"older's inequality and for the second one, Observation \ref{minmax}.
Let us now find a bound for the first factor. In fact, by Proposition \ref{acotacion.solucion}.
\begin{align*}
\int_{\Omega_{2}}(|\nabla u_1|+|\nabla u_2|)^{p(x)}\, dx & \leq 2^{p_+ - 1} \int_{\Omega_{2}}(|\nabla u_1|^{p(x)}+|\nabla u_2|^{p(x)})\, dx\le C.
\end{align*}
Observe that, by Lemma \ref{lemadesigualdad}, we are able to find a bound for the second factor.
$$
\int_{\Omega_2}\frac{|\nabla (u_1-u_2)|^2}{(|\nabla u_1|+|\nabla u_2|)^{2-p(x)}}\, dx \leq C \int_{\Omega_2} (|\nabla u_1|^{p(x)-2}\nabla u_1-|\nabla u_2|^{p(x)-2}\nabla u_2)(\nabla u_1-\nabla u_2)\, dx.
$$
Then,
\begin{align*}
\int_{\Omega_2} |\nabla (u_1-u_2)|^{p(x)}\, dx &\leq
C \left( \int_{\Omega_{2}} (|\nabla u_1|^{p(x)-2}\nabla u_2 - |\nabla u_2|^{p(x)-2}\nabla u_2) (\nabla u_1 - \nabla u_2)\, dx\right)^{\beta}\\
&\le C \left( \int_{\Omega} (|\nabla u_1|^{p(x)-2}\nabla u_2 - |\nabla u_2|^{p(x)-2}\nabla u_2) (\nabla u_1 - \nabla u_2)\, dx\right)^{\beta}\\
&\le C \|f_1-f_2\|_{-1,p'(x)}^\beta.
\end{align*}
So we can conclude that
$$
\int_{\Omega} |\nabla (u_1-u_2)|^{p(x)}\, dx \le C (\|f_1-f_2\|_{-1,p'(x)} + \|f_1-f_2\|_{-1,p'(x)}^\beta).
$$
This finishes the proof.
\end{proof}
\section{Continuity of the Dirichlet problem with respect to perturbations on the domain.}
In this section we investigate the dependence of the solutions of the Dirichlet problem $u^f_\Omega$ with respect to perturbations on the domain. We will analyze a rather general problem considering a sequence of uniformly bounded domains $\Omega_n$ converging to a limiting domain $\Omega$ in the Haussdorf complementary topology. Then we study whether $u^f_{\Omega_n}$ converges to $u^f_\Omega$ or not.
We begin this section by defining a notion of convergence of domains that will be essential for our next results.
\begin{defi}[Hausdorff complementary topology.] Let $D\subset {\mathbb {R}}^N$ be compact. Given $K_1, K_2\subset D$ compact sets, we define de Hausdorff distance $d_H$ as
$$
d_H(K_1, K_2) := \max\left\{\sup_{x\in K_1} \inf_{y\in K_2} \|x-y\|, \sup_{x\in K_2} \inf_{y\in K_1} \|x-y\|\right\}.
$$
Now, let $\Omega_1,\Omega_2\subset D$ be open sets, we define the Hausdorff complementary distance $d^H$ as
$$
d^H(\Omega_1, \Omega_2) := d_H(D \setminus \Omega_1, D \setminus \Omega_2).
$$
Finally, we say that $\{\Omega_n\}_{n\in {\mathbb {N}}}$ converges to $\Omega$ in the sense of the Hausdorff complementary topology, denoted by $\Omega_n \stackrel{H}{\to}\Omega$, if $d^H(\Omega_n, \Omega)\to 0$.
\end{defi}
For an study and properties of this topology of open sets, we refer to the book \cite{Henrot}.
We now present the one property that will be essential for our purposes.
\begin{prop}
Let $K\subset \Omega$ be a compact set. If $\Omega_n\stackrel{H}{\to}\Omega$, then $K\subset \Omega_n$ for every $n$ large enough.
\end{prop}
\begin{proof}
The proof is immediate from the definition. See \cite{Henrot}.
\end{proof}
Now we state a couple of corollaries of Proposition \ref{acotacion.solucion} that will be most useful.
\begin{corol}\label{corocontinuidad}
Let $D\subset \mathbb{R}^{N}$ be an open bounded set and let $\Omega_n \subset D$ be a sequence of open domains. Let $p \in {\mathcal{P}}^{log}(\Omega)$ such that $p_->1$. Then, $\{u_{\Omega_n}^{f}\}_{n\in{\mathbb {N}}}$ is bounded in ${ W^{1,p(x)}_{0}(D)}$.
\end{corol}
\begin{corol}\label{lemacontinuidad}
Under the same assumptions as in Corollary \ref{corocontinuidad}, we have that the sequence $\{|\nabla u_{\Omega_n}^f|^{p(x)-2} \nabla u_{\Omega_n}^f\}_{n \in {\mathbb {N}}}$ is bounded in $L^{p'(x)}(D)$.
\end{corol}
We now extend to variable exponent spaces Proposition 3.7 in \cite{Bucur}.
\begin{teo}\label{teo1}
Let us denote $u_{n} = u_{\Omega_n}^f$. Assume that $u_{n}\rightharpoonup u^{*}$ weakly in $W^{1,p(x)}_{0}(D)$. Let $\Omega \subset D$ be such that for every compact subset $K \subset \Omega$, there is an integer $n_{0}$ such that $K \subset \Omega_{n}$ for every $n\geq n_{0}$. Then, there holds that
$$
-{\mathcal{D}}elta_{p(x)}u^{*}=f \text{ in } \Omega.
$$
\end{teo}
\begin{remark}\label{loquefalta}
Observe that in order to conclude that $u^* = u_{\Omega}^f$ it remains to see that $u^*\in W^{1,p(x)}_0(\Omega)$.
\end{remark}
\begin{proof}
As $p\in {\mathcal{P}}^{log}(D)$, we need to verify that, given $\varphi \in C^{\infty}_{c}(\Omega)$, the following equality is valid:
$$
\int_{\Omega}|\nabla u^{*}|^{p(x)-2}\nabla u^{*}\nabla \varphi \, dx=\langle f,\varphi \rangle.
$$
Let $\varphi \in C^{\infty}_{c}(\Omega)$. Since $\operatorname {\text{supp}}(\varphi)\subset \Omega$ is compact, there is an integer $n_{0}$ such that $\operatorname {\text{supp}}(\varphi)\subset \Omega_{n}$ for every $n\geq n_{0}$. Therefore, $\varphi \in C^{\infty}_{c}(\Omega_{n})$ for every $n\geq n_{0}$.
Set $K=\operatorname {\text{supp}}(\varphi)$ and $K^{\varepsilon}=\{x \in \mathbb{R}^{N} \colon d(x,K)<\varepsilon \}$ with $\varepsilon$ sufficiently small to make sure that $K^{\varepsilon}\subset \subset\Omega_{n}\cap \Omega$ for every $n\geq n_{1}$.
We will, from now on, work with $n\geq \max\{n_{0},n_{1}\}$.
Let $\eta \in C^{\infty}_{c}(\Omega)$ such that $\eta=1$ in $K^{\frac{\varepsilon}{2}}$, $\eta=0$ in $(K^{\varepsilon})^{c}$ and $0\leq \eta \leq 1$.
Consider $\phi_{n}=\eta (u_{n}-u^{*})$ and since $\phi_n\in W^{1,p(x)}_0(\Omega_n)$ we have
$$
\int_{D}|\nabla u_{n}|^{p(x)-2}\nabla u_{n}\nabla \phi_{n} \, dx=\int_{\Omega_{n}}|\nabla u_{n}|^{p(x)-2}\nabla u_{n}\nabla \phi_{n} \, dx= \langle f, \phi_{n} \rangle.
$$
Standard computations now give us
$$
\int_{D}|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \eta \nabla (u_{n}-u^{*})\, dx \leq \langle f, \phi_n\rangle - \int_{D}|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \nabla \eta (u_{n}-u^{*})\, dx.
$$
Since $u_{n}\rightharpoonup u^{*}$ en $W^{1,p(x)}_{0}(D)$, $\phi_n\rightharpoonup 0$ in $W^{1,p(x)}_0(\Omega)$ and so $\langle f, \phi_n\rangle\to 0$.
On the one hand, by the compactness of the embedding $W^{1,p(x)}_0(D)\subset L^{p(x)}(D)$, we have that $u_{n}\rightarrow u^{*}$ in $L^{p(x)}(D)$, and so, by H\"older's inequality,
$$
\int_{D}|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \nabla \eta (u_{n}-u^{*})\, dx\le 2\|\nabla \eta\|_\infty \| |\nabla u_n|^{p(x)-2}\nabla u_n\|_{p'(x)} \|u_n-u\|_{p(x)} \rightarrow 0,
$$
by Corollary \ref{lemacontinuidad}. Then we can conclude that
$$
\limsup \int_{D}|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \eta \nabla (u_{n}-u^{*})\, dx\leq 0.
$$
Since $\eta=0$ en $(K^{\varepsilon})^{c}$,
\begin{equation}\label{primera}
\limsup \int_{K^{\varepsilon}}|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \eta \nabla (u_{n}-u^{*})\, dx\leq 0\
\end{equation}
On the other hand, since $\nabla u_{n}\rightharpoonup \nabla u^{*}$ en $L^{p'(x)}(K^{\varepsilon})$,
\begin{equation}\label{segunda}
\int_{K^{\varepsilon}}|\nabla u^{*}|^{p(x)-2}\nabla u^{*} \eta \nabla (u_{n}-u^{*})\, dx\rightarrow 0\
\end{equation}
By \eqref{primera} and \eqref{segunda} we have that
$$
\limsup \int_{K^{\varepsilon}}(|\nabla u_{n}|^{p(x)-2}\nabla u_{n}-|\nabla u^{*}|^{p(x)-2}\nabla u^{*})\eta \nabla (u_{n}-u^{*})\, dx\leq 0.
$$
Since $K^{\frac{\varepsilon}{2}}\subset K^{\varepsilon}$, by Lemma \ref{lemadesigualdad}, we can conclude that
$$
\lim \int_{K^{\frac{\varepsilon}{2}}}(|\nabla u_{n}|^{p(x)-2}\nabla u_{n}-|\nabla u^{*}|^{p(x)-2}\nabla u^{*})\nabla (u_{n}-u^{*})\, dx=0.
$$
Again, by Lemma \ref{lemadesigualdad}, it follows that $(|\nabla u_{n}|^{p(x)-2}\nabla u_{n} - |\nabla u^{*}|^{p(x)-2}\nabla u^{*})\nabla (u_{n}-u^{*})\to 0$ in $L^{1}(K^{\frac{\varepsilon}{2}})$ and therefore a.e. in $K^{\frac{\varepsilon}{2}}$.
From these facts, it easily follows that
\begin{equation}\label{conv.puntual}
\nabla u_{n}\rightarrow \nabla u^{*} \text{ a.e. in } K^{\frac{\varepsilon}{2}}.
\end{equation}
Finally, by Corollary \ref{lemacontinuidad}, there exists $\xi\in L^{p(x)}(K^{\frac{\varepsilon}{2}})^N$ such that $|\nabla u_{n}|^{p(x)-2}\nabla u_{n}\rightharpoonup \xi$ in $L^{p'(x)}(K^{\frac{\varepsilon}{2}})$.
From \eqref{conv.puntual}, we can conclude that $\xi=|\nabla u^{*}|^{p(x)-2}\nabla u^{*}$ in $K^{\frac{\varepsilon}{2}}$ and that
$$
\int_{K^{\frac{\varepsilon}{2}}}|\nabla u_{n}|^{p(x)-2}\nabla u_{n}\nabla \varphi \, dx\rightarrow \int_{K^{\frac{\varepsilon}{2}}}|\nabla u^{*}|^{p(x)-2}\nabla u^{*}\nabla \varphi \, dx.
$$
Since $\operatorname {\text{supp}}(\nabla \varphi)\subset K\subset K^{\frac{\varepsilon}{2}}\subset K^{\varepsilon}\subset \Omega_{n}\cap \Omega$,
$$
\int_{\Omega_{n}}|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \nabla \varphi \, dx\rightarrow \int_{\Omega}|\nabla u^{*}|^{p(x)-2}\nabla u^{*}\nabla \varphi \, dx.
$$
This finishes the proof.
\end{proof}
As we mentioned in Remark \ref{loquefalta}, in order to obtain the continuity of solutions with respect to the domain, we need to provide with conditions than ensure $u^{*}\in W^{1,p(x)}_{0}(\Omega)$. This is the content of the next theorem.
\begin{teo}\label{teo2dirichlet}
Let $D\subset {{\mathbb {R}}}^{N}$ be an open bounded set and let $\Omega_{n}, \Omega \subset D$ be open for every $n$. Let $p\in {\mathcal{P}}^{log}(D)$. If $\Omega_n \stackrel{H}{\to}\Omega$ and $\text{\rm{cap}}_{p(x)}(\Omega_{n} \setminus \Omega,D)\to 0$, then $u_{\Omega_n}^f\rightharpoonup u_{\Omega}^f$ weakly in $W^{1,p(x)}_{0}(D)$.
\end{teo}
\begin{proof}
As before, we denote $u_n = u_{\Omega_n}^f$. By Corollary \ref{corocontinuidad}, $\{u_n\}_{n\in{\mathbb {N}}}$ is bounded in $W^{1,p(x)}_{0}(D)$, therefore, we can assume that $u_{n}\rightharpoonup u^{*}$ weakly in $W^{1,p(x)}_{0}(D)$.
By Theorem \ref{teo1} and Remark \ref{loquefalta} the proof will be finished if we can prove that $u^{*}\in W^{1,p(x)}_{0}(\Omega)$. By Theorem \ref{teocaracterizacion}, it is enough to prove that $\tilde{u^{*}}=0$ $p(x)-$q.e. in $\Omega^{c}$.
Consider $\tilde{\Omega}_{j}=\operatorname{\text{dist}}isplaystyle{\cup_{n\geq j}}\Omega_{n}$ and $E=\operatorname{\text{dist}}isplaystyle{\cap_{j\geq 1}}\tilde{\Omega}_{j}$.
Since $u_{n}\rightharpoonup u^{*}$ in $W^{1,p(x)}_{0}(D)$, by Mazur's Lemma (see for instance \cite{E-T}), there is a sequence $v_{j}=\sum^{k_{j}}_{n=j}a_{n_{j}}u_{n}$ such that $a_{n_{j}}\geq 0$, $\sum^{k_{j}}_{n=j}a_{n_{j}}=1$ and $v_{j}\rightarrow u^{*}$ in $W^{1,p(x)}_{0}(D)$.
Since $u_{n}\in W^{1,p(x)}_{0}(\Omega_{n})$, by Theorem \ref{teocaracterizacion}, $\tilde{u}_{n}=0$ $p(x)-$q.e. in $\Omega_{n}^{c}$. Therefore, $\tilde{v}_{j}=\sum^{k_{j}}_{n=j}a_{n_{j}}\tilde{u}_{n}=0$ $p(x)-$q.e. in $\cap^{k_{j}}_{n=j}\Omega_{n}^{c} \supset \tilde{\Omega}_{j}^{c}$ for every $j\geq 1$. Then, $\tilde{v}_{j}=0$ $p(x)-$q.e. in $\tilde{\Omega}_{j}^{c}$ for every $j\geq 1$. As a consequence, $\tilde{v}_{j}=0$ $p(x)-$q.e. $\cup_{j\geq 1}\tilde{\Omega}_{j}^{c}=E^{c}$.
On the other hand, since $v_{j}\rightarrow u^{*}$ in $W^{1,p(x)}_{0}(D)$, by Proposition \ref{propctp} $\tilde{v}_{j_{k}}\rightarrow \tilde{u^{*}}$ $p(x)-$q.e. Then we conclude that $\tilde{u^{*}}=0$ $p(x)-$q.e. in $E^{c}$.
Since $\text{\rm{cap}}_{p(x)}(\Omega_{n} \setminus \Omega,D)$ goes to zero, passing to a subsequence, if necessary, we can assume that $\text{\rm{cap}}_{p(x)}(\Omega_{n} \setminus \Omega,D)\leq \frac{1}{2^{n}}$. Therefore,
\begin{align*}
\text{\rm{cap}}_{p(x)}(\tilde{\Omega}_{j} \setminus \Omega,D)&=\text{\rm{cap}}_{p(x)}(\cup_{n\geq j}\Omega_{n} \setminus \Omega,D)\\
& \leq \sum_{n\geq j}\text{\rm{cap}}_{p(x)}(\Omega_{n} \setminus \Omega,D)\\
& \leq \sum_{n\geq j}\frac{1}{2^{n}}=\frac{1}{2^{j-1}}.
\end{align*}
Since $E\subset \tilde{\Omega}_{j}$, we have that $E \setminus \Omega\subset \tilde{\Omega}_{j} \setminus \Omega$ for every $j\geq 1$ and so,
$$
\text{\rm{cap}}_{p(x)}(E \setminus \Omega,D)\leq \text{\rm{cap}}_{p(x)}(\tilde{\Omega}_{j} \setminus \Omega,D)\leq \frac{1}{2^{j-1}} \text{ for every } j\geq 1.
$$
Taking the limit $j\to\infty$, we have that $\text{\rm{cap}}_{p(x)}(E \setminus \Omega,D)=\text{\rm{cap}}_{p(x)}(\Omega^{c} \setminus E^{c},D)=0$. So we can conclude that $\tilde{u^{*}}=0$ $p(x)-$q.e. in $\Omega^{c}$, which completes the proof.
\end{proof}
The next result shows that the continuity of the solutions of the Dirichlet problem for the $p(x)-$laplacian with respect to the domain is independent of the second member $f$.
For constant exponents, this result was obtained in \cite[Lemma 4.1]{Bucur}. The proof that we present here, in the non-constant exponent case, follows closely the one in \cite[Theorem 3.2.5]{Henrot} where the linear case $p(x)\equiv 2$ is studied.
\begin{teo}[Independence with respect to the second member] \label{teoindep} Let $\Omega_{n},\Omega\subset D$ be open sets such that $u_{\Omega_{n}}^{1}\rightarrow u_{\Omega}^{1}$ in $L^{p(x)}(D)$. Then $u_{\Omega_{n}}^{f}\rightharpoonup u_{\Omega}^{f}$ in $W_{0}^{1,p(x)}(D)$ for every $f \in W^{-1,p'(x)}(D)$.
\end{teo}
\begin{proof}
Let us assume first that $f \in L^{\infty}(D)$. Therefore, there is a constant $M>0$ such that $-M\leq f \leq M$ a.e. We can also assume that $M>1$.
We will name $u_{n}^{f}=u_{\Omega_{n}}^{f}$ and $u^{f}=u_{\Omega}^{f}$.
Given $k>1$, since $u_{n}^{1}$ is the solution of the equation with $f \equiv 1$,
\begin{align*}
\operatorname{\text{dist}}isplaystyle{\int_{\Omega}|\nabla (k u_{n}^{1})|^{p(x)-2}\nabla (k u_{n}^{1}) \nabla \varphi \, dx} &= \operatorname{\text{dist}}isplaystyle{\int_{\Omega}k^{p(x)-1}|\nabla u_{n}^{1}|^{p(x)-2}\nabla u_{n}^{1} \nabla \varphi \, dx}\\
& \geq k^{p_--1}\operatorname{\text{dist}}isplaystyle{\int_{\Omega}|\nabla u_{n}^{1}|^{p(x)-2}\nabla u_{n}^{1} \nabla \varphi \, dx}\\
&= k^{p_--1} \operatorname{\text{dist}}isplaystyle{\int_{\Omega}\varphi \, dx}.
\end{align*}
Considering $k=M^{\frac{1}{p_--1}}$, we have that $f \leq M=k^{p_--1}\leq -{\mathcal{D}}elta_{p(x)}(k u_{n}^{1})$, therefore, $u$ is a supersolution. Since we also have that $0=u_{n}^{f}|_{\partial D}\leq k u_{n}^{1}|_{\partial D}=0$, by Proposition \ref{propmonotonia}, we can conclude that $u_{n}^{f}\leq k u_{n}^{1}$.
On the other hand, since $-{\mathcal{D}}elta_{p(x)}(-k u_{n}^{1})={\mathcal{D}}elta_{p(x)}(k u_{n}^{1})\leq -k^{p_--1}=-M\leq f$, we obtain that $-k u_{n}^{1}\leq u_{n}^{f}$.
Therefore
\begin{equation}\label{indep}
-k u_{n}^{1}\leq u_{n}^{f}\leq k u_{n}^{1}.
\end{equation}
By Corollary \ref{corocontinuidad}, $\{u_{n}^{f}\}_{n \in {\mathbb {N}}}$ is bounded in $W^{1,p(x)}_{0}(D)$. Then, by Alaoglu's Theorem, there is a subsequence, which will remaine denoted $\{u_{n}^{f}\}_{n \in {\mathbb {N}}}$ such that $u_{n}^{f}\rightharpoonup u^{*}$ en $W_{0}^{1,p(x)}(D)$.
Since, by Rellich-Kondrachov's Theorem, we know that $W_{0}^{1,p(x)}(D)$ is compactedly embbeded in $L^{p(x)}(D)$, we have that $u_{n}^{f}\rightarrow u^{*}$ in $L^{p(x)}(D)$.
Then, taking into account the convergence in $L^{p(x)}(D)$ in \eqref{indep}, we have that
$$
-k u^{1}\leq u^{*}\leq k u^{1}.
$$
Therefore, $|u^{*}|\leq k u^{1}$ and, since $u^{1} \in W_{0}^{1,p(x)}(\Omega)$ we can conclude that $u^{*} \in W_{0}^{1,p(x)}(\Omega)$.
Let us assume now that $f \in W^{-1,p'(x)}(D)$. By density, there is a sequence $\{f_{j}\}_{j \in {\mathbb {N}}} \subset L^{\infty}(D)$ such that $f_{j}\rightarrow f$ in $W^{-1,p'(x)}(D)$.
Given $\varphi \in W^{-1,p'(x)}(D)$,
$$
\langle \varphi , u_n^f - u^f \rangle = \langle \varphi, u_n^f - u_n^{f_j} \rangle + \langle \varphi, u_n^{f_j} - u^{f_j}\rangle + \langle \varphi, u^{f_j} - u^f \rangle.
$$
Now, by Theorem \ref{estabilidadDirichlet}, given $\varepsilon>0$, there exists $j_0\in {\mathbb {N}}$ such that
$$
\|\nabla u_n^f - \nabla u_n^{f_j}\|_{p(x)}\le \varepsilon \quad \text{and}\quad \|\nabla u^f - \nabla u^{f_j}\|_{p(x)}\le \varepsilon,
$$
uniformly in $n\in {\mathbb {N}}$ for every $j\ge j_0$. By the first part of the proof,
$$
\langle \varphi, u_n^{f_j} - u^{f_{j_0}}\rangle\to 0 \quad \text{as } n\to\infty.
$$
This completes the proof.
\end{proof}
\section{Extension of a result of \v{S}ver\'ak.}
In this section, we apply our results to prove the extension of the theorems of \v{S}ver\'ak discussed in the introduction. Our main result being Theorem \ref{teosverak}.
We begin by establishing some capacity estimate from below for compact connected sets. This was obtained for $p(x)\equiv 2$ by \v{S}ver\'ak in \cite{Sverak}. See the book \cite{Henrot} for a proof. For general constant exponents, this estimate was obtained in \cite{Bucur}. Our extension to variable exponents will rely on Bucur and Trebeschi's result \cite{Bucur}. In fact, we use the following proposition.
\begin{prop}[\cite{Bucur}, Lemma 5.2]\label{lemasverakdos}
Let $p>N-1$ be constant and let $K\subset {\mathbb {R}}^N$ be compact and connected. Assume that there exists a constant $a>0$ such that $2a<\operatorname {\text{diam}} K$. Then, for every $x \in K$ and $a\le r< \frac{\operatorname {\text{diam}} K}{2}$, we have the following inequality:
$$
\text{\rm{cap}}_{p}(K\cap \overline{B(x,r)},B(x,2r))\geq c,
$$
for some constant $c>0$ depending only on $p$ and $a$.
\end{prop}
The next proposition relates the relative capacity of a set for constant exponents with the one with variable exponents.
\begin{prop}\label{cotabis}
Let $p \in {\mathcal{P}}^{\log}(D)$. Then,
$$
\text{\rm{cap}}_{p_{-}}(E,D)\leq C \text{\rm{cap}}_{p(x)}(E,D)^{\beta},
$$
where$C>0$ depends on $|D|$, $p_+$ and $p_-$ and $\beta>0$ depends on $p_+$ and $p_-$.
\end{prop}
\begin{proof}
Given $\varphi\in W^{1,p(x)}_0(D)$, by H\"older's inequality and Proposition \ref{propdesigualdades}, we obtain
$$
\int_{D}|\nabla \varphi |^{p_-}\, dx \leq C\left(\int_{D}|\nabla \varphi |^{p(x)}\, dx\right)^{\beta}.
$$
So we conclude
$$
\inf_{\varphi \in S_{p(x)}(E,D)}\int_{D}|\nabla \varphi |^{p_-}\, dx \leq C \left(\inf_{\varphi \in S_{p(x)}(E,D)}\int_{D}|\nabla \varphi |^{p(x)}\, dx\right)^{\beta}.
$$
On the other hand, since $W^{1,p(x)}_{0}(D)\subset W^{1,p_-}_{0}(D)$,
$$
\inf_{\varphi \in S_{p_-}(E,D)}\int_{D}|\nabla \varphi |^{p_-}\, dx \leq C \left(\inf_{\varphi \in S_{p(x)}(E,D)}\int_{D}|\nabla \varphi |^{p(x)}\, dx\right)^{\beta}.
$$
We can conclude that $\text{\rm{cap}}_{p_{-}}(E,D)\leq C \text{\rm{cap}}_{p(x)}(E,D))^{\beta}$.
\end{proof}
From Proposition \ref{lemasverakdos} and Proposition \ref{cotabis} we obtain the following corollary.
\begin{corol}\label{propdefinitivo}
Given $K \subset D \subset {\mathbb {R}}^N$ compact and connected and $p \in {\mathcal{P}}^{log}(B(x,2r))$ such that $p_{-}\ge N-1$. Then, for every $x \in K$ and $a \leq r<\frac{\operatorname {\text{diam}} K}{2}$ for some positive constant $a$,
$$
\text{\rm{cap}}_{p(x)}(K\cap \overline{B(x,r)},B(x,2r))\geq \kappa,
$$
for some constant $\kappa>0$ depending on $|D|$, $\operatorname {\text{diam}} D$, $p_+$ and $p_-$.
\end{corol}
\begin{proof}
Just apply Proposition \ref{cotabis} to the sets $K\cap\overline{B(x,r)}$ and $B(x.2r)$, and observe that $2r<\operatorname {\text{diam}} K\le \operatorname {\text{diam}} D$. Then apply Proposition \ref{lemasverakdos}.
\end{proof}
Now we look for an extension of Theorem \ref{teo2dirichlet} in the sense that instead of requiring some capacity condition on the differences of the approximating domains with the limiting domain, we require a uniform boundary regularity in terms of capacity.
\begin{defi}
We say that $\Omega$ verifies the condition $(p(x), \alpha, r)$ if
$$
\text{\rm{cap}}_{p(x)}(\Omega^{c}\cap B(x,r),B(x,2r))\geq \alpha,\quad x \in \partial \Omega.
$$
Set $\mathcal{O}_{\alpha,r_{0}}(D)=\{\Omega \subset D \text{ open} \colon \Omega \text{ verifies the condition } (p(x), \alpha, r) \text{ for every } 0<r<r_0\}$.
\end{defi}
From now on we will need a result on uniform continuity with respect to $\Omega\in \mathcal{O}_{\alpha, r_0}(D)$ for the solutions of the Dirichlet problem, $u_\Omega^f$ with $f$ sufficiently integrable.
This result for $p(x)\equiv 2$ is classic and can be found, for instance, in \cite[Lemma 3.4.11 and Theorem 3.4.12, p.p. 109]{Henrot}. The key for its proof is to obtain the {\em Wiener conditions}, see \cite{GT}.
The extension for $1<p<N$ constant can be found in the articles \cite{GZ, KM, Mazya}. Consult the book \cite{MZ}, Theorem 4.22. The result for $p(x)$ variable was recently obtained in \cite{Lukkari}.
\begin{lema}[\cite{Lukkari}, Theorem 4.4] \label{lemawiener}
Given $\Omega \in \mathcal{O}_{\alpha,r_{0}}(D)$, $f \in L^{r}(D)$, $r>N$. Then, there are constants $M>0$ and $0<\operatorname{\text{dist}}elta<1$ such that $|u_\Omega^f(x)-u_\Omega^f(y)|\leq M |x-y|^{\operatorname{\text{dist}}elta}$.
\end{lema}
With this result we are able to prove the analogous of Theorem \ref{teo2dirichlet} for domains in $\mathcal O_{\alpha, r}$.
\begin{teo} \label{propalfa}
Given $\{\Omega_{n}\}_{n \in {\mathbb {N}}} \subset \mathcal{O}_{\alpha,r_{0}}(D)$ such that $\Omega_{n}\stackrel{H}{\rightarrow}\Omega$. Then, $u^{f}_{\Omega_{n}}\rightharpoonup u^{f}_{\Omega}$ in $W_{0}^{1,p(x)}(D)$.
\end{teo}
\begin{proof}
By Theorem \ref{teoindep}, we can assume that $f=1$ and $u^{1}_{\Omega_{n}}\rightharpoonup u^{*}$ in $W_{0}^{1,p(x)}(D)$.
In order to see that $u^{*}=u^{1}_{\Omega}$, by Theorem \ref{teo1}, it is enough to verify that $u^{*} \in W_{0}^{1,p(x)}(\Omega)$. By Theorem \ref{teocaracterizacion}, it is enough to prove that $\tilde{u^{*}}=0$ $p(x)-$q.e. in $\Omega^{c}$.
As a direct consecuence of Lemma \ref{ppiomax}, $u^{1}_{D}\geq 0$ and $u^{1}_{\Omega_{n}}\geq 0$.
By Lemma \ref{lemawiener}, given $y \in \partial D$, for every $x \notin \Omega$ we have
$$
u^{1}_{D}(x)=|u^{1}_{D}(x)-u^{1}_{D}(y)|\leq M |x-y|^{\operatorname{\text{dist}}elta}\leq M (\operatorname {\text{diam}} D)^{\operatorname{\text{dist}}elta}.
$$
By Lemma \ref{ppiomax}, $0\leq u^{1}_{\Omega_{n}}\leq u^{1}_{D}\leq M (\operatorname {\text{diam}} D)^{\operatorname{\text{dist}}elta}$. Therefore, $\{u_n\}_{n\in{\mathbb {N}}}$ is uniformly bounded.
By Lemma \ref{lemawiener}, $\{u_n\}_{n\in{\mathbb {N}}}$ is uniformly equicontinuous. Therefore, $\{u_n\}_{n\in{\mathbb {N}}}$ converges uniformily to $u^{*}$.
Given $x \in \Omega^{c}$, since $\Omega_{n}\stackrel{H}{\rightarrow}\Omega$, there is a sequence $x_{n} \in \Omega_{n}^{c}$ such that $x_{n}$ converges to $x$. By uniform convergence, we have that $u_{n}(x_{n})$ converges to $u^{*}(x)$. Since $\operatorname {\text{supp}} u_{n}\subset \bar{\Omega}_{n}$, we obtain that $u_{n}(x_{n})=0$ for every $n$ and, therefore, $u^{*}(x)=0$, which completes the proof.
\end{proof}
\begin{remark}\label{p>N}
If $p_->N$, the same proof can be applied. It is enough to observe that, by Morrey's estimates, $W^{1,p(x)}_0(D)\subset W^{1,p_-}_0(D)\subset C^\alpha(D)$ with $\alpha = 1-N/p_-$.
\end{remark}
Having presented the previous results, the proof of the extension is similar to the one given by \v{S}ver\'ak for $p=2$. We include it for the reader's convenience.
\begin{defi}
Given $l \in {\mathbb {N}}$ y $\Omega \subset D$, set $\#\Omega$ the number of connected components of $D\setminus \Omega$.
Set $\mathcal{O}_{l}(D)=\{\Omega \subset\ D \text{ open} \colon \#\Omega\leq l \}$.
\end{defi}
\begin{teo} \label{teosverak}
Given $p \in {\mathcal{P}}^{log}(D)$ such that $N-1<p_-$ and $\{\Omega_{n}\}_{n \in {\mathbb {N}}}\subset \mathcal{O}_{l}(D)$ such that $\Omega_{n}\stackrel{H}{\rightarrow}\Omega$. Then $u^{f}_{\Omega_{n}}\rightharpoonup u^{f}_{\Omega}$ in $W_{0}^{1,p(x)}(D)$.
\end{teo}
\begin{proof}
By Remark \ref{p>N}, we only have to consider the case $N-1<p_-\le N$.
By Theorem \ref{teoindep}, we can assume that $f=1$ and $u_{n}=u^{1}_{\Omega_{n}}\rightharpoonup u^{*}$ en $W_{0}^{1,p(x)}(D)$.
In order to see that $u^{*}=u^{1}_{\Omega}$, by Theorem \ref{teo1}, it is sufficient to verify that $u^{*} \in W_{0}^{1,p(x)}(\Omega)$.
Set $\bar{D} \setminus \Omega_{n}=F_{n}=F_{n}^{1}\cup F_{n}^{2}\cup...\cup F_{n}^{l}$ where each $F_{n}^{i}$ is compact and connected. Assume that $F_{n}^{j}\stackrel{H}{\rightarrow}F^{j}$ for every $1\leq j\leq l$.
Let us analyze each of the three possibilities. We will find that it is posible to disregard the first two.
(1) If $F^{j}=\emptyset$, then $F_{n}^{j}=\emptyset$ for every $n\geq n_{0}$. Set $J_{0}=\{j\in \{1,\operatorname{\text{dist}}ots,l\} \colon F_{n}^{j}=\emptyset \text{ for j large}\}$.
(2) If $F^{j}=\{x_{j}\}$, set $J_{1}=\{j\in \{1,\operatorname{\text{dist}}ots,l\} \colon F^{j}=\{x_j\} \text{ and } p(x_j)\leq N\}$. Now consider the set $\Omega^{*}=\Omega \setminus \cup_{i \in J_1}\{x_i\}$. Since $\text{\rm{cap}}_{p(x)}(\{x_i\},D)=0$, we have that $\text{\rm{cap}}_{p(x)}(\Omega^*,D)=\text{\rm{cap}}_{p(x)}(\Omega,D)$. Then, by Theorem \ref{teocaracterizacion}, $W_0^{1,p(x)}(\Omega^*)=W_0^{1,p(x)}(\Omega)$. It is enough therefore to verify that $u^* \in W_0^{1,p(x)}(\Omega^*)$.
Set $I=\{1,\operatorname{\text{dist}}ots,l\} \setminus (J_0\cup J_1)$ and consider $\Omega_n^*=D \setminus \cup_{j \in I}F_n^j\stackrel{H}{\rightarrow}\Omega^{*}$.
(3) If, for $j \in I$, $F^{j}$ contains al least two points. Let $a_{j}$ be the distance between them. These points are limits of points from $ F_{n}^{j}$ which we may assume to have a distance at least of $\frac{a_j}{2}$ between them for $n$ large enough.
Given $x \in \partial \Omega_{n}^{*}$ and $j=j(x) \in I$ such that $x \in F_{n}^{j}$, by Corollary \ref{propdefinitivo}, if $a\leq r<\frac{a_{j}}{4}$ for some positive constant $a$, then there is a universal constant $\kappa$ that verifies the following inequality:
$$
\text{\rm{cap}}_{p(x)}((\Omega_{n}^{*})^{c}\cap \overline{B(x,r)},B(x,2r))\geq \text{\rm{cap}}_{p(x)}(F_{n}^{j}\cap \overline{B(x,r)},B(x,2r))\geq \kappa>0.
$$
This shows that the open sets $\Omega_{n}^{*}$ belong to $\mathcal{O}_{\alpha,r_{0}}$ with $\alpha=\kappa$ and $r_{0}=\frac{1}{4}\min\{a_{j}\colon j \in I\}$.
Since $\Omega_{n}^{*}\stackrel{H}{\rightarrow}\Omega$, by Theorem \ref{propalfa}, we have that $u^{1}_{\Omega_{n}^{*}}\rightharpoonup u^{1}_{\Omega}$ in $W_{0}^{1,p(x)}(D)$.
On the other hand, since $\Omega_{n}\subset \Omega_{n}^{*}$, by a direct consequence of Lemma \ref{ppiomax} and Proposition \ref{propmonotonia}, we have that $0\leq u^{1}_{\Omega_{n}}\leq u^{1}_{\Omega_{n}^{*}}$. Passing to the limit $n\to\infty$, $0\leq u^{*}\leq u^{1}_{\Omega}$. We conclude then, by Lemma \ref{lemainf}, that $u^{*} \in W_{0}^{1,p(x)}(\Omega)$
If $F^{j}$ contains exactly one point $x_0$, then $p(x_0)>N$ and so $\{x_0\}$ has positive $p(x)-$capacity, the bound from below will be its capacity, which completes the proof.
\end{proof}
\end{document} | math | 64,885 |
\begin{document}
\title[Vortex Collapse]{Vortex Collapse for the $L^2$-Critical Nonlinear Schr\"odinger Equation}
\author{G. Simpson \& I. Zwiers}
\date{\today}
\begin{abstract}
The focusing cubic nonlinear Schr\"odinger equation in two dimensions
admits vortex solitons, standing wave solutions with spatial
structure, $Q^\m(r,\theta) = e^{im\theta} {(m)} athbb{R}m(r)$. In the case of spin
$m=1$, we prove there exists a class of data that collapse with the
vortex soliton profile at the log-log rate. This extends the work of
Merle and Rapha\"el, (the case $m=0$,) and suggests that the $L^2$
mass that may be concentrated at a point during generic collapse may
be unbounded. Difficulties with $m\gammaeq 2$ or when breaking the spin
symmetry are discussed.
\end{abstract}
{(m)} aketitle
\tableofcontents
\section{Introduction}
We consider the $L^2$-critical nonlinear Schr\"odinger equation in two
dimensions,
\begin{equation}\label{Eqn-NLS}
\left\{\begin{aligned}
&iu_t+\Delta u + u\abs{u}^2 = 0\\
&u(0,x) = u_0 \in H^1( {(m)} athbb{R}^2).
\end{aligned}\right.
\end{equation}
Equation (\ref{Eqn-NLS}) is locally wellposed for data $u_0\in H^1$,
\cite{GinibreVelo-NLSCauchyProb-1979,Kato-NLS-87}. That is, there
exists a solution $u \in C\left([0,T_{ {(m)} ax}),H^1\right)$ and some
fixed negative power so that $T_{ {(m)} ax} \gammaeq T_{lwp}
=\norm{u_0}_{H^1}^{-C}$. Therefore, we have the classic blowup
alternative,
\[\begin{aligned}
T_{ {(m)} ax} = +\infty &&\text{ or, }&& \lim_{t\to
T_{ {(m)} ax}}\norm{u(t)}_{H^1} = +\infty.
\end{aligned}\] Evolution of $u_0$ by equation (\ref{Eqn-NLS})
preserves the following quantities.
\begin{align}
\label{ConserveMass}
M[u_0] = M[u(t)] &= \int_{ {(m)} athbb{R}^{2}}{\abs{u(t,x)}^2\,dx}, &&
\text{(mass)}
\\
\label{ConserveEnergy}
E[u_0] = E[u(t)] &= \int{\abs{\gammarad_xu(t,x)}^2\,dx} -
\phirac{1}{2}\int{\abs{u(t,x)}^4\,dx}, && \text{(energy)}
\\
\label{ConserveMoment}
P[u_0] = P[u(t)] &= {(m)} athrm{Im}\left(\int{\overline{u}(t,x)\gammarad
u(t,x)\,dx}\right). && \text{(momentum)}
\end{align}
The associated symmetries of the equation are phase, time translation,
and spatial translation. There is a Galilean symmetry,
\[\begin{aligned}
u_{\beta_0}(t,x) = u(t,x-\beta_0
t)e^{i\phirac{\beta_0}{2}\cdot\left(x-\phirac{\beta_0}{2}t\right)},
&&\text{for any fixed }\beta_0\in {(m)} athbb{R}^2,
\end{aligned}\] and a scaling symmetry,
\[\begin{aligned}
u_{\lambdabda_0}(t,x) = \lambdabda_0 u(\lambdabda_0^2t,\lambdabda_0 x)
&&\text{for any fixed }\lambdabda_0 > 0.
\end{aligned}\] The effect of scaling on Sobolev norms is,
$\norm{u_{\lambdabda_0}}_{\dot{H}^s} =
\lambdabda_0^{-s}\norm{u}_{\dot{H}^s}$, for any reasonable $s$. Note that
only the critical norm is left invariant. By choosing $\lambdabda_0 =
\norm{u(t)}_{H^1}$ at a fixed time, and using the minimum local
wellposedness time for unit data in $H^1$, we have the scaling lower
bound for the blowup speed,
\[\begin{aligned}
u(t)\in C\left([0,T_{ {(m)} ax}), H^1\right), \text{ with }T_{ {(m)} ax}\text{
maximal, then} &&\norm{u(t)}_{H^1} \gammatrsim
\phirac{1}{\sqrt{T_{ {(m)} ax}-t}}.
\end{aligned}\]
Alternatively, the scaling lower bound can be established through
energy conservation, \cite{CW-CauchyProblemHs-90}.
Peculiar to the $L^2$-critical case, there is also the
pseudo-conformal (or lens) symmetry,
\begin{equation}\label{Defn-Eqn-PseudoConformal}
v(t,x) = \phirac{1}{T-t}u\left(\phirac{1}{(T-t)^2},\phirac{x}{T-t}\right)e^{-i\phirac{\abs{x}^2}{4(T-t)}},
\end{equation}
which acts on the virial space, $\left\{ f\in H^1 \right\}\cap\left\{
\abs{x}^2f \in L^2\right\}$. In particular, the pseudo-conformal
symmetry transforms standing wave solutions into blowup solutions with
$H^1$ norm growth $\phirac{1}{T-t}$.
\subsection{Blowup with Soliton Profile}
To find standing wave solutions of equation (\ref{Eqn-NLS}), introduce
the usual ansatz, $u(t,x) = e^{it}Q(x)$, to derive the profile
equation,
\begin{equation}\label{Eqn-Defn-Q}
\begin{aligned}
\Delta Q - Q + Q\abs{Q}^2 = 0.
\end{aligned}
\end{equation}
There is a unique real-valued positive radial solution $Q$ to equation
(\ref{Eqn-Defn-Q}), as proved by McLeod and Serrin
\cite{McLeodSerrin-Uniqueness-87}\phiootnotemark.
\phiootnotetext{Following earlier work by Coffman
\cite{Coffman-Uniqueness3DCubicGroundState-72} in 3D. Kwong
\cite{Kwong-Uniqueness-89} extended the result to all
$H^1$-subcritical nonlinearities.} This solution we call the
soliton, or the ground-state since $E(Q) = 0$.
In this paper we will focus on other solutions of equation
(\ref{Eqn-Defn-Q}), as we discuss in the next
section.
Weinstein \cite{Weinstein-NLSSharpInterpolation-82} identified the
soliton as the unique minimizer of $J[f] = \phirac{\abs{\gammarad
f}_{L^2}^2\abs{f}_{L^2}^2}{\abs{f}_{L^4}^4}$ among $H^1$
functions, thereby showing the optimal constant of the
Gagliardo-Nirenberg inequality,
\[
\norm{v}_{L^4}^4 \leq
\phirac{2}{\norm{Q}_{L^2}^2}\norm{v}_{\dot{H}^1}^2\norm{v}_{L^2}^2.
\]
Note that if $M[u_0] < M[Q]$, the Gagliardo-Nirenberg inequality gives
apriori control of the $H^1$ norm from the conservation of
energy. That is, there is global wellposedness for data with $M[u_0] <
M[Q]$.
The pseudo-conformal transformation (\ref{Defn-Eqn-PseudoConformal})
applied to the standing wave solution $e^{it}Q(x)$ gives an explicit
blowup solution with $M[u_0] = M[Q]$. We denote this explicit solution
$S(t)$; Merle \cite{Merle-BlowupSolnMinimalMass-93} showed that, up to
symmetries, it is the only blowup solution with the mass of
$Q$. Bourgain and Wang \cite{BourgainWang} proved that $S(t)$ is
stable with respect to perturbations that are exceptionally flat near
the central profile.
More generally, negative energy data in the virial space leads to
blowup, as shown by Glassey's virial identity
\cite{Glassey-BlowupUpSolnsCauchyProbNLS-77},
\[
\phirac{d^2}{dt^2}\int{\abs{x}^2\abs{u(t)}^2} =
4\phirac{d}{dt}{Im}\int{x\cdot\gammarad u\overline{u}} = 16E[u_0]
\]
Ogawa and Tsutsumi
\cite{OgawaTsutsumi-NegEnerBlowupForEnergySubcrit-91} later extended
the argument to negative energy radial data.
Let us consider ${ {(m)} athcal B}_{\alpha} = \{u_0\in H^1 : M[Q] < M[u_0]
< M[Q]+\alpha\}$, where $\alpha>0$ is some small constant. Merle and
Rapha\"el \cite{MR-UniversalityBlowupL2Critical-04} proved that there
is no solution in ${ {(m)} athcal B}_\alpha$ that blows up as predicted by
Glassey's virial identity\phiootnotemark. \phiootnotetext{There is no
solution in ${ {(m)} athcal B}_\alpha$ for which $\lim_{t\to
T_{ {(m)} ax}}\int{\abs{x}^2\abs{u(t)}^2} = 0$, in constrast to the
explicit solution $S(t)$.} They also showed
\cite{MR-SharpUpperL2Critical-03,MR-SharpLowerL2Critical-06} that
there is an open subset ${ {(m)} athcal O}\subset { {(m)} athcal B}_\alpha$,
including all the negative energy data, that lead to blowup in finite
time with the log-log rate,
\[
\norm{u(t)}_{H^1} \approx \sqrt{\phirac{\log\abs{\log(T-t)}}{T-t}}.
\]
Rapha\"el \cite{R-StabilityOfLogLog-05} proved that all solutions in
${ {(m)} athcal B}_\alpha$ that lead to blowup either belong to ${ {(m)} athcal
O}$, or blowup with at least the $H^1$ growth rate of
$S(t)$. Finally, Merle and Rapha\"el \cite{MR-ProfilesQuantization-05}
showed that all solutions in ${ {(m)} athcal B}_\alpha$ that blowup
concentrate exactly the profile $Q$ at a point, in the sense that
there are parameters $\lambdabda(t) > 0$, $\gammaamma(t)\in {(m)} athbb{R}$ and
$\overline{x}(t)\in {(m)} athbb{R}^2$ such that,
\[
u(t,x) -
\phirac{1}{\lambdabda(t)}Q\left(\phirac{x-\overline{x}(t)}{\lambdabda(t)}\right)e^{-i\gammaamma(t)}
\longrightarrow u^*(x),
\]
where the convergence is in $L^2$ as $t\to T_{ {(m)} ax}$. Moreover, the
residual profile $u^*$ identifies the blowup regime, with $u^*\not\in
H^1$ if and only if the solution belonged to ${ {(m)} athcal O}$ and
followed the log-log rate.
\subsection{Vortex Solitons}\label{SubSec-VortexSolitons}
Vortex solitons are solutions to equation (\ref{Eqn-Defn-Q}) of the
form $Q^\m(r,\theta) = e^{im\theta} {(m)} athbb{R}m(r)$, where $ {(m)} athbb{R}m$ is real-valued
and positive. That is, we seek a function $ {(m)} athbb{R}m$ that satisfies,
\begin{equation}\label{Eqn-VortexSoliton-R}
\left\{\begin{aligned}
&\Delta {(m)} athbb{R}m - \left(1+\phirac{m^2}{r^2}\right) {(m)} athbb{R}m + \left( {(m)} athbb{R}m\right)^3 = 0,\\
&\begin{aligned}
\partial_r {(m)} athbb{R}m|_{r=0} = 0, && {(m)} athbb{R}m(\abs{x})>0, && {(m)} athbb{R}m \in
H^1( {(m)} athbb{R}^2)\cap\left\{{\abs{x}}^{-1}f(x)\in L^2\right\}
\end{aligned}
\end{aligned}\right.
\end{equation}
For all $m\in {(m)} athbb{Z}$, Iaia and Warchall
\cite{IaiaWarchall-NonradialSolns-95} showed there exists a solution
to (\ref{Eqn-VortexSoliton-R}) and, analogous to the result of Kwong
\cite{Kwong-Uniqueness-89} in the case $m=0$, Mizumachi
\cite{Mizumachi-VortexSolitons-05} has shown it is unique.
Fibich and Gavish \cite[Lemma 12]{FG-TheorySingularVortex-08} have
remarked that the resulting profile $Q_m$ is the unique minimizer of
$J[f] = \phirac{\abs{\gammarad f}_{L^2}^2\abs{f}_{L^2}^2}{\abs{f}_{L^4}^4}$
among $H^1$ functions with spin $m$. We denote this space by $H^1_\m$.
Some vortex solutions are pictured in Figure \ref{f:vortex_surfaces},
and their radial profiles appear in Figure \ref{f:vortex_radial}.
\ifpdf
\begin{figure}
\caption{The real component of some vortex solutions.}
\label{f:vortex_surfaces}
\end{figure}
\begin{figure}
\caption{The radial component of some vortex solutions.}
\label{f:vortex_radial}
\end{figure}
\else (No figures in DVI mode) \phii
This variational characterization gives an optimal
Gagliardo-Nirenberg inequality
for functions in $H^1_\m$. As a consequence, for data $u_0 \in
H^1_\m$ and $L^2$ norm less than $\norm{Q^\m}_{L^2}$ there is
global wellposedness. As a second consequence, Fibich and
Gavish \cite[Corollary 16]{FG-TheorySingularVortex-08} remark
that $\norm{Q^\m}_{L^2}^2$ is a strictly increasing sequence in
$m$. Indeed, Pego and Warchall
\cite{PegoWarchall-VorticesNLS-02} showed the asymptotic form,
\[
{(m)} athbb{R}m(r) \approx \left(1 +
\phirac{m^2}{r_{ {(m)} ax}^2}\right)^\phirac{1}{2}\sqrt{2} {(m)} athrm{sech}\left(\left(1+\phirac{m^2}{r_{ {(m)} ax}^2}\right)^\phirac{1}{2}(r-r_{ {(m)} ax})\right),
\]
where $r_{ {(m)} ax} \approx \sqrt{2}m$ for $m \gammag 0$.
Therefore, $\norm{Q^\m}_{L^2}^2 \approx 4\sqrt{3} m$ for large
$m$, which Fibich and Gavish found to be a good
approximation\phiootnotemark even for small $m$.
\phiootnotetext{Error less than 3\% for $m=2$, less than 0.4\%
for $m\gammaeq 5$.}
The linearization of equation (\ref{Eqn-NLS}) near $Q^\m$ is,
\begin{equation}\label{Eqn-Linear1}\begin{aligned}
\partial_tv = -iL^\m[v], &&\text{ where,}&& L^\m[v] \equiv
\left(-\Delta + 1 - \abs{Q^\m}^2\right)v -
2\left(Q^\m\right)^2\overline{v}.
\end{aligned}\end{equation}
Written as a harmonic series,
$v=\sum_{j\in {(m)} athbb{Z}}{e^{i(m+j)\theta}f_j(r)}$,
\begin{equation}\label{Eqn-Linear2}\begin{aligned}
L^\m[v] = \sum_{j}\left(-\Delta + 1 -
\abs{Q^\m}^2\right)e^{i(m+j)\theta}f_j -
2\abs{Q^\m}^2e^{i(m-j)\theta}\overline{f_j},
\end{aligned}\end{equation}
so that it is clear the linear system excites harmonics in
pairs. In the case involving only $j=0$, that is, $v =
e^{im\theta}\left(v_1 + iv_2\right)$, we may write $-iL^\m[v]$
in matrix form as,
\begin{equation}\label{Eqn-Defn-L}\begin{aligned}
\left[\begin{matrix}0&L^\m_-\\-L^\m_+&0\end{matrix}\right]\left[\begin{matrix}v_1\\v_2\end{matrix}\right]
&&\text{ where, }\;
\begin{aligned}L^\m_+ &= -\Delta + 1 - 3\abs{Q^\m}^2,\\
L^\m_- &= -\Delta + 1 - \abs{Q^\m}^2.\end{aligned}
\end{aligned}
\end{equation}
Comparing equations (\ref{Eqn-Linear2}) and (\ref{Eqn-Defn-L})
we see that $L^\m$ takes on the form of (\ref{Eqn-Defn-L}) on
all of $H^1$ in the case of spin $m=0$. In this important
case, Weinstein \cite{Weinstein85} showed that the generalized
nullspace of $L$ has dimension $8$ and is generated by the
symmetries.
In the cases $m=1$ and $m=2$, the generalized nullspace of
$L^\m$ is generated in the same way. However, in these cases
Pego and Warchall \cite{PegoWarchall-VorticesNLS-02} found
unstable eigenvalues and additional eigenvalues in the
spectral gap (all for modes with $\abs{j} \neq 0$).
That is, there exists a function $\rho$ with spin $m=1$ such
that,
\[\begin{aligned}
&L^\m_+\left(\rho\right) = -\abs{y}^2Q_m,
&&L^\m_-\left(\abs{y}^2Q_m\right) = -4\Lambda Q_m, \\
&L^\m_+\left(\Lambda Q_m\right) = -2Q_m,
&&L^\m_-\left(Q_m\right) = 0,
\end{aligned}\] where $\Lambda = 1 + y\cdot\gammarad$ denotes the
scaling operator. The remaining Jordan chains, generated by
$\gammarad Q^\m$, consist of functions with $\abs{j} = 1$.
Instability of vortex profiles is not restricted to the cubic
nonlinearity. Mizumachi \cite{Mizumachi-InstabilityVortex-07}
has shown that there are unstable vortex profiles for any
power-type nonlinearity strictly stronger than linear.
\subsection{Blowup with Vortex Profiles}
Any vortex soliton becomes a blowup solution through the
pseudo-conformal transformation. Study of the asymptotic
profile during vortex blowup was initiated by Fibich \& Gavish
\cite{FG-TheorySingularVortex-08}, including the variational
structure referenced above. Their work includes numerical
simulations where they found data with mass slightly larger
than $Q^\m$ that blowup at exactly the scaling lower bound and
with profiles different from the vortex
soliton.
\phiootnote{In particular, they present results using
$u_0 = 1.02 Q^\m(r,\theta)$ and spin $m=2$.
The profiles identified, denoted $G_{m}$, are truncated solutions of equation (\ref{Eqn-Pmb}) with Cauchy boundary conditions and an
implied value of $b$, in this case $b\approx 0.1092$.
Our own truncated solutions of \eqref{Eqn-Pmb} are very similar.
Fibich \& Gavish have conveyed by personal communication corresponding discoveries for spin $m=1$ and data as small as $1.00001Q^\m$.
}
Our main result is that there is a class of solutions with
spin $m=1$ that blowup with exactly the vortex soliton profile
and log-log behaviour similar to that established in the case
$m=0$.
\begin{theorem}[Log-log Blowup with Vortex
Profile]\label{Thm-MainResult}
Assume the Spectral Property\phiootnotemark is true for spin
$m$. Then there exists a class of data ${\mathcal P}^\m$, open as a
subset of $H^1_\m$, such that for $u_0\in {\mathcal P}^\m$ the evolution
$u(t)$ by (\ref{Eqn-NLS}) blows up at finite time $T_{ {(m)} ax}$
with the $Q^\m$ profile and log-log rate. That is, for
$t\in[0,T_{ {(m)} ax})$ there exist continuously variable
parameters $\lambdabda(t) > 0$ and $\gammaamma(t)\in {(m)} athbb{R}$ with the
following properties: \phiootnotetext{See Proposition
\ref{Prop-SpectralProperty}, below.}
\begin{description}
\item[\it Log-log Blowup Rate:]
\begin{equation}\label{Thm-MainResult-BlowupRate}
\lim_{t\to T_{ {(m)} ax}} \norm{u(t)}_{\dot{H}^1}\sqrt{\phirac{T_{ {(m)} ax}-t}{\log\abs{\log T_{ {(m)} ax}-t}}} = C
\end{equation}
\item[\it Description of the Singularity:]
\begin{equation}\label{Thm-MainResult-L2}
\lim_{t\to T_{ {(m)} ax}} u(t,x) - \phirac{1}{\lambdabda(t)}Q^\m\left(\phirac{x}{\lambdabda(t)}\right)e^{-i\gammaamma(t)} = u^*(x) \in L^2( {(m)} athbb{R}^2).
\end{equation}
\end{description}
\end{theorem}
We will now discuss the consistency of the self-similar regime
discovered by Fibich \& Gavish and Theorem
\ref{Thm-MainResult}. Consider,
\[
B_{\alpha,m} = \left\{u_0\in H^1_\m : M[Q^\m] < M[u_0] < M[Q^\m] +
\alpha\right\}.
\]
Then, due to the variational characterization of $Q^\m$:
\begin{theorem}[``Orbital
Stability'']\label{Thm-OrbitalStability}
For $\alpha>0$ sufficiently small, let $v\in B_{\alpha,m}$
with, $E[v] \leq \alpha\norm{v}_{H^1}^2$. Then there exists
$\lambdabda_v>0, \gammaamma_v\in {(m)} athbb{R}$ such that,
\[
\norm{\lambdabda_v\,v\left(\lambdabda_vy\right)e^{i\gammaamma_v}-Q^\m(y)}_{H^1}
\leq \delta(\alpha),
\]
where $\delta(\alpha)\to 0$ as $\alpha\to 0$.
\end{theorem}
The proof is by means of concentration compactness and the
Gagliardo Nirenberg inequality in $H^1_\m$, and is not
constructive. See \cite[Theorem 6]{R-Zurich} for a clear
exposition. The class of data ${\mathcal P}^\m$ from Theorem
\ref{Thm-MainResult} belongs to $B_{\alpha,m}$, and we note that
the orbital stability of Theorem \ref{Thm-OrbitalStability} applies
to all data $v_0\in B_{\alpha,m}$ that blowup in finite time.
Indeed, we conjecture\phiootnotemark that finite time blowup
solutions from the class $B_{\alpha,m}$ either obey the
log-log blowup rate (\ref{Thm-MainResult-BlowupRate}) or the
lower bound, $\norm{v(t)}_{H^1}\gammatrsim (T_{max}-t)^{-1}$.
\phiootnotetext{We expect the analysis of
\cite{R-StabilityOfLogLog-05} to apply, and that the proof
of Theorem \ref{Thm-MainResult} may be reformulated to apply
to all $v_0\in B_{\alpha,m}$ with $\norm{v(t)}_{H^1}\in
L^1\left(t\in[0,T_{max})\right)$, as in
\cite{MR-SharpLowerL2Critical-06}.}
Collapse at the square-root rate has also been observed numerically in
the case with no spin, \cite{FGW-NewSingularSolns-05}.
These examples are an important area of continuing study.
It is possible that the threshold $\alpha$ of Theorem \ref{Thm-OrbitalStability} (and hence the applicability of Theorem \ref{Thm-MainResult}) is exceedingly small.
\begin{comment}
We conclude that either
the solutions found by Fibich \& Gavish will
collapse at the log-log rate after extreme levels of focusing, or that
the threshold $\alpha$ of Theorem \ref{Thm-OrbitalStability} (and thence the applicability of Theorem \ref{Thm-MainResult}) is exceedingly small\phiootnotemark. This unsatisfactory
conclusion simply emphasizes the importance of these
questions. \phiootnotetext{On the basis of equation
\eqref{Prop-Qb-Mass}, we believe that the mass of the
$G_{m}$ profile with $b = 0.1092$ is approximately
$1\%$ greater than the vortex soliton.
The fundamental log-log relationship $\lambdabda \approx
e^{-e^\phirac{\pi}{b}}$ implies focusing powers that are
simply unachievable.}
\end{comment}
\begin{comment}
\begin{remark}[Further Solutions Described by Theorem
\ref{Thm-MainResult}]
Define,
\[
B_{\alpha,m} = \left\{u_0\in H^1_m | \norm{Q_m}_{L^2} <
\norm{u_0}_{L^2} < \norm{Q_m}_{L^2}+\alpha\right\}.
\] There is a type of orbital stability in $B_{\alpha,
m}$, namely that for $\alpha>0$ sufficiently small, if
$u \in B_{\alpha,m}$, and $E[u] \leq
\alpha\norm{u}_{\dot{H}^1}^2$, then up to symmetries $u$
is $H^1$-close (depending on $\alpha$) to $Q_m$. The proof
is by concentration compactness and the Gagliardo
Nirenberg inequality in $H^1_m$, see \cite[Theorem
6]{R-Zurich} for a clear exposition. Note that if $u_0\in
B_{\alpha,m}$ blows up in finite time, then eventually we
can apply this orbital stability.
By entering an arbitrary $H^1$ proximity of $Q_m$, up to
symmetries, $u(t)$ has entered the domain of the implicit
function theorem argument we will use to define ${ {(m)} athcal
P}_m$, Lemma \ref{Lemma-Modulation}. Therefore, if
$u_0\in B_{\alpha,m}$ blows up in finite
time\phiootnotemark, then the blowup behaviour is described
by Theorem \ref{Thm-MainResult}. \phiootnotetext{This
should include all $u_0\in B_{\alpha,m}$ with zero
momentum and negative energy, due to Glassey's virial
identity.}
\end{remark}
\end{comment}
\subsection{Spectral Propety}
\label{s:intro_specprop}
In order to demonstrate the dynamic claimed in Theorem
\ref{Thm-MainResult}, we will attempt to parameterize the
solution in terms of the symmetries and a suitable deformation
of the profile $Q^\m$. In order for the finite-dimensional
system of parameters to capture the essential dynamics of the
solution we require two things. First, that the parameter
dynamics can be reliably predicted from a finite system of
differential inequalities. Second, that after removing the
central profile from the solution the error $\epsilonilon$ can be
estimated in terms of those parameter dynamics.
That the parameter dynamic are stable is an essential feature
of the log-log regime. Indeed, Rapha\"el showed
\cite{R-StabilityOfLogLog-05} that the relationship between a
particular ratio of parameters\phiootnotemark and a fixed
constant evolves according to a Riccati equation, with the
log-log dynamic corresponding to the stable branch.
\phiootnotetext{Namely the sign of $f_- =
\phirac{b}{\lambdabda}-d_0\sqrt{E_0}$. Parameter $b$ will be
introduced in Section \ref{SubSec-DecompModulate}.} To
control the error $\epsilonilon$ in terms of the dynamics, we will
consider the following operator, derived from the linearized
energy,
\begin{equation}\label{Eqn-Defn-H}\begin{aligned}
{(m)} athcal{H}m(\epsilonilon,\epsilonilon) =
\left\langle {(m)} athcal{L}m_1\epsilonilon_1, \epsilonilon_1\right {(m)} athrm{ran}gle +
\left\langle {(m)} athcal{L}m_2\epsilonilon_2, \epsilonilon_2\right {(m)} athrm{ran}gle,
\end{aligned}\end{equation}
where
\begin{align}
{(m)} athcal{L}m_1 = -\Delta + 3Q^\m y\cdot\gammarad Q^\mbar, &\quad
{(m)} athcal{L}m_2 = -\Delta + Q^\m y\cdot\gammarad Q^\mbar,\\
\epsilonilon_1 = e^{im\theta}\,Re\left( e^{-im\theta}\epsilonilon
\right),&\quad \epsilonilon_2 = e^{im\theta}\,Im\left(
e^{-im\theta}\epsilonilon \right).
\end{align}
This decomposition, $\epsilonilon = \epsilonilon_1 + i\epsilonilon_2$, is
powerful, as it reduces the algebraic structure of the problem
in $H^1_m$ to that of the radially symmetric problem in $H^1$.
For further discussion, see \eqref{Eqn-Defn-Notation}, below.
We will prove the following for $m=1$,
\begin{prop}[Spectral Property]
\label{Prop-SpectralProperty}
Let $\epsilonilon \in H^{1}_{(m)}$ Then there exists a universal
constant $\delta_m$ such that
\begin{equation}\label{Eqn-SpectralProperty}\begin{aligned}
{(m)} athcal{H}m(\epsilonilon,\epsilonilon) \gammaeq
&\delta_m\left(\int{\abs{\gammarad_y\epsilonilon}^2} + \int{\abs{\epsilonilon}^2e^{-\abs{y}}}\right)\\
&\quad - \phirac{1}{\delta_m} \left( \left\langle
\epsilonilon_1,Q^\m\right {(m)} athrm{ran}gle^2 +\left\langle
\epsilonilon_1,\Lambda Q^\m\right {(m)} athrm{ran}gle^2\right. \\
&\qquad \left.+\left \langle \epsilonilon_2,\Lambda
Q^\m\right {(m)} athrm{ran}gle^2 +\left\langle
\epsilonilon_2,\Lambda^2 Q^\m\right {(m)} athrm{ran}gle^2 \right).
\end{aligned}\end{equation}
\end{prop}
In the case of $L^2$-critical nonlinearity, no spin, and
dimension $N=1$, Merle and Rapha\"el \cite[Appendix
A]{MR-BlowupDynamic-05} gave an explicit proof of the Spectral
Property. In the case of $L^2$-critical nonlinearity, no spin,
and dimensions $N=2,3,4,5$, including equation \eqref{Eqn-NLS}
in the case $m=0$, Fibich, Merle and Rapha\"el
\cite{FMR-ProofOfSpectralProperty-06} have given a numerical
proof that inspires our own proof of Proposition
\ref{Prop-SpectralProperty} in Section
\ref{Section-SpectralProperty}. Details of our numerical
methods are provided in Appendix \ref{s:numerics}. Code to
reproduce our computations is available at
\url{http://www.math.toronto.edu/simpson/files/vortex_dist.tgz}.
As stated, the spectral property is false for $m=2, 3$.
\section{Proof of Log-log Blowup}
In this section, we prove Theorem \ref{Thm-MainResult} assuming
Proposition \ref{Prop-SpectralProperty}. Before decomposing the
solution, we introduce almost self-similar deformations of the vortex
profiles that simulate the effect of symmetries that do not belong to
$H^1$. The standard self-similar ansatz is, $u(t,x) =
\phirac{1}{\sqrt{2b(T-t)}}Q^\mb\left(\phirac{x}{\sqrt{2b(T-t)}}\right)e^{i\omega(t)}$,
which gives the following equation for the spatial profile,
\begin{equation}\label{Eqn-Qmb}
\Delta Q^\mb - Q^\mb +ib\Lambda Q^\mb+ Q^\mb\abs{Q^\mb}^2=0.
\end{equation}
We seek solutions with spin $m$. Remove a quadratic phase,
$e^{im\theta}P^\m_b(r) = Q^\mb\,e^{ib\phirac{r^2}{4}}$, and assume the
radial profile $P^\m_b(r)$ is real valued. We seek solutions to,
\begin{equation}\label{Eqn-Pmb}
\left\lbrace\begin{aligned}
& \Delta P^\m_b - \left(1+\phirac{m^2}{r^2}-\phirac{b^2}{4}r^2\right)P^\m_b + \left(P^\m_b\right)^3 = 0,\\
&\begin{aligned}
\lim_{r\to 0^+}r^{-m}P^\m_b(r) \neq 0,
&& \lim_{r\to 0^+}\partial_r\left(r^{-m}P^\m_b(r)\right) = 0.
\end{aligned}
\end{aligned}\right.
\end{equation}
As pointed out by Fibich and Gavish \cite[Lemma
8]{FG-TheorySingularVortex-08}, equation (\ref{Eqn-Pmb}) does not
admit solutions in either $L^2$ or $\dot{H}^1$, due to oscillations of
amplitude $r^{-1}$ outside the domain of uniform ellipticity of the
linear part. The argument is due to Johnson and Pan \cite{RX}. We
truncate a solution of (\ref{Eqn-Pmb}) at an arbitrary point, chosen
to allow close approximation to the vortex profile $Q^\m$. Define,
\begin{equation}\label{Eqn-Defn-Rb}\begin{aligned}
R_b = \phirac{1}{\abs{b}}\sqrt{2+2\sqrt{1+b^2m^2}} &\gammaeq
\phirac{2}{\abs{b}}.
\end{aligned}\end{equation}
\begin{prop}[Localized Self-Similar Profiles]\label{Prop-Qb}
Let $a > C\eta>0$ where $C>0$ is a fixed constant and $a,\eta$ are
sufficiently small parameters. Then for $\abs{b}>0$ sufficiently
small, there exists $Q^\mbT \in H^1( {(m)} athbb{R}^2)$, supported on $\abs{y} <
(1-\eta)R_b$, with the following properties.
\begin{itemize}
\item {\it Simple Profile}:
\begin{equation}\label{Prop-Qb-RealProfile}\begin{aligned}
Q^\mbT = e^{im\theta}e^{-ib\phirac{\abs{y}^2}{4}}P^\m_bT(\abs{y}),
&&\text{for }P^\m_bT\text{ real-valued, non-negative.}
\end{aligned}\end{equation}
\item {\it Algebraic Proximity to $Q^\m$}:
\begin{equation}\label{Prop-Qb-Eqn}
\DeltaQ^\mbT - Q^\mbT + ib\LambdaQ^\mbT + Q^\mbT\abs{Q^\mbT}^2 = -\Psi_b,
\end{equation}
for an error term $\Psi_b$, supported on
$(1-\eta)^2R_b<\abs{y}<(1-\eta)R_b$, that satisfies the estimate,
$\norm{P(y)\gammarad^k\Psi_b}_{L^\infty} \leq
e^{-\phirac{C(P)}{\abs{b}}},$
for $k=0,1$ and any polynomial $P$.
\item {\it Uniform Proximity to $Q^\m$}:
\begin{equation}\label{Prop-Qb-closeToQ}\begin{aligned}
\left.\norm{e^{C\abs{y}}\left(Q^\mbT-Q^\m\right)}_{C^{3}}
+\norm{e^{C\abs{y}} \left(\phirac{\partial}{\partial b}Q^\mbT +
i\phirac{\abs{y}^2}{4}Q^\m\right)}_{C^2}
\right.\longrightarrow 0 && \text{ as } && b \rightarrow 0.
\end{aligned}\end{equation}
\item {\it Supercritical Mass and Degenerate Energy}:
\begin{equation}\label{Prop-Qb-Mass}\begin{aligned}
\left. \phirac{\partial}{\partial(b^2)}\norm{Q^\mbT}^2_{L^2}\right|_{b^2=0}
= \phirac{1}{4}\int{\abs{x}^2\abs{Q^\m}^2}, && \text{ denoted }
&& d_m, && \text{ and,}
\end{aligned}\end{equation}
\begin{equation}\label{Prop-Qb-EnerMomentum}\begin{aligned}
\abs{E\left[Q^\mbT\right]} \leq
e^{-(1+C\eta)(1-a)\phirac{\pi}{\abs{b}}}.
\end{aligned}\end{equation}
\end{itemize}
\end{prop}
The proof of Proposition \ref{Prop-Qb} is similar to that given by
Merle and Rapha\"el \cite{MR-SharpUpperL2Critical-03,
MR-UniversalityBlowupL2Critical-04, MR-SharpLowerL2Critical-06} in
the case of $m=0$. An overview of the proof, and description of the
particular adaptations for $m\neq 0$, is given in Appendix
\ref{Appendix-ProofOfAlmostSelfSimilar}.
Later in the argument, Section \ref{SubSec-Lyapounov}, we will
introduce the linear radiation induced by the truncation error
$\Psi_b$. A quantity $ {(m)} athcal{G}amma_b$, related to the decay of this
radiation, will be an important dynamical quantity, measuring the rate
of mass ejection from the singular region. At the time we formally
define $ {(m)} athcal{G}amma_b$, Proposition \ref{Prop-Zb}, we will also prove the
following estimate,
\begin{equation}\label{Eqn-GammaBEstimate}\begin{aligned}
e^{-(1+C\eta)\phirac{\pi}{b}} \lesssim {(m)} athcal{G}amma_b \lesssim
e^{-(1-C\eta)\phirac{\pi}{b}}.
\end{aligned}\end{equation}
\subsection{Decomposition \& Modulation}\label{SubSec-DecompModulate}
\begin{lemma}[Modulation Near $Q^\m$]\label{Lemma-Modulation}
Suppose that $v\in H^1_\m$ is close to $Q^\m$, up to symmetries:
\begin{equation}\label{Lemma-Mod-GeoDecomp}
v(x) =
\phirac{1}{\lambdabda_v}\left(Q^\mT_{b_v}+\epsilonilon_v\right)
\left(\phirac{x}{\lambdabda}\right)e^{-i\gammaamma_v},
\end{equation}
for some symmetry parameters $\lambdabda_v > 0$, $b_v > 0$ and
$\gammaamma_v \in {(m)} athbb{R}$ such that the error is comparably small,
\begin{equation}\label{Lemma-Mod-EpsilonAssumption}
\int{\abs{\gammarad_y\epsilonilon_v(y)}^2\,dy}
+\int_{\abs{y}\leq\phirac{10}{b_v}}{\abs{\epsilonilon_v}^2e^{-\abs{y}}\,dy}
< {(m)} athcal{G}amma_{b_v}^\phirac{1}{2},
\end{equation}
where $y$ denotes $\phirac{x}{\lambdabda_v}$, and such that the deformed
profile is sufficiently close to $Q^\m$,
\begin{equation}\label{Lemma-Mod-ParamAssumption}\begin{aligned}
\lambdabda_v < \phirac{1}{10} b_v &&\text{ and, }&& b_v < \alpha^*.
\end{aligned}\end{equation}
Then there are parameters $\lambdabda_0 > 0$, $b_0 > 0$ and $\gammaamma_0
\in {(m)} athbb{R}$, nearby in the sense,
\begin{equation}\label{Lemma-Mod-NewParamAreClose}
\abs{b_0-b_v}
+\abs{\phirac{\lambdabda_0}{\lambdabda_v} - 1}
\leq {(m)} athcal{G}amma_{b_0}^\phirac{1}{5},
\end{equation}
and such that the error $\epsilonilon_0$ corresponding to these
parameters,
\begin{equation}\label{Lemma-Mod-DefnNewEpsilon}
\epsilonilon_0(y) = \lambdabda_0\,v\left(\lambdabda_0 y\right)\,e^{i\gammaamma_0} - Q^\mT_{b_0},
\end{equation}
satisfies the following orthogonality conditions\phiootnotemark:
\begin{equation}\label{Lemma-Mod-OrthogConditions}
{(m)} athrm{Re}\left\langle\epsilonilon_0,\abs{y}^2Q^\mT_{b_0}\right {(m)} athrm{ran}gle =
{(m)} athrm{Im}\left\langle\epsilonilon_0,\Lambda^2Q^\mT_{b_0}\right {(m)} athrm{ran}gle =
{(m)} athrm{Im}\left\langle\epsilonilon_0,\LambdaQ^\mT_{b_0}\right {(m)} athrm{ran}gle = 0.
\end{equation}
\end{lemma}
\phiootnotetext{ These orthogonality conditions were introduced
\cite[Lemma 6]{MR-UniversalityBlowupL2Critical-04}, and lead to a
better estimate on the phase parameter than achieved in
\cite{MR-SharpUpperL2Critical-03}. }
Let us reiterate and extend the notation alluded to by equation
(\ref{Eqn-Defn-H}),
\begin{equation}\label{Eqn-Defn-Notation}
\begin{aligned}
&\left.\begin{aligned}
&\epsilonilon_1 = e^{im\theta}\text{Re}\left(e^{-im\theta}\epsilonilon\right)\\
&\epsilonilon_2 =
e^{im\theta}\text{Im}\left(e^{-im\theta}\epsilonilon\right)
\end{aligned}\right\}
&&\Longrightarrow \epsilonilon = \epsilonilon_1 + i\epsilonilon_2,\\
&\left.\begin{aligned}
&\Sigma = e^{im\theta}\text{Re}\left(e^{-ib\phirac{\abs{y}^2}{4}}P^\m_bT\right)\\
& \Theta =
e^{im\theta}\text{Im}\left(e^{-ib\phirac{\abs{y}^2}{4}}P^\m_bT\right)
\end{aligned}\right\}
&&\Longrightarrow Q^\mbT = \Sigma + i\Theta.
\end{aligned}
\end{equation}
Products between the components of $\epsilonilon$ and $Q^\mbT$ behave as if
they were real-valued, as does the modulus, for example,
$\abs{\epsilonilon}^2 = \abs{\epsilonilon_1}^2+\abs{\epsilonilon_2}^2$. Moreover,
since $\abs{y}^2$ and the scaling operator, $\Lambda = 1 +
y\cdot\gammarad_y$, are radial operators, the algebraic relations for
$\abs{y}^2Q^\mbT$ and $\LambdaQ^\mbT$ are exactly the same as the case
$m=0$, \cite[Proposition 9 (iii)]{MR-UniversalityBlowupL2Critical-04}.
In particular, one may verify that, $ {(m)} athcal{L}m_1(\epsilonilon_1) =
\phirac{1}{2}\left[L^\m_+(\Lambda\epsilonilon_1) -
\Lambda(L^\m_+\epsilonilon_1)\right]$, is true regardless of $m$. This is
the essential relationship for Lemma \ref{Lemma-SpectralProperty2},
below. In the notation of (\ref{Eqn-Defn-Notation}), the
orthogonality conditions of equation
(\ref{Lemma-Mod-OrthogConditions}) can be written,
\[\begin{aligned}
&\left\langle\epsilonilon_1,\abs{y}^2\Sigma\right {(m)} athrm{ran}gle +
\left\langle\epsilonilon_2,\abs{y}^2\Theta\right {(m)} athrm{ran}gle = 0,\\
&\left\langle\epsilonilon_2,\Lambda^2\Sigma\right {(m)} athrm{ran}gle -
\left\langle\epsilonilon_1,\Lambda^2\Theta\right {(m)} athrm{ran}gle = 0,\\
&\left\langle\epsilonilon_2,\Lambda\Sigma\right {(m)} athrm{ran}gle -
\left\langle\epsilonilon_1,\Lambda\Theta\right {(m)} athrm{ran}gle = 0.
\end{aligned}\] These are exactly the same form as in the case
$m=0$. Indeed, the proof of Lemma \ref{Lemma-Modulation}, an implicit
function argument, is identical. See \cite[Lemma 2]{R-Zurich} for a
clear exposition. For $m=0$, the following Lemma was proven by Merle
and Rapha\"el \cite[equation
(116)]{MR-UniversalityBlowupL2Critical-04}, and the same proof applies
here.
\begin{lemma}\label{Lemma-SpectralProperty2}
Let $\epsilonilon\in H^1_\m$, and assume the Spectral Property is
true. Then,
\begin{equation}\label{Eqn-SpectralProperty2}\begin{aligned}
\left\langleL^\m_1\epsilonilon_1,\epsilonilon_1\right {(m)} athrm{ran}gle& -
\phirac{\left\langle\epsilonilon_1,L^\m_+\Lambda^2Q^\m\right {(m)} athrm{ran}gle
\left\langle\epsilonilon_1,\LambdaQ^\m\right {(m)} athrm{ran}gle}{\norm{\LambdaQ^\m}_{L^2}^2}
\gammaeq\\
&\delta_m\left(\int{\abs{\gammarad_y\epsilonilon}^2} +
\int{\abs{\epsilonilon}^2e^{-\abs{y}}}\right) - \phirac{1}{\delta_m}
\left( \left\langle \epsilonilon_1,Q^\m\right {(m)} athrm{ran}gle^2 +\left\langle
\epsilonilon_1, \abs{y}^2 Q^\m\right {(m)} athrm{ran}gle^2 \right),
\end{aligned}\end{equation}
\end{lemma}
\begin{definition}[Description of Initial Data]\label{Defn-P}
Define ${\mathcal P}^\m$ to be those functions $u_0\in H^1_\m$ for which there
are parameters $\lambdabda_0 > 0$, $b_0>0$ and $\gammaamma_0\in {(m)} athbb{R}$ that
satisfy the following conditions. Let $\epsilonilon_0$ denote the error
in approximating $u_0$ with these particular parameters,
\begin{equation}\label{Defn-P-GeoDecomp}\begin{aligned}
u_0(x) &= \phirac{1}{\lambdabda_0}\left(Q^\mT_{b_0}+\epsilonilon_0\right)
\left(\phirac{x}{\lambdabda_0}\right)e^{-i\gammaamma_0}.
\end{aligned}\end{equation}
We require that the orthogonality conditions
(\ref{Lemma-Mod-OrthogConditions}) are satisfied, that there is,
\begin{description}
\item[\it proximity to $Q^\m$,]
\begin{equation}\label{Defn-P-Proximity}\begin{aligned}
\text{in }L^2:
&&& 0 < b_0^2 + \norm{\epsilonilon}_{L^2}^2 < (\alpha^*)^2,\\
\text{in }\dot{H}^1: &&& \int{
\abs{\gammarad_y\epsilonilon_0(y)}^2\,dy}
+\int_{\abs{y}\leq\phirac{10}{b_0}}{\abs{\epsilonilon_0(y)}^2e^{-\abs{y}}\,dy}
< {(m)} athcal{G}amma_{b_0}^\phirac{6}{7},
\end{aligned}\end{equation}
\item[\it parameters consistent with the log-log rate,]
\begin{equation}\label{Defn-P-loglog}\begin{aligned}
e^{-e^\phirac{2\pi}{b_0}} < \lambdabda_0 <
e^{-e^{\phirac{\pi}{2}\phirac{1}{b_0}}}, &&\text{ and,}
\end{aligned}\end{equation}
\item[\it normalized energy,]
\begin{equation}\label{Defn-P-EnergyMomentum}
\lambdabda_0^2\abs{E_0}
< {(m)} athcal{G}amma_{b_0}^{10}.
\end{equation}
\end{description}
\end{definition}
\begin{remark}[${\mathcal P}^\m$ is Non-Empty]\label{Remark-PmNonempty}
Choose $b_0$ and $\lambdabda_0$ to satisfy (\ref{Defn-P-Proximity}) and
(\ref{Defn-P-loglog}). Let $f\in H^1_\m$ satisfy orthogonality
conditions (\ref{Lemma-Mod-OrthogConditions}) with $\norm{f}_{H^1} =
1$, $\inner{f}{Q^\m} = 1$. Such an $f$ may be computed explicitly
from $Q^\m$. Note that $\left.\partial_\nu E[Q^\m+\nu f]\right|_{\nu =
0} = - \inner{F}{Q^\m} = -1$, and therefore we may choose
$\epsilonilon_0 = \nu f$ with $\nu$ of the order of $E[Q^\mb]$ to satisfy
(\ref{Prop-Qb-EnerMomentum}).
\end{remark}
For the remainder of this section, we consider a fixed representative
$u_0 \in {\mathcal P}^\m$. By the continuity of the flow of (\ref{Eqn-NLS}) in
$H^1$, and Lemma \ref{Lemma-Modulation}, there exists continuous
functions $\lambdabda(t) > 0$, $b(t)>0$ and $\gammaamma(t)\in {(m)} athbb{R}$ and some
maximal $T_{\mathrm{hyp}}\in(0,T_{ {(m)} ax}]$ such that the following relaxations
of (\ref{Defn-P-Proximity}), (\ref{Defn-P-loglog}) and
(\ref{Defn-P-EnergyMomentum}) hold for all $t\in[0,T_{\mathrm{hyp}})$:
\begin{equation}\label{Hypo-Proximity-L2}
0 < b^2(t) + \norm{\epsilonilon(t)}_{L^2}^2 < (\alpha^*)^\phirac{1}{5},
\end{equation}
\begin{equation}\label{Hypo-Proximity-H1}
\int{ \abs{\gammarad_y\epsilonilon(t,y)}^2\,dy}
+\int_{\abs{y}\leq\phirac{10}{b(t)}}{\abs{\epsilonilon(t,y)}^2e^{-\abs{y}}\,dy}
< {(m)} athcal{G}amma_{b(t)}^\phirac{3}{4},
\end{equation}
\begin{equation}\label{Hypo-loglog}\begin{aligned}
e^{-e^\phirac{10\pi}{b(t)}} < \lambdabda(t) <
e^{-e^{\phirac{\pi}{10}\phirac{1}{b(t)}}}, &&\text{ and,}
\end{aligned}\end{equation}
\begin{equation}\label{Hypo-EnergyMomentum}
\lambdabda^2(t)\abs{E_0}
< {(m)} athcal{G}amma_{b(t)}^{2}.
\end{equation}
Note that as a consequence of these hypotheses, we may apply Lemma
\ref{Lemma-Modulation} at any $t\in[0,T_{\mathrm{hyp}})$. Therefore, one of
the following occurs:
\begin{description}
\item[Case 1:] $T_{\mathrm{hyp}} < T_{ {(m)} ax}$, and one of the hypotheses fails
at $t = T_{\mathrm{hyp}}$, or
\item[Case 2:] $T_{\mathrm{hyp}} = T_{ {(m)} ax}$, $b\to 0$ as $t\to T_{ {(m)} ax}$, and
due to (\ref{Hypo-loglog}) we have blowup.
\end{description}
In this section we will show that {\bf Case 1} cannot occur. Then,
assuming {\bf Case 2}, we will derive the conclusions of Theorem
\ref{Thm-MainResult}.
\begin{remark}[Parameters]
The parameter $\eta>0$, already introduced, relates to the cutoff
and shape of the singular profile $Q^\mbT$. Parameter $a>0$, to be
introduced in Section \ref{SubSec-Lyapounov}, will be related to a
cutoff point of the linear radiation associated with $Q^\mbT$. The
value of $\eta$ is determined by the value of $a$ so that the
argument of Subsection \ref{SubSubSec-HypoH1} is successful. At all
times, $\alpha^*>0$ is assumed sufficiently small for all the
appropriate constants to cooperate.
\end{remark}
\subsection{Conservation Laws \& Basic Estimates}
By substitution of the time-dependent version of the geometric
decomposition (\ref{Defn-P-GeoDecomp}), the conservation laws of
(\ref{Eqn-NLS}) and the orthogonality conditions
(\ref{Lemma-Mod-OrthogConditions}) lead to some basic estimates.
\begin{lemma}
\label{Lemma-PrelimEstimatesConservLaws}
For all $t\in[0,T_{\mathrm{hyp}})$,
\begin{description}
\item[\it due to conservation of mass,]
\begin{equation}\label{Eqn-prelimMassEst}
b^2 + \int{\abs{\epsilonilon}^2} \lesssim \left(\alpha^*\right)^\phirac{1}{2},
\end{equation}
\item[\it due to conservation of energy,]
\begin{equation}\label{Eqn-prelimEnerEst}\begin{aligned}
&2 {(m)} athrm{Re}\left\langle\epsilonilon,Q^\mbT-ib\LambdaQ^\mbT -
\Psi_b\right {(m)} athrm{ran}gle \sim \int{\abs{\gammarad_y\epsilonilon}^2\,dy}
-3\int_{\abs{y}\leq\phirac{10}{b}}{\abs{Q^\m\epsilonilon_1}^2}
-\int_{\abs{y}\leq\phirac{10}{b}}{\abs{Q^\m\epsilonilon_2}^2},\\
&\begin{aligned}\text{with error of the order, }&&
{(m)} athcal{G}amma_b^{1-C\eta} +
\delta(\alpha^*)\left(\epsilonNorm\right).\end{aligned}
\end{aligned}\end{equation}
\end{description}
\end{lemma}
\begin{proof}
To prove (\ref{Eqn-prelimMassEst}), expand the conservation of mass,
\[
\int{\abs{Q^\mbT}^2\,dy} - \int{\abs{Q^\m}^2\,dy}
+2Re\left\langle\epsilonilon,Q^\mbT\right {(m)} athrm{ran}gle +\int{\abs{\epsilonilon}^2}
=\int{\abs{u_0}^2} - \int{\abs{Q^\m}^2\,dy}.
\]
Recognize $\partial_{(b^2)}\norm{Q^\mbT}_{L^2}^2$, and recall
(\ref{Prop-Qb-Mass}). Use initial condition (\ref{Defn-P-Proximity})
and hypothesis (\ref{Hypo-Proximity-H1}). To prove
(\ref{Eqn-prelimEnerEst}), expand the conservation of energy, as in
\cite[eqn (188)]{MR-UniversalityBlowupL2Critical-04}.
Use the normalized energy (\ref{Hypo-EnergyMomentum}) to estimate
$\lambdabda^2E_0$. For the terms ${ {(m)} athcal O}(\epsilonilon^3)$, use the
exponential decay of $Q^\m$, the Hardy-type inequalities below, and
hypothesis (\ref{Hypo-Proximity-H1}).
\end{proof}
\begin{lemma}[Hardy-type Inequalities]
For any $\kappapa > 0$ and for all $v \in H^1( {(m)} athbb{R}^2)$,
\begin{equation}\label{Eqn-ExpDecayByGrad}
\int_{y\in {(m)} athbb{R}^2}{\abs{v(y)}^2e^{-\kappapa\abs{y}}}
\leq C(\kappapa)\left(\int{\abs{\gammarad v(y)}^2} + \int_{\abs{y} \leq 1}{\abs{v(y)}^2e^{-\abs{y}}}\right),
\end{equation}
\begin{equation}\label{Eqn-L2ByGrad}
\int_{\abs{y} \leq \kappapa}{\abs{v(y)}^2} \leq C\,\kappapa^2\log \kappapa
\left(\int{\abs{\gammarad v(y)}^2}
+ \int_{\abs{y}\leq 1}{\abs{v(y)}^2e^{-\abs{y}}}\right).
\end{equation}
\end{lemma}
\begin{proof} Equation (\ref{Eqn-L2ByGrad}) is proven {\cite[equation
(4.11)]{MR-SharpLowerL2Critical-06}}, and the same techniques
prove (\ref{Eqn-ExpDecayByGrad}).
\end{proof}
Let us reiterate the notation $y = \phirac{x}{\lambdabda(t)} \in {(m)} athbb{R}^2$, and
introduce a rescaled time,
\begin{equation}\label{Defn-Eqn-s}
\begin{aligned}
s(t) = \int_0^t{\phirac{d\,\tau}{\lambdabda^2(\tau)}} + s_0 && \text{
where } && s_0 = e^\phirac{3\pi}{4b_0} && \text{ and, } && s_1 =
s(T_{\mathrm{hyp}}) \in (s_0,\infty].
\end{aligned}\end{equation}
In these new variables, equation (\ref{Eqn-NLS}) now reads,
\begin{equation}\begin{aligned}
\label{Eqn-NLSRescaled}
ib_s\phirac{\partial}{\partial b}Q^\mbT +i\epsilonilon_s -M(\epsilonilon)
+ib\Lambda\epsilonilon = &
i\left(\phirac{\lambdabda_s}{\lambdabda}+b\right)\LambdaQ^\mbT
+\tilde{\gammaamma}_sQ^\mbT\\
& +i\left(\phirac{\lambdabda_s}{\lambdabda}+b\right)\Lambda\epsilonilon
+\tilde{\gammaamma}_s\epsilonilon\\
& +\Psi_b -R[\epsilonilon],
\end{aligned}\end{equation}
where we introduced the new variable, $\tilde{\gammaamma}(s) = -s -
\gammaamma(s),$ the term $R[\epsilonilon]$ corresponds to those terms of
$u\abs{u}^2$ that are formally ${ {(m)} athcal O}(\epsilonilon^2)$, and $M$ is
the linearized operator near $Q^\mbT$, analogous to $L^\m$,
(\ref{Eqn-Defn-L}). Using our choice of notation
(\ref{Eqn-Defn-Notation}), equation (\ref{Eqn-NLSRescaled}) has
exactly the same form as that given by Merle \& Rapha\"el \cite[Lemma
7]{MR-UniversalityBlowupL2Critical-04} in the case of $m=0$.
Indeed, the algebraic structure in $H^1_\m$ is the same, and the
arguments of \cite[Appendix C]{MR-UniversalityBlowupL2Critical-04} (or
\cite[Appendix A]{R-StabilityOfLogLog-05}) prove the following Lemma,
without modification.
\begin{lemma}
\label{Lemma-PrelimEstimatesOrthogConds}
For all $s\in[s_0,s_1)$,
\begin{description}
\item[\it due to orthogonality with $\abs{y}^2Q^\mbT$,
$\LambdaQ^\mbT$, and estimate (\ref{Eqn-prelimEnerEst}),]
\begin{equation}\label{Eqn-prelimLambda+BEst}
\abs{\phirac{\lambdabda_s}{\lambdabda}+b} +\abs{b_s}
\lesssim
{(m)} athcal{G}amma_b^{1-C\eta} + \left(\epsilonNorm\right),
\end{equation}
\item[\it due to orthogonality with $\Lambda^2Q^\mbT$,]
\begin{equation}\label{Eqn-prelimGamma+REst}
\abs{\tilde{\gammaamma}_s -
\phirac{\left\langle\epsilonilon_1,L^\m_+\Lambda^2Q^\m\right {(m)} athrm{ran}gle}{\norm{\Lambda Q^\m}_{L^2}^2} }
{
\leq
{(m)} athcal{G}amma_b^{1-C\eta} + \delta(\alpha^*)\left(\epsilonNorm\right)^\phirac{1}{2}.
}
\end{equation}
\end{description}
\end{lemma}
In order to show the coercive control (\ref{Hypo-Proximity-H1}) does
not fail, we prove the following Local Virial Identity. This estimate
was originally shown by Merle \& Rapha\"el in
\cite{MR-BlowupDynamic-05} and was inspired by the work of Martel \&
Merle \cite{MartelMerle-LiouvilleThmGKdV-00} in a proof of soliton
stability for the generalized Korteweg-de Vries equation.
\begin{lemma}[Local Virial Identity]\label{Lemma-LocalVirial}
For all $s\in[s_0,s_1)$,
\begin{equation}\label{Eqn-LocalVirial}
b_s \gammaeq \delta_1\left(\epsilonNorm\right) - {(m)} athcal{G}amma_b^{1-C\eta},
\end{equation}
where $\delta_1 > 0$ is a universal constant.
\end{lemma}
\begin{proof}[Proof Outline.]
We begin the same as the proof of (\ref{Eqn-prelimLambda+BEst}):
take the real part of the inner product of (\ref{Eqn-NLSRescaled})
by $\LambdaQ^\mbT$ and use (\ref{Eqn-prelimEnerEst}) to eliminate the
terms ${ {(m)} athcal O}(\epsilonilon)$, as in \cite[Appendix
C]{MR-UniversalityBlowupL2Critical-04}.
The interim result is,
\begin{equation}\label{Lemma-LocalVirial-Proof}\begin{aligned}
b_s\,\phirac{1}{4}\norm{yQ^\m}_{L^2}^2 \gammatrsim\;
& {(m)} athcal{H}m(\epsilonilon,\epsilonilon) - \tilde{\gammaamma}_s\left(\epsilonilon_1,\Lambda Q^\m\right)\\
&- {(m)} athcal{G}amma_b^{1-C\eta} - \delta(\alpha^*)\left(\epsilonNorm\right),
\end{aligned}\end{equation}
where we have used the preliminary estimate
(\ref{Eqn-prelimLambda+BEst}).
We have also used the proximity of $Q^\mbT$ to $Q^\m$,
(\ref{Prop-Qb-closeToQ}), to isolate the $b$-dependence from
interactions of the form $\left(Q^\mbT\right)^2\epsilonilon^2$ as a
lower-order potential, the same form as \cite[equation
(215)]{MR-UniversalityBlowupL2Critical-04}, here included as part of
the final term. To prove the Local Virial Identity, use the
preliminary estimate on $\tilde{\gammaamma}_s$ and the Spectral
Property, Proposition \ref{Prop-SpectralProperty}, as adapted by
Lemma \ref{Lemma-SpectralProperty2}.
\end{proof}
\subsection{Lyapounov Functional}\label{SubSec-Lyapounov}
We cannot hope to prove $b^2$ is monotonically decreasing, since it is
a modulation parameter, and thus cannot hope to control $\epsilonilon$ by
the local virial identity at all times. In this section we prove a
Lyapounov functional based on the mass ejection from the singular
region, to which $b^2$ is related, (\ref{Prop-Qb-Mass}), and which we
expect $b^2$ to track. To do this, we will further approximate the
central profile by including a linear radiative tail.
\begin{prop}[Linear Radiation]\label{Prop-Zb}
For $\eta > 0$ sufficiently small, and all $\abs{b}>0$ sufficiently
small depending on $\eta$, there exists a unique solution $\zeta^\m_b\in
\dot{H}^1_\m( {(m)} athbb{R}^2)$ to
\begin{equation}\label{Prop-Zb-Eqn}
\Delta\zeta^\m_b - \zeta^\m_b + ib\Lambda\zeta^\m_b = \Psi_b,
\end{equation}
where $\Psi_b$ is the truncation error given by
(\ref{Prop-Qb-Eqn}). Radiation $\zeta^\m_b\not\in L^2( {(m)} athbb{R}^2)$, and,
moreover, $\lim_{\abs{y}\to+\infty}\abs{y}\abs{\zeta^\m_b(y)}^2$
exists. We denote this decay rate as, $ {(m)} athcal{G}amma_b$.
\begin{itemize}
\item {\it Size in $\dot{H}^1$ and Derivative by $b$}:
\begin{equation}\label{Prop-Zb-smallH1}\begin{aligned}
\norm{\zeta^\m_b}_{\dot{H}^1}^2 \leq {(m)} athcal{G}amma_b^{1-C\eta}, &&\text{
and, }&& \norm{\phirac{\partial}{\partial b}\zeta^\m_b}_{C^1} \leq
{(m)} athcal{G}amma_b^{\phirac{1}{2}-C\eta}.
\end{aligned}\end{equation}
\item {\it Decay past the support of $\Psi_b$}:
\begin{equation}\begin{aligned}\label{Prop-Zb-H1L2Near}
\norm{\abs{y}\abs{\zeta^\m_b}+\abs{y}^2\abs{\gammarad\zeta^\m_b}}_{L^\infty(\abs{y}\gammaeq
R_b)} &\leq {(m)} athcal{G}amma_b^{\phirac{1}{2}-C\eta} < +\infty.
\end{aligned}\end{equation}
\item {\it Stronger decay far past the support of $\Psi_b$}:
\[\begin{aligned}
e^{-(1+C\eta)\phirac{\pi}{b}} \leq \phirac{4}{5} {(m)} athcal{G}amma_b \leq
\norm{\abs{y}^2\abs{\zeta^\m_b}^2}_{L^\infty(\abs{y}\gammaeq R_b^2)} \leq
e^{-(1-C\eta)\phirac{\pi}{b}},
\end{aligned}\] which we have already discussed, equation
(\ref{Eqn-GammaBEstimate}), and,
\begin{equation}
\label{Prop-Zb-H1Far}
\begin{aligned}
\norm{\abs{y}^2\abs{\gammarad\zeta^\m_b}}_{L^\infty(\abs{y}\gammaeq R_b^2)}
\leq C\phirac{ {(m)} athcal{G}amma_b^\phirac{1}{2}}{\abs{b}}.
\end{aligned}\end{equation}
\end{itemize}
\end{prop}
The proof of Proposition \ref{Prop-Zb} is given due to Merle and
Rapha\"el \cite[Appendix E]{MR-UniversalityBlowupL2Critical-04} and
\cite[Appendix A]{MR-SharpLowerL2Critical-06}. Brief discussion of the
necessary adapatations will be given at the end of Appendix
\ref{Appendix-ProofOfAlmostSelfSimilar}.
We denote,
\begin{equation}\begin{aligned}
\label{DefnEqn-A}
A(t) = e^{a\phirac{\pi}{b(t)}}, && \text{ so that, } &&
{(m)} athcal{G}amma_b^{-\phirac{a}{2}} \leq A \leq {(m)} athcal{G}amma_b^{-\phirac{3a}{2}},
\end{aligned}\end{equation}
where $a>0$ is a universal parameter. Let $\phi_A$ denote a smooth
cutoff function of the region, ${ {(m)} athds 1}_{\{\abs{y} > 2A\}}$.
The truncated radiation, $\zeta^\m_bT = (1-\phi_A)\zeta^\m_b$, is algebraically
close to $\zeta^\m_b$ and satisfies,
\begin{equation}\label{DefnEqn-TildeZb}\begin{aligned}
\Delta\zeta^\m_bT - \zeta^\m_bT +ib\Lambda\zeta^\m_bT = \Psi_b + F, &&\text{
where, }&& \abs{F}_{L^\infty}+\abs{y\cdot\gammarad F}_{L^\infty}
\lesssim \phirac{ {(m)} athcal{G}amma^\phirac{1}{2}_b}{A}.
\end{aligned}\end{equation}
\begin{comment}
\begin{remark}[Interpretation of
Parameters]
\label{Remark-InterpretParameters}
For smaller values of $\eta$ the central profiles $Q^\mbT$
approximate the mass of the singular region more closely at the
cost that estimates (\ref{Prop-Qb-closeToQ}) through
(\ref{Prop-Qb-EnerMomentum}) are only known for ever smaller
values of $b$. When $\eta$ is larger, to compensate for the
imperfection of our central profile we require more of the
radiative tail to get an accurate picture of mass transport,
requiring a larger choice of $a$. See \cite[page
53]{MR-SharpLowerL2Critical-06} for similar remarks on the
optimality in choice of $A(t)$.
\end{remark}
\end{comment}
We will now repeat the calculation of the local virial identity, this
time including the linear radiation $\widetilde{\zeta}_b$ as part of
the central profile. That is we write,
\begin{equation}\label{DefnEqn-epsTilde}\begin{aligned}
\epsilonTilde = \epsilonilon - \zeta^\m_bT && {(m)} athbb{R}ightarrow && u(t,x) =
\phirac{1}{\lambdabda(t)}\left(Q^\mbT + \zeta^\m_bT + \epsilonTilde\right)
\left(\phirac{x}{\lambdabda}\right)e^{-i\gammaamma(t)},
\end{aligned}\end{equation}
without affecting the parameters. This leads to a refined version of
equation (\ref{Eqn-NLSRescaled}) for $\epsilonTilde$. The proof of the
following three Lemmas is virtually identical\phiootnotemark to that of
Merle and Rapha\"el, \cite[Chapter 4]{MR-SharpLowerL2Critical-06}.
\phiootnotetext{Where Merle and Rapha\"el write, $\widetilde{\zeta}_b =
\widetilde{\zeta}_{re}+i\widetilde{\zeta}_{im}$, one should instead
read, $\zeta^\m_bT = \widetilde{\zeta}_1 + i\widetilde{\zeta}_2$, each
component with spin $m$ following the convention of equation
(\ref{Eqn-Defn-Notation}).}
\begin{lemma}[Radiative Virial Identity]\label{Lemma-RefinedVirial}
For all $s\in[s_0,s_1)$,
\begin{equation}\label{Eqn-RadiativeVirial}\begin{aligned}
\partial_s f_1 \gammaeq &\delta_2\left(\epsilonTildeNorm\right)
+ {(m)} athcal{G}amma_b - \phirac{1}{\delta_2}\int_{A\leq\abs{y}\leq
2A}{\abs{\epsilonilon}^2\,dy},
\end{aligned}\end{equation}
where $\delta_2 > 0$ is a universal constants and,
\[
f_1(s) = \phirac{b}{4}\norm{yQ^\mbT}_{L^2}^2 +
\phirac{1}{2} {(m)} athrm{Im}\left(\int{y\cdot\gammaradQ^\mbT\zeta^\m_bTbar}\right)
+ {(m)} athrm{Im}\inner{\epsilonilon}{\Lambda\zeta^\m_bT}.
\]
\end{lemma}
In the light of estimates such as (\ref{Eqn-L2ByGrad}) we cannot
expect the radiative virial identity to give a good control for
$\epsilonilon$. Let $\phi_\infty$ denote a smooth cutoff function of the
region ${ {(m)} athds 1}_{\{\abs{y} > 3A\}}$ with steady derivative
$\phi'_\infty \approx \phirac{1}{3A}$ on the region $A\leq\abs{y}\leq
2A$.
\begin{lemma}[Mass-Ejection]
\label{Lemma-MassDisperse}
\begin{equation}\label{Eqn-MassDisperse}
\partial_s\left(\int{\phi_{\infty}\left(\phirac{y}{A}\right)\abs{\epsilonilon}^2\,dy}\right)
\gammaeq \phirac{b}{400}\int_{A\leq \abs{y} \leq 2A}{\abs{\epsilonilon}^2\,dy}
- {(m)} athcal{G}amma_b^\phirac{a}{2}\int{\abs{\gammarad_y\epsilonilon}^2\,dy}
- {(m)} athcal{G}amma_b^2.
\end{equation}
\end{lemma}
\begin{remark}
As a heuristic, assume that $\epsilonilon \approx \zeta_b$ on the
region, $A \leq \abs{y} \leq 2A$. Use the definition of $ {(m)} athcal{G}amma_b$
to approximate the mass. Then with hypothesis
(\ref{Hypo-Proximity-H1}), Lemma \ref{Lemma-MassDisperse} suggests
continuous ejection of mass from the region $\abs{y} < \phirac{A}{2}$,
regardless of whether that region is growing or contracting.
\end{remark}
Together with the conservation of mass, Lemma
\ref{Lemma-RefinedVirial} and Lemma \ref{Lemma-MassDisperse} prove the
following Lemma. The argument relies on (\ref{Prop-Zb-smallH1}) and
the relation between parameters $a$ and $\eta$ stipulated by
Proposition \ref{Prop-Qb}.
\begin{lemma}[Lyapounov Functional]\label{Lemma-LyapounovFunc}
For all $s\in[s_0,s_1)$,
\begin{equation}\label{Eqn-Lyapounov}
\partial_s{ {(m)} athcal J} \leq -Cb\left(
{(m)} athcal{G}amma_b + \epsilonTildeNorm + \int_{A\leq\abs{y}\leq 2A}{\abs{\epsilonilon}^2}
\right),
\end{equation}
where $C>0$ is a universal constant,
\begin{equation}\label{DefnEqn-LyapounovFunctional}\begin{aligned}
{ {(m)} athcal J}(s) =
&\norm{Q^\mbT}_{L^2}^2 - \norm{Q^\m}_{L^2}^2\\
&+ 2\left\langle\epsilonilon_1,\Sigma\right {(m)} athrm{ran}gle +
2\left\langle\epsilonilon_2,\Theta\right {(m)} athrm{ran}gle
+\int{\left(1-\phi_{\infty}\right)\abs{\epsilonilon}^2\,dy}\\
&-\phirac{\delta_2}{800}\left( b\widetilde{f}_1(b) -
\int_0^b{\widetilde{f}_1(v)\,dv}
+b\,Im\left(\epsilonilon,\Lambda\zeta^\m_bT\right) \right),
\end{aligned}\end{equation}
and $\widetilde{f}_1$ is the principal part of $f_1$,
\[
\widetilde{f}_1(b) = \phirac{b}{4}\norm{yQ^\mbT}_{L^2}^2 +
\phirac{1}{2} {(m)} athrm{Im}\left(\int{y\cdot\gammarad\zeta^\m_bT\zeta^\m_bTbar}\right).
\]
\end{lemma}
\subsubsection{Estimates on \texorpdfstring{${ {(m)} athcal J}$}{J}}
To first order, ${ {(m)} athcal J}$ quantifies the excess mass remaining in the
singular region. After explicitly accounting for this mass, ${ {(m)} athcal J}$ is
comparable to $\norm{\epsilonilon}_{H^1}^2$, up to a power of $ {(m)} athcal{G}amma_b$
that depends on our choice of truncation of the radiation.
\begin{lemma}\label{Lemma-CrudeLyapounov}
For all $s\in[s_0,s_1)$ we have the crude estimate,
\begin{equation}\label{Eqn-CrudeLyapounovEst}
\abs{ { {(m)} athcal J} - d_mb^2 } < \delta_3b^2,
\end{equation}
where $0 < \delta_3 \ll 1$ is a universal constant, and $d_mb^2$ is
the approximate excess mass of profile $Q^\mbT$.
\end{lemma}
\begin{lemma}\label{Lemma-RefinedLyapounov}
Let $f_2$ denote those terms of ${ {(m)} athcal J}$ that are formally
${ {(m)} athcal O}(b^2)$,
\[
f_2(b) = \norm{Q^\mbT}_{L^2}^2 - \norm{Q^\m}_{L^2}^2
-\phirac{\delta_2}{800}\left( b\tilde{f}_1(b) -
\int_0^b{\tilde{f}_1(v)\,dv} \right).
\]
These are the terms concerned with the excess mass. For all
$s\in[s_0,s_1)$ we have the refined estimate,
\begin{equation}\label{Eqn-RefinedLyapounovEst} { {(m)} athcal J}(s) -
f_2(b(s)) \left\{\begin{aligned}
&\leq
{(m)} athcal{G}amma_b^{1-Ca} &+& CA^2\log A\left(\epsilonNorm\right)\\
&\gammaeq
- {(m)} athcal{G}amma_b^{1-Ca} &+& \phirac{1}{C}\left(\epsilonNorm\right).
\end{aligned}\right.
\end{equation}
\end{lemma}
\begin{proof}
The crude estimate (\ref{Eqn-CrudeLyapounovEst}) can be either
proven directly or seen as a special case of
(\ref{Eqn-RefinedLyapounovEst}) and hypothesis
(\ref{Hypo-Proximity-H1}). The estimate
(\ref{Eqn-RefinedLyapounovEst}) and its proof is exactly as given by
Merle \& Rapha\"el, \cite[equation
(5.6)]{MR-SharpLowerL2Critical-06}. The essential point is that the
most difficult term of ${ {(m)} athcal J}(s) - f_2(b(s))$ can be handled
with the conservation of energy (\ref{ConserveEnergy}), here written
in rescaled variables,
\[\begin{aligned}
2\left\langle\epsilonilon_1,\Sigma\right {(m)} athrm{ran}gle +
2\left\langle\epsilonilon_2,\Theta\right {(m)} athrm{ran}gle +
\int{(1-\phi_A)\abs{\epsilonilon}^2} = \left\langle
L^\m_+\epsilonilon_1,\epsilonilon_1\right {(m)} athrm{ran}gle +&\left\langle
L^\m_-\epsilonilon_2,\epsilonilon_2\right {(m)} athrm{ran}gle
-\int{\phi_A\abs{\epsilonilon}^2}\\
&+ E[Q^\mbT] -\lambdabda^2E_0 + { {(m)} athcal O}(\epsilonilon^3).
\end{aligned}\]
To establish the lower bound of (\ref{Eqn-RefinedLyapounovEst}) we
need $L^\m$ to be coercive. We claim that,
\begin{lemma}\label{Lemma-Maris}
For $v = v_1 + iv_2\in H^1_\m$,
\begin{equation}\label{Eqn-Coercivity-L}
\left\langle L^\m_+v_1,v_1\right {(m)} athrm{ran}gle
+\left\langle L^\m_-v_2,v_2\right {(m)} athrm{ran}gle
\gammaeq \delta_3\norm{v}_{H^1}^2 - \phirac{1}{\delta_3}\left(
\left\langle v_1,\phi_+\right {(m)} athrm{ran}gle^2
+\left\langle v_1,\gammaradQ^\m\right {(m)} athrm{ran}gle^2
+\left\langle v_2,Q^\m\right {(m)} athrm{ran}gle^2
\right),
\end{equation}
where $\phi_+$ is the normalized eigenvector corresponding to the
smallest eigenvalue of $L^\m_+$.
\end{lemma}
Merle and Rapha\"el \cite[Appendix D]{MR-SharpLowerL2Critical-06}
remark that $\phi_+$ lies in the span of $Q^\m$ and $\abs{y}^2Q^\m$,
and that (\ref{Eqn-Coercivity-L}) may be localized to,
\[
\begin{aligned}
\left\langle L^\m_+v_1,v_1\right {(m)} athrm{ran}gle +&\left\langle
L^\m_-v_2,v_2\right {(m)} athrm{ran}gle
-\int{\phi_A\abs{v}^2}\\
&\gammaeq \delta_2\norm{v}_{H^1}^2 - \phirac{1}{\delta_2}\left(
\left\langle v_1,Q^\m\right {(m)} athrm{ran}gle^2 +\left\langle
v_1,\abs{y}^2Q^\m\right {(m)} athrm{ran}gle^2
+\left\langle v_2,Q^\m\right {(m)} athrm{ran}gle^2 \right),
\end{aligned}\] assuming $A$ is sufficiently large relative to the
exponential decay of $Q^\m$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{Lemma-Maris}.]
Following the variational characterization of $Q^\m$, Weinstein
\cite[Prop 2.7]{Weinstein85} argues (in the case $m=0$) that for all
$f\inH^1_\m$, $\left.\partial^2_\epsilonilon J[Q^\m+\epsilonilon
f]\right|_{\epsilonilon=0} \gammaeq 0$. By an explicit calculation we
conclude,
\[
\inf_{f\inH^1_\m, \inner{f}{Q^\m}=0}\inner{L^\m_+f}{f} \gammaeq 0.
\]
Let $ {(m)} u_+<0$ be the lowest eigenvalue of $L^\m_+$, and $\phi_+\in
L^2$ the corresponding normalized eigenvector.
If there were two linearizely independent negative directions, then
there would be one perpendicular to $ {(m)} athbb{R}m$. Therefore,
\[
\inf_{f\inH^1_\m, \inner{f}{\phi_+}=0}\inner{L^\m_+f}{f} \gammaeq 0.
\]
The following argument due to \cite{MartelMerle01} is an improvement
on the proof of \cite[Prop 2.9]{Weinstein85}. Consider,
\[
\delta_+ = \inf\left\{\begin{aligned}\inner{L^\m_+f}{f} &&|&&
\norm{f}_{H^1_\m} = 1 && \text{ and } && \inner{f}{\phi_+} =
0\end{aligned}\right\} \gammaeq 0.
\]
Assume $\delta_+ = 0$. Then by weak convergence of a minimizing
sequence there exists a minimizer $f_+\inH^1_\m$, and there are
lagrange multipliers so that,
\[
\left(L^\m_+-\ell_1\right)f_+ = \ell_2\phi_+.
\]
An inner product with $f_+$ implies $\ell_1 = 0$, and then an inner
product by $\phi_+$ implies $\ell_2=0$. As we remarked in Section
\ref{SubSec-VortexSolitons}, Pego \& Warchall
\cite{PegoWarchall-VorticesNLS-02} found that the nullspace of
$L^\m_+$ restricted to $H^1_\m$ is empty, and we have a contradiction.
\end{proof}
\subsection{Description of the Blowup}
Let us consider the hypotheses of Section \ref{SubSec-DecompModulate}
in turn. In each case, we will show that if the solution exists for $t
= T_{hyp}$, then the hypothesis holds for some interval
$[0,T_{\mathrm{hyp}}+\delta)$. This will prove that {\bf Case 1}, introduced
in Section \ref{SubSec-DecompModulate}, cannot occur, and that,
therefore, $T_{\mathrm{hyp}} = T_{ {(m)} ax}$. This means that the dynamics
described by (\ref{Hypo-Proximity-L2}), (\ref{Hypo-Proximity-H1}),
(\ref{Hypo-loglog}) and (\ref{Hypo-EnergyMomentum}) persist for the
remaining lifetime of the solution. Indeed, we will show that that
lifetime is finite, equation (\ref{Corollary-THypSmall}), and use
these dynamics to prove the behaviour claimed for Theorem
\ref{Thm-MainResult}.
\subsubsection{Hypothesis (\ref{Hypo-Proximity-L2}).}
\noindent Preliminary estimate (\ref{Eqn-prelimMassEst}) shows that
hypothesis (\ref{Hypo-Proximity-L2}) cannot fail.
\subsubsection{Hypothesis
(\ref{Hypo-Proximity-H1}).}\label{SubSubSec-HypoH1}
\begin{lemma} For all $s\in[s_0,s_1)$,
\begin{equation}\label{Eqn-epsImproved}\begin{aligned}
\epsilonNorm \leq {(m)} athcal{G}amma_b^\phirac{4}{5},
\end{aligned}\end{equation}
which shows that hypothesis (\ref{Hypo-Proximity-H1}) cannot fail.
\end{lemma}
\begin{proof}
Consider arbitrary fixed $s \in[s_0,s_1)$.
\begin{enumerate}
\item[\bf (a)] If $\partial_sb(s) \leq 0$, then
(\ref{Eqn-epsImproved}) follows from the local virial identity,
(\ref{Eqn-LocalVirial}).
\item[\bf (b)] If $\partial_sb(s) > 0$, then there exists a largest
interval $(s_+,s)$, with $s_0\leq s_+$, on which $\partial_sb >0$.
\[
\begin{aligned}
\text{This implies, } && b(s_+) < b(s) && \text{ and either, }
&& \left.\begin{aligned}
\left({\bf c}\right) &&&s_+ = s_0, \\
\;\;\text{ or,}\\
\left({\bf d}\right) &&&\partial_sb(s_+) = 0.
\end{aligned}\right.
\end{aligned}\] In case {\bf (c)} or {\bf (d)}, either by the
initial condition (\ref{Defn-P-Proximity}) or the local virial
identity, respectively,
\[
\int{\abs{\gammarad_y\epsilonilon(s_+,y)}^2\,dy}
+\int_{\abs{y}\leq\phirac{10}{b(s_+)}}{\abs{\epsilonilon(s_+,y)}^2e^{-\abs{y}}\,dy}
\leq {(m)} athcal{G}amma_{b(s_+)}^\phirac{6}{7}.
\]
From the upper bound of refined estimate
(\ref{Eqn-RefinedLyapounovEst}), and assuming $a>0$ is
sufficiently small,
\begin{equation}\label{Proof-LowerBound-eqn5} { {(m)} athcal J}(s_+) -
f_2(b(s_+)) \leq {(m)} athcal{G}amma_{b(s_+)}^\phirac{5}{6} <
{(m)} athcal{G}amma_{b(s)}^\phirac{5}{6}.
\end{equation}
Since ${ {(m)} athcal J}$ is non-increasing,
and from the lower bound of refined estimate
(\ref{Eqn-RefinedLyapounovEst}),
\begin{equation}\label{Proof-LowerBound-eqn6}\begin{aligned}
{(m)} athcal{G}amma_{b(s)}^\phirac{5}{6} \gammaeq &
{ {(m)} athcal J}(s) - f_2(b(s_+))\\
\gammatrsim & \left(\int{\abs{\gammarad_y\epsilonilon(s,y)}^2\,dy}
+\int_{\abs{y}\leq\phirac{10}{b(s)}}{\abs{\epsilonilon(s,y)}^2e^{-\abs{y}}\,dy}
\right)\\
&\qquad- {(m)} athcal{G}amma_{b(s)}^{1-Ca} + \left(f_2(b(s)) -
f_2(b(s_+))\right).
\end{aligned}\end{equation}
As noted in the proof of crude estimate
(\ref{Eqn-CrudeLyapounovEst}), we may assume the constant
$\delta_2$ of equation (\ref{Eqn-RadiativeVirial}) is sufficiently
small relative to $d_0$, such that $0 < \left.\phirac{\partial
f_2}{\partial_{b^2}}\right|_{b^2=0} < \infty$, and proving
that $\left(f_2(b(s)) - f_2(b(s_+))\right) > 0$. Assuming $a>0$ is
sufficiently small, this proves (\ref{Eqn-epsImproved}).
\end{enumerate}
\end{proof}
\subsubsection{Hypothesis (\ref{Hypo-loglog}).}
\begin{lemma}
\label{Lemma-Upperbound}
For all $s\in[s_0,s_1)$,
\begin{equation}\label{Eqn-bLowerBound+lambdaUpperBound}\begin{aligned}
b(s) \gammaeq \phirac{3\pi}{4\log s}, &&\text{ and, }&& \lambdabda(s)
\leq \sqrt{\lambdabda_0}e^{-\phirac{\pi}{3}\phirac{s}{\log s}}.
\end{aligned}\end{equation}
\end{lemma}
\begin{proof}
Recall the bounds on $ {(m)} athcal{G}amma_b$, (\ref{Eqn-GammaBEstimate}),
hypothesis (\ref{Hypo-Proximity-H1}), inject into the local virial
identity (\ref{Eqn-LocalVirial}), carefully integrate, and recall
the clever choice of $s_0$, (\ref{Defn-Eqn-s}),
\[\begin{aligned}
\partial_se^{+\phirac{3\pi}{4b}} =
-\phirac{b_s}{b^2}\phirac{3\pi}{4}e^{+\phirac{3\pi}{4b}} \leq 1 &&
\Longrightarrow && e^{+\phirac{3\pi}{4b}} \leq s -s_0 +
e^{+\phirac{3\pi}{4b_0}} = s.
\end{aligned}\] Next, we take hypothesis (\ref{Hypo-Proximity-H1})
and preliminary estimate (\ref{Eqn-prelimLambda+BEst}) to
approximate the dynamics of $\lambdabda$,
\begin{equation}\label{Eqn-lambda-prelimDynamics}\begin{aligned}
\abs{\phirac{\lambdabda_s}{\lambdabda}+b}+\abs{b_s} <
{(m)} athcal{G}amma_b^\phirac{1}{2} && \Longrightarrow &&
-\phirac{\lambdabda_s}{\lambdabda} \gammaeq \phirac{2b}{3} \gammaeq
\phirac{\pi}{2\log s}.
\end{aligned}\end{equation}
By the initial condition on $b_0$, (\ref{Defn-P-Proximity}), we may
assume $s_0$ is sufficiently large so that,
$\int_{s_0}^s{\phirac{\pi}{2\log\sigma}d\sigma} \gammaeq
\phirac{\pi}{3}\left(\phirac{s}{\log s}-\phirac{s_0}{\log s_0}\right).$ By
the initial choice of a log-log relationship, (\ref{Defn-P-loglog}),
we may assume, $-\log\lambdabda_0 \gammaeq e^{\phirac{\pi}{2b_0}} =
s_0^\phirac{3}{2}$. That is, by integrating
(\ref{Eqn-lambda-prelimDynamics}) we have,
\[
-\log\lambdabda \gammaeq -\phirac{1}{2}\log\lambdabda_0 +
\phirac{\pi}{3}\phirac{s}{\log s}.
\]
\end{proof}
\begin{corollary}\label{Corollary-THypSmall}
\[
T_{\mathrm{hyp}} = \int_{s_0}^{s_1}{\lambdabda^2(\sigma)\,d\sigma} \leq
\lambdabda_0\int_{2}^{+\infty}{e^{-\phirac{2\pi}{3}\phirac{\sigma}{\log
\sigma}}\,d\sigma} < \alpha^*.
\]
\end{corollary}
\begin{corollary}
\[\begin{aligned}
\lambdabda \leq e^{-e^{\phirac{\pi}{5b}}}, && \text{ which shows that
half of hypothesis (\ref{Hypo-loglog}) cannot fail.}
\end{aligned}\]
\end{corollary}
\begin{proof}
Due to equation (\ref{Eqn-bLowerBound+lambdaUpperBound}), and again
assuming $s_0 > 0$ is sufficiently large,
\[
-\log\left(s\lambdabda(s)\right) \gammaeq \phirac{\pi}{3}\phirac{s}{\log s} -
\log s \gammaeq \phirac{s}{\log s}.
\]
Take the logarithm and apply equation
(\ref{Eqn-bLowerBound+lambdaUpperBound}) again,
\begin{equation}\label{Proof-Improv1-loglog-eqn2}\begin{aligned}
\log\abs{-\log\left(s\lambdabda(s)\right)} \gammaeq
\log\left(\phirac{s}{\log s}\right) \gammaeq \phirac{4}{15}\log s \gammaeq
\phirac{\pi}{5 b(s)} && \Longrightarrow && s\lambdabda(s) \leq
e^{-e^{\phirac{\pi}{5b}}}.
\end{aligned}\end{equation}
\end{proof}
\begin{lemma}
\label{Lemma-Lowerbound}
For all $s\in[s_0,s_1)$,
\begin{equation}\label{Eqn-bUpperBound}
b(s) \leq \phirac{4\pi}{3\log s}.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Lemma-Lowerbound}]
Due to the crude estimate (\ref{Eqn-CrudeLyapounovEst}) and the
Lyapounov inequality (\ref{Eqn-Lyapounov}),
\[\begin{aligned}
\partial_se^{+\phirac{5\pi}{4}\sqrt{\phirac{d_0}{{ {(m)} athcal J}}}}
\gammatrsim \phirac{b}{{ {(m)} athcal
J}}\, {(m)} athcal{G}amma_be^{\phirac{5\pi}{4}\sqrt{\phirac{d_0}{{ {(m)} athcal J}}}}
\gammaeq 1.
\end{aligned}\] where the final inequality is due to $\phirac{5}{4} >
1+C\eta$, the bound for $ {(m)} athcal{G}amma_b$, (\ref{Eqn-GammaBEstimate}), and
assumes $\alpha^*$ is sufficiently small. By integrating the
inequality,
\begin{equation}\label{Lemma-Lowerbound-ProofEqn1}
e^{+\phirac{5\pi}{4}\sqrt{\phirac{d_0}{{ {(m)} athcal J}(s)}}} \gammaeq
e^{+\phirac{5\pi}{4}\sqrt{\phirac{d_0}{{ {(m)} athcal J}(s_0)}}}
+s -s_0.\end{equation}
Finally, by the crude estimate (\ref{Eqn-CrudeLyapounovEst}) and the definition of $s_0$ (\ref{Defn-Eqn-s}),
\[
e^{+\phirac{5\pi}{4}\sqrt{\phirac{d_0}{{ {(m)} athcal J}(s_0)}}} >
e^\phirac{\pi}{b_0} > s_0,
\]
which, again with the crude estimate (\ref{Eqn-CrudeLyapounovEst}),
proves (\ref{Eqn-bUpperBound}) from
(\ref{Lemma-Lowerbound-ProofEqn1}). Finally, we note here a related
estimate that will be used in Subsection
\ref{SubSubSec-FinalDynamic}. Divide the Lyapounov inequality
(\ref{Eqn-Lyapounov}) by $\sqrt{ {(m)} athcal J}$, integrate in time, and
use the crude estimate once again to get,
\begin{equation}\label{Eqn-epsIntegral}
\int_{s_0}^s{\left(
{(m)} athcal{G}amma_{b(\sigma)} + \epsilonNorm
\right)\,d\sigma}
\lesssim \sqrt{{ {(m)} athcal J}(s_0)} - \sqrt{{ {(m)} athcal J}(s)}
\lesssim b_0.
\end{equation}
\end{proof}
\begin{corollary}
\[\begin{aligned}
e^{-e^{\phirac{5\pi}{b}}} \leq \lambdabda, && \text{ which shows the
other half of hypothesis (\ref{Hypo-loglog}) cannot fail.}
\end{aligned}\]
\end{corollary}
\begin{proof}
From the approximate dynamics of $\lambdabda$, equation
(\ref{Eqn-lambda-prelimDynamics}),
\[\begin{aligned}
-\phirac{\lambdabda_s}{\lambdabda} \leq 3b \leq \phirac{4\pi}{\log s} &&
\Longrightarrow && -\log \lambdabda(s) \leq -\log\lambdabda_0 +
4\pi\int_{s_0}^s{\phirac{1}{\log\sigma}\,d\sigma}
\end{aligned}\] Bound the integral with $4\pi (s-s_0)$, and by using
(\ref{Eqn-bUpperBound}) again, $e^{4\pi(s-s_0)} \leq
e^{4\pi\left(e^{\phirac{4\pi}{3b(s)}}-s_0\right)}$. With the
definition of $s_0$ (\ref{Defn-Eqn-s}) and initial condition
(\ref{Defn-P-loglog}),
\[
\lambdabda(s) \gammaeq \lambdabda_0e^{4\pi s_0}\,e^{-4\pi
e^\phirac{4\pi}{3b(s)}} > e^{-e^\phirac{5\pi}{b(s)}}.
\]
\end{proof}
\subsubsection{Hypothesis (\ref{Hypo-EnergyMomentum}).}
\noindent As another consequence of the approximate dynamics of
$\lambdabda$, equation (\ref{Eqn-lambda-prelimDynamics}),
\[\begin{aligned}
\phirac{d}{ds}\left(\lambdabda^2e^\phirac{5\pi}{b}\right) =
2\lambdabda^2e^\phirac{5\pi}{b}\left(\phirac{\lambdabda_s}{\lambdabda}+b-b-\phirac{5\pi
b_s}{2b^2}\right)
\leq& -\lambdabda^2be^{5\pi}{b} < 0,\\
&\Longrightarrow \lambdabda^2(t)e^{\phirac{5\pi}{b(t)}} \leq
\lambdabda_0^2e^\phirac{5\pi}{b_0}.
\end{aligned}\] Then, due to the bounds on $ {(m)} athcal{G}amma_b$,
(\ref{Eqn-GammaBEstimate}), and the initial condition
(\ref{Defn-P-EnergyMomentum}),
\[
\lambdabda^2(t)\abs{E_0} <
{(m)} athcal{G}amma_{b(t)}^4\,e^\phirac{5\pi}{b_0}\lambdabda_0^2\abs{E_0}
< {(m)} athcal{G}amma_{b(t)}^4\,e^\phirac{5\pi}{b_0} {(m)} athcal{G}amma_{b_0}^{10} \ll
{(m)} athcal{G}amma_{b(t)}^4.
\]
This shows that hypothesis (\ref{Hypo-EnergyMomentum}) cannot fail.
\subsubsection{Dynamics of Theorem
\ref{Thm-MainResult}}
\label{SubSubSec-FinalDynamic}
\begin{proof}[Proof of Log-log Rate]
By proving $T_{\mathrm{hyp}} = T_{ {(m)} ax}$, we have already shown blowup in
finite time, due to Corollary \ref{Corollary-THypSmall}. Here we
establish the rate.
\begin{comment}
By showing that $T_{\mathrm{hyp}} = T_{ {(m)} ax}$, we have also shown blowup
in finite time, due to Corollary \ref{Corollary-THypSmall}, which
implies by local wellposedness that $\lambdabda(t) \to 0$ as $t \to
T_{ {(m)} ax}$. The approximate dynamic of $\lambdabda$,
(\ref{Eqn-lambda-prelimDynamics}), implies in particular that
$\abs{\phirac{\lambdabda_s}{\lambdabda}} < 1$, which easily integrates to,
\begin{equation}\label{Eqn-sIsInfty}\begin{aligned}
\abs{\log \lambdabda(s)} + C \leq s && \Longrightarrow && s_1 =
+\infty.
\end{aligned}\end{equation}
\end{comment}
By direct calculation and a change of variable,
\begin{equation}\label{Proof-MainResult-loglogEqn1}\begin{aligned}
-\partial_t\left(\lambdabda^2\log\abs{\log\lambdabda}\right) =
&-\phirac{\lambdabda_s}{\lambdabda}\log\abs{\log\lambdabda} \left(2 +
\phirac{1}{\abs{\log\lambdabda}\log\abs{\log\lambdabda}}\right).
\end{aligned}\end{equation}
Recall the approximate dynamic $ -\phirac{\lambdabda_s}{\lambdabda} \approx
b, $ and with hypothesis (\ref{Hypo-loglog}), equation
(\ref{Proof-MainResult-loglogEqn1}) reads,
\[
\phirac{1}{C} \leq
-\partial_t\left(\lambdabda^2\log\abs{\log\lambdabda}\right) \leq C.
\]
Integrate over $[t,T_{ {(m)} ax})$.
Since $\lambdabda$ is very small we may estimate,
\begin{equation}\label{Proof-MainResult-loglogEqn2}\begin{aligned}
\phirac{1}{C}\left(\phirac{T_{ {(m)} ax}-t}{\log\abs{\log(T_{ {(m)} ax}-t)}}\right)^\phirac{1}{2}
\leq \lambdabda(t) \leq C
\left(\phirac{T_{ {(m)} ax}-t}{\log\abs{\log(T_{ {(m)} ax}-t)}}\right)^\phirac{1}{2}.
\end{aligned}\end{equation}
Moreover, the relationship between $\lambdabda(t)$ and the log-log rate
has a universal asymptotic value as $t \to T_{ {(m)} ax}$, see \cite[Prop
6]{MR-SharpLowerL2Critical-06}.
\begin{comment}
As an aside, recall that $\phirac{ds}{dt} = \phirac{1}{\lambdabda^2}$, so
that with (\ref{Proof-MainResult-BoundForLambda}) one would
conclude,
\begin{equation}\label{Proof-MainResult-BoundForS}
\phirac{1}{C}\abs{\log(T_{ {(m)} ax}-t)} \leq s(t) \leq C\abs{\log(T_{ {(m)} ax}-t)}.
\end{equation}
Then from the explicit lower and upper bounds for $b$, equations
(\ref{Eqn-bLowerBound}) and (\ref{Eqn-bUpperBound}),
\begin{equation}\label{Proof-MainResult-BoundForB}
\phirac{1}{C \log\abs{\log(T_{ {(m)} ax}-t)}}
\leq b(t)
\leq \phirac{C}{\log\abs{\log(T_{ {(m)} ax}-t)}}.
\end{equation}
\end{comment}
\end{proof}
\begin{proof}[Proof of Singularity Description in $L^2$]
The proof here is heavily inspired by that given by Merle and
Rapha\"el, \cite[Section 4]{MR-ProfilesQuantization-05}.
First, we show for any $R>0$ there exists $u^*$ such that,
\begin{equation}\label{Proof-MainResult-L2-SpatialConvergence}\begin{aligned}
\tilde{u}(t) \to u^* && \text{ in } && L^2_x\left(\abs{x} \gammaeq
R\right) && \text{ as } && t \rightarrow T_{ {(m)} ax}.
\end{aligned}\end{equation}
Second, to establish equation (\ref{Thm-MainResult-L2}), we will
prove that both,
\begin{equation}\label{Proof-MainResult-L2-NormConvergence}\begin{aligned}
u^*\in L^2( {(m)} athbb{R}^{2}), && \text{ and, } &&\int{\abs{u^*}^2} =
\lim_{t\rightarrow T_{ {(m)} ax}}\int{\abs{\tilde{u}(t)}^2}.
\end{aligned}\end{equation}
Let $\epsilonilon_0 > 0$ be arbitrary. Choose some $T_{ {(m)} ax} -
\epsilonilon_0 <t(R) < T_{ {(m)} ax}$. By hypothesis (\ref{Hypo-loglog}) we
may assume that, $u(t) = \tilde{u}$ on $\left\{\abs{x} >
\phirac{R}{4}\right\}$ for $t\in[t(R),T_{ {(m)} ax})$ and by equation
(\ref{Eqn-epsIntegral}) we may assume that, $
\int_{t(R)}^{T_{ {(m)} ax}}{\int{\abs{\gammarad\widetilde{u}}^2\,dx}\,dt} <
\epsilonilon_0. $ For a parameter $\tau > 0$, to be fixed later, we
denote,
\begin{equation}\label{Proof-MainResult-L2-DefnvTau}
v^\tau(t,x) = u(t+\tau,x) - u(t,x).
\end{equation}
Since $t(R) < T_{ {(m)} ax}$, $u(t)$ is strongly continuous in $L^2$ at
time $t(R)$. Thus, there exists $\tau_0$ such that,
\begin{equation}\label{Proof-MainResult-L2-vTauSmall}\begin{aligned}
\int{\abs{v^\tau(t(R))}^2\,dx} < \epsilonilon_0 && \text{ for all }
\tau \in [0,\tau_0].
\end{aligned}\end{equation}
Denote $\phi_R$ a smooth cutoff of the region ${ {(m)} athds
1}_{\left\{\abs{x} > R\right\}}$. By direct calculation,
\begin{equation}\label{Proof-MainResult-L2-FluxEqn1}\begin{aligned}
\phirac{1}{2}\partial_t\left(\int{\phi_R\abs{v^\tau}^2}\right)
=& Im\left(\int{\gammarad\phi_R\cdot\gammarad v^\tau \overline{v^\tau}\,dx}\right)\\
&+ Im\left(\int{\phi_R v^\tau\left(\overline{u\abs{u}^2(t+\tau)
- u\abs{u}^2(t)}\right)\,dx}\right).
\end{aligned}\end{equation}
Regarding the first RH term of (\ref{Proof-MainResult-L2-FluxEqn1}),
use H\"older, (\ref{Proof-MainResult-L2-vTauSmall}), and the choice
of $t(R)$,
\[
\int_{t(R)}^{T_{ {(m)} ax}}\abs{ Im\left(\int{\gammarad\phi_R\cdot\gammarad
v^\tau \overline{v^\tau}\,dx}\right) \,dt} \leq C
\left(\int_{t(R)}^{T_{ {(m)} ax}}{1^2\,dt}\right)^\phirac{1}{2}\epsilonilon_0^\phirac{1}{2}
< C\epsilonilon_0.
\]
Regarding the second RH term of
(\ref{Proof-MainResult-L2-FluxEqn1}), control with
$\norm{\tilde{u}}_{L^4}^4 \leq
\norm{\tilde{u}}_{L^2}^2\norm{\tilde{u}}_{\dot{H}^1}^2 \ll
\norm{\tilde{u}}_{H^1}^2$, and integrate in time to get control by
$\epsilonilon_0$. We have proven that $\tilde{u}$ is Cauchy on
$\abs{x}>R$,
\[\begin{aligned}
\int{\phi_R\abs{v^\tau(t)}^2\,dx} < C \epsilonilon_0 && \text{ for all
} \tau \in [0,\tau_0] \text{ and } t\in[t(R),T_{ {(m)} ax}-\tau).
\end{aligned}\]
We now turn our attention to
(\ref{Proof-MainResult-L2-NormConvergence}). The profile and
radiation have support of radius $R(t) = A(t)\lambdabda(t)$, which, due
to hypothesis (\ref{Hypo-loglog}), is going to zero with a bound,
$R(t) \leq (T_{ {(m)} ax}-t)^{\phirac{1}{2}-\delta}$. From the definition
of $A(t)$ and equation (\ref{Eqn-L2ByGrad}) we may bound
$\int(1-\phi_{R(t)})\abs{\widetilde{u}(t)}^2$ to prove,
\begin{comment}
Consider a family of time-variable cutoffs, $\phi_{R(t),\tau} =
\phi\left(\phirac{x}{R(t)}\right)$. By direct calculation, at a
fixed time $t<T_{ {(m)} ax}$,
\begin{equation}\label{Proof-MainResult-L2-FluxEqn2}
\begin{aligned}
\phirac{1}{2}\partial_\tau\left(\int{\phi_{R(t),\tau}\abs{u(\tau)}^2\,dx}\right)
=&
\phirac{1}{R(t)}Im\left(\int{\gammarad_x\phi_{R(t),\tau}\cdot\gammarad_xu(\tau)\overline{u(\tau)}\,dx}\right),
\end{aligned}\end{equation}
where $\gammarad_x\phi_{R(t),\tau}$ denotes
$\left.\gammarad_y\phi(y)\right|_{y=\phirac{x}{R(t)}}$.
Estimate the first RH term of (\ref{Proof-MainResult-L2-FluxEqn2})
by,
\[
\phirac{C(\phi)}{R(t)}\norm{u(\tau)}_{H^1}\norm{u_0}_{L^2} \lesssim
\phirac{1}{A(t)\lambdabda(t)\lambdabda(\tau)}.
\]
Estimate the second RH term of
(\ref{Proof-MainResult-L2-FluxEqn2}) with the same bound, using
the preliminary estimate (\ref{Eqn-prelimGamma+REst}).
Integrating $\phirac{1}{A(t)\lambdabda(t)}\phirac{1}{\lambdabda(\tau)}$ in
$\tau$ gives,
\[\begin{aligned}
\phirac{1}{\abs{\log(T_{ {(m)} ax}-t)}^C}\left(\phirac{\log\abs{\log(T_{ {(m)} ax}-t)}}{T_{ {(m)} ax}-t}\right)^\phirac{1}{2}
\int_t^{T_{ {(m)} ax}}{\left(\phirac{\log\abs{\log(T_{ {(m)} ax}-\tau)}}{T_{ {(m)} ax}-\tau}\right)^\phirac{1}{2}\,d\tau}
\leq \phirac{1}{\abs{\log(T_{ {(m)} ax}-t)}^\phirac{C}{2}},
\end{aligned}\] up to a fixed constant due to $T_{ {(m)} ax}-t <
T_{ {(m)} ax} < \alpha^*$, Corollary \ref{Corollary-THypSmall}.
Integrate (\ref{Proof-MainResult-L2-FluxEqn2}) in $\tau$,
\begin{equation}\label{Proof-MainResult-L2-EssentialResult}\begin{aligned}
\abs{\int{\phi_{R(t),T_{ {(m)} ax}}\abs{u^*}^2\,dx} -
\int{\phi_{R(t),t}\abs{u(t)}^2\,dx}} \lesssim
\phirac{1}{\abs{\log(T_{ {(m)} ax}-t)}^\phirac{C}{2}}.
\end{aligned}
\end{equation}
\end{comment}
\begin{comment}
Rigourously, to perform the integral substitute $u =
\log\abs{\log(T_{ {(m)} ax}-t)}$ to get,
\[
\int_{\log\abs{\log(T_{ {(m)} ax}-t)}}^{\infty}{\sqrt{ue^{e^u}}\phirac{d}{du}\left(e^{-e^u}\right)\,du},
\]
Then by parts we have,
\[
2\sqrt{\log\abs{\log(T_{ {(m)} ax}-t)}\,(T_{ {(m)} ax}-t)} +
\int_{\log\abs{\log(T_{ {(m)} ax}-t)}}^{\infty}{\phirac{e^{-\phirac{1}{2}e^u}}{\sqrt{u}}\,du}.
\]
This final integral is clearly much less than what we started
with. So we may approximate with
$(2+\delta)\sqrt{\log\abs{\log(T_{ {(m)} ax}-t)}\,(T_{ {(m)} ax}-t)}$.
\end{comment}
\[
\lim_{t\to T_{ {(m)} ax}}\int{\phi_{R(t)}\abs{u(t)}^2} =\lim_{t\to
T_{ {(m)} ax}}\int{\phi_{R(t)}\abs{\widetilde{u}(t)}^2} =\lim_{t\to
T_{ {(m)} ax}}\int{\abs{\widetilde{u}(t)}^2},
\]
which proves that the following limit exists,
\[
\int{\abs{u^*}^2} = \lim_{t\to
T_{ {(m)} ax}}\int{\phi_{R(t)}\abs{u(t)}^2}.
\]
This completes the proof of
(\ref{Proof-MainResult-L2-NormConvergence}) and
(\ref{Thm-MainResult-L2}).
\end{proof}
\section{The Spectral Property}
\label{Section-SpectralProperty}
We now provide a numerically assisted proof of the spectral property
for the case $m=1$. We also present some computations on higher order
vortices and discuss why they do not work. Before proving the
Spectral Property of Section \ref{s:intro_specprop}, we will establish
the following variant:
\begin{prop}
\label{p:specprop}
Let $\epsilon \in H^1_ {(m)} $ satisfy the orthogonality conditions,
\begin{equation}
\label{e:general_ortho_conds}
\inner{\epsilon_1}{Q^ {(m)} } = \inner{\epsilon_1}{\Lambda Q^ {(m)} } =
\inner{\epsilon_2}{\Lambda Q^ {(m)} } = \inner{\epsilon_2}{\Lambda^2 Q^ {(m)} } = 0.
\end{equation}
Then, for the case $m=1$, there is a universal constant $C_m > 0$,
so that,
\begin{equation}
{(m)} athcal{H}m(\epsilon,\epsilon) \gammaeq C_m \int \paren{\abs{\nabla_y \epsilon}^2 + e^{-\abs{y}}\abs{\epsilon}^2}dy.
\end{equation}
\end{prop}
Proposition \ref{Prop-SpectralProperty} is an immediate
corollary\phiootnotemark. \phiootnotetext{See the end of Subsection
\ref{Subsection-FinalProofSpectral} for details.} Following
\cite{FMR-ProofOfSpectralProperty-06, Marzuola2010}, we proceed in two
steps. First we count the number of negative eigenvalues of the
operators $ {(m)} athcal{L}^ {(m)} _1$ and $ {(m)} athcal{L}^ {(m)} _2$. We then show that the
assumed $L^2$ orthogonality conditions are sufficient to project away
from the negative directions of the bilinear forms, $ {(m)} athcal{H}^ {(m)} _1$ and
$ {(m)} athcal{H}^ {(m)} _2$, associated with $ {(m)} athcal{L}^ {(m)} _1$ and $ {(m)} athcal{L}^ {(m)} _2$.
We now restrict ourselves to $\epsilon \in H^1_ {(m)} $, $\epsilon = e^{i m \theta}
\epsilon_{{ {(m)} athrm{rad}}}$, where $\epsilon_{{ {(m)} athrm{rad}}}$ is a purely radial function,
\begin{equation}
\epsilon_{{ {(m)} athrm{rad}}} \in H^1_{{ {(m)} athrm{rad}}+} \equiv H_{ {(m)} athrm{rad}}^1( {(m)} athbb{R}^2) \cap \set{u {(m)} id
\abs{x}^{-1} u \in L^2( {(m)} athbb{R}^2)}.
\end{equation}
Given $\epsilon \in H^1_ {(m)} $, we calculate
\begin{equation}
{(m)} athcal{L}_1^ {(m)} \epsilon = e^{i m \theta} \paren{-\phirac{d^2}{dr^2 } -
\phirac{1}{r}\phirac{d}{dr} + \phirac{m^2}{r^2} + 3 R^ {(m)} y \cdot \nabla R^\pm}\epsilon_{ {(m)} athrm{rad}}
\end{equation}
This motivates defining the two operators and inner products on
$H^1_{{ {(m)} athrm{rad}}+}$
\begin{subequations}
\begin{gather}
{(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)} \equiv-\Delta_{ {(m)} athrm{rad}} + \phirac{m^2}{r^2} + 3 R^ {(m)}
y\cdot \nabla R^ {(m)} =
-\Delta_{ {(m)} athrm{rad}} + \phirac{m^2}{r^2} + {(m)} athcal{V}_{1,{ {(m)} athrm{rad}}}\\
{(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)} \equiv-\Delta_{ {(m)} athrm{rad}} + \phirac{m^2}{r^2} + R^ {(m)}
y\cdot \nabla R^ {(m)} =
-\Delta_{ {(m)} athrm{rad}} + \phirac{m^2}{r^2} + {(m)} athcal{V}_{2,{ {(m)} athrm{rad}}}\\
{(m)} athcal{H}_{1,{ {(m)} athrm{rad}}}^{(m)}(\cdot, \cdot) \equiv
\inner{ {(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)}\cdot }{\cdot},\quad
{(m)} athcal{H}_{2,{ {(m)} athrm{rad}}}^{(m)}(\cdot, \cdot) \equiv
\inner{ {(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)}\cdot }{\cdot},
\end{gather}
\end{subequations}
where $\Delta_{ {(m)} athrm{rad}}$ is the radial Laplacian, $\Delta_{ {(m)} athrm{rad}} \equiv
\phirac{d^2}{dr^2} + \phirac{1}{r}\phirac{d}{dr}$.
The orthogonality conditions, \eqref{e:general_ortho_conds}, are now
formulated as
\begin{equation}
\label{e:radial_ortho_conds}
\inner{\epsilon^{ {(m)} athrm{rad}}_1}{R^ {(m)} } = \inner{\epsilon^{ {(m)} athrm{rad}}_1}{\Lambda R^ {(m)} } =
\inner{\epsilon^{ {(m)} athrm{rad}}_2}{\Lambda R^ {(m)} } = \inner{\epsilon^{ {(m)} athrm{rad}}_2}{\Lambda^2 R^ {(m)} } = 0,
\end{equation}
where
\[
\epsilon = e^{i m \theta} \paren{\epsilon_1^{ {(m)} athrm{rad}} + i \epsilon_2^{ {(m)} athrm{rad}}}
\]
and $\epsilon \in H^1_ {(m)} $, $\epsilon_j^{ {(m)} athrm{rad}} \in H^1_{{ {(m)} athrm{rad}}+}$.
All that follows relies on the reduction to a series of one
dimensional radial problems.
\subsection{The Index of Bilinear Forms}
\begin{defn}
The {\it index} of a bilinear form $ {(m)} athcal{B}$ with respect to vector
space $V$, denoted ${ {(m)} athrm{ind}}_V {(m)} athcal{B}$, is the minimal co-dimension over
all subspaces of $V$ on which $ {(m)} athcal{B}$ is a positive.
\end{defn}
For bilinear forms induced by self-adjoint operators ({\it i.e.}
$ {(m)} athcal{B} = \inner{ {(m)} athcal{L} \cdot }{\cdot}$), the index corresponds to the
number of negative eigenvalues of the operator. To calculate the
index, we extend Theorem XIII.8 of Reed \& Simon \cite{RSv4} to:
\begin{theorem}
\label{thm:rs_idx}
Let $U$ solve,
\begin{equation*}
{(m)} athcal{L}\, U = - \phirac{d^2}{dr^2} U - \phirac{1}{r} \ddr U + {(m)} athcal{V}(r) U + \phirac{m^2}{r^2} U = 0,
\end{equation*}
with initial conditions given by the limits,
\begin{equation*}
\lim_{r\to 0} {r^{-m}}{U (r) } = 1, \quad\lim_{r\to 0} \ddr\paren{
{r^{-m}}{U (r) }} = 0,
\end{equation*}
and where the potential $ {(m)} athcal{V}$ is sufficiently smooth and decaying
at $\infty$. Then, the number of roots of $U$ not at the origin,
$N(U)$, is finite and equal to the index of the bilinear form
associated to $ {(m)} athcal{L} $ over the vector space $H^1_{{ {(m)} athrm{rad}}+}$.
\end{theorem}
\begin{proof}
The proof, which we omit, is quite similar to the proof of the
indicated Theorem of Reed \& Simon. In turn, that proof is a
generalization of the Sturm Oscillation theorem for two point
boundary value problems.
\end{proof}
\begin{prop}[Numerically Verified]
\label{p:idx_computations}
For the cases $m=1,2,3$,
\begin{equation}\begin{aligned}
{ {(m)} athrm{ind}}_{H^1_{{ {(m)} athrm{rad}}+}}\, {(m)} athcal{H}_{1,{ {(m)} athrm{rad}}}^ {(m)} = 2,&& \text{and}, &&
{ {(m)} athrm{ind}}_{H^1_{{ {(m)} athrm{rad}}+}}\, {(m)} athcal{H}_{2,{ {(m)} athrm{rad}}}^ {(m)} = 1.
\end{aligned}\end{equation}
\end{prop}
\begin{proof}
Using the methods described in Appendix \ref{s:numerics}, we solve,
\begin{subequations}
\label{e:idx_m0}
\begin{gather}
{(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^ {(m)} U^ {(m)} = 0, \quad \lim_{r\to 0} r^{-m}U^ {(m)} (0)=0,
\quad \lim_{r\to 0} \ddr
{r^{-m}}{U^ {(m)} (r) } = 0,\\
{(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^ {(m)} Z^ {(m)} = 0, \quad \lim_{r\to 0} r^{-m}Z^ {(m)} (0)=0,
\quad \lim_{r\to 0} \ddr {r^{-m}}{Z^ {(m)} (r) } = 0.
\end{gather}
\end{subequations}
Plotting the solutions in Figure \ref{f:idx}, we can see that $U^ {(m)} $
has two zero crossings and $Z^ {(m)} $ has one zero crossing. Subject to
the acceptance of these computations, Theorem \ref{thm:rs_idx}
yields the result.
\end{proof}
\ifpdf
\begin{figure}
\caption{Plots of the functions $U^{(m)}
\label{f:idx}
\end{figure}
\else (No figures in DVI) \phii
\begin{prop}
\label{prop:idx_perturbation}
There exists a constant, $\delta_0>0$, depending on $m$, such that
for $\delta\in (0, \delta_0)$, the bilinear
forms
associated with the perturbed operators,
\[
\overline{ {(m)} athcal{L}}_{j,{ {(m)} athrm{rad}}}^{(m)} \equiv { {(m)} athcal{L}}_{j,{ {(m)} athrm{rad}}}^{(m)}-
\delta e^{-\abs{y}},
\]
have the same index, {\it i.e.}
\[
{ {(m)} athrm{ind}} \, {(m)} athcal{H}_{j,{ {(m)} athrm{rad}}}^ {(m)} ={ {(m)} athrm{ind}} \, \overline{ {(m)} athcal{H}}_{j,{ {(m)} athrm{rad}}}^ {(m)} .
\]
\end{prop}
\begin{proof}
We briefly sketch the proof, which follows from three observations.
First, the of the solutions of the perturbed form of
\eqref{e:idx_m0} are continuous with respect to $\delta$. In
particular, there is $C^1_{ {(m)} athrm{loc}}$ convergence. Second, the roots of
the index functions, in the perturbed and unperturbed cases, must be
simple. For a sufficiently small $\delta_0$, we can ensure that on
any compact interval the perturbed and unperturbed solutions have
the same number of zeros. Finally, for a sufficiently large compact
interval, outside the interval the equation is approximately
``free'' (the localized potentials are negligible), and we can
ensure there are no additional zeros; this may require further
shrinking $\delta_0$.
\end{proof}
\subsection{Orthogonality Conditions and Inner Products}
To verify that orthogonality conditions \eqref{e:radial_ortho_conds}
project away from the negative subspaces, we need to compute a number
of inner products of the form $\inner{ {(m)} athcal{L}_{j,{ {(m)} athrm{rad}}}^{(m)} u}{u}$,
where $u$ solves $ {(m)} athcal{L}_{j,{ {(m)} athrm{rad}}}^{(m)} u = f$. Although these
products are computed numerically, we justify their existence:
\begin{prop}[Numerically Verified]
\label{prop:eu_bvp_2d}
Let $f$ be a continuous, radially symmetric, localized function
satisfying the bound $\abs{f(r)} \leq C e^{-\kappapa r}$ for some
positive constants $C$ and $\kappapa$. There exists a unique radially
symmetric solution,
\[\begin{aligned}
\overline{ {(m)} athcal{L}}_{j,{ {(m)} athrm{rad}}}^{(m)} u = f, &&j=1,2.
\end{aligned}\] that belongs to the class, $u \in
L^\infty([0,\infty))\cap C^2([0,\infty))$.
\end{prop}
\begin{proof}
This is Proposition 2 and 4 of
\cite{FMR-ProofOfSpectralProperty-06}, along with our computations
of the indexes in Lemma \ref{e:idx_m0}. See \cite{Marzuola2010} for
some additional details and a full proof in dimension $d=1$.
\end{proof}
\begin{remark}
The solutions in Proposition \ref{prop:eu_bvp_2d} may not vanish as
$r\to \infty$. Indeed, they can only be expected to be bounded.
\end{remark}
\begin{prop}[Numerically Verified]
\label{prop:ip_computations}
Let $U_1$, $U_2$, $Z_1$, and $Z_2$ be $L^\infty$ radially symmetric
functions solving,
\begin{subequations}
\label{e:ip_bvps}
\begin{align}
{(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)} U_1 &= R^ {(m)} ,\\
{(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)} U_2 &= \Lambda R^ {(m)} ,\\
{(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)} Z_1 &= \Lambda R^ {(m)} ,\\
{(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)} Z_2 &= \Lambda^2 R^ {(m)} .
\end{align}
\end{subequations}
Then the inner products,
\[\begin{aligned}
&K_1^{(m)}\equiv \inner{ {(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)} U_1}{U_1}, &&
&K_2^{(m)}\equiv\inner{ {(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)} U_2}{U_2},
&&K_3^{(m)}\equiv\inner{ {(m)} athcal{L}_{1,{ {(m)} athrm{rad}}}^{(m)} U_1}{U_2},\\
&J_1^{(m)}\equiv\inner{ {(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)} Z_1}{Z_1}, &&
&J_2^{(m)}\equiv\inner{ {(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)} Z_2}{Z_2},
&&J_3^{(m)}\equiv\inner{ {(m)} athcal{L}_{2,{ {(m)} athrm{rad}}}^{(m)} Z_1}{Z_2},
\end{aligned}\] take the values given in Tables \ref{t:k_ips} and
\ref{t:j_ips}.
\end{prop}
\begin{proof}
Using the methods described in Appendix \ref{s:numerics}, these are
computed numerically.
\end{proof}
\begin{table}
\centering
\caption{Inner products associated with $ {(m)} athcal{L}^ {(m)} _{1,{ {(m)} athrm{rad}}}$ for
different winding numbers.}
\label{t:k_ips}
\begin{tabular}{l l l l l }
\hline
$m$ & $K_1^{(m)}$ & $K_2^{(m)}$ & $K_3^{(m)}$ & $K_1^{(m)} K_2^{(m)} - \left(K_3^{(m)}\right)^2$\\
\hline
1 &-0.48237 &-25.798 & 1.28129&10.8025\\
2 & 0.520152&-13.1545 &1.7983 & -10.0762\\
3 &2.59249 & 5.1232&-1.54694 & 10.8888\\
\hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Inner products associated with $ {(m)} athcal{L}^ {(m)} _{2,{ {(m)} athrm{rad}}}$ for
different winding numbers.}
\label{t:j_ips}
\begin{tabular}{l l l l l }
\hline
$m$ & $J_1^{(m)}$ & $J_2^{(m)}$ & $J_3^{(m)}$ & $J_1^{(m)} J_2^{(m)} - \left(J_3^{(m)}\right)^2$\\
\hline
1 &6.6985 &163.548 & -47.7764&-1.1871e+03\\
2 & 25.1685&1319.28 &-235.186 &-2.2108e+04\\
3 &82.6396& 8426.22&-936.752 & -1.8116e+05 \\
\hline
\end{tabular}
\end{table}
As with the indices, we have stability of the inner products with
respect to perturbation by a small portential:
\begin{prop}
\label{prop:ip_perturbation}
Let $\overline{U}_l$ and $\overline{Z}_l$ denote the solutions and
$\overline{K}_l^ {(m)} $ and $\overline{J}_l^ {(m)} $ the inner products,
analogous to those of Proposition \ref{prop:ip_computations}, for
the boundary value problems with the perturbed operators,
$\overline{ {(m)} athcal{L}}_{j,{ {(m)} athrm{rad}}}^ {(m)} $. For $\delta_0>0$ sufficiently
small, the solutions and inner products are continuous with respect
to $\delta$.
\end{prop}
\begin{proof}
This follows from the invertiblity and continuity with respect to
$\delta$ of the operators.
\end{proof}
\subsection{Proof of the Spectral Property}
\label{Subsection-FinalProofSpectral}
We are now able to prove Proposition \ref{p:specprop}. The arguement
closely follows the proofs found in
\cite{FMR-ProofOfSpectralProperty-06,Marzuola2010}. The two bilinear
forms, $\overline{ {(m)} athcal{H}}_1^ {(m)} $ and $\overline{ {(m)} athcal{H}}_2^ {(m)} $, are
treated seperately. First, we will show that $L^2$ orthogonality to
$Q^ {(m)} $ and $\Lambda Q^ {(m)} $ suffices to project away from the negative
subspace of $ {(m)} athcal{H}_1^ {(m)} $. This will only be successful for
$m=1$. Later, we will show that $L^2$ orthogonality to $\Lambda Q^ {(m)} $
and $\Lambda^2 Q^ {(m)} $ projects away from the negative subspace of
$ {(m)} athcal{H}_2^ {(m)} $.
\begin{proof}[Spectral Property for $ {(m)} athcal{H}_1^ {(m)} $.]
Given an element $u\in H^1_ {(m)} $, $u = e^{i m \theta} u_{ {(m)} athrm{rad}}$,
satisfying orthogonality conditions \eqref{e:general_ortho_conds},
showing positivity of $ {(m)} athcal{H}_1^ {(m)} $ on such a $u$ is equivalent to
showing posiviity of $ {(m)} athcal{H}_{1,{ {(m)} athrm{rad}}}^ {(m)} $ on $u_{ {(m)} athrm{rad}} \in H^1_{{ {(m)} athrm{rad}}+}$
satisfying orthogonality conditions \eqref{e:radial_ortho_conds}.
By Propositions \ref{p:idx_computations} and
\ref{prop:idx_perturbation}, $\overline{ {(m)} athcal{H}}_{1,{ {(m)} athrm{rad}}}^ {(m)} $ has a
two-dimensional subspace of negative directions. Recall the
notation of equation \eqref{e:ip_bvps}. Let $ {(m)} athrm{V} = {(m)} athrm{span}
\set{\overline{U}_1, \overline{U}_2}$. We will prove that, for
$m=1$, $\overline{ {(m)} athcal{H}}^ {(m)} _{1,{ {(m)} athrm{rad}}}$ is negative on all of
$ {(m)} athrm{V}$. Indeed, consider an arbitrary element of this space,
\[
{H^1_a}t{U} = c_1 \overline{U}_1 + c_2 \overline{U}_2,
\]
and compute,
\begin{equation}\label{e:matrixH_on_V}
\begin{split}
\overline{ {(m)} athcal{H}}^ {(m)} _{1,{ {(m)} athrm{rad}}}({H^1_a}t{U},{H^1_a}t{U})
& = c_1^2 \overline{K}_1^{(m)} + 2 c_1 c_2 \overline{K}_3^{(m)}
+ c_2^2
\overline{K}_2^{(m)} \\
& = \begin{pmatrix} c_1 &
c_2\end{pmatrix} \begin{pmatrix}\overline{K}_1^{(m)}
&\overline{K}_3^{(m)}
\\ \overline{K}_3^{(m)} & \overline{K}_2^{(m)} \end{pmatrix}\begin{pmatrix} c_1 \\
c_2\end{pmatrix}.
\end{split}
\end{equation}
If the above matrix is negative definite, then the bilinear form is
negative on the two dimensional space $ {(m)} athrm{V}$. We examine the
matrix using the computations in Table \ref{t:k_ips} and elementary
properties of matrices.
For $m=1$,
\[
{(m)} athrm{tr} = \overline{K}_1^{(1)} + \overline{K}_2^{(1)} =-26.2804
+ {(m)} athrm{o}(1),
\]
where $ {(m)} athrm{o}(1)$ corresponds to taking the perturbation
parameter, $\delta$, sufficiently small. Therefore the sum of the
two eigenvalues is negative; at least one is negative. Next,
\[
{(m)} athrm{det}= \overline{K}_1^{(1)}\overline{K}_2^{(1)} -
(\overline{K}_3^{(1)})^2 = 10.8025 + {(m)} athrm{o}(1),
\]
so the two eigenvalues have the same sign. Therefore
$\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}$ is negative on $ {(m)} athrm{V}$.
Table \ref{t:k_ips} shows that this is false for $m=2,3$. We
restrict our attention to $m=1$.
Pretending that $ {(m)} athrm{V}\subset H^1_{{ {(m)} athrm{rad}}+}( {(m)} athbb{R}^2)$, we could
decompose the space as
\begin{equation}\label{e:pretendDecomp}
H^1_{{ {(m)} athrm{rad}}+}( {(m)} athbb{R}^2) = {(m)} athrm{V} \oplus_{\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}} {(m)} athrm{V}^\perp
\end{equation}
where our notation indicates that we have formed the orthogonal
complement with respect to the $\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}$
bilinear form. The non-degeneracy of the matrix
\eqref{e:matrixH_on_V} justifies this decomposition.
It follows that $\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}$ is positive on
$ {(m)} athrm{V}^\perp$. Otherwise, there would be $W \in
{(m)} athrm{V}^\perp$ such that
$\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}(W,W)<0$, which implies by
construction that, $ {(m)} athrm{span}\set{W, \overline{U}_1, \overline{U}_2}$, is
a negative definite space of $\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}$ with
dimension three. But then, given any subspace $ {(m)} athrm{U} \subset
H^1_{{ {(m)} athrm{rad}}+}$ of codimension two, $ {(m)} athrm{U} \,\cap\, {(m)} athrm{span}\set{W,
\overline{U}_1, \overline{U}_2} \neq \emptyset$, which contradicts
the index calculation.
Finally, given any function $u \in H^1_{{ {(m)} athrm{rad}}+}$ and $L^2$ orthogonal
to $R^{(1)}$ and $\Lambda R^{(1)}$, we decompose $u$ as
\[
u = c_1 \overline{U}_1 + c_2 \overline{U}_2 + u^\perp
\]
where, $u^\perp \in {(m)} athrm{V}^\perp$, again in the sense of
\eqref{e:pretendDecomp}. Then,
\[\begin{aligned}
0 = \inner{u}{R^{(1)}}_{L^2} &= c_1
\inner{\overline{U}_1}{R^{(1)}}_{L^2} +
c_2 \inner{\overline{U}_2}{R^{(1)}}_{L^2} + \inner{u^\perp}{R^{(1)}}_{L^2}\\
&= c_1 \overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}(\overline{U}_1,
\overline{U}_1) +
c_2 \overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}(\overline{U}_2, \overline{U}_1) + \overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}} (u^\perp, \overline{U}_1)\\
&= c_1 \overline{K}^{(1)}_1 + c_2 \overline{K}^{(1)}_3,\\
0 = \inner{u}{\Lambda R^{(1)}}_{L^2} &= c_1
\inner{\overline{U}_1}{\Lambda R^ {(m)} }_{L^2} +
c_2 \inner{\overline{U}_2}{\Lambda R^{(1)}}_{L^2} + \inner{u^\perp}{\Lambda R^{(1)}}_{L^2}\\
&= c_1 \overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}(\overline{U}_1,
\overline{U}_2) +
c_2 \overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}(\overline{U}_2, \overline{U}_2) + \overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}} (u^\perp, \overline{U}_2)\\
&= c_1 \overline{K}^{(1)}_3 + c_2 \overline{K}^{(1)}_2.
\end{aligned}\] Due to the non-degeneracy of \eqref{e:matrixH_on_V},
the only solution is $c_1=c_2=0$. Therefore, for all such $u$,
\begin{equation*}
\overline{ {(m)} athcal{H}}^{(1)}_{1,{ {(m)} athrm{rad}}}(u,u) \gammaeq 0.
\end{equation*}
This yields the positivity of $\overline{ {(m)} athcal{H}}^{(1)}_{1}$ on
$H^1_{(1)}$.
Of course, $\overline{U}_1$ and $\overline{U}_2$ are {\it not} in
$H^1_{{ {(m)} athrm{rad}}+}$. The above argument is made rigorous by introducing
an appropriate cutoff function and then taking limits. We refer the
reader to \cite{FMR-ProofOfSpectralProperty-06,Marzuola2010}; we
will not reproduce this here.
\end{proof}
\begin{proof}[Spectral Property for $ {(m)} athcal{H}_2^ {(m)} $.]
As in the case of $\overline{ {(m)} athcal{H}}_{1}^ {(m)} $, we will prove
positivity of $\overline{ {(m)} athcal{H}}_{2}^ {(m)} $ subject to the orthogonality
conditions, by working with the associated radial form,
$\overline{ {(m)} athcal{H}}_{2,{ {(m)} athrm{rad}}}^ {(m)} $. By Propositions
\ref{p:idx_computations} and \ref{prop:idx_perturbation},
$\overline{ {(m)} athcal{H}}^ {(m)} _{2,{ {(m)} athrm{rad}}}$ has one negative direction. Examing
Table \ref{t:j_ips}, neither $\overline{Z}_1^{(m)}$ nor
$\overline{Z}_2^{(m)}$ appears to point in the negative direction.
Define,
\begin{subequations}
\begin{align}
{H^1_a}t{R}^{(m)} &\equiv \Lambda R^ {(m)} - \phirac{\overline{J}_3^{(m)}}{\overline{J}_2^{(m)}} \Lambda^2 R^ {(m)} ,\\
{H^1_a}t{Z} &\equiv
\overline{Z}_1-\phirac{\overline{J}_3^{(m)}}{\overline{J}_2^{(m)}}\overline{Z}_2.
\end{align}
\end{subequations}
Then $ {(m)} athcal{L}_2^ {(m)} {H^1_a}t{Z} = {H^1_a}t{R}^{(m)} $ and,
\begin{equation}\label{e:H2_ZhatZhat}
\overline{ {(m)} athcal{H}}^ {(m)} _2({H^1_a}t{Z} , {H^1_a}t{Z} ) = \phirac{1}{\overline{J}_2^{(m)}}\paren{\overline{J}_1^{(m)}\overline{J}_2^{(m)}-\left(\overline{J}_3^{(m)}\right)^2}<0.
\end{equation}
Now that we have constructed a negative direction, we apply a
similar argument as in the case of $\overline{ {(m)} athcal{H}}^ {(m)} _{1,{ {(m)} athrm{rad}}}$;
however, this will hold not just for $m=1$, but also for $m=2,3$.
We decompose $H_{{ {(m)} athrm{rad}}+}^1( {(m)} athbb{R}^2)$ as
\begin{equation}
H_{{ {(m)} athrm{rad}}+}^1( {(m)} athbb{R}^2) = {(m)} athrm{span}\set{{H^1_a}t{Z} }\oplus_{\overline{ {(m)} athcal{H}}^ {(m)} _{2,{ {(m)} athrm{rad}}}} {(m)} athrm{span}\set{{H^1_a}t{Z} }^\perp
\end{equation}
Since the index of $\overline{ {(m)} athcal{H}}^ {(m)} _{2,{ {(m)} athrm{rad}}}$ is one, we are
assured that it is positive on $ {(m)} athrm{span}\set{{H^1_a}t{Z} }^\perp$. Finally,
given $v\in H_{{ {(m)} athrm{rad}}+}^1$ orthogonal to $\Lambda R^ {(m)} $ and $\Lambda^2
R^ {(m)} $, it may be decomposed as $v = c_1 {H^1_a}t{Z} + v^\perp$, and,
\[\begin{aligned}
0 = \inner{v}{{H^1_a}t{R}^{(m)} }_{L^2}
&= c_1 \overline{ {(m)} athcal{H}}^ {(m)} _2({H^1_a}t{Z} ,{H^1_a}t{Z} ) + \overline{ {(m)} athcal{H}}^ {(m)} _2(v^\perp, {H^1_a}t{Z} )\\
&= c_1 \overline{ {(m)} athcal{H}}^ {(m)} _2({H^1_a}t{Z} ,{H^1_a}t{Z} ).
\end{aligned}\] Invoking \eqref{e:H2_ZhatZhat}, this implies that,
$v = v^\perp \in {(m)} athrm{span}\set{{H^1_a}t{Z} }^\perp$. Therefore, for such
$v$,
\[
\overline{ {(m)} athcal{H}}^ {(m)} _{2,{ {(m)} athrm{rad}}}(v,v) \gammaeq 0
\]
for $m=1,2,3$. Posivitiy of $\overline{ {(m)} athcal{H}}^ {(m)} _2$ on $H^1_ {(m)} $,
subject to orthgonality to $\Lambda Q^ {(m)} $ and $\Lambda^2 Q^ {(m)} $,
follows.
\end{proof}
\begin{proof}[Proof of Proposition \ref{p:specprop}.]
Given $\epsilon = \epsilon_1 + i \epsilon_2$ satisfying the orthogonality
conditions of Proposition \ref{p:specprop} we have proven that,
\begin{equation*}
\overline{ {(m)} athcal{H}}^{(1)}(\epsilon, \epsilon) = \overline{ {(m)} athcal{H}}^{(1)}_1 (\epsilon_1, \epsilon_1) + \overline{ {(m)} athcal{H}}^{(1)}_2 (\epsilon_2,
\epsilon_2) \gammaeq 0,
\end{equation*}
from which we infer,
\begin{equation*}
{(m)} athcal{H}^{(1)} (\epsilon, \epsilon)\gammaeq \delta \int e^{-\abs{y}} \abs{\epsilon}^2 dy.
\end{equation*}
Let $\theta >0$. Then,
\begin{equation*}
(1+\theta) {(m)} athcal{H}^{(1)} (\epsilon, \epsilon) \gammaeq \theta \int \abs{\nabla \epsilon}^2 dy+ \theta
\int {(m)} athcal{V}_1 \abs{\epsilon_1}^2 + {(m)} athcal{V}_2 \abs{\epsilon_2}^2 dy + \delta \int e^{-\abs{y}} \abs{\epsilon}^2 dy.
\end{equation*}
Although the potentials are sign indefinite, for $\theta$
sufficiently small,
\begin{equation}
\theta
\int {(m)} athcal{V}_1 \abs{\epsilon_1}^2 + {(m)} athcal{V}_2 \abs{\epsilon_2}^2 dy + \delta \int
e^{-\abs{y}} \abs{\epsilon}^2 dy \gammaeq \phirac{\delta}{2}\int e^{-\abs{y}} \abs{\epsilon}^2 dy.
\end{equation}
We now have the result,
\begin{equation*}\begin{aligned}
{(m)} athcal{H}^{(1)} (\epsilon, \epsilon) &\gammaeq \phirac{\theta}{1+\theta} \int
\abs{\nabla \epsilon}^2dy +
\phirac{\delta}{2(1+\theta)}\int e^{-\abs{y}} \abs{\epsilon}^2 dy\\
&\gammaeq \delta_0 \int \abs{\nabla \epsilon}^2 + e^{-\abs{y}}
\abs{\epsilon}^2 dy.
\end{aligned}\end{equation*}
\end{proof}
\begin{proof}[Proof of Proposition \ref{Prop-SpectralProperty}.]
Let $\epsilon\in H^1_{(1)}( {(m)} athbb{R}^2)$ with $\epsilon = \epsilon_1 + i \epsilon_2$, and
further decompose this as:
\begin{subequations}
\begin{align}
\epsilon_1 &= e^{i\theta} \paren{u + c_1 R^{(1)} + c_2\Lambda R^{(1)}},\\
\epsilon_2 &= e^{i\theta} \paren{v + d_1 \Lambda R^{(1)} +
d_2\Lambda^2 R^{(1)}},
\end{align}
\end{subequations}
where $u\perp_{L^2} R^{(1)}, \Lambda R^{(1)}$ and $v\perp_{L^2}
\Lambda R^{(1)}, \Lambda^2 R^{(1)}$. Expaning,
\[\begin{aligned}
& {(m)} athcal{H}^{(m)}(\epsilon, \epsilon)
&=&&& {(m)} athcal{H}^{(m)}_1(\epsilon_1, \epsilon_1) + {(m)} athcal{H}^{(m)}_2(\epsilon_2, \epsilon_2),\\
& {(m)} athcal{H}^{(m)}_1(\epsilon_1, \epsilon_1) & =&&& {(m)} athcal{H}^{(m)}_{1,{ {(m)} athrm{rad}}}(u,u) +
2 c_1 \inner{ {(m)} athcal{L}^{(m)}_{1,{ {(m)} athrm{rad}}} u}{R^{(1)}}
+ 2 c_2 \inner{ {(m)} athcal{L}^{(m)}_{1,{ {(m)} athrm{rad}}} u}{\Lambda R^{(m)}}\\
&&&&& + c_1^2 M^{(m)}_1 + c_2^2 M^{(m)}_2 + 2 c_1 c_2 M^{(m)}_3,\\
& {(m)} athcal{H}^{(m)}_2(\epsilon_2, \epsilon_2) & =&&& {(m)} athcal{H}^{(m)}_{2,{ {(m)} athrm{rad}}}(v,v) +
2 d_1 \inner{ {(m)} athcal{L}^{(m)}_{2,{ {(m)} athrm{rad}}} v}{\Lambda R^{(m)}}
+ 2 d_2 \inner{ {(m)} athcal{L}^{(m)}_2 v}{\Lambda^2 R^{(m)}}\\
&&&&& + d_1^2 N^{(m)}_1 + d_2^2 N^{(m)}_2 + 2 d_1 d_2 N^{(m)}_3,
\end{aligned}\] where $M^{(m)}_j, N^{(m)}_j$ are fixed terms arising
from applications of the $ {(m)} athcal{H}_{j,{ {(m)} athrm{rad}}}^{(m)}$ bilinear forms
to combinations of $R^{(m)}$, $\Lambda R^{(m)}$, and $\Lambda^2
R^{(m)}$.
We now construct a lower bound. Let $\theta>0$. Then
\begin{equation}
\begin{split}
c_1 \inner{ {(m)} athcal{L}^{(m)}_1
u}{R^{(m)}}&\leq\phirac{1}{2} \paren{\theta^{-2}c_1^2 +
\theta^2\inner{ {(m)} athcal{L}^{(m)}_{1,{ {(m)} athrm{rad}}} u}{R^{(m)}}^2}\\
&\leq \phirac{1}{2}\bracket{\theta^{-2} c_1^2 +\theta^2\paren{\int \abs{u} \abs{ {(m)} athcal{L}^{(m)}_{1,{ {(m)} athrm{rad}}} R^{(m)}} }^2}\\
& \leq \phirac{1}{2}\bracket{\theta^{-2} c_1^2
+\theta^2\paren{\int \abs{u} \abs{ {(m)} athcal{L}^{(m)}_{1,{ {(m)} athrm{rad}}}
R^{(m)}}^{1/2}\abs{ {(m)} athcal{L}^{(1)}
R^{(m)}}^{1/2} }^2}\\
&\leq C\bracket{\theta^{-2} c_1^2 + \theta^2\int \abs{ {(m)} athcal{L}^{(m)}_{1,{ {(m)} athrm{rad}}} R^{(m)}}\abs{u}^2 }\\
&\leq C \paren{\theta^{-2} c_1^2 + \theta^2\int
e^{-\abs{y}}\abs{u}^2 }.
\end{split}
\end{equation}
The other terms in which $u$ or $v$ appears once are similarly
controlled. Therefore,
\[
{(m)} athcal{H}^{(m)}(\epsilon, \epsilon) \gammaeq {(m)} athcal{H}_{1,{ {(m)} athrm{rad}}}^{(m)}(u,u) +
{(m)} athcal{H}_{2,{ {(m)} athrm{rad}}}^{(m)}(v,v) - C\theta^{-2}(c_1^2 + c_2^2 + d_1^2
+d_2^2) -C \theta^2 \int e^{-\abs{y}}\abs{u+iv}^2,
\]
For the case $m=1$, we apply Proposition \ref{p:specprop} to get
\[
\begin{split}
{(m)} athcal{H}^{(1)}(\epsilon, \epsilon) \gammaeq &
C_{(1)} \int{\abs{\nabla(u+iv)}^2}\\
&+ \left(C_{(1)} - C \theta^2\right) \int{ e^{-\abs{y}}\abs{u+iv}^2 } - C\theta^{-2}(c_1^2 + c_2^2 + d_1^2 +d_2^2)\\
\gammaeq & \phirac{C_{(1)}}{2} \int \abs{\nabla(u+iv)}^2 +
e^{-\abs{y}}\abs{u+iv}^2 - C\theta^{-2}(c_1^2 + c_2^2 + d_1^2
+d_2^2),
\end{split}
\]
where we take $\theta >0$ sufficienty small. Finally,
\[\begin{aligned}
\int \abs{\nabla e^{i\theta}(u+iv)}^2 + e^{-\abs{y}}\abs{
&e^{i\theta}(u+iv)}^2\\
&\gammaeq C\paren{ \int \abs{\nabla\epsilon}^2 + e^{-\abs{y}}\abs{\epsilon}^2
} - {(m)} athrm{O}\left(c_1^2 + c_2^2 + d_1^2 + d_2^2\right).
\end{aligned}\]
\end{proof}
\appendix
\section{Almost-Self Similar Profiles}\label{Appendix-ProofOfAlmostSelfSimilar}
In this Appendix, we outline the proof of Proposition \ref{Prop-Qb},
showing modifications of the proof given in the case $m=0$,
\cite{MR-SharpUpperL2Critical-03, MR-UniversalityBlowupL2Critical-04,
MR-SharpLowerL2Critical-06}. We then briefly discuss the proof of
Proposition \ref{Prop-Zb}. Recall that for $e^{ib\phirac{r^2}{4}}Q^\mb =
e^{im\theta}P^\m_b(r)$ we have equation (\ref{Eqn-Pmb}),
\[
\Delta P^\m_b - \left(1+\phirac{m^2}{r^2}-\phirac{b^2}{4}r^2\right)P^\m_b
+ P^\m_b\abs{P^\m_b}^2 = 0.
\]
This is not a scale-invariant equation, and there is no clear
representative solution. Fibich and Gavish
\cite{FG-TheorySingularVortex-08} chose to consider the solution where
the boundary condition $\lim_{r\to 0}r^{-m}P^\m_b(r) \neq 0$ is chosen
to minimize the amplitude of the asymptotic oscillation.
Since we intend to truncate anyways, it is more convenient to choose
boundary conditions,
\begin{equation}\label{Eqn-Pmb-Conditions}
P^\m_b(r) \left\lbrace\begin{aligned}
\neq 0 &&\text{ for }&& 0 < r &< (1-\eta)R_b,\\
= 0 &&\text{ for }&& r &= (1-\eta)R_b.
\end{aligned}\right.
\end{equation}
Recall that $R_b$ was chosen, (\ref{Eqn-Defn-Rb}), so that the strong
maximum principle applies to, $\Delta -
\left(1+\phirac{m^2}{r^2}-\phirac{b^2}{4}r^2\right)$, on a region larger
than, $r \leq (1-\eta)R_b$.
\noindent {\bf Step 1:} Existence of $P^\m_b$.
Following the argument of \cite[p605-606]{MR-SharpUpperL2Critical-03},
let ${ {(m)} athcal F}_ {(m)} $ denote the space of radial profiles of functions
in $H^1_ {(m)} $. That is, radial $H^1$ functions $f(x)$ for which
$x^{-1}f(x)\in L^2$. Perform a constrained minimization of,
\[
2\,J_b[w] = \int{\abs{\gammarad w}^2} + \int{\abs{w}^2} +
m^2\int{\abs{\phirac{w}{r}}^2} -\phirac{b^2}{4}\int{\abs{rw}^2},
\]
over the subspace of finite-variance functions in ${\mathcal F}_\m$ with
$w((1-\eta)R_b) = 0$ and $\int{\abs{w}^4} = 1$, where all integrals
are taken over a larger compact set, for example $r \leq
(1-\eta^2)R_b$. Note that $J_b$ is coercive on $H^1_\m( {(m)} athbb{R}^2)$,
\begin{equation}\label{Eqn-AppA-FuncLowerBound}
J_b[w] \gammaeq C(\eta)\left(\int{\abs{\gammarad w}^2} + \int{\abs{w}^2} +
m^2\int{\abs{\phirac{w}{r}}^2}\right).
\end{equation}
This minimizing sequence can be assumed to converge weakly in
$H^1_ {(m)} $, which is simply a subspace of $H^1( {(m)} athbb{R}^2)$, and thus strongly
in $L^4$ due to Sobolev embedding on a compact domain. Here we use
that equation (\ref{Eqn-NLS}) is energy subcritical. The Lagrange
multiplier of the Frechet derivative shows that (\ref{Eqn-Pmb}) is
satisfied. Interior regularity estimates show that the weak limit is
$C^3$ on $r < (1-\eta)R_b$. The weak limit is also strictly positive
due to $w((1-\eta)R_b) = 0$ and the maximum principle.
\noindent {\bf Step 2:} $L^\infty$ Estimates, Uniform in $b$.
There exists a fixed constant $C>0$ for all $\abs{b}>0$ sufficiently
small so that,
\begin{equation}\label{Eqn-AppA-P_LInfty}
\abs{P^\m_b}_{L^\infty} \leq C.
\end{equation}
\noindent Moreover, there is uniform decay of the tail of the
solutions. For the same $b$,
\begin{equation}\label{Eqn-AppA-P_DecayTail}
\sup_{\abs{b}\sim 0}\abs{P^\m_b}_{L^\infty(r>R)} \longrightarrow 0 \text{ as }{R\to+\infty}.
\end{equation}
Both bounds are proven in \cite[p606]{MR-SharpUpperL2Critical-03}.
Equation (\ref{Eqn-AppA-P_LInfty}) is a simple consequence of the
variational characterization of Step 1, whereas to prove equation
(\ref{Eqn-AppA-P_DecayTail}), truncate to $r > R$, treat
$r^\phirac{N-1}{2}\abs{P^\m_b}$ as a one-dimensional function, and control
by the standard Sobolev embedding $H^\phirac{1}{2}( {(m)} athbb{R})
\hookrightarrow L^\infty( {(m)} athbb{R})$.
\noindent {\bf Step 3:} Local Convergence to $ {(m)} athbb{R}m$ (in $C^3$).
As $b\to 0$, $P^\m_b$ converges weakly to some positive radial function
$P$, with decay to $0$ as $r\to+\infty$, and which satisfies,
$\Delta P - \left(1+\phirac{m^2}{r^2}\right)P + P\abs{P^2}$. This
characterizes $P$ as the unique groundstate $ {(m)} athbb{R}m$,
\cite{Mizumachi-VortexSolitons-05}. Moreover, due to interior
regularity estimates, on any compact set the convergence of $P^\m_b$ is
strong in $C^3$, up to a subsequence in $b$.
\noindent {\bf Step 4:} Uniform Convergence to $ {(m)} athbb{R}m$ (in $C^3$ with
exponential weight)
Here we adapt the argument of
\cite[p658-659]{MR-UniversalityBlowupL2Critical-04}. Consider the
operator $ {(m)} athcal{K} = \Delta -
\left(1+\phirac{m^2}{r^2}-\phirac{b^2}{4}r^2 - \phirac{\eta^2}{2}\right)$,
which satisfies the maximum principle on $1 < r < (1-\eta)R_b$, for
any $\eta>0$ sufficiently small. Restate (\ref{Eqn-Pmb}) as,
\begin{equation}\label{Eqn-AppA-Pmb}
{(m)} athcal{K} P^\m_b = \phirac{\eta^2}{2}P^\m_b-\left(P^\m_b\right)^3.
\end{equation}
Consider the new function $f_{b}(r) = e^{-(1-\eta)R_b
\Theta\left(\phirac{r}{R_b}\right)}$, with,
\[
\Theta(\overline{x}i) = { {(m)} athds 1}_{0 < \overline{x}i < 1}\int{\sqrt{1 - z^2}\,dz} +
{ {(m)} athds 1}_{1 \leq \overline{x}i}\,\Theta(1)\,\overline{x}i.
\]
Note the dependence on $m$. By direct calculation,
\[\begin{aligned}
f_b^{-1} {(m)} athcal{K} f_b = (1-\eta)\phirac{ \phirac{r}{R_b^2} }{\sqrt{ 1 -
\left(\phirac{r}{R_b}\right)^2 }}
&+(1-\eta)^2\left(1-\left(\phirac{r}{R_b}\right)^2\right)\\
&- \phirac{1}{r}\sqrt{ 1 - \left(\phirac{r}{R_b}\right)^2 } -
\left(1+\phirac{m^2}{r^2}-\phirac{b^2}{4}r^2 - \phirac{\eta^2}{2}\right).
\end{aligned}\] We now approximate each term on the region
$\phirac{1}{\eta} < r \leq (1-\eta)R_b$,
\[\begin{aligned}
f_b^{-1} {(m)} athcal{K} f_b \leq \phirac{(1-\eta)^2}{R_b} &+\left((1-\eta)^2 -
1\right)\left(1-\left(\phirac{r}{R_b}\right)^2\right)
+\left(\phirac{b^2}{4}-\phirac{1}{R_b^2}\right)r^2 \\
&- \eta^\phirac{3}{2}\sqrt{2-\eta} - m^2\eta^2 + \phirac{\eta^2}{2}.
\end{aligned}\] Recall that, $R_b =
\phirac{\sqrt{2+2\sqrt{1+b^2m^2}}}{b}$. By assuming $b>0$ is
sufficiently small with respect to $\eta$, we conclude $f_b^{-1} {(m)} athcal{K}
f_b$ is strictly negative for the given range of $r$.
From Step 2, and the exponential decay of $ {(m)} athbb{R}m$, there exists a fixed
value $r_0 > \phirac{1}{\eta}$ such that for all $b>0$ sufficiently
small,
\[\begin{aligned}
\phirac{\eta^2}{2}P^\m_b - \left(P^\m_b\right)^3 > 0 &&\text{ for } && r
\in \Omega = r_0 < r < (1-\eta)R_b.
\end{aligned}\] We have shown that $ {(m)} athcal{K}\left(c\, f_b -
P^\m_b\right)<0$ for $r\in\Omega$ and any arbitrary constant
$c>0$. Now we note that,
\[
\lim_{b\to 0} f_b(r_0) = e^{-(1-\eta)r_0} > 0,
\]
so that we may choose our constant $c = 2 {(m)} athbb{R}m(r_0)e^{+(1-\eta)r_0}$
and, with our boundary condition (\ref{Eqn-Pmb-Conditions}), conclude
that,
\[
\left. c\, f_b(r) - P^\m_b(r) \right|_{\partial\Omega} > 0.
\]
The maximum principle may now be applied. The same argument can be
applied to $ {(m)} athbb{R}m$, $b=0$, and the weight $f(r) = e^{-(1-\eta)r}$.
With Step 3, this proves the first precursor of
(\ref{Prop-Qb-closeToQ}),
\begin{equation}\label{Eqn-AppA-GoalStep4}\begin{aligned}
\left.
\norm{e^{(1-C\eta)R_b\Theta\left(\phirac{r}{R_b}\right)}\left(P^\m_b- {(m)} athbb{R}m\right)}_{C^3}
\right. \longrightarrow 0 &&\text{ as } && b \rightarrow 0.
\end{aligned}\end{equation}
To prove the bound for the energy, (\ref{Prop-Qb-EnerMomentum}), note
that without loss of generality $(1+C\eta)(1-a) =
(1-\delta)<1$. Introduce a new operator $ {(m)} athcal{K}$ and function $f_b$ in
terms of $\delta$ in place of $\eta$ and argue Step 4 again. In
particular, we may assume that $r_0 < \delta R_b \ll (1-\eta)^2R_b < r
< (1-\eta)R_b \ll (1-\delta)R_b$.
\noindent {\bf Step 5:} Uniqueness of $P^\m_b$; Continuity in $b$
For fixed $b_0>0$ sufficiently small, and $b\approx b_0$, consider,
\begin{equation}\label{Eqn-AppA-Tbb}
T_{b,b_0} = \left(\phirac{R_b}{R_{b_0}}\right)P^\m_b\left(\phirac{R_b}{R_{b_0}}r\right)
\end{equation}
Then $T_{b,b_0} \in {\mathcal F}_\m$ and vanishes for $r=(1-\eta)R_{b_0}$, and
we consider the differential, $T_\Delta = T_{b,b_0} - P^\m_{b_0}$, with
the same domain. The goal is to prove,
\begin{equation}\label{Eqn-AppA-GoalStep5}
\norm{e^{im\theta}T_\Delta(r)}_{H^1( {(m)} athbb{R}^2)} \leq C\phirac{\abs{b-b_0}}{b_0},
\end{equation}
for some fixed constant $C$. To do so, consider the equation for
$T_\Delta$ written as,
\begin{equation}\label{Eqn-AppA-TDelta}\begin{aligned}
\left(L^\m_+ - \phirac{b_0^2}{4}r^2\right) T_\Delta =
&-\left(\left(1-\phirac{R_b^2}{R_{b_0}^2}\right)\left(1 -
\phirac{b_0^2}{4}r^2\right)
+\phirac{R_b^2}{R_{b_0}^2}\phirac{b_0^2 - b^2\phirac{R_b^2}{R_{b_0}^2}}{4}r^2 \right) T_\Delta\\
&-3R_ {(m)} ^2 T_\Delta +\left(T_\Delta + P^\m_{b_0}\right)^3 - \left(P^\m_{b_0}\right)^3\\
&+\left( \left(1-\phirac{R_b^2}{R_{b_0}^2}\right)\left(1 -
\phirac{b_0^2}{4}r^2\right) +\phirac{R_b^2}{R_{b_0}^2}\phirac{b_0^2
- b^2\phirac{R_b^2}{R_{b_0}^2}}{4}r^2 \right) P^\m_{b_0},
\end{aligned}\end{equation}
where $L^\m_+$ is the linerized operator from equation
(\ref{Eqn-Defn-L}). We will use ${ {(m)} athcal F}_{b,b_0}$ to denote the
final right hand term of (\ref{Eqn-AppA-TDelta}). Note that in the
case $m=0$, and thus $R_b = \phirac{2}{b}$, the final multiples of
$T_\Delta$ and $P^\m_{b_0}$ collapse. All three right hand terms of
(\ref{Eqn-AppA-TDelta}) are bounded in the same way as in
\cite[p609]{MR-SharpUpperL2Critical-03}, with only minor
adaptations\phiootnotemark.
To conclude the argument from \cite{MR-SharpUpperL2Critical-03} and
establish (\ref{Eqn-AppA-GoalStep5}) there only remains to show the
following Lemma:
\phiootnotetext{The terms due to $R_b \neq \phirac{2}{b}$ have no
effect. Part of the error term $G_1(R)$ that appears in
\cite{MR-SharpUpperL2Critical-03} has been moved to the left hand
side of (\ref{Eqn-AppA-TDelta}), so that the constant $A_0$ that
appears in \cite{MR-SharpUpperL2Critical-03} can be ignored. }
\begin{lemma}\label{Lemma-AppA-Univ212}
Let $ {(m)} u_+<0$ be the lowest eigenvalue of $L^\m_+$, and $\phi_+\in
L^2$ the corresponding normalized eigenvector. For $b>0$
sufficiently small with respect to $\eta$, and assuming $\eta>0$ is
itself sufficiently small,
\[
\inner{\left(L^\m_+-\phirac{b^2}{4}r^2\right)w}{w} \gammaeq
\delta_+\norm{w}_{H^1}^2 - \phirac{1}{\delta_+}\inner{w}{\phi_+}^2,
\]
for $\delta_+>0$ constant and any $w\in H^1_ {(m)} $ vanishing at
$r=(1-\eta)R_b$.
\end{lemma}
Lemma \ref{Lemma-AppA-Univ212} is analogous to \cite[equation
(212)]{MR-UniversalityBlowupL2Critical-04}, and is adapted from Lemma
\ref{Lemma-Maris} by using a cutoff and the exponential decay of
$\phi_+$. Details can be found,
\cite[p660]{MR-UniversalityBlowupL2Critical-04}.
\noindent {\bf Step 6:} Frechet Derivative on Fixed Domain
The aim is to prove that there exists,
\begin{equation}\label{Eqn-AppA-GoalStep6}\begin{aligned}
\left.\phirac{\partial}{\partial_b}T_{b,b_0}\right|_{b=b_0} \in
H^1_ {(m)} .
\end{aligned}\end{equation}
We will follow the argument of
\cite[p610]{MR-SharpUpperL2Critical-03}, and revisit equation
(\ref{Eqn-AppA-TDelta}). In the limit $b \to b_0$ we have, with
respect to $L^2$-norm,
\begin{equation}\label{Eqn-AppA-TDeltaTilde}\begin{aligned}
\left(L^\m_+ - \phirac{b_0^2}{4}r^2\right)\phirac{T_\Delta}{b-b_0} = 0
-3\left(\left( {(m)} athbb{R}m\right)^2 -
\left(P^\m_{b_0}\right)^2\right)\phirac{T_\Delta}{b-b_0}
+\left.\phirac{\partial}{\partial_b}{ {(m)} athcal
F}_{b,b_0}\right|_{b=b_0}.
\end{aligned}\end{equation}
Note that by direct calculation,
\[
\left.\phirac{\partial}{\partial_b}{ {(m)} athcal F}_{b,b_0}\right|_{b=b_0} =
\phirac{2}{b_0}\left( 1-\phirac{b_0^2}{4}r^2
-\phirac{1}{2}\phirac{\sqrt{1+b_0^2m^2}-1}{\sqrt{1+b_0^2m^2}}
\right)P^\m_{b_0},
\]
and clearly exists. To show equation (\ref{Eqn-AppA-GoalStep6}), we
recall from Step 5 that, for $b_0>0$ sufficiently small,
$L^\m_+-\phirac{b_0^2}{4}r^2$ is invertible over the subspace of $L^2_ {(m)} $
functions that vanish at $r=(1-\eta)R_b$.
\noindent {\bf Step 7:} Uniform Bound for
$\left.\partial_bT_{b,b_0}\right|_{b=b_0}$ (in $C^2$ with exponential
weight)
Revisit equation (\ref{Eqn-AppA-TDelta}), again in the limit $b\to
b_0$ with respect to $L^2$ norm,
\[\begin{aligned}
\left(L^\m_+ - \phirac{b_0^2}{4}r^2 + 3\left(\left( {(m)} athbb{R}m\right)^2 -
\left(P^\m_{b_0}\right)^2\right)\right)
\left.\phirac{\partial}{\partial b}T_{b,b_0}\right|_{b=b_0} &=
+\left.\phirac{\partial}{\partial_b}{ {(m)} athcal
F}_{b,b_0}\right|_{b=b_0}.
\end{aligned}\] Similar to Step 4, we apply a maximum principle
argument on the region $\phirac{1}{\eta} < r \leq (1-\eta)R_b$ to prove,
\[
\norm{ e^{(1-C\eta)R_b\Theta\left(\phirac{r}{R_b}\right)}
\left.\phirac{\partial}{\partial b}T_{b,b_0}\right|_{b=b_0}
}_{C^2(r<(1-\eta)R_b)} \lesssim \phirac{1}{b_0}.
\]
The full argument is the same as
\cite[p610-611]{MR-SharpUpperL2Critical-03} with only minor
adaptations.
\noindent {\bf Step 8:} Uniform Bound for
$\left.\partial_bP^\m_bT\right|_{b=b_0}$ (in $C^2$ with exponential
weight)
Let $P^\m_bT = \phi_bP^\m_b$ where $\phi_b$ are the smooth cutoff
functions,
\begin{equation}\label{DefnEqn-phiB}
\phi_b(r) = \left\{\begin{aligned}
1 & \text{ for } r < (1-\eta)^2R_b\\
0 & \text{ for } r > (1-\eta)R_b,
\end{aligned}\right.
\end{equation}
with the good behaviour,
$\norm{\gammarad\phi_b}_{L^\infty}+\norm{\Delta\phi_b}_{L^\infty} \to
0$, as $b\to 0$. Alternately,
\begin{equation}\label{Eqn-AppA-AlterPTilde}
P^\m_bT = \left(\phi_b - \phi_{b_0}\right)P^\m_b + \phi_{b_0}\left(P^\m_b-P^\m_{b_0}\right) + P^\mT_{b_0}.
\end{equation}
The goal is to prove that,
\begin{equation}\label{Eqn-AppA-GoalStep8}\begin{aligned}
\norm{e^{(1-C\eta)R_b\Theta\left(\phirac{r}{R_b}\right)}\phirac{\partial}{\partial_b}P^\m_bT}_{C^2( {(m)} athbb{R}^2)}
\longrightarrow 0 &&\text{ as }&& b\to 0.
\end{aligned}\end{equation}
which is the second precursor to (\ref{Prop-Qb-closeToQ}).
Regarding the first right hand term of (\ref{Eqn-AppA-AlterPTilde}),
we may re-express $P^\m_b$ in terms of $T_{b,b_0}$. Then by Step 7 and
the support of $\phi_b-\phi_{b_0}$, the contribution from that term is
neglible. The remaining term,
$\phi_{b_0}\left(P^\m_b-P^\m_{b_0}\right)$, is treated with calculations
similar to those applied to $T_\Delta$ in Steps 5, 6 and 7. Details
can be found, \cite[p611-612]{MR-SharpUpperL2Critical-03}.
\noindent {\bf Step 9:} Supercritical Mass
The proof of (\ref{Prop-Qb-Mass}) is due to \cite[Lemma
1]{MR-SharpLowerL2Critical-06}.
Here, we give a summary for the reader's convenience. To begin, note
from equation (\ref{Eqn-Pmb}) that $P^\m_bT$ is formally a function of
$b^2$. Then from Step 8 and the chain rule we conclude that, with an
exponential weight, $\partial_{(b^2)}P^\m_bT$ is bounded in $C^2$. From
equation (\ref{Eqn-Pmb}) it can be shown in the limit $b\to 0$ that,
\begin{equation}\label{Eqn-AppA-Pb2}
L_+\phirac{\partial}{\partial(b^2)}P^\m_bT = \phirac{r^2}{4}P^\m_bT.
\end{equation}
Consider then a product of (\ref{Eqn-AppA-Pb2}) by $\Lambda {(m)} athbb{R}m$,
\[\begin{aligned}
\phirac{1}{4}\int{\abs{x}^2\abs{ {(m)} athbb{R}m}^2\,dx}
&&&= -\phirac{1}{4}\inner{r^2\, {(m)} athbb{R}m}{\Lambda {(m)} athbb{R}m}\\
&&&=-\lim_{b\to 0}\inner{L_+\partial_{(b^2)}P^\m_b}{\Lambda {(m)} athbb{R}m}\\
&&&=-\lim_{b\to 0}\inner{\partial_{(b^2)}P^\m_b}{-2 {(m)} athbb{R}m} &&= \lim_{b\to
0}\partial_{b^2}\norm{P^\m_b}_{L^2( {(m)} athbb{R}^2)}.
\end{aligned}\]
This concludes our summary of the proof of Proposition \ref{Prop-Qb}.
\begin{proof}[Proof of Proposition \ref{Prop-Zb}.]
Apply the point transformation, $e^{ib\phirac{r^2}{4}}\zeta^\m_b =
e^{-im\theta}r^mZ(r)$. Then equation (\ref{Prop-Zb-Eqn}) reads,
\[
\partial_r^2Z + \phirac{2m+1}{r}\partial_rZ - Z + \phirac{b^2r^2}{4}Z =
\widetilde{\Psi}_b,
\]
where $r^m\widetilde{\Psi}_b = \Delta\phi_bP^\m_b
+\gammarad\phi_b\cdot\gammaradP^\m_b + \left(\phi_b^3-\phi_b\right)P^\m_b$. The
arguments of \cite[Appendix E]{MR-UniversalityBlowupL2Critical-04}
and \cite[Appendix A]{MR-SharpLowerL2Critical-06}, then prove a
version of Proposition \ref{Prop-Zb} for $e^{ib\phirac{r^2}{4}}Z(r)$,
as a radial function on $ {(m)} athbb{R}^{2m+2}$. By accounting for the
equivalences of norms, this proves Proposition \ref{Prop-Zb}.
\end{proof}
\section{Details of Numerical Methods}
\label{s:numerics}
Our numerical methods closely follow those detailed in
\cite{Marzuola2010}, employing the Fortran 90/95 boundary value
problem software described in \cite{shampine2006user}. We briefly
review it here.
The software is designed to solve two point boundary value problems of
the form
\begin{equation}
\phirac{d}{dr} {(m)} athbf{y} = \phirac{1}{r}S {(m)} athbf{y} + {(m)} athbf{f}(r,
{(m)} athbf{y}),
\end{equation}
by nonlinear collocation. Note that the algorithm handles $r^{-1}$
singuralities. All of our computations were performed on the domain
$[0, 50]$ with tolerance $10^{-10}$.
Codes that can be used to reproduce the computations presented here
are available at
\url{http://www.math.toronto.edu/simpson/files/vortex_dist.tgz}.
\subsection{Point Transformations}
Unfortunately, the equation for the vortex state,
\eqref{Eqn-VortexSoliton-R}, and the operators $ {(m)} athcal{L}m_{1,{ {(m)} athrm{rad}}}$ and
$ {(m)} athcal{L}m_{2,{ {(m)} athrm{rad}}}$, include $r^{-2}$ singularities. We address this
with the point transformation
\begin{equation}
R^ {(m)} (r) = r^m \tilde{R}^ {(m)} (r).
\end{equation}
Similarly, $U = e^{im\theta}r^m \tilde{U}^{(m)}(r)$ for any of the dependent
variables. With this transformation, the vortex equation becomes,
\begin{subequations}
\begin{gather}
(\tilde{R}^ {(m)} )'' + \phirac{2m+1}{r}(\tilde{R}^ {(m)} )' - \tilde{R}^ {(m)} + r^{2m} (\tilde{R}^ {(m)} )^3 = 0,\\
(\tilde{R}^ {(m)} )'(0) = 0, \quad \lim_{r\to \infty}\tilde{R}^ {(m)} (r) =
0,
\end{gather}
\end{subequations}
and the operators $ {(m)} athcal{L}m_{1,{ {(m)} athrm{rad}}}$,$ {(m)} athcal{L}m_{2,{ {(m)} athrm{rad}}}$ become,
\begin{subequations}\begin{align}
\begin{split}
{(m)} athcal{L}m_{1,{ {(m)} athrm{rad}}} U &= r^{m} \set{- \tilde{U}'' -
\phirac{2m+1}{r}\tilde{U}' + 3 r^{2m} \tilde{R}^ {(m)} (m
\tilde{R}^ {(m)} + r (\tilde{R}^ {(m)} )' )
\tilde{U}}\\
&= r^m \tilde{ {(m)} athcal{L}}_1 \tilde{U}
\end{split}\\
\begin{split}
{(m)} athcal{L}m_{2,{ {(m)} athrm{rad}}} Z &= r^{m} \set{- \tilde{U}'' -
\phirac{2m+1}{r}\tilde{U}' + r^{2m} \tilde{R}^ {(m)} (m
\tilde{R}^ {(m)} + r (\tilde{R}^ {(m)} )' )
\tilde{U}}\\
&=r^m \tilde{ {(m)} athcal{L}}_2 \tilde{Z}
\end{split}
\end{align}
\end{subequations}
The right hand sides of \eqref{e:ip_bvps} conveniently become,
\begin{align}
R^{(m)} &= r^m \tilde{R}^{(m)}\\
\Lambda R^{(m)} & = r^m\set{(m+1) \tilde{R}^{(m)}+ r
(\tilde{R}^{(m)})'}\\
\begin{split}
\Lambda^2 R^{(m)} & = r^m\set{(m+1)^2 \tilde{R}^{(m)}+(3+2m) r
(\tilde{R}^{(m)})' +
r^2 (\tilde{R}^{(m)})''}\\
&= r^m\set{\bracket{(m+1)^2+r^2 }\tilde{R}^{(m)}+2 r
(\tilde{R}^{(m)})' - r^{2(m+1)} (\tilde{R}^{(m)})^3 }
\end{split}
\end{align}
\subsection{Artificial Boundary Conditions}
As the algorithm is designed to compute on finite intervals of
$[a,b]$, we must compute on $[0, r_{ {(m)} ax}]$, where $r_{ {(m)} ax}$ is
sufficiently large. This neccessitates the introduction of an
artificial boundary condition on $\tilde{R}^{(m)}$, the vortex state,
and $U_j$ and $Z_j$ solving the boundary value problems
\eqref{e:ip_bvps}. The analogous question in the index function
computations is verifying that there are no zeros beyond $r_{ {(m)} ax}$ which
might have been missed.
To develop the artifiical boundary conditions, we examine the
asymptotic behaviour of the solutions, using that potential terms are
exponentially decaying. For the vortex state,
\begin{equation}
\label{e:vortex_abc}
\tilde{R}^{(m)}(r) \propto r^{-m -\phirac{1}{2}}e^{-r}
\end{equation}
This gives us the boundary condition at $r_{ {(m)} ax}$
\begin{equation}
(\tilde{R}^{(m)})'(r_{ {(m)} ax}) + \paren{1 + \phirac{2m+1}{2r_{ {(m)} ax}}}\tilde{R}^{(m)}(r_{ {(m)} ax}) = 0
\end{equation}
which is accurate to $ {(m)} athrm{O}(r_{ {(m)} ax}^{-2})$.
By similar analysis the solutions to the linear boundary value
problems, generically denoted by $W$, are
\begin{equation}
W(r) \propto r^{-2m}
\end{equation}
as $r\to \infty$. Thus
\begin{equation}
\label{e:bvp_abc}
\tilde{W}'(r_{ {(m)} ax}) + \phirac{2m}{r_{ {(m)} ax}}\tilde{W}(r_{ {(m)} ax})=0
\end{equation}
This too is accurate to $ {(m)} athrm{O}(r_{ {(m)} ax}^{-2})$.
\subsection{Verification of Results}
With these approximations, we solve the following sets of equations,
as single first order systems:
\begin{itemize}
\item The vortex $\tilde{R}^{(m)}$, and the index functions $U$ and
$Z$,
\item The vortex $\tilde{R}^{(m)}$, the boundary value problem
solutions $U_1$ and $U_2$, and the $K_j$ inner products.
\item The vortex $\tilde{R}^{(m)}$, the boundary value problem
solutions $Z_1$ and $Z_2$, and the $J_j$ inner products.
\end{itemize}
In computing the index functions, or alternatively the inner products,
we are actually solving mixed initial value/boundary value problems.
We now present several {\it a postiori} checks on the accuracy of our
results. All are based on checking that the behaviour of the
solutions for large $r$ is consistent with the anticipated asymptotic
behavior.
\subsubsection{Verification of the Vortex States}
Two related ways of checking that we have adequately computed the
vortex states are to examine its decay as $r$ becomes large and to see
that \eqref{e:vortex_abc} becomes small as $r\to \infty$. For the
vortices appearing in Figure \ref{f:vortex_radial}, we plot these two
metrics in Figures \ref{f:vortex_decay} and \ref{f:vortex_abc}. With
this artificial boundary condition, the exponential decay is well
captured.
\ifpdf
\begin{figure}
\caption{Behavior of the computed vortices as $r$ becomes large. We
recover the exponential decay.}
\label{f:vortex_decay}
\end{figure}
\begin{figure}
\caption{Asymptotically, the artificial boundary condition
on the vortex is well satisfied.}
\label{f:vortex_abc}
\end{figure}
\else (No figures in DVI) \phii
\subsubsection{Verification of the Index Count}
In counting the zeros of the index functions from Figure \ref{f:idx},
there is the concern that there may be another root located beyond
$r_{ {(m)} ax}$. To assess this, we note that the asympotically free behavior
of $\tilde{U}$ and $\tilde{Z}$ is
\begin{subequations}
\begin{align}
\tilde{U}^{(m)} &\sim C_0^{(m)} + C_1^{(m)} r^{-2m}\\
\tilde{Z}^{(m)} &\sim D_0^{(m)} + D_1^{(m)} r^{-2m}
\end{align}
\end{subequations}
We can estimate these constants by noting
\begin{subequations}
\begin{align}
\phirac{\tilde{U}' r^{1+2m}}{-2m} &\sim C_1^{(m)}\\
\tilde{U} + \phirac{\tilde{U}' r}{2m} &\sim C_0^{(m)}\\
\phirac{\tilde{Z}' r^{1+2m}}{-2m} &\sim D_1^{(m)}\\
\tilde{Z} + \phirac{\tilde{Z}' r}{2m} &\sim D_0^{(m)}
\end{align}
\end{subequations}
These constants are plotted in Figure \ref{f:idx_consts}. As they
show, we have certainly computed into the ``free'' equation regime.
More importantly, since $C^{(m)}_0 >0$ and $D^{(m)}_0<0$ in all cases,
we should not expect any additional zeros in the $U^{(m)}$ or
$Z^{(m)}$ functions appearing in Figure \ref{f:idx}.
\ifpdf
\begin{figure}
\caption{The asymptotical index function constants for winding
numbers $m=1,2,3$.}
\label{f:idx_consts}
\end{figure}
\else (No figures in DVI) \phii
\subsubsection{Verification of the Inner Products}
For the inner product computations, we verify that in solving the
boundary value problems, $U_l^{(m)}$, $Z_l^{(m)}$ adequately satisfy
the artificial boundary conditions \eqref{e:bvp_abc}, and that the
$K_l$, $J_l$ values are ``constant''. The check on the boundary
conditions is given in Figures \ref{f:ipp_abcs} and \ref{f:ipm_abcs}.
As these figures show, \eqref{e:bvp_abc} is well approximated.
\ifpdf
\begin{figure}
\caption{Check of the artificial boundary conditions on the
$U_l^{(m)}
\label{f:ipp_abcs}
\end{figure}
\begin{figure}
\caption{Check of the artificial boundary conditions on the
$Z_l^{(m)}
\label{f:ipm_abcs}
\end{figure}
\else (No figures in DVI) \phii
In computing the inner products, we define
\begin{equation}
k_1^{(m)}(r) \equiv \int_0^{r} U_1^{(m)} R_1^{(m)} r dr.
\end{equation}
$k_2^{(m)}$, $k_3^{(m)}$, $j_1^{(m)}$, $j_2^{(m)}$, and
$j_3^{(m)}$ are defined analogously. Clearly,
\begin{equation}
\lim_{r\to\infty}k_1^{(m)}(r) = K_1^{(m)}
\end{equation}
and analogously for the other inner product values. We
approximate,
\begin{equation}
K_1^{(m)} \approx k_1^{(m)}(r_{ {(m)} ax}),
\end{equation}
for $r_{ {(m)} ax}$ sufficiently large that these converge to their
limiting values. As Figures \ref{f:kfuncs} and \ref{f:jfuncs}
show, this is indeed the case.
\ifpdf
\begin{figure}
\caption{Convergence of the approximate inner product values
to their limiting states.}
\label{f:kfuncs}
\end{figure}
\begin{figure}
\caption{Convergence of the approximate inner product values
to their limiting states.}
\label{f:jfuncs}
\end{figure}
\else (No figures in DVI) \phii
\end{document} | math | 136,763 |
\betagin{document}
\baselineskip 18pt
\hfuzz=6pt
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{remark}[theorem]{Remark}
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\newcommand{\frac}{\fracrac}
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\newcommand{\mathbb R^n}{\mathbb R^n}
\newcommand{\sigma}{\sigmagma}
\newcommand{\gamma}{\gammamma}
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\newcommand{\fbox{Q\qt}\ }{\fracbox{Q\qt}\ }
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\newcommand{\si_{gg}}{\sigma_{gg}}
\newcommand{\si_{b2j}}{\sigma_{b2j}}
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\newcommand{\abs}[1]{\left\vert #1\right\vert}
\newcommand{\beta}{\betata}
\deltaf\mathbb RR{\mathbb R}
\deltaf\mathbb R{\mathbb R}
\title[Multilinear Multiplier Theorems]
{Multilinear Multiplier Theorems and Applications}
\author{Loukas Grafakos}
\address{Department of Mathematics, University of Missouri, Columbia MO 65211, USA}
\email{[email protected]}
\author{Danqing He}
\address{Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China}
\email{[email protected]}
\author{Hanh Van Nguyen}
\address{Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA}
\email{[email protected]}
\author{Lixin Yan}
\address{Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China}
\email{[email protected]}
\thanks{{\it Mathematics Subject Classification:} Primary 42B15. Secondary 42B25}
\thanks{{\it Keywords and phases:} Multilinear operators, multiplier operators, Calder\'on commutators.}
\thanks{The first three authors would like to acknowledge the support of Simons Foundation.
The fourth author was supported by
NNSF of China (Grant No. 11371378, 11471338 and 11521101).}
\betagin{abstract}
We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calder\'on and Calder\'on-Coifman-Journ\'e.
\end{abstract}
\maketitle
\section{Introduction}
\setcounter{equation}{0}
The theory of multilinear multipliers has made significant advances in recent years. An $n$-dimensional
$m$-linear multiplier is a bounded function $\sigma$ on $(\mathbb R^n)^m $ associated with an $m$-linear
operator
$T_\sigma$ on $\mathbb R^n\times \cdots\times \mathbb R^n$ in the following way:
\betagin{equation}\labelel{Ts}
T_\sigma(f_1,\dots , f_m)(x) =
\int_{(\mathbb R^n)^m } \!\!\!\widehat{f_1}(\xi_1) \cdots \widehat{f_m}(\xi_m) \sigma(\xi_1,\dots , \xi_m) e^{2\partiali i x\cdot (\xi_1+\cdots +\xi_m)}
d\xi_1\!\cdots \!d\xi_m,
\end{equation}
where $f_j$, $j=1,\dots , m$, are Schwartz functions in $\mathbb R^n$, and
$\widehat{f_j}(\xi_j) = \int_{\mathbb R^n} f_j(x) e^{-2\partiali i x\cdot \xi_j} dx$ is the
Fourier transform of $f_j$. A classical result of Coifman and Meyer \cite{CM-G, CM2} says that if
for all sufficiently large multiindices $\alpha_1,\dots , \alpha_m\in (\mathbb Z^+\cup \{0\})^n$ we have
\betagin{equation}\labelel{mi}
\Big| \partial_{ \xi_1}^{\alpha_1} \cdots \partial_{ \xi_m}^{\alpha_m} \sigma (\xi_1,\dots , \xi_m) \Big| \lesssim
(|\xi_1|+\cdots +|\xi_m|)^{-(|\alpha_1|+\cdots + |\alpha_m|)}
\end{equation}
for all $(\xi_1,\dots , \xi_m) \in (\mathbb R^n)^m \setminus \{(0,\dots ,0)\}$, then $T_\sigma$ admits a bounded extension from
$L^{p_1}(\mathbb R^n) \times \cdots \times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$
when $1<p_1,\dots , p_m\le \infty$, $1/p=1/p_1+\cdots +1/p_m$, and
$1\le p<\infty$. The extension of this theorem to indices $p>1/m$ was simultaneously obtained by
Kenig and Stein \cite{KS} (when $m=2$) and Grafakos and Torres \cite{GT2}.
This theorem provides an $m$-linear extension of Mikhlin's classical
linear multiplier result \cite{Mi}. H\"ormander \cite{Ho} obtained an improvement of Mikhlin's theorem
showing that when $m=1$, $T_\sigma$ maps
$L^{p_1}(\mathbb R^n)$ to $L^{p_1}(\mathbb R^n)$, $1<{p_1}<\infty$ under the weaker condition
\betagin{equation}\labelel{ho}
\sup_{j\in \mathbb Z}\big\| (I-\Delta)^{s/2} \big(\sigma(2^j \cdot ) \widehat{\Psi}\big) \big\|_{L^2(\mathbb R^n)} <\infty,
\end{equation}
where $s>n/2$ and $\widehat{\Psi}$ is a smooth function supported in an annulus centered at the origin.
Here $\Delta$ is the Laplacian and
$(I-\Delta)^{s/2}$ is an operator given on the Fourier transform side by multiplication with $(1+4\partiali^2 |\xi|^2)^{s/2}$.
H\"ormander's theorem was extended to $L^r$-based Sobolev spaces and to indices ${p_1}\le 1$, with
$L^{p_1}$ replaced by the Hardy space $H^{p_1}$, by Calder\'on and Torchinsky \cite{CT}.
The adaptation of H\"ormander's theorem to the multilinear setting was first obtained by Tomita \cite{To}. This
theorem was later extended by Grafakos and Si \cite{GrSi} to the range $p<1$ by replacing $L^2$-based Sobolev spaces
by $L^r$-based Sobolev spaces. The endpoint cases where some $p_j$ are equal to infinity were treated by
Grafakos, Miyachi, and Tomita \cite{GrMiTo}.
Fujita and Tomita \cite{FuTo} provided weighted extensions of these results and also noticed that the operator
$(I-\Delta)^{s/2}$ in $(\mathbb R^n)^m$ can be replaced by
$(I-\Delta_{\xi_1})^{s_1/2}\cdots (I-\Delta_{\xi_m})^{s_m/2}$, where $\Delta_{\xi_j}$ is
the Laplacian in the $\xi_j$th variable.
The bilinear version of the Calder\'on and Torchinsky theorem
was proved by Miyachi and Tomita \cite{MiTo}, while the $m$-linear version (for general $m$) was proved by
Grafakos and Nguyen \cite{GrNg} and Grafakos, Miyachi, Nguyen, and Tomita \cite{GrMiNgTo}.
To study certain multilinear singular integrals, such as multicommutators,
there is a need for a multilinear multiplier theorem that can handle symbols on $(\mathbb R^n)^m$ which, for instance, have
one derivative in each variable but no two derivatives in a given variable.
We notice that in the case where $s_{j} $ are positive integers for all $j $, replacing
$(I-\Delta)^{s/2}$ on $(\mathbb R^n)^m$ by $(I-\Delta_{\xi_1})^{s_1/2}\cdots (I-\Delta_{\xi_m})^{s_m/2}$, as in Fujita and Tomita \cite{FuTo},
reflects the following decay condition for the derivatives of $\sigma$
\betagin{equation}\labelel{GG1}
\big|\partial_{\xi_{1}}^{\beta_{1 }}
\partial_{\xi_{2 }}^{\beta_{2 }}
\cdots \partial_{\xi_{m }}^{\beta_{m }} \sigma (\xi_1,\dots , \xi_m) \big| \lesssim (|\xi_1|+\cdots +|\xi_m|)^{-\sum_{j=1}^m
| \beta_{j }|} ,
\end{equation}
where each multiindex $\beta_j$ satisfies $|\beta_j| \le s_j$. In this case a given coordinate of $\xi_j$ could be differentiated
as many as $s_j$ times.
In this article we study multipliers that satisfy the following coordinate-wise version of \eqref{GG1}
\betagin{equation}\betagin{split}\labelel{GG2}
\big|\partial_{\xi_{11}}^{\beta_{11}} \cdots \partial_{\xi_{1n}}^{\beta_{1n}}
\partial_{\xi_{21}}^{\beta_{21}} \cdots \partial_{\xi_{2n}}^{\beta_{2n}}
\cdots \partial_{\xi_{m1}}^{\beta_{m1}} &\cdots \partial_{\xi_{mn}}^{\beta_{mn}} \sigma (\xi_1,\dots , \xi_m) \big|
\\
& \lesssim (|\xi_1|+\cdots +|\xi_m|)^{-\sum_{j=1}^m
\sum_{\ell =1}^n \beta_{j\ell}},
\end{split} \end{equation}
where $\xi_j=(\xi_{j1}, \dots , \xi_{jn})$ and each $ \beta_{j\ell}$ is at most $ s_j /n$.
Condition \eqref{GG2}
weakens the Coifman-Meyer hypothesis \eqref{mi} and also \eqref{GG1} in the sense that
it does not allow any one-dimensional variable to be differentiated more
than an appropriate number of times.
We now state our first main result concerning the operator $T_\sigma$ in \eqref{Ts}. Here and throughout the
$i$th coordinate of the vector $\xi_j$ in $\mathbb R^n$ is denoted by $\xi_{ji}$.
We denote partial derivatives in the $\xi_{ji}$ variable by $\partial_{\xi_{ji}}$.
Also the Laplacian $\Delta_{\xi_j} $ on $\mathbb R^n$ is given by $ \partial_{\xi_{j1}}^2+\cdots + \partial_{\xi_{jn}}^2$.
We have a result that extends condition \eqref{GG2}
in the Sobolev space setting.
We define
$(I-\partial_{\xi_{i\ell}}^2)^{\frac{\gamma_{i\ell}}{2}}f(\xi)
$
as the linear operator $ ((1+4\partiali^2|\eta_{i\ell}|^2)^{\frac{\gamma_{i\ell}}{2}}\widehat f(\eta))^{\vee}(\xi)$
related to the multiplier
$(1+4\partiali^2|\eta_{i\ell}|^2)^{\frac{\gamma_{i\ell}}{2}}$.
\betagin{theorem}\labelel{1dil}
Suppose that $ 1 \le r\le 2$ and $ \gamma_{i\ell} >1/r$ for all $1\le i \le m$ and $1\le \ell \le n$. Let
$\sigmagma$ be a bounded function on $\mathbb{R}^{mn}$ such that
\betagin{equation}\labelel{dilationj}
\sup_{j\in\mathbb{Z}}\bigg\|{ \partialrod_{\substack{ 1\le i \le m \\ 1\le \ell \le n }} (I-\partial_{\xi_{i\ell}}^2)^{\frac{\gamma_{i\ell}}{2}} \big[ \sigmagma(2^j\cdot)\widehat{\Psi} \big]}\bigg\|_{L^r (\mathbb{R}^{mn})}=A<\infty,
\end{equation}
where $\widehat{\Psi}$ is a smooth function supported in the annulus $\fracrac{1}{2}\le |(\xi_1,\dots , \xi_m) |\le 2$
in $\mathbb R^{mn}$ that satisfies
$$
\sum_{j\in\mathbb{Z}}\widehat{\Psi}(2^{-j}(\xi_1,\dots , \xi_m) )=1, \quad\quaduad \textup{for
all $ (\xi_1,\dots , \xi_m) \in (\mathbb R^n)^m\setminus\{0\}.$}
$$
If $1<p_i<\infty$, $i=1,\dots , m$, satisfy $\maxL^{\infty}mits_{1\le i\le m} \maxL^{\infty}mits_{1\le \ell\le n} \fracrac{1}{\gammamma_{i\ell}} < \minL^{\infty}mits_{1\le i \le m} p_i $ and $\fracrac{1}{p}=\fracrac{1}{p_1}+\cdots+\fracrac{1}{p_m} $, then we have
\betagin{equation}\labelel{equ:TSigmaEST}
\norm{T_{\sigmagma}}_{L^{p_1}(\mathbb{R} ^{n})\times\cdots\times L^{p_m}(\mathbb{R} ^{n})\to L^p(\mathbb{R} ^{n})}\lesssim A.
\end{equation}
\end{theorem}
Taking $\gamma_{i\ell}=\gamma_i/n$ for all $\ell \in \{1,\dots , n\}$ and using simple embeddings between Sobolev spaces we
deduce the following consequence of Theorem~\ref{1dil}.
\betagin{cor}\labelel{cor1}
Let $ 1 \le r \le 2$ and suppose that $\gamma_{i} >n/r$ for all $i=1,\dots , m$. Let
$\sigmagma$ be a bounded function on $\mathbb{R}^{mn}$ such that
\betagin{equation}\labelel{dilationjj}
\sup_{j\in\mathbb{Z}}
\Big\|{ (I- \Delta_{\xi_1})^{\frac{\gamma_1}{2} } \cdots (I- \Delta_{\xi_m})^{\frac{\gamma_m}{2} }
\big[ \sigmagma(2^j\cdot)\widehat{\Psi} \big]}\Big\|_{L^r (\mathbb{R}^{mn})}=A<\infty,
\end{equation}
where $\Psi$ is as in Theorem~\ref{1dil}. Then \eqref{equ:TSigmaEST} holds where $p_i$ are as in
Theorem~\ref{1dil}.
\end{cor}
We also provide an endpoint case of Corollary~\ref{cor1} when all $p_i=1$.
Let $H^1(\mathbb R^n)$ denote the classical Hardy space on $\mathbb R^n$. We note that when $m=1$, boundedness for $T_\sigma$ is known to hold from $H^1 $ to $L^1$.
\betagin{theorem}\labelel{End}
Suppose that $ 1 \le r \le 2$ and that $\gamma_{i} >n $ for all $i=1,\dots , m$.
Let $\sigmagma$ be a bounded function on $\mathbb{R}^{mn}$ which satisfies \eqref{dilationjj}.
Then we have
\betagin{equation}\labelel{equ:TSigmaESTH1}
\norm{T_{\sigmagma}}_{H^{ 1}(\mathbb R^n)\times\cdots\times H^1(\mathbb R^n)\to L^{1/m,\infty}(\mathbb R^n)}\lesssim A.
\end{equation}
\end{theorem}
Another extension of the Coifman-Meyer multiplier theorem is in the multiparameter setting. In this case
\eqref{mi} is relaxed to
\betagin{equation}\betagin{split}\labelel{GG3}
\big|\partial_{\xi_{11}}^{\alpha_{11}} &\cdots \partial_{\xi_{1n}}^{\alpha_{1n}}
\partial_{\xi_{21}}^{\alpha_{21}} \cdots \partial_{\xi_{2n}}^{\alpha_{2n}}
\cdots \partial_{\xi_{m1}}^{\alpha_{m1}} \cdots \partial_{\xi_{mn}}^{\alpha_{mn}} \sigma (\xi_1,\dots , \xi_m) \big|
\\
& \lesssim \,\, (|\xi_{11} |
+\cdots+|\xi_{m1} |)^{-(\alpha_{11}+\cdots +\alpha_{m1})}\cdots
(|\xi_{1n} |+\cdots+|\xi_{mn} |)^{-(\alpha_{1n}+\cdots +\alpha_{mn})}
\end{split} \end{equation}
for sufficiently large indices $\alpha_{i\ell} $. Such a condition was first
considered by Muscalu, Pipher, Tao, and Thiele \cite{MuPiTaTh, MuPiTaTh2}, who obtained boundedness for the
associated operator in the case $m=2$, i.e., from
$L^{p_1}\times L^{p_2}$ to $L^p$ when $1/p_1+ 1/p_2=1/p$ and $1/2<p<\infty$.
In this article we also prove a multilinear multiplier theorem that extends condition
\eqref{GG3}. Precisely, we study multilinear multipliers that satisfy
\betagin{equation}\betagin{split}\labelel{GG4}
\big|\partial_{\xi_{11}}^{\beta_{11}} &\cdots \partial_{\xi_{1n}}^{\beta_{1n}}
\partial_{\xi_{21}}^{\beta_{21}} \cdots \partial_{\xi_{2n}}^{\beta_{2n}}
\cdots \partial_{\xi_{m1}}^{\beta_{m1}} \cdots \partial_{\xi_{mn}}^{\beta_{mn}} \sigma (\xi_1,\dots , \xi_m) \big|
\\
& \lesssim (|\xi_{11} |+\cdots+|\xi_{m1} |)^{-(\beta_{11}+\cdots +\beta_{m1})}\cdots
(|\xi_{1n} |+\cdots+|\xi_{mn} |)^{-(\beta_{1n}+\cdots +\beta_{mn})}
\end{split} \end{equation}
with $\beta_{ji} $ are restricted. To handle the case of fractional derivatives we state our
condition in terms of Sobolev spaces.
We denote by
$(I-\partial_{\xi_{j\ell}}^2)^{
\frac{\gamma_{j\ell}}{2}}
$
the operator given on the Fourier transform side
by multiplication by
$(1+4\partiali^2 |y_{j\ell}|^2) ^{
\frac{\gamma_{j\ell}}{2}}
$,
where $y_j$ is the dual variable of $\xi_j$.
We now state our multiparameter version of Theorem~\ref{1dil},
which extends the results in \cite{MuPiTaTh, MuPiTaTh2}
for H\"ormander type multipliers with minimal smoothness in a way that avoids time-frequency analysis.
\betagin{theorem}\labelel{Tensor}
Let $ 1 \le r \le 2$ and $ \gamma_{i\ell} >1/r$ for all $1\le i \le m$ and $1\le \ell\le n$. Suppose that
$\sigmagma$ is a bounded function on $\mathbb R^{mn}$ such that
$$
\sup_{k_1,\ldots,k_n\in\mathbb{Z} } \!
\bigg\| \! \partialrod_{j=1}^m (I-\partial_{\xi_{j1}}^2)^{\frac{\gamma_{j1}}{2}} \! \cdots \! (I-\partial_{\xi_{jn}}^2)^{\frac{\gamma_{jn}}{2}} \Big[ \sigmagma
\big( D_{k_1,\dots , k_n}\Xi \big)
\! \partialrod_{\ell=1}^n \!\widehat{\Psi_\ell} (\xi_{1 \ell}, \dots , \xi_{m \ell})
\Big] \bigg\|_{L^r (\mathbb{R}^{mn})}\!\!\!\! \!\! =A<\infty,
$$
where
$$
D_{k_1,\dots , k_n}\Xi =
\betagin{bmatrix}
\xi_{11} & \xi_{12} & \dots & \xi_{1n} \\
\xi_{21} & \xi_{22} & \dots & \xi_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\xi_{m1} & \xi_{m2} & \dots & \xi_{mn}
\end{bmatrix}
\betagin{bmatrix}
2^{k_1} \\
2^{k_2} \\
\vdots \\
2^{k_n}
\end{bmatrix},
$$
for some $\widehat{ \Psi_\ell }$ smooth functions on $\mathbb R^{m}$ supported
in the annulus $\fracrac{1}{2}\le |\eta |\le 2$ satisfying
\betagin{equation}\labelel{Psic}
\sum_{k\in\mathbb{Z}}\widehat{\Psi_\ell}(2^{-k}\eta)=1, \quad\quaduad \quad\textup{for all $\eta\in \mathbb R^{m}\setminus\{0\}.$}
\end{equation}
If $1<p_i<\infty$, $i=1,\dots , m$, satisfy $\maxL^{\infty}mits_{1\le i\le m} \maxL^{\infty}mits_{1\le \ell\le n} \fracrac{1}{\gammamma_{i\ell}} < \minL^{\infty}mits_{1\le i \le m} p_i $ and $\fracrac{1}{p}=\fracrac{1}{p_1}+\cdots+\fracrac{1}{p_m} $, then we have
\betagin{equation*}
\norm{T_{\sigmagma}}_{L^{p_1}(\mathbb{R} ^{n})\times\cdots\times L^{p_m}(\mathbb{R} ^{n})\to L^p(\mathbb{R} ^{n})}\lesssim A.
\end{equation*}
\end{theorem}
A version of Theorem~\ref{Tensor} was proved by
Chen and Lu \cite{CL} when $r=m=2$ and when the
differential operator
$(I-\partial_{\xi_{j1}}^2)^{\frac{\gamma_{j1}}{2}} \cdots (I-\partial_{\xi_{jn}}^2)^{\frac{\gamma_{jn}}{2}}$ is replaced by
$(I-\Delta_{\xi_j})^{\frac{\gamma_j}2}$, where $\gamma_j=\gamma_{j1} +\cdots + \gamma_{jn} $; besides allowing $r$ to be less than $2$ and $m\ge 2$,
Theorem~\ref{Tensor} improves that of Chen and Lu \cite{CL} in the sense that only a restricted number of derivatives falls on each coordinate, while in \cite{CL} all derivatives could fall on a single coordinate $\xi_j$ of the multiplier.
An immediate consequence of Theorem \ref{Tensor} is the following:
\betagin{cor}\labelel{less1}
Let $ \sigma_{\ell}(\xi_{1 \ell }, \dots , \xi_{m \ell })$ be bounded functions on $\mathbb{R}^{m }$
for $1\le \ell \le n$. Let
$\sigmagma (\xi_1, \dots , \xi_m)= \partialrod_{\ell =1}^n
\sigma_{\ell}(\xi_{1 \ell }, \dots , \xi_{m \ell })$, where $\xi_i=(\xi_{i1},\dots , \xi_{in})\in \mathbb R^n$, $1\le i\le m$.
Suppose that for some $ \gamma_{i\ell}$
and $r$ as in Theorem \ref{Tensor} we have
\betagin{equation}\labelel{dilationjQQ}
\sup_{1\le \ell\le n}\sup_{k\in\mathbb{Z}}\norm{
(I-\partial_{\xi_{1\ell}}^2)^{\frac{\gamma_{1\ell}}{2}} \cdots (I-\partial_{\xi_{m\ell}}^2)^{\frac{\gamma_{m\ell}}{2}} \Big[
\sigmagma_\ell
(2^k\cdot)\widehat{\Psi_\ell } \Big]}_{L^r (\mathbb{R}^{m})}=B<\infty
\end{equation}
where $\widehat{ \Psi_\ell}$ is a smooth function supported in an annulus in $\mathbb R^m$ that satisfies \eqref{Psic}.
Then for $\maxL^{\infty}mits_{1\le \ell \le n} (\fracrac{1}{\gammamma_{i\ell}}, 1)<p_i<\infty$
for all $i=1,\dots , m$ and $\fracrac{1}{p}=\fracrac{1}{p_1}+\cdots+\fracrac{1}{p_m} $ we have
\betagin{equation*}
\norm{T_{\sigmagma}}_{L^{p_1}(\mathbb{R} ^{n})\times\cdots\times L^{p_m}(\mathbb{R} ^{n})\to L^p(\mathbb{R} ^{n})}\lesssim B^n.
\end{equation*}
\end{cor}
\iffalse
We also have a version of Corollary \ref{less1} in which fewer than $m$ variables appear in each function $\sigma_\ell$.
\betagin{cor}\labelel{less}
For each $1\le \ell\le n$ let $m_\ell\le m$ and let $\sigma_\ell$ be bounded functions on $\mathbb R^{m_\ell}$.
If $\sigma(\xi_1,\dots,\xi_m)=\partialrod_{\ell=1}^n\sigma_\ell( \xi_{S_\ell , \ell})$
with $ \xi_{S_\ell ,\ell}=(\xi_{a_1\ell},\xi_{a_2\ell},\dots, \xi_{a_{m_\ell \ell }})$
and $S_\ell=\{a_1,a_2,\dots,a_{m_\ell}\} $ is a subset of $ \{1,2,\dots ,m\}$ of size $m_\ell$.
Suppose that for some $\vec \gamma$, {\color{blue} $\Psi$ } and $r$ as in Corollary {\color{blue} \ref{less1}} we have
$$
A=\sup_{1\le \ell \le n}\sup_{j\in \mathbb Z} \|\sigma_\ell (2^j\cdot){\color{blue} \widehat \Psi}\|_{L^r_{
{\color{blue} \gamma_{1\ell}, \dots , \gamma_{m\ell} }({\color{blue} \mathbb R^m } )}}<\infty .
$$
Then we have
$$
\norm{T_{\sigmagma}}_{L^{p_1}(\mathbb{R} ^{n})\times\cdots\times L^{p_m}(\mathbb{R} ^{n})\to L^p(\mathbb{R} ^{n})}\lesssim A^n.
$$
\fraci
As an application, we use this corollary to give a short proof of the boundedness
of Calder\'on-Coifman-Journ\'e commutators (Proposition \ref{th7.1})
where the results in \cite{CL, MuPiTaTh, MuPiTaTh2} are not applicable.
Finally, we use arrows to denote elements of $\mathbb R^{nm}$, i.e., $\vec \xi = (\xi_1,\dots , \xi_m)$,
where
$\xi_j\in \mathbb R^n$.
\section{Preliminaries}
The following lemma will be useful in the sequel.
{\betagin{lemma}\labelel{XL1}
Let $\gamma_{i\ell},\gamma_j,\gamma>0$, $1\le i \le m $, $1\le j,\ell\le n$.
Let $D^{\Gamma}$ be a differential operator on $\mathbb R^{mn}$
of one of the following three types:
\betagin{eqnarray*}
&&\partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}}; \\
&&(I-\Delta_{\xi_1})^{\frac{\gamma_1}{2}} \cdots (I-\Delta_{\xi_m})^{\frac{\gamma_m}{2}}; \\
&&(I-\Delta_{\xi_1}-\cdots -\Delta_{\xi_m})^{\frac{\gamma }{2}}.
\end{eqnarray*}
Let $1<\rho \le r<\infty $ and let $\partialhi$ be a smooth function with compact support. Then there is a constant
$C= C(\rho,r,\partialhi,n,\gamma_{i\ell},\gamma_j,\gamma)$ such that
\betagin{equation}\labelel{Lemm11ineq}
\big\| D^{\Gamma}(\partialhi f) \big\|_{L^\rho( \mathbb R^{mn})} \le C \big\| D^{\Gamma}(f) \big\|_{L^r(\mathbb R^{mn})}
\end{equation}
is valid for all Schwartz functions $f$ on $\mathbb R^{mn}$.
Moreover, if $D_\delta$ is an operator of one of the following three types:
\betagin{eqnarray*}
& & \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\delta}{2}} \\
& & (I-\Delta_{\xi_1})^{\frac{\delta}{2}} \cdots (I-\Delta_{\xi_m})^{\frac{\delta}{2}} \\
& & (I-\Delta_{\xi_1}-\cdots -\Delta_{\xi_m})^{\frac{\delta }{2}}
\end{eqnarray*}
then for $D^\Gamma$ and $D_\delta$ of the same type and $\delta>0$ we have
\betagin{equation}\labelel{Lemm11ineq2}
\big\|D^{\Gamma} D_{-\delta}(\partialhi f) \big\|_{L^1( \mathbb R^{mn})} \le C' \big\| D^{\Gamma} D_{ \delta}(f) \big\|_{L^1(\mathbb R^{mn})}
\end{equation}
for all Schwartz functions $f$ on $\mathbb R^{mn}$. Here $C'$ is a constant depending
on $ \partialhi,n,\gamma_{i\ell},\gamma_j,\gamma,\delta$.
\end{lemma}
\betagin{proof}
Estimate \eqref{Lemm11ineq} could be derived by versions
of the Kato-Ponce inequality adapted to the types of operators in question,
such versions are given in \cite[Section 5]{GrOh}.
In the case where $D^{\Gamma}= (I-\Delta )^{\frac \gammamma 2}$, the proof of \eqref{Lemm11ineq} is also
given in \cite[Lemma 7.5.7]{GrafakosMFA}. The idea in this reference also works in this setting.
We provide a sketch: we embed $D^{\Gamma}$ in the analytic family of differential operators $D^{z\Gamma}$ (in which all $\gammamma$'s are multiplied by $z$) and reduce matters to the inequality
$$
\big\| D^{z\Gamma}(\partialhi D^{-{z\Gamma}} f) \big\|_{L^\rho(\mathbb R^{mn})} \lesssim \big\| f \big\|_{L^r(\mathbb R^{mn})}\, .
$$
Let us assume that $\gamma_{i\ell}$, $\gamma_j $, $\gamma$
are rational numbers;
if the case of rational numbers is proved, then by continuity we can deduce the result for all positive numbers as follows:
on the right of the inequality we obtain a constant that is polynomial in $ \gamma_{i\ell} , \gamma_j$ or $\gamma$. But each
function $D^\Gamma (\partialhi f)$ and $D^\Gamma ( f)$ is continuous in $ \gamma_{i\ell} , \gamma_j$ or $\gamma$. Using this
continuity we obtain the conclusion for all $ \gamma_{i\ell} , \gamma_j$, $\gamma$ positive reals.
To prove \eqref{Lemm11ineq} we
interpolate between the cases where $z=it$ and $z=2N+it$, where $N$ is a natural number
and common multiple of all the denominators of
$\gamma_{i\ell}$, $\gamma_j $, $\gamma$.
At the endpoint cases $z=it$ and $z=2N+it$, the
$D^{it\Gamma}$ and $D^{-it\Gamma}$ are $L^\rho$ bounded with bounds that grow at most polynomially in $t$
(and in the $\gamma$'s), while $D^{2N\Gamma}$ is expanded via Leibniz's rule.
Applying the H\"ormander multiplier theorem and H\"older's inequality (to estimate the $L^\rho$ norm over the support of $\partialhi$ by the $L^r$ norm over the entire space) we obtain the
claimed assertion in the cases where $z=it$ and $z=2N+it$ with bounds that grow at most polynomially in $t$.
Interpolation for analytic families of operators yields the claimed conclusion.
We now turn our attention to \eqref{Lemm11ineq2} which is equivalent to
\betagin{equation}\labelel{Lemm11ineq2-2}
\big\| D^{\Gamma} D_{-\delta}\big(\partialhi D^{- \Gamma }D_{-\delta} (f)\big) \big\|_{L^1( \mathbb R^{mn})} \le C' \big\| f \big\|_{L^1(\mathbb R^{mn})}
\end{equation}
and observe that $D_{-\delta}=(D_{\delta})^{-1}$.
We embed the operator $f\mapsto D^{\Gamma} D_{-\delta}\big(\partialhi D^{- \Gamma }D_{-\delta} (f)\big)$ into the analytic family of operators
$f\mapsto D^{z\Gamma} D_{-\delta}\big(\partialhi D^{-z \Gamma }D_{-\delta} (f)\big)$ and we obtain \eqref{Lemm11ineq2-2} as a consequence
of interpolation between the points $z=it$ and $z=2N+it$, where $N$ is as before and $t$ is real.
At the endpoint $z=it$ we have that $D^{\partialm it \Gamma}D_{-\delta}$ is a convolution operator with an integrable kernel and so
$$
\| D^{it\Gamma}D_{-\delta}(\partialhi D^{-it\Gamma}D_{-\delta}(f))\|_{L^1} \lesssim \|\partialhi D^{-it\Gamma}D_{-\delta}(f)\|_{L^1}
\lesssim \| D^{-it\Gamma}D_{-\delta}(f)\|_{L^1} \lesssim \| f \|_{L^1}
$$
with constants bounded by polynomial expressions of the $\gamma$'s and $|t|$.
When $z=2N+it$ we have
\betagin{equation}\labelel{Lemm11ineq2-23}
\big\| D^{it\Gamma}D_{-\delta}D^{ 2N\Gamma}(\partialhi D^{-it}D_{-\delta}(f)) \big\|_{L^1} \lesssim
\big\| D^{ 2N\Gamma}(\partialhi D^{- 2N\Gamma-it\Gamma}D_{-\delta}(f)) \big\|_{L^1}
\end{equation}
and we expand the $D^{ 2N\Gamma}$ derivative via Leibniz's rule. Then $\| D^{ 2N\Gamma}(\partialhi G)\|_{L^1}$ is
bounded by a constant multiple of a sum of terms like $\| D^{k }(G)\|_{L^1}$ where $0\le k\le 2N\Gamma$ and
$D^{k }$ has the same type as $D^\Gamma$.
Each operator of the form $D^{k }D^{-2N\Gamma}D^{-it\Gamma} D_{-\delta}$
is given by convolution with an integrable kernel.
At the end we control the right hand side of \eqref{Lemm11ineq2-23} by a constant multiple of $\|f\|_{L^1}$, with
a constant bounded by polynomial expressions of the $\gamma$'s and $|t|$.
This concludes the sketch of proof.
\end{proof}
}
We will also need a reverse square function inequality associated with Littlewood-Paley operators
acting on each variable separately.
We denote variables in $\mathbb R^{ nl}$ by $(z_1,\dots , z_n)$, where each $z_j$ lies in $\mathbb R^l$.
Fix a smooth function $\widehat\Psi$ supported in an annulus in $\mathbb R^l$
satisfying $\sum_{j\in \mathbb Z}\widehat\Psi (2^{-j}z) =1$ for all $z\neq 0$.
For $j\in \mathbb Z$, define a Littlewood-Paley operator
$$
\Deltalta_{j }^{(k)}(f) = \big( \widehat{f} (z_1,z_2,\dots , z_n) \widehat\Psi (2^{-j}z_k) \big)\spcheck
$$
acting on functions $f$ on $\mathbb R^{ nl}$.
We need the following result.
\betagin{lemma}\labelel{GLP}
For $f\in L^p(\mathbb R^{ nl})$ with $1<p<\infty$ we have
\betagin{equation}\labelel{lp1}
\Big\|\Big(\sum_{j_1}\cdots\sum_{j_n}|\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_n}^{(n)}(f)|^2\Big)^{1/2}\Big\|_{L^p(\mathbb R^{ nl})}
\lesssim\|f\|_{L^p(\mathbb R^{ nl})}.
\end{equation}
Conversely, for $0<p<\infty$ there exists a constant $C$ such that for any $f$ in $ L^2(\mathbb R^{nl})$ satisfying
\betagin{equation}\labelel{lp0}
\|(\sum_{j_1}\cdots\sum_{j_n}|\Deltalta_{j_1}^{(1)}\cdots
\Deltalta_{j_n}^{(n)}(f)|^2)^{1/2}\|_{L^p}<\infty
\end{equation}
we have
\betagin{equation}\labelel{lp2}
\|f\|_{L^p(\mathbb R^{ nl})}\le C \Big\|\Big(\sum_{j_1}
\cdots\sum_{j_n}|\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_n}^{(n)}(f)|^2\Big)^{1/2}\Big\|_{L^p(\mathbb R^{ nl})}.
\end{equation}
\end{lemma}
\betagin{proof}[Proof of Lemma \ref{GLP}]
The proof of \eqref{lp1} is well known and is omitted; see for instance
\cite[Theorem 6.1.6]{Grafakos1} when $l=1$ but the same idea works for all $l$.
So we now focus on \eqref{lp2} which we prove inductively.
The case $n=1$ is the reverse of the Littlewood-Paley inequality when $p>1$.
When $n=1$ and $p\le 1$, then by \cite[Theorem 2.2.9]{GrafakosMFA}
there is a polynomial $Q$ on $\mathbb R^l$ such that
$$
\|f-Q\|_{H^p(\mathbb R^{ l})} \lesssim \Big\|\Big(\sum_{j_1} |\Deltalta_{j_1 }^{(1)}
(f)|^2\Big)^{1/2}\Big\|_{L^p(\mathbb R^{ l})}<\infty \, .
$$
Since $f$ lies in $ L^2(\mathbb R^{ l})$, it follows that $f-Q$ is a locally integrable
function which lies in $H^p(\mathbb R^l)$ and thus $\|f-Q \|_{L^p}\lesssim
\|f-Q\|_{H^p(\mathbb R^{ l})}<\infty$. Therefore $Q=0$ and \eqref{lp2} follows.
Assume that the assertion is valid for $n$. We will prove the
case $n+1$. Let $r_k$ be the Rademacher functions reindexed by $k\in \mathbb Z$.
Applying \eqref{lp2} to $g=\sum_kf_kr_k$ we obtain
\betagin{align*}
&\hspace{-.5in} \int_{\mathbb R^l}\cdots\int_{\mathbb R^l}
\bigg(\sum_k|f_k(x_1,\dots,x_{n})|^2\bigg)^{{ p/2}}dx_1\cdots dx_{n}\\
\lesssim &\int_{\mathbb R^l}\cdots\int_{\mathbb R^l}\int_0^1
\Big|\sum_kf_k(x_1,\dots,x_n)r_k(t_{n+1} ) \Big|^pdt_{n+1}dx_1\cdots dx_n\\
=&C\int_0^1\int_{\mathbb R^l}\cdots\int_{\mathbb R^l}|g(x_1,\dots,x_n)|^pdx_1\cdots dx_ndt_{n+1},
\end{align*}
where we used the property of Rademacher functions; see
for instance \cite[Appendix C]{Grafakos1}. By the induction hypothesis, the preceding expression is bounded by
a multiple of
\betagin{align*}
&\hspace{-.1in} \int_0^1\int_{(\mathbb R^l)^n} \bigg(\sum_{j_1}\cdots\sum_{j_n}
|\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_n}^{(n)}g(x_1,\dots,x_n)|^2 \bigg)^{p/2}dx_1\cdots dx_ndt_{n+1}\\
\lesssim&\int_0^1\!\! \int_{(\mathbb R^l)^n} \! \int_{[0,1]^{n }} \!
\Big|\sum_{j_1}\cdots\sum_{j_n}
\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_n}^{(n)}g(x_1,\dots,x_n) \partialrod_{i=1}^n r_{j_i}(t_i)
\Big|^p \! dt_1\cdots dt_n d\vec x \, dt_{n+1}\\
\approx &\int_{(\mathbb R^l)^n} \! \int_{[0,1]^{n+1 }} \!
\Big| \!\! \sum_{j_1,\dots , j_n,k} \!\!
\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_n}^{(n)}f_k(x_1,\dots,x_n)r_k(t_{n+1})\partialrod_{i=1}^n r_{j_i}(t_i)
\Big|^p \! dt_1\cdots dt_{n+1}d\vec x\\
\lesssim& \int_{(\mathbb R^l)^n} \bigg(\sum_{j_1}\cdots\sum_{j_{n }}\sum_k
|\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_{n}}^{(n)}f_k(x_1,\dots,x_{n})|^2\bigg)^{p/2}dx_1\cdots dx_{n},
\end{align*}
once again the properties of Rademacher functions were used and $d\vec x= dx_1\cdots dx_n$.
It follows that
\betagin{align*}
&\hspace{-.1in} \int_{ (\mathbb R^l)^{n+1}} |f(x_1,\dots,x_{n+1})|^pdx_1\cdots dx_{n+1}\\
\lesssim &\int_{ (\mathbb R^l)^{n+1}} \sup_{t>0} \big|[\varphi_t*f(x_1,\dots,x_{n})](x_{n+1})\big|^pdx_{n+1}dx_1\cdots dx_{n}\\
\lesssim & \int_{ (\mathbb R^l)^{n+1}} \bigg(
\sum_{j_{n+1}}|\Deltalta_{j_{n+1}}^{(n+1)}f(x_1,\dots,x_{n+1})|^2\bigg) ^{p/2}dx_{n+1}dx_1\cdots dx_{n}\\
\approx&\int_{ (\mathbb R^l)^{n+1}} \bigg(
\sum_{j_{n+1}}|\Deltalta_{j_{n+1}}^{(n+1)}f(x_1,\dots,x_{n+1})|^2\bigg) ^{p/2}dx_1\cdots dx_{n+1}\\
\lesssim&\int_{ (\mathbb R^l)^{n+1}} \! \bigg(\sum_{j_1}\cdots\sum_{j_n}\sum_{j_{n+1}}
|\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_n}^{(n)}\Deltalta_{j_{n+1}}^{(n+1)}
f(x_1,\dots,x_n,x_{n+1})|^2\bigg)^{p/2}\! dx_1\cdots dx_ndx_{n+1},
\end{align*}
where in the last step we use the inequality in the preceding alignment.
To make this argument precise, we work with
finitely many terms and then then pass to limit using Fatou's lemma.
\end{proof}
\betagin{remark}\labelel{11071}
In both \eqref{lp1} and \eqref{lp2} we do not need
the full set of variables. For example, we have
$$
\Big\|\Big(\sum_{j_1\in \mathbb Z}\cdots\sum_{j_q\in \mathbb Z}|\Deltalta_{j_1}^{(1)}\cdots\Deltalta_{j_q}^{(q)}(f)|^2\Big)^{1/2}\Big\|_{L^p(\mathbb R^{ nl})}
\approx\|f\|_{L^p(\mathbb R^{ nl})}
$$
for any $1\le q\le n$ by applying
Lemma \ref{GLP} to $f$ as a function of $(x_1,\dots, x_q)$.
\end{remark}
\betagin{remark}
As a consequence of \eqref{lp2} one can derive the following inequality:
$$
\|f\|_{L^p(\mathbb R^{nl}) }\leq C\|f\|_{H^p({\mathbb R}^{l}\times \cdots \times {\mathbb R}^{l})}\ \ \ \ {\rm for}\ \ f\in L^2(\mathbb R^{nl}) , \ 0<p\leq 1
$$
where $H^p(\underbrace{ {\mathbb R}^{l}\times \cdots \times {\mathbb R}^{l}}_{\textup{$n$ times}})$ denotes the multiparameter Hardy space; on this see \cite{HLLW}.
\end{remark}
\section{The proof of Theorem \ref{1dil}}\labelel{0926}
\betagin{proof}
For $1\le k\ne l \le m,$ we introduce sets
$$
U_{k,l} = \Big\{(\xi_1,\ldots,\xi_m)\in(\mathbb R^n)^m\ :\ \max_{j\ne k,l} |\xi_j|
\le \fracrac{11}{10}|\xi_k|\le \fracrac{11}{50m}|\xi_{l}|\Big\}
$$
and
$$
W_{k,{l}} = \Big\{(\xi_1,\ldots,\xi_m)\in(\mathbb R^n)^m\ :\ \max_{j\ne k,{l}} |\xi_j|
\le \fracrac{11}{10}|\xi_k|,\fracrac{1}{10m}|\xi_{\ell}|\le |\xi_k|\le 2|\xi_{l}|\Big\}.
$$
We now construct smooth homogeneous of degree zero functions $\Phi_{k,{l}}$
and $\Psi_{k,{l}}$ supported in $U_{k,{l}}$ and $W_{k,{l}}$, respectively, and such that
\betagin{equation}
\sum_{1\le k\ne l\le m}\Big(\Phi_{k,{l}}(\xi_1,\dots , \xi_m )+\Psi_{k,{l}}(\xi_1,\dots , \xi_m)\Big)=1
\end{equation}
for every $(\xi_1,\dots , \xi_m)$
in $\big(\mathbb R^{n}\big)^m\setminus\{0\}; $ such functions can be constructed following
the hint of Exercise 7.5.4 in \cite{GrafakosMFA}.
In the support of $\Phi_{k,l}$ the vector with the largest magnitude is $\xi_{l}$, while in the support
of $\Psi_{k,l}$ the vector with the largest magnitude is $\xi_{l} $ and the one with the second largest magnitude is
$\xi_k$. This partition of unity induces the following decomposition of $\sigma$:
\betagin{equation}\labelel{12.5.bvhuawe}
\sigma = \sum_{j=1}^m\sum_{\substack{k=1\\k\neq j}}^m
\big(\sigma \, \Phi_{j,k} + \sigma \, \Psi_{j,k} \big)\, .
\end{equation}
We will prove the required assertion for each piece of
this decomposition, i.e., for the
multipliers $ \sigma \, \Phi_{j,k}$ and $ \sigma \, \Psi_{j,k}$ for each pair $(j,k)$ in the previous sum.
In view of the symmetry of the decomposition, it suffices to consider the case of a fixed pair $(j,k)$ in the sum in \eqref{12.5.bvhuawe}.
To simplify {the} notation, we fix the {pair $(m,m-1)$;} thus, for the rest of the proof we fix $j=m$ and {$k=m-1$,} and we prove boundedness for the $m$-linear operators whose symbols are
$ \sigma_1 =\sigma \, \Phi_{m,m-1}$ and $ \sigma_2= \sigma \, \Psi_{m,m-1}$.
These correspond to the $m$-linear operators
$T_{\sigma_1}$ and $T_{\sigma_2}$, respectively. Note that $\sigma_1$ is supported in the set where
\[
\max(|\xi_1|, \dots , |\xi_{m-2}|) \le \tfrac{11}{10}\, |\xi_{m-1}|
\quad\quaduad
\textup{and}
\quad\quaduad
|\xi_{m-1}| \le \tfrac{1 }{5m}\, |\xi_m| \, .
\]
Also $\sigma_2$ is supported in the set where
\[
\max(|\xi_1|, \dots , |\xi_{m-2}|) \le \tfrac{11}{10}\, |\xi_{m-1}|
\quad\quaduad
\textup{and}
\quad\quaduad
\tfrac{1}{10m}\le \tfrac{|\xi_{m-1}|}{|\xi_m|}\le 2\, .
\]
Fix a Schwartz function $\theta$ whose Fourier transform is supported in the annulus $\fracrac{1}{2}\le |\xi|\le 2$ and
$\sum_{j\in \mathbf Z} \widehat \theta (2^{-j}\xi)=1$ for $\xi \in \mathbb R^n\setminus \{0\}. $
Associated with $\theta$ we define the Littlewood--Paley operator
$\Deltalta_j^\theta(g)= g* \theta_{2^{-j}},$ where $\theta_t(x)=
t^{-n}\theta(t^{-1}x) $ for $t>0$.
The function $\theta$ can be extended to the function $\Theta$ defined on $\mathbb R^{nm}$ by setting $\widehat{\Theta}(\vec{\xi}\,)=\widehat{\Theta}(\xi_1,\ldots,\xi_m) = \widehat{\theta}(\xi_1+\cdots+\xi_m).$
For given Schwartz functions $f_j$ we have
\betagin{align*}
\Deltalta^\theta_j \big(T_{\sigmagma_1}(f_1,&\ldots,f_m)\big)(x) \\
=& \int_{\mathbb R^{mn}}\widehat{\theta}(2^{-j}(\xi_1+\cdots+\xi_m))
\sigmagma_1(\vec{\xi}\,)\widehat{f_1}(\xi_1)\cdots\widehat{f_m}(\xi_m)
e^{2\partiali ix\cdot(\xi_1+\cdots+\xi_m)}
d\vec{\xi}\\
=& \int_{\mathbb R^{mn}}\widehat{\Theta}(2^{-j}\vec{\xi}\,)
\sigmagma_1(\vec{\xi}\,)\widehat{f_1}(\xi_1)\cdots\widehat{f_m}(\xi_m)
e^{2\partiali ix\cdot(\xi_1+\cdots+\xi_m)}
d\vec{\xi}.
\end{align*}
Note that for all $\vec \xi = (\xi_1,\ldots,\xi_m)$ in the support of the function $\widehat{\Theta}(2^{-j}\vec \xi\,)\sigmagma_1(\vec \xi\,),$ we always have $2^{j-2}\le |\xi_m|\le 2^{j+2}.$
Therefore we can take a Schwartz function $\eta$ whose Fourier transform
is supported in $\fracrac{1}{8}\le|\xi_m|\le 8$ and identical to $1$ on $\tfrac{1}{4}\leq |\xi_m|\leq 4$ and insert the factor $\widehat{\eta}(2^{-j}\xi_m)$ into the above integral without changing the outcome. More specifically
\betagin{align*}
\Deltalta^\theta_j&\big(T_{\sigmagma_1}(f_1,\ldots,f_m)\big)(x)\\
=&
\int_{\mathbb R^{mn}}\widehat{\Theta}(2^{-j}\vec{\xi}\,)
\sigmagma_1(\vec{\xi}\, )\widehat{f_1}(\xi_1)\cdots\widehat{f_{m-1}}(\xi_{m-1})\widehat{\eta}(2^{-j}\xi_m)
\widehat{f_m}(\xi_m)e^{2\partiali ix\cdot(\xi_1+\cdots+\xi_m)}d\vec{\xi}.
\end{align*}
Now define
$
\widehat{\Psi_*}(\vec \xi\,)= \sum_{|k|\le 4}\widehat{\Psi}(2^{ -k}\vec \xi\,)
$
and note that $\widehat{\Psi_*}(2^{-j}\vec \xi\,)$ is equal to $1$
on the annulus $\big\{\vec \xi\in \mathbb R^{mn} : \ 2^{j-4}\le |\vec \xi\,|\le 2^{j+4}\big\}$ which contains
the support of $\sigmagma_1\widehat{\Theta}(2^{-j}\cdot) $. Then we write
\betagin{align*}
&\Deltalta^\theta_j\big(T_{\sigmagma_1}(f_1,\ldots,f_m)\big)(x)\\
=&
\int_{\mathbb R^{mn}}\widehat{\Psi_*}(2^{-j }\vec{\xi}\,)\widehat{\Theta}(2^{-j}\vec{\xi}\,)
\sigmagma_1(\vec{\xi}\, )\widehat{f_1}(\xi_1)\cdots\widehat{f_{m-1}}(\xi_{m-1})\widehat{\eta}(2^{-j}\xi_m)
\widehat{f_m}(\xi_m)e^{2\partiali ix\cdot(\xi_1+\cdots+\xi_m)}d\vec{\xi}.
\end{align*}
Taking the inverse Fourier transform, we
obtain that
$
\Deltalta^\theta_j\big(T_{\sigmagma_1}(f_1,\ldots,f_m)\big)(x)
$
is equal to
\betagin{equation}
\labelel{eq.2A01}
\int_{(\mathbb R^n)^m}2^{mnj } (\sigmagma_1^{j }\widehat{\Psi_*}\widehat{\Theta} )\spcheck
\big(2^{j}(x-y_1), \dots ,2^{j}(x-y_m)\big)
\partialrod_{i=1}^{m-1} f_i(y_i)\,
(\Deltalta_j^\eta f_m)(y_m)\, d\vec y,
\end{equation}
where {$d\vec y= dy_1\cdots dy_m$,} and
$
\sigmagma_1^j(\xi_1,\xi_2,\dots ,\xi_m)
=\sigmagma_1(2^j\xi_1 ,2^j\xi_2,\dots ,2^j\xi_m).
$
Recall our assumptions that $\maxL^{\infty}mits_{1\le i \le m} \maxL^{\infty}mits_{1\le \ell \le n} \frac{1}{\gamma_{i\ell}}<r$ and $\maxL^{\infty}mits_{1\le i \le m} \maxL^{\infty}mits_{1\le \ell \le n} \frac{1}{\gamma_{i\ell}}<\min(p_1,\dots , p_m)$.
If $r>1$ we pick $\rho $ such that $1< \rho < 2$ and $\maxL^{\infty}mits_{1\le i \le m} \maxL^{\infty}mits_{1\le \ell \le n}\frac{1}{\gamma_{i\ell}} <\rho < \min(p_1,\dots , p_m,r ) $. If $r=1$, we set
$\rho=1$.
Define a weight for $(y_1,\dots , y_m) \in (\mathbb R^n)^m$ by setting
$$
w_{\vec \gamma}(y_1,\dots , y_m) = \partialrodL^{\infty}mits_{1\le i \le m}
\partialrodL^{\infty}mits_{1\le \ell \le n} (1+4\partiali^2 |y_{i\ell}|^2)^{\frac{\gamma_{i\ell}}{2}}\, .
$$
Let us first suppose that $\rho>1$. We have
{\alphalowdisplaybreaks
\betagin{align*}
&| \Delta_j^\theta \big( T_{\sigmagma_{1}}( f_1 ,\dots , f_{m-1} , f_m )\big)(x)|\\
\leq&
\int_{(\mathbb R^n)^m} \!\!\!\!\!
w_{\vec \gamma} \big(2^j(x-y_1),\dots ,2^j(x-y_m)\big) \,\,
| (\sigmagma_1^j\, \widehat{\Psi_*} \widehat{\Theta} )\spcheck(2^{j}(x-y_1),\dots ,2^{j}(x-y_m))|\\
&\quad\quad\quad\quad\quad\quad\quad\quad \times \fracrac{2^{mnj} |
f_1 (y_1)\cdots f_{m-1} (y_{m-1})( \Deltalta_j^\eta f_m)(y_m)|}{ w_{\vec \gamma} \big(2^j(x-y_1),\dots ,2^j(x-y_m)\big)} \, d\vec y\\
\leq&
\bigg[\int_{(\mathbb R^n)^m}\!\! \big|
\big( w_{\vec \gamma } \,
(\sigmagma_1^j\, \widehat{\Psi_*}\widehat{\Theta})\spcheck
\big) (2^{j}(x - y_1),\dots,2^{j}(x - y_m))
\big|^{\rho'}d\vec y\bigg]^{\fracrac{1}{\rho'}}\\
&\quad\quad \times 2^{mnj}\left(\int_{(\mathbb R^n)^m}\fracrac{ | f_1 (y_1)\cdots
f_{m-1} (y_{m-1}) ( \Deltalta_j^\eta f_m) (y_m)|^\rho}{w_{ \rho\vec \gamma}
\big( 2^j(x-y_1), \dots , 2^j (x-y_m) \big)} \, d\vec y
\right)^{\fracrac{1}{\rho}}\\
\leq& C\left(\int_{(\mathbb R^n)^m} \Big| w_{\vec \gamma }(y_1,\dots , y_m)
(\sigmagma_1^j\, \widehat{\Psi})
\spcheck (y_1,\dots ,y_m) \Big|^{\rho'}d\vec y\right)^{\fracrac{1}{\rho'}}\\
&\quad\quad \times \left(\int_{(\mathbb R^n)^m}\fracrac{2^{mnj}| f_1 (y_1)\cdots
f_{m-1} (y_{m-1}) (\Deltalta_j^\eta f_m) (y_m)|^\rho}{\big( \partialrod_{\ell=1}^n (1+2^j|x_\ell-y_{1\ell} |)^{ \rho \gamma_{1\ell} } \big)\cdots \big( \partialrod_{\ell=1}^n (1+2^j|x_\ell-y_{m\ell} |)^{ \rho \gamma_{m\ell} } \big) }\, d\vec y
\right)^{\fracrac{1}{\rho}}\\
\leq& C\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}} (\sigmagma_1^j\, \widehat{\Psi_*}\widehat{\Theta}) \Big\|_{L^{\rho } } \partialrod_{i=1}^{m-1} \left(\int_{\mathbb R^n}\fracrac{ 2^{jn} | f_{i} (y_{i})|^\rho}{ \partialrod_{\ell=1}^n (1+2^j|x_\ell-y_{i\ell} |)^{ \rho \gamma_{i\ell} } }\, dy_{i}\right)^{\fracrac{1}{\rho}} \\
&\quad\quadq\quad\quadq\quad\quadq\quad\quadq\times \left(\int_{\mathbb R^n}\fracrac{ 2^{jn} | (\Deltalta_j^\eta
f_m) (y_m)|^\rho}{ \partialrod_{\ell=1}^n (1+2^j|x_\ell-y_{m\ell} |)^{ \rho \gamma_{m\ell} } }\, dy_m\right)^{\fracrac{1}{\rho}}\\
\leq&C'
\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}} (\sigmagma_1^j\, \widehat{\Psi_*}\widehat{\Theta}) \Big\|_{L^{\rho } } \bigg[\partialrod_{i=1}^{m-1}
\mathcal M (|f_i|^\rho) (x) ^{\fracrac{1}{\rho}} \bigg] \mathcal M (|\Delta_j^\eta f_m|^\rho) (x) ^{\fracrac{1}{\rho}}
\end{align*}}
\noindent where $\mathcal M$ is the strong maximal function given as $\mathcal M= M^{(1)}\circ \cdots \circ M^{(n)}$,
where $M^{(j)}$ is the Hardy-Littlewood maximal operator acting in the $j$th variable.
Here we made use of the hypothesis that $\gamma_{i\ell} \rho >1$ and we used the Hausdorff-Young inequality, which is
possible since $1\le \rho<2$. Now using \eqref{Lemm11ineq} we obtain
\[
\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}} (\sigmagma_1^j\, \widehat{\Psi_*}\widehat{\Theta}) \Big\|_{L^{\rho } }
\lesssim
\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}}
(\sigmagma(2^j (\cdot)) \, \widehat{\Psi_*} ) \Big\|_{ L^{r } } \lesssim A \, .
\]
We now turn to the case where $r=1$ in which case $\rho=1$. We choose $\gamma_{i\ell}'<\gamma_{i\ell}$ and $\delta>0$ such that
$$
\frac{1}{\gamma_{i\ell} }=\frac{1}{\gamma_{i\ell}'+\delta} <\frac{1}{\gamma_{i\ell}'} <\frac{1}{\gamma_{i\ell}'-\delta} < \fracrac{1}{r}=1
$$
for all $i,\ell$. The preceding argument with $\gamma_{i\ell}'-\delta$ in place of $\gamma_{i\ell}$ yields that
is bounded by
$$
| \Delta_j^\theta \big( T_{\sigmagma_{1}}( f_1 ,\dots , f_m )\big) | \le C'
\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}'-\delta}{2}} (\sigmagma_1^j\, \widehat{\Psi_*}\widehat{\Theta}) \Big\|_{L^{1 } } \bigg[\partialrod_{i=1}^{m-1}
\mathcal M (|f_i| ) \bigg] \mathcal M (|\Delta_j^\eta f_m| ) \, .
$$
In view of \eqref{Lemm11ineq2} we obtain
\[
\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}'-\delta}{2}} (\sigmagma_1^j\, \widehat{\Psi_*}\widehat{\Theta}) \Big\|_{L^{1} }
\lesssim
\Big\| \partialrod_{i=1}^m\partialrod_{\ell=1}^n (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}'+\delta}{2}} (\sigmagma(2^j (\cdot)) \, \widehat{\Psi } ) \Big\|_{L^1} \lesssim A \, .
\]
Thus, we have obtained the estimate
\betagin{equation*}
| \Delta_j^\theta \big(T_{\sigmagma_{1}}( f_1 ,\dots , f_{m-1} , f_m ) \big)(x)|
\lesssim A
\bigg[\partialrod_{i=1}^{m-1}
\mathcal M (|f_i|^\rho) (x) ^{\fracrac{1}{\rho}} \bigg] \mathcal M (|\Delta_j^\eta f_m|^\rho) (x) ^{\fracrac{1}{\rho}}
\end{equation*}
from which it follows that
\betagin{equation*}
\bigg(\sum_{j\in \mathbb Z} | \Delta_j^\theta T_{\sigmagma_{1}}( f_1 ,\dots , f_{m-1} , f_m ) |^2\bigg)^{\frac12}
\lesssim A
\bigg[\partialrod_{i=1}^{m-1}
\mathcal M (|f_i|^\rho) ^{\fracrac{1}{\rho}} \bigg]
\bigg(\sum_{j\in \mathbb Z} \mathcal M (|\Delta_j^\eta f_m|^\rho) ^{\fracrac{2}{\rho}} \bigg)^{\frac12}\, .
\end{equation*}
The claimed bound follows by applying H\"older's inequality with exponents $p_1,\dots, p_m$ and using the
boundedness of $\mathcal M$ on $L^{p_i/\rho}$, $i=1,\dots ,m$, and the Fefferman-Stein \cite{FS} vector-valued
Hardy-Littlewood maximal function inequality on $L^{p_m/\rho}$. (Note $1<2/\rho \le 2$.)
Next we deal with $\sigmagma_2$. Using the notation introduced earlier, we write
\[
T_{\sigmagma_2}(f_1,\dots,f_{m-1},f_m)=\sum_{j\in \mathbb Z}
T_{\sigmagma_2}(f_1,\dots,f_{m-1},\Delta_j^\theta f_m )\, .
\]
We introduce another Littlewood--Paley operator {$\Delta_j^\zeta$,} which is
given on the Fourier transform by multiplying with a bump
$\widehat{\zeta}(2^{-j}\xi)$, where $\widehat{\zeta}$ is equal to one on the
annulus $\{\xi\in \mathbb R^n:\,\, \tfrac1{2^k}\le |\xi|
\le 4\}$ with $\frac1{2^k}\le\tfrac{1}{20m}$,
vanishes off the annulus $\fracrac{1}{2^{k+1}}\le |\xi|\le 8$,
and $\sum_{j}\widehat\zeta(2^{-j}\xi)=k+3$.
The key observation in this case is that
\betagin{equation}
\labelel{eq.3A4}
T_{\sigmagma_2}(f_1,\dots,f_{m-1},\Delta_j^\theta f_m )
=
T_{\sigmagma_2}\big(f_1,\dots , f_{m-2},\Deltalta_j^\zeta
f_{m-1} ,\Deltalta_j^\theta f_m \big)\, .
\end{equation}
As in the previous case, we have
\betagin{align}
\notag
&\hspace{-.7in} T_{\sigmagma_2}\big(f_1,\dots , f_{m-2},\Deltalta_j^\zeta
f_{m-1} ,\Deltalta_j^\theta f_m \big)(x)\\
\labelel{eq.2A02}
=& \int_{(\mathbf R^{ n})^m}\!\!\!\!\!\!
{
\sigmagma_2(\vec \xi \, )
\partialrod_{i=1}^{m-2} \widehat{f_i}(\xi_i) \,\,
\widehat{ \Deltalta_j^\zeta
f_{m-1} }(\xi_{m-1}) \widehat{ \Deltalta_j^\theta
f_m }(\xi_m)
e^{2\partiali ix\cdot(\xi_1 +\cdots
+\xi_m)}
}
\, d\vec \xi.
\end{align}
The integrand in the right-hand side of \eqref{eq.2A02}
is supported in
$
\fracrac{1}{2} 2^j\le|\xi_1|+\dots + |\xi_m|\le \fracrac{11m}{5} 2^j.
$
Thus one may insert the factor
$$
\widehat{\Psi_*}(2^{-j }\xi_1,\dots , 2^{-j }\xi_m)=
\sum_{|k|\le m+1}\widehat{\Psi}(2^{-j-k}\xi_1,\dots , 2^{-j-k}\xi_m)
$$
in the integrand.
A similar calculation as in the case for $\sigma_1$ yields the
estimate
$$
|T_{\sigmagma_{2}}(f_1,\dots ,f_{m-2},
\Deltalta_j^\zeta f_{m-1} ,\Deltalta_j^\theta f_m ) |
\lesssim
A \bigg(\partialrod_{i=1}^{m-2}
\mathcal M( |f_i|^\rho )^{\frac{1}{\rho}} \bigg)
\mathcal M( | \Deltalta_j^\zeta f_{m-1} |^\rho )^{\frac{1}{\rho}}
\mathcal M( | \Deltalta_j^\theta f_m|^\rho )^{\frac{1}{\rho}} \, .
$$
\noindent Summing over $j$ and taking $L^p$ norms {yields}
\betagin{equation*}
\betagin{split}
&\!\!\!\!\big\| T_{\sigmagma_{2}}(f_1,\dots ,f_{m-1},f_m)\big\|_{L^p(\mathbb R^n)}\\
\leq &C\, A \,\Big\| \bigg[ \partialrod_{i=1}^{m-2}
\mathcal M (|f_i|^\rho) ^{\fracrac{1}{\rho}} \bigg]
\sum_{j\in \mathbf Z} \mathcal {M} (|\Deltalta_j^\theta
f_{m-1} |^\rho )^{\fracrac{1}{\rho}} \mathcal {M} ( |\Deltalta_j^\eta
f_{m} |^\rho ) ^{\fracrac{1}{\rho}}
\Big\|_{L^p }\\
\leq &C\, A \,\Big\|
\bigg[ \partialrod_{i=1}^{m-2} \mathcal M (|f_i|^\rho) ^{\fracrac{1}{\rho}} \bigg]
\bigg( \sum_{j\in \mathbf Z} \mathcal {M} (|\Deltalta_j^\theta
f_{m-1} |^\rho )^{\fracrac{2}{\rho}} \bigg)^{\frac12} \bigg(\sum_{j\in \mathbf Z} \mathcal {M} ( |\Deltalta_j^\eta
f_{m} |^\rho ) ^{\fracrac{2}{\rho}} \bigg)^{\frac12}
\Big\|_{L^p(\mathbb R^n)}
\end{split}
\end{equation*}
Applying H\"older's inequality, the
boundedness of $\mathcal M$ on $L^{p_i/\rho}$, $i=1,\dots ,m-1$, and the Fefferman-Stein \cite{FS} vector-valued
Hardy-Littlewood maximal function inequality on $L^{p_{m-1}/\rho}$ or on $L^{p_m/\rho}$ (noting that
$1<2/\rho\le 2$)
concludes the proof of the theorem.
\end{proof}
\betagin{remark}\labelel{REMARK}
In case I we obtained the estimate
\betagin{equation*}
| \Delta_j^\theta \big(T_{\sigmagma_{1}}( f_1 ,\dots , f_{m-1} , f_m ) \big) | \lesssim A
\bigg[\partialrod_{i=1}^{m-1}
\mathcal M (|f_i|^\rho) ^{\fracrac{1}{\rho}} \bigg] \mathcal M (|\Delta_j^\eta f_m|^\rho) ^{\fracrac{1}{\rho}}
{\color{blue}
.}
\end{equation*}
In case II we obtained the estimate
$$
|T_{\sigmagma_2}(f_1,\dots,f_{m-1},\Delta_j^\theta f_m )| \lesssim
A \bigg(\partialrod_{i=1}^{m-2}
\mathcal M( |f_i|^\rho )^{\frac{1}{\rho}} \bigg)
\mathcal M( | \Deltalta_j^\zeta f_{m-1} |^\rho )^{\frac{1}{\rho}}
\mathcal M( | \Deltalta_j^\theta f_m|^\rho )^{\frac{1}{\rho}}
{\color{blue}
.}
$$
By symmetry for any $k_0\neq j_0$ in $\{1,\dots , m\}$ we have for $\sigma \Phi_{j_0,k_0}$
\betagin{equation*}
| \Delta_j^\theta \big(T_{\sigma \Phi_{j_0,k_0}}( f_1 ,\dots , f_m ) \big) | \lesssim A
\bigg[\partialrod_{\substack { 1\le i \le m \\ i \neq j_0 }}
\mathcal M (|f_i|^\rho) ^{\fracrac{1}{\rho}} \bigg] \mathcal M (|\Delta_j^\eta f_{j_0}|^\rho) ^{\fracrac{1}{\rho}}
\end{equation*}
and for $\sigma \Psi_{j_0,k_0}$
\betagin{equation*}
| T_{\sigma \Psi_{j_0,k_0}}( f_1 ,\dots , \Delta_j^\theta f_{j_0} , \dots , f_m ) | \lesssim A
\bigg[\partialrod_{\substack { 1\le i \le m \\ i \neq j_0 \\ i\neq k_0 }}
\mathcal M (|f_i|^\rho) ^{\fracrac{1}{\rho}} \bigg] \mathcal M (|\Delta_j^\eta f_{j_0}|^\rho) ^{\fracrac{1}{\rho}}
\mathcal M( | \Deltalta_j^\zeta f_{k_0} |^\rho )^{\frac{1}{\rho}}.
\end{equation*}
\end{remark}
\section{The proof of Theorem~\ref{End}}
\betagin{proof}
For $1\le k\ne l\le m,$ recall the sets $ U_{k,l}$ and $W_{k,l} $
and the functions $\Phi_{k,l}$ and $\Psi_{k,l}$
in the proof of Theorem \ref{1dil}.
Letting $\sigmagma^1_{k,l}=\sigmagma\Phi_{k,l}$ and $\sigmagma^2_{k,l}=\sigmagma\Psi_{k,l} $, we write
$$
\sigmagma=\sum_{1\le k\ne l\le n}\big(\sigmagma^1_{k,l}+\sigmagma^2_{k,l}\big).
$$
By the symmetry, it suffices to consider the case where $k=m-1$ and $l=m.$ We establish the claimed estimate
for $T_{\sigmagma_1}$ and $T_{\sigmagma_2}$ with
$\sigmagma_1 = \sigmagma^1_{m-1,m}$ and $\sigmagma_2 = \sigmagma^2_{m-1,m}.$
We first consider $T_{\sigmagma_1}(f_1, \dots , f_m)$, where
$f_j$ are fixed Schwartz functions.
We will prove
\betagin{equation}\labelel{equ:HpTSigmEND}
\norm{\Big(\sum_j\Deltalta_j^\theta (T_{\sigmagma_1}(f_1,\ldots,f_m))|^2\Big)^{1/2}}_{L^{1/m,\infty}(\mathbb R^n)}\lesssim A\norm{f_1}_{H^{ 1}(\mathbb R^n)}\cdots\norm{f_m}_{H^{1}(\mathbb R^n)}.
\end{equation}
Let $H^{1/m,\infty}$ denote the weak Hardy space of all bounded tempered distributions whose
smooth maximal function lies in weak $L^{1/m}$.
Given $0<p<\infty$,
for $F$ in $ L^2(\mathbb R^n)$ there is a polynomial $Q$ on $\mathbb R^n$ such that
\betagin{equation}\labelel{Q=0}
\norm{F-Q}_{L^{p,\infty}(\mathbb R^n)}\le C_{p,n} \norm{F-Q}_{H^{p,\infty}(\mathbb R^n)} \approx \Big\|\Big(\sum_j|\Delta_j(F)|^2\Big)^{1/2}\Big\|_{L^{p,\infty}(\mathbb R^n)},
\end{equation}
by a result of He \cite{He2014}.
But the fact that $F$ lies in $L^2$ implies that $Q=0$.
Applying \eqref{Q=0} with $F= T_{\sigmagma_1}(f_1,\ldots,f_m)$, for which
we observe that
$
\| T_{\sigmagma_1}(f_1,\ldots,f_m)\|_{ L^2(\mathbb R^n) }<\infty\,
$
for Schwartz functions $f_j$,
we conclude from
\eqref{equ:HpTSigmEND} that \eqref{equ:TSigmaESTH1} holds for $\sigmagma_1$.
To verify \eqref{equ:HpTSigmEND}, we recall \eqref{eq.2A01} and
set $\omega_{\gamma_i}( y) = (1+4\partiali^2 | y |^2 )^{\frac{\gamma_{i }}{2}}$ for $y\in \mathbb R^n$.
Choose $\gamma_j'$ and $\deltalta>0$ such that
$n< \gammamma_i'-\deltalta <\gammamma_i'<\gamma_i'+\delta=\gamma_i$ for all $1\le i\le m$.
Now we rewrite
\betagin{align}
& |\Deltalta^\theta_j\big(T_{\sigmagma_1}(f_1,\ldots,f_m)\big)(x) | \notag \\
\leq&
\!\! \intL^{\infty}mits_{(\mathbb R^n)^m}\!
\Big\{ \partialrod_{i=1}^m \omega_{\gamma_i'-\delta} (2^j(x-y_i)) \Big\}
| (\sigmagma_1^{j }\, \widehat{\Psi_*}\widehat{\Theta} )\spcheck(2^{{j }}(x\! -\! y_1),\dots ,2^{{j }}(x\! -\! y_m))| \notag \\
&\quad\quad\quad\quad \times \fracrac{2^{mn{j}} |f_1(y_1)|\cdots |f_{m-1}(y_{m-1})|| (\Deltalta_j^\eta f_m)(y_m)|}
{ \partialrod_{i=1}^m \omega_{\gamma_i'-\delta} (2^j(x-y_i)) } \, d\vec y \notag\\
\lesssim&\,\,\, \Big\| \Big( \partialrod_{i=1}^m
\omegaega_{ \gammamma_i'-\delta} \Big) (\sigmagma_1^{j }\, \widehat{\Psi_*}\widehat{\Theta} )\spcheck \Big\|_{L^{\infty}}
\bigg( \partialrod_{i=1}^{m-1}
M( f_i )(x) \bigg)
M( \Deltalta_j^\eta f_m )(x) \, , \labelel{MMMJJJ}
\end{align}
as a consequence of the fact that $ \gammamma_i'-\delta>n $ for all $1\le i\le m$. Here $M$ is the
uncentered Hardy-Littlewood maximal operator. In view of the Hausdorff-Young inequality, the first factor in
\eqref{MMMJJJ} is bounded by
$$
\Big\| \partialrod_{i=1}^m (I-\Delta_{\xi_i})^{\frac{\gamma_i'-\delta}{2}}
\big(\sigmagma_1^{j }\, \widehat{\Psi_*}\widehat{\Theta} \big) \Big\|_{L^1} \lesssim
\Big\| \partialrod_{i=1}^m (I-\Delta_{\xi_i})^{\frac{\gamma_i'+\delta}{2}}
\big(\sigmagma (2^{j }(\cdot)) \, \widehat{\Psi } \big) \Big\|_{L^1} \lesssim A
$$
where the penultimate inequality is a consequence of \eqref{Lemm11ineq2} and that $\gamma_i'+\delta=\gamma_i$.
Thus, we proved
\betagin{equation*}
|\Deltalta^\theta_j\big(T_{\sigmagma_1}(f_1,\ldots,f_m)\big)|
\lesssim
A \bigg( \partialrod_{i=1}^{m-1}
M( f_i ) \bigg) M( \Deltalta_j^\eta f_m ) .
\end{equation*}
Using the preceding inequality we obtain
\betagin{eqnarray*}
& &\hspace{-.7in}
\big\|T_{\sigmagma_{1}}(f_1,\dots,f_{m-1},f_m)\big\|_{H^{1/m,\infty}(\mathbb R^n)}\\
&\lesssim &\Big\| \Big\{\sum_j
| \Deltalta^\theta_j\big(T_{\sigmagma_1}(f_1,\ldots,f_m)\big) |^2 \Big\}^{\frac12}\Big\|_{L^{1/m,\infty}(\mathbb R^n)}\\
&\lesssim & A\Big\| \Big\{\sum_j M( \Deltalta_j^\eta
f_m )^{ {2} }\Big\}^{\frac12} \Big\|_{L^{1,\infty}(\mathbb R^n)}
\partialrod_{i=1}^{m-1} \big\|
M( f_i ) \big\|_{L^{1,\infty}(\mathbb R^n)} \\
&\lesssim & A\Big\| \Big\{\sum_j |\Deltalta_j^\eta
f_m | ^{ {2} }\Big\}^{\frac{1}{2}} \Big\|_{L^{1}(\mathbb R^n)}
\partialrod_{i=1}^{m-1} \|f_i \|_{L^{1}(\mathbb R^n)} \\
&\lesssim & A
\partialrod_{i=1}^{m } \|f_i \|_{H^{1}(\mathbb R^n)}\, .
\end{eqnarray*}
This proves estimate \eqref{equ:TSigmaESTH1} for $\sigma_1$.
Next we deal with $\sigmagma_2$. From \eqref{eq.3A4}, we have
\[
T_{\sigmagma_2}(f_1,\dots,f_{m-1},f_m)=\sum_{j\in \mathbb Z}
T_{\sigmagma_2}(f_1,\dots,f_{m-1},\Delta_j^\theta f_m ),
\]
where
$T_{\sigmagma_2}\big(f_1,\dots , f_{m-2},\Deltalta_j^\zeta
f_{m-1} ,\Deltalta_j^\theta f_m \big)$
is defined in \eqref{eq.2A02}.
A similar calculation as in the case for $\sigma_1$ yields the
estimate
$$
|T_{\sigmagma_{2}}(f_1,\dots ,f_{m-2},
\Deltalta_j^\zeta f_{m-1} ,\Deltalta_j^\theta f_m ) |
\lesssim\,\,
A \bigg(\partialrod_{i=1}^{m-2}
M( f_i ) \bigg)
M( \Deltalta_j^\zeta f_{m-1} )
M( \Deltalta_j^\theta f_m ) \, .
$$
Summing over $j$, taking $L^{1/m,\infty}$ quasinorms and applying the Littlewood-Paley
characterization of $H^1$ we deduce
\betagin{align*}
& \hspace{-.1in}\big\| T_{\sigmagma_{2}}(f_1,\dots ,f_{m-1},f_m)\big\|_{L^{1/m,\infty}(\mathbb R^n)}\\
& \lesssim A\Big\| \partialrod_{i=1}^{m-2}
M( f_i )
\sum_{j\in \mathbb Z} M\big( \Deltalta_j^\zeta
f_{m-1} \big)\ M\left( \Deltalta_j^\theta
f_{m} \right)
\Big\|_{L^{1/m,\infty}(\mathbb R^n) }\\
&\lesssim
{ A} \Big\| \Big\{\partialrod_{i=1}^{m-2}
M( f_i ) \Big\}
\Big\{\sum_{j\in \mathbb Z} M\big( \Deltalta_j^\zeta
f_{m-1} \big) ^{ 2 }
\Big\}^{\frac12}
\Big\{\sum_{j\in \mathbb Z} M\left( \Deltalta_j^\theta
f_{m} \right) ^{2}
\Big\}^{\frac12}
\Big\|_{L^{1/m,\infty} (\mathbb R^n)} \\
&\lesssim
{ A}\Big( \partialrod_{i=1}^{m-2} \big\|
M( f_i ) \big\|_{L^{1,\infty} } \Big)
\Big\| \Big\{\sum_{j\in \mathbb Z} M\big( \Deltalta_j^\zeta
f_{m-1} \big) ^{ 2 }
\Big\}^{\!\! \frac12} \Big\|_{L^{1,\infty} } \Big\|
\Big\{\sum_{j\in \mathbb Z} M\big( \Deltalta_j^\theta
f_{m} \big) ^{2}
\Big\}^{\!\!\frac12}
\Big\|_{L^{1 ,\infty} }\\
&\lesssim
{ A}\Big( \partialrod_{i=1}^{m-2} \big\|
f_i \big\|_{L^{1 }(\mathbb R^n)} \Big)
\Big\| \Big\{\sum_{j\in \mathbb Z} \big| \Deltalta_j^\zeta
f_{m-1} \big| ^{ 2 }
\Big\}^{\frac12} \Big\|_{L^{1 }(\mathbb R^n)} \Big\|
\Big\{\sum_{j\in \mathbb Z} \left| \Deltalta_j^\theta
f_{m} \right| ^{2}
\Big\}^{\frac12}
\Big\|_{L^{1 }(\mathbb R^n) }\\
&\lesssim
{ A} \partialrod_{i=1}^{m } \big\|
f_i \big\|_{H^{1 }(\mathbb R^n)} \, .
\end{align*}
This concludes the proof of Theorem \ref{End}.
\end{proof}
\betagin{comment}
\section{Appendix: II}
In this section we consider the special case of Proposition \ref{less}
with $n=2$, $m_1=2$ and $m_2=1$. That is to say, we prove
\betagin{prop}
If $\sigma(\xi,\eta)=\sigma_1(\xi_1,\eta_1)\sigma_2(\eta_2)$
with $\sigma_1\in L^\infty(\mathbb R^2)$ and $\sigma_2\in L^\infty(\mathbb R)$.
Suppose that for some $\gamma$ and $r$ with $r\gamma>1$
we have
$$
A=\max\{\sup_{j\in \mathbb Z} \|\sigma_1 (2^j\cdot) \widehat{ \Psi_1 }\|_{L^r_{
\gamma,\gamma} (\mathbb R^2 )},
\sup_{j\in \mathbb Z} \|\sigma_2 (2^j\cdot) \widehat{ \Psi_2 }\|_{L^r_{
\gamma} (\mathbb R )}\}<\infty ,
$$
where $\widehat{\Psi _\ell }$ ($\ell=1,2$) is a smooth function on $\mathbb R^{m_\ell}$,
where $m_1=2$ and $m_2=1$, supported
in the annulus $\fracrac{1}{2}\le |\eta |\le 2$ satisfying
\betagin{equation}\labelel{Psic2}
\sum_{k\in\mathbb{Z}}\widehat{\Psi_\ell}(2^{-k}\eta)=1, \quad\quaduad \quad\textup{for all $\eta\in \mathbb R^{m_\ell}\setminus\{0\}.$}
\end{equation}
Then we have
$$
\norm{T_{\sigmagma}}_{L^{p_1}(\mathbb{R} ^{2})\times L^{p_2}(\mathbb{R} ^{2})\to L^p(\mathbb{R} ^{2})}\lesssim A^2,
$$
where $\fracrac{1}{p}=\fracrac{1}{p_1}+\fracrac{1}{p_2} $,
$p_j>1$ for all $j=1,2$,
and
$p_j\gamma>1$.
\end{prop}
This is a bi-parameter multiplier theorem whose proof is very similar to that of
the multilinear multiplier
theorem using Sobolev spaces of dominating mixed smoothness.
\betagin{proof}
Just like before we decompose $\sigma_1(\xi_1,\eta_1)
=\sigma_{10}(\xi_1,\eta_1)+\sigma_{11}(\xi_1,\eta_1)+\sigma_{12}(\xi_1,\eta_1)$
such that $|\xi_1|\sigmam|\eta_1|$ in the support of $\sigma_{10}$,
$|\xi_1|>10|\eta_1|$ in the support of $\sigma_{11}$,
and $|\eta_1|>10|\xi_1|$ in the support of $\sigma_{12}$.
We do not decompose $\sigma_2$ for it has just one variable $\eta_2$.
For this same reason we will consider later, for a function $f$ defined on $\mathbb R^n$, the partial Fourier transform $\mathscr F_k(f)$
defined by
$$
\int_{\mathbb R} f(x)e^{-2\partiali i x_k\xi_k}dx_k.
$$
This is a function of the variable $(x_1,\dots, x_{k-1},\xi_k,x_{k+1},\dots, x_n)$.
We will need also the Littlewood-Paley operator
$\Delta_j^{k}(f)(x):=\mathscr F^{-1}(\mathscr F_k(f)\widehat \theta(2^{-j}\cdot))$
for an appropriate bump $\theta$ on $\mathbb R$ whose Fourier transform generally is
supported in the unit annulus.
We write $T=T_0+T_1+T_2$ with the obvious relation that $T_j$ is associated with the multiplier
$\sigma_{1j}\sigma_2$.
To estimate $\|T_1(f,g)\|_{L^p}$,
we consider $
\Delta_{j_1}^1\Delta_{j_2}^2(T_1(f,g))(x)$,
which could be written as
$$
\int_{\mathbb R^3}\sigma_{11}(\xi_1,\eta_1)\sigma_2(\eta_2)\mathscr F_1(f)(\xi_1)
\widehat g(\eta)\widehat\theta_{j_1}(\xi_1+\eta_1)\widehat\theta_{j_2}(\eta_2)e^{2\partiali ix_1(\xi_1+\eta_1)+x_2\eta_2}
d\xi_1d\eta,
$$
where $\eta=(\eta_1,\eta_2)$ and
$\widehat\theta_{j_1}(s)=\widehat\theta(2^{-j_1}s)$. Recalling the supports of
$\widehat\theta$ and $\sigma_{11}$,
we can
replace $\widehat\theta(\xi_1+\eta_1)$ by $\widehat \Theta(\xi_1,\eta_1)$ with changing the value of the last integral.
Moreover we can replace $f$ and $g$ by $\Delta^1_{j_1}(f)$ and $\Delta^2_{j_2}(g)$ for appropriate bumps,
and insert $\widehat{(\Psi_{1})}_{j_1}(\xi_1,\eta_1)$ and $\widehat{(\Psi_{2})}_{j_2}(\eta_2)$
taking value $1$ on the support of
$\sigma_{11}\widehat\Theta_{j_1}$ and $\sigma_2\widehat\theta_{j_2}$ respectively.
So the previous integral becomes
\betagin{align}
\int_{\mathbb R^3}&\sigma_{11}(\xi_1,\eta_1)\widehat\Theta_{j_1}(\xi_1,\eta_1)\widehat{(\Psi_{1})}_{j_1}(\xi_1,\eta_1)\\
&\widehat\theta_{j_2}(\eta_2)
\widehat{(\Psi_{2})}_{j_2}(\eta_2)\sigma_2(\eta_2)\mathscr F_1(\Delta^1_{j_1}f)(\xi_1)
\widehat {\Delta_{j_2}^2g}(\eta)e^{2\partiali ix_1(\xi_1+\eta_1)+x_2\eta_2}
d\xi_1d\eta\\
=&\int_{\mathbb R^3}\frac{2^{j_1/\rho}\Delta_{j_1}^1f(y_1,x_2)}{(1+(2^{j_1}|x_1-y_1|)^2)^{\gamma/2}}
\frac{2^{(j_1+j_2)/\rho}\Delta_{j_2}^2g(z_1,z_2)}{(1+(2^{j_1}|x_1-z_1|)^2)^{\gamma/2}(1+(2^{j_2}|x_2-z_2|)^2)^{\gamma/2}}\labelel{e10051}\\
&2^{(-2j_1-j_2)/\rho}K(x,y_1,z)(1+(2^{j_1}|x_1-y_1|)^2)^{\gamma/2}(1+(2^{j_1}|x_1-z_1|)^2)^{\gamma/2}(1+(2^{j_2}|x_2-z_2|)^2)^{\gamma/2}dy_1dz,
\end{align}
where
\betagin{multline}K(x,y_1,z)=\int_{\mathbb R^3}\sigma_{11}(\xi_1,\eta_1)\widehat\Theta_{j_1}(\xi_1,\eta_1)\widehat{(\Psi_{1})}_{j_1}(\xi_1,\eta_1)\\
\widehat\theta_{j_2}(\eta_2)
\widehat{(\Psi_{2})}_{j_2}(\eta_2)\sigma_2(\eta_2)e^{2\partiali i((x_1-y_1)\xi_1+(x-z)\cdot\eta)}d\xi_1d\eta\\
=2^{2j_1+j_2}\int_{\mathbb R^2}\sigma_{11}(2^{j_1}\xi_1,2^{j_1}\eta_1)\widehat\Theta(\xi_1,\eta_1)\widehat{\Psi_{1}}(\xi_1,\eta_1)e^{2\partiali i((x_1-y_1)\xi_1+(x_1-z_1)\eta_1)}d\xi_1d\eta_1\\
\int_{\mathbb R}\sigma_2(2^{j_2}\eta_2)\widehat\theta(\eta_2)
\widehat{\Psi_{2}}(\eta_2)e^{2\partiali i((x_2-z_2)\eta_2)}d\eta_2.
\end{multline}
Applying the H\"older inequality for $\rho\in[1,2)$ such that $ \max( \frac{1}{p_1},
\frac{1}{p_2},\frac{1}{r}) <\frac{1}{\rho}< \gamma $, we can control
\eqref{e10051} by
\betagin{multline}\labelel{e10052}
M_1(|\Delta_{j_1}^1f|^\rho)^{1/\rho}(x_1,x_2)\mathcal M(|\Delta^2_{j_2}g|^\rho)^{1/\rho}(x_1,x_2)\\
(\int_{\mathbb R^3}
|2^{(-2j_1-j_2)/\rho}K(x,y_1,z)(1+(2^{j_1}|x_1-y_1|)^2)^{\gamma/2}(1+(2^{j_1}|x_1-z_1|)^2)^{\gamma/2}(1+(2^{j_2}|x_2-z_2|)^2)^{\gamma/2}|^{\rho'} dy_1dz)^{1/\rho'},
\end{multline}
where $M_1$ is the Hardy-Littlewood maximal function with respect to the first variable
and $\mathcal M$ is the strong maximal function.
It will be easy to verify that the integral in \eqref{e10052}
is bounded by
$$
\|\sigma_{11}(2^{j_1}\cdot)\widehat\Theta\widehat\Psi_1\|_{L^r_{\gamma,\gamma}(\mathbb R^2)}
\|\sigma_2(2^{j_2}\cdot)\widehat\theta\widehat\Psi_2\|_{L^r_{\gamma}(\mathbb R)}
$$
in terms of \eqref{equ:SobqSobr}.
We know that this is bounded by $A^2$. So
\betagin{align*}
\|T_1(f,g)\|_{L^p}\le&\|\sum_{j_1}\sum_{j_2}\Delta^1_{j_1}\Delta^2_{j_2}T_1(f,g)\|_{L^p}\\
\le&A^2\|\sum_{j_1}\sum_{j_2}M_1(|\Delta^1_{j_1}f|^\rho)^{1/\rho}
\mathcal (|\Delta^2_{j_2}(g)|^{\rho})^{1/\rho}\|_{L^p}\\
\le &A^2\|f\|_{p_1}\|g\|_{p_2}.
\end{align*}
This establishes the boundedness for $T_1$, and the $T_2$ could be treated similarly.
Now we consider $T_0$ for which in the support of the related multiplier $\sigma_{10}$
we have
$|\xi_1|\sigmam|\eta_1|$.
We write $T_0(f,g)$ as $\sum_jT_0(\Delta^1_j(f),g)$. Using that $|\xi_1|\sigmam|\eta_1|$,
we can rewrite $T_0(\Delta^1_j(f),g)$ as $T_0(\Delta^1_j(f),\Delta^1_{j}(g))$.
An argument like above shows that it is bounded
by $A^2M_1(|\Delta^1_{j}f|^\rho)^{1/\rho}(x_1,x_2)\mathcal M(|\Delta^1_{j}g|^\rho)^{1/\rho}(x_1,x_2)$.
Hence
\betagin{align*}\|T_0(f,g)\|_p\le &A^2\|\sum_jM_1(|\Delta^1_{j}f|^\rho)^{1/\rho}\mathcal M(|\Delta^1_{j}g|^\rho)^{1/\rho}|\|_p\\
\le& A^2\|(\sum_jM_1(|\Delta^1_{j}f|^\rho)^{2/\rho})^{1/2}
(\sum_j\mathcal M(|\Delta^1_{j}g|^\rho)^{2/\rho})^{1/2}|\|_p\\
\le& A^2\|f\|_{p_1}\|g\|_{p_2}.
\end{align*}
This gives us the boundedness of $T_0$
and, therefore, that of $T$.
\end{proof}
\section{Appendix: The linear case}
Let
$$
K=\sup_{j\in \mathbb Z} \big\| (I-\Delta)^{\frac{s }{2}} \big[\widehat{\Psi} \sigma(2^j(\cdot))\big]\big\|_{L^r(\mathbb R^n)} <\infty\, .
$$
Suppose that $f$ is a Schwartz function on $\mathbb R^n$. Then we write
$$
\Delta_j T_\sigma(f) = \int_{\mathbb R^n} \widehat{f}(\xi) \widehat{\Psi}(2^{-j} \xi) \sigma(\xi) e^{2\partiali i x\cdot \xi}d\xi
= \int_{\mathbb R^n} (\Delta_j^{\Theta}f )\sphat{}\,(\xi) \widehat{\Psi}(2^{-j} \xi) \sigma(\xi) e^{2\partiali i x\cdot \xi}d\xi.
$$
where $\widehat{\Theta} $ looks like $\Psi$ but is equal to $1$ on its support. We write the preceding expression as
$$
2^{jn} \int_{\mathbb R^n} (\Delta_j^{\Theta}f )\sphat{}\,(2^{j} \xi') \widehat{\Psi}( \xi') \sigma(2^{j}\xi') e^{2\partiali i x\cdot 2^{j}\xi'} d\xi'
$$
which is equal to
$$
\int_{\mathbb R^n} (\Delta_j^{\Theta}f )( 2^{-j} y' ) \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat\, (2^jx-y') \, dy'
= 2^{jn} \int_{\mathbb R^n} (\Delta_j^{\Theta}f )( y ) \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat\, ( 2^j(x-y)) \, dy.
$$
We write the preceding expression as
$$
2^{jn} \int_{\mathbb R^n} \frac{ (\Delta_j^{\Theta}f )( y )}{(1+2^j|x-y|)^s} (1+2^j|x-y|)^s\big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat\, ( 2^j(x-y)) \, dy
$$
and this is bounded by
$$
\bigg(\int_{\mathbb R^n} \frac{2^{jn} |\Delta_j^{\Theta}f ( y )|^ \rho}{(1+2^j|x-y|)^{s\rho} } dy \bigg)^{\frac 1\rho }
\bigg(\int_{\mathbb R^n} 2^{jn} \big| (1+(2^j|x-y|)^2)^{\frac s 2} \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat\, ( 2^j(x-y)) \big|^{\rho'} dy \bigg)^{\frac 1{\rho'}} .
$$
Suppose that $s \rho>n$. This is in turn controlled by
$$
C\, M( |\Delta_j^{\Theta}f|^\rho)(x)^{\frac 1\rho}
\bigg(\int_{\mathbb R^n} \big| (1+|y|^2)^{\frac s 2} \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat\,\, ( y) \big|^{\rho'} dy \bigg)^{\frac 1{\rho'}}
$$
and an application of the Hausdorff-Young inequality gives the estimate
$$
C\, M( |\Delta_j^{\Theta}f|^r)(x)^{\frac 1\rho}
\bigg(\int_{\mathbb R^n} \big|\big[ (1+|\cdot |^2)^{\frac s 2} \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat \,\,\big]\spcheck ( y) \big|^{\rho } dy \bigg)^{\frac 1{\rho }} \le C\, K\, M( |\Delta_j^{\Theta}f|^\rho)(x)^{\frac 1\rho}
$$
where we used the fact that
$$
\bigg(\int_{\mathbb R^n} \big|\big[ (1+|\cdot |^2)^{\frac s 2} \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat \,\,\big]\spcheck ( y) \big|^{\rho } dy \bigg)^{\frac 1{\rho }}
\le C
\bigg(\int_{\mathbb R^n} \big|\big[ (1+|\cdot |^2)^{\frac s 2} \big[\widehat{\Psi} \sigma(2^{j}(\cdot))\big]\sphat \,\,\big]\spcheck ( y) \big|^{r } dy \bigg)^{\frac 1{r }} \le C\, K
$$
when $1<\rho<r$ (Lemma 7.5.7 in MFA).
We now let
$$
\big\| T_\sigma(f) \big\|_{L^p(\mathbb R^n)} \le C_p(n) \Big\| \Big( \sum_{j \in \mathbb Z} |\Delta_j T_\sigma (f) |^2 \Big)^{\frac12} \Big\|_{L^p}
\le C_p(n) K \Big\| \Big( \sum_{j \in \mathbb Z} M( |\Delta_j^{\Theta}f|^\rho) ^{\frac 2 \rho} \Big)^{\frac12} \Big\|_{L^p}
$$
and the conclusion follows by applying the Fefferman-Stein inequality \cite{FS} and
taking $1<\rho<r\le 2$ such that $s\rho>n$. This is possible since by assumption we have
$rs>n$.
Remarks:
1. We can replace the factor $(1+2^j|x-y|)^s$ by $(1+2^j|x_1-y_1|)^{s_1} \cdots (1+2^j|x_n-y_n|)^{s_n}$. Then the
argument will work in the same way but the requirement on the indices will be $s_1\rho>1 , \dots, s_n\rho>1$.
Then we have the condition
$$
\sup_{j\in \mathbb Z} \big\| (I-\partial_1^2)^{\frac{s_1}{2}} \cdots (I-\partial_n^2)^{\frac{s_n}{2}} \big[\widehat{\Psi} \sigma(2^j(\cdot))\big]\big\|_{L^r(\mathbb R^n)} <\infty
$$
with $\min(s_1,\dots , s_n)r>1$ instead of
$$
K=\sup_{j\in \mathbb Z} \big\| (I-\Delta)^{\frac{s }{2}} \big[\widehat{\Psi} \sigma(2^j(\cdot))\big]\big\|_{L^r(\mathbb R^n)} <\infty
$$
with $sr>n$.
2. We can replace the factor $(1+2^j|x-y|)^s$ by $(1+2^{j_1} |x_1-y_1|)^{s_1} \cdots (1+2^{j_n} |x_n-y_n|)^{s_n}$ where
$\min(s_1,\dots , s_n)r>1$. Then we obtain the estimate
$$
|\Delta_{j_1}^{(1)} \cdots \Delta_{j_n}^{(n)} T_\sigma(f)| \le C\, K \,
\Big[M^{(1)}\cdots M^{(n)} (|\Delta_{j_1}^{(1)} \cdots \Delta_{j_n}^{(n)} f|^\rho) \Big]^{\frac 1\rho}
$$
where
$$
K = \sup_{j_1\in \mathbb Z} \cdots \sup_{j_n\in \mathbb Z}
\big\| (I-\partial_1^2)^{\frac{s_1}{2}} \cdots (I-\partial_n^2)^{\frac{s_n}{2}} \big[\widehat{\partialsi}(\xi_1)\cdots \widehat{\partialsi}(\xi_n)\sigma(2^{j_1} \xi_1 , \dots ,2^{j_n} \xi_n ) \big] \big\|_{L^r(\mathbb R^n)}
$$
where $\widehat{\partialsi}$ is a smooth function on $\mathbb R$ supported around $|t|\approx 1$ and satisfying
$\sum_{k\in \mathbb Z} \widehat{\partialsi}(2^{-k} t) = 1$.
\end{comment}
\section{The proof of Theorem~\ref{Tensor}}
We provide the proof of Theorem~\ref{Tensor} next,
which is similar to the proof of Theorem \ref{1dil} but could be
read independently.
Since the detailed proof of Theorem~\ref{Tensor} is notationally cumbersome, we first present a proof in the case where $m=4$ and $n=3$, i.e., the case of $4$ variables and $3$ coordinates. This case captures all the ideas of the general case. Then we discuss the
general case at the end.
Consider the following matrix of the coordinates of all variables:
$$
\betagin{bmatrix}
\xi_{11} & \xi_{12} & \xi_{13} \\
\xi_{21} & \xi_{22} & \xi_{23} \\
\xi_{31} & \xi_{32} & \xi_{33} \\
\xi_{41} & \xi_{42} & \xi_{43}
\end{bmatrix}
=
\betagin{bmatrix}
\xi_{1 } \\
\xi_{2 } \\
\xi_{3 } \\
\xi_{4 }
\end{bmatrix}\, .
$$
Along each column we encounter two cases: the case where the largest coordinate is larger than all the other ones
(case I) and the other case where the largest coordinate is comparable to the second largest (case II).
Such a splitting along all columns produces 8 cases. We only study a representative of these 8 cases, and
in each one of those we make an
arbitrary assumption about the largest variable. The case below illustrates the general one. Assume that:
\betagin{itemize}
\item along column 1: case I (largest in modulus variable is $\xi_{41} $);
\item along column 2: case II (largest in modulus variable is $\xi_{42} $ and second largest is $\xi_{12}$);
\item along column 3: case I (largest in modulus variable is $\xi_{23} $).
\end{itemize}
We denote the symbol associated with this case by
$$
\tau=\sigma^{41,(42,12), 23}_{I,II,I}.
$$
This symbol is obtained by multiplying $\sigma$ by a function of the form
$$
\Phi \Big( \frac{|\xi_{11}| }{|\xi_{41}| } , \frac{|\xi_{21}| }{|\xi_{41}| }, \frac{|\xi_{31}| }{|\xi_{41}| } \Big)
\Phi \Big( \frac{|\xi_{12}| }{|\xi_{42}| } , \frac{|\xi_{22}| }{|\xi_{42}| }, \frac{|\xi_{32}| }{|\xi_{42}| } \Big)
\Psi \Big( \frac{|\xi_{12}| }{|\xi_{42}| } \Big)
\Phi \Big( \frac{|\xi_{13}| }{|\xi_{23}| } , \frac{|\xi_{33}| }{|\xi_{23}| }, \frac{|\xi_{43}| }{|\xi_{23}| } \Big)
$$
where $\Phi(u_1,u_2,u_3)$ is supported in
$\big[0,\frac{11}{200}]\times[0, \frac{11}{200}]\times[0, \frac{1}{20} \big]$ while
$\Psi(u )$ is supported in $\big[\frac{1}{40}, 2\big]$; see the proof of Theorem \ref{1dil} or
\cite{GrafakosMFA} (pages 570-571 or Exercise 7.5.4).
Fix a Schwartz function $\theta$ whose Fourier transform is supported in $[\fracrac{1}{2} , 2]\cup[-2,-\frac12 ]$ and satisfies
$\sum_{j\in \mathbf Z} \widehat \theta (2^{-j}v)=1$ for $v \in \mathbb R\backslash \{0\}. $
Associated with $\theta$ we define the Littlewood--Paley operator
$\Deltalta_j^{(i)}(f)= f*_i \theta_{2^{-j}},$ where $\theta_t(u)=
t^{-n}\theta(t^{-1}u) $ for $t>0$ and $*_i$ denotes the convolution in the $i$th variable.
In a Littlewood-Paley operator $\Delta_j^{(k)}$ the upper letter inside the parenthesis indicates the coordinate on which it acts, so $1\le k\le 3$.
We write
$$
T_{\tau}(f_1,f_2,f_3,f_4)= \sum_{j_1} \sum_{j_2} \sum_{j_3} T_{\tau} \big( f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big)
$$
and we have
\betagin{align*}
&\quad\quaduad T_\tau \big( f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big)(x) =\\
& \int_{\mathbb R^{12}}
\tau(\vec \xi\, )\widehat{f_1}(\xi_1) \widehat{\theta}(2^{-j_3} \xi_{23}) \widehat{f_2}(\xi_2)\widehat{f_3}(\xi_3) \widehat{\theta}(2^{-j_2} \xi_{42}) \widehat{\theta}(2^{-j_1} \xi_{41}) \widehat{f_4}(\xi_4)
e^{2\partiali ix\cdot(\xi_1+\xi_2+\xi_3+\xi_4)} d\vec{\xi} .
\end{align*}
Since $\xi_{41}$ is the largest variable among $\xi_{11},\xi _{21},\xi_{31}, \xi_{41}$, we have that
$$
|\xi_{41}| \le |\xi_{11}|+|\xi_{21}|+|\xi_{31}|+| \xi_{41}| \le \frac{232}{200} |\xi_{41}|
$$
and since $\xi_{42}$ is the largest variable among $\xi_{12},\xi _{22},\xi_{32}, \xi_{42}$, we have that
$$
|\xi_{42}| \le |\xi_{12}|+|\xi_{22}|+|\xi_{32}|+| \xi_{42}| \le \frac{232}{200} |\xi_{42}|.
$$
Likewise
$$
|\xi_{23}| \le |\xi_{13}|+|\xi_{23}|+|\xi_{33}|+| \xi_{43}| \le \frac{232}{200} |\xi_{23}| \, .
$$
We may therefore insert in the preceding integral the function
$$
\widehat{\Omega}\big(D_{-j_1,-j_2,-j_3} (\xi_1,\xi_2,\xi_3,\xi_4)\big) =
\widehat{\Theta }(2^{-j_1}(\xi_{11}+\xi_{21}+\xi_{31}+\xi_{41}))
\widehat{\Theta }(2^{-j_3}(\xi_{13}+\xi_{23}+\xi_{33}+\xi_{43})),
$$
where $\widehat{\Theta} (u) = \widehat{\theta} (u/2) +\widehat{\theta} (u) +\widehat{\theta} (2u) $; notice that $\widehat{\Theta}$
equals $1$ on the support of $\widehat{\theta}$.
We denote by $\widetilde {\Delta}_j$ the Littlewood-Paley operators associated to $\Theta$.
For the same reason we may also insert the function
$$
\betagin{aligned}
&\widehat{\Psi^*}\big( D_{-j_1,-j_2,-j_3}(\xi_1,\xi_2,\xi_3),\xi_4\big)
\\
&= \widehat{\Psi_1^*}(2^{-j_1}(\xi_{11},\xi_{21},\xi_{31},\xi_{41}))\widehat{\Psi_2^*}(2^{-j_2}(\xi_{12},\xi_{22},\xi_{32},\xi_{42}))\widehat{\Psi_3^*}(2^{-j_3}(\xi_{13},\xi_{23},\xi_{33},\xi_{43}))
\end{aligned}
$$
where
$$
\widehat{\Psi_\ell^*}(u_1,u_2,u_3,u_4)= \sum_{|k|\le 1}\widehat{\Psi_\ell}(2^{ -k} (u_1,u_2,u_3,u_4) )\, ,
$$
and $\Psi_\ell$ is as in the hypotheses of the theorem.
Let
$$
D_{j_1,j_2,j_3} (\xi_1,\xi_2,\xi_3,\xi_4)=
\left(
\betagin{bmatrix}
\xi_{11} & \xi_{12} & \xi_{13} &\xi_{14} \\
\xi_{21} & \xi_{22} & \xi_{23} &\xi_{24} \\
\xi_{31} & \xi_{32} & \xi_{33} &\xi_{34} \\
\xi_{41} & \xi_{42} & \xi_{43}&\xi_{44}
\end{bmatrix}
\betagin{bmatrix}
2^{j_1} \\
2^{j_2} \\
2^{j_3}\\
1
\end{bmatrix}
\right)
$$
and
$$
\tau^{j_1,j_2,j_3} (\xi_1,\xi_2,\xi_3,\xi_4)
= \tau \Big( D_{j_1,j_2,j_3} (\xi_1,\xi_2,\xi_3,\xi_4)\Big).
$$
Additionally, in case II there is the second largest variable which is comparable to the largest one.
Therefore we can take a Schwartz function $\eta$ whose Fourier transform
is supported in $[\fracrac{1}{256}, 8] \cup [-8,-\frac1{256}]$ and identical to $1$ on $[\fracrac{1}{128}, 4] \cup [-4,-\frac1{128}]$ and insert the factor $\widehat{\eta}(2^{-j_2}\xi_{12})$ into the above integral without changing the outcome. Let us denote the
Littlewood-Paley operator associated with $\eta$ by $\overline{\Delta}_j$.
We may therefore rewrite
\betagin{align*}
T_{\tau} \big( f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big)
\,\,= & \,\,
T_{\tau} \big(\overline{\Delta}_{j_2}^{(2)} f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big) \\
=&\,\, \widetilde\Delta_{j_1}^{(1)}\widetilde\Delta_{j_3}^{(3)}
T_{\tau} \big(\overline{\Delta}_{j_2}^{(2)} f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big)\, .
\end{align*}
Manipulations with the Fourier transform give that the above can be expressed as
\betagin{align*}
& \int_{\mathbb R^{12} } 2^{4(j_1+j_2+j_3) }\Big(\tau^{j_1,j_2,j_3} \widehat{\Psi^*} \,\,
\widehat{\Omega} \Big)\spcheck \!
\big( D_{j_1,j_2,j_3} (x-y_1,x-y_2,x-y_3,x-y_4) \big) \\
&\quad\quaduad (\overline{\Delta}_{j_2}^{(2)} f_1 )(y_1) (\Delta_{j_3}^{(3)} f_2 )(y_2) f_3(y_3)( \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4)(y_4)
d y_1 dy_2 dy_3 dy_4\, .
\end{align*}
If $r=1$, set $\rho=1$. If $r>1$ pick $\rho$
such that $1< \rho <2 $ and that
$$\maxL^{\infty}mits_{1\le i \le m} \maxL^{\infty}mits_{1\le \ell \le n}\frac{1}{\gamma_{i\ell}} <\rho < \min(p_1,\dots , p_m,r).$$
Setting $\omega_{\beta}( y) = (1+4\partiali^2 | y |^2 )^{\frac{\beta}{2}}$ for $y\in \mathbb R $, we write
{\alphalowdisplaybreaks
\betagin{align*}
& \Big| T_{\tau} \big( f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big) (x_1,x_2,x_3) \Big| \notag \\
\leq&
\intL^{\infty}mits_{\mathbb R^{12}}\!
2^{\frac{4( j_1+j_2+j_3)}{\rho'} } \Big\{ \partialrod_{i=1}^4 \partialrod_{\ell=1}^3 \omega_{\gamma_{i\ell}} (2^{j_\ell}(x_\ell -y_{i\ell})) \Big\}
(\tau^{j_1,j_2,j_3}\widehat{\Psi^*}\widehat{\Omega} )\spcheck ({ D_{j_1,j_2,j_3} (x-y_1,x-y_2,x-y_3,x-y_4)}) \\
&\quad\quaduad \frac{2^{\frac{ j_1+j_2+j_3}{\rho } } (\overline{\Delta}_{j_2}^{(2)} f_1 ) (y_1)}
{ \partialrod_{\ell=1}^3 \omega_{\gamma_{1\ell}} (2^{j_\ell}(x_\ell -y_{1\ell})) }
\quad \frac{ 2^{\frac{ j_1+j_2+j_3}{\rho } }(\Delta_{j_3}^{(3)} f_2 )(y_2) }
{ \partialrod_{\ell=1}^3 \omega_{\gamma_{2\ell}} (2^{j_\ell}(x_\ell -y_{2\ell})) } \\
&\quad\quaduad \frac{2^{\frac{ j_1+j_2+j_3}{\rho } } f_3(y_3) }
{ \partialrod_{\ell=1}^3 \omega_{\gamma_{3\ell}} (2^{j_\ell}(x_\ell -y_{3\ell})) }
\frac{2^{\frac{ j_1+j_2+j_3}{\rho } } ( \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4)(y_4)}
{ \partialrod_{\ell=1}^3 \omega_{\gamma_{4\ell}} (2^{j_\ell}(x_\ell -y_{4\ell})) }
dy_1dy_2dy_3dy_4 .
\end{align*}
}
We now apply H\"older's inequality with exponents $\rho$ and $\rho'$ to obtain the estimate
\betagin{align}\betagin{split}\labelel{Es78}
& \Big| T_{\tau} \big( \overline\Delta_{j_2}^{(2)} f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big) (x_1,x_2,x_3) \Big| \\
&
\quad\quaduad \le C A \mathcal M(|\overline\Delta_{j_2}^{(2)} f_1|^\rho)^{\frac{1}{\rho}}
\mathcal M(|\Delta_{j_3}^{(3)} f_2|^\rho)^{\frac{1}{\rho}}\mathcal M(|f_3|^\rho)^{\frac{1}{\rho}}
\mathcal M(|\Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4|^\rho)^{\frac{1}{\rho}},
\end{split} \end{align}
where we used that $\rho\gamma_{i\ell}>1$ for all $i,\ell$ and also that
\[
\Big\| \partialrod_{i=1}^4\partialrod_{\ell=1}^3 (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}} (\tau^{j_1,j_2,j_3}\, \widehat{\Psi^*}\widehat{\Omega}) \Big\|_{L^{\rho } }
\lesssim
\Big\| \partialrod_{i=1}^4\partialrod_{\ell=1}^3 (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}} (\tau^{j_1,j_2,j_3} \, \widehat{\Psi^*} ) \Big\|_{L^{\rho } }
\]
\[
\lesssim \Big\| \partialrod_{i=1}^4\partialrod_{\ell=1}^3 (I-\partial_{\xi_{i\ell} }^2)^{\frac{\gamma_{i\ell}}{2}} (\sigmagma \circ D_{j_1,j_2,j_3}) \, \widehat{\Psi^*} \Big\|_{L^{r } }
\lesssim A
\]
which is a consequence of Lemma~\ref{XL1} and of the fact that $\Psi^*$ is a finite sum of $\Psi_\ell$'s.
We now use \eqref{Es78} to estimate our operator. We write
$$
T_{\tau}(f_1,f_2,f_3,f_4)=
\sum_{j_1} \sum_{j_2} \sum_{j_3} \widetilde\Delta_{j_1}^{(1)}\widetilde\Delta_{j_3}^{(3)}
{ T_{\tau}} \big(\overline\Delta_{j_2}^{(2)} f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big)\, .
$$
Let $\mathcal M$ denote the strong maximal function.
For each $j_1$ and $j_3$ we have the pointwise estimate
\betagin{align*}
&\big| \widetilde\Delta_{j_1}^{(1)}\widetilde\Delta_{j_3}^{(3)} \sum_{j_2}
T_{\tau} \big(\overline\Delta_{j_2}^{(2)} f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big) \big|
\\
&\quad\quaduad \le C A \sum_{j_2} \mathcal M(|\overline\Delta_{j_2}^{(2)} f_1|^\rho)^{\frac{1}{\rho}}
\mathcal M(|\Delta_{j_3}^{(3)} f_2|^\rho)^{\frac{1}{\rho}}\mathcal M(|f_3|^\rho)^{\frac{1}{\rho}}
\mathcal M(|\Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4|^\rho)^{\frac{1}{\rho}} \\
&\quad\quaduad \le C A \Big( \sum_{j_2}
\mathcal M (|\overline\Delta_{j_2}^{(2)} f_1|^\rho)^{\frac{2}{\rho}}\Big)^{\frac12}
\mathcal M(|\Delta_{j_3}^{(3)} f_2|^\rho)^{\frac{1}{\rho}} \mathcal M(|f_3|^\rho)^{\frac{1}{\rho}}
\Big(\sum_{j_2} \mathcal M (|\Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4|^\rho)^{\frac{2}{\rho}}\Big)^{\frac12}
\end{align*}
We now apply Lemma~\ref{GLP} (hypothesis \eqref{lp0} is easy to check), { more precisely by Remark \ref{11071}}, to write
$$
\big\| T_{\tau}(f_1,f_2,f_3,f_4)\big\|_{L^p} \lesssim
\Big\| \Big(\sum_{j_1} \sum_{j_3} \Big|\widetilde\Delta_{j_1}^{(1)}\widetilde\Delta_{j_3}^{(3)}
\sum_{j_2} { T_{\tau}}\big(\overline\Delta_{j_2}^{(2)} f_1,\Delta_{j_3}^{(3)} f_2,f_3, \Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4\big) \Big|^2 \Big)^{\frac12} \Big\|_{L^p}
$$
and using the preceding estimate we control this expression by
$$
A \bigg\|
\Big( \sum_{j_2}
\mathcal M(|\overline\Delta_{j_2}^{(2)} f_1|^\rho)^{\frac{2}{\rho}}\Big)^{\frac12}
\Big(\sum_{j_3} \mathcal M(|\Delta_{j_3}^{(3)} f_2|^\rho)^{\frac{2}{\rho}} \Big)^{\frac12}
\mathcal M(|f_3|^\rho)^{\frac{1}{\rho}}
\Big(\sum_{j_2}\sum_{j_1}\mathcal M(|\Delta_{j_2}^{(2)}\Delta_{j_1}^{(1)}f_4|^\rho)^{\frac{2}{\rho}}\Big)^{\frac12}
\bigg\|_{L^p}.
$$
The required conclusion follows by applying H\"older's inequality, the Fefferman-Stein inequality
\cite{FS}, { and Lemma~\ref{GLP}} using the facts that $1\le\rho<2$ and $\rho<p_i$ for all $i$.
We show now how to modify { the above} proof to obtain the general case. To do so, we introduce
some notation. We consider the set $\{1,2,\dots , n\}$ that indexes the columns of the $m\times n$ matrix
$(\xi_{kl})_{ \{1\le k \le m, 1\le l \le n\} }$. We split the set $\{1,2,\dots , n\}$ into two pieces I and II, by
placing $l\in I$ if the $l$th column follows in the first case (where there the largest variable dominates all the
other ones) and placing
$l\in II$ if the $l$th column follows in the second case (where there the largest variable and the second largest are comparable). To make the notation a bit simpler, without loss of generality we suppose that
$I=\{1, \dots , q\}$ and $II=\{q+1,\dots , n\}$ for some $q$. Notice that one of these sets could be empty.
Recall the notation for the Littlewood-Paley operators $\Delta_j^{(l)}$ as in the case $m=4$, $n=3$. For the purposes
of this theorem we introduce a slightly more refined notation using two upper indices in $\Delta_j^{(k,l)}$. The first
index shows the function $f_k$ on which $\Delta_j^{(k,l)}$ acts and the second one the coordinate $\xi_{kl}$
of the variable $\xi_k$ on which $\Delta_j^{(k,l)}$ acts.
Define a map
$$
u: \{1,2,\dots , n\}\to \{1,2,\dots , m\}
$$
such that for each $l$, $u(l)$ denotes the index such that
$\xi_{u(l)l}$ is largest among $ \xi_{kl}$. Also define a map
$$
\bar u : \{q+1,\dots , n\}\to \{1,2,\dots , m\}
$$
such that
$\xi_{\bar u(l)l}$ is second largest among $ \xi_{kl}$. We always have $\bar u(l) \neq u(l)$ for all $l$ in
$\{q+1,\dots , n\}$. We also define
$$
{ \Delta}_{j }^{(u(r),r )} \vec f = { \Delta}_{j }^{(u(r),r )} (f_1,\dots , f_m) =
(f_1,\dots , { \Delta}_{j }^{(r)} f_{u(r)} ,\dots , f_m)
$$
and we extend this definition to the case where ${ \Delta}_{j_{i_1} }^{(u(i_1),i_1 )} \cdots { \Delta}_{j_{i_r} }^{(u(i_r),i_r )}$
acts on $(f_1,\dots , f_m)$. Additionally, we use the definitions of $\widetilde \Delta_j$ and $\overline \Delta_j$
as introduced in the special case $m=4$, $n=3$.
Let $\tau$ be the multilinear multiplier associated with a given fixed mapping $u$.
We write
\betagin{eqnarray*}
& & T_\tau(f_1,\dots , f_m) \\
& = &
\sum_{ j_1,\dots , j_n\in \mathbb Z } T_\tau \big[{ \Delta}_{j_1 }^{(u(1),1 )}\cdots { \Delta}_{j_n }^{(u(n),n )} (f_1,\dots , f_m) \big]\\
& = & \sum_{\substack{ j_1,\dots , j_q \in \mathbb Z }}
\widetilde{ \Delta}_{j_{1}}^{(u(1),1)} \cdots \widetilde{ \Delta}_{j_{q}}^{(u(q),q)}
\sum_{\substack{ j_{q+1},\dots , j_n\in \mathbb Z}} T_\tau \Big[
\partialrod_{\kappa=1}^q { \Delta}_{j_\kappa}^{(u(\kappa),\kappa)}
\partialrod_{\lambda=q+1}^n { \Delta}_{j_{\lambda}}^{(u(\lambda),\lambda )}
\overline{ \Delta}_{j_{ \lambda}}^{ ( \overline u(\lambda),\lambda) }
\vec f \, \Big] \, .
\end{eqnarray*}
The estimates in the case $m=4$ and $n=3$ show that
the term in the interior sum satisfies
\betagin{align*}
& \Big| \widetilde{ \Delta}_{j_{1}}^{(u(1),1)} \cdots \widetilde{ \Delta}_{j_{q}}^{(u(q),q)}
\sum_{ j_{q+1},\dots , j_n\in \mathbb Z} T_\tau \Big[
\partialrod_{\kappa=1}^q { \Delta}_{j_\kappa}^{(u(\kappa),\kappa)}
\partialrod_{\lambda=q+1}^n { \Delta}_{j_{\lambda}}^{(u(\lambda),\lambda )}
\overline{ \Delta}_{j_{ \lambda}}^{ ( \overline u(\lambda),\lambda) }
(f_1,\dots , f_m) \Big] \Big| \\
& \quad\quaduad
\lesssim A \sum_{\substack{ j_{q+1},\dots , j_n\in \mathbb Z}}
\partialrod_{i=1 }^m \mathcal M \bigg( \bigg|
\partialrod_{\substack{ 1\le \kappa \le q \\ \kappa \in u^{-1}[i]} } { \Delta}_{j_\kappa}^{(i,\kappa)}
\partialrod_{\substack{q+1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } } { \Delta}_{j_{\lambda}}^{(i,\lambda )}
\partialrod_{\substack{q+1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^\rho\bigg)^{\frac1 \rho} ,
\end{align*}
where $u^{-1}[i]=\{k \in \{1,\dots , n\}:\,\, u(k)=i\}$ and
with the understanding that if any of the index sets is empty, then the corresponding Littlewood-Paley
operators do not appear.
Applying the Cauchy-Schwarz inequality $m-q$ times successively for the indices $j_{q+1}, j_{q+1}, \dots , j_m$
we estimate the last displayed expression by
\betagin{align}
\labelel{eq.5A10}
A\partialrod_{i=1 }^m \bigg[
\sum_{\substack{ j_\lambda\in \mathbb Z\\ \lambda \in u^{-1}[i] \\ q+1\le \lambda \le n }}
\sum_{\substack{ j_\mu \in\mathbb Z\\ \mu \in \overline u^{-1}[i] \\ q+1\le \mu \le n }}
\mathcal M \bigg( \bigg|
\partialrod_{\substack{ 1\le \kappa \le q \\ \kappa \in u^{-1}[i]} } { \Delta}_{j_\kappa}^{(i,\kappa)}
\partialrod_{\substack{q+1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } } { \Delta}_{j_{\lambda}}^{(i,\lambda )}
\partialrod_{\substack{q+1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^\rho\bigg)^{\!\frac2 \rho} \bigg]^{\frac12}.
\end{align}
When $I\ne\emptyset$, we use Lemma~\ref{GLP} and \eqref{eq.5A10} to obtain
\betagin{align*}
& \big\|T_\tau(f_1,\dots , f_m) \big\|_{L^p} \\
& = \bigg\|
\sum_{\substack{ j_1,\dots , j_q \in \mathbb Z }}
\widetilde{ \Delta}_{j_{1}}^{(u(1),1)} \cdots \widetilde{ \Delta}_{j_{q}}^{(u(q),q)}
\sum_{\substack{ j_{q+1},\dots , j_n\in \mathbb Z}} T_\tau \Big[
\partialrod_{\kappa=1}^q { \Delta}_{j_\kappa}^{(u(\kappa),\kappa)}
\partialrod_{\lambda=q+1}^n { \Delta}_{j_{\lambda}}^{(u(\lambda),\lambda )}
\overline{ \Delta}_{j_{ \lambda}}^{ ( \overline u(\lambda),\lambda) }
\vec f\, \Big]
\bigg\|_{L^p} \\
& \lesssim \Bigg\|\bigg[ \sum_{\substack{ j_1,\dots , j_q \in \mathbb Z }} \Big|
\widetilde{ \Delta}_{j_{1}}^{(u(1),1)} \cdots \widetilde{ \Delta}_{j_{q}}^{(u(q),q)}
\!\!\!\!\! \sum_{\substack{ j_{q+1},\dots , j_n\in \mathbb Z}} T_\tau \Big[
\partialrod_{\kappa=1}^q { \Delta}_{j_\kappa}^{(u(\kappa),\kappa)}
\partialrod_{\lambda=q+1}^n { \Delta}_{j_{\lambda}}^{(u(\lambda),\lambda )}
\overline{ \Delta}_{j_{ \lambda}}^{ ( \overline u(\lambda),\lambda) }
\vec f\, \Big] \Big|^2 \bigg]^{\frac12} \Bigg\|_{L^p} \\
& \lesssim A\Bigg\| \bigg[\sum_{\substack{ j_1,\dots , j_q \in \mathbb Z }} \partialrod_{i=1 }^m \bigg\{ \!\!\!\!
\sum_{\substack{ j_\lambda\in \mathbb Z\\ \lambda \in u^{-1}[i] \\ q+1\le \lambda \le n }}
\sum_{\substack{ j_\mu \in\mathbb Z\\ \mu \in \overline u^{-1}[i] \\ q+1\le \mu \le n }}
\!\!\!\! \mathcal M \bigg( \bigg| \!\!\!
\partialrod_{\substack{ 1\le \kappa \le q \\ \kappa \in u^{-1}[i]} } { \Delta}_{j_\kappa}^{(i,\kappa)}
\!\!\! \partialrod_{\substack{q+1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } } { \Delta}_{j_{\lambda}}^{(i,\lambda )}
\!\!\! \partialrod_{\substack{q+1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^\rho\bigg)^{\! \frac2 \rho} \bigg\}
\bigg]^{\frac12} \Bigg\|_{L^p} \\
& \lesssim A\Bigg\| \bigg(\partialrod_{i=1 }^m
\sum_{\substack{ j_\kappa \in \mathbb Z\\ \kappa\in u^{-1}[i] \\ 1\le \kappa \le q}}
\sum_{\substack{ j_\lambda\in \mathbb Z\\ \lambda \in u^{-1}[i] \\ q+1\le \lambda \le n }}
\sum_{\substack{ j_\mu \in\mathbb Z\\ \mu \in \overline u^{-1}[i] \\ q+1\le \mu \le n }}
\!\!\! \mathcal M \bigg( \bigg|
\partialrod_{\substack{ 1\le \kappa \le q \\ \kappa \in u^{-1}[i]} } { \Delta}_{j_\kappa}^{(i,\kappa)}
\!\!\! \partialrod_{\substack{q+1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } } { \Delta}_{j_{\lambda}}^{(i,\lambda )}
\!\!\! \partialrod_{\substack{q+1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^\rho\bigg)^{\! \frac2 \rho}
\bigg)^{\frac12} \Bigg\|_{L^p} .
\end{align*}
Otherwise, when $I=\emptyset$, from \eqref{eq.5A10} we can see that $T_\tau(f_1,\dots , f_m)$ is controlled by
\betagin{align}
\labelel{eq.5A11}
A\partialrod_{i=1 }^m \bigg[
\sum_{\substack{ j_\lambda\in \mathbb Z\\ \lambda \in u^{-1}[i] \\ 1\le \lambda \le n }}
\sum_{\substack{ j_\mu \in\mathbb Z\\ \mu \in \overline u^{-1}[i] \\ 1\le \mu \le n }}
\mathcal M \bigg( \bigg|
\partialrod_{\substack{1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } }
{ \Delta}_{j_{\lambda}}^{(i,\lambda )}
\partialrod_{\substack{1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^\rho\bigg)^{\!\frac2 \rho} \bigg]^{\frac12}.
\end{align}
At this point we apply H\"older's inequality and the Fefferman-Stein inequality
\cite{FS} using the facts that $1<\rho<2$ and $\rho<p_i$ for all $i$. Then we
{
control
$\big\|T_\tau(f_1,\dots , f_m) \big\|_{L^p}$
}
by a constant multiple of
$$
A \partialrod_{i=1}^m
\Bigg\| \bigg( \sum_{\substack{j_\kappa \in \mathbb Z\\ \kappa\in u^{-1}[i] \\ 1\le \kappa \le q}}
\sum_{\substack{ j_\lambda\in \mathbb Z\\ \lambda \in u^{-1}[i] \\ q+1\le \lambda \le n }}
\sum_{\substack{ j_\mu \in\mathbb Z\\ \mu \in \overline u^{-1}[i] \\ q+1\le \mu \le n }}
\bigg|
\partialrod_{\substack{ 1\le \kappa \le q \\ \kappa \in u^{-1}[i]} } { \Delta}_{j_\kappa}^{(i,\kappa)}
\partialrod_{\substack{q+1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } } { \Delta}_{j_{\lambda}}^{(i,\lambda )}
\partialrod_{\substack{q+1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^2
\bigg)^{\frac12} \Bigg\|_{L^{p_i}}
$$
{
or
$$
A \partialrod_{i=1}^m
\Bigg\| \bigg(
\sum_{\substack{ j_\lambda\in \mathbb Z\\ \lambda \in u^{-1}[i] \\ 1\le \lambda \le n }}
\sum_{\substack{ j_\mu \in\mathbb Z\\ \mu \in \overline u^{-1}[i] \\ 1\le \mu \le n }}
\bigg|
\partialrod_{\substack{1 \le \lambda \le n \\ \lambda\in u^{-1}[i] } } { \Delta}_{j_{\lambda}}^{(i,\lambda )}
\partialrod_{\substack{1 \le \mu \le n \\ \mu\in \overline u^{-1}[i] }}
\overline{ \Delta}_{j_{ \mu}}^{ ( i,\mu) }
f_i \bigg|^2
\bigg)^{\frac12} \Bigg\|_{L^{p_i}}
$$
}
and by the Littlewood-Paley theorem the last expression is bounded by $A$ times the product of the $L^{p_i}$ norms
of the $f_i$.
{
\betagin{remark}
We see from the proof that we do not use the property
that $\xi_{kl}\in\mathbb R$, so the same argument generalizes our result
to the case when each $f_k$ is defined on $\mathbb R^{d}$
with $\xi_{kl}\in\mathbb R^d$.
This covers
\cite[Theorem 1.10]{CL}, as we claimed in the introduction.
\end{remark}
}
\section{Applications: Calder\'on-Coifman-Journ\'e commutators}
\setcounter{equation}{0}
\subsection{Calder\'on commutator}
In 1965 Calder\'on \cite{AC} introduced the (first-order) commutator
\betagin{eqnarray}\labelel{e6.1}
{ \mathcal C}_1 (f;a)(x)= \textup{p.v.}
\int_{\mathbb R} \fracrac{A(x)-A(y)}{(x-y)^2} f(y ) dy,
\end{eqnarray}
where $a$ is the derivative of a
Lipschitz function $A$ and $f$ is a test function on the real line.
It is known that ${ \mathcal C}_1$ is a bounded operator in $L^p(\Bbb R), 1<p<\infty$, if $A$ is a Lipchitz function
on $\Bbb R$ and
$$
\|{ \mathcal C}_1 (f;a)\|_{L^p({\Bbb R})}\leq C_p \|a\|_{L^\infty({\Bbb R})}\|f\|_{L^p({\Bbb R})}, \ \
1<p<\infty.
$$
See Calder\'on \cite{AC, {C78}} and Coifman-Meyer \cite{CM1} for its history.
\iffalse
In this section we apply Theorem~\ref{1dil} to deduce nontrivial
bounds for the commutator of A. Calder\'on \cite{AC}.
This operator along with its higher counterparts
first appeared in the study of the
Cauchy integral along Lipscitz curves and in fact these led to
the first proof of the $L^2$-boundedness of the latter.
The first order commutator is defined as
\betagin{eqnarray*}
{ \mathcal C}_1(f, a)(x)= \textup{p.v.} \int_{\mathbb R} {A(x)-A(y)\over (x-y)^2} f(y)dy,
\ \ \ {\rm where}\ \ A'=a.
\end{eqnarray*}
\fraci
Viewed as a bilinear operator acting on the pair $(f,a)$, then
the operator ${\mathcal C}_1$ can be written as a bilinear multiplier operator
\iffalse
\betagin{eqnarray}\labelel{e6.1}
{ \mathcal C}_1(f;a)(x)=-\textup{P.V.} \int_{\mathbb R } { f(y )\over (x-y)^2 }
\int_{x }^{y } a(u )du \,dy \, , \quad\quaduad x\in \mathbb R .
\end{eqnarray}
Using the Fourier transform $\widehat{g}(\xi) = \int_{\mathbb R } g(x) e^{-2\partiali i x\xi}dx$,
we may also write $ { \mathcal C}_1(f;a) $ as
a bilinear multiplier operator concerning $f$ and $a:$\fraci
\betagin{eqnarray}\labelel{e6.2}
\quad\quaduad { \mathcal C}_1(f;a)(x)=
-i\partiali\int_{\mathbb R}\int_{\mathbb R} \widehat{f}(\xi)\, \widehat{a}(\eta)\,
\left( \textup{sgn\,} (\eta) \Phi\big( \xi/\eta\big)\right) \, e^{2\partiali i x(\xi+\eta)} \,
d\xi d\eta\, ,
\end{eqnarray}
where $\Phi$ is the following Lipschitz function on the real line:
\betagin{eqnarray}\labelel{e6.2}
\Phi (s) = \left\{
\betagin{array}{lll}
-1, & s\le -1;\\ [6pt]
1+2s, &-1<s \le 0; \\ [6pt]
1, &s>0.
\end{array}
\right.
\end{eqnarray}
\iffalse
we can reduce the boundedness of ${ \mathscr C}_1$ to that of $T_\sigmagma$ in \eqref{Ts} with
\betagin{eqnarray}\labelel{e6.3}
\sigmagma(\xi, \eta)= \textup{sgn\,} (\eta) \Phi\big( \xi/\eta\big), \ \ \ \ (\xi, \eta)\in {\mathbb R}\times {\mathbb R}.
\end{eqnarray}
\fraci
The operator ${\mathcal C}_1$ is too singular to fall under the scope of
multilinear Calder\'on-Zygmund theory \cite{GT2}. However it was shown to be bounded
from $L^{p_1}(\mathbb RR) \times L^{p_2}(\mathbb RR)$ to $L^p(\mathbb RR)$ when
$1<p_1, p_2<\infty$ and $({1/p_1} + {1/p_2})^{-1} =p>1/2$;
see C. Calder\'on \cite{CC}. See also Coifman-Meyer \cite{CM1} and
Duong-Grafakos-Yan \cite{DGY}. The boundedness of ${ \mathcal C}_1$ on $L^p$ for $p\ge 1$
was also studied by Muscalu \cite{Mu1} via time-frequency analysis.
In this work we will apply Theorem~\ref{1dil} to
obtain a direct proof of the boundedness of ${ \mathcal C}_1 $
from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R)$ to
$L^p(\mathbb R)$ in the full range of $p>1/2$. Our proof is
based on exploiting the (limited) smoothness of the function $\Phi$, measured in terms of a Sobolev space norm.
{ A partial result using a similar idea in this direction with the restriction $p>2/3$ has been obtained by
\cite{MiTo14}.}
{ For $r\ge 1$ and $\gamma>0$, we recall the Sobolev space $L^r_\gamma(\mathbb R^n)$, $\gamma>0$ of all functions $g$ with
$\|(I-\Delta)^{\gamma/2} g\|_{L^p}<\infty $.}
{ For $\vec\gamma=(\gamma_1,\dots,\gamma_n)$,
we denote by $L^r_{\vec\gamma}(\mathbb R^n)$
the class of distributions $f$ such that
$$
\bigg\|{ \partialrod_{ 1\le \ell \le n } (I-\partial_{\ell}^2)^{\frac{\gamma_{\ell}}{2}} f }\bigg\|_{L^r (\mathbb{R}^{n})}<\infty.
$$
It is easy to verify using { multiplier theorems}
that $L^r_{\gamma}(\mathbb R^n)\subset L^r_{\vec\gamma}(\mathbb R^n)$,
where $\gamma=|\vec\gamma|=\gamma_1+\cdots+\gamma_n$.
The spaces $L^r_{\vec\gamma}(\mathbb R^n)$ are sometimes referred to as
Sobolev spaces with dominating mixed smoothness in the literature,
see \cite{STr} for more details and references.}
To begin,
we need the following characterizations of Sobolev norms, given by Stein \cite{St2},
\cite[Lemma 3, p. 136]{St1}.
\betagin{lemma}[Stein]\labelel{le6.2}
(i)
Let $0< \alphapha<1$ and ${2n/(n+ 2\alphapha)}<p<\infty$. Then $f\in L_{\alphapha}^p(\mathbb R^n)$ if and only if
$\|f\|_{L_{\alphapha}^p(\mathbb R^n)} \sigmameq \|f\|_{L^p(\mathbb R^n)} +\|I_{\alphapha}(f)\|_{L^p(\mathbb R^n)}$ where
$$
I_{\alphapha}(f)(x)=\left(\int_{\mathbb R^n} {|f(x)-f(y)|^2\over |x-y|^{n+2\alphapha} }dy\right)^{1/2}.
$$
(ii)
Let $1\leq \alphapha<\infty$ and $1<p<\infty$. Then $f\in L_{\alphapha}^p(\mathbb R^n)$ if and only if
$f\in L_{\alphapha-1}^p(\mathbb R^n)$ and for $1\le j\le n$,
${\partialartial f \over \partialartial x_j} \in L_{\alphapha-1}^p(\mathbb R^n).$
Furthermore, we have
$$\|f\|_{L_{\alphapha}^p(\mathbb R^n)} \sigmameq \|f\|_{L^p_{\alphapha-1}(\mathbb R^n)} +
\sum_{j=1}^n\left\|{\partialartial f \over \partialartial x_j}\right\|_{L^p_{\alphapha-1}(\mathbb R^n)}.
$$
\end{lemma}
\iffalse
\betagin{lemma}[Stein]\labelel{le6.3}
For $0< \alphapha<1$ and ${2n/(n+ 2\alphapha)}<p<\infty$. Then $f\in L_{\alphapha}^p(\mathbb R^n)$ if and only if
$\|f\|_{L_{\alphapha}^p(\mathbb R^n)} \sigmameq \|f\|_{L^p(\mathbb R^n)} +\|I_{\alphapha}(f)\|_{L^p(\mathbb R^n)}$ where
$$
I_{\alphapha}(f)(x)=\left(\int_{\mathbb R^n} {|f(x)-f(y)|^2\over |x-y|^{n+2\alphapha} }dy\right)^{1/2}.
$$
\end{lemma}
\fraci
Throughout this section fix a nondecreasing smooth function $h$ on $\mathbb R$ such that
\betagin{equation}\labelel{eq.func.h}
h(t)=\betagin{cases}
3, & \mbox{if } t\in [4, +\infty); \\[2pt]
\mbox{smooth}, & \mbox{if } t\in [2, 4); \\[2pt]
t, & \mbox{if } t\in [1/8, 2); \\[2pt]
\mbox{smooth}, & \mbox{if } t\in[1/32, 1/8); \\[2pt]
1/16, & \mbox{otherwise}.
\end{cases}
\end{equation}
\betagin{lemma}\labelel{var}
Let $u$ be a function supported in the rectangle
\betagin{equation}\labelel{SETU}
\{(y_1,y_2):\,\, |y_1|\le 101/100, 1/4\le y_2\le 7/4\}
\end{equation}
in $\mathbb R^2$ such that $\nabla u\in L^{\infty}(\mathbb R^2)$,
and $u(x)\in L^r_{\gamma}(\mathbb R^2)$ with $1<\gammamma<2$, $2/\gamma<r<1/(\gamma-1)$.
Define $U(y_1,y_2) = u(y_1/h(y_2),y_2).$
Then $U\in L^r_{\gamma}(\mathbb R^2)$
and
$$
\|U\|_{L^r_{\gamma}(\mathbb R^2)}\le C\big(\|\nabla u\|_{L^{\infty}(\mathbb R^2)}+\|u\|_{L^r_{\gamma}(\mathbb R^2)}\big).
$$
\end{lemma}
\betagin{proof}
Because of Lemma \ref{le6.2}, it suffices to show
for $\alpha=\gamma-1$ and $2/(1+\alphapha) <r<{1/\alphapha}$ that $U\in L^r_1(\mathbb RR^2)$, $I_{\alpha}(U)\in L^r(\mathbb RR^2)$
and
$I_{\alpha}(\partial_jU)\in L^r(\mathbb RR^2)$ with $j=1,2$.
The first assertion follows trivially by checking
the derivatives directly while the second one is verified in a way similar
to the third one, where we adapt an argument found in
Triebel \cite[Section 4.3]{Tri} with a suitable change of variables.
Next, we show that $I_{\alpha}(\partial_1U)\in L^r(\mathbb RR^2)$.
We will estimate the following expression
\betagin{eqnarray*}\labelel{eeeee}
\|I_{\alpha}(\partial_1U)\|_{L^r(\mathbb RR^2)}^r=\int_{\mathbb R^2} \left(\int_{\mathbb R^2}
{|\partialartial_1U(y)-\partialartial_1U(y')|^2\over |y-y'|^{2+2\alphapha} }\,dy\right)^{r/2} dy'.
\end{eqnarray*}
Denote by $B$ a finite ball centered at $0$ containing the support of $\partial_1U$. Then
it is easy to check that, since $\partial_1U\in L^{\infty}$, $r(1+\alpha)=r\gamma>2$,
$$
\|I_{\alpha}(\partial_1U)\|_{L^r(\mathbb RR^2)}^r\le C\left(\|\nabla u\|_{L^{\infty}}^r+\int_{3B}
\left(\int_{3B} {|\partialartial_1U(y)-\partialartial_1U(y')|^2\over |y-y'|^{2+2\alphapha} }\,dy\right)^{r/2} dy'\right),
$$
where $C$ is a constant depending on $B$.
Denote $x=(x_1, x_2)$, $y=(y_1, y_2)$.
One writes $y=\varphi(x)$ and $x=\partialsi(y)$ in the form
\betagin{eqnarray*}
\left\{
\betagin{array}{ll}
y_1&=\varphi_1(x_1, x_2)=x_1 h(x_2),\\[2pt]
y_2&=\varphi_2(x_1, x_2)=x_2
\end{array}
\right.
\end{eqnarray*}
and
\betagin{eqnarray*}
\left\{
\betagin{array}{ll}
x_1&=\partialsi_1(y_1, y_2)={ y_1 /h(y_2)},\\[2pt]
x_2&=\partialsi_2(y_1, y_2)=y_2,
\end{array}
\right.
\end{eqnarray*}
where $ h $ is a function defined in \eqref{eq.func.h}.
By the change of variables $y=\varphi(x)$ with $|{\rm det} \varphi'(x)|<C<\infty$, direct computations give
\betagin{align*}
{\partialartial_1}U(y)=& {\partialartial \over \partialartial y_1}u(\partialsi(y)) \cdot {1\over h(y_2)}=:
\partialartial_1{ u}(\partialsi(y)) \cdot {1\over h(y_2)},
\\
{\partialartial_2}U(y)=&- {\partialartial \over \partialartial y_1}
{ u}(\partialsi(y)) \cdot {y_1h'(y_2)\over h(y_2)} +{\partialartial \over \partialartial y_2} { u}(\partialsi(y))
=:- {\partialartial_1} { u}(\partialsi(y)) \cdot {y_1h'(y_2)\over h(y_2)} +{\partialartial_2} { u}(\partialsi(y)),
\end{align*}
and the fact that
$|\partialsi(y)-\partialsi(y')|\le \max\{\|\nabla\partialsi_1\|_{\infty},\|\nabla\partialsi_2\|_{\infty}\}|y-y'|,$
we have
\betagin{eqnarray}
& & \hspace{-.7in} \|I_{\alpha}(\partial_1U)\|_{L^r(\mathbb RR^2)}^r \nonumber \\
&\leq &C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+C\int_{\mathbb R^2} \left(\int_{\mathbb R^2}
{|\partialartial_1U(y)-\partialartial_1U(y')|^2\over |\partialsi(y)-\partialsi(y')|^{2+2\alphapha} }
dy\right)^{r/2} dy'\nonumber\\
&\leq &
C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+
C\int_{\mathbb R^2} \bigg[\int_{\mathbb R^2} {\left| {\partialartial_1 { u}(\partialsi(y))\over h(y_2)}
-{\partialartial_1 { u}(\partialsi(y'))\over h(y'_2)} \right|^2\over |\partialsi(y)-\partialsi(y')|^{2+2\alphapha} }
dy\bigg]^{r/2}dy'\nonumber\\
&\leq &
C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+
C\int_{\mathbb R^2} \bigg[\int_{\mathbb R^2} {\left| {\partialartial_1 { u}(x)\over h(x_2)}
-{\partialartial_1 { u}(x')\over h(x'_2)} \right|^2\over |x-x'|^{2+2\alphapha} }
|{\rm det} \varphi'(x)| dx\bigg]^{r/2} |{\rm det} \varphi'(x')| \, dx'\nonumber\\
&\leq &
C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+
C\int_{\mathbb R^2} \bigg[\int_{\mathbb R^2} {\left| {\partialartial_1 { u}(x)\over h(x_2)}
-{\partialartial_1 { u}(x')\over h(x'_2)} \right|^2\over |x-x'|^{2+2\alphapha} }
dx\bigg]^{r/2} \nonumber dx'.
\end{eqnarray}
Now take $\eta(x_1,x_2)\in C_0^{\infty}(\mathbb RR^2)$
assuming value $1$ on the support of $\partial_1u$ so that
the support of $\eta$
is just a bit larger than that of $\partial_1u$, and
$h(x_2)=x_2$
on the support of $\eta$.
Define ${\tilde h}(x_1,x_2)=\eta(x_1,x_2)/h(x_2) $ and
then write
\betagin{eqnarray*}
{\partialartial_1 { u}(x) \over h(x_2)}
-{\partialartial_1 { u}(x') \over h(x'_2)} &=& {\partialartial_1 { u}(x) {\tilde h}(x)}
-{\partialartial_1 { u}(x') {\tilde h}(x')} \\
&=& {[\partialartial_1 { u}(x)-\partialartial_1 { u}(x')] {\tilde h}(x')} +
{\partialartial_1 { u}(x)} [{ {\tilde h}(x)} -{ {\tilde h}(x')}],
\end{eqnarray*}
which
yields
\betagin{eqnarray*}
\|I_{\alpha}(\partial_1U)\|_{L^r(\mathbb RR^2)}^r \!\!\!\!\!\!
& &\leq
C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+
C\int_{\mathbb R^2} \left(\int_{\mathbb R^2} {\left| {\partialartial_1 { u}(x) }
-{\partialartial_1 { u}(x') } \right|^2\over |x-x'|^{2+2\alphapha} }
dx\right)^{r/2} dx'\\
& & \hspace{.4in}+ C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r\int_{\mathbb RR^2} \left(\int_{\mathbb R^2} {|{ {\tilde h}(x)}-{ \tilde h(x')}|^2
\over |x-x'|^{2+2\alphapha}} dx_1dx_2
\right)^{r/2} dx'_1dx'_2 \\
&&\leq
C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+
C\| {\partialartial_1 u}\|_{L^r_{\alphapha}(\mathbb R^2)}^r+C\|\nabla u\|_{L^{\infty}}^r \|\tilde h \|_{L^r_{\alphapha}(\mathbb R^2)}^r\\
&&\le C\left(\norm{\nabla u}_{L^\infty(\mathbb R^2)} + \norm{u}_{L^r_{\gammamma}(\mathbb R^2)}\right)^r.
\end{eqnarray*}
A similar argument as the one above shows that
\betagin{eqnarray*}
\|I_{\alpha}(\partial_2U)\|_{L^r(\mathbb RR^2)}^r&=& \int_{\mathbb R^2} \left(\int_{\mathbb R^2}
{|\partialartial_2U(y)-\partialartial_2U(y')|^2\over |y-y'|^{2+2\alphapha} }dy\right)^{r/2} dy'\\[3pt]
&\leq&
C\|\nabla u\|_{L^{\infty}(\mathbb R^2)}^r+
C\| {\partialartial_1 u}\|_{L^r_{\alphapha}(\mathbb R^2)}^r +
C\| {\partialartial_2 u}\|_{L^r_{\alphapha}(\mathbb R^2)}^r\\
&\le & C\left(\norm{\nabla u}_{L^\infty(\mathbb R^2)} + \norm{u}_{L^r_{\gammamma}(\mathbb R^2)} \right)^r.
\end{eqnarray*}
Also, by repeating the preceding argument we obtain,
\[
\norm{I_{\alphapha}(U)}_{L^r(\mathbb{R}^2)}\le C\left(\norm{u}_{L^\infty(\mathbb R^2)}+\norm{u}_{L^r_{\alphapha}(\mathbb{R}^2)}\right)
\le C\norm{u}_{L^r_{\gammamma}(\mathbb R^2)},
\]
where we used the Sobolev embedding theorem in the last inequality with $\gammamma r>2.$
The proof of Lemma~\ref{var} is now complete.
\end{proof}
For $g,h$ on $\mathbb R $ define a the tensor $g\otimes h$ as the following function on
$\mathbb R^2 $ by setting $(g\otimes h)(\xi,\eta) = g(\xi)h(\eta)$.
\betagin{lemma}\labelel{ten}
Let $f\in L^r_{\gamma}(\mathbb R)$ supported in $[-1,1]$,
and $\widehat{\Theta}$ is a smooth function supported in an annulus centered at $0$
with size comparable to $1$, then we have
$$
\big\| f\otimes\widehat\Theta\big\|_{ L^r_{\gamma}(\mathbb R^2)} \le C \| f\|_{ L^r_{\gamma}(\mathbb R) } \, .
$$
\end{lemma}
\betagin{proof}
We use the same idea as in the proof of Lemma \ref{var}.
It suffices to prove that
$f\otimes\widehat\Theta\in L^r_1(\mathbb R^2)$ and that $I_{\alpha}(\partial^{\betata}(f\otimes\widehat\Theta))
\in L^r(\mathbb R^2)$ with $|\betata|=1$.
It is easy to check that $\|f\otimes\widehat\Theta\|_{L^r_1}\le C\|f\|_{L^r_1}$, so we only prove that
$I_{\alpha}(\partial_{\xi}(f\otimes\widehat\Theta))
\in L^r(\mathbb R^2)$.
Note that $f\otimes\widehat\Theta$ is compactly supported and we can choose
a function $\varphi(\xi,\eta)\in C^{\infty}_0(\mathbb R^2)$ assuming $1$ on the support
of $f\otimes\widehat\Theta$ and therefore
$f\otimes\widehat\Theta=f(\xi)\varphi(\xi,\eta)\widehat\Theta(\eta)\varphi(\xi,\eta)$.
Then $\int_{\R}r|I_{\alpha}(\partial_{\xi}(f\otimes\widehat\Theta))|^r d\xi d\eta $ is split into the parts
$$
\int_{\R}r\left(\int_{\R}r\frac{|[f'(\xi)\varphi(\xi,\eta)-f'(\xi')
\varphi(\xi',\eta')]\widehat\Theta(\eta')\varphi(\xi',\eta')|^2}
{|(\xi,\eta)-(\xi',\eta')|^{2+2\alpha}}d\xi'd\eta'\right)^{r/2}d\xi d\eta
$$
and
$$
\int_{\R}r\left(\int_{\R}r\frac{|f'(\xi)\varphi(\xi,\eta)[\widehat\Theta(\eta)\varphi(\xi,\eta)-
\widehat\Theta(\eta')\varphi(\xi',\eta')]|^2}
{|(\xi,\eta)-(\xi',\eta')|^{2+2\alpha}}d\xi'd\eta'\right)^{r/2}d\xi d\eta.
$$
We prove only that the first one is finite since the latter can be proved similarly.
To prove the boundedness of the first one, we split it further via the identity
$$
f'(\xi)\varphi(\xi,\eta)-f'(\xi')
\varphi(\xi',\eta')=
(f'(\xi)-f'(\xi'))
\varphi(\xi,\eta)
+f'(\xi')(\varphi(\xi,\eta)-
\varphi(\xi',\eta')).
$$
The integral containing the second part is finite because $f'$ is bounded and
$\varphi\in L^r_{\gamma}(\mathbb R^2)$.
For the other part, a simple change of variable
$\eta'\rightarrow(\eta-\eta')/(\xi-\xi')$ shows that it is equal to
$$
C\int_{\R}r\left(\int_{\R}\frac{| f'(\xi)-f'(\xi')|^2}
{|\xi-\xi'|^{1+2\alpha}}d\xi'\right)^{r/2} \abs{\varphi(\xi,\eta)}d\xi d\eta,
$$
which, by Lemma~\ref{le6.2}, is bounded by
$\|f\|_{L^r_{\gamma}(\mathbb R)}^r$ since $\varphi\in C^{\infty}_0(\mathbb R^2)$.
\end{proof}
\betagin{lemma}\labelel{lem.PhiR1}
Let $\gammamma\in (1,2)$ and $1<r<\frac{1}{\gamma-1}$. Then
$\norm{\Phi\varphi}_{L^r_\gammamma(\mathbb{R})}<\infty,$
where $\varphi$ is a smooth function with compact support, and $\Phi$ is the function in \eqref{e6.2}.
\end{lemma}
\betagin{proof}
To obtain the claim, we need to show that
$D^{\gamma }(\varphi \Phi )= \big((1+|\xi |^2)^{\gamma/2} \widehat{ \varphi \Phi } \big)\spcheck \in L^r(\mathbb RR)$.
Since
\[
\norm{ D^\gamma (\varphi \Phi )}_{L^r(\mathbb{R})}\approx \norm{\varphi\Phi}_{L^r(\mathbb{R})}
+\norm{ \big( |\xi | ^{\gamma } \widehat{ \varphi \Phi } \big)\spcheck}_{L^r(\mathbb{R})},
\]
and trivially $\varphi \Phi \in L^r(\mathbb RR)$,
we reduce the proof to establishing
$\big\| \big( |\xi | ^{\gamma } \widehat{ \varphi \Phi } \big)\spcheck\big\|_{L^r(\mathbb{R})}<\infty.$
By the Kato-Ponce inequality for homogeneous type \cite{CW}, \cite{MuPiTaTh}, \cite{GrOh}, it suffices to show that
$ \big( |\xi | ^{\gamma } \widehat{ \Phi } \big)\spcheck$ lies in $L^r(\mathbb RR)$.
Indeed, for $\gamma \in (1,2)$ we write
\betagin{align*}
\widehat{\Phi}(\xi ) |\xi|^{\gamma} &= \frac{1}{\xi} \, \xi \, \widehat{\Phi}(\xi ) \, |\xi|^{\gamma } =
\frac{1}{2\partiali i } \frac{1}{\xi} \, \widehat{\Phi'}(\xi ) \, |\xi|^{\gamma }\\
=&-i \frac{1}{\partiali \xi } \, \widehat{\chi_{[-1,0]}}(\xi ) \, |\xi|^{\gamma }
=- i \frac{1}{\partiali \xi } \, \frac{e^{2\partiali i \xi}-1 }{2\partiali i \xi} \, |\xi|^{\gamma }\\
=& -i \frac{1}{\partiali } \, \frac{e^{2\partiali i \xi }-1 }{2\partiali i } \, |\xi|^{\gamma-2}
=-\frac{1}{2\partiali^2} (e^{2\partiali i \xi }-1 ) \, |\xi|^{\gamma-2}\, .
\end{align*}
Taking inverse Fourier transforms we obtain that
$$
\big( \widehat{\Phi}(\xi ) |\xi|^{\gamma}\big)\spcheck (x) = c_\gamma (|x+1|^{1-\gamma}-|x|^{1-\gamma})
$$
and this function lies in $L^r(\mathbb RR)$ when $1<r<\frac{1}{\gamma-1}$ and $\gamma$ is very close to $2$.
\end{proof}
The preceding result can be lifted to $\mathbb{R}^2$ as follows.
\betagin{lemma}\labelel{lem.PhiR2}
Let $\gammamma\in (1,2)$ and $1<r<\frac{1}{\gamma-1}$, and let ${\theta}$ be a
function supported in $\fracrac{1}{2}\le \abs{\xi}\le 2$ on the real line.
Define $U(\xi,\eta) = \Phi(\fracrac{\xi}{\eta})\theta(\fracrac{\xi}{\eta})\widehat{\partialsi}(\xi,\eta),$
where $\widehat{\partialsi}$ is a smooth function supported in an annulus centered at zero. Then
$\norm{U}_{L^r_\gammamma(\mathbb{R}^2)}<\infty$.
\end{lemma}
\betagin{proof}
Set
$$u(\xi,\eta) = \Phi(\xi ) \theta(\xi ) \widehat{\Psi} (\xi\eta,\eta)$$ and
$$U(\xi,\eta)= \Phi(\xi/\eta) \theta(\xi/\eta) \widehat{\Psi} (\xi,\eta).$$
Since $h(\eta)=\eta$ on the support of the function $U.$ We now apply Lemma \ref{var} to obtain
$$
\|U\|_{L^r_{\gamma}(\mathbb R^2)}\le C\big(\|\nabla u\|_{L^{\infty}(\mathbb R^2)}+\|u\|_{L^r_{\gamma}(\mathbb R^2)}\big).
$$
Thus, it is enough to show that $\|u\|_{L^r_{\gamma}(\mathbb R^2)}<\infty.$
We introduce a compactly supported
smooth function $\widehat{\Theta}(\eta)$ which is equal to $1$ on the support of
$\eta\mapsto \theta(\xi )\widehat{\Psi} (\xi\eta,\eta)$ for any $\xi$.
the Kato-Ponce inequality (\cite{KP} \cite{GrOh}) allows us to
estimate the Sobolev norm of $u$ as follows:
\betagin{align*}
\norm{u}_{L^r_\gamma(\mathbb RR^2)}
\,\,=&\,\,\big\| \Phi(\xi )\theta(\xi ) \widehat{\Theta}(\eta)\widehat{\Psi} (\xi\eta,\eta)\big\|_{L^r_\gamma(\mathbb RR^2)}\\
\lesssim&\,\,
\big\| \Phi (\xi)\theta(\xi ) \widehat{\Theta}(\eta)\big\|_{L^r_\gamma(\mathbb RR^2)}
\big\| \widehat{\Psi} (\xi\eta,\eta) \big\|_{L^\infty(\mathbb RR^2)} \\
&+
\big\| \widehat{\Psi} (\xi\eta,\eta)\big\|_{L^r_\gamma (\mathbb RR^2)}
\big\| \Phi(\xi)\theta(\xi )\widehat{\Theta}(\eta) \big\|_{L^\infty(\mathbb RR^2)}\, .
\end{align*}
We are left with establishing $ \| \Phi (\xi)\theta(\xi ) \widehat{\Theta}(\eta) \|_{L^r_\gamma(\mathbb RR^2)}<\infty$,
since all other terms on the right of the above inequality are finite.
This is achieved via Lemmas \ref{lem.PhiR1} and \ref{ten}. Thus the proof of Lemma \ref{lem.PhiR2} is
complete.
\end{proof}
Using these ideas we are able to deduce the following result concerning ${ \mathcal C}_1 $.
\betagin{prop}\labelel{C1}
The Calder\'on commutator ${ \mathcal C}_1 $ maps
$L^{p_1}(\mathbb RR) \times L^{p_2}(\mathbb RR)$ to
$L^p(\mathbb RR)$ when $1/p_1+1/p_2=1/p$, $1<p_1,p_2 <\infty$, and $1/2<p<\infty$.
\end{prop}
\betagin{proof}
Note that $\sigma(\xi,\eta)=\textup{sgn\,}(\eta)\Phi(\xi/\eta)$ has an obvious modification which is continuous
on $\mathbb R^2\backslash\{0\}$. We denote the latter by $\textup{sgn\,}(\eta)\Phi(\xi/\eta)$ as
well since there is no chance to introduce any confusion.
We introduce a smooth function with compact support $\theta$ on the real line which is supported
in two small intervals, say, of length $1/100$ centered at the points $-1$ and $0$. Then we write
$$
1= \theta (\xi/\eta) + 1-\theta (\xi/\eta)
$$
and we decompose the function $\textup{sgn\,}(\eta)\Phi(\xi/\eta) =\sigma_1(\xi,\eta)+\sigma_2(\xi,\eta)$, where
$\sigma_1(\xi,\eta)=\textup{sgn\,}(\eta)\Phi(\xi/\eta)\theta (\xi/\eta)$
and $\sigma_2(\xi,\eta)=\textup{sgn\,}(\eta)\Phi(\xi/\eta)(1-\theta (\xi/\eta))$.
Let $\widehat{\Psi}$ be a smooth bump supported in the annulus
$1/2<|(\xi,\eta)| <3/2$ in $\mathbb RR^2$.
The function $\sigma_2$ is smooth away from zero and $\sigma_2\widehat\Psi$ lies in $L^r_{\gamma}(\mathbb R^2)$ for any $r,\gamma>1$
Also, $\sigma_1\widehat\Psi$ lies in $L^r_{\gamma}(\mathbb R^2)$ with
$r\gamma>1$.
in view of Lemma \ref{lem.PhiR2}. Then Corollary \ref{cor1} implies the
required conclusion.
\end{proof}
\subsection{Commutators of Calder\'on-Coifman-Journ\'e}
Now
we focus on the boundedness
properties of the
following $n$-dimensional version of ${\mathcal C}_1 $:
\betagin{align}\betagin{split}\labelel{PTCJ}
\hspace{-29pt} & { \mathcal C}_1^{(n)}(f,a)(x)\\
& =\textup{p.v.}\!\!
\int_{\mathbb R^n} \! f(y) \! \left(
\partialrod_{l=1}^n { 1\over
(y_l-x_l)^2 } \right ) \! \int_{x_1}^{y_1} \!\!\!
\cdots \!\!\!\int_{x_n}^{y_n} a(u_1, \dots, u_n) \, du_1\cdots du_n \, dy,
\end{split}\end{align}
where $f$ is a function on $\mathbb R^n$, and
$x=(x_1,\dots, x_n)\in \mathbb R^n$, $y=(y_1, \dots , y_n) \in \mathbb R^n$.
The operator ${ \mathcal C}_1^{(n)}$ was introduced by a suggestion of Coifman
when $n=2$.
The $L^2\times L^\infty\to L^2$ bound for ${ \mathcal C}_1^{(2)}$ was studied by Aguirre \cite{Aguirre} and
Journ\'e \cite{Jo1, Jo2}, namely,
\betagin{equation}\labelel{Journe1Rev}
\|{ \mathcal C}_1^{(2)}(f,a)\|_{L^2({\mathbb R^2})}
\leq C \|a\|_{L^{\infty}(\mathbb R^2)}\|f\|_{L^{2}(\mathbb R^2)}.
\end{equation}
For general $n\geq 2$, boundedness for $\mathcal C_1^{(n)}$ from $L^{p_1}\times L^{p_2}$ to $L^p$ for $p>1/2$,
can be derived by
Muscalu's work on Calder\'on
commutators on polydiscs \cite[Theorem 6.1]{Mu3} via time-frequency analysis.
In this section we will apply Corollary~\ref{less1} to
obtain a direct proof of the boundedness of ${ \mathcal C}_1^{(n)} $
from $L^{p_1}(\mathbb R^n) \times L^{p_2}(\mathbb R^n)$ to
$L^p(\mathbb R^n)$ in the full range of $p>1/2$.
\betagin{prop}\labelel{th7.1}
Let $1<p_1, p_2<\infty$, $1/2<p<\infty$ and $1/p={1/p_1}+{1/p_2}.$
Then the operator ${ \mathcal C}_1^{(n)}(f,a)$ is bounded from $L^{p_1}({\mathbb R}^n)\times L^{p_2}({\mathbb R}^n)$
into $L^p({\mathbb R}^n)$, i.e.,
\betagin{eqnarray*}
\|{ \mathcal C}_1^{(n)}(f,a)\|_{{L^p(\mathbb RR^n)}}\leq C_p\|a\|_{L^{p_1}(\mathbb RR^n)}\|f\|_{L^{p_2}(\mathbb RR^n)}.
\end{eqnarray*}
\end{prop}
\betagin{proof}
The operator ${ \mathcal C}^{(n)}_1(f,a)$
is a bilinear operator which can also be expressed in bilinear Fourier multiplier form as
\betagin{eqnarray*}
{ \mathcal C}^{(n)}_1(f,a)(x)
= (-i\partiali)^n \iint_{\mathbb R^n\times \mathbb R^n} \widehat{f}(\xi_1,\cdots, \xi_n)\, \widehat{a}(\eta_1,\cdots, \eta_n)\,
e^{2\partiali i x\cdot(\xi+\eta) } m(\xi; \eta)
\,
d\xi d\eta\, ,
\end{eqnarray*}
where the symbol $m$ is given by
$$
m(\xi; \eta)=\partialrod_{i=1}^n\left[\textup{sgn\,} (\eta_i) \,
\Phi\Big({\xi_i\over \eta_i}\Big)\right],
$$
and $\xi=(\xi_1, \cdots, \xi_n)$ and $\eta=(\eta_1, \cdots, \eta_n).$
Since $m(\xi,\eta)=\partialrod_{i=1}^n\sigma(\xi_i,\eta_i)$
is a product of $n$ equal pieces, by Corollary~\ref{less1},
it suffices to verify that
$\sup_{k\in \mathbb Z}\|\sigma(2^k\cdot)\widehat\Psi\|_{L^r_{\gamma/2,\gamma/2}(\mathbb R^2)}
=B<\infty$.
Note that $\sigma(2^k\cdot)\widehat\Psi\in L^r_{\gamma}(\mathbb R^2)$
uniformly in $k$
by Proposition~\ref{C1}, so they are also in
$L^r_{\gamma/2,\gamma/2}(\mathbb R^2)$ uniformly
due to that $L^r_\gamma(\mathbb R^2)\subset L^r_{\gamma/2,\gamma/2}(\mathbb R^2)$.
We complete the proof of Proposition \ref{th7.1}.
\end{proof}
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\end{document} | math | 122,367 |
\begin{document}
There are at least two well-developed theories of noncommutative smoothness. Auslander regularity (generalizing Serre's homological characterization of commutative regular algebras)
is well suited to deal with questions from K-theory, derived categories and intersection theory.
Formally smooth algebras and smooth orders (generalizing Grothendieck's categorical characterization of commutative regular algebras) are well suited to deal with geometric questions such as the \'etale local structure (of the algebra and its center) and Brauer-Severi varieties.
Not surprisingly, 'real life situations' (such as stringtheory) often require the best of both worlds.
\section{Definitions}
Let $A$ be an affine $\mathbb{C}$-algebra and denote with $\wis{rep}_n~A$ the affine scheme of $n$-dimensional representations of $A$. The basechange group $GL_n$ acts on this scheme and the geometric points of the algebraic quotient $\wis{iss}_n~A = \wis{rep}_n~A// GL_n$ classify the isomorphism classes of semisimple $n$-dimensional representations of $A$. In general, $\wis{rep}_n~A$ can have several connected components and in the decomposition
\[
\wis{rep}_n~A = \bigsqcup_{\alpha}~\wis{rep}_{\alpha}~A \]
we say that $\alpha$ is a dimension vector of total dimension $| \alpha | = n$. The corresponding
algebraic quotient will be denoted by $\wis{iss}_{\alpha}~A$ and its coordinate ring
$Z_{\alpha} = \mathbb{C}[\wis{iss}_{\alpha}~A]$ is a central subring of the algebra of $GL_n$-equivariant maps
\[
\int_{\alpha}~A = M_n(\mathbb{C}[\wis{rep}_{\alpha}~A])^{GL_n} \]
from $\wis{rep}_{\alpha}~A$ to $M_n(\mathbb{C})$. The algebra $\int_{\alpha}~A$ is a Noetherian algebra and is a finite module over $Z_{\alpha}$. We define two $\alpha$-relative notions of noncommutative smoothness on $A$.
\begin{definition} $A$ is said to be $\alpha$-Auslander regular (or equivalently, $\int_{\alpha}~A$ is Auslander regular) if the following conditions are satisfied for $B = \int_{\alpha}~A$ :
\begin{enumerate}
\item{$B$ has finite global dimension, $gldim(B) < \infty$.}
\item{For every finitely generated left $B$-module $M$, every integer $j \geq 0$ and every (right)
$B$-submodule $N$ of $Ext^j_B(M,B)$ we have that $j(N) \geq j$, Here, $j(N)$ is the {\em grade number}
of $N$ which is the least integer $i$ such that $Ext^i_B(N,B) \not= 0$.}
\item{For every finitely generated left $B$-module $M$ we have the equality
\[
GKdim(M) + j(M) = GKdim(B) \]
where $GKdim$ denotes the Gelfand-Kirillov dimension, see for example \cite{KrauseLenegan}.}
\end{enumerate}
\end{definition}
A major application of this notion is that it allows us to study finitely generated $B$-modules in terms of pure modules using the spectral sequence
\[
E_2^{p,-q}(M) = Ext^p_B(Ext^q_B(M,B),B) \mathbb{R}ightarrow H^{p-q}(M) \]
where $H^0(M) = M$ and $H^i(M) = 0$ for $i \not= 0$. By property $(2)$ the second term of this sequence is triangular.
\begin{definition} $A$ is said to be $\alpha$-smooth (or equivalently, $\int_{\alpha}~A$ is a smooth order) if the following conditions are satisfied :
\begin{enumerate}
\item{ The connected component $\wis{rep}_{\alpha}~A$ is a smooth variety.}
\item{A Zariski open subset $\wis{azu}_{\alpha}~A$ (the Azumaya locus of $A$) of $\wis{rep}_{\alpha}~A$ consists of simple representations.}
\end{enumerate}
\end{definition}
By $(2)$ the quotient $\wis{rep}_{\alpha}~A \rOnto^{\pi} \wis{iss}_{\alpha}~A$ is generically a principal $PGL_n$-fibration and hence determines a central simple algebra of dimension $n^2$
(where $n = | \alpha |$) over the function field of $Z_{\alpha}$. By $(1)$, $Z_{\alpha}$ is integrally closed and therefore $\int_{\alpha}~A$ is an order in the central simple algebra having as its center $Z_{\alpha}$.
The main application of this notion is that it allows us to describe the \'etale local structure of
$\int_{\alpha}~A$ and of $Z_{\alpha}$. Let $\xi$ be a point of $\wis{iss}_{\alpha}~A$ with corresponding semi-simple $n$-dimensional representation
\[
M = S_1^{\oplus e_1} \oplus \hdots \oplus S_k^{\oplus e_k} \]
Consider the {\em local quiver setting} $(Q,\epsilonilon)$ where $Q$ is the finite quiver on $k$ vertices $\{ v_1,\hdots,v_k \}$ (corresponding
to the distinct irreducible components of $M$) such that the number of oriented arrows from $v_i$ to
$v_j$ is given by the dimension of the extension space $Ext^1_A(S_i,S_j)$. The dimension vector
$\epsilonilon$ of the quiver $Q$ is given by the multiplicities $(e_1,\hdots,e_k)$ with which these
simple components occur in $M$. To be precise, there is a $GL_n$-equivariant \'etale local isomorphism
between $\wis{rep}_{\alpha} A$ and the associated fiber bundle
\[
GL_n \times^{GL(\epsilonilon)} \wis{rep}_{\epsilonilon} Q \]
where $\wis{rep}_{\epsilonilon} Q$ is the vectorspace of $\epsilonilon$-dimensional representations of the
quiver $Q$ on which the group $GL(\epsilonilon) = GL_{e_1} \times \hdots \times GL_{e_k}$ acts by
basechange. Moreover, the embedding $GL(\epsilonilon) \rInto GL_n$ is determined by the dimensions $d_i$
of the simple components $S_i$. As a consequence, there is an \'etale local isomorphism between
$\wis{iss}_{\alpha} A$ and the quotient variety $\wis{iss}_{\epsilonilon} Q = \wis{rep}_{\epsilonilon} Q // GL(\epsilonilon)$,
the variety parametrizing isoclasses of semisimple $\epsilonilon$-dimensional representations of $Q$.
In particular this allows us to control the central singularities which were classified in low dimensions in \cite{BockLBVdW}.
\section{Auslander regularity}
In reverse geometric engineering of singularities in
stringtheory (see e.g. \cite[\S 2]{Berensteinreverse}) one is interested in the case that the non-Azumaya locus (the 'non bulk representations' in physical lingo) consists of isolated singularities.
We are able to determine the \'etale local structure of such $\alpha$-smooth orders.
\begin{lemma} \label{etaleclass} Let $A$ be an $\alpha$-smooth order such that $\{ p \}$ is an isolated singularity
which is locally the non-Azumaya locus. Then, the \'etale local structure of $\int_{\alpha}~A$ in
$p$ is determined by a quiver setting
\[
\xy 0;/r.15pc/:
\POS (0,0) *+{\txt{\tiny $1$}}="a",
(20,0) *+{\txt{\tiny $1$}}="b",
(34,14) *+{\txt{\tiny $1$}} ="c",
(34,34) *+{\txt{\tiny $1$}}="d",
(20,48) *+{\txt{\tiny $1$}} ="e",
(0,48) *+{\txt{\tiny $1$}}="f"
\POS"a" \ar@{=>}^{k_l} "b"
\POS"b" \ar@{=>}^{k_1} "c"
\POS"c" \ar@{=>}^{k_2} "d"
\POS"d" \ar@{=>}^{k_3} "e"
\POS"e" \ar@{=>}^{k_4} "f"
\POS"f" \ar@/_7ex/@{.>} "a"
\endxy
\]
where $Q$ has $l$ vertices and all $k_i \geq 1$. The central dimension
is
\[
d = \sum_i k_i + l - 1 \]
\end{lemma}
\Proof
To start, $\epsilonilon$ is the dimension vector of a simple representation of $Q$. By the results of
\cite{LBProcesi} this implies that $Q$ is a strongly connected quiver (any pair of vertices $v_i$, $v_j$
is connected by an oriented path in $Q$ starting at $v_i$ and ending in $v_j$) and that the dimension vector
$\epsilonilon$ satisfies the numerical conditions
\[
\chi_Q(\epsilonilon,\delta_i) \leq 0 \qquad \text{and} \qquad \chi_Q(\delta_i,\epsilonilon) \leq 0 \]
where $\chi_Q$ is the Euler-form of the quiver $Q$ (that is, the bilinear form on $\mathbb{Z}^k$ determined by the
$k \times k$ matrix whose $(i,j)$-entry is $\delta_{ij}-$ the number of arrows from $v_i$ to $v_j$ and
where $\delta_i$ is the basevector concentrated in $v_i$). Next, we claim that $\epsilonilon = (1,\hdots,1)$.
If not, there are $\epsilonilon$-dimensional semi-simple representations of $Q$ of representation type
\[
(1,(1,\hdots,1);e_1-1,\delta_1;\hdots;e_k-1,\delta_k) \]
(the first factor indeed corresponds to a simple representation of $Q$ as $Q$ is strongly connected)
which is impossible by $GL_n$-equivariance and the fact that the non-Azumaya locus is concentrated in $p$
(which corresponds to the point of representation type $(e_1,\delta_1;\hdots;e_k,\delta_k)$).
We claim that every oriented cycle in $Q$ has as its support all the vertices $\{ v_1,\hdots,v_k \}$
and consequently that the quiver setting $(Q,\epsilonilon)$ is of the following shape :
\[
\xy 0;/r.15pc/:
\POS (0,0) *+{\txt{\tiny $1$}} ="a",
(20,0) *+{\txt{\tiny $1$}} ="b",
(34,14) *+{\txt{\tiny $1$}} ="c",
(34,34) *+{\txt{\tiny $1$}} ="d",
(20,48) *+{\txt{\tiny $1$}} ="e",
(0,48) *+{\txt{\tiny $1$}} ="f"
\POS"a" \ar@{=>}^{k_k} "b"
\POS"b" \ar@{=>}^{k_1} "c"
\POS"c" \ar@{=>}^{k_2} "d"
\POS"d" \ar@{=>}^{k_3} "e"
\POS"e" \ar@{=>}^{k_4} "f"
\POS"f" \ar@/_7ex/@{.>} "a"
\endxy
\]
Indeed, let $C$ be an oriented cycle of minimal support in $Q$, let $\delta_{iC}=1$ iff $v_i \in supp(C_)$
and zero otherwise and let $\delta_C = (\delta_{1C},\hdots,\delta_{kC})$. Then, if $C \not= \{ v_1,\hdots,v_k \}$
there would be points of representation type
\[
(1,\delta_C;e_1-\delta_{1C},\delta_1;\hdots;e_k-\delta_{kC},\delta_k) \]
contradicting the assumptions. That is, the Euler-form of the quiver $Q$ is given by the matrix
\[
\begin{bmatrix}
1 & -k_1 & 0 & \hdots & \hdots & 0 \\
0 & 1 & -k_2 & & & 0 \\
\vdots & & \ddots & \ddots & & \vdots \\
\vdots & & & \ddots & & -k_{k-1} \\
-k_k & 0 & 0 & \hdots & \hdots & 1
\end{bmatrix}
\]
and the statement on the dimension follows again from \cite{LBProcesi}.
\par \vskip 3mm
\begin{theorem} If $\int_{\alpha}~A$ is a smooth order such that its non-Azumaya locus consists of isolated singularities, then $\int_{\alpha}~A$ is Auslander regular.
\end{theorem}
\Proof
From \cite{LBProcesi} we recall that the ring of polynomial $GL(\epsilonilon)$-invariants on
$\wis{rep}_{\epsilonilon} Q$ is generated by the traces along oriented cycles in $Q$. Therefore,
\[
\mathbb{C}[\wis{iss}_{\epsilonilon} Q] = \mathbb{C}[\wis{rep}_{\epsilonilon} Q]^{GL(\epsilonilon)} = R = \mathbb{C}[ x_{i_1}(1)x_{i_2}(2) \hdots x_{i_k}(k) ; 1 \leq i_j \leq k_j] \subset \mathbb{C}[\wis{rep}_{\epsilonilon} Q]
\]
Therefore, $p$ is a singular point of $Z_{\alpha} = \wis{iss}_{\alpha} A$ if and only if at least two of the
$k_i \geq 2$ because by the \'etale local isomorphism the completion of the coordinate ring
$\mathbb{C}[ \wis{iss}_{\alpha} A]$ at the maximal ideal determined by $p$ is isomorphic to $\hat{R}$, the
completion of $R$ at the maximal ideal generated by all traces along oriented cycles in $Q$.
Similarly, we can determine $\hat{\int_{\alpha}} A$, the completion of the smooth order $\int_{\alpha} A$ at the central maximal ideal determined by $p$ from \cite{LBProcesi},
\[
\widehat{\int_{\alpha}} A \simeq
\begin{bmatrix}
M_{d_1}(\hat{R}) & M_{d_1 \times d_2}(M_{12})& M_{d_1 \times d_3}(M_{13}) & \hdots & M_{d_1 \times d_k}(M_{1k}) \\
M_{d_2 \times d_1}(M_{21}) & M_{d_2}(\hat{R}) & M_{d_2 \times d_3}(M_{23}) & \hdots & M_{d_2 \times d_k}(M_{2k}) \\
M_{d_3 \times d_1}(M_{31}) & M_{d_3 \times d_2}(M_{32}) & M_{d_3}(\hat{R}) & \hdots & M_{d_3 \times d_k}(M_{3k}) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
M_{d_k \times d_1}(M_{k1}) & M_{d_k \times d_2}(M_{k2}) & M_{d_k \times d_3}(M_{k3}) & \hdots & M_{d_k}(\hat{R})
\end{bmatrix}
\]
Here, $d_i = dim_{\mathbb{C}}(S_i)$ and $M_{ij}$ is the $\hat{R}$-submodule of $\mathbb{C}[[x_i(j),1 \leq i \leq k_j,1\leq j \leq k]]$ generated by the oriented paths in $Q$ from $v_i$ to $v_j$.
The strategy to prove that $\hat{\int_{\alpha}} A$ is Auslander regular is to use the
$GL(\epsilonilon) = \mathbb{C}^* \times \hdots \times \mathbb{C}^*$-action on $\wis{rep}_{\epsilonilon} Q$ to
obtain central elements of $\hat{\int_{\alpha}} A$ corresponding to certain arrows. Modding
out these elements in a specific order will reduce the quiver until we are left with a
hereditary (in particular, Auslander-regular) order. We can then retrace our steps using the
result that if $B$ is a Noetherian algebra with central element $c$ such that $B/(c)$ is Auslander-regular,
then so is $B$.
Let $\{ z_1,\hdots,z_l \}$ be the vertex-indices such that $k_{z_i} \geq 2$ and after
cyclically renumbering the vertices (if needed) we may assume that $z_l = k$. For all $i \not= z_1$
set $x_i(1) = 1$ and let $\hat{R}_1$ respectively $\hat{B}_1$ be the algebra obtained from
$\hat{R}$ respectively $\hat{\int_{\alpha}} A$ using these assignments, then clearly,
\[
\hat{R} \simeq \hat{R}_1 \qquad \text{and} \qquad \widehat{\int_{\alpha}} A \simeq \hat{B}_1 \]
The advantage of this change of generators is that $\{ x_{z_1}(1),\hdots,x_{z_1}(k_{z_1}) \}$ are
generators of $\hat{R}_1$ and the quotient-algebras
\[
\overline{R}_1 = \frac{\hat{R}_1}{(x_{z_1}(2),\hdots,x_{z_1}(k_{z_1}))} \qquad \text{respectively}
\qquad \overline{B}_1 = \frac{\hat{B}_1}{(x_{z_1}(2),\hdots,x_{z_1}(k_{z_1}))}
\]
are isomorphic to the completion of the algebra of polynomial invariants (resp. equivariant maps)
of the quiver setting $(Q_1,\epsilonilon)$ where $Q_1$ has the same shape as $Q$ except that there
is just one arrow from $v_{z_1}$ to $v_{z_1+1}$. Repeat this procedure, starting with the quiver setting $(Q_1,\epsilonilon)$ with vertex $v_{z_2}$. That is, for all $i \not= z_2$ set $x_i(1)=1$ and
let $\hat{R}_2$ respectively $\hat{B}_2$ be the algebra obtained from the completion of the algebra of polynomial invariants (resp. equivariant maps)
of the quiver setting $(Q_1,\epsilonilon)$ using these assignments and let $\overline{R}_2$ respectively $\overline{B}_2$ be the quotient algebras obtained by modding out the
generators $\{ x_{z_2}(2),\hdots,x_{z_2}(k_{z_2}) \}$ of $\hat{R}_2$ and observe that these
quotients are the relevant algebras corresponding to a quiver setting $(Q_2,\epsilonilon)$ where
$Q_2$ has the same shape as $Q_1$ except that there is just one arrow from $v_{z_2}$ to
$v_{z_2+1}$ and so on.
After $l$ iterations of this procedure we arrive at the quiver setting $(Q_l,\epsilonilon)$ where
$Q_l$ is of the form
\[
\xy 0;/r.15pc/:
\POS (0,0) *+{\txt{\tiny $1$}} ="a",
(20,0) *+{\txt{\tiny $1$}} ="b",
(34,14) *+{\txt{\tiny $1$}} ="c",
(34,34) *+{\txt{\tiny $1$}} ="d",
(20,48) *+{\txt{\tiny $1$}} ="e",
(0,48) *+{\txt{\tiny $1$}} ="f"
\POS"a" \ar@{->}^{x} "b"
\POS"b" \ar@{->}^{1} "c"
\POS"c" \ar@{->}^{1} "d"
\POS"d" \ar@{->}^{1} "e"
\POS"e" \ar@{->}^{1} "f"
\POS"f" \ar@/_7ex/@{.>} "a"
\endxy
\]
from which we deduce that $\overline{R}_l \simeq \mathbb{C}[[x]]$ and that
\[
\overline{B}_l \simeq \begin{bmatrix}
M_{d_1}(\mathbb{C}[[x]]) & M_{d_1 \times d_2}(\mathbb{C}[[x]])& M_{d_1 \times d_3}(\mathbb{C}[[x]]) & \hdots & M_{d_1 \times d_k}(\mathbb{C}[[x]]) \\
M_{d_2 \times d_1}(x\mathbb{C}[[x]]) & M_{d_2}(\mathbb{C}[[x]]) & M_{d_2 \times d_3}(\mathbb{C}[[x]]) & \hdots & M_{d_2 \times d_k}(\mathbb{C}[[x]]) \\
M_{d_3 \times d_1}(x\mathbb{C}[[x]]) & M_{d_3 \times d_2}(x\mathbb{C}[[x]]) & M_{d_3}(\mathbb{C}[[x]]) & \hdots & M_{d_3 \times d_k}(\mathbb{C}[[x]]) \\
\vdots & \vdots & & \ddots & \vdots \\
M_{d_k \times d_1}(x\mathbb{C}[[x]]) & M_{d_k \times d_2}(x\mathbb{C}[[x]]) & M_{d_k \times d_3}(x\mathbb{C}[[x]]) & \hdots & M_{d_k}(\mathbb{C}[[x]])
\end{bmatrix}
\]
It is well known that $\overline{B}_l$ is an Auslander-regular algebra and as we divided out
central elements in each step (and in each step, the localizations at these central elements are Azumaya algebras with regular center hence Auslander regular), we derive using \cite[theorem III.3.6]{LiHuishiFVO} that also $\hat{\int_{\alpha}} A$
is Auslander-regular. Because Auslander-regularity is preserved under central \'etale extensions
and because $A$ is at all other points an Azumaya algebra over a commutative regular ring,
Auslander regularity of $A$ follows.
\section{Isolated singularities}
In this section we will give the \'etale local structure of an $\alpha$-smooth order in an isolated central singularity. That is, we will extend lemma~\ref{etaleclass} without the condition on the Azumaya locus. Because an $\alpha$-smooth order is locally determined by a quiver setting (see section 1) the problem reduces to classifying all quiver settings $(Q,\alpha)$ such that $\wis{iss}_{\alpha}~Q$ is an isolated singularity.
If $v$ is a vertex having no loops in $Q$ and such that $\chi_Q(\epsilonilon_v,\alpha) \geq 0$ or $\chi_Q(\alpha,\epsilonilon_v) \geq 0$, then we replace the quiver setting $(Q,\alpha)$ by $(Q',\alpha')$ where $Q'$ is the quiver obtained from $Q$ by deleting the vertex $v$ and adding arrows corresponding to $2$-paths through $v$
\[
\left[ \vcenter{
\xymatrix@=.3cm{
\vtx{u_1}&\cdots &\vtx{u_k}\\
&\vtx{\alpha_v}\ar[ul]\ar[ur]&\\
\vtx{i_1}\ar[ur]&\cdots &\vtx{i_l}\ar[ul]}}
\right]
\longrightarrow
\left[\vcenter{
\xymatrix@=.3cm{
\vtx{u_1}&\cdots &\vtx{u_k}\\
&&\\
\vtx{i_1}\ar[uu]\ar[uurr]&\cdots &\vtx{i_l}\ar[uu]\ar[uull]}}
\right].
\]
(note that some of the vertices in the picture may coincide leading to loops). The dimension vector $\alpha' = \alpha \mid \wis{supp}~Q'$. The reduction step $(Q,\alpha) \rTo (Q',\alpha')$ will be denoted by $R_I^v$.
\begin{theorem} Let $A$ be an $\alpha$-smooth order and $p$ a central isolated singularity. Then, the \'etale local structure of $\int_{\alpha}~A$ in $p$ is determined by a quiver setting $(Q,\epsilonilon)$ which can be reduced, via iterated use of $R_I^v$, to a quiver setting
\[
\xy 0;/r.15pc/:
\POS (0,0) *+{\txt{\tiny $1$}}="a",
(20,0) *+{\txt{\tiny $1$}}="b",
(34,14) *+{\txt{\tiny $1$}} ="c",
(34,34) *+{\txt{\tiny $1$}}="d",
(20,48) *+{\txt{\tiny $1$}} ="e",
(0,48) *+{\txt{\tiny $1$}}="f"
\POS"a" \ar@{=>}^{k_l} "b"
\POS"b" \ar@{=>}^{k_1} "c"
\POS"c" \ar@{=>}^{k_2} "d"
\POS"d" \ar@{=>}^{k_3} "e"
\POS"e" \ar@{=>}^{k_4} "f"
\POS"f" \ar@/_7ex/@{.>} "a"
\endxy
\]
with $l \geq 2$ vertices and all $k_i \geq 2$. The central dimension
is equal to
\[
d = \sum_i k_i + l - 1 \]
\end{theorem}
Contrary to the situation of the previous section, we can have central points $q \in \wis{iss}_{\alpha}~A$ corresponding to proper semi-simple representations
\[
M = S_1^{\oplus e_1} \oplus \hdots \oplus S_l^{\oplus e_l} \]
such that $\wis{iss}_{\alpha}~A$ is smooth in $q$, or equivalently, that the {\em local quiver setting}
$(Q_q,\epsilonilon_q)$ defined in section 1 is {\em coregular}, that is, $\wis{iss}_{\epsilonilon_q}~Q_q$ is a smooth variety. Thanks to \cite{Bocklandt2002} we have a classification of coregular quiver settings. For a quiver setting $(Q,\alpha)$ with a vertex $v$ such that $\alpha_v = 1$ and there are loops in $v$ we define the reduction step $R_{II}^v$ to be $(Q,\alpha) \rTo (Q',\alpha)$ where $Q'$ is the quiver obtained from $Q$ by removing the loops in $v$.
For a quiver setting $(Q,\alpha)$ and a vertex $v$ such that $\alpha_v = k > 1$, there is a unique loop in $v$ and the neighborhood of $Q$ in $v$ is one of the situations on the left hand side of the pictures below
\[
\left[\vcenter{
\xymatrix@=.3cm{
&\vtx{k}\ar[d]\ar[drr]\ar@(lu,ru)&&\\
\vtx{1}\ar[ur]&\vtx{u_1}&\cdots &\vtx{u_m}}}
\right]
\longrightarrow
\left[\vcenter{
\xymatrix@=.3cm{
&\vtx{k}\ar[d]\ar[drr]&&\\
\vtx{1}\ar@2[ur]^k&\vtx{u_1}&\cdots &\vtx{u_m}}}
\right],
\]
\[
\left[\vcenter{
\xymatrix@=.3cm{
&\vtx{k}\ar[dl]\ar@(lu,ru)&&\\
\vtx{1}&\vtx{u_1}\ar[u]&\cdots &\vtx{u_m}\ar[ull]}}
\right]
\longrightarrow
\left[\vcenter{
\xymatrix@=.3cm{
&\vtx{k}\ar@2[dl]_k&&\\
\vtx{1}&\vtx{u_1}\ar[u]&\cdots &\vtx{u_m}\ar[ull]}}
\right].
\]
(again, some of the vertices may be the same). Then we define a reduction step
$R_{III}^v$ which sends $(Q,\alpha) \rTo (Q',\alpha)$ where $Q'$ is $Q$ with the neighborhood of $v$ replaced by the situation on the right hand side of the pictures. The main result of \cite{Bocklandt2002} asserts that $(Q,\alpha)$ is a coregular quiver setting if and only if it can be reduced by an iterated use of the reduction steps $R_I^v,R_{II}^v$ and $R^v_{III}$ (and their inverses) to one of the three quiver settings below :
\[
\xymatrix{\vtx{k}}\hspace{2cm} \xymatrix{\vtx{k}\ar@(lu,ru)}\hspace{2cm} \xymatrix{\vtx{2}\ar@(lu,ru)\ar@(ld,rd)}.
\]
A {\em representation type} $\tau = (e_1,\beta_1;\hdots;e_l,\beta_l)$ of a quiver setting $(Q,\alpha)$ satisfies $\alpha = e_1 \beta_1 + \hdots + e_l \beta_l$ and all $\beta_i$ are dimension vectors of simple representations of $Q$ (and we have a description of those from
\cite{LBProcesi}). The local quiver setting in a point $\xi \in \wis{iss}_{\alpha}~Q$ of representation type $\tau$ depends only on $\tau$ : $Q_{\tau}$ is the quiver on $l$ vertices such that there are exactly $\delta_{ij} - \chi_Q(\beta_i,\beta_j)$ arrows (or loops) from the $i$-th to the $j$-th vertex and $\alpha_{\tau} = (e_1,\hdots,e_l)$, see \cite{LBProcesi}. The {\em stratum} $S_{\tau}$ consisting of all points in $\wis{iss}_{\alpha}~Q$ having representation type $\tau$ has dimension
\[
dim~S_{\tau} = \sum_{loop} (l_j-1)e_j^2 + 1 \]
where the sum is taken over all vertices $w_j$ having loops in $Q_{\tau}$, see \cite{LBProcesi}. If we apply this to the representation type $(\alpha_1,\epsilonilon_1;\hdots;\alpha_k,\epsilonilon_k)$ of the trivial representation we deduce :
\begin{lemma} If $(Q,\alpha)$ is a quiver setting such that $\wis{iss}_{\alpha}~Q$ is an isolated singularity, then there are no loops in $Q$.
\end{lemma}
If $(Q,\alpha) \rDotsto (Q',\alpha')$ is a sequence of reductions $R_I^v,R_{II}^v$ or $R_{III}^v$ we have that either
\[
\wis{iss}_{\alpha}~Q = \wis{iss}_{\alpha'}~Q' \qquad \text{or} \qquad \wis{iss}_{\alpha}~Q = \wis{iss}_{\alpha'}~Q' \times \mathbb{C}^z \]
for some $z$ (see \cite{Bocklandt2002}). By this and the lemma we have that any reduction of a quiver setting $(Q,\alpha)$ with $\wis{iss}_{\alpha}~Q$ an isolated singularity involves only reduction steps $R_I^v$. We will characterize the {\em reduced} settings, that is those that cannot be reduced further.
\begin{lemma} If $(Q,\alpha)$ is a reduced quiver setting with $\wis{iss}_{\alpha}~Q$ an isolated singularity, then $\alpha = \mathbf{1} = (1,\hdots,1)$.
\end{lemma}
\Proof
Assume $v$ is a vertex having maximal $\alpha_v \geq 2$. Because $(Q,\alpha)$ is reduced it follows from the definition of reduction step $R_I^w$ that for all vertices $w$ we have
\[
\chi_Q(\epsilonilon_w,\alpha) < 0 \quad \text{and} \quad \chi_Q(\alpha,\epsilonilon_w) < 0 \]
Therefore, by $\cite{LBProcesi}$ we have that $\alpha - \epsilonilon_v$ is the dimension vector of a simple representation of $Q$ and we look at the local quiver setting $(Q_{\tau},\alpha_{\tau})$ for the representation type $\tau = (1,\epsilonilon_v;1,\alpha-\epsilonilon_v)$. This is of the form
$$
\xymatrix{\vtx{1} \ar@/^/@2{->}[rr]^a & & \vtx{1} \ar@(rd, ru)[]_k \ar@/^/@2{->}[ll]^b }
$$
where $a = -\chi_Q(\epsilonilon_v,\alpha-\epsilonilon_v) = -\chi_Q(\epsilonilon_v,\alpha)+1 \geq 2$, $b = -\chi_Q(\alpha-\epsilonilon_v,\epsilonilon_v) = -\chi_Q(\alpha,\epsilonilon_v) + 1 \geq 2$ and
$k = 1 -\chi_Q(\alpha-\epsilonilon_v,\alpha-\epsilonilon_v)$. Therefore, $(Q_{\tau},\alpha_{\tau})$ is not coregular and hence $\wis{iss}_{\alpha}~Q$ is not an isolated singularity, a contradiction.
\par \vskip 6mm
\noindent
{\bf Proof of theorem 2 : } If $(Q,\alpha)$ is a quiver setting such that $\wis{iss}_{\alpha}~Q$ is an isolated singularity, we can reduce it by iterated use of $R_I^v$ to a setting $(Q',\mathbf{1})$ by the previous lemma. We now claim that $Q'$ is of the prescribed form, that is that every cycle in $Q'$ passes through all vertices. If not let
$\{ v_{i1},\hdots,v_{ip} \}$ be the vertices through which a cycle does {\em not} pass and consider the representation type
\[
\tau = (1,\mathbf{1}-\epsilonilon_{v_{i1}}- \hdots-\epsilonilon_{v_{ip}};1,\epsilonilon_{v_{i1}};\hdots;1,\epsilonilon_{v_{ip}}) \]
The local quiver $Q_{\tau}$ has $p+1$ vertices $\{ w_0,w_1,\hdots,w_p \}$ where $w_j$ corresponds to $v_{ij}$ and $w_0$ collects the remaining vertices $V$ of $Q$. Via this identification, the quiver on $\{ w_1,\hdots,w_p \}$ is identical to that of $Q$ on $\{ v_{i1},\hdots,v_{ip} \}$ and the number of arrows from (resp. to) $w_0$ to (resp. from) $w_j$ is equal to the number of arrows from (resp. to) $V$ to (resp. from) $v_{ij}$ in $Q$ and there is a number of loops in $w_0$.
If $(Q_{\tau},\mathbf{1}_{\tau})$ is reduced (after removing the loops at $w_0$), then it cannot be coregular by \cite{Bocklandt2002} as $Q_{\tau}$ has at least two vertices whence $\wis{iss}_{\alpha}~Q$ is {\em not} an isolated singularity.
If $(Q_{\tau},\mathbf{1}_{\tau})$ can be reduced (after removing the loops at $w_0$), then the only possible reduction step is $R_I^{w_0}$ as all $w_j$ ($j \geq 1$) have at least two incoming and two outgoing arrows. The following lemma asserts that $(Q_{\tau},\mathbf{1}_{\tau})$ cannot be regular whence again $\wis{iss}_{\alpha}~Q$ cannot be an isolated singularity.
\begin{lemma} A coregular quiver setting $(Q,\mathbf{1})$ with $Q$ strongly connected and having more than one vertex has at least two vertices $v$ allowing reduction step $R_I^v$.
\end{lemma}
\Proof
By induction on the number $n$ of vertices. If $n=2$, then by the classification of coregular quiver settings $(Q,\mathbf{1})$ must have the form
\[
\xymatrix{\vtx{1} \ar@/^/@{->}[rr]\ar@2@(ld, lu)^{l_2} & & \vtx{1}
\ar@2@(rd, ru)[]_{l_1} \ar@/^/@2{->}[ll]^k}.
\]
whence (after removing the loops) both vertices allow reduction $R_I$. If $n > 2$ perform one reduction $R_I^v$ (say with one outgoing arrow ending in $w$) to produce of quiver $Q'$ on $n-1$ vertices.
In $Q'$ only $w$ can change its (ir)reducible status (either way). As we removed the reducible vertex $v$ from $Q$ the number of reducible vertices in $Q'$ is less than or equal the number of reducible vertices of $Q$ but by induction $Q'$ has at least two reducible vertices.
\begin{theorem} The isolated central singularities of an $\alpha$-smooth order $A$ and a $\beta$-smooth order $B$ are \'etale equivalent if and only if their local quivers can be reduced to quiver settings
\[
\xy 0;/r.15pc/:
\POS (0,0) *+{\txt{\tiny $1$}}="a",
(20,0) *+{\txt{\tiny $1$}}="b",
(34,14) *+{\txt{\tiny $1$}} ="c",
(34,34) *+{\txt{\tiny $1$}}="d",
(20,48) *+{\txt{\tiny $1$}} ="e",
(0,48) *+{\txt{\tiny $1$}}="f"
\POS"a" \ar@{=>}^{k_l} "b"
\POS"b" \ar@{=>}^{k_1} "c"
\POS"c" \ar@{=>}^{k_2} "d"
\POS"d" \ar@{=>}^{k_3} "e"
\POS"e" \ar@{=>}^{k_4} "f"
\POS"f" \ar@/_7ex/@{.>} "a"
\endxy
\quad \text{resp.} \quad
\xy 0;/r.15pc/:
\POS (0,0) *+{\txt{\tiny $1$}}="a",
(20,0) *+{\txt{\tiny $1$}}="b",
(34,14) *+{\txt{\tiny $1$}} ="c",
(34,34) *+{\txt{\tiny $1$}}="d",
(20,48) *+{\txt{\tiny $1$}} ="e",
(0,48) *+{\txt{\tiny $1$}}="f"
\POS"a" \ar@{=>}^{m_l} "b"
\POS"b" \ar@{=>}^{m_1} "c"
\POS"c" \ar@{=>}^{m_2} "d"
\POS"d" \ar@{=>}^{m_3} "e"
\POS"e" \ar@{=>}^{m_4} "f"
\POS"f" \ar@/_7ex/@{.>} "a"
\endxy
\]
having the same number of vertices and such that the $l$-tuples $(k_1,\hdots,k_l)$ and $(m_1,\hdots,m_l)$ are the same upto a permutation.
\end{theorem}
\Proof
We give two proofs of this result. By \cite{LBProcesi} the coordinate ring of a quotient variety $\wis{iss}_{\epsilonilon}~Q$ is generated by traces along oriented cycles in the quiver $Q$. In the case of the left hand quiver settings, these invariants are easy to determine : the dimension of the power $\mathfrak{m}^i$ of the maximal graded ideal $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ is equal to
\[
M_i = \binom{k_1+i-1}{i} \hdots \binom{k_n+i-1}{i} \]
whence $M_{i+1}/M_i = (i+1)^{-n}(i+k_1) \hdots (i+k_n)$. The rational function
\[
f(x) = \frac{(x+k_1)\hdots (x+k_n)}{(x+1)^n} \]
is determined by its values on all $x \in \mathbb{N}$ whence the dimension-sequence $M_{i+1}/M_i$ determines the roots and their multiplicity (note that none of the $k_i = 1$ as the quiver setting is reduced. From this the difficult part of the result follows.
Alternatively, the result follows from the fact that $\wis{iss}_{\epsilonilon}~Q$ is the cone on the projective variety $\mathbb{P}^{k_1-1} \times \hdots \times \mathbb{P}^{k_n-1}$.
\section{Applications}
A quiver gauge theory consists of a quiver setting $(Q,\alpha)$ together with the choice of a
necklace as in \cite{LBBocklandt} (a
{\em superpotential}) $W \in \wis{dR}^0_V~\mathbb{C} Q = \mathbb{C} Q / [\mathbb{C} Q,\mathbb{C} Q]$, that is,
\[
W = \sum_j a_{i_1} \hdots a_{i_{l_j}} \]
is a sum of oriented cycles in the quiver $Q$ with arrows say $\{ a_1,\hdots,a_l \}$. Such a necklace induces a $GL(\alpha)$ invariant polynomial function
\[
W~:~\wis{rep}_{\alpha}~Q \rTo \mathbb{C} \qquad V \mapsto \sum_j Tr(V_{a_{i_1}} \hdots V_{a_{i_{l_j}}})
\]
and hence a $GL_n$-invariant function on the $\alpha$-component $GL_n \times^{GL(\alpha)} \wis{rep}_{\alpha}~Q$ of $\wis{rep}_n~\mathbb{C} Q$. From the
differential
$\wis{dR}^0_V~\mathbb{C} Q \rTo^d \wis{dR}^1_V~\mathbb{C} Q$ we can define by \cite{LBBocklandt} {\em partial
differential
operators} associated to any arrow $a$ in $Q$ with start vertex $v_i$ and end vertex $v_j$
\[
\frac{\partial}{\partial a}~:~\wis{dR}^0_{V}~\mathbb{C} Q \rTo e_i \mathbb{C} Q e_j \qquad
\text{by} \qquad df = \sum_{a \in Q_a}~\frac{\partial f}{\partial a} d a \]
To take the partial derivative of a necklace word $w$ with respect to an
arrow $a$, we run through $w$ and each time we encounter $a$ we open the
necklace by removing
that occurrence of $a$ and then take the sum of all the paths obtained.
\begin{definition} The {\em vacualgebra} of a quiver gauge theory determined by the quiver setting
$(Q,\alpha)$ and the superpotential $W$ is the Noetherian affine $\mathbb{C}$-algebra
\[
\int_{\alpha}~\partial_Q~W \qquad \text{where} \qquad \partial_Q~W = \frac{\mathbb{C} Q}{(\frac{\partial W}{\partial a_1},\hdots,
\frac{\partial W}{\partial a_l})} \]
The affine variety $\wis{rep}_{\alpha}~\partial_Q~W$ is said to be the space of {\em vacua} and the algebraic quotient $\wis{iss}_{\alpha}~\partial_Q~W$ is called the {\em moduli of superpotential vacua}, see for example \cite{LutyTaylor}.
\end{definition}
In order to get realistic models, one has to impose additional conditions, for example that the
superpotential $W$ is cubic (meaning that every arrow in $Q$ must belong to at least one oriented cycle of length $\leq 3$) or that $\wis{iss}_{\alpha}~\partial_Q~W$ is three-dimensional, see for
example \cite{Berenstein1}, \cite{DouglasGreeneMorrison}.
Applications of Auslander regularity of $\int_{\alpha}~\partial_Q~W$ are well-documented in the literature (a.o. \cite[\S 3]{Berenstein1},\cite[\S 6]{Berenstein2} or \cite{Berenstein3}). Applications
of smoothness of $\int_{\alpha}~\partial_Q~W$ (that is, that the space of vacua $\wis{rep}_{\alpha}~\partial_Q~W$ is a smooth variety) are more implicit. In comparing the algebraic quotient with the moment map description (comparing $F$-terms to $D$-terms) or
defining a K\"ahler metric one can get by using the induced properties from smoothness of
$\wis{rep}_{\alpha}~\mathbb{C} Q$. However, in comparing geometrical properties (such as flips and flops)
of related moduli spaces
(of semi-stable representations for algebraists, adding Fayet-Iliopoulos terms for physicists) one sometimes uses the stronger results of \cite{Thaddeus} and \cite{Dolgatchev} for which smoothness of the total space is crucial. Ideally, one would like to have vacualgebras having both smoothness conditions.
\begin{example}~(The conifold algebra, see \cite{Berenstein4}) The relevant quiver setting
$(Q,\alpha)$ is of the form
\[
\xymatrix{1 \ar@/^3ex/[rr]_{\txt{$x_1$}} \ar@/^4ex/[rr]^{\txt{$x_2$}} && 1
\ar@/^3ex/[ll]_{\txt{$y_1$}} \ar@/^4ex/[ll]^{\txt{$y_2$}}} \]
and the necklace (superpotential) is taken to be
\[
W = \lambda((x_1y_2-x_2y_1)^2-(y_1x_2-y_2x_1)^2) \]
Therefore, the defining equations of $\partial_Q~W$ are (taking into account that $x_ix_j=0$ and
$y_iy_j = 0$)
\begin{eqnarray*}
\frac{\partial W}{\partial x_1} &= y_1x_2y_2-y_2x_2y_1= 0 \\
\frac{\partial W}{\partial x_2} &= y_2x_1y_1-y_1x_1y_2 = 0 \\
\frac{\partial W}{\partial y_1} &= x_2y_2x_1-x_1y_2x_2 = 0 \\
\frac{\partial W}{\partial y_2} &= x_1y_1x_2-x_2y_1x_1= 0
\end{eqnarray*}
Observe that these identities are satisfied for {\em all} representations of $\wis{rep}_{\alpha}~Q$
as $\alpha = (1,1)$ and therefore
\[
\wis{rep}_{\alpha}~\partial_Q~W = GL_2 \times^{\mathbb{C}^* \times \mathbb{C}^*} \wis{rep}_{\alpha}~Q \quad
\text{and} \quad \int_{\alpha}~\partial_Q~W = \int_{\alpha}~\mathbb{C} Q \]
and therefore the vacualgebra is a smooth order.
Moreover, the quotient variety $\wis{iss}_{\alpha}~Q$ is easily seen to be the conifold singularity as the ring of invariants is generated by the primitive oriented cycles
\[
x = x_1y_1 \quad y = x_2y_2 \quad u = x_1y_2 \quad v = x_2y_1 \]
which satisfy the relation $xy = uv$. Therefore, by the theorem the {\em conifold algebra}
$\int_{\alpha}~\partial_W~Q$ is also Auslander regular. One can also check immediately that the
description of the conifold algebra given in \cite[\S 1]{Berenstein4} is the algebra of equivariant maps from $\wis{rep}_{\alpha}~Q$ to $M_2(\mathbb{C})$ (or to be more precise, if we take the relevant gauge groups into account, a ring Morita equivalent to the conifold algebra).
\end{example}
\par \vskip 3mm
By the results of \cite{BockLBVdW} we know that the only type of singularity that can occur in the center of a smooth order (in dimension three) is the conifold singularity. Therefore, in most models considered by physicists, see a.o. \cite{Berenstein1}, \cite{Greene} or \cite{Sardo} the space of superpotential vacua $\wis{rep}_{\alpha}~\partial_Q~W$ must contain singularities as the moduli space is a three dimensional quotient variety (different from the conifold) or has a one-dimensional family of singularities (which cannot happen for a three dimensional smooth order).
There is a standard way to remove (most of) the singularities in $\wis{rep}_{\alpha}~\partial_Q~W$
by restricting to semistable representations. Let us quickly run through the process. For a dimension vector $\alpha = (v_1,\hdots,v_k) \in \mathbb{N}^k$ let $U(\alpha)$ be the quotient of the
Lie group $U(v_1) \times \hdots \times U(v_k)$ by the one-dimensional central subgroup
$U(1)(1_{v_1},\hdots,1_{v_k})$. The {\em real moment map} for $\alpha$-dimensional
quiver representations of $Q$ is the map
\[
\wis{rep}_{\alpha}~Q \rTo^{\mu_{\mathbb{R}}} \wis{Lie}~U(\alpha) \qquad V \mapsto \frac{i}{2} \sum_{j=1}^l
[V_{a_j},V_{a_j}^{\dagger}] \]
There is a natural one-to-one correspondence (actually a homeomorphism)
\[
\wis{iss}_{\alpha}~Q \leftrightarrow \mu^{-1}_{\mathbb{R}}(0)/U(\alpha) \]
Let $\mu = (u_1,\hdots,u_k) \in \mathbb{Q}^k$ such that $\mu.\alpha = \sum u_iv_i = 0$, then we say that a representation $V \in \wis{rep}_{\alpha}~Q$ is $\mu$-semistable (resp.
$\mu$-stable) if for all proper subrepresentations $W \subset V$ we have that $\mu.\beta \geq 0$
(resp. $\mu.\beta > 0$) where $\beta$ is the dimension vector of $W$. If $\wis{rep}_{\alpha}^{\mu}~Q$ denotes the Zariski open set (possibly empty) of $\mu$-semistable representations of $\wis{rep}_{\alpha}~Q$, then the geometric invariant quotient
\[
\wis{moduli}^{\mu}_{\alpha}~Q = \wis{rep}_{\alpha}^{\mu}~Q // GL(\alpha) \rOnto \wis{iss}_{\alpha}~Q \]
classifies the isomorphism classes of $\alpha$-dimensional direct sums of $\mu$-stable representations and is a projective bundle over $\wis{iss}_{\alpha}~Q$. Moreover, there is a moment map description of this moduli space
\[
\wis{moduli}^{\mu}_{\alpha}~Q = \mu^{-1}_{\mathbb{R}}(\mu) / U(\alpha) \]
For more details on these matters we refer to \cite{King}.
If $I_W$ denotes the set of zeroes of the ideal of relations of $\mathbb{C}[\wis{rep}_{\alpha}~Q]$ imposed by the defining relations of $\partial_Q~W$, then
\[
\wis{iss}_{\alpha}~\partial_Q~W = (\mu^{-1}_{\mathbb{R}}(0) \cap I_W) / U(\alpha) \]
and the geometric invariant quotient of the open set of $\mu$-stable representations of $\partial_Q~W$ is a projective bundle over it
\[
\wis{moduli}^{\mu}_{\alpha}~\partial_Q~W = (\mu_{\mathbb{R}}^{-1}(\mu) \cap I_W) / U(\alpha) \rOnto
\wis{iss}_{\alpha}~\partial_Q~W \]
The moduli spaces of $\mu$-semistable representations can be covered by open sets determined by determinantal semi-invariants and consequently we can define a sheaf of noncommutative orders
\[
{\mathcal O}^{\mu}_{\partial_Q~W} \qquad \text{over} \quad \wis{moduli}^{\mu}_{\alpha}~\partial_Q~W
\]
which locally is isomorphic to $\wis{iss}_{\alpha}~A$ of a suitable algebra $A$.
In favorable situations, $\wis{rep}_{\alpha}^{\mu}~\partial_Q~W$ will be a smooth variety and the moduli space $\wis{moduli}^{\mu}_{\alpha}~\partial_Q~W$ will be a (partial) desingularization
of $\wis{iss}_{\alpha}~\partial_Q~W$. In physical terminology this process is described as 'adding a Fayet-Iliopoulos term'. An immediate consequence of the theorem then implies :
\begin{proposition} With notations as before, if $\wis{rep}_{\alpha}^{\mu}~\partial_Q~W$ is a smooth variety and if the partial desingularization $\wis{moduli}_{\alpha}^{\mu}~\partial_Q~W$ has isolated singularities as its non-Azumaya locus, then ${\mathcal O}^{\mu}_{\partial_Q~W}$ is a sheaf of Auslander regular orders over $\wis{moduli}^{\mu}_{\alpha}~\partial_Q~W$.
\end{proposition}
In physical relevant settings, the resolution $\wis{moduli}^{\mu}_{\alpha}~\partial_Q~W \rOnto
\wis{iss}_{\alpha}~\partial_Q~W$ will often be crepant meaning that the moduli space is a Calabi-Yau manifold and the sheaf of orders will be an Azumaya sheaf. However, there may be relevant situations where we only have a partial desingularization, the remaining singularities are necessarily of conifold type and the sheaf of orders is locally Morita equivalent to the conifold algebra in the singularities. This explains the importance of conifold transitions in (partial) resolutions of three dimensional quotient singularities.
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\end{document} | math | 37,597 |
\begin{document}
\begin{abstract}
A regular tree language L is locally testable if membership of a tree
in L depends only on the presence or absence of some fix set of neighborhoods in the
tree. In this paper we show that it is decidable whether a regular tree
language is locally testable. The decidability is shown for ranked trees and
for unranked unordered trees.
\end{abstract}
\title{A decidable characterization of locally testable tree languages}
\section{Introduction}
This paper is part of a general program trying to understand the expressive
power of first-order logic over trees. We say that a class of regular tree
languages has a decidable characterization if the following problem is
decidable: given as input a finite tree automaton, decide if the recognized
language belongs to the class in question. Usually a decision algorithm
requires a solid understanding of the expressive power of the corresponding
class and is therefore useful in any context where a precise boundary of this
expressive power is crucial. In particular we do not possess yet a decidable
characterization of the tree languages definable in FO($\leq$), the first-order
logic using a binary predicate $\leq$ for the ancestor relation.
We consider here the class of tree languages definable in a fragment of
FO($\leq$) known as \emph{Locally Testable} (LT). A language is in LT if
membership in the language depends only on the presence or absence of
neighborhoods of a certain size in the tree. A closely related family of
languages is the class LTT of \emph{Locally Threshold Testable} languages.
Membership in such languages is determined by counting the number of
neighborhoods of a certain size up to some threshold. The class LT is the
special case where no counting is done, the threshold is 1. In this paper we
provide a decidable characterization of the class LT over trees.
The standard approach for deriving a decidable characterization is to first
exhibit a set of closure properties that hold exactly for the languages in the
class under investigation and then show that these closure properties can be
automatically tested. This requires a formalism for expressing the desired
closure properties but also some tools, typically induction mechanisms, for
proving that the properties do characterize the class, and for proving the
decidability of those properties.
Over words one formalism turned out to be successful for characterizing many
classes of regular languages. The closure properties are expressed as
identities on the syntactic monoid or syntactic semigroup of the regular
language. The syntactic monoid or syntactic semigroup of a regular language is the
transition monoid of its minimal deterministic automaton including or not the
transition induced by the empty word. For instance the class of word languages definable in
FO$(\leq)$ is characterized by the fact that the syntactic monoid of any such
languages is aperiodic. The latter property corresponds to the identity
$x^\omega=x^{\omega+1}$ where $\omega$ is the size of the monoid. This
equation is easily verifiable automatically on the syntactic monoid.
Similarly, the classes LTT and LT have been characterized using decidable
identities on the syntactic semigroup~\cite{BS73,McN74,BP89,TW85}.
Over trees the situation is more complex and right now there is no formalism
that can easily express all the known closure properties for the classes for
which we have a decidable characterization. The most successful formalism is
certainly the one introduced in~\cite{forestalgebra} known as forest
algebras. For instance, these forest algebras were used for obtaining decidable
characterizations for the classes of tree languages definable in
EF+EX~\cite{EFEX}, EF$+$F${^{-1}}$~\cite{Boj-utl,place-csl08},
BC-$\Sigma_1(<)$~\cite{luclics08,place-csl08},
$\Delta_2(\leq)$~\cite{lucicalp08,place-csl08}. However it is not clear yet
how to use forest algebras in a simple way for characterizing the class LTT
over trees and a different formalism was used for obtaining a decidable
characterization for this class~\cite{BS09}.
We were not able to obtain a reasonable set of identities for LT either by using
forest algebras or the formalism used for characterizing LTT. Our approach is
slightly different.
There is another technique that was used on words for deciding the class LT.
It is based on the ``delay theorem''~\cite{Str85,Tilson} for computing the
required size of the neighborhoods: Given an automaton recognizing the language
$L$, a number $k$ can be computed from that automaton such that if $L$ is in LT
then it is in LT by investigating the neighborhoods of size~$k$. Once this $k$
is available, deciding whether $L$ is indeed in LT or not is a simple
exercise. On words, a decision algorithm for LT (and also for LTT) has been
obtained successfully using this approach~\cite{Boj07}. Unfortunately all
efforts to prove a similar delay theorem on trees have failed so far.
We obtain a decidable characterization of LT by combining the two approaches
mentioned above. We first exhibit a set of necessary conditions
for a regular tree language to be in LT. Those conditions are expressed using
the formalism introduced for characterizing LTT. We then show that for
languages satisfying such conditions one can compute the required size of the
neighborhoods. Using this technique we obtain a characterization of LT for
ranked trees and for unranked unordered trees.
\paragraph{\bf Other related work.}
There exist several formalisms that have been used for expressing identities
corresponding to several classes of languages but not in a decidable way. Among
them let us mention the notion of preclones introduced in~\cite{preclones} as
it is close to the one we use in this paper for expressing our necessary
conditions.
Finally we mention the class of frontier testable languages, not expressible in
FO$(<)$, that was given a decidable characterization using a specific
formalism~\cite{Wil96}.
\paragraph{\bf Organization of the paper.}
We start with ranked trees and give the necessary notations and preliminary
results in Section~\ref{section-notation}. Section~\ref{section-necessary}
exhibits several conditions and proves they are necessary for
being in LT. In Section~\ref{section-char} we show that for the languages
satisfying the necessary conditions the required size of the neighborhoods can
be computed, hence concluding the decidability of the characterization. Finally
in Section~\ref{section-unranked} we show how our result extends to unranked
trees.
\section{Notations and preliminaries}\label{section-notation}
We first investigate the case of binary trees. The case of unranked
unordered trees will be considered in Section~\ref{section-unranked}.
\paragraph{\bf Trees.}
We fix a finite alphabet $\Sigma$, and consider finite binary trees with labels
in $\Sigma$. All the results presented here extend to arbitrary ranks in a
straightforward way. In the binary case, each node of the tree is either
\emph{a leaf} (has no children) or has exactly two \emph{children}, the left
child and the right child. We use the standard terminology for trees. For
instance by the \emph{descendant} (resp. ancestor) relation we mean the
reflexive transitive closure of the child (resp. inverse of child) relation and
by \emph{distance} between two nodes we refer to the length of the shortest
path between the two nodes. A
\emph{language} is a set of trees.
Given a tree $t$ and a node $x$ of $t$ the \emph{subtree of $t$ rooted at $x$},
consisting of all the nodes of $t$ that are descendant of $x$, is denoted by
\subtree{t}{x}. A \emph{context} is a tree with a designated (unlabeled) leaf
called its {\it port} which acts as a hole. Given contexts $C$ and $C'$, their
concatenation $C \cdot C'$ is the context formed by identifying the root of
$C'$ with the port of $C$. A tree $C \cdot t$ can be obtained similarly by
combining a context $C$ and a tree $t$. Given a tree $t$ and two nodes $x,y$
of $t$ such that $y$ is a descendant (not necessarily strict) of $x$, \emph{the
context of $t$ between $x$ and $y$}, denoted by $\context{t}{x}{y}$, is
defined by keeping all the nodes of $t$ that are descendants of $x$ but not
descendants of $y$ and by placing the port at $y$.
We say that a context $C$ \emph{occurs} in $t$ if $C$ is the context of $t$
between $x$ and $y$ for some nodes $x$ and $y$ of $t$.
\paragraph{\bf Types.}
Let $t$ be a tree and $x$ be a node of $t$ and $k$ be a positive integer, the
{\em\ktype} of $x$ is the (isomorphism type of the) restriction of
\subtree{t}{x} to the set of nodes of $t$ at distance at most $k$ from $x$.
When $k$ will be clear from the context we will simply say \emph{type}. A
\ktype $\tau$ \emph{occurs} in a tree $t$ if there exists a node of $t$ of type
$\tau$. If $C$ is the context \context{t}{x}{y} for some tree $t$ and some
nodes $x,y$ of $t$, then the \ktype of a
node of $C$ is the \ktype of the corresponding node in $t$. Notice that the
\ktype of a node of $C$ depends on the surrounding tree $t$, in particular the
port of $C$ has a \ktype, the one of $y$ in $t$.
Given two trees $t$ and $t'$ we denote by \lessblocksk{t}{t'} the fact that all
\ktypes that occur in $t$ also occur in $t'$. Similarly we can speak of
\lessblocksk{t}{C} when $t$ is a tree and $C$ is \context{t'}{x}{y} for some
tree $t'$ and some nodes $x,y$ of $t'$. We denote by \sameblocksk{t}{t'} the
property that the root of $t$ and the root of $t'$ have the same \ktype and $t$
and $t'$ agree on their \ktypes: \lessblocksk{t}{t'} and \lessblocksk{t'}{t}.
Note that when $k$ is fixed the number of \ktypes is finite and hence the
equivalence relation \sameblockequiv{k} has a finite number of equivalence
classes. This property is no longer true for unranked trees and this is why we
will have to use a different technique for this case.
A language $L$ is said to be \testable{\kappa} if $L$ is a union of equivalence
classes of \sameblockequiv{\kappa}. A language is said to be \emph{locally
testable} (is in LT) if there is a $\kappa$ such that it is
\testable{\kappa}. In other words, in order to test whether a tree $t$
belongs to $L$ it is enough to check for the presence or absence of
\types{\kappa} in $t$, for some big enough $\kappa$.
\paragraph{\bf Regular Languages.} We assume familiarity with tree automata and
regular tree languages. The interested reader is referred to~\cite{tata} for
more details. Their precise definitions are not important in order to
understand our characterization. However pumping arguments will be used in the
decision algorithms.
\paragraph{\bf The problem.}
We want an algorithm deciding if a given regular language is in LT. When the
complexity is not an issue, we can assume that the language $L$ is given
as a MSO formula. Another option would be to start with a bottom-up tree
automaton for $L$ or, even better, the minimal deterministic bottom-up tree
automaton that recognize $L$. We will come back to the complexity issues in
Section~\ref{section-complexity}.
The main difficulty is to compute a bound on
$\kappa$, the size of the neighborhood, whenever such a $\kappa$ exists.
The word case is a special case of the tree case as it corresponds to trees of
rank~1. A decision procedure for LT was obtained in the word case independently
by~\cite{BS73} and ~\cite{McN74}. A language $L$ is in LT if and only if its
syntactic semigroup satisifies the equations $exe=exexe$ and $exeye=eyexe$,
where $e$ is an arbitrary idempotent ($ee=e$) while $x$ and $y$ are arbitrary
elements of the semigroup. The equations are then easily verified after
computing the syntactic semigroup.
In the case of trees, we were not able to obtain a reasonably simple set of
identities for characterizing LT. Nevertheless we can show:
\begin{thm}\label{main-theorem}
It is decidable whether a regular tree language is in LT.
\end{thm}
Our strategy for proving Theorem~\ref{main-theorem} is as follows. In a first
step we provide necessary conditions for a language to be in LT. In a second
step we show that if a language $L$ verifies those necessary conditions then we
can compute from an automaton recognizing $L$ a number $\kappa$ such that if
$L$ is in LT then $L$ is \testable{\kappa}. The last step is simple and show
that once $\kappa$ is fixed, it is decidable whether a regular language is
\testable{\kappa}. This last step follows immediately from the fact that once
$\kappa$ is fixed, there are only finitely many \testable{\kappa}
languages and hence one can enumerate them and test whether $L$ is
equivalent to one of them or not.
Given a regular language $L$, testing whether $L$ is in LT
is then done as follows: (1) compute from $L$ the $\kappa$ of the second step and (2)
test whether $L$ is \testable{\kappa} using the third step. The first step
implies that this algorithm is correct.
Before starting providing the proof details we note that there exist examples
showing that the necessary conditions are not sufficient. Such an example will
be provided in Section~\ref{section-nonsuff}. We also note that the problem of
finding $\kappa$ whenever such a $\kappa$ exists is a special case of the delay
theorem mentioned in the introduction. When applied to LT, the delay theorem
says that if a finite state automaton $A$ recognizes a language in LT then this
language must be \testable{\kappa} for a $\kappa$ computable from $A$. The
delay theorem was proved over words in~\cite{Str85} and can be used in order to
decide whether a regular language is in LT as explained in~\cite{Boj07}. We
were not able to prove such a general theorem for trees.
\section{Necessary conditions}\label{section-necessary}
In this section we exhibit necessary conditions for a regular language to be in
LT. These conditions will play a crucial role in our decision algorithm. These
conditions are expressed using the same formalism as the one used
in~\cite{BS09} for characterizing LTT.
\paragraph{\bf Guarded operations.}
Let $t$ be a tree, and $x,x'$ be two nodes of $t$ such that $x$ and $x'$ are
not related by the descendant relationship. The \emph{horizontal swap} of $t$
at nodes $x$ and $x'$ is the tree $t'$ constructed from $t$ by replacing
\subtree{t}{x} with \subtree{t}{x'} and vice-versa, see Figure~\ref{fig-H-swap}
(left). A horizontal swap is said to be \emph{$k$-guarded} if $x$ and $x'$ have
the same \ktype.
Let $t$ be a tree and $x,y,z$ be three nodes of $t$ such that $x,y,z$ are not
related by the descendant relationship and such that
$\subtree{t}{x}=\subtree{t}{y}$. The \emph{horizontal transfer} of $t$ at
$x,y,z$ is the tree $t'$ constructed from $t$ by replacing \subtree{t}{y} with
a copy of \subtree{t}{z}, see Figure~\ref{fig-H-swap} (right). A horizontal
transfer is $k$-guarded if $x,y,z$ have the same \ktype.
\begin{figure}
\caption{Horizontal Swap (left) and Horizontal Transfer (right)}
\label{fig-H-swap}
\end{figure}
Let $t$ be a tree of root $a$, and $x,y,z$ be three nodes of $t$ such that $y$
is a descendant of $x$ and $z$ is a descendant of $y$. The \emph{vertical
swap} of $t$ at $x,y,z$ is the tree $t'$ constructed from $t$ by swapping the
context between $x$ and $y$ with the context between $y$ and $z$, see
Figure~\ref{fig-V-stutter} (left). More formally let $C=\context{t}{a}{x}$,
$\Delta_1=\context{t}{x}{y}$, $\Delta_2=\context{t}{y}{z}$ and
$T=\subtree{t}{z}$. We then have $t=C \cdot \Delta_1 \cdot \Delta_2 \cdot
T$. The tree $t'$ is defined as $t'=C \cdot \Delta_2 \cdot \Delta_1 \cdot T$. A
vertical swap is \emph{$k$-guarded} if $x,y,z$ have the same \ktype.
Let $t$ be a tree of root $a$, and $x,y,z$ be three nodes of $t$ such that $y$
is a descendant of $x$ and $z$ is a descendant of $y$ such that
$\Delta=\context{t}{x}{y}=\context{t}{y}{z}$. The \emph{vertical
stutter} of $t$ at $x,y,z$ is the tree $t'$ constructed from $t$ by
removing the context between $x$ and $y$, see Figure~\ref{fig-V-stutter}
(right). A vertical stutter is $k$-guarded if $x,y,z$ have the same \ktype.
\begin{figure}
\caption{Vertical Swap (left) and Vertical Stutter (right)}
\label{fig-V-stutter}
\end{figure}
Let $L$ be a tree language and $k$ be a number. If X is any of the four
constructions above, horizontal or vertical swap, or vertical stutter or
horizontal transfer, we say that $L$ is \emph{closed under $k$-guarded X} if
for every tree $t$ and every tree $t'$ constructed from $t$ using $k$-guarded X
then $t$ is in $L$ iff $t'$ is in $L$. Notice that being closed under
$k$-guarded X implies being closed under $k'$-guarded X for $k' > k$. An
important observation is that each of the $k$-guarded operation does not
affect the set of \types{(k+1)} occurring in the trees.
If $L$ is closed under all the $k$-guarded operations described above, we say
that $L$ is \emph{\ktame.} A language is said to be \emph{\tame} if it is
\ktame for some $k$.
The following simple result shows that \tameness is a necessary condition for
LT.
\begin{prop}\label{prop-necessary}
If $L$ is in LT then $L$ is \tame.
\end{prop}
\proof Assume $L$ is in LT. Then there is a $\kappa$ such that $L$ is
\testable{\kappa}. We show that $L$ is $\kappa$-\tame. This is a
straightforward consequence of the fact that all the $\kappa$-guarded
operations above preserve \types{(\kappa+1)} and hence preserve
\types{\kappa}.\qed
A simple pumping argument shows that if $L$ is \tame then it is $k$-\tame for
$k$ bounded by a polynomial in the size of the minimal deterministic bottom-up
tree automaton recognizing $L$.
\begin{prop}\label{nec-condition-decision}
Given a regular language $L$ and $A$ the minimal deterministic bottom-up
tree automaton recognizing $L$, we have $L$ is \tame iff $L$ is $k_0$-\tame
for $k_0 = |A|^3+1$.
\end{prop}
\begin{proof}
We prove that if $X$ is one of the four operations that defines \tameness,
then if $L$ is closed under $k$-guarded $X$ for $k > k_0$, then $L$ is closed
under $k_0$-guarded $X$. This will imply that if $L$ is \ktame then it is
$k_0$-\tame.
Consider the case of $k$-guarded horizontal transfer and assume $L$ is closed
under $k$-guarded horizontal transfers. We show that $L$ is closed under
$k_0$-guarded horizontal transfers. Let $t$ be a tree and $x,y,z$ three nodes
of $t$ having the same \type{k_0} and not related by the descendant
relation such that $\subtree{t}{x}=\subtree{t}{y}$. We need to show that
replacing \subtree{t}{y} by a copy of \subtree{t}{z} does not affect
membership in $L$.
We do this in three steps, first we transform $t$ by pumping in parallel in
the subtrees of $x,y$ and $z$ until $x,y,z$ have the same \ktype, then we use
the closure of $L$ under $k$-guarded horizontal transfer in order to replace
\subtree{t}{y} by a copy of \subtree{t}{z}, and finally we backtrack the
initial pumping phase in order to recover the initial subtrees.
We let $t_1=\subtree{t}{x}$ and $t_2=\subtree{t}{z}$ and we
assume for now on that $t_1 \neq t_2$. By \emph{position} we denote a string
$w$ of $\set{0,1}^*$. A position $w$ is \emph{realized} in a tree $t$ if
there is a node $x$ of $t$ such that if $x_1,\cdots,x_n=x$ is the sequence of
nodes in the path from the root of $t$ to $x$ then for all $i\leq n$ the
$i^{th}$ bit of $w$ is zero if $x_i$ is a left child and it is one if $x_i$
is a right child. We order positions by first comparing their respective
length and then using the lexicographical order.
By hypothesis $t_1$ and $t_2$ are identical up to depth at least $k_0$. Let
$w$ be the first position such that $t_1$ and $t_2$ differ at that position.
That can be either because $w$ is realized in $t_1$ but not in $t_2$, or vice
versa, or $w$ is realized in both trees but the labels of the corresponding
nodes differ. We know that the length $n$ of $w$ is strictly greater than
$k_0$. If $n >k$, we are done with the first phase. We assume now that $n
\leq k$.
Consider the run $r$ of $A$ on $t$. The run assigns a state $q$ to each node
of $t$. From $r$ we assign to each position $w' < w$ a pair of states
$(q,q')$ such that $q$ is the state given by $r$ at the corresponding node in
$t_1$ while $q'$ is the state given by $r$ at the corresponding node in
$t_2$. Because $n > k_0 > |A|^2$, there must be two prefixes $w_1$ and $w_2$
of $w$ that were assigned the same pair of states. Consider the context
$C_1=t_1[v_1,v_2]$ where $v$ and $v'$ are the nodes of $t_1$ at position
$w_1$ and $w_2$ and the context $C_2=t_2[v'_1,v'_2]$ where $v$ and $v'$ are
the nodes of $t_2$ at position $w_1$ and $w_2$. Without affecting membership
in $L$, we can therefore at the same time duplicate $C_1$ in the two copies of $t_1$
rooted at $x$ and $y$ and $C_2$ in the copy of $t_2$ rooted at $z$.
Let $t'_1$ and $t'_2$ be the subtrees of the resulting tree, rooted
respectively at $x$ and $z$. The reader can verify that $t'_1$ and $t'_2$ now
differ at a position strictly greater than $w$.
Performing this repeatedly, we eventually arrive at a situation where the
subtree $t'_1$ rooted at $x$ and $y$ agree up to depth $k$ with the subtree
rooted at $z$. We can now apply $k$-guarded horizontal transfer and replace
one occurrence of $t'_1$ by a copy of $t'_2$. We can then replace $t'_1$ by
$t_1$ and both copies of $t'_2$ by $t_2$ without affecting membership in $L$.
The other operations are done similarly. For the horizontal swap, we pump
the subtrees at positions $x$ and $x'$ simultaneously, which is possible
because $k_0 > |A|^2$. For vertical swap, we pump the subtrees at
the positions $x$, $y$ and $z$ simultaneously, and that requires $k_0 >
|A|^3$. Finally, for vertical stutter, we pump the subtrees at the
positions $x$, $y$ and $z$ simultaneously, which again requires $k_0 > |A|^3$.
\end{proof}
Once $k$ is fixed, a brute force algorithm can check whether $L$ is \ktame or
not. Indeed, as $L$ is regular, when testing for closure under $k$-guarded $X$,
it is enough to consider all relevant states and appropriate transition functions of the
automata instead of all trees and all contexts. See for instance Lemma~12 and
Lemma~13 in~\cite{BS09}.
Therefore Proposition~\ref{nec-condition-decision} implies that \tameness is
decidable. However for deciding LT we will only need the bound on $k_0$ given by
the proposition.
\section{Deciding LT}\label{section-char}
In this section we show that it is decidable whether a regular tree language is
in LT. This is done by showing that if a regular language $L$ is in LT then
there is a $\kappa$ computable from an automaton recognizing $L$ such that $L$
is in fact \testable{\kappa}. Recall that once this $\kappa$ is computed the
decision procedure simply enumerates all the finitely many \testable{\kappa}
languages and tests whether $L$ is one of them.
Assume $L$ is in LT. By Proposition~\ref{prop-necessary}, $L$ is \tame. Even
more, from Proposition~\ref{nec-condition-decision}, one can effectively compute
a $k$ such that $L$ is \ktame. Hence Theorem~\ref{main-theorem} follows from the
following proposition.
\begin{prop}\label{prop-nec-implies-LT}
Assume $L$ is a \ktame regular tree language then $L$ is in LT iff $L$ is
\testable{\kappa} where $\kappa$ is computable from $k$.
\end{prop}
Recall that for each $k$ the number of \ktypes is finite. Let $\beta_k$ be this
number. Proposition~\ref{prop-nec-implies-LT} is an immediate consequence of
the following proposition.
\begin{prop}\label{lemma-pumping}
Let $L$ be a \ktame regular tree language. Set $\kappa=\beta_k + k + 1$.
Then for all $l>\kappa$ and any two trees $t,t'$ if
\sameblocks{t}{t'}{\kappa} then there exist two trees $T,T'$ with
\begin{enumerate}[\em(1)]
\item $t \in L\ $ ~iff~ $\ T \in L$
\item $t' \in L\ $ ~iff~ $\ T' \in L$
\item \sameblocks{T}{T'}{l}
\end{enumerate}
\end{prop}
\begin{proof}[Proof of Proposition~\ref{prop-nec-implies-LT} using
Proposition~\ref{lemma-pumping}]
Assume $L$ is \ktame and let $\kappa$ be defined as in
Proposition~\ref{lemma-pumping}. We show that $L$ is in LT iff $L$ is
\testable{\kappa}. Assume $L$ is in LT. Then $L$ is \testable{l} for some $l
\in \mathbb{N}$. We show that $L$ is actually \testable{\kappa}. For this it
suffices to show that for any pair of trees $t$ and $t'$, if
\sameblocks{t}{t'}{\kappa} then $t \in L$ iff $t' \in L$. Let $T$ and $T'$ be
the trees constructed for $l$ from $t$ and $t'$ by
Proposition~\ref{lemma-pumping}. We have \sameblocks{T}{T'}{l} and therefore
$T \in L$ iff $T' \in L$. As we also have $t \in L$ iff $T \in L$ and $t' \in
L$ iff $T' \in L$, the proposition is proved.
\end{proof}
Before proving Proposition~\ref{lemma-pumping} we need some extra terminology.
A non-empty context $C$ occurring in a tree $t$ is a \emph{loop of \ktype
$\tau$} if the \ktype of its root and the \ktype of its port is $\tau$. A
non-empty context $C$ occurring in a tree $t$ is a \kloop if there is some
\ktype $\tau$ such that $C$ is a loop of \ktype $\tau$. Given a context $C$ we call the
path from the root of $C$ to its port the \emph{principal path of
$C$}. Finally, the result of the \emph{insertion} of a \kloop $C$ at a node
$x$ of a tree $t$ is a tree $T$ such that if $t=D \cdot \subtree{t}{x}$ then
$T=D\cdot C \cdot \subtree{t}{x}$. Typically an insertion will occur only when
the \ktype of $x$ is $\tau$ and $C$ is a loop of \ktype $\tau$. In this case
the \ktypes of the nodes initially from $t$ and of the nodes of $C$ are unchanged by this operation.
\begin{proof}[Proof of Proposition~\ref{lemma-pumping}]
Suppose that $L$ is \ktame. We start by proving two lemmas that will be
useful in the construction of $T$ and $T'$. Essentially these lemmas show
that even though being \ktame does not imply being \testable{(k+1)} (recall
the remark after Theorem~\ref{main-theorem}) some of the expected behavior of
\testable{(k+1)} languages can still be derived from being \ktame. The first
lemma shows that given a tree $t$, without affecting membership in $L$, we
can replace a subtree of $t$ containing only \types{(k+1)} occurring elsewhere
in $t$ by any other subtree satisfying this property and having the same
\ktype as root. The second lemma shows the same result for contexts by
showing that a \kloop can be inserted in a tree $t$ without affecting
membership in $L$ as soon as all the \types{(k+1)} of the \kloop were
already present in $t$. After proving these lemmas we will see how to
combine them for constructing $T$ and $T'$.
\begin{lem}\label{claim-transfer-branch}
Assume $L$ is \ktame. Let $t=Ds$ be a tree where $s$ is a subtree of $t$.
Let $s'$ be another tree such that the roots of $s$ and $s'$ have the same
\ktype.
If \lessblocks{s}{D}{k+1} and \lessblocks{s'}{D}{k+1} then
$Ds\in L$ iff $Ds'\in L$.
\end{lem}
\begin{proof}
We start by proving a special case of the Lemma when $s'$ is actually another
subtree of $t$. We will use repeatedly this particular case in the proof.
\begin{claim} \label{claim-transfer-enhanced} Assume $L$ is \ktame. Let $t$
be a tree and let $x,y$ be two nodes of $t$ not related by the descendant
relationship and with the same \ktype. We write $s = \subtree{t}{x}$,
$s' = \subtree{t}{y}$ and $C$ the context such that $t = Cs$.
If $\lessblocks{s}{C}{k+1}$ then $Cs \in L$ iff $Cs' \in L$.
\end{claim}
\begin{proof} The proof is done by induction on the depth of $s$ and makes
crucial use of $k$-guarded horizontal transfer.
Assume first that $s$ is of depth less than $k$. Since $x$ and $y$ have the same
\ktype, we have $s = s'$ and the result follows.
Assume now that $s$ is of depth greater than $k$.
Let $\tau$ be the \type{(k+1)} of $x$. We assume that $s$ is a tree of the form
$a(s_1,s_2)$. Notice that the \ktype of the roots of $s_1$ and $s_2$ are completely determined
by $\tau$. Since $\lessblocks{s}{C}{k+1}$, there exists a node $z$ in $C$ of
type $\tau$. We write $s'' = \subtree{t}{z}$.
We consider several cases depending on the relationship between $x$, $y$ and $z$.
We first consider the case where $x$ and $z$ are not related by the descendant
relationship, then we reduce the other cases to this case.
Assume that $x$ and $z$ are not related by the descendant relationship. Since
$s''$ is of type $\tau$, it is of the form $a(s''_1,s''_2)$ where the roots
of $s''_1$ and $s''_2$ have the same \ktype as respectively the roots of
$s_1$ and $s_2$. By hypothesis all the \types{(k+1)} of $s_1$ and $s_2$
already appear in $C$ and hence we can apply the induction hypothesis to
replace $s_1$ by $s''_1$ and $s_2$ by $s''_2$ without affecting membership
in $L$. Notice that the resulting tree is $Cs''$, that $t=Cs \in L$ iff
$Cs'' \in L$, and that $Cs''$ contains two copies of the subtree $s''$, one
at position $x$ and one at position $z$. We now show that we can derive
$Cs'$ from $Cs''$ using $k$-guarded operations. Since $L$ is \ktame it will
follow that that $Cs'' \in L$ iff $Cs' \in L$ and thus $Cs \in L$ iff $Cs' \in L$.
Let $t''=Cs''$ and we distinguish between three cases depending on the relationship between $z$ and $y$ in
$t''$:
\tikzstyle{arr} = [line width=4pt, ->]
\tikzstyle{bag}=[minimum size=20pt,inner sep=0pt]
\tikzstyle{dot}=[draw,circle,fill,minimum size=4pt,inner sep=0pt]
\begin{enumerate}[(1)]
\item If $z$ is a descendant of $y$, let $D=t''[y,z]$ and notice that $s'=Ds''$. Since $x$, $y$ and $z$ have
the same \ktype, we use $k$-guarded vertical stutter to duplicate $D$ and a
$k$-guarded horizontal swap to move the new copy of $D$ at position $x$ (see
the picture below). The resulting tree is $Cs'$ as desired.
\begin{center}
\begin{tikzpicture}
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (1.7,1.2) -- (1.2,0.6) -- (2.2,0.6) -- (1.7,1.2);
\node[dot] at (1.7,1.2) {};
\node[bag] at (1.7,0.8) {$s''$};
\node[bag] at (1.4,1.0) {$x$};
\draw (3.3,1.2) -- (2.8,0.6) -- (3.8,0.6) -- (3.3,1.2);
\node[dot] at (3.3,1.2) {};
\draw (3.3,0.6) -- (2.8,0) -- (3.8,0) -- (3.3,0.6);
\node[dot] at (3.3,0.6) {};
\node[bag] at (3.3,0.8) {$D$};
\node[bag] at (3.3,0.2) {$s''$};
\node[bag] at (3.0,1.0) {$y$};
\node[bag] at (3.0,0.4) {$z$};
\node[bag] at (4.5,1.2) {$\Longrightarrow$};
\node[bag] at (4.5,1.8) {\footnotesize Vertical};
\node[bag] at (4.5,1.5) {\footnotesize Stutter};
\draw (6.5,2) -- (5.2,1.2) -- (7.8,1.2) -- (6.5,2);
\draw (5.7,1.2) -- (5.2,0.6) -- (6.2,0.6) -- (5.7,1.2);
\node[dot] at (5.7,1.2) {};
\node[bag] at (5.7,0.8) {$s''$};
\node[bag] at (5.4,1.0) {$x$};
\draw (7.3,1.2) -- (6.8,0.6) -- (7.8,0.6) -- (7.3,1.2);
\node[dot] at (7.3,1.2) {};
\draw (7.3,0.6) -- (6.8,0) -- (7.8,0) -- (7.3,0.6);
\node[dot] at (7.3,0.6) {};
\draw (7.3,0) -- (6.8,-0.6) -- (7.8,-0.6) -- (7.3,0);
\node[dot] at (7.3,0) {};
\node[bag] at (7.3,0.8) {$D$};
\node[bag] at (7.3,0.2) {$D$};
\node[bag] at (7.3,-0.4) {$s''$};
\node[bag] at (7.0,1.0) {$y$};
\node[bag] at (7.0,-0.2) {$z$};
\node[bag] at (8.5,1.2) {$\Longrightarrow$};
\node[bag] at (8.5,1.8) {\footnotesize Horizontal};
\node[bag] at (8.5,1.5) {\footnotesize Swap};
\draw (10.5,2) -- (9.2,1.2) -- (11.8,1.2) -- (10.5,2);
\draw (9.7,1.2) -- (9.2,0.6) -- (10.2,0.6) -- (9.7,1.2);
\node[dot] at (9.7,1.2) {};
\draw (9.7,0.6) -- (9.2,0) -- (10.2,0) -- (9.7,0.6);
\node[dot] at (9.7,0.6) {};
\node[bag] at (9.7,0.8) {$D$};
\node[bag] at (9.7,0.2) {$s''$};
\node[bag] at (9.4,1.0) {$x$};
\draw (11.3,1.2) -- (10.8,0.6) -- (11.8,0.6) -- (11.3,1.2);
\node[dot] at (11.3,1.2) {};
\draw (11.3,0.6) -- (10.8,0) -- (11.8,0) -- (11.3,0.6);
\node[dot] at (11.3,0.6) {};
\node[bag] at (11.3,0.8) {$D$};
\node[bag] at (11.3,0.2) {$s''$};
\node[bag] at (11.0,1.0) {$y$};
\end{tikzpicture}
\end{center}
\item If $z$ is an ancestor of $y$, let $D=t''[z,y]$ and notice that $s''=Ds'$. Since $y$ and $x$ have the
same \ktype, we use $k$-guarded horizontal swap followed by a $k$-guarded vertical
stutter to delete the copy of $D$ (see the picture below). The resulting tree is
$Cs'$ as desired.
\begin{center}
\begin{tikzpicture}
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (1.7,1.2) -- (1.2,0.6) -- (2.2,0.6) -- (1.7,1.2);
\node[dot] at (1.7,1.2) {};
\draw (1.7,0.6) -- (1.2,0) -- (2.2,0) -- (1.7,0.6);
\node[dot] at (1.7,0.6) {};
\node[bag] at (1.7,0.8) {$D$};
\node[bag] at (1.7,0.2) {$s'$};
\node[bag] at (1.4,1.0) {$x$};
\draw (3.3,1.2) -- (2.8,0.6) -- (3.8,0.6) -- (3.3,1.2);
\node[dot] at (3.3,1.2) {};
\draw (3.3,0.6) -- (2.8,0) -- (3.8,0) -- (3.3,0.6);
\node[dot] at (3.3,0.6) {};
\node[bag] at (3.3,0.8) {$D$};
\node[bag] at (3.0,0.4) {$y$};
\node[bag] at (3.3,0.2) {$s'$};
\node[bag] at (3.0,1.0) {$z$};
\node[bag] at (4.5,1.2) {$\Longrightarrow$};
\node[bag] at (4.5,1.8) {\footnotesize Horizontal};
\node[bag] at (4.5,1.5) {\footnotesize Swap};
\draw (6.5,2) -- (5.2,1.2) -- (7.8,1.2) -- (6.5,2);
\draw (5.7,1.2) -- (5.2,0.6) -- (6.2,0.6) -- (5.7,1.2);
\node[dot] at (5.7,1.2) {};
\node[bag] at (5.7,0.8) {$s'$};
\node[bag] at (5.4,1.0) {$x$};
\draw (7.3,1.2) -- (6.8,0.6) -- (7.8,0.6) -- (7.3,1.2);
\node[dot] at (7.3,1.2) {};
\draw (7.3,0.6) -- (6.8,0) -- (7.8,0) -- (7.3,0.6);
\node[dot] at (7.3,0.6) {};
\draw (7.3,0) -- (6.8,-0.6) -- (7.8,-0.6) -- (7.3,0);
\node[dot] at (7.3,0) {};
\node[bag] at (7.3,0.8) {$D$};
\node[bag] at (7.3,0.2) {$D$};
\node[bag] at (7.3,-0.4) {$s'$};
\node[bag] at (7.0,-0.2) {$y$};
\node[bag] at (8.5,1.2) {$\Longrightarrow$};
\node[bag] at (8.5,1.8) {\footnotesize Vertical};
\node[bag] at (8.5,1.5) {\footnotesize Stutter};
\draw (10.5,2) -- (9.2,1.2) -- (11.8,1.2) -- (10.5,2);
\draw (9.7,1.2) -- (9.2,0.6) -- (10.2,0.6) -- (9.7,1.2);
\node[dot] at (9.7,1.2) {};
\node[bag] at (9.7,0.8) {$s'$};
\node[bag] at (9.4,1.0) {$x$};
\draw (11.3,1.2) -- (10.8,0.6) -- (11.8,0.6) -- (11.3,1.2);
\node[dot] at (11.3,1.2) {};
\draw (11.3,0.6) -- (10.8,0) -- (11.8,0) -- (11.3,0.6);
\node[dot] at (11.3,0.6) {};
\node[bag] at (11.3,0.8) {$D$};
\node[bag] at (11.3,0.2) {$s'$};
\node[bag] at (11.0,0.4) {$y$};
\end{tikzpicture}
\end{center}
\item If $z$ and $y$ are not related by the descendant relation, then $x$,
$y$ and $z$ have the same \ktype and $\subtree{t''}{x} = \subtree{t''}{z}$.
We use $k$-guarded horizontal transfer to replace \subtree{t''}{x} with
\subtree{t''}{y} as depicted below.
\begin{center}
\begin{tikzpicture}
\draw (1.5,2.2) -- (0.2,1.2) -- (2.8,1.2) -- (1.5,2.2);
\draw (0.3,1.2) -- (-0.2,0) -- (0.8,0) -- (0.3,1.2);
\node[bag] at (0.3,0.4) {$s''$};
\node[bag] at (0,1.0) {$x$};
\node[dot] at (0.3,1.2) {};
\draw (1.5,1.2) -- (1.0,0) -- (2.0,0) -- (1.5,1.2);
\node[bag] at (1.5,0.4) {$s''$};
\node[bag] at (1.2,1.0) {$z$};
\node[dot] at (1.5,1.2) {};
\draw (2.7,1.2) -- (2.2,0) -- (3.2,0) -- (2.7,1.2);
\node[bag] at (2.7,0.4) {$s'$};
\node[bag] at (2.4,1.0) {$y$};
\node[dot] at (2.7,1.2) {};
\node[bag] at (5.0,1.2) {$\Longrightarrow$};
\node[bag] at (5.0,1.8) {\footnotesize Horizontal};
\node[bag] at (5.0,1.5) {\footnotesize Transfer};
\draw (8.5,2.2) -- (7.2,1.2) -- (9.8,1.2) -- (8.5,2.2);
\draw (7.3,1.2) -- (6.8,0) -- (7.8,0) -- (7.3,1.2);
\node[bag] at (7.3,0.4) {$s'$};
\node[bag] at (7,1.0) {$x$};
\node[dot] at (7.3,1.2) {};
\draw (8.5,1.2) -- (8.0,0) -- (9.0,0) -- (8.5,1.2);
\node[bag] at (8.5,0.4) {$s''$};
\node[bag] at (8.2,1.0) {$z$};
\node[dot] at (8.5,1.2) {};
\draw (9.7,1.2) -- (9.2,0) -- (10.2,0) -- (9.7,1.2);
\node[bag] at (9.7,0.4) {$s'$};
\node[bag] at (9.4,1.0) {$y$};
\node[dot] at (9.7,1.2) {};
\end{tikzpicture}
\end{center}
\end{enumerate}
This concludes the case where $x$ and $z$ are not related by the descendant
relationship in $t$. We are left with the case where $x$ is a descendant of
$z$ (recall that $z$ is outside $s$ and therefore not a descendant of $x$).
We reduce this problem to the previous case by considering two subcases:
\begin{iteMize}{$\bullet$}
\item If $y,z$ are not related by the descendant relationship, we use a
$k$-guarded horizontal swap to replace $s$ by $s'$ and vice versa. This
reverses the roles of $x$ and $y$ and as $y$ and $z$ are not related by
the descendant relationship and position $y$ now has \type{(k+1)} $\tau$
we can apply the previous case.
\begin{center}
\begin{tikzpicture}
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (1.7,1.2) -- (1.2,0.6) -- (2.2,0.6) -- (1.7,1.2);
\node[dot] at (1.7,1.2) {};
\node[bag] at (1.7,0.8) {$s'$};
\node[bag] at (1.4,1.0) {$y$};
\draw (3.3,1.2) -- (2.8,0.6) -- (3.8,0.6) -- (3.3,1.2);
\node[dot] at (3.3,1.2) {};
\draw (3.3,0.6) -- (2.8,0) -- (3.8,0) -- (3.3,0.6);
\node[dot] at (3.3,0.6) {};
\node[bag] at (3.3,0.2) {$s$};
\node[bag] at (3.0,1.0) {$z$};
\node[bag] at (3.0,0.4) {$x$};
\node[bag] at (4.5,1.2) {$\Longrightarrow$};
\node[bag] at (4.5,1.8) {\footnotesize Horizontal};
\node[bag] at (4.5,1.5) {\footnotesize Swap};
\begin{scope}[xshift=4cm]
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (1.7,1.2) -- (1.2,0.6) -- (2.2,0.6) -- (1.7,1.2);
\node[dot] at (1.7,1.2) {};
\node[bag] at (1.7,0.8) {$s$};
\node[bag] at (1.4,1.0) {$y$};
\draw (3.3,1.2) -- (2.8,0.6) -- (3.8,0.6) -- (3.3,1.2);
\node[dot] at (3.3,1.2) {};
\draw (3.3,0.6) -- (2.8,0) -- (3.8,0) -- (3.3,0.6);
\node[dot] at (3.3,0.6) {};
\node[bag] at (3.3,0.2) {$s'$};
\node[bag] at (3.0,1.0) {$z$};
\node[bag] at (3.0,0.4) {$x$};
\end{scope}
\node[bag] at (8.5,1.2) {$\Longrightarrow$};
\node[bag] at (8.5,1.8) {\footnotesize Previous};
\node[bag] at (8.5,1.5) {\footnotesize Case};
\begin{scope}[xshift=8cm]
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (1.7,1.2) -- (1.2,0.6) -- (2.2,0.6) -- (1.7,1.2);
\node[dot] at (1.7,1.2) {};
\node[bag] at (1.7,0.8) {$s'$};
\node[bag] at (1.4,1.0) {$x$};
\draw (3.3,1.2) -- (2.8,0.6) -- (3.8,0.6) -- (3.3,1.2);
\node[dot] at (3.3,1.2) {};
\draw (3.3,0.6) -- (2.8,0) -- (3.8,0) -- (3.3,0.6);
\node[dot] at (3.3,0.6) {};
\node[bag] at (3.3,0.2) {$s'$};
\node[bag] at (3.0,1.0) {$z$};
\node[bag] at (3.0,0.4) {$y$};
\end{scope}
\end{tikzpicture}
\end{center}
\item If $z$ is an ancestor of both $x$ and $y$ we use $k$-guarded vertical
stutter to duplicate the context between $z$ and $x$. This introduces a new
node $z'$ of type $\tau$ that is not related to $y$ by the descendant
relationship and we are back in the previous case.
\end{iteMize}
\begin{center}
\begin{tikzpicture}
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (2.5,1.2) -- (1.6,0.4) -- (3.4,0.4) -- (2.5,1.2);
\node[dot] at (2.5,1.2) {};
\node[bag] at (2.5,0.7) {$D$};
\node[bag] at (2.0,1.0) {$z$};
\draw (1.7,0.4) -- (1.2,-0.8) -- (2.2,-0.8) -- (1.7,0.4);
\node[dot] at (1.7,0.4) {};
\node[bag] at (1.7,-0.4) {$s'$};
\node[bag] at (1.4,0.2) {$y$};
\draw (3.3,0.4) -- (2.8,-0.8) -- (3.8,-0.8) -- (3.3,0.4);
\node[dot] at (3.3,0.4) {};
\node[bag] at (3.3,-0.4) {$s$};
\node[bag] at (3.0,0.2) {$x$};
\node[bag] at (4.5,1.2) {$\Longrightarrow$};
\node[bag] at (4.5,1.8) {\footnotesize Vertical};
\node[bag] at (4.5,1.5) {\footnotesize Stutter};
\begin{scope}[xshift=4cm]
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (2.5,1.2) -- (1.6,0.4) -- (3.4,0.4) -- (2.5,1.2);
\node[dot] at (2.5,1.2) {};
\node[bag] at (2.5,0.7) {$D$};
\node[bag] at (2.0,1.0) {$z$};
\draw (1.7,0.4) -- (1.2,-0.8) -- (2.2,-0.8) -- (1.7,0.4);
\node[dot] at (1.7,0.4) {};
\node[bag] at (1.7,-0.4) {$s'$};
\node[bag] at (1.4,0.2) {$y$};
\end{scope}
\begin{scope}[xshift=4.8cm,yshift=-0.8cm]
\draw (2.5,1.2) -- (1.6,0.4) -- (3.4,0.4) -- (2.5,1.2);
\node[dot] at (2.5,1.2) {};
\node[bag] at (2.5,0.7) {$D$};
\node[bag] at (2.0,1.0) {$z'$};
\draw (1.7,0.4) -- (1.2,-0.8) -- (2.2,-0.8) -- (1.7,0.4);
\node[dot] at (1.7,0.4) {};
\node[bag] at (1.7,-0.4) {$s'$};
\node[bag] at (1.4,0.2) {$y'$};
\draw (3.3,0.4) -- (2.8,-0.8) -- (3.8,-0.8) -- (3.3,0.4);
\node[dot] at (3.3,0.4) {};
\node[bag] at (3.3,-0.4) {$s$};
\node[bag] at (3.0,0.2) {$x$};
\end{scope}
\node[bag] at (8.5,1.2) {$\Longrightarrow$};
\node[bag] at (8.5,1.8) {\footnotesize Previous};
\node[bag] at (8.5,1.5) {\footnotesize Case};
\begin{scope}[xshift=8cm]
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (2.5,1.2) -- (1.6,0.4) -- (3.4,0.4) -- (2.5,1.2);
\node[dot] at (2.5,1.2) {};
\node[bag] at (2.5,0.7) {$D$};
\node[bag] at (2.0,1.0) {$z$};
\draw (1.7,0.4) -- (1.2,-0.8) -- (2.2,-0.8) -- (1.7,0.4);
\node[dot] at (1.7,0.4) {};
\node[bag] at (1.7,-0.4) {$s'$};
\node[bag] at (1.4,0.2) {$y$};
\end{scope}
\begin{scope}[xshift=8.8cm,yshift=-0.8cm]
\draw (2.5,1.2) -- (1.6,0.4) -- (3.4,0.4) -- (2.5,1.2);
\node[dot] at (2.5,1.2) {};
\node[bag] at (2.5,0.7) {$D$};
\node[bag] at (2.0,1.0) {$z'$};
\draw (1.7,0.4) -- (1.2,-0.8) -- (2.2,-0.8) -- (1.7,0.4);
\node[dot] at (1.7,0.4) {};
\node[bag] at (1.7,-0.4) {$s'$};
\node[bag] at (1.4,0.2) {$y'$};
\draw (3.3,0.4) -- (2.8,-0.8) -- (3.8,-0.8) -- (3.3,0.4);
\node[dot] at (3.3,0.4) {};
\node[bag] at (3.3,-0.4) {$s'$};
\node[bag] at (3.0,0.2) {$x$};
\end{scope}
\node[bag] at (12.5,1.2) {$\Longrightarrow$};
\node[bag] at (12.5,1.8) {\footnotesize Vertical};
\node[bag] at (12.5,1.5) {\footnotesize Stutter};
\begin{scope}[xshift=12cm]
\draw (2.5,2) -- (1.2,1.2) -- (3.8,1.2) -- (2.5,2);
\draw (2.5,1.2) -- (1.6,0.4) -- (3.4,0.4) -- (2.5,1.2);
\node[dot] at (2.5,1.2) {};
\node[bag] at (2.5,0.7) {$D$};
\node[bag] at (2.0,1.0) {$z$};
\draw (1.7,0.4) -- (1.2,-0.8) -- (2.2,-0.8) -- (1.7,0.4);
\node[dot] at (1.7,0.4) {};
\node[bag] at (1.7,-0.4) {$s'$};
\node[bag] at (1.4,0.2) {$y$};
\draw (3.3,0.4) -- (2.8,-0.8) -- (3.8,-0.8) -- (3.3,0.4);
\node[dot] at (3.3,0.4) {};
\node[bag] at (3.3,-0.4) {$s'$};
\node[bag] at (3.0,0.2) {$x$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{proof}
We now turn to the proof of Lemma~\ref{claim-transfer-branch}. The proof is
done by induction on the depth of $s'$. The idea is to replace $s$ with $s'$
node by node.
Assume first that $s'$ is of depth less than $k$. Then because the \ktype of
the roots of $s$ and $s'$ are equal, we have $s=s'$ and the result follows.
Assume now that $s'$ is of depth greater than $k$.
Let $x$ be the node of $t$ corresponding to the root of $s$. Let $\tau$ be
the \type{(k+1)} of the root of $s'$. We assume that $s'$ is a tree of the
form $a(s'_1,s'_2)$. Notice that the \ktype of the roots of $s'_1$ and $s'_2$
are completely determined by $\tau$. By hypothesis \lessblocks{s'}{D}{k+1},
hence there exists a node $y$ in $D$ of type $\tau$. We consider two
cases depending on the relationship between $x$ and $y$.
\tikzstyle{arr} = [line width=4pt, ->]
\tikzstyle{bag}=[minimum size=20pt,inner sep=0pt]
\tikzstyle{inner}=[draw,circle,inner sep=0pt]
\tikzstyle{dot}=[draw,circle,fill,minimum size=4pt,inner sep=0pt]
\begin{iteMize}{$\bullet$}
\item If $y$ is an ancestor of $x$, let $E$ be $t[y,x]$ and notice that $x$ and
$y$ have the same \ktype. This case is depicted below. Hence applying a $k$-guarded vertical stutter we
can duplicate $E$ obtaining the tree $DEs$. Because $L$ is \ktame, $DEs \in
L$ iff $t=Ds \in L$. Now the root of $Es$ in $DEs$ is of type $\tau$ and therefore of
the form $a(s_1,s_2)$ where the roots of $s_1$ and $s_2$ have the same \ktype
as respectively the roots of $s'_1$ and $s'_2$. By construction all the
\types{(k+1)} of $s_1$ and $s_2$ already appear in $D$ and hence we can apply
the induction hypothesis to replace $s_1$ by $s'_1$ and $s_2$ by $s'_2$
without affecting membership in $L$. Altogether this gives the desired
result.
\begin{center}
\begin{tikzpicture}
\draw (0.8,2) -- (0,1) -- (1.6,1) -- (0.8,2);
\draw (0.8,1) -- (0,0) -- (1.6,0) -- (0.8,1);
\draw (0.8,0) -- (0,-1) -- (1.6,-1) -- (0.8,0);
\node[dot] at (0.8,1) {};
\node[bag] at (1.2,0.8) {$y$};
\node[dot] at (0.8,0) {};
\node[bag] at (1.2,-0.2) {$x$};
\node[bag] at (0.8,0.4) {$E$};
\node[bag] at (0.8,-0.6) {$s$};
\node[bag] (type1) at (-1.2,0.6) {\small type $\tau$};
\draw[->,shorten >=5pt, thick] (type1) -> (0.8,1);
\node[bag] at (2.5,1.1){\footnotesize Vertical};
\node[bag] at (2.5,0.8){\footnotesize Stutter};
\node[bag] at (2.5,0.5){$\Longrightarrow$};
\draw (4.2,2) -- (3.4,1) -- (5,1) -- (4.2,2);
\draw (4.2,1) -- (3.4,0) -- (5,0) -- (4.2,1);
\draw (4.2,0) -- (3.4,-1) -- (5,-1) -- (4.2,0);
\draw (4.2,-1) -- (3.4,-2) -- (5,-2) -- (4.2,-1);
\node[dot] at (4.2,1) {};
\node[bag] at (4.2,0.4) {$E$};
\node[dot] at (4.2,0) {};
\node[bag] at (4.2,-0.6) {$E$};
\node[dot] at (4.2,-1) {};
\node[bag] at (4.2,-1.6) {$s$};
\node[bag] (type2) at (2.6,-0.35) {\small type $\tau$};
\draw[->,shorten >=5pt, thick] (type2) -> (4.2,0);
\begin{scope}[xshift=3.5cm]
\node[bag] at (2.5,0.5){$=$};
\draw (4.2,2) -- (3.4,1) -- (5,1) -- (4.2,2);
\draw (4.2,1) -- (3.4,0) -- (5,0) -- (4.2,1);
\draw (3.7,-0.6) -- (3.4,-1.5) -- (4.0,-1.5) -- (3.7,-0.6);
\draw (4.7,-0.6) -- (4.4,-1.5) -- (5,-1.5) -- (4.7,-0.6);
\node[bag] at (3.7,-1.2) {$s_1$};
\node[bag] at (4.7,-1.2) {$s_2$};
\node[dot] at (4.2,1) {};
\node[bag] at (4.2,0.4) {$E$};
\node[dot] at (4.2,0) {};
\node[inner] (a) at (4.2,-0.3) {$a$};
\node[dot] at (3.7,-0.6) {};
\node[dot] at (4.7,-0.6) {};
\draw[thick] (a) -- (3.7,-0.6);
\draw[thick] (a) -- (4.7,-0.6);
\draw[thick] (a) -- (4.2,0);
\end{scope}
\begin{scope}[xshift=7cm]
\node[bag] at (2.5,0.8){\footnotesize Induction};
\node[bag] at (2.5,0.5){$\Longrightarrow$};
\draw (4.2,2) -- (3.4,1) -- (5,1) -- (4.2,2);
\draw (4.2,1) -- (3.4,0) -- (5,0) -- (4.2,1);
\draw (3.7,-0.6) -- (3.4,-1.5) -- (4.0,-1.5) -- (3.7,-0.6);
\draw (4.7,-0.6) -- (4.4,-1.5) -- (5,-1.5) -- (4.7,-0.6);
\node[bag] at (3.7,-1.2) {$s'_1$};
\node[bag] at (4.7,-1.2) {$s'_2$};
\node[dot] at (4.2,1) {};
\node[bag] at (4.2,0.4) {$E$};
\node[dot] at (4.2,0) {};
\node[inner] (a) at (4.2,-0.3) {$a$};
\node[dot] at (3.7,-0.6) {};
\node[dot] at (4.7,-0.6) {};
\draw[thick] (a) -- (3.7,-0.6);
\draw[thick] (a) -- (4.7,-0.6);
\draw[thick] (a) -- (4.2,0);
\end{scope}
\end{tikzpicture}
\end{center}
\item Assume now that $x$ and $y$ are not related by the descendant
relationship. This case is depicted below. Let $s''$ be the subtree of $Ds$ rooted at $y$. By hypothesis
all the \types{(k+1)} of $s$ are already present in $D$ and the roots of $s$
and $s''$ have the same \ktype. Hence we can apply
Claim~\ref{claim-transfer-enhanced} and we have $Ds \in L$ iff $Ds'' \in L$.
Now the root of $s''$ is by construction of type $\tau$. Hence $s''$ is
of the form $a(s_1,s_2)$ where $s_1$ and $s_2$ have all their \types{(k+1)}
appearing in $D$ and their roots have the same \ktype as respectively
$s'_1$ and $s'_2$. Hence by induction $s_1$ can be replaced by $s'_1$ and
$s_2$ by $s'_2$ without affecting membership in $L$. Altogether this gives
the desired result.
\begin{center}
\begin{tikzpicture}
\draw (0.8,2) -- (-0.4,1) -- (2,1) -- (0.8,2);
\draw (0.2,1) -- (-0.3,0) -- (0.7,0) -- (0.2,1);
\draw (1.4,1) -- (0.9,0) -- (1.9,0) -- (1.4,1);
\node[dot] at (0.2,1) {};
\node[bag] at (0.5,0.8) {$y$};
\node[dot] at (1.4,1) {};
\node[bag] at (1.7,0.8) {$x$};
\node[bag] at (0.2,0.4) {$s''$};
\node[bag] at (1.4,0.4) {$s$};
\node[bag] (type1) at (-1.5,0.6) {type $\tau$};
\draw[->,shorten >=5pt, thick] (type1) -> (0.2,1);
\node[bag] at (3,1.3){\footnotesize Claim~\ref{claim-transfer-enhanced}};
\node[bag] at (3,1){$\Longrightarrow$};
\draw (5.6,2) -- (4.4,1) -- (6.8,1) -- (5.6,2);
\draw (5,1) -- (4.5,0) -- (5.5,0) -- (5,1);
\draw (6.2,1) -- (5.7,0) -- (6.7,0) -- (6.2,1);
\node[dot] at (5,1) {};
\node[bag] at (5.3,0.8) {$y$};
\node[dot] at (6.2,1) {};
\node[bag] at (6.5,0.8) {$x$};
\node[bag] at (5,0.4) {$s''$};
\node[bag] at (6.2,0.4) {$s''$};
\node[bag] (type2) at (8.5,0.6) {type $\tau$};
\draw[->,shorten >=5pt, thick] (type2) -> (5,1);
\draw[->,shorten >=5pt, thick] (type2) -> (6.2,1);
\node[bag] at (5.6,-0.5){$=$};
\begin{scope}[yshift=-3cm]
\draw (0.8,2) -- (-0.4,1) -- (2,1) -- (0.8,2);
\draw (0.2,1) -- (-0.3,0) -- (0.7,0) -- (0.2,1);
\node[dot] at (0.2,1) {};
\node[bag] at (0.5,0.8) {$y$};
\node[dot] at (1.4,1) {};
\node[bag] at (1.7,0.8) {$x$};
\node[bag] at (0.2,0.4) {$s''$};
\begin{scope} [xshift=-2.8cm,yshift=1cm]
\draw (3.7,-0.6) -- (3.4,-1.5) -- (4.0,-1.5) -- (3.7,-0.6);
\draw (4.7,-0.6) -- (4.4,-1.5) -- (5,-1.5) -- (4.7,-0.6);
\node[bag] at (3.7,-1.2) {$s'_1$};
\node[bag] at (4.7,-1.2) {$s'_2$};
\node[inner] (a) at (4.2,-0.3) {$a$};
\node[dot] at (3.7,-0.6) {};
\node[dot] at (4.7,-0.6) {};
\draw[thick] (a) -- (3.7,-0.6);
\draw[thick] (a) -- (4.7,-0.6);
\draw[thick] (a) -- (4.2,0);
\end{scope}
\node[bag] at (3,1.3){\footnotesize Induction};
\node[bag] at (3,1){$\Longleftarrow$};
\draw (5.6,2) -- (4.4,1) -- (6.8,1) -- (5.6,2);
\draw (5,1) -- (4.5,0) -- (5.5,0) -- (5,1);
\node[dot] at (5,1) {};
\node[bag] at (5.3,0.8) {$y$};
\node[dot] at (6.2,1) {};
\node[bag] at (6.5,0.8) {$x$};
\node[bag] at (5,0.4) {$s''$};
\begin{scope} [xshift=2cm,yshift=1cm]
\draw (3.7,-0.6) -- (3.4,-1.5) -- (4.0,-1.5) -- (3.7,-0.6);
\draw (4.7,-0.6) -- (4.4,-1.5) -- (5,-1.5) -- (4.7,-0.6);
\node[bag] at (3.7,-1.2) {$s_1$};
\node[bag] at (4.7,-1.2) {$s_2$};
\node[inner] (a) at (4.2,-0.3) {$a$};
\node[dot] at (3.7,-0.6) {};
\node[dot] at (4.7,-0.6) {};
\draw[thick] (a) -- (3.7,-0.6);
\draw[thick] (a) -- (4.7,-0.6);
\draw[thick] (a) -- (4.2,0);
\end{scope}
\end{scope}
\end{tikzpicture}
\end{center}
\end{iteMize}
\end{proof}
We now prove a similar result for \kloops.
\begin{lem}\label{lemma-insert-loop}
Assume $L$ is \ktame. Let $t$ be a tree and $x$ a node of $t$ of \ktype
$\tau$. Let $t'$ be another tree such that \sameblocks{t}{t'}{k+1} and $C$ be
a \kloop of type $\tau$ in $t'$. Consider the tree $T$ constructed from $t$
by inserting a copy of $C$ at $x$. Then $t \in L$ iff $T \in L$.
\end{lem}
\begin{proof}
The proof is done in two steps. First we use the \ktame property
of $L$ to show that we can insert a \kloop $C'$ at $x$ in $t$ such that the
principal path of $C$ is the same as the principal path of $C'$. By this we
mean that there is a bijection from the principal path of $C'$ to the
principal path of $C$ that preserves the child relation and \types{(k+1)}. In a second step we
replace one by one the subtrees hanging from the principal path of $C'$ with
the corresponding subtrees in $C$.
First some terminology. Given two nodes $y,y'$ of some tree $T$, we say that
$y'$ is a {\bf l}-ancestor of $y$ if $y$ is a descendant of the left child of
$y'$. Similarly we define {\bf r}-ancestorship.
Consider the context $C$ occurring in $t'$. Let $y_{0}, \cdots,y_{n}$ be the
nodes of $t'$ on the principal path of $C$ and $\tau_{0}, \cdots,\tau_{n}$ be
their respective \type{(k+1)}. For $0 \leq i < n$, set $c_i$ to {\bf l}
if $y_{i+1}$ is a left child of $y_i$ and {\bf r} otherwise.
From $t$ we construct using $k$-guarded swaps and $k$-vertical stutters a
tree $t_1$ such that there is a sequence of nodes $x_0,\cdots,x_n$ in $t_1$
with for all $0\leq i < n$, $x_i$ is of type $\tau_i$ and $x_i$ is an
$c_i$-ancestor of $x_{i+1}$. The tree $t_1$ is constructed by induction on
$n$ (note that this step do not require that $C$ is a \kloop).
If $n=0$ then this is a consequence of \sameblocks{t}{t'}{k+1} that one
can find in $t$ a node of type $\tau_0$. Consider now the case $n>0$. By
induction we have constructed from $t$ a tree $t'_1$ such that
$x_0,\cdots,x_{n-1}$ is an appropriate sequence in $t'_1$. By symmetry it is
enough to consider the case where $y_{n}$ is the left child of $y_{n-1}$.
Because all $k$-guarded operations preserve \types{(k+1)}, we have
\sameblocks{t}{t'_1}{k+1} and hence there is a node $x'$ of $t'_1$ of type
$\tau_n$. If $x_{n-1}$ is a {\bf l}-ancestor of $x'$ then we are done.
Otherwise consider the left child $x''$ of $x_{n-1}$ and notice that because $y_{n}$
is a child of $y_{n-1}$ and $x_{n-1}$ has the same \type{(k+1)} as
$y_{n-1}$ then $x''$, $y_n$ and $x'$ have the same \ktype.
We know that $x'$ is not a descendant of $x''$. There are two cases. If $x'$ and $x''$
are not related by the descendant relationship then by $k$-guarded swaps we
can replace the subtree rooted in $x''$ by the subtree rooted in $x'$ and we
are done. If $x'$ is an ancestor of $x''$ then the context between $x'$ and $x''$
is a \kloop and we can use $k$-guarded vertical stutter to duplicate
it. This places a node having the same \type{(k+1)} as $x'$ as the left child of $x_{n-1}$ and we are done.
\noindent This concludes the construction of $t_1$. From $t_1$ we construct using
$k$-guarded swaps and $k$-guarded vertical stutter a tree $t_2$ such that
there is a path $x_0,\cdots,x_n$ in $t_2$ with $x_i$ is of type $\tau_i$
for all $0\leq i < n$.
Consider the sequence $x_0,\cdots,x_n$ obtained in $t_1$ from the previous
step. Recall that the \ktype of $x_0$ is the same as the \ktype of $x_n$.
Hence using $k$-guarded vertical stutter we can duplicate in $t_1$ the
context rooted in $x_0$ and whose port is $x_n$. Let $t'_1$ the resulting
tree. We thus have two copies of the sequence $x_0,\cdots,x_n$ that we denote
by the \emph{top copy} and the \emph{bottom copy}. Assume $x_i$ is not a
child of $x_{i-1}$. By symmetry it is enough to consider the case where
$x_{i-1}$ is a {\bf l}-ancestor of $x_i$. Notice then that the context
between the left child of $x_{i-1}$ and $x_i$ is a \kloop. Using $k$-guarded
vertical swap (see Figure~\ref{figure-construct-t2}) we can move the top copy
of this context next to its bottom copy. Using $k$-guarded vertical stutter
this extra copy can be removed. We are left with an instance of the initial
sequence in the bottom copy, while in the top one $x_i$ is a child of
$x_{i-1}$. This construction is depicted in
figure~\ref{figure-construct-t2}.
\tikzstyle{arr} = [line width=4pt, ->]
\tikzstyle{bag}=[minimum size=20pt,inner sep=0pt]
\tikzstyle{dot}=[draw,circle,fill,minimum size=4pt,inner sep=0pt]
\begin{figure}
\caption{The construction of $t_2$, eliminating the context $D$ between
$x_{i-1}
\label{figure-construct-t2}
\end{figure}
Repeating this argument yields the desired
tree $t_2$.
Consider now the context $C'=t_2[x_0,x_n]$. It is a loop of \ktype $\tau$. Let
$T'$ be the tree constructed from $t$ by inserting $C'$ at $x$.
\begin{claim} \label{claim-reverse-swaps}
$T' \in L$ iff $t\in L$.
\end{claim}
\begin{proof}
Consider the sequence of $k$-guarded swaps and $k$-guarded vertical stutter
that was used in order to obtain $t_2$ from $t$. Because $L$ is \ktame, $t
\in L$ iff $t_2 \in L$.
We can easily identify the nodes of $t$ with the nodes of $T'$ outside of
$C'$. Consider the same sequence of $k$-guarded operations applied to $T'$.
Observe that this yields a tree $T_2$ corresponding to $t_2$ with possibly several
extra copies of $C'$. As $C'$ is a \kloop, each of the roots and the ports of these
extra copies have the same \ktype. Hence, using appropriate vertical $k$-swaps or
appropriate horizontal $k$-swaps, depending on whether two copies are related or
not by the descendant relation, they can be brought together. Two examples of
such operation is given in Figure~\ref{figure-elim-loops}.
\begin{figure}
\caption{Bringing copies of the \kloop $C'$ together in
Claim~\ref{claim-reverse-swaps}
\label{figure-elim-loops}
\end{figure}
Then, using $k$-guarded vertical stutter all but one copy can be eliminated
resulting in $t_2$. Hence $T' \in L$ iff $t_2\in L$ and the claim is
proved. See figure \ref{figure-relat-t2}.
\end{proof}
\begin{figure}
\caption{Relation with $t_2$}
\label{figure-relat-t2}
\end{figure}
It remains to show that $T' \in L$ iff $T \in L$. By construction of $T'$ we
have \lessblocks{C'}{t}{k+1}. Consider now a node $x_i$ in the principal path
of $C'$. Let $T_i$ be the subtree branching out the principal path of $C$ at
$y_i$ and $T'_i$ be the subtree branching out the principal path of $C'$ at
$x_i$. By construction $x_i$ and $y_i$ are of \type{(k+1)} $\tau_i$. Therefore
the roots of $T_i$ and $T'_i$ have the same \ktype. Because
\lessblocks{C'}{t}{k+1} all the \types{(k+1)} of $T'_i$ already appear in the
part of $T'$ outside of $C'$. By hypothesis we also have \lessblocks{T_i}{t}{k+1}.
Hence we can apply Lemma~\ref{claim-transfer-branch} and replacing $T'_i$
with $T_i$ does not affect membership in $L$. A repeated use of that
lemma eventually shows that $T' \in L$ iff $T \in L$.
\end{proof}
We return to the proof of Proposition~\ref{lemma-pumping}. Recall that
we have two trees $t,t'$ such that \sameblocks{t}{t'}{\kappa} for $\kappa=
\beta_k + k + 1$. For $l > \kappa$, we want to construct $T,T'$ such that:
\begin{enumerate}[(1)]
\item $t \in L$ iff $T \in L$
\item $t' \in L$ iff $T' \in L$
\item \sameblocks{T}{T'}{l}
\end{enumerate}
Recall that the number of \ktypes is $\beta_k$. Therefore, by choice of $\kappa$, in
every branch of a \type{\kappa} one can find at least one \ktype that is repeated.
This provides many \kloops that can be used using Lemma~\ref{lemma-insert-loop}
for obtaining bigger types.
Take $l > \kappa$, we build $T$ and $T'$ from $t$ and $t'$ by inserting \kloops
in $t$ and $t'$ without affecting their membership in $L$ using
Lemma~\ref{lemma-insert-loop}.
Let $B = \{\tau_{0},...,\tau_{n}\}$ be the set of \ktypes $\tau$ such that
there is a loop of \ktype $\tau$ in $t$ or in $t'$. For each $\tau \in B$ we
fix a context $C_\tau$ as follows. Because $\tau \in B$ there is a context
$C$ in $t$ or $t'$ that is a loop of \ktype $\tau$. For each $\tau \in B$, we
fix arbitrarily such a $C$ and set $C_\tau$ as $\underbrace{C \cdot \ldots
\cdot C}_{l}$, $l$ concatenations of the context $C$. Notice that the path
from the root of $C_\tau$ to its port is then bigger than $l$.
We now describe the construction of $T$ from $t$. The construction of $T'$
from $t'$ is done similarly. The tree $T$ is constructed by simultaneously inserting,
for all $\tau \in B$, a copy of the context $C_\tau$ at all nodes of $t$ of type $\tau$.
We now show that $T$ and $T'$ have the desired properties.
The first and second properties, $t \in L$ iff $T \in L$ and $t' \in L$ iff
$T' \in L$, essentially follow from Lemma~\ref{lemma-insert-loop}. We only
show that $t \in L$ iff $T \in L$, the second property is proved
symmetrically. We view $T$ as if it was constructed from $t$ using a sequence
of insertions of some context $C_\tau$ for $\tau \in B$. We write
$s_0,...,s_m$ the sequence of intermediate trees with $s_0=t$ and $s_m=T$. We
call $C_i$ the context inserted to get $s_{i+1}$ from $s_{i}$. We show by
induction on $i$ that (i) \sameblocks{s_i}{t}{k+1} and (ii) $s_i \in L$ iff
$s_{i+1}\in L$. This will imply $t \in L$ iff $T \in L$ as desired. (i) is
clear for $i=0$. We show that for all $i$ (i) implies (ii). Recall that $C_i$
is the concatenation of $l$ copies of a \kloop present either in $t$ or in
$t'$. We suppose without generality that the \kloop is present in $t$. Let
$s$ be the tree constructed from $t$ by duplicating the \kloop $l$
times. Hence $s$ is a tree containing $C_i$ and by construction
\sameblocks{s}{t}{k+1}. Because \sameblocks{t}{t'}{\kappa} with $\kappa >
k+1$ and \sameblocks{s_i}{t}{k+1} we have \sameblocks{s}{s_i}{k+1}. By
Lemma~\ref{lemma-insert-loop} this implies that $s_{i+1} \in L$ iff $s_{i}
\in L$. By construction we also have \sameblocks{s_{i+1}}{s_i}{k+1} and the
induction step is proved.
We now show the third property:
\begin{lem}\label{claim-sameblock}
\sameblocks{T}{T'}{l}
\end{lem}
\proof
We need to show that \lessblocks{T}{T'}{l}, \lessblocks{T'}{T}{l} and that
the roots of $T$ and $T'$ have the same \type{l}. It will be convenient for
proving this to view the nodes of $T$ as the union of the nodes of $t$ plus
some nodes coming from the \kloops that were inserted. To do this more
formally, if $x$ is a node of $t$ of \ktype not in $B$, then $x$ is
identified with the corresponding node of $T$. If $x$ is a node of $t$ whose
\ktype is in $B$ then $x$ is identified in $T$ with the port of the copy of
$C_\tau$ that was inserted at node $x$. We start with the following claim.
\begin{claim} \label{claim-identify-types} Take two nodes $x$ in $t$ and $x'$
in $t'$, such that $x$ and $x'$ have the same \type{\kappa}. Let $y$ and $y'$ be
the corresponding nodes in $T$ and $T'$. Then $y$ and $y'$ have the same
\type{l}.
\end{claim}
\begin{proof}
Let $\nu$ the \type{\kappa} of $x$ and $x'$. Consider a branch of $\nu$ of
length $\kappa$. By the choice of $\kappa$ we know that in this branch one
can find two nodes $z$ and $z'$ with the same \ktypes $\tau$, with $z$ an
ancestor of $z'$ and such that the \ktype $\tau$ of $z$ is determined by
$\nu$ ($z$ is at distance $\geq k$ from the leaves of $\nu$). Hence $\tau$ is
in $B$. Note that because the \ktype of $z$ is included in $\nu$, the
presence of a node of type $\nu$ induces the presence of a node of type
$\tau$ at the same relative position than $z$. Hence a copy of $C_\tau$ is
inserted simultaneously at the same position relative to $y$ and $y'$ during the
construction of $T$ and $T'$. Because this is true for all branches of $\nu$
and because all $C_\tau$ have depth at least $l$, then $y$ and $y'$ have the
same \type{l}.
\end{proof}
From claim~\ref{claim-identify-types} it follows that the roots of $T$ and $T'$
have the same \type{l}. By symmetry we only need to show that
\lessblocks{T}{T'}{l}. Let $y$ be a node of $T$ and $\mu$ be its \type{l}. We
show that there exists $y' \in T'$ with type $\mu$. We consider two cases:
\begin{iteMize}{$\bullet$}
\item $y$ is not a node of a loop inserted during the construction of $T$. Let
$x$ be the corresponding position in $t$ and let $\nu$ be its \type{\kappa}. Since
\sameblocks{t}{t'}{\kappa}, there is a node $x'$ of $t'$ of type $\nu$. Let $y'$
be the node of $T'$ corresponding to $y'$. By
Claim~\ref{claim-identify-types} $y$ and $y'$ have the same \type{l}.
\item $y$ is a node inside a copy of $C_\tau$ inserted to construct $T$. Let $x$ be the
node of $t$ where this loop was inserted. Let $\nu$ be the \type{\kappa} of $x$ (the \ktype of
$x$ is $\tau$). Since \sameblocks{t}{t'}{\kappa}, there is a node $x'$ of $t'$ of
type $\nu$. Since $\kappa > k$, $x$ and $x'$ have the same \type{k}, a
copy of $C_\tau$ was also inserted in $t'$ at position $x'$ during the
construction of $T'$. From Claim~\ref{claim-identify-types}, $x$ and
$x'$, when viewed as nodes of $T$ and $T'$ have the same \type{l}. Let $y'$ be the node of $T'$ in the copy of
$C_\tau$ inserted at $x'$ that corresponds to the position $y$. Since $y$ and
$y'$ are ancestors of $x$ and $x'$ that have the same \type{l}, and since the
context from $y$ to $x$ is the same as the context from $y'$ to $x'$, then $y$ and $y'$ must have the same
\type{l}.\qed
\end{iteMize}
\noindent This concludes the proof of Proposition~\ref{lemma-pumping}.
\end{proof}
\section{Unranked trees}\label{section-unranked}
\newcommand\ltype[1]{$(#1,l)$-type\xspace}
\newcommand\ltypes[1]{$(#1,l)$-types\xspace}
\newcommand\kltype{\ltype{k}}
\newcommand\kltypes{\ltypes{k}}
In this section we consider unranked unordered trees with labels in $\Sigma$.
In such trees, each node may have an arbitrary number of children but no order is
assumed on these children. In particular even if a node has only two children
we can not necessarily distinguish the left child from the right child.
Our goal is to adapt the result of the previous section and provide a decidable
characterization of locally testable languages of unranked unordered trees.
In this section by \emph{regular language} we mean definable in the logic MSO
using only the child predicate and unary predicates for the labels of the
nodes. There is also an equivalent automata model that we briefly describe
next. A tree automaton $A$ over unordered unranked trees consists essentially
of a finite set of states $Q=\{q_1,\cdots,q_k\}$, an integer $m$ denoted as the
\emph{counter threshold} in the sequel, and a transition function $\delta$
associating a unique state to any pair consisting of a label and a tuple
$(q_1, \gamma_1) \cdots (q_k,\gamma_k)$ where $\gamma_i \in \{=i~|~ i < m \}
\cup \{ \geq m \}$. The meaning is straightforward via bottom-up evaluation: A
node of label {\bf a} get assigned a state $q$ if for all $i$, the number of
its children, up to threshold $m$, that were assigned state $q_i$ is as
specified by $\delta$. In the sequel we assume without loss of generality that
all our tree automata are deterministic.
In the unranked tree case, there are several natural definitions of LT. Recall
the definition of \ktype: the \ktype of a node $x$ is the isomorphism type of
the subtree induced by the descendants of $x$ at distance at most $k$ from
$x$. With unranked trees this definition generates infinitely many \ktypes.
We therefore introduce a more flexible notion of type, \kltype, based on one
extra parameter $l$ restricting the horizontal information. It is defined by
induction on $k$. Consider an unordered tree $t$ and a node $x$ of $t$. For
$k=0$, the \kltype of $x$ is just the label of $x$. For $k>0$ the \kltype of
$x$ is the label of $x$ together with, for each \ltype{k-1}, the number, up to
threshold $l$, of children of $x$ of this type. The reader can verify that over
binary trees, the $(k,2)$-type and the \ktype of $x$ always coincide. As in the
previous section we say that two trees are $(k,l)$-equivalent, and denote this
using $\simeq_{(k,l)}$, if they have the
same occurrences of \kltypes and their roots have the same \kltype. We also use
\lessblocks{t}{t'}{(k,l)} to denote the fact that all \kltypes of $t$ also occur in $t'$.
Based on this new notion of type, we define two notions of locally testable
languages. The most expressive one, denoted ALT (A for \emph{Aperiodic}), is
defined as follows. A language $L$ is in $(\kappa,\lambda)$-ALT if it is a union of
$(\kappa,\lambda)$-equivalence classes. A language $L$ is in ALT if there is a $\kappa$ and a $\lambda$ such
that $L$ is in $(\kappa,\lambda)$-ALT.
The second one, denoted ILT in the sequel (I for \emph{Idempotent}), assumes
$\lambda=1$: A language $L$ is in ILT if there is a $\kappa$ such that $L$ is a union of
$(\kappa,1)$-equivalence classes.
The main result of this section is that we can decide
membership for both ILT and ALT.
\begin{thm}\label{theo-unranked}
It is decidable whether a regular unranked unordered tree language is ILT.
It is decidable whether a regular unranked unordered tree language is ALT.
\end{thm}
\paragraph{\bf Tameness}
The notion of \ktame is defined as in Section~\ref{section-necessary} using the
same $k$-guarded operations requiring that the swapped nodes have identical
\ktype. We also define a notion of \kltame which corresponds to our new notion
of \kltype. Consider the four operations of tameness defined in
Section~\ref{section-necessary}. A horizontal swap is said to be
$(k,l)$-guarded if $x$ and $x'$ have the same \kltype, a horizontal transfer is
$(k,l)$-guarded if $x,y,z$ have the same \kltype, a vertical swap is
$(k,l)$-guarded if $x,y,z$ have the same \kltype and a vertical stutter is
$(k,l)$-guarded if $x,y,z$ have the same \kltype. Let $L$ be a regular unranked
unordered tree language and let $m$ be the counting threshold of the minimal
automaton recognizing $L$, we say that $L$ is \kltame iff it is closed under
$(k,l)$-guarded horizontal swap, horizontal transfer, vertical swap and
vertical stutter and $l > m$ (we assume $l>m$ in order to make the
statements of the results similar to those used in the binary setting). We
first prove that over unordered trees being \ktame is the same as being
\kltame.
\begin{prop} \label{prop-kltame-ktame}
Let $L$ be an unordered unranked regular tree language, then for all integers $k$,
$L$ is \ktame iff there exists $l$ such that $L$ is \kltame. Furthermore, such
an $l$ can be computed from any automaton recognizing $L$.
\end{prop}
\begin{proof}
If there exists $l$ such that $L$ is \kltame then $L$ is obviously \ktame.
Suppose that $L$ is \ktame, and let $m$ be the counting threshold of the
minimal automaton $A$ recognizing $L$. We show that there exists $l'$ such
that $L$ is closed under $(k,l')$-guarded operations. This implies the result
as one can then take $l = max(m+1,l')$.
We need to show that $L$ is closed under $(k,l')$-guarded vertical swap, vertical
stutter, horizontal swap and horizontal transfer. The proof is similar to the
proof of \emph{Proposition 1} in~\cite{BS09}. We will use the following claim
which is proved in~\cite{BS09} using a simple pumping argument:
\begin{claim} \cite{BS09} \label{claim-kltype-ktype} For every tree automaton
$A$ there is a number $l'$, computable from $A$, such that for every $k$ if a
tree $t_1$ is $(k,l')$-equivalent to a tree $t_2$, then there are trees
$t_1',t_2'$ with $t_1'$ and $t_2'$ $k$-equivalent such that $A$ reaches the
same state on $t'_i$ as on $t_i$ for $i=1,2$.
\end{claim}
We use this claim to prove that $L$ is closed under horizontal transfer. Let
$l'$ be the number computed from $A$ by Claim~\ref{claim-kltype-ktype}. We
prove that $L$ is closed under $(k,l')$-guarded horizontal transfer. Consider
a tree $t$ and three nodes $x,y,z$ of $t$ not related by the descendant
relationship and such that $\subtree{t}{x}=\subtree{t}{y}$ and such that $x,y$
and $z$ have the same \type{(k,l')}. Let $t'$ be the horizontal transfer of $t$ at
$x,y,z$. Let $t_1 = \subtree{t}{x}$ and $t_2 = \subtree{t}{z}$ and $t_1'$,
$t_2'$ obtained from $t_1,t_2$ using Claim~\ref{claim-kltype-ktype}. Let $s$ be
the tree obtained from $t$ by replacing $\subtree{t}{x}$ and $\subtree{t}{y}$
with $t_1'$ and $\subtree{t}{z}$ with $t_2'$, and let $s'$ be the tree obtained
from $t'$ by replacing $\subtree{t'}{x}$ with $t_1'$ and $\subtree{t'}{y}$ and
$\subtree{t'}{z}$ with $t_2$. From Claim~\ref{claim-kltype-ktype} it follows
that $t \in L$ iff $s \in L$ and $t' \in L$ iff $s' \in L$. Since $L$ is
\ktame, it is closed under $k$-guarded horizontal transfer, therefore we have
$s \in L$ iff $s' \in L$, it follows that $t \in L$ iff $t' \in L$.
The closure under horizontal swap is proved using the same claim. The proofs for
vertical swap and vertical stutter uses a claim similar to
Claim~\ref{claim-kltype-ktype} but for contexts: For every tree automaton $A$
there is a number $l$ computable from $A$ such that for every $k$ if the context
$C_1$ is $(k,l)$-equivalent to the context $C_2$ (by this we mean that their roots
have the same \kltype), then there are contexts $C'_1$,
$C'_2$ with $C'_1$ $k$-equivalent to $C'_2$ such that $C'_i$ induces the same
function on the states of $A$ as $C_i$ for $i=1,2$.
\end{proof}
From this lemma we know that a regular language over unranked unordered trees
is \tame iff it is \ktame for some $k$ iff it is \kltame for some $k,l$.
Moreover, as in the binary setting, if a regular language is \tame then it is
\kltame for some $k$ and $l$ computable from an automaton recognizing $L$. The
bound on $k$ can be obtained by a straightforward adaptation of
Proposition~\ref{nec-condition-decision}. The bound on $l$ then follows from
Proposition~\ref{prop-kltame-ktame}. Hence we have:
\begin{prop}\label{nec-condition-decision-unranked}
Let $L$ be a regular language and let $A$ be its minimal deterministic bottom-up
tree automaton, we have $L$ is \tame iff $L$ is $(k_0,l_0)$-\tame
for $k_0 = |A|^3+1$ and some $l_0$ computable from $A$.
\end{prop}
\subsection{\bf Decision of ALT}
We now turn to the proof of Theorem~\ref{theo-unranked}. We begin with the
proof for $ALT$ as both the decision procedure and its proof are obtained as in
the case of binary trees. Assuming tameness we obtain a bound on $\kappa$ and
$\lambda$ such that a language is in ALT iff it is in $(\kappa,\lambda)$-ALT.
Once $\kappa$ and $\lambda$ are known, it is easy do decide if a language is
$(\kappa,\lambda)$-ALT since the number of such languages is finite. The bounds
on $\kappa$ and $\lambda$ are obtained following the same proof structure as in
the binary cases, essentially replacing \ktame by \kltame, but with several
technical modifications. Therefore, we only sketch the proofs below and only
detail the new technical material. Our goal is to prove the following
result.
\begin{prop}\label{prop-nec-implies-LT-unranked}
Assume $L$ is a \kltame regular tree language and let $A$ be its minimal
automaton. Then $L$ is in ALT iff $L$ is in $(\kappa,\lambda)$-ALT where $\kappa$
and $\lambda$ are computable from $k$, $l$ and $A$.
\end{prop}
Notice that for each $k,l$ the number of \kltypes is finite, let $\beta_{k,l}$
be this number. Proposition~\ref{prop-nec-implies-LT-unranked} is now a simple
consequence of the following proposition.
\begin{prop}\label{lemma-pumping-unranked}
Let $L$ be a \kltame regular tree language and let $A$ be the minimal
automaton recognizing $L$. Set $\lambda = |A|l + 1$ and
$\kappa=\beta_{k,l} + k + 1$. Then for all $\kappa'>\kappa$, all $\lambda'>
\lambda$ and any two trees $t,t'$ if $t \simeq_{(\kappa,\lambda)} t'$ then there
exists two trees $T,T'$ with
\begin{enumerate}[\em(1)]
\item $t \in L$ iff $T \in L$
\item $t' \in L$ iff $T' \in L$
\item $T \simeq_{(\kappa',\lambda')} T'$.
\end{enumerate}
\end{prop}
Before proving Proposition~\ref{lemma-pumping-unranked} we adapt the extra
terminology we used in the proof of Proposition~\ref{lemma-pumping} to the
unranked setting. A non-empty context $C$ occurring in a tree $t$ is a
\emph{loop of \kltype $\tau$} if the \kltype of its root and the \kltype of
its port is $\tau$. A non-empty context $C$ occurring in a tree $t$ is a
$(k,l)$-loop if there is some \kltype $\tau$ such that $C$ is a loop of \kltype
$\tau$. Given a context $C$ we call the path from the root of $C$ to its port
the \emph{principal path of $C$}. Finally, the result of the \emph{insertion}
of a $(k,l)$-loop $C$ at a node $x$ of a tree $t$ is a tree $T$ such that if
$t=D \cdot \subtree{t}{x}$ then $T=D\cdot C \cdot \subtree{t}{x}$. Typically
an insertion will occur only when the \kltype of $x$ is $\tau$ and $C$ is a loop
of \kltype $\tau$. In this case the \kltypes of the nodes initially from $t$ and
of the nodes of $C$ are unchanged by this operation.
\begin{proof}[Proof of Proposition~\ref{lemma-pumping-unranked}]
Suppose that $L$ is \kltame. As we did for the proof of the binary
case we first prove two lemmas that are crucial for the construction of $T$
and $T'$. They show that subtrees can be replaced and contexts can be
inserted as long as this does not change the $(k+1,l)$-equivalence class
of the tree. They are direct adaptations of the corresponding lemmas for
the ranked setting: Lemmas~\ref{claim-transfer-branch} and~\ref{lemma-insert-loop}.
We start with subtrees.
\begin{lem}\label{claim-transfer-branch-unranked}
Assume $L$ is \kltame. Let $t=Ds$ be a tree where $s$ is a subtree of $t$.
Let $s'$ be another tree such that the roots of $s$ and $s'$ have the same
\kltype.
If \lessblocks{s}{D}{(k+1,l)} and \lessblocks{s'}{D}{(k+1,l)} then $Ds\in L$
iff $Ds'\in L$.
\end{lem}
\begin{proof}[Proof sketch]
As in the binary setting the proof is done by first proving a restricted
version where $s'$ is actually another subtree of $t$. Before doing that, we
state a new claim, specific to the unranked setting, that will be useful later
in the induction bases of our proofs. In the binary setting, two
trees that had the same \ktype at their root and were of depth smaller than
$k$ were equal. This obviously does not extend to unranked trees and \kltypes.
However it is simple to see that equality can be replaced by indistinguishability
by the minimal tree automaton recognizing $L$.
\begin{claim} \label{base-case-unranked} Let $A$ be a tree automaton and $m$ be
its counting threshold. Let $t$ and $t'$ be two trees of depth smaller
than $k$ and whose roots have the same \type{(k,m)}. Then $t$ and $t'$
evaluate to the same state of $A$.
\end{claim}
\begin{proof}
This is done by induction on $k$. If $k = 0$, $t$ and $t'$ are leaves, it follows
from their \type{(0,m)} that $t = t'$.
Otherwise we know that $t$ and $t'$ have the same \type{(k,m)} at their root
therefore they have the same root label. Let $s$ and $s'$ be two trees that are
children of the root of $t$ or of $t'$ and have the same \type{(k-1,m)} at
their root. The depth of $s$ and $s'$ is smaller than $k-1$, therefore by
induction hypothesis $s$ and $s'$ evaluate to the same state of $A$. Now,
because the roots of $t$ and $t'$ have the same \type{(k,m)}, for each
\type{(k-1,m)} $\tau$, they have the same number of children of type $\tau$ up to
threshold $m$. From the previous remark this implies that for each state $q$ of
$A$, they have the same number of children in state $q$ up to threshold $m$. It
follows from the definition of $A$ that $t$ and $t'$ evaluate to the same state
of $A$.
\end{proof}
We are now ready to state and prove the lemma in the restricted case.
\begin{claim} \label{claim-transfer-enhanced-unranked} Assume $L$ is \kltame.
Let $t$ be a tree and let $x,y$ be two nodes of $t$ not related by the
descendant relationship and with the same \type{(k,l)}. We write
$s = \subtree{t}{x}$, $s' = \subtree{t}{y}$ and $C$ the context such that
$t = Cs$. If $\lessblocks{s}{C}{(k+1,l)}$ then $Cs \in L$ iff
$Cs' \in L$.
\end{claim}
\begin{proof}[Proof sketch] This proof only differs from its binary tree counterpart
Claim~\ref{claim-transfer-enhanced} in the details of the induction step. It
is done by induction on the depth of $s$.
Assume first that $s$ is of depth less than $k$. Since $x$ and $y$ have the same
\type{(k,l)} and since $l \geq m$ it follows from Claim~\ref{base-case-unranked} that
$s$ and $s'$ evaluate to the same state on the automaton $A$ recognizing
$L$. Hence we can replace $s$ with $s'$ without affecting membership in $L$.
Assume now that $s$ is of depth greater than $k$.
Let $\tau$ be the \type{(k+1,l)} of $x$. We write $s_1,...,s_{n}$ for the children
of $s$ and $a$ the label of its root. Since $\lessblocks{s}{C}{(k+1,l)}$,
there exists a node $z$ in $C$ of type $\tau$. We write $s'' = \subtree{t}{z}$.
We now do a case analysis depending on the descendant relationships between
$x$, $y$ and $z$. As for binary trees, all cases reduce to the case when
$x$ and $z$ are not related by the descendant relationship by simple
\kltame \!\!ness operations. Therefore we only consider this case here.
Assume that $x$ and $z$ are not related by the descendant relationship. We
show only that $Cs \in L$ iff $Cs'' \in L$. The proof that $Cs' \in L$ iff
$Cs'' \in L$ is then done exactly as for binary trees.
Since $x$ and $z$ are of same \type{(k+1,l)} $\tau$, the roots of $s'$ and
$s''$ have the same label $a$. Let $s_1'',\ldots,s_{n'}''$ be the children of
the root of $s''$. As in the binary case we want to replace the trees
$s_1,\ldots,s_n$ with these children by induction since the depth of the trees
$s_1,\ldots,s_n$ is smaller than the depth of $s$. Unfortunately for each
\kltype $\tau_i$, the number of trees whose root has type $\tau_i$ among the
children of $x$ and among the children of $z$ might not be the same. However
we know that in this case both numbers are greater than $l$. We overcome this
difficulty in two steps, first we modify the children of $x$, without affecting
membership in $L$, so that if $s_i$ and $s_j$ have the same \kltype then $s_i
= s_j$, then we use the fact that $l > m$ in order to delete or duplicate
children of $x$ until for each \kltype $\tau_i$ the number of trees of root
of type $\tau_i$ among the children of $x$ and among the children of $z$ is
the same. By definition of $A$, this does not affect membership in $L$.
Finally we replace the $s_i$ by the $s''_i$ by induction as in the binary
case.
For the first step notice that any of the $s_i$ is by definition of depth smaller
than $s$ therefore by the induction hypothesis we can replace it with any of its
siblings having the same \kltype at its root without affecting membership in $L$.
\end{proof}
We now turn to the proof of Lemma~\ref{claim-transfer-branch-unranked} in its
general statement. The proof is done by induction on the depth of $s'$. The
idea is to replace $s$ with $s'$ node by node.
Assume first that $s'$ is of depth smaller than $k$. Then because the
\kltypes of the roots of $s$ and $s'$ are the same we are in the hypothesis
of Claim~\ref{base-case-unranked} and it follows that $s$ and $s'$ evaluate
to the same state of $A$. The result follows.
Assume now that $s'$ is of depth greater than $k$.
Let $x$ be the node of $t$ corresponding to the root of $s$. Let $\tau$ be
the \type{(k+1,l)} of the root of $s'$. In the binary tree case we used a sequence
of tame operations to reduce the problem to the case where $x$ has
\type{(k+1,l)} $\tau$. Using the same operations we can also reduce the
problem to this case in the unranked setting. Then we use the induction
hypothesis to replace the children of $x$ by the children of the root of
$s'$. As in the proof of Claim~\ref{claim-transfer-enhanced-unranked}, the
problem is that the number of children might not match but this is solved
exactly as in the proof of Claim~\ref{claim-transfer-enhanced-unranked}.
\end{proof}
As in the binary tree case, we now prove a result similar to
Lemma~\ref{claim-transfer-enhanced-unranked} but for $(k,l)$-loops.
\begin{lem}\label{lemma-insert-loop-unranked}
Assume $L$ is \kltame. Let $t$ be a tree and $x$ a node of $t$ of \kltype
$\tau$. Let $t'$ be another tree such that \sameblocks{t}{t'}{(k+1,l)} and
$C$ be a $(k,l)$-loop of type $\tau$ in $t'$. Consider the tree $T$
constructed from $t$ by inserting a copy of $C$ at $x$. Then $t \in L$ iff $T \in L$.
\end{lem}
\begin{proof}[Proof sketch]
The proof is done using the same structure as Lemma~\ref{lemma-insert-loop}
for the binary case. First we use the \kltame property of $L$ to show that we
can insert a $(k,l)$-loop $C'$ at $x$ in $t$ such that the principal path of
$C$ is the same as the principal path of $C'$. By this we mean that there is
a bijection from the principal path of $C'$ to the principal path of $C$ that
preserves the child relation and \types{(k+1,l)}. In a second step we replace
one by one the subtrees hanging from the principal path of $C'$ with the
corresponding subtrees in $C$.
Let $T'$ be the tree resulting from inserting $C'$ at position $x$. We do not
detail the first step as it is done using exactly the same sequence of tame
operations we used for this step in the proof of
Lemma~\ref{lemma-insert-loop}. This yields: $t \in L$ iff $T' \in L$. We turn
to the second step showing that $T' \in L$ iff $T \in L$.
By construction of $T'$ we have \lessblocks{C'}{t}{(k+1,l)}. Consider now a
node $x'_i$ in the principal path of $C'$ and $x_i$ the corresponding node in
$C$. As in the binary tree case we replace the subtrees branching out of
the principal path of $C'$ with the corresponding trees branching out of the
principal path of $C$ using Lemma~\ref{claim-transfer-branch-unranked}. As in
the previous proof, the problem is that the numbers of children might not
match. This is solved exactly as in the proof of
Lemma~\ref{claim-transfer-branch-unranked}.
\end{proof}
We now turn to the construction of $T$ and $T'$ and prove
Proposition~\ref{lemma-pumping-unranked}.
The construction is similar to the one we did in the binary tree case.
We insert $(k,l)$-loops in $t$ and $t'$ using
Lemma~\ref{lemma-insert-loop-unranked} for obtaining bigger types. However
inserting loops only affects the depth of the types. Therefore we need to do
extra work in order to also increase the width of the types.
Assuming $t \simeq_{(k,l)} t'$ we first construct two intermediate trees $T_1$ and $T_1'$ that have
the following properties:
\begin{iteMize}{$\bullet$}
\item $t \in L\ $ ~iff~ $\ T_1 \in L$
\item $t' \in L\ $ iff $\ T_1' \in L$
\item \sameblocks{T_1}{T'_1}{(\kappa',\lambda)}
\end{iteMize}
This construction is the same as in the binary tree setting so we only briefly
describe it. Let $B = \{\tau_{0},...,\tau_{n}\}$ be the set of \kltypes $\tau$
such that there is a loop of \kltype $\tau$ in $t$ or in $t'$. For each $\tau
\in B$ we fix a context $C_\tau$ as follows. Because $\tau \in B$ there is a
context $C$ in $T_1$ or $T_1'$ that is a loop of \kltype $\tau$. For each $\tau
\in B$, we fix arbitrarily such a $C$ and set $C_\tau$ as $\underbrace{C \cdot
\ldots \cdot C}_{\kappa'}$, $\kappa'$ concatenations of the context $C$.
Notice that the path from the root of $C_\tau$ to its port is then bigger than
$\kappa'$.
$T_1$ is constructed from $t$ as follows (the construction of $T_1'$ from
$t'$ is done similarly). The tree $T_1$ is constructed by simultaneously
inserting, for all $\tau \in B$, a copy of the context $C_\tau$ at all nodes of
$t$ of type $\tau$. By Lemma~\ref{lemma-insert-loop-unranked} it follows that
$t \in L$ iff $T_1 \in L$ and $t' \in L$ iff $T_1' \in L$. Using the same proof
as that of Proposition~\ref{lemma-pumping} for the binary tree setting, we obtain
\sameblocks{T_1}{T_1'}{(\kappa',\lambda)}.
We now describe the construction of $T$ from $T_1$, the construction of $T'$
from $T'_1$ is done similarly. It will be convenient for us to view the nodes
of $T_1$ as the union of the nodes of $t$ plus some extra nodes coming from the
loops that were inserted.
Let $n$ be the maximum arity of a node of $T_1$ or of $T_1'$. We duplicate
subtrees in $T_1$ and $T_1'$ as follows. Let $x$ be a node of $T_1$, that is
not in a loop we inserted when constructing $T_1$ from $t$. For each
\type{(\kappa'-1,\lambda)} $\tau$, if $x$ has more than $\lambda$ children of
type $\tau$ we duplicate one of the corresponding subtrees until $x$ has exactly
$n$ children of type $\tau$ in total. This is possible without affecting membership in
$L$ because $\lambda > m|A|$. Indeed, because $\lambda > m|A|$, for at least one
state $q$ of $A$, there exists more than $m$ subtrees of $x$ of type $\tau$ for which
$A$ assigns that state $q$ at their root, and by definition of $A$ any of these
subtrees can be duplicated without affecting membership in $L$. The tree $T$
is constructed from $T_1$ by repeating this operation for any node $x$
of $T_1$ coming from $t$. By construction we have $T_1 \in L$ iff $T \in L$ and therefore $t
\in L$ iff $T \in L$. The same construction starting from $T'_1$ yields a tree
$T'$ such that $t' \in L$ iff $T' \in L$.
We now show that \sameblocks{T}{T'}{\kappa'}, it follows that
\sameblocks{T}{T'}{(\kappa',\lambda')} and this concludes the proof.
\begin{lem}\label{claim-sameblock-unranked}
\sameblocks{T}{T'}{\kappa'}
\end{lem}
\begin{proof}
We need to show that \lessblocks{T}{T'}{\kappa'}, \lessblocks{T'}{T}{\kappa'}
and that the roots of $T$ and $T'$ have the same \type{\kappa'}.
Recall that in $T_1$ we distinguished between two kinds of nodes, those coming
from $t$ and those coming from the loops that were inserted during the
construction of $T_1$ from $t$. We make the same distinction in $T$ by assuming
that a node generated after a duplication gets the same kind as its original copy.
Recall the definition of $B$ and of $C_\tau$ for $\tau \in B$ that was used for
defining $T_1$ and $T'_1$ from $t$ and $t'$.
As for the binary tree case it suffices to show that for any node of $T$ coming
from $t$ there is a node of $T'$ coming from $t'$ and having the same
\type{\kappa'}. Hence the result follows from the claim below that is an
adaptation of Claim~\ref{claim-identify-types}.
\begin{claim} Take two nodes $x$ in $t$ and $x'$ in $t'$, such that $x$ and
$x'$ have the same \type{(\kappa,\lambda)}. Let $z$ and $z'$ be the
corresponding nodes in $T$ and $T'$. Then $z$ and $z'$ have the same \type{\kappa'}.
\end{claim}
\begin{proof}
Let $x$ and $x'$ be two nodes of $t$ and $t'$ with the same
\type{(\kappa,\lambda)}. Let $x_1$ and $x'_1$ be the corresponding
nodes in $T_1$ and $T'_1$. The same proof as
Claim~\ref{claim-identify-types} for the binary tree case shows that $x_1$
and $x'_1$ have the same \type{(\kappa',\lambda)}.
Let $y$ be a child of $x$. Let $y_1$ be the node corresponding to $y$ in
$T_1$. Notice now that the \type{(\kappa',\lambda)} of $y_1$ in $T_1$ is
completely determined by the \type{(\kappa-1,\lambda)} $\nu$ of $y$ in $t$.
Indeed, by choice of $\kappa$, during the construction of $T_1$, a loop
of type $\tau \in B$ will be inserted between $y$ and any descendant of $y$
at distance at most $\beta_{(k,l)}-1$ from $y$. As $\kappa > \beta_{(k,l)} +
k$, the relative positions below $y$ where such a $C_\tau$ is inserted can be
read from $\nu$. As the depth of any $C_\tau$ is greater than
$\kappa'$, from $\nu$ we can compute exactly the descendants of $y_1$ in
$T_1$ up to depth $\kappa'$. Hence $\nu$ determines the
\type{(\kappa',\lambda)} of $y_1$.
It follows that two children of $x_1$ or of $x'_1$ have the same
\types{(\kappa',\lambda)} iff they had the same
\types{(\kappa-1,\lambda)} in $t$ or in $t'$.
We now construct an isomorphism between the \type{\kappa'} of $z$ and the one
of $z'$. Let $d$ be the maximal distance between $z$ and a node that is a
descendant of $z$ where a loop was inserted during the construction of $T$ from
$t$. We construct our isomorphism by induction on $d$.
If $d=0$ then the \kltype of $z$ is in $B$ and as $z$ and $z'$ have the same
\type{(\kappa',\lambda)} with $\kappa'>k$, the \kltype of $z'$ is the same as the one of $z'$.
Therefore the subtrees rooted at $z$ and $z'$ are equal up to depth $\kappa'$
as they all start with a copy of $C_\tau$ and we are done.
Otherwise, as $z$ and $z'$ have the same \type{(\kappa',\lambda)} their roots
must have the same labels. Consider now a \type{(\kappa'-1,\lambda)}
$\mu$. By construction of $T$ and $T'$, $z$ and $z'$ must have the same
number of occurrences of children of type $\mu$. Indeed from the type these
numbers must match if one of them is smaller than $\lambda$ and by
construction they are equal to $n$ otherwise. Hence we have a bijection from
the children of $z$ of type $\mu$ and the children of $z'$ of type $\mu$.
From the text above we know that the \type{(\kappa',\lambda)} of these nodes
is determined by the \type{(\kappa-1,\lambda)} of their copy in $t$ or in
$t'$. Because $x$ and $x'$ have the same \type{(\kappa,\lambda)}, the corresponding
\types{(\kappa-1,\lambda)} are all equal and hence all the nodes of type
$\mu$ actually have the same \type{(\kappa',\lambda)}. By induction they are
isomorphic up to depth $\kappa'$ and we are done.
\end{proof}
From Claim~\ref{claim-sameblock-unranked} the lemma follows as in the proof of
Lemma~\ref{claim-sameblock} for binary trees.
\end{proof}
This concludes the proof of Proposition~\ref{lemma-pumping-unranked}.
\end{proof}
\subsection{\bf Decision of ILT}
In the idempotent case we can completely characterize ILT using closure
properties. We show that membership in ILT corresponds to \tameness together
with an extra closure property denoted \emph{horizontal stutter} reflecting the
idempotent behavior. A tree language $L$ is closed under horizontal stutter
iff for any tree $t$ and any node $x$ of $t$, replacing \subtree{t}{x} with two
copies of \subtree{t}{x} does not affect membership in $L$.
Theorem~\ref{theo-unranked} for ILT is a consequence of the following theorem.
\begin{thm}\label{theorem-ILT}
A regular unordered tree language is in ILT iff it is \tame and closed under
horizontal stutter.
\end{thm}
\begin{proof}
It is simple to see that \tameness and closure under horizontal stutter are
necessary conditions. We prove that they are sufficient. Take a regular tree
language $L$ and suppose that $L$ is \tame and closed under horizontal
stutter. Then there exists $k$ and $l$ such that $L$ is \kltame. We show that
if $\sameblocks{t}{t'}{(k+1,1)}$ then $t \in L$ iff $t'\in L$. It follows
that $L$ is in ILT. We first show a simple lemma stating that if
two trees contain the same \types{(k+1,1)}, then we can pump them without
affecting membership in $L$ into two trees that contain the same
\types{(k+1,l)}.
\begin{lem} \label{lemma-pump-idem}
Let $L$ closed under horizontal stutter and let $s$ and $s'$ two trees such that
$\sameblocks{s}{s'}{(k+1,1)}$. Then there exist two trees $S$ and $S'$ such that:
\begin{iteMize}{$\bullet$}
\item $s \in L\ $ ~iff~ $\ S \in L$.
\item $s' \in L\ $ ~iff~ $\ S' \in L$.
\item $\sameblocks{S}{S'}{(k+1,l)}$
\end{iteMize}
\end{lem}
\begin{proof}
$S$ is constructed from $s$ via a bottom-up procedure. Let $x$ be a node of
$s$. For each subtree rooted at a child of $x$, we duplicate it $l$ times
using horizontal stutter. This does not affect membership in $L$.
After performing this for all nodes $x$ of $s$ we obtain a tree $S$
with the desired properties.
\end{proof}
Let $T$ and $T'$ be constructed from $t$ and $t'$ using
Lemma~\ref{lemma-pump-idem}. Let $T_1,\ldots,T_n$ the children of the root of
$T$ and $T'_1,\ldots,T'_{n'}$ the children of the root of $T'$. Let $T''$ be
the tree whose root is the same as $T$ and $T'$ and whose children is the
sequence of trees $T_1,\ldots,T_n,T'_1,\ldots,T'_{n'}$. We show that $T'' \in L$ iff
$T \in L$ and $T'' \in L$ iff $T' \in L$. It will follow that $T \in L$ iff $T'
\in L$ and by Lemma~\ref{lemma-pump-idem} that $t \in L$ iff $t'\in L$ which
ends the proof.
To show that $T'' \in L$ iff $T \in L$ we use horizontal stutter and
Lemma~\ref{claim-transfer-branch-unranked}. As the roots of $T$ and $T'$ have
the same \type{(k+1,l)}, for each $T'_i$, there exists a tree $T_j$ such that its root
has the same \kltype as $T'_i$. Fix such a pair $(i,j)$. Let $S$ be the tree
obtained by duplicating $T_j$ in $T$. By closure under horizontal stutter $T
\in L$ iff $S \in L$. But now $S = DT_j$ for some context $D$ such that
\lessblocks{T}{D}{(k+1,l)}. Altogether we have that: the roots of $T'_i$ and
$T_j$ have the same \kltype (by choice if $i$ and $j$),
\lessblocks{T'_i}{D}{(k+1,l)} (as \lessblocks{T'_i}{T'}{(k+1,l)} and $T
\simeq_{(k+1,l)} T'$) and \lessblocks{T_j}{D}{(k+1,l)} (as $T_j$ is part of $T$
hence of $D$). We can therefore apply Lemma~\ref{claim-transfer-branch-unranked} and $DT'_i\in L$ iff
$DT_j \in L$.
Repeating this argument for all $i$ eventually yields the tree $T''$. This
proves that $T'' \in L$ iff $T \in L$. By symmetry we also have $T'' \in L$ iff
$T' \in L$ which concludes the proof.
\end{proof}
\section{Tameness is not sufficient} \label{section-nonsuff}
Over strings \tameness characterizes exactly LT as vertical swap and vertical
stutter are exactly the extensions to trees of the known equations for LT
(recall Section~\ref{section-notation}). Over trees this is no longer the case.
In this section we provide an example of a language that is tame but not $LT$.
For simplifying the presentation we assume that nodes may have between 0 and
three children; this can easily be turned into a binary tree language. All
trees in our language $L$ have the same structure consisting of a root of label
{\bf a} from which exactly three sequences of nodes with only one child
(strings) are attached. The trees in $L$ have therefore exactly three leaves,
and those must have three distinct labels among $\{${\bf h$_1$}, {\bf h$_2$},
{\bf h$_3$}$\}$. The labels of two of the branches, not including the root and
the leaf, must form a sequence in the language {\bf b}$^*${\bf c}{\bf
d}$^*$. The third branch must form a sequence in the language {\bf b}$^*${\bf
c'}{\bf d}$^*$. We assume that {\bf b}, {\bf c}, {\bf c'} and {\bf d} are
distinct labels. Note that the language does not specify which leaf label among
$\{${\bf h$_1$}, {\bf h$_2$}, {\bf h$_3$}$\}$ is attached to the branch
containing {\bf c'}.
The reader can verify that $L$ is $1$-tame. We show that $L$ is not in LT. For
all integer $k$, the two trees $t$ and $t'$ depicted below are such that $t \in
L$, $t' \notin L$, while \sameblocks{t}{t'}{k}.
\tikzstyle{level 1}=[level distance=1cm, sibling distance=0.8cm]
\tikzstyle{bag} = [text width=4em, text centered]
\tikzstyle{bagc} = [red!80,fill=blue!80,text width=1em, text centered]
\begin{center}
\begin{tikzpicture}[grow=down, sloped]
\node[bag] at (0,0) {$a$}
child {
node[bag] {$b^{k}$}
child {
node[bag] {$c$}
child {
node[bag] {$d^{k}$}
child {
node[bag] {$h_{1}$}
}
}
}
}
child {
node[bag] {$b^{k}$}
child {
node[bag] {$c$}
child {
node[bag] {$d^{k}$}
child {
node[bag] {$h_{2}$}
}
}
}
}
child {
node[bag] {$b^{k}$}
child {
node[bag] {$c'$}
child {
node[bag] {$d^{k}$}
child {
node[bag] {$h_{3}$}
}
}
}
};
\node[bag] at (0,-4.5) {$t \in L$};
\node[bag] at (2.5,-2) {\sameblockequiv{k}};
\node[bag] at (5,-4.5) {$t' \notin L$};
\node[bag] at (5,0) {$a$}
child {
node[bag] {$b^{k}$}
child {
node[bag] {$c$}
child {
node[bag] {$d^{k}$}
child {
node[bag] {$h_{1}$}
}
}
}
}
child {
node[bag] {$b^{k}$}
child {
node[bag] {$c'$}
child {
node[bag] {$d^{k}$}
child {
node[bag] {$h_{2}$}
}
}
}
}
child {
node[bag] {$b^{k}$}
child {
node[bag] {$c'$}
child {
node[bag] {$d^{k}$}
child {
node[bag] {$h_{3}$}
}
}
}
};
\end{tikzpicture}
\end{center}
\section{Discussion}
We have shown a decidable characterization for the class of locally testable
regular tree languages both for ranked trees and unranked unordered trees.
\paragraph{\bf Complexity}\label{section-complexity}
The decision procedure for deciding membership in LT as described in this paper
requires a time which is a tower of several exponentials in the size of the
deterministic minimal automaton recognizing $L$. This is most likely not
optimal. In comparison, over strings, membership in LT can be performed in
polynomial time~\cite{Pin05}. Essentially our procedure requires
two steps. The first step shows that if a regular language $L$ is in LT then
it is \testable{\kappa} for some $\kappa$ computable from the minimal
deterministic automaton $A$ recognizing $L$. The $\kappa$ obtained in
Proposition~\ref{prop-nec-implies-LT} is doubly exponential in the size of $A$.
In comparison, over strings, this $\kappa$ can be shown to be polynomial. For
trees we did not manage to get a smaller $\kappa$ but we have no example where
even one exponential would be necessary.
Our second step tests whether $L$ is \testable{\kappa} once $\kappa$ is fixed.
This was easy to do using a brute force algorithm requiring several
exponentials in $\kappa$. It is likely that this can be optimized but we didn't
investigate this direction.
However for unranked unordered trees we have seen in Theorem~\ref{theorem-ILT}
that in the case of ILT it is enough to test for \tameness.
The naive procedure for deciding tameness is exponential in the size of
$A$. But the techniques presented in~\cite{BS09}
for the case of LTT, easily extend to the closure properties of tameness, and
provide an algorithm running in time polynomial in the size of $A$. Hence
membership in ILT can be tested in time polynomial in the size of the minimal
deterministic bottom-up tree automaton recognizing the language.
\paragraph{\bf Logical characterization}
There is a logical characterization of languages that are locally testable. It
corresponds to those languages definable by formulas containing the temporal
predicates {\bf G} and {\bf X} where {\bf G} stands for ``everywhere in the
tree'' while {\bf X} stands for ``child''. In the binary tree case, we also
require two predicates distinguishing the left child from the right child. In
the unranked unordered setting the logic above is closed under bisimulation
and therefore corresponds to ILT. This shows that in a sense ILT is the natural
extension of LT to the unranked setting.
\paragraph{\bf Open problem}
It would be interesting to obtain a different characterization of LT based on a
finite number of conditions such as the ones characterizing tameness. This
would be a more satisfying result and would most likely provide a more efficient
algorithm for deciding LT.
\end{document} | math | 100,662 |
\begin{document}
\title[Submanifold geometry in symmetric spaces of noncompact type]{Submanifold geometry in symmetric spaces\\of noncompact type}
\author[J.~C.~D\'iaz-Ramos]{J.~Carlos~D\'iaz-Ramos}
\address{Department of Mathematics, University of Santiago de Compostela, Spain.}
\email{[email protected]}
\author[M.~Dom\'{\i}nguez-V\'{a}zquez]{Miguel Dom\'{\i}nguez-V\'{a}zquez}
\address{Instituto de Ciencias Matem\'aticas (CSIC-UAM-UC3M-UCM), Madrid, Spain.}
\email{[email protected]}
\author[V.~Sanmart\'in-L\'opez]{V\'ictor Sanmart\'in-L\'opez}
\address{Department of Mathematics, University of Santiago de Compostela, Spain.}
\email{[email protected]}
\thanks{The authors have been supported by the project MTM2016-75897-P (AEI/FEDER, Spain). The second author acknowledges support by the ICMAT Severo Ochoa project SEV-2015-0554 (MINECO, Spain), as well as by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No.~745722.}
\subjclass[2010]{53C35, 53C40, 53B25, 58D19, 57S20}
\begin{abstract}
In this survey article we provide an introduction to submanifold geometry in symmetric spaces of noncompact type. We focus on the construction of examples and the classification problems of homogeneous and isoparametric hypersurfaces, polar and hyperpolar actions, and homogeneous CPC submanifolds.
\end{abstract}
\keywords{Symmetric space, noncompact type, homogeneous submanifold, isometric action, polar action, hyperpolar action, cohomogeneity one action, isoparametric hypersurface, constant principal curvatures, CPC submanifold.}
\maketitle
\section{Introduction}
According to the original definition given by Cartan~\cite{Ca26}, a Riemannian symmetric space is a Riemannian manifold characterized by the property that curvature is invariant under parallel translation. This geometric definition has the surprising effect of bringing the theory of Lie groups into the picture, and it turns out that Riemannian symmetric spaces are intimately related to semisimple Lie groups. To a large extent, many geometric problems in symmetric spaces can be reduced to the study of properties of semisimple Lie algebras, thus transforming difficult geometric questions into linear algebra problems that one might be able to solve.
For this reason, the family of Riemannian symmetric spaces has been a setting where many geometric properties can be tackled and tested. They are often a source of examples and counterexamples. The set of symmetric spaces is a large family encompassing many interesting examples of Riemannian manifolds such as spaces of constant curvature, projective and hyperbolic spaces, Grassmannians, compact Lie groups and more. Apart from Differential Geometry, symmetric spaces have also been studied from the point of view of Global Analysis and Harmonic Analysis, being noncompact symmetric spaces of particular relevance (see for example~\cite{He08}). They are also an outstanding family in the theory of holonomy, constituting a class of their own in Berger's classification of holonomy groups.
Our interest in symmetric spaces comes from a very general question: the relation between symmetry and shape. In a broad sense, the symmetries of a mathematical object are the transformations of that object that leave it invariant. These symmetries impose several constraints that reduce the degrees of freedom of the object, and imply a regularity on its shape. More concretely, in Submanifold Geometry of Riemannian manifolds, our symmetric objects will actually be (extrinsically) {homogeneous submanifolds}, that is, submanifolds of a given Riemannian manifold that are orbits of a subgroup of isometries of the ambient manifold. In other words, a submanifold $P$ of a Riemannian manifold $M$ is said to be homogeneous if for any two points $p$, $q\in P$ there exists and isometry $\varphi$ of $M$ such that $\varphi(P)=P$ and $\varphi(p)=q$. The symmetries of $M$ are precisely the isometries $\varphi$ in this definition. Therefore, the study of homogeneous submanifolds makes sense only in ambient manifolds with a large group of isometries, and thus, the class of Riemannian symmetric spaces is an ideal setup for this problem.
Roughly speaking (see Section~\ref{sec:symmetric}) there are three types of symmetric spaces: Euclidean spaces, symmetric spaces of compact type (in case the group of isometries is compact semisimple) and symmetric spaces of noncompact type (if the group of isometries is noncompact semisimple). Symmetric spaces of compact and noncompact type are in some way dual to each other, and some of their properties can be carried from one type to the other. An example of this is the study of totally geodesic submanifolds. However, many properties are very different. Noncompact symmetric spaces are diffeomorphic to Euclidean spaces, and thus their topology is trivial, whereas in compact symmetric spaces topology does play a relevant role. In fact, symmetric spaces of noncompact type are isometric to solvable Lie groups endowed with a left-invariant metric. In our experience, this provides a wealth of examples of many interesting concepts, compared to their compact counterparts.
Our aim when studying homogeneous submanifolds is two-fold. Firstly, we are interested in the classification (maybe under certain conditions) of homogeneous submanifolds of a given Riemannian manifold up to isometric congruence. Usually we focus on the codimension one case, that is, homogeneous hypersurfaces, but we are also interested in higher codimension under some additional assumptions, for example when the group of isometries acts on the manifold polarly. An isometric action is said to be polar if there is a submanifold that intersects all orbits orthogonally. Such a submanifold is called a section of the polar action. If the section is flat, then the action is called hyperpolar. Polar actions take their name from polar coordinates, a concept that they generalize. Sections are usually seen as sets of canonical forms~\cite{PT:tams}, as it is often the case that in symmetric spaces sections are precisely the Jordan canonical forms of matrix groups.
The second problem that we would like to address is the characterization of (certain classes of) homogeneous submanifolds. It is obvious that homogeneity imposes restrictions on the geometry of a submanifold, and this in turn has implications on its shape. The question is whether a particular property imposed on shape by homogeneity is specific of homogeneous submanifolds or, on the contrary, there might be other submanifolds having this property. For example, homogeneous hypersurfaces have constant principal curvatures, and it is known in Euclidean and real hyperbolic spaces that this property characterizes homogeneous hypersurfaces. However, this is not the case in spheres, as there are examples of hypersurfaces with constant principal curvatures that are not homogeneous. We are particularly interested in isoparametric hypersurfaces, that is, hypersurfaces whose nearby equidistant hypersurfaces have constant mean curvature. It is easy to see that homogeneous hypersurfaces are isoparametric, but we will see in this survey to what extent the converse is true.
Finally, we also study CPC submanifolds, that is, submanifolds whose principal curvatures, counted with their multiplicities, are independent of the unit normal vector. This turns out to be an interesting notion related to several other properties whose study has recently attracted our attention.
This survey is organized as follows. In Section~\ref{sec:symmetric} we review the current definition, basic properties and types of Riemannian symmetric spaces, as well as the algebraic characterization of their totally geodesic submanifolds. Then, we deal more deeply with symmetric spaces of noncompact type in Section~\ref{sec:noncompact}, giving special relevance to the so-called Iwasawa decomposition of a noncompact semisimple Lie algebra. This implies that a symmetric space of noncompact type is isometric to certain Lie group with a left-invariant Riemannian metric. The simplest symmetric spaces, apart from Euclidean spaces, are symmetric spaces of rank one, which include spaces of constant curvature. Rank one symmetric spaces of noncompact type are studied in Section~\ref{sec:rank1}, where we discuss different results regarding homogeneous hypersurfaces, isoparametric hypersurfaces and polar actions. Finally, we study symmetric spaces of higher rank in Section~\ref{sec:higher_rank}. A refinement of the Iwasawa decomposition is obtained in terms of parabolic subgroups in this section, and this is used to provide certain results in this setting, such as an extension method for submanifolds and isometric actions. Moreover, we report on what is known about polar actions in this context, and explain a method to study homogeneous CPC submanifolds given by subgroups of the solvable part of the Iwasawa decomposition of the symmetric space.
\section{A quick review on symmetric spaces}\label{sec:symmetric}
In this section we include a short introduction to symmetric spaces. We first present the notion and first properties (\S\ref{subsec:notion}), then the different types of symmetric spaces (\S\ref{subsec:types}), and we conclude with an algebraic characterization of totally geodesic submanifolds (\S\ref{subsec:totally}).
There are several references that the reader may like to consult to obtain further information on this topic. Probably, the most well-known and complete references are Helgason's book~\cite{Helgason} and Loos' books~\cite{Loos1, Loos2}. Eschenburg's survey~\cite{Eschenburg} and Ziller's notes~\cite{Ziller} are great references, especially for beginners. The books by Besse~\cite{Besse}, Kobayashi and Nomizu~\cite{Kobayashi}, O'Neill~\cite{ONeill} and Wolf~\cite{Wolf} also include nice chapters on symmetric spaces. In this section we mainly follow~\cite{Helgason} and~\cite{Ziller}.
\subsection{The notion and first properties of a symmetric space.}\label{subsec:notion}
In any connected Riemannian manifold $M$ we can consider normal neighborhoods around any point $p\in M$. If we take a geodesic ball $B_p(r)=\{q\in M: d(p,q)<r\}$, with $r$ small enough, as one of these neighborhoods, we can always consider a smooth map $\sigma_p\colon B_p(r)\to B_p(r)$ that sends each $q=\exp_p(v)$ to $\sigma_p(q)=\exp_p(-v)$, for $v\in T_pM$, $|v|<r$; hereafter, $\exp$ denotes the Riemannian exponential map. The map $\sigma_p$ is an involution, i.e.\ $\sigma_p^2=\id$, which is called geodesic reflection. If, for any $p\in M$, one can define $\sigma_p$ in the same way globally in $M$ and $\sigma_p$ is an isometry of $M$, then we say that $M$ is a \emph{(Riemannian) symmetric space}.
It follows easily from the definition that symmetric spaces are complete (since geodesics can be extended by using geodesic reflections) and homogeneous, that is, for any $p_1$, $p_2\in M$ there is an isometry $\varphi$ of $M$ mapping $p_1$ to $p_2$ (take $\varphi=\sigma_q$, where $q$ is the midpoint of a geodesic joining $p_1$ and $p_2$). A Riemannian manifold $M$ is homogeneous if and only if the group $\Isom(M)$ of isometries of $M$ acts transitively on $M$. Then, $M$ is diffeomorphic to a coset space $G/K$ endowed with certain differentiable structure. Here, $G$ can be taken as the connected component of the identity element of the isometry group of $M$, i.e.\ $G=\Isom^0(M)$, which still acts transitively on $M$ since $M$ is assumed to be connected, whereas $K=\{g\in G: g(o)=o\}$ is the isotropy group of some (arbitrary but fixed) base point $o\in M$. As the isometry group of any Riemannian manifold is a Lie group, then $G$ is also a Lie group, and $K$ turns out to be a compact Lie subgroup of $G$.
Let us define the involutive Lie group automorphism $s\colon G\to G$, $g\mapsto \sigma_o g \sigma_o$, which satisfies $G_s^0\subset K\subset G_s$, where $G_s=\{g\in G:s(g)=g\}$ and $G_s^0$ is the connected component of the identity. The differential $\theta=s_*\colon \g{g}\to\g{g}$ of $s$ is a Lie algebra automorphism called the \emph{Cartan involution} of the symmetric space (at the Lie algebra level). The isotropy Lie algebra $\g{k}$ is the eigenspace of $\theta$ with eigenvalue $1$. Let $\g{p}$ be the $(-1)$-eigenspace of~$\theta$. The eigenspace decomposition of $\theta$ then reads $\g{g}=\g{k}\oplus\g{p}$, which is called the \emph{Cartan decomposition}.
Moreover, it easily follows that $[\g{k},\g{k}]\subset \g{k}$, $[\g{k},\g{p}]\subset\g{p}$ and $[\g{p},\g{p}]\subset\g{k}$. This implies, by the definition of the Killing form $\cal{B}$ of $\g{g}$ (recall: $\cal{B}(X,Y)=\tr(\ad(X)\circ\ad(Y))$ for $X$, $Y\in\g{g}$), that $\g{k}$ and $\g{p}$ are orthogonal subspaces with respect to $\cal{B}$.
By considering the map $\phi\colon G\to M$, $g\mapsto g(o)$, one easily gets that its differential $\phi_{*e}$ at the identity induces a vector space isomorphism $\g{p}\cong T_o M$. The linearization of the isotropy action of $K$ on $M$, which turns out to be the orthogonal representation $K\to\mathsf{GL}(T_oM)$, $k\mapsto k_{*o}$, is then equivalent to the adjoint representation $K\to \mathsf{GL}(\g{p})$, $k\to \Ad(k)$. Each one of these is called the \emph{isotropy representation} of the symmetric space.
\subsection{Types of symmetric spaces.}\label{subsec:types}
If the restriction of the isotropy representation of $M\cong G/K$ to the connected component of the identity of $K$ is irreducible, we say that the symmetric space $M$ is \emph{irreducible}. This turns out to be equivalent to the property that the universal cover $\widetilde{M}$ of $M$ (which is always a symmetric space) cannot be written as a nontrivial product of symmetric spaces, unless $\widetilde{M}$ is some Euclidean space~$\ensuremath{\mathbb{R}}^n$.
A symmetric space $M\cong G/K$ is said to be of \emph{compact type}, of \emph{noncompact type}, or of \emph{Euclidean type} if $\cal{B}\rvert_{\g{p}\times\g{p}}$, the restriction to $\g{p}$ of the Killing form $\cal{B}$ of $\g{g}$, is negative definite, positive definite, or identically zero, respectively. If $M$ is irreducible, then Schur's lemma implies that $\cal{B}\rvert_{\g{p}\times\g{p}}$ is a scalar multiple of the induced metric on $\g{p}\cong T_oM$ and, according to the sign of such scalar, $M$ falls into exactly one of the three possible types. It turns out that if $M$ is of compact type, then $G$ is a compact semisimple Lie group, and $M$ is compact and of nonnegative sectional curvature; if $M$ is of noncompact type, then $G$ is a noncompact real semisimple Lie group, and $M$ is noncompact (indeed, diffeomorphic to a Euclidean space) and with nonpositive sectional curvature; and if $M$ is of Euclidean type, its Riemannian universal cover is a Euclidean space~$\ensuremath{\mathbb{R}}^n$. Moreover, in general, the universal cover of a symmetric space $M$ splits as a product $\widetilde{M}=M_0\times M_+\times M_-$, where $M_0=\ensuremath{\mathbb{R}}^n$ is of Euclidean type, $M_+$ is of compact type, and $M_-$ is of noncompact type.
Symmetric spaces of compact and noncompact type are related via the notion of duality. Being more specific, there is a one-to-one correspondence between simply connected symmetric spaces of compact type and (necessarily simply connected) symmetric spaces of noncompact type. Moreover, dual symmetric spaces have equivalent isotropy representations and, therefore, irreducibility is preserved by duality. Without entering into details, the trick at the Lie algebra level to obtain the dual symmetric space is to change $\g{g}=\g{k}\oplus\g{p}$ by the new Lie algebra $\g{g}^*=\g{k}\oplus i\g{p}$, where $i=\sqrt{-1}$. In spite of the simplicity of this procedure, dual symmetric spaces have, of course, very different geometric and even topological properties. Examples of dual symmetric spaces are the following:
\begin{enumerate}
\item The round sphere $\mathbb{S}^n=\ensuremath{\mathsf{SO}}_{n+1}/\ensuremath{\mathsf{SO}}_n$ and the real hyperbolic space $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n=\ensuremath{\mathsf{SO}}^0_{1,n}/\ensuremath{\mathsf{SO}}_n$.
\item As a extension of the previous example, the projective spaces over the division algebras (other than $\ensuremath{\mathbb{R}}$) and their dual hyperbolic spaces: the complex spaces $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{P}}^n=\ensuremath{\mathsf{SU}}_{n+1}/\ensuremath{\mathsf{S}}(\ensuremath{\mathsf{U}}_1\ensuremath{\mathsf{U}}_n)$ and $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n=\ensuremath{\mathsf{SU}}_{1,n}/\ensuremath{\mathsf{S}}(\ensuremath{\mathsf{U}}_1\ensuremath{\mathsf{U}}_n)$, the quaternionic spaces $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{P}}^n=\ensuremath{\mathsf{Sp}}_{n+1}/\ensuremath{\mathsf{Sp}}_1\ensuremath{\mathsf{Sp}}_n$ and $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n=\ensuremath{\mathsf{Sp}}_{1,n}/\ensuremath{\mathsf{Sp}}_1\ensuremath{\mathsf{Sp}}_n$, and the Cayley planes $\mathbb{O} \ensuremath{\mathsf{P}}^2=\mathsf{F}_4/\ensuremath{\mathsf{Sp}}in_9$ and $\mathbb{O} \ensuremath{\mathsf{H}}^2=\mathsf{F}_4^{-20}/\ensuremath{\mathsf{Sp}}in_9$. The spaces in this and the previous item, jointly with the real projective spaces $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{P}}^n$, constitute the so-called \emph{rank one symmetric spaces}.
\item The oriented compact Grassmannian $\mathsf{G}^+_{p}(\ensuremath{\mathbb{R}}^{p+q})=\ensuremath{\mathsf{SO}}_{p+q}/\ensuremath{\mathsf{SO}}_p\ensuremath{\mathsf{SO}}_q$ of all oriented $p$-dimensional subspaces of $\ensuremath{\mathbb{R}}^{p+q}$, and the dual noncompact Grassmannian $\mathsf{G}_{p}(\ensuremath{\mathbb{R}}^{p,q})=\ensuremath{\mathsf{SO}}^0_{p,q}/\ensuremath{\mathsf{SO}}_p\ensuremath{\mathsf{SO}}_q$, which para\-met\-rizes all $p$-dimensional timelike subspaces of the semi-Euclidean space $\ensuremath{\mathbb{R}}^{p,q}$ of dimension $p+q$ and signature $(p,q)$. This example can be extended to complex and quaternionic Grassmannians.
\item Any compact semisimple Lie group $G$ with bi-invariant metric, whose coset space is given by $(G\times G)/\Delta G$, and its noncompact dual symmetric space $G^\ensuremath{\mathbb{C}}/G$, where $G^\ensuremath{\mathbb{C}}$ is the complex semisimple Lie group given by the complexification of $G$. For instance, $\ensuremath{\mathsf{SU}}_n$ and $\ensuremath{\mathsf{SL}}_n(\ensuremath{\mathbb{C}})/\ensuremath{\mathsf{SU}}_n$ are dual symmetric spaces.
\item The space $\ensuremath{\mathsf{SU}}_n/\ensuremath{\mathsf{SO}}_n$ of all Lagrangian subspaces of $\ensuremath{\mathbb{R}}^{2n}$, and its noncompact dual space $\ensuremath{\mathsf{SL}}_n(\ensuremath{\mathbb{R}})/\ensuremath{\mathsf{SO}}_n$ of all positive definite symmetric matrices of determinant~$1$.
\end{enumerate}
The whole list of simply connected, irreducible symmetric spaces can be found, for example, in~\cite[pp.~516, 518]{Helgason}.
\begin{remark}
In some cases above we have written $M=G/K$, where the action of $G$ on $M$ is not necessarily effective (i.e.\ not necessarily $G=\Isom^0(M)$). However, in all cases such $G$-action is almost effective, that is, the ineffective kernel $\{g\in G:g(p)=p, \text{ for all } p\in M\}$ of the $G$-action on $M$ is a discrete subgroup of~$G$. Being more precise, one always considers a so-called symmetric pair $(G,K)$, where $K$ is compact, there is an involutive automorphism $s$ of $G$ such that $G_s^0\subset K\subset G_s$, and $G$ acts almost effectively on $M=G/K$. For example, the complex hyperbolic space $\ensuremath{\mathbb{C}} H^n$ is usually expressed as $\ensuremath{\mathsf{SU}}_{1,n}/\ensuremath{\mathsf{S}}(\ensuremath{\mathsf{U}}_1\ensuremath{\mathsf{U}}_n)$ instead of $(\mathsf{SU}_{1,n}/\mathbb{Z}_{n+1})/(\ensuremath{\mathsf{S}}(\ensuremath{\mathsf{U}}_1\ensuremath{\mathsf{U}}_n)/\mathbb{Z}_{n+1})$, in spite of the fact that $\ensuremath{\mathsf{SU}}_{1,n}$ has the cyclic group $\mathbb{Z}_{n+1}$ as ineffective kernel. This practice is common in the study of symmetric spaces for simplicity reasons, and because all Lie algebras involved remain the same. The symmetric pairs $(G,K)$ of compact type with $G=\Isom^0(M)$ can be found in~\cite[pp.~324-325]{Wang-Ziller}.
\end{remark}
\subsection{Totally geodesic submanifolds.}\label{subsec:totally}
Among different kinds of Riemannian submanifolds, the totally geodesic ones typically play an important role. This is particularly true~in the case of symmetric spaces. Indeed, although the classification of totally geodesic submanifolds in symmetric spaces is still an outstanding problem, these submanifolds are, intrinsically, also symmetric, and admit a neat algebraic characterization, which we recall~below.
A vector subspace $\g{s}$ of a Lie algebra $\g{g}$ is called a \emph{Lie triple system} if $[[X, Y],Z]\in\g{s}$ for any $X,Y,Z\in\g{s}$. Let now $\g{g}=\g{k}\oplus\g{p}$ be a Cartan decomposition of a symmetric space $M\cong G/K$, corresponding to a base point $o\in M$, as above. A fundamental result states that, if $\g{s}$ is a Lie triple system of $\g{g}$ contained in $\g{p}$, then $\exp_o(\g{s})$ is a totally geodesic submanifold of $M$, and it is intrinsically a symmetric space itself. And conversely, if $S$ is a totally geodesic submanifold of $M$, and $o\in S$, then $\g{s}:=T_oS\subset T_oM\cong\g{p}$ is a Lie triple system. In this situation, $\g{h}=[\g{s},\g{s}]\oplus\g{s}$ is the Cartan decomposition of the Lie algebra $\g{h}$ of the isometry group of the symmetric space $S$. Indeed, there is a one-to-one correspondence between $\theta$-invariant subalgebras of $\g{g}$ and Lie triple systems.
A consequence of the previous characterization is the fact that totally geodesic submanifolds of symmetric spaces are preserved under duality: if $\g{s}\subset\g{p}$ is a Lie triple system in $\g{g}=\g{k}\oplus\g{p}$, then $i\g{s}\subset i\g{p}$ is a Lie triple system in $\g{g}^*=\g{k}\oplus i\g{p}$. Moreover, if $\g{s}\subset \g{p}$ is a Lie triple system, then $S=\exp_o(\g{s})$ is an intrinsically flat submanifold if and only if $\g{s}$ is an abelian subspace of $\g{p}$ (i.e.\ $[\g{s},\g{s}]=0$). This follows from the Gauss equation of submanifold geometry, the property that $S$ is totally geodesic, and the fact that the curvature tensor $R$ of a symmetric space at the base point $o$ is given by
\begin{equation}\label{eq:curvature}
R(X,Y)Z=-[[X,Y],Z], \qquad X,Y,Z\in T_oM\cong \g{p}.
\end{equation}
Thus, one defines the \emph{rank} of a symmetric space $M$ as the maximal dimension of a totally geodesic and flat submanifold of $M$ or, equivalently, the dimension of a maximal abelian subspace of $\g{p}$. Clearly, the rank is an invariant that is preserved under duality.
In spite of the above algebraic characterization of totally geodesic submanifolds of symmetric spaces and the fact that, by duality, one can restrict to symmetric spaces of compact type (or of noncompact type), the classification problem remains open. In particular, one does not know any efficient procedure to classify Lie triple systems in general.
Totally geodesic submanifolds of rank one symmetric spaces are well known (see~\cite[~\S3]{Wolf:totally}). The case of rank two is much more involved, and has been addressed by Chen and Nagano~\cite{Chen-Nagano1, Chen-Nagano2} and Klein~\cite{Klein:tams, Klein:osaka}. Apart from these works, the subclass of the so-called reflective submanifolds has been completely classified by Leung~\cite{Leung:indiana, Leung:jdg}. A submanifold of a symmetric space $M$ is called \textit{reflective} if it is a connected component of the fixed point set of an involutive isometry of $M$; or, equivalently, if it is a totally geodesic submanifold such that the exponentiation of one (and hence all) normal space is also a totally geodesic submanifold. Finally, let us mention that the index of symmetric spaces (that is, the smallest possible codimension of a proper totally geodesic submanifold) has been recently investigated by Berndt and Olmos~\cite{BO:crelle, BO:blms, BO:jdg}, who proved, in particular, that the index of an irreducible symmetric space is bounded from below by the rank.
Further information on totally geodesic submanifolds of symmetric spaces can be found in~\cite[\S11.1]{BCO}.
\section{Symmetric spaces of noncompact type and their Lie group model}\label{sec:noncompact}
In this section we focus on symmetric spaces of noncompact type. Our goal will be to explain that any symmetric space of noncompact type is isometric to a Lie group endowed with a left-invariant metric. The reader looking for more information or detailed proofs can consult, for instance, Eberlein's~\cite[Chapter~2]{Eberlein}, Helgason's~\cite[Chapter~VI]{Helgason} or Knapp's books~\cite[Chapter VI, \S4-5]{Knapp}. A nice survey that includes a detailed description of the space $\ensuremath{\mathsf{SL}}_n(\ensuremath{\mathbb{R}})/\ensuremath{\mathsf{SO}}_n$ can be found in~\cite{Berndt:hyperpolar}. In this section we mainly follow \cite{Berndt:hyperpolar} and~\cite{Knapp}.
\begin{example}\label{ex:RH^2}
The real hyperbolic plane $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$ is the most basic example of symmetric space of noncompact type, and the only one of dimension at most two. It is well-known that (as any other symmetric space of noncompact type) it is diffeomorphic to an open ball, which gives rise, with an appropriate metric, to the Poincar\'e disk model for $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$. Let us consider, however, the half-space model, by regarding $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$ as the set $\{z\in \ensuremath{\mathbb{C}}: \ensuremath{\mathrm{Im}\;} z>0\}$ with metric $\langle\cdot,\cdot\rangle_{\ensuremath{\mathbb{R}}^2}/(\ensuremath{\mathrm{Im}\;} z)^2$. Then, the group $G=\ensuremath{\mathsf{SL}}_2(\ensuremath{\mathbb{R}})$ acts transitively, almost effectively and by isometries on $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$ via M\"obius transformations:
\[
\begin{pmatrix}
a &b \\ c & d
\end{pmatrix}\cdot z=\frac{az+d}{cz+d}.
\]
Then, the isotropy group $K$ at the base point $o=\sqrt{-1}$ is $\ensuremath{\mathsf{SO}}_2$, and hence $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2=\ensuremath{\mathsf{SL}}_2(\ensuremath{\mathbb{R}})/\ensuremath{\mathsf{SO}}_2$. Moreover, any matrix in $\ensuremath{\mathsf{SL}}_2(\ensuremath{\mathbb{R}})$ can be decomposed in a unique way~as
\[
\begin{pmatrix}
a &b \\ c & d
\end{pmatrix}
=
\begin{pmatrix}
\cos s & \sin s \\ -\sin s & \cos s
\end{pmatrix}
\begin{pmatrix}
\lambda &0 \\ 0 & \lambda^{-1}
\end{pmatrix}
\begin{pmatrix}
1 &u \\ 0 & 1
\end{pmatrix}, \qquad \text{where } s, u\in\ensuremath{\mathbb{R}}, \lambda>0.
\]
From an algebraic viewpoint, this decomposition turns out to encode some of the elements involved in the Gram-Schmidt process applied to the basis of $\ensuremath{\mathbb{R}}^2$ given by the column vectors of the left-hand side matrix: the orthogonal matrix is the transition matrix from the orthonormal basis produced by the method to the canonical basis, whereas the diagonal and upper triangular matrices contain the coefficients calculated in the process.
Moreover, the matrices on the right-hand side define three subgroups of $\ensuremath{\mathsf{SL}}_2(\ensuremath{\mathbb{R}})$, namely $K=\ensuremath{\mathsf{SO}}_2$, the abelian subgroup $A$ of diagonal matrices, and the nilpotent subgroup $N$ of unipotent upper-triangular matrices. This so-called Iwasawa decomposition $G=KAN$ can be extended to any symmetric space of noncompact type, as we will soon explain.
From a geometric perspective, we can get insight into the groups involved in the decomposition by looking at their isometric actions on the hyperbolic plane (see~Figure~\ref{fig:KAN}). Thus, the $K$-action fixes $o$ and the other orbits are geodesic spheres around $o$, the orbits of the $A$-action are a geodesic through $o$ and equidistant curves to such geodesic, while the $N$-action produces the horocycle foliation of $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$ centered at one of the two points at infinity of the geodesic $A\cdot o$. This description of the actions make also intuitively clear the important fact that $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2\cong G/K$ is diffeomorphic to the subgroup $AN$ of $G$.
\end{example}
\begin{figure}
\caption{Orbit foliations of the actions of the groups $K$, $A$ and $N$ on $\ensuremath{\mathbb{R}
\label{fig:KAN}
\end{figure}
We now move on to the general setting. We start by describing some important decompositions of the Lie algebra of the isometry group (\S\ref{subsec:decompositions}), and then we present the Lie group model of a symmetric space of noncompact type~(\S\ref{subsec:model}).
\subsection{Root space and Iwasawa decompositions.}\label{subsec:decompositions}
Let $M\cong G/K$ be an arbitrary symmetric space of noncompact type. Then $\g{g}$ is a real semisimple Lie algebra, which implies that its Killing form $\cal{B}$ is nondegenerate. Indeed, the Cartan decomposition $\g{g}=\g{k}\oplus\g{p}$ is $\cal{B}$-orthogonal, $\cal{B}\rvert_{\g{k}\times\g{k}}$ is negative definite (due to the compactness of $K$), and $\cal{B}\rvert_{\g{p}\times\g{p}}$ is positive definite (since $M$ is of noncompact type). Hence, by reverting the sign on $\g{k}\times\g{k}$ or, equivalently, by defining $\cal{B}_\theta(X,Y)=-\cal{B}(\theta X, Y)$, for $X,Y\in\g{g}$, we have that $\cal{B}_\theta$ defines a positive definite inner product on $\g{g}$. It is easy to check that this inner product satisfies $\cal{B}_\theta(\ad (X)Y, Z)=-\cal{B}_\theta(Y, \ad(\theta X)Z)$, $X,Y,Z\in\g{g}$.
Let $\g{a}$ be a maximal abelian subspace of $\g{p}$. One can show that any two choices of $\g{a}$ are conjugate under the adjoint action of $K$ (similar to the fact that any two maximal abelian subalgebras of a compact Lie algebra are conjugate to each other). Moreover, by definition, the rank of $M\cong G/K$ is the dimension of $\g{a}$.
For each $H\in\g{a}$, $X,Y\in\g{g}$, we have that
\[
\cal{B}_\theta(\ad(H)X,Y)=-\cal{B}_\theta(X,\ad(\theta H)Y)=\cal{B}_\theta(X,\ad(H)Y),
\]
which means that each operator $\ad(H)\in\ensuremath{\mathrm{End}}(\g{g})$ is self-adjoint with respect to $\cal{B}_\theta$. Moreover, if $H_1, H_2\in\g{a}$, then $[\ad(H_1),\ad(H_2)]=\ad[H_1,H_2]=0$, since $\ad\colon \g{g}\to \ensuremath{\mathrm{End}}(\g{g})$ is a Lie algebra homomorphism and $\g{a}$ is abelian. Thus, $\{\ad(H):H\in\g{a}\}$ constitutes a commuting family of self-adjoint endomorphisms of $\g{g}$. Therefore, they diagonalize simultaneously. Their common eigenspaces are called the \textit{restricted root spaces}, whereas their nonzero eigenvalues (which depend linearly on $H\in\g{a}$) are called the \textit{restricted roots} of $\g{g}$. In other words, if for each covector $\lambda\in\g{a}^*$ we define
\[
\g{g}_\lambda=\{X\in\g{g}: [H,X]=\lambda(H)X \text{ for all } H\in\g{a}\},
\]
then any $\g{g}_\lambda\neq 0$ is a restricted root space, and any $\lambda\neq 0$ such that $\g{g}_\lambda\neq 0$ is a restricted root. Note that $\g{g}_0$ is always nonzero, as $\g{a}\subset \g{g}_0$. If $\Sigma=\{\lambda\in\g{a}^*:\lambda\neq 0,\,\g{g}_\lambda\neq 0\}$ denotes the set of restricted roots, then we have the following $\cal{B}_\theta$-orthogonal decomposition
\begin{equation}\label{eq:root_space_decomposition}
\g{g}=\g{g}_0\oplus\biggl(\bigoplus_{\lambda\in\Sigma}\g{g}_\lambda\biggr),
\end{equation}
which is called the \emph{restricted root space decomposition} of $\g{g}$.
Observe that these definitions depend on the choice of $o\in M$ (or, equivalently, of a Cartan involution $\theta$ of $\g{g}$) and of the choice of the maximal abelian subspace $\g{a}$ of $\g{p}$. However, different choices of $o$ and $\g{a}$ give rise to decompositions that are conjugate under the adjoint action of $G$. For simplicity, in this article we will not specify this dependence and we will also omit the adjective ``restricted".
It is easy to check that the following properties are satisfied:
\begin{enumerate}[(i)]
\item $[\g{g}_\lambda,\g{g}_\mu]\subset\g{g}_{\lambda+\mu}$, for any $\lambda$, $\mu\in\g{a}^*$.
\item $\theta \g{g}_\lambda=\g{g}_{-\lambda}$ and, hence, $\lambda\in\Sigma$ if and only if $-\lambda\in \Sigma$.
\item $\g{g}_0=\g{k}_0\oplus\g{a}$, where $\g{k}_0=\g{g}_0\cap\g{k}$ is the normalizer of $\g{a}$ in $\g{k}$.
\end{enumerate}
For each $\lambda\in \Sigma$, define $H_\lambda\in\g{a}$ by the relation $\cal{B}(H_\lambda,H)=\lambda(H)$, for all $H\in\g{a}$. Then we can introduce an inner product on $\g{a}^*$ by $\langle \lambda,\mu\rangle:=\cal{B}(H_\lambda,H_\mu)$. Thus, with a bit more work one can show that $\Sigma$ is an abstract root system in $\g{a}^*$, that is, it satisfies (cf.~\cite[\S{}II.5]{Knapp}):
\begin{enumerate}[(a)]
\item $\g{a}^*=\spann\Sigma$,
\item for $\alpha,\beta\in \Sigma$, the number
$a_{\alpha\beta} =2\langle\alpha,\beta\rangle/
\langle\alpha,\alpha\rangle$ is an integer,
\item for $\alpha,\beta\in \Sigma$, we have
$\beta-a_{\alpha\beta}\,\alpha\in \Sigma$.
\end{enumerate}
This system may be nonreduced, that is, there may exist $\lambda\in\Sigma$ such that $2\lambda\in \Sigma$.
Now we can define a positivity criterion on $\Sigma$ by declaring those roots that lie at one of the two half-spaces determined by a hyperplane in $\g{a}^*$ not containing any root to be positive. If $\Sigma^+$ denotes the set of positive roots, then $\Sigma=\Sigma^+\cup(-\Sigma^+)$. As is usual in the theory of root systems, one can consider a subset $\ensuremath{\mathsf{P}}i\subset \Sigma^+$ of simple roots, that is, a basis of $\g{a}^*$ made of positive roots such that any $\lambda\in \Sigma$ is a linear combination of the roots in $\ensuremath{\mathsf{P}}i$ where all coefficients are either nonnegative integers or nonpositive integers. Of course, the cardinality of $\ensuremath{\mathsf{P}}i$ agrees with the dimension of $\g{a}$, i.e.\ with the rank of $G/K$. The set $\ensuremath{\mathsf{P}}i$ of simple roots allows to construct the Dynkin diagram attached to the root system $\Sigma$, which is a graph whose nodes are the simple roots, and any two of them are joined by a simple (respectively, double, triple) edge whenever the angle between the corresponding roots is $2\pi/3$ (respectively, $3\pi/4$, $5\pi/6$); moreover, if the system is nonreduced, two collinear positive roots are drawn as two concentric nodes.
Due to the properties of the root space decomposition, the subspace
\[
\g{n}=\bigoplus_{\lambda\in \Sigma^+}\g{g}_\lambda
\]
of $\g{g}$ is a nilpotent subalgebra of $\g{g}$. Moreover, $\g{a}\oplus\g{n}$ is a solvable subalgebra of $\g{g}$ such that $[\g{a}\oplus\g{n},\g{a}\oplus\g{n}]=\g{n}$. Any two choices of positivity criteria on $\Sigma$ give rise to isomorphic Dynkin diagrams and to nilpotent subalgebras $\g{n}$ that are conjugate by an element of $N_K(\g{a})$.
A fundamental result in what follows is the \emph{Iwasawa decomposition theorem}. At the Lie algebra level, it states that
\[
\g{g}=\g{k}\oplus\g{a}\oplus\g{n}
\]
is a vector space direct sum (but neither orthogonal direct sum nor semidirect product). Let us denote by $A$ and $N$ the connected Lie subgroups of $G$ with Lie algebras $\g{a}$ and $\g{n}$, respectively. Since $\g{a}$ normalizes $\g{n}$, the semidirect product $AN$ is the connected Lie subgroup of $G$ with Lie algebra $\g{a}\oplus\g{n}$. Then the Iwasawa decomposition theorem at the Lie group level states that the multiplication map
\[
K\times A\times N\to G,\qquad (k,a,n)\mapsto kan
\]
is an analytic diffeomorphism, and the Lie groups $A$ and $N$ are simply connected. Indeed, as $A$ is abelian and $N$ is nilpotent, they are both diffeomorphic to Euclidean spaces~\cite[Theorem~1.127]{Knapp}. Hence, the semidirect product $AN$ is also diffeomorphic to a Euclidean~space.
\subsection{The solvable Lie group model.}\label{subsec:model}
Recall the smooth map $\phi\colon G\to M$, $h\mapsto h(o)$, from the end of \S\ref{subsec:notion} . The restriction $\phi\rvert_{AN}\colon AN\to M$ is injective; indeed, if $\phi(h)=\phi(h')$ with $h,h'\in AN$, then $h^{-1}h'(o)=o$, and hence $h^{-1}h'\in K\cap AN$, which, by the Iwasawa decomposition, implies that $h^{-1}h'=e$. It is also onto: if $p\in M$, then by the transitivity of $G$ there exists $h\in G$ such that $h(p)=o$, but using the Iwasawa decomposition we can write $h=kan$, with $k\in K$, $a\in A$, $n\in N$, and then $p=h^{-1}(o)=n^{-1}a^{-1}k^{-1}(o)=(an)^{-1}(o)$. Finally, $\phi\rvert_{AN}$ is a local diffeomorphism: as $\ker\phi_{*e}=\g{k}$, we have that $(\phi\rvert_{AN})_{*e}\colon \g{a}\oplus\g{n}\to T_oM$ is an isomorphism, and, by homogeneity, any other differential $(\phi\rvert_{AN})_{*h}$ is also bijective.
Therefore, $\phi\rvert_{AN}\colon AN\to M$ is a diffeomorphism. If we denote by $g$ the Riemannian metric on $M$, we can pull it back to obtain a Riemannian metric $(\phi\rvert_{AN})^*g$ on $AN$. Hence, we trivially have that $(M,g)$ and $(AN,(\phi\rvert_{AN})^*g)$ are isometric Riemannian manifolds.
Let now $h,h'\in AN\subset G$, and denote by $L_h$ the left multiplication by $h$ in $G$. Then
\[
(h^{-1}\circ \phi\rvert_{AN}\circ L_h)(h')=h^{-1}(hh'(o))=h'(o)=\phi\rvert_{AN}(h'),
\]
from where we get $h^{-1}\circ \phi\rvert_{AN}\circ L_h=\phi\rvert_{AN}$ as maps from $AN$ to $M$. Hence, since $h^{-1}$ is an isometry of $(M,g)$, and using the previous equality, we have
\[
L_h^*(\phi\rvert_{AN})^*g=L_h^*(\phi\rvert_{AN})^*(h^{-1})^*g=(h^{-1}\circ \phi\rvert_{AN}\circ L_h)^*g=(\phi\rvert_{AN})^*g.
\]
This shows that $(\phi\rvert_{AN})^*g$ is a left-invariant metric on the Lie group $AN$.
Altogether, we have seen that \textit{any symmetric space $M\cong G/K$ of noncompact type is isometric to a solvable Lie group $AN$ endowed with a left-invariant metric}. In particular, any symmetric space of noncompact type is diffeomorphic to a Euclidean space and, since it is nonpositively curved, it is a Hadamard manifold. This allows us to regard any of these spaces as an open Euclidean ball endowed with certain metric, as happens with the ball model of the real hyperbolic space.
Moreover, it is sometimes useful to view a symmetric space of noncompact type $M$ as a dense, open subset of a bigger compact topological space $M\cup M(\infty)$ which, in this case, would be homeomorphic to a closed Euclidean ball. In order to do so, one defines an equivalence relation on the family of complete, unit-speed geodesics in $M$: if $\gamma$ and $\sigma$ are two of them, we declare them equivalent if they are asymptotic, that is, if $d(\gamma(t),\sigma(t))\leq C$, for certain constant $C$ and for all $t\geq 0$. Each equivalence class of asymptotic geodesics is called a \emph{point at infinity}, and the set $M(\infty)$ of all of them is the \emph{ideal boundary} of $M$. By endowing $M\cup M(\infty)$ with the so-called cone topology, $M\cup M(\infty)$ becomes homeomorphic to a closed Euclidean ball whose interior corresponds to $M$ and its boundary to $M(\infty)$. Two geodesics are asymptotic precisely when they converge to the same point in $M(\infty)$. We refer to~\cite[\S1.7]{Eberlein} for more details.
The Lie group model turns out to be a powerful tool for the study of submanifolds of symmetric spaces of noncompact type. The reason is that one can \textit{consider interesting types of submanifolds by looking at subgroups of $AN$} or, equivalently, at subalgebras of $\g{a}\oplus\g{n}$. For this reason, a good understanding of the root space decomposition is crucial. Of course, not every submanifold (even extrinsically homogeneous submanifold) of $M$ can be regarded as a Lie subgroup of $AN$, but very important types of examples arise in this way, sometimes combined with some additional constructions, as we will comment on in the following sections. In any case, if one wants to study submanifolds of $AN$ with particular geometric properties, one needs to have manageable expressions for the left-invariant metric on $AN$ and its Levi-Civita connection. We obtain the appropriate formulas below.
Let us denote by $\langle\cdot,\cdot\rangle_{AN}$ the inner product on $\g{a}\oplus\g{n}$ given by the left-invariant metric $(\phi\rvert_{AN})^*g$ on $AN$. Assume for the moment that $M$ is irreducible. Then, as mentioned in~\S\ref{subsec:notion}, the inner product $\phi^*g_o$ on $T_oM$ induced by the metric $g$ on $M$ is a scalar multiple of modified Killing form $\cal{B}_\theta$, i.e.\ $\phi^*g_o=k\cal{B}_\theta$, for some $k>0$. Let us define the inner product $\langle\cdot,\cdot\rangle:=k \cal{B}_\theta$ on $\g{g}$, and find the relation between $\langle\cdot,\cdot\rangle_{AN}$ and $\langle\cdot,\cdot\rangle$. Thus, if $X,Y\in\g{a}\oplus\g{n}$, and denoting orthogonal projections (with respect to $\cal{B}_\theta$) with subscripts, we have
\begin{equation}\label{eq:relation_inner}
\begin{aligned}
\langle X,Y\rangle_{AN} &{}=
(\phi\rvert_{AN})^*g_o(X_\mathfrak{k}+X_\mathfrak{p},Y_\mathfrak{k}+Y_\mathfrak{p})=g_o(\phi_*X_\mathfrak{p},\phi_* Y_\mathfrak{p})= k \cal{B}_\theta(X_\mathfrak{p},Y_\mathfrak{p})
\\
&{}= k \cal{B}_\theta \biggl(\frac{1-\theta}{2}X,\frac{1-\theta}{2}Y\biggr)
=\frac{k}{4}\,\cal{B}_\theta(2X_\mathfrak{a}+X_\mathfrak{n}-\theta X_\mathfrak{n},2Y_\mathfrak{a}+Y_\mathfrak{n}-\theta Y_\mathfrak{n})
\\
&{}=\frac{k}{4}\left(4\cal{B}_\theta(X_\mathfrak{a},Y_\mathfrak{a})+\cal{B}_\theta(X_\mathfrak{n},Y_\mathfrak{n})+\cal{B}_\theta(\theta X_\mathfrak{n},\theta Y_\mathfrak{n})\right)\\
&{}= k\bigl(\cal{B}_\theta(X_\mathfrak{a},Y_\mathfrak{a})+\frac{1}{2}\cal{B}_\theta(X_\mathfrak{n},Y_\mathfrak{n})\bigr)
\\
&{}=
\langle X_\mathfrak{a},Y_\mathfrak{a}\rangle+\frac{1}{2}\langle X_\mathfrak{n},Y_\mathfrak{n}\rangle.
\end{aligned}
\end{equation}
If $M$ is reducible, one can adapt the argument (by defining $\langle\cdot,\cdot\rangle$ as a suitable multiple of $\cal{B}_\theta$ on each factor) to prove the same formula. Note that $\langle\cdot,\cdot\rangle$ inherits from $\cal{B}_\theta$ the property
\begin{equation}
\label{eq:prop_B_theta}
\langle\ad (X)Y, Z\rangle=-\langle Y, \ad(\theta X)Z\rangle, \qquad \text{for } X,Y,Z\in\g{g}.
\end{equation}
Using Koszul formula, and relations~\eqref{eq:relation_inner} and~\eqref{eq:prop_B_theta}, one can obtain an important formula for the Levi-Civita connection $\nabla$ of the Lie group $AN$. Indeed, if $X,Y,Z\in\g{a}\oplus\g{n}$, and taking into account that $[\g{a}\oplus\g{n},\g{a}\oplus\g{n}]\subset\g{n}$, we have
\begin{equation}\label{eq:Levi-Civita}
\begin{aligned}
\langle \nabla_X Y,Z\rangle_{AN}
&{}= \frac{1}{2}\bigl(\langle [X,Y],Z\rangle_{AN} - \langle [Y,Z],X\rangle_{AN} -\langle [X,Z],Y\rangle_{AN} \bigr)
\\
&{}= \frac{1}{4}\bigl( \langle [X,Y],Z\rangle - \langle [Y,Z],X\rangle -\langle [X,Z],Y\rangle \bigr)
\\
&{}=\frac{1}{4}\langle [X,Y]+[\theta X,Y]-[X,\theta Y],Z\rangle.
\end{aligned}
\end{equation}
Note that we started and finished with different inner products. Thus, in order to obtain an explicit formula for $\nabla_X Y$ one has to impose some restrictions on $X$ and $Y$. For example, if $X$ and $Y$ do not belong to the same root space, then $[\theta X,Y]$ and $[X,\theta Y]$ are orthogonal to $\g{a}$, whence in this case $2\nabla_X Y=\bigl([X,Y]+[\theta X,Y]-[X,\theta Y])_{\g{a}\oplus\g{n}}$.
\section{Submanifolds of rank one symmetric spaces}\label{sec:rank1}
In this section we review some results about certain important types of submanifolds in the rank one symmetric spaces of noncompact type. We start by describing these spaces in further detail (\S\ref{subsec:rank1}), and then we comment on homogeneous hypersurfaces (\S\ref{subsec:rank1_homogeneous}), isoparametric hypersurfaces (\S\ref{subsec:rank1_isoparametric}), and polar actions (\S\ref{subsec:rank1_polar}) on these spaces.
\subsection{Rank one symmetric spaces}\label{subsec:rank1}
As mentioned in \S\ref{subsec:types}, the symmetric spaces of noncompact type and rank one are the hyperbolic spaces $\mathbb{F} \ensuremath{\mathsf{H}}^n$, $n\geq 2$, over the distinct division algebras, $\mathbb{F}\in\{\ensuremath{\mathbb{R}},\ensuremath{\mathbb{C}},\ensuremath{\mathbb{H}}, \mathbb{O}\}$ ($n=2$ if $\mathbb{F}=\mathbb{O}$). We observe that $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^1$, $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^1$ and $\mathbb{O}\ensuremath{\mathsf{H}}^1$ are isometric (up to rescaling of the metric) to $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$, $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^4$ and $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^8$, respectively.
The isotropy representation (i.e.\ the adjoint action of $K$ on $\g{p}$) of rank one symmetric spaces is transitive on the unit sphere of $\g{p}$. Therefore, these Riemannian manifolds are not only homogeneous, but also \textit{isotropic}, which implies that they are \textit{two-point homogeneous}. Indeed, two-point homogeneous Riemannian manifolds are symmetric and, except for Euclidean spaces, have rank one~\cite{Szabo}, \cite[\S8.12]{Wolf}, and therefore the only noncompact examples (other than Euclidean spaces) are the symmetric spaces of noncompact type and rank one. Moreover, they are precisely the \textit{symmetric spaces of strictly negative sectional curvature} (even more, their sectional curvature is pinched between $c$ and $c/4$ for some $c<0$).
The root space decomposition~\eqref{eq:root_space_decomposition} of a symmetric space $M=\mathbb{F}\ensuremath{\mathsf{H}}^n$ of rank one is rather simple. One can show that $\Sigma=\{-\alpha,\alpha\}$ if $M=\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$, and $\Sigma=\{-2\alpha,-\alpha,\alpha,2\alpha\}$ otherwise. Thus the root space decomposition can be rewritten as
\[
\g{g}=\g{k}_0\oplus\g{a}\oplus\g{g}_{-2\alpha}\oplus\g{g}_{-\alpha}\oplus\g{g}_\alpha\oplus\g{g}_{2\alpha}.
\]
Of course, $\g{a}\cong\ensuremath{\mathbb{R}}$, $\g{g}_{-2\alpha}\cong\g{g}_{2\alpha}$ and $\g{g}_{-\alpha}\cong\g{g}_{\alpha}$. Note that the connected subgroup $K_0$ of $K$ with Lie algebra $\g{k}_0$ normalizes each one of the spaces in the decomposition. See Table~\ref{table:rank_one} for the explicit description of the group $K_0$ and the spaces in the decomposition. Determining all this information involves a few linear algebra computations; see~\cite[Chapter~2]{DV:dea} for the case of the complex hyperbolic space $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$.
According to \S\ref{subsec:model}, a symmetric space $M\cong G/K$ of noncompact type is isometric to a Lie group $AN$ with a left-invariant metric. In the rank one setting, $A$ is one-dimensional, and, by declaring $\alpha$ as a positive root, $N$ can be taken to be the connected subgroup of $G$ with Lie algebra $\g{n}=\g{g}_\alpha\oplus\g{g}_{2\alpha}$.
\renewcommand\arraystretch{1.25}
\begin{table}
\begin{tabular}[h!]{llllll}
\hline
Symmetric space & $G$ & $K$ & $K_0$ & $\g{g}_\alpha$ & $\g{g}_{2\alpha}$
\\ \hline
Real hyperbolic space $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$ & $\ensuremath{\mathsf{SO}}^0_{1,n}$ & $\ensuremath{\mathsf{SO}}_n$ & $\ensuremath{\mathsf{SO}}_{n-1}$ & $\ensuremath{\mathbb{R}}^{n-1}$ & $0$
\\ \hline
Complex hyperbolic space $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ & $\ensuremath{\mathsf{SU}}_{1,n}$ & $\ensuremath{\mathsf{S}}(\ensuremath{\mathsf{U}}_1\ensuremath{\mathsf{U}}_n)$ & $\ensuremath{\mathsf{U}}_{n-1}$ & $\ensuremath{\mathbb{C}}^{n-1}$ & $\ensuremath{\mathbb{R}}$
\\ \hline
Quaternionic hyperbolic space $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$ & $\ensuremath{\mathsf{Sp}}_{1,n}$ & $\ensuremath{\mathsf{Sp}}_1\ensuremath{\mathsf{Sp}}_n$ & $\ensuremath{\mathsf{Sp}}_1\ensuremath{\mathsf{Sp}}_{n-1}$ & $\ensuremath{\mathbb{H}}^{n-1}$ & $\ensuremath{\mathbb{R}}^3$
\\ \hline
Cayley hyperbolic plane $\mathbb{O} \ensuremath{\mathsf{H}}^2$ & $\mathsf{F}_4^{-20}$ & $\ensuremath{\mathsf{Sp}}in_9$ & $\ensuremath{\mathsf{Sp}}in_{7}$ & $\mathbb{O}$ & $\ensuremath{\mathbb{R}}^7$
\\ \hline
\end{tabular}
\caption{Symmetric spaces of noncompact type and rank one}\label{table:rank_one}
\end{table}
The geometric interpretation of the groups involved in the Iwasawa decomposition of~$G$ is similar to that of $M=\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2$, described in Example~\ref{ex:RH^2} and Figure~\ref{fig:KAN}. The action of the isotropy group $K$ on $M=\mathbb{F}\ensuremath{\mathsf{H}}^{n}$ has $o$ as a fixed point, and the other orbits are geodesic spheres around $o$. The action of $A$ gives rise to a geodesic through $o$ (since $\g{a}$ is a Lie triple system), and the other orbits are equidistant curves. Note that any geodesic curve, such as $A\cdot o$, determines two points at infinity; the choice of a positivity criterion on the set $\Sigma$ of roots (equivalently, choosing $\g{n}=\g{g}_\alpha\oplus\g{g}_{2\alpha}$ or $\g{n}=\g{g}_{-\alpha}\oplus\g{g}_{-2\alpha}$) is interpreted geometrically as selecting one of the two points at infinity determined by $A\cdot o$. Thus, the orbits of the $N$-action are \textit{horospheres} centered precisely at the point at infinity $x$ determined by the choice of $\g{n}$. In other words, if $\gamma$ is a unit-speed geodesic such that $A\cdot o=\{\gamma(t):t\in\ensuremath{\mathbb{R}}\}$ and converging to $x\in M(\infty)$, then the orbits of the $N$-action are the level sets of the Busemann function $f_\gamma(p):=\lim_{t\to\infty}(d(\gamma(t),p)-t)$. See~\cite[\S2.2]{DV:dea} and~\cite[\S1.10]{Eberlein} for details.
\begin{example}\label{ex:horospheres}
We will illustrate the use of Formula~\eqref{eq:Levi-Civita} by calculating the extrinsic geometry of horospheres in $\mathbb{F}\ensuremath{\mathsf{H}}^n$. Via the Lie group model, the horosphere $N\cdot o$ is nothing but the Lie subgroup $N$ of $AN$. Thus, its tangent space at any $g\in N$ is given by the left-invariant fields of $\g{n}$ at $g$. If $B\in\g{a}$ satisfies $\langle B,B\rangle_{AN}=1$, then it defines a unit normal vector field on $N$. Hence, the shape operator $\cal{S}$ of $N$ with respect to $B$ is given by $\cal{S}X=-\nabla_X B$, for $X\in\g{n}$. If $X\in\g{g}_\lambda$, for $\lambda\in\{\alpha,2\alpha\}$, then by the comment following~\eqref{eq:Levi-Civita}, we have
\[
\cal{S}X=-\nabla_X B=-\frac{1}{2}\bigl([X,B]+[\theta X,B]-[X,\theta B]\bigr)_{\g{a}\oplus\g{n}}
=
\lambda(B)X,
\]
where we have used the definition of the root space $\g{g}_\lambda$, the fact that $[\theta X,B]\in\g{g}_{-\lambda}$ is orthogonal to $\g{a}\oplus\g{n}$, and $\theta(B)=-B$. Hence, $N\cdot o$ has two distinct constant principal curvatures, $\alpha(B)$ and $2\alpha(B)$, with respective principal curvature spaces $\g{g}_\alpha$ and $\g{g}_{2\alpha}$.
Finally, note that all horospheres in $M$ are congruent to each other. Indeed, any two horosphere foliations are congruent by an element in $K$. Moreover, the geodesic $A\cdot o$ intersects all $N$-orbits and, since $A$ normalizes $N$, any two $N$-orbits are congruent under an element~of~$A$.
\end{example}
The Lie group model of a rank one symmetric space contains some underlying additional structure that is often very helpful. Let us define a linear map $J\colon\g{g}_{2\alpha}\to\ensuremath{\mathrm{End}}(\g{g}_\alpha)$ by
\[
\langle J_ZU,V\rangle_{AN}=\langle [U,V],Z\rangle_{AN}, \qquad \text{for all }U,V\in\g{g}_\alpha, \, Z\in\g{g}_{2\alpha},
\]
or, equivalently by~\eqref{eq:relation_inner} and~\eqref{eq:prop_B_theta}, $J_ZU:=[Z,\theta U]$. Then, up to rescaling of the metric of $M$ (and hence of $\langle\cdot,\cdot\rangle_{AN}$), the endomorphism $J_Z$ satisfies (see~\cite[Proposition~1.1]{Koranyi:adv})
\[
J_Z^2=-\langle Z,Z\rangle_{AN} \Id_{\g{g}_\alpha},\qquad \text{for all }Z\in\g{g}_{2\alpha}.
\]
Thus, the map $J$ induces a representation of the Clifford algebra $\ensuremath{\mathbb{C}}l\bigl(\g{g}_{2\alpha},-\langle\cdot,\cdot\rangle_{AN}\bigr)$ on~$\g{g}_\alpha$ (see~\cite[Chapter~1]{LM} for more information on Clifford algebras and their representations). This converts $AN$ with the rescaled left-invariant metric into a so-called \emph{Damek-Ricci space}, and its nilpotent part $N$ into a \emph{generalized Heisenberg group}. These concepts were introduced by Damek and Ricci~\cite{Damek-Ricci:bams} and by Kaplan~\cite{Kaplan:tams}, respectively, and a comprehensive work for their study is~\cite{BTV}. Regarding rank one symmetric spaces of noncompact type as Damek-Ricci spaces has the advantage of allowing to use the power of the theory of Clifford modules to obtain more manageable formulas and more general arguments. For example, Formula~\eqref{eq:Levi-Civita} for the Levi-Civita connection of $AN$ adopts the form
\begin{equation}\label{eq:Levi-Civita_DR}
{\nabla}_{aB+U+X}(bB+V+Y)
=\left(\frac{1}{2}\langle U,V\rangle_{AN}\!+\langle X,Y\rangle_{AN}\!\right)\!B-\frac{1}{2}\left(bU+J_XV+J_YU\right)
+\frac{1}{2}[U,V]-bX,
\end{equation}
where $a$, $b\in\ensuremath{\mathbb{R}}$, $U$, $V\in\g{g}_\alpha$, $X$, $Y\in\g{g}_{2\alpha}$.
In what follows we review some results about certain types of submanifolds and isometric actions on symmetric spaces of noncompact type and rank one.
\subsection{Homogeneous hypersurfaces}\label{subsec:rank1_homogeneous}
A submanifold $P$ of a Riemannian manifold $M$ is said to be \emph{(extrinsically) homogeneous} if for any $p, q\in P$ there exists an isometry $\varphi$ of the ambient manifold $M$ such that $\varphi(p)=q$ and $\varphi(P)=P$. Equivalently, $P$ is a homogeneous submanifold if it is an orbit of an isometric action on $M$, i.e.\ there exists a subgroup $H$ of $\Isom(M)$ such that $P=H\cdot p$ for some $p\in P$. Moreover, $P$ is embedded if and only if $H=\{\varphi\in \Isom(M):\varphi(P)=P\}$ is closed in $\Isom(M)$, which means that the associated isometric action is proper. From now on, \emph{isometric actions will be assumed to be proper}.
\begin{remark}
The collection of orbits of an isometric action is the standard example of a singular Riemannian foliation. A \emph{singular Riemannian foliation} $\cal{F}$ on a Riemannian manifold $M$ is a decomposition of $M$ into connected, injectively immersed submanifolds $L\in\cal{F}$ (called leaves) such that they are locally equidistant to each other, and there is a collection of smooth vector fields on $M$ that spans all tangent spaces to all leaves; see~\cite{AB:book},~\cite{ABT:dga} for more information on this concept. Singular Riemannian foliations can have leaves of different dimensions: the ones of highest dimension are called regular, and the others are singular. Orbit foliations, that is, singular Riemannian foliations induced by isometric actions, are sometimes called homogeneous foliations.
\end{remark}
When an isometric action has codimension one orbits, then it is called a \emph{cohomogeneity one action}, and its codimension one orbits are \emph{homogeneous hypersurfaces}. The homogeneity property for hypersurfaces is a rather strong condition. This motivates the problem of classifying homogeneous hypersurfaces or, equivalently, cohomogeneity one actions up to orbit equivalence, in specific Riemannian manifolds, mainly in those with large isometry group. Such classification is known, for example, for Euclidean and real hyperbolic spaces (as a consequence of Segre's~\cite{Segre} and Cartan's~\cite{Cartan} works on isoparametric hypersurfaces, see~\S\ref{subsec:rank1_isoparametric}), irreducible symmetric spaces of compact type~\cite{Ko:tams}, and simply connected homogeneous $3$-manifolds with $4$-dimensional isometry group~\cite{DVM:arxiv}. Below we focus on the classification problem in symmetric spaces of noncompact type, and refer the reader to~\cite[\S6]{Berndt:notes} and~\cite[\S2.9.3 and Chapters 12-13]{BCO} for more information on cohomogeneity one actions.
As in any other Hadamard manifold, cohomogeneity one actions on symmetric spaces of noncompact type have at most one singular orbit~\cite[\S2]{BB:crelle} and no exceptional orbits~\cite[Corollary~1.3]{Ly:gd}. If there is one singular orbit, then the other orbits are homogeneous hypersurfaces which arise as distance tubes around the singular orbit. It there are no singular orbits, then all orbits are homogeneous hypersurfaces, and they define a regular Riemannian foliation of the ambient space.
Cohomogeneity one actions on hyperbolic spaces have been investigated by Berndt, Br\"uck and Tamaru in a series of papers. Berndt and Br\"uck~\cite{BB:crelle} classified cohomogeneity one actions with a totally geodesic orbit on hyperbolic spaces $M=\mathbb{F} \ensuremath{\mathsf{H}}^n$:
\begin{theorem}\label{th:BB}
Let $F$ be a totally geodesic singular orbit of a cohomogeneity one action on $\mathbb{F}\ensuremath{\mathsf{H}}^n$, $n\geq 2$. Then $F$ is congruent to one of the following totally geodesic submanifolds:
\begin{itemize}
\item in $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$: $\{o\}$, $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^1$, \dots, $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^{n-1}$;
\item in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$: $\{o\}$, $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^1,\dots,\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^{n-1},\ensuremath{\mathbb{R}}\ensuremath{\mathsf{H}}^n$;
\item in $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$: $\{o\}$, $\ensuremath{\mathbb{H}}\ensuremath{\mathsf{H}}^1,\dots,\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^{n-1},\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$;
\item in $\mathbb{O} \ensuremath{\mathsf{H}}^2$: $\{o\}$, $\mathbb{O}\ensuremath{\mathsf{H}}^1,\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^2$.
\end{itemize}
Conversely, each of these totally geodesic submanifolds arises as the singular orbit of some cohomogeneity one action.
\end{theorem}
In particular, if $M\neq \ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$, not every totally geodesic submanifold of $M$ defines homogeneous distance tubes. Moreover, it follows from Cartan's work~\cite{Cartan} that singular orbits of cohomogeneity one actions on $\ensuremath{\mathbb{R}}\ensuremath{\mathsf{H}}^n$ must be totally geodesic.
Berndt and Br\"uck~\cite{BB:crelle} also found examples of cohomogeneity one actions with a nontotally geodesic singular orbit (for $M\neq \ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$). This important construction goes as follows.~Consider the Lie algebra $\g{a}\oplus\g{g}_\alpha\oplus\g{g}_{2\alpha}$ of $AN$. Take a subspace $\g{w}$ of $\g{g}_\alpha$ and define the subalgebra
\begin{equation}\label{eq:focal}
\g{s}_\g{w}:=\g{a}\oplus\g{w}\oplus\g{g}_{2\alpha}
\end{equation}
of $\g{a}\oplus\g{n}$. Assume that the orthogonal complement $\g{w}^\perp:=\g{g}_\alpha\ominus\g{w}$ of $\g{w}$ in $\g{g}_\alpha$ is such that
\begin{equation}\label{eq:condition_normalizer}
N^0_{K_0}(\g{w}) \text{ acts transitively on the unit sphere of } \g{w}^\perp
\end{equation}
where $N^0_{K_0}(\g{w})$ denotes the connected component of the identity of the normalizer of $\g{w}$ in $K_0$. Then, if $S_\g{w}$ is the connected subgroup of $AN$ with Lie algebra $\g{s}_\g{w}$, the group $N^0_{K_0}(\g{w}) S_\g{w}$ acts with cohomogeneity one on $M$, and with $S_\g{w}\cdot o$ as singular orbit if $\dim\g{w}^\perp\geq 2$.
Berndt and Br\"uck proceeded to analyze which subspaces $\g{w}$ of $\g{g}_\alpha$ satisfy Condition~\eqref{eq:condition_normalizer}. In the case $M=\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$, they characterized this condition in terms of the so-called K\"ahler angles of $\g{w}^\perp$. Given any real subspace $V$ of a complex Euclidean space $(\ensuremath{\mathbb{R}}^{2k},J)$, where $J$ is a complex structure on $\ensuremath{\mathbb{R}}^{2k}$ (i.e.\ $J\in\g{so}(2k)$ and $J^2=-\Id$), the K\"ahler angle of a nonzero $v\in V$ with respect to $V$ is the angle $\varphi\in[0,\pi/2]$ between $Jv$ and $V$. When all unit $v\in V$ have the same K\"ahler angle $\varphi$ with respect to $V$, then we say that $V$ has \emph{constant K\"ahler angle} $\varphi$. For example, subspaces with constant K\"ahler angle $0$ or $\pi/2$ are precisely the complex and totally real subspaces, respectively. However, there are subspaces with any constant K\"ahler angle $\varphi\in(0,\pi/2)$; these can be classified, see~\cite[Proposition~7]{BB:crelle}. Now recall from Table~\ref{table:rank_one} that $\g{g}_\alpha\cong\ensuremath{\mathbb{C}}^{n-1}\cong(\ensuremath{\mathbb{R}}^{2n-2},J)$. Thus, it was proved in~\cite{BB:crelle} that $\g{w}\subset \g{g}_\alpha$ satisfies \eqref{eq:condition_normalizer} if and only if $\g{w}^\perp$ has constant K\"ahler angle $\varphi$ and $\dim\g{w}^\perp\geq 2$; moreover, the singular orbit $S_\g{w}\cdot o$ is nontotally geodesic whenever $\varphi\neq 0$.
In the case $M=\mathbb{O}\ensuremath{\mathsf{H}}^2$, by analyzing the $\ensuremath{\mathsf{Sp}}in_7$-action on $\g{g}_\alpha\cong\ensuremath{\mathbb{R}}^8$, Berndt and Br\"uck proved that $\g{w}$ satisfies \eqref{eq:condition_normalizer} if and only if $\dim\g{w}\in\{0,1,2,4,5,6\}$, where only $\g{w}=0$ yields a totally geodesic singular orbit. Interestingly, the case $M=\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$ is much more involved and, indeed, it is still open. In~\cite{BB:crelle} it was proved that Condition~(7) in this case implies that $\g{w}^\perp$ has constant quaternionic K\"ahler angle (see~\S\ref{subsec:rank1_isoparametric}, after Theorem~\ref{th:isopar_DR}, for the definition), and several subspaces with this property were found (more examples were constructed in~\cite{DRDV:adv}). However, neither a classification of subspaces of $\ensuremath{\mathbb{H}}^{k}$, $k\geq 2$, with constant quaternionic K\"ahler angle, nor the equivalence between this property and Condition~\eqref{eq:condition_normalizer} are known.
Regarding cohomogeneity one actions without singular orbits, Berndt and Tamaru~\cite{BT:jdg} proved (as a particular case of a more general result, cf.~\S\ref{subsec:cohom1}) that there are only two such actions on $M=\mathbb{F}\ensuremath{\mathsf{H}}^n$ up to orbit equivalence. One of these actions is that of the nilpotent part $N$ of the Iwasawa decomposition, giving rise to a \emph{horosphere foliation} (see Example~\ref{ex:horospheres}). The other one is given by the action of the connected subgroup $S$ of $AN$ with Lie algebra $\g{s}=\g{a}\oplus(\g{g}_\alpha\ominus\ensuremath{\mathbb{R}} U)\oplus\g{g}_{2\alpha}$, for any $U\in\g{g}_\alpha$; note that this corresponds to~\eqref{eq:focal} for the choice of a hyperplane $\g{w}$ in $\g{g}_{2\alpha}$. This $S$-action gives rise to the so-called \emph{solvable foliation} on a symmetric space of noncompact type and rank one.
Based on the results mentioned above, Berndt and Tamaru~\cite{BT:tams} were able to prove a structure result for cohomogeneity one actions on rank one symmetric spaces which states that each of these actions must be of one of the types described above.
\begin{theorem}\label{th:BT1}
Let $M=\mathbb{F}\ensuremath{\mathsf{H}}^n$ be an symmetric space of noncompact type and rank one, and let $H$ act on $M$ with cohomogeneity one. Then one of the following statements holds:
\begin{enumerate}[{\rm (1)}]
\item The $H$-orbits form a regular Riemannian foliation on $M$ which is congruent to either a horosphere foliation or a solvable foliation.
\item There exists exactly one singular $H$-orbit and one of the following two cases holds:
\begin{enumerate}[{\rm (i)}]
\item The singular $H$-orbit is one of the totally geodesic submanifolds in Theorem~\ref{th:BB}.
\item The $H$-action is orbit equivalent to the action of $N^0_{K_0}(\g{w})S_\g{w}$, where $\g{w}$ is a subspace of $\g{g}_\alpha$ such that $N^0_{K_0}(\g{w})$ acts transitively on the unit sphere of $\g{w}^\perp$, and $S_\g{w}$ is the connected subgroup of $AN$ with Lie algebra $\g{s}_\g{w}=\g{a}\oplus\g{w}\oplus\g{g}_{2\alpha}$.
\end{enumerate}
\end{enumerate}
\end{theorem}
Combining this theorem with the results in~\cite{BB:crelle}, Berndt and Tamaru derived the classification of cohomogeneity one actions up to orbit equivalence on $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$ and $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$ for $n\geq 2$, and on $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^2$ and $\mathbb{O}\ensuremath{\mathsf{H}}^2$. The classification on $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$, $n\geq 3$, remains open.
\subsection{Isoparametric hypersurfaces} \label{subsec:rank1_isoparametric}
An immersed hypersurface $P$ in a Riemannian manifold $M$ is an \emph{isoparametric hypersurface} if, locally, $P$ and its nearby equidistant hypersurfaces have constant mean curvature. An \emph{isoparametric family of hypersurfaces} or \emph{isoparametric foliation (of codimension one)} is a singular Riemannian foliation such that its regular leaves are isoparametric hypersurfaces. These objects have been studied since the beginning of the 20th century and their investigation has therefore a long and interesting history. We refer to the excellent surveys~\cite{Chi:story} and~\cite{Th:survey} for a detailed account on this history.
Segre~\cite{Segre} classified isoparametric hypersurfaces in Euclidean spaces $\ensuremath{\mathbb{R}}^n$ by proving that they must be open subsets of affine hyperplanes $\ensuremath{\mathbb{R}}^{n-1}$, spheres $\mathbb{S}^{n-1}$ or generalized cylinders $\ensuremath{\mathbb{R}}^k\times \mathbb{S}^{n-k-1}$. Cartan~\cite{Cartan} proved that, in spaces of constant curvature, a hypersurface is isoparametric if and only if it has constant principal curvatures. Then, he classified such hypersurfaces in real hyperbolic spaces $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^n$: the examples must be open subsets of totally geodesic $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^{n-1}$ or their equidistant hypersurfaces, distance tubes around totally geodesic $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^k$, $k\in\{0,\dots,n-2\}$, or horospheres. Thus, in spaces of nonpositive constant curvature, isoparametric hypersurfaces are open parts of homogeneous hypersurfaces.
Observe that homogeneous hypersurfaces are isoparametric and have constant principal curvatures. However, none of the converse implications is true. In round spheres $\mathbb{S}^n$ there are inhomogeneous isoparametric hypersurfaces (with constant principal curvatures)~\cite{FKM}. In fact, the classification problem in spheres is much more involved; for more information, we refer the reader to some of the latest advances in the topic, such as~\cite{CCJ:annals,Chi:jdg,Imm:annals,Mi:annals,Siffert}. In spaces of nonconstant curvature, the problem becomes very complicated. Apart from the results we will review below, there is a classification on complex projective spaces $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{P}}^n$, $n\neq 15$~\cite{DV:tams}, quaternionic projective spaces $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{P}}^n$, $n\neq 7$~\cite{DVG:tohoku}, the product $\mathbb{S}^2\times\mathbb{S}^2$~\cite{Urbano}, and simply connected homogeneous $3$-manifolds with $4$-dimensional isometry group~\cite{DVM:arxiv}, such as the products $\mathbb{S}^2\times\ensuremath{\mathbb{R}}$, $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^2\times\ensuremath{\mathbb{R}}$, the Heisenberg group $\textsf{Nil}_3$ or the Berger spheres. Interestingly, in all the cases mentioned so far (as well as in the rest of examples presented in this paper) an isoparametric hypersurface is always an open subset of a leaf of an isoparametric foliation of codimension one that fills the whole ambient space.
In spaces of nonconstant curvature, isoparametricity and constancy of the principal curvatures are two properties with no general theoretical relation. Berndt~\cite{Berndt:Hopf}, \cite{Berndt:HHn} classified curvature-adapted hypersurfaces with constant principal curvatures in complex and quaternionic hyperbolic spaces. Here, \emph{curvature-adapted} means that the shape operator $\cal{S}$ and the normal Jacobi operator $R_\xi=R(\cdot,\xi)\xi$ of the hypersurface commute (hereafter $\xi$ is a unit normal smooth field on the hypersurface); hence both operators diagonalize simultaneously, which simplifies calculations involving the fundamental equations of submanifolds (Gauss, Codazzi...) and Jacobi fields adapted to the hypersurface (to calculate, for example, the extrinsic geometry of equidistant hypersurfaces or focal sets, cf.~\cite[\S10.2]{BCO}). In the complex case, a hypersurface in $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$ is curvature-adapted if and only if it is \emph{Hopf}, that is, the Reeb vector field $J\xi$ is an eigenvector of the shape operator at every point, where $J$ is the K\"ahler structure of $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$. It follows from Berndt's classifications that all curvature-adapted hypersurfaces with constant principal curvatures in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ and $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$ are open subsets of homogeneous hypersurfaces. However, not all homogeneous hypersurfaces described in~\S\ref{subsec:rank1_homogeneous} are curvature-adapted: only horospheres and homogeneous tubes around totally geodesic submanifolds have this property. Without the curvature-adaptedness condition, the study of hypersurfaces with constant principal curvatures is much more convoluted, and only some partial results for $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ are known; see~\cite{DRDV:indiana} for a recent advance, and~\cite{DV:dga} for a survey.
In view of the results mentioned above and the fact that a curvature-adapted hypersurface in a rank one symmetric space is isoparametric if and only if it has constant principal curvatures~\cite[Theorem~1.4]{GTY:japan}, it follows that a curvature-adapted isoparametric hypersurface in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ or $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$ is an open part of a homogeneous hypersurface. However, again, without the curvature-adaptedness condition, basically no other results regarding isoparametric hypersurfaces in our setting were known until a few years ago.
In~\cite{DRDV:mathz}, D\'iaz-Ramos and Dom\'inguez-V\'azquez constructed the first examples of inhomogeneous isoparametric hypersurfaces in a family of symmetric spaces of noncompact type, namely in complex hyperbolic spaces. Later, the authors generalized this result to Damek-Ricci spaces and, in particular, to the other symmetric spaces of noncompact type and rank one~\cite{DRDV:adv}. This construction, which we explain below, makes use of the basic idea of Berndt-Br\"uck cohomogeneity one actions described in~\S\ref{subsec:rank1_homogeneous}.
Given a symmetric space of noncompact type and rank one, $M=\mathbb{F}\ensuremath{\mathsf{H}}^n$, consider the subalgebra $\g{s}_\g{w}=\g{a}\oplus\g{w}\oplus\g{g}_{2\alpha}$ of $\g{a}\oplus\g{n}$ defined in~\eqref{eq:focal}, where now $\g{w}$ can be \emph{any} proper vector subspace of $\g{g}_\alpha$. Let $S_\g{w}$ be the connected subgroup of $AN$ with Lie algebra $\g{s}_\g{w}$. Using Formula~\eqref{eq:Levi-Civita_DR} it is not difficult to prove that $W_\g{w}:=S_\g{w}\cdot o$ is a minimal submanifold of $M$. Then, by introducing the notion of generalized K\"ahler angle (which we explain below) and using Jacobi field theory, D\'iaz-Ramos and Dom\'inguez-V\'azquez proved the following~\cite{DRDV:adv}:
\begin{theorem}\label{th:isopar_DR}
The distance tubes around the minimal submanifold $W_\g{w}$ in a rank one symmetric space of noncompact type are isoparametric hypersurfaces, and have constant principal curvatures if and only if $\g{w}^\perp=\g{g}_\alpha\ominus\g{w}$ has constant generalized K\"ahler angle.
\end{theorem}
The concept of generalized K\"ahler angle extends both the K\"ahler angle and the quaternionic K\"ahler angle mentioned in~\S\ref{subsec:rank1_homogeneous}. Let $\g{z}\cong\ensuremath{\mathbb{R}}^m$ be a Euclidean space with inner product $\langle\cdot,\cdot\rangle$, and $\g{v}$ a Clifford module over $\ensuremath{\mathbb{C}}l(\g{z},-\langle\cdot,\cdot\rangle)$. Consider $J\colon \g{z}\to \ensuremath{\mathrm{End}}(\g{v})$ the restriction to $\g{z}$ of the Clifford algebra representation. Recall from~\S\ref{subsec:rank1} that the rank one symmetric spaces of noncompact type have a naturally associated map $J$ as above, with $\g{v}=\g{g}_\alpha$ and $\g{z}=\g{g}_{2\alpha}$. Now let $V$ be a vector subspace of $\g{v}$. For each nonzero $v\in\g{v}$, consider the map
\[
F_v\colon \g{z}\to\ensuremath{\mathbb{R}},\qquad Z\mapsto \langle (J_Zv)^V,(J_Zv)^V\rangle,
\]
where $(\cdot)^V$ denotes orthogonal projection onto $V$. Observe that $F_v$ is a quadratic form on $\g{z}$, and its eigenvalues belong to the interval $[0,|v|^2]$. Hence, such eigenvalues are of the form $|v|^2\cos^2\varphi_i(v)$, $i=1,\dots, m=\dim\g{z}$, for certain angles $\varphi_i(v)\in[0,\pi/2]$. Then, one defines the \emph{generalized K\"ahler angle} of $v$ with respect to $V$ as the ordered $m$-tuple of angles $(\varphi_1(v),\dots,\varphi_m(v))$. We say that $V$ has \emph{constant generalized K\"ahler angle} if the $m$-tuple $(\varphi_1(v),\dots,\varphi_m(v))$ is independent of the nonzero $v\in V$. Note that, if $m =1$, we recover the notion of K\"ahler angle. The concept of quaternionic K\"ahler angle introduced in~\cite{BB:crelle} agrees with that of generalized K\"ahler angle in the case where $m=3$ and $\g{v}$ is a sum of equivalent irreducible $\ensuremath{\mathbb{C}}l_3$-modules (i.e.\ $\g{v}$ is a quaternionic vector space).
Regarding complex or quaternionic hyperbolic spaces, $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ or $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$, with $n\geq 3$, most real subspaces of $\g{g}_\alpha$ ($\cong \ensuremath{\mathbb{C}}^{n-1}$ or $\ensuremath{\mathbb{H}}^{n-1}$, respectively) have nonconstant generalized K\"ahler angle; e.g.\ the orthogonal sum of a complex and a totally real subspace in $\ensuremath{\mathbb{C}}^{n-1}$ does~not have constant K\"ahler angle. Thus, Theorem~\ref{th:isopar_DR} ensures the existence of \emph{inhomogeneous isoparametric families of hypersurfaces with nonconstant principal curvatures in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ and~$\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$,~$n\geq 3$}.
The case of the Cayley plane is even more interesting. As proved in~\cite{BB:crelle} and mentioned in~\S\ref{subsec:rank1_homogeneous}, if the subspace $\g{w}$ of $\g{g}_\alpha$ has dimension $3$, the tubes around $W_\g{w}$ are not homogeneous. However, any subspace of $\g{g}_\alpha\cong\mathbb{O}$ has constant generalized K\"ahler angle; in the case $\dim\g{w}=3$, the generalized K\"ahler angle of $\g{w}^\perp$ is $(0,0,0,0,\pi/2,\pi/2,\pi/2)$. Thus, the tubes around the corresponding $W_\g{w}$ constitute an \emph{inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures in $\mathbb{O}\ensuremath{\mathsf{H}}^2$}. This is the only such example known in any symmetric space, apart from the FKM-examples in spheres~\cite{FKM}.
The homogeneous isoparametric foliations described in~\S\ref{subsec:rank1_homogeneous}, jointly with the inhomogeneous ones presented in this section, constitute an important family of examples which may encourage us to tackle the classification problem of isoparametric hypersurfaces in the rank one symmetric spaces of noncompact type. However, this is a much more complicated problem. Indeed, the only advance so far in this direction is the classification of isoparametric hypersurfaces in complex hyperbolic spaces obtained recently by the authors~\cite{DRDVSL:adv}. This constituted the first complete classification of isoparametric hypersurfaces in a complete family of symmetric spaces since Segre's~\cite{Segre} and Cartan's~\cite{Cartan} works in the~30s.
\begin{theorem}\label{th:isopCHn}
Let $M$ be a connected real hypersurface in a complex hyperbolic space $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$, $n\geq 2$. Then $M$ is isoparametric if and only if it is an open subset of one of the following:
\begin{enumerate}[{\rm (i)}]
\item A tube around a totally geodesic $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^k$, $k\in\{0,\dots,n-1\}$.
\item A tube around a totally geodesic $\ensuremath{\mathbb{R}}\ensuremath{\mathsf{H}}^n$.
\item A horosphere.
\item A leaf of a solvable foliation.
\item A tube around a submanifold $W_\g{w}$, for some subspace $\g{w}$ of $\g{g}_\alpha$ with $\dim (\g{g}_\alpha\ominus\g{w})\geq 2$.
\end{enumerate}
\end{theorem}
In particular, each isoparametric hypersurface in $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$ is an open part of a complete, topologically closed leaf of a (globally defined) isoparametric foliation on $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$. Either such foliation is regular (examples (iii) and (iv)) or has one singular orbit (examples (i), (ii) and (v)) which is minimal and homogeneous. Moreover, the homogeneous hypersurfaces in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ are precisely those in examples (i) through (iv), and those in (v) with $\g{w}^\perp=\g{g}_\alpha\ominus\g{w}$ of constant K\"ahler angle. Thus, an isoparametric hypersurface in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ is an open part of a homogeneous one if and only if it has constant principal curvatures.
The proof of Theorem~\ref{th:isopCHn} is rather involved. The starting point is to consider the Hopf map $\pi\colon \mathsf{AdS}^{2n+1}\to\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^n$ from the anti De Sitter spacetime $\mathsf{AdS}^{2n+1}$, and to prove that the preimage $\pi^{-1}(M)$ of a hypersurface $M$ in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$ is isoparametric (in a semi-Riemannian sense) if and only if $M$ is isoparametric. Since $\mathsf{AdS^{2n+1}}$ has constant curvature, $\pi^{-1}(M)$ is isoparametric precisely when it has constant principal curvatures. However, since $\pi^{-1}(M)$ is a Lorentzian hypersurface, its shape operator does not need to be diagonalizable. By analyzing each one of the four possible Jordan canonical forms for such shape operator, one can show (using elementary algebraic and geometric calculations) that three of them correspond to each one of the examples (i), (ii), (iii) above. Dealing with the fourth Jordan canonical form is much more convoluted, and requires delicate calculations with Jacobi fields and various geometric ideas. Finally, such Jordan form turns out to correspond with examples (iv) and (v) in Theorem~\ref{th:isopCHn}.
\subsection{Polar actions}\label{subsec:rank1_polar}
An isometric action on a Riemannian manifold $M$ is called \emph{polar} if there is a (a fortiori, totally geodesic) connected immersed submanifold $\Sigma$ of $M$ that intersects all orbits, and every such intersection is orthogonal. The submanifold $\Sigma$ is called a \emph{section} of the action; if $\Sigma$ is flat with respect to the induced metric, the action is called \emph{hyperpolar}. Cohomogeneity one actions constitute a particular case of hyperpolar actions.
The notion of polarity traces back at least to Dadok's classification~\cite{Dadok} of polar representations (equivalently, polar actions on round spheres): such polar actions coincide exactly with the isotropy representations of symmetric spaces, up to orbit equivalence. Later, polar actions have been studied mainly in the context of symmetric spaces of compact type: see~\cite{PT:jdg} (cf.~\cite{GK:agag}) for the classification in the rank one spaces, \cite{Ko:jdg} and~\cite{KL:blms} for the irreducible spaces of arbitrary rank, and~\cite{DR:polar} for a survey. For general manifolds, there are some topological and geometric structure results, see~\cite[Chapter~5]{AB:book} and~\cite{GZ:fixed}. Moreover, the notions of polar and hyperpolar action have been extended to the realm of singular Riemannian foliations by requiring the existence of sections through all points; see~\cite[Chapter~5]{AB:book}, \cite{ABT:dga}. Thus, homogeneous polar foliations are nothing but the orbit foliations of polar actions.
In symmetric spaces of noncompact type, very few results are known. The classification of polar actions on real hyperbolic spaces $\ensuremath{\mathbb{R}}\ensuremath{\mathsf{H}}^n$ follows from Wu's work~\cite{Wu:tams}.
\begin{theorem}\label{th:polarRHn}
A polar action on $\ensuremath{\mathbb{R}}\ensuremath{\mathsf{H}}^n$, $n\geq 2$, is orbit equivalent to one of the following:
\begin{enumerate}[{\rm (i)}]
\item The action of $\ensuremath{\mathsf{SO}}_{1,k}\times Q$, where $k\in\{0,\dots,n-1\}$ and $Q$ is a compact subgroup of $\ensuremath{\mathsf{SO}}_{n-k}$ acting polarly on $\ensuremath{\mathbb{R}}^{n-k}$.
\item The action of $N\times Q$, where $N$ is the nilpotent part of the Iwasawa decomposition of $\ensuremath{\mathsf{SO}}_{1,n}$, and $Q$ is a compact subgroup of $\ensuremath{\mathsf{SO}}_{n-k}$ acting polarly on $\ensuremath{\mathbb{R}}^{n-k}$.
\end{enumerate}
\end{theorem}
The first classification result in a symmetric space of noncompact type and nonconstant curvature was achieved by Berndt and D\'iaz-Ramos~\cite{BDR:agag} for the complex hyperbolic plane $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^2$. This classification consists of five cohomogeneity one actions and four cohomogeneity two actions, up to orbit equivalence. Interestingly, all of them can be characterized geometrically~\cite{DRDVVC:agag}.
\begin{theorem}
A submanifold of $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^2$ is isoparametric if and only if it is an open part of a principal orbit of a polar action on $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^2$.
\end{theorem}
Here, we refer to the notion of \emph{isoparametric submanifold} (of arbitrary codimension) given by Heintze, Liu and Olmos~\cite{HLO}, as a submanifold $P$ with flat normal bundle, whose parallel submanifolds have constant mean curvature in radial directions, and such that, for each $p\in P$, there is a totally geodesic submanifold $\Sigma_p$ such that $T_p\Sigma_p=\nu_p P$. Thus, an \emph{isoparametric foliation} (of arbitrary codimension) is a polar foliation whose regular leaves are isoparametric. The orbit foliations of polar actions constitute the main set of examples of isoparametric foliations.
Regarding cohomogeneity two polar actions on $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^2$, one can additionally prove~\cite{DRDVVC:jga}:
\begin{theorem}
A submanifold of $\ensuremath{\mathbb{C}}\ensuremath{\mathsf{H}}^2$ is an open subset of a principal orbit of a cohomogeneity two polar action if and only if it is a Lagrangian flat surface with parallel mean curvature. Moreover, such surfaces have parallel second fundamental form.
\end{theorem}
Coming back to the classification problem of polar actions on $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$, the case $n=2$ was extended by D\'iaz-Ramos, Dom\'inguez-V\'azquez and Kollross to all dimensions~\cite{DRDVK:mathz}.
\begin{theorem}\label{th:polarCHn}
A polar action on $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$, $n\geq 2$, is orbit equivalent to the action of the connected subgroup $H$ of $\ensuremath{\mathsf{U}}_{1,n}$ with one of the following Lie algebras:
\begin{enumerate}[{\rm (i)}]
\item $\g{h}=\g{q} \oplus \g{so}_{1,k} \subset \g{u}_{n-k}
\oplus \g{su}_{1,k}$, $k\in\{0,\dots,n\}$, where the connected subgroup $Q$ of~$\ensuremath{\mathsf{U}}_{n-k}$ with Lie algebra $\g{q}$ acts polarly with a totally real section on~$\ensuremath{\mathbb{C}}^{n-k}$.
\item $\g{h}=\g{q} \oplus \g{b} \oplus \g{w} \oplus \g{g}_{2\alpha}\subset\g{su}_{1,n}$,
where $\g{b}$ is a subspace of~$\g{a}$, $\g{w}$ is a
subspace of~$\g{g}_{\alpha}$, and $\g{q}$ is a subalgebra of~$\g{k}_0$
which normalizes~$\g{w}$ and such that the connected subgroup of $\ensuremath{\mathsf{SU}}_{1,n}$
with Lie algebra $\g{q}$ acts polarly with a totally real section on
$\g{w}^\perp=\g{g}_\alpha\ominus\g{w}$.
\end{enumerate}
\end{theorem}
In case (i), one $H$-orbit is a totally geodesic $\ensuremath{\mathbb{R}} \ensuremath{\mathsf{H}}^k$ and the other orbits are contained in the distance tubes around it. In item (ii), either $\g{b}=\g{a}$, in which case the orbit $H\cdot o$ contains the geodesic $A\cdot o$, or $\g{b}=0$, in which case $H\cdot o$ is contained in the horosphere $N\cdot o$. Moreover, in case (ii), any choice of real subspace $\g{w}\subset\g{g}_\alpha\cong\ensuremath{\mathbb{C}}^{n-1}$ gives rise to at least one polar action; the justification of this claim makes use of a decomposition theorem~\cite[\S2.3]{DRDVK:mathz} for real subspaces of a complex vector space as a orthogonal sum of subspaces of constant K\"ahler angle. Thus, whereas in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^2$ the moduli space of polar actions up to orbit equivalence is finite, in $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$, $n\geq 3$, it is uncountable infinite.
\begin{remark}
It is curious to observe that the orbit $H\cdot o$ corresponding to case (ii) in Theorem~\ref{th:polarCHn} with $\g{b}=\g{a}$ is precisely the singular leaf of the isoparametric foliations referred to in Theorems~\ref{th:isopar_DR} and~\ref{th:isopCHn}(v). In particular, it is a minimal submanifold, and the orbit foliation of the $H$-action constitutes a subfoliation of the isoparametric family of hypersurfaces given by the tubes around $H\cdot o$.
\end{remark}
\section{Submanifolds of symmetric spaces of arbitrary rank}\label{sec:higher_rank}
In this section we start by presenting some structure results for symmetric spaces of arbitrary rank, namely, their horospherical decomposition and the associated canonical extension procedure (\S\ref{subsec:horospherical}). Then we comment on the classification problem of cohomogeneity one and hyperpolar actions (\S\ref{subsec:cohom1}), and on a recent result on homogeneous CPC submanifolds~(\S\ref{subsec:CPC}).
\subsection{Horospherical decomposition and canonical extension.}\label{subsec:horospherical}
In this subsection we introduce these two important tools for the study of submanifolds in higher rank symmetric spaces. Further information can be found in~\cite[\S{}VII.7]{Knapp}, \cite[\S2.17]{Eberlein}, \cite[\S{}I.1]{BorelLi} and~\cite{DV:imrn}.
Let $M\cong G/K$ be a symmetric space of noncompact type. We follow the notation of Section~\ref{sec:noncompact}. Let $\Sigma$ be the set of roots of $M$, and $\ensuremath{\mathsf{P}}i$ a set of simple roots, $|\ensuremath{\mathsf{P}}i|=\ensuremath{\mathrm{rank}\;} M$.
Let $\ensuremath{\mathsf{P}}hi$ be any subset of $\ensuremath{\mathsf{P}}i$. Let $\Sigma_\ensuremath{\mathsf{P}}hi=\Sigma\cap\spann\ensuremath{\mathsf{P}}hi$ be the set of roots spanned by elements of $\ensuremath{\mathsf{P}}hi$, and $\Sigma_\ensuremath{\mathsf{P}}hi^+=\Sigma^+
\cap\spann\ensuremath{\mathsf{P}}hi$ the positive roots in $\Sigma_\ensuremath{\mathsf{P}}hi$. Then, we define
\[
\g{l}_\ensuremath{\mathsf{P}}hi=\g{g}_0\oplus\left(\bigoplus_{\lambda\in \Sigma_\ensuremath{\mathsf{P}}hi}\g{g}_\lambda \right), \qquad \g{n}_\ensuremath{\mathsf{P}}hi=\bigoplus_{\lambda\in \Sigma^+\setminus\Sigma^+_\ensuremath{\mathsf{P}}hi}\g{g}_\lambda,\qquad
\g{a}_\ensuremath{\mathsf{P}}hi=\bigcap_{\lambda\in\ensuremath{\mathsf{P}}hi}\ker \lambda,
\]
which are reductive, nilpotent and abelian subalgebras of $\g{g}$, respectively. Define also
\[
\g{m}_\ensuremath{\mathsf{P}}hi=\g{l}_\ensuremath{\mathsf{P}}hi\ominus\g{a}_\ensuremath{\mathsf{P}}hi,\qquad
\g{a}^\ensuremath{\mathsf{P}}hi=\g{a}\ominus\g{a}_\ensuremath{\mathsf{P}}hi=\bigoplus_{\lambda\in\ensuremath{\mathsf{P}}hi} \ensuremath{\mathbb{R}} H_\lambda.
\]
The subalgebra $\g{q}_\ensuremath{\mathsf{P}}hi=\g{l}_\ensuremath{\mathsf{P}}hi\oplus\g{n}_\ensuremath{\mathsf{P}}hi=\g{m}_\ensuremath{\mathsf{P}}hi\oplus\g{a}_\ensuremath{\mathsf{P}}hi\oplus\g{n}_\ensuremath{\mathsf{P}}hi$ is said to be the parabolic subalgebra of the real semisimple Lie algebra $\g{g}$ associated with the subset $\ensuremath{\mathsf{P}}hi\subset\ensuremath{\mathsf{P}}i$. The decompositions $\g{q}_\ensuremath{\mathsf{P}}hi=\g{l}_\ensuremath{\mathsf{P}}hi\oplus\g{n}_\ensuremath{\mathsf{P}}hi$ and $\g{q}_\ensuremath{\mathsf{P}}hi=\g{m}_\ensuremath{\mathsf{P}}hi\oplus\g{a}_\ensuremath{\mathsf{P}}hi\oplus\g{n}_\ensuremath{\mathsf{P}}hi$ are known as the Chevalley and Langlands decompositions of $\g{q}_\ensuremath{\mathsf{P}}hi$, respectively.
\begin{remark}
By considering $L_\ensuremath{\mathsf{P}}hi$ as the centralizer of $\g{a}_\ensuremath{\mathsf{P}}hi$ in $G$, an $N_\ensuremath{\mathsf{P}}hi$ as the connected subgroup of $G$ with Lie algebra $\g{n}_\ensuremath{\mathsf{P}}hi$, one can define the parabolic subgroup $Q_\ensuremath{\mathsf{P}}hi=L_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$ of $G$ associated with the subset $\ensuremath{\mathsf{P}}hi\subset\ensuremath{\mathsf{P}}i$. Geometrically speaking, parabolic subgroups of $G$ are isotropy groups of points at infinity, i.e.\ $Q_\ensuremath{\mathsf{P}}hi=\{g\in G: g(x)=x\}$ for some $x\in M(\infty)$ (except for the case $\ensuremath{\mathsf{P}}hi=\ensuremath{\mathsf{P}}i$, which gives rise to $Q_\ensuremath{\mathsf{P}}hi=G$). Thus, unlike points in $M$, isotropy groups of points at infinity are noncompact, and (except for $\ensuremath{\mathrm{rank}\;} M=1$) there are several (but finitely many, exactly $2^{\ensuremath{\mathrm{rank}\;} M}-1$) conjugacy classes of them.
\end{remark}
Consider the subspace
\[
\g{b}_\ensuremath{\mathsf{P}}hi=\g{m}_\ensuremath{\mathsf{P}}hi\cap\g{p}=\g{a}^\ensuremath{\mathsf{P}}hi\oplus\biggl(\bigoplus_{\lambda\in\Sigma_\ensuremath{\mathsf{P}}hi^+}\g{p}_\lambda\biggr),
\]
where $\g{p}_\lambda=(1-\theta)\g{g}_\lambda$ is the orthogonal projection of $\g{g}_\lambda$ onto $\g{p}$. Then $\g{b}_\ensuremath{\mathsf{P}}hi$ is a Lie triple system (see~\S\ref{subsec:totally}) in $\g{p}$. We denote by $B_\ensuremath{\mathsf{P}}hi$ the corresponding totally geodesic submanifold of $M$ which, intrinsically, is a symmetric space of noncompact type and rank $|\ensuremath{\mathsf{P}}hi|$, and is known as the \emph{boundary component} of $M$ associated with the subset $\ensuremath{\mathsf{P}}hi\subset\ensuremath{\mathsf{P}}i$. The Lie algebra of $\Isom(B_\ensuremath{\mathsf{P}}hi)$ is $\g{s}_\ensuremath{\mathsf{P}}hi:=[\g{b}_\ensuremath{\mathsf{P}}hi,\g{b}_\ensuremath{\mathsf{P}}hi]\oplus\g{b}_\ensuremath{\mathsf{P}}hi$. Thus, if $S_\ensuremath{\mathsf{P}}hi$ is the connected subgroup of $G$ with Lie algebra $\g{s}_\ensuremath{\mathsf{P}}hi$, then $B_\ensuremath{\mathsf{P}}hi=S_\ensuremath{\mathsf{P}}hi\cdot o$. It is not difficult to see that $B_\ensuremath{\mathsf{P}}hi$ can be regarded, under the isometry $M\cong AN$, as the connected subgroup of $AN$ with Lie algebra $\g{a}^\ensuremath{\mathsf{P}}hi\oplus\bigl(\bigoplus_{\lambda\in\Sigma_\ensuremath{\mathsf{P}}hi^+}\g{g}_\lambda\bigr)$.
The \emph{horospherical decomposition} theorem states that the map
\[
A_\ensuremath{\mathsf{P}}hi\times N_\ensuremath{\mathsf{P}}hi\times B_\ensuremath{\mathsf{P}}hi\to M,\qquad (a,n,p)\mapsto (an)(p),
\]
is an analytic diffeomorphism, where $A_\ensuremath{\mathsf{P}}hi$ and $N_\ensuremath{\mathsf{P}}hi$ are the connected subgroups of $G$ with Lie algebras $\g{a}_\ensuremath{\mathsf{P}}hi$ and $\g{n}_\ensuremath{\mathsf{P}}hi$, respectively.
In other words, this result implies that the connected closed subgroup $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$ of $AN$ acts isometrically and freely on $M$, and each $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$-orbit intersects $B_\ensuremath{\mathsf{P}}hi$ exactly once. Moreover, such intersection is always orthogonal (see~\cite[Proposition~4.2]{BDRT:jdg}). Thus, \emph{the $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$-action on $M$ is free and polar with section $B_\ensuremath{\mathsf{P}}hi$}. Moreover, as shown by Tamaru~\cite{Tamaru:math_ann}, \emph{all the orbits of the $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$-action are Einstein solvmanifolds and minimal submanifolds of $M$} and, actually, they are mutually congruent by elements of $S_\ensuremath{\mathsf{P}}hi$. The $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$-orbits are totally geodesic if and only if $\ensuremath{\mathsf{P}}hi$ and $\ensuremath{\mathsf{P}}i\setminus\ensuremath{\mathsf{P}}hi$ are orthogonal sets of roots.
The reinterpretation of the horospherical decomposition as a free, polar action with minimal orbits gives rise to the so-called \emph{canonical extension method}, which was introduced in~\cite{BT:crelle} for cohomogeneity one actions, and generalized in~\cite{DV:imrn} to other types of actions, foliations and submanifolds. This method allows to extend such geometric objects from a boundary component $B_\ensuremath{\mathsf{P}}hi$ to the whole symmetric space $M$, that is, from symmetric spaces of lower rank to symmetric spaces of higher rank. And, more importantly, one can do so by preserving some important geometric properties.
In order to formalize this, let $P$ be a submanifold of codimension $k$ in $B_\ensuremath{\mathsf{P}}hi$. Then
\[
A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot P:=\{h(p):h\in A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi,\, p\in P\}=\bigcup_{p\in P}A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot p
\]
is a submanifold of codimension $k$ in $M$. The mean curvature vector field of $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot P$ is $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$-equivariant and, along $P$, coincides with that of $P$. This implies that, if $P$ has parallel mean curvature, is minimal, has (globally) flat normal bundle, or is isoparametric, then $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot P$ has the same property.
One can also extend singular Riemannian foliations from $B_\ensuremath{\mathsf{P}}hi$ to $M$ by extending their leaves as above. Thus, if $\cal{F}$ is a singular Riemannian foliation on $B_\ensuremath{\mathsf{P}}hi$ that is polar, hyperpolar, or isoparametric, then the extended foliation $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot \cal{F}=\{A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot L:L\in\cal{F}\}$ has the same property. Moreover, if $\cal{F}$ is homogeneous, that is, if it is the orbit foliation of an isometric action of a subgroup $H\subset S_\ensuremath{\mathsf{P}}hi$ of isometries of $B_\ensuremath{\mathsf{P}}hi$, then the extended foliation $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi\cdot \cal{F}$ is the orbit foliation of the isometric action of $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi H$ on~$M$.
This technique plays an important role both in the construction of interesting types of submanifolds in symmetric spaces of higher rank, as well as in their classification. In~\cite{DV:imrn} it was used, for example, to extend the examples of inhomogeneous isoparametric hypersurfaces presented in~\S\ref{subsec:rank1_isoparametric} to symmetric spaces of higher rank and type $BC_r$, such as noncompact complex and quaternionic Grassmannians, or the complexified Cayley hyperbolic plane $\mathsf{E}_6^{-14}/\ensuremath{\mathsf{Sp}}in_{10}\ensuremath{\mathsf{U}}_1$. Also, it was used to construct inhomogeneous isoparametric foliations of codimension higher than one on noncompact real Grassmannians, as well as new examples of polar but nonhyperpolar actions on spaces of rank higher than one.
\subsection{Cohomogeneity one, hyperpolar and polar actions.}\label{subsec:cohom1}
By the very definition of rank, cohomogeneity one and hyperpolar actions on rank one symmetric spaces constitute the same family of actions. In higher rank, there are hyperpolar actions of greater cohomogeneity. Moreover, in any rank, there are polar actions which are not hyperpolar; for example, $A_\ensuremath{\mathsf{P}}hi N_\ensuremath{\mathsf{P}}hi$ acts polarly but not hyperpolarly on $M$, whenever $\ensuremath{\mathsf{P}}hi\neq \emptyset$. However, the classification problem of any of these types of actions is widely open.
The most general result regarding cohomogeneity one actions on symmetric spaces of noncompact type is due to Berndt and Tamaru~\cite{BT:crelle}:
\begin{theorem}\label{th:BT_higher}
Let $M\cong G/K$ be an irreducible symmetric space of noncompact type, and let $H$ be a connected subgroup of $G$ acting on $M$ with cohomogeneity one. Then one of the following statements holds:
\begin{enumerate}[{\rm (1)}]
\item The orbits form a regular foliation on $M$ and the $H$-action is orbit equivalent to the action of the connected subgroup of $AN$ with one of the following Lie algebras:
\begin{enumerate}[{\rm (i)}]
\item $(\g{a}\ominus\ensuremath{\mathbb{R}} X)\oplus\g{n}$ for some $X\in\g{a}$.
\item $\g{a}\oplus(\g{n}\ominus\ensuremath{\mathbb{R}} U)$, where $U\in \g{g}_\lambda$, for some $\lambda\in \ensuremath{\mathsf{P}}i$.
\end{enumerate}
\item There exists exactly one singular orbit and one of the following two cases holds:
\begin{enumerate}[{\rm (i)}]
\item $H$ is contained in a maximal proper reductive subgroup $L$ of $G$, the actions of $H$ and $L$ are orbit equivalent, and the singular orbit is totally geodesic.
\item $H$ is contained in a maximal proper parabolic subgroup $Q_\ensuremath{\mathsf{P}}hi$ of $G$ and the $H$-action is orbit equivalent to one of the following:
\begin{enumerate}[{\rm (a)}]
\item The canonical extension of a cohomogeneity one action with a singular orbit on the boundary component $B_\ensuremath{\mathsf{P}}hi$ of $M$.
\item The action of a group obtained by nilpotent construction.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{theorem}
Cohomogeneity one actions with no singular orbits, i.e.\ giving rise to homogeneous regular Riemannian foliations, were classified in~\cite{BT:jdg}; they correspond to case (1) in Theorem~\ref{th:BT_higher}. Note that they are induced by subgroups of $AN$.
Cohomogeneity one actions with a totally geodesic singular orbit were classified in~\cite{BT:tohoku}, and they correspond to case (2)-(i) above. Interestingly, the associated orbit foliations arise as tubes around certain reflective submanifolds, except for a few exceptional cases.
The case that is still open is that of cohomogeneity one actions with a nontotally geodesic singular orbit, case (2)-(ii) in Theorem~\ref{th:BT_higher}. The main difficulty has to do with the so-called \emph{nilpotent construction} method, which somehow extends the construction of cohomogeneity one actions with a nontotally geodesic singular orbit in rank one symmetric spaces (\S\ref{subsec:rank1_homogeneous}). We skip the explanation of the method here, and refer the reader to~\cite{BT:crelle} or to~\cite{BDV:tg}, where this method was investigated. In these papers one can also find the only complete classifications known so far on symmetric spaces of higher rank, namely on $\ensuremath{\mathsf{SL}}_3/\ensuremath{\mathsf{SO}}_3$, $\ensuremath{\mathsf{SL}}_3(\ensuremath{\mathbb{C}})/\ensuremath{\mathsf{SU}}_3$, $\mathsf{G}_2^
2/\ensuremath{\mathsf{SO}}_4$, $\mathsf{G}_2^\ensuremath{\mathbb{C}}/\mathsf{G}_2$ and $\ensuremath{\mathsf{SO}}_{2,n}^
0/\ensuremath{\mathsf{SO}}_2\ensuremath{\mathsf{SO}}_n$, $n\geq 3$.
In the more general setting of hyperpolar actions, the only known result is the classification of hyperpolar actions with no singular orbits on any symmetric space of noncompact type, up to orbit equivalence, due to Berndt, D\'iaz-Ramos and Tamaru~\cite{BDRT:jdg}. In other words, this result describes all hyperpolar homogeneous regular Riemannian foliations on symmetric spaces of noncompact type.
\begin{theorem}\label{th:BDRT}
A hyperpolar action with no singular leaves on a symmetric space of noncompact type $M$ is orbit equivalent to the hyperpolar action of the connected subgroup of $AN$ with Lie algebra
\[
(\g{a}\ominus V)\oplus\biggl(\g{n}\ominus\biggl(\bigoplus_{\lambda\in\ensuremath{\mathsf{P}}hi}\ensuremath{\mathbb{R}} X_\lambda\biggr)\biggr),
\]
where $\ensuremath{\mathsf{P}}hi\subset \ensuremath{\mathsf{P}}i$ is any subset of mutually orthogonal simple roots, and $V$ is any subspace~of~$\g{a}_\ensuremath{\mathsf{P}}hi$.
\end{theorem}
Note that the condition $\langle\lambda,\mu\rangle=0$ for any $\lambda,\mu\in\ensuremath{\mathsf{P}}hi$ implies that the associated boundary component $B_\ensuremath{\mathsf{P}}hi$ is the Cartesian product of $|\ensuremath{\mathsf{P}}hi|$ symmetric spaces of rank one, $B_\ensuremath{\mathsf{P}}hi=\prod_{\lambda\in \ensuremath{\mathsf{P}}hi}\mathbb{F}_\lambda\ensuremath{\mathsf{H}}^{n_\lambda}$, $\mathbb{F}_\lambda\in\{\ensuremath{\mathbb{R}},\ensuremath{\mathbb{C}},\ensuremath{\mathbb{H}},\mathbb{O}\}$. Thus, the intersection of the foliation described in Theorem~\ref{th:BDRT} with $B_\ensuremath{\mathsf{P}}hi$ is the product foliation of solvable foliations (cf.~\S\ref{subsec:rank1_homogeneous}) on each factor $\mathbb{F}_\lambda\ensuremath{\mathsf{H}}^{n_\lambda}$. The case $V=0$ corresponds to the canonical extension of such product foliation.
Regarding polar actions on symmetric spaces of rank higher than one, very little is known. Let us simply mention the classification of polar actions with a fixed point on any symmetric space by D\'iaz-Ramos and Kollross~\cite{DRK:dga}, and the investigation of polar actions by reductive subgroups due to Kollross~\cite{Ko:duality}.
\subsection{Homogeneous CPC submanifolds.}\label{subsec:CPC}
A submanifold $P$ of a Riemannian manifold $M$ will be called a \emph{CPC submanifold} if its principal curvatures, counted with multiplicities, are independent of the normal direction. In particular, a CPC submanifold is always austere (that is, the multiset of its principal curvatures is invariant under multiplication by~$-1$) and, hence, minimal. Although the terminology CPC comes from \emph{constant principal curvatures}, the property of being CPC is more restrictive than the one studied in~\cite{HOT91} (cf.~\cite[\S4.3]{BCO}). However, this notion is intimately related to cohomogeneity one actions. Indeed, if a cohomogeneity one action on a Riemannian manifold has a singular orbit, then the slice representation at any point of this orbit is transitive on the unit sphere of the normal space, which implies that all shape operators with respect to any unit normal vector are conjugate and, hence, the singular orbit is CPC.
The converse is not true in general. In fact, as mentioned after Theorem~\ref{th:BB}, there are totally geodesic (and, hence, CPC) submanifolds in the complex hyperbolic space whose distance tubes are not homogeneous.
In real space forms, a submanifold is CPC if and only if the distance tubes around it are isoparametric hypersurfaces with constant principal curvatures. The necessity in this equivalence is no longer true in spaces of nonconstant curvature (a counterexample is the one mentioned in the previous paragraph, in view of Theorem~\ref{th:isopCHn}), but the sufficiency holds in any Riemannian manifold for submanifolds of codimension higher than one~\cite{GT:asian}.
Moreover, totally geodesic submanifolds are examples of CPC submanifolds. Thus, the study of CPC submanifolds encompasses important problems, such as the classifications of totally geodesic submanifolds, cohomogeneity one actions, and isoparametric hypersurfaces with constant principal curvatures. Let us also emphasize that the singular leaf of the inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures on the Cayley hyperbolic plane described in~\S\ref{subsec:rank1_isoparametric} was, up to very recently, the only known example of a homogeneous, nontotally geodesic, CPC submanifold that is not an orbit of a cohomogeneity one action on a symmetric space of noncompact type.
In what follows we will report on the main results and ideas of a recent work by Berndt and Sanmart\'in-L\'opez~\cite{BS18} regarding CPC submanifods in irreducible symmetric spaces of noncompact type. One of the main goals of \cite{BS18} was precisely to provide a systematic approach to the construction of homogeneous, nontotally geodesic, CPC submanifolds, producing a large number of examples that are not orbits of cohomogeneity one actions. Another remarkable point is the introduction of an original and innovative technique based on the algebraic examination of the root system of symmetric spaces in order to calculate the shape operator of certain homogeneous submanifolds.
Let $M\cong G/K$ be an irreducible symmetric space of noncompact type; as usual, we follow the notation in~\S\ref{sec:noncompact}. Let $\alpha_0$, $\alpha_1\in \ensuremath{\mathsf{P}}i$ be two simple roots connected by a single edge in the Dynkin diagram of the symmetric space $M$. Consider a Lie subalgebra $\g{s} = \g{a} \oplus (\g{n} \ominus V)$ of $\g{a}\oplus\g{n}$, where $V$ is a subspace of $\g{g}_{\alpha_0} \oplus \g{g}_{\alpha_1}$. This implies that $V = V_0 \oplus V_1$ with $V_k \subset \g{g}_{\alpha_k}$ for $k \in \{ 0, 1\}$. Let $S$ be the connected closed subgroup of $AN$ with Lie algebra $\g{s}$. In the following lines, we will explain the approach to the classification of the CPC submanifolds of the form $S \cdot o$. Moreover, in the final part of this section, we will see that with weaker hypotheses on $\g{s}$ we still achieve the same classification result.
The orbit $S \cdot o$ is a homogeneous submanifold and therefore it suffices to study its shape operator $\ensuremath{\mathcal{S}}$ at the point $o$. Since $\ensuremath{\mathcal{S}}_{\xi} X = - (\nabla_{X} \xi)^{\top}$, where $(\cdot)^{\top}$ denotes the orthogonal projection onto $\g{s}$, the idea is to analyze carefully the terms involved in the expression \eqref{eq:Levi-Civita} for the Levi-Civita connection of $M$. Let $\xi \in V$ be a unit normal vector to $S \cdot o$ and let $X \in \g{s}$ be a tangent vector to $S \cdot o$. First, assume that $X \in \g{a}$. Then
\[
[X, \xi] + [\theta X, \xi] - [X, \theta \xi] = - [X, \theta \xi] \in \g{g}_{-\alpha_0} \oplus \g{g}_{-\alpha_1}.
\]
Hence, $[X, \theta \xi]$ has trivial projection onto $\g{a} \oplus \g{n}$. Thus, $\ensuremath{\mathcal{S}}_\xi X=-(\nabla_{X} \xi)^\top = 0$ for any tangent vector $X \in \g{a}$ and any normal vector $\xi \in V$.
Now take $\xi \in V$ and $X \in \g{g}_{\lambda}^{\top}$ with $\lambda \in \Sigma^{+}$. Using~\eqref{eq:Levi-Civita} and some other considerations that we omit for the sake of simplicity, we obtain
\begin{equation}\label{equation:shape:operator}
\ensuremath{\mathcal{S}}_{\xi} X = - \dfrac{1}{2} \left([X, \xi] -[X, \theta \xi]\right)^{\top}.
\end{equation}
Therefore, we deduce
\begin{equation}\label{equation:key:step}
\ensuremath{\mathcal{S}}_{\xi} X \in (\g{g}_{\lambda + \alpha_0} \oplus \g{g}_{\lambda + \alpha_1}) \oplus (\g{g}_{\lambda - \alpha_0}^{\top} \oplus \g{g}_{\lambda - \alpha_1}^{\top}),
\end{equation}
for each $\xi \in V$ and each $X \in \g{g}_{\lambda}^{\top}$ with $\lambda \in \Sigma^{+}$. This shows that we need to understand how the shape operator $\ensuremath{\mathcal{S}}$ relates the different positive root spaces among them.
In order to clarify this situation, we introduce a generalization of the concept of $\alpha$-string \cite[p.~152]{Knapp}. For ${\alpha_0}, {\alpha_1} \in \Sigma$ and $\lambda \in \Sigma$ we define the $({\alpha_0}, {\alpha_1})$-string containing $\lambda$ as the set of elements in $\Sigma \cup \{0\}$ of the form $\lambda + n {\alpha_0} + m {\alpha_1}$ with $n,m \in \mathbb{Z}$. This allows to define an equivalence relation on $\Sigma^{+}$. We say that two roots $\lambda_1, \lambda_2 \in \Sigma^{+}$ are $(\alpha_0, \alpha_1)$-related if $\lambda_1 - \lambda_2 = n {\alpha_0} + m {\alpha_1}$ for some $n,m \in \mathbb{Z}$. Thus, the equivalence class $[\lambda]_{(\alpha_0, \alpha_1)}$ of the root $\lambda \in \Sigma^{+}$ consists of the elements which may be written as $\lambda +n {\alpha_0} + m {\alpha_1}$ for some $n,m \in \mathbb{Z}$. We will write $[\lambda]$ for this equivalence class, taking into account that it depends on the roots $\alpha_0$ and $\alpha_1$ defining the string. Put $\Sigma^{+} / \sim$ for the set of equivalence classes. The family $\{[\lambda]\}_{\lambda \in \Delta^{+}}$ constitutes a partition of $\Sigma^{+}$. Using this notation, from~\eqref{equation:shape:operator}~and~\eqref{equation:key:step} we get that
\begin{equation}\label{eq:shape:class}
\ensuremath{\mathcal{S}}_{\xi} \left( \bigoplus_{\gamma \in [\lambda]} \g{g}_{\gamma}^{\top} \right) \subset \bigoplus_{\gamma \in [\lambda]} \g{g}_{\gamma}^{\top} \qquad\text{for all } \lambda \in \Sigma^{+}.
\end{equation}
This is the key point for studying if the orbit $S \cdot o$ is CPC. We will explain~\eqref{eq:shape:class} in words. For each $\lambda \in \Sigma^{+}$ the subspace $\bigoplus_{\gamma \in [\lambda]} \g{g}_{\gamma}^{\top}$ is a $\ensuremath{\mathcal{S}}_{\xi}$-invariant subspace of the tangent space $\g{s}$. Moreover, $S \cdot o$ is a CPC submanifold if and only if the eigenvalues of $\ensuremath{\mathcal{S}}_\xi$ are independent of the unit normal vector $\xi$ when restricted to each one of those invariant subspaces $\bigoplus_{\gamma \in [\lambda]} \g{g}_{\gamma}^{\top}$, for every $\lambda \in \Sigma^{+}$. Thus it suffices to consider the orthogonal decomposition
\begin{equation}\label{invariant:decomposition}
\g{n} \ominus V = \bigoplus_{[\lambda] \in \Sigma^{+} / \sim} \left(\bigoplus_{\gamma \in [\lambda]} \g{g}_{\gamma}^{\top} \right),
\end{equation}
and to study the shape operator when restricted to each one of these $\ensuremath{\mathcal{S}}_{\xi}$-invariant subspaces. Since $\alpha_0$ and $\alpha_1$ span an $A_2$ root system, then neither $2 \alpha_0$ nor $2 \alpha_1$ are roots. Hence, the $(\alpha_0, \alpha_1)$-string of $\alpha_0$ consists of the roots $\alpha_0$, $\alpha_1$ and $\alpha_0 + \alpha_1$. Thus, one of these subspaces is $\g{g}_{\alpha_0} \oplus \g{g}_{\alpha_1} \oplus \g{g}_{\alpha_0 + \alpha_1}$. This approach would be interesting if the rest of the subspaces respected some pattern and they could be determined explicitly. The following result addresses both questions. Recall that ${\alpha_0}$ and ${\alpha_1}$ are simple roots connected by a single edge in the Dynkin diagram. We define the level of a positive root as the sum of the nonnegative coefficients of its expression with respect to the basis $\ensuremath{\mathsf{P}}i$. Let $\lambda \in \Sigma^{+}$ be the root of minimum level in its $(\alpha_0, \alpha_1)$-string. Assume that it is not spanned by $\alpha_0$ and $\alpha_1$. Then, (taking indices modulo~$2$) we have:
\begin{enumerate}[{\rm (i)}]
\item \label{structure:strings:i} If $\langle \lambda, \alpha_0 \rangle = 0 = \langle \lambda, \alpha_1 \rangle$, then $[\lambda] = \{\lambda\}$.
\item If $|\alpha_k| \geq |\lambda|$ and $\langle \lambda, \alpha_k \rangle \neq 0$, then $[\lambda] = \{\lambda, \lambda + {\alpha_k}, \lambda + {\alpha_{k}} + {\alpha_{k+1}}\}$. \label{structure:strings:ii}
\item Otherwise, $[\lambda] = \{\lambda, \lambda + {\alpha_k}, \lambda + {\alpha_k} + {\alpha_{k+1}}, \lambda + 2{\alpha_k}, \lambda + 2 {\alpha_k} + {\alpha_{k+1}}, \lambda + 2 {\alpha_k} + 2{\alpha_{k+1}} \}$. \label{structure:strings:iii}
\end{enumerate}
The roots $\lambda$, $\alpha_0$ and $\alpha_1$ span a manageable subsystem and, roughly speaking, the proof follows from a case-by-case examination on the possible Dynkin diagrams for this subsystem. The CPC condition means that the eigenvalues of the shape operator do not depend on the normal vector when restricted to each one of the subspaces $\bigoplus_{\gamma \in [\lambda]} \g{g}_{\gamma}^{\top}$ in~\eqref{invariant:decomposition}, where $[\lambda]$ is one of the three possible types of strings above.
If $\lambda$ is under the hypotheses of case~(\ref{structure:strings:i}), then $\g{g}_{\lambda}$ belongs to the $0$-eigenspace of the shape operator. This claim follows from~(\ref{equation:key:step}) and the fact that neither $\lambda + \alpha_k$ nor $\lambda - \alpha_k$ are roots for $k \in \{0,1\}$.
We analyze case~(\ref{structure:strings:ii}) in order to give the key ideas for a nontrivial case. Let us start with some general considerations. For a fixed $l \in \{ 0,1 \}$, let $\gamma \in \Sigma^{+}$ be the root of minimum level in its $\alpha_l$-string, which consists of the roots $\gamma$ and $\gamma + \alpha_l$. Fix a normal unit vector $\xi_l \in V_{l}$ and define
\begin{equation}\label{definition:phi}
\phi_{\xi_l} = |\alpha_l|^{-1} \ad(\xi_l) \qquad \text{and} \qquad \phi_{\theta \xi_l} = -|\alpha_l|^{-1} \ad(\theta\xi_l).
\end{equation}
These maps $\phi_{\xi_l}$ and $\phi_{\theta \xi_l}$ turn out to be inverse linear isometries in the sense that $\phi_{\theta\xi_l}\circ\phi_{\xi_l}\rvert_{\g{g}_\gamma}=\id_{\g{g}_\gamma}$ and $\phi_{\xi_l}\circ\phi_{\theta\xi_l}\rvert_{\g{g}_{\gamma+\alpha_l}}=\id_{\g{g}_{\gamma+\alpha_l}}$. Moreover, for each $X \in \g{g}_{\gamma}$ we have
\begin{equation}\label{easy:shape}
\nabla_X \xi_l = - \frac{|\alpha_l|}{2} {\phi_{\xi_l}(X)} \qquad \text{and} \qquad
\nabla_{\phi_{\xi_l}(X)} \xi_l = - \frac{|\alpha_l|}{2} X.
\end{equation}
Let us come back to the study of case~(\ref{structure:strings:ii}). Write $\xi = \cos(\varphi) \xi_k + \sin(\varphi) \xi_{k+1}$ with $\varphi \in [0, \frac{\pi}{2}]$, $\xi_k \in V_k$ and $\xi_{k+1} \in V_{k+1}$. The following diagram may help to understand the situation. Note that the nodes represent root spaces and not roots.
\begin{equation*}
\begin{tikzpicture}[scale=0.4]
\draw(-15.0,0.) circle (0.5cm);
\draw(-6.,0.) circle (0.5cm);
\draw(3,0.) circle (0.5cm);
\draw[- triangle 45] (-14,0.3)-- (-7,0.3);
\draw[- triangle 45] (-7,-0.3) -- (-14,-0.3) ;
\draw[- triangle 45] (-4,0.3)-- (2,0.3);
\draw[- triangle 45] (2,-0.3) -- (-4,-0.3);
\begin{scriptsize}
\draw[color=black] (-15, 1.1) node {$\g{g}_{\lambda}$};
\draw[color=black] (-6, 1.1) node {$\g{g}_{\lambda + \alpha_k}$};
\draw[color=black] (3, 1.1) node {$\g{g}_{\lambda + \alpha_k + \alpha_{k+1}}$};
\draw[color=black] (-10.5, 0.8) node {$\phi_{\xi_k}$};
\draw[color=black] (-10.5, -1.1) node {$\phi_{\theta \xi_k}$};
\draw[color=black] (-1.5, 0.8) node {$\phi_{\xi_{k+1}}$};
\draw[color=black] (-1.5, -1.1) node {$\phi_{\theta \xi_{k+1}}$};
\end{scriptsize}
\end{tikzpicture}
\end{equation*}
Take a unit tangent vector $X \in \g{g}_{\lambda}$. Using $\ensuremath{\mathcal{S}}_{\xi} X = - (\nabla_{X} \xi)^{\top}$ and~\eqref{easy:shape} for the pair $(\gamma, \alpha_l) \in \{(\lambda, \alpha_k),(\lambda + \alpha_k, \alpha_{k+1})\}$, we can see that the $3$-dimensional vector space spanned by $X, \phi_{\xi_k}(X),(\phi_{\xi_{k+1}} \circ \phi_{\xi_k})(X)$ is $\ensuremath{\mathcal{S}}_\xi$-invariant. The matrix representation of $\ensuremath{\mathcal{S}}_{\xi}$ restricted to $\g{g}_\lambda\oplus\g{g}_{\lambda+\alpha_k}\oplus\g{g}_{\lambda+\alpha_k+\alpha_{k+1}}$ is then given by $\dim (\g{g}_{\lambda})$ blocks of the form
\begin{equation}\label{cpc:matrix}
\frac{|{\alpha_0}|}{2}
\left(\begin{array}{ccc} 0& \cos(\varphi) & 0 \\ \cos(\varphi) & 0 & \sin(\varphi) \\ 0& \sin(\varphi) & 0
\end{array}\right),
\end{equation}
with respect to the decomposition $\g{g}_{ \lambda} \oplus \phi_{\xi_k}(\g{g}_{\lambda}) \oplus (\phi_{\xi_{k+1}} \circ \phi_{\xi_k})(\g{g}_{\lambda})$. The eigenvalues of the above matrix are $0$ and $\pm \frac{|\alpha_0|}{2}$. They do not depend on $\varphi$. It is also important to note that the nonzero principal curvatures depend on the length of the root $\alpha_0$.
Case~(\ref{structure:strings:iii}) is slightly more difficult than the one we have just studied. Roughly speaking, it is necessary to generalize~(\ref{easy:shape}) having in mind the following diagram:
\begin{equation*}
\begin{tikzpicture}[scale=0.4]
\draw(-15.0,0.) circle (0.5cm);
\draw(-6.,0.) circle (0.5cm);
\draw(6,0.) circle (0.5cm);
\draw(0,3.5) circle (0.5cm);
\draw(0,-3.5) circle (0.5cm);
\draw(15.,0.) circle (0.5cm);
\draw[- triangle 45] (7,0.) -- (14,0);
\draw[- triangle 45] (-14,0.0)-- (-7,0.0);
\draw[- triangle 45] (-5,0.35)-- (-1,3.15);
\draw[triangle 45 -] (5,0.35) -- (1,3.15);
\draw[- triangle 45] (-5,-0.35)-- (-1,-3.15);
\draw[triangle 45 -] (5,-0.35)-- (1,-3.15);
\begin{scriptsize}
\draw[color=black] (-10.5, 0.5) node {$\ad(\xi_k)$};
\draw[color=black] (10.5, 0.5) node {$\ad(\xi_{k+1})$};
\draw[color=black] (-3.75, 2.5) node {$\ad(\xi_k)$};
\draw[color=black] (-4.2, -2.5) node {$\ad(\xi_{k+1})$};
\draw[color=black] (4, 2.5) node {$\ad(\xi_{k+1})$};
\draw[color=black] (3.75, -2.5) node {$\ad(\xi_k)$};
\draw[color=black] (-15, 1.1) node {$\g{g}_{\lambda}$};
\draw[color=black] (-6.2, 1.1) node {$\g{g}_{\lambda+\alpha_k}$};
\draw[color=black] (6.5, 1.1) node {$\g{g}_{\lambda+2\alpha_k + \alpha_{k+1}}$};
\draw[color=black] (0,4.3) node {$\g{g}_{\lambda+2\alpha_k}$};
\draw[color=black] (0,-4.3) node {$\g{g}_{\lambda + \alpha_k + \alpha_{k+1}}$};
\draw[color=black] (15.3, 1.1) node {$\g{g}_{\lambda +2\alpha_k + 2 \alpha_{k+1}}$};
\end{scriptsize}
\end{tikzpicture}
\end{equation*}
The principal curvatures of the shape operator do not depend on the normal vector when restricted to subspaces induced by strings of type~(\ref{structure:strings:iii}).
Thus, the problem can be reduced to studying the shape operator when restricted to
\[
\g{g}_{\alpha_0}^\top \oplus \g{g}_{\alpha_1}^\top \oplus \g{g}_{\alpha_0 + \alpha_1}.
\]
In other words, one needs to study CPC submanifolds in a symmetric space with Dynkin diagram of type $A_2$, which would conclude the classification result.
However, as mentioned above, we can state a more general result concerning CPC submanifolds. Denote by $\ensuremath{\mathsf{P}}i^{\prime}$ the set of simple roots $\alpha \in \ensuremath{\mathsf{P}}i$ with $2 \alpha \notin \Sigma$. Note that there is at most one simple root in $\ensuremath{\mathsf{P}}i$ that does not belong to $\ensuremath{\mathsf{P}}i^{\prime}$, and this happens when the restricted root system of $M$ is of type $BC_r$. Consider a Lie algebra $\g{s} = \g{a} \oplus (\g{n} \ominus V)$ with $V \subset \bigoplus_{\alpha \in \ensuremath{\mathsf{P}}i'} \g{g}_{\alpha}$. This implies that $V = \bigoplus_{\alpha \in \ensuremath{\mathsf{P}}si} V_{\alpha}$ for some set $\ensuremath{\mathsf{P}}si \subset \ensuremath{\mathsf{P}}i^{\prime}$. Similar ideas to those that led us to~(\ref{cpc:matrix}) allow to deduce that for each root $\alpha \in \ensuremath{\mathsf{P}}si$ the nonzero eigenvalues of $\ensuremath{\mathcal{S}}_{\xi_{\alpha}}$ are proportional to the length of $\alpha$. Then, if $\ensuremath{\mathsf{P}}si$ contains roots $\alpha$ and $\beta$ of different lengths, it follows that the shape operators $\ensuremath{\mathcal{S}}_{\xi_{\alpha}}$ and $\ensuremath{\mathcal{S}}_{\xi_{\beta}}$ have different eigenvalues, which implies that $S \cdot o$ is not CPC.
Moreover, assume that $\ensuremath{\mathsf{P}}si$ contains at least three roots. Then $\ensuremath{\mathsf{P}}si$ has at least two orthogonal roots, say $\alpha_0$ and $\alpha_1$. We will explain briefly why this cannot lead to a CPC submanifold $S \cdot o$. The key point is to find a positive root $\lambda \in \Sigma^{+}$ with nontrivial $\alpha_{k}$-string but trivial $\alpha_{k+1}$-string, for some $k \in \{0,1\}$ and indices modulo 2. According to~(\ref{easy:shape}), there must exist a tangent vector $X \in \g{g}_{\lambda} \oplus \g{g}_{\lambda + \alpha_k}$ such that $\ensuremath{\mathcal{S}}_{\xi_k} X = \mu X$, for a unit normal $\xi_k \in V_{\alpha_k}$ and some $\mu\neq 0$. However, from~(\ref{equation:key:step}) we deduce that $\ensuremath{\mathcal{S}}_{\xi_{k+1}} X = 0$ for a unit normal $\xi_{k+1} \in V_{\alpha_{k+1}}$. Thus, if we take a normal unit vector $\xi = \cos(\varphi) \xi_k + \sin(\varphi) \xi_{k+1}$ for $\varphi \in [0, 2 \pi]$, then we get $\ensuremath{\mathcal{S}}_{\xi} X = \cos(\varphi) \mu X$. Thus $S \cdot o$ cannot be CPC since we have an infinite family of different principal curvatures.
Finally, if $V$ is contained in a single root space $\g{g}_{\alpha}$, $\alpha\in\ensuremath{\mathsf{P}}i'$, then the $S$-action on $M$ is the canonical extension of a cohomogeneity one action with a totally geodesic orbit on the boundary component $B_{\{\alpha\}}\cong\mathbb{R}\ensuremath{\mathsf{H}}^n$ (see~\S\ref{subsec:horospherical}). Hence, if $\dim V\geq 2$, $S\cdot o$ is a singular orbit of a cohomogeneity one action, and then CPC; if $\dim V=1$, $S\cdot o$ is the only minimal orbit of an action as in Theorem~\ref{th:BT_higher}(1-ii), which also happens to be CPC. Altogether, we can state the main result of~\cite{BS18}:
\begin{theorem}\label{th:BS}
Let $\g{s} = \g{a} \oplus (\g{n} \ominus V)$ be a subalgebra of $\g{a} \oplus \g{n}$ with $V \subset \bigoplus_{\alpha \in \ensuremath{\mathsf{P}}i'} \g{g}_{\alpha}$. Let $S$ be the connected closed subgroup of $AN$ with Lie algebra $\g{s}$. Then the orbit $S \cdot o$ is a CPC submanifold of $M \cong G/K$ if and only if one of the following statements holds:
\begin{itemize}
\item[{\rm (I)}] There exists a simple root $\lambda \in \ensuremath{\mathsf{P}}i'$ with $V \subset \g{g}_{\lambda}$. \label{main:simple:examples}
\item[{\rm (II)}] There exist two nonorthogonal simple roots $\alpha_0,\alpha_1 \in \ensuremath{\mathsf{P}}i'$ with $|\alpha_0| = |\alpha_1|$ and subspaces $V_0 \subset \g{g}_{\alpha_0}$ and $V_1 \subset \g{g}_{\alpha_1}$ such that $V = V_0 \oplus V_1$ and one of the following conditions holds: \label{main:new:examples}
\begin{itemize}
\item[{\rm (i)}] $V_0 \oplus V_1 = \g{g}_{\alpha_0} \oplus \g{g}_{\alpha_1}$; \label{singular:orbit}
\item[{\rm (ii)}] $V_0$ and $V_1$ are isomorphic to $\mathbb{R}$ and $V_0 \oplus V_1$ is a proper subset of $\g{g}_{\alpha_0} \oplus \g{g}_{\alpha_1}$;
\item[{\rm (iii)}] $V_0$ and $V_1$ are isomorphic to $\mathbb{C}$, $V_0 \oplus V_1$ is a proper subset of $\g{g}_{\alpha_0} \oplus \g{g}_{\alpha_1}$ and there exists $T \in \g{k}_0$ such that $\ad(T)$ defines complex structures on $V_0$ and $V_1$ and vanishes on $[V_0, V_1]$; \label{main:complex}
\item[{\rm (iv)}] $V_0$ and $V_1$ are isomorphic to $\mathbb{H}$, $V_0 \oplus V_1$ is a proper subset of $\g{g}_{\alpha_0} \oplus \g{g}_{\alpha_1}$ and there exists a subset $\g{l} \subset \g{k}_0$ such that $\ad(\g{l})$ defines quaternionic structures on $V_0$ and $V_1$ and vanishes on $[V_0, V_1]$. \label{main:quaternionic}
\end{itemize}
\end{itemize}
Moreover, only the submanifolds given by {\rm (I)} and {\rm (II)(i)} can appear as singular orbits of cohomogeneity one actions.
\end{theorem}
\begin{remark}
One may ask whether this result is still true if $V$ is a subspace of the sum of root spaces corresponding to the roots in $\ensuremath{\mathsf{P}}i$ (instead of $\ensuremath{\mathsf{P}}i'$). However, this seems to be a more difficult problem. Indeed, it includes, in particular, the classification problem of CPC submanifolds of the type $S\cdot o$ in the quaternionic hyperbolic spaces $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$, $n\geq 3$, which turns out to be equivalent to the classification of subspaces of $\g{g}_\alpha\cong \ensuremath{\mathbb{H}}^{n-1}$ with constant quaternionic K\"ahler angle. As we mentioned in~\S\ref{subsec:rank1_homogeneous}, this is nowadays an open problem.
\end{remark}
\section{Open problems}
We include a list of open problems related to the research presented above.
\begin{enumerate}
\item In view of the exposition in~\S\ref{subsec:rank1_homogeneous}, classify homogeneous hypersurfaces in quaternionic hyperbolic spaces $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$, $n\geq 3$. More generally, classify the real subspaces $V$ of $\ensuremath{\mathbb{H}}^{n-1}$ with constant quaternionic K\"ahler angle, and, for each one of them, determine if there is a subgroup of $\ensuremath{\mathsf{Sp}}_1\ensuremath{\mathsf{Sp}}_{n-1}$ that acts transitively on the unit sphere of $V$.
\item Make further progress in the classification problem (mentioned in~\S\ref{subsec:rank1_isoparametric}) of hypersurfaces with constant principal curvatures in the complex hyperbolic spaces $\ensuremath{\mathbb{C}} \ensuremath{\mathsf{H}}^n$. This is a very difficult problem that lacks powerful ideas and techniques, apart from a clever combination of the information provided by the Codazzi and Gauss equations of submanifolds.
\item Classify curvature-adapted hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane $\mathbb{O}\ensuremath{\mathsf{H}}^2$.
\item Initiate the investigation of non-curvature-adapted hypersurfaces with constant principal curvatures in $\ensuremath{\mathbb{H}} \ensuremath{\mathsf{H}}^n$ and $\mathbb{O} \ensuremath{\mathsf{H}}^2$.
\item Is there any nonisoparametric hypersurface with constant principal curvatures in a symmetric space? In a general Riemannian setting, one can probably construct such a hypersurface for some specific ambient metric, but, even in this case, no concrete example is known to us.
\item Prove that compact embedded hypersurfaces with constant mean curvature in rank one symmetric spaces of noncompact type must be geodesic spheres. Any idea towards the solution of this problem is likely to have profound implications and many applications to other problems (such as the symmetry of solutions to the so-called overdetermined boundary value problems, see e.g.~\cite{DVPSE}).
\item Obtain a better understanding of the nilpotent construction method for cohomogeneity one actions, mentioned in Theorem~\ref{th:BT_higher}(2-ii-b). This seems to be a crucial step towards the solution of the classification problem of cohomogeneity one actions on irreducible symmetric spaces of noncompact type. Another approach may come from generalizing Theorem~\ref{th:BS} to the study of CPC submanifolds arising from arbitrary subgroups of the solvable part of the Iwasawa decomposition.
\item Construct, if possible, new examples of inhomogeneous isoparametric hypersurfaces in symmetric spaces of noncompact type and rank higher than one. All the examples known so far arise as canonical extensions of isoparametric hypersurfaces in rank one symmetric spaces.
\item Construct new examples of inhomogeneous isoparametric submanifolds of codimension higher than one on symmetric spaces of noncompact type. The only known examples are those appearing in Wu's classification for real hyperbolic spaces~\cite{Wu:tams}, and their canonical extensions to noncompact real Grassmannians. Such submanifolds are leaves of non-hyperpolar isoparametric foliations. Is there any example of an inhomogeneous hyperpolar foliation of codimension higher than one on a symmetric space of noncompact type, unlike the compact case~\cite{Christ} (cf.~\cite{Ly:gafa})?
\item Make progress in the classification problem of totally geodesic submanifolds, mentioned in~\S\ref{subsec:totally}. This problem seems nowadays infeasible in full generality. However, with the algebraic method explained in \S\ref{subsec:CPC} we are able to calculate very efficiently the shape operator of many homogeneous submanifolds. These ideas may help to obtain some classification result in certain higher rank symmetric spaces.
\end{enumerate}
\end{document} | math | 127,865 |
\begin{document}
\begin{abstract}
The logarithmic multiplicative group is a proper group object in logarithmic schemes, which morally compactifies the usual multiplicative group. We study the structure of the stacks of logarithmic maps from rational curves to this logarithmic torus, and show that in most cases, it is a product of the logarithmic torus with the space of rational curves. This gives a conceptual explanation for earlier results on the moduli spaces of logarithmic stable maps to toric varieties.
\end{abstract}
\maketitle
\section{Introduction}
A rational function on a $\mathbf P^1$ is determined up to multiplicative constant by the locations and orders of its zeroes and poles, and exists if the sum of those orders is zero. Likewise, a balanced piecewise linear function on a metric tree is determined up to addition of a constant by its slopes along unbounded edges and exists provided the sum of these slopes is zero. We observe here that both of these facts are aspects of a single phenomenon in logarithmic geometry: \textit{the space of stable genus $0$ maps to the logarithmic torus is a product of the logarithmic torus with the space of genus $0$ curves.} See Corollaries~\ref{cor: prod} and~\ref{cor: self-maps} for the precise statements of the main results, including a treatment of unstable cases.
The logarithmic torus, $\mathbf G_{\log}$ (to be defined precisely below), is a non-representable functor on logarithmic schemes that compactifies the algebraic torus, $\mathbf G_{\mathrm m}$. Despite its failure to be representable, one can make sense of its tropicalization as the undivided real line $\mathbf R$, and the fiber of its tropicalization map as the multiplicative group $\mathbf G_{\mathrm m}$. Complete toric varieties arise by pullback along the tropicalization map $\mathbf G_{\log}^n \to \mathbf R^n$ of subdivisions of $\mathbf R^n$ into fans.
The spaces of genus~$0$ stable maps to a toric variety are some of the most basic objects of logarithmic Gromov-Witten theory~\cite{AC11,Che10,GS13}. Indeed, they have been studied before, for instance in~\cite{CMR14b,CS12,R15b}. In prior work on the subject, careful polyhedral arguments play a role in determining the geometry of these spaces of maps.
Such arguments are of course necessary in order to obtain the precise geometric descriptions such as those in op.\ cit., however the results presented here suggest a simple underlying principle behind those results. An auxiliary goal of this note is to demonstrate that such mildly non-representable functors can clarify the geometry of (logarithmic) schemes. For instance, the space of logarithmic stable maps to $\mathbf P^1$ with fixed contact at $n\geq 3$ marked points is a toroidal modification of $\overline{\mathcal M}_{0,n}\times \mathbf P^1$, which is a key result in~\cite{CMR14b}. We show here that the analogous space of maps to $\mathbf G_{\log}$ is $\overline{\mathcal M}_{0,n}\times \mathbf G_{\log}$. A logarithmic modification of the target produces a logarithmic modification of the space of maps, so this provides a clear conceptual reason for this result. Analogous statements can be extracted regarding the results of~\cite{R15b}. In Section~\ref{sec: maps-of-tori} we study maps between logarithmic tori, which serves as an underlying principle for the results of~\cite{AM14,CS12}.
We note that an incarnation of $\mathbf G_{\log}$ exists with B. Parker's theory of exploded manifolds, as the exploded manifold attached to the `fan' in $\mathbf R$ with a single non-strictly convex cone equal to $\mathbf R$, see~\cite[Section 3]{Par12b}.
\section{Groundwork in the tropics}
\subsection{Functors of points}\label{sec: three-functors} Logarithmic geometry facilitates the interaction between the category of schemes and the category of convex rational polyhedral cones.
If $\Sigma$ is a rational polyhedral cone with integral lattice $L$, let $\Sigma^\vee$ be the set of vectors in the dual lattice $L^\vee$ taking nonnegative values on $\Sigma$. A logarithmic scheme $S$ has two important sheaves of monoids, $M_S$ and $\overnorm M_S / \mathcal O_S^\ast$. Let $X_\Sigma$ be the affine toric variety with fan $\Sigma$, and let $\mathcal A_\Sigma$ be the stack quotient of $X_\Sigma$ by its dense torus, which is canonically equipped with a logarithmic structure. We have two identifications:
\begin{gather}
\operatorname{Hom}_{\mathbf{LogSch}}(S, X_\Sigma) = \operatorname{Hom}(\Sigma^\vee, \Gamma(S, M_S)) \label{eqn:toric} \\
\operatorname{Hom}_{\mathbf{LogSch}}(S, \mathcal A_\Sigma) = \operatorname{Hom}(\Sigma^\vee, \Gamma(S, \overnorm M_S))
\end{gather}
\subsection{} If $\overnorm\alpha$ is a section of $\overnorm M_S^{\rm gp}$ then the fiber of $M_S^{\rm gp}$ over $\overnorm\alpha$ will be denoted $\mathcal O_S^\ast(-\overnorm\alpha)$. This is a torsor under $\mathcal O_S^\ast$ and may be completed in a unique way to an invertible sheaf $\mathcal O_S(-\overnorm\alpha)$. If $\overnorm\alpha$ is a section of $\overnorm M_S$ then the logarithmic structure gives a $\mathcal O_S^\ast$-equivariant map $\mathcal O_S^\ast(-\overnorm\alpha) \subset M_S \to \mathcal O_S$ which extends to a homomorphism $\mathcal O_S(-\overnorm\alpha) \to \mathcal O_S$.
\subsection{The logarithmic torus}
The logarithmic multiplicative group seems to have been introduced by Kato~\cite{KatoGm}. For any logarithmic scheme $S$ with logarithmic structure $M_S$, we define $\mathbf G_{\log}(S)$ by~\eqref{eqn:1}:
\begin{equation} \label{eqn:1}
\mathbf G_{\log}(S) = \Gamma(S, M_S^{\rm gp})
\end{equation}
This is a contravariant functor on logarithmic schemes. It is not representable by a scheme or even an algebraic stack with a logarithmic structure, but it does have a logarithmically \'etale cover by the toric variety $\mathbf P^1$. In similar fashion, $\mathbf G_{\log}^n$ admits a logarithmically \'etale cover by \textit{any} complete toric variety of dimension $n$. See~\cite[Section 2.2.7]{logpic} for a detailed treatment.
\subsection{} If $S$ has trivial logarithmic structure then $\mathbf G_{\log}(S) = \mathbf G_{\mathrm m}(S)$. Since the locus in any logarithmic scheme where the logarithmic structure is trivial is an open subset, we may therefore think of $\mathbf G_{\log}$ as at least a partial compactification of $\mathbf G_{\mathrm m}$.
\begin{proposition} \label{prop:p1}
The map $\mathbf P^1 \to \mathbf G_{\log}$ given in local coordinates by $(x,y) \mapsto x^{-1} y$ is a logarithmic modification, in the sense that its base change along any map $S \to \mathbf G_{\log}$ is a logarithmic modification.
\end{proposition}
\begin{proof}
A map $S \to \mathbf G_{\log}$ is given by $\alpha \in \Gamma(S, M_S^{\rm gp})$. Locally in $S$, we can represent $\alpha$ as $\beta^{-1} \gamma$ for $\beta, \gamma \in \Gamma(S, M_S^{\rm gp})$. Then $(\beta,\gamma)$ determines a map $S \to \mathbf A^2$ (with its toric logarithmic structure) by \eqref{eqn:toric}. Then $\alpha$ lifts to $S \to \mathbf P^1$ if and only if locally in $S$ we have $\alpha \in M_S$ or $\alpha^{-1} \in M_S$. This is equivalent to requiring $(\beta,\gamma)$ to lift to the blowup $Y$ of $\mathbf A^2$ at the origin, which proves that $\mathbf P^1 \mathop\times_{\mathbf G_{\log}} S = Y \mathop\times_{\mathbf A^2} S$.
\end{proof}
Since logarithmic modifications are logarithmically \'etale, the proposition shows that $\mathbf G_{\log}$ has a logarithmically \'etale cover by a logarithmic scheme and, since that logarithmic scheme is proper, it should also be regarded as proper. Unlike its toric compactification, $\mathbf G_{\log}$ is also a \emph{group object} in the category of logarithmic schemes. We emphasize that such a compactification is not possible within the category of schemes, because the only equivariant schematic compactification of $\mathbf G_{\mathrm m}$ is $\mathbf P^1$, which does not admit a group structure.
\section{Curves in the logarithmic torus}
\subsection{The space of maps} Let $\mathfrak M$ denote the stack of prestable genus $0$ logarithmic curves. If $X$ is a category fibered in groupoids over logarithmic schemes, we denote by $\frak M(X)$ the stack of logarithmic pre-stable maps from genus~$0$ logarithmic curves to $X$.
\begin{equation}
\frak M(X) = \left\{ (S, C, \xi) \: \Big| \: \parbox{2in}{$C$ is a proper logarithmic genus $0$ curve over $S$ and $\xi \in X(C)$} \right\}
\end{equation}
This applies in particular to $X = \mathbf G_{\log}$. Since $\mathbf G_{\log}$ has a group structure, $\frak M(X)$ is a sheaf of abelian groups over $\frak M$ in the \'etale topology.
Note that the category $\frak M(X)$ admits a tropicalization, following Section~\ref{sec: three-functors}. Specifically, given a pre-stable logarithmic map to $X$ over $S$, there is an associated diagram of tropical curves over $S^{\mathrm{trop}}$, together with a map to $X^{\mathrm{trop}}$.
\subsection{Piecewise linear functions}
We recall from \cite[Remark~7.3]{CCUW} how sections of the characteristic monoid of a logarithmic curve give piecewise linear functions on its tropicalization.
Let $C$ be a logarithmic curve over an algebraically closed field $S$ and let $C^{\rm trop}$ be its dual graph. Each edge $e$ of $C^{\rm trop}$ corresponds to a node where $C$ has a local equation $xy = t$ in its characteristic monoid, with $t \in M_S$. Let $\overnorm t$ be the image of $t$ in $\overnorm M_S$. Then we refer to $\overnorm t$ as the length of $e$.
Suppose that $\overnorm\alpha \in \overnorm M_C^{\rm gp}$. If $v$ is a vertex of $C^{\rm trop}$ then there is a corresponding component of $C$ on which $\overnorm M_C^{\rm gp}$ is constant with value $\overnorm M_S^{\rm gp}$. We write $\overnorm\alpha(v)$ for the constant value of $\overnorm\alpha$ on the interior of this component. If $e$ is an edge of $C$ connected vertices $v$ and $w$ then near $e$ there is a unique representation of $\overnorm\alpha$ as $\overnorm\alpha(v) + \lambda \overnorm x$, where $\overnorm x$ is the image of $x \in M_{C,e}$ in $\overnorm M_{C,e}$ and $\lambda \in \mathbf Z$. Then restricting to $w$ we find $\overnorm\alpha(w) = \overnorm\alpha(v) + \lambda \overnorm t$. This allows us to think of $\overnorm\alpha$ as a piecewise linear function on $C^{\rm trop}$ with slope $\lambda$ on the edge $e$, when directed from $v$ to $w$.
\begin{proposition}
Let $C_v$ be the component of $C$ corresponding to a vertex $v$ of $C^{\rm trop}$. Then $\mathcal O_{C_v}(\overnorm\alpha) = \mathcal O_{C_v}(\sum_{e : v \to w} \lambda_e e)$ where the sum is taken over all edges leaving $v$ and $\lambda_e$ denotes the slope of $\overnorm\alpha$ on the edge $e$.
\end{proposition}
\begin{proof}
See \cite[Proposition~2.4.1]{RSW}.
\end{proof}
\begin{proposition} \label{prop:balanced}
Suppose $C$ is a logarithmic curve over an algebraically closed field and $\overnorm\alpha \in \Gamma(C, \overnorm M_C^{\rm gp})$ lifts to $M_C^{\rm gp}$. Then $\overnorm \alpha$ is a balanced function on $C^{\rm trop}$.
\end{proposition}
\begin{proof}
Let $\alpha \in \Gamma(C, M_C^{\rm gp})$ lift $\overnorm\alpha$. Then $\alpha$ is, by definition, a section of $\mathcal O_C^\ast(-\overnorm\alpha)$. Equivalently, $\overnorm\alpha$ is a nowhere vanishing section of $\mathcal O_C(-\overnorm\alpha)$. Thus $\mathcal O_C(-\overnorm\alpha)$ is trivial and in particular has degree zero. Restricting to a component $C_v$ of $C$, we have $\mathcal O_{C_v}(-\overnorm\alpha) \simeq \mathcal O_{C_v}(\sum_{e : v \to w} \lambda_e e)$. This implies $\sum \lambda_e = 0$, which is the balancing condition.
\end{proof}
\subsection{Contact orders} Let $C$ be a logarithmic curve. A map $F: C \to \mathbf G_{\log}$ is a section of $M_C^{\rm gp}$, which in turn induces a section $\overnorm\alpha$ of $\overnorm M_C^{\rm gp}$. We regard $\overnorm\alpha$ as a linear function on the dual graph of $C$. The slopes of $\overnorm\alpha$ on the $n$ infinite legs of the dual graph of $C$ are locally constant in $\frak M(\mathbf G_{\log})$. This gives a homomorphism $\frak M(\mathbf G_{\log}) \to \mathbf Z^n$. The \emph{contact order} of the map is defined as the image of $[F]$ in $\mathbf Z^n$ under this map.
\subsection{Maps up to translation} The kernel of the homomorphism $\frak M(\mathbf G_{\log}) \to \mathbf Z^n$ consists of maps $C \to \mathbf G_{\log}$ whose associated linear function has zero slope on the infinite legs. But such a function is effectively a bounded balanced piecewise linear function on the complement of the infinite legs in the dual graph. Any such balanced function $\overnorm\alpha$ is constant. In that case, $\mathcal O_C(\overnorm\alpha)$ is the pullback of $\mathcal O_S(\overnorm\alpha)$ from the base and the map $C \to \mathbf G_{\log}$ corresponds to a trivialization of this bundle. Indeed, the fiber of $M_X^{\rm gp}$ over $\overnorm\alpha \in \overnorm M_X^{\rm gp}$ is $\mathcal O_X^\ast(-\overnorm\alpha)$, by definition, so a section of $M_X^{\rm gp}$ in the fiber over $\overnorm\alpha$ corresponds to a nowhere vanishing section of $\mathcal O_X(\overnorm\alpha)$. Since $X$ is proper over $S$ with reduced and connected fibers, all sections of $\mathcal O_X(\overnorm\alpha)$ over $X$ are pulled back from sections of $\mathcal O_S(\overnorm\alpha)$. Thus a section over $S$ of the kernel of $\frak M(\mathbf G_{\log}) \to \mathbf Z^n$ consists of pairs $(\overnorm\alpha, \alpha)$ where $\overnorm\alpha$ is a section of $\overnorm M_S^{\rm gp}$ over $S$ and $\alpha$ is a section of $\mathcal O_S^\ast(\overnorm\alpha)$. This shows that the kernel is isomorphic to $\mathbf G_{\log}$.
\begin{theorem}
Let $\mathbf Z^n_0$ be the subgroup of $\mathbf Z^n$ consisting of those $n$-tuples of integers whose sum is zero. There is an exact sequence of sheaves (in the big \'etale site) of abelian groups over $\frak M$:
\begin{equation} \label{eqn:2}
0 \to \mathbf G_{\log} \to \frak M(\mathbf G_{\log}) \to \mathbf Z^n_0 \to 0
\end{equation}
and the final map is smooth.
\end{theorem}
\begin{proof}
Note that $\frak M(\mathbf G_{\log}) \to \mathbf Z^n$ takes values in $\mathbf Z^n_0$ because by Proposition~\ref{prop:balanced} every section of $M_X^{\rm gp}$ over a rational curve $X$ induces a \emph{balanced} section of $\overnorm M_X^{\rm gp}$, which is to say that the sum of the outgoing slopes at any vertex of the dual graph is zero. This implies that the sum of the outgoing slopes along the infinite legs is also zero.
We have already proved the left exactness in the statement of the theorem. To conclude we must prove that $\frak M(\mathbf G_{\log}) \to \mathbf Z_0^n$ is a smooth surjection.
We consider the smoothness first. Since $\mathbf Z_0^n$ is \'etale over $\frak M$, it is equivalent to demonstrate that $\frak M(\mathbf G_{\log})$ is smooth over $\frak M$. Consider a first order deformation of a logarithmic curve $C \subset C'$ and a section $\alpha$ of $M_C^{\rm gp}$. Let $\overnorm\alpha$ be the image of $\alpha$ in $\overnorm M_C^{\rm gp}$. Then $\overnorm\alpha$ extends uniquely to $C'$ since $\overnorm M_C^{\rm gp} = \overnorm M_{C'}^{\rm gp}$ when their \'etale sites are identified. We can view $\alpha$ as a trivialization of $\mathcal O_C(\overnorm\alpha)$ and we wish to extend this to a trivialization of $\mathcal O_{C'}(\overnorm\alpha)$. The obstructions to doing so lie in $H^1(C, \mathcal O_C(\overnorm\alpha))$. But $\overnorm\alpha$ is a balanced function on the tropicalization of $C$, so $\mathcal O_C(\overnorm\alpha)$ has multidegree~$0$. As $C$ is a tree of rational curves, this implies $H^1(C, \mathcal O_C(\overnorm\alpha)) = 0$.
To prove the surjectivity, we fix a genus~$0$ logarithmic curve $C$ with tropicalization $\Gamma$ and a vector $\sigma \in \mathbf Z^n_0$. We can construct a unique linear function $\overnorm\alpha$ on $\Gamma$ whose slopes on the legs of $\Gamma$ are given by $\sigma$. To lift this section to an element of $\frak M(\mathbf G_{\log})$ in the fiber over $\sigma$, we must give a nowhere vanishing section of $\mathcal O_C(\overnorm\alpha)$. But $\mathcal O_C(\overnorm\alpha)$ has multidegree~$0$ and the components of $C$ are rational, so $\mathcal O_C(\overnorm\alpha) \simeq \mathcal O_C$ has a nowhere vanishing section.
\end{proof}
For a point $\Gamma$ in $\mathbf Z_0^n$, let $\mathfrak M_\Gamma(\mathbf G_{\log})$ be its fiber in the exact sequence above.
\begin{corollary}
Let $\mathfrak M_{\Gamma+1}(\mathbf G_{\log})$ denote the moduli space of $1$-marked genus~$0$ pre-stable maps to $\mathbf G_{\log}$ with contact order $\Gamma$ and one additional marked point of contact order~$0$. Then evaluation at the final marked point furnishes an isomorphism $\mathfrak M_{\Gamma+1}(\mathbf G_{\log}) \simeq \mathfrak M_{\Gamma+1} \times \mathbf G_{\log}$.
\end{corollary}
\begin{proof}
Evaluation at the new marked point with trivial contact order splits the injection in the exact sequence above, leading to the claim.
\end{proof}
\begin{corollary} \label{cor:triv-ext}
If $n \geq 3$ then the exact sequence~\eqref{eqn:2} splits and $\mathfrak M_{0,n}(\mathbf G_{\log}) \simeq \mathfrak M_{0,n} \times \mathbf G_{\log} \times \mathbf Z_0^n$.
\end{corollary}
\begin{proof}
For $n \geq 3$, let $C$ be a logarithmic curve over $S$, let $\alpha \in \Gamma(C, M_C^{\rm gp})$ give an $S$-point of $\mathfrak M_{0,n}(\mathbf G_{\log})$, and let $x$ be a marked point of $C$, viewed as a non-logarithmic section over $S$. Write $\overnorm\alpha$ for the image of $\alpha$ in $\overnorm M_C^{\rm gp}$. Then $x^{-1} \overnorm M_{C} = \mathbf N \times \overnorm M_S$, canonically, and $x^\ast \overnorm\alpha = (c(x), \overnorm\alpha(x))$ where $c$ denotes the contact order of $\alpha$ at $x$. We can view $x^\ast \alpha$ as a nowhere vanishing section of $\mathcal O_S((\overnorm\alpha(x), c(x))) = \mathcal O_S(\overnorm\alpha(x)) \otimes N_{x/X}^{\otimes c(x)}$. But, as $n \geq 3$, the universal tangent line $N_{x/C}$ is canonically isomorphic to the line bundle associated to a boundary divisor, so the $M_S^{\rm gp}$-torsor associated to $N_{x/C}^{\otimes c(x)}$ is canonically trivial. Likewise the $M_S^{\rm gp}$-torsor associated to $\mathcal O_S(\overnorm\alpha)$ is canonically trivial, so $x^\ast\alpha$ is thus identified with a section of $\mathbf G_{\log}$ and we obtain a morphism $\phi : \mathfrak M(\mathbf G_{\log}) \to \mathbf G_{\log}$. It follows from the canonical isomorphism~\eqref{eqn:3}
\begin{equation} \label{eqn:3}
\mathcal O_S(\overnorm\alpha(x) + \overnorm\alpha'(x)) \otimes N_{x/C}^{\otimes (c(x) + c'(x))} = \mathcal O_S(\overnorm\alpha(x)) \otimes N_{x/C}^{\otimes c(x)} \otimes \mathcal O_S(\overnorm\alpha'(x)) \otimes N_{x/C}^{\otimes c'(x)}
\end{equation}
that $\phi$ is a homomorphism with respect to the group structure of $\mathfrak M(\mathbf G_{\log})$. It is immediate that this homomorphism splits the inclusion of $\mathbf G_{\log}$ in $\mathfrak M(\mathbf G_{\log})$, and therefore that $\mathfrak M_{0,n}(\mathbf G_{\log}) \simeq \mathfrak M \times \mathbf Z_0^n \times \mathbf G_{\log}$.
\end{proof}
\begin{warning}
The proof of Corollary~\ref{cor:triv-ext} gives a canonical splitting of the extension~\eqref{eqn:2}, for each of the $n$ marked points of the curve. These splittings genuinely depend on the markings and distinct markings give distinct splittings.
\end{warning}
Let $\mathcal M_\Gamma(\mathbf G_{\log})$ denote genus~$0$ \emph{stable} maps to $\mathbf G_{\log}$ with fixed contact orders $\Gamma$, where stability means that if $\alpha \in \Gamma(X, M_X^{\rm gp})$ is constant on a component of $X$ then that component has at least~$3$ special points.
\begin{corollary}\label{cor: prod}
Fix a vector of $n$ contact orders and genus $0$ in the combinatorial datum $\Gamma$. We have isomorphisms:
\begin{equation*}
\mathcal M_\Gamma(\mathbf G_{\log}) \simeq \begin{cases} \varnothing & n \leq 1 \\ \overline{\mathcal M}_{0,n} \times \mathbf G_{\log} & n \geq 3\end{cases}
\end{equation*}
\end{corollary}
\begin{proof}
The statement for $n \leq 1$ is immediate because there are no nonconstant linear functions on a genus~$0$ tropical curve with only one infinite leg and for $n \geq 3$ is an immediate consequence of Corollary~\ref{cor:triv-ext}.
\end{proof}
We have not included a statement for $n = 2$ because the space `stable' maps from $2$-marked rational curves with contact orders $(r,-r)$ to $\mathbf G_{\log}$ is the nonseparated stack $(\mathbf G_{\log}/\mathbf G_{\mathrm m}) \times \mathrm B \mu_r$. The difficulty is that the unique semistable component of $\mathbf P^1$ is `contracted' by any morphism $\mathbf P^1 \to \mathbf G_{\log}$ and it is therefore necessary to contract it in the source to obtain a reasonable parameter space. We explain how this works in the next section.
\subsection{}
We can now make explicit the relationship between these results and those in~\cite{CMR14b,R15b}. The moduli space of logarithmic stable maps to $\mathbf P^1$ admits a morphism to the space of maps to $\mathbf G_{\log}$, by composing the universal map with $\mathbf P^1\to\mathbf G_{\log}$ and possibly stabilizing. The resulting map
\[
\mathcal M_\Gamma(\mathbf P^1)\to \mathcal M_\Gamma(\mathbf G_{\log})
\]
is easily seen to be logarithmically \'etale and birational. By Corollary~\ref{cor: prod}, the moduli space of logarithmic stable maps to $\mathbf P^1$ is a logarithmic modification of $\overline{\mathcal M}_{0,n} \times \mathbf G_{\log}$. The analogous statement holds for any toric variety. The specific nature of this modification is determined by the map on tropicalizations, which is a subdivision, described precisely in~\cite{CMR14b,R15b}
\section{Maps between logarithmic tori}\label{sec: maps-of-tori}
The following lemma is well known, but we include a proof for completeness.
\begin{lemma} \label{lem:Gm-units}
Let $S$ be a scheme. Then $\mathcal O_S[t,t^{-1}]^\ast = \mathcal O_S^\ast \times t^{\mathbf Z_S}$.
\end{lemma}
\begin{proof}
The assertion is straightforward to check when $S$ is integral. Let $\alpha$ be a section of $\mathcal O_S[t,t^{-1}]^\ast$. For each point $p$ of $S$, we have $\alpha(p) = u(p) t^{n(p)}$ for some $u \in k(p)^\ast$ and $n \in \mathbf Z$. Let $S_n$ be the set of points $p$ of $S$ where $n(p) = n$. It is a quick exercise to see that $S_n$ contains a constructible neighborhood of each of its points. As valuation rings are integral, it follows that $S_n$ is also stable under generization, so each $S_n$ is open.
This implies that $n(p)$ is a constructible function on $S$. Therefore $t^{-n}$ is a section of $t^{\mathbf Z_S}$ and $\alpha(p) t^{-n(p)} \in k(p)^\ast$ for every $p \in X$. This implies $\alpha \in \mathcal O_S^\ast$, as required.
\end{proof}
\begin{proposition} \label{prop:end-logGm}
Let $t$ denote the identity function on $\mathbf G_{\log}$. Any map $S \times \mathbf G_{\log} \to \mathbf G_{\log}$ can be represented uniquely as $\alpha t^n$ where $\alpha$ is a section of $M_S^{\rm gp}$ and $n : S \to \mathbf Z$ is a locally constant function.
\end{proposition}
\begin{proof}
Suppose that $\beta : S \times \mathbf G_{\log} \to \mathbf G_{\log}$ is a map. Let $\mathbf P^1 \to \mathbf G_{\log}$ be the map described in Proposition~\ref{prop:p1}. Restricting $\beta$ along this map, we obtain a section $\beta' \in \Gamma(S \times \mathbf P^1, M_{S \times \mathbf P^1}^{\rm gp})$. We note that if $j : S \times \mathbf G_{\mathrm m} \to S \times \mathbf P^1$ is the inclusion then $M_{S \times \mathbf P^1}^{\rm gp} = j_\ast M_{S \times \mathbf G_{\mathrm m}}^{\rm gp}$.
Let $q : S \times \mathbf P^1 \to S$ be the projection. We have an exact sequence~\eqref{eqn:8}:
\begin{equation} \label{eqn:8}
0 \to \mathcal O_{S \times \mathbf G_{\mathrm m}}^\ast \to M_{S \times \mathbf G_{\mathrm m}}^{\rm gp} \to j^{-1} q^{-1} \overnorm M_S^{\rm gp} \to 0
\end{equation}
Applying $j_\ast$ gives us an exact sequence~\eqref{eqn:11}:
\begin{equation} \label{eqn:11}
0 \to j_\ast \mathcal O_{S \times \mathbf G_{\mathrm m}}^\ast \to M_{S \times \mathbf P^1}^{\rm gp} \to q^{-1} \overnorm M_S^{\rm gp} \to 0
\end{equation}
Pushing forward to $S$ and applying Lemma~\ref{lem:Gm-units}, we obtain the bottom row of~\eqref{eqn:9}:
\begin{equation} \label{eqn:9}
\vcenter{\xymatrix{
0 \ar[r] & \mathcal O_S^\ast \ar[r] \ar[d] & M_S^{\rm gp} \ar[d] \ar[r] & \overnorm M_S^{\rm gp} \ar@{=}[d] \ar[r] & 0 \\
0 \ar[r] & \mathcal O_S^\ast \times t^{\mathbf Z_S} \ar[r] & q_\ast M_{S \times \mathbf P^1}^{\rm gp} \ar[r] & \overnorm M_S^{\rm gp} \ar[r] & 0
}}
\end{equation}
It follows that there is an exact sequence~\eqref{eqn:10},
\begin{equation} \label{eqn:10}
0 \to M_S^{\rm gp} \to q_\ast M_{S \times \mathbf P^1}^{\rm gp} \to t^{\mathbf Z_S} \to 0
\end{equation}
that is split by the pullback of $t$ along the second projection $S \times \mathbf P^1 \to \mathbf G_{\log}$. We therefore have $q_\ast M_{S \times \mathbf P^1}^{\rm gp} = M_S^{\rm gp} \times t^{\mathbf Z_S}$. In particular, $\beta'$ can be represented uniquely as $\alpha t^n$ where $\alpha \in M_S^{\rm gp}$ and $n : S \to \mathbf Z$ is a locally constant function.
To see that this formula actually describes $\beta$, consider $\beta \alpha^{-1} t^{-n}$. This is now a map $S \times \mathbf G_{\log} \to \mathbf G_{\log}$ whose restriction to $S \times \mathbf P^1$ is trivial. Let $f : T \to S \times \mathbf G_{\log}$ be any map. By Proposition~\ref{prop:p1}, $f^{-1}(S \times \mathbf P^1)$ is a logarithmic modification $p : T' \to T$ and $p^\ast f^\ast \beta = 1$ by construction. But $\Gamma(T, M_T^{\rm gp}) \to \Gamma(T', M_{T'}^{\rm gp})$ is an injection (in fact an isomorphism) \cite[Theorem~4.4.1]{logpic}, so we conclude $f^\ast \beta = 1$, as required.
\end{proof}
\begin{corollary}
We have $\operatorname{End}(\mathbf G_{\log}) = \mathbf Z$ and every $S$-morphism $\mathbf G_{\log} \to \mathbf G_{\log}$ is uniquely representable as the product of a translation and a homomorphism.
\end{corollary}
\begin{proof}
An endomorphism is in particular a self-map of $\mathbf G_{\log}$, so up to translation, it is given by an $n^{\mathrm{th}}$-power map as a consequence of the proposition above. Since endomorphisms must preserve the identity, the integer $n$ is the only datum distinguishing such a map.
\end{proof}
\begin{corollary}\label{cor: self-maps}
Let $\mathcal M_{0,2}(\mathbf G_{\log})$ denote the stack on logarithmic schemes whose $S$-points consist of a $\mathbf G_{\log}$-torsor $C$ on $S$ and a map $C \to \mathbf G_{\log}$. Then $\mathcal M_{0,2}(\mathbf G_{\log}) \simeq \coprod_{r \in \mathbf Z} \mathrm B \mu_r$ with the understanding that $\mathrm B \mu_0 = \mathbf G_{\log} \times \mathrm B\logGm$.
\end{corollary}
\begin{proof}
Locally in $S$, there is an isomorphism $C \simeq S \times \mathbf G_{\log}$. By Proposition~\ref{prop:end-logGm}, a map $f : C \to \mathbf G_{\log}$ is therefore representable locally as $\alpha t^r : S \times \mathbf G_{\log} \to \mathbf G_{\log}$ with $\alpha \in M_S^{\rm gp}$ and $r \in \mathbf Z$. It follows that $C \to \mathbf G_{\log}$ is equivariant with respect to the map $[r] : \mathbf G_{\log} \to \mathbf G_{\log}$. The locally constant function $r$ gives a decomposition $\mathcal M_{0,2}(\mathbf G_{\log}) = \coprod X_r$.
If $r \neq 0$, the fiber of $C \to \mathbf G_{\log}$ over the identity is therefore a torsor under $\ker \: [r] = \mu_r$. This gives a map $X_r \simeq \mathrm B \mu_r$ sending $f : C \to \mathbf G_{\log}$ to $(r, f^{-1}(1))$. Conversely, given any $\mu_r$-torsor $C_0$ on $S$ we may extend $C_0$ along $\mu_r \to \mathbf G_{\log}$ to obtain a $\mathbf G_{\log}$-torsor $C$ along with a map $C \to \mathbf G_{\log}$. These operations are easily seen to be inverse to one another.
If $r = 0$ then the map to $C \to \mathbf G_{\log}$ factors uniquely through $S$, giving a factor $\mathbf G_{\log}$. The choice of $C$ is parameterized by $\mathrm B\logGm$, yielding $X_0 = \mathbf G_{\log} \times \mathrm B\logGm$.
\end{proof}
Consider the moduli space of logarithmic stable maps from two-pointed $\mathbf P^1$ to a toric variety $X$, in the class of a one-parameter subgroup. As in the previous section, by composing such maps with $X\to \mathbf G_{\log}^n$ and stabilizing the map, we see that the moduli space of such maps is obtained from a product of copies of $\mathbf G_{\log}$ by logarithmic modification and a root construction. This is implied by the main results of~\cite{AM14,CS12}, and again, the specific logarithmic modification is determined by the tropical subdivisions that are considered explicitly there.
\end{document} | math | 28,711 |
\begin{document}
\title{
Making Differentiable Architecture Search less local
}
\begin{abstract}
Neural architecture search (NAS) is a recent methodology for automating the design of neural network architectures.
Differentiable neural architecture search (DARTS) is a promising NAS approach that dramatically increases search efficiency.
However, it has been shown to suffer from performance collapse, where the search often leads to detrimental architectures.
Many recent works try to address this issue of DARTS by identifying indicators for early stopping, regularising the search objective to reduce the dominance of some operations, or changing the parameterisation of the search problem.
In this work, we hypothesise that performance collapses can arise from poor local optima around typical initial architectures and weights.
We address this issue by developing a more global optimisation scheme that is able to better explore the space without changing the DARTS problem formulation.
Our experiments show that our changes in the search algorithm allow the discovery of architectures with both better test performance and fewer parameters.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
Designing neural network architectures improving upon the state-of-the-art requires a substantial effort of human experts.
Automating the discovery of neural network architectures by formulating it as a search problem allows us to minimise the human time spent on the search process.
Due to the large combinatorial search space of possible neural network architectures, early methods~\cite{zoph2016neural,zoph2018learning,real2019regularized} were computationally very demanding,
often requiring thousands of GPU days of computation for search, giving rise to high costs.
Many neural architecture search (NAS) works have been focused on reducing the computational cost,~\cite{liu2018progressive,bender2018understanding,elsken2017simple,pham2018efficient,cai2017efficient}.
Among them, Liu et al. \cite{liu2018darts} proposed a particularly efficient approach by making the search space of architectures differentiable (known as DARTS), which reduced
the search cost by several orders of magnitude.
Although being efficient, recent works have shown that DARTS suffers from performance collapse due to the search favouring parameter-less operations like skip connections \cite{chu2020fair,zela2019understanding}. Many follow-up works have been proposed to fix the performance collapse problem by identifying indicators for early stopping, regularising the search objective to reduce skip connections, or changing the search problem's parameterisation.
Chen \& Hsieh~\cite{pmlr-v119-chen20f} and Zela et al.~\cite{zela2019understanding} proposed to stabilise the search process by regularising the Hessian of the search objective. Chu et al.~\cite{chu2020fair} avoid the advantage of the skip connections in the search phrase by replacing the softmax with the sigmoid function for the switch among edges.
Chu et al.~\cite{chu2021darts} avoided the dominance of skip connections by changing the parameterisation of the search space.
In this paper, we hypothesise that performance collapses and the dominance of some operations observed in several works are the consequence of the existence of poor local optima around typical initial architectures and weights. Instead of identifying indicators for early stopping or tweaking the search space's parameterisation, we propose that a more global optimisation scheme should be developed that allows us to avoid bad local optima and better explore the objective over the search space to discover better solutions.
We show in experiments that even a simple scheme to make the optimisation more global reduces detrimental behaviours significantly.
Importantly,
it removes the need to stop the search early in order to avoid reaching detrimental or invalid solutions.
We show that, after searching until convergence, our method can find architectures with better test performance and fewer parameters.
\section{Empirical diagnosis}
\label{sec:empiriacl_diagnosis}
\begin{figure}
\caption{
The FairDARTS search needs to be stopped early to avoid the normal
cell having no remaining active edges following discretisation.
Shown is the mean cell weights (following the sigmoid activation function) for the normal and reduction cell, respectively, for three runs on different budgets.
The same issue persisted on every run.
The discretisation threshold used is 0.85, but the issue applies to any thresholding rule as all normal cell edge weights tend to zero.
}
\label{fig:fairdarts_sinking_ship_new}
\end{figure}
FairDARTS is a state-of-the-art DARTS variant presented in~\cite{chu2020fair}.
The method includes structural changes to the original DARTS search space,
allowing multiple edges per pair of nodes in the searched cell structure.
This was implemented by switching the softmax activation function on the weights to a sigmoid function.
Another change was adding a regularisation term (a `zero-one loss'),
encouraging the continuous edge activations to better approximate the binary discretisation of the cell happening after the search phase.
FairDARTS improved upon the DARTS method, reducing the issue with skip connections dominating,
and ultimately lead to better architectures in terms of final test performance compared to DARTS and other variants.
However, as we will demonstrate, another similar issue presents itself (still) in the FairDARTS method.
What happens is that one of the searched cell types, the `reduction cell', dominates the other (the `normal cell'), to the detriment of test performance and reliability.
In particular, if searching for longer than a small fraction of as many epochs later used for the training in the final evaluation phase,
the test performance decays, and the architectures produced quickly become invalid.
We illustrate this in \Cref{fig:fairdarts_sinking_ship_new} (the experiments were conducted on CIFAR-10 using the implementation and the setup as in~\cite{chu2020fair}).
We note that the edge weights associated with the normal cell decrease monotonically after a certain number of epochs.
If the search is not stopped early, at the right time, the weights of all operations in the normal cell become zero,
resulting in no activations being able to propagate through the cell following discretisation.
In~\cite{chu2020fair} the search was stopped after only $1 / 12$ of the number of epochs later used to train the final architecture.
Being forced to stop the search early to avoid detrimental architectures has two negative consequences.
Firstly, the right time to stop the search becomes an additional hyperparameter to tune to obtain good performance.
Secondly, it can inhibit better architectures to be found by searching for longer.
Both of these aspects are important for building a reliable NAS method for a wide range of datasets and tasks.
\section{Global Optimisation for Differentiable NAS}
DARTS~\cite{liu2018darts},
as similar to prior works~\cite{zoph2018learning,real2019regularized,liu2018progressive},
searches for a \emph{cell}, which is used as a building block for the final architecture.
The cell constitutes a directed acyclic graph of $N$ nodes.
Each node $x$ represents a latent representation
and each directed edge $(i, j)$ represents an operation $o_{i, j}$.
A node depends on all of its predecessors as $x_j = \sum_{i < j} o_{i, j}(x_i)$.
Let $\mathcal{O}$ be the set of candidate operations (e.g., convolution, max pooling, skip connection) available for each edge $(i, j)$.
FairDARTS~\cite{chu2020fair} defines the choice of operations for an edge as
$\bar{o}_{i, j}(x) = \sum_{o \in \mathcal{O}} \sigma(\alpha_{o_i,j})o(x)$,
where $\sigma(\cdot)$ is the sigmoid function.
This allows multiple operations per edge to be chosen simultaneously.
If no operations are active for a given edge this constitute the zero operation~\cite{zela2019understanding}.
Let $\bm{\alpha}$ be the concatenated vector of all operation edge weights representing the architecture,
in which the ones associated with the normal cell and the reduction cell are denoted by $\bm{\alpha}_{\text{normal}}$ and $\bm{\alpha}_{\text{reduction}}$ respectively, i.e., $\bm{\alpha} = (\bm{\alpha}_{\text{normal}}, \bm{\alpha}_{\text{reduction}})$.
Let $\bm{w}$ be the concatenated neural network parameters associated with all operations, where similarly $\bm{w} = (\bm{w}_{\text{normal}}, \bm{w}_{\text{reduction}})$.
The architecture search problem in DARTS can be stated as a bilevel optimisation problem:
\begin{mini!}|l|[2]
{\bm{\alpha}}
{
\mathcal{L}_{\text{val}}(\bm{\alpha}, \bm{w}^{\ast})
}
{}{
\label{eq:bilevel_1}
}
\addConstraint{
\bm{w}^{\ast} = \argmin_{\bm{w}} \mathcal{L}_{\text{train}}(\bm{\alpha}, \bm{w})
},
\label{eq:bilevel_2}
\end{mini!}
where $\mathcal{L}_{\text{val}}$ and $\mathcal{L}_{\text{train}}$ are the validation loss and training loss, respectively.
DARTS approximates the gradient as
$
\nabla_{\bm{\alpha}} \mathcal{L}_{\text{val}}(\bm{\alpha}, \bm{w}^{\ast})
\approx
\nabla_{\bm{\alpha}} \mathcal{L}_{\text{val}}(
\bm{\alpha},
\bm{w} -
\xi \nabla_{\bm{w}} \mathcal{L}_{\text{train}}(\bm{\alpha}, \bm{w})
)
$,
where $\xi$ is the learning rate for the inner optimisation,
and gradient-based local optimisation is performed in alternating steps.
\paragraph{Global Optimisation Scheme.}
\label{sec:globalopt}
We hypothesise that the usage of local search for the $\bm{\alpha}$ weights in the DARTS' approximation to the bilevel optimisation problem leads to convergence to local optima associated with performance collapse.
We propose an optimisation scheme that makes the search for the $\bm{\alpha}$ weights ``more global" in the sense that local valleys can be escaped using a complementary global optimisation routine.
\begin{algorithm}[t]
\caption{Doubly Stochastic Coordinate Descent (global step)}
\label{algo:doubly_coord}
\SetAlgoLined
\SetNoFillComment
\KwIn{Function $f$ defined over $\mathcal{X}$, proposal distribution $q$, initial $\bm{x}_{\text{best}}, y_{\text{best}}$}
\KwOut{$\bm{x}_{\text{best}}, y_{\text{best}}$}
\While{budget\_remaining}{
$d$ = sample\_a\_random\_dimension()\;
$\bm{x} \sim q(x|\bm{x}_{\text{best}}[d], d)$\;
$y = f(\bm{x})$\;
\If{$y < y_{\text{best}}$}{
$\bm{x}_{\text{best}} = \bm{x}$, $y_{\text{best}} = y$\;
}
}
\end{algorithm}
Our optimisation scheme consists of two types of steps: \emph{local} and \emph{global} steps.
The algorithm alternates between taking local and global steps,
similar to basin-hopping~\cite{wales1997global} for global optimisation.
A local step is a step in the gradient direction, the same as in~\cite{liu2018darts}.
A global step is taken according to the proposed doubly stochastic coordinate descent (DSCD) algorithm.
DSCD follows the stochastic coordinate descent approach~\cite{nesterov2012efficiency} and draws a random dimension of which to consider next.
In DSCD, only a single (global) step is taken each time a dimension is sampled,
and the step is stochastic,
where the new position (for the sampled dimension) is a sample from a proposal distribution.
The sample is accepted as the new position only if the objective improves upon the best lost within the last K steps
\footnote{In practice we used $K = 1000$.
Only considering the best loss within a (relatively large) window, rather than the historical best, we noted was helpful to be robust to outlier losses as a result of the mini-batching.
}.
The global step is global in the sense that there does not need to be a monotonically improving trajectory between any two positions (in terms of the loss surface), thus allowing `jumps' between valleys
\footnote{Strictly this does not need to be true for \emph{stochastic gradient descent} either, but in SGD it is still statistically unlikely to take steps in non-monotonically improving directions.}
.
The outline of DSCD is shown in Algorithm~\ref{algo:doubly_coord}.
We propose an annealing scheme for the proposal distribution. The proposal distribution is parameterised as a Beta distribution over a bounded space. At the beginning of the optimisation, the proposal distribution is uniform, and it slowly moves towards a Dirac delta centred at the current position, thus becoming increasingly local as the search progresses. The details of the annealing scheme for the proposal can be found in the appendix.
We alternate between taking local and global steps when the following is both true; $T$ consecutive steps of the same type has been taken, and the loss did not improve from the last step to the next.
In all experiments, we set $T = 50$, and noticed little to no importance of tuning this parameter.
In the appendix we assess the benefit of DSCD on multimodal functions.
\section{Experiments}
\begin{figure}
\caption{
Using the new optimisation scheme the architecture does no longer become invalid by searching for longer.
Shown is the mean cell weights (following the sigmoid activation function) for the normal and reduction cell, respectively, on three runs on different budgets.
}
\label{fig:fairdarts_using_our_search}
\end{figure}
We previously showed that all the edge weights of normal cells $\bm{\alpha}_{\text{normal}}$ tend towards zero in FairDARTS, resulting in invalid architectures.
We will now demonstrate that our optimisation scheme explores the architecture space better.
As a result, it avoids invalid architectures, discovers architectures with better test performance, and converges to good solutions without early stopping.
In the experiments, the same setup as in~\cite{chu2020fair} is used, except for ``FairDARTS + DSCD" for which we replace the local optimiser (Adam~\cite{KingmaB14}) with the proposed optimisation scheme.
In \Cref{fig:fairdarts_using_our_search} we see that the edge weights of the normal cell no longer become zero, even if searching for much longer,
and the resulting architecture can be successfully discretised.
The mean weights, after discretisation, slowly move towards the mean weights before discretisation.
Importantly, the edges that will be kept (above the $0.85$ threshold) remained the same from $1500$ epochs,
which is indicative of convergence.
\begin{table}[t]
\centering
\caption{
Comparison with FairDARTS for search and evaluation phases (accuracy in \%).
Split for $\mathcal{L}_{\text{train}}$ and $\mathcal{L}_{\text{val}}$ indicates accuracy measured on the training data for $\mathcal{L}_{\text{train}}$ and $\mathcal{L}_{\text{val}}$ respectively. Search Test indicates the accuracy on the hold-out set using the search network (undiscretised). Eval. Test indicates the test accuracy with the final architecture.
``Invalid arch." denotes no valid final architecture after discretisation.
}
\label{table:results_acc}
\resizebox{0.85\linewidth}{!}{
\begin{tabular}{ c | c c c | c }
\toprule
\multirow{2}{*}{Method} & \multicolumn{3}{|c|}{Search Phase} & Final Arch. \\
&Split for $\mathcal{L}_{\text{train}}$ & Split for $\mathcal{L}_{\text{val}}$ & Search Test & Eval. Test\\ \midrule
FairDARTS (50) & 82.02 & 75.61 & 76.15 & 97.36 \\
FairDARTS (75) & 87.35 & 78.10 & 78.65 & 97.29 \\
FairDARTS (250) & 96.95 & 81.55 & 81.52 & Invalid arch. \\
FairDARTS (500) & 99.92 & 83.49 & 83.26 & Invalid arch. \\
FairDARTS + DSCD (1500) & 100.0 & 83.12 & 83.40 & 97.50 \\
FairDARTS + DSCD (2000) & 100.0 & 84.02 & 84.71 & 97.25 \\
FairDARTS + DSCD (3000) & 100.0 & 85.51 & 85.10 & 96.92 \\
\bottomrule
\end{tabular}
}
\end{table}
In \Cref{table:results_acc} we see the accuracy of the final architectures and the searches, corresponding to \Cref{fig:fairdarts_sinking_ship_new,fig:fairdarts_using_our_search}.
Using our optimisation scheme (DSCD), the models produced become increasingly more accurate with more search,
while remaining valid.
Our method using 1500 epochs for search produces a higher test accuracy during the search phase than FairDARTS, which also results in a high test accuracy with the final architecture.
Despite the test accuracy of our method increasing with more search epochs, the test accuracy of the resulting final architectures decreases.
We argue that this is due to the fact that
the network used during the search phase is different from the network for evaluation (a network trained from scratch using the final architecture)~\cite{chu2020fair}.
Differences between the search architecture and final architecture include discretisation,
that the final architecture is larger and has auxiliary heads~\cite{chu2020fair},
as well as that the training paths are different (weights and architecture together versus weights only).
A comparison to other DARTS variants is included in the appendix.
\section{Conclusion}
Neural architecture search requires three things:
a space of models with good inductive biases,
a loss function to assess models,
and an optimisation or inference algorithm to explore the space.
In this work we focused on the optimisation algorithm, and we showed that by combining gradient-based,
local search with global optimisation techniques,
we are able to better explore the space.
\appendix
\begin{center}
{\Large Appendix}
\end{center}
\section{Comparison with other DARTS methods}
We also compared our approach with other state-of-the-art NAS methods in the DARTS family. The results are shown in Table~\ref{table:results_comp}.
\begin{table}[h]
\centering
\caption{
Comparison of state-of-the-art NAS models on CIFAR-10.
FairDARTS$\ast$ differs from FairDARTS in that the former uses additional post-processing of the edge weights after search,
with a hard limit on the number of edges kept per node pair.
}
\label{table:results_comp}
\begin{tabular}{ c c c c }
Method & Params (M) & FLOPS (M) & Accuracy (\%) \\ \hline
DARTS~\cite{liu2018darts} & 3.3 & 528 & 97.00 \\
DARTS-~\cite{chu2021darts} & 3.5 & 583 & 97.41 \\
FairDARTS$\ast$~\cite{chu2020fair} & 2.8 & 373 & 97.46 \\
FairDARTS & 6.4 & 966 & 97.36 \\
FairDARTS + DSCD & 3.6 & 532 & 97.50
\end{tabular}
\end{table}
\section{Assessment of DSCD on multimodal functions}
\label{sec:known_funcs}
\begin{figure}
\caption{
Shown is the median loss of 20 runs from uniformly sampled initial positions.
Shaded areas display the 95\% CI of the median.
The numbers following ``Adam" for each entry in the legend denote the used learning rate,
where ``schedule" denotes a linear learning rate scheduling between $0.001$ and $0.1$.
The postfix ``+ DSCD" denotes complementing the method with DSCD (Section~\ref{sec:globalopt}
\label{fig:synth_1}
\end{figure}
To confirm and quantify the beneficial effect of complementing gradient-based, local optimisation (Adam) with the proposed doubly stochastic coordinate descent (DSCD) routine, we performed comparisons with and without the routine on synthetic functions with known properties.
For reference, we compare to performing uniform sampling over the domain, as well as Covariance Matrix Adaptation Evolution Strategy (CMA-ES)~\cite{hansen2004evaluating},
a popular global optimisation method.
In Figure~\ref{fig:synth_1} we show the results on the Styblinski-Tang function~\cite{styblinski1990experiments} and the Schwefel function~\cite{jamil2013literature},
which are popular functions for benchmarking optimization methods.
Both functions have several local minima that are worse than the global minimum.
We note that for every setting of Adam with a particular learning rate,
or using a learning rate schedule,
complementing the local steps with DSCD global steps (Section~\ref{sec:globalopt}) improves the performance.
On the Styblinski-Tang function, the difference is dramatic, as all the Adam variants without DSCD become stuck in a bad local minimum at every run.
\section{Beta annealing}
\label{sec:beta}
For setting the proposal distribution, we propose an annealing scheme, which we will refer to as \emph{Beta annealing}.
The idea is that, at each step, we will sample a new (scaled) position following a Beta distribution, parameterised to have a varied concentration around the current position.
In practice, for our specific problem of setting operation edge weights going through a sigmoid, we set the proposal domain as $[-3, 3]$ for every dimension,
which accounts for the region of the domain with a significant effect on the output.
Note, however, that positions outside the domain are still possible to reach as of the local optimiser, although position outside will not be proposed in this step.
The current position we (min-max) normalize using the domain, so that it corresponds to a unit position $\upsilon_{i} \in [0, 1]$.
The new proposal unit position, which we address below, is then mapped back to the original domain before the loss evaluation.
We define a concentration parameter $\phi \in [0, 1)$,
where $\phi = 0$ correspond to an uniform distribution of the (unit) domain,
and $\phi \rightarrow 1$ tends towards a Dirac delta located at the current position.
The former represents full global exploration (of the sampled dimension), independent of the current position.
The latter represents full local exploitation at the current position.
These two extremes are represented as parameterisations of a Beta distribution,
and all the intermediate settings are as well.
During search we start with $\phi = 0$ and anneal towards $\phi = 1$ at the final epoch.
The annealing schedule used for $\phi$ is cosine annealing, typically used for learning rate scheduling~\cite{loshchilov2016sgdr}.
The proposal (unit) position is sampled as $\upsilon_{i + 1} \sim Beta(\alpha_i, \beta_i)$,
where the $\alpha_i, \beta_i$ parameters depend on $\phi$ and the current (unit) position $\upsilon_{i}$.
Specifically, $\alpha_i, \beta_i$ is derived at each step as following.
The two extremes, the uniform ($\phi = 0$) and Dirac delta ($\phi = 1$),
have known $\alpha$ and $\beta$ parameters, as we can solve for them given their respective (known) mean and standard deviation values,
\begin{multicols}{2}
\noindent
\begin{equation}
\mu_{\text{unit uniform}} = 0.5, \sigma_{\text{unit uniform}} = \nicefrac{1}{\sqrt{12}}.
\end{equation}
\begin{equation}
\mu_{\text{Dirac delta}} = \upsilon_{i}, \sigma_{\text{Dirac delta}} = 0.
\end{equation}
\end{multicols}
We linearly interpolate the mean $\mu$ and the standard deviation $\sigma$ parameters to obtain the intermediate Beta distribution parameterisations in between the two extremes,
\begin{multicols}{2}
\noindent
\begin{equation}
\mu := \phi \upsilon_{i} + (1 - \phi) \mu_{\text{uniform}}
\end{equation}
\begin{equation}
\sigma := (1 - \phi) \sigma_{\text{uniform}}.
\end{equation}
\end{multicols}
Note that the standard deviation $\sigma$ will approach (but never reach) zero as of $\phi < 1$.
We then solve for $\alpha$ and $\beta$ using the analytical mean and standard deviation of Beta distributions, resulting in
\begin{multicols}{2}
\noindent
\begin{equation}
\alpha = c_1 \beta
\end{equation}
\begin{equation}
\beta = \frac{c_1 - c_2}{c_2 (c_1 + 1)},
\end{equation}
\end{multicols}
where $c_1 = \frac{\mu}{1 - \mu}$ and $c_2 = \sigma^2 (c_1 + 1)^2$.
In the supplement we include an animation showing intermediate Beta distributions for various $\phi$ around a fixed point ($\upsilon_{i} = 0.75$).
\section{Background}
\label{sec:problem_detail}
In~\cite{zela2019understanding} it was shown that detrimental solutions, in particular solutions exhibiting an overly large number of skip connections, coincide with high validation loss curvatures.
In their work, they view these as problematic solutions within the solution set of the model.
They propose regularisation on the weight space and early stopping, which they show is helpful in avoiding reaching these solutions.
\cite{chu2020fair} instead proposes a change to the model, where different operation edges between the same nodes are not mutually exclusive,
and they also propose a regularisation term pushing edge weights towards either zero or one.
These alterations they show are beneficial for avoiding an over-reliance on skip connections, as well as reducing the approximation error resulting from the discretisation of the edge weights happening between the search and evaluation phase.
In addition, they made the solution set more expressive as of allowing multiple simultaneous operations between the same nodes of a cell.
In our work, we show that~\cite{chu2020fair} still suffers from another detrimental effect,
similar to the one it was addressing,
indicating that the issue has not yet been solved in full.
Similar to DARTS~\cite{liu2018darts} and RobustDARTS~\cite{zela2019understanding}, FairDARTS~\cite{chu2020fair} constructs the architecture from copies of a \emph{normal cell} and a \emph{reduction cell}.
What we show is that, using FairDARTS,
the search is required to be stopped early to avoid reaching solutions that are detrimental to test performance during the evaluation phase or ultimately reaching invalid solutions post-discretisation.
Notably, the architecture - as described by its operation edge weights - changes very little from very early on in the search until it is stopped.
After the epoch it would have been stopped,
the operation edge weights belonging to nodes in the normal cell all tend to zero.
Following discretisation of the edge weights, the normal cell no longer propagates activations through, making the architecture invalid.
We suggest that the cause of this problem is that the detrimental solutions correspond to local minima in the edge weights space, given typical initial positions in the neural network parameters space.
In particular,
that as a consequence of the reduction cell operations relying on fewer parameters than normal cell operations,
such solutions take up a large volume of the neural parameter space.
To see this,
let us consider a detrimental solution $\{\bm{\alpha}, \bm{w}\}_{\text{detrimental}}$,
where all $\bm{\alpha}_{\text{normal}}$ elements are close to zero.
As will be confirmed in experiments,
the neural network is sufficiently flexible to produce low loss solutions despite these elements being \emph{close} to zero.
Note that as long as activations can propagate through the normal cell, the reduction cell, being sufficiently expressive, can still represent low loss mappings.
Furthermore,
for constellations where the operations in the normal cell have little to no effect on the loss,
this directly translates into invariance to all of the associated $\bm{w}_{\text{normal}}$ neural network parameters.
In other words, such solutions are "large" in the sense that functionally equivalent solutions exist at all positions in the $\bm{w}_{\text{normal}}$ subspace.
We may think of this as an equivalent solution set.
Secondly, consider a random initial set of neural network parameter values, $\bm{w}_{\text{initial}}$.
The "larger" an equivalent solution set is, the more likely it is that $\bm{w}_{\text{initial}}$ will end up inside or "close" to it.
In general, as well known and studied in the optimisation literature,
gradient-based local optimisation is subject to finding local which are not necessarily global minima.
In many applications,
such as optimisation of neural network parameters alone,
a local minimum might be "good enough".
However, in this application, if it is applied to $\bm{\alpha}_{\text{normal}}$, it may add a bias towards local solutions, being compatible edge weights with the initial values of the neural network parameters.
\section{Differentiable Neural Architecture Search}
\subsection{Architecture}
DARTS~\cite{liu2018darts},
as similar to prior works~\cite{zoph2018learning,real2019regularized,liu2018progressive},
searches for a \emph{cell} as the building block for the final architecture.
In the case of convolutional networks, the cell is stacked, and for recurrent networks, it is recursively connected.
The cell constitutes a directed acyclic graph of $N$ nodes.
Each node $x$ represents a latent representation
and each directed edge $(i, j)$ represents an operation $o_{i, j}$.
A node depends on all of its predecessors as
\begin{equation}
x_j = \sum_{i < j} o_{i, j}(x_i).
\end{equation}
The cell is assumed to have two input nodes and a single output node.
In the case of convolutional networks, the input nodes are the outputs of the previous two layers,
and for recurrent cells, the input nodes represent the current step, and the state carried from the previous step.
The cell output is obtained by a reduction operation (e.g. concatenation) to all the intermediate nodes.
Let $\mathcal{O}$ be the set of candidate operations (e.g., convolution, max pooling, skip connection) available for each edge $(i, j)$.
\cite{liu2018darts} proposed a relaxation over the discrete operation choice using softmax
\begin{equation}
\bar{o}_{i, j}(x) =
\sum_{o \in \mathcal{O}}
\frac{
\text{exp}(\alpha_{o_i,j})
}{
\sum_{o' \in \mathcal{O}} \text{exp}(\alpha_{o'_i,j})
}
o(x),
\label{eq:softmax}
\end{equation}
where the operation weights for a pair of nodes $(i, j)$ are parameterised by a vector $\bm{\alpha}_{i, j}$ of
dimension $|\mathcal{O}|$.
Importantly, this makes the search space continuous and allows gradient-based optimisation methods.
FairDARTS~\cite{chu2020fair}, building upon~\cite{liu2018darts}, proposed replacing~Eq.~\ref{eq:softmax} with
\begin{equation}
\bar{o}_{i, j}(x) =
\sum_{o \in \mathcal{O}}
\sigma(\alpha_{o_i,j})
o(x)
\label{eq:sigmoid}
\end{equation}
where $\sigma$ is the sigmoid function.
This allows multiple operations per edge to be chosen simultaneously.
If no operations are active for a given edge, this constitutes the zero operation~\cite{zela2019understanding}.
For the case of convolutional neural networks,
on which we will focus in this paper,
both DARTS and FairDARTS searches for a normal
cell and a reduction cell to build up the final architecture.
The reduction cell,
in contrast to the 'normal' cell,
reduces the number of activation maps (or channels) out from the cell.
\subsection{Search}
\label{sec:search}
Let $\bm{\alpha}$ be the concatenated vector of all operation edge weights representing the architecture,
and $\bm{w}$ be the concatenated neural network parameters associated with all operations.
The $\bm{\alpha}$ vector contains the operation edge weights associated with both the normal cell and the reduction cell that are being searched for, i.e. $\bm{\alpha} = \{\bm{\alpha}_{\text{normal}}, \bm{\alpha}_{\text{reduction}}\}$, and the same applies to the weight parameters, $\bm{w} = \{\bm{w}_{\text{normal}}, \bm{w}_{\text{reduction}}\}$.
The architecture search problem was in~\cite{liu2018darts} stated as the bi-level optimisation problem
\begin{mini!}|l|[2]
{\bm{\alpha}}
{
\mathcal{L}_{\text{val}}(\bm{\alpha}, \bm{w}^{\ast})
}
{}{
\label{eq:bilevel_1}
}
\addConstraint{
\bm{w}^{\ast} = \argmin_{\bm{w}} \mathcal{L}_{\text{train}}(\bm{\alpha}, \bm{w})
},
\label{eq:bilevel_2}
\end{mini!}
where $\mathcal{L}_{\text{val}}$ and $\mathcal{L}_{\text{train}}$ are the validation loss and training loss, respectively.
The proposed optimisation procedure in~\cite{liu2018darts} is to approximate the gradient as
\begin{equation}
\nabla_{\bm{\alpha}} \mathcal{L}_{\text{val}}(\bm{\alpha}, \bm{w}^{\ast})
\approx
\nabla_{\bm{\alpha}} \mathcal{L}_{\text{val}}(
\bm{\alpha},
\bm{w} -
\xi \nabla_{\bm{w}} \mathcal{L}_{\text{train}}(\bm{\alpha}, \bm{w})
)
\end{equation}
and perform gradient-based local optimisation,
alternating between taking a step in the optimisation problem of $\argmin_{\bm{\alpha}} \mathcal{L}_{\text{val}}$ and of $\argmin_{\bm{w}} \mathcal{L}_{\text{train}}$.
$\bm{w}$ are the current weights and $\xi$ is the learning rate for a step in the inner optimisation problem (Eq.~\ref{eq:bilevel_2}).
This can be described as, at iteration $t$, take steps using the gradients defined at
\begin{equation}
\nabla_{\bm{\alpha}} \mathcal{L}_{\text{val}}(\bm{\alpha}_{t}, \bm{w}_{t}),
\end{equation}
followed by
\begin{equation}
\nabla_{\bm{w}} \mathcal{L}_{\text{train}}(\bm{\alpha}_{t+1}, \bm{w}_{t}),
\end{equation}
where $\bm{\alpha}_{t}$ and $\bm{w}_{t}$ is the position of respective parameter at the beginning of the iteration, and $\bm{\alpha}_{t+1}$ and $\bm{w}_{t+1}$ the updated positions, respectively.
\section{Global Optimisation Scheme}
\label{sec:optimization_scheme}
\begin{algorithm}[t]
\SetAlgoLined
\SetNoFillComment
\KwIn{Function $f$ defined over $\mathcal{X}$, initial $\bm{x}_{\text{best}}, y_{\text{best}}$, local\_step, global\_step}
\KwOut{$\bm{x}_{\text{best}}, y_{\text{best}}$}
reset schedule;\\
\While{budget remaining}{
take\_global\_step = schedule.current()\;
\uIf{take\_global\_step}{
$\bm{x}_{\text{best}}$, $y_{\text{best}}$ = global\_step($\bm{x}_{\text{best}}$, $y_{\text{best}}$)
\;
$\bm{x}_{\text{current}}, y_{\text{current}} = \bm{x}_{\text{best}}, y_{\text{best}}$\;
}
\Else{
$\bm{x}_{\text{current}}$, $y_{\text{current}}$ = local\_step($\bm{x}_{\text{current}}$, $y_{\text{current}}$)\;
\If{$y_{\text{current}} < y_{\text{best}}$}{
$\bm{x}_{\text{best}}, y_{\text{best}} = \bm{x}_{\text{current}}, y_{\text{current}}$\;
}
}
schedule.step($y_{\text{best}}$)\;
}
return $\bm{x}_{\text{best}}, y_{\text{best}}$\;
\caption{Local optimisation with global optimisation backtracking}
\label{algo:hybrid}
\end{algorithm}
In Section~\ref{sec:problem_detail} we hypothesised that the usage of local search for the $\bm{\alpha}$ weights adds bias towards solutions
compatible with $\bm{w}$ solutions that are closer to the initial position.
We will now describe a simple hybrid scheme, which makes the search for the $\bm{\alpha}$ weights "more global" in that it is less subject to the local curvature of the loss surface.
We will later evaluate this scheme empirically, contrasting it to the previous, fully local search.
The $\bm{\alpha}$ parameter,
being the collection of operation edge weights representing the architecture,
is typically vastly different than the neural network parameters $\bm{w}$
in dimensionality.
$\bm{\alpha}$ is, for the search spaced addressed, $196$-dimensional, while $\bm{w}$ has millions of parameters.
In FairDARTS and other DARTS variants, both parameters are optimised using gradient-based local optimisation, with alternating steps as described in Section~\ref{sec:search}.
However, the moderate dimensionality of the $\bm{\alpha}$ parameter makes it practically feasible to apply global optimisation techniques to optimise it.
Specifically, we will make use of the idea of coordinate descent, where one coordinate is optimised at a time,
as well as annealed sampling.
In this section, we will describe a simple hybrid between local and global optimisation,
which we later show performs well empirically.
We will first outline the general algorithm of the hybrid approach in Algorithm~\ref{algo:doubly_coord},
in turn, parameterised by functions responsible for taking a "local" step and "global" step, respectively.
At this abstraction level, we only distinguish between a global and local step, by if after taking the step, the "current position" is the same as the "best observed" position so far, in terms of smallest loss.
For brevity,
we leave out that the $global\_step$ function considers all observations so far,
without loss of generality.
The remaining components are specified in Section~\ref{sec:globalopt} and Section~\ref{sec:beta}.
\end{document} | math | 35,837 |
\begin{document}
\linespread{1.15}\normalfont\selectfont
\parskip0pt
\author{Ian Morrison}
\address{Department of Mathematics\unskip, \ignorespacesFordham University\unskip, \ignorespacesBronx, NY 10458}
\email{[email protected]}
\author{David Swinarski}
\address{Department of Mathematics\unskip, \ignorespacesFordham University\unskip, \ignorespacesBronx, NY 10458}
\email{[email protected]}
\title{Can you play a fair game of craps with a loaded pair of dice?}
\subjclass[2010]{Primary 60C05, 14Q15}
\keywords{distribution, fair, coins, dice}
\begin{abstract}
We study, in various special cases, total distributions on the product of a finite collection of finite probability spaces and, in particular, the question of when the probability distribution of each factor space is determined by the total distribution.\ifshrink\vskip-48pt\hbox{~}\fi
\end{abstract}
\maketitle
\thispagestyle{empty}
\par
\leavevmode\hbox{}
\ifshrink~$~$\vskip-96pt\else\vskip-24pt\tableofcontents\parskip3pt\fi
\section{Introduction}\label{intro}\stepcounter{subsection}
Craps is played by rolling two standard cubical dice with sides numbered $1$ to~$6$.
\ifshrink
Play is determined by the totals that show whose probabilities we call the \emph{total distribution} of the dice. For details of the rules and the pretty probability arguments computing the odds of winning, see \calcpagecite{1.craps}.
\else
For readers who may not have seen them, we review in an Appendix (\sect{craps}) the rules of the game and the pretty probability arguments needed to compute the odds of winning. Here we simply note that the basic play (ignoring common side bets) depends only on the total of the numbers showing on the two dice. We call the probabilities of these totals the \emph{total distribution} of the two dice.
\fi
The dice are called \emph{fair} if each number is equally likely to be rolled and \emph{loaded} otherwise.
The question in the title asks whether there are any pairs of loaded dice for which this total distribution matches that of a fair pair. With such a loaded pair of dice, it would be possible to play a game of craps with the same probabilities of game events as if fair dice were being used. We give three proofs (in Proposition~\ref{crapsfairno},
\ifshrink
\calcpagecite{2.solutions}
\else
Table~\ref{twodicefiftyone}
\fi
, and Corollary~\ref{crapsfairnobis}) that no such loaded dice exist. The first two proofs involve solving first partially then fully an explicit, dependent set of $11$ quadratic and $2$ linear equations in the $12$ side probabilities using a lengthy computer calculation by the mathematical software package \texttt{Magma}. The third follows by hand in a few lines from a very different formulation of the problem.
We were led to this reformulation by studying a much larger class of questions of this type that are the real focus of the paper. We consider dice of any order $k\ge 2$ whose sides may have arbitrary probabilities, and call any finite set of such dice, of possibly different orders, a \emph{sack} $\mathbf{S}$. The probabilities of seeing each possible total when the dice in a sack are rolled we call its total distribution, and we ask what we call the \emph{total-to-parts question}: When are the probability distributions of all the dice in the sack determined by its total distribution? In some cases, which we call \emph{exotic}, the answer to this more general question turns out to be negative; for example, computational evidence suggests that there are exotic pairs of dice of any order greater than or equal to $12$ (cf.~Conjecture-Problem~\ref{exoticpairsexist}). In fact, we generalize still further by allowing arbitrary complex numbers as probabilities. This is explained in \sect{totalpartssetup}, where all the terms used in this introduction are defined. Our answers are given in \sects{multpoly}{exoticsacks}.
We have taken the unusual step of leaving some of the scaffolding of our work on these questions standing in this paper. Our motivation is to provide an example to novice mathematicians of the embryology of a research project. In one direction, we hope to illustrate both how generalizing a problem often, paradoxically, places it in a simpler context. In the opposite direction, the more general context may reveal interesting new special cases whose analysis offers new avenues of attack on the original problem.
Here, very briefly is how our work illustrates these ideas. After setting up the largest class of questions that we deal with in \subsec{defnot} and \subsec{totalparts}, we explain in \subsec{geometric} how we can fit \emph{all} the part and total distributions for a fixed combinatorial type or vector of orders $\mathbf{k}$ into a function $\mathbf{F}_{\mathbf{k}}: \mathbb{C}^T \to \mathbb{C}^T$, each of whose $T$ coordinate functions is polynomial in the $T$ input variables $\null$\footnote{We use bracketed text like this to provide translations into the language of algebraic geometry, in which both authors work, for readers familiar with it, but other readers can, with no loss, simply ignore these insertions.}[or an affine morphism. In other words, the generalizations fit into algebraic families.]
Each \emph{individual} generalization (for example, the title question itself) concerns a fiber of this function---the inverse image of a \emph{single} point in its range. The answer to such questions can often be deduced for most or even all points [or generically] from geometric invariants of $\mathbf{F}_{\mathbf{k}}$ that can be much simpler to calculate. Further, these geometric invariants can often be calculated by understanding any suitably ``nice'' fiber [or specializing]. We may be able to find ``nice'' fibers that are particularly easy to analyze, and then apply our analysis to a fiber of particular interest, like the craps fiber [or generalize]. This is how we answer a number of questions stated in \subsec{totalparts} in \sect{multpoly}.
Another motivation for recapitulating our progress on the total-to-parts question is to highlight the way apparently irrelevant areas of mathematics have of unlocking problems, and the corollary value of having the broadest possible exposure to all areas of mathematics. This is already clear from the preceding paragraphs. The title question concerns probability distributions, but by having some familiarity with algebraic geometry, we make it easier. The sequel will make our point even clearer. The insight of Mike Stillman, discussed below, provides a new way to think about the maps $\mathbf{F}_{\mathbf{k}}$ that greatly simplifies all questions in \sect{multpoly}. It also leads to constructions of exotic pairs of dice in \sect{exoticsacks} that rely on understanding factorizations of cyclotomic polynomials. We hope that an exposition organized to make these precepts stand out will justify the overhead incurred.
The work discussed here developed as follows. Our interest was provoked by a talk given at Fordham by Jordan Stoyanov in which he asked whether a pair of non-standard ``dice'' might have the same total distribution as a standard pair. The quotation marks here are because he defined dice differently than we do: Stoyanov allowed the number of spots on the sides of his ``dice'' to be non-standard, but required his ``dice'' to have a uniform probability distribution. With the requirement that the number of spots be positive integers, there is, up to permutation, a unique non-standard pair of six sided dice with sides marked $(1,2,2,3,3,4)$ and $(1,3,4,5,6,8)$ (cf.~\cite{Stoyanov}*{\subsecext{12.7}}) as the reader may find it an amusing exercise to verify.
We both felt immediately that the total-to-parts question considered here was a more natural one. Indeed, during Stoyanov's talk, we worked out the case of two coins (given in Example~\ref{twocoinsfair}, but an easy exercise that we urge the reader to attempt before going further) and wrote down the equations which encode solutions for a pair of standard dice (analyzed in Example~\ref{twosixsideddice}). We then passed through a series of less special cases uncovering many pretty arguments along the way, some of which we give here. These included cases where the total distribution was that of a pair of fair dice of small orders and of a sack of fair coins (reviewed in \sect{coins}). It was this last case that first suggested to us---tardily, the reader may justly remark---considering the general total-to-parts problem. Trying to generalize our arguments and failing, we retreated to the case of a coin and a die whose analysis, reviewed in \sect{coindie}, hinted at a connection to factorizations of polynomials.
At this point, we mentioned the problem and our results to Dave Bayer, who was on sabbatical at MSRI. Dave passed it on to Mike Stillman (also visiting MSRI) over a BBQ dinner, and Mike immediately saw that there was a completely different metaphor for our general problem. When Dave emailed me Mike's idea, the first word in my reply was ``Wow!'' because it was immediately clear that this new perspective gave easy answers to many of the questions we had not yet settled, and shorter proofs of many of our results. Readers are invited to follow our progress up to the end of \sect{coindie}, and then, we hope, will share the delight we felt on seeing this work radically simplified by Mike's insight when we reveal it in \sect{multpoly}. \Sects{multpoly}{exoticsacks} use this insight to produce many exotic sacks of two dice and poses some intriguing asymptotic conjectures about them that we leave as open problems for interested readers.
Computer calculations in played an essential role in our investigation. We used the commercial software packages \texttt{Magma} and \texttt{Maple}\texttrademark\ \cites{Magma, Maple} as well as the free, open-source packages \texttt{Bertini}, \texttt{Macaulay2}, \texttt{QEPCAD-B}, and \texttt{Sage} \cites{Bertini, Macaulay, QEPCADB, SAGE}. We have posted many of our calculations online~\cite{codesamples}, typically giving both the input commands and output of an interactive session. We have tried to make these sessions informal introductions to the packages by commenting each step and hope that readers will find models in our code for calculations they want to make.
\section {First examples}\label{crapsfairsection}\stepcounter{subsection}
During Stoyanov's lecture we already worked out the simplest example, replacing the pair of six-sided dice by two coins.
\begin{ex}\label{twocoinsfair}
In tossing two fair coins, the probabilities of seeing $0$, $1$ or $2$ heads---i.e., $(HH, HT \text{~or~} TH, TT)$---are $(\frac{1}{4},\frac{1}{2},\frac{1}{4})$.
Let $p$ and $q$ be the probabilities of heads on the two possibly loaded coins, so that the probabilities of tails are $(1-p)$ and $(1-q)$. Then to obtain the total distribution of two fair coins we must have:
\begin{displaymath}
\begin{array}{rcl}
pq & = & \frac{1}{4}\unskip, \ignorespaces
p(1-q)+(1-p)q & = & \frac{1}{2}\unskip, \ignorespaces
(1-p)(1-q) & = & \frac{1}{4}\unskip, \ignorespaces
\end{array}
\end{displaymath}
Substituting the first equation into the third gives $p+q=1$. Substituting $q=1-p$ into the first equation yields $p^2-p+\frac{1}{4}=(p-\frac{1}{2})^2= 0$ so $p = q = \frac{1}{2}$.
\end{ex}
The process of substitution used above is called \emph{elimination} and is, no doubt, familiar from solving systems of linear equations. During Stoyanov's lecture, we realized that the case of two dice could be answered by these methods and wrote down the equations below. We just as quickly realized that the complexity of the calculations called for the help of one of the many computer algebra systems that implement algorithms, analogous to, but more intricate than, that of Gauss-Jordan, for elimination in systems of \emph{polynomial} equations. A few cases simple enough that elimination can be carried out by hand are treated in \sect{coindie}.
\begin{ex} \label{twosixsideddice}
Let $p_i$ and $q_j$ be the probabilities that the first and second dice come up showing each number from $1$ to $6$. Now we have $36$ possible rolls $(i,j)$, each of which would have probability $\frac{1}{6}\cdot \frac{1}{6}= \frac{1}{36}$ if the dice were fair. Grouping these rolls first by the total $i+j$ and then by the number of rolls yielding each total gives the following system of equations.
\noindent
{\setlength{\tabcolsep}{0.1em}
\begin{tabular}{>{$}r<{$}>{$}l<{$}>{$}c<{$}>{$}r<{$}>{$}l<{$}}
p_1 q_1 &=& \frac{1}{36} &=&p_6 q_6\unskip, \ignorespaces
p_1 q_2 + p_2 q_1 &=& \frac{2}{36} &=& p_5 q_6 + p_6 q_5 \unskip, \ignorespaces
p_1 q_3 + p_2 q_2 + p_3 q_1 &=& \frac{3}{36} &=&p_4 q_6 + p_5 q_5 + p_6 q_4 \unskip, \ignorespaces
\ifextracted\else\refstepcounter{equation}\label{twodiceequations}\text{\lower1pt\hbox{\textbf{(\thesubsection.\arabic{equation})}\kern38pt}}\fi
p_1 q_4 + p_2 q_3 + p_3 q_2 + p_4 q_1 &=& \frac{4}{36} &= &p_3 q_6 + p_4 q_5 + p_5 q_4 + p_6 q_3 \unskip, \ignorespaces
p_1 q_5 + p_2 q_4 + p_3 q_3 + p_4 q_2 + p_5 q_1 &= &\frac{5}{36} &=&p_2 q_6 + p_3 q_5 + p_4 q_4 + p_5 q_3 + p_6 q_2 \unskip, \ignorespaces
p_1 q_6 + p_2 q_5 + p_3 q_4 + p_4 q_3 + p_5 q_2 + p_6 q_1 &=& \frac{6}{36} &&\unskip, \ignorespaces
\displaystyle{\sum_{i=1}^{6}} p_i &=& 1 &= & \displaystyle{\sum_{i=1}^{6}} q_i\,.
\end{tabular}
}
Moreover, despite the fact that we have $13$ equations in only $12$ unknowns, we expected to find only a \emph{finite} number of solutions, because these equations are dependent: the sum of all the products on the first six lines is the product of the sums on the last. As an easy exercise, we invite the reader to write down the analogous system of $7$ equations in $6$ unknowns for triangular dice.
\end{ex}
Algebraic geometry is the study of systems of polynomial equations. However, in addition to the $13$ equations above, we also want the $p_i$ and $q_j$ to be real and to be between $0$ and $1$. Thus, we have a system of polynomial equations and \emph{inequalities}. Such systems are the subject of \emph{semialgebraic geometry}, and our first approach was to apply its techniques.
The first package we used to investigate whether fair dice give the only solution was \texttt{QEPCAD-B}~\cite{QEPCADB}. The acronym stands for \textbf{Q}uantifier \textbf{E}limination by \textbf{P}artial \textbf{C}ylindrical \textbf{A}lgebraic \textbf{D}ecomposition, and the tool combines polynomial manipulations with boolean algebra to determine whether or not positive real valued solutions exist without actually finding them. The package confirmed two coins must be fair, but crashed when we tried the analogous calculations for dice with more sides~\calcpagecite{2.qepcadb.htm}.
After this brief foray into semialgebraic geometry, we returned to our roots as algebraic geometers and analyzed the system (\ref{twodiceequations}) by elimination. We turned to \texttt{Magma}~\cite{Magma}, a sophisticated commercial computer algebra system, which performs elimination via Gr\"obner bases in a way that more directly generalizes Gaussian elimination. We will not explain Gr\"obner bases and their applications to elimination here, but we warmly recommend \cite{CoxLittleOShea}*{Chapters 2 and 3} for an introduction. We first used \texttt{Magma} to eliminate all the variables except $q_6$~\calcpagecite{2.magma-a.htm}, very much as in the example of two coins above, providing a single polynomial for $q_6$ whose roots are the possible values of $q_6$ at different solution points. In particular, only finitely many values of $q_6$ occur in the solutions of the system. We then substituted each possible positive real value for $q_6$ into the original equations and eliminated all the variables except $q_5$ to find the possible pairs $(q_5,q_6)$ with both values real and positive. By repeating this process several times, we eventually found that the only solution of these thirteen equations with all coordinates real and positive is a pair of fair dice. This constituted our first proof of the following fact:
\begin{prop}
\label{crapsfairno} You can't play a fair game of craps with a loaded pair of dice.
\end{prop}
\ifshrink
We then used \texttt{Magma}~\calcpagecite{2.magma-b.htm} to describe the solution set [affine scheme] of (\ref{twodiceequations}) geometrically, first finding that it is finite and that if the solutions are counted with multiplicities, there are $252$ [the solutions form zero-dimensional scheme of degree 252]. We then divided the solutions into groups definable by equations with rational coefficients [found the irreducible components over $\mathbb{Q}$ of our scheme] and noticed that all of the complex points had coordinates in the field $\mathbb{Q}[\sqrt{-3}]$. This allowed us to compute all $51$ complex solutions of the system exactly.
They are shown at~\calcpagecite{2.solutions}, grouped into $25$ pairs where the dice differ and the fair solution where the two dice are identical. Of the nonfair solutions, only one is real, and it contains negative entries. This piqued our interest in what happens for dice with any number of sides, and suggested that the natural context for answering these questions needed to allow arbitrary complex ``probabilities''.
\else
Although we were pleased to have an answer to the title question of the paper, we wanted to understand the system (\ref{twodiceequations}) better. We used \texttt{Magma} again in~\calcpagecite{2.magma-b.htm} to compute the dimension and degree of the solution set [affine scheme] of this system of equations, and discovered that it is zero-dimensional of degree 252. This means that the system has 252 complex solutions (counted with multiplicity). We decomposed this scheme into its irreducible components over $\mathbb{Q}$---that is, irrational solutions presented in groups definable by equations with rational coefficients--- and noticed that all of the complex points had coordinates in the subfield $\mathbb{Q}[\sqrt{-3}] \subset \mathbb{C}$. This allowed us to compute all $51$ complex solutions of the system exactly.
The solutions are shown in Table~\ref{twodicefiftyone}, where they are grouped into $25$ pairs where the dice differ, and the fair solution where the two dice are identical. Of the nonfair solutions, only the pair in the second line is real, and it contains negative entries. Also, in the table, we write the solutions in terms of the number $\zeta = \zeta_6 = e^\frac{2\pi i}{6}= \frac{1}{2}(1+\sqrt{3}\,i)$ instead of $\sqrt{-3}$; the reason we prefer this notation will become clear in \sect{multpoly}.
\begin{table}[th!]
\begin{center}
\refstepcounter{equation}\label{twodicefiftyone} \textbf{Table~\ref{twodicefiftyone}} \quad Solutions in terms of $\zeta = \frac{1+\sqrt{3}\,i}{2} = e^\frac{2\pi i}{6}$ of the system ~\ref{twodiceequations}
{\renewcommand{1.4}{1.3}\setlength{\tabcolsep}{0.2em}\small \begin{tabular}{>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} c >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$}}
p_1& p_2& p_3& p_4& p_5& p_6& & q_1& q_2& q_3& q_4& q_5& q_6 \unskip, \ignorespaces
\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}\unskip, \ignorespaces
\frac{1}{2}&\frac{-1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{-1}{2}&\frac{1}{2}&&\frac{1}{18}&\frac{1}{6}&\frac{5}{18}&\frac{5}{18}&\frac{1}{6}&\frac{1}{18}\unskip, \ignorespaces
\frac{-1}{6}&\frac{-4\zeta+1}{6}&\frac{-4\zeta+7}{6}&\frac{4\zeta+3}{6}&\frac{4\zeta-3}{6}&\frac{-1}{6}&&\frac{-1}{6}&\frac{4\zeta-3}{6}&\frac{4\zeta+3}{6}&\frac{-4\zeta+7}{6}&\frac{-4\zeta+1}{6}&\frac{-1}{6}\unskip, \ignorespaces
\frac{-\zeta-1}{9}&\frac{-5\zeta+4}{9}&\frac{-\zeta+5}{9}&\frac{\zeta+4}{9}&\frac{5\zeta-1}{9}&\frac{\zeta-2}{9}&&\frac{\zeta-2}{12}&\frac{2\zeta-1}{4}&\frac{5\zeta+5}{12}&\frac{-5\zeta+10}{12}&\frac{-2\zeta+1}{4}&\frac{-\zeta-1}{12}\unskip, \ignorespaces
\frac{-\zeta}{3}&\frac{-\zeta+3}{3}&\frac{3\zeta-2}{3}&\frac{-3\zeta+1}{3}&\frac{\zeta+2}{3}&\frac{\zeta-1}{3}&&\frac{\zeta-1}{12}&\frac{4\zeta-1}{12}&\frac{3\zeta+4}{12}&\frac{-3\zeta+7}{12}&\frac{-4\zeta+3}{12}&\frac{-\zeta}{12}\unskip, \ignorespaces
\frac{-\zeta-1}{12}&\frac{-2\zeta+1}{4}&\frac{-5\zeta+10}{12}&\frac{5\zeta+5}{12}&\frac{2\zeta-1}{4}&\frac{\zeta-2}{12}&&\frac{\zeta-2}{9}&\frac{5\zeta-1}{9}&\frac{\zeta+4}{9}&\frac{-\zeta+5}{9}&\frac{-5\zeta+4}{9}&\frac{-\zeta-1}{9}\unskip, \ignorespaces
\frac{-\zeta}{6}&\frac{-\zeta+1}{3}&\frac{-\zeta+3}{6}&\frac{\zeta+2}{6}&\frac{\zeta}{3}&\frac{\zeta-1}{6}&&\frac{\zeta-1}{6}&\frac{\zeta}{3}&\frac{\zeta+2}{6}&\frac{-\zeta+3}{6}&\frac{-\zeta+1}{3}&\frac{-\zeta}{6}\unskip, \ignorespaces
\frac{-2\zeta+1}{6}&\frac{1}{2}&\frac{2\zeta-1}{6}&\frac{-2\zeta+1}{6}&\frac{1}{2}&\frac{2\zeta-1}{6}&&\frac{2\zeta-1}{18}&\frac{4\zeta+1}{18}&\frac{2\zeta+5}{18}&\frac{-2\zeta+7}{18}&\frac{-4\zeta+5}{18}&\frac{-2\zeta+1}{18}\unskip, \ignorespaces
\frac{-2\zeta+1}{9}&\frac{-\zeta+2}{9}&\frac{-2\zeta+4}{9}&\frac{2\zeta+2}{9}&\frac{\zeta+1}{9}&\frac{2\zeta-1}{9}&&\frac{2\zeta-1}{12}&\frac{\zeta}{4}&\frac{\zeta+4}{12}&\frac{-\zeta+5}{12}&\frac{-\zeta+1}{4}&\frac{-2\zeta+1}{12}\unskip, \ignorespaces
\frac{-\zeta+1}{3}&\frac{\zeta}{3}&\frac{-\zeta+1}{3}&\frac{\zeta}{3}&\frac{-\zeta+1}{3}&\frac{\zeta}{3}&&\frac{\zeta}{12}&\frac{2\zeta+1}{12}&\frac{\zeta+3}{12}&\frac{-\zeta+4}{12}&\frac{-2\zeta+3}{12}&\frac{-\zeta+1}{12}\unskip, \ignorespaces
\frac{-\zeta+2}{3}&\frac{\zeta-1}{1}&\frac{-5\zeta+4}{3}&\frac{5\zeta-1}{3}&\frac{-\zeta}{1}&\frac{\zeta+1}{3}&&\frac{\zeta+1}{36}&\frac{2\zeta+5}{36}&\frac{\zeta+10}{36}&\frac{-\zeta+11}{36}&\frac{-2\zeta+7}{36}&\frac{-\zeta+2}{36}\unskip, \ignorespaces
\frac{-2\zeta+1}{12}&\frac{-\zeta+1}{4}&\frac{-\zeta+5}{12}&\frac{\zeta+4}{12}&\frac{\zeta}{4}&\frac{2\zeta-1}{12}&&\frac{2\zeta-1}{9}&\frac{\zeta+1}{9}&\frac{2\zeta+2}{9}&\frac{-2\zeta+4}{9}&\frac{-\zeta+2}{9}&\frac{-2\zeta+1}{9}\unskip, \ignorespaces
\frac{-\zeta+1}{4}&\frac{1}{4}&\frac{\zeta}{4}&\frac{-\zeta+1}{4}&\frac{1}{4}&\frac{\zeta}{4}&&\frac{\zeta}{9}&\frac{\zeta+1}{9}&\frac{\zeta+2}{9}&\frac{-\zeta+3}{9}&\frac{-\zeta+2}{9}&\frac{-\zeta+1}{9}\unskip, \ignorespaces
\frac{-\zeta+1}{6}&\frac{-\zeta+1}{6}&\frac{1}{3}&\frac{1}{3}&\frac{\zeta}{6}&\frac{\zeta}{6}&&\frac{\zeta}{6}&\frac{\zeta}{6}&\frac{1}{3}&\frac{1}{3}&\frac{-\zeta+1}{6}&\frac{-\zeta+1}{6}\unskip, \ignorespaces
\frac{-\zeta+2}{6}&\frac{0}{1}&\frac{\zeta+1}{6}&\frac{-\zeta+2}{6}&\frac{0}{1}&\frac{\zeta+1}{6}&&\frac{\zeta+1}{18}&\frac{\zeta+1}{9}&\frac{\zeta+4}{18}&\frac{-\zeta+5}{18}&\frac{-\zeta+2}{9}&\frac{-\zeta+2}{18}\unskip, \ignorespaces
\frac{1}{3}&\frac{-\zeta}{3}&\frac{2\zeta}{3}&\frac{-2\zeta+2}{3}&\frac{\zeta-1}{3}&\frac{1}{3}&&\frac{1}{12}&\frac{\zeta+2}{12}&\frac{\zeta+2}{12}&\frac{-\zeta+3}{12}&\frac{-\zeta+3}{12}&\frac{1}{12}\unskip, \ignorespaces
\frac{\zeta+1}{3}&\frac{-\zeta}{1}&\frac{5\zeta-1}{3}&\frac{-5\zeta+4}{3}&\frac{\zeta-1}{1}&\frac{-\zeta+2}{3}&&\frac{-\zeta+2}{36}&\frac{-2\zeta+7}{36}&\frac{-\zeta+11}{36}&\frac{\zeta+10}{36}&\frac{2\zeta+5}{36}&\frac{\zeta+1}{36}\unskip, \ignorespaces
\frac{-\zeta}{12}&\frac{-4\zeta+3}{12}&\frac{-3\zeta+7}{12}&\frac{3\zeta+4}{12}&\frac{4\zeta-1}{12}&\frac{\zeta-1}{12}&&\frac{\zeta-1}{3}&\frac{\zeta+2}{3}&\frac{-3\zeta+1}{3}&\frac{3\zeta-2}{3}&\frac{-\zeta+3}{3}&\frac{-\zeta}{3}\unskip, \ignorespaces
\frac{-2\zeta+1}{18}&\frac{-4\zeta+5}{18}&\frac{-2\zeta+7}{18}&\frac{2\zeta+5}{18}&\frac{4\zeta+1}{18}&\frac{2\zeta-1}{18}&&\frac{2\zeta-1}{6}&\frac{1}{2}&\frac{-2\zeta+1}{6}&\frac{2\zeta-1}{6}&\frac{1}{2}&\frac{-2\zeta+1}{6}\unskip, \ignorespaces
\frac{-\zeta+1}{6}&\frac{1}{3}&\frac{\zeta}{6}&\frac{-\zeta+1}{6}&\frac{1}{3}&\frac{\zeta}{6}&&\frac{\zeta}{6}&\frac{1}{3}&\frac{-\zeta+1}{6}&\frac{\zeta}{6}&\frac{1}{3}&\frac{-\zeta+1}{6}\unskip, \ignorespaces
\frac{-\zeta+1}{9}&\frac{-\zeta+2}{9}&\frac{-\zeta+3}{9}&\frac{\zeta+2}{9}&\frac{\zeta+1}{9}&\frac{\zeta}{9}&&\frac{\zeta}{4}&\frac{1}{4}&\frac{-\zeta+1}{4}&\frac{\zeta}{4}&\frac{1}{4}&\frac{-\zeta+1}{4}\unskip, \ignorespaces
\frac{-\zeta+2}{9}&\frac{\zeta+1}{9}&\frac{-\zeta+2}{9}&\frac{\zeta+1}{9}&\frac{-\zeta+2}{9}&\frac{\zeta+1}{9}&&\frac{\zeta+1}{12}&\frac{1}{4}&\frac{-\zeta+2}{12}&\frac{\zeta+1}{12}&\frac{1}{4}&\frac{-\zeta+2}{12}\unskip, \ignorespaces
\frac{1}{3}&\frac{\zeta-1}{3}&\frac{-2\zeta+2}{3}&\frac{2\zeta}{3}&\frac{-\zeta}{3}&\frac{1}{3}&&\frac{1}{12}&\frac{-\zeta+3}{12}&\frac{-\zeta+3}{12}&\frac{\zeta+2}{12}&\frac{\zeta+2}{12}&\frac{1}{12}\unskip, \ignorespaces
\frac{-\zeta+1}{12}&\frac{-2\zeta+3}{12}&\frac{-\zeta+4}{12}&\frac{\zeta+3}{12}&\frac{2\zeta+1}{12}&\frac{\zeta}{12}&&\frac{\zeta}{3}&\frac{-\zeta+1}{3}&\frac{\zeta}{3}&\frac{-\zeta+1}{3}&\frac{\zeta}{3}&\frac{-\zeta+1}{3}\unskip, \ignorespaces
\frac{-\zeta+2}{12}&\frac{1}{4}&\frac{\zeta+1}{12}&\frac{-\zeta+2}{12}&\frac{1}{4}&\frac{\zeta+1}{12}&&\frac{\zeta+1}{9}&\frac{-\zeta+2}{9}&\frac{\zeta+1}{9}&\frac{-\zeta+2}{9}&\frac{\zeta+1}{9}&\frac{-\zeta+2}{9}\unskip, \ignorespaces
\frac{-\zeta+2}{18}&\frac{-\zeta+2}{9}&\frac{-\zeta+5}{18}&\frac{\zeta+4}{18}&\frac{\zeta+1}{9}&\frac{\zeta+1}{18}&&\frac{\zeta+1}{6}&\frac{0}{1}&\frac{-\zeta+2}{6}&\frac{\zeta+1}{6}&\frac{0}{1}&\frac{-\zeta+2}{6}
\end{tabular}
}
\end{center}
\end{table}
We set up checks of
Table~\ref{twodicefiftyone}
in several tools but only one calculation completed. The open source package \texttt{Bertini}~\cite{Bertini} [which uses homtotopy continuation algorithms] impressed us by finding numerical approximations~\calcpagecite{2.bertini.htm} to all $51$ solutions.
Table~\ref{twodicefiftyone} reproves Proposition \ref{crapsfairno} revealing a bit more geometry than the proof by elimination. It piqued our interest in what happens for dice with any number of sides, and suggested that the natural context for answering these questions needed to allow arbitrary complex ``probabilities''.
\fi
\section{The total-to-parts problem}\label{totalpartssetup}
After establishing some results for small examples computationally, we formulated a much more general problem in the hopes of finding a unifying theory. We begin in \subsec{defnot} by making some preliminary definitions and establishing notation that will be used throughout the sequel. This leads to \subsec{totalparts} where we state the general problem and some attractive special cases and review what we now know about their answers.
In \subsec{geometric}, we describe a more geometric way of thinking about these problems that suggests further variations on these questions and, for the reader with no exposure to algebraic geometry, give some elementary examples that we hope make clear the notions that we draw on.
\subsection{Definitions and notation}\label{defnot}
We will index the sides of dice by $\sq{k} := \{0, 1, \ldots, k-1\}$ and \emph{sacks} or collections of dice by $\ang{n} := \{1, 2, \ldots, n\}$. Numbering sides from $0$ rather than $1$ makes possible uniform indexing and simpler formulae for total distributions, for example, in (\ref{ft}) and (\ref{multpolyformula}).
By a \emph{strict die} $\mathbf{d} = (d_0, d_1, \ldots, d_{k-1})$ of \emph{order} $k$, we mean a probability space whose underlying set is $\sq{k}$ (indexed throughout by $i$) with probability distribution given by $\mathbb{P}(i) = d_i$, subject, of course, to the standard conditions $d_i \in \mathbb{R}$, $d_i \ge 0$ for $ i\in \sq{k}$, and $ \sum_{i\in \sq{k}}d_i =1$.
The \emph{fair} $k$-die has the uniform distribution with all $d_i = \frac{1}{k}$; any other $k$-die is \emph{unfair} or \emph{loaded}.
It will be convenient to consider \emph{$k$-pseudodice} $\mathbf{p} = (p_0, p_1, \ldots, p_{k-1})$ for which the $p_i$ are pseudoprobabilities allowed to take arbitrary \emph{complex} values and required to satisfy only $\sum_{i\in \sq{k}}p_i =1$. See Remark~\ref{sumone}, for an explanation of why we retain only the ``sum to $1$'' condition for a probability distribution.
We will continue to denote by $\mathbf{d}$ a die known to be strict and by $\mathbf{p}$ a general pseudodie, and when no confusion is likely, we omit the qualifiers ``strict'' and ``pseudo''. A \emph{coin} is a die of order $2$ and a die is \emph{real} if all its probabilities are real numbers.
We define a \emph{sack} $\mathbf{S}:= (\mathbf{p}^1, \mathbf{p}^2, \dots, \mathbf{p}^n)$ to be a finite ordered set of dice (indexed throughout by $j \in \ang{n}$). We call the list $\mathbf{k}_S := (k_1, k_2, \ldots k_n)$ of orders of these dice the \emph{type} of $S$ and call the probability distributions of the individual dice in $\mathbf{S}$ (in order) its \emph{part} probabilities. To the sack $\mathbf{S}$, we associate the product sample space $K_\mathbf{S} := \prod_{j \in \ang{n}} \sq{k_j}$ (indexed throughout by $\mathbf{i}$) of order $k_\mathbf{S} := \prod_{j \in \ang{n}} k_j$. If $\mathbf{i} = (i_1, i_2, \ldots, i_n) \in K_\mathbf{S}$, we define the \emph{total} $t_{\mathbf{i}} := \sum_{j=1}^n i_j$. If we set $T_\mathbf{S} := \sum_{j \in \ang{n}} (k_j-1)$, this allows us to define a partition of $K_\mathbf{S}$ by the sets $K_t := \{\mathbf{i} \in I~|~ t_{\mathbf{i}} = t\}$ for $t =0, 1, \dots, T_\mathbf{S}$. When $\mathbf{S}$ is understood, we often omit it as a subscript and, likewise, we often omit the modifiers total or part when no confusion can result.
We will always assume that the dice in $\mathbf{S}$ are \emph{independent}. For strict sacks, this means that we have the usual product formula for the probability of the outcome $\mathbf{i}$ when the dice in the sack $\mathbf{S}$ are rolled,
$\mathbb{P}(\mathbf{i}) = \prod_{j \in \ang{n}} p_{i_j}^j$,
and we use this formula to define $\mathbb{P}(\mathbf{i})$ for any sack.
We can then define the $\thst{t}{th}$-total probability $f_t$ of $\mathbf{S}$ to be the (pseudo)probability of observing a total of $t$ when the dice in the sack $\mathbf{S}$ are rolled.
\begin{equation}\label{ft}
f_t = \sum_{\mathbf{i} \in K_t} \mathbb{P}(\mathbf{i}) = \sum_{\mathbf{i} \in K_t} \Bigl(\prod_{j \in \ang{n}} p_{i_j}^j\Bigr)\,.
\end{equation}
and define the \emph{total probability distribution} $\mathbf{f}_{\mathbf{S}} := (f_0, f_1, \cdots, f_T)$.
We call this total distribution \emph{fair} if it equals the total distribution on the sack of fair dice having the same orders as those in $\mathbf{S}$ and, if so, we call $\mathbf{S}$ (totally) \emph{fair}. Such a sack is \emph{exotic} if at least one of its dice is unfair. It is easy to check that if a strict sack of dice is totally fair, then so is the subsack obtained by removing any fair dice. Hence, any fair sack of dice contains a distinguished exotic subsack consisting of its unfair dice.
The \emph{reverse} of a die $\mathbf{d}$ is the die $\mathbf{d}'$ whose probability vector is that of $\mathbf{d}$ in reverse order. If we reverse all the dice in a sack, we get a reverse sack whose total probability vector is also reversed. A die is \emph{palindromic} if it equals its reverse, and a sack is palindromic if its total probability distribution is. A palindromic sack may contain non-palindromic dice: see Example~\ref{cointhree}.
\subsection{The total-to-parts question and special cases of it}\label{totalparts}
We can now state the most general problem we seek to investigate here, and strict and fair special cases, the last of which motivates this paper. As we will see in the sequel, the flavor of the fair version is arithmetic, of the strict version somewhat more probabilistic, and of the general version algebro-geometric.
\begin{ttsque}[General]\label{generalquestion}
When does the total probability distribution on a sack of dice determine the part probabilities of the dice, up to permutation?
\end{ttsque}
\begin{ex}\label{generalpairofcoins} To get a feel for this question, let's work out the generalization of a pair of coins from Example~\ref{twocoinsfair}. The general total distribution is given by numbers $r$, $s$ and $t$ summing to $1$, and the probabilities $p$ and $q$ of heads on the two coins must then satisfy
\begin{displaymath}
\begin{array}{rcl}
pq & = & r\unskip, \ignorespaces
p(1-q)+(1-p)q & = & s\unskip, \ignorespaces
(1-p)(1-q) & = & t\unskip, \ignorespaces
\end{array}
\end{displaymath}
From the first equation, we get $q=\frac{r}{p}$, and eliminating $q$ from second gives $p^2-(2r+s)p-r=0$. Using $r+s+t=1$, this has discriminant $D=s^2-4rt$ so $p = \frac{(2r+s)\pm \sqrt{D}}{2}$. These roots are swapped when the coins are, so the answer to Question~\ref{generalquestion} for two coins is yes.
\end{ex}
It is not hard to show that the answer to Question~\ref{generalquestion} is, in general, no (see Example~\ref{cointhree}). In the special case of sacks of any number of coins treated in \sect{coins}, the answer is positive. We will also see in Corollary~\ref{generalpartstototal} that this holds in general, \emph{only} for sacks of coins (and, of course, singleton sacks).
\begin{ttsque}[Strict]\label{strictquestion}
Does the probability distribution on a strict sack of dice determine the probabilities of the dice, up to permutation?
\end{ttsque}
In general, the answer is no---again, see Example~\ref{cointhree}. Indeed, the answer is positive only when the answer to \question{generalquestion} is. Much more subtle, as we shall see, is the following question which turns out to have an arithmetic flavor.
\begin{ttsque}[Fair]\label{fairquestion}
If the total probability distribution on a strict sack of dice is fair, must all the dice be fair? Contrapositively, do there exist exotic sacks of dice?
\end{ttsque}
Specializing to a sack of two $6$ sided dice, we get the question in the title of the paper. We confirm the negative answers of Example~\ref{twosixsideddice} and
\ifshrink
\calcpagecite{2.solutions}
\else
Table~\ref{twodicefiftyone}
\fi
by a more elegant method in Corollary~\ref{crapsfairnobis}. We also show that there are no exotic sacks consisting of a coin and a die (see \subsec{coindiefair} and Remark~\ref{coindiebis}).
Computer calculations reviewed in \sect{exoticsacks} show that totally fair sacks of two dice of certain small orders must be fair. In the other direction, for most pairs of orders, exotic sacks exist and the number of such sacks gets large as the orders do. Historically, the first we found was a pair of $13$-sided dice, but there are even simpler examples (see Table~\ref{decatab}). These examples make it clear how to produce exotic sacks with more dice and lead to some intriguing asymptotic conjectures.
\subsection{Geometric reformulations and variants}\label{geometric}
Fix a type $\mathbf{k}= (k_1, k_2, \ldots, k_n)$ of sack and set $U := \sum_{j \in \ang{n}} (k_j)$ and $T := \sum_{j \in \ang{n}} (k_j-1) = U-n$. Then we can take $p^j_i$ for $ j \in \ang{n}$ and $ i \in \sq{k_j}$ as coordinates on $\mathbb{C}^U$, and use the $n$ independent equations ${\sum_{i\in \sq{k_j}}p^j_i =1}$ to define a linear subspace $\mathbb{V} \cong \mathbb{C}^T$. Equivalently, we can use the $p^j_i$ for $ i \in \sq{k_j-1}$ as coordinates on $\mathbb{C}^T$ and use these equations to implicitly define~$p^j_{k_j}$.
Thus sacks $\mathbf{S}:= (\mathbf{p}^1, \mathbf{p}^2, \dots, \mathbf{p}^n)$ of type $\mathbf{k}$ can be identified with points of either $\mathbb{V}$ or of $\mathbb{C}^T$ [or, more algebraically, of the affine variety $\mathbb{A}^T$]. Equation~\eqref{ft} can then be viewed as defining a part-to-total mapping $\mathbf{F}_{\mathbf{k}}: \mathbb{V} \to \mathbb{C}^T$ or $\mathbf{F}_{\mathbf{k}}: \mathbb{C}^T \to \mathbb{C}^T$ by $\mathbf{S} = (\mathbf{p}^1, \mathbf{p}^2, \dots, \mathbf{p}^n)\,\to\, \mathbf{f}_\mathbf{S}$. Since the source and target are of the same dimension, we expect the map $\mathbf{F}_\mathbf{k}$ to be \emph{finite}.
Finiteness is a geometer's term that means that for \emph{general} points $\mathbf{f}$, the number of solutions of the equations $\mathbf{F}_{\mathbf{k}}(\mathbf{S}) = \mathbf{f}$---the order of the \emph{fiber} $\mathbf{f}^{-1}(\{\mathbf{f}\})$ of $\mathbf{F}_{\mathbf{k}}$ over $\mathbf{f}$---has a common finite value called the \emph{degree} of $\mathbf{F}_\mathbf{k}$. Here ``general'' is a somewhat vague but very convenient notion, indicating that there is a non-zero polynomial vanishing on the complementary set of ``special'' $\mathbf{f}$. The special fibers may be infinite, or empty, or contain a smaller finite number of points [at each of which, typically, several points of nearby general fibers come together or \emph{ramify} (cf. Example~\ref{symex} and Remark~\ref{diceramification})]. Milne's lecture notes~\cite{Milne} provide an accessible account of finiteness (Chapter~8) and of prerequisite notions (Chapters~1-7).
\begin{ex}\label{symex} A model example is the map $\mathbf{F}:\mathbb{C}^n \to \mathbb{C}^n$ sending $\mathbf{p} := (p_1, \ldots p_m)$ to $\mathbf{e}= (e_1, \ldots, e_n)$ where the $p_j$ are the roots \emph{in a fixed order} of a monic complex polynomial $P(x)$ of degree $n$ and the $e_j$ are the coefficients of $P(x)$ in degree order:
$$P(x) = \prod_{j=1}^n (x-p_j) = x^n + \sum_{i=1}^{n} e_ix^{n-i}\,.$$
Here $e_i$ is the \thst{i}{th}-elementary symmetric function of the $p_j$: that is, the sum of the $\binom{n}{i}$ squarefree monomials of degree $i$ in the $p_j$. The fundamental theorem of symmetric polynomials~\cite{Cox}*{Theorem~2.2.7} implies that the map $\mathbf{F}$ is surjective of degree $n!$ and that the fiber containing $\mathbf{p}$ is obtained by permuting the $p_j$. In particular, every fiber is finite (and non-empty). Thus, $\mathbf{e}$ general means that $P(x)$ has distinct roots or equivalently that the degree $n$ discriminant~~\cite{Lang}*{pp.161--162} is non-zero.
\end{ex}
\begin{ex}\label{finiteex} The map $f:\mathbb{C}^3 \to \mathbb{C}^3$ given by $f(x,y,z) = (a,b,c) := (yz,xz,xy)$ has degree $2$. When $xyz\not=0$, $y = \frac{c}{x}$ and $z= \frac{b}{x}$ so $a= yz = \frac{bc}{x^2}$ and hence $x = \pm \sqrt{\frac{bc}{a}}$. Fixing the sign then determines $y$ and $z$ by symmetry. However, the fiber over $(0,0,0)$ is the union of the $x$, $y$ and $z$ axes. If $a$ is non-zero, then the fiber over $(a,0,0)$ is the hyperbola $yz=a$ in the plane $x=0$ and, if $b$ is also non-zero, then the fiber $(a, b, 0)$ is empty, since $c=0$ forces $x$ or $y$ (and hence $b$ or $a$) to be $0$ too.
\end{ex}
A weaker version of~\question{generalquestion} asks whether finiteness always holds here.
\begin{ttsque}[Finite map]\label{finitequestion}
Are the part-to-total maps always finite? If so, what is the degree $\deg(\mathbf{F}_\mathbf{k})$ and what are the special fibers?
\end{ttsque}
A positive answer to this question allows, as in Example~\ref{finiteex}, for the collection of sacks $\mathbf{S}'$ having the same total distribution as $\mathbf{S}$ to be infinite, but asserts that for general $\mathbf{S}$ this collection is finite. A stronger version, though one that unlike~\question{generalquestion} allows for $\mathbf{S}'$ that are not reorderings of $\mathbf{S}$, asks whether we have a picture like that in Example~\ref{symex} with \emph{all} fibers finite and non-empty.
\begin{ttsque}[Non-empty finite fibers]\label{strongfinitequestion}
Do the equations $\mathbf{F}_\mathbf{k}(\mathbf{S}) = \mathbf{f}$ always have only a non-zero, finite number of solutions?
\end{ttsque}
\begin{rem}\label{pairofcoinsrem} Example~\ref{generalpairofcoins} shows that for a pair of coins, the answer to these questions is positive: the degree is $2$ with the total distribution determining the pair of dice up to order, and over the discriminant locus defined by the vanishing of $s^2-4rt$ we have a single pair of identical dice with head probabilities $p = \frac{2r+s}{2}$ (cf. Example~\ref{twocoinsfair}).
\end{rem}
In Theorem~\ref{strongfinitethm}, we show that the answer to \question{strongfinitequestion} (and, \textit{a fortiori}, to \question{finitequestion}) is positive in general and, in Corollary~\ref{degpartstototal}, give a multinomial formula for $\deg(\mathbf{F}_\mathbf{k})$.
\section{Sacks of coins}\label{coins}
\stepcounter{subsection}
This section should, historically, fall between \subsec{totalparts} and \subsec{geometric}: after making the extensions in the former, we studied the simplest examples, sacks of $n$ coins, which we were able to completely resolve by straightforward combinatorial arguments, and it was this solution that led to the geometric reformulation of the latter. We moved this argument back because it builds on Examples~\ref{generalpairofcoins} and~\ref{symex}.
For a sack of $n$ coins, the sample space $K_\mathbf{S}$ is simply the power set of $\ang{n}$, and the total $t$ of any roll (or toss) $\mathbf{i}$ is simply the number of $j$ for which the index $i_j$ equals $1$, so the set $K_t$ of rolls with total $t$ can be identified with subsets $J$ of $\ang{n}$ of order $t$. We can further simplify notation by writing $p_j$ for $p^j_0$ and replacing $p^j_1$ by $(1-p_j)$. Making these substitutions, equation~\eqref{ft} then becomes
\[
f_t = \sum_{\substack{J\subset \ang{n}\unskip, \ignorespaces |J|=t~}}~ \biggl[\Bigl(\prod_{j\in J} p_j\Bigr) \Bigl(\prod_{j\not\in J} (1- p_j)\Bigr)\biggr].
\]
\begin{prop} Let $\mathbf{e}_{\mathbf{S}} = (e_o,e_1, \ldots, e_n)$ where $e_i $ is the $\thst{i}{th}$ elementary function of the probabilities $p_j$ (cf. Example~\ref{symex}). Then:
\begin{enumerate}
\item For $ t \in \sq{t}$, $f_t = {\ds{\sum_{k=t}^n}} (-1)^{k-t}~{\binom{k}{t}}~ e_k$.
\item For $j \in \ang{n}$, $\operatorname{span} \{ f_n, f_{n-1}, \ldots, f_{j}\} = \operatorname{span} \{ e_n, e_{n-1}, \ldots, e_{j}\}$.
\end{enumerate}
\end{prop}
\begin{proof} The first claim follows directly from the definition of $f_t$ as follows. Fix any of the $\binom{n}{t}$ terms $P$ in the sum defining $f_t$. The second product in $P$ is a binomial of total degree $(n-t)$ containing $\binom{n-t}{k-t}$ ``interior'' terms of $p$-degree $(k-t)$ and sign $(-1)^{k-t}$. Multiplying these degree $(k-t)$ ``interior'' terms by the first product in $P$ and summing over $P$ gives a squarefree degree $k$ homogeneous symmetric polynomial in the $p_j$, which must therefore be of the form $c e_k$. Since $e_k$ contains $\binom{n}{k}$ terms, we find that
\[
c = (-1)^{k-t}\left(\frac{\binom{n}{t}\cdot \binom{n-t}{k-t}}{\binom{n}{k}}\right)\,.
\]
A straightforward calculation shows that this simplifies to $c=(-1)^{k-t}\binom{k}{t}$ but there is also a nice bijective argument for this. The denominator counts choices of a $k$ element subset and the numerator choices of such a subset in $2$ stages, by first choosing $t$ of the elements and then the other $(k-t)$. In this second count, each underlying $k$-set arises once for each of its $t$ element subsets.
The first claim shows that $\mathbf{f}_{\mathbf{S}} = U\mathbf{e}_{\mathbf{S}}$ where $U$ is an upper-triangular matrix that has ones on the diagonal. As $U$ is invertible, the second claim is immediate.
\end{proof}
The second claim of the proposition shows that $\mathbf{f}_{\mathbf{S}}$ determines $\mathbf{e}_{\mathbf{S}}$. But the elementary symmetric polynomials generate the ring of all symmetric polynomials (cf. Examples~\ref{symex}), so $\mathbf{e}_{\mathbf{S}}$ determines the $p_j$, up to permutation, and hence $\mathbf{S}$ up to permutation of the dice.
\begin{cor}\label{generalcoins} The total distribution of any sack of coins determines the part distributions of the coins, up to permutation. In other words, when all $k_j$ equal $2$, the answer to~\question{generalquestion} is positive.
\end{cor}
\section{A coin and a die}\label{coindie}
For a sack consisting of a coin and a die, we can eliminate the die probabilities to obtain an equation for the coin probability.
\subsection {The fair case}\label{coindiefair}
\begin{lem}\label{coindielem} If a coin and a die have fair total distribution, then both are fair.
\end{lem}
\begin{proof} Writing the pseudoprobabilites of the coin as $(p,1-p)$ and of the die as $(q_0, q_1, \ldots q_{k-1})$, we can express the fairness of the total distribution by the equations
\begin{center}
\begin{tabular}{>{$}l<{$}>{$}c<{$}>{$}l<{$}>{$}c<{$}>{$}c<{$}}
pq_0 & & &= &\frac{1}{2k}\unskip, \ignorespaces
pq_1 & + & (1-p)q_0&= &\frac{1}{k}\unskip, \ignorespaces
\kern6pt\vdots & & \kern18pt\vdots &&\vdots\unskip, \ignorespaces
pq_{k-1} & + & (1-p)q_{k-2}&= & \frac{1}{k}\unskip, \ignorespaces
& & (1-p)q_{k-1}&= & \frac{1}{2k}
\end{tabular}
\end{center}
Note that the first and last equations imply that $p$ and $1-p$ are both non-zero, which will allow us to use the equations in succession to solve for and eliminate the quantities $kq_j$. We start with $kq_0= \frac{1}{2p}$, then rewrite successive equations as $kq_j = \frac{1}{p} + \frac{p-1}{p}q_{j-1}$ to obtain, inductively,
\[
k q_j = \frac{1}{p}+ \frac{p-1}{p^2}+ \cdots+ \frac{(p-1)^{j-1}}{p^{j}} + \frac{1}{2}\frac{(p-1)^{j}}{p^{j+1}}\,,
\]
and finally use the last equation in the form $kq_{k-1}= -\frac{1}{2(p-1)}$ to see, after clearing denominators, that $p$ satisfies the equation
\[
\frac{1}{2}p^{k}+ (p-1)p^{k-1} + \cdots + (p-1)^{k-1}p + \frac{1}{2}(p-1)^{k}\,.
\]
If we split each interior term in two equal parts $\frac{1}{2} (p-1)^jp^{k-j}$ and group one set of these halves with the first term and the other with the last, we obtain two $k$-term geometric progressions with common ratio $r=\frac{p-1}{p}$ and respective initial terms $a'=\frac{1}{2}p^k$ and $a''=\frac{1}{2}(p-1)p^{k-1}$. This equals a single progression with ratio $r$ and initial term $a= a'+a''= (p-\frac{1}{2})p^{k-1}$. Such a progression can sum to $0$ only if either $a=0$ or $r=-1$ (and $k$ is even). In either case, we must have $p =\frac{1}{2}$, so the coin is fair, and then the total distribution equations inductively yield $q_j=\frac{1}{k}$ for all $j$, so the die is fair too.
\end{proof}
\subsection {The general case}\label{coindiegeneral}
A version of the same analysis can be carried out for a coin and a die with a general total distribution to obtain an equation for $p$ in terms of the vector $(f_0, f_1, \ldots, f_k)$ of total probabilities. It shows that
\[
p^k + \sum_{i=0}^{k-1} a_ip^i = 0 \text{\quad where \quad}
a_i = \sum_{j=0}^{i} (-1)^{k-i} \binom{k-j}{k-i} f_j\,.
\]
We will leave the details to the interested reader because it turns out that an unfair total distribution no longer determines the parts, even in the strict case. The simplest examples---there are many, as we will see in \sect{exoticsacks}---involve a coin and a three sided die.
\begin{ex}\label{cointhree} The total distribution $(\frac{1}{9}, \frac{7}{18}, \frac{7}{18}, \frac{1}{9})$ is common to the three sacks with part distributions $(\frac{1}{2}, \frac{1}{2})$ and $(\frac{2}{9}, \frac{5}{9}, \frac{2}{9})$, $(\frac{1}{3}, \frac{2}{3})$ and $(\frac{1}{3}, \frac{1}{2}, \frac{1}{6})$, and the reverse of the second.
\end{ex}
\section{The simplifying viewpoint}\label{multpoly}
\subsection{Stillman's observation and some immediate consequences} What Mike Stillman said about the part-to-total map $\mathbf{F}: \mathbb{C}^T \to \mathbb{C}^T$ for sacks of dice of a given type $\mathbf{k}$ that dropped our jaws was, ``That's polynomial multiplication''. He was seeing the problem in terms of generating functions, nicely introduced in~\cite{Concrete}*{Chapter~7}, which provide a powerful tool for assembling and relating combinatorial data. Here we associate to any pseudodie $\mathbf{p}$ of order $k$ the \emph{distribution polynomial} $\mathbf{p}(x) = \sum_{i=0}^{k-1} p_ix^i$. Conversely, every polynomial of degree $(k-1)$ or less (less, because we allow $p_i=0$) with coefficients summing to $1$ is associated to a unique die of order $k$. Likewise, the total distribution of any sack $\mathbf{S}$ yields a polynomial $\mathbf{f}_{\mathbf{S}}(x) = \sum_{i=0}^{T} f_ix^i$ of degree at most~$T$.
A moment's inspection of the total distribution equation~(\ref{ft}) shows that it can now be restated much more concisely as:
\begin{equation}\label{multpolyformula}
\mathbf{f}_{\mathbf{S}}(x) = \prod_{\mathbf{p}\in \mathbf{S}} \mathbf{p}(x)
\end{equation}
This was Stillman's insight. From it, we immediately obtain positive answers to \question{finitequestion} and \question{strongfinitequestion}.
\begin{thm}\label{strongfinitethm} For dice of a fixed type $\mathbf{k}$, all fibers of the part-to-total map $\mathbf{F}_{\mathbf{k}}$ are finite.
\end{thm}
\begin{proof} If $\mathbf{S}$ is any sack of type $\mathbf{k}$, then over the complex numbers, which are algebraically closed, $\mathbf{f}_\mathbf{S}(x)$ factors completely into (at most) $T$ monic linear factors times a non-zero scalar~\cite{Hungerford}*{Theorem~3.19}. What Stillman's observation means is that a sack $\mathbf{S}'$ has total distribution $\mathbf{f}_{\mathbf{S}}(x)$ if and only if the multiplicity of every root $z$ of $\mathbf{f}_{\mathbf{S}}(x)$---that is, the number of linear factors $(x-z)$---equals the sum over $\mathbf{p} \in \mathbf{S}'$ of the multiplicities of $z$ in $\mathbf{p}(x)$. The roots only determine each polynomial up to a non-zero homothety, but this is fixed in each case by the pseudoprobability condition that the coefficients sum to $1$. In other words, such $\mathbf{S}'$ correspond to partitions of the linear factors of $\mathbf{f}_{\mathbf{S}}(x)$ into subsets of sizes at most $k_j$. But there are only finitely many such partitions.
\end{proof}
With a bit more work, we can give a formula for the degree of $\mathbf{F}_{\mathbf{k}}$. For general sacks of type $\mathbf{k}$ all the pseudoprobabilities $p_i^j$ will be non-zero---that is, this condition fails only on the lower dimensional set where at least one of the equations $p_i^j=0$ holds, so $\mathbf{p}_j(x)$ will have degree exactly $k_j$. Likewise, for general sacks $\mathbf{S}$, all the complex roots of $\mathbf{f}_{\mathbf{S}}(x)$ will be distinct: the polynomial that vanishes if this is not true is the degree $T$ discriminant~\cite{Lang}*{pp.161--162}. So the $\mathbf{S}'$ in the same fiber as $\mathbf{S}$ will correspond bijectively to partitions of the set of $T$ roots into exactly $n$ parts of sizes $k_j-1$. Choosing these parts in succession introduces no ambiguity because, although we have not required that all the $k_j$ be distinct, we work with \emph{ordered sacks}. Thus
\begin{cor}\label{degpartstototal}
The degree of the map $\mathbf{F}_{\mathbf{k}}$ is $\displaystyle{\frac{T!}{\prod_{j=1}^n (k_j-1)!}}$.
\end{cor}
\begin{exs}For pairs of coins this degree is $\frac{2!}{(1!)^2} = 2$. The associated total polynomial, $tx^2+sx+r$, will be general and have two preimages when it has distinct roots---i.e. when the discriminant $s^2-4rt$ does not vanish (cf. Examples~\ref{generalpairofcoins} and Remark~\ref{pairofcoinsrem}). If $k_1=2$ and $k_2=3$ so $T=3$ and the three roots are distinct, then the degree is $\frac{3!}{2!1!} = 3$. Taking the roots to be $-1$, $-2$ and $-\frac{1}{2}$ gives Example~\ref{cointhree} in which the three cases correspond to which root is chosen for the coin. As the example makes clear, this kind of ambiguity is typical, with or without a restriction to the strict case. For pairs of $6$-sided dice, the degree is $\frac{10!}{(5!)^2} = 252$ confirming \texttt{Magma}'s calculation~\calcpagecite{2.magma-b.htm}.
\end{exs}
\begin{cor}\label{generalpartstototal}
The total distribution determines the parts up to permutation exactly for sacks of coins and singleton sacks.
\end{cor}
\begin{proof}
For a sack of $n$ coins, all $k_j=2$ and $T = n$ so the degree of $\mathbf{F}_{\mathbf{k}}$ is $n!$: this reproves Corollary~\ref{generalcoins}. In general, if we define $m_p$ to be the number of $j$ for which $k_j=p$ then $m_{\mathbf{k}}:= \prod_p (m_p!)$ gives the number of permutations (by exchanging parts of the same order) that preserve a set of distribution polynomials that is general in the sense discussed above. The number $m_{\mathbf{k}}$ always divides the degree of $\mathbf{F}_{\mathbf{k}}$ (because the quotient counts so-called distinguishable partitions) but the quotient is not $1$ unless all $k_j$ are equal to $2$ or there is only one die because, when we transpose a pair of roots from different factors, we only transpose the factors themselves when they are both of degree one.
\end{proof}
Over the locus of sacks where $\mathbf{f}_{\mathbf{S}}(x)$ has repeated roots---in particular, when different dice in the sack have roots in common---the fibers have smaller order. Counting these fibers [and saying what sheets come together in each] is a more intricate problem, but the arguments above make it clear that this problem is fundamentally combinatorial. We will not enter into it here, except for pairs of totally fair dice in the next subsection (cf. Lemma~\ref{faircount} and Remark~\ref{diceramification}).
Instead we now turn to the fair case, which motivated this paper and which is more subtle. A fair die $\mathbf{d}$ of order $k$ has polynomial $\frac{1}{k}\psi_k(x)$ where $\psi_k(x):= (x^{k-1} + x^{k-2} + \cdots + x + 1) = \frac{(x^k-1)}{(x-1)}$. The associated roots are the roots of unity $\zeta_{m,k} := e^{2\pi i\frac{m}{k}}$ with $-\frac{k}{2} < m \le \frac{k}{2}$ of orders dividing $k$ and not equal to $1$~\cite{Lang}*{p.116}. Of these, only $-1= \zeta_{m,2m}$ (for even $k$), corresponds to a real factor $(x+1)$. Note that, by Euler's Identity,
$\zeta_{m,k}= \cos\bigl(2\pi\frac{m}{k}\bigr) + i\sin\bigl(2\pi\frac{m}{k}\bigr)$, which yields:
\begin{lem}\label{fairfactors} The roots $\zeta_{m,k}$ and $\zeta_{-m,k}$ are complex conjugates with real part $\cos(2\pi\frac{m}{k})$ and norm $1$, and hence the monic irreducible factors over $\mathbb{R}$ of $\psi_k(x)$ are $(x+1)$ for all even $k$ and $\chi_{m,k}(x) = (x^2 - 2\cos(2\pi\frac{m}{k}) + 1)$ for $1 \le m < \frac{k}{2}$.
\end{lem}
\begin{rem}\label{coindiebis}
If a totally fair sack contains a coin and a die of order $k$, no transposition of roots is possible if $k$ is odd, and only the exchange of the factors $(x+1)$ is possible if $k$ is even. We immediately recover Lemma~\ref{coindielem}.
\end{rem}
\subsection{Pairs of totally fair dice} In this subsection, we analyze pairs $\mathbf{d}$ and $\mathbf{d}hat$ of totally fair dice of order $k$, with total polynomial $\Psi(x) =\frac{1}{k^2}\psi_k(x)^2 $ in more detail.
We first count such sacks, beginning with the observation that $\mathbf{d}(x)$ (and hence $\mathbf{d}hat(x)$) is determined by the vector $\mathbf{r} := (r_1, \ldots r_{k-1})$ giving the multiplicities of the roots $\zeta_{m,k}$ which must satisfy the multiplicity inequalities
$ 0 \le r_m \le 2, m \in \ang{k-1}$ and the degree equality $\sum_{m=1}^{k-1} r_m = k-1$. If exactly $\ell$ of the $r_m$ equal $2$, then by the equality exactly $k-1-2\ell$ of the remaining $(k-1-\ell)$ will equal $1$ and the other $\ell$ will equal $0$. The number of such $\mathbf{r}$ is simply the number of choices for the subsets of $\ang{k-1}$ with $r_m=2$ and $r_m=1$, respectively $\binom{k-1}{\ell}$ and $\binom{k-\ell-1}{k-1-2\ell}$.
\begin{lem}\label{faircount} The number of pairs of totally fair dice of order $k$ is $$\ds{\sum_{\ell=0}^{\lfloor\frac{k-1}{2}\rfloor}\binom{k-1}{\ell}\binom{k-\ell-1}{k-1-2\ell}}\,.$$
\end{lem}
For example, when $k=2$, there is a unique such sack, and when $k=6$, there are $\binom{5}{0}\binom{5}{5}+ \binom{5}{1}\binom{4}{3}+\binom{5}{2}\binom{3}{1}= 1+20+30=51$, confirming the count of solutions in
\ifshrink
\calcpagecite{2.solutions}.
\else
Table~\ref{twodicefiftyone}\footnote{In fact, the source for the typeset table was produced by using SAGE~\cite{SAGE} to list all pairs $(\mathbf{d}, \mathbf{d}hat)$ by the procedure leading to the count above, and passing this output to \texttt{Maple}\texttrademark~\cite{Maple} to have the fractions arranged in a more compact form and then add the wrappers needed to format the table.}.
\fi
\begin{rem} \label{diceramification}
A quick check on all the counts above can be obtained by working out how many of the points of a general fiber of the part-to-total map for a pair of dice of order $k$, for which the $2(k-1)$ roots of $\mathbf{f}_{\mathbf{S}}(x)$ are distinct, come together at each point of the totally fair fiber, where the roots are a multiset with $(k-1)$ distinct elements each occurring twice. To say that two $(k-1)$-element subsets of the latter multiset are equal as multisets (and hence give equal polynomials) means exactly that the corresponding vectors $\mathbf{r}$ of multiplicities are equal. When $r_m$ is $0$ or $2$, there is no ambiguity about which subset of the \thst{m}{th} pair is being chosen but when $r_m=1$ there are $2$ subsets. Thus, for a point indexed in the $\thst{\ell}{th}$ term of the sum in Lemma~\ref{faircount} where $(k-1-2\ell)$ of the $r_m$ equal $1$, there will be $2^{(k-1-2\ell)}$ points of the general fiber coming together. For example, when $k=6$ and the general fiber has $252$ elements, we obtain the check $2^5\cdot 1+2^3\cdot 20+2^1\cdot 30 = 252$.
\end{rem}
\begin{prop}\label{ddfactors} Suppose that $\mathbf{d}$ and $\mathbf{d}hat$ are a pair of strict dice of order $k$ whose total distribution is fair.
\begin{enumerate}
\item If $k$ is odd, $\mathbf{d}(x) = c \prod_m \bigl(\chi_{m,k}(x)\bigr)^{r_m}$ and $\mathbf{d}hat(x) = \widehat{c} \prod_m \bigl(\chi_{m,k}(x)\bigr)^{(2-{r_m})}$ where $0\le r_m \le 2$, $\sum_m r_m = \frac{k-1}{2}$ and $c$ and $\widehat{c}$ are non-zero scalars.
\item If $k$ is even, we get the same conclusion except that $\sum_m r_m = \frac{k-2}{2}$ and each of $\mathbf{d}(x)$ and $\mathbf{d}hat(x)$ must also contain one of the factors $(x+1)$ in $\psi_k$.
\end{enumerate}
\end{prop}
\begin{proof} For $k$ odd, this is immediate from the hypothesis $\mathbf{d}\cdot\mathbf{d}hat = (\psi_k)^2$, Lemma~\ref{fairfactors}, and the fact that $\mathbf{d}$ and $\mathbf{d}hat$ are both real. For $k$ even, the only additional observation is that parity forces each to have exactly one of the two factors $x+1$ in the right hand side.
\end{proof}
As a first application, we give a very short third proof of Proposition~\ref{crapsfairno}.
\begin{cor}\label{crapsfairnobis} You can't play a fair game of craps with a loaded pair of dice.
\end{cor}
\begin{proof}
Since $\psi_k = \chi_{1,6}(x)\chi_{2,6}(x) (x+1)$, the only sack, other than a pair of fair dice, allowed by Proposition~\ref{ddfactors} has $\mathbf{d}(x)$ and $\mathbf{d}hat(x)$ multiples of
\[
\bigl(\chi_{1,6}(x)\bigr)^2 (x+1)= (x^2-x+1)^2 (x+1) =x^5 - x^4 + x^3 + x^2 - x + 1
\]
and
\[\bigl(\chi_{2,6}(x)\bigr)^2 (x+1) = (x^2+x+1)^2 (x+1)= x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1\,.
\]
This sack is not strict: it yields the second row of
\ifshrink
\calcpagecite{2.solutions}.
\else
Table~\ref{twodicefiftyone}.
\fi
\end{proof}
\begin{rem} \label{sumone}
Theorem~\ref{strongfinitethm} does not hold without some non-degeneracy condition on our polynomials because if any $\mathbf{p}(x)=0$ so is $\mathbf{f}(x)$. The normalization that the coefficients sums to $1$ fits our probability context much more naturally than, say, requiring the $\mathbf{p}(x)$ to be monic.
\end{rem}
\section{A menagerie of exotic pairs}\label{exoticsacks}
The calculations reported through \sect{coindie} (and many we have not included) led us to suspect that the answer to~\question{fairquestion} was positive. Stillman's insight quickly led us to counterexamples.
\subsection{Existence of exotic pairs}\label{exoticpairs}
Let's start, as we did in the event, with pairs of dice of the same order. By now our strategy is clear: pick an order $k$, list all the sacks $\mathbf{d}(x)$ and $\mathbf{d}hat(x)$ permitted by Proposition~\ref{ddfactors}, and look for strict sacks. \textit{A priori}, we can think of no reason why such sacks should exist, but \textit{a posteriori}, we learn that they almost always do. Here is the first $13$-sided example we found.
\begin{ex}\label{tridecadahedralex} For simplicity, we suppress the index $13$ and the dependence on $x$. Consider the dice
$\mathbf{d} = c\,\,\chi_1\cdot\chi_2\cdot\chi_3\cdot\chi_4\cdot\chi_4\cdot\chi_6$
and
$\mathbf{d}hat = \widehat{c}\,\, \chi_1\cdot\chi_2\cdot\chi_3\cdot\chi_5\cdot\chi_5\cdot\chi_6$.
in which we have swapped the $\chi_4$ and $\chi_5$ factors of a fair pair of dice. By Lemma~\ref{fairfactors}, the total distribution of this sack is fair. Both dice are palindromic and the lower ``half'' of each probability vector is shown, approximately, in Table ~\ref{tridecatab}.
{\renewcommand{1.4}{1.1}\renewcommand{\tabcolsep}{3pt}
\begin{center} \refstepcounter{equation}\label{tridecatab}\vskip3pt\centerline{\textbf{Table~\ref{tridecatab}} Numerical probabilities for tridecahedral dice of Example~\ref{tridecadahedralex} }
\small\begin{tabular}{c>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}
& \text{\normalsize$d_0$} & \text{\normalsize$d_1$} &\text{\normalsize$d_2$} &\text{\normalsize$d_3$} &\text{\normalsize$d_4$} &\text{\normalsize$d_5$} &\text{\normalsize$d_6$} \unskip, \ignorespaces
{\normalsize $\mathbf{d}$}& 0.0992916&0.0210685&0.1381701&0.0410895&0.0693196&0.1241391&0.0138431\unskip, \ignorespaces
{\normalsize $\mathbf{d}hat$}& 0.0595938&0.1065425&0.0732460&0.0499115&0.0997570&0.0877406&0.0464172\unskip, \ignorespaces
\end{tabular}\vskip3pt
\end{center}
}
The coefficients in Table~\ref{tridecatab} are roundings to $7$ places of coefficients computed from $14$ place values for the cosines that occur in the $\chi_m$, so the positivity of these coefficients---that is, the strictness of the sack---is also unimpeachable.
\end{ex}
The \emph{smallest} exotic pair is not Example~\ref{tridecadahedralex} but is obtained from a fair pair of dice of order $10$ by swapping the $\chi_3$ and $\chi_4$ factors. As the upper half of Table~\ref{decatab} shows, this example is strict but seems a bit of a cheat because four of the $\mathbf{d}$-probabilities are zero. There are $3$ exotic pairs of order $12$. Two obtained by swapping $\chi_2$ and $\chi_3$ or $\chi_3$ and $\chi_4$ in a fair pair of dice have rational part probabilities, but again some are zero, as the reader may check. The smallest exotic pair with all probabilities positive, shown in the lower half of Table~\ref{decatab}, is obtained by swapping $\chi_4$ and $\chi_5$.
\begin{center}
\refstepcounter{equation}\label{decatab}\vskip3pt\centerline{\textbf{Table~\ref{decatab}} Probabilities of exotic pairs of dice with $10$ and $12$ sides}\nopagebreak
{\small
\begin{tabular}{>{$}c<{$}rr>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}
\text{Order} & Swap & Face & 1 & 2 & 3 & 4 & 5 & 6 \unskip, \ignorespaces
\multirow{2}{*}{$10$} & \multirow{2}{*}{$\chi_3,\chi_4$} & {\normalsize $\mathbf{d}_{10}$} & {\frac{5-\sqrt{5}}{20}} & 0 & {\frac{\sqrt{5}}{10}} & 0 & {\frac{5-\sqrt{5}}{20}}& \unskip, \ignorespaces
& & {\normalsize $\mathbf{d}hat_{10}$} & {\frac{5+\sqrt{5}}{100}} & {\frac{5+\sqrt{5}}{50}} & {\frac{1}{10}} & {\frac{5-\sqrt{5}}{50}}& {\frac{15-\sqrt{5}}{100}} & \unskip, \ignorespaces[8pt]
\multirow{2}{*}{$12$}&\multirow{2}{*}{$\chi_4,\chi_5$} & {\normalsize $\mathbf{d}_{12}$} & {\frac{2-\sqrt{3}}{4}} & {\frac{2\sqrt{3}-3}{4}} & {\frac{2\sqrt{3}-3}{4}} & \frac{2-\sqrt{3}}{4} & {\frac{2\sqrt{3}-3}{4}}& \frac{2-\sqrt{3}}{4} \unskip, \ignorespaces
&& {\normalsize $\mathbf{d}hat_{12}$} & {\frac{2+\sqrt{3}}{36}} & {\frac{1}{36}} & {\frac{4+\sqrt{3}}{36}} & {\frac{2-\sqrt{3}}{36}}& {\frac{5}{36}} & {\frac{4-\sqrt{3}}{36}}
\end{tabular}\vskip3pt
}
\end{center}
We leave the reader to find the smallest exotic sack, of type $(3,4)$. For many others, found using \texttt{Macaulay2}, \texttt{Magma}, and \texttt{Maple}\texttrademark~\cites{Macaulay, Magma, Maple} see \calctoccite{7.1}.
\subsection{Asymptotics of exotic pairs}\label{asymexoticpairs}
Similar calculations yield lots of exotic sacks of many combinatorial types. Here we will consider only sacks of $2$ dice, where we have evidence~\calctoccite{7.2} for the existence of such sacks for almost all pairs of orders, and further, for asymptotic predictions about the numbers of such sacks. We summarize our evidence in two Conjecture-Problems that we challenge interested readers to take up\footnote{If you do, please communicate your results so we can update the webpage~\calcpagecite{progress.htm} which we invite you to visit to learn about (and to avoid duplicating) work of others. We do not ourselves plan any further work, except as mentors to interested students.}. First, our calculations (and Lemma~\ref{coindielem} for the case $k=2$) confirm the list of exceptions below and suggest its completeness.
\begin{conjprob} \label{exoticpairsexist}
There are exotic sacks of $2$ dice of every pair of orders $(k,k')$ with $2 \le k \le k'$ except for $(2, k')$, $(3,3)$, $(3,6)$, $(3,9)$, $(4,4)$, $(4,8)$, $(5,5)$, $(6,6)$, $(7,7)$, $(8,8)$, $(9,9)$, and $(11,11)$.
\end{conjprob}
Inspection of our data suggests that the growth of the number of such sacks with the orders of the dice exhibits asymptotic regularities that call for explanation. We content ourselves with giving precise conjectures when one die has order~$3$.
\begin{conjprob} \label{exotictriangles} If $S_3(k)$ be the set $m$ with $1 \le m <\frac{k}{2}$ such that the sack obtained by exchanging the $\chi_m$ factor of a fair $k$-die with the $\frac{1}{3}(x^2+x+1)$ of a fair $3$-die is strict (hence exotic unless $k = 3m$), $M_3(k) := \max S_3(k)$ and $R_3(k) := \frac{M_3(k)}{k}$, then
\begin{enumerate}
\item Any $m$ between $ \lceil\frac{k}{4}\rceil$ and $ M_3(k)$ is in $S_3(k)$.
\item For $k \ge 336$, $M_3(k) \ge \lfloor\frac{5k}{12}\rfloor$.
\item $M_3(k) \le \frac{60}{143}k$ with equality exactly when $k$ is a multiple of $143$.
\item \label{exoticitem} $M_3(k+143) - M_3(k)=60$, except that there is a sequence $b_a$ with $b_1 = 0$, $b_{a+1}-b_a$ either $0$ or $1$, and such that, if $a$ is not divisible by $143$ and $k = 603a+143b_a$, then $M_3(k+143) - M_3(k)=59$.
\item The $\limsup$ as $k \to \infty$ of $R_3(k)$ equals $\frac{60}{143} \simeq 0.4195804$.
\end{enumerate}
\end{conjprob}
A few remarks are in order. Since $\frac{k}{4} \le m$ just ensures that the new $3$-die is strict, the point of these conjectures is to pin down $M_3(k)$ and $R_3(k)$. As Figure~\ref{triscatter} shows, their behavior warns us to exercise caution when making conjectures from inductive evidence. For example, $\frac{M_3(k)}{k} = \frac{5}{12}$ exactly when $k=12\ell$ and \emph{when $\ell \le 28$}! For some time we were sure, based on computations to $k=500$, that $M_3(k+143) = M_3(k)+60$ always held, and the surprising appearance of exceptions involving $603$ makes us a little nervous that others, possible with much larger modulus, may be lurking. The sequence $b_a$ and the $\liminf$ of $R_3(k)$ are mysterious. Empirically, $b_a$ takes on each value either $4$ for $5$ times, which leads to a lower bound for the $R_3(k)$ about $10^{-5}$ less than $\frac{60}{143}$ but we do not see enough of a pattern to conjecture a value for the $\liminf$. A SAGE~\cite{SAGE} notebook~\calcpagecite{7.SAGE-a.htm} is listing the exceptions up to $k=10^6$. See~\calcpagecite{progress.htm} for those up to $k=10^5$ or for word of counterexamples to Conjecture~\ref{exotictriangles}.(\ref{exoticitem}) found after going to press.
\begin{center}
\tikzstyle{background grid}=[draw, black!50,step=.5cm]
\begin{tikzpicture}[]{
\node [inner sep=0pt,above right] (image) at (0,0)
{\includegraphics[scale=0.60]{m3.pdf}};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\fill (.360,.544) circle (2pt);
\fill (.167,.990) circle (2pt);
\fill (.310,.990) circle (2pt);
\fill (.452,.990) circle (2pt);
\fill (.594,.990) circle (2pt);
\fill (.737,.990) circle (2pt);
\fill (.879,.990) circle (2pt);
\fill (.766,.783) circle (2pt);
\draw (.963,.540) node [] {$\frac{5}{12}$};
\draw (.963,.990) node [] {$\frac{60}{143}$};
\draw (.64,.28) node [] {
\begin{minipage}{6cm} \small
The circular markers at the top show the points with $k$ a multiple of $143$. The other two markers show the point with $k=336$ with smaller multiples of $12$ directly to its left, and with $\text{$k=746=603+143$}$, the first of the exceptions in Conjecture-Problem~\ref{exotictriangles}(\ref{exoticitem}). A few points with $k <20$ lie below the plot.
\end{minipage}};
\end{scope}
}
\end{tikzpicture}
\refstepcounter{equation}\label{triscatter}\centerline{\textbf{Figure~\ref{triscatter}} Scatter plot of $(k, \frac{M_3(k)}{k})$ for $k$ up to $950$}
\end{center}
What happens if, instead of a $(3,k)$-sacks, we try to enumerate $(\ell,k)$-sacks for other fixed values of $\ell$? For small $\ell$, analogues of Conjecture~\ref{exotictriangles} hold and these can, at least at first, be much simpler (if less intriguing). For example, our calculations suggest that $S_4(k)$ is the interval extending from $\lceil\frac{k}{6}\rceil$ to $\lfloor\frac{k}{3}\rfloor$.
But, complications soon arise as it becomes necessary to consider exchanges of sets of several factors $\chi_m$.
\ifshrink
\else
The swaps (each given by its $m$ and ordered first by the number swapped and then lexicographically) for $k=20$ are $[3\leftrightarrow 4]$, $[4\leftrightarrow 5]$, $[5\leftrightarrow 6]$, $[6\leftrightarrow 7]$, $[6\leftrightarrow 8]$, $[7\leftrightarrow 8]$, $[8\leftrightarrow 9]$, $[3,7 \leftrightarrow 4,6]$, $[3,7 \leftrightarrow 4,8]$, $[4,9 \leftrightarrow 5,8]$, $[5,9\leftrightarrow 6,8]$, $[6,8\leftrightarrow 7,9]$, and for $k=21$ are $[3\leftrightarrow 4]$, $[4\leftrightarrow 5]$, $[5\leftrightarrow 6]$, $[6\leftrightarrow 7]$, $[7\leftrightarrow 8]$, $[8\leftrightarrow 9]$, $[9\leftrightarrow 10]$, $[2,5\leftrightarrow 3,6]$,$[3,7\leftrightarrow 4,8]$, $[4,10\leftrightarrow 5,9]$, $[4,10\leftrightarrow 6,9]$, $[5,8\leftrightarrow 6,9]$, $[5,9\leftrightarrow 6,8]$, $[5,10\leftrightarrow 6,9]$, $[6,10\leftrightarrow 7,9]$, $[7,10\leftrightarrow 8,9]$, $[3,8,9\leftrightarrow 4,7,10]$and $[4,8,9\leftrightarrow 5,7,10]$. Roughly speaking, you can swap two sets of factors when the averages of the $m$ in each are sufficiently close.
\fi
This suggests that the growth of $E(k)$ is likely to be exponential in $k$ but the computations we have made are too limited to provide convincing qualitative evidence, let alone to suggest precise conjectures. Table~\ref{ektab} shows the first few diagonal counts of exotic pairs of $k$-dice which already show super-linear growth. We leave these questions to the interested reader.
\begin{center}\renewcommand{1.4}{1.}
\refstepcounter{equation}\label{ektab}\vskip3pt\centerline{\textbf{Table~\ref{ektab}} Number $E(k)$ of exotic pairs for small values of the order $k$ }
\begin{tabular}{r>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}
$k$ & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \unskip, \ignorespaces
$E(k)$ & 3& 2& 3& 4 & 4 & 6 & 7 & 8 & 12 & 18 & 19 & 27 & 42 & 60
\end{tabular}\vskip3pt
\end{center}
\ifshrink\else
\section{Appendix: The game of craps}\label{craps}
\stepcounter{subsection}
We first recall here how the game of craps is played, ignoring secondary aspects that affect only betting on the game and not its outcome. Then we explain how the probability of winning it may be calculated, not because this is essential to the main theme of the paper, but because the arguments provide a very pretty application of basic ideas of finite probability to an infinite sample space and are increasingly rarely covered in contemporary probability courses.
The game is played with two standard cubical dice with faces numbered from $1$ to $6$
\ifshrink
\else
\footnote{So all rolls of single die differ by $1$ from our convention in subsection~\ref{defnot}, and $2$-dice totals differ by~$2$.}
\fi
. There are two stages, in both of which, the outcome of play is determined by the total of the numbers showing on the two dice. In the first stage, consisting of single \emph{comeout} roll $t$, the player wins by throwing a $7$ or $11$ and loses by throwing a $2$, $3$ or $12$. Rolls of $4$--$6$ and $8$--$10$ lead to a second stage in which this first roll becomes the players \emph{point}. In this second stage, the player wins by re-rolling the comeout point $t$, loses by rolling a $7$ and rolls again in all other cases. So the second stage contains outcomes involving any finite number of rolls.
{\renewcommand{1.4}{1.4}
\begin{center}
\refstepcounter{equation}\label{crapstable} \textbf{Table~\ref{crapstable}} Probabilities arising in finding the chance of winning at craps.
\begin{tabular}{>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}
\text{Total $t$ on first roll} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \unskip, \ignorespaces
\mathbb{P}(t) & \frac{1}{36}& \frac{2}{36}& \frac{3}{36}& \frac{4}{36}& \frac{5}{36}& \frac{6}{36}& \frac{5}{36}& \frac{4}{36}& \frac{3}{36}& \frac{2}{36}& \frac{1}{36} \unskip, \ignorespaces
\mathbb{P}(w | t) & 0& 0& \frac{3}{9}& \frac{4}{10}& \frac{5}{11}& 1& \frac{5}{11}& \frac{4}{10}& \frac{3}{9}& 1& 0 \unskip, \ignorespaces
\mathbb{P}(t \cap w) & 0& 0& \frac{9}{324}& \frac{16}{360}& \frac{25}{396}& 1& \frac{25}{396}& \frac{16}{360}& \frac{9}{324}& 1& 0\unskip, \ignorespaces
\end{tabular}
\end{center}
}
Table~\ref{crapstable} summarizes the ingredients that go into finding the probability of a win for the player, an event that we denote $w$. Since the different totals are mutually exclusive, the probability that the player wins at craps can be computed from the table as
\[
\mathbb{P}(w) = \sum_{t=2}^{12} \mathbb{P}(t \cap w) = \frac{243}{495} \simeq 0.4909\,.
\]
How can we check the values in the table? Those in the first row are standard counts. Given the total $t$, the number $i$ on the first die determines that on the second and $i$ must be between $1$ and $t-1$ if $t \le 7$ and between $t-6$ and $6$ if $t \ge 7$. Given the numbers in the second row, those in the third follow by applying the Intersection Formula for probabilities, $\mathbb{P}(t \cap w) = \mathbb{P}(t)\cdot \mathbb{P}(w | t)$.
The conditional probabilities in the second row are more interesting. They can be computed from a tree diagram, but this diagram has the feature, seldom found in the examples treated in finite probability courses, of being infinite. An initial segment of the branch of this tree, starting from the node on the left where a comeout point of $t=9$ has just been rolled, is shown in Figure~\ref{crapsfigure}. The tree branches up to a winning leaf (shaded black) on a roll of $9$, down to a losing leaf (shaded gray) on a roll of $7$ and across on any other role. Each edge is labeled with the probability of following it from its left endpoint, and each leaf is labeled with the probability of reaching it from the root of the tree.
\begin{center}
\begin{tikzpicture}[scale=0.85]
\foreach \x in {0,1,...,3} {
\coordinate (left) at (3*\x,0);
\coordinate (top) at (3*\x+2,2);
\coordinate (right) at (3*\x+3,0);
\coordinate (bot) at (3*\x+2,-2);
\draw (left) -- (top) node [midway, fill=white] {$\frac{4}{36}$} node [above right] {$\frac{4}{36}\cdot \left(\frac{26}{36}\right)^{\x}$};
\filldraw [black] (top) circle (3pt);
\draw (left) -- (right) node [midway, fill=white] {$\frac{26}{36}$};
\draw (left) -- (bot) node [midway, fill=white] {$\frac{6}{36}$} node [below right] {$\frac{6}{36}\cdot \left(\frac{26}{36}\right)^{\x}$};
\filldraw [gray] (bot) circle (3pt);
}
\draw[dashed] (12,0) -- (13.5,0);
\end{tikzpicture}
\refstepcounter{equation}\label{crapsfigure} \textbf{Figure~\ref{crapsfigure}} Tree diagram for the game of craps \emph{after} a comeout roll of $9$.
\end{center}
The tree makes visible a formula for $\mathbb{P}(w | 9)$ as the sum $\frac{a}{1-r}=\frac{4}{10}$ of the geometric series with initial term $ a= \frac{4}{36}$ and ratio $r=\frac{26}{36}$.
But there is a much easier way to see obtain this value from the tree, by noticing that we move to a leaf on any outcome in the event $ (7 \text{~or~} 9)$ and win when this outcome is a $9$. Hence $\mathbb{P}(w | 9) = \mathbb{P}\bigl(9 |\, (7 \text{~or~} 9)\bigr) = \frac{\mathbb{P}(9)}{\mathbb{P}(7 \text{~or~} 9)}$ which the first row of Table~\ref{crapstable} gives immediately as $\frac{4}{10}$. The other non-trivial entries in the second row of the table then follow analogously from the first row.
\fi
\linespread{1.02}\normalfont\selectfont
\section*{References}
\begin{biblist}
\bibselect{totaltoparts}
\end{biblist}
\newcounter{lastbib}
\setcounter{lastbib}{\value{bib}}
\section*{Software packages referenced}
\begin{biblist}[\setcounter{bib}{\value{lastbib}}]
\bib{Bertini}{article}{
author={Bates, Daniel J.},
author={Hauenstein, Jonathan D.},
author={Sommese, Andrew J.},
author={Wampler, Charles W.},
title={\texttt{\upshape Bertini}: Software for Numerical Algebraic Geometry},
date={2012},
note={Version 1.3.1},
eprint={\neturl{bertini.nd.edu}},
doi={dx.doi.org/10.7274/R0H41PB5},
}
\bib{Macaulay}{article}{
author={Grayson, Dan},
author={Stillman, Mike},
title={\texttt{\upshape Macaulay2}: a software system for research in algebraic geometry},
date={2012},
note={Version 1.5},
eprint={\neturl{http://www.math.uiuc.edu/Macaulay2/}},
}
\bib{Magma}{article}{
author={Computational Algebra Research Group$\textrm{,}$ School of Mathematics and Statistics$\textrm{,}$ University of Sydney},
title={\texttt{\upshape MAGMA} computational algebra system},
date={2012},
note={Version 2.19-6},
eprint={\neturl{http://magma.maths.usyd.edu.au/magma/}},
}
\bib{Maple}{article}{
title={\texttt{\upshape Maple 15}},
note={Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.},
}
\bib{QEPCADB}{article}{
author={Hong, Hoon},
title={\texttt{\upshape QEPCAD}: Quantifier Elimination by Partial Cylindrical Algebraic Decomposition},
date={2011},
note={Version B 1.65},
eprint={\neturltilde{http://www.usna.edu/CS/~qepcad/B/QEPCAD.html}{http://www.usna.edu/CS/$\sim$qepcad/B/QEPCAD.html}},
}
\bib{SAGE}{article}{
author={William A. Stein et al},
title={Sage Mathematics Software},
note={Version 5.5},
publisher={The Sage Development Team},
date={2013},
eprint={\neturl{http://www.sagemath.org}},
}
\end{biblist}
\setcounter{lastbib}{\value{bib}}
\section*{Computer-assisted calculations referenced}
\begin{biblist}[\setcounter{bib}{\value{lastbib}}]
\bib{codesamples}{article}{
author={Morrison, Ian},
author={Swinarski, David},
title={Computer-assisted calculations for ``Can you play a fair game of craps with a loaded pair of dice?''},
date={2013},
note={In the body of the paper, citations of a single calculation include the name of a file (where possible as the text of a hyperlink to it) that contains the source code and output of the calculation. These files are located in the directory \neturl{http://faculty.fordham.edu/dswinarski/totaltoparts/} which also contains a descriptive table of contents \calcpage{toc.htm} numbered in parallel with this paper and linking to all the calculations. Citations of groups of calculations point to an anchor in the contents file at which links to the individual files may be found.},
}
\end{biblist}
\end{document} | math | 79,578 |
\begin{document}
\author{Mohamed El Bachraoui}
\address{Dept. Math. Sci,
United Arab Emirates University, PO Box 17551, Al-Ain, UAE}
\email{[email protected]}
\keywords{Gamma function, generalized gamma function, functional equations, special functions}
\subjclass{33B15, 11Y60, 11M06}
\begin{abstract}
We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$
in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that
many properties which $\Ga(z)$ enjoys extend in a natural way to the function $\Ga(x,z)$. Among other things we
shall provide functional equations, a multiplication formula, and analogues of the Stirling formula
with asymptotic estimates as consequences.
\end{abstract}
\date{\textit{\today}}
\title{A gamma function in two variables}
\section{Introduction}
Throughout, let $\N$, $\Z$, $\R$, and $\C$ be the sets of positive integers,
integers, real numbers, and complex numbers respectively.
Further, let $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, $\mathbb{Z}_0^{-}=\mathbb{Z}\setminus\mathbb{N}$,
$\R^{+}=\R\setminus\{r\in\R:\ r\leq 0\}$,
and $\D = \C\setminus\{x\in\R:\ x\leq 0\}$.
The gamma function $\Ga(z)$ is one of the most important special functions in mathematics with applications in many disciplines like
Physics and Statistics. It was first introduced by Euler in the integral form
\begin{equation}\label{Ga-integral}
\Ga(z) = \int_{0}^{\infty} t^{z-1} e^{-t}\ dt.
\end{equation}
Well-known equivalent definitions for the gamma function include the following three forms:
\begin{equation}\label{Ga-weierstrass}
\Ga(z) = \left(z e^{z \ga}\prod_{n=1}^{\infty}(1+\frac{z}{n})e^{-\frac{z}{n}}\right)^{-1} ,
\end{equation}
\begin{equation}\label{Ga-limit}
\Ga(z) =\lim_{n\to\infty}\frac{n^z n!}{(z)_{n+1}},
\end{equation}
\begin{equation}\label{Ga-euler}
\Ga(z) = \frac{1}{z}\prod_{n=1}^{\infty} (1+\frac{1}{n})^z (1+\frac{z}{n})^{-1},
\end{equation}
where $\ga$ is the \emph{Euler-Mascheroni constant}
\[
\ga = \lim_{n\to\infty}(1+\frac{1}{2}+\ldots+\frac{1}{n} - \log n)
\]
and $(z)_{n}$ is the \emph{Pochhammer symbol}
\[ (z)_n= \begin{cases}
1& \text{if\ } n=0, \\
z(z+1)\ldots(z+n-1)& \text{if\ } n\in\N.
\end{cases}
\]
The gamma function satisfies the basic functional equation
$\Ga(z+1) = z\Ga(z)$.
Barnes~\cite{Barnes 2} and Post~\cite{Post} investigated the theory of difference equations
of the more general form $\phi(z+1)= f(z) \phi(z)$ under conditions on the function $f(z)$ and obtained generalized gamma functions as
solutions. See also Barnes~\cite{Barnes 2} where \emph{multiple gamma functions} have been introduced.
Many mathematicians considered concrete cases of generalized gamma functions. Dilcher~\cite{Dilcher 94} introduced for
any nonnegative integer $k$ the function
\[
\Ga_k (z) := \lim_{n\to\infty}\frac{ \exp\left\{\frac{\log^{k+1}n}{k+1} z\right\}
\prod_{j=1}^n \exp\left\{ \frac{1}{k+1}\log^{k+1} j\right\}}
{\prod_{j=0}^n \exp\left\{\frac{1}{k+1}\log^{k+1} (j+z)\right\}}
\]
which for $k=0$ becomes $\Ga(z)$, see formula (\ref{Ga-limit}).
Di\'{a}z~and~Pariguan~\cite{Diaz-Pariguan} extended the integral representation~(\ref{Ga-integral}) to the function
\[
\Ga_k(z) = \int_{0}^{\infty} t^{z-1}e^{-\frac{t^k}{k}}\,dt \qquad (k\in\mathbb{R}^{+})
\]
which for $k=1$ is nothing else but $\Ga(z)$. Recently Loc~and~Tai~\cite{Loc-Tai} involved polynomials to define
\[
\Ga_f(z) = \int_{0}^{\infty}f(t)^{z-1}e^{-t}\,dt
\]
which for $f(t)=t$ clearly gives $\Ga(z)$.
In this paper we present a gamma function $\Ga(x,z)$ in two complex variables which is meromorphic in both variables
and which satisfies $\lim_{x\to 1}\Ga(x,z)= \Ga(z)$.
Our motivation is to extend the Weierstrass form (\ref{Ga-weierstrass}) in much the same way
the Hurwitz zeta function
\[ \zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(n+x)^s} \]
extends the Riemann zeta function
\[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.
\]
So our definition involves the infinite product
\[
\pr(1+ \frac{z}{n+x})^{-1} e^{\frac{z}{n+x}}\ \quad \text{rather than\quad }
\prod_{n=1}^{\infty}(1+ \frac{z}{n})^{-1}e^{\frac{z}{n}}
\]
and in order to maintain valid the analogues of properties of $\Ga(z)$ the factor $e^{-z \ga}$ will be replaced by $e^{-z \ga(x)}$,
where $\ga(x)$ is defined as follows.
\begin{definition}\label{def:ga(x)}
For $x\in \D\setminus\Z_0^{-}$ let the function $\gamma(x)$ be
\[
\gamma(x) = \lim_{n\to\infty}(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n-1} - \log n) =
\frac{1}{x}+\sum_{n=1}^{\infty}\big(\frac{1}{x+n}-\log\frac{n+1}{n}\big).
\]
\end{definition}
Note that $\gamma(1)= \gamma$ and that
$\gamma(x) = \gamma_0 (x) = -\psi(x)$ where
\[
\gamma_0 (x) =\lim_{n\to\infty}(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n} - \log (n+x))
\]
is the \emph{zeroth Stieltjes constant} and
\[
\psi(x) = \log' \Gamma(x) = \frac{\Gamma'(x)}{\Gamma(x)}
\]
is the \emph{digamma function}.
For an account of these functions we refer to Coffey~\cite{Coffey 2012} and Dilcher~\cite{Dilcher 92}.
It is easily seen that the function $\gamma(x)$ represents an analytic function on $\C\setminus\Z_0^{-}$ and that
\begin{equation}\label{gamma-functional}
\gamma(x+1) = \frac{-1}{x}+ \gamma(x).
\end{equation}
In section~2 we study the function $G(x,z)$ represented as an infinite product.
This prepares the ground for
section~3 where we introduce the gamma function $\Ga(x,z)$ along with some of its
basic properties including functional equations and a formula for the modulus $|\Ga(n+i,n+i)|$ for $n\in\N_0$.
Section~4 is devoted to the analogues of the forms~(\ref{Ga-limit})~and~(\ref{Ga-euler})
together with their consequences such as values at half-integers and residues at poles. In section~5 we give the analogue of
the Gauss' duplicate formula. Further in section~6 we present the analogue of the Stirling's formula leading to asymptotic
estimates for our function. Finally in section~7 we give series expansions in both variables and as a result we provide
recursive formulas for the coefficients of the series in terms of the Riemann-Hurwitz zeta functions.
\section{The function $G(x,z)$}
\begin{definition}\label{G(x,z)}
For $x\in\C\setminus\Z_0^{-}$ and $z\in\C$ let the function $G(x,z)$ be defined as follows
\[
G(x,z) = \pr (1+ \frac{z}{n+x}) e^{-\frac{z}{n+x}}.
\]
\end{definition}
Note that $G(x,z)$ is entire in $z$ for fixed $x \in \C\setminus\Z_0^{-}$ and that
$\lim_{z\to 0} G(x,z) = G(x,0) = 1$.
\begin{proposition}\label{G-functional}
We have:
\[ (a)\quad G(x,z-1) = (z+x-1) e^{\ga(x)} G(x,z). \]
\[ (b)\quad G(x-1,z) = \frac{z+x-1}{x-1} e^{-\frac{z}{x-1}} G(x,z). \]
\end{proposition}
\begin{proof}
(a)\ Clearly the zeros of $G(x,z)$ are $-x, -(x+1), -(x+2), \ldots$ and the zeros of $G(x,z-1)$ are
$-(x-1), -x, -(x+1), -(x+2), \ldots$. Then by the theory of Weierstrass products, we can write
\[
G(x,z-1) = e^{g(x,z)} (z+x-1) \prod_{n=0}^{\infty}(1+\frac{z}{x+n}) e^{-\frac{z}{x+n}}
\]
for an entire function $g(x,z)$. Taking logarithms and differentiating with respect to $z$ we find
\begin{equation} \label{G-help1}
\frac{d}{dz} \log G(x,z-1) = \frac{d}{dz}g(x,z) + \frac{1}{z+x-1} +
\su (\frac{1}{z+x+n}-\frac{1}{x+n}).
\end{equation}
On the other hand, from the definition of $G(x,z)$ we have
\[
\frac{d}{dz} \log G(x,z-1) = \su \left(\frac{1}{z+x+n-1}-\frac{1}{x+n}\right) \]
\[
= \frac{1}{z+x-1}-\frac{1}{x} + \su \left(\frac{1}{z+x+n}-\frac{1}{x+n}\right) + \su\left(\frac{1}{x+n}-\frac{1}{x+n+1}\right),
\]
which gives
\begin{equation} \label{G-help2}
\frac{d}{dz} \log G(x,z-1) = \frac{1}{z+x-1} + \su \left(\frac{1}{z+x+n}-\frac{1}{x+n}\right).
\end{equation}
Then the relations (\ref{G-help1}) and (\ref{G-help2}) imply that
$\frac{d}{dz} g(x,z) = 0$ and so $g(x,z)$ is independent of $z$, say $g(x,z)=g(x)$. It remains to prove that $g(x)=\ga(x)$.
From $G(x,z-1) = (z+x-1) e^{g(x)} G(x,z)$ and $G(x,0)=1$ we get
\[
e^{-g(x)} = x G(x,1) = x \pr \left(\frac{x+n+1}{x+n}\right) e^{-\frac{1}{x+n}}.
\]
Furthermore,
\[
\begin{split}
x \prod_{m=0}^{n-1} \left(\frac{x+m+1}{x+m}\right) e^{-\frac{1}{x+m}} &=
(x+n) e^{-(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n-1})} \\
&=
x e^{-(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n-1})} + n e^{-(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n-1})},
\end{split}
\]
which yields
\[
e^{-g(x)} = \lim_{n\to\infty} x \prod_{m=0}^{n-1}\left(\frac{x+m+1}{x+m}\right) e^{-\frac{1}{x+m}}
= \lim_{n\to\infty} n e^{-(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n-1})},
\]
or equivalently,
\[
g(x) = \lim_{n\to\infty}(\frac{1}{x}+\frac{1}{x+1}+\ldots+ \frac{1}{x+n-1} - \log n) = \ga (x),
\]
as desired.
Part (b) follows directly by the definition of $G(x,z)$. This completes the proof.
\end{proof}
\begin{proposition} \label{G-sin}
If $x\in\C\setminus\Z$, then
\[
G(x,-z) G(-x,z) = \frac{(z-x) \sin \pi (z-x)}{x \sin \pi x} e^{z\cot (\pi x) +\frac{z}{x}}.
\]
\end{proposition}
\begin{proof}
As the zeros of $\sin(z-x)$ are $x, \pi+x, -\pi +x, 2\pi +x, -2\pi +x,\ldots$, by the theory of Weierstrass products we have
\begin{equation} \label{G-help3}
\sin(z-x) = (z-x) e^{g(x,z)} \prod_{n=1}^{\infty}\left(1-\frac{z}{n\pi +x}\right)e^{\frac{z}{n\pi +x}}
\prod_{n=1}^{\infty}\left(1+\frac{z}{n\pi -x}\right)e^{-\frac{z}{n\pi -x}}
\end{equation}
for an entire function $g(x,z)$. We find $e^{g(x,z)}$ as follows. Setting
\[ f_n(x,z) = e^{g(x,z)} (z-x) \prod_{k=1}^{n} \left(1-\frac{z}{k\pi +x}\right)e^{\frac{z}{k\pi +x}}
\left( 1+\frac{z}{k\pi -x}\right)e^{-\frac{z}{k\pi +x}},
\]
we have $\sin(z-x) = \lim_{n\to\infty} f_n(x,z)$.
Taking logarithms and differentiating we obtain
\[
\begin{split}
\frac{f_n'(x,z)}{f_n(x,z)}
&=
\frac{d}{dz}g(x,z) + \frac{1}{z-x} + \sum_{k=1}^{n} \left(\frac{-1}{k\pi +x-z}+\frac{1}{k\pi -x+z}+\frac{1}{k\pi +x}-\frac{1}{k\pi -x}\right) \\
&=
\frac{d}{dz}g(x,z) + \frac{1}{z-x} + \sum_{k=1}^{n}\frac{2(x-z)}{(k\pi)^2 - (x-z)^2} - \sum_{k=1}^{n}\frac{2x}{(k\pi)^2 - x^2}.
\end{split}
\]
But as is well-known,
\[
\lim_{n\to\infty} \frac{f_n'(x,z)}{f_n(x,z)} = \cot (z-x)
= \frac{1}{z-x} + \sum_{n=1}^{\infty} \frac{2(z-x)}{(z-x)^2 - (n\pi)^2}.
\]
Thus
\[
\frac{d}{dz} g(x,z)= \sum_{n=1}^{\infty}\frac{2x}{(k\pi)^2 - x^2} = \frac{1}{x}-\cot x,
\]
and hence $g(x,z)= z(\frac{1}{x}-\cot x) + h(x)$. Now equation (\ref{G-help3})
gives
\[
\frac{\sin(z-x)}{z-x} = e^{h(x)} e^{\frac{z}{x}-z\cot x}
\prod_{n=1}^{\infty}\left(1-\frac{z}{n\pi +x}\right)e^{\frac{z}{n\pi +x}}
\left(1+\frac{z}{n\pi -x}\right)e^{-\frac{z}{n\pi -x}},
\]
which by letting $z\to 0$ implies
\[ e^{h(x)} = \frac{\sin x}{x}. \]
Therefore we have
\[
\frac{\sin(z-x)}{z-x} = \frac{\sin x}{x} e^{\frac{z}{x}-z \cot x}
\prod_{n=1}^{\infty}(1-\frac{z}{n\pi +x})e^{\frac{z}{n\pi +x}} (1+\frac{z}{n\pi -x})e^{\frac{-z}{n\pi -x}} .
\]
In particular,
\begin{equation}\label{G-help4}
\begin{split}
\frac{\sin \pi(z-x)}{\pi(z-x)}
&=
\frac{\sin \pi x}{\pi x}e^{\frac{z}{x}-\pi z\cot \pi x}
\prod_{n=1}^{\infty}(1-\frac{z}{n +x}) e^{\frac{z}{n +x}}
\prod_{n=1}^{\infty}(1+\frac{z}{n -x}) e^{\frac{-z}{n -x}} \\
&=
\frac{\sin \pi x}{\pi x} e^{\frac{z}{x}-\pi z\cot \pi x} G(x,-z) G(-x,z) \left(1-\frac{z}{x}\right)^{-2} e^{-\frac{2z}{x}} \\
&=
\frac{\sin \pi x}{\pi x} e^{\frac{z}{x}-\pi z\cot \pi x} \frac{x^2}{(x-z)^2} G(x,-z) G(-x,z),
\end{split}
\end{equation}
or equivalently
\[
G(x,-z) G(-x,z) = \frac{(z-x) \sin \pi (z-x)}{x \sin \pi x} e^{z\cot (\pi x) +\frac{z}{x}}.
\]
This completes the proof.
\end{proof}
\begin{corollary}
If $x\in \C\setminus\Z$, then
\[
\prod_{n=1}^{\infty}\left(1-\frac{z^2}{(n+x)^2}\right)\left(1-\frac{z^2}{(n-x)^2}\right) =
\left(\frac{x}{\sin \pi x}\right)^2 \frac{\sin^2 \pi z - \sin^2 \pi x}{z^2 -x^2}.
\]
\end{corollary}
\begin{proof}
By the first identity in (\ref{G-help4}) we have
\[
\frac{\sin \pi(z-x)}{\pi(z-x)} \frac{\sin \pi(z+x)}{\pi(z+x)} = (\frac{\sin \pi x}{\pi x})^2
\prod_{n=1}^{\infty}\left(1-\frac{z^2}{(n+x)^2}\right)\left(1-\frac{z^2}{(n-x)^2}\right),
\]
which means that
\[
\prod_{n=1}^{\infty}\left(1-\frac{z^2}{(n+x)^2}\right)\left(1-\frac{z^2}{(n-x)^2}\right)
= \frac{x^2}{z^2 - x^2}\frac{\\sin^2 \pi z - \sin^2 \pi x}{\sin^2 \pi x},
\]
which completes the proof.
\end{proof}
\section{The function $\Ga(x,z)$}
Throughout for any $x\in\C$ let
\[
S_x = \C\setminus\{-x+n:\ n\in\N_0 \cup\{-1\}\}.
\]
\begin{definition} \label{main-def}
For $x\in\C\setminus\Z_0^{-}$ and $z\in S_x$ let the function $\Ga(x,z)$ be defined as follows.
\[
\Ga(x,z) = \left( (z+x-1) e^{z \ga(x)} G(x,z) \right)^{-1}.
\]
\end{definition}
Note that for fixed $x\in \C\setminus\Z_0^{-}$ the function $\Ga(x,z)$ is meromorphic with simple poles at $z\in S_x$
and that $\lim_{x\to 1} \Ga(x,z)=\Ga(1,z) = \Ga(z)$.
\begin{proposition} \label{functional}
We have
\[
\begin{split}
(a)& \quad \Ga(x,z+1) = (z+x-1) \Ga(x,z), \qquad (x\in\C\setminus\Z_0^{-}, z+1\in S_x) \\
(b)& \quad \Ga(x+1,z) = \frac{z+x-1}{x} \Ga(x,z), \qquad (x+1\in\C\setminus\Z_0^{-}, z\in S_x) \\
(c)& \quad \Ga(x+1,z+1) = \frac{(z+x-1)(z+x)}{x} \Ga(x,z), \qquad (x+1\in\C\setminus\Z_0^{-}, z+1\in S_{x+1}).
\end{split}
\]
\end{proposition}
\begin{proof}
(a)\ We have
\[
\begin{split}
\Ga(x,z+1) &= \left( (z+x) e^{(z+1)\ga(x)} G(x,z+1) \right)^{-1} \\
&=
\left( (z+x) e^{\ga(x)} G(x,z+1) e^{z \ga(x)} \right)^{-1} \\
&=
\left( G(x,z) e^{z \ga(x)}\right)^{-1} \\
&=
(z+x-1) \Ga(x,z),
\end{split}
\]
where the fourth identity follows by Proposition~\ref{G-functional}(a).
(b)\ We have
\[
\begin{split}
\Ga(x+1,z) &= \left( (z+x) e^{z \ga(x+1)} G(x+1,z) \right)^{-1} \\
&=
\left( (z+x) e^{z(\frac{-1}{x}+ \ga(x))} G(x+1,z) \right)^{-1} \\
&=
\left( (z+x) e^{-\frac{z}{x}} G(x+1,z) e^{z \ga(x)} \right)^{-1} \\
&=
\frac{1}{x} \left( e^{z \ga(x)} G(x,z) \right)^{-1} \\
&=
\frac{z+x-1}{x} \Ga(x,z),
\end{split}
\]
where the second identity follows from the relation (~\ref{gamma-functional}) and the
fourth identity from Proposition~\ref{G-functional}(b).
(c)\ This part follows by a combination of part (a) and part (b).
\end{proof}
\begin{corollary} \label{Gamma-integers}
Let $x\in\C\setminus\Z_0^{-}$ and let $n\in\N$ . Then we have
\[
\begin{split}
(a)& \quad \Ga(x,1) = 1, \\
(b)&\quad \Ga(x,0) = \frac{1}{x-1},\quad (x\not= 1) \\
(c)&\quad \Ga(x,n) = (x)_{n-1},\quad (n\geq 2) \\
(d)&\quad \Ga(x,-n) = \frac{1}{(x-n-1)_{n+1}},\\
(e)&\quad \Ga(n,z) = \frac{(z)_{n-1}}{(n-1)!} \Ga(z),\quad (n\geq 2).
\end{split}
\]
\end{corollary}
\begin{proof}
(a)\ As $G(x,0)=1$, we have by Proposition~\ref{G-functional}(a)
\[ 1 = x e^{\ga(x)} G(x,1), \]
and thus by definition
\[ \Ga(x,1) = \left( x e^{\ga(x)} G(x,1) \right)^{-1} = 1. \]
Parts (b) and (c) follow directly from Proposition~\ref{functional}(a). As to part (d) combine part (b) and Proposition~\ref{functional}(a).
As to part (e) combine Proposition~\ref{functional}(b) with the fact that $\Ga(1,z)=\Ga(z)$.
\end{proof}
\begin{proposition}\label{Gamma-sin}
We have
\[
(a)\quad \Ga(x,1-z) \Ga(1-x, z) = \frac{- \sin \pi x}{(z-x)\sin \pi(z-x)}.
\]
\[
(b)\quad \Ga(x,z)\Ga(-x,-z) = \frac{-x \sin\pi x}{\left((z+x)^3 - (z+x)\right) \sin\pi(z+x)}.
\]
\end{proposition}
\begin{proof}
(a)\ We have
\[
\begin{split}
\Ga(x,1-z)\Ga(1-x,z)
&=
\left((-z+x) e^{(1-z)\ga(x)} (z-x) e^{z \ga(1-x)} G(x,1-z)G(1-x,z)\right)^{-1} \\
&=
\frac{- e^{-\ga(x)}e^{z\ga(x)} e^{-z(\frac{1}{x}+\ga(-x))}}{(z-x)^2}
\left( \frac{e^{-\ga(x)}}{-z+x}\frac{-x e^{\frac{-z}{x}}}{z-x} G(x,-z) G(-x,z) \right)^{-1} \\
&=
\frac{- e^{z\ga(x)} e^{-z(\frac{1}{x}+\ga(-x))}} {x e^{-\frac{z}{x}} \frac{(z-x)\sin \pi(z-x)}{x\sin\pi x} e^{z\cot \pi x-\frac{z}{x}}} \\
&=
-\frac{e^{z(\ga(x)-\ga(-x))}}{e^{z\cot \pi x-\frac{z}{x}}} \frac{\sin \pi x}{(z-x)\sin \pi(z-x)}.
\end{split}
\]
By Corollary~\ref{Gamma-integers}(a, b), the previous relation gives for $z=1$
\[
\frac{1}{x-1} = \frac{ -e^{\ga(x)-\ga(-x)-\cot \pi x +\frac{1}{x}} }{1-x} \frac{\sin \pi x}{\sin \pi(1-x)}
= \frac{ -e^{\ga(x)-\ga(-x)-\cot \pi x +\frac{1}{x}} }{1-x},
\]
which implies that
\[ e^{\ga(x)-\ga(-x)-\cot \pi x +\frac{1}{x}} = 1, \]
giving part (a).
(b)\ By Proposition~\ref{functional}(a, b) we have
\[
\Ga(1-x,z) = \frac{z-x-1}{-x} \Ga(-x,z)\ \text{and\ }
\Ga(x,1-z) = (-z+x-1)\Ga(x,-z).
\]
Then by virtue of part (a) we get
\[
\frac{z-x-1}{-x} \Ga(-x,z) (-z+x-1)\Ga(x,-z) = \frac{-\sin \pi x}{(z-x)\sin \pi (z-x)},
\]
or equivalently
\[
\Ga(-x,z) \Ga(x,-z) = \frac{- x \sin\pi x}{\bigl((z-x)^3 - (z-x) \bigr)\sin\pi (z-x)}.
\]
This completes the proof.
\end{proof}
\begin{corollary}\label{x-neg-int}
If $n\in\N_0$ and $z\not\in\N_0$, then
\[
\lim_{x\to -n}\Ga(x,z) = 0.
\]
\end{corollary}
\begin{proof}
By Proposition~\ref{Gamma-sin}(a) we have
\[
\Ga(x,z) = \Ga(x,1-(1-z)) = \frac{-\sin\pi x}{(1-z-x)\sin\pi(1-z-x)}\frac{1}{\Ga(1-x,1-z)}.
\]
Then
\[
\lim_{x\to -n}\Ga(x,z) = \frac{-\sin\pi n}{(1-z+n)\sin\pi(1-z+n)}\frac{1}{\Ga(1+n,1-z)} = 0.
\]
\end{proof}
\begin{corollary} \label{Norm}
If $n\in\N_0$, then
\[
| \Ga(n+i,n+i) |^2 = | \Ga(n-i,n-i)|^2 =
\frac{5 \prod_{k=0}^{2n-2}(4+k^2)}{\prod_{k=0}^{n-1}(1+k^2)}\frac{e^{\pi}}{10(e^{2\pi}+1)}.
\]
\end{corollary}
\begin{proof}
First note that
\begin{equation}\label{Ga-conjugate}
\Ga(\bar{x},\bar{z}) = \overline{\Ga(x,z)},
\end{equation}
from which the first identity immediately follows.
As to the second formula, using identity (\ref{Ga-conjugate}) and Proposition~\ref{Gamma-sin}(b) we obtain
\[
| \Ga(i,i) |^2 = \Ga(i,i) \overline{\Ga(i,i)} = \Ga(i,i) \Ga(\bar{i},\bar{i}) =
\frac{-i \sin \pi i}{\bigl( (2i)^3- (2i) \bigr) \sin 2\pi i} = \frac{e^{\pi}}{10(e^{2\pi}+1)},
\]
which gives the result for $n=0$. If $n>1$ we have by Proposition~\ref{functional}(c)
\[
|\Ga(n+i,n+i)|^2= \Ga(n+i,n+i) \Ga(n-i,n-i)
\]
\[
= \frac{(2i-1)(2i)\ldots(2i+2n-2)}{i(i+1)\ldots(i+n-1)} \frac{(-2i-1)(-2i)\ldots(-2i+2n-2)}{-i(-i+1)\ldots(-i+n-1)} |\Ga(i,i)|^2
\]
\[
= \frac{(-1)^{2n}(2i-1)(2i+1)(2i)(2i)(2i+1)(2i-1)\ldots(2i+2n-2)(2i-(2n-2))}
{(-1)^n (i)(i)(i+1)(i-1)\ldots(i+n-1)(i-(n-1))}|\Ga(i,i)|^2
\]
\[
= \frac{(4+1)(4)(4+1)\ldots(4+(2n-2)^2)}{1 (1+1)\ldots (1+(n-1)^2)} \frac{e^{\pi}}{10(e^{2\pi}+1)}.
\]
This completes the proof.
\end{proof}
\section{Analogues of Euler's formulas, residues, and values at half-integers}
\begin{proposition}\label{Euler-analogue}
We have
\[ (a)\quad \Ga(x,z) = \lim_{n\to\infty} \frac{n^z x(x+1)\ldots (x+n-1)}{(z+x-1)(z+x)\ldots(z+x+n-1)}
= \lim_{n\to\infty} \frac{n^z (x)_{n}}{(z+x-1)_{n+1}}. \]
\[ (b)\quad \Ga(x,z) = \frac{x}{(z+x-1)(z+x)}\prod_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^z \left(1+\frac{z}{x+n}\right)^{-1}. \]
\end{proposition}
\begin{proof}
(a) We have
\[
\begin{split}
\Ga(x,z)
&=
\lim_{n\to\infty} \left((z+x-1)e^{z(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+n-1}-\log n)}
\prod_{k=0}^{n-1} (1+\frac{z}{x+k})e^{-\frac{z}{x+k}} \right)^{-1} \\
&=
\lim_{n\to\infty} \left((z+x-1)e^{-z\log n}\prod_{k=0}^{n-1}\frac{z+x+k}{x+k}\right)^{-1} \\
&=
\lim_{n\to\infty} \left(n^{-z} \frac{(z+x-1)(z+x)\ldots (z+x+n-1)}{x(x+1)\ldots (x+n-1)}\right)^{-1} \\
&=
\lim_{n\to\infty} \frac{n^z x(x+1)\ldots (x+n-1)}{(z+x-1)(z+x)\ldots(z+x+n-1)}.
\end{split}
\]
(b) By the previous proof we have
\[
\begin{split}
\Ga(x,z)
&=
\frac{1}{z+x-1} \lim_{n\to\infty} n^z \prod_{k=0}^{n-1}\left(1+\frac{z}{x+k}\right)^{-1} \\
&=
\frac{1}{z+x-1} \lim_{n\to\infty}\prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right)^{z}
\prod_{k=0}^{n-1}\left(1+\frac{z}{x+k}\right)^{-1} \\
&=
\frac{x}{(z+x-1)(z+x)} \prod_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{z} \left(1+\frac{z}{x+n}\right)^{-1}.
\end{split}
\]
\end{proof}
\begin{corollary}\label{duplicate}
If $x, x+z \in\C\setminus\Z_0^{-}$, then
\[
\Ga(x,z) \Ga(x+z,-z) = \frac{1}{(x-1)(z+x-1)}.
\]
\end{corollary}
\begin{proof}
By Proposition~\ref{Euler-analogue}(a) we have
\[
\Ga(x,z) = \lim_{n\to\infty} \frac{n^z (x)_{n}}{(z+x-1)_{n+1}} = \frac{1}{x-1}\lim_{n\to\infty} \frac{n^z (x-1)_{n+1}}{(z+x-1)_{n+1}}
= \frac{1}{z+x-1}\lim_{n\to\infty} \frac{n^z (x)_{n+1}}{(z+x)_{n+1}},
\]
where the last identity follows since $\lim_{n\to\infty}\frac{x+n}{z+x+n} =1$.
Then
\[
\Ga(x+z,-z) = \frac{1}{x-1}\lim_{n\to\infty} \frac{n^{-z} (x+z)_{n+1}}{(x)_{n+1}}
= \frac{1}{(x-1)(z+x-1) \Ga(x,z)}.
\]
This completes the proof.
\end{proof}
\begin{corollary} \label{half-integer}
If $k, l\in\N_0$ such that $k+l\not=0$, then
\[
\Ga\left(\frac{2k+1}{2},\frac{2l+1}{2}\right) = \frac{2}{\sqrt{\pi} (2k-1)}
\frac{(2l+2)!}{(-4)^{l+1} (l+1)!}
\frac{(k+l-1)!}{(-l-\frac{1}{2})_{k+l}}
\]
\end{corollary}
\begin{proof}
On the one hand we have by Corollary~\ref{duplicate}
\[
\Ga\left(\frac{2k+1}{2},\frac{2l+1}{2}\right) \Ga\left(k+l+1,-\frac{2l+1}{2}\right) = \frac{2}{(2k-1)(k+l)}.
\]
On the other hand by Corollary~\ref{Gamma-integers}(e) and the well-known fact that
\[
\Ga(1/2 - k) = \frac{\sqrt{\pi}(-4)^k k!}{(2k)!}
\]
we have
\[
\Ga\left(k+l+1,-\frac{2l+1}{2}\right) = \frac{(-l-1/2)_{k+l}}{(k+l)!} \Ga(-l-1/2)
\]
\[
=
\frac{(-l-1/2)_{k+l}}{(k+l)!} \Ga(1/2 -(l+1)) = \frac{(-l-1/2)_{k+l}}{(k+l)!}
\frac{\sqrt{\pi} (-4)^{l+1} (l+1)!}{(2l +2)!}.
\]
Now combine these identities to deduce the required formula.
\end{proof}
\begin{corollary} \label{residues}
If $x\in\C\setminus\Z$ and $m\in\N_0\cup\{-1\}$, then the residue of $\Ga(x,z)$ at $z=-(x+m)$ is
\[
\begin{cases}
\frac{1}{(x-1)\Ga(x-1)}, & \text{if\ }m=-1 \\
\frac{(-1)^{m+1} (x)_{2m+1}}{(m+1)!}\frac{1}{\Ga(x+2m+1)},& \text{otherwise.} \\
\end{cases}
\]
\end{corollary}
\begin{proof}
Suppose first that $m=-1$. By Corollary~\ref{duplicate} and Proposition~\ref{functional}(c) we obtain
\[
\Ga(x,z) = \frac{1}{(x-1)(z+x-1)}\frac{1}{\Ga(z+x,-z)}.
\]
Then
\[
\lim_{z\to -(x-1)}(z+(x-1))\Ga(x,z) = \lim_{z\to -(x-1)} \frac{1}{(x-1)\Ga(1,x-1)} = \frac{1}{(x-1)\Ga(x-1)}.
\]
Suppose now that $m\not= -1$.
Then repeatedly application of Proposition~\ref{functional}(c) yields
\[
\Ga(x,z) = \frac{1}{(x-1)(z+x-1)}\frac{1}{\Ga(z+x,-z)}
\]
\[
=\frac{x}{(z+x-1)(z+x)}\frac{1}{\Ga(z+x+1,-z+1)}
\]
\[
= \frac{(x)_{2m+1}}{(z+x-1)_{m+2}}\frac{1}{\Ga(z+x+m+1,-z+m+1)},
\]
or equivalently,
\[
(z+x+m) \Ga(x,z) = \frac{(x)_{2m+1}}{(z+x-1)_{m+1}}\frac{1}{\Ga(z+x+m+1,-z+m+1)}.
\]
Thus
\[
\lim_{z\to -(x+m)} (z+x+m) \Ga(x,z) = \frac{(x)_{2m+1}}{(-1)^{m+1} (m+1)!}\frac{1}{\Ga(1,x+2m+1)},
\]
which implies the desired result since $\Ga(1,x+2m+1) = \Ga(x+2m+1)$.
\end{proof}
Note that if $x=1$ and $m=-1,0,1,2,\ldots$, then Corollary~\ref{residues} agrees with the well-known fact
that the residue of $\Ga(z)$ at $z=-(m+1)$ is
\[
\frac{(-1)^{m+1}}{(m+1)!}.
\]
\section{An analogues of the Gauss' multiplication formula}
\begin{proposition} \label{constant-function}
If $x\in\C\setminus\Z_0^{-}$, then the function
\[
f(x,z) = \frac{n^{nz} \Ga(x,z)\Ga(x,z+\frac{1}{n})\ldots \Ga(x,z+\frac{n-1}{n})}{n \Ga(n(x-1)+1,nz)}
\]
is independent of $z$.
\end{proposition}
\begin{proof}
By Proposition~\ref{Euler-analogue} we have
\[
f(x,z)
=
\frac{ n^{nz} \prod_{k=0}^{n-1}\lim_{m\to\infty}
\frac{m^{z +\frac{k}{n}} (x)_{m-1}}{(z+ \frac{k}{n}+x-1)(z+\frac{k}{n}+x)\ldots(z+\frac{k}{n}+x+m-2)} }
{ n \lim_{m\to\infty} \frac{ (mn)^{nz} \bigl(n(x-1)\bigr)_{mn-1} }{ \bigl(nz+n(x-1)\bigr)_{mn} }}
\]
\[
=
\lim_{m\to\infty} \frac{n^{mn-1} m^{\frac{n-1}{2}} \bigl((x)_{m-1}\bigr)^{n-1}}{ (n(x-1))_{mn-1} }
\frac{ \bigl( n(z+x-1) \bigr)_{mn} }{ \prod_{k=0}^{n-1}\prod_{j=0}^{m-1}(n(z+x-1)+k+jn) }
\]
\[
=
\lim_{m\to\infty} \frac{n^{mn-1} m^{\frac{n-1}{2}} \bigl((x)_{m-1}\bigr)^{n-1}}{ (n(x-1))_{mn-1} }
\]
where the last identity follows as
\[
\frac{ \bigl( n(z+x-1) \bigr)_{mn} }{ \prod_{k=0}^{n-1}\prod_{j=0}^{m-1}(n(z+x-1)+k+jn) } =1.
\]
This completes the proof.
\end{proof}
\begin{corollary} \label{Gauss-analogue}
We have
\[
\Ga(x,z)\Ga(x,z+\frac{1}{2})\Ga(1-x,z)\Ga(1-x,z+\frac{1}{2})
\]
\[
= 2^{2-4z}\Ga(2x-1,2z)\Ga(1-2x,2z)\frac{\tan \pi x}{x-\frac{1}{2}}.
\]
\end{corollary}
\begin{proof}
Taking $z=\frac{1}{n}$ in Proposition~\ref{constant-function} we obtain
\[
f(x,z) f(1-x,z)
=
f(x,\frac{1}{n}) f(1-x,\frac{1}{n})
\]
\[
=
\frac{ \Ga(x,\frac{1}{n}) \Ga(x,\frac{2}{n})\ldots \Ga(x,\frac{n-1}{n}) }{ \Ga(n(x-1)+1,1) }
\frac{ \Ga(1-x,\frac{1}{n}) \Ga(1-x,\frac{2}{n})\ldots \Ga(1-x,\frac{n-1}{n}) }{ \Ga(-nx+1,1) }
\]
\[
=
\Ga(x,\frac{1}{n})\Ga(1-x,\frac{n-1}{n}) \Ga(x,\frac{2}{n})\Ga(1-x,\frac{n-1}{n}) \ldots
\Ga(x,\frac{n-1}{n}) \Ga(1-x,\frac{1}{n})
\]
\[
=
\frac{ (-1)^{n-1} \sin \pi x}{\prod_{k=1}^{n-1}\left((\frac{k}{n}-x)\sin \pi(\frac{k}{n}-x)\right)},
\]
where the last identity follows by Proposition~\ref{Gamma-sin}(a). Now if $n=2$, then Proposition~\ref{constant-function}
combined with the previous formula gives
\[
2^{4z-2} \frac{\Ga(x,z)\Ga(x,z+\frac{1}{2})\Ga(1-x,z)\Ga(1-x,z+\frac{1}{2})}
{\Ga(2x-1,2z) \Ga(1-2x,2z)} =
\frac{- \sin \pi x}{(\frac{1}{2}-x) \sin (\frac{\pi}{2}-\pi x)},
\]
or equivalently,
\[
\Ga(x,z)\Ga(x,z+\frac{1}{2})\Ga(1-x,z)\Ga(1-x,z+\frac{1}{2}) = 2^{2-4z} \Ga(2x-1,2z) \Ga(1-2x,2z) \frac{\tan \pi x}{x-\frac{1}{2}}.
\]
This completes the proof.
\end{proof}
\section{An analogue of the Stirling's formula}
In this section we essentially use the same ideas as in Lang~\cite[p. 422-427]{Lang} to derive a formula for
$\log \Ga(x,z)$ leading to asymptotic formulas for $\Ga(x,z)$ which are analogues to the Stirling's formula.
For $t\in\R$, let $P(t)= t-\lfloor t\rfloor -\frac{1}{2}$ and for convenience for $z\in \D$ let
\[
I_n(z) =\int_{0}^{n}\frac{P(t)}{z+t}\,dt, \quad\text{and\quad }
I(z) = \lim_{n\to\infty}I_n(z)=\int_{0}^{\infty}\frac{P(t)}{z+t}\,dt.
\]
\begin{proposition} \label{pre-stirling}
If $x\in\R^{+}$ and $z\in\R^{+}\cap S_x$, then
\[
\log\Ga(x,z) = (z+x-\frac{3}{2})\log(z+x-1) -z + 1- (x-\frac{1}{2})\log x + I(x)- I(z+x-1).
\]
\end{proposition}
\begin{proof}
We have with the help of Euler's summation formula
\[
\log \frac{(z+x-1)(z+x)\ldots (z+x+n-1)}{x(x+1)\ldots(x+n)} =
\sum_{k=0}^n \log(z+x-1+k)-\sum_{k=0}^n \log(x+k)
\]
\[
=
\int_{0}^{n}\log(z+x-1+t)\,dt +\frac{1}{2}\bigl( \log(z+x-1+n)+\log(z+x-1) \bigr) + I_n(z+x-1)
\]
\[
- \int_{0}^{n}\log(x+t)\,dt - \frac{1}{2}\bigl( \log(x+n)+\log x \bigr) - I_n(x)
\]
\[
=
\Big[(z+x-1+t)\log(z+x-1+t)-(z+x-1+t) \Big]_0^{n} - \Big[ (x+t)\log(z+x-1+t)-(x+t)\Big]_0^{n}
\]
\[
+ \frac{1}{2}\big( \log(z+x-1+n) + \log(z+x-1)\big) - {1\over 2}\big(\log(x+n)+\log(x)\big)
+ I_n(z+x-1) - I_n(x),
\]
which after routine calculations becomes
\[
\log \frac{(z+x-1)(z+x)\ldots (z+x+n-1)}{x(x+1)\ldots(x+n)} =
\log n^z + z \log\left(1+ \frac{z+x-1}{n}\right)
\]
\[
+ (x+n-\frac{3}{2})\log\left(1+\frac{z+x-1}{n}\right)
- (z+x-\frac{3}{2})\log(z+x-1) - (x+n+\frac{1}{2})\log\left(1+\frac{x}{n}\right)
\]
\[
+ \left( x-\frac{1}{2}\right) \log x - \log n + I_n(z+x-1) - I_n(x).
\]
Equivalently,
\[
\log \frac{(z+x-1)(z+x)\ldots (z+x+n-1)}{n^z x(x+1)\ldots(x+n-1)} =
\log(x+n) + z \log\left(1+ \frac{z+x-1}{n}\right)
\]
\[
+ (x+n-\frac{3}{2})\log\left(1+\frac{z+x-1}{n}\right)
- (z+x-\frac{3}{2})\log(z+x-1) - (x+n+\frac{1}{2})\log\left(1+\frac{x}{n}\right)
\]
\[
+ \left( x-\frac{1}{2}\right) \log x - \log n + I_n(z+x-1) - I_n(x)
\]
\[
=
z \log\left(1+ \frac{z+x-1}{n}\right) + (x+n-\frac{3}{2})\log\left(1+\frac{z+x-1}{n}\right)
- (z+x-\frac{3}{2})\log(z+x-1)
\]
\[
-(x+n-\frac{1}{2})\log\left(1+\frac{x}{n}\right) + \left( x-\frac{1}{2}\right) \log x + I_n(z+x-1) - I_n(x).
\]
Now use the fact that
\[ \log\left(1 + \frac{z}{n}\right) = \frac{z}{n} + O\left(\frac{1}{n^2}\right)
\]
and Proposition~\ref{Euler-analogue}(a) and take $\lim_{n\to\infty}$ on both sides to get
\[
\log \frac{1}{\Ga(x,z)} = (z+x-1) - \left(z+x-\frac3{2} \right)\log(z+x-1) - x + (x-\frac{1}{2})\log x + I(z+x-1)-I(x),
\]
implying the required identity.
\end{proof}
\begin{corollary}\label{Strling}
Let $x\in\R^{+}$ and $z\in\R^{+}\cap S_x$. Then
(a)\ for $x\to\infty$ we have
\[
\Ga(x,z)\sim (z+x-1)^{z+x-3/2} e^{1-z} x^{1/2 - x},
\]
(b)\ for $z\to\infty$ we have
\[
\Ga(x,z) \sim (z+x-1)^{z+x-3/2} e^{1-z} x^{1/2 - x} + I(x).
\]
\end{corollary}
\begin{proof}
Combine Proposition~\ref{pre-stirling} with the fact that
$\lim_{z\to \infty} I(z) = 0$.
\end{proof}
\section{Series expansions and recursive formulas for the coefficients}
To use the property $\log (z_1 z_2)=\log z_1 + \log z_2$, we suppose in this section that $x\in\R^{+}$ and $z\in\R^{+}\cap S_x$.
\begin{proposition} \label{series-1}
If $|z|<\inf (1,|x|)$, then
\[
\log \Ga(x,z+1) = -z \ga(x) - \sum_{m=2}^{\infty} \frac{(-1)^{m-1}}{m} \zeta(m,x) z^{m}.
\]
\end{proposition}
\begin{proof}
On the one hand, we have by definition
\[
\log\Ga(x,z) = -\log(z+x-1) - z\ga(x) - \su\left(\log(1+\frac{z}{x+n}) - \frac{z}{x+n} \right).
\]
On the other hand, by Proposition~\ref{functional}(a) we have
\[
\log\Ga(x,z+1) = -\log(z+x-1) + \log\Ga(x,z).
\]
Combining these two relations we obtain
\[
\begin{split}
\log\Ga(x,z+1)
&=
- z\ga(x) - \su\left(\log(1+\frac{z}{x+n}) - \frac{z}{x+n} \right) \\
&=
- z\ga(x) - \su \left(\left(\sum_{m=1}^{\infty}\frac{-(1)^{m-1}}{m}\frac{z^m}{(x+n)^m} \right)-\frac{z}{x+n}\right) \\
&=
- z\ga(x) - \sum_{m=2}^{\infty}\frac{(-1)^{m-1}}{m} z^m \su\frac{1}{(x+n)^m} \\
&=
- z\ga(x) - \sum_{m=2}^{\infty}\frac{(-1)^{m-1}}{m} \zeta(m,x) z^m.
\end{split}
\]
\end{proof}
\begin{corollary}\label{cor-series-1}
If $|z|<\inf(1,|x|)$ and
\[
\Ga(x,z+1) = \sum_{m=0}^{\infty}a_m(x) z^m,
\]
then $a_0(x) = 1$ and for $m>0$ we have
\[
a_m(x) = \frac{1}{m}\left(-a_{m-1}(x) \ga(x) + \sum_{k=0}^{m-2}(-1)^m a_k(x)\zeta(m-k,x)\right).
\]
\end{corollary}
\begin{proof}
Clearly if $z=0$, then $\Ga(x,1)=a_0(x)=1$ by Corollary~\ref{Gamma-integers}(a).
Differentiating the power series with respect to $z$ gives
\begin{equation}\label{deriv1}
\frac{d}{dz}\Ga(x,z+1) = \sum_{m=1}^{\infty} ma_m(x) z^{m-1}.
\end{equation}
Further in Proposition~\ref{series-1} differentiating with respect to $z$ yields
\begin{equation}\label{deriv2}
\frac{d}{dz}\log \Ga(x,z+1) = \frac{\frac{d}{dz}\Ga(x,z+1)}{\Ga(x,z+1)}=
-\ga(x) - \sum_{m=2}^{\infty}(-1)^{m-1}\zeta(m,x) z^{m-1}.
\end{equation}
Next combining (\ref{deriv1}) with (\ref{deriv2}) gives
\[
\sum_{m=1}^{\infty}m a_m(x) z^{m-1} = \left(\sum_{m=0}^{\infty} a_m(x) z^m\right)
\left(-\ga(x)+ \sum_{m=2}^{\infty}(-1)^m \zeta(m,x) z^{m-1} \right).
\]
Now the desired identity follows by equating the coefficients.
\end{proof}
\begin{proposition}\label{series-2}
If $|x-1|<\inf(1,|z+1|)$, then
\[
\log\Ga(x+1,z) = \sum_{n=1}^{\infty}\left(z\log\frac{n+1}{n}-\log\frac{z+n}{n}\right)
\]
\[
+ \sum_{n=2}^{\infty}\frac{z(x-1)}{n(n+z)}+
\sum_{m=2}^{\infty} \frac{(-1)^{m}\big(\zeta(m,z)-\zeta(m)-\frac{1}{z^m}+1\big)}{m} (x-1)^{m}.
\]
\end{proposition}
\begin{proof}
Note that it is easily checked that
\begin{equation}\label{help-gamma}
-z \ga(x)+ \sum_{n=2}^{\infty}\left(\frac{z}{x+n-1}-\frac{n+z}{n}\right)
\end{equation}
\[
=
-\frac{z}{x}+\log(1+z) + \sum_{n=1}^{\infty} \left(z\log\frac{n+1}{n}-\log\frac{n+z}{n}\right).
\]
Now combining the definition of $\Ga(x,z)$ with Proposition~\ref{functional}(b) yields
\[
\log\Ga(x+1,z)
=
-\log x - z\ga(x)
\]
\[- \su\left(\log(x+n+z)-\log(x+n)-\frac{z}{x+n}\right)
\]
\[
= -z \ga(x) - \log(x+z) + \frac{z}{x}
\]
\[
-\sum_{n=2}^{\infty}\left(\log(x-1+n+z)-\log(x-1+n)-\frac{z}{x-1+n}\right)
\]
\[
= -z \ga(x) - \log(x+z) + \frac{z}{x}
\]
\[ -\sum_{n=2}^{\infty}\left(\log(n+z)+\log(1+\frac{x-1}{n+z})
-\log n -\log(1+\frac{x-1}{n})-\frac{z}{x-1+n}\right)
\]
\[
= -z \ga(x)+\sum_{n=2}^{\infty}(\frac{z}{x-1+n}-\log \frac{n+z}{n}) -\log(x+z)+\frac{z}{x}
\]
\[
-\sum_{n=2}^{\infty}\left(\log(1+\frac{x-1}{n+z})-\log(1+\frac{x-1}{n}) \right)
\]
\[
= \log(1+z)-\log(x-1+z+1)+\sum_{n=1}^{\infty} \left(z\log\frac{n+1}{n}-\log\frac{n+z}{n}\right)
\]
\[
+ \sum_{n=2}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^{m-1}}{m}\left(\frac{1}{n^m}-\frac{1}{(n+z)^m}\right) (x-1)^m
\]
\[
= -\log(1+\frac{x-1}{z+1}) + \sum_{n=1}^{\infty} \left(z\log\frac{n+1}{n}-\log\frac{n+z}{n}\right)
+ \sum_{n=2}^{\infty}\frac{z(x-1)}{n(n+z)}
\]
\[
+ \sum_{m=2}^{\infty}\frac{(-1)^{m-1}}{m}\left( -1+ \zeta(m) +\frac{1}{z^m}+\frac{1}{(z+1)^m}-\zeta(m,z)\right) (x-1)^m
\]
\[
= \sum_{n=1}^{\infty} \left(z\log\frac{n+1}{n}-\log\frac{n+z}{n}\right)
\]
\[
+ \sum_{n=2}^{\infty}\frac{z(x-1)}{n(n+z)}
+ \sum_{m=2}^{\infty}\frac{(-1)^{m-1}}{m} (-1+\zeta(m) +\frac{1}{z^m}-\zeta(m,z)) (x-1)^m,
\]
where the fifth identity follows with the help of (\ref{help-gamma}).
This completes the proof.
\end{proof}
\begin{corollary}\label{coefficients}
If $|x-1|<\inf(1,|z+1|)$ and
\[
\Ga(x+1,z) = \sum_{m=0}^{\infty} b_m(z) (x-1)^m,
\]
then $b_0(z) = z \Ga(z)$ and for $m>0$
\[
b_m (z) = \frac{1}{m}b_{m-1}(z)+ \sum_{n=2}^{\infty}\frac{z}{n(n+z)}
\]
\[
+ \frac{1}{m}\sum_{k=0}^{m-2} (-1)^{m-k} b_k(z) \bigl( \zeta(m-k,z)-\zeta(m-k)- z^{-(m-k)}+1\bigr).
\]
\end{corollary}
\begin{proof}
Taking $x=1$, we have
$b_0(z) = \Ga(2,z) = z\Ga(z)$ by Corollary~\ref{Gamma-integers}(e). Further, by Proposition~\ref{series-2} we have
\begin{equation} \label{coeff-1}
\frac{d}{dx}\log \Ga(x+1,z) = \frac{ \frac{d}{dx}\Ga(x+1,z)}{\Ga(x+1,z)}
\end{equation}
\[
=
\sum_{n=2}^{\infty}\frac{z}{n(n+z)}+\sum_{m=2}^{\infty}(-1)^m \bigl( \zeta(m,z)-\zeta(m) - z^{-m}+1\bigr) (x-1)^{m-1}.
\]
On the other hand, it follows from the assumption that
\begin{equation}\label{coeff-2}
\frac{d}{dx} \Ga(x+1,z) = \sum_{m=1}^{\infty} m b_m(z) (x-1)^{m-1}.
\end{equation}
Then from (\ref{coeff-1}) and (\ref{coeff-2}) we get
\[
\sum_{m=1}^{\infty} m b_m(z) (x-1)^{m-1} =
\left(\sum_{m=0}^{\infty} b_m (x-1)^m\right)\times
\]
\[
\left(\sum_{n=2}^{\infty}\frac{z}{n(n+z)}+\sum_{m=2}^{\infty}(-1)^m \bigl(\zeta(m,z)-\zeta(m)- z^{-m}+1\bigr) (x-1)^{m-1}\right),
\]
and the result follows by equating the coefficients.
\end{proof}
\end{document} | math | 33,669 |
\begin{document}
\def\baselinestretch{1.0}\large\normalsize
\title{ f Logarithmic vector-valued modular forms and polynomial-growth estimates of their Fourier coefficients}
\begin{abstract}
\noindent
We establish (Theorem 3.6) polynomial-growth estimates for the
Fourier coefficients of holomorphic logarithmic vector-valued modular forms.
(MSC2010: 11F12, 11F99)
\end{abstract}
\section{\Large \bf Introduction}
The present work is a natural sequel to our earlier articles
on `normal' and `logarithmic' vector-valued modular forms \cite{KM1}, \cite{KM2}, \cite{KM3}. The component functions of a normal vector-valued modular form $F$ are ordinary left-finite $q$-series with real exponents. Equivalently,
the finite-dimensional representation $\rho$ associated with $F$ has the property that
$\rho(T)$ is (similar to) a matrix that is unitary and diagonal. Here,
$T = \left(\begin{array}{cc}1 & 1 \\0 & 1\end{array}\right)$.
\begin{itemize}gskip
In the case of a general representation, $\rho(T)$ is not necessarily diagonal but may always be assumed to be in Jordan canonical form\footnote{We actually use a modified Jordan canonical form. See \cite{KM3} for details.}. This circumstance leads to \emph{logarithmic}, or \emph{polynomial} $q$-expansions for the component functions of a vector-valued modular form associated to $\rho$ (see Subsection 2.2), which take the form
\begin{eqnarray}\label{logform1}
f(\tau) = \sum_{j=0}^t (\log q)^j h_j(\tau),
\end{eqnarray}
where the $h_j(\tau)$ are ordinary $q$-series.
There follow naturally the definition of logarithmic vector-valued modular form and the concomitant notions of logarithmic meromorphic, holomorphic
(i.e., entire in the sense of Hecke) and cuspidal vector-valued modular forms (Subsection 2.3).
\begin{itemize}gskip
In \cite{KM3} we derived a number of the properties of logarithmic vector-valued modular forms (LVVMF's)
by introducing appropriate Poincar\'{e} series. In \cite{KM2} we elaborated a well-known method of
Hecke \cite{H} devised to obtain polynomial-growth estimates of the Fourier coefficients of classical (i.e.\ scalar) holomorphic modular forms, and by this means we derived analogous estimates for the coefficients of \emph{normal}
VVMF's. The purpose of the present note is to extend Hecke's method even further to establish similar
polynomial-growth estimates for the coefficients of holomorphic (i.e.\ entire in the sense of Hecke),
including cuspidal, LVVMF's. Our extension of the method here entails the assumption that the eigenvalues of
$\rho(T)$ have absolute value $1$, so that the $q$-series $h_j(\tau)$ in (\ref{logform1}) again have real exponents,
a condition that will be assumed implicitly in the remainder of the article. It requires as well a simple new estimate
(Proposition \ref{prop3}) that we apply in \S 3.2. (This same estimate is an important ingredient in our proof of convergence
of the logarithmic Poincar\'{e} series introduced in \cite{KM3}.)
\begin{itemize}gskip
The occurrence of $q$-expansions of the form (\ref{logform1}) is well known in
rational and logarithmic conformal field theory. Indeed, much of the motivation for the present work originates from a need to develop a systematic theory of vector-valued modular forms wide enough in scope to cover possible applications in such field theories. By results in \cite{DLM} and \cite{M}, the eigenvalues of $\rho(T)$ for the representations that arise in rational and logarithmic conformal field theory are indeed of absolute value $1$ (in fact, they are roots of unity). Thus this assumption is natural from the perspective of conformal field theory. Our earlier results \cite{KM1} on polynomial estimates for Fourier coefficients of entire vector-valued modular forms in the normal case have found a number of applications to the theory of rational vertex operator algebras, and we expect that the extension to the logarithmic case that we prove here will be useful in the study of $C_2$-cofinite vertex operator algebras, which constitute the algebraic underpinning of logarithmic field theory.
\begin{itemize}gskip
Other properties of logarithmic vector-valued modular forms are also of interest, from both a foundational and applied perspective. These include
a Petersson pairing, generation of the space of cusp-forms by Poincar\'{e} series, existence of a natural boundary for the component functions, and explicit formulas (in terms of Bessel functions and Kloosterman sums) for the Fourier coefficients of Poincar\'{e} series. This program was carried through in the normal case in \cite{KM2}. It is evident that the more general logarithmic case will yield a similarly rich harvest, but one must expect more complications. For example, there are logarithmic vector-valued modular forms with nonconstant component functions that may be extended to the whole of the complex plane, so that the usual natural boundary result is false \emph{per se}. Such logarithmic vector-valued modular forms are studied (indeed, classified) in \cite{KM4}. Furthermore, our preliminary calculations indicate that the explicit formulas exhibit genuinely new features. We hope to return to these questions in the future.
\section{Logarithmic vector-valued modular forms}
For the sake of completeness and clarity we present here much of the introductory material on LVVMF's that
appears in \cite{KM3}.
\subsection{Unrestricted vector-valued modular forms}\label{UVVMF}
We start with some notation that will be used throughout.
The \emph{modular group} is
\begin{eqnarray*}
\Gamma = \left\{ \left(\begin{array}{cc}a&b \\c&d\end{array}\right)\ | \ a, b, c, d \in \mathbb{Z}, \ ad-bc=1 \right\}.
\end{eqnarray*}
It is generated by the matrices
\begin{eqnarray}\label{STgens}
S = \left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right), \ T = \left(\begin{array}{cc}1&1 \\0 & 1\end{array}\right).
\end{eqnarray}
The complex upper half-plane is
\begin{eqnarray*}
\frak{H} = \{ \tau \in \mathbb{C} \ | \ \Im(\tau) > 0 \}.
\end{eqnarray*}
There is a standard left action $\Gamma \times \frak{H} \rightarrow \frak{H}$ given by
M\"{o}bius transformations:
\begin{eqnarray*}
\left( \left(\begin{array}{cc}a&b \\c&d\end{array}\right), \tau \right) \mapsto \frac{a\tau+b}{c\tau + d}.
\end{eqnarray*}
Let $\frak{F}$ be the space of holomorphic functions in $\frak{H}$.
There is a standard $1$-cocycle $j: \Gamma \rightarrow \frak{F}$ defined by
\begin{eqnarray*}
j(\gamma, \tau) = j(\gamma)(\tau) = c\tau+d, \ \ \ \gamma = \left(\begin{array}{cc}a&b \\c&d\end{array}\right).
\end{eqnarray*}
$\rho: \Gamma \rightarrow GL(p, \mathbb{C})$ will always denote a $p$-dimensional matrix representation of $\Gamma.$
An \emph{unrestricted vector-valued modular form of weight $k$ with respect to $\rho$} is a
holomorphic function $F: \frak{H} \rightarrow\mathbb{C}^p$ satisfying
\begin{eqnarray*}
\rho(\gamma)F(\tau) = F|_k \gamma (\tau), \ \ \gamma \in \Gamma,
\end{eqnarray*}
where the right-hand-side is the usual stroke operator
\begin{eqnarray}\label{vvdef}
F|_k \gamma (\tau) = j(\gamma, \tau)^{-k}F(\gamma \tau).
\end{eqnarray}
We could take $F(\tau)$ to be \emph{meromorphic} in $\frak{H}$, but we will not consider that more general situation here. Choosing coordinates, we can rewrite (\ref{vvdef}) in the form
\begin{eqnarray}\label{ract}
\rho(\gamma)
\left(\begin{array}{c}f_1(\tau) \\ \vdots \\ f_p(\tau) \end{array}\right)
= \left(\begin{array}{c}f_1|_k \gamma (\tau) \\ \vdots \\ f_p|_k (\gamma) (\tau) \end{array}\right)
\end{eqnarray}
with each $f_j(\tau) \in \frak{F}$. We also refer to $(F, \rho)$ as an unrestricted vector-valued
modular form.
\subsection{Logarithmic $q$-expansions}\label{sectqexp}
In this Subsection we consider the $q$-expansions associated to unrestricted vector-valued
modular forms.
We make use of the polynomials defined for $k \geq 1$ by
\begin{eqnarray*}
{x \choose k} = \frac{x(x-1) \hdots (x -k+1)}{k!},
\end{eqnarray*}
and with ${x \choose 0} = 1$ and ${x \choose k} = 0$ for $k \leq -1$.
\begin{itemize}gskip
We consider a finite-dimensional subspace $W \subseteq \frak{F}_k$ that is \emph{invariant} under $T$,
i.e $f(\tau +1) \in W$ whenever $f(\tau) \in W$.
We introduce the $m \times m$ matrix
\begin{eqnarray}\label{Jblock}
J_{m, \lambda} = \left(\begin{array}{cccc} \lambda &&& \\ \lambda & \ddots && \\
&\ddots & \ddots & \\& & \lambda & \lambda \end{array}\right),
\end{eqnarray}
i.e. $J_{i, j}=\lambda$ for $i = j$ or $j+1$ and $J_{i, j}=0$ otherwise.
\begin{lemma} \label{lemmaJform}There is a basis of $W$ with respect to which the
matrix $\rho(T)$ representing
$T$ is in block diagonal form
\begin{eqnarray}\label{Jform}
\rho(T) = \left(\begin{array}{ccc}J_{m_1, \lambda_1} & & \\ & \ddots & \\ & & J_{m_t, \lambda_t}\end{array}\right).
\end{eqnarray}
\end{lemma}
\begin{pf} The existence of such a representation is basically the theory of the Jordan canonical form. The \emph{usual} Jordan canonical form is similar to the above, except that the subdiagonal of each block then consists of $1$'s rather than $\lambda$'s. The $\lambda$'s that appear in (\ref{Jform})
are the eigenvalues of
$\rho(T)$, and in particular they are nonzero on account of the invertibility of $\rho(T)$. Then it is easily checked
that (\ref{Jform}) is indeed similar to the usual Jordan canonical form, and the Lemma follows.
$
\Box$
\end{pf}
\begin{itemize}gskip
We refer to (\ref{Jform}) as the \emph{modified Jordan canonical form} of $\rho(T)$,
and $J_{m_i, \lambda_i}$ as a \emph{modified Jordan block}.
To a certain extent at least, Lemma \ref{lemmaJform} reduces the study of the functions in $W$ to
those associated to one of the Jordan blocks. In this case we have the following basic result.
\begin{thm}\label{thmlogqexp} Let $W \subseteq \frak{F}_k$ be a $T$-invariant
subspace of dimension $m$.
Suppose that $W$ has an ordered basis $(g_{0}(\tau), \hdots, g_{m-1}(\tau))$
with respect to which the matrix $\rho(T)$ is a single modified Jordan block $J_{m, \lambda}$. Set $\lambda = e^{2\pi i \mu}$. Then
there are $m$ convergent $q$-expansions $h_t(\tau) = \sum_{n \in \mathbb{Z}} a_t(n)q^{n+\mu}, 0 \leq t \leq m-1,$ such that
\begin{eqnarray}\label{polyform}
g_j(\tau) = \sum_{t=0}^j {\tau \choose t}h_{j-t}(\tau), \ 0 \leq j \leq m-1.
\end{eqnarray}
\end{thm}
The case $m=1$ of the Theorem is well known. We will need it for the proof of the general case, so we state it as
\begin{lemma}\label{lemmaqexp} Let $\lambda = e^{2 \pi i \mu}$, and suppose that $f(\tau) \in \frak{F}$ satisfies $f(\tau +1) = \lambda f(\tau)$. Then $f(\tau)$
is represented by a convergent $q$-expansion
\begin{eqnarray}\label{qexp}
f(\tau) = \sum_{n \in \mathbb{Z}} a(n)q^{n+\mu}.
\end{eqnarray}
$
\Box$
\end{lemma}
Turning to the proof of the Theorem, we have
\begin{eqnarray}\label{gjrecur}
g_j(\tau +1) &=& \lambda(g_j(\tau) + g_{j-1}(\tau)), \ 0 \leq j \leq m-1,
\end{eqnarray}
where $g_{-1}(\tau) = 0$. Set
\begin{eqnarray*}
&&h_j(\tau) = \sum_{t=0}^j (-1)^t {\tau + t -1 \choose t}g_{j-t}(\tau), \ 0 \leq j \leq m-1.
\end{eqnarray*}
These equalities can be displayed as a system of equations. Indeed,
\begin{eqnarray}\label{SOE}
B_m(\tau) \left(\begin{array}{c}g_{0}(\tau) \\ \vdots \\ g_{m-1}(\tau)\end{array}\right) = \left(\begin{array}{c}h_{0}(\tau) \\ \vdots \\ h_{m-1}(\tau)\end{array}\right),
\end{eqnarray}
where $B_m(x)$ is the $m \times m$ lower triangular matrix with
\begin{eqnarray}\label{Bdef}
B_m(x)_{ij} = (-1)^{i-j}{x+i-j-1 \choose i-j} .
\end{eqnarray}
Then $B_m(x)$ is invertible and
\begin{eqnarray}\label{Bdef-1}
B_m(x)^{-1}_{ij} = {x \choose i-j}.
\end{eqnarray}
We will show that each $h_j(\tau)$ has a convergent $q$-expansion. This being the case,
(\ref{polyform}) holds and the Theorem will be proved.
Using (\ref{gjrecur}), we have
\begin{eqnarray*}
&&h_j(\tau +1) = \lambda \sum_{t=0}^j (-1)^t
{\tau + t \choose t}(g_{j-t}(\tau) + g_{j-t-1}(\tau)) \\
&=&\lambda \left\{ \sum_{t=0}^{j} (-1)^t
\left(1+ \frac{t}{\tau} \right){\tau + t -1 \choose t}g_{j-t}(\tau)
+ \sum_{t=0}^{j} (-1)^t
{\tau + t \choose t} g_{j-t-1}(\tau) \right\} \\
&=&\lambda \left\{ h_j(\tau) + \sum_{t=0}^{j} (-1)^t
{\tau + t -1 \choose t} \frac{t}{\tau} g_{j-t}(\tau) + \sum_{t=0}^{j} (-1)^t
{\tau + t \choose t} g_{j-t-1}(\tau) \right\}.
\end{eqnarray*}
But the sum of the second and third terms in the braces vanishes, being equal to
\begin{eqnarray*}
&& \sum_{t=1}^{j} (-1)^t
{\tau + t -1 \choose t} \frac{t}{\tau} g_{j-t}(\tau) + \sum_{t=1}^{j} (-1)^{t-1}
{\tau + t -1 \choose t -1} g_{j-t}(\tau) \\
&=& \sum_{t=1}^{j} (-1)^{t-1}
g_{j-t}(\tau) \left\{ {\tau +t-1 \choose t-1 } - {\tau +t-1 \choose t }\frac{t}{\tau} \right\} = 0.
\end{eqnarray*}
Thus we have established the identity $h_j(\tau +1) = \lambda h_j(\tau)$.
By Lemma \ref{lemmaqexp}, $h_j(\tau)$ is indeed represented by a $q$-expansion
of the desired shape, and the proof of Theorem \ref{thmlogqexp} is complete. $
\Box$
\begin{itemize}gskip
We call (\ref{polyform}) a \emph{polynomial}
$q$-expansion. The space of polynomials
spanned by ${x \choose t}, 0 \leq t \leq m-1$ is also spanned by the powers $x^t, 0 \leq t \leq m-1$.
Bearing in mind that $(2\pi i\tau)^t = (\log q)^t$, it follows that in Theorem \ref{thmlogqexp} we can find a basis $\{g'_j(\tau)\}$ of $W$ such that
\begin{eqnarray}\label{logform}
g'_j(\tau) = \sum_{t=0}^j (\log q)^t h'_{j-t}(\tau)
\end{eqnarray}
with $h'_t(\tau) = \sum_{n \in \mathbb{Z}} a'_t(n)q^{n+\mu}$.
We refer to (\ref{logform}) as a
\emph{logarithmic} $q$-expansion.
\subsection{Logarithmic vector-valued modular forms}
We say that a function $f(\tau)$ with a $q$-expansion (\ref{qexp}) is \emph{meromorphic at infinity}
if
\begin{eqnarray*}
f(\tau) = \sum_{n +\Re(\mu) \geq n_0} a(n)q^{n+\mu}.
\end{eqnarray*}
That is, the Fourier coefficients $a(n)$ \emph{vanish} for exponents $n+\mu$ whose \emph{real parts}
are small enough. A polynomial (or logarithmic) $q$-expansion
(\ref{polyform}) is holomorphic at infinity if each of the associated ordinary $q$-expansions
$h_{j-t}(\tau)$ are holomorphic at infinity. Similarly, $f(\tau)$ \emph{vanishes} at $\infty$ if the Fourier coefficients $a(n)$ vanish for $n+ \Re(\mu)\leq 0$; a polynomial $q$-expansion vanishes at $\infty$ if the associated ordinary $q$-expansions vanish at $\infty$. These conditions are independent of the chosen representations.
Now assume that $F(\tau) = (f_1(\tau), \hdots, f_p(\tau))^t$ is an unrestricted vector-valued modular form of weight $k$ with respect to $\rho$. It follows from (\ref{ract}) that the span $W$ of the functions $f_j(\tau)$ is a right $\Gamma$-submodule of $\frak{F}$ satisfying $f_j(\tau +1) \in W$. Choose a basis of $W$ so that $\rho(T)$ is in modified Jordan canonical form. By
Theorem \ref{thmlogqexp} the basis of $W$ consists of functions $g_j(\tau)$ which have
polynomial $q$-expansions. We call $F(\tau)$, or $(F, \rho)$, a \emph{logarithmic meromorphic, holomorphic, or cuspidal vector-valued modular form} respectively
if each of the functions $g_j(\tau)$ is meromorphic, is holomorphic, or vanishes at $\infty$, respectively.
We let $\mathcal{H}(k, \rho)$ denote the holomorphic LVVMFs
of weight $k$ with respect to $\rho$. It is finite-dimensional complex vector space (\cite{KM3}).
\section{Polynomial-growth estimate of the Fourier coefficients}
\subsection{The new estimate}\label{subsecconverge}
We state a modification and elaboration of (\cite{E}, p. 169, displays (3)-(5)) which we call \emph{Eichler's canonical representation} for elements of $\Gamma$:
\begin{lemma}\label{lemmaEprod} Let $\gamma \in \Gamma, \gamma =
\left(\begin{array}{cc}a&b \\ c&d\end{array}\right)$ with $c\not= 0$. We may assume
without loss of generality that $c>0$. Then
\begin{eqnarray}\label{Eprod}
(a)&&\mathbbox{$\gamma$ has a unique representation} \notag\\
&&\hspace{4cm}\mathbbox{ $\gamma = (ST^{l_{\nu+1}}) \hdots (ST^{l_1})(ST^{l_0})$}\\
&&\mathbbox{where $(-1)^{j-1}l_j>0$ for $1\leq j \leq \nu$ and $(-1)^{\nu}l_{\nu+1}\geq 0$.
Thus $l_1$ is positive,} \notag \\
&&\mathbbox{the $l_j$ alternate in sign for $j \geq 1$ (with the proviso that $l_{\nu+1}$ may be zero)} \notag \\
&&\mathbbox{and there is no condition on $l_0$.} \notag \\
(b)&&\mathbbox{$l_{\nu+1}\not= 0$ if, and only if, $|a/c|<1$; in the opposite case $|a/c|\geq 1$ (whence} \notag\\
&&\mathbbox{$l_{\nu+1}=0$), we have $l_{\nu} = \pm [|a/c|]$.} \notag
\end{eqnarray}
$
\Box$
\end{lemma}
\begin{rem}1. Eichler does not state (\ref{Eprod}) precisely as we have here, but his result
is the same. The proof, omitted in \cite{E}, entails repeated application of the division algorithm
in $\mathbb{Z}$. \\
2. \cite{E} makes no mention of part (b). However, that it holds is clear from the proof
mentioned in Remark 1.
\end{rem}
Now, let $\gamma \in \Gamma$ be fixed as in Lemma \ref{lemmaEprod}, with canonical
representation (\ref{Eprod}). We set
\begin{eqnarray}\label{Pdef}
P_0 &=& ST^{l_0}, \notag\\
P_{j+1} &=& (ST^{l_{j+1}})P_{j}, \ 0 \leq j \leq \nu,\notag \\
P_j &=& \left(\begin{array}{cc}a_j & b_j \\ c_j & d_j\end{array}\right), \ 0 \leq j \leq \nu+1.
\end{eqnarray}
\begin{propn}\label{prop3}
(a)\ \mathbbox{Assume $l_{\nu+1}\not= 0$. Then we have}
\begin{eqnarray}
&&|l_0l_1 \hdots l_{\nu+1}| \leq |d| \ \ \ \ \ \ \ \mathbbox{if} \ l_0 < 0; \notag \\
&&\ \ |l_1 \hdots l_{\nu+1}| \leq |d-c| \ \ \mathbbox{if} \ l_0 = 0; \label{lesta}\\
&&|l_0l_1 \hdots l_{\nu+1}| \leq |c|+ |d| \ \mathbbox{if} \ l_0 > 0. \notag
\end{eqnarray}
(b) \ \mathbbox{If $l_{\nu+1}=0$, then}
\begin{eqnarray}
&&|l_0l_1 \hdots l_{\nu-1}| \leq |d| \ \ \ \ \ \ \ \mathbbox{if} \ l_0 < 0; \notag \\
&&\ \ |l_1 \hdots l_{\nu-1}| \leq |d-c| \ \ \mathbbox{if} \ l_0 = 0; \label{lestb} \\
&&|l_0l_1 \hdots l_{\nu-1}| \leq |c|+ |d| \ \mathbbox{if} \ l_0 > 0. \notag
\end{eqnarray}
\end{propn}
\begin{pf} (a). Assume $l_0 < 0$. We will prove by induction on $j \geq 0$ that
\begin{eqnarray}\label{indonj}
&&\ \ (i) \ |l_0l_1\hdots l_j| \leq |d_j|, \\
&&\ (ii) \ (-1)^jb_jd_j \geq 0. \notag
\end{eqnarray}
Once this is established, the case $j=\nu+1$ of (\ref{indonj})(i) proves (\ref{lesta}) in this case.
Now
\begin{eqnarray*}
P_0 = \left(\begin{array}{cc}0 & -1 \\ 1 & l_0\end{array}\right),
\end{eqnarray*}
and the case $j=0$ is clear. For the inductive step, we have
\begin{eqnarray}\label{Pj+1}
P_{j+1} = \left(\begin{array}{cc}0 & -1 \\ 1 & l_{j+1} \end{array}\right)
\left(\begin{array}{cc}a_j & b_j \\ c_j& d_j\end{array}\right)
= \left(\begin{array}{cc} -c_j & -d_j \\ a_j+l_{j+1}c_j& b_j+l_{j+1}d_j\end{array}\right).
\end{eqnarray}
Thus $(-1)^{j+1}b_{j+1}d_{j+1} = (-1)^jb_jd_j+(-1)^jl_{j+1}d_j^2 \geq 0$ where the last inequality uses induction and the inequality stated in Lemma \ref{lemmaEprod}. So (\ref{indonj})(ii) holds.
As for (\ref{indonj})(i), note that because $(-1)^jb_jd_j$ and $(-1)^jl_{j+1}d_j^2 $ are both nonnegative then $b_j$ and $l_{j+1}d_j$ have the \emph{same sign}. Therefore using induction again, we have $|l_0l_1 \hdots l_{j+1}|\leq |d_jl_{j+1}|\leq|b_j|+|l_{j+1}d_j|
=|b_j+l_{j+1}d_j| = |d_{j+1}|.$ This completes the proof in the case $l_0<0$.
Assume $l_0=0$. Notice that
\begin{eqnarray*}
\gamma T^{-1}=(ST^{l_{\nu+1}})\hdots(ST^{l_1})(ST^{-1})
\end{eqnarray*}
is an instance of the first case, with $l_0=-1$. Since
\begin{eqnarray*}
\gamma T^{-1} = \left(\begin{array}{cc}a&b \\c&d\end{array}\right)\left(\begin{array}{rr}1&-1 \\0 & 1\end{array}\right) = \left(\begin{array}{cc} a&b-a \\ c&d-c\end{array}\right),
\end{eqnarray*}
it follows from the case $l_0<0$ that $|l_1\hdots l_{\nu+1}|\leq |d-c|$, as was to be
proved.
Now suppose $l_0>0$. We will prove by induction on $j$ that
\begin{eqnarray}\label{indonj1}
&&\ \ (i) \ |l_0l_1\hdots l_j| \leq |c_j|+|d_j|, \ j \geq 0 \\
&&\ (ii) \ (-1)^jb_jd_j, (-1)^ja_jc_j \geq 0, \ j \geq 1. \notag
\end{eqnarray}
Once again, the case $j=\nu+1$ of (\ref{indonj1})(i) proves the third case of (\ref{lesta}).
Now
\begin{eqnarray*}
P_0 = \left(\begin{array}{cc}0 & -1 \\ 1 & l_0\end{array}\right),
P_1 = \left(\begin{array}{cc} -1 & -l_0 \\ l_1& l_0l_1-1\end{array}\right).
\end{eqnarray*}
So when $j=0$, (\ref{indonj1})(i) is clearly true, and because $l_0, l_1>0$ we also have
\begin{eqnarray*}
-a_1c_1= l_1> 0, \ -b_1d_1= l_0(l_0l_1-1) \geq 0.
\end{eqnarray*}
So (\ref{indonj1})(ii) holds for $j=1$. As for the inductive step, $P_{j+1}$ is given by
(\ref{Pj+1}), and the proof that $(-1)^jb_jd_j\geq 0$ is the same as in the case $l_0<0$. Similarly,
$(-1)^{j+1}a_{j+1}c_{j+1} = (-1)^jc_ja_j+(-1)^{j}l_{j+1}c_j^2 \geq 0$ is the sum of two nonnegative terms and hence is itself nonnegative, so (\ref{indonj1})(ii) holds. Finally, by an argument similar to that used when $l_0<0$, we have
$|l_0 \hdots l_{j+1}| \leq |c_j+d_j||l_{j+1}| < |l_{j+1}c_j|+|l_{j+1}d_j|+|a_j|+|b_j| =|a_j+ l_{j+1}c_j|
+|b_j+ l_{j+1}d_j|= |c_{j+1}|+|d_{j+1}|.$
Part (a) of the Proposition is proved.
\begin{itemize}gskip
(b). When $l_{\nu+1}=0$, note that
\begin{eqnarray}\label{moregamma}
\gamma = -T^{l_{\nu}}(ST^{l_{\nu-1}})\hdots (ST^{l_1})(ST^{l_0})
\end{eqnarray}
and $l_{\nu}\not=0$, so that the argument of part (a) applies to $-T^{-l_{\nu}}\gamma$ rather than
to $\gamma$ itself. Since
\begin{eqnarray*}
-T^{-l_{\nu}}\gamma = \left(\begin{array}{cc}*&* \\ -c&-d\end{array}\right),
\end{eqnarray*}
we obtain the inequalities (\ref{lestb}). This completes the proof of the Proposition.
$
\Box$
\end{pf}
\begin{itemize}gskip
The \emph{Eichler length}
of $\gamma$ with canonical representation (\ref{Eprod}) is given by
\begin{eqnarray}\label{Ldef}
L(\gamma) = \left \{ \begin{array}{rl}
2\nu+4, & \ l_0l_{\nu+1}\not= 0, \\
2\nu+3, & \ l_0 = 0, l_{\nu+1}\not=0, \\
2\nu+1, & l_0\not=0, l_{\nu+1}=0, \\
2\nu, & l_0=l_{\nu+1}=0. \end{array} \right.
\end{eqnarray}
(See (\ref{moregamma}) above.)
By Lam\'{e}'s Theorem we have the estimate
\begin{eqnarray}\label{Lameest}
L(\gamma) \leq K(\log |c| +1)
\end{eqnarray}
with a positive constant $K$ independent of $\gamma$. (Cf.\ \cite{E}, p.179.)
\subsection{The matrix norm}\label{submatrixnorm}
The \emph{norm} $||\rho(\gamma)||$, defined to be
\begin{eqnarray*}
\max_{i, j} |\rho(\gamma)_{i j}|
\end{eqnarray*}
satisfies
the multiplicative condition
\begin{eqnarray}\label{mult}
||\rho(\gamma_1 \gamma_2)|| \leq p||\rho(\gamma_1)||||\rho(\gamma_2)||\ \ (\gamma_1, \gamma_2 \in \Gamma),
\end{eqnarray}
where $p = \dim \rho$. Let $\gamma \in \Gamma$ be expressed in the canonical form (\ref{Eprod}). Again there are two cases to consider, according as $l_{\nu+1}\not=0$ or $l_{\nu+1}=0$.
If
$l_{\nu+1} \not= 0$, then by (\ref{mult}),
\begin{eqnarray}\label{calc1}
||\rho(\gamma)|| \leq p^{2\nu+2}||\rho(S)||^{\nu+2}\prod_{j=0}^{\nu+1} ||\rho(T^{l_j}||.
\end{eqnarray}
If $l_{\nu+1}=0$, then (\ref{Eprod}) reduces to (\ref{moregamma}), so that (\ref{mult}) implies
\begin{eqnarray*}
||\rho(\gamma)|| \leq p^{2\nu+1}||\rho(S)||^{\nu}\prod_{j=0}^{\nu} || \rho(T^{l_j})||.
\end{eqnarray*}
Since $\rho(T^{l_{\nu+1}})=\rho(I)=I$ in this case, we obtain
the upper estimate
\begin{eqnarray}\label{ymest}
||\rho(\gamma)|| \leq Kp^{2\nu+1}||\rho(S)||^{\nu+1}\prod_{j=0}^{\nu+1} || \rho(T^{l_j})||
\end{eqnarray}
in both cases. In (\ref{ymest}), $K$ is a constant depending only on $\rho$.
\begin{lemma}\label{lemmaTlest} Let $s$ be the maximum of the sizes $m_j$ of the Jordan blocks
$J_{m_j, \lambda_j}$ of $\rho(T)$ (\ref{Jblock}), (\ref{Jform}). There is a constant $C_s$ depending only on $s$ such that for $l\not= 0$,
\begin{eqnarray}\label{Tlest}
||\rho(T^l)|| \leq C_s|l|^{s-1}.
\end{eqnarray}
\end{lemma}
\begin{pf} We have
\begin{eqnarray*}
J_{m, \lambda}^l = \lambda^lJ_{m, 1}^l = \lambda^l(I_m + N)^l = \lambda^l \sum_{i\geq 0} {l \choose i}N^i
\end{eqnarray*}
where $N$ is the nilpotent $m \times m$ matrix with each $(i, i-1)$-entry equal to $1 \ (i \geq 2),$
and all other entries zero. Note that $N^m = 0$ and the entries of $N^i$ for $1 \leq i<m$ are
$1$ on the $i$th. subdiagonal and zero elsewhere. Bearing in mind that $|\lambda| = 1$, we see that
$||J_{m, \lambda}^l||$ is majorized by the maximum of the binomial coefficients
${l \choose i}$ over the range $0 \leq i \leq m-1$. Since ${l \choose i}$ is a polynomial in $l$ of degree
$i$ then we certainly have $||J_{m, \lambda}^l|| \leq C_m|l|^{m-1}$ for a universal constant $C_m$, and since this applies to each Jordan block of $\rho(T^l)$ then the Lemma follows immediately.
$
\Box$
\end{pf}
\begin{cor}\label{lemmapolyest} There are universal constants $K_3, K_4$ such that
\begin{eqnarray}\label{polyest}
&&||\rho(\gamma)|| \leq \left \{ \begin{array}{ll}
K_3(c^2+d^2)^{K_4}, & l_{\nu+1}\not= 0, \\
K_3(c^2+d^2)^{K_4}|l_{\nu}|^{s-1}, & l_{\nu+1}= 0.
\end{array} \right.
\end{eqnarray}
Moreover the \emph{same} estimates hold for $||\rho(\gamma^{-1})||$.
\end{cor}
\begin{pf} First assume that $l_{\nu+1}\not= 0$.
From Lemma \ref{lemmaTlest} and (\ref{ymest}) we obtain
\begin{eqnarray}\label{corest}
||\rho(\gamma)|| \leq \left \{ \begin{array}{ll}
K_1^{\nu+1}\prod_{j=0}^{\nu+1} |l_j|^{s-1}, & l_0 \not= 0, \\
K_1^{\nu+1}\prod_{j=1}^{\nu+1} |l_j|^{s-1}, & l_0=0,
\end{array} \right. \notag
\end{eqnarray}
for a constant $K_1$ depending only on $\rho$. Now use (\ref{Ldef}), (\ref{Lameest})
and Proposition \ref{prop3}(a) to see that
\begin{eqnarray*}
||\rho(\gamma)|| \leq e^{(\log K_1)K_2 \log (|c|+1)}(|c|+|d|)^{s-1} \leq K_3(c^2+d^2)^{K_4}.
\end{eqnarray*}
Concerning the second assertion of the Corollary, since
\begin{eqnarray*}
\gamma^{-1} = (T^{-l_0}S)(T^{-l_1}S) \hdots (T^{-l_{\nu+1}}S)
\end{eqnarray*}
we have (\ref{ymest}) again, but with $T^{l_j}$ replaced by $T^{-l_j}$. The rest of the proof
is identical to the proof of the estimate of $||\rho(\gamma)||$, so that we indeed
obtain estimate (\ref{polyest}) for $\gamma^{-1}$ as well as $\gamma$.
\begin{eqnarray*}
||\rho(\gamma^{-1})|| \leq ||\rho(S)||^{\nu+2}\prod_{j=0}^{\nu+1} ||\rho(T)^{-l_j}||,
\end{eqnarray*}
and (\ref{Tlest}) then holds by Lemma \ref{lemmaTlest}. The rest of the proof is identical to the previous case, so that we indeed obtain the estimate (\ref{polyest}) for $\gamma^{-1}$ as well as $\gamma$.
The second case, in which $l_{\nu+1}=0$, is analogous. In this case we apply Proposition
\ref{prop3}(b) in place of Proposition \ref{prop3}(a).
$
\Box$
\end{pf}
\subsection{Application to the Fourier coefficients}
Let $F(\tau) \in \mathcal{H}(k, \rho)$ be a logarithmic vector-valued modular form
of weight $k$. We are going to show that the Fourier coefficients of $F(\tau)$
satisfy a polynomial growth condition for $n \rightarrow \infty$.
Let $F(\tau) = (f_1(\tau), \hdots, f_p(\tau))^t$ with
\begin{eqnarray*}
f_l(\tau) = \sum_{u=0}^{l} {\tau \choose u}h_{l-u}(\tau), \ \ 0 \leq l \leq m_j-1.
\end{eqnarray*}
Here, we have relabelled the components in the $j$th.\ block for notational convenience.
The proof is similar to the case treated in \cite{KM1}, but with an additional
complication due to the fact that we are dealing with polynomial $q$-expansions
rather than ordinary $q$-expansions. To deal with this we make use of the estimates that we have obtained in Subsection \ref{subsecconverge}. We continue to assume that
the eigenvalues of $\rho(T)$ are of absolute value $1$. We will sometimes drop
the subscript $j$ from the notation when it is convenient.
We write $\tau = x+iy$ for $\tau \in \frak{H}$ and let $\frak{R}$ be the usual fundamental region
for $\Gamma$. Write $z=u+iv$ for $z \in \overline{\frak{R}}$. Choose a real number
$\sigma>0$ to be fixed later, and set
\begin{eqnarray*}
g_l(\tau) = y^{\sigma}|f_l(\tau)|.
\end{eqnarray*}
Because $F(\tau)$ is holomorphic, $a_l(n)=0$ unless $n+\mu\geq 0$. It follows that
there is a constant $K_1$ such that
\begin{eqnarray}\label{glzest}
g_l(z) \leq K_1v^{\delta \sigma}, \ \ 1 \leq l \leq p, \ z \in \overline{\frak{R}},
\end{eqnarray}
where $\delta=0$ if $F(\tau)$ is a \emph{cusp-form}, and is $1$ otherwise.
Choose $\gamma = \left(\begin{array}{cc}a&b \\ c&d\end{array}\right) \in \Gamma$,
set $\tau = \gamma z\ (z \in \overline{\frak{R}})$, and write $\gamma$ in Eichler
canonical form (\ref{Eprod}). We wish to argue just as in \cite{KM1}, pp.121-122, and
to do this we need to make use of Proposition \ref{prop3}, a feature of the
proof not required in the normal case (loc.\ cit.).
We have, for $\tau \in \frak{H}$ and $\gamma, z$ as above,
\begin{eqnarray*}
g_l(\tau) &=& g_l(\gamma z) = (v|cz+d|^{-2})^{\sigma}|f_l(\gamma z)| \\
&=& v^{\sigma}|cz+d|^{k-2\sigma}|(f_l|_k \gamma)(z)| \\
&=& \mathbbox{$l^{th}$ component of}\ v^{\sigma}|cz+d|^{k-2\sigma}|\rho(\gamma)F(z)| \\
&=& v^{\sigma}|cz+d|^{k-2\sigma}\left|\sum_{m=1}^p \rho(\gamma)_{lm}f_m(z) \right| \\
&=& |cz+d|^{k-2\sigma}\left|\sum_{m=1}^p \rho(\gamma)_{lm}g_m(z) \right|.
\end{eqnarray*}
This then implies (by the triangle inequality)
\begin{eqnarray*}
g_l(\tau) \leq |cz+d|^{k-2\sigma}\sum_{m=1}^p |\rho(\gamma)_{lm}|g_m(z).
\end{eqnarray*}
Since $z \in \overline{\frak{R}}$, we also know (\cite{KM1}, display (13)) that
\begin{eqnarray}\label{ymestim}
c^2+d^2 \leq K_6|cz+d|^2
\end{eqnarray}
for a universal constant $K_6$. Using (\ref{glzest}), (\ref{ymestim}) and Corollary
\ref{lemmapolyest}, we obtain
\begin{eqnarray*}
&&g_l(\tau) \leq K_1 v^{\delta \sigma}|cz+d|^{k-2\sigma}\sum_{m=1}^p |\rho(\gamma)_{lm}| \\
&& \leq
\left \{ \begin{array}{ll}
K_2v^{\delta\sigma}|cz+d|^{k-2\sigma}(c^2+d^2)^{K_4}, & l_{\nu+1}\not= 0, \\
K_2v^{\delta\sigma}|cz+d|^{k-2\sigma}(c^2+d^2)^{K_4}|l_{\nu}|^{s-1} , & l_{\nu+1}= 0,
\end{array} \right. \\
&& \leq
\left \{ \begin{array}{ll}
K'_2v^{\delta\sigma}|cz+d|^{k-2\sigma+K_5}, & l_{\nu+1}\not= 0, \\
K'_2v^{\delta\sigma}|cz+d|^{k-2\sigma+K_5}|l_{\nu}|^{s-1} , & l_{\nu+1}= 0.
\end{array} \right.
\end{eqnarray*}
Choosing $\sigma = (k+K_5)/2$ leads to
\begin{eqnarray*}
g_l(\tau) \leq \left \{ \begin{array}{ll}
K'_2v^{\delta(k+K_5)/2}, & l_{\nu+1}\not= 0, \\
K'_2v^{\delta(k+K_5)/2}|l_{\nu}|^{s-1} , & l_{\nu+1}= 0.
\end{array} \right.
\end{eqnarray*}
In the cuspidal case we have $\delta=0$, whence
$g_l(\tau)$ is \emph{bounded} in $\frak{H}$, by a universal constant
$K_6$ if $l_{\nu+1}\not= 0$, and by $K_6|l_{\nu}|^{s-1}$ if
$l_{\nu+1}=0$. Then
\begin{eqnarray*}
|f_l(\tau)| = y^{-\sigma}g_l(\tau)= \left \{ \begin{array}{ll}
O(y^{-(k+K_5)/2}), & l_{\nu+1}\not= 0, \\
O(y^{-(k+K_5)/2})|l_{\nu}|^{s-1} , & l_{\nu+1}= 0.
\end{array} \right.
\end{eqnarray*}
In the first case, when $l_{\nu+1}\not= 0$, a standard argument, entailing
integration on the interval $\tau=x+i/n\ (n \in \mathbb{Z}^+, |x|\leq 1/2)$ implies
that the Fourier coefficients of $f_l(\tau)$ satisfy
$a(n)=O(n^{(k+K_5)/2})$ for $n \rightarrow \infty$. In the second case,
when $l_{\nu+1}=0$, an elementary argument using the location of
$z$ and $\tau\ (z \in \overline{\frak{R}}, \tau = x+i/n, n \in \mathbb{Z}^+)$
implies that $|a/c|<2$. By Lemma \ref{lemmaEprod})(b), then, if we keep in mind
that $l_{\nu}\not=0$ it follows that $|l_{\nu}| = [|a/c|]=1$. Hence the argument
used in the case $l_{\nu+1}\not=0$ implies again in this case that $a(n)=O(n^{(k+K_5)/2})$
for $n \rightarrow \infty$. $
\Box$
In the holomorphic (noncuspidal) case there is a similar argument
(cf.\ \cite{KM1}, p.\ 123) wherein the exponent is doubled. We have proved
\begin{thm} Let $\rho$ be a representation of $\Gamma$ such that all eigenvalues of
$\rho(T)$ lie on the unit circle, and suppose that $F(\tau) \in \mathcal{H}(k, \rho)$. There is
a constant $\alpha$ depending only on $\rho$ such that the Fourier
coefficients of $F(\tau)$ satisfy $a(n) = O(n^{k+\alpha})$ for $n \rightarrow \infty$.
If $F(\tau)$ is cuspidal then $a(n) = O(n^{(k+\alpha)/2})$ for $n \rightarrow \infty$.
$
\Box$
\end{thm}
\begin{itemize}gskip \noindent
Errata. We take this opportunity to correct a few typographical
errors in \cite{KM3}, upon which the present paper is based. \\
p.271, ll -14/-13. This should read as follows.
`Here $\mathcal{M}^*$ is the set of cosets of $\Gamma_{\infty}\backslash \Gamma$
distinct from $\pm \langle T \rangle$, where $\Gamma_{\infty}$ is the
stabilizer of $\infty$ in $\Gamma$, and $\hdots$' \\
p.272, l-3. This should be $P_{j+1}=(ST^{l_{j+1}})P_j, \ 0 \leq j \leq \nu$,\\
p.274, l-5. The right-hand side of display (36) should be
$p^{2\nu+2}||\rho(S)||^{\nu+2}\prod_{j=0}^{\nu+1} ||\rho(T^{l_j})||$. \\
p.274, l10. Replace $j$ by $j+1$.
\end{document} | math | 32,541 |
\begin{document}
\title{Adaptive Local (AL) Basis for Elliptic Problems with $L^\infty$-Coefficients}
\author{M. Weymuth\thanks{Institut f\"{u}r Mathematik, Universit\"{a}t Z\"{u}rich,
Winterthurerstrasse 190, CH-8057 Z\"{u}rich, Switzerland
}}
\maketitle
\begin{abstract}
\parindent0pt
We define a generalized finite element method for the discretization of elliptic partial
differential equations in heterogeneous media. In \cite{Sauter2012} a method has been introduced to set up an adaptive local finite element basis (AL basis) on a coarse mesh with mesh size $H$ which, typically, does not resolve the matrix of the media while the textbook finite element convergence rates are preserved. This method requires $O(\log(\frac{1}{H})^{d+1})$ basis functions per mesh point where $d$ denotes the spatial dimension of the computational domain. Since the continuous differential operator is involved in the construction, the method presented in \cite{Sauter2012} is only semidiscrete.
In this paper we present a fully discrete version of the method, where the AL basis is constructed by solving finite-dimensional localized problems.\\
\textbf{Keywords:} discontinuous coefficient, elliptic problem, heterogeneous media, generalized finite element method\\
\textbf{AMS subject classification} 35R05, 65N12, 65N15, 65N30
\end{abstract}
\section{Introduction}
We consider second order elliptic partial differential equations with heterogeneous and highly varying (non-periodic) coefficients.
Our emphasis is on the efficient numerical solution of problems whose coefficients contain a large number of different scales which we allow to be highly non-uniformly distributed over the domain. It is well-known that for such problems standard single scale numerical methods such as conventional finite element methods perform arbitrarily badly (see e.g. \cite{Babuska2000}).\\
Essentially there are two approaches to overcome this difficulty. One is to design (non-polynomial) generalized finite element methods where the characteristic behaviour of the solution is incorporated in the shape of the basis functions. Early papers on this topic are \cite{Babuska1983, Babuska1994} which have been further developed e.g. in \cite{Hou1997, Hughes1998}. The second approach tries to simplify the coefficient by some approximation and then employs standard finite elements. Standard methods for simplifying the coefficients are based, e.g., on homogenization methods for periodic structures (see e.g., \cite{Jikov1994, Cioranescu1999, Bensoussan1978}), or on different upscaling techniques e.g. \cite{E2007, Repin2012}. In this paper we follow the first approach.\\
Many of the existing numerical methods belonging to the first approach show promising results in practice. However, their convergence analysis usually relies on certain structural assumptions on the coefficient (e.g. periodicity or scale separation).\\
In \cite{Babuska} a generalized finite element method for general $L^\infty$-coefficient is presented where the local finite element spaces are constructed via the solution of local eigenvalue problems. This approach is based on a partition of unity method (PUM, see e.g. \cite{Babuska1997, Melenk1995, Babuska1996}) and is closely related to our approach. Further approaches for the construction and analysis of a multiscale basis for problems with high contrast without structural assumptions on the coefficient include \cite{Larson2007, Ohwadi2011, Malqvist2011, Sauter2012}.\\
In \cite{Sauter2012} a generalized finite element space has been set up as the span of the adaptive local (AL) basis. It has been proved that on a regular finite element mesh with, possibly coarse, mesh size $H$ the number $p$ of basis functions per nodal point satisfies $p=O((\log\frac{1}{H})^{d+1})$. Moreover all basis functions have local support and the accuracy of the arising Galerkin finite element method with respect to the energy norm is of order $O(H)$ without any structural assumptions on the coefficient.\\
However, the method introduced in \cite{Sauter2012} is only semidiscrete since the inverse of the continuous solution operator $L$ is involved in the construction of the basis functions. In \cite{Weymuth2013} this operator is replaced by a discrete operator $L_h$ which is obtained by a Galerkin discretization with a conforming finite-dimensional space $V_h$ on a sufficiently fine mesh. It is shown that the error estimates are preserved if the space $V_h$ satisfies the approximation property
\begin{equation*}
\sup_{f\in L^2(\Omega)\backslash\{0\}}\inf_{v\in V_h}\frac{\|L^{-1}f-v\|_{H^1(\Omega)}}{\|f\|_{L^2(\Omega)}}\leq C_{apx}H,
\end{equation*}
where $H$ denotes the coarse mesh width and the constant $C_{apx}$ is independent of $H$ and $f$. The operator $L_h^{-1}$ is a non-local fine-scale operator and the evaluation of its inverse is prohibitively expensive from the numerical point of view. In this paper we want to develop a localized version of the fully discrete method presented in \cite{Weymuth2013}.\\
The paper is structured as follows. In Section \ref{sec:problem} we formulate the model problem as well as the conditions on the coefficient. Section \ref{sec:method} is devoted to define the localized AL basis. In Section \ref{sec:regularity} we derive some $W^{1,p}$-regularity results for our model problem. These results are used in the error analysis. Finally in Section \ref{sec:error_analysis} the error analysis is presented.
\section{Model Problem}\label{sec:problem}
Let $\Omega\subset \mathbb{R}^d$, $d\in\{2,3\}$, be a bounded domain with $\partial\Omega\in C^1$. Let $\langle\cdot,\cdot\rangle$ denote the usual Euclidean scalar product on $\mathbb{R}^d$. The Sobolev space of real-valued functions in $L^2\left( \Omega\right)$ with gradients in $L^2(\Omega)$ and vanishing boundary trace is denoted by $H_0^1(\Omega)$ and its norm by $\|\cdot\|_{H^1(\Omega)}$.
We consider the following problem in variational form: Given $f\in L^{2}(\Omega)$, we are seeking $u\in H_0^1(\Omega)$ such that
\begin{equation}\label{prob_weak}
a(u,v):=\int\limits_\Omega\! \langle A\nabla u,\nabla v\rangle=\int_\Omega\! fv=: F(v)\quad\quad \forall\, v\in H_0^1(\Omega).
\end{equation}
The diffusion matrix $A\in L^{\infty}\left(\Omega,\mathbb{R}^{d\times d}_{sym}\right)$ is assumed to be uniformly elliptic, i.e.
\begin{equation}\label{coeff}
\begin{split}
0<\alpha(A,\Omega):=\essinf\limits_{x\in\Omega}\inf\limits_{v\in\mathbb{R}^d\backslash\{0\}}\frac{\langle A(x)v,v\rangle}{\langle v,v\rangle}\\
\infty>\beta(A,\Omega):=\esssup\limits_{x\in\Omega}\sup\limits_{v\in\mathbb{R}^d\backslash\{0\}}\frac{\langle A(x)v,v\rangle}{\langle v,v\rangle}.
\end{split}
\end{equation}
Since the bilinear form $a$ is symmetric, bounded and coercive, problem \eqref{prob_weak} has a unique solution.\\
We will discretize equation \eqref{prob_weak} with a conforming finite element method. For this let $\mathcal{G}$ be a conforming finite element mesh in the sense of Ciarlet \cite{Ciarlet} consisting of closed simplices $\tau$ which are the images of the reference element $\hat{\tau}$, i.e.\ the reference triangle (in 2d) or the reference tetrahedron (in 3d), under the element map $F_\tau\colon\hat{\tau}\to\tau$. We assume -- as is standard -- that the element maps of elements sharing an edge or a face induce the same parametrization on that edge or face.
Additionally, the element maps $F_\tau\colon \hat{\tau}\to\tau$ satisfy the following assumption.
\begin{assumption}\label{ass_quasi-uniform regular triangulation}
Each element map $F_\tau$ can be written as $F_\tau=R_\tau\circ A_\tau$, where $A_\tau$ is an affine map (corresponding to the scaling $\diam \tau$ of the simplex $\tau$) and $R_\tau$ is an analytic map which corresponds to the metric distortion at the possibly curved boundary and is independent of $\diam \tau$. Let $\tilde{\tau}:=A_\tau(\hat{\tau})$. The maps $R_\tau$ and $A_\tau$ satisfy for shape regularity constants $C_{affine}$, $C_{metric}$, $\gamma>0$ independent of $\diam \tau$:
\begin{align*}
&\|A_\tau'\|_{L^\infty(\hat{\tau})}\leq C_{affine}\diam \tau, && \|(A_\tau')^{-1}\|_{L^\infty(\tilde{\tau})}\leq C_{affine}(\diam \tau)^{-1}\\
&\|(R_\tau')^{-1}\|_{L^\infty(\tau)}\leq C_{metric}, && \|\nabla^nR_\tau\|_{L^\infty(\tilde{\tau})}\leq C_{metric}\gamma^nn!\quad \forall\, n\in\mathbb{N}_0.
\end{align*}
\end{assumption}
The space of continuous, piecewise linear finite elements for the mesh $\mathcal{G}$ is given by
\begin{equation*}
S:=\left\{u\in H_0^1(\Omega) :~ u|_{\tau}\circ F_\tau \in \mathbb{P}_1\ \forall\, \tau \in \mathcal{G}\right\},
\end{equation*}
where $\mathbb{P}_1$ is the space of polynomials of degree $\leq 1$. Furthermore,
let $(b_i)_{i=1}^N$ denote the usual local nodal basis of $S$ (``hat functions''), i.e.\ $b_i(x_j)=\delta_{ij}$. We denote their support by
\begin{equation*}
\omega_i:=\supp b_i.
\end{equation*}
Since $S\subset H_0^1(\Omega)$ is a finite-dimensional subspace, the abstract conforming Galerkin method to problem \eqref{prob_weak} can be formulated as: Find $u_S\in S$ such that
\begin{equation}\label{Galerkin problem}
a(u_S,v)=F(v)\quad \forall\, v\in S
\end{equation}
with $a(\cdot,\cdot)$ and $F(\cdot)$ as in \eqref{prob_weak}.
If the diffusion coefficient $A$, the right-hand side $f$ as well as the domain $\Omega$ of \eqref{prob_weak} are sufficiently smooth such that the problem is $H^2$-regular, then the unique solution $u_S$ of \eqref{Galerkin problem} satisfies the error estimate
\begin{equation*}
\|u-u_S\|_{H^1(\Omega)}\leq CH\|f\|_{L^2(\Omega)}
\end{equation*}
(see e.g. \cite{Ciarlet}). This estimate states linear convergence of the $\mathbb{P}_1$-finite element method as the mesh width $H$ tends to zero. However, the regularity assumption is not realistic for the problem class under consideration. It is well known that as long as the mesh $\mathcal{G}$ does not resolve the discontinuities and oscillations of $A$, the convergence rates of linear finite element methods are substantially reduced.\\
\section{The Adaptive Local (AL) Basis}\label{sec:method}
In this section we introduce a new generalized finite element method for the discretization of heterogeneous problems.
\subsection{Notation}
We assume that $\mathcal{G}$ is a conforming finite element mesh which is shape-regular and satisfies Assumption \ref{ass_quasi-uniform regular triangulation}. Moreover we suppose that the simplices $\tau\in\mathcal{G}$ are closed sets.
\begin{itemize}
\item [1)] Simplex layers around $\omega_i$ and corresponding meshes:\\
We define recursively
\begin{equation}\label{layers}
\begin{split}
\omega_{i,0}&:=\omega_i\\
\omega_{i,j+1}&:=\bigcup\left\{\tau :~ \tau\in\mathcal{G} \ \text{and}\ \omega_{i,j}\cap \tau\neq\emptyset\right\},\quad j=0,1,2,\dots
\end{split}
\end{equation}
Finally, we set
\begin{equation*}
\mathcal{G}_{i,j}:=\left\{\tau\in\mathcal{G} :~ \tau\subset\omega_{i,j}\right\}.
\end{equation*}
\item [2)] Local neighbourhoods around the triangle patch $\omega_{i,1}$:\\
We set
\begin{equation}\label{omega_far}
\mathcal{G}_i^{far}:=\mathcal{G}_{i,2}\backslash\mathcal{G}_{i,1}\quad \text{and}\quad \omega_i^{far}:=\interior(\omega_{i,2}\backslash\omega_{i,1}).
\end{equation}
\item [3)] (Local) mesh width:\\
We set
\begin{equation}\label{local_mesh_width}
H_i:=\max_{\tau\in\mathcal{G}_{i,2}} \diam(\tau)\qquad\text{and}\qquad H:=\max_{1\leq i\leq N}H_i.
\end{equation}
Since the mesh is assumed to be shape-regular and the number of layers is bounded by 2, we can conclude that there exist positive constants $c$, $C$ and $C_\#$ such that
\begin{align}\label{diam_leq_CH}
&\min_{\tau\in\mathcal{G}_{i,2}}\rho_\tau\geq cH_i
&&\diam \omega_{i,2}\leq CH_i\\
&\dist(\omega_{i,1},\partial\omega_{i,2}\backslash\partial\Omega)=\delta_i\geq cH_i
&&\#\mathcal{G}_{i,2}\leq C_\#\nonumber
\end{align}
holds.
$\rho_\tau$ denotes the diameter of the maximal inscribed ball in $\tau$.
\item [4)] Refinement operator:\\ Let $\mathcal{T}^{macro}$ be a fixed triangulation (with possibly curved elements at the boundary) with element maps satisfying Assumption \ref{ass_quasi-uniform regular triangulation}. We introduce a refinement operator $\mathcal{R}^1(\cdot)$. The input is a conforming finite element mesh $\mathcal{T}$ where every element is marked for refinement and the output is a new conforming finite element mesh $\mathcal{R}^1(\mathcal{T})$. Recursively we define for $t\geq 2$ the iterated refinement operator
\begin{equation}\label{refinement_op}\mathcal{R}^t(\mathcal{T}):=\mathcal{R}^1(\mathcal{R}^{t-1}(\mathcal{T})).
\end{equation}
\item [5)] Solution operator:\\
For a subdomain $D\subseteq\Omega$, let $L_D^{-1}\colon L^{2}(D)\to H_0^1(D)$ denote the solution operator associated with the (localized) variational form: Given $g\in L^{2}(D)$, find $w \in H_0^1(D)$ such that
\begin{equation}\label{local_sol_operator}
a_D(w,v):=\int_D\!\langle A\nabla w,\nabla v\rangle=\int_D\! gv=:G(v)\quad \forall\, v\in H_0^1(D).
\end{equation}
\end{itemize}
\begin{remark}\label{rem_overlap}
Note that the patches $\omega_{i,j}$, $0\leq j\leq 2$, have finite overlap. For every $\tau\in\mathcal{G}$ there exists $m_{\tau,j}\in\mathbb{N}$ such that
\begin{equation*}
\#\{i :~ \tau\in \omega_{i,j}\}= m_{\tau,j},\quad 0\leq j\leq 2.
\end{equation*}
We set
\begin{equation}\label{overlap_const}
M_j:=\max_\tau m_{\tau,j},\quad 0\leq j\leq 2.
\end{equation}
\end{remark}
\subsection{Construction of the Local Approximation Spaces}
On each patch $\omega_{i,2}$ ($1\leq i\leq N$) we will set up two low-dimensional local approximation spaces called $V_i^{near}$ and $V_i^{far}$. In order to get $V_i^{far}$ we will first construct an intermediate space $X_i^{far}$ which is the high-dimensional space of locally $L$-harmonic functions and can be approximated by a low-dimensional space.\\
We fix $i\in\mathcal{I}:=\{1,\dots,N\}$. The construction of the space $V_i^{near}$ respectively $X_i^{far}$ goes as follows.
We set
\begin{equation}\label{piecewise_constant}
S_0(\mathcal{G}):=\Span\left\{\chi_\tau :~ \tau\in\mathcal{G}\right\},
\end{equation}
where $\chi_\tau\colon \Omega\to\mathbb{R}$ is the characteristic function for the simplex $\tau\in\mathcal{G}$ and $H$ is the global mesh width. $S_0(\mathcal{G})$ is the space of piecewise constant functions on $\mathcal{G}$. Furthermore we define the space
\begin{equation}\label{S_i2h}
S_{i,2}:=\{u|_{\omega_{i,2}} :~ u\in S_{fine}\wedge \supp u\subset\omega_{i,2}\},
\end{equation}
where $S_{fine}$ is some finite-dimensional fine-scale space satisfying
\begin{equation*}
S_{fine}\subset H^1(\Omega).
\end{equation*}
The local approximation spaces are constructed by solving conventional finite element problems. For the nearfield part, i.e.\ $\tau\in\mathcal{G}_{i,1}$, we want to find $\tilde{B}_{i,\tau}^{near}\in S_{i,2}$ such that
\begin{equation}\label{nearfield_sol}
\int_{\omega_{i,2}}\!\langle A\nabla\tilde{B}_{i,\tau}^{near},\nabla v\rangle=\int_{\omega_{i,2}}\!\chi_\tau v\quad\forall\, v\in S_{i,2}.
\end{equation}
Then we set
\begin{equation*}
B_{i,\tau}^{near}:=b_i\tilde{B}_{i,\tau}^{near}
\end{equation*}
and finally our local approximation space for the nearfield part can be defined as
\begin{equation*}
V_i^{near}:=\Span\{B_{i,\tau}^{near} :~ \tau\in\mathcal{G}_{i,1}\}.
\end{equation*}
The construction of the local approximation space for the farfield part can be done analogously, but the error analysis shows that for preserving the linear convergence rate of the method we have to refine the mesh $\mathcal{G}_i^{far}$. Thus for $\tau\in\mathcal{R}^t(\mathcal{G}_i^{far})$ we are seeking $\tilde{B}_{i,\tau}^{far}\in S_{i,2}$ such that
\begin{equation*}
\int_{\omega_{i,2}}\!\langle A\nabla \tilde{B}_{i,\tau}^{far},\nabla v\rangle=\int_{\omega_{i,2}}\!\chi_\tau v\quad \forall\, v\in S_{i,2}.
\end{equation*}
The error analysis will show that the refinement parameter $t$ has to be chosen as $t=\lceil\lb\frac{1}{H_i}\rceil$. We set
\begin{equation*}
X_i^{far}:=\Span\{\tilde{B}_{i,\tau}^{far}|_{\omega_{i,1}} :~ \tau\in\mathcal{R}^t(\mathcal{G}_i^{far})\}.
\end{equation*}
\begin{remark}~
\begin{itemize}
\item[a)] In order to get a linear convergence rate in the $H^1$-norm the space $S_{fine}$ in \eqref{S_i2h} has to be chosen such that
\begin{equation*}
\sup\limits_{f\in L^{2}(\omega_{i,2})\backslash\{0\}}\inf\limits_{v\in S_{i,2}}\frac{\left\|L_{\omega_{i,2}}^{-1}f-v\right\|_{H^1(\omega_{i,2})}}{\left\|f\right\|_{L^2(\omega_{i,2})}}\leq C_{apx}H_i^2
\end{equation*}
holds, where the constant $C_{apx}$ is independent of $H_i$ and $f$.
\item[b)] The functions in $X_i^{far}$ are locally $L$-harmonic on $\interior(\omega_{i,1})$, i.e.\ any $v\in X_i^{far}$ satisfies
\begin{equation*}
\int_{\omega_{i,1}}\!\langle A\nabla v,\nabla w\rangle=0\quad \forall\, w\in S_{i,1}:=\{w|_{\omega_{i,1}} :~ w\in S_{fine}\wedge \supp w\subset\omega_{i,1}\}.
\end{equation*}
\end{itemize}
\end{remark}
\subsection{Approximation of $\boldsymbol{X_i^{far}}$}
Our goal is to approximate the space $X_i^{far}$ by a low-dimensional space $V_i^{far}$. The construction of this approximation is based on results in \cite{Bebendorf, Boerm, Sauter2012}. \\
Let $\omega_i$, $\omega_{i,1}$ as in \eqref{layers} and assume that $\omega_i\cap\partial\Omega=\emptyset$.
We introduce intermediate layers between $\omega_i$ and $\omega_{i,1}$. Therefore we set $r_{i,1}:=\dist(\omega_i,\partial\omega_{i,1})$ and
\begin{equation}\label{def_rij}
r_{i,j}:=\left(1-\frac{j-1}{\ell-1}\right)r_{i,1},\quad\quad 2\leq j\leq \ell,
\end{equation}
where $\ell$ will be fixed later. It holds $r_{i,1}>r_{i,2}>\dots>r_{i,\ell}=0$. The intermediate layers are given by
\begin{align*}
D_{i,0}&:= \omega_{i,1}\\
D_{i,j}&:=\left\{x\in\omega_{i,1} :~ \dist(x,\omega_i)\leq r_{i,j}\right\},\quad\quad 1\leq j\leq \ell,
\end{align*}
and satisfy $\omega_i=D_{i,\ell}\subset D_{i,\ell -1}\subset\cdots \subset D_{i,1} \subset D_{i,0}= \omega_{i,1}$. Note that if $\omega_i$ and $\omega_{i,1}$ are convex, then also the domains $D_{i,j}$ are convex for all $0\leq j\leq\ell$. In \cite{Bebendorf} it is shown that for any $\kappa_j\in\mathbb{N}$ there exists a subspace $V_{\kappa_j}\subset X(D_{i,j})$ such that $\dim V_{\kappa_j}\leq \kappa_j$ and the estimate
\begin{equation}\label{eq_harmonic}
\inf_{v\in V_{\kappa_j}}\|u-v\|_{L^2(D_{i,j})}\leq C\frac{\diam(D_{i,j})}{\sqrt[d]{\kappa_j}}\|\nabla u\|_{L^2(D_{i,j})}
\end{equation}
is satisfied.\footnote{$X(D_{i,j})$ denotes the space of locally harmonic functions on $D_{i,j}$. Note that the constant $C$ in \eqref{eq_harmonic} depends on Poincar\'{e}'s constant and hence on the shape of $D_{i,j}$. If $D_{i,j}$ is convex, then $C=2\sqrt[d]{2}/\pi$ (cf. \cite{Bebendorf}).} In order to construct these subspaces $V_{\kappa_j}=:\tilde{V}_{i,j}^{far}$ for $1\leq j\leq \ell $ we use $L^2$-orthogonal projections onto $X_i^{far}$.
We set $\kappa_j=:k^d$, where $k\in\mathbb{N}$ will be fixed later.
For $\rho>0$ let $\mathcal{G}_\rho$ denote a Cartesian tensor mesh on $\mathbb{R}^d$, $d\in\{2,3\}$, which consists of $d$-dimensional elements with side length $\rho$. Then define
\begin{equation*}
\mathcal{\tilde{G}}_{i,j}:=\left\{D_{i,j}\cap\tau :~ \tau\in\mathcal{G}_\rho\ \text{with}\ \rho:=\frac{\diam(D_{i,j})}{k}\right\},\quad 1\leq j\leq \ell
\end{equation*}
and
\begin{equation*}
\tilde{V}_{i,j}^{far}:=\Span \left\{(\mathcal{P}_i\chi_t)|_{\omega_i} :~ t\in \mathcal{\tilde{G}}_{i,j}\right\},
\end{equation*}
where $\mathcal{P}_i\colon L^2(\omega_{i,2})\to X_i^{far}$ is the $L^2$-orthogonal projection. We set
\begin{equation}\label{tilde_V_far}
\tilde{V}_i^{far}:=\tilde{V}_{i,1}^{far}+\tilde{V}_{i,2}^{far}+\cdots +\tilde{V}_{i,\ell}^{far}
\end{equation}
and finally,
\begin{equation*}
V_i^{far}:=\left\{b_iv :~ v\in \tilde{V}_i^{far}\right\}.
\end{equation*}
\begin{remark}
If $\omega_i\cap\partial\Omega\neq\emptyset$ we have to make the following small modifications. We set $r_{i,1}:=\dist(\omega_i,\partial\omega_{i,1}\backslash\partial\Omega)$ and $r_{i,j}$ is defined as in \eqref{def_rij}. The intermediate layers are given by
\begin{align*}
D_{i,0}&:= \omega_{i,1}\cup\{x\in\mathbb{R}^d :~ \dist(x,\omega_i)\leq r_{i,1}\}\\
D_{i,j}&:=\left\{x\in\mathbb{R}^d :~ \dist(x,\omega_i)\leq r_{i,j}\right\},\quad\quad 1\leq j\leq \ell.
\end{align*}
The remaining part of the construction is exactly the same as above.
\end{remark}
\subsection{Definition of the AL Basis}
\begin{remark}\label{remdim}
Since $b_i\in W_0^{1,\infty}(\omega_i)$ and $X_i^{far}\subset H^1(\omega_{i,1})$ we conclude that $b_iv\in H_0^1(\omega_{i})$ for all $v\in \tilde{V}_i^{far}$. Thus we can identify $b_iv$ by its extension by zero to a function (again denoted by $b_iv$) in $H_0^1(\Omega)$. In this sense we have
\begin{equation*}
V_i^{far}\subset H_0^1(\Omega),\quad\quad \dim V_i^{far}\leq \sum\limits_{j=1}^{\ell} \#\mathcal{\tilde{G}}_{i,j}\leq \sum\limits_{j=1}^{\ell} k^d=\ell k^d.
\end{equation*}
\end{remark}
\begin{definition}[AL basis]
For any support $\omega_i$ the set of \emph{AL basis} functions consists of
\begin{equation*}
V_i^{near}:=\Span\left\{b_i\tilde{B}_{i,\tau}^{near} :~ \tau\in\mathcal{G}_{i,1}\right\}
\end{equation*}
where $\tilde{B}_{i,\tau}^{near}$ is the solution of problem \eqref{nearfield_sol} and of
\begin{equation*}
V_i^{far}:=\left\{b_iv :~ v\in \tilde{V}_i^{far}\right\}.
\end{equation*}
The general notation is $b_{i,j}$, $1\leq j\leq s_i$, $1\leq i\leq N$, where $s_i:=\dim(V_i^{near}+V_i^{far})$.
The corresponding generalized finite element space $V_{AL}$ is given by
\begin{equation}\label{def:VAL-II}
V_{AL}:=\left(V_1^{near}+V_1^{far}\right)+\left(V_2^{near}+V_2^{far}\right)+\dots +\left(V_N^{near}+V_N^{far}\right).
\end{equation}
\end{definition}
The Galerkin discretization for the generalized finite element space $V_{AL}$ is given by seeking $u_{AL}^{GAL}\in V_{AL}$ such that
\begin{equation}\label{Galerkin_sol_II}
a(u_{AL}^{GAL}, v)=F(v)\quad \forall\, v\in V_{AL}.
\end{equation}
Problem \eqref{Galerkin_sol_II} has a unique solution and is equivalent to a system of linear equations of the form
\begin{equation}\label{eq_linear_system}
\sum_{i=1}^N\sum_{j=1}^{s_i}a(b_{k,\ell},b_{i,j})c_{i,j}=F(b_{k,\ell}),\quad 1\leq \ell\leq s_i,\; 1\leq k\leq N
\end{equation}
or
\begin{equation*}
Bc=F
\end{equation*}
where $B$ is the stiffness matrix, whose elements are
\begin{equation*}
B(\ell,k;j,i):=a(b_{k,\ell},b_{i,j})=\int_{\omega_i\cap\omega_k}\!\langle A\nabla b_{k,\ell},\nabla b_{i,j}\rangle
\end{equation*}
and $F$ is the load vector which is defined as
\begin{equation*}
F(k;\ell):=\int_{\omega_k}\!f b_{k,\ell}.
\end{equation*}
If $c:=\{c_{i,j}\}$ is a solution of \eqref{eq_linear_system}, then $u_{AL}^{GAL}$ can be written as
\begin{equation*}
u_{AL}^{GAL}=\sum_{i=1}^N\sum_{j=1}^{s_i}c_{i,j}b_{i,j}.
\end{equation*}
\section{$\boldsymbol{W^{1,p}}$-Regularity of the Poisson Problem with $\boldsymbol{L^\infty}$-Coefficient}\label{sec:regularity}
Let $u$ be the solution of \eqref{prob_weak}. Our goal is to derive $L^p(\Omega)$-regularity estimates for the gradient of $u$
for some $p>2$. We start from a Laplace problem, i.e. the coefficient $A$ is equal to the identity matrix and employ then a perturbation argument in order to get the desired estimates for a uniformly elliptic diffusion matrix $A\in L^\infty(\Omega,\mathbb{R}_{sym}^{d\times d})$. We will see that our estimates only depend on the size of the jumps in the coefficient. We consider the following problem: Find $w\in H_0^1(\Omega)$ such that
\begin{equation}\label{laplace}
\int_\Omega\!\langle\nabla w,\nabla v\rangle=F(v)\quad\forall\, v\in H_0^1(\Omega).
\end{equation}
\begin{theorem}[\cite{Simader1996}]\label{theo_Simader}
Let $\Omega\subset\mathbb{R}^d$, $d\geq2$, be a bounded
domain with $\partial\Omega\in C^1$. Let $1<p<\infty$. Then, for every $F\in W^{-1,p}(\Omega)$, problem \eqref{laplace} has a unique solution $w\in W_0^{1,p}(\Omega)$ which satisfies
\begin{equation*}
K_p^{-1}\|\nabla w\|_{L^p(\Omega)} \leq\| F\|_{W^{-1,p}(\Omega)}\leq \|\nabla w\|_{L^p(\Omega)}
\end{equation*}
with the Laplace $W^{1,p}$-regularity constant $K_{p}$ and
\begin{equation*}
\|F\|_{W^{-1,p}(\Omega)}:=\sup_{\substack{v\in W_0^{1,p'}(\Omega)\\ \|v\|_{W^{1,p'}(\Omega)}\leq 1}}\left|\int_\Omega\langle\nabla w,\nabla v\rangle\right|.
\end{equation*}
\end{theorem}
\begin{remark}
The constant $K_p$ is independent of $F$ (and $w$) but depends on $\Omega$, $d$ and $p$. We have $K_2=1$ and, for $p>2$, $K_p$ is
non-decreasing and continuous in $p$ (cf. \cite{Meyers1963}).
\end{remark}
\begin{figure}
\caption{The function $\eta(p)$ (left) and the function $p^{*}
\label{fig:p_star}
\end{figure}
Let $2<P<\infty$ be fixed. We define
\begin{equation*}
\eta(p):=\frac{1/2-1/p}{1/2-1/P}\qquad 2\leq p\leq P.
\end{equation*}
It can be seen in Figure \ref{fig:p_star} that $\eta(p)$ increases from the value zero at $p=2$ to the value one at $p=P$. Furthermore for any $t\in[0,1]$, we set
\begin{equation}\label{qstern}
p^*(t):=\argmax\left\{K_P^{-\eta(p)}\geq 1-t :~ 2\leq p\leq P\right\}.
\end{equation}
The function $K_P^{-\eta(p)}$ decreases from the value $1$ at $p=2$ to the value $1/K_P$ at $p=P$. The function $p^*(t)$ takes the value $2$ at $t=0$, increases then to the value $P$ at $t=1-1/K_P$ and remains constant for $t\in[1-1/K_P,1]$ (see Figure \ref{fig:p_star}).
\begin{theorem}\label{theo_gradient_estimate}
Let $\Omega\subset\mathbb{R}^d$, $d\geq 2$, be a bounded domain and let $\partial\Omega\in C^1$. If $A\in L^\infty\left(\Omega,\mathbb{R}^{d\times d}_{sym}\right)$ satisfies \eqref{coeff} and $F\in W^{-1,P}(\Omega)$ for some $P>2$, then for the solution $u\in H_0^1(\Omega)$ of $\eqref{prob_weak}$ the estimate
\begin{equation*}
\|\nabla u\|_{L^p(\Omega)}\leq C\|F\|_{W^{-1,p}(\Omega)}
\end{equation*}
holds provided $2\leq p< p^*(\alpha/\beta)$ with $p^*$ as in \eqref{qstern} and $C:=\frac{1}{\beta}\frac{K_P^{\eta(p)}}{1-K_P^{\eta(p)}\left(1-\frac{\alpha}{\beta}\right)}$.
\end{theorem}
For a proof we refer to \cite{Meyers1963, Nochetto2013, Weymuth2016}.
\begin{remark}
Let $P\in (2,\infty)$ be fixed and $K_P$ as in Theorem \ref{theo_Simader}. If the coefficient $A$ is such that $\alpha/\beta\in[1-1/K_P,1]$ and $F\in W^{-1,P}(\Omega)$, then the solution of \eqref{prob_weak} satisfies the estimate
\begin{equation*}
\|\nabla u\|_{L^p(\Omega)}\leq C\|F\|_{W^{-1,p}(\Omega)},\quad C=\frac{1}{\beta}\frac{K_P^{\eta(p)}}{1-K_P^{\eta(p)}\left(1-\frac{\alpha}{\beta}\right)}
\end{equation*}
for $2\leq p< P=p^*(\alpha/\beta)$ with $p^*$ as in \eqref{qstern}. This is due to the fact that the function $p^*$ takes the value $P$ at $1-1/K_P$ and remains constant in the interval $[1-1/K_P,1]$ (cf.\ Figure \ref{fig:p_star}).
\end{remark}
Note that for a given coefficient $A\in L^\infty\left(\Omega,\mathbb{R}^{d\times d}_{sym}\right)$ one can always determine a $P>2$ such that $\alpha/\beta\in [1-1/K_P, 1]$. The $P$ depends only on the size of the jumps in the coefficient. For constant coefficients $P$ can be chosen arbitrarily close to infinity, whereas for coefficients with large jumps $P$ is close to 2.
\section{Error Analysis}\label{sec:error_analysis}
This section analyzes the generalized finite element method which has been introduced in Section \ref{sec:method}. It is based on results in \cite{Bebendorf, Boerm, Sauter2012}.\\
The norm in $L^p(\Omega)$ will be denoted by $\|\cdot\|_{L^p(\Omega)}$. We always use the notation that, for $p\in\left[1,\infty\right]$, the number $p'\in\left[ 1,\infty\right] $ is defined via $\frac{1}{p}+\frac{1}{p'}=1$. Further we will need the Sobolev
space $W^{1,p}(\Omega)$ consisting of functions in $L^p\left(\Omega\right)$ with gradients in $L^p(\Omega)$. Its standard norm is
denoted by $\|\cdot\|_{W^{1,p}(\Omega)}$. The space of functions denoted by $W_0^{1,p}(\Omega)$ is the closure of $C_0^\infty(\Omega)$ with respect to the norm $\|\cdot\|_{W^{1,p}(\Omega)}$. We also use the space $W^{-1,p}\left(\Omega\right):=(W_0^{1,p'}\left(\Omega\right))'$ endowed with the standard dual norm $\|\cdot\|
_{W^{-1,p}(\Omega)}$. For vector and matrix valued functions, we use the same notation for the Lebesgue and Sobolev spaces as well as for the corresponding norms. For functions in $L^{2}\left( \Omega,\mathbb{R}^{d}\right) $
we set
\begin{equation*}
\|\cdot\|_{L^p(\Omega}):=\|\,\|\cdot\|_{\ell^{p}}\,\|_{L^p(\Omega)},
\end{equation*}
where $\|\cdot\|_{\ell^p}$ denotes the discrete $\ell^p$-norm in $\mathbb{R}^d$.\\
For the error analysis it is supposed that the following assumption holds.
\begin{assumption}\label{ass}
\begin{equation*}
\sup\limits_{f\in L^{2}(\omega_{i,2})\backslash\{0\}}\inf\limits_{v\in S_{i,2}}\frac{\left\|L_{\omega_{i,2}}^{-1}f-v\right\|_{H^1(\omega_{i,2})}}{\left\|f\right\|_{L^2(\omega_{i,2})}}\leq C_{apx}H_i^2,
\end{equation*}
where $S_{i,2}$ is as defined in \eqref{S_i2h}, $H_i$ is the mesh width of $\mathcal{G}_{i,2}$ (cf.\ \eqref{omega_far} and \eqref{local_mesh_width}) and the constant $C_{apx}$ is independent of $H_i$ and $f$.
\end{assumption}
\begin{notation}
Let $\tilde{L}_{\omega_{i,2}}^{-1}: L^2(\interior(\omega_{i,2}))\to S_{i,2}$ denote the discrete local solution operator: Given $g\in L^2(\interior(\omega_{i,2}))$ find $\tilde{B}_{i,\tau}\in S_{i,2}$ such that
\begin{equation*}
\int_{\omega_{i,2}}\!\langle A\nabla\tilde{B}_{i,\tau},\nabla v\rangle=\int_{\omega_{i,2}}\!g v\quad\forall\, v\in S_{i,2}.
\end{equation*}
\end{notation}
\begin{corollary}
C\'ea's lemma and Assumption \ref{ass} imply
\begin{equation}\label{assumption}
\left\|L_{\omega_{i,2}}^{-1}f-\tilde{L}_{\omega_{i,2}}^{-1}f\right\|_{H^1(\omega_{i,2})}\leq \frac{\beta}{\alpha }CH_i^2\left\|f\right\|_{L^2(\omega_{i,2})},
\end{equation}
where $\alpha,\,\beta$ are the constants from \eqref{coeff}. $C$ depends on $C_{apx}$ and on Friedrichs' constant.
\end{corollary}
\begin{proof}
By C\'ea's lemma we get
\begin{equation*}
\left\|L_{\omega_{i,2}}^{-1}f-\tilde{L}_{\omega_{i,2}}^{-1}f\right\|_{H^1(\omega_{i,2})}\leq \frac{\beta}{\alpha} C \inf_{v\in S_{i,2}}\left\|L_{\omega_{i,2}}^{-1}f-v\right\|_{H^1(\omega_{i,2})}.
\end{equation*}
Assumption \ref{ass} implies
\begin{align*}
\left\|L_{\omega_{i,2}}^{-1}f-\tilde{L}_{\omega_{i,2}}^{-1}f\right\|_{H^1(\omega_{i,2})}&\leq \frac{\beta}{\alpha}C\sup_{f\in L^2(\omega_{i,2})\backslash\{0\}} \inf_{v\in S_{i,2}}\left\|L_{\omega_{i,2}}^{-1}f-v\right\|_{H^1(\omega_{i,2})}\\
&\leq \frac{\beta}{\alpha}C H_i^2\|f\|_{L^2(\omega_{i,2})}
\end{align*}
with a constant $C$ depending on Friedrichs' constant and $C_{apx}$.
\end{proof}
\begin{remark}
The ellipticity of $L_{\omega_{i,2}}^{-1}$, the assumption \eqref{coeff} on the coefficient $A$, and the conformity of the finite element space $S_{i,2}$ imply that the approximation $\tilde{L}_{\omega_{i,2}}^{-1}$ is elliptic and
\begin{equation}\label{L}
\left\|\tilde{L}_{\omega_{i,2}}^{-1}\right\|_{H_0^1(\omega_{i,2})\leftarrow H^{-1}(\omega_{i,2})}\leq \frac{C}{\alpha},
\end{equation}
where $\alpha$ is defined in \eqref{coeff}.
\end{remark}
\begin{lemma}\label{lemma_projection}
Let $\mathcal{G}$ be a conforming finite element mesh which satisfies Assumption \ref{ass_quasi-uniform regular triangulation}. Further let $g_i\in L^2(\omega_{i,2})$ and denote by $P_i$ the $L^2$-orthogonal projection of $L^2(\omega_{i,2})$ onto $S_0(\mathcal{G})$ (cf.\ \eqref{piecewise_constant}). Then
\begin{equation*}
\|g_i-P_ig_i\|_{H^{-1}(\omega_{i,2})}\leq CH\|g_i\|_{L^2(\omega_{i,2})},
\end{equation*}
where $H$ denotes the mesh width of $\mathcal{G}$.
\end{lemma}
\begin{proof}
Using H\"{o}lder's inequality and Friedrichs' inequality we get for $\psi_i\in H_0^1(\omega_{i,2})$
\begin{align}\label{proj_est}
|\langle g_i-P_ig_i, \psi_i\rangle_{L^2(\omega_{i,2})}|&\leq \|g_i-P_ig_i\|_{L^2(\omega_{i,2})}\|\psi_i\|_{L^2(\omega_{i,2})}\nonumber\\&\leq C H \|g_i\|_{L^2(\omega_{i,2})}\|\psi_i\|_{H^1(\omega_{i,2})}.
\end{align}
By the definition of the $H^{-1}$-norm and \eqref{proj_est} we obtain \belowdisplayskip=-12pt
\begin{equation*}
\|g_i-P_ig_i\|_{H^{-1}(\omega_{i,2})}=\sup_{v\in H_0^1(\omega_{i,2})}\frac{|\langle g_i-P_ig_i, v\rangle_{L^2(\omega_{i,2})}|}{\|v\|_{H^1(\omega_{i,2})}}\leq CH\|g_i\|_{L^2(\omega_{i,2})}.\qedhere
\end{equation*}
\end{proof}
We fix some $Q\in (2,\infty)$. For $1\leq i\leq N$ let $\chi_i\colon \Omega\to\mathbb{R}$ be a cutoff function satisfying $\chi_i|_{\omega_{i,1}}=1$ and $\chi_i|_{\Omega\backslash\omega_{i,2}}=0$. Morover the following properties are fulfilled for $\frac{Q'}{3}< q< \frac{Q}{3}$.
\begin{align}
\|\chi_i\|_{L^q(\omega_i^{far})}&\leq CH_i^{\frac{d}{q}}\label{cutoff_estimate_1}\\
\|\nabla\chi_i\|_{L^q(\omega_i^{far})}&\leq CH_i^{\frac{d}{q}-1}\label{cutoff_estimate_2}\\
\|\Div(A\nabla\chi_i)\|_{L^q(\omega_i^{far})}&\leq CH_i^{\frac{d}{q}-2}\label{cutoff_estimate_3}
\end{align}
\begin{remark}
For the explicit construction of $\chi_i$ we refer to \cite{Weymuth2016}. The cutoff functions are constructed by solving homogeneous Dirichlet problems. It would be desirable to have $\chi_i\in W^{1,\infty}(\Omega)$ as well as $\Div(A\nabla\chi_i)\in L^\infty(\Omega)$. However, for $d\geq 2$ this is not possible.
\end{remark}
Let $f\in L^{p}(\Omega)$ ($p\in[2,\infty]$) be given and define $u:=L_\Omega^{-1}f$.
We set $u_i:=\chi_i(u-\bar{u}_i)$ with
\begin{equation*}
\bar{u}_i:=\begin{cases}
\frac{1}{\vol(\omega_{i}^{far})}\int\limits_{\omega_i^{far}}\!u &\text{in}\ \omega_i^{far}\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
We observe that
\begin{equation*}
u_i=L_{\omega_{i,2}}^{-1}\left(g_i\right)
\end{equation*}
with
\begin{equation*}
g_i=\begin{cases}
f &\text{in}\ \omega_{i,1}\\
\chi_if-2\langle A\nabla \chi_i,\nabla u\rangle-(u-\bar{u}_i)\Div\left(A\nabla\chi_i\right) &\text{in}\ \omega_i^{far}.
\end{cases}
\end{equation*}
We set
\begin{equation}\label{g_i_near}
g_i^{near}:=\begin{cases}
f &\text{in}\ \omega_{i,1}\\
0 & \text{in}\ \omega_i^{far}
\end{cases}
\end{equation}
and
\begin{equation}\label{g_i_far}
g_i^{far}:=\begin{cases} 0 &\text{in}\ \omega_{i,1}\\
\chi_if-2\langle A\nabla \chi_i,\nabla u\rangle-(u-\bar{u}_i)\Div\left(A\nabla\chi_i\right) &\text{in}\ \omega_i^{far}.
\end{cases}
\end{equation}
This allows us to introduce
\begin{equation}\label{splitting}
u_i=u_i^{near}+u_i^{far}:=L_{\omega_{i,2}}^{-1}\left(g_i^{near}\right)+L_{\omega_{i,2}}^{-1}(g_i^{far}).
\end{equation}
Define
\begin{equation}\label{approx_near_far}
\tilde{u}_i^{near}:=\tilde{L}_{\omega_{i,2}}^{-1}(P_ig_i^{near})\qquad \text{and}\qquad
\tilde{u}_i^{far}:=\tilde{L}_{\omega_{i,2}}^{-1}(P_i^tg_i^{far}),
\end{equation}
where $P_i$ denotes the $L^2$-orthogonal projection of $L^2(\omega_{i,1})$ onto $S_0(\mathcal{G}_{i,1})$ (cf.\ \eqref{piecewise_constant}) and $P_i^t$ is the $L^2$-orthogonal projection of $L^2(\omega_i^{far})$ onto $S_0(\mathcal{R}^t(\mathcal{G}_i^{far}))$ which is the space of piecewise constant functions on the $t$-times refined mesh ($t$ will be fixed later).\\
The following lemma is a slight modification of a result presented in \cite{Boerm, Sauter2012}.
\begin{lemma}\label{lemma_aprox_tilde_u_far}
Let $\tilde{u}_i^{far}$ as in \eqref{approx_near_far} and $\tilde{V}_i^{far}$ as in \eqref{tilde_V_far}. There exists $\hat{u}_i^{far}\in\tilde{V}_i^{far}$ such that
\begin{equation*}
\|\tilde{u}_i^{far}-\hat{u}_i^{far}\|_{H^m(\omega_i)}\leq CH_i^{3-m}\|\nabla\tilde{u}_i^{far}\|_{L^2(\omega_{i,1})}\quad m=0,1
\end{equation*}
with $H_i$ as in \eqref{local_mesh_width}.
\end{lemma}
\begin{proof}
Set
\begin{equation*}
\ell:=\max \left\{2,\left\lceil\frac{2}{\log 2}\log\frac{1}{H_i}\right\rceil\right\}\quad \text{and}\quad k:=\left\lceil\frac{2c_0\ell^2}{(\ell-1)}\right\rceil
\end{equation*}
for some $c_0=O(1).$ Choosing $p\leftarrow \ell$, $\ell\leftarrow k$, $i\leftarrow \ell$, $c\leftarrow c_0$, and $\delta\leftarrow O(H_i)$ in the second estimate of \cite[p.\,172]{Boerm} yields
\begin{equation}\label{Boerm1}
\|\tilde{u}_i^{far}-\hat{u}_i^{far}\|_{L^2(\omega_i)}\leq C H_i \left(c_0\frac{\ell}{k}\right)^\ell\|\nabla\tilde{u}_i^{far}\|_{L^2(\omega_{i,1})}.
\end{equation}
Similarly, choosing $p\leftarrow \ell$, $\ell\leftarrow k$, and $c\leftarrow c_0$ in the second last estimate of \cite[p.\,172]{Boerm} we get
\begin{equation}\label{Boerm2}
\|\nabla(\tilde{u}_i^{far}-\hat{u}_i^{far})\|_{L^2(\omega_i)}\leq \left(c_0\frac{\ell}{k}\right)^\ell\|\nabla\tilde{u}_
i^{far}\|_{L^2(\omega_{i,1})}.
\end{equation}
According to the definition of $\ell$ we have to distinguish the following two cases:
\begin{itemize}
\item Case 1: $\left\lceil\frac{2}{\log 2}\log\frac{1}{H_i}\right\rceil\leq2$\\
By definition of $\ell$ we know that $\ell=2$ and after some simple calculations we see that $H_i\geq\frac{1}{2}$. Therefore we obtain by the definition of $k$
\begin{equation}\label{case1}
\left(c_0\frac{\ell}{k}\right)^\ell=\left(\frac{\ell-1}{2\ell}\right)^\ell=\frac{1}{16}<\frac{1}{4}\leq H_i^{2}.
\end{equation}
\item Case 2: $\left\lceil\frac{2}{\log 2}\log\frac{1}{H_i}\right\rceil>2$\\
Set $\alpha:=\frac{2}{\log 2}$. Then $\ell=\lceil-\alpha\log H_i\rceil\geq-\alpha\log H_i$ and furthermore we have
\begin{align}\label{case2}
\left(c_0\frac{\ell}{k}\right)^\ell&=\left(\frac{\ell-1}{2\ell}\right)^\ell\leq 2^{-\ell}=e^{-\ell\log 2}\leq H_i^{\alpha\log 2}=H_i^{2}.
\end{align}
\end{itemize}
The assertion follows by combining \eqref{Boerm1}, \eqref{Boerm2}, \eqref{case1}, and \eqref{case2}.
\end{proof}
\begin{lemma}\label{lemma_v_near_far}
Define $d_i^{near}:=u_i^{near}-\tilde{u}_i^{near}$ and $d_i^{far}:=u_i^{far}-\hat{u}_i^{far}$ with $u_i^{near}$, $u_i^{far}$ as in \eqref{splitting}, $\tilde{u}_i^{near}$ as in \eqref{approx_near_far} and $\hat{u}_i^{far}$ as in Lemma \ref{lemma_aprox_tilde_u_far}. Set
\begin{equation*}
v^{near}:=\sum_{i=1}^Nb_id_i^{near}\qquad \text{and}\qquad v^{far}:=\sum_{i=1}^Nb_id_i^{far}.
\end{equation*}
Then the estimates
\begin{align*}
&\|\nabla v^{near}\|_{L^2(\Omega)}^2\leq 2M_0\sum\limits_{i=1}^N\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}^2+\frac{C^2}{H_i^2}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}^2\right)\\
&\|\nabla v^{far}\|_{L^2(\Omega)}^2\leq 2M_0\sum\limits_{i=1}^N\left(\left\|\nabla d_i^{far}\right\|_{L^2\left(\omega_i\right)}^2+\frac{C^2}{H_i^2}\left\|d_i^{far}\right\|_{L^2\left(\omega_i\right)}^2\right)
\end{align*}
hold with $M_0$ as in \eqref{overlap_const} and $H_i$ as in \eqref{local_mesh_width}.
\end{lemma}
\begin{proof}
Applying Cauchy--Schwarz inequality, using the Leibniz rule for products, a triangle inequality and an inverse inequality for $b_i$ we obtain the estimate
\begin{align*}
\left\|\nabla v^{near}\right\|_{L^2\left(\Omega\right)}^2&=\langle\nabla v^{near},\nabla v^{near}\rangle_{L^2\left(\Omega\right)}=\sum\limits_{i=1}^N\langle\nabla\left(b_id_i^{near}\right),\nabla v^{near}\rangle_{L^2\left(\Omega\right)}\\
&\leq \sum\limits_{i=1}^N\left\|\nabla\left(b_id_i^{near}\right)\right\|_{L^2\left(\omega_i\right)}\left\|\nabla v^{near}\right\|_{L^2\left(\omega_i\right)}\\
&\leq \sum\limits_{i=1}^N\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}+\frac{C}{H_i}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}\right)\left\|\nabla v^{near}\right\|_{L^2\left(\omega_i\right)}.
\end{align*}
A Young's inequality leads to
\begin{equation*}
\left\|\nabla v^{near}\right\|_{L^2\left(\Omega\right)}^2\leq \sum\limits_{i=1}^N\frac{\epsilon^2}{2}\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}+\frac{C}{H_i}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}\right)^2+\frac{1}{2\epsilon^2}\sum\limits_{i=1}^N\left\|\nabla v^{near}\right\|_{L^2\left(\omega_i\right)}^2.
\end{equation*}
The choice $\epsilon^2=M_0$ (cf.\ Remark \ref{rem_overlap}) yields
\begin{IEEEeqnarray}{rCl}\label{eps3}
\left\|\nabla v^{near}\right\|_{L^2\left(\Omega\right)}^2 &\leq & \sum\limits_{i=1}^N\frac{M_0}{2}\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}+\frac{C}{H_i}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}\right)^2
\nonumber\\
&& +\frac{1}{2M_0}\sum\limits_{i=1}^n\left\|\nabla v^{near}\right\|_{L^2\left(\omega_i\right)}^2
\nonumber\\
&\leq & \sum\limits_{i=1}^N\frac{M_0}{2}\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}+\frac{C}{H_i}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}\right)^2
\nonumber\\
&& +\frac{1}{2}\left\|\nabla v^{near}\right\|_{L^2\left(\Omega\right)}^2.
\end{IEEEeqnarray}
Hence, by \eqref{eps3} and a triangle inequality we get
\begin{align*}
\left\|\nabla v^{near}\right\|_{L^2\left(\Omega\right)}^2&\leq M_0\sum\limits_{i=1}^N\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}+\frac{C}{H_i}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}\right)^2\nonumber\\
&\leq 2M_0\sum\limits_{i=1}^N\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}^2+\frac{C^2}{H_i^2}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}^2\right).
\end{align*}
This shows the first estimate. The proof of the second estimate is verbatim the same.
\end{proof}
\begin{lemma}\label{lemma_est_d_near}
Let $d_i^{near}$ as in Lemma \ref{lemma_v_near_far}. If Assumption \ref{ass} holds, then
\begin{equation*}
\|\nabla d_i^{near}\|_{L^2(\omega_{i})}\leq C\left( H_i^2+H_i\right)\left\|f\right\|_{L^2(\omega_{i,1})}
\end{equation*}
and
\begin{equation*}
\| d_i^{near}\|_{L^2(\omega_i)}\leq C\left(H_i^3+H_i^2\right)\left\|f\right\|_{L^2(\omega_{i,1})}
\end{equation*}
with $H_i$ as in \eqref{local_mesh_width} and constants $C$ which depend on $\alpha$, $\beta$ (cf.\ \eqref{coeff}).
\end{lemma}
\begin{proof}
\eqref{splitting}, \eqref{approx_near_far} and a triangle inequality yield
\begin{IEEEeqnarray}{rCl}\label{nabla_d_near}
\left\|\nabla d_i^{near}\right\|_{L^2(\omega_{i,2})}& = & \left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{near})-\tilde{L}_{\omega_{i,2}}^{-1}(P_ig_i^{near})\right)\right\|_{L^2(\omega_{i,2})}
\nonumber\\
&\leq & \left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{near})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near})\right)\right\|_{L^2(\omega_{i,2})}
\nonumber\\
&& +\left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near}-P_ig_i^{near})\right)\right\|_{L^2(\omega_{i,2})},
\end{IEEEeqnarray}
where $P_i$ is the $L^2$-orthogonal projection of $L^2(\omega_{i,1})$ onto $S_0(\mathcal{G}_{i,1})$.
In order to estimate the second term of \eqref{nabla_d_near} we use \eqref{L} and Lemma \ref{lemma_projection}. This leads to
\begin{align}\label{d_near_nabla_second}
\left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near}-P_ig_i^{near})\right)\right\|_{L^2(\omega_{i,2})}&\leq\frac{C}{\alpha}\|g_i^{near}-P_ig_i^{near}\|_{H^{-1}(\omega_{i,2})}\nonumber\\
&\leq C\frac{H_i}{\alpha}\|g_i^{near}\|_{L^2(\omega_{i,2})}.
\end{align}
By \eqref{nabla_d_near}, \eqref{assumption}, \eqref{d_near_nabla_second} and the definition of $g_i^{near}$ (cf.\ \eqref{g_i_near}) we obtain
\begin{align*}
\left\|\nabla d_i^{near}\right\|_{L^2(\omega_{i,2})}&\leq \frac{\beta}{\alpha}CH_i^2\left\|g_i^{near}\right\|_{L^2(\omega_{i,2})}+C\frac{H_i}{\alpha}\left\|g_i^{near}\right\|_{L^2(\omega_{i,2})}\\
&\leq C\left( H_i^2+H_i\right)\left\|f\right\|_{L^2(\omega_{i,1})}.
\end{align*}
Since $\omega_{i}\subset\omega_{i,2}$ we also have
\begin{equation*}
\left\|\nabla d_i^{near}\right\|_{L^2(\omega_{i})}\leq C\left( H_i^2+H_i\right)\left\|f\right\|_{L^2(\omega_{i,1})} .
\end{equation*}
By \eqref{splitting}, \eqref{approx_near_far}, a triangle inequality and Friedrichs' inequality we get
\begin{IEEEeqnarray}{rCl}\label{d_near}
\left\|d_i^{near}\right\|_{L^2(\omega_i)}&=&\left\|L_{\omega_{i,2}}^{-1}(g_i^{near})-\tilde{L}_{\omega_{i,2}}^{-1}(P_ig_i^{near})\right\|_{L^2(\omega_i)}
\nonumber\\
&\leq& \left\|L_{\omega_{i,2}}^{-1}(g_i^{near})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near})\right\|_{L^2(\omega_{i})}+\left\| \tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near}-P_ig_i^{near})\right\|_{L^2(\omega_i)}
\nonumber\\
&\leq & CH_i \Big(\left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{near})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near})\right)\right\|_{L^2(\omega_{i,2})}
\nonumber\\
&&+\left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{near}-P_ig_i^{near})\right)\right\|_{L^2(\omega_{i,2})}\Big).
\end{IEEEeqnarray}
The combination of \eqref{d_near}, \eqref{assumption} and \eqref{d_near_nabla_second} leads to
\begin{align*}
\left\|d_i^{near}\right\|_{L^2(\omega_i)}&\leq C\frac{\beta}{\alpha}H_i^3\left\|g_i^{near}\right\|_{L^2(\omega_{i,2})}+C\frac{H_i^2}{\alpha}\left\|g_i^{near}\right\|_{L^2(\omega_{i,2})}
\\
&\leq C\left(H_i^3+H_i^2\right)\left\|f\right\|_{L^2(\omega_{i,1})}.
\end{align*}
In the last step we used the definition of $g_i^{near}$ (cf.\ \eqref{g_i_near}).
\end{proof}
\begin{lemma}\label{lemma_est_d_far}
Let $d_i^{far}$ as in Lemma \ref{lemma_v_near_far}. If Assumption \ref{ass} holds, then
\begin{equation*}
\|\nabla d_i^{far}\|_{L^2(\omega_{i})}\leq C \left(H_i^2+h_i\right)\|g_i^{far}\|_{L^2(\omega_i^{far})}
\end{equation*}
and
\begin{equation*}
\| d_i^{far}\|_{L^2(\omega_i)}\leq C\left(H_i^3+H_ih_i\right)\|g_i^{far}\|_{L^2(\omega_i^{far})}
\end{equation*}
with $H_i$ as in \eqref{local_mesh_width} and $h_i:=\max_{\tau \in\mathcal{R}^t(\mathcal{G}_i^{far})}\diam\tau$ is the mesh width of the refined mesh $\mathcal{R}^t(\mathcal{G}_i^{far})$ (cf.\ \eqref{omega_far} and \eqref{refinement_op}). The constants $C$ depend on $\alpha$, $\beta$ (cf.\ \eqref{coeff}).
\end{lemma}
\begin{proof}
By the definition of $d_i^{far}$, \eqref{splitting} and two triangle inequalities we get
\begin{IEEEeqnarray}{rCl}\label{nabla_d_far}
\|\nabla d_i^{far}\|_{L^2(\omega_i)}&=&\left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{far})-\hat{u}_i^{far}\right)\right\|_{L^2(\omega_i)}
\nonumber\\
&\leq & \left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{far})-\tilde{L}_{\omega_{i,2}}^{-1}(P_i^tg_i^{far})\right)\right\|_{L^2(\omega_{i})}+\|\nabla(\tilde{u}_i^{far}-\hat{u}_i^{far})\|_{L^2(\omega_i)}
\nonumber\\
&\leq &\left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{far})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far})\right)\right\|_{L^2(\omega_{i})}+\left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far}-P_i^tg_i^{far})\right)\right\|_{L^2(\omega_{i})}
\nonumber\\
&&+\|\nabla(\tilde{u}_i^{far}-\hat{u}_i^{far})\|_{L^2(\omega_i)},
\end{IEEEeqnarray}
where $P_i^t$ denotes the $L^2$-orthogonal projection of $L^2(\omega_i^{far})$ onto $S_0(\mathcal{R}^t(\mathcal{G}_i^{far}))$ and $\tilde{u}_i^{far}$ is as in \eqref{approx_near_far}.
For the first term of \eqref{nabla_d_far} we can use that $\omega_i\subset\omega_{i,2}$ and \eqref{assumption}. This leads to
\begin{align}\label{first_term_nabla_d_far}
\left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{far})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far})\right)\right\|_{L^2(\omega_{i})}&\leq \left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{far})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far})\right)\right\|_{L^2(\omega_{i,2})}\nonumber\\
&\leq \frac{\beta}{\alpha}CH_i^2\|g_i^{far}\|_{L^2(\omega_{i,2})}.
\end{align}
In order to get an estimate for the second term of \eqref{nabla_d_far} we use $\omega_i\subset\omega_{i,2}$, \eqref{L} and Lemma \ref{lemma_projection}. This yields
\begin{align}\label{second_term_nabla_d_far}
\left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far}-P_i^tg_i^{far})\right)\right\|_{L^2(\omega_{i})}&\leq \left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far}-P_i^tg_i^{far})\right)\right\|_{L^2(\omega_{i,2})}\nonumber\\
&\leq \frac{C}{\alpha}\|g_i^{far}-P_i^tg_i^{far}\|_{H^{-1}(\omega_{i,2})}\nonumber\\
&\leq C\frac{h_i}{\alpha}\|g_i^{far}\|_{L^2(\omega_{i,2})}.
\end{align}
The third term of \eqref{nabla_d_far} can be estimated by Lemma \ref{lemma_aprox_tilde_u_far}, \eqref{approx_near_far}, using that $\omega_{i,1}\subset\omega_{i,2}$, \eqref{L} and Friedrichs' inequality. Thus we have
\begin{align}\label{third_term_nabla_d_far}
\|\nabla(\tilde{u}_i^{far}-\hat{u}_i^{far})\|_{L^2(\omega_i)}&\leq CH_i^{2}\|\nabla\tilde{u}_i^{far}\|_{L^2(\omega_{i,1})}\nonumber\\
&\leq CH_i^{2}\left\|\nabla \tilde{L}_{\omega_{i,2}}^{-1}(P_i^tg_i^{far})\right\|_{L^2(\omega_{i,2})}\nonumber\\
&\leq\frac{C}{\alpha}H_i^{2}\|P_i^tg_i^{far}\|_{H^{-1}(\omega_{i,2})}\nonumber\\
&\leq\frac{C}{\alpha}H_i^{2}\|g_i^{far}\|_{L^2(\omega_{i,2})}.
\end{align}
Hence, the combination of \eqref{nabla_d_far}, \eqref{first_term_nabla_d_far}, \eqref{second_term_nabla_d_far}, \eqref{third_term_nabla_d_far} and recalling that $g_i^{far}|_{\omega_{i,1}}=0$ yields
\begin{align*}
\|\nabla d_i^{far}\|_{L^2(\omega_i)}&\leq \left(\frac{\beta}{\alpha}CH_i^2+C\frac{h_i}{\alpha}+\frac{C}{\alpha}H_i^{2}\right)\|g_i^{far}\|_{L^2(\omega_i^{far})}\\
&\leq C(H_i^2+h_i)\|g_i^{far}\|_{L^2(\omega_i^{far})}.
\end{align*}
The estimate for the $L^2$-norm of $d_i^{far}$ can be obtained similarly. By triangle inequalities, Friedrichs' inequality, Lemma \ref{lemma_aprox_tilde_u_far}, \eqref{assumption}, \eqref{L} and Lemma \ref{lemma_projection} we get
\begin{IEEEeqnarray*}{rCl}
\|d_i^{far}\|_{L^2(\omega_i)}&=&\left\|L_{\omega_{i,2}}^{-1}(g_i^{far})-\hat{u}_i^{far}\right\|_{L^2(\omega_i)}
\nonumber\\
&\leq & \left\|L_{\omega_{i,2}}^{-1}(g_i^{far})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far})\right\|_{L^2(\omega_i)}+\left\| \tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far}-P_i^tg_i^{far})\right\|_{L^2(\omega_i)}
\nonumber\\
&&+\|\tilde{u}_i^{far}-\hat{u}_i^{far}\|_{L^2(\omega_i)}
\nonumber\\
&\leq & CH_i\left\|\nabla\left(L_{\omega_{i,2}}^{-1}(g_i^{far})-\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far})\right)\right\|_{L^2(\omega_{i,2})}
\nonumber\\
&&+CH_i\left\|\nabla\left(\tilde{L}_{\omega_{i,2}}^{-1}(g_i^{far}-P_i^tg_i^{far})\right)\right\|_{L^2(\omega_{i,2})}
+CH_i^{3}\|\nabla\tilde{u}_i^{far}\|_{L^2(\omega_{i,1})}
\nonumber\\
&\leq & \left(C\frac{\beta}{\alpha}H_i^3+CH_i\frac{h_i}{\alpha}+\frac{C}{\alpha}H_i^{3}\right)\|g_i^{far}\|_{L^2(\omega_i^{far})}
\nonumber\\
&\leq & C(H_i^3+H_ih_i)\|g_i^{far}\|_{L^2(\omega_i^{far})}.
\end{IEEEeqnarray*}
\end{proof}
\begin{theorem}\label{main_theo}
Let $\Omega\subset\mathbb{R}^d$ ($d\geq 2$) be a bounded domain with $\partial\Omega\in C^1$ and let Assumption \ref{ass} be satisfied. Let $u$ denote the solution of \eqref{prob_weak} and $u_{AL}^{GAL}$ its approximation given by \eqref{Galerkin_sol_II}. Let the parameters $\ell$ and $k$ in the definition of the farfield part of $V_{AL}$ be chosen according to
\begin{equation*}
\ell:=\max \left\{2,\left\lceil\frac{2}{\log 2}\log\frac{1}{H_i}\right\rceil\right\}\quad \text{and}\quad k:=\left\lceil\frac{2c_0\ell^2}{(\ell-1)}\right\rceil
\end{equation*}
for some $c_0=O(1)$. Moreover let $Q\in (6,\infty)$ and $P\in(2Q/(Q-6),\infty)$ be fixed. Assume that $A$ satisfies \eqref{coeff} as well as $\alpha/\beta\in[\max\{1-1/K_Q,1-1/K_P\},1]$ with $K_Q$ and $K_P$ as in Theorem \ref{theo_Simader}. Further let $f\in L^{P}(\Omega)$ and assume that there exists a constant $C>0$ such that $N\leq CH^{-d}$ holds. If the refinement parameter $t$ is chosen according to
\begin{equation*}
t:=\left\lceil\lb\frac{1}{H}\right\rceil,
\end{equation*}
then the error estimate
\begin{equation}\label{main_estimate}
\left\|A^{1/2}\nabla(u-u_{AL}^{GAL})\right\|_{L^2(\Omega)}\leq CH\left\|f\right\|_{L^{p}(\Omega)}
\end{equation}
holds for any $p\in (2Q/(Q-6),P]$ with $p=2q/(q-2)$ for some $2<q< \frac{Q}{3}$. The constant $C$ depends on $\alpha$, $\beta$ and $p$.\\
For the dimension we have
\begin{equation}\label{dimension_tilde_VAL-II}
\dim V_{AL}\leq CN\ell^{d+1}\leq CH^{-d}\log^{d+1}\frac{1}{H}.
\end{equation}
\end{theorem}
\begin{proof}
Let $f\in L^P(\Omega)$ and set $u:=L_\Omega^{-1}f$. Let $u_{AL}^{GAL}\in V_{AL}$ be the Galerkin approximation of $u$ given by \eqref{Galerkin_sol_II}. By the Galerkin orthogonality we obtain for any $u_{AL}\in V_{AL}$
\begin{align*}
\|A^{1/2}\nabla(u-u_{AL}^{GAL})\|_{L^2(\Omega)}^2&=a(u-u_{AL}^{GAL},u-u_{AL}^{GAL})\\&=a(u-u_{AL}^{GAL},u-u_{AL})\\&\leq \|A^{1/2}\nabla(u-u_{AL}^{GAL})\|_{L^2(\Omega)}\|A^{1/2}\nabla(u-u_{AL})\|_{L^2(\Omega)}.
\end{align*}
Hence,
\begin{equation}\label{Galerkin_ortho}
\|A^{1/2}\nabla(u-u_{AL}^{GAL})\|_{L^2(\Omega)}\leq \|A^{1/2}\nabla(u-u_{AL})\|_{L^2(\Omega)}\quad\forall\; u_{AL}\in V_{AL}.
\end{equation}
Further let $u_i^{near}$ and $u_i^{far}$ as in \eqref{splitting}. Then it holds that
\begin{equation*}
u=\sum_{i=1}^N b_i(u_i^{near}+u_i^{far}).
\end{equation*}
Let $\tilde{u}_i^{near}$ and $\tilde{u}_i^{far}$ as in \eqref{approx_near_far}. We choose $\hat{u}_i^{far}$ as in Lemma \ref{lemma_aprox_tilde_u_far} and $u_{AL}\in V_{AL}$ by
\begin{equation*}
u_{AL}=\sum_{i=1}^N b_i(\tilde{u}_i^{near}+\hat{u}_i^{far}).
\end{equation*}
Using this notation we have
\begin{equation*}
u-u_{AL}=\sum_{i=1}^Nb_i(u_i^{near}-\tilde{u}_i^{near})+\sum_{i=1}^Nb_i(u_i^{far}-\hat{u}_i^{far}).
\end{equation*}
First we consider the nearfield part. Let $d_i^{near}:=u_i^{near}-\tilde{u}_i^{near}$ and set
\begin{equation*}
v^{near}:=\sum_{i=1}^Nb_id_i^{near}.
\end{equation*}
By Lemma \ref{lemma_v_near_far} we know that
\begin{equation*}
\left\|\nabla v^{near}\right\|_{L^2\left(\Omega\right)}^2
\leq 2M_0 \sum\limits_{i=1}^N\left(\left\|\nabla d_i^{near}\right\|_{L^2\left(\omega_i\right)}^2+\frac{C^2}{H_i^2}\left\|d_i^{near}\right\|_{L^2\left(\omega_i\right)}^2\right).
\end{equation*}
Moreover, by Lemma \ref{lemma_est_d_near} and since every simplex $\tau$ is contained in at most $M_1$ domains $\omega_{i,1}$ (cf.\ \eqref{overlap_const}) we obtain
\begin{align*}
\left\|\nabla v^{near}\right\|_{L^2(\Omega)}&\leq C\sqrt{\sum_{i=1}^N\left(H_i^4+H_i^2\right)\left\|f\right\|_{L^2(\omega_{i,1})}^2}\\
&\leq C\left(H^2+H\right)\left\|f\right\|_{L^2(\Omega)}.
\end{align*}
Since the embedding $L^p(\Omega)\hookrightarrow L^2(\Omega)$ is continuous for any $p\geq 2$, we have
\begin{equation}\label{estimate_nearfield}
\left\|\nabla v^{near}\right\|_{L^2(\Omega)}\leq C\left(H^2+H\right)\left\|f\right\|_{L^{p}(\Omega)}\leq CH\|f\|_{L^p(\Omega)}
\end{equation}
for any $p\geq 2$.
Next we consider the farfield part. Let $d_i^{far}:=u_i^{far}-\hat{u}_i^{far}$ and set
\begin{equation*}
v^{far}:=\sum_{i=1}^Nb_id_i^{far}.
\end{equation*}
Lemma \ref{lemma_v_near_far} yields
\begin{equation*}
\|\nabla v^{far}\|_{L^2(\Omega)}^2\leq 2M_0\sum\limits_{i=1}^N\left(\|\nabla d_i^{far}\|_{L^2(\omega_i)}^2+\frac{C^2}{H_i^2}\|d_i^{far}\|_{L^2(\omega_i)}^2\right).
\end{equation*}
Due to Lemma \ref{lemma_est_d_far} we finally get with a constant $C$ depending on the mesh regularity
\begin{equation}\label{nabla_vfar}
\|\nabla v^{far}\|_{L^2(\Omega)}\leq C\sqrt{\sum_{i=1}^N\left(H_i^4+h_i^2\right)\|g_i^{far}\|_{L^2(\omega_i^{far})}^2}.
\end{equation}
By the definition of $g_i^{far}$ (cf.\ \eqref{g_i_far}) we have
\enlargethispage{-3\baselineskip}
\begin{IEEEeqnarray}{rCl}\label{gifar}
\|g_i^{far}\|_{L^2(\omega_i^{far})}&\leq & \|\chi_if\|_{L^2(\omega_i^{far})}+2\|A\nabla \chi_i\nabla u\|_{L^2(\omega_i^{far})}
\nonumber\\
&&+\|(u-\bar{u}_i)\Div(A\nabla\chi_i)\|_{L^2(\omega_i^{far})}.
\end{IEEEeqnarray}
Applying general H\"{o}lder's inequality on the first term of \eqref{gifar} and by \eqref{cutoff_estimate_1} we obtain for any $2< q< Q/3$ and any $p\in(2Q/(Q-6),P]$ such that $2/q+2/p=1$ the estimate
\begin{align}\label{first_term_g_i}
\|\chi_if\|_{L^2(\omega_i^{far})}&\leq\|\chi_i\|_{L^{q}(\omega_i^{far})}\|f\|_{L^{p}(\omega_i^{far})}\nonumber\\
&\leq CH_i^{\frac{d}{q}}\|f\|_{L^{p}(\omega_i^{far})}\nonumber\\
&=CH_i^{\frac{d}{2}-\frac{d}{p}}\|f\|_{L^{p}(\omega_i^{far})}.
\end{align}
To get an estimate of the second term of \eqref{gifar} we use general H\"{o}lder's inequality, \eqref{coeff} and \eqref{cutoff_estimate_2}. For $2< q< Q/3$ and any $p\in(2Q/(Q-6),P]$ such that $2/q+2/p=1$ it holds
\begin{align}\label{second_term_g_i}
\|A\nabla\chi_i\nabla u\|_{L^2(\omega_i^{far})}&\leq \|A\|_{L^\infty(\omega_i^{far})}\|\nabla\chi_i\|_{L^{q}(\omega_i^{far})}\|\nabla u\|_{L^{p}(\omega_i^{far})}\nonumber\\
&\leq C\beta H_i^{\frac{d}{2}-\frac{d}{p}-1}\|\nabla u\|_{L^{p}(\omega_i^{far})}.
\end{align}
For the third term of \eqref{gifar} we obtain by general H\"{o}lder's inequality, using \eqref{cutoff_estimate_3} and by Poincar\'{e}'s inequality for $2< q< Q/3$ and any $p\in(2Q/(Q-6),P]$ such that $2/q+2/p=1$
\begin{align}\label{third_term_g_i}
\|(u-\bar{u}_i)\Div(A\nabla\chi_i)\|_{L^2(\omega_i^{far})}&\leq \|\Div(A\nabla\chi_i)\|_{L^{q}(\omega_i^{far})}\|u-\bar{u}_i\|_{L^p(\omega_i^{far})}\nonumber\\
&\leq CH_i^{\frac{d}{2}-\frac{d}{p}-1}\|\nabla u\|_{L^{p}(\omega_i^{far})}.
\end{align}
Next, we want to estimate the square root of $\sum_{i=1}^N (H_i^4+h_i^2)\|\chi_i f\|_{L^2(\omega_i^{far})}^2$. For this we set $\gamma_i:=(H_i^4+h_i^2)H_i^{d-2d/p}$ and $\delta_i:=\|f\|_{L^{p}(\omega_i^{far})}^2$. By \eqref{first_term_g_i} we get
\begin{align*}
\sqrt{\sum\limits_{i=1}^N (H_i^4+h_i^2)\|\chi_i f\|_{L^2(\omega_i^{far})}^2}&\leq C\sqrt{\sum\limits_{i=1}^N(H_i^4+h_i^2)H_i^{d-\frac{2d}{p}}\|f\|_{L^{p}(\omega_i^{far})}^2}\\
&= C\sqrt{\sum\limits_{i=1}^N \gamma_i\delta_i}.
\end{align*}
Applying a discrete H\"{o}lder's inequality with $r:=p/2$ and $r'=p/(p-2)$ yields
\begin{align*}
\sqrt{\sum\limits_{i=1}^N (H_i^4+ h_i^2)\|\chi_i f\|_{L^2(\omega_i^{far})}^2}&\leq C \sqrt{\|\gamma_i\|_{\ell^{r'}} \|\delta_i\|_{\ell^r}}\\
&=C\sqrt{\left(\sum\limits_{i=1}^N \gamma_i^{r'}\right)^{\frac{1}{{r'}}}\left(\sum\limits_{i=1}^N \delta_i^{r}\right)^{\frac{1}{r}}}\\
&=C\left(\sum_{i=1}^N (H_i^4+h_i^2)^{\frac{p}{p-2}}H_i^{d}\right)^{\frac{p-2}{2p}}\left(\sum\limits_{i=1}^N\|f\|_{L^p(\omega_i^{far})}^{p}\right)^{\frac{1}{p}}.
\end{align*}
Since $\omega_i^{far}\subset \omega_{i,2}$ and every simplex $\tau$ is contained in at most $M_2$ domains $\omega_{i,2}$ (cf.\ \eqref{overlap_const}) we obtain
\begin{align}\label{first_term_sum}
\sqrt{\sum\limits_{i=1}^N (H_i^4+h_i^2)\|\chi_i f\|_{L^2(\omega_i^{far})}^2}&\leq C\left(N \max_{1\leq i\leq N}(H_i^4+h_i^2)^{\frac{p}{p-2}}H_i^{d}\right)^{\frac{p-2}{2p}} \left(\sum\limits_{i=1}^N\|f\|_{L^{p}(\omega_{i,2})}^{p}\right)^{\frac{1}{p}}\nonumber\\
&\leq C (H^2+h)\|f\|_{L^{p}(\Omega)}.
\end{align}
The last inequality follows due to the assumption that $N\leq CH^{-d}$.
Now, we want to estimate the square root of $\sum_{i=1}^N (H_i^4+h_i^2)\|A\nabla\chi_i\nabla u\|_{L^2(\omega_i^{far})}^2$ in a similar way.
By \eqref{second_term_g_i} and using a discrete H\"{o}lder's inequality with $r, r', \gamma_i$ as before and $\delta_i:=H_i^{-2}\|\nabla u\|_{L^{p}(\omega_i^{far})}^2$ we get
\begin{align*}
\sqrt{\sum\limits_{i=1}^N (H_i^4+h_i^2)\|A\nabla\chi_i\nabla u\|_{L^2(\omega_i^{far})}^2}&\leq C\beta\sqrt{\sum\limits_{i=1}^N(H_i^4+h_i^2)H_i^{d-\frac{2d}{p}-2}\|\nabla u\|_{L^{p}(\omega_i^{far})}^2} \\
&\leq C\beta \sqrt{\|\gamma_i\|_{\ell^{r'}} \|\delta_i\|_{\ell^r}}\\
&\leq C\beta (H^2+h)\left(\sum\limits_{i=1}^N H_i^{-p}\|\nabla u\|_{L^{p}(\omega_i^{far})}^{p}\right)^{\frac{1}{p}}\\
&\leq C\beta (H^2+h)\left(\max_{1\leq i\leq N}H_i^{-p}\sum\limits_{i=1}^N\|\nabla u\|_{L^{p}(\omega_{i,2})}^{p}\right)^{\frac{1}{p}}\\
&\leq C\beta \left(H+\frac{h}{H}\right)\|\nabla u\|_{L^{p}(\Omega)}.
\end{align*}
By Theorem \ref{theo_gradient_estimate} we obtain\footnote{Note that \begin{align*}
\left\|F\right\|_{W^{-1,p}(\Omega)}&=\sup_{\substack{v\in W_0^{1,p'}(\Omega)\\ \|v\|W^{1,p'}(\Omega) \leq 1}}\left|\int_\Omega\! fv\right|
\leq \sup_{\substack{v\in W_0^{1,p'}(\Omega)\\ \|v\|W^{1,p'}(\Omega)\leq 1}}\left\|f\right\|_{L^p(\Omega)}\left\|v\right\|_{L^{p'}(\Omega)}
\leq \left\|f\right\|_{L^p(\Omega)}.
\end{align*}}
\enlargethispage{-3\baselineskip}
\begin{align}\label{second_term_sum}
\sqrt{\sum\limits_{i=1}^N \left(H_i^4+h_i^2\right)\|A\nabla\chi_i\nabla u\|_{L^2(\omega_i^{far})}^2}&\leq C\beta \left(H+ \frac{h}{H}\right)\|F\|_{W^{-1,p}(\Omega)}\nonumber\\
&\leq C\beta \left(H+\frac{h}{H}\right)\|f\|_{L^{p}(\Omega)}.
\end{align}
Estimate \eqref{third_term_g_i} and the same arguments as above yield
\begin{align}\label{third_term_sum}
\sqrt{\sum\limits_{i=1}^N (H_i^4+h_i^2) \|(u-\bar{u}_i)\Div(A\nabla\chi_i)\|_{L^2(\omega_i^{far})}^2}&\leq C \sqrt{\sum\limits_{i=1}^N(H_i^4+h_i^2)H_i^{d-\frac{2d}{p}-2}\|\nabla u\|_{L^{p}(\omega_i^{far})}^2}\nonumber\\
&\leq C \left(H+\frac{h}{H}\right)\|f\|_{L^{p}(\Omega)}.
\end{align}
The combination of \eqref{nabla_vfar}, \eqref{gifar}, \eqref{first_term_sum}, \eqref{second_term_sum} and \eqref{third_term_sum} yields
\begin{equation*}
\|\nabla v^{far}\|_{L^2(\Omega)}\leq C\left(H^2+H+h+\frac{h}{H}\right)\|f\|_{L^{p}(\Omega)}
\end{equation*}
for any $p\in(2Q/(Q-6),P]$ such that $2/q+2/p=1$ for some $2<q< Q/3$. The constant $C$ depends on $\alpha$, $\beta$ and $p$.
The small mesh size $h$ arises by $t$-fold refinement of the local coarse grid so that
\begin{equation}\label{choice_t_2d}
h\leq CH2^{-t}.
\end{equation}
By choosing
\begin{equation*}
t=\left\lceil\lb\frac{1}{H}\right\rceil
\end{equation*}
in \eqref{choice_t_2d} $h$ satisfies
\begin{equation*}
h\leq CH^2.
\end{equation*}
Thus it holds
\begin{equation}\label{estimate_farfield}
\|\nabla v^{far}\|_{L^2(\Omega)}\leq C\left(H^2+H+h+\frac{h}{H}\right)\|f\|_{L^{p}(\Omega)}\leq CH\|f\|_{L^{p}(\Omega)}.
\end{equation}
The combination of \eqref{Galerkin_ortho}, \eqref{estimate_nearfield} and \eqref{estimate_farfield} leads to
\begin{align*}
\|A^{1/2}\nabla(u-u_{AL}^{GAL})\|_{L^2(\Omega)}&\leq \|A^{1/2}\nabla(u-u_{AL})\|_{L^2(\Omega)}\\
&=\|A^{1/2}(\nabla v^{near}+\nabla v^{far})\|_{L^2(\Omega)}\\
&\leq \|A^{1/2}\|_{L^\infty(\Omega)}\left(\|\nabla v^{near}\|_{L^2(\Omega)}+\|\nabla v^{far}\|_{L^2(\Omega)}\right)\\
&\leq \sqrt{\beta} C H \|f\|_{L^p(\Omega)}
\end{align*}
which finally proves estimate \eqref{main_estimate}. Estimate \eqref{dimension_tilde_VAL-II} can be seen as follows: From the definition of $V_{AL}$ (cf.\ \eqref{def:VAL-II}) it is clear that
\begin{equation*}
\dim V_{AL}\leq N(\dim V_i^{near}+\dim V_i^{far})
\end{equation*}
holds. The choice $k:=\left\lceil\frac{2c_0\ell^2}{(\ell-1)}\right\rceil$ yields
\begin{align*}
k&\leq \frac{2c_0\ell^2}{\ell -1}+1\\
&=\frac{2c_0\ell(\ell -1)}{\ell-1}+\frac{2c_0(\ell-1)}{\ell-1}+\frac{2c_0}{\ell-1}+1\\
&=2c_0\ell+2c_0+\frac{2c_0}{\ell-1}+1\\
&\leq 2c_0\ell+4c_0+1\\
&\leq \ell\left(4c_0+\frac{1}{2}\right).
\end{align*}
In the last inequality we used that $\ell\geq 2$. Remark \ref{remdim} and the above computation show that
\begin{equation*}
\dim V_i^{far}\leq \ell k^d\leq \left(4c_0+\frac{1}{2}\right)^d\ell^{d+1}.
\end{equation*}
Obviously we have $\dim V_i^{near}=O(1)$. Hence,
\begin{equation*}
\dim V_{AL}\leq CN\ell^{2}\leq CH^{-d}\log^{d+1}\frac{1}{H}.
\end{equation*}
The last inequality follows by the assumption that there exists a constant $C>0$ such that $N\leq CH^{-d}$ and the choice of $\ell$.
\end{proof}
{}
\section*{Acknowledgment}
I would like to thank Prof. Dr. Stefan Sauter for many interesting and helpful discussions.
\end{document} | math | 62,760 |
\begin{document}
\title[The general linear group of degree $n$ for 3D matrices]{The general
linear group of degree $n$ for 3D matrices $GL(n,n,p;F)$}
\author{Orgest ZAKA}
\address[O. ZAKA]{Department of Mathematics, Faculty of Technical
Science, University of Vlora "Ismail QEMAL", Vlora, Albania}
\email[Orgest ZAKA]{[email protected]}
\date{December 10, 2018}
\subjclass[2000]{Primary15XX, 20XX, 47Dxx; Secondary 15A09, 15A15, 20H20, }
\keywords{3D matrix, determinant for 3D matrices, inverse of 3D matrix,
general linear group for 3D matrices}
\begin{abstract}
In this article we give the meaning of the determinant for 3D matrices with
elements from a field F, and the meaning of 3D inverse matrix. Based on my
previous work titled '3D Matrix Rings', we want to constructed the 'general
linear group of degree $n$ for 3D matrices, which i mark with $GL(n,n,p;F)$'
for 3D-matrices, analog to 'general linear group of degree $n$' known.
\end{abstract}
\maketitle
\section{Introduction, 3D matrix}
This paper comes as a continuation of the ideas that arise based on my
previous work, of the 3D matrix ring with element from an whatever field $
\mathbf{F}$ (see \cite{[OZ3DM]}). At this point we are making a brief
summary associated with 3D matrices and the 3D matrix presented in \cite
{[OZ3DM]}. Our objective is to constructed the 'general linear group of
degree $n$ for 3D matrices, which i mark with $GL(n,n,p;F)$' for
3D-matrices, analog to 'general linear group of degree $n$' known. For this
we need new notions, which we will give in the following points.
\begin{definition}
\cite{[OZ3DM]} 3-dimensional $m\times n\times p$ matrix will call, a matrix
which has: m-\textbf{horizontal layers} (analogous to m-rows), n-\textbf{
vertical page} (analogue with n-columns in the usual matrices) and p-\textbf{
vertical layers} (p-1 of which are hidden).
The set of these matrixes the write how:
\begin{equation*}
\mathcal{M}_{m\times n\times p}(\mathbf{F})=\left\{ a_{i,j,k}|a_{i,j,k}\in F-
\text{field }\forall i=\overline{1,m};\text{ }j=\overline{1,n};\text{ }k=
\overline{1,p}\right\}
\end{equation*}
\end{definition}
\subsection{ADDITION OF 3D MATRIX}
\begin{definition}
\cite{[OZ3DM]} The addition of two matrices $\mathbf{A,B}\in \mathcal{M}
_{m\times n\times p}(\mathbf{F})$ we will call the matrix:
\begin{equation*}
\mathbf{C}_{m\times n\times p}(\mathbf{F})=\left\{
c_{i,j,k}|c_{i,j,k}=a_{i,j,k}+b_{i,j,k},\text{ }\forall i=\overline{1,m};
\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}\right\}
\end{equation*}
The appearance of the addition of $m\times n\times p,$ 3D matrices will be
as in Figure 1, where matrices $\mathbf{A}$ and $\mathbf{B}$ have the
following appearance,
\begin{eqnarray*}
\mathbf{A} &=&\left\{ \left( a_{i,j,k}\right) |\text{ }\forall i=\overline{
1,m};\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}\right\} ; \\
\mathbf{B} &=&\left\{ \left( b_{i,j,k}\right) |\text{ }\forall i=\overline{
1,m};\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}\right\}
\end{eqnarray*}
\begin{figure}
\caption{The appearance of the addition of 3D matrix}
\label{fig:Fig1}
\end{figure}
\end{definition}
\begin{definition}
\cite{[OZ3DM]}The 3-D, Zero matrix $m\times n\times p,$ we will called the
matrix that has all its elements zero.
\begin{equation*}
\mathbf{O=O}_{m\times n\times p}=\left\{ \left( 0_{\mathbf{F}}\right)
_{i,j,k}|\text{ }\forall i=\overline{1,m};\text{ }j=\overline{1,n};\text{ }k=
\overline{1,p}\right\} \in \mathcal{M}_{m\times n\times p}(\mathbf{F})
\end{equation*}
\end{definition}
\begin{definition}
\cite{[OZ3DM]} The opposite matric of an matrice
\begin{equation*}
\mathbf{A}_{m\times n\times p}=\left\{ \left( a_{i,j,k}\right) |\text{ }
\forall i=\overline{1,m};\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}
\right\} \in \mathcal{M}_{m\times n\times p}(\mathbf{F})
\end{equation*}
will, called matrix
\begin{equation*}
-\mathbf{A}_{m\times n\times p}=\left\{ \left( -a_{i,j,k}\right) |\text{ }
\forall i=\overline{1,m};\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}
\right\} \in \mathcal{M}_{m\times n\times p}(\mathbf{F}).
\end{equation*}
\end{definition}
Where $\left( -a_{i,j,k}\right) $ is a opposite element of element $
a_{i,j,k}\in \mathbf{F,}$ so
\begin{equation*}
a_{i,j,k}+\left( -a_{i,j,k}\right) =0_{\mathbf{F}},\forall i=\overline{1,m}
;j=\overline{1,n};k=\overline{1,p},
\end{equation*}
and $\left( \mathbf{F,+,\cdot }\right) $ is field, which satisfies the
condition
\begin{eqnarray*}
\mathbf{A}_{m\times n\times p}+\left( -\mathbf{A}_{m\times n\times p}\right)
&=&\left\{ \left( a_{i,j,k}\right) +\left( -a_{i,j,k}\right) |\text{ }
\forall i=\overline{1,m};\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}
\right\} \\
&=&\left\{ \left( 0_{\mathbf{F}}\right) _{i,j,k}|\text{ }\forall i=\overline{
1,m};\text{ }j=\overline{1,n};\text{ }k=\overline{1,p}\right\} \\
&=&\mathbf{O}_{m\times n\times p}=\mathbf{O}
\end{eqnarray*}
\begin{theorem}
\cite{[OZ3DM]}$\left( \mathcal{M}_{m\times n\times p}(\mathbf{F}),+\right) $
is abelian grup.
\end{theorem}
\subsection{THE MULTIPLICATION OF $n\times n\times p$ MATRICES}
In the same way, as have the meaning of a 3D $3\times 3\times 3$ matrix
multiplication to \cite{[OZ3DM]}, we give the definition of 3D matrix
multiplication for \ $\mathbf{A,B}\in \mathcal{M}_{n\times n\times p}(
\mathbf{F})$.
\begin{definition}
\cite{[OZ3DM]} The multiplication of two matrices $\mathbf{A,B}\in \mathcal{M
}_{n\times n\times p}(\mathbf{F})$ we will call the matrix $\mathbf{C=A\odot
B}\in \mathcal{M}_{n\times n\times p}(\mathbf{F}),$ calculated as follows:
Matrices will normally have the appearance:
\begin{equation*}
\forall \mathbf{A=}\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1p} & a_{n2p} & \cdots & a_{nnp}
\end{array}
\right) \\
\mathbf{\vdots } \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n12} & a_{n22} & \cdots & a_{nn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n11} & a_{n21} & \cdots & a_{nn1}
\end{array}
\right)
\end{array}
\right] ;\mathbf{B=}\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
b_{11p} & b_{12p} & \cdots & b_{1np} \\
b_{21p} & b_{22p} & \cdots & b_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1p} & b_{n2p} & \cdots & b_{nnp}
\end{array}
\right) \\
\mathbf{\vdots } \\
\left(
\begin{array}{cccc}
b_{112} & b_{122} & \cdots & b_{1n2} \\
b_{212} & b_{222} & \cdots & b_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n12} & b_{n22} & \cdots & b_{nn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
b_{111} & b_{121} & \cdots & b_{1n1} \\
b_{211} & b_{221} & \cdots & b_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n11} & b_{n21} & \cdots & b_{nn1}
\end{array}
\right)
\end{array}
\right]
\end{equation*}
If we write more briefly
\begin{equation*}
\mathbf{A=}\left[
\begin{array}{c}
\mathbf{A}_{p} \\
\vdots \\
\mathbf{A}_{2} \\
\mathbf{A}_{1}
\end{array}
\right] ,\mathbf{B=}\left[
\begin{array}{c}
\mathbf{B}_{p} \\
\vdots \\
\mathbf{B}_{2} \\
\mathbf{B}_{1}
\end{array}
\right] \in \mathcal{M}_{n\times n\times p}(\mathbf{F})
\end{equation*}
where $\mathbf{A}_{i}$ and $\mathbf{B}_{i}$ are the $n\times n$ matrices $
\forall i=\overline{1,p}.$ Then
\begin{equation*}
\mathbf{C=A\odot B=\left[
\begin{array}{c}
\mathbf{A}_{p} \\
\vdots \\
\mathbf{A}_{2} \\
\mathbf{A}_{1}
\end{array}
\right] \odot }\left[
\begin{array}{c}
\mathbf{B}_{p} \\
\vdots \\
\mathbf{B}_{2} \\
\mathbf{B}_{1}
\end{array}
\right] =\mathbf{\left[
\begin{array}{c}
\mathbf{A}_{p}\ast \mathbf{B}_{p} \\
\vdots \\
\mathbf{A}_{2}\ast \mathbf{B}_{2} \\
\mathbf{A}_{1}\ast \mathbf{B}_{1}
\end{array}
\right] }
\end{equation*}
where action $"\ast "$ is the usual multiplication of matrices.
\end{definition}
\section{Multi-Scalars and Multiplication of 3D matrices with multi-scalar}
In this section we will introduce the concept of '\textit{multi-scalar}',
and we will give a clear idea of the multiplication of 3D matrices with
multi-scalar.
\begin{definition}
Multi-scalar will call one $1\times 1\times p,$ 3D matrix.
\end{definition}
\begin{remark}
A multi-scalar $\ a_{1\times 1\times p}=\left[
\begin{array}{c}
\left( \alpha _{11p}\right) \\
\vdots \\
\left( \alpha _{112}\right) \\
\left( \alpha _{111}\right)
\end{array}
\right] ,$ we will call "\textbf{absolutely}" different from zero only if $
\alpha _{11i}\neq 0_{\mathbf{F}},\forall i=\overline{1,p}.$ For the "\textbf{
absolutely zero}" multi-scalar we will use the note $\widetilde{^{a}0_{
\mathbf{F}}},$ wich is $\ \widetilde{^{a}0_{\mathbf{F}}}=\left[
\begin{array}{c}
\left( 0_{\mathbf{F},p}\right) \\
\vdots \\
\left( 0_{\mathbf{F},2}\right) \\
\left( 0_{\mathbf{F,}1}\right)
\end{array}
\right] .$
\end{remark}
Let's have a \textbf{multi-scalar \ }
$\ \ $
\begin{equation*}
a_{1\times 1\times p}=\left[
\begin{array}{c}
\left( \alpha _{11p}\right) \\
\vdots \\
\left( \alpha _{112}\right) \\
\left( \alpha _{111}\right)
\end{array}
\right] ,
\end{equation*}
and 3D matrix
$\ $
\begin{equation*}
\mathbf{A}_{m\times n\times p}\mathbf{=}\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right)
\end{array}
\right] ,
\end{equation*}
\begin{definition}
\ Multiplication of 3D-matrix $\mathbf{A}_{m\times n\times p}\in \mathcal{M}
_{m\times n\times p}(\mathbf{F})$ \ with multi-scalar $a_{1\times 1\times p}$
will we call the 3D matrix $\mathbf{B}_{m\times n\times p}\in \mathcal{M}
_{m\times n\times p}(\mathbf{F}),$ calculated as follows:
\begin{eqnarray*}
\mathbf{B}_{m\times n\times p} &\mathbf{=}&a_{1\times 1\times p}\ast \mathbf{
A}_{m\times n\times p}= \\
&=&\left[
\begin{array}{c}
\alpha _{11p} \\
\vdots \\
\alpha _{112} \\
\alpha _{111}
\end{array}
\right] \ast \left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right)
\end{array}
\right]
\end{eqnarray*}
\begin{equation*}
=\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
\alpha _{11p}\cdot a_{11p} & \alpha _{11p}\cdot a_{12p} & \cdots & \alpha
_{11p}\cdot a_{1np} \\
\alpha _{11p}\cdot a_{21p} & \alpha _{11p}\cdot a_{22p} & \cdots & \alpha
_{11p}\cdot a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha _{11p}\cdot a_{m1p} & \alpha _{11p}\cdot a_{m2p} & \cdots & \alpha
_{11p}\cdot a_{mnp}
\end{array}
\right) \\
\mathbf{\vdots } \\
\left(
\begin{array}{cccc}
\alpha _{112}\cdot a_{112} & \alpha _{112}\cdot a_{122} & \cdots & \alpha
_{112}\cdot a_{1n2} \\
\alpha _{112}\cdot a_{212} & \alpha _{112}\cdot a_{222} & \cdots & \alpha
_{112}\cdot a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha _{112}\cdot a_{m12} & \alpha _{112}\cdot a_{m22} & \cdots & \alpha
_{112}\cdot a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
\alpha _{111}\cdot a_{111} & \alpha _{111}\cdot a_{121} & \cdots & \alpha
_{111}\cdot a_{1n1} \\
\alpha _{111}\cdot a_{211} & \alpha _{111}\cdot a_{221} & \cdots & \alpha
_{111}\cdot a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha _{111}\cdot a_{m11} & \alpha _{111}\cdot a_{m21} & \cdots & \alpha
_{111}\cdot a_{mn1}
\end{array}
\right)
\end{array}
\right]
\end{equation*}
\end{definition}
So multiplication of the 3D matrix with a multi-scalar is a function
\begin{equation*}
\ast :\mathcal{M}_{1\times 1\times p}(\mathbf{F})\times \mathcal{M}_{m\times
n\times p}(\mathbf{F})\longrightarrow \mathcal{M}_{m\times n\times p}(
\mathbf{F}).
\end{equation*}
\begin{example}
Let's have a multi-scalar \
\begin{equation*}
a_{1\times 1\times 3}=\left[
\begin{array}{c}
\left( 2\right) \\
\left( 5\right) \\
\left( 3\right)
\end{array}
\right] \ \
\end{equation*}
$\ $
and 3D matrix
\begin{equation*}
\mathbf{A}_{2\times 2\times 3}=\left[
\begin{array}{c}
\left(
\begin{array}{cc}
2 & 3 \\
4 & 5
\end{array}
\right) \\
\left(
\begin{array}{cc}
5 & 0 \\
9 & 1
\end{array}
\right) \\
\left(
\begin{array}{cc}
1 & 4 \\
5 & 3
\end{array}
\right)
\end{array}
\right] .
\end{equation*}
Matrix obtained by multiplying the 3D matrix $A_{2\times 2\times 3}$ with
multi-scalar $a_{1\times 1\times 3},$ it is the matrix:
\begin{eqnarray*}
\mathbf{B}_{2\times 2\times 3} &=&a_{1\times 1\times 3}\ast \mathbf{A}
_{2\times 2\times 3} \\
&=&\left[
\begin{array}{c}
\left( 2\right) \\
\left( 5\right) \\
\left( 3\right)
\end{array}
\right] \ast \left[
\begin{array}{c}
\left(
\begin{array}{cc}
2 & 3 \\
4 & 5
\end{array}
\right) \\
\left(
\begin{array}{cc}
5 & 0 \\
9 & 1
\end{array}
\right) \\
\left(
\begin{array}{cc}
1 & 4 \\
5 & 3
\end{array}
\right)
\end{array}
\right] = \\
&=&\left[
\begin{array}{c}
2\cdot \left(
\begin{array}{cc}
2 & 3 \\
4 & 5
\end{array}
\right) \\
5\cdot \left(
\begin{array}{cc}
5 & 0 \\
9 & 1
\end{array}
\right) \\
3\cdot \left(
\begin{array}{cc}
1 & 4 \\
5 & 3
\end{array}
\right)
\end{array}
\right] =\left[
\begin{array}{c}
\left(
\begin{array}{cc}
4 & 6 \\
8 & 10
\end{array}
\right) \\
\left(
\begin{array}{cc}
25 & 0 \\
45 & 5
\end{array}
\right) \\
\left(
\begin{array}{cc}
3 & 12 \\
15 & 9
\end{array}
\right)
\end{array}
\right] .
\end{eqnarray*}
\end{example}
\subsection{DETERMINANTS OF 3D-MATRICES}
Regarding the determinant we will only talk about form matrices $\mathbf{A}
_{n\times n\times p}\in \mathcal{M}_{n\times n\times p}(\mathbf{F}),$ ie for
matrices that \emph{vertical layers} have square matrices.
\begin{definition}
Determinant of the matrix $\mathbf{A}_{n\times n\times p}\in \mathcal{M}
_{n\times n\times p}(\mathbf{F}),$ we will call the \textbf{multi-scalar}
\begin{equation*}
\det \left( \mathbf{A}_{n\times n\times p}\right) =\left[
\begin{array}{c}
\det \left( \mathbf{A}_{n\times n,p}\right) \\
\mathbf{\vdots } \\
\det \left( \mathbf{A}_{n\times n,2}\right) \\
\det \left( \mathbf{A}_{n\times n,1}\right)
\end{array}
\right]
\end{equation*}
where
\begin{equation*}
\det \left( \mathbf{A}_{n\times n,1}\right) =\det \left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n11} & a_{n21} & \cdots & a_{nn1}
\end{array}
\right) ;
\end{equation*}
\begin{equation*}
\det \left( \mathbf{A}_{n\times n,2}\right) =\det \left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n12} & a_{n22} & \cdots & a_{nn2}
\end{array}
\right) ;...;
\end{equation*}
\begin{equation*}
\det \left( \mathbf{A}_{n\times n,p}\right) =\det \left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1p} & a_{n2p} & \cdots & a_{nnp}
\end{array}
\right)
\end{equation*}
\end{definition}
Referring to the multi-scalar note '\emph{absolutely different from zero}',
we say that for a 3D matrix $\mathbf{A}_{n\times n\times p},$ \
\begin{equation*}
\det \left( \mathbf{A}_{n\times n\times p}\right) \neq \widetilde{^{a}0_{
\mathbf{F}}}\Longleftrightarrow \det \left( \mathbf{A}_{n\times n,i}\right)
\neq 0_{\mathbf{F}},\forall i=\overline{1,p}.
\end{equation*}
\begin{example}
Determinant of the matrix $\mathbf{A}_{2\times 2\times 3},$ of example 1, is
multi-scalar:
\end{example}
\begin{equation*}
\det \left( \mathbf{A}_{2\times 2\times 3}\right) =\left[
\begin{array}{c}
\det \left(
\begin{array}{cc}
2 & 3 \\
4 & 5
\end{array}
\right) \\
\det \left(
\begin{array}{cc}
5 & 0 \\
9 & 1
\end{array}
\right) \\
\det \left(
\begin{array}{cc}
1 & 4 \\
5 & 3
\end{array}
\right)
\end{array}
\right] =\left[
\begin{array}{c}
\left( -2\right) \\
\left( 5\right) \\
\left( -17\right)
\end{array}
\right]
\end{equation*}
\begin{definition}
\emph{Inverted} of the multi-scalar $a_{1\times 1\times p}=\left[
\begin{array}{c}
\left( \alpha _{11p}\right) \\
\vdots \\
\left( \alpha _{112}\right) \\
\left( \alpha _{111}\right)
\end{array}
\right] ,$ will called the multi-scalar:
\begin{equation*}
\widehat{a_{1\times 1\times p}}=\left[
\begin{array}{c}
\left( \frac{1}{\alpha _{11p}}\right) \\
\vdots \\
\left( \frac{1}{\alpha _{112}}\right) \\
\left( \frac{1}{\alpha _{111}}\right)
\end{array}
\right] .
\end{equation*}
\end{definition}
\begin{example}
\emph{Inverted} of the multi-scalar \ $a_{1\times 1\times 3}=\left[
\begin{array}{c}
\left( 2\right) \\
\left( 5\right) \\
\left( 3\right)
\end{array}
\right] ,$ is the multi-scalar:
\begin{equation*}
\widehat{a_{1\times 1\times 3}}=\left[
\begin{array}{c}
\left( \frac{1}{2}\right) \\
\left( \frac{1}{5}\right) \\
\left( \frac{1}{3}\right)
\end{array}
\right] .
\end{equation*}
\end{example}
\section{THE MULTIPLICATION GROUP OF 3D MATRICES}
Referring to \cite{[OZ3DM]}, we have a 3D Matrix Ring, and in that paper we
have shown the possibility of unitary ring. The rest of the assertions that
lead us from unitary ring to a Skew-Field are summing up in this
\begin{theorem}
The structure $\left( \mathcal{M}_{n\times n\times p}(\mathbf{F}),\odot
\right) $ is a unitary semi-group.
\end{theorem}
\begin{proof}
We show first that:
\begin{equation*}
\forall \mathbf{A},\mathbf{B},\mathbf{C}\in \mathcal{M}_{n\times n\times p}(
\mathbf{F}),\left[ \mathbf{A}\odot \mathbf{B}\right] \odot \mathbf{C=A}\odot
\left[ \mathbf{B}\odot \mathbf{C}\right]
\end{equation*}
truly,
$\left[ \mathbf{A}\odot \mathbf{B}\right] \odot \mathbf{C=}\left( \left[
\begin{array}{c}
\mathbf{A}_{p} \\
\vdots \\
\mathbf{A}_{2} \\
\mathbf{A}_{1}
\end{array}
\right] \odot \left[
\begin{array}{c}
\mathbf{B}_{p} \\
\vdots \\
\mathbf{B}_{2} \\
\mathbf{B}_{1}
\end{array}
\right] \right) \odot \left[
\begin{array}{c}
\mathbf{C}_{p} \\
\vdots \\
\mathbf{C}_{2} \\
\mathbf{C}_{1}
\end{array}
\right] =\left[
\begin{array}{c}
\mathbf{A}_{p}\cdot \mathbf{B}_{p} \\
\vdots \\
\mathbf{A}_{2}\cdot \mathbf{B}_{2} \\
\mathbf{A}_{1}\cdot \mathbf{B}_{1}
\end{array}
\right] \odot \left[
\begin{array}{c}
\mathbf{C}_{p} \\
\vdots \\
\mathbf{C}_{2} \\
\mathbf{C}_{1}
\end{array}
\right] =\left[
\begin{array}{c}
\left( \mathbf{A}_{p}\cdot \mathbf{B}_{p}\right) \cdot \mathbf{C}_{p} \\
\vdots \\
\left( \mathbf{A}_{2}\cdot \mathbf{B}_{2}\right) \cdot \mathbf{C}_{2} \\
\left( \mathbf{A}_{1}\cdot \mathbf{B}_{1}\right) \cdot \mathbf{C}_{1}
\end{array}
\right] =\left[
\begin{array}{c}
\mathbf{A}_{p}\cdot \left( \mathbf{B}_{p}\cdot \mathbf{C}_{p}\right) \\
\vdots \\
\mathbf{A}_{2}\cdot \left( \mathbf{B}_{2}\cdot \mathbf{C}_{2}\right) \\
\mathbf{A}_{1}\cdot \left( \mathbf{B}_{1}\cdot \mathbf{C}_{1}\right)
\end{array}
\right] =\left[
\begin{array}{c}
\mathbf{A}_{p} \\
\vdots \\
\mathbf{A}_{2} \\
\mathbf{A}_{1}
\end{array}
\right] \odot \left[
\begin{array}{c}
\mathbf{B}_{p}\cdot \mathbf{C}_{p} \\
\vdots \\
\mathbf{B}_{2}\cdot \mathbf{C}_{2} \\
\mathbf{B}_{1}\cdot \mathbf{C}_{1}
\end{array}
\right] $
$=\left[
\begin{array}{c}
\mathbf{A}_{p} \\
\vdots \\
\mathbf{A}_{2} \\
\mathbf{A}_{1}
\end{array}
\right] \odot \left( \left[
\begin{array}{c}
\mathbf{B}_{p} \\
\vdots \\
\mathbf{B}_{2} \\
\mathbf{B}_{1}
\end{array}
\right] \odot \left[
\begin{array}{c}
\mathbf{C}_{p} \\
\vdots \\
\mathbf{C}_{2} \\
\mathbf{C}_{1}
\end{array}
\right] \right) =\mathbf{A}\odot \left[ \mathbf{B}\odot \mathbf{C}\right] .$
From the definition of multiplication to 3D matrices it is clear that:
\begin{equation*}
\
\mathbf{I}_{n\times n\times p}=\left[
\begin{array}{c}
\mathbf{I}_{n\times n,p} \\
\vdots \\
\mathbf{I}_{n\times n,2} \\
\mathbf{I}_{n\times n,1}
\end{array}
\right] ,
\end{equation*}
is unit element of $\mathcal{M}_{n\times n\times p}(\mathbf{F})$, related to
multiplication.
\end{proof}
\begin{remark}
We mark with $\mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F}),$ 3D
matrix with determinants '\textbf{absolutely different from zero}', ie $\det
\left( \mathbf{A}_{n\times n\times p}\right) \neq \widetilde{^{a}0_{\mathbf{F
}}}$, (So \textbf{vertical layers} of 3D matrix, are 2D non-singular
matrices.).
\end{remark}
\begin{proposition}
The set of \ $\mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F}),$ is
closed regarding multiplication.
Well!
\begin{eqnarray*}
\odot &:&\mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F})\times \mathcal{M
}_{n\times n\times p}^{\ast }(\mathbf{F})\rightarrow \mathcal{M}_{n\times
n\times p}^{\ast }(\mathbf{F}),\ \\
\forall \mathbf{A,B} &\in &\mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F}
)\Rightarrow \mathbf{A\odot B}\in \mathcal{M}_{n\times n\times p}^{\ast }(
\mathbf{F}).
\end{eqnarray*}
\begin{proof}
Let's have
\begin{equation*}
\mathbf{A,B}\in \mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F}
)\Rightarrow \det \left( \mathbf{A}\right) \neq \widetilde{^{a}0_{\mathbf{F}}
}\text{ and }\det \left( \mathbf{B}\right) \neq \widetilde{^{a}0_{\mathbf{F}}
}
\end{equation*}
\begin{equation*}
\Leftrightarrow \det \left( \mathbf{A}_{n\times n,i}\right) \neq 0_{\mathbf{F
}}\text{ }and\text{ }\det \left( \mathbf{B}_{n\times n,i}\right) \neq 0_{
\mathbf{F}},,\forall i=\overline{1,p}
\end{equation*}
So all 2D matrices $\mathbf{A}_{n\times n,i}$ and $\mathbf{B}_{n\times n,i}$
, $\forall i=\overline{1,p}$ , are non-singular matrices. A well-known
result in Linear algebra, but also mentioned in \cite{[GS4Ed]}, in pp.230,
and \cite{[ShALA3ED]} in pp317, which is
\begin{equation*}
"\forall A,B\in \mathcal{M}_{n\times n}(\mathbf{F}),\ \det \left( A\cdot
B\right) =\det \left( A\right) \cdot \det \left( B\right) "
\end{equation*}
We use this result as the \textbf{vertical layers} of 3D matrix are
2D-matries, and so we have that:
$\det \left( \mathbf{A}\odot \mathbf{B}\right) \mathbf{=}\det \left( \left[
\begin{array}{c}
\mathbf{A}_{n\times n,p} \\
\vdots \\
\mathbf{A}_{n\times n,2} \\
\mathbf{A}_{n\times n,1}
\end{array}
\right] \odot \left[
\begin{array}{c}
\mathbf{B}_{n\times n,p} \\
\vdots \\
\mathbf{B}_{n\times n,2} \\
\mathbf{B}_{n\times n,1}
\end{array}
\right] \right) =\det \left[
\begin{array}{c}
\mathbf{A}_{n\times n,p}\cdot \mathbf{B}_{n\times n,p} \\
\vdots \\
\mathbf{A}_{n\times n,2}\cdot \mathbf{B}_{n\times n,2} \\
\mathbf{A}_{n\times n,1}\cdot \mathbf{B}_{n\times n,1}
\end{array}
\right] =\left[
\begin{array}{c}
\det \left( \mathbf{A}_{n\times n,p}\cdot \mathbf{B}_{n\times n,p}\right) \\
\vdots \\
\det \left( \mathbf{A}_{n\times n,2}\cdot \mathbf{B}_{n\times n,2}\right) \\
\det \left( \mathbf{A}_{n\times n,1}\cdot \mathbf{B}_{n\times n,1}\right)
\end{array}
\right] \overset{[2]}{=}\left[
\begin{array}{c}
\det \left( \mathbf{A}_{n\times n,p}\right) \cdot \det \left( \mathbf{B}
_{n\times n,p}\right) \\
\vdots \\
\det \left( \mathbf{A}_{n\times n,2}\right) \cdot \det \left( \mathbf{B}
_{n\times n,2}\right) \\
\det \left( \mathbf{A}_{n\times n,1}\right) \cdot \det \left( \mathbf{B}
_{n\times n,1}\right)
\end{array}
\right] =$
$=\left[
\begin{array}{c}
\det \left( \mathbf{A}_{n\times n,p}\right) \\
\vdots \\
\det \left( \mathbf{A}_{n\times n,2}\right) \\
\det \left( \mathbf{A}_{n\times n,1}\right)
\end{array}
\right] \odot \left[
\begin{array}{c}
\det \left( \mathbf{B}_{n\times n,p}\right) \\
\vdots \\
\det \left( \mathbf{B}_{n\times n,2}\right) \\
\det \left( \mathbf{B}_{n\times n,1}\right)
\end{array}
\right] =\det \left( \mathbf{A}\right) \odot \det \left( \mathbf{B}\right)
\neq \widetilde{^{a}0_{\mathbf{F}}}\Rightarrow $
$\Rightarrow \mathbf{A\odot B}\in \mathcal{M}_{n\times n\times p}^{\ast }(
\mathbf{F}).$
\end{proof}
\end{proposition}
\begin{theorem}
The structure $\left( \mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F}
),\odot \right) $ is a Group.
Set of 3D matrix $\mathbf{A}_{n\times n\times p}\in \mathcal{M}_{n\times
n\times p}(\mathbf{F}),$ with determinants '\textbf{absolutely different
from zero}', ie $\det \left( \mathbf{A}_{n\times n\times p}\right) \neq
\widetilde{^{a}0_{\mathbf{F}}}$, associated with ordinary multiplication is
a Group.
\begin{proof}
\textbf{1.} It is clear that the set $\mathcal{M}_{n\times n\times p}^{\ast
}(\mathbf{F})$ is sub-set of $\mathcal{M}_{n\times n\times p}(\mathbf{F})$,
and in the foregoing assertion we showed that the multiplication is closed
in this set, so we have that $\left( \mathcal{M}_{n\times n\times p}^{\ast }(
\mathbf{F}),\odot \right) $ is subsemigroup of semigroup $\left( \mathcal{M}
_{n\times n\times p}(\mathbf{F}),\odot \right) ,$ see \cite{[LSGTM]}, \cite
{[GPASG]}.
\textbf{2.} It is clear that the $\mathbf{I}_{n\times n\times p}\in \mathcal{
M}_{n\times n\times p}^{\ast }(\mathbf{F})$, after
\begin{equation*}
\det \left( \mathbf{I}_{n\times n\times p}\right) =\left[
\begin{array}{c}
\det \left( \mathbf{I}_{n\times n,p}\right) \\
\vdots \\
\det \left( \mathbf{I}_{n\times n,2}\right) \\
\det \left( \mathbf{I}_{n\times n,1}\right)
\end{array}
\right] =\left[
\begin{array}{c}
1_{\mathbf{F}} \\
\vdots \\
1_{\mathbf{F}} \\
1_{\mathbf{F}}
\end{array}
\right] \neq \widetilde{^{a}0_{\mathbf{F}}}.
\end{equation*}
\textbf{3.} Show that: $\forall \mathbf{A}_{n\times n\times p}\in \mathcal{M}
_{n\times n\times p}^{\ast }(\mathbf{F}),($such that $\det \left( \mathbf{A}
_{n\times n\times p}\right) \neq \widetilde{^{a}0_{\mathbf{F}}})$, $\exists
\mathbf{A}_{n\times n\times p}^{-1}\in \mathcal{M}_{n\times n\times p}^{\ast
}(\mathbf{F}),$ \ such that
\begin{equation*}
\mathbf{A}_{n\times n\times p}\odot \mathbf{A}_{n\times n\times p}^{-1}=
\mathbf{I}_{n\times n\times p}
\end{equation*}
Let's have
\begin{eqnarray*}
\mathbf{A}_{n\times n\times p} &=&\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p} \\
\vdots \\
\mathbf{A}_{n\times n,2} \\
\mathbf{A}_{n\times n,1}
\end{array}
\right] \in \mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F})\Rightarrow \\
&\Rightarrow &\det \left( \mathbf{A}_{n\times n\times p}\right) \neq
\widetilde{^{a}0_{\mathbf{F}}}\Leftrightarrow \det \left( \mathbf{A}
_{n\times n,i}\right) \neq 0_{\mathbf{F}},\forall i=\overline{1,p}
\end{eqnarray*}
So we have, that: $\forall \mathbf{A}_{n\times n,i}$ (as a 2D matrix), $
\exists \mathbf{A}_{n\times n,i}^{-1}$ such that $\mathbf{A}_{n\times
n,i}\cdot \mathbf{A}_{n\times n,i}^{-1}=\mathbf{I}_{n\times n,i},\forall i=
\overline{1,p}.$ From this, we can write that, the inverse of 3D matrix $
\mathbf{A}_{n\times n\times p}=\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p} \\
\vdots \\
\mathbf{A}_{n\times n,2} \\
\mathbf{A}_{n\times n,1}
\end{array}
\right] $ is a 3D matrix $\mathbf{A}_{n\times n\times p}^{-1}=\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p}^{-1} \\
\vdots \\
\mathbf{A}_{n\times n,2}^{-1} \\
\mathbf{A}_{n\times n,1}^{-1}
\end{array}
\right] ,$ because the
\begin{eqnarray*}
\mathbf{A}_{n\times n\times p}\odot \mathbf{A}_{n\times n\times p}^{-1} &=&
\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p} \\
\vdots \\
\mathbf{A}_{n\times n,2} \\
\mathbf{A}_{n\times n,1}
\end{array}
\right] \odot \left[
\begin{array}{c}
\mathbf{A}_{n\times n,p}^{-1} \\
\vdots \\
\mathbf{A}_{n\times n,2}^{-1} \\
\mathbf{A}_{n\times n,1}^{-1}
\end{array}
\right] = \\
&=&\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p}\cdot \mathbf{A}_{n\times n,p}^{-1} \\
\vdots \\
\mathbf{A}_{n\times n,2}\cdot \mathbf{A}_{n\times n,2}^{-1} \\
\mathbf{A}_{n\times n,1}\cdot \mathbf{A}_{n\times n,1}^{-1}
\end{array}
\right] \\
&=&\left[
\begin{array}{c}
\mathbf{I}_{n\times n,p} \\
\vdots \\
\mathbf{I}_{n\times n,2} \\
\mathbf{I}_{n\times n,1}
\end{array}
\right] =\mathbf{I}_{n\times n\times p}
\end{eqnarray*}
$.$
Where
\begin{equation*}
\mathbf{A}_{n\times n\times p}=\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p} \\
\vdots \\
\mathbf{A}_{n\times n,2} \\
\mathbf{A}_{n\times n,1}
\end{array}
\right] \text{\ and\ }\mathbf{A}_{n\times n\times p}^{-1}=\left[
\begin{array}{c}
\mathbf{A}_{n\times n,p}^{-1} \\
\vdots \\
\mathbf{A}_{n\times n,2}^{-1} \\
\mathbf{A}_{n\times n,1}^{-1}
\end{array}
\right] .
\end{equation*}
\end{proof}
\end{theorem}
\section{FINDING OF 3D-INVERSE MATRIX}
In this section, we provide a way to find the 3D reverse matrix of a 3D
matrix.
\begin{proposition}
3D inverse matrix of the matrix $\mathbf{A}_{n\times n\times p}\in \mathcal{M
}_{n\times n\times p}^{\ast }(\mathbf{F}),$ we called the 3D-matrix $\mathbf{
A}_{n\times n\times p}^{-1}\in \mathcal{M}_{n\times n\times p}^{\ast }(
\mathbf{F}),$ which has the following form:
\begin{equation*}
\mathbf{A}_{n\times n\times p}^{-1}=\widehat{\det \left( \mathbf{A}_{n\times
n\times p}\right) }\ast \overline{\overline{\mathbf{A}_{n\times n\times p}}}
\end{equation*}
where $\overline{\overline{\mathbf{A}_{n\times n\times p}}}$ is the \textbf{
adjugate matrix} of $\mathbf{A}_{n\times n\times p}$ (exactly page by
page)and has the form:
\begin{equation*}
\overline{\overline{\mathbf{A}_{n\times n\times p}}}=\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
A_{11p} & A_{12p} & \cdots & A_{1np} \\
A_{21p} & A_{22p} & \cdots & A_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1p} & A_{n2p} & \cdots & A_{nnp}
\end{array}
\right) ^{T} \\
\vdots \\
\left(
\begin{array}{cccc}
A_{112} & A_{122} & \cdots & A_{1n2} \\
A_{212} & A_{222} & \cdots & A_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n12} & A_{n22} & \cdots & A_{nn2}
\end{array}
\right) ^{T} \\
\left(
\begin{array}{cccc}
A_{111} & A_{121} & \cdots & A_{1n1} \\
A_{211} & A_{221} & \cdots & A_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n11} & A_{n21} & \cdots & A_{nn1}
\end{array}
\right) ^{T}
\end{array}
\right]
\end{equation*}
I give a clearer view of matrices $\mathbf{A}_{n\times n\times p}^{-1}$, as
follows:
\begin{eqnarray*}
\mathbf{A}_{n\times n\times p}^{-1} &=&\widehat{\det \left( \mathbf{A}
_{n\times n\times p}\right) }\ast \overline{\overline{\mathbf{A}_{n\times
n\times p}}}= \\
&=&\left[
\begin{array}{c}
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,p}\right\vert }\right) \\
\vdots \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,2}\right\vert }\right) \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,1}\right\vert }\right)
\end{array}
\right] \ast \left[
\begin{array}{c}
\left(
\begin{array}{cccc}
A_{11p} & A_{12p} & \cdots & A_{1np} \\
A_{21p} & A_{22p} & \cdots & A_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1p} & A_{n2p} & \cdots & A_{nnp}
\end{array}
\right) ^{T} \\
\vdots \\
\left(
\begin{array}{cccc}
A_{112} & A_{122} & \cdots & A_{1n2} \\
A_{212} & A_{222} & \cdots & A_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n12} & A_{n22} & \cdots & A_{nn2}
\end{array}
\right) ^{T} \\
\left(
\begin{array}{cccc}
A_{111} & A_{121} & \cdots & A_{1n1} \\
A_{211} & A_{221} & \cdots & A_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n11} & A_{n21} & \cdots & A_{nn1}
\end{array}
\right) ^{T}
\end{array}
\right] .
\end{eqnarray*}
\end{proposition}
\begin{proof}
Verified with ease that:
\begin{equation*}
\mathbf{A}_{n\times n\times p}\odot \mathbf{A}_{n\times n\times p}^{-1}=
\mathbf{A}_{n\times n\times p}^{-1}\odot \mathbf{A}_{n\times n\times p}=
\mathbf{I}_{n\times n\times p}.
\end{equation*}
Where
\begin{equation*}
\mathbf{I}_{n\times n\times p}=\left[
\begin{array}{c}
\mathbf{I}_{n\times n,p} \\
\vdots \\
\mathbf{I}_{n\times n,2} \\
\mathbf{I}_{n\times n,1}
\end{array}
\right] .
\end{equation*}
Let
\begin{equation*}
\mathbf{A}_{m\times n\times p}\mathbf{=}\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right)
\end{array}
\right] \in \mathcal{M}_{n\times n\times p}^{\ast }(\mathbf{F}),
\end{equation*}
and
\begin{eqnarray*}
\mathbf{A}_{n\times n\times p}^{-1} &=&\widehat{\det \left( \mathbf{A}
_{n\times n\times p}\right) }\ast \overline{\overline{\mathbf{A}_{n\times
n\times p}}}= \\
&=&\left[
\begin{array}{c}
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,p}\right\vert }\right) \\
\vdots \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,2}\right\vert }\right) \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,1}\right\vert }\right)
\end{array}
\right] \ast \left[
\begin{array}{c}
\left(
\begin{array}{cccc}
A_{11p} & A_{12p} & \cdots & A_{1np} \\
A_{21p} & A_{22p} & \cdots & A_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1p} & A_{n2p} & \cdots & A_{nnp}
\end{array}
\right) ^{T} \\
\vdots \\
\left(
\begin{array}{cccc}
A_{112} & A_{122} & \cdots & A_{1n2} \\
A_{212} & A_{222} & \cdots & A_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n12} & A_{n22} & \cdots & A_{nn2}
\end{array}
\right) ^{T} \\
\left(
\begin{array}{cccc}
A_{111} & A_{121} & \cdots & A_{1n1} \\
A_{211} & A_{221} & \cdots & A_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n11} & A_{n21} & \cdots & A_{nn1}
\end{array}
\right) ^{T}
\end{array}
\right] .
\end{eqnarray*}
to prove that it is true:
\begin{equation*}
\mathbf{A}_{n\times n\times p}\odot \mathbf{A}_{n\times n\times p}^{-1}=
\mathbf{A}_{n\times n\times p}^{-1}\odot \mathbf{A}_{n\times n\times p}=
\mathbf{I}_{n\times n\times p}?
\end{equation*}
really:
\begin{equation*}
\mathbf{A}_{n\times n\times p}\odot \mathbf{A}_{n\times n\times p}^{-1}=
\mathbf{A}_{n\times n\times p}\odot \left( \widehat{\det \left( \mathbf{A}
_{n\times n\times p}\right) }\ast \overline{\overline{\mathbf{A}_{n\times
n\times p}}}\right)
\end{equation*}
\begin{equation*}
=\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right)
\end{array}
\right] \odot \left[
\begin{array}{c}
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,p}\right\vert }\right) \\
\vdots \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,2}\right\vert }\right) \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,1}\right\vert }\right)
\end{array}
\right] \ast \left[
\begin{array}{c}
\left(
\begin{array}{cccc}
A_{11p} & A_{12p} & \cdots & A_{1np} \\
A_{21p} & A_{22p} & \cdots & A_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1p} & A_{n2p} & \cdots & A_{nnp}
\end{array}
\right) ^{T} \\
\vdots \\
\left(
\begin{array}{cccc}
A_{112} & A_{122} & \cdots & A_{1n2} \\
A_{212} & A_{222} & \cdots & A_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n12} & A_{n22} & \cdots & A_{nn2}
\end{array}
\right) ^{T} \\
\left(
\begin{array}{cccc}
A_{111} & A_{121} & \cdots & A_{1n1} \\
A_{211} & A_{221} & \cdots & A_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n11} & A_{n21} & \cdots & A_{nn1}
\end{array}
\right) ^{T}
\end{array}
\right]
\end{equation*}
\begin{equation*}
\overset{???}{=}\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right)
\end{array}
\right] \odot \left[
\begin{array}{c}
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,p}\right\vert }\right)
\cdot \left(
\begin{array}{cccc}
A_{11p} & A_{12p} & \cdots & A_{1np} \\
A_{21p} & A_{22p} & \cdots & A_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1p} & A_{n2p} & \cdots & A_{nnp}
\end{array}
\right) ^{T} \\
\vdots \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,2}\right\vert }\right)
\cdot \left(
\begin{array}{cccc}
A_{112} & A_{122} & \cdots & A_{1n2} \\
A_{212} & A_{222} & \cdots & A_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n12} & A_{n22} & \cdots & A_{nn2}
\end{array}
\right) ^{T} \\
\left( \frac{1}{\left\vert \mathbf{A}_{n\times n,1}\right\vert }\right)
\cdot \left(
\begin{array}{cccc}
A_{111} & A_{121} & \cdots & A_{1n1} \\
A_{211} & A_{221} & \cdots & A_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n11} & A_{n21} & \cdots & A_{nn1}
\end{array}
\right) ^{T}
\end{array}
\right]
\end{equation*}
\begin{equation*}
\overset{???}{=}\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right)
\end{array}
\right] \odot \left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) ^{-1} \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) ^{-1} \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right) ^{-1}
\end{array}
\right]
\end{equation*}
\begin{equation*}
=\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) \cdot \left(
\begin{array}{cccc}
a_{11p} & a_{12p} & \cdots & a_{1np} \\
a_{21p} & a_{22p} & \cdots & a_{2np} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1p} & a_{m2p} & \cdots & a_{mnp}
\end{array}
\right) ^{-1} \\
\vdots \\
\left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) \cdot \left(
\begin{array}{cccc}
a_{112} & a_{122} & \cdots & a_{1n2} \\
a_{212} & a_{222} & \cdots & a_{2n2} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m12} & a_{m22} & \cdots & a_{mn2}
\end{array}
\right) ^{-1} \\
\left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right) \cdot \left(
\begin{array}{cccc}
a_{111} & a_{121} & \cdots & a_{1n1} \\
a_{211} & a_{221} & \cdots & a_{2n1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m11} & a_{m21} & \cdots & a_{mn1}
\end{array}
\right) ^{-1}
\end{array}
\right]
\end{equation*}
\begin{equation*}
=\left[
\begin{array}{c}
\left(
\begin{array}{cccc}
1_{F} & 0_{F} & \cdots & 0_{F} \\
0_{F} & 1_{F} & \cdots & 0_{F} \\
\vdots & \vdots & \ddots & \vdots \\
0_{F} & 0_{F} & \cdots & 1_{F}
\end{array}
\right) \\
\vdots \\
\left(
\begin{array}{cccc}
1_{F} & 0_{F} & \cdots & 0_{F} \\
0_{F} & 1_{F} & \cdots & 0_{F} \\
\vdots & \vdots & \ddots & \vdots \\
0_{F} & 0_{F} & \cdots & 1_{F}
\end{array}
\right) \\
\left(
\begin{array}{cccc}
1_{F} & 0_{F} & \cdots & 0_{F} \\
0_{F} & 1_{F} & \cdots & 0_{F} \\
\vdots & \vdots & \ddots & \vdots \\
0_{F} & 0_{F} & \cdots & 1_{F}
\end{array}
\right)
\end{array}
\right] =\left[
\begin{array}{c}
\mathbf{I}_{n\times n,p} \\
\vdots \\
\mathbf{I}_{n\times n,2} \\
\mathbf{I}_{n\times n,1}
\end{array}
\right] =\mathbf{I}_{n\times n\times p}.
\end{equation*}
\end{proof}
\begin{example}
Let's have \ $A_{3\times 3\times 2}=\left[
\begin{array}{c}
\left(
\begin{array}{ccc}
3 & 1 & 5 \\
0 & 2 & 1 \\
1 & 7 & 4
\end{array}
\right) \\
\left(
\begin{array}{ccc}
1 & 2 & 4 \\
8 & 1 & 1 \\
3 & 1 & 0
\end{array}
\right)
\end{array}
\right] \in \mathcal{M}_{3\times 3\times 2}^{\ast }(\mathbb{R}).$ Find its
inverse matrix?
\begin{solution}
The inverse matrix has form:
\begin{equation*}
\mathbf{A}_{3\times 3\times 2}^{-1}=\widehat{\det \left( \mathbf{A}_{3\times
3\times 2}\right) }\ast \overline{\overline{\mathbf{A}_{3\times 3\times 2}}}
\end{equation*}
we see that
\begin{equation*}
\det \left( A_{3\times 3\times 2}\right) =\left[
\begin{array}{c}
\det \left(
\begin{array}{ccc}
3 & 1 & 5 \\
0 & 2 & 1 \\
1 & 7 & 4
\end{array}
\right) \\
\det \left(
\begin{array}{ccc}
1 & 2 & 4 \\
8 & 1 & 1 \\
3 & 1 & 0
\end{array}
\right)
\end{array}
\right] =\left[
\begin{array}{c}
\left( -6\right) \\
\left( 25\right)
\end{array}
\right] \Rightarrow \ \widehat{\det \left( \mathbf{A}_{3\times 3\times
2}\right) }=\left[
\begin{array}{c}
\left( -\frac{1}{6}\right) \\
\left( \frac{1}{25}\right)
\end{array}
\right] .
\end{equation*}
And
\begin{equation*}
\overline{\overline{\mathbf{A}_{3\times 3\times 2}}}=\left[
\begin{array}{c}
\left(
\begin{array}{ccc}
1 & 1 & -2 \\
31 & 7 & -20 \\
-9 & -3 & 6
\end{array}
\right) ^{T} \\
\left(
\begin{array}{ccc}
-1 & 3 & 5 \\
4 & -12 & 5 \\
-2 & 31 & -15
\end{array}
\right) ^{T}
\end{array}
\right] =\left[
\begin{array}{c}
\left(
\begin{array}{ccc}
1 & 31 & -9 \\
1 & 7 & -3 \\
-2 & -20 & 6
\end{array}
\right) \\
\left(
\begin{array}{ccc}
-1 & 4 & -2 \\
3 & -12 & 31 \\
5 & 5 & -15
\end{array}
\right)
\end{array}
\right] ,
\end{equation*}
Then,
\begin{eqnarray*}
\mathbf{A}_{3\times 3\times 2}^{-1} &=&\widehat{\det \left( \mathbf{A}
_{3\times 3\times 2}\right) }\ast \overline{\overline{\mathbf{A}_{3\times
3\times 2}}} \\
&=&\left[
\begin{array}{c}
\left( -\frac{1}{6}\right) \\
\left( \frac{1}{25}\right)
\end{array}
\right] \ast \left[
\begin{array}{c}
\left(
\begin{array}{ccc}
1 & 31 & -9 \\
1 & 7 & -3 \\
-2 & -20 & 6
\end{array}
\right) \\
\left(
\begin{array}{ccc}
-1 & 4 & -2 \\
3 & -12 & 31 \\
5 & 5 & -15
\end{array}
\right)
\end{array}
\right] =\left[
\begin{array}{c}
\left( -\frac{1}{6}\right) \cdot \left(
\begin{array}{ccc}
1 & 31 & -9 \\
1 & 7 & -3 \\
-2 & -20 & 6
\end{array}
\right) \\
\left( \frac{1}{25}\right) \cdot \left(
\begin{array}{ccc}
-1 & 4 & -2 \\
3 & -12 & 31 \\
5 & 5 & -15
\end{array}
\right)
\end{array}
\right]
\end{eqnarray*}
\begin{equation*}
\Rightarrow \mathbf{A}_{3\times 3\times 2}^{-1}=\left[
\begin{array}{c}
\left(
\begin{array}{ccc}
-\frac{1}{6} & -\frac{31}{6} & \frac{3}{2} \\
-\frac{1}{6} & -\frac{7}{6} & \frac{1}{2} \\
\frac{1}{3} & \frac{10}{3} & -1
\end{array}
\right) \\
\left(
\begin{array}{ccc}
-\frac{1}{25} & \frac{4}{25} & -\frac{2}{25} \\
\frac{3}{25} & -\frac{12}{25} & \frac{31}{25} \\
\frac{1}{5} & \frac{1}{5} & -\frac{3}{5}
\end{array}
\right)
\end{array}
\right]
\end{equation*}
\end{solution}
\end{example}
\end{document} | math | 45,432 |
\begin{document}
\title{Copula-based Semiparametric Regression Method for Bivariate Data under General Interval Censoring}
\author
{TAO SUN, YING DING$^\ast$ \\[4pt]
\textit{Department of Biostatistics,
University of Pittsburgh,
Pittsburgh, PA,
U.S.A.}
\\[2pt]
{[email protected]}}
\markboth
{T. SUN and Y. DING}
{Copula semiparametric model for bivariate interval-censored data}
\maketitle
\begin{abstract}
{This research is motivated by discovering and underpinning genetic causes for the progression of a bilateral eye disease, Age-related Macular Degeneration (AMD), of which the primary outcomes, progression times to late-AMD, are bivariate and interval-censored due to intermittent assessment times. We propose a novel class of copula-based semiparametric transformation models for bivariate data under general interval censoring, which includes the case 1 interval censoring (current status data) and case 2 interval censoring. Specifically, the joint likelihood is modeled through a two-parameter Archimedean copula, which can flexibly characterize the dependence between the two margins in both tails. The marginal distributions are modeled through semiparametric transformation models using sieves, with the proportional hazards or odds model being a special case. We develop a computationally efficient sieve maximum likelihood estimation procedure for the unknown parameters, together with a generalized score test for the regression parameter(s). For the proposed sieve estimators of finite-dimensional parameters, we establish their asymptotic normality and efficiency. Extensive simulations are conducted to evaluate the performance of the proposed method in finite samples. Finally, we apply our method to a genome-wide analysis of AMD progression using the Age-Related Eye Disease Study (AREDS) data, to successfully identify novel risk variants associated with the disease progression. We also produce predicted joint and conditional progression-free probabilities, for patients with different genetic characteristics.}
{Bivariate; Copula; GWAS; Interval-censored; Semiparametric; Sieve.}
\end{abstract}
\section{Introduction} \label{sec:intro}
Bivariate time-to-event endpoints are frequently used as co-primary outcomes in biomedical and epidemiological fields. For example, two time-to-event endpoints are often seen in clinical trials studying the progression (or recurrence) of bilateral diseases (e.g., eye diseases) or complex diseases (e.g., cancer and psychiatric disorders). The two endpoints are correlated as they come from the same individual. Bivariate interval-censored data arise when both events are not precisely observed due to intermittent assessment times. Therefore, the event times are only known to belong to an interval (i.e., case 2 interval-censored). A further complication is that the event status can be indeterminate (i.e., right-censored) for individuals who are event-free at their last assessment time. The special case when there exists only one assessment time, leading to the bivariate current status data (events are either left- or right-censored), can also happen for some individuals. Therefore, the bivariate data we are interested in modeling are under general interval censoring, which may include a mixture of left-, right- and interval-censored data.
Our motivating example of such bivariate general interval-censored data came from a large clinical trial studying the progression of a bilateral eye disease, Age-related Macular Degeneration (AMD), of which the two-eyes from the same patient were periodically examined for late-AMD. The study aims to discover genetic variants that are significantly associated with AMD progression, as well as to characterize both the joint and conditional risks of AMD progression. For example, the joint 5-year progression-free probability for both eyes is a clinically significant measure to group patients into different risk categories. Similarly, for patients who have one eye already progressed, the conditional 5-year progression-free probability for the non-progressed eye (given its fellow eye already progressed) is vital to both clinicians and patients. Therefore, a desired statistical method needs to characterize and predict both joint and conditional risk profiles and assess the covariate effects on them.
There are several approaches to model bivariate interval-censored data. For example, \citet{bivariate_cox_marginal}, \citet{kim2002analysis}, \citet{chen2007proportional}, \citet{tong2008regression} and \citet{chen2013linear} fitted various marginal models for multivariate interval-censored data. All these approaches model the marginal distributions based on the working independence assumption, and thus cannot produce joint or conditional distributions. Another popular method is based on frailty models (for example, \citeauthor{frailty_oaks}, \citeyear{frailty_oaks}), which are mixed effects models with a latent frailty variable applied to the conditional hazard functions. For example, \citet{chen2009frailty} and \citet{chen2014analysis} built frailty proportional hazards (PH) models with piecewise constant baseline hazards for multivariate current status data and interval-censored data, respectively. \citet{wen_chen2013} and \citet{wang2015regression} developed Gamma-frailty PH models for bivariate interval-censored data through a nonparametric maximum likelihood estimation approach and bivariate current status data through a sieve estimation approach, respectively. Recently, \citet{frailty_case_II_transformation_sieve} and \citet{frailty_case_II_transformation_NPMLE} proposed frailty-based transformation models for bivariate or multivariate interval-censored data, and obtained parameter estimates through the sieve maximum likelihood estimation and nonparametric maximum likelihood estimation, respectively. For frailty models, the covariate effects are typically interpreted on the conditional level by conditioning on the random frailty term.
The third popular approach is based on copula models \citep[for example]{Clayton}. Unlike the marginal or frailty approaches, the copula-based methods directly connect the two marginal distributions through a copula function to construct the joint distribution, of which the copula parameter determines the dependence. This unique feature makes the modeling of the margins separable from the copula function, which is attractive from both the modeling perspective and the interpretation purpose. Both joint and conditional distributions can be obtained from copula models. Several copula models have been proposed in the literature. \citet{copula_case_I_ph} used sieve estimation in a copula model with PH margins for bivariate current status data. \citet{copula_case_I_ph_piecewise} and \citet{copula_case_II_ph_pw} developed estimating equations for copula models with piecewise constant baseline marginal hazards for clustered current status and interval-censored data, respectively. \citet{HuCaseICopulaPH_2017} developed a semiparametric sieve approach for bivariate current status data using copula framework with PH margins. To date, most copula-based regression models only handle a specific interval censoring type and are often limited to the PH assumption. Also, the most frequently used copula models, such as Clayton, Gumbel, and Frank, all use only one dependence parameter, which can be lack of flexibility.
\citet{copula_vs_frailty} and \citet{wienke2010frailty} have discussed the connection and distinction between copula and frailty models. For example, the Clayton copula has the same mathematical expression as the Gamma frailty model in terms of the joint survival distribution. However, their marginal survival functions are modeled differently. Specifically, the marginal function under the Clayton model only involves the time and covariate effects, whereas the marginal function under the Gamma frailty model includes time and covariate effects, and also the frailty parameter. As a result, the joint distribution functions of the Clayton copula and Gamma frailty models are not equivalent, except when the two margins are independent. More details are discussed in the Appendix C of Supplementary Materials. In this paper, the objectives of our real study lead us to choose copula-based models, which offer a straightforward interpretation of covariate effects and dependence strength, as well as an easy generation of joint and conditional survival distributions.
We propose a class of copula-based semiparametric transformation model for bivariate data subject to general interval censoring. For the copula model, we use a two-parameter copula function that can flexibly handle dependence structure on both upper and lower tails, and the dependence strength can be quantified via Kendall's $\tau$. For the marginal model, we use the semiparametric transformation model that incorporates a variety of models including PH and PO models. We approximate the infinite-dimensional nuisance parameters using sieves with Bernstein polynomials and propose a novel maximum likelihood estimation procedure which is computationally stable and efficient. We establish the asymptotic normality and efficiency for the sieve estimators of finite-dimensional model parameters. Moreover, we develop a computationally efficient generalized score test with numerical approximations of the score function and observed Fisher information for testing a large number of covariates (e.g., millions of SNPs). Lastly, the joint distribution can be directly obtained from our model, making it applicable to estimating the joint and conditional progression profiles for patients with different characteristics.
The paper is organized as follows. Section \ref{sec_notation_likelihood} introduces the model and the joint likelihood function. Section \ref{sec_estimation_test_procedure} presents the sieve maximum likelihood estimation procedure, the asymptotic properties, and the generalized score test. Section \ref{sec_simulations} illustrates extensive simulation studies for the estimation and testing performances of our proposed methods. We analyze the AREDS data and present the findings in Section \ref{sec_real_data}. Finally, we discuss and conclude in Section \ref{sec_conclusions}. Additional simulation and analysis results, the regularity conditions, proofs and additional technical details are provided in the Supplementary Materials.
\section{Notation and Likelihood} \label{sec_notation_likelihood}
\subsection{Copula model for bivariate censored data} \label{subsec_copula}
Assume there are $n$ independent subjects in a study. For subject $i$, we observe $D_i=\{(L_{ij},R_{ij},Z_{ij}), \\ j = 1,2\}$, where $(L_{ij}, R_{ij}]$ is the time interval that the true event time $T_{ij}$ lies in and $Z_{ij}$ is the covariate vector. When $R_{ij} = \infty$, $T_{ij}$ is right-censored, and when $L_{ij} = 0$, $T_{ij}$ is left-censored. We define the marginal survival function for subject $i$ margin $j$ as $S_{j}(t_{ij} | Z_{ij})=pr(T_{ij} > t_{ij} | Z_{ij})$ and the joint survival function for subject $i$ as $S(t_{i1},t_{i2} | Z_{i1}, Z_{i2})=pr(T_{i1} > t_{i1}, T_{i2} > t_{i2} | Z_{i1}, Z_{i2})$.
By the Sklar's theorem (\citeauthor{sklar}, \citeyear{sklar}), so long as marginal survival functions $S_j$ are continuous, there exists a unique function $C_\eta$ that connects two marginal survival functions into the joint survival function: $S(t_1,t_2 | Z_1, Z_2) = C_\eta(S_1(t_1 | Z_1),S_2(t_2 | Z_2)),\ t_1, t_2 \geq 0.$ Here, the function $C_\eta$ is called a copula, which maps $[0,1]^2$ onto $[0,1]$ and its parameter $\eta$ measures the dependence between the two margins. A signature feature of the copula is that it allows the dependence to be modeled separately from the marginal distributions \citep{Nelson_2006}.
One favorite copula family for bivariate censored data is the Archimedean copula family, which usually has an explicit formula. Two frequently used Archimedean copulas are the Clayton (\citeauthor{Clayton}, \citeyear{Clayton}) and Gumbel (\citeauthor{gumbel}, \citeyear{gumbel}) copula models, which account for the lower or upper tail dependence between two margins using a single parameter.
Here, we consider a more flexible two-parameter Archimedean copula model \citep{Joe}, which is formulated as
\begin{eqnarray}
C_{\alpha,\kappa}(u,v)=[1+\{(u^{-1/\kappa}-1)^{1/\alpha} + (v^{-1/\kappa}-1)^{1/\alpha} \}^{\alpha}]^{-\kappa}, \ \alpha \in (0,1], \ \kappa \in (0,\infty), \label{two-para-c}
\end{eqnarray}
where $u$ and $v$ are two uniformly distributed margins. The two dependence parameters ($\alpha$ and $\kappa$) account for the correlation between $u$ and $v$ at both upper and lower tails, and they explicitly connect to the Kendall's $\tau$ with $\tau = 1- {2\alpha\kappa}/(2\kappa + 1)$. In particular, when $\alpha = 1$, the two-parameter copula (\ref{two-para-c}) becomes the Clayton copula, and when $\kappa \rightarrow \infty$, it becomes the Gumbel copula. Thus, the two-parameter copula model provides more flexibility in characterizing the dependence than the Clayton or Gumbel copula.
\subsection{Joint likelihood for bivariate data under general interval censoring} \label{subsec_joint_lik}
Based on the notation introduced in Section \ref{subsec_copula}, the joint likelihood function using the two-parameter copula model can be written as
\begin{eqnarray}
\label{joint_likelihood}
&&L_{n}(S_1, S_2, \alpha,\kappa \mid D) = \prod_{i=1}^n pr(L_{i1} < T_{i1} \leq R_{i1}, L_{i2} < T_{i2} \leq R_{i2} \mid Z_{i1}, Z_{i2}) \nonumber \\
& = & \prod_{i=1}^n \biggl\{pr(T_{i1} > L_{i1}, T_{i2} > L_{i2} \mid Z_{i1}, Z_{i2}) - pr(T_{i1} > L_{i1}, T_{i2} > R_{i2} \mid Z_{i1}, Z_{i2}) \nonumber \\
&& \quad - pr(T_{i1} > R_{i1}, T_{i2} > L_{i2} \mid Z_{i1}, Z_{i2}) + pr(T_{i1} > R_{i1}, T_{i2} > R_{i2} \mid Z_{i1}, Z_{i2}) \biggl\} \nonumber \\
& = & \prod_{i=1}^n\biggl[C_{\alpha,\kappa}\{S_1(L_{i1}\mid Z_{i1}),S_2(L_{i2}\mid Z_{i2})\} - C_{\alpha,\kappa}\{S_1(L_{i1}\mid Z_{i1}),S_2(R_{i2}\mid Z_{i2})\} \nonumber \\
&& \quad - C_{\alpha,\kappa}\{S_1(R_{i1}\mid Z_{i1}),S_2(L_{i2}\mid Z_{i2})\} + C_{\alpha,\kappa}\{S_1(R_{i1}\mid Z_{i1}),S_2(R_{i2}\mid Z_{i2})\} \biggr].
\end{eqnarray}
For a given subject $i$, if $T_{ij}$ is right-censored, then any term involving $R_{ij}$ becomes 0 (since $R_{ij}$ is set to be $\infty$). Then the joint survival function for subject $i$ reduces to either only one term (if both $T_{i1}$ and $T_{i2}$ are right-censored) or two terms (if one $T_{ij}$ is right-censored). The particular case of current status data can also fit into this model frame, where either $L_{ij}$ is 0 (if the event has already occurred before the examination time, which is $R_{ij}$ in this case) or $R_{ij}$ is $\infty$ (if the event has not happened upon the examination time, which is $L_{ij}$ in this case). Therefore, the likelihood function (\ref{joint_likelihood}) can handle the general form of bivariate interval-censored data.
\subsection{Semiparametric linear transformation model for marginal survival functions} \label{subsec_semipar}
We consider the semiparametric transformation models for marginal survival functions:
\begin{equation}
S_{j}(t\mid Z_j) = \exp[-G_j\{\exp(Z_j^{T}\beta_j)\Lambda_{j}(t)\}], \ j = 1,2, \label{tran_mod}
\end{equation}
where $G_j(\cdot)$ is a pre-specified strictly increasing function, $\beta_j$ is a vector of unknown regression coefficients, and $\Lambda_{j}(\cdot)$ is an unknown non-decreasing function of $t$. In model (\ref{tran_mod}), the transformation function $G_j(\cdot)$, the regression parameter $\beta_j$ and the infinite-dimensional parameter $\Lambda_j(\cdot)$ are all denoted as margin-specific (indexed by $j$) for generality. In practice, some or all of them can be the same for the two margins, and in that case, the corresponding index $j$ can be dropped.
This model (\ref{tran_mod}) contains a class of survival models. For example, when $G(x) = x$, the marginal survival function follows a PH model. When $G(x) = \log(1+x)$, the marginal function becomes a proportional odds (PO) model. In practice, the transformation function can also be ``estimated'' by the data. For example, the commonly used Box-Cox transformation $G(x) = \{(1+x)^r -1\}/r$, $r > 0$, or the logarithmic transformation $G(x) = \log(1+rx)/r$, $r > 0$, can be assumed. The PH and PO models are special cases in both transformation classes. Then the parameter $r$ in $G(\cdot)$ can be estimated together with other parameters in the likelihood, as we will demonstrate in our simulation studies.
\section{Estimation and Inference} \label{sec_estimation_test_procedure}
\subsection{Sieve likelihood with Bernstein polynomials} \label{subsec_sieve}
In our likelihood function, we are interested in estimating the unknown parameter $\theta \in \Theta$:
$$\Theta = \{ \theta = (\beta_1^T, \beta_2^T, \alpha, \kappa, \Lambda_{1}, \Lambda_{2})^T \in \mathcal{B} \otimes \mathcal{M} \otimes \mathcal{M} \}.$$
Here $\mathcal{B}=\{(\beta=(\beta_1^T,\beta_2^T)^T, \alpha,\kappa) \in R^p \times R^{(0,1]} \times R^{+},\Vert\beta\Vert + \Vert\alpha\Vert + \Vert\kappa\Vert \leq M\}$ with $p$ being the dimension of $\beta$ and $M$ being a positive constant. We denote by $\mathcal{M}$ the collection of all bounded, continuous and nondecreasing, nonnegative functions over $[c, u]$, where $0 \leq c < u < \infty$. In practice, $[c,u]$ can be chosen as the range of all $L_{ij}$ and $R_{ij}$.
In our log-likelihood function $l_{n}(\theta; D)=\log{L_n(\theta; D)}=\sum_{i=1}^{n}\log{L(\theta; D_i)} = \sum_{i=1}^{n}l(\theta; D_i)$, there are finite-dimensional parameters of interest $(\beta,\alpha,\kappa)$ and two infinite-dimensional nuisance parameters $(\Lambda_{1},\Lambda_{2})$, which need to be estimated simultaneously. Unlike the right-censored data, tools like partial likelihood and martingale can not be applied to the interval-censored data due to the absence of exact event times. Instead, following \citet{univariate_PO_sieve}, we employ the sieve approach and form a sieve likelihood. Specifically, similar to \citet{frailty_case_II_transformation_sieve}, we use Bernstein polynomials to build a sieve space $\Theta_n = \{ \theta_n = (\beta^T, \alpha, \kappa, \Lambda_{1n}, \Lambda_{2n})^T \in \mathcal{B} \otimes \mathcal{M}_{n} \otimes \mathcal{M}_{n} \}$. Here, $\mathcal{M}_n$ is the space defined by Bernstein polynomials:
$$\mathcal{M}_n = \biggl\{ \Lambda_{jn}(t) = \sum_{k=0}^{m_{n}} \phi_{jk}B_{k}(t,m_{n},c,u): \sum_{k=0}^{m_n} |\phi_{jk}| \leq M_{n}; \ 0 \leq \phi_{j0} \leq \cdot \cdot \cdot \leq \phi_{jm_{n}}; j=1,2 \biggl\},$$
where $B_{k}(t,m_{n},c,u)$ represents the Bernstein basis polynomial defined as:
\begin{equation}
B_{k}(t,m_{n},c,u) = {m_{n} \choose k} (\frac{t-c}{u-c})^{k} (1-\frac{t-c}{u-c})^{m_{n}-k}; \ k = 0,...,m_{n}, \label{Bern}
\end{equation}
with degree $m_{n} = o(n^{\nu})$ for some $\nu \in (0,1)$. We assume the basis polynomials $B_{k}(t,m_{n},c,u)$ are the same between the two margins, while the coefficients $\phi_{jk}$ can be margin-specific. In practice, one may choose $m_n$ based on model AIC values. With a pre-specified $m_n$, we solve $\phi_{jk}$ together with other parameters $(\beta, \alpha, \kappa)$. One big advantage of Bernstein polynomials is that they can achieve the non-negativity and monotonicity properties of $\Lambda_j(t)$ through re-parameterization \citep{frailty_case_II_transformation_sieve}. Another advantage of Bernstein polynomials is that they do not require the specification of interior knots, as seen from (\ref{Bern}), making them flexible for use.
With the sieve space defined above, $\Lambda_j(t)$ will be approximated by $\Lambda_{jn}(t) \in \mathcal{M}_n$. In the next section, we propose an estimation procedure to maximize $l_n(\theta; D)$ over the sieve space $\Theta_n$ to obtain the sieve maximum likelihood estimators $\hat{\theta}_n = (\hat{\beta}_n^T, \hat{\alpha}_n, \hat{\kappa}_n, \hat{\Lambda}_{1n}, \hat{\Lambda}_{2n})^T$.
\subsection{Estimation procedure for sieve maximum likelihood estimators $\hat{\theta}_n$} \label{subsec_two_step}
We develop a novel sieve maximum likelihood estimation procedure that is generally applicable to any choice of Archimedean copulas and marginal models. In principle, we can obtain the sieve maximum likelihood estimators by maximizing the joint likelihood function (\ref{joint_likelihood}) in one step. Due to the complex structure of the joint likelihood function, we recommend using a separate step to obtain appropriate initial values for all the unknown parameters. In essence, $(\beta_j, \Lambda_{jn})$ are first estimated marginally in step 1(a). Then their estimators are plugged into the joint likelihood to form a pseudo-likelihood. In step 1(b), the dependence parameters $(\alpha, \kappa)$ are estimated through maximizing the pseudo-likelihood function. Finally, using initial values from step 1(a) and 1(b), we update all the unknown parameters simultaneously under the joint log-likelihood function in step 2. The estimation procedure is described below:
\begin{enumerate}
\item Obtain initial estimates of $\theta_n$:
\begin{enumerate}
\item $(\hat{\beta}_{jn}^{(1)}, \hat{\Lambda}_{jn}^{(1)}) = \argmaxC_{(\beta_j, \Lambda_{jn})} l_{jn}(\beta_j, \Lambda_{jn})$, where $l_{jn}$ denotes the sieve log-likelihood for the marginal model, $j=1, 2$;
\item $(\hat{\alpha}_{n}^{(1)},\hat{\kappa}_{n}^{(1)})=\argmaxC_{(\alpha,\kappa)} l_n(\hat{\beta}_{n}^{(1)}=(\hat{\beta}_{1n}^{(1)},\hat{\beta}_{2n}^{(1)}), \alpha, \kappa, \hat{\Lambda}_{1n}^{(1)},\hat{\Lambda}_{2n}^{(1)})$, where $\hat{\beta}_{jn}^{(1)}$ and $\hat{\Lambda}_{jn}^{(1)}$ are the initial estimates from (a), and $l_n$ is the joint sieve log-likelihood.
\end{enumerate}
\item Simultaneously maximize the joint sieve log-likelihood to get final estimates:\\
$\hat{\theta}_n = (\hat{\beta}_n,\hat{\alpha}_n,\hat{\kappa}_n,\hat{\Lambda}_{1n},\hat{\Lambda}_{2n})=\argmaxC_{(\beta,\alpha,\kappa,\Lambda_{1n},\Lambda_{2n})} l_n(\beta,\alpha,\kappa,\Lambda_{1n},\Lambda_{2n})$ with initial values $(\hat{\beta}_{n}^{(1)},\hat{\alpha}_{n}^{(1)},\hat{\kappa}_{n}^{(1)},\hat{\Lambda}_{1n}^{(1)},\hat{\Lambda}_{2n}^{(1)})$ obtained from step 1(a) and 1(b).
\end{enumerate}
For the variance-covariance of finite-dimensional parameter estimates ($\hat{\beta}_n,\hat{\alpha}_n,\hat{\kappa}_n$), we invert the observed information matrix of all parameters including the nuisance parameters ($\phi_{jk}$) from the last iteration of step 2 and then take the corresponding block. In section \ref{subsec_asymptotics}, we establish the asymptotic normality and semiparametric efficiency for the finite-dimensional parameters. However, since the asymptotic variance form is intractable, we adopt this heuristic approach, which has been shown to work well in practice \citep{ding2011}.
Some standard optimization algorithms such as the Newton-Raphson algorithm or the conjugate gradient algorithm can be employed to obtain the maximizers and observed information matrix. Due to the complex structure of the joint sieve log-likelihood, instead of analytically deriving the first and second order derivatives, we propose to use the Richardson's extrapolation (\citeauthor{Richardson}, \citeyear{Richardson}) to approximate the score function and observed information matrix numerically. As shown in our simulations, the proposed procedure guarantees almost $100\%$ convergence and the computing speed is notably improved by using initial values from step 1.
\subsection{Asymptotic properties of sieve estimators} \label{subsec_asymptotics}
This section presents asymptotic properties of the sieve maximum likelihood estimators $\hat{\theta}_n$ with regularity conditions and proofs being supplied in Appendix D of the Supplementary Materials. Denote $P$ as the true probability measure and $\mathbb{P}_n$ as the empirical measure for $n$ independent subjects. Let $\vert v\vert$ be the Euclidean norm for a vector $v$. Define the supremum norm $\Vert f \Vert_{\infty} = sup_t\vert f(t)\vert$ for a function $f(t)$. Also define $\Vert f \Vert_{L_2(P)} = (\int \vert f \vert^2 dP)^{1/2}$ for a function $f$ under the probability measure $P$. In particular, the $L_2(P)$ norm for $\Lambda_j$ is defined as $\Vert \Lambda_j \Vert_2^2 = \int [\{\Lambda_j(l)\}^2 + \{\Lambda_j(r)\}^2 ] dF_j(l,r)$, where $F_j(l,r)$ denotes the joint cumulative distribution function of $L_{ij}$ and $R_{ij} \ (i=1,...,n; j = 1,2)$. Finally, we define the distance between $\theta_1 = (\beta_1^T,\alpha_1,\kappa_1,\Lambda_{11},\Lambda_{21})^T \in \Theta$ and $\theta_2 = (\beta_2^T,\alpha_2,\kappa_2,\Lambda_{12},\Lambda_{22})^T \in \Theta$ as
$$ d(\theta_1,\theta_2) = ( \vert \beta_1-\beta_2 \vert^2 + \vert \alpha_1-\alpha_2 \vert^2 + \vert \kappa_1-\kappa_2 \vert^2 + \Vert \Lambda_{11}-\Lambda_{12}\Vert_{2}^2 + \Vert \Lambda_{21}-\Lambda_{22} \Vert_{2}^2 )^{1/2}.$$
Let $\theta_0 = (\beta_0^T,\alpha_0,\kappa_0,\Lambda_{10},\Lambda_{20})^T$ denote the true value of $\theta \in \Theta$. The following theorems present the convergence rate, asymptotic normality, and efficiency of the sieve estimators.
\begin{theorem} \label{thm:rate}
(Convergence rate) Assume that Conditions 1-2 and 4-5 in Appendix D of the Supplementary Materials hold. Let $m_n = o(n^{\nu})$, where $\nu \in (0,1)$ and $q$ be the smoothness parameter of $\Lambda_j$ as defined in Condition 4, then we have
$$d(\hat{\theta}_n,\theta_0) = O_p\big( n^{-\min\{{q\nu}/{2}, (1-\nu)/{2}\}} \big).$$
\end{theorem}
Theorem \ref{thm:rate} states that the sieve estimator $\hat{\theta}_n$ has a polynomial convergence rate. Although this overall convergence rate is lower than $n^{-1/2}$, the following normality theorem states that the proposed estimators of the finite-dimensional parameters ($\beta, \alpha, \kappa$) are asymptotically normal and semiparametrically efficient.
\begin{theorem} \label{thm:normality}
(Asymptotic normality and efficiency) Suppose Conditions 1-5 in Appendix D hold. Define $\hat{b}_n = (\hat{\beta}_n^T, \hat{\alpha}_n, \hat{\kappa}_n)^T$ and $b_0 = (\beta_0^T, \alpha_0, \kappa_0)^T$. If $1/(2+q)< \nu < {1}/{2}$, then
$${n}^{1/2}(\hat{b}_n-b_0) = I^{-1}(b_0) n^{1/2} \mathbb{P}_n l^{*}(b_0, \Lambda_{10}, \Lambda_{20}; D) + o_p(1) \to_d N\{0, I^{-1}(b_0)\},$$
where $I(b_0) = Pl^{*}(b_0, \Lambda_{10}, \Lambda_{20}; D)^{\otimes 2}$ and $l^{*}(b_0, \Lambda_{10}, \Lambda_{20}; D)$ is the efficient score function defined in the proof. Therefore, $\hat{b}_n$ is asymptotically normal and efficient.
\end{theorem}
\subsection{Generalized score test} \label{subsec_score}
We now separate $\beta$ into two parts: $\beta_g$ and $\beta_{ng}$, where $\beta_g$ is the parameter set of interest for hypothesis testing and $\beta_{ng}$ denotes the rest of the regression coefficients. The likelihood-based tests such as the Wald, score, and likelihood-ratio tests can be constructed to test $\beta_g$, and they are asymptotically equivalent. In our motivating study, we aim to perform a GWAS on AMD progression, which contains testing millions of SNPs one-by-one. Therefore, computing speed is critical. We propose to use the generalized score test. One big advantage of the score test in a GWAS setting is, one only needs to estimate the unknown parameters once under the null model without any SNP (i.e., $\beta_g = 0$), since the non-genetic risk factors are the same no matter which SNP is being tested. Therefore, the score test is faster as compared to the Wald and likelihood ratio tests. Moreover, the Wald or likelihood ratio test needs to estimate parameters under each alternative hypothesis (a total of 6 millions in our real data application), which may fail when the estimation procedure fails to converge.
With the sieve joint likelihood, we can obtain the restricted sieve maximum likelihood estimators under $H_0$ ($\beta_g = 0$ and the rest parameters are arbitrary), and then calculate the generalized score test statistic as defined in \citet{theoretical_stat}. Similar to our estimation procedure, we also propose to use Richardson's extrapolation to numerically approximate the first and second order derivatives when calculating the score test statistic.
\section{Simulation study} \label{sec_simulations}
We first evaluated the parameter estimation of our proposed two-parameter copula sieve model (i.e., its transformation margins are approximated by sieves) for bivariate data under general interval censoring. Then we assessed the type-I error control, and power performance of the proposed generalized score test. We also evaluated the accuracy in estimating the joint survival probability using our proposed method. Finally, we evaluated the computing speed and convergence rate of our proposed method.
\subsection{Generating bivariate interval-censored times} \label{generating data}
The data were generated from various Archimedean copula models (i.e., Clayton, Frank, Ali--Mikhail--Hap (AMH) and Joe) with Loglogistic margins. We first generated bivariate true event times $T_{ij}$ using the conditioning approach described in \citet{Sun_LIDA_2018}. To obtain interval-censored data, we followed the censoring procedure in \citet{simulate_IC}, which fits the study design of AREDS data. Explicitly, we assumed each subject was assessed for $K$ times with the length between two adjacent assessment times following an Exponential distribution. In the end, for each subject $i$, $L_{ij}$ was defined as the last assessment time before $T_{ij}$ and $R_{ij}$ was the first assessment time after $T_{ij}$. The overall right-censoring rate is set to be $25\%$. For the dependence strength between margins, we set Kendall's $\tau$ at 0.2 or 0.6, indicating weak or strong dependence. We assumed that the two event times share a common baseline distribution, for example, PO model with Loglogistic baseline hazards function (scale $\lambda=1$ and shape $k=2$) or PH model with Weibull baseline hazards function (scale $\lambda=0.1$ and shape $k=2$).
We included both genetic and non-genetic covariates in the simulations. Specifically, each SNP, coded as 0 or 1 or 2, was generated from a multinomial distribution with probabilities $\{(1-p)^2,2p(1-p),p^2\}$, where $p = 40\%$ or $5\%$ is the minor allele frequency (MAF). We also included a margin-specific continuous variable, generated from $N(6,2^2)$, and a subject-specific binary variable, generated from Bernoulli ($p=0.5$).
Under all scenarios, the sample size was set as $N=500$. For simplicity, we assumed the same covariate effects for two margins, denoted as $(\beta_{ng1},\beta_{ng2},\beta_g)$, where $\beta_{ng1}$ and $\beta_{ng2}$ are marginal- and subject-specific non-genetic effects, respectively, and $\beta_g$ is the SNP effect. We set $\beta_{ng1} = \beta_{ng2}=0.1$. For estimation performance evaluation, we let $\beta_g=0$ and replicated 1,000 times. For type-I error control evaluation of testing $\beta_g = 0$, we replicated 100,000 times and evaluated at various tail levels: 0.05, 0.01, 0.001 and 0.0001, respectively. For power evaluation, we replicated 1,000 times under each SNP effect size, where a range of $\beta_g$'s were selected to represent weak to strong SNP effects.
\subsection{Simulation-I: parameter estimation} \label{Estimation}
In this section, we evaluated the estimation performance of our proposed method under both correct and misspecified settings. For the margins, we used the true linear transformation function. We assumed the same Bernstein coefficients $\phi_{1k}=\phi_{2k}$ with degree $m_n=3$ ($k=0,1,2,3$) for both $\Lambda_1$ and $\Lambda_2$. For the event time range $[c,u]$, we chose $c=0$ and set $u$ as the largest value of all $\{L_{ij}, R_{ij}\}$ plus a constant.
In Table \ref{tab:Estimation_case_II}(a), the true model is Clayton copula with Loglogistic (PO) or Weibull (PH) margins, under Kendall's $\tau = 0.6$. We fitted three models: the true copula model with parametric margins (i.e., Clayton copula with Loglogistic or Weibull margins, denoted as ``Clayton-PM''), a two-parameter copula sieve model (``Copula2-S'') and a marginal sieve model (i.e., the marginal transformation model approximated by sieves) where the variance-covariance is estimated by the robust sandwich estimator (``Marginal-S'') (a model also used in \citeauthor{frailty_case_II_transformation_sieve}, \citeyear{frailty_case_II_transformation_sieve}). We obtained estimation biases and 95\% coverage probabilities for regression coefficients and dependence parameters. Under the two-parameter copula model, the sieve maximum likelihood estimators are all virtually unbiased, and all empirical coverage probabilities are close to the nominal level. Moreover, their standard errors are virtually the same as the standard errors under the true parametric model, indicating our proposed method works well. For the robust marginal sieve model, the regression coefficient estimates are also unbiased with correct coverage probabilities, but their standard errors are apparently larger.
We further evaluated the estimation performance of the proposed model on bivariate interval-censored data generated from copula models that do not belong to the two-parameter copula family, such as Frank copula with $\tau = 0.6$, AMH copula with $\tau = 0.2$ ($\tau$ is always $<\frac{1}{3}$ for AMH copula) and Joe copula with $\tau = 0.6$. In Table \ref{tab:Estimation_case_II}(b), the regression coefficient estimates from the two-parameter copula are all unbiased with coverage probabilities close to 95\%. The biases for the $\tau$ estimates are also minimal with good coverage probabilities except in the scenario when data were generated from a Joe copula (coverage probability = 83\%). Overall, the two-parameter copula model family demonstrates good robustness to misspecification in copula functions.
In the real setting, the value of the transformation function parameter $r$ is often unknown. Therefore, we examined our methods in estimating the transformation function parameter $r$ together with the other parameters in our proposed model. The results are presented in the Table 1 of Appendix A in the Supplementary Materials, which shows satisfactory estimation performance for all parameters including the transformation parameter.
\subsection{Simulation-II, generalized score test performance} \label{test_performance}
We evaluated the type-I error control of our proposed generalized score test under Copula2-S. Specifically, we tested the SNP effect $\beta_g$ under different dependence strengths (Kendall's $\tau=0.6, \ 0.2$) and two different MAFs (40\%, 5\%). The true model is Clayton copula with Loglogistic margins. We included score tests of two misspecified copula models, one with misspecified margins but correct copula (i.e., Clayton copula with Weibull margins) and the other with misspecified copula but correct margins (i.e., Gumbel copula with Loglogistic margins). We also included the score test under the correct parametric copula model (i.e., Clayton copula with Loglogistic margins), which served as the benchmark model. Besides, we examined Wald tests from the marginal Loglogistic model with variance-covariance either from the independence estimate (i.e., the naive estimate assuming two margins are independent) or the robust sandwich estimate (i.e., accounting for the correlation between two margins).
Table \ref{tab:Type_I_error} shows type I errors under different tail levels. In the top part where Kendall's $\tau=0.6$, our proposed score test controls type-I errors as well as the correct parametric model at all tail levels and MAFs. However, type-I errors in the two misspecified copula models are inflated at all scenarios, especially when margins are wrong at MAF $ = 40\%$. It is not surprising to observe the greatest inflation occurs with the marginal approach under the independence assumption. After applying the robust variance-covariance estimate, the type-I errors are controlled at MAF = 40\% but still slightly inflated at MAF = 5\%. When Kendall's $\tau=0.2$, the proposed two-parameter model still performs as well as the correct parametric model and outperforms the other models, although the type-I error inflations from other models were smaller due to the weaker dependence.
We also compared the power performance between the score test under our Copula2-S model and score tests from two other models: the true parametric copula model and the Marginal-S model. Figure \ref{fig:Clayton_Loglog_power} presents the power curves of these three tests over a range of SNP effect sizes. Our proposed model yields the similar power performance as the true parametric model and is considerably more potent than the robust marginal sieve model.
\subsection{Simulation-III: joint survival probability estimation performance}
In addition, we evaluated the accuracy for estimating joint survival probabilities under our proposed Copula2-S model. We generated data from the Clayton copula with Weibull margins, and fitted the Clayton-Weibull (``Clayton-WB'') and Copula2-S models and obtained the average estimated joint survival probabilities $Pr(T_1 > t, T_2 > t|Z_1, Z_2)$ on a sequence of pre-specified time points given covariate values. The number of replications is $1,000$. In Appendix A of the Supplementary Materials, Figure S1 illustrates that Copula2-S produced an almost identical joint survival profile as Clayton-WB. In addition, we quantified the estimation error between the estimated and true joint survival probabilities by the mean square errors (MSE) averaged over all time points and replications, which are $0.0004$ (sd $=0.0012$) and $0.0003$ (sd $=0.0005$) for Copula2-S and Clayton-WB, respectively, indicating the probabilities are well estimated.
\subsection{Simulation-IV, convergence and computing speed} \label{Speed}
We examined the computational advantages of our proposed two-step sieve estimation procedure as compared to the one-step estimation approach (i.e., directly maximizes the joint likelihood with arbitrary initial values). Data were simulated from a Clayton copula with Loglogistic margins. For $1,000$ replications, the one-step procedure took $1,260$ seconds while our proposed procedure took $925$ seconds, saving about $27\%$ computing time. For convergence rate, the proposed procedure failed in $0.1\%$ times, whereas the one-step procedure failed in $1.6\%$ times.
We also compared the computing speed of three likelihood-based tests on testing $1,000$ SNPs under three models: the true Clayton model with Loglogistic margins, our proposed Copula2-S model and the Marginal-S model. The 1,000 genetic variants were simulated from MAF $=40\%$. The results are shown in Table S3 in the Appendix A from the Supplementary Materials. We found that the score test is about 3-5 times faster than the Wald test or the likelihood ratio test on average. Within the three score tests, although the score test under our Copula2-S model is the slowest due to model complexity, it is still faster than the Wald test under the Marginal-S model. Given its advantages in robustness, type-I error control, and power performance, we recommend the proposed Copula2-S model with the score test for the large-scale testing case.
\section{Real data analysis} \label{sec_real_data}
We implemented our proposed method to analyze the AREDS data. AREDS was designed to assess the clinical course of, and risk factors for the development and progression of AMD. DNA samples were collected from the consenting participants and genotyped by the International AMD Genomics Consortium \citep{AMD_genetic_2016}. In this study, each participant was examined every six months in the first six years and then every year after year six. To measure the disease progression, a severity score, scaled from one to twelve (with a more significant value indicating more severe AMD), was determined for each eye of each participant at every examination. The outcome of interest is the bivariate progression time-to-late-AMD, where late-AMD is defined as the stage with severity score $\ge 9$. Both phenotype and genotype data of AREDS are available from the online repository dbGap (accession: phs000001.v3.p1, and phs001039.v1.p1, respectively). By far, all the studies that analyzed the AREDS data for AMD progression treated the outcome as right-censored (e.g., \citet{AMD_prog_3}, \citet{Yan_2018}, and \citet{Sun_LIDA_2018}), and some only used data from the worst eye in each subject (e.g., \citet{Seddon_nine}).
We analyzed 2,718 Caucasian participants, including 2,295 subjects who were free of late-AMD in both eyes at the enrollment, i.e., time $0$ (bivariate data indicated as group A), and 423 subjects who had one eye already progressed to late-AMD by enrollment (univariate data indicated as group B). For the $j$th eye (free of late-AMD at time 0) of subject $i$, we observe $L_{ij}$, the last assessment time when the $j$th eye was still free of late-AMD and $R_{ij}$, the first assessment time when the $j$th eye was already diagnosed as late-AMD. For the eye that did not progress to late-AMD by the end of the study follow-up, we assigned a large number to $R_{ij}$. Since there are two groups of subjects (group A and B), we implemented a two-part model. Specifically, we created a covariate for each eye to indicate whether its fellow eye had already progressed or not at time 0 (i.e., $0$ for both eyes of group A subjects and $1$ for group B subjects). Then the joint likelihood is the product of the copula sieve model for group A subjects and the marginal sieve model for group B subjects. In addition, we performed a secondary sensitivity analysis using only group A subjects and obtained similar top SNPs as from the two-part model (Table S4 of Appendix B in the Supplementary Materials).
We included four risk factors as non-genetic covariates, including the baseline age, severity score, smoking status, and fellow-eye progression status. We checked various combinations of transformation functions (i.e., $g(x) =x$ for PH model and $g(x) = \log (1+x)$ for PO model) and Bernstein polynomial degrees $m_n$ (from 3 to 6). We chose the model that produced the smallest $\textsc{aic}$, which is the PO model (i.e., $g(x) = \log (1+x)$) with $m_n = 4$ for both margins. The $\textsc{aic}$ results are summarized in Table S5 of Appendix B in the Supplementary Materials.
We performed GWAS on 6 million SNPs (either from exome chip or imputed) with MAF $> 5\%$ across the 22 autosomal chromosomes and plotted their $-\log(p)$ in Figure S2 in the Appendix B of the Supplementary Materials. As highlighted in the figure, the \textit{PLEKHA1--ARMS2--HTRA1} region on chromosome 10 and the \textit{CFH} region on chromosome 1 have variants reaching the ``genome-wide" significance level ($p< 5\times 10^{-8}$). Previously, these two regions were found being significantly associated with AMD onset from multiple case-control studies \citep{AMD_genetic_2016}. Moreover, we identified SNPs in a previously unrecognized \textit{ATF7IP2} region on chromosome 16, showing moderate to strong association with AMD progression ($ 5 \times 10^{-8} < p < 1 \times 10^{-5}$). As a comparison, we also fitted the robust marginal sieve model (Marginal-S) and the Gamma frailty sieve model (Frailty-S) \citep{frailty_case_II_transformation_sieve}, and performed the corresponding score tests for each SNP. Overall, their results are consistent with our Copula2-S model, but the $p$-values are generally larger (as shown in Table \ref{tab:GWAS_table}). Note that the \textit{CFH} region did not reach the ``genome-wide" significance level under the Marginal-S model.
Table \ref{tab:GWAS_table} presents the top significant variants of the three regions denoted in Figure S2. Besides Copula2-S, we also present score test $p$-values from Frailty-S and Marginal-S. The odds ratio of an SNP was calculated by fitting a Copula2-S model including this SNP and those non-genetic factors. For example, $rs2284665$, a known AMD risk variant from \textit{HTRA1} region, has an estimated odds ratio of 1.66 (95\% CI $=[1.46, 1.89]$), which implies its minor allele has a ``harmful'' effect on AMD progression. Under this model, the estimated dependence parameters are $\hat{\alpha} = 0.90$ and $\hat{\kappa} = 1.00$, corresponding to $\hat{\tau} = 0.40$, which indicates moderate dependence in AMD progression between two eyes.
For variant $rs2284665$, we obtained both estimated joint and conditional survival functions from the fitted Copula2-S model. The left panel of Figure \ref{fig:3D_and_conditional} plots the joint progression-free probability contours for subjects who are smokers with the same age (= 68.6) and AMD severity score (= 3.0 for both eyes) but different genotypes of $rs2284665$. The right panel of Figure \ref{fig:3D_and_conditional} plots the corresponding conditional progression-free probability of remaining years (after year 5) for one eye, given its fellow eye has progressed by year 5. It is clearly seen that in both plots, the three genotype groups are well separated, with the $GG$ group having the largest progression-free probabilities. These estimated progression-free probabilities provide valuable information to characterize or predict the progression profiles for AMD patients with different characteristics.
\section{Conclusion and Discussion} \label{sec_conclusions}
We proposed a flexible copula-based semiparametric transformation model for analyzing and testing bivariate (general) interval-censored data. Unlike the approach proposed by \citet{HuCaseICopulaPH_2017}, which approximated the copula function by sieves, our approach kept the copula function in its parametric form but flexibly modeled the margins through semiparametric transformation models. In this way, our method guaranteed to produce consistent estimates for both regression and copula parameters, which then led to reliable joint distribution estimates. On the other hand, \citet{HuCaseICopulaPH_2017} focused on estimating regression parameters only but with possible biased estimates for the copula function. Our proposed method has the great advantage in computation and it is applicable to analyze large data sets and to perform a large number of tests. All the methods have been built into an R package \{CopulaCenR\}, which includes a variety of copula functions (e.g., Copula2, Clayton, Gumbel, Frank, Joe, AMH) and is available on CRAN at {https://cran.r-project.org/package=CopulaCenR}. The key R codes for this article can be found at https://github.com/yingding99/Copula2S.
Several model selection procedures have been proposed for copula-based methods. For example, the AIC is widely used for model selection purpose in copula models. \citet{wang_selection_2000} proposed a model selection procedure based on the nonparametric estimation of the bivariate joint survival function within Archimedean copulas. For model diagnostics, \citet{Chen_selection_2010} proposed a penalized pseudo-likelihood ratio test for copula models in complete data. Recently, \citet{Zhang2016_PIOS} developed a goodness-of-fit test for copula models using the pseudo in-and-out-of-sample method. To the best of our knowledge, there is no existing goodness-of-fit test for copula models of bivariate interval-censored data. In our real data analysis, we used AIC to guide the model selection for simplicity. However, a formal test for goodness-of-fit is desirable, especially for bivariate interval-censored data under the regression setting. It is worthwhile to investigate it as a future research direction.
We applied our method to a GWAS of AMD progression and successfully identified variants from two known AMD risk regions (\textit{CFH} on chromosome 1 and \textit{PLEKHA1--ARMS2--HTRA1} on chromosome 10) being significantly associated with AMD progression. Moreover, we also discovered variants from a region (\textit{ATF7IP2} on chromosome 16), which has not been reported before, showing moderate to strong association with AMD progression. On the contrary, we found that some known AMD risk loci (e.g., $rs12357257$ from \textit{ARHGAP21} on chromosome 10, $p=0.12$) are not associated with AMD progression. Therefore, the variants associated with risks of having AMD may not be necessarily associated with the disease progression; while some variants may be only associated with AMD progression but not with the disease onset. Our work is the first research on AMD progression which adopts a solid statistical model that appropriately handles bivariate interval-censored data. Our findings provided new insights into the genetic causes on AMD progression, which are critical for establishing novel and reliable predictive models of AMD progression to identify high-risk patients at an early stage accurately. Our proposed method applies to general bilateral diseases and complex diseases with co-primary endpoints.
\section*{Supplementary Materials} \label{Appendix}
Supplementary materials are available online at
{http://biostatistics.oxfordjournals.org}.
\begin{figure}
\caption{Simulation results for power performance of the score test under three models: Clayton-LL (top dashed curve), Copula2-S (solid curve) and Marginal-S (bottom dashed curve).}
\label{fig:Clayton_Loglog_power}
\end{figure}
\begin{figure}
\caption{Estimated progression-free probabilities for subjects with different genotypes of $rs2284665$ (smokers with age 68.6 and severity score 3.0 in both eyes). Left: joint progression-free probability contours (from top to bottom: $GG, GT, TT$); Right: conditional progression-free probability of remaining years (after year 5) for one eye, given the other eye has progressed by year 5 (from top to bottom: $GG, GT, TT$).}
\label{fig:3D_and_conditional}
\end{figure}
\begin{table}[!p]
\caption{Estimation results for bivariate interval-censored data (a) fitted with three correctly-specified models: Clayton model with parametric margins (Loglogistic for proportional odds and Weibull for proportional hazards; denoted as Clayton-PM), two-parameter copula sieve model (Copula2-S) and marginal sieve model (Marginal-S); (b) fitted with the proposed Copula2-S model (misspecified copula) where the true data are generated from Frank, AMH, and Joe copulas.}
\label{tab:Estimation_case_II}
\resizebox{1.0\linewidth}{!}{
{\tabcolsep=4.25pt
\begin{tabular}{@{}cccccccccccc@{}}
\multicolumn{1}{c}{(a)} & \multicolumn{11}{c}{} \\
\tblhead{\multicolumn{1}{c}{} & \multicolumn{3}{c}{Clayton-PM} && \multicolumn{3}{c}{Copula2-S} && \multicolumn{3}{c}{Marginal-S} \\
\cline{2-4}\cline{6-8}\cline{10-12}
\multicolumn{1}{c}{Param} & Bias & SE & SEE (CP) && Bias & SE & SEE (CP) && Bias & SE & SEE (CP)
}
\multicolumn{1}{c}{} & \multicolumn{11}{c}{{proportional odds}} \\
\rule{0pt}{3ex}
$\beta_{ng1}$ & 0.0013 & 0.0171 & 0.0163 (0.942) && 0.0003 & 0.0176 & 0.0165 (0.938) && 0.0024 & 0.0293 & 0.0300 (0.930)
\\
$\beta_{ng2}$ & 0.0120 & 0.1300 & 0.1300 (0.945) && 0.0006 & 0.1330 & 0.1310 (0.939) && 0.0110 & 0.1510 & 0.1500 (0.944)
\\
$\beta_{g}$ & -0.0007 & 0.0927 & 0.0942 (0.953) && -0.0110 & 0.0951 & 0.0947 (0.950) && 0.0012 & 0.1050 & 0.1090 (0.955)
\\
$\tau$ & -0.0005 & 0.0210 & 0.0208 (0.944) && -0.0045 & 0.0223 & 0.0221 (0.950) && NA & NA & NA
\\
\hline
\multicolumn{1}{c}{} & \multicolumn{11}{c}{{proportional hazards}} \\
\rule{0pt}{3ex}
$\beta_{ng1}$ & 0.0012 & 0.0097 & 0.0103 (0.958) && 0.0013 & 0.0099 & 0.0105 (0.957) && 0.0009 & 0.0182 & 0.0187 (0.957)
\\
$\beta_{ng2}$ & -0.0043 & 0.0780 & 0.0789 (0.952) && -0.0040 & 0.0782 & 0.0788 (0.951) && -0.0043 & 0.0960 & 0.0969 (0.957)
\\
$\beta_{g}$ & 0.0005 & 0.0606 & 0.0569 (0.935) && 0.0002 & 0.0608 & 0.0569 (0.938) && 0.0003 & 0.0722 & 0.0701 (0.945)
\\
$\tau$ & -0.0003 & 0.0220 & 0.0219 (0.952) && -0.0012 & 0.0224 & 0.0221 (0.951) && NA & NA & NA
\lastline
\end{tabular}}
}
\resizebox{1.0\linewidth}{!}{
{\tabcolsep=4.25pt
\begin{tabular}{@{}cccccccccccc@{}}
\multicolumn{1}{c}{(b)} & \multicolumn{11}{c}{} \\
\tblhead{\multicolumn{1}{c}{} & \multicolumn{3}{c}{Frank} && \multicolumn{3}{c}{AMH} && \multicolumn{3}{c}{Joe} \\
\cline{2-4}\cline{6-8}\cline{10-12}
\multicolumn{1}{c}{Param} & Bias & SE & SEE (CP) && Bias & SE & SEE (CP) && Bias & SE & SEE (CP)
}
$\beta_{ng1}$ & 0.0002 & 0.0177 & 0.0176 (0.950) && -0.0011 & 0.0262 & 0.0267 (0.953) && 0.0016 & 0.0160 & 0.0166 (0.962)
\\
$\beta_{ng2}$ & 0.0018 & 0.1480 & 0.1470 (0.944) && 0.0013 & 0.1250 & 0.1250 (0.951) && -0.0027 & 0.1388 & 0.1438 (0.954)
\\
$\beta_{g}$ & 0.0001 & 0.1050 & 0.1060 (0.952) && -0.0001 & 0.0885 & 0.0901 (0.959) && 0.0037 & 0.0984 & 0.1043 (0.962)
\\
$\tau$ & -0.0036 & 0.0219 & 0.0198 (0.937) && -0.0056 & 0.0318 & 0.0304 (0.934) && 0.0168 & 0.0195 & 0.0185 (0.830)
\lastline
\end{tabular}}
}
\end{table}
\begin{table}
\caption{Type-I error for the genetic effect $\beta_g$ at various tail levels. Six models were compared: independent marginal Loglogistic model (Indep-LL), robust marginal Loglogistic model (Robust-LL), Clayton copula with Weibull margins (Clayton-W), Gumbel copula with Loglogistic margins (Gumbel-LL), two-parameter copula with transformation margins being approximated by sieves (Copula2-S) and the true Clayton copula and Loglogistic margins (Clayton-LL).}
\label{tab:Type_I_error}
{\tabcolsep=4.25pt
\begin{tabular}{@{}clllllll@{}}
\tblhead{\multicolumn{1}{l}{{MAF}} & {Tail} & {Indep-LL} & {Robust-LL} & {Clayton-W} & {Gumbel-LL} & {Copula2-S} & {Clayton-LL} }
\multicolumn{1}{l}{} & & \multicolumn{5}{c}{{Kendall's $\tau=0.6$}} \\
\multirow{4}{*}{{40\%}} & {0.05} & 0.141 & 0.051 & 0.131 & 0.065 & 0.052 & 0.050 \\
& {0.01} & 0.053 & 0.010 & 0.041 & 0.015 & 0.010 & 0.010 \\
& {0.001} & 0.0131 & 0.0012 & 0.0074 & 0.0022 & 0.0013 & 0.0012 \\
& {0.0001} & 0.0037 & 0.0002 & 0.0012 & 0.0003 & 0.0001 & 0.0001 \\
\rule{0pt}{3ex}
\multirow{4}{*}{{5\%}} & {0.05} & 0.141 & 0.056 & 0.059 & 0.066 & 0.053 & 0.051 \\
& {0.01} & 0.053 & 0.014 & 0.012 & 0.016 & 0.012 & 0.011 \\
& {0.001} & 0.0136 & 0.0018 & 0.0013 & 0.0020 & 0.0013 & 0.0012 \\
& {0.0001} & 0.0034 & 0.0003 & 0.0002 & 0.0003 & 0.0002 & 0.0002 \\
\hline
\multicolumn{1}{l}{} & & \multicolumn{5}{c}{{Kendall's $\tau=0.2$}} \\
\multirow{4}{*}{{40\%}} & {0.05} & 0.083 & 0.051 & 0.103 & 0.061 & 0.051 & 0.050 \\
& {0.01} & 0.022 & 0.010 & 0.029 & 0.013 & 0.010 & 0.010 \\
& {0.001} & 0.0036 & 0.0012 & 0.0045 & 0.0017 & 0.0011 & 0.0010 \\
& {0.0001} & 0.0006 & 0.0002 & 0.0006 & 0.0003 & 0.0002 & 0.0002 \\
\rule{0pt}{3ex}
\multirow{4}{*}{{5\%}} & {0.05} & 0.083 & 0.056 & 0.054 & 0.060 & 0.053 & 0.052 \\
& {0.01} & 0.023 & 0.013 & 0.011 & 0.014 & 0.012 & 0.011 \\
& {0.001} & 0.0036 & 0.0017 & 0.0013 & 0.0018 & 0.0014 & 0.0013 \\
& {0.0001} & 0.0007 & 0.0003 & 0.0001 & 0.0002 & 0.0002 & 0.0001
\lastline
\end{tabular}}
\end{table}
\begin{table}[!p]
\caption{The top SNPs identified to be associated with AMD progression. The last two columns come from the gamma frailty sieve model and the robust marginal sieve model, respectively.}
\label{tab:GWAS_table}
{\tabcolsep=4.25pt
\begin{tabular}{@{}llllllll@{}}
\tblhead{ SNP & Chr & Gene & MAF & OR & $p$ (Copula2-S) & $p$ (Frailty-S) & $p$ (Marginal-S) }
$rs2284665$ & 10 & \textit{HTRA1} & 0.33 & 1.66 & $1.5 \times 10^{-14}$ & $2.7 \times 10^{-12}$ & $1.6 \times 10^{-10}$ \\
$rs2293870$ & 10 & \textit{ARMS2-HTRA1} & 0.33 & 1.65 & $2.5 \times 10^{-14}$ & $2.5 \times 10^{-12}$ & $ 2.4 \times 10^{-10}$ \\
$rs3750846$ & 10 & \textit{ARMS2-HTRA1} & 0.34 & 1.62 & $1.6 \times 10^{-13}$ & $8.5 \times 10^{-12}$ & $ 8.7 \times 10^{-10}$ \\
$rs58649964$ & 10 & \textit{PLEKHA1} & 0.24 & 1.63 & $3.0 \times 10^{-11}$ & $1.0 \times 10^{-9}$ & $ 2.0 \times 10^{-8}$ \\
$rs10922109$ & 1 & \textit{CFH} & 0.28 & 0.64 & $4.0 \times 10^{-9}$ & $7.4 \times 10^{-9}$ & $7.4 \times 10^{-8} $ \\
$rs1329427$ & 1 & \textit{CFH} & 0.28 & 0.64 & $4.4 \times 10^{-9}$ & $8.3 \times 10^{-9}$ & $ 8.1 \times 10^{-8} $ \\
$rs10801559$ & 1 & \textit{CFH} & 0.28 & 0.64 & $4.8 \times 10^{-9}$ & $9.3 \times 10^{-9}$ & $ 8.8 \times 10^{-8} $ \\
$rs1410996$ & 1 & \textit{CFH} & 0.28 & 0.64 & $5.3 \times 10^{-9}$ & $1.1 \times 10^{-8}$ & $ 1.0 \times 10^{-7} $ \\
$rs12708701$ & 16 & \textit{ATF7IP2} & 0.13 & 1.62 & $1.1 \times 10^{-7}$ & $2.5 \times 10^{-7}$ & $ 7.0 \times 10^{-7} $ \\
$rs28368872$ & 16 & \textit{ATF7IP2} & 0.13 & 1.62 & $1.3 \times 10^{-7}$ & $4.3 \times 10^{-7}$ & $ 8.7 \times 10^{-7} $
\lastline
\end{tabular}}
\end{table}
\end{document} | math | 59,928 |
\mathfrak{b}egin{document}
\title[Bessel sequences of exponentials on fractal measures]{Bessel sequences of exponentials on fractal measures}
\mathfrak{a}uthor{Dorin Ervin Dutkay}
\xdef\@thefnmark{}\@footnotetext{}
\mathfrak{a}ddress{[Dorin Ervin Dutkay] University of Central Florida\\
Department of Mathematics\\
4000 Central Florida Blvd.\\
P.O. Box 161364\\
Orlando, FL 32816-1364\\
U.S.A.\\} \email{[email protected]}
\mathfrak{a}uthor{Deguang Han}
\mathfrak{a}ddress{[Deguang Han]University of Central Florida\\
Department of Mathematics\\
4000 Central Florida Blvd.\\
P.O. Box 161364\\
Orlando, FL 32816-1364\\
U.S.A.\\} \email{[email protected]}
\mathfrak{a}uthor{Eric Weber}
\mathfrak{a}ddress{[Eric Weber]Department of Mathematics\\
396 Carver Hall\\
Iowa State University\\
Ames, IA 50011\\
U.S.A.\\} \email{[email protected]}
\thanks{}
\subjclass[2000]{28A80,28A78, 42B05} \keywords{fractal, iterated function system, frame, Bessel sequence, Riesz basic sequence, Beurling dimension}
\mathfrak{b}egin{abstract}
Jorgensen and Pedersen have proven that a certain fractal measure
$\nu$ has no infinite set of complex exponentials which form an orthonormal
set in $L^2(\nu)$. We prove that any fractal measure $\mu$ obtained from an
affine iterated function system possesses a sequence of complex exponentials which
forms a Riesz basic sequence, or more generally a Bessel sequence, in $L^2(\mu)$ such
that the frequencies have positive Beurling dimension.
\end{abstract}
\maketitle \tableofcontents
\section{Introduction}
In \mathfrak{c}ite{JP98}, Jorgensen and Pedersen prove two surprising results:
\mathfrak{b}egin{enumerate}
\item there exists a singular Borel probability measure $\sigma$ such that there exists a sequence $\{\lambda_{n}\}_{n=0}^{\infty} \subset \mathbb{R}$
such that the functions $e_{\lambda_{n}}(x) := e^{2 \pi i \lambda_{n} x}$ is an orthonormal basis for $L^2(\sigma)$; \item The Hausdorff measure
$\nu$ on the middle third Cantor set has the following property: for any three $\{ \lambda_{1}, \lambda_{2}, \lambda_{3} \} \subset \mathbb{R}$, the
set $\{e_{\lambda_{1}}, e_{\lambda_{2}}, e_{\lambda_{3}} \} \subset L^2(\nu)$ is not orthogonal.
\end{enumerate}
In both cases, the measure
arises as the (unique) invariant measure under an iterated function system \mathfrak{c}ite{Hut81}.
We prove, in contradistinction to item (ii) above, that every measure $\mu$ arising from
a suitable iterated function system possesses an infinite sequence $\{\lambda_{n} \}_{n=0}^{\infty}$ such
that the sequence $\{ e_{\lambda_{n}}\}_{n=0}^{\infty}$ is a Riesz basic sequence in $L^2(\mu)$. Moreover,
this sequence has positive Beurling dimension.
Frames were introduced by Duffin and Schaeffer \mathfrak{c}ite{DuSc52} in the context of nonharmonic Fourier series, and today they have applications in a
wide range of areas. Frames provide robust, basis-like representations of vectors in a Hilbert space. The potential redundancy of frames often
allows them to be more easily constructible than bases, and to possess better properties than those that are achievable using bases. For
example, redundant frames offer more resilience to the effects of noise or to erasures of frame elements than bases. Following Duffin and
Schaeffer a Fourier frame or frame of exponentials is a frame of the form $\{e^{2\pi i\lambda·x}\}_{\lambda\in\Lambda}$ for the Hilbert space
$L^2[0, 1]$. Fourier frames are also closely connected with sampling sequences or complete interpolating sequences \mathfrak{c}ite{OSANN}.
\mathfrak{b}egin{definition}
A sequence $\{x_n\}_{n=1}^{\infty}$ in a Hilbert space (with inner product $\langle \mathfrak{c}dot , \mathfrak{c}dot \longrightarrowngle $) is \emph{Bessel} if there exists a positive constant $B$ such that
\[ \sum_{n=1}^{\infty} | \langle v , x_n \longrightarrowngle |^2 \leq B \|v\|^2. \]
This is equivalent to the existence of a positive constant $D$ such that for every finite sequence $\{c_{1}, \mathfrak{d}ots , c_{K} \}$ of complex numbers
\[ \| \sum_{n=1}^{K} c_{n} x_n \| \leq D \sqrt{\sum_{n=1}^{K} |c_{n}|^2}. \]
Here $D^2 = B$ is called the Bessel bound.
The sequence is a frame if in addition to being a Bessel sequence there exists a positive constant $A$ such that
\[ A \| v\|^2 \leq \sum_{n=1}^{\infty} | \langle v , x_n \longrightarrowngle |^2 \leq B \|v\|^2. \]
In this case, $A$ and $B$ are called the lower and upper frame bounds, respectively.
The sequence is a Riesz basic sequence if in addition to being a Bessel sequence there exists a positive constant $C$ such that for every finite sequence $\{c_{1}, \mathfrak{d}ots , c_{K} \}$ of complex numbers
\[ C \sqrt{\sum_{n=1}^{K} |c_{n}|^2} \leq \| \sum_{n=1}^{K} c_{n} x_n \| \leq D \sqrt{\sum_{n=1}^{K} |c_{n}|^2}. \]
Here $C$ and $D$ are called the lower and upper basis bounds, respectively.
\end{definition}
The main result of Duffin and Schaeffer is a sufficient density condition for $\{e^{2\pi i\lambda\mathfrak{c}dot x}\}_{\lambda\in \Lambda}$ to be a frame for $L^2[0,1]$. Landau \mathfrak{c}ite{MR0222554}, Jaffard
\mathfrak{c}ite{Jaffard} and Seip \mathfrak{c}ite{Seip2} ``almost" characterize the frame properties of $\{e^{2\pi i\lambda\mathfrak{c}dot x}\}_{\Lambda\in \Lambda}$ in
terms of lower Beurling density:
$$
\mathcal D^-(\Lambda):= \liminf_{h\rightarrow\infty}\inf_{x\in \mathbb{R}}\frac{\#(\Lambda\mathfrak{c}ap[x-h,x+h])}{2h}.
$$
\mathfrak{b}egin{theorem}\label{th1.1} For $\{e^{2\pi i\lambda\mathfrak{c}dot x}\}_{\Lambda\in \Lambda}$ to be a frame for $L^2[0,1]$, it is necessary that $\Lambda$ is relatively separated
and $\mathcal D^-(\Lambda)\mathcal{G}eq 1$, and it is sufficient that $\Lambda$ is relatively separated and $\mathcal D^-(\Lambda)> 1$.
\end{theorem}
The property of relative separation is equivalent to the condition that the upper Beurling density $$\mathcal D^{+}(\Lambda):=
\limsup_{h\rightarrow\infty}\sup_{x\in R}\frac{\#(\Lambda\mathfrak{c}ap[x-h,x+h])}{2h}$$ is finite.
For the critical case when $\mathcal D^-(\Lambda)= 1$, the complete characterization was beautifully formulated by Joaquim Ortega-Cerd\`{a} and Kristian
Seip in \mathfrak{c}ite{OSANN} where the key step was to connect the problem to de Branges' theory of Hilbert spaces of entire functions, and this
new characterization lead to applications in a classical inequality of H. Landau and an approximation problem for subharmonic functions.
In recent years there has been a wide range of interests in expanding the classical Fourier analysis to fractal or more general probability measures \mathfrak{c}ite{MR2509326,MR2435649,MR1744572,MR1655831,MR2338387,MR2200934,MR2297038,MR1785282,MR2279556,MR2443273}. One of the central themes of this area of research involves constructive and computational bases in $L^2(\mu)$, where $\mu$ is a measure which is determined by some self-similarity property. These include classical Fourier bases, as well as wavelet and frame constructions.
For $L^2[0,1]$, a sequence of exponentials is Bessel if the frequency set $\Lambda$ has finite upper Beurling density. For a singular measure $\nu$, a necessary (but not sufficient) condition for such a sequence to be Bessel in $L^2(\nu)$ is that the upper Beurling density of $\Lambda$ is 0 (see \mathfrak{c}ite{DHSW10}). Since the measures we consider here are singular, we shall use \emph{Beurling dimension} as a replacement for Beurling density.
\mathfrak{b}egin{definition}\label{deff3}
\mathfrak{c}ite{CKS08} Let $\Lambda$ be a discrete subset of $\mathbb{R}^d$. For $r>0$, the {\it upper Beurling density corresponding to $r$} (or {\it $r$-Beurling density}) is defined by
$$\mathcal D_r^+(\Lambda):=\limsup_{h\rightarrow\infty}\sup_{x\in\mathbb{R}^d}\frac{\#(\Lambda\mathfrak{c}ap(x+h[-1,1]^d))}{h^r}.$$
The {\it upper Beurling dimension} (or simply the {\it Beurling dimension}) is defined by
$$\mathfrak{d}im^+(\Lambda):=\sup\{r>0 : \mathcal{D}^+_r(\Lambda)>0\} = \inf\{r>0 : \mathcal{D}_r^+(\Lambda)<\infty\}.$$
Given a set of exponential functions $E(\Lambda):=\{e_\lambda : \lambda\in\Lambda\}$ we also say that $\mathcal D_r^+(\Lambda)$ is the $r$-Beurling density of $E(\Lambda)$.
\end{definition}
\mathfrak{b}egin{definition}\label{defaifs}
Let $R$ be a $d\times d$ expansive integer matrix, $B\subset\mathbb{Z}^d$, with $\#B=N\mathcal{G}eq2$. Define the iterated function system
$$\tau_b(x)=R^{-1}(x+b),\quad(x\in\mathbb{R}^d).$$
For convenience, we let $S := R^{T}$.
Let $(p_b)_{b\in B}$ be a finite set of probabilities, i.e., $0<p_b<1$, $\sum_{b\in B}p_b=1$.
Define the following operator $\mathcal T$ on Borel probability measures on $\mathbb{R}^d$
\mathfrak{b}egin{equation}\label{eqtmu1}
(\mathcal{T}\mathcal{G}amma)(E)=\sum_{b\in B}p_b\mathcal{G}amma(\tau_b^{-1}(E)),
\end{equation}
for all Borel sets $E$.
Equivalently the measure $\mathcal{T}\mathcal{G}amma$ is defined by
\mathfrak{b}egin{equation}
\int f\,d\mathcal{T}\mathcal{G}amma=\sum_{b\in B}p_b\int f\mathfrak{c}irc\tau_b\,d\mathcal{G}amma,
\label{eqtmu2}
\end{equation}
for all continuous functions $f$ on $\mathbb{R}^d$.
We denote by $\mu := \mu_{B,p}$ the unique invariant measure for the operator $\mathcal{T}$, i.e. $\mathcal{T}\mu_{B,p}=\mu_{B,p}$, whose existence is guaranteed by \mathfrak{c}ite{Hut81}.
\end{definition}
\mathfrak{b}egin{definition}
For a Borel probability measure $\mathcal{G}amma$, if $\Lambda = \{\lambda_{n}\}_{n=0}^{\infty} \subset \mathbb{R}$ is such that $\{ e_{\lambda_{n}} \} \subset L^2(\mathcal{G}amma)$ is a Bessel sequence, we say $\Lambda$ is a Bessel spectrum for $\mathcal{G}amma$. Likewise, $\Lambda$ is a Riesz basic spectrum if $\{ e_{\lambda_{n}} \}$ is a Riesz basic sequence in $L^2(\mathcal{G}amma)$.
\end{definition}
In the classical Lebesgue measure case, it is relatively easy (with the help of Theorem \ref{th1.1}) to construct frames/Riesz bases or more
generally Bessel sequences/Riesz sequences $\{e^{2\pi i\lambda\mathfrak{c}dot x}\}_{\Lambda\in \Lambda}$ with $\Lambda$ having positive Burling density.
However, this is not the case anymore for fractal measures. Indeed, for the fractal measure $\mu_{B,p}$ in the case that $R = 3$, $B =
\{0,2\}$, and $p_{0} = p_{2} = 1/2$, the corresponding measure $\mu_{3}$ (which is the Hausdorff measure on the middle third Cantor) has the property that $\{
3^{n} : n=0,1,\mathfrak{d}ots \}$ is NOT a Bessel spectrum (and hence can not be a Riesz basic spectrum) \mathfrak{c}ite[Proposition 3.10]{DHSW10}. Note that this
set $\{ 3^{n} : n=0,1,\mathfrak{d}ots \}$ is very ``sparse" and in fact it has the Beurling dimension equal to $0$. One of the open problems for the
fractal measure $\mu_{3}$ is that whether frames or Riesz bases spectrum exist. In \mathfrak{c}ite[Theorem 3.5]{DHSW10} it was proved that for a fractal
measure $\mu_{B,p}$, a necessary condition for $\Lambda$ to be a Bessel spectrum is that the Beurling dimension of $\Lambda$ is at most
$\log_{R} B$, and that the Beurling dimension of $\Lambda$ is equal to $\log_{R} B$ (under a mild technical condition) in order for $\Lambda$ to
be a frame spectrum. The above example $(\Lambda = \{ 3^{n} : n=0,1,\mathfrak{d}ots \}$ ) shows that this finite Beurling dimension condition is not
sufficient for $\Lambda$ to be even a Bessel spectrum. This naturally leads to the existence problem for Bessel spectrum and Riesz basic
spectrum with positive Beurling dimensions. The main purpose of this paper is to prove that Bessel spectrum and Riesz basic spectrum with
positive Beurling dimension exists for all the fractal measures $\mu_{B,p}$. We believe that this is an important positive step toward answering
the question of whether frames or Riesz bases spectra exist for the fractal measure $\mu_{3}$.
\section{Spectra of Positive Beurling Dimension}\label{bess}
We start with our main theorem:
\mathfrak{b}egin{theorem}\label{thb1}
Let $R$ be a $d\times d$ expansive integer matrix, $0\in B\subset \mathbb{Z}^d$ , $(p_b)_{b\in B}$ a list of probabilities and let $\mu=\mu_{B,p}$ be the invariant measure associated to the iterated function system
$$\tau_b(x)=R^{-1}(x+b),\quad(x\in\mathbb{R}^d, b\in B)$$
and the probabilities $(p_b)_{b\in B}$. Then $\mu$ has an infinite Riesz basic spectrum of positive Beurling dimension.
\end{theorem}
The proof proceeds via a series of lemmas. Throughout the remainder of the paper, $R$, $B$, $p$ are fixed, and $\mu := \mu_{B,p}$.
\mathfrak{b}egin{lemma}\label{lemip}
The Fourier transform of the invariant measure $\mu$ satisfies the scaling equation
\mathfrak{b}egin{equation}
\widehat\mu(x)= m(S^{-1} x)\widehat\mu(S^{-1} x),\quad(x\in \mathbb{R}^d)
\label{eqsc}
\end{equation}
where
\mathfrak{b}egin{equation}
m(x):= \sum_{b\in b}p_be^{2\pi i b\mathfrak{c}dot x},\quad(x\in\mathbb{R}^d)
\label{eqmb}
\end{equation}
The function $\widehat\mu$ is given by the infinite product formula
\mathfrak{b}egin{equation}
\widehat\mu(x)=\prod_{k=1}^\infty m\left(S^{-k}x\right),\quad(x\in\mathbb{R}^d)
\label{eqip}
\end{equation}
The infinite product converges uniformly on compact subsets.
\end{lemma}
\mathfrak{b}egin{proof}
Apply the Fourier transform to the invariance equation \eqref{eqtmu2}. See e.g. \mathfrak{c}ite{JP98a,DJ06b} for details.
\end{proof}
\mathfrak{b}egin{lemma}\label{lemb1}
There exists $p \in \mathbb{N}$, $0<\rho<1$, and a finite set $A\subset\mathbb{Z}^d\setminus\{0\}$ with $\#A \mathcal{G}eq 2$ such that if
$M:=\max\{\|S^{-p}a-S^{-p}a'\| : a,a'\in A\mathfrak{c}up\{0\}, a\neq a'\}$, then
\mathfrak{b}egin{equation} \label{eqmrho}
\left|m\left(S^{-p} (a-a')+x\right)\right|\leq \rho
\end{equation}
for all $x$ with $\|x\|\leq \frac{M\|S^{-1}\|^{p}}{1-\|S^{-1}\|^p}$, and for all $a,a'\in A\mathfrak{c}up \{0\}$ with $a\neq a'$.
In addition, the elements of $A$ are incongruent $\operatorname{mod} S^p\mathbb{Z}^d$.
\end{lemma}
\mathfrak{b}egin{proof}
We have to stay away from the points where $|m|$ is 1. Note that $|m(x)|=1$ implies
$$\left|\sum_{b\in B}p_be^{2\pi ib\mathfrak{c}dot x}\right|=1.$$
Since $0\in B$, the term $p_0\mathfrak{c}dot 1$ appears in the sum. Using the triangle inequality we must have $e^{2\pi ib\mathfrak{c}dot x}=1$ for all $b\in B$ and therefore $b\mathfrak{c}dot x\in\mathbb{Z}$. It follows that $$\mathcal{S}_1:=\{x : |m(x)|=1\}=\{x : b\mathfrak{c}dot x\in\mathbb{Z}\}.$$
Since $\mathcal{S}_1$ has Lebesgue measure zero in $\mathbb{R}^d$, we can find two distinct points $x_0,x_1$ such that $\|x_0\|,\|x_1\|\leq 1$ with $\pm x_0,\pm x_1,\pm (x_0-x_1)\not\in\mathcal{S}_1$.
Choose $\mathfrak{d}elta'<1$ and let $c:=\|S^{-1}\|<1$ (since the matrix $S$ is expansive). We can pick $p \in \mathbb{N}$ large enough such that
\mathfrak{b}egin{equation} \label{eqpf1}
\frac{4c^p}{1-c^p}<\mathfrak{d}elta'.
\end{equation}
Moreover, since the volume of the lattice $S^{-p} \mathbb{Z}^{d}$ goes to $0$ as $p$ gets large, we may choose $p$ so that additionally there exist integers $a_0\neq a_1\in\mathbb{Z}^d$ with $\|x_0-S^{-p} a_0\|, \|x_1-S^{-p}a_1\|\leq \mathfrak{d}elta'$.
Let $A:=\{a_0,a_1\}$. Then $\|S^{-p}a_i\|\leq 1+\mathfrak{d}elta'<2$ so $M$ as defined in the hypothesis will be less than $4$. If $\|y\|<M c^p/(1-c^p)$ then for $a,a'\in A\mathfrak{c}up\{0\}$, $a\neq a'$, there exists $x,x'\in \{x_0,x_1,0\}$ such that
$$\|(S^{-p}(a-a')+y)-(x-x')\|\leq \|S^{-p}a-x\|+\|S^{-p}a'-x'\|+\|y\|\leq 3\mathfrak{d}elta'.$$
Thus, if $\mathfrak{d}elta'$ is small enough, $S^{-p}(a-a')+y$, being close to $x-x'$, stays away from the set $\mathcal{S}_1$ so by uniform continuity of $m$, there is a $\rho<1$ such that
$$|m(S^{-p}(a-a')+y)|\leq \rho,$$
for all $y$ with $\|y\|\leq Mc^p/(1-c^p)$. This proves the existence of $p,\rho$ and $A$.
The elements in $A\mathfrak{c}up\{0\}$ cannot be congruent $\operatorname{mod} S^p\mathbb{Z}^d$ because $|m(S^{-p}(a-a'))|<1$, while $m(k)=1$ for $k\in\mathbb{Z}^d$ and $m$ is $\mathbb{Z}^d$ periodic.
\end{proof}
\mathfrak{b}egin{definition}\label{defb1}
Let $p,\rho$ and $A$ be as in Lemma \ref{lemb1}. Let
$$\Lambda(A,p) := \{ a_0+S^pa_1+\mathfrak{d}ots+S^{pr}a_r : a_{i} \in A \mathfrak{c}up \{0\} \}.$$
We identify the integer $a_0+S^pa_1+\mathfrak{d}ots+S^{pr}a_r$ with the word $a_0a_1\mathfrak{d}ots a_r$ and with the infinite word $a_0a_1\mathfrak{d}ots a_r a_{r+1}\mathfrak{d}ots$, where $a_i=0$ for $i\mathcal{G}eq r+1$.
Since the elements in $A\mathfrak{c}up\{0\}$ are incongruent $\operatorname{mod} S^p\mathbb{Z}^d$, different digits means different integers.
For two such $\lambda=a_0a_1\mathfrak{d}ots$, $\lambda'=a_0'a_1'\mathfrak{d}ots$, we define the Hamming distance between them by
$$d_p(\lambda,\lambda')=\#\{ i : a_i\neq a_i'\}.$$
\end{definition}
\mathfrak{b}egin{lemma}\label{lemb2}
Let $p,\rho,A,M$ be as in Lemma \ref{lemb1}. Let $\lambda=a_0a_1\mathfrak{d}ots$, $\lambda'=a_0'a_1'\mathfrak{d}ots$ be distinct words with digits in $A$.
Then
$$|\widehat\mu(\lambda-\lambda')|\leq \rho^{d_p(\lambda,\lambda')}.$$
\end{lemma}
\mathfrak{b}egin{proof}
We use the infinite product formula for $\widehat\mu$, and we group every $p$ terms. We have
$$\widehat\mu(x)=\prod_{n=1}^\infty m^{(p)}(S^{-np}x),\quad(x\in\mathbb{R}^d)$$
where
$$m^{(p)}(x)=m(x)m(Sx)\mathfrak{d}ots m(S^{p-1}x).$$
Since $|m|\leq 1$, we get for any $I \subset \mathbb{N}$,
\mathfrak{b}egin{equation}
|\widehat\mu(x)| \leq \prod_{n \in I} |m^{(p)}(S^{-np}x)|.
\label{eqb3}
\end{equation}
Suppose $a_{n} \neq a_{n}'$. For $k \mathcal{G}eq n$, $S^{-np}(S^{kp}(a_{k} - a_{k}')) \in \mathbb{Z}^d$. Therefore, we have
$$S^{-np}(\lambda-\lambda')\equiv S^{-p}(a_{n-1}-a_{n-1}')+S^{-2p}(a_{n-2}-a_{n-2}')+\mathfrak{d}ots+S^{-np}(a_0-a_0')\operatorname{mod}\mathbb{Z}^d,$$
and since
$$\|S^{-p}(a_{n-1}-a_{n-1}')+\mathfrak{d}ots+S^{-np}(a_0-a_0')\|\leq \|S^{-p}\|(M+\|S^{-p}\|\mathfrak{c}dot M+\mathfrak{c}dots + \|S^{(-n-1)p}\| \mathfrak{c}dot M)\leq \frac{M\|S^{-1}\|^p}{1-\|S^{-1}\|^p}$$
with Lemma \ref{lemb1} we obtain
$$|m(S^{-np}(\lambda-\lambda'))|\leq \rho.$$
Thus, using \eqref{eqb3}, we obtain
$$|\widehat\mu(\lambda-\lambda')|\leq \prod_{n \in I} |m^{(p)}(S^{-np}(\lambda - \lambda')| \leq \rho^{d_p(\lambda,\lambda')},$$
where $I := \{ n : a_{n} \neq a_{n}' \}$ with $\# I = d_p(\lambda,\lambda')$.
\end{proof}
\mathfrak{b}egin{remark}\label{rem1}
By changing the set of digits $A$ and $p$ we can assume that $\rho$ is as small as we want. This is because we can replace each digit in $A$ by a repetition of it, say $l$ times. So, for example $12$ is replaced by $111222$, where $l=3$. By doing this the distance between any two words is multiplied by $l$. So if we replace $p$ by $p\mathfrak{c}dot l$ and $A$ by $A^{(l)}:=\{a^{(l)}:=\underbrace{aa\mathfrak{d}ots a}_{l\mbox{ times}} : a\in A\}$, the number $\rho$ in Lemma \ref{lemb2} is replaced by $\rho^l$, which can be made as small as needed. More precisely, we have
$$d_{p,A}(\lambda^{(l)},\mathcal{G}amma^{(l)})=l\mathfrak{c}dot d_{pl,A^{(l)}}(\lambda,\mathcal{G}amma),\mbox{ for }\lambda,\mathcal{G}amma\in A^{(l)}.$$
The distance $d_{pl,A^{(l)}}$ counts each digit $a^{(l)}$ as just one.
\end{remark}
\mathfrak{b}egin{lemma}\label{lemb3}
Let $p,\rho,A$ be as in Lemma \ref{lemb1}, and let $\Lambda \subset \Lambda(A,p)$. Suppose
$$C:=\sup_{\lambda\in\Lambda}\sum_{\lambda'\in\Lambda\setminus\{\lambda\}}\rho^{d_p(\lambda,\lambda')}<\infty.$$
Then $\Lambda$ is a Bessel spectrum with Bessel bound $1+C$.
Moreover, if $C<1$, then $\Lambda$ is a Riesz basic spectrum.
\end{lemma}
\mathfrak{b}egin{proof}
Using Lemma \ref{lemb2}, we have for all $\lambda\in\Lambda$:
$$\sum_{\lambda'\in\Lambda}|\widehat\mu(\lambda-\lambda')|\leq 1+\sum_{\lambda\neq\lambda'}\rho^{d_p(\lambda,\lambda')}\leq 1 + C.$$
An application of Schur's lemma shows that the Grammian of the set $\{e_\lambda : \lambda\in\Lambda\}$ is bounded, and, if $C<1$, it is also diagonally dominant, hence invertible (see \mathfrak{c}ite[Proposition 3.5.4]{Chr03}). This implies that $\Lambda$ is a Bessel spectrum with bound $1+C$ and is a Riesz basic spectrum when $C<1$.
\end{proof}
\mathfrak{b}egin{lemma}\label{lemb4}
Let $A$ be an alphabet with $2$ letters. Then there exists $k_{0}\mathcal{G}eq 1$ such that for every $k\mathcal{G}eq k_{0}$ and every $n$ there is
set $\Lambda_n$ containing $2^{n}$ words of length $kn$, i.e., $\Lambda_n\subset A^{kn}$,
such that the Hamming distance between any two distinct words in $\Lambda_n$ is at least $n$.
\end{lemma}
\mathfrak{b}egin{proof} This is a consequence of the Gilbert-Varshamov bound \mathfrak{c}ite{Lin99} which states that if $A_{q}(m, d)$ is the maximum possible size of a
$q$-ary code $C$ with length $m$ and minimum Hamming distance $d$ (a $q$-ary code is a code over the field ${\mathbb F}_{q}$ with $q$-elements), then
$$
A_{q}(m, d) \mathcal{G}eq \frac{q^{m}}{\sum_{j=0}^{d-1}C_{m}^{j}(q-1)^{j}},
$$
where $C_{m}^{j}$ are the binomial coefficients.
In our case, $q=2$, $d = n$ and $m = kn$. Let $H(x) = -x\log_{2}x - (1-x)\log_{2}(1-x)$. Using the argument in \mathfrak{c}ite{BIPW10} we can show that for $k > 2$
$$
\sum_{j=0}^{n}C_{m}^{j} < 2^{H(\frac{1}{k})kn}.
$$
In fact, let $u = \frac{1}{k}$. Then
$$
2^{-H(u)} = \left(\frac{u}{1-u}\right)^{u}(1 - u)
$$
and so
$$
1 = (u + (1-u))^{kn} > \sum_{j=0}^{n}C_{kn}^{j}u^{j}(1-u)^{kn-j}= \sum_{j=0}^{n}C_{kn}^{j}\left(\frac{u}{1-u}\right)^{j}(1-u)^{kn}$$
$$> \sum_{j=0}^{n}C_{kn}^{j}\left(\frac{u}{1-u}\right)^{n}(1-u)^{kn}
= \sum_{j=0}^{n}C_{kn}^{j}\left[\left(\frac{u}{1-u}\right)^{u}(1-u)\right]^{kn}= 2^{-H(u)kn}\sum_{j=0}^{n}C_{kn}^{j}$$
Thus we get $$
\sum_{j=0}^{n}C_{kn}^{j} < 2^{H(\frac{1}{k})kn},
$$
hence
$$
A_{2}(kn, n) \mathcal{G}eq \frac{2^{kn}}{\sum_{j=0}^{n-1}C_{kn}^{j}} \mathcal{G}eq \frac{2^{kn}}{\sum_{j=0}^{n}C_{kn}^{j}}\mathcal{G}eq 2^{(1-H(\frac{1}{k}))kn}.
$$
Since $1-H(\frac{1}{k})\rightarrow1$ as $k\rightarrow\infty$, there exists a $k_0$ such that for $k\mathcal{G}eq k_0$, $(1-H(\frac{1}{k}))k\mathcal{G}eq 1$ and
$$
A_{2}(kn, n) \mathcal{G}eq 2^n.
$$
\end{proof}
\mathfrak{b}egin{proof}[Proof of Theorem \ref{thb1}]
To complete the proof of Theorem \ref{thb1}, we construct a set $\Lambda$ that satisfies the hypothesis of Lemma \ref{lemb3}.
Let $A,p,\rho$ as in Lemma \ref{lemb1} where $A$ has two non-zero elements. Without loss of generality, by Remark \ref{rem1} we may assume that $\rho < 1/4$.
We choose a sequence $q_1,q_2,\mathfrak{d}ots $ of natural numbers such that $q_1+\mathfrak{d}ots+q_{n-1}+1\leq q_n$, for all $n$; for example $q_n=2^n$.
Let $k\mathcal{G}eq k_{0}$ where $k_{0}$ is as in Lemma \ref{lemb4}, let $\Lambda_{n}$ be the set of words of $A^{kq_n}$ guaranteed by Lemma \ref{lemb4} with at least $2^{q_n}$ elements and the Hamming distance between any two words is at least $q_n$.
We define $\Lambda$ by concatenating words in $\Lambda_i$ as follows
$$\Lambda:=\{\lambda_1\mathfrak{d}ots\lambda_n : \lambda_i\in \Lambda_i, n\mathcal{G}eq 1\}.$$
Fix $\lambda=\lambda_1\mathfrak{d}ots\lambda_N$; we wish to estimate
$$ \sum_{\lambda' \in \Lambda \setminus \{\lambda\}} \rho^{d_{p}(\lambda, \lambda')}. $$
For any $\lambda'=\lambda_1'\mathfrak{d}ots\lambda_m' \in \Lambda$, there exists a natural number $r(\lambda')$ which is the largest index such that $\lambda_r\neq \lambda_r'$. The Hamming distance between $\lambda_{r}$ and $\lambda'_{r}$ is at least $q_{r(\lambda')}$, so the Hamming distance between $\lambda$ and $\lambda'$ is also at least $q_{r(\lambda')}$. Thus, for a fixed $r_{0} \in \mathbb{N}$, we count how many $\lambda' \in \Lambda$ which have $r(\lambda') = r_{0}$. If $r_{0} \leq N$, the number of possibilities is
$$2^{q_1}+2^{q_1+q_2}+\mathfrak{d}ots+2^{q_1+\mathfrak{d}ots+q_{r_{0}}}\leq \sum_{i=0}^{q_1+\mathfrak{d}ots+q_{r_{0}}}2^{i}\leq 2^{q_1+\mathfrak{d}ots+q_{r_{0}}+1}\leq 2^{2q_{r_{0}}}=4^{q_{r_{0}}}.$$
If $r_{0}>N$, then $\lambda_{n+1} = \mathfrak{d}ots = \lambda_{r_{0}} = 0 = \lambda_{r_{0}+1}'$ with $\lambda_{r_{0}}' \neq 0$, so the number of possibilities is at most
$$2^{q_1+\mathfrak{d}ots+q_{r_{0}}}\leq 2^{2q_{r_{0}}}=4^{q_{r_{0}}}.$$
It follows that
$$\sum_{\lambda'\in\Lambda\setminus\{\lambda\}}\rho^{d_p(\lambda,\lambda')}\leq \sum_{r=1}^\infty 4^{q_r}\rho^{q_r}.$$
Since $\rho<1/4$, this sum converges.
We now associate words in $\Lambda$ with integers written in base $S^{p}$ with coefficients from the words in $\Lambda$ as in Definition \ref{defb1}. Combining Lemmas \ref{lemb2} and \ref{lemb3} we conclude that $\Lambda$ is a Bessel spectrum. Taking $q_1$ larger if needed, we can get the sum to be less than 1, so we obtain a Riesz basic spectrum.
It remains to prove that the Beurling dimension is positive. We will use the following lemma that can be obtained by a straightforward computation.
\mathfrak{b}egin{lemma}\label{lembe}
To compute the Beurling dimension, the unit cube $[-1,1]^d$ in Definition \ref{deff3} can be replaced by any bounded set $Q$ that contains $0$ in the interior.
\end{lemma}
Consider the elements in $\Lambda_1\mathfrak{d}ots\Lambda_n$. There are at least $2^{q_n}$ such elements, since just $\Lambda_n$ has $2^{q_n}$ elements.
The length of such a word is $kq_1+\mathfrak{d}ots+kq_n\leq 2kq_n-1$. Let $C=\max_{a\in A}\|a\|$. Then the integer represented by this word will have absolute value less than
$$C+\|S\|^pC+\mathfrak{d}ots+\|S\|^{p\mathfrak{c}dot (2kq_n-1)}C\leq D \|S\|^{2kq_np}$$
for some constant $D$.
Therefore, in the ball of radius $D\|S\|^{2kq_np}$ there are at least $2^{q_n}$ elements in $\Lambda$. Using $Q=B(0,D)$ in Lemma \ref{lembe}, $x=0$, and $h=\|S\|^{2kq_np}$ in the Definition \ref{deff3}, this implies that the Beurling dimension of $\Lambda$ is at least
$\log_{\|S\|^{2kp}}2>0$.
\end{proof}
We conclude with a remark concerning the usual Cantor middle third set. This set, and its invariant measure, is generated by the iterated function system with parameters $R = 3$, $B = \{0, 2\}$, and $p_{0} = p_{2} = \frac{1}{2}$. The invariant measure $\mu_{B,p}$ for these parameters is the measure $\nu$ mentioned in item $(ii)$ in the introduction; for this measure there are no three pairwise orthogonal complex exponentials, and hence this measure possesses no orthonormal sequence of exponentials \mathfrak{c}ite{JP98}. However, by applying Theorem \ref{thb1}, this measure on the Cantor set does possess a Riesz basic sequence of complex exponentials.
\mathfrak{b}ibliographystyle{alpha}
\mathfrak{b}ibliography{spectral}
\end{document} | math | 26,377 |
\betaegin{document}
\title{Sub-Lorentzian distance and spheres \on the Heisenberg group
ootnote{Sections 1, 2, 6--11 were written by Yu. Sachkov.
Sections 3--5 were written by E.~Sachkova.
Work by Yu. Sachkov was supported by Russian Scientific Foundation, grant 22-11-00140, https://rscf.ru/project/22-11-00140/.
Work by E. Sachkova was supported by Russian Scientific Foundation, grant 22-21-00877, https://rscf.ru/project/22-21-00877/.
}
\betaegin{abstract}
The left-invariant sub-Lo\-rent\-zi\-an problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lo\-rent\-zi\-an distance and spheres are described.
\end{abstract}
\tableofcontents
\section{Introduction}
A sub-Riemannian structure on a smooth manifold $M$ is a vector distribution $\Deltaelta \subset TM$ endowed with a Riemannian metric $g$ (a positive definite quadratic form). Sub-Riemannian geometry is a rich theory and an active domain of research during the last decades
\cite{mont, jurd_book, versh_gersh, notes, ABB, intro, UMN}.
A sub-Lo\-rent\-zi\-an structure is a variation of a sub-Riemannian one for which the quadratic form $g$ in a distribution $\Deltaelta$ is a Lorentzian metric (a nondegenerate quadratic form of index 1). Sub-Lo\-rent\-zi\-an geometry tries to develop a theory similar to the sub-Riemannian geometry, and it is still in its childhood. For example, the left-invariant sub-Riemannian structure on the Heisenberg group is a classic subject covered in almost every textbook or survey on sub-Riemannian geometry. On the other hand, the left-invariant sub-Lo\-rent\-zi\-an structure on the Heisenberg group is not studied in detail. This paper aims to fill this gap.
The paper has the following structure.
In Sec. 2 we recall the basic notions of the sub-Lo\-rent\-zi\-an geometry. In Sec. 3 we state the left-invariant sub-Lo\-rent\-zi\-an structure on the Heisenberg group studied in this paper. Results obtained previously for this problem by M. Grochowski are recalled in Sec. 4. In Sec. 5 we apply the Pontryagin maximum principle and compute extremal trajectories; as a consequence, almost all extremal trajectories (timelike ones) are parametrized by the exponential mapping. In Sec. 6 we show that the exponential mapping is a diffeomorphism and find explicitly its inverse. On this basis in Sec. 7 we study optimality of extremal trajectories and construct an optimal synthesis.
In Sec. 8 we describe explicitly the sub-Lo\-rent\-zi\-an distance, in Sec. 9 we find its symmetries, and in Sec. 10 we study in detail the sub-Lo\-rent\-zi\-an spheres of positive and zero radii. Finally, in Sec. 11 we discuss the results obtained and pose some questions for further research.
\section{Sub-Lo\-rent\-zi\-an geometry}\label{sec:SL}
A sub-Lo\-rent\-zi\-an structure on a smooth manifold $M$ is a pair $(\Delta, g)$ consisting of a vector distribution $\Delta \subset TM$ and a Lorentzian metric $g$ on $\Delta$, i.e., a nondegenerate quadratic form $g$ of index 1. Sub-Lo\-rent\-zi\-an geometry attempts to transfer the rich theory of sub-Riemannian geometry (in which the quadratic form $g$ is positive definite) to the case of Lorentzian metric $g$. Research in sub-Lo\-rent\-zi\-an geometry was started by M. Grochowski \cite{groch2, groch3, groch4, groch6, groch9, groch11}, see also \cite{grong_vas, chang_mar_vas, kor_mar, groch_med_war}.
Let us recall some basic definitions of sub-Lo\-rent\-zi\-an geometry.
A vector $v \in T_qM$, $q \in M$, is called horizontal if $v \in \Delta_q$. A horizontal vector $v$ is called:
\betaegin{itemize}
\item
timelike if $g(v)<0$,
\item
spacelike if $g(v)>0$ or $v = 0$,
\item
lightlike if $g(v)=0$ and $v \neq 0$,
\item
nonspacelike if $g(v)\leq 0$.
\end{itemize}
A Lipschitzian curve in $M$ is called timelike if it has timelike velocity vector a.e.; spacelike, lightlike and nonspacelike curves are defined similarly.
A time orientation $X$ is an arbitrary timelike vector field in $M$. A nonspacelike vector $v \in \Delta_q$ is future directed if $g(v, X(q))<0$, and past directed if $g(v, X(q))>0$.
A future directed timelike curve $q(t)$, $t \in [0, t_1]$, is called arclength paramet\-ri\-zed if $g(\dot q(t), \dot q(t)) \equiv - 1$. Any future directed timelike curve can be parametrized by arclength, similarly to the arclength parametrization of a horizontal curve in sub-Riemannian geometry.
The length of a nonspacelike curve $\gamma \in \operatorname{Lip}\nolimits([0, t_1], M)$ is
$$
l(\gamma) = \int_0^{t_1} |g(\dot \gamma, \dot \gamma)|^{1/2} dt.
$$
For points $q_1, q_2 \in M$ denote by $\Omega_{q_1q_2}$ the set of all future directed nonspacelike curves in $M$ that connect $q_1$ to $q_2$. In the case $\Omega_{q_1q_2} \neq \emptyset$ denote the sub-Lo\-rent\-zi\-an distance from the point $q_1$ to the point $q_2$ as
\betae{d}
d(q_1, q_2) = \sup \{l(\gamma) \mid \gamma \in \Omega_{q_1q_2}\}.
\end{equation}
Notice that in papers \cite{groch4, groch6} in the case $\Omega_{q_1q_2} = \emptyset$ it is set $d(q_1, q_2) = 0$. It seems to us more reasonable not to define $d(q_1, q_2)$ in this case.
A future directed nonspacelike curve $\gamma$ is called a sub-Lo\-rent\-zi\-an length maximizer if it realizes the supremum in \eq{d} between its endpoints $\gamma(0) = q_1$, $\gamma(t_1) = q_2$.
The causal future of a point $q_0 \in M$ is the set $J^+(q_0)$ of points $q_1 \in M$ for which there exists a future directed nonspacelike curve $\gamma$ that connects $q_0$ and $q_1$. The chronological future $I^+(q_0)$ of a point $q_0 \in M$ is defined similarly via future directed timelike curves $\gamma$.
Let $q_0 \in M$, $q_1 \in J^+(q_0)$. The search for sub-Lo\-rent\-zi\-an length maximizers that connect $q_0$ with $q_1$ reduces to the search for future directed nonspacelike curves $\gamma$ that solve the problem
\betae{lmax}
l(\gamma) \to \max, \qquad \gamma(0) = q_0, \quad \gamma(t_1) = q_1.
\end{equation}
A set of vector fields $X_1, \dots, X_k \in \operatorname{Vec}\nolimits(M)$ is an orthonormal frame for a sub-Lo\-rent\-zi\-an structure $(\Delta, g)$ if for all $q \in M$
\betaegin{align*}
&\Delta_q = \operatorname{span}\nolimits(X_1(q), \dots, X_k(q)),\\
&g_q(X_1, X_1) = -1, \qquad g_q(X_i, X_i) = 1, \quad i = 2, \dots, k, \\
&g_q(X_i, X_j) = 0, \quad i \neq j.
\end{align*}
Assume that time orientation is defined by a timelike vector field $X \in \operatorname{Vec}\nolimits(M)$ for which $g(X, X_1) < 0$ (e.g., $X = X_1$). Then the sub-Lo\-rent\-zi\-an problem for the sub-Lo\-rent\-zi\-an structure with the orthonormal frame $X_1, \dots, X_k$ is stated as the following optimal control problem:
\betaegin{align*}
&\dot q = \sum_{i=1}^k u_i X_i(q), \qquad q \in M, \\
&u \in U = \left\{(u_1, \dots, u_k) \in {\mathbb R}^k \mid u_1 \gammaeq \sqrt{ u_2^2 + \dots + u_k^2}\right\},\\
&q(0) = q_0, \qquad q(t_1) = q_1, \\
&l(q(\cdot)) = \int_0^{t_1} \sqrt{u_1^2 - u_2^2 - \dots - u_k^2} \, dt \to \max.
\end{align*}
\betaegin{remark}
The sub-Lo\-rent\-zi\-an length is preserved under monotone Lipschitzian time reparametrizations $t(s)$, $s \in [0, s_1]$. Thus if $q(t)$, $t \in [0, t_1]$, is a sub-Lo\-rent\-zi\-an length maximizer, then so is any its reparametrization $q(t(s))$, $s \in [0, s_1]$.
In this paper we choose primarily the following parametrization of trajectories: the arclength parametrization ($u_1^2 - u_2^2 - \cdots - u_k^2 \equiv 1$) for timelike trajectories, and the parametrization with $u_1(t) \equiv 1$ for future directed lightlike trajectories. Another reasonable choice is to set $u_1(t) \equiv 1$ for all future directed nonspacelike trajectories.
\end{remark}
\section[Statement of the sub-Lo\-rent\-zi\-an problem on the Heisenberg group]{Statement of the sub-Lo\-rent\-zi\-an problem \\on the Heisenberg group}
The Heisenberg group is the space
$M \simeq {\mathbb R}^3_{x,y,z}$ with the product rule
$$
(x_1, y_1, z_1) \cdot(x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2 + (x_1y_2 - x_2 y_1)/2).
$$
It is a three-dimensional nilpotent Lie group with a left-invariant frame
\betae{Xi}
X_1 = \pder{}{x} - \frac y2 \pder{}{z}, \qquad
X_2 = \pder{}{y} + \frac x2 \pder{}{z}, \qquad
X_3 = \pder{}{z},
\end{equation}
with the only nonzero Lie bracket $[X_1, X_2] = X_3$.
Consider the left-invariant sub-Lo\-rent\-zi\-an structure on the Heisenberg group $M$ defined by the orthonormal frame $(X_1, X_2)$, with the time orientation $X_1$. Sub-Lo\-rent\-zi\-an length maximizers for this sub-Lo\-rent\-zi\-an structure are solutions to the optimal control problem
\betaegin{align}
&\dot q = u_1 X_1 + u_2 X_2, \qquad q \in M, \label{prf1} \\
&u \in U = \{(u_1, u_2) \in {\mathbb R}^2 \mid u_1 \gammaeq |u_2|\}, \label{prf2} \\
&q(0) = q_0 = \operatorname{Id}\nolimits = (0, 0, 0), \quad q(t_1) = q_1, \label{prf3}\\
&l(q(\cdot)) = \int_0^{t_1} \sqrt{u_1^2 - u_2^2} \, dt \to \max. \label{prf4}
\end{align}
Along with this (full) sub-Lo\-rent\-zi\-an problem, we will also consider a reduced sub-Lo\-rent\-zi\-an problem
\betaegin{align}
&\dot q = u_1 X_1 + u_2 X_2, \qquad q \in M, \label{pr21} \\
&u \in \operatorname{int}\nolimits U = \{(u_1, u_2) \in {\mathbb R}^2 \mid u_1 > |u_2|\}, \label{pr22} \\
&q(0) = q_0 = \operatorname{Id}\nolimits = (0, 0, 0), \quad q(t_1) = q_1, \label{pr23}\\
&l(q(\cdot)) = \int_0^{t_1} \sqrt{u_1^2 - u_2^2} \, dt \to \max. \label{pr24}
\end{align}
In the full problem \eq{prf1}--\eq{prf4} admissible trajectories $q(\cdot)$ are future directed nonspacelike ones, while in the reduced problem \eq{pr21}--\eq{pr24} admissible trajectories $q(\cdot)$ are only future directed timelike ones.
Passing to arclength-parametrized future directed timelike trajectories, we obtain a time-maximal problem equivalent to the reduced sub-Lo\-rent\-zi\-an problem \eq{pr21}--\eq{pr24}:
\betaegin{align}
&\dot q = u_1 X_1 + u_2 X_2, \qquad q \in M, \label{pr31} \\
&u_1^2 - u_2^2 = 1, \qquad u_1 > 0, \label{pr32} \\
&q(0) = q_0 = \operatorname{Id}\nolimits = (0, 0, 0), \quad q(t_1) = q_1, \label{pr33}\\
&{t_1} \to \max. \label{pr34}
\end{align}
\section{Previously obtained results}\label{sec:groch}
The sub-Lo\-rent\-zi\-an problem on the Heisenberg group \eq{prf1}--\eq{prf4} was studied by M. Grochowski \cite{groch4, groch6}. In this section we present results of these works related to our results.
\betaegin{itemize}
\item[(1)]
Sub-Lo\-rent\-zi\-an extremal trajectories were parametrized by hyperbolic and linear functions: were obtained formulas equivalent to our formulas \eq{qc=0}, \eq{qcn0}.
\item[(2)]
It was proved that there exists a domain in $M$ containing $q_0 = \operatorname{Id}\nolimits$ in its boundary at which the sub-Lo\-rent\-zi\-an distance $d(q_0, q)$ is smooth.
\item[(3)]
The attainable sets of the sub-Lo\-rent\-zi\-an structure from the point $q_0 = \operatorname{Id}\nolimits$ were computed: the chronological future of the point $q_0$
$$
I^+(q_0) = \{(x,y,z) \in M \mid -x^2 + y^2 + 4 |z|<0, \ x > 0\},
$$
and the causal future of the point $q_0$
\betae{Jq0}
J^+(q_0) = \{(x,y,z) \in M \mid -x^2 + y^2 + 4 |z|\leq 0, \ x \gammaeq 0\}.
\end{equation}
In the standard language of control theory \cite{notes}, $I^+(q_0)$ is the attainable set of the reduced system \eq{pr21}, \eq{pr22} from the point $q_0$ for arbitrary positive time. Thus the attainable set of the reduced system \eq{pr21}, \eq{pr22} from the point $q_0$ for arbitrary nonnegative time is
$$
\mathcal{A} = I^+(q_0) \cup \{q_0\}.
$$
The attainable set of the full system \eq{prf1}, \eq{prf2} from the point $q_0$ for arbitrary nonnegative time is
$$
\operatorname{cl}\nolimits(\mathcal{A}) = J^+(q_0).
$$
The attainable set $\mathcal{A}$ was also computed in paper \cite{vinberg}, where its boundary was called the Heisenberg beak. See the set $\partial \mathcal{A}$ in Figs. \ref{fig:beak}, \ref{fig:beak1}, and its views from the $y$- and $z$-axes in
Figs.~\ref{fig:beaky} and \ref{fig:beakz} respectively.
\figout{
\onefiglabelsizen{Heis_beak3}{The Heisenberg beak $\partial \mathcal{A}$}{fig:beak}{8}
\twofiglabelsizeh
{Heis_beak_y3}{View of $\partial \mathcal{A}$ along $y$-axis}{fig:beaky}{6}
{Heis_beak_z3}{View of $\partial \mathcal{A}$ along $z$-axis}{fig:beakz}{6}
}
\item[(4)]
The lower bound of the sub-Lo\-rent\-zi\-an distance
$$
\sqrt{x^2-y^2-4|z|} \leq d(q_0, q), \qquad q = (x, y, z) \in J^+(q_0),
$$
was proved. It was also noted that an upper bound $$d (q_0, q) \leq C \sqrt{x^2-y^2-4|z|} $$ does not hold for any constant $C \in {\mathbb R}$.
\item[(5)]
It was proved that there exist non-Hamiltonian maximizers, i.e., maximizers that are not projections of the Hamiltonian vector field $\vec H$, $H = \frac 12 (h_2^2 - h_1^2)$, related to the problem.
\end{itemize}
\section{Pontryagin maximum principle}\label{sec:PMP}
In this section we compute extremal trajectories of the sub-Lo\-rent\-zi\-an problem \eq{prf1}--\eq{prf4}.
The majority of results of this section were obtained by M. Grochowski \cite{groch4, groch6} in another notation, we present these results here for further reference.
Denote points of the cotangent bundle $T^*M$ as $\lambda$. Introduce linear on fibers of $T^*M$ Hamiltonians $h_i(\lambda) = \langle\lambda, X_i\rangle$, $i = 1, 2, 3.$ Define the Hamiltonian of the Pontryagin maximum principle (PMP) for the sub-Lo\-rent\-zi\-an problem \eq{prf1}--\eq{prf4}
$$
h_u^{\nu}(\lambda) = u_1 h_1(\lambda) + u_2 h_2(\lambda) - \nu \sqrt{u_1^2 - u_2^2}, \qquad \lambda \in T^*M, \quad u \in U, \quad \nu \in {\mathbb R}.
$$
It follows from PMP \cite{PBGM, notes} that if $u(t)$, $t \in [0, t_1]$, is an optimal control in problem \eq{prf1}--\eq{prf4}, and $q(t)$, $t \in [0, t_1]$, is the corresponding optimal trajectory, then there exists a curve $\lambda_{\cdot} \in \operatorname{Lip}\nolimits([0, t_1], T^*M)$, $\pi(\lambda_t) = q(t)$\footnote{where $\map{\pi}{T^*M}{M}$ is the canonical projection, $\pi(\lambda) = q$, $\lambda \in T^*_qM$}, and a number $\nu \in \{0, -1\}$ for which there hold the conditions for a.e. $t \in [0, t_1]$:
\betaegin{enumerate}
\item
the Hamiltonian system $\dot\lambda_t = \vec{h}_{u(t)}^{\nu}(\lambda_t)$\footnote{where $\vec{h}(\lambda)$ is the Hamiltonian vector field on $T^*M$ with the Hamiltonian function $h(\lambda)$},
\item
the maximality condition $h_{u(t)}^{\nu}(\lambda_t) = \max_{v \in U} h_v^{\nu}(\lambda_t) \equiv 0$,
\item
the nontriviality condition $(\nu, \lambda_t) \neq (0, 0)$.
\end{enumerate}
A curve $\lambda_{\cdot}$ that satisfies PMP is called an extremal, and the corresponding control $u(\cdot)$ and trajectory $q(\cdot)$ are called extremal control and trajectory.
\subsection{Abnormal case}
\betaegin{theorem}\label{th:abn}
In the abnormal case $\nu = 0$ extremals $\lambda_t$ and controls $u(t)$ have the following form for some $\tau_1, \tau_2 \gammaeq 0$:
\betaegin{itemize}
\item[$(1)$]
$h_3(\lambda_t) \equiv \operatorname{const}\nolimits > 0$:
\betaegin{align*}
t \in (0, \tau_1) &\quad\Rightarrow\quad &&h_1(\lambda_t) = h_2(\lambda_t) < 0, \qquad &&u_1(t) = - u_2(t),\\
t \in (\tau_1, \tau_1+\tau_2) &\quad\Rightarrow\quad &&h_1(\lambda_t) = -h_2(\lambda_t) < 0, \qquad &&u_1(t) = u_2(t).
\end{align*}
\item[$(2)$]
$h_3(\lambda_t) \equiv \operatorname{const}\nolimits < 0$:
\betaegin{align*}
t \in (0, \tau_1) &\quad\Rightarrow\quad &&h_1(\lambda_t) = -h_2(\lambda_t) < 0, \qquad &&u_1(t) = u_2(t),\\
t \in (\tau_1, \tau_1+\tau_2) &\quad\Rightarrow\quad &&h_1(\lambda_t) = h_2(\lambda_t) < 0, \qquad &&u_1(t) = -u_2(t).
\end{align*}
\item[$(3)$]
$h_3(\lambda_t) \equiv 0$:
\betaegin{align*}
&(h_1, h_2)(\lambda_t) \equiv \operatorname{const}\nolimits \neq (0, 0), \qquad h_1(\lambda_t) \equiv -|h_2(\lambda_t)|, \\
&u(t) \equiv \operatorname{const}\nolimits, \qquad u_1(t) \equiv \pm u_2(t), \quad \pm = - \operatorname{sgn}\nolimits(h_1h_2(\lambda_t)).
\end{align*}
\end{itemize}
\end{theorem}
\betaegin{proof}
Apply the PMP for the case $\nu = 0$.
\end{proof}
\betaegin{corollary}
Along abnormal extremals $H(\lambda_t) \equiv 0$, where $H = \frac 12(h_2^2 - h_1^2)$.
\end{corollary}
\subsection{Normal case}
In the normal case ($\nu = -1$) extremals exist only for $h_1 \leq - |h_2|$.\footnote{The set $\{(h_1, h_2) \in ({\mathbb R}^2)^* \mid h_1 \leq - |h_2|\}$ is the polar set to $U$ in the sense of convex analysis.} In the case $h_1 = - |h_2|$ normal controls and extremal trajectories coincide with the abnormal ones.
And in the domain $\{ \lambda \in T^*M \mid h_1 < - |h_2|\}$ extremals are reparametrizations of trajectories of the Hamiltonian vector field $\vec H$ with the Hamiltonian $H = \frac 12(h_2^2 - h_1^2)$.
In the arclength parametrization,
the extremal controls are
\betae{u_norm}
(u_1, u_2)(t) = (-h_1(\lambda_t), h_2(\lambda_t)),
\end{equation}
and the
extremals satisfy the Hamiltonian ODE $\dot \lambda = \vec H(\lambda)$ and belong to the level surface $\{H(\lambda) = \frac 12\}$, in coordinates:
\betaegin{align*}
&\dot h_1 = - {h_2h_3}, \qquad \dot h_2 = - {h_1h_3}, \qquad \dot h_3 = 0, \\
&\dot q = \cosh \psi \, X_1 + \sinh \psi \, X_2, \\
&h_1 = - \cosh \psi, \qquad h_2 = \sinh \psi, \qquad \psi \in {\mathbb R}.
\end{align*}
We denote $c = h_3$ and obtain a parametrization of normal trajectories $q(t) = \pi \circ e^{t \vec H}(\lambda_0)$, $\lambda_0 \in H^{-1}\left(\frac 12\right) \cap T^*_{q_0}M$.
If $c = 0$, then
\betae{qc=0}
x = t \cosh \psi, \quad y = t \sinh \psi, \quad z = 0.
\end{equation}
If $c \neq 0 $, then
\betae{qcn0}
x = \frac{\sinh(\psi + ct) - \sinh \psi}{c},
\quad
y = \frac{\cosh(\psi + ct) - \cosh \psi}{c},
\quad
z = \frac{\sinh(ct) - ct}{2c^2}.
\end{equation}
Summing up, we obtain the following characterization of normal trajectories in the sub-Lo\-rent\-zi\-an problem \eq{prf1}--\eq{prf4}.
\betaegin{theorem}\label{th:norm}
Normal controls and trajectories either coincide with abnormal ones (in the case $h_1(\lambda_t) = - |h_2(\lambda_t)|$, see Th. $\ref{th:abn}$), or can be arclength parametrized to get controls \eq{u_norm} and future directed timelike trajectories \eq{qc=0} if $c = 0$, or \eq{qcn0} if $c \neq 0$.
In particular,
along each normal extremal $H(\lambda_t) \equiv \operatorname{const}\nolimits \in \left\{0, \frac 12 \right\}$.
\end{theorem}
Consequently, normal trajectories are either nonstrictly normal (i.e., simultaneously normal and abnormal) in the case $H = 0$, or strictly normal (i.e., normal but not abnormal) in the case $H = \frac 12$. Strictly normal arclength-parametrized trajectories are described by the exponential mapping
\betaegin{align}
&\map{\operatorname{Exp}\nolimits}{N}{\widetilde{\A}}, \qquad (\lambda, t) \mapsto q(t) = \pi \circ e^{t \vec{H}}(\lambda), \label{Exp}\\
&N = C \times {\mathbb R}_+, \qquad {\mathbb R}_+ = (0, + \infty), \qquad C = T^*_{\operatorname{Id}\nolimits} M \cap
H^{-1}\left(\frac 12\right)
\simeq {\mathbb R}^2_{\psi, c}, \nonumber\\
&\widetilde{\A} = \operatorname{int}\nolimits \mathcal{A} = I^+(q_0) \nonumber
\end{align}
given explicitly by formulas \eq{qc=0}, \eq{qcn0}.
In papers \cite{groch4, groch6} were obtained formulas equivalent to \eq{qc=0}, \eq{qcn0}.
\betaegin{remark}
Projections of strictly normal (future directed timelike) trajectories to the plane $(x, y)$ are:
\betaegin{itemize}
\item
either rays $y = k x$, $x \gammaeq 0$, $k \in (-1, 1)$ (for $c=0$), see Fig. $\ref{fig:xyc0}$,
\item
or arcs of hyperbolas with asymptotes $x = \pm y > 0$ (for $c \neq 0$), see Fig. $\ref{fig:xycn0}$.
\end{itemize}
\figout{
\twofiglabelsizeh
{xyc0}{Strictly normal $(x(t), y(t))$, $c = 0$}{fig:xyc0}{7}
{xycn0}{Strictly normal $(x(t), y(t))$, $c \neq 0$}{fig:xycn0}{7}
}
Projections of nonstrictly normal (future directed lightlike) trajectories to the plane $(x, y)$ are broken lines with one or two edges parallel to the rays $x = \pm y > 0$, see Fig. $\ref{fig:xyabn}$.
\figout{
\onefiglabelsizen
{xyabn}{Nonstrictly normal $(x(t), y(t))$}{fig:xyabn}{7}
}
Projections of all extremal trajectories (as well as of all admissible trajectories) to the plane $(x,y)$ are contained in the angle $\{(x, y) \in {\mathbb R}^2 \mid x \gammaeq |y| \}$, which is the projection of the attainable set $J^+(q_0)$ to this plane.
\end{remark}
\betaegin{remark}
The Hamiltonian $H = \frac 12 (h_2^2-h_1^2)$ is preserved on each extremal. On the other hand, since the problem is left-invariant, the extremals respect the symplectic foliation on the dual of the Heisenberg Lie algebra $T^*_{\operatorname{Id}\nolimits}M = \{(h_1, h_2, h_3)\}$ consisting of $2$-dimensional symplectic leaves $\{h_3 = \operatorname{const}\nolimits \neq 0\}$ and $0$-dimensional leaves $\{ h_3 = 0, \ (h_1, h_2) = \operatorname{const}\nolimits\}$. Thus projections of extremals to $T^*_{\operatorname{Id}\nolimits}M = \{(h_1, h_2, h_3)\}$ belong to intersections of the level surfaces $\left\{H = \operatorname{const}\nolimits \in \left\{0, \frac 12\right\}\right\}$ with the symplectic leaves:
\betaegin{itemize}
\item
branches of hyperbolas $h_1^2-h_2^2 = 1$, $h_1 < 0$, $h_3 \neq 0$,
\item
points $(h_1, h_2) = \operatorname{const}\nolimits$, $H \in \left\{0, \frac 12\right\}$, $h_1 \leq - |h_2|$, $h_3 = 0$,
\item
angles $h_1 = - |h_2|$, $h_3 \neq 0$.
\end{itemize}
See Figs. $\ref{fig:h123norm}$, $\ref{fig:h123abnorm}$.
\end{remark}
\figout{
\twofiglabelsizeh
{h123norm}{Strictly normal $(h_1(t), h_2(t), h_3(t))$}{fig:h123norm}{6}
{h123abnorm}{Nonstrictly normal $(h_1(t), h_2(t), h_3(t))$}{fig:h123abnorm}{6}
}
\betaegin{remark}
In the sense of work {\em \cite{groch4}},
strictly normal extremal trajectories $q(t) = \pi\circ e^{t \vec H}(\lambda)$, $\lambda \in C$, are Hamiltonian since they are projections of trajectories of the Hamiltonian vector field $\vec H$.
On the other hand, nonstrictly normal extremal trajectories given by items $(1)$, $(2)$ of Th. {\em\ref{th:abn}} are non-Hamiltonian, e.g., the broken curves
\betae{broken+-}
\betaegin{cases}
e^{t(X_1+X_2)}, & t \in [0, \tau_1],\\
e^{(t - \tau_1)(X_1-X_2)} \circ e^{\tau_1(X_1+X_2)}, & t \in [\tau_1, \tau_2],
\end{cases}
\end{equation}
and
\betae{broken-+}
\betaegin{cases}
e^{t(X_1-X_2)}, & t \in [0, \tau_1],\\
e^{(t - \tau_1)(X_1+X_2)} \circ e^{\tau_1(X_1-X_2)}, & t \in [\tau_1, \tau_2],
\end{cases}
\end{equation}
for $0 < \tau_1 < \tau_2$. See item $(5)$ in Sec. {\em\ref{sec:groch}}. Although, each smooth arc of the broken trajectories \eq{broken+-}, \eq{broken-+} is a reparametrization of projection of a trajectory of the Hamiltonian vector field $\vec H$ contained in a face of the angle $\{(h_1, h_2, h_3) \in T_{\operatorname{Id}\nolimits}^* M \mid h_1 = - |h_2|\}$, see Fig. $\ref{fig:h123abnorm}$.
\end{remark}
\section{Inversion of the exponential mapping}
\betaegin{theorem}\label{th:Exp-1}
The exponential mapping $\map{\operatorname{Exp}\nolimits}{N}{\widetilde{\A}}$ is a real-analytic diffeomorphism. The inverse mapping $\map{\operatorname{Exp}\nolimits^{-1}}{\widetilde{\A}}{N}$, $(x, y, z) \mapsto (\psi, c, t)$, is given by the following formulas:
\betaegin{align}
&z = 0 \quad\Rightarrow\quad \psi = \operatorname{artanh}\nolimits \frac yx, \quad c = 0, \quad t = \sqrt{x^2-y^2}, \label{invz0}\\
&z \neq 0 \quad\Rightarrow\quad \psi = \operatorname{artanh}\nolimits \frac yx - p, \quad c = (\operatorname{sgn}\nolimits z) \sqrt{\frac{\sinh 2p - 2p}{2z}}, \quad t = \frac{2p}{c},
\label{invzn0}
\end{align}
where $p = \beta\left(\frac{z}{x^2-y^2}\right)$, and $\map{\beta}{\left(-\frac 14, \frac 14\right)}{{\mathbb R}}$ is the inverse function to the diffeomorphism
$$
\map{\alpha}{{\mathbb R}}{\left(-\frac 14, \frac 14\right)}, \qquad
\alpha(p) = \frac{\sinh 2p-2p}{8 \sinh^2 p}.
$$
\end{theorem}
See plots of the functions $\alpha(p)$ and $\beta(z)$ in Figs. \ref{fig:alpha} and \ref{fig:beta} respectively.
\figout{
\twofiglabelsizeh
{alpha}{Plot of $\alpha(p)$}{fig:alpha}{5}
{beta}{Plot of $\beta(z)$}{fig:beta}{8}
}
\betaegin{proof}
The exponential mapping is real-analytic since the strictly normal extremals are trajectories of the real-analytic Hamiltonian vector field $\vec H$. We show that $\operatorname{Exp}\nolimits$ is bijective.
Formulas \eq{invz0} follow immediately from \eq{qc=0}.
Let $c \neq 0$. Then formulas \eq{qcn0} yield
\betaegin{align}
&x = \frac 2c \sinh p \cosh \tau, \quad
y = \frac 2c \sinh p \sinh \tau, \quad
z = \frac{1}{2c^2}(\sinh 2p - 2 p), \label{xyznew}\\
&p = \frac{ct}{2}, \qquad \tau = \psi + \frac{ct}{2}. \label{ptau}
\end{align}
Thus
\betaegin{align}
&x^2 - y^2 = \frac{4}{c^2} \sinh^2 p, \label{x2-y2}\\
&\frac{z}{x^2-y^2} = \frac{\sinh 2p - 2p}{8 \sinh^2 p} = \alpha(p). \nonumber
\end{align}
The function $\alpha(p)$ is a diffeomorphism from ${\mathbb R}$ to $\left(-\frac 14, \frac 14\right)$, thus it has an inverse function, a diffeomorphism $\map{\beta}{\left(-\frac 14, \frac 14\right)}{{\mathbb R}}$. So $p = \beta(\frac{z}{x^2-y^2})$. Now formulas \eq{invzn0} follow from \eq{xyznew}, \eq{ptau}.
So $\operatorname{Exp}\nolimits$ is a smooth bijection with a smooth inverse, i.e., a diffeomorphism.
\end{proof}
\section{Optimality of extremal trajectories}
We study optimality of extremal trajectories. The main tool is a
sufficient optimality condition (Th. \ref{th:suf_opt}) based on a field of extremals (see \cite{notes}, Sec. 17.1).
We prove optimality of all extremal trajectories (Theorems \ref{th:distIf}, \ref{th:optJ}) without apriori theorem on existence of optimal trajectories. Such a theorem was recently proved \cite{lok_pod}, and it can shorten the proof of optimality in our work.
\subsection{Sufficient optimality condition}
Let $M$ be a smooth manifold, then the cotangent bundle $T^*M$ bears the Liouville 1-form $s = p dq \in \mathcal{L}ambda^1(T^*M)$ and the symplectic 2-form $\sigma = ds = dp \wedge dq \in \mathcal{L}ambda^2(T^*M)$.
A submanifold $\mathcal{L} \subset T^*M$ is called a Lagrangian manifold if $\dim \mathcal{L} = \dim M$ and $\restr{\sigma}{\mathcal{L}} = 0$.
Consider an optimal control problem
\betaegin{align*}
&\dot q = f(q, u), \qquad q \in M, \quad u \in U,\\
&q(t_0) = q_0, \qquad q(t_1) = q_1, \\
&J[q(\cdot)] = \int_{t_0}^{t_1} \varphi(q, u) \, dt \to \min, \\
&t_0 \text{ is fixed}, \qquad t_1 \text{ is free}.
\end{align*}
Let $g_u(\lambda) = \langle \lambda, f(q, u)\rangle - \varphi(q, u)$, $\lambda \in T^*M$, $q = \pi(\lambda)$, $u \in U$, be the normal Hamiltonian of PMP. Suppose that the maximized normal Hamiltonian $G(\lambda) = \max_{u \in U} g_u(\lambda)$ is smooth in an open domain $O \subset T^*M$, and let the Hamiltonian vector field $\vec G \in \operatorname{Vec}\nolimits(O)$ be complete.
\betaegin{theorem}\label{th:suf_opt}
Let $\mathcal{L} \subset G^{-1}(0) \cap O$ be a Lagrangian submanifold such that the form $\restr{s}{\mathcal{L}}$ is exact. Let the projection $\map{\pi}{\mathcal{L}}{\pi(\mathcal{L})}$ be a diffeomorphism on a domain in $M$. Consider an extremal $\widetilde{\lam}_t = e^{t \vec G}(\lambda_0)$, $t \in [t_0, t_1]$, contained in $\mathcal{L}$, and the corresponding extremal trajectory $ \widetilde{q}(t) = \pi(\widetilde{\lam}_t)$. Consider also any trajectory $q(t) \in \pi(\mathcal{L})$, $t \in [t_0, \tau]$, such that $q(t_0) = \widetilde{q}(t_0)$, $q(\tau) = \widetilde{q}(t_1)$. Then $J[\widetilde{q}(\cdot)] < J[q(\cdot)]$.
\end{theorem}
\betaegin{proof}
Completely similarly to the proof of Th. 17.2 \cite{notes}.
\end{proof}
\subsection[Optimality in the reduced sub-Lo\-rent\-zi\-an problem]{Optimality in the reduced sub-Lo\-rent\-zi\-an problem \\on the Heisenberg group}
We apply Th. \ref{th:suf_opt} to the reduced sub-Lo\-rent\-zi\-an problem \eq{pr31}--\eq{pr34}.
For this problem
the maximized Hamiltonian $G = 1 - \sqrt{h_1^2 - h^2_2}$ is smooth on the domain $O = \{ \lambda \in T^*M \mid h_1 < - |h_2| \}$, and the Hamiltonian vector field $\vec G \in \operatorname{Vec}\nolimits(O)$ is complete. In the domain $O$ the Hamiltonian vector fields $\vec G$ and $\vec H$ have the same trajectories up to a monotone time reparametrization; moreover, on the level surface $\left\{H = \frac 12\right\} = \{G = 0\}$ they just coincide between themselves.
Define the set
\betae{L}
\mathcal{L} = \left\{ e^{t \vec G}(\lambda_0) \mid \lambda_0 \in C, \ t > 0\right\}.
\end{equation}
\betaegin{lemma}
$\mathcal{L} \subset T^*M $ is a Lagrangian manifold such that $\restr{s}{\mathcal{L}}$ is exact.
\end{lemma}
\betaegin{proof}
Consider a smooth mapping
$$
\map{\Phi}{(T^*_{\operatorname{Id}\nolimits}M \cap G^{-1}(0)) \times {\mathbb R}_+}{T^*M}, \qquad (\lambda_0, t) \mapsto e^{t\vec G}(\lambda_0).
$$
Since
\betaegin{align*}
\operatorname{rank}\nolimits \left( \pder{\Phi}{(t, \lambda_0)}\right)
&= \operatorname{rank}\nolimits \left( \vec G(\lambda), e^{t \vec G}_*\left(h_2 \pder{}{h_1} + h_1 \pder{}{h_2}\right),e^{t \vec G}_* \pder{}{h_3}\right) \\
&= \operatorname{rank}\nolimits \left( \vec G(\lambda_0), h_2 \pder{}{h_1} + h_1 \pder{}{h_2}, \pder{}{h_3}\right)\\
&= \operatorname{rank}\nolimits \left( -h_1 X_1 + h_2 X_2, h_2 \pder{}{h_1} + h_1 \pder{}{h_2}, \pder{}{h_3}\right)\\
&= 3,
\end{align*}
then $\mathcal{L}$ is a smooth 3-dimensional manifold.
Further,
$
\pi(\mathcal{L}) = \operatorname{Exp}\nolimits(N) = \widetilde{\A}$ by Th. \ref{th:Exp-1}. Moreover, since $\operatorname{Exp}\nolimits = \pi \circ \Phi$ and $\map{\operatorname{Exp}\nolimits}{N}{\widetilde{\A}}$ is a diffeomorphism by Th. \ref{th:Exp-1}, then $\map{\pi}{\mathcal{L}}{\widetilde{\A}}$ is a diffeomorphism as well.
Let us show that $\restr{\sigma}{\mathcal{L}} = 0$. Take any $\lambda = e^{t \vec G}(\lambda_0) \in \mathcal{L}$, $(\lambda_0, t) \in N$, then $T_{\lambda} \mathcal{L} = {\mathbb R} \vec G(\lambda) \oplus e^{t\vec G}_* (T_{\lambda_0}C)$. Take any two vectors $T_{\lambda} \mathcal{L} \ni v_i = r_i \vec G(\lambda) + e^{t\vec G}_* w_i$, $w_i \in T_{\lambda_0}C$, $i = 1, 2$. Then
\betaegin{align*}
\sigma(v_1, v_2) = r_1 \sigma(\vec G(\lambda_0), w_2) + r_2 \sigma(w_1, \vec G(\lambda_0)) = 0
\end{align*}
since $\sigma(w_i, \vec G(\lambda_0)) = \langle dG, w_i \rangle = 0$ by virtue of $w_i \in T_{\lambda_0} C = \{ dG = 0\}$.
So the 1-form $\restr{s}{\mathcal{L}}$ is closed. But $\widetilde{\A}$ is simply connected, thus $\mathcal{L}$ is simply connected as well. Consequently, $\restr{s}{\mathcal{L}}$ is exact by the Poincar\'e lemma.
\end{proof}
\betaegin{theorem}\label{th:optI}
For any point $q_1 \in \operatorname{int}\nolimits \mathcal{A} = I^+(q_0)$ the strictly normal trajectory $q(t) = \operatorname{Exp}\nolimits(\lambda, t)$, $t \in [0, t_1]$, is the unique optimal trajectory of the reduced sub-Lo\-rent\-zi\-an problem \eq{pr31}--\eq{pr34} connecting $q_0$ with $q_1$, where $(\lambda, t_1) = \operatorname{Exp}\nolimits^{-1}(q_1) \in N$.
\end{theorem}
\betaegin{proof}
Take any $\lambda_0 \in C$, $t_1 > t_0 > 0$. Then the Lagrangian manifold $\mathcal{L}$ \eq{L} and the extremal $\widetilde{\lam}_t = e^{t\vec G}(\lambda_0)$, $t \in [t_0, t_1]$, satisfy hypotheses of
Th. \ref{th:suf_opt}. Thus the trajectory $\widetilde{q}(t) = \pi(\widetilde{\lam}_t)$, $t \in [t_0, t_1]$, is a strict maximizer for the reduced sub-Lo\-rent\-zi\-an problem \eq{pr31}--\eq{pr34}.
Take any $\lambda_1 \in C$, $t_2 > 0$, and consider the extremal trajectory $\betaar q(t) = \operatorname{Exp}\nolimits(\lambda_1, t)$, $t \in [0, t_2]$. Take any $\widehat q \in \widetilde{\A}$. The set $\mathcal{A}$ is an attainable set of a left-invariant control system on a Lie group, thus it is a semigroup. Consequently, $\widehat q \cdot \betaar q(t)$ is an extremal trajectory contained in $\widetilde{\A}$. By the previous paragraph, this trajectory is a strict maximizer for the reduced sub-Lo\-rent\-zi\-an problem \eq{pr31}--\eq{pr34}. By left invariance of this problem, the same holds for the trajectory $\betaar q(t) $, $t \in [0, t_2]$.
\end{proof}
Denote the cost function for the equivalent reduced sub-Lo\-rent\-zi\-an problems \eq{pr21}--\eq{pr24} and \eq{pr31}--\eq{pr34}:
\betaegin{align*}
\widetilde{d}(q_1) &= \sup \{ l(q(\cdot)) \mid \text{ traj. $q(\cdot)$ of \eq{pr21}--\eq{pr24}, $q(0) = q_0$, $q(t_1) = q_1$}\}\\
&=\sup \{ t_1 > 0 \mid \exists \text{ traj. $q(\cdot)$ of \eq{pr31}--\eq{pr34} s.t. $q(0) = q_0$, $q(t_1) = q_1$}\},
\end{align*}
where $q_1 \in \operatorname{int}\nolimits \mathcal{A} = I^+(q_0)$. This function has the following description and regularity property.
\betaegin{theorem}\label{th:distI}
Let $q = (x, y, z) \in I^+(q_0)$. Then
\betae{tdq}
\widetilde{d}(q) = \sqrt{x^2-y^2} \cdot \frac{p}{\sinh p}, \qquad p = \beta\left(\frac{z}{x^2-y^2}\right).
\end{equation}
The function $\map{\widetilde{d}}{I^+(q_0)}{{\mathbb R}_+}$ is real-analytic.
\end{theorem}
\betaegin{proof}
Let $q \in I^+(q_0)$, then the sub-Lo\-rent\-zi\-an length maximizer from $q_0$ to $q$ for the reduced sub-Lo\-rent\-zi\-an problem \eq{pr31}--\eq{pr34} is described in Th. \ref{th:optI}, and the expression for $\widetilde{d}(q)$ in \eq{tdq} follows from the expression for $t$ in \eq{invzn0}.
The both functions $\sqrt{x^2-y^2}$ and $\frac{p}{\sinh p}$ are real-analytic on $ I^+(q_0)$, thus $\widetilde{d}$ is real-analytic as well.
\end{proof}
\subsection[Optimality in the full sub-Lo\-rent\-zi\-an problem]{Optimality in the full sub-Lo\-rent\-zi\-an problem \\on the Heisenberg group}
In this subsection we consider the full sub-Lo\-rent\-zi\-an problem \eq{prf1}--\eq{prf4}.
\betaegin{theorem}\label{th:distIf}
Let $q_1 \in I^+(q_0)$. Then
the sub-Lo\-rent\-zi\-an length maximizers for the full problem \eq{prf1}--\eq{prf4} are reparametrizations of the corresponding sub-Lo\-rent\-zi\-an length maximizer for the reduced problem \eq{pr31}--\eq{pr34} described in Th. {\em \ref{th:optI}}.
In particular, $\restr{d}{I^+(q_0)} = \widetilde{d}$.
\end{theorem}
\betaegin{proof}
Let $q(t)$, $t \in [0, t_1]$, be a trajectory of the full problem \eq{prf1}--\eq{prf4} such that $q(0) = q_0$, $q(t_1) = q_1$, and let $q(\cdot)$ be not a trajectory of the reduced problem \eq{pr21}--\eq{pr24} (that is, there exist $0 \leq \tau_1 < \tau_2 \leq t_1$ such that $\restr{\left(u_1 - |u_2|\right)}{[\tau_1, \tau_2]} \equiv 0$).
Let $\widetilde{q}(t)$, $t \in [0, \widetilde{t}_1]$, be the optimal trajectory in the reduced problem \eq{pr31}--\eq{pr34} connecting $q_0$ with $q_1$. We show that $l(q(\cdot)) < l(\widetilde{q}(\cdot))$. By contradiction, suppose that $l(q(\cdot)) \gammaeq l(\widetilde{q}(\cdot))$.
Let $l(q(\cdot)) = l(\widetilde{q}(\cdot))$. The trajectory $q(\cdot)$ does not satisfy the PMP for the full problem \eq{prf1}--\eq{prf4} (see Sec. \ref{sec:PMP}), thus it is not optimal in this problem. Thus there exists a trajectory $\betaar q(\cdot)$ of this problem with the same endpoints and $l(\betaar q(\cdot)) > l(\widetilde{q}(\cdot))$. The curve $\betaar q(\cdot)$ cannot be a trajectory of the reduced system since its length is greater than the maximum $ l(\widetilde{q}(\cdot))$ in this problem. So we can denote $\betaar q(\cdot))$ as $q(\cdot)$ and assume that $l(q(\cdot)) > l(\widetilde{q}(\cdot))$.
After time reparametrization we obtain that the control $u(t) = (u_1(t), u_2(t))$ corresponding to the trajectory $q(t)$, $t \in [0, t_1]$, satisfies $u_1(t) \equiv 1$, thus $|u_2(t)|\leq 1$.
For any $\delta \in (0, 1)$ define a function
$$
u_2^{\delta}(t) =
\betaegin{cases}
u_2(t) &\text{for } |u_2(t)| \leq 1 - \delta, \\
1-\delta &\text{for } u_2(t) > 1 - \delta, \\
\delta-1 &\text{for } u_2(t) < \delta-1,
\end{cases}
$$
so that
\betae{u2de}
|u_2^{\delta}(t)| \leq 1 - \delta, \quad |u_2^{\delta}(t) - u_2(t)| \leq \delta, \qquad t \in [0, t_1].
\end{equation}
Define an admissible control $u^{\delta}(t) = (1, u_2^{\delta}(t))$, $t \in [0, t_1]$, and consider the corresponding trajectory $q^{\delta}(t)$, $t \in [0, t_1]$, of the reduced problem \eq{pr21}--\eq{pr24} with $q^{\delta}(0) = q_0$. Denote its endpoint $q^{\delta}(t_1) = q_1^{\delta}$. By virtue of the second inequality in \eq{u2de},
\betaegin{align*}
&l(q^{\delta}(\cdot)) = \int_0^{t_1} \sqrt{1 - \left(u_2^{\delta}(t)\right)^2} dt \to \int_0^{t_1} \sqrt{1 - u_2^2(t)} dt =
l(q(\cdot)), \\
&\max_{t \in [0, t_1]} \|q^{\delta}(t) - q(t)\| \to 0
\end{align*}
as $\delta \to + 0$.
So for sufficiently small $\delta > 0$ we have
$$
l(q^{\delta}(\cdot)) > l(\widetilde{q}(\cdot)) \qquad \text{and} \qquad \|q_1^{\delta} - q_1\| \text{ is small},
$$
where $\|\cdot\|$ is any norm in $M \cong {\mathbb R}^3$. In particular, $q_1^{\delta} \in I^+(q_0)$ for small $\delta>0$.
Now let $\widehat{q}^{\delta}(t)$, $t \in \left[0, \widehat{t}_1^{\delta}\right]$, be the optimal trajectory in the reduced problem \eq{pr31}--\eq{pr34} with the boundary conditions $\widehat{q}^{\delta}(0) = q_0$, $\widehat{q}^{\delta}\left(\widehat{t}_1^{\delta}\right) = q_1^{\delta}$. Then for small $\delta > 0$
\betaegin{align*}
&l\left(\widehat{q}^{\delta}(\cdot)\right) \gammaeq l(q^{\delta}(\cdot)) > l(\widetilde{q}(\cdot)), \\
&\left\|q_1^{\delta} - q_1\right\| = \left\|\widehat{q}^{\delta}\left(\widehat{t}_1^{\delta}\right) - \widetilde{q}(t_1)\right\| \text{ is small}.
\end{align*}
By virtue of Th. \ref{th:distI}, the sub-Lo\-rent\-zi\-an distance $\map{\widetilde{d}}{I^+(q_0)}{{\mathbb R}_+}$ in the reduced problem \eq{pr31}--\eq{pr34} is continuous, thus for small $\delta > 0$
$$
|l\left(\widehat{q}^{\delta}(\cdot)\right) - l(\widetilde{q}(\cdot))| = |\widetilde{d}(q_1^{\delta}) - \widetilde{d}(q_1)| \text{ is small}.
$$
Summing up, for small $\delta > 0$ the difference
$$
l(q(\cdot)) - l(\widetilde{q}(\cdot)) <
\left( l(q(\cdot)) - l\left(q^{\delta}(\cdot)\right)\right) +
\left( l\left(\widehat{q}^{\delta}(\cdot)\right) - l\left(\widetilde{q}(\cdot)\right)\right)
$$
becomes arbitrarily small, a contradiction.
Thus $\widetilde{q}(\cdot)$ is optimal and $q(\cdot)$ is not optimal in the full sub-Lo\-rent\-zi\-an problem \eq{prf1}--\eq{prf4}.
\end{proof}
\betaegin{theorem}\label{th:optJ}
Let $q_1 = (x_1, y_1, z_1) \in \partial A = J^+(q_0) \setminus I^+(q_0)$, $q_1 \neq q_0$. Then an optimal trajectory in the full sub-Lo\-rent\-zi\-an problem \eq{prf1}--\eq{prf4} is a future directed lightlike piecewise smooth trajectory with one or two subarcs generated by the vector fields $X_1 \pm X_2$. In detail, up to a reparametrization:
\betaegin{itemize}
\item[$(1)$]
If $z_1 = 0$, then
$$
u(t) \equiv \operatorname{const}\nolimits = (1, \pm 1), \qquad q(t) = e^{t(X_1 \pm X_2)} = (t, \pm t, 0), \qquad t \in [0, t_1], \quad t_1 = x_1.
$$
\item[$(2)$]
If $z_1 > 0$, then
\betaegin{align*}
&t \in [0, \tau_1] \quad\Rightarrow\quad u(t) \equiv (1, - 1), \qquad q(t) = e^{t(X_1 - X_2)} = (t, -t, 0), \\
&t \in [\tau_1, \tau_1+ \tau_2] \quad\Rightarrow\quad u(t) \equiv (1, 1), \\
&\qquad\qquad\qquad q(t) = e^{(t-\tau_1)(X_1 + X_2)}e^{\tau_1(X_1 - X_2)} = (t, t - 2\tau_1, \tau_1(t-\tau_1)), \\
&\tau_1 = \frac{x_1-y_1}{2}, \qquad \tau_2 = \frac{x_1+y_1}{2}.
\end{align*}
\item[$(3)$]
If $z_1 < 0$, then
\betaegin{align*}
&t \in [0, \tau_1] \quad\Rightarrow\quad u(t) \equiv (1, 1), \qquad q(t) = e^{t(X_1 + X_2)} = (t, t, 0), \\
&t \in [\tau_1, \tau_1+ \tau_2] \quad\Rightarrow\quad u(t) \equiv (1, -1), \\
&\qquad\qquad\qquad q(t) = e^{(t-\tau_1)(X_1 - X_2)}e^{\tau_1(X_1 + X_2)} = (t, 2\tau_1 - t, -\tau_1(t-\tau_1)), \\
&\tau_1 = \frac{x_1+y_1}{2}, \qquad \tau_2 = \frac{x_1-y_1}{2}.
\end{align*}
\end{itemize}
\end{theorem}
The broken lightlike trajectories with two arcs described in items (1), (2) of Th. \ref{th:optJ} are shown in Fig. \ref{fig:S0opt}.
\betaegin{proof}
Let $q(t)$, $t \in [0, t_1]$, be a future directed nonspacelike trajectory connecting $q_0$ and $q_1$. If $q(\cdot)$ is not lightlike, then there exists a future directed timelike arc $q(t)$, $t \in [s_1, s_2]$, $0 \leq s_1 < s_2 \leq t_1$, thus $q(t_1) \in \operatorname{int}\nolimits \mathcal{A}$, a contradiction. Thus $q(\cdot)$ is lightlike, and the statement follows by direct computation of trajectories of the lightlike vector fields $X_1 \pm X_2$.
\end{proof}
\betaegin{corollary}
For any $q_1 \in J^+(q_0)$, $q_1 \neq q_0$, there is a unique, up to reparametri\-za\-tion, sub-Lo\-rent\-zi\-an length minimizer in the full problem \eq{prf1}--\eq{prf4} that connects $q_0$ and $q_1$:
\betaegin{itemize}
\item
if $q_1 \in \operatorname{int}\nolimits \mathcal{A} = I^+(q_0)$, then $q(\cdot)$ is a future directed timelike strictly normal trajectory described in Theorems $\ref{th:optI}$, $\ref{th:distIf}$.
\item
if $q_1 \in \partial \mathcal{A} = J^+(q) \setminus I^+(q_0)$, then $q(\cdot)$ is a future directed lightlike nonstrictly normal trajectory described in Th. $\ref{th:optJ}$.
\end{itemize}
\end{corollary}
\betaegin{corollary}
Any sub-Lo\-rent\-zi\-an length maximizer of problem \eq{prf1}--\eq{prf4} of positive length is timelike and strictly normal.
\end{corollary}
\betaegin{remark}
The broken trajectories described in items $(2)$, $(3)$ of Th. {\em\ref{th:optJ}} are optimal in the sub-Lo\-rent\-zi\-an problem, while in sub-Riemannian problems trajectories with angle points cannot be optimal, see {\em \cite{hak_ledon}}.
Moreover, these broken trajectories are normal and nonsmooth, which is also impossible in sub-Riemannian geometry.
\end{remark}
\section{Sub-Lo\-rent\-zi\-an distance}
Denote $d(q) := d(q_0, q)$, $q \in J^+(q_0)$.
\betaegin{theorem}\label{th:dist}
Let $q = (x, y, z) \in J^+(q_0)$. Then
\betae{dq}
d(q) = \sqrt{x^2-y^2} \cdot \frac{p}{\sinh p}, \qquad p = \beta\left(\frac{z}{x^2-y^2}\right).
\end{equation}
In particular:
\betaegin{itemize}
\item[$(1)$]
$z = 0 \iff d(q) = \sqrt{x^2 - y^2}$,\\
\item[$(2)$]
$q \in J^+(q_0) \setminus I^+(q_0) \iff d(q) = 0$.
\end{itemize}
\end{theorem}
\betaegin{remark}\label{rem:psinhp}
In the right-hand side of the first equality in \eq{dq}, we assume by continuity that $\frac{p}{\sinh p} = 1$ for $p = 0$ and $\frac{p}{\sinh p} = 0$ for $p = \infty$. See the plot of the function $\frac{p}{\sinh p} $ in Fig. $\ref{fig:pSinhp}$.
\figout{
\onefiglabelsizen
{pSinhp}{Plot of $\frac{p}{\sinh p} $}{fig:pSinhp}{2}
}
\end{remark}
\betaegin{proof}
Let $q \in I^+(q_0)$, then the sub-Lo\-rent\-zi\-an length maximizers from $q_0$ to $q$ are described in Theorem \ref{th:distIf} and the expression for $\restr{d}{\widetilde{\A}} = \widetilde{d}$ was obtained in Th. \ref{th:distI}. In particular, if $z = 0$, then $p = 0$ and $d(q) = \sqrt{x^2 - y^2}$, and vice versa.
Let $q \in J^+(q_0)\setminus I^+(q_0)$, then the sub-Lo\-rent\-zi\-an length maximizers from $q_0$ to $q$ are described in Th. \ref{th:optJ}. Thus $d(q) = 0$, which agrees with \eq{dq} since in this case $\frac{|z|}{x^2-y^2} = \frac 14$, so $p = \infty$.
\end{proof}
We plot restrictions of the sub-Lo\-rent\-zi\-an distance to several planar domains:
\betaegin{itemize}
\item
$\restr{d}{z=0}= \sqrt{x^2 - y^2}$ to the domain $J^+(q_0) \cap \{ z = 0 \} = \{ x \gammaeq |y|, \ z = 0\}$, see Fig. \ref{fig:dz=0},
\item
$\restr{d}{y=0}$ to the domain $J^+(q_0) \cap \{ y = 0 \} = \{-x^2/4 \leq z \leq x^2/4, \ y = 0\}$, see Fig. \ref{fig:dy=0},
\item
$\restr{d}{x=1}$ to the domain $J^+(q_0) \cap \{ x = 1 \} = \{y^2 + 4 |z|\leq 1, \ x = 1\}$, see Fig. \ref{fig:dx=1}.
\end{itemize}
\figout{
\twofiglabelsizeh
{dz=0}{Plot of $\restr{d}{z=0}$}{fig:dz=0}{6}
{dy=0}{Plot of $\restr{d}{y=0}$}{fig:dy=0}{10}
\onefiglabelsizen
{dx=1}{Plot of $\restr{d}{x=1}$}{fig:dx=1}{7}
}
The sub-Lo\-rent\-zi\-an distance has the following regularity properties.
\betaegin{theorem}\label{th:dreg}
\betaegin{itemize}
\item[$(1)$]
The function $d(\cdot)$ is continuous on $J^+(q_0)$ and real-analytic on $I^+(q_0)$.
\item[$(2)$]
The function $d(\cdot)$ is not Lipschitz near points $q = (x, y, z)$ with $x = |y|>0$, $z = 0$.
\end{itemize}
\end{theorem}
\betaegin{proof}
(1)
follows from representation \eq{dq}.
(2) follows from item (1) of Th. \ref{th:dist} since the function $\restr{d}{z = 0} =\sqrt{x^2-y^2}$ is not Lipschitz near points with $x = |y|>0$.
\end{proof}
\betaegin{remark}
Item $(1)$ of Th. $\ref{th:dreg}$ improves item $(2)$ of Sec. $\ref{sec:groch}$.
\end{remark}
\betaegin{remark}
Item $(2)$ of Th. $\ref{th:dreg}$ is visualized in Fig. $\ref{fig:dz=0}$ since the cone given by the plot of $\restr{d}{z = 0} = \sqrt{x^2-y^2}$ has vertical tangent planes at points $x = |y|> 0$.
Moreover, item $(2)$ of Th. $\ref{th:dreg}$ can be essentially detailed by a precise description of the asymptotics of the sub-Lo\-rent\-zi\-an distance $d(q)$ as $q \to \partial \mathcal{A}$, this will be done in a forthcoming paper {\em \cite{pop_sach}}.
\end{remark}
\betaegin{remark}
The sub-Lo\-rent\-zi\-an distance $\map{d}{J^+(q_0)}{[0, + \infty)}$ is not uniformly continuous since the same holds for its restriction $\restr{d}{z = 0} = \sqrt{x^2-y^2}$ on the angle $\{x \gammaeq |y|\}$.
\end{remark}
As was shown in \cite{groch6}, the sub-Lo\-rent\-zi\-an distance $d(q)$ admits a lower bound by the function $\sqrt{x^2 - y^2 - 4 |z|}$ and does not admit an upper bound by this function multiplied by any constant (see item (4) in Sec. \ref{sec:groch}). Here we precise this statement and prove another upper bound.
\betaegin{theorem}
\betaegin{itemize}
\item[$(1)$]
The ratio $\dfrac{\sqrt{x^2 - y^2 - 4 |z|}}{d(q)}$ takes any values in the segment $[0, 1]$ for $q =(x,y,z) \in J^+(q_0)$.
\item[$(2)$]
For any $q = (x, y, z) \in J^+(q_0)$ there holds the bound $d(q) \leq \sqrt{x^2-y^2}$,
moreover, the ratio $\dfrac{d(q)}{\sqrt{x^2-y^2}}$ takes any values in the segment $[0, 1]$.
\end{itemize}
\end{theorem}
The two-sided bound
\betae{bound}
{\sqrt{x^2 - y^2 - 4 |z|}} \leq {d(q)} \leq \sqrt{x^2-y^2}, \qquad q \in J^+(q_0),
\end{equation}
is visualized in Fig. \ref{fig:dbound}, which shows plots of the surfaces (from below to top):
$$
\sqrt{x^2-y^2} = 1, \qquad {d(q)} = 1, \qquad {\sqrt{x^2 - y^2 - 4 |z|}} = 1, \qquad q \in J^+(q_0).
$$
\figout{
\onefiglabelsizen{dbound}{Bound \eq{bound}}{fig:dbound}{6}
}
\betaegin{proof}
$(1)$
It follows from \eq{dq} that
$$
\frac{x^2 - y^2 - 4 |z|}{d^2(q)} = \frac{\sinh^2 p - \sinh p \cosh p + p}{p^2},
$$
and the function in the right-hand side takes all values in the segment $[0, 1]$ for $q \in J^+(q_0)$.
$(2)$
It follows from \eq{dq} that $\frac{d(q)}{\sqrt{x^2-y^2}} = \frac{p}{\sinh p}$. When $q \in J^+(q_0)$, the ratio $\frac{p}{\sinh p}$ takes all values in the segment $[0, 1]$, see Remark \ref{rem:psinhp} after Th. \ref{th:dist}.
\end{proof}
\section{Symmetries}
\betaegin{theorem}\label{th:sym}
\betaegin{itemize}
\item[$(1)$]
The hyperbolic rotations $X_0 = y \pder{}{x} + x \pder{}{y}$ and reflections
$\varepsilon^1 \ : \ (x, y, z) \mapsto (x, - y, z)$, $\varepsilon^2 \ : \ (x, y, z) \mapsto (x, y, -z)$ preserve $d(\cdot)$.
\item[$(2)$]
The dilations $Y = x \pder{}{x} + y \pder{}{y} + 2z \pder{}{z}$ stretch $d(\cdot)$:
$$
d(e^{sY}(q)) = e^s d(q), \qquad s \in {\mathbb R}, \quad q \in J^+(q_0).
$$
\end{itemize}
\end{theorem}
\betaegin{proof}
(1) The flow of the hyperbolic rotations
$$
e^{s X_0} \ : \ (x, y, z) \mapsto (x \cosh s + y \sinh s, x \sinh s + y \cosh s, z), \qquad
s \in {\mathbb R}, \quad (x,y,z) \in M,
$$
preserves the exponential mapping:
$$
e^{sX_0} \circ \operatorname{Exp}\nolimits(\psi, c, t) = \operatorname{Exp}\nolimits(\psi + s, c, t), \qquad (\psi, c, t) \in N, \quad s \in {\mathbb R},
$$
thus $d(e^{sX_0}(q)) = d(q)$ for $q \in I^+(q_0)$. Moreover, the flow $e^{sX_0}$ preserves the boundary $\partial \mathcal{A} = J^+(q_0) \setminus I^+(q_0)$, thus $d(e^{sX_0}(q)) = d(q) = 0$ for $q \in J^+(q_0) \setminus I^+(q_0)$.
Further, it is obvious from \eq{dq} that the reflections $\varepsilon^1$, $\varepsilon^2$ preserve $d(\cdot)$.
(2) The flow of the dilations
$$
e^{s Y} \ : \ (x, y, z) \mapsto (x e^s, ye^s, ze^{2s}), \qquad
s \in {\mathbb R}, \quad (x,y,z) \in M,
$$
acts on the exponential mapping as follows:
$$
e^{sY} \circ \operatorname{Exp}\nolimits(\psi, c, t) = \operatorname{Exp}\nolimits(\psi, ce^{-2s}, te^s), \qquad (\psi, c, t) \in N, \quad s \in {\mathbb R},
$$
thus $d(e^{sY}(q)) = e^s d(q)$ for $q \in I^+(q_0)$. The equality $d(e^{sY}(q)) = e^s d(q) = 0$ for $q \in J^+(q_0) \setminus I^+(q_0)$ follows since the flow $e^{sY}$ preserves the boundary $\partial \mathcal{A} = J^+(q_0) \setminus I^+(q_0)$.
\end{proof}
\section{Sub-Lo\-rent\-zi\-an spheres}
\subsection{Spheres of positive radius}
Sub-Lo\-rent\-zi\-an spheres
$$
S(R) = \{ q \in M \mid d(q) = R\}, \qquad R> 0,
$$
are transformed one into another by dilations:
$$
S(e^s R) = e^{sY}(S(R)), \qquad s \in {\mathbb R},
$$
thus we describe the unit sphere
\betae{S1}
S = S(1) = \{\operatorname{Exp}\nolimits(\lambda, 1) \mid \lambda \in C\}.
\end{equation}
\betaegin{theorem}
\betaegin{itemize}
\item[$(1)$]
The unit sub-Lo\-rent\-zi\-an sphere $S$ is a regular real-analytic manifold diffeomorphic to ${\mathbb R}^2$.
\item[$(2)$]
Let $q = \operatorname{Exp}\nolimits(\psi, c, 1) \in S$, $(\psi, c) \in C$, then the tangent space
\betae{TqS}
T_qS = \left\{v = \sum_{i=1}^3 v_i X_i(q) \mid - v_1 \cosh(\psi + c)+v_2 \sinh(\psi+c)+v_3c=0\right\}.
\end{equation}
\item[$(3)$]
$S$ is the graph of the function $x = \sqrt{y^2 + f(z)}$, where $f(z) = e \circ k(z)$, $e(w) = \frac{\sinh^2 w}{w^2}$, $k(z) = b(z)/2$, $b = a^{-1}$, $a(c) = \frac{\sinh c - c}{2c^2}$.
\item[$(4)$]
The function $f(z)$ is real-analytic, even, strictly convex, unboundedly and strictly increasing for $z \gammaeq 0$. This function has a Taylor decomposition $f(z) = 1 + 12 z^2 + O(z^4)$ as $z \to 0$ and
an asymptote $4|z|$ as $z \to \infty$:
\betae{fas}
\lim_{z \to \infty} (f(z) - 4 |z|) = 0.
\end{equation}
\item[$(5)$]
The function $f(z)$ satisfies the bounds
\betae{fbound}
4 |z| < f(z) < 4 |z| + 1, \qquad z \neq 0.
\end{equation}
\item[$(6)$]
A section of the sphere $S$ by a plane $\{z = \operatorname{const}\nolimits\}$ is a branch of the hyperbola $x^2-y^2 = f(z)$, $x>0$.
A section of the sphere $S$ by a plane $\{x = \operatorname{const}\nolimits>1 \}$ is a strictly convex curve $y^2+f(z) = x^2$ diffeomorphic to $S^1$.
\item[$(7)$]
The sub-Lo\-rent\-zi\-an distance from the point $q_0$ to a point
$q = (x, y, z) \in \widetilde{\A}$ may be expressed as
$d(q) = R$, where $x^2-y^2 = R^2 f(z/R^2)$.
\item[$(8)$]
The sub-Lo\-rent\-zi\-an ball $B = \{ q \in M \mid d(q) \leq 1\}$ has infinite volume in the coordinates $x, y, z$.
\end{itemize}
\end{theorem}
See in Fig. \ref{fig:plotS} a plot of the sphere $S$ (above in red) and the Heisenberg beak $\partial \mathcal{A}$ (at the bottom in blue). Different sub-Lo\-rent\-zi\-an length maximizers connecting $q_0$ and $S$ are shown in Fig. \ref{fig:Sopt}. A plot of the function $f(z)$ illustrating bound \eq{fbound} is shown in Fig. \ref{fig:f}. Sections of the sphere $S$ by the planes $\{x = 1, 2, 3\}$ are shown in Fig. \ref{fig:Sx}.
\figout{
\twofiglabelsizeh
{plotS}{The sphere $S$ and the Heisenberg beak $\partial \mathcal{A}$}{fig:plotS}{6}
{Sopt}{Maximizers connecting $q_0$ and $S$}{fig:Sopt}{6}
\twofiglabelsizeh
{f}{Plot of $f(z)$ and bound \eq{fbound}}{fig:f}{5}
{Sx}{Sections of $S$ by the planes $\{x = 1, 2, 3\}$}{fig:Sx}{5}
}
\betaegin{proof}
$(1)$
Since $\map{\operatorname{Exp}\nolimits}{C \times {\mathbb R}_+}{\widetilde{\A}}$ is a diffeomorphism, the parametrization \eq{S1} of the sphere $S$ implies that it is a smooth 2-dimensional manifold diffeomorphic to ${\mathbb R}^2$. Moreover, the exponential mapping is real-analytic, thus $S$ is real-analytic as well.
$(2)$
Let
$q = \operatorname{Exp}\nolimits(\lambda_0, 1) \in S$, $\lambda_0 = (\psi, c, q_0 ) \in C$, and let $\lambda_1 = e^{\vec H}(\lambda_0)$. Then
\betae{TqS1}
T_qS = \lambda_1^{\perp} = \{v \in T_q M \mid \langle \lambda_1, v \rangle = 0\}.
\end{equation}
Since $h_1(\lambda_1) = - \cosh(\psi+c)$, $h_2(\lambda_1) = \sinh(\psi+c)$, $h_3(\lambda_1) = c$, representation \eq{TqS} follows from \eq{TqS1}.
$(3)$
It follows from \eq{TqS} that the 2-dimensional manifold $S$ projects regularly to the coordinate plane $(y, z)$, thus it is a graph of a real-analytic function $x = F(y, z)$. Since $e^{tX_0}(S) = S$, $t \in {\mathbb R}$, then
$$
0 = \restr{X_0(F(y,z)-x)}{S} = F(y, z) \pder{F}{y}(y,z) - y.
$$
Integrating this differential equation, we get $F(y, z) = \sqrt{y^2 + f(z)}$ for a real-analytic function $f(z)$.
Since $S \cap \{ z = 0\} = \left\{x = \sqrt{y^2 + 1}, \ z = 0\right\}$, then $f(0) = 1$.
Let $z \neq 0$. Then $z = \frac{\sinh c - c}{2c^2} = a(c)$ by virtue of \eq{qcn0}. The function $\map{a}{{\mathbb R}}{{\mathbb R}}$ is a diffeomorphism, denote the inverse function $b = a^{-1}$.
By virtue of \eq{x2-y2}, we have
$f(z) = x^2 - y^2 = \frac{4}{c^2} \sinh^2 p$, whence $f(a(c)) = \frac{4}{c^2} \sinh^2 p$, thus $f(a) = e(\frac b2(a))$, where $e(x) = \frac{\sinh^2 x}{x^2}$. Item (3) follows.
$(4)$
We have already proved that $f(z)$ is real-analytic. Since $\varepsilon^1(S) = S$, then $f$ is even. Immediate computation shows that $k'(z) > 0$, $z > 0$, and $e'(x)> 0$, $x > 0$, whence $f'(z) > 0$, $z > 0$. Similarly it follows that $f''(z) > 0$ for $z > 0$. By virtue of the expansions $k(z) = 6 z + O(z^2)$, $z \to 0$ and $e(x) = 1 + \frac{x^2}{3} + O(x^4)$, $x \to 0$, we get $f(z) = 1 + 12 z^2 + O(z^4)$, $z \to 0$. Finally, it easily follows from the definition of the function $f(z)$ that $\lim_{z \to \infty}(f(z) - 4 |z|) = 0$.
$(5)$
follows from (4).
$(6)$
It is straightforward that $S \cap \{z = \operatorname{const}\nolimits\} = \{x^2 - y^2 = f(z), \ x > 0, \ z = \operatorname{const}\nolimits\}$ is a branch of a hyperbola.
The section $S \cap \{x = \operatorname{const}\nolimits > 1\} = \{y^2 + f(z) = x^2, \ x = \operatorname{const}\nolimits > 1\}$ is a smooth compact curve, thus diffeomorphic to $S^1$. If $y \gammaeq 0$, then this curve is given by the equation $y = \sqrt{x^2 - f(z)}$, which is a strictly concave function (this follows by twice differentiation).
$(7)$
Take any point $q = (x,y,z) \in \widetilde{\A}$, then there exists $s \in {\mathbb R}$ such that $e^{-s Y} (q) \in S$, i.e., $d(q) = e^s$, see item (2) of Th. \ref{th:sym}. Denoting $R = e^s$, we get $\frac xR = \sqrt{\frac{y^2}{R^2} + f\left(\frac{z}{R^2}\right)}$, and item (7) of this theorem follows.
$(8)$
The unit ball is given explicitly by
$$
B = \left\{(x, y, z) \in {\mathbb R}^3 \mid \sqrt{y^2 + 4 |z|} \leq x \leq \sqrt{y^2 + f(z)}\right\},
$$
thus its volume is evaluated by the integral
$$
V(B) = \int_{-\infty}^{+ \infty} dy \int_{-\infty}^{+ \infty} dz \left( \sqrt{y^2 + f(z)} - \sqrt{y^2 + 4 |z|}\right) = + \infty.
$$
\end{proof}
\betaegin{remark}
Thanks to bound \eq{fbound} of the function $f(z)$, the sphere $S = \left\{ x = \sqrt{y^2 + f(z)}\right\}$ is contained in the domain
$$
\left\{q = (x, y, z) \in M \mid \sqrt{y^2 + 4 |z|} < x \leq \sqrt{y^2+ 4 |z| + 1}\right\}.
$$
The bounding functions of this domain provide an approximation of the function $\sqrt{y^2 + f(z)}$ defining $S$ up to the accuracy
$$
\sqrt{y^2+ 4 |z| + 1} - \sqrt{y^2 + 4 |z|}
=
\frac{1}{\sqrt{y^2+ 4 |z| + 1} + \sqrt{y^2 + 4 |z|} }
\leq \min\left(1, \frac{2}{|y|}, \frac{1}{\sqrt{|z|}}\right).
$$
\end{remark}
\subsection{Sphere of zero radius}
Now consider the zero radius sphere
$$
S(0) = \{q \in M \mid d(q) = 0\}.
$$
\betaegin{theorem}\label{th:S0}
\betaegin{itemize}
\item[$(1)$]
$S(0) = J^+(q_0) \setminus I^+(q_0) = \partial J^+(q_0) = \partial I^+(q_0) = \partial \mathcal{A}$.
\item[$(2)$]
$S(0)$ is the graph of a continuous function $x = \Phi(y, z) := \sqrt{y^2 + 4 |z|}$, thus a $2$-dimensional topological manifold.
\item[$(3)$]
The function $\Phi(y, z)$ is even in $y$ and $z$, real-analytic for $z \neq 0$, Lipschitz near $z = 0$, $y \neq 0$, and H\"older with constant $\frac 12$, non-Lipschitz near $(y, z)= (0, 0)$.
\item[$(4)$]
$S(0)$ is filled by broken lightlike trajectories with one or two edges described in Th. $\ref{th:optJ}$, and is parametrized by them as follows:
\betaegin{multline*}
S(0) = \left\{ e^{\tau_2 (X_1-X_2)} e^{\tau_1(X_1+X_2)} = (\tau_1 + \tau_2, \tau_1-\tau_2, -\tau_1 \tau_2) \mid \tau_i \gammaeq 0 \right\} \\
\cup
\left\{ e^{\tau_2 (X_1+X_2)} e^{\tau_1(X_1-X_2)} = (\tau_1 + \tau_2, \tau_2-\tau_1, \tau_1 \tau_2) \mid \tau_i \gammaeq 0 \right\}.
\end{multline*}
\item[$(5)$]
The flows of the vector fields $Y, X_0$ preserve $S(0)$.
Moreover, the symmetries $Y$, $X_0$ provide a regular parametrization of
\betaegin{align}
S(0) \cap \{\operatorname{sgn}\nolimits z = \pm 1\} &= \left\{e^{sY} \circ e^{rX_0} (q_{\pm}) \mid r, s > 0\right\}, \label{S0par1}
\end{align}
where $q_{\pm} = (x_{\pm}, y_{\pm}, z_{\pm})$ is any point in $S(0) \cap \{\operatorname{sgn}\nolimits z = \pm 1\}$.
\item[$(6)$]
The sphere $S(0) = \left\{ 16z^2 = (x^2-y^2)^2, \ x^2 - y^2 \gammaeq 0, \ x \gammaeq 0\right\}$ is a semi-algebraic set.
\item[$(7)$]
The zero-radius sphere is a Whitney stratified set with the stratification
\betaegin{multline*}
S(0) = \betaig(S(0) \cap \{z > 0\} \betaig) \cup \betaig(S(0) \cap \{z < 0\}\betaig) \\
\cup \betaig(S(0) \cap \{z = 0, \ y > 0\}\betaig) \cup \betaig(S(0) \cap \{z = 0, \ y < 0\}\betaig) \cup \{q_0\}.
\end{multline*}
\item[$(8)$]
Intersection of the sphere $S(0)$ with a plane $\{z = \operatorname{const}\nolimits \neq 0 \}$ is a branch of a hyperbola $\{x^2-y^2 = 4 |z|, \ x > 0, z = \operatorname{const}\nolimits\}$, intersection with a plane $\{z = 0 \}$ is an angle $\{ x = |y|, z = 0\}$, intersection with a plane $\{y = k x \}$, $k \in (-1, 1)$, is a union of two half-parabolas $\{4z = \pm(1-k^2)x^2, \ x \gammaeq 0, \ y = kx\}$, and intersection with a plane $\{y = \pm x \}$ is a ray $\{y = \pm x, \ z = 0\}$.
\end{itemize}
\end{theorem}
The Heisenberg beak $S(0) = \partial \mathcal{A}$ is plotted in Figs. \ref{fig:beak}--\ref{fig:beakz} as a graph of the function $x = \sqrt{y^2 + 4|z|}$ by virtue of \eq{Jq0}, and in Fig. \ref{fig:beak1} as a parametrized surface by virtue of \eq{S0par1} with $q_{\pm} = (2, 0, \pm 1)$.
\figout{
\onefiglabelsizen
{Heis_beak1}{The Heisenberg beak $\partial \mathcal{A}$}{fig:beak1}{8}
}
\betaegin{proof}
$(1)$, $(2)$
follow from item (2) of Th. \ref{th:dist} and item (3) of Sec. \ref{sec:groch}.
$(3)$ and $(6)$--$(8)$ are obvious.
$(4)$ follows from Th. \ref{th:optJ}.
$(5)$ follows from Th. \ref{th:sym}.
\end{proof}
Lightlike maximizers filling $S(0)$ are shown in Fig. \ref{fig:S0opt}.
Sub-Lo\-rent\-zi\-an spheres or radii 0, 1, 2, 3 are shown in Fig. \ref{fig:S0123}.
\figout{
\twofiglabelsizeh
{S0opt}{Lightlike maximizers filling $S(0)$}{fig:S0opt}{6}
{S0123}{Sub-Lo\-rent\-zi\-an spheres or radii 0, 1, 2, 3}{fig:S0123}{5}
}
\betaegin{remark}
The spheres
\betaegin{align*}
&S(1) = \left\{(x, y, z) \in M \mid x = \sqrt{y^2 + f(z)}, \ y, z \in {\mathbb R}\right\}, \\
&S(0) = \left\{(x, y, z) \in M \mid x = \sqrt{y^2 + 4|z|}, \ y, z \in {\mathbb R}\right\}
\end{align*}
tend one to another as $z \to \infty$ since for any $y \in {\mathbb R}$
$$
\lim_{z \to \infty} \left(\sqrt{y^2 + f(z)} - \sqrt{y^2 + 4|z|}\right) = 0
$$
by virtue of \eq{fas}. The same holds for any spheres $S(R_1)$, $S(R_2)$, $R_i \in [0, + \infty)$.
\end{remark}
\section{Conclusion}
The results obtained in this paper for the sub-Lo\-rent\-zi\-an problem on the Heisenberg group differ drastically from the known results for the sub-Riemannian problem on the same group:
\betaegin{enumerate}
\item
The sub-Lo\-rent\-zi\-an problem is not completely controllable.
\item
Filippov's existence theorem for optimal controls cannot be immediately applied to the sub-Lo\-rent\-zi\-an problem.
\item
In the sub-Lo\-rent\-zi\-an problem all extremal trajectories are infinitely optimal, thus the cut locus and the conjugate locus for them are empty.
\item
The sub-Lo\-rent\-zi\-an length maximizers coming to the zero-radius sphere are nonsmooth (concatenations of two smooth arcs forming a corner, nonstrictly normal extremal trajectories).
\item
Sub-Lo\-rent\-zi\-an spheres and sub-Lo\-rent\-zi\-an distance are real-analytic if $d > 0$.
\end{enumerate}
It would be interesting to understand which of these properties persist for more general sub-Lo\-rent\-zi\-an problems (e.g., for left-invariant problems on Carnot groups).
\betaigskip
The authors thank A.A.Agrachev, L.V.Lokutsievskiy, and M. Grochowski for valuable discussions of the problem considered.
\alphaddcontentsline{toc}{section}{List of figures}
\listoffigures
\betaegin{thebibliography}{99}
\alphaddcontentsline{toc}{section}{References}
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distributions and variational problems. (Russian) In: {\em Itogi Nauki i
Tekhniki:
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{\betaf
16}, Dynamical Systems 7, Springer Verlag.)
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V.~Jurdjevic,
{\em Geometric Control Theory},
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R. Montgomery, {\em A tour of subriemannian geometries, their geodesics and applications}, Amer. Math. Soc., 2002.
\betaibitem{notes}
A. Agrachev, Yu. Sachkov, {\em Control theory from the geometric viewpoint},
Berlin Heidelberg New York Tokyo. Springer-Verlag, 2004.
\betaibitem{ABB}
A. Agrachev, D. Barilari, U. Boscain, {\em A Comprehensive Introduction to
sub-Riemannian Geometry
from Hamiltonian viewpoint}, Cambridge University Press, 2019.
\betaibitem{intro}
Yu. Sachkov, {\em Introduction to geometric control}, Springer, 2022.
\betaibitem{UMN}
Yu. Sachkov, Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions, {\em Russian Math. Surveys}, 77:1 (2022), 99--163
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M. Grochowski,
Geodesics in the sub-Lorentzian geometry. {\em Bull. Polish.
Acad. Sci. Math.}, 50 (2002).
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M. Grochowski,
Normal forms of germs of contact sub-Lorentzian structures on
${\mathbb R}^3$. Differentiability of the sub-Lorentzian distance. {\em J. Dynam. Control
Systems} 9 (2003), No. 4.
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M. Grochowski,
Properties of reachable sets in the sub-Lorentzian geometry, {\em J. Geom. Phys.} 59(7) (2009) 885–900.
\betaibitem{groch11}
M. Grochowski, Reachable sets for contact sub-Lorentzian metrics on ${\mathbb R}^3$. Application to control affine systems with the scalar input, {\em J. Math. Sci.} (N.Y.) 177(3) (2011) 383–394.
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M. Grochowski,
On the Heisenberg sub-Lorentzian metric on ${\mathbb R}^3$,
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BANACH CENTER PUBLICATIONS,
INSTITUTE OF MATHEMATICS,
POLISH ACADEMY OF SCIENCES,
WARSZAWA,
vol. 65,
2004.
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M. Grochowski,
Reachable sets for the Heisenberg sub-Lorentzian structure on ${\mathbb R}^3$. An estimate for the distance function.
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vol. 12,
2006,
2, 145--160.
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D.-C. Chang, I. Markina and A. Vasil'ev, Sub-Lorentzian geometry on anti-de Sitter space,
{\em J. Math. Pures Appl.}, 90 (2008), 82--110.
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A. Korolko and I. Markina, Nonholonomic Lorentzian geometry on some H-type groups, {\em J.
Geom. Anal.}, 19 (2009), 864--889.
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E. Grong, A. Vasil’ev, Sub-Riemannian and sub-Lorentzian geometry on $SU(1, 1)$ and on its universal cover, {\em J. Geom. Mech.} 3(2) (2011) 225–260.
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M. Grochowski, A. Medvedev, B. Warhurst,
3-dimensional left-invariant sub-Lorentzian contact structures,
{\em Differential Geometry and its Applications}, 49 (2016) 142--166
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H. Abels, E.B. Vinberg, On free two-step nilpotent Lie semigroups and inequalities between random variables, {\em J. Lie Theory}, 29:1 (2019), 79--87
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V. G. Boltyanskii, R. V. Gamkrelidze, E.F. Mishchenko,
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E. Hakavuori, E. Le Donne, Non-minimality of corners in subriemannian geometry, {\em Invent. Math.}, 206(3): 693--704, 2016.
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L.V. Lokutsievskiy, A.V. Podobryaev, Existence of length maximizers in sub-Lorentzian problems on nilpotent Lie groups, {\em in preparation}.
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\end{thebibliography}
\end{document} | math | 64,518 |
\begin{document}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newcommand{\forces}{\ \rule{.12mm}{2.9mm}\!\vdash}
\newcommand{\dot{\varepsilon}}{\dot{\varepsilon}}
\newcommand{
}{
pace{12pt}}
\newcommand{{\mathcal A}}{{\mathcal A}}
\newcommand{{\mathcal B}}{{\mathcal B}}
\newcommand{{\mathbb P}}{{\mathbb P}}
\newcommand{{\mathbb P} \otimes \dot{\mathbb Q}}{{\mathbb P} \otimes \dot{\mathbb Q}}
\newcommand{{\mathbb Q}}{{\mathbb Q}}
\newcommand{{\mathbb R}}{{\mathbb R}}
\newcommand{{\mathbb P}S}{{\mathbb S}}
\newcommand{{\mathbb Z}}{{\mathbb Z}}
\newcommand{\textbf{C}}{\textbf{C}}
\newcommand{{\mathcal A}ut}{\mathrm{Aut}}
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\newcommand{\mathrm{dom}}{\mathrm{dom}}
\newcommand{\mathcal{N}}{\mathcal{N}}
\newcommand{\textbf{\noindent Proof}}{\textbf{\noindent Proof}}
\title{Naturality and definability II}
\author{Wilfrid Hodges and Saharon Shelah}
\date{Draft, 13 November 2000}
\maketitle
In two papers \cite{ho:1} and \cite{hosh:1} we noted that in common practice
many algebraic constructions are defined only `up to isomorphism' rather
than explicitly. We mentioned some questions raised by this fact, and we
gave some partial answers. The present paper provides much fuller answers,
though some questions remain open. Our main result, Theorem~\ref{th:4},
says that there is a transitive model of Zermelo-Fraenkel set theory with
choice (ZFC) in which every explicitly definable construction is `weakly
natural' (a weakening of the notion of a natural transformation). A
corollary is that there are models of ZFC in which some well-known
constructions, such as algebraic closure of fields, are not explicitly
definable. We also show (Theorem~\ref{th:2}) that there is no transitive
model of ZFC in which the explicitly definable constructions are precisely
the natural ones.
Most of this work was done when the second author visited the first at Queen
Mary, London University under SERC Visiting Fellowship grant GR/E9/639 in
summer 1989, and later when the two authors took part in the Mathematical
Logic year at the Mittag-Leffler Institute in Djursholm in September 2000.
The second author proposed the approach of section \ref{se:3} on the first
occasion and the idea behind the proof of Theorem \ref{th:4} on the second.
Between 1975 and 2000 the authors (separately or together) had given some
six or seven false proofs of versions of Theorem \ref{th:4} or its negation.
The authors thank Ian Hodkinson for his invaluable help (while research
assistant to Hodges under SERC grant GR/D/33298) in unpicking some of the
earlier false proofs.
\section{Constructions up to isomorphism} \label{se:1}
To make this paper self-contained, we repeat or paraphrase some definitions
from \cite{hosh:1}.
Let $M$ be a transitive model of ZFC (Zermelo-Fraenkel set
theory with choice). By a \emph{construction} (in $M$) we mean a triple
${\bf C}=\langle \phi_1,\phi_2,\phi_3\rangle$ where
\begin{enumerate}
\item $\phi_1(x)$, $\phi_2(x)$ and $\phi_3(x)$ are formulas of
the language of set theory, possibly with parameters from $M$;
\item $\phi_1$ and $\phi_2$ respectively define first-order
languages $L$ and $L^-$ in $M$; every symbol of $L^-$ is a symbol
of $L$, and the symbols of $L \setminus L^-$ include a 1-ary relation
symbol $P$;
\item the class $\{a : M \models \phi_3(a)\}$ is in $M$ a class of
$L$-structures, called the \emph{graph} of \textbf{C};
\item if $B$ is in the graph of \textbf{C} then $P^B$, the set of
elements of $B$ satisfying $Px$, forms the domain of an $L^-$-structure
$B^-$ inside $B$; the class of all such $B^-$ as $A$ ranges over
the graph of \textbf{C} is called the \emph{domain} of \textbf{C};
\item the domain of \textbf{C} is closed under isomorphism; and
if $A, B$ are in the graph of \textbf{C} then every isomorphism
from $A^-$ onto $B^-$ extends to an isomorphism from $A$ onto $B$.
\end{enumerate}
A typical example is the construction whose domain is the class of
fields, and the structures $B$ in the graph are the algebraic closures of
$B^-$, with $B^-$ picked out by the relation symbol $P$. The algebraic
closure of a field is determined only up to isomorphism over
the field; in the terminology below, algebraic closures are
`representable' but not `uniformisable'. (What we called
definable in \cite{hosh:1} we now call \emph{uniformalisable};
the new term is longer, but it is less misleading because it
agrees better with the common mathematical use of these words.)
We say that the construction $\textbf{C}$ is $X$-\emph{representable} (in $M$) if
$X$ is a set in $M$ and all the parameters of $\phi_1$, $\phi_2$, $\phi_3$
lie in $X$. We say that $\textbf{C}$ is \emph{small} if the domain of $\textbf{C}$
(and hence also its graph) contains only a set of isomorphism types of
structures.
An important special case is where the domain of $\textbf{C}$ contains exactly
one isomorphism type of structure; in this case we say $\textbf{C}$ is
\emph{unitype}.
The map $B^- \mapsto B$ on the domain of a construction \textbf{C} is in
general not single-valued; but by clause (5) it is single-valued up to
isomorphism over $B^-$. We shall say that \textbf{C} is
\emph{uniformisable} if its graph can be uniformised, i.e.\ there is a
formula $\phi_4(x,y)$ of set theory (the \emph{uniformising formula}) such
that
\begin{quote}
for each $A$ in the domain of \textbf{C} there is a unique $B$ such
that $M \models \phi_4(A,B)$, and this $B$ is an $L$-structure in
the graph of \textbf{C} with $A = B^-$.
\end{quote}
We say that \textbf{C} is $X$-\emph{uniformisable} (in $M$) if there is such
a $\phi_4$ whose parameters lie in the set $X$.
\section{Splitting, naturality and weak naturality} \label{se:2}
Let $\nu : G \to H$ be a surjective group homomorphism. A \emph{splitting}
of $\nu$ is a group homomorphism $s: H \to G$ such that $\nu s$ is the
identity on $H$. We say that $\nu$ \emph{splits} if it has a splitting.
For our Theorem \ref{th:4} we shall need a weakening of these notions. A
stronger version of Theorem \ref{th:4} would make this unnecessary, but we
do not know whether the stronger version is true.
Let $\nu: G \to H$ be as above. By a \emph{weak splitting} of $\nu$ we mean
a map $s: H \to G$ such that
\begin{enumerate}
\item[(a)] $\nu s$ is the identity on $H$;
\item[(b)] there is a commutative subgroup $G_0$ of $G$
such that if $f_1$, \ldots, $f_k$ are elements of $H$ for which
$f_1^{\varepsilon_1} \ldots f_k^{\varepsilon_k} = 1$ (where
$\varepsilon_i$ is each either 1 or $-1$),
then $s(f_1)^{\varepsilon_1} \ldots s(f_k)^{\varepsilon_k} \in G_0$.
\end{enumerate}
If we strengthened this definition by requiring $G_0$ to be $\{1\}$, it
would say exactly that $s$ is a splitting of $\nu$. In particular every
splitting is a weak splitting. We say that $\nu$ \emph{weakly splits} if it
has a weak splitting.
Suppose $s$ is a weak splitting of $\nu$. Then there is a smallest group
$G_0$ as in (b); it is the group consisting of the words
$s(f_1)^{\varepsilon_1} \ldots s(f_k)^{\varepsilon_k}$ as in (b). This
group $G_0$ has the property that if $g$ is in $G_0$ and $f$ is in $H$ then
$s(f)^{-1}gs(f)$ is also in $G_0$. So the normaliser of $G_0$ in $G$
contains the image of $s$.
\textbf{Example 1}. Let $G$ be the multiplicative group of $3\times 3$ upper
unitriangular matrices over the ring ${\mathbb Z} /(8{\mathbb Z} )$. Let $H$ be the
corresponding group over ${\mathbb Z} /(2{\mathbb Z} )$, and let $\nu: G \to H$ be the
canonical surjection. We show that $\nu$ doesn't weakly split.
Suppose for contradiction that $s$ is a weak splitting of $\nu$. Let
$g_1, g_2$ be the two matrices
\[
g_1 = \left( \begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array} \right), \ \
g_2 = \left( \begin{array}{cccc}
1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1 \end{array} \right)
\]
in $G$, and write $f_1 = \nu(g_1)$, $f_2 = \nu(g_2)$. Now $f_1^2 = f_2^2 =
1$ in $H$, so the weak splitting property tells us that $s(f_1)^2$ and
$s(f_2)^2$ commute in $G$. But it is easily checked (using the fact that
all entries of $s(f_i) - f_i$ are divisible by $2$) that $s(f_1)^2$ and
$s(f_2)^2$ don't commute.
\textbf{Example 2}. Let $m$ and $n$ be positive integers with $n \geqslant
3$, and let $p$ be a prime with $p^m > 3$. Let $G$ (resp.\ $H$) be the
multiplicative group of invertible $n \times n$ matrices over the ring ${\mathbb Z}
/(p^{3m}{\mathbb Z})$ (resp.\ ${\mathbb Z} /(p^{m}{\mathbb Z})$), and let $\nu: G \to H$ be the
canonical surjection. We write $I$ for the identity element in $G$ and in
$H$. The kernel of $\nu$ is the group of matrices of the form $I + p^mf$
where $f$ is in $G$. For any $i,j$ with $1 \leqslant i < j \leqslant n$ let
$\delta_{ij}$ be the $n \times n$ matrix which has $1$ in the $ij$-th place
and $0$ elsewhere; then $I + \delta_{ij}$ is an element of $G$ and $\nu(I +
\delta_{ij})$ has order $p^m$. The liftings of $\nu(I + \delta_{ij})$ to
$G$ are the matrices of the form $I + \delta_{ij} + p^mf$ with $f$ in $G$.
Now we repeat a calculation from Evans, Hodges and Hodkinson \cite{evhoho:1}
Prop.\ 3.7. The element $(I + \delta_{ij} + p^mf)^{p^m}$ is
\[ I + \binom{p^m}{1}(\delta_{ij} + p^mf) + \binom{p^m}{2}(\delta_{ij} +
p^mf)^2 + \binom{p^m}{3}(\delta_{ij} + p^mf)^3 + \ldots \]
Since $\delta_{ij}\delta_{ij} = 0$, $p^{3m}x = 0$ in ${\mathbb Z} /(p^{3m}{\mathbb Z} )$ and
$p^m > 3$, this multiplies out to
\[ I + p^m\delta_{ij} + p^{2m}f + \frac{p^{2m}(p^{m}-1)}{2}(\delta_{ij}f
+ f\delta_{ij}) +
\frac{p^{2m}(p^{m}-1)(p^{m}-2)}{6}\delta_{ij}f\delta_{ij}. \]
To apply these calculations to a concrete example, take
\[ g_1 = I + \delta_{12}, \ \ g_2 = I + \delta_{23} \]
in $H$, and let $s$ be a weak splitting of $\nu$. Then
\[ s(g_1) = I + \delta_{12} + p^mf_1, \ \ s(g_2) = I + \delta_{23} +
p^mf_2 \]
for some $f_1$, $f_2$ in $G$. Since $s$ is a weak splitting,
\[ s(g_1)^{p^m}s(g_2)^{p^m} = s(g_2)^{p^m}s(g_1)^{p^m}.\]
But our calculations show at once that
\[ s(g_1)^{p^m}s(g_2)^{p^m} - s(g_2)^{p^m}s(g_1)^{p^m} =
p^{2m}\delta_{13} \neq 0. \]
This contradiction proves that $\nu$ doesn't weakly split.
Now suppose $\textbf{C}$ is a construction in the model $M$, and $B$
is a structure in the graph of \textbf{C}. Let $A$ be $B^-$. We
write ${\mathcal A}ut(A)$ and ${\mathcal A}ut(B)$ for the automorphism groups of $A$ and
$B$ respectively. By (4) in the definition of constructions, each
automorphism $g$ of $B$ restricts to an automorphism $\nu_B(g)$ of $A$.
This map $\nu_B: {\mathcal A}ut(B) \to {\mathcal A}ut(A)$ is clearly a homomorphism; by
(5) in the definition of constructions it is surjective.
We say that $B$ is \emph{(weakly) natural over} $A$ if the map $\nu_G$
(weakly) splits. We say that the construction $\textbf{C}$ is \emph{(weakly)
natural} if for every $B$ in the graph of $\textbf{C}$, $\nu_B$ (weakly) splits.
(Our paper \cite{hosh:1} explained how this terminology connects with the
notion of a natural transformation. In a related context Harvey Friedman
\cite{fr:1} used the term `naturalness' in a weaker sense.)
\textbf{Example 3}. Let $G$ and $H$ be as in Example 1. Since $n \times n$
upper triangular matrix groups are nilpotent of class $n-1$, $G$ is a finite
soluble group. So by Shafarevich \cite{sh:1} there is a Galois extension $K$
of the field $\mathbb Q$ of rationals such that $G$ is the Galois group of
$K/{\mathbb Q}$. Let $k$ be the fixed field of the kernel $G_0$ of $\nu: G
\to H$. Then $H$ is the Galois group of the extension $k/{\mathbb Q}$. One
can write a set-theoretic description of these fields---up to
isomorphism---as a construction \textbf{C} where $K$ is in the graph and $k$
is picked out within $K$ by the relation symbol $P$. This construction
\textbf{C} is small (in fact unitype) and $\emptyset$-representable in any
model of set theory, and it is not weakly natural.
\textbf{Example 4}. Let $G$ and $H$ be as in Example 2. Let $B$ (resp.\
$A$) be the direct sum of $n$ copies of the abelian group ${\mathbb Z} /(p^{3m}{\mathbb Z}
)$ (resp.\ ${\mathbb Z} /(p^m{\mathbb Z} )$), and identify $A$ with $p^{2m}B$. Let the
relation symbol $P$ pick out $A$ within $B$. Then $G$ (resp.\ $H$) is the
automorphism group of $B$ (resp.\ $A$), and $\nu: G \to H$ is the map
induced by restriction. By the result of Example 2, the construction of
$B$ over $A$, which is again unitype and $\emptyset$-representable in any
model of set theory, is not weakly natural.
In \cite{hosh:1} we conjectured that there are models of set theory
in which each representable construction is uniformisable if and only
if it is natural. See section \ref{se:7} below for some of the
background to this. Section \ref{se:3} will show that no reasonable
version of this conjecture is true. Sections \ref{se:4}--\ref{se:6}
will show that there are models in which uniformisability implies
weak naturality. Section \ref{se:7} solves some of the problems
raised in \cite{ho:1} and \cite{hosh:1}.
\section{Uniformisability} \label{se:3}
A structure $B$ is said to be \emph{rigid} if it has no nontrivial
automorphisms. We shall say that a construction $\textbf{C}$ is
\emph{rigid-based} if for every structure $B$ in the graph of $\textbf{C}$, $B^-$
has no nontrivial automorphisms. A rigid-based construction is trivially
natural.
Let $M$ be a transitive model of set theory. We shall use a device that
takes any construction $\textbf{C}$ in $M$ to a construction $\textbf{C}^r$, called
its \emph{rigidification}. Each structure $B^-$ in the domain is replaced
by a two-part structure $B^{r-}$, where the first part is $B^-$ and the
second part consists of the transitive closure of the set $P^B$ with a
membership relation $\varepsilon$ copying that in $M$. Now $B^r$ is defined
to be the amalgam of $B$ and $B^{r-}$, so that $B^{r-}$ is $(B^r)^-$. Then
$\textbf{C}^r$ is the closure of the class
\[ \{ B^r : B \textrm{ in the graph of } \textbf{C} \} \]
under isomorphism in $M$. It is clear that $\textbf{C}^r$ and the map
$B \mapsto B^r$ are definable in $M$ using no parameters beyond
those in the formulas representing $\textbf{C}$.
\begin{lemma} \label{le:1}
If $\textbf{C}$ is any construction, then $\textbf{C}^r$ is rigid-based, natural and
not small.
\end{lemma}
\textbf{Proof}. If $B^-$ is in the domain of $\textbf{C}$, then $B^{r-}$ is
rigid because its set of elements is transitively closed; so $\textbf{C}^r$ is
rigid-based. Naturality follows at once. Since the domain of $\textbf{C}$ is
closed under isomorphism, the relevant transitive closures are arbitrarily
large. ${\mathcal B}ox$
\begin{theorem} \label{th:2}
There is no transitive model of ZFC in which both the following
are true:
\begin{enumerate}
\item[(a)] Every rigid-based construction in $M$ is uniformisable.
\item[(b)] Every unitype uniformisable construction in $M$ is weakly natural.
\end{enumerate}
In particular there is no transitive model of ZFC in which the
natural constructions are exactly the uniformisable ones.
\end{theorem}
\textbf{Proof}. Suppose $M$ is a counterexample. Let $\textbf{C}$ be
a unitype non-weakly-natural construction in $M$, such as Example
2 in section \ref{se:2}. Then $\textbf{C}^r$ is rigid-based
and hence uniformisable by assumption.
But we can use the uniformising formula of
$\textbf{C}^r$ to uniformise $\textbf{C}$ with the same parameters. So
by the assumption on $M$ again, $\textbf{C}$ is weakly natural;
contradiction. ${\mathcal B}ox$
The next result gives some finer information about small constructions.
\begin{theorem} \label{th:3}
Let $M$ be a transitive model of ZFC, $Y$ a set in $M$ and $\bar{c}$ a
well-ordering of $Y$ in $M$. Assume:
\begin{quote}
In $M$, if $X$ is any set, then every unitype $X$-representable
rigid-based construction is $X \cup Y$-uniformisable.
\end{quote}
Then
\begin{quote}
In $M$, every small $\emptyset$-representable construction is
$\{\bar{c}\}$-uniformisable,
\end{quote}
and hence there are unitype $\{\bar{c}\}$-uniformisable constructions that
are not weakly natural.
\end{theorem}
\textbf{Proof}. Let $\gamma$ be the length of $\bar{c}$. Write $\bar{v}$
for the sequence of variables $(v_i : i < \gamma)$. In $M$ we can
well-order (definably, with no parameters) the class of pairs $\langle
j,\psi \rangle$ where $j$ is an ordinal and $\psi(x,y,z,\bar{v})$ is a
formula of set theory. We write $H_j$ for the set of sets hereditarily of
cardinality less than $\aleph_j$ in $M$.
Let $\textbf{C}$ be a small $\emptyset$-representable construction in $M$. Then
$\textbf{C}^r$ is an $\emptyset$-representable rigid-based construction. It is
not small; but if $B$ is any structure in the graph of $\textbf{C}$, let
$\textbf{C}_B$ be the construction got from $\textbf{C}^r$ by restricting the graph to
structures isomorphic to $B^r$. Then $\textbf{C}_B$ is a unitype and
$\{B\}$-representable rigid-based construction, so by assumption it is
$\{B\} \cup Y$-uniformisable, say by a formula $\psi_B(-,-,B,\bar{c})$ where
$B, \bar{c}$ are the parameters.
By the reflection principle in $M$ there is an ordinal $j$ such that
\begin{quote}
$M \models \exists C (C \in \textbf{C}_B \land C^- = B^{r-} \land
C$ is the unique set such that ``$H_j \models \psi_B(B^{r-},C,B,
\bar{c})$'').
\end{quote}
Hence in $M$ there is a first pair $\langle j_B, \psi_B \rangle$,
definable from $B$, such that
\begin{quote}
$M \models \exists C (C \in \textbf{C}_B \land C^- = B^{r-} \land C$ is
the unique set such that ``$H_{j_B} \models \psi_B(B^{r-},C,B,
\bar{c})$'').
\end{quote}
Since all of this is uniform in $B$, it follows that the construction
$\textbf{C}$ is $\{\bar{c}\}$-uniformisable in $M$ by the formula
\[ y = C|L \textrm{ where } H_{j_B} \models \psi_B(B^{r-},C,B,
\bar{c}). \]
The last clause of the theorem follows by choosing $\textbf{C}$ suitably,
for example as in Example 2 of section \ref{se:2}. ${\mathcal B}ox$
\section{The model} \label{se:4}
\begin{theorem} \label{th:4}
There is a transitive model $N$ of ZFC in which every
uniformisable construction is weakly natural.
\end{theorem}
The next three sections are devoted to proving this theorem. We use
forcing. The central idea is to consider a construction $\textbf{C}$ whose
parameters lie in the ground model, and introduce a highly homogeneous
generic structure $B^\star$ into the graph of $\textbf{C}$; by homogeneity
$B^\star$ must be highly symmetrical over $B^{\star -}$. Since the
parameters of a construction may lie anywhere in the set-theoretic universe,
we have to iterate this idea right up through the universe. So we need to
build $N$ by a proper class iteration.
Our forcing notation is mainly as in Jech~\cite{je:1}. Thus $p < q$ means
that $p$ carries more information than $q$. We write $\dot{x}$ for a
boolean name of the element $x$ of $N$, and $\check{x}$ for the canonical
name of an element $x$ of the ground model. If $y$ is a boolean name and
$G$ a generic set, we write $y[G]$ for the element named by $y$ in the
generic extension by $G$. Our notion of forcing is of the kind described in
Menas~\cite{me:2} as `backward Easton forcing', and we shall borrow some
technical results from Menas' paper.
We start from a countable transitive model $M$ of ZFC + GCH. In $M$,
$\Lambda$ is a definable continuous monotone increasing function from
ordinals to infinite cardinals, with the property that for any ordinal
$\alpha$, $\Lambda(\alpha + 1) > \Lambda(\alpha)^{+}$. Our notion of
forcing ${\mathbb R}_\infty$ will be defined by induction on the ordinals. We start
with a trivial ordering ${\mathbb R}_0$. At limit ordinals we take inverse limits.
For each ordinal $\alpha$ we shall define an ${\mathbb R}_\alpha$-name
$\dot{{\mathbb P}S}_\alpha$; then ${\mathbb R}_{\alpha + 1}$ will be ${\mathbb R}_\alpha \otimes
\dot{{\mathbb P}S}_\alpha$. To define this name, let $\lambda$ be an infinite
cardinal. We consider a notion of forcing, ${\mathbb P}_\lambda$. In
${\mathbb P}_\lambda$, conditions are partial maps $p: \lambda^+ \times \lambda^{+}
\times \lambda^+ \to 2$ with domain of cardinality $\leq \lambda$. Write
$TP(\lambda)$ for a set-theoretical term which defines the notion of forcing
${\mathbb P}_\lambda$. For each ordinal $\alpha$, we choose $\dot{{\mathbb P}S}_\alpha$
to be an ${\mathbb R}_\alpha$-name such that $|| \dot{{\mathbb P}S}_\alpha =
TP(\Lambda(\check{\alpha} + 1))||_{{\mathbb R}_\alpha} = 1$.
This defines a proper class notion of forcing, ${\mathbb R}_\infty =$
direct limit of $\langle{\mathbb R}_\alpha:\alpha\mbox{ ordinal}\rangle$.
\begin{lemma} \label{le:5}
For each ordinal $\alpha$, suppose $\Lambda(\alpha)$ is a cardinal
in $M^{{\mathbb R}_\alpha}$. Then with ${\mathbb R}_\alpha$-boolean value 1,
$\dot{{\mathbb P}S}_\alpha$ is $\Lambda(\alpha + 1)$-closed and satisfies
the $\Lambda(\alpha + 1)^{+}$-chain condition.
\end{lemma}
$\textbf{\noindent Proof}$. Straightforward. ${\mathcal B}ox$
\begin{lemma} \label{le:6}
\begin{enumerate}
\item[(a)] For every successor ordinal $\alpha$,
$\Lambda(\alpha)$ and $\Lambda(\alpha)^{+}$
are cardinals with ${\mathbb R}_\infty$-boolean value 1.
\item[(b)] For each successor ordinal $\alpha$, $|{\mathbb R}_{\alpha}| =
\Lambda (\alpha)^{+}$; for each limit ordinal $\delta$,
$|{\mathbb R}_\delta| \leq \Lambda (\delta)^{+}$.
\item[(c)] For each ordinal $\alpha$, ${\mathbb R}_{\alpha}$ satisfies
the $\Lambda (\alpha)^{+}$-chain condition.
\end{enumerate}
\end{lemma}
\textbf{\noindent Proof}. We prove all parts simultaneously by induction. Suppose $R_\alpha$
has cardinality $\leq \Lambda (\alpha)^{+}$ and satisfies the
$\Lambda(\alpha)^{+}$-chain condition. Put $\lambda = \Lambda (\alpha + 1)
> \Lambda(\alpha)^{+}$. All cardinals $\geq \lambda$ are cardinals with
$R_\alpha$-boolean value 1. Let $\dot{q}$ be an element of
$\dot{{\mathbb P}S}_\alpha$. Then $\dot{q}$ has cardinality $\leq \lambda$ with
$R_\alpha$-boolean value 1, and ${\mathbb R}_\alpha$ satisfies the $\lambda^+$-chain
condition, so with boolean value 1 the domain of $\dot{q}$ lies within some
$\gamma < \lambda^+$. Now $\dot{p}$ can be taken to be a map from the set
$\gamma$ to the regular open algebra $RO(R_\alpha)$, which has cardinality
$\leq (|{\mathbb R}_\alpha|^{+})^{|{\mathbb R}_\alpha|} = |{\mathbb R}_\alpha|$. The number of such
maps is at most $(|{\mathbb R}_\alpha|^{+})^\lambda = \lambda^+$. Therefore
$|{\mathbb R}_\alpha \otimes \dot{{\mathbb P}S}_\alpha| = \lambda^+$. Also $R_{\alpha + 1}$
satisfies the $\lambda^{+}$-chain condition by Lemma~\ref{le:5}, using a
standard argument on iterated forcing (Menas~\cite{me:2} Proposition 10(i)).
Now with boolean value 1, $\dot{{\mathbb P}S}_\alpha$ is $\lambda$-closed and
satisfies the $\lambda^{+}$-chain condition, so cardinals are preserved in
passing from ${\mathbb R}_\alpha$ to ${\mathbb R}_{\alpha + 1}$.
We turn to limit ordinals $\delta$. The cardinality of ${\mathbb R}_\delta$ is at
most the product of the cardinalities of ${\mathbb R}_\alpha$ with $\alpha < \delta$,
hence at most $\Lambda(\delta)^+$. It follows at once that ${\mathbb R}_\delta$
satisfies the $\Lambda(\delta)^{+}$-chain condition and preserves all
cardinals from $\Lambda(\delta)^{+}$ upwards.
It remains to show that for successor ordinals $\alpha + 1$, the cardinals
$\Lambda(\alpha + 1)$ and $\Lambda(\alpha + 1)^{+}$ are not collapsed by
${\mathbb R}_\infty$. Using the next lemma (which doesn't depend on the clause we
are now proving), ${\mathbb R}_\infty$ can be written as ${\mathbb R}_\alpha \otimes
\dot{{\mathbb P}S}_{\alpha + 1} \otimes \dot{{\mathbb R}}_{\alpha + 1,\infty}$. The first
factor satisfies the $\Lambda(\alpha + 1)$-chain condition and hence
preserves these two cardinals. The third factor preserves them since it is
$\Lambda(\alpha + 1)^{+}$-closed with boolean value 1. The middle factor is
$\Lambda(\alpha + 1)$-closed with boolean value 1, so that it preserves
$\Lambda(\alpha + 1)$ and $\Lambda(\alpha + 1)^{+}$. ${\mathcal B}ox$
\begin{lemma} \label{le:7}
For each ordinal $\alpha$ there is a proper class notion of
forcing $\dot{{\mathbb R}}_{\alpha,\infty}$ such that
\begin{enumerate}
\item[(a)] ${\mathbb R}_\infty$ is isomorphic to
${\mathbb R}_\alpha \otimes \dot{{\mathbb R}}_{\alpha,\infty}$;
\item[(b)] In $M^{{\mathbb R}_\alpha}$, $\dot{{\mathbb R}}_{\alpha,\infty}$ is the direct
limit of iterated notions of forcing $\dot{{\mathbb R}}_{\alpha,\beta}$
in such a way that for each $\beta > \alpha$, ${\mathbb R}_\beta$ is
isomorphic to ${\mathbb R}_\alpha \otimes \dot{{\mathbb R}}_{\alpha,\beta}$;
\item[(c)] For each successor ordinal $\alpha$,
$\dot{{\mathbb R}}_{\alpha,\infty}$
is $\Lambda(\alpha + 1)$-closed with $M^{{\mathbb R}_\alpha}$-boolean
value 1.
\end{enumerate}
\end{lemma}
\textbf{\noindent Proof}. As Menas~\cite{me:2} Propositions 11 and 10(i),
using the previous lemma. ${\mathcal B}ox$
The model $N$ for the theorem will be an $R_\infty$-generic
extension of $M$.
\begin{lemma} \label{le:8}
$N$ is a model of ZFC.
\end{lemma}
\textbf{\noindent Proof}. Menas~\cite{me:2} Proposition 14 derives this from the
previous lemma. ${\mathcal B}ox$
\begin{lemma} \label{le:9}
If $\dot{x}$ is an ${\mathbb R}_\infty$-name of a subset of $\Lambda(\alpha + 1)$,
and $r \in {\mathbb R}_\infty$, then there are $s \in {\mathbb R}_\infty$ and an
${\mathbb R}_\alpha$-name $\dot{y}$ such that $s \leq r$ and $s \forces_{{\mathbb R}_\infty}$
``$\dot{x} = \dot{y}$''.
\end{lemma}
\textbf{\noindent Proof}. This follows from the fact that $\dot{R}_{\alpha, \infty}$
is $\Lambda(\alpha + 1)$-closed. ${\mathcal B}ox$
\begin{lemma} \label{le:10}
Let $\alpha$ be an ordinal. Then if $\lambda$ is $\Lambda(\alpha + 1)$ or
$\Lambda(\alpha + 1)^{+}$, we have $2^\lambda = \lambda^+$ in $N$.
\end{lemma}
$\textbf{\noindent Proof}$. Suppose $\lambda = \Lambda(\alpha + 1)$. Then $2^\lambda \leq
\mu$ with $R_\alpha$-boolean value 1, where $$\mu = |RO({\mathbb R}_\alpha)|^\lambda
\leq (|{\mathbb R}_\alpha|^{\Lambda(\alpha)^{+}})^\lambda = \lambda^+.$$ With
${\mathbb R}_\alpha$-boolean value 1, ${\mathbb R}_{\alpha,\infty}$ is $\lambda$-closed and
hence adds no new subsets of $\lambda$. Similar calculations apply to the
other cases. ${\mathcal B}ox$
A notion of forcing ${\mathbb R}$ is said to be {\em homogeneous} if for any two
elements $p, q$ of ${\mathbb R}$ there is an automorphism $\sigma$ of ${\mathbb R}$ such that
$\sigma (p)$ and $q$ are compatible.
\begin{lemma} \label{le:11}
The notion of forcing ${\mathbb R}_\infty$ is homogeneous.
\end{lemma}
\textbf{\noindent Proof}. Menas~\cite{me:2} Proposition 13 proves this under the assumption
that each step of the iteration is homogeneous with boolean value 1. That
assumption holds here. ${\mathcal B}ox$
If $\alpha$ is an automorphism of the notion of forcing ${\mathbb R}$, then $\alpha$
induces an automorphism $\alpha^\star$ of the boolean universe $M^P$. Also
$\alpha$ takes any ${\mathbb R}$-generic set $G$ over $M$ to the ${\mathbb R}$-generic set
$\alpha G$.
\begin{lemma} \label{le:12}
For every element $\dot{x}$ of $M^{\mathbb R}$ we have
\[\dot{x}[G] = (\alpha^\star)\dot{x}[\alpha G]\]
\end{lemma}
\textbf{\noindent Proof}. Immediate. ${\mathcal B}ox$
To save notation we write $\alpha^\star$ as $\alpha$. We note that
$(\alpha\beta)^\star = \alpha^\star\beta^\star$, which removes one possible
source of ambiguity.
\section{The generic copies of $A$, $B$} \label{se:5}
As explained earlier, our model $N$ in the theorem will be $M[G]$ where $G$
is an ${\mathbb R}_\infty$-generic class over $M$. Henceforth $\textbf{C}$ is a
construction which is uniformisable in $N$ with uniformising formula $\phi$;
we want to show that $\textbf{C}$ is weakly natural. Let $B$ be any structure in
the graph of $\textbf{C}$. At the cost of adding $B$ as a parameter, we can
assume without loss that $\textbf{C}$ is unitype and its graph consists of
structures isomorphic to $B$. We put $A = B^{-}$. Choose an ordinal
$\alpha$ so that $A, B$ and the parameters of the formulas representing
$\textbf{C}$ all lie in $M[G \cap {\mathbb R}_\alpha]$, and $B, \mbox{Aut}(B)$ both have
cardinality $\leq \Lambda(\alpha + 1)$. We can decompose $N$ as a two-stage
extension $M[G_\alpha][G_{\alpha,\infty}]$, where $G_\alpha = G \cap
{\mathbb R}_\alpha$ and $G_{\alpha,\infty}$ is $\dot{{\mathbb R}}_{\alpha,\infty}[
G_\alpha]$-generic over $M[G_\alpha]$.
At this point we adjust our notation. We put $\lambda = \Lambda(\alpha +
1)$, and we rename $M[G_\alpha]$ as $M$. By Lemma \ref{le:6}, $\lambda$ and
$\lambda^+$ in the old $M$ are still cardinals in the new $M$. By
Lemma~\ref{le:6}, $N$ is constructed from the new $M$ by an iterated forcing
notion ${\mathbb R}_{\alpha,\infty} = \dot{{\mathbb R}}_{\alpha,\infty}[G_\alpha]$ with the
same properties as the forcing notion ${\mathbb R}_\infty$, with two differences.
First, the function $\Lambda$ must now be replaced by the function
$\Lambda_\alpha$ where $\Lambda_\alpha(\beta) = \Lambda(\alpha + \beta)$.
Second, $M$ need not satisfy the GCH everywhere; but this never matters.
(In fact it would be possible to make the GCH hold in the new $M$ and in
$N$, by adding extra factors in ${\mathbb R}_\infty$ to collapse the cardinalities of
power sets.) One can check that all the preliminary lemmas \ref{le:5} to
\ref{le:11} still hold for this notion of forcing ${\mathbb R}_{\alpha,\infty}$.
We now write ${\mathbb P}, \dot{{\mathbb Q}}$ for
$\dot{{\mathbb P}S}_\alpha[G_\alpha],
\dot{{\mathbb R}}_{\alpha + 1,\infty}[G_\alpha]$ respectively. Thus
\[{\mathbb R}_{\alpha,\infty} = {\mathbb P} \otimes \dot{\mathbb Q}. \]
We shall not need to refer to $G_\alpha$ again, and so we start afresh with
our notation for generic sets.
We shall write $N$ as $M[G_1][G_2]$ where $G_1$ is ${\mathbb P}$-generic over $M$
and $G_2$ is $\dot{{\mathbb Q}}[G_1]$-generic over $M[G_1]$.
We shall write $G$ for the ${\mathbb P} \otimes \dot{\mathbb Q}$-generic set $G_1 \otimes \dot{G}_2$ over
$M$, so that $N = M[G]$. If $x$ is an element of $N$, we write $\dot{x}$
for a boolean name for $x$ in the forcing language for ${\mathbb P} \otimes \dot{\mathbb Q}$. Note that
every ${\mathbb P}$-name over $M$ can be read as a ${\mathbb P} \otimes \dot{\mathbb Q}$-name too, so that there
is no need for a separate symbol for ${\mathbb P}$-names.
The set $\bigcup G_1$ is a total map from $\lambda^+ \times \lambda^{+}
\times \lambda^+$ to 2. For each $i < \lambda^+$ and $j < \lambda^{+}$, we
define $a_{ij} = \{ k < \lambda^+ : \bigcup G_1 (i,j,k) = 1 \}$ and $a'_i =
\{a_{ij} : j < \lambda^{+}\}$, so that $a'_i$ is a set of $\lambda^{+}$
independently generic subsets of $\lambda^+$. If $a$ and $b$ are (in
$M_1[G_2]$) sets of subsets of $\lambda^+$, we put $a \equiv b$ iff the
symmetric difference of $a$ and $b$ has cardinality $\leq \lambda$. We
write $a_i$ for the equivalence class $(a'_i)^\equiv$. The boolean names
$\dot{a}_{ij}, \dot{a}_i$ can be chosen in $M^{{\mathbb P}}$ independently of the
choice of $G$.
Consider again the structures $A$ and $B$ in $M$. Without loss we can
suppose that dom$(A)$ is an initial segment of $\lambda$. In $M[G_1]$ there
is a map $e$ which takes each element $i$ of $A$ to the corresponding set
$a_i = \dot{a}_i[G_1]$; by means of $e$ we can define a copy $A^*$ of $A$
whose elements are the sets $a_i$ ($i \in \mbox{dom}(A)$). Again the
boolean names $\dot{A}^*, \dot{e}$ can be chosen to be independent of the
choice of $G$.
Since $A, B$ and the parameters of the uniformising formula $\phi$ lie in
$M$, and the notion of forcing ${\mathbb P} \otimes \dot{\mathbb Q}$ is homogeneous by Lemma~\ref{le:11},
the statement ``$\phi$ defines a construction on the class of structures
isomorphic to $A$, which takes $A$ to $B$'' is true in $N$ independently of
the choice of $G$. In particular there are ${\mathbb P} \otimes \dot{\mathbb Q}$-boolean names
$\dot{B}^*, \dot{\varepsilon}$ such that
\begin{eqnarray} \label{eq:2}
& & || \dot{B} \mbox{ is the unique structure such that
} \phi(\dot{A}^*, \dot{B}^*) \mbox{ holds},\\
& & \dot{e} : \check{A} \to \dot{A}^* \mbox{ is
the isomorphism such that }
\dot{e}(\check{\imath}) = \dot{a}_i \mbox{ for}
\nonumber \\
& & \mbox{each } i \in \mbox{dom}(\check{A}),
\mbox{ and }
\dot{\varepsilon} : \check{B} \to \dot{B}^* \mbox{ is
an isomorphism}
\nonumber \\
& & \mbox{which extends }
\dot{e}||_{{\mathbb P} \otimes \dot{\mathbb Q}} = 1. \nonumber
\end{eqnarray}
\begin{lemma} \label{le:13}
Let $G$ be ${\mathbb P} \otimes \dot{\mathbb Q}$-generic over $M_1$. Then:
(a) {\rm Aut}$(A)^M$ =
{\rm Aut}$(A)^{M[G_1]}$ = {\rm Aut}$(A)^{M[G]}$.
(b) {\rm Aut}$(B)^M$ =
{\rm Aut}$(B)^{M[G_1]}$ = {\rm Aut}$(B)^{M[G]}$.
(c) The set of maps from {\rm Aut}$(A)$ to {\rm Aut}$(B)$ is the same
in $M$ as it is in $M[G_1]$ and $M[G]$.
\end{lemma}
\textbf{\noindent Proof}. Using Lemma~\ref{le:5} and Lemma~\ref{le:6}(c), ${\mathbb P}$ is
$\lambda$-closed over $M$, and $\dot{{\mathbb Q}}$ is $\lambda$-closed over $M[G_1]$.
Hence no new permutations of $A$ or $B$ are added since $|A| \leq |B| \leq
\lambda$ in $M$; this proves (a), (b). Likewise (c) holds since
$|\mbox{Aut}(A)| \leq |\mbox{Aut}(B)| \leq \lambda$ in $M$. ${\mathcal B}ox$
We regard ${\mathcal A}ut (A)$ as a permutation group on $\lambda^+$ by letting it fix
all the elements of $\lambda^+$ which are not in dom$(A)$.
By a {\em neat map} we mean a map $\alpha : \lambda^{+} \to {\mathcal A}ut (A)$ in $M$
which is constant on a final segment of $\lambda^{+}$; we write $\mathcal{N}$ for
the set of neat maps. We write $\pi$ for the map from $\mathcal{N}$ to ${\mathcal A}ut (A)$
which takes each neat map to its eventual value. We write $\mathcal{N}^-$ for the
set of all neat maps $\alpha$ with $\pi(\alpha) = 1$. For each ordinal $i <
\lambda^{+}$ we write $\mathcal{N}_i$ for the set of neat maps $\alpha$ such that
$\alpha(j) = 1$ for all $j < i$. We write $\mathcal{N}^-_i$ for $\mathcal{N}^- \cap
\mathcal{N}_i$.
We can regard $\alpha$ as a
permutation of the set $\lambda^+ \times \lambda^{+}
\times \lambda^+$ by putting
\[\alpha(i,j,k) = (\alpha(j)(i),j,k).\]
Then $\alpha$ induces an automorphism of ${\mathbb P}$.
\begin{lemma} \label{le:14}
If $\alpha$ and $\beta$ are distinct neat maps then they induce
distinct automorphisms of ${\mathbb P}$. Identifying each neat map with
the automorphism it induces, $\mathcal{N}$ forms a group with
subgroups $\mathcal{N}^-$, $\mathcal{N}_i$ $(i < \lambda^+)$;
the map $\pi : \mathcal{N} \to {\mathcal A}ut (A)$ is
a group homomorphism.
\end{lemma}
\textbf{\noindent Proof}. From the definitions. ${\mathcal B}ox$
The automorphism $\alpha$ can be extended to an automorphism of ${\mathbb P} \otimes \dot{\mathbb Q}$ in
many different ways, by induction on $\dot{{\mathbb R}}$ as an iterated notion of
forcing. Each factor of $\dot{{\mathbb R}}$ is with boolean value 1 the set of all
maps from $X$ to 2 of cardinality $\leq \mu$, where $X$ is $\mu^+ \times
\mu^{+} \times \mu^+$ for some cardinal $\mu$. If $\Sigma '$ is (in $M$)
the group of permutations of $X$, then an automorphism of the factor of
$\dot{{\mathbb R}}$ is determined by an element of $\Sigma '$ and a permutation of
the boolean values. For each ordinal $i$ let $\Sigma_i$ be in $M_1$ the
product of the permutation groups $\Sigma '$ for the first $i$ factors of
$\dot{{\mathbb Q}}$, and let $\Sigma$ be the direct limit of the $\Sigma_i$ in $M_i$.
Then an automorphism $\alpha$ of ${\mathbb P}$ and an element $\sigma$ of $\Sigma$
together determine an automorphism $\langle \alpha , \sigma \rangle$ of
${\mathbb P} \otimes \dot{\mathbb Q}$, and hence of $M^{{\mathbb P} \otimes \dot{\mathbb Q}}$.
\begin{lemma} \label{le:15}
The actions of the group $\cal N$ of neat maps and the group $\Sigma$ on
${\mathbb P} \otimes \dot{\mathbb Q}$ commute with each other.
\end{lemma}
\textbf{\noindent Proof}. The class ${\mathbb P} \otimes \dot{\mathbb Q}$ is ${\mathbb P} \times \hat{{\mathbb Q}}$ where $\hat{{\mathbb Q}}$ is a
class of boolean-valued subsets of a class $X$ which is definable in $M$;
the action of $\Sigma$ is through its action on $X$. Thus each element of
$\hat{{\mathbb Q}}$ is essentially a set of ordered pairs $\langle x , y \rangle$
where $x \in X$ and $y \in {\mathbb P}$. Since $\Sigma$ and $\cal N$ act
respectively on the first and second coordinates, the actions on $\hat{{\mathbb Q}}$
commute. The group $\Sigma$ keeps ${\mathbb P}$ fixed. ${\mathcal B}ox$
\begin{lemma} \label{le:16}
Suppose $\alpha : \lambda^{+} \to {\mathcal A}ut (A)$ is neat and $\alpha '$ is an
automorphism of ${\mathbb P} \otimes \dot{\mathbb Q}$ extending $\alpha$. Then the action of $\alpha '$ on
$M^{{\mathbb P} \otimes \dot{\mathbb Q}}$ setwise fixes the set $\{ \dot{a}_i : i \in {\rm dom}(A) \}$ of
canonical names of the elements of $\dot{A}^*[G]$, and it acts on this set
in the way induced by $\pi(\alpha)$ and the map $i \mapsto \dot{a}_i$. Thus
$\alpha '(\dot{a}_i) = \dot{a}_{\pi(\alpha)(i)}$.
\end{lemma}
\textbf{\noindent Proof}. Write out the names! (They lie in $M^{{\mathbb P}}$, so that the
extension from $M^{\mathbb P}$ to $M^{{\mathbb P} \otimes \dot{\mathbb Q}}$ is irrelevant.) ${\mathcal B}ox$
If $G$ is ${\mathbb P} \otimes \dot{\mathbb Q}$-generic over $M$, then so is $\langle \alpha, \sigma
\rangle G$ for every neat map $\alpha$ and every $\sigma \in \Sigma$, since
$\alpha, \sigma \in M$.
\begin{lemma} \label{le:17}
For each element $i$ of $A$, each neat map $\alpha$ and each $\sigma \in
\Sigma$, $\dot{a}_{\pi(\alpha)(i)}[\langle \alpha, \sigma \rangle G] =
\dot{a}_i[G]$. In particular $\dot{A}^*[\langle \alpha, \sigma \rangle G] =
\dot{A}^*[G]$.
\end{lemma}
$\textbf{\noindent Proof}$. By Lemma~\ref{le:16}, $\dot{a}_{\pi(\alpha)
(i)}[\langle \alpha, \sigma \rangle G] =
(\alpha \dot{a}_i)[\langle \alpha, \sigma \rangle G]$. Then by Lemma
\ref{le:12} and the fact that $\alpha \dot{a}_i$ lies in $M^{{\mathbb P}}$,
\[ (\alpha \dot{a}_i)[\langle \alpha, \sigma \rangle G] =
(\alpha \dot{a}_i)[\alpha G_1] = \dot{a}_i[G_1] = \dot{a}_i[G].
\]
${\mathcal B}ox$
We write $\dot{\varepsilon}^{-1}$ for a boolean name such that $\dot{\varepsilon}^{-1}[G] =
(\dot{\varepsilon}[G])^{-1}$ for all generic $G$.
\begin{lemma} \label{le:18}
Suppose $\alpha$ is a neat map, $\sigma \in \Sigma$ and $G$ is
${\mathbb P} \otimes \dot{\mathbb Q}$-generic over $M_1$. Then $\dot{B}^*[\langle \alpha, \sigma
\rangle^{-1} G]$ $= \dot{B}^*[G]$, and the map $(\dot{\varepsilon} ^{-1} \circ \langle
\alpha, \sigma \rangle \dot{\varepsilon})[G]$ is an automorphism of $B$ which extends
$\pi(\alpha)$.
\end{lemma}
\textbf{\noindent Proof}. Since $M[\langle \alpha, \sigma \rangle^{-1} G] = M_1[G]$ and
$\dot{A}^*[\langle \alpha, \sigma \rangle^{-1} G] = \dot{A}^*[G]$,
(\ref{eq:2}) (before Lemma \ref{le:13}) tells us that $\dot{e} [\langle
\alpha, \sigma \rangle^{-1} G] (i) = \dot{a}_i[\langle \alpha, \sigma
\rangle^{-1} G]$ for each $i \in \mbox{dom}(A)$, and that $\dot{B}^*[\langle
\alpha, \sigma \rangle^{-1} G] = \dot{B}^*[G]$ and $\dot{\varepsilon} [G]^{-1}\circ
\dot{\varepsilon} [\langle \alpha, \sigma \rangle^{-1} G]$ extends $\dot{e}[G]^{-1}\circ
\dot{e}[\langle \alpha, \sigma \rangle^{-1} G]$. Now using Lemma
\ref{le:17},
\[ \dot{e}[G]^{-1}\circ\dot{e}[\langle \alpha, \sigma \rangle^{-1} G](i)
= \dot{e}[G]^{-1}(\dot{a}_i[\langle \alpha, \sigma \rangle^{-1} G])\]
\[= \dot{e}[G]^{-1}(\dot{a}_{\pi(\alpha)(i)}[G]) = \pi(\alpha)(i).\]
${\mathcal B}ox$
\begin{lemma} \label{le:19}
For every neat map $\alpha$, each $\sigma \in \Sigma$ and all $\langle
p,\dot{q} \rangle \in {\mathbb P} \otimes \dot{\mathbb Q}$ there are $\langle p',\dot{q}' \rangle
\leqslant \langle p,\dot{q} \rangle $ and $g \in {\mathcal A}ut{B}$ such that
\[ \langle p',\dot{q}' \rangle \forces_{{\mathbb P} \otimes \dot{\mathbb Q}}
\sigma (\dot{\varepsilon}^{-1}) \circ \alpha\sigma (\dot{\varepsilon}) = \check{g}. \]
\end{lemma}
\textbf{\noindent Proof}. Let $f$ be $\pi(\alpha)$.
By Lemma \ref{le:18} we have
\[|| \sigma\dot{\varepsilon}^{-1} \circ \alpha\sigma \dot{\varepsilon} \textrm{ is an automorphism
of } B \textrm{ extending } \check{f}||_{{\mathbb P} \otimes \dot{\mathbb Q}} = 1.\]
Unpacking the existential quantifier in ``an automorphism of $B$''
gives the lemma. ${\mathcal B}ox$
Consider any $i < \lambda^{+}$. Given $\langle p,\dot{q} \rangle \in
{\mathbb P} \otimes \dot{\mathbb Q}$, define $t_{p,\dot{q},i}$ to be the set of all triples
$(f,g,\sigma)$, with $f \in {\mathcal A}ut (A)$, $g \in {\mathcal A}ut (B)$ and $\sigma \in
\Sigma$, such that for some $\alpha \in \mathcal{N}_i$, $\pi(\alpha) = f$ and
\[\langle p,\dot{q} \rangle \forces_{{\mathbb P} \otimes \dot{\mathbb Q}}\sigma(\dot{\varepsilon}^{-1}) \circ
\alpha\sigma(\dot{\varepsilon}) = \check{g}.\] Clearly if $\langle p',\dot{q}' \rangle
\leqslant \langle p,\dot{q} \rangle$ then $t_{p',\dot{q}',i} \supseteq
t_{p,\dot{q},i}$. Since there are only a set of values for $\sigma(\dot{\varepsilon})$
and $\sigma(\dot{\varepsilon}^{-1})$ with $\sigma \in \Sigma$. it follows that there is
$\langle p_i,\dot{q}_i \rangle$ such that for all $\langle p',\dot{q}'
\rangle \leqslant \langle p,\dot{q} \rangle$,
\[t_{p',\dot{q}',i} = t_{p_i,\dot{q}_i,i}.\]
We fix a choice of $p_i, \dot{q}_i$, and we write $t_i$ for the resulting
value $t_{p_i,\dot{q}_i,i}$. If $(f,g,\sigma)$ is in $t_i$, we write
$\alpha^i_{f,g,\sigma}$ for some $\alpha \in \mathcal{N}_i$ such that
\[\langle p_i,\dot{q}_i \rangle \forces_{{\mathbb P} \otimes \dot{\mathbb Q}} \sigma(\dot{\varepsilon}^{-1}) \circ
\alpha\sigma(\dot{\varepsilon}) = \check{g} \]
and $\pi(\alpha) = f$.
\begin{lemma} \label{le:20}
For each $i < \lambda^{+}$, $t_i$ is a subclass
of ${\mathcal A}ut (A) \times {\mathcal A}ut (B) \times \Sigma$ such that
\begin{enumerate}
\item[(a)] for each $(f,g,\sigma)$ in $t_i$, $g|A = f$;
\item[(b)] for each $f$ in ${\mathcal A}ut (A)$ and $\sigma$ in $\Sigma$
there is $g$ with $(f,g,\sigma)$ in $t_i$.
\end{enumerate}
(So $t_i(-,-,\sigma)$ is a first attempt at a lifting of the restriction map
from ${\mathcal A}ut (B)$ to ${\mathcal A}ut (A)$.)
\end{lemma}
\textbf{\noindent Proof}. By Lemma \ref{le:19}. ${\mathcal B}ox$
We write $t^-_{p,\dot{q},i}$ for the set of pairs $(g,\sigma)$ such that
$(1,g,\sigma)$ is in $t_{p,\dot{q},i}$. We write $\alpha^i_{g,\sigma}$ for
$\alpha^i_{1,g,\sigma}$; note that $\alpha^i_{g,\sigma}$ is in $\mathcal{N}^-_i$
by Lemma \ref{le:18}.
\begin{lemma} \label{le:21}
For each $i < \lambda^{+}$ there are $\sigma_i$ in $\Sigma$, a condition
$p'_i \leqslant p_i$ and a boolean name $\dot{v_i}$ such that
\begin{enumerate}
\item[(a)] for each $i < \lambda^{+}$, $p'_i \forces_{{\mathbb P}}
\mathrm{dom}(\sigma_i^{-1}\dot{q}_i) \subseteq \dot{v_i}$;
\item[(b)] for all $i < j < \lambda^{+}$, $||\dot{v}_i \cap
\dot{v}_j = \emptyset||_{{\mathbb P}} = 1$;
\item[(c)] for all $i < j < \lambda^{+}$,
$\sigma_i\sigma_j = \sigma_j\sigma_i$.
\end{enumerate}
\end{lemma}
\textbf{\noindent Proof}. By induction on $i < \lambda^{+}$. As we choose the $p'_i$,
$\sigma_i$ and $\dot{v_i}$, we also choose an eventually zero sequence of
ordinals $\gamma_{\mu,i} < \mu^+$ in $M_1$ so that
\[||\dot{v_i} \subseteq \prod_{\mu} (\gamma_{\mu,i} \times\mu^+
\times \mu^+)||_{{\mathbb P}} = 1.\]
Then when we have made our choices
for all $i < j$, we first extend $p_i$ to $p'_i$ forcing the domain of
$\dot{q_i}$ to lie within some set
\[X = \prod_{\mu < \mu'} (\gamma'_{\mu} \times \gamma'_{\mu}
\times \mu^{+}) \] lying in $M_1$, and we choose $\dot{w_i}$ to be a
canonical boolean name for this set $X$. Then we choose $\sigma_i$ so that
$\sigma_i^{-1}$ moves $X$ to
\[\prod_{\mu<\mu'}\left( \left[ \bigcup_{j<i}\gamma_{\mu,i},\bigcup_{j<i}
\gamma_{\mu,i}+\gamma'_{\mu} \right)\times \gamma'_{\mu} \times \mu^{+}
\right) , \]
(the product of products of three intervals), and we put $\dot{v_i} =
\sigma_i^{-1}\dot{w_i}$ and $\gamma_{\mu,j}=\bigcup_{j<i}\gamma_{\mu,i} +
\gamma'_{\mu}$. ${\mathcal B}ox$
We fix the choice of $\sigma_i$ and $\dot{v_i}$ $(i < \lambda^+)$ given by
this lemma. Without loss we extend the conditions $p_i$ to be equal to
$p'_i$.
\begin{lemma} \label{le:22}
There is a stationary subset $S$ of $\lambda^+$ such that:
\begin{enumerate}
\item[(a)] for each $i \in S$ and $j < i$,
the domain of $p_i$ is a subset of $i \times \mathrm{dom} A$;
\item[(b)] for each $i \in S$ and $j < i$, every map
$\alpha^i_{f,g} : \lambda^{+} \to {\mathcal A}ut (A)$
is constant on $[i,\lambda^{+})$;
\item[(c)] for all $i, j \in S$,
\[\{(f,g):(f,g,\sigma_i)\in t_i\}=\{(f,g):(f,g,\sigma_j)\in t_j\};\]
\item[(d)] there is a condition $p^{\star} \in {\mathbb P}$ such that for all
$i \in S$, $p_i|(i \times \mathrm{dom} A) = p^{\star}$.
\end{enumerate}
\end{lemma}
$\textbf{\noindent Proof}$. First, there is a club $C \subseteq \lambda^{+}$ on which (a) and
(b) hold. Then by F\H{o}dor's lemma there is a stationary subset $S$ of $C$
on which (c) and (d) hold. ${\mathcal B}ox$
\section{The weak lifting} \label{se:6}
In this section we use the notation $S$, $\sigma_i$, $\dot{v}_i$,
$p^{\star}$ from Lemmas \ref{le:21} and \ref{le:22}. We write $s$ for the
constant value of
\[ \{(f,g) : (f,g,\sigma_i) \in t_i\} \ (i \in S) \]
from clause (c) of Lemma \ref{le:22}, and $s^-$ for the set of all $g$ such
that $(1,g) \in s$. We write $\nu : {\mathcal A}ut(B) \to {\mathcal A}ut(A)$ for the
restriction map.
\begin{lemma} \label{le:23}
The relation $s$ is a subset of ${\mathcal A}ut(A) \times {\mathcal A}ut(B)$ that projects onto
${\mathcal A}ut(A)$, and if $(f,g)$ is in $s$ then $\nu(g) = f$.
\end{lemma}
\textbf{\noindent Proof}. Lemma \ref{le:20}. ${\mathcal B}ox$
\begin{lemma} \label{le:24}
If $(f_1,g_1)$ and $(f_2,g_2)$ are both in $s$ then $(f_1f_2,g_1g_2)$ is in
$s$.
\end{lemma}
$\textbf{\noindent Proof}$. In this and later calculations we freely use the fact (Lemma
\ref{le:15}) that the actions of $\mathcal N$ and $\Sigma$ on ${\mathbb P} \otimes \dot{\mathbb Q}$
commute. Take any $i, j \in S$ with $i < j$.
Put $\alpha_1 = \alpha^j_{f_1,g_1,\sigma_j}$, $\alpha_2 =
\alpha^i_{f_2,g_2,\sigma_i}$ and $\alpha_3 = \alpha_1\alpha_2$. Note that
$\alpha_1\alpha_2$ is in $\mathcal{N}_i$ since $i < j$.
We have
\[ \langle p_j,\dot{q}_j \rangle \forces
\sigma_j\dot{\varepsilon}^{-1} \circ \alpha_3 \sigma_j (\dot{\varepsilon}) =
\sigma_j\dot{\varepsilon}^{-1} \circ \alpha_1 \sigma_j (\dot{\varepsilon}) \circ
(\sigma_j \alpha_1(\dot{\varepsilon}))^{-1} \circ \alpha_3 \sigma_j (\dot{\varepsilon}) \]
and by assumption
\[ \langle p_j,\dot{q}_j \rangle \forces
\sigma_j\dot{\varepsilon}^{-1} \circ \alpha_1 \sigma_j(\dot{\varepsilon}) = \check{g_1}. \]
So
\[ \langle p_j,\dot{q}_j \rangle \forces
\sigma_j\dot{\varepsilon}^{-1} \circ \alpha_3 \sigma_j (\dot{\varepsilon}) =
\sigma_j\check{g_1} \circ \sigma^j(\alpha_1(\dot{\varepsilon}))^{-1} \circ
\alpha_1(\alpha_2 \sigma_j \dot{\varepsilon}). \]
Also by assumption
\[ \langle p_i,\dot{q}_i \rangle \forces
\sigma_i\dot{\varepsilon}^{-1} \circ \alpha_2 \sigma_i(\dot{\varepsilon}) = \sigma_i\check{g_2}.\]
Acting on this by $\alpha_1\sigma_j\sigma_i^{-1}$ gives
\[ \langle \alpha_1p_i,\alpha_1\sigma_j\sigma_i^{-1}\dot{q}_i\rangle
\forces \alpha_1 \sigma_j(\dot{\varepsilon}^{-1}) \circ \alpha_1\alpha_2\sigma_j\dot{\varepsilon} =
\alpha_1\sigma_j\check{g_2}. \]
Now $g_2$ is in the ground model and hence $\alpha_2\sigma_j\check{g_2} =
\check{g_2}$. Also $\alpha_1p_i = p_i$ since the support of $p_i$ lies
entirely below $j$, and $\alpha_1 = \alpha^j_{g_1,\sigma_j}$ is the identity
in this region since it lies in $\mathcal{N}_j$. So we have shown that
\[ \langle p_i, \alpha_1\sigma_j\sigma_i^{-1}\dot{q}_i \rangle
\forces \alpha_1\sigma_j\dot{\varepsilon}^{-1} \circ
\alpha_1\alpha_2\sigma_j\dot{\varepsilon} = \check{g_2}. \]
Now we note that $p_i \cup p_j$ is a condition in $P$, by (a), (d) of Lemma
\ref{le:22}. Also $p_i \cup p_j$ forces that $\mathrm{dom}
(\sigma_i^{-1}\dot{q}_i)$ is disjoint from $\mathrm{dom} (\sigma_j^{-1}\dot{q}_j)$
by Lemma \ref{le:21}, and hence also that $\mathrm{dom}
\sigma_j\sigma_i^{-1}\dot{q}_i$ is disjoint from $\mathrm{dom} r_j$. From the action
of neat maps on ${\mathbb Q}$, $\mathrm{dom} \alpha_1\sigma_j\sigma_1^{-1}= \mathrm{dom}
\sigma_j\sigma_i^{-1}$. This shows that $\langle p_i,\sigma_j\sigma_i^{-1}
\dot{q}_i \rangle$ and $\langle p_j,\dot{q}_j \rangle$ have a common
extension $\langle p',\dot{q}' \rangle$. Putting everything together, we
have that
\[\langle p',\dot{q}'\rangle\forces\sigma_j\dot{\varepsilon}^{-1}\circ\alpha_3\sigma_j
\dot{\varepsilon} = \check{g_1}\check{g_2}.\]
Since $\alpha_3$ is in $\mathcal{N}_i$, this shows that
\[ (f_1f_2,g_1g_2) \in t_{p',\dot{q}',j}. \]
Then by the maximality property of $\langle p_j,\dot{q}_j \rangle$,
\[ (f_1f_2,g_1g_2,\sigma_j) \in t_{p_j,j} \]
so that $(f_1f_2,g_1g_2)$ is in $s$.
${\mathcal B}ox$
\begin{lemma} \label{le:25}
If $g_1$ and $g_2$ are in $s^-$ then $g_1g_2 = g_2g_1$.
\end{lemma}
\textbf{\noindent Proof}. Apply the proof of Lemma \ref{le:24} to $(1,g_1)$ and $(1,g_2)$.
In the notation of that proof, $\alpha_1$ commutes with $\alpha_2$,
$\alpha_1$ is the identity below $j$ and $\alpha_2$ is the identity below
$j$ (since $i, j \in S$. But also $g_2$ lies in $s^-$, and this tells us
that $\alpha_2$ is the identity on $[j,\lambda^+)$. In particular
$\alpha_1$ commutes with $\alpha_2$.
We follow the proof of Lemma \ref{le:24} but with $g_1$ and $g_2$
transposed, starting from the observation that
\[ \langle p_i,\dot{q}_i \rangle \forces\sigma_i\dot{\varepsilon}^{-1} \circ
\alpha_3\sigma_i\dot{\varepsilon} = \sigma_i \dot{\varepsilon}^{-1} \circ \alpha_2\sigma_i \dot{\varepsilon}
\circ \alpha_2\sigma_i\dot{\varepsilon}^{-1} \circ \alpha_3\sigma_i\dot{\varepsilon}. \]
As before, we have that
\[ \langle p_i,\dot{q}_i \rangle \forces
\sigma_i\dot{\varepsilon}^{-1} \circ \alpha_2\sigma_i\dot{\varepsilon} = \check{g_2} \]
and
\[\langle \alpha_2p_j,\alpha_2\sigma_i\sigma_j^{-1}\dot{q}_j \rangle
\forces \alpha_2\sigma_i\dot{\varepsilon}^{-1} \circ \alpha_2\alpha_1\sigma_i\dot{\varepsilon} =
\alpha_3\sigma_i\check{g_1}, \]
recalling that $\alpha_1$ commutes with $\alpha_2$. Now the support of
$p_j$ lies below $i$ or within $[j,\lambda^+)\times \mathrm{dom} A$, and $\alpha_2$
is the identity in both these regions, and so $\alpha_2(p_j) = p_j$. Also
$p_i \cup p_j$ forces that $\dot{q_i}$ and
$\alpha_2\sigma_i\sigma_j^{-1}\dot{q_j}$ have disjoint domains. So as
before there is $\langle p',\dot{q}' \rangle \leqslant \langle p_i,\dot{q}_i
\rangle$ and $\leqslant \langle p_j, \alpha_2\sigma_i\sigma_j^{-1}q_j
\rangle$ such that
\[ \langle p',\dot{q}' \rangle \forces
\sigma_i\dot{\varepsilon}^{-1} \circ \alpha_3\sigma_i\dot{\varepsilon} =
\check{g_2} \check{g_1}. \]
As before, it follows that
\[ \langle p_i,\dot{q}_i \rangle \forces
\sigma_i\dot{\varepsilon}^{-1} \circ \alpha_3\sigma_i\dot{\varepsilon} =
\check{g_2} \check{g_1}, \]
and so
\[ \langle p_i, \sigma_j\sigma_i^{-1}\dot{q}_i \rangle \forces
\sigma_j\dot{\varepsilon}^{-1} \circ \alpha_3\sigma_j\dot{\varepsilon} =
\check{g_2} \check{g_1}. \]
Again there is a condition $\langle p'', \dot{q}'' \rangle \leqslant
\langle p_i, \sigma_j\sigma_i^{-1}\dot{q}_i \rangle$ and
$\leqslant \langle p_j,\dot{q}_j \rangle$. Recalling that
in the proof of Lemma \ref{le:24} we showed that
\[ \langle p_j,\dot{q}_j \rangle \forces
\sigma_j\dot{\varepsilon}^{-1} \circ \alpha_3\sigma_j(\dot{\varepsilon}) =
\check{g_2} \check{g_1}, \]
we deduce that
\[ \langle p'',\dot{q}'' \rangle \forces \check{g_1}\check{g_2}
= \check{g_2}\check{g_1}. \]
But the equation $g_1g_2 = g_2g_1$ is about the ground model, and
hence it is true. ${\mathcal B}ox$
\begin{corollary} \label{co:26}
If $(f,g_1)$ and $(f,g_2)$ are in $s$
then $g_1g_2^{-1}$ is in $\langle s^- \rangle$.
\end{corollary}
\textbf{\noindent Proof}. There is some $g' \in {\mathcal A}ut (B)$ such that $(f^{-1},g')$ is in $s$.
Then by the claim, $(1,g_1g')$ and $(1,g_2g')$ are in $s$ and so $g_1g'$,
$g_2g'$ are in $s^-$. Hence the element
\[ g_1g_2^{-1} = (g_1g')(g_2g')^{-1} \]
lies in $\langle s^- \rangle$. ${\mathcal B}ox$
\begin{corollary} \label{co:27}
Suppose $g_1, \ldots, g_k$ are elements of ${\mathcal A}ut (B)$ such that
$(\nu(g_i),g_i)$ is in $s$ for each $i$, and let each of $\varepsilon_1,
\ldots,\varepsilon_k$ be either $1$ or $-1$. If
\[\nu(g_1)^{\varepsilon_1}\ldots \nu(g_k)^{\varepsilon_k} = 1\]
then
\[g_1^{\varepsilon_1}\ldots g_k^{\varepsilon_k}\in\langle s^-\rangle.\]
\end{corollary}
\textbf{\noindent Proof}. We write $f_i$ for $\nu(g_i)$. First we show the corollary
directly in the case $k = 3$. Taking inverses, we can assume that
$\varepsilon_2 = 1$. When $\varepsilon_1 = \varepsilon_3 = 1$, the result
is immediate from Lemma \ref{le:24}. We consider next the case where
$\varepsilon_1 = 1$ and $\varepsilon_3 = -1$. Here we find $g$ such that
$(f^{-1},g)$ is in $s$. Then both of
\[ g_1^1 g_2^1 g^1, g_3^1g^1 \]
are in $s^-$ by Lemma \ref{le:24}, and so
\[ g_1^1g_2^1g_3^{-1} = (g_1^1g_2^1g^1)(g_3^1g_1)^{-1} \]
is in $\langle s^- \rangle$. By symmetry this also covers the
case where $\varepsilon_1 = -1$ and $\varepsilon_3 = 1$. Finally
when $\varepsilon_1 = \varepsilon_3 = -1$, we repeat the same moves,
noting that
\[ g_1^{-1}g_2^1 g^1 \]
is in $\langle s^- \rangle$ by the previous case.
This case also covers the case $k=2$ by adding at the end a factor $g_3^1$
where $(1,g^3)$ is in $s$. The case $k=1$ is trivial.
We prove the remaining cases by induction on $k$, assuming $k > 3$. Choose
$g$ so that $(g,f_{k-1}^{\varepsilon_{k-1}}f_k^{\varepsilon_k})$ is in $s$.
Then by induction hypothesis both the elements
\[ g_1^{\varepsilon_1} \ldots g_{k-2}^{\varepsilon_{k-2}} g^1 \]
and
\[ g^{-1}g_{k-1}^{\varepsilon^{k-1}}g_k^{\varepsilon_k} \]
lie in $\langle s^- \rangle$. Hence so does their product,
completing the proof. ${\mathcal B}ox$
Consider the subgroup $\langle s^- \rangle$ of ${\mathcal A}ut (B)$. By Lemma
\ref{le:25}, $\langle s^- \rangle$ is commutative. By Lemma \ref{le:23} and
Corollary \ref{co:27} it follows that $s$ would be a weak splitting of
$\nu$, with $\langle s^- \rangle$ as $G_0$, if for each $f$ in ${\mathcal A}ut (A)$
there was a unique $g$ with $\langle f, g \rangle$ in $s$. But we can make
this true by cutting down $s$. So $\nu$ has a weak splitting, and this
concludes the proof of Theorem \ref{th:4}.
\section{Answers to questions} \label{se:7}
The results above answer most of the problems stated in
\cite{hosh:1}. In that paper we showed:
\begin{quote}
\textbf{Theorem 3 of \cite{hosh:1}\ \ } If $\textbf{C}$ is a small natural
construction in a model of ZFC, then $\textbf{C}$ is uniformisable with
parameters.
\end{quote}
We asked (Problem A) whether it is possible to remove the restriction that
$\textbf{C}$ is small. The answer is No:
\begin{theorem} \label{th:28}
There is a transitive model of ZFC in which some $\emptyset$-represent\-able
construction is natural but not uniformisable (even with parameters).
\end{theorem}
$\textbf{\noindent Proof}$. Let $N$ be the model of Theorem~\ref{th:4}. Let $\textbf{C}$ be some
construction $\emptyset$-representable in $N$ which is not weakly natural
(such as Example 2 in section \ref{se:2}). Then by Theorem \ref{th:4},
$\textbf{C}$ is not uniformisable. The rigidifying construction $\textbf{C}^r$ of
section \ref{se:2} is $\emptyset$-representable, natural and not
uniformisable. ${\mathcal B}ox$
Problem B asked whether in Theorem 3 of \cite{hosh:1} the formula defining
$\textbf{C}$ can be chosen so that it has only the same parameters as the
formulas chosen to represent $\textbf{C}$. The answer is No:
\begin{theorem} \label{th:29}
There is a transitive model $N$ of ZFC with the following property:
\begin{quote}
For every set $Y$ there are a set $X$ and a unitype rigid-based (hence small
natural) $X$-representable construction that is not $X \cup
Y$-uniformisable.
\end{quote}
\end{theorem}
\textbf{\noindent Proof}. Take $N$ to be the model given by Theorem~\ref{th:4}. Let $Y$ be
any set in $N$. If $N$ and $Y$ are not as stated above, then for every set
$X$ and every unitype rigid-based $X$-representable construction in $N$, $X$
is $X \cup Y$-uniformisable. So the hypothesis of Theorem \ref{th:3} holds,
and by that theorem there is in $N$ a small $\{\bar{c}\}$-uniformisable
construction that is not weakly natural. But this contradicts the choice of
$N$. ${\mathcal B}ox$
Problem C asked whether there are transitive models of ZFC in which every
uniformisable construction is natural. Theorem~\ref{th:4} is the best
answer we have for this; the problem remains open.
In \cite{ho:1} one of us asked whether there can be models of ZFC in which
the algebraic closure construction on fields is not uniformisable.
\begin{theorem} \label{th:30}
There are transitive models of ZFC in which:
\begin{enumerate}
\item[(a)] no formula (with or without parameters) defines for
each field a specific algebraic closure for that field, and
\item[(b)] no formula (with or without parameters) defines for each
abelian group a specific injective hull of that group.
\end{enumerate}
\end{theorem}
\textbf{\noindent Proof}. Let the model $N$ be as in Theorem \ref{th:4}. In $N$ the
constructions of Examples 3 and 4 in section \ref{se:2} are not
uniformisable, since they are not weakly natural. So these two examples
prove (a) and (b) respectively. ${\mathcal B}ox$
One result in \cite{ho:1} was that there is no primitive recursive set
function which takes each field to an algebraic closure of that field. This
is an absolute result which applies to every transitive model of ZFC, and so
it is not strictly comparable with the consistency results proved above. In
this context we note that Garvin Melles showed \cite{me:1} that there is no
``recursive set-function'' (he gives his own definition for this notion)
which finds a representative for each isomorphism type of countable
torsion-free abelian group.
\noindent Wilfrid Hodges\\
School of Mathematical Sciences\\
Queen Mary, University of London\\
Mile End Road, London E1 4NS, England\\
\texttt{[email protected]}
\noindent Saharon Shelah\\
Institute of Mathematics, Hebrew University\\
Jerusalem, Israel\\
\texttt{[email protected]}
\end{document} | math | 58,618 |
qegin{document}
\sigmaetcounter{page}{1}
\thetaitle[Iterated Hardy-type inequalities involving suprema]{Iterated Hardy-type inequalities involving suprema}
\alphauthor[A. Gogatishvili]{Amiran Gogatishvili}
\alphaddress{Institute of Mathematics \\
Academy of Sciences of the Czech Republic \\
\v Zitn\'a~25 \\
115~67 Praha~1, Czech Republic} \email{[email protected]}
\alphauthor[R.Ch.Mustafayev]{Rza Mustafayev}
\alphaddress{Department of Mathematics \\ Faculty of Science and Arts \\ Kirikkale
University \\ 71450 Yahsihan, Kirikkale, Turkey}
\email{[email protected]}
\thetahanks{The research of A. Gogatishvili was partly supported by the grants P201-13-14743S
of the Grant Agency of the Czech Republic and RVO: 67985840, by
Shota Rustaveli National Science Foundation grants no. 31/48
(Operators in some function spaces and their applications in Fourier
Analysis) and no. DI/9/5-100/13 (Function spaces, weighted
inequalities for integral operators and problems of summability of
Fourier series). The research of both authors was partly supported
by the joint project between Academy of Sciences of Czech Republic
and The Scientific and Technological Research Council of Turkey}
\sigmaubjclass[2010]{Primary 26D10; Secondary 26D15.}
\keywords{quasilinear operators, iterated Hardy inequalities,
weights.}
qegin{abstract}
In this paper the complete solution of the restricted inequalities
for supremal operators are given. The boundedness of the composition
of supremal operators with the Hardy and Copson operators in
weighted Lebesgue spaces are characterized.
\end{abstract}
\maketitle
\sigmaection{Introduction}\lambdabel{in}
Throughout the paper we assume that $I : = (a,b)\sigmaubseteq (0,\infty)$.
By ${\mathfrak M} (I)$ we denote the set of all measurable functions on $I$.
The symbol ${\mathfrak M}^+ (I)$ stands for the collection of all $f\inftyn{\mathfrak M}
(I)$ which are non-negative on $I$, while ${\mathfrak M}^+ (I;\deltan)$ and ${\mathfrak M}^+
(I;\muparrow)$ are used to denote the subset of those functions which are
non-increasing and non-decreasing on $I$, respectively. When $I = (0,\i)$, we write simply ${\mathfrak M}^{\deltan}$ and ${\mathfrak M}^{\muparrow}$ instead of ${\mathfrak M}^+ (I;\deltan)$ and ${\mathfrak M}^+
(I;\muparrow)$, accordingly. The family of all weight functions (also called just weights) on $I$, that is,
locally integrable non-negative functions on $(0,\i)$, is given by ${\mathcal W}(I)$.
For $p\inftyn (0,\infty]$ and $w\inftyn {\mathfrak M}^+(I)$, we define the functional
$\|\cdot\|_{p,w,I}$ on ${\mathfrak M} (I)$ by
qegin{equation*}
\|f\|_{p,w,I} : = \left\{qegin{array}{cl}
\left(\inftynt_I |f(x)|^p w(x)\,dx \right)^{1/p} & \qquad\mbox{if}\qquad p<\infty \\
\omegaperatornamewithlimits{ess\,sup}_{I} |f(x)|w(x) & \qquad\mbox{if}\qquad p=\infty.
\end{array}
\right.
\end{equation*}
If, in addition, $w\inftyn {\mathcal W}(I)$, then the weighted Lebesgue space
$L^p(w,I)$ is given by
qegin{equation*}
L^p(w,I) = \{f\inftyn {\mathfrak M} (I):\,\, \|f\|_{p,w,I} < \infty\}
\end{equation*}
and it is equipped with the quasi-norm $\|\cdot\|_{p,w,I}$.
When $w\equiv 1$ on $I$, we write simply $L^p(I)$ and
$\|\cdot\|_{p,I}$ instead of $L^p(w,I)$ and $\|\cdot\|_{p,w,I}$,
respectively.
Given a operator $T:{\mathfrak M}^+ \rightarrow {\mathfrak M}^+$, for $0 < p <\infty$ and
$u\inftyn{\mathfrak M}^+$, denote by
$$
T_u (g) : = T (g u), \qquad g \inftyn {\mathfrak M}^+.
$$
Suppose $f$ be a measurable a.e. finite function on ${\mathbb R}^n$.
Then its non-increasing rearrangement $f^*$ is given by
$$
f^* (t) = \inftynf \{\lambdambda > 0: |\{x \inftyn {\mathbb R}^n:\, |f(x)| >
\lambdambda \}| \le t\}, \quad t \inftyn (0,\inftynfty),
$$
and let $f^{**}$ denotes the Hardy-Littlewood maximal function of
$f$, i.e.
$$
f^{**}(t) : = \frac{1}{t} \inftynt_0^t f^* (\thetaau)\,d\thetaau, \quad t > 0.
$$
Quite many familiar function spaces can be defined using the
non-increasing rearrangement of a function. One of the most
important classes of such spaces are the so-called classical Lorentz
spaces.
Let $p \inftyn (0,\inftynfty)$ and $w \inftyn {\mathcal W}$. Then the classical
Lorentz spaces $\Lambdambda^p (w)$ and $\Gammaamma^p (w)$ consist of all
functions $f \inftyn {\mathfrak M}$ for which $\|f\|_{\Lambdambda^p(w)} <
\inftynfty$ and $\|f\|_{\Gammaamma^p(w)} < \inftynfty$, respectively. Here it is
$$
\|f\|_{\Lambdambda^p(w)} : = \|f^*\|_{p,w,(0,\inftynfty)} \qquaduad \mbox{and}
\qquaduad \|f\|_{\Gammaamma^p(w)} : = \|f^{**}\|_{p,w,(0,\inftynfty)}.
$$
For more information about the Lorentz $\Lambdambda$ and $\Gammaamma$ see
e.g. \cite{cpss} and the references therein.
The Hardy and Copson operators are defined by
$$
H g (t) : = \inftynt_0^t g(s)\,ds, \qquad g \inftyn {\mathfrak M}^+,
$$
and
$$
H^* g (t) : = \inftynt_t^{\infty} g(s)\,ds, \qquad g \inftyn {\mathfrak M}^+,
$$
respectively. The operators $H$ and $H^*$ play a prominent role. There are other operators that are also of interest. For example, certain
specific problems such as the description of the behaviour of the
fractional maximal operator on classical Lorentz spaces \cite{ckop}
or the optimal pairing problem for Sobolev imbeddings \cite{kerp}
or various questions arising in the interpolation theory can be
handles in an elegant way with the help of the supremal operators
$$
S g (t) : = \omegaperatornamewithlimits{ess\,sup}_{0 < \thetaau \le t} g(\thetaau), \qquad g \inftyn {\mathfrak M}^+,
$$
and
$$
S^* g (t) : = \omegaperatornamewithlimits{ess\,sup}_{t \le \thetaau < \inftynfty} g(\thetaau), \qquad g \inftyn {\mathfrak M}^+.
$$
In this paper, we give complete characterization of restricted inequalities
qegin{equation}\lambdabel{supr.ineq.1}
\| S_u (f)\|_{q,w,(0,\i)} \le c\|f\|_{p,v,(0,\i)},~ f \inftyn \mathfrak M^{\deltan},
\end{equation}
qegin{equation}\lambdabel{supr.ineq.3}
\| S_u (f) \|_{q,w,(0,\infty)} \le c \| f \|_{p,v,(0.\infty)}, \, f \inftyn \mathfrak M^{\muparrow}
\end{equation}
and
qegin{equation}\lambdabel{supr.ineq.2}
\| S_u^* (f)\|_{q,w,(0,\i)} \le c\|f\|_{p,v,(0,\i)},~ f \inftyn \mathfrak M^{\muparrow},
\end{equation}
qegin{equation}\lambdabel{supr.ineq.4}
\| S_u^* (f)\|_{q,w,(0,\i)} \le c\|f\|_{p,v,(0,\i)},~ f \inftyn \mathfrak M^{\deltan}.
\end{equation}
Note that inequality \eqref{supr.ineq.1} was characterized in \cite{gop}. It should be mentioned here that it was done under some additional condition on weight function $u$, when $q < p$ (cf. \cite[Theorem 3.4]{gop}).
In particular, we characterize the validity of the iterated
Hardy-type inequalities involving suprema
qegin{equation}\lambdabel{ISI.1}
\left\|S_{u} qigg( \inftynt_0^x hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{p,v,(0,\i)},
\end{equation}
and
qegin{equation}\lambdabel{ISI.2}
\left\|S_{u} qigg( \inftynt_x^{\infty} hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{p,v,(0,\i)},
\end{equation}
where $0 < q < \inftynfty$, $1 \le p < \infty$, $u$, $w$ and $v$ are
weight functions on $(0,\inftynfty)$.
It is worth to mentoin that the characterizations of "dual"
inequalities
qegin{equation}\lambdabel{ISI.3}
\left\| S_{u}^* qigg( \inftynt_x^{\infty} hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{p,v,(0,\i)},
\end{equation}
and
qegin{equation}\lambdabel{ISI.4}
\left\|S_{u}^* qigg( \inftynt_0^x hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{p,v,(0,\i)},
\end{equation}
can be easily obtained from the solutions of inequalities
\eqref{ISI.1} - \eqref{ISI.2}, respectively, by change of variables. Note that inequality
\eqref{ISI.4} has been characterized in \cite{gop} in the case $0 < q < \inftynfty$, $1 \le p < \inftynfty$.
We pronounce that the characterizations of inequalities
\eqref{ISI.1} - \eqref{ISI.4} are important because many
inequalities for classical operators can be reduced to them (for
illustrations of this important fact, see, for instance,
\cite{GogMusPers2}). These inequalities play an important role in
the theory of Morrey spaces and other topics (see \cite{BGGM1},
\cite{BGGM2} and \cite{BO}).
The fractional maximal operator, $M_{\gamma}$, $\gamma \inftyn (0,n)$, is defined at $f \inftyn L_{\omegaperatorname{loc}}^1(\R^n)$ by
$$
(M_{\gamma} f) (x) = \sigmaup_{Q \ni x} |Q|^{ \gamma / n - 1} \inftynt_{Q} |f(y)|\,dy,\quad x \inftyn \R^n,
$$
where the supremum is extended over all cubes $Q \sigmaubset \R^n$ with sides parallel to the coordinate axes. It was shown in \cite[Theorem 1.1]{ckop} that
qegin{equation}\lambdabel{frac.max op.eq.1.}
(M_{\gamma}f)^* (t) \lesssim \sigmaup_{t \le \thetaau < \inftynfty} \thetaau^{\gamma / n - 1} \inftynt_0^{\thetaau} f^*(y)\,dy \lesssim (M_{\gamma} \thetailde{f})^* (t),
\end{equation}
for every $f \inftyn L_{\omegaperatorname{loc}}^1(\R^n)$ and $t \inftyn (0,\i)$, where $\thetailde{f} (x) = f^* (\omegamega_n |x|^n)$ and $\omegamega_n$ is the volume of $S^{n-1}$.
Thus, in order to characterize boundedness of the fractional maximal operator $M_{\gamma}$ between classical Lorentz spaces it is necessary and sufficient to characterize the validity of the weighted inequality
$$
qigg( \inftynt_0^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \thetaau^{\gamma / n - 1} \inftynt_0^{\thetaau} \varphi(y)\,dyqigg]^q w(t)\,dtqigg)^{1 / q} \lesssim qigg( \inftynt_0^{\inftynfty} [\varphi(t)]^p v(t)\,dtqigg)^{1 / p}
$$
for all $\varphi \inftyn \mathfrak M^{\deltan}$. This last estimate can be interpreted as a restricted weighted inequality for the operator $T_{\gamma}$, defined by
qegin{equation}\lambdabel{frac.max op.eq.3.}
(T_{\gamma}g) (t) = \sigmaup_{t \le \thetaau < \inftynfty} \thetaau^{\gamma / n - 1} \inftynt_0^{\thetaau} g(y)\,dy, \quad g \inftyn \mathfrak M^+ (0,\i), \quad t \inftyn (0,\i).
\end{equation}
Such a characterization was obtained in \cite{ckop} for the particular case when $1 < p \le q <\inftynfty$ and in \cite[Theorem 2.10]{o} in the case of more general operators and for extended range of $p$ and $q$. Full proofs and some further extensions and applications can be found in \cite{edop}, \cite{edop2008}.
The operator $T_{\gamma}$ is a typical example of what is called a Hardy-operator involving suprema
$$
(T_u g)(t) : = \sigmaup_{t \le s < \inftynfty} \frac{u(s)}{s} \inftynt_0^s g(y)\,dy,
$$
which combines both the operations (integration and taking the supremum).
In the above-mentioned applications, it is required to characterize a restricted weighted inequality for $T_u$. This amounts to finding a necessary and sufficient condition on a triple of weights $(u,\,v,\,w)$ such that the inequality
qegin{equation}\lambdabel{ineq.for.Tu}
qigg(\inftynt_0^{\inftynfty} qigg(\sigmaup_{t \le s < \inftynfty} u(s) f^{**}(s)qigg)^q w(t)\,dtqigg)^{1 / q} \lesssim qigg(\inftynt_0^{\inftynfty} f^*(t)^p v(t)\,dt qigg)^{1/p}
\end{equation}
holds. Particular examples of such inequalities were studied in \cite{ckop} and, in a more systematic way, in \cite{gop}. Inequality \eqref{ineq.for.Tu} was investigated in \cite{gogpick2007} in the case when $0< p \le 1$. The approach used in this paper was based on a new type reduction theorem which showed connection between three types of restricted weighted inequalities.
Rather interestingly, such operators have been recently encountered in various research projects. They have been found indispensable in the search for optimal pairs of rearrangement-invariant norms for wich a Sobolev-type inequality holds (cf. \cite{kerp}). They constitute a very useful tool for characterization of the assocaiate norm of an operator-induced norm, which naturally appears as an optimal domain norm in a Sobolev embedding (cf. \cite{pick2000}, \cite{pick2002}). Supremum operators are also very useful in limitimg interpolation theory as can be seen from their appearance for example in \cite{evop}, \cite{dok}, \cite{cwikpys}, \cite{pys}.
qegin{definition}\lambdabel{Tub.defi.1}
Let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$, $b \inftyn {\mathcal W}(0,\i)$ and $B(t) : = \inftynt_0^t b(s)\,ds$. Assume that $b$
be such that $0 < B(t) < \inftynfty$ for every $t \inftyn (0,\i)$. The operator $T_{u,b}$ is defined at $g \inftyn \mathfrak M^+ (0,\i)$ by
$$
(T_{u,b} g)(t) : = \sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \inftynt_0^{\thetaau} g(s)b(s)\,ds,\qquaduad t \inftyn (0,\i).
$$
\end{definition}
The operator $T_{\gamma}$, defined in \eqref{frac.max op.eq.3.}, is a particular example of operators $T_{u,b}$. These operators are investigated in \cite{gop} and \cite{gogpick2007}.
In this paper we give complete characterization for the inequality
qegin{equation}\lambdabel{Tub.thm.1.eq.1}
\|T_{u,b}f \|_{q,w,(0,\i)} \le c \| f \|_{p,v,(0,\i)}, \qquad f \inftyn \mathfrak M^{\deltan}(0,\infty)
\end{equation}
for $0 < q \le \inftynfty$, $0 < p < \inftynfty$ (see Theorems \ref{Tub.thm.1} and \ref{RT.SO.thm.3}).
Inequality \eqref{Tub.thm.1.eq.1} was characterized in \cite[Theorem 3.5]{gop} under additional condition
$$
\sigmaup_{0 < t < \inftynfty} \frac{u(t)}{B(t)} \inftynt_0^t \frac{b(\thetaau)}{u(\thetaau)}\,d\thetaau < \inftynfty.
$$
Note that the case when $0 < p \le 1 < q < \inftynfty$ was not considered in \cite{gop}. It is also worse to mention that in the case when $1 < p < \inftynfty$, $0 < q < p < \inftynfty$, $q \neq 1$ \cite[Theorem 3.5]{gop} contains only discrete condition. In \cite{gogpick2007} the new reduction theorem was obtained when $0 < p \le 1$, and this technique allowed to characterize inequality \eqref{Tub.thm.1.eq.1} when $b \equiv 1$, and in the case when $0 < q< p \le 1$ this paper contains only discrete condition.
The paper is organized as follows. Section \ref{pre} contains some
preliminaries along with the standard ingredients used in the
proofs. Full characterization of inequalities \eqref{supr.ineq.1} - \eqref{supr.ineq.4} and \eqref{ISI.1} - \eqref{ISI.3} are given in Sections \ref{s.3} and \ref{s.4}. Finally, solution of inequality \eqref{Tub.thm.1.eq.1} are obtained in Section \ref{s.5}.
\sigmaection{Notations and Preliminaries}\lambdabel{pre}
Throughout the paper, we always denote by $c$ or $C$ a positive
constant, which is independent of main parameters but it may vary
from line to line. However a constant with subscript such as $c_1$
does not change in different occurrences. By $a\lesssim b$,
($b\gtrsim a$) we mean that $a\leq \lambda b$, where $\lambda >0$ depends on
inessential parameters. If $a\lesssim b$ and $b\lesssim a$, we write
$a\alphapproxprox b$ and say that $a$ and $b$ are equivalent. We will denote by $qf 1$ the
function ${qf 1}(x) = 1$, $x \inftyn (0,\i)$. Unless a
special remark is made, the differential element $dx$ is omitted
when the integrals under consideration are the Lebesgue integrals.
Everywhere in the paper, $u$, $v$ and $w$ are weights.
We need the following notations:
$$
qegin{array}{ll}
V(t) : = \inftynt_0^t v, & V_*(t) : = \inftynt_t^{\infty} v,\\ [10pt] W(t) : =
\inftynt_0^t w, & W_*(t) : = \inftynt_t^{\infty} w.
\end{array}
$$
qegin{convention}\lambdabel{Notat.and.prelim.conv.1.1}
We adopt the following conventions:
{\rm (i)} Throughout the paper we put $0 \cdot \infty = 0$, $\infty / \infty =
0$ and $0/0 = 0$.
{\rm (ii)} If $p\inftyn [1,+\infty]$, we define $p'$ by $1/p + 1/p' = 1$.
{\rm (iii)} If $0 < q < p < \inftynfty$, we define $r$ by $1 / r = 1 / q - 1 / p$.
{\rm (iv)} If $I = (a,b) \sigmaubseteq \mathbb R$ and $g$ is monotone
function on $I$, then by $g(a)$ and $g(b)$ we mean the limits
$\lim_{x\rightarrow a+}g(x)$ and $\lim_{x\rightarrow b-}g(x)$, respectively.
\end{convention}
We recall some reduction theorems for positive monotone operators from \cite{GogStep} and \cite{GogMusIHI}. The following conditions will be used below:
{\rm (i)} $T(\lambda f) = \lambda Tf$ for all $\lambda \ge 0$ and $f \inftyn \mathfrak M^+$;
{\rm (ii)} $Tf(x) \le c Tg(x)$ for almost all $x \inftyn \mathbb R_+$ if $f(x)
\le g(x)$ for almost all $x \inftyn \mathbb R_+$, with constant $c > 0$
independent of $f$ and $g$;
{\rm (iii)} $T(f+ \lambda {qf 1}) \le c (T f + \lambda T {qf 1})$ for all
$f \inftyn \mathfrak M^+$ and $\lambda \ge 0$, with a constant $c > 0$ independent of
$f$ and $\lambda$.
qegin{theorem}[\cite{GogStep}, Theorem 3.1]\lambdabel{Reduction.Theorem.thm.3.1}
Let $0 < q \le \inftynfty$ and $1 \le p < \inftynfty$, and let $T: {\mathfrak M}^+ \rightarrow {\mathfrak M}^+$
be an operator. Then the inequality
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.1}
\|Tf \|_{q,w,(0,\inftynfty)} \le c \| f \|_{p,v,(0,\inftynfty)}, \qquaduad f \inftyn {\mathfrak M}^{\deltaownarrow}(0,\inftynfty)
\end{equation}
implies the inequality
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.2}
qigg\| Tqigg( \inftynt_x^{\inftynfty} h qigg)qigg\|_{q,w,(0,\inftynfty)} \le c \| h
\|_{p, V^{p} v^{1-p},(0,\inftynfty)}, \qquaduad h \inftyn {\mathfrak M}^+(0,\inftynfty).
\end{equation}
If $V(\inftynfty) = \inftynfty$ and if $T$ is an operator satisfying
conditions {\rm (i)-(ii)}, then the condition
\eqref{Reduction.Theorem.eq.1.2} is sufficient for inequality
\eqref{Reduction.Theorem.eq.1.1} to hold on the cone ${\mathfrak M}^{\deltaownarrow}$.
Further, if $0 < V(\inftynfty) < \inftynfty$, then a sufficient condition for
\eqref{Reduction.Theorem.eq.1.1} to hold on ${\mathfrak M}^{\deltaownarrow}$ is that both
\eqref{Reduction.Theorem.eq.1.2} and
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.3}
\|T {qf 1}\|_{q,w,(0,\inftynfty)} \le c \|{qf 1}\|_{p,v,(0,\inftynfty)}
\end{equation}
hold in the case when $T$ satisfies the conditions {\rm (i)-(iii)}.
\end{theorem}
qegin{theorem}[\cite{GogStep}, Theorem 3.2]\lambdabel{Reduction.Theorem.thm.3.2}
Let $0 < q \le \inftynfty$ and $1 \le p < \inftynfty$, and let $T: {\mathfrak M}^+ \rightarrow {\mathfrak M}^+$
satisfies conditions {\rm (i)} and {\rm (ii)}. Then a sufficient
condition for inequality \eqref{Reduction.Theorem.eq.1.1} to hold is
that
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.222}
qigg\| Tqigg( \frac{1}{V^2(x)} \inftynt_0^x hV qigg)qigg\|_{q,w,(0,\inftynfty)}
\le c \| h \|_{p, v^{1-p},(0,\inftynfty)}, \qquaduad h \inftyn {\mathfrak M}^+(0,\inftynfty).
\end{equation}
Moreover, \eqref{Reduction.Theorem.eq.1.1} is necessary for
\eqref{Reduction.Theorem.eq.1.222} to hold if conditions {\rm
(i)-(iii)} are all satisfied.
\end{theorem}
qegin{theorem}[\cite{GogStep}, Theorem 3.3]\lambdabel{Reduction.Theorem.thm.3.3}
Let $0 < q \le \inftynfty$ and $1 \le p < \inftynfty$, and let $T: {\mathfrak M}^+ \rightarrow {\mathfrak M}^+$
be an operator. Then the inequality
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.1.00}
\|Tf \|_{q,w,(0,\inftynfty)} \le c \| f \|_{p,v,(0,\inftynfty)}, \qquaduad f \inftyn {\mathfrak M}^{\muparrowarrow}(0,\inftynfty)
\end{equation}
implies the inequality
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.2.00}
qigg\| Tqigg( \inftynt_0^x h qigg)qigg\|_{q,w,(0,\inftynfty)} \le c \| h \|_{p,
V_*^{p} v^{1-p},(0,\inftynfty)}, \qquaduad h \inftyn {\mathfrak M}^+(0,\inftynfty).
\end{equation}
If $V_*(0) = \inftynfty$ and if $T$ is an operator satisfying the
conditions {\rm (i)-(ii)}, then the condition
\eqref{Reduction.Theorem.eq.1.2.00} is sufficient for inequality
\eqref{Reduction.Theorem.eq.1.1.00} to hold. If $0 < V_*(0) < \inftynfty$
and $T$ is an operator satisfying the conditions {\rm
(i)-(iii)}, then \eqref{Reduction.Theorem.eq.1.1.00} follows from
\eqref{Reduction.Theorem.eq.1.2.00} and
\eqref{Reduction.Theorem.eq.1.3}.
\end{theorem}
qegin{theorem}[\cite{GogStep}, Theorem 3.4]\lambdabel{Reduction.Theorem.thm.3.4}
Let $0 < q \le \inftynfty$ and $1 \le p < \inftynfty$, and let $T: {\mathfrak M}^+ \rightarrow {\mathfrak M}^+$
satisfies conditions {\rm (i)} and {\rm (ii)}. Then a sufficient
condition for inequality \eqref{Reduction.Theorem.eq.1.1.00} to hold
is that
qegin{equation}\lambdabel{Reduction.Theorem.eq.1.2222}
qigg\| Tqigg( \frac{1}{V_*^2(x)} \inftynt_x^{\inftynfty} hV_*
qigg)qigg\|_{q,w,(0,\inftynfty)} \le c \| h \|_{p, v^{1-p},(0,\inftynfty)}, \qquaduad h \inftyn
{\mathfrak M}^+(0,\inftynfty).
\end{equation}
Moreover, \eqref{Reduction.Theorem.eq.1.1.00} is necessary for
\eqref{Reduction.Theorem.eq.1.2222} to hold if conditions {\rm
(i)-(iii)} are all satisfied.
\end{theorem}
qegin{theorem}\cite[Theorem 3.1]{GogMusIHI}\lambdabel{RT.thm.main.3}
Let $0 < q \le \infty$, $1 < p < \infty$, and let $T: \mathfrak M^+ \rightarrow \mathfrak M^+$
satisfies conditions {\rm (i)-(iii)}. Assume that $u,\,w \inftyn {\mathcal W}(0,\i)$
and $v \inftyn {\mathcal W}(0,\i)$ be such that
qegin{equation}\lambdabel{RT.thm.main.3.eq.0}
\inftynt_0^x v^{1-p^{\prime}}(t)\,dt < \infty \qquad \mbox{for all} \qquad x > 0.
\end{equation}
Then inequality
qegin{equation}
\lambdabel{IHI.H.1} \left\|T qigg(\inftynt_0^x
hqigg)\right\|_{q,w,(0,\inftynfty)} \leq c
\,\|h\|_{p,v,(0,\inftynfty)}, \, h \inftyn {\mathfrak M}^+,
\end{equation}
holds iff
qegin{equation}\lambdabel{RT.thm.main.3.eq.2}
\| T_{\Phi^2} f \|_{q, w, (0,\i)} \le c \| f \|_{p, \phi,(0,\i)}, \, f \inftyn
\mathfrak M^{\deltan},
\end{equation}
holds, where
$$
\phi (x) \equiv \phiqig[v;pqig](x) : = qigg( \inftynt_0^x
v^{1-{p}^{\prime}}(t)\,dtqigg)^{- {p^{\prime}} / {(p^{\prime} + 1)}}
v^{1-{p}^{\prime}}(x)
$$
and
$$
\Phi(x) \equiv \Phi qig[v;pqig](x) = \inftynt_0^x \phi(t)\,dt = qigg( \inftynt_0^x
v^{1-{p}^{\prime}}(t)\,dtqigg)^{{1} / {(p^{\prime} + 1)}}.
$$
\end{theorem}
qegin{theorem}\cite[Theorem 3.11]{GogMusIHI}\lambdabel{RT.thm.main.8.0}
Let $0 < q \le \infty$, and let $T: \mathfrak M^+ \rightarrow \mathfrak M^+$ satisfies conditions {\rm (i)-(iii)}. Assume that $u,\,w \inftyn {\mathcal W}(0,\i)$
and $v \inftyn {\mathcal W}(0,\i)$ be such that $V(x) < \inftynfty$ for all $x > 0$.
Then inequality
qegin{equation} \lambdabel{RT.thm.main.8.0.eq.1}
\left\|T qigg(\inftynt_0^x hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{1,V^{-1},(0,\i)}, \, h \inftyn \mathfrak M^+,
\end{equation}
holds iff
qegin{equation}\lambdabel{RT.thm.main.8.0.eq.2}
\| T_{V^2} f \|_{q, w, (0,\i)} \le c \| f \|_{1, v,(0,\i)}, \, f \inftyn
\mathfrak M^{\deltan}.
\end{equation}
\end{theorem}
\sigmaection{Supremal operators on the cone of monotone functions}\lambdabel{s.3}
In this section, we give complete characterization of inequalities \eqref{supr.ineq.1} - \eqref{supr.ineq.4}.
To state the next statements we need the following notations:
$$
\omegaverline{u}(t) : = \sigmaup_{0 < \thetaau \le t} u(\thetaau), \qquaduad \munderline{u}(t) : = \sigmaup_{t \le \thetaau < \inftynfty} u(\thetaau), \qquaduad (t>0).
$$
For a given weight $v$, $0 \le a < b \le \inftynfty$ and $1 \le p < \inftynfty$, we denote
$$
\sigma_p (a,b) = qegin{cases}
qigg ( \inftynt\limits_a^b [v(t)]^{1-p'}dtqigg)^{1 / p'} & \qquaduad \mbox{when} ~ 1 < p < \inftynfty \\
\omegaperatornamewithlimits{ess\,sup}\limits_{a < t < b} \, [v(t)]^{-1} & \qquaduad \mbox{when} ~ p = 1.
\end{cases}
$$
Recall the following theorem.
qegin{theorem}\cite[Theorems 4.1 and 4.4]{gop}\lambdabel{supr.thm.101}
Let $1 \le p < \inftynfty$, $0 < q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that
$$
0 < V(x) < \infty \qquaduad \mbox{and} \qquaduad 0 < W(x) < \inftynfty \qquad \mbox{for all} \qquad x > 0.
$$
Then inequality \eqref{ISI.4}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case $c \alphapprox A_1$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [\munderline{u}]^q(x) W(x) + \inftynt_x^{\inftynfty} [\munderline{u}]^q(t) w(t)\,dtqigg)^{1 / q}\sigma_p(0,x);
$$
{\rm (ii)} $q < p$, and in this case $c \alphapprox B_1 + B_2$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} [\munderline{u}]^q(t) w(t)\,dtqigg)^{r / p} [\munderline{u}]^q(x) qigg[\sigma_p(0,x)qigg]^r w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} \munderline{u}(\thetaau) \sigma_p (0,\thetaau) qigg]^r w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
Using change of variables $x = 1/t$, we can easily obtain the following statement.
qegin{theorem}\lambdabel{supr.thm.111}
Let $1 \le p < \inftynfty$, $0 < q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that
$$
0 < V_*(x) < \infty \qquaduad \mbox{and} \qquaduad 0 < W_*(x) < \inftynfty \qquad \mbox{for all} \qquad x > 0.
$$
Then inequality \eqref{ISI.2}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case $c \alphapprox A_1$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [\omegaverline{u}]^q(x) W_*(x) + \inftynt_0^x [\omegaverline{u}]^q(t) w(t)\,dtqigg)^{1 / q}\sigma_p(x,\inftynfty);
$$
{\rm (ii)} $q < p$, and in this case $c \alphapprox B_1 + B_2$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_0^x [\omegaverline{u}]^q(t) w(t)\,dtqigg)^{r / p} [\omegaverline{u}]^q(x) qigg[\sigma_p(x,\inftynfty)qigg]^r w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W_*^{r / p}(x) qigg[\sigmaup_{0 < \thetaau \le x} \omegaverline{u}(\thetaau) \sigma_p (\thetaau,\inftynfty) qigg]^r w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
Obviously, inequality \eqref{ISI.2}
is satisfied with the best constant $c$ if and only if
qegin{equation}\lambdabel{IHI.1.1.1.1}
\left\|S_{\thetailde{u}}^* qigg( \inftynt_0^x hqigg)\right\|_{q,\thetailde{w},(0,\i)} \leq c
\,\|h\|_{p,\thetailde{v},(0,\i)},~ h \inftyn \mathfrak M^+
\end{equation}
holds, where
$$
\thetailde{u} (t) = u qigg(\frac{1}{t}qigg), ~ \thetailde{w} (t) = w qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~\thetailde{v} (t) = v qigg(\frac{1}{t}qigg)qigg(\frac{1}{t^2}qigg)^{1-p}, ~ t > 0.
$$
Using Theorem \ref{supr.thm.101}, and then applying substitution of variables mentioned above three times, we get the statement.
\end{proof}
qegin{theorem}\lambdabel{supr.thm.11}
Let $0 < p,\,q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that $0 < V_*(x) < \inftynfty$ and $0 < W_*(x) < \inftynfty$ for all $x > 0$. Then inequality \eqref{supr.ineq.1} is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + \| S_u ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{s,v,(0,\i)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [\omegaverline{u}]^q(x) W_*(x) + \inftynt_0^x [\omegaverline{u}]^q(t) w(t)\,dtqigg)^{1 / q}V^{- 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case
$c \alphapprox B_1 + B_2 + \| S_u ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{s,v,(0,\i)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_0^x [\omegaverline{u}]^q(t) w(t)\,dtqigg)^{r / p} [\omegaverline{u}]^q(x) V^{- r / p}(x) w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W_*^{r / p}(x) qigg[\sigmaup_{0 < \thetaau \le x} \omegaverline{u}(\thetaau)V^{- 1 / p}(\thetaau)qigg]^{r} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
It is easy to see that inequality \eqref{supr.ineq.1} holds if and only if
qegin{equation}\lambdabel{supr.ineq.thm.11.eq.1}
\| S_{u^p} (f)\|_{q/p,w,(0,\i)} \le c^p\|f\|_{1,v,(0,\i)},~ f \inftyn \mathfrak M^{\deltan}
\end{equation}
holds. By Theorem \ref{Reduction.Theorem.thm.3.1}, \eqref{supr.ineq.thm.11.eq.1} holds iff both
qegin{equation}\lambdabel{supr.ineq.thm.11.eq.2}
qigg\| S_{u^p} qigg(\inftynt_x^{\inftynfty} hqigg) qigg\|_{q/p,w,(0,\i)} \le c^p\|h\|_{1,V,(0,\i)},~ h \inftyn \mathfrak M^+,
\end{equation}
and
qegin{equation}\lambdabel{supr.ineq.thm.11.eq.3}
\| S_u ({qf 1})\|_{q,w,(0,\i)} \le c\|{qf 1}\|_{s,v,(0,\i)}
\end{equation}
hold. In order to complete the proof, it remains to apply Theorem \ref{supr.thm.111}.
\end{proof}
Using change of variables $x = 1/t$, we can easily obtain the following "dual" statement.
qegin{theorem}\lambdabel{supr.thm.12}
Let $0 < p,\,q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that $0 < V(x) < \inftynfty$ and $0 < W(x) < \inftynfty$ for all $x > 0$. Then \eqref{supr.ineq.2} is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + \| S_u^* ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{s,v,(0,\i)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [\munderline{u}]^q(x) W(x) + \inftynt_x^{\inftynfty} [\munderline{u}]^q(t) w(t)\,dtqigg)^{1 / q}V_*^{- 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case
$c \alphapprox B_1 + B_2 + \| S_u^* ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{s,v,(0,\i)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} [\munderline{u}]^q(t) w(t)\,dtqigg)^{r / p} [\munderline{u}]^q(x) V_*^{- r / p}(x) w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} \munderline{u}(\thetaau)V_*^{-1 / p}(\thetaau)qigg]^{r} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
It is easy to see that inequality \eqref{supr.ineq.2} is satisfied with the best constant $c$ if and only if
qegin{equation}\lambdabel{IHI.1.1.1.1.1}
\|S_{\thetailde{u}} f \|_{q,\thetailde{w},(0,\i)} \leq c
\,\|f\|_{p,\thetailde{v},(0,\i)},~ f \inftyn \mathfrak M^{\deltan}
\end{equation}
holds, where
$$
\thetailde{u} (t) = u qigg(\frac{1}{t}qigg), ~ \thetailde{w} (t) = w qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~\thetailde{v} (t) = v qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~ t > 0.
$$
Using Theorem \ref{supr.thm.11}, and then applying substitution of variables mentioned above three times, we get the statement.
\end{proof}
qegin{theorem}\lambdabel{supr.thm.23}
Let $0 < p,\,q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that $0 < V_*(x) < \inftynfty$ and $0 < W_*(x) < \inftynfty$ for all $x > 0$.
Then \eqref{supr.ineq.3} is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case $c \alphapprox A_1 + \| S_u ({qf 1}) \|_{q,w,(0,\infty)} / \| {qf 1} \|_{p,v,(0.\infty)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( qigg[\sigmaup_{0 < \thetaau \le x} \frac{u(\thetaau)^p}{V_*(\thetaau)^2}qigg]^{q / p} W_*(x) + \inftynt_0^x qigg[\sigmaup_{0 < \thetaau \le t} \frac{u(\thetaau)^p}{V_*(\thetaau)^2}qigg]^{q / p} w(t)\,dtqigg)^{1 / q}[V_*]^{ 1 / p}(x);
$$
{\rm (ii)} $0 < q < p < \inftynfty$, and in this case $c \alphapprox A_2 + \| S_u ({qf 1}) \|_{q,w,(0,\infty)} / \| {qf 1} \|_{p,v,(0.\infty)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_0^x qigg[\sigmaup_{0 < \thetaau \le t} \frac{u(\thetaau)^p}{V_*(\thetaau)^2}qigg]^{q / p} w(t)\,dtqigg)^{r / p} [V_*]^{- r / p}(x) qigg[\sigmaup_{0 < \thetaau \le x} \frac{u(\thetaau)^p}{V_*(\thetaau)^2}qigg]^{q / p} w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W_*^{r / p}(x) qigg[\sigmaup_{0 < \thetaau \le x} qigg[\sigmaup_{0 < \thetaau \le t} \frac{u(\thetaau)^p}{V_*(\thetaau)^2}qigg] V_*(\thetaau)qigg]^{r / p} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
It is easy to see that inequality \eqref{supr.ineq.3} holds if and only if
qegin{equation}\lambdabel{supr.ineq.thm.23.eq.1}
\| S_{u^p} (f)\|_{q/p,w,(0,\i)} \le c^p\|f\|_{1,v,(0,\i)},~ f \inftyn \mathfrak M^{\muparrow}
\end{equation}
holds. By Theorem \ref{Reduction.Theorem.thm.3.4} applied to the operator $S_{u^p}$, inequality \eqref{supr.ineq.thm.23.eq.1} is satisfied with the best constant $c$ if and only if
both
qegin{equation*}
\left\| S_{ u^p / V_*^2} qigg( \inftynt_{x}^{\inftynfty} hqigg) \right\|_{q/p, w, (0,\i)} \le c \|h\|_{1,1 / V_*,(0,\i)}, \, h \inftyn \mathfrak M^+,
\end{equation*}
and
qegin{equation*}
\| S_u ({qf 1}) \|_{q,w,(0,\infty)} \le c \| {qf 1} \|_{p,v,(0.\infty)}
\end{equation*}
hold. It remains to apply Theorem \ref{supr.thm.111}.
\end{proof}
The following "dual" statement holds true.
qegin{theorem}\lambdabel{supr.thm.33}
Let $0 < p,\,q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that $0 < V(x) < \inftynfty$ and $0 < W(x) < \inftynfty$ for all $x > 0$.
Then \eqref{supr.ineq.4}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case $c \alphapprox A_1 + \| S_u^* ({qf 1}) \|_{q,w,(0,\infty)} / \| {qf 1} \|_{p,v,(0.\infty)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)^2}qigg]^{q / p} W(x) + \inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)^2}qigg]^{q / p} w(t)\,dtqigg)^{1 / q} V^{ 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case $c \alphapprox B_1 + B_2 + \| S_u^* ({qf 1}) \|_{q,w,(0,\infty)} / \| {qf 1} \|_{p,v,(0.\infty)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)^2}qigg]^{q / p} w(t)\,dtqigg)^{r / p} V^{- r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)^2}qigg]^{q / p} w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)^2}qigg] V(\thetaau)qigg]^{r / p} w(x)\,dx qigg)^{1 / r}.
\end{align*}\end{theorem}
qegin{proof}
Obviously, \eqref{supr.ineq.4}
is satisfied with the best constant $c$ if and only if
qegin{equation*}
\|S_{\thetailde{u}} f \|_{q,\thetailde{w},(0,\i)} \leq c
\,\|f\|_{p,\thetailde{v},(0,\i)},~ f \inftyn \mathfrak M^{\muparrow}
\end{equation*}
holds, where
$$
\thetailde{u} (t) = u qigg(\frac{1}{t}qigg), ~ \thetailde{w} (t) = w qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~\thetailde{v} (t) = v qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~ t > 0.
$$
Using Theorem \ref{supr.thm.23}, and then applying substitution of variables mentioned above three times, we get the statement.
\end{proof}
\sigmaection{Iterated inequalities with supremal operators}\lambdabel{s.4}
In this section we characterize inequalities \eqref{ISI.1} and \eqref{ISI.3}.
The following theorem is true.
qegin{theorem}\lambdabel{supr.thm.41}
Let $1 < p < \inftynfty$, $0 < q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that
$$
0 < \inftynt_0^x v^{1-p^{\prime}}(t)\,dt < \infty \qquaduad \mbox{and} \qquaduad 0 < W_*(x) < \inftynfty \qquad \mbox{for all} \qquad x > 0.
$$
Recall that
$$
\Phi qig[v;pqig](x) = qigg( \inftynt_0^x
v^{1-{p}^{\prime}}(t)\,dtqigg)^{{1} / {(p^{\prime} + 1)}},~ x > 0.
$$
Denote by
$$
\Phi_1(x) : = \sigmaup_{0 < \thetaau \le x} u(\thetaau) \Phi^2[v;p](\thetaau), ~ x>0.
$$
Then \eqref{ISI.1}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + \| S_{u\Phi^2[v;p]} ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,\phi[v;p],(0,\i)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [\Phi_1]^q(x) W_*(x) + \inftynt_0^x [\Phi_1]^q(t) w(t)\,dtqigg)^{1 / q}\Phi[v;p]^{- 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case
$c \alphapprox B_1 + B_2 + \| S_{u\Phi^2[v;p]} ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,\phi[v;p],(0,\i)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_0^x [\Phi_1]^q(t) w(t)\,dtqigg)^{r / p} [\Phi_1]^q(x) \Phi[v;p]^{- r / p}(x) w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W_*^{r / p}(x) qigg[\sigmaup_{0 < \thetaau \le x} \Phi_1(\thetaau)\Phi[v;p]^{-1 / p}(\thetaau)qigg]^{r} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
By Theorem \ref{RT.thm.main.3} applied to the
operator $S_{u}$, inequality \eqref{ISI.1} with the best constant $c$ holds if and only if the inequality
qegin{equation}\lambdabel{RT.SC.thm.41.eq.2}
qig\| S_{u \Phi^2[v;p]} (f)qig\|_{q, w, (0,\i)} \le C \,\|
f \|_{p, \phi[v;p],(0,\i)}, \, f \inftyn \mathfrak M^{\deltan}
\end{equation}
holds. Moreover, $c \alphapprox C$. Now the statement follows by Theorem \ref{supr.thm.11}.
\end{proof}
qegin{theorem}\lambdabel{supr.thm.61}
Let $0 < q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that $0 < V(x) < \inftynfty$ and $0 < W_*(x) < \inftynfty$ for all $x > 0$. Denote by
$$
V_1(x) : = \sigmaup_{0 < \thetaau \le x} u(\thetaau) V^2(\thetaau), ~ x>0.
$$
Then
qegin{equation}\lambdabel{RT.SC.thm.51.eq.1}
\left\|S_{u} qigg( \inftynt_0^x hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{1,V^{-1},(0,\i)},
\end{equation}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + \| S_{uV^2} ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,v,(0,\i)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [V_1]^q(x) \inftynt_x^{\inftynfty} w(t)\,dt + \inftynt_0^x [V_1]^q(t) w(t)\,dtqigg)^{1 / q}V^{- 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case
$c \alphapprox B_1 + B_2 + \| S_{uV^2} ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,v,(0,\i)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_0^x [V_1]^q(t) w(t)\,dtqigg)^{r / p} [V_1]^q(x) V^{- r / p}(x) w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{0 < \thetaau \le x} [V_1](\thetaau)V^{-1 / p}(\thetaau)qigg]^{r} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
By Theorem \ref{RT.thm.main.8.0} applied to the
operator $S_{u}$, inequality \eqref{RT.SC.thm.51.eq.1}
with the best constant $c_{51}$ holds if and
only if the inequality
qegin{equation}\lambdabel{RT.SC.thm.51.eq.2}
qig\| S_{u V^2} (f)qig\|_{q, w, (0,\i)} \le C \,\| f \|_{1, v,(0,\i)}, \, f \inftyn \mathfrak M^{\deltan}
\end{equation}
holds. Moreover, $c \alphapprox C$. Now the statement follows by Theorem \ref{supr.thm.11}.
\end{proof}
The following "dual" statements also hold true.
qegin{theorem}\lambdabel{supr.thm.71}
Let $1 < p < \inftynfty$, $0 < q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that
$$
0 < \inftynt_x^{\inftynfty} v^{1-p^{\prime}}(t)\,dt < \infty \qquaduad \mbox{and} \qquaduad 0 < W(x) < \inftynfty \qquad \mbox{for all} \qquad x > 0.
$$
Recall that
$$
\Psi qig[v;pqig](x) = qigg( \inftynt_x^{\inftynfty}
v^{1-{p}^{\prime}}(t)\,dtqigg)^{{1} / {(p^{\prime}+ 1)}}, ~ x > 0.
$$
Denote by
$$
\Psi_1(x) : = \sigmaup_{x \le \thetaau < \inftynfty} u(\thetaau) \Psi^2[v;p](\thetaau), ~ x>0.
$$
Then \eqref{ISI.3} is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + \| S_{u\Psi^2[v;p]} ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,\psi[v;p],(0,\i)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [\Psi_1]^q(x) W(x) + \inftynt_x^{\inftynfty} [\Psi_1]^q(t) w(t)\,dtqigg)^{1 / q}\Psi[v;p]^{- 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case
$c \alphapprox B_1 + B_2 + \| S_{u\Psi^2[v;p]} ({qf 1})\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,\psi[v;p],(0,\i)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} [\Psi_1]^q(t) w(t)\,dtqigg)^{r / p} [\Psi_1]^q(x) \Psi[v;p]^{- r / p}(x) w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} \Psi_1(\thetaau)\Psi[v;p]^{-1 / p}(\thetaau)qigg]^{r} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
Obviously, \eqref{ISI.3}
is satisfied with the best constant $c$ if and only if
qegin{equation*}
qigg\|S_{\thetailde{u}} qigg( \inftynt_0^x hqigg) qigg\|_{q,\thetailde{w},(0,\i)} \leq c
\,\|h\|_{p,\thetailde{v},(0,\i)},~ h \inftyn \mathfrak M^+
\end{equation*}
holds, where
$$
\thetailde{u} (t) = u qigg(\frac{1}{t}qigg), ~ \thetailde{w} (t) = w qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~\thetailde{v} (t) = v qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~ t > 0.
$$
Using Theorem \ref{supr.thm.41}, and then applying substitution of variables mentioned above three times, we get the statement.
\end{proof}
qegin{theorem}\lambdabel{supr.thm.81}
Let $0 < q < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $v,\,w \inftyn {\mathcal W}(0,\i)$
be such that $0 < V_*(x) < \inftynfty$ and $0 < W_*(x) < \inftynfty$ for all $x > 0$. Denote by
$$
V_1^*(x) : = \sigmaup_{x \le \thetaau < \inftynfty} u(\thetaau) V_*^2(\thetaau), ~ x>0.
$$
Then
qegin{equation}\lambdabel{RT.SC.thm.81.eq.1}
\left\|S_{u}^* qigg( \inftynt_x^{\inftynfty} hqigg)\right\|_{q,w,(0,\i)} \leq c
\,\|h\|_{1,V_*^{-1},(0,\i)},
\end{equation}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + qig\| S_{uV_*^2}^* ({qf 1})qig\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,v,(0,\i)}$, where
$$
A_1: = \sigmaup_{x > 0}qigg( [V_1^*]^q(x) W_*(x) + \inftynt_0^x [V_1^*]^q(t) w(t)\,dtqigg)^{1 / q}V^{- 1 / p}(x);
$$
{\rm (ii)} $q < p$, and in this case
$c \alphapprox B_1 + B_2 + qig\| S_{uV_*^2}^* ({qf 1})qig\|_{q,w,(0,\i)} / \|{qf 1}\|_{p,v,(0,\i)}$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} [V_1^*]^q(t) w(t)\,dtqigg)^{r / p} [V_1^*]^q(x) V_*^{- r / p}(x) w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W_*^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} [V_1^*](\thetaau)V^{-1 / p}(\thetaau)qigg]^{r} w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
qegin{proof}
By change of variables $x = 1 / t$, it is easy to see that inequality \eqref{RT.SC.thm.81.eq.1} holds if and only if
qegin{equation}\lambdabel{IHI.1.1.1}
\left\|S_{\thetailde{u}} qigg( \inftynt_0^x hqigg)\right\|_{q,\thetailde{w},(0,\i)} \leq c
\,\|h\|_{1,\thetailde{V}^{-1},(0,\i)},~h\inftyn \mathfrak M^+
\end{equation}
holds, where
$$
\thetailde{u} (t) = u qigg(\frac{1}{t}qigg), ~ \thetailde{w} (t) = w qigg(\frac{1}{t}qigg)\frac{1}{t^2}, ~\thetailde{V} (t) = \inftynt_0^t v qigg(\frac{1}{y}qigg)\frac{1}{y^2}\,dy, ~ t > 0.
$$
Applying Theorem \ref{supr.thm.61}, and then using substitution of variables mentioned above three times, we get the statement.
\end{proof}
\sigmaection{Hardy-operator involving suprema - $T_{u,b}$}\lambdabel{s.5}
In this section we give complete characterization of inequality \eqref{Tub.thm.1.eq.1}.
\sigmaubsection{The case $1 \le p < \inftynfty$}
The following theorem is true.
qegin{theorem}\lambdabel{Tub.thm.1}
Let $0 < q \le \inftynfty$, $1 \le p < \inftynfty$ and let $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$. Assume that $b,\,v,\,w \inftyn {\mathcal W}(0,\i)$
be such that
$$
0 < B(t) < \inftynfty, ~ 0 < V(x) < \infty ~ \mbox{and} ~ 0 < W(x) < \inftynfty ~ \mbox{for all} ~ x > 0.
$$
Then inequality \eqref{Tub.thm.1.eq.1}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $1 < p \le q$, and in this case $c \alphapprox A_1 + A_2$, where
qegin{align*}
A_1: & = \sigmaup_{x > 0}qigg( qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q W(x) + \inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(t)\,dtqigg)^{1 / q}qigg(\inftynt_0^x qigg(\frac{B(y)}{V(y)}qigg)^{p'}v(y)\,dyqigg)^{1 / p'}, \\
A_2: & = \sigmaup_{x > 0}qigg( qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q W(x) + \inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q w(t)\,dtqigg)^{1 / q}qigg(\inftynt_0^x V^{p'}(y)v(y)\,dyqigg)^{1 / p'};
\end{align*}
{\rm (ii)} $1 = p \le q$, and in this case $c \alphapprox A_1 + A_2$, where
qegin{align*}
A_1: & = \sigmaup_{x > 0}qigg( qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q W(x) + \inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(t)\,dtqigg)^{1 / q}qigg(\sigmaup_{0 < y \le x} \frac{B(y)}{V(y)}qigg), \\
A_2: & = \sigmaup_{x > 0}qigg( qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q W(x) + \inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q w(t)\,dtqigg)^{1 / q}V(x);
\end{align*}
{\rm (iii)} $1 < p$ and $q < p$, and in this case $c \alphapprox B_1 + B_2 + B_3 + B_4$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q qigg(\inftynt_0^x qigg(\frac{B(y)}{V(y)}qigg)^{p'}v(y)\,dyqigg)^{r / p'} w(x)\,dx qigg)^{1/r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)}{B(y)}qigg] qigg(\inftynt_0^x qigg(\frac{B(y)}{V(y)}qigg)^{p'}v(y)\,dyqigg)^{1 / p'} qigg]^r w(x)\,dx qigg)^{1/r},\\
B_3: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q qigg(\inftynt_0^x V^{p'}(y)v(y)\,dyqigg)^{r / p'} w(x)\,dx qigg)^{1/r}, \\
B_4: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)}{V^2(y)}qigg] qigg(\inftynt_0^x V^{p'}(y)v(y)\,dyqigg)^{1 / p'} qigg]^r w(x)\,dx qigg)^{1/r}.
\end{align*}
{\rm (iv)} $q < 1 = p$, and in this case $c \alphapprox B_1 + B_2 + B_3 + B_4$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q qigg(\sigmaup_{0 < y \le x} \frac{B(y)}{V(y)}qigg)^{r} w(x)\,dx qigg)^{1/r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)}{B(y)}qigg] qigg(\sigmaup_{0 < y \le x} \frac{B(y)}{V(y)}qigg) qigg]^r w(x)\,dx qigg)^{1/r},\\
B_3: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{V^2(\thetaau)}qigg]^q V^r(x) w(x)\,dx qigg)^{1/r}, \\
B_4: & = qigg(\inftynt_0^{\inftynfty} W^{r / p}(x) qigg[\sigmaup_{x \le \thetaau < \inftynfty} qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)}{V^2(y)}qigg] V(x) qigg]^r w(x)\,dx qigg)^{1/r}.
\end{align*}
\end{theorem}
qegin{proof}
By Theorem \ref{Reduction.Theorem.thm.3.1}, \eqref{Tub.thm.1.eq.1} holds iff both
qegin{equation}\lambdabel{Tub.thm.1.eq.5}
qigg\| T_{u,b}qigg( \inftynt_x^{\infty} h qigg)qigg\|_{q,w,(0,\i)} \le c \| h
\|_{p, V^p v^{1-p},(0,\i)}, \qquad h \inftyn \mathfrak M^+(0,\infty).
\end{equation}
and
qegin{equation}\lambdabel{Tub.thm.1.eq.6}
\|T_{u,b} {qf 1}\|_{q,w,(0,\i)} \le c \|{qf 1}\|_{p,v,(0,\i)}
\end{equation}
hold.
Note that
qegin{align*}
T_{u,b}qigg( \inftynt_{t}^{\infty} h qigg) (x) & = \sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \inftynt_0^{\thetaau} qigg( \inftynt_s^{\inftynfty} h(y)\,dyqigg) b(s)\,ds \\
& \alphapprox \sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \inftynt_0^{\thetaau} h(y)B(y)\,dy + \sigmaup_{x \le \thetaau < \inftynfty} u(\thetaau) \inftynt_{\thetaau}^{\inftynfty} h(s)\,ds \\
& = S_{u/B}^* qigg(\inftynt_0^{\thetaau} hBqigg) + S_u^* qigg(\inftynt_{\thetaau}^{\inftynfty} hqigg).
\end{align*}
Hence, inequality \eqref{Tub.thm.1.eq.1}
holds iff inequalities
qegin{equation}\lambdabel{Tub.thm.1.eq.2}
qigg\|S_{u/B}^* qigg(\inftynt_0^{\thetaau} hqigg)qigg\|_{q,w,(0,\i)} \le c \| h
\|_{p, B^{-p}V^p v^{1-p},(0,\i)}, \qquad h \inftyn \mathfrak M^+(0,\infty),
\end{equation}
qegin{equation}\lambdabel{Tub.thm.1.eq.3}
qigg\| S_u^* qigg(\inftynt_{\thetaau}^{\inftynfty} hqigg) qigg\|_{q,w,(0,\i)} \le c \| h
\|_{p, V^p v^{1-p},(0,\i)}, \qquad h \inftyn \mathfrak M^+(0,\infty).
\end{equation}
and \eqref{Tub.thm.1.eq.6} hold.
Again by Theorem \ref{Reduction.Theorem.thm.3.1}, \eqref{Tub.thm.1.eq.3} with \eqref{Tub.thm.1.eq.6} is equivalent to
qegin{equation}\lambdabel{Tub.thm.1.eq.4}
\|S_u^* f \|_{q,w,(0,\i)} \le c \| f \|_{p,v,(0,\i)}, \qquad f \inftyn \mathfrak M^{\deltan}(0,\infty).
\end{equation}
Now by Theorem \ref{Reduction.Theorem.thm.3.2}, \eqref{Tub.thm.1.eq.4} is equivalent to
qegin{equation}\lambdabel{Tub.thm.1.eq.5.0}
qigg\| S_{u / V^2}^* qigg( \inftynt_0^x h qigg)qigg\|_{q,w,(0,\i)}
\le c \| h \|_{p, V^{-p}v^{1-p},(0,\i)}, \qquad h \inftyn \mathfrak M^+(0,\infty).
\end{equation}
Consequently, \eqref{Tub.thm.1.eq.1}
holds iff inequalities \eqref{Tub.thm.1.eq.2} and \eqref{Tub.thm.1.eq.5.0} hold.
{\rm (i)} $p \le q$. By Theorem \ref{supr.thm.101}, \eqref{Tub.thm.1.eq.2} and \eqref{Tub.thm.1.eq.5.0} hold if and only if both
$A_1 < \inftynfty$ and $A_2 < \inftynfty$
hold, respectively.
{\rm (ii)} $q < p$. By Theorem \ref{supr.thm.101}, \eqref{Tub.thm.1.eq.2} and \eqref{Tub.thm.1.eq.5.0} hold if and only if
$B_i < \inftynfty$, $i=1,2,3,4$
hold, respectively.
\end{proof}
\sigmaubsection{The case $0< p < 1$}
We start with a simple observation. If $0 < p \le 1$ and $t \inftyn (0,\i)$, then
qegin{equation}\lambdabel{RT.SO.eq.1.1}
\sigmaup_{0 < \thetaau \le t} f (\thetaau) B(\thetaau) \le \inftynt_0 ^t f(y)b(y)\,dy \lesssim qigg( \inftynt_0^t f(y)^p B(y)^{p-1} b(y)\,dyqigg) ^{{1} / {p}}, ~ f \inftyn \mathfrak M^{\deltan}.
\end{equation}
Since $f$ is non-increasing, the first inequality in \eqref{RT.SO.eq.1.1} is obvious. The second one follows, for instance, from the fact that (see, for instance, \cite[Theorem 3.2]{carsor1993}, cf. also \cite{ss})
$$
\sigmaup_{f\inftyn \mathfrak M^{\deltan}:\,f \not \sigmaim 0} \frac{\inftynt_0^{\inftynfty} f(x)g(x)\,dx}{qigg(\inftynt_0^{\inftynfty} f(x)^p v(x)\,dxqigg)^{1/p}} \alphapprox \sigmaup_{t > 0} qigg( \inftynt_0^t g(x)\,dx qigg(\inftynt_0^t v(x)\,dxqigg)^{- 1/ p}qigg).
$$
Our first aim is to prove a reduction theorem for the operator $T_{u,b}$. We first note that, using the monotonicity of $\inftynt_0^t fb$ and interchanging the suprema, we get
$$
(T_{u,b} g)(t) = \sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \inftynt_0^{\thetaau} g(y)b(y)\,dy = \sigmaup_{t \le \thetaau < \inftynfty} qigg(\sigmaup_{\thetaau \le x < \inftynfty}\frac{u(x)}{B(x)}qigg) \inftynt_0^{\thetaau} g(y)b(y)\,dy,\qquaduad t \inftyn (0,\i).
$$
As a consequence, we can safely assume that ${u(x)} / {B(x)}$ is non-increasing on $(0,\i)$, since otherwise we would just replace ${u(x)} / {B(x)}$ by $\sigmaup_{\thetaau \le x < \inftynfty}{u(x)} / {B(x)}$.
qegin{theorem}\lambdabel{RT.SO.thm.2}
Let $0 < p \le 1$, $0 < q < \inftynfty$. Assume that $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$ and $b,\,v,\,w \inftyn {\mathcal W}(0,\i)$ be such that $0 < B(t) < \inftynfty$ for all $x > 0$. Then the following three statements are equivalent:
qegin{align}
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}\inftynt_0^{\thetaau} f(y) b(y) \,dyqigg)^q w(t)\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}; \lambdabel{RT.SO.thm.2.eq.1} \\
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{t \le \thetaau < \inftynfty} qigg(\frac{u(\thetaau)}{B(\thetaau)}qigg)^p \inftynt_0^{\thetaau} f(y) B(y)^{p-1}b(y) \,dyqigg)^{q / p} w(t)\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau) v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}; \lambdabel{RT.SO.thm.2.eq.2} \\
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \sigmaup_{0 < y \le \thetaau} f(y) B(y) qigg)^q w(t)\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}. \lambdabel{RT.SO.thm.2.eq.3}
\end{align}
\end{theorem}
qegin{proof}
Again, in view of \eqref{RT.SO.eq.1.1}, the implications $\eqref{RT.SO.thm.2.eq.2} \mathbb Rightarrow \eqref{RT.SO.thm.2.eq.1} \mathbb Rightarrow \eqref{RT.SO.thm.2.eq.3}$ are obvious, and it just remains to show that \eqref{RT.SO.thm.2.eq.3} implies \eqref{RT.SO.thm.2.eq.2}.
Suppose, thus, that \eqref{RT.SO.thm.2.eq.3} holds. Since $u(x) / B(x)$ is non-increasing, we have
qegin{align*}
\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \sigmaup_{0 < y \le \thetaau} f(y) B(y) & = \max qigg\{ \sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \sigmaup_{0 < y \le t} f(y) B(y),\,\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} \sigmaup_{t < y \le \thetaau} f(y) B(y) qigg\} \\
& = \max qigg\{ \frac{u(t)}{B(t)} \sigmaup_{0 < y \le t} f(y) B(y),\,\sigmaup_{t \le y < \inftynfty} f(y) B(y) \sigmaup_{y \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)} qigg\} \\
& = \max qigg\{ \frac{u(t)}{B(t)} \sigmaup_{0 < y \le t} f(y) B(y),\,\sigmaup_{t \le y < \inftynfty} f(y) u(y) qigg\}.
\end{align*}
Hence, \eqref{RT.SO.thm.2.eq.3} breaks down into the following two inequalities:
qegin{align}
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{0 < y \le t} f(y) B(y) qigg)^q w(t) qigg(\frac{u(t)}{B(t)}qigg)^q\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}, \lambdabel{RT.SO.thm.2.eq.4} \\
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{t \le y < \inftynfty} f(y) u(y) qigg)^q w(t) \,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}. \lambdabel{RT.SO.thm.2.eq.5}
\end{align}
Obviously, \eqref{RT.SO.thm.2.eq.4} and \eqref{RT.SO.thm.2.eq.5} are equivalent to
qegin{align}
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{0 < y \le t} f(y) B(y)^p qigg)^{q / p} w(t) qigg(\frac{u(t)}{B(t)}qigg)^q\,dtqigg)^{p / q} & \lesssim \inftynt_0^{\inftynfty} f(\thetaau) v(\thetaau)\,d\thetaau, ~ f \inftyn \mathfrak M^{\deltan}, \lambdabel{RT.SO.thm.2.eq.4.1} \\
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{t \le y < \inftynfty} f(y) u(y)^p qigg)^{q / p} w(t) \,dtqigg)^{p / q} & \lesssim \inftynt_0^{\inftynfty} f(\thetaau) v(\thetaau)\,d\thetaau, ~ f \inftyn \mathfrak M^{\deltan}. \lambdabel{RT.SO.thm.2.eq.5.1}
\end{align}
{\rm (i)} Let $p \le q$. By Theorem \ref{supr.thm.11}, \eqref{RT.SO.thm.2.eq.4.1} holds iff both
qegin{equation}\lambdabel{RT.SO.thm.2.eq.9}
\sigmaup_{x > 0} qigg(\inftynt_0^x u(t)^q w(t)\,dt + B(x)^q\inftynt_x^{\inftynfty} qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q}V^{- 1 / p}(x) < \inftynfty
\end{equation}
and
qegin{equation}\lambdabel{RT.SO.thm.2.eq.10}
qigg( \inftynt_0^{\inftynfty} u(t)^q w(t)\,dt qigg)^{1 / q} \lesssim qigg( \inftynt_0^{\inftynfty} v(\thetaau)\,d\thetaauqigg)^{1 / p}
\end{equation}
hold.
By Theorem \ref{supr.thm.33}, \eqref{RT.SO.thm.2.eq.5.1} holds iff both
qegin{equation}\lambdabel{RT.SO.thm.2.eq.12}
\sigmaup_{x > 0}qigg( qigg[ \sigmaup_{x \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} \inftynt_0^x w(t)\,dt + \inftynt_x^{\inftynfty} qigg[ \sigmaup_{t \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} w(t)\,dtqigg)^{1 / q} V^{1 / p}(x) < \inftynfty
\end{equation}
and
qegin{equation}\lambdabel{RT.SO.thm.2.eq.13}
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{t \le \thetaau < \inftynfty} u(\thetaau)^pqigg)^{q/p} w(t)\,dt qigg)^{1 / q} \lesssim qigg( \inftynt_0^{\inftynfty} v(\thetaau)\,d\thetaauqigg)^{1 / p}
\end{equation}
hold.
On the other hand, by Theorem \ref{Tub.thm.1}, \eqref{RT.SO.thm.2.eq.2} holds iff inequalities
qegin{align}
\sigmaup_{x > 0} qigg(qigg[ \frac{u(x)}{B(x)}qigg]^q \inftynt_0^x w(t)\,dt + \inftynt_x^{\inftynfty} qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q}\sigmaup_{0 < \thetaau \le x}\frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)} & < \inftynfty \lambdabel{RT.SO.thm.2.eq.14} \\
\sigmaup_{x > 0} qigg(qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V^2(\thetaau)}qigg]^{q / p} \inftynt_0^x w(t)\,dt + \inftynt_x^{\inftynfty} qigg[ \sigmaup_{t \le \thetaau < \inftynfty}\frac{u(\thetaau)^p}{V^2(\thetaau)}qigg]^{q / p} w(t)\,dt qigg)^{1 / q} V^{ 1 / p}(x) & < \inftynfty \lambdabel{RT.SO.thm.2.eq.14.1}
\end{align}
hold.
We will thus be done if we can show that \eqref{RT.SO.thm.2.eq.9} together with \eqref{RT.SO.thm.2.eq.12} imply \eqref{RT.SO.thm.2.eq.14}. The latter can be proved as follows:
Since
$$
\sigmaup_{x > 0} qigg(\inftynt_x^{\inftynfty} qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q}\sigmaup_{0 < \thetaau \le x}\frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)} = \sigmaup_{x > 0} qigg(\inftynt_x^{\inftynfty} qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q}\frac{B(x)}{V^{ 1 / p}(x)},
$$
it remains to show that
qegin{align*}
\sigmaup_{x > 0} \frac{u(x)}{B(x)} qigg(\inftynt_0^x w(t)\,dt qigg)^{1 / q}\sigmaup_{0 < \thetaau \le x}\frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)} & \\
& \hspace{-3cm} \lesssim \sigmaup_{x > 0} \frac{u(x)}{V^{1 / p}(x)} qigg(\inftynt_0^x w(t)\,dt qigg)^{1 / q} + \sigmaup_{x > 0} \frac{B(x)}{V^{1 / p}(x)} qigg(\inftynt_x^{\inftynfty} qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q}.
\end{align*}
Interchanging the suprema, using the monotonicity of $u / B$, we get that
qegin{align*}
\sigmaup_{x > 0} \frac{u(x)}{B(x)} qigg(\inftynt_0^x w(t)\,dt qigg)^{1 / q}\sigmaup_{0 < \thetaau \le x}\frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)} & \\
& \hspace{-4cm} = \sigmaup_{\thetaau > 0} \frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)}\sigmaup_{\thetaau \le x < \inftynfty} \frac{u(x)}{B(x)} qigg(\inftynt_0^x w(t)\,dt qigg)^{1 / q} \\
& \hspace{-4cm} \lesssim \sigmaup_{\thetaau > 0} \frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)}qigg(\sigmaup_{\thetaau \le x < \inftynfty} \frac{u(x)}{B(x)} qigg)qigg(\inftynt_0^{\thetaau} w(t)\,dt qigg)^{1 / q} + \sigmaup_{\thetaau > 0} \frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)}\sigmaup_{\thetaau \le x < \inftynfty} \frac{u(x)}{B(x)} qigg(\inftynt_{\thetaau}^x w(t)\,dt qigg)^{1 / q} \\
& \hspace{-4cm} \lesssim \sigmaup_{\thetaau > 0} \frac{u(\thetaau)}{V^{ 1 / p}(\thetaau)}qigg(\inftynt_0^{\thetaau} w(t)\,dt qigg)^{1 / q} + \sigmaup_{\thetaau > 0} \frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)}\sigmaup_{\thetaau \le x < \inftynfty} qigg(\inftynt_{\thetaau}^x qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q} \\
& \hspace{-4cm} = \sigmaup_{\thetaau > 0} \frac{u(\thetaau)}{V^{ 1 / p}(\thetaau)}qigg(\inftynt_0^{\thetaau} w(t)\,dt qigg)^{1 / q} + \sigmaup_{\thetaau > 0} \frac{B(\thetaau)}{V^{ 1 / p}(\thetaau)} qigg(\inftynt_{\thetaau}^{\inftynfty} qigg[ \frac{u(t)}{B(t)}qigg]^q w(t)\,dt qigg)^{1 / q}.
\end{align*}
{\rm (ii)} Let $q < p$. By Theorem \ref{supr.thm.11}, \eqref{RT.SO.thm.2.eq.4.1} holds iff both
qegin{align*}
\inftynt_0^{\inftynfty} qigg(\inftynt_0^x u(t)^q w(t)\,dtqigg)^{r / p} u(x)^q V^{- r / p}(x) w(x)\,dx & < \inftynfty, \\
\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg(\frac{u(t)}{B(t)}qigg)^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{0 < \thetaau \le x} \frac{B(\thetaau)}{V^{1 / p}(\thetaau)}qigg]^{r}w(x) qigg(\frac{u(x)}{B(x)}qigg)^q\,dx & < \inftynfty.
\end{align*}
By Theorem \ref{supr.thm.33}, \eqref{RT.SO.thm.2.eq.5.1} holds iff inequalities
qegin{align*}
\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[ \sigmaup_{t \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} w(t)\,dtqigg)^{r / p} qigg[ \sigmaup_{x \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} V^{r / p}(x) w(x)\,dx & < \inftynfty, \\
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dt qigg)^{r / p} qigg(\sigmaup_{x \le \thetaau < \inftynfty} qigg[ \sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg] V(\thetaau) qigg)^{r / p} w(x)\,dx & < \inftynfty.
\end{align*}
On the other hand, by Theorem \ref{Tub.thm.1}, \eqref{RT.SO.thm.2.eq.2} holds iff
qegin{align*}
\inftynt_0^{\inftynfty} qigg( \inftynt_x^{\inftynfty} qigg[\frac{u(t)}{B(t)}qigg]^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)}qigg]^{r / p}w(x) qigg[\frac{u(x)}{B(x)}qigg]^q\,dx & < \inftynfty, \\
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dt qigg)^{r / p}qigg(\sigmaup_{x \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p}w(x)\,dx & < \inftynfty,\\
\inftynt_0^{\inftynfty} qigg( \inftynt_x^{\inftynfty}qigg[\sigmaup_{t \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} w(t)\,dtqigg)^{r / p}
qigg[\sigmaup_{x \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} V^{r / p}(x)w(x)\,dx & < \inftynfty, \\
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dtqigg)^{r / p} qigg( \sigmaup_{x \le \thetaau < \inftynfty}qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg] V(\thetaau)qigg)^{r / p} w(x)\,dx & \\
= \inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dtqigg)^{r / p} qigg( \sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)}qigg)^{r / p} w(x)\,dx & < \inftynfty.
\end{align*}
Obviously, it remains to show that
qegin{align*}
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dt qigg)^{r / p}qigg(\sigmaup_{x \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p}w(x)\,dx & \\
& \hspace{-5.5cm} \lesssim
\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg(\frac{u(t)}{B(t)}qigg)^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{0 < \thetaau \le x} \frac{B(\thetaau)}{V^{1 / p}(\thetaau)}qigg]^{r}w(x) qigg(\frac{u(x)}{B(x)}qigg)^q\,dx
\\
& \hspace{-5cm} +
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dtqigg)^{r / p} qigg( \sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)}qigg)^{r / p} w(x)\,dx .
\end{align*}
We will prove the assertion only in the case when $\inftynt_0^{\inftynfty} w(\thetaau)\,d\thetaau = \inftynfty$. Let $\{x_k\}$ be such that $\inftynt_0^{x_k} w(\thetaau)\,d\thetaau = 2^k$. Then
qegin{align*}
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dt qigg)^{r / p}qigg(\sigmaup_{x \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p}w(x)\,dx & \\
& \hspace{-5cm} \alphapprox \sigmaum_{k \inftyn \mathbb Z} 2^{k r / q} qigg(\sigmaup_{x_k \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p}.
\end{align*}
Note that
qegin{align*}
\sigmaup_{x_k \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)} qigg] & \alphapprox \sigmaup_{x_k \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le x_k} \frac{B(t)^p}{V(t)} qigg] +
\sigmaup_{x_k \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{x_k \le t \le \thetaau} \frac{B(t)^p}{V(t)} qigg] \\
& = qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_k} \frac{B(t)^p}{V(t)} qigg] +
\sigmaup_{x_k \le t < \inftynfty} \frac{B(t)^p}{V(t)} \sigmaup_{t \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p \\
& = qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_k} \frac{B(t)^p}{V(t)} qigg] +
\sigmaup_{x_k \le t < \inftynfty} \frac{u(t)^p}{V(t)} \\
& \alphapprox qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_{k-1}} \frac{B(t)^p}{V(t)} qigg] + qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{x_{k-1} < t \le x_k} \frac{B(t)^p}{V(t)} qigg] +
\sigmaup_{x_k \le t < \inftynfty} \frac{u(t)^p}{V(t)}
\\
& \le qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_{k-1}} \frac{B(t)^p}{V(t)} qigg] +
\sigmaup_{x_{k-1} \le t < x_k} \frac{u(t)^p}{V(t)} +
\sigmaup_{x_k \le t < \inftynfty} \frac{u(t)^p}{V(t)} \\
& \le qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_{k-1}} \frac{B(t)^p}{V(t)} qigg] +
\sigmaup_{x_{k-1} \le t < \inftynfty} \frac{u(t)^p}{V(t)}.
\end{align*}
Hence,
qegin{align*}
\inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dt qigg)^{r / p}qigg(\sigmaup_{x \le \thetaau < \inftynfty} qigg[\frac{u(\thetaau)}{B(\thetaau)}qigg]^p qigg[\sigmaup_{0 < t \le \thetaau} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p}w(x)\,dx & \\
& \hspace{-7cm} \lesssim \sigmaum_{k \inftyn \mathbb Z} 2^{k r / q} qigg(qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_{k-1}} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p} + \sigmaum_{k \inftyn \mathbb Z} 2^{k r / q} qigg(\sigmaup_{x_{k-1} \le t < \inftynfty} \frac{u(t)^p}{V(t)}qigg)^{r / p} \\
& \hspace{-7cm} \alphapprox \sigmaum_{k \inftyn \mathbb Z} qigg(\inftynt_{x_{k-1}}^{x_k} qigg(\inftynt_x^{x_k} wqigg)^{r / p}w(x)\,dxqigg) qigg(qigg[\frac{u(x_k)}{B(x_k)}qigg]^p qigg[\sigmaup_{0 < t \le x_{k-1}} \frac{B(t)^p}{V(t)} qigg]qigg)^{r / p} \\
& \hspace{-6.5cm} + \sigmaum_{k \inftyn \mathbb Z} qigg(\inftynt_{x_{k-2}}^{x_{k-1}} qigg(\inftynt_{x_k}^x wqigg)^{r / p}w(x)\,dxqigg) qigg(\sigmaup_{x_{k-1} \le t < \inftynfty} \frac{u(t)^p}{V(t)}qigg)^{r / p} \\
& \hspace{-7cm} \lesssim \sigmaum_{k \inftyn \mathbb Z} \inftynt_{x_{k-1}}^{x_k} qigg(\inftynt_x^{\inftynfty} qigg(\frac{u(t)}{B(t)}qigg)^q w(t)\,dtqigg)^{r / p}qigg[\sigmaup_{0 < \thetaau \le x} \frac{B(\thetaau)}{V^{1 / p}(\thetaau)}qigg]^{r}qigg[\frac{u(x)}{B(x)}qigg]^q w(x)\,dx \\
& \hspace{-6.5cm} + \sigmaum_{k \inftyn \mathbb Z} \inftynt_{x_{k-1}}^{x_{k+1}} qigg( \inftynt_0^x w(t)\,dtqigg)^{r / p} qigg( \sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)}qigg)^{r / p} w(x)\,dx \\
& \hspace{-7cm} \lesssim
\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg(\frac{u(t)}{B(t)}qigg)^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{0 < \thetaau \le x} \frac{B(\thetaau)}{V^{1 / p}(\thetaau)}qigg]^{r}w(x) qigg(\frac{u(x)}{B(x)}qigg)^q\,dx \\
& \hspace{-6.5cm} + \inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dtqigg)^{r / p} qigg( \sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)^p}{V(\thetaau)}qigg)^{r / p} w(x)\,dx.
\end{align*}
\end{proof}
qegin{remark}
Note that Theorem \ref{RT.SO.thm.2}, namely the fact that $\eqref{RT.SO.thm.2.eq.2} \Leftrightarrow \eqref{RT.SO.thm.2.eq.1} \Leftrightarrow \eqref{RT.SO.thm.2.eq.3}$, when $b \equiv 1$, was proved in \cite{gogpick2007}.
\end{remark}
As a corollary we obtain that for all the three operators mentioned in \eqref{RT.SO.eq.1.1}, the corresponding weighted inequalities are equivalent. It is worth noticing that this is not so when $p > 1$.
qegin{corollary}\lambdabel{RT.SO.thm.1}
Assume that $0 < p \le 1$, $0 < q < \inftynfty$, and $v,\,w \inftyn {\mathcal W}(0,\i)$. Let $b$ be a weight on $(0,\i)$
such that $0 < B(t) < \inftynfty$ for every $t \inftyn (0,\i)$. Then the following three statements are equivalent:
qegin{align}
qigg( \inftynt_0^{\inftynfty} qigg( \inftynt_0^t f(\thetaau) b(\thetaau) \,d\thetaauqigg)^q w(t)\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}; \lambdabel{RT.SO.eq.1.2} \\
qigg( \inftynt_0^{\inftynfty} qigg( \inftynt_0^t f(\thetaau)^p B(\thetaau)^{p-1}b(\thetaau) \,d\thetaauqigg)^{q / p} w(t)\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}; \lambdabel{RT.SO.eq.1.3} \\
qigg( \inftynt_0^{\inftynfty} qigg( \sigmaup_{0 < \thetaau \le t} f(\thetaau) B(\thetaau) qigg)^q w(t)\,dtqigg)^{1 / q} & \lesssim qigg( \inftynt_0^{\inftynfty} f(\thetaau)^p v(\thetaau)\,d\thetaauqigg)^{1 / p}, ~ f \inftyn \mathfrak M^{\deltan}. \lambdabel{RT.SO.eq.1.4}
\end{align}
\end{corollary}
This fact was proved in \cite[Theorem 2.1]{gogpick2007}, when $b \equiv 1$. Recently, in \cite[Theorem 3.9]{GogStep}, it was proved that $\eqref{RT.SO.eq.1.2} \Leftrightarrow \eqref{RT.SO.eq.1.4}$ for more general Volterra operators with continuous Oinarov kernels in the case when $0 < q < p \le 1$.
qegin{proof}
The proof immediately follows from Theorem \ref{RT.SO.thm.2} taking $u \equiv 1$.
\end{proof}
By the way we have proved the following statement.
qegin{theorem}\lambdabel{RT.SO.thm.3}
Let $0 < p \le 1$, $0 < q < \inftynfty$. Assume that $u \inftyn {\mathcal W}(0,\i) \cap C(0,\i)$ and $b,\,v,\,w \inftyn {\mathcal W}(0,\i)$ be such that $0 < V(t) < \inftynfty$ and $0 < B(t) < \inftynfty$ for all $x > 0$.
Then inequality \eqref{Tub.thm.1.eq.1}
is satisfied with the best constant $c$ if and only if:
{\rm (i)} $p \le q$, and in this case
$c \alphapprox A_1 + A_2$, where
qegin{align*}
A_1: & = \sigmaup_{x > 0} qigg( qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q \inftynt_0^x w(t)\,dt + \inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(t)\,dtqigg)^{1 / q} \sigmaup_{0 < y \le x} \frac{B(y)}{V^{1 / p}(y)}; \\
A_2: & = \sigmaup_{x > 0}qigg( qigg[ \sigmaup_{x \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} \inftynt_0^x w(t)\,dt + \inftynt_x^{\inftynfty} qigg[ \sigmaup_{t \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} w(t)\,dtqigg)^{1 / q} V^{1 / p}(x);
\end{align*}
{\rm (ii)} $q < p$, and in this case $c \alphapprox B_1 + B_2 + B_3 + B_4$, where
qegin{align*}
B_1: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_0^x w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{x \le \thetaau < \inftynfty} qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)}{B(y)}qigg]^q qigg( \sigmaup_{0 < y \le x} \frac{B(y)}{V^{1 / p}(y)} qigg)qigg]^r w(x)\,dx qigg)^{1 / r}, \\
B_2: & = qigg(\inftynt_0^{\inftynfty} qigg(\inftynt_x^{\inftynfty} qigg[\sigmaup_{t \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(t)\,dtqigg)^{r / p} qigg[\sigmaup_{0 < \thetaau \le x} \frac{B(\thetaau)}{V^{1 / p}(\thetaau)}qigg]^{r}qigg[\sigmaup_{x \le \thetaau < \inftynfty} \frac{u(\thetaau)}{B(\thetaau)}qigg]^q w(x)\,dx qigg)^{1 / r}, \\
B_3: & = qigg( \inftynt_0^{\inftynfty} qigg( \inftynt_0^x w(t)\,dtqigg)^{r / p} qigg( \sigmaup_{x \le \thetaau < \inftynfty}qigg[\sigmaup_{\thetaau \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg] V(\thetaau)qigg)^{r / p} w(x)\,dx qigg)^{1 / r}, \\
B_4: & = qigg(\inftynt_0^{\inftynfty} qigg( \inftynt_x^{\inftynfty}qigg[\sigmaup_{t \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} w(t)\,dtqigg)^{r / p}
qigg[\sigmaup_{x \le y < \inftynfty} \frac{u(y)^p}{V^2(y)}qigg]^{q / p} V^{r / p}(x)w(x)\,dx qigg)^{1 / r}.
\end{align*}
\end{theorem}
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\end{document} | math | 80,689 |
\begin{document}
\title[Meridian Surfaces of elliptic or hyperbolic type in the
Minkowski 4-space] {Meridian Surfaces of elliptic or hyperbolic type in the
Four-dimensional Minkowski Space}
\author{Georgi Ganchev and Velichka Milousheva}
\address{Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,
Acad. G. Bonchev Str. bl. 8, 1113 Sofia, Bulgaria}
\email{[email protected]}
\address{Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,
Acad. G. Bonchev Str. bl. 8, 1113, Sofia, Bulgaria; "L. Karavelov"
Civil Engineering Higher School, 175 Suhodolska Str., 1373 Sofia,
Bulgaria} \email{[email protected]}
\subjclass[2000]{Primary 53A35, Secondary 53A55, 53A10}
\keywords{Meridian surfaces in Minkowski space, surfaces with
constant Gauss curvature, surfaces with constant mean curvature,
Chen surfaces, surfaces with parallel normal bundle}
\begin{abstract}
We consider a special class of spacelike surfaces in the Minkowski
4-space which are one-parameter systems of meridians of the rotational
hypersurface with timelike or spacelike axis. We call these surfaces
meridian surfaces of elliptic or hyperbolic type, respectively.
On the base of our invariant theory of surfaces we study meridian surfaces with special
invariants and give the complete
classification of the meridian surfaces with
constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces and the meridian surfaces
with parallel normal bundle.
\end{abstract}
\maketitle
\section{Introduction}
One of the fundamental problems of the contemporary
differential geometry of surfaces and hypersurfaces in the
standard model spaces such as the Euclidean space $\mathbb R^n$ and the
pseudo-Euclidean space $\mathbb R^n_k$ is the investigation of the basic
invariants characterizing the surfaces. Our aim is to study and
classify various important classes of surfaces in the
four-dimensional Minkowski space $\mathbb R^4_1$ characterized by
conditions on their invariants.
An invariant theory of spacelike surfaces in $\mathbb R^4_1$ was
developed by the present authors in \cite {GM5}.
We introduced an invariant linear map $\gamma$ of Weingarten-type in the tangent plane at
any point of the surface, which generates two invariant functions $k = \det \gamma$ and $\varkappa= -\displaystyle{ \frac{1}{2}}\, \mathrm{tr} \gamma$.
On the base of the map $\gamma$ we introduced principal lines and a
geometrically determined moving frame field at each point of the
surface. Writing derivative formulas of Frenet-type for this frame
field, we obtained eight invariant functions $\gamma_1, \,
\gamma_2, \, \nu_1,\, \nu_2, \, \lambda, \, \mu, \, \beta_1,
\beta_2$ and proved a fundamental theorem of Bonnet-type, stating
that these eight invariants under some natural conditions
determine the surface up to a rigid motion in $\mathbb R^4_1$.
The basic geometric classes of surfaces in $\mathbb R^4_1$ are
characterized by conditions on these invariant functions. For
example, surfaces with flat normal connection are characterized by
the condition $\nu_1 = \nu_2$, minimal surfaces are described by
$\nu_1 + \nu_2 = 0$, Chen surfaces are characterized by $\lambda
= 0$, and surfaces with parallel normal bundle are characterized by the condition $\beta_1 = \beta_2 = 0$.
In \cite{GM2} we constructed special two-dimensional surfaces in the Euclidean 4-space $\mathbb R^4$
which are one-parameter systems of meridians of the rotational
hypersurface and called these surfaces \emph{meridian
surfaces}. We classified the meridian surfaces with
constant Gauss curvature, constant mean curvature, and constant
invariant $k$ \cite{GM2}. In \cite{GM-new} we gave the invariants $\gamma_1, \,
\gamma_2, \, \nu_1,\, \nu_2, \, \lambda, \, \mu, \, \beta_1,
\beta_2$ of the meridian surfaces and on the base of these
invariants we classified completely the Chen meridian surfaces
and the meridian surfaces with parallel normal bundle.
Similarly to the Euclidean case, in \cite{GM6} we constructed
two-dimensional spacelike surfaces in the Minkowski 4-space $\mathbb R^4_1$ which are
one-parameter systems of meridians of the rotational hypersurface with
timelike or spacelike axis. We called these surfaces \emph{meridian surfaces of elliptic type} and \emph{meridian surfaces of hyperbolic type},
respectively.
The geometric construction of the meridian surfaces is different from the construction of the standard rotational
surfaces with two-dimensional axis. Hence, the class of meridian surfaces is a new source of examples of two-dimensional surfaces in
$\mathbb R^4_1$. In \cite{GM6} we found all marginally trapped meridian surfaces of elliptic or hyperbolic type.
In \cite{GM7} we continued the study of meridian surfaces in $\mathbb R^4_1$ considering a rotational hypersurface with
lightlike axis and constructed two-dimensional surfaces
which are one-parameter systems of meridians of the rotational
hypersurface. We called these surfaces \emph{meridian surfaces of
parabolic type}. We calculated their basic invariants and found all
marginally trapped meridian surfaces of parabolic type.
In the present paper we consider meridian surfaces of elliptic or hyperbolic type in $\mathbb R^4_1$ and calculate the invariants $\gamma_1, \,
\gamma_2, \, \nu_1,\, \nu_2, \, \lambda, \, \mu, \, \beta_1,
\beta_2$ of these surfaces. Using the invariants
we describe and classify completely the meridian surfaces of elliptic or hyperbolic type with
constant Gauss curvature (Theorem \ref{T:Gauss curvature}), with constant mean curvature (Theorem \ref{T:mean curvature}), and with constant
invariant $k$ (Theorem \ref{T:constant k}). In Theorem \ref{T:Chen} we classify the Chen meridian surfaces
and in Theorem \ref{T:parallel} we give the classification of the meridian surfaces
with parallel normal bundle.
\section{Preliminaries} \label{S:Pre}
Let $\mathbb R^4_1$ be the four-dimensional Minkowski space endowed with the metric
$\langle , \rangle$ of signature $(3,1)$ and $Oe_1e_2e_3e_4$ be a
fixed orthonormal coordinate system, i.e. $\langle e_1, e_1 \rangle =
\langle e_2, e_2 \rangle = \langle e_3, e_3 \rangle = 1, \, \langle e_4, e_4 \rangle = -1$.
A surface $M^2: z = z(u,v), \, \, (u,v) \in {\mathcal D}$
(${\mathcal D} \subset \mathbb R^2$) in $\mathbb R^4_1$ is said to be
\emph{spacelike} if $\langle , \rangle$ induces a Riemannian
metric $g$ on $M^2$. Thus at each point $p$ of a spacelike surface
$M^2$ we have the following decomposition:
$$\mathbb R^4_1 = T_pM^2 \oplus N_pM^2$$
with the property that the restriction of the metric $\langle ,
\rangle$ onto the tangent space $T_pM^2$ is of signature $(2,0)$,
and the restriction of the metric $\langle , \rangle$ onto the
normal space $N_pM^2$ is of signature $(1,1)$.
Denote by $\nabla'$ and $\nabla$ the Levi Civita connections on $\mathbb R^4_1$ and $M^2$, respectively.
Let $x$ and $y$ be vector fields tangent to $M^2$ and $\xi$ be a normal vector field.
The formulas of Gauss and Weingarten give the decompositions of the vector fields $\nabla'_xy$ and
$\nabla'_x \xi$ into tangent and normal components:
$$\begin{array}{l}
\nabla'_xy = \nabla_xy + \sigma(x,y);\\
\nabla'_x \xi = - A_{\xi} x + D_x \xi,
\end{array}$$
which define the second fundamental tensor $\sigma$, the normal connection $D$
and the shape operator $A_{\xi}$ with respect to $\xi$.
The mean curvature vector field $H$ of $M^2$ is defined as
$H = \displaystyle{\frac{1}{2}\, \mathrm{tr}\, \sigma}$.
Let
$M^2: z=z(u,v), \,\, (u,v) \in \mathcal{D}$ $(\mathcal{D} \subset \mathbb R^2)$
be a local parametrization on a
spacelike surface in $\mathbb R^4_1$.
The tangent space at an arbitrary point $p=z(u,v)$ of $M^2$ is
$T_pM^2 = \mathrm{span} \{z_u,z_v\}$, where $\langle z_u,z_u \rangle > 0$, $\langle z_v,z_v \rangle > 0$.
We use the standard denotations
$E(u,v)=\langle z_u,z_u \rangle, \; F(u,v)=\langle z_u,z_v
\rangle, \; G(u,v)=\langle z_v,z_v \rangle$ for the coefficients
of the first fundamental form.
Let $\{n_1, n_2\}$ be a normal frame field of $M^2$ such that $\langle
n_1, n_1 \rangle =1$, $\langle n_2, n_2 \rangle = -1$, and the
quadruple $\{z_u,z_v, n_1, n_2\}$ is positively oriented in
$\mathbb R^4_1$.
The coefficients of the second fundamental form $II$ of the surface $M^2$ are given
by the following functions
\begin{equation} \notag
L = \displaystyle{\frac{2}{W}} \left|
\begin{array}{cc}
c_{11}^1 & c_{12}^1 \\
c_{11}^2 & c_{12}^2 \\
\end{array}
\right|; \quad
M = \displaystyle{\frac{1}{W}} \left|
\begin{array}{cc}
c_{11}^1 & c_{22}^1 \\
c_{11}^2 & c_{22}^2 \\
\end{array}
\right|; \quad
N = \displaystyle{\frac{2}{W}} \left|
\begin{array}{cc}
c_{12}^1 & c_{22}^1 \\
c_{12}^2 & c_{22}^2 \\
\end{array}
\right|,
\end{equation}
where
$$\begin{array}{lll}
c_{11}^1 = \langle z_{uu}, n_1 \rangle; & \qquad c_{12}^1 = \langle z_{uv},
n_1 \rangle; & \qquad c_{22}^1 = \langle z_{vv}, n_1 \rangle;\\
c_{11}^2 = \langle z_{uu}, n_2 \rangle; & \qquad c_{12}^2 = \langle z_{uv},
n_2 \rangle; & \qquad c_{22}^2 = \langle z_{vv}, n_2 \rangle.
\end{array} $$
The second fundamental
form $II$ is invariant up to the orientation of the tangent space
or the normal space of the surface.
The condition $L = M = N = 0$ characterizes points at which
the space $\{\sigma(x,y): x, y \in T_pM^2\}$ is one-dimensional.
We call such points \emph{flat points} of the surface.
These points are analogous to flat points in the theory of surfaces in $\mathbb R^3$ and $\mathbb R^4$ \cite{GM2}.
In \cite{GM5} we gave a local geometric description of spacelike surfaces consisting of flat points
proving that
any spacelike surface consisting of flat points whose mean
curvature vector at any point is a non-zero spacelike vector or
timelike vector either lies in a hyperplane of $\mathbb R^4_1$ or is
part of a developable ruled surface in $\mathbb R^4_1$.
Further we consider surfaces free of flat points, i.e. $(L, M, N) \neq (0,0,0)$.
Using the functions $L$, $M$, $N$ and $E$, $F$, $G$ in \cite{GM5}
we introduced a linear map $\gamma$ of Weingarten type in the
tangent space at any point of $M^2$. The map $\gamma$ is invariant with respect to
changes of parameters on $M^2$ as well as to motions in $\mathbb R^4_1$. It
generates two invariant functions
$$k = \frac{LN - M^2}{EG - F^2}, \qquad
\varkappa = \frac{EN+GL-2FM}{2(EG-F^2)}.$$
It turns out that
the invariant $\varkappa$ is the curvature of the normal
connection of the surface
(see \cite{GM5}).
As in the theory of surfaces in $\mathbb R^3$ and $\mathbb R^4$ the invariant
$k$ divides the points of $M^2$ into the following types: \emph{elliptic} ($k > 0$),
\emph{parabolic} ($k = 0$), and \emph{hyperbolic} ($k < 0$).
The second fundamental form $II$ determines conjugate, asymptotic, and
principal tangents at a point $p$ of $M^2$ in the standard way.
A line $c: u=u(q), \; v=v(q); \; q\in J \subset \mathbb R$ on $M^2$ is
said to be an \emph{asymptotic line}, respectively a
\textit{principal line}, if its tangent at any point is
asymptotic, respectively principal. The surface $M^2$ is
parameterized by principal lines if and only if $F=0, \,\, M=0.$
Considering spacelike surfaces in $\mathbb R^4_1$ whose mean
curvature vector at any point is a non-zero spacelike vector or
timelike vector, on the base of the principal lines we introduced
a geometrically determined orthonormal frame field $\{x,y,b,l\}$ at each point of such a surface \cite{GM5}.
The tangent vector fields $x$ and $y$ are collinear with the principal directions,
the normal vector field $b$ is collinear with the mean curvature vector field $H$.
Writing derivative formulas
of Frenet-type for this frame field, we obtained eight invariant
functions $\gamma_1, \, \gamma_2, \, \nu_1,\, \nu_2, \, \lambda, \, \mu,
\, \beta_1, \beta_2$, which determine the surface up to a rigid motion in $\mathbb R^4_1$.
The invariants $\gamma_1, \, \gamma_2, \, \nu_1,\, \nu_2, \, \lambda, \, \mu,
\, \beta_1$, and $\beta_2$ are determined by the geometric frame field $\{x,y,b,l\}$ as follows
\begin{equation}
\begin{array}{l} \label{E:Eq1}
\nu_1 = \langle \nabla'_xx, b\rangle, \qquad \nu_2 = \langle \nabla'_yy, b\rangle, \qquad \lambda = \langle \nabla'_xy, b\rangle,
\qquad \mu = \langle \nabla'_xy, l\rangle,\\
\gamma_1 = \langle \nabla'_xx, y\rangle, \qquad \gamma_2 = \langle \nabla'_yy, x\rangle, \qquad \beta_1 = \langle \nabla'_xb, l\rangle, \qquad
\beta_2 = \langle \nabla'_yb, l\rangle.
\end{array}
\end{equation}
The invariants $k$, $\varkappa$, and the Gauss curvature $K$ of
$M^2$ are expressed by the functions $\nu_1, \nu_2, \lambda, \mu$
as follows:
\begin{equation} \notag
k = - 4\nu_1\,\nu_2\,\mu^2, \quad \quad \varkappa = (\nu_1-\nu_2)\mu, \quad \quad
K = \varepsilon (\nu_1\,\nu_2- \lambda^2 + \mu^2),
\end{equation}
where $\varepsilon = sign \langle H, H \rangle$.
The norm $\Vert H \Vert$ of the mean curvature
vector is expressed as
\begin{equation} \notag
\Vert H \Vert = \displaystyle{ \frac{|\nu_1 + \nu_2|}{2} = \frac{\sqrt{\varkappa^2-k}}{2 |\mu |}}.
\end{equation}
If $M^2$ is a spacelike surface whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector, then
$M^2$ is minimal if and only if $\nu_1 + \nu_2 = 0$.
The geometric meaning of the invariant $\lambda$ is connected with the notion of Chen submanifolds.
Let $M$ be an $n$-dimensional submanifold of
$(n+m)$-dimensional Riemannian manifold $\widetilde{M}$ and $\xi$
be a normal vector field of $M$. B.-Y. Chen \cite{Chen1} defined
the \emph{allied vector field} $a(\xi)$ of $\xi$ by the
formula
$$a(\xi) = \displaystyle{\frac{\|\xi\|}{n} \sum_{k=2}^m \{\mathrm{tr}(A_1 A_k)\}\xi_k},$$
where $\{\xi_1 = \displaystyle{\frac{\xi}{\|\xi\|}},\xi_2, \dots, \xi_m \}$ is an
orthonormal base of the normal space of $M$, and $A_i = A_{\xi_i},
\,\, i = 1,\dots, m$ is the shape operator with respect to
$\xi_i$. The allied vector field $a(H)$ of the mean
curvature vector field $H$ is called the \emph{allied mean
curvature vector field} of $M$ in $\widetilde{M}$. B.-Y. Chen
defined the $\mathcal{A}$-submanifolds to be those submanifolds
of $\widetilde{M}$ for which
$a(H)$ vanishes identically \cite{Chen1}.
In \cite{GVV1}, \cite{GVV2} the $\mathcal{A}$-submanifolds are
called \emph{Chen submanifolds}. It is easy to see that minimal
submanifolds, pseudo-umbilical submanifolds and hypersurfaces are
Chen submanifolds. These Chen submanifolds are said to be trivial
Chen-submanifolds. In \cite{GM5} we showed that if $M^2$ is a spacelike surface
in $\mathbb R^4_1$ with spacelike or timelike mean curvature vector field then the allied mean curvature vector field of $M^2$
is
$$a(H) = \displaystyle{\frac{\sqrt{\varkappa^2-k}}{2} \,\lambda \, l}.$$
Hence, if $M^2$ is free of minimal points, then $a(H) = 0$ if and
only if $\lambda = 0$. This gives the geometric meaning of the
invariant $\lambda$: $M^2$ is a non-trivial Chen
surface if and only if the invariant $\lambda$ is zero.
Now we shall discuss the geometric meaning of the invariants $\beta_1$ and $\beta_2$. It follows from \eqref{E:Eq1} that
\begin{equation} \notag
\begin{array}{ll}
\nabla'_xb = - \nu_1\,x - \lambda\,y - \beta_1\,l; & \qquad
\nabla'_xl = - \mu\,y - \beta_1\,b;\\
\nabla'_yb = - \lambda\,x - \nu_2\,y - \beta_2\,l; & \qquad \nabla'_yl = - \mu\,x - \beta_2\,b.
\end{array}
\end{equation}
Hence, $\beta_1 = \beta_2 = 0$ if and only if $D_xb = D_yb = 0$ (or equivalently, $D_xl = D_yl = 0$).
A normal vector field $\xi$ is said to be \emph{parallel in the normal bundle} (or simply \emph{parallel}) \cite{Chen2},
if $D_x\xi = 0$ holds identically for any tangent vector field $x$.
Hence, the invariants $\beta_1$ and $\beta_2$ are identically zero if and only if the geometric normal vector fields $b$ and $l$ are parallel in the normal bundle.
Surfaces admitting a geometric normal frame field $\{b, l\}$ of parallel normal vector fields, we shall call \emph{surfaces with parallel normal bundle}.
They are characterized by the condition $\beta_1 = \beta_2 = 0$.
Note that if $M^2$ is a surface free of minimal points with parallel mean curvature vector field (i.e. $DH = 0$),
then $M^2$ is a surface with parallel normal bundle, but the converse is not true in general.
It is true only in the case $\Vert H \Vert = const$.
\section{Invariants of meridian surfaces of elliptic or hyperbolic type}
In \cite{GM2} we constructed a family of surfaces lying on a
standard rotational hypersurface in the four-dimensional Euclidean
space $\mathbb R^4$. These surfaces are one-parameter systems of
meridians of the rotational hypersurface, that is why we called
them \emph{meridian surfaces}. In \cite{GM6} we used the idea from the Euclidean case to construct
special families of two-dimensional spacelike surfaces lying
on rotational hypersurfaces in $\mathbb R^4_1$ with
timelike or spacelike axis. The construction was the following.
Let $f = f(u), \,\, g = g(u)$ be smooth functions, defined in an
interval $I \subset \mathbb R$, such that $\dot{f}^2(u) - \dot{g}^2(u) > 0,
\,\, u \in I$. We assume that $f(u)>0, \,\, u \in I$. The standard
rotational hypersurface $\mathcal{M}'$ in $\mathbb R^4_1$, obtained by
the rotation of the meridian curve $m: u \rightarrow (f(u), g(u))$
about the $Oe_4$-axis, is parameterized as follows:
$$\mathcal{M}': Z(u,w^1,w^2) = f(u)\, \cos w^1 \cos w^2 \,e_1 + f(u)\, \cos w^1 \sin w^2 \,e_2 + f(u)\, \sin w^1 \,e_3 + g(u) \,e_4.$$
The rotational hypersurface $\mathcal{M}'$ is a two-parameter system of meridians. Let $w^1 =
w^1(v)$, $w^2=w^2(v), \,\, v \in J, \, J \subset \mathbb R$. We consider the two-dimensional
surface $\mathcal{M}'_m$ lying on $\mathcal{M}'$, constructed in the following way:
\begin{equation} \notag
\mathcal{M}'_m: z(u,v) = Z(u,w^1(v),w^2(v)), \quad u \in I, \, v \in J.
\end{equation}
$\mathcal{M}'_m$ is a
one-parameter system of meridians of $\mathcal{M}'$. We call
$\mathcal{M}'_m$ a \emph{meridian surface of elliptic type}.
If we denote $l(w^1,w^2) = \cos w^1 \cos w^2 \,e_1 + \cos w^1 \sin w^2 \,e_2 + \sin w^1 \,e_3$, then the surface $\mathcal{M}'_m$ is parameterized by
\begin{equation} \label{E:Eq2}
\mathcal{M}'_m: z(u,v) = f(u) \, l(v) + g(u)\, e_4, \quad u \in I, \, v \in J.
\end{equation}
Note that $l(w^1,w^2)$ is the unit position vector of the 2-dimensional sphere $S^2(1)$ lying in the Euclidean space $\mathbb R^3 = \mathrm{span} \{e_1, e_2, e_3\}$
and centered at the origin $O$.
\vskip 2mm
In a similar way we consider meridian surfaces lying on the
rotational hypersurface in $\mathbb R^4_1$ with spacelike axis.
Let $f = f(u), \,\, g = g(u)$ be smooth functions, defined in an
interval $I \subset \mathbb R$, such that $\dot{f}^2(u) + \dot{g}^2(u)
>0$, $f(u)>0, \,\, u \in I$.
The rotational hypersurface $\mathcal{M}''$ in $\mathbb R^4_1$, obtained
by the rotation of the meridian curve $m: u \rightarrow (f(u),
g(u))$ about the $Oe_1$-axis is parameterized as follows:
$$\mathcal{M}'': Z(u,w^1,w^2) = g(u) \,e_1 + f(u)\, \cosh w^1 \cos w^2 \,e_2 + f(u)\, \cosh w^1 \sin w^2 \,e_3+ f(u)\, \sinh w^1 \,e_4.$$
If $w^1 = w^1(v), \, w^2=w^2(v), \,\, v \in J, \,J \subset
\mathbb R$, we construct a surface $\mathcal{M}''_m$ in $\mathbb R^4_1$ in
the following way:
\begin{equation} \notag
\mathcal{M}''_m: z(u,v) = Z(u,w^1(v),w^2(v)),\quad u \in I, \, v \in J.
\end{equation}
$\mathcal{M}''_m$ is a one-parameter system of
meridians of $\mathcal{M}''$. We call $\mathcal{M}''_m$ a
\emph{meridian surfaces of hyperbolic type}.
If we denote $l(w^1,w^2) = \cosh w^1 \cos w^2 \,e_2 + \cosh w^1 \sin w^2 \,e_3 + \sinh w^1 \,e_4$, then the surface $\mathcal{M}''_m$
is given by
\begin{equation} \label{E:Eq3}
\mathcal{M}''_m: z(u,v) = f(u) \, l(v) + g(u)\, e_1, \quad u \in I, \, v \in J,
\end{equation}
$l(w^1,w^2)$ being the unit position vector of the de Sitter space $S^2_1(1)$ in the Minkowski space $\mathbb R^3_1 = \mathrm{span} \{e_2, e_3, e_4\}$,
i.e. $S^2_1(1) = \{ V \in \mathbb R^3_1: \langle V, V \rangle = 1\}$.
In \cite{GM6} we found all marginally trapped meridian surfaces of elliptic or hyperbolic type.
In the present section we shall find the geometric invariant
functions $\gamma_1, \, \gamma_2, \, \nu_1,\, \nu_2, \, \lambda, \, \mu, \, \beta_1, \beta_2$ of the meridian surfaces of elliptic or hyperbolic type.
\vskip 2mm
\emph{Elliptic case:}
First we consider the surface $\mathcal{M}'_m$ parameterized by \eqref{E:Eq2}.
We assume that the smooth curve $c: l = l(v) = l(w^1(v),w^2(v)), \, v \in J$ on
$S^2(1)$ is parameterized by the arc-length, i.e. $\langle l'(v), l'(v) \rangle = 1$. Let
$t(v) = l'(v)$ be the tangent vector field of $c$. Since $\langle t(v), t(v) \rangle = 1$,
$\langle l(v), l(v) \rangle = 1$, and $\langle t(v), l(v) \rangle = 0$,
there exists a unique (up to a sign)
vector field $n(v)$, such that
$\{ l(v), t(v), n(v)\}$ is an orthonormal frame field in $\mathbb R^3$. With respect to this
frame field we have the following Frenet formulas of $c$ on $S^2(1)$:
\begin{equation} \label{E:Eq4}
\begin{array}{l}
l' = t;\\
t' = \kappa \,n - l;\\
n' = - \kappa \,t,
\end{array}
\end{equation}
where $\kappa (v)= \langle t'(v), n(v) \rangle$ is the spherical curvature of $c$.
Without loss of generality we assume that $\dot{f}^2(u) - \dot{g}^2(u) = 1$.
The tangent space of $\mathcal{M}'_m$ is spanned by the vector fields:
$$z_u = \dot{f} \,l + \dot{g}\,e_4; \qquad z_v = f\,t,$$
so, the coefficients of the first fundamental form of $\mathcal{M}'_m$ are $E = 1; \, F = 0; \, G = f^2(u) >0$.
Hence, the first fundamental form is positive
definite, i.e. $\mathcal{M}'_m$ is a spacelike surface.
Denote $X = z_u,\,\, Y = \displaystyle{\frac{z_v}{f} = t}$ and
consider the following orthonormal normal frame field:
$$n_1 = n(v); \qquad n_2 = \dot{g}(u)\,l(v) + \dot{f}(u) \, e_4.$$
Thus we obtain a frame field $\{X,Y, n_1, n_2\}$ of $\mathcal{M}'_m$, such that $\langle n_1, n_1 \rangle =1$,
$\langle n_2, n_2 \rangle =- 1$, $\langle n_1, n_2 \rangle =0$.
Taking into account \eqref{E:Eq4} we get
the following derivative formulas:
\begin{equation} \label{E:Eq5}
\begin{array}{ll}
\nabla'_XX = \qquad \qquad \qquad \qquad
\kappa_m\,n_2; & \qquad
\nabla'_X n_1 = 0;\\
\nabla'_XY = 0; & \qquad
\nabla'_Y n_1 = \displaystyle{\quad \quad \, - \frac{\kappa}{f}\,Y};\\
\nabla'_YX = \quad\quad
\quad\displaystyle{\frac{\dot{f}}{f}}\,Y; & \qquad
\nabla'_X n_2 = \kappa_m \,X;\\
\nabla'_YY = \displaystyle{- \frac{\dot{f}}{f}\,X \quad\quad +
\frac{\kappa}{f}\,n_1 + \frac{\dot{g}}{f} \, n_2}; & \qquad
\nabla'_Y n_2 = \displaystyle{ \quad \quad \quad \frac{\dot{g}}{f}\,Y},
\end{array}
\end{equation}
where $\kappa_m$ denotes the curvature of the
meridian curve $m$, i.e.
$\kappa_m (u)= \dot{f}(u) \ddot{g}(u) - \dot{g}(u) \ddot{f}(u)$.
The invariants $k$, $\varkappa$, and the Gauss curvature $K$ are given by the following formulas \cite{GM6}:
$$k = - \frac{\kappa_m^2(u) \, \kappa^2(v)}{f^2(u)}; \qquad \varkappa = 0;
\qquad K = - \frac{\ddot{f}(u)}{f(u)}.$$
The equality $\varkappa = 0$ implies the following statement.
\begin{prop}
The meridian surface of elliptic type $\mathcal{M}'_m$, defined by \eqref{E:Eq2}, is a surface with
flat normal connection.
\end{prop}
We distinguish the following three cases:
\vskip 2mm
I. $\kappa(v) = 0$, i.e. the curve $c$ is a great circle on $S^2(1)$. In
this case $n_1 = const$, and $\mathcal{M}'_m$ is a planar surface lying in
the constant 3-dimensional space spanned by $\{X, Y, n_2\}$.
\vskip 2mm
II. $\kappa_m(u) = 0$, i.e. the meridian curve $m$ is part of
a straight line. In such case $k = \varkappa = K = 0$, and $\mathcal{M}'_m$
is a developable ruled surface.
\vskip 2mm
III. $\kappa_m(u) \, \kappa(v) \neq 0$, i.e. $c$ is not a
great circle on $S^2(1)$ and $m$ is not a straight line.
\vskip 2mm
In the first two cases the surface $\mathcal{M}'_m$ consists of flat points. So, we consider the third (general) case,
i.e. we assume that $\kappa_m \neq 0$ and $\kappa \neq 0$.
It follows from \eqref{E:Eq5} that the mean curvature vector field $H$ of $\mathcal{M}'_m$ is expressed as
$$H = \frac{\kappa}{2f}\, n_1 + \frac{\dot{g} + f \kappa_m}{2f} \, n_2.$$
Using that $\dot{g}^2(u) = \dot{f}^2(u) - 1$ and $\kappa_m (u) = \displaystyle{\frac{\ddot{f}(u)}{\sqrt{\dot{f}^2(u) - 1}}}$, we get
\begin{equation} \label{E:Eq6}
H = \frac{\kappa}{2f}\, n_1 + \frac{ f \ddot{f} + \dot{f}^2 - 1}{2f \sqrt{\dot{f}^2 - 1}} \, n_2.
\end{equation}
Since $\kappa \neq 0$ the surface $\mathcal{M}'_m$ is non-minimal, i.e. $H \neq 0$.
The case $\mathcal{M}'_m$ is a marginally trapped surface, i.e. $H \neq 0$ and $\langle H, H \rangle = 0$ is described in \cite{GM6}.
So, here we consider the case $\langle H, H \rangle \neq 0$.
Note that the orthonormal frame field $\{X,Y, n_1, n_2\}$ of $\mathcal{M}'_m$ is not the geometric frame field defined in Section \ref{S:Pre}.
The principal tangents of $\mathcal{M}'_m$ are
\begin{equation}\notag
x = \displaystyle{\frac{X+Y}{\sqrt{2}}}; \qquad y = \displaystyle{\frac{- X + Y}{\sqrt{2}}}.
\end{equation}
In the case $\langle H, H \rangle > 0$, i.e. $\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2 > 0$, the geometric normal frame field $\{b,l\}$ is given by
\begin{equation}\notag
\begin{array}{l}
b = \displaystyle{\frac{1}{\sqrt{\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2}} \left( \kappa \sqrt{\dot{f}^2 - 1}\, n_1 + (f \ddot{f} + \dot{f}^2 - 1)\,n_2 \right)}; \\
l = \displaystyle{\frac{1}{\sqrt{\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2}} \left( (f \ddot{f} + \dot{f}^2 - 1) \, n_1 + \kappa \sqrt{\dot{f}^2 - 1}\,n_2 \right)}.
\end{array}
\end{equation}
In this case the normal vector fields $b$ and $l$ satisfy $\langle b, b \rangle = 1$, $\langle b, l \rangle = 0$, $\langle l, l \rangle = -1$.
In the case $\langle H, H \rangle < 0$, i.e. $\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2 < 0$, the geometric normal frame field $\{b,l\}$ is given by
\begin{equation}\notag
\begin{array}{l}
b = \displaystyle{- \frac{1}{\sqrt{(f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (\dot{f}^2 - 1)}} \left( \kappa \sqrt{\dot{f}^2 - 1}\, n_1 + (f \ddot{f} + \dot{f}^2 - 1)\,n_2 \right)}; \\
l = \displaystyle{\frac{1}{\sqrt{(f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (\dot{f}^2 - 1)}} \left( (f \ddot{f} + \dot{f}^2 - 1) \, n_1 + \kappa \sqrt{\dot{f}^2 - 1}\,n_2 \right)}.
\end{array}
\end{equation}
In this case we have $\langle b, b \rangle = - 1$, $\langle b, l \rangle = 0$, $\langle l, l \rangle = 1$.
Applying formulas \eqref{E:Eq1} for the geometric frame field $\{x,y, b, l\}$ of $\mathcal{M}'_m$ and derivative formulas \eqref{E:Eq5},
we obtain the following invariants of $\mathcal{M}'_m$:
\begin{equation} \label{E:Eq7}
\begin{array}{l}
\gamma_1 = \gamma_2 = \displaystyle{- \frac{\dot{f}}{\sqrt{2}f}};\\
\nu_1 = \nu_2 = \displaystyle{ \frac{1}{2f \sqrt{\dot{f}^2 - 1}} \sqrt{ \varepsilon(\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2)}};\\
\lambda = \displaystyle{ \varepsilon \frac{\kappa^2 (\dot{f}^2 - 1) + f^2 \ddot{f}^2 - (\dot{f}^2 - 1)^2}{2f \sqrt{\dot{f}^2 - 1} \sqrt{\varepsilon (\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2)}}}; \\
\mu = \displaystyle{\frac{\kappa \ddot{f}}{\sqrt{\varepsilon (\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2)}}}; \\
\beta_1 = \displaystyle{\frac{ - (\dot{f}^2 - 1)}{\sqrt{2} \varepsilon(\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2)}} \left( \kappa\, \frac{d}{du}\left( \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{\dot{f}^2 - 1}} \right) - \frac{d}{dv}(\kappa) \, \frac{f \ddot{f} + \dot{f}^2 - 1}{f \sqrt{\dot{f}^2 - 1}} \right); \\
\beta_2 = \displaystyle{\frac{(\dot{f}^2 - 1)}{\sqrt{2} \varepsilon(\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2)}} \left( \kappa\, \frac{d}{du}\left( \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{\dot{f}^2 - 1}} \right) + \frac{d}{dv}(\kappa) \, \frac{f \ddot{f} + \dot{f}^2 - 1}{f \sqrt{\dot{f}^2 - 1}} \right),
\end{array}
\end{equation}
where $\varepsilon = sign \langle H, H \rangle$.
\vskip 3mm
\emph{Hyperbolic case:}
Let $\mathcal{M}''_m$ be the surface parameterized by \eqref{E:Eq3}.
Assume that the curve $c: l = l(v) = l(w^1(v),w^2(v)), \, v \in J$ on
$S^2_1(1)$ is parameterized by the arc-length, i.e. $\langle l'(v), l'(v) \rangle = 1$.
Similarly to the previous case we consider an orthonormal frame field
$\{ l(v), t(v), n(v)\}$ in $\mathbb R^3_1$, such that $t(v) = l'(v)$ and $\langle n(v), n(v) \rangle = -1$.
With respect to this frame field we have the following decompositions of the vector fields $l'(v)$, $t'(v)$, $n'(v)$:
\begin{equation} \label{E:Eq8}
\begin{array}{l}
l' = t;\\
t' = - \kappa \,n - l;\\
n' = - \kappa \,t,
\end{array}
\end{equation}
which can be considered as Frenet formulas of $c$ on $S^2_1(1)$.
The function $\kappa (v)= \langle t'(v), n(v) \rangle$ is the spherical
curvature of $c$ on $S^2_1(1)$.
We assume that $\dot{f}^2(u) + \dot{g}^2(u) = 1$.
Denote $X = z_u = \dot{f} \,l + \dot{g}\,e_1, \,\, Y = \displaystyle{\frac{z_v}{f} = t}$ and
consider the orthonormal normal frame field defined by:
$$n_1 = \dot{g}(u)\,l(v) - \dot{f}(u) \, e_1; \qquad n_2 = n(v).$$
Thus we obtain a frame field $\{X,Y, n_1, n_2\}$ of $\mathcal{M}''_m$,
such that $\langle n_1, n_1 \rangle =1$, $\langle n_2, n_2 \rangle =
- 1$, $\langle n_1, n_2 \rangle =0$.
Taking into account \eqref{E:Eq8} we get
the following derivative formulas:
\begin{equation} \label{E:Eq9}
\begin{array}{ll}
\nabla'_XX = \qquad \qquad \quad - \kappa_m\,n_1; & \qquad
\nabla'_X n_1 = \kappa_m \,X;\\
\nabla'_XY = 0; & \qquad
\nabla'_Y n_1 = \displaystyle{\quad \quad \;\; \frac{\dot{g}}{f}\,Y};\\
\nabla'_YX = \quad\quad
\quad\displaystyle{\frac{\dot{f}}{f}}\,Y; & \qquad
\nabla'_X n_2 = 0;\\
\nabla'_YY = \displaystyle{- \frac{\dot{f}}{f}\,X \quad\quad -
\frac{\dot{g}}{f}\,n_1 - \frac{\kappa}{f} \, n_2}; & \qquad
\nabla'_Y n_2 = \displaystyle{ \quad \quad \quad - \frac{\kappa}{f}\,Y},
\end{array}
\end{equation}
where $\kappa_m$ is the curvature of the meridian curve $m$.
The invariants $k$, $\varkappa$, and the Gauss curvature $K$ of the meridian surface $\mathcal{M}''_m$ are expressed
by the curvatures $\kappa_m(u)$, $\kappa (v)$, and the function $f(u)$ in the same way as the invariants of the meridian surface of elliptic type, i.e.
$$k = - \frac{\kappa_m^2(u) \, \kappa^2(v)}{f^2(u)}; \qquad \varkappa = 0; \qquad K = - \frac{\ddot{f}(u)}{f(u)}.$$
The following statement holds, since $\varkappa = 0$.
\begin{prop}
The meridian surface of hyperbolic type $\mathcal{M}''_m$, defined by \eqref{E:Eq3}, is a surface with
flat normal connection.
\end{prop}
Again we have the following three cases:
\vskip 2mm
I. $\kappa(v) = 0$. In this case $n_2 = const$, and $\mathcal{M}''_m$ is a planar surface lying in
the constant 3-dimensional space spanned by $\{X, Y, n_1\}$.
\vskip 2mm
II. $\kappa_m(u) = 0$. In such case $k = \varkappa = K = 0$, and $\mathcal{M}''_m$
is a developable ruled surface.
\vskip 2mm
III. $\kappa_m(u) \, \kappa(v) \neq 0$.
\vskip 2mm
In the first two cases $\mathcal{M}''_m$ is a surface consisting of flat points. So, we consider the third (general) case,
i.e. we assume that $\kappa_m \neq 0$ and $\kappa \neq 0$.
Using \eqref{E:Eq9} we get that the mean curvature vector field $H$ of $\mathcal{M}''_m$ is
$$H = - \frac{\dot{g} + f \kappa_m}{2f}\, n_1 - \frac{\kappa}{2f}\, n_2.$$
Having in mind that $\dot{g}^2(u) = 1- \dot{f}^2(u)$ and $\kappa_m (u) = \displaystyle{- \frac{\ddot{f}(u)}{\sqrt{1 - \dot{f}^2(u)}}}$, we obtain
\begin{equation} \label{E:Eq10}
H = \frac{ f \ddot{f} + \dot{f}^2 - 1}{2f \sqrt{1 - \dot{f}^2}}\, n_1 - \frac{\kappa}{2f}\, n_2.
\end{equation}
The surface $\mathcal{M}''_m$ is non-minimal, since $\kappa \neq 0$.
The case $\mathcal{M}''_m$ is marginally trapped is described in \cite{GM6}.
So, we consider the case $\langle H, H \rangle \neq 0$, i.e. $(f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2) \neq 0$.
Similarly to the elliptic case, we find the geometric frame field $\{x,y, b, l\}$ of $\mathcal{M}''_m$.
Applying formulas \eqref{E:Eq1} for this frame field and using derivative formulas \eqref{E:Eq9},
we obtain the following invariants of $\mathcal{M}''_m$:
\begin{equation} \label{E:Eq11}
\begin{array}{l}
\gamma_1 = \gamma_2 = \displaystyle{- \frac{\dot{f}}{\sqrt{2}f}};\\
\nu_1 = \nu_2 = \displaystyle{ \frac{1}{2f \sqrt{1 - \dot{f}^2}} \sqrt{ \varepsilon( (f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2))}};\\
\lambda = \displaystyle{ \varepsilon \frac{- \kappa^2 (1 - \dot{f}^2) - f^2 \ddot{f}^2 + (1 - \dot{f}^2)^2}{2f \sqrt{1 -\dot{f}^2} \sqrt{\varepsilon ( (f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2))}}}; \\
\mu = \displaystyle{\frac{\kappa \ddot{f}}{\sqrt{\varepsilon ( (f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2))}}}; \\
\beta_1 = \displaystyle{\frac{ - (1 - \dot{f}^2)}{\sqrt{2} \varepsilon( (f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2))}} \left( \kappa\, \frac{d}{du}\left( \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{1 - \dot{f}^2}} \right) - \frac{d}{dv}(\kappa) \, \frac{f \ddot{f} + \dot{f}^2 - 1}{f \sqrt{1 - \dot{f}^2}} \right); \\
\beta_2 = \displaystyle{\frac{(1 - \dot{f}^2)}{\sqrt{2} \varepsilon((f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2))}} \left( \kappa\, \frac{d}{du}\left( \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{1 - \dot{f}^2}} \right) + \frac{d}{dv}(\kappa) \, \frac{f \ddot{f} + \dot{f}^2 - 1}{f \sqrt{1 - \dot{f}^2}} \right),
\end{array}
\end{equation}
where $\varepsilon = sign \langle H, H \rangle$.
\vskip 2mm
In the following sections, using the invariants of the meridian surfaces $\mathcal{M}'_m$ and $\mathcal{M}''_m$, we shall describe and classify some special classes of meridian surfaces
of elliptic or hyperbolic type.
\section{Meridian surfaces with constant Gauss curvature}
The study of surfaces with constant Gauss curvature is one of the
main topics in classical differential geometry. Surfaces
with constant Gauss curvature in Minkowski space have drawn the interest of many
geometers, see for example \cite{GalMarMil}, \cite{Lop}, and the references therein.
In the present section we give the classification of the meridian surfaces of elliptic or hyperbolic type in $\mathbb R^4_1$ with constant Gauss curvature.
Let $\mathcal{M}'_m$ and $\mathcal{M}''_m$ be meridian surfaces of elliptic and hyperbolic type, respectively.
The Gauss curvature in both cases depends only on the meridian curve $m$ and is expressed by the formula
\begin{equation} \label{E:Eq12}
K = - \frac{\ddot{f}(u)}{f(u)}.
\end{equation}
\begin{thm} \label{T:Gauss curvature}
Let $\mathcal{M}'_m$ (resp. $\mathcal{M}''_m$) be a meridian surface of elliptic (resp. hyperbolic) type from the general class.
Then $\mathcal{M}'_m$ (resp. $\mathcal{M}''_m$) has constant non-zero Gauss curvature $K$ if and only
if the meridian $m$ is given by
$$\begin{array}{ll}
f(u) = \alpha \cos \sqrt{K} u + \beta \sin \sqrt{K} u, & \textrm{if} \quad K >0;\\
f(u) = \alpha \cosh \sqrt{-K} u + \beta \sinh
\sqrt{-K} u, & \textrm{if} \quad K <0,
\end{array}$$
where $\alpha$ and $\beta$ are constants, $g(u)$ is defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$ in the elliptic case and
$g(u)$ is defined by $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$ in the hyperbolic case.
\end{thm}
\noindent {\it Proof:} Using \eqref{E:Eq12} we
obtain that the Gauss curvature $K = const \neq 0$ if and
only if the function $f(u)$ satisfies the following differential
equation
$$\ddot{f}(u) + K f(u) = 0.$$
The general solution of the above equation is given by
$$\begin{array}{ll}
f(u) = \alpha \cos \sqrt{K} u + \beta \sin \sqrt{K} u, & \quad \textrm{if} \quad K >0;\\
f(u) = \alpha \cosh \sqrt{-K} u + \beta \sinh
\sqrt{-K} u, & \quad \textrm{if} \quad K <0,
\end{array}$$
where $\alpha$ and $\beta$ are constants.
In the case of meridian surface of elliptic type the function $g(u)$ is
determined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$ and in the case of meridian surface of hyperbolic type $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$.
\qed
\section{Meridian surfaces with constant mean curvature}
Constant mean curvature surfaces in arbitrary spacetime
are important objects for their special role in the theory of
general relativity. The study of constant mean curvature surfaces
(CMC surfaces) involves not only geometric methods but also PDE
and complex analysis, that is why the theory of CMC surfaces is of
great interest not only for mathematicians but also for physicists
and engineers. Surfaces with constant mean curvature in
Minkowski space have been studied intensively in the last years. See for example \cite{Liu-Liu-1}, \cite{Lop-2}, \cite{Sa}, \cite{Chav-Can}, \cite{Bran}.
In this section we classify the meridian surfaces of elliptic or hyperbolic type with constant mean curvature.
Let $\mathcal{M}'_m$ and $\mathcal{M}''_m$ be meridian surfaces of elliptic and hyperbolic type, respectively.
Equality \eqref{E:Eq6} implies that the mean curvature of the meridian surface of elliptic type
$\mathcal{M}'_m$ is given by
\begin{equation} \label{E:Eq13}
|| H || = \sqrt{\frac{\varepsilon(\kappa^2 (\dot{f}^2 - 1) - (f \ddot{f} + \dot{f}^2 - 1)^2)}{4f^2 (\dot{f}^2 - 1)}}.
\end{equation}
Similarly, from \eqref{E:Eq10} it follows that the mean curvature of the meridian surface of hyperbolic type
$\mathcal{M}''_m$ is
\begin{equation} \label{E:Eq14}
|| H || = \sqrt{\frac{\varepsilon ((f \ddot{f} + \dot{f}^2 - 1)^2 - \kappa^2 (1 - \dot{f}^2))}{4f^2 (1 - \dot{f}^2)}}.
\end{equation}
\begin{thm} \label{T:mean curvature}
Let $\mathcal{M}'_m$ (resp. $\mathcal{M}''_m$) be a meridian surface of elliptic (resp. hyperbolic) type from the general class.
(i) $\mathcal{M}'_m$ has constant mean curvature $|| H || = a = const$, $a \neq 0$ if and only if
the curve $c$ on $S^2(1)$ has
constant spherical curvature $\kappa = const = b, \; b \neq 0$,
and the meridian $m$ is determined by $\dot{f} = y(f)$
where
\begin{equation} \notag
y(t) = \sqrt{1 + \frac{1}{t^2}\left(C \pm \frac{t}{2} \sqrt{b^2 - 4 a^2 t^2} \pm \frac{b^2}{4a} \arcsin \frac{2at}{b}\right)^2}, \qquad C = const,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$.
(ii) $\mathcal{M}''_m$ has constant mean curvature $|| H || = a = const$, $a \neq 0$ if and only if
the curve $c$ on $S^2_1(1)$ has constant
spherical curvature $\kappa = const = b, \; b \neq 0$,
and the meridian $m$ is determined by $\dot{f} = y(f)$
where
\begin{equation} \notag
y(t) = \sqrt{1 - \frac{1}{t^2}\left(C \pm \frac{t}{2} \sqrt{b^2 - 4 a^2 t^2} \pm \frac{b^2}{4a} \arcsin \frac{2at}{b}\right)^2}, \qquad C = const,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$.
\end{thm}
\noindent {\it Proof:}
(i) It follows from \eqref{E:Eq13} that $||H|| = a$ if
and only if
$$\kappa^2(v) = \frac{4 a^2 f^2(u)(\dot{f}^2(u) - 1) + (f(u) \ddot{f}(u) + \dot{f}^2(u) - 1)^2}{\dot{f}^2(u) - 1},$$
which implies
\begin{equation} \label{E:Eq15}
\begin{array}{l}
\kappa = const = b, \; b \neq 0;\\
4 a^2 f^2(u)(\dot{f}^2(u) - 1) + (f(u) \ddot{f}(u) + \dot{f}^2(u) - 1)^2 = b^2(\dot{f}^2(u) - 1).
\end{array}
\end{equation}
The first equality of \eqref{E:Eq15} implies that the spherical curve $c$
has constant spherical curvature $\kappa = b$,
i.e. $c$ is a circle on $S^2(1)$.
The second equality of \eqref{E:Eq15} gives the
following differential equation for the meridian $m$:
\begin{equation} \label{E:Eq16}
(f \ddot{f} + \dot{f}^2 - 1)^2 = (\dot{f}^2 - 1) (b^2 - 4 a^2 f^2).
\end{equation}
Further, if we set $\dot{f} = y(f)$ in equation \eqref{E:Eq16}, we obtain
that the function $y = y(t)$ is a solution of the following
differential equation
$$\frac{t}{2}(y^2)' + y^2 - 1 = \pm \sqrt{y^2 - 1} \sqrt{b^2 - 4 a^2 t^2}.$$
The general solution of the above equation is given by the formula
\begin{equation} \label{E:Eq17}
y(t) = \sqrt{1 + \frac{1}{t^2}\left(C \pm \frac{t}{2} \sqrt{b^2 - 4 a^2 t^2} \pm \frac{b^2}{4a} \arcsin \frac{2at}{b}\right)^2}, \qquad C = const.
\end{equation}
The function $f(u)$ is determined by $\dot{f} = y(f)$ and \eqref{E:Eq17}. The function $g(u)$ is
defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$.
(ii) Similarly to the elliptic case, from \eqref{E:Eq14} it follows that $||H|| = a$ if
and only if
the curve $c$ on $S^2_1(1)$ has constant curvature $\kappa = b$,
and the meridian $m$ is determined by the following differential equation:
\begin{equation} \label{E:Eq18}
(f \ddot{f} + \dot{f}^2 - 1)^2 = (1 - \dot{f}^2) (b^2 + 4 a^2 f^2).
\end{equation}
Setting $\dot{f} = y(f)$ in equation \eqref{E:Eq18}, we obtain
\begin{equation} \notag
y(t) = \sqrt{1 - \frac{1}{t^2}\left(C \pm \frac{t}{2} \sqrt{b^2 - 4 a^2 t^2} \pm \frac{b^2}{4a} \arcsin \frac{2at}{b}\right)^2}, \qquad C = const,
\end{equation}
In this case the function $g(u)$ is defined by $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$.
\qed
\section{Meridian surfaces with constant invariant $k$}
Let $\mathcal{M}'_m$ and $\mathcal{M}''_m$ be meridian surfaces of elliptic and hyperbolic type, respectively.
Then the invariant $k$ is given by the formula
\begin{equation} \label{E:Eq19}
k = - \frac{\kappa_m^2(u) \, \kappa^2(v)}{f^2(u)},
\end{equation}
where
$\kappa_m (u) = \displaystyle{\frac{\ddot{f}(u)}{\sqrt{\dot{f}^2(u) - 1}}}$ in the elliptic case, and
$\kappa_m (u) = \displaystyle{- \frac{\ddot{f}(u)}{\sqrt{1 - \dot{f}^2(u)}}}$ in the hyperbolic case.
In the following theorem we describe the meridian surfaces of elliptic or hyperbolic type with constant invariant $k$.
\begin{thm} \label{T:constant k}
Let $\mathcal{M}'_m$ (resp. $\mathcal{M}''_m$) be a meridian surface of elliptic (resp. hyperbolic) type from the general class.
(i) $\mathcal{M}'_m$ has constant invariant $k = const = - a^2, \; a\neq 0$ if and only if the curve $c$ on $S^2(1)$ has constant
spherical curvature $\kappa = const = b, \; b \neq 0$, and the meridian $m$ is determined by $\dot{f} = y(f)$
where
\begin{equation} \notag
y(t) = \sqrt{1 + \left(C \pm \frac{at^2}{2b}\right)^2}, \qquad C = const,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$.
(ii) $\mathcal{M}''_m$ has constant invariant $k = const = - a^2, \; a\neq 0$ if and only if the curve $c$ on $S^2_1(1)$ has constant
spherical curvature $\kappa = const = b, \; b \neq 0$,
and the meridian $m$ is determined by $\dot{f} = y(f)$
where
\begin{equation} \notag
y(t) = \sqrt{1 - \left(C \mp \frac{at^2}{2b}\right)^2}, \qquad C = const,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$.
\end{thm}
\noindent {\it Proof:}
(i) It follows from \eqref{E:Eq19} that $k = const = -
a^2, \; a \neq 0$ if and only if
$$\kappa^2(v) = \frac{a^2 f^2(u) (\dot{f}^2(u) - 1)}{\ddot{f}\,^2(u)}.$$
The last equality implies
\begin{equation} \notag
\begin{array}{l}
\kappa = const = b, \; b \neq 0;\\
a^2 f^2(u) (\dot{f}^2(u) - 1) = b^2 \ddot{f}\,^2(u).
\end{array}
\end{equation}
Hence, the curve $c$ has constant spherical curvature $\kappa = b$ and the function $f(u)$ is a solution of the following
differential equation:
\begin{equation} \label{E:Eq20}
b^2 \ddot{f}\,^2 - a^2 f^2 (\dot{f}^2 - 1) = 0
\end{equation}
Setting $\dot{f} = y(f)$ in equation \eqref{E:Eq20}, we obtain that
the function $y = y(t)$ is a solution of
\begin{equation} \notag
\frac{b}{2}(y^2)' = \pm at \sqrt{y^2 - 1}.
\end{equation}
The general solution of the above equation is given by
\begin{equation} \label{E:Eq21}
y(t) = \sqrt{1 + \left(C \pm \frac{at^2}{2b}\right)^2}, \qquad C = const.
\end{equation}
The function $f(u)$ is determined by $\dot{f} = y(f)$ and \eqref{E:Eq21}.
The function $g(u)$ is defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$.
(ii) Similarly to the elliptic case we obtain that $\mathcal{M}''_m$ has constant invariant $k = const = - a^2, \; a\neq 0$ if and only if
$c$ has constant curvature $\kappa = const = b, \; b \neq 0$,
and the meridian $m$ is determined by the following differential equation:
\begin{equation} \notag
b^2 \ddot{f}\,^2 - a^2 f^2 (1 - \dot{f}^2) = 0
\end{equation}
Again setting $\dot{f} = y(f)$ we obtain
\begin{equation} \notag
y(t) = \sqrt{1 - \left(C \mp \frac{at^2}{2b}\right)^2}, \qquad C = const.
\end{equation}
\qed
\section{Chen meridian surfaces}
Let $\mathcal{M}'_m$ and $\mathcal{M}''_m$ be meridian surfaces of elliptic and hyperbolic type, respectively.
The invariants of $\mathcal{M}'_m$ and $\mathcal{M}''_m$ are given by formulas \eqref{E:Eq7} and \eqref{E:Eq11}, respectively.
Recall that a spacelike surface in $\mathbb R^4_1$ is a non-trivial Chen surface if and only if $\lambda = 0$.
In the following theorem we classify all Chen meridian surfaces of elliptic or hyperbolic type.
\begin{thm} \label{T:Chen}
Let $\mathcal{M}'_m$ (resp. $\mathcal{M}''_m$) be a meridian surface of elliptic (resp. hyperbolic) type from the general class.
(i) $\mathcal{M}'_m$ is a Chen surface if and only if
the curve $c$ on $S^2(1)$ has
constant spherical curvature $\kappa = const = b, \; b \neq 0$,
and the meridian $m$ is determined by $\dot{f} = y(f)$
where
\begin{equation} \notag
y(t) = \frac{\pm 1}{2\,t^{\pm1}} \sqrt{4\, t^{\pm2} - a \left(t^{\pm2} - \frac{b^2}{a}\right)^2}, \qquad a = const \neq 0,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u) - 1}$.
(ii) $\mathcal{M}''_m$ is a Chen surface if and only if
the curve $c$ on $S^2_1(1)$ has constant
spherical curvature $\kappa = const = b, \; b \neq 0$,
and the meridian $m$ is determined by $\dot{f} = y(f)$
where
\begin{equation} \notag
y(t) = \frac{\pm 1}{2\,t^{\pm1}} \sqrt{4\, t^{\pm2} + a \left(t^{\pm2} - \frac{b^2}{a}\right)^2}, \qquad a = const \neq 0,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$.
\end{thm}
\noindent {\it Proof:}
(i) It follows from \eqref{E:Eq7} that $\lambda = 0$ if
and only if
\begin{equation} \notag
\kappa^2(v) = \frac{ (\dot{f}^2(u) - 1)^2 - f^2(u) \ddot{f}\,^2(u)}{\dot{f}^2(u) - 1},
\end{equation}
which implies
\begin{equation} \notag
\begin{array}{l}
\kappa = const = b, \; b \neq 0;\\
(\dot{f}^2(u) - 1)^2 - f^2(u) \ddot{f}\,^2(u) = b^2 (\dot{f}^2(u) - 1).
\end{array}
\end{equation}
Hence, the curve $c$ has constant spherical curvature $\kappa = b$ and
the function $f(u)$ is a solution of the
following differential equation:
\begin{equation} \label{E:Eq22}
(\dot{f}^2 - 1)^2 - f^2 \ddot{f}\,^2 = b^2 (\dot{f}^2 - 1).
\end{equation}
The solutions of differential equation \eqref{E:Eq22} can be found as follows. Setting $\dot{f} = y(f)$ in equation \eqref{E:Eq22}, we obtain
that the function $y = y(t)$ is a solution of the equation:
\begin{equation} \label{E:Eq23}
\frac{t^2}{4} \left( (y^2)' \right)^2 = (y^2 - 1)^2 - b^2(y^2 - 1).
\end{equation}
We set $z(t) = y^2(t) - 1$ and obtain
\begin{equation} \notag
\frac{t}{2} \,z' = \pm \sqrt{z^2 - b^2 z}.
\end{equation}
The last equation is equivalent to
\begin{equation} \label{E:Eq24}
\frac{z'}{\sqrt{z^2 - b^2 z}} = \pm \frac{2}{t}.
\end{equation}
Integrating both sides of \eqref{E:Eq24}, we get
\begin{equation} \label{E:Eq25}
\frac{b^2}{2} - z + \sqrt{z^2 - b^2 z} = c \, t^{\pm2}, \qquad c = const.
\end{equation}
It follows from \eqref{E:Eq25} that
\begin{equation} \notag
z(t) = - \frac{(a\, t^{\pm2} - b^2)^2}{4a \,t^{\pm2}}, \qquad a = 2c.
\end{equation}
Hence, the general solution of differential equation \eqref{E:Eq23} is given by
\begin{equation} \notag
y(t) = \frac{\pm 1}{2\,t^{\pm1}} \sqrt{4\, t^{\pm2} - a \left(t^{\pm2} - \frac{b^2}{a}\right)^2}, \qquad a = const \neq 0.
\end{equation}
(ii) In a similar way, in the hyperbolic case we obtain that $\lambda = 0$ if and only if
the curve $c$ has constant curvature $\kappa = b$, $b \neq 0$ and
the function $f(u)$ is a solution of
\begin{equation} \notag
(1 - \dot{f}^2)^2 - f^2 \ddot{f}\,^2 = b^2 (1 - \dot{f}^2).
\end{equation}
Doing similar calculations as in the previous case, we obtain
\begin{equation} \notag
y(t) = \frac{\pm 1}{2\,t^{\pm1}} \sqrt{4\, t^{\pm2} + a \left(t^{\pm2} - \frac{b^2}{a}\right)^2}, \qquad a = const \neq 0.
\end{equation}
\qed
\section{Meridian surfaces with parallel normal bundle}
In this section we shall describe the meridian surfaces of elliptic or hyperbolic type with parallel normal bundle.
Recall that a surface in $\mathbb R^4_1$ has parallel normal bundle if and only if $\beta_1 = \beta_2 =0$.
\begin{thm} \label{T:parallel}
Let $\mathcal{M}'_m$ (resp. $\mathcal{M}''_m$) be a meridian
surface of elliptic (resp. hyperbolic) type from the general
class.
(i) $\mathcal{M}'_m$ has parallel normal bundle if and only if
one of the following cases holds:
\hskip 10mm (a) the meridian $m$ is defined by
\begin{equation} \notag
\begin{array}{l}
f(u) = \pm \sqrt{u^2 + 2cu +d};\\
g(u) = \pm \sqrt{c^2 - d} \, \ln |u + c + \sqrt{u^2 + 2cu +d}| + a,
\end{array}
\end{equation}
where $a$, $c$, and $d$ are constants, $c^2 > d$;
\hskip 10mm (b) the curve $c$ on $S^2(1)$ has constant spherical
curvature $\kappa = const = b, \; b \neq 0$, and the meridian $m$ is
determined by $\dot{f} = y(f)$ where
\begin{equation} \notag
y(t) = \pm \frac{\sqrt{(a^2+1)\,t^2 + 2ac\,t + c^2}}{t}, \quad a = const \neq 0, \quad c = const,
\end{equation}
$g(u)$ is
defined by $\dot{g}(u) = \sqrt{\dot{f}^2(u)-1}$.
(ii) $\mathcal{M}''_m$ has parallel normal bundle if and only if
one of the following cases holds:
\hskip 10mm (a) the meridian $m$ is defined by
\begin{equation} \notag
\begin{array}{l}
f(u) = \pm \sqrt{u^2 + 2cu +d};\\
g(u) = \pm \sqrt{d - c^2} \, \ln |u + c + \sqrt{u^2 +
2cu +d}| + a,
\end{array}
\end{equation}
where $a$, $c$, and $d$ are constants, $d > c^2$;
\hskip 10mm (b) the curve $c$ on $S^2_1(1)$ has constant
spherical curvature $\kappa = const = b, \; b \neq 0$, and the
meridian $m$ is determined by $\dot{f} = y(f)$ where
\begin{equation} \notag
y(t) = \pm \frac{\sqrt{(1-a^2)\,t^2 + 2ac\,t - c^2}}{t}, \quad a = const \neq 0, \quad c = const,
\end{equation}
$g(u)$ is defined by $\dot{g}(u) = \sqrt{1 - \dot{f}^2(u)}$.
\end{thm}
\noindent {\it Proof:}
(i) Using formulas \eqref{E:Eq7} we get
that $\beta_1 = \beta_2 =0$ if and only if
\begin{equation} \label{E:Eq26}
\begin{array}{l}
\displaystyle{\kappa\, \frac{d}{du}\left( \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{\dot{f}^2 - 1}} \right) - \frac{d}{dv}(\kappa) \, \frac{f \ddot{f} + \dot{f}^2 - 1}{f \sqrt{\dot{f}^2 - 1}} = 0};\\
\displaystyle{\kappa\, \frac{d}{du}\left( \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{\dot{f}^2 - 1}} \right) + \frac{d}{dv}(\kappa) \, \frac{f \ddot{f} + \dot{f}^2 - 1}{f \sqrt{\dot{f}^2 - 1}} = 0}.
\end{array}
\end{equation}
It follows from \eqref{E:Eq26} that there are two possible cases:
\vskip 1mm
Case (a): $f \ddot{f} + \dot{f}^2 - 1 = 0$.
The general solution of this differential equation is $f(u) = \pm \sqrt{u^2 + 2cu +d}$, $c = const$, $d = const$.
Using that $\dot{g}^2 = \dot{f}^2-1$, we get $\dot{g}^2 = \displaystyle{ \frac{c^2 - d}{u^2 + 2cu +d}}$, and hence $c^2 - d >0$.
Integrating both sides of the equation
\begin{equation} \notag
\dot{g}(u) = \pm \displaystyle{ \frac{\sqrt{c^2 - d}}{\sqrt{u^2 + 2cu +d}}},
\end{equation}
we obtain $g(u) = \pm \sqrt{c^2 - d}\, \ln |u + c + \sqrt{u^2 + 2cu +d}| + a$, $a = const$.
Consequently, the meridian $m$ is defined as described in \emph{(a)}.
\vskip 1mm
Case (b):
$\displaystyle{ \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{\dot{f}^2 - 1}}} = a = const$, $a \neq 0$ and $\kappa = b = const$, $b \neq 0$.
Hence, in this case the curve $c$ has constant spherical curvature $\kappa = b$ and
the meridian $m$ is determined by the following differential
equation:
\begin{equation} \label{E:Eq27}
f \ddot{f} + \dot{f}^2 - 1 = a \sqrt{\dot{f}^2 - 1}, \qquad a = const \neq 0.
\end{equation}
The solutions of differential equation \eqref{E:Eq27} can be found in the following way. Setting $\dot{f} = y(f)$ in equation \eqref{E:Eq27}, we obtain
that the function $y = y(t)$ is a solution of the equation:
\begin{equation} \label{E:Eq28}
\frac{t}{2} \,(y^2)' + y^2 -1 = a \sqrt{y^2 - 1}.
\end{equation}
If we set $z(t) = \sqrt{y^2(t) -1}$ we get
\begin{equation} \notag
z' + \frac{1}{t}\, z = \frac{a}{t}.
\end{equation}
The general solution of the above equation is given by the formula $z(t) = \displaystyle{\frac{c + at}{t}}$, $ c = const$. Hence, the general solution of \eqref{E:Eq28} is
\begin{equation} \notag
y(t) = \pm \frac{\sqrt{(a^2+1)\,t^2 + 2ac\,t + c^2}}{t}, \quad c = const.
\end{equation}
(ii) In a similar way, considering meridian surfaces of hyperbolic type we obtain that $\beta_1 = \beta_2 =0$ if and only if
one of the following cases holds.
\vskip 1mm
Case (a): $f \ddot{f} + \dot{f}^2 - 1 = 0$. In this case we get
\begin{equation} \notag
f(u) = \pm \sqrt{u^2 + 2cu +d}; \qquad
g(u) = \pm \sqrt{d - c^2} \, \ln |u + c + \sqrt{u^2 + 2cu +d}| + a,
\end{equation}
where $a$, $c$, and $d$ are constants, $d > c^2$.
\vskip 1mm
Case (b):
$\displaystyle{ \frac{f \ddot{f} + \dot{f}^2 - 1}{\sqrt{1 - \dot{f}^2}}} = a = const$, $a \neq 0$ and $\kappa = b = const$, $b \neq 0$.
Doing similar calculations as the calculations for solving \eqref{E:Eq27}, we obtain
\begin{equation} \notag
y(t) = \pm \frac{\sqrt{(1-a^2)\,t^2 + 2ac\,t - c^2}}{t}, \quad c =
const.
\end{equation}
\qed
\vskip 3mm
Similarly to the elliptic or hyperbolic type one can study the invariants of the meridian surfaces of parabolic type.
The classes of meridian surfaces of parabolic type with constant Gauss curvature, constant mean curvature,
constant invariant $k$, the Chen meridian surfaces of parabolic type,
and the meridian surfaces of parabolic type with parallel normal bundle can be described in an analogous way.
\end{document} | math | 53,752 |
\begin{document}
\author{
Jake Grigsby \\
University of Virginia \\
\texttt{[email protected]}
\And Yanjun Qi \\
University of Virginia \\
\texttt{[email protected]}
}
\title{A Closer Look at Advantage-Filtered Behavioral Cloning in High-Noise Datasets}
\begin{abstract}
Recent Offline Reinforcement Learning methods have succeeded in learning high-performance policies from fixed datasets of experience. A particularly effective approach learns to first identify and then mimic optimal decision-making strategies. Our work evaluates this method's ability to scale to vast datasets consisting almost entirely of sub-optimal noise. A thorough investigation on a custom benchmark helps identify several key challenges involved in learning from high-noise datasets. We re-purpose prioritized experience sampling to locate expert-level demonstrations among millions of low-performance samples. This modification enables offline agents to learn state-of-the-art policies in benchmark tasks using datasets where expert actions are outnumbered nearly $65:1$.
\end{abstract}
\section{Introduction}
Reinforcement Learning (RL) promises to automate decision-making and control tasks across a wide range of applications. However, there are many real-world problems for which the typical learning and exploration process is too expensive (e.g., robotics, finance) or too dangerous (e.g., healthcare, self-driving) to make RL a viable solution. Fortunately, there is hope that we can trade RL's reliance on exploration and online feedback for a new reliance on large fixed datasets, much like those that have driven progress in supervised learning over the last decade \cite{russakovsky2015imagenet, levine2020offline, levineOfflineIAS}. In many situations, we can collect experience by recording the decisions of human demonstrators or existing control systems. For example, we can collect data for self-driving systems by mounting cameras and other sensors onto human-driven vehicles. However, as the scale of the dataset collection process grows, it becomes increasingly impractical to verify the quality of the data we are collecting. Even worse: the task may be challenging enough that no quality demonstrations could ever exist; instead, fleeting moments of optimal decision-making occur by chance in a sea of noise collected by a large number of agents.
Our goal is to build agents that can learn to pick out valuable information in large, unfiltered datasets containing experience from a mixture of sub-optimal policies. Recent work has proposed new ways to mimic the best demonstrations in a dataset \cite{nair2020accelerating, wang2020critic, wang2018exponentially, peng2019advantageweighted, chen2020bail}.
In this paper, we analyze and address some of the problems that arise in determining which demonstrations are worth learning. We build a custom dataset of experience from popular continuous control tasks to demonstrate the challenge of learning from datasets with few or zero successful trajectories and provide an easy-to-implement solution that lets us learn expert policies from millions of low-performance samples.
\section{Background and Related Work}
In the interest of space, we assume the reader is familiar with Deep RL, continuous control tasks, and the general off-policy actor-critic framework. For an introduction, please refer to \cite{lillicrap2015continuous} and \cite{fujimoto2018addressing}.
\subsection{Offline Reinforcement Learning}
Offline (or ``Batch") RL is a subfield that deals with the special case of learning from static demonstration datasets. In offline RL, our goal is to discover the best policy given a dataset of fixed experience without the ability to explore the environment. The naive application of off-policy RL methods to the fully offline setting often fails due to distribution shift between the dataset and test environment \cite{levine2020offline}. Solutions based on approximate dynamic programming are also vulnerable to exploding Q-values due to the propagation of overestimation error with no opportunity for correction in the online environment \cite{kumar2019stabilizing}.
Modern approaches to offline RL broadly involve modifications to standard off-policy algorithms to reduce overestimation and minimize distributional shift. REM uses an ensemble of critics to reduce overestimation error \cite{agarwal2020optimistic}. UWAC \cite{uwac} uses the uncertainty of value predictions to reduce error propagation. CQL \cite{kumar2020conservative} penalizes out-of-distribution Q-values to encourage in-distribution actions. MOPO \cite{yu2020mopo} is a model-based method that penalizes actions that leave the data distribution that the model was trained on.
\subsection{Advantage-Filtered Behavioral Cloning}
While much progress has been made in adapting off-policy RL to the constraints of the offline setting, these methods often struggle to outperform simple Behavioral Cloning (BC) \cite{qin2021neorl, fu2021d4rl}. In BC, we train an actor network to replicate the actions taken in the demonstration dataset:
\begin{align}
\label{bc_loss}
\mathcal{L}_{actor} = \mathop{E}_{(s, a) \sim \mathcal{D}}\left[-\text{log} \pi_{\theta}(a | s))\right]
\end{align}
Behavioral Cloning's inability to outperform its demonstrations becomes a major shortcoming when the dataset contains sub-optimal experience. Ideally, we would discard trajectories that contain low-quality demonstrations and only clone the best behavior available. This is the core idea behind Filtered Behavioral Cloning (FBC). FBC compares demonstrations to discard or down-weight the advice of sub-optimal policies. A reasonable metric for comparison is the advantage function, defined as $A^{\pi}(s, a) = Q^{\pi}(s, a) - V^{\pi}(s)$, which represents the change in expected return when taking action $a$ instead of following the current policy. Filtering experience by advantage is essentially a counterfactual query - we compute how much better action $a$ \textit{would have been} than the action chosen by our policy. We will refer to BC methods that filter experience by advantage as Advantage-Filtered Behavioral Cloning (AFBC). Because we do not have access to the true $Q^{\pi}$ or $V^{\pi}$ functions, we need to estimate the advantage. There are two main approaches:
\begin{enumerate}
\item \textbf{Monte Carlo Advantages} define an estimate of the advantage $\hat{A}^{\pi}(s, a) = \hat{\eta}(s, a) - V_{\phi}(s)$, where $\hat{\eta}(s, a)$ represents the empirical expected return of trajectories in the buffer that contain this $(s, a)$ pair and $V_{\phi}(s)$ represents a learned state value function parameterized by a neural network with weights $\phi$.
\item \textbf{Q-Based Advantages} define an estimate of the advantage $\hat{A}^{\pi}(s, a) = Q_{\phi}(s, a) - \mathop{E}_{a' \sim \pi(s)}[Q_{\phi}(s, a')] \approx Q_{\phi}(s, a) - \frac{1}{k}\mathop{\Sigma}_{i=0}^{k}Q_{\phi}(s, a' \sim \pi_{\theta}(s))$. We typically learn $Q_{\phi}$ as in standard off-policy actor-critics. The BC loss prevents the actor policy from diverging from the demonstration policy and reduces the risk of overestimation error.
\end{enumerate}
Once we have a reliable estimate of the advantage function, we add a ``filter" to the BC loss (Eq \ref{bc_loss}):
\begin{align}
\label{afbc_loss}
\mathcal{L}_{actor} = \mathop{E}_{(s, a) \sim \mathcal{D}}\left[-f(\hat{A}^{\pi}(s, a))\text{log} \pi_{\theta}(a | s))\right]
\end{align}
There are two common choices of $f$:
\begin{enumerate}
\item \textbf{Binary Filters} define $f(\hat{A}(s, a)) = \mathbbm{1}_{\{\hat{A}^{\pi}(s, a) > 0\}}$. This creates a boolean mask that eliminates samples which are thought to be worse than the current policy \cite{wang2020critic}.
\item \textbf{Exponential Filters} define $f(\hat{A}(s, a)) = \text{exp}(\beta \hat{A}^{\pi}(s, a))$, where $\beta$ is a hyperparameter. The advantage values are often clipped or normalized for numerical stability \cite{wang2018exponentially, nair2020accelerating}.
\end{enumerate}
The methods implemented and expanded upon in this paper are most directly related to CRR \cite{wang2020critic} and AWAC \cite{nair2020accelerating}, as discussed in Section \ref{implementation}. AWAC and CRR are roughly concurrent publications that arrive at a very similar method. CRR is focused on offline RL in the RL Unplugged benchmark \cite{fu2021d4rl}, while AWAC is more concerned with accelerating online fine-tuning by pre-training on smaller offline datasets. These methods are the latest iteration of the core AFBC idea that has appeared many times in recent literature. This section is partly an attempt to unify the literature and highlight a common theme that has been somewhat under-recognized. To the best of our knowledge, the first deep-learning-era AFBC implementation is MARWIL \cite{marwil}, which performed BC on samples re-weighted by their Monte Carlo advantage estimates and applied the technique to environments like robot soccer and multiplayer online battle arenas. AWR \cite{peng2019advantageweighted} is a very similar method that focuses on continuous control tasks. BAIL \cite{chen2020bail} learns an upper-envelope value function that encourages optimistic value estimates, and therefore conservative advantage estimates. Samples are filtered based on heuristics of their advantage relative to the rest of the dataset (e.g. with advantage larger than $x$ or larger than $x\%$ of all samples, where $x$ becomes a hyperparameter). A similar approach appears in the experiments of Decision Transformer \cite{chen2021decision}. In \cite{nair2018overcoming}, demonstration datasets accelerate off-policy learning by providing an expert action when its value exceeds that of the agent's policy (according to the critic network) - essentially creating a deterministic binary filter. SIL \cite{oh2018selfimitation} combines an AFBC-style loss with the standard policy gradient in an online setting, where the BC step greatly improves sample efficiency. SAIL \cite{ferret2020self} adds Q-based advantages to SIL and corrects outdated MC estimates by replacing them with parameterized estimates when they become too pessimistic. \cite{gangwani2019learning} gives another approach to self-imitating policy gradients for sample-efficient (online) continuous control. SQIL \cite{reddy2019sqil} and ORIL \cite{zolna2020offline} use standard off-policy optimization with a modified reward function that encourages the agent to stay in (or return to) $(s, a)$ pairs that are covered by the dataset, thereby helping it recover from distributional shift. We provide a rough classification of the most relevant modern literature according to the tree diagram in Figure \ref{afbc_tree}.
Imitation Learning \cite{Osa_2018} also deals with learning from sub-optimal demonstrations. While the goals and experiments are similar to the offline RL setting, the methods are typically quite different. See VILD \cite{pmlr-v119-tangkaratt20a}, IC-GAIL, and 2IWIL \cite{wu2019imitation} for recent examples.
\begin{figure}
\caption{A rough classification of the recent Advantage-Filtered Behavioral Cloning literature, which contains a surprising amount of overlap due to concurrent publication and changes in motivation/experimental focus. $^*$ SIL and SAIL use a clipped rather than an exponential advantage, but the loss is still decreasing in the magnitude of the advantage.}
\label{afbc_tree}
\end{figure}
\section{Datasets for High-Noise Offline RL}
Our goal is to investigate the performance of AFBC as the quality of the dataset decreases, and useful demonstrations are hidden in sub-optimal noise. The first step is to gather those fixed datasets and create a systematic approach to varying the quantity and quality of demonstrations provided to our agents. Learning from sub-optimal data is a widely recognized challenge for offline RL, and as such it is a key component of recent benchmarks, including D4RL \cite{fu2021d4rl}, RL Unplugged \cite{gulcehre2021rl}, and NeoRL \cite{qin2021neorl}. However, these benchmarks do not give us enough control over the demonstration dataset for our purposes. Of these, NeoRL and D4RL are closest to what we are looking for - both offer low, medium, and expert performance datasets of varying sizes but are not large enough for some of our experiments and only offer those three tiers of quality. In contrast, RL Unplugged contains enormous datasets dumped from agent replay buffers throughout training but gives us less control over the quality of any given subset of that data. In an effort to get the best of both worlds, we collect our own offline RL dataset in the following manner:
\begin{figure}
\caption{\textbf{Distribution of samples in all $9$ dataset types.}
\label{fig:tasks}
\end{figure}
\begin{enumerate}
\item We train online agents in $5$ tasks from the DeepMind Control Suite. The D4RL and RL Unplugged authors stress the importance of diverse policies and state-space coverage. With that in mind, we train multiple policies per environment using a mixture of algorithms and hyperparameters. More specifically, we use (deterministic) TD3 \cite{fujimoto2018addressing} and several variants of (stochastic) SAC \cite{haarnoja2018soft}, implemented in \cite{deepcontrol}.
\item We pause at regular intervals during training and record the policies' behavior for $50$ episodes of environment interaction. To broaden our state-action space coverage, we sample from stochastic policies (rather than taking the mean, as is standard for test-time policy evaluation). The average undiscounted return of the policy is saved alongside the ($s$, $a$, $r$, $s'$, $d$) tuples.
\item The saved experience is split into $5$ levels of performance. The DMC tasks have returns in the range $0 - 1000$, and we divide the dataset according to the average return of the policy at the time they were recorded. The final result is a large dataset with millions of samples and five levels of performance. Full size and quality information is listed in Appendix \ref{dset-details} Figure \ref{tbl:dset_splits}.
\end{enumerate}
From this raw data, we create a series of $9$ offline RL datasets for each task. The size and makeup of each dataset is listed in Figure \ref{fig:tasks}. \textit{Great-Expert} splits $1,500,000$ samples evenly across expert and high-performance demonstrations. This represents a near best-case scenario for behavioral cloning methods in which very little data needs to be actively ignored. \textit{Okay-Expert} begins a trend where the same $1,500,000$ sample budget is allocated to lower quality data, making vanilla BC less and less effective. This concludes with \textit{VeryBad-Expert}, where samples are split evenly across all $5$ performance bins. The next $4$ datasets limit expert experience to just $100,000$ samples, which our experiments show is typically enough to train successful policies without overfitting. The challenge is finding those $100,000$ samples in a buffer of millions of poor demonstrations. The most difficult of these is \textit{Signal 4.5M}, where sub-optimal data crowds the dataset at a ratio of almost $65:1$. Finally, we test the agent's ability to extract knowledge from pure noise. In the \textit{Stitching} task, the agent is given $6,500,000$ samples of experience from agents that perform only slightly better than random policies and must learn to identify the few moments of optimal decision making and combine them into a single policy.
We benchmark the challenge of Behavioral Cloning in continuous control tasks by training BC agents on our high-noise and multi-policy datasets. The results are shown in Appendix \ref{additional-results} Figure \ref{fig:sbc_results}. BC often fails to make progress, even with the \textit{Great-Expert} datasets, presumably because of distribution shift or mixed-policy learning. We see a sharp decline in BC performance as the dataset fills with noisy demonstrations.
\section{Challenges in AFBC}
\subsection{AFBC Baseline and Implementation Details}
\label{implementation}
We implement a custom AFBC baseline combining elements of AWAC and CRR. Monte Carlo advantage estimates can work well in practice but limit our ability to learn from low-performance data by creating pessimistic advantage estimates. Therefore, we adopt the Q-Based estimates from CRR and AWAC, using critic networks trained similarly to SAC \cite{haarnoja2018soft} and TD3 \cite{fujimoto2018addressing}.
\begin{align}
\mathcal{L}_{critic} &= \mathop{\mathbb{E}}_{(s, a, r, s') \sim \mathcal{D}} \left[\frac{1}{2}\sum_{i=1}^{2}(Q_{\phi, i}(s, a) - (r + \gamma(\mathop{min}_{j=1,2}Q_{\phi', j}(s', \Tilde{a}')))^2\right], \Tilde{a}' \sim \pi_{\theta}(s')
\end{align}
The advantage estimate is computed using $4$ action samples:
\begin{align}
\hat{A}^{\pi}(s, a) = \frac{1}{2}\sum_{j=1,2}Q_{\phi, j}(s, a) - \frac{1}{4}\sum_{i=0}^{4}(\frac{1}{2}\sum_{j=1,2}Q_{\phi, j}(s, a' \sim \pi_{\theta}(s)))
\end{align}
Note that we do not use distributional critics as in CRR - nor do we use the the critic weighted policy technique at test time. The policy is a tanh-squashed Gaussian distribution, implemented as in \cite{pytorch_sac}. We use a single codebase for all experiments in order to control for small implementation details \cite{henderson2019deep}, and have extensively benchmarked our code against existing results in the literature \cite{fu2021d4rl, qin2021neorl}. See Appendix \ref{appendix-implementation}
for more implementation details and a full list of hyperparameters.
\subsection{Binary vs. Exponential Filters}
While the exponential filter has been used with great success in prior work, it comes with two non-trivial implementation challenges. First, we need to deal with the magnitude of advantages across different environments. Advantage estimates are dependent on the scale of rewards, which can vary widely even across similar tasks in the same benchmark (e.g., Gym MuJoCo \cite{brockman2016openai}). There are plenty of reasonable approaches to solving this. The simplest is to clip advantages in a numerically stable range, but this runs the risk of losing the ability to differentiate between high-advantage actions. MARWIL keeps a running average and normalizes the advantages inside the filter. The AWAC codebase considers several alternatives, including softmax normalization. An interesting alternative is PopArt \cite{vanhasselt2016learning, hessel2018multitask}; by standardizing the output of our critic networks we rescale advantages and get the benefits of PopArt's stability and hyperparameter insensitivity for free. The second challenge is the temperature hyperparameter $\beta$. Prior work demonstrates significant changes in performance across similar $\beta$ values \cite{marwil}, and is often forced to use different settings in each domain \cite{nair2020accelerating}.
We demonstrate the extent of the problem by implementing $9$ reasonable variants and evaluating them on tasks from the D4RL benchmark. We use D4RL instead of our custom datasets in order to validate our implementation and compare against previously published results. The results are shown in Figure \ref{fig:exp-results}. There is very little correlation between a setting's relative return in one task and its performance in the others, making it difficult to set a high-performance default a priori\footnote{Note that all filters in our codebase are clipped for numerical stability such that exponential filters with large $\beta$ values begin to approximate binary filters when using adaptive learning rates.}. A change in $\beta$ is enough to take us from near-failure to state-of-the-art performance. We argue that this could be considered a major shortcoming when benchmarking research and dealing with real-world problems, especially in a field that already suffers from significant implementation issues \cite{engstrom2020implementation, henderson2019deep}. For this reason, all following experiments use a binary filter.
\begin{figure}
\caption{\textbf{The implementation challenges of exponential filters.}
\label{fig:exp-results}
\end{figure}
\subsection{Advantage Distributions and Effective Batch Size}
\label{problems_list}
In an effort to get a better understanding of the dynamics of the AFBC algorithm, we track the distribution of advantages in the dataset throughout training. Example results on the \textit{VeryBad-Expert} datasets are shown in Figure \ref{adv_histograms}. Using data from our experiments, we identify three challenges with current AFBC methods:
\begin{figure}
\caption{\textbf{Distribution of advantage (adv.) estimates throughout training.}
\label{adv_histograms}
\end{figure}
\paragraph{Effective Batch Size.} Regardless of dataset quality, there are many samples with negative advantage. In a typical task, the critic networks assign a negative advantage to more than half of the dataset - and that number can rise as high as $90\%$ in some cases. This means that we rely on a small fraction of each batch to compute our gradients and improve the actor. The effect is most apparent when using binary filters, where we may be entirely disregarding a large portion of each minibatch. Exponential filters attribute at least some learning signal to each sample, but we are likely to be down-weighting most of our batch. The AFBC baselines show that learning is still possible at low signal-to-noise ratios, but we are sacrificing stability by reducing our network's effective batch size. In noisy datasets (\textit{Signal-4.5M}), advantageous samples become so rare that a uniform sample of data yields batches with prohibitively high variance.
\paragraph{Noisy Labels.} Advantage estimates are highly concentrated at low absolute values. Many of the demonstrations have advantage estimates that oscillate close enough to zero that they can be labeled as positive or negative based on randomness in the estimator.
\paragraph{Static Advantage Distributions.} The advantage distribution does not change very much over time. The estimator spends the first few thousand learning steps adapting to the task's reward scale but makes few adjustments over the remaining steps. A priori, we might expect the agent to clone most or all of the dataset before becoming more confident in its ability to identify the best strategies. Instead, the estimator quickly learns to clone a small percentage of the dataset.
\subsection{Addressing Effective Batch Size with Prioritized Sampling}
As discussed above, learning from sub-optimal datasets with low-advantage actions reduces our actor network's batch size. We can improve by sampling batches that are more likely to be accepted by the advantage filter. A straightforward way to implement this is to re-purpose Prioritized Experience Relay (PER) \cite{schaul2016prioritized} to sample high-advantage actions rather than high-error bellman backups. The critic update proceeds as usual - sampling uniformly from the replay buffer and minimizing temporal difference error across the entire dataset. This gives us the opportunity to re-compute advantages and identify new useful demonstrations. During the actor update, we sample transitions from the buffer proportional to the advantage we computed when they were last used to train the critic. We can still filter low-performance samples, but this is much less likely to be necessary (see Figure \ref{fig:experience_curves}). PER also seems to offer a partial solution to the noisy labels problem; for a sample to be presented to the filter inside of the actor update, it must have been assigned a positive advantage at some point in the recent past. Before we clone the action, it then has to be labeled positive \textit{again} - reducing our ability to be fooled by the large group of near-zero-advantage actions.
\paragraph{Prioritized Experience Replay Details: } The Prioritized Experience Replay implementation is based on OpenAI Baselines \cite{baselines}. The replay is given the option of sampling uniformly from the underlying buffer or using prioritized sampling. We sample uniformly during the critic update and then update the samples' priorities according to $\text{max}(\hat{A}^{\pi}(s, a), \epsilon)$, where $\epsilon$ is a small positive constant. We also experiment with binary priority weights ($\mathbbm{1}_{\{\hat{A}^{\pi}(s, a) \geq 0\}} + \epsilon$). See Appendix \ref{additional-results} Figure \ref{fig:weight_comp} for a brief comparison. We do not use any importance sampling weights. This is a key difference between our use-case and the traditional use of PER. Normally, we prioritize samples based on a value that is highly correlated with their effect on the gradient (e.g., absolute TD error) and need to use importance weights to compensate for sampling the highest priorities. This priority system decreases our chances of sampling actions that our filter will discard but does not suffer from the same skewed gradient values. As the actor takes gradient steps in the direction of the approved experience, its action probabilities center around the provided action until the advantage drops to zero. In order to keep track of this effect, we re-compute the priorities of the sampled actions after each actor update.
\begin{figure}
\caption{\textbf{Effective batch size of AFBC agents}
\label{fig:experience_curves}
\end{figure}
This simple trick leads to a dramatic improvement in performance on the \textit{Signal} datasets. Results are listed in Figures \ref{afbc_results} and \ref{fig:hard_afbc}. Simply put: if AFBC can learn a high-quality policy, AFBC with the PER trick can maintain that performance despite an enormous amount of noise. This technique also increases performance in the \textit{Stitching} task, thanks to its enhanced ability to identify the rare optimal decisions of random or near-random policies. It also risks overfitting and instability on the high-quality datasets; this can be corrected by adjusting the $\alpha$ prioritization parameter of the experience replay. Low $\alpha$ values reduce PER to uniform sampling. These experiments use $\alpha = .6$.
\begin{figure}
\caption{\textbf{AFBC in High-Noise Continuous Control Tasks.}
\label{afbc_results}
\end{figure}
\subsection{Learning from Worst-Case Experience}
\label{mc1d-sec}
One question that arises when working with large sub-optimal datasets is what may happen in the worst-case scenario where much of the dataset is intentionally misleading. In this setting, we lose the ability to stitch together random policies and must instead learn to isolate the expert data and ignore everything else. To research this situation, we need a domain where humans can easily intuit about the optimal strategy and identify the worst possible policy. Therefore, we will take a brief detour from the high-dimensional robotic control tasks of the DMC Suite and consider the classic ``Mountain Car" task in which an under-powered car learns to gain momentum by going backward to summit a large hill. The environment is pictured in Figure \ref{fig:mc1d}a. The agent receives a large positive reward $+100$ for reaching the goal flag on top of the mountain, with a small penalty for fuel expenditure along the way. The worst possible solution is to gain the speed necessary to climb up the mountain before deciding to turn around and return to the starting position, thereby wasting as much fuel as possible. The default state space is a $(position, velocity)$ tuple, but we compress this information to a scalar representing the current position and direction of movement to plot the $Q$ and $A$ functions as a 3D surface. The action space is bounded in $[-1, 1]$, where positive actions accelerate the cart to the right, and negative actions accelerate it to the left.
The compressed state space makes the problem more difficult by hiding the velocity information necessary to manage fuel expenditure, but it is still possible to reach the goal flag. We collect a dataset of expert TD3 actions, alongside actions from a random policy. We also isolate random actions that display worst-case behavior where the cart is making progress up the mountain but reverses to move downhill. We create three offline datasets: expert demonstrations, a $9:1$ ratio of random and expert demonstrations, and a $9:1$ ratio of worst-case (or `adversarial') actions. Figure \ref{fig:mc1d}b shows the results of BC, AFBC and AFBC+PER on these datasets. BC performs as expected: it learns from expert data but is distracted by random actions and is confused by the adversarial demonstrations. Default AFBC can ignore the random actions but still succumbs to the adversarial advice. We attribute this to Q-value inflation - a hypothesis discussed in the appendix of the CRR \cite{wang2020critic} paper; in short: the adversarial advice is so concentrated in a small region of $(s, a)$ space that the bias of consistent Q-updates causes the critic to overestimate the advantage of the adversarial actions. AFBC with the PER trick is better equipped to handle this situation. By replaying experience proportional to its advantage, PER ignores actions that have a small positive advantage due to bias and clones more of the true expert data. AFBC+PER can solve the task based on the adversarial dataset, although the sparse reward has inherently high variance. The critic networks correctly minimize the advantage of the adversarial data (Figure \ref{fig:mc1d}c) and learn an accurate Q-function (Figure \ref{fig:mc1d}d).
\begin{figure}
\caption{\textbf{Adversarial Demonstrations in Modified Mountain Car.}
\label{fig:mc1d}
\end{figure}
\section{Discussions \& Conclusion}
The ability to automatically learn high-performance decision-making systems from large datasets will open up exciting opportunities to safely and effectively apply Reinforcement Learning to the real world. There are many domains where we can find enough data to train large neural network policies but cannot verify the demonstrated actions' quality. Success hinges on our ability to answer counterfactual questions about the data: are the decisions made in the dataset the correct ones, or can we find a way to do better? Advantage-Filtered Behavioral Cloning offers an intuitive way to formalize and answer this question. In this paper, we have conducted a thorough empirical study that attempts to unify existing techniques, identify critical obstacles, and provide assurance that this method can learn from unfiltered datasets of any size. However, the prioritized sampling method does not fully address the noisy label and static distribution problems discussed in Section \ref{problems_list}. We experimented with several theoretical solutions that provided somewhat underwhelming improvements on our baseline tasks. Please see Appendix \ref{future_dirs} for a thorough discussion of future directions.
Methods that can ignore or even improve using low-quality data are valuable because they simplify offline RL dataset collection by reducing the risk that additional data will damage the system. Adding more data is rarely unhelpful and is likely to increase performance. This creates an engineering situation similar to Deep Supervised Learning, where more data and a bigger model are never the wrong answer. Moving forward, we hope to combine this approach with the kinds of large network architectures and high-dimensional datasets that have spurred progress in Deep Supervised Learning to solve complicated tasks beyond simulated control benchmarks.
\printbibliography
\appendix
\section{Dataset Details}
\label{dset-details}
We provide a listing of the quantity of available Q-learning samples for each environment in Figure \ref{tbl:dset_splits}. Figure \ref{fig:tasks} shows the breakdown of how samples are distributed to generate datasets of varying quality.
\begin{figure}
\caption{\textbf{Available dataset sizes.}
\label{tbl:dset_splits}
\end{figure}
\section{Implementation Details}
\label{appendix-implementation}
The code for the experiments in this paper can be found at \textcolor{blue}{\url{https://github.com/jakegrigsby/cc-afbc}}. An updated implementation of our preferred binary-filter PER variant (along with standard online learning and a number of other training tricks from the literature) can be found at \textcolor{blue}{\url{https://github.com/jakegrigsby/super_sac}}.
\subsection{Advantage Weighted Actor-Critic and Critic Regularized Regression Baselines}
Hyperparameters and implementation details are listed in Table \ref{hparams_baseline}.
\begin{table}[h]
\centering
\begin{tabular}{|l|l|}
\hline
\textbf{Policy Log Std Range} & {[}-10, 2{]} \\
\textbf{Target Delay} & 2 \\
\textbf{Weight Decay} & None \\
\textbf{Gradient Clipping} & None \\
\textbf{Actor LR} & 1e-4 \\
\textbf{Critic LR} & 1e-4 \\
\textbf{Eval Interval} & 5000 \\
\textbf{Eval Episodes} & 10 \\
\textbf{Buffer Size} & Size of Dataset \\
\textbf{Gamma} & 0.99 \\
\textbf{Tau} & 0.005 \\
\textbf{Batch Size} & 512 \\
\textbf{Gradient Steps} & 750k \\
\textbf{Max Episode Steps} & 1000 \\
\textbf{Action Bound} & {[}-1, 1{]} \\
\textbf{Network Architecture} & (256, ReLU, 256, ReLU) \\
\hline
\end{tabular}
\caption{Default hyperparameters for the main DMC experiments.}
\label{hparams_baseline}
\end{table}
We use a tanh-squashed Gaussian distribution for the actor, implemented as in \cite{pytorch_sac}. We also ran trials on every dataset with the implementation in \cite{SpinningUp2018}, a custom implementation of the Beta distribution described in \cite{pmlr-v70-chou17a}, as well as some shorter runs with several other publicly available implementations. The results were surprisingly mixed. The distribution parameterization can be a critical implementation detail because unlike standard online approaches, the AFBC algorithm often requires us to compute log probabilities of foreign actions far outside the center of our actor's own distribution. Numerically stable log prob computations are challenging, and implementations designed for online algorithms may have had no reason to test for stability in this context. Even successful implementations typically return large log prob values early in training. This is not an issue as long as they quickly stabilize to a reasonable range, and we use some protective clipping (e.g. at clearly unstable values like $\text{log}\pi_{\theta}(a) \in (-1000, 1000)$) to minimize the damage.
\subsection{PopArt}
PopArt is implemented as described in \cite{vanhasselt2016learning} and \cite{hessel2018multitask}. We use an adaptive step size when computing the normalization statistics in order to reduce reliance on initialization. The $\nu$ value is initialized to a large positive constant to improve stability (see \cite{vanhasselt2016learning} Pg 13).
\subsection{Evaluation}
The scores displayed in the tables above are computed by:
\begin{enumerate}
\item Smoothing the learning curve with a polyak coefficient of $.65$.
\item Determining the mean and standard deviation of several (usually $5$) smoothed learning curves from different random seeds. This creates a lower-variance learning curve.
\item Reporting the average return and two standard deviations of our low-variance learning curve over the last $10$ evaluations of training.
\end{enumerate}
\section{Alternative Binary Filters, Enhanced Critic Updates, and Future Directions}
\label{future_dirs}
While prioritized sampling is an effective solution to small effective batch sizes, we would still like to learn more accurate advantage estimates and dynamic acceptance curves. In this section, we describe several enhancements in hopes of furthering future research.
\paragraph{Annealed Binary Filters with Statistical T-Tests.} Binary filters appear to rush towards an experience approval percentage that is overly pessimistic early in training. One way to address this is to introduce an overly optimistic filter and adjust its approval criteria during the learning process. A naive way to approach this is to approve samples above some low advantage threshold $a_0$ and increase that threshold over time, or to approve samples with an advantage higher than $x_0\%$ of the dataset and increase that threshold as learning progresses. However, it is challenging to set those hyperparameters across different tasks. We design a binary filter that uses a pairwise statistical T-test to create a much more task-invariant hyperparameter. We estimate the advantage of a sample $k$ times and then estimate the advantage of actions recommended by the current policy $k$ times, and only approve the dataset action for cloning if its advantage is greater than that of the current policy with statistical confidence $p_t$, where $p_t$ can be annealed from $1.0$ to high confidence $.05$. Experiments show that this does smooth the experience approval curve, but the final performance only matches that of the standard binary filter in the control tasks we study. This approach is inspired by \cite{lagoudakis2003reinforcement}, and implemented using the statistics functions in \cite{jones2001scipy}.
\paragraph{Annealed Binary Filters with Advantage Classifier Networks.} A second candidate solution to the same problem is to train an ensemble of networks that attempt to classify the advantage of actions as positive or negative. We can then use our ensemble's mean confidence and uncertainty to make informed decisions about experience approval - inspired by techniques in self-supervised learning and pseudo-labeling \cite{rizve2021defense, guo2017calibration}. This helps us deal with the noise surrounding the large fraction of samples near zero advantage because pseudo-labels have to be positive for several gradient updates before the ensemble will agree on their approval. We implement this using a similar network architecture to existing critic networks but with a sigmoid output. Once again, performance is essentially identical to the AFBC+PER baseline; the DMC tasks studied are simple enough that techniques either perform very well or fail. We think it is likely that this technique could be successfully applied to a different domain.
\paragraph{Accurate and Sample-Efficient Critics with REDQ and Weighted Bellman Updates.} The improvement of AFBC over standard BC rests on our ability to accurately estimate the advantage of $(s, a)$ pairs, which makes the method vulnerable to bias in the critic learning process. Critic overestimation and error propagation is a thoroughly investigated problem in recent work \cite{fujimoto2018addressing, lan2020maxmin, kuznetsov2020controlling, anschel2017averageddqn, kumar2020discor, agarwal2020optimistic}. REDQ \cite{chen2021randomized} trains an ensemble of critic networks on target values generated by a random subset of that ensemble and provides an effective bias-variance trade-off. It is also shown to allow for many more gradient updates per sample, something that could be useful in the offline RL context where the size of our dataset is fixed. Our main experiments are focused on high-noise datasets containing millions of samples, which makes overfitting a secondary concern, but this could be a key feature when working with small datasets. Another approach to managing overestimation error is to minimize the impact of uncertain target values in the temporal difference update. We can use REDQ's critic ensemble to estimate target uncertainty and down-weight bellman backups when $s'$ is out-of-distribution. We implement a modified version of the SUNRISE \cite{lee2020sunrise} critic loss, where we weight samples in the critic update according to a normalized uncertainty metric:
\begin{align}
\mathcal{L}_{critic} = \mathcal{L}_{critic} * \text{softmax}(\tau \text{Std}(Q_{\{\theta_i | i \in \{0, 1, \dots n\})}(s', a' \sim \pi(s')))
\end{align}
Where $\tau$ is a temperature hyperparameter. Figure \ref{redq} displays the mean Q value of the critic networks in two sample-restricted ``Walker, Walk" datasets. Interestingly, critic bias typically results in uncontrollable \textit{under}-estimation in the AFBC context because actors are restricted to in-distribution actions, and the bias is caused by the $min$ function in the target computation rather than over-exploitation of positive bias by a policy gradient update. REDQ prevents Q-function collapse and allows for more stable learning despite a very small offline dataset.
\begin{figure}
\caption{\textbf{Critic Ensembles in Low-Sample Datasets.}
\label{redq}
\end{figure}
\section{Additional Results}
This sections contains additional figures referenced in the main text that were deferred to the appendix due to the page limit.
\textbf{\begin{figure}
\caption{\textbf{A comparison of binary and clipped advantage prioritization schemes.}
\label{fig:weight_comp}
\end{figure}
}
\label{additional-results}
\begin{figure}
\caption{\textbf{AFBC in Challenging Environments where BC Fails.}
\label{fig:hard_afbc}
\end{figure}
\begin{figure}
\caption{\textbf{Behavioral Cloning in High-Noise Datasets.}
\label{fig:sbc_results}
\end{figure}
\end{document} | math | 41,563 |
\begin{document}
\title{On $p$-adic classification}
\author{Patrick Erik Bradley}
\date{\today}
\begin{abstract}
A $p$-adic modification of the split-LBG classification method is presented
in which first clusterings and then cluster centers are computed which locally
minimise
an energy function. The outcome for a fixed dataset
is independent of the prime number $p$ with
finitely many exceptions.
The methods are applied to the construction of
$p$-adic classifiers in the context of learning.
\end{abstract}
\maketitle
\section{Introduction}
The field $\mathds{Q}_p$ of $p$-adic numbers is of interest in hierarchical classification
because of its inherent hierarchical structure \cite{Perruchet1982}.
A great amount of work deals with finding $p$-adic data representation
(e.g.\ \cite{Murtagh-JoC2004, Murtagh2008}).
In \cite{Brad-JoC}, the use of more general $p$-adic numbers for encoding
hierarchical
data was advocated in order to be able to include the case of non-binary
dendrograms
into the scheme without having to resort to a larger prime number $p$.
This was applied in \cite{Brad-TCJ} to the special case of data consisting
in words over a given alphabet and where proximity of words is defined by
the length of the common initial part. There, an agglomerative
hierarchic $p$-adic
clustering algorithm was described. However, the question of finding optimal
clusterings of $p$-adic data was not raised.
Already in \cite{B-PXK2001}, the performance of
classical and $p$-adic classification algorithms was compared in the
segmentation of moving images. It was observed that the $p$-adic ones were
often more efficient. Learning algorithms using $p$-adic neural networks are
described in \cite{B-PX2007,XT2009}.
Inspired by \cite{B-PXK2001},
our main concern in this article will be a $p$-adic adaptation of the
so-called split-LBG method which finds energy-optimal
clusterings of data. The name ``LBG'' refers to the initials of the
authors of \cite{LBG1980}, where it is described first. Their method is to
find cluster
centers, and then to group the data around the centers. In the next step,
the cluster centers are split, and more clusters are obtained. This process
is repeated until the desired class number is attained.
For $p$-adic data, this approach does not make sense:
first of all, cluster centers are in general not unique; and secondly,
because the dendrogram
is already determined by data, an arbitrary choice of cluster centers is not
possible---this can lead to incomplete clusterings. Hence, we first find
clusterings by refining in the direction of highest energy reduction,
until the class number exceeds a prescribed bound.
Thereafter, candidates for cluster centers are computed: they minimise
the cluster energy. The result is a sub-optimal method for
$p$-adic classification which splits a given cluster into its maximal proper
subclusters.
A variant discards first all quasi-singletons, i.e.\ clusters of energy
below a threshold value.
The {\em a posteriori} choice of centers turns out useful for
constructing
classifiers.
A first application of some of the methods described here to event history data of building
stocks is described in \cite{Brad-buildevp}. There,
the classification algorithm is performed on different $p$-adic encodings
of the data in order to compare the dynamics of some sampled municipal building stocks.
After introducing notations in Section \ref{sec-general}, we briefly describe
the
classical split-LBG method in Section \ref{sec-LBG}.
Section \ref{sec-LBGp} reformulates the minimisation task of split-LBG in
the $p$-adic setting, and describes the corresponding algorithms.
The issue on the choice of the prime $p$ is dealt with in Section
\ref{anyprime}.
Section \ref{sec-learn} constructs classifiers and presents an adaptive
learning method in which accumulated clusters of large energy are split.
\section{Generalities} \label{sec-general}
\subsection{$p$-adic numbers}
Let $p$ be a prime number, and
$K$ a field which is a finite extension field of the field $\mathds{Q}_p$ of
rational $p$-adic numbers. We call the elements of $K$ simply $p$-adic numbers.
$K$ is a normed field whose norm $\absolute{\ }_K$ extends the $p$-adic norm
$\absolute{\ }_p$ on $\mathds{Q}_p$.
Let
$\mathcal{O}_K:=\mathset{x\in K\mid \absolute{x}_K\le 1}$ denote the local ring of integers of
$K$. Its maximal ideal $\mathfrak{m}_K=\mathset{x\in K\mid \absolute{x}_K<1}$
is generated by
a {\em uniformiser} $\pi$. It has the property $v(\pi)=\frac{1}{e}$,
where $e\in\mathds{N}$ is the ramification degree of $K/\mathds{Q}_p$.
All elements $x\in K$ have a $\pi$-adic expansion
\begin{align}
x=\sum\limits_{i\ge-m} \alpha_i\pi^i \label{pi-adic-expansion}
\end{align}
with coefficients $\alpha_i$ in some set $\mathcal{R}\subseteq K$ of
representatives for the residue field
$O_K/\mathfrak{m}_K\cong\mathds{F}_{p^f}$.
In the case $q=p$, the choice $\mathcal{R}=\mathset{0,1,\dots,p-1}$
is quite often made.
By $X$ will will always mean a finite set of data taken from $K$.
\subsection{$p$-adic clusters}
A {\em disk} in some finite set $X\subseteq K$ is a subset of the form
$$
\mathset{x\in X\mid \absolute{x-a}_K<\varepsilon}
$$
for some $a\in X$ and $\varepsilon>0$.
In particular, any singleton $\mathset{x}\subseteq X$ is a disk in $X$.
The {\em cluster property} of a subset $C$ of $p$-adic data $X\subseteq K$ is given by saying that
for any $a\in C$
it holds true that
\begin{align}
\absolute{x-a}_K<\mu(C) \Rightarrow x\in C, \label{clusterproperty}
\end{align}
where
$$
\mu(C):=\max\mathset{\absolute{x-y}_K\mid x,y\in C}
$$
is the cluster diameter.
As a consequence, a cluster is a union of disks in $X$.
We will call a disk in $X$ also a {\em verticial cluster},
because in the in the dendrogram for $X$, the vertices correspond to those
clusters which are (non-singleton)\footnote{In many definitions of
dendrograms, the data correspond to terminal vertices, but in
our definition in Section \ref{somedefs}, data are not considered
as vertices of the dendrogram. Nevertheless, we do not exlude singleton clusters
from the definition of ``vertcial''. We apologise for this inconsistency.} disks.
More to the dendrogram associated to $p$-adic
data will be said in Section \ref{somedefs}. In Figure \ref{non-clust}
the ultrametric property of dendrograms is visualised
as follows: data $b,c$ connected by a path consisting of vertical and horizontal line segments
are considered as near, if the
sum of the vertical parts is short. A third datum $a$ further away from
$b$ and $c$ is, by ultrametricity, at equal distance to $b$ and $c$.
This fact is visualised by having paths $a\leadsto b$ and $a\leadsto c$
with vertical components summing up to equal length.
\begin{exa}
Let $X=\mathset{a,b,c}$, and consider the subset $C=\mathset{a,b}$.
In Figures \ref{non-clust} and \ref{nonvertclust}, we assume two different
dendrograms
for our data $X$.
In Figure \ref{non-clust}, the disks are the singletons,
the set $\mathset{b,c}$, and the whole dataset $X$.
Hence, $C$ is not a cluster in the case of Figure
\ref{non-clust}, because it does not satisfy the cluster property (\ref{clusterproperty}): $b$ and $c$ are at distance less than the diameter which equals
the distance between $a$ and $b$, whereas $C$ contains
$b$ but not $c$.
However, in Figure \ref{nonvertclust},
all data are at equal distance, so the only disks are the singletons
and $X$.
Hence, $C$ is a cluster in Figure \ref{nonvertclust},
but not a disk,
i.e.\ not verticial.
\begin{figure}
\caption{Dendrogram in which
$b,c$ are closer to each other than to $a$. It contains a subset which is not a cluster.}
\label{non-clust}
\end{figure}
\begin{figure}
\caption{Dendrogram with equidistant data contains non-ver\-ti\-cial clusters.}
\label{nonvertclust}
\end{figure}
\end{exa}
A {\em clustering} of $X$ is a collection $\mathscr{C}$ of
disjoint clusters of $X$ whose union is the whole dataset $X$.
It is called {\em verticial}, if it consists entirely of verticial clusters.
Notice that the definition of cluster depends on the dataset $X$.
In particular, a non-verticial cluster can be made into a disk by
deleting some data from $X$. E.g.\ in Figure \ref{nonvertclust}
the removal of $c$ from the dataset turns
$C=\mathset{a,b}$ into a verticial cluster.
In general, if $\mathscr{C}$ is a clustering of $X$, and $Y\subseteq X$,
then $\mathscr{C}_Y:=\mathset{C\cap Y\mid C\in\mathscr{C}}$
is the {\em restriction of $\mathscr{C}$ to $Y$}.
This motivates us to consider only the case of verticial clusterings.
\begin{Ass}
All clusterings we consider are verticial on some specified (non-empty) subsets
of $X$.
\end{Ass}
\section{The split-LBG algorithm} \label{sec-LBG}
Here, we review briefly the classical split-LBG algorithm. Details can be
found in \cite{LBG1980}.
Let $X=\mathset{a_1,\dots,a_n}$ and $C=\mathset{c_1,\dots,c_k}$ be sets of
vectors in $\mathds{R}^m$,
where $X$ is considered as the {\em data} and $C$ are the prespecified
{\em cluster centers}. The task is classically to find a partition
$\mathscr{O}=\mathset{\Omega_c\mid c\in C}$ of $X$
into $k$ clusters $\Omega_c$ minimising the energy
$$
E(\mathscr{O},C)=\sum\limits_{c\in C}\sum\limits_{a\in\Omega_c}d(a,c),
$$
where $d(x,y)$ is Euclidean distance in $\mathds{R}^m$.
In fact, the split-LBG method works with varying $C$ by
alternatively constructing partitions and then replacing each $c\in C$ by
two new centers $c+{\bf \varepsilon},c-{\bf \varepsilon}$, where ${\bf
\varepsilon}$ is a
perturbation vector in $\mathds{R}^m$ of small norm.
From these, a new partition is constructed, etc.
\section{Split-LBG in the $p$-adic case} \label{sec-LBGp}
In \cite{B-PXK2001} it was observed that the split-LBG method has no direct
translation using the $p$-adic metric.
Here, we describe a $p$-adic modification of the task from the previous section.
Let $X=\mathset{x_1,\dots,x_n}\subseteq K$ be some data consisting of $n$
$p$-adic numbers, and fix a number $k$.
The task is to find a clustering $\mathscr{C}=\mathset{C_1,\dots,C_\ell}$
of $X$ with $\ell\le k$, and for each cluster $C\in\mathscr{C}$ a {\em center} $a_C\in C$, minimising the expression
$$
E_p(X,\mathscr{C},{\bf a}):=\sum\limits_{C\in\mathscr{C}} \sum\limits_{x\in C}\absolute{x-a_C}_K,
$$
where ${\bf a}=(a_C)_{C\in\mathscr{C}}$ is the sequence of cluster centers.
Note that, by the ultrametric
property of $\absolute{\ }_K$, cluster centers
can (and will) always be chosen within $X$. This has already been taken care
of in the definition of the task. Note further that, unlike in the Archimedean
setting, cluster centers are in general not uniquely defined by their
corresponding clusters.
The most significant difference to the Archimedean case is
given by the fact that
in the $p$-adic situation, it does not make sense to choose a cluster center
{\em a priori}, as illustrated in Example \ref{clustcentre}.
Therefore, the order is reversed: first find a good partition, and then find
corresponding
cluster centers.
\begin{exa} \label{clustcentre}
Let $\mathset{a,b,c}$ be some data with corresponding dendrogram as in Figure
\ref{non-clust}.
Then choosing $a,b$ as centers leads to the clustering
$\mathscr{C}=\mathset{\mathset{a},\mathset{b,c}}$,
whereas the choice $b,c$ leads either to
$\mathscr{C}'=\mathset{\mathset{b,c}}$,
$\mathscr{C}''=\mathset{\mathset{a,b,c}}$, or to
$\mathscr{C}'''=\mathset{\mathset{b},\mathset{c}}$. But $\mathscr{C}'$
and $\mathscr{C}'''$ are not clusterings
of $\mathset{a,b,c}$, while $\mathscr{C}''$ is.
And both $\mathscr{C}'$ and $\mathscr{C}''$ each consist of one
cluster
containing the two prescribed centers instead of two distinct clusters as
should be the case classically.
\end{exa}
Last but not least, we will not give a global solution to the task in the $p$-adic case,
but
find certain types of local minima of $E_p$ in a sense which will become clear in the
following subsection.
\subsection{Some definitions} \label{somedefs}
An important tool in the classification of $p$-adic data
$X\subseteq K$ is its
dendrogram $D(X)$. In contrast to the Archimedean situation, it is uniquely determined by the data
(cf.\ \cite{Brad-JoC,Brad-TCJ}).
We view $D(X)$ as a {\em rooted metric tree}.
This means that it has a root $v_0$, and all edges are oriented away from
$v_0$ and are assigned a length which is either positive real or infinite.
The root $v_0$ corresponds to the top cluster consisting of the whole data
$X$.
The vertices correspond to clusters containing at least
two points from $X$.
An edge $e$ of $D(X)$
connecting two vertices is always bounded.
The individual points of $X$ correspond uniquely to the {\em ends}
of the tree $D(X)$. We do not view the data $X$ as part of the tree $D(X)$, but
as its boundary. Hence, any $x\in X$ sits at the one extreme of an unbounded edge.
Our viewpoint is probably in contrast to most others
on hierarchical classification, where data correspond to terminal vertices
of dendrograms.
However, we argue in our favour
that the dendrogram should reflect hierarchic
approximations of data by clusters (vertices in $D(X)$)
or, more generally, by initial terms in some $p$-adic expansion for data
(points in $D(X)$).
We refer to
\cite{Brad-JoC,Brad-TCJ}
for a more detailed description of $p$-adic dendrograms.
Given some vertex $v$ of $D(X)$, let
${\rm ch}(v)$ denote the set of edges emanating from $v$ (i.e.\ not towards
$v_0$),
and let $\#{\rm ch}(v)$ be its cardinality. By abuse of notation, we will
identify
${\rm ch}(v)$ with the set of vertices and ends attached to the edges in ${\rm ch}(v)$.
Now, an upper bound for the contribution to $E_p$ of a cluster $C_v$, represented by some
vertex or end $v$ is
$$
\mu(v):=\mu(C_v)=\max\mathset{\absolute{x-y}_K\mid x,y\in C_v}.
$$
As a side remark, note that this is nothing but the Haar measure of $K$ evaluated in the $p$-adic
disk $D_v\subseteq K$ corresponding to $v$. In any case, if $v$ is an end then $\mu(v)=0$,
otherwise $\mu(v)>0$.
Given a set $V$ of vertices or ends of $D(X)$, we set
\begin{align}
E(V):=\sum\limits_{v\in V}(\#C_v-1)\cdot\mu(v), \label{verclusten}
\end{align}
and also write $E(v_1,\dots,v_b)$ in the case that
$V=\mathset{v_1,\dots,v_b}$.
Applying this to ${\rm ch}(v)$ for a vertex $v$, we obtain:
\begin{align}
E({\rm ch}(v))\le E(v). \label{children-diminish-energy}
\end{align}
The following remark shows that minimising $E(V)$ does make sense for our task:
\begin{rem}
Given a clustering $\mathscr{C}=\mathset{C_v\mid v\in V}$, where $V$ is the
corresponding set of vertices, for any choice of
$\alpha_v\in C_v$ it holds true that
$$
E_p(X,\mathscr{C},{\bf a})\le E(V)=:E(\mathscr{C}),
$$
where ${\bf a}=(\alpha_v)_{v\in V}$.
\end{rem}
Let $\mathfrak{X}_k(Y)$ be the set of all clusterings $\mathscr{C}$ of $X$
with cardinality $\ell\le k$ whose restriction to $Y$ is verticial. On the set
\begin{align}
\mathfrak{X}=\bigcup\limits_{k\in \mathds{N}}\bigcup\limits_{Y\subseteq
X}\mathfrak{X}_k(Y), \label{setofclusts}
\end{align}
of all clusterings, we define a partial ordering $\le$ (called {\em refinement}) as follows:
$$
\mathscr{C}'\le \mathscr{C},
$$
if all $C\in\mathscr{C}$ are of the form
$
C=\bigcup\limits_{i\in I} C'_i$
with $C'_i\in\mathscr{C}'$
$(i\in I)$.
Let $C_v$ be the smallest verticial cluster containing a given cluster $C$.
Then we can define the functional
$$
E\colon \mathfrak{X}\to\mathds{R},\;\mathscr{C}\mapsto\sum\limits_{C\in\mathscr{C}}(\#C-1)\cdot\mu(C_v),
$$
and observe that this obviously generalises $E(V)$ from (\ref{verclusten}):
\begin{lem}
If $\mathscr{C}\in\mathfrak{X}$ is verticial, then
$$
E(V)=E(\mathscr{C}),
$$
where $V$ is the vertex set associated to $\mathscr{C}$.
\end{lem}
\begin{lem} \label{monotonics}
$E$ is strictly monotonic:
\begin{align*}
\mathscr{C}'\le\mathscr{C}\Rightarrow E(\mathscr{C}')\le E(\mathscr{C}),
\end{align*}
and if $\mathscr{C}'\le\mathscr{C}$ are not equal, then $E(\mathscr{C}')< E(\mathscr{C})$.
\end{lem}
\begin{proof}
Assume $C=\bigcup\limits_{i\in I}C_i'\in\mathscr{C}$ with $C_i'\in\mathscr{C}'$.
Then
$$
\sum\limits_{i\in I}\#(C_i'-1)\cdot\mu(C'_{i,v})
\le
\sum\limits_{i\in I}\#(C_i'-1)\cdot\mu(C_v)
\le
(\#C-1)\cdot\mu(C_v),
$$
where the first inequality holds true, because all $C_i'$ are contained in
$C$. The second inequality is strict, if $I$ contains more than one element.
That is the case for some $C$, if $\mathscr{C}\neq\mathscr{C}'$.
\end{proof}
We denote by $E_{k,Y}$
the restriction of $E$ to
$\mathfrak{X}_k(Y)$
. The following is immediate:
\begin{lem}
Let $\mathscr{C}$ and $\mathscr{C}'$ minimise $E_{k,Y}$ and $E_{k',Y}$, respectively.
Then
$$
k\le k'\Rightarrow E(\mathscr{C}')\le E(\mathscr{C}).
$$
\end{lem}
\subsection{The verticial clustering algorithm} \label{sec-pclust}
The general strategy which we follow is to refine a given clustering of
$X$ in the ``direction'' which yields the lowest value of $E_p$ after
splitting
a vertex. The term ``direction'' refers to the refinement ordering on
$\mathfrak{X}$,
and we follow the possible ``gradients'' from a given point $\mathscr{C}\in\mathfrak{X}$.
Concretely, this means splitting a vertex with highest energy contribution.
In Section \ref{anyprime}, we will see that the terms in quotation marks here
can be taken ad literam.
In this subsection, we deal with
verticial clusterings only.
We can now formulate:
\begin{alg}[Verticial clustering] \rm \label{minclustp}
{\em Input}. $p$-adic data $X\subseteq K$ with $\#X \ge 2$, and upper bound
$k\ge 1$ for number of
clusters.
\noindent
{\em Step $0$}. Compute $b=\#{\rm ch}(v_0)$ and $E(v_0)=\mu(v_0)$.
\noindent
{\em Step $1$}. If $b>k$, then terminate.
Otherwise, compute $E({\rm ch}(v_0))$ which is not greater than $ E(v_0)$ by
(\ref{children-diminish-energy}). Further identify the set of vertices
$V_1:={\rm ch}(v_0)\cap {\rm Vert}(D(X))$.
\noindent
{\em Step $N$}. Assume that from the previous step, we are given some family
$\mathscr{V}_{N-1}=\mathset{V_{N-1}^{(i)}}$ of sets consisting of
$b_{N-1}^{(i)}\le k$ vertices, respectively. If for all
$i$ and all $v\in V_{N-1}^{(i)}$ it holds true that $b_v^{(i)}:=b_{N-1}^{(i)}+\#{\rm ch}(v)>k$,
then terminate.
Otherwise, find all $i$ and all $v\in V_{N-1}^{(i)}$ such that
$E(W_v^{(i)})$
is smallest possible,
where $W_v^{(i)}:={\rm ch}(v)\cup V_{N-1}^{(i)}\setminus\mathset{v}$
satisfies $\#W_v^{(i)}\le k$.
Again, by (\ref{children-diminish-energy}), it holds true that
$$
E(W_v^{(i)})\le E(V_{N-1}^{(i)}).
$$
Extract this new family $\mathscr{V}_N$ of vertex sets together with the
lower energy value $E_N=E(W)$ for $W\in\mathscr{V}_N$.
\noindent
{\em Output.} A family of clusterings $\mathset{\mathscr{C}_{i}\mid i\in I}$ (corresponding to the vertex sets in the
last step) for which $E=E(\mathscr{C})$ is locally minimal, together with the value of $E$.
\end{alg}
\subsection{$p$-adic cluster centers}
The next objective
is to find cluster centers
with respect to the energy functional. Assume that we are given a fixed cluster
$C=\mathset{a_1,\dots,a_n}\subseteq K$. We wish to find some $\alpha\in C$
which minimises
$$
\epsilon(\alpha):=E_p(C,\mathscr{C},\alpha)=\sum\limits_{a\in C}\absolute{a-\alpha}_K,
$$
where $\mathscr{C}=\mathset{C}$.
A {\em branch} $B$ of a rooted tree $(T,v)$ is a maximal subtree of $T\setminus\mathset{v}$.
It has a root $v_B$ among the vertices of ${\rm ch}(v)$.
Let $\mathcal{B}(T)$ denote the set of branches of $(T,v)$. In the case of our
dendrogram
$D(C)$, we will write $\mathcal{B}(C)$, instead of $\mathcal{B}(D(C))$.
The branches induce a natural partition of $C$:
$$
C=\bigcup\limits_{B\in\mathcal{B}(C)}C_B
$$
into a disjoint union of $C_B={\rm Ends}(B)$.
\begin{lem} \label{lem-energy-branch}
Let $\alpha\in C$, and $B_\alpha\in \mathcal{B}(C)$ the branch containing
$\alpha$ as an end, and $C_\alpha=C_{B_\alpha}$. Then
\begin{align} \label{energy-branch}
\epsilon(\alpha)=\#(C\setminus C_\alpha)\cdot\mu(v_0)+E_p(C_\alpha,\mathscr{C}_\alpha,\alpha),
\end{align}
where $\mathscr{C}_\alpha=\mathset{C_\alpha}$.
\end{lem}
\begin{proof}
Together with the identity:
$$
\sum\limits_{a\in C_\alpha}\absolute{a-\alpha}_K=E_p(C_\alpha,\mathscr{C}_\alpha,\alpha),
$$
this follows easily by looking at the tree $D(C)$.
\end{proof}
\begin{lem}
Assume the notations as in Lemma \ref{lem-energy-branch}.
It holds true that
\begin{align}
\frac{\epsilon(\alpha)}{\mu(v_0)}=N_\alpha+O(p^{\nu_\alpha}) \label{O4epsilon}
\end{align}
with $N_\alpha\in\mathds{N}$ and $\nu_\alpha<0$.
\end{lem}
Equation (\ref{O4epsilon}) means that $\frac{\epsilon(\alpha)}{\mu(v_0)}$
is a natural number plus some small term given as a multiple of $p^{\nu_\alpha}$.
\begin{proof}
Set $N_\alpha=\#(C\setminus C_\alpha)$, and notice that
\begin{align}
E_p(C_\alpha,\mathscr{C}_\alpha,\alpha)\le\#C_\alpha\cdot\mu(v_\alpha), \label{energybound}
\end{align}
where $v_\alpha$ is the root of $B_\alpha$. The claim now follows from the
obvious inequality
$\mu(v_\alpha)<\mu(v_0)$.
\end{proof}
Now, we can formulate our algorithm:
\begin{alg}[Cluster centers] \rm \label{centerclustp}
{\em Step $1$}. Find all branches $B^{(1)}\in \mathcal{B}(C)$ with largest
value of $\#C_{B^{(1)}}$. Extract
those clusters $C_{B^{(1)}}$ for which $\mu(v_{B^{(1)}})$ is minimal, and
the number
$$
c_1=\max\mathset{\#C_{B^{(1)}}\mid B^{(1)}\in \mathcal{B}(C)}.
$$
\noindent
{\em Step $N$}. Assume that in the previous step, a list of
clusters
$C_{B^{(N-1)}}$, and a number $c_{N-1}$ is produced.
Find all branches $B^{(N)}$ of the rooted trees $D(C_{B^{(N-1)}})$
with largest possible value $c_N$
of $\#C_{B^{(N)}}$.
Extract those clusters $C_{B^{(N)}}$ minimising $\mu(v_{B^{(N)}})$, together with $c_N$.
\noindent
At some point, there will be a {\em Step $N'$} in which the trees $D(C_{B^{(N)}})$ have only one
vertex each. The procedure terminates thus:
\noindent
{\em Output}. A list $(C_i)_{i\in I}$ of those clusters from Step $N'$ with
minimal value of $\mu(v_i)$, where $v_i$ is the vertex of $D(C_i)$.
\end{alg}
\begin{thm}
Let $C'=C_{N'}\subseteq C$ be a cluster produced by performing Algorithm
\ref{centerclustp}.
Then any $\alpha\in C'$ is a center of $C$ with respect to $E_p$.
\end{thm}
\begin{proof}
Let $C=C_0\supseteq C_1\supseteq \dots C_{N'}=C'$ be a strictly decreasing chain
of clusters produced by the $N'$ steps of Algorithm \ref{centerclustp}.
Let the corresponding cardinalities be $c_0,\dots,c_{N'}$.
By applying Lemma \ref{lem-energy-branch}, it holds true that
\begin{align}
\epsilon(\alpha)=c_{N'}\cdot\mu(v_{N'})+\sum\limits_{i=1}^{N'}(c_{i-1}-c_i)\cdot\mu(v_{i-1}),
\end{align}
where $v_j$ is the root of the corresponding branch from {\em Step j}.
The minimality of $\epsilon(\alpha)$ is guaranteed by (\ref{O4epsilon}),
applied to
each step. Notice, that we have used the obvious fact that for $C'$, the inequality (\ref{energybound}) is an equality.
\end{proof}
\subsection{Quasi-verticial clustering}
The two previous subsections already lead to a $p$-adic algorithm for
verticial clusterings and their centers. In this case, subdividing a cluster
$C_v$ means to make as many subclusters as there are elements in ${\rm ch}(v)$.
In the case that e.g.\ there are many singletons, this can be a disadvantage.
Hence removing singletons provides more flexibility in that the bigger
subclusters
can either be merged or kept distinct. Even greater flexibility can be achieved
if almost indistinguishable clusters are treated as singletons.
\begin{dfn}
Fix some real $\varepsilon>0$.
A verticial cluster $C_v\subseteq X$ with corresponding vertex $v$ is called a
{\em quasi-singleton}
for $\varepsilon$, if $E(v)<\varepsilon$.
\end{dfn}
When we speak of a quasi-singleton, we mean a quasi-singleton for some
$\varepsilon$ known from the context.
\begin{figure}
\caption{Dendrogram with quasi-singleton $\mathset{a,b}
\label{quasising}
\end{figure}
\begin{exa}
The dendrogram in Figure \ref{quasising} contains a quasi-singleton
$\mathset{a,b}$,
if we set $\mu(v)=p^{-\ell}$ for vertex $v$ at level $\ell$ (indicated by the
number at the left), and $p^{-1}<\varepsilon\le p^{-1}$.
For this choice of $\varepsilon$, the cluster $\mathset{c,d}$ is not
a quasi-singleton. But this is the case for larger $\varepsilon$.
\end{exa}
Clearly, any singleton is a quasi-singleton for any $\varepsilon$.
Since we are working with a fixed $p$-adic field $K$, it is possible to choose
$\varepsilon$
so small that the quasi-singletons are precisely the singletons of
our given dataset $X$.
The algorithm we propose in the following removes quasi-singletons
in order to continue with verticial clusterings.
For this, we fix some notation: When referring to a subset $Y$ of our dataset
$X$,
we will indicate this by the subscript $Y$. E.g.\
${\rm ch}_Y(v)$ means the set of edges
in $D(Y)$ going out from $v$. Simliarly, with $\mu_Y(V)$, $E_Y(V)$ etc.
\begin{alg}[Quasi-verticial clustering]\rm \label{quasivertclust}
{\em Input.} Data $X_0:=X\subseteq K$,
and numbers $k_0:=k\ge 1$, $\varepsilon>0$.
\noindent
{\em Step 1.} Remove from $D(X)$ all $v\in{\rm ch}_{X_0}(v_0)$ corresponding to
quasi-singletons for $\varepsilon$. Let $s_1$ be the number of vertices removed.
Extract corresponding reduced dataset $X_1\subseteq X_0$,
as well as
${\rm ch}_{X_1}(v_0)$, $E_{X_1}(v_0)=\mu_{X_1}(v_0)$, and $k_1:=k-s_1$.
\noindent
{\em Step $N$.} Assume that in the previous step, we are given
a quadruple of families
$$
(\mathscr{V}_{N-1},\mathscr{X}_{N-1}, E_{N-1},\mathscr{K}_{N-1})
$$
of
sets $V\in\mathscr{V}_{N-1}$ of vertices in $D(X)$,
datasets
$X(V)\in \mathscr{X}_{N-1}$,
an energy value
$E_{N-1}=E_{X(V)}(V)$, and numbers
$k_{N-1}(V)\le k$
(where $V\in\mathscr{V}_{N-1}$).
Remove for all $V\in\mathscr{V}_{N-1}$ from $D(X(V))$ all vertices in
${\rm ch}_{X(V)}(v)$ corresponding to
$s_N(v)$ quasi-singletons,
where $v\in V$. Find all $V\in\mathscr{V}_{N-1}$ and $v\in V$ such that
\begin{enumerate}
\item $k_{N-1}(V)-s_N(v)\ge 0$, and
\item $E_{X(V)}(W_v)<E_{N-1}$ is smallest possible,
\end{enumerate}
where $W_v:={\rm ch}(v)\cup V\setminus\mathset{v}$.
Extract corresponding quadruple of families
$$
(\mathscr{V}_N,\mathscr{X}_N,E_N,\mathscr{K}_N)
$$
of
new vertex sets $W_v$,
reduced datasets $X(W_v)\subseteq X(V)$,
energy value $E_N=E(W_v)$, and $k_N(W_v):=k_{N-1}(V)-s_N(v)$.
\noindent
{\em Output.}
A list of clusterings consisting of quasi-singletons for $\varepsilon$ and clusters
produced above by collecting the remnants in each step.
\end{alg}
\begin{rem}
The output clusterings of Algorithm \ref{quasivertclust} all have energy of
the form
$$
E+O(p^\alpha),
$$
where $E$ is independent of the clustering, and $\alpha<0$ is small.
\end{rem}
We can now put things together in order to find clusterings in different ways:
\begin{alg}[(Quasi-)Verticial split-LBG$_p$]\rm \label{splitLBGp}
{\em Input.} As in Algorithm \ref{minclustp} (resp.\ Algorithm \ref{quasivertclust}).
\noindent
{\em Step 1.} Perform Algorithm \ref{minclustp} (resp.\ Algorithm \ref{quasivertclust}).
\noindent
{\em Step 2.} Perform Algorithm \ref{centerclustp} for each cluster occurring
in each clustering given out in the previous step.
\noindent
{\em Output.} A list $(\mathscr{C}_i,({\bf a}_j^{(i)})_{j\in J})_{i\in I}$ of
$E$-suboptimal clusterings with corresponding list of
$E$-center vectors $({\bf a}_j^{(i)})_{j\in J}$ for clustering $\mathscr{C}_i$.
\end{alg}
Both subroutines, Algorithms \ref{minclustp} and \ref{centerclustp}, boil
down to counts and evaluations of $\mu(v)$ for vertices $v$.
Therefore, we remark:
\section{Dependence on the choice of the prime $p$} \label{anyprime}
A natural issue is, how the outputs of the algorithms introduced in the
previous sections depend
on the choice of the prime number $p$. We will prove a finiteness result.
Recall that the energy of a verticial cluster $C_V$ is of the form
\begin{align}
E(C_V)= A\cdot p^{-\nu} \label{energyterm}
\end{align}
with natural numbers $A$ and $\nu$, and is additive on disjoint unions of clusters.
Splitting a cluster is performed by replacing vertex $v$ by the vertex set
${\rm ch}(v)$, and the change in energy is given by
$$
E_{\rm new}=E_{\rm old}-E(C_v)+E({\rm ch}(v)),
$$
i.e.\ the difference is
$$
\delta_vE_p:=E(C_v)-E({\rm ch}(v)).
$$
Our approch towards minimising $E_p$ is to refine the given clustering
in the direction of largest $\delta_vE_p$. Now, the quantity $\delta_vE_p$
depends on the prime number $p$ as shown by (\ref{energyterm}).
This means that different $p$ can result in different
rankings of the vertices by the order in
which they are split.
We call this the {\em $p$-ranking} of the vertices of $D(X)$.
\begin{exa}
Assume we want to find verticial clusterings of data
$$
X=\mathset{x_1,\dots,x_{13}}
$$
having
underlying dendrogram as in Figure \ref{dendro}.
\begin{figure}
\caption{A dendrogram.}
\label{dendro}
\end{figure}
Consider the vertices $a,b,c,d$ in the underlying rooted vertex tree as depicted in
Figure \ref{tree4dendro}.
\begin{figure}
\caption{Vertex tree underlying Figure \ref{dendro}
\label{tree4dendro}
\end{figure}
Then Table \ref{vertexrank} shows the different $p$-rankings of these vertices for
$p=2$, $3$ and $5$.
\begin{table}[h]
\rm
\begin{tabular}{c|c|c}
Rank&Vertex&$\delta_vE_2$\\\hline
1.&a&\rule[-5pt]{0pt}{15pt}$\frac{11}{2}$\\\hline
2.&c&\rule[-5pt]{0pt}{15pt}$\frac{9}{4}$\\\hline
3.&b&$2$\\
&d&$2$\\\hline
\multicolumn{3}{c}{\rule[-5pt]{0pt}{20pt}$p=2$}
\end{tabular}
\begin{tabular}{c|c|c}
Rank&Vertex&$\delta_vE_3$\\\hline
1.&a&\rule[-5pt]{0pt}{15pt}$\frac{22}{3}$\\\hline
2.&c&\rule[-5pt]{0pt}{15pt}$\frac{17}{9}$\\\hline
3.&b&\rule[-5pt]{0pt}{15pt}$\frac{7}{9}$\\\hline
4.&d&\rule[-5pt]{0pt}{15pt}$\frac{16}{27}$\\\hline
\multicolumn{3}{c}{\rule[-5pt]{0pt}{20pt}$p=3$}
\end{tabular}
\begin{tabular}{c|c|c}
Rank&Vertex&$\delta_vE_5$\\\hline
1.&a&\rule[-5pt]{0pt}{15pt}$\frac{44}{5}$\\\hline
2.&c&\rule[-5pt]{0pt}{15pt}$\frac{44}{25}$\\\hline
3.&b&\rule[-5pt]{0pt}{15pt}$\frac{13}{25}$\\\hline
4.&d&\rule[-5pt]{0pt}{15pt}$\frac{26}{225}$\\\hline
\multicolumn{3}{c}{\rule[-5pt]{0pt}{20pt}$p=5$}
\end{tabular}
\caption{Vertex rankings for Figure \ref{dendro}.}\label{vertexrank}
\end{table}
\end{exa}
\begin{thm}
For all but finitely many primes, the $p$-rankings of the vertices of a given
dendrogram $D(X)$ belonging to data $X$ taken from a fixed $p$-adic field
are the same.
\end{thm}
\begin{proof}
The energy gradient for a vertex $v$ can be written as
$$
\delta_vE_p=P_v(t)|_{t=\frac{1}{p}}
$$
for some polynomial $P_v(t)$ whose coefficients are natural numbers.
By dividing off powers of $t$, we may assume that $P_v(t)$ has a non-zero
constant term, hence that $P_v(0)>0$. By the considerations from the previous
sections,
we know that
\begin{align}
0<P_v\left(\frac{1}{p}\right)<P_v(0) \label{decrease}
\end{align}
for all primes $p$. By viewing $P_v(t)$ as a continuous function on the
intervall $[0,1/2]$, we see from the right inequality in (\ref{decrease}) that
$P_v(t)$ must be decreasing on some interval $[0,x]$ with positive
$x\le\frac{1}{2}$ sufficiently small.
It follows that the sequence of values
$P_v\left(\frac{1}{p}\right)$ for prime $p\to\infty$ converges to
$P_v(0)$. Since that limit equals $E(v)$
on the maximal subtree of $D(X)$ having $v$ as its root, we have proven
$$
\lim\limits_{p\to\infty}P_v\left(\frac{1}{p}\right)=E(v).
$$
In other words, for sufficiently large prime $p$, the vertex gradient can be
approximated
by the vertex energy. Hence the ranking of the vertices is approximatively the
ranking
of the numbers
\begin{align}
\frac{E(v)}{p^{\ell(v)}}, \label{energyrank}
\end{align}
where $\ell(v)$ depends on the level of $v$ in the dendrogram.
The latter ranking does not change once $p$ is sufficently large.
Hence, for large $p$ the vertex ranking does not change.
\end{proof}
\begin{rem}
Notice from (\ref{energyrank}) that using a large prime number tends to force
splitting vertices
higher up in the hierarchy underlying the dendrogram. On the other
hand, taking a small prime number allows to split also clusters containg lots of
data
at low levels in the hierarchy.
\end{rem}
\begin{thm}
Let $C\subseteq K$ be a cluster. If $a$ is a center of $C$ with respect to
$E_p$
for some prime $p$, then it is a center for all primes.
\end{thm}
\begin{proof}
From Lemma (\ref{lem-energy-branch}) it follows that
$$
\epsilon_p(a)=E_p(C,\mathscr{C},a)=\sum\limits_{v\in V}\alpha_v\mu(v),
$$
where $V$ is the set of vertices on the path $\gamma$ from the top $v_0$ down to $a$.
As $\mu(v)=p^{-\ell(v)}$, and the $\ell(v)$ form a strictly increasing
sequence $\ell_0,\dots,\ell_M$ of natural numbers as $v$ proceeds along $\gamma$, it follows that
$\epsilon_p(a)$
is given by evaluating the polynomial
$$
F_\gamma(t)=\sum\limits_{i=0}^M\alpha^\gamma_it^{\ell_i}
$$
in $t=\frac{1}{p}$,
where $\alpha^\gamma_i>0$ equals that number $\alpha_v$ with $v$ such that $\ell(v)=\ell_i$.
Now, $\epsilon_p(a)$ being a minimum means that in the collection
$$
\mathset{F_\gamma(t)\mid\text{$\gamma$ path $v_0\leadsto X$}}
$$
the term $a_0^\gamma t^{\ell_0}$ is of
lowest
degree and that coefficient $\alpha^\gamma_0$ is smallest among those terms of
lowest degree. And this does not depend on the choice of prime $p$.
\end{proof}
\section{$p$-adic learning} \label{sec-learn}
In this section we discuss a learning situation in which some $p$-adic data
$X\subseteq K$ together with a clustering $\mathscr{C}_X$
is used as
a ``training set''. The idea is to classify
new data $Y$ taken from some $p$-adic field $L\supseteq K$ on the basis of $X$
and $\mathscr{C}$.
Without loss of generality we assume that the two $p$-adic
fields $L$ and $K$ coincide.
\subsection{$p$-adic classifiers}
Learning can be performed by using a classifier which integrates new data
$y\in Y$ into an existing dendrogram $D(X)$ in order to find a suitable
cluster for $y$. We will define such in the $p$-adic situation.
As it may happen that adjoining a point $y\in Y$ to $X$
increases the size of the smallest $p$-adic disk containing the training data $X$,
we use the point at infinity already introduced in \cite{Brad-JoC}. This
allows to classify those data in $Y$ which cannot be classified on the basis
of $(X,\mathscr{C}_X)$
as belonging to the ``cluster at infinity''.
Our method will use the extended dendrogram
$$
D_\infty(X)=D(\mathds{P}(X)),
$$
where $\mathds{P}(X)=X\cup\mathset{\infty}$\footnote{Note that $D_\infty(X)$
is
what is denoted by $D(X)$ in \cite{Brad-JoC,Brad-TCJ}.}. The datum $\infty$
will be depicted at the end of a path going upwards from $v_0$, whereas all other
data will be at the end of paths leading downwards.
\begin{exa}
\begin{figure}
\caption{Dendrogram with cluster at infinity.}
\label{learninfty}
\end{figure}
In Figure \ref{learninfty}, some datum $y$ is adjoined to a training dataset
$X=\mathset{a,b,c}$.
As it happens that the distance of $y$ to $X$ is larger than the diameter of
$X$,
the path $v_0\leadsto y$ in the dendrogram $D_\infty(X\cup\mathset{y})$ has a
portion going upwards in direction $\infty$.
\end{exa}
We call the pair $\mathcal{X}:=(X,\mathscr{C}_X)$ a {\em classification} and have a {\em
classification map}
$$
\kappa_{\mathcal{X}}\colon \mathds{P}(X)\to\mathscr{C}_X^\infty,\;x\mapsto C_x,
$$
which assigns to each $x\in \mathds{P}(X)$ the cluster $C_x$ containing $x$,
with $$\mathscr{C}_X^\infty=\mathscr{C}_X\cup\mathset{\mathset{\infty}}.$$
Now, let $Z=X\cup Y$. We have the inclusion map
$\iota\colon\mathds{P}(X)\to \mathds{P}(Z)$ which takes $x\in X$ to itself and $\infty$
to
$\infty$.
\begin{dfn}
A {\em $p$-adic classifier for $Y$ modeled on
$(X,\mathscr{C}_X)$} is a map
\begin{align*}
\lambda\colon \mathds{P}(Z)\to \mathscr{C}',
\end{align*}
where $\mathscr{C}'$ is a clustering of $\mathds{P}(Z)$, such that there
exists an injective map $\phi\colon\mathscr{C}_X^\infty\to\mathscr{C}'$ making
the diagram
$$
\xymatrix{
\mathds{P}(X)\ar[r]^\iota\ar[d]_{\kappa_{\mathcal{X}}}&\mathds{P}(Z)\ar[d]^\lambda\\
\mathscr{C}_X^\infty\ar[r]_\phi&\mathscr{C}'
}
$$
commutative. The cluster $C_\infty:=\lambda^{-1}(\phi(\mathset{\infty}))$ is
called
the {\em residue} of $\lambda$. A classifier is called saturated, if $\phi$ is bijective.
\end{dfn}
\begin{rem}
Notice that $\phi$ is unique if it exists.
\end{rem}
Our first learning algorithm constructs the classifier sequentially
by computing the distance to cluster centers for $\mathscr{C}_X$.
Let $A=\mathset{a_C\mid C\in\mathscr{C}_X}$ be the set of given cluster
centers
$a_C\in C$.
Then we have for $y\in Y$ the map
$$
d_y\colon\mathscr{C}_X\to\mathds{R},\; C\mapsto \absolute{y-a_C}_K,
$$
and let $m_y:=\min d_y(\mathscr{C}_X)$.
The vertex $v_y\in D_\infty(A\cup\mathset{y})$ nearest to $y$ can be
found e.g.\
using the $p$-adic expansions
as given by (\ref{pi-adic-expansion}).
Namely, a vertex corresponds to a disk containing two or more $p$-adic
numbers in $A\cup\mathset{y}$ having common initial terms determined by the radius of the disk.
In geometric terms, traversing along the
geodesic path $\gamma_y\colon \infty \leadsto y$ until all $a\in A$
have
branched off $\gamma_y$ yields the vertex $v_y$, and $\mu(v_y)$
is determined
by the subset $C_{v_y}\subset A$ of those elements branching off precisely in
$v_y$. The length of the path $v_0\leadsto v_y$ gives $m_y$.
And the map $d_y$ is computed:
\begin{lem}
It holds true that
$$
\mathscr{C}_y:=d_y^{-1}(m_y)=\mathset{C_a\in\mathscr{C}_X\mid a\in C_{v_y}}.
$$
\end{lem}
\begin{proof}
By what has been said above, the minimum is attained precisely for those
clusters $C\in\mathscr{C}_X$ contained in $C_{v_y}$. Hence $C=C_a$ for some $a\in C_{v_y}$.
\end{proof}
The task is now to decide into which cluster from $\mathscr{C}_y$ to put $y$.
\begin{alg}\rm \label{plearn}
{\em Input.} A classification $\mathcal{X}_0:=\mathcal{X}=(X,\mathscr{C}_X)$,
a set $A=\mathset{a_C \mid C\in\mathscr{C}_X}$ of cluster elements $a_C\in C$, and a set
$Y\subseteq K$ of cardinality $N$.
\noindent
{\em Step 0.} Set $C_\infty:=\mathset{\infty}$.
\noindent
{\em Step 1.} Take $y:=y_1\in Y$, and compute $v_y$, $m_y$, $C_{v_y}$, $\mathscr{C}_y$,
and $\mu_{v_y}$.
\noindent
{\em Case 1.} If $C_{v_y}=\mathset{a}$, then set
$C_y:=C_a\cup\mathset{y}$ and $A_1:=A$.
\noindent
{\em Case 2.} If $\#C_{v_y}>1$, then find the subset $C^y\subseteq C_{v_y}$ of all elements whose
nearest vertex in $D_\infty(C_{v_y}\cup\mathset{y})$ equals $v_y$.
If $C^y=\emptyset$, then set $C_y=\mathset{y}$ and $A_1:=A\cup\mathset{y}$.
Otherwise, find all elements $a\in C^y$ with minimal energy $E(C_{a}\cup\mathset{y})$.
If there is more than one such $a$, then $C_y:=\mathset{y}$ and
$A_1:=A\cup\mathset{y}$.
Otherwise, $C_y:=C_a\cup\mathset{y}$, and $A_1:=A$.
In any case, produce $Y_1:=Y\setminus\mathset{y}$, $A_1$ and
classification $\mathcal{X}_1:=(X_1,\mathscr{C}_{X_1})$,
where $X_1=X\cup\mathset{y}$ and
$\mathscr{C}_{X_1}:=\mathset{C_y}\cup\mathscr{C}_X\setminus\mathset{C_a}$.
Terminate, if $Y_1=\emptyset$.
\noindent
{\em Step $N$.} Assume that in the previous step, sets $Y_{N-1}$, $A_{N-1}$ and a
classification $\mathcal{X}_{N-1}$ have been
given out. Then perform Step $1$ with $\mathcal{X}:=\mathcal{X}_{N-1}$, $A:=A_{N-1}$,
and $Y:=Y_{N-1}$.
\noindent
{\em Output.} On termination in {\em Step $M$}, an optimal classifier
$$
\lambda\colon \mathds{P}(X_M)\to\mathscr{C}_{X_M},\;x\mapsto C_x,
$$
modeled on $\mathcal{X}_0$.
\end{alg}
\begin{proof}[Proof of optimality]
In each step $N$, $y_N\in Y_N$ is assigned to the cluster
$C\in\mathscr{C}_{X_N}$
with minimal energy $E(C\cup\mathset{y_N})$.
\end{proof}
\begin{thm}\label{plearnthm}
The outcome of Algorithm \ref{plearn} does not depend on the choice of the set
$A$
of cluster representatives.
\end{thm}
\begin{proof}
The outcome of {\em Step $1$} does not depend on $A$.
\end{proof}
\begin{rem}
A consequence of Theorem \ref{plearnthm} is that Algorithm \ref{plearn} does
indeed effect learning in the sense, that to any $y\in Y$ is assigned a
cluster depending on the already existing clusters. Representing a cluster
by a single element makes learning efficient.
\end{rem}
\subsection{Adaptive learning}
During the learning process\footnote{Or if for some reason one wants to
perform a variation of split-LBG$_p$ in which centers are computed after each
clustering step, instead of after termination of clustering.}, it can become
useful to subdivide
big clusters of the extended dataset $X\cup Y$. This is not a problem, as
the old cluster centers can be reused in the new clustering.
\begin{lem} \label{subclustcenter}
Let $C$ be a cluster, and $a\in C$ a center of $C$. Assume that $C'$ is a subcluster
of
$C$ containing $a$, then $a$ is a center of $C'$.
\end{lem}
\begin{proof}
Clearly, it holds true that
\begin{align}
E_p(C,\mathscr{C},a)\le E_p(C,\mathscr{C},a'), \label{trivineq}
\end{align}
where $\mathscr{C}=\mathset{C}$ and $\mathscr{C}'=\mathset{C}$.
Assume that $a'\in C'$ is a center of $C'$.
Now, inequality (\ref{trivineq}) implies that
\begin{align*}
E_p(C',\mathscr{C}',a)
+\sum\limits_{x\in C\setminus C'}\absolute{x-a}_K
&=E_p(C,\mathscr{C},a)
\\
&\le E_p(C,\mathscr{C},a')
\\
&=E_p(C',\mathscr{C}',a')+\sum\limits_{x\in C\setminus
C'}\absolute{x-a'}_K
\end{align*}
Since, by the cluster property of $C'$, it holds true that
$$
\absolute{x-a}_K=\absolute{x-a'}_K
$$
for all $x\in C\setminus C'$, it follows that
\begin{align}
E_p(C',\mathscr{C}',a)\le E_p(C',\mathscr{C}',a'), \label{lowerthancenter}
\end{align}
and, because $a'$ is a center of $C'$, this yields an equality in (\ref{lowerthancenter})
i.e.\ $a$ is a center of $C'$.
\end{proof}
\begin{rem}
Notice that Lemma \ref{subclustcenter} does not hold true, if we allow $C'$ to
be an arbitrary subset of $C$. E.g.\ assume in Figure \ref{4dendro} that
$C=\mathset{a,b,c,d}$. Then $a$ is a center of $C$, as can be verified from
the left dendrogram. However, $a$ is not a
center of $C'=\mathset{a,c,d}$, as the right dendrogram reveals.
Namely, in the first case, we compute with $\mathscr{C}=\mathset{C}$
and $\mathscr{C}'=\mathset{C'}$:
\begin{align*}
E(C,\mathscr{C},a)&=E(C,\mathscr{C},b)=\absolute{a-b}_K+2\cdot\absolute{a-c}_K\\
&<\absolute{c-d}_K+2\cdot\absolute{a-c}_K=E(C,\mathscr{C},c)=E(C,\mathscr{C},d),
\end{align*}
and in the second case:
\begin{align*}
E(C',\mathscr{C}',c)&=\absolute{a-c}_K+\absolute{d-c}_K\\
&<2\cdot\absolute{a-c}_K=E(C',\mathscr{C}',a).
\end{align*}
\begin{figure}
\caption{Dendrogram and subdendrogram.}
\label{4dendro}
\end{figure}
\end{rem}
At last, we propose the splitting of high-energy clusters accumulated during
the learning process:
\begin{alg}\rm
{\em Input.} $r\ge 0$. Otherwise, as in Algorithm \ref{plearn}.
\noindent
Perform Algorithm \ref{plearn} with modification:
\noindent
{\em Step $N'$.}
Perform {\em Step $N$}. If for $y:=y_N$ it holds true that $E(C_{y})>r$,
then split cluster $C_{y}$ into its maximal proper
subclusters,
and adjoin to $A_N$ new cluster centers using Algorithm \ref{centerclustp}.
\end{alg}
\section{Conclusion}
A straightforward translation of the split-LBG algorithm to the
situation of classifying $p$-adic
data does not exist. However, if clusterings, cluster centers and their numbers are
allowed to vary, then the minimisation problem for the $p$-adic energy
functional defined by distances to centers does make sense.
Sub-optimal
algorithmic solutions to the minimisation problem are presented, in which the choice lies
in whether or not to remove in each step
quasi-singletons, i.e.\ clusters which are almost singletons because of their
energy values being lower than a given threshold. The method is
to find rankings of vertices in the dendrogram associated to the $p$-adic data.
The outcome
depends on the prime number $p$, but it is shown that for all but finitely
many primes the rankings are identical.
The consequence for applications to data anlaysis is that for fixed prime
$p$,
the classification results do not depend on the $p$-adic representation
of the data, as long as the dendrograms are isomorphic.
Furthermore, the minimising property for
given cluster centers holds true independently of the prime. This means
that if some datum is a cluster center for one prime, it is a cluster center
for all primes (for which the corresponding cluster is not larger).
Using $p$-adic cluster centers,
one can construct classifiers from given clusterings. This can be applied to
learning situations.
\end{document} | math | 44,703 |
\begin{document}
\title{Reproducible Research: A Retrospective}
\noindent
$^1$Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health\\
\begin{abstract}
Rapid advances in computing technology over the past few decades have spurred two extraordinary phenomena in science: large-scale and high-throughput data collection coupled with the creation and implementation of complex statistical algorithms for data analysis. Together, these two phenomena have brought about tremendous advances in scientific discovery but have also raised two serious concerns, one relatively new and one quite familiar. The complexity of modern data analyses raises questions about the \textit{reproducibility} of the analyses, meaning the ability of independent analysts to re-create the results claimed by the original authors using the original data and analysis techniques. While seemingly a straightforward concept, reproducibility of analyses is typically thwarted by the lack of availability of the data and computer code that were used in the analyses. A much more general concern is the \textit{replicability} of scientific findings, which concerns the frequency with which scientific claims are confirmed by completely independent investigations. While the concepts of reproduciblity and replicability are related, it is worth noting that they are focused on quite different goals and address different aspects of scientific progress. In this review, we will discuss the origins of reproducible research, characterize the current status of reproduciblity in public health research, and connect reproduciblity to current concerns about replicability of scientific findings. Finally, we describe a path forward for improving both the reproducibility and replicability of public health research in the future.
\end{abstract}
\section{Introduction}
Scientific progress has long depended on the ability of scientists to communicate to others the details of their investigations. The exact meaning of ``details of their investigations" has changed considerably over time and in recent years has been nearly impossible to describe precisely using traditional means of communication. Rapid advances in computing technology have led to large-scale and high-throughput data collection coupled with the creation and implementation of complex statistical algorithms for data analysis. In the past, it might have sufficed to describe the data collection and analysis using a few key words and high-level language. However, with today's computing-intensive research, the lack of details about the data analysis in particular can make it impossible to re-create any of the results presented in a paper~\citep{peng:2011}. Compounding these difficulties is the impracticality of describing these myriad details in traditional journal publications using natural language. To address this communication problem, a concept has emerged known as \textit{reproducible research}, which aims to provide far more precise descriptions of an investigator's work to others in the field. As such, reproducible research is an extension of the usual communications practices of scientists, adapted to the modern era.
The notion of reproducible research, which was popularized in the early 1990s, was ultimately designed to address an emerging and serious issue at the time~\citep{schwab2000making}. Results of published findings were increasingly dependent on complex computations done on powerful computers, often implementing sophisticated algorithms on large datasets. Given the importance of computing to the generation of these results, it was surprising that consumers of scientific results had no ability to inspect or examine the details of the computations being done. Traditional forms of scientific publication allowed for extended descriptions of study design and high-level analysis approaches, but low-level details about computer code, data processing pipelines, and algorithms were not prioritized and generally left in an appendix or, with the wider availability of the internet, online supplement.
Jon Claerbout, a geophysicist at Stanford University, wrote down many of the original ideas concerning reproducibilty of computational research. His concern largely focused on developing a software system whereby the research produced by his lab could be passed on to others, including the original authors, the authors' colleagues, students, research sponsors, and the general public. He noted in particular the benefits of reproducibility to the original authors: "It may seem strange to put the author's own name at the top of the list to whom we wish to provide the reproducible research, but it often seems that the greatest beneficiary of preparing the work in a reproducible form is the original author!" It is equally notable that the public was listed last; all of the other constituencies mentioned would likely exist within the small orbit of an individual investigator~\citep{claerbout2001cd}. In Claerbout's discussion, the primary focus is on improving the transparency and productivity of the lab itself, given that much time can be lost attempting to re-create past findings for the sole purpose of understanding what was previously done.
Buckheit and Donoho introduced much of the statistical community to the concept of reproducibility with an influential paper in 1995 detailing their WaveLab software for implementing wavelets for data analysis~\citep{buck:dono:1995}. Citing Claerbout as a strong influence, their rationale for promoting reproducible research produced a useful summary of Claerbout's ideas that has since be repeated many times:
\begin{quote}
An article about computational science in a scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures.
\end{quote}
The general conclusion was that delivering a research end product such as a figure or table was no longer sufficient. Rather, the software environment and the means to create the end product must also be delivered, as those additional elements represent the actual scholarship. In order to satisfy this requirement, one would have to make available the \textbf{data} and the \textbf{computer code} used to generate the results.
\section{Reproducible Research}
The definition of reproducible research generally consists of the following elements. A published data analysis is reproducible if the analytic datasets and the computer code used to create the data analysis is made available to others for independent study and analysis~\citep{peng:2011}. This definition is sufficiently vague that it ultimately raises more questions than it answers. What is an ``analytic dataset"? What does it mean to be ``available"? What is included with the ``computer code"?
Published research can be thought of as living on a continuum up until the point of publication~\citep{peng:domi:zege:2006}. Starting from question formulation and study design, proceeding to to data collection, data processing, data analysis, and finally to presentation. Along this journey, various elements are introduced to aid in executing the research, such as computing environments, measurement instruments, and software tools. One could choose to make available to others any aspect of this sequence, depending on the practicalities of doing so and the relevance to the final published results. It is challenging to develop a universal cut point for determining which aspects of an investigation should be disseminated and which are not required. However, within various research communities, internal standards have developed and are continuously evolved to keep pace with technology~\citep[e.g.][]{brazma2001minimum,sandve2013ten}.
The analytic dataset generally contains all of the data that can be directly linked to a published result or number. For example, if a paper publishes an estimate of the rate of hospitalization for heart attacks, but the overall study also collected data on hospitalizations for influenza, the influenza data may not be part of the analytic dataset if it makes no appearance in the published result and is not otherwise relevant. While outside investigators may interested in seeing the influenza data (and the original authors may be happy to share it), it is not needed for the sake of reproducible research. The analytic computer code is any code that was used to transform the analytic dataset into results. This may include some data processing (such as variable transformations) as well as modeling or visualization. Generally, the software environment in which the analysis was conducted (e.g. R, Python, Matlab) does not need to be distributed if it is easily obtainable or open source. However, niche software which may be unfamiliar to many readers may need to be bundled with the data and code.
Upon first consideration, many see reproducibility as a non-issue. How could it be that applying the original computer code to the original dataset would \textit{not} produce the original results? The practical reality of modern research though is that many even simple results depend on a precise record of procedures and processing and the order in which they occur. Futhermore, many statistical algorithms have many tuning parameters that can dramatically change the results if they are not inputted exactly the same way as was done by the original investigators~\citep{haibe2020importance}. If any of these myriad details are not properly recorded in the code, then there is a significant possibility that the results produced by others will differ from the original investigators' results.
\subsection{A Sidebar on Terminology}
The terminology of reproducible research can be bewildering to some in the scientific community because there is little agreement about the meaning of the phrase in relation to other related concepts~\citep{barba2018terminologies,goodman2016does,plesser2018reproducibility}. In particular, one related concept with which all scientists are concerned is what we refer to here as \textit{replication}. In this review, we define replication as the independent investigation of a scientific hypothesis with newly collected data, new experimental setups, new investigators, and possibly new analytic approaches. In a thorough investigation of the terminologies of reproducible research, Lorena Barba found that some fields of study made no distinction between ``reproducible'' and ``replicable'' while some fields used those terms to mean the exact opposite of how we define them here~\citep{barba2018terminologies,national2019reproducibility}. However, a significant plurality of fields, including epidemiology, medicine, and statistics appear to adopt the definitions we use here.
A key distinction between reproducibility and replication is that reproducibility does not allow for any real variation in the results. If an independent investigator were to \textit{reproduce} the results of another investigator with the original data and code, there should not be any variation between the two investigators' results, except for some allowance for differences in machine precision. Thus, exact reproducibility is sometimes referred to as ``bitwise reproducibility"~\citep{national2019reproducibility}. However, \textit{replication} generally allows for differences in results that arise from statistical variability. Two independent investigators conducting the exact same experiment should, in theory, only differ by an amount quantified by the standard deviation of the data. More generous definitions of replication allow for slightly different study designs, analytic populations, or statistical techniques~\citep{national2019reproducibility}. In those cases, differences in results may arise beyond simple statistical variation. Patil~et al.~have devised a useful visualization of what may or may not differ when reproducing or replicating a published study~\citep{patil2019visual}. The relationship between reproducible research and replication is a topic to which we will return in greater detail in Section~\ref{sec:goals}.
It is difficult to argue that interest in exactly reproducing another investigator's work is anything but a modern phenomenon~\citep{drummond2018reproducible}. Interest in reproducibility prior to the computer and internet age was likely low or non-existent given that there was generally no expectation that investigators would share data in papers---there was simply no practical way to do that except for very small datasets. In the past, other investigators could only resort to independently replicating a published study using their own data collection and whatever high-level description of the methods that was available in the paper. In this setting, detailed descriptions of the methods of analysis were critical if others were to execute the same approach. If the process of conducting the experiment or analysis was simple enough or were sufficiently standardized, then it could be reasonably described in the confines of a journal paper. Suggesting that analyses be described with data and code is a departure from previous ways of communicating scientific results, which relied on describing experiments and analyses in more general terms to give readers the highlights of what was done. A more abstract approach could not be taken with this new form of computational research because the proper abstractions for communicating ideas and standardization of approaches were not yet available.
\subsection{Newer Developments}
The concept of reproducible research was developed to achieve arguably modest goals. Its original aims focused on providing an approach to better communicate the details of computationally intensive research to one's collaborators, colleagues, students and oneself. But two key developments over the past 30 years have changed the context around which reproducible research lives. Although the definition of reproducibility has not changed much since the 1990s, almost everything else about scientific research has.
In much of the early literature on reproducible research, the focus is on ``computationally-intensive" research which, because of its reliance on complex computer algorithms, was considered perhaps more impenetrable than other research. Fast forward 30 years and the use of computing in scientific research is ubiquitous. It is no longer the domain of niche geophysical scientists or mathematical statisticians using obscure computer packages. Now, all scientific research involves the use of powerful computers, whether it is for the data collection, the data analysis, or both. Furthermore, the increase in complexity of statistical techniques over this time period has resulted in the need for detailed descriptions of analytic approaches and data processing pipelines. We are all computational scientists now and as a result the concept of reproducibility is relevant to all scientists.
Along with computing power, another key advance over the past 30 years has been the development of the internet. Claerbout's original scheme for distributing data and code to others was via CD-ROM disks, which was a perfectly reasonable approach at the time. However, the need for a physical medium greatly limited the transferrability of information to a large audience. With the development of the internet, it became possible for academics to distribute data and code to the entire world for seemingly minimal cost. This increase in distribution reach changed the nature and importance of reproducible research from primarily improving the internal efficiency of the lab to allowing others to anonymously build on one's own work. The internet dramatically grew the size of an investigator's personal scientific community to include many members beyond one's immediate circle of collaborators. This phenomenon has provided significant benefits to science but there are some implications that could threaten the viability of maintaining and supporting reproducible research in the long-run. Before we consider these implications, we must first consider what are the goals of reproducible research and what problems we want reproducibility to solve.
\section{The Goals of Reproducible Research}
\label{sec:goals}
Beyond communicating the details of an investigation, what are the goals of making research reproducible? The stated goals achieved by making research reproducible have evolved over time since the early 1990s and have become somewhat more elusive. Originally, the goal was to better reveal the process of doing the research. Computational research added a new complexity in the form of software code and high-dimensional datasets and that complexity made understanding the research process more difficult for a reader to infer. Therefore, the solution was to simply publish every step in the process along with the data. Claerbout and colleagues were concerned that others (including themselves!) would not be able to \textit{learn} from what they have done if they did not have the details. The easiest way to do this was via the literal computer code that executed the steps. Any less precise format could risk omitting a key step that affected downstream results~\citep{haibe2020importance}.
Reproducible research comes with a few side benefits. In addition to being able to fully understand the process by which the results were obtained, readers also get the data and the computer code, both of which are valuable to the extent that they can be re-used or re-purposed for future studies or research. Some have suggested that making data and computer code available to others is a \textit{per se} goal of reproducible research, because both can be built upon and leveraged to further scientific knowledge~\citep{gent:temp:2007}. However, such an interpretation is an extension of the original ideas of reproducibility. The former view saw data and code as a medium for communicating ideas whereas the latter view sees data and code essentially as \textit{products} or \textit{digital research objects} to be used by others~\citep{stodden2015reproducing}. While converting a dataset into a data product and packaging computer code into usable software may seem like nominal tasks given that the underlying data and code already exist, there are non-negligible costs associated with the development, maintenance, and support of these products.
Another goal of reproducible research is to provide a kind of audit trail, should one be needed. In fact, one could suggest a definition of reproducible research as ``research that can be checked". Desiring an audit trail for data analyses raises the question of when such an audit trail might be used? In general, one might be interested in seeing the details of the data and the code for an analysis when there is a curiosity about how a specific result was reached. Sometimes that curiosity is raised because of suspicion of an error in the analysis, but other times there is a desire to learn the details of new techniques or methodologies~\citep{haibe2020importance}. Thus, reproducibility primarily concerns the integrity and transparency of the \textit{data analysis} for an investigation. Unlike replication, reproducibility allows for an internal check on the results and is not immediately connected to the context of the outside world.
\section{Reproducibility and Data Analysis}
One could summarize the goal of reproducible research as providing a means to answer the question, ``Do I understand and trust this data analysis?" With the computational nature of today's research, we cannot hope to answer that question without being able to look at the data and the code. In addition, we may wish to know things about the experimental or study design as well as the hypothesis being examined~\citep{hick:peng:2019}. Given the claimed results, the data, and code, one can theoretically determine the reproducibility status of a data analysis.
Reproducibility gives us the means by which we can assess our confidence and trust in an analysis, but it is important to reiterate that the mere fact of reproducibility of an analysis is not a check on validity of the analysis.
The notion of reproducibility as a binary or perhaps multi-level ``state" of a data analysis is a useful characterization in part because it is one of a few qualities of a data analysis that can be immediately verified. Unlike with replication, we do not need to wait for future studies to be conducted in order to determine the reproducibility of an analysis. However, this suggests reproducibility's usefulness is limited. What do we ultimately learn from merely reproducing the results of an analysis? For example, it may be possible to execute code on a dataset without ever looking at the code or the data. In that case, the original goal of reproducibility---to learn about the details of an investigation---has been thwarted. We have simply learned that the code produces what the authors claim the code produces. In general, executing a process and seeing that process produce the results exactly as they were expected, produces very little new information.
The statement of the question ``Do I understand and trust this data analysis?" depends critically on the perspective of the person asking the question. If the person asking is an expert in the area, they might be able to glance at the code and data and understand immediately what is going on. A non-expert in the field might be able to execute the code and produce results without ever understanding the operations of the analysis. An adjacent question that might be worth asking is ``Is this data analysis understandable and trustworthy?" However, this question is not any easier to answer because it hypothesizes underyling objective qualities of a data analysis. But opinions may still vary widely about what these underlying qualities should be depending on who is asking the question. To answer either question, one needs to look carefully at the data and the code to learn exactly what was done. But ultimately, the data and code only represent a piece of the answer. Whether an analysis can be understood or trusted depends critically on many aspects outside of the analysis itself, including the perspective of the person reading the analysis.
Nevertheless, one hope is that reproducibility can lead to higher quality data analyses. The logic is that requiring all analyses to provide data and code would put investigators on notice that their work would be scrutinized. However, one high-profile example suggests this is unlikely to be the case.
\subsection{Example: Forensic Bioinformatics}
In a now-retracted 2006 study by Potti~et~al.~\citep{potti2006genomic}, the investigators claimed to have identified genomic signatures using microarrays that could predict whether an individual responded well to chemotherapy. The analysis was conducted using data from publicly available cell lines, and so the data were in a sense available. However, subsequent attempts to reproduce the findings failed and reproducibility was only achieved when errors were deliberately inserted into the analysis code~\citep{coom:wang:bagg:2007,baggerly2009deriving}. Keith Baggerly, Kevin Coombes, and Jing Wang meticulously reconstructed the error-prone analysis and laid out all of the details in both text and code. Ironically, they were ultimately able to reproduce the analysis of Potti~et~al. after significant reverse engineering and forensic investigation. In fact, we might never have learned what mistakes were made if Baggerly and his colleagues were not able to reproduce the analysis.
The example of the Potti study is a pathological example of a reproducible analysis (after much forensic investigation) being profoundly incorrect. However, it is worth asking what role reproducibility \textit{might have played} in this case? If Potti et al.~had released the code and data that were clearly linked together, perhaps as a research compendium~\citep{gent:temp:2007}, then the errors could have been found more quickly. However, given the sheer number and complexity of the problems, it still likely would have taken some time to understand them all. Coombes~et~al. published their letter only a year after the initial publication, so the timeline might have been advanced by a few months. However, a key fact would remain---the flawed analysis was already completed. Furthermore, once the truth was ultimately revealed to the authors, it took years of further investigation by many others before the original paper was retracted.
\subsection{Reproducibility and Quality: A Prevention Model?}
Examples like the Potti paper raise the question of whether demanding or requiring reproducibility of a study beforehand can pre-emptively improve its quality. Evidence of this connection between reproducibility and quality is lacking, which is not surprising given that the question is somewhat ill-posed. What exactly are we looking for in a ``high-quality" data anlaysis? One could hypothesize that if an investigator knew in advance that the data and the code would be publicly available for scrutiny, then they would take the extra effort to make sure that the analyses were properly done. Perhaps if Potti~et~al. had been forced to make their code publicly available, they would have checked it first.
In the case of Potti~et~al.~we now know that requiring reproducibility or even just code sharing would not have made much difference. Reporting done by \textit{The Cancer Letter} showed definitively that the investigators were aware of numerous statistical and coding errors with the analysis but did not think they were serious problems~\cite{goldberg:2015}. Rather, they were considered ``differences of opinion". The notion that requiring reproducibility can lead to improved data analyses relies on the critical assumption that the investigators are able to recognize what is an error in the first place. If they do recognize the error and hide it, then that is fraud. If they do not recognize the error and publish it anyway, then that is at best careless. However, in both cases, forcing the data and code to be published would not have made any difference.
\subsection{Replication and Reproducibility}
Reproducibility does not provide a useful route to preventing poor data analyses from occurring, but it does provide the basis for a meaningful discussion about whether their might be problems in the analysis and how such problems might be fixed. Replication differs from reproducibility primarily because it addresses a different goal. Replication answers the question, ``Is this scientific claim true?" Reproducibility addresses the integrity of the data analysis that generated the evidence for a scientific claim, while replication addresses the integrity of the claim itself in the context of the outside world. Fundamentally, reproducible research has relatively little to say on the question of external validity. Claims resulting from reproducible results can be both correct and incorrect~\citep{leek:peng:2015}. Claims resulting from irreproducible results are less likely to be true, but that may depend on the reasons for the lack of reproducibility. For example, evidence generated via random algorithms may not be exactly reproducible if random number generator seeds are not saved, but the underlying evidence may still be sound. Ultimately, claims made by irreproducible studies may in fact be true, but irreproducible studies simply do not provide evidence for such claims.
\subsubsection{Example: Re-analysis of Air Pollution Studies}
In the mid 1990s, two large studies of ambient air pollution and mortality---The Six Cities Study~\citep{dockery1993association} and the American Cancer Society (ACS) Study~\citep{pope1995particulate}---were published, presenting evidence that differences in air pollution concentrations between cities were significantly associated with rates of mortaliy in these cities. Both studies came under intense scrutiny when the U.S. Environmental Protection Agency cited the results in their revision of the National Ambient Air Quality Standards for fine particles. In particular, there were demands from numerous corners that the data used in the studies should be made available. However, the data in these studies, as with most health-related studies, included personal information about the subjects and arguments were made that promises of confidentiality had to be kept. To address the impasse of making the data available, the original investigators engaged the Health Effects Institute (HEI) to serve as a kind of trusted third party to broker a reanalysis of the studies. Ultimately, HEI recruited a research team lead by investigators at the University of Ottawa to obtain the original data for both the Six Cities and ACS studies, reproduce the original findings, and conduct additional sensitivity analyses to assess the robustness of the original findings~\citep{krewski2000reanalysis}.
The extensive reanalysis found that the original studies were largely reproducible, if not perfectly reproducible. For the Six Cities study, the key result was a mortality relative risk of $1.26$, which the re-analysis team computed to be $1.28$. For the ACS study the original mortality relative risk was $1.17$, close to the re-analysis value of $1.18$.
While one could argue that these studes were strictly speaking not reproducible, such small differences are not likely to be material. In fact, we now know, after numerous follow-up studies and independent replications, that the core findings of both studies appear to be true~\citep{broo:raja:pope:2010} and that the U.S. EPA itself rates the evidence of a connection between fine particles and mortality to be ``likely causal"~\citep{isa:2009}. The re-analysis team ran many other analyses, including variables that had not been considered in the original studies. Overall, they found that the sensitivity analyses did not change any of the major conclusions. Interestingly, one of the key conclusions of the final report from HEI was that at the end of the day ``No single epidemiologic study can be the basis for determining a causal relation between air pollution and mortality"~\citep{krewski2000reanalysis}.
The HEI re-analysis of the Six Cities and ACS studies highlights the role of trust in data analysis. Prior to the re-analysis, many parties simply did not trust that the analysis was done properly or that all reasonable competing hypotheses had been considered. While making the data available might have allowed others to build that trust for themselves, allowing a neutral third party to examine the data and reproduce the findings at least ensured that one other group had seen the data. In addition, HEI's role in organizing the expert panel, conducting public outreach, and managing an open process played an important role in building trust in the community. While not all parties were completely satisified with the process, what the re-analysis did was allow fellow scientists to learn from the original studies and gain insight into the process that lead to the original findings. Ultimately, the key goals of reproducible research were achieved.
In hindsight, another lesson learned from the HEI re-analysis is that the importance of reproducibility of a given study can fade with time. Over 25 years later, there have been scores of follow-up studies and replications that have largely come to similar conclusions as the Six Cities and ACS studies. Although both studies remain seminal in the field of air pollution epidemiology, they could be deleted from the literature at this point and have little impact on our understanding of the science. This is not to say that the data and ongoing analyses do not have value, but rather the original results have been subsumed by later studies. Reproducibility was only critical when the studies were first published because of the paucity of large studies at the time.
\section{Reproducibility and Better Data Analysis}
Recent work has focused on the quality and variability of data analyses published in various fields of study~\citep[e.g.][]{open2015estimating,jager2014estimate,ioannidis2005most,patil2016should}, with some claiming the existence of a ``replication crisis" due to the wide variation between studies examining the same hypotheses~\cite{schooler2014metascience}. The causes of this variation between studies are myriad but one large category includes various aspects of the data analysis. Because of the increasing complexity of data analyses, many choices and decisions must be made by analysts in the process of obtaining a result. With these increasing complexities we also increase the risk of human error and bias in data analysis. These choices and decisions often have an unknown impact on the final estimates produced and therefore may or may not be recorded by the investigators~\citep{stodden2015reproducing}. These ``research degrees of freedom'' allow investigators to unknowingly, or perhaps knowingly, steer data analyses in directions that may support specific hypotheses rather than represent all of the evidence in the data~\citep{simmons2011false}.
What role can reproducible research play in improving the quality of data analyses across all fields? The answer can be found in part with the experience of the HEI re-analysis of the Six Cities and ACS air pollution studies. Because they were re-analyses, one could imagine the expectation was that the results would be confirmed to some reasonable degree. If there was a significant deviation from the published results, then we would have to dig into the original analysis to discover why. Because the results were largely reproduced, one could argue that little was learned. However, additional analyses were done and sensitivity analyses were conducted. As a result, we learned much about the data analysis process. The re-analysis thus produced valuable knowledge about how to analyze air pollution and health data.
For example, the re-analysis team noted that both mortality and air pollution were highly spatially correlated, a feature that was not considered in the original analysis. They noted, ``If not identified and modeled correctly, spatial correlation could cause substantial errors in both the regression coefficients and their standard errors. The Reanalysis Team identified several methods for dealing with this, all of which resulted in some reduction in the estimated regression coefficients."~\cite{krewski2000reanalysis} In addition, reproducibility helps free up time for the analysts interested in re-analyzing the data to focus on parts of the data analysis that require more human intepretation. For example, if an independent data analyst knew that an analysis was already reproducible, then more time and resources would be available to understand {\it why} a specific model was chosen, instead of {\it what} version of software was used to run this model. In the re-analysis of the data from Potti et al.~\citep{potti2006genomic}, Baggerly and Coombes noted that they had spent thousands of hours re-examining the data attempting to reproduce the original results~\citep{baggerly2010disclose,goldberg2014duke}.
There are also different degrees of reproducibility when building a data analysis and differences in audiences that may or may not be allowed to have access to these components. For example, a data analyst may choose to make the data available, but not the code (or the opposite). Others may make both the code and data available for only one audience (Audience A), but not for another audience (Audience B). There are valid reasons why an analyst might choose to do this, such as if the data analysis uses data with protected health information in a hospital setting or if the data analyst works at a business or company and cannot share the code or data with others outside of the company. It is important to note that just because an analysis is not fully reproducible to one audience (Audience B) does not mean that it is an invalid analysis with incorrect conclusions. While it does make it harder for Audience B to trust the results, it still can be a valid or correct analysis. However, the lack of reproduciblity to this audience may mean that the evidence supporting any claims is weaker. Despite these potential differences in degrees of reproducibility, as demonstrated in the HEI re-analysis, efforts made to make a data analysis more reproducible is a step in the right direction for making it a better data analysis.
Ultimately, the reproducibility of research, when possible, allows us a significant opportunity to (1)~learn from others about how best to analyze certain types of data; (2)~reduce human errors and bias as data become larger and more complex; (3)~free up time for re-analyzers to focus on parts of a data analysis that require more human interpretation; (4)~have discussions about what makes for a good data analysis in certain areas of study; and (5)~improve the quality of future data analyses. When teaching data analysis to students, it is common to talk in abstractions and theories, describing statistical methods and models in isolation. When real data is shown, it is often in the form of toy examples or in short excerpts. Increasing the reproducibility of all studies presents an opportunity to dramatically expand instruction on the craft of data analysis so that core set of elements and principles for characterizing high quality analyses can be established within a field~\citep{hick:peng:2019}.
\section{Refining Reproducibility}
In the thirty years since the idea of reproducible computational research was brought the forefront of the research community, we have learned much about its role and its value in the research enterprise. The original goal of providing a transparent means by which researchers can communicate what they have done and allow others to learn remains a primary rationale. Reproduciblity has a secondary role to play in improving the quality of data analysis in that it serves as the foundation on which people can learn how others analyze data. Without code and data, it is nearly impossible to fully understand how a given data analysis was actually done. But much about computational research has changed in the past 30 years and we can perhaps develop a more refined notion about what it means to make research ``reproducible". The two key ideas about reproducibility---data and code---are worth revisiting in greater detail.
\subsection{Data}
The sharing of data is ultimately valuable in and of itself. Data sharing, to the extent possible, reduces the need for others to collect similar data, allows for combined analyses with other datasets, and can create important resources for unforeseen future studies. Datasets can retain their value for considerable time, depending on the area and field of study. One example of the value of data sharing comes from the National Mortality, Morbidity, and Air Pollution Study, a major air pollution epidemiology study conducted in the late 1990s and early 2000s~\citep{nmmaps1,nmmaps2}. The mortality data for this study were shared on a web site and then later updated with new data. A systematic review found 67 publications had made use of the dataset, often to demonstrate the development of new statistical methodology~\citep{barnett2012benefits}. In addition, the release of the data at the time allowed for a level of transparency and trust in air pollution research that was novel for its time.
Today, many data sharing web repositories exist that allow easy distribution of data of almost any size. While in the past, an investigator interested in sharing data had to purchase and setup a web server, now investigators can simply upload to any number of services. The Open Science Framework~\citep{foster2017open}, Dataverse Project~\citep{king2007introduction}, ICPSR~\citep{swanberg2017inter}, and SRA~\citep{leinonen2010sequence} are only a handful of public and private repositories that offer free hosting of datasets. The major benefit of repositories such as these is to absorb and consolidate the cost of hosting data for possibly long periods of time.
The view of data sharing as inherently valuable is not without its challenges. Indeed, stripping data from its original context can be problematic and lead to inappropriate ``off-label" re-use by others. It has been argued that data only has value in its explicit connection to the knowledge that it produces and that we must be careful to preserve the connections between the data and the knowledge they generate~\citep{stodden2020beyond}.
Recently, best practices for sharing data have been developed. Some of these practices are specific to areas of study while some are more generic. In particular, the emergence of the concept of \textit{tidy data} has provided a generic format for many different types of data that serves as the backbone of a wide variety of analytic techniques~\citep{wickham2014tidy}. Practical guidance on sharing data via commonly used spreadsheet formats~\citep{broman2018data} and on providing relevant metadata to collaborators is now widely applicable to many kinds of data~\citep{ellis2018share}.
\subsection{Code}
The primary role of sharing code is to communicate what was done in transforming the data into scientific results. Today, almost all actions releveant to teh science will have occurred on the computer and it is essential that we have a precise way to document those actions. Computer code, via any number of programming and data analytic languages, is the most precise way to do that. The sharing of code generally represents less of a technical burden than the sharing of data. Code tends to be much smaller in size than most datasets and can easily be served by code sharing services such as GitHub, BitBucket, SourceForge, or GitLab.
While the benefits of code sharing tend to focus on the code's usability and potential for re-purposing in other applications, it is important to reiterate that code's primary purpose is to communicate what was done. In short, code is not equivalent to \textit{software}. Software is code that is specifically designed and organized for use by others in a wide variety of scenarios, often abstracting away operational details from the user. The usability of software depends critically on aspects like design, efficiency, modularity, and portability---factors that should not generally play a role when releasing research code. Sharing research code that is poorly designed and inefficient is far preferable to not sharing code at all. That said, this notion does not preclude the possibility for best practices in developing and sharing research code.
Software is often a product of research activity, particularly when new methodology is developed. In those cases, it is important that the software is carefully considered and designed well for its intended users. However, it should not be considered a requirement of reproducible research that software be a product of research. For software that is developed for distribution, there is increasing guidance for how such software should be distributed. Software package development has become easier for programming languages like R, which have robust developer and user communities~\citep{r2020}, and numerous tools have been developed to make incorporating code into packages more straightforward for non-professional programmers~\citep{devtools2020}. In addition, the concept of testing and test-based developed has been shown to be a useful framework for setting expectations for how software should perform and identifying errors or bugs~\citep{testthat2011}.
\section{Future Directions}
Technological trends over time generally favor a more open approach to science as the costs of sharing, hosting, and publishing have gone down. The continuing rapid advancement of computing technology, internet infrastructure, and algorithmic complexity will likely introduce new challenges to reproducible research. As the scientific community expands its sharing of data and code there are some important issues to consider going forward.
The rapidly evolving nature of scientific communication serves to highlight the role of reproducibility in advancing science. Without reproducibility, countless hours could be wasted simply trying to figure out what was done in a study. In situations were key decisions must be made based on scientific results, it is important that the robustness of the findings can be assessed quickly without the need for guessing or inferences about the underlying data. A stark example can be drawn from the COVID-19 pandemic. In April 2020 little was known about the disease and a study was published on medR$\chi$iv producing an estimate of the prevalence of COVID-19 in the population~\citep{bendavid2020covid}. At the time, important public health decisions had to be made in response to the pandemic and any information about the disease would have been highly relevant. Upon publication, numerous criticisms about the study's design and analysis appeared on social media and the web. However, the aspect most relevant to this review is that in many of the critiques, substantial time was taken to simply guess at what the researchers had done. Although a written statistical appendix was provided with the paper, no data or code were published along with the study. As a result, independent investigators had little choice but to infer what was done.
The urgency of decision-making based on scientific evidence can exist in a variety of situations, not just on the the minute-by-minute timescale of a worldwide pandemic. Many regulatory decisions in environmental health have to be made based on only a handful of studies. Often, there is no time to wait years for another large cohort study to replicate (or not) existing findings. In such situations where decisions need to be made, the more code and data that can be made available to assess the evidence, the better. In the interim, followup studies can be conducted and revisions to the evidence base can be made in the future if needed. The re-analyses Six Cities and ACS studies provide a clear example of this process and history has shown those results to be highly consistent across a range of replication studies.
The maintenance of code and data is generally not a topic that is discussed in the context of reproducible research. When a paper is published, it is sent to the journal and is considered ``finished'' by the investigators. Unless errors are found in the paper, one generally need not revisit a paper after publication. However, both code and data need to be maintained to some degree in order to be useful. Data formats can change and older formats can fall out of favor, often making older datasets unreadable. Code that was once highly readable can become unreadable as newer languages come to the fore and practitioners of older languages decrease in number. Maintenance of data and code is not a question of paying for computer hardware or services. Rather, it is about paying for people to periodically update and fix problems that may be introduced by the constantly changing computing environment.
Unfortunately, funding models for scientific research are aligned with the mechanism of paper publication, where one can definitively mark the end of a project (and also the end of the funding). However, with data and code, there is often no specific end point because other investigators may re-use the data or code for years into the future. Term-based project funding, which is the structure of almost all research funding, is simply not designed to provide support for maintaining materials on an uncertain timeline.
The first thirty years of reproducible research largely centered on discussions of the validity of the idea and what value it provided to the scientific community. Such discussions are largely settled now and both data and code share are practiced widely in many fields of study. However, we must now engage in a second phase of reproducible research which focuses on the continued development of infrastructure for supporting reproducibility.
\end{document} | math | 48,500 |
\begin{document}
\title{ space{-3cm}
\begin{abstract}
In this paper, we study the inverse acoustic medium scattering problem to reconstruct the unknown inhomogeneous medium from far field patterns of scattered waves.
We propose the reconstruction scheme based on the Kalman filter, which becomes possible to sequentially estimate the inhomogeneous medium.
We also show that in the linear inverse problem, the estimation for the Kalman filter is equivalent to that for the Tikhonov regularization.
Finally, we give numerical examples to demonstrate our proposed method.
\end{abstract}
\date{{\bf Key words}. Inverse acoustic scattering, Inhomogeneous medium, Far field pattern, Tikhonov regularization, Kalman filter.}
\section{Introduction}
The inverse scattering problem is the problem to determine unknown scatterers by measuring scattered waves that is generated by sending incident waves far away from scatterers.
It is of importance for many applications, for example medical imaging, nondestructive testing, remote exploration, and geophysical prospecting.
Due to many applications, the inverse scattering problem has been studied in various ways.
For further readings, we refer to the following books \cite{Cakoni, Chen, ColtonKress, Kirsch, NakamuraPotthast}, which include the summary of classical and recent progress of the inverse scattering problem.
\par
We begin with the mathematical formulation of the scattering problem.
Let $k>0$ be the wave number, and let $\theta \in \mathbb{S}^{1}$ be incident direction.
We denote the incident field $u^{inc}(\cdot, \theta)$ with the direction $\theta$ by the plane wave of the form
\begin{equation}
u^{inc}(x, \theta):=\mathrm{e}^{ikx \cdot \theta}, \ x \in \mathbb{R}^2. \label{1.1}
\end{equation}
Let $Q$ be a bounded domain and let its exterior $\mathbb{R}^2\setminus \overline{Q}$ be connected.
We assume that $q \in L^{\infty}(\mathbb{R}^2)$, which refers to the inhomogeneous medium, satisfies $\mathrm{Im}q \geq 0$, and its support $\mathrm{supp}\ q$ is embed into $Q$, that is $\mathrm{supp}\ q \Subset Q$.
Then, the direct scattering problem is to determine the total field $u=u^{sca}+u^{inc}$ such that
\begin{equation}
\Delta u+k^2(1+q)u=0 \ \mathrm{in} \ \mathbb{R}^2, \label{1.2}
\end{equation}
\begin{equation}
\lim_{r \to \infty} \sqrt{r} \biggl( \frac{\partial u^{sca}}{\partial r}-iku^{sca} \biggr)=0, \label{1.3}
\end{equation}
where $r=|x|$.
The {\it Sommerfeld radiation condition} (\ref{1.3}) holds uniformly in all directions $\hat{x}:=\frac{x}{|x|}$.
Furthermore, the problem (\ref{1.2})--(\ref{1.3}) is equivalent to the {\it Lippmann-Schwinger integral equation}
\begin{equation}
u(x, \theta)=u^{inc}(x, \theta)+k^2\int_{Q}q(y)u(y, \theta)\Phi(x,y)dy, \label{1.4}
\end{equation}
where $\Phi(x,y)$ denotes the fundamental solution to Helmholtz equation in $\mathbb{R}^2$, that is,
\begin{equation}
\Phi(x,y):= \displaystyle \frac{i}{4}H^{(1)}_0(k|x-y|), \ x \neq y, \label{1.5}
\end{equation}
where $H^{(1)}_0$ is the Hankel function of the first kind of order one.
It is well known that there exists a unique solution $u^{sca}$ of the problem (\ref{1.2})--(\ref{1.3}), and it has the following asymptotic behaviour,
\begin{equation}
u^{sca}(x, \theta)=\frac{\mathrm{e}^{ikr}}{\sqrt{r}}\Bigl\{ u^{\infty}(\hat{x},\theta)+O\bigl(1/r \bigr) \Bigr\} , \ r \to \infty, \ \ \hat{x}:=\frac{x}{|x|}. \label{1.6}
\end{equation}
The function $u^{\infty}$ is called the {\it far field pattern} of $u^{sca}$, and it has the form
\begin{equation}
u^{\infty}(\hat{x},\theta)=\frac{k^2}{4\pi}\int_{Q}\mathrm{e}^{-ik \hat{x} \cdot y} u(y, \theta)q(y)dy=:\mathcal{F}_{\theta}q(\hat{x}), \label{1.7}
\end{equation}
where the far field mapping $\mathcal{F}_{\theta}:L^{2}(Q) \to L^{2}(\mathbb{S}^{1})$ is defined in the second equality for each incident direction $\theta \in \mathbb{S}^{1}$.
For further details of these direct scattering problems, we refer to Chapter 8 of \cite{ColtonKress}.
\par
We consider the inverse scattering problem to reconstruct the function $q$ from the far field pattern $u^{\infty}(\hat{x}, \theta_n)$ for all directions $ \hat{x} \in \mathbb{S}^{1}$ and several directions $\{ \theta_n \}_{n=1}^{N}\subset \mathbb{S}^{1}$ with some $N \in \mathbb{N}$, and one fixed wave number $k>0$.
It is well known that the function $q$ is uniquely determined from the far field pattern $u^{\infty}(\hat{x}, \theta)$ for all $ \hat{x}, \theta \in \mathbb{S}^{1}$ and one fixed $k>0$ (see, e.g., \cite{bukhgeim2008recovering, novikov1988multidimensional, ramm1988recovery}), but the uniqueness for several incident plane wave is an open question.
For impenetrable obstacle scattering case, if we assume that the shape of scatterer is a polyhedron or ball, then the uniqueness for a single incident plane wave is proved (see \cite{alessandrini2005determining, cheng2003uniqueness, liu2006uniqueness, liu1997inverse}).
Recently in \cite{alberti2020infinitedimensional}, they showed the Lipschitz stability for inverse medium scattering with finite measurements $\{ u^{\infty}(\hat{x}_{i}, \theta_{j}) \}_{i,j=1,...,N}$ for large $N \in \mathbb{N}$ under the assumption that the true function belongs to some compact and convex subset of finite-dimensional subspace.
\par
Our problem for equation (\ref{1.7}) with finite measurements $\{ u^{\infty}(\cdot, \theta_n) \}_{n=1}^{N}$ is not only ill-posed, but also nonlinear, that is, the far field mappings $\mathcal{F}_{\theta}$ is nonlinear because $u(\cdot, \theta)$ in (\ref{1.7}) is a solution for the Lippmann-Schwinger integral equation (\ref{1.4}), which depends on $q$.
Existing methods for solving nonlinear inverse problem can be roughly categorized into two groups: iterative optimization methods and qualitative methods.
The iterative optimization method (see e.g., \cite{Bakushinsky, ColtonKress, Giorgi, Hohage, Kaltenbacher}) does not require many measurements, however it require the initial guess which is the starting point of the iteration.
It must be appropriately chosen by a priori knowledge of the unknown function $q$, otherwise, the iterative solution could not converge to the true function.
On the other hand, the qualitative method such as the linear sampling method \cite{ColtonKirsch}, the no-response test \cite{Honda}, the probe method \cite{Ikehata}, the factorization method \cite{KirschGrinberg}, and the singular sources method \cite{Potthast}, does not require the initial guess and it is computationally faster than the iterative method.
However, the disadvantage of the qualitative method is to require uncountable many measurements.
For the survey of the qualitative method, we refer to \cite{NakamuraPotthast}.
Recently in \cite{ito2012direct, Liu_2018}, they suggested the reconstruction method from a single incident plane wave although the rigorous justifications are lacked.
\par
If the total field $u$ in (\ref{1.7}) is replaced by the incident field $u^{inc}$, the nonlinear equation (\ref{1.7}) is transformed into the linear equation
\begin{equation}
u^{\infty}_{B}(\hat{x},\theta)=\frac{k^2}{4\pi}\int_{Q}\mathrm{e}^{-ik \hat{x} \cdot y} u^{inc}(y, \theta)q(y)dy=:\mathcal{F}_{B, \theta}q(\hat{x}), \label{1.8}
\end{equation}
which is known as the {\it Born approximation}.
The function $u^{\infty}_{B}$ is a good approximation of the far field pattern $u^{\infty}$ when $k>0$ and the value of $q$ are very small (see (\ref{1.4})).
Another interpretation is that the Born approximation is the Fr\'echet derivative of the far field mapping $\mathcal{F}$ at $q=0$.
For further readings of the inverse scattering problem with the Born approximation, we refer to \cite{Bakushinsky, Bao, ColtonKress, Kirsch2, Pike}.
In this paper, we study the linear integral equation (\ref{1.8}) instead of the nonlinear one (\ref{1.7}).
This paper is the first part of our works, and in the forthcoming paper, we will study the nonlinear integral equation (\ref{1.7}).
\par
Although the inverse scattering problem become linear by the Born approximation, the linear equation (\ref{1.8}) is ill-posed, which means that there does not generally exist the inverse $\mathcal{F}^{-1}_{B, \theta}$ of the operator $\mathcal{F}_{B, \theta}$.
A common technique to solve linear and ill-posed inverse problems is the {\it Tikhonov regularization method} (see e.g., \cite{Cakoni, Hanke, Kress, NakamuraPotthast}).
A natural approach applying regularization method to our situation is to put all available measurements $\{ u^{\infty}_{B}(\cdot, \theta_n) \}_{n=1}^{N}$ and all far field mappings $\{ \mathcal{F}_{B, \theta_n} \}_{n=1}^{N}$ into one long vectors $\vec{u}^{\infty}_{B}$ and $\vec{\mathcal{F}}_{B}$, respectively, and to apply the Tikhonov regularization method to the big system equation $\vec{u}^{\infty}=\vec{\mathcal{F}}_{B}q$.
We shall call this way the {\it Full data Tikhonov}.
\par
In this paper, we propose the reconstruction scheme based on {\it Kalman filter}.
The Kalman filter (see the original paper \cite{Kalman}) is the algorithm to estimate the unknown state in the dynamics system by using the time sequential measurements.
It has many applications such as navigations and tracking objects, and for further readings, we refer to \cite{Grewal, Jazwinski, Kalman, NakamuraPotthast}.
\par
The contributions of this paper are the following.
\begin{itemize}
\item[(A)] We propose the reconstruction algorithm for solving the linear inverse scattering problem (\ref{1.8}) based on the Kalman Filter (see (\ref{4.21})--(\ref{4.23})).
\item[(B)] We show that in the linear problem, the Full data Tikhonov is equivalent to the Kalman Filter (see Theorem \ref{equivalence}).
\end{itemize}
(A) means that we can estimate the unknown function $q$ by updating every time to give the far field pattern $u^{\infty}_{B}(\cdot, \theta_{n})$ with one incident direction $\theta_n$ without waiting for all measurements $\{ u^{\infty}_{B}(\cdot, \theta_{n}) \}_{n=1}^{N}$.
Furthermore, (B) means that the final solution of the Kalman filter coincides with the solution $q^{FT}_{N}$ of the Full data Tikhonov when the same initial guess is employed.
The advantage of the Kalman filter over the Full data Tikhonov is that we do not require to construct the big system equation $\vec{u}^{\infty}_{B}=\vec{\mathcal{F}}_{B}q$, which reduces computational costs.
Instead, we update not only state, but also the weight of the norm for the state space, which is associated with the update of the covariance matrices of the state in the statistical viewpoint (see Section \ref{Stochastic viewpoints of Kalman filter}).
\par
This paper is organized as follows. In Section \ref{Tikhonov regularization method}, we briefly recall the Tikhonov regularization theory. In Sections \ref{Full data Tikhonov}, we give the algorithm of the Full data Tikhonov. In Section \ref{Kalman Filter Section}, we give the algorithm of the Kalman filter, and show that it is equivalent to the Full data Tikhonov. In section \ref{Stochastic viewpoints of Kalman filter}, we discuss the stochastic viewpoints of Kalman filter. Finally in Section \ref{Numerical examples}, we give numerical examples to demonstrate our theoretical results.
\section{Tikhonov regularization method}\label{Tikhonov regularization method}
Tikhonov regularization is the method to provide the stable approximate solution for linear and ill-posed inverse problem.
In this section, we briefly recall the regularization approach.
For further readings,
we refer to e.g., \cite{Cakoni, Hanke, Kress, NakamuraPotthast}.
In Sections 2--5, we consider the general functional analytic situation of our inverse scattering problem.
\par
Let $X$ and $Y$ be Hilbert spaces over complex variables $\mathbb{C}$, which are associated with the state space $L^{2}(Q)$ of the inhomogeneous medium function $q$, and the observation space $L^{2}(\mathbb{S}^1)$ of the far field pattern $u^{\infty}$, respectively, and let $A:X \to Y$ be a compact linear operator from $X$ to $Y$, which is associated with the observation operator $\mathcal{F}_{B}:L^{2}(Q) \to L^{2}(\mathbb{S}^{1})$ defined in (\ref{1.8}) as the far field mapping.
We consider the following problem to determine $\varphi \in X$ given $f \in Y$.
\begin{equation}
A\varphi = f. \label{2.1}
\end{equation}
Since the observation operator $A$ is not generally invertible, the equation (\ref{2.1}) is replaced by
\begin{equation}
\alpha \varphi + A^{*}A\varphi = A^{*}f, \label{2.2}
\end{equation}
which was derived from the multiplication with the adjoint $A^{*}$ of the operator $A$ and the addition of $\alpha \varphi$ where the regularization parameter $\alpha>0$ in (\ref{2.1}).
We call the solution $\varphi_{\alpha}$ of the equation (\ref{2.2}) the regularized solution of (\ref{2.1}).
The following lemma is well known as the properties of the regularized solution $\varphi_{\alpha}$ (see e.g., Chapter 4 of \cite{ColtonKress}, Section 4 of \cite{Groetsch}, and Chapter 3 of \cite{NakamuraPotthast}).
\begin{lem}\label{lemma Tikhonov}
Let $X$ and $Y$ be Hilbert spaces and let $A:X \to Y$ be a compact linear operator from $X$ to $Y$. Then, followings hold.
\begin{description}
\item[(i)] (Theorems 4.13 in \cite{ColtonKress}) The operator ($\alpha I + A^{*}A$) is bounded invertible.
\item[(ii)] (Theorem 4.14 in \cite{ColtonKress}) There exists a unique $\varphi_{\alpha}$ such that
\begin{equation}
\alpha \left\| \varphi \right\|^{2}_{X} + \left\| f - A\varphi \right\|^{2}_{Y} = \mathrm{inf}_{\varphi \in X}\left\{\alpha \left\| \varphi \right\|^{2}_{X} + \left\| f - A\varphi \right\|^{2}_{Y} \right\}. \label{2.4}
\end{equation}
The minimizer $\varphi_{\alpha}$ is given by the unique solution of (\ref{2.2}) which has the form
\begin{equation}
\varphi_{\alpha}= (\alpha I + A^{*}A)^{-1}A^{*}f,
\end{equation}
and depends continuously on $f$.
\item[(iii)] (Lemma 3.2.2 in \cite{NakamuraPotthast} and Section 4.3 of \cite{Groetsch}) Let $X$ be finite-dimensional.
Then, we have
\begin{equation}
\varphi_{\alpha} \to A^{\dag}f, \ \alpha \to 0,
\end{equation}
if $f \in \mathrm{R}(A)$ where the operator $A^{\dag}$ is the pseudo inverse of the operator $A$ defined by $A^{\dag}:=(A^{*}A)^{-1}A^{*}$.
Furthermore, $A^{\dag}f$ is the least squares solution, which is minimizer of the following problem
\begin{equation}
\left\| A \varphi - f \right\|=\mathrm{min}_{\varphi \in X}\left\{ \left\| A \varphi -f \right\|_{Y} \right\}. \label{2.7}
\end{equation}
\item[(iv)] (Theorem 3.1.8 in \cite{NakamuraPotthast}) Let $A$ be injective, and let $f$ be of the form $f=A\varphi^{*}$.
Then, we have
\begin{equation}
\varphi_{\alpha} \to \varphi^{*}, \ \alpha \to 0.
\end{equation}
\item[(v)] (Theorem 3.1.10 in \cite{NakamuraPotthast}) Let $A$ be injective.
If $f \in \mathrm{R}(A)$, then there exists $C=C_f$ such that
\begin{equation}
\left\| \varphi_{\alpha} \right\| \leq C, \ \alpha>0, \label{2.5}
\end{equation}
and if $f \notin \mathrm{R}(A)$, then $\left\| \varphi_{\alpha} \right\|_{X} \to \infty$ as $\alpha \to 0$.
\end{description}
\end{lem}
\begin{rem}\label{Remark for Tikhonov}
We observe from (iii) that if $X$ is finite-dimensional and $f=A\varphi_{true}$ where $\varphi_{true}$ is the true solution of (\ref{2.1}), the regularized solution $\varphi_{\alpha}$ converges to the least squares solution $A^{\dag}A\varphi_{true}$.
We remark that the operator $A^{\dag}A$ is an orthogonal projection onto $\mathrm{R}(A^{*})=\mathrm{N}(A)^{\bot}$ (see Lemma 3.2.3 in \cite{NakamuraPotthast}).
Therefore, in addition if the operator $A$ is injective, then the least squares solution $A^{\dag}A\varphi_{true}$ coincides with the true solution $\varphi^{true}$.
\end{rem}
\section{Full data Tikhonov}\label{Full data Tikhonov}
The natural approach for solving the equation (\ref{1.8}) is to put all available measurements $\{ u^{\infty}_{B, n} \}_{n=1}^{N}$ and all far field mappings $\{ \mathcal{F}_{B, n} \}_{n=1}^{N}$, where the index $n$ is associated with some incident angle $\theta_n \in \mathbb{S}^{1}$, into one long vector $\vec{u}^{\infty}_{B}$ and $\vec{\mathcal{F}}_{B}$, respectively, and to employ the regularized approach discussed in the Section 2.
In order to study the above general situation, let $f_1,..., f_N \in Y$ be measurements, let $A_1,...,A_N$ be observation operators, and let us consider the problem to determine $\varphi \in X$ such that
\begin{equation}
A_n \varphi = f_n, \label{3.1}
\end{equation}
for all $n=1,...,N$.
Now, we assume that we have the initial guess $\varphi_0 \in X$, which is the starting point of the algorithm, and is appropriately determined by a priori information of the true solution $\varphi^{true}$.
Then, we consider the minimization problem of the following functional.
\begin{eqnarray}
J_{Full, N}(\varphi)&:=&\alpha \left\| \varphi - \varphi_0 \right\|^{2}_{X} + \left\| \vec{f} - \vec{A}\varphi \right\|^{2}_{Y^{N}, R^{-1}}
\nonumber\\
&=&\alpha \left\| \varphi - \varphi_0 \right\|^{2}_{X} + \sum_{n=1}^{N}\left\| f_n - A_n\varphi \right\|^{2}_{Y, R^{-1}}, \label{3.2}
\end{eqnarray}
where $\vec{f}:=\left(
\begin{array}{cc}
f_1 \\
\vdots \\
f_N
\end{array}
\right)$, and $\vec{A}:=\left(
\begin{array}{cc}
A_1 \\
\vdots \\
A_N
\end{array}
\right)$.
The norm $\left\| \cdot \right\|^{2}_{Y, R^{-1}}:=\langle \cdot, R^{-1} \cdot \rangle_{Y}$ is a weighted norm with a positive definite symmetric invertible operator $R: Y \to Y$, which is interpreted as the covariance matrices of the observation error distribution from a statistical viewpoint in the case when $Y$ is the Euclidean space (see Section \ref{Stochastic viewpoints of Kalman filter}).
With $\tilde{\varphi}=\varphi-\varphi_0$, the problem (\ref{3.1}) is transformed into
\begin{equation}
\tilde{J}_{Full, N}(\tilde{\varphi}):=\alpha \left\| \tilde{\varphi} \right\|^{2}_{X} + \left\| (\vec{f} - \vec{A}\varphi_0 ) - \vec{A}\tilde{\varphi} \right\|^{2}_{Y^{N}}. \label{3.3}
\end{equation}
By Lemma 2.1, the minimizer $\tilde{\varphi}_{\alpha}$ of (\ref{3.3}) is given by
\begin{equation}
\tilde{\varphi}_{\alpha} = (\alpha I + \vec{A}^{*}\vec{A})^{-1}\vec{A}^{*}\left(\vec{f} - \vec{A}\varphi_0 \right), \label{3.4}
\end{equation}
which implies that
\begin{equation}
\varphi^{FT}_{N} := \varphi_0 + (\alpha I + \vec{A}^{*}\vec{A})^{-1}\vec{A}^{*}\left(\vec{f} - \vec{A}\varphi_0 \right), \label{3.5}
\end{equation}
is the minimizer of (\ref{3.2}).
We call this the {\it Full data Tikhonov}.
Here, $\vec{A}^{*}$ is the adjoint operator with respect to $\langle \cdot, \cdot \rangle_{X}$ and $\langle \cdot, \cdot \rangle_{Y^{N}, R^{-1}}$.
We calculate
\begin{eqnarray}
\langle \vec{f}, \vec{A} \varphi \rangle_{Y^N, R^{-1}} &=& \sum_{n=1}^{N} \langle f_n, R^{-1}A_n \varphi \rangle_{Y}
\nonumber\\
&=&\sum_{n=1}^{N} \langle A^{H}_n R^{-1} f_n, \varphi \rangle_{X} = \langle \vec{A}^{H} R^{-1} \vec{f}, \varphi \rangle_{X} \label{3.6}
\end{eqnarray}
which implies that
\begin{equation}
\vec{A}^{*}=\vec{A}^{H} R^{-1} \label{3.7}
\end{equation}
where $A_{n}^{H}$ and $\vec{A}^{H}$ are the adjoint operator with respect to usual scalar products $\langle \cdot, \cdot \rangle_{X}$, $\langle \cdot, \cdot \rangle_{Y}$ and $\langle \cdot, \cdot \rangle_{X}$, $\langle \cdot, \cdot \rangle_{Y^{N}}$, respectively.
Then, the Full data Tikhonov solution in (\ref{3.5}) is of the form
\begin{equation}
\varphi^{FT}_{N} = \varphi_0 + \left( \alpha I + \vec{A}^{H} R^{-1}\vec{A} \right)^{-1}\vec{A}^{H} R^{-1}\left(\vec{f} - \vec{A}\varphi_0 \right). \label{3.8}
\end{equation}
\par
However, the solution (\ref{3.8}) of the Full data Tikhonov is computationally expensive when the number $N$ of measurements is increasing in which we have to construct the bigger system $\vec{A}\varphi= \vec{f}$.
So, let us consider the alternative approach based on the Kalman filter in the next section.
\section{Kalman filter}\label{Kalman Filter Section}
The Kalman filter is the algorithm to estimate the unknown state in the dynamics system by using the sequential measurements over time.
In the usual Kalman filter, the model operator to describe the process of the state in the dynamics system is defined (see e.g., Chapter 5 of \cite{NakamuraPotthast}).
In our problem, it corresponds to the identity mapping because unknown function $q$ does not develop over time.
\par
Let us formulate the Kalman filter algorithm based on the functional analytic situation using the same notations described in Sections \ref{Tikhonov regularization method} and \ref{Full data Tikhonov}.
In \cite{Freitag, NakamuraPotthast}, the similar arguments of the following was discussed in the special case when $X$ and $Y$ are the Euclidean spaces.
In this section, we discuss more general situation, that is, the Hilbert space over complex variables $\mathbb{C}$, which is applicable to our inverse scattering problem.
\par
First, we consider the following minimization problem when one measurement $f_1 \in Y$, observation operator $A_1$, and the initial guess $\varphi_0 \in X$ are given.
\begin{equation}
J_{1}(\varphi):=\alpha \left\| \varphi - \varphi_0 \right\|^{2}_{X} + \left\| f_1 - A_1 \varphi \right\|^{2}_{Y, R^{-1}}. \label{4.1}
\end{equation}
By using a weighted norm $\left\| \cdot \right\|^{2}_{X, B_{0}^{-1}}:=\langle \cdot, B_{0}^{-1} \cdot \rangle_{X}$ where $B_{0}:= \frac{1}{\alpha} I$, the functional $J_1$ can be of the form
\begin{equation}
J_{1}(\varphi)= \left\| \varphi - \varphi_0 \right\|^{2}_{X, B^{-1}_{0}} + \left\| f_1 - A_1 \varphi \right\|^{2}_{Y, R^{-1}}, \label{4.2}
\end{equation}
and its unique minimizer $\varphi_1$ is given by
\begin{equation}
\varphi_{1} := \varphi_0 + (I + A_{1}^{*}A_{1})^{-1}A_{1}^{*}\left(f - A_1\varphi_0 \right), \label{4.3}
\end{equation}
where $A^{*}_{1}$ is the adjoint operator with respect to weighted scalar products $\langle \cdot, \cdot \rangle_{X, B_{0}^{-1}}$ and $\langle \cdot, \cdot \rangle_{Y, R^{-1}}$.
We calculate
\begin{eqnarray}
\langle f, A_{1} \varphi \rangle_{Y, R^{-1}} &=& \langle f, R^{-1}A_{1} \varphi \rangle_{Y}
\nonumber\\
&=& \langle A^{H}_{1} R^{-1} f, \varphi \rangle_{X}
\nonumber\\
&=& \langle B_{0} A^{H}_{1} R^{-1} f, \varphi \rangle_{X, B_{0}^{-1}}, \label{4.4}
\end{eqnarray}
which implies that
\begin{equation}
A^{*}_{1}=B_{0}A^{H}_{1}R^{-1}, \label{4.5}
\end{equation}
where $A_{1}^{H}$ is the adjoint operator with respect to usual scalar products $\langle \cdot, \cdot \rangle_{X}$ and $\langle \cdot, \cdot \rangle_{Y}$.
Then, we have
\begin{eqnarray}
\varphi_{1} &=& \varphi_0 + (I + B_{0}A^{H}_{1}R^{-1}A_{1})^{-1}B_{0}A^{H}_{1}R^{-1}\left(f - A_1\varphi_0 \right)
\nonumber\\
&=&\varphi_0 + (B^{-1}_{0} + A^{H}_{1}R^{-1}A_{1})^{-1}A^{H}_{1}R^{-1}\left(f_1 - A_1\varphi_0 \right). \label{4.6}
\end{eqnarray}
\par
Next, we assume that one more measurement $f_2 \in Y$ and observation operator $H_2$ are given.
The functional for two measurements is given by
\begin{eqnarray}
J_{Full, 2}(\varphi)&:=& \left\| \varphi - \varphi_0 \right\|^{2}_{X, B_{0}^{-1}} + \left\| f_1 - A_1 \varphi \right\|^{2}_{Y, R^{-1}} + \left\| f_2 - A_2\varphi \right\|^{2}_{Y, R^{-1}}.
\nonumber
\\
&=& J_{1}(\varphi) + \left\| f_2 - A_2\varphi \right\|^{2}_{Y, R^{-1}}.
\label{4.7}
\end{eqnarray}
The question is whether we can find $B_1$ such that $J_{Full, 2}(\varphi)=J_2(\varphi) + c$ where $c$ is a constant number independently of $\varphi$, and the functional $J_{2}(\varphi)$ is defined by
\begin{equation}
J_{2}(\varphi) = \left\| \varphi - \varphi_1 \right\|^{2}_{X, B_{1}} + \left\| f_2 - A_2 \varphi \right\|^{2}_{Y, R^{-1}}, \label{4.8}
\end{equation}
where $\varphi_1$ is defined by (\ref{4.6}).
To answer this question, we show the following lemma.
\begin{lem}\label{lemma 4.1}
Set $B_1:=\left(B^{-1}_{0} + A^{H}_{1}R^{-1}A_{1}\right)^{-1}$. Then,
\begin{equation}
J_{1}(\varphi) = \left\| \varphi - \varphi_1 \right\|^{2}_{X, B_{1}^{-1}} + c, \label{4.9}
\end{equation}
where $c$ is some constant independently of $\varphi$.
\end{lem}
\begin{proof}
We calculate
\begin{eqnarray}
J_{1}(\varphi)&=&\left\langle \varphi - \varphi_0, B^{-1}_{0}\left(\varphi - \varphi_0 \right) \right\rangle_{X} + \left\langle f_1 - A_{1} \varphi, R^{-1}\left( f_1 - A_{1} \varphi \right) \right\rangle_{Y}
\nonumber\\
&=& \left\langle \varphi, B^{-1}_{0}\varphi \right\rangle_{X} -2 \mathrm{Re} \left\langle \varphi, B^{-1}_{0}\varphi_0 \right\rangle_{X} + \left\langle \varphi_0, B^{-1}_{0}\varphi_0 \right\rangle_{X}
\nonumber\\
&&\ \ \ + \left\langle f_1, R^{-1}f_1 \right\rangle_{Y} -2 \mathrm{Re}\left\langle \varphi, A^{H}_{1}R^{-1}f_1 \right\rangle_{X} + \left\langle \varphi, A^{H}_{1}R^{-1}A_{1}\varphi \right\rangle_{X}.
\nonumber\\
&=& \left\langle \varphi, B^{-1}_{0}\varphi \right\rangle_{X} -2 \mathrm{Re} \left\langle \varphi, B^{-1}_{0}\varphi_0 \right\rangle_{X} -2 \mathrm{Re}\left\langle \varphi, A^{H}_{1}R^{-1}f_1 \right\rangle_{X}
\nonumber\\
&&\ \ \ \ \ + \left\langle \varphi, A^{H}_{1}R^{-1}A_{1}\varphi \right\rangle_{X} + c_0
\nonumber\\
&=& \left\langle \varphi, B^{-1}_{1}\varphi \right\rangle_{X} -2 \mathrm{Re} \left\langle \varphi, B^{-1}_{0}\varphi_0 \right\rangle_{X} -2 \mathrm{Re}\left\langle \varphi, A^{H}_{1}R^{-1}f_1 \right\rangle_{X} + c_0, \nonumber\\ \label{4.10}
\end{eqnarray}
where we used $B_{1}^{-1}=\left(B^{-1}_{0} + A^{H}_{1}R^{-1}A_{1}\right)$.
By (\ref{4.6}), we have
\begin{eqnarray}
B^{-1}_{1}\left(\varphi - \varphi_{1} \right) &=& B^{-1}_{1}\varphi - B^{-1}_{1} \varphi_{1} \nonumber\\
&=&B^{-1}_{1}\varphi - \left(B^{-1}_{0} + A^{H}_{1}R^{-1}A_{1}\right) \varphi_{0} - A^{H}_{1}R^{-1}\left(f - A_1\varphi_0 \right) \nonumber\\
&=&B^{-1}_{1}\varphi - B^{-1}_{0}\varphi_0 - A^{H}_{1}R^{-1}f_1. \label{4.11}
\end{eqnarray}
By using (\ref{4.11}) and the self-adjointness of $B^{-1}_{1}$, we have
\begin{eqnarray}
&&\left\langle \varphi - \varphi_{1}, B^{-1}_{1}\left(\varphi - \varphi_{1} \right) \right\rangle_{X}
\nonumber\\
&=&\left\langle \varphi - \varphi_1, B^{-1}_{1}\varphi - B^{-1}_{0}\varphi_0 - A^{H}_{1}R^{-1}f_1 \right\rangle_{X}
\nonumber\\
&=& \left\langle B^{-1}_{1} \left(\varphi - \varphi_1\right), \varphi \right\rangle_{X} - \left\langle \varphi, B^{-1}_{0}\varphi_0 \right\rangle_{X} - \left\langle \varphi, A^{H}_{1}R^{-1}f \right\rangle_{X} + c_1
\nonumber\\
&=&\left\langle B^{-1}_{1}\varphi - B^{-1}_{0}\varphi_0 - A^{H}_{1}R^{-1}f, \varphi \right\rangle_{X}
\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ - \left\langle \varphi, B^{-1}_{0}\varphi_0 \right\rangle_{X} - \left\langle \varphi, A^{H}_{1}R^{-1}f \right\rangle_{X} + c_1
\nonumber\\
&=&\left\langle \varphi, B^{-1}_{1}\varphi \right\rangle_{X} -2 \mathrm{Re} \left\langle \varphi, B^{-1}_{0}\varphi_0 \right\rangle_{X} -2 \mathrm{Re}\left\langle \varphi, A^{H}_{1}R^{-1}f_1 \right\rangle_{X} + c_1.
\nonumber\\ \label{4.12}
\end{eqnarray}
With (\ref{4.10}) and (\ref{4.12}), $J_{1}(\varphi)$ is of the form
\begin{equation}
J_{1}(\varphi)=\left\langle \varphi - \varphi_{1}, B^{-1}_{1}\left(\varphi - \varphi_{1} \right) \right\rangle_{X} + c_2. \label{4.13}
\end{equation}
where $c_0$, $c_1$, and $c_2$ are some constant numbers independently of $\varphi$.
Lemma \ref{lemma 4.1} has been shown.
\end{proof}
This lemma tells us that $J_{Full, 2}(\varphi)$ is equivalent to $J_{2}(\varphi)$ in the sense of minimization with respect to $\varphi$.
By the same argument in (\ref{4.2})--(\ref{4.6}), its unique minimizer $\varphi_2$ is given by
\begin{equation}
\varphi_{2} := \varphi_1 + (B^{-1}_{1} + A^{H}_{2}R^{-1}A_{2})^{-1}A^{H}_{2}R^{-1}\left(f_2 - A_2\varphi_1 \right). \label{4.14}
\end{equation}
\par
We can repeat the above arguments (\ref{4.1})--(\ref{4.14}) until all measurements $f_1,...,f_n$ and all observation operators $A_1,...,A_n$ are given.
Then, we have following algorithms
\begin{equation}
\varphi_{n} := \varphi_{n-1} + K_{n}\left( f_{n}-A_{n} \varphi_{n-1} \right), \label{4.15}
\end{equation}
where the operator
\begin{equation}
K_{n}:= \left( B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n}\right)^{-1} A^{H}_{n}R^{-1}, \label{4.16}
\end{equation}
is called the {\it Kalman gain matrix}, and $B_n$ is defined by
\begin{equation}
B_n := \left(B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n}\right)^{-1}. \label{4.17}
\end{equation}
Since we have
\begin{eqnarray}
\left( B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n}\right)B_{n-1}A^{H}_{n}
&=&
A^{H}_{n} + A^{H}_{n}R^{-1}A_{n}B_{n-1}A^{H}_{n}
\nonumber\\
&=&
A^{H}_{n}R^{-1}\left(R + A_{n} B_{n-1} A^{H}_{n} \right), \nonumber
\end{eqnarray}
the Kalman gain matrix $K_{n}$ can be of the form
\begin{equation}
K_{n}= B_{n-1} A^{H}_{n}\left(R + A_{n} B_{n-1} A^{H}_{n} \right)^{-1}. \nonumber
\end{equation}
Here, we show the following lemma that the operator $B_n$ has another form.
\begin{lem}
Let $K_n$ be the Kalman gain matrix defined in (\ref{4.16}).
Then, the operator $B_n$ has the following form
\begin{equation}
B_n = \left( I - K_n A_n \right) B_{n-1}. \label{4.18}
\end{equation}
\end{lem}
\begin{proof}
By multiplying (\ref{4.16}) by $\left( B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n} \right)$ from the left hand side, and by $A_n$ from right hand side, we have
\begin{equation}
\left( B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n} \right)K_{n}A_{n} = A^{H}_{n}R^{-1} A_{n}, \label{4.19}
\end{equation}
which implies that by using (\ref{4.17})
\begin{eqnarray}
B^{-1}_{n}\left(I-K_{n}A_{n}\right)&=& \left( B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n}\right)\left(I-K_{n}A_{n}\right)
\nonumber\\
&=&\left( B^{-1}_{n-1} + A^{H}_{n}R^{-1}A_{n}\right) - A^{H}_{n}R^{-1} A_{n}
\nonumber\\
&=& B^{-1}_{n-1}. \label{4.20}
\end{eqnarray}
Multiplying (\ref{4.20}) by $B_{n}$ from the left hand side, and by $B_{n-1}$ from the right hand side, we finally get (\ref{4.18}).
\end{proof}
We summarize the update formula in the following.
\begin{equation}
\varphi^{KF}_{n}:= \varphi^{KF}_{n-1} + K_{n}\left( f_{n}-A_{n} \varphi^{KF}_{n-1} \right), \label{4.21}
\end{equation}
\begin{equation}
K_{n}:= B_{n-1} A^{H}_{n}\left(R + A_{n} B_{n-1} A^{H}_{n} \right)^{-1}, \label{4.22}
\end{equation}
\begin{equation}
B_{n}:= \left(I - K_{n} A^{H}_{n} \right)B_{n-1}, \label{4.23}
\end{equation}
for $n=1,...,N$, where $\varphi^{KF}_{0}:=\varphi_0$ and $B_{0}:=\frac{1}{\alpha}I$. We call this the {\it Kalman filter}.
\par
We observe the above algorithm.
It means that we can estimate the state $\varphi$ every time $n$ to observe one measurement $f_{n}$ without waiting all measurements $\{ f_{n} \}_{n=1}^{N}$.
It includes not only the update (\ref{4.21}) of the state $\varphi$, but also the update (\ref{4.23}) of the weight $B$ of the norm, which plays the role of keeping the information of the previous state.
In finite dimensional setting, the weight $B$ is also interpreted as the covariance matrices of the state error distribution from statistical viewpoint (see Section \ref{Stochastic viewpoints of Kalman filter}).
\par
Finally in this section, we show the equivalence of Full data Tikhonov and Kalman filter when all observation operators $A_n$ are linear.
\begin{thm} \label{equivalence}
For measurements $f_1,...,f_N$, linear operators $A_1,...,A_N$, and the initial guess $\varphi_0 \in X$, the final sate of the Kalman filter given by (\ref{4.21})--(\ref{4.23}) is equivalent to the state of the Full data Tikhonov given by (\ref{3.8}), that is
\begin{equation}
\varphi^{KF}_{N}=\varphi^{FT}_{N}. \label{4.24}
\end{equation}
\end{thm}
\begin{proof}
It is sufficient to show that
\begin{equation}
J_{Full, N}(\varphi) = \left\| \varphi - \varphi^{KF}_{N} \right\|^{2}_{X, B_{N}^{-1}} + c_N, \label{4.25}
\end{equation}
where $c_N$ is some constant independently of $\varphi$.
We will prove (\ref{4.25}) by the induction.
The case of $N=1$ has already been shown in Lemma 4.1.
\par
We assume that (\ref{4.25}) in the case of $n \in \mathbb{N}$ with $1\leq n\leq N-1$ holds, that is,
\begin{equation}
J_{Full,n}(\varphi) = \left\| \varphi - \varphi^{KF}_{n} \right\|^{2}_{X, B_{n}^{-1}} + c_{n}, \label{4.26}
\end{equation}
where $c_{n}$ is some constant.
Then, we have
\begin{eqnarray}
J_{Full, n+1}(\varphi) &=& J_{Full, n}(\varphi) + \left\| f_{n+1} - A_{n+1} \varphi \right\|^{2}_{Y, R^{-1}}
\nonumber\\
&=& \left\| \varphi - \varphi^{KF}_{n} \right\|^{2}_{X, B_{n}^{-1}} + \left\| f_{n+1} - A_{n+1} \varphi \right\|^{2}_{Y, R^{-1}} + c_{n}. \label{4.27}
\end{eqnarray}
By the same argument in Lemma \ref{lemma 4.1} replacing $B_0$, $\varphi_0$, $f_1$, $A_1$ by $B_{n}$, $\varphi_{n}$, $f_{n+1}$, $A_{n+1}$, respectively, we have that $J_{Full, n+1}(\varphi)=\left\| \varphi - \varphi^{KF}_{n+1} \right\|^{2}_{X, B_{n+1}^{-1}} + c_{n+1}$.
Theorem \ref{equivalence} has been shown.
\end{proof}
\begin{rem}\label{Remark4.2}
If $\vec{f}$ is true measurement, i.e., $\vec{A}\varphi_{true}=\vec{f}$ and $\vec{A}$ is injective, then our Kalman filter solution $\varphi_{N}^{KF}=\varphi_{N}^{FT}$, which is is equal to the Full data Tikhonov solution $\varphi_{N}^{FT}$, convergences to true $\varphi_{true}$ as $\alpha \to 0$ (see (iv) in Lemma \ref{lemma Tikhonov}).
The injectivity of $\vec{A}$ would be expected when the number $N$ of measurement is large enough.
\end{rem}
\section{Stochastic viewpoints of Kalman filter}\label{Stochastic viewpoints of Kalman filter}
In this section, we observe the Kalman filter (\ref{4.21})-(\ref{4.23}) from Bayesian viewpoints.
For simplicity, we assume that $X=\mathbb{C}^{m}$ and $Y=\mathbb{C}^{l}$, $m, l \in \mathbb{N}$, and we treat the state $\bm{\varphi} \in \mathbb{C}^{m}$ and the measurement $\bm{f} \in \mathbb{C}^{l}$ as complex random vectors.
We recall that Bayes' theorem
\begin{equation}
p(\bm{\varphi}|\bm{f})\propto p(\bm{f}|\bm{\varphi})p(\bm{\varphi})
\end{equation}
where $p(\bm{\varphi})$ is the prior probability of the state $\bm{\varphi}$ before the measurement $f$ which is modeled by information of the current state $\bm{\varphi}$, $p(\bm{f}|\bm{\varphi})$ is the probability of observing $f$ given $\bm{\varphi}$ which is called the likelihood, and $p(\bm{\varphi}|\bm{f})$ is the posterior probability of the state $\bm{\varphi}$ given the measurement $\bm{f}$.
Bayesian theory is a simple and generic approach which can be applied to inverse and ill-posed problems (see e.g., \cite{ arridge2019solving, calvetti2007introduction, Freitag, NakamuraPotthast, stuart2010inverse}).
\par
Here, we recall that complex Gaussian distribution (see e.g., \cite{gallager2008circularly, picinbono1996second, van1995multivariate}).
Let us first remind that a complex random variable $\bm{z}$ of $\mathbb{C}^{n}$ is a pair
of real random variable of $\mathbb{R}^{n}$ such that $\bm{z} = \bm{x} + i\bm{y}$.
A complex random variable $\bm{z}$ is said to be Gaussian if its real and imaginary parts $\bm{x}$ and $\bm{y}$ are jointly Gaussian.
Its distribution with zero mean is
\begin{equation}
p(\bm{z})=p(\bm{x}, \bm{y})= \frac{1}{\sqrt{(2\pi)^{2n} |\Sigma_{2n}|}} e^{-\frac{1}{2} \bm{v}^{T} \Sigma^{-1}_{2n} \bm{v} }, \label{Complex Gaussian distribution}
\end{equation}
where $\bm{v} \in \mathbb{R}^{2n}$ such that $\bm{v^{T}}=(\bm{x^{T}}, \bm{y^{T}})$, and $\bm{T}$ means transposition, and $\bm{\Sigma_{2n}} \in \mathbb{R}^{2n \times 2n}$ is the covariance matrix defined by
\begin{equation}
\bm{\Sigma_{2n}} = \left(
\begin{array}{cc}
E(\bm{x}\bm{x^{T}}) & E(\bm{x}\bm{y^{T}}) \\
E(\bm{y}\bm{x^{T}}) & E(\bm{y}\bm{y^{T}})
\end{array}
\right).
\end{equation}
If real and imaginary parts are independent Gaussian distributed random variables with mean zero and same covariance $\bm{\Sigma} \in \mathbb{R}^{n \times n}$, then (\ref{Complex Gaussian distribution}) can be computed as
\begin{equation}
p(\bm{z})= \frac{1}{(\pi)^{n} |\bm{\Sigma}|} e^{-\bm{z^{H}} \bm{\Sigma}^{-1} \bm{z} }.
\end{equation}
where $\bm{H}$ means transposition and complex conjugation.
This distribution is referred to as {\it circularly-symmetric (central) complex Gaussian distribution}, and is denoted by $\mathcal{C}\mathcal{N}(0, \Sigma)$.
\par
We assume that complex vector $\bm{\varphi^{KF}_{n-1}} \in \mathbb{C}^{m}$ and the positive definite matrix $\bm{B_{n-1}} \in \mathbb{R}^{m \times m}$ are determined in some way, and the prior $p(\bm{\varphi})$ is modeled by a circularly-symmetric complex Gaussian distribution $\mathcal{C}\mathcal{N}(\bm{\varphi^{KF}_{n-1}}, \bm{B_{n-1}})$, that is,
\begin{equation}
p(\bm{\varphi})= \frac{1}{(\pi)^{m} |\bm{B_{n-1}}|} e^{-(\bm{\varphi} - \bm{\varphi^{KF}_{n-1}})^{\bm{H}} \bm{B_{n-1}}^{-1} (\bm{\varphi} - \bm{\varphi^{KF}_{n-1}}) }.
\end{equation}
Furthermore, we assume that the observation error $\bm{f} - \bm{A}\bm{\varphi}$ is distributed from $\mathcal{C}\mathcal{N}(0, \bm{R})$ where the $\bm{R} \in \mathbb{R}^{l \times l}$ is some positive definite matrix.
Then, the likelihood $p(\bm{f}=\bm{f_{n}}|\bm{\varphi})$ is modeled by
\begin{equation}
p(\bm{f}=\bm{f_{n}}|\bm{\varphi}) = \frac{1}{(\pi)^{l} |R|} e^{-(\bm{f_{n}} - \bm{A_{n}} \bm{\varphi})^{\bm{H}} R^{-1} (\bm{f_{n}} - \bm{A_{n}} \bm{\varphi}) }
\end{equation}
Then by Bayes' theorem, the posterior $p(\varphi|\bm{f}=\bm{f_{n}})$ can be computed as
\begin{eqnarray}
p(\bm{\varphi}|\bm{f}=\bm{f_{n}}) &\propto& e^{ -\{ (\bm{\varphi} - \bm{\varphi^{KF}_{n-1}})^{\bm{H}} \bm{B_{n-1}}^{-1} (\bm{\varphi} - \bm{\varphi^{KF}_{n-1}}) + (\bm{f_{n}} - \bm{A_{n}} \bm{\varphi})^{\bm{H}} \bm{R}^{-1} (\bm{f_{n}} - \bm{A_{n}} \bm{\varphi}) \} }
\nonumber\\
&\propto& e^{- (\bm{\varphi} - \bm{\varphi^{KF}_{n}})^{H}\bm{B_{n-1}}^{-1}(\bm{\varphi} - \bm{\varphi^{KF}_{n}})}
\end{eqnarray}
where $\bm{\varphi^{KF}_{n}}$ and $\bm{B_{n}}$ is defined by (\ref{4.21}) and (\ref{4.23}), respectively.
This computation is guaranteed by the same argument in Section \ref{Kalman Filter Section}.
Therefore, posterior distribution is a circularly-symmetric complex Gaussian distribution $\mathcal{C}\mathcal{N}(\bm{\varphi^{KF}_{n}}, \bm{B_{n}})$ with mean $\bm{\varphi^{KF}_{n}} \in \mathbb{C}^{m}$ and covariance matrix $\bm{B_{n}} \in \mathbb{R}^{m \times m}$, which means that the Kalman filter update in (\ref{4.21})-(\ref{4.23}) can be interpreted as updating mean and covariance matrix of Gaussian distribution of the state in the case that the prior and likelihood are assumed to be Gaussian.
\section{Numerical examples}\label{Numerical examples}
In this section, we provide numerical examples for the Kalman filter algorithm.
Our inverse scattering problem is to solve the linear integral equation
\begin{equation}
\mathcal{F}_{B, n}q=u^{\infty}_{B}(\cdot, \theta_n), \label{5.1}
\end{equation}
for $n=1,...,N$ where the operator $\mathcal{F}_{B, n}:L^{2}(Q) \to L^{\infty}(\mathbb{S}^{1})$ is defined by
\begin{equation}
\mathcal{F}_{B, n}q(\hat{x}):=\mathcal{F}_{B}q(\hat{x},\theta_n)=\frac{k^2}{4\pi}\int_{Q}\mathrm{e}^{ik (\theta_{n} - \hat{x}) \cdot y} q(y)dy, \label{5.2}
\end{equation}
where the incident direction is given by $\theta_n:=\left(\mathrm{cos}(2\pi n/N), \mathrm{sin}(2\pi n/N) \right)$ for each $n=1,...,N$.
We assume that the support $Q$ of the function $q$ is included in the square $[-S, S]^2$ with some $S>0$.
\par
The linear integral equation (\ref{5.1}) is discretized by
\begin{equation}
\bm{\mathcal{F}_{n}} \bm{q} = \bm{u^{ \infty}_{n}},
\end{equation}
where
\begin{equation}
\bm{\mathcal{F}_{n}} = \frac{k^2 S^{2}}{4\pi M^{2}} \left(
\mathrm{e}^{ik( \theta_{n}-\hat{x}_j) \cdot y_{i,l}} \right)_{j=1,...,J, \ -M \leq i,l \leq M-1} \in \mathbb{C}^{J\times (2M)^2}. \label{5.5}
\end{equation}
where $y_{i,l}:=\left( \frac{(2i+1)S}{2M}, \frac{(2l+1)S}{2M} \right)$, and $M \in \mathbb{N}$ is a number of the division of $[0,S]$ (i.e., the function $q$ is discretized by piecewise constant on $[-S, S]^{2}$ which is decomposed by squares with the length $\frac{S}{M}$), and $\hat{x}_j:=\left(\mathrm{cos}(2\pi j/J), \mathrm{sin}(2\pi j/J) \right)$, and $J \in \mathbb{N}$ is a number of the division of $[0,2\pi]$ and
\begin{equation}
\bm{q} = \left( q(y_{i,l}) \right)_{-M \leq i,l \leq M-1} \in \mathbb{C}^{(2M)^2}, \label{5.4}
\end{equation}
and
\begin{equation}
\bm{u^{ \infty}_{n}} = \left( u^{\infty}_{B}(\hat{x}_j, \theta_{n}) \right)_{j=1,...,J} +\bm{\epsilon_{n}} \in \mathbb{C}^{J}. \label{5.3}
\end{equation}
The noise $\bm{\epsilon_{n}} \in \mathbb{C}^{J}$ is sampling from a complex Gaussian distribution $\mathcal{C}\mathcal{N}(0, \sigma^{2}\bm{I})$, which is equivalent to $\bm{\epsilon_{n}}=\bm{\epsilon^{re}_{n}}+i\bm{\epsilon^{im}_{n}}$ where $\bm{\epsilon^{re}_{n}}, \bm{\epsilon^{im}_{n}} \in \mathbb{R}^{J}$ are independently identically distributed from Gaussian distribution $\mathcal{N}(0, \sigma^{2}\bm{I})$ with mean zero and covariance matrix $\sigma^{2}\bm{I}$ where $\sigma>0$.
\par
Here, we always fix discretization parameters as $J=30$, $M=8$, $S=3$, and weight $\bm{R} \in \mathbb{R}^{J \times J}$, which is the covariance matrix of the observation error distribution, as $R=r^{2}I$, and $r=1$.
From Remarks \ref{Remark4.2} and \ref{Remark for Tikhonov}, in order to converge to true solution, the matrix $\vec{\bm{\mathcal{F}}}:=\left(
\begin{array}{cc}
\bm{\mathcal{F}_{1}} \\
\vdots \\
\bm{\mathcal{F}_{N}}
\end{array}
\right) \in \mathbb{C}^{NJ \times (2M)^{2}}$ should be injective. The necessary condition is $JN>(2M)^{2}$, so we choose the parameter $N=30$ ($NJ=30\times30=900>$ $(2M)^2=256$).
\par
We consider true functions as the characteristic function
\begin{equation}
q^{true}_{j}(x):=\left\{ \begin{array}{ll}
1 & \quad \mbox{for $x \in B_j$} \\
0 & \quad \mbox{for $x \notin B_j$}
\end{array} \right., \label{5.6}
\end{equation}
where the support $B_j$ of the true function is considered as the following two types.
\begin{equation}
B_1:=\left\{(x_1, x_2) : x^{2}_{1}+x^{2}_{2} <1.5 \right\}, \label{5.7}
\end{equation}
\begin{equation}
B_2:=\left\{(x_1, x_2):\begin{array}{cc}
(x_{1}+1.5)^2+(x_{2}+1.5)^{2} < (1.0)^{2}\ or \\
1 < x_1 < 2,\ -2 < x_2 < 2\ or \\
-2 < x_1 < 2,\ -2.0 < x_2 < -1.0
\end{array}
\right\}. \label{5.8}
\end{equation}
In Figure \ref{truefuctions}, the blue closed curve is the boundary $\partial B_j$ of the support $B_j$, and the green brightness indicates the value of the true function on each cell divided into $(2M)^2=256$ in the sampling domain $[-S, S]^2=[-3, 3]^2$.
Here, we always employ the initial guess $q_0$ as
\begin{equation}
q_0\equiv0. \label{5.9}
\end{equation}
\par
Figure \ref{comparisonKFFT} shows the reconstruction by the Kalman filter (KF) and the Full data Tikhonov (FT) discussed in (\ref{4.21})--(\ref{4.23}) and (\ref{3.8}), respectively.
The first and second column correspond to visualization of the updated state $q$ in the case when four measurements $\{u^{\infty}_{B}(\cdot, \theta_n)\}_{n=1}^{4}$ and twenty measurements $\{u^{\infty}_{B}(\cdot, \theta_n)\}_{n=1}^{20}$ are given, respectively, for different methods KF and FT, and for two different shapes $B_1$ and $B_2$.
In Figure \ref{comparisonKFFT}, the wave number and the regularization parameter are fixed as $k=3$ and $\alpha=1$, respectively, and the measurements are noisy free.
The third column corresponds to the graph of the Mean Square Error (MSE) defined by
\begin{equation}
e_{n}:=\left\|\bm{q^{true}}-\bm{q_{n}} \right\|^2, \label{5.10}
\end{equation}
where $\bm{q_{n}}$ is associated with the updated state given $n$ measurements.
The horizontal axis is with respect to number of given measurements, and the vertical axis is the value of MSE.
We observe that in Figure \ref{comparisonKFFT}, KF and FT are equivalent, which coincides with the theoretical result in Theorem \ref{equivalence}.
\par
Figures \ref{KFreconstruction} and \ref{KFreconstructionnosiy} show the reconstruction by the Kalman filter (KF) with $\sigma=0.1, 0.5$, respectively, for two different wave numbers $k=5$ and $k=1$ and two different shape $B_1$ and $B_2$.
The first and second columns correspond to visualization of the final state given full measurements ($n=30$) for different regularization parameters $\alpha=10$ and $1e-1$, respectively.
The third column corresponds to graphs of MSE, which have three evaluations with respect to $\alpha=10, 1, 1e-1$.
The case of $k=1.0$ fails to reconstruct even with small noise (see Figures \ref{KFreconstruction}), that is, the state does not converge to zero even with increasing the number of measurements and decreasing regularization parameters.
This ill-posedness is because the rank of the full far field mapping $\vec{\bm{\mathcal{F}}}=\left(
\begin{array}{cc}
\bm{\mathcal{F}_{1}} \\
\vdots \\
\bm{\mathcal{F}_{N}}
\end{array}
\right) \in \mathbb{C}^{NJ \times (2M)^{2}}$ ($NJ=30\times30=900$, $(2M)^2=256$) degenerates when the wave number $k$ decreases.
Figure \ref{rank} shows its degeneracy.
The horizontal axis is with respect to wave numbers, and the vertical axis is the number of the rank of full far field mappings $\vec{\bm{\mathcal{F}}}$ (Maximum of rank is $256$).
\section*{Acknowledgments}
This work of the first author was supported by Grant-in-Aid for JSPS Fellows (No.21J00119), Japan Society for the Promotion of Science.
\begin{figure}
\caption{true functions}
\label{truefuctions}
\end{figure}
\begin{figure}
\caption{the comparison of KF and FT, $k=3$, $\alpha=1$ (true data)}
\label{comparisonKFFT}
\end{figure}
\begin{figure}
\caption{KF reconstruction for different $k$ and $\alpha$ (nosiy $\sigma= 0.1$)}
\label{KFreconstruction}
\end{figure}
\begin{figure}
\caption{KF reconstruction for different $k$ and $\alpha$ (nosiy $\sigma= 0.5$)}
\label{KFreconstructionnosiy}
\end{figure}
\begin{figure}
\caption{the graph of the rank of $\vec{\mathcal{F}
\label{rank}
\end{figure}
\end{document} | math | 45,637 |
\begin{document}
\title{Dixmier Trace for Toeplitz Operators on Symmetric Domains}
\author{Harald Upmeier and Kai Wang}
\address{ Fachbereich Mathematik,
Universit$\ddot{a}$t Marburg, Marburg, 35032, Germany}
\email{[email protected]}
\address{School of Mathematical Sciences,
Fudan University, Shanghai, 200433, P. R. China}
\email{[email protected]}
\subjclass[2010]{32M15; 42B35; 47B35}
\keywords{bounded symmetric domain, Toeplitz operator, Dixmier trace}
\thanks{The second author was partially supported by NSFC (11271075,11420101001), the Alexander von Humboldt Foundation and Laboratory of Mathematics for Nonlinear Science at Fudan University.}
\maketitle
\begin{abstract}
For Toeplitz operators on bounded symmetric domains of arbitrary rank, we define a Hilbert quotient module corresponding to partitions of length $1$ and prove that it belongs to the Macaev class $\LL^{n,\oo}$. We next obtain an explicit formula for the Dixmier trace of Toeplitz commutators in terms of the underlying boundary geometry.\end{abstract}
\section{Introduction}
The Dixmier trace of Hilbert space operators \cite{C2}, of fundamental importance for pseudo-differential operators \cite{C1,W}, has recently found deep applications in {\bf complex analysis}, for Hankel and Toeplitz operators on strictly pseudo-convex domains \cite{AFJP,EGZ,EZ,ER,HH} and for homogeneous Hilbert quotient modules over the unit ball \cite{DTY,EE,GW,GWZ}. In these applications the underlying operators are essentially normal, i.e. commutators are compact; more precisely, belong to certain norm ideals of Schatten type.
In this paper we are concerned with operators of Toeplitz or Hankel type which are not essentially commuting. These operators arise naturally when the underlying domain $D\ic\Cl^d$ is not strictly pseudo-convex or does not have a smooth boundary. The most important case is the so-called {\bf hermitian bounded symmetric domains} $D=G/K$ of arbitrary rank $r$, which generalize the unit disk and the unit ball of rank $1.$ In this paper, we construct a suitable Hilbert quotient module of the Hardy space over the Shilov boundary $S$ and study the associated 'sub-Toeplitz' operators. Our first main result shows that commutators of such operators belong to the Macaev class $\LL^{n,\oo},$ for a suitable $n$ related to the geometry of $D.$ The second main result is an explicit formula for the Dixmier trace of products of such operators, in terms of a Jordan theoretic Grassmann-type manifold.
The results of this paper can be generalized to cover the weighted Bergman spaces instead of the Hardy space, at least for the continuous part of the Wallach set \cite{FK}. On the other hand, extending these results to all smooth functions $f\in\CL^\oo(S)$ will be more challenging, even for the basic case of rank $2$-domains (involving pseudo-differential operators on spheres \cite{BC,BCK}).
Finally, the higher strata of the boundary of $D$ give rise to a family of smooth extensions \cite{U5} and it is of interest to develop a family version of the Dixmier trace (involving cyclic cohomology) for the associated Toeplitz commutators.
\section{Symmetric domains and Toeplitz operators}
Let $D$ be an irreducible bounded symmetric domain of rank $r$ in a complex vector space $Z$ of finite dimension $d$. The unit ball $D=\Bl_d\ic\Cl^d$ corresponds to rank $r=1.$ Denote by $G=Aut(D)$ the biholomorphic automorphism group, and put
$$K:=\{g\in G: g(0)=0\}.$$
Then $D=G/K.$ It is well known \cite{FK,L} that $D$ can be realized as the open unit ball of an irreducible {\it hermitian Jordan triple} $Z$. Thus $Z$ is a complex vector space endowed with a Jordan triple product
$$u,v,w\mapsto\{uv^*w\}\in Z\qquad\forall\,u,v,w\in Z.$$
Then $K=Aut(Z)$ is the linear group of all triple automorphisms of $Z.$ Let $S$ be the Shilov boundary of $D$. Since $K$ acts transitively on $S,$ there exists a unique $K-$invariant probability measure $ds$ on $S$. Denote by $L^2(S)$ the space of $L^2$-integrable functions, with inner product
\be{5}(f|g)_S:=\I_{S}ds\,\o{f(s)}\,g(s),\ee
and define the {\bf Hardy space}
$$H^2(S)=\{\lq\in L^2(S):\,\lq\mbox{ holomorphic on }D\}.$$
For a bounded function $f$, define the {\bf Toeplitz operator}
$$T_f\lq= P_{H^2(S)}(f\lq)\qquad\forall\lq\in H^2(S).$$
In previous work \cite{U1,U2} it was shown that Toeplitz operators $T_f$ with smooth symbol function $f\in\CL^\oo(S)$, acting on
$H^2(S),$ generate a $C^*$-algebra $\TL(S)$ which is not essentially commutative (if $r>1$) but has a {\bf composition series}
$$\KL=\IL_1\ic\IL_2\ic\cdots\ic\IL_r\ic\TL(S)=\IL_{r+1},$$
starting with the compact operators $\KL,$ such that the subquotients $\IL_{k+1}/\IL_k$ are essentially commutative. More precisely,
there is a stable isomorphism
$$\IL_{k+1}/\IL_k\al\CL(S_k)\xt\KL,$$
where $S_k$ denotes the $K$-homogeneous manifold of all 'tripotents' of rank $k.$ (Similar results hold for Toeplitz operators on weighted Bergman spaces over $D,$ as shown in \cite{U4}.) An element $c\in Z$ such that $\{cc^*c\}=c$ is called a {\bf tripotent}. Every tripotent induces a {\bf Peirce decomposition}
$$Z=Z^2_c\oplus Z^1_c\oplus Z^0_c,$$
where $Z^\la_c:=\{z\in Z:\,\{cc^*z\}=2\la z\}.$
The Peirce $2$-space is a Jordan $*$-algebra with unit element $c$ and involution $z\mapsto \{cz^*c\}.$ The self-adjoint part $X_c\ic Z^2_c$ is a so-called {\bf euclidean Jordan algebra} \cite{FK}. Let $(z|w)$ denote the $K$-invariant inner product normalized by the condition $(c|c)=1$ for each minimal tripotent $c\in Z.$ Let $\PL(Z)$ be the algebra of all (holomorphic) polynomials on $Z,$ endowed with the $K$-invariant {\bf Fischer-Fock inner product}
\be{4}(p|q)_Z:=\f1{\lp^d}\I_{Z}dz\,e^{-(z|z)}\o{p(z)}\,q(z)\ee
for all $p,q\in\PL(Z).$ By \cite{FK,U3} the natural action of $K$ on $\PL(Z)$ induces a multiplicity-free {\bf Peter-Weyl decomposition}
\be{1} \PL(Z)=\S_\ll\PL_\ll(Z),\ee
where
$$\ll=\ll_1\ge\ldots\ge \ll_r\ge 0$$
runs over all integer {\bf partitions} of length $\le r.$ The decomposition \er{1} is orthogonal under \er{4}. We let $\Nl^r_+$ denote the set of all such partitions. As usual we will identify partitions that differ only by zeros. Then
$$\Nl^r_+=\bigcup_{\el=1}^r\Nl^\el_+,$$
where $\Nl^\el_+=\{\ll\in\Nl^r_+:\,\ll_{\el+1}=0\}.$ As a special type of partition we denote
$$k_\el:=(k,\ldots,k,0\ldots,0)$$
for $1\le\el\le r$ and $k\in\Nl$ repeated $\el$ times. Choose a frame $e_1,\ldots,e_r$ of minimal tripotents. The associated joint Peirce decomposition \cite{L} defines two numerical invariants $a,b$ for $Z$ such that
$$\lr:=\f dr=1+\f a2(r-1)+b.$$
For the Hilbert unit ball ($r=1$) we put $a=2$ and $b=d-1.$ Thus $b=0$ only for the unit disk. In case $b=0$ the Jordan triple $Z$ is actually a {\bf Jordan algebra} with unit element
$$e:=e_1+\cdots+e_r.$$
In this case $Z$ carries a {\bf Jordan determinant} $N=N_r$ which is normalized by $N(e)=1.$ For $1\le\el\le r$ denote by $N_\el$ the Jordan determinant polynomial for the Peirce $2$-space $Z^2_{e_1+\ldots+e_\el}.$ As shown in \cite{U3} $\PL_\ll(Z)$ has the highest weight vector
\be{2}N_\ll(z):=N_1(z)^{\ll_1-\ll_2}N_2(z)^{\ll_2-\ll_3}\cdots N_r(z)^{\ll_r}.\ee
The {\bf multi-variable Pochhammer symbol} is the product
$$(s)_\ll:=\P_{i=1}^r(s-\f a2(i-1))_{\ll_i}$$
of the usual Pochhammer symbols $(\ln)_m=\P_{i=1}^m(\ln+i-1).$ By \cite{U1,FK} the inner products \er{4} and \er{4} are related by
\be{30}(p|q)_S=\f1{(\lr)_\ll}(p|q)_Z,\qquad\forall\, p,q\in\PL_\ll(Z).\ee
We note the relation
$$\f{(\lr)_\ll}{(\lr-b)_\ll}=\P_{j=1}^r\f{(\ll_j+1+\f a2(r-j))_b}{(1+\f a2(r-j))_b}.$$
\begin{proposition}\label{nn} For $\ll\in\Nl^r_+$ we have
\be{12}\|N_\ll\|_S^2=\f{(\lr-b)_\ll}{(\lr)_\ll}\P_{1\le i<j\le r}\f{(1+\f a2(j-i-1))_{\ll_i-\ll_j}}{(1+\f a2(j-i))_{\ll_i-\ll_j}}.\ee
\end{proposition}
\begin{proof} Using the reciprocity relation
\be{6}\f{(x+b)_m}{(x)_m}=\f{(x+m)_b}{(x)_b}\ee
for integers $0\le b\le m,$ the assertion follows from \cite{U1} or (for tube domains) \cite[Proposition XI.4.3]{FK}.
\end{proof}
For any partition $\ll$ let
\be{7}P_\ll:\PL(Z)\to\PL_\ll(Z)\ee
denote the orthogonal projection. If $f_u(z)=(z|u)$ is a linear functional associated with $u\in Z,$ we simply write $T_u:=T_{f_u}.$
Moreover, $u^\dl$ denotes the directional derivative. By \cite[Theorem 2.11]{U1} we have
\be{16}T_u^*q=\S_i P_{\ll-[i]}T_u^*q=\S_{i=1}^r\f1{\ll_i+\f a2(r-i)+b}P_{\ll-[i]}u^\dl q\ee
for all $q\in\PL_\ll(Z),$ where
$$[i]=(0,\ldots,0,1,0,\ldots,0)$$
with $1$ at position $i.$ More precisely, only those terms occur where $\ll-[i]$ is again a partition.
\begin{definition} Let $\SL$ denote the set of all sequences
$$c_m=c_0+\f{c_1}{m+1}+\oL_m,$$
where $c_0,c_1$ are constants and the sequence $\{m^2\oL_m\}_m$ is bounded. Let $\SL_+$ denote the set of sequences in $\SL$ with $c_0>0$.
\end{definition}
It is clear that $\SL$ is closed under taking finite sums and products of sequences. $\SL_+$ is also closed under taking quotients.
\begin{lemma}\label{qn} Let $\la,\lg\in\Nl^{r-1}_+.$ Then $\left\{\f{\|N_{m-k,\lg}\|_S^2}{\|N_{m,\la}\|_S^2}\right\}_m\in\SL_+.$
\end{lemma}
\begin{proof} In terms of the falling Pochhammer symbol $(m)_j^*=\P_{i=1}^j(m+1-i),$ \er{12} implies
$$\f{\|N_{m-k,\lg}\|_S^2}{\|N_{m,\la}\|_S^2}=C\P_{j=1}^{r-1}\f{(m-\la_j+\f a2 j)_{k+\lg_j-\la_j}^*}{(m-\la_j+\f a2(j-1))_{k+\lg_j-\la_j}^*}$$
whenever $k+\lg_j\ge\la_j.$ Since each factor belongs to $\SL_+,$ the assertion follows.
\end{proof}
\begin{lemma}\label{i} Let $\el\le r$ and $\ll\in\Nl^\el_+.$ Then
$$T_{N_\el^k}^*N_\ll=\P_{j=1}^\el\f{(\ll_j+\f a2(\el-j))_k^*}{(\ll_j+\f a2(r-j)+b)_k^*}N_{\ll-k_\el},\qquad\forall\,k\le\ll_\el.$$
\end{lemma}
\begin{proof} Consider the Peirce $2$-space $\t Z=Z^2_{e_1+\ldots+e_\el}$ of rank $\el$ and put $\t\lr=1+\f a2(\el-1).$ Using
$N_\ll=N_\el^k\,N_{\ll-k_\el}$ and applying \er{30} to $S\ic Z$ and $\t S\ic\t Z,$ we obtain for $\lf\in\PL(\t Z)$
$$(\lr)_\ll(\lf|T_{N_\el^k}^*N_\ll)_S=(\lr)_\ll(N_\el^k\lf|N_\ll)_S=(N_\el^k\lf|N_\ll)_Z=(N_\el^k\lf|N_\ll)_{\t Z}
=(\t\lr)_\ll(N_\el^k\lf|N_\ll)_{\t S}$$
$$=(\t\lr)_\ll(\lf|N_{\ll-k_\el})_{\t S}=\f{(\t\lr)_\ll}{(\t\lr)_{\ll-k_\el}}(\lf|N_{\ll-k_\el})_{\t Z}=\f{(\t\lr)_\ll}{(\t\lr)_{\ll-k_\el}}(\lf|N_{\ll-k_\el})_Z=\f{(\t\lr)_\ll(\lr)_{\ll-k_\el}}{(\t\lr)_{\ll-k_\el}}(\lf|N_{\ll-k_\el})_S.$$
Since $\lf$ is arbitrary, it follows that
$$T_{N_\el^k}^*N_\ll=\f{(\t\lr)_\ll(\lr)_{\ll-k_\el}}{(\t\lr)_{\ll-k_\el}(\lr)_\ll}N_{\ll-k_\el}.$$
We have
\be{31}\f{(\lr)_\ll}{(\lr)_{\ll-k_\el}}=\P_{j=1}^\el(\ll_j+\f a2(r-j)+b)_k^*.\ee
Applying \er{31} to $Z$ and $\t Z,$ the assertion follows.
\end{proof}
We will now consider partitions $(m,0,\ldots,0)=m$ of length $1,$ with projection $P_m:H^2(S)\to\PL_m(Z).$ Here $\PL_m(Z)$ is spanned by the $K$-orbit of the conical polynomial $N_1^m.$ As shown in \cite{U2}, the projection
\be{8}P:=\S_m P_m\ee
on $H^2(S)$ belongs to the Toeplitz $C^*$-algebra $\TL(S).$ For a partition $\ll$ choose an orthonormal basis $p_i\in\PL_\ll(Z),$ for the inner product \er{4}. Then
$$A^\ll:=\S_i T_{p_i}PT_{p_i}^*$$
is a $K$-invariant operator, independent of the choice of orthonormal basis. Since the decomposition \er{1} is multiplicity-free, every $K$-invariant operator $T$ on $\PL(Z)$ (or $H^2(S)$) is a 'diagonal' operator. Define
$$\ll':=(\ll_2,\ldots,\ll_r)\in\Nl^{r-1}_+.$$
\begin{lemma}\label{z} Let $p\in\PL_\ll(Z)$ such that $PT_p^*N_{m,\lb}\ne 0.$ Then $\lb\le\ll\le(m,\lb).$
\end{lemma}
\begin{proof} For $\lf\in\PL_{n,0}(Z)$ the non-zero components of $p\lf$ correspond to signatures $\lm$ obtained from $\ll$ by adding a horizontal $n$-strip \cite[Proposition 5.3]{S2}. Thus
$$\lm'\le\ll\le\lm.$$
It follows that $(\lf|T_p^*N_{m,\lb})_S=(p\,\lf|N_{m,\lb})_S$ is non-zero only if $\lm=(m,\lb)$ satisfies the above condition, which leads to $\lb\le\ll\le(m,\lb).$
\end{proof}
Since
$$\mbox{Ran}(T_{p_i}P)\ic\S_{\lm'\le\ll\le\lm}P_\lm$$
by Lemma \er{z}, it follows that
\be{32}A^\ll=\S_{\lm'\le\ll\le\lm}\f{(N_\lm|A^\ll N_\lm)_S}{\|N_\lm\|_S^2}\,P_\lm,\ee
where
\be{cn}\f{(N_\lm|A^\ll N_\lm)_S}{\|N_\lm\|_S^2}=\f1{\|N_\lm\|_S^2}(N_\lm|\S_i T_{p_i}PT_{p_i}^*\,N_\lm)_S=\f1{\|N_\lm\|_S^2}\S_i\|PT_{p_i}^*N_\lm\|_S^2.\ee
\begin{proposition}\label{j} Let $\ll\in\Nl^r_+.$ Then
$\left\{\f{(N_{m,\ll'}|A^\ll N_{m,\ll'})_S}{\|N_{m,\ll'}\|_S^2}\right\}_m\in\SL_+.$
\end{proposition}
\begin{proof} The proof is by induction on the length $\el\le r$ of $\ll.$ Put $k:=\ll_\el>\ll_{\el+1}=0.$ Then
$\lg:=\ll-k_\el$ has length $<\el.$ Consider the Peirce $2$-space $\t Z:=Z^2_{e_1+\ldots+e_\el}$ of rank $\el.$ We may assume that a subfamily $p_i:i\in\t I$ is an orthonormal basis of $\PL_\ll(\t Z).$ Since $\PL_\ll(\t Z)=N_\el^k\,\PL_\lg(\t Z),$ there exists a constant $c>0$ such that $p_i=c\cdot N_\el^k\,q_i$ for all $i\in\t I,$ where $q_i\in\PL_\lg(\t Z)$ is an orthonormal basis. For
$m\ge\ll_2$ it follows from Lemma \er{i} that $T_{N_\el^k}^*N_{m,\ll'}=c_m N_{m-k,\lg'},$ where
$$c_m=\f{(m+\f a2(\el-1))_k^*}{(m+b+\f a2(r-1))_k^*}\P_{j=2}^\el\f{(\ll_j+\f a2(\el-j))_k^*}{(\ll_j+b+\f a2(r-1))_k^*}$$
belongs to $\SL_+,$ in view of the identity
$$\f{m+a}{m+b}=1+\f{a-b}m-\f{(a-b)b}{m(m+b)}.$$
For $i\in I\sm\t I$ we have $T_{p_i}^*N_{m,\ll'}=0$ since $p_i$ belongs to the ideal generated by $\t Z^\perp.$ It follows that
$$(N_{m,\ll'}|A^\ll N_{m,\ll'})_S=\S_{i\in I}(T_{p_i}^*N_{m,\ll'}|PT_{p_i}^*N_{m,\ll'})_S
=\S_{i\in\t I}(T_{p_i}^*N_{m,\ll'}|PT_{p_i}^*N_{m,\ll'})_S$$
$$=c^2\S_{i\in\t I}(T_{q_i}^*T_{N_\el^k}^*N_{m,\ll'}|PT_{q_i}^*T_{N_\el^k}^*N_{m,\ll'})_S
=c^2\cdot c_m^2\S_{i\in\t I}(T_{q_i}^*N_{m-k,\lg'}|PT_{q_i}^*N_{m-k,\lg'})_S.$$
Now consider the $K$-invariant operator
$$A^\lg=\S_{j\in J}T_{q_j}PT_{q_j}^*,$$
where $q_j,\,j\in J$ is an orthonormal basis of $\PL_\lg(Z).$ We may assume that $q_i,i\in\t I$ are a subfamily of $J.$
As above, we have $T_{q_j}^*N_{m-k,\lg'}=0$ whenever $j\in J\sm\t I.$ Therefore
$$A^\lg N_{m-k,\lg'}=\S_{j\in J}T_{q_j}PT_{q_j}^*N_{m-k,\lg'}=\S_{i\in\t I}T_{q_i}PT_{q_i}^*N_{m-k,\lg'}$$
and hence $(N_{m,\ll'}|A^\ll N_{m,\ll'})_S=c^2\cdot c_m^2(N_{m-k,\lg'}|A^\lg N_{m-k,\lg'})_S.$ Since $\lg$ has length $<\el,$ the induction hypothesis implies that $\left\{\f{(N_{m-k,\lg'}|A^\lg N_{m-k,\lg'})_S}{\|N_{m-k,\lg'}\|_S^2}\right\}_m\in\SL_+.$ It follows that the sequence
$$\f{(N_{m,\ll'}|A^\ll N_{m,\ll'})_S}{\|N_{m,\ll'}\|_S^2}
=c^2\,c_m^2\,\f{(N_{m-k,\lg'}|A^\lg N_{m-k,\lg'})_S}{\|N_{m-k,\lg'}\|_S^2}\,\f{\|N_{m-k,\lg'}\|_S^2}{\|N_{m,\ll'}\|_S^2}$$
belongs to $\SL_+,$ since Lemma \er{qn} implies that $\left\{\f{\|N_{m-k,\lg'}\|_S^2}{\|N_{m,\ll'}\|_S^2}\right\}_m\in\SL_+.$
\end{proof}
\section{Hilbert submodule and sub-Toeplitz operators}
The Hilbert sum
$$H^2_1(S)=\S_m\PL_m(Z)=\mbox{Ran}(P)$$
will be called the {\bf sub-Hardy space}. For smooth symbols $f\in\CL^\oo(S)$ define the {\bf sub-Toeplitz operator}
$$S_f:=P\,f\,P=P\,T_f\,P$$
as a bounded operator on $H^2_1(S).$ Let $\AL$ be the $*$-algebra generated by $S_p$ for polynomial symbols $p\in\PL(Z)$. For $p,q\in\PL(Z)$ we have
$$S_pS_q=S_{pq}$$
since $PT_qP^\perp=0.$ Thus it often suffices to consider linear symbols $z\mapsto(z|u)$ for some $u\in Z.$ We denote by $S_u$ the corresponding operators.
\begin{theorem}\label{r} For $\lm\in\Nl^r_+$ let $p\in\PL(Z)$ satisfy $\deg p\le|\lm'|.$ Then
$$P T_p^*T_q P=S_{T_p^*q}\qquad\forall\,q\in\PL_\lm(Z).$$
\end{theorem}
The proof is based on the following Lemma.
\begin{lemma}\label{u} Let $\lm$ be a partition and $q\in\PL_\lm(Z).$ Then we have for $u\in Z$ and each $h\in\PL_{n,0}(Z)$,
$$P_{\lm+n[1]-[i]}u^\dl P_{\lm+n[1]}h q=P_{\lm+n[1]-[i]}h P_{\lm-[i]}u^\dl q\qquad\forall\,i>1.$$
\end{lemma}
\begin{proof} Write $h q=\S_\ll P_\ll h q.$ The partitions $\ll$ occurring here satisfy $\ll\ge\lm$ and hence $\ll'\ge\lm'.$ For such $\ll$ we have
$$u^\dl P_\ll h q=\S_j P_{\ll-[j]}u^\dl P_\ll h q.$$
Now assume $\ll-[j]=\lm+n[1]-[i].$ If $j=1$ than $\ll'=\lm'-[i]\ngeq\lm'.$ Hence $j>1.$ If $j\ne i$ then $\ll'=\lm'-[i]+[j]\ngeq\lm'.$ Hence $i=j$ and therefore $\ll=\lm+n[1].$ This argument shows
\be{13}P_{\lm+n[1]-[i]}u^\dl(h q)=\S_\ll P_{\lm+n[1]-[i]}u^\dl P_\ll h q=P_{\lm+n[1]-[i]}u^\dl P_{\lm+n[1]}h q.\ee
Since $u^\dl(h q)=q(u^\dl h)+h(u^\dl q)$ and $q(u^\dl h)$ has only components $\ll\ge\lm$ which satisfy $\ll'\ge\lm'$ it follows that
\be{14}P_{\lm+n[1]-[i]}u^\dl(h q)=P_{\lm+n[1]-[i]}h(u^\dl q).\ee
We next show that
\be{15}P_{\lm+n[1]-[i]}h(u^\dl q)=\S_j P_{\lm+n[1]-[i]}h P_{\lm-[j]}u^\dl q=P_{\lm+n[1]-[i]}h P_{\lm-[i]}u^\dl q.\ee
In fact, since $h P_{\lm-[1]}u^\dl q$ cannot have a component $\ll$ with $\ll_1=\lm_1+n$ we may assume $j>1.$ If $j\ne i,$ then the components $\ll\ge\lm-[j]$ occurring in $h P_{\lm-[j]}u^\dl q$ satisfy $\ll'\ge\lm'-[j]$ which implies
$\ll'\ne\lm'-[i].$ Thus \er{15} holds. Combining equations \er{13},\er{14} and \er{15}, the assertion follows.
\end{proof}
{\bf Proof of Theorem \er{r}.} We may assume that $p(z)=(z|u_1)\cdots(z|u_k).$ Let $\ll=(\ll_1,\ll')$ be a partition such that $|\ll'|\ge k.$ Putting $[i_1,\ldots,i_k]=[i_1]+\ldots+[i_k]$ it follows from \er{16} that
$$T_{u_k}^*\ldots T_{u_1}^*\lq=\S_{i_1,\ldots,i_k}P_{\ll-[i_1,\ldots,i_k]}T_{u_k}^*\ldots P_{\ll-[i_1]}T_{u_1}^*\lq$$
for all $\lq\in\PL_\ll(Z).$ If any $i_j=1$ then $(\ll-[i_1,\ldots,i_k])'\ne 0.$ Therefore $(\ll-[i_1,\ldots,i_k])'=0$ implies that all $i_j>1.$ It follows that
\be{17}PT_{u_k}^*\ldots T_{u_1}^*\lq=P\S_{i_1>1,\ldots,i_k>1}P_{\ll-[i_1,\ldots,i_k]}T_{u_k}^*\ldots P_{\ll-[i_1]}T_{u_1}^*\lq.\ee
Moreover, if $|\ll'|>k$ we have $P T_{u_k}^*\ldots T_{u_1}^*\lq=0.$ The same argument shows
\be{18}P u_k^\dl\ldots u_1^\dl\lq=P\S_{i_1>1,\ldots,i_k>1}P_{\ll-[i_1,\ldots,i_k]}u_k^\dl\ldots P_{\ll-[i_1]}u_1^\dl\lq\ee
and $|\ll'|>k$ implies $P u_k^\dl\ldots u_1^\dl\lq=0.$ By Lemma \er{u} we have for $h\in\PL_{n,0}(Z)$
$$P_{ \lm+n[1]-[i_1]}u_1^\dl P_{\lm+n[1]}h q=P_{\lm+n[1]-[i_1]}h P_{\lm-[i_1]}u_1^\dl q.$$
Applying Lemma \er{u} to $P_{\lm-[i_1]}u_1^\dl q$, we obtain
$$P_{\lm+n[1]-[i_1,i_2]}u_2^\dl P_{\lm+n[1]-[i_1]}u_1^\dl P_{\lm+n[1]}h q=P_{\lm+n[1]-[i_1,i_2]}u_2^\dl
P_{\lm+n[1]-[i_1]}h P_{\lm-[i_1]}u_1^\dl q$$
$$=P_{\lm+n[1]-[i_1,i_2]}h P_{\lm-[i_1,i_2]}u_2^\dl P_{\lm-[i_1]}u_1^\dl q.$$
More generally,
\be{19}P_{\lm+n[1]-[i_1,\ldots,i_k]}u_k^\dl\ldots P_{\lm+n[1]-[i_1]}u_1^\dl P_{\lm+n[1]}h q
= P_{\lm+n[1]-[i_1,\ldots,i_k]}h P_{\lm-[i_1,\ldots,i_k]}u_k^\dl\ldots P_{\lm-[i_1]}u_1^\dl q.\ee
Consider
$$P h(T_{u_k}^*\ldots T_{u_1}^*q)=Ph\S_{i_1,\ldots,i_k}P_{\lm-[i_1,\ldots,i_k]}T_{u_k}^*\ldots P_{\lm-[i_1]}T_{u_1}^*q
=P\S_{i_1,\ldots,i_k}\S_\ll P_\ll h P_{\lm-[i_1,\ldots,i_k]}T_{u_k}^*\ldots P_{\lm-[i_1]}T_{u_1}^*q.$$
Note the components $\ll=(m,0)$ occurring here satisfy $\ll'=0\ge(\lm-[i_1,\ldots,i_k])'.$ Since $|\lm'|\ge k$ this implies that all
$i_j>1.$ Moreover, $m=|\ll|=n+|\lm-[i_1,\ldots,i_k]|=n+\lm_1+|\lm'-[i_1,\ldots,i_k]|=n+\lm_1.$ Therefore $\ll=(n+\lm_1,0)$ and hence
\be{7}P h(T_{u_k}^*\ldots T_{u_1}^*q)=P\S_{i_1>1,\ldots,i_k>1}P_{\lm+n[1]-[i_1,\ldots,i_k]}h P_{\lm-[i_1,\ldots,i_k]}
T_{u_k}^*\ldots P_{\lm-[i_1]}T_{u_1}^*q.\ee
We have $h q=\S_\ll P_\ll h q,$ where $\ll\ge\lm$ and the skew-partition $\ll-\lm$ is a horizontal $n$-strip. Since $\ll'\ge\lm'$ satisfies $|\ll'|\ge|\lm'|\ge k$ the condition $(\ll-[i_1,\ldots,i_k])'=0$ implies that all $i_j>1$ and in addition all terms with
$|\ll'|>k$ vanish. Assuming $|\ll'|=k$ it follows that $\ll'=\lm'$ and hence $\ll=\lm+n[1].$ This shows
$$P T_{u_k}^*\ldots T_{u_1}^*(h q)=P\S_\ll T_{u_k}^*\ldots T_{u_1}^* P_\ll h q=P T_{u_k}^*\ldots T_{u_1}^* P_{\lm+n[1]}h q.$$
With \er{17},\er{18}, \er{19} and \er{7}, we obtain
\begin{eqnarray*}P T_{u_k}^*\ldots T_{u_1}^*(h q)&=&P\S_{i_1>1,\ldots,i_k>1}P_{\lm+n[1]-[i_1,\ldots,i_k]}T_{u_k}^*\ldots P_{\lm+n[1]-[i_1]}T_{u_1}^*P_{\lm+n[1]}h q\\
&=&P\S_{i_1>1,\ldots,i_k>1}\f{P_{\lm+n[1]-[i_1,\ldots,i_k]}u_k^\dl\ldots P_{\lm+n[1]-[i_1]}u_1^\dl
P_{\lm+n[1]}h q}{\((\lm-[i_1,\ldots,i_{k-1}])_{i_k}+\f a2(r-i_k)+b\)\ldots\(\lm_{i_1}+\f a2(r-i_1)+b\)}\\
&=&P\S_{i_1>1,\ldots,i_k>1}\f{P_{\lm+n[1]-[i_1,\ldots,i_k]}h P_{\lm-[i_1,\ldots,i_k]}u_k^\dl\ldots
P_{\lm-[i_1]}u_1^\dl q}{\((\lm-[i_1,\ldots,i_{k-1}])_{i_k}+\f a2(r-i_k)+b\)\ldots\(\lm_{i_1}+\f a2(r-i_1)+b\)}\\
&=&P\S_{i_1>1,\ldots,i_k>1}P_{\lm+n[1]-[i_1,\ldots,i_k]}h P_{\lm-[i_1,\ldots,i_k]}T_{u_k}^*\ldots
P_{\lm-[i_1]}T_{u_1}^*q=P h(T_{u_k}^*\ldots T_{u_1}^*q)\end{eqnarray*}
It follows that $PT_p^*(hq)=Ph(T_p^*q).$ Since $h\in\PL_{n,0}(Z)$ is arbitrary, $PT_p^*T_qP=PT_{T_p^*q}P=S_{T_p^*q}.$ $\quad\quad \Box$\vskip 3mm
Applying Theorem \er{r} we obtain
\begin{corollary}\label{v} If $\deg p,\deg q\le|\ll'|,$ then $PT_p^*A^\ll T_qP\in\AL.$
\end{corollary}
For $\lb\in\Nl^{r-1}_+,$ consider the projections
$$P^\lb:=\S_{m\ge\lb_1}P_{m,\lb}.$$
Then $P^0=P.$
\begin{definition} Define a diagonal operator $\lL$ on $\PL(Z)$ by
\be{9}\lL p_\ll:=\ll_1\,p_\ll,\qquad\forall\,p_\ll\in\PL_\ll.\ee
\end{definition}
Let $Q_j:=\oplus_{\ll_1=j}\PL_\ll(Z)$ be the eigenvector subspace (and denote the corresponding projection by the same notation) for $\lL$ with eigenvalue $j.$ Then we have the orthogonal decomposition $H^2(S)=\oplus_{j\in\Nl}Q_j$. We call an operator $T$ of {\bf finite propagation} if there exists a positive number $l$ such that
$$T Q_j\ic\bigoplus_{|i-j|\le l}Q_i.$$
\begin{lemma}\label{fp} Suppose the operator $T$ has the finite propagation property. If $T\lL^2$ is bounded, then $\lL^2 T$ and $T^*\lL^2$ are also bounded.
\end{lemma}
\begin{proof} By assumption, we have that $T=\oplus_{-l\le i\le l}T_i$ for some number $l$, where
$$T_i=\oplus_j Q_{j+i}TQ_j$$ is an operator of degree $i$. By grading, one sees that each $\lL^2 T_i$ is bounded iff there exists a constant $C_i$ such that $\|T_ip\|\le C_i\f{\|p\|}{j^2}$ for any index $j$ and $p\in Q_j$. Indeed, if such $C_i$ exists, then for any $p=\oplus_j p_j$,
$$\|\lL^2 T_ip\|^2=\S_j\|(i+j) T_ip_j\|^2\le C^2_i\S_j\f{(i+j)^2\|p_j\|^2}{j^2}\leq C^2_i(1+l)^2\|p\|^2.$$
Using the fact that $T\lL^2$ is bounded, for each $j$ and $p\in Q_j$, we have
$$\|T\lL^2p\|^2=j^2\|\S_{-l\le i\le l}T_ip\|^2=j^2\S_{-l\le i\le l}\|T_ip\|^2\ge j^2\|T_ip\|^2$$
for each $i$. It follows that each $\lL^2 T_i$ is bounded. Therefore $\lL^2 T=\S_{-l\le i\le l}\lL^2 T_i$ is bounded. This implies that $T^*\lL^2$ is bounded.
\end{proof}
Let $\CL$ denote the $*$-algebra generated by $T_p$ with polynomial symbol $p,$ and $\f1{\lL+t}$ together with all projections $P^\lb$, where $\lb\in\Nl^{r-1}_+$ is arbitrary. Define
$$\BL:=\{B\in\CL: B\lL^2\,\,bounded\},$$
$$\BL_\lL:=\AL(\lL+1)^{-1}+\BL=\{A(\lL+1)^{-1}+B:\,A\in\AL,B\in\BL\}.$$
It is easy to check that operators in $\CL$ have the finite propagation property. Therefore Lemma \er{fp} implies that $\BL$ and $\BL_\lL$ are invariant under taking adjoints.
\begin{lemma} $\BL$ is an ideal in $\CL.$ Moreover,
$$[\CL,(\lL+t)^{-1}]\ic\BL.$$
\end{lemma}
\begin{proof} For the first assertion it suffices to show that $BT_u\in\BL$ whenever $B\in\BL.$ Define a bounded operator $R_u$ by $R_up=P_{m+1,\lb}T_up$ for $p\in\PL_{m,\lb}(Z).$ Then
$$(\lL^2T_u-T_u\lL^2)p=\S_{i=1}^r(\lL^2-m^2)P_{(m,\lb)+[i]}T_up=((m+1)^2-m^2)P_{m+1,\lb}T_u p=(2\lL-1)R_up.$$
Therefore $BT_u\lL^2=B\lL^2T_u-B(2\lL-1)R_u$ is bounded. Thus $BT_u\in\BL.$ For the second assertion it suffices to show that $[T_u,(\lL+t)^{-1}]\in\BL.$ With the previous notation, we have
\begin{eqnarray*}[T_u,(\lL+t)^{-1}]\lL^2 p&=&m^2(T_u(\lL+t)^{-1}-(\lL+t)^{-1}T_u)p=m^2\S_{i=1}^r\(\f1{m+t}-\f1{\lL+t}\)P_{(m,\lb)+[i]}T_up\\
&=&m^2\(\f1{m+t}-\f1{m+1+t}\)P_{m+1,\lb}T_up=R_u\f{\lL^2}{(\lL+t)(\lL+1+t)} p.\end{eqnarray*}
Therefore $[T_u,(\lL+t)^{-1}]\lL^2$ is bounded.
\end{proof}
\begin{lemma}\label{ba} $\BL_\lL$ is a (non-unital) $*$-algebra and an $\AL$-bimodule, i.e.,
$$\AL\BL_\lL+\BL_\lL\AL\ic\BL_\lL$$.
\end{lemma}
\begin{proof} We only show that $\BL_\lL\AL\ic\BL_\lL$. Indeed, for $A\in\AL, B\in\BL$, and $u\in Z$, we have
$$(A(\lL+1)^{-1}+B)S_u-AS_u(\lL+1)^{-1}=BS_u+A[(\lL+1)^{-1},S_u]=BS_u-AS_u(\lL+1)^{-1}(\lL+2)^{-1}\in\BL.$$
Since $AS_u\in\AL,$ it follows that $A(\lL+1)^{-1}+B\in\BL_\lL.$
\end{proof}
\begin{proposition}\label{q} For $\ll\in\Nl^r_+$ let $p,q\in\PL(Z)$ satisfy $\deg(p),\deg(q)\le|\ll'|.$ Then
$$PT_p^*P^{\ll'}T_q P\in\AL+\BL_\lL.$$
\end{proposition}
\begin{proof} The $K$-invariant operator $P^{\ll'}A^\ll P^{\ll'}$ is diagonal, and Proposition \er{j} implies that
$$P^{\ll'}A^\ll P^{\ll'}=\f{c_\ll\lL+\t c_\ll}{\lL+1}P^{\ll'}+B,$$
where $B\in\BL$ and $c_\ll>0.$ It follows that
$$P^{\ll'}=P^{\ll'}A^\ll P^{\ll'}\f{\lL+1}{c_\ll\lL+\t c_\ll}+B'$$
with $B'\in\BL.$ If $\lb\in\Nl^{r-1}_+$ satisfies $|\lb|\le|\ll'|$ and $A^\ll P^\lb$ is non-zero, then $\lb\ge\ll'$ by Lemma \er{z}. This is only possible if $\lb=\ll'.$ Therefore
$$PT_p^*A^\ll T_qP=\S_{|\lb|\le|\ll'|}PT_p^*P^\lb A^\ll P^\lb T_qP=PT_p^*P^{\ll'}A^\ll P^{\ll'}T_qP.$$
Since $\BL$ is an ideal in $\CL$ and $\[\f{\lL+1}{c_\ll\lL+\t c_\ll},T_qP\]\in\BL$ we obtain
\begin{eqnarray*}PT_p^*P^{\ll'}T_qP&=&PT_p^*P^{\ll'}A^\ll P^{\ll'}\f{\lL+1}{c_\ll\lL+\t c_\ll}T_qP+PT_p^*B'T_qP\\
&=&PT_p^*P^{\ll'}A^\ll P^{\ll'}T_qP\f{\lL+1}{c_\ll\lL+\t c_\ll}+PT_p^*P^{\ll'}A^\ll P^{\ll'}\[\f{\lL+1}{c_\ll\lL+\t c_\ll},T_qP\]
+PT_p^*B'T_qP\\
&=&PT_p^*A^\ll T_qP\f{\lL+1}{c_\ll\lL+\t c_\ll}+B'',\end{eqnarray*}
where $B''\in\BL.$ Since $PT_p^*A^\ll T_qP\in\AL$ by Corollary \er{v}, the assertion follows.
\end{proof}
\begin{proposition}\label{aab} $[\AL,\AL]\ic\BL_\lL.$
\end{proposition}
\begin{proof} In view of Lemma \er{ba} it suffices to show that $[S_u^*,S_v]\in\BL_\lL.$ We may suppose that $Z$ has rank $r>1.$ By definition,
$S_v=PT_vP=T_vP-P^1T_vP.$ Note $S_v\PL_{m,0}(Z)\ic\PL_{m+1,0}(Z)$ and $\f a2(r-1)+b=\lr-1.$ Applying \er{16} it follows that
\begin{eqnarray*}(m+\lr)S_u^*S_vP_m=(m+\lr)T_u^*S_vP_m&=&u^\dl(S_vP_m)
=u^\dl(T_vP_m-P^1 T_vP_m)\\&=&(u|v)P_m+T_v u^\dl P_m-u^\dl P^1 T_vP_m\\
&=&(u|v)P_m+(m+\lr-1)T_v S_u^*P_m-u^\dl P^1 T_vP_m\\&=&(u|v)P_m+(m+\lr-1)S_vS_u^*P_m-Pu^\dl P^1 T_v P_m.\end{eqnarray*}
Thus $S_u^*S_v(\lL+\lr)=(u|v)P+S_vS_u^*(\lL+\lr-1)-Pu^\dl P^1 T_v P$ and hence
$$[S_u^*,S_v](\lL+\lr)=(u|v)P+S_vS_u^*((\lL+\lr-1)-(\lL+\lr))-Pu^\dl P^1 T_vP=(u|v)P-S_vS_u^*-Pu^\dl P^1 T_vP.$$
By \er{16} we have
$$Pu^\dl P^1T_vP=\S_m Pu^\dl P^1T_vP_m=\S_m P_mu^\dl P_{m,1}T_vP_m=(1+\f a2(r-2)+b)\S_m P_mT_u^*P_{m,1}T_vP_m$$
$$=(1+\f a2(r-2)+b)\S_m PT_u^*P^1T_vP_m=(1+\f a2(r-2)+b)PT_u^*P^1T_vP.$$
Thus Proposition \er{q} implies that $Pu^\dl P^1T_vP\in\AL+\BL_\lL,$ and the assertion follows.
\end{proof}
\begin{lemma}\label{y} $\AL\ic\left\{\S_i S_{p_i}S_{q_i}^*+B:\,p_i,q_i\in\PL(Z),\,B\in\BL_\lL\right\}.$
\end{lemma}
\begin{proof} Since the latter set contains $S_u,S_v^*,$ it suffices to show that it is invariant under multiplication by $S_u,S_v^*$. By Proposition \er{aab} we have $[S_u^*,S_p]\in\BL_\lL$ and $[S_q^*,S_v]\in\BL_\lL.$ With Lemma \er{ba}, the assertion follows.
\end{proof}
The following technical lemma will be used in the next section.
\begin{lemma}\label{ap} Let $T\in\AL+\BL_\lL.$ Then $\left\{\f{(N_1^m|TN_1^m)_S}{\|N_1^m\|_S^2}\right\}_m\in\SL.$
\end{lemma}
\begin{proof} By Lemma \er{i}, we have
$$S_{N_1^k}^*N_1^m=\f{(m+1)_k^*}{(m+\lr)_k^*}N_1^{m-k}$$
for $0\le k\le m,$ $S_{N_1^k}^*N_1^m=0$ for $k>m$ and $S_v^*N_1^m=0$ for all $v\in Z_1^\perp.$ Thus for any $p,q\in\PL(Z)$ there exist constants $c_k(p,q),$ for $0\le k\le M(p,q):=min(\deg p,\deg q),$ such that
$$(S_p^*N_1^m|S_q^*N_1^m)_S=\S_{k=0}^{M(p,q)}c_k(p,q)\|S_{N_1^k}^*N_1^m\|_S^2=\S_{k=0}^{M(p,q)}c_k(p,q)\|\f{(m+1)_k^*}{(m+\lr)_k^*}N_1^{m-k}\|_S^2$$
for all $m\ge M(p,q).$ Since $T\in\AL+\BL_\lL$, Lemma \er{y} implies that $$T=\S_i S_{p_i}S_{q_i}^*+B_0 $$
for some polynomials
$p_i,q_i$ and $B_0=A_1(\lL+1)^{-1}+B_1\in\BL_\lL$ with $A_1\in\AL, B_1\in\BL $. Using Lemma \er{y} for $A_1$ again, there exist polynomials
$\lf_j, \lq_j$ and $ B_2\in\BL_\lL$ such that
$$T=\S_i S_{p_i}S_{q_i}^*+\(\S_j S_{\lf_j}S_{\lq_j}^*+B_2\)(\lL+1)^{-1}+B_1
=\S_i S_{p_i}S_{q_i}^*+\S_j S_{\lf_j}S_{\lq_j}^*(\lL+1)^{-1}+B,$$
where $B\in\BL.$ It follows that
$$(N_1^m|TN_1^m)_S-(N_1^m|BN_1^m)_S=\S_i(S_{p_i}^*N_1^m|S_{q_i}^*N_1^m)_S+\f1{m+1}\S_j(S_{\lf_j}^*N_1^m|S_{\lq_j}^*N_1^m)_S$$
$$=\S_i\S_{k=0}^{M(p_i,q_i)}c_k(p_i,q_i)\,\|\f{(m+1)_k^*}{(m+\lr)_k^*}N_1^{m-k}\|_S^2+\f1{m+1}\S_j\S_{k=0}^{M(\lf_j,\lq_j)}c_k(\lf_j,\lq_j)\,\|\f{(m+1)_k^*}{(m+\lr)_k^*}N_1^{m-k}\|_S^2.$$
Since
$$BN_1^m=(B(\lL+1)^2)(\lL+1)^{-2}N_1^m=\f{B(\lL+1)^2 N_1^m}{(m+1)^2}$$
the sequence $\{m^2\f{(N_1^m|BN_1^m)_S}{\|N_1^m\|_S^2}\}$ is bounded. Thus there exist finitely many sequences $\{c_k(m)\}\in\SL$ such that
$$\f{(N_1^m|TN_1^m)_S}{\|N_1^m\|_S^2}=\S_{k} c_k(m)\f{\|N_1^{m-k}\|_S^2}{\|N_1^m\|_S^2}.$$
This yields the desired result since $\left\{\f{\|N_1^{m-k}\|_S^2}{\|N_1^m\|_S^2}\right\}_m\in\SL_+$ by Lemma \er{qn}.
\end{proof}
\section{First main theorem}
\begin{theorem}\label{mr} Let $f\in\PL(Z\xx\o Z)$ be a real-analytic polynomial. Then $S_f\in\AL+\BL_\lL.$
\end{theorem}
The proof is based on a lengthy induction argument. We may assume that $f=\o p\,q$ for some $p,q\in\PL(Z).$
Let $\AL_{i,j}$ denote the set of all operators $PT_p^*T_qP,$ where $\deg p\le i,\,\deg q\le j.$ For a given $k$ we consider the following assumption
\be{20}\AL_{i,j}\ic\AL+\BL_\lL\mbox{ whenever }min(i,j)<k.\ee
We now proceed via a sequence of 'claims' which are proved under this assumption.
\begin{claim}\label{p} The assumption \er{20} implies that for each partition $\ll$ with $|\ll|<k$ there exist constants $a^\ll_\lb,b^\ll_\lb$ such that
\be{23}A^\ll-\S_{\lb\le\ll}\f{a^\ll_\lb\lL+b^\ll_\lb}{\lL+1}P^\lb\in\BL.\ee
\end{claim}
\begin{proof} For $\lb\le\ll\le(m,\lb)$ we have $N_{m,\lb}=N_{\lb_1,\lb}N_1^{m-\lb_1}$ and hence
$$\f{(N_{m,\lb}|A^\ll N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}=\S_i\f{\|PT_{p_i}^*N_{m,\lb}\|_S^2}{\|N_{m,\lb}\|_S^2}=\S_i\f{\|PT_{p_i}^*T_{N_{\lb_1,\lb}}N_1^{m-\lb_1}\|_S^2}{\|N_1^{m-\lb_1}\|_S^2}\f{\|N_1^{m-\lb_1}\|_S^2}{\|N_{m,\lb}\|_S^2}.$$
Since $\deg(p_i)=|\ll|<k,$ \er{20} implies $PT_{p_i}^*T_{N_{\lb_1,\lb}}P\in\AL+\BL_\lL.$ By Lemma \er{ap} and Lemma \er{qn}, we have that
$\left\{\f{(N_{m,\lb}|A^\ll N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}\right\}_m\in\SL$. By \er{32} there is a sequence $\oL_m,$ with $m^2\,\oL_m$ bounded, such that
$$A^\ll=\S_{\lb\le\ll\le(m,\lb)}\f{(N_{m,\lb}|A^\ll N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}\,P_{m,\lb}
=\S_{\lb\le\ll\le(m,\lb)}\(\f{a^\ll_\lb m+b^\ll_\lb}{m+1}+\oL_{m}\)P_{m,\lb}=\S_{\lb\le\ll}\f{a^\ll_\lb\lL+b^\ll_\lb}{\lL+1}P^\lb+B,$$
where we set $a^\ll_\lb=b^\ll_\lb=0$ if $\lb\ngeq\ll'.$ Thus $B-\S_{\lb\le\ll\le(m,\lb)}\oL_m\,P_{m,\lb}$ has finite rank and hence $B\in\BL$.
\end{proof}
\begin{claim}\label{m} Under the assumption \er{20} there exist constants $c^\lb_\la,d^\lb_\la$ such that
\be{24}P^\lb-\S_{\la\le\lb}\f{c^\lb_\la\lL+d^\lb_\la}{\lL+1}A^\la\in\BL,\qquad\forall\,\lb\in\Nl_+^{r-1},\,|\lb|<k.\ee
\end{claim}
\begin{proof} We use induction on $|\lb|.$ The case $\lb=0$ is trivial. Assume
\er{24} holds for all $\lb$ with $|\lb|<j<k$. Let $\lb$ satisfy $|\lb|=j.$ Then Claim \er{p} implies
\be{251}A^\lb=\f{a^\lb_\lb\lL+b^\lb_\lb}{\lL+1}P^\lb+\S_{\la<\lb}\f{a^\lb_\la\lL+b^\lb_\la}{\lL+1}P^\la+B^\lb,\ee
where $B^\lb\in\BL$, and $\la<\lb$ means that $\la\leq\lb$ and $\la\neq \lb$. Now consider the diagonal operator
$$\S_{|\ll|=j}A^\ll=\S_{\lm\in\Nl^r_+}a_\lm P_\lm.$$
If $|\ll|=j,$ then $(N_{m,\lb}|A^\ll N_{m,\lb})_S$ is non-zero only if $\ll=\lb,$ since $\lb\le\ll$ and $|\lb|=j=|\ll|.$ Therefore
$$a_{m,\lb}\,N_{m,\lb}=\S_{|\ll|=j}A^\ll N_{m,\lb}=\S_{|\ll|=j}\f{(N_{m,\lb}|A^\ll N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}\,N_{m,\lb}
=\f{(N_{m,\lb}|A^\lb N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}\,N_{m,\lb}.$$
By \cite[Theorem 1.6]{U2}, there exists a constant $c>0$ such that $a_\lm\ge c$ whenever $\lm_2+\cdots+\lm_r=j.$ For
$\lm=(m,\lb)$ this implies $\f{(N_{m,\lb}|A^\lb N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}=a_{m,\lb}\ge c$ and hence $a^\lb_\lb
=\lim_m \f{(N_{m,\lb}|A^\lb N_{m,\lb})_S}{\|N_{m,\lb}\|_S^2}\ge c>0.$
For any $|\la|<|\lb|=j,$ the induction hypothesis implies
$$P^\la=\S_{\lg\le\la}\f{c^\la_\lg\lL+d^\la_\lg}{\lL+1}A^\lg+B^\la,$$
where $B^\la\in\BL.$ Plugging into \er{251} we obtain
$$P^\lb=\f{\lL+1}{a_\lb^\lb\lL+b_\lb^\lb}\[A^\lb-\S_{\la<\lb}\f{a_\la^\lb\lL+b_\la^\lb}{a_\lb^\lb\lL+b_\lb^\lb}\(\S_{\lg\le\la}
\f{c_\lg^\la\lL+d_\lg^\la}{\lL+1}A^\lg+B^\la\)-B^\lb\].$$
It is easy to see that this expression has the desired form.
\end{proof}
\begin{claim}\label{t} The assumption \er{20} implies $\AL_{k,k}\ic\AL+\BL_\lL.$
\end{claim}
\begin{proof} Let $\deg p=\deg q=k.$ Then
$$PT_p^*T_qP=\S_{|\lb|\le k}PT_p^*P^\lb T_qP.$$
If $|\lb|=k$ then $PT_p^*P^\lb T_qP\in\AL+\BL_\lL$ by Proposition \er{q}. If $|\lb|=h<k$ and $\la\le\lb$ then \er{20} implies
$PT_p^*A^\la T_qP\in\AL_{k,h}\AL_{h,k}\ic\AL+\BL_\lL.$ It follows that
$$PT_p^*A^\la\f{c^\lb_\la\lL+d^\lb_\la}{\lL+1}T_qP=PT_p^*A^\la T_qP\f{c^\lb_\la\lL+d^\lb_\la}{\lL+1}
+PT_p^*A^\la\[\f{c^\lb_\la\lL+d^\lb_\la}{\lL+1},T_qP\]\in\AL+\BL_\lL,$$
since $\[\f{c^\lb_\la\lL+d^\lb_\la}{\lL+1},\CL\]\ic\BL$ and $\BL$ is an ideal in $\CL$. Therefore Claim \er{m} implies $PT_p^*P^\lb T_qP\in\AL+\BL_\lL.$
\end{proof}
\begin{claim}\label{w} Under the assumption \er{20}, for $T\in\CL$ and $q\in\PL(Z)$ of degree $i<k$ there exists $B\in\BL$ such that
$$PT[T_u^*,T_v]T_qP=B+\S_{|\lb|\le i}\S_{\la\le\lb}\S_{\lg\le\lb}PT A^\la[T_u^*,T_v]A^\lg T_qP
\f{c^\lb_\la c^\lb_\lg\lL+c_\la^\lb(d^\lb_\lg-c_\lg^\lb)+(d^\lb_\la-c_\la^\lb)c_\lg^\lb}{\lL+1}.$$
\end{claim}
\begin{proof} Since $\mbox{Ran}(T_qP)\ic\S_{|\lb|\le i}P^\lb$ and $[T_u^*,T_v]$ is a 'block-diagonal' operator \cite[Lemma 2.1]{U2} which commutes with each $P^\lb,$ it suffices to consider $PTP^\lb[T_u^*,T_v]P^\lb T_qP$ for $\lb\in\Nl^{r-1}_+$ satisfying $|\lb|\le i.$ By Claim \er{m} we have
$$PTP^\lb[T_u^*,T_v]P^\lb T_qP=PT\(B_1+\S_{\la\le\lb}\f{c^\lb_\la\lL+d^\lb_\la}{\lL+1}A^\la\)[T_u^*,T_v]
\(B_2+\S_{\lg\le\lb}\f{c^\lb_\lg\lL+d^\lb_\lg}{\lL+1}A^\lg\)T_qP,$$
where $B_1,B_2\in\BL.$ Since $\BL\ic\CL$ is an ideal and $\CL$ contains $PT,\,[T_u^*,T_v],\,A^\la,\,A^\lg,\,T_qP,$ the assertion follows.
\end{proof}
\begin{claim}\label{s} The assumption \er{20} implies
\be{26}PT_\lf^*[T_u^*,T_v]T_\lq P\in\AL+\BL_\lL\ee
whenever $\deg\lf,\deg\lq<k.$
\end{claim}
\begin{proof} We prove \er{26} by induction on $h=\max(\deg\lf,\deg\lq)<k$. For $h=0$, we have
$$P[T_u^*,T_v]P=PT_u^*T_vP-PT_vPT_u^*P,$$
where $PT_u^*T_vP\in\AL_{1,1}\ic\AL+\BL_\lL$ by Claim \er{t}, and $PT_vPT_u^*P\in\AL.$
For the induction step, let $\lf,\lq$ be polynomials with $\deg\lf\le h=\deg\lq<k,$ and we may assume that \er{26} holds in the case of the maximal degree less than $h.$ Then
$$PT_\lf^*[T_u^*,T_v]T_\lq P=PT_\lf^*\([T_u^*,T_{v\lq}]-T_v[T_u^*,T_\lq]\)P
=PT_{\lf u}^*T_{v\lq}P-PT_\lf^*T_{v\lq}PT_u^*P-PT_\lf^*T_v[T_u^*,T_\lq]P.$$
By the assumption \er{20}, we have $PT_\lf^*T_{v\lq}P\in\AL_{h,h+1}\ic\AL+\BL_\lL,$ and using Claim \er{t} also $PT_{\lf u}^*T_{v\lq}P\in\AL_{h+1,h+1}\ic\AL+\BL_\lL.$ For the third term we may assume that $\lq=v_{h-1}\,\cdots v_0$ for some linear functions $v_i.$ Then
$$PT_\lf^*T_v[T_u^*,T_\lq]P=\S_{i=0}^{h-1}PT_\lf^*T_{v\cdot v_{h-1}\cdots v_{i+1}}[T_u^*,T_{v_i}]T_{v_{i-1}\cdots v_0}P.$$
If $p,q,\lx,\lh$ are polynomials of degree $\le i<h$ we have
$$PT_\lf^*T_{v\cdot v_{h-1}\cdots v_{i+1}}T_pPT_q^*[T_u^*,T_{v_i}]T_\lx PT_\lh^*T_{v_{i-1}\cdots v_0}P\in\AL+\BL_\lL,$$
since \er{20} implies that $\AL+\BL_\lL$ contains $PT_\lf^*T_{v\cdot v_{h-1}\cdots v_{i+1}}T_pP\in\AL_{h,h}$ and
$PT_\lh^*T_{v_{i-1}\cdots v_0}P\in\AL_{i,i},$ and the induction hypothesis implies $PT_q^*[T_u^*,T_{v_i}]T_\lx P\in\AL+\BL_\lL.$ Thus
$$PT_\lf^*T_{v\cdot v_{h-1}\cdots v_{i+1}}A^\la[T_u^*,T_{v_i}]A^\lr T_{v_{i-1}\cdots v_0}P\ic\AL+\BL_\lL,$$
whenever $|\la|\le i$ and $|\lr|\le i.$ Now the assertion follows from Claim \er{w}
\end{proof}
The {\bf proof of Theorem \er{mr}} can now be completed as follows. Since $\AL_{m,n}^*=\AL_{n,m},$ it suffices to show that
\be{28}\AL_k:=\S_{\el\ge k}\AL_{\el,k}\ic\AL+\BL_\lL.\ee
We prove \er{28} by induction over $k\ge 0.$ The case $k=0$ is trivial. For the induction step, let $k>0$ and suppose that
$\AL_h\ic\AL+\BL_\lL$ whenever $h<k.$ This is precisely the assumption \er{20}. We prove that
\be{29}\AL_{\el,k}\ic\AL+\BL_\lL\ee
by induction over $\el\ge k.$ By Claim \er{t} we have $\AL_{k,k}\ic\AL+\BL_\lL.$ For the induction step assume that
$\AL_{\el,k}\ic\AL+\BL_\lL$ for some $\el\ge k$. Passing to $\el+1$, consider polynomials $\lf,\lq$ with $\deg\lf\le\el$ and $\deg\lq=k.$ Then we have for any linear function $u$
$$PT_{\lf\cdot u}^*T_\lq P=PT^*_\lf T_u^*T_\lq P=PT^*_\lf T_\lq PT_u^*P+PT^*_\lf[T_u^*,T_\lq]P.$$
By the induction hypothesis we have $PT_\lf^*T_\lq P\in\AL_{\el,k}\ic\AL+\BL_\lL.$ For the second term, we may assume that
$\lq=v_{k-1}\cdots v_0$ for some linear functions $v_i.$ Then
$$PT_\lf^*[T_u^*,T_\lq]P=\S_{i=0}^{k-1}PT_\lf^*T_{v_{k-1}\cdots v_{i+1}}[T_u^*,T_{v_i}]T_{v_{i-1}\cdots v_0}P.$$
If $p,q,\lx,\lh$ are polynomials of degree $\le i<k$ we have
$$PT_\lf^*T_{v_{k-1}\cdots v_{i+1}}T_pPT_q^*[T_u^*,T_{v_i}]T_\lx PT_\lh^*T_{v_{i-1}\cdots v_0}P\in\AL+\BL_\lL,$$
since the assumption \er{20} implies that $\AL+\BL_\lL$ contains $PT_\lf^*T_{v_{k-1}\cdots v_{i+1}}T_pP\in\AL_{\el,k-1}$ and
$PT_\lh^*T_{v_{i-1}\cdots v_0}P\in\AL_{i,i},$ and Claim \er{s} implies $PT_q^*[T_u^*,T_{v_i}]T_\lx P\in\AL+\BL_\lL.$ Thus
$$PT_\lf^*T_{v_{k-1}\cdots v_{i+1}}A^\la[T_u^*,T_{v_i}]A^\lg T_{v_{i-1}\cdots v_0}P\in\AL+\BL_\lL,$$
whenever $|\la|\le i$ and $|\lg|\le i.$ With Claim \er{w}, it follows that $PT_\lf^*[T_u^*,T_\lq]P\in\AL+\BL_\lL.$ Therefore $\AL_{\el+1,k}\ic\AL+\BL_\lL,$ completing the induction proof of \er{28}.
\section{Smooth extension and Dixmier trace}
Let $\KL$ denote the compact operators. By definition \cite{C2} we have
$$\LL^{n,\oo}:=\{T\in\KL:\,\lm_j(T)=O(j^{-1/n})\}$$
for $n>1,$ and
$$\LL^{1,\oo}:=\{T\in\KL:\,\S_{i=1}^j\lm_i(T)=O(\log j)\}.$$
Here $\lm_1(T)\ge\lm_2(T)\ge\cdots$ are the singular values of $T.$ We will apply these concepts to the Hilbert space $H^2_1(S).$
Using the invariants $a,b$ we put
$$n:=1+a(r-1)+b.$$
Note that $n$ is not the dimension $d=r(1+\f a2(r-1)+b)$ of the underlying domain $D,$ unless $r=1.$ We will give a geometric interpretation below.
\begin{lemma}\label{c} Consider $\lL$ as an unbounded operator on $H^2_1(S)$. Then $(\lL+1)^{-1}\in\LL^{n,\oo}.$
\end{lemma}
\begin{proof} For any partition $\ll$ it follows from \cite[Lemma 2.7 and Lemma 2.6]{U1} that
\be{10}\dim\PL_\ll(Z)=\f{(\lr)_\ll}{(\lr-b)_\ll}\P_{1\le i<j\le r}\f{\ll_i-\ll_j+\f a2(j-i)}{\f a2(j-i)}\cdot\f{(\ll_i-\ll_j+1+\f a2(j-i-1))_{a-1}}{(1+\f a2(j-i-1))_{a-1}}.\ee
Specializing \er{10} to $m=(m,0,\ldots,0)$ we obtain
$$\dim\PL_m(Z)=\f{(m+1+\f a2(r-1))_b}{(1+\f a2(r-1))_b}\P_{j=2}^r\f{m+\f a2(j-1)}{\f a2(j-1)}\cdot\f{(m+1+\f a2(j-2))_{a-1}}{(1+\f a2(j-2))_{a-1}}$$
for $m\ge b.$ It follows that asymptotically, we have
$$\dim\PL_m(Z)\sim c\cdot m^{b+a(r-1)}=c\cdot m^{n-1}$$
for some constant $c>0$ independent of $m.$ Since $(\lL+1)^{-1}$ has the eigenvalues $1/(1+m),$ with eigenspace $\PL_m(Z),$ this estimate implies that the partial sum
$$S_j((\lL+1)^{-1})=\S_{i=0}^{j}\lm_i((\lL+1)^{-1})\sim j^{1-1/n},$$
where $\lm_i(T)$ is the $i$-th eigenvalue of $T.$ This implies the assertion since, for $n>1,$ $T\in\LL^{n,\oo}$ iff $\{j^{(1/n-1)}S_j(T):\,j\ge 1\}$ is a bounded sequence \cite{C2}.
\end{proof}
\begin{theorem}\label{a} Let $f,g\in\PL(Z\xx\o Z)$ be real-analytic polynomials. Then $[S_f,S_g]\in\LL^{n,\oo}.$
\end{theorem}
\begin{proof} Let $A\in\AL,B\in\BL.$ Then Lemma \er{c} implies $A(\lL+1)^{-1}+B \in\LL^{n,\oo}$ since $A+B(\lL+1)$ is bounded. It follows that $\BL_\lL\ic\LL^{n,\oo}.$ Since $[\AL,\AL]\ic\BL_\lL$ by Proposition \er{aab} and $\BL_\lL$ is an $\AL$-bimodule we obtain
$$[\AL+\BL_\lL,\AL+\BL_\lL]\ic\BL_\lL.$$
Since $S_f,S_g\in\AL+\BL_\lL$ by Theorem \er{mr}, the assertion follows.
\end{proof}
It is well known \cite{C2} that $T_i\in\LL^{p_i,\oo}$ and $\S_{i=1}^n\f1{p_i}=1$ implies $T=T_1\cdots T_n\in\LL^{1,\oo}.$ Hence Theorem \er{a} implies
\begin{corollary} Let $f_1,g_1,\ldots f_n,g_n\in\PL(Z\xx\o Z)$ be real-analytic polynomials. Then
\be{33}[S_{f_1},S_{g_1}]\cdots[S_{f_n},S_{g_n}]\in\LL^{1,\oo}.\ee
\end{corollary}
The trace class $\LL^1$ is a proper subspace of $\LL^{1,\oo}$. For $T\in\LL^{1,\oo}$ the {\bf Dixmier trace}, denoted by
$tr_\lo(T)$, depends a priori on a choice of positive functional $\lo$ on $l^\oo(\Nl)$ vanishing on $c_0(\Nl)$. For the so-called {\bf measurable} operators $T$ the value $tr_\lo(T)$ is independent of $\lo$. More precisely, for a positive operator $T,$
$$tr_\lo(T) =\lim_{j\to\oo}\f1{\log j}{\S_{i=1}^j\lm_i(T)}$$
whenever the limit exists. It also satisfies the tracial property
$$tr_\lo(TS)=tr_\lo(ST)$$
and $tr_\lo(T)=0$ if $T\in\LL^1.$ We refer the reader to \cite{C2} for more details.
In order to determine the Dixmier trace of the operators \er{33} we consider the algebraic variety
$$Z_1^\ol:=\{z\in Z:\,rank(z)\le 1\},$$
which has (complex) dimension $\dim Z_1^\ol=1+a(r-1)+b=n,$ and is singular only at the origin.
\begin{proposition} Consider the polynomial ideal $\IL(Z_1^\ol)\ic\PL(Z)$ vanishing on $Z_1^\ol.$ Then the sub-Hardy space $H^2_1(S)$ can be identified with the Hilbert quotient module
$$H^2_1(S)=H^2(S)/\o{\IL(Z_1^\ol)}\al\o{\IL(Z_1^\ol)}^\perp.$$
\end{proposition}
\begin{proof} It suffices to show that $\IL(Z_1^\ol)$ coincides with the ideal
$$\JL=\bigoplus_{\ll_2>0}\PL_\ll(Z)\ic\PL(Z).$$
For $z\in Z_1^\ol$, we have $N_\el(z)=0$ for $\el\ge 2$ since $rank(z)\le 1.$ This implies that $\JL\ic\IL(Z_1^\ol).$ By Schur orthogonality the orthogonal projection $P_\ll$ is given by
$$\f1{d_\ll}P_\ll\,f=\S_\la\I_{K}dk\,(\lf_\la|k\cdot\lf_\la)\,(k^{-1}\cdot f)$$
for all $f\in\PL(Z),$ where $\lf_\la\in\PL_\ll(Z)$ is an orthonormal basis. It follows that the $K$-invariant ideal $\IL(Z_1^\ol)$
is invariant under all $P_\ll.$ Now suppose there exists $f\in\IL(Z_1^\ol)\sm\JL.$ Then $f=f'+f'',$ where $f''\in\JL$ and $f'\in\JL^\perp=\bigoplus_{m}\PL_m(Z)$ is non-zero. Since $\JL\ic\IL(Z_1^\ol)$ we may assume $f=f'.$ By the above, we may assume that
$f\in\PL_m(Z)$ for some $m\ge 0.$ By irreducibiliy, it follows that $\PL_m(Z)\ic\IL(Z_1^\ol),$ which is a contradiction since
$N_1^m\notin\IL(Z_1^\ol).$
\end{proof}
The unit ball $D\ui Z_1^\ol$ of $Z_1^\ol$ is a {\bf strictly pseudo-convex domain} (singular at the origin), with a $K$-homogeneous smooth boundary $S_1=\{c:\,\{cc^*c\}=c,\,rank(c)=1\}$ consisting of all minimal tripotents. Denote by $L^2(S_1)$ the $L^2$-space with respect to the $K$-invariant measure. The Hardy space $H^2(S_1)$ is the closure of the algebra $\PL(Z)$ of all polynomials on $Z,$ restricted to $S_1.$ Since $N_\el|_{S_1}=0$ for each $\el\ge 2$, it follows that
$$H^2(S_1)=\S_{m\ge 0}\PL_m^\sim(Z),$$
where $\t f=f|_{S_1}$ denotes the restriction.
\begin{lemma} Let $p,q\in\PL_m(Z).$ Then
$$(p|q)_S=\f{(ra/2)_m}{(a/2)_m}(\t p|\t q)_{S_1}.$$
Hence the transformation $U:H^2_1(S)\to H^2(S_1),$ defined by
$$Up:=\F{\f{(ra/2)_m}{(a/2)_m}}\t p\qquad\forall\,p\in\PL_m(Z),$$
is unitary.
\end{lemma}
\begin{proof} Let $X$ be the self-adjoint part of the Peirce $2$-space $Z^2_e$ of full rank $r$ \cite{FK}. For any partition $\ll\in\Nl^r_+$, the associated {\bf spherical polynomial} $\lf^\ll$ on $X\ic Z,$ normalized by $\lf^\ll(e)=1$ \cite{FK}, is given by
$$\f{\EL^\ll(t,e)}{d_\ll}=\f{\lf^\ll(t)}{(d/r)_\ll}$$
for all $t\in X$, where $d_\ll:=\dim\,\PL_\ll(Z)$ and $\EL^\ll(z,w)$ is the Fischer-Fock reproducing kernel for $\PL_\ll(Z).$ Now suppose $\ll\in\Nl^\el_+$. Then \cite[Proposition 3.7]{AU} implies
$$\lf^\ll(e_1+\ldots+e_\el)=\f{(\el a/2)_\ll}{(r a/2)_\ll}$$
and hence
$$\lf^\ll(t)=\f{(\el a/2)_\ll}{(r a/2)_\ll}\lf^\ll_\el(t)$$
for all $t\in X_\el\ic X$, where $\lf^\ll_\el$ is the spherical polynomial for the self-adjoint part $X_\el$ of the Peirce $2$-space
$Z^2_{e_1+\ldots e_\el}.$ Let $\lO_\el\ic X_\el$ be the strictly positive cone, and let $t\in\lO_\el$ be fixed. By Schur orthogonality \cite[Theorem 14.3.3]{D}, we have
$$\I_K dk\,\EL^\ll(z,\,k\F{t})\EL^\lm(k\F{t},w)=\f{\ld_{\ll,\lm}}{d_\ll}\EL^\ll(k\F{t},k\F{t})\,\EL^\ll(z,w)$$
$$=\f{\ld_{\ll,\lm}}{(d/r)_\ll}\lf^\ll(t)\,\EL^\ll(z,w)=\f{\ld_{\ll,\lm}}{(d/r)_\ll}\f{(\el a/2)_\ll}{(r a/2)_\ll}
\lf^\ll_\el(t)\EL^\ll(z,w).$$
for all $\ll,\lm\in\Nl^\el_+$ and $z,w\in Z.$ Applying this identity to $\el=r,t=e$ and $\el=1,t=e_1,$ resp., the assertion follows.
\end{proof}
Define $\t\lL\t p=m\t p$ for $p\in\PL_m(Z)$. Then Lemma \er{c} gives
$$\f1{1+\t\lL}\in\LL^{n,\oo}.$$
Let $\t T_f$ denote the Toeplitz operators $H^2(S_1).$ Then $\t T_u\t\lL=(\t\lL-1)\t T_u$ and $\t T_u^*\t\lL=(\t\lL+1)\t T_u^*.$
\begin{proposition}\label{ss} Let $u,v\in Z.$ Then $U S_u U^*-\t T_u\in\LL^{n,\oo}$ and
$$U[S_u,S_v^*]U^*-[\t T_u,\t T_v^*]\in \LL^{n/2,\oo}.$$
\end{proposition}
\begin{proof} For each $p\in\PL_m(Z)$ we have
\begin{eqnarray*}US_u U^*\t p&=&U^*PT_u\(\F{\f{(a/2)_m}{(ra/2)_m}}p\)=\F{\f{(a/2)_m}{(ra/2)_m}}U^*P(up)\\
&=&\F{\f{(a/2)_m}{(ra/2)_m}}\F{\f{(ra/2)_{m+1}}{(a/2)_{m+1}}}\w{P(up)}=\F{\f{ra/2+m}{a/2+m}}\t u\t p=\t T_u\F{\f{\t \lL+ra/2}{\t \lL+a/2}}\t p.\end{eqnarray*}
Thus we have $U S_u U^*=\t T_u\F{\f{\t\lL+ra/2}{\t\lL+a/2}}.$ This implies the first assertion. For the second assertion
\begin{eqnarray*}U[S_u,S_v^*]U^*&=&\[\t T_u\F{\f{\t\lL+ra/2}{\t\lL+a/2}},\F{\f{\t\lL+ra/2}{\t\lL+a/2}}\t T_v^*\]
=\t T_u{\f{\t\lL+ra/2}{\t\lL+a/2}}\t T_v^*-\F{\f{\t\lL+ra/2}{\t\lL+a/2}}\t T_v^*\t T_u\F{\f{\t\lL+ra/2}{\t\lL+a/2}}\\
&=&{\f{\t\lL+ra/2-1}{\t\lL+a/2-1}}\t T_u\t T_v^*-{\f{\t\lL+ra/2}{\t\lL+a/2}}\t T_v^* \t T_u
={\f{\t\lL+ra/2}{\t\lL+a/2}}[\t T_u,\t T_v^*]+ \[{\f{\t\lL+ra/2-1}{\t\lL+a/2-1}}-{\f{\t\lL+ra/2}{\t\lL+a/2}}\]\t T_u\t T_v^*.\end{eqnarray*}
Therefore
\begin{eqnarray*}U[S_u,S_v^*]U^*-[\t T_u,\t T_v^*]&=&\(\f{\t\lL+ra/2}{\t\lL+a/2}-1\)[\t T_u,\t T_v^*]
+\({\f{\t\lL+ra/2-1}{\t\lL+a/2-1}}-{\f{\t\lL+ra/2}{\t\lL+a/2}}\)\t T_u\t T_v^*\\
&=&\f{(r-1)a/2}{\t\lL+a/2}[\t T_u,\t T_v^*]+{\f{(r-1)a/2}{(\t\lL+a/2-1)(\t\lL+a/2)}}\t T_u\t T_v^*\in\LL^{n/2,\oo}\end{eqnarray*}
since $[\t T_u,\t T_v^*]$ (cf. \cite{EZ}) and $(\t\lL+a/2)^{-1}$ belong to $\LL^{n,\oo}.$
\end{proof}
To consider general symbols, we need the following algebraic lemma.
\begin{lemma}\label{sa} Suppose that the given operators $A_i,\t A_j$ satisfy that $A_i-\t A_i,\,[A_i, A_j],\,[\t A_i,\t A_j]\in\LL^{n,\oo}$ and $[A_i, A_j]-[\t A_i,\t A_j]\in\LL^{n/2,\oo}$ for $1\leq i,j\leq 4.$ Then
$$[A_1A_2,A_3A_4]-[\t A_1\t A_2,\t A_3\t A_4]\in\LL^{n/2,\oo}.$$
\end{lemma}
\begin{corollary}
For polynomials $p,q,\lf,\lq$, we have
$$U[S^*_p S_q,S^*_\lf S_\lq]U^*-[\t T_{\o p q},\t T_{\o\lf\lq}]\in\LL^{n/2,\oo}.$$
\end{corollary}
\begin{proof} Apply Lemma \er{sa} and Proposition \er{ss}.
\end{proof}
Every $f\in\CL^\oo(S)$ has a {\bf Poisson integral extension} $\h f\in\CL^\oo(D),$ which is harmonic in the sense that it is annihilated by the so-called Hua operators \cite{K,S1}. For any non-zero tripotent $c\in S_k$ there exists a continuous extension, again denoted by $\h f,$ onto the boundary component $c+D_c.$ This extension is given by
$$\h f(c+\lz)=f_c^\yi(\lz)$$
for all $\lz\in D_c^0,$ where $f_c^\yi$ denotes the Poisson extension, relative to the Shilov boundary $S_c$ of $D_c,$ for the restricted smooth function
$$f_c(\lz):=f(c+\lz),\qquad \lz\in S_c.$$
Setting $\lz=0$ the Poisson extension $\h f$ is well-defined on $S_k.$
\begin{lemma}\label{zz} For all $c\in S_k$ we have
$$\widehat{\o p q}(c)=(p_c|q_c)_{S_c}.$$
\end{lemma}
\begin{proof} Let $h(z)=\widehat{\o p q}(z)$ be the Poisson extension of $\o{p(s)}q(s)$. For all $\lz\in S_c$, we have $c+\lz\in S$ and hence
$$h_c(\lz)=h(c+\lz)=\o{p(c+\lz)}q(c+\lz)=\o{p_c(\lz)}q_c(\lz)$$
Since $h_c(\lz)$ is harmonic, the mean value property applied to the Peirce $0$-space $Z_c$ yields
$$h(c)=h_c(0)=h_c^\yi(0)=\I_{S_c}d\lz\,h_c(\lz)=\I_{S_c}d\lz\,\o{p_c(\lz)}q_c(\lz)=(p_c|q_c)_{S_c}.$$
\end{proof}
\begin{proposition}\label{df} For polynomials $f\in\PL(Z\xx\o Z)$ we have $U S_f U^*-\t T_{\h f}\in\LL^{n,\oo}$ and
$$U [S^*_f, S_f] U^*-[\t T^*_{\h f},\t T_{\h f}]\in\LL^{n/2,\oo}.$$
Here $\h f$ is the Poisson extension restricted to $S_1$.
\end{proposition}
\begin{proof} Without loss of generality we may suppose that $f=\o p q$ for $p,q\in\PL(Z).$ By Theorem \er{mr}, Proposition \er{aab} and Lemma \er{y}, there exist $B\in\BL_\lL$ and finitely many $p_i,q_i\in\PL(Z)$ such that
$$S_f=PT_p^*T_q P=B+\S_i S_{p_i}^*S_{q_i}.$$
By the definition of $\BL_\lL$, Proposition \er{aab} and Lemma (4.1), we have $S_f-\S_i S_{p_i}^*S_{q_i}\in \LL^{n,\oo}$ and
$[S^*_f,S_f]-\[\S_i S_{q_i}^*S_{p_i},\S_i S_{p_i}^*S_{q_i}\]\in\LL^{n/2,\oo}$. For any $c\in S_1,$ the symbol map in \cite[Theorem 3.12]{U2} is given by
$$(\ls_1 S_f)(c)=(1_c\xt 1_c)T_{p_c}^*T_{q_c}(1_c\xt 1_c)=(p_c|q_c)_{S_c}(1_c\xt 1_c),$$
and
$$\ls_1\(\S_i S_{p_i}^* S_{q_i}\)(c)=\S_i\o{p_i(c)}q_i(c)(1_c\xt 1_c).$$
With Lemma \er{zz} it follows that
$$\h f|_{S_1}=\S_i\o p_i q_i.$$
Since $U S_{p_i}^* S_{q_i}U^*=\t B+\t T_{p_i}^*\t T_{q_i}=\t B+\t T_{\o p_i q_i},$ it follows that
$$U S_f U^*=\t T_{\S_i\o p_i q_i}+B=\t T_{\h f}+B$$
for $B\in\LL^{n,\oo}$. Therefore $U S_f U^*-\t T_{\h f}=U S_f U^*-\S_i\t T_{\o p_i q_i}\in\LL^{n,\oo}$ and
\begin{eqnarray*}&&U\[S^*_f,S_f\]U^*-\[\t T^*_{\h f},\t T_{\h f}\]\\&=&U\(\[S^*_f,S_f\]-\[\S_i S_{q_i}^* S_{p_i},\S_i S_{p_i}^*S_{q_i}\]\)U^*-\(U\[\S_i S_{q_i}^*S_{p_i},\S_i S_{p_i}^*S_{q_i}\]U^*-\[\t T^*_{\h f},\t T_{\h f}\]\)\in\LL^{n/2,\oo}.\end{eqnarray*}
\end{proof}
The explicit computation of the Dixmier trace uses the results of \cite{EZ} on strictly pseudo-convex domains. Let $\lS=\{z\in Z:\mbox{rank}(z)=1\}.$ Then $S_1\ic\lS$ has the defining function $r(z)=(z|z)-1.$ Therefore the contact $1$-form
$\lh=(\dl r-\o\dl r)/(2i)$ on $S_1$ \cite [Section 2.1]{EZ} is given by
$$\lh_c w=\f{(w|c)-(c|w)}{2i}$$
for all $c\in S_1$ and $w\in T_c(\lS)\ic Z.$ It follows that
$$(d\lh)_c(w_1,w_2)=\f{(w_1|w_2)-(w_2|w_1)}i.$$
Since $T_c(\lS)=Z^2_c\oplus Z^1_c=\Cl\,c\oplus Z^1_c$ we may write $w=i\la\,c+v,$ with $\la\in\Cl$ and $v\in Z_c^1.$ Then
$\lh_c(i\la\,c+v)=\la.$ It follows that $Ker(\lh_c)=Z^1_c$ and the Reeb vector field $E_\perp$ \cite[p. 614]{EZ} is given by $c\mapsto ic.$ Restricted to the tangent space $T_c(S_1)=i\Rl\,c\oplus Z^1_c,$ the $2$-form $d\lh$ has the radical
$i\Rl\,c=\Rl\,E_\perp$ and is non-degenerate on $Ker(\lh_c).$ Every $\lq\in\CL^\oo(S_1)$ defines a vector field $Z_\lq\in Ker(\lh)$ such that
$$d\lh(X,Z_\lq)=X\,\lq$$
for all vector fields $X\in Ker(\lh).$ For $\lf,\lq\in\CL^\oo(S_1)$ we obtain the {\bf boundary Poisson bracket}
$$\{\lf,\lq\}_\flat=d\lh(Z_\lf,Z_\lq)=Z_\lf\,\lq.$$
\begin{theorem}\label{b} Let $f_j,g_j\in\PL(Z\xx\o Z).$ Then
$$tr_\lo[S_{f_1},S_{g_1}]\cdots[S_{f_n},S_{g_n}]=C\,\I_{S_1}ds\P_{j=1}^n\{\h f_j,\h g_j\}_\flat,$$
where $ds$ is the normalized $K$-invariant measure, $\h f$ is the Poisson extension of $f$ and $\{\lf,\lq\}_\flat$ denotes the boundary Poisson bracket. The constant
$$C=\f1{(2\lp i)^n}\I_{S_1}\lh\yi\f{(d\lh)^{n-1}}{n!}$$
will be computed in the following Proposition \er{gam}.
\end{theorem}
\begin{proof} In general, if $T_1\in\LL^{n/2,\oo}$ and $T_2,\ldots,T_n\in\LL^{n,\oo}$ then $T_1T_2\cdots T_n\in\LL^1$ since $\LL^{k,\oo}\ic\LL^{k+\Le}$ for any $\Le>0.$ By Proposition \er{df} it follows that $U[S_{f_1},S_{g_1}]\cdots[S_{f_n},S_{g_n}]U^*-T\in\LL^1,$ where
$$T:=[\t T_{\h f_1},\t T_{\h g_1}]\cdots[\t T_{\h f_n},\t T_{\h g_n}]$$
is a generalized Toeplitz operator on $H^2(S_1)$ of order $-n.$ Applying \cite[Theorem 3]{EZ} it follows that
$$Tr_\lo[S_{f_1},S_{g_1}]\cdots[S_{f_n},S_{g_n}]=Tr_\lo(T)=\f1{(2\lp)^n}\I_{S_1}\lh\yi\f{(d\lh)^{n-1}}{n!}\,\ls_{-n}(T)(x,\lh_x),$$
where $\lh$ is the contact form. By \cite[Section 4]{EZ}, $T$ has the symbol
$$\ls_{-n}(T)=\P_{j=1}^n\,\ls_{-1}[\t T_{\h f_j},\t T_{\h g_j}]=\P_{j=1}^n\,\f1i\{\ls_0\t T_{\h f_j},\ls_0\t T_{\h g_j}\}_\lS
=\P_{j=1}^n\,\f1i\{\h f_j^{(0)},\h g_j^{(0)}\}_\lS$$
in terms of the Poisson bracket of $\lS.$ Here $\lf^{(0)}(t\,c)=\lf(c)$ denotes the $0$-homogeneous extension of
$\lf\in\CL^\oo(S_1).$ Now the assertion follows, since by \cite[Corollary 8]{EZ} we have for $t=1$
$$\f1i\{\lf^{(0)},\lq^{(0)}\}_\lS=Z_\lf\lq=\{\lf,\lq\}_\flat.$$
\end{proof}
Let $V=Z^1_{e_1}$ be the Peirce $1$-space for the minimal tripotent $e_1.$ If $a\neq 2$ or $r=1,$ then $V$ is an irreducible hermitian Jordan triple. If $a=2$ and $r>1$ then $Z=\Cl^{r\xx(r+b)}$ and
$V=\Cl^{(r-1)\xx 1}\oplus\Cl^{1\xx(r+b-1)}$ is a direct sum of two hermitian Jordan triples of rank $1.$ For any irreducible hermitian Jordan triple $V$ let $\lG_V$ denote the Gindikin $\lG$-function for the radial cone $\lO_V\ic V$ \cite{FK}. Let $r_V,\,d_V,\,p_V$ denote the rank, dimension and genus of $V,$ resp.
\begin{proposition}\label{gam} If $a\neq 2$ or $r=1,$ we have
$$\f1{(2\lp)^n}\I_{S_1}\lh\yi\f{(d\lh)^{n-1}}{(n-1)!}=\f{\lG_V(p_V-\f{n-1}{r_V})}{\lG_V(p_V)};$$
If $a=2$ we have
$$\f1{(2\lp)^n}\I_{S_1}\lh\yi\f{(d\lh)^{n-1}}{(n-1)!}=\f{1}{\lG(r)\,\lG(r+b)}.$$
In the rank $r=1$ case, where $Z=\Cl^d$ and $S_1=\Sl^{2n-1},$ we have $n=d=1+b$ and both formulas imply
$$\f1{(2\lp)^n}\I_{S_1}\lh\yi(d\lh)^{n-1}=1.$$
\end{proposition}
\begin{proof} Any irreducible hermitian Jordan triple $V$ has a 'quasi-determinant' function $\lD_V(u,v)$ such that the invariant measure on its conformal compactification $M,$ containing $V$ as an open dense subset of full measure, is a multiple of $\lD_V(v,-v)^{-p_V}\,d\ll(v),$ where $d\ll(v)$ is Lebesgue measure for the normalized inner product. Moreover, by \cite{EU} we have the polar integration formula
\begin{eqnarray}\label{vol}\I_V\f{d\ll(z)}{\lp^{d_V}}\lD(z,-z)^{-p_V}=\f{\lG_V(p_V-\f{d_V}{r_V})}{\lG_V(p_V)}.\end{eqnarray}
Let $M$ denote the compact complex manifold of all Peirce $2$-spaces $U\ic Z$ of rank $1.$ There is a canonical map
$$\lp:\lS\to M$$
which maps $z\in\lS$ onto its Peirce $2$-space $Z^2_z.$ In this way, $\lS$ becomes a hermitian holomorphic line bundle over $M$ which can be identified with the tautological line bundle $\LL=\bigcup_{U\in M}U.$ The subset $S_1\ic\lS$ corresponds to the circle bundle $\bigcup_{U\in M}S_U,$ where $S_U\al\Sl^1$ is the Shilov boundary of $U\in M.$ The holomorphic map $\lp$ satisfies
$$\mbox{ker}(d_c\lp)=Z^2_c,$$
since $Z^2_c\ic\mbox{ker}(d_c\lp)$ and both spaces are $1$-dimensional. Therefore $d\lh$ vanishes on
$$T_c(S_1)\ui\mbox{ker}(d_c\lp)=i\Rl\cdot c.$$
As a consequence there exists a $K$-invariant $2$-form $\lT$ on $M$ such that $d\lh=\lp^*\lT.$ Now $\lh,$ restricted to $S_U,$ is the usual contact form on $\Sl^1$ of volume $2\lp.$ It follows that
$$\f1{(2\lp)^n}\I_{S_1}\lh\yi\f{(d\lh)^{n-1}}{(n-1)!}=\f1{(2\lp)^n}\I_{S_1}\lh\yi\f{\lp^*\lT^{n-1}}{(n-1)!}
=\f1{(2\lp)^{n-1}}\I_{M}\f{\lT^{n-1}}{(n-1)!}.$$
In order to compute this integral, let $V=Z^1_{e_1}.$ A local coordinate for $M$ is given by the map $\ls:=\lp\oc\Lt:V\to M,$ where $\Lt:V\to\lS$ is defined by
$$\Lt(v)=e_1+v+\{ve_1^*v\}.$$
The semi-simple part $K'$ of $K$ acts transitively on $M,$ and induces a 'Moebius-type' biholomorphic action on $V$ such that $\ls$ becomes $K'$-equivariant. We have
$$\Lt^*d\lh=\Lt^*(\lp^*\lT)=\ls^*\lT.$$
Since $(d_0\Lt)v=v$ at the origin $0\in V$, the pull-back $\Lt^*(d\lh)|_v(v_1,v_2)=(d\lh)_{\Lt(v)}((d_v\Lt)v_1,(d_v\Lt)v_2)$ satisfies
$$\ls^*\lT|_0(v_1,v_2)=\Lt^*(d\lh)|_0(v_1,v_2)=\f{(v_1|v_2)-(v_2|v_1)}{i}.$$
Using complex coordinates $v_j$ with respect to an orthonormal basis of $V=T_0(V)$ this means
$$\ls^*\lT|_0=\S_{j=1}^{n-1}\f{d\o v_j\yi d v_j}{i}.$$
Now assume that $a\neq 2$ or $r=1.$ Then $M$ is irreducible. Since $\lT$ is invariant under $K,$ it follows that
$\ls^*\lT$ is invariant under the Moebius action. Since $d_V=n-1,$ we obtain for the volume form
$$\f{\ls^*\lT^{n-1}}{(n-1)!}=C\cdot\lD_V(v,-v)^{-p_V}d\ll(v),$$
where $C$ is a constant. Evaluating at $0\in V$ and using
\begin{eqnarray}\label{mea}d\ll(v)=\P_{j=1}^{n-1}\f{d\o v_j\yi d v_j}{2i}=\f1{(n-1)!}\(\S_{j=1}^{n-1}\f{d\o v_j\yi d v_j}{2i}\)^{n-1}
=\f1{2^{n-1}}\f{(\ls^*\lT|_0)^{n-1}}{(n-1)!}.\end{eqnarray}
it follows that $C=2^{n-1}.$ Since $\ls$ is a Zariski dense open embedding of full measure we obtain
$$\f1{(2\lp)^{n-1}}\I_{M}\f{\lT^{n-1}}{(n-1)!}=\f1{(2\lp)^{n-1}}\I_V\f{\ls^*\lT^{n-1}}{(n-1)!}=\f{C}{2^{n-1}}\I_V\f{d\ll(v)}{\lp^{n-1}}\,\lD_V(v,-v)^{-p_V}=\f{\lG_V(p_V-\f{n-1}{r_V})}{\lG_V(p_V)}$$
by applying \er{vol} to the irreducible hermitian Jordan triple $V=Z^1_{e_1}.$
Now assume $a=2$ and $r>1.$ Then $Z=\Cl^{r\xx(r+b)}$ and $M$ is reducible. More precisely,
$$\lS=\{z\in\Cl^{r\xx(r+b)}:\,rank(z)=1\}=\{\lx_1\lx_2:\,0\neq\lx_1\in\Cl^{r\xx 1},\,0\neq\lx_2\in\Cl^{1\xx(r+b)}\}.$$
Consider the associated projective spaces $M_1=\Pl(\Cl^{r\xx 1})=\{[\lx_1]:\,0\neq\lx_1\in\Cl^{r\xx 1}\}\al\Pl^{r-1}$ and
$M_2=\Pl(\Cl^{1\xx(r+b)})=\{[\lx_2]:\,0\neq\lx_2\in\Cl^{1\xx(r+b)}\}\al\Pl^{r+b-1}.$ Then $M=M_1\xx M_2$ is a direct product via the identification $([\lx_1],[\lx_2])\mapsto Ran(\lx_1\lx_2).$ For $i\in\{1,2\},$ the map $\lp_i:\lS\to M_i$ given by $\lx_1\lx_2\mapsto[\lx_i]$ is well-defined and the canonical map $\lp:\lS\to M_1\xx M_2$ is a product
$$\lp(\lx_1\lx_2)=([\lx_1],[\lx_2])=(\lp_1(\lx_1\lx_2),\lp_2(\lx_1\lx_2)).$$
Now $\lT=\lT_1\oplus\lT_2$ is the direct sum of $K'$-invariant $2$-forms $\lT_i$ on $M_i.$ Using the binomial theorem for (commuting) $2$-forms, the corresponding volume form is
$$\f{\lT^{n-1}}{(n-1)!}=\f{\lT_1^{n_1}}{n_1!}\yi\f{\lT_2^{n_2}}{n_2!}$$
for the dimensions $n_1=r-1,n_2=r+b-1$ adding up to $n_1+n_2=2(r-1)+b=n-1.$ It follows that
$$\f1{(2\lp)^{n-1}}\I_{M}\f{\lT^{n-1}}{(n-1)!}=\f1{(2\lp)^{n_1}}\I_{M_1}\f{\lT_1^{n_1}}{n_1!}\,\f1{(2\lp)^{n_2}}\I_{M_2}\f{\lT_2^{n_2}}{n_2!}.$$
In order to compute these integrals, put $V_1:=\Cl^{(r-1)\xx 1}$ and $V_2=\Cl^{1\xx(r+b-1)}.$ Then
$$V=Z^1_{e_1}=\{\bb0{v_2}{v_1}0:\,v_i\in V_i\}\al V_1\xx V_2.$$
The local coordinate $\ls(v_1,v_2)=(\ls_1(v_1),\ls_2(v_2))$ is of product type, where $\ls_i(v_i):=[1,v_i].$ In fact, putting
$v=\bb0{v_2}{v_1}0\in V,$ we obtain
$$\Lt(v)=e_1+v+\{ve_1^*v\}=\bb{1}{v_2}{v_1}{v_1v_2}=\ba{1}{v_1}\,\ab{1}{v_2}$$
and hence
$$\ls(v)=\lp(\Lt(v))=[\ba{1}{v_1}],[\ab{1}{v_2}].$$
The semi-simple part $K'=SU(r)\xx SU(r+b)$ of $K=S(U(r)\xx U(r+b))$ acts transitively on each factor $M_i$ and induces a 'Moebius-type' biholomorphic action on $V_i$ such that $\ls_i$ becomes $K'$-equivariant. Since $\lT_i$ is invariant under $K',$ it follows that $\ls_i^*\lT_i$ is invariant under this Moebius action. This implies for the volume form
$$\f{\ls_i^*\lT_i^{n_i}}{n_i!}=C_i\cdot(1+(v_i|v_i))^{-1-n_i}\,\,d\ll_i(v_i),$$
where $C_i$ is a constant. Using the relation
$$d\ll_i(v_i)=\f1{2^{n_i}}\f{(\ls_i^*\lT_i|_0)^{n_i}}{n_i!}$$
analogous to \er{mea}, it follows that $C_i=2^{n_i}.$ Since $\ls_i:V_i\to M_i$ is a Zariski dense open embedding of full measure we obtain
$$\f1{(2\lp)^{n_i}}\I_{M_i}\f{\lT_i^{n_i}}{n_i!}=\f1{(2\lp)^{n_i}}\I_{V_i}\f{\ls_i^*\lT_i^{n_i}}{n_i!}
=\f{C_i}{2^{n_i}}\I_{V_i}\f{d\ll_i(v_i)}{\lp^{n_i}}\,(1+(v_i|v_i)^{-1-n_i}=\f{\lG(1)}{\lG(1+n_i)}=\f1{n_i!}$$
by applying \er{vol} to the irreducible hermitian Jordan triple $V_i.$
\end{proof}
Finally, let us mention a relation involving numerical invariants of the domain $D$ and $V=Z^1_{e_1}$, which would make the formulas more tractable.
\begin{lemma}\label{zv} Suppose that $a\neq 2.$ Then the rank $r_V$ and the genus $p_V$ of $V=Z^1_{e_1}$ satisfy the relation
$$r_V\,p_V=ra+b.$$
As a consequence, the $\lG$-function quotient in Proposition \er{gam} can also be expressed as
$$\f{\lG_V(p_V-\f{n-1}{r_V})}{\lG_V(p_V)}=\f{\lG_V(\f{a}{r_V})}{\lG_V(\f{ra+b}{r_V})}.$$
\end{lemma}
\begin{proof} We use the classification of hermitan Jordan triples \cite{L,N}. For $a=2,$ we obtain the Jordan triples
$Z=\Cl^{r\xx(r+b)}$ of type (I), for which the relation does not hold. The other cases are listed in the following table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
type & Z & rank & a & b & V & $r_V$ & $p_V$ \\ \hline
(II) & $\Cl^{(2r+\Le)\xx(2r+\Le)}_{\mbox{\tiny asym}} $ & r & 4 & $2\Le$ & $ \Cl^{2\xx(2(r-1)+\Le)}$ &2 & $2r+\Le$ \\ \hline
(III) & $\Cl^{r\xx r}_{\mbox{\tiny symm}} $ & r &1 & 0 & $ \Cl^{r-1}$ &1 & r \\ \hline
(IV) & $\Cl^d_{\mbox{\tiny spin}} $ & 2 & d-2 & 0& $\Cl^{d-2}_{\mbox{\tiny spin}}$ &2 &d-2 \\ \hline
(V) & $\Ol^{1\xx 2}_\Cl $ & 2 & 6 & 4 & $ \Cl^{5\xx 5}_{\mbox{\tiny asym}}$ &2 & 8\\ \hline
(VI) & $\HL_3(\Ol)\otimes \Cl $ & 3 & 8 & 0 & $ \Ol^{1\xx 2}_\Cl$ &2 & 12 \\ \hline
\end{tabular}\end{center}
\end{proof}
\end{document} | math | 60,273 |
\begin{document}
\title{Stochastic Schr\"{o}
\begin{abstract}
Firstly, the Markovian stochastic Schr\"{o}dinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples --- the damped harmonic oscillator and a two-level atom with homodyne photodetection. Then, we consider how to introduce memory effects in the stochastic Schr\"{o}dinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.
\end{abstract}
\section{Introduction}
Typically, an open quantum system is a system interacting with an external environment which experimentalists cannot control \cite{Dav76,4,28}. It is well known that the dynamics of an open quantum system can be described in one of the following ways: local and non-local master equations for the density matrix \cite{37,38,39}, Feynman's path integrals \cite{40}, stochastic Schr\"{o}dinger equations (SSE) \cite{7,28,27,Bel89,last} and quantum trajectories \cite{7,28}. In this review we give a description of the technique based on the SSE, which can be useful for the description of Markovian and non-Markovian dynamics of open quantum systems. Moreover, we shall illustrate the Markovian and non-Markovian theory by giving some simulations.
In the non-Markovian case we also use the stochastic simulations to check the validity of an analytic approximation for the mean state.
The stochastic representation of quantum Markovian processes already appeared in the fundamental work by Davies \cite{Dav76,19} and it was applied to the derivation of a photocounting formula. While the theory was originally formulated in terms of a stochastic process for the reduced density matrix, it was recognized by Barchielli and Belavkin \cite{7}, Dalibard, Castin and M\o{}lmer \cite{18} and by Dum, Zoller and Ritsch \cite{17} that it can also be formulated as a stochastic process for the state vector in the reduced system Hilbert space and that it leads to efficient numerical simulation algorithms. At the same time, there has been considerable interest in the unravelling of master equations for density operators into quantum trajectories which are the realizations of the underlying stochastic process \cite{28}. Just as different ensembles of state vectors may be represented by one density operator, one master equation may be decomposed in many different ways into SSEs.
The SSE is a differential equation for a wave-function process $\psi(t)$ which contains a stochastic term to describe the relaxation dynamics of an open quantum system. The link with the traditional master equation is given by the average property $\operatorname{\mathbb{E}}[|\psi(t)\rangle\langle\psi(t)|] = \eta(t)$, where $\operatorname{\mathbb{E}}$ denotes the ensemble average over the realizations of process $\psi(t)$ and $\eta(t)$ is the statistical operator satisfying the master equation. To find the SSE providing a given master equation by averaging is called \textit{unravelling} \cite{1}.
Also, in special situations, the SSE can be interpreted in terms of quantum measurements. In these cases, the solution $\psi(t)$ is called a \textit{quantum trajectory} \cite{11} and describes the evolution of an open system undergoing indirect continuous measurement. This interpretation is important for understanding quantum optics experiments such as direct photo-detection, spectral photo-detection, homodyning and heterodyning \cite{12,13,14,3}.
In the regime of the validity of the Markov approximation (no memory effects) \cite{26} it is known how to construct an appropriate unravelling in terms of a SSE. It is always possible to derive a linear SSE for a non-normalized vector $\phi(t)$, such that $\psi(t)=\norm{\phi(t)}^{-1}\phi(t)$. Moreover, the linear and nonlinear versions of the SSE are related by a change of probability measure, and it is this link that allows for a measurement interpretation \cite{3}. Also, these stochastic differential equations can be deduced from purely quantum evolution equations for the measured system coupled with a quantum environment, combined with a continuous monitoring of the environment itself \cite{29,Bar06,BarG13}.
In the non-Markovian case \cite{23,21,22}, to find relevant SSEs describing both non-Markovian quantum evolutions and continuous monitoring is a complex task. Other than in the Markovian case, no general theory has been developed. Nevertheless, it is possible to follow a general strategy. This strategy is first to generalize directly the Markovian SSE, second to show if it provides an unravelling of a corresponding master equation, and third to check if it has a measurement interpretation \cite{1,BarH95,2,apr,BarG12}. To work at the Hilbert space level guarantees automatically the complete positivity of the evolution of the statistical operator. It seems possible to adapt the Markovian approach by replacing white noises with more general noises and by allowing for random coefficients in the equation. We will show how to introduce memory effects in the SSE with the help of coloured noise. Specifically, we will illustrate the approach by replacing the Wiener process with the Ornstein-Uhlenbeck process. Such approaches are efficient for simulating corresponding non-Markovian evolutions. Also, the non-Markovian SSE is formulated in a way that allows for an interpretation in terms of measurements in continuous time.
The paper introduces the general theory of the SSE as well as the corresponding simulation techniques and is structured as follows. Section \mathrm{e}f{sec:SSE} describes the general theory of the SSEs in the Markovian case. It presents the general mathematical framework of the linear and nonlinear SSE. We consider a linear stochastic equation with ``multiplicative noise'' for the wave function $\phi(t)$ in the purely diffusive case. Then, we discuss how to get the physical probabilities and we derive the nonlinear SSE for the conditional states $\psi(t)$. In Section \mathrm{e}f{sec:simM} we describe the simulation techniques for SSEs and we show the simulations for two Markovian processes, the damped harmonic oscillator and a two-level atom with homodyne photodetection. Section \mathrm{e}f{sec:memory} is devoted to the introduction of coloured noise in the SSEs; we limit the presentation of this part of the theory to a restricted, but significant, class of SSEs with memory. The simulation of such non-Markovian processes is also proposed and applied as a test of other approximation techniques. In Section \mathrm{e}f{sec:concl} we briefly summarize the main results and indicate some directions of future work. Basic concepts from the theory of stochastic processes are summarized in the Appendix \mathrm{e}f{appen}.
\section{Stochastic Schr\"{o}dinger equations}\label{sec:SSE}
In this paper we will show the approach to the theory of open quantum systems based on stochastic differential equations (SDEs), with particular emphases on continuous measurements. In this theory there are four kinds of SDEs: the linear stochastic Schr\"{o}dinger equation (lSSE), a linear SDE for non-normalized vectors in the Hilbert space of the system \eqref{2.1.3}, the SSE, a nonlinear SDE for normalized vectors in the Hilbert space \eqref{SSE1}, the linear stochastic master equation \cite[Sections 3.1.2, 3.4.1]{3}, a linear SDE for positive trace-class operators, and the stochastic master equation \cite[Sections 3.5, 5.1]{3}, a nonlinear SDE for density matrices. Two kinds of noises may appear in the SSEs and characterize the jump and the diffusive cases. Here we will focus on the diffusive case. For SSEs and SMEs of the diffusive type a Wiener process $B$ appears in the linear equations and a Wiener process $W$ in the nonlinear equations; $B$ and $W$ are connected by the Girsanov transformation \eqref{newW}.
To have some hints on what we will construct, let us consider an instantaneous and pure state preserving measurement of some quantity $X$ with discrete values $\{x_k\}$. In the Hilbert space formulation of quantum mechanics, such an observation is represented by a collection of operators $\{E_k\}$ such that $\sum_kE_k^{\;\dag}E_k=\mathds1$; these operators acts on $\mathscr{H}$, the Hilbert space of the system. The map $x_k \mapsto E_k^{\;\dag}E_k$ is a (discrete) positive-operator valued measure, the modern generalization of quantum observable. Let $\varphi\in \mathscr{H}$, $\norm{\varphi}=1$, be the pre-measurement state and set $\phi_k=E_k\varphi$. Then, $\norm{\phi_k}^2\equiv \langle \varphi| E_k^{\;\dag}E_k\varphi\rangle$ is interpreted as the probability of the result $\{X=x_k\}$ in the measurement and $\psi_k=\phi_k/\norm{\phi_k}$ as the state of the system after the measurement given the result $\{X=x_k\}$. The conditional state $\psi_k$ is often called the \textit{a posteriori state} \cite{3}. For the case of measurement in continuous time the output is not discrete, but it is a whole trajectory of some observed quantity; this brings into play the stochastic processes. Apart from this complication, the lSSE is an evolution equation for the analog of the non-normalized vectors $\phi_k$, while the SSE is the evolution equation for the analog of the post-measurement states $\psi_k$. Note that the map $\varphi\mapsto \phi_k=E_k\varphi$ is linear, while the map $\phi\mapsto \psi_k$ is non-linear due to the normalization; the same difference will characterize the passage from the lSSE to the SSE.
\subsection{The linear stochastic Schr\"odinger equation}\label{sec:lSSE}
The SDEs we consider are driven by white noise. Some notions on Wiener process and stochastic calculus are given in Appendix \mathrm{e}f{appen}, but for a full presentation see \cite{KarS91, Mao97} and for a summary see \cite{3}.
First of all we work in a reference probability space
($\Omega, \mathscr{F}, \mathbb{Q}$), where $\Omega$ is the sample space, $\mathscr{F}$ the $\sigma$-algebra of events, and $\mathbb{Q}$ a reference probability. A filtration is a family $(\mathscr{F}_t)_{t\geqslant 0}$ of increasing sub-$\sigma$-algebras of $\mathscr{F}$, i.e. $\mathscr{F}_s\subset \mathscr{F}_t\subset \mathscr{F}$ for $0\leq s< t< +\infty$. Sometimes, $(\Omega, \mathscr{F}, (\mathscr{F}_t), \mathbb{Q})$ is said to be a stochastic basis. Typically, a filtration describes the accumulation of information during time: each $\mathscr{F}_t$ is the collection of all the events which we can decide whether they have been verified or not by observations up to time $t$. In the basis $(\Omega, \mathscr{F}, (\mathscr{F}_t), \mathbb{Q})$ a continuous, adapted $d$-dimensional Wiener process $B=\{B_j(t),\; t\geq 0,\; j=1,\ldots,d\}$ is defined (see Appendix \mathrm{e}f{Wiener}).
Let us start from a generic homogeneous linear SDE with ``multiplicative'' noise for the process $\phi(t)$ \cite{3}:
\begin{equation}\label{2.1.1}
\mathrm{d}\phi(t) = K(t)\phi(t)\,\mathrm{d} t +\sum_{j=1}^{d} R_j(t)\phi(t)\,\mathrm{d} B_j(t),
\end{equation}
where $\phi(0)=\psi_0$, $\psi_0\in \mathscr{H}$, the coefficients $R_j(t), K(t)$ are (non-random) linear operators on $\mathscr{H}$. The SDE \eqref{2.1.1} is to be intended in integral sense and the solution $\phi$ is the continuous, adapted It\^{o} process satisfying
\[
\phi(t)=\psi_0+\int_{0}^{t}K(s)\phi(s)\,\mathrm{d} s+\sum_{j=1}^{d}\int_{0}^{t}R_j(s)\phi(s)\,\mathrm{d} B_j(s).
\]
The last term in the above equation is a stochastic It\^{o} integral (see Appendix \mathrm{e}f{Int}).
\paragraph{The physical probability.}
To develop the theory, we need $\norm{\phi(t)}^2$ to be a probability density, cfr.\ the hints at the beginning of Section \mathrm{e}f{sec:SSE}. Precisely, let us define
\begin{equation}\label{newprob}
\mathbb{P}^t_{\psi_0}(F):=\int_F \norm{\phi(t,\omega)}^2\,\mathbb{Q}(\mathrm{d}\omega)=\operatorname{\mathbb{E}}_\mathbb{Q}[\norm{\phi(t)}^21_F], \quad \forall F\in \mathscr{F}_t,
\end{equation}
where $1_F$ is the indicator function of the set $F$. To guarantee that \eqref{newprob} defines a probability measure, we have to ask only the normalization: \begin{equation}\operatorname{\mathbb{E}}_\mathbb{Q}[\norm{\phi(t)}^2]=1, \qquad \forall t\geq 0.\end{equation}
Obviously observations in the future cannot change the probabilities on past events and to get this we need a consistency property: \begin{equation}\label{consistency}
\mathbb{P}^t_{\psi_0}(F)=\mathbb{P}^s_{\psi_0}(F), \qquad \forall F\in \mathscr{F}_s, \qquad \forall t,s:t\geq s \geq 0.
\end{equation}
This is equivalent to asking $\norm{\phi(t)}^2$ to be a $\mathbb{Q}$-\emph{martingale} (Appendix \mathrm{e}f{Martingale}). Then, its mean is a constant and the normalization for every time reduces to the normalization of the initial state $\psi_0$.
Using It\^{o}'s lemma (Appendix \mathrm{e}f{Calc}) for $\mathrm{d}\Arrowvert\phi(t)\Arrowvert^2$ we can derive as in \cite[Section 2.2.3]{3}:
\begin{multline}\label{2.1.2}
\Arrowvert\phi(t)\Arrowvert^2=\Arrowvert\psi_0\Arrowvert^2+\int_{0}^{t}\big\langle\phi(s)\big|\biggl( K(s)+K(s)^{\dag}+\sum_{j}R_j(s)^{\dag}R_j(s)\biggr)\phi(s)\big\rangle\,\mathrm{d} s \\
{}+\sum_{j=1}^{d}\int_{0}^{t}\big\langle\phi(s)\big|\left(R_j(s)+R_j(s)^{\dag}\mathrm{i}ght)\phi(s)\big\rangle\,\mathrm{d} B_j(s).
\end{multline}
In order to reduce $\Arrowvert\phi(t)\Arrowvert^2$ to a martingale, we need the integrand in the time integral in Eq.\ (\mathrm{e}f{2.1.2}) to vanish for every initial condition, i.e.
\begin{equation}
K(t)+K(t)^{\dag}+\sum_jR_j(t)^{\dag}R_j(t)=0.\nonumber
\end{equation}
Then, the operator $K(t)$ has the structure
\begin{equation}\label{K(t)}
K(t)=-\mathrm{i} H(t)-\frac 1 2 \sum_{j=1}^{d}R_j(t)^{\dag}R_j(t),
\end{equation}
where $H(t)$ is a self-adjoint operator on $\mathscr{H}$, called the \textit{effective Hamiltonian} of the system.
\paragraph{The lSSE.} Finally, the \textit{linear stochastic Schr\"{o}dinger equation} (diffusive type) is given by
\begin{gather}\label{2.1.3}
\mathrm{d} \phi(t)=\biggl(-\mathrm{i} H(t)-\frac{1}{2}\sum_{j=1}^{d}R_j(t)^{\dag}R_j(t)\biggr)\phi(t)\,\mathrm{d} t+\sum_{j=1}^{d}R_j(t)\phi(t)\,\mathrm{d} B_j(t),
\\
\phi(0)=\psi_0, \quad \psi_0\in\mathscr{H}, \quad \norm{\psi_0}=1, \qquad H(t)=H(t)^{\dag}.
\end{gather}
The linear stochastic Schr\"{o}dinger equation \eqref{2.1.3} reduces to an ordinary Schr\"{o}dinger equation $\mathrm{d}\phi(t)/\mathrm{d} t =-\mathrm{i} H(t)\phi(t)$ when we switch off the measurement and the interactions with the environment $(R_j(t) \equiv 0)$.
\subsection{The a posteriori states, the output and the master equation}
Let us consider now a finite time interval $[0,T]$; the current time $t$ will be always inside this interval. We also introduce the normalized version $\psi(t)$ of the vector $\phi(t)$:
\begin{equation}\label{apos}
\psi(t)= \frac{\phi(t)}{\norm{\phi(t)}}.
\end{equation}
Then, the interpretation of the theory is similar to the hints given at the beginning of Section \mathrm{e}f{sec:SSE} and it is given here below.
\begin{enumerate}
\item The physical probability of the events occurring up to time $T$ is $\mathbb{P}^T_{\psi_0}$. By the consistency property \eqref{consistency} the choice of $T$ is immaterial.
\item The cumulated output of the continuous measurement is the $d$-dimensional process $B$ and its distribution is given by the physical probability, so that it is no more a Wiener process. More precisely the output in any time interval $[s,t]$ is $B(t)-B(s)$, so that the instantaneous output is the formal time derivative $\dot B(t)$. The structure of the output under the physical probability is given in Eq.\ \eqref{Gir}.
\item The normalized vector $\psi(t)$ \eqref{apos} is the a posteriori state, i.e.\ the conditional state of the system at time $t$ given the observed output up to time $t$. The evolution of $\psi(t)$ is given by the SSE \eqref{SSE1}.
\end{enumerate}
Let us introduce now the average state
\begin{equation}\label{eta(t)1}
\eta(t)=\operatorname{\mathbb{E}}_{\mathbb{P}^T_{\psi_0}} [ |\psi(t)\rangle\langle \psi(t)| ] \equiv \int_\Omega |\psi(t,\omega)\rangle\langle \psi(t,\omega)| \mathbb{P}^T_{\psi_0}(\mathrm{d} \omega), \qquad T\geq t\geq 0.
\end{equation}
Note that, by construction, $\eta(t)$ is a positive operator and that, by the normalization of $\psi(t)$, one has $\operatorname{Tr}\{\eta(t)\}=1$, so that $\eta(t)$ is a statistical operator.
\begin{enumerate}\setcounter{enumi}{3}
\item The statistical operator $\eta(t)$ is the state we attribute to the system at time $t$, when the output is not known; it is called the \emph{a priori state} and satisfies the master equation \eqref{master}.
\end{enumerate}
By the consistency property \eqref{consistency} we can take $T=t$. Then, by the fact that $\mathbb{P}^T_{\psi_0}(\mathrm{d} \omega)=\norm{\phi(t,\omega)}^2 \mathbb{Q}(\mathrm{d} \omega)$ and $\norm{\phi(t,\omega)}^2 |\psi(t,\omega)\rangle\langle \psi(t,\omega)| = |\phi(t,\omega)\rangle\langle \phi(t,\omega)| $, we get the equivalent expression
\begin{equation}\label{eta(t)2}
\eta(t)=\operatorname{\mathbb{E}}_\mathbb{Q} [ |\phi(t)\rangle\langle \phi(t)| ]\equiv \int_\Omega |\phi(t,\omega)\rangle\langle \phi(t,\omega)| \mathbb{Q}(\mathrm{d} \omega) .
\end{equation}
By computing the stochastic differential of $|\phi(t)\rangle\langle \phi(t)|$ and by taking the $\mathbb{Q}$-mean of the resulting equation one gets the \emph{master equation}
\begin{subequations}\label{master}
\begin{equation}
\dot \eta(t) = \mathcal{L}(t)[\eta(t)],
\end{equation}
\begin{equation}\label{L(t)}
\mathcal{L}(t)[\varrho]= -\mathrm{i} [H(t),\varrho] + \sum_{j=1}^d \biggl(R_j(t)\varrho R_j(t)^\dag - \frac 1 2 \left\{R_j(t)^{\dag}R_j(t),\varrho\mathrm{i}ght\}\biggr).
\end{equation}
\end{subequations}
Note that the Liouville operator $\mathcal{L}(t)$ turns out to be in the usual Lindblad form.
From Eq.\ \eqref{2.1.2} with condition \eqref{K(t)} and the normalization of $\psi_0$, we get \cite[Section 2.3.1]{3}, by the rules of stochastic calculus,
\begin{equation}\label{norm2phi}
\norm{\phi(t)}^2= \exp\biggl\{\sum_j\int_0^t m_j(s)\,\mathrm{d} B_j(s) -\frac 1 2 \int_0^t m_j(s)^2\, \mathrm{d} s\biggr\},
\end{equation}
\begin{equation}\label{mj}
m_j(t)=2 \operatorname{Re} \langle \psi(t)|R_j(t) \psi(t) \rangle.
\end{equation}
Then, Girsanov theorem gives that under the probability $\mathbb{P}^T_{\psi_0}$ the process
\begin{equation}\label{newW}
W_j(t)=B_j(t) - \int_0^t m_j(s)\,\mathrm{d} s, \qquad j=1,\ldots, d, \quad t\in[0,T],
\end{equation}
is a $d$-dimensional Wiener process \cite[Sections 2.3.2 and A.5.4]{3}.
Obviously we can write
\begin{equation}\label{Gir}
B_j(t)=W_j(t)+\int_{0}^{t}m_j(s) \mathrm{d} s.
\end{equation}
Then, we can say that the instantaneous output $\dot B_j(t)$ is the sum of the white noise $\dot W_j(t)$ and the regular process $m_j(t)$ (the signal). However, let us stress that white noise and signal are not in general independent under the physical probability.
The theory of continuous measurements gives also all the correlations of the output process \cite[Section 4.3]{3}. In particular, by taking the mean of both sides in Eq.\ \eqref{Gir} and by taking into account Eqs.\ \eqref{eta(t)1} and \eqref{mj}, we get immediately the mean of the output
\begin{equation}\label{Eout}
\operatorname{\mathbb{E}}_{\mathbb{P}^T_{\psi_0}}[B_j(t)]=\operatorname{Tr}\left\{\left(R_j(t)+R_j(t)^\dagger\mathrm{i}ght)\eta(t)\mathrm{i}ght\}.
\end{equation}
This equation suggests to interpret the $j$-th output as a continuous indirect monitoring of the system quantum observable $R_j(t)+R_j(t)^\dagger$. However, the final interpretation depends on the specific model. The output $B_j$ could also represent the photocurrent in homodyne or heterodyne detection; in this case the system operator $R_j(t)$ depends on the interaction with the electromagnetic field and on the local oscillator wave. The channel $j$ could also represent a pure dissipative effect due to the environment; in this case $B_j(t)$ is not observed and the role of this channel is only for introducing a dissipative contribution into the Liouville operator \eqref{L(t)}.
\subsection{The nonlinear stochastic Schr\"odinger equation}\label{sec:nlSSE}
By using the rules of It\^o calculus and the lSSE, it is possible to compute the stochastic differential of the a posteriori state $\psi(t)=\norm{\phi(t)}^{-1}\phi(t)$. By expressing the result in terms of the new Wiener process \eqref{newW}, the final result is the SSE
\begin{multline}\label{SSE1}
\mathrm{d}\psi(t)=\sum_j\biggl[R_j(t)- \frac12\, m_j(t)\biggr]\psi(t)\,\mathrm{d} W_j(t)
\\ {}+\biggl[-\mathrm{i} H(t) - \frac 12 \sum_j R_j(t)^\dag R_j(t)+\frac 1 2 \sum_jm_j(t)R_j(t)-\frac{1}{8}\sum_j m_j(t)^2\biggr]\psi(t)\,\mathrm{d} t.
\end{multline}
As $m_j(t)$ \eqref{mj} is a bilinear function of $\psi(t)$, the SSE \eqref{SSE1} turns out to be a closed SDE for the process $\psi(t)$ under the probability $\mathbb{P}^T_{\psi_0}$ \cite[Section 2.5.1]{3}.
Let us note that the master equation \eqref{master} is invariant under the transformation $R_j(t) \to \mathrm{e}^{\mathrm{i} \theta_j}R_j(t)$. However, this is not true for the lSSE \eqref{2.1.3}, the SSE \eqref{SSE1} and the output \eqref{Gir}; indeed, $m_j(t)$ \eqref{mj} and its mean \eqref{Eout} are sensible to the phase of $R_j(t)$. So, the a posteriori states and the output depend on a phase shift in the operators of the dissipative part, while the mean dynamics is independent from such phases.
It is possible also to start from the SSE \eqref{SSE1}. In this case $W$ is a Wiener process under a probability $\mathbb{P}$, which is directly the physical probability. Then, the output is defined by Eqs.\ \eqref{Gir} and \eqref{mj} and a lSSE can be introduced by a change of normalization and of probability \cite[Section 2.5.4]{3}.
A characteristic feature of the non-linear SSEs is to preserve the normalization of the state $\psi(t)$.
\subsection{The case of a random unitary evolution}\label{sec:randomunitary}
A very particular case is when all the operators $R_j(t)$ are anti-selfadjoint:
\begin{equation}
R_j(t) = -\mathrm{i} V_j(t),\qquad V_j(t)^{\dag} = V_j(t).
\end{equation}
Then, Eqs.\ \eqref{newprob}, \eqref{apos}, \eqref{norm2phi}, \eqref{mj}, \eqref{Gir} give $m_j(t)=0$, $\norm{\phi(t)}^2=1$, $\psi(t)=\phi(t)$, $\mathbb{P}^T_{\psi_0}=\mathbb{Q}$, $W_j(t)=B_j(t)$. This means that the $B_j$ are pure noises and there is no true measurement on the system. Moreover, the lSSE and the nonlinear one coincide and give a random unitary evolution:
\begin{equation}\label{Uevol}
\mathrm{d}\psi(t)=-\mathrm{i}\biggl[H(t)\,\mathrm{d} t+\sum_{j}V_j(t)\,\mathrm{d} W_j(t)\biggr]\psi(t)-\frac{1}{2}\sum_{j}V_j(t)^2\psi(t)\,\mathrm{d} t.
\end{equation}
Formally, $H(t)+\sum_{j}V_j(t)\,\dot W_j(t)$ is the random Hamiltonian which generate the the unitary evolution. The last term is the It\^o correction due to the presence of the white noise $\dot W_j(t)$ in the formal Hamiltonian.
This class of SSEs was introduced as a model of dissipative evolution, without observation. In this case, all the physical quantities are obtained as a mean with respect to $W$ \cite{35}.
\section{Simulating SSEs for the Markovian case}\label{sec:simM}
The idea of unravelling has been a real breakthrough for simulating master equations; it is at the root of the Monte-Carlo wave function method \cite{4,23,30,31}. The basic idea of these methods is to generate independent realizations of the underlying stochastic process by a numerical algorithm and to estimate with the help of statistical means all desired expectation values from a sample of such realizations. A stochastic simulation thus amounts to perform an experiment on a computer. It yields the outcomes of single runs with their correct probabilities and provides, in addition to the mean values, estimates for the statistical errors of the quantities of interest.
Let us consider the SSE \eqref{SSE1} for the a posteriori states $\psi(t)$, with a standard Wiener process $W$ in a stochastic basis $\bigl(\Omega, \mathscr{F}, (\mathscr{F}_t), \mathbb{P}\bigr)$.
A stochastic simulation algorithm serves to generate a sample of independent realizations of the stochastic process $\psi(t)$ for the conditional wave function. Let us denote these realizations by $\psi^r(t)$, $r=1,2,...,R$, where $R$ is the number of realizations in the sample.
A quantity of interest can be thought as a real functional $F[\psi,t]$ of the a posteriori states $\psi(s)$, $s\in[0,t]$; then, let
\begin{equation}\label{Mt}
M_t=\operatorname{\mathbb{E}}_\mathbb{P}\big[F[\psi,t]\big]
\end{equation}
be its mean value.
An \emph{unbiased and consistent estimator} for the expectation value $M_t$ is provided by the \emph{sample mean}
\begin{equation}
\widehat{M}_t=\frac{1}{R}\sum_{r=1}^{R}F[\psi^r,t],
\end{equation}
where a hat is used to indicate an estimator. It is clear that the estimate is subjected to statistical errors. A natural measure of the goodness of an estimator is its \emph{mean square error}, which coincides with its variance in the case of an unbiased estimator. By the independence of the realizations we have
\begin{equation}
\text{MSE}_{\widehat M_t}= \operatorname{Var}_\mathbb{P}\big[\widehat M_t\big]=\frac{\operatorname{Var}_\mathbb{P} \big[F[\psi,t]\big]}{R},
\end{equation}
\begin{equation}
\operatorname{Var}_\mathbb{P} \big[F[\psi,t]\big]=\operatorname{\mathbb{E}}_\mathbb{P} \left[ \left(F[\psi,t]- M_t\mathrm{i}ght)^2 \mathrm{i}ght] =\operatorname{\mathbb{E}}_\mathbb{P} \left[ F[\psi,t]^2 \mathrm{i}ght] - {M_t}^2.
\end{equation}
Obviously, $\operatorname{Var}_\mathbb{P} \big[F[\psi,t]\big]$ is a theoretical quantity and needs to be estimated; its natural unbiased estimator is the \emph{sample variance}. At the end, the natural unbiased estimator of the mean square error is
\begin{equation}
\widehat{\sigma}^{\,2}_t=\widehat{\text{MSE}}_{\widehat M_t}= \frac{1}{R(R-1)}\sum_{r=1}^{R}\left(F[\psi^r,t]-\widehat{M}_t\mathrm{i}ght)^2 =\frac 1 {R-1} \biggl( \frac 1 R \sum_r F[\psi^r,t]^2 - \widehat{M}_t^{\;2}\biggr).
\end{equation}
The quantity $\widehat{\sigma}_t$ is known as the \emph{sample standard error} of the estimate of the mean value $ M_t$.
If the realizations in the sample are statistically independent, as we have assumed, and $\operatorname{Var}_\mathbb{P} \big[F[\psi,t]\big]$ is finite, the standard error $\widehat{\sigma}_t$ decreases with the square root of the sample size $R$:
\begin{equation}
\widehat{\sigma}_t\sim\frac{1}{\sqrt{R}}.
\end{equation}
Of particular interest are the a posteriori quantum expectation values of some selfadjoint operator $C$: $F[\psi, t] = \langle \psi(t)|C\psi(t)\rangle $. Note that to have these quantities for any $C$ in a basis in the space of the bounded selfadjoint operators is equivalent to give all the matrix elements of the a posteriori state $\rho(t)=|\psi(t)\rangle \langle \psi(t) |$. By Eqs.\ \eqref{eta(t)1} and \eqref{Mt} we get
\begin{equation}
M_t=\operatorname{\mathbb{E}}_\mathbb{P}[\langle\psi(t)|C\psi(t)\rangle]= \operatorname{Tr} \left\{C\eta(t)\mathrm{i}ght\}.
\end{equation}
Now the estimator of $M_t$ takes the form
\begin{equation}
\widehat{M}_t=\frac{1}{R}\sum_{r=1}^{R}\langle\psi^r(t)|C\psi^r(t)\rangle,
\end{equation}
and the estimator of its mean square error becomes
\begin{equation}
\widehat{\sigma}^{\,2}_t=\frac{1}{R(R-1)}\sum_{r=1}^{R}\left(\langle\psi^r(t)|C\psi^r(t)\rangle-\widehat{M}_t\mathrm{i}ght)^2.
\end{equation}
Let us stress that the sample standard error $\widehat{\sigma}_t$ is a measure of the statistical fluctuations, not of the numerical errors in the simulations, such that the ones due to approximations or to the discretization of the time in solving the evolution equation.
\subsection{Homodyne photodetection}\label{sec:hp}
Let us consider as a first example the stochastic Schr\"{o}dinger equation corresponding to homodyne photodetection \cite{4} of the light emitted by a two-level atom stimulated by a perfectly coherent laser in resonance with the atomic frequency \cite[Sections 8.1.3.2 and 9.2]{3}. We consider the ideal case in which all the emitted light is detected and no other dissipative contribution is present, apart from the emission of light.
Let $|1\rangle$ ($|0\rangle$) be the excited (ground) state and let $\sigma_x$, $\sigma_y$, $\sigma_z$ be the usual Pauli matrices and $\sigma_{-}$ and $\sigma_{+}$ be the lowering and rising operators; then, $\sigma_++\sigma_-=\sigma_x$, \ $\mathrm{i}\left(\sigma_-+\sigma_+\mathrm{i}ght)=\sigma_y$ and $\sigma_+\sigma_-$ is the projection on the excited state.
The model we are considering is determined by the SSE \eqref{SSE1}, \eqref{mj} with $d=1$,
\begin{subequations}\label{model1}
\begin{equation}
H(t)= \frac {\omega_0}2\, \sigma_z - \frac{\Omega_R}2\left( \mathrm{e}^{\mathrm{i} \omega_0t}\sigma_- +\mathrm{e}^{-\mathrm{i} \omega_0t}\sigma_+\mathrm{i}ght), \qquad \omega_0>0, \quad \Omega_R\geq 0,
\end{equation}
\begin{equation}
R(t)= \sqrt{\gamma}\, \mathrm{e}^{\mathrm{i} \left(\omega_0t+\theta\mathrm{i}ght)}\sigma_-\, , \qquad \gamma>0.
\end{equation}
\end{subequations}
In this model the frequencies of the atom, of the stimulating laser and of the local oscillator are equal and given by $\omega_0$; $\Omega_R$ is the \emph{Rabi frequency} ($\Omega_R^{\;2}$ is proportional to the laser intensity), $\gamma$ is the \emph{natural linewidth} of the atom ($1/\gamma$ is the relaxation time) and $\theta$ is the phase shift of the local oscillator with respect to the emitted light. Homodyne detection is sensitive to $\theta$, as discussed in Section \mathrm{e}f{sec:nlSSE}; here, we take $\theta=\pi/2$.
The explicit time dependencies can be eliminated by a unitary transformation:
\begin{equation}
\check \psi(t):= \exp\left\{\frac \mathrm{i} 2 \, \omega_0 \sigma_z t\mathrm{i}ght\} \psi(t).
\end{equation}
Then, by Eqs.\ \eqref{mj}, \eqref{SSE1}, \eqref{model1} we get the SSE in the rotating frame:
\begin{multline}\label{3.2.1}
\mathrm{d}\check\psi(t)=-\mathrm{i} H_L\check \psi(t)\,\mathrm{d} t+ \frac{\gamma}{2}\left(m_y(t)\mathrm{i} \sigma_{-}- \sigma_{+}\sigma_{-}- \frac{1}{4}\,m_y(t)^2 \mathrm{i}ght)\check \psi(t)\,\mathrm{d} t\\
{}+\sqrt{\gamma}\left(\mathrm{i}\sigma_{-}-\frac{1}{2}\,m_y(t) \mathrm{i}ght)\check\psi(t)\,\mathrm{d} W(t),
\end{multline}
\begin{equation}
H_L=-\frac{\Omega_R}{2}\,\sigma_x, \qquad m_y(t)=\langle \check \psi(t)|\sigma_y \check \psi(t) \rangle.
\end{equation}
Moreover, by Eqs.\ \eqref{Gir} and \eqref{mj}, the cumulated output (the integrated \emph{homodyne photocurrent}) is given by
\begin{equation}\label{mod1_out}
B(t)=W(t)+ \sqrt{\gamma}\int_0^t m_y(s)\,\mathrm{d} s.
\end{equation}
The master equation corresponding to the SSE \eqref{3.2.1} is
\begin{equation}
\frac{\mathrm{d}\check \eta(t)}{\mathrm{d} t}=\check \mathcal{L}\big[\check \eta(t)\big], \qquad \check \mathcal{L}[\varrho]= \frac{\mathrm{i} \Omega_R}{2}[\sigma_x,\varrho] +\gamma \sigma_-\varrho\sigma_+ -\frac\gamma 2 \left\{\sigma_+\sigma_-,\varrho\mathrm{i}ght\}.
\end{equation}
This equation can be easily solved \cite{4} and we get, with the initial condition $\eta(0)=|0\rangle \langle 0|$ and $\Omega_R^{\,2}>\gamma^2/16$, \cite[Section 8.2.2.2]{3}
\begin{equation}\label{1eta1}
\eta(t)_{11}=\langle 1|\eta(t)|1\rangle = v_+ \mathrm{e}^{-a_+ t}+v_-\mathrm{e}^{-a_- t} + \frac {\Omega_R^{\,2}}{2\Omega_R^{\,2}+\gamma^2}, \qquad \operatorname{Tr}\{\sigma_x \eta(t)\}=0,
\end{equation}
\begin{equation}\label{y}
\operatorname{Tr}\{\sigma_y \eta(t)\}=u_+\mathrm{e}^{-a_+ t}+u_-\mathrm{e}^{-a_- t} - \frac {\Omega_R\gamma}{\Omega_R^{\,2}+\gamma^2/2},
\end{equation}
\[
u_\pm=\frac{\Omega_R
\left[ \gamma \sqrt{\Omega_R^{\,2}-\gamma^2/16}\mp \mathrm{i} \left(\Omega_R^{\,2}-\gamma^2/4\mathrm{i}ght)\mathrm{i}ght]}{\sqrt{\Omega_R^{\,2}-\gamma^2/16}\left(2\Omega_R^{\,2}+\gamma^2\mathrm{i}ght)},
\]
\[
v_\pm=\frac{\Omega_R^{\,2}\left(\mp 3\mathrm{i} \gamma /4 -
\sqrt{\Omega_R^{\,2}-\gamma^2/16}\mathrm{i}ght)}{2\sqrt{\Omega_R^{\,2}-\gamma^2/16}\left(2\Omega_R^{\,2}+\gamma^2\mathrm{i}ght)}
, \qquad
a_\pm= \frac 3 4 \, \gamma \pm \mathrm{i} \sqrt{\Omega_R^{\,2}-\frac{\gamma^2}{16}}.
\]
Note that
\begin{equation}\label{meanOUT}
\operatorname{\mathbb{E}}_\mathbb{P}[B(t)]=\sqrt{\gamma}\int^t_0 \operatorname{Tr}\{\sigma_y \eta(s)\}\mathrm{d} s.
\end{equation}
To simulate this model we use the Euler algorithm to get an approximation for the state vector $\check\psi$, with a correction to maintain the normalization. We discretize the time and set $t_n= n \Delta t$; then, the algorithm takes the form
\begin{subequations}\label{recursion}
\begin{equation}\label{recur}
\psi_{n+1}=\check\psi_n+A_1(\check\psi_n)\Delta t+A_2(\check\psi_n)\Delta W_n,
\end{equation}
\begin{equation}
\check\psi_{n+1}=\frac{\psi_{n+1}}{\norm{\psi_{n+1}}},
\end{equation}
where $\Delta W_n= W(t_{n+1})-W(t_n)=Z_n \sqrt{\Delta t}$, $Z_0,\ldots,Z_n,\ldots$ is a sequence of independent random variables with standard normal distribution, and the functions $A_1, A_2$ are given by
\begin{gather}
A_1(\psi)=-\mathrm{i} H_L \psi+ \frac{\gamma}{2}\left(\langle\psi|\sigma_y\psi\rangle \mathrm{i}\sigma_{-}- \sigma_{+}\sigma_{-}- \frac{1}{4}\,\langle\psi|\sigma_y\psi\rangle^2 \mathrm{i}ght) \psi,\\
A_2(\psi)=\sqrt{\gamma}\left(\mathrm{i}\sigma_{-}-\frac{1}{2}\langle\psi|\sigma_y \psi\rangle\mathrm{i}ght)\psi.
\end{gather}
As initial condition we take the ground state
\begin{equation}
\check \psi_0 =\psi_0 =|0\rangle,
\end{equation}
\end{subequations}
\begin{figure}
\caption{A single realization of the occupation of the excited state $\rho_{11}
\label{mod}
\end{figure}
By construction, $\psi_n$ is an approximation of $\check \psi(t_n)$, so that $
\psi(t_n)\simeq \exp\left\{-\frac \mathrm{i} 2 \, \omega_0 t_n \sigma_z\mathrm{i}ght\} \psi_n$. Correspondingly, by \eqref{mod1_out}, the approximation of the integrated homodyne current is
\begin{equation}\label{approxOUT}
B(t_n)\simeq B_n= \sum_{k=0}^{n-1}\left( \Delta W_k + \sqrt{\gamma} \langle \psi_k|\sigma_y\psi_k\rangle \Delta t\mathrm{i}ght).
\end{equation}
Let us note that, by the properties of the Wiener process, $\Delta W_n/\sqrt{\Delta t}$, $n=1, 2, \ldots$, is a sequence of independent and identically distributed random variables with standard normal distribution.
The results of the simulation are shown in Figs.\ \mathrm{e}f{mod}--\mathrm{e}f{mod1}. A single realization is shown in Fig.\mathrm{e}f{mod} for the occupation of the excited state.
In Fig.\ \mathrm{e}f{modX} we plot a single realization of the output and, for comparison, its mean.
\begin{figure}
\caption{A single realization of the output $B(t)/\sqrt{t}
\label{modX}
\end{figure}
Finally, in Fig.\ \mathrm{e}f{mod1} we analyse the dependence of the simulation algorithm on the time step size. It is clearly seen that the quality of the simulation with the help of Euler algorithm decreases with increasing time step. In principle, extrapolation techniques can correct the results. However, it is more efficient to use the higher order scheme such as the Platen scheme as we shall demonstrate in the next section.
\begin{figure}
\caption{The average over 10000 realizations of the homodyne photodetection (\mathrm{e}
\label{mod1}
\end{figure}
\subsection{Damped harmonic oscillator}
Another typical example of an open system in the Markovian regime is the stochastic Schr\"{o}dinger equation \eqref{SSE1} for the damped harmonic oscillator \cite[Section 7.3.1.2]{4}:
\begin{multline}\label{3.1.1}
\mathrm{d}\psi(t)=\frac{\gamma}{2}\left(\langle a+a^\dag\rangle_{\psi(t)}a-a^\dag a-\frac{1}{4}\langle a+a^\dag\rangle^{\;2}_{\psi(t)}\mathrm{i}ght)\psi(t)\,\mathrm{d} t\\
{}+\sqrt{\gamma}\left(a-\frac{1}{2}\langle a+a^\dag\rangle_{\psi(t)}\mathrm{i}ght)\psi(t)\,\mathrm{d} W(t),
\end{multline}
\[
\langle a+a^\dag\rangle_{\psi}=\big\langle \psi\big|\left(a+a^\dag\mathrm{i}ght)\psi\big\rangle.
\]
The SSE \eqref{3.1.1} could be obtained as Eq.\ \eqref{3.2.1} by considering an harmonic oscillator with homodyning and by performing a unitary transformation. However, here the interest in this model is mainly to use it for introducing a higher order numerical scheme.
As an example, the initial state is taken to be $\psi_0=|n_0=9\rangle$ (a pure Fock state with 9 photons) and the Hilbert space has been truncated at $n_{\mathrm{max}}=12$ which means that the simulation was performed in a subspace of dimension $N=13$. The size of the time steps is $\Delta t = 0.02$.
To simulate this model we use the second-order weak scheme of Platen \cite{4}. This algorithm has the form
\begin{eqnarray}
\psi_{n+1}&=&\psi_n+\frac{1}{2}\left(D_1(\tilde{\psi}_n)+D_1(\psi_n)\mathrm{i}ght)\Delta t\nonumber\\
&&{}\ {}+\frac{1}{4}\left(D_2(\psi^{+}_n)+D_2(\psi^{-}_n)+2D_2(\psi_n)\mathrm{i}ght)\Delta W_n\nonumber\\
&&{}\ {}+\frac{1}{4}\left(D_2(\psi^{+}_n)-D_2(\psi^{-}_n\mathrm{i}ght)\{(\Delta W_n)^2-\Delta t\}\Delta t^{-1/2}\nonumber,
\end{eqnarray}
where
\begin{eqnarray}
\tilde{\psi}_n&=&\psi_n+D_1(\psi_n)\Delta t+D_2(\psi_n)\Delta W_n,\nonumber\\
\psi^{\pm}_n&=&\psi_n+D_1(\psi_n)\Delta t\pm D_2(\psi_n)\sqrt{\Delta t}.\nonumber
\end{eqnarray}
For the model under consideration the functions $D_1$ and $D_2$ are
\begin{eqnarray}
D_1(\psi)&=&\frac{\gamma}{2}\left(\langle a+a^\dag\rangle_{\psi}a-a^\dag a-\frac{1}{4}\langle a+a^\dag\rangle^2_{\psi}\mathrm{i}ght)\psi,\nonumber\\
D_2(\psi_t)&=&\sqrt{\gamma}\left(a-\frac{1}{2}\langle a+a^\dag\rangle_{\psi}\mathrm{i}ght)\psi.\nonumber
\end{eqnarray}
A single realization for the damped harmonic oscillator is shown in Fig.\ \mathrm{e}f{mod3}.
\begin{figure}
\caption{A single realization of the damped harmonic oscillator for the a posteriori expectation of $n=a^\dag a$ computed from Eq.\ (\mathrm{e}
\label{mod3}
\end{figure}
The number of photons, computed from the average of 1000 realizations is shown in Fig.\ \mathrm{e}f{mod11}.
\begin{figure}
\caption{The average over 1000 realizations of the damped harmonic oscillator by Eq.\ (\mathrm{e}
\label{mod11}
\end{figure}
\section{SSEs with memory effects}\label{sec:memory}
One of the methods for the introduction of memory effects in the system is to start from the lSSE \eqref{2.1.3}, but with random coefficients $H(t)$, $R_j(t)$ and with the white noise replaced by some coloured noise. In this way we get memory in the dynamical equations, while complete positivity of the dynamical maps and the continuous measurement interpretation are preserved \cite{1,2,apr}.
In this section we want to consider a very particular case of non-Markovian SSE and to use it to illustrate two methods of numerical approximations: the simulation of the SSE and an approximation derived in \cite{apr} based on the Nakajima-Zwanzig projection method.
Specifically, we start with a lSSE driven by a coloured noise with non-random coefficients. In this way the memory is encoded in the driving noise of the lSSE, not in the coefficients. In this case, the new lSSE will be norm-preserving, as in Section \mathrm{e}f{sec:randomunitary}, and will represent a quantum system evolving under a random Hamiltonian dynamics, while the Hamiltonian is very singular and produces dissipation.
\subsection{Coloured noise}\label{sec:colnoise}
Let us consider a one-dimensional driving noise $X(t)$ and two non-random operators $C$ and $D$ on $\mathscr{H}$. The starting point is the basic linear stochastic Schr\"{o}dinger equation
\begin{equation}\label{4.15}
\mathrm{d}\psi(t)=C\psi(t)\,\mathrm{d} t+D\psi(t)\, \mathrm{d} X(t).
\end{equation}
The simplest choice of a coloured noise is the stationary Ornstein-Uhlenbeck (O-U) process defined by
\begin{equation}\label{O-U}
X(t)=\mathrm{e}^{-k t}\,\frac{Z}{\sqrt{2k}}+\int_{0}^{t}\mathrm{e}^{-k(t-s)}\, \mathrm{d} W(s), \qquad k>0,
\end{equation}
where $W(t)$ is a one-dimensional Wiener process, defined on the stochastic basis $\big(\Omega, \mathscr{F}, (\mathscr{F}_t)$, $\mathbb{P}\big)$, and $Z$ is a standard normal random variable (mean $0$ and variance $1$); $Z$ is $\mathscr{F}_0$-measurable, which means that it is independent from the Wiener process. The O-U process $X(t)$ is a Gaussian process with zero mean and correlation function
\begin{equation}
\operatorname{\mathbb{E}}_\mathbb{P}[X(t)X(s)]=\frac{\mathrm{e}^{-k|t-s|}}{2k}.
\end{equation}
It satisfies the stochastic differential equation
\begin{equation}\label{4.1.18}
\mathrm{d} X(t)=-k X(t)\,\mathrm{d} t+\mathrm{d} W(t),\qquad X(0)=Z/\sqrt{2k}.
\end{equation}
Formally, Eq.\ (\mathrm{e}f{4.15}) is driven by the derivative of the O-U process, whose two-time correlation is no more a $\delta$-function, as in the case of white noise, but it is formally given by
\begin{equation}
\operatorname{\mathbb{E}}_\mathbb{P}[\dot X(t)\dot X(s)]=\delta (t-s)-\frac{k}{2}\,\mathrm{e}^{-k|t-s|}.\nonumber
\end{equation}
Note that the Markovian regime is recovered in the limit $k\downarrow 0$. It is then straightforward that Eq.\ (\mathrm{e}f{4.15}) can be rewritten in the form
\begin{equation}
\mathrm{d}\psi(t)=\bigl(C-kX(t)D\bigr)\psi(t)\, \mathrm{d} t+D\psi(t) \mathrm{d} W(t);
\end{equation}
the initial condition is a wave function $\psi_0\in \mathscr{H}$, such that $\Arrowvert\psi_0\Arrowvert^2=1$.
As discussed in Section \mathrm{e}f{sec:lSSE} for the Markovian case, to construct consistent probabilities we need the process $\Arrowvert\psi(t)\Arrowvert^2$ to be a martingale. By It\^{o} calculus rules (see Appendix \mathrm{e}f{Calc}), the stochastic differential of $\Arrowvert\psi(t)\Arrowvert^2$ turns out to be
\begin{multline}\label{agreement}
\mathrm{d}\langle\psi(t)|\psi(t)\rangle=\langle \mathrm{d}\psi(t)|\psi(t)\rangle+\langle \mathrm{d}\psi(t)|\mathrm{d}\psi(t)\rangle+ \langle \psi(t)|\mathrm{d}\psi(t)\rangle\\
{}=\langle\psi(t)|\left[C^{\dag}+C-k X(t)(D^{\dag}+D)+D^{\dag}D\mathrm{i}ght]\psi(t)\rangle\mathrm{d} t
+ \langle\psi(t)|(D^{\dag}+D)\psi(t)\rangle\mathrm{d} W(t).
\end{multline}
Then, the process $\Arrowvert\psi(t)\Arrowvert^2$ can be a martingale only if the term in front of $\mathrm{d} t$ is equal to zero. For it we must have
\begin{equation*}
C^{\dag}+C+D^{\dag}D=kX(t)(D^{\dag}+D),\qquad \forall t,
\end{equation*}
which implies $D^\dag+D=0$ and $C^{\dag}+C+D^{\dag}D=0$. These conditions impose that there are two self-adjoint operators $L$ and $H_0$ such that $D = -\mathrm{i} L$ and $C = -\mathrm{i} H_0 - \frac{1}{2}\,L^2$. As a consequence the initial Eq.\ (\mathrm{e}f{4.15}) becomes
\begin{equation}\label{4.1.8}
\mathrm{d}\psi(t)=-\mathrm{i}\left[\bigl(H_0-k X(t)L\bigr)\mathrm{d} t+ L \,\mathrm{d} W(t)\mathrm{i}ght]\psi(t) -\frac{1}{2}\,L^2\psi(t) \, \mathrm{d} t.
\end{equation}
Apart the further randomness introduced by the term with $X(t)$, we are in the same situation of Eq.\ \eqref{Uevol} and the evolution of the quantum system is then completely determined by the time-dependent, random Hamiltonian
\begin{equation}
H(t)=H_0+\bigl(\dot W(t)-k X(t)\bigr)L.
\end{equation}
Let us stress that it is a formal expression, due to the presence of the white noise $\dot W(t)$.
As in Section \mathrm{e}f{sec:randomunitary} the model we have constructed represents a dissipative evolution, now with memory, but without observation of the quantum system. In this case there is no change of probability, $\mathbb{P}$ is also the physical probability, and $\norm{\psi(t)}=1$, $\forall t$. The theory can be generalized \cite{2} by taking the operators $C$ and $D$ dependent on the O-U process; in this way also a true continuous observation can be introduced.
Let us stress that the class of models presented in this section is very peculiar. The process $\big(X(t),\, \psi(t)\big)_{t\geq 0}$ satisfies the couple of SDEs \eqref{4.1.18} and \eqref{4.1.8}, whose coefficients depend only on the process of time $t$; then, this composed process is Markovian.
\subsection{Projection techniques and closed master equations with memory}
As in the Markovian case, the average statistical operator (the a priori state) can be introduced:
\begin{equation}\label{aprioriM}
\eta(t)=\operatorname{\mathbb{E}}_\mathbb{P}[|\psi(t)\rangle\langle \psi(t)|].
\end{equation}
However, to get a closed equation for $\eta(t)$ is not a trivial task \cite{apr}; the final result is a generalized master equation with memory. The important point is that the complete positivity of the map $\eta(0)\mapsto \eta(t)$ is guaranteed by the stochastic representation \eqref{aprioriM}. We illustrate these techniques on the model of Section \mathrm{e}f{sec:colnoise}.
Let us define the process
\begin{equation}
\rho(t)=|\psi(t)\rangle\langle\psi(t)|.
\end{equation}
By Eq.\ \eqref{4.1.8} and It\^{o} rules, we can compute the stochastic differential of $\rho(t)$; the result is the stochastic master equation
\begin{gather}\label{eq}
\mathrm{d}\rho(t)=\mathcal{L}(t)[\rho_t]\,\mathrm{d} t+\mathcal{R}[\rho_t]\,\mathrm{d} W(t)\equiv \mathcal{L}_0[\rho_t]\,\mathrm{d} t+\mathcal{R}[\rho_t]\,\mathrm{d} X(t),
\\
\mathcal{L}(t)=\mathcal{L}_0-kX(t)\mathcal{R}, \qquad \mathcal{L}_0[\rho]=-\mathrm{i}[H_0, \rho]-\frac{1}{2}\bigl[L,\left[L,\rho\mathrm{i}ght]\bigr], \qquad \mathcal{R}[\rho]=-\mathrm{i} [L,\rho].
\end{gather}
By taking the mean of Eq.\ \eqref{eq} and by recalling that $W$ has mean zero and increments independent from the past we get
\begin{equation}\label{doteta}
\dot \eta(t)=\mathcal{L}_0[\eta(t)]-k \mathcal{R}\bigl[\operatorname{\mathbb{E}}_\mathbb{P}[X(t)\rho(t)]\bigr],
\end{equation}
which is a kind of master equation with non-Markovian effects introduced by the last term. However, this master equation is not closed, because the $X(t)$ and $\rho(t)$ are random and not independent.
A closed equation can be obtained by using the Nakajima-Zwanzig method and the generalized master equation one obtains in this way can be the starting point for some approximations \cite{apr}.
Indeed, the operation of taking the mean is a projection in the space of random trace class operators. We can think to $\eta(t)$ as the \emph{relevant} part of $\rho(t)$, while $\rho_\bot(t)=\rho(t)-\eta(t)$ is the \emph{non relevant} part. As we took a non-random initial state, we have $\rho(0)=\eta(0)$, $\rho_\bot(0)=0$. By taking the stochastic differential of $\rho_\bot(t)$ and by using Eqs.\ \eqref{eq} and \eqref{doteta}, we get
the system of equations
\begin{subequations}\label{NZ}
\begin{equation}\label{NZ1}
\dot \eta(t)=\mathcal{L}_0[\eta(t)]-k \mathcal{R}\bigl[\operatorname{\mathbb{E}}_\mathbb{P}[X(t)\rho_\bot(t)]\bigr],
\end{equation}
\begin{multline} \label{NZ2}
\mathrm{d} \rho_\bot(t)=\mathcal{L}_0\left[\rho_\bot(t)\mathrm{i}ght]\mathrm{d} t - k \mathcal{R}\bigl[X(t)\rho_\bot(t)-\operatorname{\mathbb{E}}_\mathbb{P}[X(t)\rho_\bot(t)]\bigr]\mathrm{d} t \\ {}+\mathcal{R}\left[ \rho_\bot(t)\mathrm{i}ght]\mathrm{d} W(t)+\mathcal{R}\left[\eta(t)\mathrm{i}ght]\mathrm{d} X(t).
\end{multline}
\end{subequations}
Let us introduce now the propagator of the homogeneous part of Eq.\ \eqref{NZ2}, which is defined by the SDE
\begin{equation}\label{propag}
\mathcal{V}(t,s)=\mathds1+\int_{s}^{t}\bigl(\mathcal{L}_0- k\mathcal{R}\circ \mathcal{X}(r) \bigr)\circ\mathcal{V}(r,s)\,\mathrm{d} r + \int_{s}^{t}\mathcal{R} \circ \mathcal{V}(r,s) \, \mathrm{d} W(r),
\end{equation}
where $\circ$ denotes the composition of maps and $\mathcal{X}(t)$ is the map $\rho\mapsto X(t)\rho-\operatorname{\mathbb{E}}_\mathbb{P}[X(t)\rho]$. Then, the formal solution of the Eq.\ \eqref{NZ2} with $\rho_\bot(0)=0$ can be written as
\begin{equation}\label{rhobot}
\rho_\bot(t)=-k\int_0^t \mathcal{V}(t,s)\circ\mathcal{R}[X(s)\eta(s)]\,\mathrm{d} s
+ \mathcal{V}(t,0)\circ\int_0^t \mathcal{V}(s,0)^{-1}\circ\mathcal{R}[\eta(s)]\,\mathrm{d} W(s).
\end{equation}
In the last term we used $\mathcal{V}(t,0)\circ\mathcal{V}(s,0)^{-1}$ instead of $\mathcal{V}(t,s)$ in order to have an adapted integrand in the stochastic integral, as required by the It\^o formulation. By inserting the expression \eqref{rhobot} into Eq.\ \eqref{NZ1} we get the generalized master equation for the a priori states
\begin{multline}\label{GME}
\dot{\eta}_t=\mathcal{L}_0[\eta(t)]+k^2\int_{0}^{t}\mathcal{R}\circ \operatorname{\mathbb{E}}_\mathbb{P}[X(t)X(s)\mathcal{V}(t,s)]\circ \mathcal{R}[\eta(s)]\,\mathrm{d} s\\
{}-k\operatorname{\mathbb{E}}_{\mathbb{P}}\biggl[X(t)\mathcal{R}\circ \mathcal{V}(t,0) \circ \int_0^t \mathcal{V}(s,0)^{-1}\circ\mathcal{R}[\eta(s)]\,\mathrm{d} W(s)\biggr] .
\end{multline}
Equation \eqref{GME} is very complicated, but it is useful as a starting point to find approximations. In \cite{apr} it is suggested to take the non random approximation of the propagator \eqref{propag}: $\mathcal{V}(t,s) \simeq \mathrm{e}^{\mathcal{L}_0(t-s)}$. Then, the mean values in \eqref{GME} can be computed and the generalized master equation takes the form
\begin{equation}\label{48}
\dot\eta(t)\simeq \mathcal{L}_0[\eta(t)]
+\frac{k}{2}\int_{0}^{t}\left[L,\mathrm{e}^{(\mathcal{L}_0-k) (t-s)}\bigl[\left[L,\eta(s)\mathrm{i}ght]\bigr]\mathrm{i}ght]\mathrm{d} s.
\end{equation}
\subsection{A non Markovian model: a dissipative qubit}
In this section we introduce a very simple example based on a qubit with dissipation in order to have a toy model with a non Markovian dynamics for which we can do stochastic simulations and test the approximation \eqref{48}.
Let us take a two-level system as in Section \mathrm{e}f{sec:hp} and consider the stochastic dynamics \eqref{4.1.8} with
\begin{equation}
H_0=\frac{\omega_0}2\,\sigma_z, \quad \omega_0>0, \qquad L=\sqrt{\frac\gamma 2}\,\sigma_y, \quad \gamma>0.
\end{equation}
Then, the SSE \eqref{4.1.8} becomes
\begin{subequations}\label{2SSE}
\begin{gather}
\mathrm{d} \psi_1(t) = -\frac 1 2 \left( \frac \gamma 2 +\mathrm{i} \omega_0\mathrm{i}ght) \psi_1(t) \,\mathrm{d} t -\sqrt{\frac \gamma 2}\,\psi_2(t)\,\mathrm{d} X(t),
\\
\mathrm{d} \psi_2(t) = -\frac 1 2 \left( \frac \gamma 2 -\mathrm{i} \omega_0\mathrm{i}ght) \psi_2(t) \,\mathrm{d} t +\sqrt{\frac \gamma 2}\,\psi_1(t)\,\mathrm{d} X(t).
\end{gather}
\end{subequations}
The O-U process $X(t)$ is given by Eq.\ \eqref{O-U} and its stochastic differential by \eqref{4.1.18}.
For this model we have
\begin{equation}
\mathcal{L}_0[\rho]=-\frac{\mathrm{i}\omega_0}2\left[\sigma_z, \rho\mathrm{i}ght]
-\frac{\gamma}{4}\bigl[\sigma_y,\left[\sigma_y,\rho\mathrm{i}ght]\bigr].
\end{equation}
By representing the states in the Bloch sphere, the master equation $\dot\xi(t)=\mathcal{L}_0[\xi(t)]$ can be explicitly solved and the right hand side of Eq.\ \eqref{48} can be given an explicit expression. Indeed, by writing
\begin{equation}\label{Bloch}
\eta(t)=\frac 1 2 \left[ \mathds1 + \vec x(t) \cdot \vec \sigma\mathrm{i}ght],
\end{equation}
from Eq.\ \eqref{48} we get
\begin{equation}\label{Blocheqs}
\begin{cases}\displaystyle
\dot x(t)=-\omega_0y(t) -\gamma x(t) +k\gamma \int_0^t \mathrm{e}^{-(k+\gamma)(t-s)}x(s)\,\mathrm{d} s,
\\
\dot y(t)=\omega_0x(t),
\\ \displaystyle
\dot z(t)=-\gamma z(t) +k\gamma\int_0^t \mathrm{e}^{-(k+\gamma/2)(t-s)}\left(\cos \nu (t-s) - \frac\gamma{2\nu}\,\sin \nu(t-s)\mathrm{i}ght) z(s)\,\mathrm{d} s.
\end{cases}
\end{equation}
We assume to have $\omega_0>\gamma/2$ and set $\nu=\sqrt{\omega_0^{\;2}- \gamma^2/4}$. Recall that \eqref{Bloch} and \eqref{Blocheqs} give an approximation of the a priori states.
Equations \eqref{Blocheqs} can be solved by Laplace transform techniques or, equivalently, by increasing the degrees of freedom. Let us set
\begin{subequations}
\begin{gather}
\xi(t)=\gamma\int_0^t \mathrm{e}^{-(k+\gamma/2)(t-s)}\cos \bigl(\nu (t-s)\bigr) z(s)\,\mathrm{d} s,
\\
\epsilon(t)=- \frac{\gamma^2}{2\nu}\int_0^t \mathrm{e}^{-(k+\gamma/2)(t-s)}\sin \bigl(\nu(t-s)\bigr) z(s)\,\mathrm{d} s,
\\
\zeta(t)=\gamma\int_0^t \mathrm{e}^{-(k+\gamma)(t-s)}x(s)\,\mathrm{d} s.
\end{gather}
\end{subequations}
Then, Eqs.\ \eqref{Blocheqs} reduce to the two decoupled systems of linear equations with constant coefficients
\begin{subequations}\label{approxA}
\begin{equation}\label{part1}
\begin{cases}
\dot x(t)=-\omega_0y(t) -\gamma x(t) +k\zeta(t),
\\
\dot y(t)=\omega_0x(t),
\\ \displaystyle
\dot \zeta(t)=-(k+\gamma)\zeta(t)+\gamma x(t),
\end{cases}
\end{equation}
\begin{equation}\label{part2}
\begin{cases}
\dot \xi(t)=-\left(k+\frac \gamma 2 \mathrm{i}ght)\xi(t) +\frac{2\nu^2}\gamma\, \epsilon(t)+\gamma z(t),
\\
\dot \epsilon(t)=-\left(k+\frac \gamma 2 \mathrm{i}ght)\epsilon(t)-\frac \gamma 2 \,\xi(t),
\\ \displaystyle
\dot z(t)=-\gamma z +k\bigl(\xi(t)+\epsilon(t)\bigr).
\end{cases}
\end{equation}
\end{subequations}
To get the mean state $\eta(t)$ we can now use stochastic simulations or the analytical approximation of Eqs.\ \eqref{Bloch}, \eqref{approxA}. We concentrate on the study of the occupation of the excited state $\eta(t)_{11}=\frac 1 2 \left[1+z(t)\mathrm{i}ght]$. Let us stress that it is easy to prove that $\lim_{t\to +\infty}\eta(t)_{11}=0.5$.
\begin{figure}
\caption{Plot of the mean occupation of the excited state $\eta_{11}
\label{Q1}
\end{figure}
Let us start from the analytical solution.
In Figure \mathrm{e}f{Q1} we plot $\eta(t)_{11}$ obtained by solving system \eqref{part2} by using the internal function of Mathematica ``DSolve [ ]''. The choice of parameters is $\gamma=1$, $ \omega_0=\sqrt{37}/2$ (which gives $\nu=3$) and $k=0,\; 1, \;2$; recall that $k=0$ is the Markovian case. The initial state is $\eta(0)=\begin{pmatrix} 1 & 0 \\ 0&0\end{pmatrix}$.
We can say that in this model the effect of memory (increasing $k$) is to modify and to slow down the decay.
However Eqs.\ \eqref{part2} are approximated, but we can compare this solution with the simulations based on the exact equations \eqref{2SSE}, \eqref{4.1.18}. We use the Euler algorithm applied to the Markov process $\big(X(t),\psi_1(t),\psi_2(t)\big)_{t\geq 0}$ with normalization of $\psi(t)$ at every step as in Section \mathrm{e}f{sec:hp} (10000 realizations). In Figures \mathrm{e}f{Q2} and \mathrm{e}f{Q3} the dots comes from the simulations and the solid line from the analytical approximation; we see an extremely good agreement of simulations and approximated analytical solution.
\begin{figure}
\caption{Plot of the mean occupation number of the excited state for the parameters $\gamma=1$, $ \omega_0=\sqrt{37}
\label{Q2}
\end{figure}
\begin{figure}
\caption{Plot of the mean occupation number of the excited state for the parameters $\gamma=1$, $ \omega_0=\sqrt{37}
\label{Q3}
\end{figure}
\section{Conclusions}\label{sec:concl}
The theory of linear and nonlinear SSEs has been presented in the Markovian diffusive case. Moreover we have discussed their links with the dissipative dynamics of open systems and with measurements in continuous time. Two simple cases have been used to show how to make stochastic simulations based on the SSE. A two-level atom with homodyne detection has been used to show the Euler algorithm, while the Platen algorithm was illustrated in the case of a damped harmonic oscillator.
Then, we have shown how to use coloured noise in order to construct non Markovian models. Now the average state does not satisfy the usual Markovian quantum master equation. However, by adapting the Nakajima-Zwanzig projection method, it is possible to arrive to a generalized master equation and we have shown how to get an approximate solution for this equation. On the other side, the original SDEs can be simulated and the exact solution can be obtained up to numerical errors and statistical fluctuations.
In a concrete model of a dissipative qubit we have compared the analytical approximation with the stochastic simulation of the exact equation. Such a comparison gives a strong support to the proposed approximation. This gives confidence in the possibility of studying more elaborated Markovian models, for which the two computational ways of treating them are open: analytic approximations and stochastic simulations. In the proposed model we see also some effects of the non-Markovianity: there is a slowdown of the decay and and a modification of its functional form.
\section{Appendix: some theory of random processes}\label{appen}
\subsection{The Wiener Process}\label{Wiener}
A \emph{standard Wiener process} $\{W(t)\}_{t\geq 0}$ is a continuous Gaussian process starting from 0, with independent and stationary increments,
with mean zero and variance proportional to $t$; in particular, $
\mathbb{E}[W(t)]=0$ and $ \operatorname{Cov}[W(t)W(s)]=\mathbb{E}[W(t)W(s)]=\min(t,s)$.
Due to the Gaussianity and the independence of the increments, if we take a sequence of times $0\leq t_0<t_1<\cdots<t_n$ and set $Z_k=\frac{W(t_k)-W(t_{k-1})}{\sqrt{t_k-t_{k-1}}}$, then the random variables $Z_1, Z_2,\ldots,Z_n$ are independent and identically distributed, each with standard normal distribution. This fact is used for the simulation of Wiener processes and SDEs.
Finally, a $d$-dimensional Wiener process is a collection of $d$ independent one-dimensional
Wiener processes.
\subsection{Martingales and change of measure}\label{Martingale}
Let $\big(\Omega, \mathscr{F}, (\mathscr{F}_t),\mathbb{P}\big)$ be a stochastic basis as defined at the beginning of Section \mathrm{e}f{sec:lSSE}.
An \emph{adapted process} $\{X(t)\}_{t\geq 0}$ is a stochastic process in the probability space $(\Omega, \mathscr{F}, \mathbb{P})$, such that $X(t)$ is $\mathscr{F}_t$-measurable, $\forall t\geq 0$.
A stochastic process $\{X(t)\}_{t\geq 0}$ is said to be a \emph{martingale} if (a) it is adapted, (b) $\operatorname{\mathbb{E}}[\abs{X(t)}]<+\infty$, $\forall t \geq 0$, (c) $\operatorname{\mathbb{E}}[X(t)|\mathscr{F}_s]=X(s)$, $\forall t \geq s\geq0$.
An adapted Wiener process is a martingale.
Let $Z=\{Z(t), t\geq 0\}$ be a non-negative martingale with $\mathbb{E}[Z(t)]=1$. For every fixed $t\geq 0$, the random variable $Z(t)$ can be used as a density to define a new probability measure $\mathbb{Q}_t$ on $(\Omega,\mathscr{F}_t)$:
\begin{equation}
\forall F\in \mathscr{F}_t\qquad \mathbb{Q}_t(F):=\int_F Z(t,\omega)\mathbb{P}(\mathrm{d}\omega)\equiv \mathbb{E}[Z(t)1_F].\nonumber
\end{equation}
Being $Z$ a martingale, all the probabilities $\mathbb{Q}_t$, $t\geq 0$, are consistent, in the sense that
\begin{equation}
\mathbb{Q}_t(F)=\mathbb{Q}_s(F),\qquad \forall t,s:t\geq s\geq 0,\quad \forall F\in \mathscr{F}_s\nonumber.
\end{equation}
Indeed, $1_F$ is $\mathscr{F}_s$-measurable and, by the properties of conditional expectations, one has
\[
\mathbb{Q}_t(F)=\mathbb{E}[Z(t)1_F]=\mathbb{E}\left[\mathbb{E}[Z(t)1_F|\mathcal{F}_s]\mathrm{i}ght]
=\mathbb{E}\left[\mathbb{E}[Z(t)|\mathscr{F}_s]1_F\mathrm{i}ght]=\mathbb{E}[Z(s)1_F]=\mathbb{Q}_s(F).
\]
\subsection{Stochastic integrals}\label{Int}
Let $\big(\Omega, \mathscr{F}, (\mathscr{F}_t),\mathbb{P}\big)$ be a stochastic basis, $W$ an adapted Wiener process and $F$ a continuous, adapted, stochastic process with $\operatorname{\mathbb{E}}[\abs{F(t)}^2]<+\infty$, $\forall t\in [0,T]$. Then, it is possible to define the \textit{It\^{o} integral}
\begin{equation}\label{integral}
Y(T)=\int_{0}^{T}F(t)\mathrm{d} W(t)
\end{equation}
as the mean square limit for $\Delta t \downarrow 0$ of
\begin{equation}\label{sum1}
Y_{\Delta t}(T)=\sum_{k=1}^{n-1} F(t_k)\big(W(t_{k+1})-W(t_k)\big),
\end{equation}
where $0=t_0<t_1<\cdots<t_n=T$ is a partition of $[0,T]$ and $\Delta t=\max_k\{t_{k+1}-t_k\}$. This means
\begin{equation}
\lim_{\Delta t\downarrow 0}\operatorname{\mathbb{E}}\left[\abs{Y_{\Delta t}(T)-Y(T)}^2\mathrm{i}ght]=0.
\end{equation}
By approximation techniques, the definition of the stochastic integral can be generalized to an integrand $F(t)$ such that it is adapted and $\int_0^T \operatorname{\mathbb{E}}\left[ \abs{F(t)}^2\mathrm{i}ght] \mathrm{d} t < +\infty$.
Let us consider the stochastic integral as a process $Y=\{Y(t),\, t\in[0,T]\}$. The main properties of \emph{the integral process} are that it \emph{is a martingale with vanishing mean}, $\operatorname{\mathbb{E}}[Y(t)]=0$, and that \emph{the It\^o isometry holds}:
\begin{equation}
\operatorname{\mathbb{E}}\left[\abs{Y(t)}^2\mathrm{i}ght]=\int_0^t \operatorname{\mathbb{E}}\left[ \abs{F(s)}^2\mathrm{i}ght] \mathrm{d} s.
\end{equation}
These properties are easily proved on the discrete approximation \eqref{sum1} and then it is possible to show that they survive to the limiting procedure.
The definition of stochastic integral can be extended to a larger class of integrands (now limits in probability have to be used), but it is no more guaranteed that the main properties hold; we can only say that the integral process is a \emph{local martingale}.
Other definitions of stochastic integral are possible, in particular the Stratonovich integral, whose definition starts from the discrete approximation
\begin{displaymath}\sum_{k=1}^{n-1} F\big((t_k+t_{k+1})/2\big)\big(W(t_{k+1})-W(t_k)\big).\end{displaymath} While the rules of the stochastic calculus based on the Stratonovich definition are simpler than the ones based on It\^o integral, the important properties above are lost.
\subsection{It\^{o} calculus}\label{Calc}
Let now $W$ be a $d$-dimensional Wiener process defined in the stochastic basis $(\Omega, \mathscr{F}, (\mathscr{F}_t), \mathbb{P})$. An \emph{It\^o process} $X$ is a continuous, adapted process such that $X(0)$ is $\mathscr{F}_0$-measurable and
\begin{equation*}
X(t)= X(0) +\int_0^t F(s) \,\mathrm{d} s +\sum_{j=1}^d \int_0^t G_j(s)\,\mathrm{d} W_j(s),
\end{equation*}
for some adapted process, $F$ Lebesgue integrable and $G_j$ stochastically integrable. It is usual to say that $X$ admits the \emph{stochastic differential}
\begin{equation}\label{dX(t)}
\mathrm{d} X(t)= F(t) \,\mathrm{d} t + \sum_{j=1}^d G_j(t)\,\mathrm{d} W_t(t).
\end{equation}
Take now another It\^o process with stochastic differential
\begin{equation}\label{dY(t)}
\mathrm{d} Y(t)= M(t) \,\mathrm{d} t + \sum_{j=1}^d N_j(t)\,\mathrm{d} W_t(t).
\end{equation}
The \emph{It\^o lemma} says that the product $X(t)Y(t)$ of two It\^o processes is an It\^o process with initial value $X(0)Y(0)$ and stochastic differential
\[
\mathrm{d} \big(X(t)Y(t)\big)=X(t)\, \mathrm{d} Y(t)+ Y(t)\, \mathrm{d} X(t) + \bigl(\mathrm{d} X(t)\bigr) \bigl(\mathrm{d} Y(t)\bigr),
\]
where $\mathrm{d} X(t)$, $\mathrm{d} Y(t)$ have the expressions \eqref{dX(t)}, \eqref{dY(t)}, and the \emph{It\^o correction} $\bigl(\mathrm{d} X(t)\bigr) \bigl(\mathrm{d} Y(t)\bigr)$ must be computed from the product of the two differentials by using the \emph{It\^o table}
\[
(\mathrm{d} t)^2=0, \qquad \mathrm{d} t\, \mathrm{d} W_j(t)=0, \qquad \mathrm{d} W_j(t)\, \mathrm{d} W_i(t) = \delta_{ij} \, \mathrm{d} t.
\]
This result can be generalized to polynomials in $W$ and then to smooth functions of $W$; this is the It\^o formula \cite{KarS91,Mao97}, \cite[Sections A.3.3, A.3.4]{3}.
\end{document} | math | 63,029 |
\begin{document}
\draft
\preprint{HEP/123-qed}
\title{Quantum cryptographic three party protocols}
\author{J. M{\"u}ller-Quade and H. Imai}
\address{Imai Laboratory, Institute of Industrial Science, The University of
Tokyo}
\date{October $31^{st}$, $2000$}
\maketitle
\begin{abstract}
Due to the impossibility results of Mayers and Lo/Chau it is generally
thought that a quantum channel is cryptographically strictly weaker
than oblivious transfer.
In this paper we prove that in a three party scenario a quantum channel
can be strictly stronger than oblivious transfer. With the protocol
introduced in this paper we can completely classify the cryptographic
strength of quantum multi party protocols.
\end{abstract}
\pacs{03.67.-a, 03.67.Dd, 89.70.+c}
\section{Introduction}
In a multi party protocol a set $P$ of players wants to correctly
compute a function $f(a_1,\dots,a_n)$ which depends on secret inputs
of $n$ players. Some players might collude to cheat in the protocol as
to obtain information about secret inputs of the other players or
to modify the result of the computation.
Possible collusions of cheaters are modelled by {\em adversary structures}
\begin{definition}
An adversary structure is
a monotone set ${\cal A}\subseteq 2^P$, i.\,e., for subsets
$S'\subseteq S$ of $P$ the property $S\in {\cal A}$ implies $S' \in
{\cal
A}$.
\end{definition}
The main properties of a multi party protocol are:
{\footnotesize
\begin{enumerate}
\item A multi party protocol is said to be ${\cal A}$-{\em secure} if
no single collusion from $\cal A$ is able to obtain information about
the secret inputs of other participants which cannot be derived from
the result and the inputs of the colluding players.
\item A multi party protocol is ${\cal A}$-{\em partially correct} if no
possible collusion
can let the protocol terminate with a wrong result.
\item A multi party protocol is called $\cal A$-{\em fair} if no
collusion from $\cal A$ can reconstruct the result of the multi
party computation earlier then all honest participants together. No
collusion should be able to run off with the result.
\end{enumerate}
}
We will be more strict here and demand robustness even against
disruptors.
{\footnotesize
\begin{enumerate}
\item[2'] A multi party protocol is ${\cal A}$-{\em correct} whenever no
single collusion from $\cal A$ can abort the protocol, modify its
result, or
take actions such that some player gets to know a secret value.
\end{enumerate}
}
A protocol is called $\cal A$-{\em robust} if it has all of the above
properties. Note that we will allow only one collusion to cheat, but
we think of every single player as being curious, i.\,e., even if he
is not in the collusion actually cheating he will eavesdrop all
information he can obtain without being detected cheating
With oblivious transfer all multi party protocols can be realized with
perfect security if all players are
cooperating~\cite{BeaGol89,GolLev90,CreGraTap95}. But a collusion of
players can abort the calculation, see next section.
Classically one can avoid this problem only by introducing a new
cryptographic primitive which is more powerful than oblivious
transfer~\cite{ImaMue00STOCS,FitGarMauOst00}.
This paper analyzes three party quantum protocols and how they can
cope with the problem of disruption. We prove that there are
situations where a quantum channel is strictly more powerful than
oblivious transfer. Together with the results
of~\cite{May96,LoCha96,Lo96} we can conclude that the cryptographic
power of a quantum channel is uncomparable to the power of oblivious
transfer.
\section{Impossibility of classical three party protocols}
To clearly show the advantage of quantum protocols we restate the
following impossibility result of~\cite{ImaMue00Eurocrypt}.
\begin{lemma}\label{MPNoGo}
Let $P$ be a set of players for which each pair of players is
connected by a (private) oblivious transfer channel and each player
has access to an authenticated broadcast channel. Then $\cal
A$-robust multi party computations are possible for all functions if
and only if no two sets of $\cal A$ cover $P\setminus \{ P_i\}$ for
a player $P_i\in P$ or $|P|=2$.
\end{lemma}
The basic idea to prove this impossibility result
of~\cite{ImaMue00Eurocrypt} is to have two possible collusions
$A_1,A_2$ covering $P\setminus \{ P_i\}$ (for a player $P_i$) where
either all players from $A_1$ or all players from $A_2$ refuse to
cooperate with the players of the other possible collusion. Then the
single player $P_i$ has to assist all other
players. In~\cite{ImaMue00Eurocrypt} it is proven that one cannot
avoid that the player $P_i$ learns a secret.
Especially three party protocols cannot necessarily be realized
robustly if every player is possibly cheating.
\section{Three party protocols}
If all players in a three party protocol cooperate we can use the
protocols of~\cite{ImaMue00QMPforIEICE} to implement $\cal A$-robust
quantum multi party protocols. Hence we focus on the situation where
three players (Alice, Bob, and Helen) want to perform three party
protocols and Alice and Bob are in conflict. One of the two is
refusing to cooperate with the other and it is unclear for Helen who
is cheating.
As a first step we will introduce a bit commitment protocol for Alice
and Bob. The idea is that Alice sends her quantum states via Helen and
Bob does not know which quantum data is
coming from Helen and which data is just forwarded by Helen. Hence Bob
cannot complain without reason or he risks to be detected cheating by
Helen.
Forwarding information as if it were ones own without being able to
eavesdrop is impossible classically.
One further advantage of the protocol below is that Alice can forward
all information via Helen and Bob cannot know whose information it
is: An anonymous quantum channel. This way Bob cannot distinguish
between commitments of Alice and ``pretended'' commitments of
Helen. Later we want to follow this idea with larger protocols
containing this bit commitment protocol as a subprotocol. Then
we let Alice forward all her information via Helen.
\noindent{\bf Commit}($b$)
{\footnotesize
\noindent{\tt FOR} $i\in\{1,\dots,l\}$ {\tt DO}
\begin{enumerate}
\item Alice gives a random string $r$ of qubits encoded in random
bases $s$ $\in \{+,\times\}$ to Helen.
\item Helen sends a substring to Bob
interleaved with quantum states of her own.
Helen
tells Alice which quantum states are hers without revealing
information about which substring she forwarded.
\item Bob announces to have received all quantum states.
With a probability of $1/2$ he publishes all his measurement
results.
\item Alice opens to Helen the bases she used. Now Alice is bound to
the parity bit of $r$.
\end{enumerate}
{\tt OD}
\begin{enumerate}
\item[5] Alice is now bound to the Xor of all quantum states (which
were not measured and published by Bob) Alice announces (via Helen if
needed) if this parity bit is equal to the bit $b$ she originally
wanted to commit to.
\end{enumerate}
}
\noindent{\bf Unveil}
{\footnotesize
\begin{enumerate}
\item Alice opens (via Helen if necessary) all choices she made.
\item Helen and Bob check consistency.
\end{enumerate}
}
\begin{lemma}
For ${\cal A}=\{ \{Alice\},\{Bob\},\{Helen\}\}$ the above protocol
realizes an ${\cal A}$-robust bit commitment for Alice which binds her
to Bob and to Helen even if Alice and Bob are in conflict.
\end{lemma}
\begin{proof}
We say the protocol has failed if many of the quantum states measured
and published by Bob do not match what Alice sent or what Helen sent.
If there are only very few cases with discrepancies the protocol is
considered a success.
If the protocol does not fail then the protocol is concealing to
Helen. Helen has sent some substrings to
Bob and as she could not know which substrings would be measured and
published by Bob there are some substrings which she forwarded, but
which were not tested. Hence Helen cannot measure the overall parity
bit $b$ even
after getting to know the bases. Helen cannot measure before getting
to know the bases as she will be detected cheating whenever Bob
measures and publishes a quantum state she disturbed.
If the protocol did not fail it is concealing to Bob unless Helen and
Bob collude. This is clear as Bob does not have the complete quantum
state, and can hence not measure the parity bit.
If the protocol did not fail it is binding for Alice unless she
colludes with Helen, which is impossible according to our assumption.
If the protocol failed there are two cases to be considered. First,
Bob published a lot of measurement results which do not match what
Alice sent, but Helen was not complaining about Bob, then it is clear
for Alice that Helen and Bob collude somehow, but as this is not
possible according to our assumption this will not happen. The second case is
that Helen complains about Bob, then Bob is identified as a cheater as
every player complains about Bob.
\end{proof}
As Alice is by the above protocol bound to Bob and
Helen we have bit commitment from Alice to Bob and from Alice
to Helen and from Bob to Alice and from Bob to Helen. As the quantum
channel between Alice and Helen and between Bob and Helen is working
we even have oblivious transfer from Helen to Alice and from Helen to
Bob by forcing honest measurements with bit commitment~\cite{Yao95}.
\begin{corollary}\label{CorForcingMeasurments}
The above sketched oblivious transfer protocol between Alice and Helen
and between Bob and Helen is $\cal A$-robust and becomes $\tilde{\cal
A}$-robust after it terminated for ${\cal A}=\{
\{Alice\},\{Bob\},\{Helen\}\}$ and $\tilde{\cal A} = \{
\{Alice,Bob\},\{Helen\}\}$.
\end{corollary}
\begin{proof}
The bit commitment used for forcing measurments need only be shortly
binding. It need only be binding until the honest measurement is
performed unreversibly. Hence the collusion $\{Alice,Bob\}$, which
would be able to violate the binding condition but not the concealing
condition of the above bit commitment,
cannot cheat after the measurement is performed.
\end{proof}
Next we have to define some notions which are important for multi
party protocols. Details can be looked up in~\cite{CreGraTap95}.
\begin{definition}
A {\em bit commitment with Xor} ({\em BCX}) to a bit $b$ is a
commitment to bits $b_{1L}$, $b_{2L},\dots,$ $b_{mL},$ $b_{1R},\dots,$
$b_{mR}$
such that for each $i$ $b_{iL}\oplus b_{iR}=b$.
\end{definition}
The following result about zero knowledge proofs on BCX can be found
in~\cite{CreGraTap95} and in references therein.
\begin{theorem}\label{COPYworks}
Bit commitments with Xor allow zero knowledge proofs of linear
relations among several bits a player has committed to using
BCX. Especially (in)equality of bits or a bit string being contained
in a linear code.
Furthermore BCXs can be copied, as proofs may destroy a BCX.
\end{theorem}
In a multi party scenario it is necessary that a player should be
committed to all other players. In our three party case this is given
by our bit commitment protocol which binds one player (Alice or Bob)
to the other two. For Helen the following {\em global bit commitment
with Xor} can be implemented by Corollary~\ref{CorForcingMeasurments}
and the techniques used in~\cite{CreGraTap95} as she is not in
conflict with anyone.
\begin{definition}\label{DefGBCX}
A {\em global bit commitment with Xor} ({\em GBCX}) is a BCX
commitment from a player ${\rm Alice}\in P$ to all other players such
that all players are convinced that Alice did commit to the same bit
in all the different BCX.
\end{definition}
\begin{corollary}
Zero knowledge proofs of linear relations among several GBCX are
possible. Furthermore GBCX can be copied by copying the individual
BCX.
\end{corollary}
On these commitments operates the {\em committed oblivious transfer}
protocol, defined in~\cite{CreGraTap95} which forms the basis of our
multi party protocols.
\begin{definition}
Given two players Alice and Bob where Alice is committed to bits
$b_0,b_1$ and Bob is committed to a bit $a$. Then a {\em committed
oblivious transfer} protocol ({\em COT}) is a protocol where Alice inputs her
knowledge about her two commitments and Bob will input his knowledge
about his commitment and the result will be that Bob is committed to
$b_a$.
In a {\em global committed oblivious transfer} protocol all players
are convinced of the validity of the commitments, i.e., that indeed Bob is
committed to $b_a$ after the protocol.
\end{definition}
As Helen is not actively cheating by assumption and Alice or Bob
cannot complain about Helen without being expelled from the protocol
we can realize GCOT from Helen to Alice and from Helen to Bob by
following the protocol of~\cite{CreGraTap95}. To realize GCOT from
Alice to Bob is more difficult. We will do this in two steps. First we
realize a subprotocol which we call {\em subGCOT} and second we will
observe that all other steps can be realized easily once one round of
subGCOT was successfull.
To realize subGCOT between Alice and Bob we carry out the first 7
steps of the GCOT protocol of~\cite{CreGraTap95} in a way that Bob
cannot decide if the data comes from Alice or from Helen if he then
complains without reason he risks to get in conflict with Helen, which
would prove him cheating.
{\bf subGCOT}
{\footnotesize
\begin{enumerate}
\item[2] Alice randomly picks $c_0,c_1$ from a previously agreed on
code $\cal C$ (for requirements on $\cal C$ see~\cite{CreGraTap95})
and commits to all bits of the codewords, and proves that the
codewords fulfil the linear relations of $\cal C$ (for the zero
knowledge technique used confer~\cite{CreGraTap95}).
\item[3] Bob randomly picks $I_0,I_1\subset \{1,\dots,M\}$, with
$|I_0|=|I_1| = \sigma m$ ($\sigma$ is a parameter of the code $\cal
C$), $I_1\cap I_0 =\emptyset$ and sets $b^i\leftarrow \overline b$ for
$i\in I_0$ and $b^i \leftarrow b$ for $i\not\in I_0$.
\item[4] Alice runs ${\rm OT}(c_0^i,c_1^i)(b^i)$ with Bob
(by~\cite{Yao95} and the above bit commitment which binds Bob to Helen
and Alice) who gets $w^i$ for $i\in \{1,\dots,m\}$. Bob tells
$I=I_0\cup I_1$ to Alice who opens $c_0^i,c_1^i$ for each $i\in I$.
\item[5] Bob checks that $w^i = c_{\overline b}^i$ for $i\in I_0$ and
$w^i = c_{b}^i$ for $i\in I_1$, sets $w^i\leftarrow c_b^i$, for $i\in
I_0$ and corrects $w$ using the code $\cal C$'s decoding algorithm,
commits to $w^i$ for $i\in \{1,\dots,m \}$, and proves that $w^1\dots
w^m\in {\cal C}$.
\item[6] All players together randomly pick a subset $I_2\subset
\{1,\dots,m\}$ with $|I_2|=\sigma m$, $I_2\cap I=\emptyset$ and opens
$c_0^i$ and $c_1^i$ for $i\in I_2$.
\item[7] Bob proves that $w^i = c_b^i$ for $i\in I_2$.
\end{enumerate}
} Alice and Helen play subGCOT with Bob in a way that Alice plays via
Helen using the above bit commitment protocol and the
forcing measurement technique of~\cite{Yao95} such that Bob cannot
distinguish between commitments/quantum states of Alice and
pretended commitments/quantum states of Helen, then Bob cannot destinguish
between the subGCOT protocols he plays with Alice and those which are
pretended by Helen. Two cases can occur:
\begin{enumerate}
\item After $l$ trials a subGCOT protocol was successful from Alice to
Bob and neither player complains. This subGCOT protocol can be used to
perform GCOT from Alice to Bob, as all other steps of GCOT are not
critical.
\item After $l$ trials no subGCOT protocol between Alice and Bob was
successfull. Then, as Helen and Bob do not collude and Bob cannot
distinguish between Alices and Helens data, it is clear that Alice is
cheating if Bob only complained about her data and it is clear that
Bob is cheating if he complained about Helen as well as Alice.
\end{enumerate}
Once Alice and Bob were able to run one round of subGCOT they can
complete this protocol to a GCOT protocol by the steps:
{\footnotesize
\begin{enumerate}
\item[8] Alice randomly picks and announces a privacy amplification
function $h:\{0,1\}^m\rightarrow \{0,1\}$ such that $a_0 = h(c_0)$ and
$a_1 = h(c_1)$ and proves $a_0= h(c_0^1,\dots,c_0^m)$ and $a_1=
h(c_1^1,\dots,c_1^m)$.
\item[9] Bob sets $a\leftarrow h(w)$, commits to $a$ and proves $a =
h(w^1\dots,w^m)$.
\end{enumerate}
}
Alice and Bob give their proofs (following the procedure
of~\cite{CreGraTap95}) in the above steps to Helen. Hence the
proofs must be correct as no one can risk to get into conflict with
Helen, also convincing Helen is enough as she is not colluding with
Alice or with Bob.
For the corectness of the GCOT protocol we refer to the proof
in~\cite{CreGraTap95}.
We conclude:
\begin{lemma}\label{GCOTworks}
Even if Alice and Bob are in conflict there exists an $\cal A$-robust
protocol for GCOT between Alice and Helen, Bob and Helen and Alice and
Bob. This protocol becomes $\widetilde{\cal A}$-robust after it terminated for ${\cal A}=\{
\{Alice\},\{Bob\},\{Helen\}\}$ and $\tilde{\cal A} = \{
\{Alice,Bob\},\{Helen\}\}$.
\end{lemma}
\begin{proof}
By Corollary~\ref{CorForcingMeasurments} we can have oblivious
transfer from Helen to Alice and from Helen to Bob. As Helen cannot be
in conflict with Alice or Bob (or a cheater can be identified) we can
realize GCOT from Helen to any other player by the protocols
of~\cite{CreGraTap95} with the security of the oblivious transfer
channel of Corollary\ref{CorForcingMeasurments}.
The above protocol for GCOT between Alice and Bob is still concealing
for Helen even if Alice and Bob collude, but the resulting commitments
need not be binding any more. This is no problem as we allow Alice and
Bob to collude only after the termination of the protocol (See comment
after Theorem~\ref{ThreePart}).
\end{proof}
From a bit commitment which can bind one player to the two other
players we can realize GBCX and together with a GCOT protocol working
in at least one direction between every two players we can obtain all
three party protocols, see~\cite{CreGraTap95}
(and~\cite{ImaMue00Eurocrypt} for creating a {\em distributed bit
commitment}, which is needed in~\cite{CreGraTap95}, in the presence of
conflicts). Hence we can conclude:
\begin{theorem}\label{ThreePart}
For three players Alice, Bob, and Helen all functions can be realized
by quantum multi party protocols if two players are honest. The
protocol becomes $\{ \{Alice\},\{Bob\},\{Helen\}\}\cup \{ A\}$-secure
after its execution if there is a player outside of $A$ nobody
complained about.
\end{theorem}
Note that the bit commitment used during the protocol is not
necessarily binding after Alice and Bob collude. If one wants to
implement a long binding bit commitment one has to implement it as a
multi party computation.
\section{A completeness Theorem for Quantum Multi Party Protocols}
In the paper~\cite{ImaMue00QMPforIEICE} quantum multi party protocols
were proposed which use secret sharing to force measurements to
implement oblivious transfer. Then these protocols
follow~\cite{ImaMue00Eurocrypt} to implement multi party computations
with oblivious transfer.
There was an impossibility result in~\cite{ImaMue00QMPforIEICE} that
quantum multi party protocols for all functions become impossible if
two possible collusions cover the set $P$ of players. But due to the
use of oblivious transfer this had to be weakened to the condition of
Lemma~\ref{MPNoGo}. The impossibility result of~\cite{ImaMue00QMPforIEICE}
did not seem to be sharp. Now we can prove that the result is indeed
sharp as we can implement multi party protocols even in the case not
covered by~\cite{ImaMue00QMPforIEICE}:
\begin{theorem}
$\cal A$-robust quantum multi party protocols for all functions are
possible if and only if no two collusions of $\cal A$ cover the set
$P$ of players.
These protocols become $\widetilde{\cal A}$-robust after termination
for an adversary structure $\widetilde{\cal A}$ which may contain one
and only one complement of a set of $\cal A$.
\end{theorem}
\begin{proof}
If one looks at~\cite{ImaMue00QMPforIEICE} one can see that the above
theorem is proven there for all cases but one. The case left open is
that two collusions $A_1$, $A_2$, covering $P\setminus \{ P_i\}$ for a
player $P_i$, are in conflict such that no player from $A_1$ can use
the oblivious transfer to any player of $A_2$ (and vice versa).
In this case we can proceed analogously to our three party
protocols. All commitments are made via $P_i$ (Helen) and equality of
commitments is proven to Helen. This way we obtain GBCX in a way that
whenever a player complains a cheater can be identified.
The GCOT protocol can be realized analogously to the above three party
protocol. A player Alice $\in A_1$ runs subGCOT via Helen with a player
Bob $\in A_2$ such that Bob cannot distinguish data from Helen and
data from Alice. Again Bob cannot complain without either being detected
cheating or proving that Alice cheats. On top of subGCOT GCOT can
easily be realized. With GBCX and GCOT we can realize all multi
party protocols~\cite{CreGraTap95} if one keeps in mind that {\em
distributed bit commitments} can be realized without problems even
when conflicts are present~\cite{ImaMue00Eurocrypt}.
The set $A\in {\cal A}$, for which $A^c$ may be contained in
$\widetilde{\cal A}$, can in the above case be chosen to be any set
containing Helen.
\end{proof}
The set $\widetilde{\cal A}$ can even contain more than one complement
of a set of $\cal A$ provided Helen is not in the additional collusion.
The protocol seems to be more secure than a quantum protocol where no
complains were present. This is true, due to the fact that Helen becomes a
trustable third party in the way that we know she is not colluding with
anyone. This shows that conflicts appearing during the protocol can
yield additional information which can be exploited to increase the
security.
One of the ideas of this paper, namely to anonymize oblivious
transfer, can also be applied to classical protocols. With the
primitive of {anonymous oblivious transfer} all multi party protocols
become possible with perfect security, and whenever a player tries to
abort the protocol this player is identified or the protocol
terminates correctly~\cite{ImaMue00STOCS}.
\end{document} | math | 22,275 |
\begin{document}
\begin{frontmatter}
\title{Initialization-free Distributed Algorithms for Optimal Resource Allocation with Feasibility Constraints and Application to Economic Dispatch of Power Systems\tnoteref{label0}}
\author[label1]{Peng Yi}
\address[label1]{Department of Electrical \& Computer Engineering, University of Toronto}
\ead{[email protected]}
\author[label2]{Yiguang Hong \corref{cor1}}
\address[label2]{Key Lab of Systems and Control, Academy of
Mathematics and Systems Science, Chinese Academy of Sciences,}
\ead{[email protected]}
\author[label3]{Feng Liu}
\address[label3]{Department of Electrical Engineering, Tsinghua University }
\ead{[email protected]}
\begin{abstract}
In this paper, the distributed resource allocation optimization
problem is investigated.
The allocation decisions are made to minimize the
sum of all the agents' local objective functions while satisfying
both the global network resource constraint and the local allocation
feasibility constraints. Here the data corresponding to each agent
in this separable optimization problem, such as the network
resources, the local allocation feasibility constraint, and the
local objective function, is only accessible to individual agent and
cannot be shared with others, which renders new challenges in this
distributed optimization problem. Based on either projection or
differentiated projection, two classes of continuous-time
algorithms are proposed to solve this distributed optimization
problem in an initialization-free and scalable manner. Thus, no
re-initialization is required even if the operation environment or
network configuration is changed, making it possible to achieve a
``plug-and-play" optimal operation of networked heterogeneous
agents. The algorithm convergence is guaranteed for strictly convex
objective functions, and the exponential convergence is proved for
strongly convex functions without local constraints.
Then the proposed algorithm is applied
to the distributed economic dispatch problem in power grids, to
demonstrate how it can achieve the global optimum in a scalable
way, even when the generation cost, or system load, or network
configuration, is changing.
\end{abstract}
\begin{keyword}
Resource allocation \sep Distributed optimization \sep Multi-agent system \sep Plug-and-play algorithm \sep Gradient flow \sep Projected dynamical system \sep
Economic dispatch
\end{keyword}
\tnotetext[label0]{This paper was not presented at any IFAC
meeting. This work is supported by Beijing Natural Science
Foundation (4152057), NSFC (61333001, 61573344), and Program 973
(2014CB845301/2/3).
This work is also partly supported by the National Natural Science Foundation
of China (No. 51377092), Foundation of Chinese Scholarship Council (CSC No. 201506215034).
Corresponding author: Yiguang Hong, Tel. +86(010)82541824.
Fax +86(010)62587343.}
\end{frontmatter}
\section{Introduction}
Resource allocation is one of the most important problems in network
optimization, which has been widely investigated in various areas
such as economics systems, communication networks, sensor networks,
and power grids. The allocation decisions may be made centrally by
gathering all the network data together to a decision-making center,
and then sent back to corresponding agents (referring to
\cite{RA0}). On the other hand, differing from this centralized
policy, the master-slave-type decentralized algorithms, either
price-based (\cite{RA1}) or resource-based (\cite{RA2}), are
constructed to achieve the optimal allocations by the local
computations in the slave agents under the coordinations of the
master/center through a ¡°one-to-all¡± communication architecture.
However, these methods may not be suitable or effective for the
resource allocation in large-scale networks with numerous
heterogeneous agents due to complicated network structures, heavy
communication burden, privacy concerns, unbearable time delays, and
unexpected single-point failures. Therefore, fully distributed
resource allocation optimization algorithms are highly desirable.
Distributed optimization, which cooperatively achieves optimal
decisions by the local manipulation with private data and the
diffusion of local information through a multi-agent network, has
drawn more and more research attention in recent years. To
circumvent the requirement of control center or master, various
distributed optimization models or algorithms have been developed
(\cite{Ned2}, \cite{sayd}, \cite{lou12}, and
\cite{peng2}). In light of the increasing attention to distributed
optimization and the seminal work on distributed resource
allocation in \cite{RA3}, some distributed algorithms for resource
allocation optimization have been proposed in \cite{RA4},
\cite{RA5}, \cite{RA6}, \cite{RA7}, and \cite{RA8}.
Continuous-time gradient flow algorithms have been widely
investigated for convex optimization after the pioneer work
\cite{arrow}, and detailed references can be found in
\cite{arrow2} and \cite{bhaya}. Gradient flow algorithms have been
applied to network control and optimization ( \cite{CA1},
\cite{CA2} and \cite{CA3}), neural networks (\cite{arrow2}), and
stochastic approximation (\cite{proddup2}). Recently,
continuous-time gradient flow algorithms have been adopted for
solving unconstrained distributed optimization problems (see
\cite{wang}, \cite{cort2}, \cite{mug2}, and \cite{cor1}).
Furthermore, the projection-based gradient flow dynamics have
been employed for solving the complicated constrained optimization problems
in \cite{aubin2}, \cite{prods2}, \cite{jun}, \cite{Gao} and
\cite{cortes_prods}, and the projected gradient flow ideas began to be applied to distributed constrained optimization (see
\cite{liu}, \cite{xie} and \cite{peng3}).
The economic dispatch, one of the key concerns
in power grids, is to find the optimal secure
generation allocation to balance the system loads, and hence, can be regarded as
a special resource allocation problem. In recent years, there has been increasing research
attention in solving economic dispatch problems through a multi-agent
system in a distributed manner to meet the ever growing challenges
raised by increasing penetration of renewable energies and deregulation of power infrastructure (\cite{ED_zam} and \cite{ED2}).
Mathematically, this boils down to a particular distributed resource allocation
optimization problem. Furthermore, there were various continuous-time algorithms for the Distributed Economic Dispatch Problem (DEDP). For example,
\cite{ED1} showed that the physical power grid dynamics could serve as a part of a primal-dual gradient flow
algorithm to solve the DEDP, and in fact, it
considered physical network interconnections and generator dynamics explicitly,
providing a quite comprehensive method and inspiring insights. Moreover,
\cite{ED3} solved the DEDP by combining the penalty method and the
distributed continuous-time algorithm in \cite{RA3}, and proposed a procedure to
fulfill the initialization requirement, while \cite{EDcc} constructed a novel initialization-free distributed
algorithm to achieve DEDP given one
agent knowing the total system loads.
Motivated by various practical problems, including the DEDP in power grids, we study a
Distributed Resource Allocation Optimization (DRAO) problem, where
each agent can only manipulate its private data, such as the
local objective function, Local Feasibility Constraint
(LFC), and local resource data. Such data in practice cannot be shared or known by other
agents.
As the total network resource is the sum of individual agent's local
resources, the agents need to cooperatively achieve the optimal
resource allocation in a distributed way, so that the global
objective function (as the sum of all local objective functions) is
minimized with all the constraints (including the network resource
constraint and LFCs) satisfied.
Note that the LFC is critical for the (secure) operation of practical
networks (referring to the communication system in \cite{RA10} and
\cite{RA9} as an example), even though it was not considered in most existing DRAO
works. Particularly, for the DEDP in power grids,
the generation of each generator must be limited within its box-like
capacity bounds.
The consideration of LFCs brings remarkable difficulties to existing
distributed algorithms designed for the DRAO without
LFCs, because the KKT (optimality) conditions for the DRAO with and
without LFCs are totally different (referring to Remark
\ref{kktcompare}).
So far, many DEDP works (such as \cite{ED1}, \cite{ED2} and \cite{ED3} and \cite{EDcc}) have only considered the box-like LFCs. However, the requirement from power industries, such as the secure operation of inverter-based devices in smart grids, promotes the demand to deal with non-box LFCs.
This extension is nontrivial, and we will show how to handle it systematically by using projected dynamics in this paper.
Another crucial albeit difficult problem is the initialization
coordination among all agents. Many existing results
are based on initialization coordination procedures to guarantee
that the initial allocations satisfy the network resource
constraint, which may only work well for ¡°static¡± networks.
However, for a ¡°dynamical¡± network, the resource has to be
re-allocated once the network configuration changes.
Therefore, the initialization coordination has to be
re-performed whenever these optimization algorithms re-start, which
considerably degrades their applicability. Taking the DEDP as an
example, the initialization needs to be coordinated among all agents
whenever local load demand or generation capacity/cost changes, or any
distributed generator plugs in or leaves off (see
\cite{ED3} for an initialization procedure).
This issue has to be well addressed for achieving highly-flexible power grids with the integration
of ever-increasing renewables.
The objective of this paper is to propose an initialization-free
methodology to solve the DARO with local LFCs. The main technical
contributions of this paper are highlighted as follows:
\begin{itemize}
\item By employing the (differentiated) projection operation, two fully distributed continuous-time algorithms are proposed as a kind of projected dynamics, with the local allocation of each agent kept within its own LFC set. Moreover, the algorithms ensure the network resource constraint asymptotically without requiring it being satisfied at the initial points.
Therefore, it is initialization-free, different from those given in \cite{RA4}, \cite{RA5}, \cite{RA6} and \cite{RA7},
and moreover, provides novel initialization-free algorithms different from the one given in \cite{EDcc}.
\item The convergence of the two projected algorithms is shown by the properties
of Laplacian matrix and projection operation as well as the LaSalle invariance principle.
The result can be regarded as an extension of some existing distributed optimization algorithms ( \cite{wang}, \cite{cor1}, \cite{xie}, and \cite{liu}) and an application of projected dynamics for variational inequalities (\cite{Gao} and \cite{jun}) to the DRAO problem.
\item The proposed algorithms can be directly applied to the DEDP in power systems considering generation capacity limitations.
It enables the ¡°plug-and-play¡± operation for power grids with high-penetration of flexible renewables. Our algorithms are essentially different from the ones provided in \cite{ED3} and \cite{EDcc}, and address multi-dimensional decision variables and general non-box LFCs.
Simulation results demonstrate that the algorithm effectively deals with various data and network configuration changes, and also illustrate the algorithm scalability.
\end{itemize}
The reminder of this paper is organized as follows.
The preliminaries are given and then the DRAO with LFCs is formulated
with the basic assumptions in Section 2. Then a distributed
algorithm in the form of projected dynamics is proposed with its
convergence analysis in Section 3. In Section 4, a differentiated projected algorithm is proposed with its convergence analysis for DRAO with strongly convex objective functions, and an exponential convergence rate is obtained in the case without LFCs. Moreover, the
application to the DEDP in power systems is shown in section 5 with
numerical experiments. Finally, the concluding remarks are given in
Section 6.
Notations: Denote $\mathbf{R}_{\geq 0}$ as the set of nonnegative real numbers.
Denote $\mathbf{1}_m=(1,...,1)^T \in \mathbf{R}^m$ and
$\mathbf{0}_m=(0,...,0)^T \in \mathbf{R}^m$.
Denote $col(x_1,....,x_n)=(x_1^T,\cdots, x_n^T)^T $ as the column vector stacked with vectors $x_1,...,x_n$. $I_n$ denotes the identity matrix in $\mathbf{R}^{n\times n}$.
For a matrix $A=[a_{ij}]$, $a_{ij}$ or $A_{ij}$
stands for the matrix entry in the $i$th row and $j$th column of $A$.
$A \otimes B$ denotes the Kronecker product of matrixes $A$ and $B$.
Denote $\times_{i=1,...,n}\Omega_i$ as the Cartesian product of the sets $\Omega_i,i=1,...,n$.
Denote the set of interiors of set $K$ as $int(K)$, and the boundary set of set $K$ as $\partialrtial K$.
\section{Preliminaries and problem formulation}
In this section, we first give the preliminary knowledge related to convex analysis and graph theory, and then formulate the DRAO problem of interest.
\subsection{Convex analysis and projection}
The following concepts and properties about convex functions, convex sets, and projection operations come from \cite{ob} and \cite{bersekas}.
A differentiable convex function $f: \mathbf{R}^m\rightarrow \mathbf{R}$
has the locally Lipschitz continuous gradient, if, given any compact set $Q$,
there is a constant $k^Q$ such that
$ ||\nabla f(x) - \nabla f(y) || \leq k^Q ||x-y ||, \forall x, y \in Q. $
A differentiable function $f(x)$ is called $\mu$-strongly convex on $\mathbf{R}^m$ if there
exists a constant $\mu >0$ such that, for any $x,y \in \mathbf{R}^m$,
$
f(y)\geq f(x)+ \nabla^T f(x)(y-x) + \frac{1}{2}\mu ||y-x ||^2,
$
or equivalently,
\begin{equation}\label{stronglyconvex}
(x-y)^T (\nabla f(x)- \nabla f(y)) \geq \mu ||x-y ||^2.
\end{equation}
The following notations describe the geometry properties of the convex set.
Denote $C_{\Omega}(x)$ as the normal cone of $\Omega$ at $x$, that is,
$C_{\Omega}(x) =\{v: \langle v, y-x\rangle \leq 0, \quad \forall y\in \Omega\}. $
Define $c_{\Omega}(x)$ as
$c_{\Omega}(x)=\{v: ||v||=1, \langle v, y-x \rangle \leq 0, \; \forall y\in \Omega \}$ if $x\in \partialrtial \Omega$, and $c_{\Omega}=\{\mathbf{0}\}$ if $x\in int(\Omega)$.
The feasible direction cone of $\Omega$ at $x$ is given as
$K_{\Omega}(x)= \{d: d=\beta (y-x),\;y\in \Omega,\; \beta\geq 0\}. $
The tangent cone of set $\Omega$ at $x$ is defined as
$T_{\Omega}(x)=\{ v: v = \lim_{k\rightarrow \infty} \frac{x^k-x}{\tau_k}, \tau_k\geq0, \tau_k\rightarrow 0, x^k\in \Omega, x^k\rightarrow x \}.$
Then $T_{\Omega}(x)$ is the closure of $K_{\Omega}(x)$ when $\Omega$ is a closed convex set,
and is the polar cone to $C_{\Omega}(x)$, that is, $T_{\Omega}(x)=\{y: \langle y, d\rangle \leq 0, \forall d\in C_{\Omega}(x) \}$ (referring to Lemma 3.13 of \cite{ob}).
Define the projection of $x$ onto a
closed convex set $\Omega$ by $P_{\Omega}(x)=\arg\min_{y\in \Omega} ||x-y ||$.
The basic property of projection operation is
\begin{equation}\label{projection}
\langle x-P_{\Omega}(x), P_{\Omega}(x)-y \rangle \geq 0, \forall x\in \mathbf{R}^m, \forall y\in \Omega.
\end{equation}
The following relationships can be derived from \eqref{projection},
\begin{equation}\label{projection2}
||x-P_{\Omega}(x) ||_2^2 + ||P_{\Omega}(x)-y||_2^2 \leq || x-y||_2^2, \forall x\in \mathbf{R}^m, \forall y\in \Omega,
\end{equation}
and
\begin{equation}\label{projection3}
||P_{\Omega}(x)-P_{\Omega}(y) || \leq ||x-y ||, \forall x, y \in \mathbf{R}^m.
\end{equation}
The normal cone $C_{\Omega}(x)$ can also be defined as (Lemma 2.38 of \cite{ob})
\begin{equation}
C_{\Omega}(x)=\{v: P_{\Omega}(x+v)=x\}.
\end{equation}
For a closed convex set $\Omega$, point $x\in \Omega$ and direction $v$, we define the differentiated projection operator as (\cite{proddup2} and \cite{prods2}),
\begin{equation}\label{dpoperator}
\Pi_{\Omega}(x, v) = \lim_{\deltaelta\rightarrow 0}\frac{P_{\Omega}(x+\deltaelta v)-x}{\deltaelta}.
\end{equation}
The basic properties of the differentiated projection operator are given as follows (\cite{prods4}).
\begin{lem}\label{dpoperatorbaic}
(i):If $x \in int(\Omega)$, then $\Pi_{\Omega}(x,v)=v$;
(ii): $x \in \partialrtial \Omega$, and $\max_{n \in c_{\Omega}(x)} \langle v, n \rangle \leq 0 $, then $\Pi_{\Omega}(x,v)=v$;
(iii): $x \in \partialrtial \Omega$, and $\max_{n \in c_{\Omega}(x)} \langle v, n \rangle \geq 0 $, then
$\Pi_{\Omega}(x,v)=v-\langle v, n^*\rangle n^*,$ where
$n^*=\arg\max_{n\in c_{\Omega}(x)} \langle v, n\rangle$.
Therefore, the operator $\Pi_{\Omega}(x,v)$ in \eqref{dpoperator} is equivalent with the projection of $v$ onto $T_{\Omega}(x)$, i.e.,
$$ \Pi_{\Omega}(x,v)=P_{T_{\Omega}(x)}(v).$$
\end{lem}
\subsection{Graph theory}
The following concepts of graph theory can be found in \cite{god}.
The information sharing or exchanging among the agents is described by graph
$\mathcal{G}=(\mathcal{N},\mathcal{E})$.
The edge set $\mathcal{E} \subset \mathcal{N}\times \mathcal{N} $ contains all the information interactions.
If agent $i$ can get information from agent $j$, then $(j,i) \in \mathcal{E}$ and
agent $j$ belongs to agent $i$'s neighbor set $\mathcal{N}_i=\{ j | (j,i) \in
\mathcal{E}\}$.
$\mathcal{G}$ is said to be undirected when
$(i,j)\in \mathcal{E}$ if and only if $(j,i)\in \mathcal{E}$.
A path of graph $\mathcal{G}$ is a sequence of distinct agents in
$\mathcal{N}$ such that any consecutive agents in the sequence
corresponding to an edge of graph $\mathcal{G}$. Agent $j$ is said to be
connected to agent $i$ if there is a path from $j$ to $i$.
$\mathcal{G}$ is said to be connected if any two agents are
connected.
Define adjacency matrix $A=[a_{ij}]$ of $\mathcal{G}$ with $a_{ij}=1$ if
$j\in \mathcal{N}_i$ and $a_{ij}=0$ otherwise.
Define the degree matrix
$Deg=diag\{ \sum_{j=1}^n a_{1j},..., \sum_{j=1}^n a_{nj}\}.$
Then the Laplacian of graph $\mathcal{G}$ is
$L=Deg-A.$
When $\mathcal{G}$ is a connected undirected graph, 0 is a simple
eigenvalue of Laplacian $L$ with the eigenspace $\{ \alpha \mathbf{1}_n| \alpha\in \mathbf{R}\}$, and
$L \mathbf{1}_n=\mathbf{0}_n$, $\mathbf{1}^T_{n} L=\mathbf{0}^T_n$, while all other eigenvalues are positive.
Denote the eigenvalues of $L$ in an ascending order as $0<s_2\leq \cdots \leq s_n$. Then, by the Courant-Fischer Theorem,
\begin{equation}\label{lap}
\min_{x\neq \mathbf{0},\atop{\mathbf{1}^Tx=0}} x^T L x =s_2||x||_2^2, \quad \max_{x\neq \mathbf{0}} x^T L x = s_n||x ||^2_2.
\end{equation}
\subsection{Problem formulation}\label{sec1}
Consider a group of agents with the index set $\mathcal{N}=\{1,...,n\}$ to
make an optimal allocation of network resource
under both the network resource constraint and LFCs.
Agent $i$ can decide its local allocation $x_i \in \mathbf{R}^m$,
and can access the local resource data $d_i \in \mathbf{R}^m$.
The total network resource is $\sum_{i\in \mathcal{N}} d_i$,
and therefore, the allocation should satisfy the {\bf network resource constraint:
$\sum_{i\in \mathcal{N}} x_i= \sum_{i\in \mathcal{N}} d_i$.}
Furthermore, the allocation of agent $i$ should satisfy the {\bf local feasibility constraint (LFC): $x_i\in \Omega_i$,}
where $\Omega_i \subset \mathbf{R}^m$ is a closed convex set only known by agent $i$.
Agent $i$ also has a
local objective function $f_i(x_i): \mathbf{R}^m \rightarrow \mathbf{R} $ associated with its local allocation $x_i$.
Denote $X=col(x_1,...,x_n)\in \mathbf{R}^{mn}$ as the allocation vector of the whole network.
Then the task for the agents is to collectively find the optimal allocation corresponding to the
DRAO problem as follows:
\begin{equation}\label{CRA1}
\begin{array}{l}\hline
{\bf Distributed \; Resource \; Allocation \; Optimization }: \\ \hline
\deltaisplaystyle \min_{x_i\in \mathbf{R}^m, \; i\in \mathcal{N}} f(X) =\sum_{i\in \mathcal{N}} f_i(x_i) \\
\deltaisplaystyle subject \; to \; \sum_{i\in \mathcal{N}} x_i=\sum_{i\in\mathcal{ N}} d_i, \\
\deltaisplaystyle \qquad \qquad \; x_i\in \Omega_i, \; i\in \mathcal{N}. \\ \hline
\end{array}
\end{equation}
Clearly, problem \eqref{CRA1} is an extension of the previous optimization models in \cite{RA4}, \cite{RA5} and \cite{RA6} by introducing
the additional LFCs, that is, $x_i \in \Omega_i$. Clearly, $x_i\in \Omega_i$ also generalizes previous box constraints in \cite{RA10}, \cite{RA9} and \cite{EDcc}.
The following assumptions are given for (\ref{CRA1}), which were also adopted for the distributed optimization or resource allocation in \cite{CA2}, \cite{cor1}, and \cite{liu}.
\begin{assum}\label{asumFun}
The functions $f_i(x_i),i\in \mathcal{N}$ are continuously differentiable convex functions with locally Lipschitz continuous gradients and positive definite Hessians over $\mathbf{R}^m$.
\end{assum}
Assumption \ref{asumFun} implies that $f_i(x_i)$'s are strictly convex, and hence guarantees the uniqueness of the optimal solution to (\ref{CRA1}).
\begin{assum}\label{slater}
There exists a finite optimal
solution $X^*$ to problem \eqref{CRA1}. The Slater's constraint condition is satisfied for DRAO (\ref{CRA1}), namely,
there exists $\tilde{x}_i \in int(\Omega_i),\forall i\in \mathcal{N}$,
such that $\sum_{i\in \mathcal{N}} \tilde{x}_i=\sum_{i\in\mathcal{ N}} d_i. $
\end{assum}
\begin{rem}
Define the recession cone of a convex set $\Omega$ as
$R_{\Omega}=\{d: x+\alpha d\in \Omega, \; \forall \; \alpha \geq 0, \; \forall x\in \Omega \}. $
Then the sufficient and necessary condition for the existence of finite optimal solution to \eqref{CRA1} is (referring to Proposition 3.2.2 of \cite{bersekas})
\begin{equation}\label{finiteness}
\times_{i\in \mathcal{N}} R_{\Omega_i}
\cap \{ Null(\mathbf{1}^T_n\otimes I_m)\}
\cap \times_{i\in \mathcal{N}} R_{f_i}=\mathbf{0}, \nonumber
\end{equation}
where $R_{\Omega_i}$ is the recession cone of $\Omega_i$ and $R_{f_i}$
is the recession cone of any nonempty level set of $f_i(x_i)$: $\{x_i\in \mathbf{R}^m | f_i(x_i)\leq \gamma\}$.
In many practical cases, we have $R_{f_i}=\mathbf{0}$ (taking the quadratic function as an example).
Furthermore, $R_{\Omega_i}=\mathbf{0}$ when $\Omega_i$ is compact.
Therefore, the existence of a finite solution can be easily guaranteed and verified in many practical problems.
\end{rem}
The local objective function $f_i(x_i)$, resource data $d_i$ and LFC set $\Omega_i$
are the {\bf private data} for agent $i$, which are not shared with other agents.
This makes (\ref{CRA1}) a distributed optimization problem.
To fulfill the cooperations between agents for solving \eqref{CRA1},
the agents have to share their local information through a network $\mathcal{G}=(\mathcal{N},\mathcal{E})$.
Next follows an assumption about the connectivity of $\mathcal{G}$ to guarantee that any agent's
information can reach any other agents, which is also quite standard for distributed optimization (\cite{liu}).
\begin{assum}\label{assum2}
The information sharing graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$ is undirected and connected.
\end{assum}
In a sum, the task of this paper is to design { fully distributed algorithms } for the agents
to cooperatively find the optimal resource allocation to (\ref{CRA1}) without any center.
In other words, agent $i$ needs to find its optimal allocation $x_i^*$
by manipulating its local private data $d_i$, $\Omega_i$, and $f_i(x_i)$
and by cooperations with its neighbor agents through $\mathcal{G}$.
\section{Projected algorithm for DRAO}
In this section, a distributed algorithm for \eqref{CRA1} based on projected dynamics is
proposed and analyzed.
The distributed algorithm for agent $i$ is given as follows:
\begin{equation}\label{gcd}
\begin{array}{l}\hline
{\bf Projected \; algorithm \; for \; agent \; i}: \\ \hline
\deltaisplaystyle \deltaot{x_i} = P_{\Omega_i}( x_i- \nabla f_i(x_i) +\lambda_i)-x_i \\
\deltaisplaystyle \deltaot{\lambda}_i =-\sum_{j\in \mathcal{N}_i}(\lambda_i-\lambda_j)-\sum_{j\in \mathcal{N}_i}(z_i-z_j)+(d_i-x_i) \\
\deltaisplaystyle \deltaot{z_i} =\sum_{j\in \mathcal{N}_i}(\lambda_i-\lambda_j) \\ \hline
\end{array}
\end{equation}
In algorithm \eqref{gcd}, $x_i\in \mathbf{R}^m$ is the local allocation of agent $i$, and $\lambda_i,z_i \in \mathbf{R}^m$ are two auxiliary variables of agent $i$.
Note that the algorithm (\ref{gcd}) is fully distributed because agent $i$ only needs the local data (including $d_i$, $f_i(x_i)$, and $\Omega_i$)
and the shared information $\{\lambda_j,z_j, j\in \mathcal{N}_i\}$ from its neighbor agents.
Thereby, \eqref{gcd} does not need any center to handle all the data or coordinate all the agents.
With the distributed algorithm \eqref{gcd}, each agent has the autonomy and authority to formulate its
own objective function and feasibility set, and hence, the privacy is kept within each agent.
Because each agent can instantaneously react to its local data changes, it can quickly adapt its local decision.
Therefore, the algorithm can be easily applied to large-scale networks.
The algorithm (\ref{gcd}) can be understood based on the following observations.
The duality of (\ref{CRA1}) with multiplier $\lambda \in \mathbf{R}^m$ is
\begin{equation}\label{problem2}
\max_{\lambda \in \mathbf{R}^m} q(\lambda)=\sum_{i\in \mathcal{N}} q_i(\lambda)= \sum_{i\in \mathcal{N}}\inf_{x_i\in \Omega_i}\{f_i(x_i)-\lambda^T x_i+\lambda^T d_i\}.
\end{equation}
Although some existing distributed algorithms in \cite{Ned2} and \cite{sayd} addressed the dual problem (\ref{problem2}),
they need to solve a subproblem at each time (iteration) to calculate the gradients.
In other words, two ``time scales" are needed if applying existing distributed algorithms to \eqref{CRA1}.
Here we aim to develop a simple algorithm without solving any subproblems.
To this end, we formulate a constrained optimization problem with Laplacian matrix $L$ and $\Lambda=col(\lambda_1,...,\lambda_n)\in \mathbf{R}^{mn}$ as
\begin{equation}\label{problem3}
\begin{array}{lll}
&\max_{\Lambda=(\lambda_1,...,\lambda_n)} \quad \;& Q(\Lambda)\; = \; \sum_{i\in \mathcal{N}} q_i(\lambda_i)\\
& subject \; to \; \quad\;& (L \otimes I_m) \Lambda=\mathbf{0}_{mn}.
\end{array}
\end{equation}
The augmented Lagrangian duality of (\ref{problem3}) with multipliers $Z=col(z_1,...,z_n) \in \mathbf{R}^{mn}$ is
\begin{equation}\label{problem4}
\min_{Z} \max_{\Lambda} Q(\Lambda,Z)=\sum_{i\in \mathcal{N}} q_i(\lambda_i) - Z^T (L\otimes I_m) \Lambda - \frac{1}{2} \Lambda^T (L\otimes I_m) \Lambda.
\end{equation}
Then the projected dynamics (\ref{gcd}) is derived by applying the gradient flow to (\ref{problem2}) and (\ref{problem4}) along with the projection operation to guarantee the feasibility of LFCs.
From the KKT condition, we can show that the equilibrium point of \eqref{gcd} yields the optimal solution to problem \eqref{CRA1}.
Denote $X=col(x_1,...,x_n)$, $\nabla f(X)=col(\nabla f_1(x_1),\cdots,\nabla f_n(x_n))$, $D=col(d_1,\cdots,d_n)$,
and $\Omega=\times_{i\in \mathcal{N}}\Omega_i$.
Write algorithm \eqref{gcd} in a compact form as
\begin{equation}
\begin{array}{lll}
\deltaot{X} & = & P_{\Omega}( X- \nabla f(X) +\Lambda)-X,\\
\deltaot{\Lambda} & = & -(L\otimes I_m) \Lambda-(L\otimes I_m) Z+(D-X),\\
\deltaot{Z} & = & (L\otimes I_m) \Lambda. \label{cgcd}
\end{array}
\end{equation}
\begin{thm}\label{correct}
Under Assumptions \ref{asumFun}-\ref{assum2},
if the initial point $x_i(0) \in \Omega_i, \; \forall i \in \mathcal{N}$, then $x_i(t) \in \Omega_i,\; \forall t \geq 0,\forall i \in \mathcal{N}$, and
$col(X^*,\Lambda^*,Z^*)$ is the equilibrium point of the distributed algorithm \eqref{gcd} with $X^{*}$ as the optimal solution to \eqref{CRA1}.
\end{thm}
{\bf Proof}: Note that
$\deltaot{x}_i \in T_{\Omega_i}(x_i), \; \forall x_i\in \Omega_i$ because $P_{\Omega_i}(x_i-\nabla f_i(x_i)+\lambda_i) \in \Omega_i$.
Given the initial point $x_i(0) \in \Omega_i$, by Nagumo's theorem (referring to page 174 and page 214 of \cite{aubin}), $x_i(t)\in \Omega_i, \forall t\geq 0$ (Related proofs can also be found in \cite{arrow2}).
To obtain the equilibrium point, we set $\deltaot{Z}=\mathbf{0}_{mn}$ and get $\Lambda^*= \mathbf{1}_n \otimes \lambda^*$, $\lambda^*\in \mathbf{R}^m $,
because the graph $\mathcal{G}$ is connected.
$\deltaot{\Lambda}=\mathbf{0}_{mn}$ implies that $ (L\otimes I_m) Z^* = D-X^*$. Because the graph $\mathcal{G}$ is undirected,
$\mathbf{1}_n^T L=\mathbf{0}_n^T$ and $(\mathbf{1}^T_n \otimes I_m) (L\otimes I_m) Z = (\mathbf{1}^T_n L)\otimes I_m Z =(\mathbf{1}_n^T \otimes I_m)(D-X)=\mathbf{0}_{m}$.
Hence, $\sum_{i\in \mathcal{N}} d_i=\sum_{i\in \mathcal{N}} x_i^*$.
Then, at the equilibrium point,
$$\Lambda^* =\mathbf{1}_n\otimes \lambda^*, \; \lambda^* \in \mathbf{R}^m, (L \otimes I_m) Z^*= D-X^*, \; \sum_{i\in \mathcal{N}} d_i =\sum_{i\in \mathcal{N}}x_i^*. $$
Also, at the equilibrium point, $\deltaot{x_i}=0$ implies that
$P_{\Omega_i}( x^*_i- \nabla f_i(x^*_i) + \lambda^*)=x^*_i$.
It follows from Lemma 2.38 of \cite{ob} that
$$ - \nabla f_i(x^*_i) + \lambda^* \in C_{\Omega_i}(x^*_i), i=1,...,n . $$
Therefore, the equilibrium point $col(X^*,\Lambda^*,Z^*)$ of \eqref{gcd} satisfies
\begin{equation}\label{kkt2}
\begin{array}{lll}
&&\mathbf{0}_{mn} \in \nabla f(X^*) - \mathbf{1}_n \otimes \lambda^* + C_{\Omega}(X^*), \\
&& (\mathbf{1}^T_{n} \otimes I_m) X^*= (\mathbf{1}^T_{n}\otimes I_m) D, \; \quad X^* \in \Omega,
\end{array}
\end{equation}
which is exactly the optimality condition (KKT) for DRAO \eqref{CRA1} by Theorem 3.34 in \cite{ob}.
Thus, the conclusion follows.
$\Box$
\begin{rem}
It can be shown that the equilibrium point of \eqref{gcd} has
$\lambda^*$ as the dual optimal solution to problem \eqref{problem2},
and $Z^*$ as the dual optimal solution to problem \eqref{problem3},
following a similar analysis routine of Theorem \ref{correct} and Proposition 5.3.2 in \cite{bersekas}.
It can also be shown that any $col(X^*, \mathbf{1}_m\otimes \lambda^*, Z^*)$, with $X^*$ as the optimal solution to \eqref{CRA1}, $\lambda^*$ as the
dual optimal solution to \eqref{problem2} and $Z^*$ as the dual optimal solution to \eqref{problem3}, corresponds to an equilibrium point of \eqref{gcd}.
We do not discuss their details here for space limitations.
\end{rem}
\begin{rem}\label{kktcompare}
The differences between our work and some existing ones are listed as follows:
\begin{itemize}
\item The KKT condition for DRAO without LFC is
\begin{equation}\label{kkt3}
\begin{array}{lll}
\nabla f(X^*) = \mathbf{1}_n \otimes \lambda^*, \; (\mathbf{1}^T_{n} \otimes I_m) X^*= (\mathbf{1}^T_{n}\otimes I_m) D.
\end{array}
\end{equation}
The KKT condition \eqref{kkt2} for DRAO with LFC and the condition \eqref{kkt3} for DRAO without LFC are totally different.
\eqref{kkt3} requires the optimal allocations to be the points satisfying the network resource constraint with the same marginal costs (gradients), while
\eqref{kkt2} requires the optimal allocations to be feasible in both network resource constraint and LFCs.
The optimal allocations in \eqref{kkt2} should also satisfy a variational inequality related to both the objective functions' gradients and the normal cones of the LFC sets.
In fact, the marginal costs (gradients) at the optimal allocations of \eqref{kkt2} do not necessarily reach the same levels, and the differences can be seen as the ``price of allocation feasibility".
\item
The previous algorithms (except the one in \cite{EDcc}) kept the network resource constraint satisfied (ensured its eventual feasibility) by setting feasible initial points through the initialization coordination procedure.
In other words, the network resource constraint can be guaranteed {\bf only if} it is satisfied at the initial moment.
However, the initialization for the network resource constraint is quite restrictive for large-scale dynamical networks because it involves global coordination
and has to be performed every time the network data/configuration changes.
Moreover, it is not trivial to achieve the initialization coordination with both the LFCs and the network resource constraint
(refer to \cite{ED3} for an initialization procedure with one dimensional interval constraint).
\item In fact, proportional-integral (PI)-type consensus dynamics (\cite{wang} and \cite{cort2}) and projected gradient flows (\cite{liu}), which both have been utilized for distributed optimization,
are combined together to obtain the algorithm \eqref{gcd} for the KKT condition \eqref{kkt2}.
Local $\lambda_i$ acts as the local shadow price,
and all the local shadow prices must reach consensus to be the global market clearing price.
Therefore, a second-order PI-based consensus dynamics is incorporated into (\ref{gcd})
with the integral variable $z_i$ summing up the disagreements between $\lambda_i$ and $\{\lambda_j, j \in \mathcal{N}_i\}$.
Meanwhile, the dynamics of $x_i$ adjusts the local allocation by comparing the local shadow price and the local gradient,
and also utilizes the projection operation in order to make the local allocation flowing within its LFC set all the time.
\end{itemize}
\end{rem}
Therefore, the algorithms given in \cite{RA3}, \cite{RA5}, \cite{RA6} and \cite{RA7} failed to solve \eqref{CRA1} because
they cannot ensure the LFCs given in \eqref{kkt2}.
Note that the algorithm \eqref{gcd} can ensure LFCs even during the algorithm flow with projection operations.
It only requires that each agent has its initial allocation belonging to its LFC set,
which can be trivially accomplished by each agent with one-step local projection operation.
Furthermore, algorithm \eqref{gcd} ensures the network resource constraint
asymptotically without caring about whether it is satisfied at the initial points, and therefore,
is free of initialization coordination procedure.
Due to free of any center and initialization,
algorithm \eqref{gcd} can adaptively handle online data without re-initialization whenever the network data/configuration changes.
Moreover, it can work in a ``plug-and-play" manner for dynamical networks with leaving-off or plugging-in of agents.
Next, let us analyze the convergence of \eqref{gcd}, The analysis techniques are inspired by the projected dynamical systems for variational inequalities (referring to \cite{jun} and \cite{Gao}) and distributed optimization (referring to \cite{shi1}, \cite{cor1}, \cite{xie} and \cite{liu}).
\begin{thm}\label{ascgcd}
Under Assumptions \ref{asumFun}-\ref{assum2}, and given bounded initial points $x_i(0) \in \Omega_i, \; \forall i \in \mathcal{N}$, the trajectories of the algorithm \eqref{gcd} are bounded and converge to an equilibrium point of \eqref{gcd}, namely,
agent $i$ asymptotically achieves its optimal allocation $x_i^*$ of \eqref{CRA1} with \eqref{gcd}.
\end{thm}
{\bf Proof}:
Take $m=1$ without loss of generality.
Denote $\bar{\Omega}=\Omega \times \mathbf{R}^{n}\times \mathbf{R}^{n} .$
Define a new vector $S=col(X,\Lambda,Z)$ and the vector function $F(S): R^{3n}\rightarrow R^{3n}$ as
\begin{equation}
F(S)=\left(
\begin{array}{c}
\nabla f(X)-\Lambda \\
L\Lambda+LZ-(D-X) \\
-L\Lambda \\
\end{array}
\right). \nonumber
\end{equation}
Recalling the form in \eqref{cgcd} and the fact that
$P_{\mathbf{R}^{n}}(x)=x,\; \forall x\in \mathbf{R}^n$, the dynamics of
all the agents can be written as
$$
\deltaot{S} = P_{\bar{\Omega}}(S-F(S))-S.
$$
Define $H(S)=P_{\bar{\Omega}}(S-F(S))$, and then give a Lyapunov function as
\begin{equation}
V_{g} = -\langle F(S), H(S)-S\rangle -\frac{1}{2} || H(S)-S ||_2^2+ \frac{1}{2} || S-S^* ||_2^2, \nonumber
\end{equation}
where $S^*=col(X^*,\Lambda^*,Z^*)$, $X^*$ is the optimal solution to \eqref{CRA1},
$\Lambda^*$ is the optimal solution to \eqref{problem3}, and $Z^*$ is the dual optimal solution to \eqref{problem4}.
Notice that $X^*$ is a finite point from Assumption \ref{slater}.
Because the Slater's condition is satisfied with Assumption \ref{slater},
the dual optimal solution $\lambda^*$ for \eqref{problem2} exists and is finite by
Proposition 5.3.1 of \cite{bersekas}.
Since the function $f_i(x_i)$ is strictly convex, $\nabla q_i(\lambda^*) = d_i-x_i^*$ by Theorem 2.87 of \cite{ob} and the saddle point property of \eqref{problem2}.
Then the KKT condition of \eqref{problem3} is $L\Lambda^*=\mathbf{0}$ and $D-X^*-L\Lambda^*-LZ^*=\mathbf{0}$. Hence, $LZ^*=D-X^*$ implies the finiteness of the dual optimal solution $Z^*$ for \eqref{problem3}.
Therefore, $S^*$ is a finite point.
In fact, with the KKT conditions to \eqref{problem2} and \eqref{problem3},
$ F(S^*)=col( (\nabla f(X^*) - \Lambda^*), \mathbf{0}_n, \mathbf{0}_n), $
and $-\nabla f(X^*) + \Lambda^*\in C_{\Omega}(X^*)$, we have
\begin{equation}\label{equaligcd}
H(S^*) = P_{\bar{\Omega}}(S^*-F(S^*))=S^*, \; - F(S^*) \in C_{\bar{\Omega}}(S^*).
\end{equation}
Due to \eqref{projection} and \eqref{projection2},
\begin{equation}
\begin{array}{lll}
&&-\langle F(S), H(S)-S\rangle - \frac{1}{2}\langle H(S)-S, H(S)-S \rangle \\
&&= \frac{1}{2} [ ||F(S) ||_2^2- ||F(s)+H(S)-S ||_2^2]\\
&&= \frac{1}{2}[ ||S- F(S)-S ||_2^2 - ||H(S)-(S-F(S)) ||_2^2 ]\\
&&\geq \frac{1}{2} ||S-H(S) ||_2^2. \nonumber
\end{array}
\end{equation}
Hence, $V_g\geq \frac{1}{2} ||S-H(S)||_2^2 +\frac{1}{2} ||S-S^* || \geq 0, $ and
$V_g=0$ if and only if $S=S^*$.
By Theorem 3.2 of \cite{fukusma}, any asymmetric variational inequality can be converted to a differentiable optimization problem. As a result,
\begin{equation}
\deltaot{V}_g = (F(S)-[J_F(S)-I](H(S)-S)+ S-S^*)^T (H(S)-S), \nonumber
\end{equation}
where $J_F(S)$ is the Jacabian matrix of $F(S)$ defined as
\begin{equation}
J_F(S)= \left(
\begin{array}{ccc}
\nabla^2 f(X) & -I & 0 \\
I & L & L \\
0 & -L & 0 \\
\end{array}
\right). \nonumber
\end{equation}
With Assumptions \ref{asumFun} and \ref{assum2},
$$
S^T J_F(\bar{S}) S = X^T \nabla^2 f(\bar{X}) X + \Lambda^T L \Lambda > 0, \quad \forall \bar{S} \in \bar{\Omega}, \; \forall S \neq \mathbf{0}\in \mathbf{R}^{3n}.$$
With \eqref{projection}, taking $x=S-F(S)$ and $y=S^*$ gives
$\langle S-F(S)-H(S), H(S)-S^* \rangle \geq 0$, which implies
$\langle S-H(S)-F(S), H(S)-S+S-S^* \rangle \geq 0.$
Hence,
$- ||H(S)-S ||_2^2 + \langle S-H(S), S-S^* \rangle + \langle -F(S), H(S)-S\rangle + \langle -F(S), S-S^* \rangle \geq
0 $, or equivalently,
\begin{equation}\label{proest}
\langle S-H(S), S-S^*+F(S) \rangle \geq ||H(S) -S||_2^2 + \langle F(S), S-S^* \rangle. \nonumber
\end{equation}
Consequently,
\begin{equation}
\begin{array}{lll}
\deltaot{V_g} &\leq & - (H(S)-S)^T J_F(S) (H(S)-S) + || H(S)-S||_2^2 \\
& \; &+ \langle S-S^*+F(S), H(S)-S \rangle \\
& \leq & -\langle F(S), S-S^* \rangle \\
& \leq & - \langle F(S), S-S^* \rangle + \langle F(S^*), S-S^* \rangle - \langle F(S^*), S-S^* \rangle \\
& \leq & - \langle F(S)-F(S^*), S-S^* \rangle - \langle F(S^*), S-S^* \rangle. \nonumber
\end{array}
\end{equation}
In fact,
$\langle F(S)-F(S^{'}), S-S^{'}\rangle
= \langle \nabla f(X)-\nabla f(X^{'}),X-X^{'} \rangle + \langle \Lambda-\Lambda^{'}, L(\Lambda-\Lambda^{'})\rangle
\geq 0, \; \forall S,\;S^{'}\in \bar{\Omega},$
because the local objective functions are convex, and the Laplacian matrix is positive semi-definite by Assumptions \ref{asumFun} and \ref{assum2}.
With \eqref{equaligcd}, we have $ \langle F(S^*), S-S^* \rangle \geq 0 $.
Then $\deltaot{V_g} \leq 0,$ and any finite equilibrium point $S^*$ of \eqref{gcd} is Lyapunov stable.
Furthermore, there exists a forward compact invariance set given any finite initial points,
\begin{equation}\label{compactset}
I_S=\{col(X,\Lambda,Z)\big{|}\frac{1}{2}{||S-S^* ||}\leq V_g(S(0)) \}.
\end{equation}
Therefore, with the local Lipstchitz continuity of the objective functions' gradients in Assumption \ref{asumFun} and
the non-expansive property of projection operation \eqref{projection3},
$P(S-F(S))-S$ is Lipstchitz over the compact set $I_S$ in \eqref{compactset}.
There exists a unique solution to \eqref{gcd} with time domain $[0,\infty).$
Also, the compactness and convexity of $I_S$
implies the existence of equilibrium point to dynamics \eqref{gcd} (referring to page 228 of \cite{aubin}).
By Theorem \ref{correct} and without loss of generality, the equilibrium point is assumed to be $S^*$.
Furthermore, there exists $c^*\in C_{\Omega}(X^*)$ such that $ -\nabla f(X^*)+\Lambda^* =c^*$, $LZ^*=D-X^*$, and $\Lambda^*= \mathbf{1}_n\lambda^*$.
\begin{equation}
\begin{array}{lll}
\deltaot{V_g} &\leq & - \langle F(S), S-S^* \rangle = -\langle X-X^*, \nabla f(X)-\Lambda\rangle\\
&& - \langle \Lambda-\Lambda^*, L\Lambda+LZ-(D-X^*)\rangle- \langle Z-Z^*, -L\Lambda\rangle\\
& \leq & - \langle X-X^*, \nabla f(X)-\Lambda -\nabla f(X^*)+\Lambda^* -c^* \rangle \\
&& - \langle \Lambda-\Lambda^*, L(\Lambda-\Lambda^*) \rangle
- \langle \Lambda-\Lambda^*, L(Z-Z^*) \rangle\\
&& - \langle \Lambda-\Lambda^*, LZ^*-(D-X)\rangle
- \langle Z-Z^*, -L(\Lambda-\Lambda^*)\rangle \\
&\leq & - \langle X-X^*, \nabla f(X)-\nabla f(X^*) \rangle + \langle X-X^*, c^* \rangle \\
&& - \langle \Lambda-\Lambda^*, L(\Lambda-\Lambda^*) \rangle\\
& \leq & - \langle X-X^*, \nabla f(X)-\nabla f(X^*) \rangle
-\langle \Lambda-\Lambda^*, L(\Lambda-\Lambda^*) \rangle, \nonumber
\end{array}
\end{equation}
where the last step follows from $ \langle X-X^*, c^* \rangle \leq 0$.
Denote the set of points satisfying $\deltaot{V}_g=0$ as $E_g=\{(X,\Lambda,Z) \big | \deltaot{V}_g=0 \}.$
Because the Hessian matrix of $\nabla^2 f(X)$ is positive definite,
$
\nabla f(X) = \nabla f(X^*)+ \int_{0}^1 \nabla^2 f(\tau X+(1-\tau)X^*)^T(X-X^*)d\tau
$
and the null space for $L$ imply that
$$E_g=\{(X,\Lambda,Z) \big | X=X^*,\Lambda \in span\{\alpha \mathbf{1}_n\} \}.$$
Then we claim that the maximal invariance set within the set $E_g$ is exactly the equilibrium point of \eqref{gcd}.
In fact, $\Lambda\in span\{\alpha \mathbf{1}_n\} $ implies $Z=Z^*$. Hence, $\deltaot{\Lambda}=LZ^*-(D-X^*)$. However,
$LZ^*-(D-X^*)$ must be zero; otherwise $\Lambda$ will go to infinity, which contradicts that $E_g$ is a compact set within $I_S$.
Thus, $\deltaot{\Lambda}=0$ and $\Lambda=\Lambda^*$.
By the LaSalle invariance principle and Lyapunov stability of the equilibrium point, the system \eqref{gcd} converges to its equilibrium point, which implies the conclusion.
$\Box$
\begin{rem}
Although an initialization-free algorithm has also been proposed and
investigated for the DEDP, a special case of DRAO, in \cite{EDcc},
algorithm \eqref{gcd} provides a different algorithm to
address the DRAO problem without initialization.
Additionally, algorithm \eqref{gcd} can handle general
multi-dimensional LFCs explicitly with the projection operation, while
\cite{EDcc} only addressed one-dimensional box constraints
with a penalty method. Moreover, one agent was
required to know the total network resource all the time in a
time-varying resource case in \cite{EDcc}, while each agent only
knows its local resource in \eqref{gcd}. Moreover, our
techniques introduce a variational inequality viewpoint in
addition to Lyapunov methods with the invariance principle.
\end{rem}
The following example illustrates how \eqref{gcd} ``adaptively" achieves the optimal resource allocation
without re-initialization for a dynamical network.
Notice that the following example cannot be directly addressed by the algorithm in \cite{EDcc}.
\begin{table}\label{tp1}
\caption{Parameters setting of Example \ref{exa1}}
\label{Simulation result}
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
\quad & $0\sim600s $ & $ 600s\sim1200s $ & $1200s\sim $ \\ \hline
$a_1,d_1$ & (8,2), (8,2) & (0.1,0.3), (8,2) & (0.1,0.3), (12,-3) \\ \hline
$a_2,d_2$ & (4,7), (3,4) & (-17,15), (3,4) & (-17,15), (0,7) \\ \hline
$a_3,d_3$ & (0.13,8), (3,8) & (0.13,8), (-5,12) & (3,0.7), (-5,12) \\ \hline
$a_4,d_4$ & (4,20), (10,2) & (4,20), (1,15) & (5,17), (1,15) \\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\caption{\small{The trajectories of the allocations of agent $1$ and agent $2$.}
\label{gcds1}
\end{figure}
\begin{figure}
\caption{\small{The trajectories of the allocations of agent $3$ and agent $4$.}
\label{gcds2}
\end{figure}
\begin{exa}\label{exa1}
Four agents cooperatively optimize problem \eqref{CRA1}.
The allocation variable and resource data for agent $i$ are $x_i=(x_{i,1},x_{i,2})^T \in \mathbf{R}^2,$ $d_i=(d_{i,1},d_{i,2})^T \in \mathbf{R}^2$, respectively.
The objective functions $f_i(x_i)$ are parameterized with $a_i=(a_{i,1},a_{i,2})^T \in \mathbf{R}^2$ as follows:
\begin{equation}
f_i(x_i) = (x_{i,1}+a_{i,1}x_{i,2})^2 + x_{i,1}+ a_{i,2}x_{i,2}+0.001(x^2_{i,1}+x^2_{i,2}). \nonumber
\end{equation}
The LFCs of the four agents are given as
follows: $\Omega_1=\{x_1\in \mathbf{R}^2| (x_{1,1}-2)^2+ (x_{1,2}-3)^2 \leq
25\}$, $\Omega_2=\{x_{2}\in \mathbf{R}^2 | x_{2,1}\geq 0,
x_{2,1}\geq 0, x_{2,1}+2x_{2,2}\leq 4\}$, $ \Omega_3=\{x_3 \in
\mathbf{R}^2 | 4 \leq x_{3,1}\leq 6, 2\leq x_{3,2}\leq 5\}$ and
$\Omega_4=\{x_4\in \mathbf{R}^2 | 0\leq x_{4,1}\leq 15, 0\leq
x_{4,2}\leq 20\}$, respectively, and their boundaries are shown in
Figures \ref{gcds1} and \ref{gcds2}.
The agents share information with a ring graph $\mathcal{G}$:
$$1\leftrightarrow 2 \leftrightarrow 3 \leftrightarrow 4 \leftrightarrow 1.$$
The initial allocation $x_i(0)$ of agent $i$ in algorithm
(\ref{gcd}) is randomly chosen within its LFC set, and $\Lambda,
Z$ are set with zero initial values. The data $a_i$ and $d_i$
switches as Table \ref{tp1}, while the allocation variables remain
unchanged when the data switches. The simulation results are shown
in Figures \ref{gcds1}, \ref{gcds2}, \ref{gcds3}, and \ref{gcds4}.
\begin{figure}
\caption{ \small{The trajectories of the Lagrangian multiplies $\lambda_{i}
\label{gcds3}
\end{figure}
\begin{figure}
\caption{ \small{Network resource constraint and optimality condition}
\label{gcds4}
\end{figure}
Figures \ref{gcds1} and \ref{gcds2} show that the
agents' allocation variables always remain within the corresponding
LFC sets, while Figure \ref{gcds3} shows that the Lagrangian
multipliers reach consensus after the transient processes. Figure
\ref{gcds4} demonstrates that the network resource constraint can
be satisfied asymptotically even though it is violated each time the
data/configuration changes, and Figure \ref{gcds4} also reveals that
$|| \deltaot{X}||_2^2+||\deltaot{\Lambda} ||_2^2+ || \deltaot{Z}||_2^2$ always
converges to zero, guaranteeing the optimality of the resource
allocations.
\end{exa}
\section{Differentiated projected algorithm for DRAO}
In this section, the differentiated projection operator \eqref{dpoperator} is applied to
derive an algorithm for \eqref{CRA1}.
In fact, the projected dynamics based on the operator in
$\eqref{dpoperator}$ was firstly introduced in the study of
constrained stochastic approximation in \cite{proddup2}, and later
was utilized to solve variational inequalities and constrained
optimization problems in \cite{prods2}, \cite{prods4}, and
\cite{cortes_prods}. Here the operator \eqref{dpoperator} is
applied to the construction of the distributed resource allocation
algorithm for agent $i$ given as follows:
\begin{equation}\label{cd}
\begin{array}{l}\hline
{\bf Differentiated \; projected \; algorithm \; for \; agent \; i}: \\ \hline
\deltaisplaystyle \deltaot{x}_i = \Pi_{\Omega_i}(x_i, -\nabla f_i(x_i)+\lambda_i) \\
\deltaisplaystyle \deltaot{\lambda}_i = -\sum_{j\in \mathcal{N}_i}(\lambda_i-\lambda_j)-\sum_{j\in \mathcal{N}_i}(z_i-z_j)+(d_i-x_i) \\
\deltaisplaystyle \deltaot{z_i} = \sum_{j\in \mathcal{N}_i}(\lambda_i-\lambda_j) \\ \hline
\end{array}
\end{equation}
\begin{rem}
The algorithm \eqref{cd} is a direct extension of \eqref{gcd} by differentiating the projection operator, where each agent is required to project $-\nabla f_i(x_i)+\lambda_i$ onto the tangent cone $T_{\Omega_i}(x_i)$.
Thereby, \eqref{cd} has the additional burden for the tangent cone computation compared with \eqref{gcd}.
However, for some specific convex sets such as polyhedron, Euclidean ball, and boxes, it is not hard to
get the close form of the tangent cone at any given point.
\end{rem}
Similar to the algorithm \eqref{gcd}, the algorithm \eqref{cd} is also a distributed algorithm, and
it does not need any initialization coordination procedure. Therefore, it
can efficiently process online data for dynamical networks.
Although algorithm \eqref{cd} is a discontinuous
dynamical system, the solution to \eqref{cd} is
well-defined in the Caratheodory sense (an absolutely continuous
function $col(X(t),\Lambda(t),Z(t)): [0,T]\rightarrow \mathbf{R}^{3mn}$
is a solution of \eqref{cd} if \eqref{cd} is satisfied for almost
all $t\in [0,T]$, referring to Definition 2.5 in \cite{prods2}). The
existence of an absolutely continuous solution to \eqref{cd} can be
found in Theorem 3.1 of \cite{prods3}, and the condition when the
solution can be extended to interval $[0,\infty]$ is given in
Theorem 1 of \cite{prods4}.
The following result shows the correctness of algorithm \eqref{cd}.
\begin{thm}
Suppose that Assumptions \ref{asumFun}-\ref{assum2} hold.
If the initial point $x_i(0) \in \Omega_i, \; \forall i \in \mathcal{N}$, then $x_i(t) \in \Omega_i, \; \forall t \geq 0,\forall i \in \mathcal{N}$, and there is the equilibrium point of the algorithm \eqref{cd} with $X^{*}=col(x_1^*,...,x_n^*)$ as the optimal solution to \eqref{CRA1}.
\end{thm}
{\bf Proof}: Obviously, $\deltaot{x}_i \in
T_{\Omega_i}(x_i), \; \forall x_i\in \Omega_i$ according to Lemma
\ref{dpoperatorbaic}. It follows that \eqref{cd} has an absolutely
continuous solution on interval $[0,\infty]$ by Theorem 1 of
\cite{prods4}. Moreover, Theorem 3.2 of \cite{prods3} shows that
the solution of \eqref{cd} coincides with a slow solution of a
differential inclusion. Given the
initial point $x_i(0)\in \Omega_i$, $x_i(t)\in \Omega_i, \forall
t\geq 0$ holds in light of the viability theorem in \cite{aubin}.
By Lemma \ref{dpoperatorbaic}, we have that $\Pi_{\Omega_i}(x_i,-\nabla f_i(x_i)+\lambda_i)=0$ if
at least one of the following cases is satisfied:
(i):$x_i\in int(\Omega_i),$ and $-\nabla f_i(x_i)+\lambda_i=0$;
(ii): $x_i\in \partialrtial \Omega_i$, and $-\nabla f_i(x_i)+\lambda_i=0$;
(iii): $x_i\in \partialrtial \Omega_i$, and $-\nabla f_i(x_i)+\lambda_i\in C_{\Omega_i}(x_i)$.
Hence, $\Pi_{\Omega_i}(x_i,-\nabla f_i(x_i)+\lambda_i)=0$ implies $-\nabla f_i(x_i)+\lambda_i\in C_{\Omega_i}(x_i)$.
Following similar analysis of Theorem \ref{correct}, at the equilibrium point we have
\begin{equation}
\begin{array}{lll}\label{eq2}
&& \Lambda^*=\mathbf{1}_n \otimes \lambda^*,\lambda^*\in \mathbf{R}^m, \quad \; x^*_i \in \Omega_i \\
&& (L\otimes I_m) Z^* = D-X^*, \sum_{i\in \mathcal{N}} x^*_i = \sum_{i\in \mathcal{N}} d_i, \\
&& -\nabla f_i(x_i^*)+ \lambda^* \in C_{\Omega_i}(x_i^*).
\end{array}
\end{equation}
Thus, the optimality condition \eqref{kkt2} of \eqref{CRA1} is satisfied by the equilibrium point of \eqref{cd}.
$\Box$
Next result shows the convergence of \eqref{cd} when the local objective functions are strongly convex.
\begin{thm}
Suppose that Assumptions \ref{asumFun}-\ref{assum2} hold, and
the local objective functions $f_i(x_i)$ are $\mu_i$-strongly convex functions with
$k_i$-Lipschitz continuous gradients.
Given bounded initial points $x_i(0) \in \Omega_i, \; \forall i \in \mathcal{N}$,
the trajectories of algorithm (\ref{cd}) converge to its equilibrium point.
Furthermore, if there are no LFCs (that is, $\Omega_i=\mathbf{R}^m,i=1,\cdots,n$), then algorithm (\ref{cd})
exponentially converges to its equilibrium point.
\end{thm}
{\bf Proof}: We still take $m=1$ without loss of generality.
At first, we show the convergence of \eqref{cd}.
By Lemma \ref{dpoperatorbaic},
\begin{equation}\label{eq1}
\Pi_{\Omega_i} (x_i, -\nabla f_i(x_i) + \lambda_i) = -\nabla f_i(x_i) + \lambda_i - \beta(x_i)n_{i}(x_i),
\end{equation}
where $ n_i(x_i)\in c_{\Omega_i}(x_i),\; \beta(x_i)\geq 0$.
Notice that there exist $\beta(x^*_i)\geq 0$ and $n_{i}(x^*_i)\in c_{\Omega_i}(x^*_i)$ at the equilibrium point such that
$
\nabla f_i(x^*_i) - \lambda^* = -\beta(x^*_i)n_{i}(x^*_i).
$
Define the following variables
\begin{equation}
\begin{array}{lll}\label{corc}
Y=X-X^*, \; & \theta=[r \; R]^T Y, & Y= [r \; R]\theta,\\
V=\Lambda-\Lambda^*, \; & \eta=[r \; R]^T V , & V=[r \; R] \eta,\\
W=Z-Z^*, & \deltaelta= [r \; R]^T W, & W=[r \; R]\deltaelta,\\
\end{array}
\end{equation}
with
$r=\frac{1}{\sqrt{n}} \mathbf{1}_n$ and $r^T R=\mathbf{0}^T_n$ such that $R^T R=I_{n-1}$ and $R R^T = I_n- \frac{1}{n} \mathbf{1}_n\mathbf{1}_n^T$.
We partition the variables $\theta,\eta,\deltaelta$ as $col(\theta_1,\theta_2)$, $col(\eta_1,\eta_2)$, $col(\deltaelta_1,\deltaelta_2)$ with $\theta_1,\eta_1,\deltaelta_1 \in \mathbf{R}$ and
$\theta_2,\eta_2,\deltaelta_2 \in \mathbf{R}^{n-1}$.
Then the dynamics of the variables $\theta,\eta,\deltaelta$
can be derived with \eqref{cd}, \eqref{eq2}, \eqref{eq1} and \eqref{corc} as follows,
\begin{equation}\label{corcdy}
\begin{array}{lll}
\deltaot{\theta}_1 = -r^T h+\eta_1; & \; & \deltaot{\theta}_2 = -R^T h +\eta_2;\\
\deltaot{\eta}_1 = -\theta_1 ; & \; & \deltaot{\eta}_2 = -\theta_2 -R^T L R\eta_2 - R^T L R \deltaelta_2;\\
\deltaot{\deltaelta}_1 = 0 ; & \; & \deltaot{\deltaelta}_2 = R^TLR \eta_2,
\end{array}
\end{equation}
where $h=\nabla f(Y+X^*)-\nabla f(X^*) + N_{\Omega}(X)-N_{\Omega}(X^*)$,
$N_{\Omega}(X) = col(\beta(x_1)n_{1}(x_1),..., \beta(x_n)n_n(x_n) ), $ and
$N_{\Omega}({X}^*)= col( \beta({x}^*_1)n_{1}({x}^*_1),...., \beta({x}^*_n)n_{n}({x}^*_n) )$.
Construct the following function
\begin{equation}\label{vs1}
\begin{array}{lll}
V^s_1 & = & \frac{1}{2}\alpha (\theta^T \theta+ \eta^T\eta) + \frac{1}{2}(\alpha+\gamma) \deltaelta^T_2 \deltaelta_2 \\
& + & \frac{1}{2} \gamma (\eta_2+ \deltaelta_2 )^T (\eta_2+\deltaelta_2 ),
\end{array}
\end{equation}
where $\alpha,\gamma$ are positive constants to be determined later.
Obviously,
$\frac{1}{2} \alpha ||p ||_2^2 \leq V^s_1 \leq (\frac{1}{2}(\alpha+\gamma)+\gamma) ||p ||_2^2$ where $p=col(\theta,\eta,\deltaelta_2)$.
The derivative of $V_1^s$ along (\ref{cd}) is
\begin{equation}
\begin{array}{lll}
\deltaot{V}^s_1 & = & \alpha(- Y^T h -\eta_2 R^T L R \eta_2 -\eta_2 R^T L R \deltaelta_2)\\
& + & \gamma (-\eta_2^T \theta_2- \eta_2 R^T L R \deltaelta_2 - \deltaelta_2 \theta_2 - \deltaelta_2^T R^T L R \deltaelta_2)\\
& + & (\alpha+\gamma)\deltaelta_2^T R^T L R \eta_2. \nonumber
\end{array}
\end{equation}
Because $n_i(x_i^*)\in c_{\Omega_i}(x_i^*)$ and $ \beta(x_i^*)\geq 0$, $ \beta{(x_i^*)} \langle x_i-x_i^*, n_i(x_i^*) \rangle \leq 0$.
Moreover, $n_i(x_i)\in c_{\Omega_i}(x_i)$ and $\beta(x_i)\geq 0$ imply that $\beta(x_i) \langle x_i-x_i^*,n_i(x_i)\rangle \geq 0 $.
Because the local objective functions are strongly convex,
$
Y^T h = (X-X^*)^T( \nabla f(X)-\nabla f(X^*) + N_{\Omega}( X)- N_{\Omega}(X^*))
= (X-X^*)^T( \nabla f(X)-\nabla f(X^*) )
+ \sum_{i=1}^n \langle x_i-x_i^*,+ \beta{(x_i)}n_{i}(x_i) \rangle
+ \sum_{i=1}^n \langle x_i-x_i^*, -\beta(x^*_i)n_i(x^*_i)\rangle
\geq \bar{\mu} \theta^T \theta
$
where $\bar{\mu}=\min\{\mu_1,\cdots,\mu_n\}.$
Denote $s_1\leq s_2\leq ...\leq s_n$ as the ordered eigenvalues of Laplacian matrix $L$.
Obviously, $s_1=0$ and $s_2>0$ when the graph $\mathcal{G}$ is connected.
By \eqref{lap},
$$\deltaot{V}^s_1 \leq -\alpha\bar{\mu}\theta^T \theta -\alpha s_2 \eta_2^T \eta_2 -\gamma s_2 \deltaelta_2^T \deltaelta_2
- \gamma\eta_2^T \theta_2 - \gamma\deltaelta_2 \theta_2.$$
According to
$-\gamma \eta_2^T \theta_2\leq \frac{1}{2}\gamma^2 \theta_2^T \theta_2 + \frac{1}{2} \eta_2^T \eta_2,$
and
$ - \gamma\deltaelta_2^T \theta_2 \leq \frac{1}{2} \gamma^2 \theta_2^T \theta_2 + \frac{1}{2}\deltaelta_2^T \deltaelta_2,$
we have
\begin{equation}\label{v1dot}
\deltaot{V}^s_1 \leq -(\alpha \bar{\mu} -\gamma^2) \theta^T \theta - (\alpha s_2 - \frac{1}{2}) \eta_2^T\eta_2
- (\gamma s_2 -\frac{1}{2}) \deltaelta_2^T \deltaelta_2.
\end{equation}
Take $\gamma$ and $\alpha$ such that $\gamma >\frac{1}{2s_2}$, and $\alpha > \max\{\frac{\gamma^2}{\bar{\mu}}, \frac{1}{2s_2}\}$. Then we have $\deltaot{V}^s_1<0$, which leads to the convergence of algorithm \eqref{cd}.
Next, we estimate the convergence rate of \eqref{cd} when $\Omega_i=\mathbf{R}^m, i=1,\cdots,n$.
In this case $\Pi_{\Omega_i} (x_i, -\nabla f_i(x_i) + \lambda_i) = -\nabla f_i(x_i) + \lambda_i$,
and $\beta(x_i)n_{i}(x_i)=\mathbf{0}$. Still take $V_1^s$ in \eqref{vs1}, and then \eqref{v1dot} still holds
in this case.
Take $V^s_2= \varepsilon (\theta-\eta)^T(\theta-\eta)$, and we have
\begin{equation}
\begin{array}{lll} \label{v2d}
\deltaot{V}^s_2 &=& - \varepsilon Y^T h + \varepsilon \theta^T \theta + \varepsilon \theta_2 R^T L R \eta_2 + \varepsilon \theta_2 R^T L R \deltaelta_2 \\
&&+ \varepsilon \eta^T [r, R]^T h- \varepsilon \eta^T \eta- \varepsilon \eta_2 R^T L R \eta_2- \varepsilon \eta_2 R^T L R \deltaelta_2 \\
&\leq& - (\varepsilon \overline{\mu}-\varepsilon) \theta^T \theta + \frac{1}{2} \varepsilon s_n^2 \theta^T \theta
+ \frac{1}{2}\varepsilon \eta_2^T\eta_2^T
+ \frac{1}{2} \varepsilon s_n^2 \theta^T \theta \\
&&+ \frac{1}{2} \varepsilon \deltaelta_2\deltaelta_2^T
+ \frac{1}{2} \varepsilon \eta^T\eta + \frac{1}{2} \varepsilon M^2 \theta^T \theta - \varepsilon \eta^T \eta- \varepsilon s_2\eta^T_2 \eta_2\\
&&+ \frac{1}{2} \varepsilon s_n^2 \eta_2^T\eta_2 +\frac{1}{2} \varepsilon \deltaelta^T_2\deltaelta_2 \\
&\leq& - \varepsilon( \overline{\mu}-1- s_n^2-\frac{1}{2} M^2 ) \theta^T\theta-\frac{1}{2}\varepsilon\eta^T\eta\\
&&+ \varepsilon \deltaelta_2^T\deltaelta_2- \varepsilon( s_2- \frac{1}{2}- \frac{1}{2} s_n^2 )\eta_2^T\eta_2, \nonumber
\end{array}
\end{equation}
with $M=\max \{k_1,\cdots,k_n\}$, by using the inequality $ x^Ty \leq \frac{1}{2} ||x||_2^2+|| y||_2^2 $ and \eqref{lap} in the first step of \eqref{v2d}.
With $V^s=V^s_1+V^s_2$, it is easy to see that
\begin{equation}
\begin{array}{lll}
\deltaot{V}^s&=& -(\alpha \bar{\mu} -\gamma^2 + \varepsilon( \overline{\mu}-1- s_n^2-\frac{1}{2} M^2 ) ) \theta^T \theta \\
&-& \frac{1}{2} \varepsilon \eta^T\eta -(\gamma s_2 -\frac{1}{2}- \varepsilon ) \deltaelta_2^T \deltaelta_2\\
&-& (\alpha s_2 - \frac{1}{2}+ \varepsilon( s_2- \frac{1}{2}- \frac{1}{2} s_n^2 )) \eta_2^T\eta_2. \\
\end{array}
\end{equation}
Choose $\gamma \geq \frac{ 3\varepsilon +1}{2s_2}$ such that $\gamma s_2 -\frac{1}{2}- \varepsilon \geq \frac{1}{2}\varepsilon $.
Select $\alpha$ such that
\begin{equation}\label{par111}
\alpha \bar{\mu} -\gamma^2 + \varepsilon( \overline{\mu}-1- s_n^2-\frac{1}{2} M^2 ) \geq \frac{1}{2} \varepsilon,
\end{equation}
and
\begin{equation}\label{par222}
\alpha s_2 - \frac{1}{2}+ \varepsilon( s_2- \frac{1}{2}- \frac{1}{2} s_n^2 ) \geq 0.
\end{equation}
As a result, $$ \deltaot{V}^s \leq \frac{1}{2} \varepsilon ||p ||_2^2.$$
Then, with $
\frac{1}{2} \alpha ||p ||_2^2 \leq V^s \leq (\frac{1}{2}(\alpha+\gamma)+\gamma + 2\varepsilon) ||p ||_2^2
$, we have
\begin{equation}\label{par12}
||p|| \leq \sqrt{ \frac{\alpha+3\gamma + 4\varepsilon}{\alpha}} || p(0)||e^{-\frac{2\varepsilon}{\alpha+3\gamma + 4\varepsilon}t},
\end{equation}
which leads to the exponential convergence of algorithm \eqref{cd} to its equilibrium point.
$\Box$
\begin{rem}
In fact, the exponential convergence speed $ \frac{2\varepsilon}{(\alpha+3\gamma + 4\varepsilon)}$ in \eqref{par12} can be estimated
by solving the following optimization problem
\begin{equation}
\max_{\alpha, \gamma, \varepsilon \geq 0} \frac{2\varepsilon}{(\alpha+3\gamma + 4\varepsilon)}
\; \qquad s. \; t. \; \gamma \geq \frac{ 3\varepsilon +1}{2s_2}, \; \eqref{par111},\; \eqref{par222}. \nonumber
\end{equation}
To get a simple estimation, we take $\gamma=\frac{3\varepsilon +1}{2s_2}$.
Denote $\varrho_1=(s_n^2+1-2s_2)$ and $\varrho_2= s_2^2(6+4s_n^2+2M^2-4\bar{\mu}) $. With taking
$\alpha = \max \{ \frac{1+\varepsilon\varrho_1}{2s_2}, \frac{ 9\varepsilon^2 + \varepsilon(6+\varrho_2)+1 }{4s_2^2\bar{\mu}} \} $, we have
$ \frac{2\varepsilon}{(\alpha+3\gamma + 4\varepsilon)}
\geq \min \{ \frac{2 s_2}{8+6s_2+s_n^2}, \frac{ 4s_2^2 \bar{\mu} }{ (3+2s_n^2+M^2+6\bar{\mu}) s_2^2+ 9\bar{\mu} s_2 +3\sqrt{6\bar{\mu}s_2+1} + 3 } \}$.
\end{rem}
\section{Distributed economic dispatch in power grids}
In this section, the algorithm proposed in Section 4
is applied to the DEDP in power grids to find the optimal secure
generation allocations for power balancing in a distributed manner.
Example \ref{exa2} is given to show that the distributed algorithm
can efficiently adapt to online network data/configuration changes,
including {\bf load demands, generation costs/capacities, and
plugging-in/leaving-off of buses}, while Example \ref{mexa} with a
large-scale network illustrates the {\bf scalability} of the
proposed algorithm.
Suppose that there exist control areas $\mathcal{N}=\{1,...,n\}$ with area $i$
having local generators to supply power $P_i^g\in \mathbf{R}_{\geq 0}$
and local load demands $P_i^d \in \mathbf{R}_{\geq 0}$ to be met.
The local generation must be kept within the capacity or security bounds $ \underline{P}_i \leq P_i^g \leq \bar{P}_i, \underline{P}_i, \bar{P}_i\in \mathbf{R}_{\geq 0}$.
$f_i(P_i^g): \mathbf{R}_{\geq 0}\rightarrow \mathbf{R}$ represents the local generation cost in control area $i$ with respect to its local generation $P_i^g$,
and it satisfies Assumption \ref{asumFun}.
Then the DEDP formulation can be derived in the form of (\ref{CRA1})
\begin{equation}\label{edp}
\begin{array}{l}\hline
{\bf Distributed \; Economic \; Dispatch \; Problem }: \\ \hline
\deltaisplaystyle \min_{P_i^g,i\in \mathcal{N}} \;\; \;f(P^g)= \sum_{i\in \mathcal{N}} f_i(P_i^g) \\
\deltaisplaystyle subject \; to \; \sum_{i\in \mathcal{N}} P_i^g =\sum_{i\in \mathcal{N}} P_i^d \\
\deltaisplaystyle \qquad \qquad \underline{P}_i \leq P_i^g \leq \overline{P}_i, \quad \mathcal{N}=\{1,...,n\} \\ \hline
\end{array}
\end{equation}
Here a multi-agent network is introduced to solve the DEDP (\ref{edp}) motivated by recent DEDP works like \cite{ED3}, \cite{EDcc}, \cite{ED1}, \cite{ED_zam} and \cite{ED2}.
Agent $i$ is responsible to decide the generation $P^g_i$ in control area $i$
to minimize the global cost as the sum of all individuals' generation costs, while meeting the total load demands within its capacity bounds.
In addition, each agent can react to the changing local environment in real time,
and adapt its own behavior or preference by adjusting its local data, including $P_i^d, \underline{P}_i, \bar{P}_i, f_i(P_i^g)$.
The agents can also share information
with their neighbors to facilitate the cooperations.
Then applying \eqref{cd} to \eqref{edp}, the distributed algorithm for agent $i$ is
\begin{equation}\label{cdedp}
\begin{array}{l}\hline
{\bf Distributed \; algorithm \; for \; agent \; i}: \\ \hline
\deltaisplaystyle \deltaot{P}^g_i =[-\nabla f_i(P_i^g)+\lambda_i]^{\bar{P}_i-P_i^g}_{P_i^g-\underline{P}_i} \\
\deltaisplaystyle \deltaot{\lambda}_i =-\sum_{j\in \mathcal{N}_i}(\lambda_i-\lambda_j)-\sum_{j\in \mathcal{N}_i}(z_i-z_j)+(P^d_i-P^g_i) \\
\deltaisplaystyle \deltaot{z_i} =\sum_{j\in \mathcal{N}_i}(\lambda_i-\lambda_j) \\ \hline
\end{array}
\end{equation}
where $[v]_{x-c_1}^{c_2-x}=0$ if $x-c_1=0$ and $ v \leq 0$ or $c_2-x=0$ and $v \geq 0$, otherwise $[v]_{x-c_1}^{c_2-x}=v$.
The algorithm \eqref{cdedp} ensures generation capacity bounds explicitly, and converges
without the initialization procedure, which is crucially important for the ``plug-and-play" operation in the future smart grid.
\begin{rem}
The optimization problem \eqref{CRA1} can also be applied to model the multi-period demand response in power systems. The objective functions describe the dis-utility of cutting loads in each area.
The resource constraint specifies the amount of loads to be shed in the multi-periods. Particularly, the local load shedding constraints concern with the
total power demands in the multi-periods and other specifications of that area.
Then, the previous algorithms \eqref{gcd} and \eqref{cd} can be applied to solve this multi-dimensional DRAO
with general LFCs in a distributed manner.
This issue is interesting but beyond the scope of this paper, and will be discussed elsewhere.
\end{rem}
The following two examples are presented to further show the online
adaptation property and scalability of our algorithm.
Firstly, the standard IEEE 118-bus system is adopted to illustrate the performance of
algorithm \eqref{cdedp}.
\begin{exa}[Optimality and adaptability]\label{exa2}
Consider the DEDP \eqref{edp} in the IEEE 118-bus
system with $54$ generators. Each generator has a local quadratic generation
cost function as $f_i(P_i^g)=a_i {P_i^g}^2 +b_i P_i^g+c_i$, whose
coefficients belong to the intervals $a_i\in [0.0024,0.0679](M\$/MW^2)$,
$b_i\in [8.3391,37.6968](M\$/MW)$, and $c_i\in [6.78,74.33](M\$)$. The generation
capacity bounds of the generators are drawn from $\underline{P}_i\in
[5,150](MW)$ and $\bar{P}_i\in [150,400](MW)$, while the load of each bus
varies as $P_i^d\in [0,300](MW)$.
The corresponding agents share information on an undirected ring graph with
additional undirected edges $(1,4)$, $(15,25)$, $(25,35)$, $(35,45)$
and $(45,50)$. The simulations are performed with the differentiated
projected algorithm \eqref{cdedp}. The initial generation $P_i^g$ in
algorithm (\ref{cdedp}) is set within its local capacity bounds,
while variables $\lambda_i$'s and $z_i$'s are set with zero initial values.
Next explains how the network data/configuration changes
at different times.
\begin{itemize}
\item {\bf Load variations:} At time 100s, 18 buses are randomly chosen
with randomly varying their loads by $-20 \sim +20\%$.
\item {\bf Generation capacity variations:} At time 200s, 18 generators
randomly vary their capacity lower bounds by $-50 \sim +50\%$, and another 18
generators randomly vary their capacity upper bounds by $-20
\sim +20\%$.
\item {\bf Generation cost variations:} At time 300s, 18 generators randomly
vary their $a_i$ by $0 \sim 50\%$, and another 18 generators randomly vary their $b_i$ by $-50 \sim 0\%$.
\item {\bf Leaving-off of buses:} At time 400s, generator 2 and 3 disconnect from the system, and the communication edges associated to them are also removed.
\item {\bf Plugging-in of bus:} At time 500s, generator 3 plugs in the system
with re-generated configurations
$a_i,b_i,c_i,P_i^d,\underline{P}_i$, and $\bar{P}_i$. The undirected
edge $(3,4)$ is also added to the communication graph.
\end{itemize}
\begin{figure}
\caption{Algorithm performance indexes:
(i), $||L\Lambda||$'s always decrease to zero even with different $L$'s, implying that $\lambda's$ always reach consensus.
(ii),
The power balance constraint (i.e., resource constraint) is violated if any network configuration changes.
But the power balance gap $\mathbf{1}
\label{fig2}
\end{figure}
When the data/configuration changes, each agent only projects its local generation onto to its local capacity bounds if necessary.
The trajectrories of dynamics \eqref{cdedp} are derived with a first-order Euler discretization using Matlab.
The trajectories of some algorithm performance indexes are shown in Figure \ref{fig2}.
It indicates that algorithm \eqref{cdedp} can adaptively find the optimal solutions to \eqref{edp} in a fully distributed way, even without any initialization coordination procedure, when the network data or configuration changes.
\end{exa}
\begin{exa}[Scalability]\label{mexa}
This example considers a network of ${\bf 1000}$ control areas to achieve economic dispatch during a normal day.
Control area $i$ has cost function $f_i(P^g_i)= a_i {P^g_i}^2 +b_i P^g_i$ as well as generation capacity upper bound $\bar{P}_i$ and lower bound $\underline{P}_i$.
The control areas are divided equally into two groups. The first group, named as fuel group,
is mainly supported with traditional thermal generators,
and has relatively higher generation costs and larger capacity ranges
with the nominal values of $a_i,b_i,\underline{P}_i,\bar{P}_i$ randomly drawn from the intervals, $[3,7](M\$/MW^2) $,\;
$[5,9](M\$/MW), $\;$ [2,6](MW), $\;$ [15,23](MW)$, respectively.
The second group, named as renewable group, is mainly supported with renewable energies,
and has lower generation costs and smaller capacity ranges with the nominal values of $a_i,b_i,\underline{P}_i,\bar{P}_i$ randomly drawn from
the intervals, $[\frac{1}{2},2](M\$/MW^2)$, \;
$[\frac{1}{2},4](M\$/MW) $, \;
$ [0,1](MW) $, \;
$[\frac{3}{2},7](MW)$, respectively.
\begin{figure}
\caption{The total load curve}
\label{fig_mas1}
\end{figure}
The daily 96-point load data (15 minutes for each period) of each control
area is generated from a typical load curve for a distribution
system added with certain random perturbations. The total load curve
of the network is given in Fig \ref{fig_mas1}. In each period,
10\% of the control areas in the renewable group are randomly chosen to
change their generation costs and capacities like Example
\ref{exa2} with the variations less that $\pm20\%$ of its nominal
values.
In each period, a connected graph is re-generated with random graph model $\mathcal{G}(1000,\mathbb{P})$ as the information sharing graph of that period.
In $\mathcal{G}(1000,\mathbb{P})$, every possible edge occurs independently with the probability of $\mathbb{P}$.
Here, the probability $\mathbb{P}$ in each period is randomly drawn from the interval $[0.0015,0.005]$.
For each period, the computation time is set as $80s$. Figure \ref{fig_mas2} shows the histogram of consensus error $||L\Lambda||_2$,
power balance gap $\sum P_i^g -\sum P_i^d$ and optimality condition $||\deltaot{P}^g||_2+||\deltaot{\Lambda}||_2+||\deltaot{Z}||_2$ at time $80s$.
It indicates that the agents can always find the economic power dispatch with varying loads and generation costs/capacities, and evidently demonstrates the scalability of the proposed method.
\end{exa}
\begin{figure}
\caption{ The performance indexes at time $t=80s$ in histograms: (i), The consensus error $||L\Lambda||_2$ at $t=80s$ always decreases to
a rather lower level due to the second-order proportional-integral consensus dynamics.
(ii), The power balance is almost achieved at $t=80s$ without the initialization coordination.
(iii), The optimality condition can always be satisfied at $t=80s$.
}
\label{fig_mas2}
\end{figure}
\section{Conclusions}
In this paper, a class of projected continuous-time distributed algorithms have been proposed to solve resource allocation optimization problems with the consideration of LFCs.
The proposed algorithms are scalable and free of initialization coordination procedure,
and therefore, are adaptable to working condition variations.
These salient features have important implications in the DEDP in power systems.
Firstly it allows quite general non-box LFCs,
which is crucial when inverter-based devices are involved because their LFCs are usually depicted in a quadratic form.
Secondly, our method is initialization free,
which may facilitate the implementation of the so-called ``plug-and-play" operation
for future smart grids in a dynamic environment.
Our application examples illustrate such implications, showing an appealing potential in the smart operation of future power grids.
We would like to note that many challenging DRAO problems
still remain to be investigated, including the design of algorithms for the non-smooth objective functions based on differential inclusions,
the estimation of convergence rates for the proposed algorithms with general LFCs, and the development of stochastic algorithms to achieve the DRAO with noisy data observations.
Furthermore, inspired by \cite{ED1}, we also hope to combine our algorithms with physical dynamics of
power grids to derive a more comprehensive solution for the DEDP in power systems.
\end{document} | math | 73,679 |
\begin{document}
\title{Regular Expression Matching and Operational Semantics}
\begin{abstract}
Many programming languages and tools, ranging from grep to the Java String library, contain regular expression matchers. Rather than first translating a regular expression into a deterministic finite automaton, such implementations typically match the regular expression on the fly. Thus they can be seen as virtual machines interpreting the regular expression much as if it were a program with some non-deterministic constructs such as the Kleene star. We formalize this implementation technique for regular expression matching using operational semantics. Specifically, we derive a series of abstract machines, moving from the abstract definition of matching to increasingly realistic machines. First a continuation is added to the operational semantics to describe what remains to be matched after the current expression. Next, we represent the expression as a data structure using pointers, which enables redundant searches to be eliminated via testing for pointer equality. From there, we arrive both at Thompson's lockstep construction and a machine that performs some operations in parallel, suitable for implementation on a large number of cores, such as a GPU. We formalize the parallel machine using process algebra and report some preliminary experiments with an implementation on a graphics processor using CUDA.
\end{abstract}
\section{Introduction}
Regular expressions form a minimalistic language of pattern-matching constructs. Originally defined in Kleene's work on the foundations of computation, they have become ubiquitous in computing. Their practical significance was boosted by Thompson's efficient construction~\cite{thompson1968} of a regular expression matcher based on the ``lockstep'' simulation of a Non-deterministic Finite Automaton (NFA), and the wide use of regular expressions in Unix tools such as grep and awk.
The regular expression matchers used in such tools differ in detail
from the implementation of regular expressions used in compiler construction for lexical analysis.
In compiling, lexical analyzers are typically built by constructing a Deterministic Finite Automaton (DFA), using one of the standard results of automata theory. The DFA can process input very efficiently, but its construction incurs an additional overhead before any input can be matched. Moreover, the DFA construction only works if the matching language really is a regular language, so that it can be recognized by a DFA. Many matching languages add constructs that take the language beyond what a DFA can recognize, for instance back references. (By abuse of terminology, such extended languages are sometimes still referred to as ``regexes''.)
Recently, Cox~\cite{coxregexptwo} has given a rational reconstruction of Thompson's classic NFA matcher in terms of virtual machines. In essence, a regular expression is interpreted on the fly, much as a program in an interpreted programming language. The interpreter is a kind of virtual machine, with a small set of instructions suitable for running regular expressions. For instance, the Kleene star $e^{*}$ gives a form of non-deterministic loop. Cox emphasizes that the virtual machine approach in the style of Thompson is both flexible and efficient. Once a basic virtual machine for regular expressions is set up, other constructs such as back-references can be added with relative ease. Moreover, the machine is much more efficient than other implementation techniques based on a more naive backtracking interpreter~\cite{coxregexpone}, which exhibit exponential run-time in some cases. Surprisingly, these
inefficient matchers are widely used in Java and Perl~\cite{coxregexpone}.
In this paper, we formalize the view of regular expression matchers as machines by using tools from programming language theory, specifically operational semantics. We do so starting from the usual definition of regular expressions and their meaning, and then defining increasingly realistic machines.
We first define some preliminaries and recall what it means for a string to match a regular expression
in Section~\ref{bigstep}; from our perspective, matching is a simple form of big-step semantics, and we aim to refine it into a small-step semantics.
To do so in Section~\ref{ekwmachine}, we introduce a distinction between a current expression and its continuation. We then refine this semantics by representing the regular expression as a syntax tree using pointers in memory (Section~\ref{ewpimachine}). Crucially, the pointer representation allows us to compare sub-expressions by pointer equality (rather than structurally). This pointer equality test is needed for the efficient elimination of redundant match attempts, which underlies the general lockstep NFA simulation presented in Section~\ref{lock}. We recover Thompson's machine as a sequential implementation of the lockstep construction
(Section~\ref{lockstepmachine}). Since the lockstep construction involves simulating many non-deterministic machines in parallel, we then explore a parallel version using some simple process algebra in Section~\ref{gpulockstep}. The parallel process semantics is then related to a prototype implementation we have written in CUDA~\cite{cudaintro} to run on a Graphics Processor Unit (GPU)
in
Section~\ref{cudaimplementation}. Section~\ref{conclusions} concludes with some future directions. The overall plan of the paper can be visualised as follows:
\[
\begin{tikzpicture}
\path (0, 0) node[rectangle] (regexp) {Regular expression matching as big-step semantics (Sec.~\ref{bigstep})}
++(0, -1.2) node[rectangle] (ekw) {EKW machine (Sec.~\ref{ekwmachine})}
++(0, -1.2) node[rectangle] (ewpi) {PW$\pi$ machine (Sec.~\ref{ewpimachine})}
++(0, -1.2) node[rectangle] (lockstep) {Generic lockstep construction (Sec.~\ref{lock})}
++(-3, -1.2) node[rectangle] (seq) {Sequential matcher (Sec.~\ref{lockstepmachine})}
++(+6, 0) node[rectangle] (par) {Parallel matcher (Sec.~\ref{gpulockstep})}
++(0, -1.2) node[rectangle] (cuda) {Implementation on Graphics Processor
(Sec.~\ref{cudaimplementation})};
\path[->, thick] (regexp) edge node[auto] {Small step with continuations} (ekw) ;
\path[->, thick] (ekw) edge node[auto] {Pointer representation} (ewpi) ;
\path[->, thick] (ewpi) edge node[auto] {Macro steps} (lockstep) ;
\path[->, thick] (lockstep) edge node[left] {Sequential scheduling\ \ \ \ } (seq) ;
\path[->, thick] (lockstep) edge node[right] {\ \ \ \ Parallel scheduling} (par) ;
\path[<->, thick] (par) edge node[auto] {Processes as threads in CUDA} (cuda) ;
\end{tikzpicture}
\]
\section{Regular expression matching as a big-step semantics}
\label{bigstep}
Let $\Sigma$ be a finite set, regarded as the input alphabet.
We use the following abstract syntax for regular expressions:
\[
\begin{array}{rcll}
\renewcommand\breakspace{\\ }
e &\gramto& \epsexp \breakspace
e &\gramto& a &\mbox{ where }a \in \Sigma\breakspace
e &\gramto& e^{*}\breakspace
e &\gramto& e_{1}e_{2} \breakspace
e &\gramto& e_{1} \mid e_{2}
\end{array}
\]
We let $e$ range over regular expressions, $a$ over characters, and $w$ over strings of characters. The empty string is written as $\empstring$. Note that there is also a regular expression constant $\epsexp$.
We also write the sequential composition $e_{1}\,e_{2}$ as $e_{1}\bullet e_{2}$ when we want to emphasise it as the occurrence of an operator applied to $e_{1}$ and $e_{2}$, for instance in a syntax tree.
For strings $w_{1}$ and $w_{2}$, we write their concatenation as juxtaposition $w_{1}w_{2}$. A single character $a$ is also regarded as a string of length $1$.
\begin{figure}
\caption{Regular expression matching as a big-step semantics}
\label{regexpmatch}
\end{figure}
Our starting point is the usual definition of what it means for a string $w$ to match a regular expression $e$. We write this relation as $e \matches w$, regarding it as a big-step operation semantics for a language with non-deterministic branching $e_{1} \mid e_{2}$ and a non-deterministic loop $\kleene e$. The rules are given in
Figure~\ref{regexpmatch}.
Some of our operational semantics will use lists.
We write $\cons{h}{t}$ for constructing a list with head $h$ and tail $t$. The concatenation of two lists $s$ and $t$ is written as \(\conc st\). For example, $\cons 1{[2]} = [1,2]$ and $\conc{[1,2]}{[3]} = [1,2,3]$. The empty list is written as $\emplist$.
\section{The EKW machine}
\label{ekwmachine}
The big-step operational semantics of matching in Figure~\ref{regexpmatch} gives us little information about how we should attempt to match a given input string $w$. We define a small-step semantics, called the EKW machine, that makes the matching process more explicit.
In the tradition of the SECD machine~\cite{landinmechanical}, the machine is named after its components: E for expression, K for continuation, W for word to be matched.
\begin{definition}
\label{defekw}
A configuration of the EKW machine is of the form $\ekw{e}{k}{w}$ where $e$ is a regular expression, $k$ is a list of regular expressions, and $w$ is a string. The transitions of the EKW machine are given in Figure~\ref{ekw}. The accepting configuration is $\ekw{\epsexp}{\emplist}{\empstring}$.
\begin{figure}
\caption{EKW machine transition steps}
\label{ekw}
\end{figure}
\end{definition}
Here $e$ is the regular expression the machine is currently focusing on. What remains to the right of the current expression is represented by $k$, the current continuation. The combination of $e$ and $k$ together is attempting to match $w$, the current input string.
Note that many of the rules are fairly standard, specifically the pushing and popping of the continuation stack. The machine is non-deterministic. The paired rules with the same current expressions $\kleene{e}$ or $(e_{1}\mid e_{2})$ give rise to branching in order to search for matches, where it is sufficient that one of the branches succeeds.
\begin{theorem}[Partial correctness]
\label{ekw_prop_partial}
$e\matches w$
if and only if there is a run
\[
\ekw{e}{\emplist}{w} \step \cdots \step \ekw{\epsexp}{\emplist}{\empstring}
\]
\end{theorem}
\begin{example}
\label{exampleekw}
Unfortunately, while Theorem~\ref{ekw_prop_partial} ensures that all matching strings are correctly accepted, there is no guarantee that the machine accepts all strings that it should on every run. In fact, there are valid inputs on which the machine may enter an infinite loop; an example is the configuration $\ekw{\kleene{\kleene a}}{\emplist}{a}$.
\begin{eqnarray*}
\ekw{\kleene{\kleene{a}}}{[\,]}{ a}
&\step &\ekw{\kleene{a}}{[\kleene{\kleene{a}}]}{ a}
\\
&\step &\ekw{\epsexp}{[\kleene{\kleene{a}}]}{ a}
\\
&\step &\ekw{\kleene{\kleene{a}}}{[\,]}{ a}
\\
&\step &\cdots
\end{eqnarray*}
\end{example}
Such infinite loops can be prevented by backtracking and pruning. However, backtracking implementations can still take a very long time matching expressions like $\kleene{\kleene a}$ to a string consisting of, say, 1000 occurrences of a character $\texttt a$ followed by some other $\texttt b$, due to the exponentially increasing search space~\cite{coxregexpone}.
In Thompson's matcher, such loops are avoided by means of redundancy elimination. The matcher checks whether it has encountered the same expression before. Note, however, that ``the same'' expression is to be taken in the sense of pointer equality rather than structural equality. For instance, the two occurrences of $a$ in $(a\,b) \mid (a\,c)$ would be taken as not the same, given their different positions in the syntax tree.
\section{The \pwpiname{} machine}
\label{ewpimachine}
\begin{figure}
\caption{The regular expression $\kleene{\kleene a}
\label{figurepi}
\end{figure}
We refine the EKW machine by representing the regular expression as a data structure in a heap $\heapname$, which serves as the program run by the machine. That way, the machine can distinguish between different positions in the syntax tree.
\begin{definition}
A heap $\pi$ is a finite partial function from addresses to values. There exists a distinguished address $\nullpointer$, which is not mapped to any value.
\end{definition}
In our setting, the values are syntax tree nodes, represented by an operator from the syntax of regular expressions together with pointers to the tree for the arguments (if any) of the operator. For example,
for sequential composition, we have a node containing $(p_{1} \bullet p_{2})$, where the two pointers $p_{1}$ and $p_{2}$ point to the trees of the two expressions being composed.
\begin{definition}
We write $\otimes$ for the partial operation of forming the union of two partial functions provided that their domains are disjoint. More formally, let $f_{1}: A \rightharpoonup B$ and $f_{2}: A \rightharpoonup B$ be two partial functions. Then if $\dom{f_{1}} \cap \dom{f_{2}} = \emptyset$, the function
\[
(f_{1} \otimes f_{2}): A \rightharpoonup B
\]
is defined as $f_{1} \otimes f_{2} = f_{1} \cup f_{2}$.
\end{definition}
Note that $\otimes$ is the same as the operation $*$ on heaps in separation logic~\cite{reynoldslicssep}, and hence a partial commutative monoid. We avoid the notation $*$ as it could be confused with the Kleene star. As in separation logic, we use $\otimes$ to describe data structures with pointers in memory.
\begin{definition}
We write $\pi,p \models e$ if $p$ points to the root node of a regular expression $e$ in a heap $\pi$. The relation is defined by induction on $e$ as follows:
\[
\begin{array}{llrl}
\renewcommand\breakspace{\\[.8ex]}
\pi, p &\models a &&\mbox{if } \pi(p) = a\breakspace
\pi, p &\models \epsexp &&\mbox{if } \pi(p) = \epsexp\breakspace
\pi, p &\models (e_{1} \mid e_{2}) &&\mbox{if } \pi = \pi_{0} \otimes \pi_{1} \otimes \pi_{2}
\wedge \pi_{0}(p) = (p_{1}\mid p_{2})
\breakspace
&&&
\wedge \pi_{1},p_{1} \models e_{1}
\wedge \pi_{2},p_{2}\models e_{2}\breakspace
\pi, p &\models (e_{1} \, e_{2}) &&\mbox{if } \pi = \pi_{0} \otimes \pi_{1} \otimes \pi_{2}
\wedge \pi_{0}(p) = (p_{1}\bullet p_{2})\breakspace
&&&\wedge \pi_{1},p_{1} \models e_{1}
\wedge \pi_{2},p_{2}\models e_{2}\breakspace
\pi, p &\models \kleene{e_{1}} &&\mbox{if } \pi = \pi_{0} \otimes \pi_{1}
\wedge \pi_{0}(p) = \kleene{p_{1}}
\wedge \pi_{1},p_{1} \models e_{1}
\end{array}
\]
\end{definition}
\noindent
Here the definition of $\pi,p \models e$ precludes any cycles in the child pointer chain.
As an example, consider the regular expression $e = \kleene{\kleene{a}} b$. A $\pi$ and $p_{0}$ such that $\pi,p_{0} \models e$ is given by the table in
Figure~\ref{figurepi}. The tree structure, represented by the solid arrows, is drawn on the right.
\begin{definition}
\label{defknode}
Let $\knode$ be a function
\[
\knode{} : \dom{\pi} \to (\dom{\pi} \cup \{ \nullpointer \})
\]
We write $\pi \models \knode{}$ if
\begin{itemize}
\item
If $\pi(p) = (p_{1} \mid p_{2})$, then $\knode{p_{1}} = \knode{p}$ and $\knode{p_{2}} = \knode{p}$
\item
If $\pi(p) = (p_{1} \bullet p_{2})$, then $\knode{p_{1}} = p_{2}$ and $\knode{p_{2}} = \knode{p}$
\item
If $\pi(p) = \kleene{(p_{1})}$, then $\knode{p_{1}} = p$
\item
$\knode{p_{0}} = \nullpointer$, where $p_{0}$ is the pointer to the root of the syntax tree.
\end{itemize}
\end{definition}
The function $\knode$ is uniquely determined by the tree structure layed out in $\pi$, and it is easy to compute by a recursive tree walk. We elide it when it is clear from the context, assuming that $\pi$ always comes equipped with a $\knode$ such that $\pi\models\knode$.
By treating $\knode$ as a function, we have not committed to a particular implementation;
for instance
$\knode$ could be represented as a hash table indexed by pointer values, or it could be added as another pointer field to the nodes in the heap.
In the graphical representation
in Figure~\ref{figurepi}, dashed arrows represent $\knode$. In particular, note the cycle leading downward from $p_{1}$ and up again via dashed arrows. Following such a cycle could lead to infinite loops as for the EKW machine in
Example~\ref{exampleekw}.
\begin{figure}
\caption{PW$\pi$ transitions}
\label{pwpistep}
\end{figure}
\begin{definition}
\label{pwpidef}
The \pwpiname{} machine is defined as follows.
Transitions of this machine are always relative to some heap $\pi$, which does not change during evaluation. We elide $\pi$ if it is clear from the context.
Configurations of the machine are of the form $\pwpi pw$, where $p$ is a pointer in $\pi$ and $w$ is a string of input symbols.
Given the transition relation between pointers defined in Figure~\ref{pwpistep}, the machine has the following transitions:
\[
\infer{p \astep{a} q}{\pwpi{p}{\stringconc aw} \step \pwpi{q}{w} }
\hspace{3em}
\infer{p \astep{} q}{\pwpi{p}{\stringconc w} \step \pwpi{q}{w} }{}
\]
The
accepting state of the machine is $\pwpi{\nullpointer}{\empstring}$. That is, both the continuation and the remaining input have been consumed.
\end{definition}
\begin{example}
For a regular expression $e = \kleene{\kleene{a}} b$, let $\pi$ and $p_{0}$ be such that $\pi,p_{0} \models e$. See Figure~\ref{figurepi} for the representation of $\pi$ as a tree with pointers. The diagram below illustrates two possible executions of the PW$\pi$ machine against inputs $e$ and $aab$.
\\
\begin{minipage}[!t]{0.5\linewidth}\centering
Execution - 1: Infinite loop
\begin{eqnarray*}
&\pwpi{p_{0}}{aab}\\
\longrightarrow &\pwpi{p_{1}}{aab}\\
\longrightarrow &\pwpi{p_{3}}{aab}\\
\longrightarrow &\pwpi{p_{1}}{aab}\\
\longrightarrow &\pwpi{p_{3}}{aab}\\
\longrightarrow &\pwpi{p_{1}}{aab}\\
\longrightarrow &\pwpi{p_{3}}{aab}\\
\longrightarrow &\pwpi{p_{1}}{aab}\\
\longrightarrow &\pwpi{p_{3}}{aab}\\
\longrightarrow &\pwpi{p_{1}}{aab}\\
\longrightarrow &\ldots
\end{eqnarray*}
\end{minipage}
\begin{minipage}[!t]{0.5\linewidth}\centering
Execution - 2: Successful match
\begin{eqnarray*}
&\pwpi{p_{0}}{aab}\\
\longrightarrow &\pwpi{p_{1}}{aab}\\
\longrightarrow &\pwpi{p_{3}}{aab}\\
\longrightarrow &\pwpi{p_{4}}{aab}\\
\longrightarrow &\pwpi{p_{3}}{ab}\\
\longrightarrow &\pwpi{p_{4}}{ab}\\
\longrightarrow &\pwpi{p_{3}}{b}\\
\longrightarrow &\pwpi{p_{1}}{b}\\
\longrightarrow &\pwpi{p_{2}}{b}\\
\longrightarrow &\pwpi{\nullpointer}{\empstring}\\
\end{eqnarray*}
\end{minipage}
\label{pwpiexample}
\end{example}
\begin{theorem}[Simulation]
\label{simekwpwpi}
Let $\pi$ be a heap such that $\pi,p \models e$. Then there is a run of the EKW machine of the form
\[
\ekw{e}{\emplist}{w} \steps \ekw{\epsexp}{\emplist}{\empstring}
\]
if and only if there is a run of the $PW\pi$ machine of the form
\[
\pwpi{p}{w} \steps \pwpi{\nullpointer}{\empstring}
\]
\end{theorem}
One needs to show that each step of the EKW machine can be simulated by the \pwpiname{} machine and vice versa. The invariant in this simulation
is that the stack $k$ in the EKW machine can be reconstructed by following the chain of pointers in the heap of the \pwpiname{} machine via the following function:
\[
\begin{array}{rcll}
\stackof p
&=&
\emplist
&\mbox{if } \knode p = \nullpointer
\\
\stackof p
&=& e \cons {} (\stackof q)
&\mbox{if } q= \knode p \neq \nullpointer
\\
&&&\mbox{and } \pi, q \models e
\end{array}
\]
\section{The lockstep construction in general}
\label{lock}
As we have seen, the \pwpiname{} machine is built from two kinds of steps. Pointers can be evolved via $p \astep{} q$ by moving in the syntax tree without reading any input. When a node for a constant is reached, it can be matched to the first character in the input via a step $p \astep a q$.
\begin{definition}
\label{evolved}
Let $S \subseteq\dom\pi\cup\{\nullpointer\}$ be a set of pointers. We define the evolution $\evolve S$ of $S$ as the following set:
\[
\evolve S
=
\{q \in \dom{\pi} \mid \exists p \in S. p\astep{}^{*} q \wedge \exists a.\pi(q) = a \}
\]
\end{definition}
Forming $\evolve S$ is similar to computing the $\varepsilon$-closure in automata theory.
However, this operation is not a closure operator, because $S \subseteq \evolve S$ does not hold in general.
When one computes $\evolve S$ incrementally, elements are removed as well as added. Avoiding infinite loops by adding and removing the same element is the main difficulty in the computation.
We define a transition relation analogous to Definition~\ref{pwpidef}, but as a deterministic relation
on \emph{sets} of pointers. We refer to these as macro steps, as they assume the computation of $\evolve S$ as given in a single step, whereas an implementation needs to compute it incrementally.
\begin{definition}[Lockstep transitions]
\label{locktransdef}
Let $S, S'\subseteq\dom\pi\cup\{\nullpointer\}$ be sets of pointers.
\[
\begin{array}{rcll}
S & \Astep{} & S'
& \textrm{ if } S' = \evolve S \\[1em]
S & \Astep{a} & S'
& \textrm{ if } S' = \{ q \in \dom{\pi} \mid \exists p \in S. p\astep{a} q \}
\end{array}
\]
\end{definition}
A set of pointers is first evolved from $S$ to $\evolve S$.
Then, moving from a set of pointers $\evolve S$ to $S'$ via $\evolve S \Astep{a} S'$ advances the state of the machine by advancing all pointers that can match $a$ to their continuations. All other pointers are deleted as unsuccessful matches.
\begin{definition}[Generic lockstep machine]
\label{genlockstepmachine}
The generic lockstep machine has configurations of the form $\pwpi Sw$. Transitions are defined using Definition~\ref{locktransdef}:
\[
\infer{S \Astep{a} S'}{\pwpi{S}{\stringconc aw} \macrostep \pwpi{S'}{w} }
\hspace{3em}
\infer{S \Astep{} S'}{\pwpi{S}{\stringconc w} \macrostep \pwpi{S'}{w} }{}
\]
Accepting states of the machine are of the form $\pwpi S\empstring$, where $\nullpointer\in S$.
\end{definition}
\begin{theorem}
\label{pwpithm}
For a heap $\pi, p \models e$ there is a run of the PW$\pi$ machine:
\[
\pwpi{p}{w} \steps \pwpi{\nullpointer}{\empstring}
\]
if and only if there is a run of the lockstep machine
\[
\pwpi{\{p\}}w \macrostep\ldots \macrostep \pwpi{S}{\empstring}
\]
for some set of pointers $S$
with $\nullpointer \in S$.
\end{theorem}
\section{The sequential lockstep machine}\label{lockstepmachine}
The sequential lockstep machine maintains two lists of pointers $c$, $n$ corresponding to pointers being incrementally evolved within the current macro step and pointers to be evolved in the next macro step. Another pointer list $t$ is maintained which provides support for redundancy elimination, we also introduce an auxilary function $\addunique{p}{l_{1}}{l_{2}}$ to aid in this regard:
\begin{definition}
The auxilary function $\addunique{p}{l_{1}}{l_{2}}$ is defined as:
\begin{align*}
&\addunique{p}{l_{1}}{l_{2}} = \cons{p}{l_{1}} \textrm{ if } p \notin \conc{l_{1}}{l_{2}}\\
&\addunique{p}{l_{1}}{l_{2}} = l_{1} \textrm{ if } p \in \conc{l_{1}}{l_{2}}
\end{align*}
\end{definition}
\begin{definition}
The redundancy-eliminating sequential lockstep machine has configurations of the form $\ctnw{c}{t}{n}{w}$. Its transitions are given in figure \ref{red_free_lock_step_trans}.
The accepting states are of the form $\ctnw{\cons{\nullpointer}{c'}}{t'}{n'}{\empstring}$ \begin{figure}
\caption{Sequential lockstep machine with redundancy elimination}
\label{red_free_lock_step_trans}
\end{figure}
\end{definition}
We regard this machine as a rational reconstruction of Thompson's matcher~\cite{thompson1968} in the light of Cox's elucidation as a virtual machine~\cite{coxregexptwo}. This machine uses a sequential schedule for incrementally evolving pointers, keeping a list of pointers that have been evolved already to prevent loops and search space explosion. However, our main interest is in performing this computation in parallel.
\section{Parallel lockstep semantics} \label{gpulockstep}
\begin{figure}
\caption{Process calculus}
\label{process}
\end{figure}
We now define an operational semantics where each pointer is given a dedicated thread for evolving it.
Our motivation is to leverage the large number of cores and hence threads available on GPUs. The semantics in this section is intended as an idealization of the implementation described in Section~\ref{cudaimplementation} below, capturing the essentials of the computation while abstracting from implementation details.
To describe the parallel computation, we define a simple process calculus.
Its transition rules are given in Figure~\ref{process}. Most of our calculus is a subset of CCS~\cite{milnerccs}, with one-to-one directional message passing and parallel composition. However, we also need an $n$-way synchronization with a synchronous transition inspired by Synchronous CCS~\cite{milnersynchrony}.
We let $\proc$ range over processes, $p$ over pointers that may be sent as asynchronous messages, and $a$ over input symbols, which may be used for $n$-way synchronisation. The syntax of processes is as follows:
\begin{eqnarray*}
M & \gramto& \procsend p
\\
&\mid& \proc \parcomp \proc
\\
&\mid& \ppre p\proc
\\
&\mid& \apre a\proc
\end{eqnarray*}
We impose some structural congruences $\equiv$, identifying terms up to associativity and commutativity of parallel compostion $\parcomp$.
Process transitions can be interleaved with rule \textsc{Par}.
We have CCS-style handshake communication in rule \sendrule. Here $\ppre p\proc$ receives the message $\procsend p$ and proceeds with $\proc$ afterwards. Note that receivers of the form $\ppre pM$ are not replicated (in the pi-calculus sense~\cite{milnerpibook}), so that each communication consumes the receiver. This behaviour is essential, as the processes we generate could become trapped in an infinite loop otherwise.
We also have an $n$-way synchronisation \sync. This rule is the most complex, and it is needed to implement matching to input once all pointers have been evolved. The idea is as follows:
\begin{itemize}
\item
The current process is factorized into those processes that are of the form $\apre a M_{j}$ and an $M'$ comprising everything else.
\item There are no further $\astep{}$ transitions inside $M'$, written as $M'\not\longrightarrow$.
\item If these conditions are met, then all the processes waiting to participate in an $n$-way synchronization on $a$ are advanced in one synchronous step.
\item The remaining processes in $M'$ are discarded in the same step.
\end{itemize}
Rules in this style, in which a number of processes are advanced in a single step, are sometimes referred to as ``lockstep''~\cite{milnersynchrony}. Indeed, we use this rule to implement the lockstep matching of regular expressions in the sense of Thompson and Cox.
(In practice, this rule may require a little ad-hoc protocol to implement on a given architecture.)
We translate each expression pointer $p$ in the heap $\pi$ into a process $\sem p\,\pi$ as follows:
\begin{align*}
\sem{p}\,\pi &=
\ppre p(\procsend{q_{1}} \parcomp \procsend{q_{2}})
&&\mbox{if } \pi(p) = (q_{1} \mid q_{2})
\\[1ex]
\sem{p}\,\pi &=
\ppre p\procsend{q_{1}}
&&\mbox{if } \pi(p) = (q_{1} \bullet q_{2})
\\[1ex]
\sem{p}\,\pi &=
\ppre p(\procsend{q_{1}} \parcomp \procsend{q_{2}})
&&\mbox{if } \pi(p) = \kleene{q_{1}} \mbox{ and }\knode p =q_{2}
\\[1ex]
\sem p\,\pi &= \ppre p \procsend q
&&\mbox{if } \pi(p) = \epsexp \mbox{ and } \knode p = q
\\[1ex]
\sem p\,\pi &= \ppre p \apre a\procsend q
&&\mbox{if } \pi(p) = a \mbox{ and } \knode p = q
\end{align*}
Intuitively, for each internal node in the expression tree identified by the pointer $p$, we create a dedicated little process that listens on a channel uniquely corresponding to $p$. For simplicity, we use the same name for the channel as for the pointer. The process may be activated by messages $\procsend p$ sent to it, and it may send such messages itself. These messages trigger a chain reaction that evolve the current pointer set of a macro step. There is no need for these messages to be externally visible, as their only purpose is to wake up their unique recipient.
A process $\ppre pM$ listening for $\procsend p$ is consumed by the transition that receives the message.
Processes for nodes that point to input characters $a$ at the leaves of the expression tree use a different form of communication. All these nodes synchronize on the input symbol. The symbol $a$ is visible in the resulting synchronous transition step $\astep a$, because we need it to agree with the next input symbol.
If $\dom{\pi} = \{p_{1},\ldots, p_{n}\}$, we define the translation $\sem{\pi}$ as the translation of all its pointers:
\[
\sem{p_{1}}\,\pi \parcomp \ldots \parcomp \sem{p_{n}}\,\pi
\]
If the input string is not empty, let $a$ be the first character, so that $a\,w' = w$.
The parallel machine launches processes for all the nodes in the tree, and sends a message to the process for the root. The resulting process makes a number of asynchronous transitions, followed by a synchronous move for $a$:
\[
\sem\pi \parcomp \procsend p \longrightarrow \cdots \longrightarrow \;\astep a \proc
\]
All these steps together represent one macro step. The machine then repeats the above with the next symbol $a'$ and $\proc$
\[
\sem\pi \parcomp \proc \longrightarrow \cdots \longrightarrow \;\astep {a'}\proc'
\]
The machine accepts if the remaining input is empty and the current process is of the form
\[
\procsend{\nullpointer} \parcomp \proc
\]
\begin{example}
\label{exampleprocess}
For $e = \kleene{\kleene{a}} b$, let $\pi$ and $p_{0}$ be such that $\pi,p_{0} \models e$. See Figure~\ref{figurepi} for the representation of $\pi$ as a tree with pointers. Translating the tree structure to parallel processes gives us:
\[
\sem{\pi} = (\ppre{p_{0}}\procsend{p_{1}}) \parcomp \ppre{p_{1}}(\procsend{p_{3}} \parcomp \procsend{p_{2}}) \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \ppre{p_{3}}(\procsend{p_{4}} \parcomp \procsend{p_{1}}) \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}
\]
Assume an input string of $aab$. We have the pointer evolution as follows:
\begin{align*}
&\procsend{p_{0}} \parcomp \sem{\pi}\\
\longrightarrow &\procsend{p_{0}} \parcomp \ppre{p_{0}}\procsend{p_{1}} \parcomp \ppre{p_{1}}(\procsend{p_{3}} \parcomp \procsend{p_{2}}) \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \ppre{p_{3}}(\procsend{p_{4}} \parcomp \procsend{p_{1}}) \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\procsend{p_{1}} \parcomp \ppre{p_{1}}(\procsend{p_{3}} \parcomp \procsend{p_{2}}) \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \ppre{p_{3}}(\procsend{p_{4}} \parcomp \procsend{p_{1}}) \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\procsend{p_{3}} \parcomp \procsend{p_{2}} \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \ppre{p_{3}}(\procsend{p_{4}} \parcomp \procsend{p_{1}}) \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\procsend{p_{3}} \parcomp \apre{b}\procsend{\nullpointer} \parcomp \ppre{p_{3}}(\procsend{p_{4}} \parcomp \procsend{p_{1}}) \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\apre{b}\procsend{\nullpointer} \parcomp \procsend{p_{4}} \parcomp \procsend{p_{1}} \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\apre{b}\procsend{\nullpointer} \parcomp \procsend{p_{1}} \parcomp \apre{a}\procsend{p_{3}}
\end{align*}
Since no more micro transitions are possible, we have reached the $n$-way synchronization point:
\[
\apre{b}\procsend{\nullpointer} \parcomp \procsend{p_{1}} \parcomp \apre{a}\procsend{p_{3}} \astep{a} \procsend{p_{3}}
\]
Now we feed the residual messages back into a fresh $\sem{\pi}$:
\begin{align*}
&\procsend{p_{3}} \parcomp \sem{\pi}\\
\longrightarrow &\procsend{p_{3}} \parcomp \ppre{p_{0}}\procsend{p_{1}} \parcomp \ppre{p_{1}}(\procsend{p_{3}} \parcomp \procsend{p_{2}}) \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \ppre{p_{3}}(\procsend{p_{4}} \parcomp \procsend{p_{1}}) \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\ppre{p_{0}}\procsend{p_{1}} \parcomp \ppre{p_{1}}(\procsend{p_{3}} \parcomp \procsend{p_{2}}) \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \procsend{p_{4}} \parcomp \procsend{p_{1}} \parcomp \ppre{p_{4}}\apre{a}\procsend{p_{3}}\\
\longrightarrow &\ppre{p_{0}}\procsend{p_{1}} \parcomp \ppre{p_{1}}(\procsend{p_{3}} \parcomp \procsend{p_{2}}) \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \procsend{p_{1}} \parcomp \apre{a}\procsend{p_{3}}\\
\longrightarrow &\ppre{p_{0}}\procsend{p_{1}} \parcomp \procsend{p_{3}} \parcomp \procsend{p_{2}} \parcomp \ppre{p_{2}}\apre{b}\procsend{\nullpointer} \parcomp \apre{a}\procsend{p_{3}}\\
\longrightarrow &\ppre{p_{0}}\procsend{p_{1}} \parcomp \procsend{p_{3}} \parcomp \apre{b}\procsend{\nullpointer} \parcomp \apre{a}\procsend{p_{3}}\\
\astep{a} &\procsend{p_{3}}\\
\longrightarrow &\ldots\\
\astep{b} &\procsend{\nullpointer}
\end{align*}
Therefore, we have received a $\procsend{\nullpointer}$ while the input string has become empty, resulting in a successful match.
\end{example}
We need to prove that the construction above can correctly evolve and match any set of pointers.
Let $S=\{p_{1},\ldots,p_{n}\}\subseteq\dom\pi\cup\{\nullpointer\}$ be a set of pointers in the heap. We define
\[
\procsend S = \procsend{p_{1}} \parcomp \ldots \parcomp \procsend{p_{n}}
\]
to represent this set as a parallel composition of messages.
\begin{theorem}
\label{parallelmacrostep}
Let $S, S' \subseteq\dom\pi\cup\{\nullpointer\}$.
We have
\[
S \Astep{}\Astep a S'
\]
if and only if
\[
\procsend S \parcomp \sem\pi \astep{}^{*} \astep a \procsend{S'}
\]
Moreover, each $\astep{}$ transition sequence starting from $\procsend S \parcomp \sem\pi$ is finite.
\end{theorem}
Theorem~\ref{parallelmacrostep} assures us that the parellel operational semantics correctly implements the lockstep construction. The pointers $p$ in the tree, represented as processes $\procsend p$, are evolved in parallel. Although this evolution is non-deterministic, its end result is determinate. Moreover, the cycles in the pointer chain do not lead to cyclic processes looping forever, since each receiving process becomes inactive once the node has been visited.
The correctness proof of the parallel implementation relies on a factorisation of the processes into four components. At each step $i$, we have:
\begin{itemize}
\item A set $S_{i}$ of pointers, indicating nodes that should be evolved.
\item A heap of receivers $\pi_{i}\subseteq \pi$, representing nodes that have not been visited in the current macro step.
\item A set $E_{i}$ of evolved nodes, whose process representations are of the form ready to match a character.
\item A parallel composition $D_{i}$ of messages to nodes that have already been processed.
\end{itemize}
Let $E$ be a set of pointers $E=\{p_{1},\ldots,p_{n}\}$ such that $\pi(p_{j}) =a_{j}$ and
$\knode{p_{j}} =q_{j}$.
We write
\[
\waitchar E = \apre{a_{1}}q_{1} \parcomp\ldots\parcomp\apre{a_{n}}q_{n}
\]
We need to consider transition sequences of the form
\begin{eqnarray*}
&&\procsend{S_{0}} \parcomp \sem{\pi_{0}} \parcomp \waitchar{E_{0}} \parcomp D_{0}\\
& \astep{}& \\
& \vdots& \\
& \astep{}& \procsend{S_{n}} \parcomp \sem{\pi_{n}} \parcomp \waitchar{E_{n}} \parcomp D_{n}
\end{eqnarray*}
where $\pi_{0} =\pi$ and $E_{0}=\emptyset$. The invariant we need to establish for all transition steps consists of:
\begin{eqnarray*}
\evolve{S_{0}} &=& \evolve{(S_{i} \cap \dom{\pi_{i})}} \cup E_{i}
\\
\evolve{R_{i}} & \subseteq & \evolve{(S_{i} \cap \dom{\pi_{i}})} \cup E_{i}
\\
\{p \mid \exists D.D_{i} \equiv (\procsend p \parcomp D) \}
& \subseteq & S_{i} \cup R_{i}
\end{eqnarray*}
where $R_{i} = \dom{\pi} \setminus \dom{\pi_{i}}$. The factorization of proceses at each step and the invariant are verified by case analysis on the kind of node $\pi(p)$ and hence the possible $\astep{}$ steps that its translation $\sem p\,\pi$ can make using the rules from Figure~\ref{process}.
In the final configuration we have $S_{n} \cap \dom{\pi_{n}} = \emptyset$. Hence,
\begin{eqnarray*}
\evolve{S_{0}} &=& \evolve{(S_{n} \cap \dom{\pi_{n})}} \cup E_{n}\\
&=& \evolve{\emptyset} \cup E_{n}\\
&=& E_{n}
\end{eqnarray*}
Therefore, we have $\evolve{S_{0}} = E_{n}$, as required. From that configuration, there can only be an $\astep a$ transition, exactly matching the generic lockstep transition $S \Astep{}\Astep a S'$.
\section{Implementation on a GPU}
\label{cudaimplementation}
As a proof of concept, we have written a simple regular expression matcher where the evolution of pointers is performed in parallel on a GPU.\footnote{The code is available at \url{http://www.cs.bham.ac.uk/~hxt/research/regexp.shtml}.} Programming the GPU was done via CUDA~\cite{cudaintro}.
The main points are:
\begin{itemize}
\item
The regular expression is parsed, and the syntax tree nodes are packed into an array $d$. This array represents our heap $\pi$.
A second pass through the syntax tree performs the wiring of continuation pointers, corresponding to $\knode$.
\item
Two integer vectors $c$, $n$ of the same size as the regular expression vector above are created. Here a value of $t$ - the macro step count, on $c[i]$ implies that regular expression $d[i]$ is to be simulated within the current macro step. On the other hand a value of $-t$ on $c[i]$ implies that the corresponding regular expression has already been simulated for the current macro step. This protocol realizes the semantics of a process being consumed once it has received a message. The vector $n$ is used to collect those search attempts which are able to match the current input character. A value of $t + 1$ on $n[j]$ indicates that the regular expression $d[j]$ is to be simulated on the next macro step.
\item
Each regular expression node $d[i]$ is assigned a GPU thread. This GPU thread is responsible for conditionally simulating the regular expression $d[i]$ at each invocation (depending on $c[i]$ value). While simulating an expression, a GPU thread might schedule another GPU thread / expression $d[j]$ by setting $c[j]$ to $t$ (this could happen for an example in the case of $e = e_{1} \bullet e_{2}$).
Note that one thread scheduling another thread via the $c$ vector corresponds to the sending of a message $\procsend p$ from one process to another.
\item
At each invocation of the GPU threads (called a \textit{kernel launch} in CUDA terminology), each thread which performs a successful simulation updates either of two shared flags which indicate if there were more threads activated on the $c$ or $n$ vectors during the current invocation.
A macro transition involves swapping the $c$ and $n$ vectors while incremeting the $t$ counter. It corresponds to the $n$-way synchronization transition.
\item
The initial state of the machine has only $d[0]$, the root node process, scheduled for simulation.
\end{itemize}
However, note that this description corresponds to a minimalistic GPU-based parallel lockstep machine and does not yet incorporate any optimizations from the literature~\cite{gpuirregular}, such as
\emph{persistent threads} and \emph{tasks queues}.
\section{Conclusions}
\label{conclusions}
We have derived regular expression matchers as abstract machines. In doing so, we have used a number of concepts and techniques from programming language theory. The EKW machine zooms in on a current expression while maintaing a continuation for keeping track of what to do next. In that sense, the machine is a distant relative of machines for interpreting lambda terms, such as the SECD machine~\cite{landinmechanical} or the CEK machine~\cite{felleisensecd}.
On the other hand, regular expressions are a much simpler language to interpret than lambda calculus, so that continuations can be represented by a single pointer into the tree structure (or to machine code in Thompson's original implementation).
While the idea of continuations as code pointers is sometimes advanced as a helpful intuition, the representation of continuations in CPS compiling~\cite{appel} is more complex, involving an environment pointer as well.
To represent pointers and the structures they build up, we found it convenient to use a small fragment of separation logic~\cite{reynoldslicssep}, given by just the separating conjunction and the points-to-predicate. (They are written as $\otimes$ and $\pi(p) =e$ above, to avoid clashes with other notation.)
A similar use of these connectives to describe trees in the setting of abstract machines was used in our earlier work on B+trees~\cite{btree}. Here we translate a tree-shaped data structure into a network of processes that communicate in a cascade of messages mirroring the pointers in the tree structure. The semantics of the processes is inspired by the process algebra literature~\cite{milnerccs,milnersynchrony,milnerpibook}. One reason why a process algebra is suitable for formalizing the lockstep construction with redundancy elimination is that receiving processes are eliminated once they have received a message; they are used linearly, and so are reminiscent of linearly-used continuations~\cite{LinUCHOSC}.
We intend to extend both the process algebra view and our CUDA implementation, while maintaining a close correspondence between them.
Regular expression matching is an instance of irregular parallel~\cite{gpuirregular} processing on a GPU, which presents some optimization problems.
At the moment, the parallel processing power of the GPU cores is not exercised, as each thread does little more than access the expression tree and activate threads for other nodes. We expect the load on the GPU cores to become more significant when more expensive constructs such as back-references (known to be NP-hard) are added to our matching language. It remains to be seen whether a GPU implementation will become more efficient than a sequential CPU-based one, particularly as the number of GPU cores continues to increase (it is currently in the hundreds of cores).
More generally, the operational semantics and abstract machine approach may be fruitful for reasoning about other forms of General Purpose Graphics Processing Unit (GPGPU) programming.
\end{document} | math | 42,092 |
\begin{document}
\begin{frontmatter}
\title{On a Van Kampen Theorem for Hawaiian Groups}
\author[]{Ameneh~Babaee}
\ead{[email protected]}
\author[]{Behrooz~Mashayekhy\corref{cor1}}
\ead{[email protected]}
\author[]{Hanieh~Mirebrahimi}
\ead{h\[email protected]}
\author[]{Hamid~Torabi}
\ead{[email protected]}
\address{Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of
Mashhad,\\
P.O.Box 1159-91775, Mashhad, Iran.}
\cortext[cor1]{Corresponding author}
\begin{abstract}
The paper is devoted to study the $n$th Hawaiian group $\mathcal{H}_n$, $n \ge 1$, of the wedge sum of two spaces $(X,x_*) = (X_1, x_1) \vee (X_2, x_2)$.
Indeed, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces.
First, among some results on Hawaiian groups of semilocally strongly contractible spaces, we present a structure for the $n$th Hawaiian group of the wedge sum of CW-complexes.
Second, we give more informative structures for the $n$th Hawaiian group of the wedge sum $X$, when $X$ is semilocally $n$-simply connected at $x_*$.
Finally, as a consequence, by generalizing the well-known Griffiths space for dimension $n\geq 1$, we give some information about the structure of Hawaiian groups of Griffiths spaces at any points.
\end{abstract}
\begin{keyword}
Hawaiian group\sep Hawaiian earring\sep Van Kampen Theorem\sep Griffiths space.
\MSC[2010]{55Q05, 55Q20, 55P65, 55Q52.}
\end{keyword}
\end{frontmatter}
\section{Introduction and Motivation}
In 2000, K. Eda and K. Kawamura \cite{EdaKaw} generalized the well-known Hawaiian earring space to higher dimensions $n\in \mathbb{N}$ as the following
subspace of $(n+1)$-dimensional Euclidean space ${\mathbb{R}}^{(n+1)}$
\[
\mathbb{HE}^n = \{(r_0,r_1,...,r_n)\in {\mathbb{R}}^{(n+1)}\ |\ (r_0-1/k)^2+{\sum}_{i=1}^n r_i^2 = (1/k)^2, k\in {\mathbb{N}}\}.
\]
Here $\theta = (0,0,...,0)$ is regarded as the base point of the $n$-dimensional Hawaiian earring $\mathbb{HE}^n$, and $\mathbb{S}_k^n$ denotes the $n$-sphere in $\mathbb{HE}^n$ with radius 1/k.
In 2006, U.H. Karimov and D. Repov\v{s} \cite{KarRep} defined a new notion, the $n$th Hawaiian group of
a pointed space $(X,x_0)$, denoted by ${\mathcal{H}}_n(X,x_0)$, to be the set of all pointed homotopy classes $[f]$, where
$f:({\mathbb{HE}^n},\theta)\to (X,x_0)$ is a pointed map. The operation of the $n$th Hawaiian group comes naturally from the operation of the $n$th homotopy group so that the following map
\begin{equation}
{\varphi}:{\mathcal{H}}_n(X,x_0) \to \prod_{\mathbb{N}}{\pi}_n(X,x_0), \tag{I}
\end{equation}
defined by $\varphi([f])=([f{\mid}_{\mathbb{S}_1^n}], [f{\mid}_{\mathbb{S}_2^n}],... )$ is a homomorphism, for all $n \in \mathbb{N}$. For every pointed space $(X,x_0)$, the image of $\varphi$ and also $\mathcal{H}_n(X,x_0)$ contain $\prod^W_{\mathbb{N}}{\pi}_n(X,x_0)$ as a normal subgroup (see the proof of \cite[Theorem 2.13]{1}).
One can see that ${\mathcal{H}}_n:hTop_{\ast} \to Groups $ is a covariant functor from the pointed homotopy category,
$hTop_{\ast}$, to the category of all groups, $Groups$, for any $n\geq 1$. If
$\beta:(X,x_0)\to (Y,y_0)$ is a pointed map, then
$\mathcal{H}_n(\beta)={\beta}_{\ast}:{\mathcal{H}}_n (X,x_0)\to {\mathcal{H}}_n (Y,y_0)$ defined by
${\beta}_{\ast}([f])= [\beta \circ f]$ is a homomorphism (see \cite{KarRep}).
Although the $n$th Hawaiian group functor is a pointed homotopy invariant functor on the category of all pointed topological spaces, it is not freely homotopy invariant. Because unlike other homotopy invariant functors, Hawaiian groups of contractible spaces are not necessarily trivial.
Karimov and Repov\v{s} \cite{KarRep} gave a contractible space, the cone over $\mathbb{HE}^1$, with nontrivial
$1$th Hawaiian group at some points (consisting of the points at which $C\mathbb{HE}^1$ is not locally $1$-simply connected), but with trivial homotopy, homology and cohomology groups. More precisely, it can be shown that
${\mathcal{H}}_1(C({\mathbb{HE}}^1),\theta)$ is uncountable, using \cite[Theorem 2]{KarRep}: ``{\it if $X$ is first countable at $x_0$, then countability of $n$th Hawaiian group ${\mathcal{H}}_n (X,x_0)$ implies locally $n$-simply connectedness of $X$ at $x_0$}."
Furthermore, a converse of the above statement can be found in \cite[Corollaries 2.16 and 2.17]{1}: ``{\it let $X$ be first countable at $x_0$. Then $\mathcal{H}_n(CX, \tilde{x})$ is trivial if and only if $X$ is locally $n$-simply connected at $x_0$ and it is uncountable otherwise}".
Accordingly, this functor can help us to obtain some local properties of spaces.
In addition, unlike homotopy groups, Hawaiian groups of pointed space $(X,x_0)$ depend on the behaviour of $X$ at $x_0$, and then their structures depend on the choice of the base point. In this regard, there exist some examples of path connected spaces with non-isomorphic Hawaiian groups at different points, such as the $n$-dimensional Hawaiian earring, where $n \ge 2$ (see \cite[Corollary 2.11]{1}).
Despite the above different behaviors between Hawaiian groups and homotopy groups, they have some similar behaviors. For instance, it was proved that similar to the $n$th homotopy group, the $n$th Hawaiian group of any pointed space is abelian, for $n \ge 2$ \cite[Theorem 2.3]{1}. Also, the Hawaiian groups preserve products in the category $hTop_*$ \cite[theorem 2.12]{1}.
In this paper, we investigate the Hawaiian groups of the coproduct in the category $hTop_*$ which is the wedge sum of a given family of pointed spaces. In fact, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. In Section 2, among some results on Hawaiian groups of semilocally strongly contractible spaces, we intend to present a structure for the $n$th Hawaiian group of the wedge sum of CW-complexes. A space $X$ is called semilocally strongly contractible at $x_0$ if there exists some open neighbourhood $U$ of $x_0$ such that the inclusion $i :U \hookrightarrow X$ is nullhomotopic in $X$ relative to $\{x_0\}$, in other words, $i : (U,x_0) \hookrightarrow (X,x_0)$ is nullhomotopic (see \cite{EdaKaw}).
In Section 3, we present the $n$th Hawaiian group of the wedge sum $(X,x_*) = (X_1, x_1) \vee (X_2, x_2)$ as the semidirect product of two its subgroups that are more perceptible, when $X$ is semilocally $n$-simply connected at $x_*$. Also, we prove that the Hawaiian group of a pointed space equals the Hawaiian group of every neighbourhood of the base point if all $n$-loops are small. An $n$-loop $\alpha: (\mathbb{S}^n, 1) \to (X, x)$ is called small, if for each neighbourhood $U$ of $x$, $\alpha$ has a homotopic representative in $U$ (see \cite{PasGha, Vir}).
In Section 4, generalizing the well-known Griffiths space, we define the $n$th Griffiths space for $n \ge 2$, as the wedge sum of two copies of the cone over the $n$-dimensional Hawaiian earring. For the sake of clarity, we call the well known Griffiths space as the $1$th Griffiths space. Then, using results of Sections 2 and 3, we intend to give some information about the structure of Hawaiian groups of Griffiths spaces at any points.
In this paper, all homotopies are relative to the base point, unless stated otherwise.
\section{Hawaiian Groups of Semilocally Strongly Contractible Spaces}
In this section, we investigate Hawaiian groups of wedge sum of pointed spaces which are semilocally strongly contractible at the base points. The property semilocally strongly contractibility was defined by Eda and Kawamura \cite{EdaKaw}. First, we compare semilocally strongly contractible property with some familiar properties, such as locally contractible, locally strongly contractible and semilocally contractible properties.
\begin{recall}
Let $(X, x)$ be a pointed space, then $X$ is called locally contractible at $x$ if for each open neighbourhood $U$ of $x$ in $X$, there exists some open neighbourhood $V$ of $x$ contained in $U$ such that the inclusion $V \hookrightarrow U$ is freely nullhomotopic.
We say that $X$ is locally strongly contractible at $x$ if for each open neighbourhood $U$ of $x$ in $X$, there exists some open neighbourhood $V$ of $x$ contained in $U$ such that the inclusion $V \hookrightarrow U$ is nullhomotopic relative to $\{x\}$, or briefly, $(V, x) \hookrightarrow (U, x)$ is nullhomotopic.
Moreover, $X$ is called semilocally contractible at $x$, if there exists some open neighbourhood $U$ of $x$ such that the inclusion $i :U \hookrightarrow X$ is freely nullhomotopic in $X$.
\end{recall}
One can see that there are some relations between these properties.
Locally strongly contractible property implies locally contractibility, and locally contractible property implies semilocally contractibility, but the converse statements do not necessarily hold. Moreover, locally strongly contractible property implies semilocally strongly contractibility, and semilocally strongly contractible implies semilocally contractibility, but converse statements do not necessarily hold. The following example shows that a given space can behave differently at different points.
\begin{example}
The $1$th Griffiths space $\mathcal{G}_1$ is locally strongly contractible at two vertices $v$ and $v'$. Therefore, it is locally contractible, semilocally strongly contractible and semilocally contractible at two vertices $v$ and $v'$. Moreover, $\mathcal{G}_1$ is semilocally contractible at all points except at the common point $x_*$. Also, $\mathcal{G}_1$ is semilocally contractible at any point of $A$ and $A'$, but not semilocally strongly contractible. Finally, $\mathcal{G}_1$ is neither semilocally strongly contractible, semilocally contractible nor even semilocally $1$-simply connected at $x_*$ (see \cite{Gri}).
\begin{figure}
\caption{The $1$th Griffiths space.}
\label{fi1}
\end{figure}
\end{example}
The $1$-dimensional Hawaiian earring $\mathbb{HE}^1$ is not semilocally contractible at the origin. However, it is locally strongly contractible at other points.
There exist many spaces which are semilocally contractible at each point, but not strongly at some points.
For example, consider $C\Delta$ as the cone over the Seirpeinski gasket. It is semilocally contractible at any points, but it is semilocally strongly contractible just at the vertex.
The following lemma describes the Hawaiian group of the wedge sum of two spaces by its subgroups. This construction helps us to obtain next isomorphisms.
\begin{lemma}\label{le4.1}
Let $(X_1,x_1)$ and $(X_2, x_2)$ be two pointed spaces, and $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$.
\begin{enumerate}[(i)]
\item
If $U_1$ and $U_2$ are two open neighbourhoods of $x_1$ and $x_2$ in $X_1$ and $X_2$, respectively, and $j:U_1 \vee U_2 \to X$ is the inclusion map, then
\begin{equation}\label{eq2.1nn}
\mathcal{H}_n(X, x_*) = j_* \mathcal{H}_n( U_1 \vee U_2, x_*) {\prod_{\mathbb{N}}}^W \pi_n(X,x_*).
\end{equation}
\item
If $x \in X_1 \setminus \{x_1\}$ and $\{x_1\}$ is closed in $X_1$, then
\begin{equation}\label{eq2.2nn}
\mathcal{H}_n(X, x) =
\mathcal{H}_n(X_1, x) {\prod_{\mathbb{N}}}^W \pi_n(X,x).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(i)]
\item
Let $U_1$ and $U_2$ be two open neighbourhoods of $x_1$ and $x_2$ in $X_1$ and $X_2$, respectively, $j: U_1 \vee U_2 \to X$ be the inclusion map, and
$f: (\mathbb{HE}^n, \theta) \to (X,x_*)$ be a pointed map. Since $U_1 \vee U_2$ is open in $X$,
there exists $K \in \mathbb{N}$ such that if $k \geq K$, then $im(f|_{\widetilde{\bigvee}_{k \geq K}\mathbb{S}_k^n}) \subseteq U_1 \vee U_2$. We can define $\underline{f}, \overline{f}: \mathbb{HE}^n \to (X,x)$ by
$\underline{f}|_{\bigvee_{k <K}\mathbb{S}_k^n} = f|_{\bigvee_{k <K}\mathbb{S}_k^n}$,
$\underline{f}|_{\widetilde{\bigvee}_{k \geq K}\mathbb{S}_k^n} = C|_{\widetilde\bigvee_{k \ge K}\mathbb{S}_k^n}$, $\overline{f}|_{\bigvee_{k <K}\mathbb{S}_k^n} = C|_{\bigvee_{k <K}\mathbb{S}_k^n}$, and
$\overline{f}|_{\widetilde{\bigvee}_{k \geq K}\mathbb{S}_k^n} = f|_{\widetilde\bigvee_{k \ge K}\mathbb{S}_k^n}$, where $C$ is the constant map.
Obviously, $[f] = [\underline{f}][\overline{f}] = [\overline{f}][\underline{f}]$. Moreover, $[\underline{f}]$ is an element of $\prod_{\mathbb{N}}^W \pi_n(X,x_*)$ and also, $[\overline{f}]$ is an element of $j_* \mathcal{H}_n(U_1 \vee U_2, x_*)$. Therefore, $\mathcal{H}_n(X, x_*)$ is generated by $j_*\mathcal{H}_n(U_1 \vee U_2, x_*) \cup \prod_{\mathbb{N}}^W \pi_n(X,x_*)$.
The equality \eqref{eq2.1nn} holds because $\prod_{\mathbb{N}}^W \pi_n(X, x_*) \unlhd \mathcal{H}_n(X,x_*)$ (see the proof of \cite[Theorem 2.13]{1}).
\item
Since $X_1$ is a retract of $X$, one can consider $\mathcal{H}_n(X_1, x)$ as a subgroup of $\mathcal{H}_n(X, x) $.
Let $f: (\mathbb{HE}^n, \theta) \to (X,x)$ be a pointed map. Since $\{x_1\}$ is closed in $X_1$, $X_1 \setminus \{x_1\}$ is open. Let $i: X_1 \setminus \{x_1\} \to X$ be the inclusion map. Similar to the previous part, $[f]$ can be factorized as
$[f] = [\underline{f}][\overline{f}] = [\overline{f}][\underline{f}]$, where $[\underline{f}]$ is an element of $\prod_{\mathbb{N}}^W \pi_n(X,x)$ and also $[\overline{f}]$ is an element of $i_*\mathcal{H}_n(X_1 \setminus \{x_1\}, x)$ which is a subgroup of $\mathcal{H}_n(X_1, x)$. Thus, $\mathcal{H}_n(X, x)$ is generated by $\mathcal{H}_n(X_1, x) \cup \prod_{\mathbb{N}}^W \pi_n(X,x)$. Again, equality \eqref{eq2.2nn} holds by normality of $\prod_{\mathbb{N}}^W \pi_n(X,x)$.
\end{enumerate}
\end{proof}
Note that equalities \eqref{eq2.1nn} and \eqref{eq2.2nn} may not be the same. For instance, in Example \ref{ex3.9n} one group is trivial, but not the other one, in general.
\begin{note}\label{le3.2n}
Using the property that every open neighbourhood in the wedge sum is a wedge sum of open neighbourhoods, one can rewrite the proof of Lemma \ref{le4.1} for arbitrary spaces stated as follows.
Let $(X,x_0)$ be a pointed space. Then
$\mathcal{H}_n(X,x_0) = j_* \mathcal{H}_n(U, x_0) \prod_{\mathbb{N}}^W \pi_n(X,x_0)$,
for each open neighbourhood $U$ with $x_0 \in U \subseteq X$ and $j : U \to X$ as the inclusion map.
\end{note}
A result similar to the following theorem was proved in \cite[Theorem 2.5]{1} by a slightly different argument.
\begin{theorem}\label{th2.6n}
Let $(X_1, x_1)$ and $(X_2, x_2)$ be two pointed spaces, $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$, and $n \ge 1$.
If $X_1$ and $X_2$ are semilocally strongly contractible at $x_1$ and $x_2$, respectively, then
\begin{equation*}\label{eq2.3nn}
\mathcal{H}_n(X, x_*) = {\prod_{\mathbb{N}}}^W \pi_n(X, x_*).
\end{equation*}
\end{theorem}
\begin{proof}
Since $X_1$ and $X_2$ are semilocally strongly contractible at $x_1$ and $x_2$, respectively, there exist open neighbourhoods $U_1$ of $x_1$ and $U_2$ of $x_2$ such that inclusion maps $(U_1, x_1) \hookrightarrow (X_1, x_1)$ and $(U_2, x_2) \hookrightarrow (X_2, x_2)$ are nullhomotopic. By joining these homotopies, one can see that $j:U_1 \vee U_2 \to X_1 \vee X_2$ is nullhomotopic and hence $j_* \mathcal{H}_n(U_1 \vee U_2, x_*)$ is trivial. The result holds by Lemma \ref{le4.1}.
\end{proof}
The following example shows that Theorem \ref{th2.6n} does not hold without condition semilocally strongly contractible on both of spaces.
\begin{example}\label{ex2.10nn}
Consider $x_*$ as the common point of the $1$th Griffiths space. We show that if Theorem \ref{th2.6n} holds for the $1$th Griffiths space at $x_*$, then $\pi_1(\mathcal{G}_1, x_*)$ is trivial which is a contradiction (see \cite{Gri}). Assume that $\mathcal{H}_1(\mathcal{G}_1, x_*) = \prod_{\mathbb{N}}^W \pi_1(\mathcal{G}_1, x_*)$, then the homomorphism $\varphi: {\mathcal{H}}_n(X,x_0) \to \prod_{\mathbb{N}}{\pi}_n(X,x_0)$ (see (I)) can be considered as the natural injection. If we show that the homomorphism $\varphi$ is surjective, then the natural injection $\prod^W_{\mathbb{N}} \pi_1(\mathcal{G}_1, x_*) \to \prod_{\mathbb{N}} \pi_1(\mathcal{G}_1, x_*)$ is surjective which is impossible in a nontrivial way.
To prove surjectivity of $\varphi$, let $\{[f_k]\} \in \prod_{\mathbb{N}} \pi_1(\mathcal{G}_1, x_*)$. Since any $n$-loop at $x_*$ is small, we can find a homotopic representative $f'_k$ of $[f_k]$ in $U_k$, for all $k \in \mathbb{N}$. Now we can define $f:\mathbb{HE}^1 \to \mathcal{G}_1$ by $f|_{\mathbb{S}_k^1} = f'_k$, satisfying $\varphi ([f])= \{[f_k]\}$.
\end{example}
CW-complexes are locally homeomorphic to some cells. Thus, they are semilocally strongly contractible at any point. The following result presents an isomorphism for the Hawaiian group of the wedge sum of CW-complexes.
\begin{corollary}\label{co2.5n}
Let $X_1$ and $X_2$ be two locally finite $(n-1)$-connected CW-complexes, $(X, x_*)=(X_1, x_1) \vee (X_2, x_2)$, and $n \geq 2$. Then
\begin{equation}\label{eq2.1}
\mathcal{H}_n(X,x_*) \cong\mathcal{H}_n(X_1, x_1) \oplus \mathcal{H}_n(X_2, x_2).
\end{equation}
\end{corollary}
\begin{proof}
CW-complexes are semilocally strongly contractible. Thus by Theorem \ref{th2.6n},
$\mathcal{H}_n(X,x) \cong \bigoplus_{\mathbb{N}} \pi_n(X,x)$, when $n \geq 2$.
Now by \cite[Proposition 6.36]{Swi},
$\pi_n(X,x) \cong \pi_n(X_1) \oplus \pi_n(X_2)$, and after a rearrangement,
$\mathcal{H}_n(X,x) \cong \bigoplus_{\mathbb{N}} \pi_n(X_1) \oplus \bigoplus_{\mathbb{N}}\pi_n(X_2)$. We obtain the result, using \cite[Theorem 2.5]{1}.
\end{proof}
An analogous isomorphism for \eqref{eq2.1} does not hold, when $n =1$. To obtain such an isomorphism on the $1$th Hawaiian group, we must replace direct sum by free product, because the fundamental group and the $1$th Hawaiian group are not abelian groups, in general. Using Theorem \ref{th2.6n} and the van Kampen Theorem for wedge sum we have the following result.
\begin{corollary}\label{co2.10n}
Let $X_1$ and $X_2$ be two semilocally strongly contractible spaces at $x_1$ and $x_2$, respectively, and $(X, x_*)= (X_1, x_1) \vee (X_2, x_2)$. Then
\begin{equation}\label{eq2.2n}
\mathcal{H}_1(X, x_*) \cong {\prod_{\mathbb{N}}}^W \big( \pi_1(X_1,x_1) * \pi_1(X_2,x_2) \big).
\end{equation}
\end{corollary}
Note that the isomorphism \eqref{eq2.2n} is not similar to the case $n \ge 2$, even if $X$ is a special CW-complex.
Because if we consider $\mathcal{H}_1(X, x_*) \cong\mathcal{H}_1(X_1,x_1) * \mathcal{H}_1(X_2,x_2)$, then by Theorem \ref{th2.6n}, $\mathcal{H}_1(X, x_*) \cong \prod^W_{\mathbb{N}} \pi_1(X_1,x_1) * \prod^W_{\mathbb{N}} \pi_1(X_2,x_2)$. Hence, by isomorphism \eqref{eq2.2n} we must have
\[
{\prod_{\mathbb{N}}}^W \big( \pi_1(X_1,x_1) * \pi_1(X_2,x_2) \big) \cong {\prod_{\mathbb{N}}}^W \pi_1(X_1,x_1) * {\prod_{\mathbb{N}}}^W \pi_1(X_2,x_2),
\]
which is impossible by \cite[Page 183, 6.3.10]{Rob}, in a nontrivial way.
Therefore, $\mathcal{H}_1(X, x_*)\not \cong \mathcal{H}_1(X_1,x_1) * \mathcal{H}_1(X_2,x_2)$, in a nontrivial way.
\section{Hawaiian Groups of Semilocally $n$-Simply Connected Spaces}
In this section, we study more on Hawaiian groups of the wedge sum in semilocally $n$-simply connected spaces. We present results for $n=1$ and $n \ge 2$, separately, due to the difference in group structures.
Recall that for $n \geq 1$, a space $X$ is called $n$-simply connected at $x$ if $\pi_n(X,x)$ is trivial and it is called $n$-connected at $x$ if $\pi_j(X,x)$ is trivial, for $ 1 \leq j \leq n$.
Also, $X$ is called semilocally $n$-simply connected at $x$ if there exists a neighbourhood $U$ of $x$ such that the homomorphism $\pi_n (j): \pi_n(U,x) \to \pi_n(X,x)$, induced by the inclusion, is trivial.
\begin{theorem}\label{th2.8n}
Let $(X_1,x_1)$ and $(X_2, x_2)$ be two pointed spaces and $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$. If $X$ is semilocally $1$-simply connected at $x_*$, then
\[
\mathcal{H}_1(X, x_*) \cong j_* \mathcal{H}_1 (U_1 \vee U_2) \ltimes {\prod_{\mathbb{N}}}^W \pi_1(X, x_*),
\]
for some neighbourhood $U_1 \vee U_2$ of $x_*$ with the inclusion map $j: U_1\vee U_2 \to X$.
\end{theorem}
\begin{proof}
By Lemma \ref{le4.1} part 1, $\mathcal{H}_1(X, x_*) = j_*\mathcal{H}_1(U_1 \vee U_2, x_*) \prod_{\mathbb{N}}^W \pi_1(X,x_*)$ for each open neighbourhood $U_1 \vee U_2$ of $x_*$. Let $U_1 \vee U_2$ be the neighbourhood for which $\pi_1(j):\pi_1(U_1 \vee U_2, x_*) \to \pi_1(X, x_*)$ is the trivial homomorphism and let $[f] \in j_*\mathcal{H}_1(U_1 \vee U_2, x_*) \cap \prod_{\mathbb{N}}^W \pi_1(X,x_*)$. Since $[f] \in j_*\mathcal{H}_1(U_1 \vee U_2, x_*)$, there exists $\tilde{f} :(\mathbb{HE}^1, \theta )\to (U_1 \vee U_2,x_*)$ such that $j_* [\tilde{f}] = [f]$, or equivalently, $j \circ \tilde{f} \simeq f$. Also, since $[f] \in \prod_{\mathbb{N}}^W \pi_1(X,x_*)$, $f$ can be assumed as a map with $f|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^1} = C|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^1}$ for some $K \in \mathbb{N}$. Hence $j \circ \tilde{f}|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^1} \simeq C|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^1}$. Using \cite[Lemma 2.2]{1}, one can replace $\tilde{f}$ with a map $\hat{f}$ such that $\hat{f}|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^1} \simeq C|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^1}$. Thus $[\hat{f}] \in \prod_{\mathbb{N}}^W \pi_1(U_1 \vee U_2,x_*)$ and then $[f] \in \prod_{\mathbb{N}}^W \pi_1(j)\pi_1(U_1 \vee U_2,x_*)$ which is trivial, because of the choice of $U_1 \vee U_2$.
Hence $j_*\mathcal{H}_1(U_1 \vee U_2, x_*) \cap \prod_{\mathbb{N}}^W \pi_1(X,x_*) = \langle e \rangle$, and the result holds by normality of $\prod_{\mathbb{N}}^W \pi_1(X,x_*)$ in $\mathcal{H}_1(X, x_*)$.
\end{proof}
By a similar argument, we can conclude the following result for the wedge sum of $1$-simply connected spaces.
\begin{theorem}\label{th4.2n}
Let $(X_1,x_1)$ and $(X_2, x_2)$ be two pointed spaces and $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$. If $\pi_1(X_1, x) = \langle e \rangle$ for some $x \in X_1 \setminus \{x_1\}$, then
\[
\mathcal{H}_1(X, x) \cong
\mathcal{H}_1(X_1, x) \ltimes {\prod_{\mathbb{N}}}^W \pi_1(X,x).
\]
\end{theorem}
In the following two theorems, we reconstruct isomorphisms in Theorems \ref{th2.8n} and \ref{th4.2n} for $n \ge 2$. In this case, since all groups are abelian and all subgroups are normal, semidirect product $\ltimes$ must be replaced by direct sum $\oplus$.
\begin{theorem}
Let $(X_1,x_1)$ and $(X_2, x_2)$ be two pointed spaces, $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$, and $n \ge 2$. If $X$ is semilocally $n$-simply connected at $x_*$, then
\[
\mathcal{H}_n(X,x_*) \cong j_* \mathcal{H}_n (U_1 \vee U_2) \oplus \bigoplus_{\mathbb{N}} \pi_n(X,x_*),
\]
for some neighbourhood $U_1 \vee U_2$ of $x_*$ with the inclusion map $j : U_1 \vee U_2 \to X$.
\end{theorem}
\begin{theorem}\label{th4.4}
Let $(X_1,x_1)$ and $(X_2, x_2)$ be two pointed spaces, $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$, and $n \geq 2$. If $\pi_n(X_1, x) = \langle e \rangle$ for some $x_1 \neq x \in X_1$, then
\[
\mathcal{H}_n(X, x) \cong
\mathcal{H}_n(X_1, x) \oplus \bigoplus_{\mathbb{N}} \pi_n(X,x) .
\]
\end{theorem}
\v{Z}. Virk \cite{Vir} defined small $1$-loop and studied small loop spaces. Note that a nullhomotopic loop is a small loop.
H. Passandideh and F.H. Ghane \cite{PasGha} defined and studied the notions of $n$-homotopically Hausdorffness and small $n$-loops, for $n \ge 2$. An $n$-loop $\alpha: (\mathbb{S}^n, 1) \to (X, x)$ is called small if it has a homotopic representative in every open neighbourhood of $x$.
\begin{theorem}\label{th2.5n}
Let $(X_1,x_1)$ and $(X_2, x_2)$ be two pointed spaces, $(X, x_*) = (X_1, x_1) \vee (X_2, x_2)$, and $n \ge 1$. Also, let $\{x_1\}$ be a closed subset in $X_1$, and $x_1 \neq x\in X_1$.
\begin{enumerate}[(i)]
\item
All $n$-loops in $X$ at $x_*$ are small if and only if
\begin{equation}\label{eq3.6.1}
\mathcal{H}_n(X, x_*) = j_* \mathcal{H}_n( U_1 \vee U_2, x_*),
\end{equation}
for any neighbourhoods $U_1$ and $U_2$ of $x_1$ and $x_2$ in $X_1$ and $X_2$, respectively, when $j:U_1 \vee U_2 \to X$ is the inclusion map.
\item
If all $n$-loops in $X$ at $x$ are small, then
\begin{equation}\label{eq3.6.2}
\mathcal{H}_n(X, x) =
\mathcal{H}_n(X_1, x).
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}[(i)]
\item
Let $U_1$ and $U_2$ be two arbitrary neighbourhoods of $x_1$ and $x_2$, respectively, and let $[f] \in \prod_{\mathbb{N}}^W \pi_n(X,x_*)$. We show that $[f] \in j_* \mathcal{H}_n( U_1 \vee U_2, x_*)$, and then the equality \eqref{eq3.6.1} is obtained by Lemma \ref{le4.1}.
Since $[f] \in \prod_{\mathbb{N}}^W \pi_n(X,x_*)$, one can consider $f$ as $f|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^n} = C|_{\widetilde{\bigvee}_{k \geq K} \mathbb{S}_k^n}$ for some $K \in \mathbb{N}$. Moreover, since each $n$-loop in $X$ is small at $x_*$, any $n$-loop $\alpha$ is homotopic to some $n$-loop in $U_1 \vee U_2$, say $\tilde{\alpha}:\mathbb{S}^n \to U_1 \vee U_2$, such that $j \circ \tilde{\alpha} \simeq \alpha$. By induction on finite join of $n$-loops,
$f|_{\widetilde{\bigvee}_{k < K} \mathbb{S}_k^n}$ has a homotopic representative $\tilde{f}$ in $U_1 \vee U_2$ with $j \circ \tilde{f} \simeq f|_{\widetilde{\bigvee}_{k < K}\mathbb{S}_k^n}$. Therefore, $[f] \in j_* \mathcal{H}_n( U_1 \vee U_2, x_*)$.
Conversely, let $\alpha$ be an $n$-loop in $X$ at $x_*$. Consider the map $f:(\mathbb{HE}^n, \theta) \to (X,x_*)$ so that $f|_{\mathbb{S}_1^n} = \alpha$ and $f|_{\mathbb{S}_k^n} = c$ for $k >1$. Then $[f] \in j_*\mathcal{H}_n(U_1\vee U_2, x_*)$ by equality \eqref{eq3.6.1} for any neighbourhoods $U_1$ and $U_2$. Let $[\tilde{f}]$ be the element of $\mathcal{H}_n(U_1\vee U_2, x_*)$ such that $j \circ \tilde{f} = f$. Hence, $j \circ \tilde{f}|_{\mathbb{S}_1^n} = f|_{\mathbb{S}_1^n} = \alpha$, that is $\alpha$ is homotopic to some $n$-loop in $U_1 \vee U_2$. Since $U_1$ and $U_2$ are arbitrary neighbourhoods, $\alpha$ is a small $n$-loop.
\item
Since $X_1 \setminus \{x_1\}$ is open and all $n$-loops at $x$ in $X$ are small, similar to the proof of the previous part, one can show that $\prod_{\mathbb{N}}^W \pi_n(X,x) \subseteq i_*\mathcal{H}_n(X_1 \setminus \{x_1\}, x)$, where $i: X_1 \setminus \{x_1\} \to X$ is the inclusion map. Moreover, $i_*\mathcal{H}_n(X_1 \setminus \{x_1\}, x)$ is contained in $\mathcal{H}_n(X_1, x)$ as a subgroup. Thus, by Lemma \ref{le4.1}, the equality \eqref{eq3.6.2} holds.
\end{enumerate}
\end{proof}
Since a nullhomotopic $n$-loop is a special case of small $n$-loop, the equalities \eqref{eq3.6.1} and \eqref{eq3.6.2} hold for $n$-simply connected spaces.
For example, Theorem \ref{th2.5n} holds for two cones joining at their vertices which is not only $n$-simply connected, but also contractible. Recall that the $1$th Griffiths space, the wedge sum of two cones, is not contractible, even more Griffiths \cite{Gri} proved that it is not $1$-simply connected.
\begin{example}\label{ex3.9n}
For a space $X$, put $Y= \frac{X\times [-1,1]}{X \times \{0\}}$ the wedge sum of two cones over $X$ at their vertices and $\tilde{x}_t= [(x_0, t)]$ for $t \neq 0$. One can see that $Y$ is contractible at the common point, and hence, it is $n$-simply connected. By Theorem \ref{th2.5n}, $\mathcal{H}_n(Y, \tilde{x}_t) = \mathcal{H}_n(CX , \tilde{x}_t)$.
Also, by \cite[Theorem 2.13]{1},
$\mathcal{H}_n(Y, x_0) \cong \frac{\mathcal{H}_n(X, x_0)}{\prod^W_{k \in \mathbb{N}} \pi_n(X, x_0)}$.
Let $x_*$ be the common vertex of the two cones, then $\mathcal{H}_n(Y, x_*)$ is trivial, for $Y$ is semilocally strongly contractible at $x_*$ and by Theorem \ref{th2.6n}, $\mathcal{H}_n(Y, x_*) = \prod_{\mathbb{N}}^W \pi_n(Y, x_*)$ which is trivial.
\end{example}
The following example reveals that Theorem \ref{th2.5n} does not hold, if some non small loop exists.
\begin{example}
Let $(X, x_*)= (C(\mathbb{HE}^1),\theta) \vee (\mathbb{S}^1, 1)$ (see Figure \ref{fi2}) .
If one assumes that $\mathcal{H}_1(X, x_*) = \mathcal{H}_1(C(\mathbb{HE}^1), x_*)$, then the simple $1$-loop in $\mathbb{S}^1$ must be nullhomotopic which is a contradiction.
\begin{figure}
\caption{The wedge sum of a circle and a cone on the Hawaiian earring}
\label{fi2}
\end{figure}
\end{example}
\section{Hawaiian Groups of Griffiths Spaces}
In this section, by generalizing the Griffiths space to higher dimensions and applying the results of Sections 2 and 3, we study the $n$th Hawaiian group of the $n$th Griffiths space, for $n \ge 1$.
Eda \cite{Eda} introduced the free $\sigma$-product $\times^{\sigma}_{\mathbb{N}} \mathbb{Z}$ as the group consisting of all reduced $\sigma$-words, and then proved that it is isomorphic to $\pi_1(\mathbb{HE}^1, \theta)$ \cite[Theorem A.1]{Eda}. To prove, Eda remarked that each $1$-loop in the $1$-dimensional Hawaiian earring is homotopic to some proper $1$-loop \cite[Lemma A.3]{Eda}. A $1$-loop $\alpha: (\mathbb{I}, \dot{\mathbb{I}}) \to (\mathbb{HE}^1, \theta)$ is called proper whenever for each subinterval $[a,b] \subseteq \mathbb{I}$, if $\alpha|_{[a,b]}$ is nullhomotopic, then it is constant.
Also, O. Bogopolski and A. Zastrow proved that
\[
\pi_1(\mathcal{G}_1, a) \cong \frac{\times^{\sigma}_{\mathbb{N}} \mathbb{Z}}{\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N},
\]
where $\mathbb{N}_o$ and $\mathbb{N}_e$ denote the set of odd and even numbers, respectively \cite[Theorem 3.4]{BogZas}. This isomorphism is induced by the natural embedding $\iota:\mathbb{HE}^1 \to \mathcal{G}_1$ causing epimorphism $\pi_1(\iota): \times^{\sigma}_{\mathbb{N}} \mathbb{Z} \to \pi_1(\mathcal{G}_1, x_*)$ together with $\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N$ as its kernel.
The following lemma gives a useful description of the group $\mathcal{H}_1(\mathbb{HE}^1, \theta)$ which is used in sequel.
\begin{lemma}\label{le4.1nn}
Let $\mathcal{B}$ be the subgroup of $\prod_{\mathbb{N}} \times^{\sigma}_{\mathbb{N}} \mathbb{Z}$ consisting of all countably infinite tuples of reduced $\sigma$-words that the number of components including letter of type $m$ is finite, for all $m \in \mathbb{N}$. Then
\[
\mathcal{H}_1(\mathbb{HE}^1, \theta) \cong \mathcal{B}.
\]
\end{lemma}
\begin{proof}
By \cite[Theorem 2.9]{1}, $\mathcal{H}_1(\mathbb{HE}^1, \theta)$ is isomorphic to the subgroup of $\prod_{\mathbb{N}} \pi_1(\mathbb{HE}^1, \theta)$ consisting of all sequences of homotopy classes of $1$-loops with some representative converging uniformly to the constant $1$-loop. We show that $\mathcal{B}$ equals this subgroup.
Let $\{U_l;\ l \in \mathbb{N}\}$ be the local basis at $\theta$ defined in the proof of \cite[Theorem 2.9]{1}, and also let $\{[f_k]\} \in \prod_{\mathbb{N}} \pi_1(\mathbb{HE}^1, \theta)$. Then $\{f_k\}$ converges uniformly to the constant $1$-loop if and only if for each $l \in \mathbb{N}$, there exists $K_l \in \mathbb{N}$ such that $im(f_k) \subseteq U_l$ whenever $k \ge K_l$. Assume that $f_k$ is the corresponding proper representative for all $k \in \mathbb{N}$. The image of $f_k$ is contained in $U_l$ if and only if $im(f_k) \cap \mathbb{S}_m^1= \{\theta\}$, where $(m <l)$, for $U \cap \mathbb{S}^1_m$ is contractible and $f_k$ has no trivial subpath. Therefore, $\{f_k\}$ converges uniformly to the constant $1$-loop if and only if for each $l \in \mathbb{N}$, there exists $K_l \in \mathbb{N}$ such that $im(f_k) \cap \mathbb{S}_m^1= \{\theta\}$ whenever $k \ge K_l$.
By \cite[Theorem A.1]{Eda}, $\pi_1(\mathbb{HE}^1, \theta) \cong \times^{\sigma}_{\mathbb{N}} \mathbb{Z}$, the group of reduced $\sigma$-words. Moreover, in a given reduced $\sigma$-word, the letter of type $m$ exists if and only if the $m$th circle $\mathbb{S}_m^1$ of $\mathbb{HE}^1$ appears in the corresponding proper $1$-loop. Therefore $\mathcal{H}_1(\mathbb{HE}^1, \theta)$ is isomorphic to the subgroup of $\prod_{\mathbb{N}} \times^{\sigma}_{\mathbb{N}} \mathbb{Z}$ consisting of all countably infinite tuples of reduced $\sigma$-words such that the number of components including letter of type $m$ is finite, for all $m \in \mathbb{N}$.
\end{proof}
The following theorem investigates the structure of the $1$th Hawaiian group of the $1$th Griffiths space at the common point, the two vertices and the other points. Let $i:\mathbb{HE}^1 \to \mathcal{G}_1 \setminus \{v_1, v_2\}$ be the embedding which maps $(2m-1)$th circle onto the the horizontal left $m$th circle and maps $2m$th circle onto the horizontal right $m$th circle, for $m \in \mathbb{N}$ (see Figure \ref{fi1}). Also, let $j: \mathcal{G}_1 \setminus \{v_1, v_2\} \to \mathcal{G}$ be the inclusion map .
\begin{theorem}\label{th4.3n}
Let $\mathcal{G}_1$ be the $1$th Griffiths space, $x_*$ the common point, $a \in A \cup A'$, and $x \in \mathcal{G}_1 \setminus (A \cup A' \cup \{x_*\})$. Then
\begin{align}
& \mathrm{(i)}& \mathcal{H}_1(\mathcal{G}_1, x_*) & \cong \frac{\mathcal{H}_1(\mathbb{HE}^1, \theta)}{i_*^{-1} \ker j_*} \cong \frac{\mathcal{B}}{\prod_{\mathbb{N}}^W\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N}, \label{eq1.2}
\\
&\mathrm{(ii)}& \mathcal{H}_1(\mathcal{G}_1, a)
& \cong \frac{{\mathcal{H}}_1(\mathbb{HE}^1,\theta)}{\prod_{\mathbb{N}}^W{\pi}_1(\mathbb{HE}^1,\theta)} \ltimes {\prod_{\mathbb{N}}}^W \pi_1(\mathcal{G}_1, a) \cong
\frac{\mathcal{B}}{\prod^W_{\mathbb{N}} \times^{\sigma}_{\mathbb{N}} \mathbb{Z}} \ltimes {\prod_{\mathbb{N}}}^W \frac{\times^{\sigma}_{\mathbb{N}} \mathbb{Z}}{\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N}, \label{eq3.1}
\\
&\mathrm{(iii)}& \mathcal{H}_1( \mathcal{G}_1 , x) &\cong {\prod_{\mathbb{N}}}^W \frac{\times^{\sigma}_{\mathbb{N}} \mathbb{Z}}{\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N}.\label{eq4.10n}
\end{align}
\end{theorem}
\begin{proof}
\begin{enumerate}[(i)]
\item
Since all $1$-loops in $\mathcal{G}_1$ at $x_*$ are small, Theorem \ref{th2.5n} implies that $\mathcal{H}_1 (\mathcal{G}_1 , x_*) = j_* \mathcal{H}_1 (U_1 \vee U_2, x_*)$ when $U_1$ and $U_2$ are arbitrary neighbourhoods of $x_*$ in the two cones. Suppose that $U_m$ $(m=1,2)$ is the whole of the corresponding cone except its vertex. Then $U_1 \vee U_2 = \mathcal{G}_1 \setminus \{v_1, v_2\}$ and thus
\[\mathcal{H}_1(\mathcal{G}_1, x_*) = j_* \mathcal{H}_1 (U_1 \vee U_2, x_*) = j_* \mathcal{H}_1 (\mathcal{G}_1 \setminus \{v_1, v_2\}, x_*).
\]
Moreover, the embedded Hawaiian earring $i(\mathbb{HE}^1)$ is a deformation retract of $\mathcal{G}_1 \setminus \{v_1, v_2\}$ with projection $p: \mathcal{G}_1 \setminus \{v_1, v_2\} \to \mathbb{HE}^1$ as the retraction. Therefore,
\[ \mathcal{H}_1(\mathcal{G}_1, x_*) = j_* \mathcal{H}_1 (\mathcal{G}_1 \setminus \{v_1, v_2\}, x_*) \cong j_* i_*\mathcal{H}_1 (\mathbb{HE}^1, \theta).
\]
By the First Isomorphism Theorem,
\[
j_* i_*\mathcal{H}_1 (\mathbb{HE}^1, \theta) \cong \frac{i_*\mathcal{H}_1(\mathbb{HE}^1, \theta)}{\ker j_*} \cong \frac{\mathcal{H}_1(\mathbb{HE}^1, \theta)}{i_*^{-1} \ker j_*}. \]
In the following, we show that $i_*^{-1}\ker j_*$ is mapped isomorphically onto $\prod_{\mathbb{N}}^W\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N$, by the same isomorphism which maps $\mathcal{H}_1(\mathbb{HE}^1, \theta)$ onto $\mathcal{B}$, in Lemma \ref{le4.1nn}.
Let $[g] \in i_*^{-1}\ker j_*$. Then $i_* [g] \in \ker j_*$, and therefore, $j \circ i \circ g \simeq C_{x_*}$. Hence, the $1$-loops $(j \circ i \circ g|_{S_k^1})$'s are nullhomotopic with some null convergent sequence of homotopies, say $\{H_k\}$. Thus, there exists $K\in \mathbb{N}$ such that $im H_k \subseteq \mathcal{G}_1 \setminus \{v_1, v_2\}$ for $k \geq K$. Therefore, $i \circ g|_{\mathbb{S}^1_k}$ is nullhomotopic in $\mathcal{G}_1 \setminus \{v_1, v_2\}$ for $k \geq K$ by null convergent homotopies $\{H_k\}_{k \ge K}$. Accordingly, $i \circ g|_{\widetilde{\bigvee}_{k \ge K}\mathbb{S}^1_k}$ is nullhomotpic in $\mathcal{G}\setminus \{v_1, v_2\}$. Therefore, $p \circ i \circ g|_{\widetilde{\bigvee}_{k \ge K}\mathbb{S}^1_k} = g|_{\widetilde{\bigvee}_{k \ge K}\mathbb{S}^1_k}$ is nullhomotopic in $\mathbb{HE}^1$.
Moreover, for $k <K$, $j \circ i \circ g|_{S^1_k}$ is nullhomotopic in $\mathcal{G}_1$ or equivalently $[g|_{S_k^1}] \in \ker \pi_1(\iota)$. Note that $\ker \pi_1(\iota)$ is mapped isomorphically onto $\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N$, by the same isomorphism mapping $\pi_1(\mathbb{HE}, \theta)$ onto $\times^{\sigma}_{\mathbb{N}} \mathbb{Z}$. Thus, $[g]$ is corresponded injectively to an element of $\prod_{\mathbb{N}}^W\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N$, where the corresponding is the same as the isomorphism mapping $\mathcal{H}_1(\mathbb{HE}, \theta)$ onto $\mathcal{B}$, in Lemma \ref{le4.1nn}. One can check that this correspondence is also surjective.
\item
By Theorem \ref{th4.2n},
\[
\mathcal{H}_1(\mathcal{G}_1, a) \cong
\mathcal{H}_1(C\mathbb{HE}^1, a) \ltimes {\prod_{\mathbb{N}}}^W \pi_1(\mathcal{G}_1, a) \cong \mathcal{H}_1(C\mathbb{HE}^1, a) \ltimes {\prod_{\mathbb{N}}}^W \frac{\times^{\sigma}_{\mathbb{N}} \mathbb{Z}}{\overline{\langle \times^{\sigma}_{\mathbb{N}_e} \mathbb{Z}, \times^{\sigma}_{\mathbb{N}_o} \mathbb{Z}\rangle}^N}.
\]
Now by \cite[Theorem 2.13]{1}
$\mathcal{H}_1(C\mathbb{HE}^1, a) \cong \frac{{\mathcal{H}}_1(\mathbb{HE}^1,\theta)}{\prod_{\mathbb{N}}^W{\pi}_1(\mathbb{HE}^1,\theta)}$.
By Lemma \ref{le4.1nn}, $ \mathcal{H}_1(\mathbb{HE}^1, \theta) \cong \mathcal{B}$. This isomorphism maps the subgroup $\prod_{\mathbb{N}}^W {\pi}_1(\mathbb{HE}^1,\theta)$ onto $\prod^W_{\mathbb{N}}\times^{\sigma}_{\mathbb{N}} \mathbb{Z}$. Consequently
$\mathcal{H}_1( C\mathbb{HE}^1, a) \cong \frac{\mathcal{B}}{\prod^W_{\mathbb{N}} \times^{\sigma}_{\mathbb{N}} \mathbb{Z}}$,
and hence the isomorphism \eqref{eq3.1} holds.
\item
Obviously, $\mathcal{G}_1$ is semilocally strongly contractible at $x$. Therefore, $\mathcal{H}_1 (\mathcal{G}_1, x) \cong \prod_{\mathbb{N}}^W \pi_1(\mathcal{G}_1, x)$.
\end{enumerate}
\end{proof}
We define the $n$th Griffiths space by the wedge sum of two copies of cones on $\mathbb{HE}^n$ at the origin for $n\ge 2$. In the following theorem we give some information on the structure of the $n$th Hawaiian group of the $n$th Griffiths space.
Note that by $\mathbb{Z}^k$ we mean the subgroup of $\bigoplus_{\mathbb{N}} \mathbb{Z}$ consisting of all countably infinite tuples with all zero components except the first $k$ components and thus $\mathbb{Z}^1 \leqslant \mathbb{Z}^2 \leqslant \mathbb{Z}^3 \leqslant \cdots$.
\begin{theorem}\label{th3.2n}
Let $n \ge 2$ and $\mathcal{G}_n$ be the $n$th Griffiths space, $a \in A \cup A'$, and $x \in \mathcal{G}_1 \setminus (A \cup A' \cup \{x_*\})$. Then
\begin{align*}
&\mathrm{(i)}&
\mathcal{H}_n(\mathcal{G}_n, a) &\cong
\frac{\mathcal{H}_n(\mathbb{HE}^n, \theta)}{\bigoplus_{\mathbb{N}} \pi_n(\mathbb{HE}^n, \theta)} \oplus \bigoplus_{\mathbb{N}} \pi_n(\mathcal{G}_n) \cong
\frac{\prod_{\mathbb{N}} \bigoplus_{\mathbb{N}} \mathbb{Z}}{\bigcup_{k \in \mathbb{N}} \prod_{\mathbb{N}} \mathbb{Z}^k} \oplus \bigoplus_{\mathbb{N}} \pi_n(\mathcal{G}_n)
\cr
& \mathrm{(ii)}&
\mathcal{H}_n( \mathcal{G}_n , x) &\cong \bigoplus_{\mathbb{N}} \pi_n(\mathcal{G}_n) \label{eq2.9}.
\end{align*}
\end{theorem}
\begin{proof}
\begin{enumerate}[(i)]
\item
By Theorem \ref{th4.4},
$\mathcal{H}_n(\mathcal{G}_n, a) \cong
\mathcal{H}_n(C\mathbb{HE}^n, a) \oplus \bigoplus_{\mathbb{N}} \pi_n(\mathcal{G}_n, a)$.
Using \cite[Theorem 2.13]{1},
$\mathcal{H}_n(C\mathbb{HE}^n, a) \cong \frac{{\mathcal{H}}_n(\mathbb{HE}^n,\theta)}{\bigoplus_{\mathbb{N}} \pi_n(\mathbb{HE}^n , \theta)}$. Replacing $\mathcal{H}_n(\mathbb{HE}^n, \theta)$ with $\prod_{\mathbb{N}}\bigoplus_{\mathbb{N}} \mathbb{Z}$ via \cite[Theorem 2.11]{1}, which maps $\bigoplus_{\mathbb{N}} \pi_n(\mathbb{HE}^n , \theta)$ isomorphically onto $\bigcup_{k \in \mathbb{N}} \prod_{\mathbb{N}} \mathbb{Z}^k$, the result holds.
\item
Since $\mathcal{G}_n$ is semilocally strongly contractible at $x$, the isomorphism holds by Theorem \ref{th2.6n}.
\end{enumerate}
\end{proof}
We recall that the $n$th Hawaiian group of the $n$th Griffiths space at the common point is generated by two its subgroups; $\mathcal{H}_n(\mathcal{G}_n, x_*) = j_* \mathcal{H}_n( U_1 \vee U_2, x_*) \big(\bigoplus_{\mathbb{N}} \pi_n(\mathcal{G}_n,x_*)\big)$, when $U_1$ and $U_2$ are arbitrary neighbourhoods of the origin in two cones, and $j:U_1 \vee U_2 \to \mathcal{G}_n$ is the inclusion map.
\section*{References}
\end{document} | math | 40,876 |
\begin{document}
\title{Quantum network routing and local complementation}
\author{F.\ Hahn}\affiliation{Dahlem Center for Complex Quantum Systems, Freie Universit{\"a}t Berlin, 14195 Berlin, Germany}
\author{A.\ Pappa}\affiliation{Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK}
\author{J.\ Eisert}\affiliation{Dahlem Center for Complex Quantum Systems, Freie Universit{\"a}t Berlin, 14195 Berlin, Germany}
\date{\today}
\begin{abstract}
Quantum communication between distant parties is based on suitable instances of shared entanglement. For efficiency reasons, in an anticipated quantum network beyond point-to-point communication, it is preferable that many parties can communicate simultaneously over the underlying infrastructure; however, bottlenecks in the network may cause delays. Sharing of multi-partite entangled states between parties offers a solution, allowing for parallel quantum communication. Specifically for the two-pair problem, the butterfly network provides the first instance of such an advantage in a bottleneck scenario. The underlying method differs from standard repeater network approaches in that it uses a graph state instead of maximally entangled pairs to achieve long-distance simultaneous communication. We will demonstrate how graph theoretic tools, and specifically local complementation, help decrease the number of required measurements compared to usual methods applied in repeater schemes. We will examine other examples of network architectures, where deploying local complementation techniques provides an advantage. We will finally consider the problem of extracting graph states for quantum communication via local Clifford operations and Pauli measurements, and discuss that while the general problem is known to be {\tt NP}-complete, interestingly, for specific classes of structured resources, polynomial time algorithms can be identified.
\end{abstract}
\maketitle
\subsection{Introduction}
Quantum communication schemes over optical networks necessarily
suffer from transmission losses and errors. For this reason, in order to achieve the vision of secure quantum communication over arbitrary distances, several schemes have been proposed that
are based on entanglement swapping and purification
\cite{Briegel98,RepeaterLukin,van2006hybrid,Acin07,Roadmap}. However, such existing ``quantum repeater'' approaches are based on sharing and manipulating close to
maximally entangled ``EPR'' pairs between the nodes. A lot of emphasis
has been put onto identifying efficient ways of achieving this task
\cite{RepeaterLukin,zwerger2012measurement,van2006hybrid,TeleportationReview},
amounting to challenging prescriptions. Yet, for multi-partite quantum networks going beyond point-to-point achitectures, much less is known about how to meaningfully make use of and manipulate resources.
This is particularly unfortunate since a number of protocols have been devised for tasks like secret sharing \cite{SecretSharing,bell2014experimental}, quantum voting \cite{PhysRevA.95.062306} and quantum conference key agreement \cite{PhysRevA.97.022307,ConferenceKey,EppingA}, that exploit
the genuine multi-partite character of a quantum network, having the vision of a
quantum internet in mind \cite{QuantumInternet}. In fact, one could argue that the true potential
of quantum communication is expected to lie in such multi-partite applications beyond point-to-point architectures.
Specifically in multi-partite quantum networks, it could well be preferable that the involved processes are run offline, i.e., before a request for communication is received. However, methods like the ones described in Ref.~\cite{SMIKW16} require big quantum memories, as well as a high channel capacity. Consequently, network efficiency is limited by the memory capacities of the quantum repeater stations \cite{zwerger2018long}, as well as by possible bottlenecks imposed by the quantum network architecture. What is more, in many applications, multi-partite resources are required in the first place.
In this context, new questions of \emph{quantum routing} emerge.
We use the term quantum routing as referring to the task of manipulating entangled resources in multi-partite quantum networks between arbitrary nodes, not necessarily making use of local knowledge only, as is common in classical routing, but allowing for global classical communication. The key question in this framework is how to optimally establish communication between distant nodes using the intermediate nodes of a quantum network.
In this work, we consider alternative ways for sharing entanglement between distant nodes of a network that have
favorable features both with respect to memory and channel capacity. We start from the same setting
where nodes that are connected with physical optical links share close to maximally entangled qubit pairs. By
suitable entanglement swapping steps
\cite{PhysRevLett.71.4287,PhysRevA.95.032306,TeleportationReview}, the resulting state is a \emph{graph state} \cite{Hein04,Hein06}. Methods for purifying any graph state via measurements and classical communication have been studied \cite{KMBD06} and applications in quantum networks considered \cite{pirker2017modular, Markham}. As already discussed, setting up the shared quantum state before the actual request for communication, is preferable in terms of efficiency of communication, but also allows for detection and prevention of channel or node failure.
For a given graph state and a request for communication between two distinct nodes, a straightforward solution would be to find a shortest path between the nodes, create a ``repeater'' line (by isolating the path from its environment), and then perform measurements on the intermediate nodes, thereby creating an EPR pair between the two. However, this approach is far from optimal since it requires measuring a large number of nodes and therefore diminishes the secondary use of the residual quantum state. Here, we propose another method that requires at most as many measurements as this ``repeater'' protocol, in general leaving a larger part of the graph state intact, while simultaneously solving bottleneck issues in the network.
The proposed method is based on local complementation \cite{Hein04,bouchetLCorbit} and is already underlying in the prominent bottleneck example of the butterfly scheme \cite{Butterfly,EppingA}. The painful lack of studies
in this area is due to the fact that local complementation
does not provide an advantage in classical network coding, since there is no classical equivalent to the application of local Clifford operations in order to achieve serviceable long-range correlations.
Finally, we turn towards the problem of extracting graph states from given larger graph states
via local Clifford operations and Pauli measurements. Using known results from graph theory \cite{VertexMinor, Hoyer, Beigi, RankWidth}, we discuss that while the general problem is known to be {\tt NP}-complete \cite{Dahlberg18}, for specific classes of more structured resources, polynomial time algorithms can be found. All our schemes are based on local complementation,
but are genuinely quantum, in the way that genuinely multi-partite quantum graph states are manipulated.
\subsection{Preliminaries}
A graph $G=(V,E)$ consists of a finite set of vertices $V\subsetneq \mathbb{N}$ and a set $E\subseteq V \times V$ of edges.
Vertices that are connected by an edge are called adjacent.
The set of all vertices that are adjacent to a given vertex $a$ is called the \emph{neighborhood} of $a$ and denoted by $N_a$.
We may write $|G|\mathrel{\mathop:}= |V|$ for the number of vertices.
Graphs have an adjacency matrix with entries
\begin{equation}
\left(\Gamma_G\right)_{i,j}\mathrel{\mathop:}=\begin{cases}
1, & \text{if }(i,j)\in E\\
0, & \text{if }(i,j)\not\in E
\end{cases}
\end{equation}
associated with them. In this work, we only consider \emph{simple} graphs, i.e., graphs that do not contain edges connecting a vertex to itself, or multiple edges between the same pair of vertices. Given a graph $G$, we can prepare a graph state vector $\ket{G}$ associated with it as follows. First, a qubit in
$\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$ is prepared for each of the vertices in $V$.
Subsequently, a controlled-$Z$ operation is applied to each pair of qubits that is adjacent in $G$. The resulting graph state vector can thus be written as
\begin{align}\ket{G}\mathrel{\mathop:}=\mathrm{ prob }od_{(i,j)\in E} CZ_{i,j} \ket{+}^{\otimes V}.
\end{align}
It is important to stress that graph states do not have to be prepared in this fashion. In fact, we here anticipate the states to be prepared from EPR pairs and entanglement swapping in a quantum network.
Note that local Pauli measurements on a graph state result in a different graph state up to local unitary corrections (cf. Proposition 7 in Ref.\ \cite{Hein06}). Here, we will omit these local corrections for the sake of clarity.
In this work we will make use of a graph transformation called local complementation. By $\tau_a(G)$ we denote the graph that results from locally complementing $G$ with respect to the vertex $a$.
\begin{define}[Local complementation]
A graph $G=(V,E)$ and vertex $a\in V$ define a graph $\tau_a(G)$ with adjacency matrix
\begin{equation}
\Gamma_{\tau_a(G)}\mathrel{\mathop:}=\Gamma_G+\Theta_a \mod 2,
\end{equation}
where $\Theta_a$ is the complete graph of the neighborhood $N_a$.
\end{define}
Local complementation on a graph is equivalent to applying local Clifford gates on the respective graph state \cite{VandenNest1}. In particular, the graph state that results from local complementation with respect to node $a$ of the graph state vector $\ket{G}$, is defined by $\ket{\tau_a(G)}\mathrel{\mathop:}= U_a^\tau\ket{G}$, where $U_a^\tau\mathrel{\mathop:}= ({iX_a})^{1/2} ({-i Z_{N_a}})^{1/2}$.
It is possible to verify whether two graph states can be transformed into each other via sequential local complementations in polynomial time \cite{VandenNest2}.
As we only consider local Clifford operations and Pauli measurements, the resulting states remain graph states and can be described in terms of the pre-measurement graph with the help of local complementations and $Z$-measurements \cite{Hein06}.
\subsection{Reducing the number of measurements}
We have already argued that sharing graph states between the nodes of a network allows for quicker communication with less requirements for channel capacity and memory than sharing EPR pairs between nodes. However, it is not known, given a shared graph state, what the optimal technique for entanglement sharing between nodes that are not connected via physical links is. An approach equivalent to the well-established repeater networks would be to create a ``path'' that connects the two nodes, and then, via entanglement swapping, create a long distance EPR pair.
In the following, we will prove that a ``repeater'' method is not optimal regarding the number of measurements to be performed. Having a significantly reduced number of measurements is extremely useful in quantum networks, since it allows us to ``extract'' more entanglement from the shared graph state.
The \emph{repeater protocol} entails first isolating a path between two nodes $a$ and $b$ by $Z$-measuring the neighborhood of said path (creating a repeater line) and then connecting $a$ to $b$ via $X$-measurements along the intermediate nodes of the path. The \emph{$X$-protocol} is doing the reverse, first $X$-measuring the intermediate nodes on the path between $a$ and $b$, and then $Z$-measuring everything that is left in the neighborhoods of $a$ and $b$ respectively. We specifically prove the following theorem in the Appendix.
\begin{theorem}[Creating maximally entangled pairs]\label{thm1}
We can create an EPR pair between two nodes $a$ and $b$ of an arbitrary graph state using the $X$-protocol with at most as many measurements as with the repeater protocol.
\end{theorem}
\begin{figure}
\caption{An EPR pair and a residual graph state are distilled from a cluster state with $9$ qubits using the $X$-protocol on the path $(1,2,5,6,9)$. This is visualised by considering (b) local complementations with respect to nodes $1,2,5,6,1$, followed by (c) the deletion of nodes $2,5,6$ on the graph that describes the graph state.}
\label{fig:9qubitCluster}
\end{figure}
The proof compares the number of measurements required when running the two different algorithms.
This technique decreases the number of measurements used in standard repeater scenarios, when we know a pair of nodes that intends to communicate (in this case $a$ and $b$). In particular, it allows for a larger part of the graph state to remain intact for future use.
Fig.~\ref{fig:9qubitCluster} visualizes how the $X$-protocol for a $9$ qubit cluster state allows us to communicate between the nodes $1$ and $9$ while keeping a residual graph state for simultaneous communication between any pair of nodes in $\{3,4,7,8\}$. Here, the residual graph state can be turned into the desired second EPR pair by a single measurement. Note that if we would first isolate the path between nodes $1$ and $9$ and then apply standard repeater protocols, the distillation would require the measurement of at least six nodes and thereby render the extraction of a second EPR pair impossible.
It is also beneficial to compare our protocol to the standard entanglement swapping methods based on directly sharing EPR pairs over the underlying network. To build the graph states of Fig.~\ref{fig:9qubitCluster}(c) over the underlying grid network using entanglement swapping, we need 12 EPR pairs, which is the same number required to build the cluster state in Fig.~\ref{fig:9qubitCluster}(a). The crucial difference is that, while the cluster state can accommodate more communication requests, the direct generation of the graph states in Fig.~\ref{fig:9qubitCluster}(c) via entanglement swapping limits the communication scenarios that we can implement.
If no more information about future communication requests is available, it is more resource economical
to choose the shortest path that has the minimal neighborhood. However, the following lemma will be useful in case we would like to allow more than one pair of nodes to communicate simultaneously. Specifically, it gives a visualisation of different possibilities of entanglement generation between nodes.
\begin{lemma}[Equivalence of measurements]\label{lem1}
$X$-measurements along a shortest path between two nodes are equivalent to performing a series of local complementations on the path, followed by $Z$-measurements on the intermediate nodes.
\end{lemma}
\begin{figure}
\caption{{\bf Establishment of two EPR pairs.}
\label{fig:Butterfly}
\end{figure}
Lemma \ref{lem1} allows us to transform the problem of establishing entanglement between nodes into finding suitable graphs by successive local complementations of the network graph. These repeated local complementations generate an orbit, the \emph{LC-orbit} \cite{bouchetLCorbit}. As already mentioned, if the only request for communication is between two nodes, then a shortest path with minimal neighborhood is chosen, in order to minimize the number of measurements. However, if the problem at hand is to connect more than one pair of nodes, the local complementation path will be chosen differently, according to the resulting graph.
Even if not at first apparent, this is the strategy for the well-known butterfly network scheme (Fig.~\ref{fig:Butterfly}), where in order to create EPR pairs between nodes $\{1,6\}$ and $\{2,5\}$, $X$-measurements are done on nodes $3$ and $4$. Via Lemma \ref{lem1} this is equivalent to finding a graph in the LC-orbit of the butterfly, where edges $(1, 6)$ and $(2, 5)$ exist, and no edge between sets $\{1, 6\}$ and $\{2, 5\}$ exists. This graph is found via consecutive local complementations on nodes $1$, $3$ and $4$. A $Z$-measurement on nodes $3$ and $4$ allows to extract the two required EPR pairs (Fig.~\ref{fig:Butterfly}(f)). Note that without the second request for connection of nodes $2$ and $5$, the algorithm might have chosen another path to do $X$-measurements. Similarly, the sequence of subfigures in Fig.~\ref{fig:9qubitCluster} demonstrates the equivalent process for the 9-qubit cluster state. Both this specific communication example and the one presented in the butterfly scheme, create bottlenecks in the network; this is further discussed in the following section.
\subsection{Bottleneck quantum networks}
The butterfly network is of particular interest when considering bottlenecks in the network. If the nodes can share only one EPR pair over each physical link, one of the butterfly's edges is a bottleneck if we aim to build repeater lines to create entanglement between nodes $\{1,6\}$ and $\{2,5\}$. The above method fulfills the requirement of sharing only one EPR pair per physical link, in order to build the appopriate graph state, and yet solves the communication problem by bypassing the bottleneck in the network. We can further show by exhaustive search that the butterfly network structure is uniquely minimal with respect to the number of nodes.
\begin{proposition}[No bottleneck]
There is no 5-node graph state that has a bottleneck for simultaneous communication between two pairs of nodes and that can be solved using local Cliffords and a Pauli measurement of a single node.
\end{proposition}
\begin{proposition}[Bottleneck]
There are only four 6-node graph states that have a bottleneck for simultaneous communication between two pairs of nodes and that can be solved using local Cliffords and Pauli measurements.
\end{proposition}
The only four possible 6-node graphs mentioned in the above proposition are the ones resulting from node relabeling in Fig.~\ref{fig:Butterfly}(a). Specifically, if we intend to establish EPR pairs between nodes $\{1,6\}$ and $\{2,5\}$, we obtain the four graphs by exchanging labels within the sets $\{3,4\}$ and $\{1,6\}$. Note that in allowing arbitrary local Cliffords and Pauli measurements we considered a wider class of possible algorithms than just the aforementioned $X$-protocol.
\subsection{Obtaining GHZ and other multi-partite resources}
As a further aspect, we now turn to the key question of how to extract resource states such as GHZ states from a given graph state. The more general question, whether from a given graph state vector $\ket{G}$ we can extract another graph state vector $\ket{H}$ via a sequence of local measurements, has recently been proven to be {\tt NP}-complete \cite{Dahlberg18}. This was done by solving a well-known problem in graph theory called the {\tt VERTEX-MINOR} problem, which asks whether from a graph $G$, another graph $H$ can be extracted via a sequence of (i) local complementations and (ii) deletion of vertices. Note that the {\tt NP}-completeness of deciding whether graph $H$ is a vertex-minor of $G$ has been proven for labeled graphs, which are relevant for communication scenarios, since the nodes are distinct.
Having said that, there are polynomial-time algorithms that solve the problem for important instances. A first relevant instance involves GHZ states \cite{GHZ, wallnofer2016two}, which are essential resources for multi-partite schemes in quantum networks beyond point-to-point architectures, such as \emph{quantum secret sharing}
\cite{SecretSharing,bell2014experimental}.
Building upon the method described in Theorem \ref{thm1}, we can show the following corollary.
\begin{corollary}[Extraction of GHZ3 states]\label{cor1}
We can always distill a $3$-partite GHZ state between arbitrary vertices of a connected graph state in polynomial time.
\end{corollary}
In order to obtain a $3$-partite GHZ state, we use a slightly altered version of the $X$-protocol. The proof examines different cases corresponding to distinct relative positions of the three vertices within the graph and is given in the Appendix. We now propose a sufficient criterion in order to extract $4$-partite GHZ states; note that the extraction of a complete graph of four nodes (which is a graph representing a GHZ4) is thought to be difficult in general \cite{PivotNP}.
\begin{proposition}[Extraction of GHZ4 states]\label{Prop3}
We can distill a $4$-partite GHZ state from a graph state when the underlying graph has a repeater line as vertex-minor, which contains all four nodes of the final GHZ state and at least one extra node between two pairs of the nodes.\label{lem:GHZ4}
\end{proposition}
\begin{figure}
\caption{{\bf Prototypical extraction of a GHZ4 state.}
\label{fig:GHZ4}
\end{figure}
The required criterion is very likely to be fulfilled for simple network architectures, over which the graph state will be shared. Fig.~\ref{fig:GHZ4} demonstrates this for a short-distance square-grid network, which is used to share a cluster state. Here, Fig.~\ref{fig:GHZ4}(c) visualizes the minimal repeater line that is described in Proposition \ref{Prop3}. The proof of said proposition is given in the Appendix. A more general result using the notion of rank-width \footnote{The \emph{rank-width} $k$ of a graph $G$ is the minimum width of all its rank decompositions. This amounts to $k$ being the smallest integer such that $G$ can be related to a tree-like structure by recursively splitting its vertex set so that each cut induces a matrix of rank at most $k$. The rank-width is bounded iff the clique-width is bounded \cite{Approximating}. Graphs with rank-width at most one are those where all connected induced subgraphs preserve distance \cite{VertexMinor}.} is based on
Refs.\
\cite{Dahlberg18,Wehner18,Oum17,courcelleoum,CMR00}.
\begin{observation*}[Extraction of graph states from graph states with bounded rank-width] \label{thm2} For a graph state vector $\ket{G}$ with an underlying graph of bounded rank-width, there exists a poly-time algorithm that decides if a graph state vector $\ket{H}$ can be extracted from $\ket{G}$ using local Clifford operations and $Z$-measurements, and gives the sequence of operations to be applied.
\end{observation*}
For a graph $G$, there exist algorithms with runtime $O(|G|^3)$ \cite{RankWidth} that, for a fixed $k$, either give a rank decomposition of width at most $24k$ or reply that the rank-width is larger than $k$. Then, when such a rank decomposition is given, for a fixed graph $H$, a linear time algorithm can test whether $H$ is a vertex minor of $G$ and return the sequence of local complementations and vertex deletions to be applied \cite{Wehner18}. Intuitively speaking, many structured graphs have bounded rank-width. E.g., highly sparse random graphs have a bounded rank-width \cite{RandomGraphs}, and so do graphs with a bounded tree-width \cite{oum2008rank}. For those graphs, the above observation readily applies, and it can be decided whether resource states can be extracted.
\subsection{Discussion}
As an outlook, we mention an exciting link to classical network coding theory:
Schemes have been previously studied for the teleportation of a quantum state from a set of nodes (sources) to another set (sinks), and a connection with classical network coding has been established \cite{Kobayashi09,Kobayashi11,TeleportationReview}. The $k$-pair problem in classical network routing is relevant here, where $k$ sources want to simultaneously send information to $k$ sinks. In subsequent studies, the connection with measurement-based quantum computation \cite{Oneway,gross2007novel}
has been established \cite{debeaudrap14} and subsequently, the question of sharing a general graph state over a network has been addressed \cite{EppingB}. However, the latter work has a shortcoming; the mapping of the network is done using linear codes that require the generation of two-colorable graph states at each node, and it is not straightforward to see how to make this mapping to a given network structure, where each node holds a single qubit.
In this work, we have discussed the manipulation of multi-partite entangled resources for applications in quantum routing and quantum communication across quantum networks. A key application of the strategies laid out is
in parallel quantum key distribution and notions of conference key agreement.
We have seen that via local complementation, quantum routing schemes with a reduced number of measurements
outperforming standard repeater schemes can be found, bottleneck quantum networks can be treated and the
question of extracting multi-partite resources largely addressed.
It is important to stress that, while these algorithms are classical, they apply
to true multi-partite quantum entangled states. To provide further perspective, also note that since every \emph{stabilizer state} is equivalent to some graph state
\cite{Schlingemann}, the methods laid out here are also expected to be useful in the design of quantum error correcting codes.
It is the hope that this work triggers further studies of
manipulating multi-partite entangled resources for quantum routing, which seem urgently needed in the light of the
rapid experimental progress on quantum networks.
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\section*{Appendix}
\setcounter{theorem}{0}
\begin{theorem}[Creating maximally entangled pairs]\label{thm1_a}
We can create an EPR pair between two nodes $a$ and $b$ of an arbitrary graph state using the $X$-protocol with at most as many measurements as with the repeater protocol.
\end{theorem}
Before stating the proof of Theorem \ref{thm1} we introduce some additional notation.
For two subsets $A,B \subseteq V$ of vertices we denote by
\begin{equation}
E(A,B)\mathrel{\mathop:}=\left\{(a,b):a\in A, b\in B, a\neq b\right\}
\end{equation}
the set of all possible edges between the two sets. Note that $E(A,B)$ in general contains edges that are not contained in the edgeset of an arbitrary given graph $G=(V,E)$.
For a vertex subset $W\subseteq V$ we denote by $E_{|W}$ the subset of $E$ that contains every edge that connects to at least one vertex in $W$.
We may subtract a set of edges $F$ from $E$. That is, by $E\setminus F$ we denote the set of edges in $E$ that are
not contained in $F$.
For such a second set of edges $F$ we also define the \emph{symmetric difference} of $E$ and $F$ as
\begin{equation}
E\Delta F \mathrel{\mathop:}= (E\cup F) \setminus (E\cap F).
\end{equation}
If $F$ happens to be a subset of $E$ the symmetric difference $E\Delta F$ is identical to $E \setminus F$. Otherwise the mutual edges are removed from the union of the two sets.
If the qubit associated with vertex $v\in V$ of a graph state is $X$-measured, the transformation of the corresponding edge set $E$ can be described in terms of symmetric differences. Independent of a choice $w\in N_v$, the new edge set is given by
\begin{equation}\label{xsymmetricdifference}
\left(E\Delta E_{vw}\Delta E_{v\cap w}\Delta E_{v\setminus w}\right)\setminus E_{|\{v\},}
\end{equation}
where $E_{vw}\mathrel{\mathop:}= E(N_{w},N_{v})$, $E_{v\cap w}\mathrel{\mathop:}= E(N_{w}\cap N_{v},N_{w}\cap N_{v})$ and $E_{v\setminus w}\mathrel{\mathop:}= E(\{w\},N_{v}\setminus \{w\})$ are introduced as a shorthand notation. The subtraction of the set containing only the edges that connect to the vertex itself at the end of Eq.~\eqref{xsymmetricdifference} represents the isolation of $v$ due to the measurement. Given a distinct pair of vertices $a,b\in V$, a \emph{path} of length $k$ from $a$ to $b$ is an ordered list $(v_1,v_2,\ldots, v_k)$ such that $v_1=a$, $v_k=b$, and for all $i\in[k-1]$, vertices $v_i$ and $v_{i+1}$ are adjacent. We denote by $l$ the length of a shortest path from $a$ to $b$ within the graph at hand.
In the following we will describe how the neighborhoods $N_{v_i}$ of vertices $v_i$ change due to Pauli measurements on the graph state. To indicate that the graph and therefore some neighborhoods may have changed, we make use of an additional index $t$. By $N^{(t)}_{v_i}$ we denote the neighborhood of node $v_i$ after the $t^{th}$ Pauli measurement on the initially given graph state. We carry this notation over for symmetric differences. In the expression $E^{(t)}(\cdot, \cdot)$ the $t$ indicates that all involved neigborhoods are regarded after the $t^{th}$ Pauli measurement. From the context it will always be obvious which nodes are measured in which step. In particular
\begin{equation}
N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_{k}}
\end{equation}
is the joint neighborhood of a path $(v_1,v_2,\ldots, v_k)$ in the initially given graph before any measurements are made.
In our proof we compare two measurement algorithms that both have the goal of establishing an EPR pair between the nodes $a$ and $b$ of a given graph state.
\begin{itemize}
\item The \emph{repeater protocol} selects the shortest path connecting $a$ to $b$ that has the minimum combined neighborhood. Every node that lies in the combined neighborhood of this path but not on the path itself is then $Z$-measured. This isolates the path from the rest of the graph creating a repeater line. Finally, every intermediate vertex on the line is $X$-measured yielding the EPR pair between the two nodes.
\item The \emph{X-protocol} measures the intermediate vertices along the same shortest path in the $X$ basis . Subsequently, the neighborhoods of the two nodes are $Z$-measured to create the desired EPR pair.
\end{itemize}
\noindent{}We start our comparison by counting the number of measurements required in the repeater protocol. Along the minimal neighborhood path $(v_1,v_2,\ldots, v_l)$ connecting $v_1=a$ to $v_l=b$, every neighboring vertex that does not lie on the path itself is measured in the $Z$ basis. This requires
$|N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_l}|-l$ measurements. To obtain the desired EPR pair from this newly-created repeater line we then measure the vertices $v_2, v_3, \ldots, v_{l-1}$ in the $X$ basis. These $l-2$ measurements remove the intermediate nodes
of the path one by one. The repeater protocol thus requires
\begin{equation}\label{repeaterprotocolcounting}
|N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_{l}}|-2
\end{equation}
Pauli measurements in total to establish the EPR pair.
In order to prove Theorem \ref{thm1}, we will now count the number of measurements required when using the $X$-protocol. The protocol starts by $X$-measuring along the path $(v_2,\ldots, v_{l-1})$. Here, $t$ indicates the $X$-measurement of node $v_{t+1}$ and $N^{(t)}_{v_i}$ is the neighborhood of $v_i$ after the $t^{th}$ $X$-measurement. We will need the following observation.
\begin{observation*}[Minimizing measurements]\label{equivalentclaim}
To prove Theorem \ref{thm1}, it suffices to show
\begin{equation}\label{equivalentEquation}
N^{(l-2)}_{a}\cup N^{(l-2)}_{b} \subsetneq N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_l}
\end{equation}
and that we can find at least $l-2$ elements in the neighborhood of the initial (before any measurement) path between $a$ and $b$ that are not contained in the neighborhoods of $a$ and $b$ after the $X$-measurements of the $X$-protocol.
\end{observation*}
\noindent In total, the $X$-protocol requires ($l-2$) $X$-measurements along the shortest path, and subsequently $|N^{(l-2)}_{a}\cup N^{(l-2)}_{b}|-2$
$Z$-measurements on those vertices that have connecting edges to $a$ or $b$ (we need to subtract $a$ and $b$ from the count). From Eq.~\eqref{repeaterprotocolcounting} it follows that Theorem \ref{thm1} holds if
\begin{equation}
|N^{(l-2)}_{a}\cup N^{(l-2)}_{b}| +l-2 \leq |N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_l}|.
\end{equation}
\noindent{}Therefore, in order to prove Theorem \ref{thm1}, it is sufficient to show that Eq.~\eqref{equivalentEquation} is fulfilled.
\begin{proof}
In the following we will examine how the neighborhoods of $a=v_1$ and
$b=v_l$ change with the sequence of $X$-measurements along the shortest path. The first such measurement is at vertex $v_2$. The measurement results in
a new graph state with the same set of vertices and with an edge set that can be calculated via a series of symmetric differences, according to Eq.~\eqref{xsymmetricdifference},
\begin{equation}
\left(E\Delta E^{(0)}_{v_1v_2}\Delta E^{(0)}_{v_1\cap v_2}\Delta E^{(0)}_{v_2\setminus v_1}\right) \setminus E_{|\{v_2\},} \label{zero}.
\end{equation}
By definition of $E(\cdot,\cdot)$ we find
\begin{align}
&E^{(0)}_{v_1v_2}=\left\{(x_1,x_2):x_i\in N^{(0)}_{v_i},i =1,2; x_1\neq x_2\right\},\label{one}\\
&E^{(0)}_{v_1\cap v_2}=\left\{(x,y):x,y\in N^{(0)}_{v_1}\cap N^{(0)}_{v_2} , x\neq y\right\},\label{two}\\
&E^{(0)}_{v_2\setminus v_1}=\left\{(v_1,x_2):x_2\in N^{(0)}_{v_2}\setminus \{v_1\}\right\}.\label{three}
\end{align}
In the following we analyse the consecutive symmetric differences in
Eq.~\eqref{zero} step by step.
In particular, we are interested in how the $X$-measurement on $v_2$ changes the neighborhoods of $a=v_1$ and $b=v_l$.
Since we have $v_1\in \textcolor{black}{ N^{(0)}_{v_2} }$, the set
$E^{(0)}_{v_1v_2}$ contains all edges that where connected to $v_1$ before the measurement.
From Eq.~\eqref{one} we can thus infer that
$N^{(1)}_{v_1}$ does not contain any of the elements that where previously contained in $N^{(0)}_{v_1}$.
The second symmetric difference in Eq.~\eqref{zero} does not alter the neighborhood of the starting vertex $v_1$ by virtue of Eq.~\eqref{two} and $v_1$ not being
in the intersection of $N^{(0)}_{v_1}$ and $N^{(0)}_{v_2}$. If follows that the only contribution to $N^{(1)}_{v_1}$ comes from Eq.~\eqref{three}. We find
\begin{equation}
N^{(1)}_{v_1}=N^{(0)}_{v_2}\setminus \{v_1\}\label{first}
\end{equation}
and ascertain that the new graph after the $X$-measurement on $v_2$ has a path $(v_1,v_3,v_4,\ldots,v_l)$ of length $l-1$ connecting
$a=v_1$ and $v_{l}=b$.
We note that this new, shorter path is again a shortest path between $a$ and $b$, since the $X$-measurement only alters the neighborhood of the measured node. The vertex $v_2$ is now isolated, that is, there are no edges that connect it to the remaining graph. The following measurements will remove the
other intermediate vertices from the path one by one.
The next $X$-measurement on $v_3$ yields $N^{(2)}_{v_1}=N^{(1)}_{v_{3}}\setminus \{v_1\}$
and finally after the $t^{th}$ measurement, it holds that
\begin{equation}
N^{(t)}_{v_1}=N^{(t-1)}_{v_{t+1}}\setminus \{v_1\}\label{A}.
\end{equation}
We now examine how the neighborhood of $v_{t+2}$ is changed by the $t^{th}$ measurement. Before we write down the general expression for $N^{(t)}_{v_{t+2}}$, we consider
the special case $t=1$, that is, the environment of vertex $v_3$ after the measurement on $v_2$.
Again, Eq.~\eqref{two} does not contribute to $N^{(1)}_{v_3}$,
because $v_3\in N^{(0)}_{v_1}\cap N^{(0)}_{v_2}$ would be a contradiction to $(v_1,v_2,\ldots,v_l)$ being a shortest path before the first measurement. Via Equation
\eqref{one} we add those elements of $N^{(0)}_{v_1}$ that have not previously been connected to $v_3$ and remove those that where.
Compared to $N^{(0)}_{v_{3}}$, the neighborhood $N^{(1)}_{v_{3}}$ also gains the
element $v_1$ by virtue of Eq.~\eqref{three}, since $v_3$ is certainly an element of $N^{(0)}_{v_{2}}\setminus \{v_1\}$.
To sum up this gives
$N^{(1)}_{v_{3}}=\{v_1\}\cup \left(N^{(0)}_{v_{3}}\cup N^{(0)}_{v_{1}}\right) \setminus \left(N^{(0)}_{v_{3}}\cap N^{(0)}_{v_{1}}\right)$ and thus
\begin{equation}\label{B}
N^{(t)}_{v_{t+2}}=\{v_1\}\cup \left(N^{(t-1)}_{v_{t+2}}\cup N^{(t-1)}_{v_{1}}\right) \setminus \left(N^{(t-1)}_{v_{t+2}}\cap N^{(t-1)}_{v_{1}}\right)\end{equation}
for the general case after the $t^{th}$ measurement.
For any $t=2,3,\ldots,l-2$ we can combine Eqs. \eqref{A} and \eqref{B} and obtain the expression
\begin{equation}
N^{(t)}_{v_1}=\left(N^{(t-2)}_{v_{t+1}}\cup N^{(t-2)}_{v_{1}}\right) \setminus \left(N^{(t-2)}_{v_{t+1}}\cap N^{(t-2)}_{v_{1}}\right)\label{zwoelf}
\end{equation}
In particular we can now write recursive expressions for $N^{(l-2)}_{a}$ and $N^{(l-2)}_{b}$. More specifically, we obtain
\begin{eqnarray}
N^{(l-2)}_{v_1}&=&\left(N^{(l-4)}_{v_{l-1}}\cup N^{(l-4)}_{v_{1}}\right)
\setminus \left(N^{(l-4)}_{v_{l-1}}\cap N^{(l-4)}_{v_{1}}\right),\nonumber\\
N^{(l-2)}_{v_l}&=&\{v_1\}\cup \left(N^{(l-3)}_{v_l}\cup N^{(l-3)}_{v_{1}}\right)\setminus \left(N^{(l-3)}_{v_l}\cap N^{(l-3)}_{v_{1}}\right).\nonumber\\\label{start}
\end{eqnarray}
Eqs.\ \eqref{start} contain multiple expressions of the type $N^{(t)}_{v_{t+3}}$. These expressions can easily be simplified.
For instance the case $t=1$ entails $N^{(1)}_{v_{4}}=N^{(0)}_{v_{4}}$, since $v_4\notin N_{v_i}$ for $i=1,2$. Otherwise this would be a contradiction
to $(v_1,v_2,\ldots,v_l)$ being the shortest path before the first measurement. By the same argument we recursively get
\begin{equation}
N^{(t)}_{v_{t+3}}=N^{(t-1)}_{v_{t+3}}=\ldots=N^{(0)}_{v_{t+3}}
\end{equation}
for all $t=1,2,\ldots,l-3$. Together with Eqs.\ \eqref{zwoelf} and \eqref{start}, this recursively implies that we can write $N^{(l-2)}_{a}$ and $N^{(l-2)}_{b}$ as
the union and intersection of sets of the type $N^{(0)}_{v_{i}}$ and $N^{(1)}_{v_{i}}$ where $v_i$ is a vertex on the initial shortest path. For all $v_i$, except for $i=1,2,3$, we
have $N^{(0)}_{v_{i}} = N^{(1)}_{v_{i}}$, since the negation would imply $(v_1,v_2,\ldots,v_l)$ not being the shortest path before the first measurement. The neighborhood of $v_2$ is empty after the first
measurement. Now, Eq.~\eqref{first} and $N^{(1)}_{v_{3}}=\{v_1\}\cup \left(N^{(0)}_{v_{3}}\cup N^{(0)}_{v_{1}}\right) \setminus \left(N^{(0)}_{v_{3}}\cap N^{(0)}_{v_{1}}\right)$
imply
\begin{equation}
N^{(l-2)}_{v_1}\cup N^{(l-2)}_{v_l} \subsetneq N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_l}.
\end{equation}
The subset relation is proper, because the $l-2$ vertices $v_2,v_3,\ldots,v_{l-1}$ are contained in $N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_l}$
but not in $N^{(l-2)}_{v_1}\cup N^{(l-2)}_{v_l}$. This implies
\begin{equation}
|N^{(l-2)}_{v_1}\cup N^{(l-2)}_{v_l}|+l-4 \leq |N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_l}|-2,
\end{equation}
which concludes the proof.
\end{proof}
\noindent{}Now we will prove Lemma \ref{lem1} and thereby show the equivalence of successive $X$-measurements to $Z$-measurements on a graph in the LC orbit.
\setcounter{lemma}{0}
\begin{lemma}[Equivalence of measurements]\label{lem1}
$X$-measurements along a shortest path between two nodes are equivalent to performing a series of local complementations on the path, followed by $Z$-measurements on the intermediate nodes.
\end{lemma}
\begin{proof}
An $X$-measurement of a node is equivalent to locally complementing a neighbor, then locally complementing the actual node and Z-measuring, followed by a final local complementation of the same neighbor. Suppose that the nodes $v_i$, $i=1,\dots, n$ constitute a shortest path. We denote by $X_i$ and $Z_i$ the $X$- and $Z$-measurements on node $i$ respectively, and by $LC_i$ the action of local complementation with respect to the node $v_i$. Then
\begin{equation}
X_2= LC_1~LC_2~Z_2~LC_1
\end{equation}
is a valid decomposition of $X_2$ in terms of local complementations and $Z$-measurements.
If there is no shorter path connecting $v_1$ and $v_3$, this means that the $X_2$ measurement (and more specifically $LC_2$) creates a link between $v_1$ and $v_3$. Therefore, when we measure $X_3$, we can again choose $v_1$ as a neighbor and find
\begin{equation}
X_3= LC_1~LC_3~Z_3~LC_1.
\end{equation}
Continuing along the path, we finally find that
\begin{align}
X_2\cdots X_{n-1}= LC_1~LC_2~Z_2~LC_3~Z_3~\cdots LC_{n-1}~Z_{n-1}~LC_1,
\end{align}
since two consecutive local complementations with respect to the same vertex cancel each other out.
However, $Z_i$ commutes with $LC_j$ since measuring in $Z$ removes the node and all adjacent edges. If $i\in N_j$, it does not matter whether a local complementation will connect $v_i$ with any other node or not, since all connections will disappear after the measurement. We can therefore push all $Z$-measurements to the end and obtain
\begin{align}
X_2\cdots X_{n-1}= LC_1~\cdots LC_{n-1}~LC_1~Z_2~\cdots ~Z_{n-1}
\end{align}
to conclude the proof.
\end{proof}
\noindent{}Now, building upon the $X$-protocol, we give a proof of Corollary \ref{cor1} by a short case analysis.
\begin{customcor}{1}[Extraction of GHZ3 states]\label{cor1}
We can always distill a $3$-partite GHZ state between arbitrary vertices of a connected graph state in polynomial time.
\end{customcor}
\begin{proof}
Again we take $(a=v_1, v_2, \ldots, v_{l}=b)$ to be a shortest path in the initial graph.
If $c\in N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_{l}}$ lies in the neighborhood of the chosen path, there are two subcases.
\begin{itemize}
\item{ If $c=v_i$ for some $i \in \{1,2,\ldots,l\}$, we measure the vertices $v_2,v_3,\ldots,v_{i-1}$, $v_{i+1},\ldots ,v_{l-1}$ in the $X$-basis. After these $l-3$ measurements every vertex in $N^{(l-3)}_{a}\cup N^{(l-3)}_{b}\cup N^{(l-3)}_{c}\setminus \{a,b,c\}$ is measured in the $Z$-basis.}
\item{If $c\neq v_i$ for all $i \in \{1,2,\ldots,l\}$, we measure the vertices $v_2,v_3,\ldots ,v_{l-1}$ in the $X$-basis. After these $l-2$ measurements vertex $c$ is certainly contained in $N^{(l-2)}_{a}\cup N^{(l-2)}_{b}$.
Every vertex but $c$ from this set is then measured in the $Z$-basis.}
\end{itemize}
If $c\notin N^{(0)}_{v_1}\cup N^{(0)}_{v_2} \cup \ldots \cup N^{(0)}_{v_{l}}$, we again measure the nodes $v_2,v_3,\ldots ,v_{l-1}$ in the $X$-basis. Without loss of generality let the shortest path $b=w_1, w_2, \ldots, w_{l'}=c$ be shorter than all the paths from $a$ to $c$. We continue by measuring all vertices in $N^{(l-2)}_{a}$ in the $Z$-basis followed by $w_2,w_3,\ldots ,w_{l'-1}$ in the $X$-basis. Note that $w_i \in N^{(l-2)}_{a}$ for some $i \in \{2,3,\ldots,l'\}$ would be a contradiction to the shortest path assumption.
Finally, we measure every vertex but $a$ that lies in the neighborhoods of $b$ and $c$ in the $Z$-basis. All of the above cases result in the desired $3$-partite GHZ state between $a$, $b$ and $c$ independent of the choice of paths.
\end{proof}
\noindent Finally, we turn towards the generation of $4$-partite GHZ states as stated in Proposition \ref{lem:GHZ4}.
\setcounter{proposition}{2}
\begin{proposition}[Extraction of GHZ4 states]\label{lem:GHZ4_a}
We can always distill a $4$-partite GHZ state from graph states when their underlying graph (i) is a repeater line, with at least one extra node between two pairs of the final GHZ4 nodes, or (ii) contains such a line as a vertex-minor.
\end{proposition}
\begin{proof}
To be consistent with the figure in the main text, suppose that we want to have a GHZ4 state between nodes with labels $1,2,4$ and $5$ in the original graph state. If the underlying graph has a repeater line as a vertex-minor, we may separate it from the remaining graph state via appropriate local complementations and measurements. By local complementation on the path and measurement on the nodes that are not part of the final GHZ4, we can always distill the required state, as seen in the figure in the main text. For a large subset of graph states there is however a more efficient way to generate the desired GHZ4 state in analogy to the $X$-protocol.
Without isolating the repeater line first, we may perform local complementations with respect to nodes $2,3,4$ followed by $Z$-measurements on $3$ and on every vertex that is connected to any of $1,2,4,5$ and does not lie on the repeater-line itself. If, for example, there is no shorter path connecting $1$ to $5$ initially than the one given by the repeater line, the successive local complementations result in graphs that have a subgraph like the ones displayed in the figure in the main text.\\
\end{proof}
\COMMENT{
\subsection{Workspace}
\je{Here some work space. First note that there is a figure of merit that, as a one-sided test, allows to
check whether two graphs states are LC equivalent. This the cut-rank. If the graphs of
two graph states have a different cut-rank, then they are LC inequivalent \cite{CutRank,Hein04}. The converse is not
true, two isomorphic Peterson graphs provide a counterexample \cite{Peterson}. Comment added:
Nice, but no longer relevant.}
\subsubsection{Examples for the line}
No 4-partite GHZ state on the ending nodes of a repeater line?
No overlapping EPR pairs on a repeater line? \je{Shall we settle this question before submission?
Not sure.}
\subsubsection{Examples for the ring}
Evenly distributed 4-partite GHZ state on the ring with $4k, k=2,3,\ldots$ nodes?
Can we show that the ``repeater'' scheme is the best we can do, for the 2-pair setting?
\je{Cute. But good for follow-up work.}
\subsubsection{Ideas for 4-partite GHZ states}
The following corollary is in some sense related to Theorem \ref{thm2}; having no shorter path between nodes $1$ and $5$ than the path
$1,2,3,4,5$ is essentially a local sparsity property.
}
\end{document} | math | 67,965 |
\begin{document}
\thispagestyle{empty}
\begin{abstract}
We compare several approaches to the history of mathematics recently
proposed by Bl{\aa}sj\"o, Fraser--Schroter, Fried, and others. We
argue that tools from both mathematics and history are essential for a
meaningful history of the discipline.
In an extension of the Unguru--Weil controversy over the concept of
\emph{geometric algebra}, Michael Fried presents a case against both
Andr\'e Weil the ``privileged observer'' and Pierre de Fermat the
``mathematical conqueror.'' We analyze Fried's version of Unguru's
alleged polarity between a historian's and a mathematician's history.
We identify some axioms of Friedian historiographic ideology, and
propose a thought experiment to gauge its pertinence.
Unguru and his disciples Corry, Fried, and Rowe have described
Freudenthal, van der Waerden, and Weil as Platonists but provided no
evidence; we provide evidence to the contrary. We analyze how the
various historiographic approaches play themselves out in the study of
the pioneers of mathematical analysis including Fermat, Leibniz,
Euler, and Cauchy.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\label{s1}
The recent literature features several approaches to the history of
mathematics. Thus, Michael N. Fried (\cite{Fr18}, 2018) and
Guicciardini (\cite{Gu18}, 2018) argue for versions of Unguru's
approach (see below). Bl{\aa}sj\"o (\cite{Bl14}, 2014) advocates a
rational history as opposed to an ``idiosyncraticist'' one. Fraser
and Schroter propose something of a middle course that defines the
task of the history of mathematics as ``our attempt to explain why a
certain mathematical development happened'' in (\cite{Fr19a}, 2019,
p.\;16). We illustrate the latter approach in Section~\ref{s43}, in
the context of certain developments in mathematical analysis from
Euler to Cauchy, following Fraser--Schroter (\cite{Fr20}, 2020).
We analyze the perception of mathematical historiography that posits a
polarity between a historical and a mathematical view. Such a
perception is often associated with Sabetai Unguru. Against such
Unguru polarity, we argue that tools from both disciplines are both
useful and essential.
\subsection{Unguru, Weil, van der Waerden, Freudenthal}
Sabetai Unguru (\cite{Un75}, 1975) and Andr\'e Weil (\cite{We78},
1978) famously battled one another over the relation between Greek
mathematics and the concept of \emph{geometric algebra}, a term
introduced by H. G. Zeuthen in 1885 (see Bl{\aa}sj\"o \cite{Bl16},
2016, p.\;326; H{\o}yrup \cite{Ho16}, 2016, pp.\;4--6).
We note that B.\;L.\;van der Waerden (\cite{Va76}, 1976) and Hans
Freudenthal (\cite{Fr77}, 1977) published responses to Unguru earlier
than Weil. A clarification is in order concerning the meaning of the
term \emph{geometric algebra}. Van der Waerden explained the term as
follows:
\begin{quote}
We studied the wording of [Euclid's] theorems and tried to reconstruct
the original ideas of the author. We found it \emph{evident} that
these theorems did not arise out of geometrical problems. We were not
able to find any interesting geometrical problem that would give rise
to theorems like II 1--4. On the other hand, we found that the
explanation of these theorems as arising from algebra worked well.
Therefore we adopted the latter explanation. Now it turns out
{\ldots} that what we, working mathematicians, found evident, is not
evident to Unguru. Therefore I shall state more clearly the reasons
why I feel that theorems like Euclid II 1--4 did not arise from
geometrical considerations. \cite[pp.\;203--204]{Va76} (emphasis in
the original)
\end{quote}
Further details on van der Waerden's approach can be found in
Section~\ref{s25}.
We refrain from taking a position in the debate on
the narrow issue of geometric algebra as applied to Greek mathematics,
but point out that the debate has stimulated the articulation of
various approaches to the history of mathematics. We will analyze how
the various approaches play themselves out in the study of the
pioneers of mathematical analysis including Fermat, Leibniz, Euler,
and Cauchy; see Section~\ref{s4}. On whether other historians endorse
Unguru polarity, see Section~\ref{s7b}.
\subsection{Returning from escapades}
\label{s12}
Readers familiar with the tenor of the Unguru--Weil controversy will
not have been surprised by the tone of the ``break-in" remark found in
the 2001 book by Fried and Unguru (henceforth FU):
\begin{quote}
The mathematical and the historical approaches are antagonistic.
Whoever breaks and enters typically returns from his \emph{escapades}
with other spoils than the peaceful and courteous caller.
(Fried--Unguru \cite{Fr01}, 2001, p.\;406; emphasis on
\emph{escapades} added)
\end{quote}
For readers less familiar with the controversy, it may be prudent to
clarify that the unpeaceful and uncourteous caller allegedly involved
in the break-in is, to be sure, the mathematician, not the historian.
The break-in remark is duly reproduced in a recent essay in the
\emph{Journal of Humanistic Mathematics} by Unguru's disciple Fried
(\cite{Fr18}, 2018, p.\;7). And yes, the ``escapades" are in the
original, both in FU and in Fried solo.
That Fried's mentor Unguru does not mince words with regard to Weil is
not difficult to ascertain. Thus, one finds the following phrasing:
``Betrayals, Indignities, and Steamroller Historiography: Andr\'e Weil
and Euclid'' (Unguru \cite{Un18}, 2018, p.\;26, end of Section\;I).
Clearly, neither unpeaceful break-ins nor uncourteous escapades
represent \emph{legitimate} relationships to the mathematics of the
past. The break-in remark makes the reader wonder about the precise
meaning of Fried's assurance that his plan is to catalog some of the
attitudes toward the history of mathematics ``\emph{without judgment}
as to whether they are necessarily correct or legitimate" (Fried
\cite[abstract, p.\;3]{Fr18}; emphasis added).
\subsection{Who is open-minded?}
\label{s13}
A further telling comment appears on the back cover of the FU book on
Apollonius of Perga:
\begin{quote}
Although this volume is intended primarily for historians of ancient
mathematics, its approach is fresh and engaging enough to be of
interest also to historians, philosophers, linguists, and
\emph{open-minded mathemati\-cians}. (Fried--Unguru \cite{Fr01},
2001; emphasis added)
\end{quote}
Readers will not fail to notice that, of the four classes of scholars
mentioned, the mathematician is the only class limited by the
qualifier \emph{open-minded}; here FU don't appear to imply that a
mathematician is typically characterized by the qualifier. Such a
polarizing approach (see further in Section~\ref{s52}) on the part of
FU is hardly consistent with Fried's professed idea of cataloguing
attitudes toward history ``without judgment as to whether they are
{\ldots} correct or legitimate.'' We will analyze the ideological
underpinnings of the FU approach in Section~\ref{s2}.
\section
{FU axiomatics: Discontinuity, \emph{tabula rasa}, antiplatonism}
\label{s2}
In this section, we will identify several axioms of historiography
according to the Unguru school. In the Friedian scheme of things, it
is axiomatic that the proper view of the relation between the
mathematics of the past and that of the present is that of a
\emph{dis}continuity. It is indeed possible to argue that it is (see
Section~\ref{s33} for an example in Fermat). However, Fried appears
to take it for granted that a contrary view of \emph{continuity}
between past and present necessarily amounts to whig history or, more
politely, engaging in what Oakeshott described as a ``practical past"
and Grattan-Guinness \cite{Gr04} described as ``heritage" (see Fried
\cite{Fr18}, 2018, p.\;7). Fried's attitude here is at odds with the
idea of historiography as seeking to ``explain why a certain
mathematical development happened'' (Fraser--Schroter, \cite{Fr19a},
2019, p.\;16).
\subsection{Axiom 1: Discontinuity}
\label{s31}
What emerges is the following axiom of Friedian historiographic
ideology:
\begin{quote}
\textbf{Axiom 1 (discontinuity).} The proper attitude of a historian
toward the mathematics of the past is that of a discontinuity with the
mathematics of the present.
\end{quote}
We note that, while the discontinuity view may be appropriate in
certain cases, it is an assumption that needs to be argued rather than
posited as an axiom as in Fried.
Fried presents a taxonomy of various attitudes toward the history of
mathematics. He makes it clear that he means to apply his taxonomy
rather broadly, and not merely to the historical work on Greek
mathematics:
\begin{quote}
[I]n most of the examples the mathematical past being considered by
one person or another is that of Greece. \ldots{} Nevertheless, as I
hope will be clear, the relationships evoked in the context of these
examples have little to do with the particular character of Greek
mathematics. \cite[p.\;8]{Fr18}
\end{quote}
We will examine the effectiveness of the FU approach in such a broader
context.
\subsection{Apollonius and \emph{mathematical conquerors}}
\label{s22}
Fried notes that his piece ``has been written in a light and playful
spirit" (ibid., p.\;8). Such a light(-headed) spirit is reflected in
Fried's attitude toward Fermat. Fermat's reconstruction of
Apollonius' \emph{Plane Loci} prompts Fried to place Fermat in a
category labeled ``mathematical conqueror" rather than that of a
historian. The label also covers Descartes and Vi\`ete, whose ``sense
of the past has the unambiguous character of a `practical past', again
to use Oakeshott's term'' \cite[pp.\;11--12]{Fr18}. Fried does not
spare the great magistrate the tedium of Mahoney's breezy journalese:
\begin{quote}
``Fermat was no antiquarian interested in a faithful reproduction of
Apollonius' original work; \ldots\;The Plane Loci was to serve as a
means to an end rather than an end in itself.'' (Mahoney as quoted by
Fried in \cite[p.\;11]{Fr18})
\end{quote}
Fried appears to endorse Mahoney's dismissive attitude toward Fermat's
historical work. Regrettably, Fried ignores Recasens' more balanced
evaluation of Fermat's work on Apollonius' \emph{Plane Loci},
emphasizing its classical geometric style, and contrasting it with van
Schooten's:
\begin{quote}
Fermat's demonstration of Locus II-5 is presented in the classical
geometrical style of the day, though his conception was already
algebraic; [van] Schooten's is a pure exercise of analytic geometry.
(Recasens \cite{Re94}, 1994, p.\;315)
\end{quote}
What is the source of Fried's facile dismissal of Fermat's historical
scholarship? While it is difficult to be certain, a clue is found in
the attitude of his mentor Unguru who wrote:
\begin{quote}
[Fermat] took the Greek problems away from their indigenous territory
into new and foreign lands. Interestingly (and again, I think, quite
typically), Fermat did not see in his novel and revolutionary methods
strategies intrinsically alien to Greek mathematics, thus contributing
to the creation of the pervading and pernicious myth that there are
not indeed any substantive differences between the geometrical works
of the Greeks and the algebraic treatment of Greek mathematics by
post-Vi\`etan mathematicians. (Unguru \cite{Un76}, 1976, p.\;775)
\end{quote}
Unguru describes Fermat's reading of Apollonius as contributing to a
``pernicious myth,'' and Fermat's reliance on Vieta's theory of
equations as ``alien'' to Greek mathematics; Fried apparently follows
suit.
By page 12, Fried takes on a group he labels
``mathematician-histo\-rians" whose fault is their interest in
historical continuity:
\begin{quote}
The mathematics of the past is still understood by them as continuous
with present mathematics. (Fried \cite[p.\;12]{Fr18})
\end{quote}
Again, adherence to the continuity view is cast without argument as a
fault (see Section~\ref{s31}). Yet positing discontinuity as a
working hypothesis can make scholars myopic to important aspects of
the historical development of mathematics, and have a chilling effect
on attempts to explain why certain mathematical developments happened
(the Fraser--Schroter definition of the task of a historian); see
Section~\ref{s4} for some examples.
\subsection{Axiom 2: Tabula rasa}
\label{s3}
Unguru's opposite number Weil makes a predictable appearance in (Fried
\cite{Fr18}, 2018) on page 14, under the label \emph{privileged
observer}. The label also covers Zeuthen and van der Waerden. The
fault for this particular label is the desire to take advantage of
``their mathematical ideas \ldots\;to piece together the past"
(ibid.). Fried's posture on Weil brings us to the next axiom of
Friedian historiographic ideology:
\begin{quote}
\textbf{Axiom 2 (tabula rasa).} It is both possible and proper for
historians to refrain from using modern mathematical ideas.
\end{quote}
By page 16 we learn what authentic historians of mathematics do: they
\begin{quote}
take as their working assumption, a kind of \emph{null-hypothesis},
that there is a discontinuity between mathematical thought of the past
and that of the present. \cite[p.\;16]{Fr18} (emphasis added)
\end{quote}
This formulation of Fried's discontinuity axiom (see
Section~\ref{s31}) has the advantage of being explicitly cast as a
\emph{hypothesis}. Yet nothing about Fried's tone here suggests any
intention of actually exploring the \emph{validity} of such a
hypothesis. A related objection to Fried's posture was raised by
Bl{\aa}sj\"o and Hogendijk (\cite{Bl18}, 2018, p.\;775), who argue
that ancient treatises may contain meanings and intentions that go
beyond the surface text, based on a study of Ptolemy's
\emph{Almagest}. We will analyze Fried's Axiom\;2 in
Section~\ref{s4}.
\subsection{Axiom 3: Uprooting the Platonist deviation}
\label{s24}
\begin{figure}
\caption{\textsf{Mathematical Platonism. A humorous illustration from
the book \emph{Homotopic Topology}
\label{f1}
\end{figure}
The following additional axiom is discernible in the writings of
Unguru and his students.
\begin{quote}
\textbf{Axiom 3 (Mathematicians as Platonists)} Mathematicians
interested in history are predominantly Platonists; furthermore, their
beliefs (e.g., that mathematics is eternally true and unchanging)
interfere with their functioning as competent historians.
\end{quote}
Mathematicians interested in history are repeatedly described as
Platonists (see Figure~\ref{f1}) in the writings of Fried, Rowe, and
Unguru. Thus, one finds the following comments (emphasis on
``Platonic'' added throughout):
\begin{enumerate}
[label={(Pl\,\theenumi)}]
\item
``[M]athematically minded historians {\ldots} assume tacitly or
explicitly that mathematical entities reside in the world of
\emph{Platonic} ideas where they wait patiently to be discovered by
the genius of the working mathematician" (Unguru--Rowe \cite{Un81},
1981, p.\;3, quoting \cite{Un79}). Unguru and Rowe go on at length
(pages\;5 through 10) to attack van der Waerden's interpretation of
cunei\-form tablet BM 13901.
\footnote{For a rebuttal of the Unguru--Rowe critique see Bl{\aa}sj\"o
(\cite{Bl16}, 2016, Section\;3.4).}
\item
``It has been argued that most contemporary historians of mathematics
are \emph{Platonists} in their approach. They look in the past of
mathematics for the eternally true, the unchanging, the constant"
(Unguru--Rowe \cite{Un82}, 1982, p.\;47). The problem with such an
approach is diagnosed as follows: ``If nothing changes there is no
history" (op.\;cit., p.\;48).
\item
``[T]he methodology embodied in `geometric algebra' {\ldots} is the
outgrowth of a \emph{Platonic} metaphysics that sees mathematical
ideas as disembodied beings, pure and untainted by any idiosyncratic
features'' (Fried--Unguru \cite{Fr01}, 2001, p.\;37).
\item
\label{pl4}
``That Apollonius was a skilled geometrical algebraist is clearly the
considerate opinion of Zeuthen. It is an opinion based
\emph{exclusively} on a \emph{Platonic} philosophy of mathematics,
according to which one and the same mathematical idea remains the same
irrespective of its specific manifestations'' (Fried--Unguru
\cite{Fr01}, 2001, pp.\;47--48; emphasis added).
\item
``There is one mathematics, from its pre-historical beginnings to the
end of time, irrespective of its changing appearances over the
centuries. This mathematics grows by accumulation and by a sharpening
of its standards of rigor, while, its rational, ideal, \emph{Platonic}
Kernel, remaining unaffected by the historical changes mathematics
undergoes, enjoys, as Hardy put it, immortality. In short, proven
mathematical claims remain proven forever, no matter what the changes
are that mathematics is undergoing. And since it is always possible
to present past mathematics in modern garb, ancient mathematical
accomplishments can be easily made to look modern and, therewith
seamlessly integrated into the growing body of mathematical
knowledge'' (Unguru \cite{Un18}, 2018, pp.\;19--20).
\item
``The \emph{Platonic} outlook embodied in Weil's statements, according
to which (1) mathematical entities reside in the world of
\emph{Platonic} ideas and (2) mathematical equivalence is tantamount
to historical equivalence, is inimical to history" (Unguru
\cite{Un18}, 2018, p.\;29).
\end{enumerate}
What is comical about this string of attempts to pin a
\emph{Platonist} label on scholars is the contrast between the extreme
care Unguru and his students advocate in working with primary
documents and sourcing every historical claim, on the one hand, and
the absence of such sources when it comes to criticizing scholars they
disagree with, on the other.
This is not to say that mathematicians interested in history are never
Platonists. Thus, in a recent volume by Dani--Papadopoulos, one
learns that
\begin{quote}
[T]hinkers in colonial towns in Asia Minor, Magna Graecia, and
mainland Greece, cultivated a love for systematizing phenomena on a
rational basis {\ldots} They appreciated purity, universality, a
certainty and an elegance of mathematics, the characteristics that all
other forms of knowledge do not possess. (\cite{Pa19}, 2019, p.\;216)
\end{quote}
While such attitudes do exist, it remains that claiming your opponents
are Platonists without providing evidence is no more convincing than
claiming that Greek geometry had an algebraic foundation without
providing evidence, a fault Unguru and others impute to their
opponents. Without engaging in wild-eyed accusations of Platonism
against scholars he disagreed with, Grattan-Guinness \cite{Gr96} was
able to enunciate a dignified objection to geometric algebra (for a
response see Bl{\aa}sj\"o \cite[Section\;3.10]{Bl16}).
With regard to Axiom\;3, it is worth noting that the broader the
spectrum of the culprits named by Unguru, the less credible his charge
of Platonism becomes. Consider, for example, the claim in \ref{pl4}
above that Zeuthen's opinion is ``based \emph{exclusively} on a
Platonic philosophy of mathematics'' (emphasis added). Unguru attacks
Heiberg in \cite[p.\;107]{Un75} with similar vehemence. But how
credible would be a claim that the historiographic philosophy of the
\emph{philologist} Johan Ludvig Heiberg (of the \emph{Archimedes
Palimpsest} fame) is due to mathematical Platonist beliefs, especially
if no evidence is provided?
\subsection{Corry's universals}
\label{s25}
Axiomatizing tendencies similar to those of Fried, Rowe, and Unguru
manifest themselves in the writing by Unguru's student Corry, as well.
Engagement with Platonism and its discontents appears to be a constant
preoccupation in Corry's work. He alludes to Platonism by using terms
as varied as ``eternal truth,'' ``essence of algebra,'' and
``universal properties.'' Thus, in 1997 he writes:
\begin{quote}
[Bourbaki] were extending in an unprecedented way the domain of
validity of the belief in the eternal character of mathematical
truths, from the body to the images of mathematical knowledge. (Corry
\cite{Co97}, 1997, p.\;253)
\end{quote}
In 2004 (originally published in 1996) he writes:
\begin{quote}
{\ldots} a common difficulty that has been manifest {\ldots} is the
attempt to define, by either of the sides involved, the ``essence" of
algebraic thinking throughout history. Such an attempt appears, from
the perspective offered by the views advanced throughout the present
book, as misconceived. (Corry \cite{Co04}, 2004, p.\;396).
\end{quote}
In 2013 we find:
\begin{quote}
{\ldots} the question about the ``essence of algebra'' as an
ahistorical category seems to me an ill-posed and uninteresting one.
(Corry \cite{Co13}, 2013, p.\;639)
\end{quote}
In 2007, Corry imputes to mathematicians a quest for ``universal
properties'' at the expense of historical authenticity. He appears to
endorse Unguru's view of mathematicians as Platonists when he writes:
\begin{quote}
[I]n analyzing mathematics of the past mathematicians often look for
underlying mathematical concepts, regularities or affinities in order
to conclude about historical connection. Mathematical affinity
necessarily follows from \emph{universal properties} of the entities
involved and this has often been taken to suggest a certain historical
scenario that `might be'. But, Unguru warns us, one should be very
careful not to allow such mathematical arguments led [sic] us to
mistake historical truth (i.e., the `thing that \emph{has} been') with
what is no more than mathematically possible scenarios (i.e., the
`thing that \emph{might} be'). The former can only be found by direct
\emph{historical evidence}. (Corry \cite{Co07}, 2007; emphasis on
``has'' and ``might'' in the original; emphasis on ``universal
properties'' and ``historical evidence'' added)
\end{quote}
Granted one needs direct historical evidence, as per Corry. However,
where is the evidence that instead of looking for evidence,
mathematicians interested in history look for \emph{universal
properties}? Such a view of mathematicians who are historians is
postulated axiomatically by Corry, similarly to Fried, Rowe, and
Unguru. Corry goes on to claim that
\begin{quote}
[Unguru's 1975] work immediately attracted \emph{furious} reactions,
above all from three prominent mathematicians interested in the
history of mathematics: Andr\'e Weil, Bartel L. van der Waerden, and
Hans Freudenthal. (ibid.; emphasis added)
\end{quote}
Corry's remark is specifically characterized by the attitude of
looking for ``the thing that might be" rather than the ``thing that
has been", a distinction he mentions in the passage quoted above.
Namely, once Corry postulates a \emph{universal}-seeking attitude on
the part of Weil, van der Waerden, and Freudental, it then naturally
follows, for Corry, that they would necessarily react ``furiously" to
Unguru's work. Corry does present evidence of presentist attitudes in
historiography in connection with interpreting the Pythagorean
discovery of the incommensurability of the diagonal and the side of
the square. However, Corry presents evidence of such attitudes not in
the writings of Weil, van der Waerden, or Freudental, but rather those
of{\ldots} Carl Boyer (\cite{Bo68}, 1968, p.\;80).
\footnote{In \cite{Co07}, Corry criticizes attempts to deduce a purely
historical claim merely from ``underlying mathematical affinity.''
Corry provides the following example: ``It is thus inferred that the
Pythagoreans proved the incommensurability of the diagonal of a square
with its side exactly as we nowadays prove that $\sqrt{2}$ is an
irrational number.'' Corry's example is followed by a reference to
(Boyer \cite{Bo68}, 1968, p.\;80). On that page, Boyer wrote: ``A
third explanation [of the expulsion of Hippasus from the Pythagorean
brotherhood] holds that the expulsion was coupled with the disclosure
of a mathematical discovery of devastating significance for
Pythagorean philosophy---the existence of incommensurable
magnitudes.''}
For a discussion of the shortcomings of Boyer's historiographic
approach see Section~\ref{s43}. Note that Weil is just as sceptical
as Corry about claims being made on behalf of the Pythagoreans; see
(Weil \cite{We84}, 1984, pp.\;5, 8). We will analyze Corry's
problematic criticisms of van der Waerden, Freudenthal, and Weil
respectively in Sections~\ref{s26}, \ref{s27}, and \ref{s28}.
\subsection
{van der Waerden on Diophantus and Arabic algebra}
\label{s26}
Contrary to the claims emanating from the Unguru school, some of
Unguru's opponents specifically denied being Platonist. Thus, van der
Waerden wrote:
\begin{quote}
I am simply not a Platonist. For me mathematics is not a
contemplation of essences but intellectual construction. (van der
Waerden as translated in Schappacher \cite{Sc07}, 2007, p.\;245)
\end{quote}
It is instructive to contrast Unguru's attitude toward van der Waerden
with Szab\'o's. Szab\'o's book (\cite{Sz78}, 1978) deals with van der
Waerden at length, but there is no trace there of any allegation of
Platonist deviation. On the contrary, Szab\'o's book relies on van
der Waerden's historical scholarship, as noted also by Folkerts
(\cite{Fo69}, 1969). The book does criticize van der Waerden for what
Szab\'o claims to be an over-reliance on translations. Szab\'o
discusses this issue in detail in the context of an analysis of the
meaning of the Greek
term~$\delta\acute\upsilon\nu\alpha\mu\iota\varsigma$
(\emph{dynamis}),
\footnote{At the risk of committing precisely the type of inaccuracy
criticized by Szab\'o, one could translate \emph{dynamis} roughly as
``the squaring operation''.}
as mentioned by Folkerts.
In reality, van der Waerden's 1976 article contains no sign of the
``fury'' claimed by Corry (see Section~\ref{s25}). Perhaps the most
agitated passage there is van der Waerden's rebuttal of a spurious
claim by Unguru (which is echoed thirty years later by Corry in
\cite{Co07}):
\begin{quote}
We (Zeuthen and his followers) feel that the Greeks started with
algebraic problems and translated them into geometric language.
Unguru thinks that we argued like this: We found that the theorems of
Euclid II can be translated into modern algebraic formalism, and that
they are easier to understand if thus translated, and this we took as
`the proof that this is what the ancient mathematician had in mind'.
Of course, this is nonsense. We are not so weak in logical thinking!
The fact that a theorem can be translated into another notation does
not prove a thing about what the author of the theorem had in mind.
(van der Waerden \cite{Va76}, 1976, p.\;203)
\end{quote}
What one \emph{does} find in van der Waerden's article is a specific
rebuttal of Corry's claim concerning an alleged search for
\emph{universal properties} (see Section~\ref{s25}). Here van der
Waerden was responding to Unguru, who had claimed that algebraic
thinking involves
\begin{quote}
Freedom from any ontological questions and commitments and, connected
with this, abstractness rather than intuitiveness. (Unguru
\cite{Un75}, 1975, p.\;77)
\end{quote}
Van der Waerden responded by rejecting Unguru's characterisation of
the algebra involved in his work on Greek mathematics, and pointed out
that what he is referring to is
\begin{quote}
algebra in the sense of Al-Khw\={a}rizm\={\i}, or in the sense of
Cardano's `Ars magna', or in the sense of our school algebra. (van
der Waerden \cite{Va76}, 1976, p.\;199)
\end{quote}
Thus, van der Waerden specifically endorsed a similarity between
Arabic premodern algebra and Greek mathematics, and analyzed
Diophantus specifically in \cite[p.\;210]{Va76}.
A similarity between Arabic premodern algebra and the work of
Diophantus is also emphasized by Christianidis (\cite{Ch18}, 2018).
The continuation of the passage from van der Waerden is more
problematic from the point of view of \cite{Ch18}. Here he wrote:
\begin{quote}
Algebra, then, is: the art of handling algebraic expressions
like~$(a+b)^2$ and of solving equations like \mbox{$x^2+ax=b$}. (van
der Waerden \cite{Va76}, 1976, p.\;199)
\end{quote}
The viewpoint expressed here is at odds with the emphasis in
\cite{Ch18} on the fact that premodern algebra did not deal with
equations, polynomials were not sums but rather aggregates, and the
operations stipulated in the problem were performed before the
statement of the equation. However, apart from these important
points, van der Waerden's notion of Greek mathematics as close to
Arabic premodern algebra is kindred to the viewpoint elaborated in
\cite{Ch18} and \cite{Ch19}.
\footnote{It would be more difficult to bridge the gap between the
positions of Weil and \cite{Ch18}, since Weil claims that ``there is
much, in Diophantus and in Vi\`ete's \emph{Zetetica}, which in our
view pertains to algebraic geometry'' \cite[p.\;25]{We84}, whereas
\cite{Ch18} specifically distances itself from attempts to interpret
Diophantus in terms of algebraic geometry.}
The article \cite{Ch18} elaborates a distinction between \emph{modern
algebra} and \emph{premodern algebra}. The latter term covers both
Arabic sources and Diophantus. The position presented in \cite{Ch18}
is clearly at odds with Unguru, who wrote:
\begin{quote}
With Vi\`ete algebra becomes the very language of mathematics; in
Diophantus' \emph{Arithmetica}, on the other hand, we possess merely a
refined auxiliary tool for the solution of arithmetical problems
{\ldots} (Unguru \cite{Un75}, 1975, p.\;111, note\;138)
\end{quote}
Regardless of how close the positions of van der Waerden and
\cite{Ch18} can be considered to be, there is no mention of
\emph{universal properties} in van der Waerden's work on Greek
mathematics. The \emph{universals} appear to be all Corry's, not van
der Waerden's.
\subsection{Corry's shift on Freudenthal}
\label{s27}
Similarly instructive is Corry's--as we argue--variable position on
Freu\-denthal in connection with Platonism and Bourbaki. The standard
story on Bourbaki is the one of mathematical Formalism and
\emph{structures}. In Corry's view, there is some question concerning
how different this is, in Bourbaki's case, from Platonism. Here is
what Corry wrote in his book in 1996:
\begin{quote}
The above-described mixture [in Bourbaki] of a declared formalist
philosophy with a heavy dose of \emph{actual Platonic belief} is
illuminating in this regard. The formalist imperative, derived from
that ambiguous position, provides the necessary background against
which Bourbaki's drive to define the formal concept of
\emph{structure} and to develop some immediate results connected with
it can be conceived. The \emph{Platonic} stand, on the other hand,
which reflects Bourbaki's true working habits and beliefs, has led the
very members of the group to consider this kind of conventional,
formal effort as superfluous. (Corry \cite{Co96}, 1996, p.\;311;
emphasis on ``structure'' in the original; emphasis on ``actual
Platonic belief'' and ``Platonic'' added)
\end{quote}
On page 336 in the same book, Corry quotes Freudenthal's biting
criticism of Piaget's reliance on Bourbaki and their concept of
structure as an organizing principle:
\begin{quote}
The most spectacular example of organizing mathematics is, of course,
Bourbaki. How convincing this organization of mathematics is! So
convincing that Piaget could rediscover Bourbaki's system in
developmental psychology. {\ldots} Piaget is not a mathematician, so
he could not know how unreliable mathematical system builders are.
(Freudenthal as quoted in Corry \cite{Co96}, 1996, p.\;336).
\end{quote}
The same passage is quoted in the earlier article (Corry \cite{Co92},
1992, p.\;341).
The index in Corry's book on Bourbaki contains an ample supply of
entries containing the term \emph{universal}, including
\emph{universal constructs} and \emph{universal problems}, a constant
preoccupation of Bourbaki's which can also be seen as a function of
their Platonist background philosophy in the sense Corry outlined in
\cite[p.\;311]{Co96}, where Corry speaks of Bourbaki's ``actual
Platonic belief.''
There is a clear contrast, in Corry's mind, between Platonism,
universals, Bourbaki, and Piaget, on one side of the debate, and
Freudenthal with his clear opposition to both Bourbaki and Piaget, on
the other. Freudenthal's opposition tends to undercut the idea of
Freudenthal as Platonist, which in any case is at odds with
Freudenthal's pragmatic position on mathematics education; see e.g.,
La Bastide (\cite{La15}, 2015).
On the other hand, Corry's article (\cite{Co07}, 2007) includes
Freudenthal on the list of the Platonist mathematical culprits (van
der Waerden, Freudenthal, Weil) that has been made standard by Unguru.
According to Corry 2007, these scholars are in search of
\emph{universal properties}; see Section\;\ref{s25}. This fits with
Unguru's take on Freudenthal, but is at odds with what Corry himself
wrote about Freudenthal a decade earlier, as documented above.
In fact, Freudenthal specifically sought to distance himself from
Platonism in (\cite{Fr78}, 1978, p.\;7). Freudenthal's interest in
Intuitionism is discussed in \cite[p.\;42]{La15}. He published at
least two papers in the area: \cite{Fr37a} and \cite{Fr37b}. This
interest similarly points away from Platonism, contrary to Corry's
claim.
\subsection{Weil: internalist or externalist?}
\label{s28}
Corry attacks both Weil and Bourbaki as Platonist, and dismisses
Bourbaki's volume on the history of mathematics \cite{Bo94} as
``royal-road-to-me" historiography, in (Corry \cite{Co07}, 2007).
Paumier and Aubin make a more specific claim against the Bourbaki
volume generally and Weil's historiography in particular. Namely,
they refer to the volume as ``internalist history of concepts''
\cite[p.\;185]{Pa16}, and imply that the same criticism applies to
Weil's historiography, as well. In a related vein, Kutrov\'atz casts
Unguru and Szab\'o as externalists and Weil as internalist and
Platonist in \cite{Ku02}.
To evaluate such criticisms of Weil, the Bourbaki volume is of limited
utility since it was of joint authorship. We will examine instead
Weil's own book \emph{Number theory. An approach through history.
From Hammurapi to Legendre} (\cite{We84}, 1984). Does the
\emph{internalist} criticism apply here?
To answer the question, we would need to agree first on the meaning of
\emph{internalist} and \emph{externalist}. If we posit that
historical work is \emph{externalist} if it is written by Unguru, his
disciples, and their cronies, then there is little hope for Weil.
There is perhaps hope with a less partisan definition, such as
``historiography that takes into account the contingent details of the
historical period and its social context, etc.''
It is clear that historical and social factors are important. For
instance, one obtains a distorted picture of the mathematics of
Gregory, Fermat, and Leibniz if one disregards the fierce religious
debates of the 17th century (see references listed in
Section~\ref{s41b}).
Now it so happens that Weil's book \cite{We84} does contain detailed
discussions of the historical context. Weil's book is not without its
shortcomings. For instance, when Weil mentions that Bachet
``extracted from Diophantus the conjecture that every integer is a sum
of four squares, and asked for a proof'' \cite[p.\;34]{We84}, the
reader may well feel disappointed by the ambiguity of the verb
``extracted'' and the absence of references. However, what interests
us here is the validity or otherwise of the contention (implicit in
Unguru and Corry and explicit in Paumier--Aubin and Kutrov\'atz) that
Weil was \emph{internalist}. Was Weil internalist as charged?
Weil mentions, for instance, that Euler was first motivated to look at
the problem of Fermat primes $2^{2^n}+1$ by his correspondent Goldbach
\cite[p.\;172]{We84}. To give another example, Weil mentions that
Fermat learned Vieta's symbolic algebra through his visits to
d'Espagnet's private library in Bordeaux in the 1620s
\cite[p.\;39]{We84}. Such visits took place many years before Vieta's
works were published in 1646 by van Schooten. In particular, the
Fermat--d'Espagnet contact was instrumental in Fermat's formulation of
his method of adequality (see Section~\ref{s32}) relying as it did on
Vieta's symbolic algebra. Such examples undermine the Paumier--Aubin
claims, such as the following:
\begin{enumerate}
\item
the charge of ``an `internalist history of concepts' which has only
little to say about the way in which mathematics emerged from the
interaction of groups of people in specific circumstances''
\cite[p.\;187]{Pa16};
\item
the claim that ``The focus on ideas erased much of the social dynamics
at play in the historical development of mathematics''
\cite[p.\;204]{Pa16}.
\end{enumerate}
As we showed, Weil does take the interactions and the dynamics into
account.
\section{Some case studies}
\label{s4}
We identified Fried's \emph{tabula rasa} axiom in Section~\ref{s3},
and will analyze it in more detail in this section. It seems that
while the axiom may be appropriate in certain cases, it is an
assumption that needs to be argued rather than merely postulated.
Such a need to argue the case applies to the very possibility itself
of a ``tabula rasa" attitude in the first place:
\begin{enumerate}
[label={(TR\theenumi)}]
\item
\label{i1}
Can historians of mathematics truly view the past without the lens of
modern mathematics?
\item
\label{i2}
Have historians been successful in such an endeavor?
\end{enumerate}
Whereas it may be difficult to rule out the theoretical possibility of
an affirmative answer to \ref{i1}, a number of recent studies suggest
that in practice, the answer to \ref{i2} is often negative, as we will
discuss in Section~\ref{s41b}.
\subsection{History of analysis}
\label{s41b}
Some historians of 17th through 19th century mathematical analysis,
while claiming to reject insights provided by modern mathematics in
their interpretations, turn out themselves to be \emph{privileged
observers} in Fried's sense (see Section~\ref{s3}) though still in
denial. Namely, they operate within a conceptual scheme dominated by
the mathematical framework developed by Weierstrass at the end of the
19th century, as argued in recent studies in the following cases:
\begin{itemize}
\item
Fermat, in Katz et al.\;(\cite{13e}, 2013) and Bair et
al.\;(\cite{18d}, 2018);
\item
Gregory, in Bascelli et al.\;(\cite{18f}, 2018);
\item
Leibniz, in Sherry--Katz (\cite{14c}, 2014), Bascelli et
al.\;(\cite{16a}, 2016), Bl{\aa}sj\"o (\cite{Bl17}, 2017), and Bair et
al.\;(\cite{18a}, 2018);
\item
Euler, in Kanovei et al.\;(\cite{15b}, 2015) and Bair et
al.\;(\cite{17b}, 2017);
\item
Cauchy, in Bair et al.\;(\cite{17a}, 2017); Bascelli et
al.\;(\cite{18e}, 2018); and Bair et al.\;(\cite{19a}, 2019 and
\cite{20a}, 2020).
\end{itemize}
The pattern that emerges from these studies is that some modern
historians, limited in their knowledge of modern mathematics, tend to
take a narrow view, that in some cases borders on naivete, of the work
of the great mathematical pioneers of the 17--19th centuries (see
Sections~\ref{s44} and \ref{s45} for examples).
The Fermat historian Mahoney is a case in point. Weil pointed out
numerous historical, philological, and mathematical errors in
Mahoney's work on Fermat; see \cite{We73}. Yet in the Friedian scheme
of things, Weil is neatly shelved away on the \emph{privileged
observer} shelf, whereas Mahoney's work, breezy journalese and all, is
blithely assumed to reside in that rarefied stratum called authentic
history of mathematics, and relied upon to pass judgments on the value
of the historical work by the great Pierre de Fermat (see
Section~\ref{s31}).
\subsection{Fermat's adequality}
\label{s32}
Fermat used the method of \emph{adequality} to find maxima and minima,
tangents, and solve other problems.
To illustrate Fermat's method, consider the first example appearing in
his \emph{Oeuvres} \cite[p.\;134]{Ta1}. Fermat considers a segment of
length~$B$, splits it into variable segments of length~$A$ and~$B-A$,
and seeks to maximize the product~$A(B-A)$, i.e.,
\begin{equation}
\label{e31}
BA-A^2.
\end{equation}
Next, Fermat replaces~$A$ by~$A+E$ (and~$B-A$ by~$B-A-E$). There is a
controversy in the literature as to exact nature of Fermat's~$E$, but
for the purposes of following the mathematics it may be helpful to
think of~$E$ as small. Fermat goes on to expand the corresponding
product as follows:
\begin{equation}
\label{e32}
BA-A^2+BE-2AE-E^2.
\end{equation}
In order to compare the expressions \eqref{e31} and \eqref{e32},
Fermat removes the terms independent of~$E$ from both expressions, and
forms the relation
\begin{equation}
\label{e33}
BE\adequal 2AE+E^2,
\end{equation}
also referred to as \emph{adequality}. In the original, the term
\emph{ad{\ae}quabitur} appears where we used the symbol~$\adequal$.
\footnote{A symbol similar to $\adequal$ was used several decades
later by Leibniz, interchangeably with~$=$, to denote a relation of
generalized equality.}
We will present the final part of Fermat's solution in
Section~\ref{s34}.
\subsection{van Maanen's summary}
Fermat's method is described as follows by van Maanen:
\footnote{In place of Fermat's~$E$, van Maanen uses a lower-case~$e$.
The pieces of notation~$I(x)$,~$x_M$, and~$=$ are van Maanen's.}
\begin{quote}
Fermat seems to have based his method for finding a maximum or minimum
for a certain algebraic expression~$I(x)$ on a double root argument,
but in practice the algorithm was used in the following slightly
different form. Fermat argued that if the extreme value is attained
at~$x_M$,~$I(x)$ is constant in an infinitely small neighborhood
\footnote{Describing Fermat's method in terms of the infinitely small
is not entirely uncontroversial and is subject to debate; for details
see Bair et al.\;(\cite{18d}, 2018).}
of~$x_M$. Thus, if~$E$ is very small,~$x_M$ satisfies the equation
$I(x+E)=I(x)$. \cite[p.\;52]{Va03}
\end{quote}
Fermat never actually formed an algebraic relation (using Vieta's
symbolic algebra) of \emph{ad{\ae}quabitur} between the
expressions~$I(x+E)$ and~$I(x)$. The kind of relation he did form is
illustrated in formula~\eqref{e33} in Section~\ref{s32}. Van Maanen
provides the following additional explanations:
\begin{quote}
This expression states that close to the extreme value,
\footnote{We added the comma for clarity.}
lines parallel to the~$x$-axis will intersect the graph of~$I$ in two
different points, but the extreme is characterised by the fact that
these {\ldots} parallels turn into the tangent [line] and the points
of intersections reduce to one point which counts twice. The common
terms in~$x$ are removed from the equation~$I(x+E)=I(x)$ and the
resulting equation divided by~$E$. Any remaining terms are deleted,
and~$x_M$ is solved from the resulting equation. (ibid.)
\end{quote}
While the summary by van Maanen does not mention the possibility of
dividing by $E^2$, it is important to note that in Fermat's
descriptions of the method, Fermat does envision the possibility of
dividing by higher powers of~$E$ in the process of obtaining the
extremum.
\subsection{Squaring both sides}
\label{s34}
In the example presented in Section~\ref{s32}, the term~$BE$ and the
sum~$2AE+E^2$ originally both appeared in the expression~\eqref{e32},
but appear on different sides in relation~\eqref{e33} (all with
positive sign). The remainder of Fermat's algorithm is more familiar
to the modern reader: one divides both sides by~$E$ to produce the
relation~$B\adequal 2A+E$, and discards the summand~$E$ to obtain the
solution~$A=\frac{B}{2}$.
For future reference, we note that a relation of type~\eqref{e33} can
be squared to produce a relation of type~$(BE)^2 \adequal
(2AE+E^2)^2$. In this particular example, the relation need not be
squared. However, in an example involving square roots one needs to
square both sides at a certain stage to eliminate the radicals; see
Section~\ref{s33}. Meanwhile, once one passes to the difference
$I(x+E)-I(x)$ (to use van Maanen's notation), such an opportunity is
lost.
Fermat never performed the step of carrying all the terms to the
left-hand side of the relation so as to form the difference
$I(x+E)-I(x)$;
\footnote{Fermat historian Breger did in (\cite{Br13}, 2013, p.\;27);
for details see Bair et al.\;(\cite{18d}, 2018, Section\;2.6,
p.\;573).}
nor did Fermat ever form the quotient~$\frac{I(x+E)-I(x)}{E}$ familiar
to the modern reader. In Section~\ref{s33} we will compare the
treatment of this aspect of Fermat's method by a historian and a
mathematician.
\subsection{Experiencing~$E^2$}
\label{s33}
The perspective of Unguruan polarity can lead historians to devote
insufficient attention to the actual mathematical details and
ultimately to historical error. Thus, Mahoney claimed the following:
\begin{quote}
In fact, in the problems Fermat worked out, the proviso of repeated
division by~$y$ [i.e.,~$E$] was unnecessary. But, thinking in terms
of the theory of equations, Fermat could imagine, \emph{even if he had
not experienced}, cases in which the adequated expressions contained
nothing less than higher powers of~$y$. (Mahoney \cite{Ma94}, 1994,
p.\;165; emphasis added)
\end{quote}
Mahoney assumed that Fermat ``had not experienced'' cases where
division by~$E^2$ was necessary. Meanwhile, Giusti analyzes an
example ``experienced'' by Fermat which involves radicals, and which
indeed leads to division by~$E^2$. The example (Fermat \cite{Ta1},
p.\;153) involves finding the maximum of the expression
$A+\sqrt{BA-A^2}$ (here~$B$ is fixed). In the process of solution, a
suitable relation of adequality, as in formula~\eqref{e33}, indeed
needs to be \emph{squared} (see Section~\ref{s34}). Giusti concludes:
\begin{quote}
Ce qui nous int\'eresse dans ce cas est qu'il donne en exemple une
ad\'egalit\'e o\`u les termes d'ordre le plus bas sont en~$E^2$.
Comme on sait, dans l'\'enonciation de sa r\`egle Fermat parlait de
division par~$E$ ou par une puissance de~$E$ {\ldots} Plusieurs
commentateurs ont soutenu {\ldots} que Fermat avait commis ici une
\emph{erreur} {\ldots} On doit donc penser que dans un premier moment
Fermat avait trait\'e les quantit\'es contenant des racines avec la
m\'ethode usuelle, qui conduisait parfois \`aà la disparition des
termes en~$E$, et qui ait tenu compte de cette \'event\-ualit\'e dans
l'\'enonciation de la r\`egle g\'en\'erale. (Giusti \cite{Gi09},
2009, Section\;6, pp.\;75--76; emphasis added).
\end{quote}
What Giusti is pointing out is that in this particular application of
adequality in a case involving radicals, division by~$E^2$ (and not
merely by~$E$) is required. Thus, the error is Mahoney's, not
Fermat's.
A first-rate analyst and differential geometer, Giusti was able to
appreciate the \emph{discontinuity} between Fermat's method of
adequality, on the one hand, and the modern~$\frac{I(x+E)-I(x)}{E}$
perspective, on the other, better than many a Fermat historian. More
generally, a scholar's work should be evaluated on the basis of its
own merits rather than which class he primarily belongs to, be it
historian, mathematician, or philologist.
Appreciating discontinuity is not the prerogative of Unguru's adepts,
contrary to strawman accounts found in Unguru (\cite{Un18}, 2018) and
Guicciardini (\cite{Gu18}, 2018). The portrait of a mathematician's
view of his discipline dominated by mathematical Platonism as found in
Unguru and his students (as detailed in Section~\ref{s24}) as well as
Guicciardini is similarly a strawman caricature, as when Guicciardini
elaborates on ``the perfect embodiment of the immutable laws of
mathematics written in the sky for eternity" \cite[p.\;148]{Gu18} and
claims that ``[t]he mathematician's world is the world of Urania''
(op.\;cit., p.\;150).
\subsection
{Why certain developments happened: Euler to Cauchy}
\label{s43}
Analyzing the differences between 18th and 19th century analysis,
Fraser and Schroter observe:
\begin{quote}
The decline of [Euler's] formalism stemmed mainly from its limitations
as a means of generating useful results. Moreover, as methods began
to change, an awareness of formalism's apparent difficulties and even
contradictions lent momentum to efforts to rein it in.
(Fraser--Schroter \cite{Fr20}, 2020, Section 3.3)
\end{quote}
Note that Fraser and Schroter are analyzing Euler's own work itself
here, rather than its reception by Cauchy. Fraser and Schroter
continue:
\begin{quote}
Euler had been confident that the ``out-there'' objectivity of algebra
secured the generality of his formal techniques, but Cauchy demanded
that generality be found within mathematical methods themselves. In
his [textbook] \emph{Cours d'analyse} of 1821 Cauchy rejected
formalism in favour of a fully quantitative analysis. (ibid.)
\end{quote}
Fraser and Schroter feel that the limitations and difficulties of
Euler's variety of algebraic formalism can be fruitfully analyzed from
the standpoint of considerably later developments, notably Cauchy's
``quantitative analysis.'' In their view, it is possible to comment
on the shortcomings of Euler's algebraic formalism and the reasons for
this particular development from Euler to Cauchy without running the
risk of anachronism. Meawhile, it is clear that the Fraser--Schroter
approach may run afoul of both the discontinuity axiom (see
Section~\ref{s31}) and the tabula rasa axiom (see Section~\ref{s3}).
The issue of anachronism was perceptively analyzed by Ian Hacking
(\cite{Ha14}, 2014) in terms of the distinction between the butterfly
model and the Latin model for the development of a scientific
discipline. Hacking contrasts a model of a deterministic (genetically
determined) biological development of animals like butterflies (the
egg--larva--cocoon--butterfly sequence), with a model of a contingent
historical evolution of languages like Latin. Emphasizing determinism
over contingency can easily lead to anachronism; for more details see
Bair et al.\;(\cite{19a}, 2019).
Similarly to Hacking, Fraser notes the danger for a historian in the
adoption of a model based on an analogy with the pre-determined
evolution of a biological organism. In his review of Boyer's book
\emph{The concepts of the calculus}, Fraser comments on the risks of
anachronism:
\begin{quote}
[Boyer's] focus on the development of concepts through time may
reflect as well an embrace of the metaphor of a plant or animal
organism. The concept undergoes a progressive development, moving in
a directed and pre-determined way from its origins to an adult and
completed form. {\ldots} The possibility of introducing anachronisms
is almost inevitable in such an approach, and to a certain degree this
is true of Boyer's book. (Fraser \cite{Fr19}, 2019, p.\;18)
\end{quote}
Fraser specifically singles out for criticism Boyer's teleological
view of mathematical analysis as inexorably progressing toward the
ultimate \emph{Epsilontik} achievement:
\begin{quote}
[Boyer] seemed to view the eighteenth-century work as exploratory or
approximative as the subject moved inexorably in the direction of the
arithmetical limit-based approach of Augustin-Louis Cauchy and Karl
Weierstrass. (op. cit., p.\;19)
\end{quote}
We will report on two additional cases of such teleological thinking
in the historiography of mathematics in Sections~\ref{s44} and
\ref{s45}.
\subsection{Leibnizian infinitesimals}
\label{s44}
Boyer-style, \emph{Epsilontik}-oriented teleological readings of the
history of analysis (see Section~\ref{s43}) are common in the
literature. Thus, Ishiguro interprets Leibnizian infinitesimals as
follows:
\begin{quote}
It seems that when we make reference to infinitesimals in a
proposition, we are not designating a fixed magnitude incomparably
smaller than our ordinary magnitudes. Leibniz is saying that whatever
small magnitude an opponent may present, one can assert the existence
of a smaller magnitude. In other words, we can paraphrase the
proposition with a universal proposition with an embedded existential
claim. (Ishiguro \cite{Is90}, 1990, p.\;87)
\end{quote}
What is posited here is the contention that when Leibniz wrote that
his incomparable (or inassignable)~$dx$, or~$\epsilon$, was smaller
than every given (assignable) quantity~$Q$, what he really meant was
an alternating-quantifier clause (universal quantifier~$\forall$
followed by an existential one~$\exists$) to the effect that for each
given~$Q>0$ there exists an~$\epsilon>0$ such that~$\epsilon<Q$. Such
a logical sleight of hand goes under the name of the
\emph{syncategorematic interpretation}. Here the author is
interpreting Leibniz as thinking like Weierstrass (see also
Section~\ref{sync}). For details see Bascelli et al.\;(\cite{16a},
2016) and Bair et al.\;(\cite{18a}, 2018).
\subsection{Cauchyan infinitesimals}
\label{s45}
In a similar vein, Siegmund-Schultze views Cauchy's use of
infinitesimals as a step \emph{backward}:
\begin{quote}
There has been \ldots{}\;an intense historical discussion in the last
four decades or so how to interpret certain apparent remnants of the
past or -- as compared to J. L. Lagrange's (1736--1813) rigorous
`Algebraic Analysis' -- even \emph{steps backwards} in Cauchy's book,
particularly his use of infinitesimals {\ldots} (Siegmund-Schultze
\cite{Si09}, 2009; emphasis added)
\end{quote}
Siegmund-Schultze's reader will have little trouble reconstructing
exactly which direction a \emph{step forward} may have been in.
Grabiner similarly reads Cauchy as thinking like Weierstrass; for
details see Bair et al.\;(\cite{19a}, 2019).
\subsection{History, heritage, or escapade?}
Significantly, in his essay Fried fails to mention the seminal
scholarship of Reviel Netz on ancient Greek mathematics (see e.g.,
\cite{Ne02b}, \cite{Ne01}, \cite{Ne02}). Would, for example, Netz's
detection of traces of infinitesimals in the work of Archimedes be
listed under the label of history, heritage, or ``escapade" (to quote
Fried)? Would an argument to the effect that the \emph{procedures}
(see Section~\ref{sync}) of the Leibnizian calculus find better
proxies in modern infinitesimal frameworks than in late 19th century
Weierstrassian ones, rank as history, heritage, or escapade? Would an
argument to the effect that Cauchy's definition of continuity via
infinitesimals finds better proxies in modern infinitesimal frameworks
than in late 19th century Weierstrassian ones, rank as history,
heritage, or escapade? Unfortunately, there is little in Fried's
essay that would allow one to explore such questions.
\subsection{Procedures vs ontology}
\label{sync}
The procedures/ontology distinction elaborated in B{\l}aszczyk et al.
(\cite{17d}, 2018) can be thought of as a refinement of
Grattan-Guinness' history/heritage distinction. Consider for instance
Leibnizian infinitesimal calculus. Without the procedure/ontology
distinction, interpreting Leibnizian infinitesimals in terms of modern
infinitesimals will be predictably criticized for utilizing history as
\emph{heritage}. What some historians do not appreciate sufficiently
is that, in an ontological sense, interpreting Leibniz in
Weierstrassian terms is just as much heritage. Surely talking about
Leibniz in terms of ultrafilters
\footnote{See e.g., Fletcher et al.\;\cite{17f} for a technical
explanation.}
is not writing history; however, analyzing Leibnizian
\emph{procedures} in terms of those of Robinson's procedures is better
history than a lot of what is written on Leibniz by received
historians and philosophers (who have pursued a syncategorematic
reading of Leibnizian infinitesimals; see Section~\ref{s44}), such as
Ishiguro, Arthur, Rabouin, and others; for details see Bair et
al.\;(\cite{18a}, 2018).
We summarize some of the arguments involved. \textbf{1.}\;Leibniz
made it clear on more than one occasion that his infinitesimals
violate Euclid Definition\;V.5 (Euclid\;V.4 in modern editions), which
is a version of what is known today as the Archimedean axiom; see
e.g., (Leibniz \cite{Le95b}, 1695, p.\;322). In this sense, the
procedures in Leibniz are closer to those in Robinson than those in
Weierstrass. \textbf{2.}\;If one follows Unguru's strictures and
Fried's tabula rasa, one can't exploit \emph{any} modern framework to
interpret Leibniz; however, in practice the syncategorematic society
interpret Leibniz in Weierstrassian terms, so the Unguruan objection
is a moot point as far as the current debate over the Leibnizian
calculus is concerned. \textbf{3.}\;While modern foundations of
mathematics were clearly not known to Leibniz, it is worth pointing
out that this applies both to the set-theoretic foundational
\emph{ontology} of the classical Archimedean track, and to Robinson's
non-Archimedean track. But as far as Leibniz's \emph{procedures} are
concerned, they find closer proxies in Robinson's framework than in a
Weierstrassian one. For example, Leibniz's law of continuity is more
readily understood in terms of Robinson's transfer principle than in
any Archimedean terms. \textbf{4.}\;The syncategorematic society
seems to experience no inhibitions about interpreting Leibnizian
infinitesimals in terms of alternating quantifiers (see
Section~\ref{s44}), which are conspicuously absent in Leibniz himself.
Meanwhile, Robinson's framework enables one to interpret them without
alternating quantifiers in a way closer to Leibniz's own procedures.
\textbf{5.}\;On several occasions Leibniz mentions a distinction
between \emph{inassignable} numbers like~$dx$ or~$dy$, and (ordinary)
\emph{assignable} numbers; see e.g., his \emph{Cum Prodiisset}
\cite{Le01c} and \emph{Puisque des personnes}\ldots\;\cite{Le05b}.
The distinction has no analog in a traditional Weierstrassian
framework. Meanwhile, there is a ready analog of standard and
nonstandard numbers, either in Robinson's \cite{Ro66} or in Nelson's
\cite{Ne77} framework for analysis with infinitesimals.
\subsection{Are there gaps in Euclid?}
Some of the best work on ancient Greece would possibly fail to satisfy
Fried's criteria for authentic history, such as de Risi's monumental
work (\cite{De16}, 2016), devoted to the reception of Euclid in the
early modern age. Here de Risi writes:
\begin{quote}
Euclid's system of principles has been repeatedly discussed and
challenged: A few gaps in the proofs were found \ldots{}
\cite[p.\;592]{De16}.
\end{quote}
This is a statement about Euclid and not merely its early modern
reception. Now wouldn't the claim of the existence of a what is seen
today as a ``gap" in Euclid be at odds with Fried's \emph{tabula rasa}
axiom (see Section~\ref{s3})?
\subsection{Philological thought experiment}
Fried's discussion is so general as to raise questions about its
utility. Dipert notes in his review of the original 1981 edition of
Mueller \cite{Mu06}:
\begin{quote}
It will be difficult in the coming years for anyone doing serious
research on Euclid, \emph{outside of the narrowest philological
studies}, not first to have come to grips with the present book, and
it is to be hoped that this volume will inject new vigor into
discussions of Euclid by contemporary logicians and philosophers of
mathematics. (Dipert \cite{Di81}, 1981; emphasis on ``philological
studies'' added)
\end{quote}
Inspired by Dipert's observation, we propose the following thought
experiment. Consider a hypothetical study of, say, the frequency of
Greek roots in the texts of ancient Greek mathematicians. Surely this
is a legitimate study in Philology. As far as Fried's requirements
for authentic history, such a study would meet them with flying
colors. Thus, the satisfaction of the \emph{discontinuity} axiom (see
Section~\ref{s31}) is obvious. The satisfaction of the \emph{tabula
rasa} axiom (see Section~\ref{s3}) is evident, seeing that no modern
mathematics is used at all in such a study. The risk of a Platonist
deviation (see Section~\ref{s24}) is infinitesimal. Freudenthal, van
der Waerden, and Weil may well have written on interpreting the
classics; but by Fried's ideological criteria, our hypothetical
philological study would constitute legitimate mathematical history,
surpassing anything that such ``privileged observers'' may have
written. Yet it seems safe to surmise, following Dipert, that the
audience for such a philological study among those interested in the
history of mathematics would be limited.
\section{Evolution of Unguru polarity}
Fried admits in his 2018 essay that when he was a graduate student
under Unguru, it was axiomatic that there are only two approaches to
the history of mathemathics: that of a historian, and that of a
mathematician, as we show in Section~\ref{s51}.
\subsection{Fried's upgrade}
\label{s51}
In his 2018 article, Fried writes:
\begin{quote}
By the time I finished my Ph.D., I could make some distinctions: I
could divide historians of mathematics into a mathematician type, such
as Zeuthen or van der Waerden, a historian type, like Sabetai Unguru,
and, perhaps, a postmodern type {\ldots} (\cite[p.\;4]{Fr18})
\end{quote}
The latter ``type'' is quickly dismissed as ``not in fact a serious
option'' leaving us with onlys two options, historian and
mathematician. Fried goes on to relate in his essay that he came to
appreciate that the historiographic picture is more complex, resulting
in the novel labels of \emph{mathematical conquerors},
\emph{privileged observers}, and the like. Such a more complex
picture is something of a departure from Unguruan orthodoxy, as we
analyze in Section~\ref{s52}.
\subsection{Unguru polarity}
\label{s52}
In Section~\ref{s51} we summarized Fried's upgrade of Unguru's
original framework. Meanwhile, Unguru himself sticks to his guns as
far as the original dichotomy of mathematician versus historian is
concerned. In his 2018 piece, Unguru reaffirms the idea that there
are only two approaches to the history of mathematics:
\begin{quote}
The paper deals with two \emph{polar-opposite} approaches to the study
of the history of mathematics, that of the mathematician, tackling the
history of his discipline, and that of the historian. (Unguru 2018,
\cite{Un18}, p.\;17; emphasis added)
\end{quote}
Unguru proceeds to reveal further details on the alleged polarity:
\begin{quote}
[S]ince it is always possible to present past mathematics in modern
garb, ancient mathematical accomplishments can be easily made to look
modern and, therewith seamlessly integrated into the growing body of
mathematical knowledge. That this is a historical \emph{calamity} is
not the mathematician's worry
\ldots\;Never mind that this procedure is tantamount to the
\emph{obliteration} of the history of mathematics as a
\emph{historical} discipline. Why, after all, should this concern the
mathematician? \cite[p.\;20]{Un18} (emphasis on \emph{historical} in
the original; emphasis on \emph{calamity} and \emph{obliteration}
added)
\end{quote}
It seems to us that Unguru's assumption that a mathematician does not
care about a possible ``obliteration'' of the history of mathematics
as a historical discipline, is unwarranted. We note that ``calamity''
and ``obliteration'' are strong terms to describe the work of
respected scholars such as van der Waerden, Freudenthal, and others.
We will examine the issue in more detail in Section~\ref{s43b}.
\subsection{Polarity-driven historiography}
\label{s43b}
What is the driving force behind Unguru's historiographic ideology,
including his readiness to describe the two approaches as
``antagonistic'' (see Section~\ref{s12})? The ideological polarity
postulated in Unguru's approach appears to involve a perception of
class struggle, as it were, between historians (H-type, our notation)
and mathematicians (M-type, our notation) with their ``antagonistic''
class interests. As we already noted in Section~\ref{s13}, M-type as
a class does not fare very well relative to the attribute of
\emph{open-mindedness} in the FU ideology. For a detailed study of
polarity-driven historiography as applied to, or more precisely
against, Felix Klein and (in Unguru's words) ``the obliteration of the
history of mathematics as a historical discipline'' by Mehrtens, see
Bair et al.\;(\cite{18b}, 2017).
\footnote{Mehrtens (\cite{Me90}, 1990), in an avowedly marxist
approach, postulates the existence of two polar-opposite attitudes
among German mathematicians at the beginning of the 20th century:
modern (M-type, our notation) and countermodern (C-type, our
notation). Felix Klein had the bad luck of being pigeonholed as a
C-type, along with unsavory types like Ludwig Bieberbach and the
\emph{SS-Brigadef\"uhrer} Theodor Vahlen. The value of such crude
interpretive frameworks is limited.}
Unguru seeks to forefront the struggle between H-type and M-type as
\emph{the} fundamental ``antagonism'' in terms of which all historical
scholarship must be evaluated, in an attitude reminiscent of the
classic adage ``The history of all hitherto existing society is the
history of class struggles.'' How fruitful is such a historiographic
attitude? We will examine the issue in the context of a case study in
Section~\ref{s7}.
\subsection{Is exponential notation faithful to Euclid?}
\label{s7}
As a case study illustrating his historiographic ideology, Unguru
proposes an examination of Euclid's Proposition IX.8, dealing with
what would be called today a geometric progression of lengths starting
with the unity. Unguru already focused on this example over forty
years ago in \cite{Un75}. Paraphrased in modern terms, the
proposition asserts that in the geometric progression, every other
term is a square, every third term is a cube, etc. Unguru \cite{Un18}
objects to reformulating the proposition in terms of the algebraic
properties of the exponential notation~$1,a,a^2, a^3,\ldots$, echoing
the criticisms he already made in \cite{Un75}.
Why does Unguru feel that exponential notation must not be used to
reformulate Proposition IX.8? He provides a detailed explanation in
the following terms:
\begin{quote}
A proposition for the proof of which Euclid has to toil subtly and
painstakingly, and in the course of whose proof he had to rely on many
previous propositions and definitions (e.g., VIII.22 and 23,
def.\;VII.20) becomes a \emph{trivial commonplace}, which is an
immediate outgrowth, a trite after-effect, of our symbolic
notation:~$1$,~$a$,~$a^2$,~$a^3$,
$a^4, a^5, a^6, a^7, \ldots$ As a matter of fact, if we use modern
symbolism, this ceases altogether to be a proposition and its
truthfulness is an immediate and \emph{trivial application} of the
definition of a geometric progression in the particular case when the
first member equals~$1$ and the ratio,~$q=a$, is a positive integer
(for Euclid)! \cite[p.\;27]{Un18} (emphasis added)
\end{quote}
Unguru claims that using exponential notation causes Euclid's
proposition to become a trivial commonplace severed from Euclid's
``previous propositions,'' and a trivial application of the definition
of a geometric progression. In this connection, Bl{\aa}sj\"o points
out that
\begin{quote}
Unguru \ldots\;mistakenly believes that certain algebraic insights are
somehow built into the notation itself. (Bl{\aa}sj\"o \cite{Bl16},
2016, p.\;330)
\end{quote}
Namely,
\begin{quote}
The fact that, for example,~$a^4$ is a square is not by any means
implied by the symbolic notation itself. The fact
that~$a^{xy}=(a^x)^y$ is a contingent fact, a result that needs
proving. It is not at all obvious from the very notation itself
{\ldots} (ibid.)
\end{quote}
Thus, contrary to Unguru's claim, Euclid's Proposition IX.8 is
\emph{not} severed from Euclid's ``previous propositions'' which are
similarly more accessible to modern readers when expressed in modern
notation, whose properties require proof just as Euclid's propositions
do.
For instance, Proposition VIII.22 mentioned by Unguru asserts the
following: ``If three numbers are in continued proportion, and the
first is square, then the third is also square.'' In modern
terminology this can be expressed as follows: if~$a^2:b=b:c$, then
$a^2c=b^2$, and therefore~$c=x^2$ for some~$x$. Put another way,
$a^2rr=(ar)^2$.
\footnote{Here the first term in the progression is a square~$a^2$ by
hypothesis. The second term is~$a^2r$ and the third term
is~$(a^2r)r$. The identity $a^2rr=(ar)^2$ enables one to conclude
that the third term is also a square.}
This is not a triviality but rather an identity that requires proof.
Such an identity could possibly be used in the proof of special cases
of~$a^{xy}=(a^x)^y$. For more details see Mueller \cite{Mu06}.
Unguru's ideological opposition to using modern exponential notation
in this case has little justification.
It is a pity that (Unguru \cite{Un18}, 2018) chose not to address
Bl{\aa}sj\"o's rebuttal of his objections. Note that the rebuttal
(Bl{\aa}sj\"o \cite{Bl16}, 2016) appeared two years earlier than
Unguru's piece.
\subsection{What is an acceptable meta-language?}
\label{s61}
In his 2018 piece, Unguru reiterates a sweeping claim he already made
in 1979:
\begin{quote}
The only acceptable meta-language for a historically sympathetic
investigation and comprehension of Greek mathematics seems to be
ordinary language, not algebra. \cite[p.\;30]{Un18}.
\end{quote}
Given such a stance, it is not surprising that Unguru opposes any and
all use of algebraic notation (including exponential notation) in
dealing with Euclid (see Section~\ref{s7}) and Apollonius (see
Section~\ref{s22}). However, Berggren notes in his review of (Unguru
\cite{Un79}, 1979) that the reason
\begin{quote}
why modern words, with the concepts they embody, are acceptable as
analytic tools where Renaissance (or even Arabic) algebra is
forbidden,
\footnote{We added the comma for clarity.}
is never explained [by Unguru]. (Berggren \cite{Be79}, 1979)
\end{quote}
The position of some other historians with regard to Unguru's claims
is discussed in Section~\ref{s7b}.
\section{Do historians endorse Unguru polarity?}
\label{s7b}
Unguru's positing of a polarity of historian \emph{vs} mathematician
tends to obscure the fact that a number of distinguished
\emph{historians} have broken ranks with Unguru on the methodological
issues in question, such as the following scholars.
\begin{enumerate}
\item
Kirsti (M\o ller Pedersen) Andersen wrote a negative review of
Unguru's polarity manifesto \cite{Un75} for Mathematical Reviews,
noting in particular that Unguru ``underestimates the historians'
[e.g., Zeuthen's] understanding of Greek mathematics'' (\cite{An75},
1975).
\item
C. M. Taisbak notes that Unguru and Rowe ``are being ridiculously
unfair, to say the least, towards Heath at this point [concerning
interpretation of the \emph{Elements}, items I44 and I45], to say
nothing of others'' (\cite{Ta81}, 1981).
\item
\'Arp\'ad Szab\'o receives the strongest endorsement from Unguru in
\cite[pp.\;78, 81]{Un75}. Yet when Szab\'o analyzed \emph{Elements}
Book\;V \cite[p.\;47]{Sz78}, he employed symbolic notation introduced
in the 19th century by Hermann Hankel.
\footnote{In more detail, Hankel (\cite{Ha74}, 1874, pp.\;389--404)
introduced algebraic notation in an account of Euclid's
\emph{Elements} book\;V. Furthermore, Heiberg (\cite{He83}, 1883,
vol.\;II, s.\;3) employed Hankel's notation in his translation of
Book\;V into Latin. For more details see B{\l}aszczyk \cite[p.\;3,
notes 5, 11]{Bl13}.}
Such a practice is clearly contrary to Unguru's position on modern
algebraic notation; see Section~\ref{s61}. Unlike Unguru, Szab\'o
treats van der Waerden's scholarship with respect and even relies on
it (see Section~\ref{s24}).
\item
Christianidis \cite[p.\;36]{Ch18} proposes a distinction between
premodern algebra and modern algebra and argues that Diophantus can be
legitimately analyzed in terms of the former category (see
Section~\ref{s26}).
\end{enumerate}
The present article is \emph{not} a defense of the mathematician as
mathematical historian. The main thrust of this article is the
following. The postulation of an ideological polarity of historian
\emph{vs} mathematician (the latter routinely suspected of a Platonist
deviation) does more harm than good in that it obscures the only
possible basis for evaluating work in the history of mathematics,
namely competent scholarship. A mathematician who wishes to write
about a historical figure, but is insufficiently familiar with the
historical period and/or the primary documents, should be criticized
as much as a historian insufficiently familiar with the mathematics to
appreciate the fine points, and indeed the implicit aspects (as
detailed e.g., in Bl{\aa}sj\"o--Hogendijk \cite{Bl18}), of what the
historical figure actually wrote. The axioms of discontinuity and
tabula rasa and the positing of a polarity between mathematicians and
historians are of questionable value to the task of the history of
mathematics.
\section*{Acknowledgments}
We are grateful to John T. Baldwin, Viktor Bl{\aa}sj\"o, Piotr
B{\l}aszczyk, Robert Ely, Jens Erik Fenstad, Yvon Gauthier, Karel
Hrbacek, Vladimir Kanovei, and David Sherry for helpful comments on
earlier versions of the article. We thank Professor A. T. Fomenko,
academician, Moscow State University for granting permission to
reproduce his illustration in Figure~\ref{f1} in Section~\ref{s24}.
The influence of Hilton Kramer (1928--2012) is obvious.
\end{document} | math | 73,476 |
\begin{document}
\title{{\bf An FPT Algorithm Beating 2-Approximation for $k$-Cut}}
\author{ Anupam Gupta\thanks{Supported in part by NSF awards
CCF-1536002, CCF-1540541, and CCF-1617790. This work was done in
part when visiting the Simons Institute for the Theory of
Computing. } \and Euiwoong Lee\thanks{Supported by NSF award CCF-1115525, Samsung scholarship, and Simons award for graduate students in TCS.}
\and Jason Li\thanks{{\tt [email protected]} }}
\date{Computer Science Department \\ Carnegie Mellon University \\ Pittsburgh, PA 15213.}
\thispagestyle{empty}
\maketitle
\begin{abstract}
In the $k$\textsc{-Cut}\xspace problem, we are given an edge-weighted graph $G$ and an
integer $k$, and have to remove a set of edges with minimum total
weight so that $G$ has at least $k$ connected components. Prior work
on this problem gives, for all $h \in [2,k]$, a $(2-h/k)$-approximation
algorithm for $k$-cut that runs in time $n^{O(h)}$. Hence to get a
$(2 - \varepsilon)$-approximation algorithm for some absolute constant
$\varepsilon$, the best runtime using prior techniques is
$n^{O(k\varepsilon)}$. Moreover, it was recently shown that getting a
$(2 - \varepsilon)$-approximation for general $k$ is NP-hard, assuming the
Small Set Expansion Hypothesis.
If we use the size of the cut as the parameter, an FPT algorithm to
find the exact $k$\textsc{-Cut}\xspace is known, but solving the $k$\textsc{-Cut}\xspace problem exactly
is $W[1]$-hard if we parameterize only by the natural parameter of
$k$. An immediate question is: \varepsilonmph{can we approximate $k$\textsc{-Cut}\xspace better
in FPT-time, using $k$ as the parameter?}
We answer this question positively. We show that for some absolute
constant $\varepsilon > 0$, there exists a $(2 - \varepsilon)$-approximation
algorithm that runs in time $2^{O(k^6)} \cdot \widetilde{O} (n^4) $. This is the first FPT
algorithm that is parameterized only by $k$ and strictly improves the
$2$-approximation.
\varepsilonnd{abstract}
\setcounter{page}{1}
\section{Introduction}
\langlebel{sec:introduction}
We consider the $k$\textsc{-Cut}\xspace problem: given an edge-weighted graph $G =
(V,E,w)$ and an integer $k$, delete a minimum-weight set of edges so
that $G$ has at least $k$ connected components. This problem is a
natural generalization of the global min-cut problem, where the goal is
to break the graph into $k=2$ pieces.
Somewhat surprisingly, the problem has poly-time algorithms for any
constant $k$: the current best result gives an $\tilde{O}(n^{2k})$-time
deterministic algorithm~\cite{Thorup08}. On the approximation algorithms
front, several $2$-approximation algorithms are
known~\cite{SV95, NR01, RS02}. Even a trade-off result is known: for any
$h \in [1,k]$, we can essentially get a $(2-\frac{h}{k})$-approximation in
$n^{O(h)}$ time~\cite{XCY11}. Note that to get $(2-\varepsilon)$ for some
absolute constant $\varepsilon > 0$, this algorithm takes time $n^{O(\varepsilon k)}$,
which may be undesirable for large $k$. On the other hand, achieving a
$(2-\varepsilon)$-approximation is NP-hard for general $k$, assuming the
Small Set Expansion Hypothesis (SSEH)~\cite{Manurangsi17}.
What about a better \varepsilonmph{fine-grained result} when $k$ is small?
Ideally we would like a runtime of $f(k) \mathrm{poly}(n)$ so it scales better
as $k$ grows --- i.e., an FPT algorithm with parameter $k$. Sadly, the
problem is $W[1]$-hard with this parameterization~\cite{DEFPR03}. (As an
aside, we know how to compute the optimal $k$\textsc{-Cut}\xspace in time $f(|\varepsilonnsuremath{\mathsf{Opt}}\xspace|) \cdot
n^{2}$~\cite{KT11, Chitnis}, where $|\varepsilonnsuremath{\mathsf{Opt}}\xspace|$ denotes the cardinality of
the optimal $k$\textsc{-Cut}\xspace.)
The natural question suggests itself: can we give a better approximation
algorithm that is FPT in the parameter $k$?
Concretely, the question we consider in this paper is: \varepsilonmph{If we
parameterize $k$\textsc{-Cut}\xspace by $k$, can we get a $(2-\varepsilon)$-approximation for
some absolute constant $\varepsilon > 0$ in FPT time---i.e., in time
$f(k) \mathrm{poly}(n)$?} (The hard instances which show $(2-\varepsilon)$-hardness
assuming SSEH~\cite{Manurangsi17} have $k = \Omega(n)$, so such an FPT
result is not ruled out.) We answer the question positively.
\begin{theorem}[Main Theorem]
\langlebel{thm:kcut-main}
There is an absolute constant $\varepsilon > 0$ and an a
$(2-\varepsilon)$-approximation algorithm for the $k$\textsc{-Cut}\xspace problem on general
weighted graphs that runs in time $2^{O(k^6)} \cdot \tilde{O}(n^4)$.
\varepsilonnd{theorem}
Our current $\varepsilon$ satisfies $\varepsilon \geq 0.0003$ (see the calculations in
\S\ref{sec:conclusion}). We hope that our result will serve as a
proof-of-concept that we can do better than the factor of~2 in FPT$(k)$
time, and eventually lead to a deeper understanding of the trade-offs
between approximation ratios and fixed-parameter tractability for the
$k$\textsc{-Cut}\xspace problem. Indeed, our result combines ideas from approximation
algorithms and FPT, and shows that considering both settings
simultaneously can help bypass lower bounds in each individual setting,
namely the $W[1]$-hardness of an exact FPT algorithm and the
SSE-hardness of a polynomial-time $(2-\varepsilon)$-approximation.
To prove the theorem, we introduce two variants of $k$\textsc{-Cut}\xspace.
\text{Laminar}kcut{k} is a special case of $k$\textsc{-Cut}\xspace where both the graph and the
optimal solution are promised to have special properties, and \textsc{Partial VC}\xspacelong
(\textsc{Partial VC}\xspace) is a variant of $k$\textsc{-Cut}\xspace where $k - 1$ components are required to be
singletons, which served as a hard instance for both the exact
$W[1]$-hardness and the $(2 - \varepsilon)$-approximation SSE-hardness. Our
algorithm consists of three main steps where each step is modular,
depends on the previous one: an FPT-AS for \textsc{Partial VC}\xspace, an algorithm for
\text{Laminar}kcut{k}, and a reduction from $k$\textsc{-Cut}\xspace to \text{Laminar}kcut{k}. In the
following section, we give more intuition for our three steps.
\subsection{Our Techniques}
\langlebel{sec:techniques}
For this section, fix an optimal $k$-cut
${\cal S}^* = \{ S^*_1, \dots, S^*_k\}$, such that
$w(\partial{S^*_1}) \leq \dots \leq w(\partial{S^*_k})$. Let the
optimal cut value be
$\varepsilonnsuremath{\mathsf{Opt}}\xspace := w(E(S^*_1, \dots, S^*_k)) = \sum_{i=1}^k w(\partial{S^*_i}) /
2$; here $E(A_1,\cdots, A_k)$ denotes the edges that go between
different sets in this partition. The $(2 - 2/k)$-approximation iterative
greedy algorithm by Saran and Vazirani~\cite{SV95} repeatedly computes
the minimum cut in each connected component and takes the cheapest one
to increase the number of connected components by $1$. Its
generalization by Xiao et al.~\cite{XCY11} takes the minimum $h$-cut
instead of the minimum $2$-cut to achieve a $(2 - h/k)$-approximation in
time $n^{O(h)}$.
\subsubsection{Step I: \textsc{Partial VC}\xspacelong}
\langlebel{sec:overview-pvc}
The starting point for our algorithm is the $W[1]$-hardness result of
Downey et al.~\cite{DEFPR03}: the reduction from $k$-clique results in a
$k$\textsc{-Cut}\xspace instance where the optimal solution consists of $k-1$ singletons
separated from the rest of the graph. Can we approximate such instances
well? Formally, the \textsc{Partial VC}\xspace problem asks: given a edge-weighted graph, find
a set of $k-1$ vertices such that the total weight of edges hitting
these vertices is as small as possible? Extending the result of
Marx~\cite{Marx07} for the maximization version, our first conceptual
step is an FPT-AS for this problem, i.e., an algorithm that given a
$\delta >0$, runs in time $f(k,\delta)\cdot \mathrm{poly}(n)$ and gives a
$(1+\delta)$-approximation to this problem.
\subsubsection{Step II: Laminar $k$-cut}
\langlebel{sec:overview-lam}
The instances which inspire our second idea are closely related to the
hard instances above. One instance on which the greedy algorithm of
Saran and Vazirani gives a approximation no better than $2$ for large
$k$ is this: take two cliques, one with $k$ vertices and unit edge
weights, the other with $k^2$ vertices and edge weights $1/(k+1)$, so
that the weighted degree of all vertices is the same. (Pick one vertex
from each clique and identify them to get a connected graph.) The
optimal solution is to delete all edges of the small clique, at cost
$\binom{k}{2}$. But if the greedy algorithm breaks ties poorly, it will
cut out $k-1$ vertices one-by-one from the larger clique, thereby
getting a cut cost of $\approx k^2$, which is twice as large. Again we
could use \textsc{Partial VC}\xspace to approximate this instance well. But if we replace each
vertex of the above instance itself by a clique of high weight edges,
then picking out single vertices obviously does not work. Moreover, one
can construct recursive and ``robust'' versions of such instances where
we need to search for the ``right'' (near-)$k$-clique to break
up. Indeed, these instances suggest the use of dynamic programming (DP),
but what structure should we use DP on?
One feature of such ``hard'' instances is that the optimal $k$\textsc{-Cut}\xspace
${\cal S}^* = \{S_1^*, \ldots, S_k^*\}$ is composed of near-min-cuts in the
graph. Moreover, no two of these near-min-cuts cross each other. We now
define the \text{Laminar}kcut{k} problem: find a $k$\textsc{-Cut}\xspace on an instance where
none of the $(1+\varepsilon)$-min-cuts of the graph cross each other, and where
each of the cut values $w(\partial{S_i^*})$ for $i = 1,\ldots, k-1$ are
at most $(1+\varepsilon)$ times the min-cut. Because of this laminarity (i.e.,
non-crossing nature) of the near-min-cuts, we can represent the
near-min-cuts of the graph using a tree $\mathcal T$, where the nodes of $G$
sit on nodes of the tree, and edges of $\mathcal T$ represent the near-min-cuts
of $G$. Rooting the tree appropriately, the problem reduces to
``marking'' $k-1$ incomparable tree nodes and take the near-min-cuts
given by their parent edges, so that the fewest edges in $G$ are
cut. Since all the cuts represented by $\mathcal T$ are near-min-cuts and
almost of the same size, it suffices to mark $k-1$ incomparable nodes to
maximize the number of edges in $G$ both of whose endpoints lie below a
marked node. We call such edges \varepsilonmph{saved} edges. \agnote{Any
figures?} In order to get a $(2-\varepsilon)$-approximation for \text{Laminar}kcut{k},
it suffices to save $\approx \varepsilon k \mathsf{Mincut}$ weight of edges.
Note that if $\mathcal T$ is a star with $n$ leaves and each vertex in $G$ maps
to a distinct leaf, this is precisely the \textsc{Partial VC}\xspace problem, so we do not
hope to find the optimal solution (using dynamic programming, say).
Moreover, extending the FPT-AS for \textsc{Partial VC}\xspace to this more general setting
does not seem directly possible, so we take a different approach. We
call a node an \varepsilonmph{anchor} if has some $s$ children which when marked
would save $\approx \varepsilon s \mathsf{Mincut}$ weight. We take the following
``win-win'' approach: if there were $\Omega(k)$ anchors that were
incomparable, we could choose a suitable subset of $k$ of their children
to save $\approx \varepsilon k \mathsf{Mincut}$ weight. And if there were not, then all
these anchors must lie within a subtree of $\mathcal T$ with at most $k$
leaves. We can then break this subtree into $2k$ paths and guess which
paths contain anchors which are parents of the optimal solution. For
each such guess we show how to use \textsc{Partial VC}\xspace to solve the problem and save a
large weight of edges. Finally how to identify these anchors? Indeed,
since all the mincuts are almost the same, finding an anchor again
involves solving the \textsc{Partial VC}\xspace problem!
\subsubsection{Step III: Reducing $k$\textsc{-Cut}\xspace to \text{Laminar}kcut{k}}
\langlebel{sec:overview-redn}
\begin{wrapfigure}{L}{0.38\textwidth}
\centering
\includegraphics[width=0.35\textwidth]{conform}
\caption{\langlebel{fig:conform} The blue set on the right,
formed by $S_5^* \cup S_7^* \cup S_{11}^*$, conforms to the
algorithm's partition ${\cal S}$ on the left.}
\varepsilonnd{wrapfigure}
We now reduce the general $k$\textsc{-Cut}\xspace problem to \text{Laminar}kcut{k}. This
reduction is again based on observations about the
graph structure in cases where the iterative greedy algorithms do not
get a $(2 - \varepsilon$)-approximation. Suppose ${\cal S} = \{ S_1, \dots,
S_{k'} \}$ be the connected components of $G$ at some point of an
iterative algorithm ($k' \leq k$). For a subset $\varepsilonmptyset \neq U
\subsetneq V$, we say that $U$ {\varepsilonm conforms} to partition ${\cal S}$ if
there exists a subset $J \subsetneq [k']$ of parts such that $U =
\cup_{j \in J} S_j$. One simple but crucial observation is the
following: if there exists a subset $\varepsilonmptyset \neq I \subsetneq [k]$ of
indices such that $\cup_{i\in I} S^*_i$ conforms to ${\cal S}$
(i.e., $\cup_{i\in I} S^*_i = \cup_{j \in J} S_j$), we can
``guess'' $J$ to partition $V$ into the two parts $\cup_{i \in I} S^*_i$
and $\cup_{i \notin I} S^*_i$. Since the edges between these two parts
belong to the optimal cut and each of them is strictly smaller than $V$,
we can recursively work on each part without any loss.
Moreover, the number of choices for $J$ is at most
$2^{k'}$ and each guess produces one more connected component, so the
total running time can be bounded by $f(k)$ times the running time of
the rest of the algorithm, for some function $f(\cdot)$. Therefore, we
can focus on the case where none of $\cup_{i \in I} S^*_i$ conforms to
the algorithm's partition ${\cal S}$ at any point during the algorithm's
execution.
\begin{wrapfigure}{R}{0.38\textwidth}
\centering
\includegraphics[width=0.35\textwidth]{histogram}
\caption{\langlebel{fig:histo} The blue curve shows cut sizes
for algorithm's cuts, red curve shows $w(\partial{S^*_i})$
values. The blue area (and in fact all the area below
$w(\partial{S_1^*})$ and above the algorithm's curve) makes the first
inequality loose. The grey area (and in fact all the area above
$w(\partial{S_1^*})$ and below OPT's curve) makes the second
inequality loose.}
\varepsilonnd{wrapfigure}
Now consider the iterative min-cut algorithm of Saran and Vazirani, and
let $c_i$ be the cost of the min cut in the $i^{th}$ iteration ($1 \leq
i \leq k - 1$). By our above assumption about non-conformity, none of
$\cup_{i \in I} S^*_i$, and in particular the subset $S^*_1$, conform to
the current components. This implies that deleting the remaining edges
in $\partial S^*_1$ is a valid cut that increases the number of
connected components by at least $1$, so $c_i \leq
w(\partial{S^*_1})$. Then we have the following chain of inequalities:
\[
\sum_{i=1}^{k-1} c_i \leq k \cdot w(\partial{S^*_1}) \leq \sum_{i=1}^k w(\partial{S^*_i}) = 2\varepsilonnsuremath{\mathsf{Opt}}\xspace.
\]
If the iterative min-cut algorithm could not get a $(2 -
\varepsilon)$-approximation, the two inequalities above must be essentially
tight. Hence almost all our costs $c_i$ must be close to
$w(\partial{S^*_1})$ and almost all $w(\partial{S^*_i})$ must be close
to $w(\partial{S^*_1})$.
Slightly more formally, let $\mathfrak{a} \in [k]$ be the smallest integer such
that
$c_{\mathfrak{a}} \gtrsim w(\partial{S^*_1})$
---so that the first $\mathfrak{a} - 1$ cuts are ones where
we pay ``much'' less than $\partial{S^*_1}$ and make the first
inequality loose. And let $\mathfrak{b} \in [k]$ be the smallest number such that
$w(\partial{S^*_{\mathfrak{b}}}) \gtrsim w(\partial{S^*_1})$
--- so that the last
$k - \mathfrak{b}$ cuts in OPT are much larger than $\partial{S^*_1}$ and make
the second inequality loose. Then if the iterative min-cut algorithm is
no better than a $2$-approximation, we can imagine that $\mathfrak{a} = o(k)$ and
$\mathfrak{b} \geq k - o(k)$.
For simplicity, let us assume that $\mathfrak{a} = 1$ and $\mathfrak{b} = k$ here.
Indeed, instead of just considering min-cuts, suppose we also consider
min-4-cuts, and take the one with better edges cut per number of new
components. The arguments of the previous paragraph still hold, so
$\mathfrak{a} = 1$ implies that
the best min-cuts and best min-4-way cuts (divided by 3) are roughly
at least $w(\partial{S^*_1})$ in the original $G$.
Since the min-cut is also at most $w(\partial{S^*_1})$,
the weight of the min-cut is
roughly $w(\partial{S^*_1})$ and none of the near-min-cuts
cross (else we would get a good 4-way cut). I.e., the
near-min-cuts in the graph form a laminar family.
Together with the fact that $\partial{S^*_1}, \dots, \partial{S^*_{k - 1}}$ are near-min-cuts (we assumed $\mathfrak{b} = k$),
this is precisely an instance of \text{Laminar}kcut{k}, which completes
the proof!
\iffalse
\alert{I would stop here and leave the rest of the details to the main
body, maybe just move on to explaining laminar-cut.}
\varepsilonlnote{I am fine with this except that the another promise of Laminar ensure that all but one optimal components are near min cuts. }
Let $S^*_{\geq \mathfrak{b}} := \cup_{i \geq \mathfrak{b}} S^*_{i}$ so that in the
$\mathfrak{b}$-cut $\{ S^*_1, \dots, S^*_{\mathfrak{b} - 1}, S^*_{\geq \mathfrak{b}} \}$, all but
one component satisfy that the weight of their boundary is close to
$w(\partial{S^*_1})$. Another application of the conformity argument
ensures that among the current components $S_1, \dots, S_{\mathfrak{a}}$,
\varepsilonlnote{this is another cute idea... should we not explain it here? If
we do, we definitely need a figure explaining what happens when two
$S_j$'s cross one $S^*_i$. } \agnote{I would say we postpone this idea
for now, we might lose people.} there is exactly one component (say
$S_1$) intersecting every $S^*_1, \dots, S^*_{\mathfrak{b} - 1}, S^*_{\geq \mathfrak{b}}$
and all the others are strictly contained in one $S^*_i$ (or $S^*_{\geq
\mathfrak{b}}$). In particular, note that $S_1$ intersects both $S^*_1$ and $V
\setminus S^*_1$, so $\mathsf{Mincut}(G[S_1]) \leq w(\partial{S^*_1})$.
Then we can see that $G[S_1]$ satisfies the promises of
$\text{Laminar}cut{\mathfrak{b}}{\varepsilonilon'}$: there exists a $\mathfrak{b}$-cut where all but
one component are close to the min cut and no two near-min cuts cross.
We guess $S_1$, run our algorithm for $\text{Laminar}cut{\mathfrak{b}}{\varepsilonilon'}$ for
$G[S_1]$ to get $\mathfrak{b} - 1$ more components. It results in $\mathfrak{a} + \mathfrak{b} -
1$ components, and if it is still less than $k$, we finally perform the
iterative greedy algorithm (here the min $2$-cut suffices) to get $k -
\mathfrak{a} - \mathfrak{b} + 1$ more components. The conformity still ensures that we pay
at most $w(\partial{S^*_1})$ in each iteration.
The total cost is the sum of (1) the cost to get the first $\mathfrak{a}$ components (2) the cost of $\text{Laminar}cut{\mathfrak{b}}{\varepsilonilon_1}$, and (3) the cost to get the final $k - \mathfrak{a} - \mathfrak{b} + 1$ components.
Since $\mathfrak{a}$ and $k - \mathfrak{b}$ are very small compared to $k$, (1) and (3) do not contribute much, so we can beat the $2$-approximation if we do for $\text{Laminar}cut{\mathfrak{b}}{\varepsilonilon_1}$.
\fi
\paragraph{Roadmap.} After some related work and preliminaries, we first
present the details of the reduction from $k$\textsc{-Cut}\xspace to \text{Laminar}kcut{k} in
Section~\ref{sec:reduction}. Then in Section~\ref{sec:laminar} we give
the algorithm for \text{Laminar}kcut{k} assuming an algorithm for \textsc{Partial VC}\xspace. Finally
we give our FPT-AS for \textsc{Partial VC}\xspace in Section~\ref{sec:partial-vc}.
\subsection{Other Related Work}
\langlebel{sec:related}
The $k$\textsc{-Cut}\xspace problem has been widely studied. Goldschmidt and Hochbaum gave
an $O(n^{(1/2- o(1))k^2})$-time algorithm~\cite{GH94}; they also showed
that the problem is NP-hard when $k$ is part of the input. Karger and
Stein improved this to an $O(n^{(2-o(1))k})$-time randomized Monte-Carlo
algorithm using the idea of random edge-contractions~\cite{KS96}.
After Kamidoi et al.~\cite{KYN06} gave an $O(n^{4k + o(1)})$-time
deterministic algorithm based on divide-and-conquer,
Thorup gave an $\tilde{O}(n^{2k})$-time deterministic algorithm based on
tree packings~\cite{Thorup08}.
Small values of $k \in [2, 6]$ also have been separately studied~\cite{NI92, HO92, BG97, Karger00, NI00, NKI00, Levine00}.
On the approximation algorithms front, a $2(1-1/k)$-approximation was
given by Saran and Vazirani~\cite{SV95}. Naor and Rabani~\cite{NR01},
and Ravi and Sinha~\cite{RS02} later gave $2$-approximation algorithms
using tree packing and network strength respectively. Xiao et
al.~\cite{XCY11} completed the work of Kapoor~\cite{Kapoor96} and Zhao
et al.~\cite{ZNI01} to generalize Saran and Vazirani to essentially give
an $(2 - h/k)$-approximation in time $n^{O(h)}$. Very recently,
Manurangsi~\cite{Manurangsi17} showed that for any $\varepsilon > 0$, it is
NP-hard to achieve a $(2 - \varepsilon)$-approximation algorithm in time
$\mathrm{poly}(n,k)$ assuming the Small Set Expansion Hypothesis.
\varepsilonmph{FPT algorithms:} Kawarabayashi and Thorup give an
$f(\varepsilonnsuremath{\mathsf{Opt}}\xspace) \cdot n^{2}$-time algorithm~\cite{KT11} for unweighted
graphs. Chitnis et al.~\cite{Chitnis} used a randomized color-coding
idea to give a better runtime, and to extend the algorithm to weighted
graphs. In both cases, the FPT algorithm is parameterized by the
cardinality of edges in the optimal $k$\textsc{-Cut}\xspace, not by $k$. For a
comprehensive treatment of FPT algorithms, see the excellent
book~\cite{FPT-book}, and for a survey on approximation and FPT
algorithms, see~\cite{Marx07}.
\varepsilonmph{Multiway Cut:} A problem very similar to $k$\textsc{-Cut}\xspace is the
\textsc{Multiway Cut} problem, where we are given $k$ terminals and want
to disconnect the graph into at least $k$ pieces such that all terminals
lie in distinct components. However, this problem behaves quite
differently: it is NP-hard even for $k=3$ (and hence an $n^{f(k)}$
algorithm is ruled out); on the other hand several algorithms are
known to approximate it to factors much smaller than~$2$ (see,
e.g.,~\cite{BuchbinderSW17} and references therein). FPT algorithms
parameterized by the size of $\varepsilonnsuremath{\mathsf{Opt}}\xspace$ are also known; see~\cite{CaoCF14}
for the best result currently known.
\section{Notation and Preliminaries}
\langlebel{sec:prelims}
For a graph $G = (V,E)$, and a subset $S \subseteq V$, we use $G[S]$ to
denote the subgraph induced by the vertex set $S$. For a collection of
disjoint sets $S_1, S_2, \ldots, S_t$, let $E(S_1, \ldots, S_t)$ be the
set of edges with endpoints in some $S_i, S_j$ for $i \neq j$. Let
$\partial S = E(S, V \setminus S)$. We say two cuts $(A, V\setminus A)$
and $(B,V\setminus B)$ \varepsilonmph{cross} if none of the four sets $A
\setminus B, B \setminus A, A \cap B$, and $V \setminus (A \cup B)$ is
empty.
$\mathsf{Mincut}$ and $\text{\sf{Min-4-cut}}$ denote the weight of the min-2-cut
and the min-4-cut respectively.
A cut $(A, V \setminus A)$ is called $(1 + \varepsilon)$-mincut if
$w(A, V \setminus A) \leq (1 + \varepsilon) \mathsf{Mincut}$.
\begin{restatable}[\textsc{Laminar $k$-Cut}$(\varepsilon_1)$]{definition}{LamDef}
\langlebel{def:laminarcut}
The input is a graph $G = (V,E)$ with edge weights, and two parameters
$k$ and $\varepsilon_1$, satisfying two promises: (i)~no two $(1+\varepsilon_1)$-mincuts
cross each other, and (ii)~there exists a $k$-cut ${\cal S}' = \{S_1',
\ldots, S_k'\}$ in $G$ with $w(\partial(S_i')) \le (1+\varepsilon_1)\mathsf{Mincut}(G)$
for all $i \in [1,k-1]$. Find a $k$-cut with the total weight.
The approximation ratio is defined as the ratio of
the weight of the returned cut to the weight of the $k$\textsc{-Cut}\xspace ${\cal S}'$
(which can be possibly less than $1$).
\varepsilonnd{restatable}
\begin{definition}[\textsc{Partial VC}\xspacelong]
\langlebel{def:pvc}
Given a graph $G = (V,E)$ with edge and vertex weights, and an integer
$k$, find a vertex set $S \subseteq V$ with $|S| = k$ nodes, minimizing the
weight of the edges hitting the set $S$ plus the weight of all
vertices in $S$.
\varepsilonnd{definition}
\section{Reduction to $\text{Laminar}cut{k}{\varepsilon_1}$}
\langlebel{sec:reduction}
In this section we give our reduction from $k$\textsc{-Cut}\xspace to
$\text{Laminar}cut{k}{\varepsilon_1}$, showing that if we can get a better-than-2
approximation for the latter, we can beat the factor of two for the
general $k$\textsc{-Cut}\xspace problem too. We assume the reader is familiar with the
overview in Section~\ref{sec:overview-redn}. Formally, the main theorem
is the following.
\begin{theorem}
\langlebel{thm:reduction1}
Suppose there exists a $(2 - \varepsilon_2)$-approximation algorithm for
$\text{Laminar}cut{k}{\varepsilon_1}$ for some $\varepsilon_1 \in (0, 1/4)$ and
$\varepsilon_2 \in (0, 1)$ that runs in time $f(k) \cdot g(n)$. Then
there exists a $(2 - \varepsilon_3)$-approximation algorithm for $k$\textsc{-Cut}\xspace
that runs in time $2^{O(k^2 \log k)} \cdot f(k) \cdot (n^4 \log^3 n + g(n))$ for some constant
$\varepsilon_3 > 0$.
\varepsilonnd{theorem}
\begin{algorithm}
\caption{$\text{Main}(G = (V, E, w), k)$}
\langlebel{alg:main}
\begin{algorithmic}[1]
\State $k' = 1$, $S_1 \gets V$
\While {$k' < k$ }
\mathbb For {$\boldsymbol{\mathrm{r}} \in [k]^{k'}$ } \langlebel{line:start-of-check} \Comment {Further partition each $S_i$ into $r_i$ components by Laminar}
\State $|\boldsymbol{\mathrm{r}}| \gets \sum_{j=1}^{k'} r_j$; $\{ C_1, \dots,
C_{|\boldsymbol{\mathrm{r}}|} \} \gets \cup_{i \in [k']}
\text{Laminar}(\inducedG{S_i}, r_i)$.
\If{$|\boldsymbol{\mathrm{r}}| \geq k$} $C_k \gets C_k \cup \dots \cup C_{|\boldsymbol{\mathrm{r}}|}$
\Else \State $\{C_1, \dots, C_{k}\} \gets \text{Complete}(G, k, C_1, \dots, C_{|\boldsymbol{\mathrm{r}}|})$ \langlebel{line:complete}
\EndIf
\State \mathbb Record($\text{Guess}(\{C_1, \dots, C_k \})$)
\EndFor \langlebel{line:end-of-check}
\State {} \Comment {Split some $S_i$ by a mincut or a min-4-cut}
\If{$k' > k - 3$ or $\min_{i \in [k']} \mathsf{Mincut} (\inducedG{S_{i}}) \leq
\min_{i \in [k']} \text{\sf{Min-4-cut}} (\inducedG{S_{i}}) / 3$} \langlebel{line:start-extend}
\State $i \gets \min_{i} \mathsf{Mincut} (\inducedG{S_{i}})$; $\{ T_1, T_2 \} \gets \text{Mincut}(\inducedG{S_{i}})$
\State $S_i \gets T_1$; $S_{k' + 1} \gets T_2$; $c_{k'} \gets \mathsf{Mincut} (\inducedG{S_{i}})$; $k' \gets k' + 1$
\Else
\State $i \gets \arg\min_{i} \text{\sf{Min-4-cut}} (\inducedG{S_{i}})$; $\{ T_1, \dots, T_4 \} \gets \text{Min-4-cut}(\inducedG{S_{i}})$;
$S_{i} \gets T_1$
\State $S_{k' + 1}, S_{k' + 2}, S_{k' + 3} \gets T_2, T_3, T_4$;
$c_{k'}, c_{k' + 1}, c_{k' + 2} \gets \text{\sf{Min-4-cut}} (\inducedG{S_{i}}) / 3$;
$k' \gets k' + 3$
\EndIf \langlebel{line:end-extend}
\EndWhile
\State let ${\cal S} = \{S_1, \ldots, S_k\}$ be the final reference $k$-partition.
\State \mathbb Record($\text{Guess}(G, k, {\cal S})$) \langlebel{line:lastupdate}
\State Return the best recorded $k$-partition.
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
\begin{algorithm}
\caption{$\text{Complete}(G = (V, E, w), k, {\cal C} = \{C_1, \ldots, C_\varepsilonll\})$}
\langlebel{alg:complete}
\begin{algorithmic}[1]
\While {$\varepsilonll < k$}
\State $i \gets \min_{i \in [\varepsilonll]} \mathsf{Mincut}(\inducedG{C_i})$; $T_1, T_2 \gets \text{Mincut}(\inducedG{C_i})$
\State $C_i \gets T_1$; $C_{\varepsilonll + 1} \gets T_2$; $\varepsilonll \gets \varepsilonll + 1$
\EndWhile
\State Return ${\cal C} := \{C_1, \dots, C_k\}$.
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
\begin{algorithm}
\caption{$\text{Guess}(G = (V, E, w), k, {\cal C} = \{C_1, \dots, C_k\})$}
\langlebel{alg:guess}
\begin{algorithmic}[1]
\State \mathbb Record($C_1, \dots, C_k$) \Comment{Returned
partition no worse than starting partition}
\mathbb For {$\varepsilonmptyset \neq J \subsetneq [k]$ }
\mathbb For {$k' = 1, 2, \dots, k - 1$ }
\State $L \gets \cup_{j \in J} C_j$; $R \gets V
\setminus L$ \Comment {Divide $S_i$ into two groups,
take union of each group}
\State $D_1, \dots, D_{k'} \gets \text{Main}(\inducedG{L},
k')$ \Comment{and recurse}
\State $D_{k'+1}, \dots, D_k \gets \text{Main}(\inducedG{R}, k - k')$
\State \mathbb Record($D_1, \dots, D_k$)
\EndFor
\EndFor
\State Return the best recorded $k$-partition among all these guesses.
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
The main algorithm is shown in Algorithm~\ref{alg:main} (``\text{Main}'').
It maintains a ``reference'' partition ${\cal S}$, which is initially the
trivial partition where all vertices are in the same part. At each
point, it guesses how many pieces each part $S_i$ of this reference partition
${\cal S}$ should be split into using the ``Laminar'' procedure, and then
extends this to a $k$-cut using greedy cuts if necessary
(Lines~\ref{line:start-of-check}--\ref{line:end-of-check}).
It then
extends the reference partition by either taking the best min-cut or the
best min-4-cut among all the parts
(Lines~\ref{line:start-extend}--\ref{line:end-extend}).
Every time it
has a $k$-partition, it guesses (using ``Guess'') if the union of some
of the parts equals some part of the optimal partition, and uses that to
try get a better partition.
If one of the guesses is right, we strictly increase the number of
connected components by deleting edges in the optimal $k$-cut, so we can
recursively solve the two smaller parts. If none of our guesses was
right during the algorithm, our analysis in
Section~\ref{subsec:approx_factor} shows that there exist values of $k',
\boldsymbol{\mathrm{r}}$ such that ${\cal C} = \{C_1, \dots, C_k\}$ in
Line~\ref{line:complete}, obtained from the reference partition ${\cal S} =
\{ S_1, \dots, S_{k'} \}$ by running Laminar($G[S_{i}], r_i$) for each $i \in [k']$
and using Complete if necessary to get $k$ components, beats the $2$-approximation. Finally, a couple words about
each of the subroutines.
\begin{itemize}
\item Mincut$(G = (V, E, w))$ (resp.\ Min-4-cut$(G)$)
returns the minimum $2$-cut (resp.\ $4$-cut) as a partition of $V$
into $2$ (resp. $4$) subsets.
\item The subroutine ``Laminar'' returns a $(2-\varepsilon_2)$-approximation
for \text{Laminar}cut{$k$}{$\varepsilon_1$}, using the algorithm from
Theorem~\ref{thm:laminar}. Recall the definition of the problem in Definition~\ref{def:laminarcut}.
\item The operation ``\mathbb Record(${\cal P}$)'' in \text{Guess}\ and \text{Main}\ takes a
$k$-partition ${\cal P}$ and compares the weight of edges crossing this
partition to the least-weight $k$-partition recorded thus far (within
the current recursive call). If the current partition has less weight,
it updates the best partition accordingly.
\item Algorithm~\ref{alg:complete}(``\text{Complete}'') is a simple algorithm
that given an $\varepsilonll$-partition ${\cal P}$ for some $\varepsilonll \leq k$, outputs a
$k$-partition by iteratively taking the mincut in the current graph.
\item Algorithm~\ref{alg:guess}(``\text{Guess}''), when given an
$\varepsilonll$-partition ${\cal P}$ ``guesses'' if the vertices belonging to some
parts of this partition $\{ S_j \}_{j \in J}$ coincide with the union
of some $k'$ parts of the optimal partition. If so, we have made
tangible progress: it recursively finds a small $k'$-cut in the graph
induced by $\cup_{j \in J} S_j$, and a small $k-k'$ cut in the
remaining graph. It returns the best of all these guesses.
\varepsilonnd{itemize}
\subsection{The Approximation Factor}
\langlebel{subsec:approx_factor}
\begin{lemma}[Approximation Factor]
\langlebel{lem:apx-main}
$\text{Main}(G, k)$ achieves a $(2 - \varepsilon_3)$ approximation for some
$\varepsilon_3 > 0$ that depends on $\varepsilon_1, \varepsilon_2$ in
Theorem~\ref{thm:reduction1}.
\varepsilonnd{lemma}
\begin{proof}
We prove the lemma by induction on $k$. The value of $\varepsilon_3$ will be
determined later. The base case $k = 1$ is trivial. Fix some value
of $k$, and a graph $G$. Let ${\cal S} = \{S_1, \dots, S_k\}$ be the
final reference partition generated by the execution of $\text{Main}(G, k)$,
and let $c_1, \dots, c_{k - 1}$ be the values associated with it. From the
definition of the $c_i$'s in Procedure~\text{Main}, $\sum_{i=1}^{k - 1} c_i =
w(E(S_1, \dots, S_k))$. The $k$-partition returned by $\text{Main}(G, k)$ is
no worse than this partition ${\cal S}$ (because of the update on
line~\ref{line:lastupdate}), and hence has cost at most $\sum_{i=1}^{k-1}
c_i = w(E(S_1, \dots, S_k))$. Let us fix an optimal $k$-cut ${\cal S}^*
= \{ S^*_1, \dots, S^*_k\}$, and let $w(\partial{S^*_1}) \leq \dots
\leq w(\partial{S^*_k})$. Let $\varepsilonnsuremath{\mathsf{Opt}}\xspace := w(E(S^*_1, \dots, S^*_k)) =
\sum_{i=1}^k w(\partial{S^*_i}) / 2$.
\begin{definition}[Conformity]
\langlebel{def:conform}
For a subset $\varepsilonmptyset \neq U \subsetneq V$, we say that $U$ {\varepsilonm
conforms} to partition ${\cal S}$ if there exists a subset $J
\subsetneq [k]$ of parts such that $U = \cup_{j \in J} S_j$. (See Figure~\ref{fig:conform}.)
\varepsilonnd{definition}
The following claim shows that if there exists a subset $\varepsilonmptyset
\neq I \subsetneq [k]$ of indices such that $\cup_{i\in I} S^*_i$
conforms to ${\cal S}$, the induction hypothesis guarantees a $(2 -
\varepsilon_3)$-approximation.
\begin{claim}
Suppose there exists a subset $\varepsilonmptyset \neq I \subsetneq [k]$ such
that $\cup_{i \in I} S_i^*$ conforms to ${\cal S}$. Then $\text{Main}(G, k)$
achieves a $(2 - \varepsilon_3)$-approximation.
\langlebel{claim:good}
\varepsilonnd{claim}
\begin{proof}
Since $S^*_I := \cup_{i \in I} S^*_i$ conforms to ${\cal S}$, during
the run of $\text{Guess}(G, k, {\cal S})$ it will record the $k$-partition
$(\text{Main}(\inducedG{S^*_I}, |I|), \text{Main}(\inducedG{V \setminus S^*_I},
k - |I|) )$, and hence finally output a $k$-partition which cuts no
more edges than this starting partition. By the induction
hypothesis, $\text{Main}(\inducedG{S^*_I}, |I|)$ gives a $|I|$-cut of
$\inducedG{S^*_I}$ whose cost is at most $(2 - \varepsilon_3)$ times
$w(E(S^*_i)_{i \in I})$, and $\text{Main}(\inducedG{V \setminus S^*_I}, k
- |I|)$ outputs a $(k - |I|)$-cut of $\inducedG{V\setminus S^*_I}$
of cost at most $(2 - \varepsilon_3)$ times $w(E(S^*_i)_{i \notin
I})$. Thus, the value of the best $k$-partition returned by
$\text{Main}(G, k)$ is at most
\begin{align*}
& w(E(S^*_I, V \setminus S^*_I)) + (2 - \varepsilon_3) \left(
w(E(S^*_i)_{i \in I}) + w(E(S^*_i)_{i \notin I}) \right) \\
\leq & \ (2 - \varepsilon_3) w(E(S^*_1, \dots, S^*_k)) = (2 -
\varepsilon_3) \mathsf{Opt}. \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
Therefore, to prove Lemma~\ref{lem:apx-main}, it suffices to assume
that no collection of parts in \varepsilonnsuremath{\mathsf{Opt}}\xspace conforms to our partition at any
point in the algorithm. I.e.,
\begin{leftbar}
\As{1}: for every subset $\varepsilonmptyset \neq I \subsetneq [k]$, $\cup_{i
\in I} S^*_i$ does not conform to ${\cal S} = \{ S_1, \dots, S_k\}$.
\varepsilonnd{leftbar}
Next, we study how $\varepsilonnsuremath{\mathsf{Opt}}\xspace$ is related to $w(\partial{S_1^*})$. Note
that $\varepsilonnsuremath{\mathsf{Opt}}\xspace \geq (k/2) \cdot w(\partial{S_1^*})$. The next claim shows
that we can strictly improve the $2$-approximation if $\varepsilonnsuremath{\mathsf{Opt}}\xspace$ is even
slightly bigger than that.
\begin{claim}
For every $i = 1, \dots, k-1$, $c_i \leq w(\partial{S^*_1})$. Moreover,
if $\varepsilonnsuremath{\mathsf{Opt}}\xspace \geq (k - 1)w(\partial{S_1^*}) / (2 - \varepsilon_3)$, $\text{Main}(G,
k)$ achieves a $(2 - \varepsilon_3)$-approximation.
\langlebel{claim:notgood}
\varepsilonnd{claim}
\begin{proof}
Consider the beginning of an arbitrary iteration of the while loop
of $\text{Main}(G, k)$. Let $k'$ and ${\cal S}' = \{ S_1, \dots, S_{k'} \}$
be the values at that iteration. By \As{1}, set $S_1^*$ does not
conform to ${\cal S}'$ (because ${\cal S}'$ only gets subdivided as the
algorithm proceeds, and $S_1^*$ does not conform to the final partition
${\cal S}$). So there exists some $i \in [k']$ such that $S_i$
intersects both $S^*_1$ and $V \setminus S^*_1$. If we consider
$\inducedG{S_i}$ and its mincut,
\[
\mathsf{Mincut}(\inducedG{S_i}) \leq
w(E(S_i \cap S^*_1, S_i \setminus S^*_1))
\leq w(\partial{S^*_1}).
\]
Now the new $c_j$ values created in this iteration of the while loop
are at most the smallest mincut value, so we have that each $c_j
\leq w(\partial{S_1^*})$. Therefore,
\[
w(E(S_1, \dots, S_k)) = \sum_{i=1}^{k - 1} c_i \leq (k - 1)\cdot
w(\partial{S_1^*}),
\]
and $\text{Main}(G, k)$ achieves a $(2 - \varepsilon_3)$-approximation if $(k
- 1) w(\partial{S^*_1}) \leq (2 - \varepsilon_3) \varepsilonnsuremath{\mathsf{Opt}}\xspace$.
\varepsilonnd{proof}
Consequently, it suffices to additionally assume that $\varepsilonnsuremath{\mathsf{Opt}}\xspace$ is
close to $(\nicefrac{k}{2}) \, w(\partial{S^*_1})$. Formally,
\begin{leftbar}
\As{2}: $ \varepsilonnsuremath{\mathsf{Opt}}\xspace < w(\partial{S^*_1}) \cdot \frac{k - 1}{2 - \varepsilon_3} $.
\varepsilonnd{leftbar}
Recall that $\varepsilon_1, \varepsilon_2 > 0$ are the parameters such that
there is a $(2 - \varepsilon_2)$-approximation algorithm for
$\text{Laminar}cut{k}{\varepsilon_1}$. Let $\mathfrak{a} \in [k]$ be the smallest
integer such that $c_{\mathfrak{a}} > w(\partial{S^*_1}) (1 -
\nicefrac{\varepsilon_1}{3})$ (set $\mathfrak{a} = k$ if there is no such integer).
(See Figure~\ref{fig:histo}.)
In other words, $\mathfrak{a}$ is the value of $k'$ in the while loop of
$\text{Main}(G, k)$ when both $\min_i \mathsf{Mincut}(\inducedG{S_i})$ and $\min_i
\text{\sf{Min-4-cut}}(\inducedG{S_i}) / 3$ are bigger than $w(\partial{S^*_1}) (1 -
\nicefrac{\varepsilon_1}{3})$ for the first time. Let $\varepsilon_4 > 0$ be a constant
satisfying
\begin{equation}
\langlebel{eq:para_1}
(2/3) \cdot \varepsilon_1 \varepsilon_4 \geq \varepsilon_3.
\varepsilonnd{equation}
The next claim shows that we are done if $\mathfrak{a}$ is large.
\begin{claim}
If $\mathfrak{a} \geq \varepsilon_4 k$, $\text{Main}(G, k)$ achieves a $(2 -
\varepsilon_3)$-approximation.
\langlebel{clm:left_tail}
\varepsilonnd{claim}
\begin{proof}
If $\mathfrak{a} \geq \varepsilon_4 k$, we have
\begin{align*}
\sum_{i=1}^{k-1} c_i &\leq
(\mathfrak{a} - 1) (1 - \nicefrac{\varepsilon_1}{3})\cdot w(\partial{S^*_1}) +
(k - \mathfrak{a})\cdot w(\partial{S^*_1}) \\
& \leq k \cdot w(\partial{S^*_1})\cdot (1 - \nicefrac{\varepsilon_1 \varepsilon_4}{3})
\leq (2 - (\nicefrac23) \varepsilon_1 \varepsilon_4) \varepsilonnsuremath{\mathsf{Opt}}\xspace \leq (2 - \varepsilon_3)
\varepsilonnsuremath{\mathsf{Opt}}\xspace. \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
Thus, we can assume that our algorithm finds very few cuts appreciably
smaller than $w(\partial{S^*_1})$.
\begin{leftbar}
\As{3}: $\mathfrak{a} < \varepsilon_4 k$.
\varepsilonnd{leftbar}
Let $\mathfrak{b} \in [k]$ be the smallest number such that
$w(\partial{S^*_{\mathfrak{b}}}) > w(\partial{S^*_1}) (1 +
\nicefrac{\varepsilon_1}{3})$; let it be $k$ if there is no such
number. (Again, see Figure~\ref{fig:histo}.) Observe that $\mathfrak{a}$ is
defined based on our algorithm, whereas $\mathfrak{b}$ is defined based on the
optimal solution. Let $\varepsilon_5 > 0$ be a constant satisfying:
\begin{equation}
\langlebel{eq:para_2}
\frac{1}{2 - \varepsilon_3} \leq \frac{1 + \nf{\varepsilon_1 \varepsilon_5}{3}}{2}
~~~\Leftrightarrow~~~ (1 + \nf{\varepsilon_1 \varepsilon_5}{3})(2 -
\varepsilon_3) \geq 2.
\varepsilonnd{equation}
The next claim shows that $\mathfrak{b}$ should be close to $k$.
\begin{claim}
$\mathfrak{b} \geq (1 - \varepsilon_5)k$.
\varepsilonnd{claim}
\begin{proof}
Suppose that $\mathfrak{b} <(1 - \varepsilon_5) k$. We have
\begin{align*}
& \ \frac{ k \cdot w(\partial{S^*_1}) }{2 - \varepsilon_3} \stackrel{\As{2}}{>} \varepsilonnsuremath{\mathsf{Opt}}\xspace = \frac{1}{2} \sum_{i = 1}^k w(\partial{S^*_i}) \\
\geq& \ \frac{w(\partial{S^*_1})}{2} \left( (1 - \varepsilon_5)k +
\varepsilon_5 k (1 + \varepsilon_1 / 3) \right)
=
\frac{k\cdot w( \partial{S^*_1} )}{2} \left( 1 + \nf{\varepsilon_1 \varepsilon_5}{3} \right),
\varepsilonnd{align*}
which contradicts~\varepsilonqref{eq:para_2}.
\varepsilonnd{proof}
Therefore, we can also assume that very few cuts in \varepsilonnsuremath{\mathsf{Opt}}\xspace
are appreciably larger than $w( \partial{S^*_1})$.
\begin{leftbar}
\As{4}: $\mathfrak{b} \geq (1 - \varepsilon_5) k$.
\varepsilonnd{leftbar}
\textbf{Constructing an Instance of Laminar Cut:} In order to
construct the instance for the problem, let $S^*_{\geq \mathfrak{b}} = \cup_{i = \mathfrak{b}}^{k}
S^*_i$ be the union of these last few components from ${\cal S}^*$ which
have ``large'' boundary. Consider the iteration of the while loop
when $k' = \mathfrak{a}$ and consider $S_1, \dots, S_{\mathfrak{a}}$ in that
iteration. By its definition, $c_{\mathfrak{a}} > w(\partial{S_1^*}) (1 -
\nicefrac{\varepsilon_1}{3})$. Hence
\begin{gather}
\min_i \mathsf{Mincut}(\inducedG{S_i}) > w(\partial{S_1^*}) (1 -
\nicefrac{\varepsilon_1}{3}),
\langlebel{eq:stop1} \\
\min_i \text{\sf{Min-4-cut}}(G[S_i]) > 3 w(\partial{S_1^*}) (1 -
\nicefrac{\varepsilon_1}{3}).
\langlebel{eq:stop2}
\varepsilonnd{gather}
In particular, \varepsilonqref{eq:stop2} implies that no two near-min-cuts
cross, since two crossing near-min-cuts will result in a $4$-cut of
weight roughly at most $2 w(\partial{S_1^*})$.
However, we are not yet done, since we need to factor out the effects
of the $\mathfrak{a} - 1$ ``small'' cuts found by our algorithm. For this, we need
one further idea.
Let $\boldsymbol{\mathrm{r}} = (r_1, r_2, \ldots, r_{\mathfrak{a}}) \in [k]^{\mathfrak{a}}$ be such that
$r_i$ is the number of sets
$S^*_1, \dots, S^*_{\mathfrak{b} - 1}, S^*_{\geq \mathfrak{b}}$ that intersect with
$S_i$, and let $|\boldsymbol{\mathrm{r}}| := \sum_{i = 1}^{\mathfrak{a}} r_i$. If we consider the
bipartite graph where the left vertices are the algorithm's
components $S_1, \dots, S_{\mathfrak{a}}$, the right vertices are
$S^*_1, \dots, S^*_{\mathfrak{b} - 1}, S^*_{\geq \mathfrak{b}}$, and two sets have an
edge if they intersect, then $|\boldsymbol{\mathrm{r}}|$ is the number of edges. Since
there is no isolated vertex and the graph is connected (otherwise
there would exist $\varepsilonmptyset \neq I \subsetneq [k']$ and
$\varepsilonmptyset \neq J \subsetneq [k]$ with
$\cup_{i \in I}S_i = \cup_{j \in J} S^*_j$ contradicting~\As{1}),
the number of edges is $|\boldsymbol{\mathrm{r}}| \geq \mathfrak{a} + \mathfrak{b} - 1$.
\begin{claim}
\langlebel{clm:promises}
For each $i$ with $r_i \geq 2$,
the graph $\inducedG{S_i}$ satisfies the two promises of the problem
$\text{Laminar}cut{r_i}{\varepsilon_1}$.
\varepsilonnd{claim}
\begin{proof}
Fix $i$ with $r_i \geq 2$. Let
$J := \{ j \in [\mathfrak{b} - 1] \mid S_i \cap S^*_j \neq \varepsilonmptyset \}$ be
the sets $S_j^*$ among the first $\mathfrak{b} - 1$ sets in the optimal
partition that intersect $S_i$. Since $|J| \geq r_i - 1$ and $r_i \geq 2$, $|J| \geq 1$.
Note that
$(1 - \nf{\varepsilon_1}3)\cdot w(\partial{S^*_1}) <
\mathsf{Mincut}(\inducedG{S_i})$ by~\varepsilonqref{eq:stop1}.
For every $j \in J$,
\[
\mathsf{Mincut}(\inducedG{S_i}) \leq
w(E(S_i \cap S^*_j, S_i \setminus S^*_j)) \leq w(\partial{S^*_j})
\leq (1 + \varepsilon_1 / 3)\; w(\partial{S^*_1}) \leq (1 + \varepsilon_1)
\;\mathsf{Mincut}(\inducedG{S_i}).
\]
The first and second inequality hold since both parts
$S_i \cap S^*_j$ and $S_i \setminus S^*_j$ are nonempty, and hence
deleting all the edges in $\partial{S_j^*}$ would separate
$G[S_i]$.
The third inequality is by the choice of $\mathfrak{b}$, and the last
inequality uses~(\ref{eq:stop1}) and the fact that
$(1 + \nicefrac{\varepsilon_1}{3}) \leq (1 + \varepsilon_1)(1 -
\nicefrac{\varepsilon_1}{3})$ when $\varepsilon_1 < 1/4$.
This implies that in $\inducedG{S_i}$, for every $j \in J$,
$(S_i \cap S^*_j, S_i \setminus S^*_j)$ is a $(1 + \varepsilon_1)$-mincut.
Furthermore, in $\inducedG{S_i}$, no two $(1+\varepsilon_1)$-mincuts cross
because it will result a 4-cut of cost at most
\[
2(1+\varepsilon_1)\; \mathsf{Mincut}(\inducedG{S_i}) \leq 2(1+\varepsilon_1) (1 + \varepsilon_1 / 3)
\; w(\partial{S_1^*}),
\]
contradicting~\varepsilonqref{eq:stop2}. (Note that $2(1 +
\varepsilon_1) (1 + \nicefrac{\varepsilon_1}{3}) \leq 3(1 - \nicefrac{\varepsilon_1}{3})$ when
$\varepsilon_1 < 1/4$.) Hence, in $\inducedG{S_i}$, the two promises for
$\text{Laminar}cut{r_i}{\varepsilon_1}$ are satisfied.
\varepsilonnd{proof}
Our algorithm $\text{Main}(G, k$) runs $\text{Laminar}(\inducedG{S_i}, r_i)$ for
each $i \in [\mathfrak{a}]$ when it sets $k' = \mathfrak{a}$ and the vector $\boldsymbol{\mathrm{r}}$ as
defined above. As in the algorithm, let
${\cal C} = \{C_1, \dots, C_{k}\}$ be the partition obtained in
Line~\ref{line:complete}. In other words, to obtain the $k$ sets
$C_1, \dots, C_k$ from the set $V$, we take the reference partition
$S_1, \dots, S_{\mathfrak{a}}$ and further partition these sets using Laminar
to get $|\boldsymbol{\mathrm{r}}|$ parts $C_1, \dots, C_{|\boldsymbol{\mathrm{r}}|}$. If $|\boldsymbol{\mathrm{r}}| \geq k$, we
can merge the last $|\boldsymbol{\mathrm{r}}| - k + 1$ parts to get exactly $k$ parts if
we want (but we will not take any edge savings into account in this
calculation). If $|\boldsymbol{\mathrm{r}}| < k$, we get $k - |\boldsymbol{\mathrm{r}}|$ more parts using
the Complete procedure.
The total cost of this solution ${\cal C}$ is $w(E(C_1, \dots, C_k))$,
which is $\sum_{j=1}^{\mathfrak{a} - 1} c_j \leq (\mathfrak{a} - 1) w(\partial{S^*_1})$
plus the cost of $\text{Laminar}(\inducedG{S_i}, r_i)$ for all
$i \in [\mathfrak{a}]$ and the cost of $\text{Complete}$. Since
Claim~\ref{clm:promises} considers the partition of each
$\inducedG{S_i}$ obtained by cutting edges belonging to the optimal
$k$-partition,
the sum of the cost of the $r_i$-partition we compare to in
each \text{Laminar}kcut{r_i} is exactly $\varepsilonnsuremath{\mathsf{Opt}}\xspace$.
Hence the cost of the
solution given by $\text{Laminar}(\inducedG{S_i}, r_i)$ summed over
$i \in [\mathfrak{a}]$ is bounded by $(2 - \varepsilon_2) \varepsilonnsuremath{\mathsf{Opt}}\xspace$, by the approximation
assumption in Theorem~\ref{thm:reduction1}.
If $\cup_{i \in I} S^*_i$ for some $\varepsilonmptyset \neq I \subsetneq [k]$
conforms to ${\cal C}$, then since \text{Main}\ also records $\text{Guess}({\cal C})$,
the proof of Claim~\ref{claim:good} guarantees that $\text{Main}(G, k)$
gives a $(2 - \varepsilon_3)$ approximation using the induction hypothesis.
Otherwise, $S^*_1$ does not conform to ${\cal C}$, so the arguments used
in the proof of Claim~\ref{claim:notgood} show that the cost of
$\text{Complete}$ is at most $(k - |\boldsymbol{\mathrm{r}}|)\, w(\partial{S^*_1})$ if
$|\boldsymbol{\mathrm{r}}| \leq k$, and $0$ otherwise. Since $|\boldsymbol{\mathrm{r}}| \geq \mathfrak{a} + \mathfrak{b} - 1$,
the total cost $w(E(C_1, \dots, C_k))$ is then bounded
by
\begin{align*}
& (\mathfrak{a} - 1 ) w(\partial{S^*_1}) + (2 - \varepsilon_2) \varepsilonnsuremath{\mathsf{Opt}}\xspace +
(k - \mathfrak{a} - \mathfrak{b} + 1) w(\partial{S^*_1}) \\
= & \ (2 - \varepsilon_2) \varepsilonnsuremath{\mathsf{Opt}}\xspace + (k - \mathfrak{b}) w(\partial{S^*_1}) \\
\leq & \ (2 - \varepsilon_2) \varepsilonnsuremath{\mathsf{Opt}}\xspace + \varepsilon_5 k \cdot
w(\partial{S^*_1}) \tag{by \As{4}}\\
\leq & \ (2 - \varepsilon_2 + 2 \varepsilon_5) \varepsilonnsuremath{\mathsf{Opt}}\xspace.
\varepsilonnd{align*}
Therefore, if
\begin{equation}
\varepsilon_3 \leq \varepsilon_2 -2 \varepsilon_5,
\langlebel{eq:para_4}
\varepsilonnd{equation}
then $\text{Main}(G, k)$ gives a $(2 - \varepsilon_3)$ approximation in every
possible case. We set $\varepsilon_3, \varepsilon_4, \varepsilon_5 > 0$ so that they
satisfy the three conditions~\varepsilonqref{eq:para_1}, \varepsilonqref{eq:para_2},
and~\varepsilonqref{eq:para_4}, namely,
\[
(2/3) \cdot \varepsilon_1 \varepsilon_4 \geq \varepsilon_3, \quad (1 +
\varepsilon_1 \varepsilon_5 / 3)(2 - \varepsilon_3) \geq 2,
\quad \varepsilon_3 \leq \varepsilon_2 - 2 \varepsilon_5.
\]
(For instance, setting $\varepsilon_4 = \varepsilon_5 = \min(\varepsilon_1,
\varepsilon_2) / 3$ and $\varepsilon_3 = \varepsilon_4^2$ works.)
\varepsilonnd{proof}
\subsection{Running Time}
We prove that this algorithm also runs in FPT time, finishing the proof
of Theorem~\ref{thm:reduction1}.
\begin{lemma}
Suppose that $\text{Laminar}(G, k)$ runs in time $f(k) \cdot g(n)$.
Then Main$(G, k)$ runs in time $2^{O(k^2 \log k)} \cdot f(k) \cdot (g(n) + n^4 \log^3 n)$.
\varepsilonnd{lemma}
\begin{proof}
Let $\mathsf{Time}(\text{P})$ denote the running time of a procedure \text{P}.
Here each procedure is only parameterized by the number of sets it outputs (e.g., $\text{Main}(k), \text{Guess}(k), \text{Complete}(k), \text{Laminar}(k)$).
We use the fact that the global min-cut can be computed in time $O(n^2 \log^3 n)$~\cite{KS96} and the min-$4$-cut can be computed in $O(n^4 \log^3 n)$~\cite{Levine00}.
First, $\mathsf{Time}(\text{Complete}(k)) = O(kn^2 \log^3 n)$. For $\text{Guess}$ and $\text{Main}$,
\[
\mathsf{Time}(\text{Guess}(k)) \leq k \cdot 2^{k + 1} \cdot ( \mathsf{Time}(\text{Main}(k - 1)) + O(n) ),
\]
and
\begin{align*}
\mathsf{Time}(\text{Main}(k)) & \leq k^k \cdot (\mathsf{Time}(\text{Laminar}(k)) +
\mathsf{Time}(\text{Guess}(k)) + \mathsf{Time}(\text{Complete}(k))) + O(k n^4 \log^3 n) \\
& \leq 2^{O(k \log k)} \cdot f(k) \cdot (g(n) + O(n^4 \log^3 n)) +
2^{O(k \log k)} \cdot \mathsf{Time}(\text{Main}(k - 1)).
\varepsilonnd{align*}
We can conclude $\mathsf{Time}(\text{Main}(k)) \leq 2^{O(k^2 \log k)} \cdot f(k) \cdot (g(n) + n^4 \log^3 n)$.
\varepsilonnd{proof}
\section{An Algorithm for \text{Laminar}kcut{k}}
\langlebel{sec:laminar}
Recall the definition of the \text{Laminar}kcut{k} problem:
\LamDef*
Let $\mathcal O_{\varepsilon_1}$ contain all partitions $S_1,\ldots,S_k$ of $V$ with
the restriction that the boundaries of the first $k-1$ parts is
small---i.e., $w(\partial{S_i}) \le (1+\varepsilon_1)\mathsf{Mincut}(G)$ for all $i \in
[k-1]$. We emphasize that the weight of the last cut, i.e.,
$w(\partial{S_k})$, is unconstrained. In this section, we give an
algorithm to find a $k$-partition (possibly not in $\mathcal O_{\varepsilon_1}$) with
total weight
\[ w(E(S_1,\ldots,S_k)) \le (2 - \varepsilon_2) \min\limits_{\{S_i'\}\in
\mathcal O_{\varepsilon_1}} w(E(S_1',\ldots,S_k')). \]
Formally, the main theorem of this section is the following:
\begin{theorem}[Laminar Cut Algorithm]
\langlebel{thm:laminar}
Suppose there exists a $(1+\delta)$-approximation algorithm for
$\PartialVC{k}$ for some $\delta \in (0, 1/24)$ that runs in time
$f(k) \cdot g(n)$. Then, for any $\varepsilon_1\in(0,1/6-4\delta)$, there
exists a $(2 - \varepsilon_2)$-approximation algorithm for
$\text{Laminar}cut{k}{\varepsilon_1}$ that runs in time $2^{O(k)}f(k)(\tilde O(n^4) + g(n))$
for some constant $\varepsilon_2 > 0$.
\varepsilonnd{theorem}
In the rest of this section we present the algorithm and the analysis.
For a formal description, see the pseudocode in
Appendix~\ref{sec:pseudocode-laminar}.
\subsection{Mincut Tree}
The first idea in the algorithm is to consider the structure of a laminar family of cuts. Below, we introduce the concept of a \textit{mincut tree}. The vertices of the mincut tree are called \textit{nodes}, to distinguish them from the vertices of the original graph.
\begin{definition}[Mincut Tree]
A tree $\mathcal T = (V_{\mathcal T}, E_{\mathcal T}, w_{\mathcal T})$ is a \textbf{$(1+\varepsilon_1)$-mincut tree} on a graph $G=(V,E,w)$ with mapping $\phi : V \to V_{\mathcal T}$ if the following two sets are equivalent:
\begin{enumerate}
\item The set of all $(1+\varepsilon_1)$-mincuts of $G$.
\item Cut a single edge $e \in E_{\mathcal T}$ of the tree, and let $A_e
\subset V_{\mathcal T}$ be the nodes on one side of the cut. Define $S_e :=
\phi^{-1}(A_e) = \{v \mid \phi(v) \in A_e\}$ for each $e\in E_{\mathcal T}$,
and take the set of cuts $ \{ (S_e, V \setminus S_e) : e \in E_{\mathcal T}
\}$.
\varepsilonnd{enumerate}
Moreover, for every pair of corresponding $(1+\varepsilon_1)$-mincut $(S_e, V
\setminus S_e)$ and edge $e \in E_{\mathcal T}$, we have $w_{\mathcal T}(e) = w(E(S_e,
V\setminus S_e))$.
\varepsilonnd{definition}
We use the term \textit{mincut tree} without the $(1+\varepsilon_1)$ when the
value of $\varepsilon_1$ is either implicit or irrelevant.
For the rest of this section, let
\[ {\mu} := \mathsf{Mincut}(G) \] for brevity. Observe that the last condition
implies that ${\mu}\le w_{\mathcal T}(e)\le(1+\varepsilon_1){\mu}$ for all $e\in
E_{\mathcal T}$. The existence of a mincut tree (and the algorithm for it)
assuming laminarity, is standard, going back at least to Edmonds and
Giles~\cite{EG75}.
\begin{theorem}[Mincut Tree Existence/Construction]
\langlebel{thm:mincutTreeExistence}
If the set of $(1+\varepsilon_1)$-mincuts of a graph is laminar, then an
$O(n)$-sized $(1+\varepsilon_1)$-mincut tree always exists, and can
be found in $O(n^3)$ time.
\varepsilonnd{theorem}
\begin{proof}
We refer the reader to~\cite[Section~2.2]{KV12}. Fix a vertex $v \in
V$, and for each $(1+\varepsilon_1)$-mincut $(S,V\setminus S)$, pick the side
that contains $v$; this family of subsets of $V$ satisfies the laminar
condition in Proposition~2.12 of that book. Corollary~2.15 proves that
this family has size $O(n)$, and the construction of $T$ in
Proposition~2.14 gives the desired mincut tree. Furthermore, we can
compute the mincut tree in $O(n^3)$ time as follows: first precompute
whether $X \subset Y$ for every two sets $X$ and $Y$ in the family,
and then compute $T$ following the construction in the proof of
Proposition~2.14.
\varepsilonnd{proof}
\begin{definition}[Mincut Tree Terminology] Let $\mathcal T$ be a rooted mincut
tree. For $a \in V_{\mathcal T}$, define the following terms:
\begin{OneLiners}
\item[1.] $\ensuremath{\mathrm{children}}(a)$: the set of children of node $a$ in the rooted tree.
\item[2.] $\ensuremath{\mathrm{desc}}(a)$: the set of descendants of $a$, i.e., nodes $b \in V_{\mathcal T} \setminus a$ whose path to the root includes $a$.
\item[3.] $\ensuremath{\mathrm{anc}}(a)$: the set of ancestors of $a$, i.e., nodes $b \in V_{\mathcal T} \setminus a$ on the path from $a$ to the root.
\item[4.] $\ensuremath{\mathrm{subtree}}(a)$: vertices in the subtree rooted at $a$, i.e., $\{a\} \cup \ensuremath{\mathrm{desc}}(a)$.
\varepsilonnd{OneLiners}
\varepsilonnd{definition}
For the set of partitions $\mathcal O_{\varepsilon_1}$ (as defined at the beginning
of this section), we observe the following.
\begin{claim}[Representing Laminar Cuts in $\mathcal T$]
Let $\mathcal T = (V_{\mathcal T}, E_{\mathcal T}, w_{\mathcal T})$ be a $(1+\varepsilon_1)$-mincut tree of
$G=(V,E,w)$, and consider a partition $\{S_1, \ldots, S_k\} \in
\mathcal O_{\varepsilon_1}$. Then, there exists a root $r \in V_{\mathcal T}$ and nodes
$a_1, \ldots, a_{k-1} \in V_{\mathcal T} \setminus r$ such that if we root
the tree $\mathcal T$ at $r$,
\begin{enumerate}
\item For any two nodes in $\{a_1, \ldots, a_{k-1}\}$, neither is an
ancestor of the other. (We call two such nodes
\textbf{incomparable}).
\item For each $v_i$, let $A_i := \ensuremath{\mathrm{subtree}}(a_i)$, and let $A_k =
V_{\mathcal T} \setminus \bigcup_{i=1}^{k-1} A_i$ (so that $r \in A_k$). We
have the two equivalences $\{\phi^{-1}(A_i) \mid i \in [k-1]\} = \{S_1,
\ldots, S_{k-1}\}$ and $\phi^{-1}(A_k) = S_k$. In other words, the
components $A_i \subset V_{\mathcal T}$, when mapped back by $\phi^{-1}$,
correspond exactly to the sets $S_i \subset V$, with the additional
guarantee that $A_k$ and $S_k$ match.
\varepsilonnd{enumerate}
\varepsilonnd{claim}
\begin{proof}
Since $S_i$ is a $(1+\varepsilon_1)$-mincut for each $i \in [k-1]$, there
exists an edge $e_i \in E_{\mathcal T}$ such that the set $A_i'$ of nodes on
one side of $e_i$ satisfies $\phi^{-1}(A_i') = S_i $. The sets $A_i'$
for $i\in[k-1]$ are necessarily disjoint, and they cannot span all
nodes in $V_{\mathcal T}$, since $S_k$ is still unaccounted for. If we root
$\mathcal T$ at a node $r$ not in any $A_i'$, then each $A_i'$ is a subtree
of the rooted $\mathcal T$. Altogether, the roots of the subtrees $A_i'$
satisfy condition~(1) of the lemma, and the $A_i'$ themselves satisfy
condition~(2).
\varepsilonnd{proof}
For a graph $G=(V,E,w)$ and mincut tree $\mathcal T=(V_{\mathcal T},E_{\mathcal T},w_{\mathcal T})$
with mapping $\phi:V\to V_{\mathcal T}$, define $E_G(A,B)$ for $A,B \subset
V_{\mathcal T}$ as $E\left(\phi^{-1}(A), \phi^{-1}(B)\right)$, i.e., the total
weight of edges crossing the sets corresponding to $A$ and $B$ in $V$.
\begin{observation}
Given a root $r\in V_{\mathcal T}$ and incomparable nodes $a_1, \ldots,
a_{k-1} \in V_{\mathcal T} \setminus r$, we can bound the corresponding
partition $S_1,\ldots,S_k$ as follows:
\begin{align*}
w(E(S_1, \ldots, S_k)) &= \textstyle \sum_{i=1}^{k-1}
w(\partial(S_i)) - \sum_{i<j \le k-1} w(E(S_i, S_j)) \\ &=
\textstyle \sum_{i=1}^{k-1} w_{\mathcal T}(e_i) - \sum_{i<j\le k-1}
w(E_G(\ensuremath{\mathrm{subtree}}(a_i),\ensuremath{\mathrm{subtree}}(a_j))) ,
\varepsilonnd{align*}
where $e_i$ is the parent edge
of $v_i$ in the rooted tree.
\varepsilonnd{observation}
Note that ${\mu} \le w_{\mathcal T}(e) \le (1+\varepsilon_1){\mu}$ for all $e\in
E_{\mathcal T}$, so to approximately minimize the above expression for a fixed
root $r$, it suffices to approximately maximize
\begin{align*}
\textstyle \mathsf{Saved}(a_1,\ldots,a_{k-1}) := \sum\limits_{i<j\le k-1}
w(E_G(\ensuremath{\mathrm{subtree}}(a_i),\ensuremath{\mathrm{subtree}}(a_j))), \langlebel{eq:saved}
\varepsilonnd{align*}
which we think of as the edges \textit{saved} in the double counting of
$\sum_{i=1}^{k-1}w_{\mathcal T}(e_i)$. The actual approximation factor is made
precise in the proof of Theorem~\ref{thm:laminar}.
To maximize the number of saved edges over all partitions in
$\mathcal O_{\varepsilon_1}$, it suffices to try all possible roots $r$ and take the
best partition. Therefore, for the rest of this section, we focus on
maximizing $\mathsf{Saved}(a_1,\ldots,a_{k-1})$ for a fixed root
$r$.
Let $\varepsilonll^*(r)$ be that maximum value for root $r$, and let $\mathsf{Opt}(r) =
\{a_1^*, \ldots, a_{k-1}^*\} \subset V_{\mathcal T}$ be the solution that
attains it.
\subsection{Anchors}
Root the mincut tree $\mathcal T$ at $r$, and let $a_1^*,\ldots,a_{k-1}^*$ be incomparable nodes in the solution $\varepsilonnsuremath{\mathsf{Opt}}\xspace(r)$.
First, observe that we can assume w.l.o.g.\ that for each node $a_i^*$,
its parent node is an ancestor of some $a_j^* \neq a_i^*$: if not, we can replace $a_i^*$ with its parent, which can only increase $\mathsf{Saved}(a_1^*,\ldots,a_{k-1}^*)$.
\begin{observation}
Consider nodes $a_1^*,\ldots,a_s^* \in \mathsf{Opt}(r)$ which share the same parent $a \notin \mathsf{Opt}(r)$, and assume that $a$ has no other descendants. If we replace $a_1^*,\ldots,a_s^*$ in $\mathsf{Opt}(r)$ with $a$, then we lose at most $\mathsf{Saved}(a_1^*,\ldots,a_s^*)$ in our solution.\footnote{The new solution may no longer have $k-1$ nodes, but we will fix this problem in the proof of Theorem~\ref{thm:laminar}. For now, assume that we are allowed to choose any number up to $k-1$ nodes.}
\varepsilonnd{observation}
If $\mathsf{Saved}(a_1^*,\ldots,a_s^*)$ is small, i.e., compared to $(s-1){\mu}$, then we do not lose too much. This idea motivates the idea of anchors.
\begin{definition}[Anchors]
Let $\mathcal T=(V_{\mathcal T},E_{\mathcal T},w_{\mathcal T})$ be a rooted tree.
For a fixed constant $\varepsilon_3 > 0$, define an \textbf{$\varepsilon_3$-anchor} to be a node $a\in V_{\mathcal T}$ such that there exists $s \in [2,k-1]$ and $s$ children $a_1,\ldots,a_s$ such that $\mathsf{Saved}(a_1,\ldots,a_s) \ge \varepsilon_3(s-1){\mu}$. When the value of $\varepsilon_3$ is implicit, we use the term anchor, without the $\varepsilon_3$.
\varepsilonnd{definition}
We now claim that we can transform any solution to another
well-structured solution, with only a minimal loss.
\begin{lemma}[Shifting Lemma]
\langlebel{lem:anchor}
Let $a_1,\ldots,a_{k-1}$ be a set of incomparable nodes of a
$(1+\varepsilon_1)$-mincut tree $\mathcal T$. Then, there exists a set
$b_1,\ldots,b_s$ of incomparable nodes, for $1\le s\le k-1$, such that
\begin{enumerate}
\item The parent of every node $b_i$ is either an $\varepsilon_3$-anchor, or is
an ancestor of some node $b_j \neq b_i$ whose parent is an anchor.
\item $\mathsf{Saved}(b_1,\ldots,b_s) \ge \mathsf{Saved}(a_1,\ldots,a_{k-1}) -
\varepsilon_3(k-s){\mu}$.
\varepsilonnd{enumerate}
In particular, if $\{a_1, \ldots, a_{k-1}\} = \mathsf{Opt}(r)$, condition~(2)
implies $\mathsf{Saved}(b_1,\ldots,b_s) \ge \varepsilonll^*(r) - \varepsilon_3(k-1){\mu}$.
\varepsilonnd{lemma}
\begin{proof}
We begin with the solution $b_i=a_i$ for all $i$, and iteratively
shift non-anchors in the solution while maintaining the potential
function $\Phi:=\mathsf{Saved}(b_1,\ldots,b_s) - \mathsf{Saved}(a_1,\ldots,a_{k-1}) +
\varepsilon_3(k-s){\mu}$ nonnegative.
At the beginning, $\Phi=0$. Suppose there is a
node $b_i$ not satisfying condition (1). Choose one such $b_i$ of
maximum depth in the tree, and let $b'$ be its non-anchor parent. Then
the only descendants of $b'$ in the current solution are siblings of
$b_i$. Replace $b_i$ and its $s'$ siblings in the solution by
$b'$. Since $b'$ is not an anchor, $\mathsf{Saved}(b_1,\ldots,b_s)$ drops by
at most $\varepsilon_3(s'-1){\mu}$. This drop is compensated by the decrease
of the solution size from $s$ to $s-(s'-1)$.
\varepsilonnd{proof}
Hence, at a loss of $\varepsilon_3(k-1){\mu}$, it suffices to focus on a
solution $\mathsf{Opt}'(r)$ which fulfills condition (1) of
Lemma~\ref{lem:anchor} and has $\mathsf{Saved}$ value $\varepsilonll'(r) \geq \varepsilonll^*(r) -
\varepsilon_3(k-1){\mu}$.
The rest of the algorithm splits into two cases. At a high level, if
there are enough anchors in a mincut tree $\mathcal T$ that are incomparable
with each other, then we can take such a set and be done. Otherwise, the
set of anchors can be grouped into a small number of paths in $\mathcal T$, and
we can afford to try all possible arrangements of anchors. But first we
show how to find all the anchors in $\mathcal T$.
\subsection{Finding Near-Anchors}
\newcommand{\ensuremath{\mathrm{anc}}hors}{\mathcal{A}}
\begin{lemma}[Finding (Near-)Anchors]
\langlebel{lem:compute-anchors}
Assume access to a $(1+\delta)$-approximation algorithm for $\PartialVC k$
running in time $f(k) \cdot g(n)$. Then, there is an algorithm running
in time $O(n \cdot (n^2 + k \cdot f(k) \cdot g(n)))$ that computes a
set $\ensuremath{\mathrm{anc}}hors$ of ``near''-anchors in $\mathcal T$, i.e., vertices $a \in
V_{\mathcal T}$ for which there exists an integer $s \in [2,k-1]$ and $s$
children $b_1, \ldots, b_s$ such that $\mathsf{Saved}(b_1,\ldots,b_s) \geq
\varepsilon_3(s-1){\mu} - \delta(1+\varepsilon_1)s{\mu}$.
\varepsilonnd{lemma}
\begin{proof}
To determine if a node $a$ is an anchor or not, for each integer $s
\in [2, k-1]$ we wish to compute the maximum value of
$\mathsf{Saved}(b_1,\ldots,b_s)$ for $b_1,\ldots,b_s \in
\ensuremath{\mathrm{children}}(a)$. Consider the following weighted, complete graph with
vertex and edge weights: for each $b \in \ensuremath{\mathrm{children}}(a)$ create a vertex
$x_b$, and the edge $(x_{b_1}, x_{b_2})$ has weight
$\mathsf{Saved}(b_1,b_2)$. Each vertex $x_b$ also has weight $(1+\varepsilon_1){\mu}
- w(\partial x_b)$, where $w(\partial x_b)$ is the sum of the weights
of edges incident to $x_b$. Note that this graph is
$(1+\varepsilon_1){\mu}$-regular, if we include vertex weights in the
definition of vertex degree.
Observe that $w(\partial x_b)
\le \partial\left(\phi^{-1}(\ensuremath{\mathrm{subtree}}(b))\right) \le (1+\varepsilon_1){\mu}$,
since every edge in $G$ that contributes to $\mathsf{Saved}(b,b')$ for another
child $b'$ also contributes to the cut
$\partial\left(\phi^{-1}(\ensuremath{\mathrm{subtree}}(b))\right)$, which we know is $\le
(1+\varepsilon_1){\mu}$.
Therefore, each vertex has a nonnegative weight.
Also, a partial vertex cover on this graph with vertices $x_{b_1},
\ldots, x_{b_s}$ has weight exactly $(1+\varepsilon_1)s{\mu} - \mathsf{Saved} (b_1,
\ldots, b_s)$.
Let $b_1^*,\ldots,b_s^* \in \ensuremath{\mathrm{children}}(a)$ be the solution with maximum
$\mathsf{Saved}(b_1^*,\ldots,b_s^*)$. To compute this maximum, we can build
the above graph and run the $(1+\delta)$-approximate partial vertex
cover algorithm from Theorem~\ref{thm:pvc}. The solution
$b_1,\ldots,b_s$ satisfies
\[ (1+\varepsilon_1)s{\mu} - \mathsf{Saved}(b_1,\ldots,b_s) \le
(1+\delta)\left((1+\varepsilon_1)s{\mu} - \mathsf{Saved}(b_1^*,\ldots,b_s^*)\right),
\]
so that
\begin{align*}
\mathsf{Saved}(b_1,\ldots,b_s) & \ge (1+\delta)\,\mathsf{Saved}(b_1^*,\ldots,b_s^*)
- \delta(1+\varepsilon_1)s {\mu} \\
& \ge \mathsf{Saved}(b_1^*,\ldots,b_s^*) - \delta(1+\varepsilon_1)s{\mu}.
\varepsilonnd{align*}
We run this subprocedure for the vertex $a$ for each integer $2 \leq s
\le \min\{|\ensuremath{\mathrm{children}}(a)|, k-1\}$, and mark vertex $a$ if there exists
an integer $s$ such that the weight of saved edges is at least
$\varepsilon_3(s-1){\mu} - \delta(1+\varepsilon_1)s{\mu}$. The set $\ensuremath{\mathrm{anc}}hors$ of
near-anchors is exactly the set of marked vertices.
As for running time, for each node $a$, it takes $O(n^2)$ time to construct the $\textsc{Partial VC}\xspace$ graph and $O(k) \cdot f(k) \cdot g(n)$ time to solve $\PartialVC s$ for each $s \in [2,k-1]$. Repeating the above for each of the $O(n)$ nodes achieves the promised running time.
\varepsilonnd{proof}
\subsection{Many Incomparable Near-Anchors}
\begin{lemma}[Many Anchors]
\langlebel{lem:incomparable}
Suppose we have access to a $(1+\delta)$-approximation algorithm for
$\PartialVC k$ running in time $f(k) \cdot g(n)$. Suppose the set
$\ensuremath{\mathrm{anc}}hors$ of near-anchors contains $k-1$ incomparable nodes from the
mincut tree $\mathcal T$. Then, there is an algorithm computing a solution
with $\mathsf{Saved}$ value
$\ge \frac14\varepsilon_3(k-1){\mu} - \delta(1+\varepsilon_1)(k-1){\mu}$ for any
$\delta>0$, running in time
$O(n \cdot (n^2 + k \cdot f(k) \cdot g(n)))$.
\varepsilonnd{lemma}
\begin{proof}
First, we compute the set $\ensuremath{\mathrm{anc}}hors$ in $O(n \cdot (n^2 + k \cdot f(k)
\cdot g(n))$ time, according to Lemma~\ref{lem:compute-anchors}. If
$\ensuremath{\mathrm{anc}}hors$ contains $k-1$ incomparable nodes, we can \textit{find}
them in $O(n^2)$ time by greedily choosing nodes in a topological,
bottom-first order (see lines 4--11 in
Algorithm~\ref{alg:laminarRooted}). Each of these $k-1$ marked nodes
$a_1, \ldots, a_{k-1}$ has an associated value $s_i$, indicating that
$a_i$ has some $s_i$ children whose $\mathsf{Saved}$ value is at least
$\varepsilon_3(s_i-1){\mu} - \delta(1+\varepsilon_1)s_i{\mu}$. If we consider a
subset $A \subset [k-1]$ and choose the $s_i$ children for each $a_i$
with $i\in A$, then we get a set with $\sum_{i\in A} s_i$ nodes, whose
total $\mathsf{Saved}$ value at least \[\varepsilon_3\left( \sum_{i\in
A}(s_i-1)\right){\mu} - \delta(1+\varepsilon_1)\left( \sum_{i\in A}s_i
\right){\mu}.\] Assuming that $\sum_{i\in A}s_i \le k-1$, i.e., we
choose at most $k-1$ children, the second
$\delta(1+\varepsilon_1)\left(\sum_{i\in A}s_i\right){\mu}$ term is at most
$\delta(1+\varepsilon_1) (k-1){\mu}$. To optimize the $\varepsilon_3\left( \sum_{i\in
A}(s_i-1)\right){\mu}$ term, we reduce to the following knapsack
problem: we have $k-1$ items $i \in [k-1]$ where item $i$ has size
$s_i \in [2,k-1]$ and value $s_i-1$, and our bag size is $k-1$. A
knapsack solution of value $Z:=\sum_{i\in A}(s_i-1)$ translates to a
solution with $\mathsf{Saved}$ value $\ge \varepsilon_3{\mu} \cdot Z -
\delta(1+\varepsilon_1)(k-1){\mu}$. By Lemma~\ref{lemma:knapsack}, when $k
\ge 5$, we can compute a solution $A \subset [k-1]$ of value $\ge
(k-1)/4$ in $O(k)$ time. (If $k\le4$, we can use the exact $\tilde O(n^4)$
$k$\textsc{-Cut}\xspace algorithm from~\cite{Levine00}.) Selecting the children of each
$u_i$ with $i\in A$ gives a total $\mathsf{Saved}$ value of at least
$\frac14\varepsilon_3(k-1){\mu} - \delta(1+\varepsilon_1)(k-1){\mu}$.
\varepsilonnd{proof}
\subsection{Few Incomparable Near-Anchors}
\begin{figure}
\centering
\begin{tikzpicture} [xscale=.5, yscale=.3]
\tikzstyle{every node}=[circle, fill, scale=.3];
\node (v1) at (-1,4) {};
\node (v2) at (-2,2) {};
\node (v12) at (0.5,2) {};
\node (v3) at (-3,0) {};
\node (v10) at (-1.5,0) {};
\node (v11) at (0,0) {};
\node (v13) at (1,0) {};
\node (v14) at (2,0) {};
\node (v4) at (-4,-2) {};
\node (v7) at (-2,-2) {};
\node (v5) at (-5,-4) {};
\node (v6) at (-3.5,-4) {};
\node (v8) at (-2.5,-4) {};
\node (v9) at (-1,-4) {};
\node (v15) at (1,-2) {};
\node (v16) at (3,-2) {};
\draw [line width=.5pt] (v1) edge (v2);
\draw [line width=.5pt] (v2) edge (v3);
\draw [line width=.5pt] (v3) edge (v4);
\draw [line width=.5pt] (v4) edge (v5);
\draw [line width=.5pt] (v4) edge (v6);
\draw [line width=.5pt] (v7) edge (v3);
\draw [line width=.5pt] (v8) edge (v7);
\draw [line width=.5pt] (v9) edge (v7);
\draw [line width=.5pt] (v10) edge (v2);
\draw [line width=.5pt] (v11) edge (v12);
\draw [line width=.5pt] (v12) edge (v1);
\draw [line width=.5pt] (v13) edge (v12);
\draw [line width=.5pt] (v12) edge (v14);
\draw [line width=.5pt] (v15) edge (v14);
\draw [line width=.5pt] (v14) edge (v16);
\draw (v3) circle (.3);
\draw (v14) circle (.3);
\draw (v16) circle (.3);
\draw (v5) circle (.3);
\draw (v8) circle (.3);
\draw (v9) circle (.3);
\draw [line width=.5pt] (v5) edge (-5.5,-5);
\draw [line width=.5pt] (v5) edge (-4.5,-5);
\draw [line width=.5pt] (v8) edge (-3,-5);
\draw [line width=.5pt] (v8) edge (-2.25,-5);
\draw [line width=.5pt] (v9) edge (-1.25,-5);
\draw [line width=.5pt] (v9) edge (-0.5,-5);
\varepsilonnd{tikzpicture}
\qquad
\begin{tikzpicture} [xscale=.5, yscale=.3]
\tikzstyle{every node}=[circle, fill, scale=.5];
\node (v1) at (-1,4) {};
\node (v3) at (-3,0) {};
\node (v7) at (-2,-2) {};
\node (v5) at (-5,-4) {};
\node (v8) at (-2.5,-4) {};
\node (v9) at (-1,-4) {};
\node (v16) at (3,-2) {};
\draw [white,line width=.5pt] (v5) edge (-5.5,-5);
\draw [white,line width=.5pt] (v5) edge (-4.5,-5);
\draw [white,line width=.5pt] (v8) edge (-3,-5);
\draw [white,line width=.5pt] (v8) edge (-2.25,-5);
\draw [white,line width=.5pt] (v9) edge (-1.25,-5);
\draw [white,line width=.5pt] (v9) edge (-0.5,-5);
\draw [line width=1pt] (v1) edge (v3);
\draw [line width=1pt] (v3) edge (v5);
\draw [line width=1pt] (v3) edge (v7);
\draw [line width=1pt] (v7) edge (v8);
\draw [line width=1pt] (v9) edge (v7);
\draw [line width=1pt] (v1) edge (v16);
\varepsilonnd{tikzpicture}
\qquad
\begin{tikzpicture} [xscale=.5, yscale=.3]
\tikzstyle{every node}=[circle, fill, scale=.3];
\node[fill=brown] (v1) at (-1,4) {};
\node[fill=blue] (v2) at (-2,2) {};
\node[fill=green] (v12) at (0.5,2) {};
\node[fill=blue] (v3) at (-3,0) {};
\node[fill=] (v10) at (-1.5,0) {};
\node[fill=] (v11) at (0,0) {};
\node[fill=] (v13) at (1,0) {};
\node[fill=green] (v14) at (2,0) {};
\node[fill=red] (v4) at (-4,-2) {};
\node[fill=purple] (v7) at (-2,-2) {};
\node[fill=red] (v5) at (-5,-4) {};
\node[fill=] (v6) at (-3.5,-4) {};
\node[fill=orange] (v8) at (-2.5,-4) {};
\node[fill=yellow] (v9) at (-1,-4) {};
\node[fill=] (v15) at (1,-2) {};
\node[fill=green] (v16) at (3,-2) {};
\draw [blue,line width=1pt] (v1) edge (v2);
\draw [blue,line width=1pt] (v2) edge (v3);
\draw [red,line width=1pt] (v3) edge (v4);
\draw [red,line width=1pt] (v4) edge (v5);
\draw [line width=1pt] (v4) edge (v6);
\draw [purple,line width=1pt] (v7) edge (v3);
\draw [orange,line width=1pt] (v8) edge (v7);
\draw [yellow,line width=1pt] (v9) edge (v7);
\draw [line width=1pt] (v10) edge (v2);
\draw [line width=1pt] (v11) edge (v12);
\draw [green,line width=1pt] (v12) edge (v1);
\draw [line width=1pt] (v13) edge (v12);
\draw [green,line width=1pt] (v12) edge (v14);
\draw [line width=1pt] (v15) edge (v14);
\draw [green,line width=1pt] (v14) edge (v16);
\draw [white,line width=.5pt] (v5) edge (-5.5,-5);
\draw [white,line width=.5pt] (v5) edge (-4.5,-5);
\draw [white,line width=.5pt] (v8) edge (-3,-5);
\draw [white,line width=.5pt] (v8) edge (-2.25,-5);
\draw [white,line width=.5pt] (v9) edge (-1.25,-5);
\draw [white,line width=.5pt] (v9) edge (-0.5,-5);
\varepsilonnd{tikzpicture}
\caption{\langlebel{figure:branches}
Establishing the set of branches $\mathcal B$. The circled nodes on the left are the near-anchors. The middle graph is the tree $\mathcal T'$. On the right, each non-black color is an individual branch; actually, the branches only consist of nodes, but we connect the nodes for visibility. Also, note that the root is its own branch. The red, orange, yellow, and green branches form an incomparable set.}
\varepsilonnd{figure}
If the condition in Lemma~\ref{lem:incomparable} does not hold, then
there exist $\le k-2$ paths from the root in $\mathcal T$ such that every node
in the near-anchor set $\ensuremath{\mathrm{anc}}hors$ lies on one of these paths. If we view
the union of these paths as a tree $\mathcal T'$ with $\le k-2$ leaves, then we
can partition the nodes in tree $\mathcal T'$ into a collection $\mathcal B$ of at
most $2k-3$ \textit{branches}. Each branch $B$ is a collection of
vertices obtained by taking either a leaf of $\mathcal T'$ or a vertex of
degree more than two, and all its immediate degree-2 ancestors; see
Figure~\ref{figure:branches}. Note that it is possible that the root
node is its own branch. Hence, given two branches $B_1, B_2 \in \mathcal B$,
either every node from $B_1$ is an ancestor of every node from $B_2$ (or
vice versa), or else every node from $B_1$ is incomparable with every
node from $B_2$.
Let $A' \subseteq \ensuremath{\mathrm{anc}}hors$ be the set of anchors with at least one child in
$\mathsf{Opt}'(r)=\{a_1^*,\ldots,a_s^*\}$; recall that $\mathsf{Opt}'(r)$ was produced
by the shifting procedure in Lemma~\ref{lem:anchor}. Let $A^* \subseteq A'$
be
the \textit{minimal} anchors in $A'$, i.e., every anchor in $A'$ that is
not an ancestor of any other anchor in $A'$. We know that every anchor in
$A^*$ falls inside our set of branches, although the algorithm does not
know where. Moreover, by condition~(1) of Lemma~\ref{lem:anchor}, the
parent of every $a_i^* \in \mathsf{Opt}'(r)$ either lies in $A^*$, or is an
ancestor of an anchor in $A^*$.
As a warm-up, consider the case where all the anchors in $A'$ are
contained within a single branch.
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.25]
\node [circle, fill=red, scale=.3] (v3) at (0,5) {};
\node [circle, fill=red, scale=.3] (v2) at (2.5,7.5) {};
\node [circle, fill=red, scale=.3] (v1) at (5,10) {};
\node [circle, fill=red, scale=.3] (v4) at (-2.5,2.5) {};
\node [circle, fill=red, scale=.3] (v5) at (-5,0) {};
\node [circle, fill=red, scale=.3, label=left:$a^*$] (v6) at (-7.5,-2.5) {};
\draw (v1) edge (v2);
\draw (v2) edge (v3);
\draw (v3) edge (v4);
\draw (v4) edge (v5);
\draw (v5) edge (v6);
\node [circle, fill=blue, scale=.3] (v7) at (5,8) {};
\node [circle, fill=blue, scale=.3] (v8) at (6.5,8) {};
\node [circle, fill=blue, scale=.3] (v9) at (2.5,5.5) {};
\node [circle, fill=blue, scale=.3] (v10) at (4,5.5) {};
\node [circle, fill=blue, scale=.3] (v11) at (0,3) {};
\node [circle, fill=blue, scale=.3] (v12) at (1.5,3) {};
\node [circle, fill=blue, scale=.3] (v13) at (-2.5,0.5) {};
\node [circle, fill=blue, scale=.3] (v14) at (-1,0.5) {};
\node [circle, fill=blue, scale=.3] (v15) at (-5,-2) {};
\node [circle, fill=blue, scale=.3] (v16) at (-3.5,-2) {};
\node [circle, fill=blue, scale=.3] (v17) at (-9,-4.5) {};
\node [circle, fill=blue, scale=.3] (v18) at (-7.5,-4.5) {};
\node [circle, fill=blue, scale=.3] (v19) at (-6,-4.5) {};
\draw (v1) -- (v7) -- (4.5,6.5) -- (5.5,6.5) -- (v7) (v1) -- (v8) -- (6,6.5) -- (7,6.5) -- (v8) (v2) -- (v9) -- (2,4) -- (3,4) -- (v9) (v2) -- (v2) -- (v10) -- (3.5,4) -- (4.5,4) -- (v10) (v3) -- (v11) -- (-0.5,1.5) -- (0.5,1.5) -- (v11) (v3) -- (v12) -- (1,1.5) -- (2,1.5) -- (v12) (v4) -- (v13) -- (-3,-1) -- (-2,-1) -- (v13) (v4) -- (v14) -- (-1.5,-1) -- (-0.5,-1) -- (v14) (v5) -- (v15) -- (-5.5,-3.5) -- (-4.5,-3.5) -- (v15) (v5) -- (v16) -- (-4,-3.5) -- (-3,-3.5) -- (v16) (v6) -- (v17) -- (-9.5,-6) -- (-8.5,-6) -- (v17) (v6) -- (v18) -- (-8,-6) -- (-7,-6) -- (v18) (v6) -- (v19) -- (-6.5,-6) -- (-5.5,-6) -- (v19);
\varepsilonnd{tikzpicture}
\qquad
\begin{tikzpicture}[scale=0.25]
\node [circle, fill=black, scale=.3] (v3) at (0,5) {};
\node [circle, fill=black, scale=.3] (v2) at (2.5,7.5) {};
\node [circle, fill=black, scale=.3] (v1) at (5,10) {};
\node [circle, fill=red, scale=.3] (v4) at (-2.5,2.5) {};
\node [circle, fill=red, scale=.3] (v5) at (-5,0) {};
\node [circle, fill=red, scale=.3] (v6) at (-7.5,-2.5) {};
\draw (v1) edge (v2);
\draw (v2) edge (v3);
\draw (v3) edge (v4);
\draw[red, line width=2pt] (v4) edge (v5);
\draw[red, line width=2pt] (v5) edge (v6);
\node [circle, fill=black, scale=.3] (v7) at (5,8) {};
\node [circle, fill=black, scale=.3] (v8) at (6.5,8) {};
\node [circle, fill=black, scale=.3] (v9) at (2.5,5.5) {};
\node [circle, fill=black, scale=.3] (v10) at (4,5.5) {};
\node [circle, fill=black, scale=.3] (v13) at (-2.5,0.5) {};
\node [circle, fill=black, scale=.3](v14) at (-1,0.5) {};
\node [circle, fill=black, scale=.3](v15) at (-5,-2) {};
\node [circle, fill=black, scale=.3] (v16) at (-3.5,-2) {};
\node [circle, fill=black, scale=.3] (v17) at (-9,-4.5) {};
\node [circle, fill=black, scale=.3](v18) at (-7.5,-4.5) {};
\node [circle, fill=black, scale=.3](v19) at (-6,-4.5) {};
\draw (v1) -- (v7) -- (4.5,6.5) -- (5.5,6.5) -- (v7);
\draw (v1) -- (v8) -- (6,6.5) -- (7,6.5) -- (v8);
\draw (v2) -- (v9) -- (2,4) -- (3,4) -- (v9);
\draw (v2) -- (v10) -- (3.5,4) -- (4.5,4) -- (v10);
\draw (v4) -- (v13) -- (-3,-1) -- (-2,-1) -- (v13) (v4) -- (v14) -- (-1.5,-1) -- (-0.5,-1) -- (v14) (v5) -- (v15) -- (-5.5,-3.5) -- (-4.5,-3.5) -- (v15) (v5) -- (v16) -- (-4,-3.5) -- (-3,-3.5) -- (v16) (v6) -- (v17) -- (-9.5,-6) -- (-8.5,-6) -- (v17) (v6) -- (v18) -- (-8,-6) -- (-7,-6) -- (v18) (v6) -- (v19) -- (-6.5,-6) -- (-5.5,-6) -- (v19);
\node [circle, fill=red, scale=.3] (v11) at (2,2.5) {};
\node [circle, fill=red, scale=.3] (v21) at (4,0) {};
\node [circle, fill=red, scale=.3] (v28) at (6,-2.5) {};
\node [circle, fill=black, scale=.3](v12) at (0.5,0.5) {};
\node [circle, fill=black, scale=.3](v20) at (2,0.5) {};
\node [circle, fill=black, scale=.3](v22) at (2.5,-2) {};
\node[circle, fill=black, scale=.3] (v23) at (4,-2) {};
\node[circle, fill=black, scale=.3] (v26) at (6,-4.5) {};
\node [circle, fill=black, scale=.3](v24) at (4.5,-4.5) {};
\node [circle, fill=black, scale=.3](v27) at (7.5,-4.5) {};
\draw [red, line width=2pt] (v11) edge (v21);
\draw [red, line width=2pt] (v21) edge (v28);
\draw (v3) edge (v11);
\draw (v11) -- (v12) -- (0,-1) -- (1,-1) -- (0.5,0.5);
\draw (v11) -- (v20) -- (1.5,-1) -- (2.5,-1) -- (v20);
\draw (v21) -- (v22) -- (2,-3.5) -- (3,-3.5) -- (v22) (v21) -- (v23) -- (3.5,-3.5) -- (4.5,-3.5) -- (v23);
\draw (v28) -- (v24) -- (4,-6) -- (5,-6) -- (v24) (v28) -- (v26) -- (5.5,-6) -- (6.5,-6) -- (6,-4.5) (6,-2.5) -- (v27) -- (7,-6) -- (8,-6) -- (v27);
\draw [green, line width=1pt] plot[smooth, tension=.7] coordinates {(-9,-5.5) (-8.5,-5) (-7.5,-5.5)};
\draw [green, line width=1pt] plot[smooth, tension=.7] coordinates {(-6,-5.5) (-4.5,-5.5) (-3.5,-3)};
\draw [green, line width=1pt] plot[smooth, tension=.7] coordinates {(-5,-3) (-4,-0.5) (-1,-0.5)};
\draw [green, line width=1pt] plot[smooth, tension=.7] coordinates {(0.5,-0.5) (1.5,-4.5) (4.5,-5.5)};
\draw [green, line width=1pt] plot[smooth, tension=.7] coordinates {(2.5,-3) (3.5,-2.5) (4,-3)};
\draw [blue, line width=1pt] plot[smooth, tension=1] coordinates {(-2.5,-0.5) (-0.5,-2) (2,-0.5)};
\draw [blue, line width=1pt] plot[smooth, tension=.7] coordinates {(-6,-5) (-1.5,-5) (2.5,-6.5) (6,-5.5)};
\draw [blue, line width=1pt] plot[smooth, tension=.7] coordinates {(2.5,4.5) (3.5,4) (4,4.5)};
\draw [blue, line width=1pt] plot[smooth, tension=.7] coordinates {(6.5,7) (6,2.5) (7.5,-2) (7.5,-5.5)};
\draw [blue, line width=1pt] plot[smooth, tension=.7] coordinates {(4,5) (5,5.5) (5,7)};
\varepsilonnd{tikzpicture}
\caption{\langlebel{figure:single-branch}Left (Claim~\ref{claim:singleBranch}): The red nodes form our branch $B$, and the blue nodes form the set $\ensuremath{\mathrm{children}}((\{a^*\}
\cup \ensuremath{\mathrm{anc}}(a^*)) \cap B)$. The triangles are the subtrees participating in the $\textsc{Partial VC}\xspace$ instance.
Right (Lemma~\ref{lem:chains}): The red nodes form our two incomparable branches. The green edges are internal edges, while the blue edges are external.}
\varepsilonnd{figure}
\begin{claim}[Warm-up]
\langlebel{claim:singleBranch}
Assume there exists a $(1+\delta)$-approximation algorithm for $\PartialVC k$ running
in time $f(k) \cdot g(n)$. Suppose the set of anchors $A'$ with at least one child in
$\mathsf{Opt}'(r)$ is contained within a single branch $B$. Then there is an
algorithm computing a solution with $\mathsf{Saved}$ value at least $\varepsilonll'(r)
- \delta(1+\varepsilon_1)(k-1){\mu}$, running in time $O(n \cdot (n^2 + f(k) \cdot g(n)))$.
\varepsilonnd{claim}
\begin{proof}
If all of $A'$ lies on $B$, the minimal anchor $a^* \in A^*$ must also
be in $B$. Moreover, for every $a_i^*\in\mathsf{Opt}'(r)$, its parent is
either $a^*$ or an ancestor of $a^*$, which means that $\mathsf{Opt}'(r) \subseteq
\ensuremath{\mathrm{children}}((\{a^*\} \cup \ensuremath{\mathrm{anc}}(a^*)) \cap B)$. Since the nodes in $\ensuremath{\mathrm{children}}((\{a^*\}
\cup \ensuremath{\mathrm{anc}}(a^*)) \cap B)$ are incomparable (see Figure~\ref{figure:single-branch}), we can construct the same
graph as the one in Lemma~\ref{lem:compute-anchors} on all these nodes
in $\ensuremath{\mathrm{children}}((\{a^*\} \cup \ensuremath{\mathrm{anc}}(a^*)) \cap B)$ and run the \textsc{Partial VC}\xspace-based
algorithm to get the same $\mathsf{Saved}$ guarantees (see
Algorithm~\ref{alg:subtreePVC}).
Therefore, the algorithm guesses the location of $a^*$ inside $B$ by
trying all possible $|B|=O(n)$ nodes, and for each choice of $a^*$,
runs the $(1-\delta)$-approximate \textsc{Partial VC}\xspace-based algorithm from
Lemma~\ref{lem:compute-anchors} on the corresponding graph (see
Algorithm~\ref{alg:singleBranch}).
\varepsilonnd{proof}
Now for the general case. Consider $\mathsf{Opt}'(r)$ and the set of all
branches $\mathcal B$. Let $\mathcal B^* \subseteq \mathcal B$ be the incomparable branches that
contain the minimal anchors, i.e., those in $A^*$. We classify the
$\varepsilonll(r')$ saved edges in $\mathsf{Opt}'(r)$ into two groups (see Figure~\ref{figure:single-branch}): if an edge is
saved between the subtrees below $a_i^*,a_j^* \in \mathsf{Opt}'(r)$ whose
parent(s) belong to the same branch in $\mathcal B^*$, then call this
\textit{an internal edge}. Otherwise, it is an \textit{external edge}:
these are saved edges in $\mathsf{Opt}'(r)$ that either go between two subtrees
in different branches, or between subtrees in the same branch in $\mathcal B
\setminus \mathcal B^*$. One of the two sets has $\ge \frac12\varepsilonll'(r)$ saved
edges, and we provide two separate algorithms, one to approximate each
group.
\begin{lemma}\langlebel{lem:chains}
Assume there exists a $(1+\delta)$-approximation algorithm for $\PartialVC k$ running
in time $f(k) \cdot g(n)$. Suppose that all anchors of $\mathsf{Opt}'(r)$ are contained in a set $\mathcal B$ of
$\le 2k-3$ branches. Then there is an algorithm that computes a
solution with $\mathsf{Saved}$ value $\ge \frac12\varepsilonll'(r) -
\delta(1+\varepsilon_1)(k-1){\mu}$, running in time $ 2^{O(k)} \cdot (n^2+f(k)\cdot g(n)) $.
\varepsilonnd{lemma}
\begin{proof}
\varepsilonmph{Case I: internal edges $\ge\frac12\varepsilonll'$.} For each branch
$B\in\mathcal B$ and each $s\in [k-1]$, compute a solution of $s$ nodes that
maximizes the number of internal edges \textit{within branch
$\mathcal B$}, in the same manner as in
Claim~\ref{claim:singleBranch}; this takes time $O(k^2n \cdot (n^2 + f(k) \cdot g(n)))$. Finally, guess all possible $\le
2^{2k-3}$ subsets of incomparable branches; for each subset
$\mathcal B'\subseteq\mathcal B$, try all vectors $\mathbf i \in [k-1]^{\mathcal B'}$ with
$\sum_{B\in\mathcal B'} i_B \le k-1$, look up the solution using $i_B$
vertices in branch $B$, and sum up the total number of internal
edges. Actually, trying all vectors $\mathbf i \in [k-1]^{\mathcal B'}$ takes $k^{O(k)}$ time, but we can speed up this step to $\mathrm{poly}(k)$ time using dynamic programming. Since one
of the guesses $\mathcal B'$ will be $\mathcal B^*$, the best solution will save at
$\ge\frac12\varepsilonll'(r)-\delta(1+\varepsilon_1)(k-1){\mu}$ edges. The total running time for this case is $O(k^2 \cdot f(k) \cdot g(n) + 2^{2k}\cdot\mathrm{poly}(k))$.
\varepsilonmph{Case II: external edges $\ge\frac12\varepsilonll'$.} Again, we guess the
set $\mathcal B^*\subset\mathcal B$ of incomparable branches containing minimal
anchors $A^*$. For a branch $B \in \mathcal B^*$, let $a_B:=(a \in B : B
\setminus a \subseteq \ensuremath{\mathrm{desc}}(a))$ be the ``highest'' node in $B$, that is an
ancestor of every other node in $B$. For each branch, we can replace
all nodes in $\mathsf{Opt}'(r)$ that are descendants of $a_B$ with just $a_B$;
doing can only increase the number of external edges. The new solution
has all nodes contained in the set
\[ \ensuremath{\mathrm{children}}\bigg(\ensuremath{\mathrm{anc}}\bigg(\bigcup_{B\in\mathcal B^*}\{a_B\}\bigg)\bigg), \]
which is a set of incomparable nodes. Therefore, we can construct the
graph of Lemma~\ref{lem:compute-anchors} and use the \textsc{Partial VC}\xspace-based
algorithm with this node set instead. This gives a solution with $\ge
\frac12\varepsilonll'(r)-\delta(1+\varepsilon_1)(k-1){\mu}$ saved edges. The total running time for this case is $O(2^{2k}\cdot (n^2 + f(k) \cdot g(n)))$.
\varepsilonnd{proof}
\subsection{Combining Things Together}
Putting things together, we conclude with Theorem~\ref{thm:laminar}. We
refer the reader to Algorithm~\ref{alg:laminar} for the pseudocode of
the entire algorithm.
\begin{proof}[Proof (Theorem~\ref{thm:laminar}).]
Let the original graph be $G=(V,E,w).$ We compute a $(1+\varepsilon_1)$-mincut
tree $\mathcal T=(V_{\mathcal T},E_{\mathcal T},w_{\mathcal T})$ with mapping $\phi:V\to V_{\mathcal T}$ in time $O(n^3)$, following Theorem~\ref{thm:mincutTreeExistence}.
Then, by running the two algorithms in Lemma~\ref{lem:incomparable} and
Lemma~\ref{lem:chains}, we compute a solution with $s \le k-1$
vertices with $\mathsf{Saved}$ value at least
\begin{align*}
& \max\left\{
\frac14\varepsilon_3(k-1){\mu} - \delta(1+\varepsilon_1)(k-1){\mu},\
\frac12\varepsilonll'(r) - \delta(1+\varepsilon_1)(k-1){\mu} \right\} \\
= & \max\left\{ \frac14\varepsilon_3(k-1){\mu}, \ \frac12\varepsilonll'(r) \right\} - \delta(1+\varepsilon_1)(k-1){\mu}
\varepsilonnd{align*}
for each root $r \in V_{\mathcal T}$ (see
Algorithm~\ref{alg:laminarRooted}). Using $\max\{p, q\} \geq
(4p+2q)/6$ and
$\varepsilonll'(r) \ge \varepsilonll^*(r) - \varepsilon_3(k-1){\mu}$
we get a solution with $\mathsf{Saved}$ value at least
\begin{align*}
& \frac16 \left( 4 \cdot \frac14\varepsilon_3(k-1){\mu} + 2 \cdot \frac12 \left[ \varepsilonll^*(r) - \varepsilon_3(k-1){\mu} \right] \right) - \delta(1+\varepsilon_1)(k-1){\mu} \\
\ge & \frac16\varepsilonll^*(r) - 2\delta(k-1){\mu},
\varepsilonnd{align*}
using that $\varepsilon_1 \leq 1$. In particular, the best solution
$v_1,\ldots,v_s \in V_{\mathcal T}$ over all $r$ satisfies
\[ \mathsf{Saved}(v_1,\ldots,v_s) \ge \frac16 \varepsilonll^* - 2\delta(k-1){\mu} ,\]
where $\varepsilonll^*(r)$ was replaced by $\varepsilonll^*$.
Let $v_1,\ldots,v_s \in V_{\mathcal T}$ be our solution with
$\mathsf{Saved}(v_1,\ldots,v_s) \ge \frac16 \varepsilonll^* - 2\delta(k-1){\mu}$. Let
$S_1,\ldots,S_s \subset V$ be the corresponding subsets in $V$, i.e.,
$S_i := \phi^{-1}(\ensuremath{\mathrm{subtree}}(v_i))$. Then, add the complement set
$S_{s+1}:=V \setminus \bigcup_{i\in[s]}S_i$ to the solution, so that
the sets $S_i$ partition $V$, and
\[w(E(S_1,\ldots,S_{s+1})) \le s(1+\varepsilon_1){\mu} - \left(\frac16 \varepsilonll^* -
2\delta(k-1){\mu}\right). \] Then, extend the solution to a
$k$-partition using Algorithm~\ref{alg:complete}. We now claim that every additional
cut that Algorithm~\ref{alg:complete} makes is a $(1+\varepsilon_1)$-mincut.
To see this, observe that $S_1^*, \ldots, S_{k-1}^*$ are all $(1+\varepsilon_1)$-mincuts and one
of them, say $S_j^*$, has to intersect some $S_i$. Then, the cut $(S_i \cap S_j^*, S_i \setminus S_j^*)$
is a $(1+\varepsilon_1)$-mincut in $S_i$. We can repeat this argument as long as we have $< k$ components $S_i$.
At the end, we have a
solution $S_1', \ldots, S_k'$ satisfying
\begin{align*}
w(E(S_1', \ldots, S_k')) & \le w(E(S_1,\ldots,S_s)) +
(k-1-s)(1+\varepsilon_1){\mu}
\\
& \le (k-1)(1+\varepsilon_1){\mu}- \left(\frac16 \varepsilonll^* -
2\delta(k-1){\mu}\right)
\varepsilonnd{align*}
Let $S^*_1,\ldots,S^*_k$ be the optimal partition in $\mathcal O_{\varepsilon_1}$
satisfying $\phi(r) \in S^*_k$, and let $\varepsilonll^*$ be the maximum of
$\mathsf{Saved}(v_1^*,\ldots,v_{k-1}^*)$ over incomparable
$v_1^*,\ldots,v_{k-1}^*$. Our solution has approximation ratio
\begin{align*}
\frac{w(E(S_1,\ldots,S_k))}{w(E(S_1^*,\ldots,S_k^*))} & \le
\frac{(k-1)(1+\varepsilon_1){\mu} - \frac16\varepsilonll^* +
2\delta(k-1){\mu}}{(k-1){\mu} - \varepsilonll^*} \\
& = \frac{(k-1)(1+\varepsilon_1){\mu} - \frac16\varepsilonll^* }{(k-1){\mu} -
\varepsilonll^*} + \frac{ 2\delta(k-1){\mu}}{(k-1){\mu} - \varepsilonll^*} \\
& \le 2(1+\varepsilon_1) - \frac16 + 4\delta,
\varepsilonnd{align*}
with the worst case achieved at $\varepsilonll^*=\frac12(k-1){\mu}$, which is
the highest $\varepsilonll^*$ can be. Setting $\varepsilon_2:=1/6 - 2\varepsilon_1-4\delta$
concludes the proof.
As for running time, we run the algorithms in Lemma~\ref{lem:incomparable} and Lemma~\ref{lem:chains} sequentially, and the final running time is $\tilde 2^{O(k)}f(k)(\tilde O(n^4) + g(n))$. (The $\tilde O(n^4)$ comes from the case when $k\le 4$, in which we solve the problem exactly in $\tilde O(n^4)$ time.)
\varepsilonnd{proof}
\newcommand{\mathsf{Wdeg}}{\mathsf{Wdeg}}
\section{An FPT-AS for \textsc{Partial VC}\xspacelong}
\langlebel{sec:partial-vc}
Recall the \textsc{Partial VC}\xspacelong (\textsc{Partial VC}\xspace) problem: the input is a graph $G = (V,E)$
with edge and vertex weights, and an integer $k$. For a set $S$, define
$E_S$ to be the set of edges with at least one endpoint in $S$. The goal
of the problem is to find a set $S$ with size $|S| = k$, minimizing the
weight $w(E_S) + w(S)$, i.e., the weight of all edges hitting $S$ plus
the weight of all vertices in $S$. Our main theorem is the following.
\begin{theorem}[\textsc{Partial VC}\xspacelong]
\langlebel{thm:pvc}
There is a randomized algorithm for \textsc{Partial VC}\xspace on weighted graphs that, for
any $\delta \in (0,1)$, runs in $O(2^{k^6/\delta^3} (m +
k^8/\delta^3)\,n \log n)$ time and outputs a
$(1+\delta)$-approximation to \textsc{Partial VC}\xspace with probability $1 - 1/\mathrm{poly}(n)$.
\varepsilonnd{theorem}
We first extend a result of Marx~\cite{Marx07} to give a
$(1+\delta)$-approximation algorithm for the case where $G$ has edge
weights being integers in $\{1, \ldots, M\}$ and no vertex weights, and
then show how to reduce the general case to this special case, losing
only another $(1+\delta)$-factor.
\subsection{Graphs with Bounded Weights}
\begin{lemma}
\langlebel{lem:pvc-simple}
Let $\delta \leq 1$. There is a randomized algorithm for the \textsc{Partial VC}\xspace
problem on simple graphs with edge weights in $\{1, \ldots, M\}$ (and
no vertex weights) that runs in $O(m+Mk^4/\delta)$ time, and outputs
a $(1+\delta)$-approximation with probability at least
$2^{-(Mk^2/\delta)}$.
\varepsilonnd{lemma}
\begin{proof}
This is a simple extension of a result for the
maximization case given by Marx~\cite{Marx07}. We give two algorithms: one for the case when the
optimal value is smaller than $\tau := Mk^2/\delta$ (which returns the
correct solution in time, but with probability $2^{-(Mk^2/\delta)}$),
and another for the case of the optimal value being at least $\tau$
(which deterministically returns a $(1+\delta)$-approximation in
linear time). We run both and return the better of the two solutions.
First, the case when the optimal value is at least $\tau$. Let the
\varepsilonmph{weighted degree} of a node $v$, denoted $w(\partial v)$ be
defined as $\sum_{e: v \in e} w(e)$. Observe that for
any set $S$ with $|S| \leq k$,
\[ 0 \leq \sum_{v \in S} w(\partial v) - w(E_S) \leq M\cdot
\binom{k}{2}. \] Hence, if $S^*$ is the optimal solution and
$w(E_{S^*}) \geq \tau$, then picking the set of $k$ vertices with the
least weighted degrees is a $(1+\delta)$-approximation.
Now for the case when the optimal value is at most $\tau$. In this case,
the optimal set $S^*$ can have at most $\tau$ edges incident to it, since
each edge must have weight at least $1$. Consider the color-coding
scheme where we independently and uniformly colors the vertices of $G$
with two colors (red and blue). With probability $2^{-(\tau+k)}$, all the
vertices in $S^*$ are colored red, and all the vertices in $N(S^*)
\setminus S^*$ are colored blue. Consider the ``red components'' in
the graph obtained by deleting the blue vertices. Then $S^*$ is the
union of one or more of these red components. To find it, define the
``size'' of a red component $C$ as the number of vertices in it, and
the ``cost'' as the total weight of edges in $G$ that are incident to
it (i.e., cost $= \sum_{e \in E: e \cap C \neq \varepsilonmptyset} w(e)$.)
Now we can use dynamic programming to find a collection of red
components with total size equal to $k$ and minimum total cost: this
gives us $S$ (or some other solution of equal cost). Indeed, if we
define the ``type'' of each component to be the tuple $(s,c)$ where $s
\in [1\ldots k]$ is the size (we can drop components of size greater
than $k$) and $c \in [1\ldots \tau]$ is the cost (we can drop all
components of greater cost). Let $T(s,c)$ be the number of copies of
type $(s,c)$, capped at $k$. Assume the types are numbered $\tau_1,
\tau_2, \ldots, \tau_{k\tau}$. Now if $C(i,j)$ is the minimum cost we can
have with components of type $\leq \tau_i = (s,c)$ whose total size is
$j$, then
\[ C(i,j) = \min_{0 \leq \varepsilonll \leq T(s,c)} C(i-1, j - \varepsilonll s) + \varepsilonll
c. \] Finally, we return the component achieving $C(k\tau,k)$. This can
all be done in $O(m + k^2\tau)$ time.
\varepsilonnd{proof}
Repeating the algorithm $O(2^{\tau + k} \log n) = O(2^{Mk^2/\delta + k} \log n)$ times and outputting
the best set found in these repetitions gives an algorithm that finds a
$(1+\delta)$-approximation with probability $1 - 1/\mathrm{poly}(n)$.
\subsection{Solving The General Case}
We now reduce the general \textsc{Partial VC}\xspace problem, where we have no bounds on the
edge weights (and we have vertex weights), to the special case from the
previous section.
The idea is simple: given a graph $G = (V,E)$ with edge and vertex
weights, we construct a collection of $|V|$ simple graphs $\{ H_v \}_{v
\in V}$, each defined on the vertex set $V$ plus a couple new nodes,
and having $O(|V|+|E|)$ edges, with each edge-weight $w'(e)$ being an
integer in $\{1, \ldots, M\}$ and $M = O(k/\delta)^2$, and with no vertex
weights. We find a $(1+\delta/2)$-approximate \textsc{Partial VC}\xspace solution on each $H_v$,
and then output the set $S$ which has the smallest weight (in $G$) among
these. We show how to ensure that $S \subseteq V$ and that it is a
$(1+\delta)$-approximation of the optimal solution in $G$.
\begin{proof}[Proof of Theorem~\ref{thm:pvc}]
Let $S^*$ be an optimal solution on $G$. Define the \varepsilonmph{extended
weighted degree} of a vertex $v$, denoted by $\mathsf{Wdeg}(v)$, to be its
vertex weight plus the weight of all edges adjacent to it. I.e.,
$\mathsf{Wdeg}(v) := w(v) + w(\partial v)$.
Firstly, assume we know a vertex $v^* \in S^*$ with the largest
$\mathsf{Wdeg}(v^*)$; we just enumerate over all vertices to find this vertex.
We now proceed to construct the graph $H_{v^*}$. Let $L =
\mathsf{Wdeg}(v^*)$, and delete all vertices $u$ with $\mathsf{Wdeg}(u) > L$. Note
that (a)~any solution containing $v^*$ has total weight at least $L$,
and (b)~each remaining edge and vertex has weight $\leq L$.
Assume that $G$ is simple, since we can combine parallel edges
together by summing their weights. Create two new vertices $p, q$,
and add an edge of weight $L k^2$ between them; this ensures that
neither of these vertices is ever chosen in any near-optimal
solution.
Let $\delta' > 0$ be a parameter to be fixed later; think of $\delta'
\approx \delta$. For each edge $e = (u,v)$ in the edge set $E$ that
has weight $w(e) < L\delta'/k^2$, remove this edge and add its weight
$w(e)$ to the weight of both its endpoints $u,v$. Finally, when there
are no more edges with $w(e) < L\delta'/k^2$, for each vertex $u$ in
$V$, create a new edge $\{u,p\}$ with weight being equal to the
current vertex weight $w(u)$, and zero out the vertex weight. Let the
new edge set be denoted by $E'$. We claim that for any set $S
\subseteq V$ of size $\le k$,
\[ \left( \sum_{e \in E': e \cap S} w(e)\right) - \left( \sum_{e \in
E: e \cap S} w(e) + \sum_{v \in S} w(v) \right) \leq \delta' L. \]
Indeed, the only change comes because of edges with weight $w(e) <
L\delta'/k^2$ and with both endpoints within $S$---these edges
contributed once earlier, but replacing them by the two edges means we
now count them twice. Since there are at most $\binom{k}{2}$ such
edges, they can add at most $\delta' L$.
At this point, all edges in the original edge set $E$ have weights in
$[L \delta'/k^2, Lk^2]$; the only edges potentially having weights $<
L \delta'/k^2$ are those between vertices and the new vertex $p$. For
any such edge with weight $< L \delta'/k$, we delete the edge. This
again changes the optimal solution by at most an additive $L \delta'$,
and ensure all edges in the new graph have weights in $[L \delta'/k^2,
Lk^2]$. Note that since the optimal solution has value at least $L$
by our guess, these additive changes of $L \delta'$ to the optimal
solution mean a multiplicative change of only $(1+\delta')$.
Finally, discretize the edge weights by rounding each edge weight to
the closest integer multiple of $L \delta'^2/k^2$. Since each edge
weight $\geq L \delta'/k^2$, each edge weight incurs a further
multiplicative error at most $1+\delta'$. Note that $M =
k^4/\delta'^2$. Now use Lemma~\ref{lem:pvc-simple} to get a
$(1+\delta')$-approximation for \textsc{Partial VC}\xspace on this instance with high
probability. Setting $\delta' = O(\delta)$ ensures that this solution
is within a factor $(1+\delta)$ of that in $G$.
\varepsilonnd{proof}
\section{Conclusion and Open Problems}
\langlebel{sec:conclusion}
Putting the sections together, we conclude with a proof of our main theorem.
\begin{proof}[Proof of Theorem~\ref{thm:kcut-main}]
Fix some $\delta \in (0,1/24)$. By Theorem~\ref{thm:pvc}, there is a
$(1+\delta)$-approximation algorithm for $\PartialVC k$ running in
time $O(2^{k^6/\delta^3} (m + k^8/\delta^3)\,n \log n) = 2^{O(k^6)}
n^4$ time. Plugging in $f(k) := 2^{O(k^6)}$ and $g(n) := n^4$ into
Theorem~\ref{thm:laminar}, we get a $(2-\varepsilon_2)$-approximation algorithm
to $\text{Laminar}cut k{\varepsilon_1}$ in time $2^{O(k)}f(k)(n^3 + g(n)) =
2^{O(k^6)} n^4$, for a fixed $\varepsilon_1 \in (0,1/6-4\delta)$. Plugging in
$f(k):=2^{O(k^6)}$ and $g(n):=n^4$ into Theorem~\ref{thm:reduction1}
gives a $(2-\varepsilon_3)$-approximation for $$k$\textsc{-Cut}\xspace$ in time $2^{O(k^2 \log k)} \cdot
f(k) \cdot (n^4 \log^3 n + g(n)) = 2^{O(k^6)} n^4 \log^3 n$.
Finally, for our approximation factor. Theorem~\ref{thm:laminar} sets
$\varepsilon_2:=1/6 - 2\varepsilon_1 - 4\delta$ for any small enough $\delta$. We can
take $\varepsilon_1$ and $\varepsilon_2$ to be equal, so that $\varepsilon_1 = \varepsilon_2 = 1/18 -
\nicefrac43\cdot\delta$. Finally, setting
$\varepsilon_4=\varepsilon_5=\min(\varepsilon_1,\varepsilon_2)/3$ and $\varepsilon_3:=\varepsilon_4^2$ in
Theorem~\ref{thm:reduction1} gives $\varepsilon_3 = 1/54^2 - \delta'$ for some
arbitrarily small $\delta'>0$. In other words, our approximation
factor is $2 - 1/54^2 + \delta'$, or $1.9997$ for an appropriately
small $\delta'$.
\varepsilonnd{proof}
Our result
combines ideas from approximation algorithms and FPT algorithms and
shows that considering both settings simultaneously can help bypass
lower bounds in each individual setting, namely the $W[1]$-hardness of
an exact FPT algorithm and the SSE-hardness of a polynomial-time
$(2-\varepsilon)$-approximation. While our improvement is quantitatively modest,
we hope it will prove qualitatively significant. Indeed, we hope these
and other ideas will help resolve whether an $(1+\varepsilon)$-approximation
algorithm exists in FPT time, and to show a matching lower and upper
bound.
\paragraph{Acknowledgments.} We thank Marek Cygan for generously giving
his time to many valuable discussions.
{\small
}
\appendix
\section{Pseudocode for \text{Laminar}cut{$k$}{$\varepsilon_1$}}
\langlebel{sec:pseudocode-laminar}
\begin{algorithm}
\caption{SubtreePartialVC$(G,\mathcal T,A,s,\delta)$}
\langlebel{alg:subtreePVC}
\begin{algorithmic}
\If {$|A| < s$}
\State \mathbb Return None
\EndIf
\mathbb For {$a \in A$}
\State $C_a \gets V(a) \cup \displaystyle\bigcup\limits_{a' \in \ensuremath{\mathrm{desc}}(a)}V(a')$
\EndFor \Comment{\textbf{Assert}: $C_a$ are all disjoint}
\
\State $\mathcal C \gets \{ C_a : a \in A\}$
\State $H \gets \text{Contract}(G, \mathcal C)$ \Comment{For each $C_a \in \mathcal C$, contract all vertices in $C_a$ into a single vertex in $H$}
\
\mathbb For {$i \in [k-1]$}
\State $P_{i} \gets \text{PartialVC}(H,i)$ \Comment{$P_{i} \in V(H)^i$}
\State $\mathcal S_i \gets \text{Expand}(H,P_i)$ \Comment { \parbox[t]{.5\linewidth}{ Map each $v \in P_i$ to the set of vertices in $V$ which contract to $v$ in $H$, and call the result $\mathcal S_i \in \left(2^V\right)^i$ } }
\EndFor
\State \mathbb Return $\{\mathcal S_{i} : i \in [s]\}$
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
\begin{algorithm}
\caption{SingleBranch$(G,\mathcal T,B,k,\delta)$}
\langlebel{alg:singleBranch}
\begin{algorithmic}
\mathbb For{$a \in B$}
\State $\mathbb Record(\text{SubtreePartialVC}(G, \mathcal T, \ensuremath{\mathrm{children}} \left((\{a\} \cup \ensuremath{\mathrm{anc}}(a)) \cap B \right), k-1, \delta))$
\EndFor
\State Return the best recorded solution $\{v_1, \ldots, v_{k-1}\} \in V_{\mathcal T}$.
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
\begin{algorithm}
\caption{Laminar$(G=(V,E,w),\mathcal T,k,\varepsilon_1,\delta)$}
\langlebel{alg:laminar}
\begin{algorithmic}
\State $\mathcal T=(V_{\mathcal T},E_{\mathcal T},w_{\mathcal T}) \gets \text{MincutTree}(G)$.
\mathbb For{$r \in V_{\mathcal T}$}
\State Root $\mathcal T$ at $r$.
\State $\mathbb Record(\text{LaminarRooted}(G,\mathcal T,r,k,\varepsilon_1,\delta))$
\EndFor
\State Return the best recorded $k$-partition.
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
\begin{algorithm}
\caption{LaminarRooted$(G=(V,E,w),\mathcal T,r,k,\delta_1,\delta)$}
\langlebel{alg:laminarRooted}
\begin{algorithmic}
\mathbb For {$a \in V(\mathcal T)$}
\State $\{S_{a,i} : i \in [k-1]\} \gets \text{SubtreePartialVC}(G, \mathcal T, \ensuremath{\mathrm{children}}(a), k-1, \delta)$ \Comment{$S_{a,i} \in \left(2^V\right)^i$}
\EndFor
\
\State $A \gets \varepsilonmptyset$ \Comment{$A \subset V(\mathcal T) \times [k]$ is the set of \textit{anchors}}
\mathbb For {$a \in V(\mathcal T)$ in topological order from leaf to root}
\State $\varepsilon_3 \gets \frac{1-\delta}4-2\varepsilon_1$ \Comment {The optimal value of $\varepsilon_3$}
\State $I_a \gets \{i \in [k-1] : \text{Value}(P_{a,i}) \ge \varepsilon_3(1-\delta)(i-1){\mu} \}$
\If {$I_a \ne \varepsilonmptyset$ \textbf{and} $\nexists (a',i) \in A : a' \in \ensuremath{\mathrm{desc}}(a)$} \Comment{Only take \textit{minimal} anchors}
\State $A \gets A \cup \{(a, \max I_a)\}$
\EndIf
\EndFor
\
\If {$|A| \ge k-1$} \Comment{\textbf{Case (K)}: Knapsack}
\State $A' \gets \text{Knapsack}(A)$ \Comment{The Knapsack algorithm as described in Lemma~\ref{lem:incomparable}}
\State $\mathcal S \gets \displaystyle\bigcup\limits_{(a,i) \in A} \{S_{a,i}\}$ \Comment{The partition for Case (K), to be computed. \textbf{Assert}: $|S| \le k-1$}
\State $\mathbb Record(\text{Complete}(G,k,\mathcal S))$
\Else
\State $\mathcal B \gets \text{Branches}(A)$ \Comment{$\mathcal B \subset \left( 2^{V(\mathcal T)} \right)^r$ for some $k-1 \le r \le 2k-3$}
\
\mathbb For {$B \in \mathcal B$} \Comment {\textbf{Case (B1)}: Compute branches independently}
\State $\{P_{B,i} : i \in [k-1]\} \gets \text{SingleBranch}(G, \mathcal T, B, k-1, \delta)$ \Comment{$P_{B,i} \in V^i$}
\EndFor
\State $(\mathcal B^*,\mathbf i^*) \gets \operatornamewithlimits{argmin}\limits_{ \substack {\mathcal B' \subset \mathcal B \text{ incomparable},\\ \mathbf i \in [k-1]^{\mathcal B'} : \ \sum_{B} i_B\ =\ k-1} } \ \displaystyle\sum\limits_{B \in \mathcal B'} w(E(P_{B,i_B}))$ \Comment{Computed by brute force}
\State $\mathcal S_1 \gets \bigcup_{B \in \mathcal B^*} \{P_{B, i_B}\}$ \Comment{The partition in Case (B1)}
\State $\mathbb Record(\text{Complete}(G,k,\mathcal S_1))$
\
\mathbb For {$B \in \mathcal B$} \Comment{\textbf{Case (B2)}: Guess the branches with the anchors}
\State $a_B \gets (a \in B : B \setminus a \subset \ensuremath{\mathrm{desc}}(a))$ \Comment{$a_B$ is the common ancestor of branch $B$}
\EndFor
\mathbb For {$\mathcal B' \subset \mathcal B$ s.t.\ $\nexists B_1,B_2\in\mathcal B' : B_1 \subset \ensuremath{\mathrm{desc}}(B_2)$} \Comment{Subsets whose branches are incomparable}
\State $A_{\mathcal B'} \gets \ensuremath{\mathrm{children}}\left(\bigcup_{B \in \mathcal B'} \left( \{a_B\} \cup \ensuremath{\mathrm{anc}}(a_B) \right) \right)$
\State $\mathcal S_{2,\mathcal B'} \gets \text{SubtreePartialVC}(G,\mathcal T,A_{\mathcal B'},k-1,\delta)$ \Comment{The partition for $\mathcal B'$ in Case (B2)}
\State $\mathbb Record(\text{Complete}(G,k,\mathcal S_{2,\mathcal B'}))$
\EndFor
\EndIf
\
\State Return the best recorded $k$-partition.
\varepsilonnd{algorithmic}
\varepsilonnd{algorithm}
\section{Missing Proofs}
\langlebel{sec:missing-proofs}
\begin{lemma} \langlebel{lemma:knapsack}
Consider the knapsack instance of $k-1$ items $i \in [k-1]$ where item $i$ has size $s_i \in [2,k-1]$ and value $s_i-1$. There is an algorithm achieving value $\ge (k-1)/4$ for $k \ge 5$, running in $O(k)$ time.
\varepsilonnd{lemma}
\begin{proof}
Consider the greedy
knapsack solution where we always choose the heaviest item, if still
possible. Let $A \in [k-1]$ be our solution. If our total size
$\sum_{i\in A}s_i$ is at least $k - 1 - \sqrt k$, then our value is
at least $\sum_{i\in A}(s_i-1) \ge \sum_{i\in A}s_i/2 \ge (k-1-\sqrt
k)/2$. Otherwise, since we could not fit the next item of size at
least $\sqrt k$ into our solution, all of our items have size at
least $\sqrt k$. Furthermore, our total solution size is at least
$(k-1)/2$, so $\sum_{i \in A}(s_i-1) \ge \sum_{i\in A}(1-1/\sqrt
k)s_i \ge (1-1/\sqrt k)(k-1)/2$. When $k \ge 5$, the
value is $\ge (1-1/\sqrt 5)(k-1)/2 \ge (k-1)/4$.
\varepsilonnd{proof}
\varepsilonnd{document} | math | 111,472 |
\begin{document}
\begin{frontmatter}
\title{\vspace*{-25pt}Matched detectors as definers of force\\[-40pt]}
\author[madjid]{F. Hadi Madjid} and
\author[myers]{John M. Myers}
\address[madjid]{82 Powers Road, Concord, Massachusetts 01742}
\address[myers]{Gordon McKay Laboratory, Division of Engineering and
Applied Sciences, Harvard University, Cambridge, Massachusetts 02138}
\begin{abstract}
Although quantum states nicely express interference effects, outcomes
of experimental trials show no states directly; they indicate
properties of probability distributions for outcomes. We prove
categorically that probability distributions leave open a choice of
quantum states and operators and particles, resolvable only by a move
beyond logic, which, inspired or not, can be characterized as a guess.
By recognizing guesswork as inescapable in choosing quantum states and
particles, we free up the use of particles as theoretical inventions
by which to describe experiments with devices, and thereby replace the
postulate of state reductions by a theorem. By using the freedom to
invent probe particles in modeling light detection, we develop a
quantum model of the balancing of a light-induced force, with
application to models and detecting devices by which to better
distinguish one source of weak light from another. Finally, we
uncover a symmetry between entangled states and entangled detectors, a
dramatic example of how the judgment about what light state is
generated by a source depends on choosing how to model the detector of
that light.
\end{abstract}
\begin{keyword}
Quantum mechanics \sep Modeling \sep Detection \sep Metastability \sep
Agreement
\PACS 03.65.Ta \sep 03.65.Nk \sep 84.30.Sk
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{sec:1}
Connecting the bench of experiment and the blackboard of theory offers
physicists opportunities for creativity that we propose to make
explicit. Traditional views underplay the physicist's role in making
these connections. Although physicists have wished for mathematics
that would connect directly to experiments on the bench, the equations
of quantum mechanics express quantum states and operators not directly
visible in spectrometers or other devices. Here we look into quantum
mechanics as mathematical language used to model behaviors of devices
arranged on the laboratory bench. After separating models as
mathematical objects from any assertion that a certain model describes
a given experiment with devices, we ask: given a certain form of
model, which models, if any, fit the behavior of some particular
devices on the bench? In contrast to any hope for a seamless, unique
blackboard description of devices on a laboratory bench, we argue,
based on mathematical proofs presented in Sec. 3, that no matter what
experimental trials are made, if a quantum model generates calculated
probabilities that match given experimentally determined relative
frequencies, there are other quantum models that match as well but
that differ in their predictions for experiments not yet performed.
The proofs demonstrate what before could only be suspected:
between the two pillars of calculation and measurement must stand a
third pillar of choice making, involving personal creativity beyond
logic, so there can be no reason to expect or demand that any two
people choose alike.
What does recognizing choice mean for physicists? In physics, as in
artistic work, pleasure and joy come from the choices one makes that
lead to something interesting. Looking back, physicists can hardly
help noticing that their proudest accomplishments, whether theoretical
or experimental, have involved choices made by reaching beyond logic
on the basis of intuition, hunches, analogies---some kind of guess,
perhaps inspired, but still outside of logic. To understand the proofs
is to see opportunities for making guesses. Although
hunches and guesses and intuition can be as personal as dreams,
the recognition of guesswork as a permanent pillar of physics has
more than personal impact:
\begin{enumerate}
\item Describing device behavior will be recognized in Sec.~4 as a
bi-lingual enterprise, with a language of wrenches and lenses for
experimental trials on the bench and a different language of states
and operators for the blackboard, linked by metaphors as guesswork.
We show how freedom to choose particles as constituents of models of
devices both helps in modeling devices and allows us to replace a
widespread but questionable postulate of ``state reductions'' by a
theorem.
\item The need for bi-lingual descriptions bridging bench and
blackboard gives local color to certain concepts. In Sec. 5 we
develop a notion of {\em force} in the context of light detection that
gives meaning both at the blackboard and the bench not only to
expectation values of light forces, but also to higher-order
statistics associated with them, with application to models and
detecting devices by which to better distinguish one source of weak
light from another.
\item
In Sec. 6 we uncover a symmetry, pertaining to entanglement, to make
vivid the way judgments about how to model light are interdependent
with judgments about how to model light-detectors.
\end{enumerate}
\vfil
\section{Models mathematically distinct from experiments}\label{sec:2}
Experimental records can hold: (1) numerals interpreted as the
settings of knobs that control an experiment, and (2) numerals
interpreted as experimental outcomes, thought of as the clicks and
flashes and electronically tallied pulses by which the devices used in
the laboratory respond to and measure whatever is at issue. As an
abstraction by which one can model experimental outcomes, quantum
theory offers what we shall call {\em theoretical outcomes} (referred
to in the literature variously as {\em outcomes, results, the finding
of particles} and {\em the finding of states}). Probabilities
of theoretical outcomes are expressed in terms of states and operators
by what we shall call {\em quantum models}. We discuss
these first, and then distinguish the probabilities expressed
by models from relative frequencies of experimental outcomes.
\subsection{Definition of quantum-mechanical models}\label{subsec:2.1}
The propositions about linking models to devices (Sec. 3) can be
proved using any formulation of quantum mechanics that includes
probabilities of theoretical outcomes. Here is a standard formulation
taken from Dirac \cite{dirac} and von Neumann \cite{vN}, bulwarked by
a little measure theory \cite{rudin}; however, as discussed in Sec.\
\ref{sec:4}, we invoke no postulate of state reductions.
Let $H$ be a Hilbert space, let $\rho$ be any self-adjoint
operator of unit trace on $H$ (otherwise known as a density operator),
and let $M$ be a $\sigma$-algebra of subsets of a set $\Omega$ of
possible \textit{theoretical outcomes}. By a \textit{theoretical
outcome} we mean a number or a list of numbers as a mathematical
object, in contrast to an experimental outcome extracted from an
experimental record. Let $E$ be any projective resolution on $M$ of
the identity operator on $H$ (which implies that for any $\omega \in
M$, $E(\omega)$ is a self-adjoint projection \cite{rudin2}). Let
$U(t)$ be a unitary time-evolution operator (typically defined by a
Schr\"odinger equation or one of its relativistic
generalizations). These mathematical objects can be combined to define
a probability distribution $\mu$ on $M$, parameterized by $t$:
\begin{equation}\mu(t)(\omega) \stackrel{\rm def}{=}
\mathrm{Tr}[U(t)\rho U^\dag(t)E(\omega)],
\label{eq:mu} \end{equation} where $\mu(t)(\omega)$ is the probability of
an outcome in the subset $\omega$ of $\Omega$, for the parameter value
$t$.
For a probability $\mu(t)(\omega)$ to be compared to anything
experimental, one needs to make explicit the dependence of the $\rho$
and $E$ that generate it on the experimentally controllable
parameters. It is convenient to think of these parameters as the
settings of various knobs; to express them we let $A$ and $B$ be
mathematically arbitrary sets (interpreted as sets of knob
settings). Let $\mathcal{D}$ be the set of functions from $A$ to
density operators acting on $H$. Let $\mathcal{E}$ be the set of
functions from $B$ to projective resolutions of the identity on $M$ of
the identity operator on $H$. Then what we shall call {\em a specific
quantum model} is a triple of functions $(\rho,E,U)$ together
with $\rho \in \mathcal{D}$, $E \in \mathcal{E}$, and $U(t)$ a unitary
evolution operator. By the basic rule of quantum mechanics, such a
specific quantum model generates a probability-distribution as a
function of knob settings and time:
\begin{eqnarray}
&&(\forall\ a \in A,\ b \in B,\ t \in I,\ \omega \in
M)\nonumber\\[3pt]
&&\qquad\mu(a,b,t) (\omega)=\mathrm{Tr}[U(t)
\rho(a)U^\dag(t)E(b)(\omega)].
\label{eq:muab} \end{eqnarray}
Often one needs something less specific than a triple $(\rho,E,U)$.
By a {\em model} we shall mean a set of properties of that limit but
need not fully specify $\rho$, $U$, and~$E$. For example, in modeling
entangled light, we might construct a model in this sense by specifying
relevant symmetry properties of $E$, $U$, and $\rho$, leaving many
fine points unspecified.
\noindent\textbf{Remarks}
\begin{enumerate}
\item An element $a \in A$ can be a list: $a=\{a^{(1)},\dots
,a^{(n)}\}$; similarly an element $b \in B$ can be a list.
\item
By the {\em domain} of a model, we mean the cartesian product of
the sets $A \times B \times I \times M$. Models of a given domain
can differ as to the functions $\rho$, $E$, and $U$ defined on the
sets $A$, $B \times M$, and $I$, respectively, so that different
models with a given domain can differ in the states, operators, and
probabilities that they assert.
\item
In defining the domain of a model, we view $\mu$ as a function from $A
\times B \times I \times M \rightarrow [0,1]$, but in writing the left
side of Eq.\ (\ref{eq:muab}) we take the alternative view of $\mu$ as a
function $A\times B \times I \rightarrow \{\mbox{probability measures
on }M\}$.
\item
Different specific models can be distinguished by labels: a model
$\alpha$ consisting of $(\rho_\alpha,U_\alpha,E_\alpha)$ generates a
probability function $\mu_\alpha$. Because $\mu_\alpha$ is invariant
under any unitary transformation applied to all three of
$\rho_\alpha$, $E_\alpha$, and $U_\alpha$, an important feature of
the states asserted by a specific model $\alpha$ is the
unitary-invariant overlap between pairs of them. The measure of
overlap convenient to Sec.\ \ref{sec:3} is
\begin{equation} \mathrm{Overlap}(\rho_\alpha(a_1),\rho_\alpha(a_2))
\mathop{\stackrel{\mathrm{def}}{=}}\nolimits \mathrm{Tr}[\rho_\alpha(a_1)^{1/2}\rho_\alpha(a_2)^{1/2}].
\label{eq:overlap}\end{equation}
We express the difference in state preparations asserted by specific
models $\alpha$ and $\beta$ having the same domain for elements $a_1,
a_2 \in A_\alpha \cap A_\beta$ by
$|\mathrm{Overlap}(\rho_\alpha(a_1),\rho_\alpha(a_2)) -
\mathrm{Overlap}(\rho_\beta(a_1),\rho_\beta(a_2))|$.
\item We call a model $\beta$ a {\em restriction} of a model $\alpha$ if
the domain of model $\beta$ is a subset of the domain of model $\alpha$.
\end{enumerate}
\subsection{Domains of models and of experimental records}
\label{subsec:2.2}
So much for quantum models as mathematical objects; how do we compare
probabilities from these models with results of an experiment with
lasers and lenses and other devices? First one contrives to view the
experiment as consisting of trials, each for certain settings of some
knobs, yielding at each trial one of several possible experimental
outcomes. By tallying the experimental outcomes for various knob
settings, one extracts from the experimental record the relative
frequencies of experimental outcomes as a function on a domain of
experimental knob settings and outcomes.
To compare experimental relative frequencies with probabilities
calculated from a model, both viewed as functions on domains of knob
settings and outcome bins, it is necessary to identify the experimental
domain as a subset of the model domain. This entails associating to
each experimental outcome $c$ some model outcome $\omega_c \in M$.
For the experimental relative-frequency of outcome $c$ for each
setting of knobs $(a,b,t)$ in the experimentally covered subset of $A
\times B \times I$ we write $\nu_r(a,b,t)(c)$; this is the ratio of
the number of trials with knob settings and time $(a,b,t)$ and an
experimental outcome in $c$ to the number of trials with knob settings
and time $(a,b,t)$ regardless of the outcome. Letting $C$ denote
the set of experimental outcomes, one has
\begin{equation}(\forall\ t \in I,\ a \in A,\ b \in B)\quad
\sum_{c\in C}
\nu_r(a,b,t)(c) = 1.\label{eq:nudef}
\end{equation}
By virtue of the mapping $c \mapsto \omega_c \subset \Omega$, one can
compare the experimental relative-frequency function $\nu_r$ with the
probability function $\mu_{\alpha}$ asserted by any model~$\alpha$
having a domain containing a subset identified with the domain of the
experimental relative frequencies. Because of this need for
identification, a choice of model domain constrains the design, or at
least the interpretation, of experimental records to which models of
that domain can be compared. In compensation, committing oneself to
thinking about an experimental endeavor in terms of a particular model
domain makes it possible to:
\begin{enumerate}
\item organize experiments to generate data that can be compared with
models having that domain;
\item express the results of an experiment mathematically without
having to assert that the results fit any particular model;
\item pose the question of whether the experimental data fit one model
having that domain better than they fit another model \cite{ams}.
\end{enumerate}
\section{Choosing a model to fit given probabilities}\label{sec:3}
What can relative frequencies of outcomes of experimental trials of
devices tell us about how to model those devices? At best, from
relative frequencies of experimental outcomes as a function of knob
settings, as in Eq.\ (\ref{eq:nudef}), one abstracts an approximation
to a probability-distribution function of knob settings. By ignoring
statistical and other error, we picture an ideal case of arriving at
some $\mu_\alpha(a,b)(\omega)$, but without any further information
concerning the states or operators that generate it. This raises a
question inverse to the text-book task of calculating probabilities
from states and operators: given the probability function
$\mu_\alpha(a,b)(\omega)$, what states and operators generate it via
the rule of Eq.\ (\ref{eq:muab})? Put another way, what are the
constraints and freedoms on a density-operator function $\rho$, an
evolution operator $U(t)$, and a measurement-function $E$ if these are
to generate a given probability function $\mu_\alpha(a,b)(\omega)$ via
the rule of Eq.\ (\ref{eq:muab})? Here are some answers.
\subsection{Constraint on density operators}\label{subsec:3.1}
If for some values $a_1$, $a_2$, $b$, and $\omega$ one has
$\mu(a_1,b,t)(\omega)$ large and $\mu(a_2,b,t)(\omega)$ small, then
Eq.\ (\ref{eq:muab}) implies that $\rho(a_1)$ is significantly
different from $\rho(a_2)$. This can be quantified in terms of the
\textit{overlap} of two density operators $\rho(a_1)$ and $\rho(a_2)$
defined in Eq.\ (\ref{eq:overlap}).
\noindent\textbf{Proposition 1}:\ \ For a specific model $(\rho,E,U)$ to
be consistent with a probability function $\mu$ in the sense of Eq.\
(\ref{eq:muab}), the overlap of density operators for distinct knob
settings $a_1$ and $a_2$ has an upper bound given by
\begin{eqnarray}
\lefteqn{(\forall\ a_1, a_2 \in A)}\quad\nonumber\\[3pt]
&&\mathrm{Tr}[\rho(a_1)^{1/2}\rho(a_2)^{1/2}]\leq \min_{b,t,\omega}
\{[\mu(a_2,b,t)(\omega)]^{1/2} +
[1-\mu(a_1,b,t)(\omega)]^{1/2}\}.\nonumber\\
\label{eq:upper}
\end{eqnarray}
\noindent\textit{Proof}:\ \ For purposes of the proof, abbreviate
$\rho(a_1)$ by $a_1$, $\rho(a_2)$ by $a_2$, and
$U^\dag(t)E(b)U(t)(\omega)$ by~$E$. Because $E$ is a projection, $E =
E^2$. Then the Schwarz inequality\footnote{For any operators $F$ and
$G$ for which the traces exist, $|\mathrm{Tr}(FG^\dag)| \leq
[\mathrm{Tr}(FF^\dag)]^{1/2} [\mathrm{Tr}(GG^\dag)]^{1/2}$.} and a
little algebra implies
\begin{eqnarray} |\mathrm{Tr}(a_1^{1/2}a_2^{1/2})| &=&
|\mathrm{Tr}(a_1^{1/2}Ea_2^{1/2}) +
\mathrm{Tr}(a_1^{1/2}(1-E)a_2^{1/2})| \nonumber \\[3pt] & \leq &
|\mathrm{Tr}(a_1^{1/2}Ea_2^{1/2})| +
|\mathrm{Tr}(a_1^{1/2}(1-E)a_2^{1/2})| \nonumber \\[3pt]
\noalign{\goodbreak}
& \leq &
(\mathrm{Tr}\,a_1)^{1/2}
[\mathrm{Tr}(Ea_2^{1/2}a_2^{1/2}E)]^{1/2}\nonumber\\[3pt]
&&\mbox{} +
[\mathrm{Tr}(a_1^{1/2}(1-E)(1-E)a_1^{1/2})]^{1/2}
(\mathrm{Tr}\,a_2)^{1/2} \nonumber \\[3pt]
&= &
[\mathrm{Tr}(Ea_2^{1/2}a_2^{1/2}E)]^{1/2} +
[\mathrm{Tr}(a_1^{1/2}(1-E)(1-E)a_1^{1/2})]^{1/2} \nonumber \\[3pt] &=&
[\mathrm{Tr}(a_2E)]^{1/2} + [ 1 - \mathrm{Tr}(a_1E)]^{1/2}
. \end{eqnarray}
Expanding the notation, we have
\begin{eqnarray}
&&(\forall\ a_1, a_2 \in A)\nonumber\\[3pt]
&&\qquad\mathrm{Tr}[\rho(a_1)^{1/2}\rho(a_2)^{1/2}]\leq
\min_{b,t,\omega}\{[\mathrm{Tr}(\rho(a_2)U^\dag(t)E(b)(\omega)U(t))]^{1/2}
\nonumber \\[3pt] &&\hskip1.85in\mbox{} +
[1-\mathrm{Tr}(\rho(a_1)U^\dag(t)E(b)(\omega)U(t))]^{1/2}\},
\end{eqnarray}
which, with Eq.\ (\ref{eq:muab}), completes the proof. $\Box$
\noindent\textit{Example}:\ \ For $0 \leq \epsilon, \delta \ll 1$, if for
some $b$ and $\omega$, $\mu(a_2,b,t)(\omega) = \epsilon$ and
$\mu(a_1,b,t)(\omega)$ $= 1 - \delta$, then it follows that
$\mathrm{Tr}[\rho(a_1)^{1/2}\rho(a_2)^{1/2}] \leq \epsilon^{1/2} +
\delta^{1/2}$. If, in addition, $\rho(a_1) = |a_1\rangle\langle a_1|$
and $\rho(a_2) = |a_2\rangle\langle a_2|$, then we have $|\langle
a_1|a_2\rangle|^2 \leq \epsilon^{1/2} + \delta^{1/2}$.
\subsection{Freedom of choice for density operators}\label{subsec:3.2}
The preceding proof of an upper bound invites the question: is there a
corresponding positive lower bound? The answer turns out to be ``no.''
\noindent\textbf{Proposition 2}:\ \ For any specific model
$(\rho_\alpha,E_\alpha,U_\alpha)$ and any knob settings $a_1, a_2 \in
A$, regardless of Overlap$(\rho_\alpha(a_1),\rho_\alpha(a_2))$ there
is a specific model $(\rho_\beta,E_\beta,U_\beta)$ with $\mu_\beta =
\mu_\alpha$ and Overlap$(\rho_\beta(a_1),\rho_\beta(a_2)) = 0$.
\noindent\textit{Proof by construction}:\ \
Let $H_\beta$ be the direct sum of three Hilbert spaces
$H_0$, $H_1$, and $H_2$, each a copy of the Hilbert space $H_\alpha$ of
model $\alpha$: $H_\beta = H_0 \oplus H_1 \oplus H_2$. Let
$E_\beta(b)(c)$ be the direct sum of three copies of $E_\alpha(b)(c)$,
one for each of the $H_j$; similarly, let $U_\beta$ be the direct sum
of three copies of $U_\alpha$. Define $\rho_\beta$ by
\begin{equation} \rho_\beta(a) = \begin{cases}
\ \rho_\alpha \oplus 0 \oplus 0 &\mbox{\ if \ }a \neq a_1 \mbox{\ and\ }
a \neq a_2; \\
\ 0 \oplus \rho(a_1) \oplus 0 &\mbox{\ if \ }a = a_1;\\
\ 0 \oplus 0 \oplus \rho(a_2) &\mbox{\ if \ }a = a_2.\end{cases}
\end{equation}
This defines a model $\beta$ of the form of Eq.\ (\ref{eq:muab})
for which we have
\begin{equation}(\forall\ a \in A,\ b \in B,\ t \in I,\ \omega \in M)
\hskip1em \mu_\alpha(a,b,t) (\omega) = \mu_\beta(a,b,t) (\omega),
\end{equation}
but for the Overlap as defined in Eq.\ (\ref{eq:overlap}),
$\mathrm{Overlap}(\rho_\beta(a_1),\rho_\beta(a_2)) = 0$,
regardless of the value of
$\mathrm{Overlap}(\rho_\alpha(a_1),\rho_\alpha(a_2))$. $\Box$
This proof shows the impossibility of establishing by experiment a
positive lower bound on state overlap without reaching outside of
logic to make an assumption, or, to put it baldly, to {\em guess}
\cite{ams}. The need for a guess, no matter how educated, has the
following interesting implication. Any experimental demonstration of
quantum superposition depends on showing that two different settings
of the $A$-knob produce states that have a positive overlap. For
example, a superposition $|a_3\rangle = (|a_1\rangle +
e^{i\phi}|a_2\rangle)/\sqrt{2}$ has a positive overlap with state
$|a_1\rangle$. Because, by Proposition 2, no positive overlap is
experimentally demonstrable without guesswork, we have the following:
\noindent\textbf{Corollary to Proposition 2}:\ \ Experimental
demonstration of the super\-position of states requires resort to
guesswork.
\subsection{Constraint on resolutions of the identity}\label{subsec:3.3}
Much the same story of constraint and freedom holds for resolutions of
the identity. For the norm of an operator $A$ we take $\|A\| =
\sup_u\|Au\|$, where $u$ ranges over all unit vectors. Then we have:
\noindent\textbf{Proposition 3}:\ \ In order for a specific model
$(\rho,E,U)$ to generate a given probability function
$\mu$, the resolution of the identity must satisfy the constraint
\begin{displaymath}
\| E(b_1)(\omega_c) -
E(b_2)(\omega_c) \| \ge \max_{a \in
A,\,t \in I}|\mu(a,b_1,t)(c) - \mu(a,b_2,t)(c)|.
\end{displaymath}
\noindent\textit{Proof}:\ \ Replacing unit vectors by density operators
in the definition of the norm results in the same norm, from which we
have
\begin{eqnarray*}
\| E(b_1)(\omega_c) - E(b_2)(\omega_c) \|&\ge&
\max_{a \in A,\,t\in
I}|\mathrm{Tr}[U(t)\rho(a)U^\dag(t)
E(b_1) (\omega_c)]\\[3pt] &&\mbox{} -
\mathrm{Tr}[\rho(a)E(b_2)(\omega_c)]|\\[5pt] &
=&\max_{a \in A,\,t \in I}|\mu(a,b_1)(c) - \mu(a,b_2)(c)|.\\
&&\hskip2.9in \Box
\end{eqnarray*}
\subsection{Freedom of choice for resolutions of the identity}
\label{subsec:3.4}
The difference of any two commuting projections has norm less than or
equal to 1. Can requiring a model $\alpha$ to generate a given
probability function $\mu$ impose any upper bound less than 1 on
$\|E_\alpha(b_1)(\omega_c) - E_\alpha(b_2)(\omega_c)\|$?
\noindent\textbf{Proposition 4}: For any specific model
$(\rho_\alpha,E_\alpha,U_\alpha)$ and any knob settings $b_1$ and
$b_2$, regardless of $\|E_\alpha(b_1)(\omega_c) -
E_\alpha(b_2)(\omega_c)\|$ there is a specific model
$(\rho_\beta,E_\beta,U_\beta)$ with $\mu_\beta = \mu_\alpha$ and
$\|E_\beta(b_1)(\omega_c) - E_\beta(b_2)(\omega_c)\| = 1$.
\noindent\textit{Proof by construction}:\ \ Let the Hilbert space for
model $\beta$ be $H_\beta = H_\alpha \oplus H_\perp$ where $H_\perp$
is a space orthogonal to $H_\alpha$. Let $E_\beta(b_1)(\omega_c) =
E_\alpha(b_1)(\omega_c)\oplus \mathbf{1}_\perp$ and
$E_\beta(b_2)(\omega_c) = E_\alpha(b_2)(\omega_c)\oplus
\mathbf{0}_\perp$, where $\mathbf{1}_\perp$ is the unit operator on
$H_\perp$ and $\mathbf{0}_\perp$ is the zero operator on $H_\perp$.
Let $\rho_\beta = \rho_\alpha \oplus \mathbf{0}_\perp$. Then,
$\|E_{\alpha}(b_1)(\omega_c)- E_{\alpha}(b_2)(\omega_c)\|$\break $ = 1$.
$\Box$
\noindent\textbf{Remark}:
Suppose, as displayed in the proof of Proposition 2, models $\alpha $
and $\beta$ have the same domain and $\mu_\alpha$ = $\mu_\beta$, but
the $\alpha$-states have overlaps differing from those of the
$\beta$-states. Then there is always a resolution of the identity $E'$
outside the range of $E_\alpha$ and $E_\beta$, such that for some $a$,
$\,\mbox{Tr}[\rho_\alpha(a)E'] \ne \mbox{Tr}[\rho_\alpha(a)E']$. In
this sense the models $\alpha$ and $\beta$ conflict concerning their
predictions \cite{JOptB}.
\section{Impact on quantum physics}\label{sec:4}
The connection of any specific quantum model to experiments is via a
probability function. This and the proofs of Propositions 2 and 4
show something that experiments cannot show, namely that modeling an
experiment takes guesswork, and that a model, once guessed, is subject
to surprises arising in experiments not yet performed. Some guesses
get tested (one speaks of {\em hypotheses}), but testing a guess
requires other guesses not tested. By way of example, to guide
the choice of a density operator by which to model the light emitted by
a laser, one sets up the laser, filters, and a detector on a bench to
produce experimental outcomes. But to arrive at any but the
coarsest properties of a density operator one needs, in addition to these
outcomes, a model of the detector, and concerning this model, there must
always be room for doubt; we can try to characterize the detector better,
but for that we have to assume a model for one or more sources of light.
When we link bench and blackboard, we work in the high branches of a tree
of assumptions, holding on by metaphors, where we can let go of one
assumption only by taking hold of others. Because of the guesswork
needed to bridge between models and experiments, describing device
behavior is forever a bi-lingual enterprise, with a language of wrenches
and lenses for the bench and a different language of states and operators
for the blackboard.
We will show how some words work as metaphors, straddling bench and
blackboard, where by `words' we mean to include whatever mathematical
symbols are used to describe devices. We consider the mathematics of
quantum mechanics not in contrast to words but as blackboard language,
words of which are sometimes borrowed for use at the bench to describe
devices. By showing some choices of metaphorical uses of the words
{\em state, operator, spacetime, outcome, and particle}, we promote
freedom to invent particles as needed to describe interesting
features of device behavior. Recognizing choices in word use reflects
back on how we formulate quantum mechanics: the notion of repeated
measurements `of a state' will be revealed as neither necessary nor
sensible, and the so-called postulate of state reductions will evaporate,
leaving in its place a theorem.
\subsection{Word use at blackboard and lab bench}\label{sec:4.1}
We start by looking at several related but distinct uses of {\em
spacetime coordinates}. In the laboratory one uses clocks and rulers
to assign coordinates to acts of setting knobs, transmitting signals,
recording detections, {\em etc}, and one thinks of these
experimentally generated coordinates as points of a
spacetime---something mathematical. We call this a `linear spacetime'
to distinguish it from a second spacetime, that we call `cyclic,' onto
which the linear spacetime is folded, like a thread wound around a
circle, so that experimental outcomes for different trials can be
tallied in bins labeled by coordinates. Distinct from linear and
cyclic spacetimes, any quantum-mechanical model involves a third
spacetime on which are defined solutions of a Schr\"odinger equation
(or one of its relativistic generalizations), and it is with
reference to this spacetime that particles as theoretical constructs
are defined.
Any quantum model written in terms of particles generates probabilities,
and if the probabilities of the model fit the relative frequencies of
experimental outcomes well enough, one is tempted to say that one has
``seen the particles''; however, because particles in their mathematical
sense are creatures of models, and multiple, conflicting models are
consistent with any given experimental data, this ``seeing of particles''
stands on guesswork and metaphor, needed, for example, to bind the {\em
electron} as a solution of the Dirac equation defined on a model
spacetime to a flash from a phosphor on a screen. This metaphorical role
of {\em electron, photon, etc.}, though habitual and easily overlooked,
can be noticed when a surprise prompts one to make a change in the use of
the word {\em electron} at the blackboard while leaving the use at the
bench untouched, or {\em vice versa}.\looseness=-1
Next we address notions of (a) components of a theoretical outcome, (b) a
distinction between signal particles and probe particles, and (c)
various measurement times. We take these in order.
\subsubsection{Multiple components of an outcome}
The term {\em theoretical outcome} pertains to a vector space of
multi-particle wave functions defined on a model spacetime. This vector
space is a tensor product of factors, one factor for each particle. For a
resolution of the identity that factors accordingly, we shall view each
theoretical outcome for this resolution as consisting of a list of
components, one component for each of the factors. A probability
density for such multi-component outcomes can be viewed as a joint
probability density for the component parts of the outcome, modeling the
joint statistics of the detection of many particles.
\subsubsection{Signal and probe particles}
We who model are always free to shift the boundary between states (as
modeled by density operators) and measuring devices (as modeled by
resolutions of the identity) so as to include more of the measuring
devices within the scope of the density-operator part of the model
\cite{vN}. Consider for example a coarse model $\alpha$ that portrays a
detecting device by a resolution of the identity. While a resolution of
the identity has no innards, detecting devices do. To model, say, a
photo-diode and its accompanying circuitry, we can replace model
$\alpha$ by a more detailed model $\beta$: the quantum state asserted
by model $\alpha$ becomes what we call a {\em signal} state, a factor
in a tensor product (or more generally a sum of tensor products)
accompanied by factors for one or more {\em probe-particle states}.
According to this model $\beta$, the signal state is measured only
indirectly, via a probe state with which it has interacted, followed
by a measurement of the probe states, as modeled by ``a resolution of the
identity'' that works on the probe factor, not the signal factor.
\subsubsection{Measurement times}
Recognizing probe-states as free choices in modeling clarifies a
variety of times relevant to quantum measurements. For any quantum
model $\alpha$, the form of Eq.\ (\ref{eq:muab}) links an outcome
(whether single- or multi-component) to some point time $t_\alpha$;
however, the use of such a model is to describe an actual or
anticipated experiment, and for this, as described above, one is
always free to choose a more detailed model $\beta$, in which the
state of model $\alpha$ appears as the {\em signal} state that
interacts with a probe state, followed by a measurement of the probe
state at some time $t_\beta$ after the interaction. Thus model
$\beta$ replaces the point time $t_\alpha$ by a time stretch during
which the signal and probe states interact, thus separating the time
during which the signal state interacts with the probe from the time
at which the probe state is measured.
In more complex models involving more probe states, a
succession of ``times of measurement'' in the sense of interactions
can be expressed by a single resolution of the identity.
Finally, in modeling spatially dispersed signal states that interact with
entangled probe particles \cite{aharonov}, one can notice a prior
``probe-interaction time'' during which the probe particles must
interact with one another, in order to have become entangled.
\subsection{State reduction as a theorem, not a postulate}
\label{subsec:4.2}
Recognizing choice in modeling allows one to sidestep a long-troubling
issue in formulating quantum language. In logical conflict with
the Schr\"{o}dinger equation as the means of describing time evolution
\cite{tai}, Dirac and other authors introduce {\em state reductions}
by a special postulate that asserts an effect on a quantum state of a
resolution of the identity; allegedly needed to express repeated
measurements of a system. Once we recognize the modeling freedom to
make signal-probe interactions explicit, we can always replace any
story about devices involving a ``state to be measured repeatedly'' by
a model in which a {\em signal} state interacts with a succession of
probe states, followed by a simultaneous measurement of all the probe
states, as expressed by a single resolution of the identity
and a composite state that incorporates both signal and probe states.
Thus any apparent need for a postulate to do with ``repeated
measurements'' evaporates, and with it the unfortunate appearance
of state reductions in a postulate.
Although inconsistent as a postulate, state reduction still works in
many cases as a trick of calculation, as justified by the following
theorem.
\noindent\textbf{Theorem}:
Assume any specific model of the form Eq.\ (\ref{eq:muab}), and assume
$\omega = (j,k)$ is an outcome with components $j$ and~$k$. If
$E(b)(j,k)$ is a tensor product $E(b)(j,k) = E_A(b)(j) \otimes E_B(b)$,
then for any density-operator function $\rho(a)$, the joint probability
distribution $\mu(a,b)(j,k)$ induces a conditional probability
distribution for $k$ given $j$ that matches the quantum probability of
$k$ obtained using a ``reduced density operator'' obtained by the
usual rule for state reduction applied to~$\rho$.
\noindent\textit{Proof}: Streamline notation by suppressing the
dependence on $a$ and $b$, and incorporate $U$ into $\rho$, so that the
relevant form is that of Eq.\ (\ref{eq:mu}). For any state $\rho$ it
follows from Eq.\ (\ref{eq:mu}) that $\Pr(j,k) = \mbox{Tr}[\rho
(E_A(j)\otimes E_B(k))]$. The conditional probability of $k$ given
$j$ is defined by Bayes rule \cite{feller}:
\begin{eqnarray}\Pr(k|j) & = & \frac{\Pr(j,k)}{\Pr(j)} =
\frac{\Pr(j,k)}{\sum_{k'}\Pr(j,k')}
=\frac{\mbox{Tr}[\rho (E_A(j)\otimes E_B(k))]}{\mbox{Tr}[\rho
(E_A(j)\otimes \sum_{k'}E_B(k'))]}.
\end{eqnarray}
By the definition of a resolution of the identity, we have
$\sum_{k'}E_B(k') = 1$; recalling that $E_B(k)$ is a
projection that commutes with $E_A(j)$, one then has
\begin{equation}
\Pr(k|j) = \mbox{Tr}[\rho_{\rm red}E_B(k)],
\end{equation}
for an operator
\begin{equation}\rho_{\rm red} \stackrel{\rm def}{=} \frac{E_A(j)\rho
E_A(j)}{\mbox{Tr}[E_A(j)\rho E_A(j)]}.
\end{equation}
This $\rho_{\rm red}$ matches the `reduced density operator' obtained
by the usual rule of state reduction. Q.E.D.
\noindent\textbf{Remarks}:
\begin{enumerate}
\item In case $\rho = |\psi\rangle\langle\psi|$ is a pure state,
then $\rho_{\rm red} = |\psi_{\rm red}\rangle\langle\psi_{\rm red}|$,
where the reduced state
\begin{equation}|\psi_{\rm red}\rangle = \frac{E_A(j)|\psi
\rangle}{\|E_A(j)|\psi \rangle\|},
\end{equation}
which is one form of the usual rule for state reduction, but here
obtained by calculation with no need for any postulate.
\item Either of the outcome components can be a composite, so the
theorem applies to cases involving more than two outcome components.
\item In relativistic formulations of quantum mechanics, detections at
spatially separated locations $A$ and $B$ can be modeled by
projections of the form assumed.
\end{enumerate}
\section{Balancing forces in the detection of weak light}\label{sec:5}
Of interest in particle physics, astrophysics, and emerging practical
applications, sources of weak light are characterized experimentally
by the experimental outcomes of detectors \cite{sobo}. Because
detector outcomes are statistical, trial-to-trial differences in
outcomes can arise both from trial-to-trial irregularity in
the sources and from quantum indeterminacy in their detection.
As we shall see, detecting devices work in two parts,
one of which balances a light-induced force against some reference. By
taking advantage of the freedom to invent probe particles when we
model particle detectors, we are led to a quantum mechanical expression
of force in the context of balancing devices, with application to
models and detecting devices by which to better distinguish one source
of weak light from another.
In Newtonian physics, the word {\em force} is used both on the
blackboard and with balancing devices. In quantum physics, {\em
force}, as used at the blackboard, gets re-defined in terms of the
expectation values pertaining to dynamics of wave functions
\cite{feynmanIII}. We will find useful a concept of force in
characterizing light. Because of its employment in experimental
work, our concept of {\em force} necessarily takes on local coloring
special to one or another experimental bench; we develop a notion of
{\em force} in the context of light detection that gives meaning both
at the blackboard and the bench not only to expectation values of
forces, but also to higher-order statistics associated with them.
These higher-order statistics allow the expression, within quantum
mechanics, of the teetering of a balance that happens when forces are
nearly equal. We begin by reviewing some details of detector behavior.
\subsection{Balancing in detectors}\label{subsec:5.1}
Under circumstances to be explored, particle detectors employed to
decide among possible quantum states produce unambiguous experimental
outcomes. Seen up close, a detecting device consists of two components.
The first is a {\em transducer} such as a photo-diode that responds to
light by generating a small current pulse. To tally a transducer
response as corresponding to one or another theoretical outcome in the
sense of quantum mechanics, one has to {\em classify} the response using
some chosen criterion. As phrased in the engineering language of inputs
and outputs, the response of the transducer is fed as an input to a
second component of the detector, in effect an unstable balance
implemented as a flip flop (made of transistors organized into a
cross-coupled differential amplifier). The flip-flop produces an out{\em
put} intended to announce a decision between two possible experimental
out{\em comes}, say 0 and~1.
If we think classically, we picture the flip-flop as a ball and two
bins, one bin for each possible outcome, separated by a barrier, the
height of which can be adjusted, as shown in Fig.\ \ref{fig:1}. The
ball, starting in bin 0 is kicked by the transducer; an outcome of 1 is
recorded if and only if the balance is tipped and the ball rolls past the
barrier into bin~1. This ball-and-bin technique avoids ambiguity by
virtue of a convention that gives the record a certain leeway: it does
not matter if the ball is a little off center in its bin, so long as the
ball does not teeter on the barrier between bins.
Although usually producing an unambiguous outcome, the flip-flop, seen
up close, can teeter in its balancing, perhaps for a long time, before
slipping into one bin or the other \cite{gray,chaney}. Absent some special
intervention, two parties (people or machines) to which a teetering
output fans out can differ in how they classify this output as an
outcome: one finds a 0, the other finds a 1. To reduce the risk of
disagreement, the two parties have to delay their reading of the
output, hopefully until the ball slips into one or the other bin.
Ugly in this classical cartoon are two related features: (1) the ball
can teeter forever, so that waiting is no help, and (2) the mean time
for teetering to end is entirely dependent on some {\em ad hoc}
assumption about ``noise'' \cite{gray,chaney}. Although we have described
the flip-flop classically, it is built of silicon and glass presumably
amenable to quantum modeling. To see what quantum models offer us,
the first step is to recall that a quantum model implies probabilities
to be related to an experiment, so that inventing a quantum model and
choosing an experimental design go hand in hand. Thirty years ago,
thinking not in quantum but in circuit terms, we designed and carried
out an experiment to measure teetering of the output of a
flip-flop. As we recognized only recently, the record of this
experiment is compatible with some quantum models. A quantum
model to be offered shortly describes the experiment already
performed and serves as a guide for designing future experiments to
exploit what can be called a statistical texture of force, previously
obscured by the ``noise'' invoked in classical analysis.
\subsection{Experimental design}\label{subsec:5.2}
To experiment with the teetering of a detecting device comprised of a
transducer $D_0$ (for diode) connected to a flip-flop
$F_0$, shown in Fig.~\ref{fig:2}, we replace the
transducer $D_0$ by a laboratory generator of weak electrical pulses
to drive $F_0$.\footnote{Rather than a separate transducing
photo-diode as input to $F_0$, one could replace one transistor of
flip-flop $F_0$ by a photo-transistor.} Putting the flip-flop $F_0$
into a teetering state takes very sensitive adjustment of the pulse
generator, achieved by feedback from a running average of the outcomes
produced by $F_1$ and $F_2$ to the pulse generator. The output of
$F_0$ is made to fan out, as shown in Fig.~\ref{fig:3}, to a matched
pair of flip-flops, $F_1$ and $F_2$, each of which acts as an
auxiliary detector, not of the incoming light but of the output of
$F_0$. The flip-flops $F_1$ and $F_2$ are clocked at a time $T$ later
than is $F_0$. The experimental outcome consists of two binary
components, one from $F_1$ and the other from $F_2$. If after the
waiting time $T$ the output of $F_0$ is still teetering, the
flip-flops $F_1$ and $F_2$ can differ, one registering a 1 while the
other registers a 0; the disagreement between the two flip-flops
registers the teetering of the output of $F_0$. The measured relative
frequency of disagreements between $F_1$ and $F_2$ is shown in
Fig.~\ref{fig:4}.
\subsection{Quantum modeling of balancing in detectors}\label{subsec:5.3}
We want to model the teetering statistics that we will use to discriminate
among various sources of weak signals. Traditional analyses of
solid-state detecting devices and their flip-flops invoke quantum
mechanics only to determine parameters for classical stories involving
voltage and current. Analyzed that way, teetering in a
photo-diode-based detector that employs a flip-flop made of
transistors arises in two ways: first, there can be teetering in the
entry of electrons and holes into the conduction band of the
photo-diode; second, there can be teetering in the response of $F_0$
to whatever amplified pulse comes from the photo-diode. Although both
these teeterings involve electrons and holes going into a conduction
band, the statistical spread of outputs for a given state is blamed on
{\em noise} unconnected with the signal, and known analyses of a
flip-flop invoke {\em noise} to evade the embarrassment of a possible
infinite hesitation.
Avoiding the invocation of `noise,' we picture $F_0$ quantum
mechanically as a pair of probe particles. Light acting via the
transducer applies a force to the two probe particles. This scattering
process transforms an initially prepared state of the light and probes
to an out-state consisting of a sum (or integral) of products, each of
which has a factor for the light and a factor for the two probe
particles of $F_0$. After some waiting time of evolution $T$, the two
probe particles are measured, as expressed by a resolution of the
identity that ignores the signal state; thus the probabilities of
theoretical outcomes of the detection after the interaction are
expressible by a reduced density operator obtained by tracing over the
signal states. In this view, teetering shows up in the probability of
detecting the two probe particles on different sides of a reference;
we will show how this probability depends on both the signal detected
and a waiting time $T$, and how in this dependence Planck's constant
enters.
In the model presented here, we simplify the effect of the signal
state to that of preparing, at time 0, a pair of probe-particle wave
functions. For simplicity, the probe-particle wave functions have
only one space dimension. Let $x$ be the space coordinate for one
probe particle and $y$ be the space coordinate for the other. The
difference between one possible signal state and another is reflected
by concentrating the initial probe-state wave functions
slightly to one side or the other of an energy hump centered at the
origin $x=0$, $y = 0$. (As might be expected, teetering is most
pronounced in the borderline case of a signal state that puts the
initial probe-particle wave functions evenly over the energy hump.) We
model the flip-flop $F_1$ as a resolution of the identity that has an
theoretical outcome of 1 for $x > 0$ and 0 for $x < 0$; similarly
$F_2$ is modeled as a resolution of the identity for $y$. Thus the
two-component theoretical outcomes are 00 for $x,y<0$; 01 for $x<0$,
$y>0$; 10 for $x>0$, $y<0$; and 11 for $x,y>0$. By assuming a
coupling between the two probe particles, we will model how increasing
the waiting until time $T$ to detect the probe particles decreases the
probability of disagreement between $F_1$ and $F_2$, {\em i.e.}\
diminishes the probability of $x$ and $y$ being measured with
different signs.
Thinking of the $x$-probe particle as a wave-function concentrated
near an energy hump, assuming that the long-time behavior
of the particle depends only on the hump curvature,
and for the moment neglecting coupling between the two probe
particles, we express the dynamics of the $x$-particle by the
Schr\"odinger equation for an unstable oscillator:
\begin{equation}i\hbar\,\frac{\partial \;}{\partial t}\,\psi(x,t) =
\left(-\frac{\hbar^2}{2m}\,\frac{\partial^2 \;}{\partial x^2} -
\frac{k x^2}{2}\right)\psi(x,t),
\label{eq:single}\end{equation}
where the instability comes from the minus sign in the term
proportional to $x^2$. We express the $y$ probe particle similarly. In
order to produce growth over time in the correlation of the detection
probabilities, we put in the coupling term $\frac{1}{4}k\lambda(x-y)^2$.
This produces the following two-particle Schr\"odinger equation which is
the heart of our model:
\begin{eqnarray}i\hbar\,\frac{\partial \;}{\partial t}\,\psi(x,y,t) &=&
-\frac{\hbar^2}{2m} \left(\frac{\partial^2 \;}{\partial x^2}
+\frac{\partial^2 \;}{\partial y^2}\right)\psi(x,y,t)\nonumber\\
&&\mbox{} +
\frac{k}{2}\left(-x^2 - y^2 + \frac{\lambda}{2}
(x-y)^2\right)\psi(x,y,t). \label{eq:pairphys}
\end{eqnarray}
The natural time parameter for this equation is $\omega^{-1}$ defined
here by $\omega \mathop{\stackrel{\mathrm{def}}{=}}\nolimits$\break $\sqrt{k/m}$; similarly there is
a natural distance parameter $\sqrt{\hbar/m\omega}$.
\subsection{Initial conditions}\label{subsec:5.4}
{}For the initial condition, we will explore a wave packet of the form:
\begin{equation}
\psi(x,y,0) = \frac{1}{\pi^{1/2}b}\exp[-(x-c)^2/2b^2]\exp[-(y-c)^2/2b^2].
\label{eq:initxy}
\end{equation}
For $c=0$, this puts the recording device exactly on edge, while
positive or negative values of $c$ bias the recording device toward 1 or
0, respectively.
\subsection{Solution}\label{subsec:5.5}
As discussed in Appendix~A, the solution to this model is
\begin{eqnarray}
|\psi(x,y,t)|^2 &=& \frac{1}{\pi B_1(t)B_2(t)}\nonumber\\
&&\mbox{}\times
\exp\left\{-\left(\frac{x+y}{\sqrt{2}} - c\sqrt{2}\cosh
t\right)^2\Bigg/B^2_1(t)- \frac{(x-y)^2}{2B^2_2(t)}\right\},\nonumber\\
\label{eq:jointxyphys}
\end{eqnarray}
with
\begin{eqnarray}
B_1^2(t) &=& b^2 \left[1 + \left(\frac{\hbar^2}{\omega^2 m^2 b^4}
+1\right)\sinh^2 t\right],
\nonumber \\
B_2^2(t) &=& b^2\left[1 + \left(\frac{\hbar^2}{\omega^2 m^2
b^4(\lambda-1)}-1\right)
\sin^2 \sqrt{\lambda-1}\,t\right].
\label{eq:Bphysdef}
\end{eqnarray}
The probability of two detections disagreeing is the integral of this
density, $|\psi(x,y,t)|^2$, over the second and fourth quadrants of the
$(x,y)$-plane. For the especially interesting case of $c =0$, this
integral can be evaluated explicitly as shown in Appendix A:
\begin{eqnarray}
\lefteqn{\Pr(F_1\mbox{ and }F_2\mbox{ disagree at }t)}\qquad\nonumber\\
&=&
\frac{2}{\pi}\tan^{-1}\left(
\frac{\displaystyle 1 + \left[\frac{\hbar^2}{\omega^2 m^2
b^4(\lambda-1)}-1\right]\sin^2\sqrt{\lambda-1}\,\omega t}{\displaystyle 1
+\left(\frac{\hbar^2}{\omega^2 m^2 b^4}+1\right)\sinh^2 \omega
t}\right)^{1/2}.
\label{eq:edge1phys}
\end{eqnarray}
This formula works for all real $\lambda$. For $\lambda > 1$, it shows
an oscillation, as illustrated in Fig.~\ref{fig:4}. For the case $0 <
\lambda <1$, the numerator takes on the same form as the denominator, but
with a slower growth with time and lacking the oscillation, so that the
probability of disagreement still decreases with time, but more
slowly. Picking values of $b$ and $\lambda$ to fit the experimental
record, we get the theoretical curve of Fig.~\ref{fig:4}, shown in
comparison with the relative frequencies (dashed curve) taken from the
experimental record. For the curve shown, $\,\lambda = 1.81$ and $b =
0.556$ times the characteristic distance $\sqrt{\hbar/\omega m}$.
According to this model $\alpha$, a design to decrease the half-life
of disagreement calls for making both $k/m$ and $\lambda$ large.
Raising $\lambda$ above 1 has the consequence of the oscillation,
which can be stronger than that shown in Fig.~\ref{fig:4}. When the
oscillation is pronounced, the probability of disagreement, while
decreasing with the waiting time $t$, is not monotonic, so in some
cases judging sooner has less risk of disagreement than judging later.
\subsection{An alternative to model $\alpha$}\label{sec:5.6}
As Proposition 2 of Sec.~\ref{sec:3} suggests, one can construct
alternatives to the above model $\alpha$ of a flip-flop. Instead of
initial probe states specified by ``locating blobs,'' expressed in the
choice of the value of $c$ in Eq.\ (\ref{eq:initxy}), a model $\beta$
can employ initial probe states specified by momenta. In this
``shooting of probe particles at an energy hump,'' the initial wave
functions are concentrated in a region $x,y < 0$ and propagate toward
the energy saddle at $x,y = 0$. Writing a 0 is expressed by an
expectation momentum for the initial state less than that for the
initial state that corresponds to writing a 1. Hints for this
approach are in the paper of Barton \cite{barton}, which contains a
careful discussion of the energy eigenfunctions for the single
inverted oscillator of Eq.\ (\ref{eq:phiprob}), as well as of wave
packets constructed from these eigenfunctions. Such a model $\beta$
based on an energy distinction emphasizes the role of a flip-flop as a
decision device: it ``decides'' whether a signal is above or below the
energy threshold.
\subsection{The dependence of probability of disagreement on
$\hbar$}\label{subsec:5.7}
{}For finite $b$, the limit of Eq.\ (\ref{eq:edge1phys}) as $\hbar
\rightarrow 0$ is
\begin{equation}
\Pr(F_1\mbox{ and }F_2\mbox{ disagree at }t) =
\frac{2}{\pi}\tan^{-1}\left(\left|
\frac{\cos \sqrt{\lambda-1}\,\omega t}{\cosh \omega t}\right|\right).
\label{eq:edgeh0}
\end{equation}
This classical limit of model $\alpha$ contrasts with the
quantum-mechanical Eq.\ (\ref{eq:edge1phys}) in how the disagreement
probability depends on $\lambda$. Quantum behavior is also evident in
entanglement exhibited by the quantum-mechanical model. At $t = 0$ the
wave function is the unentangled product state of Eq.\
(\ref{eq:initxy}). Although it remains in a product state when viewed in
$(u,v)$-coordinates discussed in Appendix A, as a function of
$(x,y)$-coordinates it becomes more and more entangled with time, as
it must to exhibit correlations in detection probabilities for the
$x$- and $y$-particles. By virtue of a time-growing entanglement and
the stark contrast between Eq.\ (\ref{eq:edge1phys}) and its
classical limit, the behavior of the 1-bit recording device exhibits
quantum-mechanical effects significantly different from any classical
description.
The alternative model $\beta$ based on energy differences can be
expected to depend on a \textit{sojourn time} with its interesting
dependence on Planck's constant, as discussed by Barton \cite{barton}.
Both models $\alpha$ and $\beta$ thus bring
Planck's constant into the description of decision and recording
devices, not by building up the devices atom by atom, so to speak, but
by tying quantum mechanics directly to the experimentally recorded
relative frequencies of outcomes of uses of the devices.
\subsection{Balancing and the characterization of light force}
\label{subsec:5.8}
Detection of teetering in a detector of weak light pulses allows
finer distinctions by which to characterize a source of that light.
Without teetering, a first measure of a weak light source is its mean
intensity, expressed operationally as the fraction of 1's detected in
a run of trials in which it illuminates a detector. Now comes a
refinement that draws on teetering. If the detector's balancing
flip-flop $F_0$ fans out to auxiliary flip-flops $F_1$ and $F_2$, two
sources $A$ and $B$ that produce the same fraction of 1's can be
tested for a finer-grained distinction as follows. For each source,
using feedback to stabilize the relation between the source and $F_0$,
so that the running fraction of 1's detected is held nearly steady,
conduct one run of trials for a fixed waiting time $T_1$, a second run
of trials for a fixed waiting time $T_2$, {\em etc.} Let $\nu_A(T_k)$
denote the fraction of trials of source $A$ for which the outcome
components from $F_1$ and $F_2$ disagree in the run of trials with
waiting time $T_k$; similarly, $\nu_B(T_k)$ denotes this fraction for
source $B$. These additional data express additional statistical
``texture'' by which to compare source $A$ against source $B$. Even
when they produce the same overall fractions of 1's, they are still
measurably distinguishable if they consistently show strong
differences, for some $T_k$, between $\nu_A(T_k)$ and $\nu_B(T_k)$.
For example, if source $B$ has more classical jitter than source
$A$, so that the $B$ quantum state bounces up and down in expectation
energy from pulse to pulse, then source $B$ is more apt than source
$A$ to push the probe particles both over the hill or neither over the
hill. In other words, source $A$ will produce more teeterings, and
hence we would find $\nu_A(T_k)$ significantly greater than
$\nu_B(T_k)$.
\section{Entangled signals {\em vs}.\ entangled balances}
With the the freedom to invoke probe particles as developed in
Secs.\ 4 and 5, we can show a striking additional freedom of choice
in modeling, resolvable only by going beyond the application of logic
to experimental data. This freedom pertains to entanglement.
Suppose that experimental trials yield outcomes consistent with a
model, according to which a source of entangled weak light
illuminates a pair of unentangled detectors at two separate locations,
$A$ and $B$. Models of detectors detailed enough to invoke
probe-particle states, as in Sec. 5, must specify how these
probes are prepared. As discussed in a different context long ago
\cite{aharonov}, there is the possibility of entangling the probe
particles, amounting to entangling the detectors. This possibility of
an entangled pair of detectors points to a symmetry relation between
entanglement of signals and entanglement of probes.
To see how this works, consider first modeling a single detector
involving a probe state prepared by choosing some $|p(q)\rangle $,
where $q$ is a parameter such as the expectation momentum for this
state. Consider also a set of possible signal states $|s(q)\rangle$.
Assume that the detector model calls for detecting the probe particle
after its interaction with the signal particle, as expressed
by a measurement operator $E$ acting on the probe alone. Then the
probability of outcome $j$ resulting from an initial signal $|s(q)\rangle$
interacting with a probe $|p(q')\rangle $ has the form of the square
of a complex amplitude that depends on both $s(q)$ that labels the
signal state and $p(q')$ that labels the probe state:
\begin{equation}
\Pr(j|s(q),p(q')) =\|\mbox{Amp}(j|s(q),p(q'))\|^2,
\label{eq:pj}
\end{equation}
with
\begin{equation}
\mbox{Amp}(j|s(q),p(q')) = (1_s \otimes E(j))U (|s(q)\rangle
\otimes |p(q') \rangle).
\end{equation}
Here $U$, acting on the combined signal-probe space of wave functions,
expresses the signal-probe interaction. Assume for the moment that
the probability in Eq.\ (\ref{eq:pj}) is symmetric under interchange
of signal and probe:
\begin{equation}
\Pr(j|s(q),p(q')) = \Pr(j|s(q')p(q)),
\label{eq:symP}
\end{equation}
which implies that for some real-valued phase-function $\phi$,
\begin{equation}
\mbox{Amp}(j|s(q),p(q')) = e^{i\phi(q,q';j)}\mbox{Amp}(j|s(q')p(q)).
\label{eq:symA}
\end{equation}
To see how this symmetry impacts on modeling a pair of detectors,
consider two such detectors, one at location $A$, the other at $B$,
having identical initial probe states $|p_A(q_0)\rangle$ and
$|p_B(q_0)\rangle$, respectively. For unentangled signals having
initial states $|s_A(q_1)\rangle$ and $|s_B(q_2)\rangle$, the
amplitude for the joint outcome $j_A$ at $A$ and $j_B$ at $B$ is then
\begin{eqnarray}
\lefteqn{\mbox{Amp}(j_A,j_B|s_A(q_1)s_B(q_2);p_A(q_0)p_B(q_0))=}
\hskip1in\nonumber\\[8pt] & & [(1_{sA}\otimes
E_A(j))\otimes(1_{sB}\otimes E_B(k))](U_A\otimes U_B) \nonumber \\[8pt]
&&
\quad [(|s_A(q_1)\rangle |p_A(q_0)\rangle)\otimes (|s_B(q_2)\rangle
|p_B(q_0)\rangle)]
\label{eq:joe}
\end{eqnarray}
(which can be written as a product of $A$- and $B$-factors, expressing the
lack of correlation when there is no entanglement). Now consider the
same pair of detectors responding to an entangled signal state
\begin{equation}
\mathcal{N}[s_{A1}s_{B2} + e^{i\theta}
s_{A2} s_{B1}],
\label{eq:entstate}
\end{equation}
where $\mathcal{N}$ is a normalizing constant, dependent on $q$ and
$q'$, and we have condensed the notation by writing $p_{A1}$ for
$|p_A(q_1)\rangle$, {\em etc}. The combined signal-probe state written
with the tensor products in the order assumed in Eq.\ (\ref{eq:joe})
is then
\begin{equation}
\mathcal{N}[s_{A1}p_{A0}s_{B2}p_{B0} + e^{i\theta}
s_{A2}p_{A0} s_{B1}p_{B0}];
\label{eq:entstate3}
\end{equation}
thus the joint probability of outcomes for the entangled signal state is
\begin{eqnarray}\lefteqn{
\Pr(j_A,j_B|\mathcal{N}[s_{A1}s_{B2}+e^{i\theta}s_{A2}s_{B1}],p_{A0}
p_{B0}) =}
\hspace{1.0in} \nonumber \\[8pt] & & \|\mathcal{N}[(1_{sA}\otimes
E_A(j))\otimes(1_{sB}\otimes E_B(k))](U_A\otimes U_B)\nonumber \\[8pt]& &
\;\;[s_{A1}p_{A0}s_{B2}p_{B0} + e^{i\theta} s_{A2}p_{A0}
s_{B1}p_{B0}]\|^2.
\label{eq:joe1}
\end{eqnarray}
Assuming the invariance up to phase of Eq.\ (\ref{eq:symA}), the
exchange of signal and probe parameters results only in
phases that leave the probability unaffected, leading to the relation
\begin{eqnarray}
\lefteqn{\Pr(j_A,j_B|\mathcal{N}[s_{A1}s_{B2}+e^{i\theta}s_{A2}s_{B1}],p_{A0}
p_{B0})}\quad\nonumber\\[8pt]
&=&\Pr(j_A,j_B|s_{A0}s_{B0},\mathcal{N}[p_{A1}p_{B2}+
e^{i\theta}p_{A2}p_{B1}]),
\end{eqnarray}
so that outcome probabilities are the same for an entangled state
measured by untangled detectors as they are for an unentangled
state measured by entangled detectors.
Without assuming Eqs.\ (\ref{eq:symP}), (\ref{eq:symA}), one can still
ask: given a model $\alpha$ that ascribes probabilities of outcomes to
an entangled signal measured via an unentangled probe state
$p_{A0}p_{B0}$, is there an alternative model $\beta$ that ascribes
the same probabilities to an unentangled signal state measured via an
entangled probe state? We conjecture that the answer is ``yes,'' in
which case no experiment can distinguish between a model of it that
asserts entangled signal states measured via unentangled probes and a
model that asserts unentangled signal states measured via entangled
probes.
\section{Acknowledgments}
Tai Tsun Wu called our attention to the conflict between the
Schr\"odinger equation and state reductions. We thank Howard E. Brandt
for discussions of quantum mechanics. We thank
Dionisios Margetis for analytic help.
This work was supported in part by the Air Force Research Laboratory and
DARPA under Contract F30602-01-C-0170 with BBN Technologies.
\appendix
\section*{Appendix A.\ \ Solution to Model $\alpha$ of a 1-bit recording
device}\label{sec:App}
\newcounter{appendix}
\setcounter{appendix}{1}
\def\Alph{appendix}{\Alph{appendix}}
\def\theappendix.\arabic{equation}{\Alph{appendix}.\arabic{equation}}
Starting with Eq.\ (\ref{eq:pairphys}), and writing $t$ as
the time parameter times a dimensionless ``$t$'' and $x$ and $y$ as
the distance parameter times dimensionless ``$x$'' and ``$y$,''
respectively, we obtain
\begin{equation}i\,\frac{\partial \;}{\partial t}\,\psi(x,y,t) =
\frac{1}{2}
\left(-\frac{\partial^2 \;}{\partial x^2} -\frac{\partial^2
\;}{\partial y^2} - x^2 - y^2 + \frac{\lambda}{2}
(x-y)^2\right)\psi(x,y,t). \label{eq:pair}
\end{equation}
This equation is solved by introducing a non-local coordinate change:
\begin{equation} u = \frac{x +y}{\sqrt{2}}
\quad\mbox{and}\quad v = \frac{x-y}{\sqrt{2}}.
\end{equation}
With this change of variable, Eq.\ (\ref{eq:pair}) becomes
\begin{equation}i\,\frac{\partial \;}{\partial t}\,\psi(u,v,t) =
\frac{1}{2}\left(-\frac{\partial^2 \;}{\partial u^2} -\frac{\partial^2
\;}{\partial v^2} - u^2 + (\lambda-1) v^2\right)\psi(u,v,t),
\label{eq:uvpair}
\end{equation} for which separation of variables is immediate,
so the general solution is a sum of products, each of the form
\begin{equation}
\psi(u,v,t) = \phi(u,t)\chi(v,t).
\label{eq:factor}
\end{equation}
The function $\phi$ satisfies its own Schr\"odinger
equation,
\begin{equation}
\left(i\,\frac{\partial \;}{\partial t} + \frac{1}{2}\,\frac{\partial^2
\;}{\partial u^2} + \frac{u^2}{2}\right)\phi(u,t) = 0,
\label{eq:phieq}
\end{equation}
which is the quantum-mechanical equation for an unstable harmonic
oscillator, while $\chi$ satisfies
\begin{equation}
\left(i\,\frac{\partial \;}{\partial t} + \frac{1}{2}\,
\frac{\partial^2 \;}{\partial v^2} + \frac{1- \lambda}{2}\,
v^2\right)\chi(v,t) = 0 , \label{eq:chieq}
\end{equation}
which varies in its interpretation according to the value of
$\lambda$, as follows: (a) for $\lambda < 1$, it expresses an unstable
harmonic oscillator; (b) for $\lambda = 1$, it expresses a free particle;
and (c) for $\lambda > 1$, it expresses a stable harmonic oscillator. The
last case will be of interest when we compare behavior of the model
with an experimental record.
By translating Eq.\ (\ref{eq:initxy})\ into $(u,v)$-coordinates, one
obtains initial conditions
\begin{eqnarray}
\phi(u,0) &=&
\pi^{-1/4}b^{-1/2}\exp\left\{-\frac{(u-\sqrt{2}c)^2}{2b^2}\right\}, \\
\chi(v,0) &=& \pi^{-1/4}b^{-1/2}\exp\left\{-\frac{v^2}{2b^2}
\right\}.
\end{eqnarray}
The solution to Eq.\ (\ref{eq:phieq}) with these
initial conditions is given by Barton \cite{barton}; we deal with
$\phi$ and $\chi$ in order. From (5.3) of Ref.~\cite{barton}, one has
\begin{equation}
|\phi(u,t)|^2 = \frac{1}{\pi^{1/2}B_1(t)}\,\exp\left\{\frac{-(u -
c\sqrt{2}\cosh
t)^2}{B_1^2(t)}\right\},
\label{eq:phiprob}
\end{equation}
where
\begin{equation}
B_1^2(t) = b^2 \left[1 + \left(\frac{1}{b^4} +1\right)\sinh^2 t\right].
\label{eq:Bdef}
\end{equation}
Similarly, integrating the Green's function for the stable oscillator
($\lambda > 1)$ over the initial condition for $\chi$ yields
\begin{equation}
|\chi(v,t)|^2 =
\frac{1}{\pi^{1/2}B_2(t)}\,\exp\left\{\frac{-v^2}{B_2^2(t)}\right\},
\end{equation}
where
\begin{equation}
B_2^2(t) = b^2\left[1 + \left(\frac{1}{b^4(\lambda-1)}-1\right)
\sin^2\sqrt{\lambda-1}\,t\right].
\label{eq:B2def}
\end{equation}
Multiplying these and changing back to $(x,y)$-coordinates yield the
joint probability density
\begin{eqnarray}
|\psi(x,y,t)|^2 &=& \frac{1}{\pi B_1(t)B_2(t)}\nonumber\\[8pt]
&&\mbox{}\times \exp\left\{-\left(\frac{x+y}{\sqrt{2}} - c\sqrt{2}\cosh
t\right)^2\Bigg/ B^2_1(t)- \frac{(x-y)^2}{2B^2_2(t)}\right\}.\nonumber\\
\label{eq:jointxy}
\end{eqnarray}
The probability of two detections disagreeing is the integral of this
density over the second and fourth quadrants of the $(x,y)$-plane.
This is most conveniently carried out in $(u,v)$-coordinates. For the
especially interesting case of $c =0$ (and $\lambda > 1)$, this
integral can be transformed into
\begin{eqnarray}
\lefteqn{\Pr(F_1\mbox{ and }F_2\mbox{ disagree at
}t)}\qquad\nonumber\\[6pt] &=&
\frac{1}{\pi B_1(t) B_2(t)}\int^{\infty}_{-\infty} dv \int^v_{-v}du\,
\exp\left\{\frac{-u^2}{B_1^2(t)} - \frac{v^2}{B_2^2(t)}\right\} \nonumber
\\[6pt] &=&
\frac{4}{\pi}\int_0^{\infty}e^{-V^2}\, dV\int_0^{B_2(t) V/B_1(t)}dU\,
e^{-U^2} \nonumber \\[6pt] \noalign{\goodbreak} &=&
\frac{2}{\pi}\tan^{-1}\left(\frac{B_2(t)}{B_1(t)}\right)
\nonumber \\[6pt]\noalign{\goodbreak}
&=& \frac{2}{\pi}\tan^{-1}\left(
\frac{\displaystyle 1 +
\left[\frac{1}{b^4(\lambda-1)}-1\right]\sin^2\sqrt{\lambda-1}\,
t}{\displaystyle 1 +\left(\frac{1}{b^4}+1\right)\sinh^2 t}\right)^{1/2}.\qquad
\label{eq:edge1}
\end{eqnarray}
It is easy to check that this formula works not only when $\lambda >
1$ but also for the case $\lambda < 1$. For $0 < \lambda <1$,
the numerator takes on the same form as the denominator, but with
a slower growth with time, so that the probability of disagreement
still decreases with time exponentially, but more slowly.
Converting Eq.\ (\ref{eq:jointxy}) from dimensionless back to physical
time and distance variables results in Eq.\ (\ref{eq:jointxyphys})
with Eqs.\ (\ref{eq:Bphysdef}), and similarly Eq.\ (\ref{eq:edge1})
leads to Eq.\ (\ref{eq:edge1phys}).
\renewcommand\thefigure{\arabic{figure}}
\setcounter{figure}{0}
\begin{figure}
\caption{Detector as probe particle in a double well.}
\label{fig:1}
\end{figure}
\eject
\begin{figure}
\caption{Flip-flop exposed to race between signal going high and clock
going low. }
\label{fig:2}
\end{figure}
\eject
\begin{figure}
\caption{Flip-flops $F_1$ and $F_2$ used to read $F_0$ after a delay
$T$. }
\label{fig:3}
\end{figure}
\eject
\begin{figure}
\caption{Probability of disagreement vs.\ settling time.}
\label{fig:4}
\end{figure}
\end{document} | math | 65,613 |
\begin{document}
\doi{10.1080/0950034YYxxxxxxxx}
\issn{1362-3044}
\issnp{0950-0340} \jvol{00} \jnum{00} \jyear{2010} \jmonth{10 January}
\title{Sideband cooling of small ion Coulomb crystals in a Penning trap}
\author{G. Stutter$^1$\thanks{$^1$Present Address: Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark.} , P. Hrmo, V. Jarlaud , M. K. Joshi, J. F. Goodwin$^2$\thanks{$^2$Present Address: Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom} and R. C. Thompson$^{\ast}$\thanks{$^\ast$Corresponding author. Email: [email protected]
}\\
{\em{Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2AZ, United Kingdom} }
\\
\received{v8 released \today}}
\maketitle
\begin{abstract}
We have recently demonstrated the laser cooling of a single $^{40}$Ca$^+$ ion to the motional ground state in a Penning trap using the resolved-sideband cooling technique on the electric quadrupole transition S$_{\nicefrac{1}{2}} \leftrightarrow$ {D}$_{\nicefrac{5}{2}}$. Here we report on the extension of this technique to small ion Coulomb crystals made of two or three $^{40}$Ca$^+$ ions. Efficient cooling of the axial motion is achieved outside the Lamb-Dicke regime on a two-ion string along the magnetic field axis as well as on two- and three-ion planar crystals. Complex sideband cooling sequences are required in order to cool both axial degrees of freedom simultaneously. We measure a mean excitation after cooling of $\bar n_\text{COM}=0.30(4)$ for the centre of mass mode and $\bar n_\text{B}=0.07(3)$ for the breathing mode of the two-ion string with corresponding heating rates of {11(2)}\,s$^{-1}$ and \SI{1(1)}{\per\second} at a trap frequency of \SI{162}{\kilo\hertz}. The ground state occupation of the axial modes is above 75\% for the two-ion planar crystal and the associated heating rates 0.8(5) s$^{-1}$ at a trap frequency of \SI{355}{\kilo\hertz}.
\begin{keywords}Penning trap; trapped ions;
sideband cooling; ion Coulomb crystals
\end{keywords}
\end{abstract}
\maketitle
\section{Introduction}
Ion Coulomb crystals (ICCs) consisting of cold, trapped atomic ions are a widely used and highly versatile experimental platform \cite{Thompson2015}. The level of control achievable with ICCs makes them a suitable choice for many applications in atomic and molecular physics, including quantum computation and simulation~\cite{Britton2012,Debnath2016,Monz2016}, cavity QED \cite{Herskind2009}, atomic clocks~\cite{Chou2010} and precision measurements~\cite{Leanhardt2011}. Penning traps use a combination of static electric and magnetic fields to confine charged particles. Such traps are often used in experiments where large magnetic fields are required,
for example in quantum simulation \cite{Britton2012}, non-neutral plasma physics \cite{Bollinger2003}, precision measurements of
masses and magnetic moments \cite{Blaum2010}, and for experiments on particle beamlines (where their large, open electrode structures and large trap depths are an advantage)
\cite{Smorra2015,Murbock2016}.
In quantum information experiments, the common modes of motion of ICCs provide a convenient mechanism for transmitting information between ions~\cite{CZ95}. Radio-frequency, linear Paul traps are prevalent in this kind of work, typically using one-dimensional crystals that align along the radio frequency null of the trap~\cite{Lanyon2013}. Such a system with three ions in a line has been used to simulate the frustration that occurs when three spins are arranged in a triangle configuration \cite{Kim2010}. In contrast, planar ICCs in a Penning trap naturally form in a triangular lattice, which provides a suitable platform for quantum simulation of frustrated systems~\cite{Bohnet2016}, without the need for highly specialised trap designs~\cite{Yoshimura2015,Mielenz2016}.
In order to conduct many of these types of experiments, the ions must be in the Lamb-Dicke regime, where the amplitude of the motional mode is much less than the wavelength of the interacting light. This often requires the ions to be cooled below the Doppler limit, most commonly achieved in ion traps using the resolved-sideband technique.
Here we present results on resolved-sideband laser cooling of the axial motion of two- and three-ion $^{40}$Ca$^+$ ICCs in a Penning trap.
We dedicate this article to the memory of our colleague Professor Danny Segal whose inspiration and insight led to many of the ideas that are described here. Over many years Danny designed and built the apparatus and developed the techniques for working with ground-state cooled ions in the Penning trap and he anticipated the efficient sideband cooling and long coherence times that we have now demonstrated for small ion Coulomb crystals. Sadly he was only partially able to see his ideas come to fruition as he passed away in September 2015 before this work had been completed.
\section{Background}
To confine charged particles in all directions of space, a Penning trap uses a strong magnetic field aligned with the trap axis ($\vec{B}=B\hat{z}$) and a static cylindrically-symmetric quadrupolar electric potential. Expressed in cylindrical coordinates, the potential has the form $\phi=A(2z^2-\rho^2)$ where $A$ is a constant depending on the voltage applied on the electrodes of the trap and the geometry of the trap. In the axial direction (along $\hat{z}$), a single particle will undergo a simple harmonic motion with a frequency $\nu_z=\sqrt{4qA/M}/2\pi$, where $q$ is the charge of the particle and $M$ is its mass. Typically this frequency lies in the range 150 to 400\,kHz in our trap. Confinement in the radial plane is ensured by the magnetic field, resulting in stable orbits about the axial direction. The motion in this plane is characterised by the modified cyclotron and the magnetron modes whose frequencies $\nu_+$ and $\nu_-$ are given by $\nu_{\pm}=\nu_c/2\pm\sqrt{\nu_c^2/4-\nu_z^2/2}$, where the cyclotron frequency, $\nu_c$, is given by $\nu_c=qB/(2\pi M)$. In our trap, the cyclotron frequency is around 715\,kHz.
Multiple ions in a trap exert a repulsive Coulomb force on one another which counteracts the trapping force. If these ions are sufficiently cold, they form a structure known as an ion Coulomb crystal, where the individual ions exhibit small amplitude motion around their equilibrium positions \cite{Thompson2015}. An ICC consisting of $N$ particles has a total of 3$N$ of these collective modes of motion. Due to the cylindrical symmetry of the Penning trap potentials and the presence of the axial magnetic field, one of the radial modes corresponds to a bulk rotation of the crystal and for stable radial confinement the frequency of this rotation, $\nu_r$, must be between $\nu_-$ and $\nu_+$.
\begin{figure}
\caption{EMCCD images of two ions in a string (left) and in a planar crystal (right) as seen from the radial plane. The planar crystal appears as a blurred line due to its rotation about the magnetic field axis. }
\label{ModesDiagram}
\end{figure}
The effective radial trapping strength in the crystal's frame of reference is described by the frequency $\nu_{\text{eff}}$, which can be expressed as a function of $\nu_r$ by
\begin{equation}
\nu_{\text{eff}}=\sqrt{\nu_r (\nu_c - \nu_r) -\tfrac{1}{2}\nu_z^2}\ ,
\label{freqEff}
\end{equation}
which is maximum when the rotation frequency is half the cyclotron frequency, i.e., $\nu_r = \frac{1}{2} \nu_c$. For a given rotation frequency, an increase in the axial trapping strength (i.e., an increase in the endcap voltage) leads to a decrease in the effective radial trapping strength and a change in the shape of the ICC~\cite{Mavadia2013}. At low axial frequency the ions can be made to line up along the magnetic field axis (an `axial string'), and by increasing the axial frequency the ions can be made to lie in the radial plane (a `planar crystal'). These two configurations are of particular interest, as the axial and radial modes are separable and the simpler axial modes can be addressed independently. Only the axial modes of ICCs are discussed in this paper.
Figure \ref{ModesDiagram} shows the modes of two-ion ICCs in the string and planar configurations. A two-ion string has an axial centre of mass (COM) mode at a frequency $\nu_z$, and a breathing mode (where the ions move in opposite directions) at $\sqrt{3}\nu_z$. For planar crystals there is the COM mode at $\nu_z$ and a `tilt' mode at $\nu_{\text{tilt}}=\sqrt{\nu_z^2- \nu_{\text{eff}}^2}$. In the tilt mode the ions have equal and opposite axial displacements. For a three-ion planar ICC there are two degenerate tilt modes at $\nu_{\text{tilt}}$. The degeneracy of these two modes could be lifted by the application of a rotating wall drive, which breaks the cylindrical symmetry of the system~\cite{Hasegawa2005}.
The interaction Hamiltonian for the axial motion of a two-level, multi-ion crystal uniformly illuminated by a plane electromagnetic wave of frequency $\omega_L/2\pi$ is given by
\begin{equation}
H_I= \frac{\hbar}{2} \sum_{j=1}^{N} \Omega_0 (\sigma_j^++\sigma_j^-) e^{-i \sum_k \eta_k^{(j)} ( a_k^\dagger + a_k) } e^{i\omega_Lt}+ \text{h.c.}
\label{hamiltonian}
\end{equation}
where $\Omega_0$ is the Rabi frequency in the absence of ion motion, $a_k^\dagger$ and $a_k$ are the phonon raising and lowering operators for the $k$-th mode and $\eta_k^{(j)}$ is the Lamb-Dicke parameter for the $j$-th ion in the $k$-th mode, which takes into account the different amplitude of motion for different ions in a particular mode.
\begin{figure}
\caption{Normalised coupling strength at the carrier (black), first red sideband (red) and second red sideband (magenta) for an axial frequency of \SI{162}
\label{sidebands}
\end{figure}
Let us first consider what happens in the case of a single ion with oscillation frequency $\nu_z$. In this case the Lamb-Dicke parameter is given by
\begin{equation}
\eta_0=\frac{1}{\lambda}\sqrt{\frac{h}{2M\nu_z}}\ ,
\end{equation}
where $\lambda$ is the wavelength and $h$ the Planck constant. The Lamb-Dicke regime is defined by the condition that $\eta_0\sqrt{2n+1}\ll1$, where $n$ is the quantum number of the axial motion.
The axial motion gives rise to a set of sidebands around the carrier frequency spaced at the corresponding oscillation frequency ($\nu_z$), where the sideband at frequency $m\nu_z$ corresponds to transitions from state $n$ to state $n'=n+m$. The relative Rabi frequencies associated with each sideband (and the carrier, with $m=0$) as a function of phonon number $n$ are proportional to associated Laguerre polynomials \cite{Leibfried2003}. This is shown in Figure~\ref{sidebands} for a trapping frequency of $\nu_z=\SI{162}{\kilo\hertz}$,
where it can be seen that there are values of $n$ at which the amplitude of a sideband becomes very close to zero. We will refer to these points as `minima'. These points give rise to what are in effect dark states at particular values of $n$, where there is very little interaction between the ion and the laser when it is tuned to that sideband. In particular, in this case, there is a minimum for the carrier around $n=24$ and for the first sideband around $n=63$. These minima move to lower values of $n$ as $\eta_0$ rises. The figure also shows the thermal distribution at the Doppler limit expressed in terms of the relative occupation probability $P(n)$ for each Fock state $n$ (normalised to unity for $n=0$).
At low trapping frequencies, and therefore at large Lamb-Dicke parameters, an ion after Doppler cooling will be left in a thermal distribution with significant parts of the population at phonon numbers higher than the lowest minimum of the first red sideband. The standard sideband cooling technique, which consists of exciting only the first red sideband, will leave this part of the population `trapped' at the minimum, where only a very small rate of cooling can occur, and a high ground state occupation probability will therefore not be achieved. However, because the minima are located at a different phonon number for each sideband, sideband cooling well outside the Lamb-Dicke regime remains possible but requires different red sidebands (typically the first and second alternately) to be addressed~\cite{Poulsen2012}. This is the procedure that we adopted in the sideband cooling of a single ion reported in Ref. \cite{Goodwin2016}.
We now consider the extension to the case of two ions. In this situation, the Lamb-Dicke parameters for both ions are always equal to each other. The centre of mass mode of a two-ion crystal always has a Lamb-Dicke parameter of $\eta_{\text{COM}}=\eta_0/\sqrt{2}$ because the mass of the crystal is twice that of a single ion. To a good approximation the Lamb-Dicke parameter for the tilt mode of a radial crystal also has this value in our experiments. However, the breathing mode of the axial crystal has $\eta_{\text{B}}=\eta_0/\sqrt{2\sqrt{3}}$ \cite{Morigi1999}.
The Hamiltonian in Equation \ref{hamiltonian} gives rise to a spectrum of sidebands at the harmonics of each mode and at the frequencies of all the corresponding intermodulation products. Outside the Lamb-Dicke regime, where sidebands beyond first order are significant, the spectrum can become very complicated. The Rabi frequency for a sideband transition between initial phonon states $(n_1,n_2)$ and final phonon states $(n'_1,n'_2)$ is given by
\begin{equation}
\Omega_{n_1',n_2',n_1,n_2} = \Omega_0 \prod_{k=1}^2\left[\sqrt{\frac{\min{\{n_k,n'_k\}}!}{\max{\{n_k,n'_k\}}!}} \eta_k^{|n'_k-n_k|} e^{-\eta_k^2/2} L_{\min{\{n_k,n'_k\}}}^{|n'_k-n_k|}(\eta_k^2)\right]\label{Rabi}
\end{equation}
where $L$ is a Laguerre polynomial. Here $k=1$ refers to the COM mode and $k=2$ refers to the breathing mode. This equation shows that
the strength of any sideband described by $\Omega_{n_1',n_2',n_1,n_2}$ depends on all four quantum numbers. In particular, the strength of any sideband of one motion is proportional to the strength of the \emph{carrier} of the other motion.
\begin{figure}
\caption{Rabi frequency of various sidebands for a two-ion axial string at $\nu_z=\SI{162}
\label{TwoIonStrength}
\end{figure}
As a result, the simultaneous cooling of the two motional modes of a two-ion axial string outside the Lamb-Dicke regime is significantly more difficult than for a single ion. We will discuss here the case of a two-ion string, which is more challenging to cool than the planar structure, but the reasoning would be similar for the latter. As in the case of a single ion, sideband cooling of two ions becomes very slow at some phonon numbers where the coupling strength of the first red sideband of either motional mode approaches zero. This problem can be addressed by cooling on higher-order sidebands in a similar fashion to the single-ion case. Additionally, sideband cooling on a given mode will also become extremely slow for phonon numbers where the carrier strength of the \emph{other} mode is at a minimum.
However there are some combinations of quantum numbers that give population trapping where the carrier strength for \emph{both} modes is at a minimum. Such `dark' regions can be identified in a two-dimensional phonon space where each dimension corresponds to one of the motional modes.
These regions are represented in Figure \ref{TwoIonStrength}. The top left quadrant (a) shows the collective strength of the first three red sidebands of the centre of mass mode, representative of the cooling rate for a sequence that alternates between these sidebands. The darker regions correspond to phonon numbers where the carrier strength of the breathing mode is at a minimum. A similar plot for the breathing mode (b), where only the two first red sidebands need to be considered, shows an analogous situation. It can be seen, by overlapping the two plots, that the brighter areas of one eliminate most of the dark regions of the other. This means that by alternating cooling between the two modes, most of the population trapping can be avoided. However, there remains a dark area at the intersection of the minima of the carriers of the two modes. This region is centred around $n_\text{COM}=49$ and $n_\text{B}=86$. The population trapped at this intersection can only be brought to the ground state by including an intermodulation product sideband in the cooling sequence, that is, a sideband that changes the phonon number for both modes at the same time. A plot of the strength of the second red COM sideband of the first red breathing mode sideband is shown on Figure \ref{TwoIonStrength}(c), which features a bright region at the dark crossing area of (a) and (b). This sideband therefore cools population that has accumulated in the dark region efficiently to lower phonon numbers, allowing further cooling on the other sidebands to continue. On the last quadrant (d) where the three other plots are overlapped, no dark region remains, thus preventing population trapping over the complete cooling sequence. More detailed simulations we have carried out allow us to optimise the cooling sequence to obtain the highest ground-state population in both modes~\cite{Joshi2017}.
Once the ions are cooled to the Lamb-Dicke regime, the interaction Hamiltonian (Equation~\ref{hamiltonian}) can be greatly simplified in the interaction picture by only considering terms to low orders in $\eta_k$. With this approximation, the Hamiltonian for two ions can be written as:
\begin{eqnarray}
H_I&=& \frac{ \Omega_0\hbar}{2} \sum_{j=1}^{2} e^{-i\delta_j t
}\ket{\downarrow_j}\bra{\uparrow_j} \times \nonumber \\
&& [(1-\eta_1^2(n_1+\tfrac{1}{2})-\eta_2^2(n_2+\tfrac{1}{2}))\ket{n_1,n_2}\bra{n_1,n_2}+ \nonumber \\
&& i\eta_1(e^{i\nu_1t}\sqrt{n_1}\ket{n_1-1,n_2}\bra{n_1,n_2}+e^{-i\nu_1t}\sqrt{n_1+1}\ket{n_1+1,n_2}\bra{n_1,n_2}) + \nonumber \\
&& i\eta_2(e^{i\nu_2t}\sqrt{n_2}\ket{n_1,n_2-1}\bra{n_1,n_2}+e^{-i\nu_2t}\sqrt{n_2+1}\ket{n_1,n_2+1}\bra{n_1,n_2})]
+ \text{h.c.}
\label{TwoIonHam}
\end{eqnarray}
In this equation the second line describes the carrier, the third line the red and blue sidebands of mode 1 and the fourth line the red and blue sidebands of mode 2.
Note that the detuning $\delta_j$ is not necessarily constant across all ions $j$, despite using a single laser to excite the whole crystal. This becomes important in the case of the axial string of two ions, where their carrier frequencies differ by approximately \SI{2}{\kilo\hertz} in our experiment due to the axial magnetic field inhomogeneity.
\section{Methods}
The apparatus used to perform this experiment is largely the same as that previously used to sideband cool single ions to the quantum ground state~\cite{Goodwin2016} and for observing and controlling the spatial configurations of ICCs~\cite{Mavadia2013}.
The Penning trap consists of stacked cylindrical electrodes with an internal diameter of \SI{21.6}{\milli\meter}, housed in a vacuum system and inserted into the bore of superconducting magnet producing a magnetic field strength of approximately \SI{1.865}{\tesla} at the field centre. DC voltages are applied to two of the electrodes which act as end-caps, and `axialisation drive' voltages applied to a segmented ring electrode produce an oscillating quadrupole potential to aid Doppler cooling of the radial motion~\cite{Powell2002a}. A cloud of $^{40}$Ca$^+$ ions is loaded into the trap via three-photon non-resonant photoionisation of an atomic beam using a frequency-doubled pulsed Nd:YAG laser. The ions are then cooled to close to the Doppler limit by addressing $\textrm{S}_{\nicefrac{1}{2}}\leftrightarrow\textrm{P}_{\nicefrac{1}{2}}$ electric dipole transitions at \SI{397}{\nano\metre}, as well as relevant repumping transitions~\cite{Mavadia2014}. Doppler cooling beams propagate both along and perpendicular to the direction of the magnetic field to ensure that all of the motional modes of the crystal are cooled~\cite{Itano1982,Mavadia2013}. The radial cooling beam is offset from the trap centre, resulting in an intensity gradient across the crystal and giving coarse control of the crystal rotation frequency~\cite{Itano1988,Asprusten2014}. Note that good control of the radial beam position and intensity is paramount for the stability of the crystal. Unlike for a single ion, the axialisation drive has to be fairly strong for ICCs to maintain the crystal configuration and achieve efficient Doppler cooling. The results presented here for ICCs were obtained with \SIrange{2}{4}{\volt} of axialisation while a \SIrange{50}{200}{\milli\volt} signal is typically enough to efficiently cool the radial motion of a single ion in our trap.
The crystals are sideband cooled using broadly the same method used for single ions~\cite{Goodwin2016}, addressing specific motional sidebands of the $\textrm{S}_{\nicefrac{1}{2}}(m_j=-\frac{1}{2})\leftrightarrow\textrm{D}_{\nicefrac{5}{2}}(m_j=-\frac{3}{2})$ transition using a narrow-linewidth laser at \SI{729}{\nano\metre} propagating along the trap axis. The frequency of this laser is controlled using an acousto-optic modulator and a direct digital synthesis (DDS) frequency generation system,
which permits sequential addressing of up to seven sidebands during each cooling sequence. An additional `quench' laser at \SI{854}{\nano\metre} is applied to artificially increase the effective decay rate from the $\textrm{D}_{\nicefrac{5}{2}}$ state and therefore increase the cooling rate~\cite{Marzoli1994}.
After sideband cooling the ions are prepared in the $\textrm{S}_{\nicefrac{1}{2}}(m_j=-\frac{1}{2})$ state by optical pumping and then probed on the $\textrm{S}_{\nicefrac{1}{2}}(m_j=-\frac{1}{2})\leftrightarrow\textrm{D}_{\nicefrac{5}{2}}(m_j=-\frac{3}{2})$ spectroscopy transition (the same as that used for sideband cooling). We then perform a projective measurement of the state of the ions by turning on the Doppler cooling lasers and collecting fluorescence from the ions using either an EMCCD camera or photomultiplier tubes \cite{Mavadia2014}. The state of each of the individual ions in an axial string can be determined using the EMCCD camera, which spatially resolves the ions. For planar ICCs the bulk rotation of the crystal prevents resolution of different ions using side-on imaging (as seen on Figure \ref{ModesDiagram}) and the total number of photons received must therefore be used to distinguish between different numbers of bright ions. This cooling, probe and detection cycle is repeated many times (typically 200) at a particular probe laser frequency to measure the excitation probability, before stepping the frequency to obtain a full sideband spectrum.
\section{Results and discussion}
Before we attempt to sideband cool or perform spectroscopy on an ICC, we first observe it on an EMCCD camera during continuous Doppler cooling and adjust a combination of the endcap voltage, radial beam offset and axialisation drive amplitude until the crystal remains stable in the desired configuration. For an axial string to form we require that the axial trapping strength is sufficiently weaker than the radial trapping strength (with the converse being true for planar crystals). However, the effective radial trapping strength depends on the rotation frequency of the crystal. It is maximum when the crystal rotates at half the cyclotron frequency. This defines a critical trapping voltage above which the two-ion crystal will always lie in the radial plane. However, below this voltage the crystal may still lie in the radial plane because the radial trapping strength (described by $\nu_{\text{eff}}$) is reduced for different rotation frequencies. Indeed, there is always some rotation frequency for which $\nu_{\text{eff}}<\nu_z$ so the crystal cannot be forced into the axial string configuration solely by relaxing the axial potential, without some control over the crystal rotation frequency. We find that the end-cap voltage must be tuned significantly below the calculated critical value in order to achieve axial strings that remain stable for the duration of an experiment.
\subsection{Two-ion axial string}
Prior to sideband cooling it is essential to pre-cool the ions to their Doppler limit. Figure \ref{2ionAxialDoppler} shows a spectrum obtained after Doppler cooling of a two-ion string at a trapping frequency (COM) $\nu_z=\SI{162}{\kilo\hertz}$. At this frequency, the Lamb-Dicke parameter for both modes is fairly high: 0.17 for the COM mode and 0.13 for the breathing mode, consistent with the many sidebands visible. Due to the complexity of the spectrum, a fit to the data was not attempted. Instead, a spectrum was simulated and the parameters adjusted so that the simulated curve approaches the data points.
In this simulated spectrum, the average phonon number is set to be $\bar{n}_\text{COM}$ = 87 and $\bar{n}_\text{B}$ = 51 (these values are approximately 1.5 times the Doppler limit). The occupancy of each phonon state $\ket{n_\text{COM}}\otimes\ket{n_\text{B}}$ is calculated and the excitation strength is evaluated numerically from the two-dimensional Rabi strength in Equation \ref{Rabi}. The Rabi frequency and the pulse length are fixed to the values that are used to perform the experiment. The contribution due to each phonon state and motional sideband is summed at a particular laser detuning and thus a spectrum is generated which matches the experimental data reasonably well.
Using the parameters of the simulation, and assuming a thermal distribution, the probability for $n_\text{COM}$ to remain above the first minimum of the red sideband after Doppler cooling is about 24\,\% while it is less than 1.4\,\% for the breathing mode.
\begin{figure}
\caption{Spectrum of the excitation probability of one ion in a two-ion string after Doppler cooling at a trapping frequency $\nu_z=\SI{162}
\label{2ionAxialDoppler}
\end{figure}
To sideband cool both modes simultaneously and avoid population trapping, several sidebands are addressed sequentially with laser pulses of a few hundreds of microseconds. Table~\ref{twoIonCoolSeq} shows the pulse sequence used to sideband cool the two-ion string. It is the result of both theoretical studies and empirical adjustments to obtain the lowest final temperature. Figure~\ref{2ionAxialSBC} shows the spectrum after sideband cooling. It can be seen that the sideband structure simplifies greatly compared to the Doppler spectrum. The strongest visible components are now just the carrier and the first blue sidebands of the COM and breathing modes. There are small peaks present at the positions of the first red sidebands of the two motions.
The large asymmetry between the blue and red sidebands indicates a high ground state occupation probability.
In order to extract the average phonon number, we assume a thermal distribution among the lowest five states of each mode (i.e., $n=0$ to 4). Numerical solutions of the Schr\"{o}dinger equation are constructed using the Hamiltonian (Equation \ref{TwoIonHam}) for the carrier and first red and blue sidebands of both modes. This numerical solution correctly takes into account the entanglement created between the two ions, based on the method discussed in \cite{Homethesis}. Using the spatial information from the camera, the fluorescence data of each ion was recorded separately. The two spectra were fitted independently to the excitation probabilities of the ions calculated using our model, taking into account the small frequency difference of the two carriers due to the magnetic field inhomogeneity.
The extracted parameters were then averaged to obtain the final mean phonon numbers $\bar{n}_\text{COM}=0.30(4)$ for the COM mode and $\bar{n}_\text{B}=0.07(3)$ for the breathing mode. We measured the heating rates by inserting delay periods after sideband cooling but before probing the ions. We obtain heating rates of {11(2)}\,s$^{-1}$ for the COM and \SI{1(1)}{\per\second} for the breathing mode.
\begin{table}
\begin{center}
\tbl{Sequence of red sidebands addressed to sideband cool a two ion axial string (COM = centre of mass mode; B = breathing mode).}
{\begin{tabular}{l c c}
\toprule
Sideband & Pulse time ($\upmu$s) & Sequence repeats\\
\hline
\hline
2\textsuperscript{nd} COM & 500 &\\
2\textsuperscript{nd} B & 500&\\
3\textsuperscript{rd} COM & 300& 15\\
1\textsuperscript{st} B & 200&\\
2\textsuperscript{nd} COM 1\textsuperscript{st} B & 500&\\
\hline
1\textsuperscript{st} B & 200&\\
2\textsuperscript{nd} COM & 500 & 2\\
1\textsuperscript{st} COM & 500 &\\
2\textsuperscript{nd} COM 1\textsuperscript{st} B & 500&\\
\hline
2\textsuperscript{nd} COM 1\textsuperscript{st} B & 500&\\
1\textsuperscript{st} COM & 2000 & 1\\
1\textsuperscript{st} B & 500&\\
\botrule
\end{tabular}}
\label{twoIonCoolSeq}
\end{center}
\end{table}
\begin{figure}
\caption{Spectrum of the excitation probability of one ion in a sideband-cooled two-ion axial crystal at an axial frequency of 162\,kHz, showing a fit to the carrier and the first-order sidebands of both motions (see text for details).
}
\label{2ionAxialSBC}
\end{figure}
\subsection{Planar crystals}
Working with planar crystals is a somewhat simpler proposition as the axial frequency is necessarily higher than for an axial crystal having the same number of ions. This results in a lower value of $\eta$ and therefore there are fewer visible sidebands after cooling to the Doppler limit. In contrast to axial crystals, there are trapping voltages at which the crystal is always stable in the radial plane.
One of the main limiting factors is our ability to precisely control the rotation frequency using only the torque provided by an offset radial cooling beam. Greater control could be achieved using a quadrupolar or hexapolar rotating wall drive~\cite{Hasegawa2005,Khan2015}.
Figure~\ref{2ionPlanarDoppler} shows the spectrum of a two-ion planar ICC after Doppler cooling. This spectrum is taken at a trapping frequency of $\nu_z=\SI{353}{\kilo\hertz}$ (corresponding to $\eta_\text{COM}=0.115$). The rotation frequency of the crystal is such that the tilt mode is very close in frequency to the COM mode and their corresponding sidebands are not distinguishable on the spectrum.
Owing to the relatively small Lamb-Dicke parameter and the near degeneracy of the COM and tilt modes, the pulse sequence for sideband cooling of the planar crystal is much simpler than for the string. Pulses are only applied on the COM red sidebands -- the tilt mode is excited off-resonantly -- and a maximum of just two sidebands (first and second red) need to be addressed to prevent any significant population trapping.
Figure~\ref{2ionPlanarSBC} shows the spectrum obtained after sideband cooling of a two-ion planar ICC at $\nu_z=\SI{346}{\kilo\hertz}$ for $\SI{10}{\milli\second}$ on the first red sideband only. The laser probe was set to a low power in order to resolve the COM and tilt modes (Rabi frequency on the carrier of $\sim\SI{14}{\kilo\hertz}$
). A clear asymmetry can be seen between the blue sidebands and the red sidebands, which are almost completely suppressed. A noticeable feature of the spectrum is that the tilt mode peak does not reach the height of the centre of mass peak and appears broader. This is most likely due to the instability of the rotation frequency of the crystal during the acquisition of the spectrum. To account for this factor, and to take into account the effect of decoherence, the fitting function in the vicinity of the tilt mode and the centre of mass mode peaks was convoluted with a normalised Gaussian function, where the standard deviation was left as a free parameter. The fitting model is the same as for the two ions in an axial string, but there is now no relative detuning between the two ions, and the Lamb-Dicke parameters of the two modes are the same.
Since the fluorescence of each ion cannot be recorded individually for the radial crystal, we set the detection threshold to correspond to excitation of both ions simultaneously, because experimentally we can only reliably distinguish between both ions being dark and at least one ion fluorescing. Note that the red sideband of both motions is very strongly suppressed because it is necessary for the mode to be in the state $n=2$ or higher for the red sideband to appear in the two-ion spectrum. We find that the average phonon number for both modes is $\bar{n} < 0.15$ with a Gaussian broadening of the centre of mass mode and the tilt mode of \SI{0.7}{\kilo\hertz} and \SI{1.0}{\kilo\hertz} respectively. From the position of the sidebands, the rotation frequency can be estimated using Equation \ref{freqEff} which yields {$\nu_r=\SI{106}{\kilo\hertz}$} (approximately 9\,kHz greater than the magnetron frequency). This rotation speed is consistent with measurements of the size of the image of the crystal.
We observe a heating rate of 0.8(5)\,s$^{-1}$ for the COM mode, consistent with our single ion rate.
\begin{figure}
\caption{Spectrum of a Doppler-cooled, two-ion planar crystal at a trapping frequency of $\nu_z=\SI{353}
\label{2ionPlanarDoppler}
\end{figure}
\begin{figure}
\caption{Spectrum of a two-ion planar crystal after \SI{10}
\label{2ionPlanarSBC}
\end{figure}
A three-ion planar crystal was sideband cooled with a sequence of three laser pulses on the first, second and first red sidebands respectively. Figure~\ref{3ionPlanarSBC} shows the spectrum obtained for the three-ion planar crystal after sideband cooling for a total of \SI{30}{\milli\second} (\SI{10}{\milli\second} on each pulse) at a trapping frequency $\nu_z=\SI{379}{\kilo\hertz}$. It can be seen that the amplitude of the red sideband of the COM mode is close to zero while the blue sideband almost reaches one. This large asymmetry suggests a high ground state occupation probability, although the complexity of the spectrum prevents us from fitting the data points and obtaining a more accurate figure. Due to the misalignment of the laser beam, many radial features can also be observed and identified on the spectrum. {This includes sidebands on each of the main axial peaks due to the crystal rotation in the radial plane. The fact that the red radial sidebands are much more prominent than the blue ones implies a radial offset of the probe beam, as well as an angular misalignment}.
\begin{figure}
\caption{Spectrum of a three-ion planar crystal after \SI{30}
\label{3ionPlanarSBC}
\end{figure}
It is of interest to note that while the ICCs in this paper have been cooled close to the quantum ground state in the axial direction, the tangential speed of the ions due to the bulk rotation of the ICC remains on the order of \SI
{10}{\meter\per\second} in the laboratory frame. This demonstrates that the two bulk modes of this ion crystal, one of which is in the quantum mechanical ground state with low heating rate, have
energies differing by four orders of magnitude, a situation we believe to be unique to this type of system.
\section{Conclusion}
We have demonstrated the sideband cooling of small ICCs held in a Penning trap. Ground state confinement of all axial modes was achieved with high probability, for both linear and planar ICCs. When cooling multi-ion crystals from far beyond the Lamb-Dicke regime, sequential addressing of several sidebands including at least one intermodulation product is necessary to prevent population trapping.
Scaling up the number of ions, Penning traps are not well suited to confinement of much longer ion strings, as the very low axial trapping frequencies required make Lamb-Dicke confinement impossible. However, the techniques developed in this work could be readily extended to larger planar crystals or even three-dimensional ICCs, geometries which are difficult to obtain in a Paul trap. Cooling these structures to the Lamb-Dicke regime is a prerequisite for performing high fidelity entangling operations on ICCs and for producing strong and low-noise optical dipole forces in quantum simulators~\cite{Bohnet2016}. Such quantum simulations are typically performed on significantly larger ICCs of many hundreds of ions. Here the complexity of the mode structure may make EIT cooling~\cite{Lechner2016} a more practical means of achieving sub-Doppler temperatures, but resolved sideband techniques will remain useful for the direct thermometry and manipulation of individual motional modes. Given the richness of physics that has been demonstrated with simple one dimensional strings of ions, it is very interesting to consider the possibilities presented by cooling more complex structures to their motional ground state.
\end{document} | math | 36,945 |
\begin{document}
\title[Loops and the Lagrange property]{Loops and the Lagrange property}
\author[O.~Chein]{Orin Chein}
\address{Department of Mathematics,
Temple University,
1805 N. Broad St.,
Philadelphia, PA 19122 USA}
\email{[email protected]}
\urladdr{http://math.temple.edu/\symbol{126}orin}
\author[M.~K.~Kinyon]{Michael K. Kinyon}
\address{Department of Mathematics,
Western Michigan University,
Kalamazoo, MI~49008-5248 USA}
\email{[email protected]}
\urladdr{http://unix.cc.wmich.edu/\symbol{126}mkinyon}
\author[A.~Rajah]{\\ Andrew Rajah}
\address{School of Mathematical Sciences,
Universiti Sains Malaysia,
11800 USM Penang, Malaysia}
\email{[email protected]}
\urladdr{http://www.mat.usm.my/math/staff2.htm{\#}AR}
\author[P.~Vojt\v{e}chovsk\'y]{Petr Vojt\v{e}chovsk\'y}
\address{Department of Mathematics,
Iowa State University,
Ames, IA~50011 USA}
\email{[email protected]}
\urladdr{http://www.vojtechovsky.com}
\subjclass{20N05}
\keywords{Lagrange property, Moufang loop, Bol loop, A-loop}
\begin{abstract}
Let $\mathcal{F}$ be a family of finite loops closed under subloops and factor loops.
Then every loop in $\mathcal{F}$ has the strong Lagrange property if and only if every
simple loop in $\mathcal{F}$ has the weak Lagrange property. We exhibit several such
families, and indicate how the Lagrange property enters into the problem of
existence of finite simple loops.
\end{abstract}
\maketitle
\thispagestyle{empty}
\noindent
The two most important open problems in loop theory, namely the existence of a
finite simple Bol loop and the Lagrange property for Moufang loops, have been
around for more than 40 years. While we certainly have not solved these
problems, we show that they are closely related. Some of the ideas developed
here have been present in the loop-theoretical community, however, in a rather
vague form. We thus felt the need to express them more precisely and in a more
definite way.
We assume only basic familiarity with loops, not reaching beyond the
introductory chapters of \cite{pf}. All loops mentioned below are finite.
We begin with the crucial notion: the Lagrange property. A loop $L$ is said to
have the \emph{weak Lagrange property} if, for each subloop $K$ of $L$, $|K|$
divides $|L|$. It has the \emph{strong Lagrange property} if every subloop $K$
of $L$ has the weak Lagrange property.
A loop may have the weak Lagrange property but not the strong Lagrange
property. Four of the six nonisomorphic loops of order $5$ have elements of
order $2$ and hence fail to satisfy the weak Lagrange property. Let $K$ be one
of these loops. As noted in \cite[p.\ 13]{pf}, if $L$ is a loop of order $10$
having $K$ as a subloop and satisfying the property that every proper subloop
of $L$ is contained in $K$, then $L$ will have the weak but not the strong
Lagrange property. It is not difficult to construct a multiplication table for
such a loop.
Our main result depends on the following lemma, which is a restatement of
\cite[Lemma V.2.1]{br}.
\begin{lemma}\label{lem:bruck}
Let $L$ be a loop with a normal subloop $N$ such that
\begin{enumerate}
\item[(i)] $N$ has the weak $($resp.\ strong$)$ Lagrange property,
and
\item[(ii)] $L/N$ has the weak $($resp.\ strong$)$ Lagrange property.
\end{enumerate}
Then L has the weak $($resp.\ strong$)$ Lagrange property.
\end{lemma}
There are some classes of loops studied in the literature to which the lemma
applies directly. For each of these, the normal subloop in question is
associative. For instance, an easy induction shows that any solvable loop
satisfies the strong Lagrange property. In particular, any (centrally)
nilpotent loop satisfies this property as well.
Before we turn to more specific examples, we recall a few definitions. Let $L$
be a loop and $x\in L$. When $x$ has a two-sided inverse, we denote it by
$x^{-1}$. A loop $L$ has the \emph{automorphic inverse property} if
$x^{-1}y^{-1}=(xy)^{-1}$ holds for every $x$, $y\in L$. A loop $L$ is
\emph{$($right$)$ Bol} (resp.\ \emph{left Bol}), if $((xy)z)y=x((yz)y)$ (resp.
$(x(yx))z=x(y(xz))$) holds for every $x, y, z\in L$. \emph{Moufang loops} are
loops that are both right Bol and left Bol. Since the concepts of right Bol
loop and left Bol loop are anti-isomorphic to each other, right Bol and left
Bol loops share the same algebraic properties. Thus, in what follows, when we
refer to Bol loops, the reader may think of left Bol or right Bol as he or she
sees fit. A Bol loop which has the automorphic inverse property is called a
\emph{Bruck loop}. Bruck loops of odd order are called \emph{B-loops}\cite[p.\
376]{gl1}. An \emph{A-loop} is a loop all of whose inner mappings are
automorphisms \cite{bp}. Finally, an \emph{$M_k$ loop} is a Moufang loop $L$
for which $L/\mathrm{Nuc}(L)$ has exponent $k-1$ and no smaller exponent, where
$\mathrm{Nuc}(L)$ is the nucleus of $L$ \cite{pf1, cp}.
\begin{example}
Let $L$ be a Moufang loop with an associative normal subloop $K$ such that
$L/K$ satisfies the strong Lagrange property. By the lemma, $L$ has the strong
Lagrange property.
As an example, if $L$ is an $M_k$ loop where $k = 2^m + 1$ and if $K =
\mathrm{Nuc}(L)$, then $L/K$ is a Moufang $2$-loop which, by \cite{gw} is
centrally nilpotent (cf. \cite{c1}).
As another example of this type, let $L$ be a Moufang loop with an associative
normal subloop $K$ such that $|L/K|$ is odd. In this case $L/K$ has the strong
Lagrange property by \cite[Thm.\ 2]{gl2}. For instance, if $L$ is an $M_k$ loop
with $k$ even, then $K = \mathrm{Nuc}(L)$ is an associative normal subloop such
that $L/K$ is of odd exponent, $k-1$, and so $L/K$ must have odd order (cf.
\cite{c1}).
Since every Moufang $A$-loop is an $M_4$ loop \cite[Cor.\ 2]{kkp}, and every
commutative Moufang loop is a Moufang $A$-loop \cite{bp}, it follows that every
Moufang $A$-loop, and in particular, every commutative Moufang loop has the
strong Lagrange property.
\end{example}
That every commutative Moufang loop has the strong Lagrange property is, in
fact, a well-known folk result, and follows from the central nilpotence of
these loops and the lemma.
\begin{example}
Let $L$ be a Bruck loop with an associative normal subloop $K$ such that
$|L/K|$ is odd. Since $L/K$ is a B-loop, it follows from \cite[Cor.\ 4]{gl1}
that $L/K$ has the strong Lagrange property. By the lemma, so does $L$. For
instance, since a commutative Moufang loop of odd order is obviously a B-loop,
this gives an alternative proof that every such loop has the strong Lagrange
property.
\end{example}
\begin{example}
The lemma also applies directly to those loops $L$ for which the derived
subloop $L'$ (i.e., the smallest subloop $L'$ such that $L/L'$ is an abelian
group) has the strong Lagrange property. For instance, let $L$ be a ``central''
Bol loop in the terminology of Kreuzer \cite{kr}, i.e., a Bol loop $L$ such
that $L'$ is contained in the center. These are just centrally nilpotent Bol
loops of nilpotence class 2, and thus these loops have the strong Lagrange
property. As another example, Bruck and Paige \cite{bp} showed that an A-loop
$L$ has the property that all of its loop isotopes are A-loops if and only if
$L/\mathrm{Nuc}(L)$ is a group, in other words, if and only if $L$ is nuclearly
nilpotent of class $2$. By the lemma, such an $L$ has the strong Lagrange
property.
\end{example}
We now come to our main result---the connection between simple loops and loops
satisfying the Lagrange property.
\begin{theorem}\label{thm:main}
Let $\mathcal{F}$ be a nonempty family of finite loops such that
\begin{enumerate}
\item If $L \in \mathcal{F}$ and $N \triangleleft L$, then $N \in \mathcal{F}$;
\item If $L \in \mathcal{F}$ and $N \triangleleft L$, then $L/N \in \mathcal{F}$;
\item Every simple loop in $\mathcal{F}$ has the weak Lagrange property.
\end{enumerate}
Then every loop in $\mathcal{F}$ has the weak Lagrange property.
\end{theorem}
\begin{proof}
We proceed by induction on the order of loops in $\mathcal{F}$. Note that (1) implies
that $\mathcal{F}$ contains the trivial loop $\langle 1 \rangle$, for which the desired
conclusion is trivial. Now fix $L \in \mathcal{F}$ and assume that the result holds for
all loops in $\mathcal{F}$ of order less than $|L|$. If $L$ is simple, we are finished
by (3). Thus assume that $L$ is not simple, so that $L$ has a nontrivial proper
normal subloop $N$. Since $|N| < |L|$, (1) and the induction hypothesis imply
that $N$ has the weak Lagrange property. Since $|L/N| < |L|$, (2) and the
induction hypothesis imply that $L/N$ has the weak Lagrange property. By Lemma
\ref{lem:bruck}, $L$ has the weak Lagrange property.
\end{proof}
\begin{corollary}\label{coro:1}
Let $\mathcal{F}$ be a nonempty family of finite loops such that
\begin{enumerate}
\item[(1')] If $L \in \mathcal{F}$ and $K \leq L$, then $K \in \mathcal{F}$;
\item[(2)] If $L \in \mathcal{F}$ and $N \triangleleft L$, then $L/N \in \mathcal{F}$;
\item[(3)] Every simple loop in $\mathcal{F}$ has the weak Lagrange property.
\end{enumerate}
Then every loop in $\mathcal{F}$ has the strong Lagrange property.
\end{corollary}
\begin{proof}
Since (1') implies (1), Theorem \ref{thm:main} implies that every loop in $\mathcal{F}$
has the weak Lagrange property. But then (1') yields the desired result.
\end{proof}
\begin{corollary}\label{coro:2}
Let $\mathcal{V}$ be a variety of loops such that every simple loop in $\mathcal{V}$ has the
weak Lagrange property. Then every loop in $\mathcal{V}$ has the strong Lagrange
property.
\end{corollary}
Corollary \ref{coro:2} is of particular interest for those varieties of loops
for which there exists a classification of all simple loops. A prominent
example is the variety of Moufang loops, where it is known that every simple
nonassociative Moufang loop is isomorphic to a Paige loop (cf.\ \cite{paige},
\cite{liebeck}). It follows from Corollary \ref{coro:2} that if the weak
Lagrange property can be established for each of the Paige loops, then every
Moufang loop will have the strong Lagrange property.
There is one Paige loop for every finite field $GF(q)$; its order is
$q^3(q^4-1)$ when $q$ is even, and $q^3(q^4-1)/2$ when $q$ is odd \cite{paige}.
The weak Lagrange property for the smallest 120-element Paige loop has been
established in \cite{mgpm} and \cite{vo}. The next smallest Paige loop has
order $1080$. Thus based on published literature, we can state this result.
\begin{corollary}\label{coro:3}
Every Moufang loop of order less than $1080$ has the strong Lagrange property.
\end{corollary}
\begin{proof}
A simple Moufang loop of order less than $1080$ is a group or the smallest
Paige loop. The result follows from Corollary \ref{coro:1}.
\end{proof}
Incidentally, since no Paige loop is commutative, it follows from Corollary
\ref{coro:2} that, once again, every commutative Moufang loop has the strong
Lagrange property.
The authors have been informed by G.~E.~Moorehouse \cite{em} that by a computer
search, he has found that the Paige loop of order $1080$ satisfies the weak
Lagrange property. If we assume this to be correct, then we may state the
following.
\begin{corollary}\label{coro:4}
Every Moufang loop of order less than $16320$ has the strong Lagrange property.
\end{corollary}
Altogether, we have demonstrated that a loop has the strong Lagrange property
whenever it belongs to one of the following classes: Moufang loops with an
associative normal subloop of odd index, Bruck loops with an associative normal
subloop of odd index, loops whose derived subloops have the strong Lagrange
property, and Moufang loops of order less than 1080.
We conclude this paper with a couple of remarks on a potential application for
Corollary \ref{coro:2} to the existence of finite simple non-Moufang Bol loops.
This can be split into two problems: the existence of a finite simple Bruck
loop and the existence of a finite simple proper (non-Moufang, non-Bruck) Bol
loop.
First, to establish the existence of a finite simple Bruck loop, it would be
sufficient to find a Bruck loop which violated the weak Lagrange property, for
then Corollary \ref{coro:2} would imply the existence of a simple Bruck loop
which is not a cyclic group.
On the other hand, Corollary 2 might apply to the problem of the existence of
finite simple proper Bol loops if the weak Lagrange property is established for
all Paige loops. It would then be sufficient to find a Bol loop which violated
the weak Lagrange property, for by Corollary 2, there would exist a simple Bol
loop which violated that property. If all simple Moufang loops have the weak
Lagrange property, then the simple Bol loop in question will not be Moufang.
\end{document} | math | 12,671 |
\begin{document}
\def\mathbb{K}{\mathbb{K}}
\def\mathbb{R}{\mathbb{R}}
\def\mathbb{C}{\mathbb{C}}
\def\mathbb{Z}{\mathbb{Z}}
\def\mathbb{Q}{\mathbb{Q}}
\def\mathbb{D}{\mathbb{D}}
\def\mathbb{N}{\mathbb{N}}
\def\mathbb{T}{\mathbb{T}}
\def\mathbb{P}{\mathbb{P}}
\renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{cond}{C}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\title{Geometric generalizations in Kresin-Maz'ya
Sharp Real-Part Theorems \thanks{AMS classification number:30A10.}
\thanks{Keywords: Sharp constants, Real Part Theorems.}}
\author {Lev Aizenberg \\
Department of Mathematics, Bar-Ilan University,\\
52900 Ramat-Gan, Israel,
email: [email protected] \\
and\\
Alekos Vidras,\\
Department of Mathematics and Statistics,\\
Univ.of Cyprus, Nicosia 1678, Cyprus, \\email: [email protected]}
\maketitle
\begin{abstract}
In the present article we give geometric generalizations
of the estimates from Chapters 5,6,7 from \cite{krem:gnus},
while extending their sharpness to new cases.
\end{abstract}
\section{Preliminaries }
\setcounter{equation}{0}
G.Kresin and V.Maz'ya, in their recently published, remarkable
research monograph \cite{krem:gnus}, have collected in one place
generalizations and different modifications of theorems, which the
authors called {\it{real-part theorems}} honoring the known
theorem of Hadamard (1892). All of them are formulated with sharp
constants.\par
In the present article, our starting point is the content of the
three chapters of the above monograph, namely, Chap.5,{\it{ Estimates for the
derivatives of analytic functions,}} Chap.6, {\it{Bohr's type real
estimates}}, Chap.7, {\it{Estimates for the increment of
derivatives of analytic functions}}. Methods, used in
\cite{aiz:gnus}, allow a geometric generalization for some of the
results in those chapters
and the sharpness of the results is extended to new cases. We
remark, that there are two methods to prove the sharpness of the
corresponding estimates. The first one, which is shorter, uses
a number of facts from the monograph \cite{krem:gnus}, the
second, somewhat longer, but independent of \cite{krem:gnus}. To
illustrate the point we use the approach in \S2, and the second
one in \S3.
\section{Estimates for the derivatives of holomorphic functions}
\setcounter{equation}{0} We begin by formulating a theorem,
inspired by Th.5.1, from \cite{krem:gnus}
\begin{theorem}
{\rm{(G.Kresin-V.Maz'ya)}}
Let $f$ be holomorphic in
$\mathcal{D}_R=\{z\in\mathbb{C}:\; \vert z\vert <R\}$ and let
$z=re^{i\theta}$, $a=r_ae^{i\theta}$, $0\leq r_a\leq r<R$. Then
the inequality
\begin{eqnarray}
\vert f^{(n)}(z)\vert \leq
{2n!R(R-r_a)\over(R-r)^{n+1}(R+r_a)}\mathcal{Q}_a(f)
\end{eqnarray}
holds for every $n\geq 1$ with the best possible constant , where
$\mathcal{Q}_a(f) $ is each of the following expressions\\
{\rm{i)} }$\sup_{\vert \zeta \vert <R}\mathbb{R}e f(\zeta)-\mathbb{R}e f(a)$. In
this case the claim of the {\rm Theorem 2.1} is a generalization
of the
Hadamard real-part theorem.\\
{\rm{ii)}} $\sup_{\vert \zeta \vert <R}\vert \mathbb{R}e f(\zeta)\vert
-\vert \mathbb{R}e f(a)\vert $. In this case the claim of the {\rm Theorem
2.1} is a
generalization of the Landau type inequality.\\
{\rm ii)} $\sup_{\vert \zeta \vert <R}\vert f(\zeta)\vert -\vert
f(a)\vert $. In this case the claim of the {\rm Theorem 2.1} is a
generalization of the Landau inequality.\\
{\rm iv)} $\mathbb{R}e f(a)$, if $\mathbb{R}e f>0$ on $\mathcal{D}_R$. In this
case the claim of the {\rm Theorem 2.1} is a generalization of the
Caratheodory inequality.
\end{theorem}
We remark that the estimate $(2.1)$ has
a meaning in the cases {\rm(i)}, {\rm(ii)}, {\rm(iii)} only if the
corresponding $sup$ is finite. In the case $a=z$ the inequalities
are due to S.Ruscheweyh \cite{rush:gnus}. Other references on
different type of inequalities preceding the inequality $(2.1)$
are to be found in \cite{krem:gnus}.\par
Denote now by $\widetilde G$ the convex
hull of the domain $G\subset\mathbb{C}$. A point $p\in\partial G$
is called a {\em point of convexity } if $p\in \partial \widetilde
G$. A point of convexity $p$ is called {\em regular} if there
exists a disk $\mathcal {D}^{\prime}\subset G$ so that $p\in
\mathcal {D}^{\prime}$.
\begin{theorem}
Let $f$ be holomorphic function in $\mathcal{D}_R$,
$f(\mathcal{D}_R)\subset G$, where $G$ is a domain $\mathbb{C}$, so
that $\widetilde G\not=\mathbb{C}$. Let also $z=re^{i\theta}$,
$a=r_ae^{i\theta}$ be complex numbers so that $0\leq r_a\leq r<R$.
Then the inequality
\begin{eqnarray}
\vert f^{(n)}(z)\vert \leq {2n!R(R-r_a)\over
(R-r)^{n+1}(R+r_a)}dist(f(a),\partial \widetilde G)
\end{eqnarray}
holds for $n\geq 1$. If $\partial G$ contain at least one regular
point of convexity, then the constant in $(2.2)$ is sharp.
\end{theorem}
\begin{remark}
We point out that the cases {\rm(i)} and {\rm(ii)} in
{\rm Theorem 2.1} are the cases when
\begin{eqnarray*}
G=\{z\in\mathbb{C}:\; \mathbb{R}e z<\sup_{\vert \zeta\vert <R}\mathbb{R}e f(\zeta)\}
\end{eqnarray*}
is half-plane or
\begin{eqnarray*}
G=\{z\in\mathbb{C}:\; \vert z\vert <\sup_{\vert \zeta\vert <R}\vert
\mathbb{R}e f(\zeta)\vert\}
\end{eqnarray*}
is a strip. The case {\rm(iii)} in {\rm Theorem 2.1} corresponds to the set
\begin{eqnarray*}
G=\{z\in\mathbb{C}:\; \vert z\vert <\sup_{\vert \zeta\vert <R}\vert
f(\zeta)\vert\}
\end{eqnarray*}
being a disc. The case {\rm(iv)} of this theorem corresponds to
the right half-plane case
\begin{eqnarray*}
G=\mathbb{P}i=\{z\in\mathbb{C}:\; \mathbb{R}e f( z)>0\}.
\end{eqnarray*}
\end{remark}
{\bf{Proof:}} Let us consider the case ${\rm (iv)}$ in Theorem
2.1. If $\mathbb{R}e f(z)>0$, $z $ on $\mathcal{D}_R$, then for every
$n\geq 1$ one has
\begin{eqnarray}
\vert f^{(n)}(z)\vert \leq {2n!R(R-r_a)\over
(R-r)^{n+1}(R+r_a)}dist(f(a),\partial \mathbb{P}i),
\end{eqnarray}
where $\mathbb{P}i $ is the right half-plane. By translation and rotation,
this half plane can be transformed into any half-plane $\mathbb{P}i _1$.
The same transformations can be applied to holomorphic in
$\mathcal{D}_R$ functions, that is $f(z)\longrightarrow
(f(z)+c)e^{i\phi }$. Under such transformations of both, the
half-plane and the functions, at the same time the conclusion
$(2.3)$ does not alter.\par
Now, let $G$ be a domain in
$\mathbb{C}$, such that $\widetilde G\not=\mathbb{C}$. Let also
the distance from the right hand-side of the inequality $(2.2)$ is
realized at the point $p\in\partial\widetilde G$. Then there exist
a line of support to $\widetilde G$ at $p$, bounding the
half-plane $\mathbb{P}i_1 $, such that $G\subset \mathbb{P}i_1$. For this
particular half-plane $\mathbb{P}i _1$ we apply the inequality $(2.3)$ and
using the fact
\begin{eqnarray*}
dist(f(a),\partial \widetilde G)=dist(f(a),p)=dist(f(a),\mathbb{P}i_1),
\end{eqnarray*}
we obtain $(2.2)$. \par
Assume now that $\partial G$ contains at
least one regular point of convexity $p\in \partial G\cap\partial
\widetilde G\cap\partial \mathcal{D}^{\prime}$, where
$\mathcal{D}^{\prime}$ is some disc $\mathcal{D}^{\prime}\subset
G$. Let $\mathcal{D}_{\beta}$ be a disk, in which the functions
from {\rm(iii)} of Theorem 2.1 take their values:
\begin{eqnarray*} \mathcal{D}_{\beta}=\{z\in\mathbb{C}:\vert
z\vert <\beta =\sup_{\vert \zeta \vert <R}\vert f(\zeta)\vert \}.
\end{eqnarray*}
For this disc $\mathcal{D}_{\beta}$ the constant in $(2.1)$ is
sharp. That is, there exists a family of functions, which we take
from \S5.7,\cite{krem:gnus},
\begin{eqnarray}
g_{\xi}(z)={\xi \over z-\xi}+{\vert \xi\vert^2\over \vert \xi
\vert ^2-R^2},
\end{eqnarray}
depending on complex parameter $\xi=\rho e^{i\theta}$, $\rho >R$,
for which this constant is attained. Put $z=x<0$, $\xi =\rho >0 $,
the sharpness of the constant in \cite{krem:gnus} was proved by
passing to the limit when $\rho\downarrow R$. Then the first
summand in $(2.4)$ is negative and remains bounded while $\rho
\downarrow R$. The second summand in $(2.4)$ tends to $+\infty $ while $\rho
\longrightarrow R$. Hence, when $\rho $ is sufficiently close to
$R$, the function $g_{\rho}(x)$ is positive. This implies that
$g_{\rho}(a)$ is also positive.\par Of crucial importance in our
reasoning is the fact that one can transform the disk
$\mathcal{D}_{\beta}$ into the disc $\mathcal{D}^{\prime}$ by
using homothety and translation. At the same time , we apply both
transformations to the family of functions
$g_{\xi}(z)\longrightarrow \widetilde g_{\xi}(z)=\alpha
g_{\xi}(z)+c$. Then the inequality $(2.1)$ in the case {\rm (iii)}
of the Theorem 2.1 does not change. Furthermore, if the for the
family of functions $\{g_{\xi}(z)\}_{\xi }$ the sharpness of the
constant was attained before the applications of the
transformations under the assumption $f(\mathcal{D}_R)\subset
\mathcal{D}_{\beta}$, then the family of functions obtained after
the transformation illustrates the sharpness of the constant under
the assumption $f(\mathcal{D}_R)\subset \mathcal{D}^{\prime}$.\par
In the \S5.7 of \cite{krem:gnus}, for the proof of the sharpness of the given constant only the modula of $\vert
g_{\xi}(z)\vert $ and $\vert g_{\xi}(a)\vert $ were used.
Therefore, instead of the family $\{g_{\rho}(z)\}_{\rho }$ one can
use the family $\{e^{i\phi }g_{\rho}(z)\}_{\rho }$ in order to
show the required sharpness of the constant, provided that $\phi $ is chosen in a such a manner that the point
$e^{i\phi }g_{\rho}(a)$ on the radius eminnating from the center of the disc $ \mathcal{D}_{\beta}$ to
the point $p^{\prime}\in\partial \mathcal{D}_{\beta}$, where $p^{\prime}$ is the pre-image of the point $p$ under the
above mentioned homothety and translation of the disc $ \mathcal{D}_{\beta}$ into the disc $\mathcal{D}^{\prime}$.
Then, after the transformation, the point $\widetilde g_{\rho}(a)$
will lie on the radius eminnating from the center of the disc
$\mathcal{D}^{\prime}$ to the point $p$, and therefore
\begin{eqnarray*}
dist(\widetilde g_{\rho}(a),\partial \widetilde G)=dist(\widetilde
g_{\rho}(a),p).
\end{eqnarray*}
This completes the proof of the theorem. $\diamondsuit $\par
For $a=0$ one has the following
\begin{corollary}
Let $f$ be holomorphic in the disc $ \mathcal{D}_{R}$ and $f(\mathcal{D}_{R})\subset
G$, where $G$ is a domain in $\mathbb{C}$ whose convex hull
$\widetilde G $ is not equal to $\mathbb{C}$.Then the inequality
\begin{eqnarray}
\vert f^{(n)}(z)\vert\leq {2n!R\over (R-r)^{n+1}}dist (f(0),
\partial \widetilde G)
\end{eqnarray}
holds for every $n\geq 1$.
If $\partial \widetilde G $ contains at least one regular point
of convexity , then the constant in $(2.5)$ is sharp.
\end{corollary}
\section{Bohr's type real part estimates}
\setcounter{equation}{0} The results, contained in the Theorems
6.1-6.4 in \cite{krem:gnus}, are collected in the following
\begin{theorem}
({\rm G.Kresin-V.Maz'ya}) Let the function
\begin{eqnarray}
f(z)=\sum\limits_{n=0}^{\infty}c_nz^n
\end{eqnarray}
be holomorphic in the disc $ \mathcal{D}_{R}$ and $q>0$, $m\geq
1$, $\vert z\vert =r <R $. Then the inequality
\begin{eqnarray}
\left(\sum\limits_{n=m}^{\infty}\vert c_nz^n\vert
^q\right)^{{1\over q}}\leq {2r^m\over R^{m-1}(R^q-r^q)^{{1\over
q}}}\mathcal{R}(f)
\end{eqnarray}
holds with the best possible constant in the cases when
$\mathcal{R}(f)$ is each one of the following expressions:\\
{\rm i)} $\sup_{\vert \zeta \vert <R}(\mathbb{R}e (f(\zeta)-\mathbb{R}e f(0))$.\\
{\rm ii)} $\sup_{\vert \zeta \vert <R}(\vert \mathbb{R}e f(\zeta)\vert
-\vert
\mathbb{R}e f(0)\vert )$. \\
{\rm iii)} $\sup_{\vert \zeta \vert <R}(\vert f(\zeta)\vert -\vert
f(0)\vert )$. \\
{\rm iv)} $\mathbb{R}e f(0)$, if $\mathbb{R}e f>0$ on $\mathcal{D}_R$.
\end{theorem}
We remark here that the case ${\rm(iii)}$ of the above theorem
gives for $m=q=1$ the classical theorem of Bohr, \cite{bohr:gnus}
(for related references see \cite{aiz:gnus}) for $r={R\over 3}$.
Similarly to the previous section, we state the geometric
generalization of this theorem.
\begin{theorem}
Let the function $(3.1)$ be holomorphic in the disc $\mathcal{D}_R$,
$q>0$, $m\geq 1$, $\vert z\vert =r<R$, and $f(\mathcal{D}_R)\subset
G$, where $G$ be a domain in $\mathbb{C}$ and $\widetilde
G\not=\mathbb{C}$. Then the inequality
\begin{eqnarray}
\left(\sum\limits_{n=m}^{\infty}\vert c_nz^n\vert
^q\right)^{{1\over q}}\leq {2r^m\over R^{m-1}(R^q-r^q)^{{1\over
q}}}dist(c_0,\partial \widetilde G),
\end{eqnarray}
holds. If the boundary $\partial G$ contains at least one regular
point of convexity, then the constant in $(3.3)$ is sharp.
\end{theorem}
{\bf{Proof:}} The estimate $(3.3)$ is proven in exactly the same
way as in the previous section. The sharpness of the constant in
$(3.3)$ can also be proven in the same manner, but we prefer to
give an independent proof.\par The main point of our approach is
the simple observation that convex hull of the domain between to
discs, when the smaller one is contained in the larger one and
their boundaries have exactly one common point, is the larger
disc.\par
To be more specific, for $a>0$ we consider
\begin{eqnarray*}
D_1&=& \{z\in {\bf C}: \vert z-ai\vert < a\}\\
D_2&=&\{z\in {\bf C}: \vert z-2ai\vert < 2a\}\\
\end{eqnarray*}
be two discs. It is clear that $\partial D_1\cap \partial
D_2=\{0\}$. Define the domain
\begin{eqnarray*}
G=D_1^c\cap D_2,
\end{eqnarray*}
where, as usual, $D_1^c $ denotes the complement of the disc $D_1$
in $\mathbb{C}$.\par
It is obvious that the convex hull
$\widetilde G$ of $G$ is equal to $D_2$. The conformal map
$f(\zeta)={1\over \zeta }$ maps $G$ onto a strip
\begin{eqnarray*}
T=\{z\in {\bf C}: -{i\over 2a}<\Im z<-{i\over 4a}\}
\end{eqnarray*}
The $\partial T$ consists of two parallel lines, on which $z$ moves
in the opposite directions. The point $z=\infty $ is a double point.
The width of the strip is equal to ${1\over 4a}$. The map
$w=f_1(z)=z+{i\over 2a}$ shifts the strip up. The new strip $
T+{i\over 2a}$ has the real axis as its lower bound, while the width
remains the same. Then the map $\omega=f_2(w)=e^{4a\pi w}$
transforms con-formally the strip onto the upper half plane
\begin{eqnarray*}
H=\{\omega \in {\bf C}: \Im \omega >0\}
\end{eqnarray*}
Finally, we transfer $H$ by translation $\psi=f_3(\omega)=\omega
+p$, where $p\in {\bf C}$, $\Im p <0 $. All the above maps are
invertible. Thus we have a map
\begin{eqnarray*}
F&:& H+p\longrightarrow G\\
F(\psi)&=&{1\over {1\over 4a\pi }\ln (\psi-p)-{i\over 2a}},\\
\end{eqnarray*}
which is the inverse of the composition of $f_i$. Its expansion in
the disc $D(0,\vert p\vert)$ is given by
\begin{eqnarray*}
c_n&=&{a_n-c_0b_n-c_1b_{n-1}-\dots-c_{n-1}b_1\over
b_0},\;\rm{where}\\
a_n&=&0,\; \forall n\geq 1,\;a_0=1,\\
b_n&=&{({1\over 4a\pi}\ln(\psi
-p)-{i\over 2a})^{(n)}\over n!}\vert _{\psi =0},\;n\geq 1,\\
\end{eqnarray*}
and $c_0=F(0)$. Or equivalently,
\begin{eqnarray*}
c_n&=&{(-1)^nb_1^n\over b_0^{n+1}}, \;\rm{where}\\
b_1&=&{1\over 4a\pi}{1\over (-p)},\;b_0={1\over
4a\pi}\ln(-p)-{i\over 2a}\\
\end{eqnarray*}
Thus
\begin{eqnarray*}
c_n={4a\pi\over p^n (\ln (-p)-{i\over 2a})},\;n\geq 1
\end{eqnarray*}
If $p=-i$ then $c_0={1\over {1\over 4a\pi}\ln (i) -{i\over
2a}}={8a\over 3}i$
and hence
\begin{eqnarray*}
dist(c_0,\tilde G)={4a\over 3}
\end{eqnarray*}
Assume now, that the best constant in $(3.3)$ is denoted by $C(r)$.
For $\vert z\vert =r $, $0<r<p$, one has
\begin{eqnarray*}
\sum\limits_{n=1}^{\infty}\vert c_n z^n\vert
^q&=&\sum\limits_{n=1}^{\infty}({4\pi a\over \vert \ln(-p)-{i\over
2a}\vert})^q{r^{qn}\over p^{qn}}
\end{eqnarray*}
The last power series is geometric one with ratio $({r\over p})^q$.
Therefore, for $m\geq 1$, one has
\begin{eqnarray*}
({4a\pi \over \vert \ln(-p)-{i\over
2a}\vert})^q\sum\limits_{n=m}^{\infty}({r\over p})^{nq}&=&({4a\pi
\over \vert \ln(-p)-{i\over 2a}\vert})^q{r^{mq}p^q\over
p^{mq}(p^q-r^q)}\\
&=&2{4a\over 3}{r^{mq}p^q\over p^{mq}(p^q-r^q)}
\end{eqnarray*}
Thus, the estimate from above is
\begin{eqnarray*}
{8a\over 3} {r^{m}\over p^{m-1}}{1\over ((p)^q-r^q)^{{1\over
q}}}\leq C(r) dist(c_0,\tilde G)=C(r){4a\over 3}
\end{eqnarray*}
or
\begin{eqnarray*}
{2r^{m}\over R^{m-1}((R^q-r^q))^{{1\over q}}}\leq C(r)
\end{eqnarray*}
Thus the exactness is proven for $m=1,2,\dots $. \par
This
particular example shows that in the case of bounded, simply
connected domain $G$, whose boundary contains a point of regular
convexity, the constant in the Theorem 3.2 is sharp. Actually if
$\zeta _0$ is a point of regular convexity, then it means that
$\zeta_0\in
\partial \widetilde {\mathcal G}$ and there is a disc $U$ of radius
$\rho$, contained in $ G$ so that $\partial U\cap\partial
{\mathcal G}=\{\zeta _0\}$. We inscribe in the disc $U$, the disc
$U_1$ whose center lies in the diameter of $U$, whose end-points
are $\zeta _0, \zeta_0^{\prime}$. The radius of the disc $U_1$ is
${1\over 2}\rho$ and its center $k_1$ satisfies $\vert
k_1-\zeta_0^{\prime}\vert ={\rho\over 2}$. Then for the domain
${\mathcal U}=U\cap U_1^c $ we repeat the construction of the
above example. The key fact here is that $\widetilde {\mathcal
U}=U $ and that the distance $d(c_0, \widetilde {\mathcal U})$ is
realized at the point $\zeta _0\in \widetilde G$. So the sharpness
of the constant is proven.$\diamondsuit $\par
Furthermore, the Theorem 6.5, \cite{krem:gnus}, states
\begin{theorem}
({\rm G.Kresin-V.Maz'ya})
Let $f(z)$ be a function holomorphic in
the disc $\mathcal {D}_R$ and assume that in the neighborhood of
the point $a \in \mathcal {D}_R$ the expansion
\begin{eqnarray}
f(z)=\sum\limits_{k=0}^{\infty}c_k(a)(z-a)^k
\end{eqnarray}
is valid. Then for every $z\in \mathcal {D}_R$, $\vert z-a\vert =r
<d_a =dist(a, \partial \mathcal {D}_R)$ the following inequality
\begin{eqnarray*}
\sum\limits_{k=1}^{\infty}\vert c_k(a)(z-a)^k\vert \leq {2Rr\over
(2R-d_a)(d_a-r)}\mathcal{Q}_a(f)
\end{eqnarray*}
holds with the best possible constant and where $\mathcal{Q}_a(f)$
is each of the following expressions {\rm(i)}, {\rm(ii)},
{\rm(iii)}, {\rm(iv)} from the {\rm Theorem 2.1}.
\end{theorem}
Similarly one can prove
\begin{theorem}
Let $f(z)$ be a function holomorphic in the disc $\mathcal {D}_R$
and assume that in the neighborhood of the point $a \in \mathcal
{D}_R$ the expansion $(3.4)$ is valid. Assume also that
$f(\mathcal {D}_R)\subset G$, where $G$ is a domain in
$\mathbb{C}$ such that $\widetilde G\not=\mathbb{C}$. Then for
every $z\in \mathcal {D}_R$, $\vert z-a\vert =r <d_a =dist(a,
\partial \mathcal {D})$ the following inequality
\begin{eqnarray}
\sum\limits_{k=1}^{\infty}\vert c_k(a)(z-a)^k\vert \leq {2Rr\over
(2R-d_a)(d_a-r)}dist(f(a), \partial \widetilde G)
\end{eqnarray}
holds. If $\partial G $ contains at least one regular point of
convexity, then the constant in $(3.5)$ is sharp.
\end{theorem}
\section{Estimates for the increment of derivatives of holomorphic functions}
\setcounter{equation}{0} In this section we will be using the
notation $\mathbb{D}elta g(z)=g(z)-g(0)$ to describe the increment of a
function $g$ at $z=0$. We will formulate the results of the
Corollaries 7.2-7.5 from the book \cite{krem:gnus}in the following
manner.
\begin{theorem}
({\rm G.Kresin-V.Maz'ya}) Let $f(z)$ be a function holomorphic in
the disc $\mathcal {D}_R$. Then, for any fixed $z$, $\vert z\vert
=r<R$, the inequality
\begin{eqnarray*}
\vert f^{(n)}(z)-f^{(n)}(0)\vert \leq
{2n!(R^{n+1}-(R-r)^{n+1})\over (R-r)^{n+1}R^n}\mathcal{R}(f)
\end{eqnarray*}
holds with the best constant for every $n\geq 0$ and where
$\mathcal{R}(f)$ is each of the expression {\rm(i)}-{\rm (iv)}
from the {\rm Theorem 3.1}.
\end{theorem}
Analogously to the previous sections one can prove the following
\begin{theorem}
Let $f(z)$ be a function holomorphic in the disc $\mathcal {D}_R$.
Assume also that $f(\mathcal {D}_R)\subset G$, where $G$ is a
domain in $\mathbb{C}$ such that $\widetilde G\not=\mathbb{C}$.
Then for every fixed $z\in \mathcal {D}_R$, $\vert z\vert =r $ the
following inequality
\begin{eqnarray}
\vert f^{(n)}(z)-f^{(n)}(0)\vert \leq
{2n!(R^{n+1}-(R-r)^{n+1})\over (R-r)^{n+1}R^n}dist
(f(0),\partial\widetilde G)
\end{eqnarray}
holds for every $n\geq 0$. If $\partial G $ contains at least one
regular point of convexity, then the constant in $(4.1)$ is the
best one.
\end{theorem}
We conclude the article with the following
\begin{remark}
1) It seems that it is possible to formulate a
geometric
variant of the results from \cite{kres:gnus}. \\
2) Do the constants in the above cited geometric generalizations
of results of Kresin-Maz'ya remain sharp, if one assumes
that $\partial G$ does not contain any regular point of convexity?
\end{remark}
\end{document} | math | 20,851 |
\begin{document}
\newcommand{\ket}[1]
{\left|#1\right\rangle}
\newcommand{\bra}[1]
{\left\langle #1\right|}
\newcommand{\braket}[2]
{\left\langle #1\right|\left.#2\right\rangle}
\title{Displacement-enhanced entanglement distillation of single-mode-squeezed entangled states}
\author{Anders Tipsmark, Jonas S. Neergaard-Nielsen,$^{\ast}$ and Ulrik L. Andersen}
\address{Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kgs. Lyngby, Denmark}
\email{$^{\ast}[email protected]}
\begin{abstract}
It has been shown that entanglement distillation of Gaussian entangled states by means of local photon subtraction can be improved by local Gaussian transformations. Here we show that a similar effect can be expected for the distillation of an asymmetric Gaussian entangled state that is produced by a single squeezed beam. We show that for low initial entanglement, our largely simplified protocol generates more entanglement than previous proposed protocols. Furthermore, we show that the distillation scheme also works efficiently on decohered entangled states as well as with a practical photon subtraction setup.
\end{abstract}
\ocis{(270.5585) Quantum information and processing;
(270.5290) Photon statistics;
(270.6570) Squeezed states;
(000.6800) Theoretical physics}
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\end{thebibliography}
\section*{}
Continuous variable (CV) entanglement is a valuable resource for many quantum informational protocols~\cite{Braunstein2005,Andersen2010,Weedbrook2012}. However, the performance of these protocols is often limited due to the difficulty in generating CV states with a high degree of entanglement. Moreover, even if a large degree of entanglement can be produced using a highly efficient nonlinear parametric process, the distribution of it (e.g. among two parties in a network) will inevitably lead to dissipation, rendering the state weakly entangled. To improve the performance of quantum information processing, it is therefore important to devise a protocol that increases the amount of entanglement between two distant parties by means of local quantum transformations and classical communication. This can be done by the process of entanglement distillation.
Distillation of non-Gaussian entanglement (either a pure or a mixed non-Gaussian entangled state)
can be implemented using simple linear optics~\cite{Browne2003} as demonstrated in~\cite{Dong2008,Hage2008}. On the contrary, the distillation of Gaussian states (either pure or mixed Gaussian states) is challenging as it will inevitably require some non-Gaussian transformations~\cite{Eisert2002,Giedke2002,Fiurasek2002} enabled by a very strong Kerr nonlinearity~\cite{Duan2000a,Fiurasek2003}, using a non-Gaussian measurement~\cite{Opatrny2000,Cochrane2002,Olivares2003}, using non-Gaussian resources~\cite{Xiang2010} or using a combination of photon addition and subtraction~\cite{Yang2009,Fiurasek2010,Lee2011c,Navarrete-Benlloch2012}. An intriguing scheme for entanglement distillation of Gaussian states was suggested by Opatrn\'y et al. \cite{Opatrny2000} and involves local photon subtraction of a two-mode entangled state. The scheme was recently implemented by Takahashi et al~\cite{Takahashi2010a} (entanglement distillation by non-local photon subtraction has also been demonstrated~\cite{Ourjoumtsev2007b}). This photon subtraction scheme can however be improved by local Gaussian operations: Zhang and van Loock~\cite{Zhang2011} showed that local squeezing operations prior to photon subtraction improve the performance of distillation in terms of producing states with a higher degree of entanglement and with higher success rate. Secondly, it was realized by Fiur\'a\v{s}ek~\cite{Fiurasek2011a} that by using the much simpler operations of local phase space displacements, a similar improvement can be achieved.
In all these previous proposals on entanglement distillation, the considered Gaussian entangled state was produced by interfering two single mode squeezed states on a beam splitter. However, in practice it is much simpler to generate entanglement from a single squeezed mode that is split on a balanced beam splitter, as was done in \cite{Takahashi2010a}. It was found in \cite{Tipsmark2012,Cernotik2012} that -- surprisingly -- in some cases the usage of a single mode squeezed beam for the generation of Gaussian entangled states, the degree of entanglement after distillation is higher than if a two-mode squeezed state was used. In this paper, we further analyze the displacement enhanced entanglement distillation with a focus on the initial single mode squeezed state, which will be closer to an experimental realization. We investigate the photon number distribution of the distilled states to gain further insight into how displacement helps, and we study the influence of losses in the distribution channels on the attainable entanglement and optimal displacements.
We consider the entanglement distillation setup shown in Fig.~\ref{fig:setup}. The entangled state is simply produced by dividing a single mode squeezed state on a balanced beam splitter, and the resulting modes are sent through lossy channels to the two sites, denoted A and B. At these two sites a Gaussian displacement transformation as well as single photon subtractions are applied to distill the quantum state. In the following we first consider the distillation protocol when the channels are loss-free and the photon subtraction is ideal, and secondly we consider the more realistic scenario where the channels are lossy and the photon subtraction process is non-ideal.
\begin{figure}
\caption{a) Schematic of the proposed setup. $S$ denotes the single mode squeezed state and $D(\alpha),D(\beta)$ are displacement operations. b) Two equivalent setups for displacement controlled photon subtraction where $\beta' = \sqrt{1-T}
\label{fig:setup}
\end{figure}
Our starting point is a single mode squeezed state mixed with vacuum on a balanced beam splitter (written in the Fock state representation):
\begin{eqnarray}
\ket{\zeta_1}~&=&~U_{BS}(1-\gamma_1^2)^{1/4}\sum_{n=0}^\infty\frac{\sqrt{(2n)!}}{n!}\left(\frac{\gamma_1}{2}\right)^n\ket{2n}|0\rangle, \label{SMS}
\end{eqnarray}
where $U_{BS}$ is the beam splitter unitary transformation, $\gamma_1=\tanh(s_1)$ and $s_1$ is the squeezing parameter.
In the weak squeezing limit ($\gamma\ll 1$) the state can be truncated to
\begin{eqnarray}
\ket{\zeta_1}~&\approx&~\ket{00}+\frac{\gamma_1}{\sqrt{2}}\left(\frac{1}{2}\ket{20}+\frac{1}{\sqrt{2}}\ket{11}+\frac{1}{2}\ket{02}\right). \label{SMS2}
\end{eqnarray}
The entanglement of a pure state like this is usually quantified by the entropy of entanglement; however, since we will later compare with mixed states for which the entropy is not a proper measure, we adopt instead as our entanglement measure the logarithmic negativity (LN). The LN is defined as the binary logarithm of the trace norm of the partially transposed density matrix $(\ket{\zeta_1}\bra{\zeta_1})^{T_A}$ \cite{Vidal2002}. The logarithmic negativity of the above state as a function of the average photon number $\langle n \rangle = \gamma_1^2/(1-\gamma_1^2)$ is plotted in Fig.~\ref{fig:negativity}(a) by the black dotted line.
The idea is now to increase the entanglement of the state, and this can be done by removing the vacuum contribution and equalizing the weight of the low order excitation terms. For weak squeezing (as in Eq.~(\ref{SMS2})), it is straightforwardly realized that such an equalization can be obtained simply by performing single photon subtraction on just one of the modes A or B. For single photon subtraction of mode A, enabled by the bosonic annihilation operator $a_A$, the result is
\begin{eqnarray}
\ket{\zeta_1}~&\rightarrow&~\hat{a}_A\otimes\mathbb{I}_B\ket{\zeta_1}\\
~&=&~\frac{\gamma_1}{2}\left(\ket{10}+\ket{01}\right), \label{SPS}
\end{eqnarray}
which is maximally entangled (in the two-dimensional subspace, thus neglecting higher order terms). The resulting entanglement is plotted in Fig.~\ref{fig:negativity}(a) with the black dash-dotted line, and it is shown that for very low squeezing degrees the LN reaches the maximum value of 1 for a 2D Hilbert space. However, the state should be described in a larger Hilbert space in which the state is not maximally entangled.
We next consider the simultaneous subtraction of single photons at both sites as described by the operator $\hat{a}_A\otimes\hat{a}_B$ and denoted 2PS (2-Photon-Subtraction). The resulting LN is plotted in Fig. \ref{fig:negativity}(a) by the black dashed line. In contrast to the previous protocol (with a single photon being subtracted, 1PS), this protocol is not very effective for low average photon numbers but for higher numbers (larger than about 0.21, corresponding to 3.9 dB of squeezing) it becomes more effective.
The entanglement can however be further enhanced for all degrees of initial squeezing by applying a Gaussian transformation prior to photon subtraction. The transformation that leads to this enlargement is the simple phase space displacements, $D(\alpha)=\exp(\alpha a_A^\dagger-\alpha^* a_A)$ and $D(\beta)=\exp(\beta a_B^\dagger-\beta^* a_B)$, where $\alpha$ and $\beta$ are the complex excitations of the displacements. By implementing these displacements in mode A and B prior to and after the two photon subtractions (see Fig.~\ref{fig:setup}(b)), the state reads
\begin{eqnarray}
\ket{\Psi}~&=&~\left(\alpha\beta+\frac{\gamma_1}{2}\right)\ket{00}+\frac{\gamma_1}{\sqrt{2}}(\beta+\alpha)\frac{\ket{10}+\ket{01}}{\sqrt{2}}\nonumber\\ &&+\frac{\gamma_1}{\sqrt{2}}\alpha\beta\left(\frac{1}{2}\ket{20}+\frac{1}{\sqrt{2}}\ket{11}+\frac{1}{2}\ket{02}\right)+O(\gamma_1^2).\nonumber
\end{eqnarray}
It is clear from this expression that the vacuum and single-photon contributions, which prevent the state from being strongly entangled, can be removed by setting
\begin{eqnarray}
\alpha~=~-\beta~=~\sqrt{\frac{\gamma_1}{2}},
\label{disp}
\end{eqnarray}
which can be shown to give
\begin{eqnarray}
\ket{\Psi}~=~-\frac{\gamma_1^2}{2\sqrt{2}}\left(\frac{1}{2}\ket{20}+\frac{1}{\sqrt{2}}\ket{11}+\frac{1}{2}\ket{02}\right)+O(\gamma_1^3). \label{2PSD}
\end{eqnarray}
This state contains an entanglement of $E_N=1.54$ for low initial squeezing degrees. This is not maximally entangled in the 3-dimensional Hilbert space due to the unequal weights but it is close to - the maximally entangled state would have $E_N=1.585$.
As can be seen from Fig.~\ref{fig:negativity}(b), the displacements in Eq.~(\ref{disp}) are maximizing the LN only for low degrees of initial squeezing. For larger squeezing degrees, the optimum displacement is larger than the one in Eq.~(\ref{disp}), and thus it is not optimized by the removal of the vacuum term. The single-photon components are however always cancelled by the choice of $\alpha=-\beta$.
The resulting optimized LN is plotted in Fig.~\ref{fig:negativity}(a) (both by black solid curves).
Clearly, the pre-Gaussian processing improves the entanglement for all average photon numbers~\cite{Tipsmark2012}. From Fig.~\ref{fig:negativity}(b) we furthermore see that the entanglement is highly sensitive to the exact value of the displacement amplitude for small initial photon numbers, but less so for increasing photon numbers.
\begin{figure}
\caption{(a) Logarithmic negativity as a function of the average number of photons of the initial squeezed state. 1(2)MSV: Single (two) mode squeezed vacuum states used initially. 1(2)PS: Single (two) photon subtraction. D2PS: Displacement based two-photon subtraction with optimal displacement.
(b) The logarithmic negativities obtained in the 1MSV D2PS setting for varying displacement amplitudes $\alpha=-\beta$. The color scale goes from $E_N=0$ (blue) to $E_N=2.1$. The dashed black curves indicate the $\alpha$ values that optimize $E_N$ (which results in the entanglement curve in (a)), while the solid line follows the optimal value in the low-squeezing limit of Eq. (\ref{disp}
\label{fig:negativity}
\end{figure}
For comparison, we also briefly consider the distillation of a two-mode squeezed state as was treated in \cite{Fiurasek2011a}. Here the starting point is
\begin{eqnarray}
\ket{\zeta_2}_{AB}~&=&~\sqrt{1-\gamma_2^2}\sum_{n=0}^\infty\gamma_2^n\ket{n}_A\ket{n}_B
\label{TMS}
\end{eqnarray}
where $\gamma_2=\tanh(s_2)$ and $s_2$ is the two-mode squeezing parameter. This state, which has an average photon number $\langle n \rangle = 2\gamma_2^2/(1-\gamma_2^2)$, can be produced by interfering and phase locking two single mode squeezed vacua on a balanced beam splitter. The implementation of a local single photon subtraction, either on one or both sites, transforms the entangled state into another entangled state with higher LN as illustrated in Fig. \ref{fig:negativity}(a) by yellow curves. Note that for identical initial $\langle n \rangle$, $\gamma_2$ is smaller than $\gamma_1$.
By displacing the state with excitations of $\alpha_A~=~-~\alpha_B~=~\sqrt{\gamma_2}$ before subtracting two photons, the distilled state reads $\ket{\psi^{\gamma_2}}~=~\gamma_2^{3/2}(\ket{10}-\ket{01})+O(\gamma_2^2)$ for weak initial entanglement ($\gamma_2 \ll 1$). This state is also maximally entangled in the two-dimensional sub space for low initial squeezing as was the case for the 1PS single mode squeezed state in Eq. (\ref{SPS}).
For larger initial squeezing levels, the optimal displacement is lower than $\sqrt{\gamma_2}$, as seen from Fig. \ref{fig:negativity}(c). The sensitivity to the displacement amplitude is however less than for the single-mode squeezed input state.
In particular, we note that the degree of entanglement after distilling the two-mode state using the displacement-enhanced protocol is lower than the one obtained for the single mode scheme if the average photon number is low. That is, using the largely simplified protocol with a single mode squeezer split on a beam splitter (rather than interfering and phase locking two single-mode squeezers) the distillation transformation produces an entangled state that contains more entanglement. However, this conclusion only holds for a low average photon number. For higher photon numbers, the usage of a two-mode squeezed state results in a state with larger entanglement than if a single-mode squeezed state was used. But the low-gain regime is often of interest, since entanglement in this region is fairly easy to prepare, manipulate and maintain.
\begin{figure}
\caption{Distribution of the different eigenstates of the entangled state that is produced from a single squeezed mode. The distributions are for the initial state (before distillation), the single photon subtracted state (1PS), the two photon subtracted state (2PS) and the displacement-enhanced protocol.}
\label{distri}
\end{figure}
It is known that entanglement is optimized when the dominant eigenstates of a given state have equal weights as is the case in Eq.~(\ref{SPS}) and partially in Eq.~(\ref{2PSD}). To get some physical insight into the formation of stronger entanglement, in Fig.~\ref{distri} we plot the weights of the different eigenstates resulting from the different distillation protocols using a single mode squeezed state. (These, as well as most of our other numerical calculations were done by the package \cite{Johansson2012}.) Due to the very large vacuum contribution in the initial entangled state, it is evident that this state is not highly entangled. However, by implementing the 1PS scheme, the vacuum term is basically split in two new eigenstates ($|01\rangle$ and $|10\rangle$) leading to a large increase in the entanglement. By subtracting two photons (2PS scheme) rather than a single photon, the balancing of the eigenstates is destroyed for low photon numbers as the vacuum term again becomes dominant, and thus the entanglement is lower. However, by implementing the displacement-enhanced scheme (D2PS), the weights between the eigenstates are partially re-balanced, thereby producing strong entanglement.
Based on this discussion, we can also clarify the reason behind a somewhat unintuitive observation from Fig. \ref{fig:negativity}(a), namely that a large amount of entanglement in some cases can be extracted from even very weakly squeezed initial states. As we just saw, the subtraction of a single photon or two photons following a displacement actually \textsl{adds} non-local photons to the final, heralded state. This is of course only possible because the photon subtraction is a highly probabilistic process -- see the discussion on success probabilities later. With the two-mode squeezed vacuum as initial state, a delocalized photon is also obtained in the weak squeezing limit with displacement and two-photon subtraction, giving a large entanglement. For the single-photon subtraction, however, no entanglement is obtained -- in stark contrast to the 1MSV case. The origin of this difference is the different form of the initial states: For 1MSV, the state in Eq.~(\ref{SMS2}) is a delocalized 2-photon state (plus vacuum), which results in a delocalized single photon after subtraction. For 2MSV, Eq.~(\ref{TMS}) is a twin-photon state (plus vacuum), which turns into a \textsl{localized} single photon upon detection in one of the modes.
\begin{figure}
\caption{Two-mode squeezing: Variance of the $x$-quadrature difference (left),
$p$-quadrature sum (center) and their sum (right) for the same states
as studied in Fig. \ref{fig:negativity}
\label{fig:variance}
\end{figure}
The photon subtraction is a highly non-Gaussian process, so although the initial entangled state is Gaussian this is not the case for the distilled state. A commonly used necessary and sufficient criterion for entanglement of two-mode Gaussian states (after suitable local transformations) is that of Simon \cite{Simon2000} and Duan et al \cite{Duan2000}: The state is entangled when the sum of the variances of the $x$-quadrature difference and $p$-quadrature sum of the two modes (the two-mode squeezing) is below the corresponding vacuum noise level, $\langle \Delta x_-^2 \rangle + \langle \Delta p_+^2 \rangle < 2$. Here, $x=(a^\dag+a)/\sqrt2$, $p=i(a^\dag-a)/\sqrt2$ and $x_-=x_A-x_B$, $p_+=p_A+p_B$. This criterion does not apply to non-Gaussian states, but it may still be relevant to see how the distilled states evaluate on this variance metric. For continuous variable quantum teleportation, for example, the performance is determined by the amount two-mode squeezing. We therefore plot the two-mode squeezing and its $x$ and $p$ components in Fig. \ref{fig:variance}.
One can see the simpler structure of the single-mode squeezed state in that $\langle \Delta x_-^2 \rangle$ is at the vacuum level for all cases, with and without photon subtraction. It is noteworthy from the right hand graph that, while the initial Gaussian states and the states distilled by non-displaced two-photon subtraction are always squeezed below the vacuum level, the situation is much more complex for the cases of one-photon subtraction and two-photon subtraction with displacement. When just considering the second-order moments as we do here, these states appear very ``noisy'' for low initial photon numbers, while they become squeezed for higher photon numbers. This is in spite of their large entanglement as quantified by the logarithmic negativity in Fig. \ref{fig:negativity}. Most remarkably, the 1MSV D2PS state which contains the most entanglement for low initial photon numbers is also by far the state with the largest variance due to its non-Gaussian character. This means that, although it is highly entangled, it will not be the optimal choice for all protocols. It may, however, be possible to turn the state more Gaussian through a protocol like the one in \cite{Browne2003}.
We now investigate the performance of the protocol under dissipation. This is of interest as in most protocols the entangled state is distributed in a network connected by lossy channels, thereby rendering the entangled states in any practical realization impure. The logarithmic negativities after distillation using the different strategies outlined above for the single- and two-mode squeezed states are depicted in Fig.~\ref{fig:loss}(a) for $\langle n\rangle=0.1$ as a function of the channel attenuation. It is assumed that the two channels possess identical attenuations.
Of course, the LN decrease with losses, but it appears that the displacement-distilled single-mode squeezed vacuum is more fragile to attenuation than the two-mode squeezing. While it is the most strongly entangled state in the ideal case, the exposure to losses soon makes it less entangled than the two-photon subtracted two-mode squeezing, both with and without displacement. Moreover, as the losses increase further, a point is reached where it is no longer advantageous to do displacement before the photon subtraction -- the kink on the curve around 55\% loss. This behaviour is not observed for the 2MSV input.
\begin{figure}
\caption{(a) The logarithmic negativity is plotted as a function of the loss of the two channels. The attenuation is identical in the two channels. See caption of Fig.~\ref{fig:negativity}
\label{fig:loss}
\end{figure}
\begin{figure}
\caption{(a) Optimal displacement as a function of channel losses for varying losses. The losses are increasing for lower-lying curves. (b) Maximum attainable logarithmic negativity for varying losses. The losses are the same as for the solid curves in (a). The shaded areas in gray (yellow) designate the ranges of initial photon numbers for which the 1MSV (2MSV) states are superior.}
\label{fig:loss_squeezing}
\end{figure}
To investigate it further, we calculate in Fig.~\ref{fig:loss}(b) the LN for the displacement distillation as a function of $\alpha$ for a range of different losses. The trends are qualitatively different for the 1MSV and 2MSV states. Whereas the 2MSV states have distinct optima that gradually tend towards zero as losses increase, the location of the peaks of the 1MSV curves go more slowly towards smaller amplitudes. On the other hand, for high losses the height of the peaks dip below the values for zero displacement -- this is where the kink in the 1MSV D2PS curve in (a) comes from. Consequently, we discover that this curve in fact consists of two separate regimes: To the left of the kink on the low-loss side, the entangled states have been distilled \textsl{with} displacement, while on the high-loss side, the distillation took place \textsl{without} displacement. We would therefore expect the states to be of very different character, for example in terms of their quadrature variance -- referring to Fig. \ref{fig:variance}(c), the two different regimes are, respectively, above and below the vacuum level -- and this is indeed the case (not displayed here).
In Fig.~\ref{fig:loss_squeezing}(a) we take a complementary look at the optimal distillation strategy. The optimal displacement amplitudes (those maximizing the LN) are plotted as a function of initial photon number for the same range of losses as in Fig.~\ref{fig:loss}(b). For zero loss, the 1MSV and 2MSV curves are identical to those in Figs.~\ref{fig:negativity}(b,c). We see the same pattern as observed before: For 2MSV initial state, the optimal displacement decreases gradually to zero with incresing losses. Interestingly, though, for large initial squeezing levels the displacement ceases to be effective at around 50-80\% channel attenuation, while for lower squeezing levels it is always beneficial to implement a displacement prior to photon detection.
While this latter observation is also true for the 1MSV case, the transition from the displacement-enhanced regime to the regime where it is optimal to avoid the displacement (the ``kink'') happens drastically, rather than smoothly as for 2MSV. The transition takes place for channel losses just above 50\% as seen from the extra dashed curves.
If we disregard the added complexity of using an initial 2MSV state instead of 1MSV, it is of interest to consider which of the two initial state preparations provide the most entanglement after the distillation process. As we have seen, for small to moderate photon numbers the 1MSV state can be distilled to the highest level of logarithmic negativity. If the distribution channels are lossy, however, we saw from Fig.~\ref{fig:loss}(a) that the 1MSV degrades faster. In Fig.~\ref{fig:loss_squeezing} we plot the attainable LN for our representative range of losses. It is evident that with increasing attenuation, using 2MSV rapidly becomes advantageous for almost all squeezing levels. Moreover, it also appears that for almost any amount of loss, there is no benefit at all in terms of LN of increasing the squeezing/photon number of the 1MSV state -- the levels are essentially constant. Of course, the event rate will still be higher with higher initial photon numbers.
In the above study we have assumed that a single photon is perfectly removed from each of the entangled modes, and we have modelled this by the annihilation operator. This is however an idealization. In practice a photon is usually subtracted by reflecting a small part of the beam on an asymmetric beam splitter, and subsequently measuring the presence of a single photon in the reflected mode. The most practical detector for this purpose is an avalanche photodiode (APD) which is an on/off detector that discriminates between zero photons and some photons. Assuming that the reflectivity of the asymmetric beam splitter as well as the initial average photon number are low, the reflected beam will contain much less than a single photon on average, and thus the APD will effectively work as a single photon counter. On the other hand, for a finite reflectivity and a large initial squeezing, in some rare (although non-negligible) cases two photons will impinge onto the detector which cannot be discriminated from a single photon event. This causes an error.
\begin{figure}
\caption{Performance of the distillation protocol for realistic photon subtraction. Logarithmic negativity (a) and success probability (b) as a function of the initial average photon number. The tap-off beamsplitter reflectivity for photon subtraction is 5\%. See caption of Fig.~\ref{fig:negativity}
\label{neg_onoff}
\end{figure}
To simulate the realistic setup, we have used a more rigourous model in which the initial state (either Eq.~(\ref{SMS}) or Eq.~(\ref{TMS})) is transformed through a beam splitter with a reflectivity of 5\%, displaced and finally measured with the projector $\Pi=1-|0\rangle\langle 0|$. The results (both for the single mode and the two-mode squeezed state) are shown in Fig.~\ref{neg_onoff}. It is obvious from the figures that the trend of the LN as a function of the initial average photon numbers is identical to the trend for the ideal distillation scheme in Fig.~\ref{fig:negativity}(a), but the amount of entanglement is slightly lower in the former case.
Another crucial parameter for characterizing the performance of the distillation protocols is the success probability. This is plotted in Fig.~\ref{neg_onoff}(b). At first it seems counter-intuitive that the success probability decreases when displacement is included, but it can be understood as follows.
The rate of the initial photon detection in mode A is indeed increased when displacement is introduced. However, photon subtraction from a squeezed state \textsl{increases} its average photon number, so the displacement (which is experimentally implemented by admixture of a coherent state) leads to a \textsl{lower} increase in the photon number of mode B after the subtraction in mode A. Furthermore, the displaced photon subtraction results in a state which has a small displacement in phase space (as opposed to the zero-mean of the initial state). The subsequent displacement of mode B before the photon detection there is opposite in direction of the state's displacement, leading to destructive interference in the detected mode. As a result of these two effects, the success probability of the second photon detection is considerably lower with displacement than without, outweighing the increased probability of the first detection.
In conclusion, we have theoretically investigated a displacement-enhanced distillation scheme of entangled states that are produced by a single squeezed mode. We have found that a simple Gaussian displacement operation prior to photon subtraction increases the entanglement of the distilled state. Similar conclusion has been found for the two-mode squeezed state scheme, but in contrast to the previous proposals, the experimental realization of our scheme is much simpler as it does not require the control and phase locking of two independent squeezed beams. An experimental realization is therefore feasible with current technology~\cite{Tipsmark2011,Neergaard-Nielsen2010,Lee2011}.
On the other hand, our analysis also shows that if the entanglement distribution channels are sufficiently lossy, it is still advantageous to use two-mode squeezing at the initial stage. This may also be required if a Gaussian-like two-mode squeezing is required for a given protocol.
\section*{Acknowledgments}
We acknowledge financial support from the Danish Agency for Science Technology and Innovation.
\end{document} | math | 36,107 |
\begin{document}
\title{Monogamy equality in $2\otimes 2 \otimes d$ quantum systems}
\author{Dong Pyo Chi}
\affiliation{
Department of Mathematical Sciences,
Seoul National University, Seoul 151-742, Korea
}
\author{Jeong Woon Choi}
\affiliation{
Department of Mathematical Sciences,
Seoul National University, Seoul 151-742, Korea
}
\author{Kabgyun Jeong}
\affiliation{
Nano Systems Institute (NSI-NCRC),
Seoul National University, Seoul 151-742, Korea
}
\author{Jeong San Kim}
\affiliation{
Institute for Quantum Information Science,
University of Calgary, Alberta T2N 1N4, Canada
}
\author{Taewan Kim}
\affiliation{
Department of Mathematical Sciences,
Seoul National University, Seoul 151-742, Korea
}
\author{Soojoon Lee}
\affiliation{
Department of Mathematics and Research Institute for Basic Sciences,
Kyung Hee University, Seoul 130-701, Korea
}
\date{\today}
\begin{abstract}
There is an interesting property about multipartite entanglement,
called the monogamy of entanglement.
The property can be shown by the monogamy inequality,
called the Coffman-Kundu-Wootters inequality~[Phys. Rev. A {\bf 61}, 052306 (2000);
Phys. Rev. Lett. {\bf 96}, 220503 (2006)],
and
more explicitly
by the monogamy equality
in terms of the concurrence
and the concurrence of assistance,
$\mathcal{C}_{A(BC)}^2=\mathcal{C}_{AB}^2+(\mathcal{C}_{AC}^a)^2$,
in the three-qubit system.
In this paper,
we consider the monogamy equality
in $2\otimes 2 \otimes d$ quantum systems.
We show that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$ if and only if $\mathcal{C}_{AC}^a=0$,
and also show that
if $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^a$ then $\mathcal{C}_{AB}=0$,
while there exists a state in a $2\otimes 2 \otimes d$ system such that
$\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)}>\mathcal{C}_{AC}^a$.
\end{abstract}
\pacs{
03.65.Ud,
03.67.Mn
}
\maketitle
Entanglement provides us with a lot of useful applications
in quantum communications,
such as quantum key distribution and teleportation.
In order to apply entanglement to more various and useful quantum information processing,
there are several important things which we should take into account.
One is to quantify the degree of entanglement,
and another one is to know about more properties of entanglement.
We here consider two measures of entanglement,
and investigate some properties of entanglement related to the two entanglement measures
in multipartite systems,
especially $2\otimes 2 \otimes d$ quantum systems.
Wootters' {\em concurrence}~\cite{Wootters}, $\mathcal{C}$
has been considered as one of the simplest measure of entanglement,
although there does not in general exist its explicit formula.
For any pure state $\ket{\phi}_{AB}$,
it is defined as $\mathcal{C}(\ket{\phi}_{AB})=\sqrt{2(1-\mbox{$\mathrm{tr}$}\rho_A^2)}$,
where $\rho_A=\mbox{$\mathrm{tr}$}_B\ket{\phi}_{AB}\bra{\phi}$.
Note that $\sqrt{2(1-\mbox{$\mathrm{tr}$}\rho_A^2)}=2\sqrt{\det\rho_A}$ in $2\otimes d$ systems.
For any mixed state $\rho_{AB}$,
it is defined as
\begin{equation}
\mathcal{C}(\rho_{AB})=\min \sum_{k} p_k \mathcal{C}(\ket{\phi_k}_{AB}),
\label{eq:concurrence}
\end{equation}
where the minimum is taken over its all possible decompositions,
$\rho_{AB}=\sum_k p_k \ket{\phi_k}_{AB}\bra{\phi_k}$.
Recently, another measure of entanglement has been presented,
and it is called the {\em concurrence of assistance} (CoA)~\cite{LVE},
which is defined as
\begin{equation}
\mathcal{C}^a(\rho_{AB})=\max \sum_{k} p_k \mathcal{C}(\ket{\phi_k}_{AB}),
\label{eq:CoA}
\end{equation}
where the maximum is taken over all possible decompositions of $\rho_{AB}$.
In multiqubit systems,
there is an interesting property about multipartite entanglement,
which is called the {\em monogamy of entanglement} (MoE).
Coffman, Kundu, and Wootters (CKW)
first proposed the monogamy inequality~\cite{CKW}, which states
the MoE in the 3-qubit system,
\begin{equation}
\mathcal{C}_{A(BC)}^2\ge\mathcal{C}_{AB}^2+\mathcal{C}_{AC}^2,
\end{equation}
and then its generalization was proved by Osborne and Verstraete~\cite{OV}.
Symmetrically, its dual inequality in terms of the CoA for 3-qubit states,
\begin{equation}
\mathcal{C}_{A(BC)}^2\le(\mathcal{C}_{AB}^a)^2+(\mathcal{C}_{AC}^a)^2,
\end{equation}
and its generalization into $n$-qubit cases have been also shown in~\cite{GMS,GBS}.
In particular, for 3-qubit states,
it can be readily proved that the monogamy equality~\cite{LJK,YS},
\begin{equation}
\mathcal{C}_{A(BC)}^2=\mathcal{C}_{AB}^2+\left(\mathcal{C}_{AC}^a\right)^2
\label{eq:ME222}
\end{equation}
holds.
We note that this monogamy equality shows the MoE
more explicitly than the CKW inequality.
Thus, it could be important to investigate
whether the monogamy equality would be possible
in any higher dimensional tripartite quantum systems,
and could be helpful for us to understand multipartite entanglement.
In this paper, we consider the monogamy equality in $2\otimes 2 \otimes d$ systems.
We show that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$ if and only if $\mathcal{C}_{AC}^a=0$,
and also show that if $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^a$ then $\mathcal{C}_{AB}=0$,
whereas there exists a state in a $2\otimes 2 \otimes d$ system such that
$\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)}>\mathcal{C}_{AC}^a$.
Now, we present the first main theorem.
\begin{Thm}\label{Thm:1}
Let $\ket{\Psi}_{ABC}$ be a state in a $2\otimes 2 \otimes d$ system.
Then the followings are equivalent.
\begin{enumerate}
\item[\em (i)] $\ket{\Psi}$ is of the form
$\ket{\phi}_A \otimes \ket{\psi}_{BC}$ or $\ket{\phi'}_C \otimes \ket{\psi'}_{AB}$.
\item[\em (ii)] $\mathcal{C}_{AC}^{a}=0$.
\item[\em (iii)] $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$.
\end{enumerate}
\end{Thm}
In order to prove Theorem~\ref{Thm:1},
we introduce the following lemma,
which is called the {\em Lewenstein-Sanpera decomposition}
for two-qubit states~\cite{LS}.
\begin{Lem}\label{Lem:LS_decomposition}
Let $\rho$ be a density matrix on $\mathbb{C}^2\otimes \mathbb{C}^2$.
Then $\rho$ has a unique decomposition in the form
$\rho=\lambda\rho_s+(1-\lambda)P_e$,
where $\rho_s$ is a separable density matrix,
$P_e=\ket{\Psi_e}\bra{\Psi_e}$ for a pure entangled state $\ket{\Psi_e}$,
and $\lambda\in [0,1]$ is maximal.
\end{Lem}
We now give the proof of the first main theorem.
\begin{proof}[Proof of Theorem~\ref{Thm:1}]
We first prove that (i) is equivalent to (ii).
Since $\rho_{AC}$ is in the form of
$\ket{\psi}_A\bra{\psi}\otimes\sigma_C$ or $\ket{\psi}_C\bra{\psi}\otimes\sigma_A$,
it is trivial that $\mathcal{C}_{AC}^{a}=0$.
Conversely, suppose that
$\rho_{AC}$ is not in the form of
$\ket{\psi}_A\bra{\psi}\otimes\sigma_C$ or $\ket{\psi}_C\bra{\psi}\otimes\sigma_A$.
Then $\rho_A$ and $\rho_C$ have at least rank 2.
Since $\mathcal{C}_{AC}^a = 0$,
\begin{equation}
\rho_{AC}=\sum_i {p_i}\ket{\phi_i}_A\bra{\phi_i}\otimes\ket{\psi_i}_C\bra{\psi_i},
\label{eq:rho_AC}
\end{equation}
and there exists at least one pair $(i,j)$ such that
$|\inn{\phi_i}{\phi_j}|\neq 1$ and $|\inn{\psi_i}{\psi_j}|\neq 1$.
By Hughston-Jozsa-Wootters (HJW) theorem~\cite{HJW},
$\rho_{AC}=\sum_k{q_k}\ket{\Phi_k}_{AC}\bra{\Phi_k}$ such that at least one
\begin{equation}
\ket{\Phi_k}_{AC}=\alpha\ket{\phi_i}_A\ket{\psi_i}_C
+\beta\ket{\phi_j}_A\ket{\psi_j}_C
\label{eq:Phi_k}
\end{equation}
is entangled ($\alpha\neq 0$ and $\beta\neq 0$),
and hence $\mathcal{C}_{AC}^{a} > 0$.
Since if $\ket{\Psi}_{ABC}$ is in the form
$\ket{\phi}_A \otimes \ket{\psi}_{BC}$ or $\ket{\phi'}_C \otimes \ket{\psi'}_{AB}$
then it is clear that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$,
the final one for completing the proof of this theorem,
is to show that if $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$
then $\ket{\Psi}_{ABC}$ has the form
$\ket{\phi}_A \otimes \ket{\psi}_{BC}$ or $\ket{\phi'}_C \otimes \ket{\psi'}_{AB}$.
We assume that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}\neq 0$
(If $\mathcal{C}_{A(BC)} = \mathcal{C}_{AB} = 0$ then
$\ket{\Psi}_{ABC}$ is of the form $\ket{\phi}_A \otimes \ket{\psi}_{BC}$,
and so this theorem is trivially true).
Then we clearly have $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}^a=\mathcal{C}_{AB}$,
that is, the average concurrence of any decomposition of $\rho_{AB}$ is equal to
$\mathcal{C}_{A(BC)}$.
By Lemma~\ref{Lem:LS_decomposition},
$\rho_{AB}=\lambda\rho_s + (1-\lambda)P_e$,
where $\rho_s$ is separable and $P_e$ is purely entangled.
Then since $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}=(1-\lambda)\mathcal{C}_{AB}(P_e)$,
we can see that $\rho_s$ is in the form of $\ket{0}_A\bra{0}\otimes \sigma_B$ or
$(x\ket{0}_A\bra{0}+y\ket{1}_A\bra{1})\otimes \ket{0}_B\bra{0}$
up to local unitary operations.
We now let
\begin{equation}
\begin{pmatrix}
a & b \\
b^{\ast} & c \\
\end{pmatrix}
\equiv (1-\lambda)\mbox{$\mathrm{tr}$}_B(P_e).
\label{eq:matrix01}
\end{equation}
Then $(1-\lambda)\mathcal{C}_{AB}(P_e)=2\sqrt{ac-|b|^2}$,
and
\begin{equation}
\rho_A=
\begin{pmatrix}
\lambda+a& b\\
b^*& c
\end{pmatrix}
\hbox{ or }
\begin{pmatrix}
\lambda{x}+a& b\\
b^*& \lambda{y}+c
\end{pmatrix}.
\label{eq:matrix02}
\end{equation}
Thus, since $\mathcal{C}_{A(BC)}=2\sqrt{(\lambda+a)c-|b|^2}$
or $2\sqrt{(\lambda{x}+a)(\lambda{y}+c)-|b|^2}$,
it is obtained that $\lambda=0$, that is, $\rho_{AB}=P_e$.
Therefore, we can conclude that $\ket{\Psi}_{ABC}$ is of the form
$\ket{\phi}_C \otimes \ket{\psi}_{AB}$.
\end{proof}
We now present the second main theorem.
\begin{Thm}\label{Thm:2}
If $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^{a}$ then $\mathcal{C}_{AB}=0$.
\end{Thm}
For the proof of Theorem~\ref{Thm:2},
we introduce the two following lemmas.
One is as follows.
\begin{Lem}\label{Lem:rank}
If $\rho$ and $\sigma$ are $2\times 2$ positive matrices with
$\mathrm{rank}(\rho) =1$ and $\mathrm{rank}(\sigma)=2$, respectively
then
for any $\lambda_j\ge 0$,
$\sqrt{\det(\lambda_0 \rho + \lambda_1 \sigma)}\ge \lambda_1 \sqrt{\det\sigma}$,
where the equality holds if and only if $\lambda_{0}=0$ or $\lambda_{1}=0$.
If $\rho$ and $\sigma_j$ are $2\times 2$ positive matrices with
$\mathrm{rank}(\rho) =1$ and $\mathrm{rank}(\sigma_j)=2$, respectively
then
for any $\alpha, \beta_j \ge 0$,
\begin{equation}
\sqrt{\det(\alpha \rho + \beta_0 \sigma_0 + \beta_1 \sigma_1)} \geq
\sqrt{\det(\beta_0\sigma_0+\beta_1\sigma_1)},
\label{eq:Lem2}
\end{equation}
where the equality holds if and only if $\alpha=0$ or $\beta_j=0$.
\end{Lem}
\begin{proof}
To begin with, we show the first statement.
Without loss of generality, we may assume that
$\rho=\ket{\psi}\bra{\psi}$ and $\sigma=a \ket{0}\bra{0}+ b \ket{1}\bra{1}$,
where $\ket{\psi}=x\ket{0}+y\ket{1}$ and $a, b >0$.
Then
\begin{eqnarray}
\sqrt{\det(\lambda_0 \rho + \lambda_1 \sigma)}
&=& \sqrt{\lambda_{0}\lambda_{1}(|x|^{2}b+|y|^{2}a)+\lambda_{1}^{2}ab}
\nonumber\\
&\ge& \sqrt{\lambda_{1}^{2}ab} \nonumber \\
&=&\lambda_1 \sqrt{\det\sigma}.
\label{eq:rank12}
\end{eqnarray}
It is clear that
the equality in (\ref{eq:rank12}) holds if and only if $\lambda_{0}=0$ or $\lambda_{1}=0$.
Similarly, we can show the second statement.
\end{proof}
The other lemma is called the {\em Minkowski determinant inequality theorem}~\cite{HJ}.
\begin{Lem}\label{Lem:MDI}
If $n\times n$ matrices $A$, $B$ are positive definite, then
\begin{equation}
\left[\det(A+B)\right]^{1/n}\ge\left(\det A\right)^{1/n}+\left(\det B\right)^{1/n}.
\label{eq:MDI}
\end{equation}
The equality in (\ref{eq:MDI}) holds if and only if $B=cA$ for some $c\ge 0$.
\end{Lem}
In the proof of the second main theorem,
we will use Lemma~\ref{Lem:MDI}
just in the case of $n=2$.
\begin{proof}[Proof of Theorem~\ref{Thm:2}]
We first let
\begin{equation}
\rho_{AC}=\sum_{i\in I} \lambda_i \ket{\psi_i}_{AC}\bra{\psi_i}
\label{eq:optimal}
\end{equation}
be an optimal decomposition of $\rho_{AC}$ for the CoA, $\mathcal{C}_{AC}^a$.
Then we can consider the three cases
according to the rank of $\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})$;
(i)~$\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})\right]=1$ for all $i\in I$,
(ii)~there exist two nonempty subsets $I_1$ and $I_2=I-I_1$ of $I$ such that
$\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})\right]=1$ for all $i\in I_1$
and $\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_j}_{AC}\bra{\psi_j})\right]=2$ for all $j\in I_2$,
(iii)~$\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})\right]=2$ for all $i\in I$.
We now prove this theorem case by case.
(Case i) Assume that $\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})\right]=1$
for all $i\in I$.
Then since $\ket{\psi_i}_{AC}\bra{\psi_i}$'s are all pure and separable states,
$\mathcal{C}_{AC}^{a}=0$ and so $\mathcal{C}_{A(BC)}=0$.
Thus, we have $\ket{\Psi}_{ABC}$ is of the form $\ket{\phi}_A \otimes \ket{\psi}_{BC}$,
and it immediately follows that $\mathcal{C}_{AB}=0$.
(Case ii) Assume that
there exist two nonempty subsets $I_1$ and $I_2=I-I_1$ of $I$ such that
$\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})\right]=1$ for all $i\in I_1$
and $\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_j}_{AC}\bra{\psi_j})\right]=2$ for all $j\in I_2$.
The by Lemma~\ref{Lem:rank} and Lemma~\ref{Lem:MDI},
we can obtain the following inequality.
\begin{widetext}
\begin{eqnarray}
\mathcal{C}_{A(BC)}
&=&
2\sqrt{\det\left[\sum_{i\in I_1}\lambda_i \mbox{$\mathrm{tr}$}_C(\ket{\psi_i}_{AC}\bra{\psi_i})
+\sum_{j\in I_2}\lambda_j \mbox{$\mathrm{tr}$}_C(\ket{\psi_j}_{AC}\bra{\psi_j})\right]}
\ge
2\sqrt{\det\left[\sum_{j\in I_2}\lambda_j \mbox{$\mathrm{tr}$}_C(\ket{\psi_j}_{AC}\bra{\psi_j})\right]}
\nonumber \\
&\ge& 2\sum_{j\in I_2}\lambda_j\sqrt{\det(\mbox{$\mathrm{tr}$}_C(\ket{\psi_j}_{AC}\bra{\psi_j}))}
= \mathcal{C}_{AC}^{a}.
\label{eq:rank1122}
\end{eqnarray}
\end{widetext}
Since $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^{a}$,
the equality in the first inequality should hold,
and hence $\lambda_i=0$ for all $i\in I_1$
or $\lambda_j=0$ for all $j\in I_2$ by Lemma~\ref{Lem:rank}.
This means that it is sufficient to consider the cases (i) and (iii).
(Case iii) We assume that
$\mathrm{rank} \left[\mbox{$\mathrm{tr}$}_{C}(\ket{\psi_i}_{AC}\bra{\psi_i})\right]=2$ for all $i\in I$.
Then by Lemma~\ref{Lem:MDI},
\begin{eqnarray}
\mathcal{C}_{A(BC)}
&=& 2\sqrt{\det\left[\mbox{$\mathrm{tr}$}_{C}\sum_{i\in I}\lambda_{i}\ket{\psi_i}_{AC}\bra{\psi_i}\right]}
\nonumber \\
&\ge& 2\sum_{i\in I}\lambda_{i}\sqrt{\det\left(\mbox{$\mathrm{tr}$}_{C}\ket{\psi_i}_{AC}\bra{\psi_i}\right)}
\nonumber \\
&=& \mathcal{C}_{AC}^{a},
\label{eq:ineq2}
\end{eqnarray}
and the equality in the inequality (\ref{eq:ineq2}) holds if and only if
$\mbox{$\mathrm{tr}$}_{C}\ket{\psi_i}_{AC}\bra{\psi_i}=\rho_{A}$
for all $i\in I$.
Let $\rho_A=\mu_{0}\ket{0}_A\bra{0}+\mu_{1}\ket{1}_A\bra{1}$
be its spectral decomposition.
By the Gisin-Hughston-Jozsa-Wootters theorem~\cite{HJW,Gisin}, for $0, 1\in I$,
there is a unitary operator $U$ such that
\begin{eqnarray}
\ket{\psi_{0}}_{AC}&=&\sqrt{\mu_{0}}\ket{0}_{A}\ket{0}_{C}+\sqrt{\mu_{1}}\ket{1}_{A}\ket{1}_{C},
\nonumber \\
\ket{\psi_{1}}_{AC}&=&\sqrt{\mu_{0}}\ket{0}_{A}U\ket{0}_{C}+\sqrt{\mu_{1}}\ket{1}_{A}U\ket{1}_{C}.
\label{eq:Unitary}
\end{eqnarray}
Let
$\rho_{AC}=\nu_0\ket{\phi_0}_{AC}\bra{\phi_0}+\nu_1\ket{\phi_1}_{AC}\bra{\phi_1}$
be the spectral decomposition of $\rho_{AC}$.
Then since $\mathrm{rank} (\rho_{AC})=2$,
the eigenvectors $\ket{\tilde{\phi_{0}}}=\sqrt{\nu_0}\ket{\phi_0}$
and $\ket{\tilde{\phi_{1}}}=\sqrt{\nu_1}\ket{\phi_1}$
should be linear combinations of $\ket{\psi_{0}}$ and $\ket{\psi_{1}}$.
It follows that
\begin{equation}
\ket{\Psi}_{ABC}=\ket{\tilde{\phi_{0}}}_{AC}\ket{0}_B+\ket{\tilde{\phi_{1}}}_{AC}\ket{1}_B,
\label{eq:psi}
\end{equation}
where $\ket{\tilde{\phi_{0}}}=x_0\ket{\psi_{0}}+x_1\ket{\psi_{1}}$
and $\ket{\tilde{\phi_{1}}}=y_0\ket{\psi_{0}}+y_1\ket{\psi_{1}}$.
Let
\begin{equation}
\ket{\Psi'}_{ABC}=\ket{\tilde{\phi_{0}}'}_{AC}\ket{0}_B+\ket{\tilde{\phi_{1}}'}_{AC}\ket{1}_B,
\label{eq:psi1}
\end{equation}
where $\ket{\tilde{\phi_{0}}'}=x_1^*\ket{\psi_{0}}+x_0^*\ket{\psi_{1}}$ and
$\ket{\tilde{\phi_{1}}'}=y_1^*\ket{\psi_{0}}+y_0^*\ket{\psi_{1}}$.
Then, by tedious but straightforward calculations,
we can check that
the partial transpose $\rho_{AB}^{T_B}$ of $\rho_{AB}$ is equal to
$\rho'_{AB}=\mbox{$\mathrm{tr}$}_C(\ket{\Psi'}_{ABC}\bra{\Psi'})$,
and thus $\rho_{AB}$ has positive partial transposition (PPT).
Therefore, $\mathcal{C}_{AB}=0$.
\end{proof}
So far, we have seen the case that the monogamy equality holds in $2\otimes 2\otimes d$ systems.
We now exhibit a counterexample that the monogamy equality does not hold,
in particular, $\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)} > \mathcal{C}_{AC}^{a}$.
\begin{Exam}
Consider two orthogonal states in the $2\otimes 3$ quantum system,
$\ket{x}=(\ket{02}+\sqrt{2}\ket{10})/{\sqrt{3}}$,
$\ket{y}=(\ket{12}+\sqrt{2}\ket{01})/{\sqrt{3}}$.
We now take into account the following state in the $2\otimes 2\otimes 3$ quantum system,
\begin{eqnarray}
\ket{\Psi}_{ABC}&=& \frac{1}{\sqrt{2}}\ket{x}_{AC}\ket{0}_B
+\frac{1}{\sqrt{2}}\ket{y}_{AC}\ket{1}_B
\nonumber \\
&=&\frac{1}{\sqrt{6}}\ket{002}_{ABC}+\frac{1}{\sqrt{3}}\ket{100}_{ABC}
\nonumber \\
&&+\frac{1}{\sqrt{6}}\ket{112}_{ABC}+\frac{1}{\sqrt{3}}\ket{011}_{ABC}.
\label{eq:ex_Psi}
\end{eqnarray}
Then since $\rho_{A}=(\ket{0}_{A}\bra{0}+\ket{1}_{A}\bra{1})/2$,
it is clear that $\mathcal{C}_{A(BC)}=1$,
and since $\rho_{AC}=(\ket{x}_{AC}\bra{x}+\ket{y}_{AC}\bra{y})/2$,
by the HJW theorem, for any decompositions $\rho_{AC}=\sum_{i}p_{i}\ket{\phi_{i}}_{AC}\bra{\phi_{i}}$,
$\sqrt{p_{i}}\ket{\phi_{i}}_{AC}=(c_{i1}\ket{x}_{AC}+c_{i2}\ket{y}_{AC})/\sqrt{2}$
for some unitary operator $(c_{ij})$ with $2p_{i}=|c_{i1}|^{2}+|c_{i2}|^{2}$.
Then
\begin{equation}
2p_{i}\mbox{$\mathrm{tr}$}_{C}(\ket{\phi_{i}}_{AC}\bra{\phi_{i}})
=\frac{1}{3}
\begin{pmatrix}
|c_{i1}|^{2}+2|c_{i2}|^{2} & c_{i1}c_{i2}^{*} \\
c_{i2}c_{i1}^{*} & |c_{i2}|^{2}+2|c_{i1}|^{2}
\end{pmatrix},
\label{eq:decomp_matrix}
\end{equation}
and hence
\begin{equation}
\mbox{$\mathrm{tr}$}_{C}(\ket{\phi_{i}}_{AC}\bra{\phi_{i}})
=\frac{1}{3}I_{A}+\frac{1}{3}\ket{\psi_{i}}_{A}\bra{\psi_{i}}
\label{eq:ex_trC}
\end{equation}
with $\ket{\psi_{i}}=(c_{i2}^*\ket{0}+c_{i1}^*\ket{1})/{\sqrt{2p_{i}}}$.
Thus we obtain that $\mathcal{C}_{AC}=\frac{2\sqrt{2}}{3}=\mathcal{C}_{AC}^{a}$.
Since
\begin{equation}
\rho_{AB}=\frac{1}{6}
\begin{pmatrix}
1 & 0 & 0 & 1 \\
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
1 & 0 & 0 & 1
\end{pmatrix}
\label{eq:ex_rhoAB}
\end{equation}
clearly has PPT,
$\mathcal{C}_{AB}=0$.
Therefore,
there exists a quantum state in the $2\otimes 2 \otimes 3$ system
such that $\mathcal{C}_{AB}=0$,
but $\mathcal{C}_{A(BC)}=1 > \frac{2\sqrt{2}}{3}=\mathcal{C}_{AC}^{a}$.
\end{Exam}
In conclusion,
we have considered the monogamy equality
in $2\otimes 2 \otimes d$ quantum systems.
We have shown that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$ if and only if $\mathcal{C}_{AC}^a=0$,
and have also shown that
if $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^a$ then $\mathcal{C}_{AB}=0$,
while there exists a state in a $2\otimes 2 \otimes d$ system such that
$\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)}>\mathcal{C}_{AC}^a$.
However, in $2\otimes 2 \otimes d$ quantum systems,
the monogamy inequality in terms of the concurrence and the CoA,
$\mathcal{C}_{A(BC)}^2\ge\mathcal{C}_{AB}^2+\left(\mathcal{C}_{AC}^a\right)^2$,
has been still unknown.
D.P.C. was supported by the Korea Science and Engineering Foundation
(KOSEF) grant funded by the Korea government (MOST) (No.~R01-2006-000-10698-0),
J.S.K was supported by Alberta's informatics Circle of Research Excellence (iCORE), and
S.L. was supported by the Korea Research Foundation Grant funded by the Korean Government
(MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00049).
\end{document} | math | 19,790 |
\begin{document}
\title[Automorphisms of nonsplit coverings]
{Automorphisms of nonsplit coverings of $PSL_2(q)$ \\ in odd characteristic dividing $q-1$}
\author{Andrei V. Zavarnitsine}
\address{Andrei V. Zavarnitsine
\newline\indent Sobolev Institute of Mathematics,
\newline\indent 4, Koptyug av.
\newline\indent 630090, Novosibirsk, Russia
} \email{[email protected]}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } We classify the nonsplit extensions of elementary abe\-lian $p$-groups
by $\operatorname{PSL}_2(q)$, with odd $p$ dividing $q-1$, for an irreducible induced action,
calculate the relevant low-dimensional cohomology groups, and describe the automorphism groups of such extensions.
\noindent{\sc Keywords:} Automorphism group, nonsplit extension, cohomology.
\end{quote}
}
\section{Introduction}
Given a short exact sequence of groups
\begin{equation}\label{g}
0\to V \to G \to L\to 1,
\end{equation}
where $V$ is abelian (written additively), we say that $G$ is {\em an extension} of $V$ by~$L$, or
{\em a covering} of $L$ with kernel $V$.
Such extensions arise naturally in inductive arguments
or when constructing minimal examples and counterexamples. We will be interested in the case
where $G$ is finite and nonsplit and $V$ acquires the structure of an irreducible $FL$-module (for a
suitable finite field $F$ of characteristic $p$) from the conjugation in $G$.
Such extensions can only exist if $p$ divides $|L|$.
We also restrict ourselves to the case $L\cong\operatorname{PSL}_2(q)$.
Extensions of this form for $p=2$ and $q$ odd were explicitly constructed in \cite{00Buri}, and their automorphism groups were described \cite{21RevZav}. Some results in the case of $q$ being a power of $p$ were obtained in~\cite{15Bur}.
The aim of this paper is to classify such extensions in the case $2\ne p\mid (q-1)$ and describe their automorphism groups.
In this case, we can use the fact that the natural permutation $FL$-module arising from the action of $L$ on the projective
line over $\mathbb{F}_q$ is completely reducible. This is not so if $2\ne p\mid (q+1)$
which case will be a subject of future research. We now state the main results.
\begin{thm} \label{nonsplit} Up to isomorphism there is a unique nonsplit extension
of an ele\-men\-ta\-ry abelian $p$-group~$V$ by
$L=\operatorname{PSL}_2(q)$ with irreducible induced action of $L$ on~$V$, where $2\ne p\mid (q-1)$.
In this extension, $|V|=p^q$.
\end{thm}
The group $V$ from Theorem \ref{nonsplit} as an $\mathbb{F}_pL$-module can be identified with the unique
nonprincipal irreducible module in the principal $p$-block of $L$.
The low-dimensional co\-ho\-mo\-lo\-gy of $V$ is as follows.
\begin{thm}\label{2coh}
In the above notation, we have
$H^1(L,V)\cong H^2(L,V)\cong \mathbb{F}_p$.
\end{thm}
Recall that $P\Gamma L_2(q)$ denotes the extension of $PGL_2(q)$ by its field auto\-mor\-phisms.
The automorphism group of the nonsplit extension from Theorem \ref{nonsplit} is described~by
\begin{thm} \label{aut}
Let $G$ fit in the nonsplit exact sequence $(\ref{g})$, where $V$ is an irreducible $\mathbb{F}_pL$-module
for $L=\operatorname{PSL}_2(q)$ and $2\ne p\mid (q-1)$. Then there is a short exact sequence
\begin{equation}\label{autg}
\qquad 0\to W \to \operatorname{Aut}(G) \to P\Gamma L_2(q)\to 1,
\end{equation}
where $W$ is elementary abelian of order $p^{q+1}$.
\end{thm}
\section{Auxiliary facts}
Basic notation and facts of homological algebra can be found in \cite{70Gru,94Wei}. For abelian groups $A$ and $B$, we denote $\operatorname{Hom}(A,B) = \operatorname{Hom}_\mathbb{Z}(A,B)$ and $\operatorname{Ext}(A,B) = \operatorname{Ext}_\mathbb{Z}^1(A,B)$.
\begin{lem}[The Universal Coefficient Theorem for Cohomology]\cite[Ch.\,3, The\-o\-rem 3]{70Gru} \label{ucoe} For all $i\geqslant 1$, every group $G$, and every trivial $G$-module $A$,
$$
H^i(G,A)\cong\operatorname{Hom}(H_i(G,\mathbb{Z}),A)\oplus\operatorname{Ext}(H_{i-1}(G,\mathbb{Z}),A).
$$
\end{lem}
\begin{lem}\cite[\S 3.5]{70Gru}\label{coh1} For a trivial $G$-module $A$, we have
\begin{enumerate}
\item[$(i)$] $H^1(G,A)\cong \operatorname{Hom}(G/G',A)$.
\item[$(ii)$] $H_1(G,A)\cong G/G'\otimes_\mathbb{Z} A$.
\end{enumerate}
\end{lem}
\begin{lem}[Shapiro's lemma]\cite[\S 6.3]{94Wei} \label{shap} Let $H\leqslant G$ with $|G:H|$ finite. If $V$ is an $H$-module and $i\geqslant 0$ then $H^i(G,V^G)\cong H^i(H,V)$, where $V^G$ is the induced $G$-module.
\end{lem}
\begin{lem}\cite[p. 322]{18Deo}\label{ezmn} $\operatorname{Ext}(\mathbb{Z}_m,\mathbb{Z}_n)\cong \mathbb{Z}_d$, where $d=(m,n)$.
\end{lem}
\begin{lem}\cite[Proposition 3.3.4]{94Wei}\label{esum}.
$\operatorname{Ext}_R^i(A, B_1\oplus B_2)\cong \operatorname{Ext}_R^i(A,B_1)\oplus \operatorname{Ext}_R^i(A,B_2)$ for all rings $R$, $R$-modules $A,B_1,B_2$, and all $i\geqslant 0$.
\end{lem}
The Schur multiplier of a group $G$ is denoted by $\operatorname{Sch}(G)$.
If $A$ is a finite abelian group and $p$ a prime then $A_{(p)}$ denotes the $p$-primary component of $A$.
Henceforth, we assume that $G$ is finite.
\begin{lem}\cite[Theorem 25.1]{67Hup} \label{tim} Let $p$ be a prime and let $S\in\operatorname{Syl}_p(G)$.
Then $\operatorname{Sch}(G)_{(p)}$ is isomorphic to a subgroup of $\operatorname{Sch}(S)$.
\end{lem}
\begin{lem}\cite{76Gag}\label{lgag}
Let $F$ be a field of characteristic $p>0$ and let $V$ be an irreducible
$FG$-module that does not belong to the principal $p$-block of $G$. Then $H^n(G,V) = 0$ for all $n\geqslant 0$.
\end{lem}
Let $\theta$ be an irreducible character of $G$.
If $\operatorname{Z}(G)=1$ then $G\trianglelefteqslant \operatorname{Aut}(G)$
and we may speak of the inertia group
$I_{\operatorname{Aut}(G)}(\theta)=\{g\in \operatorname{Aut}(G)\mid \theta^g=\theta\}$.
\begin{prop}\cite[Proposition 4]{21RevZav}\label{normim}
Let $F$ be a field and $\mathcal{X}$ a faithful irreducible $F$-representation of a group $G$
with Brauer character $\theta\in\operatorname{iBr}_F(G)$ of degree $n$. Suppose that
$\operatorname{Z}(G)=1$ and denote
$$N=N_{\operatorname{GL}_n(F)}(\mathcal{X}(G))\quad \text{and} \quad Z=C_{\operatorname{GL}_n(F)}(\mathcal{X}(G)).$$ Then
$N/Z\cong I_{\operatorname{Aut}(G)}(\theta).$
\end{prop}
\section{Isomorphic extensions}
Let $Q$ be a group, $K$ a commutative ring with $1$, and $M$ a right $KQ$-module. The pair
$(\nu,\mu)\in \operatorname{Aut}(Q)\times \operatorname{Aut}_K(M)$ is {\it compatible} if
$$
(mg)\mu = (m\mu)(g\nu)
$$
for all $m\in M$, $g\in Q$. The set of all compatible pairs forms a group $\operatorname{Comp}(Q,M)$ under composition. Given $\tau\in Z^2(Q,M)$, one can define
\begin{equation}\label{cact}
\tau^{(\nu,\mu)}(g,h)=\tau(g\nu^{-1},h\nu^{-1})\mu
\end{equation}
for all $g,h\in Q$. Then the map $\tau\mapsto \tau^{(\nu,\mu)}$ is an action of $\operatorname{Comp}(Q,M)$ on $Z^2(Q,M)$
which preserves $B^2(Q,M)$ and so yields an action on $H^2(Q,M)$.
A {\em $KQ$-module extension} on $M$ by $Q$ is a group $E$ that fits in the short exact sequence
\begin{equation}\label{kge}
0\to M \stackrel{\iota}{\to} E \stackrel{\pi}{\to} Q \to 1
\end{equation}
so that the conjugation of $M$ (identified with $M\iota$) by elements of $E$ agrees with the $KQ$-module structure of $M$,
i.\,e. $m^e=m(e\pi)$ for all $m\in M$, $e\in E$.
\begin{prop}\cite[\S 2.7.4]{05HolEicObr} \label{comp} The classes of those isomorphisms of $KQ$-module extensions of $M$ by $Q$
that leave $M$ invariant as a $K$-module are in
a one-to-one correspondence with the orbits of $\operatorname{Comp}(Q,M)$ on $H^2(Q,M)$.
\end{prop}
In Proposition \ref{comp}, an isomorphism leaving $M$ invariant as a $K$-module means one that induces on $M$ an element of $\operatorname{Aut}_K(M)$.
The $KQ$-module structure on $M$ gives rise to the representation homomorphism $\mathcal{C}:Q\to \operatorname{Aut}_K(M)$ by the rule $\mathcal{C}(g):m\mapsto mg$ for all $m\in M$, $g\in Q$. Let $C$ be the centraliser of $\mathcal{C}(Q)$ in $\operatorname{Aut}_K(M)$. Then $(1,\gamma)\in \operatorname{Comp}(Q,M)$ for every $\gamma \in C$, because
$$(mg)\gamma=m\mathcal{C}(g)\gamma = m\gamma\mathcal{C}(g)=(m\gamma)g$$
for all $m\in M$, $g\in Q$. Hence, we also have an action of $C$ on both $Z^2(Q,M)$ and $H^2(Q,M)$ by setting
$\tau^\gamma=\tau^{(1,\gamma)}$ for $\tau \in Z^2(Q,M)$, $\gamma \in C$, i.\,e. $\tau^\gamma(g,h)=\tau(g,h)\gamma$. By Proposition \ref{comp}, this yields the following:
\begin{lem} \label{h2gscal} The elements of $H^2(Q,M)$ that are in the same $C$-orbit correspond to isomorphic
$KQ$-module extensions.
\end{lem}
In particular, we have the following fact, where two elements of $H^2(Q,M)$ are called scalar multiples if they differ by a factor in~$K^\times$.
\begin{cor} \label{h2scal} $KQ$-module extensions of $M$ by $Q$ corresponding to scalar multiples in $H^2(Q,M)$
are isomorphic.
\end{cor}
\section{Automorphisms of extensions}\label{sec:ae}
Fix an extension
\begin{equation}\label{ext}
\bm{e}: \qquad 0\to M \stackrel{\iota}{\longrightarrow}E \to Q \to 1
\end{equation}
with abelian kernel $M$. Let $\mathcal{C}:Q\to \operatorname{Aut}(M)$ be the induced representation and
let $\overline{\varphi}\in H^2(Q,M)$ be the element that corresponds to $\bm{e}$.
We assume that $\mathcal{C}$ is faithful.
In particular, $Q\cong \mathcal{C}(Q)$ and the conjugation of $\mathcal{C}(Q)$ by any $\mu\in N_{\operatorname{Aut}(M)}(\mathcal{C}(Q))$
induces an element $\mu'\in \operatorname{Aut}(Q)$, i.\,e. $\mathcal{C}(g)^\mu=\mathcal{C}(g\mu')$ for
all $g\in Q$.
One defines an action of $N_{\operatorname{Aut}(M)}(\mathcal{C}(Q))$ on $H^2(Q,M)$ given by
\begin{equation}\label{act}
\overline{\psi}\mapsto(\mu')^{-1}\overline{\psi}\mu
\end{equation}
for every $\mu \in N_{\operatorname{Aut}(M)}(\mathcal{C}(Q))$ and $\overline{\psi}\in H^2(Q,M)$,
which should be understood modulo $B^2(Q,M)$ for representative cocycles, see \cite{82Rob} for details.
We denote by
$N_{\operatorname{Aut}(M)}^{\,\overline{\varphi}}(\mathcal{C}(Q))$ the stabiliser of
$\overline{\varphi}$ with respect to this action.
Let $\operatorname{Aut}(\bm{e})$ denote
the group of those automorphisms of $E$ that leave $M\iota$ invariant as a set.
\begin{prop}\cite[Statements (4.4),(4.5)]{82Rob}\label{Rob}
Let the extension $(\ref{ext})$ have an abelian kernel $M$ and let it
determine an element $\overline{\varphi}\in H^2(Q,M)$
and an injective induced representation $\mathcal{C}:Q\to \operatorname{Aut}(M)$.
Then there exists a short exact sequence of groups
\begin{equation}\label{ZAN}
0\to Z^1(Q,M)\to \operatorname{Aut}(\bm{e})\to N_{\operatorname{Aut}(M)}^{\,\overline{\varphi}}(\mathcal{C}(Q))\to 1.
\end{equation}
\end{prop}
\noindent
{\em Remark.} It is easy to see that, in the notation above, there is an embedding $N_{\operatorname{Aut}(M)}(\mathcal{C}(Q))\to \operatorname{Comp}(Q,M)$, $\mu\mapsto (\mu',\mu)$, where
we view $M$ as a $\mathbb{Z}Q$-module, under which action (\ref{act}) becomes a particular case of (\ref{cact}), and that this embedding is in fact an isomorphism in case $\mathcal{C}$ is faithful (which we assume).
\section{Cohomology of $PSL_2(q)$ in characteristic dividing~$q-1$}
The aim of this section is to classify up to group isomorphism nonsplit ex\-ten\-sions~(\ref{g}),
where $L=\operatorname{PSL}_2(q)$, $V$ is an elementary abelian $p$-group with irreducible induced action of $L$,
and $p\ne 2$ is a divisor of $q-1$.
By Lemma \ref{lgag}, $V$ must belong to the the principal $p$-block of $L$. This block
contains only one nonprincipal module with Brauer character $\chi$, see \cite{76Bur}.
The values of characters in the principal block are shown in Table \ref{bt}.
\begin{table}[htb]
\centering
\caption{Brauer $p$-modular characters of $L=PSL_2(q)$ in the principal block, where $2\ne p\mid(q-1)$.
Notation: $q=l^m$, $l$ prime, $d=(2,q-1)$,
$x,y\in L$, $|x|=\frac{1}{d}(q-1)_{p'}$, $|y|=\frac{1}{d}(q+1)$.\label{bt}}
\begin{tabular}{c|rrrrrr}
$q$ odd & $1a$ & $2a$ & $la$ & $lb$ & $(x^r)^L$ & $(y^t)^L$ \\
\hline
$1_L^{\vphantom{A^A}}$ & $1$ & $ 1$ & $1$ & $1$ & $1$ & $1$ \\
$\chi$ & $q $ & $-1$ & $0$ & $0$ & $1$ & $-1$
\end{tabular}
\qquad
\begin{tabular}{c|rrrr}
$q$ even & $1a$ & $2a$ & $(x^r)^L$ & $(y^t)^L$ \\
\hline
$1_L^{\vphantom{A^A}}$ & $1$ & $ 1$ & $1$ & $1$ \\
$\chi$ & $q $ & $0$ & $1$ & $-1$
\end{tabular}
\end{table}
We first note that $V$ is not the principal module. Indeed, extension (\ref{g}) would otherwise be central,
but $\operatorname{Sch}(L)$ has no $p$-torsion, because
\begin{equation}\label{Sch}
\operatorname{Sch}(L)=\left\{
\begin{array}{rl}
\mathbb{Z}_2, & q\ne 9\ \text{odd or}\ q=4;\\
\mathbb{Z}_6, & q=9;\\
1, & q\ne 4 \ \text{even}
\end{array}
\right.
\end{equation}
as follows from \cite{85Atlas}. Therefore, $V$ must be the
$\mathbb{F}_pL$-module with character $\chi$.
We can now prove Theorem \ref{2coh} stated in the introduction.
\begin{proof} Let $P$ be the permutation $\mathbb{F}_pL$-module of dimension $q+1$ that
corresponds to the natural permutation action of $L$ on the projective line over $\mathbb{F}_q$.
We have $P=I_L\oplus V$, where $I_L$ is
the principal $\mathbb{F}_pL$-module. This can be deduced either by
considering the Brauer character $\chi$ of $V$ or from \cite[Table 1]{78Mor}. In particular,
by Lemma \ref{esum}, we have
\begin{equation}\label{hadd}
H^i(L,P)\cong H^i(L,I_L) \oplus H^i(L,V)
\end{equation}
for $i=1,2$, since $H^i(L,B)\cong \operatorname{Ext}^i_{\mathbb{F}_pL}(\mathbb{F}_p,B)$
for every $\mathbb{F}_pL$-module $B$, see \cite[Exercise 6.1.2]{94Wei}.
Since $P$ is a permutation module, we have $P\cong (I_H)^L$, where $I_H$ is the principal
$\mathbb{F}_pH$-module for a point stabiliser $H\leqslant L$. Hence, Lemma \ref{shap} implies
\begin{equation}\label{shs}
H^i(L,P)\cong H^i(H,I_H)
\end{equation}
for $i=1,2$. By Lemma \ref{coh1}$(i)$, we have
\begin{equation}\label{h1l}
H^1(L,I_L)\cong \operatorname{Hom}(L/L',I_L)=0,
\end{equation}
since $L=L'$. Also,
\begin{equation}\label{h1h}
H^1(H,I_H)\cong \operatorname{Hom}(H/H',I_H)\cong \mathbb{F}_p,
\end{equation}
since $I_H\cong \mathbb{F}_p$, $H\cong \mathbb{F}_q \leftthreetimes \mathbb{Z}_{(q-1)/(2,q-1)}$, and $p\mid (q-1)$. By Lemma \ref{ucoe}, we have
$$
H^2(L,I_L)\cong \operatorname{Hom}(H_2(L,\mathbb{Z}),I_L)\oplus \operatorname{Ext}(H_1(L,\mathbb{Z}),I_L),
$$
where the first summand vanishes, since $H_2(L,\mathbb{Z})\cong \operatorname{Sch}(L)$ has no $p$-torsion by~(\ref{Sch}), and the second summand vanishes by Lemma \ref{coh1}$(ii)$, since $L/L'=1$. Thus
\begin{equation}\label{h2lil}
H^2(L,I_L)=0.
\end{equation}
Finally, Lemma \ref{ucoe} also yields
\begin{equation}\label{h2hihe}
H^2(H,I_H)\cong \operatorname{Hom}(H_2(H,\mathbb{Z}),I_H)\oplus \operatorname{Ext}(H_1(H,\mathbb{Z}),I_H).
\end{equation}
By Lemma \ref{tim}, the $p$-part of $H_2(H,\mathbb{Z})\cong \operatorname{Sch}(H)$ is isomorphic to a subgroup of $\operatorname{Sch}(S)$ for a $p$-Sylow subgroup $S$ of $H$. However, $S$ is cyclic and
cyclic groups have trivial Schur multiplier. Thus, the first summand in (\ref{h2hihe}) vanishes, because $I_H\cong \mathbb{F}_p$. Since $H_1(H,\mathbb{Z})\cong H/H'\cong \mathbb{Z}_{(q-1)/d}$ and
$\operatorname{Ext}(\mathbb{Z}_{(q-1)/d},\mathbb{F}_p)\cong \mathbb{F}_p$ by Lemma \ref{ezmn}, we have
\begin{equation}\label{h2hih}
H^2(H,I_H)\cong \mathbb{F}_p.
\end{equation}
The claim follows by combining (\ref{hadd}) through (\ref{h2hih}).
\end{proof}
We can now prove Theorem \ref{nonsplit} stated in the introduction.
\begin{proof} As we explained in the beginning of this section, $V$ viewed as an $\mathbb{F}_pL$-module
must be the unique nonprincipal module in the principal $p$-block of $L$. This module has dimension $q$
and can be written over $\mathbb{F}_p$, since it is a direct summand of a permutation module. Therefore,
$|V|=p^q$. By Theorem \ref{2coh}, we have $H^2(V,L)\cong \mathbb{F}_p$ and so all nonzero elements of
$H^2(V,L)$ are scalar multiples of one another. By Corollary \ref{h2scal}, they correspond to isomorphic nonsplit extensions. The claim follows.
\end{proof}
\section{The automorphism group}
In this section, we prove that the structure of the automorphism group of the unique nonsplit
extension from Theorem \ref{nonsplit} is as stated in Theorem \ref{aut}.
\begin{proof}
Consider the extension $\bm{e}$ given by (\ref{g}). Theorem \ref{nonsplit} implies that
$G$ is unique up to isomorphism and $V$ has order $p^q$. Moreover, viewed as an $\mathbb{F}_p L$-module, $V$ has Brauer character $\chi$ from Table \ref{bt}.
By Proposition \ref{Rob}, we have the short exact sequence
\begin{equation}\label{ZAN1}
0\to Z^1(L,V)\to \operatorname{Aut}(\bm{e})\to N_{\operatorname{Aut}(V)}^{\,\overline{\varphi}}(\mathcal{X}(L))\to 1,
\end{equation}
where the representation $\mathcal{X}:L\to \operatorname{Aut}(V)$
and the element $\overline{\varphi}\in H^2(L,V)$ are determined by
(\ref{g}). First, note that $\operatorname{Aut}(\bm{e})=\operatorname{Aut}(G)$ as $V$ is characteristic in~$G$. Denote $W=Z^1(L,V)$. Since $B^1(L,V)\cong V/C_V(L)$ and $L$ acts on $V$ irreducibly and nontrivially, we have $C_V(L)=0$ and $B^1(L,V)\cong V$. Now, since $H^1(L,V)=Z^1(L,V)/B^1(L,V)$, we have
$|Z^1(L,V)|=p^{q+1}$ in view of Theorem \ref{2coh}.
Denote $N=N_{\operatorname{GL}(V)}(\mathcal{X}(L))$ and $Z=C_{\operatorname{GL}(V)}(\mathcal{X}(L))$.
By Proposition \ref{normim}, we have $N/Z\cong I_{\operatorname{Aut}(L)}(\chi)$.
Since $\chi$ is the only irreducible character of $L$ of dimension~$q$,
it must be invariant under any automorphism; in particular,
$I_{\operatorname{Aut}(L)}(\chi)=\operatorname{Aut}(L)$.
By \cite{85Atlas}, $\operatorname{Aut}(L)\cong P\Gamma L_2(q)$.
Since $V$ is absolutely irreducible as an $\mathbb{F}_pL$-module, by Schur's lemma, we see that
$Z\cong \mathbb{F}_p^\times\cong \mathbb{Z}_{p-1}$ consists of scalars.
In order to determine the structure of the stabiliser $N_0=N_{\operatorname{Aut}(V)}^{\,\overline{\varphi}}(\mathcal{X}(L))$,
we consider the action of $N$ on $H^2(L,V)$ as explained in Section \ref{sec:ae}. Let $H^\times$ denote the set
of $p-1$ nonzero elements of $H^2(L,V)$. The elements of $H^\times$ correspond to nonsplit extensions and so we have an action
homomorphism $\alpha: N\to \operatorname{Sym}(H^\times)$ to the symmetric group on $H^\times$.
Since all nonsplit extension of $V$ by $L$
are isomorphic by Theorem~\ref{nonsplit}, we may assume that $\overline{\varphi}$ is an arbitrary element of $H^\times$.
The subgroup $Z\leqslant N$ acts on $H^\times$ by
scalar multiplication, cf. Corollary \ref{h2scal}, and so the image $\alpha(Z)$ is a cyclic subgroup
of $\operatorname{Sym}(H^\times)$ generated by a full cycle of length $p-1$.
Since $Z$ is central in $N$, $\alpha(N)$ must centralise $\alpha(Z)$. However, a full cyclic subgroup is self-centralising in
$\operatorname{Sym}(H^\times)$ and so $\alpha(Z)$ must be the entire image $\alpha(N)$. Thus, $\operatorname{Ker}(\alpha)$ is a normal
subgroup of $N$ of index $p-1$ which intersects trivially with $Z$ and is thus isomorphic to $N/Z\cong P\Gamma L_2(q)$.
Furthermore, $\operatorname{Ker}(\alpha)$ coincides with the stabiliser of every element of $H^\times$ which yields
$N=N_0\times Z$ and $N_0\cong P\Gamma L_2(q)$ as claimed.
\end{proof}
It also follows from this proof that the representation $\mathcal{X}:L\to \operatorname{Aut}(V)$
with character $\chi$ extends to a representation of $I_{\operatorname{Aut}(L)}(\chi)\cong P\Gamma L_2(q)$. This fact does
not hold in general for a simple group $L$ and its irreducible character $\chi$, see \cite[Example 1]{21RevZav}.
{\em Acknowledgement.\/} The work was supported by the RAS Fundamental Research Program, project FWNF-2022-0002.
\end{document} | math | 19,974 |
\begin{document}
\title[On approximation of solutions of operator-differential
equations $\ldots $]
{On approximation of solutions of operator-differential
equations with their entire solutions of exponential type}
\author{V. M. Gorbachuk}
\address{National Technical University "KPI", 37 Peremogy Prosp., Kyiv, 06256, Ukraine}
\email{[email protected]}
\subjclass[2000]{Primary 34G10}
\date{01/04/2016;\ \ Revised 12/04/2016}
\keywords{Hilbert and Banach spaces, differential-operator equation,
weak solution, $C_{0}$-semigroup of linear operators, entire
vector-valued function, entire vector-valued function of exponential
type, the best approximation, direct and inverse theorems of the
approximation theory.}
\begin{abstract}
We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0,
\infty)$, where $A$ is a nonnegative self-adjoint operator in a
Hilbert space. We give direct and inverse theorems on approximation
of solutions of this equation with its entire solutions of
exponential type. This establishes a one-to-one correspondence
between the order of convergence to $0$ of the best approximation of
a solution and its smoothness degree. The results are illustrated with
an example, where the operator $A$ is generated by a second
order elliptic differential expression in the space $L_{2}(\Omega)$
\ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth
boundary) and a certain boundary condition.
\end{abstract}
\maketitle
\vspace*{-5mm}
{\bf 1.} \ Let $A$ be a nonnegative self-adjoint operator in a Hilbert
space $\mathfrak{H}$ with a scalar product $(\cdot, \cdot)$. Denote by
$C_{\{1\}}(A)$ the set of all its exponential type entire vectors (see
[4]), namely,
$$
\begin{aligned}
C_{\{1\}}(A) = \Big\{f \in C^{\infty}(A) = \bigcap_{n = 1}^{\infty}\mathcal{D}(A^{n})\big|
\exists \alpha > 0, \ \exists c & = c(f) > 0:
\\
\|A^{n}f\| & \leq c\alpha^{n},
\ n \in \mathbb{N}_{0} = \mathbb{N}\cup \{0\}\Big\}
\end{aligned}
$$
(everywhere in the sequel $c$ denotes various numerical constants
corresponding to the situations under consideration, $\mathcal{D}(A)$ is
a domain of $A$ and $\|f\| = \sqrt{(f, f)}$). The number
$$
\sigma (f, A) = \inf \left\{\alpha > 0 \bigl| \exists c > 0, \ \forall n \in \mathbb{N}_{0}: \|A^{n}f\| \leq c\alpha^{n}\right\}
$$
is called the type of the vector $f$ with respect to the operator $A$.
As has been shown in [3] that
$$
C_{\{1\}}(A) = \left\{f \in \mathfrak{H}\bigl| f = E(\lambda)g, \ \forall \lambda > 0, \ \forall g \in \mathfrak{H}\right\},
$$
where $E(\lambda) = E([0, \lambda])$ is the spectral measure of $A$.
Now, consider the equation
$$
y'(t) + Ay(t) = 0, \quad t \in \mathbb{R}_{+} = [0, \infty). \leqno (1)
$$
By a weak solution of this equation we mean a continuous vector-valued
function $y(t):\mathbb{R}_{+} \mapsto \mathfrak{H}$ such that for any
$t \in \mathbb{R}_{+}$,
\vspace*{-1mm}
$$
\int_{0}^{t} y(s)\,ds \in \mathcal{D}(A) \quad \text{and} \quad y(t) = -A \int_{0}^{t} y(s)\,ds + y(0).
$$
\vspace*{-2mm}
Denote by $S$ the set of all weak solutions of (1). As it was established
in [1],
$$
S = \left\{y(t):\mathbb{R}_{+} \mapsto \mathfrak{H}\big| \ y(t) = e^{-At}f, f \in \mathfrak{H} \right\}, \leqno (2)
$$
where
$$
e^{-At}f = \int_{0}^{\infty} e^{-\lambda t}\,dE(\lambda)f.
$$
Note that the set of all strong (or simply) solutions of (1) is given
by formula (2) with $f$ ranging over whole $\mathcal{D}(A)$.
It is not difficult to verify that $S$ is a Hilbert space with the norm
$$
\|y\|_{S} = \sup_{t \in \mathbb{R}_{+}} \left\|e^{-At}f\right\| = \|f\|. \leqno (3)
$$
If the operator $A$ is bounded, then each weak solution $y(t)$ of
equation (1) can be extended to an entire $\mathfrak{H}$-valued vector
function $y(z)$ of exponential type,
$$
\sigma (y) = \inf\left\{\alpha > 0: \|y(z)\| \leq ce^{\alpha |z|}\right\}.
$$
But it is not the case if $A$ is unbounded. The set $S_{0}$ of all
weak solutions of (1) admitting an extension to an entire vector
function of exponential type is described by the following theorem.
\begin{theo} A weak solution $y(t)$ of equation (1) belongs to $S_{0}$
if and only if it can be represented in form (2) with $f \in
C_{\{1\}}(A)$. The set $S_{0}$ is dense in $S$, and $\sigma (y) =
\sigma (f, A)$.
\end{theo}
\begin{proof}
Let $f \in C_{\{1\}}(A)$. Then $f = E(\alpha)f$, where $\alpha =
\sigma (f, A)$. By (2),
$$
y (t) = \int_{0}^{\alpha} e^{-\lambda t}\,dE(\lambda)f.
$$
From this it follows that $y(t)$ can be extended to an entire
vector-valued function $y(z)$ and
$$
\|y(z)\|^{2} = \int_{0}^{\alpha} e^{-2\text{Re}z\lambda}\,d(E(\lambda)f, f) \leq e^{2\sigma(f, A)|z|}\|f\|^{2},
$$
that is, $y(z)$ is an entire $\mathfrak{H}$-valued function of exponential type $\sigma (y) \leq \sigma (f, A)$.
Conversely, if $y(t) = e^{-At}f$ admits an extension to an entire
vector-valued function of exponential type $\sigma (y)$, then, by
virtue of
$$
\|y(-t)\|^{2} = \int_{0}^{\infty} e^{2\lambda t}\,d(E(\lambda)f, f) \leq ce^{2\sigma(y)t}, \quad t \geq 0,
$$
we have
$$
\int_{\sigma (y)}^{\infty} e^{2t(\lambda - \sigma (y))}\,d(E(\lambda)f, f) \leq c.
$$
Passing to the limit under the integral sign as $t \to \infty$, we
conclude, on the basis of the Fatou theorem, that the measure
generated by the monotone function $(E(\lambda)f, f)$ is concentrated
on the interval $[0, \sigma (y)]$. Hence, $\sigma (f, A) \leq \sigma
(y)$.
Density of $S_{0}$ in $S$ follows from the density in $\mathfrak{H}$
of the set $\{E([0, \alpha])f, \forall \alpha >0, \forall f \in
\mathfrak{H}\}$.
\end{proof}
In view of Theorem 1, it is reasonable to ask whether it is possible
to approximate an arbitrary weak solution of equation (1) with its
exponential type entire solutions. An answer to the question is given
below. We prove direct an inverse theorems which ascertain the
relationship between the degree of smoothness of a solution and the
rate of convergence to $0$ of its best approximation. In doing so, the
operator approach developed in [7, 4, 5] plays an important role.
{\bf 2.} \ Recall some definitions and notations of the
approximation theory required to formulate further results.
For $y \in S $ and a number $r > 0$, we put
$$
\mathcal{E}_{r}(y) = \inf_{y_{0} \in S_{0}: \sigma (y_{0}) \leq r} \|y - y_{0}\|_{S}.
$$
Thus, $\mathcal{E}_{r}(y)$ is the best approximation of a weak
solution $y(t)$ of equation (1) with its entire solutions of exponential
type not exceeding $r$. If $y$ is fixed, the function
$\mathcal{E}_{r}(y)$ does not increase and, since $\overline{{S}_{0}}
= S, \ \mathcal{E}_{r}(y) \to 0$ as $r \to \infty$.
For $f \in \mathfrak{H}$, set also
$$
\mathcal{E}_{r}(f, A) = \inf_{f_{0} \in C_{\{1\}}(A): \sigma (f_{0}, A) \leq r} \|f - f_{0}\| = \|(I - E(r))f\|.
$$
If $y(t) = e^{-At}f$, then, by virtue of (3),
$$
\mathcal{E}_{r}(y) = \mathcal{E}_{r}(f, A). \leqno (4)
$$
Besides, for any $k \in \mathbb{N}_{0}$, we introduce the function
$$
\omega_{k}(t, y) = \sup_{|h| \leq t}\sup_{s \in \mathbb{R}_{+}} \Big\| \sum_{j = 0}^{k}
(-1)^{k - j}C_{k}^{j}y(s + jh)\Big\|, \quad k \in \mathbb{N}; \quad \omega_{0}(t, y) \equiv \|y\|_{S}, \quad t > 0.
$$
Taking into account (2) and the equality $e^{-As}y(t) = y (t + s)$, we conclude that
$$
\forall k \in \mathbb{N}_{0}:\omega_{k}(t, y) = \sup_{|h| \leq t}\left\|\left(e^{-Ah} - I\right)^{k}y\right\|_{S}
$$
($I$ is the identity operator).
The following theorem establishes a relation between
$\mathcal{E}_{r}(y)$ and $\omega_{k}(t, y)$, and it is an analog of
the well-known Jackson's theorem on approximation of a continuous
periodic function by trigonometric polynomials.
\begin{theo} Let $y \in S$. Then
$$
\forall k \in \mathbb{N}, \ \exists c_{k} > 0: \mathcal{E}_{r}(y) \leq c_{k}\omega_{k}\left(\frac{1}{r}, y\right), \quad r > 0. \leqno (5)
$$
\end{theo}
\begin{proof}
By (2), $y(t) = e^{-At}f, \ f \in \mathfrak{H}$. From (3), (4), it follows that
$$
\begin{aligned}
\omega_{k}^{2}(t, y) = & \sup_{0 \leq s \leq t} \left\|\left(e^{-As} - I\right)^{k}y\right\|_{S}^{2} \geq \left\|\left(e^{-At} - I\right)^{k}y\right\|_{S}^{2} = \sup_{s \in \mathbb{R}_{+}} \left\|\left(e^{-At} - I\right)^{k}e^{-As}f\right\|^{2}
\\
= & \left\|\left(e^{-At} - I\right)^{k}f\right\|^{2} = \int_{0}^{\infty}\left(e^{-\lambda t} - 1\right)^{2k}\, d(E(\lambda)f, f)
\\
\geq & \int_{\frac{1}{t}}^{\infty}\left(e^{-\lambda t} - 1\right)^{2k}\, d(E(\lambda)f, f)
\geq \left(1 - e^{-1}\right)^{2k}\mathcal{E}_{\frac{1}{t}}(y).
\end{aligned}
$$
So,
$$
\forall t > 0: \mathcal{E}_{\frac{1}{t}}(y) \leq \left(1 - e^{-1}\right)^{k}\omega_{k}(t, y).
$$
Setting $r = \frac{1}{t}$ and $c_{k} = \left(1 - e^{-1}\right)^{k}$, we obtain (5).
\end{proof}
Denote by $C^{n}(\mathbb{R}_{+}, \mathfrak{H})$ the set of all $n$
times continuously differentiable on $\mathbb{R}_{+}$
$\mathfrak{H}$-valued vector-valued functions. Since the operator $A$
is closed, the inclusion $y \in S \cap C^{n}(\mathbb{R}_{+},
\mathfrak{H})$ implies that $y^{(k)} \in S, \ k = 1, 2, \dots, n$.
\begin{theo} Suppose that $y \in C^{n}(\mathbb{R}_{+}, \mathfrak{H}), \ n
\in \mathbb{N}_{0}$. Then
$$
\forall r > 0, \ \forall k \in \mathbb{N}_{0}: \mathcal{E}_{r}(y) \leq \frac{c_{k + n}}{r^{n}}\omega_{k}\left(\frac{1}{r}, y^{(n)}\right),
$$
where the constants $c_{k}$ are the same as in Theorem 2.
\end{theo}
\begin{proof}
Let $y \in C^{n}(\mathbb{R}_{+}, \mathfrak{H}), r > 0$ and $0 \leq t
< \frac{1}{r}$. Using properties of a contraction $C_{0}$-semigroup,
we get
$$
\left\|\left(e^{-At} - I\right)^{k + n}y(s)\right\| = \left\|\left(e^{-At} - I\right)^{n}\left(e^{-At} - I\right)^{k}y(s)\right\|
$$
\vspace*{-2mm}
$$
\leq \int_{0}^{t} \dots \int_{0}^{t} \left\|e^{-A(s_{1} + \dots s_{n})}\right\|\left\|\left(e^{-At} - I\right)^{k}A^{n}y(s)\right\|\,ds_{1} \dots ds_{n}
$$
\vspace*{-2mm}
$$
\leq t^{n}\left\|\left(e^{-At} - I\right)^{k}y^{(n)}(s)\right\|,
$$
whence
$$
\omega_{k + n}\left(\frac{1}{r}, y\right) \leq \frac{1}{r^{n}} \omega_{k}\left(\frac{1}{r}, y^{(n)}\right)
$$
and, because of (5),
$$
\mathcal{E}_{r}(y) \leq c_{k + n}\omega_{k + n}\left(\frac{1}{r}, y\right) \leq \frac{c_{k + n}}{r^{n}}\omega_{k}\left(\frac{1}{r}, y^{(n)}\right),
$$
which is what had to be proved.
\end{proof}
Setting, in Theorem 3, $k = 0$ and taking into account that
$\omega_{0}(t, y^{(n)}) = \|y^{(n)}\|_{S}$, we arrive at the following
assertion.
\begin{cor} Let $y \in C^{n}(\mathbb{R}_{+}, \mathfrak{H}), \ n \in
\mathbb{N}$. Then
$$
\forall r > 0: \mathcal{E}_{r}(y) \leq \frac{c_{n}}{r^{n}}\|y^{(n)}\|_{S}.
$$
\end{cor}
For numbers $h > 0$ and $k \in \mathbb{N}_{0}$, we put
$$
\Delta_{h}^{k} = \left(e^{-Ah} - I\right)^{k} = \sum_{j = 0}^{k}(-1)^{k - j}C_{k}^{j}e^{-Ajh}.
$$
\begin{lm} If $y \in S_{0}$ and $\sigma (y) = \alpha$, then
$$
\forall h > 0, \ \forall k, n \in \mathbb{N}_{0}: \left\|\Delta_{h}^{k}y^{(n)}\right\|_{S} \leq (\alpha h)^{k}\alpha^{n}\|y\|_{S}. \leqno (6)
$$
\end{lm}
\noindent{\it Proof}.
It follows from the inequality
$$
1 - \lambda h - e^{-\lambda h} \leq 0 \quad (\lambda \geq 0, \ h > 0)
$$
and the representation $y(t) = e^{-At}f$ that
$$
\begin{aligned}
\left\|\Delta_{h}^{k}y^{(n)}\right\|^{2} = &
\int_{0}^{\alpha} \left(1 - e^{-\lambda h}\right)^{2k}e^{-2\lambda
t}\lambda^{2n}\,d(E(\lambda)f,f)
\\
\leq & \int_{0}^{\alpha}(\lambda h)^{2k}\lambda^{2n}\,d(E(\lambda)f, f) \leq (\alpha h)^{2k}\alpha^{2n}\|f\|.
\end{aligned}
$$
This and (3) imply
$$
\qquad\qquad\qquad\qquad\quad\left\|\Delta_{h}^{k}y^{(n)}\right\|_{S} \leq (\alpha h)^{k}\alpha^{n}\|f\| = (\alpha h)^{k}\alpha^{n}\|y\|_{S}.\qquad\qquad\qquad\qquad\quad\qed
$$
Taking in (6) $k = 0$, we arrive at an analog of Bernstein's
inequality, namely \vspace*{-1mm}
$$
\forall n \in \mathbb{N}: \left\|y^{(n)}\right\| \leq \alpha^{n}\|y\|_{S}. \leqno (7)
$$
Putting there $n = 0$, we obtain
$$
\forall n \in \mathbb{N}: \left\|\Delta_{h}^{k}y\right\|_{S} \leq (\alpha h)^{k}\|y\|_{S} = (\alpha h)^{k}\alpha^{n}\|y\|_{S}.
$$
It should be noted that the inequality
$$
\mathcal{E}_{r}(y) \leq \frac{c}{r^{n}}, \quad r > 0, \quad n \in \mathbb{N}, \leqno (8)
$$
does not yet imply the inclusion $y \in C^{n}(\mathbb{R}_{+},
\mathfrak{H})$. Nevertheless, the following statement, inverse to
Theorem 3, is valid.
\begin{theo} Suppose that $y \in S$, and let $\omega (t)$ be a continuity
module type function, i.e.,
$1) \ \omega (t)$ is continuous and nondecreasing on
$\mathbb{R}_{+}$;
$2) \ \omega (0) = 0$;
$3) \ \exists c > 0, \ \forall t > 0: \omega (2t) \leq c\omega (t)$. \\
In order that $y \in C^{n}(\mathbb{R}_{+}, \mathfrak{H})$, it is sufficient that there exist a number $m > 0$ such that \vspace*{-1mm}
$$
\forall r > 0, \ \forall n \in \mathbb{N}: \mathcal{E}_{r}(y) \leq \frac{m}{r^{n}}\omega \left(\frac{1}{r}\right). \leqno (9)
$$
\end{theo}
\begin{proof}
Assume that, for $y \in S$, condition (9) is fulfilled. Then there
exists a sequence $y_{i} \in S: \sigma (y_{i}) < 2^{i} \ (i \in
\mathbb{N})$ such that
$$
\|y - y_{i}\|_{S} \to 0 \quad \text{as} \quad i \to \infty.
$$
In view of (7) and the inequality $ \sigma (y_{i} - y_{i - 1}) < 2^{i}
\ (i \in \mathbb{N})$, we have
$$
\begin{aligned}
\left\|y_{i}^{(n)} - y_{i - 1}^{(n)}\right\|_{S} \leq & 2^{in}\|y_{i} - y_{i - 1}\|_{S} \leq 2^{in}(\|y - y_{i} \|_{S} + \|y - y_{i - 1}\|_{S})
\\
\leq & 2^{in}\left(\frac{m}{2^{in}}\omega\left(\frac{1}{2^{i}}\right) +
\frac{m}{2^{(i - 1)n}}\omega\left(\frac{1}{2^{i - 1}}\right)\right).
\end{aligned}
$$
From this it follows that
$$
\left\|y_{i}^{(n)} - y_{i - 1}^{(n)}\right\|_{S} \leq m2^{n}\left(\frac{1}{2^{n}} + c\right)\omega\left(\frac{1}{2^{i}}\right),
$$
and therefore
$$
\left\|y_{i}^{(n)} - y_{i - 1}^{(n)}\right\|_{S} \to 0 \quad \text{as} \quad i \to \infty.
$$
Since the space $S$ is complete, there exists $\widetilde{y} \in S$ such that
$$
\left\|y_{i}^{(n)} - \widetilde{y}\right\|_{S} \to 0 \quad \text{when} \quad i \to \infty.
$$
Thus, $y_{i} \to y, \ y_{i}^{(n)} \to \widetilde{y} \ (i \to \infty)$
in the space S. Taking into account that the operator
$\frac{d^{n}}{dt^{n}}$ is closed in $S$, we conclude that $y \in
C^{n}(\mathbb{R}_{+}, \mathfrak{H})$ and $y^{(n)}(t) \equiv
\widetilde{y}(t)$.
\end{proof}
Replacing in inequality (8) $n$ by $n + \varepsilon$ and thus
strengthening it, we shall arrive at the following consequence.
\begin{cor} Let, for $y \in S$,
$$
\exists c > 0, \ \exists \varepsilon > 0: \mathcal{E}_{r}(y) \leq \frac{c}{r^{n + \varepsilon}}.
$$
Then $y \in C^{n}(\mathbb{R}_{+}, \mathfrak{H})$.
\end{cor}
{\bf 3.} \ Now let $\{m_{n}\}_{n \in \mathbb{N}_{0}}$ be a
nondecreasing sequence of numbers (there is no loss of generality in
assuming that $m_{0} = 1$). We put
$$
C_{\{m_{n}\}} = C_{\{m_{n}\}}(\mathbb{R}_{+}, \mathfrak{H}) = \bigcup_{\alpha > 0}C_{m_{n}}^{\alpha},
\quad C_{(m_{n})} = C_{(m_{n})}(\mathbb{R}_{+}, \mathfrak{H}) = \bigcap_{\alpha > 0}C_{m_{n}}^{\alpha},
$$
where
$$
\begin{aligned}
C_{m_{n}}^{\alpha} & = C_{m_{n}}^{\alpha}(\mathbb{R}_{+}, \mathfrak{H})
\\
& = \Big \{y \in C^{\infty}(\mathbb{R}_{+}, \mathfrak{H})\bigl| \exists c = c(y) > 0,
\ \forall k \in \mathbb{N}_{0}:\sup_{t \in \mathbb{R}_{+}}
\Big\|y^{(k)}(t)\Big\| \leq cm_{k}\alpha^{k}\Big\}
\end{aligned}
$$
is a Banach space with respect to the norm
$$
\|y\|_{C_{m_{n}}^{\alpha}} = \sup_{k \in \mathbb{N}}\frac{\sup_{t \in \mathbb{R}_{+}}\left\|y^{(k)}(t)\right\|}{\alpha^{k}m_{k}}.
$$
The spaces $C_{\{m_{n}\}}$ and $C_{(m_{n})}$ are equipped with the
topologies of inductive and projective limits of the spaces
$C_{m_{n}}^{\alpha}$, respectively. Note that the spaces $C_{\{n!\}}, \
C_{(n!)} \ (m_{n} = n!)$ and $C_{\{1\}} \ (m_{n} \equiv 1)$ are
nothing that, respectively, the spaces of bounded on $\mathbb{R}_{+}$
with all their derivatives analytic, entire, and entire of exponential
type $\mathfrak{H}$-valued vector functions.
In what follows, we assume in addition that $\{m_{n}\}_{n \in
\mathbb{N}_{0}}$ satisfies the condition
$$
\forall \alpha > 0, \ \exists c = c(\alpha): m_{n} \geq c\alpha^{n} \leqno (10)
$$
and put
$$
\tau (\lambda) = \sum_{n = 0}^{\infty} \frac{\lambda^{n}}{m_{n}}. \leqno (11)
$$
It is clear that $\tau (\lambda)$ is entire, $\tau (\lambda) \geq 1$ for $\lambda \geq 0$, and $\tau (\lambda) \uparrow \infty$ as $\lambda \to \infty$.
\begin{theo} Suppose the condition
$$
\exists c > 0, \ \exists h > 1, \ \forall n \in \mathbb{N}_{0}: m_{n + 1} \leq ch^{n}m_{n} \leqno (12)
$$
to be fulfilled for the sequence $\{m_{n}\}_{n \in
\mathbb{N}_{0}}$. Then the following equivalence relations hold:
$$
\begin{array}{rcl}
y \in C^{\infty}(\mathbb{R}_{+}, \mathfrak{H}) & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(y) = O\left(\frac{1}{r^{\alpha}}\right) \quad (r \to \infty), \\
y \in C_{\{m_{n}\}} & \Longleftrightarrow & \exists \alpha > 0:
\mathcal{E}_{r}(y) = O\left(\tau^{-1}(\alpha r)\right) \quad (r \to \infty), \\
y \in C_{(m_{n})} & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(y) = O\left(\tau^{-1}(\alpha r)\right) \quad (r \to \infty) \\
\end{array}
$$
$(\tau (\lambda)$ is defined by (11)).
\end{theo}
\begin{proof}
Let $C^{\infty}(A)$ denote the set of all infinitely differentiable
vectors of the operator $A$,
$$
C^{\infty}(A) = \bigcap_{n \in \mathbb{N}_{0}} \mathcal{D}(A^{n}).
$$
For a number $\alpha > 0$, we put
$$
C_{m_{n}}^{\alpha}(A) = \left\{f \in C^{\infty}(A)\bigl| \exists c = c(f) > 0, \ \forall n \in \mathbb{N}_{0}: \left\|A^{n}f\right\| \leq c\alpha^{n}m_{n}\right\}.
$$
The set $C_{m_{n}}^{\alpha}(A)$ is a Banach space with respect to the norm
$$
\|f\|_{C_{m_{n}}^{\alpha}(A)} = \sup_{n \in \mathbb{N}_{0}} \frac{\left\|A^{n}f\right\|}{\alpha^{n}m_{n}}.
$$
Then
$$
C_{\{m_{n}\}}(A) = \bigcup_{\alpha > 0}C_{m_{n}}^{\alpha}(A) \ \text{and} \ C_{(m_{n})}(A) = \bigcap_{\alpha > 0}C_{m_{n}}^{\alpha}(A)
$$
are linear locally convex spaces with topologies of the inductive and
the projective limits, respectively.
Let
$$
\mathcal{E}_{r}(f, A) = \inf_{f_{0} \in C_{\{1\}}(A)} \|f - f_{0}\|.
$$
As it has been shown in [4], the following equivalence relations take place:
$$
\begin{array}{rcl}
f \in C^{\infty}(A) & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(f, A) = O\left(\frac{1}{r^{\alpha}}\right) \quad (r \to \infty), \\
f \in C_{\{m_{n}\}}(A) & \Longleftrightarrow & \exists \alpha > 0:
\mathcal{E}_{r}(f, A) = O\left(\tau^{-1}(\alpha r)\right) \quad (r \to \infty), \\
f \in C_{(m_{n})}(A) & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(f, A) = O\left(\tau^{-1}(\alpha r)\right) \quad (r \to \infty). \\
\end{array} \leqno (13)
$$
Consider the map $F: \mathfrak{H} \mapsto S$, \vspace*{-1mm}
$$
Ff = e^{-At}f.
$$
Since, for $y \in S$, there exists a unique vector $f \in \mathfrak{H}$
such that $y = e^{-At}f$, this transformation is one-to-one. From (3)
it follows that $F$ maps $\mathfrak{H}$ onto $S$ isometrically and
$$
F\left(C^{\infty}(A)\right) = C^{\infty}(\mathbb{R}_{+}, \mathfrak{H}), \quad F\left(C_{\{m_{n}\}}(A)\right) = C_{\{m_{n}\}}, \quad F\left(C_{(m_{n})}(A)\right) = C_{(m_{n})}. \leqno (14)
$$
The proof of the theorem follows from (13), (14) because of
$\mathcal{E}_{r}(f, A) = \mathcal{E}_{r}\left(e^{-At}f\right)$.
\end{proof}
If $m_{n} = n^{n\beta} \ (\beta > 0)$, then $\tau (r) =
e^{-r^{1/\beta}}$ and Theorem 5 yields the following assertion.
\begin{cor} The following equivalence relations are valid:
$$
\begin{array}{rcl}
y \in C_{\{n^{n\beta}\}}(A) & \Longleftrightarrow & \exists \alpha > 0:
\mathcal{E}_{r}(y) = O\left(e^{-\alpha r^{1/\beta}}\right) \quad (r \to \infty), \\
y \in C_{(n^{n\beta})}(A) & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(y) = O\left(e^{-\alpha r^{1/\beta}}\right) \quad (r \to \infty). \\
\end{array}
$$
\end{cor}
Recall that an entire $\mathfrak{H}$-valued vector function
$x(\lambda)$ has a finite order of growth if
$$
\exists \gamma > 0, \ \forall \lambda \in \mathbb{C}: \|x(\lambda)\| \leq \exp(|\lambda|^{\gamma}).
$$
The greatest lower bound $\rho (x)$ of such $\gamma$ is the order of
$x(\lambda)$. The type of an entire vector-valued function
$x(\lambda)$ of an order $\rho$ is determined as
$$
\sigma (x) = \inf \left\{a > 0: \|x(\lambda)\| \leq \exp(a|\lambda|^{\rho})\right\}.
$$
Since the semigroup $\left\{e^{-At}\right\}_{t \geq 0}$ is analytic,
every weak solution $y(t)$ of equation (1) is analytic on $(0,
\infty)$. It is not difficult to show that it is analytic on $[0,
\infty)$ if and only if $y \in C_{\{n^{n}\}}$.
By Corollary 3,
$$
\exists \alpha > 0: \mathcal{E}_{r}(y) = O\left(e^{-\alpha r}\right) \quad (r \to \infty).
$$
As for the extendability of $y(t)$ to an entire vector function of
order $\rho$ and finite type, the answer to the question gives the
next theorem.
\begin{theo} In order that a weak solution of equation (1) admit an
extension to an entire $\mathfrak{H}$-valued vector function $y(z)$,
it is necessary and sufficient that
$$
\forall \alpha > 0: \mathcal{E}_{r}(y) = O\left(e^{-\alpha r}\right) \quad (r \to \infty). \leqno (15)
$$
The extension $y(z)$ is of finite order $\rho$ and finite type if and only if
\vspace*{-1mm}
$$
\exists \alpha > 0: \mathcal{E}_{r}(y) = O\left(e^{-\alpha r^{1/\beta}}\right) \quad (r \to \infty),
$$\vspace*{-1mm}
where $\beta$ and $\rho$ are connected with each other by the formula
\vspace*{-0mm}
$$
\beta = \frac{\rho - 1}{\rho} < 1.
$$
\end{theo}
Note that we may always suppose $\rho > 1$. Indeed, if $\rho \leq 1$
and the type is finite, then $y \in S_{0}$ and it has no sense to
approximate a solution from $S_{0}$ by solutions from the same space.
\vspace*{1mm}
\noindent{\it Proof of Theorem 6.}
Let $y(t)$ be a weak solution of equation (1). By Corollary 3, $y
\in C_{(n^{n})}$, so $y(t)$ admits an extension to an entire vector
function if and only if relation (15) is fulfilled.
Assume that $y(t)$ admits an extension to an entire
$\mathfrak{H}$-valued vector function $y(z)$ which has an order
$\rho$ and a finite type $\sigma$. Then
$$
\forall \sigma_{1} > \sigma, \ \exists c = c(\sigma_{1}): \|y(z)\| \leq ce^{\sigma_{1}|z|^{\rho}}.
$$
Hence, \vspace*{-2mm}
$$
\forall r > 0, \ \forall n \in \mathbb{N}_{0}: \left\|y^{(n)}(z)\right\| \leq \frac{n!}{2\pi} \int_{|z - \zeta| = r} \frac{\|y(\zeta)\|}{|z - \zeta|^{n + 1}}\,d\zeta \leq \frac{c\sigma_{1}n!}{r^{n}}\exp\left(2\sigma_{1}r^{\rho}\right).
$$
Taking into account that the function
$\frac{\exp\left(ar^{\rho}\right)}{r^{n}}$ reaches its minimum at the
point $\left(\frac{n}{a\rho}\right)^{1/\rho}$ and using Stirling's
formula
$$
n! = n^{n}e^{-n}\sqrt{2\pi n}\left(1 + O\left(\frac{1}{n}\right)\right) \quad (n \to \infty),
$$
we get \vspace*{-1mm}
$$
\left\|y^{(n)}(z)\right\| \leq c\left(2e^{1 - \rho}\sigma_{1}\rho\right)^{\frac{1}{\rho}}n^{\frac{\rho - 1}{\rho}n},
$$
which shows that $y \in C_{\{n^{n\beta}\}}$, where
$$
\beta \leq \frac{\rho - 1}{\rho} \Longleftrightarrow \rho \geq \frac{1}{1 - \beta}, \quad \beta < 1. \leqno (16)
$$
By Corollary 3,
$$
\exists \alpha > 0: \mathcal{E}_{r}(y) = O\left(e^{-\alpha r^{1/\beta}}\right) \quad (r \to \infty). \leqno (17)
$$
Conversely, let (17) hold true. Then Corollary 3 implies that $y \in
C_{\{n^{n\beta}\}} \ (0 < \beta < 1)$ is an entire vector-valued
function and it can be represented by the series $\sum_{k =
0}^{\infty}\frac{y^{(k)}(0)}{k!}z^{k}$. For its order of growth
$\rho$ we have
$$
\rho = \varlimsup_{n \to \infty} \frac{n\ln n}{\ln \frac{n!}{\left\|y^{(n)}(0)\right\|}} \leq
\varlimsup_{n \to \infty} \frac{n\ln n}{\ln \left(n!n^{-n\beta}\right)} = \frac{1}{1 - \beta},
$$
that is,
$$
\rho \leq \frac{1}{1 - \beta}. \leqno (18)
$$
It follows from (16) and (18) that
$$
\qquad\qquad\qquad\qquad\qquad\qquad\quad\rho = \frac{1}{1 - \beta}, \quad 0 < \beta < 1.\qquad\qquad\qquad\qquad\qquad\qquad\quad \qed
$$
{\bf 4.} \ The direct and inverse theorems of the
approximation theory are usually formulated for Banach space. Proving
them is slightly more complex than in the case of a Hilbert
space. Below we show how, for example, Theorem 5 can be reformulated
in a Banach space. To this end we introduce the following notations:
$$
\mathfrak{H}^{n} = \mathcal{D}(A^{n}), \|f\|_{\mathfrak{H}^{n}} = \left(\|f\|^{2} + \|A^{n}f\|^{2}\right)^{1/2}.
$$
The space $\mathfrak{H}^{n}$ is continuously and densely embedded into
$\mathfrak{H}$. Denote by $\mathfrak{H}^{-n}$ the completion of
$\mathfrak{H}$ in the norm $\|f\|_{\mathfrak{H}^{-n}} = \left\|\left(A
+ I\right)^{-n}f\right\|$, the so called negative space associated
with the positive space $\mathfrak{H}^{n}$ in the chain
$$
\mathfrak{H}^{n} \subseteq \mathfrak{H} \subseteq \mathfrak{H}^{-n}
$$
(see [2]). By a suitable choice of the norms in $\mathfrak{H}^{n}$ and $\mathfrak{H}^{-n}$, one can attain
the relation
$$
\|f\|_{\mathfrak{H}^{-n}} \leq \|f\| \leq \|f\|_{\mathfrak{H}^{n}}.
$$
\begin{theo} Let $\mathfrak{B}$ be a Banach space inside the chain
$$
\mathfrak{H}^{k_{1}} \subseteq \mathfrak{B} \subseteq \mathfrak{H}^{-k_{2}} \leqno (19)
$$
of continuously and densely embedded into each other spaces with some
$k_{1}, k_{2} \in \mathbb{N}$, and let a sequence $\{m_{n}\}_{n \in
\mathbb{N}_{0}}$ possess the properties (10) and (12). Then for a
weak solution $y(t)$ of equation (1), the following equivalence
relations hold true:
$$
\begin{array}{rcl}
y \in C^{\infty}(\mathbb{R}_{+}, \mathfrak{H}) & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(y, \mathfrak{B}) = O\left(r^{-\alpha}\right) \quad (r \to \infty), \\
y \in C_{\{m_{n}\}} & \Longleftrightarrow & \exists \alpha > 0,:
\mathcal{E}_{r}(y, \mathfrak{B}) = O\left(\tau^{-1}(\alpha r)\right) \quad (r \to \infty), \\
y \in C_{(m_{n})} & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{r}(y, \mathfrak{B}) = O\left(\tau^{-1}(\alpha r)\right) \quad (r \to \infty),\\
\end{array}
$$
where
$$
\mathcal{E}_{r}(y, \mathfrak{B}) = \inf_{y_{0} \in S_{0}: \sigma (y_{0}) \leq r} \sup_{s \in \mathbb{R}_{+}} \|y(s) - y_{0}(s)\|_{\mathfrak{B}},
$$
$\tau (\lambda)$ is defined by (11), $\|\cdot\|_{\mathfrak{B}}$ is the norm in $\mathfrak{B}$.
\end{theo}
\begin{proof}
Show first that the spaces $C_{\{m_{n}\}}(A)$ and $C_{(m_{n})}(A)$
considered as subspaces of $\mathfrak{H}$ coincide with the
corresponding subspaces $C_{\{m_{n}\}}^{k}(A)$ and
$C_{(m_{n})}^{k}(A)$ constructed in the Hilbert space
$\mathfrak{H}^{k}$ from the restriction $A\upharpoonright
\mathfrak{H}^{k}$ which is a nonnegative self-adjoint operator
in~$\mathfrak{H}^{k}$.
So, let $f \in C_{\{m_{n}\}}(A)$. Then
$$
\exists \alpha > 0, \ \exists c > 0: \left\|A^{i}f\right\|_{\mathfrak{H}^{k}} = \left(\left\|A^{i}f\right\|^{2} + \left\|A^{i + k}f\right\|^{2}\right)^{1/2}
$$
$$
\qquad \qquad \qquad \leq c\left(\alpha^{2i}m_{i}^{2} + \alpha^{2(i +
k)}m_{k + i}^{2}\right)^{1/2} \leq \widetilde{c}\left(\alpha
h^{k}\right)^{i}m_{i}, \leqno (20)
$$
i.e., $ C_{\{m_{n}\}}(A) \subseteq C_{\{m_{n}\}}^{k}(A)$. From (19) we
also have the embedding $ C_{(m_{n})}(A) \subseteq
C_{(m_{n})}^{k}(A)$. The inverse embeddings are consequences of the
estimate $\left\|A^{i}f\right\|_{\mathfrak{B}} \leq
\left\|A^{i}f\right\|_{\mathfrak{H}^{k}}$. Thus we have
$$
C_{\{m_{n}\}}(A) = C_{\{m_{n}\}}^{k}(A), \quad C_{(m_{n})}(A) = C_{(m_{n})}^{k}(A). \leqno (21)
$$
It is also evident that $C_{k}^{\infty}(A) = C^{\infty}(A)$.
Since for a vector $g \in C_{\{1\}}(A) \ (m_{n} \equiv 1)$ of type
$\sigma(g, A) \leq k$, the inequality
$$
\left\|A^{i}f\right\|_{\mathfrak{H}^{k}} = \left(\left\|A^{i}g\right\|^{2} + \left\|A^{i + k}g\right\|^{2}\right)^{1/2} \leq c\left(\alpha^{2i} + \alpha^{2(i + k)}\right)^{1/2} \leq \widetilde{c}\alpha^{i}, \quad \widetilde{c} = \left( 1 + \alpha^{2k}\right)^{1/2},
$$
is valid, the space $C_{\{1\}}(A)$ coincides with
$C_{\{1\}}^{k}(A)$. Moreover, the type of $g$ for the operator $A$ in
the space $C_{\{1\}}(A)$ is the same as the one for $A\upharpoonright
\mathfrak{H}^{k}$ in the space $C_{\{1\}}^{k}(A)$.
Denote by $\widetilde{A}$ the closure of $A$ in the space
$\mathfrak{H}^{-k}$. It is easy to make sure that $\widetilde{A}$ is a
nonnegative self-adjoint operator in $\mathfrak{H}^{-k}$ and if the
space $\mathfrak{H}$ is considered as a subspace of
$\mathfrak{H}^{-k}$, then we arrive at the previous situation. For
this reason,
$$
C_{\{m_{n}\}}(A) = C_{\{m_{n}\}}^{-k}(A), \quad C_{(m_{n})}(A) = C_{(m_{n})}^{-k}(A), \leqno (22)
$$
and
$$
\forall g \in C_{\{1\}}(A) = C_{\{1\}}^{k}(A): \sigma(g, A) = \sigma(g, \widetilde{A}).
$$
Taking into account that the restriction (extension) of the semigroup
$\left\{e^{-At}\right\}_{t \in \mathbb{R}_{+}}$ to the space
$\mathfrak{H}^{k_{1}}, \ k_{1} > 0,$ (to $\mathfrak{H}^{-k_{2}}, \
k_{2} > 0)$ is an analytic contractive $C_{0}$-semigroup in
$\mathfrak{H}^{k_{1}}$ (in $\mathfrak{H}^{-k_{2}}$), the embeddings
$C_{\{m_{n}\}}^{k}(A) \subseteq \mathfrak{B}, \ C_{(m_{n})}(A)
\subseteq \mathfrak{B}$ and the chain (19), we obtain for $y(t) =
e^{-At}f, \ f \in C_{\{m_{n}\}}(A)$ and $y_{0}(t) = e^{-At}g, \ g \in
C_{\{1\}}(A)$ that
$$
\left\|e^{-At}f - e^{-As}g\right\|_{\mathfrak{B}} \leq \left\|e^{-At}f - e^{-As}g\right\|_{\mathfrak{H}^{k_{1}}} \leq \|f - g\|_{\mathfrak{H}^{k_{1}}},
$$
whence
$$
\mathcal{E}_{r}(y, \mathfrak{B}) = \inf_{y_{0} \in S_{0}: \sigma (y_{0}) \leq r} \sup{s \in \mathbb{R}_{+}} \left\|e^{-As}f - e^{-As}g\right\|_{\mathfrak{B}} \leq \|f - g\|_{\mathfrak{H}^{k_{1}}},
$$
that is, \vspace*{-1mm}
$$
\mathcal{E}_{r}(y, \mathfrak{B}) \leq \mathcal{E}_{r}(f, A\upharpoonright \mathfrak{H}^{k_{1}}). \leqno (23)
$$
From (19) it follows that
$$
\forall t \in \mathbb{R}_{+}: \left\|e^{-At}f - e^{-At}g\right\|_{\mathfrak{H}^{-k_{2}}} \leq \left\|e^{-At}f - e^{-At}g\right\|_{\mathfrak{B}}.
$$
This implies that
$$
\|f - g\|_{\mathfrak{H}^{-k_{2}}} = \sup_{t \in \mathbb{R}_{+}} \left\|e^{-At}f - e^{-At}g\right\|_{\mathfrak{H}^{-k_{2}}} \leq \sup_{t \in \mathbb{R}_{+}} \left\|e^{-At}f - e^{-At}g\right\|_{\mathfrak{B}}
$$
and, hence,\vspace*{-2mm}
$$
\inf_{g \in C_{\{1\}}(A): \sigma (g) \leq r}\|f - g\|_{\mathfrak{H}^{-k_{2}}} \leq
\inf_{y_{0} \in S_{0}: \sigma (y_{0}) \leq r} \sup_{t \in \mathbb{R}_{+}} \|y(t) - y_{0}(t)\|_{\mathfrak{B}}.
$$
Thus,\vspace*{-1mm}
$$
\mathcal{E}_{r}(f, \widetilde{A}) \leq \mathcal{E}_{r}(y, \mathfrak{B}). \leqno (24)
$$
Inequalities (23) and (24), with regard to (19), (21), (22) and
Theorem 5, complete the proof of Theorem 7.
\end{proof}
\vspace*{-2mm}
{\bf 5.} \ Let $A$ be a self-adjoint operator in
$\mathfrak{H}$ whose spectrum is discrete. Assume that its eigenvalues
$\lambda_{k} = \lambda_{k}(A), \ k \in \mathbb{N},$ satisfy the
condition $\sum_{k = 1}^{\infty} \lambda_{k}^{-p} < \infty$
with some $p > 0$. Suppose also that $\lambda_{k}$ are enumerated in
ascending order and each one is counted according to its multiplicity
and denote by $\{e_{n}\}_{n \in \mathbb{N}}$ the orthonormal basis in
$\mathfrak{H}$ consisting of eigenvectors of $A$. Then the spectral
function $E(\lambda)$ of the operator $A$ has the form
$$
E(\lambda)f = \sum_{\lambda_{k} \leq \lambda} f_{k}e_{k},
$$
where $f_{k} = (f_{k}, e_{k})$ are the Fourier coefficients of $f$, and
$$
\mathcal{E}_{r}(f, A) = \sum_{\lambda_{k} > r} f_{k}e_{k}.
$$
As it has been shown in [6], the following assertion holds true.
\begin{pr} The following equivalence relations are valid:
$$
\begin{array}{rcl} f \in C^{\infty}(A) & \Longleftrightarrow & \forall \alpha > 0, \ \exists c = c(\alpha) > 0:
|f_{k}| \leq c \lambda_{k}^{-\alpha}, \\
f \in C_{\{1\}}(A) & \Longleftrightarrow & \exists n_{0} \in \mathbb{N}: f_{k} = 0 \ \text{as} \ k \geq n_{0}, \\
f \in C_{\{m_{n}\}}(A) & \Longleftrightarrow & \exists \alpha > 0, \ \exists c > 0: |f_{k}| \leq c \tau^{-1}(\alpha
\lambda_{k}), \\
f \in C_{(m_{n})}(A) & \Longleftrightarrow & \forall \alpha > 0, \ \exists c = c(\alpha)> 0: |f_{k}|
\leq c \tau^{-1}(\alpha \lambda_{k}). \\
\end{array}
$$
(The function $\tau (\lambda)$ was defined in (11)).
\end{pr}
Let now $y(t)$ be a weak solution of (1). Then
$$
y(t) = \sum_{k = 1}^{\infty} e^{-\lambda_{k}t} f_{k}e_{k}, \quad \sum_{k = 1}^{\infty} |f_{k}|^{2} < \infty. \leqno (25)
$$
The solution $y(t)$ is an entire vector-valued function of exponential type $(y \in S_{0})$ if and only if
$$
\exists n_{0} \in \mathbb{N}: f_{k} = 0 \ \text{as} \ k \geq n_{0}.
$$
Proposition 1, Theorems 3, 5 and (13) imply the following.
\begin{theo} The following equivalence relations take place:
$$
\begin{array}{rcl}
y \in C^{n}(\mathbb{R}_{+}, \mathfrak{H}) & \Longleftrightarrow & \mathcal{E}_{\lambda_{k}}(y)
= o\left(\lambda_{k + 1}^{-n}\right) \quad (n \to \infty), \\
y \in C^{\infty}(\mathbb{R}_{+}, \mathfrak{H}) & \Longleftrightarrow & \forall \alpha > 0:
\mathcal{E}_{\lambda_{k}}(y) = O \left(\lambda_{k + 1}^{-\alpha}\right) \quad (n \to \infty), \\
y \in C_{\{m_{n}\}} & \Longleftrightarrow & \exists \alpha > 0: \mathcal{E}_{\lambda_{k}}(y) =
O \left(\tau^{-1}(\alpha\lambda_{k + 1})\right) \quad (n \to \infty), \\
y \in C_{(m_{n})} & \Longleftrightarrow & \forall \alpha > 0: \mathcal{E}_{\lambda_{k}}(y) =
O \left(\tau^{-1}(\alpha\lambda_{k + 1})\right) \quad (n \to \infty). \\
\end{array}
$$
\end{theo}
\vspace*{-1mm}
{\bf 6.} Put $\mathfrak{H} = L_{2}(\Omega)$, where $\Omega$
is a bounded domain in $\mathbb{R}^{q}$ with piecewise smooth boundary
$\partial\Omega$, and denote by $B'$ the operator generated in
$L_{2}(\Omega)$ by the differential expression \vspace*{-3mm}
$$
(\mathcal{L}u)(x)
= - \sum_{i = 1}^{q}\sum_{k = 1}^{q}\frac{\partial}{\partial x_{i}}\left(a_{ik}(x)\frac{\partial u(x)}{\partial x_{k}}\right) + c(x)u(x), \leqno (26)
$$
on\vspace*{-1mm}
$$
\mathcal{D}(B') = \left\{u \in C^{2}(\overline{\Omega})\bigl|
u\upharpoonright_{\partial\Omega} = 0\right\}. \leqno (27) $$
It is
assumed that $a_{ik}(x), c(x) \in C^{\infty}(\overline{\Omega}), \
c(x) \geq 0$. Suppose also the expression (26) to be of elliptic type
in $\overline{\Omega}$. In this case all the eigenvalues $\mu_{i}(x),
\ i = 1, \dots q$, of the matrix $\|a_{ik}(x)\|_{i,k = 1}^{q}, \ x \in
\overline{\Omega}$, have the same sign; without loss of generality we
may assume $\mu_{i}(x) > 0, \ x \in \overline{\Omega}$.
It is not hard to make sure that $B'$ is a positive definite Hermitian
operator with dense domain in $L_{2}(\Omega)$. So, $B'$ admits a
closure to a positive definite selfadjoint operator $B$ on
$L_{2}(\Omega)$. We shall call $B$ the operator generated by (26),
(27). The spectrum of $B$ is discrete, and for its eigenvalues,
$\lambda_{1}(B) < \lambda_{2}(B) < \dots < \lambda_{n}(B) < \dots$, the
estimate
$$
c_{1}n^{2/q} \leq \lambda_{n}(B) \leq c_{2}n^{2/q}, \quad 0 < c_{i} = {\rm {const}}, \quad i = 1, 2. \leqno (28)
$$
is valid (see [8]). Denote by $e_{n}(x), \ n \in \mathbb{N}$, the
orthonormal basis in $L_{2}(\Omega)$, consisting of eigenfunctions of
$B$.
In the case where $\Omega$ is a $q$-dimensional cube, $0 < x_{k} < a, \
k = 1, \dots, q, \ a > 0$, and $\mathcal{L} = -\sum_{i =
1}^{q}\frac{\partial^{2}}{\partial x_{i}^{2}}$, the following
formulas for the eigenvalues $\lambda_{n_{1}\dots n_{q}}, \ n_{k} \in
\mathbb{N}$, and eigenfunctions $e_{n_{1}\dots n_{q}}(x)$ of the
operator $B$ hold: \vspace*{-2mm}
$$
\lambda_{n_{1}\dots n_{q}} = \frac{\pi^{2}}{a^{2}}\sum_{k = 1}^{q}n_{k}^{2}; \quad e_{n_{1}\dots n_{q}}(x) = \left( \frac{2}{a}\right)^{q/2}\prod_{k = 1}^{q} \sin\frac{\pi}{a}x_{k}.
$$
Let $y(t) = u(t, x) \in C(\mathbb{R}_{+}, L_{2}(\Omega))$ be a weak
solution of the problem
$$
\left(\frac{\partial}{\partial t} - \sum_{k = 1}^{q}\sum_{i = 1}^{q} \frac{\partial}{\partial x_{k}}\left(a_{ki}(x)\frac{\partial}{\partial x_{i}}\right) + c(x)\right)u(t,x) = 0, \leqno (29)
$$
$$
\forall t > 0, \ \forall x \in \partial \Omega: u(t, x) = 0, \leqno (30)
$$
where the conditions on $a_{ki}(x)$ and $c(x)$ are the same as
before. Then $u(t, x)$ admits a representation of form (25). Set
$$
L_{2}^{n} = \mathcal{D}(B^{n}), \quad \|f\|_{L_{2}^{n}} = \left( \|f\|^{2} + \left\|B^{n}f\right\|^{2}\right)^{1/2},
$$
where $B$ is an operator generated in the space $L_{2}(\Omega)$ by
expression (26) and boundary value condition (27). The space
$L_{2}^{n}$ is continuously and densely embedded into
$L_{2}(\Omega)$. Denote by $L_{2}^{-n}$ the negative space
corresponding to the positive one $L_{2}^{n} \subset
L_{2}(\Omega)$. In the case where $\mathcal{L} = -\sum_{k =
1}^{q} \frac{\partial^{2}}{\partial x_{k}^{2}}$, \ $L_{2}^{n}$ is
none other than the well-known Sobolev space
$\stackrel{\circ}{W}_{2}^{2n}(\Omega)$.
Using estimate (28) for $\lambda_{k}(B)$ and Theorem 8, we obtain in a
way analogous to that used in the proof of Theorem 7 the following
assertion.
\begin{theo} Let $\mathfrak{B}$ be a Banach space and let
$$
L_{2}^{n_{1}} \subseteq \mathfrak{B} \subseteq L_{2}^{-n_{2}}, \quad n_{1}, n_{2} \in \mathbb{N},
$$
be a chain of continuously and densely embedded into each other
spaces. Suppose also the sequence $\left\{m_{n}\right\}_{n =
1}^{\infty}$ to satisfy (10) and (12). Then
$$
\begin{array}{rcl}
y(t) = u(t, x) \in C^{\infty}(\mathbb{R}_{+}, L_{2}(\Omega)) & \Longleftrightarrow & \alpha > 0:\mathcal{E}_{\lambda_{k}}^{\mathfrak{B}}(y) = O\left(\frac{1}{(k + 1)^{\alpha}}\right), \\
y(t) \in C_{\{m_{n}\}} & \Longleftrightarrow & \exists \alpha > 0: \mathcal{E}_{\lambda_{k}}^{\mathfrak{B}}(y) = O \left(\tau^{-1}\left(\alpha(k + 1)^{2/q}\right)\right), \\
y(t) \in C_{(m_{n}} & \Longleftrightarrow & \forall \alpha > 0: \mathcal{E}_{\lambda_{k}}^{\mathfrak{B}}(y) = O \left(\tau^{-1}\left(\alpha(k + 1)^{2/q}\right)\right), \\
\end{array}
$$
where
$$
\mathcal{E}_{\lambda_{k}}^{\mathfrak{B}}(y) = \inf_{y_{0} \in S_{0}: \sigma(y_{0})\leq \lambda_{k}} \sup_{s \in \mathbb{R}_{+}} \|y(s) - y_{0}(s)\|_{\mathfrak{B}}.
$$
\end{theo}
It is relevant to remark that in the case where $a_{ki}(x) =
\delta_{ki}$ and $c(x) \equiv 0$, by virtue of the embedding theorems
for Sobolev spaces, one can take the space $C(\overline{\Omega})$ of
continuous in $\overline{\Omega}$ functions or $L_{p}(\Omega), \ 1
\leq p < \infty,$ as $\mathfrak{B}$ and consider not only the
Dirichlet but some other boundary value problems, in particular, the
Neumann problem.
\end{document} | math | 38,194 |
\begin{document}
\begin{titlepage}
\title{Competitive equilibrium \\and the double auction
\thanks{For helpful comments and useful discussions, I would like to thank Jesper Akesson, Miguel Ballester, Adam Brzezinski, David Van Dijcke, Dan Friedman, Steve Gjerstad, Bernhard Kasberger, Erik Kimbrough, John Ledyard, Luke Milsom, Heinrich Nax, Charlie Plott, David Porter and Jasmine Theilgaard. I would also like to thank audiences at Oxford and Michigan State's Quantitative Economics Workshop. Finally, I am grateful to
Aniket Chakravorty for excellent research assistance and to the George Webb Medley Fund for generous financial support.
}}
\author{Itzhak Rasooly\thanks{Sciences Po, the Paris School of Economics, and the University of Oxford.} }
\date{\today}
\maketitle
\vspace*{-0.4cm}
\begin{center}
\end{center}
\vspace*{-0.8cm}
\begin{abstract}
\noindent In this paper, we revisit the common claim that double auctions necessarily generate competitive equilibria. We begin by observing that competitive equilibrium has some counterintuitive implications: specifically, it predicts that monotone shifts in the value distribution can leave prices unchanged. Using experiments, we then test whether these implications are borne out by the data. We find that in double auctions with stationary value distributions, the resulting prices can be far from competitive equilibria. We also show that the effectiveness of our counterexamples is blunted when traders can leave without replacement as time progresses. Taken together, these findings suggest that the `Marshallian path' is crucial for generating equilibrium prices in double auctions.
\noindent \\
\\
\noindent \textsc{Keywords:} double auction, competitive equilibrium, Marshallian path
\\
\noindent\textsc{JEL Codes:} C92, D01, D02, D90\\
\end{abstract}
\setcounter{page}{0}
\thispagestyle{empty}
\end{titlepage}
\pagebreak
\section{Introduction}\label{introduction}
In his `Introduction to Economic Science', \cite{fisher} wrote that `If you want to make a first-class economist, catch a parrot and teach him to say ``supply and demand'' in response to every question you ask him.' Apparently, this joke was considered dated even in 1910 --- it is attributed to a critic of economics writing from a `long time ago' --- and it thus serves to illustrate the dominance of supply and demand in the century before Fisher's publication. However, supply and demand analysis remained popular in Fisher's time; and indeed one of the purposes of Fisher's book was to expound and refine such analysis.
More than a century later, the idea of supply and demand --- or `competitive equilibrium', as we might now call it --- remains pervasive throughout economic theory. Its extension to multiple markets in the form of general equilibrium theory has provided a basis for celebrated welfare theorems \citep{arrow1951extension, debreu1954valuation}, as well as the foundation for much of modern macroeconomics (starting with \cite{lucas1977understanding}, \cite{kydland1982time}, etc.). In addition, partial equilibrium models have been applied to areas as diverse as discrimination \citep{becker1957economics}, marriage \citep{becker1973theory, becker1974theory} and location choice \citep{glaeser2007economics}. Thus, although the notion of competitive equilibrium may not have quite the dominance that it did in Fisher's time --- supplanted in part by the rise of non-cooperative game theory --- it surely remains one of the foundational concepts of economic theorising.
In part, the pervasiveness of competitive equilibrium may be due to the perception that its predictions have been experimentally vindicated by a series of double auction experiments starting with \cite{smith1962experimental}. As \cite{plott1981theories} puts it in an early review of the literature: `The overwhelming result [from these experiments] is that these markets converge to the competitive equilibrium even with very few traders'. In a more recent study of two thousand classroom experiments, \cite{lin2020evidence} reach a similar conclusion, declaring that competitive equilibrium convergence in double auctions appears to be `as close to a culturally universal, highly reproducible outcome as one is likely to get in social science'. They add that such convergence should be considered `as reproducible as the kinds of experiments that are done in a college chemistry laboratory to demonstrate universal chemistry principles'.\footnote{Indeed, existing experimental results appear to be surprisingly robust to changing the number of bidders \citep{smith1965experimental}, changing the cultural context \citep{kachelmeier1992culture}, and even introducing `extreme earnings inequality' at equilibrium \citep{holt1986market, smith2000boundaries, kimbrough2018testing}. Thus, while it may be possible to at least slow convergence to competitive equilibrium through large changes to the underlying environment --- e.g. by allowing for resale as in \cite{dickhaut2012commodity} or market power as in \cite{kimbrough2018testing} --- the existing literature suggests fast convergence to competitive equilibrium in double auction environments similar to those first considered by \cite{smith1962experimental}. See also \cite{ikica2018competitive} for a large-scale replication of competitive equilibrium convergence in standard double auction environments.}
In this paper, we revisit this conclusion. Our starting point is that competitive equilibrium can make highly counterintuitive, but previously unstudied, predictions. To see the basic idea, consider a market with 99 buyers, each with unit demand and with valuations of £1.01, £2.01, £3.01, ..., up to £99.01. Meanwhile, suppose that there are 99 sellers, each possessing just one unit to sell and with valuations (or `costs') of £0.99, £1.99, £2.99, ... up to £98.99. Under such assumptions, one can check that the (essentially unique\footnote{Depending on how one handles ties, prices of £49.99 and £50.01 can also clear the market.}) market-clearing price is £50: at such a price, 50 buyers want to purchase (those with valuations above £50) and 50 sellers want to sell (those with valuations below £50). Imagine now that we decrease the valuations of all sellers whose initial valuation was below £50 by an arbitrary amount, say to £0. Intuitively, one would expect this to drive down the price, both by inducing sellers to accept lower offers and by allowing them to profitably submit lower offers themselves. Despite this, however, the shift does not change the competitive equilibrium price: a price of £50 still generates demand from 50 buyers (since demand has not changed), and still generates supply from the same 50 sellers (who are now even more eager to sell).
In Section \ref{CE preserving shifts}, we begin by generalising the example just given. That is, we identify a broad class of downward shifts to the distributions of buyer and seller valuations which preserve the set of competitive equilibrium prices. As discussed, it is highly counterintuitive that such downward shifts would in fact leave market prices unaltered. As a result, such shifts provide a challenging test for competitive price theory.
We then conduct experimental double auctions to investigate whether such shifts do in fact depress observed prices. One important feature of our initial set of double auction experiments is that they hold the value distributions fixed through use of a queue of buyers and sellers: every time a buyer or seller exits the market, a new buyer or seller takes their place (see \cite{brewer2002behavioral} for a similar methodology). We use this queueing procedure for two reasons. First, it can be justified on grounds of realism: in actual markets, trade does not end after a couple of periods once all willing buyers and sellers have traded. Instead, the market is continually replenished by a new stock of traders. Second, and much more importantly, our design necessarily holds competitive equilibrium fixed, thereby allowing us to rigorously study if double auctions outcomes converge to \textit{the} set of competitive equilibrium prices. In contrast, standard designs do not possess this feature: every time a pair of traders drop out of the market, the supply and demand schedules shift, something which may (or may not) change the set of competitive equilibrium prices.\footnote{Remarkably, this issue was noticed in the first-ever experimental study of double auctions: see footnote 6 of \cite{smith1962experimental}. Despite conceding that `the supply and demand functions continually alter as the trading process occurs', Smith asserts that it is `the \textit{initial} [supply and demand] schedules prevailing at the opening of each trading period' that are of interest to `the theorist'. However, Smith does not give us any reason for privileging the initial demand and supply schedules over any others; and in the absence of such a reason, such privileging would appear to be entirely arbitrary.}
Our initial set of experiments yields three main findings. First, contrary to the predictions of competitive equilibrium theory, our downward shifts in the valuations do markedly decrease observed prices. Second, and partly as a result of the first finding, prices are almost never at the competitive equilibrium. Remarkably, this is the case even for our symmetric treatment, which might have been expected to yield competitive equilibrium prices. Third, prices show very little sign of converging to competitive equilibrium. Taken together, these findings imply that competitive equilibrium is not a good description of double auctions with stationary value distributions.
We then discuss why decreasing valuations and costs might decrease observed prices. While we do not attempt to identify one model as definitively `correct', we do identify a number of models that can rationalise our finding. To take one example, within a zero intelligence model \citep{gode1993allocative}, decreasing valuations leads to stochastically lower distributions of bids and asks, thereby lowering expected prices. More interestingly, perhaps, this effect is also generated by optimising models, including those based both on myopic pay-off maximisation \citep{gjerstad1998price} and more sophisticated optimal stopping \citep{friedman1991simple}.
Our first set of experiments establishes that, if value distributions are held fixed, double auctions need not produce competitive equilibria; and that double auction outcomes are sensitive to the kind of value shifts described previously. It is natural to wonder, however, whether our shifts are still able to move observed prices in more standard double auction environments in which players are allowed to drop out of the market (without replacement) as trade progresses. To investigate this question, we also run a series of double auctions without queues, including some very long sessions involving over an hour's trading in order to give the auctions the best possible chance of equilibrating.
Our findings from these more conventional experiments are more mixed. On the one hand, there is still some evidence that our shifts depress the observed prices; and some of the sessions we run fail to equilibrate even after many periods of trading. On the other hand, there is now a marked tendency towards equilibrium, and some of our sessions do converge to equilibrium despite the marked asymmetry in the designs. Thus, these sessions reveal both the power of our shifts as well as the equilibrating forces first observed in \cite{smith1962experimental}.
Our central pair of findings --- that one can easily `break' competitive equilibrium in environments with a queue of buyers and sellers but much less easily in an environment without a queue --- suggests that the question of whether the value distribution is held fixed is of critical importance. We provide a theoretical explanation as to why this should be the case. We assume that trade follows a `Marshallian path' \citep{brewer2002behavioral}, which means that (i) at any point in time, trade takes place between the active buyer with the highest valuation and the active seller with the lowest cost, and (ii) trade occurs if and only if it is mutually beneficial (we provide a formal definition in Section \ref{The Marshallian Path}). We prove that in standard double auction formats, a Marshallian path implies that final trades must take place at equilibrium prices; and we discuss why this result might also extend to non-final trades. Importantly, this result does \textit{not} extend to double auctions with fixed value distributions, which can help explain why such auctions lack the standard equilibrating tendencies. Therefore, we identify the Marshallian path dynamic as a key driver of equilibration in standard double auctions, thus helping to solve the `scientific mystery' introduced by Vernon Smith 60 years ago.
The remainder of this article is structured as follows. Section \ref{CE preserving shifts} generalises and formalises the idea of competitive equilibrium preserving shifts. Section \ref{Experimental Design} outlines the design of experiments aimed at testing the impacts of such shifts; Section \ref{Results} contains the results of such experiments; and Section \ref{Understanding the monotonicity} discusses which models can rationalise our findings. Section \ref{The Marshallian Path} uses a combination of theory and further experimentation to study whether our findings change once value distributions are no longer held stationary. Finally, Section \ref{Concluding remarks} concludes with an outline of some new areas of research opened up by this work.
\section{CE preserving shifts}\label{CE preserving shifts}
In this section, we generalise and formalise the example from the introduction in order to obtain a better understanding of its structure. To this end, let us consider a unit mass of buyers, each indexed by $i \in [0, 1]$.\footnote{While we work with a continuum of buyers and sellers for convenience, similar results are available in the discrete case.} Each buyer has a valuation $v_i \in [0, \bar{v}_b]$ (where $\bar{v}_b > 0$ is the maximum buyer valuation) and chooses to buy (exactly) one unit of the good if and only if their valuation is at least the market price $p$. The distribution of buyer valuations is described by the cumulative distribution function $F \colon \mathbb{R} \rightarrow [0, 1]$. For simplicity, we assume that (i) $F$ has full support on the interval $[0, \bar{v}_b]$ (ii) $F$ is continuous.
Let $d(p)$ denote market demand at price $p$. Then
\begin{equation}
d(p) = \int_0^1 \mathbbm{1}(v_i \geq p) di = \mathbb{P}(v_i \geq p) = 1 - F(p)
\end{equation}
where $\mathbbm{1}(v_i \geq p)$ is an indicator function. Since, $d(p) = 1 - F(p)$, $d(0) = 1 - F(0) = 1$ and $d(\bar{v}_b) = 1 - F(\bar{v}_b) = 0$. That is, demand starts at $1$ (at a price of $0$) and eventually falls to $0$ (at a price of $\bar{v}_b$). In addition, observe that $d$ is continuous (since it inherits the continuity of $F$) and strictly decreasing over the interval $[0, \bar{v}_b]$ (since $F$ has full support on this interval). In summary, then, we obtain a continuous and strictly decreasing demand function which starts at 1 before falling to 0 when the price equals the maximum valuation.
We treat sellers entirely symmetrically. That is, we have a unit mass of sellers, indexed by $i\in [0, 1]$; and each seller has a valuation (or `cost') $v_i \in [0, \bar{v}_s]$ (where $v_s > 0$ is the maximum seller valuation). Seller $i$ chooses to sell their one unit of the good if and only if $v_i \leq p$. The distribution of seller valuations is described by the cumulative distribution function $G \colon \mathbb{R} \rightarrow [0, 1]$. As before, we assume that (i) $G$ has full support on the interval $[0, \bar{v}_s]$ (ii) $G$ is continuous.
Let $s(p)$ denote market supply at price $p$. Then
\begin{equation}
s(p) = \int_0^1 \mathbbm{1}(v_i \leq p) di = \mathbb{P}(v_i \leq p) = G(p)
\end{equation}
where $\mathbbm{1}(v_i \leq p)$ is again an indicator function. Since $s(p) = G(p)$, observe that $s(0) = G(0) = 0$ and $s(\bar{v}_s) = G(\bar{v}_s) = 1$. Moreover, $s$ is continuous (since $G$ is continuous) and strictly increasing over the interval $[0, \bar{v}_s]$ (since $G$ has full support on that interval).
We define a \textit{competitive equilibrium price} as a price $p^* \in \mathbb{R^+}$ such that $d(p^*) = s(p^*)$. According to competitive price theory, this is the price which will prevail in a market; and the associated quantity traded will be $d(p^*) = s(p^*)$. As an aside, we notice that, although competitive price theory gives us a clear prediction as to the price which should prevail in a market, it does not provide us with an explanation as to \textit{why} such a price should arise. Whether such an explanation can be provided is itself an interesting question.\footnote{\cite{aumann1964markets} and \cite{cripps2006efficiency} are particularly influential attempts to provide a foundation to competitive equilibrium theory. The result we prove in Section \ref{The Marshallian Path} provides a rather different (and much simpler) foundation, albeit one that is most directly applicable to final period trades.}
As our first observation, let us note that, under our assumptions, there is exactly one competitive equilibrium price $p^*$. To see this, define excess demand by $e(p) = d(p) -s(p)$ and note that excess demand is positive at a price of zero (specifically, $e(0) = d(0) - s(0) = 1 - 0 = 1$). Meanwhile, $e(\bar{v}_b) = d(\bar{v}_b) - s(\bar{v}_b) = 0 - G(\bar{v}_b)$, so excess demand becomes negative at a price of $v_b$. Given that excess demand inherits the continuity of $d$ and $s$, this means that there must be some $p^*$ such that $e(p^*) = 0$, i.e there exists an equilibrium price. Moreover, since $e$ is strictly decreasing, we see that $p^*$ is unique.
We now define the central concept of this section.
\begin{definition}
A \textit{competitive equilibrium preserving demand contraction} is a transformation $T_b \colon [0, \bar{v}_b] \rightarrow \mathbb{R}^+$ such that
\begin{enumerate}
\item $T(V_b) \leq V_b$ for all $V_b \leq p^* - \epsilon^-$
\item $T(V_b) = V_b$ for all $V_b \in (p^* - \epsilon^-, p^* + \epsilon^+)$
\item $T(V_b) \in [p^* + \epsilon^+, V_b]$ for all $V_b \geq p^* + \epsilon^+$
\end{enumerate}
for some $\epsilon^+, \epsilon^- > 0$.
\end{definition}
As stated above, competitive equilibrium preserving demand contractions are downward shifts to the distribution of buyer valuations that satisfy three conditions. First, we require that low valuations (specifically those below $p^* - \epsilon^-$) are weakly decreased. Second, we require that there is some (possibly asymmetric) $\epsilon$-ball around $p^*$ at which valuations remain unchanged. Finally, we require that high valuations (those above $p^* + \epsilon^+$) are reduced, but not reduced so much that they are brought below $p^* + \epsilon^+$. Altogether, this amounts to a stochastic reduction to the distribution of buyer valuations, although one that leaves the ranking (i.e. percentile) of valuations in a neighbourhood of $p^*$ unchanged.
We define a competitive equilibrium preserving decrease in the seller valuations analogously.
\begin{definition}
A \textit{competitive equilibrium preserving supply expansion} is a transformation $T_s \colon [0, \bar{v}_s] \rightarrow \mathbb{R}^+$ such that
\begin{enumerate}
\item $T(V_s) \leq V_s$ for all $V_s \leq p^* - \epsilon^-$
\item $T(V_s) = V_s$ for all $V_s \in (p^* - \epsilon^-, p^* + \epsilon^+)$
\item $T(V_s) \in [p^* + \epsilon^+, V_s]$ for all $V_s \geq p^* + \epsilon^+$
\end{enumerate}
for some $\epsilon^+, \epsilon^- > 0$.
\end{definition}
\begin{figure}
\caption{A CE preserving shift}
\label{shifts}
\label{fig2}
\end{figure}
Figure \ref{shifts} plots a competitive equilibrium preserving demand contraction and supply expansion. As can be seen, we have decreased the buyer and seller valuations when the values were previously low. This corresponds to a downward shift in the demand and supply functions in the leftward portion of the diagram. In addition, we have left the buyer and seller valuations around the equilibrium price unchanged, which means that the demand and supply schedules remain unchanged in a neighbourhood of $p^*$. Finally, we have decreased valuations that were initially very high, corresponding to a downward shift in supply and demand in the rightward portion of Figure \ref{shifts}. As can be seen from the diagram, the unique equilibrium price remains at $p^*$ despite these downward shifts.\footnote{As will be clear from Figure \ref{shifts}, we hold valuations fixed within an $\epsilon$-ball of $p^*$ purely to preserve the uniqueness of the competitive equilibrium price. In fact, only one half of the ball is required for this purpose, although we retain the full ball for simplicity of exposition.} We now verify that this result holds in general.\footnote{All proofs are collected in Appendix \ref{proofs}.}
\begin{proposition}\label{prop1}
Let $p^*$ denote the competitive equilibrium price when buyer and seller valuations are distributed according to $V_b$ and $V_s$ respectively. Then if $T_b$ (respectively, $T_s$) is a competitive equilibrium preserving demand contraction (supply expansion), $p^*$ remains the unique competitive equilibrium price when buyer and seller valuations are distributed according to $V_b' = T_b(V_b)$ and $V_s' = T_s(V_s)$.
\end{proposition}
We have thus seen that there is a wide class of downward shifts to the buyer and seller valuations that leave competitive equilibrium predictions unaffected.\footnote{Unsurprisingly, one can also define an analogous class of \textit{upward} shifts to the buyer and seller valuations that leave equilibria unaffected. More generally, any shift that preserves the ranking of valuations in a neighbourhood of $p^*$ unaffected will preserve the unique equilibrium price $p^*$ (although we focus on everywhere upward or everywhere downward shifts since these generate the most counterintuitive implications).} Intuitively, one might expect such shifts to depress prices, either because they encourage buyers to offer lower prices or accept lower offers from sellers, or otherwise because they encourage sellers to offer lower prices or accept lower prices from buyers. That such shifts should leave the predictions of competitive equilibrium unchanged may thus come as a surprise. Whether these counterintuitive predictions are borne out by the data is a topic that we take up in the next section.
\section{Experimental design}\label{Experimental Design}
In order to examine the effect of competitive equilibrium preserving shifts, we ran a series of double auction experiments in Oxford in early June 2022.\footnote{The experiments received IRB Approval from the University of Oxford (ECONCIA21-22-44) and were pre-registered on the AEA registry: \url{https://www.socialscienceregistry.org/trials/9547}} The basic idea of the experiment was straightforward. Buyers and sellers were first endowed with their own (private) values and costs using the technique of induced valuations (see \cite{smith1976experimental} for discussion and elaboration). They then participated in a series of double auctions. In such auctions, buyers may, at any point in time, make `bids' to purchase at a particular price or accept offers that have been made by sellers. Similarly, at any point in time, sellers may offer to sell at a particular price (`making an ask') or accept a bid made by a buyer.
We opted to conduct a series of oral double auctions, which means that subjects made bids/asks verbally instead of submitting them electronically. While this resulted in somewhat slower data collection than one would have obtained from a computerised experiment, it yielded several important advantages. First, based on some pilot experiments, it seemed that subjects found oral double auctions more engaging, and also found the structure of oral double auctions rather easier to understand. Second, using oral double auctions ensured that our results were maximally comparable to classic studies like \cite{smith1962experimental}, \cite{smith1965experimental} and so forth. For this reason, we kept as close as possible to classical experimental economics protocols (see \cite{plott_document} for a helpful document outlining how such experiments were run and \cite{kimbrough2018testing} for a more recent experiment that adheres closely to such protocols).
In this initial set of experiments, we employed a queue in order to keep supply and demand schedules stable over time. This meant that every time a trade was executed, a new buyer and seller entered the market with the valuation and cost of the just departed buyer and seller. As discussed in the introduction, we did this for two reasons. First, such a design is arguably more realistic: actual markets do not typically dissolve after several trades have occurred (as in standard double auction experiments). Instead, they are continually replenished by a steady flow of new buyers and sellers. Second and much more importantly, our queue ensured that the set of competitive equilibria remained fixed over time, allowing us to rigorously study whether prices approached \textit{the} competitive equilibrium set. Standard experiments do not necessarily have this property.
To implement the queue experimentally, we recruited a group of four buyers and four sellers for every session who began each trading period as `inactive' (so they could not engage in market activity). As trade progressed, buyers and sellers were successively drawn from the queue into the main trading area, and explicitly adopted the value and cost of the buyer and seller who had just departed (by sitting down in their place and inspecting the back of the value/cost card that had been left on the table). This was done in full view of the other experimental participants so as to emphasise that the distribution of the values and costs had remained unchanged.
To study the impact of competitive equilibrium preserving shifts, we used two different treatments, each with five `active' buyers and five `active' sellers (in addition to the eight initially inactive traders in the queue). In the `symmetric' treatment, buyer valuations were £12, £32, £52, £72, £92; and seller valuations were £8, £28, £48, £68, £88. One can check that this yields an interval of competitive equilibrium prices from £48 to £52.\footnote{The competitive equilibrium prices of £48 and £52 are only \textit{weak} competitive equilibria: at such prices, there exists at least one trader who lacks any strict incentive to act in a way that clears the market. However, prices of £49, £50 and £51 are strict equilibria.} Our second treatment (the `low values' treatment) was obtained by decreasing the valuations and costs in the symmetric treatment as aggressively as possible in a way that preserves the set of competitive equilibrium prices. To this end, we changed the buyer valuations of £12 and £32 to £0 and reduced the buyer valuations of £72 and £92 to £52. Meanwhile, we reduced the seller valuations of £8 and £28 to £0 and reduced the seller valuations of £68 and £58 to £52. This yielded the new vector of buyer and seller valuations, namely £0, £0, £52, £52, £52 for the buyers and £0, £0, £48, £52, £52 for the sellers. One can verify directly that these distributions yield the very same set of equilibrium prices, namely £48 -- £52.\footnote{Since the low values treatment yields most of the surplus to sellers at the competitive equilibrium, it is reminiscent of the extreme earnings inequality design that has been studied in the literature (see, for example, \cite{smith2000boundaries}). However, it differs in at least two important respects. First, our design has an equal number of buyers and sellers: in contrast, the classic design generates extreme earnings inequality through an imbalance in the number of buyers and sellers. Second, our design generates an interval of strict competitive equilibrium prices (namely, £49, £50 and £51). In contrast, the standard extreme earnings design leads to the non-existence of strict competitive equilibria (the weak competitive equilibrium is computed by finding the price at which one side of the market is indifferent between trading and not trading).}
To control for subject fixed effects, we conducted both treatments sequentially within every experimental session. To get a handle on order effects, we conducted two sessions (Sessions 1 and 2) and varied the order of the treatments within these sessions. We ran the symmetric treatment first in Session 1; and ran the low values treatment first in Session 2 (see Table \ref{overview} for an overview of all experimental sessions).
At the start of each session, the auction rules were presented in written form (see Appendix \ref{instructions} for the rules with which subjects were presented). The rules were then further emphasised through an extensive oral quiz; and subjects were asked if they had any outstanding questions about the auction rules. Finally, subjects were asked to engage in a mock round of trading (which would not be used to calculate payoffs) for didactic purposes. As a result of these measures, subjects' understanding of the rules seemed to be excellent. As one indication of this, 99.5\% of bids and asks made in our experiments were `individually rational' in the sense that they would have made a (non-negative) profit if accepted. This compares favourably to existing auction datasets: for example, the dataset used by \cite{lin2020evidence} involves individual rationality violations in 90\% of rounds (see \cite{ledyard_unpublished} for discussion).\footnote{In our main analysis, we drop the handful of bids and asks that violate individual rationality from our dataset. However, our results are almost entirely unaffected if we include such data points.}
Once the illustrative trading round had concluded, the real trading began. Within each round of trading, active buyers and sellers were free to make or accept offers (`bids' or `asks') at any time. All offers were repeated by the auctioneer and recorded on a whiteboard.\footnote{To make or accept an offer, a buyer would say (for example) `Buyer 2 bids 30' or `Buyer 2 accepts 60'; and this would be duly repeated by the auctioneer. An analogous comment applies to sellers. All offers and acceptances were recorded by a research assistant and double-checked using an audio recording of the experimental sessions.} Trading used the standard improvement rule, which meant that bids needed to get successively higher and asks needed to get successively lower until a transaction occurred (at which point everything was reset and all bids and asks became permissible). Each round continued until the queue had been exhausted (i.e. until four trades had occurred); and we conducted five rounds of trading for each of the two sets of value distributions. In line with recent recommendations \citep{charness2016experimental, azrieli2018incentives} and double auction experiments \citep{ikica2018competitive}, subjects were only paid for one randomly chosen round within a session.\footnote{On average, subjects received £18.36, with a mean absolute deviation of £12.09. In the experiments reported in Section \ref{The Marshallian Path}, average earnings were £16.98 with a mean absolute deviation of £11.35. In general, sessions took around 1.5 hours, about half an hour of which was devoted to carefully explaining the rules to subjects, with the remaining time devoted to trading.}
\section{Results}\label{Results}
We begin by examining the transactions that occurred in our first two experimental sessions. Panel A of Figure \ref{sessions_1_2} displays the buyer valuation (top line), price (middle line) and cost (bottom line) associated with each of the transactions in Session 1. The band of equilibrium prices (£48--£52) is indicated by the dotted lines, and the end of each of the five rounds is indicated by a break. The left half of Panel A shows the transactions from the first half of the experiment (i.e., the symmetric treatment); whereas the right half of Panel A shows the transactions from the second half of the experiment (i.e. the low values treatment). Analogously, Panel B of Figure \ref{sessions_1_2} displays the valuations, prices and costs associated with the transactions from Session 2, which began with the low values treatment and proceeded to the symmetric treatment.
\begin{figure}
\caption{Valuations, prices, and costs}
\label{sessions_1_2}
\end{figure}
Three results are apparent. First, it is clear that shifting valuations and costs downward lowers observed prices, in violation of competitive equilibrium. Comparing the first halves of the separate sessions --- which is perhaps the cleanest comparison since it is uncomplicated by order effects --- we see that average prices are £56.0 (in the symmetric treatment) as opposed to £32.3 (in the low value treatment) ($p < 0.0001$).\footnote{The $p$-values in this and the next paragraph are generated by unpaired $t$-tests of the hypothesis of equal means.} We also see a similar trend within sessions. In the first session, shifting values and costs downward reduces average bids from £56.0 to £42.9 ($p < 0.0001$). In the second session, shifting values and costs upward increased average bids from £32.3 to £40.0 ($p < 0.01$). Therefore, we obtain strong evidence that competitive equilibrium preserving shifts do in fact shift prices, though the effects are substantially larger when comparing across sessions than when comparing within sessions (as one might expect given the `price stickiness' observed in the data).
While Figure \ref{sessions_1_2} solely displays data for the transactions, we can also see a similar pattern when examining the data on bids and asks. Comparing the first halves of the two sessions, we see that average bids/asks are £47.0/£67.9 in the symmetric treatment, as opposed to £26.6/£58.9 in the low values treatment ($p < 0.0001$, $p = 0.03$).\footnote{In all our analyses of the bid and ask data, we remove the handful of (rather optimistic) asks that exceed £1,000 (e.g. we drop one participant's offer to sell for £1 million). Including these asks only strengthens our conclusions.} In the first session, shifting values downward reduces average bids from £47.0 to £38.0 ($p < 0.01$) and reduces average asks from £67.9 to £48.5 ($p < 0.0001$). In the second session, shifting values up increases average bids from £26.6 to £31.7 ($p = 0.03$) and increases average asks from £59.0 to £153.2 ($p = 0.02$). We therefore conclude that, just as shifting valuations and costs downward reduces observed transaction prices, it also tends to reduce the bids and asks made by traders.
Our second main finding, which partially although not entirely follows from the first, is that prices are almost never at competitive equilibrium. In the first session (again, see Figure \ref{sessions_1_2}), prices start consistently above equilibrium, and then fall persistently below it. Moreover, not merely does competitive equilibrium fail as a literal description of the observed prices, but we can also reject a stochastic version of competitive equilibrium that allows for independent errors in every period ($p < 0.0001$, $p < 0.0001$).\footnote{To test this, we evaluate the null hypothesis that the price data were i.i.d. draws from a normal distribution with a mean of the closest competitive equilibrium price and a variance to be estimated from the data. Observe that both of these choices --- choosing the closest competitive equilibrium price along with allowing the variance to be fit ex post --- substantially stacks the deck in competitive equilibrium's favour, which then makes the clear rejection of competitive equilibrium even more striking.} The prices in our second session are also almost never at equilibrium, but in a different way. Now, prices are persistently below even the lowest competitive equilibrium price. Again, not only can we reject a rather literal interpretation of competitive equilibrium, but we can also reject a stochastic version that allows for independent errors ($p < 0.0001$, $p < 0.01$).
Our third main result is that prices do not seem to be converging to competitive equilibrium over time. In Session 1, there is little indication that prices are trending downwards (in the first half) or upwards (in the second half). Indeed, the average price changes are close to zero (£0.17 and -£0.14 in the first and second half respectively) and neither are statistically different from zero ($p = 0.84$, $p = 0.89$). In Session 2, there is a little more indication of an upward drift in prices, but again this is very weak: average changes are again close to zero (£0.83, £0.24) and again statistically insignificant ($p = 0.61$, $p = 0.94$). Even more strikingly, in neither session does competitive equilibrium seem to be an absorbing state. For example, although the price starts at equilibrium in Session 1 (with a first transaction price of £50), prices quickly drift upwards away from the equilibrium set. Similarly, although prices hit competitive equilibrium briefly in the third round of the second half of Session 2, they again move away from it. Thus, there is little appearance of convergence in about an hour's worth of trading.
In summary, we see that in double auctions with stationary value distributions (a property ensured through our use of a queue), resulting prices can remain far from competitive equilibrium even after long periods of trading, and show little sign of converging to equilibrium. Consistent with this, our competitive equilibrium preserving shifts substantially depress observed prices. In the next section, we turn to the question of what might explain this phenomenon.
\section{Understanding the monotonicity}\label{Understanding the monotonicity}
There are a number of double auction models which can rationalise the monotonicity documented in the previous section (i.e., that lower values/costs generate lower prices). To start with, consider the zero intelligence (ZI) model introduced by \cite{gode1993allocative}: buyers bid uniformly between 0 and their valuation, sellers bid uniformly between their valuation and some maximum, and trade occurs when the market bid and market ask cross (at a price equal to the earlier of the two offers). Under such assumptions, decreasing buyer and seller valuations leads to buyer bid distributions and seller ask distributions that are stochastically lower (in the sense of first-order stochastic dominance). As a result, it will tend to decrease observed prices.
To see how this works quantitatively, we conducted an extensive simulation of ZI trading under both of our experimental treatments. To operationalise ZI trading, one needs to specify a maximum ask; and we set this maximum at £100. We also assumed that at every point in time, one trader (either a buyer or seller) is chosen randomly to make an offer; and we then simulated a sequence of 10 million such offers (leading to about 800,000 market prices).\footnote{If the value distributions are held fixed as in our experiment, then nothing changes under ZI trading at the conclusion of a round. Thus, it is easiest to simply simulate a very large number of offers (and examine the resulting prices when these offers lead to trade) instead of simulating a large number of rounds.} The simulation reveals that, as one would expect, average ZI prices in the symmetric treatment are around £50. Meanwhile, average ZI prices in the low values treatment are around £35. Thus, the ZI model predicts that shifting values and costs down in the way done in our experiments should very substantially depress average prices.
While ZI can rationalise the monotonicity we observe in our data, it is doubtful that postulating random bids and asks can be said to explain the source of the monotonicity in any meaningful way. Fortunately, however, such monotonicity is also generated by optimising models. For instance, consider the model developed in \cite{gjerstad1998price}: buyers and sellers choose bids and asks in a way that maximises this period's expected pay-off. Ignoring integer constraints, the optimal bid/ask satisfies a first order condition, inspection of which reveals that optimal bids/asks are increasing in valuation/costs. Thus, the Gjerstad/Dickhaut model also predicts the monotonicity observed in our data.
Finally, we observe that this monotonicity also arises in more complicated optimising models. For example, consider the model of \cite{friedman1991simple}, in which buyers and sellers optimally choose reservation prices so as to balance the benefit of waiting for better bids/offers against the costs of running out of time. As Friedman remarks (p. 60), optimal reservation prices are monotone in valuations: this means, for example, that buyers with lower valuations are happy to accept lower offers. As a result, Friedman's optimal stopping logic also predicts the monotonicity that we observe.
In this section, our goal is not to select one model which is the `true' explanation for the observed monotonicity; and still less to discuss which model can best explain all aspects of double auction experiments (for efforts in this direction, see \cite{cason1996price} and \cite{ledyard_unpublished}). Rather, our goal is simply to argue that the observed monotonicity is nothing very mysterious: indeed, it is a simple consequence of both non-optimising as well as optimising double auction models.
\section{The Marshallian path}\label{The Marshallian Path}
Although the previous results demonstrate that double auctions need not generate equilibrium prices, one might suspect that this has something to do with the queuing procedure used in order to ensure that the value distribution remains fixed. Indeed, if traders are allowed to drop out without replacement as time progresses, then one may expect that prices approach competitive equilibrium due to a Marshallian path dynamic. While this route to equilibration has been discussed informally (see e.g. \cite{brewer2002behavioral}), we now formalise the dynamic in order to obtain a more rigorous understanding of its properties.
To this end, return to the environment in Section \ref{CE preserving shifts}, recalling that $F$ and $G$ denote the distribution of buyer and seller valuations respectively, and that $p^*$ denotes the unique competitive equilibrium price. Consider now a sequence of trades, indexed by $t \in [0, T]$. Let $v_b(t)$, $v_s(t)$ and $p(t)$ denote the buyer valuation, seller valuation, and price associated with trade $t$; and (with some abuse of notation) denote the corresponding functions by $v_b$, $v_s$ and $p$. We can now formalise the concept of a Marshallian path.
\begin{definition}
A \textit{Marshallian path} is a triple $(v_b, v_s, p)$ such that
\begin{enumerate}
\item For all $t \in [0, T]$, $v_b(t) = F^{-1}(1-t)$ and $v_s(t) = G^{-1}(t)$.
\item For all $i \in [0, 1]$, $i \in [0, T]$ if and only if $F^{-1}(1-i) \geq G^{-1}(i)$.
\item For all $t \in [0, T]$, $v_s(t) \leq p(t) \leq v_b(t)$.
\end{enumerate}
\end{definition}
As stated above, a Marshallian path is a sequence of trades that satisfies three conditions. To understand the first condition, start with the simpler equation $v_s(t) = G^{-1}(t)$ and invert it to get $ t = G(v_s(t))$. This says that the fraction of values that are below $v_s(t)$ is $t$; so at time $t$, the valuation of the seller about to engage in trade is at the $t$-th percentile of the seller value distribution.\footnote{Our usage of the word `percentile’ differs from the normal usage by a factor of 100: for example, we say `0.2-th percentile’ to mean the 20th percentile.} Likewise, the other equation reads $v_b(t) = F^{-1}(1-t)$, which may be inverted to yield $t = 1 - F[v_b(t)]$. This says that the fraction of buyer valuations that are \textit{above} $v_b(t)$ is $t$; so at time $t$, the valuation of the buyer about to engage in trade is at the $(1-t)$-th percentile of the buyer value distribution. Taken together, these assumptions say that trade takes place `in order’: at every point in time, trade occurs between the buyer with the highest valuation and the seller with the lowest valuation out of those `active' traders who remain in the market.
The second condition says that for any possible unit, that unit is traded if and only if the buyer's valuation for that unit exceeds the seller's valuation for that unit (assuming that these units are traded `in order’, in line with condition 1). In other words, trades take place if and only if there is some price at which they would be mutually beneficial. We will discuss the plausibility of this claim when we analyse the experimental results of this section.
The final condition says that, in actual fact, trade is mutually beneficial: the price of every trade lies between the valuation of the relevant buyer and the valuation (i.e. `cost') of the relevant seller. Observe that, although there is just one path of trades satisfying conditions 1 and 2, there are multiple mutually beneficial price paths. Thus, as far as prices are concerned, the Marshallian path is indeterminate; which is why we may, for reasons of pedantry, choose to speak of `a' rather than `the' Marshallian path.\footnote{On the other hand, there is a single path of \textit{trades} which counts as Marshallian, which is presumably why one finds discussion of ‘the’ Marshallian path in prior literature \citep{brewer2002behavioral, plott2013marshall}.}
\begin{proposition}\label{prop2}
If $(v_b, v_s, p)$ is a Marshallian path, then $p(T) = p^*$.
\end{proposition}
Proposition \ref{prop2} says that, if trade follows a Marshallian path, then the final trade must be transacted at the equilibrium price. To see why this is true, consider Figure \ref{fig:prop2}. Recall that the demand curve can be interpreted as the valuation of the marginal buyer (once buyers have been sorted from highest valuation to lowest), and the supply curve can be interpreted as the valuation of the marginal seller (assuming sellers have been sorted in order from lowest to highest valuation). As a result, to assume a Marshallian path is simply to assume that trade takes place in the rightward direction, starting with the buyer with the highest valuation and the seller with the lowest cost (condition 1). At every point in time, the price must lie between the buyer and seller valuation (condition 3), which means that the price path must lie between the demand and supply functions. Furthermore, trade occurs if and only if it could be mutually beneficial (condition 2), which means that the final trade is at the intersection of the demand and supply curves. At such a trade, the only mutually beneficial price is $p(T) =p^*$, so we obtain final period equilibration.\footnote{Incidentally, it is unclear whether this argument for equilibration actually appears anywhere in the work of Marshall. \cite{marshall1890principles} does present a theory of equilibration (see Book V, Chapter 3, Section 6), but this theory involves a quantity adjustment at disequilibrium prices. At the risk of giving Marshall credit for yet another idea that he did not originate (see \cite{humphrey1996marshallian}), we retain the phrase `Marshallian path' in deference to previous literature on this issue \citep{brewer2002behavioral, plott2013marshall}.}
\begin{figure}
\caption{A Marshallian path}
\label{fig:prop2}
\end{figure}
The previous result establishes that if trade follows a Marshallian path, then final period prices must be at competitive equilibrium. However, it does not address why trade should take place in exactly the order required for a Marshallian path (condition 1). While we will not address this issue in detail, there are two reasons why it might seem reasonable to make this assumption. First, prior literature shows that at least as far as experimental double auctions go, condition 1 is indeed a rough approximation of reality (for instance, see \cite{plott2013marshall}). Indeed, as we later show, this assumption matches up very well with our own experimental data. Second, condition 1 should arise for precisely the reasons discussed in Section \ref{Understanding the monotonicity}: it is, after all, simply an assertion that offers are monotone in valuations.
We should perhaps also emphasise that this result does \textit{not} extend to settings with stationary value distributions (like those studied experimentally in Section \ref{Results}). In such settings, Marshallian path logic would suggest that trade should take place between the buyer with the highest value (here, £52) and the seller with the lowest cost (here, £0): and indeed this is precisely what we observe in the low values treatment.\footnote{This phenomenon is also observed, although a little less strikingly, in the symmetric treatments.} However, as far as the logic of Proposition \ref{prop2} goes, there is no reason why they should trade at equilibrium prices if they are continually replaced once they leave the market. Moreover, under anything like an equal division of the surplus, we would expect prices to be considerably below equilibrium (as documented in Section \ref{Results}). Thus, not only does this result provide us with a reason to expect final period equilibration, but it provides us with a reason that is conspicuously absent in the case of stationary value distributions.
Motivated by our result, we now report the results of a series of additional experimental sessions held in order to determine whether the Marshallian path is indeed crucial for equilibrium convergence. These experiments followed exactly the same procedure as those outlined in Section \ref{Experimental Design}, with the exception that we removed the queue of buyers and sellers (so traders could drop out without replacement as time progressed). As in classical double auction experiments, each round now concluded once there were no longer any active traders who wanted to make or accept an offer.\footnote{Similarly to \cite{kimbrough2018testing}, the auctioneer asked: `Sellers, would any of you like to make any offers or accept the current market bid? Buyers, would any of you like to make any bids or accept the current market ask? Going once… going twice… the round has concluded'. See Appendix \ref{instructions} for the full instructions for the experiment.}
\begin{figure}
\caption{Valuations, prices, and costs}
\label{sessions_3_4}
\end{figure}
As before, we ran two sessions (Sessions 3 and 4), varying the order in which the sessions were presented. Session 3 began with the symmetric treatment, whereas Session 4 began with the low values treatment. Panels A and B of Figure \ref{sessions_3_4} display the main results for Session 3 and Session 4 respectively. As usual, the buyer valuation, price, and seller cost associated with each transaction are denoted by the top, middle, and bottom lines respectively. The conclusion of a round is indicated by a break; and the set of equilibrium prices is indicated by the dotted lines.
Two features of the panels are especially noteworthy. First, we still obtain evidence that our shifts depress prices, although the evidence is now more mixed. Comparing across the first halves of each session, which is again the cleanest comparison since it is uncomplicated by order effects, we see that average bids are £46.4 in the symmetric treatment as opposed to £35.3 in the low values treatment ($p < 0.001)$. Thus, shifting the values downward appears to substantially depress prices, at least in the short run. Looking within periods, we see that shifting the values and costs upward increased the average bids in Session 4, from £35.4 to £45.8 ($p < 0.001$). On the other hand, no such effect is observed within Session 3: average prices are actually a little higher in the low value treatment, although the difference is not statistically significant ($p = 0.36$).
Our second finding, which again is related to the first, is that the double auctions now demonstrate strong equilibrating tendencies. In Session 3, prices reach equilibrium by the last trade of the second round of the symmetric treatment and remain within the band of equilibrium prices for the rest of the session (even after valuations are shifted downward). In Session 4, prices failed to equilibrate despite half an hour's trading in the low values treatment. However, they do show a clear upward drift, and we do obtain an equilibrium price for one transaction. Moreover, switching to the symmetric treatment leads prices to more or less equilibrate by the end of the second round.
Given the previous findings, it is natural to conjecture that the low values treatment might equilibrate (even without the help of the symmetric treatment) if it were simply given enough time to do so. To check this conjecture, we ran two further experimental sessions (Sessions 5 and 6) which solely studied the low value treatment.\footnote{These experimental sessions were not contained in our original pre-registration.} To give our auctions the best chance of equilibration, we conducted nine rounds within each session (which corresponded to around one hour of trading time per session). To prevent subjects with unfavourable valuations from becoming bored by their lack of profitable trading opportunities, we re-shuffled the values and costs twice (once at the beginning of the fourth round, and once at the beginning of the seventh). We did not explicitly inform subjects that the distribution of values and costs had remained the same, though we also gave them no indication that it had changed.
\begin{figure}
\caption{Valuations, prices, and costs}
\label{sessions_5_6}
\end{figure}
Panels A and B of Figure \ref{sessions_5_6} display the results of our final two sessions. Several features of the data are evident. First, trade very clearly follows the order assumed by the Marshallian path. We find that trade follows the exact Marshallian order in 17 out of the 18 rounds (the corresponding figure is 47 out of 48 rounds if one includes the low valuation treatments from Sessions 3 and 4). These results are consistent with previous evidence on the order of trade within double auctions (see, e.g., \cite{cason1996price}, \cite{plott2013marshall}, \cite{lin2020evidence} and \cite{sherstyuk2021randomized}).\footnote{While other studies observe a similar phenomenon in terms of trading order, their results on trading order are rather less pronounced. Presumably, this is due to the fact that the two highest valuations are the same, and the two lowest costs are also the same, making it much easier to obtain a Marshallian trading order.}
Second, we observe striking evidence of third period equilibration. As predicted by our theory, \textit{all} third trades occur at a competitive equilibrium price. Moreover, whilst not all rounds lead to a third trade, a good portion of them do. As a result, we obtain final period equilibration in 12 out of 18 of the rounds (this corresponding figure is 16 out of 38 rounds if one includes the low values treatments from Sessions 3 and 4). Consistent with our theory, prices are much less likely to be at equilibrium for non-third period trades. For example, in Session 5, we \textit{never} observe equilibrium prices for first and second period trades.
Third, we also obtain evidence that equilibration is possible even for non-final periods. This is certainly not evident in Session 5, where (as noted) we obtain equilibrium prices only for third period trades. On the other hand, in Session 6, we obtain equilibrium prices for all trades in the fifth, sixth, eighth and ninth rounds. We discuss why this may be below.
Altogether, our results are highly consistent with the theoretical result presented above. In essentially all periods, the first trades happen between buyers with the highest valuations (here £52) and sellers with the lowest cost (here £0): this validates condition 1 of the Marshallian path. Once these two trades have been executed, the only remaining mutually beneficial traders are a buyer with a valuation of £52 and a seller with a cost of £48. Thus, if all mutually beneficial trades are going to occur, a deal must be struck in the equilibrium price range of £48 -- £52. This is exactly what we see in our data (again, see Figure \ref{sessions_5_6}): it is as if a force, namely a large jump in the valuation of the marginal seller, is `squeezing' the realised price into the equilibrium set.
While our results are broadly consistent with our theory, they do leave at least two unanswered questions. The first question is why not all mutually beneficial trades are made within certain rounds. From a mechanical point of view, one can see that the obstacle is essentially on the buyers' side. In all but one of the third period trades, seller offers fall to the equilibrium range, and it is the market bid which potentially fails to rise high enough for a deal to be struck (see Figure \ref{bids_asks} for the evolution of the market bids and asks in such situations). The question is then why third period buyers with a valuation of £52 refuse to trade at equilibrium prices, thereby `leaving money on the table'.
While we do not attempt to definitively answer this question, at least two explanations are suggested by comments made by buyers in post-experimental discussion. First, buyers may have had a strategic motive: by refusing to pay equilibrium prices, they may have been attempting to signal a reluctance to pay such prices in future rounds, thereby encouraging lower future offers from sellers.\footnote{Ironically, this would be a reason why repetition of rounds might actually hinder equilibration: such an effect would not be present in a single round double auction.} Second, buyers may have refused to accept sellers' offers of equilibrium prices on the mistaken belief that such offers would deliver almost all the surplus to the relevant seller. In rounds that generated only two trades, prices tended to be in the £35 -- £40 range, with an average of about £37. As a result, buyers might have reasonably formed the expectation that sellers' values are rather low (about £22 under the assumption of equal surplus sharing) and therefore become quite frustrated when the remaining seller started to demand competitive equilibrium prices. In such a situation, a buyer might be willing to forfeit a few pounds in profit in return for the pleasure of punishing what they take to be greediness on a seller's part (see \cite{rabin1993incorporating} and \cite{fehr1999theory} for formal models in this vein).
A final question is why prices can potentially become `stuck' at equilibrium, even for non-third period trades. For concreteness, let us consider Session 6 (see panel B of Figure \ref{sessions_5_6}) since this phenomenon does not arise in Session 5. By Marshallian path logic, one would expect all third period trades to be at equilibrium, and indeed this is precisely what we observe. However, there is no special reason, at least so far as the logic of Proposition \ref{prop2} goes, to expect equilibrium prices for earlier trades; and under anything like equal surplus sharing, one would expect the realised prices to be substantially lower. Strikingly, however, we see that once prices have been pushed to equilibrium repeatedly by the force of the Marshallian path, they may persist at equilibrium (or may not, as in Session 5). The question is then what might explain this.
While we do not attempt to answer this question formally, it is not too hard to see how the Marshallian path logic might extend to trades that do not occur in the final period. Suppose that the first two trades occur at reasonably low prices, say £35, and the third period price jumps (by the logic of Proposition \ref{prop2}) to a competitive equilibrium price. Over time, traders should notice this pattern --- and indeed many traders reported noticing this in post-experimental discussions. Given that they expect the price to rise in the final period, buyers should be especially keen to trade earlier than their competitors. This in turn should induce them to submit higher bids and to accept higher asks for early period offers. Similarly, given that the price is expected to jump in the third period, sellers should be keen to trade late, which should induce them to only offer very high asks or accept very high bids in the early periods. Taken together, these forces explain how the final period equilibration delivered by the Marshallian path might eventually drag earlier round prices up to equilibrium levels.\footnote{Importantly, a similar argument can apply even in rounds which lack a third period trade. Such third periods typically involve a substantial increase in buyer bids and seller asks relative to previous periods. As a result, even those periods that do not lead to a trade can facilitate equilibration.}
\section{Concluding remarks}\label{Concluding remarks}
In this paper, we revisit the link between competitive equilibrium and the double auction. We begin by showing that competitive equilibrium generates a set of highly counterintuitive predictions. Specifically, it predicts that prices can remain unchanged following a certain class of downward shifts to buyer and seller valuations. We find that in double auctions with stationary value distributions, these predictions are strongly falsified; and more generally that competitive equilibrium is a poor description of outcomes in such auctions. On the other hand, we also find that the effectiveness of our counterexamples is blunted in double auctions in which traders may drop out (without replacement) as time progresses. Taken together, this pair of findings imply that whether value distributions are held stationary is a crucial determinant of whether prices converge to competitive equilibrium; which in turn suggests that the Marshallian path is a key driver of equilibration in double auctions.
Despite the long history of double auction research, we believe that our findings open up several new avenues for investigation. First, further experimental work on the potential importance of the stationarity of the value distribution is clearly needed. Indeed, we are aware of only one other paper on this issue \citep{brewer2002behavioral}; and that paper reaches the rather contrasting conclusion that double auctions can converge even with stationary value distributions.\footnote{In part, the apparent difference between this paper and ours may be somewhat illusory. Examining Figures 7 -- 9 in \cite{brewer2002behavioral}, one sees that prices are persistently above equilibrium, and then (following the shifts) persistently below equilibrium. While we would not classify this as `convergence to equilibrium', it does count as convergence under the relatively undemanding definition used by \cite{brewer2002behavioral} (see p. 191 for details).} Given the paucity of studies on this issue and the apparently mixed nature of the evidence, it is vital that further experiments are conducted to verify whether the issue of stationarity is indeed as critical as we claim.
On the theoretical front, it may also be useful to explicitly develop models of double auction behaviour in stationary environments. Classical double auction models, e.g. \cite{easley1986theories}, \cite{friedman1991simple} and \cite{gjerstad1998price}, explicitly address the situation in which traders drop out without replacement as time progresses, and it would be interesting to investigate the extent to which their insights carry over to the stationary setting. Indeed, the stationary setting would appear to be rather more tractable from a modelling perspective, since it may be viewed (at least very roughly) as a repeated version of the much simpler $k$-double auction. As a result, the insights of \cite{chatterjee1983bargaining}, \cite{jackson2005existence}, \cite{reny2006toward}, \cite{fudenberg2007existence} and others may prove highly relevant.
Finally, it might be valuable to further study what precisely hinders equilibration in the `low values treatment' investigated in this paper. We have informally sketched two alternatives: that buyers reject equilibrium offers on the (mistaken) belief that they allocate almost all the surplus to sellers, or otherwise that buyers reject these offers strategically as a means of generating higher offers in subsequent rounds. While we have not attempted to learn which of these is the key driver, this would seem straightforward to check experimentally. For example, the second channel is possible only when rounds are repeated, so can be easily ‘turned off’ by conducting double auctions with a single incentivised round.
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\begin{appendices}
\section{Proofs} \label{proofs}
\begin{proof}[Proof of Proposition \ref{prop1}]
We begin by arguing that $p^*$ remains \textit{an} equilibrium price when valuations are distributed according to $T_b(V_b) = V_b'$ and $T_s(V_s) = V_s'$. To this end, define $\mathcal{B} = (p^* - \epsilon^-, p^* + \epsilon^+)$ and fix some $p \in \mathcal{B}$. By the law of total probability,
\begin{equation}\label{totalprob}
\begin{split}
P(V'_b \leq p) &= P(V'_b \leq p|V_b \leq p^* - \epsilon^-)P(V_b \leq p^* - \epsilon^-) + P(V'_b \leq p|V_b \in \mathcal{B} )P(V_b \in \mathcal{B} ) \\ &+ P(V'_b \leq p |V_b \geq p^* + \epsilon^+)P(V_b \geq p^* + \epsilon^+)
\end{split}
\end{equation}
Since $T_b(V_b) = V_b'$, where $T_b$ is a CE preserving demand contraction,
\begin{itemize}
\item If $V_b \leq p^* - \epsilon^-$, then $V'_b \leq V_b \leq p^* - \epsilon^- \leq p$, so $P(V'_b \leq p|V_b \leq p^* - \epsilon^-) = 1$
\item If $V_b \in \mathcal{B}$, then $V_b' = V_b$ and so $P(V'_b \leq p|V_b \in \mathcal{B} ) = P(V_b \leq p|V_b \in \mathcal{B} )$
\item If $V_b \geq p^* + \epsilon^+$, then $V_b' \geq p^* + \epsilon^+ > p$, so $P(V'_b \leq p |V_b \geq p^* + \epsilon^+) = 0$
\end{itemize}
Inserting these equalities into (\ref{totalprob}), we obtain
\begin{equation}
\begin{split}
P(V'_b \leq p) &= P(V_b \leq p^* - \epsilon^-) + P(V_b \leq p|V_b \in \mathcal{B} )P(V_b \in \mathcal{B} ) \\
&= P(V_b \leq p^* - \epsilon^-) + P(V_b \leq p \wedge V_b \in \mathcal{B} ) \\
&= P(V_b \leq p^* - \epsilon^-) +P(p^* - \epsilon^- < V_b \leq p) \\
&= P(V_b \leq p)
\end{split}
\end{equation}
Let $F'$ and $G'$ denote the distributions of $V_b'$ and $V_s'$; let $d'(p)$ and $s'(p)$ denote the demand and supply functions generated by these distributions; and define $e'(p) \equiv d'(p) - s'(p)$.
The argument just given reveals that $F'(p) = F(p)$ for any $p \in \mathcal{B}$. Therefore, $d'(p) = 1 - F'(p) = 1 - F(p) = d(p)$ (for $p \in \mathcal{B}$). In particular, $d'(p^*) = d(p^*)$. Likewise, if we replace $F$ with $G$ in the previous argument, we see that $G'(p) = G(p)$ for any $p \in \mathcal{B}$. Hence, $s'(p) = G'(p) = G(p) = s(p)$ at such prices; and in particular, $s'(p^*) = s(p^*)$. Since $d(p^*) = s(p^*)$ by definition, this means that $d'(p^*) = s'(p^*)$, i.e. $p^*$ remains a CE.
To see that the CE remains unique, fix some $p < p^*$ and consider the following cases:
Case 1: $p \in \mathcal{B}$. Then $e'(p) = e(p)$ (shown above), and $e(p) > e(p^*) = 0$ (since $p < p^*$ and $e$ is strictly decreasing). This means that $e'(p) > 0$, i.e. no such prices clear the market.
Case 2: $p \notin \mathcal{B}$. Since $p < p^*$, this means that $p \leq p^* - \epsilon^-$. Now define $\bar{p} \equiv p^* - 0.5\epsilon^-$. Plainly, $\bar{p} \in \mathcal{B}$ and $p < p^*$, so $e'(\bar{p}) = e(\bar{p}) > 0$. Moreover, since $e'(p) = 1 - F'(p) - G'(p)$ where $F'$ and $G'$ are CDFs, $e'$ is weakly decreasing over $\mathbb{R}^+$. Thus, if $p \leq p^* - \epsilon^- < \bar{p}$, then $e'(p) \geq e'(\bar{p}) = e(\bar{p}) > 0$, so no such prices can clear the market either.
Since these cases exhaust the possibilities, we see that no prices $p < p^*$ can be CE. By an analogous argument, one can also rule out prices $p > p^*$. We conclude that $p^*$ must remain the unique competitive equilibrium price.\end{proof}
\begin{proof}[Proof of Proposition \ref{prop2}]
First, we show that $v_b(T) = v_s(T)$. To see this, recall condition 2 of the Marshallian path definition: for every $i \in [0, 1]$, $i \in [0, T]$ if and only if $F^{-1}(1-i) \geq G^{-1}(i)$. Define $\phi(i) = F^{-1}(1-i) - G^{-1}(i)$, so our inequality is $\phi(i) \geq 0$. Observe that $\phi(0) = F^{-1}(1) - G^{-1}(0) = \bar{v}_b$ and $\phi(1) = F^{-1}(0) - G^{-1}(1) = -\bar{v}_s$. Also, $\phi$ is continuous and strictly decreasing in $i$. Therefore, letting $i^*$ denote the unique root of $\phi$, the set of $i \in [0, 1]$ such that $\phi(i) \geq 0$ is the set $[0, i^*]$. Hence, $T = i^*$ and so $\phi(T) = 0$, or $F^{-1}(1-T) = G^{-1}(T)$. Using condition 1, this finally yields $v_b(T) = v_s(T)$.
By condition 3, we see that $v_b(T) \geq p(T) \geq v_s(T)$. However, $v_b(T) = v_s(T)$. Therefore, $p(T) = v_b(T) = v_s(T)$.
Finally, invert condition 1 to get $F[v_B(T)] = 1-T$ and $G[p(T)] = T$. Using $p(T) = v_b(T) = v_s(T)$, we infer that $F[p(T)] = 1-T$ and $G[p(T)] = T$, and so
\begin{equation}G[p(T)] = 1 - F[p(T)] \end{equation}
But this is precisely the equation whose unique solution is $p^*!$ Hence, $p(T) = p^*$.\end{proof}
\section{Additional tables and figures} \label{tables_figures}
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\begin{table}[H]
\centering
\caption{Overview of the experimental sessions}\label{overview}
\begin{threeparttable}[h]
\begin{tabular}{cclcc}
\hline
Session & Queue? & \hspace{3em}Treatments & Rounds & Participants \\ \hline
1 & Yes & Symmetric then low values & 10 & 18 \\
2 & Yes & Low values then symmetric & 10 & 18 \\
3 & No & Symmetric then low values & 10 & 10 \\
4 & No & Low values then symmetric & 10 & 10 \\
5 & No & Low values only & 9 & 10 \\
6 & No & Low values only & 9 & 10 \\ \hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \hspace{-0.2em}\textit{Notes}: This table describes the differences between our experimental treatments. The second column specifies whether a treatment used a queue of buyers and sellers, and the third column specifies which treatments were run. The fourth and fifth columns specify the number of rounds and participants in the session.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}
\caption{Market bids and asks}
\label{bids_asks}
\end{figure}
\section{Experimental instructions} \label{instructions}
\subsection{Instruction for Buyers (Sessions 1 and 2)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your buyer number. B1 means `Buyer 1', B2 means ‘Buyer 2’, and so forth. Please wear your ID card visibly at all times.
\end{itemize}
\textit{Roles}
\begin{itemize}
\item There are two types of buyers in this experiment: active buyers and pending buyers. If your buyer number is between 1 and 5 inclusive, then you are an active buyer and will be sitting in the main trading area. On the other hand, if your buyer number is 6 or higher, then you are a pending buyer and will be sitting in the queue.
\item Active buyers are free to trade from the very start of a trading period. However, pending buyers may only trade after active buyers have made a trade and dropped out of the market (see elaboration below).
\end{itemize}
\textit{Valuations}
\begin{itemize}
\item If you are an active buyer, you will have a card marked ‘Valuation’ in front of you. Examine the number on the back of this card, being careful not to let any other participants see this number. This number is your ‘valuation’ for the fictitious commodity to be traded. Memorise your valuation, turn your ID card back face-down (so the valuation is hidden), and do not reveal your valuation to anybody else!
\item If you are a pending buyer, then you have not yet been assigned a valuation. However, you will acquire a valuation as soon as an active buyer has dropped out and you have taken that buyer’s place (and valuation).
\item If you manage to make a trade, you will receive your valuation minus the price that you agreed to pay (this assumes that the trade ‘counts’ — see discussion below). For example, if your valuation is £55 and you agree to a price of £20, your net earnings are £35.
\item Notice that your valuation represents the most that you should be conceivably willing to pay to make a trade. For example, if your valuation is £40, then you should never pay more than £40 to buy a unit: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, active buyers may offer to buy at a particular price – this is called making a ‘bid’. To make a bid, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you want to bid. For example, if you are buyer 3 and you want to bid £40, say ‘buyer 3 bids 40’.
\item In the sequence of market activity leading up to a trade, each bid must be higher than the current bid. For example, if one buyer bids £20, then all subsequent buyers need to bid more than £20.
\item All bids need to be whole numbers. For example, while you can bid £30, you cannot bid £30.14.
\item Just as active buyers may make bids (at any point in time), active sellers can make ‘asks’ (at any point in time). For example, if a seller ‘asks’ for £70, that means that she is willing to sell for £70. Each new ask must be lower than the current ask, so asks must ‘improve’ over time.
\item At any stage, active buyers may accept an ask that has been made by a seller. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current ask. For example, if you are buyer 2 and want to accept an ask of £70, say ‘buyer 2 accepts 70’.
\item If multiple buyers want to make a bid or accept an ask, priority will be given to the buyer who has raised their hand first.
\item If you have made a trade, you become inactive and cannot make any further trades in that round. At this point, you should move to the inactive area and allow your place to be taken by the buyer at the front of the queue (or go to the back of the queue if you were previously a pending buyer). That new buyer will acquire your valuation (so should examine the card on the desk to see what that valuation is).
\item While each bid needs to be higher than the previous bid, everything is reset following a trade. In other words, once a trade has been made, active buyers are free to submit any bid that they choose – even if that bid is lower than previously submitted bids.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Buyer 3 wants to bid £40 so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 3 bids 40’.
\item Buyer 1 wants to outbid her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 1 bids 43’.
\item Seller 4 wants to offer to sell for £95, so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 4 asks 95’.
\item Buyer 2 wants to accept the ask, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 2 accepts 95’. A trade has occurred (at a price of £95).
\item Since buyer 2 has just traded with seller 4, they both become inactive and should move to the inactive area. Two traders from the queue take their place and acquire their valuation and cost respectively. Participants then continue to bargain over prices. Since a trade has just occurred, subsequent bids are no longer constrained to be above £43 (the previous leading bid).
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until the auctioneer chooses to end the round.
\item Within each round, you are only allowed to purchase (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new valuation for the fictitious commodity. So even if your current valuation is rather low, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There are two types of buyers: active buyers (IDs 1-5) and pending buyers (IDs 6 or higher). Once active buyers have made a trade, the pending buyer at the front of the queue takes their place and their valuation.
\item Active buyers may make bids or accept asks at any point in time. Similarly, active sellers may make asks or accept bids at any point in time.
\item If you want to make a bid or accept an ask, you need to first raise your hand.
\item Bids must be whole numbers; and new bids must be greater than previous bids (until a trade is made).
\item If you manage to make a trade, you earn your valuation minus the price you agreed to pay (assuming that this trade is selected to ‘count’ for your earnings).
\item You can purchase up to one unit within every round.
\end{itemize}
\subsection{Instructions for Sellers (Sessions 1 and 2)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your seller number. S1 means ‘Seller 1’, S2 means ‘Seller 2’, and so forth. Please wear your ID card visibly at all times.
\end{itemize}
\textit{Roles}
\begin{itemize}
\item There are two types of sellers in this experiment: active sellers and pending sellers. If your seller number is between 1 and 5 inclusive, then you are an active seller and will be sitting in the main trading area. On the other hand, if your seller number is 6 or higher, then you are a pending seller and will be sitting in the queue.
\item Active sellers are free to trade from the very start of a trading period. However, pending sellers may only trade after active sellers have made a trade and dropped out of the market (see elaboration below).
\end{itemize}
\textit{Costs}
\begin{itemize}
\item If you are an active seller, you will have a card marked ‘Cost’ in front of you. Examine the number on the back of this card, being careful not to let any other participants see this number. This number is how much it would cost you to produce and sell a unit of the fictitious commodity to be traded. Memorise your cost, turn your ID card back face-down (so the cost is hidden), and do not reveal your cost to anybody else!
\item If you are a pending seller, then you have not yet been assigned a cost. However, you will acquire a cost as soon as an active seller has dropped out and you have taken that seller’s place (and cost).
\item If you manage to make a trade, you will receive the price paid by the buyer minus your cost (this assumes that the trade ‘counts’ — see discussion below). For example, if you sell for a price of £55 and your cost is £20, your net earnings are £35.
\item Notice that your cost represents the least that you should be conceivably willing to sell for. For example, if your cost is £40, then you should never sell for less than £40: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, active sellers may offer to sell at a particular price – this is called making an ‘ask’. To make an ask, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you are asking for. For example, if you are seller 3 and you want to ask for £60, say ‘seller 3 asks 60’.
\item In the sequence of market activity leading up to a trade, each ask must be lower than the current ask. For example, if one seller asks £80, then all subsequent sellers need to ask for less than £80.
\item All asks need to be whole numbers. For example, while you can ask £70, you cannot ask £70.14.
\item Just as active sellers may make asks (at any point in time), active buyers can make bids (at any point in time). For example, if a buyer bids £70, that means that she is willing to pay £70. Each new bid must be higher than the current bid, so bids must ‘improve’ over time.
\item At any stage, active sellers may accept a bid that has been made by a buyer. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current bid. For example, if you are seller 2 and want to accept a bid of £70, say ‘seller 2 accepts 70’.
\item If multiple sellers want to make an ask or accept a bid, priority will be given to the seller who has raised their hand first.
\item If you have made a trade, you become inactive and cannot make any further trades in that round. At this point, you should move to the inactive area and allow your place to be taken by the seller at the front of the queue (or go to the back of the queue if you were previously a pending seller). That new seller will acquire your cost (so should examine the card on the desk to see what that cost is).
\item While each ask needs to be lower than the previous ask, everything is reset following a trade. In other words, once a trade has been made, you are free to submit any ask that you choose – even if that ask is higher than previously submitted asks.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Seller 3 wants to ask for £60 so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 3 asks 60’.
\item Seller 1 wants to undercut her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 1 asks 57’.
\item Buyer 4 wants to bid £5, so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 4 bids 5’.
\item Seller 2 wants to accept the bid, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 2 accepts 5’. A trade has occurred (at a price of £5).
\item Since seller 2 has just traded with buyer 4, they both become inactive and should move to the inactive area. Two traders from the queue take their place and acquire their valuation and cost respectively. Participants then continue to bargain over prices. Since a trade has just occurred, subsequent asks are no longer constrained to be below £57 (the previous lowest).
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until the auctioneer chooses to end the round.
\item Within each round, you are only allowed to sell (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new cost for the fictitious commodity. So even if your current cost is rather high, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There are two types of sellers: active sellers (IDs 1-5) and pending sellers (IDs 6 or higher). Once active sellers have made a trade, the pending seller at the front of the queue takes their place and their cost.
\item Active sellers may make asks or accept bids at any point in time. Similarly, active buyers may make bids or accept asks at any point in time.
\item If you want to make an ask or to accept a bid, you need to first raise your hand.
\item Asks must be whole numbers; and new asks must be lower than previous asks (until a trade is made).
\item If you manage to make a trade, you earn the price paid by the buyer minus your cost (assuming that this trade is selected to ‘count’ for your earnings).
\item You can sell up to one unit within every round.
\end{itemize}
\subsection{Instructions for Buyers (Sessions 3 -- 6)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your buyer number. B1 means ‘Buyer 1’, B2 means ‘Buyer 2’, and so forth. Please keep your ID card visible at all times.
\end{itemize}
\textit{Valuations}
\begin{itemize}
\item Please examine the number of the back of your ID card, being careful not to let any other participants see this number. This number is your ‘valuation’ for the fictitious commodity to be traded. Memorise your valuation, turn your ID card back face-down (so the valuation is hidden), and do not reveal your valuation to anybody else!
\item If you manage to make a trade, you will receive your valuation minus the price that you agreed to pay (this assumes that the trade ‘counts’ — see discussion below). For example, if your valuation is £55 and you agree to a price of £20, your net earnings are £35.
\item Notice that your valuation represents the most that you should be conceivably willing to pay to make a trade. For example, if your valuation is £40, then you should never pay more than £40 to buy a unit: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, you may offer to buy at a particular price – this is called making a ‘bid’. To make a bid, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you want to bid. For example, if you are buyer 3 and you want to bid £40, say ‘buyer 3 bids 40’.
\item In the sequence of market activity leading up to a trade, each bid must be higher than the current bid. For example, if one buyer bids £20, then all subsequent buyers need to bid more than £20.
\item All bids need to be whole numbers. For example, while you can bid £30, you cannot bid £30.14.
\item Just as buyers may make bids (at any point in time), sellers can make ‘asks’ (at any point in time). For example, if a seller ‘asks’ for £70, that means that she is willing to sell for £70. Each new ask must be lower than the current ask, so asks must ‘improve’ over time.
\item At any stage, you may accept an ask that has been made by a seller. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current ask. For example, if you are buyer 2 and want to accept an ask of £70, say ‘buyer 2 accepts 70’.
\item If multiple buyers want to make a bid or accept an ask, priority will be given to the buyer who has raised their hand first.
\item Once you have made a trade, you become inactive and cannot make any further trades in that round. Likewise, each seller is only able to make at most one trade within a round – so it is as if they possess just one unit of the fictitious commodity.
\item While each bid needs to be higher than the previous bid, everything is reset following a trade. In other words, once a trade has been made, you are free to submit any bid that you choose – even if that bid is lower than previously submitted bids.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Buyer 3 wants to bid £40 so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 3 bids 40’.
\item Buyer 1 wants to outbid her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 1 bids 43’.
\item Seller 4 wants to offer to sell for £95, so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 4 asks 95’.
\item Buyer 2 wants to accept the ask, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 2 accepts 95’. A trade has occurred (at a price of £95).
\item Since buyer 2 has just traded with seller 4, they both become inactive and remain silent for the rest of the round. Meanwhile, other participants continue to bargain over prices. Since a trade has just occurred, subsequent bids are no longer constrained to be above £43 (the previous leading bid).
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until there is a pause in market activity for roughly 20 seconds. At that point, the auctioneer will ask buyers if they would like to make any new bids, or would like to accept the current market ask. The auctioneer will then ask sellers if they would like to make any new asks, or would like to accept the current market bid. If all traders remain silent, then the auctioneer will close the market and conclude the round.
\item Within each round, you are only allowed to purchase (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new valuation for the fictitious commodity. So even if your current valuation is rather low, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There is a group of buyers, and a group of sellers. Buyers may make bids or accept asks at any point in time. Similarly, sellers may make asks or accept bids at any point in time.
\item If you want to make a bid or accept an ask, you need to first raise your hand.
\item Bids must be whole numbers; and new bids must be greater than previous bids (until a trade is made).
\item If you manage to make a trade, you earn your valuation minus the price you agreed to pay (assuming that this trade is selected to ‘count’ for your earnings).
\item You can purchase up to one unit within every round.
\end{itemize}
\subsection{Instructions for Sellers (Sessions 3 -- 6)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your seller number. S1 means ‘Seller 1’, S2 means ‘Seller 2’, and so forth. Please keep your ID card visible at all times.
\end{itemize}
\textit{Costs}
\begin{itemize}
\item Please examine the number on the back of your ID card, being careful not to let any other participants see this number. This number is how much it would cost you to produce and sell one unit of the fictitious commodity to be traded. Memorise your cost, turn your ID card back face-down (so the cost is hidden), and do not reveal your cost to anybody else!
\item If you manage to make a trade, you will receive the price paid by the buyer minus your cost (this assumes that the trade ‘counts’ — see discussion below). For example, if you sell for a price of £55 and your cost is £20, your net earnings are £35.
\item Notice that your cost represents the least that you should be conceivably willing to sell for. For example, if your cost is £40, then you should never sell for less than £40: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, you may offer to sell at a particular price – this is called making an ‘ask’. To make an ask, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you are asking for. For example, if you are seller 3 and you want to ask for £60, say ‘seller 3 asks 60’.
\item In the sequence of market activity leading up to a trade, each ask must be lower than the current ask. For example, if one seller asks £80, then subsequent sellers need to ask for less than £80.
\item All asks need to be whole numbers. For example, while you can ask for £70, you cannot ask for £70.14.
\item Just as sellers may make asks (at any point in time), buyers can make bids (at any point in time). For example, if a buyer bids £70, that means that she is willing to pay £70. Each new bid must be higher than the current bid, so bids must ‘improve’ over time.
\item At any stage, you may accept a bid that has been made by a buyer. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current bid. For example, if you are seller 2 and want to accept a bid of £70, say ‘seller 2 accepts 70’.
\item If multiple sellers want to make an ask or accept a bid, priority will be given to the seller who has raised their hand first.
\item Once you have made a trade, you become inactive and cannot make any further trades in that round --- so it is as if you possess just one unit of the fictitious commodity. Likewise, each buyer is only able to make at most one trade within a round.
\item While each ask needs to be lower than the previous ask, everything is reset following a trade. In other words, once a trade has been made, you are free to submit any ask that you choose – even if that ask is higher than previously submitted asks.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Seller 3 wants to ask for £60 so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 3 asks 60’.
\item Seller 1 wants to undercut her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 1 asks 57’.
\item Buyer 4 wants to bid £5, so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 4 bids 5’.
\item Seller 2 wants to accept the bid, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 2 accepts 5’. A trade has occurred (at a price of £5).
\item Since seller 2 has just traded with buyer 4, they both become inactive and remain silent for the rest of the round. Meanwhile, other participants continue to bargain over prices. Since a trade has just occurred, subsequent asks are no longer constrained to be below £57 (the previous lowest ask).
\item
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until there is a pause in market activity for roughly 20 seconds. At that point, the auctioneer will ask buyers if they would like to make any new bids, or would like to accept the current market ask. The auctioneer will then ask sellers if they would like to make any new asks, or would like to accept the current market bid. If all traders remain silent, then the auctioneer will close the market and conclude the round.
\item Within each round, you are only allowed to sell (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new cost for the fictitious commodity. So even if your current cost is rather high, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There is a group of sellers, and a group of buyers. Sellers may make asks or accept bids at any point in time. Similarly, buyers may make bids or accept asks at any point in time.
\item If you want to make an ask or to accept a bid, you need to first raise your hand.
\item Asks must be whole numbers; and new asks must be lower than previous asks (until a trade is made).
\item If you manage to make a trade, you earn the price paid by the buyer minus your cost (assuming that this trade is selected to ‘count’ for your earnings).
\item You can sell up to one unit within every round.
\end{itemize}
\end{appendices}
\end{document} | math | 98,162 |
\begin{document}
\title{Holomorphicity of slice-regular functions}
\tableofcontents
\section{Introduction}
The theory of slice-regular functions of a quaternionic variable poses itself as a possible generalization of the theory of holomorphic functions of a complex variable; the definition of slice-regular functions was given by Gentili and Struppa in \cites{GS1,GS2}.
Given $q\in{\mathbb H}$, we write $q=x+vy$ with $x,y\in{\mathbb R}$ and $v\in{\mathbb H}$ such that $v^2=-1$, i.e. an imaginary unity; for $\Omega\subset {\mathbb H}$ an axially symmetric domain, we call $f:U\to{\mathbb H}$ a left slice-regular function at $x+vy$ if it is real differentiable and
$$\frac{\partialartial f}{\partialartial x}(x+vy)+v\frac{\partialartial f}{\partialartial y}(x+vy)=0$$
for all imaginary units $v$. Some variations are possible, but all of them amount to saying that the function $f$ satisfies a sort of Cauchy-Riemann equations on each complex plain; such a condition implies a certain amount of symmetry, namely the map $v\mapsto f(x+vy)$ is linear affine.
This means that if we know the value of a slice-regular function $f$ at $x+vy$ for some imaginary unit $v$, then we can calculate it to all the other quaternions of the form $x+wy$, varying the imaginary unit $w$; given ${\mathcal U}\subseteq{\mathbb R}^2$, we define $U\subseteq{\mathbb H}$ as the set of all quaternions $x+vy$ with $(x,y)\in{\mathcal U}$, then any slice regular function $f:U\to{\mathbb H}$ is given by $f(x+vy)=\alpha(x,y)+v\beta(x,y)$, where $\alpha,\beta:{\mathcal U}\to{\mathbb H}$ satisfy the Cauchy-Riemann equations
$$\left\{\begin{array}{l}\partialartial_x\alpha=\partialartial_y\beta\\\partialartial_y\alpha=-\partialartial_x\beta\;.\end{array}\right.$$
In \cite{GP1}, Ghiloni and Perotti built on the ideas of Fueter, Qian, Sce and Sommen \cites{F,Q,S, So} and proposed a different viewpoint, involving an actual holomorphic function. Let ${\mathbb C}$ be the field of complex numbers, with imaginary unit $\iota$, and, for every imaginary unit $v$ of ${\mathbb H}$, consider the map $\rho_v:{\mathbb C}\to{\mathbb H}$ given by $\rho_v(x+\iota y)=x+vy$; this map can be extended to the tensor product ${\mathbb H}\otimes{\mathbb C}$ by setting
$$\rho_v(z\otimes q)=\rho_v(z)q$$
and extending it linearly.
So, given ${\mathcal U}$ and $U$ as above, for every map $F:{\mathcal U}\to{\mathbb H}\otimes{\mathbb C}$ such that $F(\overline{z})=\overline{F(z)}$ there exists a unique map $f:U\to{\mathbb H}$ such that the diagram
$$\xymatrix{{\mathcal U}\ar[r]^{\!\!\!\!\! F}\ar[d]_{\rho_v}& {\mathbb H}\otimes{\mathbb C}\ar[d]^{\rho_v}\\U\ar[r]_{f}&{\mathbb H}}$$
commutes for every imaginary unit $v$. Now, ${\mathbb H}\otimes{\mathbb C}$ can be naturally identified with ${\mathbb C}^4$; then, $f$ is slice-regular if and only if $F:{\mathcal U}\to{\mathbb C}^4$ is holomorphic.
It is reasonable to expect that the holomorphicity of $F$ implies the properties of slice-regular functions that mimic more closely the behaviour of classical holomorphic functions; however, this direction of investigation has not been pursued further.
One reason could be the fact that the map $\rho_v$, which allows us to pass from an element of ${\mathbb C}^4$ to an element of ${\mathbb H}$, somehow ruins the holomorphicity. So, for instance, there is no evident way of relating the zeros of $f$ to the zeros of $F$.
For an account of the theory of slice-regular functions and more generaly hypercomplex analysis, the interested reader can consult a variety of texts, for instance \cites{ACS, CSS1, CSS2, GSS}.
The aim of this work is to show how many results about slice-regular functions follow flawlessly from the properties of holomorphic functions.
The first problem we consider is to investigate the zeros of $f$ in terms of the values of $F$: we show that the set of points $p$ in ${\mathbb C}^4$ such that $\rho_v(p)=0$ for some $v$ is a complex analytic set, fact related to the holomorphicity of $f^s$, the symmetrization of $f$.
We go on, studying the nature of the zeros of slice-regular functions and deducing the analogues of Rouché theorem and argument principle for slice-regular functions; the generalization to meromorphic functions is also straightforward.
The Cauchy formula for $f$ is deduced from the one for $F$; in general, if we have an integral kernel for an operator on holomorphic functions, we can extend it to an integral kernel for slice-regular functions, if some conditions are satisfied.
Finally, we study the relations between the supremum norm and the $L^2$ norm of $f$ and $F$; this allows us to obtain the maximum modulus principle for slice-regular functions and to use the previous observations on integral kernels to define a Bergman kernel for slice-regular functions.
We end the paper with a short section with some comments and examples on the possible extensions to the case of Clifford algebras, which will be the subject of a future paper \cite{Mo}.
\section{General setting}
Let ${\mathbb H}$ denote the algebra of quaternions and ${\mathbb S}$ the $2$-sphere of imaginary units, i.e.
$${\mathbb S}=\{q\in{\mathbb H}\ :\ q^2=-1\}\;.$$
For a fixed $v\in{\mathbb S}$, we denote by ${\mathbb C}_v$ the (real) vector space generated by $1$ and $v$; obviously ${\mathbb C}_v$ is isomorphic to the usual field of complex numbers, through the map taking $x+vy$ to $x+\iota y$. Let $\rho_v:{\mathbb C}\to{\mathbb C}_v$ be the inverse of such map.
We extend the map $\rho_v$ to the tensor product ${\mathbb H}\otimes{\mathbb C}$ by setting $\rho_v(z\otimes q)=\rho_v(z)q$ and imposing linearity; in what follows, we will identify ${\mathbb H}\otimes{\mathbb C}$ and ${\mathbb C}^4$, using the base $\{1,i,j,k\}$ of ${\mathbb H}$.
For $q\in{\mathbb H}$, we write $q=q_0+q_1i+q_2j+q_3k$ with $q_0,\ldots, q_3\in{\mathbb R}$; we denote by $\underline{q}$ the vector part, i.e. $q-q_0$.
Let $\partiali:{\mathbb R}^2\times {\mathbb S}\to{\mathbb H}$ be given by $\partiali(x,y,v)=x+vy$; if $q\in{\mathbb H}$ is not real, then $\partiali^{-1}(q)$ contains $2$ points, whereas, for $x\in{\mathbb R}$, $\partiali^{-1}(x)=\{(x,0)\}\times{\mathbb S}$.
Let ${\mathcal U}\subset{\mathbb R}^2$ be an open domain, invariant with respect to the map $(x,y)\mapsto (x,-y)$ and define $U=\partiali_1({\mathcal U}\times{\mathbb S})$ - the sets $U$ obtained this way are called spherically symmetric. We can identify ${\mathbb R}^2$ with ${\mathbb C}$ and interpret ${\mathcal U}$ as an open domain in ${\mathbb C}$, invariant by conjugation.
\begin{defin}A (left) slice-regular function on $U$ is a map $f:U\to{\mathbb H}$ such that $f\circ \partiali(x,y,v)=\alpha(x,y)+v\beta(x,y)$ where $\alpha,\beta:{\mathcal U}\to{\mathbb H}$ are such that
$$\alpha(x,-y)=\alpha(x,y)\qquad \beta(x,-y)=-\beta(x,y)$$
$$\partialartial_x\alpha=\partialartial_y\beta \qquad \partialartial_y\alpha=-\partialartial_x\beta\;.$$
\end{defin}
\begin{lemma}\label{lemma_rappr} Given a (left) slice-regular function $f:U\to{\mathbb H}$, there exists a unique holomorphic map $F:{\mathcal U}\to{\mathbb C}^4$ such that $F(\bar{z})=\overline{F(z)}$ and $f\circ \partiali(x,y,v)=\rho_v(F(x+\iota y))$.\end{lemma}
\begin{proof} As $f$ is slice-regular, there exist functions $\alpha,\beta:{\mathcal U}\to{\mathbb H}$ such that $f\circ \partiali(x,y,v)=\alpha(x,y)+v\beta(x,y)$.
If we write $\alpha=\alpha_0+i\alpha_1+j\alpha_2+k\alpha_3$ and similarly $\beta=\beta_0+i\beta_1+j\beta_2+k\beta_3$, $\alpha_m, \beta_m:{\mathcal U}\to{\mathbb R}$ are functions with the same symmetries and, in pairs, they respect Cauchy-Riemann equations, so we obtain that the functions $f_m(x+\iota y)=\alpha_m(x,y)+\iota\beta_m(x,y)$ for $m=0,1,2,3$ are holomorphic functions from ${\mathcal U}\subseteq{\mathbb C}$ to ${\mathbb C}$, with the property
$$f_m(\bar{z})=\overline{f_m(z)}\quad \forall z\in {\mathcal U}\;$$
and
$$f\circ\partiali(x,y,v)=\rho_v(f_0(x+\iota y), f_1(x+\iota y), f_2(x+\iota y), f_3(x+\iota y))\;.$$
Thus, defining $F:{\mathcal U}\to{\mathbb C}^4$ by $F=(f_0,f_1,f_2,f_3)$, the conclusion follows.\end{proof}
In what follows, we will denote by lower case letters the slice-regular functions and by upper case letters the corresponding holomorphic maps.
The study of the sets ${\mathbb P}i^{-1}(q)$ is related to the link between the values taken by $f$ and the values taken by $F$.
For $q\in{\mathbb H}$, we define
$$Z(q)=\{z\in{\mathbb C}^4 \ :\ \exists v\in{\mathbb S} \textrm{ s.t. } {\mathbb P}i(z,v)=q\}$$
and, for $q\in{\mathbb H}$ and $v\in{\mathbb S}$,
$$Z_v(q)=\{z\in{\mathbb C}^4\ :\ {\mathbb P}i(z,v)=q\}$$
Also, for short, $Z(0)=Z$, $Z_v(0)=Z_v$. Obviously
$$Z=\bigcup_{v\in{\mathbb S}} Z_v$$
and
$${\mathbb P}i^{-1}(q)=\bigcup_{v\in{\mathbb S}}Z_v\times\{v\}\;.$$
\begin{rem}We will always consider left slice-regular functions in this paper; however, to obtain right slice-regular function it is enough to consider ${\mathbb C}\otimes{\mathbb H}$ and to extend $\rho_v$ as $\rho_v(q\otimes z)=q\rho_v(z)$.\end{rem}
\section{Geometry of the set $Z$}
We start by noticing the following
\begin{lemma} For $v\in{\mathbb S}$, $Z_v$ is a complex vector subspace of ${\mathbb C}^4$ of dimension $2$.\end{lemma}
\begin{proof} Let us write $z_m=z_m+\iota y_m$ for $m=0,\ldots, 3$ and $v=ai+bj+ck\in{\mathbb S}$. We compute
$${\mathbb P}i(z, v)=x_0+ay_0i+by_0j+cy_0k+x_1i-ay_1-by_1k+cy_1j+x_2j+$$
$$+ay_2k-by_2-cy_2i+x_3k-ay_3j+by_3i-cy_3=$$
$$=x_0-ay_1-by_2-cy_3+(ay_0+x_1-cy_2+by_3)i+(by_0+cy_1+x_2-ay_3)j+$$
$$+(cy_0-by_1+ay_2+x_3)k\;,$$
so, the set $Z_v$ is described by the four equations
$$\left\{\begin{array}{ll}x_0-ay_1-by_2-cy_3&=0\\
ay_0+x_1-cy_2+by_3&=0\\
by_0+cy_1+x_2-ay_3&=0\\
cy_0-by_1+ay_2+x_3&=0\end{array}\right.$$
which are clearly linear and independent. Therefore $Z_v$ is a real vector subspace of ${\mathbb C}^4$ and ${\rm d}im_{\mathbb R} Z_v=4$.
Moreover,
$$x_0-ay_1-by_2-cy_3+\iota(a(ay_0+x_1-cy_2+by_3)+b(by_0+cy_1+x_2-ay_3)+$$
$$+c(cy_0-by_1+ay_2+x_3)=$$
$$=x_0+\iota y_0+\iota ax_1-ay_1+\iota bx_2-by_2+\iota cx_3-cy_3=z_0+\iota (az_1+bz_2+cz_3)\;.$$
Hence we can describe $Z_v$ by the following system of (complex linear) equations
$$\left\{\begin{array}{ll}
-\iota z_0+az_1+bz_2+cz_3&=0\\
-az_0-\iota z_1+cz_2-bz_3&=0\\
-bz_0-cz_1-\iota z_2+az_3&=0\\
-cz_0+bz_1-az_2-\iota z_3&=0\end{array}\right.$$
whose rank on ${\mathbb C}$ is $2$; this means that $Z_v$ is the kernel of
\begin{equation}\label{eq_matrice}A_v=\begin{pmatrix}-\iota&a&b&c\\-a&-\iota&c&-b\\-b&-c&-\iota&a\\-c&b&-a&-\iota\end{pmatrix}\end{equation}
hence a complex vector subspace of dimension $2$.
\end{proof}
We consider the Grassmannian of $2$-planes in ${\mathbb C}^4$, $\mathrm{Gr}(4,2)$, so that we have a map $v\mapsto Z_v$ from ${\mathbb S}$ to $\mathrm{Gr}(4,2)$. We want to study the image of such map, so we embedd the Grassmannian into ${\mathbb C}P^5$ via the Pl\"ucker embedding.
Given $L\in\mathrm{Gr}(4,2)$, let $\{z,\tilde{z}\}$ be a base for $L$; we associate to $L$ the point in ${\mathbb C}P^5$ with homogeneous coordinates $[w_0,\ldots, w_5]$ given by
$$w_0=\minor{0}{1}\quad w_1=\minor{0}{2}\quad w_2=\minor{0}{3}$$
$$w_3=\minor{1}{2}\quad w_4=\minor{1}{3}\quad w_5=\minor{2}{3}\;.$$
The set of points $[w_0, \ldots, w_5]\in{\mathbb C}P^5$ obtained this way is the projective hypersurface
$$W=\{[w_0,\ldots, w_5]\in{\mathbb C}P^5\ : \ w_0w_5-w_1w_4+w_2w_3=0\}\;.$$
\begin{lemma}The image of the map $v\mapsto Z_v$ is a complex subspace of $\mathrm{Gr}(4,2)$.\end{lemma}
\begin{proof}We want to use the Pl\"ucker embedding, but in order to do this we would need a basis of $Z_v$; we note that, as $Z_v$ is the kernel of $A_v$, then $Z_v^\partialerp$ (which still belongs to $\mathrm{Gr}(4,2)$) is the range of $\overline{A}_v^t$, where $A_v$ is given by \ref{eq_matrice}.
Suppose, without loss of generality, that $a\neq \partialm1$, so that $b^2+c^2\neq 0$, then $Z_v$ is the kernel of
$$\begin{pmatrix}-\iota&a&b&c\\-a&-\iota&c&-b\end{pmatrix}$$
and, consequently, $Z_v^\partialerp$ is the range of
$$\begin{pmatrix}\iota&-a\\a&\iota\\b&c\\c&-b\end{pmatrix}\;,$$
so that the point of $W$ corresponding to $Z_v^\partialerp$ is
$$[-1+a^2:ab+\iota c:ac-\iota b:ac-\iota b:-ab-\iota c:-b^2-c^2]\;.$$
It is easy to see that every such point is contained in
$$\mathcal{S}=\{[w_0:\ldots:w_5]\in W\ :\ w_0-w_5=w_1+w_4=w_2-w_3=0\}\;,$$
which is a complex projective non-degenerate conic in ${\mathbb C}P^5$, i.e. a copy of ${\mathbb C}P^1$.
The same result is obtained if one supposes $b\neq \partialm1$ or $c\neq \partialm1$.
To complete the proof, we have to show that $\mathcal{S}$ is indeed the image of the map $v\mapsto Z_v^\partialerp$.
We take $w\in \mathcal{S}$, i.e.
$$w=[w_0:w_1:w_2:w_2:-w_1:w_0]\qquad \textrm{with }w_0w_5-w_1w_4+w_2w_3=0$$
and we suppose $w_0\neq 0$ (which corresponds to $a\neq \partialm 1$), so we can write $\zeta_1=w_1/w_0$, $\zeta_2=w_2/w_0$; then, taking $v=ai+bj+ck$ with
$$a=\frac{{\mathbb R}e \zeta_2}{\mathsf{Im} \zeta_1}$$
$$b=\frac{(\mathsf{Im} \zeta_2)^2-({\mathbb R}e\zeta_1)^2}{\mathsf{Im}\zeta_2}$$
$$c=\frac{({\mathbb R}e\zeta_2)^2-(\mathsf{Im} \zeta_1)^2}{\mathsf{Im} \zeta_1}$$
we have that the Pl\"ucker coordinates for $Z_v^\partialerp$ are exactly
$$[1:\zeta_1:\zeta_2:\zeta_2:-\zeta_1:1]=[w_0:w_1:w_2:w_2:-w_1:w_0]\;.$$
As $w$ was taken in $W$, we have $w_0w_5-w_1w_4+w_2w_3=0$, which, on $\mathcal{S}$, is equivalent to $w_0^2+w_1^2+w_2^2=0$; so $1+\zeta_1^2+\zeta_2^2=0$. From this relation it is a simple matter of computation to verify that $a^2+b^2+c^2=1$.
\end{proof}
We are now in the position for stating and proving the geometric description of $Z$.
\begin{teorema}\label{teo_zeroset}The set $Z$ is a complex hypersurface in ${\mathbb C}^4$, described by the equation
$$z_0^2+z_1^2+z_2^2+z_3^2=0\;.$$
In particular, it is the only critical level of the function ${\mathbb P}hi(z)=z_0^2+z_1^2+z_2^2+z_3^2$ and its only critical point is the origin.
\end{teorema}
\begin{proof} We consider the analytic set $\mathfrak{Z}\subseteq {\mathbb C}^4\times{\mathbb C}P^5$ defined by
$$\mathfrak{Z}=\{(z,w)\in{\mathbb C}^4\times \mathcal{S}\ :\ z_0w_3-z_1w_1+z_2w_0=z_0w_4-z_1w_2+z_3w_0=$$
$$=z_0w_5-z_2w_2+z_3w_1=z_1w_5-z_2w_4+z_3w_3=0\}\;.$$
One easily checks that $\mathfrak{Z}$ has complex dimension $3$.
Let $p_1:{\mathbb C}^4\times{\mathbb C}P^5$ be the projection on the first factor; then $Z=p_1(\mathfrak{Z})$. We introduce $Z^*=Z\setminus\{0\}$ and $\mathfrak{Z}^*=p_1^{-1}(Z^*)$ and we claim that
$$p_1\vert_{\mathfrak{Z}^*}:\mathfrak{Z}^*\to Z^*$$
is a biholomorphism. Indeed, given $z\in Z^*$, a pair $(z,w)$ belongs to $\mathfrak{Z}^*$ if and only if $w\in\ker B$, with
$$B=\begin{pmatrix}z_2&-z_1&0&z_0&0&0\\z_3&0&-z_1&0&z_0&0\\
0&z_3&-z_2&0&0&z_0\\0&0&0&z_3&-z_2&z_1\\1 &0 &0&0&0&-1\\0&1&0&0&1&0\\0&0&1&-1&0&0\end{pmatrix}\;.$$
Computing the minors of $B$, one has that if $\mathrm{rk} B<5$ then $z=0$; so, for $z\in Z^*$, there exists a unique $w\in {\mathbb C}P^5$ such that $p_1(z,w)=z$. It follows that $p_1$ is a biholomorphism between $\mathfrak{Z}^*$ and $Z^*$; therefore, $Z$ is a complex analytic set, by the Remmert-Stein theorem. Moreover, ${\rm d}im_{\mathbb C} Z=3$, so there exists a global equation for it in ${\mathbb C}^4$.
If we explicitly solve $Bw=0$ as a function of $z$ and we require that the solution lies on $W$, we obtain the following equation
$$z_0^2+z_1^2+z_2^2+z_3^2=0\;.$$
By irreducibility, we conclude that $Z=\{z\in{\mathbb C}^4\ :\ z_0^2+z_1^2+z_2^2+z_3^2=0\}$; if we consider the holomorphic function ${\mathbb P}hi:{\mathbb C}^4\to{\mathbb C}$, ${\mathbb P}hi(z)=z_0^2+z_1^2+z_2^2+z_3^2$, then $Z$ is the preimage of $0$ and the only critical level set. Moreover, the only singular point of $Z$ is the origin. \end{proof}
\begin{rem}We note that the previous construction gives a diffeomorphism between ${\mathbb S}$ and $\mathcal{S}\cong{\mathbb C}P^1$, thus inducing a complex structure on ${\mathbb S}$; moreover, $\mathcal{S}$ being a conic inside $W$, this map can be seen as taking values in the Grassmannian of $2$-planes in ${\mathbb C}^4$, so it gives a (holomorphic) fiber bundle on ${\mathbb S}$. The set $\mathfrak{Z}$ is the total space of such a bundle and $Z$ is the space obtained by contracting the zero section to a point. \end{rem}
\begin{rem}The sets $\mathfrak{Z}$ and $Z$, being described by equations with real coefficients, are invariant under conjugation of all variables. This reflects the fact that $x+vy=x+(-v)(-y)$.\end{rem}
We conclude this section with some observations on the properties of $Z$ and $Z(q)$.
\begin{corol}\label{corol_prop}\begin{enumerate}
\item The set $Z(q)$ is the zero locus of the function ${\mathbb P}hi_q(z)=(z_0-q_0)^2+(z_1-q_1)^2+(z_2-q_2)^2+(z_3-q_3)^2\;.$
\item The fundamental group of ${\mathbb C}^4\setminus Z(q)$ is ${\mathbb Z}$, for every $q\in{\mathbb H}$.
\item If $f:U\to{\mathbb H}$ is a slice-regular function and $F:\mathcal{U}\to{\mathbb C}^4$ is the map given by Lemma \ref{lemma_rappr}, then $F(\mathcal{U})\cap Z(q)$ is discrete for every $q\in{\mathbb H}$.
\end{enumerate}
\end{corol}
\begin{proof} Statement $(1)$ is obvious. As for statement $(2)$, we restrict our attention to the case $q=0$, the general case being equivalent up to translation.
The function ${\mathbb P}hi$ has non-vanishing gradient on ${\mathbb C}^4\setminus Z$, hence it induces a fibration, by Ehresmann theorem (see Theorem 9.3 and Remark 9.4 in \cite{Vo}); the generic fiber is
$$\{z\in{\mathbb C}^4\ :\ z_0^2+z_1^2+z_2^2+z_3^2=c\}$$
for $c\in{\mathbb C}^*$. It is well known that such a set is diffeomorphic to the total space of the tangent bundle to ${\mathbb S}^3$, hence simply connected. From the long exact sequence for the homotopy of a fibration, we get
$$\partiali_1({\mathbb C}^4\setminus Z)\cong\partiali_1({\mathbb C}^*)\cong{\mathbb Z}\;.$$
To prove $(3)$, again we only consider the case $q=0$. By construction, the open Riemann surface $F(\mathcal{U})$ intersects ${\mathbb R}^4\subset{\mathbb C}^4$ in a set with positive linear measure; however, $Z$ intersects ${\mathbb R}^4\subset{\mathbb C}^4$ only in one point; this means that $F(\mathcal{U})$ is not contained in $Z$, so their intersection must be an analytic set of dimension $4-3-1=0$, i.e. a discrete set (possibly empty).
\end{proof}
\section{Zeros of slice-regular functions}
Let $f:U\to{\mathbb H}$ be a slice-regular function and $F:{\mathcal U}\to{\mathbb C}^4$ be the holomorphic map obtained from Lemma \ref{lemma_rappr}; suppose $q\in U$ is a point where $f(q)=0$ and write $q=x+vy$. Then $F(x+\iota y)$ and $F(x-\iota y)$ belong to $Z\subset{\mathbb C}^4$.
If $y=0$, then we have a real $x$ such that $F(x)\in Z$, but $F(x)\in{\mathbb R}^4\subset{\mathbb C}^4$ and $Z\cap{\mathbb R}^4=\{(0,0,0,0)\}$, so this implies that $F(x)=(0,0,0,0)$. Similarly, if there are two points $q, q'$, with $q=x+vy$ and $q'=x+v'y$, where $f(q)=f(q')=0$, then $F(x+\iota y)=F(x-\iota y)=(0,0,0,0)$: indeed, let $F(x+\iota y)=z\in{\mathbb C}^4$, then there should be $w, w'\in\mathcal{S}$, with $w'\neq \bar{w}$, such that $p_1(z,w)=p_1(z,w')=0$, but this is possible only if $z=(0,0,0,0)$.
Combining these observations with Corollary \ref{corol_prop}, we immediately obtain that the zeros of $f$, generic, real or spherical, are a discrete set and no two isolated zeros are on the same sphere.
From this geometric perspective, an immediate consequence of Theorem \ref{teo_zeroset} is the following.
\begin{propos}\label{cor_equiv}Let $f:U\to{\mathbb H}$ be a slice-regular function; a point $x+vy\in U$ is a zero of $f$ if and only if $x+\iota y$ is a zero for the holomorphic function ${\mathbb P}hi\circ F$. Moreover,
\begin{enumerate}
\item if $x+vy$ is an isolated non-real zero of multiplicity $k$ for $f$, then $x+\iota y$ and $x-\iota y$ are zeros of multiplicity $k$ for ${\mathbb P}hi\circ F$;
\item if $[x+vy]$ is a spherical zero of multiplicity $k$ for $f$, then $x+\iota y$ and $x-\iota y$ are zeros of multiplicity $2k$ for ${\mathbb P}hi\circ F$
\item if $x$ is an isolated real zero of multiplicity $k$ for $f$, then $x$ is a zero of multiplicity $2k$ for ${\mathbb P}hi\circ F$. \end{enumerate}
\end{propos}
\begin{proof} The only part needing a proof is the one about multiplicities, which follows easily if one notices that $(0,0,0,0)$ is a singular point for ${\mathbb P}hi$, where $\nabla {\mathbb P}hi$ vanishes, but $\nabla^2{\mathbb P}hi$ is non-degenerate: both real zeros and spherical zeros correspond to points $z\in{\mathcal U}$ where $F(z)=(0,0,0,0)$.
So, for an isolated non-real zero, the multiplicity of the intersection between $F({\mathcal U})$ and $Z$ is the multiplicity of zero of ${\mathbb P}hi\circ F$, whereas for a real or spherical zero, the multiplicity gains a factor $2$ because $(0,0,0,0)$ is a singular point of $Z$, namely, a double point.\end{proof}
The previous result means that we can use all the techniques, which we employ to study the zeros of holomorphic functions, to understand the zeros of slice-regular functions. But there is more to it: such zeros are actually fully characterized as intersection points between two complex analytic sets in ${\mathbb C}^4$ (a Riemann surface and the quadric cone).
Some of the results we present in the following were already obtained by \cite{V}, however, we hope that this presentation can shed new light on the nature of such results.
\begin{rem}The function ${\mathbb P}hi\circ F$ is linked to the sphericization of $f$, namely
$$f^s\circ\partiali(x,y,v)=\rho_v\circ{\mathbb P}hi\circ F(x+\iota y)\;.$$
\end{rem}
\begin{teorema}[Counting zeros - I] \label{teo_contazeri1}Let $f$ be a slice-regular function defined on a neighbourhood of the closure of $U$, suppose that $f$ does not vanish on $bU$. If $f$ has $k$ isolated (real or non-real) zeros and $m$ spherical zeros in $U$ (counted with multiplicities), then
$$\frac{1}{2\partiali\iota}\int_{b{\mathcal U}}\frac{({\mathbb P}hi\circ F)'(z)}{({\mathbb P}hi\circ F)(z)}dz=2k+4m\;,$$
where $z=x+\iota y$ and $bU$, $b{\mathcal U}$ are supposed to be regular enough for the integration to make sense.
\end{teorema}
\begin{proof}It is enough to apply the known result for holomorphic function and to keep in mind Corollary \ref{cor_equiv}.\end{proof}
For a slice-regular function as in the hypotheses of the previous result, we call $k+2m$ the \emph{weighted number of zeros} in $U$.
We can specialize this observation to entire function and recover the classical theory of distribution of zeores.
In particular,given $f:{\mathbb H}\to{\mathbb H}$, in accordance with Theorem \ref{teo_contazeri1}, we define
$$n_f(t)=k_f(t)+2m_f(t)$$
where $k_f(t)$ is the number of isolated zeros of $f$ in the ball $B(0,t)=\{q\in{\mathbb H}\ :\ |q|<t\}$ and $m_f(t)$ is the number of spherical zeros of $f$ in the same ball. This is the same counting function defined in \cite{CSS1}.
Then, recalling the classical Jensen formula, we obtain that
$$\int_0^R\frac{n_f(t)}{t}dt={\rm d}frac{1}{4\partiali}\int_0^{2\partiali}\log\left|{\mathbb P}hi\circ F(Re^\iota \theta)\right|d\theta - \log\left|{\mathbb P}hi\circ F(0)\right|\;,$$
which is the same result obtained in \cite{CSS1}.
The following version of Rouché theorem is also readily obtained by the standard statement of one complex variable.
\begin{teorema}[Rouché theorem for slice-regular functions]
Suppose that $f,g$ are slice-regular functions defined on a neighbourhood of the closure of $U$. Suppose that, for each $z\in b{\mathcal U}$,
$$|({\mathbb P}hi\circ F)-({\mathbb P}hi\circ G)|<|({\mathbb P}hi\circ F)| + |({\mathbb P}hi\circ G)|\;.$$
Then $f$ and $g$ have the same weighted number of zeros in $U$.
\end{teorema}
\begin{proof} It is enough to apply Rouché theorem \cite{C}*{Theorem 3.8 -- p. 125} to the functions ${\mathbb P}hi\circ F$ and ${\mathbb P}hi\circ G$.
\end{proof}
It is interesting to remark that Corollary \ref{corol_prop} ensures that we are not losing any topological information about the winding number by using the function ${\mathbb P}hi\circ F$ in place of the function $F$ (or the function $f$).
So far, the integration was carried out on the boundary of sets which are symmetric with respect to the real axis; the next theorem considers the case where this condition no longer holds.
\begin{teorema}[Counting zeros - II]\label{teo_conteggio2}
Let $f:U\to{\mathbb H}$ be a slice-regular function; consider $\Omega\subset {\mathcal U}$ and let
\begin{itemize}
\item $k_0$ be the number of isolated non-real zeros $q=\partiali(x,y,v)$ of $f$ with $z=x+\iota y\in\Omega$ and $\bar{z}\in\Omega$
\item $m_0$ be the number of spherical zeros $[q]=\partiali(\{(x,y)\}\times {\mathbb S})$ of $f$ with $z=x+\iota y\in\Omega$ and $\bar{z}\in\Omega$
\item $k_1$ be the number of isolated non-real zeros $q=\partiali(x,y,v)$ of $f$ with $z=x+\iota y\in\Omega$ and $\bar{z}\not\in\Omega$
\item $m_1$ be the number of spherical zeros $[q]=\partiali(\{(x,y)\}\times {\mathbb S})$ of $f$ with $z=x+\iota y\in\Omega$ and $\bar{z}\not\in\Omega$
\item $r$ be the number of real zeros $x$ of $f$, with $x\in\Omega$.
\end{itemize}
Then we have
$$\frac{1}{2\partiali\iota}\int_{b\Omega}\frac{({\mathbb P}hi\circ F)'(z)}{({\mathbb P}hi\circ F)(z)}dz=2k_0+k_1+2r+4m_0+2m_1\;.$$
\end{teorema}
Cunningly varying the set $\Omega$, we can include or exclude every species of zeros in the counting and thus obtain various linear equations, in order to calculate the number of isolated non-real, spherical and real zeros of $f$ in a given spherically symmetric open set.
Finally, the classical Hurwitz theorem easily implies its version for slice-regular functions.
\begin{teorema}Let $\{f_n\}_{n\in{\mathbb N}}$ be a sequence of non-vanishing slice-regular functions $f_n:U\to{\mathbb H}$ which converges uniformly to $f:U\to{\mathbb H}$. Then $f$ is either constantly zero or non-vanishing.\end{teorema}
\begin{proof} It is enough to apply the classical Hurwitz theorem to the sequence of functions ${\mathbb P}hi\circ F_n$, once one checks that their sequence converges to ${\mathbb P}hi\circ F$; the latter is easy: if $g:U\to{\mathbb H}$ is slice-regular and $G$ is the holomorphic map given by Lemma \ref{lemma_rappr}, then, once we write $g\circ \partiali(x,y,v)=\alpha(x,y)+v\beta(x,y)$, then
$$|G(x+\iota y)|^2=|\alpha(x,y)|^2+|\beta(x,y)|^2\leq\max_{v\in{\mathbb S}}|g\circ\partiali(x,y,v)|^2\;,$$
so if $\|f_n-f\|_{\infty}\to0$, then $\|F_n-F\|_{\infty}\to 0$ and, as ${\mathbb P}hi$ is also holomorphic, the conclusion follows.\end{proof}
\section{$\star$-products and poles}
Let $f,g:U\to{\mathbb H}$ be slice-regular functions; it is well known that their punctual product is not slice-regular and, to overcome this difficulty, another product, called $\star$-product, is defined. If we write
$$f\circ \partiali(x,y,v)=\alpha(x,y)+v\beta(x,y) \qquad g\circ \partiali(x,y,v)=\gamma(x,y)+v{\rm d}elta(x,y)$$
then
\begin{equation}\label{eq_starprod}(f\star g)\circ\partiali(x,y,v)=(\alpha\gamma-\beta{\rm d}elta)(x,y)+v(\alpha{\rm d}elta+\beta\gamma)(x,y)\;.\end{equation}
We define a (${\mathbb R}$-homogeneous) product on ${\mathbb C}^4$ by identifying ${\mathbb C}^4$ with ${\mathbb H}\otimes_{\mathbb R}{\mathbb C}$ and we denote such product by $\star$.
\begin{lemma} \label{lemma_omom}If $f,g,h:U\to{\mathbb H}$ are slice-regular functions such that $f\star g=h$, then $H=F\star G$ and ${\mathbb P}hi\circ H=({\mathbb P}hi\circ F)({\mathbb P}hi\circ G)$.\end{lemma}
\begin{proof}
The $\star$-product on ${\mathbb C}^4$ works as follows: it is ${\mathbb R}$-homogeneous and
$$\iota\star\iota=i\star i=j\star j=k\star k=-1$$
$$i\star j= -j\star i = k\qquad j\star k= -k\star j= -i \qquad k\star i= - i\star k= j$$
where $\{1,i,j,k\}$ is the standard (${\mathbb C}$-)base for ${\mathbb C}^4$ and $1$ and $\iota$ commute with everything.
So
$$F\star G=({\mathbb R}e F)\star({\mathbb R}e G) - (\mathsf{Im} F)\star(\mathsf{Im} G) + \iota (({\mathbb R}e F)\star(\mathsf{Im} G) + (\mathsf{Im} F)\star({\mathbb R}e G))$$
where the $\star$-product now is between vectors with real entries, so it is the usual quaternionic product. This is the same formula as in \eqref{eq_starprod}, which shows that $H=F\star G$.
Now, the equality $|ab|^2=|a|^2|b|^2$ for $a,b\in{\mathbb H}$, being an algebraic identity, holds also in ${\mathbb C}^4$ with the $\star$-product and the function ${\mathbb P}hi$ in place of the squared norm:
$${\mathbb P}hi(A\star B)={\mathbb P}hi(A){\mathbb P}hi(B)\qquad \textrm{for } A,B\in{\mathbb C}^4\;.$$
This concludes the proof.
\end{proof}
An easy consequence of this lemma is the following proposition.
\begin{propos}\label{propos_prodzero}Let $f,g:U\to{\mathbb H}$ be slice-regular functions; if either $f$ or $g$ have a zero on the sphere $[q]$, then also $f\star g$ has a zero on that sphere.\end{propos}
\begin{proof}Let $h=f\star g$. First of all, we note that
$${\mathbb P}hi\circ H=({\mathbb P}hi\circ F)({\mathbb P}hi\circ G)=({\mathbb P}hi\circ G)({\mathbb P}hi\circ F)\;,$$
so we can suppose, without loss of generality, that the function vanishing on a point of the sphere $[q]$ is $f$.
Let $q=x+vy$, then ${\mathbb P}hi\circ F(x+\iota y)=0$, which obviously implies that ${\mathbb P}hi\circ H(x+\iota y)=0$, so there exists at least one $v'\in{\mathbb S}$ such that $h(x+v'y)=0$.
\end{proof}
\begin{defin}Let $f:U\to{\mathbb H}$ be a slice-regular function; for $q\in U$, $q=x+vy$, we define the order (of zero) of $f$ on the sphere $[q]$ as
$${\rm ord}_f([q])=\left\{\begin{matrix}{\rm ord}_{{\mathbb P}hi\circ F}(x+\iota y) & \textrm{if }y\neq 0\\ & \\{\rm ord}_{{\mathbb P}hi\circ F}(x)/2&\textrm{if }y=0\end{matrix}\right.$$
where the order of zero of ${\mathbb P}hi\circ F$ is defined as usual for a holomorphic function.
\end{defin}
\begin{propos}Let $f,g:U\to{\mathbb H}$ be slice-regular functions and $q\in U$. Then
$${\rm ord}_{f\star g}([q])={\rm ord}_{f}([q]){\rm ord}_g([q])\;.$$
\end{propos}
\begin{proof}
The identity follows from the proof of Proposition \ref{propos_prodzero}.
\end{proof}
A slice-meromorphic function is of the form $f^{-\star}\star g$, where $f,g:U\to{\mathbb H}$ are slice-regular functions.
\begin{propos}Let $f:U\to{\mathbb H}$ be a slice-regular function and define $\mathcal{V}=\{z\in{\mathcal U}\ :\ F(z)\neq 0\}$ and $V=\partiali(\mathcal{V}\times{\mathbb S})$. We set $h=f^{-\star}$, so that $h:V\to{\mathbb H}$ is slice-regular.
Then, ${\mathbb P}hi\circ H$ extends to a meromorphic function on ${\mathcal U}$; moreover, if ${\rm ord}_f([q])=k$, then ${\mathbb P}hi\circ H$ has a pole of order $k$ in $x+\iota y$, with $q=x+vy$.
\end{propos}
\begin{proof} As noted before, ${\mathcal U}\setminus\mathcal{V}$ is a discrete set, so ${\mathbb P}hi\circ H$ is meromorphic on $U$; moreover, by Lemma \ref{lemma_omom}, we have that
$${\mathbb P}hi\circ H=\frac{1}{{\mathbb P}hi\circ F}$$
so, the poles of ${\mathbb P}hi\circ H$ are the zeros of ${\mathbb P}hi\circ F$ and the last part of the statement follows.
\end{proof}
So, we can extend the definition of order to zeros and poles of a meromorphic function: if $h=f^{-\star}\star g$, then we set
$${\rm ord}_h([q])={\rm ord}_g([q])-{\rm ord}_f([q])\;.$$
However, order zero does not mean a removable singularity. For example, the function
$$f(q)=\frac{1}{q^2+1}(q^2+q(i-j)-k)$$
has ${\rm ord}_f([i])=0$, because it has a double isolated zero on the sphere, which is also a pole of order $2$, but it does not extend to a slice-regular function on ${\mathbb H}$. On the other hand, the corresponding holomorphic function ${\mathbb P}hi\circ F$ is the constant function $1$.
Theorem \ref{teo_conteggio2} can be extended to meromorphic functions in the usual way.
\begin{teorema}Let $f:U\to{\mathbb H}$ be a slice-meromorphic function and let $V\subset U$ be the maximal axially symmetric open set such that $f$ is slice-regular on $V$. Consider $\Omega\subset{\mathcal U}$ such that $f$ does not have zeros or poles on $b\Omega$. The quantities $k_0, k_1, m_0, m_1, r$ are as in Theorem \ref{teo_conteggio2}, moreover denote by
\begin{itemize}
\item $p_0$ the number of poles $[q]=\partiali(\{(x,y)\}\times{\mathbb S})$ of $f$ with $z=x+\iota y\in\Omega$ and $\bar{z}\in\Omega$
\item $p_1$ the number of poles $[q]=\partiali(\{(x,y)\}\times{\mathbb S})$ of $f$ with $z=x+\iota y\in\Omega$ and $\bar{z}\not\in\Omega$.
\end{itemize}
Then we have
$$\frac{1}{2\partiali\iota}\int_{b\Omega}\frac{({\mathbb P}hi\circ F)'(z)}{({\mathbb P}hi\circ F)(z)}dz=2k_0+k_1+2r+4m_0+2m_1-2p_0-p_1\;.$$
\end{teorema}
We have so far characterized the poles of $f$ only in terms of poles of ${\mathbb P}hi\circ F$; however, given that a pole is always spherical, the behaviour of $F$ around a pole is also quite easy.
\begin{propos} Let $f:U\to{\mathbb H}$ be a slice-meromorphic and $V\subset U$ the maximal open set such that $f$ is slice-regular on $V$. If $w\in{\mathcal U}\setminus\mathcal{V}$, then
$$\lim_{z\to w} |f_m(z)|=\infty\qquad m=0,1,2,3\;,$$
where $F=(f_0,f_1,f_2,f_3)$.\end{propos}
So, the poles of $f$ correspond to points where all the components of $F$ have a pole, in accordance with the fact that they are spherical zeros of the denominator, i.e. zeros where all the components of the holomorphic map go to zero.
\section{Integral kernels}
In the two previous sections, we employed the holomorphicity of the function ${\mathbb P}hi\circ F$ in order to obtain some information about the zeros of the slice-regular function $f$. However, as we noted earlier, studying the function ${\mathbb P}hi\circ F$ is equivalent to studying the function $f^s$, which amounts to a loss of information, because different slice-regular functions can give rise to the same symmatrization.
If we want to encode all the information about $f$, we have to look at the holomorphic map $F:{\mathcal U}\to{\mathbb C}^4$. This presents a greater complexity, but allows us to prove more.
As an example, the classical Cauchy formula for vector-valued holomorphic functions of one variable implies that
\begin{equation}\label{eq_cauchy1}\frac{1}{2\partiali\iota}\int_{b{\mathcal U}}{\rm d}frac{1}{\zeta-z}F(\zeta)d\zeta=F(z)\end{equation}
for all $z\in{\mathcal U}$. From this identity, we immediately obtain a way to determine $f(q)$ in terms of the values of $F$ on the boundary of ${\mathcal U}$; however, we would prefer an integral formula (like the Cauchy formula) expressing $f(q)$ in terms of the values of $f$ on some $bU\cap{\mathbb C}_v$, $v\in{\mathbb S}$.
\begin{lemma}\label{lmm_commute}For every $p(z), q(z)\in{\mathbb R}[z]$, we have
$$\rho_v(p(z)q^{-1}(z))=p(\rho_v(z))q^{-1}(\rho_v(z))\;.$$
\end{lemma}
\begin{proof}
It is quite obvious that
$$\rho_v(z^k)=(\rho_v(z))^k$$
and also, for $a\in{\mathbb R}$,
$$\rho_v(az)=a\rho_v(z)\;,$$
therefore $p(\rho_v(z))=\rho_v(p(z))$.
We note that $|\rho_v(z)|=|z|$ (where the first is computed in ${\mathbb H}$ and the second in ${\mathbb C}$),so, given a converging power series
$$\sum_{n=0}^\infty z^na_n$$
we have that
$$\rho_v\left(\sum_{n=0}^\infty z^na_n\right)=\sum_{n=0}^\infty\rho_v(z)^na_n\;.$$
Moreover, if $q(z)$ has real coefficients, we have
$$p(\rho_v(z))q^{-1}(\rho_v(z))=q^{-1}(\rho_v(z))p(\rho_v(z))$$
and also
$$p(\rho_v(z))q^{-1}(\rho_v(z))=p(\rho_v(z))\rho_v(q^{-1}(z))\;.$$
Finally, $p(z)q^{-1}(z)$ can be expressed as a converging power series around some point $z_0$ and we can assume it to be the origin, so
$$\rho_v(p(z)q^{-1}(z))=p(\rho_v(z))q^{-1}(\rho_v(z))\;.$$
\end{proof}
In the formula \eqref{eq_cauchy1}, the complex variable is $z$, $\zeta$ being the integration variable, so we would like to write everything in terms of power series of $z$ with real coefficients. As we already know that the components of $F$ have this property, the integral in \eqref{eq_cauchy1} has to produce functions whose power series have real coefficients, so we turn our attention to the part depending explicitly on $z$:
$${\rm d}frac{1}{\zeta-z}=-{\rm d}frac{1}{z-\zeta}=-{\rm d}frac{z-\overline{\zeta}}{z^2-2{\mathbb R}e\zeta z + |\zeta|^2}\;.$$
Hence, by Lemma \ref{lmm_commute}, we write
\begin{eqnarray*}f(q)&=&{\rm d}frac{1}{2\partiali \iota}\int_{b{\mathcal U}}{\rm d}frac{\overline{\zeta}-q}{q^2-2{\mathbb R}e\zeta q + |\zeta|^2}f_0(\zeta)d\zeta\\
&+&{\rm d}frac{1}{2\partiali \iota}\int_{b{\mathcal U}}{\rm d}frac{\overline{\zeta}-q}{q^2-2{\mathbb R}e\zeta q + |\zeta|^2}f_1(\zeta)d\zeta i\\
& +& {\rm d}frac{1}{2\partiali \iota}\int_{b{\mathcal U}}{\rm d}frac{\overline{\zeta}-q}{q^2-2{\mathbb R}e\zeta q + |\zeta|^2}f_2(\zeta)d\zeta j\\
&+ &{\rm d}frac{1}{2\partiali \iota}\int_{b{\mathcal U}}{\rm d}frac{\overline{\zeta}-q}{q^2-2{\mathbb R}e\zeta q + |\zeta|^2}f_3(\zeta)d\zeta k\end{eqnarray*}
and, reordering the terms in each integral, we can write
$$f(q)={\rm d}frac{1}{2\partiali \iota}\int_{b{\mathcal U}}{\rm d}frac{\overline{\zeta}-q}{q^2-2{\mathbb R}e\zeta q + |\zeta|^2}d\zeta (f_0(\zeta)+f_1(\zeta)i+f_2(\zeta)j+f_3(\zeta)k)\;.$$
Now, as the components of $F$ have power series expansions with real coefficients, the integral itself cannot depend on $\iota$; indeed, if one rearranges the elements correctly, one obtains an expression that doesn't depend on $\iota$ even if the latter is not assumed to commute with the three quaternionic imaginary units. The correct order involves also the choice of the side on which we multiply by the inverse of the denominator: this can be worked out observing that it is should be right slice-regular in the variable $\zeta$, therefore the quotient becomes a multiplication on the left by the inverse of the denominator.
So, we can substitute $\zeta$ with a quaternionic variable $s$ varying on $bU\cap {\mathbb C}_v$ for some $v\in{\mathbb S}$. We have proven the following.
\begin{teorema}Let $f:U\to{\mathbb H}$ be a slice-regular function which extends continuously to the boundary, then, for every $q\in U$,
$$f(q)={\rm d}frac{1}{2\partiali}\int_{bU\cap{\mathbb C}_v}S^{-1}_L(q,s){\rm d}frac{ds}{v}f(s)\;,$$
where $S^{-1}_L(q,s)=-(q^2-2{\mathbb R}e s q+|s|^2)^{-1}(q-\overline{s})$.\end{teorema}
With analogous arguments, we can obtain many related results, like the integral formulas for the derivatives and the Cauchy estimates for them.
In general, let us consider a function $K:{\mathcal U}\times b{\mathcal U}\to{\mathbb C}$ of the form
$$K(z,w)=\sum_{n=0}^\infty\sum_{m=-\infty}^\infty a_{nm}z^nw^m$$
with $a_{nm}\in{\mathbb R}$.
It defines an integral operator on holomorphic functions on ${\mathcal U}$ by
$$Kh(z)=\int_{b{\mathcal U}}K(z,w)h(w)dw\;.$$
This integral operator can be extended to slice-regular functions in two meaningful ways:
\begin{enumerate}
\item by formally replacing $z$ and $w$ with quaternionic variables in $K$ and writing
$$K_{\mathbb H} f(q)=\int_{U\cap{\mathbb C}_v} K(q,s){\rm d}frac{1}{v}dsf(s)$$
where $K(q,s)=\sum_{n=0}^\infty\sum_{m=-\infty}^\infty a_{nm}q^ns^m$;
\item by applying the operator $K$ to the four components of $F$.
\end{enumerate}
\begin{rem}\label{rem_integkernel}The computations and considerations carried out for the Cauchy formula apply to such a general case and make us conclude that these two extensions give us the same integral operator on slice-regular functions, namely
$$\rho_v\left(Kf_0(x+\iota y)\right)+\rho_v\left(Kf_1(x+\iota y)\right)i+\rho_v\left(Kf_2(x+\iota y)\right)j+$$
$$+\rho_v\left(Kf_3(x+\iota y)\right)k=K_{\mathbb H} f(\partiali(x,y,v))\;.$$
\end{rem}
Moreover, the same can be said when integrating on the whole set.
\begin{rem}\label{rem_integkernel2}If we define
$$Kh(z)=\int_{{\mathcal U}} K(z,w)h(w)d\mu$$
where $\mu$ is the Lebesgue measure on ${\mathcal U}$, we reach the same conclusion as in the previous remark. The operator on slice-regular functions will be then defined as
$$K_{\mathbb H} f(q)=\int_{U\cap{\mathbb C}_v}K(q,s)f(s)d\mu\;.$$
\end{rem}
\section{Norms}
As a last application, we look at the relations between the $L^\infty$ and $L^2$ norms of $f$ and of $F$.
\begin{propos}Let $f:U\to{\mathbb H}$ be a slice-regular function. Suppose there exists $q\in U$, $q=s+v_0t$, such that $|f|$ has a local maximum at $q$; then $|F|$ has a local maximum in $s+\iota t$.\end{propos}
\begin{proof}Let $f\circ\partiali(x,y,v)=\alpha(x,y)+v\beta(x,y)$; we note that
\begin{equation}\label{eq_max}\max_{v\in{\mathbb S}}|f\circ \partiali(x,y,v)|=|\alpha(x,y)|+|\beta(x,y)|\end{equation}
whereas
$$|F(x+\iota y)|=\sqrt{|\alpha(x,y)|^2+|\beta(x,y)|^2}\;,$$
so, in general these two values do not coincide and the latter is the lower. However, let us suppose that $V$ is a neighbourhood of $q$ in $U$ such that for every $p\in V$, $|f(p)|\leq|f(q)|$ and, without loss of generality, let us assume that $\alpha(s,t)=0$.
By continuity, if $(x,y)$ is close enough to $(s,t)$, then there is a $v\in {\mathbb S}$ close enough to $v_0$ which realizes the maximum in \eqref{eq_max}. So, we can suppose, up to shrinking $V$, that for every $p\in V$, the point on $[p]$ that realizes the maximum in \eqref{eq_max} is in $V$.
Then, for every $x+\iota y$ such that $\partiali(\{(x,y)\}\times{\mathbb S})\cap V\neq \empty$, let $p=x+vy$ where the maximum in \eqref{eq_max} is attained; we have
$$|F(x+\iota y)|\leq|f(p)|=|\alpha(x,y)|+|\beta(x,y)|\leq|\alpha(s,t)|+|\beta(s,t)|=$$
$$=|\beta(s,t)|=\sqrt{|\alpha(s,t)|^2+|\beta(s,t)|^2}=|F(s+\iota y)|\;.$$
This concludes the proof.\end{proof}
\begin{teorema}Let $f:U\to{\mathbb H}$ be a slice-regular function such that $|f|:U\to{\mathbb R}$ has a maximum in a (interior) point $q\in U$. Then $f$ is constant.\end{teorema}
\begin{proof}By the previous proposition, if the point $q=x+v y$ is a maximum point for $|f|$, then $x+\iota y\in{\mathcal U}$ is a maximum point for $|F|$. The latter is a vector-valued holomorphic map, so its norm attains the maximum on the boundary, unless the map itself is constant; but, if $F$ is constant, so is $f$, which proves our result.
\end{proof}
In general, $|f(q)|$ and $|F(x+\iota y)|$ are different, but we have that
\begin{equation}\label{eq_norms}\min_{v\in{\mathbb S}}|f(x+vy)|\leq |F(x+\iota y)|\leq \max_{v\in{\mathbb S}}|f(x+vy)|\;.\end{equation}
So, a number of results that hold for holomorphic functions can be adapted to slice-regular functions.
\begin{propos}Let $f:{\mathbb H}\to{\mathbb H}$ be a slice-regular function such that $|f(q)|\sim |q|^n$ when $|q|\to\infty$. Then $f$ is a polynomial of degree at most $n$.\end{propos}
\begin{proof}From \eqref{eq_norms}, we conclude that also the components of $F$ are entire holomorphic functions that grow as a power of $|z|$; by a standard result of one complex variable, all the components of $F$ are polynomials of degree at most $n$, which easily implies the conclusion.
\end{proof}
As we have just seen, the norm of the map $F$ is controlled by the norm of $f$ on a sphere; if we look at the $L^2$ norm of these functions, however, we find a closer connection.
\begin{lemma}Let $f:U\to{\mathbb H}$ be a slice-regular function. For $q\in U$, $q=x+vy$, we have
$$\int_{{\mathbb S}}|f(x+v'y)|^2d\sigma(v')=4\partiali|F(x+\iota y)|^2\;,$$
where $d\sigma$ is the area measure on the unit sphere ${\mathbb S}$.
\end{lemma}\begin{proof}We write
$$f\circ\partiali(x,y,v')=\alpha(x,y)+v'\beta(x,y)$$
so
$$|f(x+v'y)|^2=|\alpha(x,y)|^2+|\beta(x,y)|^2+\alpha(x,y)\overline{\beta(x,y)v'}+v'\beta(x,y)\overline{\alpha(x,y)}\;.$$
Now, $x,y$ are fixed, so we set $\alpha(x,y)=a$ and $\beta(x,y)=b$. The functions $a\overline{bv'}$ and $v'b\overline{a}$, as functions of $v'$, are odd, so their integral on ${\mathbb S}$ vanishes. Hence
$$\int_{{\mathbb S}}|f(x+v'y)|^2d\sigma(v')=\int_{{\mathbb S}}(|a|^2+|b|^2)d\sigma(v')=4\partiali(|a|^2+|b|^2)$$
and the last quantity is exactly $4\partiali|F(x+\iota y)|^2$.
\end{proof}
From this lemma one immediately obtains the following result.
\begin{corol}Let $f:U\to{\mathbb H}$ be a slice-regular function, then
$$\int_{U}|f(q)|^2dx_0dx_1dx_2dx_3=4\partiali\int_{{\mathcal U}}y^2|F(x+\iota y)|^2dxdy$$
$$\int_{U\cap {\mathbb C}_v}|f(x+vy)|^2dxdy=\int_{{\mathcal U}}|F(x+\iota y)|^2dxdy$$
for any $v\in{\mathbb S}$.\end{corol}
So, the Hilbert space of slice-regular functions on $U$ with the $L^2$-norm computed on a slice and the Hilbert space of $L^2$ holomorphic maps from ${\mathcal U}$ to ${\mathbb C}^4$ are isometric. Therefore, in view of Remark \ref{rem_integkernel2}, we can extend the Bergman projection to the quaternionic setting.
\begin{teorema}Let $K_{{\mathcal U}}$ be the classical Bergman kernel on ${\mathcal U}$. We define $K_U$ as the extention of $K_{{\mathcal U}}$ as a slice-regular function. Then
\begin{enumerate}
\item $K_U(q,s)=\overline{K_U(s,q)}$,
\item $K_U(\cdot, \cdot)$ is slice-regular in the first variable and slice-antiregular in the second one,
\item $K_U$ is a reproducing kernel, i.e.
$$f(q)=\int_{U\cap{\mathbb C}_v}K(q,s)f(s)d\mu$$
for every $v\in{\mathbb S}$ and every $q\in U$.
\end{enumerate}
\end{teorema}
\begin{proof} The listed properties are well-known for the classical Bergman kernel $K_{{\mathcal U}}$, hence the first two follow from the fact that $K_U$ is the extension of $K_{{\mathcal U}}$. The third one follows by Remark \ref{rem_integkernel2}, applying the reproducing kernel property of $K_{{\mathcal U}}$ with the components of $F$.
\end{proof}
\section{Clifford Algebras}
As stated in \cite{GP1}, the definition of slice-regular functions via stem functions works in the same way for a real alternative algebra. However, all the computations we did in the beginning of this paper cannot be carried over verbatim to the general case.
Let us consider, for instance, the case of ${\mathbb R}_3$, the Clifford algebra on $3$ generators; through the standard basis
$$\{1,e_0,e_1,e_2,e_0e_1,e_0e_2,e_1e_2,e_0e_1e_2\}$$
we identify ${\mathbb R}_3$ with ${\mathbb R}^8$ as vector spaces. For $a\in{\mathbb R}_3$, we denote by $a_\ell$, $\ell=0,\ldots,7$ its components as an element of ${\mathbb R}^8$ and by $\|a\|$ its Euclidean norm.
We define the set of imaginary units as
$$S=\{u\in{\mathbb R}_3\ :\ u_0=0,\ \|u\|=1,\ u^2=-1\}\;.$$
Given a holomorphic function
$$F:{\mathcal U}\to{\mathbb C}\otimes{\mathbb R}_3\cong{\mathbb R}^8$$
we obtain a slice-regular function $f:U\to{\mathbb R}_3$, where $U=\{x+uy\ :\ (x+\iota y)\in{\mathcal U},\ u\in S\}$.
Our object of study is now the set
$$Z=\{z\in{\mathbb C}^8\ :\ \exists u\in S \textrm{ s.t. } {\mathbb P}i(z,u)=0\}$$
where ${\mathbb P}i:{\mathbb C}^8\times S\to{\mathbb R}_3$ is defined in analogy with what we did in Section 2.
Some calculations show that
$$S=\{ u\in{\mathbb R}^8\ :\ u_0=u_7=0,\ u_1^2+\cdots+u_6^2=1,\ u_1u_6-u_2u_5+u_3u_4=0\}$$
and
$$Z=\{ z\in{\mathbb C}^8\ :\ z_0z_7-z_1z_6+z_2z_5-z_3z_4=0,\ z_0^2+\cdots+z_7^2=0\}\;.$$
Defining ${\mathbb P}hi:{\mathbb C}^8\to{\mathbb C}$ as ${\mathbb P}hi(z)=z_0^2+\cdots+z_7^2$, we note that, if $z\in Z$ and $w\in{\mathbb C}^8$, then ${\mathbb P}hi(z\star w)={\mathbb P}hi(z){\mathbb P}hi(w)$, where $\star$ is the product induced on ${\mathbb C}^8$ by the isomorphism with ${\mathbb C}\otimes{\mathbb R}_3$.
We can thus obtain all the considerations about the geometry of the zeros of slice-regular functions; however, as $Z$ is not an hypersurface anymore, we cannot replicate the results in the spirit of Rouché theorem without changes. For example, from the argument principle we do not obtain a way of counting zeros exactly, but only an upper bound on the number of zeros.
If we consider ${\mathbb R}_n$ with $n>3$, the codimension of the set $Z$ increases with $n$, making the direct computation of the equation impossible for the general case. However, this set and the set $S$ are linked to geometric and algebraic properties of ${\mathbb R}_n$; we will explore this connection in a future paper \cite{Mo}.
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\end{document} | math | 52,698 |
\begin{document}
\title{Cut Finite Elements for Convection in Fractured Domains}
\begin{abstract}
We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain which is a union of manifolds of different dimensions such that a $d$ dimensional component always resides on the boundary of a $d+1$ dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.
\end{abstract}
\section{Introduction}
\paragraph{Fractured Domains.}
Transport phenomena in media with complicated microstructure occur in several
applications for instance transport in porous media and composite materials. The
properties of the microstructure may have different characteristics ranging from
stochastic to highly structured or a combination of these. In this work we focus on
problems where the microstructure consists of embedded surfaces and their
intersections. The surfaces can be used to model fractures or thin embedded
sheets with different transport properties. We refer to such domains as fractured
domains, see examples in Figure~\ref{fig:fractured-schematic}.
\begin{figure}
\caption{Two example fractured domains in 2D.}
\label{fig:fractured-schematic}
\end{figure}
\paragraph{New Contributions.}
A fractured domain in $\mathbb{R}^n$ is a disjoint union of smooth manifolds of dimension
$d=0,\dots,n$, constructed in such a way that a $d$ dimensional component
always reside on the boundary of a $d+1$ dimensional component. These domains
are also called mixed-dimensional or stratified domains. See the recent work
\cite{BoNoVa17} where a similar description is used to study the pressure problem.
On such a domain we consider a first order system of hyperbolic equations which
models transport in fractured media.
Introducing convenient multi-dimensional directional derivative and divergence
operators the problem may be formulated in an abstract form similar to the standard
one field transport problem on a domain in $\mathbb{R}^n$.
We develop a cut finite element method, see \cite{BurClaHanLarMas15} for an introduction,
which is based on embedding the composite domain into a fixed background mesh and then
for each of the components we define the active mesh as the set of all elements that
intersect the component. Note that in this way we obtain one active mesh for each
component of the domain and thus certain elements will appear in several meshes.
The active meshes are each equipped with a continuous finite element space and
the finite element method is obtained by stabilizing the variational formulation using
certain stabilization terms. Other methods, for instance the
discontinuous Galerkin method, may
be used as well but here we stay in the simplest framework of
continuous finite element spaces and Galerkin least-squares stabilization.
Using the abstract framework the formulation of the method is straightforward and
the basic coercivity result also follows. Combining coercivity with
the consistency of the finite element method and applying interpolation
error estimates for cut finite element methods on embedded manifolds
\cite{BurHanLarMas16}, we obtain a priori error estimates that are
optimal in the sense typical for stabilized finite element methods
applied to the transport equation.
\paragraph{Earlier Work.}
The computation of flows in fractured media has received increasing
attention lately. For modelling of the equations of flow and transport
in porous media we refer to \cite{MR1911534,MR2359657} and
in particular the mixed dimensional models presented in
\cite{NoBo17}, and \cite{BoNoVa17}.
Finite volume approaches have been proposed
\cite{2016arXiv160106977B,MR3489127,MR3671645} and virtual elements
in \cite{FumKei17}. For stochastic
methods we refer to \cite{MR3372046,MR3595292}. Various model reduction
techniques have been proposed such as
\cite{MR3489127,MR3307584}. Other work considers meshed fractures
\cite{MR2400465} or particle methods \cite{MR3437907}.
Compared to meshed methods cut finite element methods have the advantage
that we do not need to construct a mesh that fits a possibly complex arrangement
of fractures. For surface
surface partial differential equations this approach, also called trace finite
elements, was first introduced in \cite{OlReGr09} and has then been developed
in different directions including stabilization \cite{BurHanLar15} higher order approximations in \cite{Reu15},
discontinuous Galerkin methods \cite{BurHanLarMas17}, transport problems
\cite{OlsReuXu14-b,2015arXiv151102340B}, embedded membranes \cite{CenHanLar16},
coupled bulk-surface problems \cite{BurHanLarZah16} and \cite{GroOlsReu15},
minimal surface problems \cite{CenHanLar15}, and time
dependent problems on evolving surfaces \cite{HanLarZah16,OlsReu14,OlsReuXu14}, and \cite{Zah17}. We also refer to the overview article
\cite{BurClaHanLarMas15} and the references therein. For the present
work we also draw on experiences from the paper \cite{BurHanLarMas16},
where CutFEMs on on embedded manifolds of arbitrary codimensions was
considered and \cite{MR3709202} for the design of CutFEMs on
composite surfaces.
\paragraph{Outline.} In Section 2 we introduce the notion of fractured domains, the
abstract differential operators on these domains, and formulate an integration by parts
formula, and formulate the model problem both in componentwise and abstract form.
In Section 3 we formulate the finite element method. In Section 4 we derive a priori
error estimates. In Section 5 we present numerical results. In Section 6 we draw some
conclusions and mention directions for future work.
\section{The Model Problem}
\subsection{The Domain and Function Spaces}
\label{sec:defs}
We here introduce the notation needed to describe a fractured domain and define
the appropriate function spaces on such a domain.
\paragraph{Composite Domain.} Let $\Omega$ be a
domain in $\mathbb{R}^n$ such that
\begin{itemize}
\item There is a partition $\mathcal{O} = \{\Omega_d\}_{d=0}^n$,
\begin{equation}
\Omega = \cup_{d=0}^n \Omega_d
\end{equation}
\item For each $\Omega_d \in \mathcal{O}$ there is a partition $\mathcal{O}_d = \{ \Omega_{d,i} \}_{i=1}^{n_d}$,
\begin{equation}
\Omega_d = \cup_{i=1}^{n_d} \Omega_{d,i}
\end{equation}
where each $\Omega_{d,i}$ is a smooth $d$-dimensional manifold with
boundary $\partial \Omega_{d,i}$.
\item The partition satisfies
\begin{equation}
\partial \Omega_{d,i} \subset \cup_{l=1}^{d-1} \Omega_{l} \qquad i = 1,\dots, n_d, \quad d =0,\dots,n
\end{equation}
\item We define the boundary operators
\begin{equation}
\partial_d \mathcal{O}_{d} = \bigsqcup_{i=1}^{n_d} \partial \Omega_{d,i}
\qquad
\partial \mathcal{O} = \bigsqcup_{d=0}^n \partial \mathcal{O}_{d}
\end{equation}
where $\sqcup$ denotes the disjoint union.
\end{itemize}
The notation introduced here for partitions in $\mathcal{O}$ is illustrated in Figure~\ref{fig:notation}.
\begin{figure}
\caption{Illustration of notation used for components of different dimension in a fractured domain in $\mathbb{R}
\label{fig:notation}
\end{figure}
\paragraph{Function Spaces on $\mathcal{O}$ and $\partial \mathcal{O}$.}
\begin{itemize}
\item Let $H^s(\Omega_{d,i})$ be the Sobolev space on the manifold $\Omega_{d,i}
\in \mathcal{O}$ of order $s$ with scalar product $(v,w)_{H^s(\Omega_{d,i})}$,
and define
\begin{equation}
H^s(\mathcal{O}_{d}) = \bigoplus_{i=1}^{n_d} H^s(\Omega_{d,i}) ,
\qquad
H^s(\mathcal{O}) = \bigoplus_{d=0}^n H^s(\Omega_d)
\end{equation}
with scalar products
\begin{equation}
(v_d,w_d)_{H^s(\mathcal{O}_d)} = \sum_{i=1}^{n_d} (v,w)_{H^s(\Omega_{d,i})},
\qquad
(v,w)_{H^s(\mathcal{O})} = \sum_{d=0}^n (v_d,w_d)_{H^s(\Omega_d)}
\end{equation}
and inner product norms $\| v_d \|_{H^s(\mathcal{O}_d)}$ and $\|v\|_{H^s(\mathcal{O})}$. For
$d=0$, $H^s(\Omega_{0,i}) = \mathbb{R}$ and is equipped with the usual absolute value
$\| v \|^2_{H^s(\Omega_{0,i})} = v^2$.
\item In the case $s=0$ we use the notation $L^2(\mathcal{O}_d)=H^0(\mathcal{O}_d)$
and $L^2(\mathcal{O})=H^0(\mathcal{O})$ with scalar products $(v_d,w_d)_{\mathcal{O}_d}$
and $(v,w)_\mathcal{O}$ and norms $\|v\|_{\mathcal{O}_d}$ and $\| v \|_\mathcal{O}$
\item On $\partial \mathcal{O}$ we define
$L^2(\partial \mathcal{O}) = \bigoplus_{d=1}^n \bigoplus_{i=1}^{n_d} L^2(\partial \Omega_{d,i})$ and
we equip the components in $\partial \mathcal{O}_d$ with the natural $d-1$ dimensional
measure and thus all components of dimension less or equal to $d-2$ has measure
zero which means that
\begin{equation}
(v,w)_{\partial \mathcal{O}_d} = \sum_{i=1}^{n_d} (v,w)_{\partial \Omega_{d,i}} =
\sum_{i=1}^{n_d} (v,w)_{\partial \Omega_{d,i}\cap \Omega_{d-1}}
\end{equation}
\end{itemize}
\paragraph{Tangential and Normal Vector Fields.}
\begin{itemize}
\item We say that $a= \oplus_{d=0}^n a_{d}$ is a tangential vector field on $\mathcal{O}$
if $a_d = \oplus_{i=1}^{n_d} a_{d,i}$ and each $a_{d,i}$ is a tangential vector field
on the manifold $\Omega_{d,i} \in \mathcal{O}_d$.
\item We define the unit exterior normal vector field $\nu$ on $\partial \mathcal{O}$
by $\nu|_{\partial \Omega_{d,i}} = \nu_{d,i}$, where $\nu_{d,i}$ is the unit
tangential vector field on $\Omega_{d,i}$ which is orthogonal to $\partial \Omega_{d,i}$
and exterior to $\Omega_{d,i}$, see Figure~\ref{fig:normals}.
\item The pointwise dot product $a \cdot b$ of two tangential vector fields $a$
and $b$ on $\mathcal{O}$ is the scalar field $(a\cdot b)_d = a_d \cdot b_d$ on each
$\mathcal{O}_d$, $d=0,\dots,n$.
\end{itemize}
\paragraph{Tangential Gradient.}
\begin{itemize}
\item For $\delta>0$ let $U^n_\delta(\Omega_{d,i}) = \cup_{x\in \Omega_{d,i}} B_\delta(x) \subset \mathbb{R}^n$, where $B_\delta(x)$ is the open ball of radius $\delta$ withe center $x$, be an open neighborhood of $\Omega_{d,i}$. Then there is a continuous extension operator
$E:v \in H^s(\Omega_{d,i}) \rightarrow H^s(U^n_\delta(\Omega_{d,i}))$, see
\cite{BuHaLaLaMa15} for the construction necessary to handle the fact that
$\Omega_{d,i}$ has a boundary. We employ the
shorthand notation $Ev = v^e$ when necessary for clarity otherwise we simplify
further and write $v = v^e$.
\item Let $\nabla_d$ be the tangential gradient on $\Omega_d$ and
\begin{equation}
\nabla v = \oplus_{d=1}^{n} \nabla_d v_d
\end{equation}
where for each $x \in \Omega_{d,i}$, $(\nabla_d v)|_x = (P_d \nabla_{\mathbb{R}^n} v^e)|_x $
and $P_d|_x: \mathbb{R}^n \rightarrow T_x(\Omega_{d,i})$ is the projection onto the tangent plane $T_x(\Omega_{d,i})$.
\item Given a tangential vector field $\beta$ let
\begin{equation}
V_{\beta} = \{ v \in L^2(\mathcal{O}) : \|\beta \cdot \nabla v\|_{\mathcal{O}} \lesssim 1 \}
\end{equation}
In other words we for $v\in V_{\beta}$ in each component $\Omega_{d,i}\in\mathcal{O}$ have that $v|_{\Omega_{d,i}} \in L^2(\Omega_{d,i})$ and $v|_{\Omega_{d,i}} \in H^1(\omega)$ where $\omega$ is any $d-1$ dimensional manifold $\omega$ tangential to $\beta|_{\Omega_{d,i}}$.
\end{itemize}
\begin{figure}
\caption{Fractured domain}
\label{fig:domain-ex-a}
\caption{Exterior unit normal field}
\label{fig:domain-ex-b}
\caption{Illustration of the exterior unit normal vector field on a fractured domain. (a) A fractured domain with $n=2$, $n_0=8$, $n_1=10$, and $n_2=3$. (b) The exterior unit normal field for the domain in (a).}
\label{fig:normals}
\end{figure}
\subsection{Abstract Differential Operators}
In this section we introduce jump operators used for coupling between different subdomains and
also differential operators that enable formulation of the convection problem in a compact form.
\paragraph{Jump Operators.} To express the coupling between subdomains we use the following operators:
\begin{itemize}
\item The jump operator $\llbracket \cdot\rrbracket_d:L^2(\partial \mathcal{O}_{d+1} )
\rightarrow L^2(\mathcal{O}_d)$ is defined by $\llbracket \cdot\rrbracket_n
= 0$ and for $d=0,\dots,n-1$,
\begin{equation}
\llbracket v_{d+1} \rrbracket_{d}|_{\Omega_{d,i}}
= \sum_{j=1}^{n_{d+1}} v_{d+1,j} |_{\partial \Omega_{d+1} \cap \Omega_{d,i}}
\end{equation}
We then have the identity
\begin{equation}
(v_{d+1},w_d)_{\partial \mathcal{O}_{d+1}} = (\llbracket v_{d+1} \rrbracket_d, w_d)_{\mathcal{O}_d}
\qquad w_d \in L^2(\mathcal{O}_d)
\end{equation}
and we also note that
\begin{equation}
\llbracket v_{d+1} w_d \rrbracket_d = \llbracket v_{d+1} \rrbracket_d w_d \qquad w_d \in L^2(\mathcal{O}_d)
\end{equation}
\item The jump operator $[\cdot ]_d: L^2(\mathcal{O}_{d-1}) \times L^2(\partial \mathcal{O}_{d})
\rightarrow L^2(\partial \mathcal{O}_{d})$ is defined by $[v]_0 = 0$ and for $d=1,\dots,n$,
\begin{equation}
[v]_{d,i} |_{\partial \Omega_{d,i}} = v_{d,i}|_{\partial \Omega_{d,i}} - \sum_{j=1}^{n_{d-1}}v_{d-1,j}|_{\partial\Omega_{d,i}\cap\Omega_{d-1,j}}
\end{equation}
\end{itemize}
Note that the jump operators provide the coupling between the different
subdomains and that only neighboring subdomains with difference in dimension
equal to one couple to each other.
\paragraph{The Directional Derivative and Divergence Operators.}
Let $\beta$ be a smooth tangential vector field on $\mathcal{O}$, i.e.
$(\beta)_{d,i}$ is a smooth tangential vector field on each $\Omega_{d,i} \in \mathcal{O}$,
and let $\nu$ be the unit exterior normal vector field on $\partial \mathcal{O}$ defined
in Section \ref{sec:defs}.
\begin{itemize}
\item Let the derivative $D_\beta$ in the direction $\beta$ be defined by
\begin{equation}
( D_{\beta} v )_n = \beta_n \cdot \nabla_n v_n,
\qquad
(D_{\beta} v )_0 = \sum_{i=1}^{n_1} \nu_{1,i} \cdot \beta_{1,i} (v_{0} - v_{1,i})
\end{equation}
and for $d=1,\dots,n-1,$ let
\begin{equation}
(D_{\beta} v)_d
= \beta_d \cdot \nabla_d v_d + \sum_{i=1}^{n_{d+1}}
\nu_{d+1,i} \cdot \beta_{d+1,i} (v_{d} - v_{d+1,i})
\end{equation}
or equivalently in terms of the jump operators
\begin{equation}
(D_\beta v)_d = \beta_d \cdot \nabla_d v_d - \llbracket \nu_{d+1} \cdot \beta_{d+1} [v]_{d+1}\rrbracket_{d}
\end{equation}
\item Let the divergence $\Div \beta$ be defined by
\begin{equation}\label{eq:Div}
(\Div\beta)_d = \nabla_d \cdot \beta_d - \sum_{i=1}^{n_{d+1}}
\nu_{d+1,i} \cdot \beta_{d+1,i}
\end{equation}
or equivalently in terms of the jump operators
\begin{equation}\label{eq:Div-jump}
(\Div\beta)_d = \nabla_d \cdot \beta_d
- \llbracket \nu_{d+1} \cdot \beta_{d+1} \rrbracket_d
\end{equation}
\end{itemize}
In order to formulate a partial integration formula for $D_\beta$ we introduce the
notation
\begin{equation}
\partial \mathcal{O}_B = \partial \mathcal{O} \cap \partial \Omega
=
\sqcup_{i,d} (\partial \Omega_{i,d} \cap \partial \Omega),
\qquad
\partial \mathcal{O}_I = \partial \mathcal{O} \setminus \partial \Omega
= \sqcup_{i,d} (\partial \Omega_{i,d} \setminus \partial \Omega)
\end{equation}
to denote the components in $\partial \mathcal{O}$ which belong to the boundary and the interior respectively. We end this section by stating a lemma from \cite{BuHaLaLa18-b}.
\begin{lem}\label{lem:partial-integration} {\bf(Partial Integration)}
For a smooth tangential vector field $\beta$ on $\mathcal{O}$ and $v\in V_\beta$,
\begin{equation}\label{eq:Div-formula}
\Div(\beta v ) = D_\beta v + (\Div\beta) v
\end{equation}
and for $v,w\in V_\beta$,
\begin{equation}\label{eq:Dbeta-partial-integration}
(D_\beta v, w)_\mathcal{O} = - (v,D_\beta w)_{\mathcal{O}} -((\Div\beta) v,w)_{\mathcal{O}}
+ (\nu \cdot \beta [v],[w])_{\partial \mathcal{O}_I}
+ (\nu \cdot \beta v, w)_{\partial \mathcal{O}_B}
\end{equation}
where $\nu$ is the exterior unit normal vector field on $\partial \mathcal{O}$.
\end{lem}
\subsection{The Model Problem}
In this section we introduce our model convection problem on a fractured domain.
\paragraph{Componentwise Formulation.}
Find $u_{d,i}:\Omega_{d,i} \rightarrow \mathbb{R}$ such that
\begin{alignat}{3}\label{eq:strong-form-bulk}
\nabla_{d,i}\cdot (\beta_{d,i} u_{d,i} ) + \alpha_{d,i} u_{d,i}
- \llbracket \nu_{d+1} \cdot \beta_{d+1} u_{d+1}\rrbracket_{d,i}
&= f_{d,i}& \qquad &\text{in $\Omega_{d,i}$}
\\ \label{eq:strong-form-inflow-internal}
(\nu_{d,i} \cdot \beta_{d,i})_- [ u ]_{d,i} &= 0 & \qquad &\text{on $\partial \Omega_{d,i} \setminus \partial \Omega$}
\\ \label{eq:strong-form-inflow-boundary}
(\nu_{d,i} \cdot \beta_{d,i})_- (u_{d,i} - g_{d,i}) &=0 & \qquad &\text{on
$ \partial \Omega_{d,i} \cap \partial \Omega$}
\end{alignat}
where
$(v)_- = \min(v,0)$ denotes the negative part of
$v$.
\paragraph{Abstract Formulation.}
We note that using the definition (\ref{eq:Div}) of the divergence we may rewrite
(\ref{eq:strong-form-bulk}) as follows
\begin{align}\nonumber
&\nabla_{d,i}\cdot (\beta_{d,i} u_{d,i} ) + \alpha_{d,i} u_{d,i}
- \llbracket \nu_{d+1} \cdot \beta_{d+1} u_{d+1}\rrbracket_{d,i}
\\
&\qquad =
\beta_{d,i} \cdot \nabla_{d,i} u_{d,i}
+ (\nabla_{d,i}\cdot \beta_{d,i} ) u_{d,i} + \alpha_{d,i} u_{d,i}
\\ \nonumber
&\qquad \qquad - \llbracket \nu_{d+1} \cdot \beta_{d+1} (u_{d+1} - u_d) \rrbracket_{d,i}
- \llbracket \nu_{d+1} \cdot \beta_{d+1} u_d \rrbracket_{d,i}
\\
&\qquad =
\beta_{d,i} \cdot \nabla_{d,i} u_{d,i} - \llbracket \nu_{d+1} \cdot \beta_{d+1} [u]_{d+1} \rrbracket_{d,i}
\\ \nonumber
&\qquad \qquad + (\nabla_{d,i}\cdot \beta_{d,i} ) u_{d,i} - \llbracket \nu_{d+1} \cdot \beta_{d+1} \rrbracket_{d,i} u_d
+ \alpha_{d,i} u_{d,i}
\\
&\qquad = (D_\beta u + \Div \beta + \alpha )_{d,i}
\end{align}
where we essentially added and subtracted $\llbracket \nu_{d+1} \cdot \beta_{d+1} u_d \rrbracket$
and rearranged the terms. Thus in terms of the abstract operators (\ref{eq:strong-form-bulk})
takes the form
\begin{equation}
D_\beta u + (\alpha + \Div\beta )u = f
\end{equation}
Thus we obtain the problem: find $u \in V$ such that
\begin{alignat}{3}\label{eq:problem-a}
D_\beta u + \gamma u &=f & \qquad &\text{in $\mathcal{O}$}
\\ \label{eq:problem-b}
(\nu \cdot \beta)_- [ u ] &= 0 & \qquad &\text{on $\partial \mathcal{O}_I$}
\\ \label{eq:problem-c}
(\nu \cdot \beta)_- ( u - g) &= 0 & \qquad &\text{on $\partial \mathcal{O}_B$}
\end{alignat}
where $\gamma = \alpha + \Div \beta$ or in component form
\begin{equation}
\gamma_d = \alpha_d + \nabla_d \cdot \beta_d - \llbracket \nu_{d+1} \cdot \beta_{d+1} \rrbracket_{d}
\end{equation}
\paragraph{Weak Formulation.} Find $u \in V_\beta$ such that
\begin{equation}\label{eq:weak-problem}
a(u,v) = l(v) \qquad \forall v \in V_\beta
\end{equation}
where the forms are defined by
\begin{align}
a(v,w) &= (D_\beta v,w)_\mathcal{O} + (\gamma v,w)_\mathcal{O}
+ (|\nu \cdot \beta|_- [ v ],[w])_{\partial \mathcal{O}_I}
+ (|\nu \cdot \beta|_- v , w)_{\partial \mathcal{O}_B}
\\
l(w) &= (f,w)_\mathcal{O} + (|\nu \cdot \beta|_- g, w)_{\partial \mathcal{O}_B}
\end{align}
and we used the simplified notation $| v |_- = |(v)_-|$, for the absolute value of the
negative part. Using Lemma \ref{lem:partial-integration} we may derive the following stability
result.
\begin{lem}\label{lem:coercivity}{\bf (Coercivity)}
If there is a constant $c_0>0$ such that
\begin{equation}\label{eq:assumption-coeff}
c_0 \leq \| 2 \alpha + \Div \beta \|_{L^\infty(\Gamma)}
\end{equation}
then
\begin{equation}\label{eq:coercivity}
\| v \|^2_{\mathcal{O}}
+ \|[v]\|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_I}
+ \| v \|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_B}
\lesssim
a(v,v)
\qquad \forall v\in V_\beta
\end{equation}
where we introduced the norms
\begin{equation}
\| w \|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_J} = \left\| ( |\nu \cdot \beta|_{-} )^{1/2} w \right\|^2_{\partial \mathcal{O}_J} ,\quad J\in\{I,B\}
\end{equation}
\end{lem}
\section{The Cut Finite Element Method}
\subsection{The Mesh and Finite Element Spaces}
\begin{itemize}
\item Let $\Omega_0 \in \mathbb{R}^n$ be a polygonal domain such that $\Omega\subset \Omega_0$
and let $\{ \mathcal{T}_{h,0},\, h\in (0,h_0] \}$ for some constant $h_0 > 0$ be a family of quasi-uniform
meshes with mesh parameter $h$ of $\Omega_0$ om shape regular elements $T$.
\item Let $V_{h,0}$ be a finite element space of continuous piecewise polynomial functions on $\mathcal{T}_{h,0}$. We consider low order elements with linear polynomials or tensor product polynomials. Adaption to higher order elements is outlined in the next section.
\item For each $\Omega_{d,i} \in \mathcal{O}$ let the active mesh be defined by
\begin{equation}\label{eq:mesh-active}
\mathcal{T}_{h,d,i} = \{ T \in \mathcal{T}_{h,0} : T\cap \Omega_{d,i} \neq \emptyset \}
\end{equation}
and define the associated finite element space $V_{h,d,i} = V_{h,0} |_{\Omega_{d,i}}$, see Figure~\ref{fig:meshes}. Note that in most cases it is not necessary to introduce active meshes on components without a source term that constitute part of the boundary. This is due to the solution in those parts being directly given by either the boundary condition or the coupling to a higher dimensional component. For simplicity, we therefore from this point on assume all components $\Omega_{d,i} \in \mathcal{O}$ satisfy $\Omega_{d,i} \cap \partial\mathcal{O}_B = \emptyset$.
\item Define the finite element space on $\mathcal{O}$ as the direct sum
\begin{equation}
V_h = \bigoplus_{d=0}^n V_{h,d}, \qquad V_{h,d} = \bigoplus_{i=1}^{n_d} V_{h,d,i}
\end{equation}
\end{itemize}
\begin{figure}
\caption{$d=2$}
\label{fig:meshes-codim0}
\caption{$d=1$}
\label{fig:meshes-codim1}
\caption{$d=0$}
\label{fig:meshes-codim2}
\caption{Meshes for an example geometry in 2D consisting of three bulk domains ($d=2$), three cracks ($d=1$), and one bifurcation point ($d=0$). The colored parts are the active meshes $\{\mathcal{T}
\label{fig:meshes}
\end{figure}
\subsection{The Method}
We consider a finite element method based on the weak formulation (\ref{eq:weak-problem}) which takes care of the coupling between the different domains. Using a conforming
finite element space we will need to stabilize
the convection term and furthermore since we are using a cut finite element method
we need to stabilize in order to control the variation of the solution orthogonal to
$\Omega_{d,i}$. For simplicity, we will consider piecewise linear elements and use
standard Galerkin Least Squares (GLS) method together with so called full gradient
stabilization for the cut elements developed in \cite{BuHaLaMaZa16}. The full gradient
stabilization adds control of the variation of the finite element solution in the direction
orthogonal to the manifold $\Omega_{d,i}$ and also provides control of the resulting
condition number of the linear system of equations. The full gradient stabilization is not
consistent and we scale it in such a way that we do not lose order of convergence.
Essentially, for linear elements we obtain an artificial tangent diffusion of order $h^{3/2}$.
In the case of higher order elements we may use a weaker full gradient stabilization or
preferably a more refined stabilization which is consistent (on exact geometry) such
as the recently developed normal stabilization, \cite{BurHanLarMas16} and \cite{GraLehReu16}, or
the combined normal-face stabilization \cite{LaZa17}.
\paragraph{Galerkin Least Squares (GLS).}
Find $u_h \in V_h$ such that
\begin{equation}\label{eq:fem}
a_h(u_h,v) = l_h(v)\qquad \forall v \in V_h
\end{equation}
where
\begin{align}
a_h(v,w) &= \sum_{d=0}^n \sum_{i=1}^{n_d} a_{h,d,i}(v_{d,i},w_{d,i})
+ (|\nu\cdot \beta|_- [v],[w])_{\partial \mathcal{O}_I} + (|\nu\cdot \beta|_- v,w)_{\partial \mathcal{O}_B}
\\
l_h(v) &= \sum_{d=0}^n \sum_{i=1}^{n_d} l_{h,d,i}(v_{d,i})
\end{align}
The forms $a_{h,d,i}$ and $l_{d,h,i}$ are linear forms on $V_{h,d,i}$ defined by
\begin{align}
a_{h,d,i}(v,w)
&=
(L_d v, w)_{\Omega_{d,i}} + (\tau_1 h L_d v, L_d w )_{\Omega_{d,i}}
+ s_{h,d,i}(v,w)
\\
l_{h,d,i}(v) &= (f_{d,i}, v_{d,i})_{\Omega_{d,i}}
+ (\tau_1 h f_{d,i}, L_d v)_{\Omega_{d,i}}
+ (|\nu\cdot \beta|_{-} g,v)_{\partial \Omega_{d,i}}
\end{align}
where $\tau_1>0$ is a parameter
\begin{align}
L_{d,i} v &= (D_\beta v + \gamma v)|_{d,i}
\\
&=\beta_{d,i}\cdot \nabla_{d,i} v_{d,i}
+ ( (\nabla_{d,i} \cdot (\beta_{d,i} ) + \alpha_{d,i}) v_{d,i}
- \llbracket \nu_{d+1} \cdot \beta_{d+1} v_{d+1}\rrbracket_{d,i}
\end{align}
and $s_{h,d,i}$ is the stabilization form
\begin{equation}
s_{h,d,i}(v,w) = \tau_2 h^{3 - (n-d)} (\nabla_{\mathbb{R}^n} v, \nabla_{\mathbb{R}^n} w )_{\mathcal{T}_{h,d,i}}
\end{equation}
where $\tau_2$ is a parameter and $\nabla_{\mathbb{R}^n}$ denotes the gradient in $\mathbb{R}^n$.
We also note that $n-d$ is the codimension of $\Omega_{d,i}$ and thus the scaling
factor $h^{-(n-d)}$ compensates for the fact that we integrate over the $n$ dimensional
set $\mathcal{T}_{h,d,i}$. We will see that the additional $h^3$ scaling ensures that we do not
lose order of convergence when adding $s_{h}$.
\section{Error Estimates}
We prove a basic error estimate in the natural energy norm associated with
the GLS method. We assume that the geometry is represented exactly and
that all integrals are computed exactly. In this situation the proof is done using
the standard techniques combined with an interpolation error estimate for
manifolds of arbitrary codimension. Estimates of the geometric error can be
done using a generalization of the approach developed in \cite{BurHanLarMas16}.
\subsection{Coercivity and Continuity}
Let
\begin{equation}
V^e = \{v^e = Ev \,:\, v \in V_\beta \} \,, \qquad W = V^e + V_h
\end{equation}
where $E$ is the extension operator defined in Section~\ref{sec:defs} when introducing the tangential gradient.
Define the energy norm
\begin{equation}\label{eq:energy-norm}
|\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|^2_h = \| v \|^2_\mathcal{O} + h \| L v \|^2_\mathcal{O} + \| v \|^2_{s_h}
+ \|[v]\|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_I}
+ \| v \|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_B},
\qquad v \in W
\end{equation}
and the norm
\begin{equation}\label{eq:energy-norm-cont}
|\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|^2_{h,*} = h^{-1} \| v \|^2_\mathcal{O}+|\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|^2_h,
\qquad v \in W
\end{equation}
which we will need in the statement of continuity.
\begin{lem}
The form $a_h$ is continuous
\begin{equation}\label{eq:Ah-continuity}
a_h(v, w) \lesssim |\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|_{h,*} |\kern-1pt|\kern-1pt| w |\kern-1pt|\kern-1pt|_h, \qquad v,w \in W
\end{equation}
and if (\ref{eq:assumption-coeff}) holds coercive
\begin{equation}\label{eq:Ah-coercivity}
|\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|_h^2 \lesssim a_h(v,v)\qquad v,w \in W
\end{equation}
\end{lem}
\begin{proof} The continuity (\ref{eq:Ah-continuity}) follows by
first applying the Cauchy-Schwarz inequality in all the symmetric
terms of $a_h$,
\begin{equation}
a_h(v, w) \lesssim (D_\beta v,w)_{\mathcal{O}} + |\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|_h |\kern-1pt|\kern-1pt| w |\kern-1pt|\kern-1pt|_h
\end{equation}
Using the integration by parts formula in the first term of the right
hand side yields
\begin{align}
(D_\beta v,w)_{\mathcal{O}} &= -((\Div\beta) v,w)_{\mathcal{O}} - (v,D_\beta w)_{\mathcal{O}}
+ (\nu \cdot \beta [v],[w])_{\partial \mathcal{O}_I}
+ (\nu \cdot \beta v, w)_{\partial \mathcal{O}_B}
\\
&\leq (v,D_\beta w)_{\mathcal{O}} + C |\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|_h |\kern-1pt|\kern-1pt| w |\kern-1pt|\kern-1pt|_h
\\
&\leq h^{-1/2} \|v\|_\mathcal{O} h^{1/2} \|L w\|_\mathcal{O} + C |\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|_h |\kern-1pt|\kern-1pt| w |\kern-1pt|\kern-1pt|_h
\\
&\lesssim |\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|_{h,*} |\kern-1pt|\kern-1pt| w |\kern-1pt|\kern-1pt|_h
\end{align}
where we used the uniform bound $\|\Div\beta \|_{L^\infty(\mathcal{O})} \lesssim 1$
and the definition of the norm $ |\kern-1pt|\kern-1pt| \cdot |\kern-1pt|\kern-1pt|_{h,*}$ in the last step.
The coercivity (\ref{eq:Ah-coercivity}) follows by observing that
\begin{equation}
a_h(v,w) = a(v,w) + (\tau_1 h L v, L w)_{\mathcal{O}} + s_h(v,w)
\end{equation}
and thus
\begin{align}
a_h(v,v) &= a(v,v) + \tau_1 h \| L v \|^2_\mathcal{O} + \| v \|^2_{s_h}
\\
&\gtrsim \| v \|^2_\mathcal{O} + \|[v]\|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_I}
+ \| v \|^2_{|\nu \cdot \beta|,\partial \mathcal{O}_B}
+ \tau_1 h \| L v \|^2_\mathcal{O} + \| v \|^2_{s_h}
\\
&= |\kern-1pt|\kern-1pt| v |\kern-1pt|\kern-1pt|^2_h
\end{align}
where we used Lemma \ref{lem:coercivity}.
\end{proof}
\subsection{Interpolation Error Estimates}
There is an interpolation operator $\pi_h:L^2(\Omega_{d,i}) \rightarrow V_{h,d,i}$ such
that the following interpolation error estimate holds
\begin{equation}\label{eq:interpol-energy}
|\kern-1pt|\kern-1pt| v - \pi_h v |\kern-1pt|\kern-1pt|^2_*\lesssim h^{3} \| v \|^2_{H^{k+1}(\mathcal{O})}
\end{equation}
We define $\pi_h$ by
\begin{equation}
\pi_h v = \pi_{h,Cl} v^e
\end{equation}
where $\pi_{h,Cl}: L^2(\mathcal{T}_{h,d,i}) \rightarrow V_{h,d,i}$ is the usual Clement interpolator.
We refer to \cite{BurHanLarMas16} for further details including a proof of the basic interpolation
estimate
\begin{equation}
\| u - \pi_h u \|^2_{\Omega_{d,i}} + h^2 \| \nabla_d ( u - \pi_h u )\|^2_{\Omega_{d,i}}
\lesssim h^4 \| u \|^2_{H^2(\Omega_{d,i})}
\end{equation}
which is the used to derive (\ref{eq:interpol-energy}).
\subsection{Error Estimates}
\begin{thm} If $u$ is the solution to (\ref{eq:weak-problem}) satisfies $u \in H^2(\mathcal{O})$
and $u_h$ is the finite element approximation defined by (\ref{eq:fem}), then
\begin{equation}
|\kern-1pt|\kern-1pt| u - u_h |\kern-1pt|\kern-1pt|_h^2 \lesssim h^{3} \| u \|^2_{H^{2}(\mathcal{O})}
\end{equation}
\end{thm}
\begin{proof} Using coercivity
\begin{align}
|\kern-1pt|\kern-1pt| u - u_h |\kern-1pt|\kern-1pt|_h^2 &\lesssim a_h(u - u_h, u - u_h)
\\
&\lesssim a_h(u - u_h, u - \pi_h u) + a_h(u - u_h, \pi_h u - u_h)
\\
&\lesssim |\kern-1pt|\kern-1pt| u - u_h |\kern-1pt|\kern-1pt|_h \|u - \pi_h u|\kern-1pt|\kern-1pt|_{h,*} + a_h(u, \pi_h u - u_h) - l_h(\pi_h u - u_h)
\\
&\lesssim \delta |\kern-1pt|\kern-1pt| u - u_h |\kern-1pt|\kern-1pt|^2_h + \delta^{-1} |\kern-1pt|\kern-1pt| u - \pi_h u|\kern-1pt|\kern-1pt|^2_{h,*} + s_h(u^e, \pi_h u - u_h)
\end{align}
for $\delta>0$. Next
\begin{align}
s_h (u, \pi_h u - u_h) &= s_h (u, \pi_h u - u ) + s_h (u, u - u_h )
\\
&\leq \| u \|_{s_h} \| \pi_h u - u \|_{s_h} + \|u\|_{s_h} \| u - u_h \|_{s_h}
\\
&\leq \| u \|^2_{s_h}
+ \underbrace{\| \pi_h u - u \|^2_{s_h}}_{\leq |\kern-1pt|\kern-1pt| u - \pi_h u |\kern-1pt|\kern-1pt|^2_{h,*}}
+\delta^{-1} \|u\|^2_{s_h}
+ \delta\| u - u_h \|^2_{s_h}
\end{align}
Using kick back and taking $\delta>0$ small enough we arrive at
\begin{equation}
|\kern-1pt|\kern-1pt| u - u_h |\kern-1pt|\kern-1pt|_h^2 \lesssim \|u - \pi_h u|\kern-1pt|\kern-1pt|^2_{h,*} + \| u \|^2_{s_h}
\lesssim h^3 \| u \|_{H^2(\mathcal{O})}^2 + h^3 \| u \|^2_{H^1(\mathcal{O})}
\end{equation}
where we used the interpolation error bound
(\ref{eq:interpol-energy}) for the first term and the second was estimated as follows
\begin{equation}
\| u \|^2_{s_h}
= \sum_{d=0}^n \sum_{i=1}^{n_d} \tau_2 h^{3 - (n-d)} \| \nabla_{\mathbb{R}^n} u^e \|^2_{\mathcal{T}_{h,d,i}}
\lesssim \sum_{d=0}^n \sum_{i=1}^{n_d} h^3 \| \nabla_d u \|^2_{\Omega_{d,i}}
\end{equation}
where we used the estimate
\begin{equation}
\| \nabla_{\mathbb{R}^n} u^e \|^2_{\mathcal{T}_{h,d,i}}
\lesssim \| \nabla_d u \|^2_{\mathcal{T}_{h,d,i}}
\lesssim h^{n-d} \| \nabla_d u \|^2_{\Omega_{d,i}}
\end{equation}
which completes the proof.
\end{proof}
\section{Numerical Examples}
\paragraph{Implementation.}
To generate numerical examples we implemented the method \eqref{eq:fem} in 2D, i.e. $n=2$, which means that in our examples the fractured domains may consist of bulk domains ($d=2$), cracks ($d=1$), and bifurcation points ($d=0$).
We first generate a background triangle mesh $\mathcal{T}_{h,0}$ embedding the complete geometry and from this mesh we extract an active mesh for each bulk domain, crack domain and bifurcations point, see Figure~\ref{fig:meshes}. On each active mesh we then define a finite element space consisting of linear elements. Note that, while we do generate an active mesh and corresponding linear finite element space for each bifurcation point, this is actually not required as the solution there will only be a point value making it redundant to define a finite element.
\paragraph{Parameters and Meshes.} In all our examples below we use the Galerkin least squares parameter $\tau_1=10^{-2}$, stabilization parameter $\tau_2=10^{-3}$ and $\alpha_{d,i}=0$. Also, in all examples the background mesh $\mathcal{T}_{h,0}$ is a triangulation of the unit square $\Omega_0 = [0,1]^2$ with mesh parameter $h=0.1$. The resulting active meshes for Example 1--3 are presented in Figure~\ref{fig:ex1-3-mesh} while the active meshes for Example 4, which also includes a bifurcation point, are presented in Figure~\ref{fig:ex4-mesh}.
\paragraph{Example 1: Crack with in-flow.} This simple example is outlined in Figure~\ref{fig:ex1} where a crack divides the unit square in half. Here the vector fields $\{\beta_{2,i}\}$ in the bulk domains only goes into the crack resulting in the solution on the crack being effected by the bulk solutions but not the other way around. For this example we can actually derive an exact solution where $u=1$ in the bulk domains and $u=2y$ on the crack. As this solution lies in $V_h$ our numerical approximation coincides with the exact solution.
\paragraph{Example 2: Crack with out-flow.} In this example presented in Figure~\ref{fig:ex2} we revert the bulk vector fields in Example 1, yielding a crack with only out-flow to the bulk. As expected the solution in the bulk is affected by the solution on the crack but not the other way around. Also in this case we can derive the exact solution, $u=e^{-2y}$, which is well approximated by our numerical solution.
\paragraph{Example 3: Flow crossing a crack.} In Figure~\ref{fig:ex3} we consider the same geometry as in previous examples but with diagonal bulk vector fields passing through the crack. First, in Figure~\ref{fig:ex3-num1} we consider the case where the vector field in the crack is zero which results in there being no transport in the crack. The presence of the crack in this case actually doesn't effect the solution at all which gives some modeling possibilities as the presence of a crack also allow for discontinuous solutions. Increasing the crack vector field, we note in Figures~\ref{fig:ex3-num2}--\ref{fig:ex3-num3}, that the solution is transported along the crack when passing to the other side.
\paragraph{Example 4: Cracks with a bifurcation point.} This example is presented in Figure~\ref{fig:case4-illustration} and the active meshes used are presented in Figure~\ref{fig:ex4-mesh}. In contrast to previous examples we here also include a bifurcation point where the crack splits. From the top and bottom bulk domains we have flow into the crack while we from the third bulk domain have flow out of the crack. First, in Figure~\ref{fig:case4-same}, we consider the case where the vector fields on the cracks all are unit vectors in the tangential direction. We note that the solution flowing into the bifurcation point is then evenly divided between the two cracks flowing out of the bifurcation point.
In Figure~\ref{fig:case4-diff} we change the relation of the vector fields between the top and bottom cracks, i.e. the cracks flowing out of the bifurcation point, yielding a slightly different distribution. This change also effects the in-flow from the bulk regions which is clear by inspecting the crack solutions further away from the bifurcations point.
\paragraph{Example 5: System of cracks.}
As a final example we in Figure~\ref{fig:ex-tree} consider a system of cracks affected by in-flow from bulk domains. In this case the vector fields on the cracks are again unit vectors in the tangential direction. Thus, at each bifurcation point the sum of the crack solutions flowing into a bifurcation point will equal the sum of the crack solutions flowing out of the bifurcation point.
\section{Conclusions}
We develop a cut finite element method for a convection problem on a fractured domain.
The upshot of the method is that the mesh does not need to conform to the embedded manifolds,
which in practice is very convenient. The cut elements are handled using certain stabilization
terms which leads to a stable method with optimal order convergence properties. Different methods
may be used to discretize the PDE, and we have here chosen to study a least squares stabilized
formulation which is convenient to implement and analyze. Some directions for future work
include existence and uniqueness results for convection problems on fractured domains,
extensions to convection diffusion problems, higher order methods, time dependent problems,
and coupled problems with both flow equations and transport.
\begin{figure}
\caption{$d=2$}
\label{fig:case1-mesh-bulk}
\caption{$d=1$}
\label{fig:case1-mesh-crack}
\caption{Active meshes ($h=0.1$) used for Examples 1--3 where a single crack divides the unit square into two equal parts.}
\label{fig:ex1-3-mesh}
\end{figure}
\begin{figure}
\caption{Set-up}
\label{fig:ex1-illustration}
\caption{Numerical solution}
\label{fig:ex1-num}
\caption{Crack with in-flow (Example 1).
(a) The set-up for this example is $\beta_{2,1}
\label{fig:ex1}
\end{figure}
\begin{figure}
\caption{Set-up}
\label{fig:ex2-illustration}
\caption{Numerical solution}
\label{fig:ex2-num}
\caption{Crack with out-flow (Example 2).
(a) The set-up for this example is $\beta_{2,1}
\label{fig:ex2}
\end{figure}
\begin{figure}
\caption{Set-up}
\label{fig:ex3-illustration}
\caption{$\beta_{1,1}
\label{fig:ex3-num1}
\caption{$\beta_{1,1}
\label{fig:ex3-num2}
\caption{$\beta_{1,1}
\label{fig:ex3-num3}
\caption{Flow crossing a crack (Example 3).
(a) The set-up for this example is $\beta_{2,1}
\label{fig:ex3}
\end{figure}
\begin{figure}
\caption{$d=2$}
\label{fig:case4-mesh-bulk}
\caption{$d=1$}
\label{fig:case4-mesh-crack}
\caption{$d=0$}
\label{fig:case4-mesh-point}
\caption{Active meshes used in Example 4 ($h=0.1$). (a) Meshes for the bulk domains. (b) Meshes for the cracks. (c) Mesh for the bifurcation point.}
\label{fig:ex4-mesh}
\end{figure}
\begin{figure}
\caption{Set-up}
\label{fig:case4-illustration}
\caption{$\beta_{1,2}
\label{fig:case4-same}
\caption{$\beta_{1,2}
\label{fig:case4-diff}
\caption{Cracks with a bifurcation point (Example 4).
(a) Here $\beta_{2,1}
\label{fig:ex4}
\end{figure}
\begin{figure}
\caption{Set-up}
\label{fig:tree-illustration}
\caption{Numerical solution}
\label{fig:tree-num}
\caption{System of cracks with in-flow (Example 5).
(a) Starting in the lower left corner and traversing the bulk domains clockwise $\beta_{2,i}
\label{fig:ex-tree}
\end{figure}
\footnotesize{
}
\noindent
\footnotesize {\bf Acknowledgements.}
This research was supported in part by the Swedish Foundation
for Strategic Research Grant No.\ AM13-0029, the Swedish Research
Council Grants Nos.\ 2013-4708, 2017-03911, and the Swedish
Research Programme Essence. EB was supported by EPSRC research grants EP/P01576X/1 and EP/P012434/1.
\noindent
\footnotesize {\bf Authors' addresses:}
\noindent
Erik Burman, \quad
Mathematics, University College London, UK\\
{\tt [email protected]}
\noindent
Peter Hansbo, \quad
Mechanical Engineering, J\"onk\"oping University, Sweden\\
{\tt [email protected]}
\noindent
Mats G. Larson, \quad
Department of Mathematics and Mathematical Statistics, Ume{\aa}~University, SE-901\,87 Ume{\aa}, Swedenshort\\
{\tt [email protected]}
\noindent
Karl Larsson, \quad
Department of Mathematics and Mathematical Statistics, Ume{\aa}~University, SE-901\,87 Ume{\aa}, Swedenshort\\
{\tt [email protected]}
\end{document} | math | 40,494 |
\betagin{document}
\title{\parbox{14cm}
\betagin{abstract}
In this paper we prove that the set of tuples of edge lengths in $K_1\times K_2$ corresponding to a finite tree has non-empty interior, where $K_1,K_2\subsetbset \ensuremath{\mathbb{R}}$ are Cantor sets of thickness $\tau(K_1)\cdot \tau(K_2) >1$.
Our method relies on establishing that the pinned distance set is robust to small perturbations of the pin.
In the process, we prove a nonlinear version of the classic Newhouse gap lemma, and show that if $K_1,K_2$ are as above and $\phi: \ensuremath{\mathbb{R}}^2\times \ensuremath{\mathbb{R}}^2 \rightarrow \ensuremath{\mathbb{R}} $ is a function satisfying some mild assumptions on its derivatives, then there exists an open set $S$ so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.
\end{abstract}
\title{\parbox{14cm}
\section{Introduction}
It is a simple consequence of the Lebesgue density theorem that subsets of $\ensuremath{\mathbb{R}}^n$ of positive Lebesgue measure contain
a translated and scaled copy of every finite point set for an interval worth of scalings \cite{Steinhaus20}.
Under more general assumptions on $E$,
a problem of great current interest is that of describing the set of configurations that exists within $E$. This includes
finding conditions on the structure or size of $E$ that guarantee the existence of various patterns within $E$,
see, for instance, \cite{Bourgain86, CLP, FKW90, IosLiu, IosMag, Krause, FY21, Ziegler06},
as well as the more quantitative question of describing the size of the set of similar copies of a given configuration \cite{GGIP, GIP, GIT21, GIT19, M21}.
We focus on the latter question.
A particularly simple setting involves two point configurations and distances, which we describe using the following notation.
\betagin{defn}[Distance sets]
Given a set $E\subsetbset \ensuremath{\mathbb{R}}^d$, define the distance set of $E$ to be the set
\[
\Deltalta(E)=\{|x-y|:x,y\in E\}.
\]
For $x\in \ensuremath{\mathbb{R}}^d$, define the \thetaxtbf{pinned distance set} of $E$ at $x$ to be
\[
\Deltalta_x(E)=\{|x-y|:y\in E\}.
\]
\end{defn}
A classic problem in harmonic analysis and geometric measure theory is the Falconer distance problem, which asks how large the Hausdorff dimension of a set $E\subsetbset \ensuremath{\mathbb{R}}^d$ must be to ensure that $\Deltalta(E)$ has positive Lebesgue measure \cite{Fal85}, or more generally non-empty interior.
Falconer proved that $\dim_{\rm{H}}(E) >\frac{d}{2}$ is necessary and $\dim_{\rm{H}}(E) >\frac{d+1}{2}$ suffices for positive measure.
The best known result in the plane improves this threshold to $\dim_{\rm{H}}(E) >\frac{5}{4}$ \cite{GIOW}. In the non-empty interior direction, a result of Mattila and Sj\"olin shows that $\Deltalta(E)$ has non-empty interior provided that $\dim_{\rm{H}}(E)>\frac{d+1}{2}$; for a simple Fourier analytic proof of this fact, see also \cite{IMT12}. The first known results on the interior of pinned distance sets appeared in \cite[Corollary 2.12 \& Theorem 2.15]{STinterior}.
\\
To pose questions about more complex patterns, the language of graph theory is useful. We pause to state some basic definitions.
\betagin{defn}[Graphs]
A (finite) \thetaxtbf{graph} is a pair $G=(V,E)$, where $V$ is a (finite) set and $E$ is a set of $2$-element subsets of $V$. If $\{i,j\}\in E$ we say $i$ and $j$ are \thetaxtbf{adjacent} and write $i\psim j$.
\end{defn}
Throughout this paper, we will always consider our vertex set to be $\{1,\dots,k+1\}$. We are particularly interested in the following types of graphs.
\betagin{defn}[Chain and tree graphs]
The \thetaxtbf{$k$-chain} is the graph on vertex set $\{1,\dots,k+1\}$ with $i\psim j$ if and only if $|i-j|=1$. A \thetaxtbf{tree} is a connected, acyclic graph; equivalently, a tree is a graph in which any two vertices are connected by exactly one path. If $T$ is a tree, the \thetaxtbf{leaves} of $T$ are the vertices which are adjacent to exactly one other vertex of $T$.
\end{defn}
Note that all chains are trees. We record the basic structural properties of trees as a proposition.
\betagin{prop}[Tree structure]
\lambdabel{treestructure}
If $T$ is a tree with $k+1$ vertices, then $T$ has $k$ edges. Moreover, given such a tree $T$ there is a sequence of trees $T_1,...,T_k,T_{k+1}$ such that $T_1=T$, $T_{k+1}$ consists of only one vertex, and each $T_{i+1}$ is obtained from $T_i$ by removing one leaf and its corresponding edge.
\end{prop}
\betagin{defn}[$G$ distance sets]\lambdabel{Gdist_defn}
Let $G$ be a graph on the vertex set $\{1,\dots,k+1\}$ with $m$ edges, and let $\psim$ denote the adjacency relation on $G$.
Define the \thetaxtbf{$G$ distance set} of $E$ to be
\[
\Deltalta_G(E)=\{(|x^i-x^j|)_{i\psim j}:x^1,...,x^{k+1}\in E, x^i\neq x^j\},
\]
where $(a_{i,j})_{i\psim j}$ denotes a vector in $\ensuremath{\mathbb{R}}^m$ with coordinates indexed by the edges of $G$.
\end{defn}
If $G$ is the $1$-chain, the set $\Deltalta_G(E)$ essentially coincides with the set $\Deltalta(E)$ (our definition of $G$ distance sets excludes degenerate configurations, so really $\Deltalta(E)=\Deltalta_G(E)\cup \{0\}$ for this choice of $G$).
\\
With these definitions in place, the question becomes: what structural conditions on $E$ are needed to ensure $\Deltalta_G(E)$ has positive measure/non-empty interior?
When $G$ is a $k$-chain, the problem was studied by Bennett, Iosevich, and the second author of this paper in \cite{BIT16}. They prove that if the Hausdorff dimension of $E$ is greater than $\frac{d+1}{2}$, then $\Deltalta_G(E)$ has non-empty interior.
Moreover, there is an interval $I\subsetbset\ensuremath{\mathbb{R}}$ such that $I^k \subsetbset \Deltalta_G(E)$, where $I^k$ denotes the $k$-fold Cartesian product of $I$.
This result was later generalized by Iosevich and the second author to the case when $G$ is a tree \cite{ITtrees}.
For additional progress on relating the Hausdorff dimension of a set to the interior of the set of configurations that it contains, see \cite{GIT21, GIT19}.
\\
In \cite{OuT20}, Ou and the second listed author show that when $T$ is a tree and $\dim_{\rm H}(E)>5/4$ for $E\subsetbset \ensuremath{\mathbb{R}}^2$, then $\Deltalta_T(E)$ has positive Lebesgue measure. However, when $\dim_H(E)<3/2$ it is not known whether $\Deltalta_T(E)$ has non-empty interior.
We make progress on this question by replacing the Hausdorff dimension with an alternate notion of structure, namely that of Newhouse thickness. Our precise definitions are as follows.
\betagin{defn}[Cantor sets]
A \thetaxtbf{Cantor set} is a subset of $\ensuremath{\mathbb{R}}^d$ which is compact, perfect, and totally disconnected.
\end{defn}
When $K\subsetbset \ensuremath{\mathbb{R}}$ is a Cantor set, we have the following notion of structure (also see see \cite{PTbook}).
\betagin{defn}[Thickness]
A \thetaxtbf{gap} of a Cantor set $K\subsetbset \ensuremath{\mathbb{R}}$ is a connected component of the compliment $\ensuremath{\mathbb{R}}\setminus K$. If $u$ is the right endpoint of a bounded gap $G$, for $b\in \ensuremath{\mathbb{R}}\cup \{\infty\}$, let $(a,b)$ be the closest gap to $G$ with the property that $u<a$ and $|G|\le b-a$. The interval $(u,a)$ is called the \thetaxtbf{bridge} at $u$ and is denoted $B(u)$. Analogous definitions are made when $u$ is a left endpoint. The \thetaxtbf{thickness} of $K$ at $u$ is the quantity
\[
\tau(K,u):=\frac{|B(u)|}{|G|}.
\]
Finally, the thickness of the Cantor set $K$ is the quantity
\[
\tau(K):=\inf_u \tau(K,u),
\]
the infimum being taken over all gap endpoints $u$.
\end{defn}
Our goal is to prove Falconer type results for sets in the plane of the form $K\times K$, where $K$ is a Cantor satisfying $\tau(K)>1$.
Before stating our results, we comment on the relationship between thickness and Hausdorff dimension.
One can easily construct a Cantor set $K$ with arbitrarily small thickness and Hausdorff dimension arbitrarily close to 1.
This is due to the fact that thickness is defined using an infimum, so one can construct a thin Cantor set by simply ensuring one bridge is much smaller than the corresponding gap.
More precisely, for any $ \partialta>0$ and $1< N< \partialta^{-1}$ we can construct $K$ as a subset of $[0, \partialta]\cup[N \partialta,1]$.
It is clear that Cantor sets of this form can attain any Hausdorff dimension in $[0,1]$.
Considering the gap $( \partialta, N \partialta)$ and corresponding bridge $[0, \partialta]$, we conclude that $\tau(K)\leq \frac{1}{N-1}$. \\
On the other hand, large thickness implies large Hausdorff dimension. Specifically, one can prove the bound (see \cite[pg. 77]{PTbook}):
$$\dim_{\rm H}(K) \geq \frac{\log{2}}{\log{\left( 2 + \frac{1}{\tau(K)} \right) }}.$$
In particular, when $K$ is a Cantor set of thickness $\tau(K)>1$ then we have
\[
\dim_{\rm H}(K\times K)> \frac{2\cdot\log 2}{\log 3}\approx 1.26
\]
\subsetbsection{\thetaxtit{Main results}}
Recall that if $E\subsetbset \ensuremath{\mathbb{R}}^2$ and $\dim_{\rm H}(E)>\frac{5}{4}$, then $\Deltalta_T(E)$ has positive Lebesgue measure.
However, when $\dim_H(E)<\frac{3}{2}$ it is not known whether $\Deltalta_T(E)$ has non-empty interior.
Our first result is as follows.
\betagin{thm}[Interior of tree distance sets]
\lambdabel{distancetrees}
Let $K_1,K_2$ be Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2) > 1$
For any finite tree $T$, the set $\Deltalta_T(K_1\times K_2)$ has non-empty interior.
\end{thm}
We note that Theorem \ref{distancetrees} is an extension of the work in \cite{STinterior} of Simon and the second listed author, where it is shown $\Deltalta_x(K\times K)$ has non-empty interior provided that $K$ is a Cantor set satisfying $\tau(K)> 1$.
Theorem \ref{distancetrees} also holds if the Euclidean norm is replaced with more general norms; see Theorem \ref{phi_distancetrees} below.\\
Beyond trees,
the existence of patterns in thick subsets of $\ensuremath{\mathbb{R}}^d$ with rigid structure was investigated in \cite{Yav20} when $d=1$, and in \cite{FY21} when $d\geq 1$.
In \cite{Yav20}, it is shown that, given any compact set $C$ in $\mathbb{R}$ with thickness
$\tau$, there is an explicit number $N(\tau)$ such that $C$ contains a
translate of all sufficiently small
similar copies of every finite set in $\mathbb{R}$ with at most $N(\tau)$
elements.
Higher dimensional analogues of these results are subsequently given in \cite{FY21}.
The only drawback is that the theorems in \cite{Yav20, FY21} assume very large thickness; moreover, the threshold depends on the size of the configuration one wants to find.
For instance, in order to ensure $N(\tau)\geq 3$, one needs $\tau$ at least on the order of $10^9$.
In contrast, our results for trees apply to any Cantor sets of thickness greater than $1$, regardless of how large the tree is.
\\
Another Falconer type problem which has received much attention is obtained by replacing the Euclidean distance with other geometric quantities, notably dot products.
We make the following definition.
\betagin{defn}[Dot product sets]
Given $E\subsetbset\ensuremath{\mathbb{R}}^d$, the \thetaxtbf{dot product set} of $E$ is the set
\[
\Pi(E)=\{x\cdot y: x,y\in E\}.
\]
We also consider the \thetaxtbf{pinned dot product set}
\[
\Pi_x(E)=\{x\cdot y:y\in E\}.
\]
Finally, given a graph $G$ on vertices $\{1,...,k+1\}$, define
$$\Pi_{G}(E)=\{(x^i\cdot x^j)_{i\psim j}:x^1,...,x^{k+1}\in E\}.$$
\end{defn}
When $E\subsetbset \ensuremath{\mathbb{R}}^d$ is a set of sufficient Hausdorff dimension, the dot product set is treated in \cite[Theorem 1.8]{EIT11}.
In particular, it is shown there that if $\dim_{\rm H}(E) >\frac{d+1}{2}$, then $\Pi(E)$ has positive measure. The related set $\{x^\perp\cdot y:x,y\in E\}$, where $x^\perp=(-x_2,x_1)$ when $d=2$, is the set of (signed) areas of parallelograms spanned by points of $E$. Similar to the above definition, for any graph $G$ one can consider the vector which encodes all areas determined by points $x^i,x^j$ such that $i\psim j$. This problem was investigated by the first author in \cite{M21} in the case where $G$ is a complete graph, and the analogous problem in higher dimensions was studied by the first author and Galo in \cite{GM21}. \\
In the setting where one is considering sets $E$ with large Hausdorff dimension, the proofs of distance and dot product results are generally similar in complexity. However, in the setting where $E= K\times K$, where $K$ is a sufficiently thick Cantor set, the dot product problem is considerably more straightforward than to the distance problem. We nevertheless record the result here and provide its proof in Section \ref{KeyLemmas} as a demonstration of how our techniques vary in these two regimes.
\betagin{thm}[Interior of tree dot product sets]
\lambdabel{dotproducttrees}
Let $K$ be a Cantor set satisfying $\tau(K)\geq 1$. For any finite tree $T$, the set $\Pi_{T}(K\times K)$ has non-empty interior.
\end{thm}
Our next main theorem concerns the standard middle thirds Cantor set, which we will denote $C_{1/3}$ throughout this paper. Note that Theorem \ref{distancetrees} does not apply to $C_{1/3}$, as the hypothesis of that theorem is $\tau(K)>1$ and clearly $\tau(C_{1/3})=1$. While we do not expect that Theorem \ref{distancetrees} can be extended to the $\tau(K)=1$ case in general, this weaker thickness condition together with the self similarity of $C_{1/3}$ allow us to modify the proof in that case. The result is as follows.
\betagin{thm}[Interior of $T$ distance sets in the middle third Cantor set]
\lambdabel{trees_thrm_middle_third}
For any finite tree $T$, the set $\Deltalta_{T}(C_{1/3}\times C_{1/3})$ has non-empty interior.
\end{thm}
\vskip.125in
\subsetbsection{\thetaxtit{General distance trees}}
Having established results for the Euclidean distance and dot products, we turn to the more general setting of $(G,\phi)$ distance trees.
\betagin{defn}[$ (G,\phi)$ distance sets]\lambdabel{Gdist_defn}
Let $G$ be a graph on the vertex set $\{1,\dots,k+1\}$ with $m$ edges, and let $\psim$ denote the adjacency relation on $G$.
Given a function $\phi:\ensuremath{\mathbb{R}}^d\times \ensuremath{\mathbb{R}}^d\rightarrow \ensuremath{\mathbb{R}}$, define the \thetaxtbf{$(G,\phi)$-distance set} of $E$ to be
\[
\Deltalta_{(G, \phi)}(E)=\{ (\phi( x^i,x^j))_{i\psim j}:x^1,...,x^{k+1}\in E, x^i\neq x^j\}.
\]
\end{defn}
We require the following derivative condition on $\phi$.
\betagin{defn}[Derivative condition]
\lambdabel{deriv_cond}
Let $\phi: \ensuremath{\mathbb{R}}^2 \times \ensuremath{\mathbb{R}}^2\rightarrow \ensuremath{\mathbb{R}}$ be a $C^1$ function
on $A\times B$, for open sets $A, B\subsetbset \ensuremath{\mathbb{R}}^2$.
We say that $\phi$ satisfies the \thetaxtbf{derivative condition} on $A\times B$ if for each $x\in A$, if $\varphi_x(y)= \phi(x,y)$, then
the partial derivatives of $\varphi$ are bounded away from zero on $B$.
\end{defn}
\betagin{note}
Note that the derivative condition is satisfied, for instance, by $\phi(x,y) = |x-y|_p$, the $p$-norm, whenever $p\geq1$, and $\phi(x,y) = x\cdot y $ for appropriate choices of $A$ and $B$.
\end{note}
\betagin{thm}[Interior of $ (T,\phi)$ distance sets]
\lambdabel{phi_distancetrees}
Let $K_1,K_2$ be Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2) > 1$.
Suppose $\phi: \ensuremath{\mathbb{R}}^2 \times \ensuremath{\mathbb{R}}^2\rightarrow \ensuremath{\mathbb{R}}$
satisfies the derivative condition on $A\times B$, for open $A,B\subsetbset \ensuremath{\mathbb{R}}^2$, each of which intersects $K_1\times K_2$. Then,
for any finite tree $T$, the set $\Deltalta_T(K_1\times K_2)$ has non-empty interior.
\end{thm}
The proof of Theorem \ref{phi_distancetrees} is given in Section \ref{general_proof} and relies on the mechanism in Section \ref{mechanismproof}.
\subsetbsection{\thetaxtit{Method of proof}}
We now discuss the strategy for proving our results. The starting point is the following classical result known as the Newhouse Gap Lemma \cite[pg. 61]{PTbook}.
\betagin{lem}[Newhouse Gap Lemma]
Let $K_1,K_2\subsetbset \ensuremath{\mathbb{R}}$ be Cantor sets satisfying $\tau(K_1)\tau(K_2)\geq 1$. Suppose further that neither of the sets $K_1,K_2$ is contained in a single gap of the other. Then, $K_1\cap K_2\neq\emp$.
\end{lem}
In practice, if $K_2$ is contained in the convex hull of $K_1$ it can be difficult to check whether $K_2$ is contained in a gap of $K_1$. We will often use a special case of this condition which is easier to check. To do this we first introduce some terminology.
\betagin{defn}[Linked sets]
Two open, bounded intervals $I,J\subsetbset\ensuremath{\mathbb{R}}$ are said to be \thetaxtbf{linked} if they have non-empty intersection, but neither is contained in the other. Bounded (not necessarily open) intervals are linked if their interiors are linked. Finally, two bounded sets $K_1,K_2\subsetbset\ensuremath{\mathbb{R}}$ are linked if their convex hulls are linked.
\end{defn}
\betagin{prop}[Special case of Newhouse Gap Lemma]
\lambdabel{specialcase}
Let $K_1,K_2\subsetbset\ensuremath{\mathbb{R}}$ be linked Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2)\geq 1$. Then, $K_1\cap K_2\neq\emp$.
\end{prop}
Now, given a fixed point $x\in \ensuremath{\mathbb{R}}^2$ and distance $t\in \ensuremath{\mathbb{R}}$, we have $|x-y|=t$ if $y_2=g_{x,t}^{\thetaxt{dist}}(y_1)$, where
\[
g_{x,t}^{\thetaxt{dist}}(z)=x_2+\sqrt{t^2-(z-x_1)^2}.
\]
Likewise, we have $x\cdot y=t$ if $y_2=g_{x,t}^{\thetaxt{dot}}(y_1)$, where
\[
g_{x,t}^{\thetaxt{dot}}(z)=\frac{t}{x_2}-\frac{x_1}{x_2}z.
\]
We are therefore interested in applying the Newhouse Gap Lemma to find a point in the intersection $K\cap g(K)$ for an appropriate function $g$. In the case of the dot product this is simple, since affine transformations preserve thickness and we therefore only have to find values of $x,t$ for which $K$ and $g_{x,t}^{\thetaxt{dot}}(K)$ are linked. Matters are more difficult for distances since, in general, smooth functions do not necessarily preserve thickness. In \cite{STinterior} it is proved that if $g$ is continuously differentiable and $I$ is a sufficiently small interval on which $g'$ is bounded away from zero, then the thickness of $g(K\cap I)$ is not too much smaller than that of $K$. This allows one to prove that the pinned sets $\Deltalta_x(K\times K)$ and $\Pi_x(K\times K)$ have non-empty interior if $\tau(K)>1$. \\
Using this strategy, we prove that there exists an interval $I$ which is contained in $\Deltalta_x(K\times K)$, and such that $I$ remains in $\Deltalta_x(K\times K)$ if the pin $x$ is perturbed a small amount. Indeed, let $g_{x,t}$ denote either of the functions defined above. For fixed $z$, the quantity $g_{x,t}(z)$ is continuous in the parameters $x$ and $t$. Therefore, the condition that $K$ and $g_{x,t}(K)$ are linked is an open condition. In Section \ref{KeyLemmas} we prove these ``pin wiggling" lemmas for each of the functions needed in our theorems. Once we have proved such a lemma, we can convert it into a theorem about trees using a mechanism which is discussed in Section \ref{mechanismproof}.
\section{Converting Pin Wiggling Lemmas into Tree Theorems}
\lambdabel{mechanismproof}
In Section \ref{KeyLemmas}, we will prove lemmas showing that not only do pinned distance and dot product sets contain intervals, but that there is a single interval which works for all such sets obtained by wiggling the pin a small amount. The goal of this section is to state and prove a theorem which gives us a mechanism to convert pin wiggling lemmas to our main theorems. The setup is as follows. Let $\phi:\ensuremath{\mathbb{R}}^2\times \ensuremath{\mathbb{R}}^2\to \ensuremath{\mathbb{R}}$ be any function; for example, to prove Theorem \ref{distancetrees} we use the function $\phi(x,y)=|x-y|$. Given a point $x$ and a set $E$, we use the notation
\[
\phi(x,E):=\{\phi(x,y):y\in E\}.
\]
A \thetaxtbf{pin wiggling lemma} is a lemma which says, under some assumptions on $E$, that there is a single interval $I$ contained in $\phi(x,E)$ for a range of $x$; equivalently, the set
\[
\bigcap_{x\in S}\phi(x,E)
\]
has non-empty interior for some neighborhood $S$ of pins. Now, given such a function $\phi$ and a tree $T$ on vertices $\{1,\dots,k+1\}$, define
\[
\Phi(x^1,\dots,x^{k+1})=(\phi(x^i,x^j))_{i\psim j}.
\]
Thus, the sets $\Deltalta_T(E)$ and $\Pi_T(E)$ are the images of $E^{k+1}$ under $\Phi$ for $\phi(x,y)=|x-y|$ and $\phi(x,y)=x\cdot y$, respectively. Our main theorems are therefore giving conditions under which the sets $\Phi(E^{k+1})$ have non-empty interior. With this setup, the conversion mechanism is as follows.
\betagin{thm}[Tree building mechanism]
\lambdabel{mechanism}
Fix a map $\phi:\ensuremath{\mathbb{R}}^2\times \ensuremath{\mathbb{R}}^2\to\ensuremath{\mathbb{R}}$ and a tree $T$ on vertices $\{1,\dots,k+1\}$, and consider the map $\Phi:(\ensuremath{\mathbb{R}}^2)^{k+1}\to\ensuremath{\mathbb{R}}^k$ defined by
\[
\Phi(x^1,\dots,x^{k+1})=(\phi(x^i,x^j))_{i\psim j},
\]
where $\psim$ denotes the adjacency relation of the graph $T$.
Let $K_1,K_2$ be Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2) > 1$,
and let $x^1,\dots,x^{k+1}\in K_1\times K_2$ be distinct points. Suppose that for any Cantor sets $\widetilde{K_j} \subsetbset K_j$,
there exist open neighborhoods $S_i$ of $x^i$ such that the set
\[
\bigcap_{x\in S_i}\phi(x,\widetilde{K_1}\times \widetilde{K_2})
\]
has non-empty interior. Then, $\Phi((K_1\times K_2)^{k+1})$ has non-empty interior. Moreover, $\Phi(x^1,\dots,x^{k+1})$ is in the closure of $\Phi((K_1\times K_2)^{k+1})^\circ$.
\end{thm}
\betagin{proof}
For the purpose of ensuring non-degeneracy,
let $2\epsilon>0$ denote the minimal distance:
$$\epsilon= \frac{1}{2} \min \left\{ |x^i- x^{j}|: i \neq j \in \{ 1, 2, \dots, k+1\} \right\}>0,$$
and, for each $i=1,2, \dots, k+1$, define the $\epsilon$-box about $x^i$ by
\betagin{align*}
C(x^i, \epsilon)
=& x^i + [- \epsilon,\epsilon]^2\\
=& [x^i_1 - \epsilon, x^i_1+\epsilon] \times [x^i_2- \epsilon, x^i_2+\epsilon]\\
=& C_1(x^i, \epsilon) \times C_2(x^i, \epsilon),
\end{align*}
where $C_1(x^i, \epsilon), C_2(x^i, \epsilon)$ are the closed $\epsilon$-intervals about the coordinates of $x^i$ (Figure \ref{boxes}).
\betagin{figure}[ht]
\centering
\betagin{tikzpicture}
\betagin{axis}[xmin=-.1,xmax=5,ymin=-.1,ymax=5,axis x line=center, axis y line=center, xtick={0}, xticklabels={}, ytick={0},yticklabels={}]
\addplot[mark=*] coordinates {(1,2)};
\addplot[mark=*] coordinates {(3,2.5)};
\addplot[mark=*] coordinates {(4,.5)};
\addplot[mark=*] coordinates {(4.5,3)};
\addplot[mark=*] coordinates {(1.5,4)};
\addplot[dashed] coordinates {(1,2)(1.5,4)};
\addplot[dashed] coordinates {(1,2)(3,2.5)};
\addplot[dashed] coordinates {(3,2.5)(4.5,3)};
\addplot[dashed] coordinates {(3,2.5)(4,.5)};
\addplot[thick] coordinates {(.8,1.8)(.8,2.2)(1.2,2.2)(1.2,1.8)(.8,1.8)};
\addplot[thick] coordinates {(2.8,2.3)(2.8,2.7)(3.2,2.7)(3.2,2.3)(2.8,2.3)};
\addplot[thick] coordinates {(3.8,.3)(3.8,.7)(4.2,.7)(4.2,.3)(3.8,.3)};
\addplot[thick] coordinates {(4.3,2.8)(4.3,3.2)(4.7,3.2)(4.7,2.8)(4.3,2.8)};
\addplot[thick] coordinates {(1.3,3.8)(1.3,4.2)(1.7,4.2)(1.7,3.8)(1.3,3.8)};
\end{axis}
\end{tikzpicture}
\caption{Boxes $C(x^i,\epsilon)$ around points $x^1,...,x^5$}
\lambdabel{boxes}
\end{figure}
Next, choose any leaf of $T$; without loss of generality we may assume we have labeled the vertices so that $k+1$ is our leaf. Let $i$ denote the unique vertex which satisfies $i\psim k+1$. Let $\widetilde{K_j} = K_j \cap C_j(x^{k+1}, \epsilon)$. By assumption, there exists a neighborhood $S_{i}$ of $x^{i}$ so that the set
\betagin{equation}\lambdabel{first_iter_eq}
\bigcap_{x\in S_{i}}\phi(x,\widetilde{ K_1 }\times \widetilde{K_2})
\end{equation}
has non-empty interior. Further, we may assume $S_{i}\subsetbset C(x^{i},\epsilon)$, which guarantees that the points in $S_{i}$ and points in $\widetilde{K_1}\times \widetilde{K_2}\subsetbset C(x^{k+1}, \epsilon)$ are distinct.
Moreover, we can choose $\epsilon_{2} \in (0, \epsilon]$ so that $C(x^i,\epsilon_{2}) \subsetbset S_{i}$, and hence \eqref{first_iter_eq} still holds with $C(x^{i},\epsilon_2) $ in place of $S_{i}$. For simplicity, we replace each of the $\epsilon$-boxes about $x^{1}, \dots, x^{k+1}$ by potentially smaller boxes $C(x^{j}, \epsilon_2)$ for each $j \in \{1, \dots, k+1\}$.
\\
To conclude, let $E_{i} = C(x^{i},\epsilon_2)\cap (K_1\times K_2)$, let $T_2$ be the tree obtained from $T$ by removing the vertex $k+1$ and its corresponding edge, and let $\Phi_2$ be the function as in the statement of the theorem, corresponding to the tree $T_2$. We have proved there exists a non-empty open interval $I_{1}$ so that
\[
\Phi(E_1\times\cdots\times E_{k+1})\subsetpset \Phi_2(E_1\times\cdots\times E_k)\times I_1.
\]
Running this argument successively on each of the trees $T_1,T_2,...,T_k$ as in Proposition \ref{treestructure}, we conclude that $\Phi((K_1\times K_2)^{k+1})$ contains a set of the form $I_1\times \dots\times I_k$ for non-empty open intervals $I_1, \dots, I_k$. By construction, it is clear that $\Phi(x^1,...,x^{k+1})$ is in the closure of $I_1\times\cdots\times I_k$.
\end{proof}
From the statement of Theorem \ref{mechanism}, we see that we may start with any points $x^1,\dots,x^{k+1}\in K_1\times K_2$ and obtain an open box near $\Phi(x^1,\dots,x^{k+1})$, provided we can prove pin wiggling lemmas around those points. We will refer to the starting points $x^1,\dots,x^{k+1}$ as a \thetaxtbf{skeleton}.
\betagin{remark}
It would be interesting to find an interval $I$ such that $I^k\subsetbset \Deltalta_T(K\times K)$, as is done in \cite{BIT16, ITtrees} in the large Hausdorff dimension context. By Theorem \ref{mechanism} below, this amounts to showing there exists a skeleton $x^1,...,x^{k+1}\in K\times K$ such that to two points share a coordinate, and the distances $|x^i-x^j|$ are constant for $i\psim j$. It is not clear how to do this in general. In the special case where $T$ is a $k$-chain, it is sufficient (but not necessary) that $K$ contains a length $k+1$ arithmetic progression. Given an arithmetic progression $a_1,...,a_{k+1}\in K$, we could then take $x^i=(a_i,a_i)$. A result of Yavicoli \cite{Yav20} shows that long arithmetic progressions exist in (very) thick Cantor sets. However, the required lower bound on thickness is much larger then the $\tau(K)>1$ assumption in our results; to ensure even a $3$-term arithmetic progression, one needs $\tau(K)$ at least on the order of $10^9$. Moreover, let $C_\epsilon$ denote the middle-$\epsilon$ Cantor set, obtained by starting with the unit interval and at each stage deleting the middle $\epsilon$ proportion from the remaining intervals. Broderick, Fishman, and Simmons \cite{BFS19} prove that there is no arithmetic progression in $C_\epsilon$ of length greater than $\frac{1}{\epsilon}+1$ for $\epsilon$ sufficiently small. Therefore, for any $k\in \ensuremath{\mathbb{Z}}$ sufficiently large, the set $C_{2/k}$ is a Cantor set with thickness $\frac{k-2}{4}$ and no $(k+1)$-term arithmetic progression.
This means there is no hope for a thickness threshold which is uniform in $k$. One can still hope to find a common interval by using the $2$-dimensionality of $K\times K$ instead of hoping to take a sequence of points along the diagonal, but it is not clear how to find the necessary skeleton.
\end{remark}
\section{Pin Wiggling Lemmas in Various Contexts}
\lambdabel{KeyLemmas}
The proofs in this section are presented in increasing order of complexity.
\subsetbsection{\thetaxtit{Proof of Theorem \ref{dotproducttrees}}}
We begin by proving our result on dot product trees. As discussed in the introduction, dot products are much simpler than distances because thickness is preserved under affine transformations. As a consequence, Theorem \ref{dotproducttrees} is the simplest of our results.\\
Theorem \ref{dotproducttrees} is an immediate consequence of Theorem \ref{mechanism} and the following lemma.
\betagin{lem}[Pin Wiggling for Dot Products]
\lambdabel{dotproductkeylemma}
Let $K_1,K_2$ be Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2) \geq 1$. Let $\ell_j$ denote the length of the convex hull of $K_j$.
\betagin{enumerate}[(i)]
\item For any $x=(x_1,x_2)\in \ensuremath{\mathbb{R}}^2$ with both coordinates nonzero, the set $\Pi_x(K_1\times K_2)$ contains an interval of length at least $\ell\cdot\min(|x_1|,|x_2|)$
\item Let $x^0=(x_1^0,x_2^0)\in \ensuremath{\mathbb{R}}^2$ be a point with both coordinates nonzero. Let $Q$ be the square centered at $x^0$ with side length $2 \partialtalta$, and assume $ \partialtalta<\frac{1}{3}\min(|x_1^0|,|x_2^0|)$. The set
\[
\bigcap_{x\in Q}\Pi_x(K_1\times K_2)
\]
contains an interval of length at least $\ell\cdot(\min(|x_1^0|,|x_2^0|)-3 \partialtalta)$.
\end{enumerate}
\end{lem}
\betagin{proof}
For any $x=(x^1, x^2)\in \ensuremath{\mathbb{R}}^2$, we have $t\in \Pi_x(K_1\times K_2)$ if and only if $(t-x_1K_1)\cap (x_2K_2)\neq\emp$. Since $\tau(t-x_1K_1)\cdot \tau(x_2K_2)=\tau(K_1)\cdot \tau(K_2)\geq 1$, by the Newhouse Gap Lemma, this intersection will be non-empty for any $x$ and $t$ such that the sets $(t-x_1K_1)$ and $x_2K_2$ are linked. Denote the convex hull of $x_jK_j$ by $[a_j,a_j+\ell|x_j|]$, and without loss of generality assume $|x_1|\geq |x_2|$. The sets $t-x_1K$ and $x_2K$ are linked whenever
\[
a_1+a_2+\ell |x_1|<t<a_1+a_2+\ell |x_1|+\ell |x_2|.
\tag{$*$}
\]
The set of $t$ satisfying ($*$) is an interval of length $\ell |x_2|$, and so (i) follows immediately. To prove (ii), assume $(x_1,x_2)\in Q$ and therefore $|x_j-x_j^0|< \partialtalta$ for each $j$. The value $t$ satisfies ($*$) for all such $x_1,x_2$ provided
\[
a_1+a_2+\ell |x_1^0|+\ell \partialtalta<t<a_1+a_2+\ell |x_1^0|+\ell |x_2^0|-2\ell \partialtalta.
\]
This inequality determines an interval of length $\ell(|x_2^0|-3 \partialtalta)$.
\end{proof}
Note that when we apply Theorem \ref{mechanism}, we can start with any skeleton $x^1,\dots,x^{k+1}\in K\times K$ such that none of the points $x^i$ are on the axes.
\subsetbsection{\thetaxtit{Proof of Theorem \ref{distancetrees}}}
As in the previous section, the proof will rely on the mechanism established in Theorem \ref{mechanism} coupled with a pin wiggling lemma. The difference is that the lemma of this section will not follow directly from the linear theory and some preliminary set up is required.
First, observe that given a pin $x\in \ensuremath{\mathbb{R}}^2$ and distance $t\in \ensuremath{\mathbb{R}}$, we have
$t\in \Deltalta_x(K\times K)$ whenever $y_2=g_{x,t}(y_1)$ for some $y = (y_1, y_2) \in K\times K$, where
\[
g_{x,t}(z)=x_2+\sqrt{t^2-(z-x_1)^2}.
\]
We would like to apply the Newhouse Gap Lemma to $K$ and $g_{x,t}(K)$ to prove a pin wiggling lemma for distance sets (Lemma \ref{distanceskeylemma} below), then conclude that Theorem \ref{distancetrees} follows (by Theorem \ref{mechanism}). However, there is no thickness assumption on $K$ which would guarantee $\tau(g_{x,t}(K))\geq 1$, so we cannot apply Newhouse directly. However, if $I$ is a sufficiently small interval about a non-singular point of $g_{x,t}$, then the thickness of $g_{x,t}(K\cap I)$ is not too much smaller than that of $K$. This can be proved using a generalization of thickness which was introduced in \cite{STinterior}, which we describe here.
\betagin{defn}[$\epsilonilon$-thickness]
Let $K\subsetbset\ensuremath{\mathbb{R}}$ be a Cantor set, let $u$ be a right endpoint of a bounded gap, and let $\epsilon>0$. Let $(a,b)$ be the closest gap to $G$ with the property that $a>u$ and $(b-a)>(1-\epsilon)|G|$ The \thetaxtbf{$\epsilon$-bridge} of $u$, denoted $B_\epsilon(u)$, is the interval $(u,a)$. We make analogous definitions for left endpoints. The \thetaxtbf{$\epsilon$-thickness} of $K$ at $u$ is the quantity
\[
\tau_\epsilon(K,u):= \frac{|B_\epsilon(u)|}{|G|}.
\]
Finally, the $\epsilon$-thickness of the Cantor set $K$ is the quantity
\[
\tau_\epsilon(K):=\inf_u \tau_\epsilon(K,u),
\]
the infimum being taken over all gap endpoints $u$.
\end{defn}
We record some easily verifiable properties of $\epsilon$-thickness in the following proposition.
\betagin{prop}[$\epsilon$-thickness converges to regular thickness]
\lambdabel{basics}
Let $K\subsetbset \ensuremath{\mathbb{R}}$ be a Cantor set.
\betagin{enumerate}[(i)]
\item If $\epsilon_1<\epsilon_2$ then $\tau_{\epsilon_1}(K)\geq \tau_{\epsilon_2}(K)$.
\item $\tau_\epsilon(K)\to \tau(K)$ as $\epsilon\to 0$.
\end{enumerate}
\end{prop}
With these definitions in place, we can prove that the image of a thick Cantor set must at least contain a thick Cantor set. More precisely, we have the following lemma, which is essentially Lemma 3.8 in \cite{STinterior}. We include a proof here for completeness.
\betagin{lem}[Thickness of the image is nearly preserved]
\lambdabel{imagethickness}
Let $K\subsetbset\ensuremath{\mathbb{R}}$ be a Cantor set, let $u$ be a right endpoint of some gap of $K$, and let $g$ be a function which is continuously differentiable on a neighborhood of $u$ and satisfies $g'(u)\neq 0$. For every $\epsilon>0$, there exists $ \partialtalta>0$ such that
\[
\tau(g(K\cap [u,u+ \partialtalta]))>\tau_\epsilon(K)(1-\epsilon).
\]
\end{lem}
\betagin{proof}
Fix $\epsilon>0$. By continuity of $g'$, we may choose $ \partialtalta$ such that for all $x_1,x_2\in[u,u+ \partialtalta]$ we have
\[
\left|\frac{g'(x_1)}{g'(x_2)}-1\right|<\epsilon.
\]
Note that our choice of $ \partialtalta$ guarantees that $g$ is monotone on the interval $[u,u+ \partialtalta]$, so for any subinterval $I$ the mean value theorem guarantees the existence of some $x_I\in I$ such that $|g(I)|=|I|\cdot |g'(x_I)|$. Let $v$ be the endpoint of some gap $G$ in $K\cap [u,u+ \partialtalta]$. We first observe $g(B_\epsilon(v))\subsetbset B_{\epsilon^2}(g(v))$. To prove this, note that any gap in $g(K\cap [u,u+ \partialtalta])$ is the image of a gap in $K\cap [u,u+ \partialtalta]$. Therefore, it suffices to prove that any gap $H\subsetbset B_\epsilon(v)$ satisfies $g(H)<(1-\epsilon^2)g(G)$. We have
\betagin{align*}
|g(H)|&=|H|\cdot |g'(x_H)| \\
&<(1-\epsilon)|G|\cdot\frac{|g'(x_H)|}{|g'(x_G)|}\cdot |g'(x_G)| \\
&<(1-\epsilon)|G|\cdot (1+\epsilon)\cdot |g'(x_G)| \\
&=(1-\epsilon^2)|g(G)|.
\end{align*}
It follows that the thickness of our image at the point $g(v)$ satisfies
\betagin{align*}
\tau_{\epsilon^2}(g(K\cap[u,u+ \partialtalta]),g(v)) &\geq \frac{|g(B_\epsilon(v))|}{|g(G)|}\\
&=\frac{|B_\epsilon(v)|}{|G|}\cdot \left|\frac{g'(x_{B_\epsilon(v)})}{g'(x_G)}\right| \\
&\geq \tau_\epsilon(K\cap[u,u+ \partialtalta],v)\cdot(1-\epsilon).
\end{align*}
Taking the infimum over $v$, we have
\[
\tau(g(K\cap[u,u+ \partialtalta]))>\tau_{\epsilon^2}(g(K\cap[u,u+ \partialtalta]))\geq \tau_\epsilon(K)(1-\epsilon).
\]
This is the first claim in the statement of the theorem. The second then follows from Proposition \ref{basics} and the assumption that $\tau(K)>1$.
\end{proof}
We are now prepared to state and prove the key lemma for distances.
\betagin{lem}[Pin Wiggling for Distances]
\lambdabel{distanceskeylemma}
Let $K_1,K_2$ be Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2) > 1$. For any $x^0\in \ensuremath{\mathbb{R}}^2$, there exists an open $S$ about $x^0$ so that
$$\bigcap_{x\in S} \Deltalta_{x}(K_1\times K_2)$$
has non-empty interior.
\end{lem}
\betagin{proof}
For $(x,t)\in\ensuremath{\mathbb{R}}^2\times (0,\infty)$, define
\betagin{equation}\lambdabel{baby_g}
g_{x,t}(z)=x_2+\sqrt{t^2-(z-x_1)^2},
\end{equation}
and observe that $t\in\Deltalta_{x}(K_1\times K_2)$ provided $K_2\cap g_{x,t}(K_1)\neq\emp$. Let $u_j$ be a right endpoint of a bounded gap of $K_j$, and without loss of generality assume $u_j>x_j^0$, where $x^0 = (x^0_1, x^0_2)$. We choose small subsets $\widetilde{K_j}\subsetbset K_j$ with left endpoint $u_j$, and focus on the box $\widetilde{K_1}\times \widetilde{K_2}$ (Figure \ref{box}). Let $t_0=|x^0-(u_1,u_2)|$. Let $\widetilde{K_j}=K_j\cap[u_j,u_j+ \partialtalta_j]$ for some small $ \partialtalta_1, \partialtalta_2$. In particular, we choose $ \partialtalta_j>0$ small enough that $\tau(\widetilde{K_2})\cdot \tau(g(\widetilde{K_1}))>1$ (this is possible by Lemma \ref{imagethickness}), and so that $\widetilde{K_1}$ is in the domain of $g_{x,t}$ whenever $(x,t)$ is sufficiently close to $(x^0,t_0)$. We will also assume $u_j+ \partialtalta_j\in K_j$.
\betagin{figure}[ht]
\centering
\betagin{minipage}[b]{0.45\linewidth}
\betagin{tikzpicture}
\betagin{axis}[xmin=0,xmax=5,ymin=0,ymax=5,axis x line=center, axis y line=center, xtick={2,3,4}, xticklabels={$x_1^0$,$u_1$,$u_1+ \partialtalta_1$}, ytick={1,3,4},yticklabels={$x_2^0$,$u_2$,$u_2+ \partialtalta_2$}]
\addplot[mark=*] coordinates {(2,1)};
\addplot[dashed] coordinates {(3,0)(3,5)};
\addplot[dashed] coordinates {(4,0)(4,5)};
\addplot[dashed] coordinates {(0,3)(5,3)};
\addplot[dashed] coordinates {(0,4)(5,4)};
\addplot[thick] coordinates {(3,3)(4,3)(4,4)(3,4)(3,3)};
\end{axis}
\end{tikzpicture}
\caption{The box containing $\widetilde{K_1}\times \widetilde{K_2}$}
\lambdabel{box}
\end{minipage}
\qquad
\betagin{minipage}[b]{0.45\linewidth}
\betagin{tikzpicture}
\betagin{axis}[xmin=0,xmax=5,ymin=0,ymax=5,axis x line=center, axis y line=center, xtick={2,3,4}, xticklabels={$x_1^0$,$u_1$,$u_1+ \partialtalta_1$}, ytick={1,3,4},yticklabels={$x_2^0$,$u_2$,$u_2+ \partialtalta_2$}]
\addplot[domain=0:5,samples=100,red] {1+sqrt(5-(x-2)^2)};
\addlegendentry{$g_{x^0,t_0}$}
\addplot[domain=0:5,samples=100,blue] {1+sqrt(5.56-(x-2)^2)};
\addlegendentry{$g_{x^0,t}$}
\addplot[dashed] coordinates {(3,0)(3,3.13)};
\addplot[dashed] coordinates {(4,0)(4,2.25)};
\addplot[dashed] coordinates {(0,2.25)(4,2.25)};
\addplot[dashed] coordinates {(0,3.13)(3,3.13)};
\addplot[mark=*] coordinates {(2,1)};
\addplot[] coordinates {(3,3)(4,3)(4,4)(3,4)(3,3)};
\end{axis}
\end{tikzpicture}
\caption{Graphs of $g$}
\lambdabel{graphs}
\end{minipage}
\end{figure}
By the Newhouse Gap Lemma (Proposition \ref{specialcase}), we will have $\widetilde{K_2}\cap g_{x,t}(\widetilde{K_1})\neq\emp$ whenever the parameters $(x,t)$ are such that $\widetilde{K_2}$ and $g_{x,t}(\widetilde{K_1})$ are linked. To this end, consider the set
\[
U=\{(x,t)\in\ensuremath{\mathbb{R}}^2\times\ensuremath{\mathbb{R}}:g_{x,t}(u_1+ \partialtalta_1)<u_2<g_{x,t}(u_1)<u_2+ \partialtalta_2\}.
\]
By construction, for any $(x,t)\in U$ we have $K_2$ and $g_{x,t}(K_1)$ linked (Figure \ref{graphs}) and hence $t\in\Deltalta_x(K_1\times K_2)$. We claim that $U$ is an open set containing a point of the form $(x^0,t)$ for some $t$. Lemma \ref{distanceskeylemma} follows from this claim, as we can then take open neighborhoods $S,T$ of $x^0,t$ respectively such that
\[
T\subsetbset \bigcap_{x\in S} \Deltalta_{x}(K_1\times K_2).
\]
To prove the claim, we first observe that $U$ is open, since for fixed $z$ the quantity $g_{x,t}(z)$ is a continuous function of $(x,t)$. To finish the proof, we must find a $t$ such that $(x^0,t)\in U$. By construction, we have $g_{x^0,t_0}(u_1)=u_2$. Since the quantity $g_{x,t}(z)$ is strictly increasing in $t$, for any $t>t_0$ we will have $g_{x^0,t}(u_1)>u_2$. On the other hand, by continuity in $t$ we will also have $g_{x^0,t}(u_1+ \partialtalta_1)<u_2$ and $g_{x^0,t}(u_1)<u_2+ \partialtalta_2$ whenever $t$ is sufficiently close to $t_0$. Therefore, we can choose $t$ with the property that $(x^0,t_0)\in U$.
\end{proof}
Theorem \ref{distancetrees} follows from Lemma \ref{distanceskeylemma} and Theorem \ref{mechanism}. Note that when we apply Theorem \ref{mechanism}, we can start with any skeleton $x^1,\dots,x^{k+1}\in K_1\times K_2$ provided no two points $x^i,x^j$ share a coordinate.
\subsetbsection{\thetaxtit{Proof of Theorem \ref{phi_distancetrees}}}\lambdabel{general_proof}
As in the previous two sections, Theorem \ref{phi_distancetrees} on $\phi$ distance trees
is an immediate consequence of Theorem \ref{mechanism} and the following lemma.
\betagin{lem}[Pin Wiggling for $\phi$ distance trees]
\lambdabel{phi_distanceskeylemma}
Let $K_1,K_2$ be Cantor sets satisfying $\tau(K_1)\cdot \tau(K_2) > 1$
Suppose $\phi:\ensuremath{\mathbb{R}}^2 \times \ensuremath{\mathbb{R}}^2 \rightarrow \ensuremath{\mathbb{R}}$
satisfies the derivative condition of Definition \ref{deriv_cond} on $A\times B$ for open $A,B\subsetbset \ensuremath{\mathbb{R}}^2$, each of which intersects $K_1\times K_2$.
For any $x^0 \in A$,
there exists an open $S$ about $x^0$ so that
$$\bigcap_{x\in S} \Deltalta_{\phi, x}(K_1\times K_2)$$
has non-empty interior,
where $\Deltalta_{\phi, x}(K_1\times K_2) = \{ \phi(x,y): y \in K_1\times K_2\}.$
\end{lem}
The strategy for establishing Lemma \ref{phi_distanceskeylemma} is as follows:
Given a pin $x\in \ensuremath{\mathbb{R}}^2$ and distance $t\in \ensuremath{\mathbb{R}}$, we note that $t\in \Deltalta_{\phi,x}(K_1\times K_2)$ whenever
$\phi(x,y) = t$ for some $y=(y_1,y_2)\in K_1\times K_2$.
We then use the implicit function theorem to solve for $y_2$ in terms of $(x, y_1,t)$ and call the resulting function $g$.
Observing that $g$ behaves like the function $g_{x,t}$ introduced in the proof of Lemma \ref{distanceskeylemma} from the previous section,
the lemma then follows from the exact proof used for Euclidean distances. The only real effort of the proof then is setting up the implicit function theorem.
\\
\betagin{proof}
Let $\varphi_x(y) = \phi(x,y)$ as in Definition \ref{deriv_cond} so that for all $(x,y) \in A\times B$,
\betagin{equation}\lambdabel{deriv_varphi}
\frac{\partial (\varphi_x)}{\partial y_i} (y)\neq 0 \thetaxt{ for } i =1,2.
\end{equation}
Define $F:\ensuremath{\mathbb{R}}^2\times \ensuremath{\mathbb{R}}^2 \times \ensuremath{\mathbb{R}} \rightarrow \ensuremath{\mathbb{R}}$ by
\betagin{equation}\lambdabel{Feq}
F(x, y,t) = \phi(x,y) -t.
\end{equation}
Then $F$ is continuously differentiable on $A\times B\times \ensuremath{\mathbb{R}}$. Moreover, it follows by \eqref{deriv_varphi} that the partial derivative of $F$ in $y_2$ is non-vanishing on $A\times B\times \ensuremath{\mathbb{R}}$.
Choose
$x^0 \in A$ and $u=(u_1,u_2) \in B\cap (K_1 \times K_2)$. Set
$$t_0 = \phi( x^0, u)$$
so that
$$F(x^0,u, t_0) = 0.$$
By the implicit function theorem,
there exists a real-valued function $g$ and a $ \partialta$-ball about the point $(x^0, u_1, t_0)$, denoted by
$B_ \partialta= B_{ \partialta}(x^0, u_1, t_0)$, so that
\betagin{itemize}
\item $g$ is continuously differentiable on $B_{ \partialta}$;
\item $g(x^0,u_1, t_0) = u_2$;
\item if $(x, y_1, t) \in B_{ \partialta}$, then $(x, y) \in A\times B$ for all $y= (y_1, g(x,y_1, t) )$;
\item $\phi(x, y) = t$ for all $y= (y_1, g(x,y_1, t) ) $ when $(x, y_1, t) \in B_{ \partialta}$.
\end{itemize}
Moreover, if $g_{x,t}(y_1) = g(x,y_1, t)$, then
$$g_{x,t}' (y_1) =
-\left( \frac{\partial (\varphi_x)}{\partial y_1}(y_1, g(x, y_1, t))\right) /
\left( \frac{\partial (\varphi_x)}{\partial y_2}(y_1, g(x, y_1, t))\right) \neq 0
$$
for $(x, y_1, t) \in B_ \partialta= B_{ \partialta}(x^0,u_1, t_0)$.
In other words, there is a neighborhood of $(x^0, t_0) $ so that the derivative of $g_{x,t}$ is non-vanishing in a neighborhood of $u_1$.
Replacing the function in \eqref{baby_g} of Lemma \ref{distanceskeylemma} with this new $g_{x,t}$, the proof proceeds as in the proof of Lemma \ref{distanceskeylemma}.
\end{proof}
\subsetbsection{\thetaxtit{Distance Trees in the Middle Thirds Cantor Set}}
In this section, we prove Theorem \ref{trees_thrm_middle_third}.
The new twist in this context is that the middle thirds Cantor set has thickness equal to 1, not strictly greater than 1.
Our method uses Lemma \ref{imagethickness} which controls how much smaller the thickness of $g(\widetilde{K})$ can be compared to $K$, where $\widetilde{K}$ is a choosen subset of $K$.
If $\tau(K)=1$, then we cannot assume $\tau(g(\widetilde{K}))\geq 1$ and therefore cannot apply the Newhouse Gap Lemma directly. However, the self similarity of the middle thirds Cantor set provides a tool that allows one to adapt the proof of the Newhouse Gap Lemma in this special setting. This observation was used in \cite{STinterior} to prove that the pinned distance set $\Deltalta_x(C_{1/3}\times C_{1/3})$ has non-empty interior. We use this idea to prove a pin wiggling lemma for the middle thirds Cantor set. \\
Before proceeding, we introduce some terminology. Let $C_{1/3}$ denote the standard middle thirds Cantor set in the interval $[0,1]$. The standard construction of this set is given by defining $C_{1/3}$ as the intersection of a family of sets $\{C_n\}$, where each $C_n$ is the union of $2^n$ closed intervals of length $1/3^n$. Define a \thetaxtbf{section} of $C_{1/3}$ to be the intersection of $C_{1/3}$ with any one of the intervals making up any of the sets $C_n$.
\betagin{lem}
\lambdabel{CantorSetImage}
Let $K_1,K_2$ be sections of the standard middle thirds Cantor $\mathcal{C}$, and let $g$ be a continuously differentiable monotone function which satisfies $1<g'<3$ on the convex hull of $K_1$. If $K_2$ and $g(K_1)$ are linked, then $K_2\cap g(K_1)\neq \emp$.
\end{lem}
\betagin{proof}
Let $I_1,I_2$ denote the convex hulls of $K_1, K_2$, respectively.
By the mean value theorem, for any interval $J$ there exists $x_J\in J$ such that $|g(J)|=|J|\cdot|g'(x_J)|$. In particular, if $U$ and $V$ are bounded gaps of $K_1$ and $|U|>|V|$, we have $|U|\geq 3|V|$ and therefore
\betagin{align*}
|g(U)|&=|U|\cdot |g'(x_U)| \\
&> 3|V|\cdot 1 \\
&> |g'(x_V)|\cdot |V| \\
&= |g(V)|.
\end{align*}
It follows that the bridges of the images are the images of the bridges. More precisely, the bridge of gap $g(U)$ in $g(K_1)$ is $g(B)$, where $B$ is the bridge next to the gap $U$. \\
To prove the theorem, we proceed by contradiction and assume $K_2\cap g(K_1)=\emp$. The strategy of the proof is a variant of the strategy used to prove the original Newhouse Gap Lemma. We construct sequences of gaps $U_n, V_n$ satisfying the following conditions:
\betagin{itemize}
\item $U_n$ is a bounded gap of $K_2$ and $g(V_n)$ is a bounded gap of $g(K_1)$
\item $U_n$ and $g(V_n)$ are linked for every $n$
\item For every $n$, either
\betagin{enumerate}[(a)]
\item $U_{n+1}=U_n$ and $|g(V_{n+1})|<|g(V_n)|$, or
\item $V_{n+1}=V_n$ and $|U_{n+1}|<|U_n|$
\end{enumerate}
\end{itemize}
Thus, at each stage we are replacing one of the gaps $U_n,g(V_n)$ with a strictly smaller one and leaving the other unchanged. In particular, this means that one of the two gap sequences must have a subsequence which is strictly decreasing in length. Since gaps of different sizes are automatically disjoint and the total length of the gaps is bounded, we must have either $|U_n|\to 0$ or $|g(V_n)|\to 0$ as $n\to\infty$. However, because $U_n$ and $g(V_n)$ are linked, the closure of $U_n$ contains points of both $K_2$ and $g(K_1)$, and similarly for $g(V_n)$. This implies that the distance between $K_2$ and $g(K_1)$ is zero, which contradicts the assumption $K_2\cap g(K_1)= \emp$. \\
We construct our sequences $\{U_n\}$ and $\{V_n\}$ recursively. First, we construct $U_1$ and $V_1$. Recall that $I_i$ denotes the convex hull of $K_i$ for $i=1,2$. Since $I_2$ and $g(I_1)$ are linked by assumption, there is a bounded gap $U_1$ of $K_2$ which is a subset of $I_2\cap g(I_1)$. Let $u$ be an endpoint of $U_1$. In particular we have $u\in K_2$, and therefore, by our assumption that $K_2\cap g(K_1)=\emp$, it follows that $u$ is in some bounded gap $g(V_1)$ of $g(K_1)$.
\\
\betagin{figure}
\betagin{tikzpicture}
\draw[ultra thick] (0,0)--(5,0);
\draw[ultra thick, blue] (5,0)--(6,0);
\draw[ultra thick, red] (6,0)--(8,0);
\draw[ultra thick] (8,0)--(12,0);
\draw [fill, blue] (5,0) circle [radius=0.1];
\draw [fill, blue] (6,0) circle [radius=0.1];
\draw [fill,] (8,0) circle [radius=0.1];
\node [below, blue] at (5.5,0) {$g(B_n^V)$};
\node [below, red] at (7,0) {$g(V_n)$};
\draw[ultra thick] (0,1)--(4,1);
\draw[ultra thick, red] (4,1)--(7,1);
\draw[ultra thick, blue] (7,1)--(10,1);
\draw[ultra thick] (10,1)--(12,1);
\draw [fill, blue] (7,1) circle [radius=0.1];
\draw [fill, blue] (10,1) circle [radius=0.1];
\draw [fill] (4,1) circle [radius=0.1];
\node [above, red] at (5.5,1) {$U_n$};
\node [above, blue] at (8.5,1) {$B_n^U$};
\end{tikzpicture}
\caption{Construction of gap sequence (gaps are red, bridges are blue)}
\lambdabel{gapsequence}
\end{figure}
Next, suppose we have constructed $U_n$ and $V_n$ satisfying the properties in the first two bullet points above. We construct $U_{n+1}$ and $V_{n+1}$ satisfying the third bullet point.
By construction, $U_n$ and $g(V_n)$ are linked. Let $B_n^U$ be the bridge corresponding to $U_n$ on the same side as $g(V_n)$ (see Figure \ref{gapsequence}), and let $g(B_n^V)$ be the bridge of $g(V_n)$ on the same side as $U_n$. We claim that one of the following two inequalities must hold:
\betagin{align*}
|g(B_n^V)|&>|U_n| \\
|B_n^U|&>|g(V_n)|.
\end{align*}
This follows because we are working with the middle thirds Cantor set, so bounded gaps and corresponding bridges have the same length, i.e., we have $|B_n^U|=|U_n|$ and $|B_n^V|=|V_n|.$
The middle thirds Cantor set also has the property that if $U$ and $V$ are gaps with $|U|>|V|$, we automatically have $|U|\geq 3|V|$. Finally, by our assumptions on $g$, we have $|J|< |g(J)|< 3|J|$ for every interval $J\subsetbset I_1$. The claim then follows, as $|U_n|>|V_n|$ implies the second inequality and $|U_n|\leq |V_n|$ implies the first. \\
Assume the second inequality holds (an analogous argument will apply when the first holds). This means one endpoint of $g(V_n)$ is in $U_n$ and the other is in $B_n^U$ (see Figure \ref{gapsequence} again). Recall that we are proving the lemma by contradiction, assuming $K_2\cap g(K_1)=\emp$. This assumption means the endpoint of $g(V_n)$ which is in $B_n^U$ must be contained in some gap $U_{n+1}$ of $K_2$, and by definition of a bridge we must have $|U_{n+1}|<|U_n|$. We then take $V_{n+1}=V_n$. This completes the construction.
\end{proof}
\betagin{figure}[ht]
\centering
\betagin{tikzpicture}
\betagin{axis}[xmin=0,xmax=1.2,ymin=0,ymax=1.2,axis x line=center, axis y line=center, xtick={1/9,2/9,3/9,6/9,7/9,8/9,1}, xticklabels={$\frac{1}{9}$,$\frac{2}{9}$,$\frac{3}{9}$,$\frac{6}{9}$,$\frac{7}{9}$,$\frac{8}{9}$,$1$}, ytick={1/9,2/9,3/9,6/9,7/9,8/9,1},yticklabels={}]
\addplot[mark=*] coordinates {(0,0)};
\addplot[red] coordinates {(0,0)(3,3)};
\addplot[red] coordinates {(0,0)(3,1)};
\addplot[thick] coordinates {(0,0)(0,1/9)(1/9,1/9)(1/9,0)(0,0)};
\addplot[thick] coordinates {(2/9,0)(2/9,1/9)(3/9,1/9)(3/9,0)(2/9,0)};
\addplot[thick] coordinates {(0,2/9)(0,3/9)(1/9,3/9)(1/9,2/9)(0,2/9)};
\addplot[thick] coordinates {(2/9,2/9)(2/9,3/9)(3/9,3/9)(3/9,2/9)(2/9,2/9)};
\addplot[thick] coordinates {(6/9,0)(6/9,1/9)(7/9,1/9)(7/9,0)(6/9,0)};
\addplot[thick] coordinates {(8/9,0)(8/9,1/9)(9/9,1/9)(9/9,0)(8/9,0)};
\addplot[thick] coordinates {(6/9,2/9)(6/9,3/9)(7/9,3/9)(7/9,2/9)(6/9,2/9)};
\addplot[thick] coordinates {(8/9,2/9)(8/9,3/9)(9/9,3/9)(9/9,2/9)(8/9,2/9)};
\addplot[thick] coordinates {(0,6/9)(0,7/9)(1/9,7/9)(1/9,6/9)(0,6/9)};
\addplot[thick] coordinates {(2/9,6/9)(2/9,7/9)(3/9,7/9)(3/9,6/9)(2/9,6/9)};
\addplot[thick] coordinates {(0,8/9)(0,9/9)(1/9,9/9)(1/9,8/9)(0,8/9)};
\addplot[thick] coordinates {(2/9,8/9)(2/9,9/9)(3/9,9/9)(3/9,8/9)(2/9,8/9)};
\addplot[thick] coordinates {(6/9,6/9)(6/9,7/9)(7/9,7/9)(7/9,6/9)(6/9,6/9)};
\addplot[thick] coordinates {(8/9,6/9)(8/9,7/9)(9/9,7/9)(9/9,6/9)(8/9,6/9)};
\addplot[thick] coordinates {(6/9,8/9)(6/9,9/9)(7/9,9/9)(7/9,8/9)(6/9,8/9)};
\addplot[thick] coordinates {(8/9,8/9)(8/9,9/9)(9/9,9/9)(9/9,8/9)(8/9,8/9)};
\end{axis}
\end{tikzpicture}
\caption{The wedge of acceptable boxes}
\lambdabel{wedge}
\end{figure}
\betagin{thm}
Given a point $x^0\in \ensuremath{\mathbb{R}}^2$, let $W_{x^0}$ denote the open wedge
\[
W_{x^0}=\{x\in\ensuremath{\mathbb{R}}^2:(x_2-x_2^0)<(x_1-x_1^0)<3(x_2-x_2^0)\}.
\]
Let $K_1,K_2$ be sections of the middle thirds Cantor set, and suppose $K_1\times K_2\subsetbset W_{x^0}$ (Figure \ref{wedge}). Then, there exists an open neighborhood $S$ of $x^0$ such that
\[
\bigcap_{x\in S}\Deltalta_x(K_1\times K_2)
\]
has non-empty interior.
\end{thm}
\betagin{proof}
For $x\in\ensuremath{\mathbb{R}}^2$ and $t\in\ensuremath{\mathbb{R}}$, consider the function
\[
g_{x,t}(z)=x_2+\sqrt{t^2-(z-x_1)}.
\]
We have
\[
g_{x,t}'(z)=-\frac{z-x_1}{g_{x,t}(z)-x_2},
\]
so $1<|g_{x,t}'(z)|<3$ whenever $(z,g_{x,t}(z))\in W_x$. For $j=1,2$, let $u_j=\min K_j$, so that $u=(u_1,u_2)$ denotes the lower left corner of $K_1\times K_2$, and let $t_0=|x^0-u|$ (thus, we have $g_{x^0,t_0}(u_1)=u_2$). For every $ \partialtalta>0$, define $\widetilde{K_j}=K_j\cap [u_1,u_1+ \partialtalta]$ and consider the set
\[
U_ \partialtalta=\{(x,t) :\widetilde{K_2}\thetaxt{ and }g_{x,t}(\widetilde{K_1}) \thetaxt{ are linked, and }(z,g_{x,t}(z))\in W_x \thetaxt{ for all }z\in[u_1,u_1+ \partialtalta]\}.
\]
By Lemma \ref{CantorSetImage}, if $(x,t)\in U_ \partialtalta$ then $t\in \Deltalta_x(K_1\times K_2)$. Clearly $U_ \partialtalta$ is open, so as in the previous proofs it suffices to show that $U_ \partialtalta$ contains a point of the form $(x^0,t)$. We first observe that if $ \partialtalta$ is sufficiently small, the sets $\widetilde{K_2}$ and $g_{x,t}(\widetilde{K_1})$ are linked for all $t\in(t_0,t_0+ \partialtalta)$. This is because for any $t>t_0$ we have $g_{x^0,t}(u_1)>u_2$, and for any $t<u_1+ \partialtalta$ we will also have $g_{x^0,t}(u_1)<u_2+ \partialtalta$ and $g_{x^0,t}(u_1+ \partialtalta)<u_2$. Finally, since $u\in W_{x^0}$ by assumption, if $ \partialtalta'$ is sufficiently small we will have $(z,g_{x^0,t}(z))\in W_{x^0}$ for all $z\in [u_1,u_1+ \partialtalta]$ and all $t\in (t_0,t_0+ \partialtalta')$.
\end{proof}
\betagin{proof}[Proof of Theorem \ref{trees_thrm_middle_third}]
By Theorem \ref{mechanism}, it suffices to find a skeleton $x^1,...,x^{k+1}\in C_{1/3}\times C_{1/3}$ such that whenever $i<j$ we have $x^j\in W_{x^i}$. Wedge membership is transitive in the sense that $x^3\in W_{x^2}$ and $x^2\in W_{x^1}$ implies $x^3\in W_{x^1}$, so it suffices to construct our skeleton so that $x^{i+1}\in W_{x^i}$ for every $i$. We construct such a sequence recursively as follows. Let $x^1$ be the origin. One can check that $W_{x^1}$ contains the box $[8/9,1]\times [6/9,7/9]$ (refer again to Figure \ref{wedge}). Let $x^2=(8/9,6/9)$. Since $x^2$ is the lower left corner of a similar copy of $C_{1/3}\times C_{1/3}$, one can run the same argument by symmetry and take $x^3$ to be the lower left corner of a box contained in $W_{x^2}$. This process can be repeated as many times as needed.
\end{proof}
\end{document} | math | 54,987 |
\begin{equation}gin{document}
\newtheorem{lem}{Lemma}[section]
\newtheorem{pro}[lem]{Proposition}
\newtheorem{thm}[lem]{Theorem}
\newtheorem{rem}[lem]{Remark}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{df}[lem]{Definition}
\title{f Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian}
\begin{equation}gin{center}
{\small
\noindent $^{a}$ Dipartimento di Matematica della Universit\'a di Roma Tor Vergata, \\
Via della Ricerca Scientifica 1, 00133 Roma, ITALY.
}
\
\
\
\
{\small
\noindent $^{b}$ Department of Mathematics of Howard University \\ Annex 3, Graduate School of Arts and Sciences, \# 217 \\ DC 20059 Washington, USA.
}
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{\small
\noindent
}
\end{center}
\footnotetext[1]
{
E-mail: [email protected],
[email protected].
\\
\thanks
{
\\ C. B. Ndiaye was partially supported by NSF grant DMS--2000164
\\ M.Mayer has been supported by the Italian MIUR Department of Excellence grant CUP E83C18000100006.
}
}
\
\
\begin{equation}gin{center}
{\bf Abstract}
\end{center}
We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei\cite{kmw1} in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper \cite{martndia2} to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in \cite{nss} to show solvability of the fractional Yamabe problem for conformal infinities of dimension \;$3$ and fractional parameter in \;$(\frac{1}{2}, 1)$\; to a global case left by previous works.
\begin{equation}gin{center}
\
\
\noindent{\bf Key Words:} Fractional scalar curvature, Fractional conformal Laplacian, Poincar\'e-Einstein manifolds, Poisson kernel, Green's function, Fermi-coordinates.
\centerline{\bf AMS subject classification: 53C21, 35C60, 58J60, 55N10.}
\end{center}
\begin{equation}gin{center}
$_{}$
\tableofcontents
\end{center}
\section{Introduction and statement of the results}
In the last decades there has been a lot of study about fractional order operators in Analysis and Geometric Analysis as well. In both fields, the recurrent themes are existence, regularity and sharp estimates, see \cite{cafroq}, \cite{cafsyl}, \cite{cafsoug}, \cite{cafval}, \cite{cafvas}, \cite{cg}, \cite{fg}, \cite{fabkenser}, \cite{qr}, \cite{gz}, \cite{guil}, \cite{gms}, \cite{gq}). In this paper we are interested in the issue of existence, regularity and sharp estimates in the context of Conformal Geometry. Precisely, we study the issue of existence, regularity and sharp asymptotics of the Poisson and Green's functions of the fractional conformal Laplacian on conformal infinities of asymptotically hyperbolic manifolds.
\noindent
To introduce the fractional conformal Laplacian, we first recall some definitions in the theory of asymptotically hyperbolic metrics. Given \;$X=X^{n+1}$\; a smooth manifold with boundary $M=M^n$\; and \;$n\geq 2$ we say that \;$\varrho$\; is a defining function of the boundary \;$M$\; in \;
$X$,\; if
$$
\varrho>0\;\;\text{ in }\;\;X,\;\;\varrho=0\;\;\text{ on }\;\;M\;\;\text{ and }\;\;d\varrho\neq 0\;\;\text{ on }\;\; M.
$$
A Riemannian metric \;$g^+$\; on \;$X$\; is said to be conformally compact, if for some defining function \;$\varrho$, the Riemannian metric
\begin{equation}gin{equation}\label{eq:cmetric}
g:=\varrho^2g^+
\end{equation}
extends to \;$\overline{X}:=X\cup M$\; so that \;$(\overline{X}, \, g)$\; is a compact Riemannian manifold with boundary \;$M$ and interior \;$X$. Clearly this induces a conformal class of Riemannian metrics
$$ [h]=[ g|_{TM}]$$ on \;$M$, where $TM$ denotes the tangent bundle of \;$M$, when the defining functions \;$\varrho$\; vary and the resulting conformal manifold \;$(M, [h])$\; is called conformal infinity of \;$(X, \;g^+)$. Moreover a Riemannian metric \;$g^+$\; in \;$X$\; is said to be asymptotically hyperbolic, if it is conformally compact and its sectional curvature tends to \;$-1$\; as one approaches the conformal infinity of \;$(X, \;g^+)$,
which is equivalent to
\begin{equation}gin{equation*}
|d\varrho|_{\begin{eqnarray}r g}=1
\end{equation*}
on\;$M$, see \cite{mazzeo1}, and in such a case \;$(X, \;g^+)$\; is called an asymptotically hyperbolic manifold. Furthermore a Riemannian metric \;$g^+$\; on \;$X$\; is said to be conformally compact Einstein or Poincar\'e-Einstein (PE), if it is asymptotically hyperbolic and satisfies the Einstein equation
\begin{equation}gin{equation*}
Ric_{g^+}=-ng^+,
\end{equation*}
where $Ric_{g^+}$\; denotes the Ricci tensor of \;$(X, \;g^+)$.
\noindent
On one hand for every asymptotically hyperbolic manifold \;$(X, \;g^+)$\; and every choice of the representative \;$h$\; of its conformal infinity \;$(M, [h])$, there exists a geodesic defining function \;$ y $\; of \;$M$\; in \;$X$\; such that in a tubular neighborhood of \;$M$\; in \;$X$, the Riemannian metric \;$g^+$\; takes the following normal form
\begin{equation}gin{equation}\label{eq:uniqdef}
g^+=\frac{d y ^2+h_{ y }}{ y ^2},
\end{equation}
where \;$h_{ y }$\; is a family of Riemannian metrics on \;$M$\; satisfying \;$h_0=h$\; and \;$ y $\; is the unique such a one in a tubular neighborhood of $M$. Furthermore we say that the conformal infinity \;$(M, \;[\hat h])$\; of an asymptotically hyperbolic manifold \;$(X, \;g^+)$\; is locally flat, if \;$h$\; is locally conformally flat, and clearly this is independent of the representative \;$h$\; of \;$[h]$. Moreover we say that \;$(M, [h])$\; is umbilic, if \;$(M, h)$\; is umbilic in \;$(X, \; g)$ \;where \;$g$\; is given by \eqref{eq:cmetric} and \;$ y $\; is the unique geodesic defining function given by \eqref{eq:uniqdef}, and this is clearly independent of the representative \;$h$\; of \;$[h]$, as easily seen from the uniqueness of the normal form \eqref{eq:uniqdef} or Lemma 2.3 in \cite{gq}. Similarly we say that \;$(M, [h])$\; is minimal if \;$H_{g}=0$\; with \;$H_{g}$\; denoting the mean curvature of \;$(M, \;h)$\; in \;$(\overline X, \;g)$ with respect to the inward direction, and this is again clearly independent of the representative of \;$h$\; of \;$[h]$, as easily seen from Lemma 2.3 in \cite{gq}. Finally we say that \;$(M, [h])$\; is totally geodesic, if \;$(M, [h])$\; is umbilic and minimal.
\begin{equation}gin{rem}\label{eq:minimal}
We remark that in the conformally compact Einstein case, \;$h_{ y }$\; as in \eqref{eq:uniqdef} has an asymptotic expansion which contains only even powers of \;$ y $, at least up to order \;$n$, see \cite{cg}.
In particular the conformal infinity \;$(M, [h])$\; of any Poincar\'e-Einstein manifold \;$(X, g^+)$\; is totally geodesic. \end{rem}
\begin{equation}gin{rem}
As every \;$2$-dimensional Riemannian manifold is locally conformally flat, we will say locally flat conformal infinity of a Poincar\'e-Einstein manifold to mean just the conformal infinity of a Poincar\'e-Einstein manifold when the dimension is either \;$2$\; or which is further locally flat if the dimension is bigger than \;$2$.
\end{rem}
\noindent
On the other hand, for any asymptotically hyperbolic manifold \;$(X, g^+)$\; with conformal infinity \;$(M, [h])$, Graham-Zworsky\cite{gz}\; have attached a family of scattering operators \;$S(s)$\; which is a meromorphic family of pseudo-differential operators on \;$M$ defined on \;$\mathbb{C}$, by considering Dirichlet-to-Neumann operators for the scattering problem for \;$(X, \;g^+)$\; and a meromorphic continuation argument. Indeed it follows from \cite{gz} and \cite{mazmel} that for every \;$f\in C^{\infty}(M)$, and for every \;$s\in \mathbb{C}$\; such that \;$Re(s)>\frac{n}{2}$\;and\; \;$s(n-s)$ is not an \;$L^2$-eigenvalue of \;$-\D_{g^+}$, the following generalized eigenvalue problem
\begin{equation}gin{equation}\label{eq:geneig}
-\D_{g^+}u-s(n-s)u=0\,\;\;\text{ in }\;\;X
\end{equation}
has a solution of the form
$$
u=F y ^{n-s}+G y ^s, \;\;\; F,\;G\in C^{\infty}(\overline X), \;\;\;F|_{ y =0}=f,
$$
where \;$ y $\; is given by \eqref{eq:uniqdef} and for those values of \;$s$\; the scattering operator \;$S(s)$\; on \;$M$\; is defined as
\begin{equation}gin{equation}\label{eq:dtnscat}
S(s)f=G|_{M}.
\end{equation}
Furthermore using a meromorphic continuation argument, Graham-Zworsky\cite{gz} extend \;$S(s)$\; defined by \eqref{eq:dtnscat} to a meromorphic family of pseudo-differential operators on \;$M$\; defined on all \;$\mathbb{C}$\; and still denoted by $S(s)$\;with only a discrete set of poles including the trivial ones \;$s=\frac{n}{2}, \frac{n}{2}+1, \cdots, $\; which are simple poles of finite rank, and possibly some others corresponding to the \;$L^2$-eigenvalues of \;$-\D_{g^+}$. Using the regular part of the scattering operators \;$S(s)$, to any \;$\gamma\in (0, 1)$\; such that
\begin{equation}gin{equation*}
\left(\frac{n}{2}\right)^2-\gamma^2<\l_1(-\D_{g^+})
\end{equation*}
with \;$\l_1(-\D_{g^+})$\: denoting the first eigenvalue of \;$-\D_{g_+}$,\; Chang-Gonzalez\cite{cg} have attached the following fractional order pseudo-differential operators referred to as fractional conformal Laplacians or fractional Paneitz operators
\begin{equation}gin{equation}\label{eq:fracopd}
P^{\gamma}[g^+, \;h]:=-d_\gamma S\left(\frac{n}{2}+\gamma\right),
\end{equation}
where \;$d_\gamma$\; is a positive constant depending only on \;$\gamma$\; and chosen such that the principal symbol of \;$P^{\gamma}[g^+, h]$\; is exactly the same as the one of the fractional Laplacian \;$(-\D_{h})^{\gamma}$, when
\begin{equation}gin{equation*}
X=\mathbb{R}^{n+1}_{+}, \;M=\mathbb{R}^{n},\;h=g_{\mathbb{R}^{n}} \;\;\text{ and }\;\; g^{+}=g_{\mathbb{H}^{n+1}}.
\end{equation*}
When there is no possible confusion with the metric \;$g^+$, we just use the simple notation
\begin{equation}gin{equation*}
P^{\gamma}_{h}:=P^{\gamma}[g^+, \;h].
\end{equation*}
Similarly to the other well studied conformally covariant differential operators, Chang-Gonzalez\cite{cg} associate to each \;$P^{\gamma}_{h}$\; the curvature quantity
\begin{equation}gin{equation*}
Q^{\gamma}_{h}:=P^{\gamma}_{h}(1).
\end{equation*}
The \;$Q^\gamma_{h}$\; are referred to as fractional scalar curvatures, fractional \;$Q$-curvatures or simply \;$Q^\gamma$-curvatures. Of particular importance to conformal geometry is the conformal covariance property verified by \;$P^{\gamma}_{h}$
\begin{equation}gin{equation}\label{eq:confinv}
P^{\gamma}_{h_u}(v)=v^{-\frac{n+2\gamma}{n-2\gamma}}P^{\gamma}_{h}(uv)
\;\;\text{ for }\;\; h_v=v^{\frac{4}{n-2\gamma}} \;\;\text{ and }\;\; 0<v\in C^{\infty}(M).
\end{equation}
The fractional Yamabe problem is the problem of finding conformal metrics of with constant \;$Q^\gamma$-curvature. As in the classical Yamabe problem, see \cite{sc}, its study deeply depends on the existence, regularity and sharp asymptotic of the Green's function of \;$P^{\gamma}_{h}$.
\noindent
In this paper, we show existence, regularity and sharp asymptotics of the Poisson kernel \;$K_g$\; and Green's functions \;$\Gamma_g$\; under weighted Neumann boundary conditions of the Chang-Gonzalez\cite{cg} extension problem associated to \;$P^{\gamma}_{h}$ and the Green's function\;$G_h$\; of \;$P^{\gamma}_{h}$. Indeed we prove:
\begin{equation}gin{thm}
\label{Greens_function_asymptoticsgamma1}$_{}$\\
Let \;$(X, \;g^{+})$\; be an asymptotically hyperbolic manifold with
conformal infinity \;$(M, [h])$\; of dimension \;\;$n\geq 2$. If
\begin{equation}gin{equation*}
\frac{1}{2}\neq \gamma \in (0,1) \;\;\text{ and } \;\,\;
\lambda_{1}(-\Delta_{g^{+}})>s(n-s)\;\;
\text{ for } s=\frac{n}{2}+\gamma,
\end{equation*}
then the Poison kernel \;$K_{g}$\; and the Green's functions \;$\Gamma_{g}$\; and \;$G_{h}$\;
respectively for
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}U=0 \;\;\text{ in }\;\; X \\
U=f \,\,\;\text{ on } \;\;M
\end{cases}
\quad\quad
\begin{equation}gin{cases}
D_{g}U=0\;\; \text{ in } \;\;X \\
-d_{\gamma}^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}U=f \;\;\text{ on }\;\; M
\end{cases}
\quad \text{and}\quad
\begin{equation}gin{cases}
P_h^\gamma u=f\;\;\text{ on }\;\;M
\end{cases}
\end{equation*}
exist and we may expand in \;$g$-normal
Fermi-coordinates
around $\xi \in M$
\begin{equation}gin{enumerate}[label=(\roman*)]
\item
\quad
$
K_{g}(z,\xi)
\;\in
\eta_{\xi}(z)
\left( p_{n, \gamma}
\frac{y^{2\gamma}}{\vert z
\vert^{n+2\gamma}}+\sum^{2m+5-2\gamma}_{l=-n-2\gamma}y^{2\gamma}H_{1+l}(z)
\right)
+
y^{2\gamma}C^{2m,\alpha}(X)
$
\item
\quad
$
\Gamma_{g}(z,\xi)
\;\in
\eta_{\xi}(z)
\left(
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+\sum^{2m+3}_{l=-n}H_{1+2\gamma+l}(z)
\right)
+
C^{2m,\alpha}(X)
$
\item \quad
$
G_{h}(x,\xi)
\;\in
\eta_{\xi}(x)
\left(
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+\sum^{2m+3}_{l=-n}H_{1+2\gamma+l}(x)
\right)
+
C^{2m,\alpha}(M)
$
\end{enumerate}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n+1}_+\setminus \{0\})$\; being homogeneous of order
\;$l$, \;$\eta_{\xi}$\; as in \eqref{etaxi}, \;$p_{n, \gamma}$\; is as in \eqref{pngamma}, and \;$g_{n, \gamma}$\; is as in \eqref{gngamma}, provided \;$H_{g}=0$.
\end{thm}
\noindent
In the case of locally flat conformal infinities of Poincare-Einstein manifolds, we have:
\begin{equation}gin{thm}
\label{cor_kernels_for_poincare_einstein_metrics1}$_{}$\\
\noindent
Let \;$(X, \;g^{+})$\; be a Poincar\'e-Einstein manifold with
conformal infinity \;$(M, [h])$\; of dimension \;$n= 2$\; or \;$n\geq 3$\; and \;$(M, [h])$\; is locally flat. If
\begin{equation}gin{equation*}
\frac{1}{2}\neq \gamma \in (0,1) \;\;\text{ and } \;\,\;
\lambda_{1}(-\Delta_{g^{+}})>s(n-s)\;\;
\text{ for } s=\frac{n}{2}+\gamma,
\end{equation*}
then the Poisson kernel \;$K_{g}$\; and the Green's functions \;$\Gamma_{g}$\; and \;$G_{h}$\;
respectively for
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}U=0 \;\;\text{ in }\;\; X \\
U=f \,\,\;\text{ on } \;\;M
\end{cases}
\quad\quad
\begin{equation}gin{cases}
D_{g}U=0\;\; \text{ in } \;\;X \\
-d_{\gamma}^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}U=f \;\;\text{ on }\;\; M
\end{cases}
\quad \text{and}\quad
\begin{equation}gin{cases}
P_h^\gamma u=f\;\;\text{ on }\;\;M
\end{cases}
\end{equation*}
are respectively of class \;$y^{2\gamma}C^{2,\alpha}$\; and \;$C^{2,\alpha}$\; away from the
singularity and admit for every \;$a\in M$ locally in \;$g_{a}$-normal
Fermi-coordinates an expansion around $a$
\begin{equation}gin{enumerate}[label=(\roman*)]
\item \quad
$
K_{a}(z)\;
\in
p_{n, \gamma}\frac{y^{2\gamma}}{\vert z \vert^{n+2\gamma}}
+
y^{2\gamma}H_{-2\gamma}(z)
+
y^{2\gamma}H_{1-2\gamma}(z)
+
y^{2\gamma}H_{2-2\gamma}(z)
+
y^{2\gamma}C^{2,\alpha}(X)
$
\item \quad
$
\Gamma_{a}(z)\;
\in
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+H_{2\gamma}(z)+H_{1+2\gamma}(z)
+
C^{2,\alpha}(X)
$
\item \quad
$
G_{a}(x)\;
\in
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+H_{2\gamma}(x)+H_{1+2\gamma}(x)
+
C^{2,\alpha}(M),
$
\end{enumerate}
where \;$g_a$\;is as in \eqref{dfmga}, \;$K_{a}=K_{g_{a}}(\cdot, a)$, \;$\Gamma_{a}=\Gamma_{g_{a}}(\cdot,
a)$ and \;$G_a=G_{h_a}(\cdot,a)$\; and \;$H_{k}\in C^{\infty}(\overlineerline{\mathbb{R}^{n}_{+}}\setminus \{0\})$\; are
homogeneous of degree \;$k$.
\end{thm}
\noindent
To prove Theorem \ref{Greens_function_asymptoticsgamma1} and Theorem \ref{cor_kernels_for_poincare_einstein_metrics1}, we use the method of Lee-Parker\cite{lp} of killing deficits successively. However difficulties arise due the the rigidity involved in the problem (see \eqref{eq:uniqdef}) and the lack of classical regularity theory. To overcome the rigidity issue, we work with the space of homogeneous functions rather than the one of polynomials as done in \cite{lp}. To handle the regularity issue, we show some higher order regularity results for the Dirichlet problem and the weighted Neumann boundary problem of the Chang-Gonzalez\cite{cg} extension problem for \;$P_h^{\gamma}$\; which are of independent interest, see Proposition \ref{poisson_regularity} and Proposition \ref{green_regularity}. We point out that even if the estimates in Proposition \ref{poisson_regularity} and Proposition \ref{green_regularity} are weak, they are enough for our purpose and in turn get improved by the estimates of the Poisson kernel and Green's function in Theorem \ref{Greens_function_asymptoticsgamma1} and Theorem \ref{cor_kernels_for_poincare_einstein_metrics1} that they imply. On the other hand, we would like to emphasize that (ii) of Theorem \ref{cor_kernels_for_poincare_einstein_metrics1} answers the first part of the Conjecture of Kim-Musso-Wei\cite{kmw1} about the asymptotics of \;$\Gamma_a$\; and gives the definition of the fractional mass, see our work \cite{martndia3}, Definition 4.3 and Lemma 4.1.
\noindent
The structure of the paper is as follows: In Section \ref{notation_and_prelimiaries} we fix some notations. In Section \ref{nonhom} we develop a non-homogeneous extension of some aspects of the works of Chang-Gonzalez\cite{cg} and Graham-Zworsky\cite{gz}. It is divided in two subsections. In the first one, namely Subsection \ref{eq:nhscatdeg}, we develop a non-homogeneous scattering theory, define the associated non-homogeneous fractional operator and its relation to a non-homogeneous uniformly degenerate boundary value problem. In Subsection \ref{confnonhom} we discuss the conformal property of the non-homogeneous fractional operator. We point out that Section \ref{nonhom} even being of independent interest contains estimates which are used in Section \ref{pceinsten} and in \cite{martndia3}, and a regularity result that we use in \cite{martndia3} . Section \ref{fundso} is concerned with the study of the Poisson kernel \;$K_g$ and the Green's function \;$\Gamma_g$\; under weighted Neumann boundary conditions of the Chang-Gonzalez extension problem \;of $P_h^\gamma$, and the Green's function \;$G_h$\; of $P_h^\gamma$\; all in the general case of asymptotically hyperbolic manifolds with minimal conformal infinity. In Section \ref{pceinsten} we sharpen the results obtained in Section \ref{fundso} in the particular case of a locally flat conformal infinity of a Poincar\'e-Einstein manifold.
\section{Notations and preliminaries }\label{notation_and_prelimiaries}
In this section we fix some notations. First of all let \;$X=X^{n+1}$\; be a manifold of dimension \;$n+1$\; with boundary \;$M=M^{n}$\; and closure \;$\overline{X}$\; with\;$n\geq 2$.
\noindent
In the following, for any Riemannian metric \;$\begin{eqnarray}r h$\; defined on \;$M$, $a\in M$\; and \;$r>0$, we use the notation \;$B^{\begin{eqnarray}r h}_{r}(a)$\; to denote the geodesic ball with respect to $\begin{eqnarray}r h$\; of radius \;$r$\;and center \;$a$. We also denote by \;$d_{\begin{eqnarray}r h}(x,y)$\; the geodesic distance with respect to \;$\begin{eqnarray}r h$\; between two points \;$x$\;and \;$y$\; of \;$M$. $inj_{\begin{eqnarray}r h}(M)$\;stands for the injectivity radius of \;$(M, \begin{eqnarray}r h)$. $dV_{\begin{eqnarray}r h}$\;denotes the Riemannian measure associated to the metric\;$\begin{eqnarray}r h$\; on \;$M$. For \;$a\in M$ we use the notation \;$\exp^a_{\begin{eqnarray}r h}$\; to denote the exponential map with respect to \;$\begin{eqnarray}r h$\; on \;$M$.
\noindent
Similarly for any Riemannian metric \;$\begin{eqnarray}r g$\; defined on \;$\overline{X}$, $a\in M$\; and \;$r>0$ we use the notation \;$B^{\begin{eqnarray}r g, + }_{r}(a)$\; to denote the geodesic half ball with respect to \;$\begin{eqnarray}r g$\; of radius \;$r$\;and center \;$a$. We also denote by \;$d_{\begin{eqnarray}r g}(x,y)$\; the geodesic distance with respect to \;$\begin{eqnarray}r g$\; between two points \;$x\in M$\;and \;$y\in \overline {X}$. $inj_{\begin{eqnarray}r g }(\overline{X})$\;stands for the injectivity radius of \;$(\overline{X}, \begin{eqnarray}r g)$. $dV_{\begin{eqnarray}r g}$\;denotes the Riemannian measure associated to the metric\;$\begin{eqnarray}r g$\; on \;$\overline{X}$. For \;$a\in M^{n}$\; we use the notation \;$\exp_a^{\begin{eqnarray}r g, +}$\; to denote the exponential map with respect to \;$\begin{eqnarray}r g$\; on \;$\overline{X}$.
\noindent
$\mathbb{N}$\;denotes the set of nonnegative integers, $\mathbb{N}^*$\; the set of positive integers and for $k\in \mathbb{N}^*$, $\mathbb{R}^k$\;stands for the standard $k$-dimensional Euclidean space, $\mathbb{R}^k_+$ the open positive half-space of $\mathbb{R}^k$, and $\begin{eqnarray}r \mathbb{R}^k_+$ its closure in $\mathbb{R}^k$. For simplicity we use the notation \;$\mathbb{R}_+:=\mathbb{R}^1_+$, and $\begin{eqnarray}r \mathbb{R}_+:=\begin{eqnarray}r \mathbb{R}^1_+$. For $r>0$
we denote respectively
\begin{equation}gin{equation*}
B^{\mathbb{R}^k}_r(0) \;\; \text{ and }\;\; B^{\mathbb{R}^k_+}_r(0)=B_r^{\mathbb{R}^{k}}(0)\cap \mathbb{R}^k_+\simeq ]0, r[\times B_r^{\mathbb{R}^{k-1}}(0)
\end{equation*}
the open and open upper half ball of \;$\mathbb{R}^k$\; of center \;$0$\; and radius \;$r$, and set
\;$B_r=B_r^{\mathbb{R}^n}$\; and $B_r^+=B_r^{\mathbb{R}^{n+1}_+}$.
\noindent
For \;$p\in \mathbb{N}^*$, let \;$M^p$\; denotes the Cartesian product of \;$p$\; copies of \;$M$. We define \;$(M^2)^*:=M^2\setminus Diag(M^2)$, where \;$Diag(M^2)=\{(a, a): \,a\in M\}$\; is the diagonal of \;$M$.
\noindent
For \;$1\leq p\leq \infty,\;k\in \mathbb{N}$, $s\in \mathbb{R}_+$, $\begin{equation}ta\in ]0, 1[$ and \;$\begin{eqnarray}r h$\; a Riemannian metric defined on \;$M$, \begin{equation}gin{equation*}
L^p(M, \begin{eqnarray}r h), \;W^{s, p}(M, \begin{eqnarray}r h), \;C^k(M, \begin{eqnarray}r h)\;\;\text{ and }\;\;C^{k, \begin{equation}ta}(M, \begin{eqnarray}r h)
\end{equation*}
stand respectively for the standard $p$-Lebesgue and \;$(s, p)$-Sobolev space, $k$-continuously differentiable space and \;$k$-continuously differential space of H\"older exponent \;$\begin{equation}ta$, all on \;$M$ and with respect to \;$\begin{eqnarray}r h$, if the definition required a metric structure. Similarly for\;$1\leq p\leq \infty,\;k\in \mathbb{N}$, $s\in \mathbb{R}_+$, $\begin{equation}ta\in ]0, 1[$ and \;$\begin{eqnarray}r g$\; a Riemannian metric defined on \;$\overline{X}$,
\begin{equation}gin{equation*}
L^p_{f}(\overline{X}, \begin{eqnarray}r g), \;W^{s, p}_{f}(\overline{X}, \begin{eqnarray}r g ), \;C^k(\overline{X}, \begin{eqnarray}r g)\text{ and }\;C^{k, \begin{equation}ta}(\overline{X}, \begin{eqnarray}r g)
\end{equation*}
stand respectively for the weighted \;$p$-Lebesgue and \;$(s, p)$-Sobolev space, \;$k$-continuously differentiable space and \;$k$-continuously differential space of H\"older exponent \;$\begin{equation}ta$, all on \;$\overline{X}$, and as above with respect to \;$\begin{eqnarray}r g$\; and a measurable function \;$f>0$\; on \;$X$\;, if required. For precise definitions and properties see \cite{aubin}, \cite{dinpalval}, \cite{gold}, \cite{gt} and \cite{s}.
$C_0^{\infty}(X)$ means element in \;$C^{\infty}(X)$\; vanishing on \;$M$\; to infinite order.
\noindent
For \;$\varepsilonilon>0$\; and small \;$o_{\varepsilonilon}(1)$\; means quantities which tend to \;$0$\; as \;$\varepsilonilon$\; tends to \;$0$. $O(1)$ stands for quantities which are bounded. For \;$x\in \mathbb{R}$\; we use the notation \;$O(x)$\; and
\;$o_{\varepsilonilon}(x)$ to mean respectively \;$|x|O(1)$\; and \; $|x|o_{\varepsilonilon}(1)$. Large positive constants are usually denoted by \;$C$\; and the value of \;$C$\; is allowed to vary from formula to formula and also within the same line. Similarly small positive constants are denoted by \;$c$\; and their values may vary from formula to formula and also within the same line.
\noindent
We define
\begin{equation}gin{equation}\label{dsgamma}
d_{\gamma}^*=\frac{d_\gamma}{2\gamma},
\end{equation}
cf. \eqref{eq:fracopd}. Furthermore, we set
\begin{equation}gin{equation}\label{defc3}
c_{n, 3 }^{\gamma}=\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|x|^2}\right)^{\frac{n+2\gamma}{2}}dx,\;\;
\end{equation}
and
\begin{equation}gin{equation}\label{pngamma}
p_{n,\gamma}=\frac{1}{c_{n,3}^{\gamma}}
\end{equation}
\noindent
Let\;$(X, g^+)$ be an asymptotically hyperbolic manifold of dimension \;$n+1$\; with \;$n\geq 2$\; and minimal conformal infinity \;$(M, [h])$. Then, because of \eqref{eq:uniqdef} and minimality of the conformal infinity, we can consider a geodesic defining function \;$y$\; splitting the metric
\begin{equation}gin{equation*}
g=y^{2}g^{+}, \;\; g=dy^{2}+h_{y}\;\;\text{near}\;\;M\;\;\text{ and } \;\;h=h_{y}\lfloor_{M}
\end{equation*}
in such a way, that \;$H_{g}=0$.
Moreover using the existence of conformal normal coordinates, cf. \cite{gun}, there exists for every \;$a\in M$\; a conformal factor
\begin{equation}gin{equation}\label{conformal_factor_properties}
0<u_{a}\in C^{\infty}(M)\;\;\text{ satisfying }\;\;\frac{1}{C}\leq u_{a}\leq
C, \;\;u_{a}(a)=1\;\;\text{ and }\;\;\nabla u_{a}(a)=0,
\end{equation}
inducing a conformal normal coordinate system close to \;$a$\; on \;$M$, in
particular in normal coordinates with respect to
$$h_{a}=u_{a}^{\frac{4}{n-2\gamma}}h$$ we have for some small \;$\varepsilonilon>0$
\begin{equation}gin{equation*}
h_{a}=\delta elta+O(\vert x \vert^{2}), \;\;\delta et h_{a}\equiv 1\;\;\text{ on } \;\;
B_{\varepsilonilon}^{h_a}(a).
\end{equation*}
As clarified in
Subsection \ref{confnonhom} the conformal factor \;$u_{a}\;$ then
naturally extends onto \;$X$ via
$$u_{a}=(\frac{y_{a}}{y})^{\frac{n-2\gamma}{2}},$$
where \;$y_{a}$\; close to the
boundary \;$M$ is the unique geodesic defining function, for which
\begin{equation}gin{equation*}
g_{a}=y_{a}^{2}g^{+}, \;\; g_{a}=dy_{a}^{2}+h_{a, y_a}\;\;\text{near}\;\;M\;\;\text{ with }
\;\;h_{a}=h_{a, y_a}\lfloor_{M}
\end{equation*}
and there still holds \;$H_{g_{a}}=0$. Consequently
\begin{equation}gin{equation*}
g_{a}=\delta elta+O(y+\vert x \vert^{2})\;\;\text{ and }\;\; \delta et g_{a}=1+O(y^{2}) \;\;\text{in
} \;\;B_{\varepsilonilon}^{g_a, +}(a).
\end{equation*}
\section{Non-homogeneous scattering theory}\label{nonhom}
In this section we extend some aspects of the works of Chang-Gonzalez\cite{cg} and Graham-Zworsky\cite{gz} to a non-homogeneous setting and in the general framework of asymptotically hyperbolic manifolds. It is of independent interest, but in it we derive estimates that are used in Section \ref{pceinsten} and \cite{martndia3}, and an existence and regularity result used in \cite{martndia3} to construct barrier solutions in order to compare different types of bubbles via maximum principle. We divide this section in two subsections.
\subsection{Scattering operators and uniformly degenerate equations}\label{eq:nhscatdeg}
In this subsection we extend some parts of the works of Chang-Gonzalez\cite{cg} and Graham-Zworski\cite{gz} to a non-homogeneous setting in the context of asymptotically hyperbolic manifolds. First of all let \;$(X, g^{+})$\; be an asymptotically hyperbolic manifold with conformal infinity \;$(M, [h])$\; and \;$y$\; the unique geodesic defining function associated to \;$h$\; given by \eqref{eq:uniqdef}. Then we have the normal form
\begin{equation}gin{equation*}\begin{equation}gin{split}
\;y^{2}g^{+}=g=dy^{2}+h_{y} \;\;\text{ near }\;\;M
\end{split}\end{equation*}
with
\;$
y>0\;\; \text{ in } \;\;X,\;\;y=0\;\;\text{ on }\;\; M\;\; \text{ and }\;\;\vert dy \vert_{g}=1 \;\;\text{ near }\;\; M.
$ Furthermore let
\begin{equation}gin{equation*}
\square_{g^{+}}=-\Delta_{g^{+}}-s(n-s),
\end{equation*}
where by definition
\begin{equation}gin{equation*}
s=\frac{n}{2}+\gamma,\;\;\gamma\in (0,1), \;\,\gamma\neq \frac{1}{2}\;\; \text{and}\; \;\;s(n-s)\in (0,\frac{n^{2}}{4}).
\end{equation*}
According to Mazzeo and Melrose \cite{mazzeo1}, \cite{mazzeo2}, \cite{mazmel}
\begin{equation}gin{equation*}
\sigma(-\Delta_{g^{+}})=\sigma_{pp}(-\Delta_{g^{+}})\cup [\frac{n^{2}}{4}, \infty),
\; \sigma_{pp}(-\Delta_{g^{+}})\subset (0,\frac{n^{2}}{4}),
\end{equation*}
where \;$\sigma(-\Delta_{g^{+}})$\; and \;$\sigma_{pp}(-\Delta_{g^{+}})$ are respectively the spectrum and the pure point spectrum of \;$L^{2}$-eigenvalues of \;$-\Delta_{g^{+}}$.
Using the work of Graham-Zworski\cite{gz}, see equation (3.9) therein, we may solve
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
\square_{g^{+}}u=f\;\;\text{ in }\;\;X \\
y^{s-n}u=\underline{v} \;\;\text{ on } \;\;M
\end{cases}
\end{split}\end{equation*}
for \;$s(n-s)\not \in \sigma_{pp}(-\Delta_{g^{+}})$\; and \;$f\in y^{n-s+1}C^{\infty}(\overline X)+y^{s+1}C^{\infty}(\overline X)$\; in the form
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
u=y^{n-s}A+y^{s}B \;\;\text{ in }\;\;X\\
A,\; B\in C^{\infty}(\overline X),\;\;\,A=\underline{v} \;\;\text{ on } \;\;M.
\end{cases}
\end{split}\end{equation*}
As in the case \;$f=0$, which corresponds to the generalized eigenvalue problem of Graham-Zworsky\cite{gz}, this gives rise to a Dirichlet-to-Neumann map \;$S_{f}(s)$\; via
\begin{equation}gin{equation*}\begin{equation}gin{split}
\underline{v}=A\lfloor _{M}\longrightarrow\-B\lfloor _{M}=\overlineerline{v},
\end{split}\end{equation*}
which we refer to as non-homogeneous scattering operator and denote it by \;$S_{f}(s)$. Clearly $S_0(s)=S(s)$ and $S_{f}(s)$ is invertible, since the standard scattering operator \;$S_0(s)$\;
is invertible, cf. equation (1.2) in \cite{jb}. We define the non-homogeneous fractional operators by
$$
P^{\gamma}_{f,h}=-d_\gamma S_{f}(s),
$$
where \;$d_\gamma$\; is as in \eqref{eq:fracopd}. Following \cite{gq} we find by conformal covariance of the conformal Laplacian that
\begin{equation}gin{equation}\begin{equation}gin{split}\label{transformation_scattering_extension}
\square_{g^{+}}u=f
\xLeftrightarrow{U=y^{s-n}u} D_{g}U=y^{-s-1}f,
\end{split}\end{equation}
where
\begin{equation}gin{equation}\label{eqdg}
\begin{equation}gin{split}
D_{g}U=-div_{g}(y^{1-2\gamma}\nabla_{g}U)+E_{g}U
\end{split}
\end{equation}
and with \;$L_g=-\Delta_{g}+\frac{R_{g}}{c_{n}}$\; denoting the conformal Laplacian on \;$(X, g)$
\begin{equation}gin{equation}\begin{equation}gin{split}\label{Eg_general}
E_{g} :=y^{\frac{1-2\gamma}{2}}L_{g}y^{\frac{1-2\gamma}{2}} -(\frac{R_{g^{+}}}{c_{n}}+s(n-s))y^{(1-2\gamma)-2}
,\;\;
c_{n}=\frac{4n}{n-1}.
\end{split}
\end{equation}
Thus we find for \;$\phi,\psi \in C^{\infty}(\overline X)$, that
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
\square_{g^{+}}u=y^{n-s+1}\phi +y^{s+1}\psi\;\;\text{ in }\;\;X \\
y^{s-n}u=\underline{v} \;\; \text{ on } \;\;M
\end{cases}
\xLeftrightarrow{U=y^{s-n}u}
\begin{equation}gin{cases}
D_{g}U=y^{-2\gamma}\phi + \psi\;\;\text{ in }\;\;X \\
U=\underline{v} \;\;\text{ on } \;\;M
\end{cases}.
\end{split}\end{equation*}
Note, that such a solution \;$U$\; is of the form
\begin{equation}gin{equation*}
\begin{equation}gin{split}
U=A+By^{2\gamma}=\sum A_{i}y^{i}+\sum B_{i}y^{i+2\gamma}+U_0
\end{split}
\end{equation*}
for some \;$U_0\in C^{\infty}_0(X)$\; and has principal terms
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
\underline{v}+\overlineerline{v} y^{2\gamma}\;\; \text{ for }\;\; \gamma<\frac{1}{2}\\
\underline{v}+A_{1}y+\overlineerline{v} y^{2\gamma}\;\; \text{ for }\;\; \gamma>\frac{1}{2}.
\end{cases}
\end{equation*}
As for the case $\gamma>\frac{1}{2}$, expanding the boundary metric
\;$h_y$,\; we find
\begin{equation}gin{equation*}
\begin{equation}gin{split}
h_{y}=h_{0}+h_{1}y+O(y^{2}) \;\;\text{ with }\;\; h_{1}=2\Pi_g
\end{split}
\end{equation*}
and \;$\Pi_g$\; denoting the second fundamental form of \;$(M, h)$\; in $(\overline{X}, g)$. Still according to \cite{gz} we may solve
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
\square_{g^{+}}u=\square_{g^{+}}u=y^{n-s+2}\phi +y^{s+1}\psi\;\;\text{ in }\;\;X \\
y^{s-n}u=\underline{v} \;\;\text{on} \;\;M
\end{cases}
\end{split}\end{equation*}
for \;$\phi,\psi \in C^{\infty}(\overline X)$ in the form
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
u=y^{n-s}A+y^{s}B \;\;\text{ in }\;\;X\\
A, \;B\in C^{\infty}(\overline X),\;\;A=\underline{v} \;\;\text{ on } \;\;M
\end{cases}
\end{split}\end{equation*}
with asymptotic
\begin{equation}gin{equation*}
\begin{equation}gin{split}
A=\sum A_{i}y^{i}, \;\; A_{0}=\underline{v}, \;\;A_{1}=0
\end{split}
\end{equation*}
at a point, where \;$H_g=0$, i.e. the mean curvature vanishes. Thus for \;$\gamma>\frac{1}{2}$
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
\square_{g^{+}}u=y^{n-s+2}\phi +y^{s+1}\psi\;\;\text{ in }\;\;X \\
y^{s-n}u=\underline{v} \;\;\text{ on } \;\; M
\end{cases}
\xLeftrightarrow{U=y^{s-n}u}
\begin{equation}gin{cases}
D_{g}U=y^{1-2\gamma}\phi+\psi\;\;\text{ in }\;\;X \\
U= \underline{v} \;\;\text{ on } \;\;M
\end{cases}
\end{split}\end{equation*}
with principal terms
\begin{equation}gin{equation*}
U=\underline{v}+\overlineerline{v} y^{2\gamma}+o(y^{2\gamma})
\end{equation*}
at a point with \;$H_{g}=0$ - just like in the case \;$\gamma<\frac{1}{2}$ - and there holds
$
\overlineerline{v}=\frac{1}{2\gamma}\lim_{y\to 0}y^{1-2\gamma}\partial_{y}U.
$
\noindent
We summarize the latter discussion in the following proposition.
\begin{equation}gin{pro}\label{prop_scattering}
Let \;$(X, g^{+})$\; be a \;$(n+1)$-dimensional asymptotically hyperbolic manifold with conformal infinity \;$(M, [h])$\; of dimension \;$n\geq 2$ \;being minimal in case \;$\gamma\in(\frac{1}{2},1)$\; and \;$y$ the unique geodesic defining function associated to \;$h$\; given by \eqref{eq:uniqdef}. Assuming that $$s=\frac{n}{2}+\gamma,\;\;\gamma\in (0,1), \;\,\gamma\neq \frac{1}{2}, \; \;\;s(n-s)\not \in \sigma_{pp}(\Delta_{g^{+}})$$
and \;$f\in y^{n-s+2}C^{\infty}(\overline {X})+y^{s+1}C^{\infty}(\overline {X})$, then for every \;$\underline{v}\in C^{\infty}(M)$
$$
P^{\gamma}_{f,h}(\underline{v})=-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}U^{f},
$$
where \;$U^{f}$\; is the unique solution to
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}U=y^{-s-1}f\;\;\text{in}\;\;X
\\
U=\underline{v}\;\;\text{on} \;\;M
\end{cases}
\end{equation*}
and \;$d_\gamma^*$ is as in \eqref{dsgamma}. Moreover $U^{f}$\; satisfies
$$U^{f}=A+y^{2\gamma}B,\; \;\;A, \;B\in C^{\infty}(\overline{X})$$ and \;$A$\; and \,$B$\; satisfy the asymptotics
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
A=\sum A_{i}y^{i}, \;\;\,A_{i}\in C^{\infty}(M),\;\; A_{0}=\underline{v}\; \text{ and }\;\; A_{1}=0 \\
B=\sum B_{i}y^{i}, \;\;B_{i}\in C^{\infty}(M)\; \;\;\text{ and }\;\; -d_\gamma B_{0}=-d_\gamma\overlineerline{v}=P^{\gamma}_{f,h}(\underline{v}),
\end{cases}
\end{equation*}
where \;$d_\gamma$\; is as in \eqref{eq:fracopd}, hence
$U^{f}=\underline{v}+\overlineerline{v} y^{2\gamma}+o(y^{2\gamma}).$
\end{pro}
\subsection{Conformal property of the non-homogeneous scattering operator}\label{confnonhom}
In this subsection we study the conformal property of the non-homogeneous scattering operator \;$P_{h, f}^{\gamma}$\; of the previous subsection. To this end we first
consider as background data \;$(X, g^{+})$\; with conformal infinity \;$(M, [h])$\; with \;$n\geq 2$\; and \;$y$\; the associated unique geodesic definition function such that
\begin{equation}gin{equation*}
g=y^{2}g^{+}, \;\; g=dy^{2}+h_{y}\;\; \text{ close to \;$M$\; and }\;\;h=g\lfloor_{M}
\end{equation*}
as in \eqref{eq:uniqdef}. From \eqref{Eg_general} it is easy to see, that in \;$g$-normal Fermi coordinates \;$(y, x)$
\begin{equation}gin{equation}\label{Eg_local}
E_{g}=\frac{n-2\gamma}{2}\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^{-2\gamma}\;\;
\text{ close to }\;\; M.
\end{equation}
We assume further that \;$(M, [h])$\; is minimal and \;$\square_{g^+}$\; is positive, i.e.
\begin{equation}gin{equation*}
H_{g}=0 \;\; \text{ and }\;\; \lambda_{1}(-\Delta_{g^{+}})>s(n-s).
\end{equation*}
Then \;$\partial_{y}\sqrt{g}=0$\; on \;$M^{n+1}$\; and we may assume \;
\begin{equation}gin{equation}\label{sqrt_g_sim_yC_infty}
\partial_{y}\sqrt{g}\in
yC^{\infty}(\overline {X})
\end{equation} whence
$D_{g}$\; is well defined on
$$
W^{1,2}_{y^{1-2\gamma}}=W^{1,2}_{y^{1-2\gamma}}(X,g)
=
\overlineerline{C^{\infty}(X)}^{\Vert \cdot \Vert_{W^{1,2}_{y^{1-2\gamma}}(X,g)}},
\;
\Vert u \Vert_{W^{1,2}_{y^{1-2\gamma}}(X,g)}^{2}
=
\int_{X} y^{1-2\gamma}(\vert du \vert^{2}_{g}+u^{2})dV_{g}
$$
and becomes positive under Dirichlet condition, cf. \eqref{transformation_scattering_extension}, so
\begin{equation}gin{equation*}
\partial_{y}\sqrt{g}\in yC^{\infty}(\overline{X})\;\;\text{ and }\;\;\langle\cdot,\cdot \rangle_{D_{g}} \simeq \langle \cdot,\cdot \rangle_{W^{1,2}_{y^{1-2\gamma}}}.
\end{equation*}
Let us consider now a conformal metric \;$\tilde h=\varphi^{\frac{4}{n-2\gamma}}h$\; on \;$M$. We then find a unique geodesic defining function \;$\tilde y>0$, precisely unique in a tubular neighborhood of \;$M$, such that
\begin{equation}gin{equation*}
\tilde g=d\tilde y^{2}+\tilde h_{y}\;\;\text{ close to }\;\;M, \;\;\tilde{y}^{-2}\tilde g=g^{+}=y^{-2}g
\;\;\text{ and }\;\;
\tilde h=\varphi^{\frac{4}{n-2}}h=(\frac{\tilde y}{y})^{2}h\;\;\text{ on }\;\;M.
\end{equation*}
So we may naturally extend \;$\varphi=(\frac{\tilde y}{y})^{\frac{n-2\gamma}{2}}$\; onto \;$X$\; and
by the conformal relation
\begin{equation}gin{equation*}
\tilde g=(\frac{\tilde y}{y})^{2}g=\varphi^{\frac{4}{n-2\gamma}}g,
\end{equation*}
we still have \;$\langle \cdot,\cdot \rangle_{D_{\tilde g}}\simeq \langle \cdot,\cdot \rangle_{W^{1,2}_{\tilde y^{1-2\gamma}}}$. Putting \;$\tilde y=\alpha y$, the equation
\begin{equation}gin{equation*}
\vert d y\vert^{2}_{g}
=
1
=
\vert d \tilde y\vert^{2}_{\tilde g}
=
1
+
2\frac{y}{\alpha}\langle d\alpha,dy\rangle_{g}+ (\frac{y}{\alpha})^{2}\vert d\alpha \vert_{g}^{2}
\end{equation*}
for the geodesic defining functions implies
$\partial_{y}\alpha=-\frac{1}{2}\frac{y}{\alpha}\vert d\alpha\vert^{2}_{g}$. Since $\tilde g =\alpha^{2}g$ by definition, we firstly find
$H_{g}=0 \Longrightarrow H_{\tilde g}=0$, i.e. minimality is preserved as already observed by Gonzalez-Qing\cite{gq}, and secondly \;$\tilde y=\alpha_{0}y+O(y^{3})$.
Thus on the one hand side the properties
\begin{equation}gin{equation*}
\partial_{\tilde y}\sqrt{\tilde g}\in \tilde y C^{\infty}
\text{ and }
\langle \cdot,\cdot \rangle_{D_{\tilde g}}\simeq \langle \cdot,\cdot \rangle_{W^{1,2}_{\tilde y^{1-2\gamma}}}\;\;
\end{equation*}
are preserved under a conformal change of the metric on the boundary. Moreover we obtain a conformal transformation for the extension operators \;$D_{\tilde g}$\; and $\;D_{g}$\; subjected to Dirichlet and weighted Neumann boundary conditions. Put \;$\tilde u=(\frac{y}{\tilde y})^{n-s}u$.
As for the Dirichlet case, \eqref{transformation_scattering_extension} directly shows
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}u=f \;\;\text{ in } \;\;X \\
u= v\;\;\text{ on } \;\;M
\end{cases}
\Longleftrightarrow
\begin{equation}gin{cases}
D_{\tilde g}\tilde u=(\frac{y}{\tilde y})^{s+1}f \;\;\text{ in } \;\;X \\
\tilde u=(\frac{y}{\tilde y})^{n-s}v \;\;\text{ on } \;\;M.
\end{cases}
\end{equation*}
Moreover there holds
$$
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=v
\Longleftrightarrow
\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y}\tilde u=(\frac{y}{\tilde y})^{n-s+2\gamma}v,
$$
since \;$\tilde y=\alpha_{0}y+O(y^{3})$, whence for the weighted Neumann case we obtain
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}u=f \;\;\text{ in } \;\;X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=v \;\;\text{ on } \;\;M
\end{cases}
\Longleftrightarrow
\begin{equation}gin{cases}
D_{\tilde g}\tilde u=(\frac{y}{\tilde y})^{s+1}f \;\;\text{ in } \;\;X \\
\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y}\tilde u=(\frac{y}{\tilde y})^{n-s+2\gamma}v \;\;\text{ on } \;\;M.
\end{cases}
\end{equation*}
We may rephrase this via \;$\varphi=(\frac{\tilde y}{y})^{\frac{n-2\gamma}{2}}=(\frac{\tilde y}{y})^{n-s}$\; as
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}(\varphi u)=\varphi^{\frac{s+1}{n-s}} f\;\; \text{ in } \;\;X \\
\varphi u=\varphi v \;\,\;\text{ on } \;\,\;M
\end{cases}
\Longleftrightarrow
\begin{equation}gin{cases}
D_{\tilde g} u=f \;\,\;\text{in} \;\;X \\
u=v\;\;\text{on} \;\;M
\end{cases}
\end{equation*}
and
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}(\varphi u)=\varphi^{\frac{s+1}{n-s}}f \;\;\text{in}\;\; X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}(\varphi u)=\varphi^{\frac{n+2\gamma}{n-2\gamma}}v \;\;\text{ on } \;\;M^{n}
\end{cases}
\Longleftrightarrow
\begin{equation}gin{cases}
D_{\tilde g}\tilde u=f \;\;\text{ in } \;\;X \\
\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y} u=v \;\;\text{ on } \;\;M.
\end{cases}
\end{equation*}
Noticing \;$\frac{s+1}{n-s}=\frac{n+2+2\gamma}{n-2\gamma}$\; we thus have
shown
\begin{equation}gin{equation*}
\xymatrix@R=6ex@C=-11ex
{
*+[l]
{
P^{\gamma}_{f,\tilde h}(\underline v)=\overlineerline{v}
}
\ar@{<=>}[d] \ar@{<=>}[r]
&
*+[r]
{
{
\begin{equation}gin{cases}
D_{\tilde g}u=f \;\;\text{ in } \;\;X \\
u=\underline v \;\;\text{ on } \;\;M \\
-d_\gamma^*\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y} u=\overlineerline v \;\;\text{ on } \;\;M
\end{cases}
}
}
\ar@{<=>}[d]
\\
*+[l]
{
P^{\gamma}_{\varphi^{\frac{s+1}{n-s}}f,h }(\varphi \underline v)=\varphi^{\frac{n+2\gamma}{n-2\gamma}}\overlineerline v
}
\ar@{<=>}[r]
&
*+[r]
{
{
\begin{equation}gin{cases}
D_{ g}(\varphi u)=\varphi^{\frac{n+2+2\gamma}{n-2\gamma}}f \text{ in } X \\
\varphi u=\varphi \underline v \text{ on } M \\
-d_\gamma^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{ y} (\varphi
u)=\varphi^{\frac{n+2\gamma}{n-2\gamma}}\overlineerline v \text{ on } M
\end{cases}
}
}
}
\end{equation*}
Therefore the non-homogeneous fractional operator verifies the conformal property
$$
P^{\gamma}_{f,\tilde h}(\underline v)=\varphi^{-\frac{n+2\gamma}{n-2\gamma}}P^{\gamma}_{\varphi^{\frac{s+1}{n-s}}f,h }(\varphi \underline v)
\;\text{ for} \;
\tilde h=\varphi^{\frac{4}{n-2\gamma}}h
$$
or equivalently
$$P^{\gamma}_{\tilde f,\tilde h}(\underline v)
=
\varphi^{-\frac{n+2\gamma}{n-2\gamma}}P^{\gamma}_{f,h }(\varphi \underline v)
\; \text{ for } \;
\tilde h=\varphi^{\frac{4}{n-2\gamma}}h\;\text{ and }\;\tilde f=\varphi^{\frac{-s-1}{n-s}}f,
$$
hence extending the conformal property of the homogeneous fractional operator to the non-homogeneous setting. We remark that $$P^{\gamma}_{h}=P^{\gamma}_ {0,h}.$$
\section{Fundamental solutions in the asymptotically hyperbolic case}\label{fundso}
In this section, keeping the notations of the previous one, for an asymptotically hyperbolic manifold\;$(X, g^+)$ with conformal infinity
\;$(M, [h])$, we study the existence and asymptotic behavior of the Poisson kernel \;$K_g:=K_{g}^{\gamma}$ \; of \;$D_g$, the Green's functions \;$\Gamma_g:=\Gamma_{g}^{\gamma}$\; of \;$D_g$\; under weighted normal boundary condition and \;$G_{h}:=G^{\gamma}_{h}$\; of the fractional conformal Laplacian \;$P^{\gamma}_{h}$, i.e.
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
D_{g}K_g(\cdot, \xi)=0\;\; \text{ in }\;\; X \;\;\text{ and for all }\;\; \xi\in M\\
\lim_{y\rightarrow 0}K_g(y, x, \xi)=\delta _{x}(\xi)\;\;\text{ and for all }\;\; x,\;\xi\in M
\end{cases}
\end{split}\end{equation*}
and
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
D_{g}\Gamma_g(\cdot, \xi)=0\; \text{ in } X \;\;\text{ and for all }\;\; \xi\in M\\
-d_\gamma^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_y\Gamma_g(y, x, \xi)=\delta _{x}(\xi)\;\;\text{ and for all }\;\; x,\xi\in M,
\end{cases}
\end{split}\end{equation*}
where \;$d_\gamma^*$ is given by \eqref{dsgamma}, and
$P^\gamma_h G^{\gamma}_{h}(x, \xi)=\delta _{x}(\xi),\;\;x\in M.
$
So by definition
\begin{equation}gin{equation*}\begin{equation}gin{split}
K_{g}:(\overlineerline X \times M)\setminus Diag(M)\longrightarrow\mathbb{R}_{+}
\end{split}\end{equation*}
is the Green's function to the extension problem
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
D_{g}U=0\; \text{ in } \;\;X \\
U=\underline{v}\;\; \text{on} \;\;M,
\end{cases}
\end{split}\end{equation*}
while
\begin{equation}gin{equation*}\begin{equation}gin{split}
\Gamma_{g}:(\overlineerline X\times M)\setminus Diag(M)\longrightarrow\mathbb{R}
\end{split}\end{equation*}
is the Green's function to the dual problem
\begin{equation}gin{equation*}\begin{equation}gin{split}
\begin{equation}gin{cases}
D_{g}U=0\;\;\text{ in }\;\;X\\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}U=\overlineerline{v} \;\;\text{ on } \;\;M
\end{cases}
\end{split}\end{equation*}
and
\begin{equation}gin{equation*}\begin{equation}gin{split}
G_{h}:(M\times M)\setminus Diag(M)\longrightarrow\mathbb{R}.
\end{split}\end{equation*}
is the Green's function of the nonlocal problem
$
P^{\gamma}_{h} \underline{v}=\overlineerline{v} \;\;\text{ on } \;\;M.
$
They are linked via
\begin{equation}gin{equation}\label{regreen}
\Gamma_{g}=K_{g}*G_{h},
\end{equation}
where \;$*$\; denotes the standard convolution operation.
\subsection{Study of the Poisson kernel for \;$D_g$\; }
In this subsection we study the Poisson kernel \;$K_g$\; focusing on the existence issue and its asymptotics. We follow the method of Lee-Parker\cite{lp} of killing deficits successively. However, due to the rigidity property involved in the problem, see the normal form \eqref{eq:uniqdef}, we have to work close to the boundary in Fermi coordinates rather than normal ones. To compensate this we are forced to pass from the space of polynomials used in \cite{lp} to the space of homogeneous functions. We start with recalling some related facts in the case of the standard Euclidean space \;$\mathbb{R}^{n+1}_+$. According to \cite{cafsyl} on \;$\mathbb{R}^{n+1}_+$
\begin{equation}gin{equation}\label{Poisson_kernel_flat}
\begin{equation}gin{split}
K(y,x,\xi)=K^\gamma(y, x, \xi)= p_{n, \gamma}\frac{y^{2\gamma}}{(y^{2}+\vert x-\xi\vert^{2})^{\frac{n+2\gamma}{2}}},
\end{split}
\end{equation}
where \;$p_{n, \gamma}$\; is as in \eqref{pngamma}, is the Poisson kernel of the operator
\begin{equation}gin{equation*}\label{Extension_Operator_flat}
D=-div(y^{1-2\gamma}\nabla (\,\cdot\,)),
\end{equation*}
namely the Green's function of the extension problem
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
Du=0 \;\;\text{ in }\;\; \mathbb{R}^{n+1}_+ \\
u=f \;\;\text{ on }\;\;\mathbb{R}^{n},
\end{cases}
\end{equation*}
i.e.
\begin{equation}gin{equation}\label{poisson_defining_equation}
\begin{equation}gin{cases}
DK(y,x,\xi)=0 \;\;\text{ in }\;\;\mathbb{R}^{n+1}_+\;\; \\
K(y,x,\xi)\rightarrow\delta elta_{x}(\xi)\; \;\;\text{ for }\;\; y\rightarrow 0.
\end{cases}
\end{equation}
We will construct the Poisson kernel for \;$D_{g}$, cf. \eqref{eqdg}, namely the Green's function of the analogous extension problem
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}u=0 \;\;\text{in} \;\;X \\
u=f \;\;\text{ on } \;\;M,
\end{cases}
\end{equation*}
i.e. \;$K_g$\; solves for \;$z\in X$\; and \;$\xi \in M$
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}K_{g}(z,\xi)=0\;\;\text{ in } \;\;X\\
K(z,\xi)\rightarrow\delta elta_{x}(\xi)\;\;\text{ for } \;\;y\rightarrow 0,
\end{cases}
\end{equation*}
where \;$z=(y,x)\in X$\; for \;$z$\; close to \;$M$. To that end we identify
\begin{equation}gin{equation*}
\xi \in M\cap U\subset U\cap X \text{ with }0\in B^{\mathbb{R}^{n+1}}_{\varepsilonilon}(0)\cap
\mathbb{R}^{n}\subset B_{\varepsilonilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}
\end{equation*}
for some open neighborhood \;$U$\; of \;$\xi$\; in \;$X$\; and small \;$\varepsilonilon>0$, and write \;$K(z)=K(z,0)$. We then have
\begin{equation}gin{equation}\label{poisson_local_equation_for_the_poisson_kernel}
D_{g}K=-\frac{\partial_{p}}{\sqrt{g}}(\sqrt{g}g^{p,q}y^{1-2\gamma}\partial_{q}
K)+E_{g}K=f\in
y H_{-n-2\gamma-1}C^{\infty}
\end{equation}
on $\;B_{\varepsilonilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}$\; due \eqref{sqrt_g_sim_yC_infty}, which relies on minimality \;$H_{g}=0$,
where by definition
\begin{equation}gin{equation}\label{definition_H_l}
H_{l}=\{\varphi\in C^{\infty}(\overlineerline{\mathbb{R}^{n+1}_{+}}\setminus \{0\})\mid
\varphi \text{ is homogeneous of degree } l\}.
\end{equation}
The next lemma allows us to solve homogeneous deficits
homogeneously.
\begin{equation}gin{lem}
\label{poisson_homogeneous_solvability_dirichlet}$_{}$\\
For $\frac{1}{2}\neq \gamma \in (0, 1)$ and \;$f_{l}\in yH_{l-1}, \;l\in \mathbb{N}-n-2\gamma$\; there exists \;$K_{1+2\gamma+l}\in
y^{2\gamma}H_{l+1}$\; such, that
$$DK_{1+2\gamma+l}=f_{l}.$$
\end{lem}
\begin{equation}gin{pf}
First of all the Stone-Weierstra{\ss} Theorem implies
\begin{equation}gin{equation*}
\langle Q^{k}_{l}(y,x)=y^{2\gamma+2k}P_{l}(x)\mid k,l\in \mathbb{N}\text{ and }P_{l}\in
\Pi_{l}\rangle \underset{\text{dense}}{\subset} y^{2\gamma}C^{0}(\overlineerline
B_{1}(0)\cap\mathbb{R}^{n+1}_{+})
\end{equation*}
and an easy induction argument shows, that we have a unique representation
\begin{equation}gin{equation*}
Q^{k}_{l}=\sum \vert z \vert^{2i}A_{2k+l-2i}
\end{equation*}
with \;$D$-harmonics of the form \;$A_{m}(y,x)=\sum y^{2\gamma+2l}P_{m-2l}(x),\;
DA_{m}=0$. Since
\begin{equation}gin{equation*}
y^{2\gamma}C^{0}(S^{n}_{+})\underset{\text{dense}}{\subset}
L^{2}_{y^{1-2\gamma}}(S^{n}_{+}),
\end{equation*}
we thus obtain a \;$D$-harmonic basis \;$E=\{e^{i}_{k}\}$\; for
\;$L^{2}_{y^{1-2\gamma}}(S^{n}_{+})$\; with
\begin{equation}gin{equation*}
De^{i}_{k}=0, \;\;k=deg(e^{i}_{k}), \;\; k\in \mathbb{N} +2\gamma\text{ and } \;\;i\in
\{1,\delta ots,d_{k}\},
\end{equation*}
where \;$d_{k}$\; denotes the dimension of the space of \;$D$-harmonics of degree
$k$. We may assume, that \;$e^{i}_{k},e^{j}_{k}$ for $i\neq j$\; are orthogonal
with respect to the scalar product on \;$L^{2}_{y^{1-2\gamma}}(S^{n}_{+})$.
Moreover on \;$S^{n}_{+}$\; we have
\begin{equation}gin{equation*}
\begin{equation}gin{split}
0
= &
-De^{i}_{k}
=
\partial_{y}(y^{1-2\gamma}\partial_{y}e^{i}_{k})+y^{1-2\gamma}\Delta_{x}e^{i}_{k
}
=
\nabla y^{1-2\gamma}\nabla e^{i}_{k}+y^{1-2\gamma}\Delta e^{i}_{k} \\
= &
\nabla y^{1-2\gamma} \nabla
e^{i}_{k}+y^{1-2\gamma}\frac{\Delta_{S^{n}}}{r^{2}}e^{i}_{k}+y^{1-2\gamma}[
\partial_{r}^{2}+\frac{n\partial_{r}}{r}]e^{i}_{k} \\
= &
\nabla_{S^{n}_{+}}^{\perp} y^{1-2\gamma}\nabla_{S^{n}_{+}}^{\perp} e^{i}_{k}
+
div_{S^{n}_{+}}(y^{1-2\gamma}\nabla_{S^{n}_{+}}e^{i}_{k})
+
k(k+n-1)y^{1-2\gamma}e^{i}_{k},
\end{split}
\end{equation*}
whence due to
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nabla_{S^{n}_{+}}^{\perp} y^{1-2\gamma}\nabla_{S^{n}_{+}}^{\perp} e^{i}_{k}
= &
\langle \nabla y^{1-2\gamma}, \nu_{S^{n}_{+}}\rangle \langle
\nu_{S^{n}_{+}}, \nabla e^{i}_{k}\rangle
=
(1-2\gamma)y^{-2\gamma}\langle e_{n+1}, \nu_{S^{n}_{+}}\rangle
r\partial_{r}e^{i}_{k}
=
(1-2\gamma)ky^{1-2\gamma}e^{i}_{k}
\end{split}
\end{equation*}
there holds for\;
$D_{S^{n}_{+}}=-div_{S^{n}_{+}}(y^{1-2\gamma}\nabla_{S^{n}_{+}}\,\cdot\,)$
\begin{equation}gin{equation*}
D_{S^{n}_{+}}e^{i}_{k}=k(k+n-2\gamma)y^{1-2\gamma}e^{i}_{k}.
\end{equation*}
Therefore \;$E=\{e^{i}_{k}\}$\; is an orthogonal basis of
\;$y^{2\gamma-1}D_{S^{n}_{+}}$-eigenfunctions with eigenvalues
$$\lambda_{k}=k(k+n-2\gamma).$$
By the same argument solving
\begin{equation}gin{equation}\label{poisson_homogeneous_to_solve}
\begin{equation}gin{cases}
Du=f\in L^{2}_{y^{2\gamma-1}}(\mathbb{R}^{n+1}_{+})\;\; \text{ in } \;\;\mathbb{R}^{n+1}_{+}\\
u=0 \;\;\text{ on } \;\; \mathbb{R}^{n}
\end{cases}
\end{equation}
with homogeneous \;$f, \;u$\; of degree \;$\lambda, \;\lambda+1+2\gamma$\; is equivalent to
solving
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{S^{n}_{+}}u=f+(\lambda+1+2\gamma)(\lambda+n+1)y^{1-2\gamma}u\;\;\text{ in
}\;\;S^{n}_{+} \\
u=0 \;\;\text{ on }\;\; \partial S^{n}_{+}=S^{n-1}
\end{cases}
\end{equation*}
and thus, writing
$
u=\sum a_{i,k}e^{i}_{k}, \,y^{2\gamma-1}f=\sum
b_{j,l}e^{j}_{l}
$
, also equivalent to solving
\begin{equation}gin{equation*}
\sum a_{i,k}(k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1))e^{i}_{k}=\sum
b_{j,l}e^{j}_{l}
\end{equation*}
and the latter system is always solvable in case
\begin{equation}gin{equation}\label{solvability_homogeneous_poisson}
k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1)\neq 0 \;\;\text{ for all } \;\;
k, \;n, \;\lambda\in \mathbb{N}.
\end{equation}
This observation allows us to prove the lemma, by whose assumptions
\begin{equation}gin{equation*}
deg(f_{l})=\lambda=m-n-2\gamma, \;\; m\in \mathbb{N}.
\end{equation*}
And we know
\begin{equation}gin{equation*}
deg(e^{i}_{k})=k=m^{\prime}+2\gamma, \;\; m^{\prime}\in \mathbb{N}.
\end{equation*}
Plugging these values into \eqref{solvability_homogeneous_poisson}, solvability
of \eqref{poisson_homogeneous_to_solve} is a consequence of
\begin{equation}gin{equation*}
(m^{\prime}+2\gamma)(m^{\prime}+n)-(m-n+1)(m+1-2\gamma)\neq 0\;\;\text{ for all } \;\;
m^{\prime}, \; \;n,m\in \mathbb{N}
\end{equation*}
and this holds true for \;$\frac{1}{2}\neq\gamma \in (0,1)$. Thus we have
proven solvability of
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
DK_{1+2\gamma+l}=f_{l} \;\;\text{ in }\;\;\mathbb{R}^{n+1}_{+} \\
K_{1+2\gamma+l}=0 \;\;\text{ on } \;\;\mathbb{R}^{n}\setminus \{0\}
\end{cases}
\end{equation*}
with \;$K_{1+2\gamma+l}$\; being homogeneous of degree \;$1+2\gamma+l$. We are left
with showing \;$K_{1+2\gamma+l}\in y^{2\gamma}H_{l+1}$. But this follows easily from
Proposition \ref{poisson_regularity} below.
\end{pf}
\noindent
Now recalling \eqref{poisson_local_equation_for_the_poisson_kernel} we may use
Lemma \ref{poisson_homogeneous_solvability_dirichlet} to solve
\eqref{poisson_defining_equation} successively, since
\begin{equation}gin{equation*}
D_{g}K_{1+2\gamma+l}=f_{l}+(D_{g}-D)K_{1+2\gamma+l}\in f_{l}+ yH_{l}C^{\infty}
\end{equation*}
due to \eqref{sqrt_g_sim_yC_infty} and \;$K_{1+2\gamma+l}\in y^{2\gamma}H_{l+1}$. With a suitable cut-off function
\begin{equation}gin{equation}\label{etaxi}
\eta_{\xi}:\overlineerline X\longrightarrow \mathbb{R}^{+}, \;\;supp(\eta_{\xi})=B_{\varepsilonilon}^{+}(\xi)=B_{\varepsilonilon}^{g, +}(\xi)\;\;\text{
for } \;\;M\ni \xi \sim 0 \in \mathbb{R}^{n} \;\;\text{and}\;\;\varepsilonilon>0\;\;\text{ small}
\end{equation}
and for the meaning of \;$B_{\varepsilonilon}^{g, +}(\xi)$\; see Section \ref{notation_and_prelimiaries},
we then find
\begin{equation}gin{equation*}
K_{g}=\eta_{\xi}(K+\sum_{l=-n-2\gamma}^{m+2-2\gamma} K_{1+2\gamma+l})+\kappa_{m}
\end{equation*}
for \;$m\in \mathbb{N}$\; and a weak solution
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}\kappa_{m}=-D_{g}\left(\eta_{\xi}(K+\sum_{l=-n-2\gamma}^{m+2-2\gamma}
K_{1+2\gamma+l})\right)=h_{m} \;\;\text{ in }\;\,\;X \\
\kappa_{m}=0\;\;\text{ on } \;\;M
\end{cases}
\end{equation*}
with \;$h_{m} \in yC^{m,\alpha}$.
The following weak regularity statement will be sufficient for our purpose.
\begin{equation}gin{pro}
\label{poisson_regularity} $_{}$\\
Let \;$h\in yC^{2k+3,\alpha}(X)$\; and \;$u\in W^{1,2}_{y^{1-2\gamma}}(X)$\; be a
weak solution of
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}u=h\;\;\text{ in }\;\;X \\
u=0\;\; \text{ on } \;\;M.
\end{cases}
\end{equation*}
Then \;$u$\; is of class \;$y^{2\gamma}C^{2k,\begin{equation}ta}(X)$, provided \;$H_{g}=0$.
\end{pro}
\noindent
Putting these facts together before giving the proof of Proposition \ref{poisson_regularity}, we have the existence of \;$K_g$\; and can describe its asymptotic.
\begin{equation}gin{cor}
\label{Poisson_kernel_asymptotics}$_{}$\\
Let \;$\frac{1}{2}\neq \gamma \in (0,1)$. Then \;$K_g$\; exists and we may expand in \;$g$-normal
Fermi-coordinates around $\xi\in M$
\begin{equation}gin{equation*}
K_{g}(z,\xi)
\;\in
\eta_{\xi}(z)
\left( p_{n, \gamma}
\frac{y^{2\gamma}}{\vert z
\vert^{n+2\gamma}}+\sum^{2m+5-2\gamma}_{l=-n-2\gamma}y^{2\gamma}H_{1+l}(z)
\right)
+
y^{2\gamma}C^{2m,\alpha}(X)
\end{equation*}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n+1}_+\setminus \{0\})$\; being homogeneous of order
$l$\; and \;$p_{n, \gamma}$\; is as in \eqref{pngamma}, provided \;$H_{g}=0$.
\end{cor}
\noindent
\begin{equation}gin{pfn}{ of Proposition \ref{poisson_regularity}}
\noindent
We use the Moser iteration argument. First let
$
p,q=1,\ldots,n+1\text{ and } i,j=1,\ldots,n
$
such, that $g_{n+1,i}=g_{y,i}=0$. The statement clearly holds by standard local
regularity away from the boundary, since $D_{g}$ is strongly elliptic there.
Now fixing a point $\xi\in M$ and a cut-off function
\begin{equation}gin{equation*}
\eta\in C^{\infty}_{0}(B^{+}_{r_{2}}(0),\mathbb{R}_{+}),\,\; \eta\equiv 1\text{ on }
B^{+}_{r_{1}}(0)\;\;\text {for }\; 0<r_{1}<r_{2}\ll 1,\;\text{ where } \;\;M\ni \xi \sim 0\in \mathbb{R}^{n},
\end{equation*}
we pass to $g$-normal Fermi-coordinates around $\xi$ and estimate for some \;$\lambda\geq 2$\; and \;$\alpha\in \mathbb{N}^{n}$
\begin{equation}gin{equation}\label{poisson_kernel_moser_iteration_1}
\begin{equation}gin{split}
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{z}(\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\eta)\vert^{2}
\leq &
2\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{z}\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
+
2\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \partial^{\alpha}_{x}u\vert^{\lambda}\vert
\nabla_{z}\eta\vert^{2}
\end{split}
\end{equation}
and
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\underset{\mathbb{R}^{n+1}_{+}}{\int}& y^{1-2\gamma}\vert \nabla_{z}\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
=
\frac{\lambda^{2}}{4}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma} \nabla_{z} \partial^{\alpha}_{x}u
\nabla_{z} \partial^{\alpha}_{x}u \vert
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}\\
= &
\frac{\lambda^{2}}{4(\lambda-1)}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma} \nabla_{z}
\partial^{\alpha}_{x}u
\nabla_{z} (\partial^{\alpha}_{x}u \vert
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2})
-
\frac{\lambda^{2}}{2(\lambda-1)}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma} \nabla_{z}
\partial^{\alpha}_{x}u
\partial^{\alpha}_{x}u \vert
\partial^{\alpha}_{x}u\vert^{\lambda-2}\nabla_{z}\eta \eta \\
\leq &
\frac{\lambda^{2}}{4(\lambda-1)}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma} \nabla_{z}
\partial^{\alpha}_{x}u
\nabla_{z} (\partial^{\alpha}_{x}u \vert
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}) \\
& +
\frac{\lambda^{2}}{8}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma} \vert \nabla_{z}
\partial^{\alpha}_{x}u\vert^{2}
\vert \partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}
+
\frac{\lambda^{2}}{2(\lambda-1)^{2}}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}
\vert \partial^{\alpha}_{x}u\vert^{\lambda}\vert \nabla_{z}\eta \vert^{2}.
\end{split}
\end{equation*}
Absorbing the second summand above this implies
\begin{equation}gin{equation}\label{poisson_kernel_moser_iteration_3}
\begin{equation}gin{split}
\underset{\mathbb{R}^{n+1}_{+}}{\int}
\hspace{-2pt}
y^{1-2\gamma}&\vert \nabla_{z}(\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}})\vert^{2}\eta^{2}
\leq
\frac{\lambda^{2}}{2(\lambda-1)}\underset{\mathbb{R}^{n+1}_{+}}{\int}
\hspace{-2pt}
D(\partial^{\alpha}_{x}u)
\partial^{\alpha}_{x}u \vert \partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}
+
\frac{\lambda^{2}}{(\lambda-1)^{2}}\underset{\mathbb{R}^{n+1}_{+}}{\int}
\hspace{-2pt}
y^{1-2\gamma}\vert
\partial^{\alpha}_{x}u\vert^{\lambda}\vert \nabla_{z}\eta\vert^{2}
\end{split}
\end{equation}
Due to
$
D(\partial^{\alpha}_{x}u)=\partial^{\alpha}_{x}Du,
$
and the structure of the metric
\begin{equation}gin{equation}\label{poisson_kernel_moser_iteration_4}
\begin{equation}gin{split}
\underset{\mathbb{R}^{n+1}_{+}}{\int} &\partial_{x}^{\alpha}(D u)\partial^{\alpha}_{x}u\vert
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}
=
\underset{\mathbb{R}^{n+1}_{+}}{\int} \partial_{x}^{\alpha}(D_{g}u)\partial^{\alpha}_{x}u\vert
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}
-
\underset{\mathbb{R}^{n+1}_{+}}{\int} \partial_{x}^{\alpha}((D_{g}-D)u)\partial^{\alpha}_{x}u\vert
\partial^{\alpha}_{x}u\vert^{\lambda-2} \eta^{2} \\
= &
\underset{\mathbb{R}^{n+1}_{+}}{\int}
\partial_{x}^{\alpha}
[
h
+
\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{1-2\gamma}g^{p,q}\partial_{q}u
+
y^{1-2\gamma}\partial_{i}((g^{i,j}-\delta elta^{i,j})\partial_{j}u) \\
& \quad \quad\quad\quad\quad\quad\quad\quad\quad -
\frac{n-2\gamma}{2}\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^{-2\gamma}u
]
\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}
=
I_{1}+\ldots+I_{4}.
\end{split}
\end{equation}
We may assume \;$\nabla^{k}_{z}\eta\leq \frac{C}{\varepsilonilon^{k}}$\; for \;$k=0,1,2$,
where $\varepsilonilon=r_{2}-r_{1}$. Then
\begin{equation}gin{enumerate}[label=(\roman*)]
\item \quad
$
\vert I_{1} \vert
\leq
C\int_{\mathbb{R}^{n+1}_{+}}\vert \nabla ^{\vert \alpha \vert}_{x}h\vert
\vert \nabla^{\vert \alpha \vert}_{x}u\vert^{\lambda-1} \eta^{2}
$
\item using integrations by parts and \eqref{sqrt_g_sim_yC_infty}
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\vert I_{2}\vert
\leq &
\vert
\underset{\mathbb{R}^{n+1}_{+}}{\int}
y^{1-2\gamma}\partial_{q}\partial^{\alpha}_{x}(\frac{\partial_{p}\sqrt{g}}{\sqrt
{g}}g^{p,q}u)\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert
^{\lambda-2} \eta^{2}\vert +
\vert \underset{\mathbb{R}^{n+1}_{+}}{\int}
y^{1-2\gamma}\partial^{\alpha}_{x}(\partial_{q}(\frac{\partial_{p}\sqrt{g}}{
\sqrt{g}}g^{p,q})u)\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert
^{\lambda-2} \eta^{2}\vert \\
\leq &
\vert \underset{\mathbb{R}^{n+1}_{+}}{\int}
y^{1-2\gamma}\partial_{y}\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{\sqrt
{g}}u)\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert ^{\lambda-2}
\eta^{2} \vert
+
\frac{C_{\vert \alpha \vert}}{\lambda}\sum_{m\leq \vert \alpha \vert}\underset{\mathbb{R}^{n+1}_{+}}{\int}
y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert \vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda-2}{2}}
\vert \nabla_{x} \vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert
\eta^{2} \\
& +
C_{\vert \alpha \vert}\sum_{m\leq \vert \alpha \vert} \underset{\mathbb{R}^{n+1}_{+}}{\int} y^{1-2\gamma}\vert
\nabla_{x}^{m}u\vert \vert \partial^{\alpha}_{x}u\vert^{\lambda-1} [\vert
\nabla_{x}\eta\vert \eta+\eta^{2}] \\
\leq &
\frac{C_{\vert \alpha \vert}}{\lambda}\sum_{m\leq \vert \alpha \vert}
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert^{\frac{\lambda}{2}}
\vert \nabla_{z} \vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert
\eta^{2}
+
C_{\vert \alpha \vert}\sum_{m\leq \vert \alpha \vert} \underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert
\nabla_{x}^{m}u \vert^{\lambda} [\vert \nabla_{z}\eta\vert \eta+\eta^{2}] \\
\end{split}
\end{equation*}
\item using integration by parts and recalling $i,j=1,\ldots,n$
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\vert I_{3} \vert
\leq &
\frac{C}{\lambda} \underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert
\partial^{\alpha}_{x}((g^{i,j}-\delta elta^{i,j})\partial_{j}u)\vert \vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda-2}{2}}\vert \partial_{i}\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert \eta^{2} \\
& +
C\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert
\partial^{\alpha}_{x}((g^{i,j}-\delta elta^{i,j})\partial_{j}u)\vert \vert
\partial^{\alpha}_{x}u\vert^{\lambda-1}\vert \partial_{i}\eta \vert \eta \\
\leq &
\frac{C}{\lambda^{2}}\sup_{B^{+}_{r_{2}}}\vert g^{i,j}-\delta elta^{i,j}\vert \underset{\mathbb{R}^{n+1}_{+}}{\int}
y^{1-2\gamma}
\vert \partial_{i}\vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert
\vert \partial_{j}\vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert
\eta^{2} \\
& +
\frac{C_{\vert \alpha \vert}}{\lambda}\sum_{m\leq \vert \alpha }
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert \vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda-2}{2}}\vert \nabla_{x}\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert \eta^{2}\\
& +
\frac{C}{\lambda}\sup_{B^{+}_{r_{2}}}\vert g^{i,j}-\delta elta^{i,j}\vert \underset{\mathbb{R}^{n+1}_{+}}{\int}
y^{1-2\gamma} \vert \partial_{j}\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert \vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert\partial_{i} \eta \vert
\eta
+
C_{\vert \alpha \vert}\sum_{m\leq \vert \alpha }
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert^{\lambda} \vert
\nabla_{x}\eta\vert \eta
\end{split}
\end{equation*}
\item \quad
$
\vert I_{4} \vert
\leq
C_{\vert \alpha \vert}\sum_{m\leq \vert\alpha\vert}\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma} \vert
\nabla^{m}_{x}u\vert^{\lambda}
\eta^{2}
$
using \eqref{sqrt_g_sim_yC_infty}.
\end{enumerate}
Applying H\"older's and Young's inequality to (i)-(vi) we obtain
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\sum^{4}_{i=1}\vert I_{i}\vert
\leq &
\frac{C\sup_{B^{+}_{r_{2}}}\vert g-\delta elta\vert}{\lambda^{2}}
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{z}\vert \partial^{\alpha}_{x}u
\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
+
\frac{C_{\vert \alpha\vert}}{\varepsilonilon^{2}}\sum_{k\leq \vert \alpha \vert}
\Vert
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda} \\
& +
C\Vert y^{2\gamma-1}\nabla^{\vert \alpha
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1}
.
\end{split}
\end{equation*}
We may assume \;$C\sup_{B^{+}_{r_{2}}}\vert g-\delta elta\vert<\frac{1}{2}$, whence in
view of \eqref{poisson_kernel_moser_iteration_3} and
\eqref{poisson_kernel_moser_iteration_4}
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{z}\vert
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
\leq &
\frac{C_{\vert \alpha \vert}\lambda}{\varepsilonilon^{2}}
\sum_{k\leq \vert \alpha \vert}
\Vert
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda}
\\
& +
C\lambda\Vert y^{2\gamma-1}\nabla^{\vert \alpha
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1},
\end{split}
\end{equation*}
so \eqref{poisson_kernel_moser_iteration_1} implies
\begin{equation}gin{equation}\label{poisson_kernel_moser_iteration_2}
\begin{equation}gin{split}
\underset{\mathbb{R}^{n+1}_{+}}{\int}y^{1-2\gamma}\vert \nabla_{z}(\vert \partial^{\alpha}_{x}
u\vert^{\frac{\lambda}{2}}\eta)\vert^{2}
\leq &
\frac{C_{\vert \alpha \vert}\lambda}{\varepsilonilon^{2}}
\sum_{k\leq \vert \alpha \vert}
\Vert
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda}
\\
& +
C\lambda\Vert y^{2\gamma-1}\nabla^{\vert \alpha
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1}.
\end{split}
\end{equation}
The weighted Sobolev inequality of Fabes-Kenig-Seraponi \cite{fabkenser} Theorem 1.2 with \;$\kappa=\frac{n+1}{n}$\;
then shows
\begin{equation}gin{equation*}
\begin{equation}gin{split}
r_{2}^{-\frac{n+2\gamma}{n+1}}\Vert \partial^{ \alpha }_{x}u \Vert_{L^{\kappa
\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{1}})}^{\lambda}
\leq &
\frac{C_{\vert \alpha \vert}\lambda}{\varepsilonilon^{2}}
\sum_{k\leq \vert \alpha \vert}
\Vert
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda}
\\
& +
C\lambda\Vert y^{2\gamma-1}\nabla^{\vert \alpha
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1}
.
\end{split}
\end{equation*}
By rescaling we may assume for some \;$0<\varepsilonilon_{0}\ll 1$, that
\begin{equation}gin{equation}\label{poisson_rescaling_deficit}
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}+\sum^{\vert \alpha \vert}_{k=0}\Vert
y^{2\gamma-1}\nabla^{k}_{x}h\Vert_{L^{\infty}_{y^{1-2\gamma}}(B^{+}_{(2+\vert
\alpha \vert)\varepsilonilon_{0}})}= 1,
\end{equation}
and putting \;$\lambda_{i}=2(\frac{n+1}{n})^{i}$\; and
\;$\rho_{i}=\varepsilonilon_{0}(1+\frac{1}{2^{i}})$\; we obtain
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\Vert \nabla^{ \vert \alpha \vert }_{x}u
\Vert_{L^{\lambda_{i+1}}_{y^{1-2\gamma}}(B^{+}_{\rho_{i+1}})}
\leq
\sqrt[\lambda_{i}]{C_{\vert\alpha\vert,\varepsilonilon_{0}}\lambda_{i}2^{2i}}
\cdot
\sup_{m\leq \vert \alpha \vert}
[
& \Vert \nabla^{m}_{x}u\Vert_{L^{\lambda_{i}}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
+
\Vert
\nabla^{m}_{x}u\Vert_{L^{\lambda_{i}}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\frac{1}{
2}}
],
\end{split}
\end{equation*}
where we have used \;$\frac{1}{2}\leq \frac{\lambda_{i}-1}{\lambda_{i}}<1$.
Iterating this inequality then shows
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\Vert \nabla^{ \vert \alpha \vert}_{x}u
\Vert_{L^{\infty}_{y^{1-2\gamma}}(B^{+}_{\varepsilonilon_{0}})}
\leq
C_{\alpha,\varepsilonilon_{0}}(1+\sup_{m\leq \vert \alpha \vert}
\Vert \nabla^{m}_{x}u \Vert_{L^{2}_{y^{1-2\gamma}}(B^{+}_{2\varepsilonilon_{0}})})
\leq
C_{\alpha,\varepsilonilon_{0}},
\end{split}
\end{equation*}
where the last inequality follows from iterating
\eqref{poisson_kernel_moser_iteration_2} with \;$\lambda=2$
and \eqref{poisson_rescaling_deficit}. Rescaling back we conclude
\begin{equation}gin{equation}\label{poisson_kernel_moser_iteration_5}
\begin{equation}gin{split}
\sum^{m}_{k=0}\Vert \nabla^{ k }_{x}u
\Vert_{L^{\infty}_{y^{1-2\gamma}}(B^{+}_{\varepsilonilon_{0}})}
\leq
C_{m,\varepsilonilon_{0}}
[
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}
+
\sum^{m}_{k=0}\Vert y^{2\gamma-1}\nabla^{ k }_{x}h
\Vert_{L^{\infty}_{y^{1-2\gamma}}}
].
\end{split}
\end{equation}
Note, that
$
D(\partial^{\alpha}_{x}u)
=
\partial_{x}^{\alpha}h
+
\partial^{\alpha}_{x}((D-D_{g})u),
$
where
\begin{equation}gin{equation}\label{poisson_x_deriviative_versus_difference_of_operators}
\begin{equation}gin{split}
\partial^{\alpha}_{x}((D& -D_{g})u)
=
\partial^{\alpha}_{x}
[
\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{1-2\gamma}g^{p,q}\partial_{q}u
+
\partial_{i}(y^{1-2\gamma}(g^{i,j}-\delta elta^{i,j})\partial_{j}u)
-
\frac{n-2\gamma}{2}\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^{-2\gamma}u
]
\\
= &
\partial_{q}\partial_{x}^{\alpha}(\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{
1-2\gamma}g^{p,q}u)
-
\partial^{\alpha}_{x}(\partial_{q}(\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{
1-2\gamma}g^{p,q})u) \\
& +
\partial_{i}\partial^{\alpha}_{x}(y^{1-2\gamma}(g^{i,j}-\delta elta^{i,j})\partial_{j
}u)
-
\frac{n-2\gamma}{2}\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^
{-2\gamma}u).
\end{split}
\end{equation}
In particular, since
$
-\partial_{p}(y^{1-2\gamma}g^{p,q}\partial_{q}v)
=
Dv
-
\partial_{i}(y^{1-2\gamma}(g^{i,j}-\delta elta^{i,j})v)
$
we may write
\begin{equation}gin{equation*}
\partial_{p}(y^{1-2\gamma}g^{p,q}\partial_{q}\partial^{\alpha}_{x}u)
=
\partial^{\alpha}_{x}h
+
h^{\alpha}
+
\sum \partial_{p}h^{\alpha}_{p},
\end{equation*}
where \;$h^{\alpha}, \;h^{\alpha}_{p}$\; depend only on \;$x-$derivatives of \;$u$\; of
order up to \;$\vert \alpha \vert$, and
due to \eqref{sqrt_g_sim_yC_infty}, \;\eqref{poisson_kernel_moser_iteration_5},
there holds
\begin{equation}gin{equation*}
\sum^{m}_{\vert \alpha \vert=0}\Vert
\frac{h^{\alpha}}{y^{1-2\gamma}}, \frac{h^{\alpha}_{p}}{y^{1-2\gamma}}\Vert_{L^{
\infty}_{y^{1-2\gamma}}(B^{+}_{\varepsilonilon_{0}})}
\leq
C_{m,\varepsilonilon_{0}}
[
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}
+
\sum^{m}_{k=0}\Vert y^{2\gamma-1}\nabla^{ k }_{x}h
\Vert_{L^{\infty}_{y^{1-2\gamma}}}
]
\;\;\text{for all}\;\; m\in \mathbb{N}.
\end{equation*}
Then Zamboni\cite{Zamboni} Theorem 5.2 shows H\"older regularity, i.e. for all \;$m\in \mathbb{N}$
\begin{equation}gin{equation}\label{poisson_zamboni}
\sum^{m}_{k=0}\Vert
\nabla^{k}_{x}u\Vert_{C^{0,\alpha}(B^{+}_{\frac{\varepsilonilon_{0}}{2}})}
\leq
C_{m,\varepsilonilon_{0}}
[
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}
+
\sum^{m}_{k=0}\Vert y^{2\gamma-1}\nabla^{ k }_{x}h
\Vert_{L^{\infty}_{y^{1-2\gamma}}}
].
\end{equation}
This allows us to integrate the equation directly. Indeed from
\eqref{poisson_x_deriviative_versus_difference_of_operators} we have
\begin{equation}gin{equation*}
D(\partial^{\alpha}_{x}u)
=
\partial^{\alpha}_{x}h
+
\partial^{\alpha}_{x}((D-D_{g})u)
=
\partial^{\alpha}_{x}h
+
\partial_{y}(y^{2-2\gamma}f^{\alpha}_{1})+y^{1-2\gamma}f^{\alpha}_{2},
\end{equation*}
where by definition
\;$
f^{\alpha}_{1}=\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{y\sqrt{g}}u)
$\;
and
\begin{equation}gin{equation*}
\begin{equation}gin{split}
f_{2}^{\alpha}
= &
\partial_{i}\partial_{x}^{\alpha}(\frac{\partial_{j}\sqrt{g}}{\sqrt{g}}g^{i,j}u)
-
\partial^{\alpha}_{x}(y^{2\gamma-1}\partial_{q}(\frac{\partial_{p}\sqrt{g}}{
\sqrt{g}}y^{1-2\gamma}g^{p,q})u) \\
& +
\partial_{i}\partial^{\alpha}_{x}((g^{i,j}-\delta elta^{i,j})\partial_{j}u)
-
\frac{n-2\gamma}{2}\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{y\sqrt{g}}
u)
.
\end{split}
\end{equation*}
This implies
\begin{equation}gin{equation*}
-\partial_{y}(y^{1-2\gamma}\partial_{y}\partial^{\alpha}_{x}u)
=
\partial_{y}(y^{2-2\gamma}f^{\alpha}_{1})
+
y^{1-2\gamma}(f^{\alpha}_{2}+\Delta_{x}\partial^{\alpha}_{x}u+y^{2\gamma-1}
\partial^{x}_{\alpha}h)
\end{equation*}
and we obtain
\begin{equation}gin{equation}\label{poisson_equation_integrated_1}
\begin{equation}gin{split}
\partial^{\alpha}_{x}u(y,x)
=
y^{2\gamma}\begin{eqnarray}r u_{0}^{\alpha}(x)
-
\int^{y}_{0}\sigma \tilde f^{\alpha}_{1}(\sigma,x)d\sigma
-
\int^{y}_{0}\sigma^{2\gamma-1}\int^{\sigma}_{0}\tau^{1-2\gamma} \tilde
f^{\alpha}_{2}(\tau,x)d\tau d\sigma,
\end{split}
\end{equation}
where by definition we may write with smooth coefficients \;$f_{i,\begin{equation}ta}$
\begin{equation}gin{equation}\label{poisson_tilde_f_structure}
\tilde f^{\alpha}_{1}=\sum _{\vert \begin{equation}ta\vert \leq \vert \alpha
\vert}f_{1,\begin{equation}ta}\partial^{\begin{equation}ta}_{x}u
\;\text{ and }\;
\tilde f^{\alpha}_{2}=\frac{\partial^{\alpha}_{x}h}{y^{1-2\gamma}}+\sum _{\vert
\begin{equation}ta\vert \leq \vert \alpha \vert+2}f_{2,\begin{equation}ta} \partial_{x}^{\begin{equation}ta}u.
\end{equation}
Let \;$h\in yC^{l,\lambda^{\prime}}$. Then \eqref{poisson_zamboni} shows
\begin{equation}gin{equation*}
\forall \vert \alpha \vert\leq l \;: \;\nabla^{\vert \alpha\vert}_{x}u \in
C^{0,\lambda},
\end{equation*}
whence \;$\forall\; \vert \alpha\vert \leq l-2 \;: \;\tilde
f_{i}^{\alpha}\in C^{0,\lambda}$\; due to \eqref{poisson_tilde_f_structure}. In
particular \eqref{poisson_equation_integrated_1} implies
\begin{equation}gin{equation*}
\partial^{\alpha}_{x}u(y,x)
=
y^{2\gamma}\begin{eqnarray}r u_{0}^{\alpha}(x) +o(y^{2\gamma}),
\end{equation*}
so \;$\begin{eqnarray}r u_{0}^{\alpha}\in C^{l+2,\lambda}$\; anyway by interior regularity. We define
\begin{equation}gin{equation}\label{poisson_y_minus_2_gamma_redefinition}
\begin{eqnarray}r u^{\alpha}=y^{-2\gamma}\partial^{\alpha}_{x}u
,\;\;
\begin{eqnarray}r f_{1}^{\alpha}=y^{-2\gamma}f_{1}^{\alpha}
,\;\;
\begin{eqnarray}r f_{2}^{\alpha}=y^{-2\gamma}\tilde f^{\alpha}_{2}.
\end{equation}
We then find from \eqref{poisson_equation_integrated_1}, that
\begin{equation}gin{equation}\label{poisson_equation_integrated}
\begin{equation}gin{split}
\begin{eqnarray}r u^{\alpha}(y,x)
= &
\begin{eqnarray}r u_{0}^{\alpha}(x)
-
y^{-2\gamma}\int^{y}_{0}\sigma^{1+2\gamma} \begin{eqnarray}r f^{\alpha}_{1}(\sigma,x) d\sigma
-
y^{-2\gamma}\int^{y}_{0}\sigma^{2\gamma-1}\int^{\sigma}_{0}\tau\begin{eqnarray}r
f^{\alpha}_{2}(\tau,x)d\tau d\sigma\\
= &
\begin{eqnarray}r u_{0}^{\alpha}(x) + \begin{eqnarray}r u_{1}^{\alpha}(y,x) +\begin{eqnarray}r u_{2}^{\alpha}(y,x),
\end{split}
\end{equation}
where according to
\eqref{poisson_tilde_f_structure}, \eqref{poisson_y_minus_2_gamma_redefinition}
we may write with smooth coefficients \;$f_{i,\begin{equation}ta}$
\begin{equation}gin{equation}\label{poisson_bar_f_structure}
\begin{eqnarray}r f^{\alpha}_{1}=\sum _{\vert \begin{equation}ta\vert \leq \vert \alpha
\vert}f_{1,\begin{equation}ta}\begin{eqnarray}r u^{\begin{equation}ta}
\;\;\text{ and }\;\;
\begin{eqnarray}r f^{\alpha}_{2}=\frac{\partial^{\alpha}_{x}h}{y}+\sum _{\vert \begin{equation}ta\vert
\leq \vert \alpha \vert+2}f_{2,\begin{equation}ta}\begin{eqnarray}r u^{\begin{equation}ta}.
\end{equation}
Then \eqref{poisson_equation_integrated_1} and \;$\forall \;\vert
\alpha\vert \leq l-2 \;: \;\tilde f_{i}^{\alpha}\in C^{0,\lambda}$\; already show
\begin{equation}gin{equation*}
\forall \vert \alpha\vert \leq l-2 \; :\;\begin{eqnarray}r u^{\alpha}\in C^{0,\lambda}
\end{equation*}
and we may assume
$
\forall \vert \alpha\vert\leq l-2-2m\;:\;\partial^{2m}_{y}\begin{eqnarray}r u^{\alpha}\in
C^{0,\lambda}
$
inductively, whence according to \eqref{poisson_bar_f_structure}
\begin{equation}gin{equation*}
\forall \vert \alpha \vert\leq l-2-2(m+1) \;:\;\partial^{2m}_{y}\begin{eqnarray}r f^{\alpha}_{i}\in
C^{0,\lambda}.
\end{equation*}
Then \eqref{poisson_equation_integrated} implies via Taylor expansion
\begin{equation}gin{equation*}
\forall \vert \alpha \vert\leq l-2-2(m+1)\; :\;\partial_{y}^{2m+2}\begin{eqnarray}r
u^{\alpha}_{i}, \;\partial_{y}^{2m+2}\begin{eqnarray}r u^{\alpha}\in C^{0,\alpha}.
\end{equation*}
Thus we have proven
$\forall\; \vert \alpha \vert\leq l-2-2m \; : \;\partial^{2m}_{y}\partial^{\alpha}_{x}
u\in C^{0,\lambda}$\; for some \;$\lambda>0$. However, since there are only even
powers in the \;$y$-derivative, we only find
\;$u\in C^{l-3,\lambda}$\; for $\;l\in 2\mathbb{N}$.
The proof is thereby complete.\end{pfn}
\subsection{Green's function for \;$D_g$\; under weighted Neumann boundary condition}\label{subsec:Greens_function}
In this subsection we study the Green's function \;$\Gamma_g$. As in the previous one we consider the existence and asymptotics issue. To do that we use the method of Lee-Parker\cite{lp} and have the same difficulties to overcome as in the previous subsection. We first note that on \;$\mathbb{R}^{n+1}_+$
\begin{equation}gin{equation}\label{gngamma}
\Gamma(y,x,\xi)=\Gamma^\gamma(y,x,\xi)=\frac{g_{n, \gamma}}{(y^{2}+\vert x-\xi\vert^{2})^{\frac{n-2\gamma}{2}}}, \;\; (y, x)\in \mathbb{R}^{n+1}_+, \;\;\xi\in \mathbb{R}^n
\end{equation}
for some \;$g_{n, \gamma}>0$\; is the Green's function to the dual problem
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
Du=0\;\;\text{ in }\;\;\mathbb{R}^{n+1}_+ \\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}u(y,\cdot)=f\;\; \text{ on }\;\; \mathbb{R}^{n},
\end{cases}
\end{equation*}
i.e.
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D\Gamma(, \xi)=0\;\;\text{ in }\;\;\mathbb{R}^{n+1}_+, \;\;\xi\in \mathbb{R}^n \\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\Gamma(y,x,\xi)=\delta elta_{x}(\xi), \;\;x, \;\xi\in \mathbb{R}^n.
\end{cases}
\end{equation*}
We will construct the Green's function \;$\Gamma_g$\; for the analogous problem
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}u=0\;\; \text{ in }\;\;X \\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}u(y,\cdot)=f \;\;\text{ on }\;\;M
\end{cases}
\end{equation*}
for \;$D_{g}=-div_{g}(y^{1-2\gamma}\nabla_{g} (\,\cdot\,))+E_{g}$, i.e. for \;$z\in X$\; and \;$\xi \in M$
\begin{equation}gin{equation}\label{green_defining_equation}
\begin{equation}gin{cases}
D_{g}\Gamma_{g}(\cdot,\xi)=0 \;\;\text{ in }\;\; X\\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\Gamma_{g}(z,\xi)=\delta elta_{x}(\xi),
\end{cases}
\end{equation}
where \;$z=(y,x)\in X$\;in $g$-normal Fermi-coordinates close to \;$M$. To that end we identify
\begin{equation}gin{equation*}
\xi \in M\cap U\subset U\cap X \;\;\text{ with }\;\;0\in B_{\varepsilonilon}^{\mathbb{R}^{n+1}}(0)\cap
\mathbb{R}^{n}\subset B_{\varepsilonilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}
\end{equation*}
as in the previous subsection, and write \;$\Gamma(z)=\Gamma(z,0)$. On \;$B_{\varepsilonilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}$\; we
then have
\begin{equation}gin{equation}\label{green_local_equation_for_the_greens_function}
D_{g}\Gamma=-\frac{\partial_{p}}{\sqrt{g}}(\sqrt{g}g^{p,q}y^{1-2\gamma}\partial_
{ q}\Gamma)+E_{g}\Gamma
=
f
\in
y^{1-2\gamma}H_{-n+2\gamma-1}C^{\infty}.
\end{equation}
Again we may solve homogeneous deficits homogeneously.
\begin{equation}gin{lem}
\label{green_homogeneous_solvability_neumann}$_{}$\\
For $\frac{1}{2}\neq \gamma \in (0,1)$ and $f_{l}\in y^{1-2\gamma}H_{l+2\gamma-1}, \; l\in \mathbb{N}-n$\; there exists
\;$\Gamma_{1+2\gamma+l}\in H_{1+2\gamma+l}$\; such, that
$$D\Gamma_{1+2\gamma+l}=f_{l}\;\;\text{ in }\;\mathbb{R}^{n+1}\;\;\text{ and } \;\;\lim_{y\rightarrow 0}
y^{1-2\gamma}\partial_{y}\Gamma_{1+2\gamma+l}=0\text{ on } \mathbb{R}^{n}\setminus
\{0\}.$$
\end{lem}
\begin{equation}gin{pf}
This time we use
\begin{equation}gin{equation*}
\langle Q^{k}_{l}(y,x)=y^{2k}P_{l}(x)\mid \;\,k, \;l\in \mathbb{N}\;\;\text{ and }\;\;P_{l}\in
\Pi_{l}\rangle \underset{\text{dense}}{\subset} C^{0}(\overlineerline
B_{1}^{\mathbb{R}^{n+1}}(0)\cap\mathbb{R}^{n+1}_{+}),
\end{equation*}
to obtain a orthogonal basis \;$E=\{e^{i}_{k}\}$\; for\;
$L^{2}_{y^{1-2\gamma}}(S^{n}_{+})$
consisting of $D$-harmonics of the form
\begin{equation}gin{equation*}
e^{i}_{k}=A_{m}\lfloor_{S^{n}_{+}}, \;\;\,A_{m}(y,x)=\sum
y^{2l}P_{k-2l}(x), \;\;DA_{m}=0
\end{equation*}
and we have \;$
D_{S^{n}_{+}}e^{i}_{k}=k(k+n-2\gamma)y^{1-2\gamma}e^{i}_{k}.
$
Then for homogeneous \;$f, \;u$\; of degree \;$\lambda,\; \lambda+1+2\gamma$ solving
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
Du=f\in L^{2}_{y^{2\gamma-1}}(\mathbb{R}^{n+1}_{+})\;\;\text{ in }\;\;\mathbb{R}^{n+1}_{+} \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=0 \;\;\text{ on } \;\; \mathbb{R}^{n}
\end{cases}
\end{equation*}
is, when writing
$
u=\sum a_{i,k}e^{i}_{k}\,,y^{2\gamma-1}f=\sum b_{j,l}e^{j}_{l},
$
equivalent to solving
\begin{equation}gin{equation*}
\sum a_{i,k}(k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1))e^{i}_{k}=\sum
b_{j,l}e^{j}_{l}
\end{equation*}
and the latter system is always solvable in case
\begin{equation}gin{equation}\label{solvability_homogeneous_neumann}
k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1)\neq 0 \,\,\text{ for all
}\;\;k, \;n, \;\lambda \in \mathbb{N}.
\end{equation}
As for proving the lemma there holds
\begin{equation}gin{equation*}
deg(f_{l})=\lambda=m-n\;\;
\text{ and }\;\;
deg(e^{i}_{k})=k=m^{\prime}
\;\;\text{ for some }\,\;
m, \;m^{\prime}\in \mathbb{N}
\end{equation*}
and plugging this into
\eqref{solvability_homogeneous_neumann} we verify for $\frac{1}{2}\neq \gamma \in (0,1)$
\begin{equation}gin{equation*}
m^{\prime}(m^{\prime}+n-2\gamma)-(m-n+1+2\gamma)(m+1)\neq 0\;\;\text{ for all } \;\;
n \;,m, \;m^{\prime}\in \mathbb{N}.
\end{equation*}
This shows homogeneous
solvability, whereas regularity of the solution follows from Proposition
\ref{green_regularity}.
\end{pf}
\noindent
Analogously to the case of the Poisson kernel we may solve
\eqref{green_defining_equation} successively using Lemma
\ref{green_homogeneous_solvability_neumann} and obtain
\begin{equation}gin{equation*}
\Gamma_{g}=\eta_{\xi}(\Gamma+\sum_{l=-n}^{m} \Gamma_{1+2\gamma+l})+\gamma_{m}
\end{equation*}
for \;$m\geq 0$, where \;$\eta_{\xi}$\; is as in \eqref{etaxi} and a weak solution
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}\gamma_{m}=-D_{g}\left(\eta_{\xi}(\Gamma+\sum_{l=-n}^{m}
\Gamma_{1+2\gamma+l})\right)=y^{1-2\gamma}h_{m}\;\; \text{ in }\;\;X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}\gamma_{m}=0 \,\;\text{ on } \;\;M
\end{cases}
\end{equation*}
with \;$h_{m} \in C^{m,\alpha}$. As in the previous subsection a weak regularity statement is
sufficient for our purpose.
\begin{equation}gin{pro}
\label{green_regularity}$_{}$\\
Let \;$h\in y^{1-2\gamma}C^{2k+3,\alpha}(X)$\; and \;$u\in
W^{1,2}_{y^{1-2\gamma}}(X)$\; be a weak solution of
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}u=h\;\;\text{ in }\;\;X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=0 \,\;\text{ on } \;\;M.
\end{cases}
\end{equation*}
Then \;$u$\; is of class \;$C^{2k,\begin{equation}ta}(X)$, provided \;$H_{g}=0$.
\end{pro}
\noindent
As in the previous subsection, putting these facts together before presenting the proof of Proposition \ref{green_regularity}, we have the existence of \;$\Gamma_g$\; and can describe its asymptotics.
\begin{equation}gin{cor}
\label{Greens_function_asymptotics}$_{}$\\
Let \;$\frac{1}{2}\neq \gamma\in (0,1)$. Then \;$\Gamma_g$\; exists and we may expand in \;$g$-normal
Fermi-coordinates around $\xi \in M$
\begin{equation}gin{equation*}
\Gamma_{g}(z,\xi)
\;\in
\eta_{\xi}(z)
\left(
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+\sum^{2m+3}_{l=-n}H_{1+2\gamma+l}(z)
\right)
+
C^{2m,\alpha}(X)
\end{equation*}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n+1}_+\setminus \{0\})$\; being homogeneous of order
\;$l$\; and \;$g_{n, \gamma}$\; is as in \eqref{gngamma}, provided \;$H_{g}=0$.
\end{cor}
\noindent
\begin{equation}gin{pfn}{ of Proposition \ref{green_regularity}}
\noindent
As in the previous subsection we use the Moser iteration argument. Indeed by exactly the same arguments as the ones used when proving Proposition
\ref{poisson_regularity} we recover H\"older regularity \eqref{poisson_zamboni}
and integrating the equation directly we find the analogue of
\eqref{poisson_equation_integrated_1}, namely
\begin{equation}gin{equation}\label{green_equation_integrated}
\begin{equation}gin{split}
\partial^{\alpha}_{x}u(y,x)
= &
u_{0}^{\alpha}(x)
-
\int^{y}_{0}\sigma \tilde f^{\alpha}_{1}(\sigma,x)d\sigma
-
\int^{y}_{0}\sigma^{2\gamma-1}\int^{\sigma}_{0}\tau^{1-2\gamma} \tilde
f^{\alpha}_{2}(\tau,x)d\tau d\sigma \\
= &
u^{\alpha}_{0}(x)+u^{\alpha}_{1}(y,x)+u^{\alpha}_{2}(y,x),
\end{split}
\end{equation}
where \;$\tilde f_{1}, \;\tilde f_{2}$\; are given by
\eqref{poisson_tilde_f_structure}.
Let \;$h\in y^{1-2\gamma}C^{l,\lambda^{\prime}}$. Then
\eqref{poisson_zamboni} and \eqref{poisson_tilde_f_structure} show
\begin{equation}gin{equation*}
\forall \vert \alpha\vert \leq l-2 \;:\,\;\tilde
f_{i}^{\alpha}\in C^{0,\lambda}.
\end{equation*}
In particular \eqref{green_equation_integrated} implies
\begin{equation}gin{equation*}
\partial^{\alpha}_{x}u(y,x)
=
u_{0}^{\alpha}(x) +O(y),
\end{equation*}
so \;$u_{0}^{\alpha}\in C^{l+2,\lambda}$\; anyway by interior regularity
and we may assume inductively
\begin{equation}gin{equation*}
\forall \vert \alpha\vert\leq l-2-2m\; : \;\partial^{2m}_{y}\partial^{\alpha}_{x}u\in
C^{0,\lambda},
\end{equation*}
whence according to \eqref{poisson_tilde_f_structure}
\begin{equation}gin{equation*}
\forall \vert \alpha \vert\leq l-2-2(m+1)\; : \;\partial^{2m}_{y}\tilde
f^{\alpha}_{i}\in C^{0,\lambda}.
\end{equation*}
Then \eqref{green_equation_integrated} implies via Taylor expansion
\begin{equation}gin{equation*}
\forall \vert \alpha \vert\leq
l-2-2(m+1) \; : \;\partial_{y}^{2m+2}u_{i}^{\alpha}, \;\partial_{y}^{2m+2}\partial^{\alpha
}_{x}u \in C^{0,\lambda}
\end{equation*}
Thus we have proven \;$
\forall \vert \;\alpha \vert\leq
l-2-2m \; : \;\partial_{y}^{2m}\partial^{\alpha
}_{x}u \in C^{0,\lambda}
$
for some \;$\lambda>0$. However, since there are only even powers in
the \;$y$-derivative, we only find \;$u\in C^{l-3,\lambda}$\; for \;$l\in 2\mathbb{N}$.
The proof is thereby complete.
\end{pfn}
\subsection{Green's function for the fractional conformal Laplacian }
In this short subsection we study the Green's function \;$G_h^\gamma$\; of \;$P_h^\gamma$. We derive its existence and asymptotics as a consequence of the results of the previous subsections and formula \eqref{regreen}.
\begin{equation}gin{cor}
\label{Greens_function_asymptoticsgamma}$_{}$\\
Let \;$\frac{1}{2}\neq \gamma\in (0,1)$. Then \;$G_h$\; exists and we may expand in \;$h$-normal-coordinates
around $\xi \in M$
\begin{equation}gin{equation*}
G_{h}(x,\xi)
\;\in
\eta_{\xi}(x)
\left(
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+\sum^{2m+3}_{l=-n}H_{1+2\gamma+l}(x)
\right)
+
C^{2m,\alpha}(M)
\end{equation*}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n}\setminus \{0\})$\; being homogeneous of order
\;$l$, provided \;$H_{g}=0$.
\end{cor}
\noindent
To end this section, we give the proof of Theorem \ref{Greens_function_asymptoticsgamma1}.\\\\
\begin{equation}gin{pfn}{ of Theorem \ref{Greens_function_asymptoticsgamma1}}\\
It follows directly from Corollary \ref{Poisson_kernel_asymptotics}, Corollary \ref{Greens_function_asymptotics}, and Corollary \ref{Greens_function_asymptoticsgamma}.
\end{pfn}
\section{Locally flat conformal infinities of PE-manifolds}\label{pceinsten}
In this section we sharpen the results of Section \ref{fundso} in the case of Poincar\'e-Einstein manifold \;$(X, g^+)$\; with locally flat conformal infinity \;$(M, [h])$.
\subsection{Fermi-coordinates in this particular case}
By our assumptions we have
\begin{equation}gin{enumerate}[label=(\roman*)]
\item a geodesic defining function \;$y$\; splitting the metric
\begin{equation}gin{equation*}
g=y^{2}g^{+}, \; g=dy^{2}+h_{y}\;\;\text{ near }\;\;M\;\;\text{ and }\;\; h=h_{y}\lfloor_{M}
\end{equation*}
and for every \;$a\in M$\; a conformal factor as in \eqref{conformal_factor_properties},
whose conformal metric \;$h_{a}=u_{a}^{\frac{4}{n-2\gamma}}h$\; close to \;$a$\; admits an Euclidean
coordinate system, \;$h_{a}=\delta elta$\; on $B_{\varepsilonilon}^{h_a}(a)$. As clarified in
subsection \ref{confnonhom} and recalling Remark \ref{eq:minimal}, this gives rise to a geodesic defining
function \;$y_{a}$, for which
\begin{equation}gin{equation}\label{dfmga}
g_{a}=y_{a}^{2}g^{+}, \;\; g_{a}=dy_{a}^{2}+h_{a,y_{a}}\;\;\text{near}\;\;M\;\;\text{ with }\;\;
h_{a}=h_{a, y_a}\lfloor_{M}\;\;\text{ and }\;\;\delta elta=h_{a}\lfloor_{
B_{\varepsilonilon}^{h_a}(a)},
\end{equation}
the boundary \;$(M, [h_a])$\; is totally geodesic and the extension operator \;$D_{g_{a}}$\; is positive.
\item
as observed by Kim-Musso-Wei\cite{kmw1} in the case \;$n\geq 3$, cf. Lemma 43 in \cite{kmw1}, and for \;$n=2$\; due to Remark \ref{eq:minimal} and the existence of isothermal coordinates we have
\begin{equation}gin{equation}\label{flatness}
g_{a}=\delta elta+O(y_{a}^{n}) \;\;\text{ on } \;\;B_{\varepsilonilon}^{g_a, +}(a)
\end{equation}
in \;$g_{a}$-normal
Fermi-coordinates around $a$for some small \;$\varepsilonilon >0$.
Therefore the previous results on the fundamental
solutions in the case of an asymptotically hyperbolic manifold with minimal conformal infinity of Section \ref{fundso} are applicable. We collect them in the following subsection.
\end{enumerate}
\subsection{Fundamental solutions in this particular case}
In this subsection we sharpen the results of Section \ref{fundso} in the case of a Poincar\'e-Einstein manifold \;$(X, g^+)$\; with locally flat conformal infinity \;$(M, [h])$.
\noindent
To do that
let us first recall that \;$K_{a}=K_{g_{a}}(\cdot, a)$, \;$\Gamma_{a}=\Gamma_{g_{a}}(\cdot,
a)$ and \;$G_a=G_{h_a}(\cdot,a)$.
From \eqref{flatness} we then find
\begin{equation}gin{equation*}
D_{g_{a}}K_{a}\in yH_{-2\gamma-2}C^{\infty},
\;\,\;\;
D_{g_{a}}\Gamma_{a}\in y^{1-2\gamma}H_{2\gamma-2}C^{\infty}
\end{equation*}
for the lowest order
deficits in \eqref{poisson_local_equation_for_the_poisson_kernel} and
\eqref{green_local_equation_for_the_greens_function}.
Then in view of Lemmas \ref{poisson_homogeneous_solvability_dirichlet}, \ref{green_homogeneous_solvability_neumann} the corresponding expansions given
by Corollaries
\ref{Poisson_kernel_asymptotics}, \ref{Greens_function_asymptotics}, \ref{Greens_function_asymptoticsgamma}
are
\begin{equation}gin{enumerate}[label=(\roman*)]
\item \quad
$
K_{a}(z)
\in
\eta_{\xi}(z)
\left(p_{n, \gamma}
\frac{y^{2\gamma}}{\vert z
\vert^{n+2\gamma}}+\sum^{2m+6}_{l=0}y^{2\gamma}H_{l-2\gamma}(z)
\right)
+
y^{2\gamma}C^{2m,\alpha}(X)
$
\item \quad
$
\Gamma_{a}(z)
\;\in
\eta_{\xi}(z)
\left(
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+\sum^{2m+4}_{l=0}H_{l+2\gamma}(z)
\right)
+
C^{2m,\alpha}(X)
$
\item \quad
$
G_{a}(x)
\;\in
\eta_{\xi}(x)
\left(
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+\sum^{2m+4}_{l=0}H_{l+2\gamma}(x)
\right)
+
C^{2m,\alpha}(M).
$
\end{enumerate}
Recalling \eqref{definition_H_l} there holds \;$y^{2\gamma}H_{l-2\gamma}\subset
C^{m,\alpha}$\; for \;$l>m $\; and
$H_{l+2\gamma}\subset C^{m,\alpha}$\; for \;$l\geq m$. We have therefore proven the following result.
\begin{equation}gin{cor}
\label{cor_kernels_for_poincare_einstein_metrics}$_{}$\\
\noindent
Let \;$(X, \;g^{+})$\; be a Poincar\'e-Einstein manifold with
conformal infinity \;$(M, [h])$\; of dimension \;$n= 2$\; or \;$n\geq 3$\; and \;$(M, [h])$\; is locally flat. If
\begin{equation}gin{equation*}
\frac{1}{2}\neq \gamma \in (0,1) \;\;\text{ and } \;\,\;
\lambda_{1}(-\Delta_{g^{+}})>s(n-s)\;\;
\text{ for } s=\frac{n}{2}+\gamma,
\end{equation*}
then the Poison kernel \;$K_{g}$\; and the Green's functions \;$\Gamma_{g}$\; and \;$G_{h}$\;
respectively for
\begin{equation}gin{equation*}
\begin{equation}gin{cases}
D_{g}U=0 \;\;\text{ in }\;\; X \\
U=f \,\,\;\text{ on } \;\;M
\end{cases}
\quad\quad
\begin{equation}gin{cases}
D_{g}U=0\;\; \text{ in } \;\;X \\
-d_{\gamma}^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}U=f \;\;\text{ on }\;\; M
\end{cases}
\quad \text{and}\quad
\begin{equation}gin{cases}
P_h^\gamma u=f\;\;\text{ on }\;\;M
\end{cases}
\end{equation*}
are respectively of class \;$y^{2\gamma}C^{2,\alpha}$\; and \;$C^{2,\alpha}$\; away from the
singularity and admit for every \;$a\in M$ locally in \;$g_{a}$-normal
Fermi-coordinates an expansion around $a$
\begin{equation}gin{enumerate}[label=(\roman*)]
\item \quad
$
K_{a}(z)\;
\in
p_{n, \gamma}\frac{y^{2\gamma}}{\vert z \vert^{n+2\gamma}}
+
y^{2\gamma}H_{-2\gamma}(z)
+
y^{2\gamma}H_{1-2\gamma}(z)
+
y^{2\gamma}H_{2-2\gamma}(z)
+
y^{2\gamma}C^{2,\alpha}(X)
$
\item \quad
$
\Gamma_{a}(z)\;
\in
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+H_{2\gamma}(z)+H_{1+2\gamma}(z)
+
C^{2,\alpha}(X)
$
\item \quad
$
G_{a}(x)\;
\in
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+H_{2\gamma}(x)+H_{1+2\gamma}(x)
+
C^{2,\alpha}(M),
$
\end{enumerate}
where \;$g_a$\;is as in \eqref{dfmga} and \;$H_{k}\in C^{\infty}(\overlineerline{\mathbb{R}^{n}_{+}}\setminus \{0\})$\; are
homogeneous of degree \;$k$.
\end{cor}
\noindent
Finally, we give the proof of Theorem \ref{cor_kernels_for_poincare_einstein_metrics1}.\\\\
\begin{equation}gin{pfn}{ of Theorem \ref{cor_kernels_for_poincare_einstein_metrics1}}\\
It is exactly the statement of Corollary \ref{cor_kernels_for_poincare_einstein_metrics}.
\end{pfn}
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\end{document} | math | 105,031 |
\begin{document}
\title{Hamiltonian Simulation Using Linear Combinations of Unitary Operations}
\author{Andrew M. Childs}
\affiliation{Department of Combinatorics \& Optimization, University of Waterloo, Ontario N2L 3G1, Canada}
\affiliation{Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada}
\author{Nathan Wiebe}
\affiliation{Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada}
\begin{abstract}
We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.
\end{abstract}
\maketitle
\section{Introduction}
Simulating the time evolution of quantum systems is a major potential application of quantum computers. While quantum simulation is apparently intractable using classical computers, quantum computers are naturally suited to this task. Even before a fault-tolerant quantum computer is built, quantum simulation techniques can be used to prove equivalence between Hamiltonian-based models of quantum computing (such as adiabatic quantum computing~\cite{ADKLLR07} and continuous-time quantum walks~\cite{Chi09a}) and to develop novel quantum algorithms~\cite{FGGS00,CCDFGS03,FGG07,HHL09,B10}.
In recent years there has been considerable interest in optimizing quantum simulation algorithms. The original approach to quantum simulation, based on product formulas, was proposed by Lloyd for time-independent local Hamiltonians~\cite{lloyd}. This work was later generalized to give efficient simulations of sparse time-independent Hamiltonians that need not have tensor-product structure~\cite{CCDFGS03,ATS03}. Further refinements of these schemes have yielded improved performance~\cite{Chi04,BACS07,PaZh10,CK11} and the techniques have been extended to cover time-dependent Hamiltonians~\cite{WBHS11,PQS+11}. Recently a new paradigm has been proposed that uses quantum walks rather than product formulas~\cite{Chi09,BC09}. This approach is superior to the product formula approach for simulating sparse time-independent Hamiltonians with constant accuracy (and in addition, can be applied to non-sparse Hamiltonians), whereas the product formula approach is superior for generating highly accurate simulations of sparse Hamiltonians.
The performance of simulation algorithms based on product formulas is limited by the fact that high-order approximations are needed to optimize the algorithmic complexity. The best known high-order product formulas, the Lie--Trotter--Suzuki formulas, approximate the time evolution using a product of unitary operations whose length scales exponentially with the order of the formula \cite{Suz91}. In contrast, classical methods known as multi-product formulas require a sum of only polynomially many unitary operations to achieve the same accuracy~\cite{BCR99} (although of course the overall cost of classical simulations based on multi-product formulas remains exponential in the number of qubits used to represent the Hilbert space). However, these methods cannot be directly implemented on a quantum computer because unitary operations are not closed under addition.
Our work addresses this by presenting a non-deterministic algorithm that can be used to perform linear combinations of unitary operators on quantum computers. We achieve high success probabilities provided the operators being combined are near each other.
We apply this tool to quantum simulation and thereby improve upon existing quantum algorithms for simulating Hamiltonian dynamics. Our main result is as follows.
\begin{theorem}\label{thm:mainresult}
Let the system Hamiltonian be $H=\sum_{j=1}^m H_j$ where each $H_j\in \mathbb{C}^{2^n\times 2^n}$ is Hermitian and satisfies $\norm{H_j} \le h$ for a given constant $h$. Then the Hamiltonian evolution $e^{-iHt}$ can be simulated on a quantum computer with failure probability and error at most $\epsilon$ as a product of linear combinations of unitary operators. In the limit of large $m,ht,1/\epsilon$, this simulation uses
\begin{equation}\label{eq:mainresult}
\tilde O\left(m^2 h te^{1.6\sqrt{\log(m h t/\epsilon)}}\right)
\end{equation}
elementary operations and exponentials of the $H_j$s.
\end{theorem}
Although we have not specified the method used to simulate the exponential of each $H_j$, there are well-known techniques to simulate simple Hamiltonians. In particular, if $H_j$ is $1$-sparse (i.e., has at most one non-zero matrix element in each row and column), then it can be simulated using $O(1)$ elementary operations~\cite{ATS03,CCDFGS03}, so \eq{mainresult} gives an upper bound on the complexity of simulating sparse Hamiltonians.
Our simulation is superior to the previous best known simulation algorithms based on product formulas. Previous methods have scaling of the same form, but with the coefficient 1.6 replaced by 2.54 \cite[Theorem 1]{BACS07} or 2.06 \cite[Theorem 1]{WBHS10}.
Also note that Theorem 1 of~\cite{PaZh10} gives a similar scaling as in \cite{WBHS10}, except the term in the exponential depends on the second-largest $\norm{H_j}$ rather than $h$.
Perhaps more significant than the quantitative improvement to the complexity of Hamiltonian simulation is that our approach demonstrates a new class of simulation protocols going beyond the Lie--Trotter--Suzuki paradigm, the approach used in most previous simulation algorithms. It remains unknown how efficiently one can perform quantum simulation as a function of the allowed error $\epsilon$, and we hope our work will lead to a better understanding of this question.
The remainder of this article is organized as follows.
In \sec{add}, we provide a general method for implementing linear combinations of unitary operators using quantum computers and lower bound its success probability.
This method is optimal among a large class of such protocols, as shown in the appendix.
In \sec{mpf}, we provide a brief review of Lie--Trotter--Suzuki formulas and multi-product formulas and then show how to implement multi-product formulas on quantum computers.
Error bounds and overall success probabilities of our simulations are derived in \sec{mpferror}. We then bound the number of quantum operations used in our simulation in \lem{nexp}, from which \thm{mainresult} follows.
We conclude in \sec{conclusions} with a summary of our results and a discussion of directions for future work.
\section{Adding and Subtracting Unitary Operations Using Quantum Computers}\label{sec:add}
In this section we describe basic protocols for implementing linear combinations of unitary operations. \lem{main:2case} shows that a quantum computer can nearly deterministically perform a weighted average of two nearby unitary operators. (Our approach is reminiscent of a technique for implementing fractional quantum queries using discrete queries \cite{CGMSY09}.) We build upon \lem{main:2case} in \thm{main:nonunitary}, showing that a quantum computer can non-deterministically implement an arbitrary linear combination of a set of unitary operators.
\begin{lemma}\label{lem:main:2case}
Let $U_a,U_b \in \mathbb{C}^{2^n\times 2^n}$ be unitary operations and let $\Delta=\norm{U_a-U_b}$. Then for any $\kappa\ge 0$, there exists a quantum algorithm that can implement an operator proportional to $\kappa U_a + U_b$ with failure probability at most $\Delta^2\kappa/(\kappa+1)^2\le 4\kappa/(\kappa+1)^2$.
\end{lemma}
\begin{proof}
Let
\begin{equation}
V_\kappa\colonequals
\begin{pmatrix}\sqrt{\frac{\kappa}{\kappa+1}} & \frac{-1}{\sqrt{\kappa+1}}\\
\frac{1}{\sqrt{\kappa+1}} & \sqrt{\frac{\kappa}{\kappa+1}}\end{pmatrix}.
\end{equation}
Our protocol for implementing the weighted average of $U_a$ and $U_b$ works as follows (see \fig{main:addition}). First, we perform $V_{\kappa}$ on an ancilla qubit. Second, we perform a zero-controlled $U_a$ gate and a controlled $U_b$ gate on the state $\ket{\psi}$ using the ancilla as the control. Finally, we apply $V_\kappa^\dagger$ to the ancilla qubit and measure it in the computational basis. This protocol performs the following transformations:
\begin{align}
\ket{0}\ket{\psi}
&\mapsto
\left(\sqrt{\frac{\kappa}{\kappa+1}}\ket{0}+\frac{1}{\sqrt{\kappa+1}}\ket{1} \right)\ket{\psi}\nonumber\\&\mapsto
\left(\sqrt{\frac{\kappa}{\kappa+1}}\ket{0}U_a\ket{\psi}+\frac{1}{\sqrt{\kappa+1}}\ket{1}U_b\ket{\psi} \right)\nonumber\\
&\mapsto \ket{0}\left(\frac{\kappa}{\kappa+1} U_a + \frac{1}{\kappa+1} U_b \right)\ket{\psi}+\ket{1}\frac{\sqrt{\kappa}}{\kappa+1}(U_b-U_a)\ket{\psi}.\label{eq:thm1:last}
\end{align}
If the first qubit is measured and a result of $0$ is observed, then this protocol performs
$\ket{\psi}\mapsto (\kappa U_a+U_b)\ket{\psi}$ (up to normalization). If the measurement yields $1$ then the algorithm fails. The probability of this failure, $P_+$, is
\begin{equation}
P_+\le \frac{\norm{U_b-U_a}^2\kappa}{(\kappa+1)^2} = \frac{\Delta^2\kappa}{(\kappa+1)^2}.
\end{equation}
Since $\Delta \le 2$, this is at most $\frac{4\kappa}{(\kappa+1)^2}$.
\end{proof}
By substituting $U_b\rightarrow -U_b$, \lem{main:2case} also shows that unitary operations can be subtracted. (Alternatively, replacing $V_\kappa^\dagger$ with $V_\kappa$ in \fig{main:addition} also simulates $U_a-U_b$.) Similarly, we could make the weights of each unitary complex by multiplying $U_0$ and $U_1$ by phases, although we will not need to make use of this freedom.
\begin{figure}
\caption{Quantum circuit for non-deterministically performing an operator proportional to $\kappa U_a + U_b$ given a measurement outcome of zero.\label{fig:main:addition}
\label{fig:main:addition}
\end{figure}
General linear combinations of unitary operators can be performed by iteratively applying \lem{main:2case}. The following theorem gives constructs such a simulation and provides bounds on the probabilities of failure.
\begin{theorem}\label{thm:main:nonunitary}
Let $V\colonequals\sum_{q=1}^{k+1} C_q U_q$ for $k\ge 1$ where $C_q\ne 0$, $\norm{U_q}=1$, and $\max_{q\ne q'}\norm{U_q-U_{q'}} \le \Delta$. Let $\kappa\colonequals (\sum_{q:~C_q>0} C_q)/(\sum_{q:~C_q<0} |C_q|)$. Then there exists a quantum algorithm that implements an operator proportional to $V$ with probability of failure $P_+ + P_-$ with
\begin{align}
P_+&\le \frac{k\Delta^2}{4}, \\
P_-&\le \frac{4\kappa}{(\kappa+1)^2},
\end{align}
where $P_+$ is the probability of failing to add some pair of operators and $P_-$ is the probability of failing to perform the subtraction.
\end{theorem}
\begin{proof}
Let
\begin{align}
A&\colonequals\frac{1}{\sum_{q:C_q>0} C_q}\sum_{q\colon C_q>0}C_q U_q,\\
B&\colonequals\frac{1}{\sum_{q:C_q<0} |C_q|}\sum_{q\colon C_q<0}|C_q| U_q.
\end{align}
We implement $V\propto \kappa A-B$ using circuits that non-deterministically implement operators proportional to $A$ and $B$. We recursively perform these sums by using the circuits given in \lem{main:2case}, except that we defer measurement of the output until the algorithm is complete.
We then implement $\kappa A -B$ by using our addition circuit, with $U_a$ taken to be the circuit that implements $A$ and $U_b$ taken to be the circuit that implements $-B$. Then we measure each of the control qubits for the addition steps. Finally, if all the measurement results are zero (indicating success), we measure the control qubit for the subtraction step. If that qubit is zero, then we know from prior analysis that the non-unitary operation $V$ is implemented successfully.
The operators $A$ and $B$ are implemented by recursively adding terms. For example, the sum $U_1+U_2+U_3+U_4$ is implemented as $(((U_1+U_2)+U_3)+U_4)$. Implementing the sums in $A$ and $B$ requires $k-1$ addition operations, so $V$ can be implemented using $k-1$ addition operations and one subtraction operation (assuming that all the control qubits are measured to be zero at the end of the protocol).
According to \lem{main:2case}, the probability of failing to implement $V$, given that we successfully implement $A$ and $B$, is
\begin{equation}
P_-\le \frac{4\kappa}{(\kappa+1)^2}.
\end{equation}
The probability of failing to perform the $k-1$ sums needed to construct $A$ and $B$ obeys
\begin{equation}
P_+
\le (k-1)\frac{\Delta^2\kappa}{(\kappa+1)^2}\le \frac{k\Delta^2}{4},
\end{equation}
where the last step follows from the fact that we can take $\kappa \ge 1$ without loss of generality.
\end{proof}
These results show that we can non-deterministically implement linear combintations of unitary operators with high probability provided that $\kappa\gg 1$ and $\Delta\ll 1$. As we will see shortly, this situation can naturally occur in quantum simulation problems.
Note that it is not possible to increase the success probability of the algorithm by replacing the single-qubit unitary $V_\kappa$ with a different unitary, even if that unitary is allowed to act on all of the ancilla qubits simultaneously. We present this argument in \app{opt}.
\section{Implementing Multi-Product Formulas on Quantum Computers}\label{sec:mpf}
In this section we present a new approach to quantum simulation: we approximate the time evolution using a sequence of non-unitary operators that are each a linear combination of product formulas. Such sums of product formulas are known in the numerical analysis community as multi-product formulas, and can be more efficient than product formulas for classical computations~\cite{BCR99,Chin10}. We show how to implement multi-product formulas using quantum computers by leveraging our method for non-deterministically performing linear combinations of unitary operators.
\subsection{Review of Lie--Trotter--Suzuki and Multi-Product Formulas}
Product formula approximations can be used to accurately approximate an operator exponential as a product of operator exponentials that can be easily implemented. Apart from their high degree of accuracy, these approximations are useful because they approximate a unitary operator with a sequence of unitary operators, making them ideally suited for quantum computing applications.
The most accurate known product formula approximations are the Lie--Trotter--Suzuki formulas, which approximate $e^{-iHt}$ for $H=\sum_{j=1}^m H_j$ as a product of the form
$$
e^{-iHt}\approx\prod_{k=1}^{N_{\exp}} e^{-i H_{j_k} t_k}.
$$
These formulas are recursively defined for any integer $\chi>0$ by~\cite{Suz91}
\begin{align}
S_1(t)&=\prod_{j=1}^m e^{-i H_j t/2}\prod_{j=m}^1 e^{-i H_j t/2},\nonumber\\
S_\chi(t)&=\left(S_{\chi-1}(s_{\chi-1}t)\right)^2S_{\chi-1}([1-4s_{\chi-1}]t)\left(S_{\chi-1}(s_{\chi-1}t)\right)^2\label{eq:1_2},
\end{align}
where $s_p=(4-4^{1/(2p+1)})^{-1}$ for any integer $p>0$.
This choice of $s_p$ is made to ensure that the Taylor series of $S_\chi$ matches that of $e^{-iHt}$ to $O(t^{2\chi +1})$.
Consequently, the approximation can be made arbitrarily accurate for suitably large values of $\chi$ and small values of $t$.
The advantage of these formulas is clear: they are highly accurate and approximate $U(t) \colonequals e^{-iHt}$ by a sequence of unitary operations, which can be directly implemented using a quantum computer. The primary disadvantage is that they require $O(5^k)$ exponentials to construct an $O(t^{2k+1})$ approximation. This scaling leads to quantum simulation algorithms with complexity $(\norm{H}t)^{1+o(1)}$. A product formula requiring significantly fewer exponentials could result in a substantial performance improvement over existing product formula-based quantum simulation algorithms.
In the context of classical simulation, multi-product approximations were introduced to address these problems~\cite{BCR99,Chin10}. Multi-product formulas generalize the approximation-building procedure used to construct $S_\chi$ to allow sums of product formulas. The resulting formulas are simpler since it is easier to construct a Taylor series by adding polynomials than by multiplication alone. Specifically, multi-product formulas only need $O(k^2)$ exponentials to construct an approximation of $U(t)$ to $O(t^{2k+1})$.
We consider multi-product formulas of the form
\begin{equation}
M_{k,\chi}(t)=\sum_{q=1}^{k+1} C_q S_\chi(t/\ell_q)^{\ell_q},\label{eq:mkdef}
\end{equation}
where the formula is accurate to $O(t^{2(k+\chi)+1})$.
Here $\ell_1,\ldots,\ell_{k+1}$ are distinct natural numbers and $C_1,\ldots,C_{k+1} \in \mathbb{R}$ satisfy $\sum_{q=1}^{k+1} C_q=1$. Explicit expressions for the coefficients $C_q$ are known for the case $\chi=1$~\cite{Chin10}:
\begin{equation}
C_q=\prod_{j=\{1,\ldots,k+1\}\setminus q} \frac{\ell_q^2}{\ell_q^2-\ell_j^2}.\label{eq:cqgen}
\end{equation}
We will show later that the same expressions for $C_q$ also apply for $\chi>1$.
For classical simulations, the most numerically efficient formulas correspond to $\ell_q=q$ and $\chi=1$. The simplest example of a multi-product formula is the Richardson extrapolation formula for $S_1$~\cite{Ri1911}:
\begin{equation}
U(t)=\frac{4S_1(t/2)^2-S_1(t)}{3}+O(t^5).
\end{equation}
Even though the above expression is non-unitary, it is very close to a unitary operator. In fact, there exists a unitary operator within distance $O(t^{10})$ of the multi-product formula. In general, Blanes, Casas, and Ros show that if a multi-product formula $M_{k,\chi}$ is accurate to $O(t^{2(k+\chi)+1})$ then it is unitary to $O(t^{4(k+\chi)+2})$~\cite{BCR99}; therefore such formulas are practically indistinguishable from unitary operations in many applications~\cite{Chin10,BCR99}.
The principal drawback of these formulas is that they are less numerically stable than Lie--Trotter--Suzuki formulas. Because they involve sums that nearly perfectly cancel, substantial roundoff errors can occur in their computation. Such errors can be mitigated by using high numerical precision or by summing the multi-product formula in a way that minimizes roundoff error.
An additional drawback is that linear combinations of unitary operators are not natural to implement on a quantum computer. Furthermore, our previous discussion alludes to a sign problem for the integrators: the more terms of the multi-product formula that have negative coefficients, the lower the success probability of the implementation of \thm{main:nonunitary}. This sign problem cannot be resolved completely because, as shown by Sheng~\cite{She89}, it is impossible to construct a high-order multi-product formula of the form in~\eq{mkdef} without using negative $C_q$. Nevertheless, we show that this problem is not fatal and that multi-product formulas can be used to surpass what is possible with previous simulations based on product formulas.
\subsection{Implementing Multi-Product Formulas Using Quantum Computers}
We now discuss how to implement multi-product formulas using quantum computers. The main obstacle is that the multi-product formulas most commonly used in classical algorithms have a value of $\kappa$ (as defined in \thm{main:nonunitary}) that approaches $1$ exponentially quickly as $k$ increases. Thus the probability of successfully implementing such multi-product formulas using \thm{main:nonunitary} is exponentially small.
Instead, we seek a multi-product formula $M_{k,\chi}$, with a large value of $\kappa$, such that $\norm{M_{k,\chi}(\lambda)-U(\lambda)} \in O(\lambda^{2(k+\chi)+1})$. Although many choices are possible, we take our multi-product formulas to be of the following form because they yield a large value of $\kappa$ while consisting of relatively few exponentials.
\begin{definition}\label{def:MPF}
Let $k \ge 0$ and $\chi \ge 1$ be integers, $\gamma$ a real number such that $e^{\gamma (k+1)}$ is an integer, and $S_\chi(\lambda)$ a symmetric product formula approximation to $U(\lambda)$ obeying $\norm{S_\chi(\lambda)-U(\lambda)} \in O(\lambda^{2\chi+1})$. Then for any $t\in\mathbb{R}$ we define the multi-product formula $M_{k,\chi}(t)$ as
\begin{equation}
M_{k,\chi}(t) \colonequals \sum_{q=1}^{k+1} C_q S_\chi(t/\ell_q)^{\ell_q}\label{eq:MkDef}
\end{equation}
where
\begin{equation}
\ell_q\colonequals\begin{cases}q&\textrm{if $q\le k$}\\
e^{\gamma (k+1)}& \textrm{if $q=k+1$}\end{cases}\label{eq:lqDef}
\end{equation}
and
\begin{equation}
C_q\colonequals \begin{cases}\frac{q^2}{q^2-e^{2\gamma (k+1)}} \prod_{j\ne q}^{k} \frac{q^2}{q^2-j^2}&\text{if $q\le k$}\\ \prod_{j=1}^{k} \frac{e^{2\gamma (k+1)}}{e^{2\gamma (k+1)}-j^2}& \text{if $q=k+1$}.\end{cases} \label{eq:cq}
\end{equation}
\end{definition}
We choose these values of $\ell_q$ because for sufficiently large $\gamma$ they guarantee that $C_{k+1}$ is much larger in absolute value than all other coefficients. This ensures a high success probability in \thm{main:nonunitary} because $\kappa \ge |C_{k+1}|/\sum_{q=1}^k |C_{q}|$ is
large if $|C_{k+1}|$ exceeds the sum of all other $|C_q|$.
The following lemma shows that $M_{k,\chi}$ is a higher-order integrator than $S_k$. Quantitative error bounds are proven in the next section.
\begin{lemma}\label{lem:MPFHigherOrder}
Let $M_{k,\chi}$ be a multi-product formula constructed according to \defn{MPF}. Then for $\lambda\ll 1$ we have
\begin{equation}
\norm{M_{k,\chi}(\lambda)-U(\lambda)} \in O(\lambda^{2(k+\chi)+1}).
\end{equation}
\end{lemma}
\begin{proof}
We follow the steps outlined in Chin's proof for the case where $\chi=1$~\cite{Chin10}. As shown in~\cite{BCR99}, a sufficient condition for a multi-product formula of the form $M_{k,\chi}(\lambda)=\sum_p C_p S_\chi(\lambda/\ell_p)^{\ell_p}$ to satisfy $\norm{U(\lambda)-M_{k,\chi}(\lambda)}\in O(\lambda^{2(k+\chi)+1})$ is for $C$ to satisfy the following matrix equation:
\begin{equation}
\begin{pmatrix}
1&1&1&\cdots &1\\
\ell_1^{-2\chi}&\ell_2^{-2\chi}&\ell_3^{-2\chi}&\cdots&\ell_{k+1}^{-2\chi}\\
\ell_1^{-2\chi-2}&\ell_2^{-2\chi-2}&\ell_3^{-2\chi-2}&\cdots&\ell_{k+1}^{-2\chi-2}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
\ell_1^{-2(k+\chi-1)}&\ell_2^{-2(k+\chi-1)}&\ell_3^{-2(k+\chi-1)}&\cdots&\ell_{k+1}^{-2(k+\chi-1)}
\end{pmatrix}
\begin{pmatrix}C_1\\mathbb{C}_2\\mathbb{C}_3\\\vdots\\mathbb{C}_{k+1}\end{pmatrix}
= \begin{pmatrix} 1\\0\\0\\\vdots\\0\end{pmatrix}.
\end{equation}
This ensures that the sum of all $C_q$ is $1$ and the coefficients of all the error terms in the multi-product formula are zero up to $O(\lambda^{2(k+\chi)-1})$. Denoting the matrix in the above equation by $V$, the vector $C$ is then the first column of $V^{-1}$. The matrix $V$ is a generalized Vandermonde matrix, which can be explicitly inverted~\cite{Moa03}. The entries of $C$ correspond to~\eq{cqgen}, which coincides with the values of $C_q$ given in~\eq{cq} for the values of $\ell_q$ in~\eq{lqDef}. Therefore the result of \cite{BCR99} shows that these values of $C_q$ extrapolate an $O(\lambda^{2\chi+1})$ symmetric product formula into an $O(\lambda^{2(k+\chi)+1})$ multi-product formula, as claimed.
\end{proof}
The following upper bound on the coefficients of the multi-product formula will be useful.
\begin{lemma}\label{lem:cqbound}
If $e^{2\gamma (k+1)} \ge 2k^2$, then for all $1 \le q < k+1$, the coefficients $C_q$ from \defn{MPF} satisfy
\begin{align}
|C_q|\le \sqrt{2} \, k^{3/2} e^{2k(1+\log(\eta)/2-\gamma)}
\label{eq:cqexpansion8}
\end{align}
where
\begin{equation}
\eta \colonequals \max_{\lambda\in [0,1)} \frac{\lambda^2}{(1+\lambda)^{1+\lambda}(1-\lambda)^{1-\lambda}} \approx 0.3081.
\label{eq:eta}
\end{equation}
\end{lemma}
\begin{proof}
For any $q<k+1$, \defn{MPF} gives
\begin{align}
C_q
&= \frac{q^2}{q^2-e^{2\gamma (k+1)}} \prod_{j\in \{1,\ldots,k\}\setminus q} \frac{q^2}{q^2-j^2} \nonumber \\
&= \frac{q^2}{q^2-e^{2\gamma (k+1)}} \prod_{j\in \{1,\ldots,k\}\setminus q} \frac{q^2}{(q+j)(q-j)} \nonumber \\
&= \frac{q^2}{q^2-e^{2\gamma (k+1)}} q^{2(k-1)} \frac{2q \cdot q!}{(q+k)!} \frac{(-1)^{k-q}}{(q-1)!(k-q)!} \nonumber \\
&= \frac{(-1)^{k-q}2q^{2k+2}}{(k+q)!(k-q)! (q^2-e^{2\gamma (k+1)})}\label{eq:cqexpansion2}.
\end{align}
Using $e^{2\gamma (k+1)} \ge 2k^2 \ge 2q^2$, we have the bound
\begin{align}
|C_q| &\le \frac{4q^{2k+2}e^{-2\gamma (k+1)}}{(k+q)!(k-q)!}\label{eq:cqexpansion3}.
\end{align}
We proceed by using the lower bound \cite{AbSt64}
\begin{equation}
n!\ge \sqrt{2\pi n}\,n^{n} e^{-(n+1/13)}\label{eq:factorialbd}.
\end{equation}
We will use this bound differently to estimate $|C_q|$ for the cases where $q<k$ and $q=k$. Using~\eq{factorialbd}, we
lower bound both factorial functions in~\eq{cqexpansion3} as follows:
\begin{align}
|C_q| &\le \frac{4q^{2k+2}e^{-2\gamma(k+1)+2k+2/13}}{2\pi \sqrt{k}(k+q)^{k+q}(k-q)^{k-q}}
\label{eq:cqexpansion4}.
\end{align}
Here we have used the fact that $\sqrt{(k+q)(k-q)}\ge \sqrt{k}$ for $q\le k-1$.
Now introduce a parameter $\lambda$ such that $q=k\lambda$. We simplify~\eq{cqexpansion4} using this substitution and
divide the numerator and denominator of~\eq{cqexpansion4} by $k^{2k}$ to find
\begin{align}
|C_q| &\le \frac{2k^{3/2}e^{2k(1-\gamma)-2\gamma+2/13}}{\pi}\left(\frac{\lambda^{2}}{(1+\lambda)^{1+\lambda}(1-\lambda)^{1-\lambda}} \right)^k\nonumber\\
&\le \frac{2k^{3/2}e^{2k(1-\gamma)-2\gamma+2/13}}{\pi}\eta^k
\label{eq:cqexpansion5}.
\end{align}
We then simplify this expression and find that
\begin{equation}
|C_q|\le\frac{2k^{3/2}e^{2k(1+\log(\eta)/2 -\gamma)}}{\pi e^{2\gamma-2/13}},
\end{equation}
which for $\gamma>0$ is bounded above by
\begin{align}
|C_q|\le {{k^{3/2}e^{2k(1+\log(\eta)/2-\gamma)}}}
< \sqrt{2} \, {{k^{3/2}e^{2k(1+\log(\eta)/2-\gamma)}}}.\label{eq:cqexpansion7}
\end{align}
Our bound for the case where $q=k$ is found using similar (but simpler) reasoning. We first substitute $q=k$ into~\eq{cqexpansion3},
use the lower bound in~\eq{factorialbd} to remove the factorial function, and simplify the result to find
\begin{equation}
|C_{k}|\le 2{k^{3/2}e^{2k(1-\log(2)-\gamma)}}/\sqrt{\pi}
< \sqrt{2} \, {k^{3/2}e^{2k(1-\log(2)-\gamma)}},
\end{equation}
which is less than the value in~\eq{cqexpansion7} because $-0.6931 \approx -\log 2 < \log (\eta)/2\approx -0.5886$.
Therefore \eq{cqexpansion8} holds for all $q\le k$ as required.
\end{proof}
The value of $\gamma$ used in $M_{k,\chi}$ can be chosen to minimize the probability of a subtraction error. The following lemma relates the value of $\kappa$ to $\gamma$, allowing us to use \thm{main:nonunitary} to find a value of $\gamma$ that ensures a sufficiently small probability of a subtraction error.
Henceforth we assume that $k\ge 1$. This is because $k=0$ corresponds to an ordinary product formula and the bounds that we prove for multi-product formulas can tightened by excluding this case, which is also already well analyzed~\cite{BACS07,Suz91}.
\begin{lemma}\label{lem:1}
Let $M_{k,\chi}$ be a multi-product formula as in \defn{MPF} and let $\kappa$ be defined for $M_{k,\chi}$ as in \thm{main:nonunitary}. Then if $2k^2 \le e^{2\gamma (k+1)}$ and $k>0$, we have
\begin{equation}
\kappa \ge 2^{-1/2}e^{-2k(1+\log(\eta)/2-\gamma)-\log(k^{5/2})}\label{eq:lem1:kappabound}
\end{equation}
where $\eta$ is defined in \eq{eta}.
\end{lemma}
\begin{proof}
According to \thm{main:nonunitary}, we have
$
\kappa = {\Sigma_+}/{\Sigma_-}
$,
where $\Sigma_+$ is the sum of all $C_q$ with positive coefficients and $\Sigma_-$ is the absolute value of the corresponding negative sum. A lower bound on $\kappa$ is therefore found by dividing a lower bound on $\Sigma_+$ by an upper bound on $\Sigma_-$.
Using the expression for $C_q$ from \defn{MPF}, we have
\begin{align}
C_{k+1}
= \prod_{j=1}^{k} \frac{e^{2\gamma (k+1)}}{e^{2\gamma (k+1)}-j^2}
= \prod_{j=1}^{k} \frac{1}{1-j^2e^{-2\gamma (k+1)}}.
\label{eq:lem5:rhs19}
\end{align}
The denominators on the right hand side of~\eq{lem5:rhs19} are positive under the assumption that $2k^2\le e^{2\gamma (k+1)}$, which ensures that $k^2e^{-2\gamma (k+1)}<1$ and simplifies the subsequent results of \cor{gamma}. We also have $C_{k+1}\ge 1$ because each denominator is less than $1$. Since $C_{k+1} > 0$, we have $\Sigma_+\ge C_{k+1} \ge 1$, and therefore
\begin{equation}
\kappa \ge \frac{1}{\Sigma_-}.\label{eq:lem2:sigmaminus}
\end{equation}
Next we provide an upper bound for $\Sigma_-$. Since $C_{k+1} > 0$, we have
\begin{equation}
\Sigma_- \le \sum_{q=1}^{k} |C_q|.\label{eq:lem5:sigma-bd}
\end{equation}
An upper bound for $\Sigma_-$ can then be obtained directly from upper bounds for $\max_{q<k+1}|C_q|$.
Using \lem{cqbound}, we have
\begin{equation}
\Sigma_-\le \sum_{q=1}^{k} {{\sqrt{2}\,k^{3/2}e^{2k(1+\log(\eta)/2-\gamma)}}}< \sqrt{2}\,k^{5/2}e^{2k(1+\log(\eta)/2-\gamma)}.\label{eq:lem2:sumabsCq}
\end{equation}
We substitute this inequality into~\eq{lem2:sigmaminus} to obtain
\begin{equation}
\kappa\ge 2^{-1/2}k^{-5/2}e^{-2k(1+\log(\eta)/2-\gamma)}
\end{equation}
as claimed.
\end{proof}
In fact, the bound of \lem{1} is nearly tight, in that $\kappa$ decays exponentially with $k$ if $\gamma < 1+\log(\eta)/2$, as we discuss in more detail below. Thus the success probability of our algorithm decays exponentially if $\gamma$ is too small. Consequently, we will find that unlike the classical case, our quantum algorithm does not provide poly-logarithmic error scaling.
The following corollary provides a sufficient value of $\gamma$ to ensure that the probability of our algorithm making a subtraction error is small.
\begin{corollary}\label{cor:gamma}
Let $M_{k,\chi}$ be a multi-product formula as in \defn{MPF}, let $\kappa$ be defined for $M_{k,\chi}$ as in \thm{main:nonunitary}, and let $\delta\le 1$. Furthermore, suppose $k> 0$ and
$$
\gamma\ge 1+\frac{\log(\eta)}{2}+\frac{1}{2k} \log\left(\frac{(2k)^{\frac{5}{2}}}{\delta}\right).
$$
Then $P_-\le \delta$ and $k^2e^{-2\gamma (k+1)}\le 1/2$.
\end{corollary}
\begin{proof}
Without loss of generality, we can take $\kappa\ge 1$ for our subtraction step because $\sum_q C_q =1$ and hence $\kappa=\Sigma_+/\Sigma_- \ge 1$. This observation and
the result of \thm{main:nonunitary} imply that the probability of failing to perform the subtraction step in our implementation of $M_{k,\chi}$ satisfies
\begin{equation}
P_-\le 4/\kappa.\label{eq:P-bd}
\end{equation}
Eq.~\eq{P-bd} and the bounds on $\kappa$ in \lem{1} give $P_-\le \delta$ provided
\begin{equation}
4\sqrt{2}\,e^{2k(1+\log(\eta)/2-\gamma)+\log(k^{5/2})}\le \delta.\label{eq:lemgamma:errbd}
\end{equation}
We obtain our sufficient value of $\gamma$ by solving~\eq{lemgamma:errbd} for $\gamma$, giving
\begin{equation}
\gamma\ge 1+\log(\eta)/2+\frac{1}{2k} \log\left(\frac{(2k)^{\frac{5}{2}}}{\delta}\right).\label{eq:gammabd}
\end{equation}
Eq.~\eq{gammabd} also implies that $k^2e^{-2\gamma (k+1)}\le 1/2$. For $0<\delta\le 1$ and $\gamma$ saturating~\eq{gammabd}, it is easy to see that $k^2 e^{-2\gamma (k+1)}$ is a monotonically decreasing function of $k$, and therefore achieves its maximum value at $k=1$, the smallest possible value of $k$. We find that $k^2e^{-2\gamma (k+1)}<1/2$ at $k=1$ for $\delta=1$, and therefore the condition $k^2e^{-2\gamma (k+1)}\le 1/2$ is automatically implied by our choice of $\gamma$ in~\eq{gammabd}.
\end{proof}
\begin{figure}
\caption{Scaling of $\kappa$ with $k$ for three values of $\gamma$ centered around $\gamma_c \colonequals 1+\log(\eta)/2$. The data show that $\kappa$ approaches $1$ polynomially quickly if $\gamma=\gamma_c$, whereas a slight increase in $\gamma$ causes $\kappa$ to grow exponentially and a slight decrease causes $\kappa$ to converge to $1$ exponentially with $k$.
}
\label{fig:etadata}
\end{figure}
The value of $\gamma$ given by \cor{gamma} is tight up to $O(k^{-1} \log k)$. This is illustrated in \fig{etadata}, which shows $\kappa$ as a function of $k$ for $\gamma=\gamma_c\colonequals 1+\log(\eta)/2$ and two slightly perturbed values of $\gamma$ centered around $\gamma_c$. We see that small deviations away from $\gamma=\gamma_c$ lead to either exponential growth of $\kappa$ or exponential convergence of $\kappa$ to $1$. Thus our lower bound for $\gamma$ cannot be significantly improved.
\section{Analysis of Simulation and Errors in Multi-product Formulas\label{sec:mpferror}}
The results of the previous section show how $\kappa$ scales with the number of terms in the multi-product formula used in the simulation. We now expand on these results by bounding the approximation errors incurred by using the multi-product formula. We also present an error correction method to ensure that our implementation fails at most a constant fraction of the time. Our error bounds are established as follows. First, we estimate the error in multi-product formulas that utilize high-order Lie--Trotter--Suzuki formulas. Second, we discuss how these multi-product formulas are implemented. Finally, we estimate the inversion error for the resulting multi-product formulas and bound the average error resulting from a given step.
\begin{lemma}\label{lem:MPFSuzError}
Let $M_{k,k}(\lambda)$ satisfy \defn{MPF} for evolution time $\lambda\ge 0$, let $|C_q|\le 2$ for all $q=1,\ldots,k+1$, and let $h\lambda\le \frac{3\log(2)}{4mk(5/3)^{k-1}}$. Then
\begin{equation}
\norm{U(\lambda)-M_{k,k}(\lambda)} \le (2m(5/3)^{k-1}h \lambda)^{4k+1}.
\end{equation}
\end{lemma}
The proof of \lem{MPFSuzError} requires upper bounds on the remainder terms of Taylor series expansions. Let $\mathbb{R}em_\ell(f)$ denote the remainder term of the Taylor series of a function $f$ truncated at order $\ell$. The following lemma bounds the remainder term for an operator exponential.
\begin{lemma}\label{lem:Rl}
Let $a_j \in \mathbb{R}$ for $j=1,\ldots,M$ and suppose $\norm{H_j} \le h$. Then
\begin{equation}
\Norm{\mathbb{R}em_{\ell}\left(\prod_{j=1}^M e^{-i a_j H_j t}\right)} \le \frac{(\sum_{j=1}^M |a_j| h t)^{\ell+1}}{(\ell+1)!}\exp\left(\sum_{q=1}^M |a_q| h t\right).
\end{equation}
\end{lemma}
\begin{proof}
Using the triangle inequality and sub-multiplicativity of the norm,
we find
\begin{align}
\Norm{\mathbb{R}em_{\ell}\left(\prod_{j=1}^M e^{-i a_j H_j t}\right)}
&= \Norm{\mathbb{R}em_{\ell}\left(\prod_{j=1}^M \left( \sum_{p=0}^\infty (-i a_j H_j t)^p/p!\right)\right)} \\
&\le \mathbb{R}em_{\ell}\left(\prod_{j=1}^M \left( \sum_{p=0}^\infty (|a_j| h t)^p/p!\right)\right) \nonumber \\
&= \mathbb{R}em_{\ell}\left(\exp\left(\sum_{j=1}^M|a_j| h t \right)\right) \nonumber \\
&=\sum_{p=\ell+1}^\infty \frac{\left(\sum_{j=1}^M|a_j| h t \right)^p}{p!}\nonumber\\
&\le\frac{\left(\sum_{j=1}^M|a_j| h t \right)^{\ell+1}}{(\ell+1)!}\exp\left(\sum_{j=1}^M|a_j| h t \right)
\end{align}
as claimed.
\end{proof}
Now we are ready to prove \lem{MPFSuzError}.
\begin{proofof}{\lem{MPFSuzError}}
\lem{MPFHigherOrder} implies that
\begin{equation}
\norm{M_{k,k}(\lambda)-U(\lambda)} \in O(\lambda^{4k+1}),
\end{equation}
so the approximation error is entirely determined by the terms of order $\lambda^{4k+1}$ in $M_{k,k}$ and $U$. If we remove the terms in the Taylor series of $M_{k,k}$ and $U$ that cancel, the remainders, $\mathbb{R}em_{4k}(M_{k,k}(\lambda))$ and $\mathbb{R}em_{4k}(U(\lambda))$, determine the error via
\begin{align}
\norm{M_{k,k}(\lambda)-U(\lambda)}
&=\norm{\mathbb{R}em_{4k}(M_{k,k}(\lambda))-\mathbb{R}em_{4k}(U(\lambda))} \nonumber \\
&\le \norm{\mathbb{R}em_{4k}(M_{k,k}(\lambda))}+\norm{\mathbb{R}em_{4k}(U(\lambda))}.
\label{eq:lem8:triangle}
\end{align}
\lem{Rl} implies that
\begin{align}
\norm{\mathbb{R}em_{4k}(U(\lambda))}
&\le \frac{(mh\lambda)^{4k+1}}{(4k+1)!}e^{mh\lambda} \nonumber \\
&< \left(\frac{4}{3}m (5/3)^{k-1} h \lambda\right)^{4k+1}.\label{eq:R4kU}
\end{align}
The second inequality in~\eq{R4kU} follows from the assumption that $h\lambda\le \frac{3\log(2)}{4mk(5/3)^{k-1}}$, which implies that $\exp(mh\lambda)/(4k+1)!\le 2^{3/4}/5!< 1$.
The definition of $M_{k,\chi}$ implies
\begin{align}
\norm{\mathbb{R}em_{4k}(M_{k,k}(\lambda))}
&\le \sum_{q=1}^{k+1} |C_q|\Norm{\mathbb{R}em_{4k}(S_k(\lambda/\ell_q)^{\ell_q})}.
\label{eq:lem8:r4kMkk}
\end{align}
Thus we upper bound $\norm{\mathbb{R}em_{4k}(S_k(\lambda/p)^{p})}$.
This bound follows similar logic to the bound for $U(\lambda)$, but the calculation is slightly more complicated because $S_k$ is the product of many exponentials. Specifically,
\begin{equation}
S_k(\lambda/p)=\prod_{\ell=1}^{2m5^{k-1}} e^{-iH_{j_\ell} q_{k,\ell} \lambda/p},
\end{equation}
where $q_{k,\ell}$ is the ratio between $\lambda/p$ and the duration of the $\ell^{\rm th}$ exponential in $S_k$.
\lem{Rl} gives
\begin{equation}
\norm{\mathbb{R}em_{4k}(S_k(\lambda/p)^p)} \le \frac{(2m5^{k-1} \max q_{k,\ell} h \lambda)^{4k+1}}{(4k+1)!}e^{2m5^{k-1} \max q_{k,\ell} h \lambda}.\label{eq:lem8:R4kbd2}
\end{equation}
Using the upper bound $q_{k,\ell}\le {2k}/{3^k}$ from Appendix A of~\cite{WBHS10}, we have
\begin{equation}
\norm{\mathbb{R}em_{4k}(S_k(\lambda/p)^p)} \le \frac{(\frac{4}{3}mk (5/3)^{k-1} h \lambda)^{4k+1}}{(4k+1)!} e^{\frac{4}{3}mk (5/3)^{k-1} h \lambda}.\label{eq:lem8:51}
\end{equation}
This bound can be simplified using $(4k+1)!\ge k^{4k+1}5!$ for $k\ge 1$ (a consequence of~\eq{factorialbd}) and the hypothesis that $\frac{4}{3}mk (5/3)^{k-1} h \lambda\le \log(2)$, giving
\begin{equation}
\norm{\mathbb{R}em_{4k}(S_k(\lambda/p)^p)} \le \frac{2}{5!}\left(\frac{4}{3}m (5/3)^{k-1} h \lambda\right)^{4k+1}.\label{eq:Skremainder}
\end{equation}
Using this result and the assumption that $|C_q|\le 2$ in \eq{lem8:r4kMkk} gives
\begin{equation}
\norm{\mathbb{R}em_{4k}(M_{k,k}(\lambda))}
\le \frac{8(k+1)}{5!}\left(\frac{4}{3}m (5/3)^{k-1} h \lambda\right)^{4k+1}\label{eq:lem8:r4kMkk2}.
\end{equation}
Combining~\eq{lem8:triangle},~\eq{R4kU}, and~\eq{lem8:r4kMkk2} gives
\begin{align}
\norm{U(\lambda)-M_{k,k}(\lambda)} &\le \left(1+\frac{8(k+1)}{5!}\right)\left(\frac{4}{3}m (5/3)^{k-1} h \lambda\right)^{4k+1}\nonumber\\
&\le (2m (5/3)^{k-1} h \lambda)^{4k+1},
\label{eq:lem8:final}
\end{align}
proving the lemma.
\end{proofof}
A useful consequence of performing $M_{k,k}$ using a single subtraction step is that if a subtraction error occurs, the simulator performs the operation
\begin{equation}
E_k(\lambda) \colon \ket{\psi} \mapsto \frac{\sum_{q} |C_{q}| S_k(\lambda/\ell_q)^{\ell_q}\ket{\psi}}{\norm{\sum_{q} |C_q| S_{k}(\lambda/\ell_q)^{\ell_q}\ket{\psi}}}.
\label{eq:ek}
\end{equation}
This error operation can be approximately corrected because, as Blanes et al.\ proved~\cite[Theorem 1]{BCR99}, \begin{equation}E_k(-\lambda)E_k(\lambda)=\openone+O(\lambda^{4k+2}).\label{eq:blanesinv}\end{equation}
Since the coefficients $|C_q|$ are all positive, \thm{main:nonunitary} shows that the approximate correction operation $E_k(-\lambda)$ can be performed with success probability close to $1$ provided $\Delta$ is small. The following lemma states that the error incurred by approximately correcting subtraction errors is at most equal to our upper bound for the approximation error for $M_{k,k}(\lambda)$.
\begin{lemma}\label{lem:ekinv}
Let $E_k(\lambda)$ act as in \eq{ek}, where $C_q$ and $\ell_q$ are given in \defn{MPF}. If $2mk(5/3)^{k-1}h\lambda\le 1/2$, then \rm
\begin{equation}
\max_{\ket{\psi}}\norm{\left(\openone -E_k(-\lambda)E_k(\lambda)\right)\ket{\psi}}
\le \left(2mk(5/3)^{k-1}h \lambda\right)^{4k+2}.
\end{equation}
\end{lemma}
\begin{proof}
By \eq{ek} and \eq{blanesinv},
\begin{align}
\max_{\ket{\psi}}\norm{\left(\openone -E_k(-\lambda)E_k(\lambda)\right)\ket{\psi}}
&= \max_{\ket{\psi}}\norm{\mathbb{R}em_{4k+1}(E_k(-\lambda)E_k(\lambda)\ket{\psi})}\nonumber\\
&\le \frac{\Norm{\mathbb{R}em_{4k+1}\left(\sum_p \sum_q |C_p| |C_q| S_k(-\lambda/p)^pS_k(\lambda/q)^q\right)}}{\min_{\ket{\phi}}\norm{\sum_p |C_p| S_k(\lambda/\ell_p)^{\ell_p}\ket{\phi}}^2}.\label{eq:ekinv:sk1}
\end{align}
We then follow the same reasoning used in the proof of \lem{Rl}.
By the triangle inequality, the norm of the remainder of a Taylor series is upper bounded by the sums of the norms of the individual terms in the remainder. This can be bounded by replacing the exponent of each exponential in $S_k$ with its norm. We use similar reasoning to that used in~\eq{lem8:51} to find
\begin{equation}
\Norm{\mathbb{R}em_{4k+1}\left(S_k(-\lambda/p)^pS_k(\lambda/q)^q\right)} \le \mathbb{R}em_{4k+1}\left( e^{\frac{8}{3}mk (5/3)^{k-1} h \lambda} \right),
\end{equation}
so the numerator of \eq{ekinv:sk1} satisfies
\begin{align}
\Norm{\mathbb{R}em_{4k+1}\left(\sum_p \sum_q |C_p| |C_q| S_k(-\lambda/p)^pS_k(\lambda/q)^q\right)}
&\le \sum_p \sum_q |C_p| |C_q| \mathbb{R}em_{4k+1}\left( e^{\frac{8}{3}mk (5/3)^{k-1} h \lambda}\right) \nonumber \\
&\le \norm{C}_1^2 \frac{ \left(\tfrac{8}{3}mk(5/3)^{k-1}h\lambda\right)^{4k+2}e^{\frac{8}{3}mk(5/3)^{k-1}h\lambda}}{(4k+2)!}
\end{align}
where $C$ is a vector with entries $C_p$, so $\norm{C}_1 = \sum_p |C_p|$.
Our assumptions imply $\frac{8}{3}mk(5/3)^{k-1}h\lambda\le 2/3 < \log(2)$, so
\begin{align}
\Norm{\mathbb{R}em_{4k+1}\left(\sum_p \sum_q |C_p| |C_q| S_k(-\lambda/p)^pS_k(\lambda/q)^q\right)}
&\le \norm{C}_1^2 \frac{2 \left(\frac{8}{3}mk(5/3)^{k-1}h\lambda\right)^{4k+2}}{(4k+2)!} \nonumber \\
&\le \norm{C}_1^2 \frac{2 \left(\frac{2e}{3}m(5/3)^{k-1}h\lambda\right)^{4k+2}}{\sqrt{12\pi}e^{25/26}}
\label{eq:ekinv:sk3},
\end{align}
where the last inequality results from using Stirling's approximation as given in~\eq{factorialbd}.
The denominator of \eq{ekinv:sk1} can be lower bounded as follows:
\begin{align}
\min_{\ket{\phi}}{\Norm{\sum_p |C_p| S_k(\lambda/\ell_p)^{\ell_p}\ket{\phi}}}
&= \min_{\ket{\phi}}{\Norm{\sum_p |C_p| (e^{-iHt}-(e^{-iHt}-S_k(\lambda/\ell_p)^{\ell_p}))\ket{\phi}}}\nonumber\\
&\ge \norm{C}_1 \left(1-\max_p \norm{e^{-iHt}-S_k(\lambda/\ell_p)^{\ell_p}}\right)\nonumber\\
&= \norm{C}_1 (1-\norm{e^{-iHt}-S_k(\lambda)}).
\end{align}
Since $2mk(5/3)^{k-1}h\lambda \le 1/2 < 3/(4\sqrt{2})$, Theorem 3 of~\cite{WBHS10} implies that $\norm{e^{-iHt}-S_k(\lambda)} \le 2(2 mk(5/3)^{k-1}h\lambda)^{2k+1}$.
Using $2mk(5/3)^{k-1}h\lambda\le 1/2$ and $k\ge 1$ we have that
\begin{equation}
\norm{e^{-iHt}-S_k(\lambda)} \le \frac{1}{4},
\end{equation}
implying
\begin{equation}
\min_{\ket{\phi}}
\Norm{\sum_p |C_p| S_k(\lambda/\ell_p)^{\ell_p} \ket{\phi}}
\ge \frac{3}{4} \norm{C}_1.
\end{equation}
Combining this with our upper bound on the numerator gives
\begin{align}
\max_{\ket{\psi}}
\norm{(\openone -E_k(-\lambda)E_k(\lambda))\ket{\psi}}
&\le \frac{2(4/3)^2}{\sqrt{12\pi}e^{25/26}}\left(\frac{2e}{3}m(5/3)^{k-1}h\lambda\right)^{4k+2} \nonumber \\
&\le (2m(5/3)^{k-1}h\lambda)^{4k+2}\label{eq:ekinv:sk4}
\end{align}
as claimed.
\end{proof}
We simulate $U(t)$ using $r$ iterations of $M_{k,k}(t/r)$ for some sufficiently large $r$.
Our next step is to combine \lem{MPFSuzError} and \lem{ekinv} to find upper bounds on $r$ such that $U(t)$ is approximated to within some fixed error.
We take $\delta=1/2$, i.e., we accept a maximum failure probability of $1/2$ for each multi-product formula. We then sum the cumulative errors and use the Chernoff bound to show that, with high probability, the simulation error is at most $\epsilon$. These results are summarized in the following lemma.
\begin{lemma}\label{lem:r}
Let $M_{k,k}$ be a multi-product formula given by \defn{MPF} with $|C_q|\le 2$ for all $q\le k+1$. Let $\gamma$ be chosen as in \cor{gamma} with $\delta=1/2$ and let the integer $r$ satisfy
\begin{equation}
r\ge \frac{(2m (5/3)^{k-1}h t)^{1+1/4k}}{(\epsilon/5)^{1/4k}}\label{eq:lem11:rbound}
\end{equation}
for $\epsilon\le m h tk^{-4k}$.
Then a quantum computer can approximately implement $U(t)$ as $M_{k,k}(t/r)^r$ with error at most $\epsilon$ and with probability at least $1-e^{-r/13}$, assuming that no addition errors occur during the simulation, while utilizing no more than $5r$ subtraction attempts and approximate inversions.
\end{lemma}
\begin{proof}
First we bound the probability of successfully performing the subtraction steps given a fixed maximum number of attempts.
We simplify our analysis by assuming that the simulation uses exactly $3r$ subtractions, corresponding to the worst-case scenario in which $2r$ inversions are used. We want to find the probability that a randomly chosen sequence of subtractions contains at least $r$ successes, correctly implementing the multi-product formula. The probability that a sequence is unsuccessful is exponentially small in $r$ because for $\delta=1/2$, the mean number of failures is $\mu = 3r/2$, which is substantially smaller than our tolerance of $2r$ failures. By the Chernoff bound, the probability of having more than $2r$ failures satisfies
\begin{equation}
\Pr(X>2r)\le e^{-\mu((1+\alpha)\log(1+\alpha)-\alpha)}<e^{-r/13},
\end{equation}
where $1+\alpha = 2r/\mu = 4/3$.
If we attempt the subtraction steps in our protocol $3r$ times and fail $2r$ times, then $5r$ subtractions and approximate inversions must be performed, because every failure requires an approximate inversion. We bound the resulting error using \lem{MPFSuzError}, \lem{ekinv}, and the subadditivity of errors. These lemmas apply because the requirement $\frac{4}{3}mk(5/3)^{k-1}ht/r\le\log(2)$ is implied by our choice of $r$ and the assumption $\epsilon\le mhtk^{-4k}$. By this argument, the simulation error satisfies
\begin{equation}
\norm{\tilde M_{k,k}(t/r)^r-U(t)} \le 5r(2m(5/3)^{k-1}ht/r)^{4k+1}\label{eq:lem11:errorbound}
\end{equation}
with probability at least $1-e^{-r/13}$, where $\tilde M_{k,k}(t/r)$ denotes the operation performed by our non-deterministic algorithm for the multi-product formula $M_{k,k}$.
The assumption $\epsilon\le m h t k^{-4k}$ and the value of $r$ from~\eq{lem11:rbound} imply that $\norm{\tilde M_{k,k}(t/r)^r-U(t)} \le \epsilon$ as required.
\end{proof}
\lem{r} assumes an error tolerance of at most $mhtk^{-4k}$, which may appear to be very small. However, our ultimate simulation scheme has $k\in O(\sqrt{\log (mht/\epsilon})$, so in fact the error tolerance is modest.
Now we are ready to prove a key lemma that provides bounds on the number of exponentials used by the simulation in the realistic scenario where both addition and subtraction errors may occur. Our main result, \thm{mainresult}, follows as a simple consequence.
\begin{lemma}\label{lem:nexp}
Let $M_{k,k}$ be a multi-product formula for $H=\sum_{j=1}^m H_j$ as in \defn{MPF}, with $k\ge 1$. Let $\tilde{M}_{k,k}$ be the implementation of $M_{k,k}$ described above. Let $\epsilon$ be a desired error tolerance and let $\beta$ be a desired upper bound on the failure probability of the algorithm. Then there is a simulation of $U(t) = e^{-iHt}$ that has error at most $\epsilon$ with probability at least $1-\beta$ using
\begin{equation}
N_{\exp}\le 1000m5^{k-1}k^{9/4}e^{(1+\log(\eta)/2)k}r \label{eq:thmmain:nexp}
\end{equation}
exponentials of the form $e^{-iH_j t}$, where
\begin{enumerate}
\item $\gamma=\frac{1}{k}\log\lceil \exp([ 1+\log(\eta)/2+\frac{1}{2k}\log(2(2k)^{5/2})]k)\rceil$,
\item $\tilde\epsilon= \min(1,\epsilon, \beta, m h t k^{-4k})$,
\item $r= \left\lceil\max \left\{\frac{(4m(5/3)^{k-1}h t)^{1+1/4k}}{(\tilde\epsilon/5)^{1/4k}},13\log (2/\beta) \right\}\right\rceil$.
\end{enumerate}
\end{lemma}
\begin{proof}
We approximate $U(t)$ as a product of $r$ multi-product formulas, each implemented using the operation $\tilde M_{k,k}$.
We can suppose that the sequence of $r$ multi-product formulas is implemented using at most $5r$ subtraction and inversion steps according to~\lem{r}. As $k$ unitary operations are combined in each step, we must implement a total of $5r k$ unitary operations. \defn{MPF} implies that each of these unitaries is composed of at most $e^{\gamma (k+1)}$ Suzuki integrators $U_q$. Each $U_q$ is a product of $2m5^{k-1}$ exponentials of elements from $\{H_j\}$. Thus, if the algorithm succeeds after performing this maximum number of subtractions and inversions, we have
\begin{align}
N_{\exp}&\le 10m5^{k-1}ke^{\gamma (k+1)}r\nonumber\\
&\le 10m 5^{k-1}k(2^{11/4}e^2k^{5/4}e^{(1+\log(\eta)/2)k}+1)r\nonumber\\
&< 1000 m 5^{k-1}k^{9/4}e^{(1+\log(\eta)/2)k}r
\end{align}
where we have used $\gamma \le 2$ and $e^{\gamma k}\le 2(2^{7/4}k^{5/4}e^{k(1+\log(\eta)/2)})$.
Equation \eq{thmmain:nexp} then follows by substituting the value of $r$ assumed by the lemma, which guarantees that the simulation error is less than $\epsilon$ when the simulation is successful because it exceeds the value of $r$ from \lem{r}. We choose $r$ to be larger because it simplifies our results and guarantees that the probability of an addition error is at most
$\tilde \epsilon/2$.
\lem{r} requires $C_q\le 2$, which we have not explicitly assumed. Substituting the above value of $\gamma$ into the upper bound for $\Sigma_-$ in~\eq{lem2:sumabsCq} shows that $\sum_{q=1}^{k}|C_q|\le 1$, so $|C_q|\le 1$ for all $q=1,\ldots,k$. Our multi-product formula satisfies $\sum_q C_q=1$, so our choice of $\gamma$ ensures $C_{k+1}\le 2$. Thus $C_q\le 2$ for all $q$.
\lem{r} implies that the probability of the simulation failing due to too many subtraction errors is at most $e^{-r/13}$. However, it does not address the possibility of the algorithm failing due to addition errors. There are at most $5r$ addition steps, so by \thm{main:nonunitary} and the union bound, the probability of an addition error is at most
\begin{equation}
5r P_+\le \frac{5\Delta^2 kr}{4}.\label{eq:mainthm:Delta1}
\end{equation}
By the definition of $\Delta$ in \thm{main:nonunitary},
\begin{align}
\Delta&= \Norm{\max_{q,q'}\mathbb{R}em_{2k}\left(S_k(t/k_qr)^{k_q}-S_k(t/k_{q'}r)^{k_{q'}}\right) }.
\end{align}
Using \lem{Rl} (similarly as in \eq{lem8:51}), the triangle inequality, and $\frac{4}{3} mk(5/3)^{k-1} h t/r\le \log(2)$, we have
\begin{align}
\Delta&\le 4\frac{(\frac{4}{3}mk (5/3)^{k-1} h t/r)^{2k+1}}{(2k+1)!}.\label{eq:mainthm:Delta2}
\end{align}
Substituting the assumed value of $r$ into~\eq{mainthm:Delta2} gives
\begin{equation}
\Delta\le \frac{20k^{2k+1}}{(2k+1)!3^{2k+1}}\left(\frac{\tilde \epsilon }{4m(5/3)^{k-1}ht} \right)^{\frac{1}{2}+\frac{1}{4k}}.\label{eq:mainthm:Delta3}
\end{equation}
Substituting this bound into~\eq{mainthm:Delta1} and using~\eq{factorialbd}, we find that the total probability of an addition error is at most
\begin{align}
\frac{20k^{4k+3}\tilde \epsilon}{((2k+1)!)^23^{4k+2}}\left(\frac{\tilde \epsilon}{4mk(5/3)^{k-1}ht} \right)^{1/4k}&\le \frac{20k\tilde \epsilon}{6\pi (6/e)^{4k+2}e^{-2/13}}\left(\frac{\tilde \epsilon}{4mk(5/3)^{k-1}ht} \right)^{1/4k}\label{eq:mainthm:Delta4}.
\end{align}
Since $\tilde \epsilon\le mhtk^{-4k}$, this implies
\begin{equation}
P_+\le \frac{20\tilde \epsilon}{6\pi (6/e)^{4k+2}e^{-2/13}}\left(\frac{1}{4k(5/3)^{k-1}} \right)^{1/4k} < \frac{\tilde \epsilon}{130}< \frac{\tilde \epsilon}{2}.
\end{equation}
Here the second inequality follows from the first by substituting $k=1$, since the middle expression is a monotonically decreasing function of $k$.
The total probability of success $P_s$ satisfies
\begin{equation}
P_s \ge 1-\frac{\tilde \epsilon}{2}-e^{-r/13}.\label{eq:mainthm:psbound1}
\end{equation}
Using $r\ge 13\log(2/\beta)$ and $\tilde \epsilon\le \beta$, we find $P_s \ge 1-\beta$ as claimed.
\end{proof}
As in~\cite{BACS07}, there is a tradeoff between the exponential improvement in the accuracy of the formula and the exponential growth of $M_{k,k}$ with $k$. To see this, note that apart from terms that are bounded above by a constant function of $k$, $N_{\exp}$ is the product of two terms:
\begin{align}
N_{\exp}&\in O\left(\left[m^2k^{9/4} e^{(1+\log(\eta)/2 +\log(25/3))k}\right]\left[m h t/\tilde\epsilon \right]^{1/4k} \right)\label{eq:scale:nexpscale}\\
&\in O\left(\left[m^2k^{9/4} e^{2.54k}\right]\left[\frac{m h t}{\min(\epsilon,\beta)} \right]^{1/4k} \right).
\end{align}
Here we have not included the $O(\log(1/\beta))$ term from $r$ because $\log(1/\beta)\in O(1/\beta)^{1/4k}$.
For comparison, the results of~\cite{BACS07} and~\cite{WBHS10} have the following complexities:
\begin{align}
N_{\exp}&\in O\left(\left[m^2e^{3.22 k}\right]\left[m h t/\epsilon \right]^{1/2k} \right),\\
N_{\exp}&\in O\left(\left[m^2k e^{2.13 k}\right]\left[m h t/\epsilon \right]^{1/2k} \right),
\end{align}
respectively. The tradeoff between accuracy and complexity as a function of $k$ is more favorable in our setting than in either of these approaches.
Finally, we give a detailed analysis of the tradeoff.
\begin{proofof}{\thm{mainresult}}
Neglecting polynomially large contributions in~\eq{scale:nexpscale}, we see that the dominant part of~\eq{scale:nexpscale} is
\begin{equation}
e^{(1+\log(\eta)/2+\log(25/3))k+\frac{1}{4k}\log(mh t/\tilde\epsilon)}.\label{eq:nexpdominant}
\end{equation}
It is natural to choose $k$ to minimize~\eq{nexpdominant}. The minimum is achieved by taking $k=k_{\rm opt}$, where
\begin{equation}
k_{\rm opt}=\left\lceil\frac{1}{2}\sqrt{\frac{\log(mh t/\tilde\epsilon)}{1+\log(\eta)/2+\log(25/3)}}~\right\rceil\approx
0.3142 \sqrt{\log(mh t/\tilde\epsilon)}.
\end{equation}
Using this $k$, we find that
\begin{equation}
N_{\exp}\in O\left(k_{\rm opt}^{9/4}m^2h te^{1.6\sqrt{\log(mh t/\tilde\epsilon)}}\right).\label{eq:scale:MPF}
\end{equation}
We have $\tilde \epsilon = \min(1,\epsilon,\beta,mhtk^{-4k})$, so $\tilde\epsilon$ depends implicitly on $k$. However, we now show that this term can be neglected in the limit of large $mht/\tilde \epsilon$. In this limit, we have
\begin{equation}
\frac{k_{\rm opt}^{4k_{\rm opt}}}{mht}\in O\left(\frac{{\log(mht/\tilde \epsilon)}^{0.16\sqrt{\log(mht/\tilde \epsilon)}}}{mht} \right)\subset o(1/\tilde \epsilon).
\end{equation}
The above follows since $\lim_{x\rightarrow \infty} \log(x)^{c\sqrt{\log(x)}}/x=0$ for any $c>0$.
Correspondingly, $mhtk_{\rm opt}^{-4k_{\rm opt}}\in \omega (\tilde\epsilon)$. We therefore conclude that this term can be neglected asymptotically and that $\tilde\epsilon\in \Omega(\epsilon)$, so we can replace $\tilde \epsilon$ with $\epsilon$ asymptotically.
The result then follows by substituting $k_{\mathrm{opt}}$ into~\eq{scale:MPF} and dropping all poly-logarithmic factors.
\end{proofof}
The parameters $\epsilon$ and $\beta$ can be decreased to improve the simulation fidelity and success probability, respectively. The cost of such an improvement is relatively low, although it is not poly-logarithmic in $1/\epsilon$ and $1/\beta$. If the initial state of a simulation can be cheaply prepared and the result of the computation can be easily checked, it may be preferable to use large values of $\epsilon$ and $\beta$ and repeat the simulation an appropriate number of times. Then the Chernoff bound implies that a logarithmic number of iterations is sufficient to achieve success with high probability. However, this may not be possible if the simulation is used as a subroutine in a larger algorithm (e.g., as in~\cite{HHL09}).
\section{Conclusions}\label{sec:conclusions}
We have presented a new approach to quantum simulation that implements Hamiltonian dynamics using linear combinations, rather than products, of unitary operators. The resulting simulation gives better scaling with the simulation error $\epsilon$ than any previously known algorithm and scales more favorably with all parameters than simulation methods based on product formulas.
Aside from the quantitative improvement to simulation accuracy, this work provides a new way to address the errors that occur in quantum algorithms. Specifically, our work shows that approximation errors can be reduced by coherently averaging the results of different approximations. It is common to perform such averages in classical numerical analysis, and we hope that the techniques presented here may have applications beyond quantum simulation.
It remains an open problem to further improve the performance of quantum simulation as a function of the error tolerance $\epsilon$. In particular, we would like to determine whether there is a simulation of $n$-qubit Hamiltonians with complexity $\poly(n, \log \frac{1}{\epsilon})$. Classical simulation algorithms based on multi-product formulas achieve scaling polynomial in $\log\frac{1}{\epsilon}$, but they are necessarily inefficient as a function of the system size. Our algorithms fail to provide such favorable scaling in $\frac{1}{\epsilon}$ due to the sign problem discussed in \sec{mpf}. A possible resolution to this problem could be attained by finding multi-product formulas with only positive coefficients and backward timesteps. Such formulas are not forbidden by Sheng's Theorem~\cite{She89} since that result only applies to multi-product formulas that are restricted to use forward timesteps~\cite{Suz91}. New approximation-building methods might use backward timesteps to give multi-product formulas that are easier to implement with our techniques. Conversely, a proof that Hamiltonian simulation with $\poly(n,\log \frac{1}{\epsilon})$ elementary operations is impossible would also show that Lie--Trotter--Suzuki and multi-product formulas with positive coefficients cannot be constructed with polynomially many exponentials, answering an open question in numerical analysis.
Finally, we have focused on the case of time-independent Hamiltonian evolution. One can consider multi-product formulas that are adapted to handle time-dependent evolution, as discussed in~\cite{CG11}. It is nontrivial to use such formulas to generalize our results to the time-dependent case because of difficulties that arise when the Hamiltonian is not a sufficiently smooth function of time. Further investigation of these issues could lead to developments in numerical analysis as well as quantum computing.
\begin{acknowledgments}
This work was supported in part by MITACS, NSERC, the Ontario Ministry of Research and Innovation, QuantumWorks, and the US ARO/DTO.
\end{acknowledgments}
\appendix
\section{Optimality of the linear combination procedure}\label{app:opt}
The goal of this appendix is to show that no protocol for implementing linear combinations of unitary operations in a large family of such protocols can have failure probability less than
\begin{equation}
\frac{4\kappa}{(\kappa+1)^2},\label{eq:appendix:a1}
\end{equation}
where $\kappa$ is defined in \thm{main:nonunitary}.
Specifically, we consider protocols of the form shown in \fig{appendix:generalcircuit}.
Our result shows that the protocol of \thm{main:nonunitary} is optimal among such all protocols in the limit of small $\Delta$ (i.e., when the unitary operations being combined are all similar).
\begin{theorem}\label{thm:opt}
Any protocol for implementing $V=\sum_{q=0}^{k} C_q U_q$ using a circuit of the form of \fig{appendix:generalcircuit} must fail with probability at least $4\kappa/(\kappa+1)^2$.
\end{theorem}
\begin{figure}
\caption{A general circuit for implementing a linear combination of $k+1$ unitary operators using unitary operations $A$ and $B$. We assume for simplicity that $k+1$ is an integer power of $2$. This circuit corresponds to preparing the ancilla states in an arbitrary state (specified by $A$) and measuring them in an arbitrary basis (specified by $B$).
\label{fig:appendix:generalcircuit}
\label{fig:appendix:generalcircuit}
\end{figure}
\begin{proof}
For convenience, we take $k$ to be an integer power of $2$. We can generalize the subsequent analysis to address the case where $k+1$ is not a power of $2$ by replacing $k+1$ by $k'+1=2^{\lceil \log_2 (k+1) \rceil}$ and taking $C_p=0$ for $p > k$.
Observe that the circuit in \fig{appendix:generalcircuit} acts as follows:
\begin{align}
\ket{0^{ \log_2 k }}\ket{\psi}&\mapsto \sum_{m=0}^{k} A_{m,0} \ket{m}\ket{\psi}\nonumber\\
&\mapsto \sum_m A_{m,0} \ket{m} U_{m} \ket{\psi}\nonumber\\
&\mapsto \sum_{n,m} B_{n,m}A_{m,0} \ket{n}U_{m}\ket{\psi}.\label{eq:appendix:a2}
\end{align}
Furthermore, we can modify $B$ to include a permutation such that desired transformation occurs when the first register is measured to be zero.
(Orthogonality prevents us from having more than one successful outcome, as we show below.)
The implementation is successful if there exists a constant $K>0$ such that
\begin{equation}
B_{0,m}A_{m,0}=K{C_{m}}\label{eq:appendix:a3}
\end{equation}
for all $m$.
Our goal is to maximize the success probability, which is equivalent to maximizing $K$ over all choices of the unitary operations $A$ and $B$ satisfying \eq{appendix:a3}.
We can drop the implicit normalization of the matrix elements of $B$ and $A$ by defining coefficients $b_{0,m}$ and $a_{m,0}$ such that
\begin{align}
\frac{b_{0,m}}{\sqrt{\sum_{j} |b_{0,j}|^2}}&\colonequals B_{0,m} \\ \frac{a_{m,0}}{\sqrt{\sum_{j} |a_{j,0}|^2}}&\colonequals A_{m,0}.
\end{align}
Using these variables, we have
\begin{equation}
|K C_{m}|
= \frac{|b_{0,m}a_{m,0}|}{\sqrt{(\sum_j |a_{j,0}|^2)(\sum_j |b_{0,j}|^2)}}
\le\frac{|b_{0,m}a_{m,0}|}{\sum_j |a_{j,0}b_{0,j}|}
=\frac{|C_{m}|}{\sum_j |C_j|},
\label{eq:appendix:ambmsum}
\end{equation}
where the bound follows from the Cauchy-Schwarz inequality.
This bound is tight because it can be saturated by taking $a_{m,0}=b_{0,m}=\sqrt{C_m}$.
The probability of successfully implementing the multi-product formula is
\begin{equation}
1-P_-
= \Big\|{\sum_{j} K C_j U_j\ket{\psi}}\Big\|^2
\le \left( \frac{\sum_{j} C_j }{\sum_{j} |C_j|} \right)^2
=\left(\frac{\Sigma_+ -\Sigma_-}{\Sigma_+ +\Sigma_-}\right)^2,
\label{eq:appendix:pbound0}
\end{equation}
where $\Sigma_+$ and $\Sigma_-$ are defined in \lem{1}. We have $\kappa\colonequals\Sigma_+/\Sigma_-$, so
\begin{equation}
1-P_-\le \left(\frac{\kappa-1}{\kappa+1}\right)^2,\label{eq:appendix:pbound}
\end{equation}
which implies that the failure probability satisfies $P_- \ge 4\kappa/(\kappa+1)^2$ as claimed.
It remains to see why it suffices to consider a single successful measurement outcome. In principle, we could imagine that many different measurement outcomes lead to a successful implementation of $V$. Assume that there exists a measurement outcome $v \ne 0^{\log_2 k}$ such that the protocol also gives the same multi-product formula on outcome $v$.
If both outcomes are successful then, up to a constant multiplicative factor, the coefficients of each $U_q$ must be the same. This occurs if there exists a constant $\Gamma\ne 0$ such that for all $q$,
\begin{equation}\label{eq:appendix:Gamma}
B_{0,q}A_{q,0}=\Gamma B_{v,q}A_{q,0},
\end{equation}
i.e., if $B_{0,q}= \Gamma B_{v,q}$ (note that $A_{q,0}$ must be nonzero provided $C_q \ne 0$). This is impossible because $B$ is unitary and hence its columns are orthonormal. Consequently, we cannot obtain the same multi-product formula from different measurement outcomes.
\end{proof}
The above proof implicitly specifies an optimal protocol for implementing linear combinations of unitaries. However, we do not use this protocol in \thm{main:nonunitary} because it is difficult to perform a correction if the implementation fails. When the method of \lem{main:2case} fails to implement a difference of nearby unitaries, the desired correction operation is a sum of nearby unitaries, so it can be implemented nearly deterministically. The correction operation may not have such a form if we use the protocol implicit in the above proof.
One simple generalization of the form of the protocol shown in \fig{appendix:generalcircuit} is to enlarge the ancilla register and allow each unitary in the linear combination to be performed conditioned on a higher-dimensional subspace of the ancilla states. It can be shown that a certain class of protocols of this form also do not improve the success probability. Whether protocols of another form could achieve a higher probability of success remains an open question.
\end{document} | math | 63,792 |
\begin{document}
\title{Stability, fragility, and Rota's Conjecture\footnote{Parts of this paper were previously published in the third author's PhD thesis \cite{vZ09}. The research of all authors was partially supported by a grant from the Marsden Fund of New Zealand. The first author was also supported by a FRST Science \& Technology post-doctoral fellowship. The third author was also supported by the Netherlands Organisation for Scientific Research (NWO).}}
\author{Dillon Mayhew\thanks{School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand. E-mail: \url{[email protected]}, \url{[email protected]}} \and Geoff Whittle\footnotemark[2] \and Stefan H. M. van Zwam\thanks{Centrum Wiskunde en Informatica, Postbus 94079, 1090 GB Amsterdam, The Netherlands. E-mail: \url{[email protected]}}}
\maketitle
\abstract{
Fix a matroid $N$. A matroid $M$ is $N$-\emph{fragile} if, for each element $e$ of $M$, at least one of $M\ensuremath{\!\Oldsetminus\!} e$ and $M\ensuremath{\!/} e$ has no $N$-minor. The Bounded Canopy Conjecture is that all $\GF(q)$-representable matroids $M$ that have an $N$-minor and are $N$-fragile have branch width bounded by a constant depending only on $q$ and $N$.
A matroid $N$ \emph{stabilizes} a class of matroids over a field $\mathds{F}$ if, for every matroid $M$ in the class with an $N$-minor, every $\mathds{F}$-representation of $N$ extends to at most one $\mathds{F}$-representation of $M$.
We prove that, if Rota's conjecture is false for $\GF(q)$, then either the Bounded Canopy Conjecture is false for $\GF(q)$ or there is an infinite chain of $\GF(q)$-representable matroids, each not stabilized by the previous, each of which can be extended to an excluded minor.
Our result implies the previously known result that Rota's Conjecture holds for $\GF(4)$, and that the classes of near-regular and sixth-roots-of-unity have a finite number of excluded minors. However, the bound that we obtain on the size of such excluded minors is considerably larger than that obtained in previous proofs. For $\GF(5)$ we show that Rota's Conjecture reduces to the Bounded Canopy Conjecture.
}
\section{Introduction}\label{sec:intro}
Rota's Conjecture, widely regarded as the most important open problem in matroid theory, is as follows.
\begin{conjecture}[\citet{Rot71}]\label{con:rota1}
For all prime powers $q$, the class of matroids representable over $\GF(q)$ can be characterized by a finite set of excluded minors.
\end{conjecture}
Progress on this conjecture has been intermittent. It has been settled completely only for $q \leq 4$ \citep{Tut65,Bix79,Sey79,GGK}. \citet{GGW06} showed that an excluded minor contains no large projective geometry. Another partial result towards Rota's Conjecture is the following:
\begin{theorem}[\citet{GW02}]\label{thm:rotabw}
Let $\mathds{F}$ be a finite field and $k \in \mathds{N}$. Let $\mathcal{M}$ be a minor-closed class of $\mathds{F}$-representable matroids. Then finitely many excluded minors for $\mathcal{M}$ have branch width $k$.
\end{theorem}
In 1996, \citet{SW96} introduced matroids representable over \emph{partial fields}. Anticipating some of the definitions in Section~\ref{sec:partialfields}, we say a partial field $\ensuremath{\mathbb{P}}$ is \emph{finitary} if there exists a homomorphism $\varphi:\ensuremath{\mathbb{P}}\rightarrow\GF(q)$ for some prime power $q$. We denote by $\matset(\ensuremath{\mathbb{P}})$ the set of $\ensuremath{\mathbb{P}}$-representable matroids. Since homomorphisms preserve representability, $\matset(\ensuremath{\mathbb{P}})\subseteq\matset(\GF(q))$ for some prime power $q$ if $\parf$ is finitary. Conjecture~\ref{con:rota1} can then be generalized as follows:
\begin{conjecture}\label{con:rota2}
For every finitary partial field $\ensuremath{\mathbb{P}}$, $\matset(\ensuremath{\mathbb{P}})$ can be characterized by a finite set of excluded minors.
\end{conjecture}
Like Rota's Conjecture, this conjecture has been settled for only a handful of partial fields. In particular, it is known for the regular, sixth-roots-of-unity, and near-regular partial fields \citep{Tut65,GGK,HMZ11}.
At the moment Geelen, Gerards, and Whittle are carrying out a project aimed at proving that $\matset(\GF(q))$ is well-quasi-ordered with respect to the minor-order (see, for instance, \citet{GGW06b}). That result, when combined with a proof of Conjecture~\ref{con:rota1}, would imply Conjecture~\ref{con:rota2}, since proper minor-closed classes of $\matset(\GF(q))$ would be characterized by a finite set of excluded minors. In this paper we set the stage for a proof of Rota's Conjecture for $q = 5$, by reducing it to a conjecture that should be a consequence of the structure theory being developed for the matroid minors project.
To state our main result we need to introduce a few concepts. We say that a matroid $N$ \emph{stabilizes} a matroid $M$ over a partial field $\parf$ if, for each minor $M'$ of $M$ isomorphic to $N$, each $\parf$-representation of $M'$ extends to at most one $\parf$-representation of $M$. A matroid $N$ is a \emph{stabilizer} for a class of matroids $\matset$ if $N$ stabilizes each 3-connected member of $\matset$. We will be more precise in Definition \ref{def:stab}. Stabilizers were introduced by Whittle \cite{Whi96b}, who proved that checking if a matroid is a stabilizer requires a finite amount of work.
A second concept we need is \emph{fragility}. Let $N$, $M$ be matroids. Then $M$ is \emph{$N$-fragile} if, for all $e \in E(M)$, at least one of $M\ensuremath{\!\Oldsetminus\!} e, M\ensuremath{\!/} e$ has no minor isomorphic to $N$. If $M$ is $N$-fragile and $N$ is a minor of $M$ then $M$ is \emph{strictly $N$-fragile}. A slightly more general definition will be given in Section~\ref{sec:fragility}. Note that fragility has been studied previously under a different name. If $\mathcal{M}$ is a minor-closed class of matroids, then a matroid $M$ is \emph{almost-$\mathcal{M}$} if, for each $e \in E(M)$, at least one of $M\ensuremath{\!\Oldsetminus\!} e$ and $M\ensuremath{\!/} e$ is in $\mathcal{M}$. See, for instance, \cite{Oxl90, KL02}.
A third concept, already mentioned in Theorem \ref{thm:rotabw}, is branch width. Roughly speaking, a matroid with high branch width cannot be decomposed into small pieces along low-order separations. It is closely related to the notion of tree width in graphs. We will define the branch width of a matroid, denoted by $\bw(M)$, in Section~\ref{sec:conn}.
\begin{definition}
Let $\matset$ be a class of matroids. Then $N$ has \emph{bounded canopy} over $\matset$ if there exists an integer $l$ such that, for all strictly $N$-fragile matroids $M \in \matset$, $\bw(M) \leq l$.
\end{definition}
Finally,
\begin{definition}\label{def:well-closed}
A class of matroids is \emph{well-closed} if it is closed under isomorphism, duality, taking minors, direct sums, and 2-sums.
\end{definition}
Our main result now is the following:
\begin{theorem}\label{thm:rotapf}
Let $\ensuremath{\mathbb{P}}$ be a finitary partial field, let $\matset$ be a well-closed class of $\parf$-representable matroids, each of which has bounded canopy over $\matset$, and let $N \in \matset$ be such that
\begin{enumerate}
\item $N$ is 3-connected and not binary;
\item $N$ stabilizes $\matset$ over $\parf$;
\item all 3-connected $\parf$-representable matroids, which have an $N$-minor and are stabilized by $N$, are in $\matset$.
\end{enumerate}
Then there are finitely many excluded minors for $\matset$ having an $N$-minor.
\end{theorem}
\newcounter{rotasavecounter}
\setcounter{rotasavecounter}{\value{theorem}}
Of course the set $\matset$ we are most interested in is $\matset(\parf)$, but it might be possible to establish by other means that certain $\parf$-representable matroids do not occur as minors of some excluded minor. Then Theorem \ref{thm:rotapf} can be applied to a more restricted class.
The condition that the matroids in $\matset$ have bounded canopy is needed because our result depends crucially on Theorem~\ref{thm:rotabw}. At first it may seem like a rather strong restriction. However, it is expected that, if $\ensuremath{\mathbb{P}}$ is a finitary partial field, \emph{every} matroid $N$ has bounded canopy over $\matset(\ensuremath{\mathbb{P}})$. The following is a weaker version of Conjecture 5.9 in \citet{GGW06b}.
\begin{conjecture}\label{con:boundcanopy}
Let $N$ be a $\GF(q)$-representable matroid. There is an integer $l$, depending only on $N$ and $q$, such that, if $M$ is a $\GF(q)$-representable matroid with $\bw(M) > l$ and $N$ is a minor of $M$, then there exists an $e \in E(M)$ for which both $M\ensuremath{\!\Oldsetminus\!} e$ and $M\ensuremath{\!/} e$ have a minor isomorphic to $N$.
\end{conjecture}
The difference with Geelen et al.'s conjecture is that they require that both $M\ensuremath{\!\Oldsetminus\!} e$ and $M\ensuremath{\!/} e$ have a \emph{fixed} $N$-minor. Our conjecture is clearly implied by theirs.
Our main application of Theorem \ref{thm:rotapf} is the following result:
\begin{theorem}\label{thm:gf5bcc}
Rota's Conjecture for $\GF(5)$ is implied by Conjecture \ref{con:boundcanopy}.
\end{theorem}
Unfortunately we cannot make a similar statement for bigger finite fields, since our proof relies on the fact that 3-connected quinary matroids have a bounded number of inequivalent representations, a property that is not shared by bigger fields \cite{OVW95}.
Theorem \ref{thm:rotapf} comes very close to the following conjecture:
\begin{conjecture}
Let $\ensuremath{\mathbb{P}}$ be a partial field. If $\matset(\ensuremath{\mathbb{P}})$ has infinitely many excluded minors, then there is an infinite chain of matroids $N_1, N_2, \ldots$ such that $N_i$ has at least $i$ inequivalent representations over $\ensuremath{\mathbb{P}}$, and such that $N_i$ is a minor of some excluded minor.
\end{conjecture}
The catch is in the observation that a matroid may not be stabilized by $N$ yet have fewer representations than $N$. We can, however, deduce the following:
\begin{corollary}\label{cor:infichain}
Let $\ensuremath{\mathbb{P}}$ be a partial field. If $\matset(\parf)$ has infinitely many excluded minors, but Conjecture \ref{con:boundcanopy} holds for $\parf$, then there is an infinite chain $N_1, N_2, \ldots$, with $N_i$ a minor of $N_{i+1}$ and $N_{i+1}$ not stabilized by $N_i$.
\end{corollary}
The paper is built up as follows. First, in Section \ref{sec:partialfields}, we give an overview of the theory of matroid representation over partial fields. Next, in Section \ref{sec:conn} we recall some standard results on connectivity. Section \ref{ssec:2seps} contains a few new results on $2$-separations. Section \ref{sec:fragility} contains a number of observations concerning fragility. In Section \ref{sec:incriminate} we use \emph{deletion pairs} to create a matrix over a partial field $\parf$ that should represent a matroid $M$ having an $N$-minor, if $M$ were representable over $\parf$. We introduce an \emph{incriminating set} which indicates where this particular representation fails. Deletion pairs and incriminating sets dictate the basic structure of the proof, in Section \ref{sec:thestrongproof}, of a weaker version of Theorem \ref{thm:rotapf}, in which $N$ is required to be a \emph{strong} stabilizer. In Section \ref{sec:theproof}, then, we show how to prove Theorem \ref{thm:rotapf} from this weaker version, and prove Corollary \ref{cor:infichain}. We conclude in Section \ref{sec:examples} with a number of applications of our result.
Unexplained notation follows \citet{oxley}. We write $\si(M)$ for the simplification of $M$ and $\co(M)$ for the cosimplification of $M$. We write $N\ensuremath{\preceq} M$ if $N$ is isomorphic to a minor of $M$. The smallest member of $\mathds{N}$ is 0.
\section{Partial fields and representations}\label{sec:partialfields}
We start with the definition of a partial field. In this section we omit proofs, all of which can be found in at least one of \citep{SW96,PZ08conf,PZ08lift}. All proofs are also collected in Van Zwam \cite{vZ09}.
\begin{definition}\label{def:pf}
A \emph{partial field} is a pair $(R, G)$, where $R$ is a commutative ring and $G$ is a subgroup of the group of units of $R$ such that $-1 \in G$.
\end{definition}
In some contexts (for instance in Definition \ref{def:hom}) we may implicitly identify $\parf$ with the set $G \cup \{0\}$. Likewise, we say that $p$ is an \emph{element of} $\ensuremath{\mathbb{P}}$ (notation: $p \in \ensuremath{\mathbb{P}}$) if $p = 0$ or $p \in G$. We define $\ensuremath{\mathbb{P}}^* := G$. Clearly, if $p,q \in \ensuremath{\mathbb{P}}$ then also $p\cdot q \in \ensuremath{\mathbb{P}}$, but $p+q$ need not be an element of $\ensuremath{\mathbb{P}}$.
\begin{definition}\label{def:hom}
Let $\ensuremath{\mathbb{P}}_1,\ensuremath{\mathbb{P}}_2$ be partial fields. A function $\varphi:\ensuremath{\mathbb{P}}_1\rightarrow\ensuremath{\mathbb{P}}_2$ is a \emph{partial-field homomorphism} if
\begin{enumerate}
\item\label{it:hom1} $\varphi(1) = 1$;
\item\label{it:hom2} For all $p,q \in \ensuremath{\mathbb{P}}_1$, $\varphi(pq) = \varphi(p)\varphi(q)$;
\item\label{it:hom3} For all $p,q,r \in \ensuremath{\mathbb{P}}_1$ such that $p+q = r$, $\varphi(p) + \varphi(q) = \varphi(r)$.
\end{enumerate}
\end{definition}
Recall that $\parf$ is \emph{finitary} if there is a partial-field homomorphism $\parf\rightarrow\GF(q)$ for some prime power $q$. We single out some special homomorphisms:
\begin{definition}\label{def:pfisom}
Let $\ensuremath{\mathbb{P}}_1,\ensuremath{\mathbb{P}}_2$ be partial fields and let $\varphi:\ensuremath{\mathbb{P}}_1\rightarrow\ensuremath{\mathbb{P}}_2$ be a homomorphism. Then $\varphi$ is an \emph{isomorphism} if\index{isomorphism!between partial fields|see{partial field}}
\begin{enumerate}
\item $\varphi$ is a bijection;
\item $\varphi(p)+\varphi(q) \in \ensuremath{\mathbb{P}}_2$ if and only if $p+q \in \ensuremath{\mathbb{P}}_1$.
\end{enumerate}
\end{definition}
\begin{definition}
A partial-field \emph{automorphism} is an isomorphism $\varphi:\ensuremath{\mathbb{P}}\rightarrow\ensuremath{\mathbb{P}}$.\index{partial field!automorphism|defi}
\end{definition}
We introduce some notation related to matrices. Recall that formally, for linearly ordered sets $X$ and $Y$, an $X\times Y$ matrix $A$ over a partial field $\parf$ is a function $A:X\times Y \rightarrow \parf$. If $X=(1,2,\ldots,k)$ then we say that $A$ is a $k\times Y$ matrix.
If $X'\subseteq X$ and $Y'\subseteq Y$, then we denote by $A[X',Y']$ the submatrix of $A$ obtained by deleting all rows and columns in $X\ensuremath{-} X'$, $Y\ensuremath{-} Y'$. If $Z$ is a subset of $X\cup Y$ then we define $A[Z] := A[X\cap Z, Y \cap Z]$. Also, $A-Z := A[X\ensuremath{-} Z, Y\ensuremath{-} Z]$.
Let $A_1$ be an $X\times Y_1$ matrix over a partial field $\parf$ and $A_2$ an $X\times Y_2$ matrix over $\parf$, where $Y_1\cap Y_2 = \emptyset$. Then $A := [A_1 \ A_2]$ denotes the $X\times (Y_1 \cup Y_2)$ matrix with $A_{xy} = (A_1)_{xy}$ for $y \in Y_1$ and $A_{xy} = (A_2)_{xy}$ for $y \in Y_2$. If $X$ is an ordered set, then $I_X$ is the $X\times X$ identity matrix. If $A$ is an $X\times Y$ matrix over $\mathds{F}$, then we use the shorthand $[I\ A]$ for $[I_X \ A]$.
Note that, for our purposes, the ordering of $X$ and $Y$ is only significant for the sign of determinants. And since the sign is irrelevant to the underlying matroid structure, we will freely permute rows and columns, always along with their labels, throughout the paper.
\begin{definition}\label{def:weakPmatrix}
Let $\ensuremath{\mathbb{P}} = (R, G)$ be a partial field and let $A$ be a matrix with entries in $R$. Then $A$ is a \emph{$\ensuremath{\mathbb{P}}$-matrix} if, for each square submatrix $D$ of $A$, $\det(D) \in \ensuremath{\mathbb{P}}$.
\end{definition}
In particular, all entries of $A$ are in $\ensuremath{\mathbb{P}}$.
\begin{proposition}\label{prop:parfmatroid}
Let $\ensuremath{\mathbb{P}} = (R, G)$ be a partial field, let $A$ be an $r\times E$ $\ensuremath{\mathbb{P}}$-matrix, and define
\begin{align*}
\mathcal{B} := \big\{\,X \subseteq E \,:\, |X| = r, \det(A[r,X]) \neq 0\,\big\}.
\end{align*}
If $\mathcal{B} \neq \emptyset$ then $\mathcal{B}$ is the set of bases of a matroid.
\end{proposition}
Following the notation for matroids representable over fields, we denote the matroid of Proposition~\ref{prop:parfmatroid} by $M[A]$. Some more terminology:
\begin{definition}\label{def:representable}
Let $M$ be a matroid. We say $M$ is \emph{representable} over a partial field $\ensuremath{\mathbb{P}}$ (or, shorter, $\ensuremath{\mathbb{P}}$-representable) if there exists a $\ensuremath{\mathbb{P}}$-matrix $A$ such that $M = M[A]$. Moreover, we refer to $A$ as a \emph{representation matrix} of $M$ and say $M$ is \emph{represented by} $A$.
\end{definition}
\begin{proposition}\label{prop:pmatops}Let $A$ be a $\ensuremath{\mathbb{P}}$-matrix. Then $A^T$ and $[I\ A]$ are also $\ensuremath{\mathbb{P}}$-matrices. Let $\varphi:\parf\rightarrow\parf'$ be a partial-field homomorphism. Then $\varphi(A)$ is a $\parf'$-matrix and $M[I\ A] = M[I \ \varphi(A)]$.
\end{proposition}
We will sometimes refer to the rank of a $\ensuremath{\mathbb{P}}$-matrix.
\begin{definition}
Let $A$ be an $X\times Y$ $\ensuremath{\mathbb{P}}$-matrix. The \emph{rank} of $A$ is
\begin{align*}
\rank(A) := \max \big\{\, k\in \mathds{N} \, :\, & \textrm{ there are } X'\subseteq X, Y'\subseteq Y \textrm{ with } |X'| = |Y'| = k, \\ & \textrm{ and } \det(A[X',Y']) \neq 0 \,\big\}.
\end{align*}
\end{definition}
It is not hard to verify that the rank function is preserved by partial-field homomorphisms, and that it corresponds to the usual rank function if $\parf$ is a field.
\begin{definition}\label{def:pivot}Let $A$ be an $X\times Y$ matrix over a ring $R$ and let $x \in X, y \in Y$ be such that $A_{xy} \in R^*$. Then we define $A^{xy}$ to be the $(X\ensuremath{-} x)\cup y \times (Y\ensuremath{-} y)\cup x$ matrix with entries
\begin{align*}
(A^{xy})_{uv} = \left\{ \begin{array}{ll}
(A_{xy})^{-1} \quad & \textrm{if } (u,v) = (y,x)\\
(A_{xy})^{-1} A_{xv} & \textrm{if } u = y, v\neq x\\
-A_{uy} (A_{xy})^{-1} & \textrm{if } v = x, u \neq y\\
A_{uv} - A_{uy} (A_{xy})^{-1} A_{xv} & \textrm{otherwise.}
\end{array}\right.
\end{align*}
\end{definition}
We say that $A^{xy}$ is obtained from $A$ by \emph{pivoting} over $xy$. To give some intuition for this definition, we remark that it corresponds to row reduction in the matrix $[I_X\ A]$, as follows. Multiply row $x$ with $(A_{xy})^{-1}$, then add multiples of row $x$ to the other rows so the other entries in column $y$ become zero. Finally, exchange columns $x$ and $y$, and relabel row $x$ to $y$. The resulting matrix is $[I_{(X\ensuremath{-} x)\cup y} \ A^{xy}]$. The next lemma formalizes this.
\begin{lemma}\label{lem:pivotmat}
Let $A, x, y$ be as in Definition \ref{def:pivot}. Define $a := A_{xy}$, $b := A[X\ensuremath{-} x,y]$, $X' := X\ensuremath{-} x$, and
\begin{align}
F := \kbordermatrix{ & x & X'\\
y & a^{-1} & 0 \cdots 0\\
& \phantom{0} &\\
X' & -a^{-1} b & I_{X'}\\
& \phantom{0} &
}.\label{eq:Finverse}
\end{align}
Let $P$ be the $(X\cup Y) \times (X\cup Y)$ permutation matrix swapping $x$ and $y$. Then
\begin{align*}
F [I\ A] P = [I\ A^{xy}].
\end{align*}
\end{lemma}
Note that $F$ is the inverse of $[I\ A][X,(y\cup X)\ensuremath{-} x]$.
\begin{proposition}\label{prop:pivotproper}
Let $A$ be an $X\times Y$ $\ensuremath{\mathbb{P}}$-matrix and let $x \in X, y \in Y$ be such that $A_{xy} \neq 0$. Then $A^{xy}$ is a $\ensuremath{\mathbb{P}}$-matrix.
\end{proposition}
We introduce some notions of equivalence of $\parf$-matrices.
\begin{definition}
Let $A$, $A'$ be matrices with entries in a partial field $\parf$.
\begin{enumerate}
\item If $A'$ can be obtained from $A$ by repeatedly scaling rows and columns by elements of $\parf$, then we say that $A$ and $A'$ are \emph{scaling-equivalent}.
\item If $A'$ can be obtained from $A$ by repeatedly scaling rows or columns, permuting rows, permuting columns, or pivoting, then we say that $A$ and $A'$ are \emph{geometrically equivalent}.
\item If $\varphi(A')$ is geometrically equivalent to $A$ for some partial-field automorphism $\varphi$, then we say that $A'$ and $A$ are \emph{algebraically equivalent}.
\end{enumerate}
\end{definition}
Note that in all operations, labels are exchanged along with their rows and columns. It is easy to verify that the defined relations are indeed equivalence relations, and that equivalent matrices represent the same matroid, as follows.
\begin{lemma}
Let $A$, $A'$ be algebraically equivalent $\parf$-matrices. Then $M[I\ A] = M[I\ A']$.
\end{lemma}
From this definition it is clear that there is a choice in how to count representations of a matroid. When we say ``$M$ has $k$ inequivalent representations'', we mean that $M$ has $k$ \emph{algebraically} inequivalent representations. In contrast, for the definition of a stabilizer below we use geometric equivalence.
In the remainder of the section we introduce some tools to help us to recognize when matrices are equivalent.
\begin{definition}\label{def:matroidgraph}
Let $M$ be a matroid, $B$ a basis of $M$, and $D := E(M)\ensuremath{-} B$. Then $G(M,B)$ is the bipartite graph with vertices $B\cup D$ and edges $\{bd : B \triangle \{b,d\} \textrm{ is a basis of } M\}$.
\end{definition}
The graph $G(M,B)$ is the \emph{$B$-fundamental cocircuit incidence graph} of $M$ with respect to $B$ (cf. \cite[page 194]{oxley}). It has the following properties:
\begin{lemma}\label{lem:bipconn}
Let $M$ be a matroid and $B$ a basis of $M$.
\begin{enumerate}
\item\label{bip:conn} $M$ is connected if and only if $G(M,B)$ is connected.
\item\label{bip:threeconn} If $M$ is 3-connected, then $G(M,B)$ is 2-connected.
\end{enumerate}
\end{lemma}
\begin{definition}\label{def:graph}
Let $A$ be an $X\times Y$ matrix. Then $G(A)$ is the bipartite graph with vertices $X\cup Y$ and edges $\{xy : A_{xy}\neq 0\}$.
\end{definition}
\begin{lemma}\label{lem:graphsequal}
Let $A$ be an $X\times Y$ $\parf$-matrix and $M := M[I\ A]$. Then $G(M,X) = G(A)$.
\end{lemma}
The following is a straightforward generalization of a well-known result by \citet{BL76} to partial fields \citep[see also][Theorem 6.4.7]{oxley}.
\begin{lemma}\label{lem:bippropertiesunique}
Let $A$, $A'$ be matrices with entries in a partial field $\parf$. If $A'$ is scaling-equivalent to $A$ and $A'_{e} = A_{e}$ for all edges $e$ of a maximal spanning forest of $G(A)$, then $A' = A$.
\end{lemma}
Some more terminology: if $A_e = 1$ for all edges $e$ of a maximal spanning forest $T$ of $G(A)$, then we say $A$ is $T$-\emph{normalized}.
Finally, if two matrices are geometrically equivalent and have the same row labels, they are scaling-equivalent:
\begin{proposition}\label{prop:scalepivotcommute}
Let $A$, $A'$ be geometrically equivalent $X\times Y$ $\ensuremath{\mathbb{P}}$-matrices, where $X$, $Y$ are disjoint sets. Then $A$ is scaling-equivalent to $A'$.
\end{proposition}
\begin{proof}
Since $A$ is geometrically equivalent to $A'$, we have
\begin{align}
[I_X\ A'] = F[I_X\ A]D\label{eq:scalepivotcommuteeq}
\end{align}
for an invertible matrix $F$ and a diagonal $(X\cup Y)\times (X\cup Y)$ matrix $D$, by Lemma \ref{lem:pivotmat}. From \eqref{eq:scalepivotcommuteeq} we conclude that
\begin{align*}
I_X = F I_X D[X,X].
\end{align*}
This implies that $F$ is a diagonal matrix. But then $A$ is scaling-equivalent to $A'$, as desired.
\end{proof}
\subsection{Stabilizers}
\label{sub:stabilizers}
We now give a more precise definition of stabilizers.
\begin{definition}\label{def:stab}
Let $\parf$ be a partial field, $M$ a matroid, $X$ a basis of $M$, $Y := E(M)\ensuremath{-} X$, $S\subseteq X$, $T\subseteq Y$, and $N := M\ensuremath{\!/} S \ensuremath{\!\Oldsetminus\!} T$.
If, for all $X\times Y$ $\parf$-matrices $A_1, A_2$ such that
\begin{enumerate}
\item $M = M[I\ A_1] = M[I\ A_2]$
\item $A_1[X\ensuremath{-} S,Y\ensuremath{-} T]$ is scaling-equivalent to $A_2[X\ensuremath{-} S,Y\ensuremath{-} T]$,
\end{enumerate}
we have that $A_1$ is scaling-equivalent to $A_2$, then we say that $N$ \emph{stabilizes} $M$.
\end{definition}
\begin{definition}\label{def:strongstab}
If $N$ stabilizes $M$ over $\parf$, and \emph{every} representation of $N$ extends to a representation of $M$, then we say $N$ \emph{strongly stabilizes} $M$ over $\parf$.
\end{definition}
If $N$ has a unique representation over $\parf$ and $N$ stabilizes $M$, then $N$ is necessarily a strong stabilizer. Strong stabilizers were introduced by \citet{GOVW98}.
We say that $N$ stabilizes a set of matroids $\matset$ over a partial field $\parf$ if, for each 3-connected $M \in \matset$, every minor $M'$ isomorphic to $N$ stabilizes $M$ over $\parf$. The following is easily verified:
\begin{lemma}\label{lem:sistab}
Let $M$ and $N$ be $\parf$-representable matroids such that $N\ensuremath{\preceq} M$ and $N$ stabilizes $\si(M)$ over $\parf$. Then $N$ stabilizes $M$ over $\parf$.
\end{lemma}
\section{Connectivity and branch width}\label{sec:conn}
\subsection{The connectivity function}
Recall the standard definition of the connectivity function:
\begin{definition}
Let $M$ be a matroid with ground set $E$. The connectivity function of $M$, $\lambda_M:2^E\rightarrow \mathds{N}$ is defined by
\begin{align*}
\lambda_M(Z) := \rank_M(Z) + \rank_M(E\ensuremath{-} Z) - \rank(M).
\end{align*}
\end{definition}
As usual, a $k$-\emph{separation} of $M$ is a partition $(X,Y)$ of $E(M)$ with $|X|, |Y| \geq k$ and $\lambda_M(X) < k$. A matroid is $k$-\emph{connected} if it has no separations of order $k-1$ or less.
We start with some elementary and well-known properties of the connectivity function.
\begin{lemma}
The function $\lambda_M$ is self-dual, submodular, and monotone under taking minors.
\end{lemma}
For representable matroids, the following lemma reformulates the connectivity function in terms of the ranks of certain submatrices of $A$.
\begin{lemma}[\citet{TruI}]\label{lem:matrixconn}
Suppose $A$ is an $(X_1\cup X_2)\times(Y_1\cup Y_2)$ $\ensuremath{\mathbb{P}}$-matrix (where $X_1, X_2, Y_1, Y_2$ are pairwise disjoint). Then
\begin{align*}
\lambda_{M[I\ A]}(X_1\cup Y_1) = \rank(A[X_1,Y_2]) + \rank(A[X_2,Y_1]).
\end{align*}
\end{lemma}
To keep track of the connectivity of minors of $M$ it is convenient to introduce some extra notation.
\begin{definition}\label{def:basisminor}
Let $M$ be a matroid, $B$ a basis of $M$, and $Y = E(M)\ensuremath{-} B$. If $Z \subseteq E(M)$ then $M_B[Z] := M\ensuremath{\!/} (B\ensuremath{-} Z) \ensuremath{\!\Oldsetminus\!} (Y\ensuremath{-} Z)$ and $M_B - Z := M_B[E\ensuremath{-} Z]$.
\end{definition}
The following is easily seen:
\begin{lemma}
If $M = M[I\ A]$ for an $X\times Y$ $\ensuremath{\mathbb{P}}$-matrix $A$, sets $X$ and $Y$ are disjoint, and $Z \subseteq X\cup Y$, then $M_X[Z] = M[I\ A[Z]]$.
\end{lemma}
To counter the stacking of subscripts we introduce alternative notation for the connectivity function. This definition generalizes Lemma~\ref{lem:matrixconn} to arbitrary matroids $M$ and to arbitrary minors of $M$. It is equivalent to the definition found in \citet{GGK}.
\begin{definition}\label{def:connectivityB}
Let $M$ be a matroid and $B$ a basis of $M$. Then $\lambda_B:2^{E(M)}\times 2^{E(M)} \rightarrow \mathds{N}$ is defined as
\begin{align*}
\lambda_B(X,Y) := \rank_{M\ensuremath{\!/} (B\ensuremath{-} Y)}(X\ensuremath{-} B) + \rank_{M\ensuremath{\!/} (B\ensuremath{-} X)}(Y\ensuremath{-} B)
\end{align*}
for all $X,Y\subseteq E(M)$.
\end{definition}
The following lemma shows that this is indeed the connectivity function of a minor of $M$ when $X$ and $Y$ are disjoint. Once again we omit the straightforward proof.
\begin{lemma}\label{lem:connectivityoldnew}
Let $M$ be a matroid, $B$ a basis of $M$, and $X,Y$ disjoint subsets of $E(M)$. Then
\begin{align*}
\lambda_{B}(X,Y) = \lambda_{M_B[X\cup Y]}(X).
\end{align*}
\end{lemma}
The following two results can be found in \citet[{Proposition 4.3.6, Corollary 11.2.1}]{oxley}.
\begin{theorem}\label{thm:2connchain}
Let $M$ and $N$ be connected matroids, $N\ensuremath{\preceq} M$, with $|E(N)|<|E(M)|$. Then there is an $e \in E(M)$ such that some $M' \in \{M\ensuremath{\!\Oldsetminus\!} e, M\ensuremath{\!/} e\}$ is connected with $N\ensuremath{\preceq} M'$.
\end{theorem}
\begin{theorem}[Splitter Theorem]\label{thm:3connchain}
Let $M$ and $N$ be $3$-connected matroids, $N\ensuremath{\preceq} M$, with $|E(M)| > |E(N)| \geq 4$, such that $M$ is not isomorphic to a wheel or a whirl. Then there is an $e \in E(M)$ such that some $M' \in \{M\ensuremath{\!\Oldsetminus\!} e, M\ensuremath{\!/} e\}$ is $3$-connected with $N\ensuremath{\preceq} M'$.
\end{theorem}
\subsection{Blocking sequences}
The following definitions are from \citet{GGK}.
\begin{definition}
Let $M$ be a matroid on ground set $E$, $M'$ a minor of $M$ on ground set $E'\subseteq E$, and $(Z_1',Z_2')$ a $k$-separation of $M'$. We say that $(Z_1', Z_2')$ is \emph{induced} in $M$ if there exists a $k$-separation $(Z_1,Z_2)$ of $M$ with $Z_1' \subseteq Z_1$ and $Z_2' \subseteq Z_2$.
\end{definition}
Let $B$ be a basis of $M$ such that $M' = M_B[E']$.
\begin{definition}\label{def:blockseq}
Let $M$, $M'$, $E$, $E'$, $Z_1'$, and $Z_2'$ be as in the previous definition. A \emph{blocking sequence}\index{blocking sequence} for $(Z_1',Z_2')$ is a sequence of elements $v_1, \ldots, v_t$ of $E\ensuremath{-} E'$ such that
\begin{enumerate}
\item\label{bls:first} $\lambda_B(Z_1', Z_2' \cup v_1) = k$;
\item\label{bls:middle} $\lambda_{B}(Z_1'\cup v_i, Z_2'\cup v_{i+1}) = k$ for $i = 1, \ldots, t-1$;
\item\label{bls:last} $\lambda_{B}(Z_1'\cup v_t,Z_2') = k$; and
\item no proper subsequence of $v_1, \ldots, v_t$ satisfies the first three properties.
\end{enumerate}
\end{definition}
Blocking sequences find their origin in Seymour's work on regular matroid decomposition \citep[Section 8]{Sey80}. The first general formulation was due to \citet{TruIII}, but blocking sequences truly took off with the publication of the proof of Rota's Conjecture for $\GF(4)$ \citep{GGK}. We have opted to use their notation rather than the notation used in, for instance, \citet{GHW05}, because Definition~\ref{def:blockseq} clearly exhibits the symmetry.
The following theorem illustrates the usefulness of blocking sequences:
\begin{theorem}[\citet{GGK}, Theorem 4.14]\label{thm:blockseq}
Let $M$ be a matroid on ground set $E$, $B$ a basis of $M$, $M' := M_B[E']$ for some $E'\subseteq E$, and $(Z_1',Z_2')$ an exact $k$-separation of $M'$. Exactly one of the following holds:\index{k-separation@$k$-separation}
\begin{enumerate}
\item There exists a blocking sequence for $(Z_1',Z_2')$;
\item $(Z_1',Z_2')$ is induced in $M$.
\end{enumerate}
\end{theorem}
In the first case we say that $(Z_1',Z_2')$ is \emph{bridged} in $M$.
Another useful property of blocking sequences is the following:
\begin{lemma}[\citet{GGK}, Proposition 4.15(iv)]\label{lem:blockseqalternating}If $v_1, \ldots, v_t$ is a blocking sequence for the $k$-separation $(Z_1',Z_2')$, then $v_i\in B$ implies $v_{i+1} \in E\ensuremath{-} B$ and $v_i \in E\ensuremath{-} B$ implies $v_{i+1} \in B$ for $i = 1, \ldots, t-1$.
\end{lemma}
We will use the following lemma:
\begin{lemma}[\citet{GGK}, Proposition 4.16(i)]\label{lem:blseqlem}
Let $v_1, \ldots, v_t$ be a blocking sequence for $(Z_1', Z_2')$. If $Z_2'' \subseteq Z_2'$ is such that $|Z_2''| \geq k$ and $\lambda_B(Z_1',Z_2'') = k-1$, then $v_1, \ldots, v_{t-1}$ is a blocking sequence for the exact $k$-separation $(Z_1',Z_2''\cup v_t)$.
\end{lemma}
\subsection{Branch width}
A graph $T = (V,E)$ is a \emph{cubic tree} if $T$ is a tree in which each vertex has degree exactly one or three. We denote the leaves of $T$ by $L(T)$.
\begin{definition}
Let $M$ be a matroid. A \emph{partial branch decomposition} of $M$ is a pair $(T, l)$, where $T$ is a cubic tree and $l:V(T)\rightarrow 2^{E(M)}$ a function assigning a subset of $E(M)$ to each vertex of $T$, such that $\{l(v) : v \in V(T)\}$ partitions $E(M)$.
\end{definition}
If $T$ is a tree and $e=vw \in E(T)$, then we denote by $T_v$ the component of $T\ensuremath{\!\Oldsetminus\!} e$ containing $v$.
\begin{definition}
Let $M$ be a matroid and let $(T,l)$ be a partial branch decomposition of $M$. We define $w_{(T,l)}: V^2\rightarrow \mathds{N}$ as
\begin{align*}
w_{(T,l)}(v,w) & = \left\{ \begin{array}{cl}
\lambda_M\left(\bigcup_{u \in V(T_v)} l(u)\right) + 1 & \textrm{ if } vw \in E(T);\\
\phantom{\bigg(} 0 & \textrm{ otherwise.}
\end{array}\right.
\end{align*}
\end{definition}
In words, $w_{(T,l)}(v,w)$ is the degree of the separation of $M$ displayed by the edge $vw$. Note that $(\bigcup_{u \in V(T_v)} l(u), \bigcup_{u \in V(T_w)} l(u))$ is a partition of $E(M)$, so $w_{(T,l)}(v,w) = w_{(T,l)}(w,v)$. Hence, for $e = vw \in E(T)$, we will write $w_{(T,l)}(e)$ as shorthand for $w_{(T,l)}(v,w)$.
\begin{definition}
Let $M$ be a matroid and let $(T,l)$ be a partial branch decomposition of $M$. The \emph{width} of $(T,l)$ is
\begin{align*}
w(T,l) := \left\{ \begin{array}{ll}
\max_{e \in E(T)} w_{(T,l)}(e) & \textrm{ if } E(T) \neq \emptyset\\
1 & \textrm{ otherwise.}
\end{array}\right.
\end{align*}
\end{definition}
\begin{definition}
Let $M$ be a matroid. A \emph{branch decomposition} of $M$ is a partial branch decomposition such that $|l(v)| \leq 1$ for all $v \in L(T)$, and such that $l(v) = \emptyset$ for all $v \in V(T)\ensuremath{-} L(T)$.
\end{definition}
\begin{definition}
Let $M$ be a matroid. A \emph{reduced branch decomposition} of $M$ is a branch decomposition such that $|l(v)| = 1$ for all $v \in L(T)$.
\end{definition}
We denote the set of reduced branch decompositions of $M$ by $\mathcal{D}_M$.
\begin{definition}
Let $M$ be a matroid. The \emph{branch width} of $M$ is
\begin{align*}
\bw(M) := \min_{(T,l) \in \mathcal{D}_M} w(T,l).
\end{align*}
\end{definition}
We start with some elementary and well-known observations. We omit the proofs.
\begin{lemma}
Let $(T,l)$ be a branch decomposition of a matroid $M$. There is a reduced branch decomposition $(T',l')$ of $M$ such that $w(T,l) = w(T',l')$.
\end{lemma}
\begin{proposition}\label{prop:bwmonotone}
Let $M$ be a matroid and $e \in E(M)$. Then
\begin{align*}
\bw(M\ensuremath{\!\Oldsetminus\!} e) \leq \bw(M) \leq \bw(M\ensuremath{\!\Oldsetminus\!} e) + 1.
\end{align*}
\end{proposition}
Series and parallel classes do not have an effect on the branch width of a matroid:
\begin{proposition}Let $M$ be a matroid with $\bw(M)\geq 2$. Then $\bw(M) = \bw(\si(M))$.
\end{proposition}
\citet[Theorem 1.4]{GHW05} proved the following result, which states that a blocking sequence does not increase branch width by much:
\begin{theorem}\label{thm:blseqbranchwidth}
Let $M$ be a matroid having basis $B$, and let $Z \subseteq E(M)$. Suppose that $M_B[Z]$ has a $k$-separation $(X,Y)$, and that $v_1, \ldots, v_t$ is a blocking sequence for $(X,Y)$ in $M$. Then $\bw(M_B[Z\cup\{v_1, \ldots, v_t\}]) \leq \bw(M_B[Z]) + k$.\index{blocking sequence}
\end{theorem}
We note one particular case for the examples in Section \ref{sec:examples}:
\begin{lemma}\label{lem:bwwhirl}
For all $n \geq 2$, $\bw(\whirl{n}) = 3$.\index{whirl}
\end{lemma}
\subsection{Results on $2$-separations}\label{ssec:2seps}
We will need to bound the number of $2$-separations in small extensions of a $3$-connected matroid. The following lemma does just that.
\begin{lemma}\label{lem:2sepbound}
If $M$ is a connected matroid, $N \ensuremath{\preceq} M$, $N$ is $3$-connected, $|E(N)| \geq 4$, and $|E(M)|-|E(N)| \leq k$, then the number of 2-separations in $M$ is at most $2^{k+1}$.
\end{lemma}
\begin{proof}
Let $t_k$ denote the maximum number of $2$-separations of a $k$-element extension of a $3$-connected matroid. We argue by induction on $k$. By Theorem~\ref{thm:2connchain} there exist a basis $B$ of $M$, a subset $X$ of $E(M)$, and an ordering $e_1, \ldots, e_k$ of the elements of $E(M)\ensuremath{-} X$ such that $N \cong M_B[X]$ and $M_B[X\cup \{e_1, \ldots, e_i\}]$ is connected for all $i \in \{1, \ldots, k\}$.
If $k = 1$ then $e_1$ can be in series or in parallel with at most one element of $M_B[X]$, and it cannot be both in series and in parallel. Hence $t_1 = 1$.
By duality we may assume $e_k \not \in B$. Let $(Z_1,Z_2)$ be a $2$-separation of $M$, with $e_k \in Z_1$. If $|Z_1| \geq 3$ then $\lambda_{M\ensuremath{\!\Oldsetminus\!} e_k}(Z_2) \leq 1$, and connectivity of $M\ensuremath{\!\Oldsetminus\!} e_k$ implies that equality holds. Hence $(Z_1\ensuremath{-} e_k, Z_2)$ is a $2$-separation of $M\ensuremath{\!\Oldsetminus\!} e_k$. This leads to at most two $2$-separations of $M$: $(Z_1,Z_2)$ and $(Z_1\ensuremath{-} e_k, Z_2\cup e_k)$.
If a $2$-separation of $M$ is not an extension of a $2$-separation of $M\ensuremath{\!\Oldsetminus\!} e_k$, then we must have $|Z_1| = 2$. There is one of these for each $f \in E(M)\ensuremath{-} \{e_k\}$ such that $e_k, f$ are in series or in parallel. But $e_k$ can, again, be in series or in parallel with at most one element of $X$, as well as with each of $e_1, \ldots, e_{k-1}$, so it follows that
\begin{align*}
t_k \leq 2 t_{k-1} + k.
\end{align*}
Define $t'_k := 2^{k+1} - k - 2$. We claim that $t_k \leq t_k'$. Indeed: $t'_1 = t_1 = 1$, and if the claim is valid for $k-1$, then
\begin{align*}
t_k \leq 2 t_{k-1} + k \leq 2 t'_{k-1} + k = 2(2^k- (k-1)-2)+k = 2^{k+1} -k - 2 = t'_k.
\end{align*}
Obviously $t'_k \leq 2^{k+1}$, and the result follows.
\end{proof}
The following definitions are from \citet{GGK}.
\begin{definition}
Let $M$ be a matroid and let $(X_1,X_2)$ and $(Y_1,Y_2)$ be $2$-separations of $M$. If $X_i \cap Y_j\neq \emptyset$ for all $i,j \in \{1,2\}$, then we say that $(X_1,Y_1)$ and $(X_2,Y_2)$ \emph{cross}.
\end{definition}
\begin{definition}
Let $M$ be a matroid and let $(X_1,X_2)$ be a $2$-separation of $M$. We say that $(X_1,X_2)$ is \emph{crossed} if there exists a $2$-separation $(Y_1,Y_2)$ of $M$ such that $(X_1,X_2)$ and $(Y_1,Y_2)$ cross. Otherwise we say $(X_1,X_2)$ is \emph{uncrossed}.
\end{definition}
Crossing $2$-separations have previously been studied by \citet{CE80}. \citet{OSW04} characterized crossing $3$-separations in $3$-connected matroids, and those results have been generalized to crossing $k$-separations by \citet{AO08}. The proof of the following lemma is an instance of the technique of ``uncrossing'' from those papers.
\begin{lemma}\label{lem:uncrossed2sep}
Let $M$ be a connected, nonbinary matroid. If $M$ has a 2-separation, then $M$ must have an uncrossed 2-separation.
\end{lemma}
\begin{proof}
Since $M$ is non-binary, $M$ has a $U_{2,4}$-minor. Fix such a minor, say with elements $\{a,b,c,d\}$. If $(X,Y)$ is a $2$-separation of $M$, then either $|X\cap\{a,b,c,d\}| \leq 1$ or $|Y \cap \{a,b,c,d\}| \leq 1$. Let $(X',Y')$ be a $2$-separation of $M$ such that $Y'$ is maximal subject to $|Y' \cap \{a,b,c,d\}| \leq 1$. Let $(U,V)$ be a $2$-separation that crosses $(X',Y')$, and assume $|V\cap\{a,b,c,d\}|\leq 1$. Then $X'\cap U$ has at least two elements from $\{a,b,c,d\}$. Now
\begin{align*}
2 = \lambda_M(X') + \lambda_M(U) \geq \lambda_M(X'\cap U) + \lambda_M(X'\cup U),
\end{align*}
so we must have $\lambda_M(X'\cap U) = 1 = \lambda_M(Y' \cup V)$. Since $|(X'\cap U)\cap \{a,b,c,d\}| \geq 2$, it follows that $|(Y'\cup V)\cap \{a,b,c,d\}| \leq 1$. But $|Y'\cup V| > |Y'|$, a contradiction.
\end{proof}
Uncrossed $2$-separations are relevant because they can be bridged without introducing new $2$-separations:
\begin{lemma}[\citet{GGK}, Proposition 4.17]\label{lem:block2sep}
Let $M$ be a matroid, $B$ a basis of $M$, $E' \subseteq E$, and $(Z_1',Z_2')$ an uncrossed $2$-separation of \index{2-separation@$2$-separation} $M_B[E']$. Let $v_1, \ldots, v_t$ be a blocking sequence for $(Z_1',Z_2')$. If $(Z_1,Z_2)$ is a $2$-separation of $M_B[E'\cup \{v_1, \ldots, v_t\}]$ then $Z_i' \cup \{v_1, \ldots, v_t\} \subseteq Z_j$ for some $i, j \in \{1,2\}$.
\end{lemma}
\begin{corollary}\label{cor:block2sep}
Let $M$ be a matroid, $B$ a basis of $M$, $E' \subseteq E$, and $(Z_1',Z_2')$ an uncrossed $2$-separation of the connected matroid $M_B[E']$. Let $v_1, \ldots, v_t$ be a blocking sequence for $(Z_1',Z_2')$. Then $M_B[E'\cup \{v_1, \ldots, v_t\}]$ has strictly fewer $2$-separations than $M_B[E']$.
\end{corollary}
\begin{proof}
Let $(Z_1,Z_2)$ be a $2$-separation of $M_B[E'\cup\{v_1,\ldots,v_t\}]$. Possibly after relabelling, Lemma~\ref{lem:block2sep} implies that $Z_2'\cup \{v_1,\ldots,v_t\} \subseteq Z_2$. Therefore we know that $|Z_2\ensuremath{-}\{v_1,\ldots,v_t\}| \geq 2$. Also $|Z_1| \geq 2$ so, since $M_B[E']$ is connected, $1 \leq \lambda_B(Z_1,Z_2\ensuremath{-} \{v_1, \ldots, v_t\}) \leq \lambda_B(Z_1,Z_2) = 1$. Hence $(Z_1, Z_2\ensuremath{-}\{v_1,\ldots,v_t\})$ is a $2$-separation of $M_B[E']$, and the result follows.
\end{proof}
\subsection{Excluded minors for well-closed classes}
We omit the easy proofs of the observations in this section. In all results, $\matset$ is a well-closed class of matroids.
\begin{lemma}\label{lem:exmindual}
Let $M$ be an excluded minor for $\matset$. Then $M^*$ is an excluded minor for $\matset$.
\end{lemma}
\begin{lemma}\label{lem:exmin3c}
Let $M$ be an excluded minor for $\matset$. Then $M$ is $3$-connected.
\end{lemma}
\begin{lemma}\label{lem:exminrankbound}
Suppose all matroids in $\matset$ are representable over some finite field $\GF(q)$. Let $r \in \mathds{N}$. Then there are finitely many rank-$r$ excluded minors for $\matset$.
\end{lemma}
\section{Fragility}\label{sec:fragility}
In the introduction we defined fragility for a single matroid. A slightly more general definition is the following:
\begin{definition}\label{def:fragile}
Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is $\mathcal{N}$-\emph{fragile} if, for all $e \in E(M)$, at least one of $M\ensuremath{\!\Oldsetminus\!} e$ and $M\ensuremath{\!/} e$ has no minor isomorphic to a member of $\mathcal{N}$. Moreover, an $\mathcal{N}$-fragile matroid $M$ is \emph{strictly} $\mathcal{N}$-fragile if some minor of $M$ is isomorphic to a member of $\mathcal{N}$.
\end{definition}
Let $N$ be a matroid. We say that a matroid $M$ is $N$-\emph{fragile} if $M$ is $\{N\}$-fragile. We establish a few basic properties of $\mathcal{N}$-fragile matroids. The following is easy to see from the definition:
\begin{lemma}
If $M$ is $\mathcal{N}$-fragile and $M'\ensuremath{\preceq} M$ then $M'$ is $\mathcal{N}$-fragile.
\end{lemma}
The following proposition is well-known; see, for instance, \citet[Corollary 2.4]{GW01} for a proof technique.
\begin{proposition}\label{pro:twosepminorintersection}
Let $M$ be a matroid with a 2-separation $(A,B)$ and let $N$ be a 3-connected minor of $M$. Assume $|E(N)\cap A| \geq |E(N)\cap B|$. Then $|E(N)\cap B| \leq 1$. Moreover, unless $B$ consists of a parallel class or series class, there is an $e\in B$ such that both $M\ensuremath{\!\Oldsetminus\!} e$ and $M\ensuremath{\!/} e$ have a minor isomorphic to $N$.
\end{proposition}
An immediate corollary is the following. It was also proven by Kingan and Lemos \cite[Proposition 3.1]{KL02}.
\begin{proposition}\label{prop:almostconn}
Let $\mathcal{N}$ be a set of 3-connected matroids with $|E(N)| \geq 4$ for all $N \in \mathcal{N}$, and let $M$ be a strictly $\mathcal{N}$-fragile matroid. Then $M$ is $3$-connected up to series and parallel classes.
\end{proposition}
Some more terminology:
\begin{definition}
Let $\mathcal{N}$ be a set of matroids, let $M$ be a matroid, and let $e \in E(M)$.
\begin{enumerate}
\item If $M\ensuremath{\!/} e$ has a minor isomorphic to a member of $\mathcal{N}$ then $e$ is \emph{$\mathcal{N}$-contractible};
\item If $M\ensuremath{\!\Oldsetminus\!} e$ has a minor isomorphic to a member of $\mathcal{N}$ then $e$ is \emph{$\mathcal{N}$-deletable};
\item If neither $M\ensuremath{\!\Oldsetminus\!} e$ nor $M\ensuremath{\!/} e$ has a minor isomorphic to a member of $\mathcal{N}$ then $e$ is \emph{$\mathcal{N}$-essential}.
\end{enumerate}
\end{definition}
We will drop the prefix ``$\mathcal{N}$-'' if it is clear from the context which set is intended. For readers familiar with the work of \citet{TruVI} this definition may cause some confusion: Truemper defines a \emph{con} element $e$ to be such that $M\ensuremath{\!/} e$ has no $F_7$-minor and no $F_7^*$-minor, and a \emph{del} element $e$ to be such that $M\ensuremath{\!\Oldsetminus\!} e$ has no $F_7$- and no $F_7^*$-minor. The reasoning behind his choice is clear: rather than studying $\{F_7,F_7^*\}$-fragile binary matroids, he studies \emph{almost regular} binary matroids. Hence losing the minor is a good thing for him. For us the elements of $\mathcal{N}$ will be stabilizers, so we want to keep a member of $\mathcal{N}$ by all means. We use the following notation:
\begin{definition}
Let $\mathcal{N}$ be a set of matroids and let $M$ be a matroid.
\begin{align*}
\mathbf{C}_{\mathcal{N},M} & := \{\, e \in E(M) : e \textrm{ is } \mathcal{N}\textrm{-contractible}\,\};\\
\mathbf{D}_{\mathcal{N},M} & := \{\, e \in E(M) : e \textrm{ is } \mathcal{N}\textrm{-deletable}\,\};\\
\mathbf{E}_{\mathcal{N},M} & := \{\, e \in E(M) : e \textrm{ is } \mathcal{N}\textrm{-essential}\,\}.
\end{align*}
\end{definition}
We conclude the section with a number of elementary properties of $\mathcal{N}$-fragile matroids. We omit the straightforward proofs.
\begin{lemma}\label{lem:almostpartition}
Let $\mathcal{N}$ be a set of matroids, and let $M$ be an $\mathcal{N}$-fragile matroid.
\begin{enumerate}
\item $\mathbf{C}_{\mathcal{N},M}$, $\mathbf{D}_{\mathcal{N},M}$, $\mathbf{E}_{\mathcal{N},M}$ are pairwise disjoint and partition $E(M)$.
\item Let $\mathcal{N}^* := \{N^* : N \in \mathcal{N}\}$. Then $M^*$ is $\mathcal{N}^*$-fragile with $\mathbf{C}_{\mathcal{N}^*, M^*} = \mathbf{D}_{\mathcal{N},M}$, $\mathbf{D}_{\mathcal{N}^*, M^*} = \mathbf{C}_{\mathcal{N},M}$, and $\mathbf{E}_{\mathcal{N}^*, M^*} = \mathbf{E}_{\mathcal{N},M}$.
\item Let $M'\ensuremath{\preceq} M$.
\begin{enumerate}
\item If $e \in E(M')$ and $e\in \mathbf{C}_{\mathcal{N},M}$ then $e \in \mathbf{C}_{\mathcal{N},M'}\cup \mathbf{E}_{\mathcal{N},M'}$;
\item If $e \in E(M')$ and $e\in \mathbf{D}_{\mathcal{N},M}$ then $e \in \mathbf{D}_{\mathcal{N},M'}\cup \mathbf{E}_{\mathcal{N},M'}$;
\item If $e \in E(M')$ and $e\in \mathbf{E}_{\mathcal{N},M}$ then $e \in \mathbf{E}_{\mathcal{N},M'}$.
\end{enumerate}
\item\label{it:parpairdel} If $N$ is 3-connected and $|E(N)|\geq 4$ for all $N\in\mathcal{N}$, and if $\rank_M(\{e,f\}) = 1$, then $e$ and $f$ are both deletable.
\item If $N$ is 3-connected and $|E(N)|\geq 4$ for all $N\in\mathcal{N}$, and if $\rank^*_M(\{e,f\}) = 1$, then $e$ and $f$ are both contractible.
\end{enumerate}
\end{lemma}
\section{Deletion pairs and incriminating sets}\label{sec:incriminate}
The results in this section form part of the basic strategy of our proof. They are closely related to results in \citet{GGK} and \citet{HMZ11}. Our first ingredient is an easy corollary of a theorem by \citet{Whi96b}. We start by defining a \emph{deletion pair}.
\begin{definition}\label{def:delpair}
Let $M$ be a matroid having an $N$-minor. Then $\{u,v\} \subseteq E(M)$ is a \emph{deletion pair preserving} $N$ if $M\ensuremath{\!\Oldsetminus\!} \{u,v\}$ is connected and $\co(M\ensuremath{\!\Oldsetminus\!} u)$, $\co(M\ensuremath{\!\Oldsetminus\!} v)$, $\co(M\ensuremath{\!\Oldsetminus\!} \{u,v\})$ are $3$-connected and have an $N$-minor.
\end{definition}
A deletion pair is guaranteed to exist, provided that $M$ is sufficiently large and 3-connected:
\begin{theorem}[\citet{Whi96b}, Theorem 3.2]\label{thm:secondelt}
Let $M$, $N$ be matroids such that $N\ensuremath{\preceq} M$, $\rank(M) -\rank(N) \geq 3$, and both $M$ and $N$ are $3$-connected. If there exists a $u \in E(M)$ such that $\si(M\ensuremath{\!/} u)$ is $3$-connected and has an $N$-minor, then there exists a $v \in E(M)$, $v\neq u$, such that $\si(M\ensuremath{\!/} v)$ and $\si(M\ensuremath{\!/} \{u,v\})$ are both $3$-connected, and $\si(M\ensuremath{\!/}\{u,v\})$ has an $N$-minor.
\end{theorem}
\begin{corollary}\label{cor:delpairexists}
Let $M$ and $N$ be $3$-connected matroids, with $N\ensuremath{\preceq} M$, and suppose $M$ is not a wheel or a whirl. If $\rank(M) - \rank(N) \geq 3$ and $\rank(M^*) - \rank(N^*) \geq 3$, then for some $(M',N') \in \{(M,N), (M^*, N^*)\}$, $M'$ has a deletion pair $\{u,v\}$ preserving $N'$. Moreover, $\{u,v\}$ can be chosen such that $M'\ensuremath{\!\Oldsetminus\!} u$ is $3$-connected.
\end{corollary}
\begin{proof}
By the Splitter Theorem there is a $u \in E(M)$ such that either $M\ensuremath{\!\Oldsetminus\!} u$ is $3$-connected with an $N$-minor, or $M\ensuremath{\!/} u$ is $3$-connected with an $N$-minor. Using duality we may assume, without loss of generality, that the former holds. Then the dual of Theorem~\ref{thm:secondelt} implies the existence of a $v \in E(M)\ensuremath{-} u$ such that $\co(M\ensuremath{\!\Oldsetminus\!} v)$ and $\co(M\ensuremath{\!\Oldsetminus\!} \{u,v\})$ are $3$-connected with an $N$-minor. To ensure that $\{u,v\}$ is a deletion pair we need to prove that $M\ensuremath{\!\Oldsetminus\!} \{u,v\}$ is connected. But $M\ensuremath{\!\Oldsetminus\!} \{u,v\} = (M\ensuremath{\!\Oldsetminus\!} u)\ensuremath{\!\Oldsetminus\!} v$, and since $M\ensuremath{\!\Oldsetminus\!} u$ is $3$-connected, $M\ensuremath{\!\Oldsetminus\!} \{u,v\}$ is $2$-connected.
\end{proof}
In the remainder of this section $\ensuremath{\mathbb{P}}$ will be a partial field, $\matset$ will be a well-closed class of $\parf$-representable matroids, $N \in \matset$ will be a $3$-connected $\ensuremath{\mathbb{P}}$-representable matroid that is a strong $\ensuremath{\mathbb{P}}$-stabilizer for $\matset$, $M$ will be a $3$-connected matroid with an $N$-minor, and $\{u,v\}\subseteq E(M)$ will be a deletion pair preserving $N$.
Next we employ the deletion pair to create a candidate $\ensuremath{\mathbb{P}}$-representation for $M$ when $M\ensuremath{\!\Oldsetminus\!} u$ and $M\ensuremath{\!\Oldsetminus\!} v$ are $\parf$-representable.
\begin{lemma}\label{lem:delpairscale}
Let $D$, $D'$ be $X\times Y$ matrices with entries in a partial field $\ensuremath{\mathbb{P}}$. Let $u,v \in Y$ be such that
\begin{enumerate}
\item $D-u$ is scaling-equivalent to $D'-u$ and $D-v$ is scaling-equivalent to $D'-v$;
\item $G(D-\{u,v\})$ is connected.
\end{enumerate}
Then $D$ is scaling-equivalent to $D'$.
\end{lemma}
\begin{proof}
If one of $D[X,u]$ and $D[X,v]$ is an all-zero column then the result is trivially true, so we assume this is not the case. Now let $T'$ be a spanning tree for $G(D-\{u,v\})$ and let $T := T' \cup \{xu, x'v\}$ for some $x, x' \in X$ with $D_{xu} \neq 0$, $D_{x'v} \neq 0$. Then $T$ is a spanning tree for $G(D) = G(D')$. Assume, without loss of generality, that $D$ and $D'$ are $T$-normalized. Then $D-u$ and $D'-u$ are $(T\ensuremath{-} xu)$-normalized, and hence, by Lemma~\ref{lem:bippropertiesunique}, $D-u = D'-u$. Likewise $D-v = D'-v$. But then $D = D'$, and the result follows.
\end{proof}
\begin{theorem}\label{thm:uniquematrix}
Let $D$ be an $X_N\times Y_N$ $\ensuremath{\mathbb{P}}$-matrix such that $N = M[I\ D]$. Choose sets $B, E_N \subseteq E(M)$ such that $B$ is a basis of $M\ensuremath{\!\Oldsetminus\!} \{u,v\}$, $E_N\subseteq E(M)\ensuremath{-}\{u,v\}$ is such that $M_B[E_N] = N$, and $X_N\subseteq B$. Suppose $M\ensuremath{\!\Oldsetminus\!} u, M\ensuremath{\!\Oldsetminus\!} v \in \matset$.
Then there is a $B\times (E(M)\ensuremath{-} B)$ matrix $A$ with entries in $\ensuremath{\mathbb{P}}$ such that
\begin{enumerate}
\item $A-u$ and $A-v$ are $\ensuremath{\mathbb{P}}$-matrices;
\item $M[I\ (A-u)] = M\ensuremath{\!\Oldsetminus\!} u$ and $M[I\ (A-v)] = M\ensuremath{\!\Oldsetminus\!} v$;
\item $A[E_N]$ is scaling-equivalent to $D$.
\end{enumerate}
Moreover, $A$ is unique up to scaling of rows and columns.
\end{theorem}
\begin{proof}
Suppose $D$, $B$, $E_N$ are as in the theorem. Let $T$ be a spanning tree for $G(M,B)$ having $u$ and $v$ as leaves; $T$ exists since $\{u,v\}$ is a deletion pair. The fact that $N$ is a strong $\ensuremath{\mathbb{P}}$-stabilizer for $\matset$, together with the dual of Lemma \ref{lem:sistab}, shows that there is a unique $(T\ensuremath{-} u)$-normalized $\ensuremath{\mathbb{P}}$-matrix $A'$ such that $A'[E_N]$ is scaling-equivalent to $D$ and $M\ensuremath{\!\Oldsetminus\!} u = M[I\ A']$, and a unique $(T\ensuremath{-} v)$-normalized $\ensuremath{\mathbb{P}}$-matrix $A''$ such that $A''[E_N]$ is scaling-equivalent to $D$ and $M\ensuremath{\!\Oldsetminus\!} v = M[I\ A'']$. Since $N$ is a strong $\ensuremath{\mathbb{P}}$-stabilizer, also $A'-v = A''- u$. Now let $A$ be the matrix obtained from $A'$ by appending column $A''[B,v]$. Then $A$ satisfies all properties of the theorem. Uniqueness follows from Lemma~\ref{lem:delpairscale}.
\end{proof}
Most of the time we will apply Theorem~\ref{thm:uniquematrix} to matrices $D$ that do not extend to a representation of $M$. If a matrix with entries in a partial field does not represent a matroid, then it must have one of three problems, described by the next definition.
\begin{definition}\label{def:incrimi}
Let $B$ be a basis of $M$ and let $A$ be a $B\times (E(M)\ensuremath{-} B)$ matrix with entries in $\ensuremath{\mathbb{P}}$. A set $Z\subseteq E(M)$ \emph{incriminates} the pair $(M,A)$ if $A[Z]$ is square and one of the following holds:\index{incriminating set|defi}
\begin{enumerate}
\item\label{it:incrimi1} $\det(A[Z])\not\in\ensuremath{\mathbb{P}}$;
\item\label{it:incrimi2} $\det(A[Z]) = 0$ but $B\triangle Z$ is a basis of $M$;
\item\label{it:incrimi3} $\det(A[Z]) \neq 0$ but $B\triangle Z$ is dependent in $M$.
\end{enumerate}
\end{definition}
The proof of the following lemma is obvious and therefore omitted.
\begin{lemma}\label{lem:incriminate}
Let $A$ be an $X\times Y$ matrix, where $X$ and $Y$ are disjoint and $X\cup Y = E(M)$. Exactly one of the following statements is true:
\begin{enumerate}
\item $A$ is a $\ensuremath{\mathbb{P}}$-matrix and $M = M[I\ A]$;
\item some $Z\subseteq X\cup Y$ incriminates $(M,A)$.
\end{enumerate}
\end{lemma}
For the remainder of this section we will assume that $A$ is an $X\times Y$ matrix with entries in $\ensuremath{\mathbb{P}}$ such that $X$ and $Y$ are disjoint, $X\cup Y = E(M)$, and $u,v\in Y$.
It is often desirable to have a small incriminating set. If we have some information about minors of $A$ then this can be achieved by pivoting.
\begin{theorem}\label{thm:incriminatingsetsmall}
Suppose $A-u$, $A-v$ are $\ensuremath{\mathbb{P}}$-matrices and $M\ensuremath{\!\Oldsetminus\!} u = M[I\ (A-u)]$, $M\ensuremath{\!\Oldsetminus\!} v = M[I\ (A-v)]$. Suppose $Z \subseteq X\cup Y$ incriminates $(M,A)$. Then there exists an $X'\times Y'$ matrix $A'$ and $a,b \in X'$, such that $u,v \in Y'$, $A-u$ is geometrically equivalent to $A'-u$, such that $A-v$ is geometrically equivalent to $A'-v$, and such that $\{a,b,u,v\}$ incriminates $(M,A')$.
\end{theorem}
\begin{proof}
Suppose the theorem is false. Let $X,Y,A,u,v,M,Z$ form a counterexample, and suppose the counterexample was chosen such that $|Z\cap Y|$ is minimal. Clearly $u,v \in Z$. Suppose $y \in Z$ for some $y \in Y\ensuremath{-}\{u,v\}$.
\begin{claim}
Some entry of $A[X\cap Z, y]$ is nonzero.
\end{claim}
\begin{subproof}
Suppose all entries of $A[X\cap Z, y]$ equal zero. Then $\det(A[Z]) = 0$. Since $Z$ incriminates $(M,A)$, this implies that $X\triangle Z$ is a basis of $M$. Now there is an $x \in Z\cap X$ such that $B := X\triangle \{x,y\}$ is a basis of $M$. But since $u,v \not \in B$, $B$ is also a basis of $M\ensuremath{\!\Oldsetminus\!} \{u,v\}$. Since $M\ensuremath{\!\Oldsetminus\!} u = M[I\ (A-u)]$, this implies that $A_{xy} \neq 0$, a contradiction.
\end{subproof}
Now pick $x \in X\cap Z$ such that $A_{xy} \neq 0$, let $X' := X\triangle \{x,y\}$, let $Y' := Y\triangle \{x,y\}$, $A' := A^{xy}$, and let $Z' := Z\ensuremath{-} \{x,y\}$. Since $A^{xy}-u = (A-u)^{xy}$, the matrix $A'-u$ is a $\ensuremath{\mathbb{P}}$-matrix and $M\ensuremath{\!\Oldsetminus\!} u = M[I\ (A'-u)]$. Likewise $A'-v$ is a $\ensuremath{\mathbb{P}}$-matrix and $M\ensuremath{\!\Oldsetminus\!} v = M[I\ (A'-v)]$.
\begin{claim}
$Z'$ incriminates $(M,A')$.
\end{claim}
\begin{subproof}
Note that $\det(A'[Z']) = \pm A_{xy}^{-1} \det(A[Z])$. Therefore, if $\det(A[Z]) \not \in \ensuremath{\mathbb{P}}$, then certainly $\det(A'[Z']) \not \in \ensuremath{\mathbb{P}}$ and the claim follows. Otherwise, observe that $X'\triangle Z' = X \triangle Z$, so $X'\triangle Z'$ is a basis of $M$ if and only if $X\triangle Z$ is a basis. Moreover, $\det(A'[Z']) = 0$ if and only if $\det(A[Z]) = 0$. The claim now follows from Definition \ref{def:incrimi}.
\end{subproof}
But $Z' \cap Y' = (Z\cap Y) \ensuremath{-} y$, contradicting minimality of $|Z\cap Y|$.
\end{proof}
For the remainder of this section we assume $A-u$, $A-v$ are $\ensuremath{\mathbb{P}}$-matrices, $M\ensuremath{\!\Oldsetminus\!} u = M[I\ (A-u)]$, $M\ensuremath{\!\Oldsetminus\!} v = M[I\ (A-v)]$, and $M\ensuremath{\!\Oldsetminus\!} u, M\ensuremath{\!\Oldsetminus\!} v \in \matset$. We also assume that $a,b \in X$ are such that $\{a,b,u,v\}$ incriminates $(M,A)$.
Pivots were used to create a small incriminating set, but they may destroy it too. We identify some pivots that don't.
\begin{definition}\label{def:allowable}
If $x \in X, y \in Y\ensuremath{-}\{u,v\}$ are such that $A_{xy} \neq 0$, then a pivot over $xy$ is \emph{allowable} if there are $a', b' \in X\triangle \{x,y\}$ such that $\{a',b',u,v\}$ incriminates $(M,A^{xy})$.
\end{definition}
\begin{lemma}\label{lem:allowableab}
If $x \in \{a,b\}, y \in Y\ensuremath{-}\{u,v\}$ are such that $A_{xy} \neq 0$, then $\{a,b,u,v\}\triangle\{x,y\}$ incriminates $(M,A^{xy})$.
\end{lemma}
\begin{proof}
By symmetry we may assume $x = a$. Let $Z := \{a,b,u,v\}$ and $Z' := \{y,b,u,v\}$. First suppose $\det(A[Z]) \not\in\ensuremath{\mathbb{P}}$, but $\det(A^{ay}[Z'])\in \ensuremath{\mathbb{P}}$. Then $A^{ay}[Z\cup y]$ is a $\ensuremath{\mathbb{P}}$-matrix. Indeed: all entries are in $\ensuremath{\mathbb{P}}$, $\det(A^{ay}[\{y,b,a,u\}])\in\ensuremath{\mathbb{P}}$, and $\det(A^{ay}[\{y,b,a,v\}]) \in \ensuremath{\mathbb{P}}$. This is clearly impossible, since $(A^{ay})^{ya}$ is scaling-equivalent to $A$, after which Proposition~\ref{prop:pivotproper} implies that $A[Z\cup y]$ is a $\ensuremath{\mathbb{P}}$-matrix. Hence $\det(A^{ay}[Z'])\not\in\ensuremath{\mathbb{P}}$, and the lemma follows.
Next suppose that $\det(A[Z]) = 0$ and that $X\triangle Z$ is a basis of $M$. Consider $M' := M_X[Z\cup y]$. Since $\det(A[Z])\in \ensuremath{\mathbb{P}}$, $A[Z\cup y]$ is a $\ensuremath{\mathbb{P}}$-matrix. Let $N' := M[I\ A[Z\cup y]]$. We have $N' \neq M'$, since $\{u,v\}$ is a basis of $M'$ yet dependent in $N'$. But since $\{u,v\}$ is dependent in $N'$, we have $\det(A^{ay}[Z']) = 0$. Since $X\triangle Z = (X\triangle \{a,y\})\triangle Z'$, the lemma follows.
The final case, where $\det(A[Z]) \in \ensuremath{\mathbb{P}}^*$ and $B\triangle Z$ is dependent in $M$, is similar to the second and we omit the proof.
\end{proof}
\begin{lemma}\label{lem:allowablezeros}
If $x \in X\ensuremath{-} \{a,b\}, y \in Y\ensuremath{-}\{u,v\}$ are such that $A_{xy} \neq 0$ and either $A_{xu} = A_{xv} = 0$ or $A_{ay} = A_{by} = 0$, then $\{a,b,u,v\}$ incriminates $(M,A^{xy})$.
\end{lemma}
\begin{proof}
Let $Z := \{a,b,u,v\}$ and define $X' := X\triangle\{x,y\}$.
Since $A^{xy}[Z] = A[Z]$, we have $\det(A^{xy}[Z])\in \ensuremath{\mathbb{P}}$ if and only if $\det(A[Z])\in\ensuremath{\mathbb{P}}$. Therefore we only need to prove the two cases where $\det(A[Z]) \in \ensuremath{\mathbb{P}}$. Define $M' := M_X[Z\cup \{x,y\}]$.
\begin{claim}
$x$ and $y$ are either in series or in parallel in $M'$.
\end{claim}
\begin{subproof}
If $A_{ay} = A_{by} = 0$ then $x$ and $y$ are clearly in parallel, since they are in parallel in $M'\ensuremath{\!\Oldsetminus\!} v = M[I\ A[\{x,a,b,y,u]]$. Now assume $A_{xu} = A_{xv} = 0$. If $x$ and $y$ are not in series, then $\{x,y,z\}$ is a cobasis of $M'$ for some $z \in Z$. Clearly $\{y,u,v\}$ is a cobasis of $M'$, so $\{x,y,u'\}$ is a cobasis of $M'$ for some $u'\in\{u,v\}$. Without loss of generality, assume $u' = u$. But then a pivot over $xv$ should be possible in $M'\ensuremath{\!\Oldsetminus\!} u = M[I\ A[\{x,a,b,y,v\}]]$, contradicting $A_{xv} = 0$.
\end{subproof}
But now it follow that $\{x,u,v\}$ is a basis of $M'$ if and only if $\{y,u,v\}$ is a basis of $M'$, and hence that $X\triangle Z$ is a basis of $M$ if and only if $X'\triangle Z$ is a basis of $M$. The lemma follows.
\end{proof}
The next theorem gives sufficient conditions under which a certain minor of $M$ can be shown to be outside $\matset$.
\begin{theorem}\label{thm:incriminatingsetinminor}
Let $N'$ be a strong stabilizer for $\matset$ and suppose $C \subseteq E(M)$ is such that $M_X[C]$ is strictly $N'$-fragile. If there exist subsets $Z,Z_1, Z_2\subseteq E(M)$ such that
\begin{enumerate}
\item $u \in Z_1\ensuremath{-} Z_2$, $v \in Z_2\ensuremath{-} Z_1$;
\item $C\cup \{a,b\}\subseteq Z \subseteq Z_1\cap Z_2$;
\item $M_X[Z]$ is connected;
\item $M_X[Z_1]$ is $3$-connected up to series and parallel classes;
\item $M_X[Z_2]$ is $3$-connected up to series and parallel classes;
\item $\{a,b,u,v\}$ incriminates $(M_X[Z_1\cup Z_2],A[Z_1\cup Z_2])$;
\end{enumerate}
then $M_X[Z_1\cup Z_2]$ is not strongly $\parf$-stabilized by $N'$.
\end{theorem}
\begin{proof}
Let $C$, $Z_1$, and $Z_2$ be as in the theorem. Suppose that, contrary to the result claimed, $M_X[Z_1\cup Z_2]$ \emph{is} strongly $\ensuremath{\mathbb{P}}$-stabilized by $N'$. Then $M_X[Z_1\cup Z_2] = M[I\ A']$, where $A'$ is an $(X\cap (Z_1\cup Z_2)) \times (Y\cap (Z_1\cup Z_2))$ $\ensuremath{\mathbb{P}}$-matrix. Since $N'$ is a strong stabilizer for $\matset$, we may assume that $A'$ was chosen so that $A'[C] = A[C]$. By Lemma \ref{lem:sistab} and its dual, then, $A'[Z_1]$ is scaling-equivalent to $A[Z_1]$ and $A'[Z_2]$ is scaling-equivalent to $A[Z_2]$. Since $Z\subseteq Z_1 \cap Z_2$, also $A'[Z\cup u]$ is scaling-equivalent to $A[Z\cup u]$ and $A'[Z\cup v]$ is scaling-equivalent to $A[Z\cup v]$.
Since $M_X[Z]$ is connected, it follows from Lemma~\ref{lem:delpairscale} that $A'[Z\cup \{u,v\}]$ is scaling-equivalent to $A[Z\cup \{u,v\}]$. But then $\det(A'[\{a,b,u,v\}]) = p \det(A[\{a,b,u,v\}])$ for some $p \in \parf^*$, and hence $\{a,b,u,v\}$ incriminates $(M_X[Z_1\cup Z_2],A')$, a contradiction.
\end{proof}
\section{Excluded minors containing a strong stabilizer}\label{sec:thestrongproof}
The main step in our proof of Theorem \ref{thm:rotapf} is the following result:
\begin{theorem}\label{thm:rotapfN}
Let $s, t$ be positive integers, let $\ensuremath{\mathbb{P}}$ be a finitary partial field, let $\matset$ be a well-closed class of $\parf$-representable matroids, and let $\mathcal{N}$ be a set of $\parf$-representable matroids such that, for each $N' \in \mathcal{N}$,
\begin{enumerate}
\item\label{it:pfN1} $N'$ is 3-connected and non-binary;
\item\label{it:pfN2} $N'$ is a stabilizer for $\matset(\parf)$;
\item\label{it:pfN3} $N'$ is a strong stabilizer for $\matset$.
\end{enumerate}
Let $N \in \mathcal{N}$ be a matroid with the following additional property.
\begin{enumerate}\addtocounter{enumi}{3}
\item\label{it:pfN4} If $M'$ is an excluded minor for $\matset$ having an $N$-minor and $M'$ is $\parf$-representable, then either $M'$ is not strongly stabilized by $N$ or $M'$ has branch width at most $s$.
\end{enumerate}
If all strictly $\mathcal{N}$-fragile matroids have branch width at most $t$, then there is a constant $l$ depending only on $s,t, \parf, \matset, \mathcal{N},N$, such that an excluded minor $M$ for $\matset$, with $N\ensuremath{\preceq} M$, has branch width at most $l$.
\end{theorem}
Note that \eqref{it:pfN4} is trivially satisfied if $\matset$ contains all 3-connected $\parf$-representable matroids strongly stabilized by $N$. In the applications in this paper this will always be the case. Moreover, within this paper we will only apply this result with $|\mathcal{N}| = 1$. We expect that the more general version will be useful in other contexts.
The proof can be summarized as follows. First, we pick an excluded minor having an $N$-minor but big branch width, and we select a deletion pair $\{u,v\}$ preserving $N$. We construct a matrix $A$ that is close to representing $M$ and locate a small incriminating set, $\{a,b,u,v\}$. Then we identify a $3$-connected minor $M'$ using $\{a,b,u,v\}$ such that $M'\ensuremath{\!/} \{a,b\}\ensuremath{\!\Oldsetminus\!} \{u,v\}$ is $\mathcal{N}$-fragile. Now $\{u,v\}$ may not be a deletion pair for $M'$ since the connectivity of $\co(M'\ensuremath{\!\Oldsetminus\!} u)$, $\co(M'\ensuremath{\!\Oldsetminus\!} v)$, $\co(M'\ensuremath{\!\Oldsetminus\!} \{u,v\})$ may be too low. We count the $1$- and $2$-separations and find that the number does not depend on $\mathcal{N}$ or $\ensuremath{\mathbb{P}}$. But then only a constant number of blocking sequences need to be added back to $M'$ to repair the connectivity. The resulting matroid, $M''$ say, has branch width bounded by the branch width of $M'$ plus some constant. But $M''$ still has a strong stabilizer $N' \in \mathcal{N}$ as minor, and we can show $M'' \not \in \matset$, which leads to a contradiction.
\begin{proof}
Let $\parf$, $\matset$, $\mathcal{N}$, $N$, $s$, $t$ be as in the theorem. Let $r$ be an integer such that the excluded minors $M$ for $\matset$ with $\min\{\rank(M)-\rank(N), \rank(M^*)-\rank(N^*)\} < 3$ have branch width at most $r$. By Lemmas \ref{lem:exmindual} and \ref{lem:exminrankbound} there are finitely many such $M$, so $r$ exists. Let $l := \max\{r,s,t+4109\}$.
Suppose that $M$ is an excluded minor for $\matset$ having an $N$-minor, but $\bw(M) > l$. Then $\rank(M)-\rank(N)\geq 3$ and $\rank(M^*)-\rank(N^*) \geq 3$. Let $E$ be the ground set of $M$. By Corollary~\ref{cor:delpairexists}, some $M' \in \{M,M^*\}$ has a deletion pair $\{u,v\}$ such that $M'\ensuremath{\!\Oldsetminus\!} u$ is $3$-connected. By swapping $N$ with $N^*$ and $M$ with $M^*$ if necessary, we may assume $M' = M$. Pick sets $B, E_N$ such that $B$ is a basis of $M$ and $E_N \subseteq E\ensuremath{-} \{u,v\}$ is such that $M_B[E_N] \cong N$.
By \eqref{it:pfN4} and the fact that $\bw(M) > s$, $M$ is either not $\parf$-representable or $M$ is not strongly stabilized by $N$. In the latter case it follows from \eqref{it:pfN2} that $M$ is stabilized by $N$. So in both cases there must be some representation of $N$ that does not extend to a representation of $M$. Fix an $(E_N\cap B)\times (E_N\ensuremath{-} B)$ $\parf$-matrix $D$ with $N = M[I\ D]$ such that $D$ does not extend to a representation of $M$, and let $A'$ be the matrix described in Theorem~\ref{thm:uniquematrix}.
It follows that some $S\subseteq E$ incriminates $(M,A')$. Clearly $u,v \in S$. By Theorem~\ref{thm:incriminatingsetsmall}, there exists an $X\times Y$ matrix $A$ geometrically equivalent to $A'$ such that $a,b \in X$, $u,v \in Y$, and $\{a,b,u,v\}$ incriminates $(M,A)$. By Proposition \ref{prop:scalepivotcommute}, $A$ is unique up to scaling.
Let $C\subseteq E\ensuremath{-} \{u,v\}$ be a smallest possible set such that $M_X[C]$ has a minor isomorphic to a member of $\mathcal{N}$. Since $M\ensuremath{\!\Oldsetminus\!} \{u,v\}$ has an $N$-minor, $C$ exists.
\begin{claim}
$M_X[C]$ is $3$-connected.
\end{claim}
\begin{subproof}
For all $x \in C$, $M_X[C\ensuremath{-} x]$ has no minor in $\mathcal{N}$. Hence, if $x\in C\cap X$ then $x \not\in \mathbf{C}_{\mathcal{N},M}$, and if $x \in C\cap Y$ then $x \not \in \mathbf{D}_{\mathcal{N},M}$. It follows that $M_X[C]$ is strictly $\mathcal{N}$-fragile. Clearly $M_X[C]$ has no loops or coloops. By Proposition~\ref{prop:almostconn}, $M_X[C]$ is $3$-connected up to series and parallel classes. Suppose $M_X[C]$ is not $3$-connected, and let $\{e,f\}$ be a parallel pair. By Lemma~\ref{lem:almostpartition}\eqref{it:parpairdel}, $e,f \in \mathbf{D}_{\mathcal{N},M}$. Since $X$ is a basis of $M$ and $\rank_M(\{e,f\}) = 1$, $|X\cap \{e,f\}| \leq 1$, say $f \not \in X$. But then $M_X[C\ensuremath{-} f]$ has a minor in $\mathcal{N}$, a contradiction. The same argument shows that $M_X[C]$ has no series pairs.
\end{subproof}
Be aware that $M_X[C]$ may have no $N$-minor. However, it still contains \emph{some} strong stabilizer as minor. Let $N'$ be a minor of $M_X[C]$ such that $N' \in \mathcal{N}$. By our assumptions we have $\bw(M_X[C]) \leq t$.
We now refine the choice of our small incriminating set. By $d_X(U,W)$ we denote the minimal distance between the vertices indexed by $U$ and the vertices indexed by $W$ in $G(M,X)$.
\begin{assumption}\label{ass:lexmin}
$X,a,b,C$ were chosen such that $(d_X(a,C), d_X(b,C))$ is lexicographically minimal.
\end{assumption}
We now start constructing sets $Z$, $Z_1$, $Z_2$ having the properties in Theorem~\ref{thm:incriminatingsetinminor}.
\begin{claim}\label{cl:Z}
There exists a set $Z \subseteq E\ensuremath{-}\{u,v\}$, with $C\cup \{a,b\} \subseteq Z$, such that $M_X[Z]$ is connected. Moreover, $Z$ can be chosen so that $|Z| \leq |C|+8$.
\end{claim}
\begin{subproof}
Let $P_a$ be a shortest $a-C$ path in $G(M,X)$. Suppose $|P_a| = k > 3$, say $P_a = (a,x_1, x_2, x_3, \ldots, x_k)$, where $x_k \in C$. Then $x_2$ labels a row of $A$. Also $A_{x_2c} = 0$ for all $c \in C$, and $A_{ax_3} = A_{bx_3} = 0$. It follows that a pivot over $x_2x_3$ is allowable and $A^{x_2x_3}[C] = A[C]$. However, $d_{X\triangle\{x_2,x_3\}}(a,C) < d_{X}(a,C)$, a contradiction to Assumption~\ref{ass:lexmin}.
Similarly, if $P_b$ is a shortest $b-(C\cup P_a)$ path, then $|P_b| \leq 3$. Now $M_X[C\cup P_a \cup P_b]$ is connected, and the result follows.
\end{subproof}
Let $Z$ be as in Claim~\ref{cl:Z}. Note that $\bw(M_X[Z]) \leq \bw(M_X[C]) + 8$, by Proposition~\ref{prop:bwmonotone}. Since $\{u,v\}$ is a deletion pair, $\co(M\ensuremath{\!\Oldsetminus\!} v)$ is $3$-connected.
\begin{claim}\label{cl:contractoutsideC}
There is a set $S \subseteq (X \ensuremath{-} Z) \cup \{a,b\}$ such that $M_X[E\ensuremath{-} (S\cup v)]$ is $3$-connected and isomorphic to $\co(M\ensuremath{\!\Oldsetminus\!} v)$.
\end{claim}
\begin{subproof}
Let $S_1$ be a series class in $M\ensuremath{\!\Oldsetminus\!} v$. At most one element of $S_1$ is not in $X$. It follows that we can obtain a matroid isomorphic to $\co(M\ensuremath{\!\Oldsetminus\!} v)$ by contracting only elements from $X$. Let $S \subset X$ be such that $\co(M\ensuremath{\!\Oldsetminus\!} v) \cong M\ensuremath{\!/} S\ensuremath{\!\Oldsetminus\!} v$, and suppose $S$ was chosen such that $|S\cap (Z\ensuremath{-} \{a,b\})|$ is minimal. Let $x \in (X \ensuremath{-} (C\cup \{a,b\}))\cap Z$. Then $x$ is in a shortest $a-C$ path or in a shortest $b-C$ path. In either case $A[x,Y\ensuremath{-} v]$ has at least two nonzero entries. Likewise, if $x \in X\cap C$ then $A[x,Y\ensuremath{-} v]$ has at least two nonzero entries, since $M_X[C]$ is $3$-connected. It follows that, if $x \in (Z\ensuremath{-} \{a,b\})\cap S$, then also $y\in X$ for all $y$ such that $x,y$ are in series. Clearly $y \not \in Z\ensuremath{-} \{a,b\}$, as $M_X[Z\ensuremath{-} \{a,b\}]$ has no series classes. There is such a $y$ that is not in $S$. But then $M_X[Z\ensuremath{-} (S\cup v)] \cong M_X[Z\ensuremath{-} (S\triangle\{x,y\}\cup v)]$, contradicting minimality of $|S\cap (Z\ensuremath{-}\{a,b\})|$.
\end{subproof}
Let $S$ be as in Claim~\ref{cl:contractoutsideC}.
\begin{claim}\label{cl:bridgedelv}
Let $Z_0' \subseteq E\ensuremath{-} (v \cup S)$ be such that $(Z\ensuremath{-} S)\cup u \subseteq Z_0'$ and such that
$M_X[Z_0']$ has exactly $k$ distinct $2$-separations. Then there exists a set $Z_0 \subseteq E\ensuremath{-} (v\cup S)$ such that $Z_0 \supseteq Z_0'$, $M_X[Z_0]$ is $3$-connected and such that $\bw(M_X[Z_0])\leq \bw(M_X[Z_0']) + 2 k$.
\end{claim}
\begin{subproof}
The result is obvious if $k = 0$, so we suppose $k > 0$. Since $M_X[Z_0']$ is a minor of the $3$-connected matroid $M\ensuremath{\!/} S \ensuremath{\!\Oldsetminus\!} v$, no $2$-separation of $M_X[Z_0']$ is induced. Since each matroid in $\mathcal{N}$ is non-binary, $U_{2,4} \ensuremath{\preceq} N'$. It then follows from Lemma \ref{lem:uncrossed2sep} that $M_X[Z_0']$ has an uncrossed $2$-separation, say $(W_1,W_2)$. Let $v_1, \ldots, v_t$ be a blocking sequence for $(W_1, W_2)$. By Theorem~\ref{thm:blseqbranchwidth}, $\bw(M_X[Z_0'\cup \{v_1, \ldots, v_t\}]) \leq \bw(M_X[Z_0']) + 2$. By Corollary~\ref{cor:block2sep}, the number of $2$-separations in $M_X[Z_0'\cup\{v_1, \ldots, v_t\}]$ is strictly less than $k$. The result now follows by induction.
\end{subproof}
Pick $Z_0' = (Z\ensuremath{-} S) \cup u$. Then $|Z_0'| - |C| \leq 9$, by Claim \ref{cl:Z}.
By Lemma~\ref{lem:2sepbound}, $M_X[Z_0']$ has at most $\sepbound{9}$ distinct $2$-separations. Then Claim \ref{cl:bridgedelv} proves the existence of a set $Z_0\supseteq Z_0'$ such that $M_X[Z_0]$ is $3$-connected and such that $\bw(M_X[Z_0]) \leq \bw(M_X[Z_0']) + 2\cdot \sepbound{9}$.
Define $Z_1 := Z_0\cup\{a,b\}$. For all $x \in S\cap \{a,b\}$, $Z_0 \cup x$ is either $3$-connected or has a series pair. It follows that $M_X[Z_1]$ is $3$-connected up to series classes. Also, $\bw(M_X[Z_1]) \leq \bw(M_X[Z_0]) + 2$.
\begin{claim}\label{cl:bridgedelu}
Let $Z_2' \subseteq E\ensuremath{-} u$ be such that $Z\cup v \subseteq Z_2'$ and such that $M_X[Z_2']$ has exactly $k$ distinct $2$-separations. Then there exists a set $Z_2\subseteq E\ensuremath{-} u$ such that $Z_2 \supseteq Z_2'$, $M_X[Z_2]$ is $3$-connected, and $\bw(M_X[(Z_1 \ensuremath{-} u)\cup Z_2])\leq \bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2']) + 2 k$.
\end{claim}
\begin{subproof}
The result is obvious if $k = 0$, so we suppose $k > 0$. Since $M_X[Z_2']$ is a minor of the $3$-connected matroid $M\ensuremath{\!\Oldsetminus\!} u$, no $2$-separation of $M_X[Z_2']$ is induced. Again it follows from Lemma \ref{lem:uncrossed2sep} that $M_X[Z_2']$ has an uncrossed $2$-separation, say $(W_1,W_2)$. If $(W_1,W_2)$ is bridged in $M_X[(Z_1\ensuremath{-} u)\cup Z_2']$ then we set $T = \emptyset$. Otherwise let $(W_1', W_2')$ be a $2$-separation of $M_X[(Z_1\ensuremath{-} u)\cup Z_2']$ such that $W_1 \subseteq W_1'$ and $W_2 \subseteq W_2'$. Let $v'_1, \ldots, v'_{p'}$ be a blocking sequence for $(W_1', W_2')$ and set $T := \{v'_1, \ldots, v'_{p'}\}$.
Now $(W_1, W_2)$ is bridged in $M_X[(Z_1\ensuremath{-} u)\cup Z_2' \cup T]$, so there is a blocking sequence $v_1, \ldots, v_t$ contained in $Z_1\ensuremath{-} u \cup T$. By Theorem~\ref{thm:blseqbranchwidth}, $\bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2'\cup \{v_1, \ldots, v_t\}]) \leq \bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2' \cup T]) \leq \bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2']) + 2$. By Corollary~\ref{cor:block2sep}, the number of 2-separations in $M_X[Z_2'\cup\{v_1, \ldots, v_t\}]$ is strictly less than $k$. The result now follows by induction.
\end{subproof}
Pick $Z_2' := Z \cup v$. Then $|Z_2'| - |C| \leq 9$, by Claim~\ref{cl:Z}. By Lemma \ref{lem:2sepbound}, $M_X[Z_2']$ has at most $\sepbound{9}$ distinct $2$-separations. Then Claim~\ref{cl:bridgedelu} proves the existence of a set $Z_2\supseteq Z_2'$ such that $M_X[Z_2]$ is $3$-connected and such that $\bw(M_X[Z_1 \cup Z_2]) \leq \bw(M_X[(Z_1\ensuremath{-} u) \cup Z_2]) + 1 \leq \bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2']) + 2\cdot \sepbound{9}+1$.
It now follows from Theorem~\ref{thm:incriminatingsetinminor} that $M_X[Z_1\cup Z_2]$ is not strongly stabilized by $N'$, and hence $M_X[Z_1\cup Z_2]\not\in\matset$. But $M$ is an excluded minor for $\matset$, so we must have $M = M_X[Z_1\cup Z_2]$. By liberal application of Proposition~\ref{prop:bwmonotone} we can now deduce
\begin{align}
\bw(M) & = \bw(M_X[Z_1\cup Z_2])\label{eq:final1}\\
& \leq \bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2]) + 1\label{eq:final2}\\
& \leq \bw(M_X[(Z_1\ensuremath{-} u)\cup Z_2']) + 2 \cdot \sepbound{9} + 1\label{eq:final3}\\
& \leq \bw(M_X[Z_1\ensuremath{-} u]) + 2 \cdot \sepbound{9} + 2\label{eq:final4}\\
& \leq \bw(M_X[Z_1]) + 2 \cdot \sepbound{9} + 2\label{eq:final5}\\
& \leq \bw(M_X[Z_0]) + 2 \cdot \sepbound{9} + 4 \label{eq:final6}\\
& \leq \bw(M_X[Z_0']) + 4 \cdot \sepbound{9} + 4 \label{eq:final7}\\
& \leq \bw(M_X[Z_0'\ensuremath{-} u]) + 4\cdot \sepbound{9} + 5 \label{eq:final8}\\
& \leq \bw(M_X[Z]) + 4 \cdot \sepbound{9} + 5 \label{eq:final9}\\
& \leq \bw(M_X[C]) + 4 \cdot \sepbound{9} + 13 \label{eq:final10}\\
& \leq t + 4 \cdot \sepbound{9} + 13\label{eq:final11},
\end{align}
where \eqref{eq:final3} follows from Claim~\ref{cl:bridgedelu}, \eqref{eq:final4} holds because $Z_2'\ensuremath{-} (Z_1\ensuremath{-} u) = \{v\}$, \eqref{eq:final6} holds because $Z_1\ensuremath{-} Z_0 \subseteq \{a,b\}$, \eqref{eq:final7} follows from Claim~\ref{cl:bridgedelv}, \eqref{eq:final9} holds because $Z\ensuremath{-} (Z_0'\ensuremath{-} u) \subseteq \{a,b\}$, and \eqref{eq:final10} follows from Claim~\ref{cl:Z}. But this contradicts our choice of $M$, and our proof is complete.
\end{proof}
\section{Proof of Theorem \ref{thm:rotapf} and Corollary \ref{cor:infichain}}\label{sec:theproof}
\begin{proof}[Proof of Theorem \ref{thm:rotapf}]
Let $\parf$ be a finitary partial field and let $\matset$ be a well-closed class of $\parf$-representable matroids, each of which has bounded canopy. Suppose that Theorem \ref{thm:rotapf} is false for a matroid $N$. Then $N$ satisfies all conditions of the theorem, yet occurs in an infinite number of excluded minors for $\matset$. Choose $N$ with as few algebraically inequivalent representations over $\parf$ as possible.
If $N$ has a unique representation over $\parf$ then $N$ is clearly a strong stabilizer. If we apply Theorem \ref{thm:rotapfN} with $\mathcal{N}=\{N\}$ then we find that there is a constant $l$ such that excluded minors for $\matset$ with an $N$-minor have branch width at most $l$. Then Theorem \ref{thm:rotabw} implies the result.
Therefore $N$ has at least two algebraically inequivalent representations over $\matset$. Let $\matset_N \subseteq \matset$ be the smallest well-closed class containing $N$ and all matroids that are strongly stabilized by $N$. If we apply Theorems \ref{thm:rotapfN} and \ref{thm:rotabw} to $\matset_N$, again with $\mathcal{N} = \{N\}$, then we find that there are finitely many excluded minors for $\matset_N$ having an $N$-minor.
Let $N'$ be such an excluded minor. Then either $N'$ is also an excluded minor for $\matset$, or $N'\in\matset$ but $N'$ is not strongly stabilized by $N$. Assume the latter holds. We know that $N'$ is stabilized by $N$, so $N'$ must have strictly fewer algebraically inequivalent $\parf$-representations than $N$. Hence, by induction, $N'$ is contained in a finite number of excluded minors for $\matset$. It follows that $N$ is contained in only a finite number of excluded minors for $\matset$, a contradiction.
\end{proof}
A similar argument proves Corollary \ref{cor:infichain}:
\begin{proof}[Proof of Corollary \ref{cor:infichain}]
Let $\parf$ be a finitary partial field. Suppose the Bounded Canopy Conjecture holds for $\parf$, yet $\parf$ has infinitely many excluded minors. First consider the excluded minors with no $U_{2,4}$-minor. Either this set is empty (i.e. $\matset(\parf)$ contains all binary matroids) or it is $\{F_7, F_7^*\}$ (since matroids with no minor in $\{U_{2,4}, F_7, F_7^*\}$ are regular and hence certainly $\parf$-representable). Hence infinitely many excluded minors contain $U_{2,4}$.
Now consider the following algorithm. Initially, define $\mathcal{S} := \{ U_{2,4}\}$. While $\mathcal{S} \neq \emptyset$, do the following. Take $N \in \mathcal{S}$. Let $\matset_N$ be the smallest well-closed class in $\matset(\parf)$ such that every $\parf$-representable matroid stabilized by $N$ is in $\matset_N$. By Theorem \ref{thm:rotapf}, finitely many excluded minors for $\matset_N$ have an $N$-minor. Let $\{M_1, \ldots, M_k\}$ be these excluded minors, and let $\{M_{i_1}, \ldots, M_{i_l}\}$ be the subset that is representable over $\parf$. By definition of $\matset_N$, none of these is stabilized by $N$. Replace $\mathcal{S}$ by $(\mathcal{S}\ensuremath{-} \{N\}) \cup \{M_{i_1}, \ldots, M_{i_l}\}$ and continue.
Since $\matset(\parf)$ has infinitely many excluded minors, this algorithm does not terminate. It is now straightforward to extract an infinite chain as in the corollary.
\end{proof}
\section{Applications}\label{sec:examples}
In all examples presented here we will have a strong stabilizer at our disposal, so we can apply Theorem \ref{thm:rotapfN}. An advantage of this is that we only need $N$ to have bounded canopy, which we can actually prove in a few cases.
\subsection{Excluded minors for the classes of near-regular and $\ensuremath{\sqrt[6]{1}}$ matroids}
Near-regular matroids were introduced in \cite{Whi95} as the class of matroids representable over a certain partial field that we denote here by $\mathds{U}_1$. It turns out that the class of near-regular matroids is exactly the class of matroids representable over all fields of size at least $3$. These representations can be obtained from partial-field homomorphisms, so $\mathds{U}_1$ is finitary. We apply Theorem~\ref{thm:rotapfN} to give an alternative proof of the following result:\index{excluded minor}
\begin{theorem}[\citet{HMZ11}]\label{thm:exminnreg}
The class $\matset(\mathds{U}_1)$ has a finite number of excluded minors.
\end{theorem}
First we need to find the structure of $U_{2,4}$-fragile matroids.
\begin{lemma}\label{lem:U24fragile}
Let $M$ be a $3$-connected $U_{2,4}$-fragile matroid that has no minor isomorphic to $U_{2,6}$ or $U_{4,6}$. Then exactly one of the following holds.
\begin{enumerate}
\item $M$ has rank or corank two;
\item $M$ has a minor isomorphic to $F_7^-$ or $(F_7^-)^*$;
\item $M$ has rank at least 3 and is a whirl.
\end{enumerate}
\end{lemma}
The proof follows easily from the following result:
\begin{lemma}[\citet{GGK}, Lemma 3.3]\label{lem:notwhirlthingy}
Let $M$ be a $3$-connected, non-binary matroid that is not a whirl. Then $M$ has a minor in the set
\begin{align*}
\{U_{2,5}, U_{3,5}, F_7^-, (F_7^-)^*, P_7, P_7^*, O_7, O_7^*\}.
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:U24fragile}]
Suppose that the lemma is false, and let $M$ be a matroid that is not in one of the classes mentioned. Then $M$ must have rank and corank at least $3$. It is easily checked that each of $P_7$, $O_7$, and their duals has an element that is both deletable and contractible, so by Lemma \ref{lem:notwhirlthingy}, $M$ must have a $U_{2,5}$- or $U_{3,5}$-minor.
By the Splitter Theorem, $M$ must have a one-element extension of $U_{n-2,n}$ or a one-element coextension of $U_{2,n}$ as a minor, where $n \geq 5$. It is readily checked that $M$ then has a minor in $P_6, Q_6, U_{3,6}$, each of which has an element that is both deletable and contractible, a contradiction.
\end{proof}
\begin{lemma}\label{lem:nregU24}
Let $M$ be an excluded minor for $\matset(\mathds{U}_1)$. If $M \not\in \{F_7,F_7^*\}$, then $M$ has a $U_{2,4}$-minor.
\end{lemma}
\begin{proof}
It is readily checked that $F_7$ is an excluded minor for $\matset(\mathds{U}_1)$. But if $M$ has no minor in $\{F_7, F_7^*, U_{2,4}\}$, then $M$ is regular and hence certainly near-regular.
\end{proof}
\begin{lemma}\label{lem:nregalmostU24whirl}
If $M \in \matset(\mathds{U}_1)$ is $3$-connected and strictly $U_{2,4}$-fragile, then $M$ is a whirl.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:nregalmostU24whirl}]
The matroids $U_{2,5}$, $F_7^-$, and their duals are not near-regular. The result follows from Lemma \ref{lem:U24fragile}.
\end{proof}
\begin{lemma}[\citet{GOVW98}]
The matroid $U_{2,4}$ is a strong stabilizer for $\matset(\mathds{U}_1)$.
\end{lemma}
\begin{proof}
Since $U_{2,4}$ has no near-regular $3$-connected single-element extensions or coextensions, the stabilizer theorem from \cite{Whi96} immediately implies that $U_{2,4}$ is a stabilizer. Since $U_{2,4}$ is uniquely representable over $\mathds{U}_1$, it is strong.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:exminnreg}]
Lemma~\ref{lem:nregU24} implies that finitely many excluded minors have no $U_{2,4}$-minor. But $U_{2,4}$ is non-binary, $3$-connected, a strong stabilizer, and has bounded canopy over $\mathds{U}_1$ (by Lemma~\ref{lem:nregalmostU24whirl} and Lemma~\ref{lem:bwwhirl}). Hence Theorems \ref{thm:rotapfN} and \ref{thm:rotabw} imply that finitely many excluded minors do have a $U_{2,4}$-minor, so the result follows.
\end{proof}
Let $\mathds{S}$ be the sixth-roots-of-unity partial field introduced by Whittle \cite{Whi97}. He showed that $\matset(\mathds{S})$ equals the set of matroids representable over both $\GF(3)$ and $\GF(4)$. All results above remain valid if we replace $\mathds{U}_1$ by $\mathds{S}$. Hence we also have the following result by \citet{GGK}:
\begin{theorem}\label{thm:exminsru}
The class $\matset(\mathds{S})$ has a finite number of excluded minors.
\end{theorem}
\subsection{Excluded minors for the class of quaternary matroids}
Using almost the same arguments as in the previous section we can give an alternative proof of the following result by \citet{GGK}:
\begin{theorem}[\citet{GGK}]\label{thm:exminGF4}
The class $\matset(\GF(4))$ has a finite number of excluded minors.
\end{theorem}
\begin{lemma}\label{lem:GF4U24}
Let $M$ be an excluded minor for $\matset(\GF(4))$. Then $M$ has a $U_{2,4}$-minor.
\end{lemma}
\begin{proof}
If $M$ has no $U_{2,4}$-minor then $M$ is binary and hence certainly $\GF(4)$-representable.
\end{proof}
\begin{lemma}
The matroid $U_{2,4}$ is a strong stabilizer for $\matset(\GF(4))$.
\end{lemma}
\begin{proof}
\citet{Whi96b} proved that $U_{2,4}$ is a $\GF(4)$-stabilizer. Since $U_{2,4}$ is uniquely representable over $\GF(4)$ (cf. \citet{Ka88}), it is also strong.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:exminGF4}]
Lemma~\ref{lem:GF4U24} implies that all excluded minors have a $U_{2,4}$-minor. But $U_{2,4}$ is non-binary, $3$-connected, a strong stabilizer, and has bounded canopy over $\GF(4)$ (by Lemma~\ref{lem:U24fragile}, the fact that $F_7^-$ and $(F_7^-)^*$ themselves are excluded minors for $\matset(\GF(4))$, and Lemma~\ref{lem:bwwhirl}). Hence Theorems \ref{thm:rotapfN} and \ref{thm:rotabw} imply that finitely many excluded minors do have a $U_{2,4}$-minor, so the result follows.
\end{proof}
\section{On Rota's Conjecture for quinary matroids}
We will now prove Theorem \ref{thm:gf5bcc} from the introduction. First we need to deal with certain degenerate cases. We will use the following explicit excluded-minor characterizations:
\begin{theorem}[\citet{Tut65}]
The excluded minors for the class of regular matroids are $U_{2,4}$, $F_7$, and $F_7^*$.
\end{theorem}
\begin{theorem}[\citet{Bix79,Sey79}]
The excluded minors for $\matset(\GF(3))$ are $U_{2,5}$, $U_{3,5}$, $F_7$, and $F_7^*$.
\end{theorem}
\begin{theorem}[\citet{HMZ11}]
The excluded minors for the class of near-regular matroids are $U_{2,5}$, $U_{3,5}$, $F_7$, $F_7^*$, $F_7^-$, $(F_7^-)^*$, $P_8$, $\AG(2,3)\ensuremath{\!\Oldsetminus\!} e$, $(\AG(2,3)\ensuremath{\!\Oldsetminus\!} e)^*$, and $\varDelta_T(\AG(2,3)\ensuremath{\!\Oldsetminus\!} e)$.
\end{theorem}
\begin{lemma}\label{lem:noU25U35}
Conjecture \ref{con:boundcanopy} implies that finitely many excluded minors for $\matset(\GF(5))$ have no minor isomorphic to $U_{2,5}$ and $U_{3,5}$.
\end{lemma}
\begin{proof}
Let $M$ be an excluded minor for $\matset(\GF(5))$ having no minor isomorphic to $U_{2,5}$ and no minor isomorphic to $U_{3,5}$. It is well-known that $F_7$ and $F_7^*$ are excluded minors for $\matset(\GF(5))$, so assume $M$ does not have a minor isomorphic to these two matroids either. Then $M$ is ternary. The class of matroids representable over both $\GF(3)$ and $\GF(5)$ is the class of dyadic matroids. Hence $M$ is an excluded minor for this class.
If $M$ has no minor in $\{F_7^-, (F_7^-)^*, P_8, \AG(2,3)\ensuremath{\!\Oldsetminus\!} e, (\AG(2,3)\ensuremath{\!\Oldsetminus\!} e)^*, \varDelta_T(\AG(2,3)\ensuremath{\!\Oldsetminus\!} e)\}$ then $M$ is near-regular, and hence certainly quinary. Of this list, only the first three matroids are quinary. But each of these is a stabilizer for the class of dyadic matroids (see \citet{PZ08conf}), so Theorem \ref{thm:rotapf} implies that finitely many excluded minors have these as a minor, provided that Conjecture \ref{con:boundcanopy} is true for $\GF(3)$ or for $\GF(5)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:gf5bcc}]
Suppose Conjecture~\ref{con:boundcanopy} holds for $\GF(5)$. By Lemma \ref{lem:noU25U35} all but finitely many excluded minors for $\matset(\GF(5))$ have no minor isomorphic to $U_{2,5}$.
Now $U_{2,5}$ is a stabilizer for $\matset(\GF(5))$ (see \citet{Whi96b}), so finitely many excluded minors for $\matset(\GF(5))$ have a $U_{2,5}$-minor, by Theorem~\ref{thm:rotapf}. This concludes the proof.
\end{proof}
\paragraph{Acknowledgements.} We thank the anonymous referee for meticulously reading the manuscript, and for helpful suggestions to improve the clarity.
\renewcommand{\Dutchvon}[2]{#1}
\end{document} | math | 91,108 |
\begin{equation}gin{document}
\begin{equation}gin{abstract}
We are concerned with the instability of a generic compressible two--fluid
model in the whole space $\mathbb{R}^3$, where the capillary pressure $f(\alphalpha^-\rhoho^-)=P^+-P^-\neq 0$ is taken into account.
For the case that the capillary pressure is a strictly decreasing
function near the equilibrium, namely, $f'(1)<0$,
[Evje--Wang--Wen, Arch Rational Mech Anal
221:1285--1316, 2016] established global stability of the constant equilibrium state for the three--dimensional
Cauchy problem under some
smallness assumptions. Recently, [Wu--Yao--Zhang, arXiv:2204.10706] proved global stability of the constant equilibrium state for the case $P^+=P^-$ (corresponding to $f'(1)=0$).
In this work, we investigate the instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, $f'(1)>0$.
First, by employing Hodge decomposition technique and making detailed analysis of the Green's function for the corresponding linearized system, we construct solutions
of the linearized problem that grow exponentially in time in the Sobolev space $H^k$, thus leading to a global
instability result for the linearized problem.
Moreover, with the help of the global
linear instability result and a local existence theorem of classical solutions to the original nonlinear system, we can then
show the instability of the nonlinear problem in the sense of Hadamard by making a delicate analysis on the properties of the semigroup.
Therefore, our result shows that for the case $f'(1)>0$, the constant equilibrium state of the two--fluid model is linearly globally unstable and nonlinearly locally unstable
in the sense of Hadamard, which is in contrast to
the cases $f'(1)<0$ ([Evje--Wang--Wen, Arch Rational Mech Anal
221:1285--1316, 2016]) and $P^+=P^-$ (corresponding to $f'(1)=0$) ([Wu--Yao--Zhang, arXiv:2204.10706]) where the constant equilibrium state of the two--fluid model equationref{1.5} is nonlinearly globally stable.
\end{equation}d{abstract}
\maketitle
\section{\leftline {\bf{Introduction.}}}
\setcounter{equation}{0}
\subsection{Background and motivation}
As is well--known, most of the flows in nature are multi--fluid
flows. Such a terminology includes the flows of non--miscible fluids
such as air and water; gas, oil and water. For the flows of miscible
fluids, they usually form a ``new" single fluid possessing its own
rheological properties. One interesting example is the stable
emulsion between oil and water which is a non--Newtonian fluid, but
oil and water are Newtonian ones.\partialar
One of the classic examples of multi--fluid flows is
small amplitude waves propagating at the interface between air and
water, which is called a separated flow. In view of modeling, each
fluid obeys its own equation and couples with each other through the
free surface in this case. Here, the motion of the fluid is governed
by the pair of compressible Euler equations with free surface:
\begin{equation}gin{align}
\partialartial_{t} \rhoho_{i}+\nablabla \cdot\left(\rhoho_{i} v_{i}\rhoight) &=0, \quad i=1,2,\label{1.1} \\
\partialartial_{t}\left(\rhoho_{i} v_{i}\rhoight)+\nablabla \cdot\left(\rhoho_{i} v_{i} \otimes v_{i}\rhoight)+\nablabla p_i &=-g\rhoho_{i}
e_3\partialm F_D.\label{1.2}
\end{equation}d{align}
In above equations, $\rhoho_1$ and $v_1$ represent the density and
velocity of the upper fluid (air), and $\rhoho_2$ and $v_2$ denote
the density and velocity of the lower fluid (water). $p_{i}$ denotes
the pressure. $-g\rhoho_{i} e_3$ is the gravitational force with the
constant $g>0$ the acceleration of gravity and $e_3$ the vertical
unit vector, and $F_D$ is the drag force. As mentioned before, the
two fluids (air and water) are separated by the unknown free surface
$z=\eta(x, y, t)$, which is advected with the fluids according to
the kinematic relation:
\begin{equation}gin{equation}\partialartial_t\eta=v_{1,z}-v_{1,x}\partialartial_x \eta-v_{1, y}\partialartial_y \eta\label{1.3}\end{equation}d{equation}
on two sides of the surface $z=\eta$ and the pressure is continuous
across this surface.\partialar When the wave's amplitude becomes large
enough, wave breaking may happen. Then, in the region around the
interface between air and water, small droplets of liquid appear in
the gas, and bubbles of gas also appear in the liquid. These
inclusions might be quite small. Due to the appearances of collapse
and fragmentation, the topologies of the free surface become quite
complicated and a wide range of length scales are involved.
Therefore, we encounter the situation where two--fluid models become
relevant if not inevitable. The classic approach to simplify the
complexity of multi--phase flows and satisfy the engineer's need of
some modeling tools is the well--known volume--averaging method (see
\cite{Ishii1, Prosperetti} for details). Thus, by performing such a
procedure, one can derive a model without surface: a two--fluid
model. More precisely, we denote $\alphalpha^{\partialm}$ by the volume
fraction of the liquid (water) and gas (air), respectively.
Therefore, $\alphalpha^++\alphalpha^-=1$. Applying the volume--averaging
procedure to the equations equationref{1.1} and equationref{1.2} leads to the
following generic compressible two--fluid model:
\begin{equation}gin{equation}\label{1.4}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t}\left(\alphalpha^{\partialm} \rhoho^{\partialm}\rhoight)+\operatorname{div}\left(\alphalpha^{\partialm} \rhoho^{\partialm} u^{\partialm}\rhoight)=0, \\
\partialartial_{t}\left(\alphalpha^{\partialm} \rhoho^{\partialm} u^{\partialm}\rhoight)+\operatorname{div}\left(\alphalpha^{\partialm} \rhoho^{\partialm} u^{\partialm} \otimes u^{\partialm}\rhoight)
+\alphalpha^{\partialm} \nablabla P^\partialm=-g\alphalpha^{\partialm}\rhoho^{\partialm} e_3\partialm F_D.
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
\partialar
We have already discussed the case of water waves, where a separated
flow can lead to a two--fluid model from the viewpoint of practical
modeling. As mentioned before, two--fluid flows are very common in
nature, but also in various industry applications such as nuclear
power, chemical processing, oil and gas manufacturing. According to
the context, the models used for simulation may be very different.
However, averaged models share the same structure as equationref{1.4}. By
introducing viscosity effects and capillary pressure effects, one can generalize the above system
equationref{1.4} to
\begin{equation}gin{equation}\label{1.5}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t}\left(\alphalpha^{\partialm} \rhoho^{\partialm}\rhoight)+\operatorname{div}\left(\alphalpha^{\partialm} \rhoho^{\partialm} u^{\partialm}\rhoight)=0, \\
\partialartial_{t}\left(\alphalpha^{\partialm} \rhoho^{\partialm} u^{\partialm}\rhoight)+\operatorname{div}\left(\alphalpha^{\partialm} \rhoho^{\partialm} u^{\partialm} \otimes u^{\partialm}\rhoight)
+\alphalpha^{\partialm} \nablabla P^{\partialm}\left(\rhoho^{\partialm}\rhoight)=\operatorname{div}\left(\alphalpha^{\partialm} \thetaau^{\partialm}\rhoight), \\
P^{+}\left(\rhoho^{+}\rhoight)-P^{-}\left(\rhoho^{-}\rhoight)=f\left(\alphalpha^{-}
\rhoho^{-}\rhoight),
\end{equation}d{array}\rhoight.
\end{equation}d{equation} where
$\rhoho^{\partialm}(x, t) \geqq 0, u^{\partialm}(x, t)$ and
$P^{\partialm}\left(\rhoho^{\partialm}\rhoight)=A^{\partialm}\left(\rhoho^{\partialm}\rhoight)^{\bar{\gamma}^{\partialm}}$
denote the densities, the velocities of each phase, and the two
pressure functions, respectively. $\bar{\gamma}^{\partialm} \geqq 1,
A^{\partialm}>0$ are positive constants. In what follows, we set
$A^{+}=A^{-}=1$ without loss of any generality. As in \cite{Evje9},
we assume that the capillary pressure $f$ belongs to $C^{3}([0, \infty))$. Moreover,
$\thetaau^{\partialm}$ are the viscous stress tensors
\begin{equation}gin{equation}\label{1.6}
\thetaau^{\partialm}:=\mu^{\partialm}\left(\nablabla u^{\partialm}+\nablabla^{t}
u^{\partialm}\rhoight)+\lambda^{\partialm} \operatorname{div} u^{\partialm} \mathrm{Id},
\end{equation}d{equation}
where the constants $\mu^{\partialm}$ and $\lambda^{\partialm}$ are shear and
bulk viscosity coefficients satisfying the physical condition:
$\mu^{\partialm}>0$ and $2 \mu^{\partialm}+3 \lambda^{\partialm} \geqq 0,$ which
implies that $\mu^{\partialm}+\lambda^{\partialm}>0.$ For more information
about this model, we refer to \cite{Brennen1, Bresch1, Bresch2,
Friis1, Ishii1, Prosperetti, Raja} and references therein.
However, it is well--known
that as far as mathematical analysis of two--fluid model is
concerned, there are many technical challenges. Some of them
involve, for example:
\begin{equation}gin{itemize}
\item The two--fluid model is a partially dissipative system.
More precisely, there is no dissipation on the mass conservation
equations, whereas the momentum equations have viscosity
dissipations;
\item The corresponding linear system of the model has
zero eigenvalue, which makes mathematical analysis (well--posedness
and stability) of the model become quite difficult and complicated;
\item Transition to single--phase regions, i.e, regions where the mass
$\alphalpha^{+} \rhoho^{+}$ or $\alphalpha^{-} \rhoho^{-}$ becomes zero, may
occur when the volume fractions $\alphalpha^{\partialm}$ or the densities
$\rhoho^{\partialm}$ become zero;
\item The system is non--conservative, since the non--conservative terms $\alphalpha^{\partialm} \nablabla
P^{\partialm}$ are involved in the momentum equations. This brings various
mathematical difficulties for us to employ methods used
for single phase models to the two--fluid model.
\end{equation}d{itemize}\partialar
For the case that the capillary pressure is a strictly decreasing
function near the equilibrium, namely, $f'(1)<0$, Evje--Wang--Wen \cite{Evje9} obtained global stability of the constant equilibrium state for the three--dimensional
Cauchy problem of the two--fluid model equationref{1.5} under the assumption that the initial perturbation is small in $H^2$-norm and bounded in $L^1$-norm. It should be noted that as pointed out by Evje--Wang--Wen in \cite{Evje9}, the assumption $f'(1)<0$
played a crucial role in their analysis and appeared to have an essential stabilization effect on the
model in question.
Bretsch et al. in
the seminal work \cite{Bresch1} considered a model similar to
equationref{1.5}. More specifically, they made the following assumptions:
\begin{equation}gin{itemize} \item $P^{+}=P^{-}$ (particularly, $f'(1)=0$ in this case);\\
\item inclusion of viscous terms of the form equationref{1.2} where $\mu^{\partialm}$
depends on densities $\rhoho^{\partialm}$ and $\lambda^{\partialm}=0$;\\
\item inclusion of a third order derivative of $\alphalpha^{\partialm}
\rhoho^{\partialm}$, which are so--called internal capillary forces
represented by the well--known Korteweg model on each phase.
\end{equation}d{itemize}
They obtained the global weak solutions in the periodic domain with
$1<\overlineerline{\gamma}^{\partialm}< 6$. It is worth
mentioning that the method of \cite{Bresch1} doesn't work for the case without
the internal capillary forces. Later, Bresch--Huang--Li
\cite{Bresch2} established the global existence of weak solutions in
one space dimension without the internal capillary forces when
$\overlineerline{\gamma}^{\partialm}>1$ by taking
advantage of the one space dimension. However, the method of \cite{Bresch2} relies
crucially on the advantage of one space dimension, and particularly cannot be applied for high
dimensional problem. Recently, Wu--Yao--Zhang \cite{WYZ} showed
the global stability of the constant equilibrium state in three space dimension by
exploiting the
dissipation structure of the model (with $P^+=P^-$ and without internal capillary forces) and making full use of several key observations.
For the case of
the special density-dependent viscosities with equal viscosity coefficients and the case of general constant
viscosities, Cui--Wang--Yao--Zhu
\cite{c1} and Li--Wang--Wu--Zhang \cite {LWWZ} proved the global stability of the constant equilibrium state
for the three--dimensional
Cauchy problem with the internal capillary forces, respectively.
\partialar
To sum up, the works \cite{Evje9} and \cite{WYZ} rely essentially on the assumption $f'(1)<0$ and
$P^+=P^-$ (corresponding to $f'(1)=0$). Therefore, a natural and
important problem is that what will happen for the case that the capillary pressure is a strictly increasing
function near the equilibrium, namely, $f'(1)>0$. That is to say,
what about the stability of three--dimenional
Cauchy problem to the two--fluid model equationref{1.5} with $f'(1)>0$. The main purpose of this work is to give a definite answer to this issue.
More precisely, we first employ Hodge decomposition technique and make detailed analysis of the Green's function for the corresponding linearized system to construct solutions
of the linearized problem that grow exponentially in time in the Sobolev space $H^k$, thus leading to a global
instability result for the linearized problem.
Then, based on the global
linear instability result and a local existence theorem of classical solutions to the original nonlinear system, we can
prove the instability of the nonlinear problem in the sense of Hadamard by making a delicate analysis on the properties of the semigroup.
Therefore, our result shows that for the case $f'(1)>0$, the constant equilibrium state of the two--fluid model equationref{1.5} is linearly globally unstable and nonlinearly locally unstable
in the sense of Hadamard, which is in contrast to
the cases $f'(1)<0$ (\cite{Evje9}) and $P^+=P^-$ (corresponding to $f'(1)=0$) (\cite{WYZ}) where the constant equilibrium state of the two--fluid model equationref{1.5} is nonlinearly globally stable.
\subsection{New formulation of system equationref{1.5} and Main Results}
In this subsection, we devote ourselves to reformulating the system
equationref{1.5} and stating the main results. To begin with, noting
the relation between the pressures of equationref{1.5}$_3$, one has
\begin{equation}gin{equation}\label{1.7}
\mathrm{d} P^{+}-\mathrm{d} P^{-}=\mathrm{d} f\left(\alphalpha^{-}
\rhoho^{-}\rhoight),
\end{equation}d{equation}
where $P^{\partialm}:=P^{\partialm}\left(\rhoho^{\partialm}\rhoight).$ It is clear that
\[
\mathrm{d} P^{+}=s_{+}^{2} \mathrm{d} \rhoho^{+}, \quad \mathrm{d}
P^{-}=s_{-}^{2} \mathrm{d} \rhoho^{-}, \quad \thetaext { where }
s_{\partialm}^{2}:=\fracrac{\mathrm{d} P^{\partialm}}{\mathrm{d}
\rhoho^{\partialm}}\left(\rhoho^{\partialm}\rhoight)=\bar{\gamma}^{\partialm}
\fracrac{P^{\partialm}\left(\rhoho^{\partialm}\rhoight)}{\rhoho^{\partialm}}.
\]
Here $s_{\partialm}$ represent the sound speed of each phase respectively.
Motivated by \cite{Bresch1}, we introduce the fraction densities
\begin{equation}gin{equation}\label{1.8}
R^{\partialm}=\alphalpha^{\partialm} \rhoho^{\partialm},
\end{equation}d{equation}
which together with the fact that $\alphalpha^++\alphalpha^-=1$ gives
\begin{equation}gin{equation}\label{1.9}
\mathrm{d} \rhoho^{+}=\fracrac{1}{\alphalpha_{+}}\left(\mathrm{d}
R^{+}-\rhoho^{+} \mathrm{d} \alphalpha^{+}\rhoight), \quad \mathrm{d}
\rhoho^{-}=\fracrac{1}{\alphalpha_{-}}\left(\mathrm{d} R^{-}+\rhoho^{-}
\mathrm{d} \alphalpha^{+}\rhoight).
\end{equation}d{equation}
By virtue of equationref{1.7} and equationref{1.9}, we finally get
\begin{equation}gin{equation}\label{1.10}
\mathrm{d} \alphalpha^{+}=\fracrac{\alphalpha^{-} s_{+}^{2}}{\alphalpha^{-}
\rhoho^{+} s_{+}^{2}+\alphalpha^{+} \rhoho^{-} s_{-}^{2}} \mathrm{d}
R^{+}-\fracrac{\alphalpha^{+} \alphalpha^{-}}{\alphalpha^{-} \rhoho^{+}
s_{+}^{2}+\alphalpha^{+} \rhoho^{-}
s_{-}^{2}}\left(\fracrac{s_{-}^{2}}{\alphalpha^{-}}+f^{\partialrime}\rhoight)
\mathrm{d} R^{-}. \end{equation}d{equation}
Substituting equationref{1.10} into equationref{1.9}, we deduce
the following expressions:
\[
\mathrm{d} \rhoho^{+}=\fracrac{\rhoho^{+} \rhoho^{-}
s_{-}^{2}}{R^{-}\left(\rhoho^{+}\rhoight)^{2}
s_{+}^{2}+R^{+}\left(\rhoho^{-}\rhoight)^{2} s_{-}^{2}}\left(\rhoho^{-}
\mathrm{d} R^{+}+\left(\rhoho^{+}+\rhoho^{+} \fracrac{\alphalpha^{-}
f^{\partialrime}}{s_{-}^{2}}\rhoight) \mathrm{d} R^{-}\rhoight),
\]
and
\[
\mathrm{d} \rhoho^{-}=\fracrac{\rhoho^{+} \rhoho^{-}
s_{+}^{2}}{R^{-}\left(\rhoho^{+}\rhoight)^{2}
s_{+}^{2}+R^{+}\left(\rhoho^{-}\rhoight)^{2} s_{-}^{2}}\left(\rhoho^{-}
\mathrm{d} R^{+}+\left(\rhoho^{+}-\rhoho^{-} \fracrac{\alphalpha^{+}
f^{\partialrime}}{s_{+}^{2}}\rhoight) \mathrm{d} R^{-}\rhoight),
\]
which together with equationref{1.7} gives the pressure differential
$\mathrm{d} P^{\partialm}$
\[
\mathrm{d} P^{+}=\mathcal{C}^{2}\left(\rhoho^{-} \mathrm{d}
R^{+}+\left(\rhoho^{+}+\rhoho^{+} \fracrac{\alphalpha^{-}
f^{\partialrime}}{s_{-}^{2}}\rhoight) \mathrm{d} R^{-}\rhoight) ,\] and
\[
\mathrm{d} P^{-}=\mathcal{C}^{2}\left(\rhoho^{-} \mathrm{d}
R^{+}+\left(\rhoho^{+}-\rhoho^{-} \fracrac{\alphalpha^{+}
f^{\partialrime}}{s_{+}^{2}}\rhoight) \mathrm{d} R^{-}\rhoight) ,\] where
\[
\mathcal{C}^{2}:=\fracrac{s_{-}^{2} s_{+}^{2}}{\alphalpha^{-} \rhoho^{+}
s_{+}^{2}+\alphalpha^{+} \rhoho^{-} s_{-}^{2}}.\]\partialar \noindent Next, by noting the
fundamental relation: $\alphalpha^++\alphalpha^-=1$, we can get the
following equality:
\begin{equation}gin{equation}\label{1.11}
\fracrac{R^+}{\rhoho^+}+\fracrac{R^-}{\rhoho^-}=1, ~~\hbox{and thus}~~
\rhoho^-=\fracrac{R^-\rhoho^+}{\rhoho^+-R^+}.\end{equation}d{equation} Then, it holds
from the pressure relation $equationref{1.5}_3$ that
\begin{equation}gin{equation}\label{1.12}
\varphi(\rhoho^+, R^+,
R^-):=P^+(\rhoho^+)-P^-{\left(\fracrac{R^-\rhoho^+}{\rhoho^+-R^+}\rhoight)}-f({R^-})=0.
\end{equation}d{equation}
\noindent Thus, we can employ the implicit function theorem to
define $\rhoho^{+}$. To see this, by differentiating the above
equation with respect to $\rhoho^{+}$ for given $R^{+}$ and $R^{-}$,
we get
\[
\fracrac{\partialartial\varphi}{\partialartial\rhoho^+}(\rhoho^+, R^+,
R^-)=s_{+}^{2}+s_{-}^{2} \fracrac{R^{-}
R^{+}}{\left(\rhoho^{+}-R^{+}\rhoight)^{2}},
\]
which is positive for any $\rhoho^{+}\in(R^+, +\infty)$ and
$R^{\partialm}>0.$ This together with the implicit function theorem
implies that $\rhoho^{+}=\rhoho^{+}\left(R^{+}, R^{-}\rhoight)
\in\left(R^{+},+\infty\rhoight)$ is the unique solution of the
equation equationref{1.12}. By virtue of equationref{1.8}, equationref{1.12} and
the
fundamental fact that $\alphalpha^++\alphalpha^-=1$, $\rhoho^{-}$ and $\alphalpha^{\partialm}$ can be defined by
\[
\begin{equation}gin{aligned}
\rhoho^{-}\left(R^{+}, R^{-}\rhoight) &=\fracrac{R^{-} \rhoho^{+}\left(R^{+}, R^{-}\rhoight)}{\rhoho^{+}\left(R^{+}, R^{-}\rhoight)-R^{+}}, \\
\alphalpha^{+}\left(R^{+}, R^{-}\rhoight) &=\fracrac{R^{+}}{\rhoho^{+}\left(R^{+}, R^{-}\rhoight)}, \\
\alphalpha^{-}\left(R^{+}, R^{-}\rhoight)
&=1-\fracrac{R^{+}}{\rhoho^{+}\left(R^{+},
R^{-}\rhoight)}=\fracrac{R^{-}}{\rhoho^{-}\left(R^{+}, R^{-}\rhoight)}.
\end{equation}d{aligned}
\]
We refer the readers to [\cite{Bresch2}, P. 614] for more details.
\partialar
Therefore, we can rewrite system equationref{1.5} into the following
equivalent form:
\begin{equation}gin{equation}\label{1.13}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t} R^{\partialm}+\operatorname{div}\left(R^{\partialm} u^{\partialm}\rhoight)=0, \\
\partialartial_{t}\left(R^{+} u^{+}\rhoight)+\operatorname{div}\left(R^{+} u^{+} \otimes u^{+}\rhoight)+\alphalpha^{+} \mathcal{C}^{2}\left[\rhoho^{-} \nablabla R^{+}+\left(\rhoho^{+}+\rhoho^{+} \fracrac{\alphalpha^{-} f^{\partialrime}}{s_{-}^{2}}\rhoight) \nablabla R^{-}\rhoight] \\
\hspace{2.5cm}=\operatorname{div}\left\{\alphalpha^{+}\left[\mu^{+}\left(\nablabla u^{+}+\nablabla^{t} u^{+}\rhoight)
+\lambda^{+} \operatorname{div} u^{+} \operatorname{Id}\rhoight]\rhoight\}, \\
\partialartial_{t}\left(R^{-} u^{-}\rhoight)+\operatorname{div}\left(R^{-} u^{-} \otimes u^{-}\rhoight)+\alphalpha^{-} \mathcal{C}^{2}\left[\rhoho^{-} \nablabla R^{+}+\left(\rhoho^{+}-\rhoho^{-} \fracrac{\alphalpha^{+} f^{\partialrime}}{s_{+}^{2}}\rhoight) \nablabla R^{-}\rhoight] \\
\hspace{2.5cm}=\operatorname{div}\left\{\alphalpha^{-}\left[\mu^{-}\left(\nablabla
u^{-}+\nablabla^{t} u^{-}\rhoight)+\lambda^{-} \operatorname{div} u^{-}
\operatorname{Id}\rhoight]\rhoight\}.
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
In the present paper, we consider the initial value problem to
equationref{1.13} in the whole space $\mathbb R^3$ subject to the initial condition
\begin{equation}gin{equation}\label{1.14} (R^{+}, u^{+}, R^{-}, u^{-})(x,
0)=(R_{0}^{+}, u_{0}^{+}, R_{0}^{-},
u_{0}^{-})(x)\rhoightarrow(R_{\infty}^{+}, \overlineerrightarrow{0}, R_{\infty}^{-}, \overlineerrightarrow{0}) \quad
\hbox{as}\quad |x|\rhoightarrow\infty \in \mathbb{R}^{3},
\end{equation}d{equation}
where $R^{\partialm}_\infty>0$ denote the background doping profile, and
for simplicity, are taken as 1 in this paper.
In this work, we investigate the instability of the constant equilibrium state for the Cauchy problem equationref{1.13}--equationref{1.14} in the case that $f^{\partialrime}(1)>0$, which should be kept in mind throughout the rest of this paper. Taking
\[ n^{\partialm}=R^{\partialm}-1,
\]
then we can rewrite equationref{1.13} in terms of the varaibles $(n^+, u^+, n^-, u^-)$:
\begin{equation}gin{equation}\label{1.15}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t} n^++\operatorname{div}u^+=F_1, \\
\partialartial_{t}u^{+}+\alphalpha_1\nablabla n^++\alphalpha_2\nablabla
n^--\nu^+_1\Delta u^+-\nu^+_2\nablabla\operatorname{div} u^+=F_2, \\
\partialartial_{t} n^-+\operatorname{div}u^-=F_3, \\
\partialartial_{t}u^{-}+\alphalpha_3\nablabla n^++\alphalpha_4\nablabla
n^--\nu^-_1\Delta u^--\nu^-_2\nablabla\operatorname{div} u^-=F_4, \\
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
where $\nu_{1}^{\partialm}=\fracrac{\mu^{\partialm}}{\rhoho^{\partialm}(1,1)}$,
$\nu_{2}^{\partialm}=\fracrac{\mu^{\partialm}+\lambda^{\partialm}}{\rhoho^{\partialm}(1,1)}>0$,
$\alphalpha_{1}=\fracrac{\mathcal{C}^{2}(1,1)
\rhoho^{-}(1,1)}{\rhoho^{+}(1,1)}$,
$\alphalpha_{2}=\mathcal{C}^{2}(1,1)+\fracrac{\mathcal{C}^{2}(1,1)
\alphalpha^{-}(1,1) f^{\partialrime}(1)}{s_{-}^{2}(1,1)}$,
$\alphalpha_{3}=\mathcal{C}^{2}(1,1)$,
$\alphalpha_{4}=\fracrac{\mathcal{C}^{2}(1,1)
\rhoho^{+}(1,1)}{\rhoho^{-}(1,1)}-\fracrac{\mathcal{C}^{2}(1,1)
\alphalpha^{+}(1,1) f^{\partialrime}(1)}{s_{+}^{2}(1,1)}$,
and
the nonlinear terms are given by
\begin{equation}gin{align}
\label{1.16}F_{1}=&-\operatorname{div}\left(n^{+} u^{+}\rhoight), \\
F_{2}^{i}=&-g_+\left(n^{+}, n^{-}\rhoight) \partialartial_{i} n^{+}-\bar{g}_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{i} n^{-}
-\left(u^{+} \cdot \nablabla\rhoight) u_{i}^{+} \nonumber\\
&+\mu^{+} h_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{j}
n^{+} \partialartial_{j} u_{i}^{+}+\mu^{+} k_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{j} n^{-} \partialartial_{j} u_{i}^{+} \nonumber\\
\label{1.17}&+\mu^{+} h_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{j}
n^{+} \partialartial_{i} u_{j}^{+}+\mu^{+} k_{+}\left(n^{+}, n^{-}\rhoight)
\partialartial_{j} n^{-}
\partialartial_{i} u_{j}^{+}\\
&+\lambda^{+} h_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{i} n^{+}
\partialartial_{j} u_{j}^{+}+\lambda^{+} k_{+}\left(n^{+},
n^{-}\rhoight) \partialartial_{i} n^{-} \partialartial_{j} u_{j}^{+} \nonumber\\
&+\mu^{+} l_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{j}^{2} u_{i}^{+}+\left(\mu^{+}+\lambda^{+}\rhoight) l_{+}\left(n^{+}, n^{-}\rhoight) \partialartial_{i}
\partialartial_{j} u_{j}^{+}, \nonumber\\
\label{1.18}F_{3}=&-\operatorname{div}\left(n^{-} u^{-}\rhoight), \\
F_{4}^{i}=&-g_-\left(n^{+}, n^{-}\rhoight) \partialartial_{i} n^{-}-
\bar{g}_{-}\left(n^{+}, n^{-}\rhoight) \partialartial_{i} n^{+}-\left(u^{-} \cdot \nablabla\rhoight) u_{i}^{-}\nonumber \\
&+\mu^{-} h_{-}\left(n^{+}, n^{-}\rhoight) \partialartial_{j} n^{+} \partialartial_{j} u_{i}^{-}+\mu^{-} k_{-}\left(n^{+}, n^{-}\rhoight)
\partialartial_{j} n^{-} \partialartial_{j} u_{i}^{-} \nonumber\\
\label{1.19}&+\mu^{-} h_{-}\left(n^{+}, n^{-}\rhoight) \partialartial_{j}
n^{+}
\partialartial_{i} u_{j}^{-}+\mu^{-} k_{-}\left(n^{+}, n^{-}\rhoight)
\partialartial_{j} n^{-} \partialartial_{i} u_{j}^{-} \\
&+\lambda^{-} h_{-}\left(n^{+}, n^{-}\rhoight) \partialartial_{i} n^{+} \partialartial_{j} u_{j}^{-}+\lambda^{-} k_{-}\left(n^{+}, n^{-}\rhoight)
\partialartial_{i} n^{-} \partialartial_{j} u_{j}^{-}\nonumber \\
&+\mu^{-} l_{-}\left(n^{+}, n^{-}\rhoight) \partialartial_{j}^{2}
u_{i}^{-}+\left(\mu^{-}+\lambda^{-}\rhoight) l_{-}\left(n^{+},
n^{-}\rhoight) \partialartial_{i} \partialartial_{j} u_{j}^{-},\nonumber
\end{equation}d{align}
where
\begin{equation}gin{equation}\label{1.20}
\left\{\begin{equation}gin{array}{l}
g_{+}\left(n^{+}, n^{-}\rhoight)=\fracrac{\left(\mathcal{C}^{2} \rhoho^{-}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{\rhoho^{+}\left(n^{+}+1, n^{-}+1\rhoight)}-\fracrac{\left(\mathcal{C}^{2} \rhoho^{-}\rhoight)(1,1)}{\rhoho^{+}(1,1)}, \\
g_{-}\left(n^{+}, n^{-}\rhoight)=\fracrac{\left(\mathcal{C}^{2} \rhoho^{+}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{\rhoho^{-}\left(n^{+}+1, n^{-}+1\rhoight)}-\fracrac{\left(\mathcal{C}^{2} \rhoho^{+}\rhoight)(1,1)}{\rhoho^{-}(1,1)}-\fracrac{f^{\partialrime}\left(n^{-}+1\rhoight)\left(\mathcal{C}^{2} \alphalpha^{+}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{s_{+}^{2}\left(n^{+}+1, n^{-}+1\rhoight)} \\
\hspace{2.2cm}+\fracrac{f^{\partialrime}(1)\left(\mathcal{C}^{2} \alphalpha^{+}\rhoight)(1,1)}{s_{+}^{2}(1,1)}, \\
\end{equation}d{array}\rhoight.\end{equation}d{equation}
\begin{equation}gin{equation}\label{1.21}
\left\{\begin{equation}gin{array}{l}
\bar{g}_{+}\left(n^{+}, n^{-}\rhoight)=\mathcal{C}^{2}\left(n^{+}+1, n^{-}+1\rhoight)-=\mathcal{C}^{2}\left(1, 1\rhoight)
+\fracrac{f^{\partialrime}\left(n^{-}+1\rhoight)\left(\mathcal{C}^{2} \alphalpha^{-}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{s_{-}^{2}\left(n^{+}+1, n^{-}+1\rhoight)}\\
\hspace{2.2cm}-\fracrac{f^{\partialrime}(1)\left(\mathcal{C}^{2} \alphalpha^{-}\rhoight)(1,1)}{s_{-}^{2}(1,1)},\\
\bar{g}_{-}\left(n^{+}, n^{-}\rhoight)=\mathcal{C}^{2}\left(n^{+}+1, n^{-}+1\rhoight)-\mathcal{C}^{2}(1,1),\\
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
\begin{equation}gin{equation}\label{1.22}
\left\{\begin{equation}gin{array}{l}
h_{+}\left(n^{+}, n^{-}\rhoight)=\fracrac{\left(\mathcal{C}^{2}\alphalpha^{-}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{(n^++1)s_{-}^{2}\left(n^{+}+1, n^{-}+1\rhoight)},\\
h_{-}\left(n^{+}, n^{-}\rhoight)=-\fracrac{\left(\mathcal{C}^{2} \rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{(\rhoho^-s_{-}^{2})\left(n^{+}+1, n^{-}+1\rhoight)},
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
\begin{equation}gin{equation}\label{1.23}
\left\{\begin{equation}gin{array}{l}
k_{+}\left(n^{+}, n^{-}\rhoight)=-\left[\fracrac{\mathcal{C}^{2}\left(n^{+}+1, n^{-}+1\rhoight)}{(n^++1)(s_{+}^{2}\rhoho^+)\left(n^{+}+1, n^{-}+1\rhoight)}+\fracrac{f^{\partialrime}(n^-+1)\mathcal{C}^{2}\left(n^{+}+1, n^{-}+1\rhoight)}{(\rhoho^+\rhoho^-s_{+}^{2}s_{-}^{2})\left(n^{+}+1, n^{-}+1\rhoight)}\rhoight],\\
k_{-}\left(n^{+}, n^{-}\rhoight)=-\fracrac{\left(\alphalpha^+\mathcal{C}^{2}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{(n^-+1)s_{+}^{2}\left(n^{+}+1, n^{-}+1\rhoight)}+\fracrac{f^{\partialrime}(n^-+1)\left(\alphalpha^+\mathcal{C}^{2}\rhoight)\left(n^{+}+1, n^{-}+1\rhoight)}{(\rhoho^-s_{+}^{2}s_{-}^{2})\left(n^{+}+1, n^{-}+1\rhoight)},\\
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
\begin{equation}gin{equation}\label{1.24}
l_{\partialm}(n^+, n^-)=\fracrac{1}{\rhoho_{\partialm}\left(n^{+}+1, n^{-}+1\rhoight)}-\fracrac{1}{\rhoho_{\partialm}\left(1, 1\rhoight)}.
\end{equation}d{equation}
Taking change of variables by
\[
n^{+} \rhoightarrow \alphalpha_{1} n^{+}, \quad u^{+} \rhoightarrow \sqrt{\alphalpha_{1} u^{+}}, \quad n^{-} \rhoightarrow \alphalpha_{4} n^{-}, \quad u^{-} \rhoightarrow \sqrt{\alphalpha_{4} u^{-}},
\]
and setting
\[
\begin{equation}ta_{1}=\sqrt{\alphalpha_{1}}, \quad \begin{equation}ta_{2}=\fracrac{\alphalpha_{2} \sqrt{\alphalpha_{1}}}{\alphalpha_{4}},
\quad \begin{equation}ta_{3}=\fracrac{\alphalpha_{3} \sqrt{\alphalpha_{4}}}{\alphalpha_{1}}, \quad \begin{equation}ta_{4}=\sqrt{\alphalpha_{4}}
\]
and
\[
\begin{equation}ta^{+}=\sqrt{\fracrac{\begin{equation}ta_{1}}{\begin{equation}ta_{2}}}, \quad \begin{equation}ta^{-}=\sqrt{\fracrac{\begin{equation}ta_{4}}{\begin{equation}ta_{3}}},
\]
the Cauchy problem equationref{1.13} and equationref{1.14} can be reformulated as
\begin{equation}gin{equation}\label{1.25}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t} n^{+}+\begin{equation}ta_{1} \operatorname{div} u^{+}=\mathcal{F}_{1}, \\
\partialartial_{t} u^{+}+\begin{equation}ta_{1} \nablabla n^{+}+\begin{equation}ta_{2} \nablabla n^{-}-v_{1}^{+} \Delta u^{+}-v_{2}^{+} \nablabla \operatorname{div} u^{+}=\mathcal{F}_{2}, \\
\partialartial_{t} n^{-}+\begin{equation}ta_{4} \operatorname{div} u^{-}=\mathcal{F}_{3}, \\
\partialartial_{t} u^{-}+\begin{equation}ta_{3} \nablabla n^{+}+\begin{equation}ta_{4} \nablabla n^{-}-v_{1}^{-} \Delta u^{-}-v_{2}^{-} \nablabla \operatorname{div} u^{-}=\mathcal{F}_{4},
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
subject to the initial condition
\begin{equation}gin{equation}\label{1.26}
\left(n^{+}, u^{+}, n^{-}, u^{-}\rhoight)(x, 0)=\left(n_{0}^{+}, u_{0}^{+}, n_{0}^{-}, u_{0}^{-}\rhoight)(x) \rhoightarrow(0, \overlineerrightarrow{0}, 0, \overlineerrightarrow{0}), \quad \thetaext { as }|x| \rhoightarrow+\infty,
\end{equation}d{equation}
where the nonlinear terms are given by
\[
\mathcal{F}_{1}=\alphalpha_{1} F_{1}\left(\fracrac{n^{+}}{\alphalpha_{1}}, \fracrac{u^{+}}{\sqrt{\alphalpha_{1}}}\rhoight), \quad \mathcal{F}_{2}=\sqrt{\alphalpha_{1}} F_{2}
\left(\fracrac{n^{+}}{\alphalpha_{1}}, \fracrac{u^{+}}{\sqrt{\alphalpha_{1}}}, \fracrac{n^{-}}{\alphalpha_{4}}, \fracrac{u^{-}}{\sqrt{\alphalpha_{4}}}\rhoight),
\]
and
\[
\mathcal{F}_{3}=\alphalpha_{4} F_{3}\left(\fracrac{n^{-}}{\alphalpha_{4}}, \fracrac{u^{-}}{\sqrt{\alphalpha_{4}}}\rhoight), \quad \mathcal{F}_{4}=\sqrt{\alphalpha_{4}} F_{4}\left(\fracrac{n^{+}}{\alphalpha_{1}}, \fracrac{u^{+}}{\sqrt{\alphalpha_{1}}}, \fracrac{n^{-}}{\alphalpha_{4}}, \fracrac{u^{-}}{\sqrt{\alphalpha_{4}}}\rhoight).
\]
Noticing that
\begin{equation}gin{equation}\label{1.27}
\begin{equation}ta_{1} \begin{equation}ta_{4}-\begin{equation}ta_{2} \begin{equation}ta_{3}=-\fracrac{\mathcal{C}^{2}(1,1) f^{\partialrime}(1)}{\sqrt{\alphalpha_{1} \alphalpha_{4}} \rhoho^{+}(1,1)}<0,
\end{equation}d{equation}
it is clear that $\begin{equation}ta^+\begin{equation}ta^-<1$. Before stating our main results, let us state the
corresponding linearized system of equationref{1.25} as follows:
\begin{equation}gin{equation}\label{1.28}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t} \thetailde n^{+}+\begin{equation}ta_{1} \operatorname{div} \thetailde u^{+}=0, \\
\partialartial_{t}\thetailde u^{+}+\begin{equation}ta_{1} \nablabla\thetailde n^{+}+\begin{equation}ta_{2} \nablabla\thetailde n^{-}-v_{1}^{+} \Delta\thetailde u^{+}-v_{2}^{+} \nablabla \operatorname{div}\thetailde u^{+}=0, \\
\partialartial_{t}\thetailde n^{-}+\begin{equation}ta_{4} \operatorname{div}\thetailde u^{-}=0, \\
\partialartial_{t}\thetailde u^{-}+\begin{equation}ta_{3} \nablabla\thetailde n^{+}+\begin{equation}ta_{4} \nablabla\thetailde n^{-}-v_{1}^{-} \Delta\thetailde u^{-}-v_{2}^{-} \nablabla \operatorname{div}\thetailde u^{-}=0.
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
Now, we are in a position to state our main results. The first one is concerned with
the linear instability, which is stated in the following theorem.
\begin{equation}gin{Theorem}[Linear instability]\label{2mainth} Let $\thetaheta=\fracrac{\sqrt{(\nu^+\begin{equation}ta_4^2+\nu^-\begin{equation}ta_1^2)^2+4\nu^+\nu^-(\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4-\begin{equation}ta_1^2\begin{equation}ta_4^2)}
-(\nu^+\begin{equation}ta_4^2+\nu^-\begin{equation}ta_1^2)}{2\nu^+\nu^-}$ which is positive due to equationref{1.27}, where $\nu^\partialm=\nu_1^\partialm+\nu_2^\partialm.$ Then for any $\vartheta>0$, the linearized system equationref{1.28} admits an unstable solution $(\thetailde n^+_\vartheta,\thetailde u^+_\vartheta,\thetailde n^-_\vartheta,\thetailde u^-_\vartheta)$ satisfying
$$\thetailde n^\partialm_\vartheta \in C^0(0, \infty; H^{2}(\mathbb{R}^3))\cap C^1(0, \infty; H^{1}(\mathbb{R}^3)),\quad \thetaext{and}\quad \thetailde u^\partialm_\vartheta\in C^0(0, \infty; H^{2}(\mathbb{R}^3))\cap C^1(0, \infty; L^{2}(\mathbb{R}^3)),$$
and
\begin{equation}gin{equation}\label{1.29}\left\|\thetailde n^+_{0,\vartheta}\rhoight\|_{L^2}\left\|\thetailde u^+_{0,\vartheta}\rhoight\|_{L^2}\left\|\thetailde n^-_{0,\vartheta}\rhoight\|_{L^2}\left\|\thetailde u^-_{0,\vartheta}\rhoight\|_{L^2}>0.
\end{equation}d{equation}
Moreover, the solution satisfies the following estimate:
\begin{equation}gin{equation}\thetaext{e}^{(\thetaheta-\vartheta) t} \|\thetailde{n}^\partialm_{0,\vartheta}\|_{L^2}\le \|\thetailde{n}^\partialm_\vartheta(t)\|_{L^2}\le \thetaext{e}^{\thetaheta t}\|\thetailde{n}^\partialm_{0,\vartheta}\|_{L^2} \quad \thetaext{and}\quad \thetaext{e}^{(\thetaheta-\vartheta) t}\|\thetailde{u}^\partialm_{0,\vartheta}\|_{L^2}\le\|\thetailde{u}^\partialm_\vartheta(t)\|_{L^2}\le\thetaext{e}^{\thetaheta t} \|\thetailde{u}^\partialm_{0,\vartheta}\|_{L^2}.\label{1.30}\end{equation}d{equation}
\end{equation}d{Theorem}
\begin{equation}gin{remark} For any $\epsilon>0$ which may be small enough, it is direct to check that $(\epsilon\thetailde n^+,\epsilon\thetailde u^+,\epsilon\thetailde n^-,\epsilon\thetailde u^-)$ is still a solution of system equationref{1.28}. This solution is obvious unstable due to equationref{1.29} and equationref{1.30}.
\end{equation}d{remark}
The second result is concerned with nonlinear instability, which is stated in the following theorem.
\begin{equation}gin{Theorem}[Nonlinear instability]\label{3mainth} The steady state $(0, \overlineerrightarrow{0}, 0, \overlineerrightarrow{0})$ of the system equationref {1.25} is unstable in the Hadamard sense, that is, there exist positive constants $\thetaheta$, $\vartheta$, $\epsilon_0$ and $\delta_0$, and functions $( \thetailde n_{0,\vartheta}^+,\thetailde u_{0,\vartheta}^+,\thetailde n_{0,\vartheta}^-,\thetailde u_{0,\vartheta}^-)\in H^4(\mathbb R^3)$, such that for any $\epsilon\in(0,\epsilon_0)$ and the initial data
\begin{equation}gin{equation}\label{1.32}( n_0^+, u_0^+, n_0^-,u_0^-)\thetariangleq\epsilon( \thetailde n_{0,\vartheta}^+,\thetailde u_{0,\vartheta}^+,\thetailde n_{0,\vartheta}^-,\thetailde u_{0,\vartheta}^-),\end{equation}d{equation}
the Cauchy problem equationref{1.25} and equationref{1.32} admits a unique strong solution satisfying
$$ n^\partialm \in C^0(0, T^{\max}; H^{4}(\mathbb{R}^3))\cap C^1(0, T^{\max}; H^{3}(\mathbb{R}^3))\quad \thetaext{and}\quad u^\partialm\in C^0(0, T^{\max}; H^{4}(\mathbb{R}^3))\cap C^1(0, T^{\max}; H^{2}(\mathbb{R}^3)),$$
and
\begin{equation}gin{equation}\label{1.33}\left\| (n^+,u^+,n^-,u^-)(T^\varepsilon)\rhoight\|_{H^4}\ge \delta_0.
\end{equation}d{equation}
for some escape time $T^\varepsilon\in [0,T^{\max})$, where $T^{\max}$ denotes the maximal time of existence of the solution.
\end{equation}d{Theorem}
\begin{equation}gin{remark} Theorem \rhoef{2mainth} and Theorem \rhoef{3mainth} show that for the case $f'(1)>0$, the constant equilibrium state of the two--fluid model is linearly globally unstable and nonlinearly locally unstable
in the sense of Hadamard, which is in contrast to
the cases $f'(1)<0$ in Evje--Wang--Wen \cite{Evje9} and $P^+=P^-$ (corresponding to $f'(1)=0$) in Wu--Yao--Zhang \cite{WYZ} where the constant equilibrium state of the two--fluid model equationref{1.5} is nonlinearly globally stable.
\end{equation}d{remark}
Now, let us sketch the main ideas in the proofs of Theorem \rhoef{2mainth} and Theorem \rhoef{3mainth}.
For the proof of Theorem \rhoef{2mainth}, we need construct a solution to the linearized system equationref{1.28} that has a growing $H^k$
norm for any $k$ and the proof can be outlined as follows.
First, we exclude the stabilizing part of the linearized system by employing the Hodge decomposition technique firstly
introduced by Danchin \cite{Dan1} to split the linearized system into three systems (see equationref {2.1} and equationref {2.2} for details). One is a $4\thetaimes 4$ system and
its characteristic polynomial possesses four distinct roots, the other two systems are the heat equation.
This key observation allows us to construct an unstable solution.
Second, we assume a growing mode ansatz, i.e., $$\widehat{\thetailde{n}^+}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{n}^+_0},\
\widehat{\thetailde{\varphi}^+}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{\varphi}^+_0},\
\widehat{\thetailde{n}^-}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{n}^-_0},\
\widehat{\thetailde{\varphi}^-}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{\varphi}^-_0}, ~~\hbox{for some}~\lambda,$$
and submit this ansatz into the Fourier transformation of the $4\thetaimes 4$ system to get
a time--independent system for $\lambda$.
Third, we solve the time--independent system by making careful analysis and using several key observations. Indeed, noticing that
the characteristic polynomial $F(\lambda)$ defined in equationref{2.6} is a strictly increasing function on $(0, \infty)$, and
$F(\thetaheta)>0$ for $\thetaheta>0$ defined in Theorem \rhoef{2mainth}, we show that $0<\lambda_1<\thetaheta$ is the unique positive root of the characteristic equation $F(\lambda)=0$, and $\thetaheta>0$ in Theorem \rhoef{2mainth} is the largest possible growth rate since $Re(\lambda_i)\leq \thetaheta$ with $1\leq i\leq 4$. Therefore, the growing mode constructed in Theorem \rhoef{2mainth} actually does grow in time at
the fastest possible rate.\partialar
For the proof of Theorem \rhoef{3mainth}, we deduce the nonlinear instability. Compared to \cite{Guo1,Jang,Jiang1,Jiang2,WangT} where nonlinear energy estimates and a careful bootstrap argument are employed to prove stability and instability, we need to develop new
ingredients in the proof to handle with the difficulties arising from the strong interaction of two fluids,
which requires some new thoughts. Indeed, since the strong
coupling terms are involved in the right--hand of the system equationref{1.25}, it seems impossible to follow the energy methods of \cite{Guo1,Jang,Jiang1,Jiang2,WangT}
to get the lyapunov--type inequality: $\fracrac{d}{dt}\mathcal{E}(t)\leq \thetaheta \mathcal{E}(t)$ to prove the largest possible growth rate.
Therefore, we must pursue another route by resorting to semigroup methods to capture the largest possible growth rate, but the cost is that
we need the higher regularity of the solutions. More precisely,
with the help of the global linear instability result of Theorem \rhoef{2mainth} and a local existence theorem of classical solutions to the original nonlinear system, we can make delicate spectral analysis for the linearized system and apply Duhamel's principle to prove the nonlinear instability
result stated in Theorem \rhoef{2mainth}.
\subsection{Notations and conventions.}
Throughout this paper, we denote $H^k(\mathbb R^3)$ by the usual Sobolev spaces with norm $\|\cdot\|_{H^k}$ and
denote $L^p$, $1\leq p\leq \infty$ by the usual $L^p(\mathbb R^3)$
spaces with norm $\|\cdot\|_{L^p}$. We drop the domain $\mathbb R^3$ in integrands over $\mathbb R^3$. For the sake of conciseness, we
do not precise in functional space names when they are concerned
with scalar--valued or vector--valued functions, $\|(f, g)\|_X$
denotes $\|f\|_X+\|g\|_X$. We will employ the notation $a\lesssim
b$ to mean that $a\leq Cb$ for a universal constant $C>0$ that only
depends on the parameters coming from the problem. We denote
$\nablabla=\partialartial_x=(\partialartial_1,\partialartial_2,\partialartial_3)$, where
$\partialartial_i=\partialartial_{x_i}$, $\nablabla_i=\partialartial_i$ and put
$\partialartial_x^\ell f=\nablabla^\ell f=\nablabla(\nablabla^{\ell-1}f)$. Let
$\Lambda^s$ be the pseudo differential operator defined by
\begin{equation}gin{equation}\Lambda^sf=\mathfrak{F}^{-1}(|{\bf \xi}|^s\widehat f),~\hbox{for}~s\in \mathbb{R},\nonumber\end{equation}d{equation}
where $\widehat f$ and $\mathfrak{F}(f)$ are the Fourier transform
of $f$.
\section{\leftline {\bf{Linear instability.}}}
\setcounter{equation}{0}
To construct a solution to the linearized system equationref{1.28} that has growing $H^k$--norm for any positive integer $k$, by using a real method
as in \cite{Kowalczyk}, one need to make a detailed
analysis on the properties of the semigroup. To exclude the stabilizing part, we will employ the Hodge decomposition technique firstly
introduced by Danchin \cite{Dan1} to split the linear system into three systems. One only has four
equations and its characteristic polynomial possesses four distinct roots, the other two systems
are the heat equation. This key observation allows us to construct a unstable solution.
To see this, let
$\varphi^{\partialm}=\Lambda^{-1}{\rhom div}\thetailde{u}^{\partialm}$
be the ``compressible part" of the velocities $\thetailde{u}^{\partialm}$, and denote $\partialhi^{\partialm}=\Lambda^{-1}{\rhom curl}\thetailde{u}^{\partialm}$ (with $({\rhom curl} z)_i^j
=\partialartial_{x_j}z^i-\partialartial_{x_i}z^j$) by the ``incompressible part"
of the velocities $\thetailde{u}^{\partialm}$. Setting $\nu^{\partialm}=\nu^{\partialm}_1+\nu^{\partialm}_2$, the system equationref{1.28} can be
decomposed into
the following three systems:
\begin{equation}gin{equation}\label{2.1}
\begin{equation}gin{cases}
\partialartial_t{\thetailde{n}^+}+\begin{equation}ta_1\Lambda{\varphi^+}=0,\\
\partialartial_t{\varphi^+}-\begin{equation}ta_1\Lambda{\thetailde{n}^+}-\begin{equation}ta_2\Lambda{\thetailde{n}^-}+\nu^+\Lambda^2{\varphi^+}=0,\\
\partialartial_t{\thetailde{n}^-}+\begin{equation}ta_4\Lambda{\varphi^-}=0,\\
\partialartial_t{\varphi^-}-\begin{equation}ta_3\Lambda{\thetailde{n}^+}-\begin{equation}ta_4\Lambda{\thetailde{n}^-}+\nu^-\Lambda^2{\varphi^-}=0,\\
\end{equation}d{cases}
\end{equation}d{equation}
and
\begin{equation}gin{equation}\label{2.2}
\begin{equation}gin{cases}
\partialartial_t\partialhi^++\nu^+_1\Lambda^2\partialhi^+=0,\\
\partialartial_t\partialhi^-+\nu^-_1\Lambda^2\partialhi^-=0.
\end{equation}d{cases}
\end{equation}d{equation}
We see that Eqs. equationref{2.2}$_1$ and equationref{2.2}$_2$ are the standard parabolic equations with good stability. Thus, the onset of instabilities of system equationref{1.28} comes from equationref{2.1}. Taking the Fourier transform to the system equationref{2.1}, one has
\begin{equation}gin{equation}\label{2.3}
\begin{equation}gin{cases}
\partialartial_t\widehat{\thetailde{n}^+}+\begin{equation}ta_1|\xi|\widehat{\varphi^+}=0,\\
\partialartial_t\widehat{\varphi^+}-\begin{equation}ta_1|\xi|\widehat{\thetailde{n}^+}-\begin{equation}ta_2|\xi|\widehat{\thetailde{n}^-}+\nu^+|\xi|^2\widehat{\varphi^+}=0,\\
\partialartial_t\widehat{\thetailde{n}^-}+\begin{equation}ta_4|\xi|\widehat{\varphi^-}=0,\\
\partialartial_t\widehat{\varphi^-}-\begin{equation}ta_3|\xi|\widehat{\thetailde{n}^+}-\begin{equation}ta_4|\xi|\widehat{\thetailde{n}^-}+\nu^-|\xi|^2\widehat{\varphi^-}=0.\\
\end{equation}d{cases}
\end{equation}d{equation}
To construct a solution to the linearized equations equationref{2.3} that has growing $H^k$--norm for any $k$, we shall make a growing normal mode ansatz of solutions, i.e.,
$$\widehat{\thetailde{n}^+}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{n}^+_0},\
\widehat{\thetailde{\varphi}^+}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{\varphi}^+_0},\
\widehat{\thetailde{n}^-}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{n}^-_0},\
\widehat{\thetailde{\varphi}^-}=\thetaext{e}^{\lambda(|\xi|)t}\widehat{\thetailde{\varphi}^-_0}.$$
Substituting this ansazt into equationref{2.3}, one obtains the time--independent system
\begin{equation}gin{equation}\label{2.4}
\begin{equation}gin{cases}
\lambda{\widehat{\thetailde{n}^+_0}}+\begin{equation}ta_1|\xi|\widehat{{\varphi^+_0}}=0,\\
\lambda\widehat{{\varphi^+_0}}-\begin{equation}ta_1|\xi|\widehat{{\thetailde{n}^+_0}}-\begin{equation}ta_2|\xi|\widehat{{\thetailde{n}^-_0}}+\nu^+|\xi|^2\widehat{{\varphi^+_0}}=0,\\
\lambda\widehat{{\thetailde{n}^-_0}}+\begin{equation}ta_4|\xi|\widehat{{\varphi^-_0}}=0,\\
\lambda\widehat{{\varphi^-_0}}-\begin{equation}ta_3|\xi|\widehat{{\thetailde{n}^+_0}}-\begin{equation}ta_4|\xi|\widehat{{\thetailde{n}^-_0}}+\nu^-|\xi|^2\widehat{{\varphi^-_0}}=0.\\
\end{equation}d{cases}
\end{equation}d{equation}
After a series of tedious but direct calculations, we can conclude from equationref{2.4} that
\begin{equation}gin{equation}\begin{equation}gin{split}\label{2.5}
&[\lambda^4+(\nu^+|\xi|^2+\nu^-|\xi|^2)\lambda^3+(\begin{equation}ta_1^2|\xi|^2+\begin{equation}ta_4^2|\xi|^2+\nu^+\nu^-|\xi|^4)\lambda^2\\&+(\nu^+\begin{equation}ta_4^2|\xi|^4+\nu^-\begin{equation}ta_1^2|\xi|^4)\lambda
+\begin{equation}ta_1^2\begin{equation}ta_4^2|\xi|^4-\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4|\xi|^4]\widehat{{\varphi^-_0}}=0.
\end{equation}d{split}\end{equation}d{equation}
Therefore, the system equationref{2.4} has non--zero solutions if the characteristic equation
\begin{equation}gin{equation}\begin{equation}gin{split}\label{2.6}
F(\lambda)=&\lambda^4+(\nu^+|\xi|^2+\nu^-|\xi|^2)\lambda^3+(\begin{equation}ta_1^2|\xi|^2+\begin{equation}ta_4^2|\xi|^2+\nu^+\nu^-|\xi|^4)\lambda^2\\&+(\nu^+\begin{equation}ta_4^2|\xi|^4+\nu^-\begin{equation}ta_1^2|\xi|^4)\lambda
+\begin{equation}ta_1^2\begin{equation}ta_4^2|\xi|^4-\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4|\xi|^4=0
\end{equation}d{split}\end{equation}d{equation}
has a real characteristic root.
\begin{equation}gin{Lemma}\label{lemma2.1}
There exists a positive constant $\eta_1\gg 1 $, such that for
$|\xi|\ge \eta_1$, the characteristic equation equationref{2.6} admits a real positive solution satisfying the following Taylor series
expansion
\begin{equation}gin{equation}\lambda_1=\thetaheta+\mathcal {O}(|\xi|^{-1}).\label{2.7}\end{equation}d{equation}
Moreover, the following estimate holds
\begin{equation}gin{equation}\lambda_1<\thetaheta \quad \thetaext{for any}\quad \xi\in\mathbb R^3.\label{2.8}\end{equation}d{equation}
\end{equation}d{Lemma}
\begin{equation}gin{proof}Employing
the similar argument of Taylor series expansion as in
\cite{Mat1}, then equationref{2.7} follows from some tedious but direct calculations. It is noticed that
$F(\lambda)$ is a strictly monotonically increasing function if $\lambda>0$. Furthermore,
\begin{equation}gin{equation}\nonumber\begin{equation}gin{split}F(\thetaheta)>\nu^+\nu^-|\xi|^4\thetaheta^2+(\nu^+\begin{equation}ta_4^2|\xi|^4+\nu^-\begin{equation}ta_1^2|\xi|^4)\thetaheta
+\begin{equation}ta_1^2\begin{equation}ta_4^2|\xi|^4-\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4|\xi|^4=0,\end{equation}d{split}\end{equation}d{equation}
therefore equationref{2.8} holds and the proof of lemma is completed.
\end{equation}d{proof}
Let $\partialhi\in C_0^\infty(\mathbb
R^3_{{\bf \xi}})$ be a radial function satisfying $\partialhi({\bf \xi})=1$ when $\fracrac{3}{2}\eta\le |{\bf
\xi}|\leq 3\eta$ and $\partialhi({\bf \xi})=0$ when $|{\bf
\xi}|\le \eta$ and $|{\bf
\xi}|\ge 4\eta$. From equationref{2.4}, we set
$${\widehat{\thetailde{n}^+_0}}=\partialhi({\bf \xi}),\ \widehat{{\varphi^+_0}}=-\fracrac{\lambda_1(|\xi|)}{\begin{equation}ta_1|\xi|}\partialhi({\bf \xi}),\ {\widehat{\thetailde{n}^-_0}}=-\fracrac{\lambda^2_1(|\xi|)+\begin{equation}ta_1^2|\xi|^2+\nu^+\lambda_1(|\xi|)|\xi|^2}{\begin{equation}ta_1\begin{equation}ta_2|\xi|^2}\partialhi({\bf \xi})$$
and
$$\widehat{{\varphi^-_0}}=\fracrac{\lambda^3_1(|\xi|)+\begin{equation}ta_1^2\lambda_1(|\xi|)|\xi|^2+\nu^+\lambda^2_1(|\xi|)|\xi|^2}{\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_4|\xi|^3}\partialhi({\bf \xi}).$$
Then, it is direct to check that $(\widehat{\thetailde{n}^+_0},\widehat{\thetailde\varphi^+_0},\widehat{\thetailde{n}^-_0},\widehat{\thetailde\varphi^-_0})$ is a solution of the system equationref{2.4}. Thus, we conclude the following proposition, which implies Theorem \rhoef{2mainth}.
\begin{equation}gin{Proposition}\label{Prop2.2} Let
$${\thetailde{n}^\partialm}=\mathfrak{F}^{-1}\left(\thetaext{e}^{\lambda_1t}\widehat{\thetailde{n}^\partialm_0}\rhoight)\quad \thetaext{and}\quad
{\thetailde{u}^\partialm}=-\Lambda^{-1}\nablabla\mathfrak{F}^{-1}\left(\thetaext{e}^{\lambda_1(|\xi|)t}\widehat{\thetailde{\varphi}^\partialm_0}\rhoight).$$
Then $(\thetailde{n}^+,\thetailde u^+,\thetailde{n}^-,\thetailde u^-)$ is a solution of equationref{1.29} and
satisfies
\begin{equation}gin{equation}\thetaext{e}^{(\thetaheta-\vartheta) t} \|\thetailde{n}^\partialm_0\|_{L^2}\le \|\thetailde{n}^\partialm(t)\|_{L^2}\le \thetaext{e}^{\thetaheta t}\|\thetailde{n}^\partialm_0\|_{L^2} \quad \thetaext{and}\quad \thetaext{e}^{(\thetaheta-\vartheta) t}\|\thetailde{u}^\partialm_0\|_{L^2}\le\|\thetailde{u}^\partialm(t)\|_{L^2}\le\thetaext{e}^{\thetaheta t} \|\thetailde{u}^\partialm_0\|_{L^2},\label{2.9}\end{equation}d{equation}
if $\eta_1$ large enough.
\end{equation}d{Proposition}
\begin{equation}gin{proof} Set $\partialhi^\partialmequationuiv0$. As the definition of $\varphi^\partialm$ and $\partialhi^\partialm$, and the relation
$$\thetailde u^\partialm=-\Lambda^{-1}\nablabla\varphi^\partialm-\Lambda^{-1}\thetaext{div}\partialhi^\partialm,$$
it is easy to prove that $(\thetailde{n}^+,\thetailde u^+,\thetailde{n}^-,\thetailde u^-)$ is a solution of equationref{1.29}. Moreover, in virtue of Plancherel theorem, we have
\begin{equation}gin{equation}\begin{equation}gin{split}\|\thetailde u^\partialm(t)\|_{L^2}^2=&\|\widehat{\thetailde u^\partialm}(t)\|_{L^2}^2\\
=&\int\thetaext{e}^{2\lambda_1(|\xi|)t}|\widehat{\thetailde u^\partialm_0}|^2\mathrm{d}\xi\\
=&\int_{\eta\le|\xi|\le 4|\eta|}\thetaext{e}^{2\lambda_1(|\xi|)t}|\widehat{\thetailde u^\partialm_0}|^2\mathrm{d}\xi\\
\ge &~\thetaext{e}^{2(\thetaheta-\vartheta) t}\|\thetailde u^\partialm_0(t)\|_{L^2}^2,
\end{equation}d{split}\end{equation}d{equation}
if $\eta$ is large enough. Performing the similar procedures, we can prove $\|\thetailde{u}^\partialm(t)\|_{L^2}\le\thetaext{e}^{\thetaheta t} \|\thetailde{u}^\partialm_0\|_{L^2}$ and $\thetaext{e}^{(\thetaheta-\vartheta) t} \|\thetailde{n}^\partialm_0\|_{L^2}\le\|\thetailde{n}^\partialm(t)\|_{L^2}\le\thetaext{e}^{\thetaheta t} \|\thetailde{n}^\partialm_0\|_{L^2}$. The proof of proposition is complete.
\end{equation}d{proof}
\section{Spectral analysis and linear $L^2$--estimates}\label{1section_appendix}
In this section, we are devoted to deriving the linear $L^2$--estimates, by using a real method
as in \cite{Mat1}, one need to make a detailed analysis on the
properties of the semigroup.
\subsection{Spectral analysis for system equationref{2.1}} We consider the Cauchy problem of equationref{2.1} with the initial data
\begin{equation}gin{equation}\label{3.1}(\thetailde{n}^+, \varphi^+, \thetailde{n}^-, \varphi^-)\big|_{t=0}=({n}^+_0, \Lambda^{-1}{\rhom div}\thetailde{u}^{+}_0, {n}^-_0, \Lambda^{-1}{\rhom div}\thetailde{u}^{-}_0)(x)
\end{equation}d{equation}
In terms of the semigroup theory, we may represent the IVP equationref{2.1} and equationref{3.1} for $\mathcal U=(\thetailde{n}^+, \varphi^+, \thetailde{n}^-, \varphi^-)^t$ as
\begin{equation}gin{equation}
\begin{equation}gin{cases}
\mathcal U_t=\mathcal B_1\mathcal U,\\
\mathcal U\big|_{t=0}=\mathcal U_0,
\end{equation}d{cases} \label{3.2}
\end{equation}d{equation}
where the operator $\mathcal B_1$ is defined
by
\begin{equation}gin{equation}\nonumber\mathcal B_1=\begin{equation}gin{pmatrix}
0&-\begin{equation}ta_1\Lambda&0&0\\
\begin{equation}ta_1\Lambda&-\nu^+\Lambda^2&\begin{equation}ta_2\Lambda&0\\
0&0&0&-\begin{equation}ta_4\Lambda\\
\begin{equation}ta_3\Lambda&0&\begin{equation}ta_4\Lambda&-\nu^-\Lambda^2
\end{equation}d{pmatrix}.\end{equation}d{equation}
Taking the Fourier transform to the system equationref{3.2}, we obtain
\begin{equation}gin{equation}
\begin{equation}gin{cases}
\widehat {\mathcal U}_t=\mathcal A_1(\xi)\widehat {\mathcal U},\\
\widehat {\mathcal U}\big|_{t=0}=\widehat {\mathcal U}_0,
\end{equation}d{cases} \label{3.3}
\end{equation}d{equation}
where $\widehat {\mathcal U}(\xi,t)=\mathfrak{F}({\mathcal U}(x,t))$ and $\mathcal A_1(\xi)$ is given by
\begin{equation}gin{equation}\nonumber\mathcal A_1(\xi)=\begin{equation}gin{pmatrix}
0&-\begin{equation}ta|\xi|&0&0\\
\begin{equation}ta_1|\xi|&-\nu^+|\xi|^2&\begin{equation}ta_2|\xi|&0\\
0&0&0&-\begin{equation}ta_4|\xi|\\
\begin{equation}ta_3|\xi|&0&\begin{equation}ta_4|\xi|&-\nu^-|\xi|^2
\end{equation}d{pmatrix}.\end{equation}d{equation}
We compute the eigenvalues of the matrix $\mathcal A_1(\xi)$ from the determinant
\begin{equation}gin{equation}\begin{equation}gin{split}\label{3.4}&{\rhom det}(\lambda{\rhom I}-\mathcal A_1(\xi))\\
&=\lambda^4+(\nu^+|\xi|^2+\nu^-|\xi|^2)\lambda^3+(\begin{equation}ta_1^2|\xi|^2+\begin{equation}ta_4^2|\xi|^2+\nu^+\nu^-|\xi|^4)\lambda^2+(\nu^+\begin{equation}ta_4^2|\xi|^4+\nu^-\begin{equation}ta_1^2|\xi|^4)\lambda\\
&\quad+\begin{equation}ta_1^2\begin{equation}ta_4^2|\xi|^4-\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4|\xi|^4\\
&=0,
\end{equation}d{split}\end{equation}d{equation}
which is the same as characteristic equation equationref{2.6} and implies that the matrix $\mathcal A_1(\xi)$ possesses four different
eigenvalues:
\begin{equation}gin{equation}\nonumber
\lambda_1=\lambda_1(|\xi|),\quad \lambda_2=\lambda_2(|\xi|),\quad
\lambda_3=\lambda_3(|\xi|),\quad \lambda_4=\lambda_4(|\xi|).
\end{equation}d{equation}
Consequently, the semigroup $e^{t\mathcal A_1}$ can be decomposed
into
\begin{equation}gin{equation}\label{2.21}
\thetaext{e}^{t\mathcal A_1(\xi)}=\sum_{i=1}^4\thetaext{e}^{\lambda_it}P_i(\xi),
\end{equation}d{equation}
where the projector $P_i(\xi)$ is defined by
\begin{equation}gin{equation}\label{2.22}
P_i(\xi)=\partialrod_{j\neq i}\fracrac{\mathcal A_1(\xi)-\lambda_jI}{\lambda_i-\lambda_j}, \quad i,j=1,2,3,4.
\end{equation}d{equation}
Thus, the solution of IVP equationref{3.3} can be expressed as
\begin{equation}gin{equation}
\widehat {\mathcal U}(\xi,t)=\thetaext{e}^{t\mathcal A_1(\xi)}\widehat {\mathcal U}_0(\xi)=\left(\sum_{i=1}^4
\thetaext{e}^{\lambda_it}P_i(\xi)\rhoight)\widehat {\mathcal U}_0(\xi).\label{2.23}
\end{equation}d{equation}
To derive long time properties of the semigroup
$\thetaext{e}^{t\mathcal A_1}$ in $L^2$--framework, one need to analyze
the asymptotical expansions of $\lambda_i$, $P_i$ $(i =1, 2, 3, 4)$
and $\thetaext{e}^{t\mathcal A_1(\xi)}$. Employing the similar argument of Taylor series expansion as in
\cite{Mat1}, we have the following lemmas from tedious
calculations.
\begin{equation}gin{Lemma}\label{lemma3.1}
There exists a positive constant $\eta_2\ll 1 $ such that, for
$|\xi|\leq \eta_2$, the spectral has the following Taylor series
expansion:
\begin{equation}gin{equation}\label{3.8-1}
\left\{\begin{equation}gin{array}{lll}\displaystyle \lambda_1=-\left[\fracrac{\nu^++\nu^-}{4}-\fracrac{\nu^+(\begin{equation}ta_1^2-\begin{equation}ta_4^2)+\nu^-(\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1}\rhoight]|\xi|^2+\sqrt{\kappa_1-\kappa_2}|\xi|+\mathcal O(|\xi|^3),\\
\displaystyle\lambda_2=-\left[\fracrac{\nu^++\nu^-}{4}-\fracrac{\nu^+(\begin{equation}ta_1^2-\begin{equation}ta_4^2)+\nu^-(\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1}\rhoight]|\xi|^2-\sqrt{\kappa_1-\kappa_2}|\xi|+\mathcal O(|\xi|^3),
\\ \displaystyle \lambda_3=-\left[\fracrac{\nu^++\nu^-}{4}+\fracrac{\nu^+(\begin{equation}ta_1^2-\begin{equation}ta_4^2)+\nu^-(\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1}\rhoight]|\xi|^2
+\sqrt{\kappa_2+\kappa_1}\thetaext{i}|\xi|+\mathcal O(|\xi|^3),
\\ \displaystyle \lambda_4=-\left[\fracrac{\nu^++\nu^-}{4}+\fracrac{\nu^+(\begin{equation}ta_1^2-\begin{equation}ta_4^2)+\nu^-(\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1}\rhoight]|\xi|^2-\sqrt{\kappa_2+\kappa_1}\thetaext{i}|\xi|+\mathcal O(|\xi|^3),
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
where $\kappa_1=\sqrt{\fracrac{(\begin{equation}ta_1^2-\begin{equation}ta_4^2)^2}{4}+\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4}$ and $\displaystyle\kappa_2=\fracrac{\begin{equation}ta_1^2+\begin{equation}ta_4^2}{2}$.
\end{equation}d{Lemma}
For $|\xi|\leq\eta_2$, from Lemma \rhoef{lemma3.1}, a direct computation gives
{\small \begin{equation}gin{equation}\label{3.9}\begin{equation}gin{split}P_1(\xi)=&\begin{equation}gin{pmatrix}
\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}&
\fracrac{\begin{equation}ta_1(2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1\sqrt{\kappa_1-\kappa_2}}&\fracrac{-\begin{equation}ta_1\begin{equation}ta_2}{4\kappa_1}&\fracrac{-\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_1-\kappa_2}}\\
\fracrac{\begin{equation}ta_1(\begin{equation}ta_1^2-\begin{equation}ta_4^2-2\kappa_1)+2\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4}{8\kappa_1
\sqrt{\kappa_1-\kappa_2}}&\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}&-\fracrac{\begin{equation}ta_2\sqrt{\kappa_1-\kappa_2}}{4\kappa_1}&\fracrac{-\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1}\\
\fracrac{-\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1}&\fracrac{-\begin{equation}ta_1\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_1-\kappa_2}}&\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}{8\kappa_1}&\fracrac{\begin{equation}ta_4(2\kappa_1
+\begin{equation}ta_1^2-\begin{equation}ta_4^2)}{8\kappa_1\sqrt{\kappa_1-\kappa_2}}\\
-\fracrac{\begin{equation}ta_3\sqrt{\kappa_1-\kappa_2}}{4\kappa_1}&\fracrac{-\begin{equation}ta_1\begin{equation}ta_3}{4\kappa_1}&\fracrac{\begin{equation}ta_4(\begin{equation}ta_4^2-\begin{equation}ta_1^2-2\kappa_1)+2\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3}{8\kappa_1\sqrt{\kappa_1-
\kappa_2}}&\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}{8\kappa_1}
\end{equation}d{pmatrix}+\mathcal O(|\xi|),\end{equation}d{split}\end{equation}d{equation}
\begin{equation}gin{equation}\label{3.10}\begin{equation}gin{split}P_2(\xi)=&\begin{equation}gin{pmatrix}
\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}&\fracrac{-\begin{equation}ta_1(2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1\sqrt{\kappa_1-\kappa_2}}&
\fracrac{-\begin{equation}ta_1\begin{equation}ta_2}{4\kappa_1}&\fracrac{\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_1-\kappa_2}}\\
-\fracrac{\begin{equation}ta_1(\begin{equation}ta_1^2-\begin{equation}ta_4^2-2\kappa_1)+2\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4}{8\kappa_1\sqrt{\kappa_1-\kappa_2}}&
\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}&\fracrac{\begin{equation}ta_2\sqrt{\kappa_1-\kappa_2}}{4\kappa_1}&\fracrac{-\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1}\\
\fracrac{-\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1}&\fracrac{\begin{equation}ta_1\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_1-\kappa_2}}&\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}
{8\kappa_1}&-\fracrac{\begin{equation}ta_4(2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2)}{8\kappa_1\sqrt{\kappa_1-\kappa_2}}\\
\fracrac{\begin{equation}ta_3\sqrt{\kappa_1-\kappa_2}}{4\kappa_1}&\fracrac{-\begin{equation}ta_1\begin{equation}ta_3}{4\kappa_1}&-\fracrac{\begin{equation}ta_4(\begin{equation}ta_4^2-\begin{equation}ta_1^2-2\kappa_1)+2\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3}
{8\kappa_1\sqrt{\kappa_1-\kappa_2}}&\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}{8\kappa_1}
\end{equation}d{pmatrix}+\mathcal O(|\xi|),\end{equation}d{split}\end{equation}d{equation}
\begin{equation}gin{equation}\label{3.11}\begin{equation}gin{split}P_3(\xi)=&\begin{equation}gin{pmatrix}
\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}{8\kappa_1}&\fracrac{\begin{equation}ta_1(2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2)}
{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&\fracrac{\begin{equation}ta_1\begin{equation}ta_2}{4\kappa_1}&\fracrac{\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}\\
-\fracrac{\begin{equation}ta_1(\begin{equation}ta_1^2-\begin{equation}ta_4^2+2\kappa_1)+2\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4}{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}
{8\kappa_1}&-\fracrac{\begin{equation}ta_2\sqrt{\kappa_2+\kappa_1}}{4\kappa_1}{i}&\fracrac{\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1}\\
\fracrac{\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1}&\fracrac{\begin{equation}ta_1\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&
\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}&\fracrac{\begin{equation}ta_4(2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}\\
-\fracrac{\begin{equation}ta_3\sqrt{\kappa_2+\kappa_1}}{4\kappa_1}{i}&\fracrac{\begin{equation}ta_1\begin{equation}ta_3}{4\kappa_1}&-\fracrac{\begin{equation}ta_4(\begin{equation}ta_4^2-\begin{equation}ta_1^2+2\kappa_1)
+2\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3}{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}
\end{equation}d{pmatrix}+\mathcal O(|\xi|),\end{equation}d{split}\end{equation}d{equation}
and
\begin{equation}gin{equation}\label{3.12}\begin{equation}gin{split}P_4(\xi)=&\begin{equation}gin{pmatrix}
\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}{8\kappa_1}&-\fracrac{\begin{equation}ta_1(2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2)}
{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&\fracrac{\begin{equation}ta_1\begin{equation}ta_2}{4\kappa_1}&-\fracrac{\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}\\
\fracrac{\begin{equation}ta_1(\begin{equation}ta_1^2-\begin{equation}ta_4^2+2\kappa_1)+2\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4}{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&\fracrac{2\kappa_1+\begin{equation}ta_1^2-\begin{equation}ta_4^2}
{8\kappa_1}&\fracrac{\begin{equation}ta_2\sqrt{\kappa_2+\kappa_1}}{4\kappa_1}{i}&\fracrac{\begin{equation}ta_2\begin{equation}ta_4}{4\kappa_1}\\
\fracrac{\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1}&-\fracrac{\begin{equation}ta_1\begin{equation}ta_3\begin{equation}ta_4}{4\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&
\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}&-\fracrac{\begin{equation}ta_4(2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2)}{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}\\
\fracrac{\begin{equation}ta_3\sqrt{\kappa_2+\kappa_1}}{4\kappa_1}{i}&\fracrac{\begin{equation}ta_1\begin{equation}ta_3}{4\kappa_1}&\fracrac{\begin{equation}ta_4(\begin{equation}ta_4^2-\begin{equation}ta_1^2+2\kappa_1)+2\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3}
{8\kappa_1\sqrt{\kappa_2+\kappa_1}}{i}&\fracrac{2\kappa_1+\begin{equation}ta_4^2-\begin{equation}ta_1^2}{8\kappa_1}
\end{equation}d{pmatrix}+\mathcal O(|\xi|),\end{equation}d{split}\end{equation}d{equation}}
\begin{equation}gin{Lemma}\label{lemma3.2}
For $\eta_2\le |\xi|\le \eta_1$, there exists a positive constant $C$ such that
\begin{equation}gin{equation}\label{3.13}\thetaext{Re}(\lambda_i)\le \thetaheta \quad \thetaext{and} \quad \left | P_i\rhoight |\le C,
\end{equation}d{equation}
for $1\le i\le 4.$
\end{equation}d{Lemma}
\begin{equation}gin{Lemma}\label{lemma3.3}
There exists a positive constants $\eta_1\gg 1 $ such that, for
$|\xi|\geq \eta_1$, the spectral has the following Taylor series
expansion:
\begin{equation}gin{equation}\label{3.8}
\left\{\begin{equation}gin{array}{lll}\displaystyle \lambda_1=\thetaheta+\mathcal O(|\xi|^{-1}),\\
\displaystyle\lambda_2=\fracrac{-(\nu^+\begin{equation}ta_4^2+\nu^-\begin{equation}ta_1^2)-\kappa_3
}{2\nu^+\nu^-}+\mathcal O(|\xi|^{-1}),
\\ \displaystyle \lambda_3=-\nu^+|\xi|^2+\fracrac{\begin{equation}ta_1^2}{\nu^+}
+\mathcal O(|\xi|^{-1}),
\\ \displaystyle \lambda_4=-\nu^-|\xi|^2+\fracrac{\begin{equation}ta_4^2}{\nu^-}+\mathcal O(|\xi|^{-1}),
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
where $\kappa_3=\sqrt{(\nu^+\begin{equation}ta_4^2+\nu^-\begin{equation}ta_1^2)^2+4\nu^+\nu^-(\begin{equation}ta_1\begin{equation}ta_2\begin{equation}ta_3\begin{equation}ta_4-\begin{equation}ta_1^2\begin{equation}ta_4^2)}$.
\end{equation}d{Lemma}
For $|\xi|\geq\eta_1$, from Lemma \rhoef{lemma3.3}, a direct computation gives
\begin{equation}gin{equation}\label{3.15}\begin{equation}gin{split}P_1(\xi)=&\begin{equation}gin{pmatrix}
\fracrac{\nu^+\begin{equation}ta_4^2-\nu^-\begin{equation}ta_1^2+\kappa_3}{2\kappa_3}&
0&-\fracrac{\begin{equation}ta_1\begin{equation}ta_2\nu^-}{\kappa_3}&0\\
0&0&0&0\\
\fracrac{-\begin{equation}ta_3\begin{equation}ta_4\nu^+}{\kappa_3}&0&\fracrac{\nu^-\begin{equation}ta_1^2-\nu^+\begin{equation}ta_4^2+\kappa_3}{2\kappa_3}&0\\
0&0&0&0
\end{equation}d{pmatrix}+\mathcal O(|\xi|^{-1}),\end{equation}d{split}\end{equation}d{equation}
\begin{equation}gin{equation}\label{3.16}\begin{equation}gin{split}P_2(\xi)=&\begin{equation}gin{pmatrix}
\fracrac{\nu^-\begin{equation}ta_1^2-\nu^+\begin{equation}ta_4^2+\kappa_3}{2\kappa_3}&
0&\fracrac{\begin{equation}ta_1\begin{equation}ta_2\nu^-}{\kappa_3}&0\\
0&0&0&0\\
\fracrac{\begin{equation}ta_3\begin{equation}ta_4\nu^+}{\kappa_3}&0&\fracrac{\nu^+\begin{equation}ta_4^2-\nu^-\begin{equation}ta_1^2+\kappa_3}{2\kappa_3}&0\\
0&0&0&0
\end{equation}d{pmatrix}+\mathcal O(|\xi|),\end{equation}d{split}\end{equation}d{equation}
\begin{equation}gin{equation}\label{3.17}\begin{equation}gin{split}P_3(\xi)=&\begin{equation}gin{pmatrix}
0&0&0&0\\
0&1&0&0\\
0&0&0&0\\
0&0&0&0\\
\end{equation}d{pmatrix}+\mathcal O(|\xi|),\end{equation}d{split}\end{equation}d{equation}
and
\begin{equation}gin{equation}\label{3.18}\begin{equation}gin{split}P_4(\xi)=&\begin{equation}gin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
\end{equation}d{pmatrix}+\mathcal O(|\xi|).\end{equation}d{split}\end{equation}d{equation}
With the help of Lemmas \rhoef{lemma3.1}--\rhoef{lemma3.3}, we can have the following proposition which is concerned with
long time properties of $L^2$--norm for the
solution.
\begin{equation}gin{Proposition}[$L^2$--theory]\label{Prop3.4} It holds that
\begin{equation}gin{equation}\|\thetaext{e}^{t\mathcal B_1}\mathcal U(0)\|_{L^2}\lesssim \thetaext{e}^{\thetaheta t}\| {\mathcal U}(0)\|_{L^2},\label{2.30}\end{equation}d{equation}
for any $t\geq 0$.
\end{equation}d{Proposition}
\subsection{Spectral analysis for system equationref{2.2}} We consider the Cauchy problem of equationref{2.2} with the initial data
\begin{equation}gin{equation}\label{3.20}(\partialhi^+, \partialhi^-)\big|_{t=0}=(\Lambda^{-1}{\rhom curl}\thetailde{u}^{+}_0, \Lambda^{-1}{\rhom curl}\thetailde{u}^{-}_0)(x).
\end{equation}d{equation}
From the classic theory of the heat equation, it is clear that the solution $\mathcal V=(\partialhi^+, \partialhi^-)^t$ to the IVP equationref{2.2} and equationref{3.20} satisfies the following decay
estimates.
\begin{equation}gin{Proposition}[$L^2$--theory]\label{Prop3.5} It holds that
\begin{equation}gin{equation}\|\thetaext{e}^{-\nu^\partialm t\Lambda^2}\mathcal V(0)\|_{L^2}\lesssim
\| {\mathcal V}(0)\|_{L^2},\nonumber\end{equation}d{equation}
for any $t\geq 0$.
\end{equation}d{Proposition}
We consider the Cauchy problem of equationref{1.28} with the initial data
\begin{equation}gin{equation}\label{3.21}
\left(\thetailde n^{+}, \thetailde u^{+}, n^{-},\thetailde u^{-}\rhoight)(x, 0)=\left(n_{0}^{+}, \thetailde u_{0}^{+}, n_{0}^{-},\thetailde u_{0}^{-}\rhoight)(x) \rhoightarrow(0, \overlineerrightarrow{0}, 0, \overlineerrightarrow{0}), \quad \thetaext { as }|x| \rhoightarrow+\infty,
\end{equation}d{equation}
By virtue of the definition of $\varphi^{\partialm}$ and $\partialhi^{\partialm}$, and the
fact that the relations
$$\thetailde{u}^{\partialm}=-\wedge^{-1}\nablabla\varphi^{\partialm}-\wedge^{-1}\thetaext{div}\partialhi^{\partialm},$$
involve pseudo--differential operators of degree zero, the estimates
in space $H^k(\mathbb R^3)$ for the original function $\thetailde{u}^{\partialm}$ will
be the same as for $(\varphi^{\partialm}, \partialhi^{\partialm})$. Combining Propositions
\rhoef{Prop3.4} and \rhoef{Prop3.5}, we have the
following result concerning long time properties for the solution
semigroup $\thetaext{e}^{t\mathcal{A}}$.
\begin{equation}gin{Proposition}\label{Prop3.6} The global solution $\thetailde{U}=(\thetailde{n}^+,\thetailde{u}^+,\thetailde{n}^-,\thetailde{u}^-)^t$ of the IVP equationref{1.28} and equationref{3.21} satisfies
\begin{equation}gin{equation}\| \thetaext{e}^{t\mathcal{A}}\thetailde{U}(0)\|_{L^2}\lesssim \thetaext{e}^{\thetaheta t}
\| \thetailde{U}(0)\|_{L^2}.\label{3.22}\end{equation}d{equation}
\end{equation}d{Proposition}
\section{Nonlinear instability}\label{1section_appendix-2}
We mention that the local existence of strong solutions to a generic compressible
two--fluid model can be established by using the standard iteration arguments as in \cite{Wen1} whose
details are omitted. We can arrive at the following conclusion:
\begin{equation}gin{Proposition}\label{Prop4.1} Assume that the notations and hypotheses in Theorem \rhoef{3mainth} are in force. For any given initial data $\left(n_0^+,u_0^+,n_0^-,u_0^-\rhoight)\in H^4(\mathbb R^3)$ satisfying $\inf_{x\in\mathbb R^3}\{n_0^\partialm+1\}>0$, there exist a $T>0$ and a unique strong solution $(n^+,u^+,n^-,u^-)\in C^0([0,T];H^4(\mathbb R^3))$ to the Cauchy problem equationref{1.26}--equationref{1.27}. Moreover, the strong solution satisfies
\begin{equation}gin{equation}\nonumber
\begin{equation}gin{split}
\mathcal E(t)
\leq C(T) \mathcal E(0),
\end{equation}d{split}
\end{equation}d{equation}
where $\mathcal E(t)=\left\|\left(n^+,u^+,n^-,u^-\rhoight)(t)\rhoight\|_{H^4}$.
\end{equation}d{Proposition}
\thetaextbf{\thetaextit{Proof of Theorem \rhoef{3mainth}.}} Now we are in a position to prove Theorem \rhoef{3mainth} by adopting the basic ideas in \cite{Guo1,Jang,Jiang1,Jiang2,WangT}.
In view of Theorem \rhoef{2mainth}, we can construct a linear solution $\left(\thetailde n^+_\vartheta,\thetailde u^+_\vartheta,\thetailde n^-_\vartheta,\thetailde u^-_\vartheta\rhoight)\in C^0([0,\infty);H^4(\mathbb R^3))$ to the linear system equationref{1.28}. Moreover, without loss of generality, we suppose that
\begin{equation}gin{equation}\mathcal E\left(\thetailde n^+_{0,\vartheta},\thetailde u^+_{0,\vartheta},\thetailde n^-_{0,\vartheta},\thetailde u^-_{0,\vartheta}\rhoight)=\left\|\left(\thetailde n^+_{0,\vartheta}, \thetailde u^+_{0,\vartheta},\thetailde n^-_{0,\vartheta}, \thetailde u^-_{0,\vartheta}\rhoight)\rhoight\|_{H^4}=1.\nonumber
\end{equation}d{equation}
Denote $\left(n^{+,\varepsilon}_{0,\vartheta},u^{+,\varepsilon}_{0,\vartheta},n^{-,\varepsilon}_{0,\vartheta},u^{-,\varepsilon}_{0,\vartheta}\rhoight)\overlineerset{\thetariangle}=\varepsilon \left(\thetailde n^+_{0,\vartheta},\thetailde u^+_{0,\vartheta},\thetailde n^-_{0,\vartheta},\thetailde u^-_{0,\vartheta}\rhoight)$. Then, by virtue of Proposition \rhoef{Prop4.1}, there is a positive constant $\varepsilon_0$ which may be quite small such that for any $\varepsilon<\varepsilon_0$, there is a unique local strong solution $\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)\in C^0([0,T];H^4(\mathbb R^3))$ to the Cauchy problem equationref{1.26}--equationref{1.27}, emanating from the initial data $\left(n^{+,\varepsilon}_{0,\vartheta},u^{+,\varepsilon}_{0,\vartheta},n^{-,\varepsilon}_{0,\vartheta},u^{-,\varepsilon}_{0,\vartheta}\rhoight)$ with $\mathcal E\left(n^{+,\varepsilon}_{0,\vartheta},u^{+,\varepsilon}_{0,\vartheta},n^{-,\varepsilon}_{0,\vartheta},u^{-,\varepsilon}_{0,\vartheta}\rhoight)=\varepsilon$.
We fix $\varepsilon_0>0$ which may be small enough, then for any $\varepsilon\in(0,\varepsilon_0)$. Define
$$T^*=\sup\left\{t\in(0,T^{\max})\big|\sup\limits_{\thetaau\in[0,t]}\mathcal E\left(\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)(\thetaau)\rhoight)\le \varepsilon_0\rhoight\}$$
and
$$T^{**}=\sup\left\{t\in(0,T^{\max})\big|\sup\limits_{\thetaau\in[0,t]}\left\|\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)(\thetaau)\rhoight\|_{L^2}\le \varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta t}\rhoight\}$$
where $T^{\max}$ denotes the maximal time of existence. Obviously, $T^*T^{**}>0$, and furthermore,
\begin{equation}gin{equation}\mathcal E\left(\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)(T^*)\rhoight)= \varepsilon_0\quad \thetaext{if} \quad T^*<\infty,\nonumber
\end{equation}d{equation}
and
\begin{equation}gin{equation}\left\|\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)(T^{**})\rhoight\|_{L^2}= \varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta T^{**}}\quad \thetaext{if} \quad T^{**}<\infty.\label{4.1}
\end{equation}d{equation}
Assume $T^*=\infty$, otherwise let $T^\varepsilon=T^*$ and $\delta_0=\varepsilon_0$, we can prove Theorem \rhoef{3mainth} immediately.
Let
\begin{equation}gin{equation}\label{4.2}T^\varepsilon=\fracrac{1}{\thetaheta}\ln\fracrac{2\varepsilon_0}{\varepsilon}\left(\thetaext{i.e.,}\ \varepsilon\thetaext{e}^{\thetaheta T^\varepsilon}=2\varepsilon_0\rhoight)\quad \thetaext{and}\quad \vartheta=\fracrac{1}{T^\varepsilon}.
\end{equation}d{equation}
Set $\left(n^+_{d},u^+_{d},n^-_{d},u^-_{d}\rhoight)=\left(n^{+,\varepsilon}_{\vartheta},u^{+,\varepsilon}_{\vartheta},n^{-,\varepsilon}_{\vartheta},u^{-,\varepsilon}_{\vartheta}\rhoight)-\varepsilon(\thetailde n^+_\vartheta,\thetailde u^+_\vartheta,\thetailde n^-_\vartheta,\thetailde u^-_\vartheta)$. Noticing that $$\left(n^+_{l},u^+_{l},n^-_{l},u^-_{l}\rhoight)=\varepsilon(\thetailde n^+_\vartheta,\thetailde u^+_\vartheta,\thetailde n^-_\vartheta,\thetailde u^-_\vartheta)$$ is also a solution to the linear system equationref{1.28} with the initial data $\left(n^{+,\varepsilon}_{0,\vartheta},u^{+,\varepsilon}_{0,\vartheta},n^{-,\varepsilon}_{0,\vartheta},u^{-,\varepsilon}_{0,\vartheta}\rhoight)\in H^2(\mathbb R^3)$, it is clear that $\left(n^+_{d},u^+_{d},n^-_{d},u^-_{d}\rhoight)$ is a solution to the system
\begin{equation}gin{equation}\label{4.3}
\left\{\begin{equation}gin{array}{l}
\partialartial_{t} n^{+}_d+\begin{equation}ta_{1} \operatorname{div} u^{+}_d=\mathbb{F}_{1}, \\
\partialartial_{t} u^{+}_d+\begin{equation}ta_{1} \nablabla n^{+}_d+\begin{equation}ta_{2} \nablabla n^{-}_d-v_{1}^{+} \Delta u^{+}_d-v_{2}^{+} \nablabla \operatorname{div} u^{+}_d=\mathbb{F}_{2}, \\
\partialartial_{t} n^{-}_d+\begin{equation}ta_{4} \operatorname{div} u^{-}_d=\mathbb{F}_{3}, \\
\partialartial_{t} u^{-}_d+\begin{equation}ta_{3} \nablabla n^{+}_d+\begin{equation}ta_{4} \nablabla n^{-}_d-v_{1}^{-} \Delta u^{-}_d-v_{2}^{-} \nablabla \operatorname{div} u^{-}_d=\mathbb{F}_{4},
\end{equation}d{array}\rhoight.
\end{equation}d{equation}
subject to the initial condition
\begin{equation}gin{equation}\label{4.4}
\left(n^{+}_d, u^{+}_d, n^{-}_d, u^{-}_d\rhoight)(x, 0)=0,
\end{equation}d{equation}
where the nonlinear terms are given by
\[
\mathbb{F}_{1}=\alphalpha_{1} F_{1}\left(\fracrac{n^{+,\varepsilon}_\vartheta}{\alphalpha_{1}}, \fracrac{u^{+,\varepsilon}_\vartheta}{\sqrt{\alphalpha_{1}}}\rhoight), \quad \mathbb{F}_{2}=\sqrt{\alphalpha_{1}} F_{2}
\left(\fracrac{n^{+,\varepsilon}_\vartheta}{\alphalpha_{1}}, \fracrac{u^{+,\varepsilon}_\vartheta}{\sqrt{\alphalpha_{1}}}, \fracrac{n^{+,\varepsilon}_\vartheta}{\alphalpha_{4}}, \fracrac{u^{+,\varepsilon}_\vartheta}{\sqrt{\alphalpha_{4}}}\rhoight),
\]
and
\[
\mathbb{F}_{3}=\alphalpha_{4} F_{3}\left(\fracrac{n^{+,\varepsilon}_\vartheta}{\alphalpha_{4}}, \fracrac{u^{+,\varepsilon}_\vartheta}{\sqrt{\alphalpha_{4}}}\rhoight), \quad \mathbb{F}_{4}=\sqrt{\alphalpha_{4}} F_{4}\left(\fracrac{n^{+,\varepsilon}_\vartheta}{\alphalpha_{1}}, \fracrac{u^{+,\varepsilon}_\vartheta}{\sqrt{\alphalpha_{1}}}, \fracrac{n^{+,\varepsilon}_\vartheta}{\alphalpha_{4}}, \fracrac{u^{+,\varepsilon}_\vartheta}{\sqrt{\alphalpha_{4}}}\rhoight).
\]
Now, we claim that
\begin{equation}gin{equation}\label{4.5} T^\varepsilon=\min\{T^\varepsilon,T^{**}\},
\end{equation}d{equation}
provided that $\varepsilon_0$ is small enough. Indeed, if $T^{**}=\min\{T^\varepsilon,T^{**}\}$, then $T^{**}<\infty$.
By defining $U=(n^+_d, u^+_d, n^-_d, u^-_d)^t$ and $\mathcal
F=(\mathcal{F}^1,\mathcal{F}^2,\mathcal{F}^3,\mathcal{F}^4)^t$, it holds from Duhamel's principle that
\begin{equation}gin{equation}\nonumber U=\int_0^t\thetaext{e}^{(t-\thetaau)\mathcal{A}}\mathcal F(\thetaau)\mathrm{d}\thetaau.
\end{equation}d{equation}
By virtue of Proposition \rhoef{Prop3.6} and equationref{4.2}, we
have after a complicated but straightforward computation that
\begin{equation}gin{equation}\label{4.6}\begin{equation}gin{split}\|U(T^{**})\|_{L^2}\lesssim& \int_0^{T^{**}}\left\|\thetaext{e}^{(t-\thetaau)\mathcal{A}}\mathcal F(\thetaau)\rhoight\|_{L^2}\mathrm{d}\thetaau\\ \lesssim &\int_0^{T^{**}}\thetaext{e}^{\thetaheta(t-\thetaau)}\left\|\mathcal F(\thetaau)\rhoight\|_{L^2}\mathrm{d}\thetaau
\\ \lesssim &\int_0^{T^{**}}\thetaext{e}^{\thetaheta(t-\thetaau)}\left(\left\|\left(n^{+,\varepsilon}_{\vartheta},
u^{+,\varepsilon}_{\vartheta},n^{-,\varepsilon}_{\vartheta},u^{-,\varepsilon}_{\vartheta}\rhoight)(\thetaau)\rhoight\|_{L^2}\left\|\nablabla\left(n^{+,\varepsilon}_{\vartheta},
u^{+,\varepsilon}_{\vartheta},n^{-,\varepsilon}_{\vartheta},u^{-,\varepsilon}_{\vartheta}\rhoight)(\thetaau)\rhoight\|_{W^{1,\infty}}\rhoight.
\\&+\left.\left\|\nablabla\left(n^{+,\varepsilon}_{\vartheta},
u^{+,\varepsilon}_{\vartheta},n^{-,\varepsilon}_{\vartheta},u^{-,\varepsilon}_{\vartheta}\rhoight)(\thetaau)\rhoight\|_{L^4}^2\rhoight)\mathrm{d}\thetaau
\\ \lesssim &~\int_0^{T^{**}}\thetaext{e}^{\thetaheta(t-\thetaau)}\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta \thetaau}\left(\left(\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta \thetaau}\rhoight)^{\fracrac{1}{6}}\varepsilon_0^{\fracrac{5}{6}}+\left(\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta \thetaau}\rhoight)^{\fracrac{1}{8}}\varepsilon_0^{\fracrac{7}{8}}\rhoight)\mathrm{d}\thetaau
\\ \lesssim &~\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta T^{**}}\left(\left(\varepsilon\thetaext{e}^{\thetaheta T^{**}}\rhoight)^{\fracrac{1}{6}}\varepsilon_0^\fracrac{7}{9}+\left(\varepsilon\thetaext{e}^{\thetaheta T^{**}}\rhoight)^{\fracrac{1}{8}}\varepsilon_0^\fracrac{5}{6}\rhoight)\\ \lesssim &~\varepsilon_0^\fracrac{17}{18}\left(\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta T^{**}}\rhoight), \end{equation}d{split}\end{equation}d{equation}
where, by H\"older's inequality and Sobolev's inequality, we used the facts
$$\|\nablabla f\|_{L^\infty}\lesssim \|f\|_{L^2}^\fracrac{1}{6}\|\nablabla^3 f\|_{L^2}^\fracrac{5}{6},$$
$$\|\nablabla^2 f\|_{L^\infty}\lesssim \|f\|_{L^2}^\fracrac{1}{8}\|\nablabla^4 f\|_{L^2}^\fracrac{7}{8}$$
and
$$\|\nablabla f\|_{L^4}\lesssim \|f\|_{L^2}^\fracrac{9}{16}\|\nablabla^4 f\|_{L^2}^\fracrac{7}{16}.$$
If $\varepsilon_0$ is small enough, by Proposition \rhoef{Prop2.2} and equationref{4.6}, we see that
\begin{equation}gin{equation}\nonumber\left\|\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)(T^{**})\rhoight\|_{L^2}
\le C\left( \varepsilon\thetaext{e}^{\thetaheta T^{**}} +\varepsilon_0^\fracrac{17}{18}\left(\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta T^{**}}\rhoight)\rhoight)<\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta T^{**}},
\end{equation}d{equation}
which contradicts with equationref{4.1}.
Finally, performing the similar procedure as in equationref{4.6} and using Proposition \rhoef{Prop2.2}, we deduce that
\begin{equation}gin{equation}\begin{equation}gin{split}&\left\|\left(n^{+,\varepsilon}_\vartheta,u^{+,\varepsilon}_\vartheta,n^{-,\varepsilon}_\vartheta,u^{-,\varepsilon}_\vartheta\rhoight)(T^{\varepsilon})\rhoight\|_{L^2}
\\ \ge&~\thetaext{e}^{(\thetaheta-\vartheta)T^\varepsilon}\varepsilon \left\|\left(\thetailde n^+_{0,\vartheta},\thetailde u^+_{0,\vartheta},\thetailde n^-_{0,\vartheta},\thetailde u^-_{0,\vartheta}\rhoight)\rhoight\|_{L^2}-C\varepsilon_0^\fracrac{17}{18}\left(\varepsilon\varepsilon_0^{-\fracrac{1}{3}}\thetaext{e}^{\thetaheta T^{\varepsilon}}\rhoight)\\ \ge&~\fracrac{2\varepsilon_0m_0}{\thetaext{e}}-C\varepsilon_0^\fracrac{29}{18}\\ \ge&~\fracrac{\varepsilon_0m_0}{\thetaext{e}},
\end{equation}d{split}\end{equation}d{equation}
if $\varepsilon_0$ is small enough, where $m_0=\left\|\left(\thetailde n^+_{0,\vartheta},\thetailde u^+_{0,\vartheta},\thetailde n^-_{0,\vartheta},\thetailde u^-_{0,\vartheta}\rhoight)\rhoight\|_{L^2}$. This completes the proof of Theorem \rhoef{3mainth} by defining
$\delta_0=\min\left\{\varepsilon_0,\fracrac{\varepsilon_0m_0}{\thetaext{e}}\rhoight\}$.
$\Box$
{\bf Statement:} No conflict of interest exists in the submission of this
manuscript, and
the datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
\section*{Acknowledgments}
Yinghui Zhang' research is
partially supported by Guangxi Natural Science Foundation
$\#$2019JJG110003, $\#$2019AC20214 and Key Laboratory of Mathematical and Statistical Model (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region. Lei Yao's research is
partially supported by National Natural Science Foundation of China
$\#$12171390, $\#$11931013, $\#$11571280 and Natural Science Basic Research Plan
for Distinguished Young Scholars in Shaanxi Province of China (Grant
No. 2019JC-26).
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\end{equation}d{document} | math | 86,217 |
{\beta}egin{document}
\thetaetaitle[critical points]{On the number of critical points of solutions of semilinear equations in $\mathbb{R}^2$}
\thetaetahanks{This work was supported by Prin-2015KB9WPT, by Universit\'a di Roma "La Sapienza" and partially supported by Indam-Gnampa}
{\alpha}uthor[Gladiali]{Francesca Gladiali}
{\alpha}ddress{Dipartimento di Chimica e Farmacia, Universit\`a di Sassari, via Piandanna 4 - 07100 Sassari, e-mail: {{\sigma}f [email protected]}.}
{\alpha}uthor[Grossi]{Massimo Grossi }
{\alpha}ddress{Dipartimento di Matematica, Universit\`a di Roma ``La Sapienza", P.le A. Moro 2 - 00185 Roma, e-mail: {{\sigma}f [email protected]}.}
\maketitle
{\beta}egin{abstract}
In this paper we construct families of bounded domains $\Omega_{\varepsilon}$ and solutions $u_{\varepsilon}$ of
\[{\beta}egin{cases}
-\Deltaelta u_{\varepsilon}=1&\thetaetaext{ in }\ \Omega_{\varepsilon}\\
u_{\varepsilon}=0&\thetaetaext{ on }\ \partial\Omega_{\varepsilon}
{\varepsilon}nd{cases}\]
such that, for any integer $k\gammae2$, $u_{\varepsilon}$ admits at least $k$ maximum points for small enough ${\varepsilon}psilon$. The domain $\Omega_{\varepsilon}$ is ``not far'' to be convex in the sense that it
is starshaped, the curvature of $\partial\Omega_{\varepsilon}$ vanishes at exactly $two$ points and the minimum of the curvature of $\partial\Omega_{\varepsilon}$ goes to $0$ as ${\varepsilon}\thetaetao0$.
{\varepsilon}nd{abstract}
{\sigma}ezione{Introduction}{\lambda}abel{s0}
The computation of the number and of the nature of critical points of positive solution of the problem
{\beta}egin{equation}{\lambda}abel{i0}
{\beta}egin{cases}
-\Deltaelta u=f(u)&\thetaetaext{ in }\ \Omega\\
u=0&\thetaetaext{ on }\ \partial\Omega
{\varepsilon}nd{cases}
{\varepsilon}nd{equation}
where $\Omega{\sigma}ubset\mathbb{R}^n$, $n\gammae2$ is a smooth bounded domain and $f$ is a smooth nonlinearity, is a classic and fascinating problem.\\
Many techniques and important results were developed in the literature (Morse theory, degree theory, etc.) to address this problem. In these few lines is impossible to mention all these contributions, so we will limit ourselves to recall some of them that are closer to the purpose of this paper.\\
One of the first major results concerns the case $f(u)={\lambda} u$, so $u$ is the first eigenfunction of the Laplacian with zero Dirichlet boundary condition. It was proved by
Brascamp and Lieb \cite{bl} and Acker, Payne and Phillippin \cite{app} in dimension $n=2$ that if $\Omega{\sigma}ubset\mathbb{R}^n$ is strictly convex then $-{\lambda}og u$ is convex, so that the superlevel sets are convex and $u$ admits a unique critical point in $\Omegaega$. Other results on the shape of level sets for various nonlinearities $f$ can be found in \cite{Ko83}, \cite{CS82}, \cite{k}, \cite{Ka86}, \cite{Ga55}, \cite{Ga57}, \cite{Cslin94} and references therein.
\\
A second seminal result that we want to mention is the fundamental theorem by Gidas, Ni and Nirenberg \cite{gnn}, which holds in domains which are convex in the direction $x_i$ for any $i=1,..,n$. We have that a domain is convex in the direction $x_1$ (say) if $P=(p_1,x')i^{*}n\Omegaega$ and $Q=(q_1,x')i^{*}n\Omegaega$ then the line segment $\omegaverline{PQ}$ is contained in $\Omegaega$.
\\
{\beta}egin{theorem*}[{\beta}f Gidas, Ni, Nirenberg]
Let $\Omega{\sigma}ubset\mathbb{R}^n$ a bounded, smooth domain which is symmetric with respect to the plane $x_i=0$ for any $i=1,..,n$ and convex in the $x_i$ direction for $i=1,..,n$. Suppose that $u$ is a positive
solution to {\varepsilon}qref{i0}
where $f$ is a locally Lipschitz nonlinearity. Then
{\beta}egin{itemize}
i^{*}tem $u$ is symmetric with respect to $x_1,..,x_n$. (Symmetry)
i^{*}tem $\frac {\partial u}{\partial x_i}<0$ for $x_i>0$ and $i=1,\partialltaots,n$. (Monotonicity)
{\varepsilon}nd{itemize}
{\varepsilon}nd{theorem*}
\noindent An easy consequence of the symmetry and monotonicity properties in the previous theorem is that \[
{\sigma}um_{i=1}^nx_i\frac{\partial u}{\partial x_i}<0 \ \ f_0rall x\ne0\] that is all the superlevel sets are $starshaped$ with respect to the origin.\\
This theorem holds in symmetric domains. Although it is expected that the uniqueness of the critical point (as well as the starlikeness of superlevel sets) holds in more general convex domains, this is a very difficult hypothesis to remove. \\
Next we mention another important result which holds for a wide class of nonlinearities $f$ without the symmetry assumption on $\Omega$ and for semi-stable solutions. To this end we recall that a solution $u$ to {\varepsilon}qref{i0} is semi-stable if the linearized operator at $u$ admits a nonnegative first eigenvalue.
{\beta}egin{theorem*}[{\beta}f Cabr\'e, Chanillo \cite{cc}]
Assume $\Omegaega$ is a smooth, bounded and convex domain of $\mathbb{R}^2$ whose boundary has positive curvature. Suppose $f\gammae0$ and $u$ is a semi-stable positive solution to {\varepsilon}qref{i0}.
Then $u$ has a unique critical point, which is non-degenerate.
{\varepsilon}nd{theorem*}
As a consequence the superlevel sets of $u$ are strictly convex in a neighborhood of the critical point and in a neighborhood of the boundary. It is thought that they are all convex, but this is certainly not true for suitable nonlinearities like in the following surprising result:
{\beta}egin{theorem*}[{\beta}f Hamel, Nadirashvili, Sire \cite{hns}]
In dimension $n=2$ there are some smooth bounded convex domains $\Omegaega$ and some $C^{i^{*}nfty}$ functions $f:[0,+i^{*}nfty)\thetaetao \mathbb{R}$ for which problem {\varepsilon}qref{i0} admits a solution $u$ which is not quasiconcave.
{\varepsilon}nd{theorem*}
We recall that a function is called quasiconcave if its superlevel sets are all convex. We can then conclude that the convexity of the domain is not always preserved by the superlevel sets. Nevertheless by the Gidas, Ni, Nirenberg theorem, being the domain $\Omegaega$ in \cite{hns} symmetric, the superlevel sets in this example are still starshaped and the maximum point of the solution is unique.\\
\noindent The previous results suggest the following questions:
\vskip0cm\noindentkip0.2cm
\noindent {{\beta}f Question 1}: {{\varepsilon}m Assume $\Omegaega$ is starshaped. Are the superlevel sets of any positive solution to {\varepsilon}qref{i0} starshaped?}
\vskip0cm\noindentkip0.2cm
\noindent {{\beta}f Question 2}: {{\varepsilon}m Assume that $u$ is a positive solution to {\varepsilon}qref{i0}
in a smooth bounded domain $\Omega{\sigma}ubset\mathbb{R}^2$ whose curvature is negative somewhere. What about the number of critical points of $u$?}
\vskip0cm\noindentkip0.2cm
\noindent Of course interesting examples deal with contractible domains $\Omegaega$, otherwise it is not difficult to construct examples of solution $u$ to {\varepsilon}qref{i0} with many critical points.
Some results in the direction to prove Question 1 were obtained for non-symmetric domains, in a perturbative setting, by Grossi and Molle \cite{gm} and Gladiali and Grossi \cite{gg1,gg2}.\\
In this paper we answer Question 1 showing that the starlikeness of the domain is not maintained by the superlevel sets.
Moreover we consider also Question 2 showing that in general there is no bound on the number of critical points. \\
Of course this last result is very sensitive to the shape of $\Omega$. In a recent paper \cite{lr} it was showed that if $\partial\Omega$ is contained in ${\lambda}eft\{zi^{*}n\mathbb{C}:|z|^2=f(z)+\omegaverline{f(z)}\rhoight\}$ where $f(z)$ is a rational function, then, differently than our case, there is a bound on the number of the critical points. We refer to \cite{am} for other results in these direction.
\\
Actually we will construct a family of domains $\Omega_{\varepsilon}$
starshaped with respect to an interior point and solutions $u_{\varepsilon}$ of the classical torsion problem, namely
{\beta}egin{equation}{\lambda}abel{eq:torsion}
{\beta}egin{cases}
-\Deltaelta u=1&\thetaetaext{ in } \ \Omega\\
u=0&\thetaetaext{ on }\ \partial\Omega
{\varepsilon}nd{cases}
{\varepsilon}nd{equation}
with an arbitrary large number of maxima and of disjoint superlevel sets.
Moreover the curvature of $\partial\Omega_{\varepsilon}$ vanishes at exactly two points and its minimum value goes to $0$ as ${\varepsilon}\thetaetao0$.
In some sense our domains $\Omega_{\varepsilon}$ are not ``far'' to be convex.
More precisely our result is the following,
{\beta}egin{theorem}{\lambda}abel{i1}
For any integer $k\gammaeq 2$ there exists a family of smooth bounded domains $\Omega_{{\varepsilon},k}{\sigma}ubset\mathbb{R}^2$ and smooth functions $u_{{\varepsilon},k}:\Omega_{{\varepsilon},k}\thetaetao\mathbb{R}^+$ which solves the torsion problem {\varepsilon}qref{eq:torsion} in $\Omegaega_{{\varepsilon},k}$, such that for ${\varepsilon}$ small enough,
{\beta}egin{itemize}
i^{*}tem $\Omega_{{\varepsilon},k}$ is starshaped with respect to an interior point.
$(P0)$
i^{*}tem The set $u_{{\varepsilon},k,}$ $\{u_{{\varepsilon},k}>c\}$ is non-empty and has at least
$k$ connected components; in particular $u_{{\varepsilon},k}$ has at least $k$ maximum points.
$(P1)$
i^{*}tem If $S$ is the strip $S=\{(x,y)i^{*}n\mathbb{R}^2\hbox{ such that }|y|<1\}$
and $Q$ is any compact set of $\mathbb{R}^2$ then \ $\Omega_{{\varepsilon},k}\cap Q\xrightarrow[{\varepsilon}\thetaetao0]\ S\cap Q$.\hskip5cm(P2)
i^{*}tem The curvature of $\partial\Omega_{{\varepsilon},k}$ changes sign and vanishes exactly at two points. Moreover $\min\Big(Curv_{\partial\Omega_{{\varepsilon},k}}\Big)\xrightarrow[{\varepsilon}\thetaetao0]\ 0$
.
$(P3)$
{\varepsilon}nd{itemize}
{\varepsilon}nd{theorem}
A picture of $\Omega_{{\varepsilon},2}$ for ${\varepsilon}$ small is given in Fig.1.
{\beta}egin{figure}[h]
\centering
i^{*}ncludegraphics[scale=0.15]{graf.png}
\caption{Domain $\Omega_{{\varepsilon},2}$ with level set $\{u_{{\varepsilon},2}=c\}$}
{\varepsilon}nd{figure}
Of course $(P2)$ implies that the superlevel set $\{u_{{\varepsilon},k}>c\}$ is not starshaped.
We recall that every solution to {\varepsilon}qref{eq:torsion} is positive by the Maximum principle and semi-stable as in \cite{cc}.
We point out that the solution $u_{{\varepsilon},k}$ will be explicitly provided and the domain $\Omega_{{\varepsilon},k}$ will be the superlevel set $\{u_{{\varepsilon},k}>0\}$.\\
In some sense our result shows that the assumption on the positivity of the curvature of $\partial\Omega$ in
Cabr\'e and Chanillo's Theorem cannot be relaxed because it is enough that the curvature of $\partial\Omega_{{\varepsilon},k}$ satisfies $(P3)$ to imply that there exists a semi-stable solution of a (simple) PDE with an arbitrary number of critical points. By $(P2)$ our domain is 'locally''' close to a strip and $u_{{\varepsilon},k}\xrightarrow[{\varepsilon}\thetaetao0]\ \frac12-\frac{y^2}2$ in $C^2_{loc}(\mathbb{R}^2)$. Note that the function $\frac12-\frac{y^2}2$, which solves $-\Deltaelta u=1$ in the strip $S$, was also used in Hamel, Nadirashvili and Sire \cite{hns}.
We point out that when ${\varepsilon}$ is small enough, the domain $\Omegaega_{{\varepsilon},k}$ in Theorem \rhoef{i0} looks like the one in \cite{hns} even if it has negative curvature somewhere.
Before describing the construction of the solution $u_{{\varepsilon},k}$ let us make some remarks on $(P2)$. It proves that the starlikeness of $\Omega_{{\varepsilon},k}$ is not enough to guarantee that the superlevel sets are starshaped proving Question 1. To our knowledge this is the first example with this property.
Theorem {\varepsilon}qref{i1} also shows that it cannot exist a starshaped rearrangement which associates to a smooth function $u$ another function $u^*$ with starshaped superlevel sets verifying the standard properties of rearrangements, i.e.
{\beta}egin{equation}{\lambda}abel{i2}
i^{*}nt_{\Omega^*}|u^*|^p= i^{*}nt_\Omega|u|^p\quadf_0rall p\gammae1\qquad\hbox{ and }\qquadi^{*}nt_{\Omega^*}|\nablabla u^*|^2{\lambda}e i^{*}nt_\Omega|\nablabla u|^2.
{\varepsilon}nd{equation}
A starshaped rearrangement which verifies, under additional assumptions, properties {\varepsilon}qref{i2} was introduced by Kawohl in \cite{k1} and \cite{k}. This implies that, jointly with {\varepsilon}qref{i2},
{\beta}egin{equation}{\lambda}abel{i99}
i^{*}nf{\lambda}imits_{ui^{*}n H^1_0(\Omega)}\frac12i^{*}nt_\Omega|\nablabla u|^2-i^{*}nt_\Omega u
{\varepsilon}nd{equation}
is achieved at a unique function $u$ with starshaped superlevel sets. However this type of rearrangement does not always exist, it is depending on the shape of $\Omega$.
An example (see \cite{k} and \cite{g}) is the so-called {{\varepsilon}m Grabmüller's long nose''} \cite{g}. Since our pair $(\Omega_{{\varepsilon},k},u_{{\varepsilon},k})$ satisfies {\varepsilon}qref{i99} and $u_{{\varepsilon},k}$ has some superlevel sets which are not starshaped, the requested starshaped rearrangement cannot exists for $\Omega_{{\varepsilon},k}$.\\
Finally we remark that in Makar-Limanov \cite{ml} it was proved that if $\Omegaega$ is a smooth bounded strictly convex domain of $\mathbb{R}^2$ and $u$ solves the torsion problem in $\Omegaega$ then the superlevel sets are strictly convex too. It seems then that the torsion problem is a {{\varepsilon}m``good'' } problem in which the properties of $\Omegaega$ are maintained by the superlevel sets. It is then even more unexpected that this does not hold for the starlikeness.
Next we say some words about the construction of $u_{{\varepsilon},k}$. The starting point is given by the function
$$\phi(y)=\frac 12-\frac12y^2$$
which solves
{\beta}egin{equation}
{\beta}egin{cases}
-\Deltaelta\phi=1&\thetaetaext{ in }\ |y|<1\\
u=0&\thetaetaext{ on }\ y=\pm1.
{\varepsilon}nd{cases}
{\varepsilon}nd{equation}
Our function $u_{{\varepsilon},k}$ is a perturbation of $\phi$ with suitable $harmonic$ functions. The choice of the harmonic functions is quite delicate: let us consider the holomorphic function $F_k:\mathbb{C}\thetaetao\mathbb{C}$,
{\beta}egin{equation}{\lambda}abel{eq:F}
F_k(z)=-\Pi_{i=1}^{2k}(z-x_i)
{\varepsilon}nd{equation}
for arbitrary real numbers $x_1<x_2<..<x_{2k}$ and define
$$v_k(x,y)=Re\Big(F_k(z)\Big).$$
Next we define $u_{{\varepsilon},k}$ as
$$u_{{\varepsilon},k}(x,y)=\frac 12-\frac12y^2+{\varepsilon}(y^3-3yx^2)+{\varepsilon}^\frac32v_k(x,y).$$
We have trivially that $-\Deltaelta u_{{\varepsilon},k}=1$ and the proof of Theorem \rhoef{i1} reduces to show that for ${\varepsilon}$
small enough the set $\Omega_{{\varepsilon},k}=\{u_{{\varepsilon},k}>0\}$ is a bounded smooth domain which verifies $(P0)-(P3)$. Although the function $u_{{\varepsilon},k}$ is explicitly provided, the proof of Theorem \rhoef{i1} involves delicate computations. Note that the power $\frac32$ appearing in the definition of $u_{{\varepsilon},k}$ can be replaced with any real number ${\alpha}lphai^{*}n(1,2)$. However ${\alpha}lpha=2$ is not allowed for technical reasons (``bad'' interactions occur).\\
There is a flexibility in the choice of the holomorphic function $F_k$; indeed it can be replaced by another one such that the restriction to the real line has $k$ maxima points and verifies some suitable growth condition at $\pmi^{*}nfty$. \\
Theorem \rhoef{i0} can be extended to semi-stable solutions of more general nonlinear problems. Let us consider a solution $u$ to
{\beta}egin{equation}{\lambda}abel{f1}
{\beta}egin{cases}
-\Deltaelta u={\lambda}ambda f(u)&\hbox{in }\Omega\\
u>0&\hbox{in }\Omega\\
u=0&\hbox{on }\partial\Omega
{\varepsilon}nd{cases}
{\varepsilon}nd{equation}
where $\Omega{\sigma}ubset\mathbb{R}^2$ is a bounded smooth domain, $f:\mathbb{R}^+\thetaetao\mathbb{R}$ is a smooth nonlinearity (say $C^1$) with $f(0)>0$ and $u_{\lambda}$ is a family of solutions of {\varepsilon}qref{f1} satisfying
{\beta}egin{equation}{\lambda}abel{f2}
||u_{\lambda}||_i^{*}nfty{\lambda}e C\quad\hbox{for ${\lambda}$ small},
{\varepsilon}nd{equation}
with $C$ independent of ${\lambda}$.
A classical example of solutions satisfying {\varepsilon}qref{f1} and {\varepsilon}qref{f2} was given by Mignot and Puel \cite{mp} when $f$ is a positive, increasing and convex nonlinearity and $0<{\lambda}<{\lambda}^*$.
See \cite[Theorem 10]{gg1} for some results about convexity and uniqueness of the critical point to solutions to {\varepsilon}qref{f1}.
Then we have the following result,
{\beta}egin{theorem}{\lambda}abel{i3}
Let ${\varepsilon}>0$, $k\gammae 2$ and $\Omegaega_{{\varepsilon},k}$ be as in Theorem \rhoef{i0}. Then there exists ${\beta}ar {\lambda}$ (depending on ${\varepsilon}$) such that if $u_{{\lambda},{\varepsilon},k}$
is a solution to {\varepsilon}qref{f1} in $\Omegaega_{{\varepsilon},k}$ that satisfies {\varepsilon}qref{f2}, we have that, for any $0<{\lambda}<{\beta}ar {\lambda}$, $u_{{\lambda},{\varepsilon},k}$
is semi-stable and satisfies $(P1)$-$(P3)$.
{\varepsilon}nd{theorem}
{\sigma}ection{The holomorphic function $F(z)$}
Here and in the next sections, to simplify the notation we omit the index $k$ when we define the functions $v(x,y)$, $u_{\varepsilon}(x,y)$ and the domains $\Omegaega_{\varepsilon}$.\\
For $k\gammae2$ let us consider arbitrary real numbers
$x_1<x_2<..<x_{2k}$ and the holomorphic function $F:\mathbb{C}\thetaetao\mathbb{C}$ in {\varepsilon}qref{eq:F} given by
{\beta}egin{equation}{\lambda}abel{b1}
F(z)=-\Pi_{i=1}^{2k}(z-x_i)=-{\sigma}um_{i=0}^{2k}a_iz^i
{\varepsilon}nd{equation}
where of course $a_{2k}=1$.\\
Let us denote by $f$ the restriction of $F$ to the {{\varepsilon}m real line}. We immediately get that $f(x_1)=..=f(x_{2k})=0$ and that $f$ has $k$ maximum points.
Let us consider the function $v:\mathbb{R}^2\thetaetao\mathbb{R}$ defined as
{\beta}egin{equation}{\lambda}abel{eq:v-re}
v(x,y)=Re\Big(F(z)\Big)
{\varepsilon}nd{equation}
which is harmonic in $\mathbb{R}^2$ and satisfies $v(x,0)=f(x)$. By construction
{\beta}egin{equation}{\lambda}abel{eq:v-pj}
v(x,y)=-{\sigma}um _{j=0}^{2k} a_jP_j(x,y){\varepsilon}nd{equation}
with $a_{2k}=1$ and where $P_j$ are homogeneous harmonic polynomials of degree $j$. Finally we introduce the function {\beta}egin{equation}{\lambda}abel{eq:u-epsilon}
{\beta}oxed{u_{\varepsilon}(x,y)=\frac 12-\frac12y^2+{\varepsilon}(y^3-3x^2y)+{\varepsilon}^\frac 32v(x,y)}
{\varepsilon}nd{equation}
which satisfies
\[-\Deltaelta u_{\varepsilon}=1\ \ \thetaetaext{ in }\mathbb{R}^2.\]
The function $v$ coincides with $f(x)$ along the $x$-axis, while $u_{\varepsilon}(x,0)=\frac 12-f(x)$. We end this section with a brief comment on the term ${\varepsilon}(y^3-3x^2y)$: it appears in the definition of $u_{\varepsilon}$ to have that the curvature of our domain vanishes exactly at two points. It
breaks the symmetry of the domain with respect to $y$, otherwise we would have that the curvature vanishes at $four$ points.
{\sigma}ection{Proof of Theorem \rhoef{i0}}
In this section we show that the function $u_{\varepsilon}$ in {\varepsilon}qref{eq:u-epsilon} verifies the claim of Theorem \rhoef{i0}. In the rest of the paper we let $o(1)$ be a quantity that goes to zero as ${\varepsilon}$ goes to zero and $k\gammaeq 2$.
{\beta}egin{theorem}{\lambda}abel{b2}
For ${\varepsilon}$ small enough the function $u_{\varepsilon}(x,y)$ in {\varepsilon}qref{eq:u-epsilon}
admits a connected component (that we call $\Omegaega_{\varepsilon}$) of the superlevel set
\[
\{(x,y)i^{*}n\mathbb{R}^2\hbox{ such that }u_{\varepsilon}(x,y)>0\}
\]
which satisfies:\\
i) $\Omegaega_{{\varepsilon}}$ is a smooth bounded domain;\\
ii) $\Omegaega_{{\varepsilon}}$ is starshaped with respect one of its points;\\
iii) $\Omegaega_{{\varepsilon}}$ contains $k$ disjoint connected components $Z_{1,{\varepsilon}},..,Z_{k,{\varepsilon}}$ of the superlevel set
$\{(x,y)i^{*}n\mathbb{R}^2\hbox{ such that } u_{\varepsilon}(x,y)>\frac 12\}$.
{\varepsilon}nd{theorem}
{\beta}egin{proof}
{{\beta}f Step 1:} Let $x_{\varepsilon}={\lambda}eft(\frac 3{{\varepsilon}^\frac 32}\rhoight)^\frac 1{2k}$. We want to show that
\[u(\pm x_{\varepsilon},y){\lambda}eq -2 \ \ \thetaetaext{ for } |y|<1+h\]
when ${\varepsilon}$ is small enough and $0<h<1$. In {\varepsilon}qref{eq:v-pj}
let us consider the polynomial of degree $2k$, namely
\[P_{2k}(x,y)={\sigma}um_{j=0}^k b_j x^{2k-2j}y^{2j}\]
for some suitable coefficients $b_j$ such that $b_0=b_{k}=1$. Then
\[{\varepsilon}^\frac 32 P_{2k}(x_{\varepsilon},y)={\varepsilon}^\frac 32 {\sigma}um_{j=0}^k b_j {\lambda}eft(\frac 3{{\varepsilon}^\frac 32}\rhoight)^\frac {2k-2j}{2k} y^{2j}
=3+o(1)\quad\hbox{as }{\varepsilon}\thetaetao0\]
uniformly with respect to $-1-h<y<1+h$.\\
In a very similar manner, for any $0{\lambda}e j{\lambda}e2k-1$ we have that
\[{\varepsilon}^\frac 32 P_j(x_{\varepsilon},y)=o(1)\]
and
\[{\lambda}eft| {\varepsilon} (y^3-x_{\varepsilon}^2y)\rhoight|=O{\lambda}eft({\varepsilon}^\frac {2k-3}{2k}\rhoight)\]
for ${\varepsilon}\thetaetao 0$ uniformly with respect to $-1-h<y<1+h$.
Considering all these estimates we obtain
\[{\lambda}eft| u_{\varepsilon}(x_{\varepsilon},y)+\frac 52+\frac 12 y^2\rhoight|=o(1)\quad\hbox{as }{\varepsilon}\thetaetao0.
\]
The very same computation shows also that
\[ u_{\varepsilon}(-x_{\varepsilon},y){\lambda}eq -2
\]
when ${\varepsilon}$ is small enough and concludes the proof.
\
\noindent {{\beta}f Step 2:} We show that $u_{\varepsilon}(x,y)< 0$ on the segments
$$T_{\pm h}=\{(x,y)i^{*}n\mathbb{R}^2\hbox{ such that } y=\pm(1+ h), xi^{*}n[-x_{\varepsilon},x_{\varepsilon}]\}$$
for some $0<h<1$ when ${\varepsilon}$ is small enough.\\
First let us observe that for
$(x,y)i^{*}n T_{\pm h}$,
\[{\lambda}eft|{\varepsilon}(y^3-3x^2y)\rhoight|{\lambda}eq {\varepsilon} (8+6x_{\varepsilon}^2){\lambda}eq 8{\varepsilon}+12{\varepsilon}^\frac{2k-3}{2k}=O{\lambda}eft({\varepsilon}^\frac{2k-3}{2k}\rhoight)\]
when ${\varepsilon}\thetaetao 0$.
Next, note that, by {\varepsilon}qref{eq:v-pj}
\[v(x,\pm(1+h))=-{\sigma}um_{j=1}^{2k}a_jP_j(x,\pm( 1+h))\]
and since $a_{2k}=1$ we get that
\[{\sigma}up_{xi^{*}n \mathbb{R}}v(x,\pm(1+h))=Ci^{*}n\mathbb{R}.\]
Then we obtain
\[u_{\varepsilon}(x,\pm(1+h))=-\frac 12h^2-h+{\varepsilon} ((\pm(1+h))^3\pm3x^2(1+h))+{\varepsilon} ^\frac 32 v(x,\pm(1+h))<-\frac 12 h^2<0\]
for ${\varepsilon} $ small enough.\\
\noindent {{\beta}f Step 3:}
We have proved that
for every ${\varepsilon}$ small enough $u_{\varepsilon}(x,y)<0$
on the boundary of the rectangle $R_{\varepsilon}=\{(x,y)i^{*}n\mathbb{R}^2\hbox{ such that } -x_{\varepsilon}{\lambda}e x{\lambda}e x_{\varepsilon}, -(1+h){\lambda}e y{\lambda}e1+h \}$. Since $u_{\varepsilon}(x_1,0)=\frac 12+{\varepsilon} v(x_1,0)=\frac 12+{\varepsilon} f(x_1)=\frac 12$
this implies that there is a connected component of the superlevel set $u_{\varepsilon}(x,y)>0$
, that we call $\Omegaega_{{\varepsilon}}$, which is contained in the interior of $R_{\varepsilon}$ and contains the point $(x_1,0)$.
Since $u_{\varepsilon}$ is continuous then $\Omegaega_{{\varepsilon}}$ is a connected
open set with non empty interior.\\
Furthermore when ${\varepsilon}$ satisfies
{\beta}egin{equation}{\lambda}abel{cond-ep-1}
{\varepsilon}<{\lambda}eft(\frac 1{2{\sigma}up_{xi^{*}n[x_1,x_{2k}] }(-f(x))}\rhoight)^\frac 23
{\varepsilon}nd{equation}
then all the segment $[x_1,x_{2k}]\thetaetaimes \{0\}$ belongs to $\Omegaega_{\varepsilon}$.\\
\noindent {{\beta}f Step 4:}
In this step we prove that when ${\varepsilon}$ is small enough $\Omegaega_{{\varepsilon}}$ is $smooth$ and $starshaped$ with respect to the point $(x_1,0)$, which is equivalent to show that
$$(x-x_1,y)\cdot\nu(x,y){\lambda}e-{\alpha}lpha<0\hbox{ for any }(x,y)i^{*}n\partial \Omegaega_{{\varepsilon}},$$
where $\nu(x,y)$ is the outer normal of $\partial \Omegaega_{\varepsilon}$ at the point $(x,y)$. In particular we will show that
{\beta}egin{equation}\nonumber
(x-x_1)\frac{\partial u_{\varepsilon}}{\partial x}+y\frac{\partial u_{\varepsilon}}{\partial y}{\lambda}e-{\alpha}lpha\quadf_0rall (x,y)i^{*}n R_{\varepsilon}\hbox{ such that } u_{\varepsilon}(x,y)=0.
{\varepsilon}nd{equation}
It is easily seen that
\[(x-x_1)\frac{\partial u_{\varepsilon}}{\partial x}+y\frac{\partial u_{\varepsilon}}{\partial y}=-y^2+{\varepsilon}{\lambda}eft( -9x^2y+6xx_1y+3y^3\rhoight) +{\varepsilon}^\frac 32 {\lambda}eft((x-x_1)\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}\rhoight).\]
On the other hand since $u_{\varepsilon}(x,y)=0$ on $\partial \Omegaega_{{\varepsilon}}$ we get that
\[-y^2=-1-2{\varepsilon}(y^3-3x^2y) -2{\varepsilon}^\frac 32v(x,y)\]
and
{\beta}egin{equation}{\lambda}abel{eq:stell}\nonumber
(x-x_1)\frac{\partial u_{\varepsilon}}{\partial x}+y\frac{\partial u_{\varepsilon}}{\partial y}=-1+{\varepsilon}{\lambda}eft(y^3-3x^2y+6xx_1y \rhoight)
+{\varepsilon}^\frac 32 {\lambda}eft((x-x_1)\frac{\partial v}{\partial x}+y\frac {\partial v}{\partial y}-2v(x,y)\rhoight).
{\varepsilon}nd{equation}
By {\varepsilon}qref{eq:v-pj} and Euler Theorem we get
{\beta}egin{equation}\nonumber
x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}=-{\sigma}um_{j=0}^{2k}a_j{\lambda}eft(
x\frac{\partial P_j}{\partial x}+
y\frac{\partial P_j}{\partial y}\rhoight)=- {\sigma}um_{j=1}^{2k} j a_j P_j(x,y)
{\varepsilon}nd{equation}
and so, recalling that $a_{2k}=1$,
\[
-2v(x,y)+(x-x_1)\frac{\partial v}{\partial x}+y\frac {\partial v}{\partial y}=- {\sigma}um_{j=0}^{2k} (j-2) a_j P_j(x,y)+x_1{\sigma}um_{j=1}^{2k} a_j\frac{\partial P_j}{\partial x}\xrightarrow[|x|\thetaetaoi^{*}nfty]\ -i^{*}nfty
\]
uniformly for $yi^{*}n[-1-h,1+h]$. Hence
\[{\sigma}up_{(x,y)i^{*}n (-i^{*}nfty,i^{*}nfty)\thetaetaimes [-1-h,1+h]}{\lambda}eft(-2v(x,y)+(x-x_1)\frac{\partial v}{\partial x}+y\frac {\partial v}{\partial y}\rhoight)=d<i^{*}nfty.\]
In addition
\[{\sigma}up_{(x,y)i^{*}n [-x_{\varepsilon},x_{\varepsilon}]\thetaetaimes [-1-h,1+h]} {\lambda}eft(y^3-3x^2y+6xx_1y\rhoight){\lambda}e Cx_{\varepsilon}^2=O{\lambda}eft({\varepsilon}^{-\frac 3{2k}}\rhoight)\]
as ${\varepsilon}\thetaetao 0$, so that
\[{\varepsilon} {\lambda}eft(y^3-3x^2y+6xx_1y\rhoight)2=O{\lambda}eft({\varepsilon}^{\frac {2k-3}{2k}}\rhoight)\]
in the rectangle $R_{\varepsilon}$.
Summarizing again we have that
\[{\sigma}up_{\partial \Omegaega_{\varepsilon}{\sigma}ubset R_{\varepsilon}}{\lambda}eft( (x-x_1)\frac{\partial u_{\varepsilon}}{\partial x}+y\frac{\partial u_{\varepsilon}}{\partial y}\rhoight){\lambda}e -1+o(1)<-\frac12\]
for ${\varepsilon}\thetaetao 0$ which gives the claim.\\
Of course $(x-x_1)\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\neq 0$ on $\partial \Omegaega_{{\varepsilon}}$ implies that $\partial \Omegaega_{{\varepsilon}}$ is a smooth curve. \\
\noindent {{\beta}f Step 5:} Here we prove that the superlevel set
\[L_{{\varepsilon}}:=
\{(x,y)i^{*}n \mathbb{R}^2 \hbox{ such that }
u_{\varepsilon}(x,y)> \frac 12
\}
\]
admits in $\Omegaega_{\varepsilon}$ at least $k$ disjoint components $Z_{1,{\varepsilon}},\partialltaots, Z_{k,{\varepsilon}}$.\\
Since $f(x_j)=0$ for $j=1,\partialltaots,2k$ and $f(x)\thetaetao -i^{*}nfty$ as $|x|\thetaetao i^{*}nfty$, there exist points $s_ji^{*}n (x_{2j},x_{2j+1})$ for $j=1,\partialltaots,k-1$ and points ${\beta}ar s_ji^{*}n (x_{2j+1},x_{2j+2})$ for $j=0,\partialltaots,k$ such that
{\beta}egin{align*}
f(s_j)=\min_{xi^{*}n [ x_{2j},x_{2j+1}]}f(x)<0 & \thetaetaext{ for }j=1,\partialltaots,k-1\\
f({\beta}ar s_j)=\max_{xi^{*}n [ x_{2j+1},x_{2j+2}]}f(x)>0 & \thetaetaext{ for }j=0,\partialltaots,k-1
{\varepsilon}nd{align*}
First observe that
\[u_{\varepsilon}({\beta}ar s_j,0)=\frac 12 +{\varepsilon}^\frac 32 v({\beta}ar s_j,0)=\frac 12 +{\varepsilon}^\frac 32 f({\beta}ar s_j)>\frac 12\]
for $j=0,\partialltaots,k$ so that the points $({\beta}ar s_j,0)$ are contained in $L_{{\varepsilon}}$ for every ${\varepsilon}$.
Next we want to prove that
{\beta}egin{equation}{\lambda}abel{eq:u-delta}
u_{\varepsilon}(s_j,y)<\frac 12
{\varepsilon}nd{equation}
for $j=1,\partialltaots,k-1$ and $-1-h<y<1+h$.\\
In this way since ${\beta}ar s_j<x_{2j}<s_{j+1}$ and the segment $[x_1,x_{2k}]\thetaetaimes \{0\}$ is contained in $\Omegaega_{\varepsilon}$ by Step 3, we also obtain that the superlevel set $L_{{\varepsilon}}$ admits at least $k$ disjoint components.\\
To prove {\varepsilon}qref{eq:u-delta} we argue by contradiction and assume that there exists
a sequence ${\varepsilon}_n\thetaetao 0$ and points $y_ni^{*}n [-1-h,1+h]$ such that
{\beta}egin{equation}{\lambda}abel{eq:passaggio}
u_{{\varepsilon}_n}(s_j,y_n)=\frac 12-\frac 12 y_n^2+{\varepsilon}_n(y_n^3-3s_j^2y_n)+{\varepsilon}_n^\frac 32 v(s_j,y_n)\gammae\frac 12
{\varepsilon}nd{equation}
for $n\thetaetao i^{*}nfty$, for a fixed value of $j$. Formula {\varepsilon}qref{eq:passaggio} easily implies that $y_n\thetaetao 0$ as $n\thetaetao i^{*}nfty$ since $(y_n^3-3s_j^2y_n)$ and $v(s_j,y_n)$ are uniformly bounded and ${\varepsilon}_n\thetaetao 0$. Next we observe that since $v(s_j,0)=f(s_j)<0$
then $v(s_j,y_n){\lambda}e \frac {f(s_j)} 2<0$ for $n$ large enough. Moreover, using that $-\frac 12y_n^2+{\varepsilon}_n y_n^3<-\frac 14y_n^2$ for $n$ large enough we have
\[{\beta}egin{split}
u_{{\varepsilon}_n}(s_j,y_n)&=\frac 12-\frac 12 y_n^2+{\varepsilon}_n(y_n^3-3s_j^2y_n)+{\varepsilon}_n^\frac 32 v(s_j,y_n)\\
&{\lambda}e \frac 12-\frac 14y_n^2-3{\varepsilon}_n s_j^2y_n+{\varepsilon}_n^\frac 32 \frac {f(s_j)}2 <\frac 12
{\varepsilon}nd{split}
\]
since ${\varepsilon}_n^\frac 32{\lambda}eft( \frac {f(s_j)}2+9{\varepsilon}_n^\frac 12s_j^4\rhoight)<0$
for $n$ large enough. This contradiction ends the proof.
{\varepsilon}nd{proof}
Next aim is to derive additional information about the shape of $\Omegaega_{\varepsilon}$, in particular regarding the oriented curvature of $\partial \Omegaega_{\varepsilon}$. Since $\partial \Omegaega_{\varepsilon}$ is a level curve of $u_{\varepsilon}(x,y)$ then its oriented curvature at the point $(x,y)$ is given by
{\beta}egin{equation} {\lambda}abel{eq:curv}
Curv_{\partial \Omegaega_{\varepsilon}}(x,y)=-\frac{(u_{\varepsilon})_{xx}(u_{\varepsilon})_y^2-2(u_{\varepsilon})_{xy}(u_{\varepsilon})_x(u_{\varepsilon})_y+(u_{\varepsilon})_{yy}(u_{\varepsilon})_x^2}{{\lambda}eft((u_{\varepsilon})_x^2+(u_{\varepsilon})_y^2\rhoight)^\frac32}
{\varepsilon}nd{equation}
In particular, we want to prove
the following result
{\beta}egin{lemma}{\lambda}abel{lem:curv-2}
The oriented curvature of $\partial \Omegaega_{\varepsilon}$ vanishes exactly at two points when ${\varepsilon}$ is small enough.
{\varepsilon}nd{lemma}
Let us start examining the behavior of some points $(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})i^{*}n \partial \Omegaega_{\varepsilon}$ when ${\varepsilon}$ goes to zero.
{\beta}egin{lemma}{\lambda}abel{lem:behavior}
Let $(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})$ be a point on $\partial \Omegaega_{\varepsilon}$. Then if $|\zeta_{\varepsilon}|\thetaetao i^{*}nfty$ we have
{\beta}egin{equation}{\lambda}abel{e2}
|\zeta_{\varepsilon}|= {\lambda}eft(\frac 12 (1-{\varepsilon}ta_{\varepsilon}^2)\rhoight)^{\frac 1{2k}} {\varepsilon}^{-\frac 3{4k}}(1+o(1)).
{\varepsilon}nd{equation}
{\varepsilon}nd{lemma}
{\beta}egin{proof}
First
$(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})i^{*}n \partial \Omegaega_{\varepsilon}$ implies that
{\beta}egin{equation}{\lambda}abel{e3}
{\varepsilon} ({\varepsilon}ta_{\varepsilon}^3-3\zeta_{\varepsilon}^2{\varepsilon}ta_{\varepsilon})+{\varepsilon}^\frac 32 v(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=\frac 12 ({\varepsilon}ta_{\varepsilon}^2-1).
{\varepsilon}nd{equation}
Next we observe that, since ${\beta}ar \Omegaega_{\varepsilon}{\sigma}ubset R_{\varepsilon}$, where $R_{\varepsilon}$ is the rectangle introduced in Step 3 in the proof of Theorem \rhoef{b2}, then $|\zeta_{\varepsilon}|<3^\frac 1{2k}{\varepsilon}^{-\frac 3{4k}}$ and this implies that
{\beta}egin{equation}
{\lambda}abel{e4}
{\varepsilon}({\varepsilon}ta_{\varepsilon}^3-3\zeta_{\varepsilon}^2{\varepsilon}ta_{\varepsilon})=O{\lambda}eft({\varepsilon}^\frac{2k-3}{2k}\rhoight)
{\varepsilon}nd{equation}
for ${\varepsilon}\thetaetao 0$ and {\varepsilon}qref{e3} becomes, since $|\zeta_{\varepsilon}|\thetaetao i^{*}nfty$
\[{\varepsilon}^\frac 32 v(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=\frac 12 ({\varepsilon}ta_{\varepsilon}^2-1)+O{\lambda}eft({\varepsilon}^\frac{2k-3}{2k}\rhoight).\]
Finally by {\varepsilon}qref{eq:v-pj}, when $|\zeta_{\varepsilon}|\thetaetao i^{*}nfty$
\[v(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=-\zeta_{\varepsilon}^{2k}(1+o(1))\]
which jointly with the previous estimate gives
{\beta}egin{equation}{\lambda}abel{eq:exp-x}
{\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k}=\frac 12 (1-{\varepsilon}ta_{\varepsilon}^2)(1+o(1))
{\varepsilon}nd{equation}
when ${\varepsilon}\thetaetao 0$, from which {\varepsilon}qref{e2} follows.
{\varepsilon}nd{proof}
\vskip0cm\noindentkip0.3cm
{\beta}egin{proof}[Proof of Lemma \rhoef{lem:curv-2}]
Here let us denote by $(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})i^{*}n\partial\Omega_{\varepsilon}$ a point such that $Curv_{\partial\Omega_{\varepsilon}}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=0$. By {\varepsilon}qref{eq:curv}
{\beta}egin{equation}{\lambda}abel{e9}\nonumber
Curv_{\partial\Omega_{\varepsilon}}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=-\frac {N_{1,{\varepsilon}}+N_{2,{\varepsilon}}+N_{3,{\varepsilon}}}{D_{\varepsilon}^\frac32}
{\varepsilon}nd{equation}
with
{\beta}egin{align*}
N_{1,{\varepsilon}}&=(-6{\varepsilon} {\varepsilon}ta_{\varepsilon}+{\varepsilon}^\frac32v_{xx}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon}))
{\lambda}eft(-{\varepsilon}ta_{\varepsilon}+3{\varepsilon}({\varepsilon}ta_{\varepsilon}^2-\zeta_{\varepsilon}^2)+{\varepsilon}^\frac32v_y(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})\rhoight)^2\\
N_{2,{\varepsilon}}&=-2(- 6{\varepsilon} \zeta_{\varepsilon}+{\varepsilon}^\frac32v_{xy}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon}))\Big(-6{\varepsilon} \zeta_{\varepsilon} {\varepsilon}ta_{\varepsilon}+{\varepsilon}^\frac32v_x(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})\Big)\\
&(-{\varepsilon}ta_{\varepsilon}+3{\varepsilon}({\varepsilon}ta_{\varepsilon}^2-\zeta_{\varepsilon}^2)+{\varepsilon}^ \frac32v_y(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon}))\\
N_{3,{\varepsilon}}&=(-1+6 {\varepsilon} {\varepsilon}ta_{\varepsilon}+{\varepsilon}^\frac32v_{yy}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon}))\Big(-6{\varepsilon} \zeta_{\varepsilon}{\varepsilon}ta_{\varepsilon}+{\varepsilon}^\frac32v_x(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})\Big)^2\\
D_{\varepsilon}&= {\beta}ig(-6{\varepsilon}\zeta_{\varepsilon} {\varepsilon}ta_{\varepsilon}+{\varepsilon}^\frac32v_x(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon}){\beta}ig)^2+{\beta}ig(-{\varepsilon}ta_{\varepsilon}+3{\varepsilon}(\zeta_{\varepsilon}^2-{\varepsilon}ta_{\varepsilon}^2)+{\varepsilon}^\frac32v_y(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon}){\beta}ig)^2.
{\varepsilon}nd{align*}
We divide the proof in some steps.
\vskip0cm\noindentkip0.2cm
{{\beta}f Step 1: $|\zeta_{\varepsilon}|\thetaetao+i^{*}nfty$}
\vskip0cm\noindentkip0.1cm
We reason by contradiction. If the claim does not hold we can take sequences ${\varepsilon}_n$, $\zeta_n$, ${\varepsilon}ta_n$ such that ${\varepsilon}_n\thetaetao 0$, $\zeta_n\thetaetao \zeta_0$, ${\varepsilon}ta_n\thetaetao {\varepsilon}ta_0$ (since $|{\varepsilon}ta_{\varepsilon}|<2$ by definition of $R_{\varepsilon}$) and such that
$Curv_{\partial\Omega_{{\varepsilon}_n}}(\zeta_n,{\varepsilon}ta_n)=0$. Since $(\zeta_n,{\varepsilon}ta_n)i^{*}n \partial \Omegaega_n$ then {\varepsilon}qref{e3} holds and, passing to the limit, we have that ${\varepsilon}ta_n\thetaetao \pm1$ and
{\beta}egin{equation}{\lambda}abel{eq:somma-N}
{\beta}egin{split}
&0=\frac{N_{1,{\varepsilon}_n}+N_{2,{\varepsilon}_n}+N_{3,{\varepsilon}_n}}{\varepsilon}_n=\\
&-6{\beta}ig({\varepsilon}ta_n^3+o(1){\beta}ig)+72{\varepsilon}_n{\beta}ig(\zeta_0^2{\varepsilon}ta_n^2+o(1){\beta}ig)-36{\varepsilon}{\beta}ig(\zeta_0^2{\varepsilon}ta_n^2+o(1){\beta}ig)=-6{\beta}ig(\pm1+o(1){\beta}ig)
{\varepsilon}nd{split}
{\varepsilon}nd{equation}
which gives a contradiction.
\vskip0cm\noindentkip0.2cm
{{\beta}f Step 2}: We have that there exists $two$ values $\zeta_{\varepsilon}$ given by
$
\zeta_{\varepsilon}^\pm{\sigma}im\pm{\lambda}eft(\frac3{k(2k-1){\varepsilon}^\frac12}\rhoight)^\frac1{2k-2}
$
\vskip0cm\noindentkip0.1cm
First, observe that, by {\varepsilon}qref{b1}, {\varepsilon}qref{eq:v-re}, {\varepsilon}qref{eq:v-pj}
and Step 1 we have that
{\beta}egin{align*}
&v_x=-2k\zeta_{\varepsilon}^{2k-1}(1+o(1)) & v_y=c_k \zeta_{\varepsilon}^{2k-2} {\varepsilon}ta_{\varepsilon}(1+o(1))\\
& v_{xx}=-2k(2k-1) \zeta_{\varepsilon}^{2k-2}(1+o(1)) & v_{xy}=c'_k \zeta_{\varepsilon}^{2k-3} {\varepsilon}ta_{\varepsilon}(1+o(1)) \\
& v_{yy}=c_k \zeta_{\varepsilon}^{2k-2}(1+o(1))
{\varepsilon}nd{align*}
where $c_k,c'_k\ne0$ are constants depending on $k$.
Using {\varepsilon}qref{e2} and ${\varepsilon} \zeta_{\varepsilon}^2\thetaetao 0$ as ${\varepsilon} \thetaetao 0$, we obtain (denoting again by ${\varepsilon}ta_0={\lambda}im {\varepsilon}ta_{\varepsilon}$)
\[{\beta}egin{split}
&N_{1,{\varepsilon}}{\sigma}im {\lambda}eft(-6{\varepsilon}{\varepsilon}ta_{\varepsilon}-2k(2k-1){\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-2}\rhoight){\lambda}eft(-{\varepsilon}ta_{\varepsilon}-3{\varepsilon} \zeta_{\varepsilon}^2+c_k{\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-2}{\varepsilon}ta_{\varepsilon}
\rhoight)^2(1+o(1))\\
&={\beta}egin{cases}
{\varepsilon}ta_0^2 {\lambda}eft( -6{\varepsilon} {\varepsilon}ta_0-2k(2k-1){\varepsilon}^\frac 32\zeta_{\varepsilon}^{2k-2}\rhoight)(1+o(1)) & \thetaetaext{ when }{\varepsilon}ta_0\neq 0\\
o({\varepsilon}+{\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-2}) & \thetaetaext{ when }{\varepsilon}ta_0=0
{\varepsilon}nd{cases}
{\varepsilon}nd{split}\]
Using that ${\varepsilon}^\frac 32\zeta_{\varepsilon}^{2k}=O(1)$ (see {\varepsilon}qref{eq:exp-x}),
\[{\beta}egin{split}
&N_{2,{\varepsilon}}{\sigma}im -2 {\lambda}eft(-6{\varepsilon} \zeta_{\varepsilon}+c'_k {\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-3}{\varepsilon}ta_{\varepsilon}\rhoight){\lambda}eft(-6{\varepsilon}\zeta_{\varepsilon}{\varepsilon}ta_{\varepsilon}-2k {\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-1}\rhoight)\cdot\\
&{\lambda}eft( -{\varepsilon}ta_{\varepsilon} -3{\varepsilon} \zeta_{\varepsilon}^2+c_k{\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-2}{\varepsilon}ta_{\varepsilon}\rhoight)(1+o(1))\\
&={\beta}egin{cases}
2{\varepsilon}ta_0^2 {\lambda}eft( 12k {\varepsilon}^{1+\frac 32} \zeta_{\varepsilon}^{2k}-2k c'_k{\varepsilon}^3 \zeta_{\varepsilon}^{4k-4}{\varepsilon}ta_{\varepsilon}\rhoight) (1+o(1)) & \thetaetaext{ when }{\varepsilon}ta_0\neq 0\\
o({\varepsilon}+{\varepsilon}^3 \zeta_{\varepsilon}^{4k-4}) & \thetaetaext{ when }{\varepsilon}ta_0=0
{\varepsilon}nd{cases}
{\varepsilon}nd{split}\]
\[{\beta}egin{split}
&N_{3,{\varepsilon}}{\sigma}im {\lambda}eft( -1+6{\varepsilon} {\varepsilon}ta_{\varepsilon}+c_k{\varepsilon}^\frac 32\zeta_{\varepsilon}^{2k-2}\rhoight){\lambda}eft( -6{\varepsilon} \zeta_{\varepsilon}{\varepsilon}ta_{\varepsilon}-2k{\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-1}\rhoight)^2(1+o(1))\\
&=-1{\lambda}eft(4k^2{\varepsilon}^3\zeta_{\varepsilon}^{4k-2}+12 {\varepsilon}^{1+\frac 32}\zeta_{\varepsilon}^{2k}{\varepsilon}ta_{\varepsilon}^2\rhoight)(1+o(1)).
{\varepsilon}nd{split}\]
Hence if ${\beta}oxed{{\lambda}im{\lambda}imits_{{\varepsilon}\thetaetao0}{\varepsilon}ta_{\varepsilon}={\varepsilon}ta_0\neq \pm1}$, then ${\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-2}{\sigma}im {\varepsilon}^{\frac 3{2k}}{\lambda}eft( \frac 12 (1-{\varepsilon}ta_0^2)\rhoight)^{\frac {k-1}k}$ and
\[{\beta}egin{split}
N_{1,{\varepsilon}}+N_{2,{\varepsilon}}+N_{3,{\varepsilon}}
&={\varepsilon}^{\frac 3{2k}}{\lambda}eft( -{\varepsilon}ta_0^2 2k(2k-1) {\lambda}eft( \frac 12 (1-{\varepsilon}ta_0^2)\rhoight)^{\frac {k-1}k}\rhoight.\\
&{\lambda}eft. -4k^2{\lambda}eft( \frac 12 (1-{\varepsilon}ta_0^2)\rhoight)^{\frac {2k-1}k}\rhoight)(1+o(1))<0
{\varepsilon}nd{split}\]
showing that the curvature is {{\varepsilon}m strictly positive} in this case.\\
So we necessarily have that ${\beta}oxed{{\lambda}im{\lambda}imits_{{\varepsilon}\thetaetao0} {\varepsilon}ta_{\varepsilon}=\pm 1}$ and by {\varepsilon}qref{e2} we have that ${\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k}=o(1)$. This implies that
\[
N_{1,{\varepsilon}}={\lambda}eft( \mp 6{\varepsilon}-2k(2k-1){\varepsilon}^\frac 32 \zeta_{\varepsilon}^{2k-2}\rhoight)(1+o(1));\]
\[
N_{2,{\varepsilon}}=O{\lambda}eft( {\varepsilon}^2\zeta_{\varepsilon}^2+{\varepsilon}^{1+\frac 32}\zeta_{\varepsilon}^{2k}+{\varepsilon}^3\zeta_{\varepsilon}^{4k-4}\rhoight);\]
\[N_{3,{\varepsilon}}=O{\lambda}eft( {\varepsilon}^3\zeta_{\varepsilon}^{4k-2} +{\varepsilon}^3\zeta_{\varepsilon}^{4k-4}\rhoight)
\]
and, since ${\varepsilon}^2\zeta_{\varepsilon}^2,{\varepsilon}^{1+\frac 32}\zeta_{\varepsilon}^{2k}=o({\varepsilon})$ and ${\varepsilon}^3 \zeta_{\varepsilon}^{4k-4},{\varepsilon}^3\zeta_{\varepsilon}^{4k-2}
=o({\varepsilon}^\frac 32\zeta_{\varepsilon}^{2k-2})$ then $N_{2,{\varepsilon}},N_{3,{\varepsilon}}
=o(N_{1,{\varepsilon}})$ and
{\beta}egin{equation}{\lambda}abel{eq:curv-fin}
N_{1,{\varepsilon}}+N_{2,{\varepsilon}}+N_{3,{\varepsilon}}
={\lambda}eft(\mp6{\varepsilon}-2k(2k-1){\varepsilon}^\frac32\zeta_{\varepsilon}^{2k-2}\rhoight)(1+o(1)).
{\varepsilon}nd{equation}
Since $Curv_{\partial \Omegaega_{\varepsilon}}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=0$ we deduce
{\beta}egin{equation}{\lambda}abel{e10}
{\lambda}eft(\mp6{\varepsilon}-2k(2k-1){\varepsilon}^\frac32\zeta_{\varepsilon}^{2k-2}\rhoight)(1+o(1))=0
{\varepsilon}nd{equation}
as ${\varepsilon}\thetaetao 0$.
First we get that if ${\varepsilon}ta_{\varepsilon}\thetaetao1$ {\varepsilon}qref{e10} does not have solutions. Hence ${\varepsilon}ta_{\varepsilon}\thetaetao-1$ and {\varepsilon}qref{e10} becomes
{\beta}egin{equation}
{\lambda}eft(6-2k(2k-1){\varepsilon}^\frac12\zeta_{\varepsilon}^{2k-2}\rhoight)(1+o(1))=0
{\varepsilon}nd{equation}
that implies
{\beta}egin{equation}
{\varepsilon}^\frac12\zeta_{\varepsilon}^{2k-2}=\frac3{k(2k-1)}(1+o(1)).
{\varepsilon}nd{equation}
Correspondingly we get $two$ solutions $\zeta_{\varepsilon}$ whose behavior is given by
{\beta}egin{equation}
\zeta_{\varepsilon}^\pm{\sigma}im\pm{\lambda}eft(\frac3{k(2k-1){\varepsilon}^\frac12}\rhoight)^\frac1{2k-2}.
{\varepsilon}nd{equation}
\vskip0cm\noindentkip0.2cm
{{\beta}f Step 3: conclusion}
\vskip0cm\noindentkip0.1cm
We end the proof showing that, corresponding to $\zeta_{\varepsilon}^+$ there exists only one ${\varepsilon}ta_{\varepsilon}^+$ which verifies $0=Curv_{\partial\Omega_{\varepsilon}}(\zeta_{\varepsilon},{\varepsilon}ta_{\varepsilon})=N_{1,{\varepsilon}}+N_{2,{\varepsilon}}+N_{3,{\varepsilon}}$ and the same is true for $\zeta_{\varepsilon}^-$. We apply the implicit function theorem to $u_{\varepsilon}$.
We have that $u_{\varepsilon}(\zeta_{\varepsilon}^+,{\varepsilon}ta_{\varepsilon}^+)=0$ and, recalling that ${\varepsilon}ta_{\varepsilon}^+\thetaetao-1$,
\[
(u_{\varepsilon})_y(\zeta_{\varepsilon}^+,{\varepsilon}ta_{\varepsilon}^+)=-{\varepsilon}ta_{\varepsilon}+3{\varepsilon}(\zeta_{\varepsilon}^2{\varepsilon}-{\varepsilon}ta_{\varepsilon}^2)+{\varepsilon}^\frac32v_y=1+o(1)\]
for ${\varepsilon}\thetaetao 0$.
So by the implicit function theorem we deduce that the equation $u_{\varepsilon}(x,y)=0$ has only one solution for $x=\zeta^+_{\varepsilon}$ and $y$ close to $-1$ which ends the proof.
{\varepsilon}nd{proof}
{\beta}egin{proof}[Proof of Theorem \rhoef{i0}]
The existence of the family of solutions $u_{{\varepsilon},k}$ to {\varepsilon}qref{eq:torsion} and of the domains $\Omegaega_{{\varepsilon},k}$, as well as the properties $(P0)$ and $(P1)$ follow by Theorem \rhoef{b2}.\\
Property $(P2)$ is a consequence of the definition of $u_{{\varepsilon},k}(x,y)$ and that locally $u_{{\varepsilon},k}(x,y)\thetaetao \frac 12 (1-y^2)$ as ${\varepsilon}\thetaetao 0$.\\
Concerning $(P3)$, the curvature of $\partial\Omegaega_{{\varepsilon},k}$ does change sign because, denoting by $q_{\varepsilon}=(0,{\beta}eta_{\varepsilon})i^{*}n\partial\Omegaega_{{\varepsilon},k}$ with ${\beta}eta_{\varepsilon}\thetaetao-1$, we have
$$Curv_{\partial\Omega_{{\varepsilon},k}}(q_{\varepsilon})={\beta}ig(-6+o(1){\beta}ig)<0.$$
Next the fact that the curvature of $\partial \Omegaega_{{\varepsilon},k}$ vanishes exactly at two points follows by Lemma \rhoef{lem:curv-2}.
To prove that $\min{\lambda}eft(Curv_{\partial\Omega_{{\varepsilon},k}}\rhoight)\thetaetao 0$ as ${\varepsilon}\thetaetao 0$
we proceed as in the proof of Lemma \rhoef{lem:curv-2}. Denote by $(\thetaetailde\zeta_{\varepsilon},\thetaetailde{\varepsilon}ta_{\varepsilon})i^{*}n\partial\Omegaega_{\varepsilon}$ a point which achieves the minimum of the curvature of $\partial\Omegaega_{\varepsilon}$ (recall that $|\thetaetailde{\varepsilon}ta_{\varepsilon}|{\lambda}e C $). If, up to some subsequence, $\thetaetailde\zeta_{{\varepsilon}_n}\thetaetao\thetaetailde\zeta_0 $, since $\Omegaega_{\varepsilon}$ converges to a strip on compact set, the claim follows. On the other hand, if $|\thetaetailde\zeta_{{\varepsilon}_n}|\thetaetao+i^{*}nfty$, repeating step by step the computation in Case $2$ of Lemma \rhoef{lem:curv-2} we again get $Curv_{\partial\Omega_{{\varepsilon},k}}(\thetaetailde\zeta_{{\varepsilon}_n},\thetaetailde{\varepsilon}ta_{{\varepsilon}_n})\thetaetao0$. This ends the proof.
{\varepsilon}nd{proof}
{\sigma}ection{More general nonlinearities}
In this section we consider solutions to {\varepsilon}qref{f1} which satisfy {\varepsilon}qref{f2}. The existence is guaranteed for example if the assumptions in \cite{mp} are satisfied. Next lemma studies the behavior as ${\lambda}\thetaetao0$.
{\beta}egin{lemma}{\lambda}abel{lem:convergence}
Let $u_{\lambda}$ be a family of solutions to {\varepsilon}qref{f1} satisfying {\varepsilon}qref{f2}. Then we have that
{\beta}egin{equation}{\lambda}abel{f4}
\frac{u_{\lambda}}{{\lambda} f(0)}\thetaetao u_0\quad\hbox{as ${\lambda}\thetaetao0$ in }C^2(\Omega)
{\varepsilon}nd{equation}
where $u_0$ is a solution to
{\beta}egin{equation}{\lambda}abel{f6}
{\beta}egin{cases}
-\Deltaelta u=1&\hbox{in }\Omega\\
u=0&\hbox{on }\partial\Omega.
{\varepsilon}nd{cases}
{\varepsilon}nd{equation}
{\varepsilon}nd{lemma}
{\beta}egin{proof}
Let us show that
{\beta}egin{equation}{\lambda}abel{f3}
|u_{\lambda}|{\lambda}e C{\lambda}\quad\hbox{in }\Omega
{\varepsilon}nd{equation}
where $C$ is a constant independent of ${\lambda}$. By the Green representation formula we have that
\[
|u_{\lambda}(x)|{\lambda}e{\lambda}i^{*}nt_\Omega G(x,y){\lambda}eft|f{\beta}ig(u_{\lambda}(y){\beta}ig)\rhoight|dy{\lambda}e{\lambda}\max{\lambda}imits_{si^{*}n[0,C]}|f(s)|i^{*}nt_\Omega G(x,y)dy{\lambda}e C{\lambda}
\]
where $C$ is independent of ${\lambda}$. Next by {\varepsilon}qref{f2}, {\varepsilon}qref{f3} and the standard regularity theory we derive that
\[
u_{\lambda}\thetaetao0\quad\hbox{in }C^2(\Omega)
\]
as ${\lambda}\thetaetao 0$ so that $f(u_{\lambda})\thetaetao f(0)$.
Finally the standard regularity theory, applied to $\frac{u_{\lambda}}{{\lambda} f(0)}$, and
{\varepsilon}qref{f1} gives the claim.
{\varepsilon}nd{proof}
Theorem \rhoef{i3} is a straightforward consequence of the previous lemma
{\beta}egin{proof}[Proof of Theorem \rhoef{i3}]
Assume ${\varepsilon}$ is small enough to satisfy the assumptions of Theorem \rhoef{i0}. By Lemma \rhoef{lem:convergence} $\frac {u_{\lambda}}{{\lambda} f(0)}\thetaetao u_{{\varepsilon},k}$ as ${\lambda}\thetaetao 0$. Then the claim follows by the $C^2$ convergence of $u_{\lambda}$ to $u_{{\varepsilon},k}$ and the semi-stability of all solutions to {\varepsilon}qref{f6}.
{\varepsilon}nd{proof}
{\beta}ibliography{GladialiGrossiFinal.bib}
{\beta}ibliographystyle{abbrv}
{\varepsilon}nd{document} | math | 47,984 |
\begin{document}
\title{Instantons: the next frontier}
\begin{abstract}
Instantons, emerged in particle physics, have been intensely studied since the 1970's and had an enormous impact in mathematics since then. In this paper, we focus on one particular way in which mathematical physics has guided the development of algebraic geometry in the past 40+ years. To be precise, we examine how the notion of \textit{(mathematical) instanton bundles} in algebraic geometry has evolved from a class of vector bundles over $\cp3$ both to a class of torsion free sheaves on projective varieties of arbitrary dimension, and to a class of objects in the derived category of Fano threefolds. The original results contained in this survey focus precisely on the latter direction; in particular, we prove that the classical rank 2 instanton bundles over the projective space are indeed instanton objects for any suitable chamber in the space of Bridgeland stability conditions.
\end{abstract}
\tableofcontents
\section{Introduction}
Let $(M,g)$ be a 4-dimensional, oriented Riemannian manifold, and let $E\to M$ be a complex vector bundle over $M$. A \textit{connection} on $E$ is a $\mathbb{C}$-linear map
$$ \nabla ~\colon~ \Gamma(E) \longrightarrow \Gamma(E)\otimes\Omega^1_M $$
satisfying the \textit{Leibniz rule}: given a smooth function $f\in C^{\infty}(M)$ and a global section $\sigma\in\Gamma(E)$, we have
$$ \nabla(f\cdot\sigma) = f\cdot \nabla(\sigma) + \sigma\otimes df; $$
here, $\Gamma(E)$ denotes the $C^{\infty}(M)$-module of smooth global sections of the vector bundle $E$, and $\Omega^p_M$ denotes the space of smooth $p$-forms on $M$.
The composition $F_\nabla:=\nabla\circ\nabla$ leads to a $C^{\infty}(M)$-linear map
$$ F_\nabla ~\colon~ \Gamma(E) \longrightarrow \Gamma(E)\otimes\Omega^2_M $$
which is called the \textit{curvature} of the connection $\nabla$. In other words, $F_\nabla$ can be regarded as a 2-form on $M$ with values in the endomorphism bundle $\End(E)$: $F_\nabla\in\Gamma(\End(E))\otimes\Omega^2_M$.
A connection $\nabla$ is called an \textit{instanton connection} on the vector bundle $E\to M$ if $F_\nabla$ is anti-self-dual with respect to the Hodge star operator $*:\Omega^2_M\to\Omega^2_M$ (recall that $*^2=1$), that is
\begin{equation} \label{eq:asdym}
*F_\nabla = - F_\nabla.
\end{equation}
which is known as the \textit{anti-self-dual Yang--Mills equation}. When $M$ is not compact, one usually also imposes a finiteness condition on the total $L^2$-norm of the curvature, that is,
$$ \|F_\nabla\|_{L^2}^2 := \int_M {\rm tr}(F_\nabla\wedge*F_\nabla) < \infty. $$
Instantons have been intensely studied since the 1970's, providing the initial ideas for the rise of a new area of mathematics, namely \textit{gauge theory}, that had great influence in more traditional areas like differential topology, differential geometry, mathematical physics, representation theory, and algebraic geometry. In this paper, we focus on the latter, by explaining how the notion of \textit{(mathematical) instanton bundles} in algebraic geometry has evolved in the past 40+ years.
The starting point was the \textit{Atiyah--Ward correspondence}, which is described in Section \ref{section:AWcorrespondece} below; it was the first fundamental link between mathematical physics and algebraic geometry to be discovered, (another one is the \textit{Hitchin--Kobayashi correspondence}), and leads to the highly influential work of Atiyah, Hitchin Drinfeld and Manin \cite{ADHM}. Roughly speaking, the Atiyah--Ward correspondence transformed the differential geometric problem of finding solutions of the anti-self-dual Yang--Mills equation \ref{eq:asdym} over $M=\mathbb{R}^4$ into an algebraic geometric problem of constructing vector bundles over $\cp3$ with certain properties. These vector bundles were subsequently studied by mathematicians (namely, Barth, Hulek, Le Potier, Okonek, Schneider, Spindler) who coined the expression \textit{(mathematical) instanton bundles} in the 1980's. What initially was a class of vector bundles over $\cp3$ became, after the work of Salamon \cite{S84} and Donaldson \cite{DFlag}, a class of vector bundles on odd dimensional projective spaces \cite{OS} and on flag manifolds \cite{MMPL}.
In Section \ref{section:perverse} we explain why it is natural to consider a further generalization of instanton bundles on projective spaces both to arbitrary projective spaces and to non locally free sheaves. The notion of (non necessarily locally free) \textit{instanton sheaves} was introduced in \cite{J-inst}, and lead also to the notion of \textit{perverse instanton sheaves}, see Definition \ref{defn:instantons} and Definition \ref{defn:perverse}, respectively. These notions arise when one tries to compactify the moduli space of instanton bundles when regarded as an open subset either of the Gieseker moduli space of stable sheaves, or the moduli space of stable representations of a certain quiver.
In Section 4 and Section 5 we illustrate how it is possible to extend the definition of instanton to algebraic varieties beyond projective spaces.
In Section 4, we focus on the specific case of Fano threefolds of Picard rank one. Also in this setting, we start treating the case of rank 2 vector bundles: we summarize their main properties and those of their families presented by Faenzi in \cite{F}.
Once again, we are lead to take into account a wider family of sheaves which includes sheaves of arbitrary rank and that are not necessarily locally free. This is what motivated
\cite{CJ}, where the definition of instanton sheaves on Fano threefolds is provided and their main features are collected.
An investigation of rank 2 instanton bundles on a Fano threefold $X$ (in particular on a Fano $X$ of index 2) has also been conducted by Kuznetsov in \cite{K}, but this time the language chosen
to present features of instantons is the one of derived categories. Kuznetsov establishes indeed
a correspondence between instanton bundles on $X$ and objects in the triangulated category ${\mathcal B}_X:=\langle {\mathcal O}_X, {\mathcal O}_X(1)\rangle ^{\perp}$, named \textit{acyclic extensions} of instantons, and shows how the categorical properties of these latter ``reflect'' the sheaf-theoretical properties of instantons. We end Section 4 with a brief survey of the main results of \cite{K}.
Section \ref{section:h-inst} is dedicated to review the definition and main properties of $h$-instanton sheaves, which have been recently introduced by Antonelli and Casnati in \cite{AC}. Their notion seeks to generalize the previous definitions of instanton sheaves to arbitrary projective schemes $X$ endowed with an ample and globally generated line bundle ${\mathcal O}_X(h)$.
For an $n$ dimensional scheme $X$, the line bundle ${\mathcal O}_X(h)$ induces a finite map from $X$ to $\mathbb{P}^n$. The direct image of an ordinary $h$-instanton sheaf on $X$
by this map is indeed an instanton sheaf on ${\mathbb{P}^n}$ in the sense of \cite{J-inst}, and conversely the pullback of an instanton sheaf on ${\mathbb{P}^n}$ is an ordinary $h$-instanton sheaf on $X$.
One of the noticeable features of an instanton sheaf is that it can be constructed as the cohomology of a monad. An $h$-instanton sheaf on $X$ has such a monadic presentation if $X$ is ACM with respect to the line bundle ${\mathcal O}_X(h)$. We then focus on some particular cases of $X$, where the monads for ordinary $h$-instantons are more neatly presented. Those monads overlap some known monadic presentations on ${\mathbb{P}^n}$ and some smooth quadrics, and provide some new monadic presentations on scrolls.
Next, we review the construction of rank 2 $h$-instanton bundles on some smooth varieties of low dimension, namely curves, surfaces, Fano threefolds of Picard rank $1$ and scrolls. In particular, we compare the rank $2$ $h$-instanton bundles on a Fano threefold with the instanton bundles in Section \ref{section:Fano}. We will see that these two notions coincide in most cases.
In the end, we review an example of an $h-$instanton bundle on the image of the Segre embedding $\p1\times \p1 \times \p1 \hookrightarrow \p3$ to illustrate that $h$-instanton bundles on varieties of higher Picard rank can have several pathologies comparing to the classical instanton bundles.
Finally, Section \ref{section:bridgeland} contains the original results presented in this paper. We return to the setup of Section \ref{section:Fano} and introduce the notion of ${\mathcal C}$-instanton object in the derived category of a Fano threefold of Picard number one. This notion is highly inspired by the works of Faenzi \cite{F}, Kuznetsov \cite{K}, and Comaschi--Jardim \cite{CJ}, and the examples of Bridgeland semistable objects in the projective space provided in \cite{JMM}. The idea is to find a chamber ${\mathcal C}$ in the stability manifold so that semistability in this chamber alone already provides the correct cohomology vanishing conditions. In the cases of the projective space and the quadric threefold, we prove that there is a chamber ${\mathcal C}$ such that ${\mathcal C}$-instantons have monad-type descriptions. Examples of ${\mathcal C}$-instanton objects are provided, including some that do not fit any previous definitions. Additionally, we prove that the classical rank 2 instanton bundles in the projective space are indeed ${\mathcal C}$-instanton objects for any suitable chamber ${\mathcal C}$. Acyclic extensions are proved to exist for stable ${\mathcal C}$-instantons on any Fano threefold of index 2, and moreover such acyclic extensions are again ${\mathcal C}$-instanton objects.
\subsection*{Acknowledgments}
The authors would like to express their gratitude to Daniele Faenzi for numerous conversations on the topic of this article, and in particular for suggesting the invariance under the twisted duality functor \eqref{dualityFunctor} as the right way to fix the Chern character of an instanton object, which allowed us to find the correct $(\alpha,s)$-slice for defining ${\mathcal C}$-instanton objects in Section \ref{section:bridgeland}. GC is supported by the FAPESP post-doctoral grant number 2019/21140-1 and the BEPE grant number 2022/09063-4. MJ is supported by the CNPQ grant number 302889/2018-3 and the FAPESP Thematic Project \textit{Gauge theory and Algebraic Geometry} number 2018/21391-1. CM is supported by the FAPESP post-doctoral grant number 2020/06938-4, which is part of the FAPESP Thematic Project \textit{Gauge theory and Algebraic Geometry} number 2018/21391-1. DM is supported by the FAPESP post-doctoral grant number 2020/03499-0.
\section{Atiyah--Ward correspondence and mathematical instanton bundles}\label{section:AWcorrespondece}
We will now consider our Riemannian manifold $(M,g)$ as being
the four dimensional sphere $S^4$ equipped with the usual round metric; recall that this is conformal to the usual euclidean metric on $\mathbb{R}^4\simeq S^4\setminus\{\infty\}$; by virtue of the Uhlenbeck removable singularities theorem, any instanton connection on $\mathbb{R}^4$ extends to an instanton on $S^4$.
Furthermore, $S^4$ can be identified with the quaternionic projective line $\hp1$ as follows: any $p\in\mathbb{R}^4$ is associated to the point $[p:1]\in\hp1$, while $\infty$ goes to $[1:0]\in\hp1$. One can then consider the smooth map $\tau:\cp3\to\hp1$ given by
$$ \tau[x:y:z:w] = [x+jy:z+jw] . $$
Note that the fibers of $\tau$ are isomorphic to $\cp1$: the pre-image of the point $[u+jv:1]\in\hp1$ is the line $[u:v:1:0]$, while $\tau^{-1}([1:0])=[u:v:0:0]$.
This so-called \textit{twistor map} has the following fantastic property, first noticed by Atiyah and Ward in \cite{AW}. If $(E,\nabla)$ is a complex vector bundle on $S^4\simeq\hp1$ equipped with an instanton connection, then the curvature $F_{\tau^*\nabla}$ of the pull-back connection $\tau^*\nabla$ is a 2-form with values in $\tau^*\End(E)=\End(\tau^*E)$ of type $(1,1)$. This means that $F_{\tau^*\nabla}$ induces a holomorphic structure on the pulled-back vector bundle $\tau^*E$; let us denote this by ${\mathcal E}$.
Observe that ${\mathcal E}$ must satisfy some obvious properties. First, its restriction to the fibers $\tau^{-1}(p)$ must always be holomorphically trivial. Second, note that
$$ \tau[x:y:z:w]=\tau[-y:x:-w:z], $$
since their images only differ by multiplication by $j$ on the left; therefore, the pulled-back bundle $\tau^*E$ must be invariant under the involution $\iota:\cp3\to\cp3$ given by $\iota[x,y,z,w]=[-y:x:-w:z]$, ie. $\iota^*{\mathcal E}\simeq{\mathcal E}$.
The non-trivial key property satisfied by ${\mathcal E}$ is provided by the \textit{Penrose transform}, which provides an isomorphism between the kernel of the Laplacian operator $\Delta$ coupled to the instanton connection $\nabla$, and the sheaf cohomology group $H^1(\cp3,{\mathcal E}(-2))$, where ${\mathcal E}(-2)={\mathcal E}\otimes{\mathcal O}_{\cp3}(-2)$ as usual. It turns out that $\ker\Delta$ is empty precisely because $S^4$ has positive scalar curvature and $F_\nabla$ is anti-self-dual. Therefore, we must have that $H^1(\cp3,{\mathcal E}(-2))=0$.
The \textit{Atiyah--Ward correspondence} correspondence essentially says that the original smooth instanton connection on $S^4$ can be reconstructed from the associated holomorphic vector bundle ${\mathcal E}$ satisfying the properties described above. (Since we would like to get into algebraic geometry as soon as possible, we are actually omitting many details here and this claim is an oversimplification of the actual theorems; the interested reader should look at \cite{A,WW}).
The Atiyah--Ward correspondence was the first fundamental link between mathematical physics an algebraic geometry to be discovered, and it has sparked a flurry of intense activity among algebraic geometers; Hartshorne \cite{H78} and Barth--Hulek \cite{BH} are perhaps the first algebraic geometry papers on bundles on projective spaces motivated by the works of Atiyah and collaborators (especially \cite{AW,ADHM}). However, to our knowledge, the first reference that uses the expression \textit{``instanton bundle"} is \cite[p. 370]{OSS}; in this reference, the authors define a \textit{complex instanton bundle} on $\cp3$ as a stable rank 2 holomorphic bundle ${\mathcal E}$ with $c_1({\mathcal E})=0$ satisfying $H^2({\mathcal E}(-2))=0$.
At this point (circa 1977-78), Barth noticed that every such complex instanton bundle $E$ can be realized as the cohomology of a \textit{monad}, that is, a complex of sheaves
\begin{equation} \label{monad1}
{\mathcal O}_{\cp3}(-1)^{\oplus c} \stackrel{\alpha}{\longrightarrow} {\mathcal O}_{\cp3}^{\oplus 2+2c} \stackrel{\beta}{\longrightarrow} {\mathcal O}_{\cp3}(1)^{\oplus c}
\end{equation}
for which $\alpha$ is injective, $\beta$ is surjective, and such that $E\simeq\ker\beta/\im\alpha$, and where $c=h^1({\mathcal E}(-1))$. Since the morphisms $\alpha$ and $\beta$ can be regarded as matrices whose entries are linear polynomials, this allowed to translate the classification of complex instanton bundles into a problem in linear algebra. This is essentially the crucial point explored in the seminal paper \cite{ADHM} by Atiyah, Drinfeld, Hitchin and Manin, where the full classification of $SU(2)$ instantons on $S^4$ was given.
In 1984, Salamon presented a higher dimensional version of the Atiyah--Ward correspondence \cite{S84}; see also \cite{MCS}. He considers a map $\tau:\cp{2k+1}\to\hp{k}$ given by
$$ \tau[x_0:y_0:\cdots:x_k:y_k]=[x_0+jy_0:\cdots:x_k+jy_k], $$
thus generalizing the twistor map defined above; the fibers of $\tau:\cp{2k+1}\to\hp{k}$ are also isomorphic to $\cp1$. Now let $E\to\hp{k}$ be a complex vector bundle equipped with a connection $\nabla$; this is said to be a \textit{quaternionic instanton} if its curvature $F_\nabla$ is of type $(1,1)$ with respect to any choice of almost complex structure in $\hp{k}$; when $k=1$, this is equivalent to the usual definition of an instanton on a 4-dimensional manifold.
It turns out that this condition is just what is needed to prove that the curvature $F_{\tau^*\nabla}$ of the pulled-back connection on $\tau^*E$ is of type $(1,1)$ on $\cp{2k+1}$, and therefore induces a holomorphic structure on $\tau^*E$.
This motivated the definition of \textit{mathematical instanton bundles} by Okonek and Spindler, see \cite{OS} in the following year. To be precise, a mathematical instanton bundle is a rank $2k$ holomorphic bundle ${\mathcal E}$ on $\cp{2k+1}$ satisfying the following conditions
\begin{enumerate}
\item $E$ is simple, ie. $\operatorname{Hom}({\mathcal E},{\mathcal E})=\mathbb{C}$;
\item its Chern polynomial is given by $c_t({\mathcal E})=(1-t^2)^{-c}$;
\item it has natural cohomology in the rank $-2k-1\le p\le 0$, that is, for each $p$ in the specified rank, at most one of the cohomology groups $H^p({\mathcal E}(l))$ can be non trivial;
\item ${\mathcal E}$ has \textit{trivial splitting type}, ie. ${\mathcal E}|_l$ is trivial for at least one line $\ell\subset\cp{2k+1}$;
\item ${\mathcal E}$ admits a symplectic structure, meaning that there exists an isomorphism $\phi:{\mathcal E}\to{\mathcal E}^*$ such that $\phi^*=-\phi$.
\end{enumerate}
Later, Ancona and Ottaviani noticed in \cite{AO} that conditions (2) and (3) imply condition (1). Mathematical instanton bundles were the subject of several articles in 1980's and 1990's. One fact that will be relevant later on is that conditions (2) and (3) imply that any mathematical instanton bundle is isomorphic to the cohomology of a monad
\begin{equation} \label{monad2}
{\mathcal O}_{\cp{2k+1}}(-1)^{\oplus c} \stackrel{\alpha}{\longrightarrow} {\mathcal O}_{\cp{2k+1}}^{\oplus 2k+2c} \stackrel{\beta}{\longrightarrow} {\mathcal O}_{\cp{2k+1}}(1)^{\oplus c}.
\end{equation}
Let $\wp:=\{y_k=0\}\subset\cp{2k+1}$ be a hyperplane, and note that the restriction $\tau|_{\wp}:\wp\to\hp{k}$ of the twistor map is surjective and provides a bijection between the affine subsets
$$ \wp\supset\{x_k\ne0\}=\mathbb{C}^{2k} ~~ \mapsto ~~ \mathbb{H}^k=\{q_k\ne0\}\subset\hp{k} $$
$$ (x_0,y_0,\cdots,x_{k-1},y_{k-1}) \mapsto (x_0+jy_0,\cdots,x_{k-1}+jy_{k-1}). $$
So lifting a quaternionic instanton connection on $\hp{k}$ to a mathematical instanton bundle on $\cp{2k+1}$ and restricting it to the hyperplane $\wp$ provides an injective map from the moduli space of quaternionic instantons on $\hp{k}$ to the moduli space of rank $2k$ holomorphic bundles on $\wp$ that arise as cohomology of a linear monad similar to the one in display \eqref{monad2}. For the case $k=1$, Donaldson used the so-called \textit{ADHM construction} to show that this map is actually and isomorphism \cite{D}; however, it is unknown to the authors whether the same is true for $k>1$. However, this observation can be regarded as a motivation to consider holomorphic bundles over even dimensional complex projective spaces which arise as the cohomology of a linear monad, see Definition \ref{defn:instantons} below.
\section{Instanton and perverse instanton sheaves}\label{section:perverse}
From this point onwards, we will shift to denoting sheaves by capital roman letters, while $\mathbb{P}^n$ means $\mathbb{CP}^n$, as it is more usual in algebraic geometry.
We start by recalling the characterization of the family of sheaves on projective spaces that can be represented as the cohomology of a monad as in display \eqref{monad2}.
\begin{theorem}{\label{thm:monadic presentation}}
A torsion free sheaf $E$ on ${\mathbb{P}^n}$ is the cohomology of a monad of the form
$$ U\otimes{\mathcal O}_{\mathbb{P}^n}(-1) \stackrel{\alpha}{\longrightarrow} V\otimes{\mathcal O}_{\mathbb{P}^n} \stackrel{\beta}{\longrightarrow} W\otimes{\mathcal O}_{\mathbb{P}^n}(1) , $$
where $U$, $V$ and $W$ are vector spaces, if and only if
\begin{enumerate}
\item[(1)] $H^0(E(-1)) = H^n(E(-n)) = 0$ for $n\ge2$;
\item[(2)] $H^1(E(-2)) = H^{n-1}(E(1-n)) = 0$ for $n\ge3$;
\item[(3)] $H^p(E(k)) = 0$ for every $k$ and $2\le p\le n-2$, when $n\ge4$.
\end{enumerate}
\end{theorem}
The proof is given in \cite[Theorem 3]{J-inst}; the \textit{if} part is an application the Beilinson spectral sequence, after some further cohomological vanishings are established; the converse claim is an easy calculation with long exact sequences.
Since the locally free sheaves on odd and on even dimensional complex projective spaces arising from quaternionic instantons on quaternionic projective spaces have $c_1=0$, the following definition definition was proposed in \cite{J-inst}.
\begin{definition} \label{defn:instantons}
An instanton sheaf on ${\mathbb{P}^n}$ is a torsion free sheaf $E$ with $c_1(E)=0$ satisfying the following cohomological conditions
\begin{enumerate}
\item[(1)] $H^0(E(-1)) = H^n(E(-n)) = 0$ for $n\ge2$;
\item[(2)] $H^1(E(-2)) = H^{n-1}(E(1-n)) = 0$ for $n\ge3$;
\item[(3)] $H^p(E(k)) = 0$ for every $k$ and $2\le p\le n-2$, when $n\ge4$.
\end{enumerate}
The number $c:=h^1(E(-1))=c_2(E)$ is called the \textit{charge} of $E$; this is also often called the \textit{quantum number} of $E$. The trivial bundle ${\mathcal O}_{\mathbb{P}^n}^{\oplus r}$ is regarded as an instanton sheaf of charge 0.
\end{definition}
Mathematical instanton bundles, as defined in the previous section, are simply locally free instanton sheaves of rank $2k$ on $\mathbb{P}^{2k+1}$. Therefore, the previous definition generalizes the notion of mathematical instanton bundles to include possibly non locally free sheaves of arbitrary rank on projective spaces of any dimension.
The first observation is that instanton sheaves of rank $r$ less than $n-1$ on ${\mathbb{P}^n}$ are necessarily trivial, i.e. ${\mathcal O}_{\mathbb{P}^n}^{\oplus r}$ \cite[Corollary 6]{J-inst}. Moreover, notice that we do not impose any condition on the stability of $E$; in fact, letting $E$ be a non trivial instanton sheaf of rank $r$ on ${\mathbb{P}^n}$, one can show that:
\begin{enumerate}
\item if $E$ is reflexive and $r\ge n-1$, then $E$ is $\mu$-semistable;
\item if $E$ is locally free and $r\ge 2n-1$, then $E$ is $\mu$-semistable.
\item when $r=2$ and $n=3$, then $E$ is Gieseker stable.
\end{enumerate}
The first two claims are contained in \cite[Theorem 15]{J-inst}, while the third was established in \cite[Theorem 4]{JMaT}.
Thus, in general, it is not in principle clear how to construct a moduli space of instanton sheaves of arbitrary rank and charge. This issue is addressed in \cite{HLa,HJM} using the ADHM construction of framed instanton bundles on ${\mathbb{P}^n}$, and more recently in \cite{JS} using representations of quivers. Let us comment on both approaches.
\subsection{The ADHM construction and perverse instanton sheaves}
Let $V$ and $W$ be vector spaces of dimension $c$ and $r$, respectively, and consider matrices
$$ A,B \in \End(V)\otimes H^0(\op{n-2}(1)) $$
$$ I \in\operatorname{Hom}(W,V)\otimes H^0(\op{n-2}(1)) ~~,~~ J \in\operatorname{Hom}(V,W)\otimes H^0(\op{n-2}(1)) $$
where $n\ge2$; these are the so-called \textit{ADHM matrices}. Let $\mathbb{X}_n(r,c)$ denote the set of all ADHM matrices as above satisfying the \textit{ADHM equation}:
$$ \mathcal{X}_n(r,c) := \Big\{ (A,B,I,J) ~\Big|~ [A,B]+IJ=0 \Big\} $$
The group $GL(V)$ acts on $\mathcal{X}_n(r,c)$ as follows
$$ g\cdot(A,B,I,J) = (gAg^{-1},gBg^{-1},gI,Jg^{-1}). $$
One can then consider the GIT quotient
$$ \mathcal{F}_n(r,c) := \mathcal{X}_n(r,c)/\!\!/ GL(V) ; $$
a quadruple $(A,B,I,J)\in\mathcal{X}_n(r,c))$ is GIT stable if there is no proper subspace $S\subset V$ for which the inclusions $A(S),B(S),I(W)\subset S\otimes H^0(\op{n-2}(1))$, see \cite[Section 2.3]{HJM} and \cite{HLa} for the case $n=3$.
Each point $(A,B,I,J)\in\mathbb{X}_n(r,c)$ can be used to construct a complex of sheaves on ${\mathbb{P}^n}$ as follows, where $X=(A,B,I,J)$:
\begin{equation} \label{eq:fpi}
E^\bullet_{X} ~:~ V\otimes{\mathcal O}_{\mathbb{P}^n}(-1) \stackrel{\alpha}{\longrightarrow} \big( V\oplus V\oplus W\big) \stackrel{\beta}{\longrightarrow} V\otimes{\mathcal O}_{\mathbb{P}^n}(1)
\end{equation}
where the maps $\alpha$ and $\beta$ are given by
$$ \alpha = \left( \begin{array}{c}
A + x\mathbf{1}_V \\
B + y\mathbf{1}_V \\
J
\end{array} \right) ~~ ~~
\beta = \left( \begin{array}{lcr}
-B - y\mathbf{1}_V &
A + x\mathbf{1}_V &
I
\end{array} \right) . $$
To be more clear, let $[z_0:\cdots:z_{n-2}]$ be homogeneous coordinates on $\mathbb{P}^{n-2}$ and $[z_0:\cdots:z_{n-2}:x:y]$ denote homogeneous coordinates on ${\mathbb{P}^n}$. Then the first line of the morphism $\alpha$ can be written in the following way
$$ A_0z_0 + \cdots + A_{n-2}z_{n-2} + x\mathbf{1}_V, $$
where $\mathbf{1}_V$ denotes the identity in $\End(V)$; the other entries of $\alpha$ and $\beta$ can be interpreted in a similar way.
Notice that $\beta\alpha=0$ precisely because the ADHM equation holds. Moreover, $\alpha$ is injective, while $\coker\beta$ has codimension at least 2, since $\beta$ is surjective along the line $\ell=\{z_0=\cdots=z_{n-2}=0\}$; in fact, $\beta$ is surjective if and only if for each $p\in\p{n-2}$, there is no proper $S\subset V$ for which the inclusions $A(p)(S),B(p)(S),I(p)(W)\subset S$, a condition that implies the GIT stability of $X=(A,B,I,J))$.
Finally, two complexes $E_X$ and $E_{X'}$ are isomorphic if and only if $X'=g\cdot X$.
These observations motivates the following definition.
\begin{definition} \label{defn:perverse}
A \textit{perverse instanton sheaf} is an object $E\in D^b({\mathbb{P}^n})$ quasi-isomorphic to a complex of the form
$$ {\mathcal O}_{\mathbb{P}^n}(-1)^{\oplus c} \longrightarrow {\mathcal O}_{\mathbb{P}^n}^{\oplus r+2c} \longrightarrow {\mathcal O}_{\mathbb{P}^n}(1)^{\oplus c} $$
satisfying the following conditions
\begin{itemize}
\item ${\mathcal H}^{p}(E)=0$ for $p\ne0,1$;
\item ${\mathcal H}^0(E)$ is a torsion free sheaf;
\item ${\mathcal H}^1(E)$ is a torsion sheaf of codimension at least 2.
\end{itemize}
Note that $r=\rk\big({\mathcal H}^0(E)\big)$, and this is called the rank of $E$; a \textit{rank 0 instanton sheaf} is just a perverse instanton sheaf of rank 0. The integer $c=\operatorname{ch}_2(E)$ is called the \textit{charge} of $E$.
In addition, a \textit{framing} on $E$ is a choice of isomorphism $\varphi:{\mathcal H}^0(E)|_{\ell}\stackrel{\sim}{\to}{\mathcal O}_{\ell}^{\oplus r}$. The pair $(E,\varphi)$ is called a\textit{ framed perverse instanton sheaf}.
\end{definition}
Therefore, for any $X\in\mathcal{X}_n(r,c)$ the complex $E_X$ presented in display \eqref{eq:fpi} is a perverse instanton sheaf of rank $r$ and charge $c$, equipped with a framing $\varphi:{\mathcal H}^0(E)|_{\ell}\stackrel{\sim}{\to}W\otimes{\mathcal O}_{\ell}$. The GIT quotient ${\mathcal F}_n(r,c)$ can then be interpreted as the moduli space of \textit{GIT stable} framed perverse instanton sheaves of rank $r$ and charge $c$.
Every instanton sheaf in the sense of Definition \ref{defn:instantons} is a perverse instanton sheaf with ${\mathcal H}^1(E)=0$; and every perverse instanton sheaf with ${\mathcal H}^1(E)=0$ is just an instanton sheaf. Therefore, every framed instanton sheaf is semistable, in the sense that it corresponds to a GIT semistable ADHM datum.
The main example of a perverse instanton sheaf that is not a sheaf is the derived dual of a non locally free instanton sheaf. In general, the $0^{\rm th}$-cohomology of a perverse instanton sheaf may not be an instanton sheaf.
\subsection{Instantons as representations of a quiver}\label{sec:repQuiver}
The information contained in a linear monad of the form
\begin{equation} \label{monad3}
V\otimes{\mathcal O}_{\mathbb{P}^n}(-1) \stackrel{\alpha}{\longrightarrow} W\otimes{\mathcal O}_{\mathbb{P}^n} \stackrel{\beta}{\longrightarrow} U\otimes{\mathcal O}_{\mathbb{P}^n}(1)
\end{equation}
can be neatly packaged as a representation of the quiver
\begin{equation} \label{Q}
\mathbf{Q} := \left\{\begin{tikzcd}
\underset{-1}{\bullet} \arrow[rr,bend left,"\alpha_0"] \arrow[rr,bend right,swap,"\alpha_n"] &\vdots & \underset{0}{\bullet} \arrow[rr,bend left,"\beta_{0}"] \arrow[rr,bend right,swap,"\beta_n"] &\vdots &\underset{1}{\bullet}
\end{tikzcd}\right\}
\end{equation}
with $n+1$ arrows between each vertex, satisfying the relations
\begin{equation}\label{eq:relations}
\beta_j\alpha_i+\beta_i\alpha_j=0 ~~{\rm with}~~ 0\le i,j \le n.
\end{equation}
Indeed, we place the vector spaces $V$, $W$ and $U$ on the vertices $-1$, $0$, and $1$, respectively. Set $[x_0:\dots:x_n]$ as homogeneous coordinates on ${\mathbb{P}^n}$; the morphisms $\alpha$ and $\beta$ can then be written as follows
$$ \alpha = \sum_{i=0}^n A_ix_i ~~{\rm and} ~~ \beta = \sum_{i=0}^n B_ix_i $$
where $\alpha_i\in\operatorname{Hom}(V,W)$ and $\beta_i\in\operatorname{Hom}(W,U)$. To complete the representation of the quiver $\mathbf{Q}$, we attach the matrices $A_i$ to the arrows $\alpha_i$, while the matrices $B_i$ are attached to the matrices $\beta_i$. Finally, the fact that $\beta\alpha=0$ implies that the relations in display \eqref{eq:relations} are satisfied. The injectivity of $\alpha$ and surjectivity of $\beta$ impose further (open) conditions on the set of representations of $\mathbf{Q}$ that come from linear monads; further details and generalizations can be found in \cite{JP} and in \cite{JS}.
Turning back our attention to the moduli space of instanton sheaves, we observe that the dimension vector of a representation of $\mathbf{Q}$ associated to the monad for an instanton sheaf of rank $r$ and charge $c$ is given by $(c,r+2c,c)$. One can then consider the King moduli space ${\mathcal R}_\theta(r,c)$ of $\theta$-semistable representations of $\mathbf{Q}$ with fixed dimension vector $(c,r+2c,c)$; here, the stability parameter $\theta$ is given by
$$ \theta = \big( \alpha , -(\alpha+\gamma)\frac{c}{r+2c} , \gamma \big) ~~{\rm with}~~ \alpha,\gamma\in\mathbb{R}.$$
It is not difficult to see that ${\mathcal R}_\theta(r,c)$ is empty whenever $\alpha>0$ and $\gamma<0$, see \cite[Lemma 7]{JS}.
The case of rank 2 instanton sheaves was studied in detail in \cite{JS}. One can show that if $E$ is an instanton sheaf, then there is $\theta$ such that the corresponding representation of $\mathbf{Q}$ is $\theta$-stable \cite[Proposition 8]{JS}; moreover, there is a wall in the $\alpha\gamma$-plane that destabilizes every representation corresponding to a non locally free instanton sheaf, so that perverse instanton sheaves do correspond to certain $\theta$-stable representation of $\mathbf{Q}$.
In summary, there are at least two ways to construct reasonable (i.e., projective) moduli spaces of instantons sheaves of arbitrary rank and charge: via the ADHM construction, or via moduli spaces of representations of quivers. However, both constructions include more complicated objects in the derived category of sheaves, like the perverse instantons sheaves considered in Definition \ref{defn:perverse}.
\section{Instanton sheaves on Fano threefolds}\label{section:Fano}
In section \ref{section:perverse}, we saw how to generalize the ``classical" notion of instanton extending the definition to torsion free sheaves of arbitrary rank and even to objects belonging to the derived category $D^b({\mathbb{P}^n})$. Another possible direction is to construct instantons on projective varieties beside projective spaces. For the particular cases of Fano threefolds of Picard rank one, this was done in \cite{F,K,CJ}.
\subsection{Instanton bundles on Fano threefolds}
The rank 2 locally free instantons on the Fano threefolds of Picard rank one are the main subject \cite{F}. We summarize here the main results of Faenzi's work.
To begin with we consider a Fano threefold $X$ of Picard rank one and we denote by $H_X$ the ample generator of $\Pic(X)\simeq\mathbb{Z}$. We then write the anticanonical class $K_X$ as $K_X=-i_X H_X$. We have that $i_X$ is a positive (since $X$ is Fano) integer, referred to as the \textit{index} of $X$, that takes values in $i_X\in \{1,2,3,4\}$.
We set $i_X=2q_X+e_X$, where $q_X$ and $e_X$ are integers such that $q_X\ge 0$ and $0\leq e_X\leq 1$. The definition of instanton presented in \cite{F} is the following:
\begin{definition}\label{defn:inst-F}
An instanton bundle on $X$ is a rank 2 stable bundle with $c_1=-e_X$, and such that
$$ E\simeq E^*(-e_X), \hspace{3mm} H^1(E(-q_X))=0.$$
\end{definition}
\noindent Note that the \textit{instantonic condition} $H^1(E(-q_X))=0$ is the analogue of the vanishing $H ^1(E(-2))$ holding for instantons on $\p3$.
Using the Serre's correspondence and the stability assumption it can be shown that the instantons satisfy the following cohomological vanishing:
\begin{lemma}\label{lem:vanish_bundle}
If $E$ is a rank 2 instanton bundle on $X$ we then have:
\begin{equation}\label{orth-bundle}
H^i(E(-q_X))=0\ \text{and}\ \operatorname{Ext}^i(E,{\mathcal O}_X(-q_X-e_X))=0,\ \text{for all}\ i;
\end{equation}
and $H^1(E(-q_X-t))=0, \: H^2(E(-q_X+t))=0$, for all $t\ge 0$.
\end{lemma}
\subsection{Non-emptiness of moduli spaces of instantons }
The first main result of \cite{F}, concerns the non-emptiness of the moduli space ${\mathcal I}(n)$ of instanton bundles of charge $n$ on all Fano threefolds $X$ of Picard rank one and index $i_X>1$ and on non-hyperelliptic Fano threefolds of index one (which means that $-K_X$ is very ample) containing a line $\ell\subset X$ with normal bundle ${\mathcal O}_{\ell}\oplus {\mathcal O}_{\ell}(-1)$.
\begin{theorem}
The moduli space ${\mathcal I}(n)$ has a generically smooth irreducible component whose dimension is the number $\delta$ below:
\begin{center}
\begin{tabular}{@{}l|cccc@{}}
\toprule
$i_X$ & $4$ & $3$ & $2$ & $1$ \\ \hline
$\delta$ & $8n-3$ & $6n-6$ & $4n-3$ & $2n-g_X-2$\\
\bottomrule
\end{tabular}
\end{center}
and ${\mathcal I}(n)$ is empty when $i_X=2$ and $n=1$, and when $i_X=1$ and $2n<g_X+2$.
\end{theorem}
The integer $g_X$ appearing in the statement of the theorem is the \textit{genus} of $X$; we recall that this parameter, defined on Fano threefolds of index one, is the genus of a general codimension 2 plane section of $X$.
\begin{remark}\label{rmk:et}\textbf{Existence of instantons on Fano threefolds of index 2.}
In the particular case of Fano threefolds of index 2, the proof of the non-emptiness of the moduli spaces ${\mathcal I}(n)$, for $n> 2$ relies on the construction of divisors in $\overline{{\mathcal I}(n)}$ parameterizing non-locally free sheaves $E$ that still satisfy the cohomological vanishing $H^{\bullet}(E(-1))=0$ (note that all these conditions are, by semicontinuity, open).
To be more precise, these sheaves $E$ are \textit{elementary transformations} of instanton bundles $F$ of charge $n-1$ along structure sheaves ${\mathcal O}_{\ell}$ of lines ${\ell}\subset X$. This means that the sheaves $E$ fit into short exact sequences of the form:
\begin{equation}\label{singular-line}
0\longrightarrow E \longrightarrow F\longrightarrow {\mathcal O}_{\ell}\longrightarrow 0
\end{equation}
(from which we learn in particular that $F\simeq E^{**}$ and $\operatorname{Sing}(F)=\ell$).
The non-emptiness of ${\mathcal I}(n)$ can then be proved applying an induction argument. To begin with we show the existence of a generically smooth irreducible component of ${\mathcal I}(2)$ (this is done applying the well-known Serre's correspondence relating locally complete intersection curves on $X$ with rank 2 bundles). The induction step consists then in showing that for a general pair $(F, \ell)$ with $[F]\in{\mathcal I}(n-1)$ and $\ell\subset X$ a line on $X$, the general deformation of a sheaf $E$ fitting into a short exact sequence of the form (\ref{singular-line}) is an instanton bundle of charge $n$.
This procedure suggested that, more generally, the investigation on families of rank 2 non-locally free sheaves $E$ with $H^i(E(-q_X))=0,$ might contribute to get a better understanding of $\overline{{\mathcal I}(n)}$.
\end{remark}
\
\subsection{Monadic representations of instantons}
Faenzi focuses then his attention on instantons defined over Fano threefolds $X$ such that $H ^3(X)=0$. On these threefolds, the instanton bundles share another common feature with instantons on the projective space: they can still be represented as cohomology of monads. This property is remarkable for the following reasons: in the first place it provides us with a ``recipe" to construct instantons, in the second place it allows us to construct their moduli as GIT quotients. As it turns out, the condition $H^3(X)=0$ is indeed equivalent to the fact that $X$ admits a full strong exceptional collection; this allows to prove analogues of the Beilinson's theorem on ${\mathbb{P}^n}$. More specifically, on a Fano threefold $X$ such that $H^3(X)=0$, there exist vector bundles ${\mathcal E}_i, \ i=0,\dots, 3$, satisfying
$$ {\mathcal E}_0\simeq{\mathcal O}_X(-q_X-e_X), \hspace{2mm} {\mathcal E}_3^*(-e_X)\simeq{\mathcal E}_1, \hspace{2mm}
{\mathcal E}_2^*(-e_X)\simeq{\mathcal E}_2$$
and such that $D^b(X)=\langle {\mathcal E}_0,{\mathcal E}_1,{\mathcal E}_2,{\mathcal E}_3\rangle$.
Denoting then by $\langle{\mathcal F}_0,{\mathcal F}_1,{\mathcal F}_2,{\mathcal F}_3\rangle$ the dual collection, we have that each coherent sheaf $E$ on $X$ is the cohomology of a complex ${\mathcal C}^{\bullet}_{E}$
with ${\mathcal C}^{j}_E=\oplus_i H^i(F\otimes {\mathcal F}_{j-i+3})\otimes {\mathcal E}_{j-i+3}$
where the index $i$ runs between $\max\{0,j\}$ and $\min\{3,j+3\}$.
In the particular case in which $E$ is an instanton bundle, the complex ${\mathcal C}^{\bullet}_E$ is a monad whose terms can be described, in further detail, as follows.
For an integer $n$ let us fix vector spaces $I$ and $W$ whose dimensions are subjected to the following constraints:
\begin{center}\label{charge-monads}
\begin{tabular}{@{}lccc@{}}
\toprule
$i_X$& $n$& $\dim(I)$ & $\dim(W)$ \\ \midrule
$4$ & $n\ge 1$ & $n$ & $2n+2$ \\
$3$ & $n\ge 2$ & $n-1$ & $n$ \\
$2$ & $n \ge 2$ & $n$ & $4n+2$\\
$1$ & $n \ge 8$ & $n-7$ & $3n-20$ \\
\bottomrule
\end{tabular}
\end{center}
and let us denote by $U$ the vector space $U:=\operatorname{Hom}({\mathcal E}_2,{\mathcal E}_3).$
We fix then an isomorphism $D:W\to W^*$ such that $D^t=(-1)^{e_X+1}D$ and we consider the locally closed subvariety $\overset{\circ}{{\mathcal D}_{X,n}}$ of the vector space $\operatorname{Hom}(W^*\otimes {\mathcal E}_2, I\otimes {\mathcal E}_3)\simeq I\otimes W\otimes U$ defined as
$$\overset{\circ}{{\mathcal D}_{X,n}}:=\{ A\in \operatorname{Hom}(W^*\otimes {\mathcal E}_2, I\otimes {\mathcal E}_3)\mid A\:D\:A^t=0 \ \text{and A is surjective}\}.$$
Finally, we write $G(W,D)$ for the symplectic group $Sp(W,D)$, or for the orthogonal group $O(W,D)$, depending on whether $e_X=0,1$; the group $G_n:=GL(I)\times G(W,D)$ acts then on $\overset{\circ}{{\mathcal D}_{X,n}}$ via $(\zeta,\eta)\cdot A= (\zeta A \eta^t)$.
The second main result of \cite{F} is the following:
\begin{theorem}{\label{thm:monadFano}}
Let $X$ be a smooth Fano threefold of Picard rank one and such that $H^3(X)=0$. Let $I,\: W,\: D,\: {\mathcal E}_i$ as above. Then an instanton $E$ of charge $n$ on $X$ is the cohomology of a monad of the form
$$ I^*\otimes {\mathcal E}_1\xrightarrow{DA^t}W^*\otimes {\mathcal E}_2\xrightarrow{A} I\otimes {\mathcal E}_3,$$
and conversely the cohomology of such a monad is an instanton of charge $n$.
The moduli space ${\mathcal I}(n)$ of instanton bundles of charge $n$ is isomorphic to the geometric quotient $\overset{\circ}{{\mathcal D}_{X,n}}/G_n$.
\end{theorem}
\subsection{Instanton sheaves on Fano threefolds.}
The main properties of rank 2 instanton bundles on Fano threefolds, illustrated in \cite{F}, appear as ``natural generalizations" of the properties of mathematical instantons on the projective space.
We might then wonder if something similar still happens if we extend our study to sheaves of arbitrary rank and that are not necessarily locally free; in other words we might try to adapt to the Fano threefolds besides $\p3$ the approach adopted by Jardim in \cite{J-inst}.
This issue had been dealt in \cite{CJ} where the following definition of \textit{instanton sheaf} is presented (the notations adopted are the one we introduced in the previous section):
\begin{definition}\label{defn:inst-CJ}
Let $X$ be a Fano threefold of Picard rank one and index \mbox{$i_X=2q_X+e_X$}, where $q_X$, $e_X$ are integers such that $q_X\ge0$ and $0\le e_X\le 1$.
An \textit{instanton sheaf} $E$ on $X$ is a torsion free $\mu$-semistable sheaf with first Chern class $c_1(E)=-e_X$ and such that:
\begin{equation}
H^1(E(-q_X))=H^2(E(-q_X))=0.
\end{equation}
The \textit{charge} of $E$ is defined to be $c_2(E)$.
\end{definition}
\begin{remark}{\label{rmk:mu-unstable}}
Note that this definition appears to be more restrictive than the one adopted in \cite{J-inst} since this latter does not necessarily implies $\mu$-semistability (see e.g. \cite[Example 3]{J-inst}).
\end{remark}
Moving to this more general setting, some of the cohomological characterizations of instantons presented in Lemma \ref{lem:vanish_bundle} still hold:
\begin{lemma}\label{lem:vanish_bundle2}
Let $E$ be an instanton sheaf. Then:
$$H^i(E(-q_X))=\operatorname{Ext}^i(E,{\mathcal O}_X(-q_X-e_X))=0 \ \ \text{for all}\ \ i.$$
\end{lemma}
A conspicuous part of \cite{CJ} is devoted to the study of the non-locally free instanton sheaves. An efficient way to produce sheaves of such a kind is performing \textit{elementary transformations} of instantons along rank 0 instantons.
\begin{definition}\label{defn:rank-0}
A rank 0 instanton sheaf on $X$ is a 1-dimensional sheaf $T$ satisfying
$H^i(T(-q_X))=0, \ i=0,1$.
\end{definition}
\begin{remark}\label{rmk:pure-dim}
The vanishing of $H^0(T(-q_X))$ implies that $H^0(T(-n))=0$ for $n\gg 0$.
Accordingly a rank 0 instanton must have pure dimension 1 (that is to say it admits no zero-dimensional subsheaf).
\end{remark}
The notion of elementary transformation had already been introduced in Remark \ref{rmk:et}: we say that $E$ is the elementary transformation of an instanton $F$ along a rank 0 instanton $T$ if $E$ fits into a short exact sequence of the form:
\begin{equation}\label{eq:et}
0\longrightarrow E\longrightarrow F\longrightarrow T\longrightarrow 0.
\end{equation}
From this short exact sequence we can easily verify that the sheaf $E$ is indeed an instanton that moreover satisfies $E^{**}\simeq F^{**}$; in particular if ever $F$ is reflexive, $F\simeq E^{**}$.
Notice therefore that the non-locally free sheaves $E$ constructed in Remark \ref{rmk:et} and belonging to the boundary $\partial\overline{{\mathcal I}(n)}$ are instanton sheaves: they are indeed obtained performing elementary transformation of rank 2 instanton bundles $F$ (so that, in particular, $F\simeq E^{**}$) along structure sheaves of lines ${\mathcal O}_{\ell}$ (these latter are rank 0 instantons on Fano varieties of index 2 since $H^i({\mathcal O}_{\ell}(-1))=0$ for $i=0,1$).
Via the technique described above, we can thus construct families of non-reflexive instantons with 1-dimensional singular locus.
The main properties of non-reflexive instantons are summarized in the following proposition:
\begin{proposition}\label{prop:non-relf}
Let $E$ be a non-reflexive instanton sheaf of rank $r>0$. Then the following hold:
\begin{itemize}
\item $T_E:=E^{**}/E$ has pure dimension one;
\item $E$ has homological dimension one;
\item $E^{**}$ is an instanton if and only if $T_E$ is a rank 0 instanton.
\end{itemize}
\end{proposition}
Summing up, we can always construct families of non-reflexive instantons via elementary transformation of reflexive instantons along rank 0 instantons but, from Proposition \ref{prop:non-relf} we learn that, in general, not all non-reflexive instanton are obtained in this way.
Nevertheless this last assertion holds true if we restrict to the rank two case.
\begin{theorem}\label{thm:classification-rk2}
Let $E$ be a rank 2 instanton sheaf. Then $E^{**}$ is an instanton bundle and $T_E:=E^{**}/E$ is a rank 0 instanton whenever $T_E\ne 0$.
\end{theorem}
\begin{remark}\label{rmk:sing-rk2 }
From \ref{rmk:pure-dim} and Theorem \ref{thm:classification-rk2} we learn
the following: a rank 2 instanton $E$ either does not present singularities or it has purely one dimensional singular locus $\operatorname{Sing}(E)=\operatorname{Supp}(E^{**}/E)$.
\end{remark}
In the rank one case the investigation of non-reflexive instantons even lead to a complete classification of the rank one instantons.
\begin{proposition}\label{prop:rank-one}
Let $L$ be a rank 1 instanton sheaf of charge $n$ on a Fano threefold $X$ with Picard rank one. The following hold:
\begin{itemize}
\item if $i_X=3,4$ then $n=0$ and $L\simeq {\mathcal O}_X(-e_X)$;
\item if $i_X=1,2$, we have $L\simeq {\mathcal O}_X(-e_X)$ whenever $n=0$ whilst for $n>0$, $L$ always fits in a short exact sequence of the form:
$$0\rightarrow L\rightarrow L'\rightarrow {\mathcal O}_{\ell}(-e_X)\rightarrow 0$$
for a line $\ell\subset X$ and a rank one instanton $L'$ of charge $n-1$.
\end{itemize}
\end{proposition}
\begin{comment}
Each rank one instanton $L$ of charge $n$ on a Fano threefold $X$ of index $i_X=1$ or $2$ is therefore always isomoprhic to ${\mathcal I}_C(-e_X)$ for $C$ a l.c.m. curve $C$ of degree $n$ satisfying $H^i({\mathcal O}_C(-1))=0, \ i=0,1$.
Curves of such a kind cab be constructed "inductively" from extensions:
$$ 0\rightarrow {\mathcal O}_l\rightarrow calo_C\rightarrow {\mathcal O}_{C'}\rightarrow 0$$
for $C'$ a l.c.m. curve of degree $n-1$ and satisfying the same cohomological conditions $H^i({\mathcal O}_{C'}(-1))=0$.
\end{comment}
\subsection{Instanton bundles on Fano threefolds of index 2}
The rank 2 instanton bundles on a Fano threefold $X$ of index 2 are also the main subject of Kuznetsov's work \cite{K}. In the article the author's attention is mainly drawn to
the behavior of these bundles seen as objects in the derived category $D^b(X)$ of $X$.
The definition of instanton provided by Kuznetsov is the following:
\begin{definition}\label{defn:kutz}
Let $X$ be a Fano threefold of index 2. An \textit{instanton of charge n} is a stable vector bundle $E$ of rank 2 with $c_1(E)=0, \ c_2(E)=n$ and such that $H^1(E(-1))=0.$
\end{definition}
We notice therefore that on $X$, the definition of instanton adopted by Kuznetsov coincides the one presented by Faenzi.
Kuznetsov's investigation of instanton bundles on $X$ starts with the computation of their cohomology table.
\begin{lemma}\label{lem:cohom-inst}
Let $E$ be an instanton bundle of charge $n$ on a Fano threefold $X$ of index 2. Then the cohomology table of $E$ has the following shape:
\begin{center}
\begin{tabular}{@{}l|lcccccc@{}}
\toprule
$t$& $\cdots $& $-3$ & $-2$ & $-1$ & $0$ & $1$ & $\cdots $ \\ \hline
&&&&&&& \\
$h^3(E(t))$ & $\cdots$ & $\ast$ & $0$ & $0$ & $0$ & $0$ & $\cdots $ \\
&&&&&&& \\
$h^2(E(t))$ & $\cdots$ & $\ast$ & $n-2$ & $0$ & $0$ & $0$ &$\cdots$ \\
&&&&&&& \\
$h^1(E(t))$ & $\cdots$ & $0$ & $0$ & $0$ & $n-2$ & $\ast$ &$\cdots$ \\
&&&&&&& \\
$h^0(E(t))$ & $\cdots$ & $0$ & $0$ & $0$ & $0$ &$\ast$ &$\cdots$ \\
\bottomrule
\end{tabular}
\end{center}
\end{lemma}
As an immediate corollary we also get the following:
\begin{corollary}
The charge of an instanton is greater or equal than $2$.
\end{corollary}
\begin{remark}
\noindent
\begin{itemize}
\item By \cite[Theorem 3.1]{F}, we know that instantons of charge 2 indeed exist so that 2 is actually the minimal value of the charge of an instanton bundle on a Fano threefold of index 2. However, ${\mathcal I}_\ell\oplus{\mathcal O}_X$ is an example of a non locally free instanton sheaf of charge 1.
\item By the table displayed in Lemma \ref{lem:cohom-inst}, we see that since $H^0(E)=0$, the Gieseker stability of an instanton bundle $E$ actually coincides with its slope-stability.
\item Because of the stability assumption required in Definition \ref{defn:kutz}, we see that a vector bundle that is an instanton in the sense of Kuznetsov (or equivalently of Faenzi) is clearly an instanton in the sense of Definition \ref{defn:inst-CJ}. Nevertheless the converse implication does not hold: it is indeed shown in \cite{CJ} that Fano threefolds of index 2 admit strictly $\mu$-semistable rank 2 vector bundles $E$ with $ch(E)=(2,0,-n,0)$ and $H^i(E(-1))=0$, for all $i$. These bundles are therefore instantons according to Definition \ref{defn:inst-CJ} but not in the sense of Definitions \ref{defn:kutz} and \ref{defn:inst-F}. In loc.cit. it is shown that actually the Fano threefolds of index 2 indeed are the only ones carrying families of strictly $\mu$-semistable rank 2 instanton bundles of charge $n>0$ and that moreover each instanton $E$ of such a kind fits into a short exact sequence:
$$ 0\longrightarrow {\mathcal O}_X\longrightarrow E\longrightarrow L\longrightarrow 0 $$
with $L$ a rank one instanton of charge $n$.
\end{itemize}
\end{remark}
\subsection{The acyclic extension of instantons} As we mentioned before, one of Kuznetsov's main aims is to describe the properties of instantons using the language of derived categories.
We recall that for a Fano threefold $X$ of index 2, the collection of line bundles ${\mathcal O}_X, \ {\mathcal O}_X(1)$ is exceptional; accordingly we obtain the following semiorthogonal decompositon of the derived category $D^b(X)$:
\begin{equation*}
D^b(X)= \langle {\mathcal B}_X, {\mathcal O}_X,{\mathcal O}_X(1)\rangle, \hspace{2mm} {\mathcal B}_X:=\langle {\mathcal O}_X, {\mathcal O}_X(1)\rangle ^{\perp}.
\end{equation*}
Starting from an instanton $E$, as $E\in \langle{\mathcal O}_X(1)\rangle^{\perp}$ (this is due to Lemma \ref{lem:cohom-inst}), we can construct an object $\tilde{E}\in {\mathcal B}_X$ performing a \textit{left mutation through ${\mathcal O}_X$}. We recall that the left mutation through ${\mathcal O}_X$ is the functor $\mathbb{L}_{{\mathcal O}}:D^b(X)\rightarrow \langle{\mathcal O}_X\rangle^{\perp}$ sending an object $F\in D^b(X)$ to the cone of the evaluation morphism $\operatorname{Ext}^{\bullet}({\mathcal O}_X,F)\otimes {\mathcal O}_X\to F$.
Since for an instanton bundle $E$, the complex $\operatorname{Ext}^{\bullet}({\mathcal O}_X,E)\otimes {\mathcal O}_X$ is concentrated in degree $-1$ (once again, this is due to Lemma \ref{lem:cohom-inst}), $\tilde{E}:=\mathbb{L}_{{\mathcal O}}(E)$ is actually a sheaf object that fits into a short exact sequence:
$$
0\longrightarrow E\longrightarrow\tilde{E}\longrightarrow {\mathcal O}_X^{n-2}\longrightarrow 0.
$$
The sheaf $\tilde{E}$ is referred to as the \textit{acyclic extension of $E$}.
\begin{lemma}\label{lem:acyclic-inst}
The acyclic extension of an instanton $E$ is a simple slope-semistable vector bundle $\tilde{E}$ with $\operatorname{ch}(\tilde{E})=(n, \: 0, \: -n,\:0)$ and such that $
H^{\bullet}(\tilde{E})=H^{\bullet}(\tilde{E}(-1))=0$. Moreover $h^{0}(\tilde{E})=h^{1}(\tilde{E})=n-2$ and $h^{2}(\tilde{E})=h^{3}(\tilde{E})=0.$
\end{lemma}
Recall now that since an instanton $E$ (in the sense of Definition \ref{defn:kutz}) has rank 2 and first Chern class 0, it is \textit{self-dual}; this property implies in particular a ``generalized self-duality" of its acyclic extension.
Consider indeed the functor \mbox{$D:D^b(X)\to D^b(X), \ F\mapsto \mathbb{L}_{{\mathcal O}}(R\inhom(F,{\mathcal O}_X))$}. It is not difficult to prove that the functor $D$ satisfies the following properties:
\begin{lemma}
\noindent
\begin{itemize}
\item There exists a natural isomorphism $\delta:D^2\xrightarrow{\sim} id$;
\item the category ${\mathcal B}_X$ is preserved by $D$.
\end{itemize}
\end{lemma}
Once we have defined the functor $D$ we can state the self-duality property of acyclic extensions.
\begin{proposition}\label{prop:acyclic-selfdual}
Let $\tilde{E}$ be the acyclic extension of an instanton bundle $E$. Then there exists a skew-symmetric isomorphism $D(\tilde{E})\xrightarrow{\phi} \tilde{E}$, in the sense that it fits into a commutative diagram:
$$
\begin{tikzcd}
& D^2(\tilde{E}) \arrow{dr}{\delta_{\tilde{E}}} \\
D(\tilde{E}) \arrow{ur}{D(\phi)} \arrow{rr}{-\phi} && \tilde{E}.
\end{tikzcd}
$$
\end{proposition}
As it turns out, an instanton bundle $E$ can be ``reconstructed" from its acyclic extension. Each vector bundle $F$ satisfying the properties of both Proposition \ref{prop:acyclic-selfdual} and Lemma \ref{lem:acyclic-inst} is indeed the acyclic extension of a \textit{unique} instanton bundle.
\begin{theorem}\label{thm:reconstruction}
Let $F$ be a vector bundle on $X$ with $\operatorname{ch}(F)=(n, \: 0, \: -n,\:0)$ and such that $
H^{\bullet}(F)=H^{\bullet}(F(-1))=0$. Then $h^i(F^*)=0$ for $i>1$ and \mbox{$h^0(F^*)=h^1(F^*)\le n-2$}; if moreover $h^0(F^*)=n-2$, then there exists a unique instanton bundle $E$ of charge $n$ such that $F\simeq \tilde{E}$.
\end{theorem}
It is worth mentioning that also in this setting, we have that the ideal sheaves of lines ${\mathcal I}_{\ell}, \ \ell\subset X$, share several common features with instantons, or better to say, with their acyclic extensions.
It is immediate to prove that ${\mathcal I}_{\ell}\in{\mathcal B}_X$; moreover the following holds
\begin{proposition}
Ideal sheaves ${\mathcal I}_{\ell}$ of lines $\ell\subset X$ are fixed by $D$: $D({\mathcal I}_{\ell})\simeq{\mathcal I}_{\ell}$. Moreover this isomorphism is skew-symmetric (in the sense of Proposition \ref{prop:acyclic-selfdual}).
\end{proposition}
\section{$h$-instanton sheaves on projective varieties}\label{section:h-inst}
More recently, several authors have further extended the notion of instanton bundles to beyond Fano 3-folds. This was, once again, initially motivated by gauge-theory: recall that $\p2$, just like $S^4$ also has the structure of a quarternionic K\"ahler manifold, and its twistor space is the full flag manifold $F(0,1,2)$ of points and lines in $\cp2$; the Atiyah--Ward correspondence provides in this case a correspondence between quaternionic instantons on $\cp2$ and a class of holomorphic bundles on $F(0,1,2)$, and these are again called instanton bundles.
Following \cite{DFlag}, Marchesi, Malaspina and Pons-Llopis provided a mathematical treatment of the instanton bundles on $F(0,1,2)$ in \cite{MMPL}. This case study further motivated the introduction of the notion of instanton bundles on other Fano threefolds with Picard rank larger than 1, see for instance \cite{AM20,ACG,ACG21,CCGM}.
In the latest development, Antonelli and Casnati defined a class of sheaves on a projective scheme $X$ with respect to an ample and globally generated line bundle $\mathcal{O}_X(h)$ via certain cohomological vanishing conditions that generalize the example that have been previously studied (projective space, Fano threefolds of Picard rank 1); they show that these sheaves can be constructed via monads, so it is reasonable to call them instanton sheaves on $X$. In this section, we review the definition and constructions of these instanton sheaves following \cite{AC}; let us start with the definition proposed by these authors.
\begin{definition}{\label{defn:h-instanton}}{\cite[Definition 1.3 and Theorem 1.4]{AC}}
Let $X$ be an irreducible projective scheme of dimension $n$ ($n\geq 1$) endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$. If $\mathcal{E}$ is a coherent sheaf on $X$, $k$ a non–negative integer and $\delta \in \{0, 1\}$, then $\mathcal{E}$ is called an $h$-instanton sheaf if the following assertions hold:
\begin{enumerate}
\item $h^0(\mathcal{E}(-h))=h^n(\mathcal{E}((\delta-n)h))=0$;
\item $h^i(\mathcal{E}(-(i+1)h))=h^{n-1}(\mathcal{E}((\delta-n+i)h))= 0$ if $1\leq i\leq n-2$;
\item $\delta h^i(\mathcal{E}(-ih))=0$ for $2\leq i\leq n-2$;
\item $h^1(\mathcal{E}(-h))=h^{n-1}(\mathcal{E}((\delta-n)h))=k$;
\item $\delta(\operatorname{ch}i(\mathcal{E})-(-1)^n\operatorname{ch}i(\mathcal{E}(-nh))))=0$.
\end{enumerate}
\end{definition}
The following chart gives the cohomologies of an $h-$instanton sheaf $\mathcal{E}$ with defect $\delta$ and quantum number $k$ when $n\geq 4$.
\begin{center}
\begin{tabular}{@{}l|ccccccccc@{}}
\toprule
$t$ &$\cdots$& $-n-1$ & $-n$ & $-n+1$ & $\cdots$ & $-2$& $-1$& $0$&$\cdots$\\
\hline
$h^n(\mathcal{E}(t))$&$\cdots$&$\ast$&\diagbox{$\ast$}{$0$}&$0$&$\cdots$&$0$&$0$&$0$&$\cdots$\\
\hline
$h^{n-1}(\mathcal{E}(t))$&$\cdots$&$\ast$&\diagbox{$k$}{$\ast$}&\diagbox{$0$}{$k$}&$\cdots$&$0$&$0$&$0$&$\cdots$\\
\hline
$h^{n-2}(\mathcal{E}(t))$&$\cdots$&$0$&$0$&$0$&$\cdots$&$0$&$0$&$0$&$\cdots$\\
$\vdots$& & & & & & & & & \\
$h^2(\mathcal{E}(t))$&$\cdots$&$0$&$0$&$0$&$\cdots$&$0$&$0$&$0$&$\cdots$\\
$h^1(\mathcal{E}(t))$&$\cdots$&$0$&$0$&$0$&$\cdots$&$0$&$k$&$\ast$&$\cdots$\\
$h^0(\mathcal{E}(t))$&$\cdots$ &$0$&$0$&$0$&$\cdots$&$0$&$0$&$\ast$&$\cdots$\\
\bottomrule
\end{tabular}
\end{center}
A cell $\begin{tabular}{|c|}\hline\diagbox{a}{b}\\\hline\end{tabular}$ in the above chart means that the cohomology is equal to $b$ when $\delta=0$, and is equal to $a$ when $\delta=1$.
An $h$-instanton sheaf $\mathcal{E}$ with $\delta=0$ is called an ordinary instanton, and $\mathcal{E}$ is called non-ordinary if $\delta=1$.
\begin{remark}{\label{rmk:h-inst unstable}}
Definition \ref{defn:h-instanton} doesn't require stability for an $h-$instanton sheaf. Even for the case that $X$ is smooth and $\mathcal{E}$ is an $h-$instanton bundle of rank $2$, $\mathcal{E}$ could be strictly $\mu-$semistable or $\mu-$unstable which is similar to the situation for an instanton sheaf on ${\mathbb{P}^n}$ (Remark \ref{rmk:mu-unstable}). We refer to \cite[Proposition 8.4]{AC} for more details.
\end{remark}
\subsection{$h$-instanton sheaves on projective spaces and projective schemes}{\label{subsection:h-instOnProj}}
When $X\cong \mathbb{P}^n$, we choose the ample line bundle to be $\mathcal{O}_{\mathbb{P}^n}(h):= \mathcal{O}_{\mathbb{P}^n}(1)$. Then the definition of an ordinary $h-$instanton sheaf coincides with Definition \ref{defn:instantons} in section \ref{section:perverse}, meaning that if $\mathcal{E}$ is an ordinary $h$-instanton sheaf with rank $r$ and charge $c$, then $\mathcal{E}$ is an instanton sheaf in the sense of \cite{J-inst}. Thanks to Theorem \ref{thm:monadic presentation}, $\mathcal{E}$ has the following monadic presentation, and conversely, the cohomology of such a presentation is an $h-$instanton sheaf provided that it's torsion free.
$$
0\to \mathcal{O}^{\oplus c}_{\mathbb{P}^n}(-1)\to \mathcal{O}^{\oplus 2r+c}_{\mathbb{P}^n} \to \mathcal{O}^{\oplus c}_{\mathbb{P}^n}(1)\to 0
$$
This property generalizes to non-ordinary $h-$instanton sheaves. It is proved in \cite[Proposition 3.2]{AC} that a non-ordinary $h-$instanton sheaf is the cohomology of the monad given below, and the cohomology of such a monad is a non-ordinary $h-$instanton provided $b_1, b_2\geq \operatorname{ch}i(\mathcal{E})$.
$$
0\to \mathcal{M}^{-1}\to \mathcal{M}^{0}\to \mathcal{M}^{1}\to 0,
$$
in which
\begin{align*}
\mathcal{M}^{-1}&:=\mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus b_1-\operatorname{ch}i(\mathcal{E})}\\
\mathcal{M}^{0}&:=\begin{cases} \mathcal{O}^{\oplus b_0}_{\mathbb{P}^n}\oplus \Omega^1_{\mathbb{P}^n}(1)^{\oplus k}\oplus \Omega^{n-1}_{\mathbb{P}^n}(n-1)^{\oplus k}\oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus b_1}& \text{if} \ n\geq 3\\
\mathcal{O}^{\oplus b_0}_{\mathbb{P}^n}\oplus \Omega^1_{\mathbb{P}^n}(1)^{\oplus k}\oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{b_1}& \text{if} \ n=2
\end{cases}\\
\mathcal{M}^{1}&:=\mathcal{O}_{\mathbb{P}^n}^{\oplus b_0-\operatorname{ch}i(\mathcal{E})}.
\end{align*}
For a general projective scheme $X$ endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$, the full linear series of $\mathcal{O}_X(h)$ induces a finite map $\varphi_{|\mathcal{O}_X(h)|}\colon X\to \mathbb{P}^N$ for some $N\in \mathbb{Z}_{>0}$. Then projecting from $N-n$ general points in $\mathbb{P}^N$ induces a finite map $p\colon X\to \mathbb{P}^n$ with the property that $p^*(\mathcal{O}_{\mathbb{P}^n}(1))=\mathcal{O}_X(h)$. Using the fact that $R^ip_*(\mathcal{E})=0$ ($i\geq 1$) for any finite map $p$ and the projection formula, it is proved (\cite[Theorem 1.4]{AC}) that if $\mathcal{E}$ is an $h$-instanton sheaf on $X$, then $p_*(\mathcal{E})$ is a $\mathcal{O}_{\mathbb{P}^n}(1)-$instanton sheaf on $\mathbb{P}^n$. More generally, an $h$-instanton sheaf is preserved by a push-forward along a finite map.
\subsection{Monadic presentations of h-instanton bundles}
We have seen that monadic presentations exist for instanton sheaves on ${\mathbb{P}^n}$ (Theorem \ref{thm:monadic presentation}), perverse instanton sheaves on ${\mathbb{P}^n}$ (Definition \ref{defn:perverse}), and for instanton sheaves on some Fano threefolds (Theorem \ref{thm:monadFano}).
For a smooth n-fold $X$ ($n\geq 3$), endowed with a very ample line bundle ${\mathcal O}_X(h)$, assume that $X$ is ACM with respect to this line bundle ${\mathcal O}_X(h)$. It is proved in \cite[Theorem 1.7]{AC} that a vector bundle $\mathcal{E}$ on $X$ is an $h-$instanton bundle with defect $\delta\in \{0,1\}$ and quantum number $k\in \mathbb{Z}_{\geq 0}$ if and only if it is the cohomology of a monad of certain kind. In this subsection, we recall the monadic presentations for $h-$instanton bundles on such a scheme $X$ with an additional constraint that $h^0(X, \omega_X((n-1)h))=0$. With this extra condition, a monad will be more neatly presented, and it satisfies some duality property. We refer to \cite[Theorem 1.7]{AC}, \cite[Theorem 3.3]{JVM-10} and \cite{CM} for technical details on the construction, and the construction on a general ACM scheme and on a quadric hypersurface.
Firstly, let us recall that a smooth variety $X$ with a very ample line bundle $\mathcal{O}_X(h)$ is ACM if
\begin{enumerate}
\item $h^i(\mathcal{O}_X(th))=0$ for $i=1,2,..., n-1$, $t\in \mathbb{Z}$;
\item $h^i(\mathcal{I}_{X|\mathbb{P}^N}(t))=0$ (where the embedding is $|\mathcal{O}_X(h)|: X\hookrightarrow \mathbb{P}^N$).
\end{enumerate}
A sheaf $\mathcal{E}\in Coh(X)$ is called Ulrich if
\begin{enumerate}
\item $h^0(\mathcal{E}(-(t+1)h))=h^n(\mathcal{E}((t-n)h))=0$ for $t\geq 0$
\item $h^i(\mathcal{E}(t))=0$ for $i=1,2,...,n-1$ and $t\in \mathbb{Z}$
\end{enumerate}
If $X$ is an ACM scheme, and it satisfies an additional vanishing condition $h^0(X, \omega_X((n-1))h)=0$, then the monadic presentation of an ordinary h-instanton bundle $\mathcal{E}$ will be in the following form (\cite[Corollary 7.2]{AC})
\begin{equation}{\label{monadulrich}}
0\to \mathcal{C}^{U, h}\to \mathcal{B}\to \mathcal{C} \to 0
\end{equation}
where $\mathcal{C}=\mathcal{O}_X^{\oplus k}$, and $\mathcal{B}$ is a Ulrich bundle.
The sheaf $\mathcal{C}^{U, h}$ is the Ulrich dual sheaf of $\mathcal{C}$ in the sense that $\mathcal{C}^{U, h}:= \mathcal{C}^{\vee}((n+1)h+K_X)$.
Indeed, for a smooth scheme $X$, the vanishing condition $h^0(X, \omega_X((n-1))h)=0$ holds only when $X$ falls into the following three cases: $X\cong \mathbb{P}^n$; $X$ is a smooth quadric hypersurface; or $X$ is a scroll over a smooth curve $B$.
\subsubsection{}
If $X=\mathbb{P}^n$, the monadic presentations for both ordinary (section \ref{section:perverse}) and non-ordinary h-instanton sheaves (section \ref{subsection:h-instOnProj}) are shown in the previous sections.
\subsubsection{}
If $X\subset \mathbb{P}^{n+1}$ is a smooth quadric hypersurface, let $\mathcal{O}_X(h):=\mathcal{O}_{\mathbb{P}^n}(1)|_{X}$, $\mathcal{S}$ (for $n$ odd) and $\mathcal{S}'$, $\mathcal{S}''$ (for $n$ even) be the spinor bundles. Then, depending on the parity of $n$, monad (\ref{monadulrich}) can be written explicitely as one of the following two monads
$$0\to \mathcal{O}_X^{\oplus k} \to S(h)^{\oplus s}\to \mathcal{O}_X(h)^{\oplus k}\to 0 \quad \text{for\ }n= \text{odd} $$
in which $s=h^0(\mathcal{E}\otimes \mathcal{S})-h^1(\mathcal{E}\otimes\mathcal{S})+2^{\left[\frac{n-1}{2}\right]}k$
$$0\to \mathcal{O}_X^{\oplus k} \to S'(h)^{\oplus s'}\oplus S''(h)^{\oplus s''}\to \mathcal{O}_X(h)^{\oplus k}\to 0 \quad \text{for\ } n=\text{even}$$
in which $s'$ and $s''$ are given as follows:
\begin{align*}
s'&=\begin{cases}
h^0(\mathcal{E}\otimes \mathcal{S}')-h^1(\mathcal{E}\otimes\mathcal{S}')+2^{\left[\frac{n-1}{2}\right]}k& \text{if} \ n \equiv 0 \ (\text{mod} 4)\\
h^0(\mathcal{E}\otimes \mathcal{S}'')-h^1(\mathcal{E}\otimes\mathcal{S}'')+2^{\left[\frac{n-1}{2}\right]}k& \text{if} \ n \equiv 2 \ (\text{mod} 4)
\end{cases},\\
s''&=\begin{cases}
h^0(\mathcal{E}\otimes \mathcal{S}'')-h^1(\mathcal{E}\otimes\mathcal{S}'')+2^{\left[\frac{n-1}{2}\right]}k& \text{if} \ n \equiv 0 \ (\text{mod} 4)\\
h^0(\mathcal{E}\otimes \mathcal{S}')-h^1(\mathcal{E}\otimes\mathcal{S}')+2^{\left[\frac{n-1}{2}\right]}k& \text{if} \ n \equiv 2 \ (\text{mod} 4)
\end{cases}.
\end{align*}
For the cases $n=3,4,5$, the above monads coincide with the ones in \cite{F}, \cite{SA} and \cite{O}.
\subsubsection{}
If $X$ is a scroll, i.e., $X=\mathbb{P}(\mathcal{G})$ where $\mathcal{G}$ is a locally free sheaf on a smooth curve $B$, define $\mathcal{O}_X(h):=\mathcal{O}_{\mathbb{P}(\mathcal{G})}(1)$, and let $f$ be the fiber of the projection $\pi\colon X=\mathbb{P}(\mathcal{G})\to B$ at a closed point. In this case, $X$ is a variety of minimal degree in $\mathbb{P}^N$, and Ulrich bundles on such varieties are described in \cite{AHMP}. Furthermore, $X$ is ACM with respect to $\mathcal{O}_X(h)$ only when $B\cong \mathbb{P}^1$. We review two examples for dimensions $n=3$ and $4$ below, and we refer to \cite[Example 7.7 and 7.8]{AC} for more details.
\begin{example}
For $n=3$, monad (\ref{monadulrich}) for an ordinary $h$-instanton bundle will be in the following form
$$
0\to \mathcal{O}_X((d-2)f)^{\oplus k}\to \mathcal{B} \to \mathcal{O}_X(h)^{\oplus k} \to 0
$$
where $\mathcal{B}$ is a Ulrich bundle on $X$. Thanks to \cite[Theorem 4.7]{AHMP}, we know that the Ulrich bundle $\mathcal{B}$ fits into the short exact sequence
$$
0\to \mathcal{B}_2 \to \mathcal{B} \to \mathcal{O}_X((d-1)f)^{\oplus s_3}\to 0,
$$
where $\mathcal{B}_2$ is given as an extension
$$
0\to \mathcal{O}_X(h-f)^{\oplus s_1}\to \mathcal{B}_2 \to \Omega_{X|\mathbb{P}^1}^1(2h-f)^{\oplus s_2} \to 0
$$
for some $s_1, s_2, s_3\in \mathbb{Z}_{>0}$.
\end{example}
\begin{example} For $n=4$, let $X$ be the image of the Segre embedding $\mathbb{P}^1\times\mathbb{P}^3\hookrightarrow\mathbb{P}^7$. Then $X$ is a rational normal scroll with $\displaystyle\mathcal{G}\cong \bigoplus_{i=0}^{3}\mathcal{O}_{\mathbb{P}^1}(1)$. Let $p\colon X\to \mathbb{P}^3$ be the projection, and we have $\Omega^1_{X|\mathbb{P}^1}\cong p^*(\Omega^1_{\mathbb{P}^3})$.
If $\mathcal{E}$ is an ordinary $h$-instanton bundle with rank $r$ and quantum number $k$, then monad (\ref{monadulrich}) becomes
$$0\to \mathcal{O}_X(2f)^{\oplus k}\to \mathcal{B} \to \mathcal{O}_X(h)^{\oplus k} \to 0$$
where $\mathcal{B}$ fits into the short exact sequence
$$0\to \mathcal{O}_X(h-f)^{\oplus s_1}\oplus p^*\Omega^1_{\mathbb{P}^3}(2h-f)^{\oplus s_2} \to \mathcal{B}\to p^*\Omega^1_{\mathbb{P}^3}(3h-f)^{\oplus s_2}\oplus \mathcal{O}_X(3f)^{\oplus s_4} \to 0$$
for some $s_1, s_2, s_3, s_4 \in \mathbb{Z}_{>0}$.
\end{example}
\subsection{Examples of $h$-instanton bundles}
In this subsection, we review some constructions of (orientable) $h$-instanton bundles on smooth varieties of low dimensions ($n \leq 3$) and on scrolls. Firstly, recall that a rank 2 $h$-instanton bundle $\mathcal{E}$ on $X$ is orientable if it has defect $\delta\in \{0,1\}$ and $\displaystyle c_1(\mathcal{E})=(n+1-\delta)h+K_X$ in $A^1(X)\cong \Pic(X)$.
\subsubsection{} When $X$ is a smooth curve endowed with a globally generated ample line bundle $\mathcal{O}_X(h)$, an $h$-instanton sheaf $\mathcal{E}$ is necessarily locally free. Moreover, let $g$ be the genus of $X$, and choose a non-effective divisor $\theta\in \Pic^{g-1}(X)$. Then, by definition $\mathcal{O}_X(\theta+h)^r$ and $(\mathcal{O}_X(\theta)\oplus \mathcal{O}_X(\theta+h))^r$ are respectively, an ordinary $h$-instanton bundle of rank $r$ and an non-ordinary $h$-instanton bundle of rank $2r$.
In particular, if $X\cong\mathbb{P}^1$, an $h$-instanton sheaves $\mathcal{E}$ is
$$\mathcal{E}=\left\{
\begin{array}{ll}
\mathcal{O}_{\mathbb{P}^1}^{\oplus \operatorname{ch}i(\mathcal{E})}& \text{if} \ \delta=0 \\
(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(-1))^{\oplus \operatorname{ch}i(\mathcal{E})}& \text{if} \ \delta=1 \\
\end{array}
\right.
$$
The quantum number is this case is $k=\delta\operatorname{ch}i(\mathcal{E})$.
\subsubsection{} When $X$ is a smooth surface, rank 2 orientable $h$-instanton bundles with defect $\delta\in \{0, 1\}$ and large quantum number always exist. Next, we briefly recall their construction for the case that the Kodaira dimension $\kappa(X)$ is $-\infty$ (\cite[Example 6.10]{AC}), and we refer to \cite[Example 6.10]{AC} and \cite{ESW} for the cases when $\kappa(X)=0,1,2$.
For $\kappa(X)=-\infty$, let $\mathcal{O}_X(h)$ be a very ample line bundle on $X$ with $h^0(X, \mathcal{O}_X(h))=N+1$. A rank $2$ $h$-instanton bundle can be constructed by the Serre correspondence in the following extension
$$
0\to \mathcal{O}_X \to \mathcal{E} \to \mathcal{I}_{Z|X}((1-\delta)h-K_X) \to 0,
$$
where $Z\subset X$ is a $0-$dimensional scheme with degree$(Z)\geq (1-\delta)(N+1)+1$.
\subsubsection{} When $X$ is a Fano threefold of Picard rank $1$, define $\mathcal{O}_X(h):=\mathcal{O}_X(H)$ where the divisor $H$ is an ample generator of $\Pic(X)$ (we call $\mathcal{O}_X(H)$ the fundamental line bundle in this case). Denote the index of $X$ by $i_X$ ($i_X=1,2,3,4$). According to \cite[Definition 1.1]{ACG}, we call a rank 2 vector bundle $\mathcal{E}$ on $X$ a \textit{classical instanton bundle} if
\begin{enumerate}
\item $c_1(\mathcal{E})=-\epsilon h$ with $\epsilon\in \{0,1\}$
\item $h^0(\mathcal{E})=h^1(\mathcal{E}(-q^{\epsilon}_Xh))=0$ where $\displaystyle q^{\epsilon}_X:=\left[\frac{i_X+1-\epsilon}{2}\right]$.
\end{enumerate}
\begin{remark}
Comparing to the other two notions of instanton sheaves on a Fano variety (Definitions \ref{defn:inst-F} and \ref{defn:inst-CJ}), we don't require that $c_1(\mathcal{E})=-e_X$ for $\mathcal{E}$ being a \textit{classical instanton bundle} in this context. There is no constraint on stability of $\mathcal{E}$ either (Remark \ref{rmk:h-inst unstable}).
\end{remark}
The following proposition shows the relation between \textit{classical instanton bundles} and $h$-instanton bundles of rank $2$.
\begin{proposition} \cite[Proposition 8.6]{AC}
Let $X$ be a Fano threefold of Picard rank 1, endowed with a very ample fundamental line bundle $\mathcal{O}_X(h)$. If $\mathcal{E}$ is a rank 2 vector bundle with $c_1(\mathcal{E})=(4-\delta-i_X)h$ where $\delta =\{0, 1\}$, then the following assertions hold:
\begin{enumerate}
\item If $\mathcal{E}$ is an h-instanton bundle, then its defect is $\delta$ and
\begin{enumerate}
\item if $(i_X, \delta)\not\in \{(4,0), (4,1), (3,1)\}$, then $\mathcal{E}_{norm, h}$ is a classical instanton bundle, where $\mathcal{E}_{norm, h}:=\mathcal{E}\left(-\left[\frac{c_1(\mathcal{E})+1}{2}\right]h\right)$.
\item if $(i_X, \delta)\in \{(4,0), (4,1), (3,1)\}$, then $\mathcal{E}_{norm, h}$ is a classical instanton bundle if and only if $h^0(\mathcal{E})=0$.
\end{enumerate}
\item If $\mathcal{E}_{norm, h}$ is a classical instanton bundle and
\begin{enumerate}
\item if $(i_X, \delta)\not\in \{(1, 0)\}$, then $\mathcal{E}$ is an h-instanton bundle.
\item if $(i_X, \delta)\in \{(1, 0)\}$, then $\mathcal{E}$ is an h-instanton bundle with defect $\delta$ if and only if $h^0(\mathcal{E}_{norm, h})=0$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\subsubsection{} When $X$ is a scroll of dimension $n\geq 3$ on a smooth curve $B$, rank 2 ordinary $h$-instanton bundles on $X$ with quantum number $k$ can be constructed via the Serre correspondence. Following \cite[Section 10]{AC}, we briefly recall the construction below.
Let $\mathcal{G}$ be a locally free sheaf of rank $r\geq 3$ on $B$. Define $\mathcal{O}_X(h):=\mathcal{O}_{\mathbb{P}(\mathcal{G})}(1)$, and assume that $\mathcal{O}_X(h)$ is ample and globally generated.
For each $k\in \mathbb{Z}_{\geq 0}$, take $k$ general points $b_i\in B$ ($i=1,2,\dots,k$), and let $L_i=\pi^{-1}(b_i)\cong \mathbb{P}^{n-1}$ be the fibers of the morphism $\pi\colon X=\mathbb{P}(\mathcal{G})\to B$. Let $\theta\in \Pic^{g-1}(B)$ be a non-effective $\Theta$-characteristic of $B$, and $D$ be a divisor on $B$ such that $\mathcal{O}_B(D)=\det(\mathcal{G})$.
Then the rank 2 vector bundle $\mathcal{E}$ in the following sequence is proved to be an ordinary orientable $\mu$-semistable $h$-instanton with quantum number $k$:
$$
0\to \mathcal{O}_X(\pi^*(D+\theta))\to \mathcal{E}\to \mathcal{I}_{Z|X}(h+\pi^*(\theta)) \to 0.
$$
\begin{remark}
If $X$ is a Fano threefold with Picard number $\rho_X\geq 2$, then $h$-instanton bundles can behave differently from the classical instanton bundle. We review the case when $X$ is the image of the Segre embedding $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1\hookrightarrow \mathbb{P}^7$. We refer to \cite{CCGM}, \cite{MMPL} and \cite{AM} for classical instanton bundles on Fano varieties of higher Picard rank, and \cite[Section 9]{AC} for other examples of pathologies of $h$-instanton bundles on varieties with higher Picard rank.
Let $X$ be the image of the Segre embedding $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1\hookrightarrow \mathbb{P}^7$, and $p_i\colon X\to \mathbb{P}^1$ ($i=1,2,3$) be the three projections. Define $\mathcal{O}_X(h_i):=p_i^*(\mathcal{O}_{\mathbb{P}^1}(1))$, and $\mathcal{O}_X(h):= \mathcal{O}_X(h_1+h_2+h_3)$. Let $L\subset X$ be the intersection of general sections of $|h_2|$ and $|h_3|$, then $L\cong \mathbb{P}^1$. Choose $s\geq 0$ disjoint such curves, and let $Z$ be their union. There are rank 2 vector bundles $\mathcal{E}$ that fit the following sequence, and one can check that such vector bundles are orientable, ordinary, and simple $h$-instanton bundles but are $\mu$-unstable:
$$
0\to \mathcal{O}_X(h_1+3h_3)\to \mathcal{E}\to \mathcal{I}_{Z|X}(h_1+2h_2-h_3) \to 0.
$$
\end{remark}
\section{Instanton complexes via Bridgeland stability} \label{section:bridgeland}
For the remainder of the article we continue with the assumptions of Section \ref{section:Fano}, i.e., $X$ will denote a smooth projective Fano threefold of Picard number one with $\Pic(X)=\mathbb{Z}H$. As before, we write $K_X=-i_XH$ and $i_X=2q_X+e_X$, where $i_X,q_X,e_X\in\mathbb{Z}$, $i_X>0$, $q_X\geq 0$ and $e_X\in \{0,1\}$. For an object $E\in D^b(X)$ and $\beta\in \mathbb{R}$ we define the numerical twisted Chern character as
$$
v_{\beta}(E)=(\operatorname{ch}_0^{\beta}(E)H^3,\operatorname{ch}_1^{\beta}(E)H^2,\operatorname{ch}_2^{\beta}(E)H,\operatorname{ch}_3^{\beta}(E)),
$$
where
\begin{align*}
\operatorname{ch}_0^{\beta}(E)&= \operatorname{ch}_0(E) \\
\operatorname{ch}_1^{\beta}(E)&= \operatorname{ch}_1(E)-\beta \operatorname{ch}_0(E)H\\
\operatorname{ch}_2^{\beta}(E)&= \operatorname{ch}_2(E)-\beta\operatorname{ch}_1(E)H+\frac{\beta^2}{2}\operatorname{ch}_0(E)H^2 \\
\operatorname{ch}_3^{\beta}(E)&=\operatorname{ch}_3(E)-\beta\operatorname{ch}_2(E)H+\frac{\beta^2}{2}\operatorname{ch}_1(E)H^2-\frac{\beta^3}{6}\operatorname{ch}_0(E)H^3.
\end{align*}
As shown in \cite{Li}, for every $\beta,\alpha,s\in\mathbb{R}$ with $\alpha,s>0$, the function
$$
Z_{\beta,\alpha,s}(E)=-\operatorname{ch}_3^{\beta}(E)+\left(s+\frac{1}{6}\right)\operatorname{ch}_1^{\beta}(E)H^2+i\left(\operatorname{ch}_2^{\beta}(E)H-\frac{\alpha^2}{2}\operatorname{ch}_0(E)H^3\right)
$$
is the central charge of a stability condition $\sigma_{\beta,\alpha,s}=(Z_{\beta,\alpha,s},\mathcal{A}^{\beta,\alpha})$, whose supporting heart is constructed by the two-step tilting process described below.
\begin{enumerate}
\item[(a)] Start by considering the Mumford slope $$
\mu_{\beta}(E)=\begin{cases}\frac{\operatorname{ch}_1^{\beta}(E)H^2}{\operatorname{ch}_0(E)H^3} & \text{if}\ \operatorname{ch}_0(E)\neq 0\\
+\infty & \text{otherwise.}\end{cases}
$$
The following full additive subcategories of $\mathbb{C}oh(X)$ form a torsion pair:
\begin{align*}
\mathcal{F}_{\beta}&=\{E\in\mathbb{C}oh(X)\colon \mu_{\beta}(F)\leq 0\ \text{for all subsheaves}\ F\hookrightarrow E\}\\
\mathcal{T}_{\beta}&=\{E\in\mathbb{C}oh(X)\colon \mu_{\beta}(Q)> 0\ \text{for all quotient sheaves}\ E\twoheadrightarrow Q\}.
\end{align*}
Tilting $\mathbb{C}oh(X)$ with respect to this torsion pair, we obtain the heart of a bounded t-structure on $D^b(X)$:
$$
\mathbb{C}oh^{\beta}(X):=\langle \mathcal{F}_{\beta}[1],\mathcal{T}_{\beta}\rangle.
$$
\item[(b)] Consider now the tilt slope $$
\nu_{\beta,\alpha}(E)=\begin{cases}\frac{\operatorname{ch}_2^{\beta}(E)H-\frac{\alpha^2}{2}\operatorname{ch}_0(E)H^3}{\operatorname{ch}_1^{\beta}(E)H^2} & \text{if}\ \operatorname{ch}_1^{\beta}(E)H^2\neq 0\\
+\infty & \text{otherwise.}\end{cases}
$$
As before, we have the torsion pair
\begin{align*}
\mathcal{F}_{\beta,\alpha}&=\{E\in\mathbb{C}oh^{\beta}(X)\colon \nu_{\beta,\alpha}(F)\leq 0\ \text{for all subobjects}\ F\hookrightarrow E\ \text{in}\ \mathbb{C}oh^{\beta}(X)\}\\
\mathcal{T}_{\beta,\alpha}&=\{E\in\mathbb{C}oh^{\beta}(X)\colon \nu_{\beta,\alpha}(Q)> 0\ \text{for all quotients}\ E\twoheadrightarrow Q\ \text{in}\ \mathbb{C}oh^{\beta}(X)\}.
\end{align*}
Tilting $\mathbb{C}oh^{\beta}(X)$ with respect to this torsion pair, we obtain the desired heart
$$
\mathcal{A}^{\beta,\alpha}:=\langle \mathcal{F}_{\beta,\alpha}[1],\mathcal{T}_{\beta,\alpha}\rangle.
$$
\end{enumerate}
We denote the corresponding Bridgeland slope on $\mathcal{A}^{\beta,\alpha}$ by
$$
\lambda_{\beta,\alpha,s}(E)=\frac{\operatorname{ch}_3^{\beta}(E)-\left(s+\frac{1}{6}\right)\alpha^2\operatorname{ch}_1^{\beta}(E)H^2}{\operatorname{ch}_2^{\beta}(E)H-\frac{\alpha^2}{2}\operatorname{ch}_0(E)H^3}.
$$
We want to propose a new definition of instanton object that ``extends'' the definitions by Faenzi \cite{F}, Kuznetsov \cite{K}, and Comaschi--Jardim \cite{CJ} included in Section \ref{section:Fano} and that can be satisfied by some objects in $D^b(X)$. We hope that our categorical approach will allow for a more systematical way of studying instanton moduli spaces. The idea will be to replace the $\mu$-semistablity and the vanishing conditions by some type of Bridgeland stability. To determine the appropriate stability condition recall that in the classical case of rank 2 instanton bundles on $\mathbb{P}^3$, the vanishing conditions giving the monad description are obtained after combining $\mu$-stability with the vanishing of one cohomology group, Serre duality, and the fact that a rank 2 vector bundle with trivial first Chern class is self-dual.
Let us start by consider the functor
\begin{equation}\label{dualityFunctor}
E\mapsto E^D:=R\mathcal{H}om(E,\mathcal{O}(-e_X))[2].
\end{equation}
We have the following:
\begin{proposition}\label{duality} \cite[Proposition 6.12]{JMM}.
Suppose that $\nu_{\beta,\alpha}(E)\neq 0$, then
$$
E\ \text{is}\ \sigma_{\beta,\alpha,s}-\text{semistable}\ \Longleftrightarrow\ E^D\ \text{is}\ \sigma_{-\beta-e_X,\alpha,s}-\text{semistable}.
$$
In particular, if $\beta_0:=-e_X/2$ and $\nu_{\beta_0,\alpha}(E)\neq 0$, then $E$ is $\sigma_{\beta_0,\alpha.s}$-semistable if and only if $E^D$ is $\sigma_{\beta_0,\alpha,s}$-semistable.
\end{proposition}
\begin{proposition}
If $\operatorname{ch}(E)=\operatorname{ch}(E^D)$ then $v_{\beta_0}(E)=(-R,0,D,0)$. Additionally, if $E\in \mathcal{A}^{\beta_0,\alpha}$ for all $\alpha>0$ then $R,D\geq 0$.
\end{proposition}
\begin{proof}
If $v=(v_0,v_1,v_2,v_3)$, define $v^{\vee}:=(v_0,-v_1,v_2,-v_3)$. Then it is clear that
$$
v_{\beta}(E)^{\vee}=v_{-\beta}(E^{\vee}).
$$
Thus
$$
v_{\beta_0}(E)^{\vee}=v_{-\beta_0-e_X}(E^{\vee}\otimes \mathcal{O}(-e_X))=v_{\beta_0}(E^D)=v_{\beta_0}(E).
$$
Additionally, if $E\in\mathcal{A}^{\beta_0,
\alpha}$ for all $\alpha>0$ then
$$
D+\frac{\alpha^2}{2}R\geq 0\ \ \text{for all}\ \alpha>0,
$$
implying that $R,D\geq 0$.
\end{proof}
\begin{proposition}\label{stab:linebundles}
The line bundle ${\mathcal O}(q_X)$ is $\sigma_{\beta_0,\alpha,s}$-stable for $\alpha<i_X/2$ and $s>0$.
\end{proposition}
\begin{proof}
This is a straightforward computation as line bundles on a threefold of Picard rank 1 are tilt stable and Bridgeland stable as long as they belong to $\mathbb{C}oh^{\beta_0}(X)$ and $\mathcal{A}^{\beta_0,\alpha}$, respectively. Notice that
$$
v_{\beta_0}({\mathcal O}(q_X))=\left(H^3,\frac{i_X}{2}H^3,\frac{1}{2}\left(\frac{i_X}{2}\right)^2H^3,\frac{1}{6}\left(\frac{i_X}{2}\right)^3H^3\right),
$$
and so
$$
\mu_{\beta_0}({\mathcal O}(q_X))=\frac{i_X}{2},\ \ \nu_{\beta_0,\alpha}({\mathcal O}(q_X))=\frac{\frac{1}{2}\left(\frac{i_X}{2}\right)^2-\frac{\alpha^2}{2}}{\frac{i_X}{2}}.
$$
Thus ${\mathcal O}(q_X)\in \mathbb{C}oh^{\beta_0}(X)$ because ${\mathcal O}(q_X)$ is $\mu_{\beta_0}$-stable. Therefore, ${\mathcal O}(q_X)$ is also $\nu_{\beta_0,\alpha}$-stable. Since $\nu_{\beta_0,\alpha}({\mathcal O}(q_X))>0$ if and only if $\alpha<i_X/2$, then for these values ${\mathcal O}(q_X)\in \mathcal{A}^{\beta_0,\alpha}$.
\end{proof}
Consider the region
$$
\mathcal{U}:=\left\{(\alpha,s)\colon \alpha,s>0,\ \left(s+\frac{1}{6}\right)\alpha^2<\frac{1}{6}\left(\frac{i_X}{2}\right)^2\right\}.
$$
We see $\mathcal{U}\subset \mathrm{Stab}(X)$ via the identification $(\alpha,s)\leftrightarrow \sigma_{\beta_0,\alpha,s}$. We refer to the set of stability conditions $\{\sigma_{\beta_0,\alpha,s}\}_{\alpha,s>0}$ as the $(\alpha,s)$-slice. From the computations above, it is clear that $\lambda_{\beta_0,\alpha,s}({\mathcal O}(q_X))>0$ for all $(\alpha,s)\in \mathcal{U}$.
\begin{definition}\label{instantonComplex}
Fix a numerical twisted Chern character $v_{\beta_0}=(-R,0,D,0)$ with $R\geq 0$, $D>0$, and let $\mathcal{C}$ be a chamber for $v_{\beta_0}$ such that $\mathcal{C}\cap\mathcal{U}\neq \emptyset$. Let $E\in D^b(X)$ with $v_{\beta_0}(E)=v_{\beta_0}$, we say that $E$ is a $\mathcal{C}$-instanton object if $E$ is Bridgeland semistable for a stability condition in $\mathcal{C}$.
\end{definition}
\begin{remark}
It was proven in \cite[Section 5]{JMM} that for the numerical twisted Chern character $v_{\beta_0}=(-R,0,D,0)$, the wall and chamber decomposition of the $(\alpha,s)$-slice is finite. Moreover, an algorithm to compute the walls was provided.
\end{remark}
\begin{lemma}\label{limit:inst}
Let $E$ be a ${\mathcal C}$-instanton that is also Bridgeland semistable in the outermost chamber of the $(\alpha,s)$-slice. Then if $(\alpha_0,s_0)\in{\mathcal C}$, $E$ is $\sigma_{\beta_0,\alpha,s}$-semistable for all $\alpha\geq \alpha_0$, i.e., in all the chambers to the right of ${\mathcal C}$.
\end{lemma}
\begin{proof}
Since the walls are disjoint and each destabilizing wall for $E$ intersects the vertical line $\{(\alpha_0,s)\colon s>0\}$ then it is enough to prove that no subobject of $E$ in the category $\mathcal{A}^{\beta_0,\alpha_0}$ can destabilize $E$ for $s>s_0$. Indeed, if $A\hookrightarrow E$ is such subobject then
\begin{align*}
\lambda_{\beta_0,\alpha_0,s}(A)&\leq \lambda_{\beta_0,\alpha_0,s}(E)\ \ \text{in the outermost chamber, and}\\
\lambda_{\beta_0,\alpha_0,s}(A)&=\lambda_{\beta_0,\alpha_0,s}(E)=0\ \ \text{at the wall produced by}\ A.
\end{align*}
Thus, $\lambda_{\beta_0,\alpha_0,s_0}(A)>\lambda_{\beta_0,\alpha_0,s_0}(E)$ because the numerator of $\lambda_{\beta_0,\alpha_0,s}$ is linear in $s$. A contradiction, unless $A$ never really destabilizes $E$, i.e., $\lambda_{\beta_0,\alpha_0,s}(A)=0$ for all $s\geq s_0$.
\end{proof}
\begin{theorem}\label{mainVanishing}
Let $E\in D^b(X)$ be an object with $v_{\beta_0}(E)=(-R,0,D,0)$ and $\mathcal{C}$ be a chamber for this Chern character in the $(\alpha,s)$-slice such that $\mathcal{C}\cap\mathcal{U}\neq \emptyset$. Then
\begin{enumerate}
\item $E$ is a $\mathcal{C}$-instanton object if and only if $E^D$ is a $\mathcal{C}$-instanton object.
\item If $E$ is a $\mathcal{C}$-instanton object then $E\in \langle{\mathcal O}(q_X)\rangle^{\perp}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Part (1) is a direct consequence of Proposition \ref{duality} and the fact that $v_{\beta_0}(E)=v_{\beta_0}(E^D)$. For part (2) notice that
$$
\operatorname{Hom}({\mathcal O}(q_X),E)=0
$$
since for stability conditions on $\mathcal{C}$, ${\mathcal O}(q_X)$ is stable with
$$
\lambda_{\beta_0,\alpha,s}({\mathcal O}(q_X))>0=\lambda_{\beta_0,\alpha,s}(E).
$$
Notice that
\begin{align*}
\operatorname{Ext}^i({\mathcal O}(q_X),E)&=\operatorname{Ext}^{3-i}(E,{\mathcal O}(q_X-i_X))^*\\
&=\operatorname{Ext}^{3-i}({\mathcal O},E^{\vee}\otimes {\mathcal O}(-e_X)\otimes {\mathcal O}(q_X-i_X+e_X))^*\\
&=\operatorname{Ext}^{3-i}({\mathcal O},E^D\otimes{\mathcal O}(-q_X)[-2])^*\\
&=\operatorname{Ext}^{1-i}({\mathcal O}(q_X),E^D)^*.
\end{align*}
Combining this computation with part (1) we get $\operatorname{Ext}^1({\mathcal O}(q_X),E)=0$. For other values of $i$ we get a negative Ext between objects in the same heart, which is impossible. Therefore $E\in \langle{\mathcal O}(q_X)\rangle^{\perp}$.
\end{proof}
\begin{example}
It was shown in \cite[Proposition 5.3]{JMM} that the only Bridgeland semistable objects of twisted Chern character $v_{\beta_0}=(0,0,D,0)$ with $D>0$ in the outermost chamber of the corresponding finite wall and chamber decomposition of the $(\alpha,s)$-slice are precisely the (twisted) Gieseker semistable sheaves. Moreover, if a (twisted) Gieseker semistable sheaf $T$ with $v_{\beta}(T)=(0,0,D,0)$ is also a $\mathcal{C}$-instanton object, i.e., does not get destabilized at any point before the potential numerical wall produced by ${\mathcal O}(q_X)$, then Theorem \ref{mainVanishing} shows that $E$ is a 1-dimensional instanton sheaf.
\end{example}
\begin{example}\label{spinorEx}
Let $X=Q^3\subset \mathbb{P}^4$ be a quadric hypersurface and $\iota\colon X\hookrightarrow \mathbb{P}^4$ the corresponding inclusion. The spinor bundle $S$ on $X$ is defined by the short exact sequence
$$
0\longrightarrow {\mathcal O}_{\p3}(-1)^{\oplus 4}\stackrel{M}{\longrightarrow} {\mathcal O}_{\p3}^{\oplus 4}\stackrel{N}{\longrightarrow} \iota_*S\longrightarrow 0,
$$
where the matrix $M$ satisfies
$$
M^2=(x_0^2+x_1x_2+x_3x_4)I.
$$
Moreover, restricting $N$ to $X$ produces the short exact sequence
\begin{equation}\label{spinor}
0\longrightarrow S(-1)\longrightarrow {\mathcal O}_X^{\oplus 4}\longrightarrow S\longrightarrow 0.
\end{equation}
The sheaf $S(-1)$ is an instanton sheaf of rank 2 and minimal charge. Notice that in this case $i_X=3$ and so $q_X=1=e_X$. Thus $\beta_0=-1/2$ and a simple computation using the exact sequence \eqref{spinor} leads to
$$
v_{\beta_0}(S(-1)[1])=\left(-2H^3,0,\frac{H^3}{4},0\right)=\left(-4,0,\frac{1}{2},0\right).
$$
Notice that since $S(-1)$ is slope stable with $\mu_{\beta_0}(S(-1))=0$ then $S(-1)[1]\in \mathbb{C}oh^{\beta_0}(X)$, and since $\nu_{\beta_0,\alpha}(S(-1)[1])=+\infty$ then $S(-1)[1]\in \mathcal{A}^{\beta_0,\alpha}$ for all $\alpha>0$.
Now, it follows from \cite[Theorem 3.1]{JMM} that $S(-1)[1]$ is asymptotically $\lambda_{\beta_0,\alpha,s}$-stable and so Bridgeland stable in the outermost chamber of the $(\alpha,s)$-slice. Moreover, as proven in \cite[Section 5.3]{JMM}, if $E$ is an object with $v_{\beta_0}(E)=(-R,0,D,0)$ and $A\hookrightarrow E$ is a destabilizing subobject producing a 1-dimensional wall in the $(\alpha,s)$-slice, then if $v_{\beta_0}(A)=(r,c,d,e)$ we must have
\begin{align}
& 0<d<D,\label{ineq:d}\\
&0<c(6e)\leq \min\{(2d)^2,(2D-2d)^2\},\label{ineq:c6e}\\
&-\frac{c}{6e}(2D-2d)-R\leq r\leq \frac{c}{6e}2d.\label{ineq:r}
\end{align}
Besides, in our case we also have
\begin{equation}\label{integralcond}
\operatorname{ch}_1(A)H^2\in 2\mathbb{Z},\ \operatorname{ch}_2(A)H\in \mathbb{Z},\ r\in 2\mathbb{Z},\ 4d\in\mathbb{Z},\ \text{and}\ 24e\in\mathbb{Z}.
\end{equation}
Thus, using $v_{\beta_0}(S(-1)[1])=\left(-4,0,\frac{1}{2},0\right)$, a straightforward computation shows that the only possibilities for the Chern character of a destabilizing subobject of $S(-1)[1]$ are
$$
v_{\beta_0}(A)=\left(r,1,\frac{1}{4},\frac{1}{24}\right),\ \ r=-6,\ 2,
$$
which produce the only potential destabilizing wall:
$$
W=\left\{(\alpha,s)\colon \left(s+\frac{1}{6}\right)\alpha^2=\frac{1}{24}\right\}.
$$
Thus, if $\mathcal{C}_{out}$ denotes the outermost chamber for $v_{\beta_0}(S(-1)[1])$ in the $(\alpha,s)$-slice, then $\mathcal{C}_{out}\cap \mathcal{U}\neq \emptyset$ and so $S(-1)[1]$ is a $\mathcal{C}_{out}$-instanton object.
\end{example}
\begin{example}\label{ex:idealsheaf}
Let $X$ be a Fano threefold of Picard number one, index 2 and degree $H^3>1$. In this case, $e_X=0$ and $q_X=1$. If $\ell\subset X$ is a line then $\operatorname{ch}(\mathcal{I}_{\ell}[1])=(-1,0,\ell,0)$ and so
$$
v_0(\mathcal{I}_{\ell}[1])=(-H^3,0,1,0).
$$
It follows from \cite[Example 3.4]{JMM} and the Gieseker stability of $\mathcal{O}_{\ell}$ that $\mathcal{I}_{\ell}[1]$ is $\sigma_{0,\alpha,s}$-stable for all $\alpha\gg 0$. On the other hand, if $\mathcal{I}_{\ell}[1]$ is ever unstable in the $(\alpha,s)$-slice then there should be a destabilising sub-object $A\hookrightarrow \mathcal{I}_{\ell}[1]$ with $v_0(A)=(r,c,d,e)$ satisfying inequalities \eqref{ineq:d}, \eqref{ineq:c6e}, and \eqref{ineq:r}. Thus, we should have
$$
0<2d<2
$$
and so $2d=1$, $c=1$, and $6e=1$. However, $c=\operatorname{ch}_1(A)H^2\neq 1$ because $H^3>1$ and so such $A$ can not exist. Therefore, $\mathcal{I}_{\ell}[1]$ is a $\mathcal{C}$-instanton object for each chamber $\mathcal{C}$ such that $\mathcal{C}\cap \mathcal{U}\neq \emptyset$.
\end{example}
\begin{example}
It was noticed in \cite[Example 6.15]{JMM} that the object $A\in D^b(\mathbb{P}^3)$ defined by the exact triangle
\begin{equation}\label{nonsheaf:inst}
{\mathcal O}_H(-1)[1] \longrightarrow A \longrightarrow {\mathcal O}_H(2),
\end{equation}
where $H$ denotes a hyperplane in $\p3$, is Bridgeland stable in the stability chamber that contains the rank 0 instanton sheaves as stable objects. Thus, such $A$ is a $\mathcal{C}$-instanton object in our definition. We would like to directly check that $A$ has indeed a ``monadic'' presentation.
Our starting point is the Euler sequence for the cotangent bundle of $H$:
$$ 0 \longrightarrow \Omega^1_H(2) \longrightarrow {\mathcal O}_H(1)^{\oplus 3} \longrightarrow {\mathcal O}_H(2) \longrightarrow 0 $$
Composing the epimorphism above with ${\mathcal O}_{\p3}(1)^{\oplus 3}\twoheadrightarrow{\mathcal O}_H(1)^{\oplus 3}$ we obtain an exact sequence
\begin{equation}\label{goodG}
0 \longrightarrow G \longrightarrow {\mathcal O}_{\p3}(1)^{\oplus 3} \longrightarrow {\mathcal O}_H(2) \longrightarrow 0,
\end{equation}
with the sheaf $G$ being given by the following extension
$$ 0 \longrightarrow {\mathcal O}_{\p3}^{\oplus 3} \longrightarrow G \longrightarrow \Omega_H^1(2) \longrightarrow 0. $$
Now there is a composed epimorphism $\op3^{\oplus 3}\twoheadrightarrow{\mathcal O}_H^{\oplus 3}\twoheadrightarrow\Omega_H^1(2)$
whose kernel $K$ is given by the exact sequence
$$ 0 \longrightarrow \op3(-1)^{\oplus 3} \longrightarrow K \longrightarrow {\mathcal O}_H(-1) \longrightarrow 0, $$
and also satisfies the sequence
\begin{equation}
0\longrightarrow K \longrightarrow \op3^{\oplus 6} \longrightarrow G\longrightarrow 0.
\end{equation}
We then obtain a monomorphism
$$ \alpha\colon \op3(-1)^{\oplus 3} \hookrightarrow K \hookrightarrow \op3^{\oplus 6} $$
and a morphism
$$ \beta\colon \op3^{\oplus 6} \twoheadrightarrow G \hookrightarrow \op3(1)^{\oplus 3} $$
whose cokernel is precisely ${\mathcal O}_H(2)$. Moreover, $\beta\alpha=0$, and $(\ker\beta/\im\alpha)\simeq{\mathcal O}_H(-1)$, since $\ker\beta=K$ and $\im\alpha\simeq\op3(-1)^{\oplus 3}$.
In summary, the object $A$ is quasi-isomorphic to the complex
$$ \op3(-1)^{\oplus 3} \stackrel{\alpha}{\longrightarrow} \op3^{\oplus 6} \stackrel{\beta}{\longrightarrow} \op3(1)^{\oplus 3}. $$
\end{example}
\begin{remark}
The $\mathcal{C}$-instanton object $A$ defined by the exact triangle \eqref{nonsheaf:inst} is a new type of instanton. Indeed, the only instanton complexes previously defined were the perverse instantons and $A$ does not fall into this category since $\mathcal{H}^{-1}(A)={\mathcal O}_H(-1)$ is not a torsion free sheaf (see Definition \ref{defn:perverse}).
\end{remark}
\subsection{Monad descriptions from quiver regions} In this subsection we will study two instances, namely $\mathbb{P}^3$ and the quadric $Q^3\subset \mathbb{P}^4$, in which we have full strong exceptional collections producing quiver regions intercepting the corresponding $(\alpha,s)$-slice. This will then lead to monad-type descriptions for some $\mathcal{C}$-instanton objects. We remark that similar techniques may be used to study other Fano threefolds with full strong exceptional collections.
\subsubsection{$X=\mathbb{P}^3$} In this case we have $e_X=0$ and so $\beta_0=0$. A simple computation as in \cite[Lemma 6.10]{JMM} shows that for
$$
(\alpha,s)\in \mathcal{R}=\left\{(\alpha,s)\colon 1<(6s+1)\alpha^2<\frac{4-3\alpha^2}{2-\alpha^2},\ 0<\alpha<1,\ s>0\right\}
$$
we have ${\mathcal O}_{\p3}(-2)[2],{\mathcal O}_{\p3}(-1)[2],{\mathcal O}_{\p3}[1],{\mathcal O}_{\p3}(1)\in\mathcal{A}^{0,\alpha}$ and moreover
$$
\lambda_{0,\alpha,s}({\mathcal O}_{\p3}(-2)[2])<\lambda_{0,\alpha,s}({\mathcal O}_{\p3}(1))\leq 0=\lambda_{0,\alpha,s}({\mathcal O}_{\p3}[1])\leq \lambda_{0,\alpha,s}({\mathcal O}_{\p3}(-1)[2]).
$$
Thus, for each $(\alpha,s)\in \mathcal{R}$ we can choose $t_{\alpha,s}\in\mathbb{R}$ such that
$$
\lambda_{0,\alpha,s}({\mathcal O}_{\p3}(-2)[2])<t_{\alpha,s}<\lambda_{0,\alpha,s}({\mathcal O}_{\p3}(1)),
$$
so that tilting $\mathcal{A}^{0,\alpha}$ with respect to the torsion pair
\begin{align*}
\mathcal{F}_{0,\alpha,s}&=\{E\in\mathcal{A}^{0,\alpha}\colon \lambda_{0,\alpha,s}(F)\leq t_{\alpha,s}\ \text{for all subobjects}\ F\hookrightarrow E\ \text{in}\ \mathcal{A}^{0,\alpha}\}\\
\mathcal{T}_{0,\alpha,s}&=\{E\in \mathcal{A}^{0,\alpha}\colon \lambda_{0,\alpha,s}(Q)> t_{\alpha,s}\ \text{for all quotients}\ E\twoheadrightarrow Q\ \text{in}\ \mathcal{A}^{0,\alpha}\},
\end{align*}
we obtain the heart $\langle {\mathcal O}_{\p3}(-2)[3],{\mathcal O}_{\p3}(-1)[2],{\mathcal O}_{\p3}[1],{\mathcal O}_{\p3}(1)\rangle$.
Now, notice that every Bridgeland semistable object $E$ with $\operatorname{ch}(E)=(-R,0,D,0)$ is in the subcategory $\mathcal{T}_{0,\alpha,s}$ and so it is quasi-isomorphic to a complex of the form
\begin{equation}\label{quiver:p3}
0\longrightarrow {\mathcal O}_{\p3}(-1)^{\oplus D}\longrightarrow {\mathcal O}_{\p3} ^{\oplus 2D+R}\longrightarrow {\mathcal O}_{\p3}(1)^{\oplus D}.
\end{equation}
Moreover, it was also established in \cite[Lemma 6.6]{JMM} that the Bridgeland stability of $E$ is equivalent to King stability of the complex \eqref{quiver:p3} with respect to the vector $\Theta_0=(-1,0,1)$. In particular, none such objects can be Bridgeland unstable in the region $\mathcal{R}$. Therefore there is only one Bridgeland chamber $\mathcal{C}$ for $v_0=(-R,0,D,0)$ intersecting the region $\mathcal{R}$, and all the $\mathcal{C}$-instanton objects in $\p3$ have the monad-type description \eqref{quiver:p3}.
\begin{center}
\begin{figure}
\caption{The quiver region $\mathcal{R}
\label{fig:quiverregion}
\end{figure}
\end{center}
\begin{remark}\label{vector:quiverR}
As explained in Section \ref{sec:repQuiver}, in \cite{JS} the authors studied in detail the variation of GIT for representations of the quiver
\begin{equation}\label{quiverP3}
\begin{tikzcd}
\bullet \arrow[rr,bend left,"\alpha_{0}"] \arrow[rr,bend right,swap,"\alpha_{3}"] &\vdots & \bullet \arrow[rr,bend left,"\beta_{0}"] \arrow[rr,bend right,swap,"\beta_{3}"] &\vdots &\bullet
\end{tikzcd}
\end{equation}
for the dimension vector $\vec{n}=(1,3,1)$, i.e., $R=2$ and $D=1$. They proved that King's space of stability parameters $\vec{n}^{\perp}$ has a finite wall and chamber decomposition where only two chambers have associated nonempty moduli spaces. These chambers share a wall given by the vector $\Theta_0:=(-1,0,1)\in \vec{n}^{\perp}$. Moreover, they proved that the $\Theta_0$-semistable representations are S-equivalent to complexes \eqref{quiver:p3} quasi-isomorphic to shifts of instanton sheaves (both locally and non-locally free) or to perverse instantons.
\end{remark}
\begin{example}
Consider the unique non split extension
$$
0\longrightarrow {\mathcal O}_{\p3}(-2)[2]\longrightarrow E\longrightarrow {\mathcal O}_{\p3}(2)\longrightarrow 0.
$$
At the ``instanton wall'' given by $\lambda_{0,\alpha,s}({\mathcal O}_{\p3}(2))=0$, the object $E$ is strictly semistable and so it is stable in a chamber ${\mathcal C}_0$ intersecting the region $\mathcal{U}$. Notice that even though the object $E$ is self-dual, i.e., $E^D=E$, this is not a ${\mathcal C}_0$-instanton in our definition since its Chern character $\operatorname{ch}(E)=(2,0,2,0)$ does not satisfy the usual Bogomolov inequality. Moreover, the chamber ${\mathcal C}_0$ can not intersect the quiver region $\mathcal{R}$ since such $E$ does not have a monad-type description. Indeed, if that were the case then $E$ would be quasi-isomorphic to a complex of the form
$$
{\mathcal O}_{\p3}(-1)^{\oplus 2}\longrightarrow {\mathcal O}_{\p3}^{\oplus 2}\longrightarrow {\mathcal O}_{\p3}(1)^{\oplus 2},
$$
but this would imply that there is a surjective map ${\mathcal O}_{\p3}(1)^{\oplus 2}\twoheadrightarrow {\mathcal O}_{\p3}(2)$, which is impossible.
\end{example}
\begin{example}
Let $E$ be a rank 2 instanton bundle in $\p3$ as defined in Section 2, in particular $E$ is $\mu$-stable. Then the classical monad presentation of $E$ tells us that $E[1]$ is quasi-isomorphic to a complex of the form
$$
{\mathcal O}_{\p3}^{\oplus c}\longrightarrow {\mathcal O}_{\p3}^{\oplus 2+2c}\longrightarrow {\mathcal O}_{\p3}(1)^{\oplus c}.
$$
Moreover, by \cite[Proposition 6]{JS} this representation of the quiver $\mathbf{Q}$ in display \eqref{Q} is stable with respect to the vector $\Theta_0=(-1,0,1)$. On the other hand, due to the $\mu$-stability and the locally freeness of $E$, the object $E[1]$ is stable in the outermost chamber as it satisfies the hypotheses of \cite[Theorem 3.1]{JMM}. Therefore, Lemma \ref{limit:inst} implies that $E[1]$ is a ${\mathcal C}$-instanton object for every chamber ${\mathcal C}$ such that ${\mathcal C}\cap\mathcal{U}\neq \emptyset$. Clearly, the same argument shows that if $E$ is $\mu$-stable locally-free instanton sheaf on $\p3$ then $E[1]$ is a ${\mathcal C}$-instanton object.
\end{example}
\subsubsection{$X=Q^3\subset\p4$} In this case we have $e_X=1$ and so $\beta_0=-1/2$. We have the full strong exceptional collection
$$
{\mathcal O}_X(-1),\ S(-1),\ {\mathcal O}_X,\ {\mathcal O}_X(1),
$$
where $S(-1)$ is the spinor bundle defined in Example \ref{spinorEx}. Notice that since $H^3=2$ then
$$
v_{\beta_0}({\mathcal O}_X(n))=\left(2,2\left(n+\frac{1}{2}\right),\left(n+\frac{1}{2}\right)^2,\frac{1}{3}\left(n+\frac{1}{2}\right)^3\right).
$$
Thus, computing the signs of the Mumford and tilt slopes we get
\begin{align*}
{\mathcal O}_X(-1)[1],\ {\mathcal O}_X,\ {\mathcal O}_X(1)\in \mathbb{C}oh^{\beta_0}(X),&\\
{\mathcal O}_X(-1)[2]\in \mathcal{A}^{\beta_0,\alpha}\ \ \text{for}\ \alpha\leq \frac{1}{2}&,\\
{\mathcal O}_X\in \mathcal{A}^{\beta_0,\alpha}\ \ \text{for}\ \alpha< \frac{1}{2}&,\\
{\mathcal O}_X(1)\in \mathcal{A}^{\beta_0,\alpha}\ \ \text{for}\ \alpha< \frac{3}{2}&.
\end{align*}
For $(\alpha,s)$ in the region
$$
\mathcal{R}=\left\{(\alpha,s)\colon \frac{1}{4}<(6s+1)\alpha^2<\frac{9}{4},\ \ 0<\alpha<\frac{1}{2},\ \ s>0\right\}
$$
we have
$$
\lambda_{\beta_0,\alpha,s}({\mathcal O}_X)<\lambda_{\beta_0,\alpha,s}(S(-1)[1])=0<\lambda_{\beta_0,\alpha,s}({\mathcal O}_X(-1)[2]),\ \ \text{and}\ \ \lambda_{\beta_0,\alpha,s}({\mathcal O}_X(1))>0,
$$
and from Example \ref{spinorEx}, it follows that all these objects are Bridgeland semistable in $\mathcal{R}$. Additionally, since within $\mathcal{R}$, $\lambda_{\beta_0,\alpha,s}({\mathcal O}_X(1))$ approaches zero as we get closer to $(6s+1)\alpha^2=\frac{9}{4}$ while $\lambda_{\beta_0,\alpha,s}({\mathcal O}_X(-1)[2])$ approaches zero as we get closer to $(6s+1)\alpha^2=\frac{1}{2}$, then the region
$$
\mathcal{R}_0=\left\{(\alpha,s)\colon \lambda_{\beta_0,\alpha,s}({\mathcal O}_X(-1)[2])<\lambda_{\beta_0,\alpha,s}({\mathcal O}_X(1)),\ \alpha,s>0\right\}
$$
has a nonempty intersection with $\mathcal{R}$. Therefore, for each $(\alpha,s)\in \mathcal{R}\cap\mathcal{R}_0$ we can choose $t_{\alpha,s}\in\mathbb{R}$ such that
$$
\lambda_{\beta_0,\alpha,s}({\mathcal O}_X(-1)[2])<t_{\alpha,s}<\lambda_{\beta_0,\alpha,s}({\mathcal O}_X(1)),
$$
and tilting the heart $\mathcal{A}^{\beta_0,\alpha}$ with respect to the torsion pair
\begin{align*}
\mathcal{F}_{\beta_0,\alpha,s}&=\{E\in\mathcal{A}^{\beta_0,\alpha}\colon \lambda_{\beta_0,\alpha,s}(F)\leq t_{\alpha,s}\ \text{for all subobjects}\ F\hookrightarrow E\ \text{in}\ \mathcal{A}^{\beta_0,\alpha}\}\\
\mathcal{T}_{\beta_0,\alpha,s}&=\{E\in \mathcal{A}^{\beta_0,\alpha}\colon \lambda_{\beta_0,\alpha,s}(Q)> t_{\alpha,s}\ \text{for all quotients}\ E\twoheadrightarrow Q\ \text{in}\ \mathcal{A}^{\beta_0,\alpha}\},
\end{align*}
we obtain the heart $\langle {\mathcal O}_X(-1)[3],S(-1)[2],{\mathcal O}_X[1],{\mathcal O}_X(1)\rangle$.
Therefore, it follows that if $E$ is a $\lambda_{\beta_0,\alpha,s}$-semistable object for $(\alpha,s)\in \mathcal{R}\cap\mathcal{R}_0$ with $v_{\beta_0}(E)=(-R,0,D,0)$, then $E[1]$ is quasi-isomorphic to a complex of the form
$$
{\mathcal O}_X(-1)^{\oplus a}\longrightarrow S(-1)^{\oplus b} \longrightarrow {\mathcal O}_X^{\oplus c}\longrightarrow {\mathcal O}_X(1)^{\oplus n}.
$$
A straightforward computation of the Chern characters of this complex gives us
$$
n=0,\ a=c=D-\frac{R}{8}=\operatorname{ch}_2(E)H,\ \text{and}\ b=c+\frac{R}{4}.
$$
Thus, if $\mathcal{C}$ is a chamber for $v_{\beta_0}=(-R,0,D,0)$ intersecting the region $\mathcal{R}\cap\mathcal{R}_0$, then every $\mathcal{C}$-instanton object has a monad-type description of the form
$$
{\mathcal O}_X(-1)^{\oplus c}\longrightarrow S(-1)^{\oplus c+\frac{R}{4}} \longrightarrow {\mathcal O}_X^{\oplus c}.
$$
\subsection{Acyclic extensions revisited} Let $X$ be a Fano threefold of Picard number one and index 2. In this case $q_X=1$ and $e_x=0$. As in the case of rank 2 instanton bundles we have the following.
\begin{lemma}\label{vanishing:Cinst}
Let $E$ be an stable $\mathcal{C}$-instanton object with $v_0(E)=(-R,0,D,0)$. Then
$$
\dim\operatorname{Ext}^{i}({\mathcal O}_X[1],E)=\begin{cases} D-\frac{R}{H^3} &\text{if}\ i=1\\
0&\text{otherwise}.
\end{cases}
$$
\end{lemma}
\begin{proof}
A simple verification of slopes as the one in Proposition \ref{stab:linebundles} will give us ${\mathcal O}_{X}(2)\in\mathcal{A}^{0,\alpha}$ for all $\alpha^2<2$. Moreover, ${\mathcal O}_X(2)$ is stable for each stability condition in the $(\alpha,s)$-slice with $\alpha^2<2$.
Now, notice that
\begin{align*}
\operatorname{Ext}^i({\mathcal O}_X[1],E)&\cong \operatorname{Ext}^{3-i}(E,{\mathcal O}_X(-2)[1])^*\\
&=\operatorname{Ext}^{4-i}({\mathcal O}_X(2),E^{\vee})^*\\
&=\operatorname{Ext}^{2-i}({\mathcal O}_X(2),E^D)^*.
\end{align*}
Since $E^D$ is again a stable $\mathcal{C}$-instanton object and $\lambda_{0,\alpha,s}({\mathcal O}_X(2))>0$ on the region $\mathcal{U}$, then
$$
\operatorname{Ext}^i({\mathcal O}_X[1],E)=0\ \ \text{for}\ \ i\geq 2.
$$
Likewise, $\operatorname{Ext}^i({\mathcal O}_X[1],E)=0$ for $i\leq 0$ because ${\mathcal O}_X[1]$ is stable for every stability condition in the $(\alpha,s)$-slice and $\lambda_{0,\alpha,s}({\mathcal O}_X[1])=\lambda_{0,\alpha,s}(E)=0$. Thus
$$
\dim\operatorname{Ext}^{1}({\mathcal O}_X[1],E)=\operatorname{ch}i(E)=D-\frac{R}{H^3}.
$$
\end{proof}
\begin{proposition}
Let $E$ be an stable $\mathcal{C}$-instanton object, then there exists a $\mathcal{C}$-instanton object $\tilde{E}$ fitting into an exact sequence
$$
0\longrightarrow E\longrightarrow \tilde{E} \longrightarrow {\mathcal O}_X^{\oplus D-\frac{R}{H^3}}[1]\longrightarrow 0,
$$
and satisfying $\operatorname{Hom}^{\bullet}({\mathcal O}_X[1],\tilde{E})=0$.
\end{proposition}
\begin{proof}
Since $\operatorname{Hom}^{\bullet}({\mathcal O}_X[1],E)\otimes {\mathcal O}_X[1]={\mathcal O}_X^{\oplus D-\frac{R}{H^3}}$ by Lemma \ref{vanishing:Cinst}, then $\tilde{E}$ is just the cone of the evaluation map
$$
\operatorname{Hom}^{\bullet}({\mathcal O}_X[1],E)\otimes {\mathcal O}_X[1]\rightarrow E.
$$
The semistability of $\tilde{E}$ follows from the facts that
\begin{equation}\label{acyclic:Cinst}
0\longrightarrow E\longrightarrow \tilde{E} \longrightarrow {\mathcal O}_X^{\oplus D-\frac{R}{H^3}}[1]\longrightarrow 0
\end{equation}
is a short exact sequence in $\mathcal{A}^{0,\alpha}$ whenever $E\in \mathcal{A}^{0,\alpha}$, and that
$$
\lambda_{0,\alpha,s}(E)=\lambda_{0,\alpha,s}({\mathcal O}_X[1])=0.
$$
The vanishing of $\operatorname{Hom}^{\bullet}({\mathcal O}_X[1],\tilde{E})$ follows from applying this functor to the short exact sequence \eqref{acyclic:Cinst}.
\end{proof}
\begin{example}
Assume that $H^3>1$ and let $\ell\subset X$ be a line. Then ${\mathcal O}_{\ell}$ is a rank 0 instanton sheaf and also a $\mathcal{C}$-instanton. In fact, ${\mathcal O}_{\ell}$ is a Gieseker stable sheaf and so it is $\lambda_{0,\alpha,s}$-semistable for all $\alpha\gg 0$ (see \cite[Proposition 5.3]{JMM}), and if ${\mathcal O}_{\ell}$ was ever unstable in the $(\alpha,s)$-slice then it would have to be destabilized by a sub-object $A\hookrightarrow {\mathcal O}_{\ell}$ with $v_0(A)=(r,c,d,e)$ satisfying
$$
2d=1,\ \text{and}\ \ 0<c(6e)\leq 1,
$$
which is impossible since $H^3>1$. The acyclic extension of ${\mathcal O}_{\ell}$ is exactly $\mathcal{I}_{\ell}[1]$, which is a strictly semistable $\mathcal{C}$-instanton (see Example \ref{ex:idealsheaf}). Notice that unlike the case of rank 2 instanton bundles whose acyclic extensions are sheaves, the acyclic extension of a rank 0 instanton sheaf that is also an stable $\mathcal{C}$-instanton is necessarily a complex.
\end{example}
\end{document} | math | 106,528 |
\begin{document}
\title{Efficiency in Quantum Key Distribution protocols
with entangled gaussian states}
\begin{abstract}
Quantum key distribution (QKD) refers to specific quantum
strategies which permit the secure distribution of a secret key
between two parties that wish to communicate secretly. Quantum
cryptography has proven unconditionally secure in ideal
scenarios and has been successfully implemented using quantum
states with finite (discrete) as well as infinite (continuous)
degrees of freedom. Here, we analyze the efficiency of QKD
protocols that use as a resource entangled gaussian states and
gaussian operations only. In this framework, it has already
been shown that QKD is possible \cite{Navascues05} but the
issue of its efficiency has not been considered. We propose a
figure of merit (the efficiency $E$) to quantify the number of
classical correlated bits that can be used to distill a key
from a sample of $N$ entangled states. We relate the
efficiency of the protocol to the entanglement and purity of
the states shared between the parties.
\end{abstract}
\section{Introduction}
Quantum cryptography relies on the possibility of establishing a
secret random key between two distant parties traditionally
denoted as Alice and Bob. If the key is securely distributed, the
algorithms used to encode and decode any message can be made
public without compromising security. The key consists typically
in a random sequence of bits which both, Alice and Bob, share as a
string of classically correlated data. The superiority of quantum
cryptography comes from the fact that the laws of quantum
mechanics permit to the legitimate users (Alice and Bob) to infer
if an eavesdropper has monitored the distribution of the key and
has gained information about it. If this is the case, Alice and
Bob will both agree in withdrawing the key and will start the
distribution of a new one. In contrast, classical key
distribution, no matter how difficult the distribution from a
technological point of view is, can always be intercepted by an
eavesdropper without Alice and Bob realizing it. In quantum
cryptography, there exist several protocols that Alice and Bob can
use in order to establish a secret key. Some of them, like Ekert91
\cite{Ekert91}, use as a resource shared entanglement between the
two parties, while in others, like BB84 \cite{BB84}, the key is
established by sending non entangled quantum states between the
parties and communicating classically. If Alice and Bob share a
collection of distillable entangled states, they can always obtain
from them a smaller number of maximally entangled states from
which they can establish a secure key \cite{Deutsch96}. The number
of singlets (maximally entangled states) that can be extracted
from a quantum state using only Local Operations and Classical
Communication (LOCC) is referred to as the Entanglement of
Distillation $E_D$. For establishing a key, another important
concept is the number of secret bits $K_D$, that can be extracted
from a quantum state using LOCC. Since a secret bit can always be
extracted from a maximally entangled state, $E_D \leq K_D$. There
exist also quantum states which are entangled but cannot be
distilled, {\it i.e.}, have $E_D=0$. They are usually referred to
as bound entangled states since its entanglement is bound to the
state. Nevertheless, for some of those states it has been shown
that $K_D \neq 0$, and thus, they can be used to establish a
secret key \cite{Horodecki05}.
A particular case of states that cannot be ``distilled'' by
``normal'' procedures are continuous variables gaussian states,
{\it e.g.}, thermal, coherent, and squeezed states of light. By
``normal'' procedures we mean operations that preserve the
gaussian character of the state (gaussian operations). They
correspond {\it e.g.}, to beam splitters, squeezers, mirrors, etc.
Thus, in the gaussian scenario all entangled gaussian states
posses bound entanglement. Quantum cryptography with gaussian
states using gaussian operations has been experimentally
implemented using ``prepare and measure schemes'' with either
squeezed or coherent states \cite{Preskill01, Grangier02}. Those
schemes do not demand entanglement between the parties.
Recently, Navascu{\'e}s {\it et al.} \cite{Navascues05} have shown
that it is also possible with only gaussian operations to extract
a secret key {\it {\`a} la} Ekert91 from entangled gaussian states, in
spite the fact that these states are not gaussian distillable. In
other words, it has been shown that in the gaussian scenario all
entangled gaussian states fulfill $GK_D>0$ (where the letter $G$
stands for gaussian) while $GE_D=0$. Alice and Bob can extract a
list of classically correlated bits from a set a of $1 \times 1$
entangled modes as follows: i) they agree on a value $x_0 > 0$,
ii) Alice (Bob) measures the quadrature of each of her (his) modes
$\hat X_A (\hat X_B)$, iii) they accept only outputs such that
$|x_A|=|x_B|=x_0$, iv) they associate {\it e.g.}, the classical
value $0$(1) to $x_i=+x_0(-x_0)$, $i=A, B$ and thus establish a
list of classically correlated bits. From there, they can apply
Advantage Distillation \cite{Maurer93} to establish the secret
key. This protocol is secure against individual eavesdropper
attacks. Since the protocol is based on output coincidences of the
measurements of the quadratures which, by definition, are
operators with a continuous spectrum, the protocol has zero
efficiency \cite{Navascues05}.
Here, we study the consequences of relaxing the above condition to
a more realistic scenario. We assume that Alice and Bob can
extract a list of sufficiently correlated classical bits obtained
by accepting measurement outputs that do not coincide but are
bound within a range. We ask ourselves which is the possibility
that Alice and Bob can still distribute the key in a secure way
under individual and coherent attacks. We obtain that there exists
always a finite interval for which the protocol can be implemented
successfully. The length of this interval depends on the
entanglement and on the purity of the shared states, and increases
with increasing entanglement.
The paper is organised as follows. In Sect. $2$, we present the
formalism needed to tackle this problem. In Sect. $3$, we first
review the previous protocol \cite{Navascues05} and present our
new results. Finally, we present our conclusions in Sect. $4$.
\section{Formalism}\label{Formalism}
Systems of continuous variables are often expressed in terms of
modes, where each mode has two associated canonical degrees of
freedom (``position'' and ``momentum'') which fulfill the
canonical commutation relations (CCR). The CCR for a quantum
system with $n$ modes can be compactly expressed {\it via} the
symplectic matrix. Denoting the canonical coordinates by
\begin{equation}
\hat{\mathbf{R}}^T =
(\hat{q}_1,\hat{p}_1,\ldots,\hat{q}_n,\hat{p}_n) \equiv
(\hat{R}_1,\ldots,\hat{R}_{2n}),\nonumber
\end{equation}
the CCR simply read $[\hat{R}_i,\hat{R}_j] = \mathrm{i}(J_n)_{ij}$, where
$i,j=1,\ldots,2n$ and
\begin{equation}
\mathbf{J}_n = \bigoplus_{i=1}^n \mathbf{J}, \quad \quad
\mathbf{J} \equiv
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}.
\end{equation}
The symplectic matrix $\mathbf{J}$ defines the symplectic scalar
product and describes the geometry of the phase space. There is a
bijective map between a quantum state described by a density
matrix $\hat \rho$ (in an infinite Hilbert space) and its
corresponding characteristic function $\chi_{\rho}$, which is
given by the Fourier-Weyl transform:
\begin{equation}\label{Characteristic}
\chi_{\rho}(\boldsymbol{\zeta}) \equiv \rm{tr} \{\hat{\rho}
\hat{W}_{(\boldsymbol{\zeta})} \},
\end{equation}
\begin{equation}
\hat{\rho} \equiv \frac{1}{(2 \pi)^n} \int d^{2n}
\boldsymbol{\zeta} \, \chi_{\rho}(\boldsymbol{\zeta})
\hat{W}_{(-\boldsymbol{\zeta})},
\end{equation}
where $\boldsymbol{\zeta} \in \mathbb{R}^{2n}$ and
$\hat{W}_{\boldsymbol{\zeta}} = e^{\mathrm{i}
\boldsymbol{\boldsymbol{\zeta}}^T \mathbf{J}_n
\hat{\boldsymbol{R}}}$ are the so-called Weyl operators. Gaussian
states are characterized by a gaussian $\chi_\rho$ function,
\begin{equation}\label{Characteristic2}
\chi_\rho(\boldsymbol{\zeta}) = e^{\mathrm{i} \boldsymbol{\zeta}^T
\cdot \mathbf{J}_n \cdot \mathbf{d} - \frac{1}{4}
\boldsymbol{\zeta}^T \mathbf{J}^T_n \cdot \boldsymbol{\gamma}
\cdot \mathbf{J}_n \boldsymbol{\zeta}},
\end{equation}
where $\mathbf{d}$ is a $2n$ real vector, called displacement
vector (DV), and $\boldsymbol{\gamma}$ is a $2n \times 2n$
symmetric real matrix, denoted as covariance matrix (CM). A
convenient representation of gaussian quantum states is given in
terms of the Wigner quasi-distribution function $\mathcal{W}_\rho$
\cite{Wigner32}, which is related to the characteristic function
by the symplectic Fourier transform which preserves the gaussian
character,
\begin{eqnarray}
\mathcal{W}_{\rho}(\boldsymbol{\zeta})& =& \frac{1}{(2 \pi)^{2n}}
\int d^{2n} \boldsymbol{\eta} \, \chi_{\rho}(\boldsymbol{\eta})
e^{-\mathrm{i} \boldsymbol{\eta}^T \cdot \mathbf{J}_n \cdot
\boldsymbol{\zeta}}, \\
\chi_{\rho}(\boldsymbol{\eta}) &=&
\int d^{2n} \boldsymbol{\zeta}
\mathcal{W}_{\rho}(\boldsymbol{\zeta}) e^{\mathrm{i} \boldsymbol{\eta}^T
\cdot \mathbf{J}_n \cdot \boldsymbol{\zeta}},
\end{eqnarray}
where $\boldsymbol{\eta} \in \mathbb{R}^{2n}$. Thus, a gaussian
quantum state can equivalently be defined as a quantum state whose
Wigner function is gaussian,
\begin{equation}
\mathcal{W}_\rho(\boldsymbol{\zeta}) = \frac{1}{\pi^n \sqrt{{\rm
det} \boldsymbol{\gamma}}} e^{-(\boldsymbol{\zeta}-\mathbf{d)^T
\cdot \frac{1}{\boldsymbol{\gamma}} \cdot (\boldsymbol{\zeta}
-\mathbf{d})}}.
\end{equation}
$\mathbf{d}$ and $\boldsymbol{\gamma}$ are defined as:
\begin{equation}
d_i = {\rm tr} (\hat{\rho} \hat{R}_i),
\end{equation}
\begin{equation}
\gamma_{ij} = {\rm tr} (\hat{\rho} \{ \hat{R}_i-d_i \hat{\mathbb{I}},
\hat{R}_j-d_j \hat{\mathbb{I}} \}),
\end{equation}
and are computed {\it via} the first and second moments of the
characteristic function,
\begin{equation}
\left. d_i' = -\mathrm{i} \frac{\partial}{\partial \zeta_i} \chi_{\rho}
(\boldsymbol{\zeta}) \right|_{\boldsymbol{\zeta} = 0} = {\rm
tr} (\hat{\rho} \hat{R'}_i),
\end{equation}
\begin{equation}
\left. \frac{\gamma'_{ij}}{2} + d_i'd_j' = (-\mathrm{i})^2
\frac{\partial^2}{\partial \zeta_i \partial \zeta_j} \chi_{\rho}
(\boldsymbol{\zeta}) \right|_{\boldsymbol{\zeta} = 0} = \frac{1}
{2}{\rm tr}(\hat{\rho} \{ \hat{R'}_i, \hat{R'}_j \}),
\end{equation}
where $\hat{R}'_i = J_{ij} \hat{R}_j$, $d'_i = J_{ij} d_j$ and
$\gamma'_{ij} = J^T_{ik} \gamma_{kl} J_{lj}$.
In analogy with classical probability theory, the displacement
vector $\mathbf{d}$ plays the role of the mean value $\mu_i = {\rm
E}[x_i]$, and the covariance matrix elements $\boldsymbol{\gamma}$
play the role of the covariances $C_{ij} = {\rm Cov}(x_i,x_j) =
{\rm E}[(x_i-\mu_i)(x_j-\mu_j)]$ of a classical probability
distribution. So only relative displacement vectors have physical
meaning and only the non block-diagonal terms of the covariance
matrix tell us about the quantum correlations present in the
state.
Since the density matrix is a semidefinite positive operator,
$\hat\rho \geq 0$, the corresponding covariance matrix must
fulfill: $\boldsymbol{\gamma} + \mathrm{i} \mathbf{J}_n \geq 0$. One can
also define the fidelity between continuous gaussian states in
terms of Wigner functions. We use here the Bures-Uhlmann fidelity
between two arbitrary states $\hat\rho_1$ and $\hat\rho_2$ defined
as \cite{Barnum96}
\begin{equation}
\mathcal{F}(\hat{\rho}_1, \hat{\rho}_2) = \left [ {\rm tr}
\sqrt{\sqrt{\hat{\rho}_1} \hat{\rho}_2 \sqrt{\hat{\rho}_1}} \right
]^2,\nonumber
\end{equation}
which coincides with the so called Hilbert-Schmidt fidelity
\begin{equation}
\mathcal{F}(\hat{\rho}_1, \hat{\rho}_2) = {\rm tr} (\hat{\rho}_1
\hat{\rho}_2),
\end{equation}
whenever at least one of the states is pure. At the level of CM,
using the Quantum Parseval relation \cite{Holevo82}, the
Hilbert-Schmidt fidelity between two gaussian states can be
written as:
\begin{equation}
\begin{split}
\mathcal{F} (\hat{\rho}_1, \hat{\rho}_2) &= (2\pi)^{n} \int
d^{2n} \boldsymbol{\zeta} \, \mathcal{W}_1( \boldsymbol{\zeta})
\mathcal{W}_2 ( \boldsymbol{\zeta}) =\\
&= \frac{1}{\sqrt{{\rm det} \left( \frac{\boldsymbol{\gamma}_1 +
\boldsymbol{\gamma}_2}{2} \right) }} e^{-(\mathbf{d}_2 -
\mathbf{d}_1)^T \cdot \left( \frac{1}{\boldsymbol{\gamma}_1 +
\boldsymbol{\gamma}_2} \right) \cdot(\mathbf{d}_2-\mathbf{d}_1)}
\end{split}
\end{equation}
where $\boldsymbol{\gamma}_{1(2)}$ and $\mathbf{d}_{1(2)}$ belong
to $\hat{\rho}_{1(2)}$. Clearly, only relative DVs are of physical
significance. The purity of the state translates to
$\mathcal{P}(\boldsymbol{\gamma}) = {\rm tr}(\hat{\rho}^2) = {\rm
det}(\boldsymbol{\gamma})^{-1/2} \leq 1$. It is important to
notice that gaussian states always admit a purification. Thus, any
mixed gaussian state of $n$ modes can be expressed as the
reduction of a pure gaussian state of $2n$ modes of the form:
\begin{equation}\nonumber
\boldsymbol{\gamma}_{2n} =
\begin{pmatrix}
\boldsymbol{\gamma}_{n} & \mathbf{C}_n\\
\mathbf{C}_n^T & \boldsymbol{\theta}_n \boldsymbol{\gamma}_{n}
\boldsymbol{\theta}_n^T
\end{pmatrix}, \quad \quad
\mathbf{C}_n = \boldsymbol{J}_n \sqrt{-(\boldsymbol{J}_n
\boldsymbol{\gamma}_n)^2-\mathbb{I}} \, \boldsymbol{\theta}_n, \quad
\quad \boldsymbol{\theta}_n = \bigoplus_{i=1}^n
\boldsymbol{\theta},
\end{equation}
such that the mixed state can be obtained after tracing out $n$
modes from $\boldsymbol{\gamma}_{2n}$. Here $ \boldsymbol{\theta}
= \bigl(
\begin{smallmatrix}
1 & 0\\
0 & -1
\end{smallmatrix}
\bigr)$, which is the momentum reflection in phase-space, is the
associated symplectic operation.
For what follows it is also important to study entanglement
properties in the formalism of covariance matrices. A necessary
and sufficient condition for separability for an arbitrary
bipartite state is given in \cite{Giedke01}. For the case $1
\times 1$ and $1 \times N$ modes a necessary and sufficient
condition for separability is provided by the PPT criterion
\cite{Werner01} (for the rest of the states this criterion is only
necessary but not sufficient). The PPT criterion tells us that a
state is entangled if and only if the state $\hat{\rho}$ has non
positive partial transposition (NPPT): $\hat{\rho}^{T_A} < 0$. In
terms of CMs this criterion reads $ \boldsymbol{\theta}_A
\boldsymbol {\gamma}\boldsymbol{\theta}_A^T + \mathrm{i} \boldsymbol{J} <
0$. The second property, particularly relevant in what follows, is
that any NPPT gaussian state can be mapped by Gaussian Local
Operations and Classical Communication (GLOCC) to an NPPT
symmetric state of $1 \times 1$ modes.
To quantify the entanglement of our states, we will use the
logarithmic negativity as entanglement measure; ${\rm LN}
(\hat\rho) = \log_2 ||\hat\rho^{T_A}||_1$, where $||\hat A||_1 =
{\rm tr} \sqrt{\hat A^\dag \hat A}$ can be easily computed
through the sum of the singular values of $\hat A$. One can extend
this measure to gaussian states of $n$ modes through
\cite{Plenio07}:
\begin{equation}
{\rm LN} (\boldsymbol{\gamma}_n) = -\sum_{i=1}^n\log_2
{\rm min}(\tilde{\mu}_i,1)
\end{equation}
where $\{ \pm\tilde{\mu}_i \} = {\rm spec}(-i \boldsymbol{J}_n
\boldsymbol{\gamma}_n^{T_A})$ {\it i.e.}, the symplectic spectrum
of the partial transposed CM.
With this formalism at hand we now move to
the presentation of our calculations and results.
\section{Results}\label{Results}
First, we summarize the main steps of the protocol used in
\cite{Navascues05}. Without loosing generality, and by virtue of
the properties of gaussian states, one should only consider the
case in which Alice and Bob share many copies of a quantum system
of $1 \times 1$ symmetric NPPT gaussian state $\hat \rho_{AB}$. To
extract a list of classically correlated bits to establish a
secret key, each party measures the quadratures of her/his mode
$\hat X_{A,B}$ and accepts only those outputs $x_{A,B}$ for which
both parties have a consistent result $|x_A|=|x_B|=x_0$. With
probability $p(i,j)$, each party associates the
classical bit $i=0(1)$ to her/his outcome $+x_0(-x_0)$. The
probability that their symbols do not coincide is given by
$\epsilon_{AB}=(\sum_{i\neq j}p(i,j)) / (\sum_{i,j}p(i,j))$.
Having fixed a string of $N$ classical correlated values, they can
apply classical advantage distillation \cite{Maurer93}. To this
aim, Alice generates a random bit $b$ and encodes her string of $N$
classical bits into a vector $\vec b$ of length $N$ such that
$b_{Ai}+b_i=b \mbox{ mod }2$. Bob checks that for his symbols all
results $b_{Bi}+b_i=b' \mbox{ mod }2$ are consistent, and in this
case accepts the bit $b$. The new error probability is given by
\begin{equation}
\epsilon_{AB,N} = \frac{(\epsilon_{AB})^N}{(1 - \epsilon_{AB}
)^N+(\epsilon_{AB})^N} \leq \left( \frac{\epsilon_{AB}}{1 -
\epsilon_{AB}} \right)^N,
\end{equation}
which tends to zero for sufficiently large $N$. The most general
scenario for eavesdropping is to assume that Eve has access to the
states before their distribution. Hence, the states that Alice and
Bob share correspond to the reduction of a pure 4-mode state. Now
security with respect to individual attacks from the eavesdropper
Eve, can be established if
\begin{equation}
\left(\frac{\epsilon_{AB}}{1 - \epsilon_{AB}}\right)^N <
|\braket{e_{++}}{e_{--}}|^N,
\end{equation}
where $\ket{e_{\pm \pm}}$ denotes the state of Eve once Alice and
Bob have projected their states onto $\ket{\pm x_0}$. Notice that
Eve can gain information if the overlap between her states after
Alice and Bob have measured coincident results is sufficiently
small. The above inequalities come from the fact that in the case
of individual attacks the error on Eve's estimation of the final
bit $b$ is bound from below by a term proportional to
$|\braket{e_{++}} {e_{--}}|^N$ \cite{Navascues05}. Therefore,
Alice and Bob can establish a key if
\begin{equation}\label{Security}
\frac{\epsilon_{AB}}{1 - \epsilon_{AB}} < |\braket{e_{++}}
{e_{--}}|.
\end{equation}
In \cite{Navascues05} it was shown that any $1 \times 1$ NPPT
state fulfills the above inequality and thus any NPPT gaussian
state can be used to establish a secure key in front of individual
eavesdropper attacks.
Let us now present our results. Notice that since security relies
on the fact that Alice and Bob have better correlations than the
information the eavesdropper can learn about their state, perfect
correlation is not a requirement to establish a secure key. We
denote Alice's outputs by $x_{0A}$ and we calculate which are the
outputs Bob can accept so that the correlation established between
Alice and Bob outputs can be used to extract a secret bit.
We use the standard form of a bipartite $1 \times 1$ mode gaussian
state,
\begin{equation}
\gamma_{AB} = \begin{pmatrix}
\lambda_A & 0 & c_x & 0\\
0 & \lambda_A & 0 & -c_p\\
c_x & 0 & \lambda_B & 0\\
0 & -c_p & 0 & \lambda_B\\
\end{pmatrix}
\end{equation}
with $\lambda_{A,B} \geq 0$, and $c_x \geq c_p \geq 0$ (we can shift
the DV to 0). The gaussian state is called symmetric if $\lambda_A
= \lambda_B = \lambda$ and fully symmetric if also $c_x = c_p$. We
shall deal with mixed symmetric states. The positivity
condition reads $(\lambda - c_x)(\lambda + c_p) \geq 1$, while the
entanglement NPPT condition is given by $(\lambda - c_x)(\lambda - c_p)
< 1$. As in \cite{Navascues05}, we impose that the global state
including Eve is pure (she has access to all degrees of freedom
outside Alice an Bob) while the mixed symmetric state, shared by
Alice and Bob is just its reduction,
\begin{equation}
\gamma_{ABE} =
\begin{pmatrix}
\gamma_{AB} & C\\
C^T & \theta \gamma_{AB} \theta^T
\end{pmatrix},
\end{equation}
\begin{equation}
C = J_{AB} \sqrt{-(J_{AB} \gamma_{AB})^2 - \mathbb{I}_2} \, \theta_{AB} =
\begin{pmatrix}
0 & -\textsc{X} & 0 & -\textsc{Y}\\
-\textsc{X} & 0 & -\textsc{Y} & 0\\
0 & -\textsc{Y} & 0 & -\textsc{X}\\
-\textsc{Y} & 0 & -\textsc{X} & 0\\
\end{pmatrix},
\end{equation}
\begin{equation}
\theta_{AB} = \theta_A \oplus \theta_B, \quad \quad J_{AB} = J_A
\oplus J_B,
\end{equation}
where
$$\textsc{X}=\frac{\sqrt{a+b} + \sqrt{a-b}}{2},$$
$$\textsc{Y}=\frac{\sqrt{a+b} - \sqrt{a-b}}{2},$$
and $a = \lambda^2 -c_x c_p - 1$, $b = \lambda(c_x - c_p)$.
Performing a measurement with uncertainty $\sigma$, the
probability that Alice finds $\pm |x_{0A}|$ while Bob finds $\pm
|x_{0B}|$, is given by the overlap between the state of Alice and
Bob, $\hat\rho_{AB}$, and a pure product state $\hat \rho_{A,i}
\otimes \hat \rho_{B,j}$ (with $i,j=0,1$) of gaussians centered at
$\pm |x_{0A}|(\pm |x_{0B}|)$ respectively with $\sigma$ width
(notice $\hat\rho_{A,0} \equiv \ketbra{+|x_{0A}|}{+|x_{0A}|}$). We
use here the Hilbert-Schmidt fidelity which leads to:
\begin{equation}\label{Prob00}
\begin{split}
p(0,0) &= p(1,1) = {\rm tr}[\hat \rho_{AB} (\hat \rho_{A,0}
\otimes \hat \rho_{B,0})] =\\
&= (2\pi)^4 \int d^4
\boldsymbol{\zeta}_{AB} \, \mathcal{W}_{\rho_{AB}}
(\boldsymbol{\zeta}_{AB}) \mathcal{W}_{\rho_{A,0} \otimes
\rho_{B,0}} (\boldsymbol{\zeta}_{AB}) =\\
&= K(\sigma) \exp \left( \frac{2|x_{0A}| |x_{0B}| c_x - (\lambda
+ \sigma^2)(x_{0A}^2 + x_{0B}^2)}{(\lambda + \sigma^2)^2 -
c_x^2} \right),
\end{split}
\end{equation}
for the probability that their symbols do coincide and,
\begin{equation}\label{Prob01}
p(0,1) = p(1,0) = K(\sigma) \exp \left( \frac{-2|x_{0A}| |x_{0B}|
c_x - (\lambda + \sigma^2)(x_{0A}^2 + x_{0B}^2)}{(\lambda +
\sigma^2)^2 - c_x^2} \right),
\end{equation}
for the probability that they do not coincide, where
\begin{equation}
K(\sigma) = \frac{4\sigma^2}{\sqrt{(\lambda + \sigma^2)^2 -
c_x^2}\sqrt{(\lambda \sigma^2 + 1)^2 - c_p^2 \sigma^4}}.
\end{equation}
The error probability for $\sigma \rightarrow 0$ reads
\begin{equation}\label{Errab}
\epsilon_{AB} = \lim_{\sigma \to 0} \frac{\sum_{i \neq
j}p\,(i,j)}{\sum_{i,j} p\,(i,j)} = \frac{1}{1 + \exp \left(
\frac{4c_x |x_{0A}||x_{0B}|}{\lambda^2 - c_x^2} \right)}.
\end{equation}
Let us calculate the state of Eve $\ket{e_{\pm \pm}}$ after Alice
has projected onto $\ket{\pm |x_{0A}|}$ and Bob onto $\ket{\pm
|x_{0B}|}$:
\begin{equation}
\gamma_{++} = \gamma_{--} =
\begin{pmatrix}
\gamma_{x} & 0\\
0 & \gamma_{x}^{-1}
\end{pmatrix}, \quad \quad \gamma_{x} =
\begin{pmatrix}
\lambda & c_x\\
c_x & \lambda
\end{pmatrix},
\end{equation}
\begin{equation}
d_{\pm\pm} = \mp
\begin{pmatrix}
0\\
0\\
A \delta x_0 - B \Delta x_0\\
A \delta x_0 + B \Delta x_0
\end{pmatrix},
\end{equation}
where $A = \frac{\sqrt{a+b}}{\lambda + c_x}$, $B
=\frac{\sqrt{a-b}}{\lambda - c_x}$, $\Delta x_0 = |x_{0B}| -
|x_{0A}|$ and $\delta x_0 = |x_{0B}| + |x_{0A}|$. The overlap
between the two states of Eve is given by:
\begin{multline}\label{Eveov}
|\braket{e_{++}}{e_{--}}|^2 = \exp \Bigg(\frac{-4}{\lambda^2 -
c_x^2} \Bigg[ \left( \frac{x_{0A}^2 + x_{0B}^2}{2} \right)
(\lambda^2 - c_x^2-1)\lambda + \\ + |x_{0A}| |x_{0B}| \left(
c_x - c_p(\lambda^2-c_x^2) \right) \Bigg] \Bigg).
\end{multline}
Substituting Eqs. \eqref{Errab} and \eqref{Eveov} into
\eqref{Security} one can check, after some algebra, that the
inequality \eqref{Security} reduces to:
\begin{equation}\label{SecurityIneq}
\left(\frac{x_{0A}^2 + x_{0B}^2}{2}\right)(\lambda^2 -
c_x^2 - 1)\lambda + |x_{0A}| |x_{0B}| \left( -c_x - c_p
(\lambda^2 - c_x^2) \right) < 0.
\end{equation}
Notice that condition \eqref{SecurityIneq} imposes both,
restrictions on the parameters defining the state ($\lambda, c_x,
c_p$), and on the outcomes of the measurements ($x_{0A},
x_{0B}$). The constraints on the state parameters are equivalent
to demand that the state is NPPT and satisfies
\begin{equation}\label{Constrain}
(\lambda - c_x)(\lambda + c_x) \geq 1.
\end{equation}
Nevertheless, as $c_x \ge c_p$, any positive state fulfills this
condition. Hence for any NPPT symmetric state, there exists, for a
given $x_{0A}$, a range of values of $x_{0B}$ such that secret
bits can be extracted (Eq. \eqref{Security} is fulfilled). This
range is given by
\begin{equation}
\Delta x_0 = |x_{0B}| - |x_{0A}| \in {\mathfrak D}_\alpha =
\left[ \frac{2}{-\sqrt \alpha-1} , \frac{2}{\sqrt \alpha-1}
\right] |x_{0A}|,
\end{equation}
where
\begin{equation}
\alpha = \left( \frac{c_x -
\lambda}{c_x + \lambda} \right) \left[ \frac{1 - (\lambda +
c_x)(\lambda + c_p)}{1- (\lambda - c_x)(\lambda - c_p)}
\right].
\end{equation}
After Alice communicates $|x_{0A}|$ to Bob, he will accept only
measurement outputs within the above interval. The interval
$\Delta x_0$ is well defined if $\alpha \geq 1$, which equivals to
fulfill Eq. \eqref{Constrain}. Notice also that the interval is
not symmetric around $|x_{0A}|$ because the probabilities
calculated in Eqs. \eqref{Prob00} and \eqref{Prob01} do depend on
this value in a non symmetric way. The length of the interval of
valid measurements outputs for Bob is given by
\begin{equation}
D_\alpha = \frac{4 \sqrt \alpha}{\alpha-1} |x_{0A}|.
\end{equation}
It can be observed that maximal $D_\alpha\rightarrow\infty$
($\alpha=1$) corresponds to the case when Alice and Bob share a
pure state (Eve is disentangled from the system) and thus
condition \eqref{Security} is always fulfilled. On the other
hand, any mixed NPPT symmetric state ($\alpha > 1$) admits a
finite $D_\alpha$. This ensures a {\it finite} efficiency on
establishing a secure secret key in front of individual attacks.
If we assume that Eve performs more powerful attacks, namely
finite coherent attacks, then security is only guaranteed if
\cite{Navascues05}:
\begin{equation}\label{Security2}
\frac{\epsilon_{AB}}{1 - \epsilon_{AB}} < |\braket{e_{++}}
{e_{--}}|^2.
\end{equation}
This condition is more restrictive than \eqref{Security}. With a
similar calculation as before we obtain that now security is not
guaranteed for all mixed entangled symmetric NPPT states, but only
for those that also satisfy:
\begin{equation}\label{Constrain2}
\lambda - (\lambda+c_x)(\lambda-c_x)(\lambda-c_p) > 0.
\end{equation}
For such states, and given a measurement result $x_{0A}$ of Alice,
Bob will only accept outputs within the range:
\begin{equation}
\Delta x_0 = |x_{0B}| - |x_{0A}| \in {\mathfrak D}_\beta =
\left[ \frac{2}{-\sqrt \beta-1} , \frac{2}{\sqrt \beta-1}
\right] |x_{0A}|,
\end{equation}
where
\begin{equation}
\beta = \frac{2\lambda(\lambda^2-c_x^2-1)}{\lambda-
(\lambda+c_x)(\lambda-c_x)(\lambda-c_p)} \geq 1.
\end{equation}
Eqs. \eqref{Constrain} and \eqref{Constrain2} already guarantee
that $\beta \geq 1$.
Let us now focus on the efficiency issue. We define the efficiency
$E(\gamma_{AB})$ of the protocol for a given state $\gamma_{AB}$,
as the average probability of obtaining a classically correlated
bit. Explicitly,
\begin{equation}\label{protocolefficiency}
E(\gamma_{AB}) = \int_{\Delta x_0 \in {\mathfrak D}} dx_{0A} dx_{0B}
(1-\epsilon_{AB}) {\rm tr} (\hat\rho_{AB} \ketbra{x_{0A},
x_{0B}}{x_{0A},x_{0B}}).
\end{equation}
The marginal distribution in phase-space is easily computed by
integrating the corresponding Wigner function in momentum space
\cite{Lee95}:
\begin{equation}
\begin{split}
{\rm tr} (\hat\rho_{AB} \ketbra{x_{0A},x_{0B}}{x_{0A},x_{0B}})
&= \int \int dp_A dp_B \mathcal{W}_{\rho_{AB}} (\boldsymbol
{\zeta}_{AB}) =\\
&= \frac{\exp \left( \frac{2 c_x x_{0A} x_{0B}
- \lambda (x_{0A}^2+x_{0B}^2)}{\lambda^2-c_x^2} \right) }
{\pi \sqrt{\lambda^2-c_x^2}},
\end{split}
\end{equation}
but the final expression of \eqref{protocolefficiency} has to be
calculated numerically. Note that if Alice and Bob share as a
resource $N$ identical states (NPPT states for individual attacks,
and NPPT states fulfilling \eqref{Constrain2} for finite coherent
attacks), the number of classically correlated bits that can be
extracted from them is $\sim~N \times E(\gamma_{AB})$. The
efficiency \eqref{protocolefficiency} increases with increasing
$D$ and decreasing $\epsilon_{AB}$. In particular, for the
protocol given in \cite{Navascues05}, $D=0$, and therefore
$E(\gamma_{AB})=0$ for any state.
We investigate now the dependence of $E(\gamma_{AB})$ on the
entanglement of the NPPT mixed symmetric state used for the
protocol as well as on the purity of the state. As a measure of
the entanglement between Alice and Bob we compute the logarithmic
negativity
\begin{equation}
{\rm LN} (\boldsymbol{\gamma}_{AB}) = \log_2 \left( \frac{1}
{\sqrt{(\lambda-c_x)(\lambda-c_p)}} \right) > 0.
\end{equation}
\begin{figure}
\caption{Protocol efficiency (quantified by $E(\gamma_{AB}
\label{fig2}
\end{figure}
In Fig. \ref{fig2}, we display the efficiency of the protocol
(assuming individual attacks) versus entanglement shared between
Alice and Bob for different states $\gamma_{AB}$. There is not a
one-to-one correspondence between $E(\gamma_{AB})$ and
entanglement, since states with the same entanglement can have
different purity, which can lead to different efficiency. This is
so because there are two favorable scenarios to fulfill
\eqref{Security}. The first one is to demand large correlations so
that the relative error $\epsilon_{AB}$ of Alice and Bob is small.
The second scenario happens when Alice and Bob share a state with
high purity, {\em i.e.}, Eve is very disentangled. In this case,
independently of the error $\epsilon_{AB}$, \eqref{Security} can
be fulfilled more easily.
Despite the fact that efficiency generally increases with
increasing entanglement, this enhancement, as depicted in the
figure, is a complex function of the parameters involved.
Nevertheless, one can see that there exist an entanglement
threshold (around ${\rm LN}(\gamma_{AB}) \simeq 0.2$) below which
the protocol efficiency diminishes drastically no matter how mixed
are the states shared between Alice and Bob.
It is also illustrative to examine the dependence of $\alpha$
(which determines the interval length $D_\alpha$) on the
entanglement of the states shared by Alice and Bob.
\begin{figure}
\caption{Entanglement of the states shared between Alice and Bob
measured in terms of the logarithmic negativity ${\rm
LN}
\label{fig1}
\end{figure}
In Fig. \ref{fig1} we plot the logarithmic negativity of a given
state versus the parameter $\alpha$. States with the same
entanglement but different purity are associated to quite
different values of $\alpha$. Nevertheless states with high
entanglement permit a large interval length (small $\alpha$) and,
thus, high efficiency.
In both, Fig. \ref{fig2} and Fig. \ref{fig1}, we have observed
that states with different entanglement give the same efficiency.
However it is important to point out that to extract the key's
bits, classical advantage distillation \cite{Maurer93} stills
needs to be performed. The efficiency of Maurer's protocol,
strongly increases with decreasing $\epsilon_{AB}$, and,
therefore, the states with higher entanglement will provide a
higher key rate.
\section{Conclusions}\label{Conclusions}
Efficiency is a key issue for any experimental implementation of
quantum cryptography since available resources are not unlimited.
Here, we have shown that the sharing of entangled gaussian
variables and the use of only gaussian operations permits
efficient quantum key distribution against individual and finite
coherent attacks. All mixed NPPT symmetric states can be used to
extract secret bits under individual attacks whereas under finite
coherent attacks and additional condition has to be fulfilled. We
have introduced a figure of merit (the efficiency $E$) to quantify
the number of classical correlated bits that can be use to distill
a key from a sample of $N$ entangled states. We have observed that
this quantity grows with the entanglement shared between Alice and
Bob. This relation it is not one-to-one due to the fact that
states with less entanglement but purer (Eve more disentangled)
can be equally efficient. Nevertheless as we have pointed out,
these states would be, inefficient in the distillation of the key.
Finally, we would like to remark that our study is not restricted
to quantum key distribution protocols, but can be extended to any
other protocol that uses as a resource entangled continuous
variable states to establish a set of classically correlated bits
between distant parties \cite{rodo2}.
{\it Acknowledgments} -- We thank A. Ac{\'\i}n, A. Monras, and J. Bae
for discussions. We acknowledge support from ESF PESC QUDEDIS, MEC
(Spanish Government) under contracts EX2005-0830, AP2005-0595,
CIRIT (Catalan Government) under contracts CSG-00185,
FIS2005-01369 and Consolider-Ingenio 2010 CSD2006-0019.
\end{document} | math | 33,461 |
\begin{document}
\begin{center}
{\Large \bf Odd length for even hyperoctahedral groups and\\
signed generating functions (-1)^{\ell(\sigma)}x^{L(\sigma)}ootnote{2010 Mathematics Subject
Classification: Primary 05A15; Secondary 05E15, 20F55.}}
Francesco Brenti \\
Dipartimento di Matematica \\
Universit\`{a} di Roma ``Tor Vergata''\\
Via della Ricerca Scientifica, 1 \\
00133 Roma, Italy \\
{\em [email protected] } \\
Angela Carnevale (-1)^{\ell(\sigma)}x^{L(\sigma)}ootnote{Partially supported by German-Israeli Foundation for Scientific Research and Development, grant no. 1246.} \\
Fakultat f\"ur Mathematik \\
Universit\"at Bielefeld\\
D-33501 Bielefeld, Germany \\
{\em [email protected] } \\
\end{center}
\begin{abstract}
We define a new statistic on the even hyperoctahedral
groups which is a natural analogue of the odd length statistic recently
defined and studied on Coxeter groups of types $A$ and $B$. We compute the signed (by length)
generating function of this statistic over the whole group and over
its maximal and some other quotients and show that it always factors nicely.
We also present some conjectures.
\end{abstract}
\section{Introduction}
The signed (by length) enumeration of the symmetric group, and other finite Coxeter groups by
various statistics is an active area of research (see, e.g., \cite{AGR, Bia, BC, Cas, DF, L, Man, Mon, Rei, Siv, Wa}). For example, the
signed enumeration of classical Weyl groups by major index was carried out by
Gessel-Simion in \cite{Wa} (type $A$), by Adin-Gessel-Roichman in \cite{AGR} (type $B$)
and by Biagioli in \cite{Bia} (type $D$), that by descent by Desarmenian-Foata in \cite{DF}
(type $A$) and by Reiner in \cite{Rei} (types $B$ and $D$), while that by excedance by Mantaci
in \cite{Man} and independently by Sivasubramanian in \cite{Siv} (type $A$) and by Mongelli in \cite{Mon}
(other types).
In \cite{KV}, \cite{VS} and \cite{VS2} two statistics were introduced on the symmetric and hyperoctahedral
groups, in connection with the enumeration of partial flags in a quadratic space and the
study of local factors of representation zeta functions of certain groups, respectively
(see \cite{KV} and \cite{VS2}, for details). These statistics combine combinatorial and
parity conditions and have been called the ``odd length'' of the respective groups.
In \cite{KV} and \cite{VS2} it was conjectured that the signed (by length) generating functions
of these statistics over all the quotients of the corresponding groups always factor in a
very nice way, and this was proved in \cite{BC} (see also \cite{CarT}) for types $A$ and $B$
and independently, and in a different way, in \cite{L} for type $B$.
In this paper we define a natural analogue of these statistics for the even hyperoctahedral
group and study the corresponding signed generating functions. More precisely, we show that
certain general properties that these signed generating functions have in types $A$ and $B$
(namely ``shifting'' and ``compressing'') continue to hold in type $D$. We then show that these
generating functions factor nicely for the whole group (i.e., for the trivial quotient) and
for the maximal quotients. As a consequence of our results we
show that the signed generating function over the whole even hyperoctahedral group is
the square of the one for the symmetric group.
The organization of the paper is as follows. In the next section we recall some definitions, notation, and results that are used in the sequel. In \S 3 we define a new statistic on the even
hyperoctahedral group which is a natural analogue of the odd length statistics that have already been defined in types $A$ and $B$ in \cite{KV} and \cite{VS2}, and study
some general properties of the corresponding signed generating functions. These
include a complementation property, the identification of subsets of the quotients over
which the corresponding signed generating function always vanishes, and operations on a quotient that leave the corresponding signed generating function unchanged. In \S 4 we
show that the signed generating function over the whole even hyperoctahedral group factors
nicely. As a consequence of this result we obtain that this signed generating function is the
square of the corresponding one in type $A$. In \S 5 we compute the signed generating functions of the maximal, and some other, quotients and show that these also always factor nicely. Finally, in \S 6, we present some conjectures naturally arising from
the present work, and the evidence that we have in their favor.
\section{Preliminaries}
In this section we recall some notation, definitions, and results that are used in the sequel.
\noindent
We let $\mathbb P:=\{1,\,2,\ldots\}$ be the set of positive integers and $\mathbb N:= \mathbb{P} \cup \{0\}$. For all $m,\,n \in \mathbb{Z}$, $m\leq n$ we let $[m,n]:= \{m,\,m+1,\ldots,\,n\}$ and $[n]:=[1,\,n]$. Given a set $I$ we denote by $|I|$ its cardinality.
For a real number $x$ we denote by $\left\lfloor x \right\rfloor$ the greatest integer less than or equal to $x$ and by $\left\lceil x \right\rceil$ the smallest integer greater than or equal to $x$. Given $J \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$ there are unique integers $a_1 < \cdots < a_s$ and $b_1 < \cdots < b_s$ such that $J = [a_1,b_1] \cup \cdots \cup [a_s,b_s]$ and $a_{i+1} - b_{i} >1$ for $i=1, \ldots , s-1$. We call the intervals $[a_1,b_1], \ldots , [a_s,b_s]$ the {\em connected components} of $J$.
\noindent
For $n_1,\ldots,\,n_k \in \mathbb N$ and $n:=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{i=1}^k n_i $, we let $(-1)^{\ell(\sigma)}x^{L(\sigma)}ootnotesize \left[ \begin{array}{c} n \\
n_1,\ldots,n_k \end{array} \right]_{q}$ \normalsize denote the {\em $q$-multinomial coefficient}
\[\left[ \begin{array}{c} n \\
n_1,\ldots,n_k \end{array} \right]_{q}:= (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{[n]_{q}!}{[n_1]_q !\cdot \ldots\cdot [n_k]_q !},\]
where
\[[n]_q := (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{1-q^n}{1-q},\qquad\qquad [n]_q ! := \noindentrod_{i=1}^{n} [i]_q \qquad\qquad\mbox{and}\qquad [0]_q!:= 1.
\]
The symmetric group $S_n$ is the group of permutations of the set $[n]$. For $\sigma \in S_n$ we use both the one-line notation $\sigma=[\sigma(1),\,\ldots,\,\sigma(n)]$ and the disjoint cycle notation.
We let $s_1,\ldots,\,s_{n-1}$ denote the standard generators of $S_n$, $s_i=(i,\,i+1)$.
The hyperoctahedral group $B_n$ is the group of signed permutations, or permutations $\sigma$ of the set $[-n,n]$ such that $\sigma(j)=-\sigma(-j)$.
For a signed permutation $\sigma$ we use the window notation $\sigma = [\sigma(1), \ldots ,\sigma(n)]$ and the disjoint cycle notation.
The standard generating set of $B_n$ is $S=\{s_0,\,s_1,\,\ldots,\,s_{n-1}\}$, where $s_0=[-1,\,2,\,3,\ldots,\,n]$ and $s_1,\ldots,\,s_{n-1}$ are as above.
We follow $\cite{BB}$ for notation and terminology about Coxeter groups. In particular, for $(W,S)$ a Coxeter system we let $\ell$ be the Coxeter length and for
$I\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq S$ we define the quotients:
\[
W^{I} := \{w\in W \;:\; D(w)\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq S\setminus I\},
\]
and
\[
^{I}W := \{w\in W \;:\; D_L(w)\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq S\setminus I\},
\]
where $D(w)=\{s\in S \;:\; \ell(ws)<\ell(w)\}$, and
$D_L(w)=\{s\in S \;:\; \ell(sw)<\ell(w)\}$. The parabolic subgroup $W_I$
is the subgroup generated by $I$. The following result is well known (see, e.g., \cite[Proposition 2.4.4]{BB}).
\begin{pro}
Let $(W,S)$ be a Coxeter system, $J \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq S$, and $w \in W$. Then there exist
unique elements $w^J \in W^J$ and $w_J \in W_J$ (resp., $^Jw \in ^JW$ and
$_Jw \in W_J$) such that $w= w^J w_J$ (resp., $_Jw ^Jw$). Furthermore
$\ell(w)= \ell(w^J)+\ell(w_J)$ (resp., $\ell(_Jw)+\ell(^Jw)$).
\end{pro}
It is well known that $S_n$ and $B_n$, with respect to the above generating
sets, are Coxeter systems and that the following results hold (see, e.g.,
\cite[Propositions 1.5.2, 1.5.3, and \S 8.1]{BB}).
\begin{pro}
Let $\sigma \in S_n$. Then
$
\ell_A(\sigma)=| \{ (i,j) \in [n]^2 : i<j , \sigma(i) > \sigma(j) \} |
$
and
$
D(\sigma) = \{ s_i : \sigma(i) > \sigma(i+1) \}.
$
\end{pro}
For $\sigma \in B_n$ let
\begin{align*}\noindent \ \inv(\sigma):=& |\{(i,j)\in [n]\times[n] \; : \; i<j,\,\sigma(i)>\sigma(j)\}|, \\ \negg(\sigma):=& |\{ i\in [n]\; : \; \sigma(i)<0\}|, \\ \nsp(\sigma):=& |\{(i,j)\in [n]\times[n] \; : \; i<j, \, \sigma(i)+\sigma(j)<0\}|.\end{align*}
\begin{pro}
Let $\sigma \in B_n$. Then
\[
\ell_B(\sigma)= (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{1}{2} | \{ (i,j) \in [-n,n]^2 : i<j , \sigma(i) > \sigma(j) \} |=\ \inv(\sigma)+\negg(\sigma)+\nsp(\sigma)
\]
and
$
D(\sigma) = \{ s_i : i \in [0,n-1] , \sigma(i) > \sigma(i+1) \}.
$
\end{pro}
The group $D_n$ of even-signed permutations is the subgroup of $B_n$ of elements with an even number of negative entries in the window notation:
\[D_n=\{\sigma \in B_n\,:\, \negg(\sigma)\equiv 0 \noindentmod 2\}.\]
This is a Coxeter group of type $D_{n}$, with set of generators $S=\{s_0 ^D,s^D _1,\ldots ,s^D _{n-1}\}$, where $s_0 ^D:=[-2,-1,3,\ldots n]$ and $ s^D_{i}:=s_i$ for $i\in [n-1]$.
Moreover, the following holds (see, e.g., \cite[Propositions 8.2.1 and 8.2.3]{BB}).
\begin{pro}
\label{combD}
Let $\sigma \in D_n$. Then
$
\ell_D(\sigma)=\inv(\sigma)+\nsp(\sigma)
$
and
$
D(\sigma) = \{ s^D_i : i \in [0,n-1] , \sigma(i) > \sigma(i+1) \},
$
where
$\sigma(0):=\sigma(-2)$.
\end{pro}
Thus, for a subset of the generators $I\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq S$, that we identify with the corresponding subset $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$, we have the following description of the quotient
$$D_n^I=\{\sigma \in D_n \,:\, \sigma(i)<\sigma(i+1) \mbox{ for all } i \in I\}$$
where $\sigma(0):=-\sigma(2)$.
The following statistic was first defined in \cite{KV}. Our definition is not the
original one, but is equivalent to it (see \cite[Definition 5.1 and Lemma 5.2]{KV})
and is the one that is best suited for our purposes.
\begin{defn}
Let $n\in {\mathbb P}$. The statistic $L_{A}:S_n \rightarrow \mathbb N$ is defined as follows. For $\sigma \in S_n$
\[
L_{A}(\sigma):= |\{(i,j) \in [n]^2 \; : \; i<j,\,\sigma(i)>\sigma(j),\,i\not\equiv j \noindentmod{2}\}|.
\]
\end{defn}
The following statistic was introduced in \cite{VS} and \cite{VS2}, and is a natural
analogue of the statistic $L_A$ introduced above, for Coxeter groups of type $B$.
\begin{defn}
\label{defLB}
Let $n\in {\mathbb P}$. The statistic $L_{B}:B_n \rightarrow \mathbb N$ is defined as follows. For $\sigma \in B_n$
\[
L_{B}(\sigma):= (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{1}{2} |\{(i,j)\in [-n,\,n]^2 \, : \, i<j,\,\sigma(i)>\sigma(j),\,i\not\equiv j \noindentmod{2}\}|.\]
\end{defn}
\noindent For example, if $n=4$ and $\tau=[-2,4,3,-1]$ then $L_B (\tau)= (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{1}{2}|\{(-4,-3),\,(-4,1),\,(-3,-2),$ $\,(-1,0),\,(-1,4),\,(0,1),\,(2,3),\,(3,4)\}|=4$.
We call these statistics $L_A$ and $L_B$ the {\em odd length} of the symmetric and hyperoctahedral groups, respectively.
Note that if $\sigma \in S_n \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bset B_n $ then $ L_B (\sigma)=L_A(\sigma)$.
The odd length of an element $\sigma \in B_n$ also has a description in terms of statistics of the window notation of $\sigma$.
Given $\sigma \in B_n$ we let
\begin{align*}\noindent \ \oinv(\sigma):=& |\{(i,j)\in [n]^2 \; : \; i<j,\,\sigma(i)>\sigma(j),\,i\not\equiv j \noindentmod{2}\}|, \\ \oneg(\sigma):=& |\{ i\in [n]\; : \; \sigma(i)<0,\,i\not\equiv 0 \noindentmod{2}\}|, \\ \onsp(\sigma):=& |\{(i,j)\in [n]^2 \; : \; \sigma(i)+\sigma(j)<0,\,i\not\equiv j \noindentmod{2}\}|.\end{align*}
The following result appears in \cite[Proposition 5.1]{BC}.
\begin{pro}\label{LB}
Let $\sigma \in B_n$. Then
$
L_B(\sigma)= \oinv(\sigma)+ \oneg(\sigma)+ \onsp(\sigma).$
\end{pro}
The signed generating function of the odd length factors very nicely both on quotients of $S_n$ and of $B_n$. The following result was conjectured in \cite[Conjecture C]{KV} and
proved in \cite{BC}.
\begin{thm}
\label{Aquot}
Let $n \in {\mathbb P}$, $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [n-1]$, and $I_{1}, \ldots , I_{s}$ be the connected components of $I$. Then
\begin{align}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in S_{n}^{I}} (-1)^{\ell_A (\sigma )} x^{L_A(\sigma )} &=\mc \noindentrod_{k=2m+2}^n \left(1+(-1)^{k-1}x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{k}{2}\right\rfloor}\right)
\end{align}
where $m := \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{k=1}^{s} \left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{|I_{k}|+1}{2} \right\rfloor $.
\end{thm}
In particular, for the whole group we have the following.
\begin{cor}\label{wgpA}
Let $n \in {\mathbb P},\, n\geq 2$. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in S_n} (-1)^{\ell_A(\sigma)}x^{L_A(\sigma)} = \noindentrod_{i=2}^{n} \left(1+(-1)^{i-1}x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{i}{2}\right\rfloor}\right).
\]
\end{cor}
For $J\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$ we define $J_0\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq J$ to be the connected component of $J$ which contains $0$, if $0\in J$, or $J_{0} := \emptyset$ otherwise. Let $J_1,\ldots,J_s$ be the remaining ordered connected components. The following result was conjectured in \cite[Conjecture 1.6]{VS2} and proved in \cite{BC} and independently in \cite{L}.
\begin{thm}
\label{Bquot}
Let $n\in \mathbb P$, $J \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$, and $J_0,\ldots, J_s$ be the connected components of $J$ indexed as just described. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}mb=(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{\noindentrod\limits_{j=a+1}^{n}(1-x^j)}{\noindentrod\limits_{i=1}^{m}(1-x^{2i})} \mcb
\]
where $m:=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{i=1}^s \left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{|J_{i}|+1}{2}\right\rfloor$ and $a:=\min\{ [0, \,n-1] \setminus J\}$.
\end{thm}
\section{Definition and general properties}
In this section we define a new statistic, on the even
hyperoctahedral group $D_n$, which is a natural analogue of the odd length statistics that have already been defined and studied in types $A$ and $B$, and study
some of its basic properties.
Given the descriptions of $L_A$ and $L_B$ in terms of odd inversions, odd negatives and odd negative sum pairs, and the relation between the Coxeter lengths of the Weyl groups of types
$B$ and $D$ (see, e.g., \cite[Propositions 8.1.1 and 8.2.1]{BB}), the following definition is
natural.
\begin{defn}
Let $\sigma \in D_n$. We let
\[L_D(\sigma):=L_B(\sigma)- \oneg(\sigma)= \oinv(\sigma)+ \onsp(\sigma). \]
\end{defn}
For example let $n=5$, $\sigma=[2,-1,5,-4,3]$. Then $L_D(\sigma)=5$.
We call $L_D$ the {\em odd length} of type $D$. Note that the statistic $L_D$ is well defined also on $S_n$ (where it coincides with $L_A$) and on $B_n$.
In fact, the signed distribution of $L_D$ over any quotient of $D_{n}$ and over
its ``complement'' in $B_n$, is exactly the same, as we now show. For $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$
let $(B_{n} \setminus D_{n})^{I}:= \{ \sigma \in B_n \setminus D_n : \; \sigma (i) <
\sigma (i+1) \mbox{ for all } i \in I \}$ where $\sigma (0) := - \sigma (2)$. Note that
$(B_n \setminus D_n)^{I}=B_{n}^{I} \setminus D_n^I$ if $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [n-1]$.
\begin{lem}\label{compl}
Let $n\in \mathbb P$ and $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$. Then
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I}}{y^{\ell_D(\sigma)}x^{L_D(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in (B_n \setminus D_n)^{I}}{y^{\ell_D(\sigma)}x^{L_D(\sigma)}}. \]
In particular, $\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I}}{(-1)^{\ell_D(\sigma)}x^{L_D(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in (B_n \setminus D_n)^{I}}{(-1)^{\ell_D(\sigma)}x^{L_D(\sigma)}}$. \end{lem}
\begin{proof}
Left multiplication by $s_0$ (that is, changing the sign of $1$ in the window notation) is a bijection between $D_n^{I}$ and $(B_n\setminus D_n)^{I}$. Moreover, (odd) inversions and (odd) negative sum pairs are preserved by this operation so
$L_D(s_0 \sigma )=L_D(\sigma)$, and $ \ell_D(s_0 \sigma )=\ell_D(\sigma)$,
for all $ \sigma \in D_{n}$ and the result follows.
\end{proof}
In what follows, since we are mainly concerned with distributions in type $D$, we omit the subscript and write just $\ell$ and $L$ for the length and odd length, respectively, on $D_n$.
We now show that the generating function of $(-1)^{\ell(\cdot)}x^{L(\cdot)}$
over any quotient of $D_{n}$ such that $s_{0}^{D} \in D_{n}^{I}$ can be reduced to elements for which the maximum (or the minimum) is in certain positions. More precisely, we prove that, for a given quotient, our generating function is zero over all elements for which the maximum (or minimum) is sufficiently far from $I$. For a subset $I\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$ we let $\delta_0 (I)=1$ if $0 \in I$ and $\delta_0 (I)=0$ otherwise.
\begin{lem}\label{zerod}
Let $n\in \mathbb P$, $n\geq 3$, $I\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$. Let $a \in [2+\delta_0 (I),n-1]$ be such that $[a-2,a+1]\cap I=\emptyset$. Then
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^I :\\\sigma(a)=n\}}}(-1)^{\ell(\sigma)}x^{L(\sigma)} =\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^I :\\\sigma(a)=-n\}}}(-1)^{\ell(\sigma)}x^{L(\sigma)}=0.\]
\end{lem}
\begin{proof}
In our hypotheses, if $\sigma \in D_n ^I$ then also $\sigma^{a}:= \sigma (-a-1,-a+1)
(a-1,a+1)$ is in the same quotient. Clearly $(\sigma^a)^a=\sigma$ and $|\ell(\sigma)-\ell(\sigma^a)|=1$, while, since $\sigma(a)=n$, $L(\sigma^a)=L(\sigma)$. Therefore we have that
\begin{align*}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in {D}^{I}_{n}: \\ \sigma(a)=n \} }} (-1)^{\ell (\sigma )}x^{L(\sigma )}&=& \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in {D}^{I}_n: \sigma(a)=n, \\ \sigma (a-1) < \sigma (a+1) \} }}\left( (-1)^{\ell (\sigma )}x^{L(\sigma )} + (-1)^{\ell (\sigma^a )}x^{L(\sigma^a )}\right) =0.\end{align*}
The proof of the second equality is exactly analogous and is therefore omitted.
\end{proof}
Although we do not know of any definition of our (or of any other) odd length statistics
in Coxeter theoretic language, it is natural to expect that the only automorphism of
the Dynkin diagram of $D_n$ preserves the corresponding signed generating function.
This is indeed the case, as we now show.
\begin{pro}\label{zerouno}
Let $n \in \mathbb P$, $n\geq 2$, and $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [2,n-1]$. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I \cup \{0\}}}{y^{\ell(\sigma)}x^{L(\sigma)}}=
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I \cup \{1\}}}{y^{\ell(\sigma)}x^{L(\sigma)}}.
\]
In particular, $\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I \cup \{0\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I \cup \{1\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}$.
\end{pro}
\begin{proof}
Right multiplication by $s_{0}$ (i.e., changing the sign of the leftmost element in the window notation) is a bijection between $D_{n}
^{I \cup \{ 0 \}}$ and $(B_{n} \setminus D_{n} )^{I \cup \{ 1 \}}$. Furthermore, if $\sigma \in D_{n}$, then
\begin{eqnarray*}
\oinv (\sigma s_{0}) & = & \oinv (\sigma ) - |\{ i \in [2,n]: i \equiv 0 \noindentmod{2}, \; \sigma (1)>\sigma (i) \} | \\
& & + |\{ i \in [2,n]: i \equiv 0 \noindentmod{2}, \; -\sigma (1) > \sigma (i) \} | ,\\
\onsp (\sigma s_{0}) & = & \onsp (\sigma ) - |\{ i \in [2,n]: i \equiv 0 \noindentmod{2}, \; \sigma (1) + \sigma (i) <0 \} | \\
& & + | \{ i \in [2,n]: i \equiv 0 \noindentmod{2}, \; -\sigma (1)+ \sigma (i) <0 \} | ,\\
\inv (\sigma s_{0} )& =& \inv (\sigma ) -|\{ i \in [2,n]: \; \sigma (1) > \sigma (i) \} | + |\{ i \in [2,n]: \; - \sigma (1)>
\sigma (i) \} |,\\
\mbox{and} \quad \qquad& &
\\
\nsp (\sigma s_{0} ) &=& \nsp (\sigma ) -|\{ i \in [2,n]: \; \sigma (1) + \sigma (i)<0 \} | + |\{ i \in [2,n]: \; - \sigma (1)+
\sigma (i) <0 \} |. \end{eqnarray*}
Therefore $L(\sigma s_{0})=L(\sigma )$ and $\ell (\sigma s_{0})=\ell (\sigma )$. Hence
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in (B_{n} \setminus D_{n}) ^{I \cup \{ 1\}}}y^{\ell (\sigma )} \; x^{L(\sigma )} =
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m _{\sigma \in D_{n}^{I \cup \{ 0 \}}}y^{\ell (\sigma s_{0})} \; x^{L(\sigma s_{0})} =
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m _{\sigma \in D_{n}^{I \cup \{ 0 \}}}y^{\ell (\sigma )} \; x^{L(\sigma )} ,
\]
and the result follows from Lemma \ref{compl}.
\end{proof}
We conclude by showing
that when $I$ does not contain $0$, each connected component can be shifted to the left or to the right, as long as it remains a connected component, without changing the generating function over the corresponding quotient. The proof is identical to that of \cite[Proposition 3.3]{BC}, and is therefore omitted.
\begin{pro}\label{shd}
Let $I\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [n-1]$, $i\in \mathbb P$, $k\in \mathbb N$ be such that $[i,\,i+2k]$ is a connected component of $I$ and $i+2k+2 \notin I$. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^I}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I\cup \tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{\tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}
\]
where $\tilde{I}:=(I\setminus\{i\})\cup\{i+2k+1\}$.
\end{pro}
Shifting is also allowed when $I$ contains $0$, but only for connected components which are sufficiently far from it, as stated in the next result.
\begin{pro}
Let $I\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$, $i\in \mathbb P$, $i>2$ and $k\in \mathbb N$ such that $[i,\,i+2k]$ is a connected component of $I$ and $i+2k+2 \notin I$. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^I}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{I\cup \tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n ^{\tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}
\]
where $\tilde{I}:=(I\setminus\{i\})\cup\{i+2k+1\}$.
\end{pro}
\begin{proof}
The proof is analogous to that of \cite[Proposition 3.3]{BC} noting that , since $i>2$, $\sigma \in D_n^{I}$ if and only if $\sigma(i+2j,i+2k+2)(-i-2j,-i-2k-2)\in D_n^I$.
\end{proof}
\section{Trivial quotient}
In this section, using the results in the previous one, we compute the generating function of $(-1)^{\ell(\cdot)}x^{L(\cdot)}$ over the whole even hyperoctahedral group $D_n$. In particular, we obtain that this generating function is
the square of the corresponding one in type $A$ (i.e., for the symmetric group).
\begin{thm}\label{sq}
Let $n\in \mathbb{P}$, $n\geq 2$. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n}{(-1)^{\ell (\sigma)}x^{L(\sigma)}}=\noindentrod_{j=2}^{n}(1+(-1)^{j-1}x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2}\right\rfloor})^2 .
\]
\end{thm}
\begin{proof}
We proceed by induction on $n$. By Lemma \ref{zerod}, the sum over all elements for which $n$ or $-n$ appears in positions different from $1$ and $n$ is zero. So the generating function over $D_n$ reduces to
\begin{align*}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}&=
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n: \\ \sigma(1)=n\}}}
{(-1)^{\ell(\sigma)}x^{L(\sigma)}} +\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n: \\ \sigma(n)=n\} }
}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}
\\
&+
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n: \\ \sigma(1)=-n\} }}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n: \\ \sigma(n)=-n\}
}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \\
&= \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell(\tilde{\sigma})}x^{L(\tilde{\sigma})}} +
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \\
&+ \!\!\!\!\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}\!\!{(-1)^{\ell(\hat{\sigma})}x^{L(\hat{\sigma})}}+\!\!\!\!\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}\!\!{(-1)^{\ell(\check{\sigma})}x^{L(\check{\sigma})}} ,
\end{align*}
\normalsize
where $\tilde\sigma:=[n,\sigma(1),\ldots,\sigma(n-1)]$, $\hat\sigma:=[-n,\sigma(1),\ldots,\sigma(n-1)],$
and $\check\sigma:=[\sigma(1),\ldots,\sigma(n-1),-n]$.
But, by our definition and Proposition 8.2.1 of \cite{BB}, we have that
\begin{align}
L(\tilde\sigma)&
=L(\sigma)+m, \qquad \ell(\tilde\sigma)=\ell(\sigma)+n-1\\ \label{ha}
L(\hat\sigma)&
=L(\sigma)+m, \qquad \ell(\hat\sigma)=\ell(\sigma)+n-1\\ \label{che}
L(\check\sigma)&
=L(\sigma)+2m,\:\: \quad \ell(\check\sigma)=\ell(\sigma)+2(n-1),
\end{align}
where $m := \left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n}{2} \right\rfloor $. Therefore
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell(\tilde{\sigma})}x^{L(\tilde{\sigma})}}=(-1)^{n-1}x^m\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell({\sigma})}x^{L({\sigma})}}\]
and, similarly,
\begin{eqnarray*}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell(\hat{\sigma})}x^{L(\hat{\sigma})}}&=&(-1)^{n-1}x^m\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell({\sigma})}x^{L({\sigma})}}\\
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell(\check{\sigma})}x^{L(\check{\sigma})}}&=&x^{2m}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell({\sigma})}x^{L({\sigma})}}.
\end{eqnarray*}
So by Lemma \ref{compl} and our induction hypothesis we obtain that
\begin{eqnarray*}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}&=&(1+(-1)^{n-1}x^m) \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} +\\
&+&((-1)^{n-1}x^m +x^{2m})\!\!\!\!\!\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}{\!\!\!\!\!\!(-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&=&(1+2(-1)^{n-1}x^{m} +x^{2m})\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma\in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&=&\left(1+(-1)^{n-1}x^{\left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n}{2} \right\rfloor }\right)^2\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}} (-1)^{\ell(\sigma)}x^{L(\sigma)},\\
\end{eqnarray*}
and the result follows by induction.
\end{proof}
As an immediate consequence of Theorem \ref{sq} and of Corollary \ref{wgpA} we obtain the
following result.
\begin{cor}
\label{DA}
Let $n \in \mathbb{P}$, $n \geq 2$. Then
\[ \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m _{\sigma \in D_{n}} (-1)^{\ell (\sigma )} \, x^{L}(\sigma ) =
\left( \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m _{\sigma \in S_{n}} (-1)^{\ell _{A}(\sigma )} \, x^{L_{A}(\sigma )} \right)^{2}.
{\cal B}ox \]
\end{cor}
It would be interesting to have a direct proof of Corollary \ref{DA}.
\section{Maximal and other quotients}
In this section we compute, using the results in \S 3, the signed generating function of the odd length over the maximal, and some other,
quotients of $D_{n}$. In particular, we obtain that these generating functions always factor nicely.
\begin{thm}\label{maxquod}
Let $n\in \mathbb P$, $n\geq 3$ and $i\in [0,n-1]$. Then
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n^{\{i\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=(1-x^2)\noindentrod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2.
\]
\end{thm}
\begin{proof}
By Propositions \ref{zerouno} (with $I = \emptyset$) and \ref{shd}, we may assume $i=1$. We proceed by induction on $n \geq 3$.
By Lemma \ref{zerod} we have that the sum over $\sigma\in D^{\{1\}}_n$ such that $n$ or $-n$ appear in the window in any position but $1,2,3$, or $n$ is zero. Furthermore, if $\sigma \in D_n^{\{1\}}$ then $\sigma^{-1}(n)\neq 1$ and $\sigma^{-1}(-n)\neq 2$. Moreover, the map $\sigma
\mapsto \sigma (1,3) (-1,-3)$ is a bijection of $\{ \sigma \in D_{n}^{\{ 1\} }: \; \sigma (2)
=n \}$ in itself. But $L (\sigma (1,3)(-1,-3))=L(\sigma )$ and $\ell
(\sigma ) \not \equiv \ell (\sigma (1,3)(-1,-3)) \noindentmod{2} $ for all $\sigma \in D_{n}^{\{
1\} }$ such that $\sigma (2)=n$ so the sum is zero also over this kind of elements. Thus we have that:
\begin{align*}
&\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(n)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} +\\
&+ \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(1)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(n)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}
\end{align*}
\begin{align*}&=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} + \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}}{ (-1)^{\ell(\hat\sigma)}x^{L(\hat\sigma)}}+\\
&+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in (B_{n-1}\setminus D_{n-1})^{\{ 1\}}}{ (-1)^{\ell(\check\sigma)}x^{L(\check\sigma)}}
\end{align*}
where $\hat\sigma:=[-n,\sigma(1),\ldots,\sigma(n-1)]$ and $\check\sigma:=[\sigma(1),\ldots,\sigma(n-1),-n]$. Now
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\bar\sigma)}x^{L(\bar\sigma)}}\]
where $\bar\sigma:=[\sigma(2),\sigma(1),n,\sigma(3),\ldots,\sigma(n-1)]$. But
$\ell(\bar\sigma)= \inv(\sigma)+n-4+\nsp(\sigma)=\ell(\sigma)+n-4$, and
$L(\bar\sigma)= \oinv(\sigma)-1+\left\lceil(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-3}{2}\right\rceil+ \onsp(\sigma)=L(\sigma)+\left\lceil(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-5}{2}\right\rceil=L(\sigma)+m-2,$
where $m := \left\lceil (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-1}{2} \right\rceil$, so
\begin{align*} &\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=(-1)^n x^{m-1}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&=(-1)^n x^{m-2}\left(\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}(-1)^{\ell(\sigma)}x^{L(\sigma)} -\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}} (-1)^{\ell(\sigma)}x^{L(\sigma)}\right) .
\end{align*}
Similarly,
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in B_{n-1}\setminus D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\bar{\bar{\sigma}})}x^{L(\bar{\bar{\sigma}})}}\]
where $\bar{\bar{\sigma}}:=[\sigma(2),\sigma(1),-n,\sigma(3),\ldots,\sigma(n-1)]$ and
$\ell(\bar{\bar{\sigma}})= \inv(\sigma)+1+\nsp(\sigma)+n-1=\ell(\sigma)+n$, $L(\bar{\bar{\sigma}})= \oinv(\sigma)+ \onsp(\sigma)+1+\left\lceil(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-3}{2}\right\rceil=L(\sigma)+\left\lceil(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-1}{2}\right\rceil=L(\sigma)+m$.
So, by Lemma \ref{compl},
\begin{align*}
&\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=(-1)^n x^{m} \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in B_{n-1}\setminus D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&=(-1)^{n}x^{m}\left(\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma\in B_{n-1} \setminus D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}-\!\!\!\!\!\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{ (B_{n-1} \setminus D_{n-1})^{\{ 1\}}}{\!\!\! (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right)\\
&=(-1)^{n}x^{m}\left(\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}-\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{ \sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right).
\end{align*}
Moreover, by \eqref{ha} and \eqref{che} we have
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}} {(-1)^{\ell(\hat\sigma)}x^{L(\hat\sigma)}}=(-1)^{n-1}x^m \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in B_{n-1}\setminus D_{n-1}} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}\]
and
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in (B_{n-1} \setminus D_{n-1})^{\{ 1\}}} {(-1)^{\ell(\check\sigma)}x^{L(\check\sigma)}}=x^{2m} \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in (B_{n-1}\setminus D_{n-1})^{\{ 1 \}}} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}.
\]
Thus we get, again by Lemma \ref{compl},
\sigma^{-1}all
\begin{eqnarray*}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}&=&
(-1)^n x^{m-2}\left(\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}-\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right)\\
&+&\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+(-1)^{n-1}x^{m}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&+&(-1)^{n}x^{m}\left(\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}-\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right) \\
&+&x^{2m}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{ 1 \}}}{ \!\!\!(-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&=&(-1)^n x^{m-2}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\
&+&\left(1+(-1)^{n-1}x^{m-2}+(-1)^{n-1}x^{m}+x^{2m}\right)\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma\in D^{\{1\}}_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}
\end{eqnarray*}
\normalsize
and the result follows by Theorem \ref{sq} and our induction hypothesis.
\end{proof}
We note the following consequence of Theorems \ref{sq} and \ref{maxquod}.
\begin{cor}
Let $n \in \mathbb{P}$, $n \geq 3$, and $i \in [0,n-1]$. Then
\[ \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m _{\sigma \in D_{n}}(-1)^{\ell (\sigma )} \, x^{L(\sigma )}=(1-x^{2}) \, \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m _{\sigma
\in D_{n}^{\{ i\}}}(-1)^{\ell (\sigma )} \, x^{L(\sigma )} . \]
\end{cor}
\begin{proof}
This follows immediately from Theorems \ref{sq} and \ref{maxquod}.
\end{proof}
The results obtained up to now compute
$\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L(\sigma )}$ when $|I| \leq 1$.
A natural next step is to try to compute these generating functions if $|I \setminus \{ 0 \}| \leq 1$. We are able to do this for $I=\{ 0,1 \}$, and $I=\{ 0,2 \}$.
The computation
for $I=\{ 0,2 \}$ follows easily from results that we have already obtained.
\begin{cor}
\label{pro02}
Let $n \in {\mathbb P}$, $n \geq 4$. Then
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in
D_{n}^{\{ 0, \, 2 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} =
(1-x^2) \noindentrod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2.
\]
\end{cor}
\begin{proof}
By Proposition \ref{zerouno} and Proposition \ref{shd} we have
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in
D_{n}^{\{ 0,2 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} =\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in
D_{n}^{\{ 1,2 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} = \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in
D_{n}^{\{ 1 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}, \]
and the result follows by Theorem \ref{maxquod}.
\end{proof}
We conclude this section by computing
$\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L(\sigma )}$ when $I =\{ 0,1 \}$.
\begin{thm}
\label{pro01}
Let $n \in {\mathbb P}$, $n \geq 3$. Then
\[\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in
D_{n}^{\{ 0, \, 1 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} =
(1+x^2) \noindentrod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2.
\]
\end{thm}
\begin{proof}
We proceed by induction on $n \geq 3$.
By Lemma \ref{zerod} we have that the sum over $\sigma\in D^{\{0,\,1\}}_n$ such that $n$ or $-n$ appear in the window in any position but $1,2,3$, or $n$ is zero; moreover for $\sigma \in D_n^{\{0,\,1\}}$ we always have $\sigma^{-1}(\noindentm n)\neq 1$ and $\sigma^{-1}(-n)\neq 2$. Also, the map $\sigma
\mapsto \sigma (1,3) (-1,-3)$ is a bijection of $\{ \sigma \in D_{n}^{\{ 0,\,1\} }: \, \sigma (2)
=n \}$ in itself, so by the same argument as in the proof of Theorem \ref{maxquod} the sum over this kind of elements is zero. Thus we have that:
\begin{align*}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_n^{\{0,\,1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}&=
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{0,\,1\}}: \\ | \sigma(3) |=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ | \sigma(n) |=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}.
\end{align*}
By \eqref{che} and Lemma \ref{compl} we have that
\begin{align*}
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ | \sigma(n) |= n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} & = \left(1+x^{2\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n}{2}\right\rfloor}\right)\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{0,\,1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} \\
& = \left(1+x^{2\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n}{2}\right\rfloor}\right) (1+x^2) \noindentrod_{j=4}^{n-1} (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2
\end{align*}
by our induction hypothesis. Moreover,
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{\{\sigma \in D_n^{\{0,\,1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}\setminus D_{n-1}^{\{0,\,1\}}}{ (-1)^{\ell(\bar\sigma)}x^{L(\bar\sigma)}}
\]
where $\bar\sigma:=[-\sigma(2),-\sigma(1),n,\sigma(3),\ldots,\sigma(n-1)]$.
But
\begin{eqnarray*}
\inv (\bar\sigma )& =& \inv ([n,\sigma (3), \ldots, \sigma (n-1)]) + |\{ j \in [3,n-1]: \; - \sigma (1) > \sigma (j) \} | \\ & & + |\{ j \in [3,n-1]: \; - \sigma (2)>
\sigma (j) \} |,\\
\nsp (\bar\sigma ) &=& \nsp ([n,\sigma (3), \ldots, \sigma (n-1)]) +
|\{ j \in [3,n-1]: \; - \sigma (1) + \sigma (j) <0 \} | \\ & & +
|\{ j \in [3,n-1]: \; - \sigma (2)+ \sigma (j) <0 \} |, \\
\oinv (\bar\sigma ) & = & \oinv ([n,\sigma (3), \ldots, \sigma (n-1)] ) + |\{ j \in [3,n-1]: j \equiv 0 \noindentmod{2}, \; \sigma (1)+\sigma (j) <0 \} | \\
& & + |\{ j \in [3,n-1]: j \equiv 1 \noindentmod{2}, \; \sigma (2) + \sigma (j) <0 \} | ,\\
\mbox{and} \quad \qquad& &
\\
\onsp (\bar\sigma ) & = & \onsp ([n,\sigma (3), \ldots, \sigma (n-1)] ) + |\{ j \in [3,n-1]: j \equiv 0 \noindentmod{2}, \; \sigma (1) > \sigma (j) \} | \\
& & + | \{ j \in [3,n-1]: j \equiv 1 \noindentmod{2}, \; \sigma (2) > \sigma (j) \} |. \end{eqnarray*}
Therefore,
$\ell(\bar\sigma)=\ell(\sigma)+n-4$, and
$L(\bar\sigma)= \oinv(\sigma)-1+\left\lceil(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-3}{2}\right\rceil+ \onsp(\sigma)=L(\sigma)+m-2,$
where $m := \left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n}{2} \right\rfloor$.
Similarly,
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in B_{n-1}\setminus D_{n-1}: \\ \sigma(1)<\sigma(2)<-\sigma(1)\}}}{ (-1)^{\ell(\bar{\bar{\sigma}})}x^{L(\bar{\bar{\sigma}})}}
\]
where $\bar{\bar{\sigma}}:=[-\sigma(2),-\sigma(1),-n,\sigma(3),\ldots,\sigma(n-1)]$ and
$\ell(\bar{\bar{\sigma}})=\ell(\sigma)+n$, $L(\bar{\bar{\sigma}})= \oinv(\sigma)+ \onsp(\sigma)+1+\left\lceil(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{n-3}{2}\right\rceil=L(\sigma)+m$.
But $\{ \sigma \in B_{n-1} \setminus D_{n-1} \, : \,\sigma (1) < \sigma (2) < - \sigma (1) \} = (B_{n-1} \setminus D_{n-1})^{ \{ 1 \} } \setminus (B_{n-1} \setminus D_{n-1})^{ \{ 0,1 \} }$, so by Lemma \ref{compl}, Theorem \ref{maxquod} and our induction hypothesis
\begin{align*}
\hspace{-2.5em}\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bstack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ | \sigma(3) |= n\}}}\!\!\!{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}& = (-1)^n x^ {m-2} (1+x^2) \left(\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{1\}}}{\! (-1)^{\ell(\sigma)}x^{L(\sigma)}}-\!\!\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n-1}^{\{0,\,1\}}}\!\!{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right)\\
&=2(-1)^{n-1}x^m (1+x^2) \noindentrod_{j=4}^{n-1} (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2
\end{align*}
Thus
\[
\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n}^{\{ 0, \, 1 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} =
(1+(-1)^{n-1}x^m)^2 (1+x^2) \noindentrod_{j=4}^{n-1} (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2
\]
and the result follows.
\end{proof}
\section{Open problems}
In this section we present some conjectures naturally arising from the present work
and the evidence that we have in their favor.
In this paper we have given closed product formulas for $\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L(\sigma )}$ when $|I| \leq 1$, $I=\{0,1\}$ and $I=\{0,2\}$.
We feel that such formulas always exist. In particular, if $|I \setminus \{ 0,1 \}| \leq 1$, we feel that the following holds. For $n \in {\mathbb P}$ and $I \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$
let, for brevity, $D_{n}^{I}(x) := \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L (\sigma )}$.
\begin{con}
\label{con0i}
Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then
\[
D_{n}^{\{ 0, \, i \}}(x) =
\noindentrod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2.
\]
\end{con}
\begin{con}
\label{con01i}
Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then
\[
D_{n}^{\{ 0,1, \, i \}} (x) =
(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{1+x^2}{1-x^2} \; \noindentrod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2.
\]
\end{con}
We have verified these conjectures for $n \leq 8$. Note that, by Proposition \ref{shd}, it
is enough to prove Conjectures \ref{con0i} and \ref{con01i} for $i=3$.
Note that, by Theorem \ref{Aquot}, Conjectures
\ref{con0i} and \ref{con01i} may
be formulated in the following equivalent way. For $n \in {\mathbb P}$ and $J \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [n-1]$
let $S_{n}^{J}(x) := \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{\sigma \in S_{n}^{J}} (-1)^{\ell_A (\sigma )} x^{L_{A}(\sigma )}$.
\begin{con}
\label{con0ib}
Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then
\[
D_{n}^{\{ 0, \, i \}} (x) = (S_n^{ \{ i \} } (x))^2.
\]
\end{con}
\begin{con}
\label{con01ib}
Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then
\[
D_{n}^{\{ 0,1,i \}} (x) =
(1-x^4) \, (S_n^{ \{ 1, \, i \} } (x))^2.
\]
\end{con}
We feel that the presence of the factor
$\noindentrod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor(-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor})^2$ in Conjectures
\ref{con0i} and \ref{con01i} is not a coincidence. More generally, we feel that
the following holds.
\begin{con}
Let $n \in {\mathbb P}$, $n \geq 3$, and $J \sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}bseteq [0,n-1]$. Let $J_0, J_1, \ldots , J_s$
be the connected components of $J$ indexed as described before Theorem \ref{Bquot}. Then
there exists a polynomial $M_J (x) \in {\mathbb Z}[x]$ such that
\[ D_{n}^{J} (x) =
M_J (x) \; \noindentrod_{j=2m+2}^{n} (1+(-1)^{j-1} x^{\left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{j}{2} \right\rfloor })^2 ,
\]
where $m:=\sum_{\sigma \in {S_n^{I}}} (-1)^{\ell(\sigma)} x^{L(\sigma)}m_{i=0}^s \left\lfloor (-1)^{\ell(\sigma)}x^{L(\sigma)}rac{|J_{i}|+1}{2}\right\rfloor$. Furthermore,
$M_J(x)$ only depends on $( |J_0|, |J_1|, \ldots , |J_s| )$ and is a symmetric function of
$ |J_1|, \ldots , |J_s| $.
\end{con}
This conjecture has been verified for $n\leq 8$.
\end{document} | math | 50,541 |
\begin{document}
\title[]{On the KPZ equation with fractional diffusion: global regularity and existence results}
\author[B. Abdellaoui, I. Peral, A. Primo, F. Soria]{Boumediene Abdellaoui, Ireneo Peral${}^{(*)}$, Ana Primo, Fernando Soria }
\address{\hbox{\partialarbox{5.7in}{\text{\rm{meas }}skip\noindent {$*$ Laboratoire d'Analyse Nonlin\'eaire et Math\'ematiques
Appliqu\'ees.
\break\indent D\'epartement de Math\'ematiques, Universit\'e Abou Bakr Belka\"{\i}d, Tlemcen,
\break\indent Tlemcen 13000,
Algeria.}}}}
\address{\hbox{\partialarbox{5.7in}{\text{\rm{meas }}skip\noindent{Departamento de Matem\'aticas,\\ Universidad Aut\'onoma de Madrid,\\
28049, Madrid, Spain. \\[3pt]
\em{E-mail:\,}{\tt [email protected], [email protected], [email protected], [email protected]
}.}}}}
\date{\today}
\thanks{ This work was partially supported by research grants MTM2016-80474-P and PID2019-110712GB-I00, MINECO, Spain.
The first author has been also partially supported by an Erasmus grant from Autonomous University of Madrid and by the DGRSDT, Algeria. \\
During the final stages of this work, Ireneo Peral sadly passed away after a very short period of illness. He worked very hard and made fundamental contributions to this project, but the responsibility for the correctness of the presentation rests solely and exclusively with the other three authors. The paper is dedicated to him as a tribute for his guidance and teachings.}
\keywords{Fractional
heat equations, Global regularity, Nonlinear term in the gradient, Kardar-Parisi-Zhang equation, general comparison principle.
\\
\indent 2000 {\it Mathematics Subject Classification: MSC 2000: 35K59,47G20, 47J35. }}
\begin{abstract}
In this work we analyze the existence of solutions to the fractional quasilinear problem,
$$
(P) \left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u &=&|\nabla u|^{\alpha}+ f &\mbox{ in } \Omegaega_T\equiv\Omegaega\times (0,T),\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
u(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
$$
where $\Omegaega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2}<s<1$. We will assume that $f$ and $u_0$ are non negative functions
satisfying some additional hypotheses that will be specified later on.
Assuming certain regularity on $f$, we will prove the existence of a solution to problem $(P)$ for values $\alpha<\dfrac{s}{1-s}$, as well as the non existence of such a
solution when $\alpha>\dfrac{1}{1-s}$. This behavior clearly exhibits a deep difference with the local case.
\vskip 2mm
{\it To Ireneo, our teacher, colleague and friend, in memoriam.}
\end{abstract}
\maketitle
{\footnotesize
\tableofcontents
}
\section{Introduction}
In the paper {\cal C}_0^{\iy}\,te{KPZ}, Kardar, Parisi and Zhang describe the following model for the growth of surfaces
$$u_t-\Deltalta u=c\sqrt{1+|\nabla u|^2}+f, $$
where $f$ represents in general a stochastic process. After a Taylor expansion for small size of the gradient, they consider instead the so called KPZ equation,
$$u_t-\Deltalta u=c|\nabla u|^2+g.$$
It is important to say that, from a physical point of view, the KPZ equation is a relevant case of study because, among other things, it defines a new \textit{ universality class} for a lot of models in Statistical
Mechanics (see for instance {\cal C}_0^{\iy}\,te{barstan} and {\cal C}_0^{\iy}\,te{cor}).
In that sense, the behavior of the so called Hopf-Cole class of solutions has been deeply researched in the seminal paper by M. Hairer {\cal C}_0^{\iy}\,te{Hairer}.
\
We will restrict ourselves to the deterministic setting, that is, when the source term is a function with a suitable summability.
In the local critical case $\a=2$ with $s=1$ there is a large literature of results. For instance, in the paper {\cal C}_0^{\iy}\,te{ADP} (see also the references therein) a classification of the
solutions was found showing in particular an extreme case of non-uniqueness.
\
The relevant facts in the KPZ model are that the growth is driven in the direction of the gradient of the interface while the diffusion comes from the classical Laplacian
(remember that behind all this one finds always a Brownian motion).
\
There is another question to take into account, which is that the diffusion to consider may change according with the medium. For instance in the paper by
Barenblatt, Bertsch, Chertock, and Prostokishin, {\cal C}_0^{\iy}\,te{baren}, the growth in a porous medium was considered. This model has been studied for instance in the
papers, {\cal C}_0^{\iy}\,te{APW} and {\cal C}_0^{\iy}\,te{APW1}. The diffusion for a power law in the gradient (the p-Laplacian) has been also studied, see for instance {\cal C}_0^{\iy}\,te{ADPS},
{\cal C}_0^{\iy}\,te{BG} and the references therein.
\
The main goal of this work is to study a non local version of the Kardar-Parisi-Zhang equation. More precisely, the idea is to consider diffusion driven by the
\textit{fractional Laplacian} (so that behind it what one finds is a Levy process). More specifically, we deal with the problem
\begin{equation}\label{grad}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla u|^{\alpha}+ f & \mbox{ in } \Omegaega_T\equiv\Omegaega\times (0,T),\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
u(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
where $\Omegaega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2}<s<1$. We suppose that $f$ and $u_0$ are non negative functions
satisfying some hypotheses that we will precise later.
\
By $(-\Deltalta)^s$ we mean the fractional Laplacian operator of order $2s$ given as the multiplier of the Fourier transform with symbol $|\timesi|^{2s}$. That is, for every $u\in \mathscr{S}(\mathbb{R}^N)$, the Schwartz class, $$(-\Deltalta)^{s}u=\mathcal{F}^{-1}(|\timesi|^{2s}\mathcal{F}(u)),\, \timesi\in\mathbb{R}^{N}, s\in(0,1),$$
where $\mathcal{F}$ denotes Fourier transform and $\mathcal{F}^{-1}$ its inverse.
\
As was indicated by M. Riesz in his foundational paper {\cal C}_0^{\iy}\,te{Riesz}, since the formal homogeneous kernel corresponding to the multiplier $|\timesi|^{2s}$ is a constant multiple of
$|x|^{-N-2s}$, therefore not in $L^1_{loc}$, the definition cannot be a convolution but rather a principal value given by the following expression
\begin{equation}\label{fraccionario}
(-\Deltalta)^{s}u(x):=a_{N,s}\mbox{ P.V. }\int_{\mathbb{R}^{N}}{\frac{u(x)-u(y)}{|x-y|^{N+2s}}\, dy},\quad s\in(0,1),
\end{equation}
where
$$a_{N,s}:=2^{2s-1}\partiali^{-\frac N2}\frac{\Gamma(\frac{N+2s}{2})}{|\Gamma(-s)|}$$
is the normalization constant related to the definition through the Fourier transform. This formula is obtained by analytic continuation of the Riesz
potentials in the complex plane. See the details for instance in {\cal C}_0^{\iy}\,te{Landkof} and {\cal C}_0^{\iy}\,te{PSbook}.
The hypothesis $s>\frac 12$ is a natural assumption to allow the presence of a power of the gradient as a nonlinear perturbation.
The stationary problem associated to problem \eqref{grad} has recently been studied in {\cal C}_0^{\iy}\,te{CV2} and {\cal C}_0^{\iy}\,te{AP}.
\
The interest of the fractional Laplacian is motivated, in addition to the mathematical relevance, by the fact that it has recently been used in a number of equations
modeling concrete phenomena. Among others, we mention crystal dislocation {\cal C}_0^{\iy}\,te{DP_P_V_2015}, {\cal C}_0^{\iy}\,te{DP_F_V_2014}, mathematical finances {\cal C}_0^{\iy}\,te{Apple_2004} and
quantum mechanics, see {\cal C}_0^{\iy}\,te{L_2000}.
For the nonlocal case $s\in (\frac 12,1)$ and for regular data, the authors in {\cal C}_0^{\iy}\,te{W,W1} proved the existence of a regular solution using semi-group theory and probabilistic tools. More precisely, the authors in {\cal C}_0^{\iy}\,te{W} treat the case $f=0$ and $\O={I\!\!R}n$, under regular assumption on $u_0$, in order to get global estimates. In their approach, the fact that $\O={I\!\!R}n$ turns out to be a fundamental key, which is lost in the case of a bounded domain. As we will see later, the work on bounded domains, $\O$, generates a loss of regularity near the boundary and, as a consequence, non existence results holds for large values of $q$.
\
The main goal of this paper is to consider a general class of data. It is important to remark that monotony arguments have serious limitations in order to pass to
the limit in the approximating problems. To overcome these difficulties we will follow the arguments used for the elliptic case in {\cal C}_0^{\iy}\,te{AP}. In particular, we will use
apriori estimates and the Schauder fixed point theorem, which in the stationary case are inspired by results in {\cal C}_0^{\iy}\,te{CON2} and {\cal C}_0^{\iy}\,te{CON1} for local operators.
\
We briefly sketch now the main results in the paper. First,
via a fixed point argument we obtain the following results for general data.
\begin{Theorem}\label{maria}
Suppose in the problem \eqref{grad} that $\alpha<\dfrac{N+2s}{N+1}$, then { there exists $T:=T(\O,s)>0$ such that } for
all $(f, u_0)\in L^m(\O_T)\times L^1(\O)$ with $1\leqslantm<\dfrac{1}{s}$, problem
\eqref{grad} has a solution $u\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$ and $T_k(u)\in L^{2}(0,T;H^s_0(\Omegaega))$ for all $k>0$, { where $T_k(\s)$ is given by
\begin{gather*}\label{f-trun}
T_k(\s)=\left\{\begin{array}{cl}
\s\,,&\hbox{ if }|\s|\leqslantk\,;\\[2mm]
k\frac{\s}{|\s|}\,,&\hbox{ if }|\s|> k\,;
\end{array}\right.
\end{gather*}
}
\end{Theorem}
Notice that, even for $f\in L^m(\O_T)$ with $1<m<\dfrac{1}{s}$, the existence result holds with the same assumption on $\a$ as in $L^1$ data. This assumption does
not appear in the local case $s=1$ where the relation $\a\leftrightarrow m$ is strictly increasing. In the non local case, this limitation is due to the fact
that the global regularity for the gradient term imposes many restrictions on the parameters $s,m,\a$ and makes a fundamental difference between the local and the
nonlocal case.
In the case of $L^1$-data the above existence result is optimal, in the sense that if $\a>\frac{N+2s}{N+1}$, then we can find $f\in L^1(\O_T)$ or $u_0\in
L^1(\O)$ such that problem \eqref{grad} has no solution in the space $L^\a(0,T; W^{1,\a}_0(\O))$.
\
For large value of $\a$ a serious limitation appears as a consequence of the lack
of regularity for the linear problem near the boundary.
This loss of regularity allows us to get the following non existence
result which makes more significant the difference between the local and
the nonlocal case. However, this is coherent with the local case; indeed,
one sees in the threshold that $\dfrac 1{1-s}\to \infty$ as $s\to 1$.
\begin{Theorem}\label{nonint}
Suppose that $\a>\dfrac{1}{1-s}$, and let $(f,u_0)\in L^\infty(\O_T)\times L^\infty(\O)$ be nonnegative functions with $(f,u_0)\neq (0,0)$. Then, the problem
$$
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla u|^{\alpha}+ f & \mbox{ in } \Omegaega_T\equiv\Omegaega\times (0,T),\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
u(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
$$
has no solution $u$ such that $u\in L^\a(0,T;W^{1,\a}_0(\O))$.
\end{Theorem}
In the same direction we prove a general non existence result of weak solutions in a suitable weighted Sobolev space for a range of the parameter $\a$. This
gives a fundamental difference related to the local case, $s=1$, where existence of a solution is proved for all $\a>1$ under suitable regularity assumptions on
the data, see for example {\cal C}_0^{\iy}\,te{BS}.
\
For the existence result, we will distinguish two types of problems according to the integrability of the gradient term. In the first case we look for the global integrability of the gradient term in the whole domain $\Omegaega_T$.
\begin{Theorem}\label{hhh}
Assume that $\dfrac{2s-1}{1-s}>\dfrac{(N+2s)^2}{N+1}$ and that $\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{2s-1}{(1-s)(N+2s)}$.
Suppose that $u_0=0$, $f\in L^{m}(\Omegaega_T)$ with $m\geqslant\dfrac{1}{s}$ satisfies one of the following conditions:
\begin{enumerate}
\item[(I)] either $\dfrac{N+2s}{2s-1}\leqslantm$,
\item[(II)] or $\dfrac{N+2s}{\a'}\dfrac{1}{(2s-1)-(1-s)(N+2s)}<m<\dfrac{N+2s}{2s-1}$,
\end{enumerate}
then {there exists $T:=T(\O,s)>0$ such that} problem \eqref{grad} has a solution $u\in L^{\a}(0,T;W_{0}^{1,\a}(\Omegaega))$ and moreover $u\in L^{\g}(0,T;W_{0}^{1,\g}(\Omegaega))$ for all $\g<\dfrac{1}{1-s}$ if $(I)$ holds and $u\in L^{\g}(0,T;W_{0}^{1,\g}(\Omegaega))$ for all
$\g<\dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}$ if $(II)$ holds.
\end{Theorem}
Notice that the hypothesis on $s$ means that $s$ must be close to $1$ and $\alpha<<\dfrac{s}{1-s}$. The above conditions are needed to ensure the global integrability of the gradient term in the whole $\O_T$.
\noindent For the complete range of the parameter $\alpha$, that is $\dfrac{N+2s}{N+1}\leqslant\alpha<\dfrac{s}{1-s}$ and without any restriction in the order of the operator $s>\dfrac 12$, to have a weak solution to the problem requires a natural weight that in fact is a power of the distance to the boundary of $\Omegaega$. { For simplicity, throughout this paper we denote $\d(x):=\text{dist}(x,\partial\O)$, with $x\in \O$, such a distance.} Hence, the existence of a distributional solution in this case will be obtained in a weighted Sobolev space.
More precisely we have the following result.
\begin{Theorem}\label{fix001}
For every $s\in(\dfrac 12, 1)$, assume that $\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{s}{1-s}$. Let $f$ be a nonnegative function such that $f\in L^m(\O)$ with
$m>\max\bigg\{\dfrac{N+2s}{s(2s-1)}, \dfrac{N+2s}{s-\a(1-s)}\bigg\}$, then {{there exists $T:=T(\O,s)>0$ such that}} problem \eqref{grad} has
a distributional solution
$u\in L^{\a}(0,T;W^{1,\a}_{loc}(\O))\cap L^1(0,T;W^{1,1}_0(\O))$.
Moreover $u\d^{1-s}\in L^{m\a}(0,T;W^{1,m\a}_0(\O)).$
\end{Theorem}
\
In the common values of $\alpha$ in Theorem {I\!\!R}f{hhh} we require $s$ to be close to $1$. Also, the integrability of the datum, $m$, is bigger than in Theorem {I\!\!R}f{fix001}, where there is no restriction in $s$. The optimality of the results remain open.
\
If the source term { $f$ is null, then we can prove the existence of a solution using a suitable change of function and Theorem {I\!\!R}f{hhh}. More precisely we have }
\begin{Theorem}\label{hhh2}
In the problem \eqref{grad}, let us consider $f=0$. Assume that $\dfrac{2s-1}{1-s}>\dfrac{(N+2s)^2}{N}$ and that $\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{2s}{(1-s)(N+2s)+1}$.
Let $u_0$ be a nonnegative measurable function such that $u_0\in L^\s(\O)$ with $\s>\dfrac{(\a-1)N}{(2s-\a)-\a(1-s)(N+2s)}$, then {there exists $T:=T(\O,s)>0$ such that} the problem
\begin{equation}\label{gradu0}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla u|^{\alpha} &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& u_0(x) & \mbox{ in }\Omegaega,
\end{array}\right.
\end{equation}
has a solution $u\in L^{\g}(0,T;W_{0}^{1,\g}(\Omegaega))$, for all $\g< \dfrac{\s(N+2s)}{(1-s)\s(N+2s)+N+\s}$.
\end{Theorem}
\
A similar result with weights could be obtained by application of Theorem {I\!\!R}f{fix001}. This problem will be studied in a forthcoming paper, in which we will also look for the asymptotic behavior of the solutions with respect to $t$.
\
As a direct application of the arguments developed, we will treat the problem with \textit{drift}, that is, the nonlinearity on the gradient is substituted by a term of the form $B(x,t)\,\cdot\, \nabla u$.
The existence of a solution is known in the literature under
regularity conditions on the data $f$ and $u_0$.
Here we will prove the existence of a solution under natural
condition on the field $B$ and general data $(f,u_0)\in L^1(\O_T)\times L^1(\O)$. Using a suitable Harnack inequality we are able to prove a
comparison principle and then the uniqueness of the solution follows
for the linear problem with drift. This will be the key in order to show
the uniqueness of a \emph{good solution} to problem \eqref{grad}.
\
{
\begin{remark}
Notice that the above existence results also hold true without any additional assumption on $T$ if we alternatively assume that the corresponding norm of the data are small. This can be seen if we substitute the data $f,u_0$ with $\lambda f,\lambda u_0$ with $|\l|$ small.
However in the case of Theorem {I\!\!R}f{maria}, using the uniqueness result proved in Theorem {I\!\!R}f{uniqapr}, we are able to show the existence of a minimal solution (and in some cases a unique solution) without any rescription on the norm of the data. See Theorem {I\!\!R}f{uniqq} below.
\end{remark}
}
The paper is organized as follows.
In Section {I\!\!R}f{sec2} we collect some tools that will be used systematically in the paper. We begin by specifying the sense in which the solutions are
understood, and we state some tools as the Kato's inequality and the gradient regularity for an associated elliptic problem.
In order to prove gradient regularity for the solution based on the representation formula, we need to show gradient regularity for the fractional heat kernel. This is done in Section {I\!\!R}f{sec3} where we also consider the general linear fractional heat equation with $L^1$ data. In this case we are able to show the uniqueness of the solution and to prove the strong convergence of the solution of the approximating problems in a suitable Sobolev space without using the Landes regularizing approximation. General regularity results of the gradient and the Hardy-Sobolev term $\dfrac{u}{\deltalta^s}$ are obtained, in particular, close to $\partialartial \Omegaega\times(0,T)$.
This will be useful in order to complete the regularity schema of this class of equations, as is done in {\cal C}_0^{\iy}\,te{AP} for the elliptic equation.
As a consequence of the results in Section 3, Section 4 is devoted to prove the surprising nonexistence result. We consider the Problem \eqref{grad} in Section {I\!\!R}f{sec4}. We begin with the case of $L^1$ data and for all $\a<\a_0=\dfrac{N+2s}{N+1}$ and prove the existence of a weak solution with suitable regularity. We also
prove that the condition on $\a$ is optimal in the sense that for $\a>\a_0$, then there exists $f\in L^1(\O_T)$ such that problem \eqref{grad} has no solution.
Problem \eqref{grad} with general $\alpha<\dfrac{s}{1-s}$ is treated in Subsection {I\!\!R}f{sub:sec41}. Under suitable hypotheses on $f$, we are able to show the
existence of a weak solution. We also treat the case where $f\equiv 0$ and $u_0\gneqq 0$. Following closely the ideas as in the case $f\gneqq 0$, we
prove the existence of a solution that lives in a suitable Sobolev space.
In the last section we consider the linear problem with \textit{drift}. The existence of a weak solution is proved for all data in $L^1$.
According to additional hypotheses on the \textit{drift} term, we are able to show the uniqueness of the weak solution. As a consequence we prove a general comparison principle, using a suitable singular Gronwall-Bellman inequality, that allows us to show the existence of a minimal solution to problem \eqref{grad} under suitable hypothesis on $\alpha$.
{ We thank A. Younes for pointing us some misprints in an earlier version of this work.}
\section{Preliminaries and functional setting}\label{sec2}
Let us begin by some results from fractional Sobolev spaces that will be used in this paper. We refer to {\cal C}_0^{\iy}\,te{dine} for more details and proofs.
Assume that $s\in (0,1)$ and $p>1$. For a measurable $\O\subset {I\!\!R}n$, the fractional Sobolev Space $W^{s,p}(\Omegaega)$ is defined by
$$
W^{s,p}(\Omegaega)\equiv \Big\{ \partialhi\in L^p(\O):\displaystyle\int_{\O}\displaystyle\int_{\O}|\partialhi(x)-\partialhi(y)|^pd\nu<+\infty\Big\},
$$
where, for simplicity, we set
\begin{equation}\label{thekernel}
d\nu=\displaystyle\frac{dxdy}{|x-y|^{N+ps}}.
\end{equation}
Notice that $W^{s,p}(\O)$ is a Banach Space endowed with the norm
$$
\|\partialhi\|_{W^{s,p}(\O)}= \Big(\displaystyle\int_{\O}|\partialhi(x)|^pdx\Big)^{\frac 1p} +\Big(\displaystyle\int_{\O}\displaystyle\int_{\O}|\partialhi(x)-\partialhi(y)|^pd\nu\Big)^{\frac 1p}.
$$
The space $W^{s,p}_{0} (\O)$ is defined as the completion of $\mathcal{C}^\infty_0(\O)$ with respect to the previous norm.
If $\O$ is a bounded regular domain, we can endow $W^{s,p}_{0}(\O)$ with the equivalent norm
$$
||\partialhi||_{W^{s,p}_{0}(\O)}= \Big(\int_{\O}\displaystyle\int_{\O}|\partialhi(x)-\partialhi(y)|^pd\nu\Big)^{\frac 1p}.
$$
The next Sobolev inequality is proved in {\cal C}_0^{\iy}\,te{Adams}, see also {\cal C}_0^{\iy}\,te{dine} and {\cal C}_0^{\iy}\,te{Ponce} for an elementary proof.
\begin{Theorem} \label{Sobolev}(Fractional Sobolev inequality)
Assume that $0<s<1, p>1$ satisfy $ps<N$. There exists a positive constant $S\equiv S(N,s,p)$ such that for all $v\in C_{0}^{\infty}({I\!\!R}n)$,
$$
\iint_{{I\!\!R}^{2N}} \dfrac{|v(x)-v(y)|^{p}}{|x-y|^{N+ps}}\,dxdy\geqslantS \Big(\displaystyle\int_{\mathbb{R}^{N}}|v(x)|^{p_{s}^{*}}dx\Big)^{\frac{p}{p^{*}_{s}}},
$$
where $p^{*}_{s}= \dfrac{pN}{N-ps}$.
\end{Theorem}
We will denote by $H^s({I\!\!R}n)$ the Hilbert space $W^{s,2}({I\!\!R}n)$. If $u\in H^s({I\!\!R}n)$, we define
$$
(-\Deltalta)^s u(x)={ P.V. } \displaystyle\int_{{I\!\!R}n}\dfrac{u(x)-u(y)}{|x-y|^{N+2s}}{dy}.
$$
For $w, v\in H^s({I\!\!R}n)$, we have
$$
\langle (-\Deltalta)^s w,v\rangle =\dfrac 12\iint_{{I\!\!R}^{2N}}\dfrac{(w(x)-w(y))(v(x)-v(y))}{|x-y|^{N+2s}}{dxdy}.
$$
If $H^s_0(\O)$ is the closure of $\mathcal{C}_0^\infty(\Omegaega)$ with respect to the norm $H^s({I\!\!R}n)$ and if $w, v\in H^s_0(\O)$, then
$$
\langle (-\Deltalta)^s w,v\rangle =\dfrac 12\iint_{D_\O}\dfrac{(w(x)-w(y))(v(x)-v(y))}{|x-y|^{N+2s}}{dxdy},
$$
where $D_{\O}=({I\!\!R}n\times {I\!\!R}n)\setminus (\mathcal{C}\O\times \mathcal{C}\O)$.
Since we are considering parabolic problems, we need to define the corresponding parabolic spaces.
For $q\geqslant1$, the space $L^{q}(0,T; W^{s,q}_0(\O))$ is defined as the set of functions $\partialhi$ such that
$\partialhi\in L^q(\O_T)$ with $||\partialhi||_{L^{q}(0,T; W^{s,q}_0(\O))}<\infty$ where
$$
||\partialhi||_{L^{q}(0,T; W^{s,q}_0(\O))}=\Big(\int_0^T\iint_{D_{\O}}|\partialhi(x,t)-\partialhi(y,t)|^qd\nu\,dt\Big)^{\frac 1q}.
$$
It is clear that $L^{q}(0,T; W^{s,q}_0(\O))$ is a Banach Space.
Consider now the linear problem
\begin{equation}\label{eq:def-0}
\left\{
\begin{array}{rcll}
u_t+(-\D)^s u&=& \displaystyle f & \text{ in } \O_{T}=\Omegaega \times (0,T) , \\ u&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\ u(x,0)&=&u_0(x)& \mbox{ in }\O,
\end{array}
\right.
\end{equation}
where $\O\subset {I\!\!R}n$ is a bounded regular domain. If the data $(f,u_0)\in L^2(\O_T)\times L^2(\O)$, then we can deal with energy solution. More precisely we
have the next definition.
\begin{Definition}\label{energy}
Assume $(f,u_0)\in L^2(\O_T)\times L^2(\O)$, then we say that $u$ is an energy solution to problem \eqref{eq:def-0} if $u\in L^{2}(0,T; H^s_0(\O))\cap
\mathcal{C}([0,T], L^2(\O))$, $u_t\in L^{2}(0,T; H^{-s}(\O))$, and for all $v\in L^{2}(0,T; H^s_0(\O))$ we have
\begin{equation*}
\begin{array}{lll}
&\displaystyle\int_0^T\langle u_t, v\rangle dt +\dfrac 12\int_0^T\iint_{D_{\O}}(u(x,t)-u(y,t))(v(x,t)-v(y,t))d\nu\ dt\\ &\displaystyle=\iint_{\O_T} fv dx\,dt
\end{array}
\end{equation*}
and $u(x,.)\to u_0$ strongly in $L^2(\O)$, as $t\to 0$.
\end{Definition}
Notice that the existence of energy solution follows using classical arguments, see for instance {\cal C}_0^{\iy}\,te{LPPS}.
If the datum lies in $L^1$, we need to define a more general concept of solution. Let us begin by the next definitions.
Assume that $\alpha,\beta\in (0,1)$, we define the set
\begin{equation*}\begin{split}
\mathcal{T}:=\{&\partialhi:\mathbb{R}^N\times [0,T]\rightarrow\mathbb{R},\,\hbox{ s.t. }-\partialhi_t+(-\Deltalta)^s\partialhi=\varphi,\, \varphi\in L^\infty(\Omegaega\times (0,T))\cap
\mathcal{C}^{\alpha, \beta}(\Omegaega\times (0,T)),\\ &\partialhi=0\mbox{ in } ({I\!\!R}n\setminus \Omegaega)\times {(0,T]},\partialhi(x,T)=0 \mbox{ in } \Omegaega \}.
\end{split}\end{equation*}
From {\cal C}_0^{\iy}\,te{LPPS}, we know that if $\partialhi\in \mathcal{T}$, then $\partialhi\in L^\infty(\Omegaega\times (0,T))$ and $\partialhi\in \mathcal{T}$ satisfies the equation in a
pointwise sense.
We are now able to state the meaning of weak solution.
\begin{Definition}\label{veryweak} Assume that $(f,u_{0})\in L^1(\O_T)\times L^{1}(\Omegaega)$.
We say that $u\in \mathcal{C}([0,T); {L}^{1}(\O))$, is a weak solution to problem \eqref{eq:def-0} if for all $\partialhi\in \mathcal{T}$ we have
\begin{equation}\label{eq:subsuper}
\iint_{\O_T}\,u\big(-\partialhi_t\, +(-\Deltalta)^{s}\partialhi\big)\,dx\,dt=\\ \iint_{\O_T}\,f\partialhi\,dxdt +\int_\Omegaega{u_0(x)\partialhi(x,0)\,dx}.
\end{equation}
\end{Definition}
The next existence result is proved in {\cal C}_0^{\iy}\,te{LPPS}, ({ see also {\cal C}_0^{\iy}\,te{AABP} and {\cal C}_0^{\iy}\,te{BWZ} for some different approaches.})
\begin{Theorem}\label{th1}
Suppose that $(f,u_0)\in L^1(\O_T)\times L^1(\O)$, then problem \eqref{eq:def-0} has a unique weak solution $u$ such that $u\in \mathcal{C}([0,T];L^1(\O))\cap
L^m(\O_T)$ for all $m\in [1, \frac{N+2s}{N})$, \;$|(-\D)^{\frac{s}{2}} u|\in L^{r}(\O_T)$ for all $r\in[1, \frac{N+2s}{N+s})$ and $T_k(u)\in L^2(0,T,H^s_{
0}(\Omegaega))$ for all $k>0$ where $T_k(\s)=\max \{-k, \min\{k,\s\}\}$. Moreover $u\in {L^q(0,T,W^{s,q}_{ 0}(\O))}$ for all $1\leqslantq<\frac{N+2s}{N+s}$. In addition
we have
\begin{equation}\label{main-estim}
\begin{array}{lll}
&\displaystyle ||u||_{\mathcal{C}([0,T],L^1(\O))}+ ||u||_{L^m(\O_T)}+||(-\D)^{\frac{s}{2}} u||_{L^{r}(\O_T)}+ ||u||_{L^q(0,T,W^{s,q}_{ 0}(\O))}\\ &
\displaystyle \le
C(\O_T)\bigg(||f||_{L^1(\O_T)}+||u_0||_{L^1(\O)}\bigg).
\end{array}
\end{equation}
\end{Theorem}
\begin{remark}\label{optimal}
The regularity condition obtained in Theorem {I\!\!R}f{th1} is optimal in the sense that if $m\geqslant\dfrac{N+2s}{N}$, then we can find $f\in L^1(\O_T)$ and $u_0\in
L^1(\O)$ such that $u^m\notin L^1(\O_T)$. This fact will be used in Theorem {I\!\!R}f{maria} in order to show the optimality of the condition imposed on $\a$.
\end{remark}
{In the case of having data in $L^1_{loc}$ the natural concept is
the usual \textit{distributional solution}, that is, given by the following
definition.
\begin{Definition}\label{distribu}
Assume that $(f,u_0)\in L^1_{loc}(\O_T)\times L^1_{loc}(\O)$. We say that $u\in L^1(\O_T)\cap\,\mathcal{C}([0,T], L^1_{loc}(\O))$ is a distributional solution to Problem \eqref{eq:def-0} if for all $\varphi \in \mathcal{C}^\infty_0(\O_T)$, for all $\eta\in \mathcal{C}^\infty_0(\O)$, we have
$$
\iint_{\O_T} u(-\varphi_t+(-\Deltalta)^s\varphi)\,dx\,dt=\iint_{\O_T}f(x,t)\varphi(x,t)\,dx\,dt, $$
and
$$
\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\,\eta(x)dx\to \int_{\O}u_0(x)\,\eta(x)\,dx \mbox{ as }t\to 0.
$$
\end{Definition}
}
\
\
Finally, the next Kato type inequality will be useful in order to show apriori estimates and the positivity of the solution. { The proof follows exactly as in the elliptic case proved in {\cal C}_0^{\iy}\,te{CV1}. { (See also {\cal C}_0^{\iy}\,te{PSbook}).}
\begin{Theorem}\label{Conv}
Let $\partialhi\in \mathcal{C}^{2}({I\!\!R})$ be a convex function. Assume $u\in L^{2}(0,T; H^s_0(\O))\,\cap\, \mathcal{C}([0,T], L^2(\O))$. Define $v=\partialhi(u)$ and suppose
that $|v_t+(-\D)^s v|\in L^1(\O_T)$, then
\begin{equation}\label{ka}
v_t+(-\D)^s v\leqslant\partialhi'(u) (u_t+(-\D)^s (u)).
\end{equation}
\end{Theorem}
\section{Gradient regularity of the solutions to the linear problem with $\frac 12<s<1$.}\label{sec3}
In this section the principal goal is to study the gradient regularity of the solution to the linear problem
\begin{equation}\label{eq:def}
\left\{
\begin{array}{rcll}
u_t+(-\D)^s u&=& \displaystyle f & \text{ in } \O_{T}=\Omegaega \times (0,T) , \\ u&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\ u(x,0)&=&u_0(x)& \mbox{ in }\O,
\end{array}
\right.
\end{equation}
where $\O\subset {I\!\!R}n$ is a bounded regular domain and assuming that $s\in (\frac 12,1)$.
We emphasize that the hypothesis $\frac 12< s<1$ is assumed across the whole section.
Under this hypothesis the heat kernel lies in the space $L^q((0,T), W^{1,q}_0(\O))$ for suitable $q$ as we will see later. This regularity is the starting point to study the problem \eqref{eq:def}.
Denote by $P_{\Omegaega}$ the kernel of the heat equation $\dfrac{d}{dt}+(-\Deltalta)^s$ in $\Omegaega\times(0,T)$ with Dirichlet boundary condition. The solution, $u$, to problem \eqref{eq:def} is represented by
$$
u(x,t)=\displaystyle\int_{\Omegaega} u_0(y) P_{\Omegaega} (x,y, t)\,dy+ \iint_{\O_t}f(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma,
$$
where $\O_t=\O\times (0,t)$.
As in the elliptic case, in order to get apriori estimates on the gradient of $u$, we need some sharp properties of the heat kernel $P_{\Omegaega}$. These properties
are resumed in the next lemma whose proof can be found in {\cal C}_0^{\iy}\,te{BJ}, {\cal C}_0^{\iy}\,te{CZ} and {\cal C}_0^{\iy}\,te{KS}.
\begin{Lemma}\label{estimmm}
Assume that $s\in (\frac 12,1)$, then for all $x,y\in \O$ and for all $0<t<T$,
\begin{equation}\label{green1}
P_{\Omegaega}(x,y,t)\backsimeq \Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{t}}\Big)\times \Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t}}\Big)\times
\Big(t^{-\frac{N}{2s}}\wedge \frac{t}{|x-y|^{N+2s}}\Big)
\end{equation}
and
\begin{equation}\label{green2}
|\nabla_x P_{\Omegaega}(x,y,t)|\leqslantC \Big( \dfrac{1}{\deltalta(x) \wedge t^{\frac{1}{2s}}}\Big)\, P_{\Omegaega}(x,y,t).
\end{equation}
Setting
\begin{equation}\label{green0}
\displaystyle\int_{0}^{\infty} P_{\Omegaega} (x,y,
t)\,dt=\mathcal{G}_s(x,y)
\end{equation}
where $\mathcal{G}_s(x,y)$ is the Green function of the fractional Laplacian operator with Dirichlet condition, that is,
\begin{equation}\label{green00}
\mathcal{G}_s(x,y)\simeq \frac{1}{|x-y|^{N-2s}}\bigg(\frac{\d^s(x)}{|x-y|^{s}}\wedge 1\bigg) \bigg(\frac{\d^s(y)}{|x-y|^{s}}\wedge 1\bigg).
\end{equation}
\end{Lemma}
\begin{remark} Notice that
$$ P_{\Omegaega} (x,y,t)\leqslantt^{-\frac{N}{2s}}\wedge \frac{t}{|x-y|^{N+2s}},$$
therefore
\begin{equation}\label{upper-kernel}
P_{\Omegaega} (x,y,t)\leqslant\dfrac{2t}{t^\frac{N+2s}{2s}+|x-y|^{N+2s}}\leqslantC(s,N)\dfrac{t}{(t^\frac 1{2s}+|x-y|)^{N+2s}}.
\end{equation}
Hereafter we will call
\begin{equation}\label{upper-kernel11}
H(x,t):=\dfrac{t}{(t^\frac 1{2s}+|x|)^{N+2s}}.
\end{equation}
\end{remark}
We begin by proving the following basic result.
\begin{Proposition}\label{propo1}
Assume that $s\in (\frac 12,1)$, then for all $q<\dfrac{N+2s}{N+1}$, we have
$$
\displaystyle\int_{0}^{T} \displaystyle\int_{\Omegaega} \displaystyle\int_{\Omegaega}|\nabla_x P_{\Omegaega}|^q\, dx\,dy\,dt\leqslantC(\O)(T^{\frac{N+2s-q(N+s)}{2s}}+T^{\frac{N+2s-q(N+1)}{2s}}).
$$
\end{Proposition}
\begin{proof}
From \eqref{green1} and \eqref{green2} we obtain that $$ |\nabla_x P_{\Omegaega}(x,y,t)|\leqslantC \Big( \dfrac{1}{\deltalta(x) \wedge t^{\frac{1}{2s}}}\Big)\,\Big( 1\wedge
\frac{\deltalta^s(x)}{\sqrt{t}}\Big)\times \Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t}}\Big)\times \dfrac{t}{\bigg(t^{\frac{1}{2s}}+|x-y|\bigg)^{N+2s}}.
$$
Hence, according with \eqref{upper-kernel},
\begin{equation}\label{rrr}
|\nabla_x P_{\Omegaega} (x,y,t)|\leqslant\left\{\begin{array}{rcl}
\Big( 1\wedge \dfrac{\deltalta^s(y)}{\sqrt{t}}\Big)\cdot \dfrac{\sqrt{t}}{(\deltalta(x))^{1-s}} \cdot \dfrac{C}{(t^{\frac{1}{2s}} +|x-y|)^{(N+2s)}}, \mbox{ if }
\deltalta(x)<t^{\frac{1}{2s} },\\
\\
C \Big( 1\wedge \dfrac{\deltalta^s(y)}{\sqrt{t}}\Big)\cdot \dfrac{t^{\frac{2s-1}{2s}}}{(t^{\frac{1}{2s}}+ |x-y|)^{N+2s}}, \mbox{ if } \deltalta(x)\geqslantt^{\frac{1}{2s}
}.
\end{array}
\right.
\end{equation}
Thus
\begin{eqnarray*}
&\displaystyle \!\!\!\! \iiint_{\O_T\times \O}|\nabla_x P_{\Omegaega}|^q\, dx\,dy\,dt\leqslant\\ \\ &\displaystyle \iiint_{\{(0,T)\times \O\times \O\}\cap
\{\deltalta(x)<t^{\frac{1}{2s}}\}}|\nabla_x P_{\Omegaega}|^q\, dx\,dy\,dt+\!\! \iiint_{\{(0,T)\times \O\times \O\}\cap \{\deltalta(x)\geqslantt^{\frac{1}{2s}}\}}|\nabla_x
P_{\Omegaega}|^q\, dx\,dy\,dt:= I_1+I_2.
\end{eqnarray*}
Using \eqref{rrr}, it holds that
$$
\begin{array}{rcl}
I_1 &\leq& \displaystyle\int_{\Omegaega}\dfrac {dx}{(\deltalta (x))^{(1-s)q}}\Bigg(\iint_{\O_T}\dfrac{t^{\frac{q}{2}}\,dydt} {(t^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}
\Bigg)\\
\\
&\leq& \displaystyle\int_{\Omegaega}\dfrac {dx}{(\deltalta (x))^{(1-s)q}} \Bigg(\displaystyle\int_{0}^T \displaystyle\int_{{ {I\!\!R}n}} \dfrac{t^{\frac{q}{2}- \frac{1}{2s} q(N+2s)}\,dydt}
{\Big(1+\frac{|x-y|}{t^{\frac{1}{2s}}}\Big)^{q(N+2s)}} \Bigg)\\
\\
&=& C\displaystyle\int_{\Omegaega}\dfrac {dx}{(\deltalta (x))^{(1-s)q}}\cdot \displaystyle\int_{0}^{T} t^{\frac{N}{2s}} t^{\frac{qs-q(N+2s)}{2s}}\,dt\cdot \displaystyle\int_{0}^{\infty}
\dfrac{\theta^{N-1}}{(1+\theta) ^{(N+2s)q}}\,d\theta,
\end{array}
$$
with $\theta=\dfrac{|x-y|}{t^{\frac{1}{2s}}}$.
Since $q<\dfrac{N+2s}{N+s}<\dfrac{1}{1-s}$, then $(1-s)q<1$ and $\dfrac{N}{2s}+\dfrac{qs-q(N+2s)}{2s}>-1$. Hence
$$
I_1\leqslantC(\O)T^{\frac{N+2s-q(N+s)}{2s}}.
$$
Respect to $I_2$, we have
$$
\begin{array}{rcl}
I_2 &\leq& C \displaystyle\int_{0}^{T} \displaystyle\int_{\Omegaega} \displaystyle\int_{\Omegaega} \dfrac{t^{q\frac{2s-1}{2s}}}{(t^{\frac{1}{2s}}+ |x-y|)^{(N+2s)q}}\,dxdy dt\\ &&\\ &=& C \Big(
\displaystyle\int_{0}^{T} t^{\frac{q(2s-1)+N-(N+2s)q}{2s}}\,dt\Big) \displaystyle\int_{0}^{\infty} \dfrac{\theta^{N-1}}{(1+\theta)^{q(N+2s)}}\,d\theta.
\end{array}
$$
Since $q<\frac{N+2s}{N+1}$, then $\frac{q(2s-1)+N-(N+2s)q}{2s}> -1$. Hence
$$
I_2\leqslantC(\O)T^{\frac{N+2s-q(N+1)}{2s}}.
$$
Combining the above estimates on $I_1$ and $I_2$, the result follows.
\end{proof}
We start with the following elementary result about the linear problem \eqref{eq:def} without source term.
\begin{Proposition}\label{first11}
Suppose that $f\equiv 0$ and $u_0\in L^\rho(\O)$. If $u$ is the unique weak solution to problem \eqref{eq:def}, then for all $r\geqslant\rho$ and for all $t>0$, we
have
\begin{equation}\label{sem100}
||u(\cdot,t)||_{L^r(\O)}\leqslantC(\O)t^{-\frac{N}{2s}(\frac{1}{\rho}-\frac{1}{r})}||u_0||_{L^\rho(\O)}.
\end{equation}
\end{Proposition}
\begin{proof}
We have the representation formula for the solution,
$$
u(x,t)=\displaystyle\int_{\Omegaega}u_0(y) P_{\Omegaega} (x,y, t)\,dy.
$$
From Theorem {I\!\!R}f{th1} we obtain that $u\in L^m(\O_T)$ for all $m<\frac{N+2s}{N}$.
To prove the estimate \eqref{sem100} we take advantage of the linearity of the problem and use a duality argument.
Let $\partialhi\in \mathcal{C}^\infty_0(\O)$, then
$$
||u(.,t)||_{L^r(\O)} =\displaystyle \sup_{\{||\partialhi||_{L^{r'}(\O)}\leqslant1\}}\int\lim\limit_{n\to\infty}tss_\Omega \partialhi(x) u(x,t)dx= \displaystyle
\sup_{\{||\partialhi||_{L^{r'}(\O_T)}\leqslant1\}}\int\lim\limit_{n\to\infty}tss_\Omega \partialhi(x) \displaystyle\int_{\Omegaega}u_0(y) P_{\Omegaega} (x,y, t)\,dy \,dx.
$$
Using estimate \eqref{green1}, it holds that
$$
\begin{array}{rcl}
||u(.,t)||_{L^r(\O)} &\le& \displaystyle \sup_{\{||\partialhi||_{L^{r'}(\O)}\leqslant1\}}\iint_{\O\times \O}
|u_0(y)||\partialhi(x)|\dfrac{t}{(t^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,dx\\ &= & \displaystyle \sup_{\{||\partialhi||_{L^{r'}(\O)}\leqslant1\}} \iint_{\O\times \O} |\partialhi(x)||u_0(y)|
H(x-y,t)dydx
\end{array}
$$
with $H(x,\s)$ defined in \eqref{upper-kernel11}. Using Young inequality, we get
$$
||u(.,t)||_{L^r(\O)}\leqslantC\displaystyle \sup_{\{||\partialhi||_{L^{r'}(\O)}\leqslant1\}}||\partialhi||_{L^{r'}(\O)} ||u_0||_{L^\rho(\O)}||H(., t)||_{L^a(\O)}
$$
with $\frac{1}{r'}+\frac{1}{\rho}+\frac{1}{a}=2$. By a direct computations we reach that
$$
||H(.,t)||^a_{L^a(\O)}\leqslant||H(.,t)||^a_{L^a({I\!\!R}n)}= t^{a-\frac{a(N+2s)}{2s}}\displaystyle \displaystyle\int_{{I\!\!R}n}
\dfrac{1}{\Big(1+\dfrac{|x|}{t^{\frac{1}{2s}}}\Big)^{a(N+2s)}}\,dx.
$$
Setting $z=\dfrac{x}{t^{\frac{1}{2s}}}$, then $||H(.,t)||^a_{L^a(\O)}\leqslantC t^{a+\frac{N}{2s}-\frac{a(N+2s)}{2s}}.$
Thus
\begin{equation}\label{L-a-H}
||H(., t)||_{L^a(\O)}\leqslantC(\O) t^{\frac{N}{2s}(\frac{1}{a}-1)}=C t^{\frac{N}{2s}(\frac{1}{r}-\frac{1}{\rho})}.
\end{equation}
Hence
$$
||u(.,t)||_{L^r(\O)} \leqslantC(\O) t^{\frac{N}{2s}(\frac{1}{r}-\frac{1}{\rho})}||u_0||_{L^\rho(\O)}.
$$
\end{proof}
Next we state the following compactness result that could be seen as the parabolic extension of the result in {\cal C}_0^{\iy}\,te{CV2} and which have an interest in itself.
\begin{Theorem} \label{gradiente}
Assume that $(f,u_0)\in L^1(\O_T)\times L^{1}(\Omegaega)$. Let $u$ be the unique solution to problem \eqref{eq:def}, then for all $q<\frac{N+2s}{N+1}$,
$$
||u||_{\mathcal{C}([0,T],L^1(\O))}+ ||\nabla u||_{L^{q}(\Omegaega_T)}\leqslantC(q,\Omegaega_T)\bigg(||f||_{L^{1}(\Omegaega_T)}+||u_0||_{L^1(\O)}\bigg).
$$
Moreover, for $q<\frac{N+2s}{N+1}$ fixed, setting $\hat{K}: L^{1}(\Omegaega_T)\times L^1(\O)\to L^q(0,T; W_{0}^{1,q}(\Omegaega))$, $\hat{K}(f, u_0)=u$, the unique
solution to problem \eqref{eq:def}, then $\hat{K}$ is a compact operator.
\end{Theorem}
\begin{proof}
Without loss of generality we can assume that the data $u_0,f$ are nonnegative, since thanks to the linearity of the operator the general case can be obtained by decomposing the datum into its positive
and negative parts and then dealing with two data separately.
Since $(u_0,f)\in L^1(\O)\times L^{1}(\Omegaega_T)$, then $u\in L^{m}(\Omegaega_T)$ for all $m<\frac{N+2s}{N}$ (see {\cal C}_0^{\iy}\,te{LPPS}). From the representation formula, setting $\O_t=\O\times (0,t)$ with $t<T$, we get
$$
u(x,t)=\displaystyle\int_{\Omegaega}u_0(y) P_{\Omegaega} (x,y, t)\,dy\,+ \displaystyle \iint_{\O_t} f(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma.
$$
Hence
$$
\begin{array}{rcl}
|\nabla u(x,t)|^q&\leq&
C(\O_T) \Big(\displaystyle\int_{\Omegaega} u_0(y)|\nabla_x P_{\Omegaega} (x,y, t)|\,dy\,+
\iint_{\O_t} f(y,\sigma) |\nabla_x P_{\Omegaega} (x,y,t-\sigma)|\,dy\,d\sigma\Big)^q\\
\\ &\leq& C(\O_T) \Big(\displaystyle\int_{\Omegaega} u_0(y)\frac{|\nabla_x P_{\Omegaega} (x,y, t)|}{P_{\Omegaega} (x,y, t)}\,P_{\Omegaega} (x,y, t) dy\,\\
\\
&+ & \displaystyle \iint_{\O_t} f(y,\sigma) \frac{|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|}{P_{\Omegaega} (x,y, t-\sigma)}P_{\Omegaega} (x,y,
t-\sigma)\,dy\,d\sigma\Big)^q\\ \\
&\leq& C(\O_T) \Big(\displaystyle\int_{\Omegaega} u_0(y) h(x,y,t){P_{\Omegaega} (x,y, t)}\,dy\Big)^q\\
\\
&+ & \displaystyle C(\O_T)\Big(\iint_{\O_t} f(y,\sigma) h(x,y,t-\s){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q \\ \\ &:=& J_1(x,t)+J_2(x,t)
\end{array}
$$
with $h(x,y,t)=\dfrac{|\nabla_x P_{\Omegaega} (x,y, t)|}{P_{\Omegaega} (x,y, t)}\leqslantC \Big( \dfrac{1}{\deltalta(x) \wedge t^{\frac{1}{2s}}}\Big)$.\\
To estimate $J_1$ we decompose the integral as follows
\begin{eqnarray*}
J_1(x,t) & = & C \Big(\displaystyle\int_{\Omegaega} u_0(y) h(x,y,t){P_{\Omegaega} (x,y, t)}\, dy\Big)^q\\ \\
& \leqslant& C \Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)>t^{\frac{1}{2s}}\}\}} u_0(y) h(x,y,t){P_{\Omegaega} (x,y, t)}\,dy\Big)^q \\ & + &
\Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y) h(x,y,t){P_{\Omegaega} (x,y, t)}\, dy\Big)^q\\
&\leqslant& \frac{C}{t^{\frac{q}{2s}}}\Big(\displaystyle\int\lim\limit_{n\to\infty}tss_{\{\Omegaega\cap \{\d(x)>t^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\Big)^q +
\frac{C}{\d^q(x)}\Big(\displaystyle\int\lim\limit_{n\to\infty}tss_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\Big)^q\\
& = & J_{11}(x,t)+ J_{12}(x,t).
\end{eqnarray*}
By estimate \eqref{green1} and H\"older inequality, we obtain that
\begin{eqnarray*}
J_{11}(x,t) & = & \frac{C}{t^{\frac{q}{2s}}}\Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)>t^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\Big)^q \\
& \leqslant& C\Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)>t^{\frac{1}{2s}}\}\}} u_0(y)\frac{t^{1-\frac{1}{2s}}}{(t^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\Big)^q\\
&\leqslant& C||u_0||^{q-1}_{L^1(\O)}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\frac{t^{q(1-\frac{1}{2s})}}{(t^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}\,dy.
\end{eqnarray*}
Thus
$$
\iint_{\O_T}J_{11}(x,t) dxdt \leqslantC||u_0||^{q-1}_{L^1(\O)}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\bigg(\iint_{\O_T}\frac{t^{q(1-\frac{1}{2s})}}{(t^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}\,dx\,dt\bigg)\,dy.
$$
Setting, $z=\dfrac{x-y}{t^{\frac{1}{2s}}}$, then
\begin{eqnarray*}
\displaystyle \iint_{\O_T}J_{11}(x,t) dxdt & \leqslant& C||u_0||^{q}_{L^1(\O)}\int_0^T t^{q(1-\frac{1}{2s})-\frac{q(N+2s)}{2s}+\frac{N}{2s}} \int_{{I\!\!R}n}\frac{dz}{(1+|z|)^{q(N+2s)}}dz\,dt\\
&\leqslant& C||u_0||^{q}_{L^1(\O)}\int_0^T t^{\g_1} dt
\end{eqnarray*}
where $\g_1=q(1-\frac{1}{2s})-\frac{q(N+2s)}{2s}+\frac{N}{2s}=\frac{N}{2s}-q\frac{N+1}{2s}$. Since $q<\frac{N+2s}{N+1}$, then $\g_1>-1$. Thus
$$
\displaystyle \iint_{\O_T}J_{11}(x,t) dxdt\leqslantCT^{\g_1+1}||u_0||^{q}_{L^1(\O)}.
$$
We deal now with $J_{12}$ which is more involved.
By estimate \eqref{green1}, we obtain that $P_{\Omegaega} (x,y, t)\leqslantCt^{-\frac{N}{2s}}$.
Hence $\bigg(t^{\frac{N}{2s}}P_{\Omegaega} (x,y, t)\bigg)^{q-1}\leqslantC$ and then
\begin{equation}\label{one0}
P^q_{\Omegaega} (x,y, t)\leqslantCt^{-(q-1)\frac{N}{2s}}P_{\Omegaega} (x,y, t)\leqslant\frac{C}{(\d(x))^{(q-1)N}}P_{\Omegaega} (x,y, t)\:\: \mbox{ if }\d(x)\leqslantt^{\frac{1}{2s}}.
\end{equation}
Therefore using H\"older inequality, it holds that
\begin{equation}\label{newww}
\begin{array}{lll}
J_{12}(x,t)& = & \dfrac{C}{\d^q(x)}\Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\Big)^q\\
&\leqslant& \dfrac{C||u_0||^{q-1}_{L^1(\O)}}{\d^q(x)}\displaystyle\int_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y){P^q_{\Omegaega} (x,y, t)}\,dy\\
& \leqslant& \dfrac{C||u_0||^{q-1}_{L^1(\O)}}{\d^q(x)}\displaystyle\int_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y)\frac{1}{(\d(x))^{(q-1)N}}P_{\Omegaega} (x,y, t)dy\\
& \leqslant& C||u_0||^{q-1}_{L^1(\O)}\displaystyle\int_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y)\frac{P_{\Omegaega} (x,y, t)}{(\d(x))^{q(N+1)-N}}dy.
\end{array}
\end{equation}
Thus
$$
\iint_{\O_T}J_{12}(x,t)dxdt\leqslantC||u_0||^{q-1}_{L^1(\O)}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\int\lim\limit_{n\to\infty}tss_\Omega \frac{1}{(\d(x))^{q(N+1)-N}}\bigg(\int_0^T P_{\Omegaega} (x,y, t)dt\bigg)dxdy.
$$
Recall that, from \eqref{green0},
$$
\displaystyle\int_{0}^{\infty} P_{\Omegaega} (x,y,
t)\,dt=\mathcal{G}_s(x,y),
$$
the Green function of the fractional Laplacian in $\Omegaega$.
Then
$$
\iint_{\O_T}J_{12}(x,t)dxdt\leqslantC||u_0||^{q-1}_{L^1(\O)}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\bigg(\int\lim\limit_{n\to\infty}tss_\Omega \frac{\mathcal{G}_s(x,y)}{(\d(x))^{q(N+1)-N}}dx\bigg)dy.
$$
Let $\varphi(x):=\displaystyle\int\lim\limit_{n\to\infty}tss_\Omega \frac{\mathcal{G}_s(x,y)}{\d^{q(N+1)-N}}dx$, then $\varphi$ is the unique solution to the problem
\begin{equation}\label{varphi}
\left\{
\begin{array}{rcll}
(-\D)^s \varphi &=& \dfrac{1}{(\d(x))^{q(N+1)-N}} & \text{ in } \O, \\
\varphi &=&0 & \text{ in }({I\!\!R}n\setminus\O). \\
\end{array}
\right.
\end{equation}
Since $q(N+1)-N<2s$, from {\cal C}_0^{\iy}\,te{AP}, see also {\cal C}_0^{\iy}\,te{Adm}, it follows that $\varphi\in L^\infty(\O)$. Hence we conclude that
$$
\iint_{\O_T}J_{12}(x,t)dxdt\leqslantC||u_0||^{q}_{L^1(\O)}.
$$
Combing the above estimate we deduce that
$$
\iint_{\O_T}J_1(x,t) dxdt \leqslantC||u_0||^{q}_{L^1(\O)}.
$$
We treat now $J_2$. As in the previous estimates, we have
\begin{eqnarray*}
J_2(x,t) & = & \Big(\iint_{\{\O\times (0,t)\}}f(y,\sigma) h(x,y,t-\s){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q\\
&=& \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}f(y,\sigma) h(x,y,t-\s){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q\\
&+ & \displaystyle \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}\}}f(y,\sigma) h(x,y,t-\s){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q\\
&\leqslant& C\Big(\iint_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}f(y,\sigma) \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{1}{2s}}}\,dy\,d\sigma\Big)^q\\
&+ & \frac{C}{\d^q(x)}\displaystyle \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}\}}f(y,\sigma){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q\\ \\
&=& J_{21}(x,t) + J_{22}(x,t).
\end{eqnarray*}
Using estimate \eqref{green1} and by H\"older inequality, we get
\begin{eqnarray*}
J_{21}(x,t) &=& \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}f(y,\sigma) \frac{(t-\s)^{1-\frac{1}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} \,dy\,d\sigma\Big)^q\\
& \leqslant & C||f||^{q-1}_{L^1(\O_T)}\iint_{\{\O\times (0,t)\}\}}f(y,\sigma) \frac{(t-\s)^{q(1-\frac{1}{2s})}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}} \,dy\,d\sigma.
\end{eqnarray*}
Integrating in $\O_T$, following closely the computations used in the estimate of the term $J_{11}$, we deduce that
$$
\displaystyle \iint_{\O_T}J_{21}(x,t) dxdt\leqslantCT^{\g_1+{{2}}}||f||^{q}_{L^1(\O_T)}
$$
with $\g_1=\dfrac{N}{2s}-q\dfrac{N+1}{2s}>-1$.
Now respect to $J_{22}$, using estimate \eqref{one0} and by H\"older inequality, it follows that
\begin{equation}\label{newww0}
\begin{array}{lll}
J_{22}(x,t)& \leqslant& \dfrac{C||f||^{q-1}_{L^1(\O_T)}}{\d^q(x)}\displaystyle\iint_{\{\O\times (0,t)\}\}}f(y,\sigma){P^q_{\Omegaega} (x,y, t-\s)}\,dy,\,d\s\\
& \leqslant& \dfrac{C||f||^{q-1}_{L^1(\O_T)}}{\d^q(x)}
\displaystyle\iint_{\{\O\times (0,t)\}\}}\frac{f(y,\sigma)}{(\d(x))^{(q-1)N}}P_{\Omegaega} (x,y, t-\s)dy\,d\sigma \\
& \leqslant& C||f||^{q-1}_{L^1(\O_T)} \displaystyle \iint_{\{\O\times (0,t)\}\}}f(y,\sigma)
\frac{P_{\Omegaega} (x,y, t-\s)}{(\d(x))^{q(N+1)-N}}dy.
\end{array}
\end{equation}
Integrating in $\O_T$ as above, there results that
$$
\iint_{\O_T}J_{22}(x,t)dxdt\leqslantC||f||^{q-1}_{L^1(\O_T)}\iint_{\O_T}f(y,\s)\Big(\int\lim\limit_{n\to\infty}tss_\Omega \frac{1}{(\d(x))^{q(N+1)-N}}\int_{\s}^T P_{\Omegaega} (x,y, t-\s)dt\,dx\Big)dy\,d\sigma.
$$
It is clear that, from \eqref{green0},
$$
\displaystyle\int_{\s}^{T} P_{\Omegaega} (x,y,
t-\s)\,dt= \displaystyle\int_{0}^{T-\s} P_{\Omegaega} (x,y,
\eta)\,d\eta\leqslant\mathcal{G}_s(x,y).
$$
Thus
$$
\iint_{\O_T}J_{22}(x,t)dxdt\leqslantC||f||^{q-1}_{L^1(\O_T)}\iint_{\O_T}f(y,\s)\,\varphi(y)dy,
$$
where $\varphi$ is the unique solution to problem \eqref{varphi}. Since $q(N+1)-N<2s$, we have that $\varphi\in L^{\infty}(\O)$ and then
$$
\iint_{\O_T}J_{22}(x,t)dxdt\leqslantC||f||^{q}_{L^1(\O_T)}.
$$
Hence
$$
\iint_{\O_T}J_2(x,t) dxdt \leqslantC||f||^{q}_{L^1(\O_T)}.
$$
Therefore we conclude that
\begin{equation}\label{main001}
\begin{array}{lll}
\displaystyle \Big(\iint_{\O_T}|\nabla u(x,t)|^q\, dx\,dt \Big)^{\frac{1}{q}} & \leqslant& \displaystyle
\Big(\iint_{\O_T} (J_1(x,t)+J_2(x,t))\, dx\,dt \Big)^{\frac{1}{q}}\\ \\ & \leqslant&
C(\O,T)\bigg(\|f\|_{L^{1}(\Omegaega_T)}+||u_0||_{L^1(\O)}\bigg).
\end{array}
\end{equation}
Fixed $q_0<\frac{N+2s}{N+1}$, we define
$$\hat{K}: L^{1}(\Omegaega_T)\times L^1(\O)\to L^{q_0}(0,T; W_{0}^{1,q_0}(\Omegaega)),$$
by $\hat{K}(f,u_0)=u$, where $u$ is
the unique solution to \eqref{eq:def}. From \eqref{main001} we deduce that $\hat{K}$ is well defined and continuous. Let show that $\hat{K}$ is a compact operator.
Denote $w$ and $\tilde{w}$ the unique solutions to the problems
\begin{equation}\label{ww11}
\left\{
\begin{array}{rcll}
w_t+(-\D^s) w &=& 0 & \text{ in } \O_{T} , \\ w&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\
w(x,0)&=& u_0 & \mbox{ in }\O,
\end{array}
\right.
\end{equation}
and
\begin{equation}\label{ww111}
\left\{
\begin{array}{rcll}
\tilde{w}_t+(-\D^s) \tilde{w} &=& f & \text{ in } \O_{T} , \\ \tilde{w}&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\
\tilde{w}(x,0)&=& 0 & \mbox{ in }\O,
\end{array}
\right.
\end{equation}
respectively. It is clear that $w+\tilde{w}=u$.
From Proposition {I\!\!R}f{first11}, we know that $w\in L^{m}(\Omegaega_T)$ for all $m<\frac{N+2s}{N}$ and for all $\theta>1$,
$$
t^{\frac{N}{2s}(1-\frac{1}{\theta})}||w(.,t)||_{L^{\theta}}\leqslantC||u_0||_{L^1(\O)}.
$$
Then
\begin{equation}\label{ff1}
\sup_{t\in [0,T]} t^{\frac{N}{2s}(\theta-1)} \int\lim\limit_{n\to\infty}tss_\Omega w^\theta(x,t)dx\leqslantC(\O_T)||u_0||^\theta_{L^1(\O)}.
\end{equation}
Now, going back to the definition of $J_{11}$, we get
$$
J_{11}(x,t)=\frac{C}{t^{\frac{q}{2s}}}\Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)>t^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\Big)^q\leqslant\frac{C}{t^{\frac{q}{2s}}}w^q(x,t).
$$
Choosing $\theta=q$ in \eqref{ff1},
\begin{eqnarray*}
\iint_{\O_T} J_{11}(x,t) dxdt & \leqslant& C\iint_{\O_T}t^{-\frac{q}{2s}} w^q(x,t) t^{\frac{N(q-1)}{2s}} t^{-\frac{N(q-1)}{2s}}dxdt\\
& \leqslant& C\int_0^T t^{-\frac{q}{2s}-\frac{N(q-1)}{2s}}\bigg(\int\lim\limit_{n\to\infty}tss_\Omega w^q(x,t) t^{\frac{N(q-1)}{2s}} dx\bigg)dt.
\end{eqnarray*}
By hypothesis $q<\dfrac{N+2s}{N+1}$, that is, $\dfrac{q}{2s}+\dfrac{N(q-1)}{2s}<1$, then since
$$\bigg(\int\lim\limit_{n\to\infty}tss_\Omega w^q(x,t) t^{\frac{N(q-1)}{2s}} dx\bigg)\in L^\infty(0,T),$$ there exists $a>1$ such that $a(\dfrac{q}{2s}+\dfrac{N(q-1)}{2s})<1$ and
\begin{equation}\label{j11}
\iint_{\O_T} J_{11}(x,t) dxdt\leqslantC \bigg(\int_0^T\bigg(\int\lim\limit_{n\to\infty}tss_\Omega w^q(x,t) t^{\frac{N(q-1)}{2s}} dx\bigg)^{a'}dt\bigg)^{\frac{1}{a'}}.
\end{equation}
Respect to the term $J_{21}$, we have
\begin{eqnarray*}
J_{21}(x,t)& = & C\Big(\iint_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}f(y,\sigma) \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{1}{2s}}}\,dy\,d\sigma\Big)^q\\
&\leqslant& C\Big(\iint_{\{\O\times (0,t) \}}f(y,\sigma)P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\Big)^{q-1}\iint_{\{\O\times (0,t)\}}f(y,\sigma) \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{q}{2s}}}\,dy\,d\sigma\\
\\
&\leqslant& C {{\tilde{w}}}^{q-1}(x,t) \iint_{\{\O\times (0,t) \}}f(y,\sigma) \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{q}{2s}}}\,dy\,d\sigma.
\end{eqnarray*}
Thus integrating in $\O_T$ and by estimate \eqref{green1}, we obtain that
\begin{eqnarray*}
\iint_{\O_T} J_{21}(x,t)dxdt &\leqslant& C \iint_{\O_T}f(y,\sigma) \bigg(\iint_{\{\O\times (\s,T) \}}{{\tilde{w}}}(x,t)^{q-1}\frac{(t-\s)^{1-\frac{q}{2s}}}{((t-\s)^{\frac{1}{2s}}-|x-y|)^{N+2s}}dxdt\bigg)dy\,d\sigma.
\end{eqnarray*}
Since $q<\dfrac{N+2s}{N+1}$ and { using the fact that $s>\frac 12$}, we get the existence of $\frac{2s-1}{N+1}<r<\dfrac{N+2s}{N}$ such that $q<\dfrac{N+2s+2sr}{N+2s+r}$. Hence $r>q-1$ and
$$
\frac{r}{r-(q-1)}\bigg(\frac{2s-q}{2s}-\frac{N+2s}{2s}\bigg)+\frac{N}{2s}>-1.
$$
Therefore, using H\"older inequality
\begin{eqnarray*}
&\displaystyle \iint\lim\limit_{n\to\infty}tss_{\O_T} J_{21}(x,t)dxdt\leqslant\\
& \displaystyle C \iint\lim\limit_{n\to\infty}tss_{\O_T}f(y,\sigma) \bigg(\iint\lim\limit_{n\to\infty}tss_{\O_T}{{\tilde{w}}}^r(x,t) dxdt\bigg)^{\frac{q-1}{r}}
\bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O\times (\s,T)\}}\frac{(t-\s)^{(\frac{2s-q}{2s})
\frac{r}{r-(q-1)}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{(N+2s)\frac{r}{r-(q-1)}}}dxdt
\bigg)^{\frac{r-(q-1)}{r}}dyd\sigma.
\end{eqnarray*}
Setting $z=\dfrac{x-y}{(t-\s)^{\frac{1}{2s}}}$, it holds that
\begin{eqnarray*}
&\displaystyle \iint_{\O_T} J_{21}(x,t)dxdt\leqslant\\
& \displaystyle C ||{{\tilde{w}}}||^{q-1}_{L^r(\O_T)}\iint_{\O_T}f(y,\sigma) \bigg\{\bigg(\int_{\s}^T(t-\s)^{\gamma_2} \int_{\re^N} \frac{1}{(1+|z|)^{(N+2s)\frac{r}{r-(q-1)}}}
dzdt\bigg)^{\frac{r-(q-1)}{r}}\bigg\}dy d\sigma,
\end{eqnarray*}
where $\g_2=\dfrac{r}{r-(q-1)}\bigg(\dfrac{2s-q}{2s}-\dfrac{N+2s}{2s}\bigg)+\dfrac{N}{2s}$. Since $\g_2>-1$, then
\begin{equation}\label{j21}
\iint_{\O_T} J_{21}(x,t)dxdt\leqslantC T^{\frac{(r-(q-1))(\g_2+1)}{r}}||f||_{L^1(\O_T)}||{{\tilde{w}}}||^{q-1}_{L^r(\O_T)}.
\end{equation}
Respect to the terms $J_{12}$ and $J_{22}$ we have that
\begin{equation}\label{j12}
J_{12}(x,t)=\frac{C}{\d^q(x)}\Big(\displaystyle\int_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\Big)^q=C\frac{w^q(x,t)}{\d^q(x)},
\end{equation}
and
\begin{equation}\label{j22}
J_{22}(x,t)=\frac{C}{\d^q(x)}\displaystyle \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslantt^{\frac{1}{2s}}\}\}\}}f(y,\sigma){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q=C\frac{\tilde{w}^q(x,t)}{\d^q(x)}.
\end{equation}
Let $\{f_n, u_{n0}\}_n$ be a bounded sequence in $L^{1}(\Omegaega_T)\times L^1(\O)$ and define $u_n=\hat{K}(f_n, u_{n0})$. Using the previous
estimates, it follows that, for all $q<\dfrac{N+2s}{N+1}$,
$$
||\nabla u_n||_{L^{q}(\Omegaega_T)}\leqslantC(q,\Omegaega)(||f_n||_{L^{1}(\Omegaega_T)}+||u_{n0}||_{L^{1}(\Omegaega)})\leqslantC. $$
Hence, there exists $u\in L^q(0,T; W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$ such that, up to a subsequence, $u_n\rightharpoonup u$ weakly in $L^q(0,T; W_{0}^{1,q}(\Omegaega))$, $u_n\to u$ strongly in $L^\s(\Omegaega_T)$ for all $\s<\dfrac{N+2s}{N}$ and $u_n\to u$ a.e in $\O_T$.
Fixing the above subsequence, we define $(w_n,\tilde{w}_n)$ be the solutions to problems \eqref{ww11} and \eqref{ww111} with data $u_{n0}$ and $f_n$ respectively. Then the sequences $\{w^q_n(x,t) t^{\frac{N(q-1)}{2s}}\}_n$, $\{w_n\}_n$ and $\{\dfrac{\tilde{w}_n}{\d}\}_n$ are bounded in $L^\infty((0,T);L^\s(\O))$, $L^r(\O_T)$ and in $L^\s(\O_T)$ for all $\s<\dfrac{N+2s}{N+1}$ and $r<\dfrac{N+2s}{N}$ respectively. Hence using Vitali's lemma we deduce that the sequences $\{w^q_n(x,t) t^{\frac{N(q-1)}{2s}}\}_n$, $\{w_n\}_n$ and $\{\dfrac{\tilde{w}_n}{\d}\}_n$ are strongly converging in $L^a((0,T);L^\s(\O))$, in $L^r(\O_T)$ and in $L^\s(\O_T)$ for all $\s<\dfrac{N+2s}{N+1}$, for all $r<\dfrac{N+2s}{N}$ and for all $a>1$.
By the linearity of the operator, it follows that $(u_i-u_j)$
solves
\begin{equation}\label{eq:def00}
\left\{
\begin{array}{rcll}
(u_i-u_j)_t+(-\D^s) (u_i-u_j)&=& \displaystyle f_i-f_j & \text{ in } \O_{T}, \\ u_i-u_j&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\ (u_i-u_j)(x,0)&=&
u_{i0}-u_{j0} & \mbox{ in }\O.
\end{array}
\right.
\end{equation}
Going back to the first formula in \eqref{main001} and by estimates \eqref{j11}, \eqref{j21}, \eqref{j12}, \eqref{j22}, we get
\begin{eqnarray*}
& \displaystyle\Big(\iint_{\O_T}|\nabla (u_i-u_j)(x,t)|^q\, dx\,dt \Big)^{\frac{1}{q}} \leqslant\\
& C(\O,T)\displaystyle \bigg(\bigg(\int_0^T\bigg(\int\lim\limit_{n\to\infty}tss_\Omega |w_i(x,t)-w_j(x,t)|^q t^{\frac{N(q-1)}{2s}} dx\bigg)^{a'}dt\bigg)^{\frac{1}{a'q}} +
\|f_i-f_j\|_{L^{1}(\Omegaega_T)}||\tilde{w}_i-\tilde{w}_j||^{q-1}_{L^r(\O_T)} \\
& + ||\dfrac{w_i-w_j}{\d}||_{L^q(\O_T)} + ||\dfrac{\tilde{w}_i-\tilde{w_j}}{\d}||_{L^q(\O_T)}\bigg).
\end{eqnarray*}
Letting $i,j\to \infty$, it holds that
$$
\Big(\iint_{\O_T}|\nabla (u_i-u_j)(x,t)|^q\, dx\,dt \Big)^{\frac{1}{q}}\to 0.
$$
Then the operator $\hat{K}$ is compact.
\end{proof}
\
\begin{remark}\label{mainrr}
\
\begin{enumerate}
\item Thanks to the above computations, we can prove that the constant $C(T,\O)$, that appears in estimate \eqref{main001} satisfies $C(T,\O)\to 0$ as $T\to
0$. This fact will be used below in order to show existence result
for problem \eqref{grad} using a \textit{Fixed Point Theorem}.
\item Using an approximation argument and by the linearity of the operator, we can prove that the
result of Theorem {I\!\!R}f{gradiente} holds if $f$ is a bounded Radon measure.
\item It is worthy to point out that the same arguments give the elliptic case by a slightly different method as in {\cal C}_0^{\iy}\,te{CV2}.
\end{enumerate}
\end{remark}
Following the same representation argument as above we get the next technical regularity result.
\begin{Proposition}
Assume that $(f,u_0)\in L^1(\O_T)\times L^1(\O)$ and let $u$ be the unique solution to the problem \eqref{eq:def}, then $T_k(u)\in L^\s(0,T;W^{1,\s}_0(\O))$ for
all $1\leqslant\s<2s$. Moveover we have
$$
||T_k(u)||^\s_{L^\s(0,T;W^{1,\s}_0(\O))}\leqslantC(\O,T)k^{\s-1}(||u_0||_{L^1(\O)}+||f||_{L^1(\O_T)}).
$$
In addition, if $u_n=\hat{K}(f_n, u_{n0})$, then, up to a subsequence, it follows that $T_k(u_n)\to T_k(u)$ strongly in $L^\s(0,T;W^{1,\s}_0(\O))$, for all $1\le
\s<2s$.
\end{Proposition}
\begin{proof}
Without loss of generality we can assume that $f\geqslant0$ in $\O_T$ and $u_0\geqslant0$ in $\O$. Fix $1<\rho<2s$, as in the proof of Theorem {I\!\!R}f{gradiente}, we have
$$
\begin{array}{rcl}
& & \displaystyle |\nabla u(x,t)|^\rho \leqslantC\Big(\displaystyle\int_{\Omegaega}\, u_0(y) P_{\Omegaega} (x,y, t) \,dy\Big)^{\frac{\rho}{\rho'}} \Big(\displaystyle\int_{\Omegaega} h^\rho(x,y,t) |u_0(y)|P_{\Omegaega} (x,y,
t)\,dy\Big)\\ \\ &+ & \displaystyle C\Big(\iint_{\O_t}\, f(y,\s) P_{\Omegaega} (x,y, t-\sigma) \,dy\,d\sigma \Big)^{\frac{\rho}{\rho'}} \Big(
\iint_{\O_t} h^\rho(x,y,t-\s)f(y,\s) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\Big)\\ \\ &\leqslant& \displaystyle C\, u^{\frac{\rho}{\rho'}}(x,t) \Big(\displaystyle\int_{\Omegaega} h^\rho(x,y,t)
|u_0(y)|P_{\Omegaega} (x,y, t)\,dy + \iint_{\O_t} h^\rho(x,y,t-\s) f(y,\s) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\Big).
\end{array}
$$
Thus,
$$
|\nabla T_k(u)|^{\rho}\leqslantC\, k^{\rho -1}\Big(\displaystyle\int_{\Omegaega} h^\rho(x,y,t) |u_0(y)|P_{\Omegaega} (x,y, t)\,dy + \iint_{\O_t}h^\rho(x,y,t-\s) f(y,\s) P_{\Omegaega} (x,y,
t-\sigma)\,dy\,d\sigma\Big)\chi_{\{u<k\}}.
$$
Hence
\begin{equation}\label{mainii}
\begin{array}{lll}
&\,&\displaystyle \iint_{\O_T}|\nabla T_k(u(x,t))|^\rho\, dx\,dt \\
&\leqslant& \displaystyle C(\O,T)\, k^{\rho -1}\Big(\int\lim\limit_{n\to\infty}tss_\Omega u_0(y) \bigg(\iint_{\O_T}h^\rho(x,y,t)P_{\Omegaega} (x,y,
t)\,dxdt\bigg)dy\\ &+& \displaystyle \iint_{\O_T}f(y,\s)\bigg(\int_\s^T\int\lim\limit_{n\to\infty}tss_\Omega h^\rho(x,y,t-\s) P_{\Omegaega} (x,y, t-\s)\,dx\,dt\bigg)dy\\ &\leqslant& Ck^{\rho-1}( J_1+J_2).
\end{array}
\end{equation}
Recall that $h(x,y,t)\leqslantC \Big( \dfrac{1}{\deltalta(x) \wedge t^{\frac{1}{2s}}}\Big)$, hence using the fact that $\rho<2s$, we reach that
\begin{eqnarray*}
J_1&= & \displaystyle \int\lim\limit_{n\to\infty}tss_\Omega u_0(y) \bigg(\iint_{\{\O_T\cap \{\d(x)\geqslantt^{\frac{1}{2s}}\}\}}h^\rho P_{\Omegaega} (x,y,
t)\,dxdt\bigg)dy \\ & + & \displaystyle \int\lim\limit_{n\to\infty}tss_\Omega u_0(y) \bigg(\iint_{\{\O_T\cap \{\d(x)< t^{\frac{1}{2s}}\}\}}h^\rho(x,y,t) P_{\Omegaega} (x,y,
t)\,dxdt\bigg)dy\\ &= & I_1+I_2.
\end{eqnarray*}
Now, by Lemma {I\!\!R}f{estimmm}, it holds that
$$
I_1\leqslantC \int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\iint_{\O_T}\dfrac{1}{t^{\frac{\rho}{2s}}}\dfrac{t}{ (t^{\frac{1}{2s}}+|x-y|) ^{N+2s}}\,\,dxdt\, dy.
$$
Setting, $z=\dfrac{x-y}{t^{\frac{1}{2s}}}$, it follows that
\begin{eqnarray*}
I_1 & \leqslant& C ||u_0||_{L^1(\O)}\displaystyle\int_{0}^{T} t^{1-\frac{\rho}{2s}-\frac{N+2s}{2s}+\frac{N}{2s}}dt\displaystyle\int_{{I\!\!R}n} \frac{dz}{(1+|z|)^{(N+2s)}}\\
& \leqslant& C||u_0||_{L^1(\O)}\displaystyle\int_{0}^{T} t^{-\frac{\rho}{2s}}dt.
\end{eqnarray*}
Since $\frac{\rho}{2s}<1$, then
$$ I_1\le
CT^{1-\frac{\rho}{2s}}||u_0||_{L^1(\O)}.$$
We deal now with $I_2$. We have
\begin{eqnarray*}
I_2 & \leqslant& C \int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\iint_{\O_T}\dfrac{1}{\d^\rho(x)}\,P_{\Omegaega} (x,y,
t)\,dxdt\, dy\\
&\leqslant& C \int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\displaystyle\int_{\Omegaega} \dfrac{1}{\d^\rho(x)}\,(\displaystyle\int_{0}^{\infty} P_{\Omegaega} (x,y,
t)\,dt)dx\, dy.
\end{eqnarray*}
Recall that
$$
\displaystyle\int_{0}^{\infty} P_{\Omegaega} (x,y,
t)\,dt=\mathcal{G}_s(x,y),
$$
then
$$
I_2\leqslantC \int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\iint_{\O_T}\dfrac{1}{\d^\rho(x)}\,\mathcal{G}_s(x,y) dx\, dy= C \int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\varphi (y) dy,
$$
where $\varphi$ is the solution to problem \eqref{varphi}. Since $\rho<2s$, then $\varphi\in L^\infty(\O)$, hence
$$ I_2\le
C||u_0||_{L^1(\O)}.$$
Hence we conclude that $J_1\le
C(\O,T)||u_0||_{L^1(\O)}.$
In the same way we obtain that
$$
J_2\leqslantC(\O,T)||f||_{L^1(\O_T)}.
$$
Therefore
$$
\iint_{\O_T}|\nabla T_k(u(x,t))|^\s\, dx\,dt\le\displaystyle C(\O,T)\, k^{\sigma -1}(||u_0||_{L^1(\O)}+||f||_{L^1(\O_T)}).
$$
Define $u_n=\hat{K}(f_n, u_{n0})$, then according with the proof of Theorem {I\!\!R}f{gradiente}, we know that, up to a subsequence, $u_n\to u$ strongly in $L^q(0,T;
W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$. Now, using the fact that the sequence $\{T_k(u_n)\}_n$ is bounded in $L^\s(0,T;W^{1,\s}_0(\O))$ for all $1\le
\s<2s$ and using Vitali's Lemma, it follows that $T_k(u_n)\to T_k(u)$ strongly in $L^\s(0,T;W^{1,\s}_0(\O))$.
\end{proof}
\begin{remark}
If $u_0=0$ and $f\in L^m(\O_T)$ with $m>1$, we can improve the regularity results obtained previously. Notice that, related to the
fractional Laplacian, a kind of regularity holds in the local gradient where we are close to the boundary. This can be seen in the explicit example
$w(x)=(1-|x|^2)_+^s$ which solves
$$
(-\D)^s w=1 \mbox{ in }B_1(0) \mbox{ and }w=0 \mbox{ in }{I\!\!R}n\backslash B_1(0).
$$
However, as it was observed in {\cal C}_0^{\iy}\,te{AP}, we can show a complete regularity schema using in a suitable weighted Sobolev space. The main tool will be a universal control of the term $\dfrac{u}{\d^s}$ that holds for any $s\in (0,1)$ and without using the classical Hardy-Sobolev inequality.
\end{remark}
Before considering the case $m>1$, we enunciate the next regularity result that will be used through the paper and that clarifies the regularity of the solution in the space for fixed time.
\begin{Proposition}\label{more-tregu}
Suppose that $f\in L^1(\O_T)$ and $u_0=0$. Let $u$ be the unique weak solution to problem \eqref{eq:def}, then for all $1<q<\dfrac{N+2s}{N+1}$ and for all $\eta>0$,
\begin{equation}\label{time-regu}
\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^q dx \leqslantC(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}-\eta} dy\,d\sigma,
\end{equation}
where $\hat{\g}:=\dfrac{N}{2s}-q\dfrac{N+1}{2s}\in (-1,0)$. In particular, we obtain that for all $\eta>0$,
\begin{equation}\label{time-regu1}
\bigg(\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^q dx\bigg)^{\frac{1}{q}}\leqslantC(\O_T)\bigg(\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}-\eta} dy\,d\sigma\bigg)^{\frac{1}{q}}.
\end{equation}
\end{Proposition}
\begin{proof}
From Theorem {I\!\!R}f{gradiente}, we know that $u\in L^q((0,T), W^{1,q}_0(\O))$ for all $q<\frac{N+2s}{N+1}$. Then we have
$$
u(x,t)=\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma.
$$
Hence
\begin{eqnarray*}
|\nabla u(x,t)| & \leqslant& \displaystyle \displaystyle\int\lim\limit_{n\to\infty}tss_{0}^{t} \displaystyle\int_{\Omegaega}| f(y,\sigma)| \frac{|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|}{P_{\Omegaega} (x,y, t-\sigma)}P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\\&\leqslant& C\iint\lim\limit_{n\to\infty}tss_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}|f(y,\sigma)| \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{1}{2s}}}\,dy\,d\sigma\\
&+ & \displaystyle\Bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O\times (0,t) \cap\{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}\}}|f(y,\sigma)|{P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Bigg)\,\frac{C}{\d(x)}.
\end{eqnarray*}
Therefore, fixing $1<q<\dfrac{N+2s}{N+1}$, we reach that
\begin{equation}\label{vvv0}
\begin{array}{lll}
|\nabla u(x,t)|^q &\leqslant& \displaystyle C\Big(\iint\lim\limit_{n\to\infty}tss_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}|f(y,\sigma)| \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{1}{2s}}}\,dy\,d\sigma\Big)^q\\
&+ & \displaystyle \frac{C}{\d^q(x)}\displaystyle \Big(\iint\lim\limit_{n\to\infty}tss_{\{\O\times (0,t) \cap\{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}\}}|f(y,\sigma)|{P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q \\ \\ &=& I_1(x,t) + I_{2}(x,t).
\end{array}
\end{equation}
Using estimate \eqref{upper-kernel} and by H\"older inequality, we get
\begin{eqnarray*}
I_{1}(x,t) &=& \Big(\iint\lim\limit_{n\to\infty}tss_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}|f(y,\sigma)| \frac{(t-\s)^{1-\frac{1}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} \,dy\,d\sigma\Big)^q\\
& \leqslant & C||f||^{q-1}_{L^1(\O_t)}\iint\lim\limit_{n\to\infty}tss_{\{\O\times (0,t)\}\}}|f(y,\sigma)| \frac{(t-\s)^{q(1-\frac{1}{2s})}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}} \,dy\,d\sigma.
\end{eqnarray*}
Integrating in $\O$, following the same change of variable as in the proof of Theorem {I\!\!R}f{gradiente},
$$
\begin{array}{lll}
\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega I_{1}(x,t) dx & \leqslant& \displaystyle C||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|\int\lim\limit_{n\to\infty}tss_\Omega \frac{(t-\s)^{q(1-\frac{1}{2s})}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}} dx \,dy\,d\sigma\\
&\leqslant& \displaystyle C||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|(t-\s)^{\frac{N}{2s}-q\frac{N+1}{2s}}dy\,d\sigma.
\end{array}
$$
Setting $\hat{\g}:=\dfrac{N}{2s}-q\dfrac{N+1}{2s}\in (-1,0)$, then
\begin{equation}\label{ii1}
\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega I_{1}(x,t) dx \leqslant\displaystyle C||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|(t-\s)^{\hat{\g}}dy\,d\sigma.
\end{equation}
We treat now $I_2$. As above, we get
\begin{eqnarray*}
I_{2}(x,t)& \leqslant& \frac{C||f||^{q-1}_{L^1(\O_t)}}{\d^q(x)}\iint_{\O_t}|f(y,\sigma)|{P^q_{\Omegaega} (x,y, t-\s)}\,dy,\,d\s\\
& \leqslant& \frac{C||f||^{q-1}_{L^1(\O_t)}}{\d^q(x)}
\iint_{\{\O_t\cap \{|x-y|<\frac 12 \d(x)\}\}}|f(y,\sigma)|P^q_{\Omegaega} (x,y, t-\s)dy\,d\sigma \\
& + & \frac{C||f||^{q-1}_{L^1(\O_t)}}{\d^q(x)} \iint_{\{\O_t \cap\{|x-y|\geqslant\frac 12 \d(x)\}\}}|f(y,\sigma)|
P^q_{\Omegaega} (x,y, t-\s) dy \,d\sigma=I_{21}(x,t)+ I_{22}(x,t).
\end{eqnarray*}
We begin by estimating $I_{21}$. Notice that $|\d(y)-\d(x)|\leqslant|x-y|$, then $\d(y)\leqslant\dfrac 32 \d(x)$. Thus $\dfrac{\d(y)}{(t-\s)^{\frac{1}{2{ s}}}}\leqslantC(\O_T)$. Hence
using \eqref{upper-kernel},
\begin{eqnarray*}
P^q_{\Omegaega} (x,y, t-\s) & \leqslant& C(\O_T)\frac{(t-\s)^q}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}.
\end{eqnarray*}
Since $|x-y|\leqslant\frac 13\d(x)$, then
$$
\frac{P^q_{\Omegaega} (x,y, t-\s)}{\d^q(x)}\leqslant C(\O_T)\frac{(t-\s)^q}{|x-y|^q((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}.
$$
Going back to the definition of $I_{21}$, setting $z=\dfrac{x-y}{(t-\s)^{\frac{1}{2s}}}$ and integrating in $\O$, there results that
\begin{eqnarray*}
\int\lim\limit_{n\to\infty}tss_\Omega I_{21}(x,t)dx & \leqslant& C(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|\bigg(\int\lim\limit_{n\to\infty}tss_\Omega \frac{(t-\s)^q}{|x-y|^q((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}dx\bigg)\,dy\, d\sigma\\
& \leqslant& C(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}}\bigg(\int_{\re^N}\frac{dz}{|z|^q(1+|z|)^{q(N+2s)}}\bigg) dy\,d\sigma,
\end{eqnarray*}
where, as above, $\hat{\g}=\dfrac{N}{2s}-q\dfrac{N+1}{2s}\in (-1,0)$. Using the fact that $q<N$ we deduce that
\begin{equation}\label{ii21}
\int\lim\limit_{n\to\infty}tss_\Omega I_{21}(x,t)dx \leqslantC(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}}dy\,d\sigma.
\end{equation}
Respect to $I_{22}$, since $\d(x)\leqslant2 |x-y|$, we get $\d(y)\leqslantC|x-y|$, then, in this case
$$
P^q_{\Omegaega} (x,y, t-\s)\leqslantC(\O_T) \Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{(t-\s)}}\Big)^q\times \Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{(t-\s)}}\Big)^q
\frac{(t-\s)^q}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}.
$$
Since, for all $\theta\in (0,1)$,
$$
\Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{(t-\s)}}\Big)^q\leqslant\Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{(t-\s)}}\Big)^{q\theta}\mbox{ and }
\Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{(t-\s)}}\Big)^q \leqslant\Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{(t-\s)}}\Big)^{q\theta}.
$$
Choosing $\theta=\dfrac{1}{q}$, we deduce that
\begin{eqnarray*}
P^q_{\Omegaega} (x,y, t-\s) & \leqslant& C(\O_T) \frac{\deltalta^s(x)}{\sqrt{(t-\s)}}\times \frac{\deltalta^s(y)}{\sqrt{(t-\s)}}
\frac{(t-\s)^q}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}\\
&\leqslant& C(\O_T)\deltalta^s(x)\deltalta^s(y)\frac{(t-\s)^{q-1}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}}\\
&\leqslant& \frac{C(\O_T)\deltalta^s(x)\deltalta^s(y)}{|x-y|^{N}} \frac{(t-\s)^{q-1}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)-N}}\\
&\leqslant& \frac{C(\O_T)\deltalta^s(x)\deltalta^s(y)}{|x-y|^{N}}(t-\s)^{-\frac{N}{2s}(q-1)-1}.\\
\end{eqnarray*}
Thus, for $\eta\in (0,1)$ small enough,
\begin{eqnarray*}
\frac{P^q_{\Omegaega} (x,y, t-\s)}{\d^q(x)} &\leqslant& \frac{C(\O_T)\deltalta^s(x)\deltalta^s(y)}{|x-y|^{N}} \frac{1}{(\d(x))^{q +2s-q-2s\eta}}(\d(x))^{2s-q-2s \eta}(t-\s)^{-\frac{N}{2s}(q-1)-1}\\
& \leqslant& \frac{C(\O_T)\deltalta^s(x)\deltalta^s(y)}{|x-y|^{N}} \frac{1}{(\d(x))^{2s(1-\eta)}}
(t-\s)^{\frac{2s-q-2s\eta}{2s}}(t-\s)^{-\frac{N}{2s}(q-1)-1}\\
& \leqslant& \frac{C(\O_T)\deltalta^s(x)\deltalta^s(y)}{|x-y|^{N}} \frac{1}{(\d(x))^{2s(1-\eta)}}(t-\s)^{\frac{N}{2s}-\frac{q(N+1)}{2s}-\eta}.
\end{eqnarray*}
From \eqref{green00}, we deduce that, in this case,
$$
\mathcal{G}_s(x,y)\simeq \frac{\d^s(x)\d^s(y)}{|x-y|^{N}}.
$$
Hence
$$
\frac{P^q_{\Omegaega} (x,y, t-\s)}{\d^q(x)}\leqslantC(\O_T)\frac{\mathcal{G}_s(x,y)}{(\d(x))^{2s(1-\eta)}}(t-\s)^{\hat{\g}-\eta}.
$$
Therefore
\begin{eqnarray*}
\int\lim\limit_{n\to\infty}tss_\Omega I_{22}(x,t)dx & \leqslant& C(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}-\eta}\bigg(\int\lim\limit_{n\to\infty}tss_\Omega \frac{\mathcal{G}_s(x,y)}{(\d(x))^{2s(1-\eta)}}dx\bigg)\,dy\, d\sigma\\
& \leqslant& C(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}-\eta}\varphi(y) dy\,d\sigma,
\end{eqnarray*}
where $\varphi(y):=\displaystyle\int\lim\limit_{n\to\infty}tss_\Omega \frac{\mathcal{G}_s(x,y)}{(\d(x))^{2s(1-\eta)}}dx$. It is clear that $\varphi$ solves the problem
\begin{equation*}
\left\{
\begin{array}{rcll}
(-\D)^s \varphi &=& \dfrac{1}{(\d(x))^{2s(1-\eta)}} & \text{ in } \O, \\
\varphi &=&0 & \text{ in }({I\!\!R}n\setminus\O). \\
\end{array}
\right.
\end{equation*}
Since $2s(1-\eta)<2s$, then $\varphi\in L^\infty(\O)$. Thus, for all $\eta>0$,
\begin{equation}\label{ii22}
\int\lim\limit_{n\to\infty}tss_\Omega I_{22}(x,t)dx \leqslantC(\O_T)||f||^{q-1}_{L^1(\O_t)}\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\s)|(t-\s)^{\hat{\g}-\eta}dy\,d\sigma.
\end{equation}
Combing estimates \eqref{ii1},\eqref{ii21},\eqref{ii22}, going back to \eqref{vvv} and integrating in $\O$, we conclude that for all $\eta>0$,
$$
\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^qdx\leqslantC(\O_T) ||f||^{q-1}_{L^1(\O_t)} \int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|\bigg((t-\s)^{\hat{\g}-\eta}+ (t-\s)^{\hat{\g}}\bigg)dy\,d\sigma.
$$
Hence
$$
\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^qdx\leqslantC(\O_T) ||f||^{q-1}_{L^1(\O_t)} \int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|(t-\s)^{\hat{\g}-\eta} dy\,d\sigma.
$$
Now using the fact that
$$
||f||_{L^1(\O_t)} \leqslantC(\O_T)\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|(t-\s)^{\hat{\g}-\eta}dy\,d\sigma,
$$
we conclude that
$$
\Big(\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^qdx\Big)^{\frac{1}{q}}\leqslantC(\O_T) \int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |f(y,\sigma)|(t-\s)^{\hat{\g}-\eta}dy\,d\sigma.
$$
\end{proof}
If $m>1$, we can prove a regularity result in a suitable weighted Sobolev space whose weight is a power of the distance to the boundary.
This result will be a consequence of the next two Theorems.
\begin{Theorem}\label{hardy0}
Assume that $u_0\equiv0$, $f\in L^m(\O_T)$ for some $m>1$ and let $u$ be the unique weak solution to problem \eqref{eq:def}, then $\dfrac{u}{\d^s}\in L^\theta(\O_T)$ for
all $\theta>1$ such that $\dfrac{1}{\theta}>\dfrac{1}{m}-\dfrac{s}{N+2s}$. Moreover,
\begin{equation}\label{hardyeq}
\Big|\Big|\frac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)}\leqslantC ||f||_{L^m(\O_T)},
\end{equation}
with
$$
\left\{
\begin{array}{lll}
\theta &<& \infty \mbox{ if }m\geqslant\dfrac{N+2s}{s},\\
&\,&\\
\theta &< & \dfrac{m (N+2s)}{N+2s-ms}\mbox{ if } m<\dfrac{N+2s}{s}.
\end{array}
\right.
$$
\end{Theorem}
\begin{proof}
Without loss of generality, we can assume that $f\gvertneqq 0$, hence $u\gvertneqq 0$ in ${I\!\!R}n\times (0,T)$. By the representation formula we have that
$$
u(x,t)=\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma,
$$
then using the properties of $P_\O$, and \eqref{upper-kernel} it holds that
$$
\begin{array}{rcl}
\dfrac{u(x,t)}{\d^s(x)} &\leq& C \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}f(y,\s) \dfrac{(t-\s)^{{\frac 12}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.\\
\end{array}
$$
As above, consider $\partialhi\in \mathcal{C}^\infty_0(\O_T)$, then
$$
\begin{array}{rcl}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)} &=& \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \iint_{\O_T} \partialhi(x,t) \frac{u(x,t)}{\deltalta^s(x)}dxdt\\ &\leqslant& \displaystyle
\sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \iint_{\O_T}|\partialhi(x,t)|\displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}f(y,\s) \dfrac{(t-\s)^{{\frac 12}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma dxdt\\ &\leqslant& \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T\int_0^t \int\lim\limit_{n\to\infty}tss_\Omega \int\lim\limit_{n\to\infty}tss_\Omega |\partialhi(x,t)|
H(x-y,t-\s)f(y,\s)dydx d\sigma dt,
\end{array}
$$
where
$$
H(|x-y|,\s)=\dfrac{(t-\s)^{{\frac 12}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}.
$$
Using Young inequality, it holds that
\begin{equation}\label{tt}
\begin{array}{rcl}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)} &\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)} \int_0^t
||f(.,\s)||_{L^m(\O)}||H(., t-\s)||_{L^a(\O)}d\sigma dt,
\end{array}
\end{equation}
with $\dfrac{1}{\theta'}+\dfrac{1}{m}+\dfrac{1}{a}=2$.
As in the computation of the term defined in \eqref{upper-kernel} and by \eqref{L-a-H}, we get
$$
||H(., t-\s)||_{L^a(\O)}\leqslantC(t-\s)^{-{{\frac 12}}+\frac{N}{2s a}-\frac{N}{2s}}.
$$
Substituting in \eqref{tt},
$$
\begin{array}{rcl}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)} &\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)} \int_0^t
||f(.,\s)||_{L^m(\O)} (t-\s)^{-{{\frac 12}}+\frac{N}{2s a}-\frac{N}{2s}} d\sigma dt
\\[5mm]
&\leqslant& { C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)} ||f(.,\s)||_{L^m(\O)} |t-\s|^{-{{\frac 12}}+\frac{N}{2s a}-\frac{N}{2s}} d\sigma dt.}
\end{array}
$$
Setting
$$
\hat{H}(|t-\s|)=|t-\s|^{-{{\frac 12}}+\frac{N}{2s a}-\frac{N}{2s}},
$$
then using again Young inequality, we get
$$
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)}\leqslantC\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} ||\partialhi(.,t)||_{L^{\theta'}(\O_T)} ||f||_{L^m(\O_T)}\bigg(\int_0^T \hat{H}^\gamma(t)dt\bigg)^{\frac{1}{\g}},
$$
where $\g\geqslant1$ and $\dfrac{1}{\theta'}+\dfrac{1}{m}+\dfrac{1}{\g}=2$. It is clear that $a=\g$.
Hence
$$
\int_0^T \hat{H}^\gamma(t)=\int_0^T t^{\gamma( -{{\frac 12}}+\frac{N}{2s a}-\frac{N}{2s})}dt.
$$
The previous integral is finite if and only if
$$
\gamma( -{{\frac 12}}+\frac{N}{2s \g}-\frac{N}{2s})>-1.
$$
Hence $\g<\frac{N+2s}{N+s}$. Using the fact that
$\dfrac{1}{\theta'}+\dfrac{1}{m}+\dfrac{1}{ \g}=2$, it holds that $\dfrac{1}{\theta}>\dfrac{1}{m}-\dfrac{s}{N+2s}$ and hence $\theta<\dfrac{m (N+2s)}{(N+2s-sm)_+}$. Then
we conclude.
\end{proof}
\begin{Theorem}\label{regu-g}
Assume that the conditions of Theorem {I\!\!R}f{hardy0} hold. Let $u$ be the unique weak solution to problem \eqref{eq:def}, then $|\nabla u|\d^{1-s}\in L^p(\O_T)$ for all
$p\geqslant1$ such that $\dfrac{1}{p}>\dfrac{1}{m}-\dfrac{2s-1}{N+2s}$. Moreover
\begin{equation}\label{gradeq1}
\big\| |\nabla u| \d^{1-s} \big\|_{L^p(\O_T)}\leqslantC ||f||_{L^m(\O_T)},
\end{equation}
with
$$
\left\{
\begin{array}{rcll}
p &<& \infty &\mbox{ if } m\geqslant\dfrac{N+2s}{2s-1},\\
p &< & \dfrac{m (N+2s)}{N+2s-m(2s-1)} &\mbox{ if } m<\dfrac{N+2s}{2s-1}.
\end{array}
\right.
$$
\end{Theorem}
\begin{proof}
We follow the same technique as in the proof of Theorem {I\!\!R}f{hardy0}. By the representation formula and by using \eqref{green2} and setting $\O_t=\O\times (0,t)$, we have
\begin{equation}\label{madrid20}
\begin{array}{rcll}
& & |\nabla u(x,t)|\leqslantC \displaystyle \iint_{\O_t} f(y,\sigma) |\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma\\ \\
&\leqslant& C \displaystyle\iint_{\O_t} f(y,\sigma) \frac{|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|}{P_{\Omegaega} (x,y, t-\sigma)}P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\\ \\ &\leqslant& \dfrac{C}{\d(x)}\displaystyle \iint_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f(y,\s) P_{\Omegaega} (x,y, t-\sigma) \,dy\,d\sigma \\ \\
& + & C\displaystyle \iint_{\{\O_t\cap \{\d(x)\geqslant(t-\s)^{\frac{1}{2s}}\}\}}f(y,\s)
\dfrac{(t-\s)^{\frac{2s-1}{2s}}\,dy\,d\sigma }{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\\ \\ &\leqslant& C\dfrac{u(x,t)}{\d(x)}+C
\displaystyle\iint_{\O_t}f(y,\s)
\dfrac{(t-\s)^{\frac{2s-1}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.
\end{array}
\end{equation}
Hence
\begin{equation}\label{rrt}
\begin{array}{rcl}
|\nabla u(x,t)| \d^{1-s}(x)&\leq& C\dfrac{u(x,t)}{\d^s(x)}+C\displaystyle \iint_{\O_t}f(y,\s)
\dfrac{(t-\s)^{\frac{2s-1}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ &:= & J_1(x,t)+J_2(x,t).
\end{array}
\end{equation}
From Theorem {I\!\!R}f{hardy0}, it holds that $J_1\in L^\theta(\O_T)$ for all $\theta<\dfrac{m (N+2s)}{(N+2s-ms)_+}$ and
\begin{equation}\label{h000}
||J_1||_{L^\theta(\O_T)}\leqslantC||f||_{L^m(\O_T)}.
\end{equation}
We deal with $J_2$. As above, we will use a duality argument. Let $\partialhi\in \mathcal{C}^\infty_0(\O_T)$, and define
$$\bar{H}(x,t):=\dfrac{t^{\frac{2s-1}{2s}}}{(t^{\frac{1}{2s}}+|x|)^{N+2s}},$$
then
$$
\begin{array}{lll}
||J_2||_{L^p(\O_T)} &= & \displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \iint_{\O_T}\partialhi(x,t)J_2(x,t) dxdt\\ &\leqslant& \displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}}
\iint_{\O_T}|\partialhi(x,t)| \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}f(y,\s) \dfrac{(t-\s)^{\frac{2s-1}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma dxdt\\ &\leqslant& \displaystyle
\sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T\int_0^t \int\lim\limit_{n\to\infty}tss_\Omega \int\lim\limit_{n\to\infty}tss_\Omega |\partialhi(x,t)| \bar{H}(x-y,t-\s)f(y,\s)dydx d\sigma dt.
\end{array}
$$
Hence using Young inequality, we obtain that
\begin{equation}\label{rrr-1}
||J_2||_{L^p(\O_T)}=\displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{p'}(\O)} \int_0^t ||\bar{H}(.,t-\s)||_{L^a(\O)} ||f(.,\s)||_{L^m(\O)}
d\sigma dt,
\end{equation}
with $\dfrac{1}{p'}+\dfrac{1}{m}+\dfrac{1}{a}=2$.
By direct computations, we have
$$
||\bar{H}(.,t-\s)||_{L^a(\O)}\leqslantC (t-\s)^{\frac{2s-1}{2s} + \frac{N}{2s a}-\frac{(N+2s)}{2s}}={ C (t-\s)^{\frac{-1}{2s} + \frac{N}{2s a}-\frac{N}{2s}}.}
$$
Going back to \eqref{rrr-1}, we conclude that
$$
\begin{array}{lll}
||J_2||_{L^p(\O_T)} &\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{p'}(\O)} \int_0^t ||f(.,\s)||_{L^m(\O)} (t-\s)^{\frac{-1}{2s} +
\frac{N}{2sa}-\frac{N}{2s}} d\sigma dt\\
&\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T \int_0^T||\partialhi(.,t)||_{L^{p'}(\O)} \int_0^t ||f(.,\s)||_{L^m(\O)} |t-\s|^{\frac{-1}{2s} +\frac{N}{2sa}-\frac{N}{2s}} d\sigma dt\\
\end{array}
$$
Thus, using again Young inequality, we get
$$
\begin{array}{lll}
||J_2||_{L^p(\O_T)} &\leqslant& C ||f||_{L^m(\O_T)} \displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} ||\partialhi||_{L^{p'}(\O_T)} \bigg(\int_0^T
|t|^{\g(\frac{-1}{2s} +\frac{N}{2sa}-\frac{N}{2s})}
dt\bigg)^{\frac{1}{\g}},
\end{array}
$$
where $\dfrac{1}{p'}+\dfrac{1}{m}+\dfrac{1}{\g}=2$. Hence $\g=a$.
It is clear that the last integral is finite if and only if
$\g<\frac{N+2s}{N+1}$. Since
$\dfrac{1}{p'}+\dfrac{1}{m}+\dfrac{1}{\g}=2$, then $\dfrac{1}{p}>\dfrac{1}{m}-\dfrac{2s-1}{N+2s}$. Hence
\begin{equation}\label{k000-1}
||J_2||_{L^p(\O_T)}\leqslantC||f||_{L^m(\O_T)}.
\end{equation}
If $m\geqslant\dfrac{N+2s}{2s-1}$, then the above condition holds for all $p>1$.
Since $s\in(\dfrac 12, 1)$, $\dfrac{N+2s}{2s-1}>\dfrac{N+2s}{s}$, and then
combining \eqref{h000} and \eqref{k000-1},
we conclude that, for all $p<\infty$,
\begin{equation}\label{k0001}
|||\nabla u| \d^{1-s}||_{L^p(\O_T)}\leqslantC||f||_{L^m(\O_T)}.
\end{equation}
If $m<\dfrac{N+2s}{2s-1}$, then $p<\dfrac{m (N+2s)}{N+2s-m(2s-1)}$.
Since $\dfrac{m (N+2s)}{(N+2s-m(2s-1))}<\dfrac{m (N+2s)}{(N+2s-ms)_+}$, using
\eqref{h000} and \eqref{k000-1}, we obtain that
\begin{equation}\label{k000}
||\nabla u| \d^{1-s}||_{L^p(\O_T)}\leqslantC||f||_{L^m(\O_T)}
\end{equation}
for all $p$ which satisfies $\dfrac{1}{p}>\dfrac{1}{m}-\dfrac{2s-1}{N+2s}$.
\end{proof}
{\begin{remark} Let $u(x,t)=t w$ where $w(x)=(1-|x|^2)_+^s$, solves the problem
$$
(-\D)^s w=1 \mbox{ in }B_1(0) \mbox{ and }w=0 \mbox{ in }{I\!\!R}n\backslash B_1(0).
$$
Then
$$
u_t+(-\D)^s u=w+t:=f(x,t) \mbox{ in }B_1(0)\times (0,T).
$$
Notice that $f\in \mathcal{C}(\overline{\O_T})$, however $|\nabla u(x)|=2st|x|(1-|x|^2)_+^{s-1}$ in $B_1(0)\times (0,T)$, then $|\nabla u|\d^\a\in L^\infty(B_1(0)\times (0,T))$ if and only if $\a\geqslant1-s$ which show in some way the optimality of the regularity result obtained in Theorem {I\!\!R}f{regu-g}.
\end{remark}
}
\begin{Corollary}\label{cor11}
\
\begin{enumerate}
\item By the result of Theorems {I\!\!R}f{hardy0}
and {I\!\!R}f{regu-g} and since $|\nabla \d|=1$ a.e. in $\Omegaega$, it holds that if $u$ is the unique weak solution to problem \eqref{eq:def}, then $(u \d^{1-s})\in L^p(0,T; W^{1,p}_0(\O))$ for all $p$ such that $\dfrac{1}{p}>\dfrac{1}{m}-\dfrac{2s-1}{N+2s}$ and
$$
||u \d^{1-s} ||_{L^p(0,T; W^{1,p}_0(\O))}\leqslantC ||f||_{L^m(\O)}.
$$
Moveover, if $m<\dfrac{N+2s}{2s-1}$, then the above estimate holds for all $p<\dfrac{m (N+2s)}{N+2s-m(2s-1)}$.
\item Assume that $f\in L^m(\O_T)$ for some $m>1$ and let $u$ be the unique weak solution to problem \eqref{eq:def}, then
\begin{enumerate}
\item If $m\ge\dfrac{N+2s}{2s-1}$, then $\displaystyle \iint_{\O_T}|\nabla u|^a dx<\infty$ for all $a<\dfrac{1}{1-s}$.
\item If $\dfrac{1}{s}\leqslantm<\dfrac{N+2s}{2s-1}$,
then $\displaystyle \int_0^T\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u|^a dx<\infty$,
for all $a<\check{P}:=\dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}$.
\item If $1<m<\dfrac{1}{s}$, then $\displaystyle
\iint_{\O_T}|\nabla u|^a dx<\infty$ for all $a<\dfrac{N+2s}{N+1}$.
\end{enumerate}
\end{enumerate}
\end{Corollary}
\begin{proof}
{ Let us begin with the first statement. Define $v(x,t)=u(x,t)\d^{1-s}(x)$, then
$$
\nabla v(x,t)=\d^{1-s}(x)\nabla u +(1-s)\frac{u(x,t)}{\d^s(x)}\nabla \d(x).
$$
Using the fact that $|\nabla \d(x)|=1$ a.e. in $\O$, it holds that
$$
|\nabla v(x,t)|\leqslant\d^{1-s}(x)|\nabla u(x,t)| +(1-s)\frac{|u(x,t)|}{\d^s(x)}.
$$
Hence the desired estimate follows combining the two estimates obtained in Theorems {I\!\!R}f{hardy0} and
{I\!\!R}f{regu-g}. }
\
We prove the second point that provides a global regularity for the gradient term
without using any weight. It is clear that $u\in L^a(0,T;W^{1,a}_0(\O))$ for all
$a<\dfrac{N+2s}{N+1}$. Now, using Theorem {I\!\!R}f{regu-g}, we reach that
$|\nabla u|\d^{1-s}\in L^p(\O_T)$ with $p\geqslant1$ which satisfies
$\dfrac{1}{p}>\dfrac{1}{m}-\dfrac{2s-1}{N+2s}$.
Hence using H\"older inequality, we get
$$
\begin{array}{lll}
\displaystyle \iint_{\O_T}|\nabla u|^a dx dt&= &\displaystyle \iint_{\O_T}\bigg(|\nabla u|\d^{1-s}\bigg)^a \d^{-a(1-s)}dx dt\\ &\leqslant& \displaystyle \leqslant\bigg(\iint_{\O_T}(|\nabla u|\d^{1-s})^p dx dt
\bigg)^{\frac{a}{p}}\bigg(\iint_{\O_T}\d^{-\frac{pa}{p-a}(1-s)}dx dt \bigg)^{\frac{p-a}{p}},
\end{array}
$$
where $p>a$ to be chosen later. The last integral is finite if and
only if $\dfrac{ap}{p-a}(1-s)<1$, that is, if $a<\dfrac{p}{(1-s)p+1}$.
Notice that in particular, $a<\dfrac{1}{1-s}$.
\noindent If $m\geqslant\dfrac{N+2s}{2s-1}$, then by Theorem {I\!\!R}f{regu-g}, we know that
$|\nabla u|\d^{1-s}\in L^p(\O_T)$ for all $p<\infty$. Hence the condition
$\dfrac{ap}{p-a}(1-s)<1$ holds if $a(1-s)<1$ and then we conclude.
\noindent Assume that $m<\dfrac{N+2s}{2s-1}$, since $p<\dfrac{m (N+2s)}{N+2s-ms}$,
we get $$a<\breve{P}:=\dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}.$$
It is clear that $\breve{P}\geqslant\dfrac{N+2s}{N+1}$ if $m\geqslant\dfrac{1}{s}$.
Thus we conclude.
\end{proof}
{In the next result, under suitable hypotheses on $s,m$ and $p$, we improve the integrability of the gradient of the solution without the degenerate weight or with a less degenerate weight at the boundary. }
{
\begin{Theorem}\label{regu-glast}
Assume that the conditions of Theorem {I\!\!R}f{hardy0} hold. Let $u$ be the unique weak solution to problem \eqref{eq:def}, then:
\begin{enumerate}
\item Let $m_1=\min\{\dfrac{s}{1-s}, m\}$. Then $u\in L^p(0,T;W^{1,p}_0(\O))$ for $p<\dfrac{m_1(N+2s)}{N+s+(1-s)m_1}$, and
$$
||\nabla u||_{L^p(\O_T)}\leqslantC(\O_T,s,N,p)||f||_{L^m(\O_T)}
$$
\item If $\dfrac{m(N+2s)}{N+s+(1-s)m}\leqslantp <\min\{\dfrac{m(N+2s)}{N+s}, \dfrac{m(N+2s)}{N+2s-m(2s-1)}\}$, then for all $\a\in (0,1)$ such that
$$
\a>\frac{1}{1-s}\bigg((\frac{1}{m}-\frac{1}{p})(N+2s)+(1-s)-\frac{s}{m}\bigg),
$$
we have $|\nabla u|\d^{\a(1-s)}\in L^p(\O_T)$ and
$$
|| |\nabla u|\, \d^{\a(1-s)}||_{L^p(\O_T)}\leqslantC(\O_T,s,N,p)||f||_{L^m(\O_T)}.
$$
\end{enumerate}
\end{Theorem}
}
\begin{proof} As in the proof of Theorem {I\!\!R}f{regu-g}, from estimate \eqref{madrid20}, we have
\begin{equation}\label{madrid21}
\begin{array}{rcll}
|\nabla u(x,t)|&\leqslant&
\displaystyle \dfrac{C}{\d(x)}\displaystyle \iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f(y,\s) P_{\Omegaega} (x,y, t-\sigma) \,dy\,d\sigma \\ \\ &+& C\displaystyle \iint\lim\limit_{n\to\infty}tss_{\O_t}f(y,\s) \dfrac{(t-\s)^{\frac{2s-1}{2s}}\,dy\,d\sigma }{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\\ \\
& :=&\hat{J}_1(x,t)+ \hat{J}_2(x,t).
\end{array}
\end{equation}
The estimate of the term $\hat{J}_2$ is similar to the estimate of the term
$J_2$ in the proof of Theorem {I\!\!R}f{regu-g}. Precisely,
it holds that $\hat{J}_2\in L^p(\O_T)$ for $p\geqslant1$ satisfying
$\dfrac{1}{p}>\dfrac{1}{m}-\dfrac{2s-1}{N+2s}$ and
\begin{equation}\label{k000-1last}
||\hat{J}_2||_{L^p(\O_T)}\leqslantC||f||_{L^m(\O_T)}.
\end{equation}
We deal with $\hat{J}_1$, in this case we follow the argument as in the proof of Theorem {I\!\!R}f{gradiente}.
Since $f\in L^m(\O_T)$ with $m>1$, then using H\"older inequality we deduce that
\begin{equation}\label{last0}
\begin{array}{lll}
\hat{J}_1(x,t)& \leqslant& \dfrac{C}{\d(x)}\displaystyle\bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f^m(y,\sigma){P_{\Omegaega} (x,y, t-\s)}\,dy\,d\s\bigg)^{\frac{1}{m}}\\
& \times & \displaystyle \bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,{P_{\Omegaega} (x,y, t-\s)}\,dy\,d\s\bigg)^{\frac{1}{m'}}\\
& \leqslant& \dfrac{C}{(\d(x))^{1-\frac{s}{m'}}}\displaystyle\bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f^m(y,\sigma){P_{\Omegaega} (x,y, t-\s)}\,dy\,d\s\bigg)^{\frac{1}{m}},
\end{array}
\end{equation}
where we have used the fact that
$$
\iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\, {P_{\Omegaega} (x,y, t-\s)}\,dy\,d\s\leqslantC(\O,T)\d^s(x) \mbox{ for }(x,t)\in \O_T.
$$
Fix $p>m$ to be chosen later, then
\begin{equation}\label{last1}
\begin{array}{lll}
\hat{J}^p_1(x,t)& \leqslant& \dfrac{C}{(\d(x))^{p(1-\frac{s}{m'})}}\displaystyle\bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f^m(y,\sigma){P_{\Omegaega} (x,y, t-\s)}\,dy\,d\s\bigg)^{\frac{p}{m}}\\ \\
& \leqslant& \dfrac{C}{(\d(x))^{p(1-\frac{s}{m'})}}\displaystyle\bigg(\iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f^m(y,\sigma){P^{\frac{p}{m}}_{\Omegaega} (x,y, t-\s)}\,dy\,d\s\bigg)
\bigg(\iint\lim\limit_{n\to\infty}tss_{\O_t}\,f^m(y,\sigma)\,dy\,d\s\bigg)^{\frac{p-m}{m}}\\ \\
& \leqslant& \dfrac{C ||f||^{p-m}_{L^m(\O_T)}}{(\d(x))^{p(1-\frac{s}{m'})}}\displaystyle \iint\lim\limit_{n\to\infty}tss_{\{\O_t\cap \{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}}\,f^m(y,\sigma){P^{\frac{p}{m}}_{\Omegaega} (x,y, t-\s)}\,dy\,d\s.
\end{array}
\end{equation}
Recall that $t^{\frac{N}{2s}}P_{\Omegaega} (x,y, t)\leqslantC:=C(\O,s,N,T)$, thus $\bigg(t^{\frac{N}{2s}}P_{\Omegaega} (x,y, t)\bigg)^{\frac{p}{m}-1}\leqslantC$. Hence, if $\d(x)\leqslant(t-\s)^{\frac{1}{2s}}$, it follows that
$$
P^{\frac{p}{m}}_{\Omegaega} (x,y, t-\s)\leqslantC(t-\s)^{-(\frac{p}{m}-1)\frac{N}{2s}}P_{\Omegaega} (x,y, t-\s)\leqslant\frac{C}{(\d(x))^{(\frac{p}{m}-1)N}}P_{\Omegaega} (x,y, t-\s).
$$
Therefore we obtain that
\begin{equation}\label{last12}
\begin{array}{lll}
\hat{J}^p_1(x,t)& \leqslant& \dfrac{C ||f||^{p-m}_{L^m(\O_T)}}{(\d(x))^{p(1-\frac{s}{m'})+N(\frac{p}{m}-1)}}\displaystyle\iint_{\O_t}f^m(y,\sigma){P}_{\Omegaega} (x,y, t-\s)\,dy\,d\s.
\end{array}
\end{equation}
Let $\beta=p(1-\dfrac{s}{m'})+N(\dfrac{p}{m}-1)$, then according to the value of $\beta$, we will consider two cases:
\
{\bf The first case $\beta<2s$:} Notice that in this case if $m_1<\dfrac{s}{1-s}$ then $m_1<p<\dfrac{m_1(N+2s)}{N+s+(1-s)m_1}$. Moreover, $\dfrac{m_1(N+2s)}{N+s+(1-s)m_1}<\dfrac{m_1(N+2s)}{N+2s-m_1(2s-1)}$ defined in Theorem {I\!\!R}f{regu-g}.
Integrating in $\O_T$, there results that
$$
\iint_{\O_T}\hat{J}^p_{1}(x,t)dxdt\leqslantC||f||^{p-m_1}_{L^{m_1}(\O_T)}\iint_{\O_T}f^{m_1}(y,\s)\Big(\int\lim\limit_{n\to\infty}tss_\Omega \frac{1}{\d^\beta(x)}\int_{\s}^T P_{\Omegaega} (x,y,t-\s)dt\,dx\Big)dy\,d\sigma.
$$
Recall that
$$
\displaystyle\int_{\s}^{T} P_{\Omegaega} (x,y,
t-\s)\,dt\leqslant\displaystyle\int_{0}^{T-\s} P_{\Omegaega} (x,y,
\eta)\,d\eta\leqslant\mathcal{G}_s(x,y).
$$
Hence
$$
\iint_{\O_T}\hat{J}^p_{1}(x,t)dxdt\leqslantC||f||^{p-m_1}_{L^{m_1}(\O_T)}
\iint_{\O_T}f^{m_1}(y,\s)\varphi(y) dy\,d\sigma.
$$
where $\varphi(y)=\displaystyle\int\lim\limit_{n\to\infty}tss_\Omega \frac{\mathcal{G}_s(x,y)}{(\d(x))^{\beta_0}}dx$.
Since $\beta<2s$, then $\varphi\in L^{\infty}(\O)$ and then
$$
\iint_{\O_T}\hat{J}^p_{1}(x,t)dxdt\leqslantC||f||^{p}_{L^{m_1}(\O_T)}.
$$
If $m>\dfrac{s}{1-s}$ the final estimate follows by the H\"{o}lder inequality.
Notice that in this case, $p<\dfrac{s}{1-s}$.
{\bf The second case $\beta\geqslant2s$}: Consider
$\dfrac{m(N+2s)}{N+s+(1-s)m}\leqslantp <\min\{\dfrac{m(N+2s)}{N+s}, \dfrac{m(N+2s)}{N+2s-m(2s-1)}\}$.
Since $\beta\geqslant2s$,
$$
(\dfrac{1}{m}-\dfrac{1}{p})(N+2s)<\dfrac{s}{m}.
$$
Let
$$
{\Upsilon}:= (\frac{1}{m}-\frac{1}{p})(N+2s)+(1-s)-\frac{s}{m},
$$
then $0<\Upsilon<1-s$. Fix $0<\a<1$ such that $\frac{\Upsilon}{1-s}<\a<1$, thus we reach that $\beta-p\a(1-s)<2s$.
Going back to \eqref{last12}, it holds that
\begin{equation}\label{last1211}
\begin{array}{lll}
\hat{J}^p_1(x,t) (\d(x))^{p\a(1-s)}& \leqslant& \dfrac{C ||f||^{p-m}_{L^m(\O_T)}}{(\d(x))^{p(1-\frac{s}{m'})+N(\frac{p}{m}-1)-p\a(1-s)}}\displaystyle\iint_{\O_t}f^m(y,\sigma){P}_{\Omegaega} (x,y, t-\s)\,dy\,d\s.
\end{array}
\end{equation}
Setting $\hat{\beta}=p(1-\dfrac{s}{m'})+N(\dfrac{p}{m}-1)-p\a(1-s)$, then
$\hat{\beta}=\dfrac{p}{m}(s+(1-s)m+N-\alpha m(1-s))-N$.
By the above condition on $p$ and $\a$, we deduce that $\hat{\beta}<2s$. Repeating the argument used in the first case, it holds that
$$
\iint_{\O_T}\hat{J}^p_{1}(x,t)(\d(x))^{p\a(1-s)}dxdt\leqslantC||f||^{p}_{L^m(\O_T)},
$$
and then we conclude.
\end{proof}
\begin{remark}
Notice that $u\in L^{2s}(0,T;W^{1,2s}_0(\O))$ if $m>\dfrac{2s(N+2s)}{N+2s^2}$.
\end{remark}
Now, if $f\in L^1(\O_T)\cap L^m(K\times (0,T))$, where $m>1$ and $K\subset\subset \O$ is any compact set of $\O$, then the regularity result of Theorem {I\!\!R}f{regu-g} holds locally in $\O\times (0,T)$. More precisely we have
\begin{Proposition}\label{key2-locc}
{ Suppose that $m>1$ and assume that $f\in L^1(\O_T)\cap L^m(K\times (0,T))$ for every compact set $K\subset\subset \O$.} Let $u$ be the unique weak solution to
problem \eqref{eq:def} and consider $\O_1\subset\subset \O$ with $\text{dist}(\O_1,\partial\O)>0$. Let $K_1\subset\subset \O$ be a compact set of $\O$ such that
$\O_1\subset\subset K$. Then $u\in L^\theta(\O_1\times (0,T))$ for all $\theta<\dfrac{m (N+2s)}{(N+2s-ms)_+}$ and $|\nabla u|\in L^p(\O_1\times (0,T))$ for all
$p<\dfrac{m (N+2s)}{(N+2s-m(2s-1))_+}$. Moveover,
\begin{equation}\label{eqq1}
||u||_{L^\theta(\O_1\times (0,T))}\leqslantC (||f||_{L^m(K_1\times (0,T))}+||f||_{L^1(\O_T)}),
\end{equation}
and
\begin{equation}\label{eqq2}
\|\nabla u\|_{L^p(\O_1\times (0,T))}\leqslantC (||f||_{L^m(K_1\times (0,T))}+||f||_{L^1(\O_T)}),
\end{equation}
where $C:=C(K_1,\O_1, \O, T,N,m)$.
\end{Proposition}
\begin{proof}
Since $f\in L^1(\O_T)$, then $|\nabla u|\in L^{q}(\O_T)$ for all $q<\dfrac{N+2s}{N+1}$. Now we closely follow the proofs of Theorem {I\!\!R}f{hardy0} and Theorem
{I\!\!R}f{regu-g}. We have
$$
\begin{array}{rcl}
\dfrac{u(x,t)}{\d^s(x)} &\leq& C \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.\\
\end{array}
$$
Fix $\O_1\subset\subset \O$ with $\text{dist}(\O_1,\partial\O)=c_0>0$ and let $K_1$ be a compact set of $\O$ such that $\O_1\subset\subset K_1\subset\subset \O$. Let
$x\in \O_1$, then
$$
\begin{array}{rcl}
u(x,t) &\leqslant& C(\O_1, c_0, C)\displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ &\leqslant& C(\O_1, c_0,
C)\bigg\{\displaystyle\int_{0}^{t}\displaystyle\int_{K_1}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ & + &
\displaystyle\int_{0}^{t}\displaystyle\int_{(\O\backslash K_1)}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\bigg\}.
\end{array}
$$
Since $x\in \O_1\subset\subset K_1$, then for all $y\in \O\backslash K_1, |x-y|>\hat{c}>0$. Thus
$$
\begin{array}{rcl}
& &\displaystyle\int_{0}^{t}\displaystyle\int_{(\O\backslash K_1)}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ & \leqslant&
\displaystyle\int_{0}^{t}\displaystyle\int_{(\O\backslash K_1)}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+\hat{c})^{N+2s}}\,dy\,d\sigma\\ & \leqslant& C
||f||_{L^1(\O_T)}.
\end{array}
$$
Therefore we conclude that
$$
\begin{array}{rcl}
u(x,t) &\leqslant& C(\O_1, c_0, C)\bigg(\displaystyle\int_{0}^{t}\displaystyle\int_{K_1}f(y,\s) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma
+||f||_{L^1(\O_T)}\bigg).
\end{array}
$$
Since $f\in L^m(K_1\times (0,T))$, then the rest of the proof follows exactly from the same duality argument as in the proof of Theorem {I\!\!R}f{hardy0}. Hence, estimate
\eqref{eqq1} holds. In a similar way, we prove estimate \eqref{eqq2}.
\end{proof}
\begin{remarks}\label{rm001}
\
\begin{enumerate}
\item For $a\geqslant1$, we define the space $L^{a}_{loc}(\Omegaega_T)$ as the set of measurable functions $u$ such that $u\eta\in L^{a}(\Omegaega_T)$, for any $\eta\in \mathcal{C}^\infty_0(\O)$. Then the result of Proposition {I\!\!R}f{key2-locc} affirms that if $u$ is the unique solution to problem \eqref{eq:def}, then
$u\in L^\theta_{loc}(\O_T)$ for all $\theta<\dfrac{m (N+2s)}{(N+2s-ms)_+}$ and $|\nabla u|\in L_{loc}^p(\O_T)$ for all $p<\dfrac{m (N+2s)}{(N+2s-m(2s-1))_+}$.
\item The result of Proposition {I\!\!R}f{key2-locc} will be useful in order to get $\mathcal{C}^1$ regularity using a bootstrap argument if, in addition, we have global
bounds in $L^1$ and a local family of bounds in a suitable $L^m_{loc}$ space.
\end{enumerate}
\end{remarks}
\noindent We deal now with the case $f\equiv 0$ and $u_0\in L^\theta(\O)$ with $\theta\geqslant1$. Following the same computations as above, we get the next results.
\begin{Theorem}\label{u0}
Suppose that $f\equiv 0$ and $u_0\in L^\rho(\O)$ with $\rho\geqslant1$. If $u$ is the unique weak solution to problem \eqref{eq:def}, then $\dfrac{u}{\d^s}\in
L^\theta(\O_T)$ for all ${\theta<\dfrac{\rho(N+2s)}{N+s\rho}}$ and $|\nabla u|\d^{1-s}\in L^{p}(\O)$ for all $p<\dfrac{\rho(N+2s)}{N+\rho}$. Moveover,
$$
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)}+|||\nabla u|\d^{1-s}||_{L^p(\O_T)}\leqslantC(\O_T, p,\theta) ||u_0||_{L^\rho(\O)}.
$$
\end{Theorem}
\begin{proof}
For the reader convenience, we include here some details for the estimate of the term $\dfrac{u}{\d^s}$.
$$
\begin{array}{rcl}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)} &=& \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}}(\O_T)\leqslant1\}} \iint_{\O_T}\partialhi(x,t) \frac{u(x,t)}{\d^s(x)}dxdt\\ &\leqslant& \displaystyle
\sup_{\{||\partialhi||_{L^{\theta'}}(\O_T)\leqslant1\}} \iint_{\O_T}|\partialhi(x,t)|\displaystyle\int_{\Omegaega}u_0(y) \dfrac{t^{{\frac 12}}}{(t^{\frac{1}{2s}}+|x -y|)^{N+2s}}\,dy\,dxdt\\
& \leqslant& \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T\int\lim\limit_{n\to\infty}tss_\Omega \int\lim\limit_{n\to\infty}tss_\Omega |\partialhi(x,t)|
H(x-y,t)u_0(y)dydx dt
\end{array}
$$
with $\theta'=\dfrac{\theta}{\theta-1}$. Similarly to the proofs above, by Young's inequality, we have that
\begin{equation*}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)}\leqslantC\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)} ||u_0||_{L^\rho(\O)}||H(., t)||_{L^a(\O)}d\sigma dt,
\end{equation*}
where
$$
H(|x-y|,t)= \dfrac{t^{{\frac 12}}}{(t^{\frac{1}{2s}}+|x-y|)^{N+2s}}
$$
and $\dfrac{1}{\theta'}+\dfrac{1}{\rho}+\dfrac{1}{a}=2$.
Notice that
$$
||H(., t)||_{L^a(\O)}\leqslantC t^{-\frac{1}{2}+\frac{N}{2s a}-\frac{N}{2s}}.
$$
Therefore,
$$
\begin{array}{rcl}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)} &\leqslant& C ||u_0||_{L^\rho(\O)} \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)}
t^{-\frac 12+\frac{N}{2s a}-\frac{N}{2s}}dt
\end{array}
$$
and by using the H\"older inequality, we get
$$
\begin{array}{lll}
\Big|\Big|\dfrac{u}{\d^s}\Big|\Big|_{L^\theta(\O_T)} &\leqslant& C ||u_0||_{L^\rho(\O)} \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} ||\partialhi||_{L^{\theta'}(\O_T)}
\displaystyle\left(\int_0^T t^{\theta(-\frac 12+\frac{N}{2s a}-\frac{N}{2s})}dt\displaystyle\right)^{\frac{1}{\theta}}.
\end{array}
$$
The last integral is finite if and only if $\theta(-\dfrac 12+\dfrac{N}{2s a}-\dfrac{N}{2s})>-1$, thus
$$ \dfrac{N}{2s}(1-\frac{1}{a})<\frac{1}{\theta}-\frac 12.
$$
Since $\dfrac{1}{\rho}+\dfrac{1}{a}=1+\dfrac{1}{\theta}$
then $1-\dfrac{1}{a}=\dfrac{1}{\rho}-\dfrac{1}{\theta}$.
Substituting in the previous inequality, we conclude that
${{\theta<\dfrac{\rho (N+2s)}{N+\rho s}}}$.
To estimate the gradient term we consider that, by the representation formula, we have
$$
\begin{array}{rcl}
|\nabla u(x,t)|&\leq&
C(\O_T) \displaystyle\int_{\Omegaega} u_0(y)|\nabla_x P_{\Omegaega} (x,y, t)|\,dy\leqslantC(\O_T)\displaystyle\int_{\Omegaega} u_0(y)\frac{|\nabla_x P_{\Omegaega} (x,y, t)|}{P_{\Omegaega} (x,y, t)}\,P_{\Omegaega} (x,y, t) dy\,\\
&\leq& C(\O_T)\displaystyle\int_{\Omegaega} u_0(y) h(x,y,t){P_{\Omegaega} (x,y, t)}\,dy
\end{array}
$$
with $h(x,y,t)=\dfrac{|\nabla_x P_{\Omegaega} (x,y, t)|}{P_{\Omegaega} (x,y, t)}\leqslantC \Big( \dfrac{1}{\deltalta(x) \wedge t^{\frac{1}{2s}}}\Big)$.
Hence
\begin{eqnarray*}
|\nabla u(x,t)| & \leqslant& \frac{C}{t^{\frac{1}{2s}}}}{ \chi_{\{\d(x)>t^{\frac{1}{2s}}\}}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y){P_{\Omegaega} (x,y, t)}\,dy +
\frac{C}{\d(x)}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y){P_{\Omegaega} (x,y, t)}\,dy\\
& = & J_{11}(x,t)+ J_{12}(x,t).
\end{eqnarray*}
We again use the duality argument, starting by the estimation of the term $J_{11}$.
By estimate \eqref{green1} and H\"older inequality, we obtain that
$$
\begin{array}{rcl}
||J_{11}||_{L^p(\O_T)} &=& \displaystyle \sup_{\{||\partialhi||_{L^{p'}}(\O_T)\leqslant1\}} \iint_{\O_T}\partialhi(x,t) J_{11}(x,t)dxdt\\ &\leqslant& \displaystyle
\sup_{\{||\partialhi||_{L^{p'}}(\O_T)\leqslant1\}} \iint_{\O_T}|\partialhi(x,t)|\displaystyle\int_{\Omegaega}u_0(y) \dfrac{t^{1-\frac 1{2s}}}{(t^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,dxdt\\
& \leqslant& \displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T\int\lim\limit_{n\to\infty}tss_\Omega \int\lim\limit_{n\to\infty}tss_\Omega |\partialhi(x,t)|
H(x-y,t)u_0(y)dydx dt
\end{array}
$$
with $H(x,t)=\dfrac{t^{1-\frac 1{2s}}}{(t^{\frac{1}{2s}}+|x|)^{N+2s}}$.
Using Young inequality, it holds that
\begin{equation*}
||J_{11}||_{L^p(\O_T)}\leqslantC\displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{p'}(\O)} ||u_0||_{L^\rho(\O)}||H(., t)||_{L^a(\O)}d\sigma dt,
\end{equation*}
with $\dfrac{1}{\theta'}+\dfrac{1}{\rho}+\dfrac{1}{a}=2$.
Notice that by a direct calculation
$$
||H(., t)||_{L^a(\O)}\leqslantC t^{(1-\frac 1{2s})+\frac{N}{2s a}-\frac{(N+2s)}{2s}}.
$$
Therefore we reach that
$$
\begin{array}{rcl}
||J_{11}||_{L^\theta(\O_T)} &\leqslant& C ||u_0||_{L^\rho(\O)} \displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{p'}(\O)}
t^{(1-\frac 1{2s})+\frac{N}{2s a}-\frac{(N+2s)}{2s}}dt.
\end{array}
$$
Using H\"older inequality, we get
$$
\begin{array}{lll}
||J_{11}||_{L^p(\O_T)} &\leqslant& C ||u_0||_{L^\rho(\O)} \displaystyle \sup_{\{||\partialhi||_{L^{p'}(\O_T)}\leqslant1\}} ||\partialhi||_{L^{p'}(\O_T)}
\displaystyle\left(\int_0^T t^{p((1-\frac 1{2s})+\frac{N}{2s a}-\frac{(N+2s)}{2s})}dt\displaystyle\right)^{\frac{1}{p}}.
\end{array}
$$
The last integral is finite if and only if $p((1-\dfrac 1{2s})+\dfrac{N}{2s a}-\dfrac{(N+2s)}{2s})>-1$.
Thus
$$\dfrac 1{2s}+\dfrac{N}{2s}(1-\frac{1}{a})<\frac{1}{p}.
$$
Since $\dfrac{1}{p'}+\dfrac{1}{\rho}+\dfrac{1}{a}=2$ we have
$\dfrac{1}{\rho}+\dfrac{1}{a}=1+\dfrac{1}{p}$.
Then $1-\dfrac{1}{a}=\dfrac{1}{\rho}-\dfrac{1}{p}$.
Going back to the previous inequality, we conclude that
$$
\dfrac{1}{2s}+\dfrac{1}{\rho} \dfrac{N}{2s}<\dfrac{1}{p}(1+\dfrac{N}{2s}).
$$
Hence $J_{11}\in L^p(\O_T)$ for all $p<\dfrac{\rho (N+2s)}{N+\rho}$.
We deal now with $J_{12}$.
We have
$$
\begin{array}{lll}
J_{12}(x,t) &= & \dfrac{C}{\d(x)}\displaystyle\int\lim\limit_{n\to\infty}tss_{\{\Omegaega\cap \{\d(x)\leqslantt^{\frac{1}{2s}}\}\}} u_0(y){P_{\Omegaega} (x,y, t)}\,dy\\
& \leqslant& C \frac{C}{\d^{1-s}(x)}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\dfrac{t^{1-{ \frac 1{2}}}}{(t^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy.\\
\end{array}
$$
Hence
$$
J_{12}(x,t)\d^{1-s}(x)\leqslantC\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\dfrac{t^{{{\frac 12}}}}{(t^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy,
$$
that is the same term estimated above. Hence we conclude that $J_{12}\d^{1-s}\in L^p(\O_T)$ for all $p<\dfrac{\rho (N+2s)}{N+s\rho}$.
Since $\dfrac{\rho (N+2s)}{N+s\rho}>\dfrac{\rho (N+2s)}{N+\rho}$, then using the fact that $\O_T$ is bounded, it follows that $|\nabla u|\d^{1-s}(x)
\in L^p(\O_T)$ for all $p<\dfrac{\rho (N+2s)}{N+\rho}$.
\end{proof}
As in Corollary {I\!\!R}f{cor11}, we have the next regularity for the gradient.
\begin{Corollary}\label{cor1100}
Assume that $f\equiv 0$ and $u_0\in L^\rho(\O)$ with $\rho\geqslant1$. If $u$ is the unique weak solution to problem \eqref{eq:def}, then $|\nabla u|\in L^{a}(\O)$ for all $a<\check{U}:=\dfrac{\rho(N+2s)}{(1-s)\rho(N+2s)+N+\rho}$. Moveover $\check{U}>\dfrac{N+2s}{N+1}$ if $\dfrac{N+2s}{N}<\frac{1}{1-s}$ and $\rho>\dfrac{N}{sN-2s(1-s)}$.
\end{Corollary}
{ As a conclusion, from Proposition {I\!\!R}f{first11} and using the same approach as in the proof of Theorem {I\!\!R}f{u0}, keeping the power in the time variable, we get the next results. }
\begin{Proposition}\label{pro:lp2}
Suppose that $f\equiv 0$ and $u_0\in L^\rho(\O)$. If $u$ is the unique weak solution to problem \eqref{eq:def}, then for all $r\geqslant1$ and for all $t>0$, we
have
\begin{equation}\label{sem1}
||u(\cdot,t)||_{L^r(\O)}\leqslantCt^{-\frac{N}{2s}(\frac{1}{\rho}-\frac{1}{r})}||u_0||_{L^\rho(\O)},
\end{equation}
\begin{equation}\label{sem001}
\Big\|\frac{u(\cdot,t)}{\d^s}\Big\|_{L^r(\O)}\leqslantCt^{-\frac{N}{2s}(\frac{1}{\rho}-\frac{1}{r})-\frac{1}{2s}}||u_0||_{L^\rho(\O)},
\end{equation}
and
\begin{equation}\label{sem111}
||\nabla u(\cdot,t)\d^{1-s}||_{L^r(\O)}\leqslantCt^{-\frac{N}{2s}(\frac{1}{\rho}-\frac{1}{r})-\frac{1}{2s}}||u_0||_{L^\rho(\O)}.
\end{equation}
Moreover, $u\in L^{\s}(\O_T)$ for all $\s<\frac{\rho(N+2s)}{N}$ and $\dfrac{u}{\d^s}, |\nabla u|\d^{1-s}\in L^{\gamma}(\O_T)$ for all $\gamma<\frac{\rho(N+2s)}{N+\rho}$.
\end{Proposition}
{ Within the same framework of the Theorem {I\!\!R}f{regu-glast}, in order to get regularity for the gradient term without any degenerate weight we have the next result.
\begin{Theorem}\label{lastu0}
Suppose that $f\equiv 0$ and $u_0\in L^\rho(\O)$ and let $\rho_1$ be defined by $\rho_1=\min\{\rho, 2s\}$. If $u$ is the unique weak solution to problem \eqref{eq:def}, then $|\nabla u|\in L^{q}(\O)$ for all $q<\dfrac{\rho_1(N+2s)}{N+\rho_1}$. Moveover,
$$
|||\nabla u|||_{L^q(\O_T)}\leqslantC(\O_T, p,\theta) ||u_0||_{L^\rho(\O)}.
$$
\end{Theorem}
}
\begin{proof}
To estimate the gradient term we consider that, by the representation formula, we have
\begin{eqnarray*}
& & |\nabla u(x,t)|\leq
C(\O_T) \displaystyle\int_{\Omegaega} u_0(y)|\nabla_x P_{\Omegaega} (x,y, t)|\,dy\leqslantC(\O_T)\displaystyle\int_{\Omegaega} u_0(y)\frac{|\nabla_x P_{\Omegaega} (x,y, t)|}{P_{\Omegaega} (x,y, t)}\,P_{\Omegaega} (x,y, t) dy\,\\
& & \leqslant\frac{C}{t^{\frac{1}{2s}}}\chi_{\{\d(x)>t^{\frac{1}{2s}}\}}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y){P_{\Omegaega} (x,y, t)}\,dy +
\frac{C}{\d(x)}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}}\int\lim\limit_{n\to\infty}tss_\Omega u_0(y){P_{\Omegaega} (x,y, t)}\,dy\\
& &= \breve{J}_{1}(x,t)+ \breve{J}_{2}(x,t).
\end{eqnarray*}
As in the proof of Theorem {I\!\!R}f{u0}, using the duality argument we obtain that, for all $q<\dfrac{\rho(N+2s)}{N+\rho}$,
$$
||\breve{J}_{1}||_{L^q(\O_T)}\leqslantC ||u_0||_{L^\rho(\O)}.
$$
We treat now $\breve{J}_{2}$. We have
{
$$
\begin{array}{lll}
\breve{J}_{2}(x,t) &= & \displaystyle \dfrac{C}{\d(x)}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \int\lim\limit_{n\to\infty}tss_\Omega u_0(y){P_{\Omegaega} (x,y, t)}\,dy\\
&\leqslant& \displaystyle \dfrac{C}{\d(x)}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y){P_{\Omegaega} (x,y, t)}\,dy\bigg)^{\frac{1}{\rho}}
\bigg(\int\lim\limit_{n\to\infty}tss_\O{P_{\Omegaega} (x,y, t)}\,dy\bigg)^{\frac{1}{\rho'}}\\
& \leqslant& \displaystyle \dfrac{C}{\d(x)} \chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y){P_{\Omegaega} (x,y, t)}\,dy\bigg)^{\frac{1}{\rho}},
\end{array}
$$
where we have used the fact that
$$
\int\lim\limit_{n\to\infty}tss_\O{P_{\Omegaega} (x,y, t)}\,dy\leqslantC\frac{\d^s(x)}{\sqrt{t}}\leqslantC \mbox{ in the set } \{\d(x)\leqslantt^{\frac{1}{2s}}\}.
$$
}
Hence, as in the estimate of the term $\hat{J}_1$ in Theorem {I\!\!R}f{regu-glast}, we deduce that
$$
\begin{array}{lll}
\breve{J}^q_{2}(x,t) &\leqslant& \displaystyle \dfrac{C}{\d^q(x)}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y){P_{\Omegaega} (x,y, t)}\,dy\bigg)^{\frac{q}{\rho}}\\
&\leqslant& \displaystyle\dfrac{C}{\d^q(x)} \chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y){P^{\frac{q}{\rho}}_{\Omegaega} (x,y, t)}\,dy\bigg)\bigg(\int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y) dy \bigg)^{\frac{q-\rho}{\rho}}\\
&\leqslant& \displaystyle \dfrac{C||u_0||^{\frac{q-\rho}{\rho}}_{L^\rho(\O)}}{\d^q(x)} \chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y){P^{\frac{q}{\rho}}_{\Omegaega} (x,y, t)}\,dy\bigg)\\
&\leqslant& \displaystyle \dfrac{C||u_0||^{\frac{q-\rho}{\rho}}_{L^\rho(\O)}}{(\d(x))^{q+(\frac{q}{\rho}-1)N}}\chi_{\{\d(x)\leqslantt^{\frac{1}{2s}}\}} \int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y){P}_{\Omegaega} (x,y, t)\,dy.\\
\end{array}
$$
Setting $\beta_0=q+(\frac{q}{\rho}-1)N$, then $\beta<2s$ if and only if $q<\frac{\rho(N+2s)}{N+\rho}$. It is clear that $\frac{\rho(N+2s)}{N+\rho}>\rho$ if and only if $\rho<2s$. Thus assuming the above assumptions and integrating in $\O_T$, we obtain that
$$
\iint_{\O_T}\breve{J}^q_{2}(x,t) dxdt \leqslantC||u_0||^{\frac{q-\rho}{\rho}}_{L^\rho(\O)} \int\lim\limit_{n\to\infty}tss_\Omega u^\rho_0(y)\partialhi (y) dy,
$$
where, as above, $\partialhi(y)=\displaystyle\int\lim\limit_{n\to\infty}tss_\Omega \frac{\mathcal{G}_s(x,y)}{(\d(x))^{\beta_0}}dx\leqslantC$.
Hence
$$
\iint_{\O_T}\breve{J}^q_{2}(x,t) dxdt \leqslantC||u_0||^{\frac{q}{\rho}}_{L^\rho(\O)}.
$$
Thus $u\in L^q((0,T); W^{1,q}_0(\O))$ and
$$
|| |\nabla u|||_{L^q(\O_T)}\leqslantC(\O_T,N,s,p) ||u_0||_{L^\rho(\O)}.
$$
\end{proof}
Notice that, from {\cal C}_0^{\iy}\,te{BKW}, see also {\cal C}_0^{\iy}\,te{W}, working in the whole space ${I\!\!R}n$, the above regularity result on the gradient holds globally. However, when working in a bounded domain, the term $\deltalta^{1-s}$ appears in a natural way describing the gradient's behavior.
\
Under the local summability condition $f\d^\beta\in L^1(\O_T)$ for some $\beta<2s-1$, it is possible to show the existence of weak solution with the same range of regularity. This will be the key in order to analyze problem \eqref{grad} for large value of $\a$.
More precisely, we have the following existence result.
\begin{Theorem} \label{key}
Assume that $f, u_0$ are measurable functions such that $f\d^\beta \in L^1(\O_T)$ for some $\beta<(2s-1)$ and $u_0\in L^1(\O)$.
Then problem \eqref{eq:def} has a
unique weak solution $u$ such that for all $q<\dfrac{N+2s}{N+\beta+1}$,
$$
||u||_{\mathcal{C}([0,T],L^1(\O, \d^\beta dx))}+ ||\nabla u||_{L^{q}(\Omegaega_T)}\leqslantC(q,\beta,\Omegaega_T)\bigg(||f\d^\beta
||_{L^{1}(\Omegaega_T)}+||u_0||_{L^1(\O)}\bigg).
$$
Moreover, for $q<\dfrac{N+2s}{N+\beta+1}$ fixed, setting $\hat{K}: L^{1}(\Omegaega_T, \d^\beta(x)dxdt)\times L^1(\O)\to L^q(0,T; W_{0}^{1,q}(\Omegaega))$, $\hat{K}(f,
u_0)=u$, the unique solution to problem \eqref{eq:def}, then $\hat{K}$ is a compact operator.
\end{Theorem}
\begin{proof}
Without loss of generality, we can assume that $u_0=0$ and that $f\gneqq 0$ in $\O_T$. We begin by proving the existence part. Let $f_n=T_n(f)$ and define $u_n$
to be the unique energy solution to the approximating problem
\begin{equation}\label{apprd}
\left\{
\begin{array}{rcll}
u_{nt}+(-\D)^s u_n&=& \displaystyle f_n & \text{ in } \O_{T}, \\ u_n&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\ u(x,0)&=& 0& \mbox{
in }\O.
\end{array}
\right.
\end{equation}
It is clear that $\{u_n\}_n$ is an increasing sequence in $n$. Let $\partialsi$ be the solution to the problem
\begin{equation}\label{adim}
\left\{
\begin{array}{rcll}
(-\Deltalta)^s \partialsi&=& \dfrac{1}{\d^{2s-\beta}} & \text{ in } \O, \\ \partialsi &=&0 & \text{ in }{I\!\!R}n\setminus\O,
\end{array}
\right.
\end{equation}
whose existence is a consequence of {\cal C}_0^{\iy}\,te{CV2}. By the results in {\cal C}_0^{\iy}\,te{abatan} and {\cal C}_0^{\iy}\,te{Adm} extending the results in {\cal C}_0^{\iy}\,te{quim}, we find that $\partialsi\backsimeq C \d^\beta$. Hence, by an approximation argument, we can use $\partialsi$ as a test function in \eqref{apprd} to obtain that
$$
\sup_{t\in [0,T]}\int\lim\limit_{n\to\infty}tss_\Omega u_n(x,t) \d^\beta(x) dx +\int_0^T\int\lim\limit_{n\to\infty}tss_\Omega \frac{u_n(x,t)}{\d^{2s-\beta}(x)}dxdt \leqslantC(\O,\beta,s) \int_0^T\int\lim\limit_{n\to\infty}tss_\Omega f\d^\beta(x)dxdt.
$$
Hence there exists a measurable function $u\in L^\infty(0,T;L^1(\O, \d^\beta dx))\cap L^1(\O_T)$, $\dfrac{u}{\d^{2s-\beta}}\in L^1(\O_T)$, such that
$\dfrac{u_n}{\d^{2s-\beta}}\to \dfrac{u}{\d^{2s-\beta}}$ strongly in $L^1(\O_T)$.
We claim that the sequence $\{u_n\}_n$ is bounded in $L^\theta(\O_T)$, for all $\theta<\frac{N+2s}{N+\beta}$.
To prove the claim consider the representation formula,
$$
u_n(x,t)=\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f_n(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma,
$$
then using the properties of $P_\O$, it holds that
$$
\begin{array}{rcl}
u_n(x,t) &\leq& C \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}\Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t-\s}}\Big) f(y,\s)
\dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ &\leqslant& C \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}\Big( 1\wedge
\frac{\deltalta^s(y)}{\sqrt{t-\s}}\Big)\frac{1}{\d^\beta(y)} \, f(y,\s)\d^\beta(y) \dfrac{(t-\s)}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.\\
\end{array}
$$
Since
$$
\Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t-\s}}\Big)\frac{1}{\d^\beta(y)}\leqslantC(\O_T,s,\beta)(t-\s)^{-\frac{\beta}{2s}} \mbox{ in }\O_T,
$$
we conclude that
$$
u_n(x,t)\leqslantC \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}\, f(y,\s)\d^\beta(y) \dfrac{(t-\s)^{\frac{2s-\beta}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.
$$
Setting $g(y,\s)=f(y,\s)\d^\beta(y)$, then $g\in L^1(\O_T)$. We use the duality argument as in the proof of Theorem {I\!\!R}f{hardy0}. For the reader convenience we
include here some details.
Let $\partialhi\in \mathcal{C}^\infty_0(\O_T)$, then
$$
\begin{array}{rcl}
||u_n||_{L^\theta(\O_T)} &=& \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \iint_{\O_T}\partialhi(x,t) u_n(x,t)dxdt\\ &\leqslant& \displaystyle
\sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \iint_{\O_T}|\partialhi(x,t)| \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}\, g(y,\s)\d^\beta(y)
\dfrac{(t-\s)^{\frac{2s-\beta}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ &\leqslant& \displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}}
\int_0^T\int_0^t \int\lim\limit_{n\to\infty}tss_\Omega \int\lim\limit_{n\to\infty}tss_\Omega |\partialhi(x,t)| H(x-y,t-\s)g(y,\s)dydx d\sigma dt,
\end{array}
$$
where $H(x,\s)=\dfrac{\s^{\frac{2s-\beta}{2s}}}{(\s^{\frac{1}{2s}}+|x|)^{N+2s}}$.
Using Young inequality, it holds that
\begin{equation}\label{ttbeta}
\begin{array}{rcl}
||u_n||_{L^\theta(\O_T)} &\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)} \int_0^t ||g(.,\s)||_{L^1(\O)}||H(.,
t-\s)||_{L^a(\O)}d\sigma dt,
\end{array}
\end{equation}
with $\dfrac{1}{\theta'}+\dfrac{1}{a}=1$ { and then $a=\theta$.}
By a direct computation we deduce that $||H(.,t-\s)||_{L^a(\O)} \leqslantC(t-\s)^{\frac{2s-\beta}{2s}+\frac{N}{2s a}-\frac{(N+2s)}{2s}}$. Thus
$$
\begin{array}{rcl}
||u_n||_{L^\theta(\O_T)} &\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T ||\partialhi(.,t)||_{L^{\theta'}(\O)} \int_0^t ||g(.,\s)||_{L^1(\O)}
(t-\s)^{\frac{2s-\beta}{2s}+\frac{N}{2s a}-\frac{(N+2s)}{2s}} d\sigma dt\\ &\leqslant& C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} \int_0^T
||g(.,\s)||_{L^1(\O)}\int_{\s}^T||\partialhi(.,t)||_{L^{\theta'}(\O)} (t-\s)^{\frac{2s-\beta}{2s}+\frac{N}{2s a}-\frac{(N+2s)}{2s}} d\sigma dt.
\end{array}
$$
Using H\"older inequality, we get
$$
\begin{array}{lll}
& ||u_n||_{L^\theta(\O_T)}\le\\ & C\displaystyle \sup_{\{||\partialhi||_{L^{\theta'}(\O_T)}\leqslant1\}} ||\partialhi||_{L^{\theta'}(\O_T)} \int_0^T ||g(.,\s)||_{L^1(\O)}\bigg(\int_{\s}^T
(t-\s)^{\theta[\frac{2s-\beta}{2s}+\frac{N}{2s a}-\frac{(N+2s)}{2s}]} d\s\bigg)^\frac{1}{\theta} dt.
\end{array}
$$
The last integral is finite if and only if $\theta[\dfrac{2s-\beta}{2s}+\dfrac{N}{2s a}-\dfrac{(N+2s)}{2s}]>-1$. Since $\dfrac{1}{\theta'}+\dfrac{1}{a}=1$, then the
above condition holds if $\theta<\dfrac{N+2s}{N+\beta}$ and then the claim follows. Thus
\begin{equation}\label{eq:beta1}
||u_n||_{L^\theta(\O_T)}\leqslantC(\O_T,s,\beta)||g||_{L^1(\O_T)}=C||f\d^\beta||_{L^1(\O_T)}.
\end{equation}
In the same way we can prove that the sequence $\{\dfrac{u_n}{\d^s}\}_n$ is bounded in $L^\theta(\O_T)$ for all $\theta<\dfrac{N+2s}{N+s+\beta}$.
To show that the sequence $\{|\nabla u_n|\}_n$ is bounded in $L^p(\O_T)$ for all $1\leqslantp<\dfrac{N+2s}{N+\beta+1}$ we use the same arguments as in the proof of Theorem {I\!\!R}f{regu-g}. We have
$$
\begin{array}{rcl}
& & |\nabla u_n(x,t)|\leqslantC \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f_n(y,\sigma) |\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma\\ \\ &\leq& C \displaystyle \displaystyle\iint_{\{\O\times (0,t)\}}f_n(y,\sigma) \frac{|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|}{P_{\Omegaega} (x,y, t-\sigma)}P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\\ \\ &\leqslant& C\Big(\displaystyle\iint_{\{\O\times (0,t) \cap\{\d(x)>t^{\frac{1}{2s}}\}\}\}} f_n(y,\s)\d^\beta(y)
\frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{\b+1}{2s}}}\,dy\,d\sigma \\
& + & \displaystyle \dfrac{1}{\d(x)}\displaystyle\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslantt^{\frac{1}{2s}}\}\}\}} f_n(y,\s) P_{\Omegaega} (x,y, t-\sigma) \,dy\,d\sigma \bigg)\\ \\ &=& I_{1n}(x,t)+I_{2n}(x,t).
\end{array}
$$
Let us begin by estimating $I_{1n}$. Using estimate \eqref{green1} and by H\"older inequality, we get
\begin{eqnarray*}
I^q_{1n}(x,t) &=& \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)>t^{\frac{1}{2s}}\}\}\}}f_n(y,\sigma)\d^\beta(y) \frac{(t-\s)^{1-\frac{1+\beta}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} \,dy\,d\sigma\Big)^q\\
& \leqslant & C||f_n\d^\beta||^{q-1}_{L^1(\O_T)}\iint_{\{\O\times (0,t)\}\}}f_n(y,\sigma)\d^\beta(y) \frac{(t-\s)^{q\frac{2s-\beta-1}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{q(N+2s)}} \,dy\,d\sigma.
\end{eqnarray*}
Integrating in $\O_T$, as in the proof of Theorem {I\!\!R}f{regu-g}, we deduce that
$$
\displaystyle \iint_{\O_T}I^q_{1n}(x,t) dxdt\leqslantCT^{\g_1+1}||f_n\d^\beta||^{q}_{L^1(\O_T)},
$$
with $\g_1=\dfrac{N}{2s}-q\dfrac{N+\beta+1}{2s}>-1$.
Now respect to $I_{2n}$, using estimate \eqref{one0} and by H\"older inequality, it follows that
\begin{eqnarray*}
I^q_{2n}(x,t)& =& \dfrac{C}{\d^q(x)}\displaystyle\bigg(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslantt^{\frac{1}{2s}}\}\}\}} f_n(y,\s) P_{\Omegaega} (x,y, t-\sigma) \,dy\,d\sigma \bigg)^q\\ & = & \dfrac{C}{\d^q(x)}\displaystyle\bigg(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslantt^{\frac{1}{2s}}\}\}\}} f_n(y,\s)\d^\beta(y) \frac{P_{\Omegaega} (x,y, t-\sigma)}{\d^\beta(y)}\,dy\,d\sigma \bigg)^q\\
&\leqslant& \frac{C||f_n\d^\beta||^{q-1}_{L^1(\O_T)}}{\d^q(x)}\iint_{\{\O\times (0,t)\}\}}f_n(y,\sigma)\d^\beta(y)\frac{P^q_{\Omegaega} (x,y, t-\s)}{\d^{q\beta(y)}}\,dy,\,d\s\\
& \leqslant& C||f_n\d^\beta||^{q-1}_{L^1(\O_T)}
\iint_{\{\O\times (0,t)\}\}}\frac{f_n(y,\sigma)\d^\beta(y)}{(\d(x))^{(q-1)N+q}}\frac{P_{\Omegaega} (x,y, t-\s)}{\d^{q\beta(y)}}dy\,d\s.
\end{eqnarray*}
Integrating in $\O_T$,
$$
\iint\lim\limit_{n\to\infty}tss_{\O_T}I^q_{2n}(x,t)dxdt\leqslantC||f_n\d^\beta||^{q-1}_{L^1(\O_T)}\iint\lim\limit_{n\to\infty}tss_{\O_T}\frac{f_n(y,\s)\d^\beta(y)}{\d^{q\beta}(y)}\Big(\int\lim\limit_{n\to\infty}tss_\Omega \frac{1}{(\d(x))^{q(N+1)-N}}\int_{\s}^T P_{\Omegaega} (x,y, t-\s)dt\,dx\Big)dy\,d\sigma.
$$
Recalling that
$$
\displaystyle\int_{\s}^{T} P_{\Omegaega} (x,y,
t-\s)\,dt\leqslant\displaystyle\int_{0}^{T-\s} P_{\Omegaega} (x,y,
\eta)\,d\eta\leqslant\mathcal{G}_s(x,y),
$$
we find that
$$
\iint_{\O_T}I^q_{2n}(x,t)dxdt\leqslantC||f_n\d^\beta||^{q-1}_{L^1(\O_T)}\iint_{\O_T}f_n(y,\s)\d^\beta(y)\,\frac{\varphi(y)}{\d^{q\beta}(y)}dy,
$$
where $\varphi$ is the unique solution to problem \eqref{varphi}. Since $s<q(N+1)-N<2s$ then from {\cal C}_0^{\iy}\,te{abatan} and {\cal C}_0^{\iy}\,te{Adm}, it follows that $\varphi\in L^{\infty}(\O)$ and $\varphi(y)\simeq (\d(y))^{2s-(q(N+1)-N)}$. Thus $\dfrac{\varphi(y)}{\d^{q\beta}(y)}\simeq (\d(y))^{2s-(q(N+1)-N)-q\beta}$.
It is clear that $2s-(q(N+1)-N)-q\beta>0$ if and only if $q<\dfrac{N+2s}{N+1+\beta}$, which is the hypothesis. Thus
$$
\iint_{\O_T}I^q_{2n}(x,t)(x,t)dxdt\leqslantC||f_n\d^\beta||^{q}_{L^1(\O_T)}.
$$
As a conclusion, we have proved that for all $q<\frac{N+2s}{N+1+\beta}$,
$$
||\nabla u_n||_{L^q(\O_T)}\leqslantC(\O_T)||f_n\d^\beta||_{L^1(\O_T)}.
$$
Hence there exists a solution $u$ to problem \eqref{eq:def} in the
sense of distributions such that
$$u\in L^q(0,T; W_{0}^{1,q}(\Omegaega))\cap\mathcal{C}([0,T], L^1(\O, \d^\beta dx)),
\hbox{ for all } q<\frac{N+2s}{N+\beta+1}.$$
It is clear that if $u_1,u_2\in L^q(0,T; W_{0}^{1,q}(\Omegaega))\cap\mathcal{C}([0,T], L^1(\O,\d^\beta dx))$ are solutions
to \eqref{eq:def}, then the difference $v=u_1-u_2$, satisfies $v\in L^q(0,T;
W_{0}^{1,q}(\Omegaega))\cap\mathcal{C}([0,T], L^1(\O, \d^\beta dx))$ and
\begin{equation}\label{vvv}
\left\{
\begin{array}{rcll}
v_{t}+(-\D)^s v&=& \displaystyle 0 & \text{ in } \O_{T}, \\ v&=&0 & \text{ in }({I\!\!R}n\setminus\O) \times (0,T), \\ v(x,0)&=&0& \mbox{ in }\O.
\end{array}
\right.
\end{equation}
Using Kato inequality as in Theorem {I\!\!R}f{Conv}, we reach that $v_+\in L^q(0,T; W_{0}^{1,q}(\Omegaega))\cap\mathcal{C}([0,T], L^1(\O, \d^\beta dx))$ satisfies
\begin{equation}\label{v++}
(v_{+})_t+(-\D)^s v_+\leqslant0 \text{ in } \O_{T}.
\end{equation}
Hence using $\partialhi_1$, the positive first eigenfunction of the fractional Laplacian in $\O$, as a test function in \eqref{v++}, we reach that $v_+=0$. In the same
way and since $-v$ is also a solution to \eqref{vvv}, we conclude that $v_-=0$. Thus $v=0$.
Now setting $$\hat{\mathcal{K}}: L^{1}(\Omegaega_T, \d^\beta(x)dxdt)\times L^1(\O)\to L^q(0,T; W_{0}^{1,q}(\Omegaega)), q<\dfrac{N+2s}{N+\beta+1},$$
where $\hat{\mathcal{K}}(f, u_0)=u$ is
the unique solution to problem \eqref{eq:def}, then as in the proof
of Theorem {I\!\!R}f{gradiente}, taking advantage of the linearity of the operator,
we conclude that $\hat{\mathcal{K}}$ is a compact operator.
\end{proof}
As in proposition {I\!\!R}f{key2-locc}, if
$f\in L^{1}(\Omegaega_T, \d^\beta(x)dxdt)\cap L^m(K\times (0,T))$, with $m>1$
and $K\subset\subset \Omegaega$ is any compact set of $\Omegaega$, then we have the next general regularity result.
\begin{Proposition}\label{key2-beta}
{ Let $m>1$. Assume that $f\in L^{1}(\Omegaega_T, \d^\beta(x)dxdt)\cap L^m(K\times (0,T))$ for any compact set $K$ of $\O$.} Define $u$ to be the
unique weak solution to problem \eqref{eq:def} and let $\O_1\subset\subset \O$ with $\text{dist}(\O_1,\partial\O)>0$. Consider $K_1\subset\subset \O$, a compact set of
$\O$ such that $\O_1\subset\subset K$.
Then $u\in L^\theta(\O_1\times (0,T))$ for all $\theta<\dfrac{m (N+2s)}{(N+2s-m(2s-\beta))_+}$ and $|\nabla u|\in
L^p(\O_1\times (0,T))$ for all $p<\dfrac{m (N+2s)}{(N+2s-m(2s-1-\beta))_+}$. Moveover
\begin{equation}\label{eqq1beta}
||u||_{L^\theta(\O_1\times (0,T))}+ \|\nabla u\|_{L^p(\O_1\times (0,T))}\leqslantC (||f||_{L^m(K_1\times (0,T))}+||f\d^{\beta}||_{L^1(\O_T)}),
\end{equation}
where $C:=C(K_1,\O_1, \O, T,N,m)$.
\end{Proposition}
\begin{proof}
Since $f\d^\beta\in L^1(\O_T)$, then $|\nabla u|\in L^{q}(\O_T)$ for all $q<\frac{N+2s}{N+\beta+1}$. As in the proof of Proposition {I\!\!R}f{key2-locc}, we have
$$
u_n(x,t)\leqslantC \displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}\, f(y,\s)\d^\beta(y) \dfrac{(t-\s)^{\frac{2s-\beta}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.
$$
Fix $\O_1\subset\subset \O$ with $\text{dist}(\O_1,\partial\O)=c_0>0$ and let $K_1$ be a compact set of $\O$ such that $\O_1\subset\subset K_1\subset\subset \O$. Let
$x\in \O_1$, then
$$
\begin{array}{rcl}
u(x,t) &\leqslant& C(\O_1, c_0, C)\bigg\{\displaystyle\int_{0}^{t}\displaystyle\int_{K_1}f(y,\s)\d^\beta(y) \dfrac{(t-\s)^{\frac{2s-\beta}{2s}}
}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\\ & + & \displaystyle\int_{0}^{t}\displaystyle\int_{\O\backslash K_1}f(y,\s)\d^\beta(y)
\dfrac{(t-\s)^{\frac{2s-\beta}{2s }}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\bigg\}.
\end{array}
$$
If $(x,y)\in (\O_1\subset\subset K_1)\times (\O\backslash K_1)$, then $|x-y|>\hat{c}>0$. Hence
$$
\begin{array}{rcl}
\displaystyle\int_{0}^{t}\displaystyle\int_{\O\backslash K_1}f(y,\s)\d^\beta(y) \dfrac{(t-\s)^{\frac{2s-\beta}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma\leqslantC
||f\d^\beta||_{L^1(\O_T)}.
\end{array}
$$
Thus
$$
\begin{array}{rcl}
u(x,t) &\leqslant& C(\O_1, c_0, C)\bigg(\displaystyle\int_{0}^{t}\displaystyle\int_{K_1}f(y,\s)\d^\beta(y) \dfrac{(t-\s)^{\frac{2s-\beta}{2s}}
\,dy\,d\sigma}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} +||f\d^\beta ||_{L^1(\O_T)}\bigg).
\end{array}
$$
Since $f\d^\beta\in L^m(K_1\times (0,T))$, then we exactly follow the same duality argument as in the proof of Theorem {I\!\!R}f{hardy0}. Hence estimate
\eqref{eqq1beta} follows. In a similar way we prove estimate in the gradient.
\end{proof}
\begin{remarks}\label{rm00}
To obtain the above regularity result we have used the fact that if
$$
g(x,t):=\displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}f(y,\s)\dfrac{(t-\s)^a}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma
$$
with $f\in L^m(\O_T), m\geqslant1$ and $a>0$, then $g\in L^\gamma(\O_T)$ where $\g$ satisfies
$$
\frac{1}{\g}>\frac{1}{m}-\frac{2s a}{N+2s}.
$$
This result will be used systematically in what follows.
\end{remarks}
\section{Non existence result.}
In the local case, $s=1$, existence of solution holds for all $\a>0$ at least for $f\in L^\infty(\O_T)$ and $u_0\in L^\infty(\O)$, see for instance {\cal C}_0^{\iy}\,te{BS}, {\cal C}_0^{\iy}\,te{GGK} and {\cal C}_0^{\iy}\,te{QS}. However in the nonlocal case, $s<1$, and by the lack of regularity near of the boundary $\partialartial\Omegaega$, a threshold on $\alpha $ appears for the existence.
This behavior represents a deep difference with the local case, though it is stable when $s\to 1$.
Let us recall and prove the non existence result stated in the introduction.
\begin{Theorem}\label{non1}
Assume that $\a\geqslant\dfrac{1}{1-s}$, then for all nonnegative data $(f,u_0)\in L^\infty(\O_T)\times L^\infty(\O)$ with $(f,u_0)\neq (0,0)$, the problem
\begin{equation}\label{grad-nn}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla u|^{\alpha}+ f & \mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
u(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
has no weak solution $u$ in the sense of Definition {I\!\!R}f{veryweak} with $u\in L^\a(0,T;W^{1,\alpha}_0(\O))$.
\end{Theorem}
Before starting with the proof of Theorem {I\!\!R}f{non1}, we need the following
result that extends to the fractional framework the one proved in {\cal C}_0^{\iy}\,te{Mart}
for the heat equation. The result is proved in {\cal C}_0^{\iy}\,te{BM} using apriori estimates. We give here a different proof using the properties of the Dirichlet heat kernel.
\begin{Proposition}\label{diss}
Assume that the condition on $(f,u_0)$ of the above Theorem holds. Let $w$ be the unique solution to the problem
\begin{equation}\label{grad-nn11}
\left\{
\begin{array}{rcll}
w_t+(-\Deltalta )^s w&=& f & \mbox{ in } \Omegaega_T,\\ w(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ w(x,0)&=&u_{0}(x) &
\mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
with either $u_0\gneqq 0$ or $f\gneqq 0$ in $\O\times (0,t_0)$, $t_0<T$ being fixed. Then there exists $C:=C(t_0,\O,u_0,f)>0$ such that
\begin{equation}\label{contr}
w(x,t_0)\geqslantC \d^s(x) \mbox{ in }\O.
\end{equation}
\end{Proposition}
\begin{proof}
Notice that
\begin{eqnarray*}
w(x,t) &= & \displaystyle\int_{\Omegaega}u_0(y) P_{\Omegaega} (x,y, t)\,dy\,+ \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\\ &\geqslant&
\displaystyle\int_{\Omegaega}u_0(y) P_{\Omegaega} (x,y, t)\,dy.
\end{eqnarray*}
{ Suppose in the first case that $u_0\gneqq 0$, then
$$
w(x,t) \geqslant\displaystyle\int_{\Omegaega}u_0(y) P_{\Omegaega} (x,y, t)\,dy.
$$
}
Now fix $t_0\in (0,T)$, using the estimate on $P_\O$ given in \eqref{green1}, we deduce that
$$
w(x,t_0)\geqslantC\Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{t_0}}\Big)\ \int\lim\limit_{n\to\infty}tss_\Omega \Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t_0}}\Big)\times \Big(t_0^{-\frac{N}{2s}}\wedge
\frac{t_0}{|x-y|^{N+2s}}\Big) u_0(y) dy.
$$
Notice that for all $x, y\in \O$, we have
$$
\Big(t_0^{-\frac{N}{2s}}\wedge \frac{t_0}{|x-y|^{N+2s}}\Big)\geqslantC\dfrac{t_0}{(t_0^{\frac{1}{2s}}+|x-y|)^{N+2s}}\ge
C\dfrac{t_0}{(t_0^{\frac{1}{2s}}+\text{diam}(\O))^{N+2s}},
$$
and
$$
\Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{t_0}}\Big)\geqslant\d^s(x)\min\{\frac{1}{\sqrt{t_0}}, \frac{1}{\deltalta^s(x)}\}\geqslant\d^s(x)\min\{\frac{1}{\sqrt{t_0}},
\frac{1}{\text{diam}^s(\O)}\}={{\d^s(x)}}C(t_0,\O).
$$
Hence
$$
w(x,t_0)\geqslantR(t_0)\deltalta^s(x)\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\d^s(y)dy,
$$
with
$$
R(t)=\bigg(\dfrac{t_0}{(t_0^{\frac{1}{2s}}+\text{diam}(\O))^{N+2s}} (\min\{\frac{1}{\sqrt{t_0}},
\frac{1}{\text{diam}^s(\O)}\})^2\bigg).
$$
Notice that
$$
R(t)\ge
\left\{
\begin{array}{lll}
\dfrac{C(\O)}{t^{\frac{N+2s}{2s}}} & \mbox{ if } & t\geqslant(\text{diam}(\O))^{2s}\\ \\
C(\O)t & \mbox{ if } & t\leqslant(\text{diam}(\O))^{2s}.
\end{array}
\right.
$$
Thus $w(x,t_0)\geqslantC(t_0,u_0,\O)\d^s(x)$ which is the desire estimate.\\
{ Consider now the case where $f\gneqq 0$ in $\O\times (0,t_0)$. As above we deduce that
$$
w(x,t) \geqslant\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma.
$$
}
For $t_0>0$ fixed, using the properties of the kernel $P_\O$, given in \eqref{green1}, it holds that
$$
w(x,t_0)\geqslantC \displaystyle\int_{0}^{t_0} \displaystyle\int_{\Omegaega} \Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{t_0-\s}}\Big)\, \Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t_0-\s}}\Big)\times \Big((t_0-\s)^{-\frac{N}{2s}}\wedge
\frac{t_0-\s}{|x-y|^{N+2s}}\Big) f(y,\s)dy\,d\s.
$$
As in the first case, it follows that
$$
w(x,t_0)\geqslantC \d^s(x)\displaystyle\int_{0}^{t_0} \displaystyle\int_{\Omegaega} R(t_0-\s)f(y,\s)dy\,d\s.
$$
Since $R_0(t_0-\s)\geqslantC(T,\O)(t_0-\s)$, then using the fact that $f\gneqq 0$ in $\O\times (0,t_0$, it follows that
$$
\displaystyle\int_{0}^{t_0} \displaystyle\int_{\Omegaega} R(t_0-\s)f(y,\s)dy\,d\s=C(t_0,f,\O)>0.
$$
Thus we conclude.
\end{proof}
\begin{remark}
It is clear that if $t\in (t_1,t_2)\subset \subset (0,T)$, by
Proposition {I\!\!R}f{diss} we have that, for all $(x,t)\in \O\times (t_1,t_2)$,
$$
w(x,t)\geqslantC(t_0,t_1,\O) \deltalta^s(x)\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\d^s(y)dy.
$$
\end{remark}
We are now in position to prove the non existence result in Theorem {I\!\!R}f{non1}.
\noindent{\bf Proof of Theorem {I\!\!R}f{non1}}. We argue by contradiction. Assume that $\a\geqslant\dfrac{1}{1-s}$ and suppose that problem \eqref{grad-nn} has a solution $u\in
L^\a(0,T; W^{1,\a}_0(\O))$. Fix $(t_1,t_2)\subset \subset (0,T)$, then by Proposition {I\!\!R}f{diss}, we reach that
$$
u(x,t)\geqslantC(t_0,t_1\O) \deltalta^s(x)\int\lim\limit_{n\to\infty}tss_\Omega u_0(y)\d^s(y)dy \mbox{ for all }(x,t)\in \O\times (t_1,t_2).
$$
Since $u\in L^\a(0,T; W^{1,\a}_0(\O))$, using Hardy inequality it holds that $\dfrac{u}{\d}\in L^\a(\O\times (t_1,t_2))$. Thus
$$
C(t_0,t_1\O) \int_{t_1}^{t_2}\int\lim\limit_{n\to\infty}tss_\O\frac{\deltalta^{\alpha s}(x)}{\d^\a(x)}dx\leqslantC\int_{t_1}^{t_2} \int\lim\limit_{n\to\infty}tss_\Omega \frac{u^\a(x,t)}{\d^\a(x)}dx<\infty.
$$
Since $\a\geqslant\dfrac{1}{1-s}$, then $\a(1-s)\geqslant1$ and then we reach a
contradiction. Hence we conclude. $\Box$
If we are dealing with weak solution in the sense of Definition {I\!\!R}f{veryweak},
we can prove a non existence result for a suitable range
of $\a$. Before starting with the non existence result,
we recall the weighted Hardy inequality proved in {\cal C}_0^{\iy}\,te{Ne} (see also {\cal C}_0^{\iy}\,te{LE}).
\begin{Proposition}\label{hardygg}
Assume that $\alpha>1$ and $0<\s<\a-1$, then for all $\partialhi\in W^{1,\a}_0(\O)$, we have
\begin{equation}\label{eq:super-hardy}
\int\lim\limit_{n\to\infty}tss_\Omega \dfrac{|\partialhi(x)|^\a}{\d^{\a-\s}(x)}dx\leqslantC\int\lim\limit_{n\to\infty}tss_\Omega |\nabla \partialhi(x)|^\alpha \d^\s(x)\, dx,
\end{equation}
where $C:=C(\O,p,N)$.
\end{Proposition}
Define the space
$$
\widehat{W}_{\a,\s}(\O):=\{\varphi\in W^{1,1}_0(\O) \mbox{ with }\int\lim\limit_{n\to\infty}tss_\Omega |\nabla \varphi|^\alpha \d^\sigma dx<\infty\}.
$$
If $\s+1<\a$, then using H\"older inequality, the space $\widehat{W}_{\a,\s}(\O)$ can be endowed with the norm
$$
||\varphi||_{\widehat{W}_{\a,\s}(\O)}=\bigg(\int\lim\limit_{n\to\infty}tss_\Omega |\nabla \varphi|^\alpha \d^\sigma dx\bigg)^{\frac{1}{\a}}.
$$
Let $\widehat{H}_{\a,\s}(\O)$ be the completion of $\mathcal{C}^\infty_0(\O)$ with respect to the norm of $\widehat{W}_{\a,\s}(\O)$.
Notice that $\d^\s$ belongs to the Muckenhoupt class $A_\a$ if { $0<(\s+1)<\a$, see for instance Theorem 3.1 in {\cal C}_0^{\iy}\,te{Dur}}.
Hence by the results of {\cal C}_0^{\iy}\,te{ZZ}, we reach that
$\widehat{H}_{\a,\s}(\O)=\widehat{W}_{\a,\s}(\O)$. Thus,
for all $\varphi\in \widehat{W}_{\a,\s}(\O)$ and by Proposition {I\!\!R}f{hardygg}, for all
$\varphi\in \widehat{W}_{\a,\s}(\O)$, we have
\begin{equation}\label{super-hardy}
C(\O,N) \int\lim\limit_{n\to\infty}tss_\Omega \dfrac{|\varphi(x)|^\a}{\d^{\a-\s}(x)}dx \le\int\lim\limit_{n\to\infty}tss_\Omega |\nabla \varphi|^\alpha \d^\sigma dx.
\end{equation}
As a consequence, we have the next general non existence result.
\begin{Theorem}\label{non100}
Assume that $\a\geqslant\dfrac{1+\beta}{1-s}$ for some $\beta>0$. Then for all nonnegative data $(f,u_0)\in L^\infty(\O_T)\times L^\infty(\O)$ with $(f,u_0)\neq (0,0)$, the problem \eqref{grad-nn} has no weak solution $u$ in the sense of
Definition {I\!\!R}f{veryweak} such that $u\in L^1(0,T;W^{1,1}_0(\O))$ with $|\nabla u|^\alpha \d^\b\in L^1(\O_T)$.
\end{Theorem}
\begin{proof} We argue by contradiction. Assume that the above conditions hold and that problem \eqref{grad} has a weak solution $u$ with $u\in L^1(0,T;
W^{1,1}_0(\O))$ and $|\nabla u|^\alpha \d^{\b}\in L^1(\O_T)$. Since $\beta+1<\a$, then $u\in \widehat{W}_{\a,\beta}(\O)$.
Fix $(t_1,t_2)\subset \subset (0,T)$, then by Proposition {I\!\!R}f{diss},
we reach that
\begin{equation}\label{rrrp}
u(x,t)\geqslantC(t_0,t_1,\O) \deltalta^s(x)\mbox{ for all }(x,t)\in \O\times (t_1,t_2).
\end{equation}
Since $u\in L^\a(0,T; \widehat{W}_{\a,\beta}(\O))$, using the weighted Hardy
inequality \eqref{super-hardy}, it holds that
$$
C(\O,N) \int\lim\limit_{n\to\infty}tss_\Omega \dfrac{u^\a(x,t)}{\d^{\a-\b}(x)}dx\leqslant\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^\alpha \d^{\b} dx \mbox{ a.e. in } (0,T).
$$
Integrating in the time,
$$
C(\O,N) \int_{t_0}^{t_1}\int\lim\limit_{n\to\infty}tss_\Omega \dfrac{u^\a(x,t)}{\d^{\a-\b}(x)}dx \leqslant\int_{t_0}^{t_1}\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^\alpha \d^{\b} dx<\infty.
$$
Hence by estimate \eqref{rrrp} we reach that
$$
C(t_0,t_1,\O) \int\lim\limit_{n\to\infty}tss_\Omega \frac{1}{(\deltalta(x))^{\a(1-s)-\b}} dx\leqslant\int_{t_0}^{t_1}\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u(x,t)|^\alpha \d^{1-s} dx<\infty.
$$
Using the fact that $\a(1-s)-\b\geqslant1$, we reach a contradiction.
\end{proof}
\begin{Corollary}
\
\begin{enumerate}
\item Assume that $\a\geqslant\dfrac{1+s}{1-s}$, then problem \eqref{grad-nn}
has no weak solution $u$ in the sense of Definition {I\!\!R}f{veryweak} with $u\in
L^1(0,T;W^{1,1}_0(\O))$ and $|\nabla u|^\alpha \d^s\in L^1(\O_T)$.
\item Assume that $\a\geqslant\dfrac{2s}{1-s}$, then problem \eqref{grad-nn} has non
weak solution $u$ in the sense of Definition {I\!\!R}f{veryweak} with
$u\in L^1(0,T;W^{1,1}_0(\O))$ and $|\nabla u|^\alpha \d^\b\in L^1(\O_T)$ for some
$\beta<2s-1$.
\end{enumerate}
\end{Corollary}
\begin{remarks}
The above non existence results make a significative difference with respect
to the local case $s=1$, where an existence result holds for all $\a>1$ under
suitable condition of the data. See for example {\cal C}_0^{\iy}\,te{BS} and the reference
therein.
\end{remarks}
\section{The existence results.}\label{sec4}
The goal of this section is to prove an existence result for problem
\eqref{grad} under suitable condition on $\alpha$ and the data $f$ as was established in the Introduction.
\subsection{Existence result for $L^1$ data and $\a<\dfrac{N+2s}{N+1}$.}
In this subsection we will prove Theorem {I\!\!R}f{maria}. We assume that $(f,u_0)\in L^m(\O_{\overline{T}})\times L^1(\O)$ where
$1\leqslantm<\dfrac{1}{s}$. By the regularity result in Theorem {I\!\!R}f{gradiente} and
the second point in Corollary {I\!\!R}f{cor11}, it follows that the condition
$\a<\dfrac{N+2s}{N+1}$ is natural in order to get the existence of a
solution to problem \eqref{grad}.
\noindent{\bf Proof of Theorem {I\!\!R}f{maria}.} {Let $T<\overline{T}$ to be chosen later} and define
$E_q(\O_T)=L^q(0,T; W_{0}^{1,q}(\Omegaega))$ for $q\geqslant1$. Assume that
$1\leqslantm<\dfrac{1}{s}$ and fix $\a<\frac{N+2s}{N+1}$. Let $l>0$ to be chosen later. Define the set
$$
E(\O_T)=\{v\in E_1(\O_T)\mbox{ such that }v\in E_r(\O_T) \mbox{ with }\alpha<r<\frac{N+2s}{N+1} \mbox{ and }\|v\|_{E_r(\O_T)}\leqslantl^{\frac{1}{\alpha}}\}.
$$
It is easy to check that $E(\O_T)$ is a closed convex set of $E_1(\O_T)$. For $(f,u_0)\in L^m(\O_T)\times L^1(\O)$ fixed, we consider the operator
$$
\begin{array}{rcl}
K:E(\O_T) &\rightarrow& E_1(\O_T)\\
v&\rightarrow&T(v)=u
\end{array}
$$
where $u$ is the unique solution to problem
\begin{equation}\label{gradff}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla v|^{\alpha}+ f & \mbox{ in } \Omegaega_T\equiv\Omegaega\times (0,T),\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=&
u_0(x) & \mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation}
Since $\alpha<r$, then $|\nabla v|^{\alpha}+f \in L^1(\O_T)$, thus the existence of $u$ is a consequence of {\cal C}_0^{\iy}\,te{LPPS} and moreover $u\in L^q(0,T;
W_{0}^{1,q}(\Omegaega))\cap \mathcal{C}([0,T];L^1(\O))$ for all $q<\dfrac{N+2s}{N+1}.$ Hence $K$ is well defined.
We claim that:
\begin{enumerate}
\item There exists $l>0$ such that $K(E(\O_T))\subset E(\O_T)$,
\item $K$ is a continuous and compact operator on $E(\O_T)$.
\end{enumerate}
\textit{ Proof of the claim.}
First, we prove that we can find $l>0$ such that $K(E(\O_T))\subset E(\O_T)$. In fact, thanks to Theorem
{I\!\!R}f{gradiente}, we have that $u\in E_q(\O_T)$ for all $q<\dfrac{N+2s}{N+1}$. In
particular, $u\in E_r(\O_T)$. Now, fixed $q<\dfrac{N+2s}{N+1}$, it follows that
\begin{eqnarray*}
\|u\|_{E_q(\O_T)} &\leqslant& C(T,\O) \Big(|| f+|\nabla v|^\alpha||_{L^{1}(\Omegaega_T)}+ ||u_0||_{L^1(\O)}\Big)\\ &\leqslant& C(T,\O)\bigg(||f||_{L^{1}(\Omegaega_T)}+||\nabla
v||^\alpha_{L^{\alpha}(\Omegaega_T)}+||u_0||_{L^1(\O)}\bigg)\\ &\leqslant& C(T,\O)\bigg( ||f||_{L^{1}(\Omegaega_T)}+||\nabla
v||^{\a}_{L^{r}(\Omegaega_T)}+||u_0||_{L^1(\O)}\bigg)\\ &\leqslant& C(T,\O)\bigg( ||f||_{L^{1}(\Omegaega_T)}+||v||^\alpha_{E_r}+||u_0||_{L^1(\O)}\bigg)\\ &\le
& C(T,\O)\bigg( ||f||_{L^{1}(\Omegaega_T)}+||u_0||_{L^1(\O)}+ l\bigg).
\end{eqnarray*}
Recall that, by Remark {I\!\!R}f{mainrr}, we know that $C(\O,T)\to 0$ as $T\to 0$,
then choosing $T$ small, there exists $l>0$ such that $C(T,\O)\bigg(
||f||_{L^{1}(\Omegaega_T)}+||u_0||_{L^1(\O)}+l\bigg)\leqslantl^{\frac{1}{\alpha}}$. Hence
$$
\|u\|_{E_r(\O_T)}\leqslantl^{\frac{1}{\alpha}},
$$
and then $u\in E(\O_T)$. Thus $K(E(\O_T))\subset E(\O_T)$. {Hence, from now, we fix $T<\overline{T}$ such that the above conclusions hold.}
To prove the continuity of $K$ with respect to the topology of $E_1(\O_T)$,
we consider $\{v_n\}\subset E(\O_T)$ such that $v_n\rightarrow v$ strongly in
$E_1(\O_T)$. Define $u_n=K(v_n)$, $u=K(v)$ and $w_n=u_n-u$. We have to show that $u_n\rightarrow u$ strongly in $E_1(\O_T)$.
Since
$$
w_{nt}+(-\Deltalta )^s w_n=|\nabla v_n|^{\alpha}-|\nabla v|^\alpha,
$$
to show that $w_n\to 0$ strongly in $E_1(\O_T)$, we have to prove that $||\nabla v_n-\nabla v||_{_{L^{\alpha}(\Omegaega_T)}}\to 0$ as $n\to \infty.$
Recall that $\{v_n\}_n\subset E(\O_T)$ and $||v_n-v||_{E_1(\O_T)}\to 0$ as $n\to \infty$, then $\nabla v_n\to \nabla v$ strongly in $(L^1(\Omegaega_T))^N$ and $||\n
v_n||_{L^{r}(\Omegaega_T)}\leqslantC$.
Since $1<\alpha<r$, then using Hölder inequality, we reach that
\begin{eqnarray*}
||\nabla v_n-\nabla v||_{L^{\alpha}(\Omegaega_T)} & \leqslant& ||\nabla v_n-\nabla v||^{\frac{r-\alpha}{1+r}}_{L^{1}(\Omegaega_T)} ||\nabla v_n-\nabla v||^{\frac{\alpha}{1+r}}_{L^{r}(\Omegaega_T)}\\
&\leqslant& C||\nabla v_n-\nabla v||^{\frac{r-\alpha}{1+r}}_{L^{1}(\Omegaega)}\to 0\mbox{ as }n\to \infty.
\end{eqnarray*}
Now, by using the definition of $u_n$ and $u$, there results that $u_n\to u$ strongly in $E_1(\O_T)$. Thus $K$ is continuous.
To finish we have just to show that $K$ is a compact operator with respect to the topology of $E_1(\O_T)$.
Let $\{v_n\}_n\subset E(\O_T)$ be such that $||v_n||_{E_1(\O_T)}\leqslantC$. Since $\{v_n\}_n\subset E_r(\O_T)$, then $||v_n||_{E_r}\leqslantC$ and therefore up to a
subsequence, $v_{n}\rightharpoonup v$ weakly in $E_r(\O_T)$.
Since $\a<r$, then there exists $\d>0$ such that the sequence $\{|\nabla v_n|^{\alpha}\}_n$ is bounded in $L^{1+\deltalta}(\Omegaega_T)$. Thus, up to a subsequence,
$$
|\nabla v_n|^{\alpha}\rightharpoonup g\mbox{ weakly in }L^{1+\deltalta}(\Omegaega_T).
$$
Let $u$ to be the unique solution to the problem
\begin{equation}\label{grad011}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=& g+f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& u_0(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
using the compactness result of Theorem {I\!\!R}f{gradiente}, we conclude that, up to a subsequence, $u_{n}\to u$ strongly in $E_q(\O_T)$ for all
$q<\frac{N+2s}{N+1}$. In particular $u_{n}\to u$ strongly in $E_r(\O_T)$, hence the claim follows.
As a conclusion and using the Schauder Fixed Point Theorem, there exists $u\in E(\O_T)$ such that $K(u)=u$, then $u\in L^q(0,T; W_{0}^{1,q}(\Omegaega))$ and $u$
solves \eqref{grad}. It is not difficult to show that
$T_k(u)\in L^{2}(0,T;H^s_0(\Omegaega))$ for all $k>0$.\hskip 10.5cm {$\square$}
\begin{remark}
\
\begin{enumerate}
\item Using a suitable approximation argument we can prove that the
existence result of Theorem {I\!\!R}f{maria} holds if $f$ is a bounded Radon measure.
\item The existence result of Theorem {I\!\!R}f{maria} is optimal in the sense that if $\a>\dfrac{N+2s}{N+1}$, then we can find $f\in L^1(\O_T)$ or $u_0\in L^1(\O)$ such that
problem \eqref{grad} has no solution in the space $L^\a(0,T; W^{1,\a}_0(\O))$. To see the optimality condition, we will use Remark {I\!\!R}f{optimal}.\\
Fix $0\leqslantf\in L^1(\O_T)$ or $0\leqslantu_0\in L^1(\O)$ such that problem
\eqref{eq:def} has a solution $v$ with $v^m\notin L^1(\O_T)$, being
$m=\dfrac{N+2s}{N}$. Suppose now that problem \eqref{maria} has a solution
$u$ with $\a\geqslant\dfrac{N+2s}{N+1}$. By the comparison principle we easily reach
that $u\geqslantv$. Since $u\in L^\a(0,T; W^{1,\a}_0(\O))\cap \mathcal{C}([0,T]; L^1(\O))$, then we obtain that $u\in L^\s(\O_T)$ where $\s=\a\dfrac{N+1}{N}$.
Since $v\leqslantu$, then $v\in L^\s(\O_T)$. It is clear that $\s>\dfrac{N+2s}{N}$, hence we reach a contradiction with the hypotheses on $v$.
\item For the uniqueness and the global existence in the time, we refer to Theorem {I\!\!R}f{uniqq}.
\end{enumerate}
\end{remark}
\subsection{Existence results for $\dfrac{N+2s}{N+1}\leqslant\a$.
Proofs of Theorem {I\!\!R}f{hhh} and Theorem {I\!\!R}f{fix001}}\label{sub:sec41}
In this subsection we assume that $\dfrac{N+2s}{N+1}\leqslant\a$. According to the regularity of $f$ and $u_0$, we are able to show the existence of a solution that is in a suitable Sobolev space, under suitable condition on $m$ and $s$. For simplicity of presentation we will consider two cases:
\begin{itemize}
\item $f\neq 0$, $u_0=0$
and
\item $f=0$, $u_0\neq 0$.
\end{itemize}
{\bf Proof of Theorem {I\!\!R}f{hhh}.}
Recall that $\dfrac{2s-1}{1-s}>\dfrac{(N+2s)^2}{N+1}$ and $\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{2s-1}{(1-s)(N+2s)}$.
Assume that $f\in L^m(\O_T)$ with $m\geqslant\dfrac{1}{s}$.
We present a proof in the case $m<\dfrac{N+2s}{2s-1}$. The other case follows in a similar way.
Since $\frac{N+2s}{\a'}\frac{1}{(2s-1)-(1-s)(N+2s)}<m$, then $\a<\dfrac{(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}$. It is clear that $m>\dfrac{2s-1}{2s-1-(1-s)(N+2s)}$. Hence using the fact that
$$\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)},$$ we get the existence of $r$ such that $m\a<r<\dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}=\check{P}$ defined in Corollary {I\!\!R}f{cor11}.
Recall that $E_\s(\O_T)\equiv L^\s(0,T; W_{0}^{1,\s}(\Omegaega))$. Define the set
\begin{equation}\label{set}
E(\O_T)=\{v\in E_1(\O_T)\mbox{ such that }v\in E_{r}(\O_T) \mbox{ with } \|v\|_{E_{r}(\O_T)}\leqslantl^{\frac{1}{\alpha}}\}.
\end{equation}
Then $E(\O_T)$ is a closed convex set of $E_1(\O_T)$. Setting
$$
\begin{array}{rcl}
K:E(\O_T) &\rightarrow& E_1(\O_T)\\
v&\rightarrow&K(v)=u
\end{array}
$$
where $u$ is the unique solution to problem
\begin{equation}\label{gradss}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla v|^{\alpha}+ f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& 0 & \mbox{ in }\Omegaega.
\end{array}\right.
\end{equation}
Since $m\a<r< \dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-ms}$, using the regularity result in Corollary {I\!\!R}f{cor11} and as in the proof of Theorem {I\!\!R}f{maria}, there exists $T>0$ such that $K(E(\O_T))\subset E(\O_T)$ and that $K$ is a continuous, compact operator on $E(\O_T)$. Using the Schauder Fixed Point Theorem, we
get the existence of $u\in E(\O_T)$ such that $K(u)=u$, then $u\in L^r(0,T; W_{0}^{1,r}(\Omegaega))$ and $u$ solves \eqref{grad}. \cqd
\begin{remarks}
Notice that
\begin{enumerate}
\item If $m>\dfrac{N+2s}{2s-1}$, then the critical growth range $\a=2s$ is covered by the existence result of Theorem {I\!\!R}f{hhh} if
$
\dfrac{2s-1}{(1-s)}>2s(N+2s), \mbox{ which holds at least for $s$ close to $1$}.
$
\item If $\dfrac{1}{s}<m\leqslant\dfrac{N+2s}{2s-1}$, then the critical growth range $\a=2s$ is covered by the existence result of Theorem {I\!\!R}f{hhh} if
\begin{equation}\label{tmm}
m>\dfrac{N+2s}{2s}\dfrac{2s-1}{(2s-1)-(1-s)(N+2s)}.
\end{equation}
Since in this case $m\leqslant\dfrac{N+2s}{2s-1}$, then we deduce that from \eqref{tmm}, we have $\dfrac{2s-1}{(1-s)}>2s(N+2s)$.
\end{enumerate}
\end{remarks}
We deal now with the complete range of the parameter $\a$, $\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{s}{1-s}$. In this case, under suitable hypothesis on $m$, we will show the existence of a distributional solution that is in a suitable weighted Sobolev space. This result shows the influence of the singularity of the kernel on the boundary, which is a relevant difference with the heat equation.
{\bf Proof of Theorem {I\!\!R}f{fix001}.} Without loss of generality we can assume that $N\geqslant2$.
Fix $\a$ such that $\dfrac{N+2s}{N+1}\leqslant\a<\dfrac{s}{1-s}$ and suppose that $f\in L^m(\O_T)$ with
$$m>\max\left\{\dfrac{N+2s}{s(2s-1)}, \dfrac{N+2s}{s-\a(1-s)}\right\}.$$
Recall that if $v$ solves the problem
$$
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v &=& f &\mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ v(x,0)&=& 0 & \mbox{ in }\Omegaega,\\
\end{array}
\right.
$$
then by Theorem {I\!\!R}f{regu-g}, for all $p<\infty$, we have
\begin{equation}\label{fix2}
|||\nabla v|\,\d^{1-s}||_{L^p(\Omegaega_T)}\leqslantC_0||f||_{L^m(\O_T)},
\end{equation}
and then
\begin{equation}\label{fix25}
||\nabla (v\,\d^{1-s})||_{L^p(\Omegaega_T)}\leqslant\hat{C}_0||f||_{L^m(\O_T)}.
\end{equation}
Since $s>\dfrac 12$, we can chose $T>0$ such that for some universal constant $\bar{C}$ depending only on $\O_T, N,s$, there exists $l>0$ such
that
\begin{equation}\label{llc}
\bar{C}(l+||f||_{L^m(\O_T)})=l^{\frac{1}{2s}}.
\end{equation}
Fix $T, l>0$ as above and define the set
\begin{equation}\label{sett}
E=\{v\in E_1(\O): v\, \d^{1-s}\in L^{m\a}(0,T; W^{1,m\a}_0(\O))\mbox{ and } \bigg(\iint_{\O_T} |\nabla (v\, \d^{1-s})|^{m\a} dxdt\bigg)^{\frac{1}{m\a}}\le
l^{\frac{1}{\a}}\}.
\end{equation}
It is clear that $E$ is a closed convex set of $E_1(\O_T)$. Using Hardy inequality we reach that if $v\in E$, then
$$
\bigg(\iint_{\O_T} |\nabla v|^{m\a}\, \d^{m\a(1-s)}dx\bigg)^{\frac{1}{m\a}}\leqslant\hat{C}_0 l^{\frac{1}{\a}}.
$$
Consider the operator
$$
\begin{array}{rcl}
\mathcal{T}:E &\rightarrow& E_1(\O_T)\\
v&\rightarrow&T(v)=u
\end{array}
$$
where $u$ is the unique solution to problem
\begin{equation}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u &=&|\nabla v|^{\a}+ f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=&u_{0}(x)
& \mbox{ in }\Omegaega.\\
\end{array}
\right. \label{fix1}
\end{equation}
{\bf First step:} We prove that $\mathcal{T}$ is well defined. By Theorem {I\!\!R}f{key}, to get the desired result, we just have to show the existence of $\b<2s-1$ such
that $|\nabla v|^{\a}\d^\b\in L^1(\O_T)$. Since $v\in E$, then $|\nabla v|^{\a}\in L^1_{loc}(\O)$. We have
\begin{eqnarray*}
& & \displaystyle \iint_{\O_T}|\nabla v|^{\a}\d^\beta dxdt=\iint_{\O_T}|\nabla v|^{\a}\d^{\a(1-s)}\d^{\beta-\a(1-s)}dxdt\\
\\
& \leqslant& \displaystyle \bigg(\iint_{\O_T}|\nabla v|^{m\a}\d^{(1-s)m\a}dxdt\bigg)^{\frac{1}{m}} \bigg(\iint_{\O_T}\d^{(\beta-\a(1-s))m'}dxdt\bigg)^{\frac{1}{m'}}.
\end{eqnarray*}
If $\a(1-s)<2s-1$, we can chose $\beta<2s-1$ such that $\a(1-s)<\beta$. Hence $\displaystyle\int_\Omegaega (\d(x))^{(\beta-\a(1-s))m'}dx<\infty$.
Assume that $\a(1-s)\geqslant2s-1$, then $s\in \Big(\dfrac{1}{2}, \dfrac{\a+1}{\a+2}\Big]$. Since $m>\dfrac{N+2s}{s-\a(1-s)}$ and $\a<\dfrac{s}{1-s}$, then $(\a(1-s)-(2s-1))m'<1$.
Hence we get easily the existence of $\beta<2s-1$ such that $(\a(1-s)-\beta)m'<1$. Thus we conclude.
Therefore, since $|\nabla v|^{\a}\d^\b+f \in L^1(\O_T)$, using Theorem {I\!\!R}f{key}, there exists $u$ that solves problem \eqref{fix1} with $u\in
E_a(\O_T)$ for all $a<\dfrac{N+2s}{N+\beta+1}$. Hence $\mathcal T$ is well defined.
{\bf Second step:}
We claim that:
\begin{enumerate}
\item For $l>0$ as above, $\mathcal{T}(E)\subset E$,
\item $\mathcal{T}$ is a continuous and compact operator on $E$.
\end{enumerate}
\textsc{Proof of (1)}.
We have
\begin{eqnarray*}
u(x,t) & = & \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} |\nabla v(y,\s)|^{\a}P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma+ \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f(y,\s)P_{\Omegaega} (x,y,
t-\sigma)\,dy\,d\sigma\\ &= & J_1(x,t)+J_2(x,t).
\end{eqnarray*}
Let us begin by estimating $J_1$. Notice that, for all $\theta\in [0,1]$,
\begin{equation}\label{gh1}
\Big( 1\wedge \frac{\deltalta^s(y)}{\sqrt{t-\s}}\Big)\le\d^{s\theta}(y)(t-\s)^{-\frac{\theta}{2}} \mbox{ in }\O_T,
\end{equation}
and
\begin{equation}\label{gh2}
\Big( 1\wedge \frac{\deltalta^s(x)}{\sqrt{t-\s}}\Big)\leqslant\d^{s(1-\theta)}(x)(t-\s)^{-\frac{(1-\theta)}{2}} \mbox{ in }\O_T.
\end{equation}
Choosing $\theta=\dfrac{\a(1-s)}{s}<1$ and using the properties of $P_\O$, it holds that
$$
\begin{array}{rcl}
J_1(x,t)&\leq& C(\d(x))^{s(\a+1)-\a}\displaystyle\int_{0}^{t}\displaystyle\int_{\Omegaega}|\nabla v(y,\s)|^{\a} \d^{\a(1-s)}(y) \dfrac{(t-\s)^{\frac
12}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}\,dy\,d\sigma.
\end{array}
$$
Since $|\nabla v|^{\a} \d^{\a(1-s)}\in L^m(\O_T)$, then by Remark {I\!\!R}f{rm00} and since $m>\dfrac{N+2s}{s}$, then $\dfrac{J_1}{\d^{s-\a(1-s)}}\in L^\gamma(\O_T)$ for
all $\gamma<\infty$ and
$$
\Big|\Big| \dfrac{J_1}{\d^{s-\a(1-s)}}\Big|\Big|_{L^\gamma(\O_T)}\leqslantC || |\nabla v|^{\a} \d^{\a(1-s)}||_{L^m(\O_T)}.
$$
As in the proof of Theorem {I\!\!R}f{hardy0} and since $f\in L^m(\O_T)$, we obtain that
$$
\Big|\Big| \dfrac{J_2}{\d^{s-\a(1-s)}}\Big|\Big|_{L^\gamma(\O_T)}\leqslantC || f||_{L^m(\O_T)}.
$$
Hence we conclude that
\begin{equation}\label{tlm1}
\Big|\Big| \dfrac{u}{\d^{s-\a(1-s)}}\Big|\Big|_{L^\gamma(\O_T)}\leqslantC \bigg(|| |\nabla v|^{\a} \d^{\a(1-s)}||_{L^m(\O_T)} +||f||_{L^m(\O_T)}\bigg).
\end{equation}
We deal now with the gradient term.
We have that $$
\begin{array}{rcl}
& |\nabla u(x,t)|\\ & \displaystyle \leqslantC \bigg\{ \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} |\nabla v(y,\sigma)|^\alpha |\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma + \displaystyle\int_{0}^{t}
\displaystyle\int_{\Omegaega} f(y,\sigma) |\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma\bigg\}\\ \\ & =I_1(x,t)+I_2(x,t).
\end{array}
$$
Notice that following the same computation as in the proof of Theorem {I\!\!R}f{regu-g} and using the fact that $m>\dfrac{N+2s}{s(2s-1)}$, we find that
\begin{equation}\label{II2}
\bigg\| I_2 \d^{1-s} \bigg\|_{L^\g(\O_T)}\leqslantC ||f||_{L^m(\O_T)},
\end{equation}
for all $\g<\infty$. Hence we have just to estimate $I_1$. Notice that
$$
I_1(x,t)\leqslant\displaystyle \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} |\nabla v(y,\sigma)|^\alpha \Big( \dfrac{1}{\deltalta(x) \wedge (t-\s)^{\frac{1}{2s}}}\Big)\, P_{\Omegaega}(x,y,t-\s)dyd\s.
$$
Thus
$$
\begin{array}{lll}
& \displaystyle I_1(x,t)\d^{1-s}(x)\leqslant\displaystyle \d^{1-s}(x)\iint_{\Omegaega_T} |\nabla v(y,\sigma)|^\alpha \Big( \dfrac{1}{\deltalta(x) \wedge (t-\s)^{\frac{1}{2s}}}\Big)\,
P_{\Omegaega}(x,y,t-\s)dyd\s\\ &\leqslant\displaystyle C\bigg(\frac{1}{\d^s(x)}\iint_{\O_T\cap \{\d(x)<(t-\s)^{\frac{1}{2s}}\}}|\nabla v(y,\sigma)|^\alpha P_{\Omegaega}(x,y,t-\s)dyd\sigma \\ \\
&+ \displaystyle \d^{1-s}(x)\iint_{\O_T\cap \d(x)\geqslant(t-\s)^{\frac{1}{2s}}\}}|\nabla v(y,\sigma)|^\a\frac{ P_{\Omegaega}(x,y,t-\s)}{(t-\s)^{\frac{1}{2s}}}dyd\s\bigg)\\ \\ & =
L_1(x,t)+L_2(x,t).\\
\end{array}
$$
By the estimates \eqref{gh1} and \eqref{gh2}, we deduce that
$$
\begin{array}{lll}
L_1(x,t)\leqslant\displaystyle \frac{1}{\d^{s(1-\theta)}(x)} \iint\lim\limit_{n\to\infty}tss_{\O_T\cap \{\d(x)<(t-\s)^{\frac{1}{2s}}\}}|\n
v(y,\sigma)|^{\a}\d^{\a(1-s)}(y)\dfrac{(t-\s)^{\frac{2s-\a(1-s)}{2s}-\frac \theta 2}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} dyd\s.
\end{array}
$$
Setting
$$
g(x,t):=\iint_{\O_T\cap \{\d(x)<(t-\s)^{\frac{1}{2s}}\}}|\nabla v(y,\sigma)|^{\a}\d^{\a(1-s)}(y)\dfrac{(t-\s)^{\frac{2s-\a(1-s)}{2s}-\frac \theta
2}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} dyd\s,
$$
then by Remark {I\!\!R}f{rm00} and since $|\nabla v(y,\sigma)|^{\a}\d^{\a(1-s)}(y)\in L^{m}(\O_T)$, it follows that $g\in L^{\g_\theta}(\O_T)$ with $\g_\theta$ that satisfies
\begin{equation}\label{gama2}
\dfrac{1}{\g_\theta}>\dfrac{1}{m}-\dfrac{2s(\dfrac{2s-\a(1-s)}{2s}-\dfrac{\theta}{2})}{N+2s}=\frac{1}{m}-\frac{2s-\a(1-s)-s\theta}{N+2s}.
\end{equation}
It is clear that for all $\e>0$, we can chose $\theta$ closed to $1$ such that
$L_1(x,t)\in L^{\g_\theta-\e}(\O_T)$. Hence to show that $L_1(x,t)\in L^{m\a}(\O_T)$, we have
just to show that $\g_\theta>m\a$ for $\theta$ close to 1.
Since $m>\dfrac{N+2s}{s-\a(1-s)}$, then $\dfrac{1}{m\a}>\dfrac{1}{m}-\dfrac{2s-\a(1-s)-s}{N+2s}$. Hence,
for any $\theta<1$, we have $\dfrac{1}{m\a}>\dfrac{1}{m}-\dfrac{2s-\a(1-s)-\theta s}{N+2s}$ and then we conclude. As a consequence we deduce that $L_1\in
L^{m\a+\rho}(\O_T)$ for some $\rho>0$ and
\begin{equation}\label{l1}
||L_1||_{L^{m\a+\rho}(\O_T)}\leqslantC(\O_T) || |\n
v|^\a\d^{\a(1-s)}||_{L^m(\O_T)}.
\end{equation}
We deal now with $L_2(x,t)$.
\begin{eqnarray*}
L_2(x,t) &=& \displaystyle \d^{1-s}(x)\iint_{\O_T\cap \d(x)\geqslant(t-\s)^{\frac{1}{2s}}\}}|\nabla v(y,\sigma)|^\a\frac{ P_{\Omegaega}(x,y,t-\s)}{(t-\s)^{\frac{1}{2s}}}dyd\s\\ \\ &
= & \displaystyle \d^{1-s}(x)\iint_{\O_T\cap \d(x)\geqslant(t-\s)^{\frac{1}{2s}}\cap \{|x-y|\leqslant\frac12 \d(x)\}} + \d^{1-s}(x)\iint_{\O_T\cap \d(x)\ge
(t-\s)^{\frac{1}{2s}}\cap \{\frac12 \d(x)<|x-y|\leqslant\d(x)\}}\\ & + & \displaystyle \d^{1-s}(x)\iint_{\O_T\cap \d(x)\geqslant(t-\s)^{\frac{1}{2s}}\cap \{\d(x)<|x-y|\}} \\ \\
&= & L_{21}(x,t) + L_{22}(x,t) + L_{23}(x,t).
\end{eqnarray*}
Let us begin by estimating $L_{21}$. Since $\d$ is a Lipschitz function, it holds that, if $(x,y)\in \{|x-y|<\frac 12 \d(x)\}$, then $|\d(y)-\d(x)|\leqslant|x-y|\leqslant\frac
12 \d(x).$ Thus $ \frac 12 \d(x)\leqslant\d(y)\leqslant\frac 32 \d(x). $ Hence we obtain that
\begin{eqnarray*}
L_{21}(x,t) &\leqslant& C\iint\lim\limit_{n\to\infty}tss_{\O_T\cap \d(x)\geqslant(t-\s)^{\frac{1}{2s}}\cap |x-y|\leqslant\frac12 \d(x)\}}|\nabla v(y,\sigma)|^\a\d^{(1-s)+\theta
s}(y)\dfrac{(t-\s)^{1-\frac{1}{2s}-\frac \theta 2}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}dyd\s.
\end{eqnarray*}
Choosing $\theta=\dfrac{(\a-1)(1-s)}{s}<1$, we get
\begin{eqnarray*}
L_{21}(x,t) &\leqslant& C\iint\lim\limit_{n\to\infty}tss_{\O_T\cap \d(x)\geqslant(t-\s)^{\frac{1}{2s}}\cap |x-y|\leqslant\frac12 \d(x)\}}|\n
v(y,\sigma)|^\a\d^{\a(1-s)}(y)\dfrac{(t-\s)^{\frac{s-\a(1-s)}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}}dyd\s.
\end{eqnarray*}
Using Remark {I\!\!R}f{rm00}, we conclude that $L_{21}\in L^\g(\O)$ for all $\g$ such that $\dfrac{1}{\g}>\dfrac 1m-\dfrac{s-\a(1-s)}{N+2s}$ and
\begin{equation}\label{l21}
||L_{21}||_{L^\g(\O_T)}\leqslantC(\O_T) || |\n
v|^\a\d^{\a(1-s)}||_{L^m(\O_T)}.
\end{equation}
Since $m>\dfrac{N+2s}{s-\a(1-s)}$, then the above estimate holds for all $\g$ and then we conclude.
We analyze now the term $L_{22}$. We set
$$
A:=\O_T\cap \{\d(x)\geqslant(t-\s)^{\frac{1}{2s}}\}\cap \{\frac12 \d(x)<|x-y|\leqslant\d(x)\},
$$
then as above choosing $\theta=\dfrac{\a(1-s)}{s}<1$, we obtain that
$$
L_{22}(x,t)\leqslant\displaystyle \d^{1-s}(x)\iint\lim\limit_{n\to\infty}tss_{A} |\n
v(y,\sigma)|^{\a}\d^{\a(1-s)}(y)\dfrac{(t-\s)^{\frac{(2s-1)-\a(1-s)}{2s}}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{N+2s}} dyd\s.
$$
It is clear that $2s-1-\a(1-s)>0$ if and only if $s>\dfrac{\a+1}{\a+2}$. Hence, in any case, using H\"older inequality, we get
\begin{eqnarray*}
& L_{22}(x,t)\leqslant\\ & \displaystyle \d^{1-s}(x)\bigg(\iint\lim\limit_{n\to\infty}tss_{A} |\nabla v(y,\sigma)|^{m\a}(\d(y))^{m\a(1-s)}dyd\s\bigg)^{\frac{1}{m}} \bigg(\iint\lim\limit_{n\to\infty}tss_{A}
\dfrac{(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{m'(N+2s)}}dyd\sigma \bigg)^{\frac{1}{m'}}.
\end{eqnarray*}
By the definition of the set $A$, we obtain
\begin{eqnarray*}
& \displaystyle \iint\lim\limit_{n\to\infty}tss_{A} \dfrac{(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{m'(N+2s)}}dyd\s\\ & \displaystyle =\int_{\{(t-\s)\leqslant\d^{2s}(x)\}}
(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))}\int_{\{\frac12 \d(x)<|x-y|\leqslant\d(x)\}}\dfrac{dy\, d\s}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{m'(N+2s)}}\\ \\ &=\displaystyle
C(N)\int_{\{(t-\s)\leqslant\d^{2s}(x)\}}(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))+\frac{N}{2s}-\frac{m'(N+2s)}{2s}}\int_{\frac12\frac{
\d(x)}{(t-\s)^{\frac{1}{2s}}}}^{\frac{ \d(x)}{(t-\s)^{\frac{1}{2s}}}} \dfrac{r^{N-1}dr \, d\s}{(1+r)^{m'(N+2s)}} \\ \\ &\leqslant\displaystyle C(N,s) \int_{\{(t-\s)\le
\d^{2s}(x)\}}(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))+\frac{N}{2s}-\frac{m'(N+2s)}{2s}} \times \frac{(t-\s)^{\frac{1}{2s}(m'(N+2s)-N)}}{\d^{m'(N+2s)-N}(x)} d\s\\ \\
&\displaystyle \leqslant\frac{C(N,s)}{(\d(x))^{m'(N+2s)-N}} \int_{\{(t-\s)\le
\d^{2s}(x)\}}(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))+\frac{N}{2s}-\frac{m'(N+2s)}{2s}+\frac{1}{2s}(m'(N+2s)-N)}d\s\\ \\ &\displaystyle \le
\frac{C(N,s)}{(\d(x))^{m'(N+2s)-N}} \int_0^{\d^{2s}(x)}\rho^{\frac{m'}{2s}((2s-1)-\a(1-s))}d\rho.
\end{eqnarray*}
If $s\geqslant\dfrac{\a+1}{\a+2}$, then $\dfrac{m'}{2s}((2s-1)-\a(1-s))\geqslant0$. If $\dfrac 12<s<\dfrac{\a+1}{\a+2}$, since $m'<\dfrac{N+2s}{N+s+\a(1-s)}$, then
$\dfrac{m'}{2s}((2s-1)-\a(1-s))>-1$. Thus
$$
\displaystyle \frac{C(N,s)}{(\d(x))^{m'(N+2s)-N}} \int_0^{\d^{2s}(x)}\rho^{\frac{m'}{2s}((2s-1)-\a(1-s))}d\rho\leqslant(\d(x))^{N+2s-m'(N+1+\a(1-s))}.
$$
Going back to the estimate of $L_{22}$ it holds that
$$
L_{22}(x,t)\leqslantC (\d(x))^{s-\a(1-s)-\frac{N+2s}{m}} \bigg(\iint\lim\limit_{n\to\infty}tss_{A} |\nabla v(y,\sigma)|^{m\a}(\d(y))^{m\a(1-s)}dyd\s\bigg)^{\frac{1}{m}}.
$$
Since $m>\dfrac{N+2s}{s-\a(1-s)}$, we reach that $s-\a(1-s)-\dfrac{N+2s}{m}>0$. Thus $L_{22}\in L^\infty(\O_T)$ and for all $\g>1$,
\begin{equation}\label{l22}
||L_{22}||_{L^\g(\O_T)}\leqslantC(\O_T)\bigg(\iint\lim\limit_{n\to\infty}tss_{\O_T} |\nabla v(y,\sigma)|^{2sm}\d^{2sm(1-s)}(y)dyd\s\bigg)^{\frac{1}{m}}.
\end{equation}
Respect to $L_{23}$, setting
$$
\bar{A}:=\O_T\cap \{\d(x)\geqslant(t-\s)^{\frac{1}{2s}}\}\cap \{|x-y|\geqslant\d(x)\}\}
$$
and following the same computations as above, it holds that
\begin{eqnarray*}
& L_{23}(x,t)\le\\ & \displaystyle \d^{1-s}(x)\bigg(\iint\lim\limit_{n\to\infty}tss_{\bar{A}} |\nabla v(y,\sigma)|^{m\a}\d^{\alpha m(1-s)}(y)dyd\s\bigg)^{\frac{1}{m}} \bigg(\iint\lim\limit_{n\to\infty}tss_{\bar{A}}
\dfrac{(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))}}{((t-\s)^{\frac{1}{2s}}+|x-y|)^{m'(N+2s)}}dyd\sigma \bigg)^{\frac{1}{m'}}\\ \\ &\leqslant\displaystyle
\d^{1-s}(x)\bigg(\iint\lim\limit_{n\to\infty}tss_{\bar{A}} |\nabla v(y,\sigma)|^{m\a}\d^{m\a(1-s)}(y)dyd\s\bigg)^{\frac{1}{m}}\\ \\ &\times \displaystyle \bigg( \int_{\{(t-\s)\le
\d^{2s}(x)\}}(t-\s)^{\frac{m'}{2s}((2s-1)-\a(1-s))+\frac{N}{2s}-\frac{m'(N+2s)}{2s}} \int_{\frac{ \d(x)}{(t-\s)^{\frac{1}{2s}}}}^{\infty}
\dfrac{r^{N-1}}{(1+r)^{m'(N+2s)}}dr \, d\s\bigg)^{\frac{1}{m'}}.
\end{eqnarray*}
Thus, as above, we conclude that
$$
L_{23}(x,t)\leqslantC (\d(x))^{s-\a(1-s)-\frac{N+2s}{m}}\bigg(\iint\lim\limit_{n\to\infty}tss_{\bar{A}} |\nabla v(y,\sigma)|^{m\a}\d^{m\a(1-s)}(y)dyd\s\bigg)^{\frac{1}{m}}.
$$
Then as in the estimate of $L_{22}$, we reach that $L_{23}\in L^\infty(\O_T)$ and for all $\g>1$,
\begin{equation}\label{l23}
||L_{23}||_{L^\g(\O_T)}\leqslantC\bigg(\iint\lim\limit_{n\to\infty}tss_{\O_T} |\nabla v(y,\sigma)|^{2sm}\d^{2sm(1-s)}(y)dyd\s\bigg)^{\frac{1}{m}}.
\end{equation}
As a conclusion and combing estimates \eqref{l21}, \eqref{l22} and \eqref{l23} we reach that if $u$ solves \eqref{fix1} with $v\in E$, then
\begin{eqnarray*}
& \displaystyle \bigg(\iint_{\O_T} |\nabla u|^{\g}\, \d^{\g(1-s)}) dxdt\bigg)^{\frac{1}{\g}}\\ & \leqslant\displaystyle C(\O_T,N,s)\bigg(\bigg(\iint\lim\limit_{n\to\infty}tss_{\O_T} |\n
v(y,\sigma)|^{m\a}\d^{m\a(1-s)}(y)dyd\s\bigg)^{\frac{1}{m}} +||f||_{L^m(\O_T)}\bigg)
\end{eqnarray*}
and
$$
\Big|\Big| \dfrac{u}{\d^{s-\a(1-s)}}\Big|\Big|_{L^\gamma(\O_T)}\leqslantC \bigg(|| |\nabla v|^{\a} \d^{\a(1-s)}||_{L^m(\O_T)} +||f||_{L^m(\O_T)}\bigg),
$$
for some $\g>m\a$. Recall that $T$ is fixed such that \eqref{llc} holds. Thus $u\in E$ and then $\mathcal{T}(E)\subset E$.
\
Now we proof (2). Let begin by proving the continuity of $\mathcal{T}$ respect to the topology of $L^{1}(0,T; W^{1,1}_0(\O))$. Consider $\{v_n\}_n\subset E$ such that
$v_n\to v$ strongly in $L^{1}(0,T; W^{1,1}_0(\O))$. Define $u_n=\mathcal{T}(v_n), u=\mathcal{T}(v)$.
We have to show that $u_n\to u$ strongly in $L^{1}(0,T; W^{1,1}_0(\O))$. Using Theorem {I\!\!R}f{key}, to show the desired result, we have just to prove that
$$||\nabla v_n -\nabla v||_{_{L^{\a}(\d^\beta dx, \O_T)}}\to 0 \hbox{ as } n\to \infty,$$ for some $\beta<2s-1$. As in the proof of the fist step, we get the existence
of $\beta<2s-1$ such that
$\displaystyle\int_{\O_T}|\nabla v_n|^{\a}\d^\beta dx\leqslantC$ for all $n$.
Since $m\a>1$, then setting $a=\dfrac{\a(m-1)}{m\a-1}<1$, it follows that $\dfrac{\a-a}{1-a}=m\a$. Hence by Hölder inequality, we conclude that
\begin{eqnarray*}
||\nabla v_n -\nabla v||_{_{L^{\a}(\d^\beta dx, \O_T)}} & \leqslant& ||\nabla v_n-\nabla v||^{\frac{a}{\a}}_{L^{1}(\d^{\beta} dx, \O_T)} ||\nabla v_n-\n
v||^{\frac{\a-a}{\a}}_{L^{\frac{\a-a}{1-a}}(\d^{1-s} dx, \O_T)}\\ &\leqslant& C||\nabla v_n-\nabla v||^{\frac{a}{\a}}_{L^{1}(\O_T)}\to 0\mbox{ as }n\to \infty.
\end{eqnarray*}
Now, using the definition of $u_n$ and $u$, there results that $u_n\to u$ strongly in $L^{\s}(0,T; W^{1,\s}_0(\O))$ for some $\s>1$ and then we conclude.
We prove now that $\mathcal{T}$ is compact. Let $\{v_n\}_n\subset E$ be such that $||v_n||_{L^{1}(0,T; W^{1,1}_0(\O))}\leqslantC$.
Since $\{v_n\}_n\subset E$, then $||\nabla (v_n \d^{1-s})||_{L^{m\a}(\O_T)}\leqslantC$. Therefore, up to a subsequence,
$$v_{n_k}\d^{1-s}\rightharpoonup v\d^{1-s} \hbox{ weakly in } L^{m\a}(0,T; W^{1,m\a}_0(\O)).$$
It is clear that $v_{n_k}\rightharpoonup v$ weakly in $L^{m\a}(0,T; W^{1,m\a}_{loc}(\O))$.
Let
$$
F_n=|\nabla v_n|^{\a}+f, F=|\nabla v|^{\a}+f,
$$
then, as in the first step, we can prove that $F_n \d^\beta$ is bounded in $L^{1+a}(\O)$ for some $a>0$ and then $F_n \d^\beta\rightharpoonup F \d^\beta$ weakly in
$L^{1+a}(\O)$. Using the compactness result in Theorem {I\!\!R}f{key}, we conclude that, up to a subsequence, $u_{n_k}\to u$ strongly in $L^{q}(0,T; W^{1,q}_0(\O))$
for all $q<\dfrac{N+2s}{N+\beta+1}$ and then the result follows.
\noindent Hence we are in position to use the Schauder Fixed Point Theorem and then there exists $u\in E$ such that $\mathcal{T}(u)=u$. Thus $u\d^{1-s}\in L^{m\a}(0,T;
W^{1,m\a}_0(\O))$ and $u$ solves \eqref{grad} in the sense of distribution. \cqd
\begin{remarks}
The above arguments can be used to treat the problem
\begin{equation}\label{grada}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta)^s u +|\nabla u|^\alpha &= & f & \text{ in }\Omegaega_T ,
\\ u(x,t) &=& 0 &\hbox{ in } \mathbb{R}^N\setminus\Omegaega\times (0,T),\\
u(x,0)&=&0 &\hbox{ in }\Omegaega,
\end{array}
\right.
\end{equation}
where $\a<\dfrac{s}{1-s}$ and $f\in L^m(\O_T)$ with $m>\max\{\dfrac{N+2s}{s(2s-1)}, \dfrac{N+2s}{s-\a(1-s)}\}$.
Then for $T$ small, problem \eqref{grada} has a distributional solution $u\in
L^{\a}(0,T;W^{1,\a}_{loc}(\O))\cap L^1(0,T;W^{1,1}_0(\O))$ with $u\d^{1-s}\in L^{m\a}(0,T;W^{1,m\a}_0(\O))$.
\end{remarks}
In the case where $\dfrac{s}{1-s}\leqslant\a$, then we consider the modified problem
$$
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta)^s u &= & \d^{\a(1-s)-s}(x)|\nabla u|^{\a}+f & \text{ in }\Omegaega_T ,
\\ u(x,t) &=& 0 &\hbox{ in } \mathbb{R}^N\setminus\Omegaega\times (0,T),\\
u(x,0)&=&0 &\hbox{ in }\Omegaega,
\end{array}
\right.
$$
where $m>\max\{\dfrac{N+2s}{\a'(2s-1)}, \dfrac{N+2s}{\a(1-s)-s}\}$, then as in the proof of the previous Theorem, existence of solution holds using the Schauder fixed point Theorem in the set
$$
\hat{E}(\O_T)=\{v\in E_1(\O_T): v\, \d^{1-s}\in E_{m}(\O_T) \mbox{ with } \|v \d^{1-s}\|_{E_{m}(\O_T)}\leqslantl^{\frac{1}{\alpha}}\}.
$$
\
\
If $f=0$ and $u_0\neq 0$, we have the result in Theorem {I\!\!R}f{hhh2} whose proof follows.
{\bf Proof of Theorem {I\!\!R}f{hhh2}. }
Recall that $\dfrac{2s-1}{1-s}>\dfrac{(N+2s)^2}{N}$ and that $\frac{N+2s}{N+1}\leqslant\a<\frac{2s}{(1-s)(N+2s)+1}$.
Define $\partialsi$ to be the unique solution to problem
\begin{equation}\label{inter1}
\left\{
\begin{array}{rcll}
\partialsi_t+(-\Deltalta )^s \partialsi&=& 0 &\mbox{ in } \Omegaega_T,\\ \partialsi(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ \partialsi(x,0)&=& u_0(x) & \mbox{ in }\Omegaega.
\end{array}\right.
\end{equation}
Recall that $u_0\in L^\s(\O)$ where $\s>\dfrac{(\a-1)N}{(2s-\a)-\a(1-s)(N+2s)}$, then by the regularity results in Proposition {I\!\!R}f{pro:lp2} and Corollary {I\!\!R}f{cor1100}, we obtain that $|\nabla \partialsi|\in L^\theta(\O_T)$ for all $\theta<m_1=\dfrac{\s(N+2s)}{(1-s)\s(N+2s)+N+\s}$.
Notice that if $v$ solves the problem
\begin{equation}\label{gradss-n}
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v&=&|\nabla (v+\partialsi)|^{\alpha} &\mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ v(x,0)&=& 0 & \mbox{ in }\Omegaega,
\end{array}\right.
\end{equation}
then $u\equiv v+\partialsi$ is a solution to \eqref{gradu0}. Hence we have just to prove the existence of $v$. Notice that
$$
|\nabla (v+\partialsi)|^{\alpha}\leqslantC_1|\nabla v|^\a+C_2|\nabla \partialsi|^\a.
$$
Define $f\equiv C_2|\nabla \partialsi|^\a$, then $f\in L^m(\O_T)$ for any $m<\dfrac{m_1}{\a}=\dfrac{\s(N+2s)}{\a[(1-s)\s(N+2s)+N+\s]}$. It is clear that $m<\dfrac{\s(N+2s)}{\alpha \s}<\dfrac{N+2s}{2s-1}$.
Hence, using the fact that $\frac{N+2s}{N+1}\leqslant\a<\frac{2s}{(1-s)(N+2s)+1}$ and $\s>\frac{(\a-1)N}{(2s-\a)-\a(1-s)(N+2s)}$, then we get the existence of $\frac{1}{s}\leqslantm<\frac{N+2s}{2s-1}$ such that $m\a<\dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}=\check{P}$ defined in Corollary {I\!\!R}f{cor11}.
Hence we can fix $r>1$ such that $m\a<r<\dfrac{m(N+2s)}{(N+2s)(m(1-s)+1)-m(2s-1)}$. Now following the same argument as in the proof of Theorem {I\!\!R}f{hhh} we get the desired existence result. \cqd
\begin{remarks}
\
\begin{enumerate}
\item { Let us consider now the case where $f\gneqq 0$ and $u_0 \gneqq 0$ simultaneously . As in the proof of Theorem {I\!\!R}f{hhh2}, define $\partialsi$ to be the unique solution to problem \eqref{inter1} and let $\vartheta$ the solution to the problem
\begin{equation}\label{gradss-nnn}
\left\{
\begin{array}{rcll}
\vartheta_t+(-\Deltalta )^s \vartheta &=&|\nabla (\vartheta+\partialsi)|^{\alpha}+f &\mbox{ in } \Omegaega_T,\\ \vartheta(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ \vartheta(x,0)&=& 0 & \mbox{ in }\Omegaega,
\end{array}\right.
\end{equation}
then $u\equiv \vartheta+\partialsi$ solves the problem
\begin{equation*}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u &=&|\nabla u|^{\alpha}+f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& 0 & \mbox{ in }\Omegaega,
\end{array}\right.
\end{equation*}
Taking into consideration that
$$
|\nabla (\vartheta+\partialsi)|^{\alpha}\leqslantC_1|\nabla \vartheta|^\a+C_2|\nabla \partialsi|^\a,
$$
we can reproduce the same approach as in the proof of Theorem {I\!\!R}f{hhh2}, with $\tilde{f}=C_2|\nabla \partialsi|^\a,+ f$, to get the existence of a solution $\vartheta$ to problem {I\!\!R}f{gradss-nnn} combining the regularity of $f$ and $u_0$. Hence we conclude.
}
\
\
\item In a forthcoming paper we will treat the problem
\begin{equation}\label{gradss01}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla u|^{\alpha}&\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& u_0(x) & \mbox{ in }\Omegaega.
\end{array}\right.
\end{equation}
under the general condition $\a<\dfrac{s}{1-s}$. Existence of solutions will proved in a suitable parabolic weighted Sobolev space. Global existence in time or blow-up in finite time will be also analyzed.
\end{enumerate}
\end{remarks}
\section{Comparison principle and a partial uniqueness result for a problem
with a drift term. Applications to the quasi-linear problem \eqref{grad}}
We will study in this Section an equation with a \textit{ drift}, that is, we substitute the nonlinear term in the gradient by a term of the form $B(x,t)\,\cdot\, \nabla u$.
Then the existence of a solution is obtained under natural conditions of the field $B$ and the data $f, u_0$. Using the previous arguments, we can prove the existence for the largest class of the data $f, u_0$, under the natural condition on $B$.
Let us begin by considering the next problem
\begin{equation}\label{grad1}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=& B(x,t)\,\cdot\, \nabla u+ f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& u_0(x)
&\mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
where $B\in (L^{m}(\O_T))^N$ with $m>\dfrac{N+2s}{2s-1}$. We are able to prove the next existence result that extends the one obtained in {\cal C}_0^{\iy}\,te{W1}.
\begin{Theorem}\label{th2}
Assume that $\O\subset {I\!\!R}n$ is a bounded regular domain and $s\in (\dfrac 12, 1)$. Suppose that $B\in (L^{m}(\O_T))^N$ with $m>\dfrac{N+2s}{2s-1}$, then for all $(f,u_0)\in
L^{1}(\Omegaega_T)\times L^{1}(\Omegaega)$, the problem \eqref{grad1} has a weak solution $u\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega)), \, q<\dfrac{N+2s}{N+1}$ and $T_k(u)\in
L^{2}(0,T;H^s_0(\Omegaega))$ for all $k>0$.
Furthermore, If $u_0\geqslant0$ in $\O$ and $f\geqslant0$ in $\O_T$, then $u\geqslant0$ in $\O_T$.
Moreover, if $B$ satisfies one of the following assumption:
\begin{enumerate}
\item $B$ does not depend on $t$,
\item $B\in (L^{m}(\O_T))^N$ with $m>\max\{q', \dfrac{2s}{(N+2s)-q(N+1)}\}>\dfrac{N+2s}{2s-1}$ for some $1<q<\dfrac{N+2s}{N+1}$,
\end{enumerate}
then the solution obtained above is unique.
\end{Theorem}
\begin{proof}
As in the proof of Theorem {I\!\!R}f{th1}, we define the operator
$$
\begin{array}{rcl}
K:\widetilde{E}(\O_T)&\rightarrow& E_1(\O_T)\\
v&\rightarrow&T(v)=u,
\end{array}
$$
where
$$
\widetilde{E}(\O_T)=\{v\in E_1(\O_T)\mbox{ such that }v\in E_r(\O_T) \mbox{ with }\s'<r<\frac{N+2s}{N+1} \mbox{ and }\|v\|_{E_r(\O_T)}\leq
l^{\frac{1}{\alpha}}\},
$$
and $u$ is the unique solution to problem
\begin{equation}\label{grad00}
\left\{
\begin{array}{rcl}
u_t+(-\Deltalta )^s u&=& B(x,t)\,\cdot\, \nabla v+ f\mbox{ in } \Omegaega_T\equiv\Omegaega\times (0,T),\\ u(x,t)&=&0\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=&
u_0(x)\mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation}
Let us begin by proving that $K$ is well defined.
Since $B\in (L^{m}(\O_T))^N$ with $m>\dfrac{N+2s}{2s-1}$, then $m'<\dfrac{N+2s}{N+1}$. Fix $r$ such that $m'<r<\dfrac{N+2s}{N+1}$, then for $v\in
\widetilde{E}(\O_T)$, using H\"older inequality, we conclude that
$$
\displaystyle\int_{0}^{T}\displaystyle\int_{\Omegaega} |B(x,t)\,\cdot\, \nabla v|dxdt\leqslantC(T,\O)\Big(\displaystyle\int_{0}^{T}\displaystyle\int_{\Omegaega}|B(x,t)|^m
dxdt\Big)^{\frac{1}{m}}\Big(\displaystyle\int_{0}^{T}\displaystyle\int_{\Omegaega}|\nabla v|^r dxdt\Big)^{\frac{1}{r}}.
$$
Hence $|B(x,t)\,\cdot\, \nabla v|\in L^1(\O_T)$ and then $K$ is well defined. Now, the existence result follows using the same compactness arguments as in the proof
of Theorem {I\!\!R}f{th1}.
Assume now that $f\geqslant0$. To get the existence of a nonnegative solution we consider the next variation of the operator $K$. Namely we define $u$ to be the unique
solution to the problem
\begin{equation}\label{grad001}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=& B(x,t)\,\cdot\, \nabla v_+ + f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& u_0(x)
&\mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation}
It is clear that, with the new definition, if $u$ is a fixed point of $K$, then $u$ solves
$$
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=& B(x,t)\,\cdot\, \nabla u_+ + f &\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=& u_0(x) &
\mbox{ in }\Omegaega.\\
\end{array}\right.
$$
Using $T_k(u_-)$ as a test function in the previous equation we reach that $T_k(u_-)=0$ for all $k$. Hence $u_-=0$ and then $u\geqslant0$ in $\O_T$.
It is clear that in a symmetric way, if $f\leqslant0$ in $\O_T$ and $u_0\leqslant0$ in $\O$, then we get the existence of a $u\leqslant0$ in
$\O_T$.
We prove now that the solution is unique. Assume that $u_1, u_2$ are solutions to problem \eqref{grad1}, setting $v=u_1-u_2$, then $v$ solves
\begin{equation}\label{grad001-un}
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v&=& B(x,t)\,\cdot\, \nabla v &\mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 &\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times (0,T),\\ v(x,0)&=& 0 &\mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation}
Notice that $v \in L^{a}(0,T;W_{0}^{1,a}(\Omegaega))$ for all $a<\dfrac{N+2s}{N+1}$ and $T_k(v)\in L^{2}(0,T;H^s_0(\Omegaega))$ for all $k>0$.
Define $g(x,t)=B(x,t)\,\cdot\, \nabla v(x,t)$, since $B\in (L^m(\O_T))^N$ with $m>\dfrac{N+2s}{2s-1}$, then we can fix $1<q<\dfrac{N+2s}{N+1}$ such that $q'<m$.
Notice that
$$
v(x,t)=\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} g(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma,
$$
and
$$
|\nabla v(x,t)|\leqslant\displaystyle \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega}| g(y,\sigma)| |\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma.
$$
From Proposition {I\!\!R}f{more-tregu} we deduce that for all $\eta>0$,
$$
\Big(\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla v(x,t)|^qdx\Big)^{\frac{1}{q}}\leqslantC(\O_T) \bigg(\int_0^t\int\lim\limit_{n\to\infty}tss_\Omega |g(y,\sigma)|(t-\s)^{\hat{\g}-\eta}dy\,d\sigma\bigg)^{\frac{1}{q}}.
$$
Recall that $|g(x,t)|\leqslant|B(x,t)| |\nabla v(x,t)|$, thus
$$
\int\lim\limit_{n\to\infty}tss_\Omega |g(y,\s)| dy \leqslant\bigg(\int\lim\limit_{n\to\infty}tss_\Omega |\nabla v(y,\s)|^q dy \bigg)^{\frac{1}{q}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega |B(y,\s)|^{q'} dy \bigg)^{\frac{1}{q'}}.
$$
Let $\a=1+\hat{\g}-\eta\in (0,1)$, then
\begin{eqnarray*}
\Big(\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla v(x,t)|^qdx\Big)^{\frac{1}{q}} & \leqslant& C(\O_T) \int_0^t(t-\s)^{\a-1}\bigg(\int\lim\limit_{n\to\infty}tss_\Omega |\nabla v(y,\s)|^q dy \bigg)^{\frac{1}{q}} \bigg(\int\lim\limit_{n\to\infty}tss_\Omega |B(y,\s)|^{q'} dy \bigg)^{\frac{1}{q'}}\,d\sigma. \\
\end{eqnarray*}
Setting $Y(t)=\bigg(\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla v(x,t)|^qdx\bigg)^{\frac{1}{q}}$, then $Y\in L^q(0,T)$ and
$$
Y(t)\leqslantC(\O_T)\int_0^t (t-\s)^{\a-1}K(\s) Y(\s) d\sigma,
$$
with $K(\s)=\bigg(\displaystyle\int_\Omegaega |B(y,\s)|^{q'} dy \bigg)^{\frac{1}{q'}}$.
Assume that $B$ depends only on $x$, then $K\in L^\infty(0,T)$. From the singular Bellman-Gronwall inequality proved in {\cal C}_0^{\iy}\,te{Hen}, Lemma 7.7.1, we deduce that $Y=0$ in $L^1(0,T)$. Thus
$$\displaystyle\int_0^T\bigg(\displaystyle \int\lim\limit_{n\to\infty}tss_\Omega |\nabla v(x,t)|^qdx\bigg)^{\frac{1}{q}}dt=0.$$ Since $v\in L^q((0,T);W^{1,q}_0(\O))$, we obtain that $v=0$ and the result follows.
Now, in second hypothesis, if $B\in (L^{m}(\O_T))^N$ with $m>\max\{q', \dfrac{2s}{(N+2s)-q(N+1)}\}>\dfrac{N+2s}{2s-1}$, then for $\eta$ small enough, $m\a>1$. We set $\hat{K}(t,\s)=(t-\s)^{\a-1}K(\s)$, then
there exists $\theta\in \Big(1, \dfrac{1}{1-\a}\Big)$ such that
$$
\int_0^T \bigg(\int_0^t (\hat{K}(t,\s))^\theta d\s\bigg)^{\frac{1}{\theta-1}} dt<\infty.
$$
Since
$$
Y(t)\leqslantC(\O_T)\int_0^t \hat{K}(\t,\s)Y(\s) d\sigma,
$$
then from {\cal C}_0^{\iy}\,te{Kwa}, we obtain that $Y=0$ in $L^1(0,T)$. Hence as above we conclude that $v=0$.
\end{proof}
\begin{remarks}
Under the additional hypothesis on $B$ that ensures the uniqueness of the solution to problem \eqref{grad1}, we can also see $u$ as mild solution to problem \eqref{grad1} in the sense that, if we denote by $\hat{P}_{\Omegaega}$, the Dirichlet heat kernel associated to the operator
$$
L(v):=\partial_t v+(-\Deltalta )^s v -B(x,t)\,\cdot\, \nabla v,
$$
then
$$
u(x,t)=\displaystyle\int_{\Omegaega}u_0(y) \hat{P}_{\Omegaega} (x,y, t)\,dy\,+ \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} f(y,\sigma) \hat{P}_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma.
$$
Notice that from {\cal C}_0^{\iy}\,te{JS1} (see Example 3, page 335), we get that $B\in \mathcal{K}(\eta,Q)$, defined in
{\cal C}_0^{\iy}\,te{JS1}. Then $\hat{P}_{\Omegaega}\backsimeq P_{\Omegaega}$.
\end{remarks}
\begin{remarks}
Related to the linear problem \eqref{grad001-un}, we can show that $v\in \mathcal{C}^{\tau,2s}_{t,x}(\Omegaega\times (0,T))$ for some $\tau\in
(0,1)$. Effectively, let us begin by proving that $g\in L^{m}_{loc}(\Omegaega_T)$ (where $L^{m}_{loc}(\Omegaega_T)$ is defined in Remark {I\!\!R}f{rm001}).
Recall that $m>\dfrac{N+2s}{2s-1}$, then, using H\"older inequality it holds that $g\in L^{l_1}(\Omegaega_T)$ for $1<l_1<\dfrac{\hat{q}\sigma}{\sigma+\hat{q}}$ where $\hat{q}=\frac{N+2s}{N+1}$ . Fix $l_1$
as above, by Proposition {I\!\!R}f{key2-locc}, we obtain that $|\nabla v|\in L^{r_1}_{loc}(\Omegaega_T)$ with $r_1=\dfrac{l_1(N+2s)}{N+2s-l_1(2s-1)}$. Using again H\"{o}lder
inequality, we reach that $g\in L^{l_2}_{loc}(\Omegaega_T)$ with $l_2=\dfrac{r_1\sigma}{\sigma+r_1}$. Using again Proposition {I\!\!R}f{key2-locc}, it follows that $|\n
w|\in L^{r_2}_{loc}(\Omegaega_T)$ with $r_2=\dfrac{l_2(N+2s)}{N+2s-l_2(2s-1)}$.
Hence, we define the two sequences $\{l_n\}_n$ and $\{r_n\}_n$ by
$$
\left\{
\begin{array}{rcll}
1< & l_1 & <\hat{q}=\dfrac{N+2s}{N+1},\\ \\ r_i &=& \dfrac{l_i (N+2s)}{N+2s-l_i(2s-1)}, i\geqslant1,\\ \\ l_{i+1} & = & \dfrac{r_i \s}{\s+r_i}, i\geqslant1, i\geqslant1.
\end{array}\right.
$$
Thus
$$
r_{i+1}=\frac{m(N+2s)r_i}{m(N+2s)-r_i((2s-1)m-(N+2s))}.
$$
It is clear that $r_{i+1}>r_i$. Let show that there exists $i_0$ such that $r_{i_0}\geqslant\dfrac{\sigma (N+2s)}{(2s-1)m-(N+2s)}$. We argue by contradiction.
Assume that $r_i<\dfrac{m (N+2s)}{(2s-1)m-(N+2s)}$ for all $i$. Since $\{r_i\}_i$ is an increasing sequence, there exists a $\bar{r}$ such that $r_i\uparrow \bar{r}\leqslant\dfrac{m (N+2s)}{(2s-1)m-(N+2s)}$.
Thus $\bar{r}=\dfrac{m(N+2s)\bar{r}}{m(N+2s)-\bar{r}((2s-1)m-(N+2s))}$, hence $\bar{r}=0$,
a contradiction with the fact that $\{r_i\}_i$ is an increasing sequence.\\
Therefore, there exists $i_0\in {I\!\!N}$ such that $r_{i_0}\geqslant\dfrac{m (N+2s)}{(2s-1)m-(N-2s)}$.Thus, using H\"older inequality, we conclude that $g\in L_{loc}^{\frac{N+2s}{2s-1}}(\Omegaega_T)\cap L^1(\O_T)$. Hence by the result of Proposition {I\!\!R}f{key2-locc},
we obtain that $|\nabla v|\in L^{a}_{loc}(\Omegaega_T)$ for all $a>1$. Thus $g\in L^{m}_{loc}(\Omegaega_T)$ and the claim follows.
\noindent Therefore, by the regularity results in {\cal C}_0^{\iy}\,te{CF}, {\cal C}_0^{\iy}\,te{Gr} and {\cal C}_0^{\iy}\,te{JS1}, we obtain that then $v\in \mathcal{C}^{\tau,2s}_{t,x}(\Omegaega\times (0,T{]})$ for some $\tau\in (0,1)$.
\end{remarks}
Under the hypotheses
\begin{equation}\label{covid}f\d^\beta\in L^1(\O),\, B\d^\beta \in (L^{\s}(\O_T))^N \hbox{ with } \s>\frac{N+2s}{2s-\beta-1} \hbox{ for some } 0<\beta<2s-1,
\end{equation}
then as
in Theorem {I\!\!R}f{th2}, we have the next existence result.
\begin{Theorem}\label{th2beta}
Assume that the hypotheses \eqref{covid} on $f$ and $B$ hold and $u_0\in L^1(\O)$. Then the problem \eqref{grad1} has a distributional solution $u\in
L^{a}(0,T;W_{0}^{1,a}(\Omegaega)), \, a<\dfrac{N+2s}{N+\beta+1}$. Moreover, if in addition $B$ satisfies one of the following conditions:
\begin{enumerate}
\item $B$ does not depends on $t$,
\item $B\in (L^{m}(\O_T))^N$ with $m>\max\{q', \dfrac{2s}{(N+2s)-q(N+\beta+1)}\}>\dfrac{N+2s}{2s-\beta-1}$ for some $1<q<\dfrac{N+2s}{N+\beta+1}$,
\end{enumerate}
then the solution is unique.
\end{Theorem}
\begin{proof}
As in the proof of Theorem {I\!\!R}f{th2}, consider the set
$$
E_\beta(\O_T)=\{v\in E_1(\O_T)\mbox{ such that }v\in E_r(\O_T) \mbox{ with }\s'<r<\frac{N+2s}{N+\beta+1} \mbox{ and }\|v\|_{E_r(\O_T)}\leq
l^{\frac{1}{\alpha}}\}.
$$
We define the operator
$$
\begin{array}{rcl}
K:E_\beta(\O_T)&\rightarrow& E_1(\O_T)\\
v&\rightarrow&T(v)=u
\end{array}
$$
where $u$ is the unique solution to problem
\begin{equation}\label{grad00beta}
\left\{
\begin{array}{rcl}
u_t+(-\Deltalta )^s u&=& B(x,t)\,\cdot\, \nabla v+ f\mbox{ in } \Omegaega_T,\\ u(x,t)&=&0\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=&
u_0(x)\mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation}
Since, for $v\in E_\beta(\O_T)$,
$$
\iint_{\O_T}|B(x,t)| |\nabla v| \d^\beta dxdt \leqslantC(T,\O)\Big(\displaystyle\int_{0}^{T}\displaystyle\int_{\Omegaega}|B(x,t)|^\s
\d^{\beta\s}dxdt\Big)^{\frac{1}{\s}}\Big(\displaystyle\int_{0}^{T}\displaystyle\int_{\Omegaega}|\nabla v|^r dxdt\Big)^{\frac{1}{r}},
$$
we obtain that $|\langle B(x,t), \nabla v\rangle+ f|\d^\beta\in L^1(\O_T)$. Thus using Theorem {I\!\!R}f{key}, we get the existence of a unique $u\in E_a(\O_T)$ for all
$a<\dfrac{N+2s}{N+\beta+1}$ and
$$
\begin{array}{lll}
||u||_{E_a(\O_T)} &\leqslant& \displaystyle C(\O_T)\bigg(||f\d^\beta||_{L^1(\O_T)}+\Big(\iint_{\O_T}|B(x,t)|^\s
\d^{\beta\s}dxdt\Big)^{\frac{1}{\s}}\Big(\iint_{\O_T}|\nabla v|^r dxdt\Big)^{\frac{1}{r}}\bigg)\\ &\leqslant&
C(\O_T)\bigg(||f\d^\beta||_{L^1(\O_T)}+l\bigg)\leqslantl^{\frac{1}{r}}.
\end{array}
$$
Hence we conclude that $K$ is well defined and that $K(E_\beta(\O_T))\subset E_\beta(\O_T)$. Now the rest of the proof follows exactly as the proof of Theorem
{I\!\!R}f{th2}.
To prove the uniqueness part under the additional hypotheses on $B$, we follow exactly the same arguments as in the proof of uniqueness part in Theorem {I\!\!R}f{th2}.
\end{proof}
\subsection{Applications}
In this subsection we will obtain some applications of Theorem {I\!\!R}f{th2} in order to prove a comparison principle and, as a consequence, a uniqueness result for some particular cases of the quasi-linear problem. We begin by showing the next comparison principle.
\begin{Theorem}\label{compa0}(Comparison Principle)
Let $w_1, w_2\in L^q(0,T;W^{1,q}_0(\O))$ for all $q<\frac{N+2s}{N+1}$, be such that
$$\begin{array}{ll}
\begin{cases}(w_1)_t+(-\Deltalta)^s w_1 = H_1(x,t,w_1,\nabla w_1) \hbox{ in }\Omegaega_T, \\
w_1 = 0 \mbox{ in } (\mathbb{R}^{N}\setminus\Omegaega)\times (0,T),\\ w(x,0) = u_{0}(x) \mbox{ in }\Omegaega,
\end{cases}
&
\begin{cases}(w_2)_t+(-\Deltalta)^s w_2=H_2(x,t,w_2,\nabla w_2)\hbox{ in }\O_T, \\
w_2=0\mbox{ in } (\mathbb{R}^{N}\setminus\Omegaega)\times (0,T),\\ w_2(x,0)=\hat{u}_0(x)\mbox{ in }\Omegaega,
\end{cases}
\end{array}$$
\
where
\begin{enumerate}
\item $H_1(x,t,w_1,\nabla w_1), H_1(x,t,w_2,\nabla w_2)\in L^1(\O_T)$ and $u_{0}, \hat{u}_{0} \in L^1(\O)$. \item $ H_1(x,t,w_1,\nabla w_1)-H_1(x,t,w_2,\nabla w_2)=\langle
B(x,t,w_1,w_2), \nabla (w_1-w_2)\rangle + f(x,t,w_1,w_2) \mbox{ in }\O_T$ where $B\in (L^{m}(\O_T))^N$ with
$m>\max\{q', \dfrac{2s}{(N+2s)-q(N+1)}\}>\dfrac{N+2s}{2s-1}$ for some $1<q<\dfrac{N+2s}{N+1}$ and $f\leqslant0$ in $\O_T$.
\item $u_{0}\leqslant\hat{u}_{0} \in L^1(\O)$.
\end{enumerate}
Then $w_1\leqslantw_2$ in $\O_T$.
\end{Theorem}
\begin{proof}
Let $v=(u_1-u_2)$, then $v$ solves
\begin{equation}\label{op}
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v& = & B(x,t,w_1,w_2)\,\cdot\, \nabla v + f(x,t,w_1,w_2) & \mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times (0,T),\\ v(x,0)&=&v_0=u_{0}-\hat{u}_{0} & \mbox{ in }\Omegaega.
\end{array}\right.
\end{equation}
Since $B$ satisfies the hypotheses of Theorem {I\!\!R}f{th2}, it follows that problem \eqref{op} has a unique solution. Now, using the fact that $f\leqslant0$ in $\O_T$ and $v_0\leqslant0$, then by Theorem {I\!\!R}f{th2}, we reach that $v\leqslant0$ in $\O_T$ and then we conclude.
\end{proof}
As a consequence, we get the next comparison result for approximated problems.
\begin{Theorem}\label{uniqapr}
Assume that $a>0$ and $\a>1$. Then for all $(f,u_0)\in L^1(\O_T)\times L^1(\O)$, the problem
\begin{equation}\label{apr0}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&\dfrac{|\nabla u|^{\alpha}}{a+|\nabla u|^{\alpha}}+ f & \mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
has a unique solution $u_a$ such that $u_a\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$ and $T_k(u_a)\in L^{2}(0,T;H^s_0(\Omegaega))$, for all
$k>0$. Moreover, if $0<a_1<a_2$, then $u_{a_1}\geqslantu_{a_2}$.
\end{Theorem}
\begin{proof}
Since the dependance on the gradient term is bounded, then the existence follows using the arguments as in the proof of Theorem \eqref{gradiente}. Let show the comparison result which gives apriori the uniqueness part. Let $0<a_1\leqslanta_2$ and consider $u_{a_1}, u_{a_2}$ the solutions to \eqref{apr0} with $a=a_1,
a_2$ respectively. Setting $v=(u_{a_1}-u_{a_2})$, then $v\in L^\tau((0,T),W^{1,\tau}_0(\O))$, for all $\tau<\dfrac{N+2s}{N+1}$, and $v$ solves
$$
v_t+(-\Deltalta )^s v=\dfrac{|\nabla u_1|^{\alpha}}{a_1+|\nabla u_1|^{\alpha}}-\dfrac{|\nabla u_2|^{\alpha}}{a_2+|\nabla u_2|^{\alpha}}.
$$
Let $H(\rho)=\dfrac{\rho^\a}{a_1+\rho^\a}, \,\, \rho\geqslant0$, then
$$
v_t+(-\Deltalta )^s v=H(|\nabla u_1|)-H(|\nabla u_2|) -\bigg(\dfrac{|\nabla u_2|^{\alpha}}{a_1+|\nabla u_2|^{\alpha}}-\dfrac{|\nabla u_2|^{\alpha}}{a_2+|\nabla u_2|^{\alpha}}\bigg).
$$
It is clear that $h(x,t):=\bigg(\dfrac{|\nabla u_2|^{\alpha}}{a_1+|\nabla u_2|^{\alpha}}-\dfrac{|\nabla u_2|^{\alpha}}{a_2+|\nabla u_2|^{\alpha}}\bigg)\geqslant0$ a.e. in $\O_T$. Thus $v$ satisfies
\begin{equation*}
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v& = &B(x,t)\,\cdot\, \nabla v +h(x,t)& \mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
v(x,0)&=&0 & \mbox{ in }\Omegaega,
\end{array}\right.
\end{equation*}
where
$$
B(x,t)= \left\{
\begin{array}{rcll}
0 & & \mbox{ if } & |\nabla u_{a_1}-\nabla u_{a_2}|=0,\\ \\ (H(|\nabla u_{a_1}|)-H(|\nabla u_{a_2}|))\dfrac{\nabla u_{a_1}-\nabla u_{a_2}}{|\nabla u_{a_1}-\nabla u_{a_2}|^2} & & \mbox{ if
} & |\nabla u_{a_1}-\nabla u_{a_2}|\neq 0.
\end{array}
\right.
$$
One can easily see that $|B(x,t)|\leqslantC$. Therefore by Theorem {I\!\!R}f{th2} we obtain that $v\leqslant0$ and then we conclude.
\end{proof}
\begin{Theorem}\label{uniqq}
Consider the problem
\begin{equation}\label{uniq-grad}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u&=&|\nabla u|^{\alpha}+ f & \mbox{ in } \Omegaega_T,\\ u(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
u(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
where $\Omegaega\subset {I\!\!R}n$ is a bounded regular domain with $N> 2s$ and $\dfrac{1}{2}<s<1$. Assume that $\a<\dfrac{N+2s}{N+1}$ and $(f,u_0)\in L^1(\O_T)\times L^1(\O)$ are non negative functions. Then problem \eqref{uniq-grad} has a minimal solution $u\in
L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$. {In addition, if
$\a<\dfrac{(N+2s)^2+2s(N+1)}{(N+1)(N+4s)}<\dfrac{N+2s}{N+1}$, then the solution is unique.}
\end{Theorem}
\begin{proof}
Using Theorem {I\!\!R}f{maria} we get the existence of $T_0\leqslantT$ such that problem \eqref{uniq-grad} has a solution $u\in
L^{q}(0,T_0;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$.
To show the existence of a minimal solution we consider $f_n=T_n(f)$ and $u_{0n}=T_n(u_0)$. Define $u_n$ to be the unique solution to problem
\begin{equation}\label{apr00n}
\left\{
\begin{array}{rcll}
u_{nt}+(-\Deltalta )^s u_n&=&\dfrac{|\nabla u_n|^{\alpha}}{1+{\frac 1n}|\nabla u_n|^{\alpha}}+ f_n & \mbox{ in } \Omegaega_{T_0}\equiv\Omegaega\times (0,T_0),\\ u_n(x,t)&=&0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T_0),\\ u_n(x,0)&=&u_{0n}(x) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
It is clear that $\{u_n\}_n$ is an increasing sequence of $n$. If $\hat{u}$ is a nonnegative solution to problem \eqref{uniq-grad} with $\hat{u}\in
L^{q}(0,T_0;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$, then by the comparison principle in Theorem {I\!\!R}f{uniqapr} we deduce that $u_n\leqslant\hat{u}$ for all $n$. Hence we get the existence of $u=\lim\limit_{n\to\infty}ts_{n\to \infty}u_n$ such that $u\leqslant\hat{u}$. Thus $u\in L^r(\O_{T_0})$ for all $r<\dfrac{N+2s}{N}$. To finish we have just to show that $u$ is a solution to problem \eqref{uniq-grad}.
We claim that the sequence $\bigg\{\dfrac{|\nabla u_n|^{\alpha}}{1+{\frac 1n}|\nabla u_n|^{\alpha}}\bigg\}_n$ is bounded in $L^1(\O_{T_0})$.
For simplicity of tipping we set $g_n(x,t)=\dfrac{|\nabla u_n|^{\alpha}}{1+{\frac 1n}|\nabla u_n|^{\alpha}}$ and we define $w_n$ to be the unique solution to the problem
\begin{equation*}
\left\{
\begin{array}{rcll}
w_{nt}+(-\Deltalta )^s w_n&=& f_n & \mbox{ in } \Omegaega_{T_0}\equiv\Omegaega\times (0,T_0),\\ w_n(x,t)&=&0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T_0),\\ w_n(x,0)&=&u_{0n}(x) & \mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation*}
By Theorem {I\!\!R}f{th1}, we obtain that the sequence $\{w_n\}$ is bounded in $L^{q}(0,T_0;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$ and that $w_n\uparrow w$, the unique weak solution to the problem
\begin{equation*}
\left\{
\begin{array}{rcll}
w_t+(-\Deltalta )^s w&=& f & \mbox{ in } \Omegaega_{T_0},\\ w(x,t)&=&0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T_0),\\ w(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation*}
Thus
$$
u_n(x,t)=\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega} g_n(y,\sigma) P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma +w_n(x,t)
$$
and
\begin{equation*}
|\nabla u_n(x,t)|\leqslant\displaystyle \displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega}g_n(y,\sigma)|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma +|\nabla w_n(x,t)|.
\end{equation*}
Fixed $q\in \Big(\a, \dfrac{N+2s}{N+1}\Big)$, then
\begin{equation}\label{nnn}
|\nabla u_n(x,t)|^q\leqslant\displaystyle C\bigg(\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega}g_n(y,\sigma)|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma\bigg)^q +C |\nabla w_n(x,t)|^q.
\end{equation}
Setting
$$
D_n(x,t)=\bigg(\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega}g_n(y,\sigma)|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|\,dy\,d\sigma\bigg)^q,
$$
and taking into consideration that $\{|\nabla w_n|^q\}_n$ is bounded in $L^1(\O)$, to prove the claim we have just to show that $\{D_n\}$ is bounded in $L^1(\O_T)$.
As in the proof of the compactness part in Theorem {I\!\!R}f{th1}, we have
\begin{eqnarray*}
D_n(x,t) & = & \bigg(\displaystyle\int_{0}^{t} \displaystyle\int_{\Omegaega}g_n(y,\sigma)\frac{|\nabla_x P_{\Omegaega} (x,y, t-\sigma)|}{P_{\Omegaega} (x,y, t-\sigma)} P_{\Omegaega} (x,y, t-\sigma)\,dy\,d\sigma\bigg)^q\\
&\leqslant& C\Big(\iint_{\{\O\times (0,t) \cap\{\d(x)>(t-\s)^{\frac{1}{2s}}\}\}\}}g_n(y,\sigma) \frac{P_{\Omegaega} (x,y, t-\sigma)}{(t-\s)^{\frac{1}{2s}}}\,dy\,d\sigma\Big)^q\\
&+ & \frac{C}{\d^q(x)}\displaystyle \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslant(t-\s)^{\frac{1}{2s}}\}\}\}}g_n(y,\sigma){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q\\ \\
&=& D_{n1}(x,t) + D_{n2}(x,t).
\end{eqnarray*}
Similar to estimating the terms $J_{21}$ and $J_{22}$ in \eqref{j21}, \eqref{j21} respectively, we reach that
\begin{equation}\label{D21n}
\iint_{\O_T} D_{n1}(x,t)dxdt\leqslantC(\O_T)T_0^{\frac{(r-(q-1))(\g_2+1)}{r}}||g_n||_{L^1(\O)}||\hat{u}||^{q-1}_{L^r(\O_{T_0})}
\end{equation}
and
\begin{equation}\label{dn22}
D_{n2}(x,t)=\frac{C}{\d^q(x)}\displaystyle \Big(\iint_{\{\O\times (0,t) \cap\{\d(x)\leqslantt^{\frac{1}{2s}}\}\}\}}g_n(y,\sigma){P_{\Omegaega} (x,y, t-\sigma)}\,dy\,d\sigma\Big)^q\leqslantC\frac{\hat{u}^q(x,t)}{\d^q(x)},
\end{equation}
where $r<\frac{N+2s}{N}$. Thus
$$
\iint_{\O_{T_0}} D_{n}(x,t)dxdt\leqslantC(\O_{T_0})\iint_{\O_{T_0}}\frac{\hat{u}^q(x,t)}{\d^q(x)} dxdt + T^{\frac{(r-(q-1))(\g_2+1)}{r}}||g_n||_{L^1(\O)}||\hat{u}||^{q-1}_{L^r(\O_{T_0})}.
$$
Using the fact that $\dfrac{\hat{u}^q(x,t)}{\d^q(x)}\in L^1(\O_{T_0})$ for all $q<\frac{N+2s}{N+1}$, $\hat{u}\in L^r(\O_{T_0})$ for all $r<\frac{N+2s}{N}$, and going back to \eqref{nnn}, it holds that
$$
\iint_{\O_{T_0}} |\nabla u_n(x,t)|^q dxdt\leqslantC(\O_{T_0})||g_n||_{L^1(\O)}+C(\O_{T_0})\leqslantC(\O_{T_0}) \iint_{\O_{T_0}}|\nabla u_n(x,t)|^\alpha dxdt + C(\O_{T_0}).
$$
Using the fact that $\a<q$ and by Young inequality we reach that $\displaystyle\iint_{\O_T} |\nabla u_n(x,t)|^q dxdt\leqslantC(\O_{T_0})$ for all $n$. Hence $\{g_n\}_n$ is bounded in $L^1(\O_{T_0})$ and the claim follows. Therefore, using Theorem {I\!\!R}f{th1} we conclude that, up to a subsequence, $u_n\to u$ strongly in $L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$. Thus $u$ is the minimal solution to problem \eqref{uniq-grad} in $\O_{T_0}$.
{We prove now that the minimal solution $u$ can be defined in the set $\O_T$. According to Theorem {I\!\!R}f{maria}, the existence result holds for $L^1$ data in the set $\O\times (t_1,t_2)$ if $t_2-t_1\leqslant\ddot{C}:=\ddot{C}(\O,s,N)$. Let $u$ the minimal solution obtained above in the set $\O\times (0,T_0)$ and suppose that $T_0<T$. Consider $T_1=T_0-\e$ with $\e>0$ is chosen such that $0<\e<\ddot{C}$. Then $u(.,T_1)\in L^1(\O)$ and then the problem
\begin{equation}\label{globTT}
\left\{
\begin{array}{rcll}
v_{t}+(-\Deltalta )^s v&=&|\nabla v|^{\alpha}+ f & \mbox{ in } \Omegaega\times (T_1,T_1+\ddot{C}),\\ v(x,t)&=&0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [T_1,T_1+\ddot{C}),\\ v(x,T_1)&=&u(x,T_1) & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
has a minimal solution $v$. It is clear that $u$ is a solution of the same problem as $v$ in the set $\O\times [T_1,T_0)$. Hence $v=u$ in the set $\O\times [T_1,T_0)$. Setting
$$
\overline{u}(x,t)=
\left\{
\begin{array}{lll}
u(x,t) &\mbox{ if }& (x,t)\in \O\times [0, T_1]\\
v(x,t) &\mbox{ if }& (x,t)\in \O\times [T_1, T_1+\ddot{C})\\
\end{array}
\right.
$$
then $\overline{u}$ is the minimal solution to the problem
\begin{equation}\label{globTTTT}
\left\{
\begin{array}{rcll}
\overline{u}_{t}+(-\Deltalta )^s \overline{u}&=&|\nabla \overline{u}|^{\alpha}+ f & \mbox{ in } \Omegaega\times (0, T_1+\ddot{C}),\\ \overline{u}(x,t)&=& 0 &
\mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times (0,T_1+\ddot{C}),\\ \overline{u}(x,0)&=&u_{0}(x) & \mbox{ in }\Omegaega.\\
\end{array}\right.
\end{equation}
Repeating the above argument in a finite time of steps we get the existence of a minimal solution $u$ to the problem \eqref{uniq-grad} defined in the set $\O_T$ with $u\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$.\\
}
{
Finally to show the uniqueness under the condition $\a<\dfrac{(N+2s)^2+2s(N+1)}{(N+1)(N+4s)}$. Notice that $\dfrac{(N+2s)^2+2s(N+1)}{(N+1)(N+4s)}<\dfrac{N+2s}{N+1}$ if and only if $2s>1$ which is our main hypothesis.\\
We will use the comparison principle in Theorem {I\!\!R}f{compa0}. If $u_1,u_2$ are two solutions to problem \eqref{uniq-grad} with $u_1,u_2\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ for all $q<\dfrac{N+2s}{N+1}$. Then $v=u_1-u_2$ solves
\begin{equation}\label{uniq-grad000}
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v&=& \langle B(x,t), \nabla v\rangle & \mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
v(x,0)&=& 0 & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
where $v\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$, $q<\dfrac{N+2s}{N+1}$ and $|B(x,t)|\leqslantC(|\nabla u_1|^{\a-1}+|\nabla u_2|^{\a-1})$. It is clear that $B\in L^m(\O_T)$ for all $m<\dfrac{N+2s}{(N+1)(\a-1)}$. Recall that $\a<\dfrac{N+2s}{N+1}$, then $\a'<\dfrac{N+2s}{(N+1)(\a-1)}$. Since $\a<\dfrac{(N+2s)^2+2s(N+1)}{(N+1)(N+4s)}$, then
$$
\dfrac{N+2s}{(N+1)(\a-1)} >\max\{\a', \dfrac{2s}{(N+2s)-\a(N+1)}\}>\dfrac{N+2s}{2s-1}.
$$
Hence we can chose $m<\frac{N+2s}{(N+1)(\a-1)}$ such that $m>\max\{\a', \dfrac{2s}{(N+2s)-\a(N+1)}\}>\dfrac{N+2s}{2s-1}$. Hence by the comparison principle in Theorem {I\!\!R}f{compa0} we deduce that $v=0$ and then we conclude.
}
\end{proof}
{ Under additional regularity hypothesis on the solution, we can prove the next uniqueness result.
\begin{Theorem}\label{uniiq}
Assume that $\a>1$, then the problem \eqref{uniq-grad} has at most one solution $u$ such that $u\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ with $\dfrac{q}{\a-1}>\max\{\b', \dfrac{2s}{(N+2s)-\b(N+1)}\}$ for some $1<\beta<\dfrac{N+2s}{N+1}$. In particular, problem \eqref{uniq-grad} has at most one solution $u\in \mathcal{C}^{1}(\overline{\O_T})$.
\end{Theorem}
\begin{proof}
If $u_1,u_2$ are two solution with the above regularity, then wetting $v=u_1-u_2$, it holds that $v$ solves the problem
\begin{equation}\label{uniq-grad33}
\left\{
\begin{array}{rcll}
v_t+(-\Deltalta )^s v&=& B(x,t)\,\cdot\, \nabla v & \mbox{ in } \Omegaega_T,\\ v(x,t)&=&0 & \mbox{ in }(\mathbb{R}^N\setminus\Omegaega)\times [0,T),\\
v(x,0)&=& 0 & \mbox{ in }\Omegaega,\\
\end{array}\right.
\end{equation}
where $v\in L^{q}(0,T;W_{0}^{1,q}(\Omegaega))$ and $|B(x,t)|\leqslantC(|\nabla u_1|^{\a-1}+|\nabla u_2|^{\a-1})$. According to the regularity hypothesis on $u_1,u_2$, we obtain that $B\in L^m(\O_T)$ with $m=\frac{q}{(\a-1)}>\max\{\b', \dfrac{2s}{(N+2s)-\b(N+1)}\}$ for some $1<\beta<\dfrac{N+2s}{N+1}$.
Thus using the comparison principle in Theorem {I\!\!R}f{compa0} we obtain that $u_1=u_2$ and then we conclude.
\end{proof}
}
\subsection{Some remarks on asymptotic behavior}
We now deal with asymptotic behavior of the solutions. Let us begin by the next global existence result for the Cauchy problem given in {\cal C}_0^{\iy}\,te{DI}.
\begin{Theorem}\label{cauchy}
Assume that $\a<\dfrac{N+2s}{N+1}$ and $u_0\in W^{1,\infty}({I\!\!R}n)\cap L^1({I\!\!R}n)$, then the problem
\begin{equation}\label{cauchyeq}
\left\{
\begin{array}{rcll}
u_t+(-\Deltalta )^s u& = & |\nabla u|^\alpha & \mbox{ in } {I\!\!R}n\times (0,T),\\ u(x,0)&=&u_0(x) & \mbox{ in }{I\!\!R}n,
\end{array}\right.
\end{equation}
has a unique global solution $u$ such that $u\in \mathcal {C}([0,T], W^{1,\infty}({I\!\!R}n))$ for all $T>0$. Moreover if $\a>\dfrac{N+2s}{N+1}$ and either
$||u_0||_{L^1({I\!\!R}n)}$ or $||\nabla u_0||_{\infty}$ is small, then $||u(.,t)||_{L^1(\O)}\leqslantC$ for all $t$.
\end{Theorem}
It is clear that if $u$ is a solution to problem \eqref{cauchyeq} with $u_0$ satisfying the conditions of Theorem {I\!\!R}f{cauchy}, then $u$ is globally defined in $t$. We refer to {\cal C}_0^{\iy}\,te{DI} and {\cal C}_0^{\iy}\,te{W} for the proof.
In our case and according to the value of $\a$, we can prove the next partial blow up result.
\begin{Theorem}\label{blowup1}
Assume that $s\in (\dfrac{\sqrt{5}-1}{2}, 1]$ and suppose that $1+s<\a<\dfrac{s}{1-s}$. Then for all data $f\in L^\infty(\O\times (0,\infty))$, $f\geqslant0$, the solution $u$ to
problem \eqref{grad} obtained in Theorem {I\!\!R}f{fix001} blows-up in a finite time in the sense that
$$
\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\d^s(x)dx\to \infty\mbox{ for }t\to T^*.
$$
\end{Theorem}
\begin{proof} Since $s>\dfrac{\sqrt{5}-1}{2}$ the interval of $\alpha$ is non empty.
We will use a convexity argument. Let $\partialhi_1$ be the first positive bounded eigenfunction of the fractional Laplacian, then $\partialhi_1$ satisfies
\begin{equation*}
\left\{
\begin{array}{rcll}
(-\Deltalta )^s \partialhi_1&=& \l_1\partialhi_1 & \mbox{ in } \O,\\ \partialhi_1&>& 0 & \mbox{ in }\O,\\ \partialhi_1 &=& 0 & \mbox{ in } \mathbb{R}^N\setminus\Omegaega,
\end{array}\right.
\end{equation*}
and $\partialhi_1(x)\backsimeq \d^s(x)$. Using $\partialhi_1$ as a test function in the problem of $u$ and integrating in $x$, we reach that
$$
\frac{d}{dt}\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx+\l_1 \int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx =\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u|^\a\partialhi_1(x) dx\geqslantC\int\lim\limit_{n\to\infty}tss_\Omega |\nabla u|^\a\d^s(x)dx.
$$
Since $s<\a-1$, then using the weighted Hardy inequality \eqref{eq:super-hardy}, we conclude that
$$
\displaystyle \frac{d}{dt}\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx+\l_1 \int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx \geqslantC(\O,s)\int\lim\limit_{n\to\infty}tss_\Omega \frac{u^{\alpha}(x,t)}{\d^{\alpha-s}(x)}dx.
$$
On the other hand $\a>s$, hence
$\displaystyle
\int\lim\limit_{n\to\infty}tss_\Omega \frac{u^{\alpha}(x,t)}{\d^{\alpha-s}(x)}dx \geqslantC(\O,s)\int\lim\limit_{n\to\infty}tss_\Omega u^{\a}(x,t)\partialhi_1(x) dx.
$
Therefore using Jensen's inequality it follows that
\begin{eqnarray*}
\displaystyle \frac{d}{dt}\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx+\l_1 \int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx & \geqslant& \displaystyle C(\O,s)\int\lim\limit_{n\to\infty}tss_\Omega u^{\alpha}(x,t)\partialhi_1(x)dx\\
& \geqslant& \displaystyle C(\O,s,\alpha)\left(\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx\right)^\alpha.
\end{eqnarray*}
Define $Y(t)=\int\lim\limit_{n\to\infty}tss_\Omega u(x,t)\partialhi_1(x) dx$, then
$$
Y'(t)+\l_1 Y(t)\geqslantC(\O,s,\alpha)Y^\a(t).
$$
A simple convex argument allows us to get the existence of $A$ such that if $Y(0)>A$, then $Y(t)\to \infty$ if $t\to T^*$ depending only on the data. Hence we
conclude.
\end{proof}
\end{document} | math | 205,825 |
\begin{document}
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\preprint{APS/123-QED}
\title{Minimal informationally complete measurements \\ for probability representation of quantum dynamics}
\author{V.I. Yashin}
\affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\affiliation{Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia}
\affiliation{Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia}
\author{E.O. Kiktenko}
\affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\affiliation{Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia}
\affiliation{Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia}
\author{A.S. Mastiukova}
\affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\affiliation{Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia}
\author{A.K. Fedorov}
\affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\affiliation{Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia}
\date{\today}
\begin{abstract}
In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions,
which are obtained via minimal informationally complete quantum measurements.
The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems.
A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs).
We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism.
Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator.
We apply the MIC-POVM-based probability representation to the digital quantum computing model.
In particular, for the case of spin-$1/2$ evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics.
One of the most important findings is that the MIC-POVM-based probability representation
gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach.
Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum (NISQ) devices.
\end{abstract}
\maketitle
\section{Introduction}
The problem of the description of quantum dynamics plays a significant role both in studying fundamental aspects of quantum physics~\cite{Schroeck1996} and exploring potential applications~\cite{Lukin2014}.
In the latter case, it is crucial to highlight the role of non-classical phenomena and understand the origin of advantages of the use of quantum systems in various applications, such as quantum communication and quantum computing~\cite{NielsenChuang}.
This question is quite non-trivial, in particular, due to the fact that the commonly used descriptions of quantum states drastically differ from the language of statistical physics, which uses probability distributions.
Several attempts to describe quantum systems using quantum analogues of probability distributions, such as the Wigner function~\cite{Wigner1932},
have been made~\cite{Wigner1932,Glauber1963,Sudarshan1963,Glauber1969,Agarwal1970,Husimi1940,Kano1965,Ferrie2009}.
Although the Wigner function cannot be interpreted as the probability distribution since it takes negative values,
its negativity can be linked to the resource providing quantum speed-up in solving computational problems~\cite{Galvao2006,Spekkens2008,Ferrie2011,Gottesman2012,Gottesman2014,Howard2014,Raussendorf2015,Pashayan2015}.
Quasi-probability distributions of the other type such the Glauber-Sudarshan~\cite{Glauber1963,Sudarshan1963} and Husimi~\cite{Husimi1940} functions are also actively used for the description of quantum systems.
Quantum phenomena can be also described on the language of tomographic distributions~\cite{Manko1996,Manko1997,Manko2010,Fedorov2013}, which are parametrized family of probability distributions.
Quantum tomograms are related to Wigner functions via the Radon transformation~\cite{Lvovsky2009}.
Quantum tomography is essentially related to the question of the completeness of quantum measurements~\cite{Lvovsky2009,Busch1991}.
Advances in understanding the role of various types of quantum measurements have formulated several new concepts.
In particular, quantum systems can be described via a probability distribution, which is obtained via informationally complete (IC) quantum measurements~\cite{Busch1991,Busch1995}.
Since measurements in the quantum domain are represented by positive operator valued measures (POVMs),
the full determination of quantum states requires the use of so-called informationally complete POVMs (IC-POVMs)~\cite{Busch1991,Busch1995,Caves2004}.
Importantly, this question is linked to the idea of using single measurement for quantum state characterization~\cite{Busch1995},
and to the concept of the Husimi representation the quantum systems with continuous variables systems~\cite{Husimi1940}.
We also note that probability structures behind quantum theory have been widely studied~\cite{Busch1991,Busch1995,Holevo2012}.
An important special case of IC-POVM is its symmetric version, which is known as symmetric IC-POVMs (SIC-POVMs), where all pairwise inner products between the POVM elements are equal.
SIC-POVMs are explored in various applications including tomographic measurements~\cite{Manko2010,Caves2002}, quantum cryptography~\cite{Fuchs2003}, and measurement-based quantum computing~\cite{Jozsa205}.
In addition, the idea of SIC-POVMs is actively used in quantum Bayesianism reformulation of quantum mechanics~\cite{Caves2002,Caves20022,Caves2004,Fuchs2013}.
The quantum part of a classical probability simplex, which is achievable by measurements obtained via SIC-POVM (SIC-POVM measurements), is referred to as a qplex (i.e. a `quantum simplex')~\cite{Appleby2017}.
The SIC-POVM formalism can be further extended for the description of dynamics of finite-dimensional quantum systems.
The main difference with the quasi-probability representation is that quantum systems are described via (positive and normalized) probability distributions, which are obtained by SIC-POVMs~\cite{Kiktenko2020}.
These probability distributions evolve under the action of pseudostochastic maps --- stochastic maps, which are described by matrices that may have {\it negative} elements.
This idea, in a sense, changes the paradigm of revealing the distinction between quantum and classical dynamics.
Indeed it allows linking `quantumness' with negative probabilities can be extended to the study of non-classical properties of quantum dynamics and measurement processes~\cite{Wetering2017}.
Quantum dynamical equations both for unitary evolution of the density matrix governed by the von Neumann equation and dissipative evolution governed by Markovian master equation,
which is governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator, can be derived~\cite{Kiktenko2020}.
Moreover, practical measures of non-Markovianity of quantum processes can be obtained and applied for studying existing quantum computing devices.
However, this approach has a number of challenging aspects, which are related in particular to the problem of SIC-POVM existence, which is considered analytically and numerically just for a number of cases (for a review, see Ref.~\cite{Fuchs2017}).
Thus, this representation is based on probability distributions, which are given by SIC-POVM measurements, is hardly applicable to the analysis of multi-qubit systems with an arbitrary number of qubits.
This limits applications of such an approach, in particular, for the analysis of quantum information processing devices.
\begin{figure}
\caption{Quantum dynamics in terms of the commonly accepted density matrix approach (a quantum channel $\Phi(\cdot)$ acts on a density matrix $\rho$, and the measurement process is described by POVM $M$) and
the probabilistic description via MIC-POVM (a pseudostochastic map $\mathbf{S}
\label{fig:sch}
\end{figure}
In this work, we present a generalization of the probability representation of quantum dynamics using minimal informationally complete positive operator-valued measures based (MIC-POVM) measurements,
which are an important class of quantum measurements~\cite{Weigert2000,Weigert2006,DeBrota2020,Smania2020,Planat2018}.
For a $d$-dimensional Hilbert space an IC-POVM is said to be MIC-POVM if it contains exactly $d^2$ linearly independent elements.
Here we construct pseudostochastic maps that act on probability distributions, which are obtained by MIC-POVM measurements.
We demonstrate that this approach is a generalization of the SIC-POVM-based representation~\cite{Kiktenko2020}, and it has a number of important features.
First, such an approach allows for preserving the tensor product structure, which is important for the description of multi-qubit systems.
Second, MIC-POVMs are easier to construct in comparison with SIC-POVMs.
Using the MIC-POVM-based probability representation, we derive quantum dynamical equations both for the unitary von-Neumann evolution and the Markovian dissipative evolution,
which is governed by the GKSL generator.
It allows us to generalize previously obtained results on the description of the von-Neumann evolution of quantum systems on the probability language~\cite{Weigert2000}.
We demonstrate how the suggested approach can be applied for the analysis of NISQ computing processes and obtain pseudostochastic maps for various single-qubit decoherence channels, as well as single-qubit and multi-qubit quantum gates.
This gives an interpretation of quantum computations as actions of pseudostochastic maps on bitstring,
where the nature of quantum speedup is linked to the negative elements in pseudostochastic matrices that are corresponding to the quantum algorithm (as a sequence of gates and projective measurements).
Our work is organized as follows.
In Sec.~\ref{sec:construction}, we construct a probability representation via MIC-POVMs.
In Sec.~\ref{sec:dynamics}, we derive quantum dynamical equations in the MIC-POVM representation.
In Sec.~\ref{sec:decoherence}, we use the probability representation to study a dissipative dynamics of a spin-1/2 particle.
One of the key findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach.
In Sec.~\ref{sec:qcomp}, we demonstrate the applicability of the MIC-POVM-based probability representation for the analysis of quantum computing processes.
We illustrate our approach by considering Grover's algorithm.
We summarize the main results and conclude in Sec.~\ref{sec:conclusion}.
\section{Probability representation via MIC-POVMs}\label{sec:construction}
Here we construct a MIC-POVM-based probability representation of quantum mechanics in the case of finite-dimensional systems.
For this purpose, we first consider the representation of states, then study their evolution described by quantum channel, consider quantum measurements, and finally discuss a transition between representations defined by different MIC-POVMs.
We also highlight here an important feature of the MIC-POVM probability representation, which is the simple tensor product structure.
The summary of results is presented in Table~\ref{tbl:1}.
\begin{table*}[t]
\begin{tabular}{p{0.17\linewidth}|p{0.34\linewidth}|p{0.36\linewidth}}
& Standard formalism & MIC-POVM formalism \\ \hline \hline
State & Density matrix $\rho\in \States{\mathcal{H}}$ & Probability vector $p \in \Distributions{E}$ \\
Channel & CPTP map $\Phi: \States{\mathcal{H}^{\rm in}} \rightarrow \States{\mathcal{H}^{\rm out}} $ & Pseudostochastic matrix ${\bf S} \in \Mat{d^{2}}{d^{2}}{\mathbb{R}}$ \\
Measurement & POVM $M=\{M_{i}\}_{i=1}^{m}$, $M_{i} \in \Effects{\mathcal{H}}$ & Pseudostochasitc matrix ${\bf M} \in \Mat{m}{d^{2}}{\mathbb{R}}$ \\\hline
Tensor product rules & $\rho^{AB}=\rho^{A}\otimes\rho^{B}$ & $p^{AB}=p^{A} \otimes p^{B}$ \\
& $\Phi^{AB}=\Phi^{A}\otimes \Phi^{B}$ & ${\bf S}^{AB}={\bf S}^{A} \otimes {\bf S}^{B}$ \\
& $M^{AB}=M^{A}\otimes M^{B}=\{M^{A}_{i}\otimes M^{B}_{j}\}_{i=1,j=1}^{m_{1},m_{2}}$ & ${\bf M}^{AB} = {\bf M}^{A} \otimes {\bf M}^{B}$
\end{tabular}
\caption{The correspondence between standard and MIC-POVM-based formalisms.}
\label{tbl:1}
\end{table*}
\subsection{Definitions and notations}\label{sec:preliminaries}
We start our consideration by introducing basic definitions and notations. Let $\mathcal{H}$ be a $d$-dimensional Hilbert space, where $d$ is finite.
We introduce $\Bound{\mathcal{H}}$ as an algebra of bounded operators on $\mathcal{H}$.
We also introduce the space of trace class operators $\TrClass{\mathcal{H}}$ and space of $n \times k$-matrices over the field $\mathbb{F}$, which we refer to as $\Mat{n}{k}{\mathbb{F}}$.
Standardly, we use density operators $\rho \in \TrClass{\mathcal{H}}$ for the description of quantum states, where $\rho\geq0$ and $\Tr{\rho} = 1$.
The convex space of quantum states is denoted as $ \States{\mathcal{H}} $.
Extreme points of this space are called \emph{pure states}, and we denote them as $\StatesPure{\mathcal{H}} $.
An operator $B \in \Bound{\mathcal{H}}$ is called an \emph{effect}, if $0\leq{B}\leq {\bf I}_{d}$, where we use ${\bf I}_{n}$ to denote $n$-dimensional identity operator.
The space of all effects is denoted by $\Effects{\mathcal{H}}$.
The \emph{channel} $\Phi :\States{\mathcal{H}^{\rm in}}\rightarrow\States{\mathcal{H}^{\rm out}} $ is a trace-preserving, completely-positive (CPTP) linear map between states on Hilbert spaces $\mathcal{H}^{\rm in}$ and $\mathcal{H}^{\rm out}$.
A set of effects $E=\{E_1,\ldots,E_m\}$ with $E_k\geq 0$ and that satisfies the condition $\sum_k E_k = {\bf I}_{d} $, is known as POVM (positive operator-valued measure).
The Born rule implies that a given state $\rho$ defines a probability distribution which we treat as a column-vector
\begin{equation}
p=\begin{bmatrix} p_1 & \ldots & p_m \end{bmatrix}^\top, \quad p_k=\Tr{\rho E_k},
\end{equation}
where $\top$ the denotes a standard transposition.
The POVM $E$ is the MIC-POVM if it forms the basis of $ \Bound{\mathcal{H}} $.
In this case, $E$ contains $d^2$ elements, and every state $\rho$ is fully described by the corresponding probability vector $p$.
We denote a set of possible probability vectors with fixed MIC-POVM $E$ and varying $\rho$ as $\Distributions{E} $.
We note that SIC-POVMs are a particular class of MIC-POVMs.
The elements of SIC-POVM $E^{\rm sym}$ have the following form:
\begin{equation}
E_k^{\rm sym} = \frac{1}{d} \ket{\psi^{\rm sym}_k}\bra{\psi^{\rm sym}_k}, \quad k = 1,\ldots,d^{2}.
\end{equation}
Here vectors $\ket{\psi^{\rm sym}_k}$ satisfy the following condition:
\begin{equation}
|\langle{\psi^{\rm sym}_k}|\psi^{\rm sym}_l\rangle|^2 = \frac{d\delta_{kl}+1}{d+1},
\end{equation}
and $\delta_{kl}$ is the Kronecker symbol.
As it is noted above, MIC-POVM and SIC-POVM measurements give probability distributions that fully describe quantum states.
Therefore, in order to describe the dynamics of quantum systems, one has to find corresponding maps acting on these probability distributions.
We note that stochastic maps are not sufficient for the description of quantum dynamics.
As it is shown in Ref.~\cite{Wetering2017}, corresponding maps for a description of dynamics of quantum states, which are presented by a probability distribution, are quasistochastic or pseudostochastic.
In line with Ref.~\cite{Chruscinski2013, Chruscinski2015, Kiktenko2020} in our work we prefer to use a term pseudostochastic to emphasize that we deal with classical probability distributions rather than quasi-probabilities.
We remind that a stochastic matrix $M\in\Mat{n}{k}{\mathbb{R}} $ is a matrix, for which $ M_{ij} \geq 0, \sum_{i=1}^{n} M_{ij} = 1 $ for every $j=1,\ldots,k$.
It is called \emph{bistochastic}, if $n=k$ and also $\sum_j M_{ij} = 1$ for every $i = 1,\ldots,n$.
Under bistochastic map, a fully chaotic state $\begin{bmatrix} 1/n & \ldots & 1/n \end{bmatrix}$ remains the same.
For the description of quantum dynamics it is necessary to introduce pseudostochastic maps which can be presented as a matrix $M \in \Mat{n}{k}{\mathbb{R}}$ with $\sum_{i=1}^{n} M_{ij} = 1$ but without the restriction on the positivity of matrix elements.
The square $n\times n$ matrix is pseudobistochastic if for any $j\in\{1,\ldots n\}$ one has $\sum_{j=1}^{n} M_{ij} = 1$ as well (again, some elements of $M_{ij} $ may be negative).
\subsection{Representation of states}\label{sec:states}
We consider a MIC-POVM $E=\{E_k\}_{k=1}^{d^2}$ in the $d$-dimensional Hilbert space $\mathcal{H}$.
There is a canonical duality between spaces $\Bound{\mathcal{H}}$ and $\TrClass{\mathcal{H}}$, which is given by the bilinear form $(B,\rho) \mapsto \Tr{\rho B}$ for $\rho\in\TrClass{\mathcal{H}}$ and $B\in\Bound{\mathcal{H}}$.
It means that any linear functional on $\Bound{\mathcal{H}}$ can be represented as $\Tr{\rho~\cdot}$, and any functional on $\TrClass{\mathcal{H}}$ can be represented as $\Tr{\cdot~B}$.
Since MIC-POVM $E$ forms a linear basis in $\Bound{\mathcal{H}}$ one can construct a basis $e = \{e_i\}_{i=1}^{d^{2}} $ in $\TrClass{\mathcal{H}}$, such that $\Tr{E_l e_k} = \delta_{l,k}$.
This basis is usually referred to as a \emph{dual} basis to $E$.
Explicitly, the elements of this basis are as follows:
\begin{equation}
e_l = \sum_{k=1}^{d^2} ({\bf T}^{-1})_{lk} E_k , \quad {\bf T}_{nm} = \Tr{E_n E_m}.
\end{equation}
Then an arbitrary state $\rho \in \States{\mathcal{H}}$ can be represented in the following form:
\begin{equation}\label{rho}
\rho = \sum_{k=1}^{d^2}{p_k e_k}, \quad p_{k}={\rm Tr}(\rho E_{k}).
\end{equation}
We note that in the SIC-POVM case we have
\begin{equation}
\begin{aligned}
{\bf T}_{nm} &= \frac{d\delta_{nm}+1}{d^{2}(d+1)}, \\
({\bf T}^{-1})_{nm} &= d(d+1)\delta_{nm} - 1
\end{aligned}
\end{equation}
Let $s$ and $p$ be two probability vectors corresponding to density operators $\rho$ and $\sigma$.
Then we can introduce a probability representation of the Hilbert--Schmidt product:
\begin{equation}\label{eq:HSproduct}
{\rm Tr} (\rho \sigma) = \sum_{n,m=1}^{d^{2}} p_{n} s_{m} {\rm Tr} (e_{n} e_{m}) = s^\top {\bf T}^{-1} p,
\end{equation}
and an analog of the matrix-matrix multiplication:
\begin{multline}
\left(s*p\right)_k \equiv \Tr{\sigma \rho E_k}\\
=\sum_{n,m=1}^{d^{2}} \Tr{e_n e_m E_k} s_n p_m = p^{\top} \Lambda^{(k)} s.
\end{multline}
Here $\Lambda^{(k)}$ is a $d^{2} \times d^{2}$ matrix with elements $\Lambda^{(k)}_{nm} = \Tr{e_n e_m E_k}$.
One can check that operators $e_k$ have a unit trace, but they are not necessarily positive.
Therefore, not any probability vector $p$ corresponds to a quantum state.
The set of possible distributions $\Distributions{E}$ has a form
\begin{equation}\label{Distr}
\Distributions{E} = \Set{p}{\sum_{k=1}^{d^2}{p_k}= 1, \: \sum_{k=1}^{d^2}{p_k e_k}\geq 0}.
\end{equation}
which is referred to as qplex (see Ref.~\cite{Appleby2017}).
The set of distributions corresponding to pure states is as follows:
\begin{equation}\label{DistrPure}
\DistributionsPure{E}=\Set{p}{\sum_{k=1}^{d^2} p_k = 1, \: p * p = p}.
\end{equation}
The convex hull of this set is a set of distributions corresponding to all states $\Distributions{E}$.
We show schematic diagram of relations between sets $\Distributions{E}$, $\DistributionsPure{E}$, and the full $d^{2}$-dimensional simplex $\mathfrak{X}$ in Fig.~\ref{fig:Qplex}.
Since $\Distributions{E}$ does not occupy the full space of $d^{2}$-dimensional simplex $\mathfrak{X}$ it is valuable to have a method for checking whether given distribution $p$ belongs to $ \Distributions{E}$.
A straightforward way to cope with this task is to apply Eq.~\eqref{rho} to reconstruct $\rho$ and check whether $\rho\geq 0$.
However, in the present work, we are interested in a method that does not require a transition to the standard formalism.
Consider a characteristic polynomial of a density operator $\rho$
\begin{equation}
\chi(\lambda) = \prod_{k=1}^d (\lambda - \lambda_k),
\end{equation}
where $\{ \lambda_n \}_{n=1}^d $ is the spectrum of $\rho$.
Let us define a set $\{a_n\}_{n=1}^d$ with the following elements:
\begin{equation}
a_n := \Tr{\rho^n} = \sum_l (p^{*n})_l = \sum_{k=1}^{d} \lambda_k^n, \quad n=1,\ldots,d.
\end{equation}
In order to check that $ p \in \Distributions{E} $, it is necessary and sufficient to check that $ \lambda_k \geq 0 $ for all $k$.
Using the Newton--Girard identities the characteristic polynomial can be rewritten in the form
\begin{equation}
\chi(\lambda) = \sum_{m=0}^d (-1)^m b_m \lambda^{d-m},
\end{equation}
where $b_{0}= 1$ and
\begin{equation}
b_n = \frac{1}{n}\sum_{i=1}^n (-1)^{i-1} b_{n-i} a_i, \quad n=1,\ldots,d.
\end{equation}
If starting from some $d'\in\{1,\ldots,d\}$
\begin{equation}
b_{d'}=b_{d'+1}=\ldots=b_{d}=0,
\end{equation}
then $\chi(\lambda)$ has $d-d'+1$ zero roots.
It is convenient to remove them from consideration by resetting
\begin{equation}
\chi(\lambda) := \chi(\lambda) / \lambda^{d-d'+1}.
\end{equation}
Otherwise we set $d':=d$.
Then we suggest using the Routh--Hurwitz criterion in order to verify that every root of the polynomial $\chi(\lambda)$ is nonnegative.
Let
\begin{equation}
\widetilde{\chi}(\lambda) = \prod_{k=1}^{d'}(\lambda + \lambda_k) = (-1)^{d'} \chi(-\lambda) = \sum_{m=0}^{d'} b_m \lambda^{d'-m}.
\end{equation}
One can see that nonnegative roots of $\chi(\lambda)$ imply nonpositive roots of $\widetilde{\chi}(\lambda)$.
The Routh--Hurwitz criterion states that every root of $\widetilde{\chi}(\lambda)$ is negative if and only if the principal minors $\{\Delta_i\}_{i=1}^{d'}$ of the Hurwitz matrix
\begin{equation}
\mathscr{H}=\begin{bmatrix}
b_1 & b_3 & b_5 & \cdots & 0 & 0\\
b_0 & b_2 & b_4 & \cdots & 0 & 0\\
0 & b_1 & b_3 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ldots & b_{d'-2} & b_d'
\end{bmatrix}
\end{equation}
are positive:
\begin{equation}
\Delta_1 > 0, \: \Delta_2 > 0, \: \Delta_3 > 0, \: \ldots , \: \Delta_{d'} > 0.
\end{equation}
These relations form the constructive way for checking whether $p \in \Distributions{E}$.
\begin{figure}
\caption{Schematic two-dimensional diagram showing relations, and points of contact to the probability simplex $\mathfrak{X}
\label{fig:Qplex}
\end{figure}
\subsection{Representation of tensor products}\label{Tensor}
The use of MIC-POVM probability vectors allows one to employ a simpler description of tensor products.
This is an important advantage in comparison to the SIC-POVM case~\cite{Kiktenko2020}.
Let $ \mathcal{H}^A$ and $\mathcal{H}^B $ be Hilbert spaces with MIC-POVMs $E^A=\{E_i^A\}_{i=1}^{d_A^2}$ and $E^B=\{E_i^B\}_{i=1}^{d_B^2}$, correspondingly.
We then take MIC-POVM
\begin{equation}
E^{AB}=\{E_i^A\otimes E_j^B\}_{i,j=1}^{d_A^2,d_B^2}\equiv E^A \otimes E^B
\end{equation}
on the space $\mathcal{H}^A \otimes \mathcal{H}^B$.
If $ e^A $ and $ e^B $ are dual bases for $ E^A$ and $E^B $, then $ e^{AB} = e^A \otimes e^B $ is dual to $ E^{AB} $.
If $ \rho^A \in \States{\mathcal{H}^A}, \rho^B \in \States{\mathcal{H}^B} $ are states and $p^A, p^B$ are corresponding probability vectors, then the probability vectors of $\rho^A \otimes \rho^B$ is as follows:
\begin{equation}
p^{AB}_{(kl)} = \Tr{ ( \rho^A \otimes \rho^B) (E^A_k \otimes E^B_l) } = p^A_k p^B_l,
\end{equation}
Here we use notation $(\alpha,\beta)$ with $\alpha\in\{1,\ldots,d_{A}\}$ and $\beta\in\{1,\ldots,d_{B}\}$ to define a multiindex.
One can think that $(\alpha,\beta)\equiv(d_{A}-1)\alpha + \beta$ according the the standard Kronecker product rules.
In the vector form, we have $p^{AB}=p^{A}\otimes p^{B}$.
\subsection{Representation of channels}
The next step is to obtain the probability representation of quantum channels.
Let $\mathcal{H}^{\rm in}$, $\mathcal{H}^{\rm out}$ be Hilbert spaces with MIC-POVMs $E^{\rm in}, E^{\rm out}$ and $\Phi : \States{\mathcal{H}^{\rm in}} \rightarrow \States{\mathcal{H}^{\rm out}}$ be a quantum channel (CPTP map).
Consider a state $\rho^{\rm in} \in \States{\mathcal{H}^{\rm in}}$ and let $\rho^{\rm out}= \Phi(\rho^{\rm in})\in \States{\mathcal{H}^{\rm out}}$.
Denote the probability vectors corresponding to $\rho^{\rm in}$ and $\rho^{\rm out}$ as $p^{\rm in}$ and $p^{\rm out}$ respectively.
Then the channel $\Phi$ can be characterized with a matrix $\bf{S}$ such that
\begin{equation}
p^{\rm out} = {\bf{S}} p^{\rm in}, \quad {\bf{S}}_{lk} = \Tr{ E^{\rm out}_l \Phi(e_k^{\rm in}) }.
\end{equation}
This matrix is generally pseudostochastic (i.e. $\sum_k {\bf{S}}_{kl} = 1$), but it is not necessarily stochastic.
An action of the channel $\Phi$ on the state $\rho^{\rm in}$ can be written as follows:
\begin{equation}\label{eq:channel-from-matr}
\Phi: \rho^{\rm in} \mapsto \sum_{k,l=1}^{d_{\rm out},d_{\rm in}} {\bf{S}}_{kl} e^{\rm out}_k \Tr{ E_l^{\rm in} \rho^{\rm in} }.
\end{equation}
In turn, an action of the dual channel $\Phi^{*}: \TrClass{\mathcal{H}^{\rm out}} \rightarrow \TrClass{\mathcal{H}^{\rm in}} $ is then given by
\begin{equation}\label{eq:dual-channel-from-matr}
\Phi^{*}: M^{\rm out} \mapsto \sum_{k,l=1}^{d_{\rm in},d_{\rm out}} {\bf{S}}_{kl} E^{\rm in}_k \Tr{ e_l^{\rm out} M^{\rm out} }.
\end{equation}
In the case of Kraus representation where the operation of the channel is defined in the form
\begin{equation}\label{eq:Kraus}
\Phi(\rho){=}\sum_n V_n \rho V_n^{\dagger},
\end{equation}
the corresponding elements of the pseudostochastic matrix are
\begin{equation}
{\bf{S}}_{lk} = \sum_n \Tr{E^{\rm out}_l V_n e_k^{\rm in} V_n^{\dagger}}.
\end{equation}
The representation of tensor products for quantum channels can be used similarly to Sec.~\ref{Tensor}.
If
\begin{equation}
\Phi^A : \States{\mathcal{H}^{{\rm in},A}} \rightarrow \States{\mathcal{H}^{{\rm out},A}}
\end{equation}
and
\begin{equation}
\Phi^B : \States{\mathcal{H}^{{\rm in},B}} \rightarrow \States{\mathcal{H}^{{\rm out},B}}
\end{equation}
are quantum channels with corresponding pseudostochastic matrices ${\bf{S}}^A$ and ${\bf{S}}^B$, then
\begin{multline}
{\bf{S}}^{AB}_{(kl)(nm)} = \\
={\rm Tr} \left(
\left\{
E^{{\rm out},A}_k \otimes E^{{\rm out},B}_l
\right\}
\left\{
\Phi^A \otimes \Phi^B
\right\}
\left\{
e^{{\rm in},A}_{n} \otimes e^{{\rm in},B}_{m}
\right\}
\right)
\\=\Tr{ E^{{\rm out},A}_k \Phi^A(e_n^{{\rm in},A}) \otimes E^{{\rm out},B}_l \Phi^B(e_{m}^{{\rm in},B}) }\\ = {\bf{S}}^A_{(k,n)} \otimes {\bf{S}}^B_{(l,m)}.
\end{multline}
Thus, the tensor product of two quantum channels maps to the tensor product of two corresponding matrices.
The channel of a partial trace
\begin{equation}
\TrPart{\mathcal{H}^{B}}{\cdot}:\rho^{AB} \mapsto \TrPart{\mathcal{H}^{B}}{\rho^{AB}}
\end{equation}
taking an input $\rho^{AB} \in \States{\mathcal{H}^{A} \otimes \mathcal{H}^{B}}$
then corresponds to the matrix in the form:
\begin{multline}
{\bf{S}}_{l(nm)} = \Tr{E^A_l \TrPart{B}{e^A_n \otimes e^B_m}} \\
= \Tr{E^A_l e^A_n} = \delta_{ln}.
\end{multline}
As in the case of states, not any pseudostochastic matrix ${\bf{S}}$ corresponds to a (physical) quantum channel $\Phi$.
In order to formulate a criterion, we use the Choi--Jamio{\l}kowski duality~\cite{Jamiolkowski1972,Choi1975}.
Let $ \Phi : \States{\mathcal{H}^{\rm in}} \rightarrow \States{\mathcal{H}^{\rm out}} $ be a trace-preserving map.
By fixing the orthonormal basis $\{\ket{n}\}_{n=1}^{d_{\rm in}}$ in $\mathcal{H}^{\rm in}$ ($d_{\rm in}={\rm dim} \mathcal{H}^{\rm in}$), we define a state $\sigma \in \States{\mathcal{H}'^{\rm in} \otimes \mathcal{H}^{\rm in}}$ with $\mathcal{H}'^{\rm in}=\mathcal{H}^{\rm in}$ of the following form:
\begin{equation} \label{eq:max-ent-state}
\sigma=\frac{1}{d_{\rm in}} \sum_{n,m=1}^{d_{\rm in}} \ket{n}\bra{m} \otimes \ket{n}\bra{m}.
\end{equation}
One can see that it is a density matrix of the pure state
\begin{equation}
\ket{\phi}= \frac{1}{\sqrt{d_{\rm in}}}\sum_{k=1}^{d_{\rm in}} \ket{k} \otimes \ket{k}.
\end {equation}
We then call \emph{Choi state} an operator $ \rho_\Phi $
\begin{equation}
\rho_\Phi = ({\rm Id} \otimes \Phi) (\sigma) = \frac{1}{d_{\rm in}} \sum_{n,m=1}^{d_{\rm in}} \ket{n}\bra{m} \otimes \Phi(\ket{n}\bra{m}),
\end{equation}
where ${\rm Id}$ is an identical map.
The Choi--Jamio{\l}kowski isomorphism says that the operator $ \rho_\Phi $ is a quantum state if and only if $\Phi$ is a quantum channel.
One can reconstruct an action of $\Phi$ on an arbitrary input using the following formula:
\begin{equation}\label{eq:purestate-for-Choi}
\Phi(\rho) = \TrPart{\mathcal{H}'^{\rm in}}{ ( \rho^{\rm in\top} \otimes {\bf I}_{d_{\rm in}}) \rho_\Phi}.
\end{equation}
The Choi--Jamio{\l}kowski isomorphism can be naturally formulated in the probability representation.
Let $ {\bf{S}} $ be a matrix corresponding to a trace-preserving map $ \Phi $, and $ s $ be a vector corresponding to $\sigma$.
We assume that $s$ is obtained with MIC-POVM $E^{\rm in} \otimes E^{\rm in}$.
Then the \emph{Choi} probability vector has the form
\begin{equation}\label{eq:Choi-prob}
p_{\bf{S}} = \left( {\bf I}_{d_{\rm in}^{2}} \otimes {\bf{S}} \right) s
\end{equation}
One can see that $ {\bf{S}} $ corresponds to the quantum channel only in case $ p_{\bf{S}} \in \Distributions{E^{\rm in} \otimes E^{\rm out}} $.
In order to reconstruct $ {\bf{S}} $ via the vector $p_{\bf{S}}$, one can use the following relation:
\begin{multline}
{\bf{S}}_{lk} = \Tr{E^{\rm out}_l \Phi(e_k^{\rm in})} = \Tr{\rho_\Phi (e^{\rm in \top}_k \otimes E^{\rm out}_l)} \\
= \sum_{n,m=1}^{d_{\rm in}^{2}} p_{{\bf S}{(nm)}} \Tr{ e_n^{\rm in} e^{\rm in \top}_k } \Tr{ e^{\rm out}_m E^{\rm out}_l} \\
= \sum_{n=1}^{d_{\rm in}^{2}} \Tr{ e^{\rm in}_n e^{\rm in \top}_k } p_{{\bf S}{(n,l)}}.
\end{multline}
It is useful to define $s$ in terms of probability vectors without the notion of the Hilbert space $\mathcal{H}^{\rm in}$ and the state $\sigma$.
Consider random pure state $\ket{\psi_{1}}\bra{\psi_{1}} \in \States{\mathcal{H}^{\rm in}}$ and denote its probability vector as $p^{(11)}$.
Let us construct as a set of orthonormal probability vectors $\{p^{(kk)}\}_{k=1}^{d_{\rm in}}$ using the following equations for each $k = 2,\ldots,d_{\rm in}$:
\begin{equation}\label{eq:set-for-orthonormal-set}
\begin{aligned}
&\sum_{l=1}^{d_{\rm in}} p^{(kk)}_l = 1; \\
&p^{(kk)}*p^{(kk)} = p^{(kk)};\\
&\sum_{l=1}^{d_{\rm in}} (p^{(kk)} * p^{(nn)})_l = 0, \quad n = 1,\ldots,k-1,
\end{aligned}
\end{equation}
where we consider orthonormality with respect to the Hilbert-Schmidt product~\eqref{eq:HSproduct}.
One can think about $p^{(kk)}$ as a probability vector of state $\ket{\psi_{k}}\bra{\psi_{k}}$ taken from an orthonormal basis constructed from $\ket{\psi_{1}}\bra{\psi_{1}}$.
Of course, Eq.~\eqref{eq:set-for-orthonormal-set} has infinite number of solutions.
Then the vectors $p^{(nm)}$ with $n\neq m$, corresponding to states $\ket{\psi_{n}}\bra{\psi_{m}}$, can be obtained using straightforward multiplicative relations.
For example, vector $ p^{(12)} $ can be obtained as a solution of the following equations:
\begin{equation}
\begin{aligned}
p^{(12)} * p^{(kk)} &= 0,~~k\neq2; \quad &p^{(kk)} * p^{(12)} &= 0,~~k\neq1; \\
p^{(12)} * p^{(11)} &= 0; \quad &p^{(11)} * p^{(12)} &= p^{12}; \\
p^{(12)} * p^{(12)} &= 0; \quad &p^{(12)} * p^{(12)} &= 0; \\
p^{(12)} * p^{(22)} &= p^{12}; \quad &p^{(22)} * p^{(12)} &= 0. \\
\end{aligned}
\end{equation}
By finding $ p^{(nm)} $ for all $n,m=1,\ldots,d_{\rm in}$, we obtain the Choi distribution in the form
\begin{equation}
p_{\bf{S}} = \frac{1}{d_{\rm in}} \sum_{n,m=1}^{d_{\rm in}} p^{(nm)} \otimes {\bf{S}} p^{(nm)}.
\end{equation}
We also would like to mention a special case where $E^{\rm in}=\{\ket{\psi_{i}^{\rm sym}}\bra{\psi_{i}^{\rm sym}}\}_{i=1}^{d_{\rm in}}$ is a SIC-POVM.
Let $\overline{E}^{\rm in}=\{\ket{\overline{\psi}_{i}^{\rm sym}}\bra{\overline{\psi}_{i}^{\rm sym}}\}_{i=1}^{d_{\rm in}}$ with
\begin{equation}
\ket{\overline{\psi}_{i}^{\rm sym}} = \sum_{i=1}^{d_{\rm in}} \ket{i} \overline{\bra{i} \psi^{\rm sym}_{i} \rangle} = \sum_{i=1}^{d_{\rm in}} \ket{i} \bra{\psi^{\rm sym}_{i} }i \rangle,
\end{equation}
where $\overline{x}$ stands for complex conjugate of $x$, and $\{\ket{n}\}_{n=1}^{d_{\rm in}}$ is a computational basis as usual.
Then the probability vector of the state $\sigma = \ket{\phi}\bra{\phi}$ (see Eq.~\eqref{eq:purestate-for-Choi}) takes the following form with respect to the MIC-POVM $\overline{E}^{\rm in} \otimes E^{\rm in}$:
\begin{multline}
s_{(nm)} = \Tr{ \ket{\phi} \bra{\phi} (\overline{E}^{\rm in}_n \otimes E^{\rm in}_m) } = \\
= \frac{1}{d_{\rm in}^2} \sum_{k,l} \langle k\ket{\overline{\psi}^{\rm sym}_n} {\bra{\overline{\psi}^{\rm sym}_n}} l \rangle \langle k \ket{\psi^{\rm sym}_m}\bra{\psi^{\rm sym}_m} l \rangle = \\
= \frac{1}{d_{\rm in}} \Abs{ \langle{\psi_m}|\psi_n\rangle }^2 = \frac{d_{\rm in} \delta_{nm} + 1}{d_{\rm in}^2 (d_{\rm in}+1)}.
\end{multline}
It then can be substituted to Eq.~\eqref{eq:Choi-prob} in order to obtain a Choi probability vector and verify that it corresponds to valid quantum state.
\subsection{Representation of measurements}
Here we consider a MIC-POVM-based probability representation of an arbitrary measurement with the finite number of outcomes.
In the general case it is given by a POVM $M=\{M_{i}\}_{i=1}^{m}$ with some finite $m$.
Note that $M$ may not belong to MIC class.
According to the Born rule the probability to obtain $i$th outcome for an input state $\rho$ is given by
\begin{equation}
q_{i} = \Tr{\rho M_{i}}, \quad i=1,\ldots,m
\end{equation}
(here we assume that $M$ and $\rho$ are defined with respect to the same $d$-dimensional Hilbert space $\mathcal{H}$).
Taking $\rho$ in the probability representation given by Eq.~\eqref{rho}, we obtain the following expression for the probability vector:
\begin{equation}
\begin{split}
q&=\begin{bmatrix}q_{1} & \ldots & q_{m}\end{bmatrix}^{\top}, \\
q&= {\bf M}p, \quad {\bf M}_{ij} = \Tr{M_{i}e_{j}}.
\end{split}
\end{equation}
One can see that ${\bf M}$ is $m\times d^{2}$ pseudostochastic matrix because of normalization condition $\sum_{i=1}^{m}M_{i}={\bf I}_{d}$.
We note that given matrix ${\bf M}$, the effects of the POVM in the standard formalism are given by
\begin{equation}
M_k = \sum_{l=1}^{d^{2}} \Tr{M_k e_l} E_l = \sum_{l=1}^{d^{2}} {\bf M}_{kl} E_l.
\end{equation}
Next, we consider a problem of the verification that a given $m\times d^{2}$ pseudostochastic matrix ${\bf M}$ corresponds to some valid POVM $M$ with $m$ outcomes.
An idea behind such a test is very similar to the case of states, which is considered in Sec.~\ref{sec:states}, with the main difference that we swap the basis $E$ and the dual basis $e$.
Consider two operators $X,Y \in \TrClass{\mathcal{H}}$.
Using a dual basis $\{e_{i}\}_{i=1}^{d}$ one can represent them with row-vectors
$\lambda=\begin{bmatrix} \lambda_{1} & \ldots \lambda_{d} \end{bmatrix}$ and $\mu = \begin{bmatrix} \mu_{1} & \ldots \mu_{d} \end{bmatrix}$ according to the following expressions:
\begin{equation}
\begin{aligned}
X &= \sum_{i=1}^{d^{2}}\lambda_{i} E_{i}, \quad &\lambda_{i} &= \Tr{ e_{i} X}, \quad i=1,\ldots,d^{2} ;\\
Y &= \sum_{j=1}^{d^{2}}\mu_{j} E_{j} \quad &\mu_{j} &= \Tr{ e_{j} Y}, \quad j=1,\ldots,d^{2}.
\end{aligned}
\end{equation}
Note that the trace operation takes the form:
\begin{equation}
\Tr{X} = \sum_{i=1}^{d^{2}} \lambda_{i} \Tr{E_{i}} = \lambda \kappa
\end{equation}
with $\kappa = \begin{bmatrix} \Tr{E_{1}} & \ldots & \Tr{E_{d^{2}}} \end{bmatrix}^{\top}$.
Then we can introduce a `multiplication' of vectors $\mu$ and $\lambda$, denoted by $\circledast$, as follows:
\begin{multline}
(\lambda \circledast \mu)_{k} \equiv \Tr{XY e_{k}} \\
=\sum_{n,m=1}^{d^{2}} \Tr{E_n E_m e_k} \lambda_n \mu_m = \lambda \widetilde{\Lambda}^{(k)} \mu^{\top},
\end{multline}
where $ \widetilde{\Lambda}^{(k)}$ is $d^{2} \times d^{2}$ matrix with elements
\begin{equation}\label{eq:lambda-tilde}
\widetilde{\Lambda}^{(k)}_{nm} = \Tr{E_n E_m e_k}.
\end{equation}
Now we are ready to describe the verification algorithm.
The normalization condition $\sum_{i=1}^{m}M_{i}={\bf I}_{d}$ follows from the fact that ${\bf M}$ is pseudostochastic.
So the only remaining issue is to check the semi-positivity condition $M_{i} \geq 0$.
We note that in the case of states we derived expressions for $\Tr{\rho}$, \ldots, $\Tr{\rho^{d}}$ from the probability representation of $\rho$ and then substitute them into Routh--Hurwitz-like criterion.
Here we act in a similar manner.
For each $i$th row of the matrix ${\bf M}$ set $\lambda^{(i)}:=\begin{bmatrix} {\bf M}_{i,1} & \ldots & {\bf M}_{i,d^{2}} \end{bmatrix}$ and compute
\begin{equation}
a^{(i)}_{j} := \lambda^{(i) \circledast j} \kappa = \Tr{M_{i}^{j}}.
\end{equation}
Then proceed with same steps as in the case of states replacing ${\{a_{n}\}_{n=1}^{d}}$ with ${\{a_{i}\}_{n=1}^{d}}$.
If the positivity condition is fulfilled for all $i=1,\ldots,m$, then ${\bf S}$ corresponds to valid `physical' quantum measurement.
Finally, we consider the case a measurement given by some Hermitian operator $O=O^{\dagger}\in \TrClass{\mathcal{H}}$, also known as an \emph{observable}.
To obtain its probability representation we first take it spectral decomposition in the form
\begin{equation}
O = \sum_{i=1}^{m} x_{i} \Pi_{i},
\end{equation}
where $\{x_{i}\}_{i=1}^{m}$ are physical quantities which can be observed, and $\{\Pi_{i}\}_{i=1}^{m}$ are complete set of orthogonal (self-adjoint) projectors: $\sum_{i=1}^{m}\Pi_{i} = {\bf I}_{d}$, $\Pi_{i}\Pi_{j}=\delta_{ij}$.
One can consider a POVM $M_{O}$ with effects $\{\Pi_{i}\}_{i=1}^{m}$ and its corresponding pseudostochastic matrix ${\bf M}$ keeping in mind that each $i$th outcome corresponds the quantity $x_{i}$.
However, it is important to note if one is interested in mean value for some state $\rho$ is given by $\langle O \rangle = \Tr{O \rho}$, then one can consider a row-vector
\begin{equation}
{\bf O}^{\rm mean} := \begin{bmatrix} x_{1} & \ldots & x_{m} \end{bmatrix}{\bf M}
\end{equation}
and compute mean value as $\langle O \rangle = {\bf O}_{\rm mean} p$, where $p$ is a probability vector of $\rho$.
Finally, we note that the rules of the tensor product remain the same as in the case of quantum channels:
Pseudostochastic matrix of measurements on several physical subsystems is given by a tensor product of pseudostochastic measurements on each of subsystems.
\subsection{Transitions between MIC-POVM-based representations}
Up to this point the MIC-POVM, which determines the probability representation, was fixed.
Here we consider a question of how to make a transition between representations determined by different MIC-POVMs.
Let $E$ and $F$ be two MIC-POVMs defined with respect to the same $d$-dimensional Hilbert space $\mathcal{H}$, and let $e$ and $f$ be their corresponding dual bases.
We use superscript $[E]$ or $[F]$ to emphasize that given probability vector or pseudostochastic matrix of a channel/measurement is given in $E$- or $F$-based representation.
Consider elements of a pseudostochastic matrix of the measurement in $E$ given in $F$-based representation:
\begin{equation}
\left( {\bf M}_{E}^{[F]} \right)_{m,n} = \Tr{E_{m} f_{n}}, \quad m,n=1,\ldots,d^{2}.
\end{equation}
One can see the pseudostochastic matrix of the measurement in $F$ given in $E$-based representation is given by ${\bf M}_{F}^{[E]} = \left( {\bf M}_{E}^{[F]} \right)^{-1}$
Then we come to the following relations:
\begin{eqnarray}
&p^{[E]} &= {\bf M}_{E}^{[F]} p^{[F]}, \\
&{\bf S}^{[E]} &= {\bf M}_{E}^{[F]} {\bf S}^{[F]} {\bf M}_{F}^{[E]}, \\
&{\bf M}^{[E]} &= {\bf M}^{[F]} {\bf M}^{[E]}_{F},
\end{eqnarray}
where $p^{[\cdot]}$, ${\bf S}^{[\cdot]}$, and ${\bf M}^{[\cdot]}$ corresponds to some state, channel, and POVM, correspondingly.
\section{Dynamical equations}\label{sec:dynamics}
Here we apply the developed MIC-POVM-based representation to quantum evolution equations (master equations).
We consider two conceptually important cases.
The first is the Liouville-von Neumann equation corresponding to the unitary evolution of quantum states.
The second is the dissipative evolution governed by a Markovian master equation, which is governed by the GKSL generator.
In the both cases we restrict ourself with a condition that generators are time-independent.
\subsection{Liouville-von Neumann equation}
Consider a $d$-dimensional dimensional Hilbert space $\mathcal{H}$.
The evolution of a quantum state under the Hamiltonian $H=H^{\dagger} \in \TrClass{\mathcal{H}}$ is described by the Liouville-von Neumann equation
\begin{equation}
\dot{\rho}(t) = - \frac{\imath}{\hbar} \Commutator{H}{\rho(t)},
\end{equation}
where $ \Commutator{\cdot}{\cdot}$ denotes commutator.
In what follows we use dimensionless units and set $\hbar\equiv1$.
Using Eq.~\eqref{rho}, then left multiplying by $E_l$ and taking trace, the equation takes the form
\begin{equation}
\dot{p}_l(t) = - \imath \sum_{k=1}^{d^{2}} \Tr{E_l \Commutator{H}{e_k}} p_k(t).
\end{equation}
Therefore, the Liouville-von Neumann equation takes the form of the ordinary linear differential equation with generator $\textbf{H}$
\begin{equation}\label{eq:vNeq}
\dot{p}(t) = \textbf{H} p(t), \quad \textbf{H}_{l,k} = \imath \Tr{H \Commutator{E_l}{e_k}}.
\end{equation}
We note that such a form of the equation for the unitary dynamics on the probability language has been obtained in Ref.~\cite{Weigert2000}.
In the following section we present the generalization of this results for the case of the GKSL equation.
The $d^{2} \times d^{2}$ matrix $ \textbf{H}$ is a probabilistic representation of the Hamiltonian, which has following properties.
\begin{enumerate}
\item The matrix $\textbf{H}$ is real: $\textbf{H} \in \Mat{d^{2}}{d^{2}}{\mathbb{R}}$.
\item The sum of each column is zero: $\sum_{l=1}^{d^{2}} \textbf{H}_{l,k} = 0$.
\end{enumerate}
The first property follows from the fact that $\Commutator{E_{l}}{e_{k}}$ is Hermitian, and second fact come from the normalization condition on MIC-POVM effects $E_{i}$.
It is worth to note that if $E$ is a SIC-POVM, then $\textbf{H}$ becomes antisymmetric ($\textbf{H}^{\top}=-\textbf{H}$), and thus all its diagonal elements are zero, and all rows also sum to zero (see Ref.~\cite{Kiktenko2020} for more details).
The solution to Eq.~\eqref{eq:vNeq} can be presented in the form
\begin{equation}\label{eq:unitary-stoch}
p(t) = {\bf U}(t) p_{0}\quad {\bf U}(t)=e^{\textbf{H} t},
\end{equation}
where $p_{0}$ is a probability vector at $t=0$.
Note that ${\bf U}(t)$ is pseudostochastic.
The unitarity of the evolution governed by the Liouville-von Neumann equation implies preserving the Hilbert-Schmidt product between two arbitrary vectors during their evolution according to Eq.~\eqref{eq:vNeq}.
Taking into account the probability representation of the Hilbert-Schmidt product is given by Eq.~\eqref{eq:HSproduct} we obtain the identity
\begin{equation}
{\bf T}^{-1} = {\bf U}(t)^{\top} {\bf T}^{-1} {\bf U}(t).
\end{equation}
Considering small times $t=\delta t \ll 1$ and expanding exponent of the evolution operator in Eq.~\eqref{eq:unitary-stoch} into Taylor series we arrive at the 3rd property of the MIC-POVM-based representation of a Hamiltonian.
\begin{enumerate}
\item[3.] The following identity holds:
\begin{equation}\label{eq:extra-cond}
\textbf{H}^{\top} {\bf T}^{-1} + {\bf T}^{-1} \textbf{H} =0.
\end{equation}
\end{enumerate}
Consider a matrix $\widetilde{\textbf{H}}:={\bf T}^{-1} \textbf{H}$.
One can see that since ${\bf T}$ is symmetric, $\widetilde{\textbf{H}}$ is antisymmetric: $\widetilde{\textbf{H}}^{\top} + \widetilde{\textbf{H}}=0$.
Combining this fact with the property 2 one has $\sum_{i=1}^{d^{2}}({\bf T} \widetilde{\textbf{H}})_{ij}=0$.
Antisymmetric matrix $\widetilde{\textbf{H}}$ possessing these properties can be defined with $(d^{2}-2)(d^{2}-1)/2$ independent parameters.
Note that for $d>2$ this quantity is larger than $d^{2}-1$ -- the number of independent parameters required to define physical properties of a Hamiltonian (the term -1 comes from the fact that energy is always defined up to some constant).
So the properties 2 and 3 are insufficient to determine a set of possible probability representations of Hamiltonians.
In what follows we consider a necessary and sufficient condition on matrix $\textbf{H} \in \Mat{d^{2}}{d^{2}}{\mathbb{R}}$ to be a probability representation of some Hamiltonian $H$.
In order to proceed, we first need to introduce the ``vectorised'' notation for operators.
If $ A \in \TrClass{\mathcal{H}}$ is an operator, then it can be written in a form
\begin{equation}
A = \sum_{n,m=1}^{d} A_{nm} \ket{n}\bra{m}, \quad A_{nm} = \bra{n} A \ket{m}.
\end{equation}
Then the bra- and ket-representations of $A$ will be denoted as
\begin{equation}
\kett{A} = \sum_{n,m} A_{nm} \ket{n} \otimes \ket{m}, ~~
\braa{A} = \sum_{n,m} \overline{A_{nm}} \bra{n} \otimes \bra{m}.
\end{equation}
These representations may be understood as raising or lowering index.
The inner product between such vectors yields $ \braa{A}\kett{B} = \Tr{A^\dagger B} $.
We also note that $B=UAV^{\dagger}$ transforms into $\kett{B} = U\otimes \overline{V} \kett{A}$ with $U,V \in \TrClass{\mathcal{H}}$.
We introduce tensors $\textbf{e}$ and $\textbf{E}$ in the form
\begin{equation}
\textbf{e} =
\begin{bmatrix} \kett{ e_1 } & \cdots & \kett{ e_{d^2} } \end{bmatrix},
\quad
\textbf{E} =
\begin{bmatrix} \braa{E_1} \\ \vdots \\ \braa{E_{d^2}} \end{bmatrix}.
\end{equation}
Then one has
\begin{equation}
\kett{\rho} = \textbf{e} p,\quad \textbf{E} \textbf{e} = \mathbf{1}, \quad \textbf{e} \textbf{E} = \sum_{i=1}^{d^2} \kett{e_i}\braa{E_i}.
\end{equation}
By using this notation, the matrix $\textbf{H}$ can be written as follows:
\begin{equation}
\textbf{H} = - \imath \textbf{E} \left( H \otimes \mathbf{I}_{d} - \mathbf{I}_{d} \otimes \overline{H} \right) \textbf{e}.
\end{equation}
Let $ \{ \sigma^{(i)} \}_{i=1}^{d^2-1} $ be a linearly independent set of operators, satisfying following conditions:
(i) operators are traceless $\Tr{\sigma^{(i)}}=0$;
(ii) operators are Hermitian $ \sigma^{(i)} = (\sigma^{(i)})^*$;
(iii) $ \Tr{\sigma^{(i)} \sigma^{(j)}} = 2 \delta_{ij} $.
In a two-dimensional Hilbert space ($d=2$) these such a set can be presented with Pauli matrices.
Then the Hamiltonian can be represented as follows:
\begin{equation}
H = \nu_0 \mathbf{I}_{d} + \sum_{i=1}^{d^2-1} \nu_i \sigma^{(i)},
\end{equation}
where $ \nu_0 = \Tr{H}/d$ and $\nu_i = \Tr{H \sigma^{(i)}}/2 $.
Let $\{\textbf{H}^{(i)}\}_{i=1}^{d^{2}}$ be a set of matrices corresponding to $ \{ \sigma^{(i)} \}_{i=1}^{d^2-1} $:
\begin{equation}
\textbf{H}^{(i)}_{lk} = \imath \Tr{\sigma^{(i)} \Commutator{E_l}{e_k}}.
\end{equation}
We note the probability representation of a Hamiltonian equal to identity matrix gives zero matrix.
Therefore, we have
\begin{equation}
\textbf{H} = \sum_{i=1}^{d^2-1} \nu_i \textbf{H}^{(i)}.
\end{equation}
Next, we can see that
\begin{multline}
\Tr{ \textbf{H}^{(i)} \textbf{H}^{(j)} } \\
= -\Tr{ \sigma^{(i)} \sigma^{(j)} \otimes \mathbf{I}_{d} + \mathbf{I}_{d} \otimes \overline{\sigma^{(i)}} \overline{\sigma^{(j)}} } = -4 d \delta_{ij}.
\end{multline}
Thus, it is possible to define the projector $\mathcal{P}_{\rm unit}(\cdot)$ on the space $\Mat{d^{2}}{d^{2}}{\mathbb{R}}$ that correspond to Hamiltonians in the following explicit form:
\begin{equation}
\mathcal{P}_{\rm unit}(\mathbf{M}) = -\frac{1}{4 d} \sum_{i=1}^{d^2-1} \Tr{ \mathbf{M} \cdot \mathbf{H}^{(i)} } \mathbf{H}^{(i)}
\end{equation}
(here $\mathbf{M}$ is some $d^{2}\times d^{2}$ matrix ).
Finally, the matrix $\textbf{H}$ corresponds to a Hamiltonian, if and only if it satisfies the identity
\begin{equation}
\mathcal{P}_{\rm unit}(\textbf{H}) = \textbf{H}.
\end{equation}
It also worth to note the $\mathcal{P}_{\rm unit}$ turns to be a useful tool for studying experimental data in quantum process tomography experiments, since it allows extracting the unitary part of generator for a reconstructed process~\cite{Kiktenko2020}.
\subsection{GKSL equation}
Here we generalize the results of the previous section on the case of the GKSL equation~\cite{Gorini1976,Linblad1976}.
Consider the Markovian master equation in the form $\dot{\rho}(t)=L(\rho)$ with
\begin{multline}\label{eq:GKSL}
L(\rho) = -\imath \Commutator{H}{\rho (t)} + \\
\sum_k\left(A_k\rho(t) A_k^{\dagger} - \frac{1}{2} \AntiCommutator{A_k^{\dagger}A_k}{\rho(t)} \right),
\end{multline}
where $\AntiCommutator{\cdot}{\cdot}$ is an anticommutator, and $A_k$ are some arbitrary operators describing dissipative evolution, also known as noise operators.
Let us introduce a CP map $\Psi: M \mapsto \sum_{k}A_{k} M A_{k}^{\dagger}$.
One can think about $\Psi(\cdot)$ as a quantum channel without trace-preserving property.
The second term of Eq.~\eqref{eq:GKSL} can be written in the following form:
\begin{equation}
\Psi(\rho) - \frac{1}{2} \AntiCommutator{\Psi^*(\mathbf{I}_{d})}{\rho} \equiv D(\rho),
\end{equation}
where $\Psi^{*}$ is a map dual to $\Psi$.
Let ${\bf S}\in\Mat{d^{2}}{d^{2}}{\mathbb{R}}$ be MIC-POVM-based representation of $\Psi$, i.e. ${\bf S}_{ij}=\Tr{E_{i} \Psi(e_{j})}$.
Then by using Eqs.~\eqref{eq:channel-from-matr}, \eqref{eq:dual-channel-from-matr}, and \eqref{eq:lambda-tilde} one obtains
\begin{equation}
\Tr{E_{i} D(e_{j})} = {\bf S}_{ij} - \sum_{k,l=1}^{d^{2}}{\bf S}_{kl} \frac{\widetilde\Lambda_{i,l}^{(j)} + \widetilde\Lambda_{l,i}^{(j)}}{2} \equiv {\bf D}_{ij}
\end{equation}
for $i,j=1,\ldots,d^{2}$.
Using the probability representation of the first term of \eqref{eq:GKSL} from the previous section, we obtain the GKSL equation in the MIC-POVM-based probability as follows:
\begin{equation}
\dot{p}(t) = \textbf{L} p(t), \quad \textbf{L} = \textbf{H} + \textbf{D}.
\end{equation}
One can easily verify that $\textbf{D} \in \Mat{d^{2}}{d^{2}}{\mathbb{R}}$ and $\sum_{i=1}^{d^{2}}\textbf{D}_{ij}=0$ for every $j=1,\ldots,d^{2}$.
We also discuss a necessary and sufficient properties of ${\bf L}$ to be corresponded to some Liouvillian $L(\cdot)$.
It is known (see Ref.~\cite{Wolf2012}) that $L(\cdot)$ is a generator of CPTP maps if and only if
\begin{equation} \label{eq:L-check}
\overline{P}_{\sigma} ({\rm Id} \otimes L)(\sigma) \overline{P}_{\sigma} \geq 0,
\end{equation}
where $\sigma \in \States{\mathcal{H} \otimes \mathcal{H}}$ is a maximally entangled state [see e.g. Eq.~\eqref{eq:max-ent-state}] and $\overline{P}_{\sigma}=I_{d^{2}}-\sigma$.
We then consider $d^{4}$-dimensional vectors $s$ and $p_{s}$ with the following components:
\begin{equation}
\begin{aligned}
s_{(ij)} &= \Tr{E_{i}\otimes E_{j} \sigma}, \\
\overline{p}_{s,(ij)} &= \Tr{E_{i}} \Tr{E_{j}} - s_{(ij)}.
\end{aligned}
\end{equation}
The probability representation of the expression in left-hand side of Eq.~\eqref{eq:L-check} takes the form
\begin{equation}
\overline{p}_{s} * ({\bf I}_{d^{2}} \otimes {\bf D}) s * \overline{p}_{s} \equiv p_{\bf D}.
\end{equation}
Thus, one can employ the algorithm from Sec.~\ref{sec:states} to check that $p_{\bf D}$ corresponds to non-normalized state with respect to the MIC POVM $E\otimes E$, and thus condition~\eqref{eq:L-check} is fulfilled.
\subsection{Heisenberg picture}
Up to this point, we described evolution equation for states from the viewpoint of the Schr{\"o}dinger picture.
Here we show how to adapt the MIC-POVM-based probability representation to the Heisenberg picture, where measurement operators evolve.
As a master equation, we consider GKSL equation from the previous section.
Consider a POVM $M=\{M_{i}\}_{i=1}^{m}$.
Remember, that in the MIC-POVM-based representation it is defined by $m \times d^{2}$ pseudostocastic matrix ${\bf M}$.
The probability vector of measurement outcomes at time $t\geq 0$ is given by
\begin{equation}
{\bf M}p(t) = {\bf M}e^{{\bf L}t}p_{0}.
\end{equation}
Now we can introduce a Heisenberg representation of ${\bf M}$: ${\bf M}^{\rm Heis}(t):={\bf M}e^{{\bf L}t}$, that is solution of the equation
\begin{equation}\label{eq:Heis-GKSL}
\frac{d}{dt}{\bf M}^{\rm Heis}(t) = {\bf M}^{\rm Heis}(t){\bf L}, \quad {\bf M}^{\rm Heis}(0) = {\bf M}(0).
\end{equation}
The probabilities of outcomes are given by $q(t) = {\bf M}^{\rm Heis}(t)p_{0}$
We note that Eq.~\eqref{eq:Heis-GKSL} can be also adapted to an evolution of a particular effect (row of ${\bf M}$) instead of full matrix ${\bf M}$.
In the case of Hermitian observable one can write the following equation for an row-vector allowing to compute mean value of $\langle O(t)\rangle = {\bf O}_{\rm mean}^{\rm Heis}(t)p_{0}$:
\begin{equation}\label{eq:Heis-GKSL}
\frac{d}{dt}{{\bf O}}_{\rm mean}^{\rm Heis}(t) = {\bf O}_{\rm mean}^{\rm Heis}(t) {\bf L}, \quad {\bf O}_{\rm Heis}^{\rm mean}(0) = {\bf O}_{\rm mean}.
\end{equation}
\section{MIC-POVM probability representation and quantum-to-classical transition}\label{sec:decoherence}
One of the important questions that can be addressed in the probability representation of quantum dynamics via pseudostochastic maps is how to quantify an aspect of `non-classicality' of a particular quantum dynamics.
As we see, quantum dynamics is essentially different from classical stochastic dynamics by the possibility of negative conditional probabilities.
The study of negative elements in pseudostochastic matrices seems to be very important in particular for understanding the origin of the complexity of simulating quantum dynamics of large-scale quantum systems.
We note that the fact of the complexity of efficient simulation of the behaviour of quantum systems by classical stochastic systems is indeed a widely believed but unproven conjecture.
At the same time, the presence of decoherence drastically changes the nature of the dynamics of quantum-mechanical systems.
We note that in Ref.~\cite{Kiktenko2020} it has been shown that decoherence process accompanying a unitary quantum dynamics in the framework of quantum Markovian master equation can eliminate negative elements
in the resulting pseudostochastic matrix making it purely stochastic and looking like a classical stochastic process.
Here we study a decay quantum features of a dissipative quantum Markovian dynamics within a MIC-POVM representation and compare the cases of SIC-POVM-based and general MIC-POVM-based representations.
First of all, we note that appearance of negative conditional probabilities in pseudostochastic matrix ${\bf S}(t)=e^{{\bf L}t}$ is determined by negative non-diagonal elements of the generator ${\bf L}$.
On the one hand, if there exist at least one negative non-diagonal element ${\bf L}_{i^{\star}j^{\star}}<0$ ($i^{\star}\neq j^{\star})$,
then there appear negative elements in ${\bf S}(t)$ at least for small enough time $t>0$ for which ${\bf S}(t)\approx {\bf I}+{\bf L}t$.
On the other hand, if all non-diagonal elements of ${\bf L}$ are non-negative, then considering the identity
\begin{equation}
{\bf S}(t) = \lim_{n\rightarrow\infty} \left( {\bf I}+{\bf L}\frac{t}{n} \right)^{n}
\end{equation}
we come to the conclusion that all elements of ${\bf S}(t)$ are non-negative for any $t>0$ since all elements of ${\bf I}+{\bf L}t/n$ are non-negative.
Let us have in mind that (i) for every generator of the GKSL equation one has $\sum_{i}{\bf L}_{ij}=0$, and (ii) in the case of purely unitary evolution one has the diagonal elements being equal to zero.
From these two facts it directly follows that every non-trivial generator of decoherence-free evolution ${\bf L}={\bf H}\neq 0$ has at least one negative element for sure.
As it was already mentioned, an appearance of a dissipator term ${\bf D}$ in the generator ${\bf L}={\bf H}+{\bf D}$ can change the situation, and negative elements can disappear from the generator.
We then consider the case of a spin-1/2 particle processing in a magnetic field and exposed by one the following standard dissipative process: depolarizing, dephasing and amplitude damping.
From the viewpoint of the standard formalism, we consider a Hamiltonian in the following form:
\begin{equation}
H_{\theta} = \frac{1}{2}(\sigma^{(1)}\sin\theta + \sigma^{(3)}\cos\theta),
\end{equation}
and the following variants of noise-operators (Lindblad operators):
\begin{eqnarray}
&A_{\tau,i}^{\rm depol}&= \frac{1}{2\sqrt{\tau}}\sigma^{(i)}, \quad i=1,2,3;\\
&A_{\tau,0}^{\rm deph}&= \frac{1}{\sqrt{\tau}}\sigma^{(3)},\\
&A_{\tau,0}^{\rm damp}&= \frac{1}{\sqrt{\tau}}\sigma^{-}, \quad \sigma^{-}=\frac{1}{2}(\sigma^{(1)}-\imath\sigma^{(2)}).
\end{eqnarray}
Here $\sigma^{(1)}, \sigma^{(2)}, \sigma^{(3)}$ are standard $x$-, $y$-, $z$- Pauli matrices correspondingly, $\theta$ is a real polar angle which determines a direction of the magnetic field (azimuthal angle is set to zero), and $\tau>0$ is a decoherence process characteristic time.
Thus we consider the GKSL equation $\dot{\rho}= L_{\theta,\tau}^{\rm dec}[\rho]$ with generator
\begin{multline}\label{eq:mainGenerator}
L_{\theta,\tau}^{\rm dec}[\rho]=-\imath \Commutator{H_{\theta}}{\rho} + \\\sum_k\left(A_{\tau,k}^{\rm dec}\rho A_{\tau,k}^{\rm dec\dagger} - \frac{1}{2} \left[A_{\tau,k}^{\rm dec\dagger}A_{\tau,k}^{\rm dec}\rho + \rho A_{\tau,k}^{\rm dec\dagger}A_{\tau,k}\right]\right)
\end{multline}
with ${\rm dec}\in\{ {\rm depol}, {\rm deph}, {\rm damp} \}$.
We note that Larmor frequency is fixed and equal to 1 and all three decoherence processes are invariant under rotation around $z$-axis.
Next, we study a pseudostochastic matrix of the quantum channel, which maps an initial state at the moment $t=0$ to the final state at some $t>0$ according to Eq.~\eqref{eq:mainGenerator}.
In particular, we are interested in the conditions on the parameters $\theta$ and $\tau$ which make the resulting evolution matrix to be purely stochastic for any time $t>0$.
As it was already mentioned, this kind of quantum-to-classical transition is determined by the form of the generator.
In turn, it depends not only on the values of $\theta$ and $\tau$, but also on the particular MIC-POVM used for its representation.
\begin{figure*}
\caption{Results of numerical calculations of the characteristic time $\tau_{\rm crit}
\label{fig:tau_crit}
\end{figure*}
Let ${\bf L}^{[E]}$ be a probability representation of generator~\eqref{eq:mainGenerator} with respect to a MIC-POVM $E$.
As a reference point, we take a SIC-POVM $E^{\rm sym}$ with the following effects:
\begin{equation}\label{eq:SIC-POVM}
\begin{aligned}
E_1^{\rm sym} &= \frac{1}{4} {\bf I}_{2} + \frac{\sqrt{3}}{12} (-\sigma^{(1)} + \sigma^{(2)} + \sigma^{(3)}), \\
E_2^{\rm sym} &= \frac{1}{4} {\bf I}_{2} + \frac{\sqrt{3}}{12} (\sigma^{(1)} - \sigma^{(2)} + \sigma^{(3)}), \\
E_3^{\rm sym} &= \frac{1}{4} {\bf I}_{2} + \frac{\sqrt{3}}{12} (\sigma^{(1)} + \sigma^{(2)} - \sigma^{(3)}), \\
E_4^{\rm sym} &= \frac{1}{4} {\bf I}_{2} + \frac{\sqrt{3}}{12} (- \sigma^{(1)} - \sigma^{(2)}- \sigma^{(3)}).
\end{aligned}
\end{equation}
In the ${\bf E}^{\rm sym}$-based representation the generator term corresponding to the unitary evolution takes the form
\begin{multline}
{\bf H}_{\theta}^{[E^{\rm sym} ]} = \frac{\sin\theta}{4}
\begin{bmatrix}
0 & -1 & 1 & 0 \\
1 & 0 & 0 & -1 \\
-1 & 0 & 0 & 1 \\
0 & 1 & -1 & 0 \\
\end{bmatrix}
+ \\\frac{\cos\theta}{4}
\begin{bmatrix}
0 & -1 & 0 & 1 \\
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
\end{bmatrix}.
\end{multline}
The generators of depolarization, dephazing and damping are
\begin{eqnarray}
&{\bf D}^{{\rm depol} [E^{\rm sym}]}_{\tau} &= \frac{1}{4\tau}
\begin{bmatrix}
-3 & 1 & 1 & 1 \\
1 & -3 & 1 & 1 \\
1 & 1 & -3 & 1 \\
1 & 1 & 1 & -3 \\
\end{bmatrix},\\
&{\bf D}^{{\rm deph}[E^{\rm sym}]}_{\tau} &= \frac{1}{2\tau}
\begin{bmatrix}
-1 & 1 & 0 & 0 \\
1 & -1 & 0 & 0 \\
0 & 0 & -1 & 1 \\
0 & 0 & 1 & -1 \\
\end{bmatrix},\\
&{\bf D}^{{\rm damp}[E^{\rm sym}]}_{\tau} &=
\frac{1}{4\tau}
\begin{bmatrix}
-2 & 0 & 1 & 1 \\
0 & -2 & 1 & 1 \\
1& 1 & -2 & 0 \\
1 & 1 & 0 & -2 \\
\end{bmatrix} + \\&&\frac{1}{4\sqrt{3}\tau}
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
-1& -1 & -1 & -1 \\
-1 & -1 & -1 & -1 \\
\end{bmatrix},
\end{eqnarray}
respectively.
The transition of the generators to an arbitrary MIC-POVM $E$-based representation can be realized via the transformation
\begin{equation}
{\bf L}_{\theta,\tau}^{{\rm dec}[E]} = {\bf M}_E^{[E_{\rm sym}]} {\bf L}_{\theta,\tau}^{{\rm dec}[{E^{\rm sym} }]} {\bf M}_{E^{\rm sym}}^{[E]},
\end{equation}
where ${\bf L}_{\theta,\tau}^{\rm dec[\cdot]}={\bf H}_{\theta}^{[\cdot]}+{\bf D}_{\tau}^{\rm dec[\cdot]}$ and ${\bf M}^{[E_{2}]}_{E_{1}}$ is a pseudostochastic matrix of a measurement with MIC-POVM $E_{1}$ in MIC-POVM $E_{2}$-based representation.
Let $\mathcal{N}({\bf L}^{[E]})$ be a sum of negative elements in ${\bf L}^{[E]}$:
\begin{equation}
\mathcal{N}({\bf L}^{[E]}) = \sum_{i,j} \max \left(0, -{\bf L}^{[E]}_{ij}\right), \quad i \neq j.
\end{equation}
This quantity characterizes an extent of non-classicality of a quantum Markovian generator with respect to the probability representation with MIC-POVM $E$.
The crucial point is that $\mathcal{N}({\bf L}^{[E]})$ in the case of the considered spin-1/2 evolution can depend not only on physical parameters $\tau$ and $\theta$, but also on the employed MIC-POVM $E$.
In order to reduce dependence on particular MIC-POVM for quantifying non-classicality, we introduce the following quantity:
\begin{equation}
\mathcal{N}_{\Omega}(L) = \min_{E\in\Omega}\mathcal{N}({\bf L}^{[E]}),
\end{equation}
where $\Omega$ is the set of MIC-POVMs, $L$ is the generator of the quantum Markovian dynamics (one can think of it as a super-operator from the right-hand side of the GKSL equation written in the standard form),
and ${\bf L}^{[E]}$ is the $E$-based probability representation of $L$.
Finally, we introduce a quantity
\begin{equation}
\tau_{\rm crit}^{\rm dec}(\theta; \Omega) = \sup\Set{\tau}{\mathcal{N}_{\Omega}(L_{\theta,\tau}^{\rm dec})=0},
\end{equation}
which shows a minimal strength of decoherence process ${\rm dec}\in\{{\rm depol}, {\rm deph}, {\rm damp}\}$, which is characterized by corresponding characteristic time, such that the resulting evolution looks ``classical-like''
(i.e. has stochastic evolution matrix) from the viewpoint of optimal probability representation with MIC-POVM taken from some set $\Omega$.
We note that this value also depends on the angle $\theta$ which determines a unitary evolution process.
We calculate $\tau_{\rm crit}^{\rm dec}(\theta; \Omega)$ numerically for three different sets $\Omega$:
(i) the set of all possible SIC-POVM;
(ii) the set of projective MIC-POVMs (pMIC-POVMs), that is MIC-POVMs consisted of rank-1 projectors only; and
(iii) the general set of all possible MIC-POVMs.
One can see that these three cases go from a special one to the most general one.
The results of our numerical calculations are presented in Fig.~\ref{fig:tau_crit}.
As one may expect, for the depolarization model [see Fig.~\ref{fig:tau_crit}(a)] a dependence on $\theta$ is absent due to the symmetry of the corresponding generator.
The obtained values of $\tau_{\rm crit}^{\rm depol}$ for SIC-POVM, pMIC-POVM, and MIC-POVM cases are 0.5, 0.6, and 0.61 correspondingly.
So the transition from SIC-POVM-based representation to MIC-POVM-based representation allows decreasing a tolerable strength of depolarization process at which the whole quantum process looks similar to a classical stochastic process.
Dependence on the angle $\theta$ appears in the dephasing process.
Fig.~\ref{fig:tau_crit}(b) shows that MIC-POVMs provide the same results as MIC-POVMs, while SIC-POVMs allows obtaining the stochastic form of the evolution matrix only in the small neighbourhood of $\theta=0$ and $\theta=\pi$.
In the case of damping [see Fig.~\ref{fig:tau_crit}(c)] we see that pMIC-POVMs and MIC-POVMs provide results better than SIC-POVMs,
meanwhile we obtain $\tau_{\rm crit}^{\rm damp} = 0.5$ for any $\theta$ whereas results for pMIC-POVMs depend on $\theta$ with some plateau-like behavior around $\theta = 0.5$.
On the basis of the obtained results, we can conclude that turning from the SIC-POVM representation to the MIC-POVM representation allows pushing back a `quantum-classical border' from the side of classical processes,
and thus makes it easier to employ a theory of classical stochastic processes to quantum dynamics.
These results are relevant to future investigation of emulating quantum dynamics with randomized algorithms run on a classical computer and, correspondingly, to the quantum advantage problem~\cite{Martinis2019,Wisnieff2020,Chen2020,Lidar2020}.
\section{MIC-POVM-based representation for quantum computing}\label{sec:qcomp}
In this section, we discuss how the MIC-POVM representation allows taking a fresh look at the process of quantum computing within a (digital) gate-based quantum computing model with qubits.
In this model, execution of quantum algorithm can be divided into three basic steps: (i) initialization of a qubit register, (ii) manipulation with qubit states, and (iii) read-out measurement.
The state initialization essentially represents the preparation of a state $\ket{\psi^{\rm init}}=\ket{0}^{\otimes n}$,
where $n$ is a number of qubits (hereinafter we use standard notation $\{\ket{0}, \ket{1}\}$ for the computational basis vectors of Hilbert spaces corresponding to each qubit).
The state manipulation is represented as a sequence of single-qubit and two-qubit gates, i.e. a set of unitary operators acting in corresponding Hilbert spaces.
It is a well-known fact that any quantum algorithm can be efficiently decomposed into a sequence of gates from some finite universal gate set (see Ref.~\cite{NielsenChuang}),
which consists of a single two-qubit gate, such as controlled-NOT (CNOT) or controlled-phase gates:
\begin{equation}
CX = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{bmatrix}, \quad
CZ = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
\end{bmatrix}, \quad
\end{equation}
and number of single-qubit gates, e.g. Hadamard gate $H$ and $T$-gate:
\begin{equation}
H=\frac{1}{\sqrt{2}}\begin{bmatrix}
1 & 1\\
1 & -1
\end{bmatrix}, \quad
T=\begin{bmatrix}
1 & 0 \\
0 & e^{\imath \pi/4}
\end{bmatrix}.
\end{equation}
The read-out measurement is essentially a projective measurement in the computation basis performed on some subset (or the full set) of $n$ qubits.
One can think about the read-out measurement as a sampling of a random variable according to a distribution determined by the state $\ket{\psi^{\rm fin}}=U^{\rm circ}\ket{\psi^{{\rm init}}}$, where $U^{\rm circ}$ is unitary operator of all applied gates.
\subsection{Initialization}
\begin{figure}
\caption{Evolution of the single qubit state $\ket{0}
\label{fig:had_gate_impl}
\end{figure}
\begin{figure*}
\caption{Construction of the cluster state in the probability representation. Here $(ij)$ outcome corresponds to $E_{i}
\label{fig:cz_gate_impl}
\end{figure*}
Let us consider a process of quantum computations from the viewpoint of MIC-POVM based probability representation.
Within this section we fix a single MIC-POVM $E$ constructed with tensor product operation from SIC-POVM effects~\eqref{eq:SIC-POVM}:
\begin{equation}\label{eq:symmetric-MIC-POVM}
E = \{E_{i_{1}}^{\rm sym} \otimes \ldots \otimes E_{i_{n}}^{\rm sym} \}_{i_{1},\ldots,i_{n}\in\{1,2,3,4\}}.
\end{equation}
The initial $4^{n}$-dimensional probability vector corresponding to $\ket{\psi^{\rm init}}$ then takes the form
\begin{equation}
p^{\rm init} = p^{(0)}\otimes \ldots \otimes p^{(0)},
\end{equation}
where
\begin{equation}
p^{(0)}=\begin{bmatrix} \frac{3+\sqrt{3}}{12} & \frac{3+\sqrt{3}}{12} & \frac{3-\sqrt{3}}{12} & \frac{3-\sqrt{3}}{12} \end{bmatrix}^\top
\end{equation}
is probability vector corresponding to the state $\ket{0}$.
We note that in the considered probability representation each qubit corresponds to a classical two-bit string, and the whole $n$-qubit state can be considered as a probability distribution over all possible values of $2n$-bit string.
\subsection{Single-qubit gates}
We consider a representation of single-qubit and two-qubit gates.
A pseudostochastic matrix of a single-qubit gate $U_{1}$ in the SIC-POVM representation reads:
\begin{equation}\label{eq:singlequbit}
{\bf S}({U_{1}})=3{\bf s}({U_{1}})-2{\bf J}_{1},
\end{equation}
where
\begin{equation}
\begin{split}
{\bf s}({U_{1}})_{ij} = 2{\rm Tr}(E^{\rm sym}_{i}U_{1}E^{\rm sym}_{j}U_{1}^\dagger), \\ \quad i,j=1,2,3,4,
\end{split}
\end{equation}
and ${\bf J}_{1}$ is a $4\times 4$ matrix with all entities equal to 1/4.
It is easy to see that ${\bf s}({U_{1}})$ is a bistochastic matrix (note that $\rho_{j}$ can be considered as a fair quantum state).
The matrix ${\bf J}_{1}$ is also bistochastic and outputs maximally chaotic state for any input probability vector.
Pseudostochastic matrices for some common single-qubit gates are provided in Appendix~\ref{apx:gates}.
We also demonstrate an application of the Hadamard gate to the state $\ket{0}$ in Fig.~\ref{fig:had_gate_impl}.
\subsection{Multi-qubit gates}
In the case of $n\geq 2$ qubits, the pseudostochastic matrix corresponding to an action of $U_{1}$ on a particular qubit can be obtained by tensor product with identity matrix (matrices).
As an illustrative example consider the case where $U_{1}$ acts on the second qubit among $n=4$ qubits.
Since an absence of operation corresponds to identity stochastic matrix the resulting pseudostochastic matrix reads
\begin{multline}\label{eq:exampleofmultiplequbits}
{\bf I}_{4} \otimes {\bf S}({U_{1}}) \otimes {\bf I}_{4}\otimes {\bf I}_{4}\\=3 {\bf I}_{4} \otimes {\bf s}({U_{1}})\otimes {\bf I}_{16}-2{\bf I}_{4} \otimes {\bf J}_{1}\otimes {\bf I}_{16}.
\end{multline}
We note that the right-hand side of Eq.~\eqref{eq:exampleofmultiplequbits} is a difference of two scaled stochastic matrices.
We see that the probability vector resulting from the action of the considered pseudostochastic matrix appears to be a linear combination of two probability vectors obtained from the initial one by acting with different bistochastic matrices.
The sum of coefficients of this linear combination is equal to one thus forming affine combinations.
Due to the negative elements inside, it is very different from a convex hull typical for a classical randomization process.
The situation with a two-qubit entangling gate $U_{2}$ (e.g. $ CX$ or $CZ$) is a bit more complex.
The corresponding pseudostochastic matrix reads
\begin{equation}\label{eq:twoqubit}
{\bf S}(U_{2})= 9{\bf s}_{\rm I}(U_{2}) - 12 {\bf s}_{\rm II}(U_{2}) + 4 {\bf J}_{2},
\end{equation}
where
\begin{equation}
\begin{aligned}
&{\bf s}_{\rm I}(U_{2})_{(ij),(kl)}=4{\rm Tr}(E^{\rm sym}_{i}\otimes E^{\rm sym}_{j} U_{2}E^{\rm sym}_{k}\otimes E^{\rm sym}_{l}U_{2}^{\dagger}), \\
&{\bf s}_{\rm II}(U_{2})_{(ij),(kl)}={\rm Tr}(E^{\rm sym}_{i}\otimes E^{\rm sym}_{j} U_{2}\widetilde{\rho}_{kl}U_{2}^{\dagger}), \\
&\widetilde{\rho}_{kl}= E^{\rm sym}_{k}\otimes\rho^{\rm mix} + \rho^{\rm mix} \otimes E^{\rm sym}_{l}, \\
&\quad i,j,k,l=1,2,3,4,
\end{aligned}
\end{equation}
where $\rho^{\rm mix}={\bf I}_{2}/2$ is a maximally mixed state, ${\bf J}_{2}$ is a matrix with all entities equal to 1/16.
One can check that all matrices ${\bf s}_{\rm I}(U_{2})$, ${\bf s}_{\rm II}(U_{2})$, and ${\bf J}_{2}$ are bistochastic.
We observe again a linear combination of stochastic matrices with coefficient giving a total of unity and having negative elements.
As an example we show the construction of a two-node cluster state in Fig.~\ref{fig:cz_gate_impl}.
We also provide pseudostochastic matrices corresponding to different two-qubit gates in Appendix~\ref{apx:gates}.
The pseudostochastic matrix of two-qubit operation acting on a particular pair of $n\geq 3$ qubits can be obtained by employing tensor product with identity matrix (matrices) in a similar way as in the case of the single-qubit gate.
\subsection{Measurements}
The pseudostochastic matrix of the single-qubit projective measurement in the computation basis is given by the following expression:
\begin{equation}\label{eq:projmeas}
{\bf M}_{\rm pr} = 3{\bf m}_{\rm pr} - 2{\bf J}_{\rm pr},
\end{equation}
where
\begin{equation}
{\bf m}_{\rm pr} = \frac{1}{2}
\begin{bmatrix}
1+\frac{1}{\sqrt{3}} & 1+\frac{1}{\sqrt{3}} & 1-\frac{1}{\sqrt{3}} & 1-\frac{1}{\sqrt{3}} \\
1-\frac{1}{\sqrt{3}} & 1-\frac{1}{\sqrt{3}} & 1+\frac{1}{\sqrt{3}} & 1+\frac{1}{\sqrt{3}}
\end{bmatrix}
\end{equation}
and ${\bf J}_{\rm pr}$ is $2\times 4$ stochastic matrix with all elements equal to 1/2.
One can see that the form of Eq.~\eqref{eq:projmeas} is similar to Eq.\eqref{eq:singlequbit}.
However, the important difference is that within the projective measurement we have a reduction of probability space dimensionality.
An example of the projective measurement of a state $\ket{1}$ in the probability representation is shown in Fig.~\ref{fig:proj_meas}.
In the case of $n>1$ qubits the pseudostochastic matrix of the single-qubit measurement can be obtained in a similar fashion as in the case single-qubit gate [see example in Eq.~\eqref{eq:exampleofmultiplequbits}]
A pseudostochasitc matrix of the projective measurement of several qubits can be obtained as a sequence of measurements on individual qubits.
\begin{figure}
\caption{Projective measurement in the probability representation. Here 0 and 1 stand for standard outcome labels.}
\label{fig:proj_meas}
\end{figure}
We see that the running of $n$-qubit quantum circuit can be considered as kind of random walk of $2n$-bit string.
In each step corresponding to an implementation of single-qubit or two-qubit quantum gates a single or a pair of 2-bit chunks are affected (each chunk consists of $(2k)$th and $(2k+1)$th position in the string with $k=0,1,\ldots,n-1$).
The final step of readout measurement corresponds to a compressive random mapping of each 2-bit chunk, corresponding to measured qubit, to 1 bit value.
The quantum nature of this randomized process manifests itself by the fact that all steps are described with pseudostochastic matrices given by Eqs.~\eqref{eq:singlequbit}, \eqref{eq:twoqubit}, and \eqref{eq:projmeas}.
On the one hand the negative conditional probabilities of pseudostochastic matrices prevent us from straightforward emulating of quantum processes within quantum computation with classical randomized algorithms,
and on the hand they actually underlie the advantage of quantum computers over classic ones.
\begin{figure*}
\caption{Demonstration of Grover's algorithm in the probability representation.
In (a) the circuit for Grover's algorithm is illustrated.
In (b) the step-by-step implementation of Grover's algorithm in the probability representation is shown.}
\label{fig:Grover}
\end{figure*}
\subsection{Grover's algorithm}
In order to provide another illustrative example of the probabilistic representation of quantum computing processes, we consider a two-qubit Grover's algorithm~\cite{Grover1996} with a classical oracle function
\begin{equation}
\chi(10)=1, \quad \chi(00) = \chi(01) = \chi(11) = 0.
\end{equation}
The corresponding circuit which allows finding the `secret string' 10 by a single query to the two-qubit quantum oracle
\begin{equation}
U_{\chi}\ket{xy}=(-1)^{\chi(xy)}\ket{xy}, \quad x,y\in\{0,1\}
\end{equation}
is shown in Fig.~\ref{fig:Grover}(a).
The evolution of the probability representation of corresponding four-bit string is shown in Fig.~\ref{fig:Grover}(b).
One can observe how the information about the secret string first gets into the state after applying the oracle and then is extracted with the diffusion gate and projective measurement.
\section{Conclusion and outlook}\label{sec:conclusion}
Here we summarize the main results of our paper.
We have developed the MIC-POVM-based probability representation, which generalizes the SIC-POVM-based approach.
We have demonstrated advantages of this approach with the focus on the description of multi-qubit systems.
We have derived quantum dynamical equations both for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator.
We have also discussed applications of the suggested approach for the analysis of NISQ computing processes and obtain pseudostochastic maps for various decoherence channels and quantum gates.
In particular, we have demonstrated that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach.
These results seem to be relevant to future investigation of emulating quantum dynamics with randomized algorithms run on a classical computer and, correspondingly,
to the quantum advantage problem~\cite{Martinis2019,Wisnieff2020,Chen2020,Lidar2020}.
Our approach seems to be promising in the context of investigating dynamics of open quantum systems with initially correlated states of a system and bath, which has been studied in details in Ref.~\cite{Wiseman2019}.
We note that there is an interesting connection between MIC representations (and SIC representations particularly) with Lie algebras and Lie groups~\cite{Fuchs2011,Fuchs2015,Manko2010}.
The properties of a concrete MIC-POVM may be studied using the structure constants of a generated Lie algebra. For example, it was shown, that the structure constants are completely antisymmetric exactly in the case of SIC-POVMs \cite{Fuchs2015}.
We also would like to note that the representation may be considered as a faithful functor from the category of channels to the category of pseudostochastic maps (see Ref.~\cite{Wetering2017}), which can be explored in future in more details.
Another interesting direction for the further research is related to tensor networks~\cite{Carrasquilla2019, Luchnikov2019}, which are also important from the perspective of quantum computing.
\section*{Acknowledgments}
We thank D. Chru{\'s}ci{\'n}ski, A.E. Teretenkov, and V.I. Man'ko for useful comments.
Results of Secs.~\ref{sec:construction},~\ref{sec:dynamics}, and ~\ref{sec:qcomp} were obtained by E.O.K. and V.I.Y with the support from the Russian Science Foundation Grant No. 19-71-10091.
Results of Sec.~\ref{sec:decoherence} were obtained by A.K.F. and A.S.M. with the support from the Grant of the President of the Russian Federation (Project No. MK923.2019.2).
Corresponding authors: E.O.K ([email protected]) and A.K.F ([email protected]).
\appendix
\section{Pseudostochastic matrices of common single-qubits channels and single- and two-qubit gates}\label{apx:gates}
We provide explicit form of pseudostochastic matrices for some common types of quantum channels in the case spin 1/2 particle (qubit) in Table~\ref{tbl:generators-matrices}.
For constructing probability representation of all channels, SIC-POVM ~\eqref{eq:SIC-POVM} is used.
\begin{table*}[h!]
\caption{Pseudostochastic matrices corresponding to some common channels in SIC-POVM-based formalism. }
\begin{tabular}{|p{0.2\linewidth}|p{0.28\linewidth}|p{0.45\linewidth}|}
\hline
Quantum channel & Standard formalism & Pseudostochastic matrix \\ \hline
Identity
& $ \Phi_t[\rho] = \rho $
& $
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $\\
\hline
Depolarization
& $ \Phi_t[\rho] = e^{-t/\tau} \rho + (1-e^{-t/\tau}) \chi $
& $ \frac{1}{4}
\begin{bmatrix}
1 + 3 e^{-t/\tau} & 1 - e^{-t/\tau}& 1 - e^{-t/\tau} & 1 - e^{-t/\tau} \\
1 - e^{-t/\tau} & 1 +3e^{-t/\tau}& 1 - e^{-t/\tau} & 1 - e^{-t/\tau} \\
1 - e^{-t/\tau} & 1 - e^{-t/\tau}& 1 +3e^{-t/\tau} & 1 - e^{-t/\tau} \\
1 - e^{-t/\tau} & 1 - e^{-t/\tau}& 1 - e^{-t/\tau} & 1 +3e^{-t/\tau}
\end{bmatrix} $
\\
\hline
Dephasing
& $ \Phi_t[\rho] =
\begin{bmatrix}
\rho_{00} & e^{-t/\tau} \rho_{01} \\
e^{-t/\tau} \rho_{10} & \rho_{11}
\end{bmatrix} $
& $\frac{1}{2}
\begin{bmatrix}
1 + e^{-t/\tau} & 1 - e^{-t/\tau} & 0 & 0 \\
1 - e^{-t/\tau} & 1 + e^{-t/\tau} & 0 & 0 \\
0 & 0 & 1 + e^{-t/\tau} & 1 - e^{-t/\tau} \\
0 & 0 & 1 - e^{-t/\tau} & 1 + e^{-t/\tau}
\end{bmatrix} $
\\
\hline
Damping
& $ \Phi_t[\rho] =
\begin{bmatrix}
1 - e^{-t/\tau} \rho_{11} & e^{-t/2\tau} \rho_{01} \\
e^{-t/2\tau} \rho_{10} & e^{-t/\tau} \rho_{11}
\end{bmatrix} $
&
$\begin{bmatrix}
a & b & c & c \\
b & a & c & c \\
d & d & e & f \\
d & d & f & e
\end{bmatrix}$,
where
$a=\alpha+\frac{e^{-t/2\tau}}{2}, b =\alpha-\frac{e^{-t/2\tau}}{2}, c=\alpha-\frac{e^{-t/\tau}}{2} ,$ \\
& &
$d= -\alpha+\frac{1}{2},
e=\alpha+\frac{e^{-t/\tau}}{2\sqrt{3}}+\frac{e^{-t/2\tau}}{2}-\frac{1}{2\sqrt{3}} ,$ \\
& &
$f=\alpha+\frac{e^{-t/\tau}}{2\sqrt{3}}-\frac{e^{-t/2\tau}}{2}-\frac{1}{2\sqrt{3}} ,$ and $ \alpha=\frac{e^{-t/\tau}}{4}-\frac{e^{-t/\tau}}{4\sqrt{3}}+\frac{1}{4}+\frac{1}{4\sqrt{3}}$\\
\hline
Rotation around $x$-axis
& $ \Phi_t[\rho] = e^{- \frac{i \omega t}{2} \sigma^{(1)}} \rho e^{\frac{i \omega t}{2} \sigma^{(1)}} $
& $\frac{1}{2}
\begin{bmatrix}
2\cos^2(\omega t/2) & -\sin(\omega t) & \sin(\omega t) & 2\sin^2(\omega t/2) \\
\sin(\omega t) & 2\cos^2(\omega t/2) & 2\sin^2(\omega t/2) & -\sin(\omega t) \\
-\sin(\omega t) & 2\sin^2(\omega t/2) & 2\cos^2(\omega t/2 & \sin(\omega t) \\
2\sin^2(\omega t/2) & \sin(\omega t) & -\sin(\omega t) & 2\cos^2(\omega t/2)
\end{bmatrix}$
\\
\hline
Rotation around $y$-axis
& $ \Phi_t[\rho] = e^{- \frac{i \omega t}{2} \sigma^{(2)}} \rho e^{\frac{i \omega t}{2} \sigma^{(2)}} $
& $\frac{1}{2}
\begin{bmatrix}
2\cos^2(\omega t/2 & -\sin(\omega t) & 2\sin^2(\omega t/2) & \sin(\omega t) \\
\sin(\omega t) & 2\cos^2(\omega t/2 & -\sin(\omega t) & 2\sin^2(\omega t/2) \\
2\sin^2(\omega t/2) & \sin(\omega t) & 2\cos^2(\omega t/2 & -\sin(\omega t) \\
-\sin(\omega t) & 2\sin^2(\omega t/2) & \sin(\omega t) & 2\cos^2(\omega t/2
\end{bmatrix}$
\\
\hline
Rotation around $z$-axis
& $ \Phi_t[\rho] = e^{- \frac{i \omega t}{2} \sigma^{(3)}} \rho e^{\frac{i \omega t}{2} \sigma^{(3)}} $
& $\frac{1}{2}
\begin{bmatrix}
2\cos^2(\omega t/2 & 2\sin^2(\omega t/2) & \sin(\omega t) & -\sin(\omega t) \\
2\sin^2(\omega t/2) & 2\cos^2(\omega t/2 & -\sin(\omega t) & \sin(\omega t) \\
-\sin(\omega t) & \sin(\omega t) & 2\cos^2(\omega t/2 & 2\sin^2(\omega t/2) \\
\sin(\omega t) & -\sin(\omega t) & 2\sin^2(\omega t/2) & 2\cos^2(\omega t/2
\end{bmatrix} $ \\
\hline
\end{tabular}
\label{tbl:generators-matrices}
\end{table*}
We also demonstrate pseudostochastic matrices for some single- and two-qubit gates in Fig.~\ref{fig:gates}.
Pseudostochastic matrices of singe-qubit gates are obtained with SIC-POVM $E^{\rm sym}$ given in Eq.~\eqref{eq:SIC-POVM}.
For constructing probability representation of two-qubit gates, the MIC-POVM $E^{\rm sym}\otimes E^{\rm sym}$ is used.
One can observe an appearance of negative elements in pseudostochastic matrices of Hadamard gate, $T$ gate, $S=T^{2}$ gate; $CZ$ gate, CNOT ($CX$) gate, and iSWAP gate.
We note that negative elements are absent for Pauli-X gate and SWAP gate.
\begin{figure*}
\caption{Pseudostochastic matrices of a single- and two-qubit gates: (a) Hadamard gate, b) X-gate, c) T-gate, d) S-gate, e) CZ-gate, f) CNOT-gate, g) SWAP gate, and h) iSWAP gate.}
\label{fig:gates}
\end{figure*}
\end{document} | math | 87,677 |
\begin{document}
\begin{flushright}
This work concludes a research cycle, \\but not the friendship that has tied us.\\ You left us, dear Mimmo, too soon.\\ The disease has won, but your memories\\ will always be with us.
\end{flushright}
\begin{center}
\Large \bf Multi-integrals of finite variation
\\
\vskip.5cm \large
D. Candeloro, L. Di Piazza, K. Musia{\l},
A.R. Sambucini
\vskip.5cm
\end{center}
{\bf $$\supset$bset$mall \noindent Abstract}
The aim of this paper is to investigate different types of multi-integrals of finite variation
and to obtain decomposition results.\\
{\noindent \bf $$\supset$bset$mall Keywords} Finite interval variation, multivalued integral, decomposition of multifunctions\\
{\noindent \bf $$\supset$bset$mall MSC}
28B20, 26E25, 26A39, 28B05, 46G10, 54C60, 54C65\\
$$\supset$bset$ection{Introduction}\label{intro}
In \cite{f1994} was proved that a Banach space valued function is McShane integrable if and only if it is Pettis and Henstock integrable. That result has been then generalized to compact valued multifunctions $\Gamma$ (see \cite{dp}), weakly compact valued multifunctions (see \cite{cdpms2016b}) and bounded valued multifunctions (see \cite{cdpms2018a}). Di Piazza and Marraffa \cite{dp-ma} presented an example of a Pettis and variationally Henstock integrable function that is not variationally McShane integrable (= Bochner integrable in virtue of \cite[Lemma 2]{dpm-ill}). It turns out that Fremlin's theorem can be formulated for variational integrals if and only if the variation of the integral is finite in the following sense:
$$
$$\supset$bset$up \left\{$$\supset$bset$um_i \left\|\int_{I_i} \Gamma \right\|\colon \{I_1,\ldots,I_n\} \;\mbox{\rm is a finite partition of } [0,1] \right\}\, < +\infty.
$$
Finally, in the last section, using $DL$ or $Db$ conditions we are able to prove that the scalar integrability of a multifunction can be obtained as a traslation of the Pettis integrability (Theorem \ref{p5}), while its Henstock integrability under $DL$ condition is obtained using Birkhoff integrability (Theorem \ref{aB}),
both results with integrals of finite variation.\\
This article is the last in which Domenico Candeloro was able to cooperate and to give his personal contribution, always precious, and we want to dedicate it to him, in his memory.
$$\supset$bset$ection{Preliminaria}
\label{sec:1}
Throughout $X$ is a Banach space with norm $\| \cdot \|$ and its dual $X^*$.
The closed unit ball of $X$ is denoted by $B_X$. The symbol
$c(X)$ denotes the collection of all
nonempty closed convex subsets of $X$ and $cb(X),\,cwk(X)$ and $ck(X)$ denote respectively
the family of all bounded, weakly compact and compact members of $c(X)$. For every $C \in c(X)$ the
{\it support function of} $\, C$ is denoted by $s( \cdot, C)$ and
defined on $X^*$ by $s(x^*, C) = $$\supset$bset$up \{ \langle x^*,x \rangle \colon \ x
\in C\}$, for each $x^* \in X^*$. $\|C\|_h= d_H(C, \{ 0\}) :=$$\supset$bset$up\{\|x\|: x\in{C}\}$ and $d_H$ is the Hausdorff metric on the hyperspace $cb(X)$.
The map $i:cb(X)\to \ell_{\infty}(B_{X^*})$
given by $i(A):=s(\cdot, A)$ is the R{\aa}dstr\"{o}m embedding (see, for example, \cite[Theorem 3.2.9 and Theorem 3.2.4(1)]{Beer},
\cite[Theorem II-19]{CV}, or \cite{L1}).\\
${\mathcal I}$ is the collection of all closed subintervals of the unit interval $[0,1]$.
All functions investigated are defined on the unit interval $[0,1]$
endowed with Lebesgue measure $\lambda$ and Lebesgue measurable sets $\mathcal{L}$.\\
A map $\Gamma: [0,1]\to c(X)$ is called a {\it multifunction}.
In the sequel, given a multifunction $\Gamma:[0,1]\to c(X)$, we set
$D_{\Gamma}(t):=\mbox{diam }(\Gamma(t)),$
for all $t\in [0,1]$. We say that $\Gamma$ satisfies the
\begin{description}
\item[({\em $Db$-condition})] if ${\rm sup\,ess}_t D_{\Gamma}(t)<\infty$;
\item[({\em $DL$-condition})] if $\overlineerline{\int}_0^1 D_{\Gamma}(t)dt<+\infty$ (where $\overlineerline{\int}$ denotes the upper integral).
\end{description}
We recall that a multifunction $\Gamma:[0,1]\to{c(X)}$ is said to be {\it integrably bounded} if there is a function $h\in{L_1[0,1]}$ such that $\|\Gamma(t)\|_h \leq |h(t)|$ for almost all $t\in[0,1]$.
We have always $D_{\Gamma}(t)\leq 2|\Gamma(t)|$. Hence, if $\Gamma$ is integrably bounded, then $\Gamma$ satisfies $DL$. If $\Gamma(t)\ni{0}$ for almost every $t\in[0,1]$, then $|\Gamma(t)|\leq D_{\Gamma}(t)$ a.e. Each function $g:[0,1]\to{X}$, considered as a $ck(X)$-valued multifunction, trivially satisfies the $Db$ property. \\
\begin{comment}
\marginpar{$$\supset$bset$mall {\color{green!40!blue} $\mathcal{L}$} instead of $\Sigma$}
A map $M: {\color{green!40!blue} \mathcal{I}} \rightarrow cb(X)$ is an {\it additive interval multimeasure}, if $M(A\cup{B})=M(A) \oplus {M(B)}$ for every pair of disjoint elements of ${\color{green!40!blue} \mathcal{I}}$. An additive map $M: \mathcal{L} \rightarrow cb(X)$ is called a {\it multimeasure} if
$s(x^*,M(\cdot))$ is a finite measure, for every $x^*\in{X^*}$.
If $M$ is a point map, then we talk about measure.
\\
\end{comment}
We recall that if $\Phi:\mathcal{L} \to Y$ is an additive vector measure with values in a normed space $Y$, then the {\em variation} of $\Phi$ is the extended non negative function $|\Phi|$ whose value on a set $E \in \mathcal{L}$ is given by
$|\Phi|(E) = $$\supset$bset$up_{\pi} $$\supset$bset$um_{A \in \pi} \|\Phi(A)\|, $
where the supremum is taken over all partitions $\pi$ of $E$ into a finite number of pairwise disjoint members of $\mathcal{L}$. If $|\Phi| < \infty$, then $\Phi$ is called a measure of finite variation.
\\
If the measure $\Phi$ is defined only on ${\mathcal I}$, the finite partitions considered in the definition of variation are composed by intervals. In this case we will speak of {\em finite interval variation} and we will use the symbol $\widetilde{\Phi}$, namely:
$$
\widetilde{\Phi}([0,1])=$$\supset$bset$up \{ $$\supset$bset$um_i\|\Phi(I_i)\| \colon \{I_1,\ldots,I_n\}\, \mbox{is a finite interval partition of }[0,1]\}.
$$
If $Y$ is a metric space, for example $(cb(X),d_H)$, which is a near vector space in the sense of \cite{L1}, and
$\Phi:{\mathcal I}\to{cb(X)}$ is additive we consider in its interval variation the distance $d_H (\Phi(A), \{0\})$ instead of $\|\Phi(A)\|$.\\
We recall here briefly the definitions of integrals involved in this article.
A scalarly integrable multifunction $\Gamma:[0,1]\to c(X)$ is {\it Dunford integrable} in a non-empty family $\mathcal C$$\supset$bset$ubset{c(X^{**})}$, if
for every set $A \in {\mathcal L}$ there
exists a set $M_{\Gamma}^D(A)\in {\mathcal C}$ such that
\[
s(x^*,M_{\Gamma}^D(A))=\int_A s(x^*,\Gamma)\,d\lambda\,,\;\mbox{for every}\; x^*\in X^*.
\]
If $M_{\Gamma}^D(A)$$\supset$bset$ubset{X}$ for every $A\in{\mathcal L}$, then $\Gamma$ is called {\it Pettis integrable}.
We write it as $(P)\int_A\Gamma\,d\mu$ or $M_{\Gamma}(A)$.
We say that a Pettis integrable $\Gamma: [0,1]\to c(X)$ is {\it strongly Pettis
integrable}, if $M_{\Gamma}$ is an $h$-multimeasure (i.e. it is countably additive in the Hausdorff metric).
\\
A multifunction $\Gamma:[0,1]\to cb(X)$ is said to be {\it Henstock} (resp. {\it McShane})
integrable on $[0,1]$, if there exists $\Phi{}_{\Gamma}([0,1]) \in cb(X)$
with the property that for every $\varepsilon > 0$ there exists a gauge $\delta: [0,1] \to \mathbb{R}^+$
such that for each Perron partition (resp. partition)
$\{(I{}_1,t{}_1), \dots,(I{}_p,t{}_p)\}$ of
$[0,1]$ with $I_i $$\supset$bset$ubset [t_i - \delta(t_i), t_i + \delta(t_i)]$ for all $i$ ( i.e. $\delta$--fine),
we have
\begin{eqnarray}\label{e14}
d_H \left(\Phi_{\Gamma}([0,1]),$$\supset$bset$um_{i=1}^p\Gamma(t_i)\lambda (I_i)\right)<\varepsilon.
\end{eqnarray}
If the gauges above are taken to be measurable, then we
speak of $\mathcal H$ (resp. Birkhoff)-integrability on $[0,1]$.
If $I\in{\mathcal I}$, then $\Phi_{\Gamma}(I):=\Phi_{\Gamma\chi_I}[0,1]$.
Finally if, instead of formula (\ref{e14}), we have
\begin{eqnarray}\label{var}
$$\supset$bset$um_{i=1}^p d_H \left(\Phi_{\Gamma}( I_i),\Gamma(t_i)\lambda (I_i)\right)<\varepsilon.
\end{eqnarray}
we speak about variational {\it Henstock} (resp. {\it McShane})
integrability on $[0,1]$. In all the cases $\Phi_{\Gamma}: {\mathcal I} \to cb(X)$ is an additive interval multimeasure.\\
Thanks to the R{\aa}dstr\"{o}m embedding, a multifunction $\Gamma$ is "gauge" integrable (in one of the previous types) if and only if its image $i \circ \Gamma$ in $l_{\infty}(B_{X^*})$ is integrable in the same way.\\
A multifunction $\Gamma:[0,1]\to{cb(X)}$ is said to be Henstock-Kurzweil-Pettis (or HKP) integrable in $cb(X)$ if it is scalarly Henstock-Kurzweil (or HK)-integrable and for each $I\in{\mathcal I}$ there exists a set $N_{\Gamma}(I)\in cb(X)$ such that
$
s(x^*,N_{\Gamma}(I))=(HK) \int_Is(x^*,\Gamma)\quad$ for every $x^*\in{X^*}.$
If an HKP-integrable $\Gamma$ is scalarly integrable, then it is called {\it weakly McShane integrable} (or wMS).
We recall that a function $f:[0,1]\to{\mathbb{R}}$ is Denjoy-Khintchine (DK) integrable (\cite[Definition 11]{gordon-p}), if there exists an ACG function (cf. \cite{Gor}) F such that its approximate derivative is almost everywhere equal to $f$. \\
A multifunction $\Gamma:[0,1]\to{cb(X)}$ is Denjoy-Khintchine-Pettis (DKP)
integrable in
a non empty family $\mathcal C$ in $cb(X)$, if for each $x^*\in{X^*}$ the function $s(x^*,\cdot)$ is Denjoy-Khintchine integrable and for every $I\in{\mathcal I}$ there exists $C_I\in\mathcal C$ with $(DK)\int_Is(x^*,\Gamma)=s(x^*,C_I)$, for every $x^*\in{X^*}$.
\\
As regards other definitions
of measurability and integrability that will be treated here and are not explained and the known relations among them, we refer to \cite{cdpms2016,cdpms2016a,cdpms2016b,cdpms2019a,cs2014,ccgs,ccgs2019,ckr2,eh,dms,nara}, in order do not burden the presentation.
$$\supset$bset$ection{Multimeasures of finite variation}\label{sec:2}
We begin with a known fact.
\begin{comment}
\begin{lem}\label{L3}
Let $\Phi:{\mathcal I}\to{cb(X)}$ be additive. If its distribution function $\varphi:[0,1]\to{X}$ defined by $\varphi(x)=\Phi([0,x])$ is continuous and
$$
\widetilde\Phi[0,1]=$$\supset$bset$up\{$$\supset$bset$um_i\|\Phi(I_i)\|, \{I_1,\ldots, I_n\} \;\mbox{\rm is a finite partition of } [0,1]\}<\infty
$$
then $V_{\Phi}(I)=\widetilde\Phi(I)$, for each $I\in{\mathcal I}$.
\end{lem}
\begin{proof}
It is known that $V_{\Phi}(I)\leq\widetilde\Phi(I)$, for each $I\in{\mathcal I}$. Fix $\varepsilon>0$ and a partition $\{I_1,\ldots,I_n\}$ of $[0,1]$ such that $$$\supset$bset$um_i\|\Phi(I_i)\|>\widetilde\Phi[0,1]-\varepsilon$. If $\delta$ is an arbitrary gauge on $[0,1]$, then any $\delta$-fine partition $\{(J_1,t_1),\ldots,(J_m,t_m)\}$ of $[0,1]$ that refines $\{I_1,\ldots,I_n\}$ satisfies the inequality
$$$\supset$bset$um_j\|\Phi(J_j)\|\geq $$\supset$bset$um_i\|\Phi(I_i)\|$. It follows that $Var({\Phi},\delta,[0,1])\geq $$\supset$bset$um_i\|\Phi(I_i)\|>\widetilde\Phi[0,1]-\varepsilon$. Consequently, we have also $V_{\Phi}[0,1]\geq\widetilde\Phi[0,1]-\varepsilon$.\\
The general case follows in the same way or one may argue also as follows: If $[0,1]=I\cup{J}$ is a partition, then $\widetilde\Phi[0,1]=\widetilde\Phi(I)+\widetilde\Phi(J)\geq V_{\Phi}(I)+V_{\Phi}(J)=V_{\Phi}[0,1]=\widetilde\Phi[0,1]$. It follows that $V_{\Phi}(I)=\widetilde\Phi(I)$.
\qed
\end{proof}
\end{comment}
\begin{lem}\label{l1}
If $f:[0,1]\to{\mathbb{R}}$ is the Denjoy-Khintchine integrable and the interval variation
of its integral is finite, then $f$ Lebesgue integrable.
\end{lem}
\begin{proof}
Let $F$ be the Denjoy-Khintchine primitive of
$f: [a,b] \to {\mathbb{R}}$. Then $F$ is an ACG function and, according to \cite[Theeorem 15.8]{Gor},
$F$ is continuous on $[a,b]$. So $F$ satisfies the condition (N) of Lusin on in $[a,b]$ (see \cite[Theorem 6.12]{Gor}). Since $F$ is also BV, an application of \cite[Theorem 6.15]{Gor} gives that $F$ is also AC on on $[a,b]$. So $f$ is Lebesgue integrable.\qed
\end{proof}
\begin{thm}\label{t1}
Let $\Phi:{\mathcal I}\to{cb(X)}$ be the DKP-integral
of $\Gamma:[0,1]\to{cb(X)}$. If $$$\supset$bset$up_{x^*\in{B_X}}\widetilde{\langle{x^*,\Phi}\rangle}([0,1])<\infty$, then $\Gamma$ is weakly McShane integrable in $cb(X)$ and Gelfand integrable in $cw^*k(X^{**})$. If $\widetilde\Phi([0,1])<\infty$, then $\Phi$ is strongly Pettis integrable in $cb(X)$.
\end{thm}
\begin{proof}
By Lemma \ref{l1} $\Gamma$ is wMS-integrable in $cb(X)$. According to \cite[Theorem 3.2]{cdpms2018a} it is Gelfand integrable in $cw^*k(X^{**})$. Denote the Gelfand integral by $\Psi$. \\
Assume now that $\widetilde\Phi ([0,1])<\infty$.
If $\{I_i : i \in \mathbb{N} \}$ is a sequence of non-overlapping subintervals of $[0,1]$, then
$$
$$\supset$bset$um_i\|\Phi(I_i)\|_h \leq \widetilde \Phi([0,1])<\infty
$$
and so, due to the completeness of $cb(X)$ under Hausdorff distance,
the series $$$\supset$bset$um_i\Phi(I_i)$ is convergent in $cb(X)$.\\
But for each $x^*\in{X^*}$ the function $s(x^*,\Psi)$ is a measure and so
$$$\supset$bset$um_is(x^*,\Phi(I_i))=s(x^*,\Psi(\bigcup_iI_i))$. Since the sum of $$$\supset$bset$um_i\Phi(I_i)$ is uniquely determined, we have
$$
\Psi(\bigcup_i I_i)=$$\supset$bset$um_i\Phi(I_i)\in{cb(X)}\,.
$$
It follows that $\Psi$ is $$$\supset$bset$igma$-additive (in the Hausdorff metric) on the algebra $\mathfrak J$ generated by ${\mathcal I}$. Hence, also $i\circ\Psi$ is $$$\supset$bset$igma$-additive on
$\mathfrak J$.
But $\widetilde{i\circ\Psi}([0,1])=\widetilde{\Psi}([0,1])=\widetilde\Phi([0,1])<\infty$ and so due to \cite[Proposition I.15]{du}, $i\circ\Psi$ restricted to $\mathfrak J$ is strongly additive. \\
It is a consequence of \cite{katz} or \cite[Theorem I.5.2]{du} that $i\circ\Psi$ is a measure on the $$$\supset$bset$igma$-algebra of Borel subsets of $[0,1]$. But $i\circ\Psi(E)=0$, provided Lebesgue measure vanishes on $E$ and consequently, $i\circ\Psi$ is measure on ${\mathcal L}$. Since $i(cb(X))$ is a closed cone also $\Psi$ is a measure in the Hausdorff metric of $cb(X)$ and therefore $\Gamma$ is strongly Pettis integrable on ${\mathcal L}$.
\qed
\end{proof}
\begin{cor}
If $\Gamma\colon [0,1]\to{c(X)}$ is Pettis integrable in $cb(X)$, $M_{\Gamma}$ is its indefinite Pettis integral and $|M_{\Gamma}|([0,1])<\infty$, then $\Gamma$ is strongly Pettis integrable.
\end{cor}
\begin{proof} It is easily seen that due to the finite variation of $M_{\Gamma}$, the multifunction $\Gamma$ takes a.e. bounded values. Without loss of generality we may assume that $\Gamma:[0,1]\to{cb(X)}$.
We have $\widetilde{M_{\Gamma}}([0,1])\leq |M_{\Gamma}|([0,1])<\infty$ and so we may apply Theorem \ref{t1}. \qed
\end{proof}
Under stronger assumptions one obtains stronger results. We proved in \cite{cdpms2018a} the following
\begin{thm}\label{T11}
Let $\Gamma:[0,1]\to{cb(X)}$ be Henstock (or $\mathcal H$) integrable and let $\Phi_{\Gamma}$ be its Henstock ($\mathcal H$)-integral. If $\widetilde{\Phi}_{\Gamma}[0,1]<\infty$, then $\Gamma$ is McShane (Birkhoff) integrable.
\end{thm}
Finally, we can formulate the characterization of variationally McShane integral in terms of the variational Henstock integral.
\begin{thm}\label{t2}
A multifunction $\Gamma:[0,1]\to{cb(X)}$ is variationally McShane integrable
if and only if it is variationally Henstock integrable and the interval variation of the Henstock integral is finite.
\end{thm}
\begin{proof}
We need to prove only that each vH-integrable multifunction $\Gamma:[0,1] \to {cb(X)}$ with integral of finite interval variation is
variationally McShane integrable.
We know already from Theorem \ref{t1} that $\Gamma$ is Pettis integrable. Since $i\circ\Gamma$ is vH-integrable it is strongly measurable. If $M_{\Gamma}$ is the Pettis integral of $\Gamma$, then $i\circ{M_{\Gamma}}$ is a measure of finite variation and $ i\circ{M_{\Gamma}}(I)=(vH)\int_Ii\circ\Gamma$. It follows that $i\circ\Gamma$ is Bochner integrable. Now we may apply
\cite[Proposition 3.6]{cdpms2016} to obtain variational McShane integrability of $\Gamma$. \qed
\end{proof}
In case of vector valued functions $f: [0,1] \to X$, by the properties of the Pettis and the Bochner integrals, it follows at once that if $f$ is strongly measurable, Pettis integrable and its Pettis integral has finite variation, then $f$ is Bochner integrable. The next result is the multivalued version of this result.
\begin{thm}\label{new}
Let $\Gamma: [0,1] \to cb(X)$ be
Bochner measurable, Pettis integrable, and its Pettis integral has finite variation. Then
$\Gamma$ is integrably bounded.
\end{thm}
\begin{proof} Since $\Gamma$ is Bochner measurable, it is a.e. limit of simple multifunctions. It follows that $i\circ\Gamma$ is strongly measurable. Let us assume that $Y:=\overline{span}(i\circ{\Gamma([0,1])})$ is a closed separable subspace of $ \ell_{\infty} (B_{X^*})$. Then, we follow the proof of \cite[Proposition 3.5]{ckr0}.
If $e_{x^*}\in{B_{\ell_{\infty}(B_{X^*})^*}}$ is defined by $\langle{e_{x^*},g}\rangle:=g(x^*)$ for every $g\in\ell_{\infty}(B_{X^*})$, then the set $B:=\{e_{x^*}|Y:x^*\in{B_{X^*}}\}$$\supset$bset$ubset B_{Y^*}$ is norming. By the Pettis integrability of $\Gamma$
the family ${\mathcal Z}_{i\circ\Gamma,B}:=\{\langle{e_{x^*},i\circ\Gamma}\rangle: x^*\in{B_{X^*}}\}=\{s(x^*,\Gamma):x^*\in{B_{X^*}}\}$ is uniformly integrable.
\begin{comment}
nota per me: il fatto che ${\mathcal Z}_{i\circ\Gamma,B}$ sia uniformemente integrabile, unito al fatto che $la $i \circ \Gamma$ è a valori in un sottoinsieme separabile $Y$ di $ \ell_{\infty} (B_{X^*})$ mi assicura la sua integrabilità alla Pettis in tale spazio.
\end{comment}
Consequently, $i\circ\Gamma$ is a Pettis integrable function. Moreover,
$i((P)\int_A\Gamma\,d\lambda)=(P)\int_Ai\circ\Gamma\,d\lambda$ for every $A\in{\mathcal L}$ (see the proof of \cite[Proposition 3.5]{ckr0}). By the assumption the variaton of $(P)\int{i\circ\Gamma\,d\lambda}$ is finite and so $i\circ\Gamma$ is Bochner integrable. Consequently, $\Gamma$ is integrably bounded. \qed
\end{proof}
Then by \cite[Proposition 3.6]{cdpms2016} (formulated for $cb(X)$-valued multifunctions)
and Theorem \ref{new} we get the following
\begin{prop}
Let $\Gamma:[0,1]\to{cb(X)}$ be a scalarly measurable multifunction. Then the following conditions are equivalent:
\begin{enumerate}
\item
$\Gamma$ is variationally McShane integrable;
\item
$i(\Gamma) \in L_1(\lambda, \ell_{\infty} (B_{X^*}))$;
\item
$\Gamma$ is Bochner measurable and integrably bounded;
\item $\Gamma$ is Bochner measurable, Pettis integrable, and its Pettis integral has finite variation.
\end{enumerate}
\end{prop}
\begin{proof}
It is an immediate consequence of Theorem \ref{new} if we proceed analogously to \cite[Proposition 3.6]{cdpms2016}.
\qed
\end{proof}
$$\supset$bset$ection{Decompositions}
In the study of the integrability of multifunctions it is important to decompose a multifunction as a sum of a selection that is integrable in the same sense and a multifunction that is integrable in a stronger sense than the original one
(see for example \cite{ft,dpm-ill,dm2,cdpms2016,cdpms2016a,cdpms2016b,cdpms2018a}).
Using $Db$ or $DL$ conditions we are able to extend
decomposition results and to write integrable multifunctions as a translation of a multifunction with its integral of finite variation.
\begin{thm}\label{p5}
Let $\Gamma:[0,1]\to{c(X)}$ be integrable in $cb(X)$ ($cwk(X)$ or $ck(X)$) in the sense of one of the scalarly defined integrals. If $\Gamma$ possesses at least one selection integrable in the same way, then the following conditions are equivalent:
\begin{enumerate}
\item
$\Gamma$ satisfies the $DL$-condition (or $Db$ condition);
\item
$\Gamma=G+f$, where $f$ is a properly integrable selection of $\Gamma$, $G$ is Pettis integrable in $cb(X)$ ($cwk(X)$ or $ck(X)$) and $\overline{\int}_0^1D_G(t)\,dt<\infty$ ( and $G$ is bounded). In particular the indefinite integral of $G$ is of finite variation.
\end{enumerate}
\end{thm}
\begin{proof}
Assume that $\Gamma$ is DP-integrable. Due to \cite[Theorem 3.5]{cdpms2018a} $\Gamma=G+f$, where $G$ is Pettis integrable, $f$ is Denjoy integrable and $G$ satisfies the condition DL. It is obvious that the Pettis integral of $G$ is of finite variation.
\qed
\end{proof}
\begin {rem}\label{r1}
\rm
Unfortunately, even if $G:[0,1]\to{ck(X)}$ is a positive multifunction that is Pettis integrable and its integral is of finite variation, the multifunction $G$ may not satisfy the DL condition. To see it let $X=\ell_2[0,1]$ and let $\{e_t:t\in(0,1]\}$ be its orthonormal system. If $G(t):=conv\{0,{e_t}/t\}$, then $s(x,G)=0$ a.e. for each separate $x\in\ell_2[0,1]$ and so the integral and its variation are equal zero. However, $diam\{G(t)\}=1/t$ and so the DL-condition fails.
Moreover, $G$ is not Henstock integrable.
Indeed, let $\delta$ be any gauge and $\{(I_1,t_1),\ldots, (I_n,t_n)\}$ be a $\delta$-fine Perron partition of $[0,1]$.
Assume that $0\in{I_1}$, then $t_1\leq|I_1|$. Hence $\lambda(I_1)/{t_1}\geq1$ for $t_1>0$ and so
$
\left\|$$\supset$bset$um_{i\leq{n}} \frac{e_i}{t_i}\lambda(I_i)\right\|\geq 1.
$
Consider now the multifunction given by $H(t):=conv\{0,e_t\}$, where $X$ is as above. We are going to prove that $H$ is Birkhoff-integrable. Given $\varepsilon>0$, let $n\in{\mathbb{N}}$ be such that $1/{$$\supset$bset$qrt{n}}<\varepsilon$ and $\delta$ be any gauge, pointwise less than $1/n$. If $\{(I_1,t_1),\ldots,(I_m,t_m)\}$ is a $\delta$-fine partition of $[0,1]$ and $\{J_1,\ldots,J_n\}$ is the division of $[0,1]$ into closed intervals of the same length, then
\begin{eqnarray*}
\biggl \|$$\supset$bset$um_{i\leq{m}}e_i \lambda(I_i) \biggr\|&=& \biggl\|$$\supset$bset$um_{i\leq{m}}$$\supset$bset$um_{k\leq{n}}e_i \lambda(I_i\cap{J_k})\biggr\|=\biggl\|$$\supset$bset$um_{k\leq{n}}$$\supset$bset$um_{i\leq{m}}e_i \lambda(I_i\cap{J_k})\biggr\|\\
&=&
\biggl($$\supset$bset$um_{k\leq{n}}$$\supset$bset$um_{i\leq{m}} \lambda(I_i\cap{J_k})^2\biggr)^{1/2}\leq1/{$$\supset$bset$qrt{n}}<\varepsilon\,.
\end{eqnarray*}
(We apply here the inequality $$$\supset$bset$um_{i}a_i^2\leq ($$\supset$bset$um_{
}a_i)^2$. For each fixed $k\leq{n}$ we take as $a_i$ the number $\lambda(I_i\cap{J_k})$
). If $\delta$ is measurable, then we get Birkhoff integrability of $H$.
\end{rem}
Some additional results will be given now, in order to get decompositions with gauge integrable multifunctions.
\begin{thm}\label{aB}
Let $\Gamma:[0,1]\to cwk(X)$ satisfy $DL$-condition, and assume that $\Gamma$ is $\mathcal{H}$-integrable (or H-integrable). Then we have
$\Gamma=G+f,$
where $f\in\mathcal{S_H}(\Gamma)$ ($f\in\mathcal{S}_H(\Gamma)$) is arbitrary and $G$ is an abs-Birkhoff integrable multifunction.
In particular the integral of $G$ has finite variation.
If $\Gamma$ is Bochner measurable, then $G$ is also variationally Henstock integrable.
\end{thm}
\begin{proof}
Assume that $\Gamma$ is $\mathcal{H}$-integrable. It is known (see \cite[Theorem 3.1]{dp}) that $\Gamma$ has an $\mathcal{H}$-integrable selection $f$. Thanks to \cite[Theorem 4]{nara}, both $i \circ \Gamma$ and $f$ are Riemann-measurable. So, if $G:=\Gamma-f$, it is clear that $i \circ G$ is Riemann-measurable too. Moreover, thanks to the $DL$-condition, the function $t\mapsto \|i \circ G(t)\|$ is integrably bounded, i.e.
$\overlineerline{\int}_0^1 \|i \circ G(t)\| dt<+\infty$
since
$\|i \circ G(t)\|=$$\supset$bset$up\{\|u\|_X: u\in G(t)\}=$$\supset$bset$up\{\|v-f(t)\|_X: v\in \Gamma(t)\}\leq \mbox{diam} (\Gamma(t))$.
So, $i \circ G$ is Riemann-measurable and integrably bounded, which means that $i \circ G$ (and so $G$) is absolutely Birkhoff integrable, thanks to \cite[Theorem 2]{ncm}.
\\
Assume now that $\Gamma$ is $H$-integrable. Then, according to \cite[Theorem 3.5]{cdpms2018a} $\Gamma=G+f$, where $G$ is Birkhoff integrable. By the assumption $G$ satisfies the DL-condition. Hence again the function $t\mapsto \|i \circ G(t)\|$ is integrably bounded. Consequently, $i\circ{G}$ is absolutely Birkhoff integrable and hence also $G$. The vH-integrability of $G$ follows from \cite[Corollary 4.1]{bdm}, since $G$ is Pettis integrable.
\qed \end{proof}
A similar result can be given also for Birkhoff integrable functions $\Gamma:[0,1]\to{cwk(X)}$: the proof is essentially the same but instead of \cite{dp} we invoke \cite[Theorem 3.4]{cdpms2016}.
\begin{prop}\label{df1}
Let $\Gamma:[0,1]\to cwk(X)$ satisfy $DL$-condition, and assume that $\Gamma$ is Birkhoff integrable. Then we have
$\Gamma=G+f,$
where $f$ is any Birkhoff integrable selection of $\Gamma$, and $G$ is an abs-Birkhoff integrable multifunction. In particular the integral of $G$ has finite variation.
\end{prop}
\begin{que} \rm
Assume that $f:[0,1]\to{X}$ is Birkhoff integrable and the classical variation of the indefinite integral is finite. Is $f$ absolutely Birkhoff integrable? That is, do we have
$\overline{\int}_0^1\|f(t)\|\,dt<\infty$?
A partial answer is contained in \cite[Corollary 2]{ncm}. \\
Another way, does there exist a Birkhoff integrable $f$ that is scalarly equivalent to zero and $\overline{\int}_0^1\|f(t)\|\,dt=\infty$?
Recall that $G$ from Remark \ref{r1} is not Birkhoff integrable. \\
Fremlin proved that a Birkhoff integrable function is properly measurable in the Talagrand sense. It is known that $f$ is Talagrand integrable if and only if $f$ is properly measurable and $\overline{\int}\|f\|\,d\lambda<\infty$. Then, it is known that $f$ is absolutely Birkhoff integrable if and only if it is Riemann measurable and $\overline{\int}\|f\|\,d\lambda<\infty$ if and only if $f$ is Birkhoff integrable and $\overline{\int}\|f\|\,d\lambda<\infty$. Thus, if $f$ is absolutely Birkhoff integrable, then $f$ is also Talagrand integrable. The converse result fails by \cite[Example 3C]{FM} (where a function $f: [0,1] \to \ell_{\infty}(\mathbb{N})$ is shown, which is Talagrand but not even McShane integrable).
\end{que}
It is also possible to obtain decompositions where the multifunction $G$ turns out to be variationally McShane integrable, as follows.
\begin{prop}\label{df2}
Let $\Gamma:[0,1]\to cwk(X)$ satisfy $DL$-condition, and assume that $\Gamma$ is Bochner measurable. Then we have
$\Gamma=G+f,$
where $f$ is any strongly measurable selection of $\Gamma$, and $G$ is a variationally McShane integrable multifunction.
\end{prop}
\begin{proof}
Let $f$ be any strongly measurable selection of $\Gamma$, and set $G=\Gamma-f$. Then clearly $G$ is Bochner measurable. Moreover, since $\Gamma$ satisfies the $DL$ condition and $f$ is a selection from $\Gamma$, $i \circ G$ is integrably bounded. Then $i \circ G$ is strongly measurable and integrably bounded, and therefore variationally McShane integrable.
Of course this implies that also $G$ is integrable.
\qed
\end{proof}
\begin{prop}\label{p9}
If $\Gamma: [0,1] \to cwk(X)$ is Henstock ($\mathcal H$, variationally Henstock, Pettis, McShane, Birkhoff) integrable, then $\Gamma$ cannot be, in general, written as $G + f$, where $G $ is variationally McShane integrable and $f$ is integrable in the same way as $\Gamma$.
\end{prop}
\begin{proof}
Take $f$ as in \cite{dp-ma}; then $f$ is vH and Pettis integrable, but not Bochner integrable. Let $\Gamma = {\rm conv} \{0, f(t) \}$, then $\Gamma$ is vH-integrable, Pettis but not Bochner integrable, as shown in \cite[Example 4.7]{cdpms2016}. By \cite[Theorem 4.3 (a) and (c)]{cdpms2016}, then $\Gamma$ is also McShane integrable and then Birkhoff integrable, since $\Gamma$ is Bochner measurable. Following the same motivations of \cite[Remark 5.4]{cdpms2016b} the multifunction $G = \Gamma - f $ is not variationally McShane integrable.
\qed
\end{proof}
Almost nothing is known on possible decompositions of a Pettis integrable multifunctions. We have only the following negative result:
\begin {ex}\label{ex17}
\rm
Let $\Gamma:[0,1]\to cwk(X)$ be Pettis integrable. Assume that $M_{\Gamma}({\mathcal L})$ is not relatively compact in the Hausdorff metric. Then $\Gamma$ cannot be represented as
$\Gamma=G+f$, where $G$ is McShane integrable and
$f$ is a Pettis integrable selection of $\Gamma$. In such a case $i\circ\Gamma$ is not Pettis integrable.
\end{ex}
\begin{proof}
Suppose that such a decomposition exists. Then, since $G$ is McShane integrable, the function $i \circ G$ is also McShane integrable and consequently it has relatively norm compact range.That is however equivalent to the norm relative compactness of $M_G({\mathcal L})$ in $d_H$. But then $G$ can be approximated by simple functions (see \cite[Theorem 2.3]{mu8}). Since the integral of $f$ is norm relatively compact (because Lebesgue measure is perfect) also $f$ can be approximated by simple functions in the Pettis norm (see \cite[Theorem 9.1]{mu}). As a result the multifunction $\Gamma$ can be approximated by simple multifunctions, which is impossible, since its range $M_{\Gamma}({\mathcal L})$ is not relatively compact in the Hausdorff metric. (see \cite[Theorem 2.3]{mu8}). The non-integrability of $i\circ\Gamma$ is a consequence of perfectness of Lebesgue measure. Indeed, the range of the integral of a Pettis integrable function on $[0,1]$ (or on any perfect measure space) is norm relatively compact (\cite[3J]{ft}).
\qed
\end{proof}
$$\supset$bset$ection*{Acknowledgements}
This research was partially supported by Grant "Metodi di analisi reale per l'appros\-simazione attraverso operatori discreti e applicazioni” (2019) of GNAMPA -- INDAM (Italy), by University of Perugia -- Fondo Ricerca di Base 2019 ``Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni``, by University of Palermo -- Fondo Ricerca di Base 2019 and by Progetto Fondazione Cassa di Risparmio cod. nr. 2018.0419.021 (title: Metodi e Processi di Intelligenza
artificiale per lo sviluppo di una banca di immagini mediche per fini diagnostici (B.I.M.)).
$$\supset$bset$mall
Domenico Candeloro and Anna Rita Sambucini,
Department of Mathematics and Computer Sciences, University of Perugia 1, Via Vanvitelli - 06123, Perugia (Italy),
orcid ID: 0000-0003-0526-5334, 0000-0003-0161-8729
\email{[email protected], [email protected]}\\
Luisa Di Piazza,
Department of Mathematics and Computer Sciences, University of Palermo, Via Archirafi 34, 90123 Palermo (Italy) orcid ID: 0000-0002-9283-5157 \email{[email protected]}
\\
Kazimierz Musia\l,
Institute of Mathematics, Wroc{\l}aw University, Pl. Grunwaldzki 2/4, 50-384 Wroc{\l}aw (Poland) orcid ID: 0000-0002-6443-2043
\email{[email protected]}
\end{document} | math | 30,873 |
\begin{document}
\title{An Envy-Free Online UAV Charging Scheme with Vehicle-Mounted Mobile Wireless Chargers}
\begin{abstract}
In commercial unmanned aerial vehicle (UAV) applications, one of the main restrictions is UAVs' limited battery endurance when executing persistent tasks.
With the mature of wireless power transfer (WPT) technologies, by leveraging ground vehicles mounted with WPT facilities on their proofs, we propose a mobile and collaborative recharging scheme for UAVs in an on-demand manner. Specifically, we first present a novel air-ground cooperative UAV recharging framework, where ground vehicles cooperatively share their idle wireless chargers to UAVs and a swarm of UAVs in the task area compete to get recharging services. Considering the mobility dynamics and energy competitions, we formulate an energy scheduling problem for UAVs and vehicles under practical constraints. A fair online auction-based solution with low complexity is also devised to allocate and price idle wireless chargers on vehicular proofs in real time. We rigorously prove that the proposed scheme is strategy-proof, envy-free, and produces stable allocation outcomes. The first property enforces that truthful bidding is the dominant strategy for participants, the second ensures that no user is better off by exchanging his allocation with another user when the auction ends, while the third guarantees the matching stability between UAVs and UGVs. Extensive simulations validate that the proposed scheme outperforms benchmarks in terms of energy allocation efficiency and UAV's utility.
\keywords{UAV recharging; WPT; air-ground collaboration; dynamic energy scheduling; envy-freeness}
\end{abstract}
\section{Introduction}
\label{Introduction}
The emerging unmanned aerial vehicles (UAVs) have gained significant success in various applications such as crop surveys, search and rescue, and infrastructure inspection \cite{9849496,9732222,9456851,wang2023survey}.
Thanks to their low cost, flexible deployment, and controllable maneuverability, UAVs mounted with rich onboard sensors can be fast dispatched to enable autonomous and on-demand mission execution (e.g., sensing and communication recovery) anytime and anywhere \cite{9453820,10106022,9152148}.
However, commercial UAVs such as quadrotors generally have stringent space and weight limitations, causing inherent constrained battery endurance to support long-duration missions. For example, most mini-UAVs (powered by lithium-ion or lithium polymer batteries) only afford up to 90 minutes of endurance \cite{9743346}. Besides, in executing complex and persistent tasks, relevant compute-intensive operations with video streaming and image processing may consume a considerable amount of UAV's battery energy \cite{10106022,9696188}.
Notably, increasing UAV's battery capacity beyond a certain point can degrade its flight time due to excessive weight \cite{9420719}.
Hence, it is crucial to design effective battery recharging approaches to sustain the life cycle of a UAV flight.
A number of research efforts have been made to address the UAV's battery recharging issue, which can be mainly divided into three types: \emph{energy harvesting} \cite{9060991} from the environment (e.g., solar and wind energy), \emph{battery hotswapping} \cite{6701199} at battery swap stations, and \emph{wireless charging} \cite{9488324} using wireless chargers. In energy harvesting, the energy output of outfitted photovoltaic (PV) arrays or turbine generators can be intermittent and uncertain and highly rely on weather conditions. Besides, the additionally added size and weight on the UAV may raise difficulty in safely landing at dedicated locations.
In battery hotswapping, it generally involves human labors to replace UAV's depleted battery with a fully charged one, affecting autonomous UAV operations in inaccessible or hazardous places \cite{9420719}.
Moreover, it can incur high round-trip energy costs for frequent battery replacement operations.
With the recent breakthrough in wireless power transfer (WPT) techniques, UAVs can be conveniently charged by distributed wireless chargers in a fully automatic manner \cite{9420719,9488324,8846189,9209109}.
As reported, the commercial WPT product of Powermat company can transfer 600\,W of wireless power over a distance of up to 150\,mm to small or medium UAVs with over $90\%$ energy efficiency and high misalignment tolerance \cite{WPTDronePowermat}.
However, deploying and maintaining such static wireless chargers at large-scale task areas (e.g., survivor rescue in disaster sites) can be costly and time-consuming, especially in environmentally harsh terrains.
Besides, the solutions built on static wireless chargers usually lack feasibility and on-demand energy supply capabilities for UAVs, as well as restricting UAVs' operations within specific geographical areas.
\begin{figure}
\caption{An example scenario of vehicle-mounted mobile wireless chargers for on-demand UAV recharging in the task area.}
\label{fig:intro1}
\end{figure}
In this paper, as shown in Fig.~\ref{fig:intro1}, we focus on an on-demand and cost-effective solution by leveraging unmanned ground vehicles (UGVs) with controllable mobility, where UGVs equipped with wireless charging facilities are deployed in task areas and collaboratively offer sufficient wireless energy supply to prolong the lifetime of the UAV network.
In academia and industry, research works \cite{7535886,9058225,9462603} and companies such as Renault \cite{Renault2014} and DSraider \cite{DSraider2021} have developed such mobile and collaborative platforms for efficient UAV launching, recycling, and recharging using a special compartment in the UGV's roof.
Despite the fundamental contributions on system and protocol design of existing literature \cite{7535886,9058225,9462603,Renault2014,DSraider2021}, the double-side energy scheduling along with user fairness in UG\underline{V}-assisted \underline{w}ireless \underline{r}echargeable \underline{U}AV \underline{n}etworks (VWRUNs) are rarely studied, which motivates our work.
On one hand, compared with fixed chargers, the size of charging (also landing) pads on roofs of UGVs are comparably smaller \cite{8660495}, thereby restricting the number of concurrent charging UAVs. Moreover, in complex missions (e.g., large-scale surveillance), it usually depends on the coordination among multiple UAVs due to the limited capacity (e.g., sensing range) of a single UAV \cite{6701199,9488324}.
Consequently, in highly dynamic VWRUNs, efficient real-time charging scheduling among multiple UAVs and multiple UGVs is of necessity to motivate their energy cooperation.
On the other hand, as UAVs and UGVs are self-interested agents and mutually distrustful, they may behave strategically to maximize their gains and even perform market manipulation by diminishing the legitimate interests of others \cite{8902165}. For example, strategic UGVs may collude to overclaim their energy costs for higher payments from UAVs. Besides, the envy-freeness (i.e., no agent envies the allocation of another agent) \cite{9496271}, as an essential metric to ensure market fairness, is neglected in most of existing works. The violation of envy-freeness may raise low user willingness and acceptance, eventually lowering energy allocation efficiency.
Therefore, it remains an open and vital issue to design a real-time and envy-free charging strategy among UAVs and UGVs while preventing strategic behaviors and motivating their dynamic cooperation in VWRUNs.
To address the above issues, we adopt a market-based approach by formulating the double-side charging scheduling problem among multiple UAVs and UGVs in VWRUNs as an online sealed-bid auction, where both UAVs (i.e., buyers) and UGVs (i.e., sellers) are allowed to send their sealed bidding information (including trading time, valuation, supply/demand energy volume, etc.) anytime to the auctioneer (i.e., the ground station).
The auctioneer collects the bids within a maximum waiting time and publishes the auction outcome when the auction ends. UAVs that fail to match a desired UGV can participate the next-round auction or alternatively fly to a nearby static energy swap/charging station to replenish energy.
Besides, our proposed scheme allows UAVs and UGVs to dynamically join and exit the auction process.
Using rigorous theoretical analysis, we prove that the proposed battery recharging auction is \emph{strategy-proof} (i.e., able to resist strategic agents) and produces \emph{envy-free} allocations (i.e., ensuring market fairness).
The main contributions of this paper are summarized as below.
\begin{itemize}
\item We propose an on-demand and collaborative UAV recharging framework by employing idle wireless chargers mounted on mobile UGVs to replenish energy for multiple UAVs in executing long-term tasks. An optimization problem for energy scheduling is formulated in VWRUNs based on the UGV type model and UAV's state-of-charge (SoC) model under practical constraints.
\item We devise an online auction-based approach to solve the real-time energy scheduling problem with low complexity, consisting of the winner determination phase and pricing phase. We theoretically analyze the equilibrium strategy of participants and rigorously prove its strategy-proofness, envy-freeness, and stability.
\item We carry out extensive simulations to evaluate the feasibility and effectiveness of the proposed scheme. Numerical results demonstrate the superiority of the proposed scheme in terms of energy allocation efficiency, UAV's utility, and social surplus, in comparison with conventional schemes.
\end{itemize}
The remainder of this paper is organized as follows. Section \ref{sec:RELATEDWORK} surveys the related literature, and Section \ref{sec:SYSTEMMODEL} introduces the system model. The detailed design of the proposed scheme is presented in Section \ref{sec:FRAMEWORK}, and its performance is evaluated in Section \ref{sec:SIMULATION} using simulations. Finally, this paper is concluded in Section \ref{sec:CONSLUSION}.
\section{Related Works}\label{sec:RELATEDWORK}
In this section, we review related literature on static/mobile wireless charging solutions and charging scheduling approaches in UAV networks.
\subsection{Static and Mobile Wireless Charging Solutions for UAVs}\label{subsec:relatedwork1}
In modern UAV applications, limited battery capacity poses a significant operational challenge to UAVs' flight durations, especially in executing large-scale persistent missions.
Compared with contact-based conductive charging techniques, the promising WPT technology offers a contact-free and fully automatic wireless charging solution for UAVs while withstanding challenging weather conditions \cite{8846189,9488324,9420719}.
Based on the transmission range, current WPT techniques for UAVs can be categorized into two types \cite{9420719}: \emph{near-field} and \emph{far-field}. The former mainly includes magnetic resonance coupling (MRC) \cite{8846189} and capacitive coupling \cite{8267698}, while the latter usually refers to as non-directive radio frequency (RF) radiation such as laser charging \cite{8995773} and WISP-reader charging \cite{9488324}.
Existing WPT-based UAV charging approaches are mainly built on static wireless charging pads located at building rooftops, power poles, cell towers, etc. For large-scale UAV missions, it highly relies on and bears the costly deployment/maintenance fee of additional wireless charging infrastructures.
In the literature, few works have attempted to design mobile wireless chargers to build a feasible and on-demand solution to sustain large-scale persistent UAV operations. Wu \emph{et al}. \cite{9058225} implemented a collaborative UAV-UGV recharging system, where the UGV equips with an object tracking camera (to automatically maneuver towards the UAV) and a Qi charger on the landing pad (to wirelessly transfer energy to the UAV after landing).
To address the misalignment issues between transmit and receiving coils in WPT-based UAV recharging systems, Rong \emph{et al}. \cite{9349168} designed an optimized coupling mechanism for UAVs with high misalignment tolerance based on the genetic algorithm. A real implementation shows that the design UAV recharging system can transfer a maximum power of 100W with the WPT efficiency of 92.41\%.
Ribeiro \emph{et al}. \cite{9462603} investigated the route planning problem for multiple mobile charging platforms (which can travel to different locations) to support long-duration UAV operations. In \cite{9462603}, the routing problem is formulated as a mixed-integer linear programming (MILP) model and solved using a genetic algorithm together with a construct-and-adjust heuristic method.
{There have been several recent studies leveraging WPT for wireless services and UAV services. Wu \emph{et al}. \cite{9718086} proposed a non-orthogonal multiple access (NOMA)-aided federated learning (FL) framework with WPT, where the base station uses WPT to recharge end devices that perform local training and data transmission in FL. A layered algorithm was also designed in \cite{9718086} to minimize FL convergence latency and overall energy consumption under practical constraints.
Shen \emph{et al}. \cite{9440683} studied a UAV-aided flexible radio resource slicing mechanism in 5G uplink radio access networks (RANs), where the joint 3D placement of UAVs and UAV-device association problem was formulated via an interference-aware graph model. In addition, a lightweight approximation algorithm and an upgraded clique method were devised in \cite{9440683} for reduced complexity.}
{However, existing works mainly focus on the system design and trajectory planning of VWRUNs, whereas the double-side energy scheduling among UAVs and UGVs along with the user fairness in the energy charging market are rarely studied.}
\subsection{WPT Charging Scheduling Methods for UAVs}\label{subsec:relatedwork2}
In the literature, there has been an increasing interest in designing {WPT-based} charging scheduling methods for UAVs.
Zhao \emph{et al}. \cite{8995773} proposed a power optimization method in a static laser charging system for a rotary-wing UAV. A non-convex optimization problem is formulated with coupling variables and practical mobility, transmission, and energy constraints, and two algorithms are devised to search the optimal strategy with guaranteed convergence to stationary solutions.
Li \emph{et al}. \cite{9488324} designed an energy-efficient charging time scheduling algorithm to turn on static wireless chargers (SWCs) in scheduled time periods with the aim to minimize SWCs' energy waste in wirelessly charging UAVs. In their scheme, UAVs' continuous flight trajectories are discretized in both temporal and spatial dimensions, and a pruning-based exhaustive method is devised for near-optimal solution searching.
Yu \emph{et al}. \cite{8460819} studied the problem of route planning for a mini-UAV to visit multiple sensing sites within its battery lifetime, where UGVs acting as mobile recharging stations can offer battery recharging services for the UAV along its tour.
Shin \emph{et al}. \cite{8660495} presented a second-price auction-based charging time scheduling method between multiple UAVs and a ground vehicle, where the ground vehicle serves as a mobile charging station and the auctioneer. In their auction, multiple UAVs bid for the vehicle's charging time slots, and the UAV with the highest valuation is assigned as the winner.
{One can observe that existing works on UAV charging mainly focus on the one-to-one charging pattern \cite{8995773} or many-to-one charging pattern \cite{9488324,8660495}, which is not applicable in our considered scenario with multiple UAVs and multiple UGVs. Besides, due to the high dynamics of VWRUNs and potential strategic entities during charging scheduling, real-time and fair energy scheduling should be enforced, which is rarely studied.} Distinguished from existing works, we design an envy-free online auction mechanism for efficient many-to-many charging scheduling among multiple UAVs and UGVs with consideration of their mobility dynamics, double-side competition, and practical constraints.
\section{System Model}\label{sec:SYSTEMMODEL}
In this section, we first elaborate on the system model including the network model (in Sect.~\ref{subsec:networkmodel}) and UAV energy consumption and wireless charging model (in Sect.~\ref{subsec:WPTmodel}). {The notations used in this paper is summarized in Table~\ref{table0}.}
\begin{table}[!t]
{
\caption{{Summary of Notations}}\label{table0}\centering
\resizebox{1.02\linewidth}{!}{
\begin{tabular}{|c|l|}
\hline
\textbf{Notation} & \textbf{Description} \\ \hline
$\mathcal{I}$&Set of UAVs to be recharged. \\
$\mathcal{J}$&Set of UGVs with WPT facilities. \\
$\Lambda$&The GS that coordinates UAVs and UGVs. \\
${\mathcal{I}}'$&Set of low-battery UAVs with $s_i[t] \leq s_{\min}$. \\
${\mathcal{J}}'$&Set of UGVs with idle WPT facilities in time window $\tau$. \\
${\mathcal{W}}$&Winner set of UAVs in the auction. \\
$\mathbb{A}$&UAV recharging auction. \\
$T$&Finite time horizon containing $N$ time slots. \\
$\Delta_t$&Duration of each time slot. \\
$\mathbf{l}_i[t]$&Instant 3D location of UAV $i$ at $t$-th time slot. \\
$R_i$&Radius of sensing spot of UAV $i$. \\
$\theta_i$&Maximum detection angle of UAV $i$'s sensor. \\
$z_{\max}$&Maximum flight altitude of UAV $i$. \\
$v_i$&Flying velocity of UAV. \\
${v_{\max}}$&Maximum velocity of UAV. \\
$s_i[t]$&SoC of UAV $i$'s battery at $t$-th time slot. \\
$C_i$&Battery capacity of UAV $i$. \\
$s_{\min}$&Minimum reserved battery energy. \\
$s_i^{\mathrm{sat}}$&Satisfactory SoC level of UAV $i$. \\
$\rho_i[t]$&Charging urgency of UAV $i$. \\
$C_j$&Total wireless energy supply of UGV $j$. \\
$s_j[t]$&Remaining wireless energy supply of UGV $j$. \\
$q_j[t]$&QoRS of UGV $j$ at $t$-th time slot. \\
$P_i^{\mathrm{fly}}$&Flying power of UAV $i$. \\
$P_i^{\mathrm{hov}}$&Hovering power of UAV $i$. \\
$P_i^{\mathrm{d}}$&Power of UAV $i$ in the descending process. \\
$P_i^{\mathrm{a}}$&Power of UAV $i$ in the ascending process. \\
$P_j^{\mathrm{e}}$&Wireless power transferred by the UGV $j$. \\
$\alpha _i^u$&State variable of UAV $i$. \\
$\eta_j$&Wireless power efficiency of UGV $j$. \\
${\boldsymbol{{b}}}$&Bid profile of all UAVs. \\
$\Phi_i[t]$&Valuation or reserve price of UAV $i$. \\
$\mathcal{U}_i$&Utility function of UAV $i$. \\
$\beta_{i,j}$&Binary allocation outcome. \\
$\bar{\Phi}_i$&Average valuation of UAV $i$ during time window $\tau$. \\
$p_j$&Payment to UGV $j$. \\
$\mathcal{U}_j$&Utility function of UGV $j$. \\
$\mathcal{S}$&Social surplus of involved entities. \\
$g(i)$&Identity of allocated UGV to UAV $i$ in the auction. \\
$\tau$&Time window of the auction. \\
\hline
\end{tabular} } }
\end{table}
\subsection{Network Model}\label{subsec:networkmodel}
As depicted in Fig.~\ref{fig:intro1}, we consider a typical scenario of VWRUN in a given investigated area, which mainly consists of a swarm of $I$ UAVs, a fleet of $J$ UGVs, and a ground station (GS).
\emph{\textbf{UAVs}.} Due to the limited sensing coverage and energy supply of a single UAV, a swarm of UAVs, denoted by the set $\mathcal{I} = \{1,\cdots, i,\cdots,I\}$, are dispatched to collaboratively execute a common mission (e.g., air quality monitoring and geographic surveying) in the given task area \cite{9035635}. The sensing spot of UAV $i \in \mathcal{I}$ is denoted as a circle with center $\left(x_i,y_i,0\right)$ and radius $R_i$. UAVs can communicate with each other using air-to-air (A2A) communications \cite{9599638}.
Let $T$ denote the finite time horizon, which is evenly divided into $N$ time slots with duration $\Delta_t$ \cite{10106022}, i.e., $T = N\cdot \Delta_t$. The instant 3D location of UAV $i \in \mathcal{I}$ at $t$-th time slot ($1\le t \le T$) is denoted by $\mathbf{l}_i[t] = (x_i[t],y_i[t],z_i[t])$, where $(x_i[t],y_i[t])$ is its instant horizontal coordinate.
The instant altitude $z_i[t]$ of UAV $i$ satisfies
\begin{equation}\label{eq3-1}
R_i \cot(\theta_i) \le z_{i}[t] \le z_{\max},
\end{equation}
where $\theta_i$ is the maximum detection angle of UAV $i$'s sensor and $z_{\max}$ is its maximum flight altitude. Besides, $\left||\mathbf{l}_i[t+1] - \mathbf{l}_i[t]|\right| \leq {v_{\max}} \Delta_t$, $\forall 1\le t \le T-1$, where ${v_{\max}}$ is the maximum velocity of UAV.
The state-of-charge (SoC) of UAV $i$'s on-board battery at $t$-th time slot is $s_i[t]$, which satisfies $s_{\min}\leq s_i[t] \leq C_i$. Here, $C_i$ is the battery capacity of UAV~$i$ and $s_{\min}$ is the minimum reserved battery energy to prolong the battery lifetime \cite{9632356}. For UAV $i$, when its remaining battery SoC $s_i[t]$ is below the alert level $s_{\min}$, it leaves its sensing spot for recharging and another UAV can cooperatively replace this low-battery UAV $i$ at the target sensing spot to offer uninterrupted sensing service \cite{8648453}. The charging urgency of each UAV $i$ is computed as
\begin{equation}\label{eq3-1-2}
\rho_i[t] = 1 - \frac{s_i[t] - s_{\min}}{C_i}, \,\mathrm{and}~ \rho_i[t] \in [0,1].
\end{equation}
\emph{\textbf{UGVs}.} A fleet of UGVs equipped with wireless charging facilities on the vehicular roofs are deployed in the investigated area to collaboratively offer on-demand wireless energy supply for low-battery UAVs \cite{Renault2014,DSraider2021}. The set of UGVs is denoted as $\mathcal{J} = \{1,\cdots, j,\cdots,J\}$. UGVs are smart vehicles integrated with various advanced sensors to allow self-driving to the rendezvous and perform automatically UAV tracking, launching, and recharging operations on the charging pad on vehicular roofs. Let $C_j$ and $s_j[t]$ be the total/remaining wireless energy supply of UAV recharging {of UGV $j$}, respectively. It is assumed that $s_j[t] \geq \max\{s_i^{\mathrm{sat}} - s_i[t], \forall i \in \mathcal{N}\}$.
Besides, UGVs generally have diverse quality of recharging service (QoRS), which is affected by various factors such as the wireless charging rate and the driving distance to the task area. Let $q_j[t]$ denote the QoRS of UGV $j$ at $t$-th time slot. A higher QoRS indicates the higher charging preference of UAVs.
\emph{\textbf{GS}.} In VWRUN, the aerial UAV subnetwork and the ground vehicular subnetwork are coordinated by the GS (denoted by $\Lambda$) \cite{7317860}. The GS is located at a micro base station and can perform flight planning, flying control, and task assignment for UAVs via ground-to-air (G2A) links. Moreover, after receiving recharging requests from UAVs, the GS can schedule the UGVs with idle wireless chargers in its communication range via infrastructure-to-vehicle (I2V) links \cite{9631953} to offer on-demand recharging services.
\subsection{UAV Energy Consumption and Wireless Charging Model}\label{subsec:WPTmodel}
To avoid collisions and save energy in the flight, UAVs need to horizontally fly over the task area and hover above the assigned task spot to perform sensing missions \cite{10155496}. In energy recharging process, for simplicity, each UAV $i$ vertically descends to the target UGV's proof and vertically ascends to a preset altitude after reaching the satisfactory SoC level $s_i^{\mathrm{sat}}$ \cite{8758340}. According to \cite{7991310}, the required flying power at a constant speed $v_i$ can be approximately attained as:
\begin{equation}\label{eq3-2}
P_i^{\mathrm{fly}} (v_i) = \kappa_1 v_i^3 + (\kappa_2 + \kappa_3) \Psi^{3/2},
\end{equation}
where $\kappa_1, \kappa_2, \kappa_3$ are constant UAV-related factors, $\Psi$ is the thrust of UAV \cite{8758340}. The hovering power of UAV~$i$ is $P_i^{\mathrm{hov}} = (\kappa_2 + \kappa_3)(m g)^{3/2}$, where $m$ is UAV's mass and $g=9.8\,\mathrm{m/s^2}$. Given the fixed descending speed $v_i^{\mathrm{d}}$ and ascending velocity $v_i^{\mathrm{a}}$, the required power of UAV in the descending process and ascending process can be separately expressed as \cite{8758340,7991310}:
\begin{equation}\label{eq3-3}
P_i^{\mathrm{d}} (v_i^{\mathrm{d}}) \!=\! \epsilon_1 m g \left[\sqrt{\frac{(v_i^{\mathrm{d}})^2}{4} \!+\! \frac{m g}{(\epsilon_2)^2}} - \frac{v_i^{\mathrm{d}}}{2} \right] + \kappa_3(m g)^{3/2}\!,\!
\end{equation}
\begin{equation}\label{eq3-4}
P_i^{\mathrm{a}} (v_i^{\mathrm{a}}) \!=\! \epsilon_1 m g \left[\sqrt{\frac{(v_i^{\mathrm{a}})^2}{4} \!+\! \frac{m g}{(\epsilon_2)^2}} + \frac{v_i^{\mathrm{a}}}{2} \right] + \kappa_3(m g)^{3/2}\!,\!
\end{equation}
where $\epsilon_1, \epsilon_2$ are constant UAV-related factors.
Let $P_j^{\mathrm{e}}$ denote the wireless power transferred by the UGV $j$. Then, the battery dynamics of UAV $i$ can be described as a linear model, i.e.,
\begin{equation}\label{eq3-5}
s_i[t+1] \!=\! s_i[t] +
\left[ \begin{matrix}{}
\alpha _i^1\!&\! \alpha _i^2\!&\! \alpha _i^3\!&\! \alpha _i^4\!&\! \alpha _i^5\\
\end{matrix} \right] \!\times\! \left[ \begin{array}{c}
- \eta_i P_i^{\mathrm{fly}}\\
- \eta_i P_i^{\mathrm{hov}}\\
- \eta_i P_i^{\mathrm{d}}\\
- \eta_i P_i^{\mathrm{d}}\\
\eta_i \eta_j P_j^{\mathrm{e}}\\
\end{array} \right]
\!,\!
\end{equation}
where $\alpha _i^u =\{0,1\},u=\{1,\cdots,5\}$ are binary variables, denoting the state of UAV $i$. $\eta_j$ is the wireless power efficiency of UGV $j$. Here, $\alpha _i^u=1$ means that UAV $i$ is in the corresponding state (i.e., horizontally flying, hovering, vertically descending, vertically ascending, or wireless charging); otherwise, $\alpha _i^u=0$.
\begin{figure}
\caption{Illustration of online auction-based charging scheduling process among UAVs and UGVs (\ding{172}
\label{fig:Auction}
\end{figure}
\section{The Proposed Scheme}\label{sec:FRAMEWORK}
This section first formulates the online charging scheduling and pricing problem for UAVs and UGVs in VWRUNs (in Sect.~\ref{subsec:scheme1}). Then, an auction-based solution with strategy-proofness and envy-freeness is designed, followed by the theoretical analysis of its properties (in Sect.~\ref{subsec:scheme2}).
\subsection{Online Charging Scheduling and Pricing (OCSP) Problem}\label{subsec:scheme1}
As shown in Fig.~\ref{fig:Auction}, the auction-based UAV charging scheduling process is carried out by the GS in an \emph{online} manner, where UAVs are allowed to bid at anytime and the bid collection phase finishes until a maximum waiting time $\tau$ elapses.
Let ${\boldsymbol{{b}}} = \left(b_1,\cdots,b_i,\cdots,b_{I(\tau)} \right)$ denote the bid profile of all UAVs in ${\mathcal{I}}(\tau) \subseteq \mathcal{I}$. Here, ${\mathcal{I}}' = {\mathcal{I}}(\tau)$ is the set of low-battery UAVs with $s_i[t] \leq s_{\min}$, $\forall i \in \mathcal{N}, \forall t \in \tau$. ${\boldsymbol{{b}}_{-i}}$ is the bid profile of other UAVs except UAV $i$.
In the auction, the valuation (i.e., reserve price) of UAV $i$ in a recharging service is associated to its energy state and charging urgency, i.e., $\Phi_i[t] = \Phi_i(\rho_i[t])$. Generally, the higher charging urgency, the larger valuation. Besides, the higher charging urgency, the larger marginal valuation. Hence, $\frac{\mathrm{d} \Phi_i(\rho_i[t])}{\mathrm{d} \rho_i[t]} \!>\! 0$, $\frac{\mathrm{d}^2 \Phi_i(\rho_i[t])}{\mathrm{d} \rho_i[t]^2} \!\geq\! 0$. In the following, we define utility functions of UAVs and UGVs, as well as the social surplus.
\begin{definition}[UAV Utility]\label{definition1}
The utility function of UAV $i \in \mathcal{N}$ is the revenue minuses its payment:
\begin{equation}\label{eq4-1}
\mathcal{U}_i = \left\{\begin{array}{cl}
\sum\nolimits_{j\in \mathcal{J}}{\beta_{i,j} \left[q_j \bar{\Phi}_i - p_j( {\boldsymbol{{b}}})\right]}, & i \in {\mathcal{I}}',\\
0, & i \!\in\! {\mathcal{I}}\backslash{\mathcal{I}}'.
\end{array}\right.
\end{equation}
\end{definition}
\emph{Remark.} In Eq. (\ref{eq4-1}), the binary variable $\beta_{i,j}=\{0,1\}$ indicates the allocation outcome, where $\beta_{i,j}=1$ if UAV $i$ is allocated to get charged at UGV $j$. Otherwise, $\beta_{i,j}=0$. $q_j$ is the QoRS of UGV $j$, which is assumed to remain unchanged during the time window $\tau$. $\bar{\Phi}_i$ is the average valuation of UAV $i$ during the time window $\tau$, which is computed as $\bar{\Phi}_i = \lfloor \frac{\tau}{t} \rfloor^{-1} \cdot \sum_{t\in \tau}{\Phi}_i[t]$. $p_j({\boldsymbol{{b}}})$ is the payment to UGV $j$.
\begin{definition}[UGV Utility]\label{definition2}
The utility function of UGV $j \in \mathcal{J}$ is associated with its payment, i.e.,
\begin{equation}\label{eq4-2}
\mathcal{U}_j = \left\{\begin{array}{cl}
\sum\nolimits_{i\in \mathcal{I}'} {\beta_{i,j} p_j({\boldsymbol{{b}}})}, & j \in {\mathcal{J}}',\\
0, & j \in {\mathcal{J}}\backslash{\mathcal{J}}'.
\end{array}\right.
\end{equation}
\end{definition}
\emph{Remark.} In Eq. (\ref{eq4-2}), ${\mathcal{J}}'={\mathcal{J}}(\tau)$ denotes the set of UGVs with idle WPT facilities during time window $\tau$, where ${\mathcal{J}}(\tau)\subseteq \mathcal{J}$.
\begin{definition}[Social Surplus]\label{definition3}
The social surplus is defined as the overall utility of involved entities \cite{8902165}, i.e.,
\begin{equation}\label{eq4-3}
\mathcal{S} = \sum\limits_{i\in \mathcal{I}} {\mathcal{U}_i} + \sum\limits_{j\in \mathcal{J}} {\mathcal{U}_j} = \sum\limits_{i\in \mathcal{I}'}{\sum\limits_{j\in \mathcal{J}'} {{\beta_{i,j} q_j \bar{\Phi}_i }}}.
\end{equation}
\end{definition}
Besides, the UAV recharging auction should be strategy-proof and envy-free to prevent strategic entities and ensure market fairness, whose formal definitions are given as below.
\begin{definition}[Strategy Proofness]\label{definition4}
The UAV recharging auction $\mathbb{A}$ satisfies strategy proofness if the following two properties hold \cite{10155496}:
\begin{itemize}
\item (i) Individual rationality (IR): both UAVs and UGVs acquire non-negative utilities in the auction, i.e., $\mathcal{U}_i\geq 0, \forall i \in \mathcal{I}'$ and $\mathcal{U}_j\geq0, \forall j \in \mathcal{J}$.
\item (ii) Individual compatibility (IC): each UAV can obtain its maximum utility when truthfully choosing its bid strategy, i.e., $\mathcal{U}_i(\bar{\Phi}_i,\boldsymbol{b}_{-i}) \geq \mathcal{U}_i(b_i',\boldsymbol{b}_{-i}), \forall b_i' \neq \bar{\Phi}_i, i \in\mathcal{I}'$.
\end{itemize}
\end{definition}
\begin{definition}[Envy Freeness]\label{definition5}
The UAV recharging auction $\mathbb{A}$ is energy-free if no UAV is happier to exchange its allocation with another UAV to improve its utility when the auction ends \cite{9496271}, i.e.,
\begin{equation}\label{eq4-4}
\mathcal{U}_i(g(i),p_{g(i)}) \geq \mathcal{U}_i (g(k),p_{g(k)}), k\neq i, \forall i,k\in \mathcal{I}',
\end{equation}
where $g(i)$ is the identity of allocated UGV to UAV $i$ and $p_{g(i)}$ is the corresponding payment of UAV $i\in \mathcal{I}'$.
\end{definition}
In VWRUNs, the online charging scheduling and pricing (OCSP) problem is to maximize the social surplus while meeting practical constraints, i.e.,
\begin{equation}\label{eq4-5}
\mathbf{P}1:
\mathop {\max }\limits_{{\boldsymbol{{\beta}}},\,\boldsymbol{p}} \sum\limits_{i\in \mathcal{I}'}{\sum\limits_{j\in \mathcal{J}'} {{\beta_{i,j} q_j \bar{\Phi}_i }}},
\end{equation}
\begin{numcases}{{\rm{s.t.}}}
\sum\nolimits_{i \in {\mathcal{I}'}} {{\beta_{i,j}}} \le 1 \label{eq:cons1}
\\
\sum\nolimits_{j \in \mathcal{J}'} {{\beta_{i,j}}} \le 1 \label{eq:cons2}
\\
{\beta_{i,j}} = \left\{0,1\right\},{p_j({\boldsymbol{{b}}})} \ge 0, \forall i \!\in\! {\mathcal{I}'},\forall j \!\in\! \mathcal{J}' \label{eq:cons3}
\\
\sum\nolimits_{u=1}^{5} \alpha_i^u = 1, \forall i \in \mathcal{I}' \label{eq:cons4}
\\
\mathcal{U}_i\geq 0, \mathcal{U}_j\geq 0, \forall i \in \mathcal{I}' ,\forall j \in \mathcal{J}' \label{eq:cons5}
\\
\mathcal{U}_i(\bar{\Phi}_i,\boldsymbol{b}_{-i}) \!\geq\! \mathcal{U}_i(b_i',\boldsymbol{b}_{-i}), \forall b_i' \!\neq\! \bar{\Phi}_i, i \!\in\! \mathcal{I}' \label{eq:cons6}
\\
\mathcal{U}_i(g(i),p_{g(i)}) \!\geq\! \mathcal{U}_i (g(k),p_{g(k)}), \forall k \neq i. \label{eq:cons7}
\end{numcases}
\emph{Remark.} In $\mathbf{P}1$, ${\boldsymbol{{\beta}}} = [{\beta_{i,j}}]_{I(\tau) \times J(\tau)}$ and ${\boldsymbol{p}}=(p_1,\cdots,p_{J(\tau)})$ are decision variables. Constraint (\ref{eq:cons1}) means that a UGV can only offer charging service for at most one UAV during $\tau$. Constraint (\ref{eq:cons2}) implies that a UAV can only recharge at most one UGV during $\tau$. Constraint (\ref{eq:cons4}) is the state constraint of UAV $i$. Constraint (\ref{eq:cons5}) is the IR constraint, and constraint (\ref{eq:cons5}) is the IC constraint.
Both constraints (\ref{eq:cons5})--(\ref{eq:cons6}) refer to the strategy proofness, and constraint (\ref{eq:cons7}) corresponds to the envy freeness.
\begin{theorem}\label{theorem0}
The OCSP problem $\mathbf{P}1$ is NP-hard.
\end{theorem}
\begin{proof}
Based on \cite{7293173}, the relaxed version of problem $\mathbf{P}1$ with constraints (\ref{eq:cons1})--(\ref{eq:cons4}) and the constant payment ${\boldsymbol{p}}$ is a typical set cover problem and is proved to be NP-hard. As such, the raw OCSP problem $\mathbf{P}1$ is NP-hard. Theorem~\ref{theorem0} is proved.
\end{proof}
\subsection{Strategy-Proof and Envy-Free Auction Mechanism}\label{subsec:scheme2}
As the OCSP problem $\mathbf{P}1$ is NP-hard, in this subsection, we devise a practical heuristic auction mechanism to derive its near-optimal solution with polynomial complexity, while satisfying strategy proofness and envy freeness.
The key phases in our proposed online auction mechanism is presented {in Algorithm~\ref{Algorithm1}.}
\begin{algorithm}[t!]\begin{footnotesize}
\caption{\small{{Online Strategy-Proof and Envy-Free Auction Algorithm for UAVs and UGVs in VWRUNs}}}\label{Algorithm1}
{\begin{algorithmic}[1]
\STATE \emph{\textbf{{Phase 1}} (Type Evaluation for UAVs and UGVs):}
\STATE Initialize the time window $\tau$, UAV type (i.e., charging urgency) $\rho_i[t]$, UGV type (i.e., QoRS) $q_j[t]$, the set of UAVs with charging desires (i.e., $\mathcal{I}'$), and the set of UGVs with idle charging facilities (i.e., $\mathcal{J}'$). Then, UAV's average valuation $\bar{\Phi}_i$, $\forall i \in \mathcal{I}'$ can be calculated.
\STATE Initialize all elements in ${\boldsymbol{{\beta}}}$ and ${\boldsymbol{{p}}}$ with $0$.
\STATE Initialize $\mathcal{W}=\emptyset$, $\mathcal{L}=\mathcal{I}'$, and $\mathcal{D}=\mathcal{J}'$.
\STATE \emph{\textbf{{Phase 2}} (Biding and Allocation):}
\STATE Each UAV $i\in {\mathcal{I}'}$ determines its sealed-bid strategy $b_i$ based on its valuation and sends it to the GS.
\STATE The GS obtains the latest types of UGVs in $\mathcal{J}'$, and re-sorts the UGVs in descending order of their types, i.e., $q_1 \geq q_2 \geq \cdots \geq q_{J(\tau)}>0$. The sorted set of $\mathcal{J}'$ is denoted by ${\mathcal{J}''}$.
\STATE UAVs in set ${\mathcal{I}'}$ are sorted in descending order of their valuations, i.e., $\bar{\Phi}_1 \geq \bar{\Phi}_2 \geq \cdots \geq \bar{\Phi}_{I(\tau)}\geq 0$. The new sorted set is denoted as ${\mathcal{I}''}$.
\STATE The GS sorts the UAVs in ${\mathcal{I}''}$ in descending order of their submitted bids, and allocates the WPT facilities mounted on UGVs in ${\mathcal{J}''}$ to the UAVs starting from the maximum UGV type. The maximum type of UGV is allocated to the UAV with the largest bid (i.e., $g(1)$), the second highest type of UGV to $g(2)$, and so on, down to the type of UGV $K$, where $K = \min\{I(\tau),J(\tau)\}$, $I(\tau) =|{\mathcal{I}}'|$, and $J(\tau) =|{\mathcal{J}}'|$.
\STATE Update $\mathcal{W} = \mathcal{W}\cup \{i|i\leq K, \forall i \in {\mathcal{I}''}\}$, $\mathcal{L} = \mathcal{L}\backslash \{i|i\leq K, \forall i \in {\mathcal{I}''}\}$, $\mathcal{D} = \mathcal{D}\cup \{j|j\leq K, \forall j \in {\mathcal{J}''}\}$.
\STATE \emph{\textbf{{Phase 3}} (Payment Determination):}
\STATE The payment of the last UAV $g(K)\in{\mathcal{W}}$ is
\begin{equation}\label{eq4-6}
p_{g(K)} \!=\! \left\{\begin{array}{cl}
q_{J(\tau)}b_{g(J(\tau)+1)}, & J(\tau) \!<\! I(\tau),\\
0, & otherwise.
\end{array}\right.
\end{equation}
\STATE The payment of UAV $g(j)\in{\mathcal{W}}$ with $j<\min\{I(\tau),J(\tau)\}$ that wins to charge at UGV $j\in{\mathcal{J}''}$ is determined based on the negative externality that it imposes on others, i.e.,
\begin{align}\label{eq4-7}
&p_{g(j)}({\boldsymbol{{b}}})= \mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''} - \mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''\backslash\{j\}} \nonumber \\
&=\Big[ \sum\nolimits_{k=1}^{j-1}{q_k b_{g(k)}} + \sum\nolimits_{k=j+1}^{K}{q_{k-1} b_{g(k)}} \Big]\nonumber \\
& ~~~~- \Big[ \sum\nolimits_{k=1}^{j-1}{q_k b_{g(k)}} + \sum\nolimits_{k=j+1}^{K}{q_{k} b_{g(k)}} \Big]\nonumber \\
&= \sum\nolimits_{k=j}^{K-1}{(q_k - q_{k+1}) b_{g(k+1)}}.
\end{align}
\STATE \emph{\textbf{{Phase 4}} (Wireless Charging of Winning UAVs):}
\STATE Each winner UAV $i=g(j)$ and its allocated UGV $j$ drive to their jointly produced rendezvous for battery recharging, and the winner UAV $i=g(j)$ pays the corresponding payment to the UGV $j$.
\STATE For loser UAVs, they can participate in the next-round auction, together with the new UAVs with charging desires, to compete for battery charging. Alternatively, losing UAVs can also fly to a nearby static swap/charging station to replenish energy.
\end{algorithmic}}\end{footnotesize}
\end{algorithm}
{Specifically, phase 1 (lines 1--4) evaluates the type information of UAVs and UGVs in the time window $\tau$; phase 2 (lines 5--10) determines the bidding strategy of each UAV and the auction winners; phase 3 (lines 11--13) determines the payment of each UAV in the winner set ${\mathcal{W}}$; and phase 4 (lines 14--16) performs wireless charging for each pair of matched UAV and UGV in the winner set and enforces the financial settlement.}
\emph{Remark.} In Eq.~(\ref{eq4-7}), $\mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''}$ means the social surplus when UAV $i$ leaves the auction, and $\mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''\backslash\{j\}}$ is the actual social surplus when UAV $i$ participates in the auction without UGV $j$. Via recursive operations, the payment for UGV $j<\min\{I(\tau),J(\tau)\}$ can be rewritten as:
\begin{equation}
p_{g(j)}({\boldsymbol{{b}}}) = (q_j - q_{j+1})b_{g(k+1)} + p_{g(j+1)}({\boldsymbol{{b}}}).
\end{equation}
In the following, we analyze the desirable properties of the proposed auction mechanism in terms of strategy-proofness, envy-freeness, allocation stability, and computational complexity in the following theorems and corollaries.
\begin{theorem}\label{theorem1}
In the proposed auction mechanism, participants always attain non-negative utilities; and the truth-telling bidding strategy is the dominant equilibria for all participating UAVs, i.e., $\mathbb{A}$ is strategy-proof.
\end{theorem}
\begin{proof}
According to Definition~\ref{definition4}, it suffices to prove that both IR and IC hold in $\mathbb{A}$. We first prove the IR. Obviously, according to Eqs. (\ref{eq4-2})--(\ref{eq4-3}), the utilities of UAVs and UGVs equal to zero if they do not participate in the auction $\mathbb{A}$. For the participating UGVs, as the payments to them are always non-negative, their utilities are no less than zero. For any participating UAV $i=g(j)$, its utility can be reformulated as:
\begin{align}
\mathcal{U}_{g(j)}&={q_j b_{g(j)}} + \mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''\backslash\{j\}} - \mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''} \nonumber \\
&= \sum\nolimits_{l=1}^{|\mathcal{J}''|}{q_l b_{g(l)}} - \mathcal{S}_{{\mathcal{I}''}\backslash\{g(j)\}}^{\mathcal{J}''} \nonumber \\
&= \sum\nolimits_{l=j}^{|\mathcal{J}''|-1}{q_l (b_{g(l)} - b_{g(l+1)})} + {q_{|\mathcal{J}''|} b_{g(|\mathcal{J}''|)}} \nonumber \\
&\geq 0.
\end{align}
Thereby, for all participating UAVs and UGVs, their utilities are always non-negative. Next, we prove the IC. Notably, the payment decision strategy in our proposed auction mechanism follows the Vickrey–Clarke–Groves (VCG) mechanism. As truth-telling is a well-known property of the VCG mechanism \cite{Benjamin2007Internet}, our auction $\mathbb{A}$ also satisfies the truth-telling property (i.e., IC). Theorem~\ref{theorem1} is proved.
\end{proof}
{\emph{Remark.} Theorem~2 shows that our proposed auction algorithm can resist strategic UAVs/UGVs and prevent market manipulation in practical energy recharging services by enforcing IC constraints. Besides, as IR constraints are satisfied, individual UAVs/UGVs can be motivated to join the energy recharging system to gain benefits.
}
\begin{theorem}\label{theorem2}
The proposed auction $\mathbb{A}$ is envy-free, if (i) $b_{g(j)} \in \left[\bar{\Phi}_{g(j)}, \bar{\Phi}_{g(j-1)} \right]$, $1<j\leq K$, and (ii) ${g(j)} = j$, $1\leq j\leq K$ hold.
\end{theorem}
\begin{proof}
According to Definition~\ref{definition5}, it suffices to prove that any UAV ${g(j)}\in \mathcal{W}$ is indifferent between charging at UGV $j$ at price $p_{g(j)}$ and charging at UGV $l$ at price $p_{g(l)}$, where $j\ne l$. Without loss of generality, we consider the following two cases.
\underline{Case 1}: $j>l$. In this case, as $b_{g(j)} \geq \bar{\Phi}_{g(j)}$ and $q_j\leq q_l$, we have $\left( q_j-q_l \right) \bar{\Phi} _{g\left( j \right)} \geq \left( q_j-q_l \right) b _{g\left( j \right)}$ and $\left( q_{j-2}-q_{j-1} \right)b _{g\left( j-1 \right)} \geq \left( q_{j-2}-q_{j-1} \right)b _{g\left( j \right)}$. The utility difference of UAV ${g(j)}$ between charging at UGV $j$ and charging at UGV $l$ is:
\begin{equation}\label{eq-p2-1}
\begin{aligned}
&\Delta\mathcal{U}_{j,l} = q_j\bar{\Phi} _{g\left( j \right)}-p_{g\left( j \right)}-\left[ q_l\bar{\Phi} _{g\left( j \right)}-p_{g\left( l \right)} \right]\\
&=\sum_{k=l}^{j-1}{\left( q_k-q_{k+1} \right) b_{g\left( k+1 \right)}}+\left( q_j-q_l \right) \bar{\Phi} _{g\left( j \right)}\\
&\ge \sum_{k=l}^{j-1}{\left( q_k-q_{k+1} \right) b_{g\left( k+1 \right)}}+\left( q_j-q_l \right) b_{g\left( j \right)}.
\end{aligned}
\end{equation}
If $j = l + 1$, the following inequality holds:
\begin{equation}
\Delta\mathcal{U}_{j,l}\geq \left( q_{j-1}-q_j \right) b_{g( j )} + \left( q_j-q_{j-1} \right) b_{g( j )}\geq 0.
\end{equation}
If $j > l + 1$, via recursive operations, we have
\begin{equation}\label{eq-p2-2}
\begin{aligned}
&\sum\nolimits_{k=l}^{j-1}{\left( q_k-q_{k+1} \right) b_{g\left( k+1 \right)}}+\left( q_j-q_l \right) b_{g\left( j \right)} \\
&\ge \sum_{k=l}^{j-2}{\left( q_k-q_{k+1} \right) b_{g\left( k+1 \right)}}+\left( q_{j-1}-q_l \right) b_{g\left( j \right)}\\
&\ge \cdots \ge \left( q_l-q_l \right) b_{g\left( j \right)}=0.
\end{aligned}
\end{equation}
Hence, it can be concluded that $\Delta\mathcal{U}_{j,l}\geq 0$.
\underline{Case 2}: $j<l$. In this case, as $b_{g(j)} \leq \bar{\Phi}_{g(j-1)}$ and $q_j\geq q_l$, we have $\left( q_j-q_{l} \right) \bar{\Phi}_{g(j)}\geq \left( q_j-q_{l} \right) b_{g\left( j+1 \right)}$ and $\left( q_l-q_{l-1} \right) b_{g\left( l \right)}\geq \left( q_l-q_{l-1} \right) b_{g\left( l-1 \right)}$.
Then, we can obtain:
\begin{equation}\label{eq-p2-3}
\begin{aligned}
&\Delta\mathcal{U}_{j,l} = q_j\bar{\Phi} _{g\left( j \right)}-p_{g\left( j \right)}-\left[ q_l\bar{\Phi} _{g\left( j \right)}-p_{g\left( l \right)} \right]\\
&=\sum_{k=j}^{l-1}{\left( q_{k+1}-q_k \right) b_{g\left( k+1 \right)}}+\left( q_j-q_l \right) \bar{\Phi} _{g\left( j \right)}\\
&\ge \sum_{k=j}^{l-1}{\left( q_{k+1}-q_k \right) b_{g\left( k+1 \right)}}+\left( q_j-q_l \right) b_{g\left( j+1 \right)}.
\end{aligned}
\end{equation}
If $j = l - 1$, the following inequality holds:
\begin{equation}
\Delta\mathcal{U}_{j,l}\geq \left( q_{j+1}-q_j \right) b_{g( j+1 )} + \left( q_j-q_{j+1} \right) b_{g( j+1 )}\geq 0.
\end{equation}
Otherwise, if $j < l + 1$, via recursive operations, we have
\begin{align}
&\sum\nolimits_{k=j}^{l-1}{\left( q_{k+1}-q_k \right) b_{g\left( k+1 \right)}} \nonumber \\
&\geq \sum_{k=j}^{l-2}{\left( q_{k+1}-q_k \right) b_{g\left( k+1 \right)}} + \left( q_{l}-q_{l-1} \right)b_{g\left( l-1 \right)}\nonumber \\
&\geq \sum_{k=j}^{l-3}{\left( q_{k+1}-q_k \right) b_{g\left( k+1 \right)}} + \left( q_{l}-q_{l-2} \right)b_{g\left( l-2 \right)}\nonumber \\
&\geq \cdots \geq \left( q_l-q_j \right) b_{g\left( j+1 \right)}.
\end{align}
Hence, $\Delta\mathcal{U}_{j,l} \geq 0$ holds. Theorem~\ref{theorem2} is proved.
\end{proof}
\begin{corollary}\label{corollary1}
In the proposed auction mechanism, the truth-telling equilibrium is also an envy-free Nash equilibrium (EFNE).
\end{corollary}
\begin{proof}
As the truth-telling equilibrium satisfies $b_{g(j)} = \bar{\Phi}_{g(j)}$ and $b_{g(j)} < \bar{\Phi}_{g(j-1)}$, given ${g(j)} = j$ ($1\leq j\leq K$), then we have $\Delta\mathcal{U}_{j,l}\geq 0$ if $j>l$ and $\Delta\mathcal{U}_{j,l}> 0$ if $j<l$. Thereby, the truth-telling equilibrium is an EFNE. Corollary~\ref{corollary1} is proved.
\end{proof}
\begin{corollary}\label{corollary2}
The outcome of the proposed auction mechanism is a stable assignment.
\end{corollary}
\begin{proof}
It suffices to prove that no UAV can gain an improved profit by aborting the assigned UGV in the auction and re-matching with another UGV for battery recharging. According to the Theorem~\ref{theorem2}, in both cases, we can derive $q_j\bar{\Phi} _{g\left( j \right)}-p_{g\left( j \right)}-\left[ q_l\bar{\Phi} _{g\left( j \right)}-p_{g\left( l \right)} \right] \geq 0$ under given constraints. Hence, the assignment outcome produced by our proposed auction mechanism is stable. Corollary~\ref{corollary2} is proved.
\end{proof}
\begin{theorem}\label{theorem5}
The overall computational complexity of the proposed auction mechanism yields $\mathcal{O}\big(I(\tau)\log(I(\tau)) + J(\tau)\log(J(\tau)) + |W(\tau)|^2\big)$.
\end{theorem}
\begin{proof}
In the phase 1 of the auction mechanism, the computational complexity for type evaluation of UAVs and UGVs is $\mathcal{O}(I(\tau) + J(\tau))$. In the next phase 2, the sorting process of UGVs' types and UAVs' valuations yields a complexity of $\mathcal{O}(I(\tau)\log(I(\tau)) + J(\tau)\log(J(\tau)))$, while the allocation process has a complexity of $\mathcal{O}(K)$, where $K = \min\{I(\tau),J(\tau)\}$. In the last phase 3, the payment determination process for winners has a complexity of $\mathcal{O}(|W(\tau)|^2)$, where $|W(\tau)|$ is the number of winners in an auction with time window $\tau$. Thereby, the overall computational complexity yields $\mathcal{O}(I(\tau)\log(I(\tau)) + J(\tau)\log(J(\tau)) + |W(\tau)|^2)$. Theorem~\ref{theorem5} is proved.
\end{proof}
\section{PERFORMANCE EVALUATION}\label{sec:SIMULATION}
In this section, we first introduce the simulation settings, then we discuss the numerical results.
\subsection{Simulation Setup}\label{subsec:evalution1}
We consider a 3D simulation area of $5000\times5000\times10\, \mathrm{m}^3$, where the sensing spot locates at the center of the area and the sensing task area is a circle with radius 200\,m. UAVs are flying over the sensing task area with constant altitudes in $[5,10]$\,m. UGVs are located outside the sensing task area, and their distances to the sensing spot are uniformly distributed in $[0.3,2.5]$\,km. One base station with location $(3000,3000,50)$ offers wireless communication services for UAVs and UGVs at the considered area. The time window in the auction is set as $\tau=8$ seconds. UAV's battery capacity is set as $C_i = 97.58$\,Wh \cite{8758340}, and the alert battery SoC level is $s_{\min}=20\%$. The current battery SoC of UAVs follows the uniform distribution in $[30\%,100\%]$. The wireless energy efficiency of UGVs is set as $\eta_j = 80\%$. The velocity of UGVs varies between $20$\,km/h and $60$\,km/h. The maximum flying velocity of UAV is set as $v_{\max} = 10$\,m/s. UAV's flying power parameters and channel parameters are set according to \cite{7991310,8758340}. The normalized QoRS is adopted, and UGV's QoRS is computed according to its normalized distance to the sensing spot. The rendezvous for each matched pair of UAV and UGV is generated at the midpoint between the sensing spot and the corresponding UGV. The linear function is adopted for UAV's valuation modelling in recharging, i.e., $\Phi_i[t] = \mu_0 + \mu_1 \rho_i[t]$, where the parameters are set as $\mu_0=1$ and $\mu_1=5$.
We compare the proposed scheme with the following two conventional schemes.
\begin{itemize}
\item \emph{Exhaustive optimal scheme:} it exhaustively searches the optimal allocation outcomes for UAVs and UGVs in the OCSP problem $\mathbf{P}1$ without the envy-free constraint (\ref{eq:cons7}).
\item \emph{Static WPT scheme:} it replaces the UGVs with static WPT facilities in the simulation area for UAV charging services. The auction process between UAVs and static WPT facilities is similar to our proposed auction between UAVs and UGVs.
\end{itemize}
\subsection{Numerical Results}\label{subsec:evalution2}
In Figs.~\ref{fig:simu01}--\ref{fig:simu03}, we evaluate the satisfaction level of UAVs, utility of UAVs, and social surplus in the proposed scheme, compared with the conventional schemes. Then, we evaluate the effect of auction time window in Fig.~\ref{fig:simu03}. After that, in Tables~\ref{simutable1}--\ref{simutable2}, we evaluate the strategy proofness and envy freeness of the proposed scheme. The \emph{satisfaction level of UAVs} is defined as:
\begin{align}
SL \!=\! \sum_{i=1}^{I(\tau)}{\sum_{j=1}^{J(\tau)}{\beta_{i,j}q_j \rho_i(\tau)} },\,\rho_i(\tau) \!=\! \lfloor \frac{\tau}{t} \rfloor^{-1} \cdot \sum_{t\in \tau}{\rho}_i[t].
\end{align}
Besides, the \emph{non-envy ratio} is adopted to measure the envy freeness of the assignment outcomes, which is defined as the number of UAVs that does not envy the other UAV's assignment to the total number of participating UAVs in the auction.
\begin{figure}
\caption{Satisfaction level of UAVs vs. number of UGVs (i.e., $J(\tau)$) in three schemes, where $I(\tau)=10$.}
\label{fig:simu01}
\end{figure}
\begin{figure}
\caption{Utility of UAVs vs. number of UGVs (i.e., $J(\tau)$) in three schemes, where $I(\tau)=10$.}
\label{fig:simu02}
\end{figure}
Fig.~\ref{fig:simu01} and Fig.~\ref{fig:simu02} show the satisfaction level and utility of UAVs in three schemes, respectively, when the number of UGVs in the auction grows from $6$ to $14$. In these two simulations, the number of UAVs in the auction is set as $10$.
As seen in these two figures, the proposed scheme outperforms the static WPT scheme in attaining a smaller gap with the exhaustive optimal approach. It can be explained as follows.
Compared with the static WPT facilities, our proposed scheme utilizes mobile UGVs to dynamically generate the rendezvouses for UAV launching and battery recharging, thereby saving the flying cost for UAVs. Besides, in both figures, UAVs' satisfaction level and total utility are increasing when the number of UGVs is increasing.
The reason is when more UGVs participating in the auction, each UAV can have a higher chance to match a more preferred UGV for battery recharging. Thereby, UAVs can enjoy a higher satisfaction level of charging service and obtain higher utilities.
\begin{figure}
\caption{Social surplus vs. number of UGVs (i.e., $J(\tau)$) in three schemes, where $I(\tau)=10$.}
\label{fig:simu03}
\end{figure}
\begin{figure}
\caption{Utility of UAVs vs. auction time window $\tau$ under different numbers of UGVs (i.e., $J(\tau)$), where $I(\tau)=10$.}
\label{fig:simu04}
\end{figure}
Fig.~\ref{fig:simu03} depicts the social surplus in three schemes, where the number of UGVs in the auction varies between $6$ and $14$. Here, the number of UAVs in the auction is $10$. From Fig.~\ref{fig:simu03}, it can be observed that the proposed scheme attains a higher social surplus than the static WPT scheme. The reason is that in the static WPT scheme, the WPT facilities in the simulation area are static, causing a higher round-trip cost for UAVs than our UGV-assisted WPT scheme. Additionally, given more UGVs, as the chances for both UAVs and UGVs to match their more preferred parter can be higher, the overall utility of UAVs and UGVs (i.e., social surplus defined in Eq. (\ref{eq4-3})) is greater.
Fig.~\ref{fig:simu04} illustrates the utility of UAVs in the proposed scheme when both the auction time window $\tau$ and number of UGVs vary. As seen in Fig.~\ref{fig:simu04}, the longer auction time window can result in higher UAV utilities when the number of UGVs is fixed. The reason is that, in our proposed auction scheme, the bid collection process ends if the time window $\tau$ elapses. As such, given the longer auction time window, more bids of UAVs and UGVs can be included in the current auction to help them make better energy matching choices. Besides, the utility of UAVs grows with the number of UGVs, which has been analyzed in Fig.~\ref{fig:simu02}.
\begin{table}[!t]\setlength{\abovecaptionskip}{0.1cm}
\centering
\caption{Comparison of the utility of a randomly selected UAV under truthful bidding and untruthful bidding in different auction sizes.}\label{simutable1}
\begin{tabular}{|c|c|c|}
\hline
{Auction size} & \textbf{\begin{tabular}[c]{@{}c@{}}Truthful\\bidding\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Untruthful\\bidding\end{tabular}} \\ \hline
($I(\tau)=5,J(\tau)=5$) & \textbf{3.874} & 2.0148 \\ \hline
($I(\tau)=20,J(\tau)=20$) & \textbf{28.234} & 22.041 \\ \hline
\end{tabular}
\end{table}
\begin{table}[!t]\setlength{\abovecaptionskip}{0.1cm}
\centering
\caption{Comparison of the ratio of non-envy UAVs in three schemes under small-scale and large-scale auctions.}\label{simutable2}
\begin{tabular}{|c|c|c|}
\hline
Auction size & \textbf{\begin{tabular}[c]{@{}c@{}}Exhaustive \\ optimal\end{tabular}} & \multicolumn{1}{l|}{\textbf{Ours}} \\ \hline
($I(\tau)=5,J(\tau)=5$) & 40\% & \textbf{100\%} \\ \hline
($I(\tau)=20,J(\tau)=20$) & 30\% & \textbf{100\%} \\ \hline
\end{tabular}
\end{table}
Table~\ref{simutable1} compares the UAV utility under truthful bidding and untruthful bidding in the proposed auction. As seen in Table~\ref{simutable1}, the utility of the randomly selected UAV when bidding truthfully is greater than that in untruthfully bidding under both small-scale and large-scale auctions. It indicates that bidding truthfully is the dominant strategy for UAVs, which validates the strategy proofness of our auction mechanism and conforms to Theorem~\ref{theorem1}.
Table~\ref{simutable2} compares the ratio of non-envy UAVs in two schemes under small-scale auction (i.e., $I(\tau)=5,J(\tau)=5$) and large-scale auction (i.e., $I(\tau)=20,J(\tau)=20$). As observed in Table~\ref{simutable2}, the proposed scheme outperforms the exhaustive optimal approach and enforces envy freeness for all UAVs under both small-scale and large-scale auctions, which also conforms to the theoretical results in Theorem~\ref{theorem2}.
\section{CONCLUSION}\label{sec:CONSLUSION}
UAV's limited flight endurance is one of the main impediments to modern UAV applications. By leveraging ground vehicles mounted with WPT facilities on their proofs, this paper has proposed a mobile and collaborative recharging scheme for UAVs to facilitate on-demand wireless battery recharging. An energy scheduling problem for multiple UAVs and multiple vehicles has been formulated under practical constraints and energy competitions in the highly dynamic network. We have also devised an online auction-based solution with low complexity to allocate and price idle wireless chargers on vehicular proofs in real time. Theoretical analyses have proved that the proposed scheme produces strategy-proof, envy-free, and stable allocation outcomes. Lastly, numerical results validate the effectiveness of the proposed scheme in delivering more satisfactory UAV charging services. For the future work, we plan to improve the auction efficiency and investigate the charging scheduling mechanism under more complex situations.
\end{document} | math | 56,618 |
\begin{document}
\title{Distinguishing photon blockade in a $\mathcal{PT}$-symmetric optomechanical system}
\author{Dong-Yang Wang}
\affiliation{Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China}
\author{Cheng-Hua Bai}
\affiliation{Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China}
\author{Shutian Liu\footnote{E-mail: [email protected]}}
\affiliation{Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China}
\author{Shou Zhang\footnote{E-mail: [email protected]}}
\affiliation{Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China}
\author{Hong-Fu Wang\footnote{E-mail: [email protected]}}
\affiliation{Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China}
\begin{abstract}
We study the effects of parity-time($\mathcal{PT}$)-symmetry on the photon blockade and distinguish the different blockade mechanisms in a double-cavity optomechanical system. By studying the light statistics of the system, we find the completely different photon blockade behaviors when the $\mathcal{PT}$-symmetry is broken or unbroken, which is related to the $\mathcal{PT}$ phase transition. Furthermore, an interesting phenomenon that the two cavities are blocked at the same time is found with the appropriate system parameters. Those statistical phenomenons are all analyzed in detail and demonstrated by analytically solving the Schr\"{o}dinger equation and numerically simulating the master equation, respectively. Finally, we also consider the non-$\mathcal{PT}$ symmetric situations which further reveal the physical essence of the photon blockade by comparing those results. Different from the usual photon blockade, our proposal is feasible even with weak parameter mechanism, i.e., the proposal neither requires the strong optomechanical coupling nor the large tunneling coupling between cavities.
\pacs{42.50.Wk, 07.10.Cm, 42.50.Ar}
\keywords{optomechanics, $\mathcal{PT}$-symmetry, photon blockade}
\end{abstract}
\maketitle
\section{Introduction}\label{sec.1}
Cavity optomechanics~\cite{RevModPhys.86.1391,andp.525.215,Science.321.1172,PhysToday.65.29,Physics.2.40} has made great success in the exploration of macroscopic quantum effects and the study of interaction between electromagnetic radiation and micromechanical motion over the past decades. Various cavity optomechanical systems~\cite{PhysRevLett.110.193602,NatComm.6.6981,OptLett.41.2422,PhysRevA.90.043831,NatComm.7.11338,arXiv:1810.03709} have been proposed and become the remarkable platforms to study the questions of quantum mechanics on the macroscopic scale, such as the ground-state cooling~\cite{Nature.524.325,PhysRevLett.99.093901,PhysRevLett.99.093902,Nature.463.72,Nature.444.67,PhysRevA.98.023816}, mechanical quadrature squeezing~\cite{Science.349.952,PhysRevX.3.031012,PhysRevLett.103.213603,PhysRevA.88.013835}, macroscopic entanglement~\cite{PhysRevLett.98.030405,PhysRevLett.99.250401,PhysRevLett.88.120401,arXiv:1811.06227}, quantum superposition state of the mechanical resonator~\cite{PhysRevLett.116.163602,PhysRevLett.110.160403,PhysRevA.88.023817}, etc. In addition, the electromagnetic field (optical part) of the cavity optomechanical system is also affected by the motion of the mechanical resonator. Some researches have attracted significant attention, such as the optomechanically induced transparency~\cite{PhysRevA.81.041803,Science.330.1520,Nature.472.69}, normal-mode splitting~\cite{PhysRevLett.101.263602}, photon blockade~\cite{PhysRevLett.107.063601,PhysRevLett.107.063602,PhysRevA.87.025803,PhysRevA.88.023853,PhysRevA.92.033806,PhysRevA.93.063860,arXiv:1802.09254}, etc. Of these studies, the photon blockade is a particularly important nonclassical light statistic effect and can be used to generate the single photon source for those fundamental studies in quantum information processing and quantum optics fields.
To this end, there are two general ideas: (i) the anharmonicity of eigenenergy spectrum coming from kinds of nonlinearities, which is called as the conventional photon blockade (CPB); (ii) the destructive quantum interference between two different transition paths, which is called as the unconventional photon blockade (UPB). The first physical mechanism for achieving blockade has been studied theoretically~\cite{PhysRevLett.79.1467,PhysRevA.49.R20,PhysRevA.46.R6801,PhysRevB.87.235319,PhysRevA.90.023849} and realized experimentally in various systems~\cite{Nature.436.87,PhysRevLett.107.053602,NatPhys.4.859}. Besides, due to the nonlinear optomechanical interaction, the optomechanical systems can also be used to achieve the photon blockade effect~\cite{PhysRevLett.107.063601,PhysRevLett.107.063602,PhysRevA.87.025803,PhysRevA.88.023853,PhysRevA.92.033806,PhysRevA.93.063860,arXiv:1802.09254}. However, it is worth noting that the study of photon blockade in an optomechanical system requires the strong optomechanical coupling condition, which still poses major technological challenges. On the other hand, the physical mechanism of the destructive quantum interference for achieving blockade has been proposed~\cite{PhysRevLett.104.183601} and studied extensively in previous years~\cite{PhysRevA.83.021802,PhysRevA.90.033809,PhysRevA.91.063808,PhysRevA.92.023838,PhysRevA.92.053837,OE.23.32835,JPB.51.035503,PhysRevA.96.053810,PhysRevA.97.043819,PhysRevA.98.023856}. Recently, it has been observed experimentally in coupled quantum-dot-cavity system~\cite{PhysRevLett.121.043601}. Based on the theory, the UPB has also been studied in various double-cavity optomechanical systems~\cite{PhysRevLett.109.013603,PhysRevA.87.013839,JPB.46.035502,PhysRevA.98.013826}. It is worth noting that the UPB usually requires the large coupling strength between the two components~\cite{PhysRevA.96.053810}, which structure the different transition paths. In a word, the usual photon blockade requires strong parameter mechanism, i.e., large nonlinearity or strong coupling in different blockade mechanism. So how to break the limitation of strong parameter mechanism will be beneficial to the experimental realization.
Moreover, the $\mathcal{PT}$-symmetric system exists plenty of interesting physical behaviors~\cite{PhysRevLett.80.5243,RepProgPhys.70.947} and has been studied in various physical fields~\cite{PhysRevLett.103.093902,PhysRevLett.103.123601,PhysRevLett.106.213901,Nature.488.167,NatMat.12.108,NatPhys.6.192,PhysRevA.96.043810,Science.346.975,NatPhotonics.8.524,Nature.548.187,Science.363.eaar7709}. Ref.~\cite{PhysRevA.92.053837} has studied the enhancement of photon blockade in a $\mathcal{PT}$-symmetric coupled-cavity system. Naturally, the study of the $\mathcal{PT}$-symmetry in cavity optomechanics has also attracted much attention in recent years, such as the phonon laser~\cite{PhysRevLett.113.053604}, chaos~\cite{PhysRevLett.114.253601}, optomechanically induced transparency~\cite{SR.5.9663,SR.6.31095,PhysRevA.98.033832}, metrology~\cite{PhysRevLett.117.110802}, etc. Of these proposals, the $\mathcal{PT}$-symmetric optomechanical systems usually consist of two optical cavities and one mechanical mode; namely, the linear double-cavity optomechanical system, where the passive optical cavity is coupled to the mechanical resonator. In this way, studying the effects of $\mathcal{PT}$-symmetry on the photon blockade in optomechanical system will be interesting and important, and has not yet been reported so far.
In this paper, we focus on studying the effects of $\mathcal{PT}$-symmetry on the photon blockade in a double-cavity optomechanical system, where the optomechanical cavity is passive and the other one is an active cavity. We obtain the equal-time second-order correlation functions of the photon via analytically solving the Schr\"{o}dinger equation or numerically simulating the master equation, where the analytical and numerical results agree with each other very well. We find that the two cavities can be blocked simultaneously with appropriate system parameters and the photon blockade behaviors are completely different in the different $\mathcal{PT}$ phase regions, i.e., the correlation has three dips in the unbroken $\mathcal{PT}$-symmetric region, while in the broken $\mathcal{PT}$-symmetric region, there is only one dip. To explore and distinguish the deeply physical mechanisms of the different photon blockade behaviors, we analyze the $\mathcal{PT}$-symmetric double-cavity optomechanical system by utilizing the theories of CPB and UPB, respectively. We find that, in the unbroken $\mathcal{PT}$-symmetric region, the three dips come from the anharmonicity of eigenenergy spectrum and the destructive quantum interference between paths, which correspond to the CPB and UPB, respectively. However, in the broken $\mathcal{PT}$-symmetric region, the only one dip belongs to the UPB, which comes from the destructive quantum interference between paths. In addition, the present UPB is slightly different from the usual theory, where the interference paths are not affected by the $\mathcal{PT}$-symmetry. Finally, we also take into account the situations of non-$\mathcal{PT}$ symmetry: (i) double passive cavity; (ii) unequal detunings (different cavity resonance frequencies). We find that the balance gain-loss ratio is the main reason for the perfect photon blockade and the different detunings just change the location of the perfect photon blockade. Furthermore, the perfect photon blockade would not be achieved under the weak parameter mechanism if the balance is broken.
The paper is organized as follows: In Sec.~\ref{sec.2}, we derive the Hamiltonian of the $\mathcal{PT}$-symmetric double-cavity optomechanical system. In Sec.~\ref{sec.3}, we analytically and numerically solve the equal-time second-order correlation functions and analyze the effects of $\mathcal{PT}$ phase transition on the photon blockade. In Sec.~\ref{sec.4}, we discuss the photon blockade behavior when the system is non-$\mathcal{PT}$ symmetric. Finally, a conclusion is given in Sec.~\ref{sec.5}.
\section{System and Hamiltonian}\label{sec.2}
\begin{figure}
\caption{Schematic diagram of the $\mathcal{PT}
\label{fig:double cavity optomechangcs}
\end{figure}
Inspired by the studies about the $\mathcal{PT}$-symmetric optomechanical systems~\cite{PhysRevLett.113.053604,PhysRevLett.114.253601,SR.5.9663,SR.6.31095,PhysRevA.98.033832,PhysRevLett.117.110802}, we consider a double-cavity optomechanical system, as depicted in Fig.~\ref{fig:double cavity optomechangcs}, where one optical cavity is passive and coupled to the mechanical resonator with single photon optomechanical coupling strength $g$, and the other one is an active cavity coupled to the passive cavity with tunneling coupling strength $J$, which can be controlled by changing the distance between the whispering-gallery resonators~\cite{Nat.Phys.10.394}. In the presence of the external driving field, the total Hamiltonian for the system is written as ($\hbar=1$)
\begin{eqnarray}\label{e01}
H&=&\omega_{1}a_{1}^{\dagger}a_{1}+\omega_{2}a_{2}^{\dagger}a_{2}
+\omega_{m}b^{\dagger}b+J(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger})\cr\cr
&&-ga_{1}^{\dagger}a_{1}(b^{\dagger}+b)+(Ea_{1}^{\dagger}e^{-i\omega_{l}t}+E^{\ast}a_{1}e^{i\omega_{l}t}),
\end{eqnarray}
where $a_{j}~(b)$ and $a_{j}^{\dagger}~(b^{\dagger})$ represent the annihilation and creation operators for the $j$-th optical (mechanical) mode with frequency $\omega_{j}~(\omega_{m})$, respectively ($j=1,2$). $E=\sqrt{2\kappa_{1}P/(\hbar\omega_{l})}$ is the driving amplitude of the input laser with frequency $\omega_{l}$ and power $P$, and $\kappa_{1}$ is the decay rate of the passive cavity 1. The gain rate of the active cavity (cavity 2) is $\kappa_{2}$ which can be fabricated by pumping Er$^{3+}$ ions in microtoroid resonator of Er$^{3+}$-doped silica~\cite{Nat.Phys.10.394,Science.346.328,IEEEJQE.46.1626}. In the rotating frame defined by $V_{1}=\exp[-i\omega_{l}t(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})]$, the transformed Hamiltonian $H_{1}=V_{1}^{\dagger}HV_{1}-iV_{1}^{\dagger}\dot{V_{1}}$ reads
\begin{eqnarray}\label{e02}
H_{1}&=&\Delta_{1}a_{1}^{\dagger}a_{1}+\Delta_{2}a_{2}^{\dagger}a_{2}
+\omega_{m}b^{\dagger}b+J(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger})\cr\cr
&&-ga_{1}^{\dagger}a_{1}(b^{\dagger}+b)+(Ea_{1}^{\dagger}+E^{\ast}a_{1}),
\end{eqnarray}
where $\Delta_{j}=\omega_{j}-\omega_{l}$ is the $j$-th cavity-laser detuning. In the displacement representation of the mechanical mode, defined by the canonical transformation $V_{2}=\exp[g/\omega_{m}a_{1}^{\dagger}a_{1}(b^{\dagger}-b)]$, the transformed Hamiltonian $H_{2}=V_{2}^{\dagger}H_{1}V_{2}$ is
\begin{eqnarray}\label{e03}
H_{2}&=&\Delta_{1}a_{1}^{\dagger}a_{1}+\Delta_{2}a_{2}^{\dagger}a_{2}
+\omega_{m}b^{\dagger}b-\frac{g^{2}}{\omega_{m}}(a_{1}^{\dagger}a_{1})^{2}\cr\cr
&&+J[a_{1}^{\dagger}a_{2}e^{-\frac{g}{\omega_{m}}(b^{\dagger}-b)}+a_{1}a_{2}^{\dagger}e^{\frac{g}{\omega_{m}}(b^{\dagger}-b)}]\cr\cr
&&+[Ea_{1}^{\dagger}e^{-\frac{g}{\omega_{m}}(b^{\dagger}-b)}+E^{\ast}a_{1}e^{\frac{g}{\omega_{m}}(b^{\dagger}-b)}].
\end{eqnarray}
Under the weak optomechanical coupling condition ($g\ll\omega_{m}$), those exponential factors of Eq.~(\ref{e03}) can be approximately omitted (all the approximates will be validated later via numerical simulation). Then the Hamiltonian can be rewritten as
\begin{eqnarray}\label{e04}
H_{3}&=&\Delta_{1}a_{1}^{\dagger}a_{1}+\Delta_{2}a_{2}^{\dagger}a_{2}+\omega_{m}b^{\dagger}b
-\frac{g^{2}}{\omega_{m}}(a_{1}^{\dagger}a_{1})^{2}\cr\cr
&&+J(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger})+(Ea_{1}^{\dagger}+E^{\ast}a_{1}),
\end{eqnarray}
which indicates that the mechanical resonator is decoupled from the system; namely, the evolutions of optical and mechanical parts are independent. So the state of the system is a separable state, which can be written as $|\psi\rangle|\hat{n}_{m}\rangle=\sum_{n_{1},n_{2}}C_{n_{1}n_{2}}|n_{1},n_{2}\rangle|\hat{n}_{m}\rangle$, where $n_{j}$ represents the photon number in $j$-th cavity and $|\hat{n}_{m}\rangle=V_{2}^{\dagger}|n_{m}\rangle$ is the mechanical displaced Fock state related to the photon number of cavity 1. In Hamiltonian (\ref{e04}), the part of the mechanical resonator can be ignored when we only care about the optical properties of the system. The reduced Hamiltonian reads
\begin{eqnarray}\label{e05}
H_{4}&=&\Delta_{1}a_{1}^{\dagger}a_{1}+\Delta_{2}a_{2}^{\dagger}a_{2}
-\frac{g^{2}}{\omega_{m}}(a_{1}^{\dagger}a_{1})^{2}\cr\cr
&&+J(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger})+(Ea_{1}^{\dagger}+E^{\ast}a_{1}),
\end{eqnarray}
which is similar to the coupled cavity system consisting of a linear cavity and a Kerr-type nonlinear cavity~\cite{PhysRevA.80.065801,arXiv:1803.06642}, where the Kerr-type nonlinear strength is related to the mechanical resonator.
\section{Photon blockade in the $\mathcal{PT}$-symmetric optomechanical system}\label{sec.3}
\subsection{Analytical solution}\label{subsec.3A}
Here, the system we consider is a $\mathcal{PT}$-symmetric double-cavity optomechanical system, which includes a passive cavity and an active cavity. Naturally, the non-Hermitian Hamiltonian can be written by adding phenomenologically the imaginary decay and gain terms into the original Hamiltonian. Furthermore, due to the added imaginary terms have no effect on the previous calculation process, the reduced non-Hermitian Hamiltonian can be directly written as
\begin{eqnarray}\label{e06}
H_{\mathrm{NM}}&=&(\Delta_{1}-i\frac{\kappa_{1}}{2})a_{1}^{\dagger}a_{1}+(\Delta_{2}+i\frac{\kappa_{2}}{2})a_{2}^{\dagger}a_{2}
-\frac{g^{2}}{\omega_{m}}(a_{1}^{\dagger}a_{1})^{2}\cr\cr
&&+J(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger})+(Ea_{1}^{\dagger}+E^{\ast}a_{1}).
\end{eqnarray}
To satisfy the $\mathcal{PT}$-symmetry and simplify the next calculation, we set $\Delta_{1}=\Delta_{2}$ and $\kappa_{1}=\kappa_{2}$. The dynamical evolution of the optical components is calculated via the Schr\"{o}dinger equation $i\partial|\psi(t)\rangle/\partial t=H_{\mathrm{NM}}|\psi(t)\rangle$, where $|\psi(t)\rangle$ is the time-dependent photon state of the coupled cavities. Under the weak driving condition, we can restrict the total photon number within the low-excitation subspace up to 2. At this time, the time-dependent photon state $|\psi(t)\rangle$ can be expressed as
\begin{eqnarray}\label{e07}
|\psi(t)\rangle&=&C_{00}(t)|0,0\rangle+C_{10}(t)|1,0\rangle+C_{01}(t)|0,1\rangle\cr\cr
&&+C_{20}(t)|2,0\rangle+C_{11}(t)|1,1\rangle+C_{02}(t)|0,2\rangle,
\end{eqnarray}
where $C_{n_{1}n_{2}}(t)$ is the probability amplitude for the coupled cavities being in the state $|n_{1},n_{2}\rangle$. Utilizing the above Schr\"{o}dinger equation, we can get a set of linear differential equations for the probability amplitudes
\begin{eqnarray}\label{e08}
i\frac{\partial C_{10}}{\partial t}&=&(\Delta_{1}-i\frac{\kappa_{1}}{2}-\frac{g^{2}}{\omega_{m}})C_{10}+JC_{01}+E+\sqrt{2}E^{\ast}C_{20},\cr\cr
i\frac{\partial C_{01}}{\partial t}&=&(\Delta_{1}+i\frac{\kappa_{1}}{2})C_{01}+JC_{10}+E^{\ast}C_{11},\cr\cr
i\frac{\partial C_{20}}{\partial t}&=&2(\Delta_{1}-i\frac{\kappa_{1}}{2}-\frac{2g^{2}}{\omega_{m}})C_{20}+\sqrt{2}JC_{11}+\sqrt{2}EC_{10},\cr\cr
i\frac{\partial C_{11}}{\partial t}&=&(2\Delta_{1}-\frac{g^{2}}{\omega_{m}})C_{11}+\sqrt{2}J(C_{20}+C_{20})+EC_{01},\cr\cr
i\frac{\partial C_{02}}{\partial t}&=&2(\Delta_{1}+i\frac{\kappa_{1}}{2})C_{02}+\sqrt{2}JC_{11}.
\end{eqnarray}
Here, it has the fact $\{C_{20},C_{11},C_{02}\}\ll\{C_{10},C_{01}\}\ll C_{00}$ and we thus can set $C_{00}\simeq1$ due to the weak driving assumption. Next, by ignoring higher-order terms $E^{\ast}C_{20}$ and $E^{\ast}C_{11}$, the steady-state solution can be obtained approximatively
\begin{eqnarray}\label{e09}
C_{10}&=&\frac{2\omega_{m}E(2\Delta_{1}+i\kappa_{1})}{2g^{2}(2\Delta_{1}+i\kappa_{1})-\omega_{m}(4\Delta_{1}^{2}+\kappa_{1}^{2}-4J^{2})},\cr\cr
C_{01}&=&\frac{-4J\omega_{m}E}{2g^{2}(2\Delta_{1}+i\kappa_{1})-\omega_{m}(4\Delta_{1}^{2}+\kappa_{1}^{2}-4J^{2})},\cr\cr
C_{20}&=&2\sqrt{2}\omega_{m}^{2}E^{2}(2\Delta_{1}+i\kappa_{1})^{2}(g^{2}-2\Delta_{1}\omega_{m})/M,\cr\cr
C_{11}&=&16J\omega_{m}^{2}E^{2}(2\Delta_{1}+i\kappa_{1})(\Delta_{1}\omega_{m}-g^{2})/M,\cr\cr
C_{02}&=&16\sqrt{2}J^{2}\omega_{m}^{2}E^{2}(g^{2}-\Delta_{1}\omega_{m})/M,
\end{eqnarray}
with
\begin{eqnarray}\label{e10}
M&=&[2g^{2}(2\Delta_{1}+i\kappa_{1})-\omega_{m}(4\Delta_{1}^{2}+\kappa_{1}^{2}-4J^{2})]\cr\cr
&&\times[4g^{4}(2\Delta_{1}+i\kappa_{1})+2\Delta_{1}\omega_{m}^{2}(4\Delta_{1}^{2}+\kappa_{1}^{2}-4J^{2})\cr\cr
&&-g^{2}\omega_{m}(20\Delta_{1}^{2}+8i\Delta_{1}\kappa_{1}+\kappa_{1}^{2}-8J^{2})].
\end{eqnarray}
Normally, the photon blockade effect is usually characterized by the equal-time second-order correlation function $g_{j}^{(2)}(0)=\langle a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}\rangle/\langle a_{j}^{\dagger}a_{j}\rangle^{2}$, which characterizes the probability of detecting two photons at the same time. Here, we have the photon bunching effect for $g_{j}^{(2)}(0)>1$ and the photon antibunching effect for $g_{j}^{(2)}(0)<1$. On the other hand, the cross correlation function between the two cavities can be also calculated by $g_{12}^{(2)}(0)=\langle a_{1}^{\dagger}a_{2}^{\dagger}a_{2}a_{1}\rangle/(\langle a_{1}^{\dagger}a_{1}\rangle\langle a_{2}^{\dagger}a_{2}\rangle)$, which represents the probability that each cavity has one photon at the same time. Then the steady-state correlation functions of the two cavities can be analytically obtained via the steady-state solution of the Schr\"{o}dinger equation, respectively, and are given by
\begin{eqnarray}\label{e11}
g_{1}^{(2)}(0)&=&\frac{2|C_{20}|^{2}}{(|C_{10}|^{2}+|C_{11}|^{2}+2|C_{20}|^{2})^{2}}\simeq\frac{2|C_{20}|^{2}}{|C_{10}|^{4}},\cr\cr
g_{2}^{(2)}(0)&=&\frac{2|C_{02}|^{2}}{(|C_{01}|^{2}+|C_{11}|^{2}+2|C_{02}|^{2})^{2}}\simeq\frac{2|C_{02}|^{2}}{|C_{01}|^{4}},\cr\cr
g_{12}^{(2)}(0)&=&|C_{11}|^{2}/[(|C_{10}|^{2}+|C_{11}|^{2}+2|C_{20}|^{2})\cr\cr
&&\times(|C_{01}|^{2}+|C_{11}|^{2}+2|C_{02}|^{2})]\cr\cr
&\simeq&\frac{|C_{11}|^{2}}{|C_{10}|^{2}|C_{01}|^{2}}=\frac{2|C_{02}|^{2}}{|C_{01}|^{4}},
\end{eqnarray}
where $g_{1}^{(2)}(0)$ and $g_{2}^{(2)}(0)$ are the steady-state correlation functions of the passive cavity and the active cavity, respectively. $g_{12}^{(2)}(0)$ is the cross correlation function of the two cavities, which equals $g_{2}^{(2)}(0)$ approximately. Combining the above expressions with Eq.~(\ref{e09}), it is easy to find that the perfect photon blockade can be achieved in the $\mathcal{PT}$-symmetric double-cavity optomechanical system when the detuning $\Delta_{1}$ equals a determined value, which is related to the single photon optomechanical coupling strength and mechanical frequency. That is to say, when the detuning satisfies $\Delta_{1}=g^{2}/(2\omega_{m})$, the amplitude $C_{20}=0$, showing that the perfect photon blockade of the passive cavity 1 occurs. Similarly, when the detuning is set as $\Delta_{1}=g^{2}/\omega_{m}$, the correlation functions $g_{2}^{(2)}(0)=g_{12}^{(2)}(0)=0$ are obtained.
\subsection{Numerical simulation}
The exact solution of the photon blockade can also be obtained via numerical simulation, which utilizes the quantum master equation with the initial Hamiltonian $H_{1}$. The dynamics of the system is described by the quantum master equation as
\begin{eqnarray}\label{e12}
\dot{\rho}&=&-i[H_{1},\rho]+\kappa_{1}\mathcal{L}[a_{1}]\rho-\kappa_{2}\mathcal{L}[a_{2}]\rho\cr\cr
&&+\gamma_{m}(n_\mathrm{th}+1)\mathcal{L}[b]\rho+\gamma_{m}n_\mathrm{th}\mathcal{L}[b^{\dagger}]\rho,
\end{eqnarray}
where $\mathcal{L}[o]\rho=o\rho o^{\dagger}-(o^{\dagger}o\rho+\rho o^{\dagger}o)/2$ is the standard Lindblad operator for the arbitrary system operator $o$. The negative sign in front of $\kappa_{2}$ indicates that the cavity 2 is an active cavity. $\gamma_{m}$ is the damping rate of the mechanical resonator. $n_\mathrm{th}=\{\exp[\hbar\omega_{m}/(k_{B}T)]-1\}^{-1}$ is the mean thermal excitation number of the mechanical resonator at temperature $T$, where $k_{B}$ is the Boltzmann constant. Under the limit of a long time, we can obtain the steady-state density matrix $\rho_{s}$ of the $\mathcal{PT}$-symmetric double-cavity optomechanical system. Meanwhile, the steady-state correlation functions are given by
\begin{eqnarray}\label{e13}
g_{1}^{(2)}(0)&=&\frac{\mathrm{Tr}(a_{1}^{\dagger}a_{1}^{\dagger}a_{1}a_{1}\rho_{s})}{[\mathrm{Tr}(a_{1}^{\dagger}a_{1}\rho_{s})]^{2}},\cr\cr
g_{2}^{(2)}(0)&=&\frac{\mathrm{Tr}(a_{2}^{\dagger}a_{2}^{\dagger}a_{2}a_{2}\rho_{s})}{[\mathrm{Tr}(a_{2}^{\dagger}a_{2}\rho_{s})]^{2}},\cr\cr
g_{12}^{(2)}(0)&=&\frac{\mathrm{Tr}(a_{1}^{\dagger}a_{2}^{\dagger}a_{2}a_{1}\rho_{s})}{\mathrm{Tr}(a_{1}^{\dagger}a_{1}\rho_{s})\mathrm{Tr}(a_{2}^{\dagger}a_{2}\rho_{s})}.
\end{eqnarray}
\begin{figure}
\caption{The equal-time second-order correlation functions versus the detuning (left column) or versus both the detuning and the optomechanical coupling strength (right column). In the left column, the solid black lines are the analytical solutions of the correlation functions defined as Eq.~(\ref{e11}
\label{fig:blockade-Delta-g}
\end{figure}
Next, we strictly validate the validity of our previous calculations by comparing all the results of correlation functions, which come from the analytical solutions and the numerical simulations with the Hamiltonian $H_{1}$ under the weak driving condition (see the left column in Fig.~\ref{fig:blockade-Delta-g}). The numerical simulations are in good agreement with the analytical solutions for the correlation functions $g_{1}^{(2)}(0)$, $g_{2}^{(2)}(0)$, and $g_{12}^{(2)}(0)$. In addition, the correlation functions versus both the detuning and the optomechanical coupling strength are shown in the right column of Fig.~\ref{fig:blockade-Delta-g}. The results indicate that the perfect photon blockade can be achieved when the detuning satisfies the optimal condition: $\Delta_{1}=g^{2}/(2\omega_{m})$ for $g_{1}^{(2)}(0)$ and $\Delta_{1}=g^{2}/\omega_{m}$ for $g_{2}^{(2)}(0)$ and $g_{12}^{(2)}(0)$ (see the dashed white line in the right column of Fig.~\ref{fig:blockade-Delta-g}). It is worth noting that the perfect photon blockade can be achieved with the weak single photon optomechanical coupling strength ($g\ll\omega_{m}$), which breaks the limits of strong coupling in the usual single-cavity optomechanical systems~\cite{PhysRevLett.107.063601,PhysRevLett.107.063602,PhysRevA.87.025803,PhysRevA.88.023853,PhysRevA.92.033806,PhysRevA.93.063860,arXiv:1802.09254}. Moreover, there has an interesting phenomenon due to the similar behaviors of the correlation functions $g_{2}^{(2)}(0)$ and $g_{12}^{(2)}(0)$, i.e., $g_{2}^{(2)}(0)=g_{12}^{(2)}(0)=0$ when the detuning satisfies $\Delta_{1}=g^{2}/\omega_{m}$. The excitations of both cavities are blocked simultaneously when the state of the coupled cavities is $|0,1\rangle$. That means the system no longer absorbs energy to excite the optical cavities until the state of the coupled cavities is changed. That is to say, when the only photon is transmitted to cavity 1, the system might be possible to be further excited through the cavity 1. On the other hand, for $\Delta_{1}=g^{2}/(2\omega_{m})$, the perfect blockade occurs only in the cavity 1. This means that when the state of the coupled cavities is $|1,0\rangle$, the system can be further excited only through the cavity 2. However, when the state of the coupled cavities is $|0,1\rangle$, the system can be further excited through any one of the two cavities. The phenomenon is clearly shown in Fig.~\ref{fig:Blockade}. It is worth noting that the imperfect photon blockade of any cavity can still be obtained in the vicinity of the optimal detuning due to the fact that all the correlation functions are less than 1.
\begin{figure}
\caption{The perfect photon blockade with the different detunings.}
\label{fig:Blockade}
\end{figure}
Finally, to characterize the nature of the photon emission, we also calculate the delayed second-order correlation functions in the steady state. The delayed second-order correlation functions are defined by $g_{j}^{(2)}(\tau)=\langle a_{j}^{\dagger}(0)a_{j}^{\dagger}(\tau)a_{j}(\tau)a_{j}(0)\rangle/\langle a_{j}^{\dagger}(0)a_{j}(0)\rangle^{2}$, where $\tau$ is a finite-time delay. The results of the delayed second-order correlation function for different cavities are shown in Fig.~\ref{fig:twotime}. One can see from Fig.~\ref{fig:twotime} that the value of the delayed second-order correlation function gradually increases with the increase of the time delay and it always exceeds 1 when the time delay is long enough. The main reason for this is the existence of the gain in $\mathcal{PT}$-symmetric double-cavity optomechanical system. While for the system consisting only decay cavities without gain, the delayed second-order correlation function of the no-gain system is finally stabilized at 1 with the increase of the time delay~\cite{PhysRevA.90.023849,PhysRevA.92.023838,PhysRevA.96.053810}. The deeper physical mechanism of the present results needs to be further studied and explored in the future.
\begin{figure}
\caption{The delayed second-order correlation functions for different cavities versus the time delay $\kappa_{1}
\label{fig:twotime}
\end{figure}
\subsection{The effect of $\mathcal{PT}$ phase transition}
As we all know, the $\mathcal{PT}$-symmetric systems exist the interesting spontaneous $\mathcal{PT}$-symmetry breaking behaviour, which comes from the variation of the system parameters and it is determined via checking the eigenvalue. In order to study the effect of $\mathcal{PT}$ phase transition on the photon blockade, we should first find the threshold of the spontaneous $\mathcal{PT}$-symmetry breaking. To this end, we expand the non-Hermitian Hamiltonian without the driving terms, $H_{\mathrm{ND}}=(\Delta_{1}-i\kappa_{1}/2)a_{1}^{\dagger}a_{1}+(\Delta_{2}+i\kappa_{2}/2)a_{2}^{\dagger}a_{2}
-g^{2}/\omega_{m}(a_{1}^{\dagger}a_{1})^{2}+J(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger})$, by the vector $a=[a_{1},a_{2}]^{T}$. Ignoring the weak nonlinear term $g^{2}/\omega_{m}(a_{1}^{\dagger}a_{1})^{2}$, the result reads
\begin{eqnarray}\label{e14}
h=\left(\begin{array}{cc}
\Delta_{1}-i\frac{\kappa_{1}}{2}~&~J\\
J & \Delta_{2}+i\frac{\kappa_{2}}{2}
\end{array}\right),
\end{eqnarray}
where $a^{\dagger}ha$ is the approximate Hamiltonian. Under the condition of satisfying $\mathcal{PT}$-symmetry, i.e., $\Delta_{1}=\Delta_{2}$ and $\kappa_{1}=\kappa_{2}$, the eigenvalues of the matrix $h$ are $\varepsilon_{\pm}=\Delta_{1}\pm\sqrt{J^{2}-(\kappa_{1}/2)^{2}}$, which are real only when $J\geqslant\kappa_{1}/2$. So the $\mathcal{PT}$ phase transition point is approximately $J=\kappa_{1}/2$, which is also called as exceptional point. Meanwhile, the $\mathcal{PT}$ symmetry can be roughly divided into two different regions: the unbroken ($J>\kappa_{1}/2$) and broken ($J<\kappa_{1}/2$) $\mathcal{PT}$-symmetric regions.
\begin{figure}
\caption{The equal-time correlation function of the passive cavity 1 in the different $\mathcal{PT}
\label{fig:blockade-Delta-J}
\end{figure}
In Fig.~\ref{fig:blockade-Delta-J}, we show the equal-time second-order correlation function of the passive cavity 1 in the different $\mathcal{PT}$-symmetric regions, where Fig.~\ref{fig:blockade-Delta-J}(a) belongs to the broken $\mathcal{PT}$-symmetric region and Fig.~\ref{fig:blockade-Delta-J}(b) belongs to the unbroken $\mathcal{PT}$-symmetric region. It is easy to find that the photon blockade behaviors are different in the different $\mathcal{PT}$-symmetric regions. One can see from Fig.~\ref{fig:blockade-Delta-J}(a) that there is one dip in the broken $\mathcal{PT}$-symmetric region, while three dips occur in the unbroken $\mathcal{PT}$-symmetric region, as shown in Fig.~\ref{fig:blockade-Delta-J}(b). To clearly show the effect of the $\mathcal{PT}$ phase transition on the correlation function of the passive cavity 1, we also show the correlation function changing with the detuning and the photon tunneling coupling strength in Fig.~\ref{fig:blockade-Delta-J}(c). It is clear to find that, except for the dip located at the optimal detuning (the middle one), the other two dips on both sides of the optimal detuning occur only when the system is in the unbroken $\mathcal{PT}$-symmetric region.
To explain the photon blockade effect in the $\mathcal{PT}$-symmetric double-cavity optomechanical system, we expand the non-Hermitian Hamiltonian without the driving term in different excitation subspaces. In the zero-excitation subspace, the eigenvalue equation of the system is $H_{\mathrm{ND}}|\phi_{0}\rangle=\varepsilon_{0}|\phi_{0}\rangle$, where $|\phi_{0}\rangle$ is the eigenstate and $\varepsilon_{0}=0$ is the eigenvalue. Similarly, in the single-excitation subspace $\{|1,0\rangle,|0,1\rangle\}$, the eigenvalue equation of the system is $H_{\mathrm{ND}}|\phi_{1\pm}\rangle=\varepsilon_{1\pm}|\phi_{1\pm}\rangle$ and the matrix form of $H_{\mathrm{ND}}$ is written as
\begin{eqnarray}\label{e15}
H_{\mathrm{ND}}=\left(\begin{array}{cc}
\Delta_{1}-i\frac{\kappa_{1}}{2}-\frac{g^{2}}{\omega_{m}}~&~J\\
J & \Delta_{2}+i\frac{\kappa_{2}}{2}
\end{array}\right).
\end{eqnarray}
Under the condition of satisfying $\mathcal{PT}$-symmetry, i.e., $\Delta_{1}=\Delta_{2}$ and $\kappa_{1}=\kappa_{2}$, the eigenvalues $\varepsilon_{1\pm}$ can be given approximatively as
\begin{eqnarray}\label{e16}
\varepsilon_{1\pm}\simeq\Delta_{1}-\frac{g^{2}}{2\omega_{m}}\pm\sqrt{J^2-(\frac{\kappa_{1}}{2})^{2}}.
\end{eqnarray}
\begin{figure}
\caption{Level diagram of the photon state for the $\mathcal{PT}
\label{fig:level}
\end{figure}
According to the theory of CPB, for the resonant transitions between the zero-excitation and single-excitation states [see Fig.~\ref{fig:level}(a)], we can obtain the optimal relations $\varepsilon_{1\pm}-\varepsilon_{0}=0$ for the photon blockade. Moreover, the photon transitions of the higher-excitation state is off-resonant due to the anharmonicity of the eigenenergy spectrum. So the optimal photon blockade occurs when the system parameters satisfy the optimal relations. This can be verified via showing the probability amplitude of the single-excitation state, as shown in Fig.~\ref{fig:blockade-Delta-J}(d). We can see that, when the single-excitation is resonant, the amplitude of the state $|1,0\rangle$ reaches the peaks, which correspond to the resonance of different single-excitation eigenvalues, respectively. For the same photon tunneling coupling $J$, the locations of the peaks for the amplitude of the single-excitation state correspond to the locations of the optimal CPB [see Fig.~\ref{fig:blockade-Delta-J}(b,d)]. The optimal relations for the single-excitation resonance are also shown in Fig.~\ref{fig:blockade-Delta-J}(c) (see the dashed white line) in which they agree very well with the results of the photon blockade. Therefore, we can conclude that the two dips of the correlation function located at both side come from the single-excitation resonance, and belong to the CPB and only occur in the unbroken $\mathcal{PT}$-symmetric region.
\begin{figure}
\caption{The equal-time correlation function of the passive cavity 1 versus the detuning with different $J$. The other parameters are the same as in Fig.~\ref{fig:blockade-Delta-g}
\label{fig:ga1-Delta-diff-J}
\end{figure}
On the other hand, the middle dip of the correlation function $g_{1}^{(2)}(0)$ is located at the optimal detuning $\Delta_{1}=g^{2}/(2\omega_{m})$ whether the $\mathcal{PT}$-symmetry is broken or not, as shown in Fig.~\ref{fig:blockade-Delta-J}(a-c). To explore the reason for the photon blockade located at the optimal detuning, we draw the energy level of the bare photon state in Fig.~\ref{fig:level}(b). According to the theory of UPB, for the destructive interference between different paths of two-photon excitation, we show the different paths with different color arrows. For the $\mathcal{PT}$-symmetric double-cavity optomechanical system, the two paths suffer from different effects according to the $\mathcal{PT}$-symmetry, i.e., photon loss in the path $|1,0\rangle\rightarrow|2,0\rangle$ and photon gain in the path $|1,0\rangle\rightarrow|0,1\rangle\rightarrow|1,1\rangle\rightarrow|2,0\rangle$. That is different from the usual UPB. Through the calculation in Sec.~\ref{subsec.3A}, we find that the perfect photon blockade of arbitrary cavity can be achieved when the detuning is chosen properly. Although it is slightly different from the usual UPB, the perfect photon blockade located at the optimal detuning also come from the destructive interference between different paths, where the photon gain enhances the interference path resulting in the perfect photon blockade occurring even with the weak parameter mechanism. That can be further found in Fig.~\ref{fig:ga1-Delta-diff-J}, which show that the correlation function of the passive cavity 1 $g_{1}^{(2)}(0)$ changes with the detuning for different photon tunneling coupling strengths ($J/\kappa_{1}=0,~0.1,~0.4,~0.6$). We can see that, when the system is reduced to single-cavity optomechanical system ($J/\kappa_{1}=0$), the perfect photon blockade is non-existent (see the dashed black line and black pentagram line in Fig.~\ref{fig:ga1-Delta-diff-J}). But when the second active cavity is added, the perfect photon blockade occurs at the optimal detuning. And the width of dip located at optimal detuning increases with the increase of the photon tunneling coupling strength. When the coupling strength continues to increase to be larger than the $\mathcal{PT}$ phase transition point, the other two dips of the correlation function occur just as in the unbroken $\mathcal{PT}$-symmetric region (see the pecked magenta line and magenta triangle in Fig.~\ref{fig:ga1-Delta-diff-J}). In contrast to the usual UPB, the present photon blockade is slightly different, where the interference paths are affected by $\mathcal{PT}$-symmetry and the perfect photon blockade can be achieved even with the weak parameter mechanism.
\section{Non-$\mathcal{PT}$ symmetry}\label{sec.4}
In the above section, we study the photon blockade phenomenon in $\mathcal{PT}$-symmetric double-cavity optomechanical system and find the different blockade behaviors when the $\mathcal{PT}$ phase transition occurs. Here, as a comparison, we focus on the study of the photon blockade in non-$\mathcal{PT}$ symmetric double-cavity optomechanical system. There are two kinds of non-$\mathcal{PT}$ symmetric situations: $\kappa_{1}\neq\kappa_{2}$ or $\Delta_{1}\neq\Delta_{2}$.
\begin{figure}
\caption{The equal-time correlation function of the passive cavity 1 versus the detuning in non-$\mathcal{PT}
\label{fig:ga1-Delta1-non-PT}
\end{figure}
For the first kind of the non-$\mathcal{PT}$ symmetric double-cavity optomechanical system $\kappa_{1}\neq\kappa_{2}$, we calculate the correlation function of the cavity 1 via simulating the master equation [Eq.~(\ref{e12})] numerically, and show the results in Fig~\ref{fig:ga1-Delta1-non-PT}(a). The results indicate that the perfect photon blockade located at the optimal detuning is achieved only when the gain-loss ratio is balance [see the black pentagram line in Fig.~\ref{fig:ga1-Delta1-non-PT}(a)]. However, when both of the cavities are passive, the photon blockade phenomenon in the cavity 1 is unobvious [see the red square line in Fig.~\ref{fig:ga1-Delta1-non-PT}(a)]. And the corresponding analytical solution is given in Appendix \ref{App1}, which proves that the perfect photon blockade cannot be achieved with the weak parameter mechanism when the two cavities are all passive. In addition, when the cavity 2 is an active cavity, the photon blockade phenomenon becomes more obvious for some certain detunings even with unbalance gain-loss ratio [see the blue circle line and the magenta triangle line in Fig.~\ref{fig:ga1-Delta1-non-PT}(a)]. Therefore, the balance gain-loss ratio is a necessary condition for the occurrence of the perfect photon blockade with the weak parameter mechanism.
On the other hand, the non-$\mathcal{PT}$ symmetric double-cavity optomechanical system can also be studied by setting $\Delta_{1}\neq\Delta_{2}$. The photon blockade phenomenon of the passive cavity 1 is studied and shown in Fig.~\ref{fig:ga1-Delta1-non-PT}(b) with different detunings of the active cavity 2. We find that the perfect photon blockade can also be achieved even when the detunings of the two cavities are different. However, the location of the perfect photon blockade is $\Delta_{1}=g^{2}/\omega_{m}-\Delta_{2}$ (see Appendix \ref{App2}), which is related to the optomechanical coupling strength, the mechanical frequency, and the detuning of the active cavity 2. The perfect photon blockade is located at $\Delta_{1}=0$ when the detuning of cavity 2 is chosen as $\Delta_{2}=\Delta_{1}+g^{2}/\omega_{m}$ [see the red square line in Fig.~\ref{fig:ga1-Delta1-non-PT}(b)]. Those results indicate that the main reason for the occurrence of the perfect photon blockade is the balance gain-loss ratio in the $\mathcal{PT}$-symmetric double-cavity optomechanical system and the different detunings of the two cavities only affect the location of the perfect photon blockade.
Finally, we briefly discuss the experimental feasibility of the present proposal. The $\mathcal{PT}$-symmetric coupled cavity systems have been widely studied in the experiment~\cite{Nature.488.167,NatMat.12.108,NatPhotonics.8.524,Science.346.975}. However, the single photon optomechanical coupling strength $g$ is still difficult to reach the coupling region ($g>\kappa_{1}$) for current experiments, e.g., $g\sim10^{-3}\kappa_{1}$ in photonic crystals~\cite{Nature.478.89} and microresonators~\cite{Nature.482.63}. Though the present proposal can achieve the photon blockade with $g\ll\omega_{m}$, which has relaxed the experimental requirements for the usual optomechanical systems~\cite{PhysRevLett.107.063601,PhysRevLett.107.063602,PhysRevA.87.025803,PhysRevA.88.023853,PhysRevA.92.033806,PhysRevA.93.063860,arXiv:1802.09254}, it still requires the optomechanical coupling is comparable to the cavity decay $g\sim\kappa_{1}$, which is still difficult for current experimental techniques. On the other hand, the effect of $\mathcal{PT}$ symmetry on photon blockade might be proved in the $\mathcal{PT}$-symmetric coupled system including a Kerr-type nonlinear medium, where the required Kerr-type nonlinear strength is just about $10^{-1}\kappa_{1}$ and can be implemented experimentally.
\section{Conclusions}\label{sec.5}
In conclusion, we have studied the photon blockade effects in $\mathcal{PT}$-symmetric double-cavity optomechanical system, where the optomechanical cavity is passive and the other one is an active cavity. Through the analytical solution and numerical simulation, we respectively obtain the equal-time second-order correlation functions, which are used to describe the photon blockade effects and agree with each other very well. We find an interesting phenomenon, where both of the cavities can be blocked at the same time when the detuning of the system equals to an appropriate value. Furthermore, we find that the photon blockade phenomenons are completely different in the broken and unbroken $\mathcal{PT}$-symmetric regions of the system. The CPB induced by the anharmonicity of the eigenenergy spectrum occurs only in the unbroken $\mathcal{PT}$-symmetric region. However, the UPB coming from interference between paths with gain and loss can occur whether the system is in the broken or unbroken $\mathcal{PT}$-symmetric region.
We strictly analyze and discuss the reason for the occurrence of the perfect photon blockade via the CPB and UPB theories, respectively, and find that the present UPB is slightly different from the usual UPB theory. Moreover, to further explain the photon blockade mechanism, we study the correlation function with the non-$\mathcal{PT}$ symmetric situations, e.g., unbalance gain-loss ratio or unequal detunings. Those results indicate that the perfect photon blockade can be achieved when the gain-loss ratio is balance and the unequal detunings just change the location of the perfect photon blockade. Compared with the traditional optomechanical system, the photon blockade in the $\mathcal{PT}$-symmetric double-cavity optomechanical system does not require the strong optomechanical coupling strength and can be achieved with the weak parameter mechanism, i.e., $g\ll\omega_{m}$ and $J<\kappa_{1}$. Therefore, our proposal might be more feasible in experiment and would contribute to the generation of the single photon source.
\begin{center}
{\bf{ACKNOWLEDGMENTS}}
\end{center}
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61822114, 61465013, 61575055, and 11465020, and the Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project under Grant No. 20160519022JH.
\appendix
\section{double-passive-cavity optomechanical system}\label{App1}
Here, as a comparison, we study the photon blockade of the double-passive-cavity optomechanical system; namely, $\kappa_{2}=-\kappa_{1}$, where the system is non-$\mathcal{PT}$ symmetric. Similar to the calculation in Sec.~\ref{subsec.3A}, we can obtain the steady-state probability amplitudes
\begin{eqnarray}\label{Ae01}
C_{10}&=&\frac{2\omega_{m}E(2\Delta_{1}-i\kappa_{1})}{2g^{2}(2\Delta_{1}-i\kappa_{1})-\omega_{m}[(2\Delta_{1}-i\kappa_{1})^{2}-4J^{2}]},\cr\cr
C_{01}&=&\frac{-4J\omega_{m}E}{2g^{2}(2\Delta_{1}-i\kappa_{1})-\omega_{m}[(2\Delta_{1}-i\kappa_{1})^{2}-4J^{2}]},\cr\cr
C_{20}&=&2\sqrt{2}\omega_{m}^{2}E^{2}(2\Delta_{1}-i\kappa_{1})^{2}[g^{2}-\omega_{m}(2\Delta_{1}-i\kappa_{1})]/M,\cr\cr
C_{11}&=&8J\omega_{m}^{2}E^{2}(2\Delta_{1}-i\kappa_{1})[2g^{2}-\omega_{m}(2\Delta_{1}-i\kappa_{1})]/M,\cr\cr
C_{02}&=&8\sqrt{2}J^{2}\omega_{m}^{2}E^{2}[2g^{2}-\omega_{m}(2\Delta_{1}-i\kappa_{1})]/M,
\end{eqnarray}
with
\begin{eqnarray}\label{Ae02}
M&=&\{2g^{2}(2\Delta_{1}-i\kappa_{1})-\omega_{m}[(2\Delta_{1}-i\kappa_{1})^{2}-4J^{2}]\}\cr\cr
&&\times\{4g^{4}(2\Delta_{1}-i\kappa_{1})-g^{2}\omega_{m}[5(2\Delta_{1}-i\kappa_{1})^{2}-8J^{2}]\cr\cr
&&+\omega_{m}^{2}(2\Delta_{1}-i\kappa_{1})[(2\Delta_{1}-i\kappa_{1})^{2}-4J^{2}]\}.
\end{eqnarray}
To achieve the photon blockade of the cavity 1, we need to choose a set of system parameters to make $C_{20}=0$. That is to say, the system parameters should satisfy the follow relations
\begin{eqnarray}\label{Ae03}
\begin{cases}
12\Delta_{1}^{2}-4\Delta_{1}\frac{g^{2}}{\omega_{m}}=\kappa_{1}^{2}>0,\\
4\Delta_{1}(g^{2}-4\Delta_{1}\omega_{m})^{2}=0.
\end{cases}
\end{eqnarray}
It is easy to find that the above relations cannot be satisfied at the same time. That is to say that the perfect photon blockade of the cavity 1 cannot be achieved with the weak parameter mechanism in the double-passive-cavity optomechanical system. In addition, the parameter conditions of $C_{02}=0$ and $C_{11}=0$ are also unsatisfed due to $J\neq0$. So we can conclude that the perfect photon blockade cannot be achieved in the current situation. However, when the optomechanical coupling strength is enhanced~\cite{PhysRevA.87.013839} or the location of the driving field is changed~\cite{JPB.46.035502}, it will become possible to achieve the perfect photon blockade with the strong system parameters. Here, we would not discuss them in detail.
\section{unequal detunings}\label{App2}
For the non-$\mathcal{PT}$ symmetric system with different detunings, $\Delta_{1}\neq\Delta_{2}$, we can also obtain the probability amplitudes via solving the Schr\"{o}dinger equation. Here, the probability amplitude $C_{20}$ is
\begin{eqnarray}\label{Be01}
C_{20}&=&2\sqrt{2}\omega_{m}^{2}E^{2}(2\Delta_{2}+i\kappa_{1})^{2}[g^{2}-\omega_{m}(\Delta_{1}+\Delta_{2})]/\cr\cr
&&\{[4J^{2}\omega_{m}+(2\Delta_{2}+i\kappa_{1})(2g^{2}-\omega_{m}(2\Delta_{1}-i\kappa_{1}))]\cr\cr
&&\times[4J^{2}\omega_{m}(2g^{2}-\omega_{m}(\Delta_{1}+\Delta_{2}))+(2\Delta_{2}+i\kappa_{1})\cr\cr
&&\times(g^{2}-\omega_{m}(\Delta_{1}+\Delta_{2}))(4g^{2}-\omega_{m}(2\Delta_{1}-i\kappa_{1}))]\},\cr\cr
&&
\end{eqnarray}
where $C_{20}=0$ can be obtained when $\Delta_{1}+\Delta_{2}=g^{2}/\omega_{m}$; namely, the perfect photon blockade is located at the point of $\Delta_{1}=g^{2}/\omega_{m}-\Delta_{2}$. That means that the perfect photon blockade can be achieved even when the system is non-$\mathcal{PT}$ symmetry and the location of the perfect photon blockade is related to the system detunings.
\end{document} | math | 46,779 |
\begin{document}
\title{One particle self correlations for a separable representation
of the singlet spin state beyond standard Quantum Mechanics}
\author{Carlos L\'opez\\
Department of Mathematics and Physics, UAH\\
28873 Alcal\'a de Henares (Madrid) SPAIN\\
[email protected]}
\date{\today}
\maketitle
\centerline{Abstract}
A new pure quantum state, isotropic in spin variables, is defined in an extended spin phase space beyond standard quantum mechanics. It allows to represent the entangled singlet state in separable form. The statistical correlations between Alice and Bob measurements become self correlations between hidden spin values for each particle, together with perfect anti correlation between spin values on the pair. Alice determines through measurement on her particle the value of spin in some direction. Spin in another direction is inferred from Bob measurement on the companion. Bell's inequalities are violated because of the wave like behaviour of quantum systems. In full analogy with the two slit experiment, interference terms between spin field components appear determining the contextual character of quantum distributions of probability.
\vskip 0.7 truecm
PACS: 03.65Ca, 03.65Ta, 03.65Ud
\vskip 0.4 truecm
Keywords: Singlet state, extended QM, double field solution
\section{\label{sec1}After Bell}
Violations of Bell's type inequalities in spatially se\-pa\-rated measurements\cite{Aspect} have been empirically tested beyond any reasonable doubt\cite{tests,t3,t4,t5}. All relevant loopholes have been satisfactorily closed, and the predicted quantum correlations confirmed. It is time to look for physical evidence of the non local influence between measurement events. However, known interactions are mediated by physical systems, either particles following time--like or light--like paths or distributed fields evolving relativistically.
Space--like curves as paths of particles are discarded because they have a frame dependent time orien\-ta\-tion, according to relativity.
The value of a field at a space time event depends on its values along a spatial sheet inside the past light cone; in other words, it commutes with values of the field at spatially separated events. Even in case of time--like separation between measurements it is unlikely the existence of a media\-ting system connecting them without observable decay for increasing distance and with other systems in between that do not shield its propagation.
We can, alternatively, go beyond the standard formulation of Quantum Mechanics (SQM) and develop an explicitly local, separable description of entangled states for composites.
The celebrated EPR paper\cite{EPR} about incompleteness of SQM, the quantum potential in Bohm mechanics\cite{Bohm}, the
analysis of Renninger\cite{ren} of the wave particle duality in an interferometer, inconsistency between SQM and the action reaction principle\cite{ARP,ARP2}, among other considerations\cite{other,o2,o3}, are enough arguments to explore the possibility of a formulation of Quantum Mechanics in extended phase spaces\cite{EQM}(EQM).
States of quantum systems could be described by some $[x,\Phi]$, $x$ commuting and non commuting variables of a corpuscular subsystem and
$\Phi$ an accompanying de Broglie\cite{dBro} (or pre--quantum,
sub quantum\cite{subq,s3})
field. In the double field $\Phi$--$\Psi$ model, the distribution of amplitude
$\Psi = R exp(i\theta)$ is a statistical representation of an ensemble of composite systems, $R^2$ distribution of probability for variables $x$ of the particle and
$\bigtriangleup \theta$ relative phases between field components.
No go theorems\cite{nogo,n2,n3,n4,n5} are dead end paths for the pursuit of an extended phase space in EQM. The existence of global, non contextual distributions of probability $P(x)$ in spaces of (so called) hidden variables for the particle and whose marginals match the quantum distributions are mathematically forbidden. But these distributions would ignore the accompanying field and its interaction with the particle. Obviously, models that do not fulfil some hypothesis of these no go theorems are not ruled out\cite{EQM}.
The empirical fact is that entanglement appears exclusively after local interaction in the past between the correlated systems, or it is conditional to some intermediate interaction if both systems of interest have not been in contact in the past.
In this letter, a local description for the quantum co\-rre\-la\-tions in the singlet spin state is formulated in the framework of an extended phase space for spin variables. In section II, quantum distributions of amplitude, not classical distributions of probability, describe all standard pure quantum states, each one eigenstate of some spin ope\-ra\-tor, and a new pure quantum state isotropic in spin, which has no counterpart in SQM. It is this ingre\-dient of the formalism, the use, as in SQM, of quantum amplitudes in the extended space instead of classical proba\-bi\-li\-ties, which allows to overcome the thesis of no go theorems. The calculation of marginal amplitudes through projection over the standard phase spaces, followed by Born rule, reproduces the results of SQM, that is, the correlations between a previously known (the eigenvalue) and a measured value of spin. Correlations between hidden va\-lues of spin in two arbitrary directions can be consistently computed in the isotropic state. In the singlet state of a composite, section III, two values of spin can be determined for each particle. One is obtained through direct measurement and the other is inferred from the perfect anti correlation with the measured companion. The state of the composite is separable, each particle is in the new isotropic spin state, with its associated individual self correlations. This formalism can be relevant for the ontological interpretation of Quantum Mechanics, the ensemble character of pure quantum states.
\section{\label{sec2}The isotropic spin state}
If locality is assumed for joint and spatially separated measurements on the singlet spin state, perfect anti correlation between outputs for any common, freely and independently chosen by Alice and Bob, direction ${\bf n}$ of measurement implies that a complete representation of the physical state is characterised by all values of spin. If,
to avoid mathematical complications, we consider a finite set of directions
$\{{\bf n}_1,{\bf n}_2,\ldots,{\bf n}_N \}$, values
$(s_1^{\alpha},s_2^{\alpha},\ldots,s_N^{\alpha})_{\nu}$ and
$(s_1^{\beta},s_2^{\beta},\ldots,s_N^{\beta})_{\nu}$ for each pair of jointly generated particles $(\alpha_{(\nu)},\beta_{(\nu)})$,
$\nu \in \{1,2,\ldots\}$, are fixed from the generation event, fulfilling
$s_{j(\nu)}^{\alpha} +$ $s_{j(\nu)}^{\beta} = 0$.
Three independent values as $s_x$, $s_y$, $s_z$
do not determine the other variables of spin, e.g. $s_{\theta}$
for the magnitude (operator) $S_{\theta} = \cos(\theta) S_x + \sin(\theta) S_y$.
The functional relations between non commuting operators are not fulfilled by
their eigenvalues, $s_{\theta} \neq \cos(\theta) s_x + \sin(\theta) s_y$. In SQM, the dimension of the phase space is lower than in Classical Hamiltonian Mechanics, e.g. position and momentum variables $\{(q,p)\}$ are restricted to $\{q\}$ (resp. $\{p\}$) in the position (momentum) representation. The phase space of EQM has higher dimensions than its classical counterpart, according to the infinite degrees of freedom of the accompanying field.
Let us consider the extended spin phase space
${\cal P}h=\{(s_1,\ldots,s_N)|s_k=\pm\}$, $|{\cal P}h|=2^N$, associated to an elementary spin $1/2$ particle. Bell's inequalities state that for $N>2$ there are not global, non contextual distributions of probability
on ${\cal P}h$, describing a classical statistical ensemble from which the quantum probabilities for the singlet could be obtained.
\[
P_{QM}(s_1^{\alpha},s_2^{\beta})=\frac {1}{4}(1-s_1^{\alpha}s_2^{\beta}
{\bf n}_1\cdot{\bf n}_2) =
\]
\[
= \frac {1}{4}(1+s_1^{\alpha}s_2^{\alpha}
{\bf n}_1\cdot{\bf n}_2) = P_{EQM}(s_1^{\alpha},s_2^{\alpha})
\]
can not be reproduced by a global distribution of pro\-ba\-bility
$P_{Cl}(s^{\alpha}_1,$ $\ldots,s^{\alpha}_N)$ through marginals
$\sum _{l\neq1,2}\sum _{s_l}$ $P_{Cl}(s^{\alpha}_1,$ $\ldots,s^{\alpha}_N)$.
The existence of a classical probabilistic mixture $P_{Cl}$ of physical states with hidden variables, representing an ensemble quantum state, is a ``natural'' hypothesis
systematically considered in the literature of no go theorems. However, it is not
unavoidable, and interference phenomena as in the paradigmatic two slit experiment
point to the need of other mathematical tools.
An alternative algorithm must be applied in
${\cal P}h$, able to reproduce the quantum distributions for a
statistical sample of measurements over the same pure/ensemble quantum state.
Let us apply the ``quantum way'', a distribution of amplitude of probability
$Z(s_1,\ldots,s_N)$, $Z: {\cal P}h \to K$ (in the spin phase space, $K$ will be
the set of ima\-gi\-na\-ry quaternions). We can mimic the paradigmatic two slit experiment
and obtain marginals for the distribution of amplitude $Z$
\[
Z(s_j) = \sum _{l\neq j}\left( \sum _{s_l} Z(s_1,\ldots,s_N) \right)
\]
Applying now Born rule, we get the probabilities
\[
P(s_j) = \frac {|Z(s_j)|^2}{|Z(+_j)|^2+|Z(-_j)|^2} \, ,
\]
where there will appear generically interference terms in the
squared sum of amplitudes. Compare it with
\[
\Psi(x_0,y_0) = \Psi _L(x_0,y_0) + \Psi_R(x_0,y_0)
\]
\[
P(x_0,y_0) = \frac {|\Psi(x_0,y_0)|^2}{\sum _{(x,y)}|\Psi(x,y)|^2} \, ,
\]
where $(x,y)$ are the position variables at the final screen of the two slit experiment,
and $\Psi(x_0,y_0) = \Psi _L(x_0,y_0) + \Psi_R(x_0,y_0)$ is the marginal amplitude,
sum of left and right slit field components. Interference terms in $|\Psi(x_0,y_0)|^2$ are here responsible of the diffraction pattern. Similarly, interference terms in
$|Z(s_j)|^2$ are responsible of the contextual character of quantum distributions of probability, i.e., its dependence on the field components allowed by the physical context.
Formal distributions of probability for correlated va\-lues of spin in two arbitrary
directions ${\bf n}_j$ and ${\bf n}_k$ can be similarly determined, although in different, alternative ways.
One of these values remains necessarily counterfactual because of the incompatibility of joint measurements; measurement of $s_j$ unavoidably perturbs the previous value of $s_k$. From the marginals
\[
Z(s_j,s_k) = \sum _{l\neq j,k}\left( \sum _{s_l} Z(s_1,\ldots,s_N) \right)
\]
we could formally define the joint, unobservable distribution
\[
\Pi (s_j,s_k) = \frac {|Z(s_j,s_k)|^2}
{\sum _{s'_j,s'_k}|Z(s'_j,s'_k)|^2} \, ;
\]
and the same definition can be generalized to $\Pi(s_j,$ $s_k,s_l)$, etc.
Generically, $P(s_j)$ $\neq$ $\Pi(s_j,+_k)$ $+$ $\Pi(s_j,-_k)$
because of the interference terms when Born rule is applied to a sum of
amplitudes
\[
|Z(s_j,+_k)+Z(s_j,-_k)|^2=|Z(s_j,+_k)|^2+|Z(s_j,-_k)|^2 +
\]
\[
+ \Big( Z^*(s_j,+_k)Z(s_j,-_k)+Z^*(s_j,-_k)Z(s_j,+_k)b \Big) _{\rm interf}
\]
We can interpret $P(s_j)$ and $\Pi(s_j,s_k)$ as corresponding to incompatible physical contexts, as in the two slit experiment.
Alternatively, we can also define conditional probabilities
\[
\Pi(s_k|s_j) = \frac {|Z(s_j,s_k)|^2}
{\sum _{s'_k}|Z(s_j,s'_k)|^2}
\]
from which
\[
\Pi (s_j;s_k) = P(s_j) \Pi(s_k|s_j)
\]
and similarly for $\Pi (s_k;s_j)$.
Now, $P(s_j)=$ $\Pi(s_j;+_k)$ $+$ $\Pi(s_j;-_k)$ but generically
$\Pi (s_j;s_k)$ $\neq$ $\Pi (s_j,s_k)$ $\neq$ $\Pi (s_k;s_j)$.
When values $s_j$ and $s_k$ are jointly observable, as in the singlet state, it must happen that $\Pi(s_j;s_k)=$
$\Pi (s_j,s_k)=$ $\Pi(s_k;s_j)$, matching the observed $P(s_j,s_k)$. This will happen if the physical contexts associated to $P(s_j)$ and $P(s_j,s_k)$ are compatible.
Let us consider the quaternion
\[
{\bf N}[{\bf n}]= ({\bf n}\cdot{\bf i}){\bf I} +
({\bf n}\cdot{\bf j}){\bf J} + ({\bf n}\cdot{\bf k}){\bf K} \, ,
\]
with null real part, associated to a unit vector $\bf n$. Each spin state
$(s_1,\ldots,s_N)$ will have a fixed as\-so\-cia\-ted amplitude $Z$, sum of elementary amplitudes $s_j{\bf N}_j\equiv$
$s_j{\bf N}[{\bf n}_j]$, in analogy with the elementary amplitudes
$e^{iS[{\rm path}]/\hbar}$ for virtual paths in the path integral formalism,
$Z(s_1,\ldots,s_N) \equiv \sum _j s_j {\bf N}_j$
The physical context determines which {\it virtual spin} states are considered, in the same way that different phy\-si\-cal configurations determine the virtual paths to be taken into account, e.g. in the two slit experiment. The SQM state $|+_1>$, spin up in direction
${\bf n}_1$, can be prepared using a Stern--Gerlach apparatus
that splits the incoming trajectory into up and down spin output paths. The up path
does not have down spin field components, so that in the extended formalism
the ensemble state $|+_1\!>$ has associated distribution of amplitude
$Z_{+_1}(s_1,s_2,\ldots,s_N)$ where $Z_{+_1}(-_1,\ldots)\equiv 0$.
The marginals become $Z_{+_1}(-_1)=0$, $Z_{+_1}(+_1) = 2^{N-1}{\bf N}_1$,
$Z_{+_1}(s_2) = 2^{N-2}({\bf N}_1+s_2{\bf N}_2)$. When $s_2$ is measured
$\pm_j$ terms interfere for $j\geq 3$.
These marginal amplitudes determine the associated observable probabilities, $P_{+_1}(-_1)=0$, $P_{+_1}(+_1)=1$, as well as $P_{+_1}(s_2)=(1+s_2{\bf n}_1\cdot{\bf n}_2)/2$, where
the relations
\[
{\bf N}^*=-{\bf N} \quad {\bf N}^2=-1 \quad {\bf N}^*_1{\bf N}_2=
{\bf n}_1\cdot{\bf n}_2 - {\bf N}[{\bf n}_1\times {\bf n}_2]
\]
have been used. The SQM distributions are reproduced.
When the context allows both spin up and down field components in all directions the corresponding state $S_0$ becomes isotropic, with distribution of amplitude $Z_0$ containing all components in ${\cal P}h$, $Z_0\equiv Z$, and distributions of probability $P_0(s_j)=1/2$ for all $j$. This quantum state has no counterpart in the Hilbert space of SQM, where every vector of state is up eigenstate for the spin operator in some direction. A classical mixture like
\[
\rho=\frac {1}{2} |+_1><+_1| + \frac {1}{2} |-_1><-_1|
\]
reproduces the isotropic distribution too, but it
has different ontological content; $\rho$ represents two sub--ensembles of pure states,
$|-_1>$ and $|+_1>$, each one lacking the other field components, while $S_0$ contains all of them which can interfere. $S_0$ and $\rho$ are associated to different physical contexts.
Formal distributions of probability for two or more va\-lues of spin are obtained through marginal amplitudes and Born rule,
\[
\Pi_0(s_1,s_2) = (1+s_1s_2{\bf n}_1\cdot{\bf n}_2)/4 \, ,
\]
proportional to the (not normalized) squared marginal amplitude
\[
| 2^{N-2} \left( s_1 {\bf N}_1 + s_2 {\bf N}_2 \right) |^2
\]
as well as $\Pi_0(s_1,s_2,s_3) = $
\[
\frac {1}{24}(3 + 2s_1s_2 {\bf n}_1\cdot{\bf n}_2
+ 2s_1s_3 {\bf n}_1\cdot{\bf n}_3 + 2s_2s_3 {\bf n}_2\cdot{\bf n}_3)
\]
proportional to
\[
|2^{N-3} \left( s_1 {\bf N}_1 + s_2 {\bf N}_2 + s_3 {\bf N}_3 \right) |^2
\]
$\Pi _0(s_1,s_2)$ is not observable, but it is consistently defined: $\Pi_0(s_1,+_2)+$
$\Pi_0(s_1,-_2)$ $=P_0(s_1)$,
and $\Pi_0(s_1;s_2)$ $=\Pi_0(s_1,s_2)$, so that we can consider a ``classical'' distribution $P_0(s_1,s_2)$. On the other hand, a
$P(s_1,s_2,s_3)$ is not consistently defined,
\[
\Pi_0(s_1,s_2,+_3) + \Pi_0(s_1,s_2,-_3) \neq P_0(s_1,s_2)
\]
Notice the analogy with the two slit experiment
\[
P(x,y,L) + P(x,y,R) \neq P(x,y)
\]
\section{\label{sec3}The singlet state}
Two particles $\alpha$ and $\beta$ are jointly generated in the singlet spin state
\[
|S_{\rm singlet}> =
|+_1>^{\alpha}\otimes |-_1>^{\beta}
- |-_1>^{\alpha}\otimes |+_1>^{\beta}
\]
Each particle (marginal) density is isotropic in spin va\-ria\-bles, $P(s_j^{\alpha})=$ $1/2$ for all directions $j$; no individual pure quantum state of $\alpha$ in the two dimensional Hilbert spin space of SQM can represent it. In the usual interpretation of SQM, with one to one correspondence between physical and pure quantum states, a separable description of the singlet is not possible. On the other hand, in the extended phase space where pure quantum states, distributions of quaternion amplitudes, represent ensembles of physical states, the $\alpha$-$\beta$ correlation applies to jointly generated pairs
$(\alpha_{(\nu)},\beta_{(\nu)})$ and not to isotropic spin states $S_0^{\alpha}$ and
$S_0^{\beta}$, which describe statistical ensembles for each particle separately.
It is obvious that there is not correlation between pairs of outputs for Alice and Bob measurements over particles $\alpha_{(\nu)}$ and $\beta_{(\nu')}$ belonging to different pairs
$\nu\neq\nu'$. Correlations apply to jointly generated particles $s_{j(\nu)}^{\alpha}+s_{j(\nu)}^{\beta}=0$.
The singlet state is expressed in separable form $S_{\rm singlet} =$
$S_0^{\alpha}\otimes_{\rm corr}S_0^{\beta}$ if
$\otimes_{\rm corr}$ is understood as the perfect anti correlation between
jointly generated pairs. Each particle, if we ignore the companion, is in the pure state $S_0$ of EQM.
Equivalently, a distribution $Z_{\rm singlet}$ can be defined on the subset
${\cal P}h_{\rm corr}\subset$ ${\cal P}h_{\alpha}\times$ ${\cal P}h_{\beta}$ defined by the correlation equations, or
\[
Z_{\rm singlet}\Big(
(s_1^{\alpha},\ldots,s_N^{\alpha}),
(s_1^{\beta},\ldots,s_N^{\beta})\Big)\equiv 0
\]
outside ${\cal P}h_{\rm corr}$ ($s_{j}^{\alpha}+s_{j}^{\beta}\neq0$ for some $j$) and
\[
Z_{\rm singlet}\Big(
(s_1^{\alpha},\ldots,s_N^{\alpha}),
(-s_1^{\alpha},\ldots,-s_N^{\alpha})\Big)\equiv
\]
\[
\equiv Z_0(s_1^{\alpha},\ldots,s_N^{\alpha}) = - Z_0(s_1^{\beta},\ldots,s_N^{\beta})
\]
The $\alpha$ ($\beta$) marginal of $Z_{\rm singlet}$, when projecting
from ${\cal P}h_{\rm corr}$ onto ${\cal P}h_{\alpha}$ (${\cal P}h_{\beta}$), becomes trivially
(there is only one non vanishing term in the fibre of the projection) the isotropic
$S_0^{\alpha}$ ($S_0^{\beta}$), i.e., they are pure quantum states and not mixtures as
the marginals of the density $\rho_{\rm singlet}=$ $|S_{\rm singlet}><S_{\rm singlet}|$.
The formal distribution $\Pi _0 (s^{\alpha}_{j(\nu)},s^{\alpha}_{k(\nu)})$ is observable.
Recall it is consistent with $\Pi _0 (s^{\alpha}_{j(\nu)};s^{\alpha}_{k(\nu)})$ and
$\Pi _0 (s^{\alpha}_{k(\nu)};s^{\alpha}_{j(\nu)})$, defining a classical distribution. The second value of spin is inferred from the output of Bob measurement over $\beta _{(\nu)}$, without perturbing the state of $\alpha _{(\nu)}$. It means we observe (infer) the
value of $s^{\alpha}_{k(\nu)}$ previous to measurement of $s^{\alpha}_{j(\nu)}$.
Phy\-si\-cal splitting into $\pm_k$ spin field components of
$\beta _{(\nu)}$, at Bob's apparatus, does not perturb Alice's $\alpha _{(\nu)}$ particle. Splitting of the $\alpha _{(\nu)}$ $\pm_j$ spin field components maintains on each branch both $\pm_k$ (and other $\pm_l$) spin components of the
total spin field, which interfere.
Both ${\bf n}_j$ and ${\bf n}_k$ are freely and independently chosen by Alice and Bob. The correlations in each individual isotropic spin state $S_0$ are an inner property of each particle separately. The predicted distributions of probability obtained from $S_0$
through marginal amplitudes and Born rule, observable because of perfect anti correlation,
match the SQM predictions for the entangled singlet state.
When considering a third direction, interference in
\[
|Z_0(s_1,s_2,+_3)+Z_0(s_1,s_2,-_3)|^2
\]
does not vanish. A global classical distribution
of probability $P_{Cl}(s_1,\ldots)$ does not exist, according to Bell's inequalities. As in the two slit experiment,
field components of hidden, not measured magnitudes are superposed and interfere.
The only relevant distinction between both physical processes is that $x$ and $y$ position coordinates at the final screen commute and can be jointly measured on an individual particle, while $s_1$ and $s_2$ do not commute and one of them can only be inferred from measurement on the correlated companion. Counterfactual values are widespread in Physics, and our degree of confidence in them is linked to our confidence (empirically grounded) in the applied theory, in this case Quantum Mechanics. The property of consistency depends on the quantum state, here the isotropic spin $S_0$. We could calculate in an orthodox $Z_{+_1}$ state
formal joint or conditional correlations between $s_2$ and $s_3$, but $\Pi_{+_1}(s_2;s_3)$
$\neq \Pi_{+_1}(s_2,s_3)$ $\neq \Pi_{+_1}(s_3;s_2)$ are incompatible. $s_2$ and $s_3$ variables are not jointly observable.
The contextual character of the quantum distributions is already present in the paradigmatic two slit ex\-pe\-ri\-ment. Two non vanishing probabilities $P_R(x,y)$ and $P_L(x,y)$, applied each to the physical context with one slit open and the other closed, do not add to the distribution $P(x,y)$ in the third context with both slits open. Wave superposition and interference, a typical phenomenon for distributed fields, is behind this contextual behaviour of the quantum probabilities, and suggest to interpret
elementary particles as composites
made of a corpuscular system and a distributed, relativistically (locally, causally) evolving field.
The same phenomenon, applied to spin field components, is found in the isotropic spin state,
which is a pure quantum state in EQM without counterpart in SQM.
Other entangled composites in SQM could also find
a local, separable and contextual representation through new states in
adequate extended phase spacse of EQM.
\section{\label{ack}Acknowledgements}
Financial support from project MTM2015-64166-C2-1-P (Spain) is acknowledged.
\end{document} | math | 21,863 |
\begin{document}
\renewcommand{1.07}{1.07}
\title[Punctured Limit Sets]
{Fatou Components with Punctured Limit Sets}
\author{Luka Boc-Thaler}
\author{John Erik Forn\ae ss}
\author{Han Peters}
\subjclass[2000]{32E20, 32E30, 32H02}
\date{April 15, 2013}
\keywords{}
\vfuzz=2pt
\vskip 1cm
\begin{abstract}
We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^2$. In the recurrent case these components were classified by Sibony and the second author in 1995. In 2008 Ueda completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and the third author classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of $\mathbb{C}^2$ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.
\epsilonnd{abstract}
\maketitle \vfuzz=2pt
\section{Introduction}
Let $F$ be a holomorphic endomorphism of $\mathbb{P}^2$. Recall that $z \in \mathbb{P}^2$ lies in the \epsilonmph{Fatou set} if there exists a neighborhood $U(z)$ on which the family of iterates $\{F^n\}$ is normal. A connected component of the Fatou set is called a \epsilonmph{Fatou component}.
Fatou components for rational functions acting on the Riemann sphere have been precisely described. Sullivan \cite{Sullivan1985} proved in 1985 that every Fatou component is (pre-) periodic. Invariant Fatou components were classified by Fatou, who showed that an invariant Fatou component is either the basin of an attracting fixed point, the basin of a parabolic fixed point, or a rotation domain conformally equivalent to either the unit disk or an annulus. The existence of rotation domains equivalent to a disk was later shown by Siegel, and the existence of rotation domains equivalent to an annulus was shown by Herman in 1979.
Fatou components in higher dimensional projective space were studied in \cite{FS1994, FS1995b, HP1994, JL2003, LP2012, Ueda1994, U2008, Weickert2003}. The following definition is due to Bedford and Smillie \cite{BS1991a}.
\begin{definition}
An invariant Fatou component $\Omega$ is called \epsilonmph{recurrent} if there exists an orbit in $\Omega$ with an accumulation point in $\Omega$.
\epsilonnd{definition}
It follows that if $\Omega$ is non-recurrent then all orbits in $\Omega$ converge to the boundary $\partial \Omega$. Recurrent Fatou components were classified in 1995 by Sibony and the second author \cite{FS1995}:
\begin{theorem}[Forn{\ae}ss-Sibony]\label{thm:recurrent}
Let $F$ be a holomorphic endomorphism with a recurrent invariant Fatou component $\Omega$. Then one of the following holds.
\begin{enumerate}
\item $\Omega$ is the basin of an attracting fixed point $p \in \Omega$.
\item \label{case:two} All orbits in $\Omega$ converge to a closed invariant $1$-dimensional submanifold $\Sigma \subset \Omega$. The map $F$ acts on $\Sigma$ as an irrational rotation, and $\Sigma$ is biholomorphically equivalent to either the disk, the punctured disk, or an annulus.
\item $\Omega$ is a \epsilonmph{Siegel domain}: There exists a sequence $(n_j)$ so that $F^{n_j}$ converges uniformly on compact subsets of $\Omega$ to the identity map.
\epsilonnd{enumerate}
\epsilonnd{theorem}
The punctured disk in Case \epsilonqref{case:two} was ruled out in 2008 by Ueda \cite{U2008}.
\begin{theorem}[Ueda]\label{thm:Ueda}
The invariant submanifold $\Sigma$ in Case \epsilonqref{case:two} of Theorem \ref{thm:recurrent} cannot be equivalent to a punctured disk.
\epsilonnd{theorem}
All other cases are known to occur, which makes the classification of recurrent Fatou components for holomorphic endomorphisms of $\mathbb{P}^2$ complete.
The situation is more complicated in the non-recurrent case. If $\Omega$ is a non-recurrent invariant Fatou component then by normality there exists an increasing sequence $(n_j)$ such that the maps $f^{n_j}$ converge to a limit map $h: \Omega \rightarrow \partial \Omega$. The main difficulty in dealing with non-recurrent Fatou components arises because it is not known whether the limit set $h(\Omega)$ is always of the sequence $(n_j)$. If $h(\Omega)$ is independent of $(n_j)$ then we say that $\Omega$ has a \epsilonmph{unique limit set}. The following was proved in 2012 by Lyubich and the third author \cite{LP2012}.
\begin{theorem}[Lyubich-Peters]\label{thm:MishaHan}
Let $F$ be a holomorphic endomorphism of $\mathbb{P}^2$ of degree at least $2$, and let $\Omega$ be a non-recurrent invariant Fatou component with a unique limit set. Then $h(\Omega)$ is either a fixed point, or $h(\Omega)$ is equivalent to the unit disk, an annulus, or a punctured disc, and $F$ acts on $h(\Omega)$ as an irrational rotation.
\epsilonnd{theorem}
Examples where $h(\Omega)$ is a fixed point, a rotating disk or a rotating annulus were known to exist, but whether the punctured disk could also exist remained open. In light of the aforementioned result by Ueda one might expect the punctured disk not to exist in the non-recurrent case either. In Theorem \ref{main} we give an explicit construction of a non-recurent Fatou component with a unique limit set equivalent to a punctured disk.
In the last section we will give a general construction for many more punctured limit sets.
\begin{theorem}\label{thm:regular}
Let $V \in \mathbb{C}^2$ be a pure one-dimensional analytic set. Then there exist a holomorphic endomorphism $F$ of $\mathbb{C}^2$ such that for every irreducible component $V_1$ of $V$ the map $F$ has a non-recurrent Fatou component $\Omega$ on which all orbits converge to $V_1 \setminus \mathrm{Sing}(V)$.
\epsilonnd{theorem}
The idea is the following. We will construct a map $F = \mathrm{Id} + G$, where $G$ vanishes on the analytic set $V$. By our construction the map $F$ will have a \epsilonmph{parabolic curve} attached to each point $(z,w) \in \mathrm{Reg}(V)$, and these curves vary continuously with $(z,w)$. The union of these curves will be contained in an inviariant Fatou component whose orbits converge to $V$. The fact that the singular points of $V$ are not contained in the limit set then follows from the following result from \cite{LP2012}.
\begin{theorem}[Lyubich-Peters]\label{smoothness}
Let $X$ be a $2$-dimensional complex manifold and $F: X \rightarrow X$ a holomorphic endomorphism. Let $\Omega \subset X$ be an invariant Fatou component and suppose that the sequence $(f^{n_j})$ converges uniformly on compact subsets of $\Omega$ to a rank $1$ limit map $h: \Omega \rightarrow \partial \Omega$. Then $h(\Omega)$ is an injectively immersed Riemann surface.
\epsilonnd{theorem}
The local dynamics of maps tangent to the identity plays an important role in our results. Let us recall the basic notions here. Let $F : (\mathbb{C}^2, p) \rightarrow (\mathbb{C}^2, p)$ be the germ of a holomorphic map. If $DF(p) = \mathrm{Id}$ then we say that $F$ is \epsilonmph{tangent to the identity} at the point $p$. In other words, after changing coordinates by a translation $F$ takes the form
\begin{equation*}
F = \mathrm{Id} + F_k + F_{k+1} + \ldots,
\epsilonnd{equation*}
where each $F_j$ is a homogeneous polynomial of degree $j$, with \epsilonmph{order} $k \ge 2$. Following Hakim \cite{Hakim1998} we say that $v \in \mathbb{C}^2$ is a \epsilonmph{characteristic direction} for $F$ if there exists a $\lambda \in \mathbb{C}$ so that
\begin{equation*}
F_k(v) = \lambda v,
\epsilonnd{equation*}
If $\lambda = 0$ then $v$ is said to be \epsilonmph{degenerate}, while if $\lambda \neq 0$ then $v$ is \epsilonmph{non-degenerate}. An orbit $\{F^n(z)\}$ is said to converge to the origin \epsilonmph{tangentially} to $v$ if $F^n(z) \rightarrow 0$ and $[F^n(z)] \rightarrow [v]$ in $\mathbb{P}^1$.
A \epsilonmph{parabolic curve} for $F$ tangent to $[v]\in \mathbb{P}^{1}$ is an injective holomorphic map $\varphi: \mathbb{D}\rightarrow\mathbb{C}^2\backslash\{0\}$, satisfying the following properties:
\begin{itemize}
\item $\varphi$ is continuous at $1\in\partial \mathbb{D}$ and $\varphi(1)=0$,
\item $\varphi(\mathbb{D})$ is $F$-invariant and $(F|_{\varphi(\mathbb{D})})^n\rightarrow 0$ uniformly on compact subsets,
\item $[\varphi(\zeta)]\rightarrow[v]$ as $\zeta\rightarrow 1$ in $\mathbb{D}$.
\epsilonnd{itemize}
\begin{theorem}[Hakim]\label{thm:Hakim} Let $F: (\mathbb{C}^2,0) \rightarrow (\mathbb{C}^2,0)$ be a holomorphic germ tangent to the identity of order $k \ge 2$. Then for any non-degenerate characteristic direction $v$ there exist (at least) $k-1$ parabolic curves for $F$ tangent to $[v]$.
\epsilonnd{theorem}
The layout of the paper is as follows. In Section (2) we construct a holomorphic endomorphism of $\mathbb{P}^2$ with an invariant Fatou component $\Omega$ where the limit set is a punctured disk in the boundary of $\Omega$. In Section (3) we construct a large class of holomorphic and polynomial maps for which there exist non-recurrent Fatou components with limit sets equal to the regular parts of analytic sets.
\begin{remark}
The endomorphism of $\mathbb{P}^2$ that we construct in Section (2) is a special case of a one-resonant biholomorphism, which were studied by Bracci and Zaitsev in \cite{BZ2013}. We note that our example is not parabolically attracting, and their main result does not hold for our construction. The maps we study in Section (2) are examples of maps tangent to the identity on one-dimensional analytic subset. Such maps were studied in great detail in \cite{BM2003} and \cite{ABT2004}.
\epsilonnd{remark}
{\bf Acknowledgement.}
The third author was supported by a SP3-People Marie Curie Actionsgrant in the project Complex Dynamics (FP7-PEOPLE-2009-RG, 248443).
\section{Construction of a punctured disk}
Throughout this section we let $f$ be the polynomial endomorphism of $\mathbb{C}^2$ given by
\begin{equation*}
f(z,w) = (\lambda z + z^3, \lambda^{-1} (w + zw^2) + w^3).
\epsilonnd{equation*}
Here $\lambda = e^{2\pi i \theta}$, where $\theta \in \mathbb{R} \setminus \mathbb{Q}$ is chosen such that the maps $z \mapsto \lambda z + z^3$ and $w \mapsto \lambda^{-1}w + w^3$ are linearizable in a neighborhood of the origin. Observe that the polynomial map $f$ extends to a holomorphic endomorphism of $\mathbb{P}^2$, given in homogeneous coordinates by
\begin{equation*}
F[Z:W:T] = [\lambda ZT^2 + Z^3: \lambda^{-1} (WT^2 + ZW^2) + W^3: T^3]
\epsilonnd{equation*}
Our main result is the following.
\begin{theorem}\label{main}
The map $F$ has an invariant Fatou component $\Omega$ on which all orbits converge to an embedded punctured disk $D^\star \subset \partial \Omega$, and $F$ acts on $D^\star$ as an irrational rotation.
\epsilonnd{theorem}
The set $D$ will actually be a bounded simply connected subset of the $z$-axis, hence it will be sufficient to only consider the map $f$ and work in Euclidean coordinates. We write $(z_n, w_n)=f^n(z_0, w_0)$ and define
\begin{equation*}
\varphi_n(z,w) = (z, \lambda^n w),
\epsilonnd{equation*}
and
\begin{align*}
G_n & = \varphi_n \circ f \circ \varphi_{n-1}^{-1}\\
& = (\lambda z+ z^3, w + \lambda^{1-n} zw^2 + \lambda^{-2n+3} w^3).
\epsilonnd{align*}
We also write
\begin{equation*}
g_n(w) = w + \alpha_{n-1} w^2 + \lambda^{-2n+3} w^3,
\epsilonnd{equation*}
where $\alpha_n = \lambda^{-n}z_n$. Since we have chosen $\lambda$ such that the map $z \mapsto \lambda z + z^3$ is linearizable, there exists a local change of coordinates $\epsilonta(z)=z+z^2h(z)$, conjugating our map $z \mapsto \lambda z + z^3$ to the linear $\zeta \mapsto \lambda \zeta$. Let $A>0$ be such that $|h(z)|<A$ for all $z$ sufficiently close to the origin, so that we obtain
\begin{equation*}
|\lambda^n \epsilonta(z)- \epsilonta (\lambda^{n}z)|=|z^2 h(z)-z^2\lambda^nh(\lambda^n z)|<2A|z|^2.
\epsilonnd{equation*}
Since we can also bound $|(\epsilonta^{-1})'(z)|< B$ for $z$ sufficiently close to $0$, it follows that
\begin{equation*}
|\alpha_n -z_0| =| \lambda^{-n} \epsilonta^{-1}( \lambda^n \epsilonta(z_0))-z_0| =| \epsilonta^{-1}( \lambda^n \epsilonta(z_0))- \epsilonta^{-1}( \epsilonta (\lambda^{n}z_0))|\leq C | z_0|^2
\epsilonnd{equation*}
where $C>0$ does not depend on $n$ and $z_0$. Hence for $z_0$ in a sufficiently small disk $D \subset \mathbb{C}_z$ centered at the origin we have that
\begin{equation}\label{eq:smallball}
|\alpha_n - z_0| \le \frac{|z_0|}{2}
\epsilonnd{equation}
holds for all $n \in \mathbb{N}$.
\begin{lemma}
For all $z_0 \in D \setminus \{0\}$ there exists an open set $\mathcal{C}_{z_0} \subset \{z = z_0\}$ on which
\begin{equation*}
\|f^n(z_0,w) - f^n(z_0, 0)\| \rightarrow 0,
\epsilonnd{equation*}
uniformly on compact subsets of $\mathcal{C}_{z_0}$.
\epsilonnd{lemma}
\begin{proof}
Notice that $f^n = \varphi_n^{-1} \circ G_n \circ \ldots G_1$. Since we are interested in the set of $w$-values for which $w_n$ converges to $0$, and the map $\varphi_n$ preserves distances to $(z_n,0)$, it is equivalent to consider the $w$-values for which
\begin{equation*}
\pi_2 (G_n \circ \ldots \circ G_1(z_0,w)) = g_n \circ \ldots \circ g_1(w)
\epsilonnd{equation*}
converges to $0$. We will write $g(n)$ for the composition $g_n \circ \ldots \circ g_1$. Let us also write
\begin{equation*}
u_n = \frac{1}{g(n)(w)}.
\epsilonnd{equation*}
We have that
\begin{equation*}
u_{n+1} = \frac{1}{g_{n+1}(\frac{1}{u_n})} = u_n - \alpha_n + O(\frac{1}{|u_n|}).
\epsilonnd{equation*}
Having chosen $z_0$ sufficiently close to the origin so that Equation \epsilonqref{eq:smallball} holds, it follows that $|u_n| \rightarrow \infty$ if the initial value $u_0$ lies in a halfplane of the form
\begin{equation}\label{halfplane}
\mathbb{H}_{z_0} = \{u \in \mathbb{C} \mid \mathrm{Re}(u \bar{z_0}) < - K(z_0)\}.
\epsilonnd{equation}
Hence for all $u\in \mathbb{H}_{z_0}$ we have
\begin{equation*}
\|f^{n}(z_0,u^{-1})-f^n (z_0,0)\| \rightarrow 0.
\epsilonnd{equation*}
The statement of the lemma therefore holds for the set
\begin{equation*}
\mathcal{C}_{z_0} = \left\{(z_0, w) \mid \; \frac{1}{w} \in \mathbb{H}_{z_0} \right\}.
\epsilonnd{equation*}
\epsilonnd{proof}
In Equation \epsilonqref{halfplane} the constant $K(z)$ can be chosen to vary continiously with $z$ in the punctured disk $D \setminus \{0\}$, and therefore the sets $\mathcal{C}_{z_0}$ also vary continuously with $z \in D \setminus \{0\}$. Let $U \subset D$ be an $f$-invariant neighborhood of the origin and write
\begin{equation*}
V= \{(z_0, w_0) \mid z_0 \in U \setminus \{0\}, \; w_0 \in C_{z_0} \}.
\epsilonnd{equation*}
We then define the $f$-invariant open connected set $\Lambda$ by
\begin{equation*}
\Lambda=\bigcup_{n=0}^{\infty}f^n(V).
\epsilonnd{equation*}
Observe that $\{f^n\}_n$ is normal familly on $\Lambda$, hence $\Lambda$ is contained in some invariant Fatou component $\Omega$.
\begin{lemma}
Every orbit in $\Omega$ converges to the plane $\mathbb{C}_z$.
\epsilonnd{lemma}
\begin{proof}
Let $(f^{n_j})$ be a sequence that converges uniformly on compact subsets of $\Omega$ to a map $h$. Then $h(\Lambda) \subset \{w = 0\}$. Since $h$ is holomorphic and $\Lambda$ has interior, it follows that $h(\Omega) \subset \{w = 0\}$.
\epsilonnd{proof}
\begin{lemma}
Let $(f^{n_j})$ be a convergent subsequence with limit $h: \Omega \rightarrow \partial \Omega$. Then $h(\Omega)$ is contained in the Siegel disk in the plane $\mathbb{C}_z$, and is independent of the sequence $(n_j)$.
\epsilonnd{lemma}
\begin{proof}
Lemma 13 of \cite{LP2012} states that the restriction of the maps $\{f^n\}$ to the set $h(\Omega)$ must also form a normal family. Hence $h(\Omega)$ is contained in the Siegel disk in the $z$-plane centered at the origin, which we call $V$. Suppose that $k: \Omega \rightarrow \bar{\Omega}$ is any other limit map of the sequence $(f^n)$. By the same argument as above we have that $k(\Omega)$ lies in the Siegel disk $V$. Therefore it follows from the skew-product structure of $f$ that $k = \rho \circ h$, where $\rho: V \rightarrow V$ is a limit map of the sequence $(f^n)$ restricted to $V$. By invariance of $\Omega$ it follows that $h(\Omega) = k(\Omega)$.
\epsilonnd{proof}
\begin{lemma}
The Fatou component $\Omega$ is non-recurrent and the limit set $h(\Omega)$ is a punctured disk.
\epsilonnd{lemma}
\begin{proof}
It follows from the skew-product structure of $f$, and the fact that the restriction of $f$ to $\{z = 0\}$ is linearizable in a neighborhood of the origin, that no orbits in $\Omega$ converge to $(0,0)$. But $h(\Omega)$ does contain any point $(z,0)$ with $z \neq 0$ sufficiently small. Hence $h(\Omega)$ is a $1$-dimensional submanifold of $\mathbb{C}^2$ that is not equivalent to either the unit disk or to an annulus. Therefore it follows from Theorems \ref{thm:recurrent} and \ref{thm:Ueda} that $\Omega$ must be a non-recurrent Fatou component, and Theorem \ref{thm:MishaHan} implies that $h(\Omega)$ is an embedded punctured disk.
\epsilonnd{proof}
With this lemma we have completed the proof of Theorem \ref{main}.
\begin{remark}
One easily sees that the Siegel disk centered at $0$ at in the $z=0$ plane lies in the Julia set. Indeed, suppose that a point $(0,w)$ lies in the Fatou set, and let $U$ be a neighborhood of $(0,w)$ on which the family $\{f^n\}$ is normal. Since $\frac{\partial f^n}{\partial z}(0,w)$ and $\frac{\partial f^n}{\partial w}(0,w)$ are bounded away from $0$, and by our assumption that the iterates $(f^n)$ form a normal family, the union of the forward images of $U$ contains a tubular neighborhood of the $\omega$-limit set of $(0,w)$, which is a Jordan curve in the $z=0$ plane whose interior contains the origin. Hence for $|c|$ sufficiently small the intersection of this tubular neighborhood with the fiber $\{z = c\}$ contains an annulus. But then the family of iterates of $f$ restricted to the area enclosed by this annulus must be bounded, and thus a normal family. But this area includes the origin, which leads to a contradiction.
\epsilonnd{remark}
\section{Regular limit sets}\label{cusp}
Let $V \subset \mathbb{C}^2$ be any analytic set of pure dimension one. Observe that there exist an open cover $U_n\subset \mathbb{C}^2$ of $V$ and a collection of minimal defining functions $g_n\in{\epsilonnsuremath{\mathcal{O}}}(U_n)$ for sets $V\cap U_n$, i.e. $\{g_n=0\}=V\cap U_n$ and $\{g_n=dg_n=0\}=\mathrm{Sing}(V\cap U_n)$. Using the fact that the Cousin II problem is always solvable on any one-dimesional Stein space one can prove the existence of a minimal defining function $g\in{\epsilonnsuremath{\mathcal{O}}}(\mathbb{C}^2)$ for the set $V$. Let us define the following map
\begin{equation}\label{map}
\quad\quad\quad\quad\quad\quad\quad F = (z,w) + g^k(z,w)(P(z,w),Q(z,w)),\quad\quad\quad (k\geq 2),
\epsilonnd{equation}
where $P$ and $Q$ are holomorphic functions on $\mathbb{C}^2$ and $g$ is aminimal defining function of $V$. Observe that $F$ is tangent to identity on $V$. If we can find $P$ and $Q$ such that for each point in $\mathrm{Reg}(V)$ there exists a non-degenerate characteristic direction, then by Hakim's Theorem $\ref{thm:Hakim}$ there will be a parabolic curve attached to each point $(z,w)$ from the regular part of $V$. These curves vary continuously with the base point. The union of these curves will be contained in an inviariant Fatou component whose orbits converge to $V$. The fact that the singular points of $V$ are not contained in the limit set then follows from Theorem $\ref{smoothness}$.
Let us pick a point $(z_0,w_0)\in\mathrm{Reg}(V)$. After conjugating $F$ with the right translation map we note that the existence of a non-degenerate characteristic direction corresponds to
\begin{equation}\label{condition}
g_z(z_0,w_0)P(z_0,w_0)+g_w(z_0,w_0)Q(z_0,w_0)\neq0,
\epsilonnd{equation}
where $g_z$ and $g_w$ are the partial derivatives with respect to $z$ and $w$. We see immediately that in order to have non-characteristic directions it is necessary that the gradient of $g$ does not vanish on $\mathrm{Reg}(V)$. Observe that this condition is satisfied if and only if $g$ is a minimal defining function of set $V$. In order to prove Theorem \ref{thm:regular} we first have to prove the existence of holomorphic functions $P$ and $Q$ for which (\ref{condition}) holds.
\begin{definition}Let $V$ be an analytic set and let $\mathrm{Sing}(V)$ be the set of all singular points of $V$. A function $f:V \setminus \mathrm{Sing}(V) \rightarrow \mathbb{C}$ is
said to be \epsilonmph{weakly holomorphic} on $V$, if $f$ is holomorphic on $V \setminus \mathrm{Sing}V$ and locally bounded along $\mathrm(V)$. The sheaf of weakly holomorphic functions on $V$ is denoted by $\tilde{{\epsilonnsuremath{\mathcal{O}}}}_V$.
\epsilonnd{definition}
\begin{definition} Let $V$ be an analytic set in some open set $U$ in $\mathbb{C}^n$. A holomorphic function $f$ on $U$ is called a \epsilonmph{universal denominator} for $V$ at the point $z\in V$, if $f_z\cdot\tilde{{\epsilonnsuremath{\mathcal{O}}}}_{V,z}\subset {\epsilonnsuremath{\mathcal{O}}}_{V,z}$.
\epsilonnd{definition}
\begin{lemma}\label{lem:PQ}
Let $g$ be the minimal defining function of an analytic set $V\subset\mathbb{C}^2$. There exist holomorphic functions $P$ and $Q$ on $\mathbb{C}^2$ such that $g_zP+g_wQ$ does not vanish on $\mathrm{Reg}(V)$.
\epsilonnd{lemma}
\begin{proof} The partial derivatives of $g$ with respect to $z$ and $w$ will be denoted by $g_z$ and $g_w$ respectively.
Recall that an analytic set $V=\{g(z,w)=0\}$ is a Stein space. Let us denote by ${\epsilonnsuremath{\mathcal{O}}}_V$ the coherent sheaf of holomorphic functions on $V$.
If $V$ is a manifold, then the partial derivatives $g_z|_V$ and $g_w|_V$ do not have common zeros and so they generate ${\epsilonnsuremath{\mathcal{O}}}_V$. By Cartan's Division Theorem (Corollary 2.4.4., \cite{For}) there are $p,q\in{\epsilonnsuremath{\mathcal{O}}}_V$ such that $p\cdot g_z|_V+q\cdot g_w|_V=1$. Cartan's Extension Theorem (Corollary 2.4.3., \cite{For}) gives that every holomorphic function on $V$ can be extended to a holomorphic function on $\mathbb{C}^2$, which proves the existence of desired holomorphic functions $P$ and $Q$.
In general we can not expect $V$ to be a manifold. Let us denote the (discrete) set of singular points of $V$ by $\{a_n\}_{n\geq1}$. By our assumption on $g$ we have that $\mathrm{Sing}(V)= \{g = dg = 0\}$. Since $V$ is one-dimension variety, it has a normalization given by a non-compact Riemann surface $M$ together with a holomorphic map $\pi:M\rightarrow V$, \cite{Cir}. We can lift $g_z$ and $g_w$ to holomorphic functions on $M$ denoted by $\varphi$ and $\psi$ respectively. Since $M$ is a non-compact Riemann surface, by Weierstrass Theorem (Theorem 26.7., \cite{Fo}), we can find a holomorphic function $\theta$ on $M$ with prescribed zeros, such that $\varphi/\theta$ and $\psi/\theta$ are holomorphic on $M$ and without common zeros. As before, $M$ is a Stein manifold and $\varphi/\theta$, $\psi/\theta$ generate ${\epsilonnsuremath{\mathcal{O}}}_M$, so we can find $\tilde{p},\tilde{q}\in{\epsilonnsuremath{\mathcal{O}}}_M$ such that $$\frac{\varphi}{\theta}\tilde{p}+\frac{\psi}{\theta}\tilde{q}=1,$$ hence
\begin{equation*}
\varphi\tilde{p}+\psi\tilde{q}=\theta.
\epsilonnd{equation*}
The functions $\tilde{p}$ and $\tilde{q}$ induce weakly holomorphic functions $p,q\in \tilde{{\epsilonnsuremath{\mathcal{O}}}}_V$. For every $a_n\in \mathrm{Sing}(V)$ there exists an integer $m_n>0$ such that $g_z^{m_n}$ and $g_w^{m_n}$ are universal denominators in some neighborhood $a_n\in U_n\subset\mathbb{C}^2$ (Corollary $3.$ \cite{N1966}, p.$59.$). By expanding $(g_zp+g_wq)^{2m_n+1}$ in to series we obtain
\begin{equation*}
\label{1} (g_zp+g_wq)^{2m_n+1}=g_zP_n+g_wQ_n,
\epsilonnd{equation*}
with
\begin{align*}
P_n & =g_z^{m_n}\sum_{k=0}^{m_n} \binom{2m_n+1}{k} p^{2m_n+1-k}g_z^{m_n-k}(qg_w)^k, \; \; \mathrm{and}\\
Q_n & =g_w^{m_n}\sum_{k=0}^{m_n} \binom{2m_n+1}{k} q^{2m_n+1-k}g_w^{m_n-k}(pg_z)^k.
\epsilonnd{align*}
Since $g_z^{m_n}$ and $g_w^{m_n}$ are universal denominators we can conclude that $P_n$ and $Q_n$ are holomorphic on $U_n$.
From the construction it follows that $g_zp+g_wq$ is non-zero on $\mathrm{Reg}(V)\cap U_n$, and the same holds for the function $h_n:=g_zP_n+g_wQ_n$. We can take $\{U_n\}_{n\geq 1}$ to be pairwise disjoint. Let us take one more open set $U_0\subset\mathbb{C}^2$ such that $W_0=U_0\cap V\subset\mathrm{Reg}(V)$ and that $\{W_n=U_n\cap V\}_{n\geq0}$ covers $V$. The function $h_0=g_zp+g_wq$ is non-zero holomorphic on $W_0$. Note that $\{(h_n,W_n)\}_{n\geq0}$ is a Cousin II distribution and the Cousin II problem is always solvable on any one-dimensional Stein space (\cite{GR}, p.148). Hence there exists a holomorphic function $H\in {\epsilonnsuremath{\mathcal{O}}}_V$ with the property
\begin{equation*}
H|_{W_n}=h_n\varphi_n,
\epsilonnd{equation*}
where the holomorphic functions $\varphi_n$ are non-zero on $W_n$. If we define $\tilde{P_n}=P_n\varphi_n$ and $\tilde{Q_n}=Q_n\varphi_n$ (where $P_0=p$ and $Q_0=q$) observe that $$H|_{W_n}=\tilde{P_n}g_z+\tilde{Q_n}g_w.$$
Let $\mathcal{J}_V\triangleleft {\epsilonnsuremath{\mathcal{O}}}_V $ be the ideal sheaf generated by $g_z$ and $g_w$. Since $\mathcal{J}_V$ is a coherent analytic sheaf we are given a short exact sequence
$$0\rightarrow{\epsilonnsuremath{\mathcal{R}}}\stackrel{i}{\hookrightarrow}{\epsilonnsuremath{\mathcal{O}}}^2_V\stackrel{\tau}{\rightarrow}\mathcal{J}_V\rightarrow 0$$
where $i$ is an inclusion map and $\tau$ maps $(f_1,f_2)$ into $f_1g_z+f_2g_w$. Now we can form a long exact sequence of cohomology grups
$$0\rightarrow \Gamma(V,{\epsilonnsuremath{\mathcal{R}}})\stackrel{i^*}{\rightarrow}\Gamma(V,{\epsilonnsuremath{\mathcal{O}}}^2_V)\stackrel{\tau^*}{\rightarrow}\Gamma(V,\mathcal{J}_V)\rightarrow H^1(V,{\epsilonnsuremath{\mathcal{R}}})\rightarrow\cdots$$
and since $V$ is Stein we know that $H^1(V,{\epsilonnsuremath{\mathcal{R}}})=0$, hence $\tau^*$ is surjective. Observe that $H$ induces a section of $\mathcal{J}_V$. Since $\tau^*$ is surjective this section can belifted to a section of ${\epsilonnsuremath{\mathcal{O}}}_V^2$ (Section 7.2 \cite{Kra}), hence there exist holomophic functions $P$, $Q$ on $V$ such that $Pg_z+Qg_w=H$ on $V$. By Cartan's Extension Theorem we can extend $P$ and $Q$ to holomorphic functions on $\mathbb{C}^2$.
\epsilonnd{proof}
From now on let $F$ be of the form given in (\ref{map}), where $P$ and $Q$ are as in Lemma \ref{lem:PQ}.
\begin{lemma}\label{lemma:local}
For any $(z,w) \in \mathrm{Reg}(V)$ we can find a local change of coordinates that maps $(z,w)$ to $(0,0)$ so that in the new coordinates $F$ takes the form
\begin{equation}\label{germ}
\left\{ \begin{aligned}
x_1 & = x - x^k + x^k O(x,u),\\
u_1 & = u + x^k O(x,u).
\epsilonnd{aligned} \right.
\epsilonnd{equation}
\epsilonnd{lemma}
\begin{proof}
Near any regular point of $V$ we can change coordinates so that $V$ is given by $\{(z,w) \mid z = 0\}$. By its definition the map $F$ now takes the form
\begin{equation*}
\left\{ \begin{aligned}
z_1 & = z + z^kp(z,w),\\
w_1 & = w + z^kq(z,w).
\epsilonnd{aligned} \right.
\epsilonnd{equation*}
Note that if we write $F = Id + F_k + F_{k+1} + \ldots$, where $F_j$ is homogeneous of degree $j$, then the fact that $dg \neq 0$ on $V$ implies that $F_k$ is non-zero at $(0,0)$. The assumption that $g_zP+g_wQ$ does not vanish on $\mathrm{Reg}(V)$ implies that there is a non-degenerate characteristic direction at each point in $\mathrm{Reg}(V)$, in particular at $(0,0)$. This direction cannot be $(0,1)$, which is degenerate, hence we may write $(1,\lambda)$ for the non-degenerate characteristic direction. We can now find a linear change of coordinates that fixes $V = \{z = 0\}$ and changes the non-degenerate characteristic direction to the tangent vector $(1,0)$. Therefore in these new coordinates $F$ takes on the form
\begin{equation*}
\left\{ \begin{aligned}
x_1 & = x + \beta x^k + x^k O(x,u),\\
u_1 & = u + x^k O(x,u).
\epsilonnd{aligned} \right.
\epsilonnd{equation*}
By conjugating with a linear map $(x,u) \mapsto ((- \beta)^{\frac{1}{1-k}}x, u)$ we obtain the required form.
\epsilonnd{proof}
Let us rewrite Equation \epsilonqref{germ} as
\begin{equation}\label{germ2}
\left\{ \begin{aligned}
x_1 & = x - x^k(1 + u \varphi(u)) + O(x^{k+1}),\\
u_1 & = u + x^k O(x,u).
\epsilonnd{aligned} \right.
\epsilonnd{equation}
We note that for a fixed value of $u$ the real attracting directions of the map $x \mapsto x_1$ satisfy
\begin{equation*}
\mathrm{Arg}(x) = \mathrm{Arg}(x^k(1 + u\varphi(u))) = k \mathrm{Arg}(x) + \mathrm{Arg}(1 + u\varphi(u)),
\epsilonnd{equation*}
hence the $k-1$ attracting real directions satisfy
\begin{equation*}
\mathrm{Arg}(x) = \theta_u(m) = \frac{2m\pi - \mathrm{Arg}(1 + u \varphi(u)) }{k-1},
\epsilonnd{equation*}
for $m = 1 , \ldots , k-1$. For $\varepsilon >0 $ sufficiently small we define the regions
\begin{equation*}
R_\varepsilon(m) := \left\{(x,u) \mid 0 < |x| < \varepsilon, \; |u| < 2 \varepsilon, \; |\mathrm{Arg}(x) - \theta_u(m)| < \frac{\pi}{2k-2} \right\}.
\epsilonnd{equation*}
\begin{lemma}\label{lemma:attracting}
For $\varepsilon>0$ sufficiently small the iterates $F^{n}$ converge uniformly on each $R_\varepsilon(m)$ to the axis $\{x = 0\}$.
\epsilonnd{lemma}
\begin{proof}
For a fixed $u$-value the convergence of the $x$-coordinate to $0$ follows from standard estimates. Two estimates guarantee that variation in the $u$-coordinate does not break this convergence. On the one hand we have for sufficiently small $x, u$ that satisfy
\begin{equation}\label{eq:middle}
\frac{1}{3}\frac{\pi}{2k-2} \le \left|\mathrm{Arg}(x) - \theta_u(m)\right| \le \frac{5}{3}\frac{\pi}{2k-2}
\epsilonnd{equation}
the estimate
\begin{equation}\label{eq:rotation}
\left|\mathrm{Arg}(x_1) - \theta_{u_1}(m)\right| < \left|\mathrm{Arg}(x) - \theta_u(m)\right|,
\epsilonnd{equation}
and even
\begin{equation}\label{eq:rotation2}
|x|\left(\left|\mathrm{Arg}(x) - \theta_u(m)\right| - \left|\mathrm{Arg}(x_1) - \theta_{u_1}(m)\right|\right) \gg |u_1 - u|.
\epsilonnd{equation}
On the other hand we have for sufficiently small $x, u$ that satisfy
\begin{equation}\label{eq:center}
\left|\mathrm{Arg}(x) - \theta_u(m)\right| < \frac{1}{2}\frac{\pi}{2k-2}
\epsilonnd{equation}
that
\begin{equation}\label{eq:attract}
|x_1| < |x|,
\epsilonnd{equation}
and even
\begin{equation}\label{eq:attract2}
|x| - |x_1| \gg |u_1 - u|.
\epsilonnd{equation}
The statement of the lemma follows.
\epsilonnd{proof}
Let us write
\begin{equation*}
U_\varepsilon = \left\{(x,u) \mid |x| < \varepsilon, \; |u| < 2\varepsilon \right\},
\epsilonnd{equation*}
and define the nested sequence
\begin{align*}
W_0(m) & = R_\varepsilon(m), \; \; \; \mathrm{and} \\
W_{n+1}(m) & = (W_n(m) \cup f^{-1} W_{n}(m)) \cap U_\varepsilon.
\epsilonnd{align*}
Finally we define
\begin{equation*}
W(m) = \bigcup_{n \in \mathbb{N}} W_n(m).
\epsilonnd{equation*}
Since $f$ is invertible in a neighborhood of $(0,0)$, for sufficiently small $\varepsilon$ the sets $W(m)$ are open, connected and disjoint. Moreover we have the following.
\begin{lemma}\label{lemma:separatrix}
Suppose $x_0$ and $u_0$ satisfy $|x_0| < \varepsilon$ and $|u_0| < \varepsilon$, and $(x_0, u_0)$ does not lie in one of the sets $W(m)$. Then there is an $N \in \mathbb{N}$ such that the orbit $(z_n) = ((x_n, u_n))$ satisfies
\begin{enumerate}
\item[$\bullet$] For $n \le N$ we have $|x_n| < \varepsilon$ and $|u_n| < 2 \varepsilon$, and
\item [$\bullet$] $|x_N| \ge \varepsilon$.
\epsilonnd{enumerate}
\epsilonnd{lemma}
\begin{proof}
Considering Equations \epsilonqref{eq:center}, \epsilonqref{eq:attract} and \epsilonqref{eq:attract2} for $F^{-1}$ gives for sufficiently small $x, u$ that satisfy
\begin{equation*}
\left|\mathrm{Arg}(x) - (\theta_u(m) + \frac{\pi}{k-1})\right| < \frac{1}{2}\frac{\pi}{2k-2}
\epsilonnd{equation*}
the estimate
\begin{equation*}
|x_1| > |x|,
\epsilonnd{equation*}
and even
\begin{equation*}
|x_1| - |x| \gg |u_1 - u|.
\epsilonnd{equation*}
The combination with Equations \epsilonqref{eq:middle}, \epsilonqref{eq:rotation} and \epsilonqref{eq:rotation2} implies the statements of the lemma.
\epsilonnd{proof}
We can now complete the proof of the main result of this section.
\begin{proof}[{\bf Proof of Theorem \ref{thm:regular}}]
Let $g$ be a minimal defining function of $V$ and let $P$ and $Q$ be as in Lemma \ref{lem:PQ}. Let $k\geq 2$ and define
\begin{equation*}
F = (z,w) + g^k(z,w)(P(z,w),Q(z,w)).
\epsilonnd{equation*}
By Lemma \ref{lemma:local} we can find a local change of coordinates near any point in $(z,w)\in\mathrm{Reg}(V)$ to obtain the form \epsilonqref{germ}. In particular by Theorem \ref{thm:Hakim} we obtain $k-1$ parabolic curves on which we have attraction towards (in original coordinates) the point $(z, w)$.
Let us first assume that $\mathrm{Reg}(V)$ is connected. By Lemma \ref{lemma:local} and Lemma \ref{lemma:attracting} there exist, in a neighborhood of the point $(x,y)$, $k-1$ open regions $R_\varepsilon(m)$ on which all orbits converge uniformly to $V$. These regions vary smoothly with the point $(x,y)$ and each region $R_\varepsilon(m)$ intersects at least one of the $k-1$ parabolic curves at $(x,y)$. Consider the union over all base points $(x,y)$ of the regions $R_\varepsilon(m)$, and let $\tilde{\Omega}$ be one of the connected components of this union. Then $F^n$ converges uniformly on $\tilde{\Omega}$ to $\mathrm{Reg}(V)$. Hence $\tilde{\Omega}$ is contained in a Fatou component for $F$, which we denote by $\Omega$.
By the aforementioned non-empty intersection with the parabolic curves, there exist for each point $z \in \mathrm{Reg}(V)$ corresponding points in $\Omega$ whose orbits converge to $z$. Hence on $\Omega$ the iterates $F^n$ converges to a holomorphic map whose image contains $\mathrm{Reg}(V)$. By Lemma \ref{lemma:separatrix} it follows that $\mathrm{Reg}(V)$ cannot be contained in $\Omega$, hence must lie in $\partial \Omega$. Thus $\Omega$ is a non-recurrent Fatou component. By Theorem \ref{smoothness} it follows that the limit set must be exactly equal to $\mathrm{Reg}(V)$.
Now suppose that $\mathrm{Reg}(V)$ is not connected and let $V_1$ be an irreducible component of $V$. By the above argument there is a Fatou component $\Omega$ on which all orbits converge to $V_1 \setminus \mathrm{Sing}(V_1)$. It remains to be shown that there are no points converging to intersection points of $V_1$ with other irreducible components of $V$. Let $V_2 \neq V_1$ be a connected component of $V$ and assume that there is a point $z \in V_1 \cap V_2$. Let us recall an argument that was used in the proof of Theorem \ref{smoothness}. Suppose for the purpose of a contradiction that there exists a point $x \in \Omega$ whose orbit converges to $z$. Denote the limit of the sequence $(F^n)$ on $\Omega$ by $h$. Let $D$ be a holomorphic disk through $x$ so that $h(D) = h(U)$ for a small neighborhood $U$ of $x$. Then $F^n(D)$ intersects $V_2$ for $n$ large enough. But as we noted earlier, $V_2$ lies in the Julia set while $F^n(D)$ lies in the Fatou set, which gives a contradiction. This completes the proof. \epsilonnd{proof}
Let $\Omega$ be a Fatou component on which the orbits converge to the regular partner of an irreducile component $V_1 \subset V$. Then by Lemmas \ref{lemma:attracting} and \ref{lemma:separatrix} the orbit of any point in $\Omega$ eventually lands in one of the sets $R_\varepsilon(m)$. Vice versa, any point whose orbit eventually lands in $R_\varepsilon(m)$ is contained in $\Omega$. In particular the parabolic curve at $z$ corresponding to $R_\varepsilon(m)$ must be contained in $\Omega$.
\begin{proposition}
The orbit of any point in $\Omega$ is evetually mapped onto a parabolic curves contained in $\Omega$.
\epsilonnd{proposition}
\begin{proof}
What remains to be shown is that any point whose orbit lands in a set $R_\varepsilon(m)$ for $\varepsilon$ sufficiently small is contained in parabolic curve. Hence it is sufficient to show that, in the local coordinates given by Equation \epsilonqref{germ}, any point $(x,0)$ with $|x|$ sufficiently small and
\begin{equation}\label{eq:angle}
|\mathrm{Arg}(x) - \theta_u(m)| < \frac{\pi}{2k-2}
\epsilonnd{equation}
lies on a parabolic curve. By Hakim's proof of Theorem \ref{thm:Hakim} in \cite{Hakim1998} we know that, for $\varepsilon>0$ sufficiently small, the parabolic curve at $(0,u)$ is a graph over
\begin{equation*}
\{x \in \mathbb{C} \mid |x| < \varepsilon, \; \;|\mathrm{Arg}(x) - \theta_u(m)| < \frac{\pi}{2k-2} \},
\epsilonnd{equation*}
of the form $v_u(x) = u + x^k h_u(x)$, with $\|h_u\| < 1$. By Hakim's proof these graphs vary continuously with $h_u$, hence for $|x|$ small enough any point $(x,0)$ satisfying Equation \epsilonqref{eq:angle} lies on one of the parabolic curves.
\epsilonnd{proof}
If $k \ge 3$ then for each point in $\mathrm{Reg}(V)$ there exist $k-1$ parabolic curves. We investigate whether these curves lie in distinct Fatou components.
Note that the parabolic curves vary continuously with the basepoint in $\mathrm{Reg}(V)$. To each of these parabolic curves corresponds to a real tangent vector $\alpha$ for which
\begin{equation}
\mathrm{d}F \alpha = - \lambda \alpha,
\epsilonnd{equation}
with $\lambda \in \mathbb{R}^+$. The vector $\alpha$ is unique up to multiplication in $\mathbb{R}^+$. Let $z_0, z_1 \in \mathrm{Reg}(V)$ with corresponding parabolic curves $C_0$ and $C_1$ and real attracting tangent vectors $\alpha_0$ and $\alpha_1$. We say that $(z_0, C_0) \sim (z_1, C_1)$ if there exists a continuous map $\varphi$ from $[0,1]$ to the set of pairs $(z, \alpha)$, with $z \in \mathrm{Reg}(V)$ and $\alpha$ a real attracting tangent vector to $z$, so that $\varphi(0) = (v_0, \alpha_0)$ and $\varphi(1) = (v_1, \alpha_1)$. Clearly if $(z_0, C_0) \sim (z_1, C_1)$ then $C_0$ and $C_1$ lie in the same Fatou component. The converse also holds.
\begin{lemma}
The parabolic curves $C_0$ and $C_1$ lie in the same Fatou component if and only if $(z_0, C_0) \sim (z_1, C_1)$.
\epsilonnd{lemma}
\begin{proof}
We only need to show that if $C_0$ and $C_1$ lie in the same Fatou component, then $(z_0, C_0) \sim (z_1, C_1)$. Assume therefore that $w_1 \in C_1$ and $w_2 \in C_2$, and suppose that $w_1$ and $w_2$ lie in the same Fatou component $\Omega$. Let $\gamma$ be a
continuous real curve in $\Omega$, starting at $w_1$ and ending at $w_2$. Since $\gamma$ is compact we know by Theorem \ref{thm:regular} that the sequence $(F^n)$ converges uniformly on $\gamma$ to a limit set $h(\gamma)$ contained in $\mathrm{Reg}(V)$.
Since $h(\gamma)$ is compact we can find a uniform $\varepsilon >0$ for which the statements in Lemmas \ref{lemma:attracting} and \ref{lemma:separatrix} hold. Let $N \in \mathbb{N}$ be such that $\|F^n - h\|_\gamma < \varepsilon$ for all $n \ge N$. The curve $F^N(\gamma)$ lies in the invariant Fatou component $\Omega$, and still starts at a point in $C_0$ and ends at a point in $C_1$. It follows from Lemma \ref{lemma:separatrix} that $F^N(\gamma)$ must lie in the union of the open sets $W(m)$. Hence we can follow the real attracting direction of the sets $W(m)$, starting with the direction corresponding to $C_0$ and ending with the direction corresponding to $C_1$.
\epsilonnd{proof}
We give an simple family of polynomial endomorphisms of $\mathbb{C}^2$ for which we can easily determine whether the parabolic curves lie in distinct Fatou components. We let $p$ and $q$ be relatively prime, $k \ge 2$ be an integer, and define
\begin{equation*}
F(z,w) = (z,w) + (z^p-w^q)^k \cdot(z, -w).
\epsilonnd{equation*}
As noted in the proof of Theorem \ref{thm:regular} every point $(t^q, t^p) \in V = \{z^p - w^q = 0\}$ has a non-degenerate characteristic direction $(t^q, -t^p)$, and there are exactly $k-1$ parabolic curves tangent to this characteristic direction. Whether the curves $C_1, \ldots C_{k-1}$ all lie in distinct Fatou components or not depends on the values of $k, p$ and $q$.
\begin{proposition}
If $p\cdot q$ is divisible by $k-1$ then the curves $C_1, \ldots, C_{k-1}$ all lie in distinct Fatou components. If $p\cdot q$ and $k-1$ are relatively prime then $C_1, \ldots ,C_{k-1}$ all lie in the same Fatou component.
\epsilonnd{proposition}
\begin{proof}
We note that the orbits on the parabolic curves converge to $(t^p, t^q)$ along real directions $\alpha \cdot (t^q - t^p)$ satisfying
\begin{equation*}
G_k(\alpha \cdot (t^q, - t^p)) = \alpha^k (p-q)^k t^{kpq} (t^p, - t^q) = - \alpha \cdot (t^q, - t^p),
\epsilonnd{equation*}
which gives
\begin{equation*}
\alpha^{k-1} = \frac{-1}{(p-q)^k t^{kpq}}.
\epsilonnd{equation*}
Hence we see that if $p\cdot q$ and $k-1$ are relatively prime, and therefore $kpq$ and $k-1$ are also relatively prime, then if $t$ moves in a full circle around the origin then the $k-1$ parabolic curves have been permuted in a full cycle. As the parabolic curves vary smoothly this implies that the $k-1$ parabolic curves lie in the same Fatou component.
Suppose on the other hand that $pq$ is divisible by $k-1$. Then following the parabolic curves as $t$ makes one full circle around the origin leaves the parabolic curves invariant. However, loops around the origin generate the fundamental group of $\mathrm{Reg}(V)$, hence the monodromy group is trivial. It follows that two parabolic curves $C_1$ and $C_2$ touching $V$ at the same point $z$ lie in the same Fatou component if and only if $C_1 = C_2$.
\epsilonnd{proof}
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\epsilonnd{document} | math | 45,644 |
\begin{document}
\title[Dualizable algebras]{Dualizable algebras with parallelogram terms}
\author{Keith A. Kearnes}
\address[Keith Kearnes]{Department of Mathematics\\
University of Colorado\\
Boulder, CO 80309-0395\\
USA}
\email{[email protected]}
\author{\'Agnes Szendrei}
\address[\'Agnes Szendrei]{Department of Mathematics\\
University of Colorado\\
Boulder, CO 80309-0395\\
USA}
\email{[email protected]}
\thanks{This material is based upon work supported by
the Hungarian National Foundation for Scientific Research (OTKA)
grant no.\ K83219 and K104251.
}
\subjclass[2010]{Primary: 08C20; Secondary: 08A05, 08A40}
\keywords{natural duality, dualizable algebra, parallelogram term, cube term}
\begin{abstract}
We prove that if $\m a$ is a finite algebra
with a parallelogram term that satisfies
the split centralizer condition,
then $\m a$ is dualizable. This yields
yet another proof of the dualizability
of any finite algebra with a near unanimity term,
but more importantly proves that every finite
module, group or ring in a residually small
variety is dualizable.
\end{abstract}
\maketitle
\section{Introduction}\label{intro}
Natural Duality Theory investigates categorical
dualities that are mediated by finite
algebras.
One of the main goals of the theory is to
identify those finite algebras that can serve
as character algebras for a duality. Such algebras are
called \emph{dualizable}. Although general criteria
for dualizability
are known, the problem of identifying the finite
algebras satisfying the criteria is still difficult.
The broadest natural class of algebras
where the problem has been solved is the class of
finite algebras $\m a$ such that the prevariety
$\mathsf{S}\mathsf{P}(\m a)$ has tame congruence-theoretic
typeset
contained in $\{\btyp, \ltyp\}$.
According to the Big NU Obstacle Theorem
from~\cite{davey-heindorf-mckenzie}
a finite algebra satisfying this assumption is
dualizable if and only if it has a near
unanimity term operation.
This paper probes the broader class of
finite algebras $\m a$ such that the prevariety
$\mathsf{S}\mathsf{P}(\m a)$ has tame congruence-theoretic
typeset contained in $\{\atyp, \btyp, \ltyp\}$.
There is a natural analogue of a near unanimity term
operation that is likely to be involved in this context,
called a ``parallelogram term operation''. Recently
M.~Moore \cite{moore} has announced an extension of one direction
of the Big NU Obstacle Theorem, namely,
under the assumption that $\m a$
is finite and
$\mathsf{S}\mathsf{P}(\m a)$ has tame congruence-theoretic
typeset contained in $\{\atyp, \btyp, \ltyp\}$,
it is the case that if $\m a$ is dualizable then it
must have a parallelogram term operation. Known examples
show that an unmodified converse
to this statement cannot hold. Thus, we expect that
there is some condition $\varepsilon$ such that,
if $\m a$
is finite and
$\mathsf{S}\mathsf{P}(\m a)$ has tame congruence-theoretic
typeset contained in $\{\atyp, \btyp, \ltyp\}$, then
$\m a$ is dualizable if and only if $\m a$
has a parallelogram term operation and condition $\varepsilon$ holds.
In this paper we produce a candidate for condition
$\varepsilon$ under the additional assumption that
$\m a$ lies in a residually small variety.
We call the condition the ``split centralizer condition''.
Specifically, we operate under the global assumptions
that $\m a$
is a finite algebra from a residually small variety and
$\mathsf{S}\mathsf{P}(\m a)$ has tame congruence-theoretic
typeset contained in $\{\atyp, \btyp, \ltyp\}$.
We conjecture that, under the global assumptions,
$\m a$ will be dualizable if and only if
$\m a$ has a parallelogram term
operation and satisfies the split centralizer condition.
What we prove in this paper is the ``if'' part of this conjecture.
Let us reformulate the statement that we prove.
It is a fact that a finite algebra with a parallelogram term
automatically has the property that
$\mathsf{S}\mathsf{P}(\m a)$ has tame congruence-theoretic
typeset contained in $\{\atyp, \btyp, \ltyp\}$.
Thus, our result is that
if $\m a$ is finite, lies in a residually small variety,
has a parallelogram term, and satisfies the split centralizer
condition, then $\m a$ is dualizable.
Interestingly, our result is strong enough to
prove the dualizability of the dualizable algebras
in residually small varieties with a parallelogram term
in all the known cases,
and in many new cases, and yet neatly avoids
an obstacle to dualizability identified by Idziak in 1994.
Let us state the split centralizer condition.
Let $\m b$ be a finite algebra and let $\mathcal Q$
be a quasivariety containing $\m b$. A \emph{$\mathcal Q$-congruence}
on $\m b$ is a congruence $\kappa$ of $\m b$
such that $\m b/\kappa\in \mathcal Q$.
Let $\delta$ be a meet irreducible congruence on $\m b$
with upper cover $\theta$, and let $\nu = (\delta:\theta)$
be the centralizer of $\theta$ modulo $\delta$.
We will say that the triple of congruences $(\delta,\theta,\nu)$
which arises in this way
is \emph{split} (relative to $\mathcal Q$)
by a triple $(\alpha,\beta,\kappa)$ of congruences of $\m b$ if
\begin{enumerate}
\item[(i)] $\kappa$ is a $\mathcal Q$-congruence on $\m b$,
\item[(ii)] $\beta\leq \delta$,
\item[(iii)] $\alpha\wedge\beta = \kappa$,
\item[(iv)] $\alpha\vee\beta = \nu$, and
\item[(v)]
$\alpha/\kappa$ is abelian (or equivalently, $[\alpha,\alpha]\leq\kappa$).
\end{enumerate}
Now, if
$\theta/\delta$
is nonabelian, then
$\nu=(\delta:\theta) = \delta$, so
$(\delta,\theta,\nu)$ will be split by
$(\alpha,\beta,\kappa):=(0,\delta,0)$. Therefore
splitting is only in question when
$\theta/\delta$
is abelian. Henceforth we shall focus only on this case
and call a triple $(\delta,\theta,\nu)$
of congruences on $\m b$ \emph{relevant}
if
\begin{enumerate}
\item[(a)] $\delta$ is completely meet irreducible,
\item[(b)] $\theta$ is the upper cover of $\delta$,
\item[(c)] $\nu = (\delta:\theta)$, and
\item[(d)]
$\theta/\delta$ is abelian.
\end{enumerate}
The
\emph{split centralizer condition}
for a finite algebra
$\m a$ is the condition that, for $\mathcal Q := \mathsf{S}\mathsf{P}(\m a)$
and for any subalgebra $\m b\leq\m a$, each
relevant triple $(\delta,\theta,\nu)$ of $\m b$
is split (relative to $\mathcal Q$) by
some triple $(\alpha,\beta,\kappa)$.
The relationships between
these congruences of $\al B$
is
depicted in Figure~1.
It is not hard to show that if a finite algebra
with a parallelogram term satisfies
the split centralizer condition, then it generates a residually
small variety (combine Corollary~\ref{cor-ssc-rs} and
Theorem~\ref{thm-parterm-cm}).
Therefore our main result can be restated as follows.
\begin{thm}
\label{thm-main}
If $\al A$ is a finite algebra with a parallelogram term and $\al A$
satisfies the split centralizer condition, then $\al A$ is dualizable.
\end{thm}
Section~2 summarizes preliminaries.
Sections~3 through 5 are devoted to proving
Theorem~\ref{thm-main}.
In Section~6, we apply Theorem~\ref{thm-main} to prove that
if $\var{V}$ is a variety with a parallelogram term
in which every finite subdirectly irreducible algebra is either
abelian or neutral or almost neutral, then every finite algebra in
$\var{V}$ is dualizable (Theorem~\ref{maincor}).
This generalizes the following known results:
\begin{enumerate}
\item[(1)] every
finite algebra with a near unanimity term
is dualizable~\cite{davey-werner},
\item[(2)] every paraprimal algebra is dualizable~\cite{davey-quackenbush},
and
\item[(3)]
every finite commutative ring $R$ with unity whose Jacobson radical
squares to zero is dualizable~\cite{commutative_rings}.
\end{enumerate}
Theorem~\ref{maincor}
also implies the new results that
\begin{enumerate}
\item[(4)] every finite module is dualizable,
and
\item[(5)] every finite ring (commutative or not, unital or not)
in a residually small variety is dualizable.
\end{enumerate}
We also show how to apply Theorem~\ref{thm-main}
to deduce that
every finite group with abelian Sylow subgroups is
dualizable \cite{nickodemus}.
This last theorem is not a consequence of
Theorem~\ref{maincor}.
\section{Preliminaries}\label{prelim}
Algebras will be denoted by boldface letters, their universes by the
same letters in italics. For arbitrary algebras $\al A$ and $\al B$,
$\Aut(\al A)$ denotes the automorphism group of $\al A$,
$\Con(\al A)$ the congruence lattice of $\al A$, and $\Hom(\al A,\al B)$
the set of all homomorphisms $\al A\to\al B$.
The top and bottom elements of $\Con(\al A)$ are denoted $1$ and $0$,
respectively, and the identity map on any set $A$ is denoted $\id_A$.
The variety generated by an algebra $\al A$ is denoted by $\var{V}(\al A)$.
We will write $\al B\le\al A$ to indicate that $\al B$ is a subalgebra of
$\al A$. By a \emph{section} of $\al A$ we mean a quotient of a subalgebra
of $\al A$.
Let $\vartheta$ be an equivalence relation on a set $A$.
The $\vartheta$-class of an element $a\in A$ is denoted by $a/\vartheta$,
and the number of
equivalence classes of $\vartheta$ may be referred to as
\emph{the index of $\vartheta$}.
We will often write $a\equiv_{\vartheta} b$ instead of $(a,b)\in\vartheta$.
For $B\subseteq A$ the restriction of
$\vartheta$ to
$B$ is denoted by $\vartheta_B$. If $\al B\le\al A$ and
$\vartheta$ is a congruence
of $\al A$, we may write $\vartheta_{\al B}$ for $\vartheta_B$, which is
a congruence of $\al B$.
Let $\vartheta$ be a congruence of an algebra $\al A$, and let
$\al B$ be a subalgebra of $\al A$.
We say that $\al B$ is \emph{saturated with respect
to $\vartheta$}, or $\al B$ is a \emph{$\vartheta$-saturated subalgebra}
of $\al A$, if $b\in\al B$ and $b\equiv_{\vartheta}a$ imply
$a\in\al B$ for all $a\in\al A$.
In other words, $\al B$ is $\vartheta$-saturated if and only if
its universe is a union of $\vartheta$-classes of $\al A$.
For arbitrary subalgebra $\al B$ of $\al A$ there exists a smallest
$\vartheta$-saturated subalgebra of $\al A$ that contains $\al B$, which we
denote by $\al B[\vartheta]$; the universe of $\al B[\vartheta]$
is $B[\vartheta]:=\bigcup_{b\in B}b/\vartheta$. By the second
isomorphism theorem the map
$\al B/\vartheta_{\al B}\to\al B[\vartheta]/\vartheta_{\al B[\vartheta]}$,
$b/\vartheta_{\al B}\mapsto b/\vartheta_{\al B[\vartheta]}(=b/\vartheta)$
is an isomorphism, so the indices of $\vartheta_{\al B}$ and
$\vartheta_{\al B[\vartheta]}$ are equal.
For every natural number $m$ we will use the notation $[m]$ for the set
$\{1,2,\dots,m\}$.
\subsection{The Modular Commutator and Residual Smallness}
For the definition and basic properties of the commutator operation
$[\phantom{n},\phantom{n}]$ on congruence lattices of algebras in
congruence modular varieties the reader is referred to \cite{freese-mckenzie}.
To avoid confusion, intervals in congruence lattices will be denoted by
$\interval{\phantom{n}}{\phantom{n}}$.
Recall that a congruence $\alpha\in\Con(\al A)$ of an algebra $\al A$ is called
\emph{abelian} if $[\alpha,\alpha]=0$, and an interval
$\interval{\beta}{\alpha}$ in $\Con(\al A)$
is called abelian if $[\alpha,\alpha]\le\beta$,
or equivalently, if the congruence $\alpha/\beta\in\Con(\al A/\beta)$
is abelian.
For two congruences $\beta\le\alpha$ in $\Con(\al A)$ \emph{the centralizer of
$\alpha$ modulo $\beta$}, denoted $(\beta:\alpha)$, is the largest congruence
$\gamma\in\Con(\al A)$ such that $[\alpha,\gamma]\le\beta$.
The \emph{center} of $\al A$ is the congruence $\zeta:=(0:1)$.
For a cardinal $c$,
a variety $\var{V}$ is called \emph{residually less than $c$}
if every subdirectly irreducible algebra in $\var{V}$ has cardinality
$<c$; $\var{V}$ is called \emph{residually small} if it is residually
less than some cardinal.
It is proved in \cite[Theorem~10.14]{freese-mckenzie} that for a congruence
modular variety $\var{V}$ to be residually small, it is necessary that
the congruence lattice of every algebra $\al A\in\var{V}$ satisfy
the commutator identity
\begin{align}
[x\wedge y,y]&= x\wedge[y,y],\tag{C1}
\end{align}
which can also be expressed by the implication $x\le[y,y]\to x=[x,y]$
(see \cite[Theorem~8.1]{freese-mckenzie}).
For finitely generated varieties the converse is also true, as the following
theorem states.
\begin{thm}[{{}From \cite[Theorem~10.15]{freese-mckenzie}}]
\label{thm-rs}
Let $\al A$ be a finite algebra that generates a congruence modular variety
$\var{V}(\al A)$. Then the following conditions are equivalent:
\begin{enumerate}
\item[{\rm(a)}]
$\var{V}(\al A)$ is residually small,
\item[{\rm(b)}]
$\var{V}(\al A)$ is residually ${}<q$ for some natural number $q$,
\item[{\rm(c)}]
{\rm(C1)} holds in the congruence lattice of every subalgebra of $\al A$.
\end{enumerate}
\end{thm}
If $(\delta,\theta,\nu)$ is a relevant triple of an algebra $\al B$
such that (C1) holds in $\Con(\al B)$,
then $[\theta\wedge\nu,\nu]=[\theta,\nu]\le\delta$ implies that
$\theta\wedge[\nu,\nu]\le\delta$, and
since $\theta>\delta$ and $\delta$ is
completely meet irreducible, it must be the case that
$[\nu,\nu]\le\delta$.
Conversely, it is not hard to show (see the first paragraph of the proof of
\cite[Theorem~10.14]{freese-mckenzie}) that if (C1) fails in $\Con(\al B)$,
then there is a failure (with $x=\alpha$, $y=\beta$)
of the following form:
\[
[\alpha,\beta]\le\delta<\theta\le\alpha\le[\beta,\beta]\,(\le\beta)
\]
where $\delta$ is completely meet irreducible and $\theta$ is its upper cover.
It follows that
$[\theta,\theta]\le[\alpha,\beta]\le\delta$, so $\theta/\delta$ is abelian.
Moreover, $\nu:=(\delta:\theta)\ge\beta$ since
$[\theta,\beta]\le[\alpha,\beta]\le\delta$, which implies that
$[\nu,\nu]\ge[\beta,\beta]\ge\theta$.
Consequently, $(\delta,\theta,\nu)$ is a relevant triple such that
$[\nu,\nu]\not\le\delta$.
This shows that conditions
(a)--(c) in Theorem~\ref{thm-rs} are also equivalent to the condition
\begin{enumerate}
\item[(d)]
$[\nu,\nu]\le\delta$
for every relevant triple $(\delta,\theta,\nu)$ of a subalgebra of $\al A$.
\end{enumerate}
\begin{cor}
\label{cor-ssc-rs}
Let $\al A$ be a finite algebra that generates a congruence modular
variety $\var{V}(\al A)$.
If $\al A$ satisfies the split centralizer condition,
then {\rm(C1)} holds in
the congruence lattice of every subalgebra of $\al A$, so
$\var{V}(\al A)$ is residually small.
\end{cor}
\begin{proof}
If $\al A$ satisfies the split centralizer condition, and
$(\delta,\theta,\nu)$ is a relevant triple of a subalgebra $\al B$ of $\al A$
which is split by the triple $(\alpha,\beta,\kappa)$, then
$\interval{\delta}{\nu}\subseteq\interval{\beta}{\nu}
\searrow\interval{\kappa}{\alpha}$ with the last interval abelian,
therefore the other two intervals are also abelian.
This proves that condition (d) holds for $\al A$.
Our statement now follows from the equivalence of (d) to conditions
(c) and (b) in Theorem~\ref{thm-rs}.
\end{proof}
\subsection{Compatible Relations, Entailment, and Dualizability}
Let $A$ be a set. By a \emph{relation} on $A$ we mean a subset $\rho$ of $A^n$
for some positive integer $n$, which we call the \emph{arity} of $\rho$.
For any nonempty subset $I$ of $[n]$,
\emph{projection onto the coordinates in $I$} is the map
$\mathrm{proj}_I\colon A^n\to A^I$, $(a_i)_{i\in[n]}\mapsto(a_i)_{i\in I}$.
For any algebra $\al A$, an $n$-ary relation $\rho$ on $A$ is called
a \emph{compatible relation of $\al A$} if $\rho$ is the universe
of a subalgebra of $\al A^n$
(or equivalently, $\rho$ is a nonempty subuniverse of $\al A^n$).
The set of all compatible relations of
$\al A$ of arity $\le n$ will be denoted by $\rel{R}_n(\al A)$,
and $\rel{R}(\al A)$
will stand for the set $\bigcup_{n>0}\rel{R}_n(\al A)$
of all compatible relations of $\al A$.
It is straightforward to check that if $R$ is a set of compatible relations
of an algebra $\al A$, then so is every relation that can be obtained,
in finitely many steps, from
relations in $R\cup\{{=}\}$ by the following constructs:
\begin{itemize}
\item
nonempty
intersection of relations of the same arity,
\item
direct product of two relations,
\item
permutation of coordinates of a relation, and
\item
projection of a relation onto a nonempty subset of its coordinates.
\end{itemize}
The relations that can be obtained in this way are exactly the
nonempty
relations that are definable by primitive positive formulas (with $=$)
using the relations in $R$.
The relations that can be obtained, in finitely many steps, from
relations in $R\cup\{{=}\}$ by the first three types of constructs
are exactly the
nonempty
relations that are definable by
quantifier free primitive positive formulas (with $=$)
using the relations in $R$.
A \emph{critical relation} of an algebra $\al A$ is a compatible relation
$\rho$ of $\al A$ that is
\begin{enumerate}
\item[(i)]
completely $\cap$-irreducible in
the
lattice of subuniverses
of $\al A^n$ where $n$ is the arity of $\rho$, and
\item[(ii)]
directly indecomposable as a relation, that is, $[n]$ cannot be partitioned
into two nonempty sets $I$ and $J$ such that $\rho$ and
$\mathrm{proj}_I(\rho)\times\mathrm{proj}_J(\rho)$ differ only by a permutation of
coordinates.
\end{enumerate}
More informally, a compatible relation of $\al A$ is critical if it cannot be
obtained in a nontrivial way from other compatible relations
using only the first three of the four types of constructs above.
In the theory of natural dualities there is an entailment concept,
which we will denote by $\models_{\text{\rm d}}$ ($d$ stands for
`$\underline{\textrm{d}}$uality').
We refer the reader interested in the definition to Definition~4.1
in \cite[Chapter~2]{clark-davey}.
For the purposes of this paper an algebraic characterization of $\models_{\text{\rm d}}$,
which we state in Theorem~\ref{thm-dentails} below, will be sufficient.
We need some definitions beforehand.
Let $B$ be a compatible relation of an algebra $\al A$, say $B$ is $n$-ary,
and let $B':=\mathrm{proj}_I(B)$ for some nonempty $I\subseteq[n]$.
Then the projection map $\mathrm{proj}_I\colon\al B\to\al B'$ is a surjective
homomorphism
between the algebras $\al B\,(\le\al A^n)$ and $
\al B'\,(\le\al A^I)$ with universes $B$ and $B'$.
Following \cite{clark-davey},
we call $\mathrm{proj}_I\colon\al B\to\al B'$ a \emph{retractive projection}
if it is a retraction, that is, if there exists a homomorphism
$\phi\colon\al B'\to\al B$ such that $\mathrm{proj}_I\circ\phi=\id_{B'}$.
An important special case is a \emph{bijective projection}
$\mathrm{proj}_I\colon\al B\to\al B'$, when the retractive projection
is bijective.
Accordingly, we say that $B'$ is obtained from $B$ by
retractive projection (or bijective projection) onto its coordinates in $I$.
From now on we will restrict to finite algebras.
The following theorem is a consequence of Theorems~2.2 and 2.6
in \cite[Chapter~9]{clark-davey}.
\begin{thm}[{}From \cite{clark-davey}]
\label{thm-dentails}
Let $\al A$ be a finite algebra, and let $R\subseteq\rel{R}(\al A)$.
For a
nonempty
relation $\rho$ on $A$, we have
$R\models_{\text{\rm d}}\rho$ if and only if
$\rho$ can be obtained, in finitely many steps, from
relations in $R\cup\{{=}\}$ by the following constructs:
\begin{itemize}
\item
nonempty
intersection of relations of the same arity,
\item
product of two relations,
\item
permutation of coordinates of a relation, and
\item
retractive projection of a relation onto a nonempty subset of its coordinates.
\end{itemize}
\end{thm}
The four types of constructs in this theorem will be referred to as
\emph{$\models_{\text{\rm d}}$-constructs}.
Notice that the only difference between the list
of $\models_{\text{\rm d}}$-constructs and the earlier list of
constructs for compatible relations is that among the $\models_{\text{\rm d}}$-constructs
projections are restricted to retractive projections.
Two immediate consequences are worth mentioning:
(i)~Since every $\models_{\text{\rm d}}$-construct occurs on the earlier list, it follows
that if $R\subseteq\rel{R}(\al A)$ and
$R\models_{\text{\rm d}}\rho$, then $\rho\in\rel{R}(\al A)$.
(ii)~Since the first three constructs on the two lists are the same,
our earlier remark implies that if $R\subseteq\rel{R}(\al A)$
and $\rho$
is a nonempty relation on $A$ that
is definable by a quantifier-free primitive positive formula
(with ${=}$) using relations in $R$, then $R\models_{\text{\rm d}}\rho$.
Observation (i) above allows us to extend $\models_{\text{\rm d}}$ to subsets of
$\rel{R}(\al A)$ as follows: for $R,R'\subseteq\rel{R}(\al A)$
let $R\models_{\text{\rm d}} R'$ mean that $R\models_{\text{\rm d}}\rho$ for every $\rho\in R'$.
It is easy to check that the relation $\models_{\text{\rm d}}$ is transitive on subsets
of $\rel{R}(\al A)$, that is, if $R,R',R''\subseteq\rel{R}(\al A)$
satisfy $R\models_{\text{\rm d}} R'$ and $R'\models_{\text{\rm d}} R''$, then $R\models_{\text{\rm d}} R''$.
The most powerful general criterion for dualizability is the following theorem
of Willard and Z\'adori.
\begin{thm}[See \cite{willard},\cite{zadori}]
\label{thm-WZ}
Let $\al A$ be a finite algebra.
If $\rel{R}_n(\al A)\models_{\text{\rm d}}\rel{R}(\al A)$
for some integer $n\ge1$, then $\al A$ is dualizable.
\end{thm}
In fact, it is shown in \cite{zadori} that
if $\rel{R}_n(\al A)\models_{\text{\rm d}}\rel{R}(\al A)$
holds for some $n\ge1$, then
it also holds with the restriction that
retractive projections among the $\models_{\text{\rm d}}$-constructs are limited to
bijective projections.
\subsection{Algebras with Parallelogram Terms}
Let $m$ and $n$ be positive integers
and let $k= m+n$. The concept of an
{\em $(m,n)$-parallelogram term}
(or {\em $k$-parallelogram term})
for a variety $\var{V}$ was introduced in
\cite{parallelogram} to mean
a term
${\sf P}$
such that the identities represented by the rows of the following
matrix equation hold in $\var{V}$:
\begin{equation}\label{p}
{\sf P}
\left(
\begin{array}{ccc|}
x&x&y\\
x&x&y\\
&\vdots&\\
x&x&y\\
\hline
y&x&x\\
&\vdots&\\
y&x&x\\
y&x&x\\
\end{array}
\begin{array}{cccccccc}
z&y&\cdots&y&y&\cdots&y&y\\
y&z&&y&y&&y&y\\
\vdots&&\ddots& &&&&\vdots\\
y&y&&z&y&&y&y\\
y&y&&y&z&&y&y\\
\vdots&&&&&\ddots&&\vdots\\
y&y&&y&y&&z&y\\
y&y&\cdots&y&y&\cdots&y&z\\
\end{array}
\right) =
\left(\begin{matrix}
y\\
y\\
\vdots\\
y\\
y\\
\vdots\\
y\\
y
\end{matrix}
\right).
\end{equation}
Here $\sf P$ is $(k+3)$-ary, the rightmost block
of variables is a $k\times k$ array, the upper left
block is $m\times 3$ and the lower left block is $n\times 3$.
An $(m,n)$-parallelogram term (or $k$-parallelogram term) for an algebra
$\al A$ is defined to be
an $(m,n)$-parallelogram term (or $k$-parallelogram term)
for the variety $\var{V}(\al A)$ it generates.
It is easy to see from these definitions that a $k$-parallelogram term
that is independent of its last $k$ variables is a Maltsev term, and
a $k$-parallelogram term that is independent of its first $3$ variables
is a $k$-ary near unanimity term.
It was proved in \cite[Theorem~3.5]{parallelogram} that
if $m,n,m',n'$ are positive integers such that $m+n=k=m'+n'$, then a variety
has an $(m,n)$-parallelogram term if and only if it has an
$(m',n')$-parallelogram term; this justifies referring to them as
$k$-parallelogram terms. In addition, \cite[Theorem~3.5]{parallelogram}
also shows that a variety has a $k$-parallelogram term if and only if it has
a term, called a $k$-cube term, introduced in \cite[Definition~2.4]{bimmvw}.
It follows from \cite[Theorem~2.7]{bimmvw}
that every variety with
a $k$-cube term is congruence modular.
So, by combining all these results, or by a direct argument using
\cite[Theorem~3.2]{dent-kearnes-szendrei} we get the following conclusion.
\begin{thm}[See \cite{bimmvw},\cite{parallelogram};\cite{dent-kearnes-szendrei}]
\label{thm-parterm-cm}
Every variety with a parallelogram term is congruence modular.
\end{thm}
Our starting point for the proof of Theorem~\ref{thm-main}
will be the structure theorem in \cite{parallelogram}
(see \cite[Theorems~4.1 and 2.5]{parallelogram})
on the critical relations
of a finite algebra with a parallelogram term.
If $C$ is an $n$-ary critical relation of $\al A$, let
$\al C$ denote the subalgebra of $\al A^n$ with universe
$C$, and let $\al A_i:=\mathrm{proj}_i(\al C)$ for each $i\in[n]$.
Furthermore, let $\delta_i\in\Con(\al A_i)$ ($i\in[n]$)
be such that
$\delta:=\prod_{i=1}^n\delta_i$
is the largest product congruence of
$\prod_{i=1}^n\al A_i$ with the property that $\al C$ is a $\delta$-saturated
subalgebra of $\prod_{i=1}^n\al A_i$.
Then $\bar{\al C}:=\al C/\delta_{\al C}$ is a subdirect product
of the algebras $\bar{\al A}_i:=\al A_i/\delta_i$ $(i\in[n])$, which we call
\emph{the reduced representation of $\al C$}.
The next theorem contains
those parts of the structure theorem on critical relations
that we will need later on; we
retain the numbering from \cite[Theorem~2.5]{parallelogram}.
\begin{thm}[From \cite{parallelogram}]
\label{thm-paralg}
Let $C$ be an $n$-ary critical relation of a finite algebra
$\al A$ with a $k$-parallelogram term,
and let $\bar{\al C}$ be its reduced representation.
If $n\ge k\,(>1)$, then the following hold.
\begin{enumerate}
\item[{\rm(1)}]
$\bar{\al C}\le\prod_{i=1}^n\bar{\al A}_i$ is a representation of
$\bar{\al C}$ as a subdirect product of subdirectly irreducible
algebras $\bar{\al A}_i$.
\item[{\rm(7)}]
If $n>2$, then each $\bar{\al A}_i$ has abelian monolith $\mu_i$
$(i\in[n])$.
\item[{\rm(8)}]
For the centralizers $\rho_\ell:=(0:\mu_\ell)$ of the monoliths $\mu_\ell$
$(\ell\in[n])$, the image of the composite map
\[
\bar{\al C}\hookrightarrow\prod_{\ell=1}^n\bar{\al A}_\ell
\twoheadrightarrow\bar{\al A}_i/\rho_i\times\bar{\al A}_j/\rho_j
\]
is the graph of an isomorphism
for any $i,j\in[n]$.
\end{enumerate}
\end{thm}
Since $\bar{\al A}_i=\al A_i/\delta_i$ $(i\in[n])$ is subdirectly irreducible,
$\delta_i\in\Con(\al A_i)$ is completely meet irreducible,
and the monolith of $\bar{\al A}_i$ is
$\mu_i=\theta_i/\delta_i$ where $\theta_i\in\Con(\al A_i)$
is the unique cover of $\delta_i$.
Furthermore, $(0:\mu_i)=\rho_i=\nu_i/\delta_i$ where
$\nu_i=(\delta_i:\theta_i)$, and hence $\bar{\al A}_i/\rho_i\cong\al A_i/\nu_i$.
Thus, with our current terminology of a relevant triple,
Theorem~\ref{thm-paralg} (restricted to the case $n>2$)
can be restated as follows.
\begin{cor}
\label{cor-paralg}
Let $C$ be an $n$-ary critical relation of a finite algebra
$\al A$ with a $k$-parallelogram term,
and let $\al C$ be the subalgebra of $\al A^n$ with universe $C$.
If $n\ge\max(3,k)$, then the following hold.
\begin{itemize}
\item
$\al C$ is a subdirect product of the subalgebras
$\al A_i:=\mathrm{proj}_i(\al C)$ $(i\in[n])$ of $\al A$.
\item
If $\delta=\prod_{i=1}^n\delta_i$ $\bigl(\delta_i\in\Con(\al A_i)\bigr)$
is the largest product congruence
of $\prod_{i=1}^n\al A_i$ for which $\al C$ is $\delta$-saturated,
then each $\delta_i$ $(i\in[n])$ is the first component
of a relevant triple $(\delta_i,\theta_i,\nu_i)$ of $\al A_i$.
\item
The assignment
\[
\iota_{ij}\colon\al A_i/\nu_i\to\al A_j/\nu_j,\ \
c_i/\nu_i\mapsto c_j/\nu_j\ \
\text{whenever}\ \ (c_1,\dots,c_n)\in C
\]
is an isomorphism for any $i,j\in[n]$.
\end{itemize}
\end{cor}
\section{Reduction to Abelian Congruences of Bounded Index}\label{reduc}
Our main tool for proving Theorem~\ref{thm-main} will be
Theorem~\ref{thm-WZ}.
If $\al A$ is a finite algebra satisfying the assumptions of
Theorem~\ref{thm-main}, then Corollary~\ref{cor-ssc-rs} and
Theorem~\ref{thm-parterm-cm} imply that (C1) holds in the congruence lattice
of every subalgebra of $\al A$.
Hence $\nu'/\delta'$ is abelian for every relevant triple
$(\delta',\theta',\nu')$ of a subalgebra $\al A'$ of $\al A$.
Therefore,
if $C$ is an $n$-ary critical relation of $\al A$
such that, with the notation of Corollary~\ref{cor-paralg},
$\delta_i=0$ for every $i\in[n]$, then $\nu_i$ is abelian for every
$i\in[n]$, and hence the product congruence $\nu:=\prod_{i=1}^n\nu_i$
of $\prod_{i=1}^n\al A_i$ restricts to $\al C$ as an abelian congruence
$\nu_{\al C}$.
Moreover, by the last item in Corollary~\ref{cor-paralg},
$\nu_{\al C}$ has index $<|A|$.
In this situation we can use modules associated to abelian congruences,
as explained in Sections~\ref{modules}--\ref{mainthm-sec}, to show that
$C$ is entailed by compatible relations of bounded arity.
However, this argument does not work if, instead of an abelian interval
$\interval{0}{\nu_{\al C}}$, we have an abelian interval
$\interval{\delta_{\al C}}{\nu_{\al C}}$ with $\delta_{\al C}\not=0$.
In the general case when
$\delta_{\al C}$ may be a nontrivial congruence
of $\al C$,
we will use the assumption that $\al A$ satisfies
the split centralizer condition to replace $C$ by another compatible relation
$B$, which is not weaker than $C$ with respect to entailment, but
has an abelian interval
$\interval{0}{\tilde{\alpha}_{\al C}}$
corresponding to $\interval{\delta_{\al C}}{\nu_{\al C}}$
at the bottom of the congruence lattice, moreover,
the index of $\tilde{\alpha}_{\al C}$
remains bounded by a number independent of $n$
(though might be much bigger than the index of $\nu_{\al C}$,
which is $<|A|$).
The purpose of this section is to construct such a relation $B$ for every
critical relation $C$ of $\al A$.
Given a finite algebra $\al A$, we define several constants associated to
$\al A$ as follows.
\begin{itemize}
\item
Let $\mathsf{a}$ be the maximum of all
$|\Aut(\al S/\nu)|$ where $\al S\le\al A$ and $(\delta,\theta,\nu)$
is a relevant triple of $\al S$.
($\mathsf{a}$ stands for `$\underline{\text{a}}$utomorphisms'.)
\item
Let $\mathsf{s}$ be the number of distinct pairs $(\al S,\delta)$
such that $\al S\le\al A$ and $(\delta,\theta,\nu)$
is a relevant triple of $\al S$.
($\mathsf{s}$ stands for `$\underline{\text{s}}$ubdirectly irreducible
$\underline{\text{s}}$ections $\al S/\delta$'.)
\item
Let $\mathsf{i}:=|A|^{\mathsf{a}\mathsf{s}}$.
($\mathsf{i}$ stands for `$\underline{\text{i}}$ndex';
see Theorem~\ref{thm-reduction} below.)
\item
Let $\mathsf{p}$ be the least positive integer with the property that
for every subalgebra $\al S\le\al A$ every relevant triple of
$\al S$ is split by a triple $(\alpha,\beta,\kappa)$ such that
$\al S/\kappa$ embeds into $\al A^p$ for some $p\le\mathsf{p}$.
($\mathsf{p}$ stands for `$\underline{\text{p}}$ower'.)
\end{itemize}
\begin{thm}
\label{thm-reduction}
Let $\al A$ be a finite algebra with a $k$-parallelogram term
such that $\al A$ satisfies the split centralizer condition.
For every $n\ge \max(3,k)$ and for every $n$-ary critical relation
$C$ of $\al A$, there exists a compatible relation $B$ of $\al A$
which has the following two properties:
\begin{enumerate}
\item[$(*)$]
There exist
\begin{enumerate}
\item[{\rm(I)}]
subalgebras $\al B_i\le\al A^{p_i}$ with $p_i\le\mathsf{p}$
for each $i\in[n]$
such that $\al B_i$ is isomorphic to a section of $\al A$, and
\item[{\rm(II)}]
nontrivial abelian congruences
$\tilde\alpha_i\in\Con(\al B_i)$ $(i\in[n])$
\end{enumerate}
such that
\begin{enumerate}
\item[{\rm(III)}]
$B$ is the universe of
a subdirect product $\al B$ of
$\al B_1,\dots,\al B_n$, and
\item[{\rm(IV)}]
the product congruence
$\tilde\alpha:=\prod_{i=1}^n \tilde\alpha_i$ of $\prod_{i=1}^n\al B_i$
restricts to $\al B$ as a congruence
$\tilde\alpha_{\al B}$
of index $\le\mathsf{i}$.
\end{enumerate}
\item[$(**)$]
$\rel{R}_{1+\mathsf{p}}(\al A)\cup\{B\}\models_{\text{\rm d}} C$.
\end{enumerate}
\end{thm}
\begin{proof}
Corollary~\ref{cor-paralg} describes the structure of $C$.
Using the notation of Corollary~\ref{cor-paralg}, we first
define two equivalence relations, $\approx$ and $\sim$,
on $[n]$
as follows:
\begin{itemize}
\item
$i\approx j$ iff $\al A_i=\al A_j$ and
$(\delta_i,\theta_i,\nu_i)=(\delta_j,\theta_j,\nu_j)$;
\item
$i\sim j$ iff $i\approx j$ and
$\iota_{ij}$ is the identity isomorphism.
\end{itemize}
To give an upper bound for the indices of $\approx$ and $\sim$,
notice that if $i,j\in[n]$ are such that
$\al A_i=\al A_j$ and $\delta_i=\delta_j$, then it follows that
$(\delta_i,\theta_i,\nu_i)=(\delta_j,\theta_j,\nu_j)$, so $i\approx j$.
This implies that $\approx$ has at most $\mathsf{s}$ classes, that is,
$\approx$ has index $|[n]/{\approx}|\le\mathsf{s}$.
The definition of $\sim$ shows that
every class of ${\approx}/{\sim}$ has size $\le\mathsf{a}$.
Hence $\sim$ has at most $\mathsf{a}\mathsf{s}$ classes, that is,
$\sim$ has index $|[n]/{\sim}|\le\mathsf{a}\mathsf{s}$.
Our assumption that $\al A$ satisfies the split centralizer condition
ensures that for each $i\in[n]$ the relevant triple
$(\delta_i,\theta_i,\nu_i)$ of $\al A_i\ (\le\al A)$ is split
by a triple $(\alpha_i,\beta_i,\kappa_i)$.
These choices could differ coordinate by coordinate, but we can
choose a transversal $T_\approx$ for the classes of $\approx$ and
redefine the triples in coordinates $i\notin T_\approx$ to arrange that
$(\alpha_i,\beta_i,\kappa_i)=(\alpha_j,\beta_j,\kappa_j)$
whenever $i\approx j$ ($i,j\in[n]$).
Let $\alpha$, $\beta$, $\delta$, and $\kappa$
denote the product congruences $\prod_{i=1}^n \alpha_i$,
$\prod_{i=1}^n \beta_i$, $\prod_{i=1}^n \delta_i$,
and $\prod_{i=1}^n \kappa_i$ of
$\prod_{i=1}^n\al A_i$. We have
$\kappa\le\alpha$,
$\kappa\le\beta\le\delta$, and
$\kappa=\alpha\wedge\beta$,
because
$\kappa_i\le\alpha_i$,
$\kappa_i\le\beta_i\le\delta_i$, and
$\kappa_i=\alpha_i\wedge\beta_i$
for all $i$.
By Corollary~\ref{cor-paralg},
$\al C$ is $\delta$-saturated, and hence also
$\beta$- and $\kappa$-saturated.
Now let $D$ denote the set
of all tuples $(d_1,\dots,d_n)\in C$
such that
\begin{enumerate}
\item[$(\dagger)$]
$d_i\equiv_{\alpha_j} d_j$ whenever $i\sim j$.
\end{enumerate}
Note that $i\sim j$ implies that $\alpha_i=\alpha_j$,
so $\alpha_i$ and $\alpha_j$ are interchangeable in $(\dagger)$.
\begin{clm}
\label{clm-c-d}
Let $T$ be a transversal for the classes of $\sim$.
For every $(c_1,\dots,c_n)\in C$
\begin{enumerate}
\item[{\rm(1)}]
there exists
$(d_1,\dots,d_n)\in D$ such that
\begin{enumerate}
\item[$(\ddagger)$]
$(d_1,\dots,d_n)\equiv_\beta(c_1,\dots,c_n)$ and $d_t=c_t$ for all $t\in T$;
\end{enumerate}
\item[{\rm(2)}]
the tuples $(d_1,\dots,d_n)\in D$ satisfying $(\ddagger)$
are uniquely determined modulo $\kappa$.
\end{enumerate}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-c-d}.}
Choose $(c_1,\dots,c_n)\in C$.
For (1),
define $d_t:=c_t$ for $t\in T$.
Our aim is to show that
\begin{multline}
\label{eq-dtuple}
\text{\qquad for each $t\in T$ and for all $i\sim t$ with $i\not=t$}\\
\text{there exist $d_i\in A_i$ such that
$c_i\equiv_{\beta_i} d_i\equiv_{\alpha_t} d_t$.\qquad}
\end{multline}
This will complete the proof of (1) for the following reason.
Let $(d_1,\dots,d_n)$ be a tuple obtained in this way.
By its construction, $(d_1,\dots,d_n)$ will satisfy condition $(\ddagger)$.
It will also satisfy condition $(\dagger)$, because
whenever $i\sim j$, we have a $t\in T$ with $i,j\sim t$,
so $d_i,d_j\equiv_{\alpha_t} d_t$; since $\alpha_j=\alpha_t$, we get that
$d_i\equiv_{\alpha_j} d_j$.
Now, since $C$ is $\beta$-saturated, the fact that $(\ddagger)$
holds for $(d_1,\dots,d_n)$
ensures that $(d_1,\dots,d_n)\in C$.
Hence
the fact that $(\dagger)$ also holds for $(d_1,\dots,d_n)$
implies that $(d_1,\dots,d_n)\in D$.
To prove \eqref{eq-dtuple},
choose $t\in T$ and $i\sim t$ with $i\not=t$.
Then $i\sim t$ implies that $\al A_i=\al A_t$, $\beta_i=\beta_t$,
$\nu_i=\nu_t$, and $c_i/\nu_i=c_t/\nu_t$. Hence,
$c_i\equiv_{\nu_t} c_t$.
By our assumptions, $(\alpha_t,\beta_t,\kappa_t)$
splits $(\delta_t,\theta_t,\nu_t)$, so
$\alpha_t\vee\beta_t=\nu_t$ and
$\alpha_t/\kappa_t$ is an abelian congruence of
$\al A_t/\kappa_t$.
Since $\al A_t/\kappa_t$ lies in a congruence modular variety,
we have that $\alpha_t/\kappa_t$ permutes with all congruences of
$\al A_t/\kappa_t$ (see \cite[Theorem~6.2]{freese-mckenzie}).
Hence, $\nu_t=\alpha_t\vee\beta_t=\beta_t\circ\alpha_t$.
Thus, the fact that
$c_i\equiv_{\nu_t} c_t$ holds implies that
there exists $d_i\in A_t=A_i$ such that
$c_i\equiv_{\beta_t} d_i\equiv_{\alpha_t} c_t=d_t$. Since $\beta_i=\beta_t$,
this $d_i$ is the element we wanted to find.
For (2),
assume that $(d_1,\dots,d_n),\,(d_1',\dots,d_n')\in D$
both satisfy condition $(\ddagger)$.
We want to show that
$(d_1,\dots,d_n)\equiv_{\kappa}(d_1',\dots,d_n')$.
Since $\kappa=\alpha\wedge\beta$, this is equivalent to showing that
$(d_1,\dots,d_n)\equiv_{\alpha}(d_1',\dots,d_n')$ and
$(d_1,\dots,d_n)\equiv_{\beta}(d_1',\dots,d_n')$.
The latter follows from the assumption that
both tuples satisfy condition $(\ddagger)$.
To prove the former, consider any $i\in[n]$, and let
$t\in T$ be such that $i\sim t$.
Combining the second part of condition $(\ddagger)$ with
condition $(\dagger)$ in the definition of $D$, we get that
$c_t=d_t\equiv_{\alpha_i} d_i$, and similarly,
$c_t=d_t'\equiv_{\alpha_i} d_i'$, hence
$d_i\equiv_{\alpha_i} d_i'$.
Thus, $(d_1,\dots,d_n)\equiv_{\alpha}(d_1',\dots,d_n')$, completing the
proof of (2).
$\diamond$
\begin{clm}
\label{clm-saturation}
$D$ is the universe of a subalgebra $\al D$ of
$\al C$ with the following properties:
\begin{enumerate}
\item[{\rm(1)}]
$\al D$ is a subdirect product of $\al A_1,\dots,\al A_n$,
\item[{\rm(2)}]
$\mathrm{proj}_{T}(\al D)=\mathrm{proj}_{T}(\al C)$ for every transversal $T$ for
the classes of $\sim$,
\item[{\rm(3)}]
$\al D[\beta]=\al C$, and
\item[{\rm(4)}]
$\al D[\kappa]=\al D$.
\end{enumerate}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-saturation}.}
Consider a
transversal $T$ for the classes of $\sim$.
Then $D\subseteq C$ implies that
$\mathrm{proj}_{T}(D)\subseteq\mathrm{proj}_{T}(C)$, and
Claim~\ref{clm-c-d}~(1) shows that
$\mathrm{proj}_{T}(D)\supseteq\mathrm{proj}_{T}(C)$.
Thus, $\mathrm{proj}_{T}(D)=\mathrm{proj}_{T}(C)$.
This implies that
$\mathrm{proj}_t(D)=\mathrm{proj}_t(C)=A_t$ for every $t\in T$.
Since every element of $[n]$ occurs in some transversal $T$,
we get that $\mathrm{proj}_i(D)=A_i$ for all $i\in[n]$.
In particular, we see that $D\not=\emptyset$.
The definition of $D$ shows that $D\subseteq C$ and $D$
is definable by a primitive positive
formula, using the compatible relations
$\alpha_1,\dots,\alpha_n$ and $C$ of $\al A$.
Hence, $D$ is a compatible relation of $\al A$.
These arguments prove that $D$ is the universe of a subalgebra
$\al D$ of $\al C$ satisfying (1)--(2).
For (3), observe that
since $\al D\le\al C$ and $\al C$ is $\beta$-saturated, it follows that
$\al D[\beta]\le \al C[\beta]$ and $\al C[\beta]=\al C$.
Thus, $\al D[\beta]\le\al C$.
However, we get from Claim~\ref{clm-c-d}~(1) that
$C\subseteq D[\beta]$.
Hence, $\al D[\beta]=\al C$, as claimed.
Finally, we prove (4).
Clearly, $\al D\le\al D[\kappa]$.
The reverse inclusion $D[\kappa]\subseteq D$ can be established as follows:
\[
D[\kappa]
\subseteq D[\alpha]\cap D[\beta]
= D[\alpha]\cap C
\subseteq D,
\]
where the last inclusion
is an immediate consequence of the definition of $D$.
This completes the proof of
Claim~\ref{clm-saturation}.
\phantom{m}
$\diamond$
\begin{clm}
\label{clm-index-of-alpha}
For every transversal $T$
for the classes of $\sim$,
\[
|\al D/\alpha_{\al D}|\le|\mathrm{proj}_{T}(\al D)|\le\mathsf{i}.
\]
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-index-of-alpha}.}
Let $T$ be a transversal for the classes of $\sim$.
From the inequality $|[n]/{\sim}|\le\mathsf{a}\mathsf{s}$ proved earlier
we get that $|T|\le\mathsf{a}\mathsf{s}$.
Thus,
$|\mathrm{proj}_{T}(\al D)|
\le|A|^{|T|}
\le|A|^{\mathsf{a}\mathsf{s}}=\mathsf{i}$, which establishes the second
inequality.
It follows from the definition of $D$ that
the kernel of the projection map
$\mathrm{proj}_T\colon\al D\to\mathrm{proj}_T(\al D)$
is contained in $\alpha$. This implies the
first inequality in Claim~\ref{clm-index-of-alpha}.
$\diamond$
Now we are ready to define the algebras $\al B_1,\dots,\al B_n$, $\al B$,
and the congruences $\tilde{\alpha}_i\in\Con(\al B_i)$ for which
the conclusions $(*)$--$(**)$ of Theorem~\ref{thm-reduction} hold.
Since each $\kappa_i$ is a $\var{Q}$-congruence of $\al A_i$
for $\var{Q}=\mathsf{S}\mathsf{P}(\al A)$, we have that
there exist
subalgebras $\al B_i\le\al A^{p_i}$ with $p_i\le\mathsf{p}$
for each $i$, and surjective homomorphisms $\phi_i\colon\al A_i\to\al B_i$
with kernels $\kappa_i$ $(i\in[n])$ such that
$\phi_i$ induces an isomorphism
$\bar\phi_i\colon\al A_i/\kappa_i\to\al B_i$.
Recall that we chose the splitting
triples so that $(\alpha_i,\beta_i,\kappa_i)=(\alpha_j,\beta_j,\kappa_j)$
whenever $i\approx j$ (and hence $\al A_i=\al A_j$).
With the same reasoning, we can arrange that
the algebras $\al B_i$ and the
homomorphisms $\phi_i$ are selected so that
$\al B_i=\al B_j$ and $\phi_i=\phi_j$ whenever
$i\approx j$.
For every $i\in[n]$,
define $\tilde{\alpha}_i\in\Con(\al B_i)$ to be the image of
$\alpha_i\in\Con(\al A_i)$ under the homomorphism $\phi_i$,
which is the same as the image of
$\alpha_i/\kappa_i\in\Con(\al A_i/\kappa_i)$ under the isomorphism
$\bar{\phi}_i$; so $\tilde{\alpha}_i$ is indeed a congruence of
$\al B_i$.
Furthermore, define $\al B$ to be the image of $\al D$ under
the product homomorphism
$\phi:=\prod_{i=1}^n\phi_i\colon\prod_{i=1}^n\al A_i\to\prod_{i=1}^n\al B_i$.
\begin{clm}
\label{clm-Bs-alphas}
Conditions {\rm(I)--(IV)} hold for
the algebras $\al B_1,\dots,\al B_n$ and $\al B$, and
for the congruences $\tilde{\alpha}_1,\dots,\tilde{\alpha}_n$
defined above.
Moreover, $\al D$ is the full inverse image of $\al B$ under the
homomorphism $\phi$.
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-Bs-alphas}.}
(I) follows from the choice of the $\al B_i$'s.
(II) holds, because for every $i\in[n]$,
$\tilde{\alpha}_i=\bar{\phi}_i[\alpha_i/\kappa_i]$
where $\bar{\phi}_i\colon\al A_i/\kappa_i\to\al B_i$
is an isomorphism and
$\alpha_i/\kappa_i$ is a nontrivial abelian congruence of $\al A_i$.
(The latter follows from the fact that
$(\alpha_i,\beta_i,\kappa_i)$ splits the relevant
triple $(\delta_i,\theta_i,\nu_i)$ of $\al A_i$.)
For (III), recall that
$\al D$ is a subdirect product of $\al A_1,\dots,\al A_n$
by Claim~\ref{clm-saturation}(1), and the homomorphisms
$\phi_i\colon\al A_i\to\al B_i$ ($i\in[n]$) are onto.
Since $\phi=\prod_{i=1}^n\phi_i$, it follows that
$\al B=\phi[\al D]$
is a subdirect product of $\al B_1,\dots,\al B_n$.
To verify (IV) notice that
$\phi:=\prod_{i=1}^n\phi_i\colon\prod_{i=1}^n\al A_i\to\prod_{i=1}^n\al B_i$
is a surjective homomorphism with kernel
$\kappa=\prod_{i=1}^n\kappa_i$, therefore
$\phi$ decomposes as shown by the first line of the array below:
\begin{equation}
\label{eq-phi}
\begin{matrix}
\phi\colon & \prod_{i=1}^n\al A_i &
\stackrel{\textrm{nat}}{\to} & \bigl(\prod_{i=1}^n\al A_i\bigr)/\kappa &
\stackrel{\cong}{\to} & \prod_{i=1}^n(\al A_i/\kappa_i) &
\stackrel{\bar{\phi}}{\to} & \prod_{i=1}^n\al B_i & \phantom{m}\\[6pt]
& \alpha &
\mapsto & \alpha/\kappa &
\mapsto & \prod_{i=1}^n\alpha_i/\kappa_i &
\mapsto & \tilde{\alpha}\\[6pt]
\phi{\restriction}_{\al D}\colon & \al D &
\stackrel{\textrm{nat}}{\to} & \al D/\kappa_{\al D} &
& \stackrel{\bar{\phi}\circ\cong}{\longrightarrow} &
& \al B\\[6pt]
& \alpha_{\al D} &
\mapsto & \alpha_{\al D}/\kappa_{\al D} &
& \mapsto &
& \tilde{\alpha}_{\al B}\\
\end{matrix}
\end{equation}
The leftmost factor of $\phi$ is the natural map,
the middle factor is the natural isomorphism, and
the rightmost factor is the isomorphism $\bar{\phi}:=\prod_{i=1}^n\bar{\phi}_i$.
Since $\tilde{\alpha_i}=\phi[\alpha_i]$ for every $i\in[n]$, $\phi$ maps
the congruence $\alpha=\prod_{i=1}^n\alpha_i$ of $\prod_{i=1}^n\al A_i$
onto the congruence
$\tilde{\alpha}=\prod_{i=1}^n\tilde{\alpha}_i$ of $\prod_{i=1}^n\al B_i$, via
this factorization, as indicated by the second line of
\eqref{eq-phi}.
Therefore, combining the two isomorphisms among the factors of $\phi$ and
restricting $\phi$ to $\al D$ yields that $\phi{\restriction}_{\al D}$ factors
and acts on $\alpha_{\al D}$ as the third and fourth lines
of \eqref{eq-phi} show.
Thus,
\[
\al D/\alpha_{\al D}
\cong (\al D/\kappa_{\al D})/(\alpha_{\al D}/\kappa_{\al D})
\cong \al B/\tilde\alpha_{\al B}.
\]
This implies that the index of $\tilde{\alpha}_{\al B}$ in $\al B$
is $|\al B/\tilde\alpha_{\al B}|=|\al D/\alpha_{\al D}|$, which is $\le\mathsf{i}$
by Claim~\ref{clm-index-of-alpha}.
For the last statement of the claim, $B=\phi[D]$ implies that
$\phi^{-1}[B]\supseteq D$. Since the kernel of $\phi$ is $\kappa$,
the equality $B=\phi[D]$ also implies that $\phi^{-1}[B]\subseteq D[\kappa]$.
But we know from Claim~\ref{clm-saturation} that
$D[\kappa]=D$, so the equality $\phi^{-1}[B]=D$ we wanted to prove follows.
$\diamond$
\begin{clm}
\label{clm-C-entailed}
For every $i\in[n]$, $A_i$, $B_i$,
the (graph of the) homomorphism $\phi_i$, and
\[
\beta_i\circ\phi_i:=\{(x,y)\in A_i\times B_i: x\equiv_{\beta_i} z
\ \text{and}\ y=\phi_i(z)\ \text{for some $z\in A_i$}
\}
\]
are compatible relations of $\al A$. Moreover,
\begin{equation}
\label{eq-C-entailed}
\{A_i,\,B_i,\,\phi_i,\,\,\beta_i\circ\phi_i:i\in[n]\}\cup\{B\}
\models_{\text{\rm d}} C.
\end{equation}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-C-entailed}.}
Choose any $i\in[n]$. Since $\al A_i\le\al A$ and $\al B_i\le\al A^{p_i}$,
we have that $A_i\in\rel{R}_1(\al A)$ and
$B_i\in\rel{R}_{p_i}(\al A)$.
The fact that $\phi_i$ is a homomorphism $\al A_i\to\al B_i$
implies that
its graph is the universe of a subalgebra of
$\al A_i\times\al B_i\le\al A^{1+p_i}$.
Thus, $\phi_i\in\rel{R}_{1+p_i}(\al A)$.
The definition of $\beta_i\circ\phi_i$ shows that
$\beta_i\circ\phi_i\not=\emptyset$ and
$\beta_i\circ\phi_i$ is definable by a primitive positive
formula, using the compatible relations
$A_i, B_i, \beta_i, \phi_i\in\rel{R}(\al A)$,
therefore $\beta_i\circ\phi_i\in\rel{R}(\al A)$.
In fact, $\beta_i\circ\phi_i$ is also the universe
of a subalgebra of $\al A_i\times\al B_i\le\al A^{1+p_i}$, so
$\beta_i\circ\phi_i\in\rel{R}_{1+p_i}(\al A)$.
This proves the first statement of the claim.
To prove the second statement, let us fix a transversal $T$ for the
classes of $\sim$, and define $X$ to be the
set of all tuples
\[(c_1,\dots,c_n,b_1,\dots,b_n)\
\in \prod_{i=1}^n A_i\times\prod_{i=1}^n B_i\,\Bigl(\subseteq A^{n+\sum p_i}\Bigr)
\]
such that
\begin{itemize}
\item
$(b_1,\dots,b_n)\in B$,
\item
$(c_i,b_i)\in\beta_i\circ\phi_i$ for all $i\in[n]$, and
\item
$b_t=\phi_t(c_t)$ for all $t\in T$.
\end{itemize}
Our main goal is to show that
\begin{enumerate}
\item[(i)]
$\mathrm{proj}_{[n]}(X)=C$ and
\item[(ii)]
the projection map $\mathrm{proj}_{[n]}\colon X\to C$ is one-to-one,
\end{enumerate}
because we can deduce \eqref{eq-C-entailed} from
statements (i)--(ii) as follows.
We have $X\not=\emptyset$ by (i), and
$X$ is definable by a quantifier-free primitive positive formula
using the relations $A_i$, $B_i$, $B$, $\phi_t$, and $\beta_i\circ\phi_i$,
therefore
\[
\{A_i,\,B_i,\,\phi_i,\,\,\beta_i\circ\phi_i:i\in[n]\}\cup\{B\}
\models_{\text{\rm d}} X.
\]
Furthermore, by (i) and (ii),
$C$ can be obtained from $X$ by bijective projection, which is an
$\models_{\text{\rm d}}$-construct, so $\{X\}\models_{\text{\rm d}} C$.
Hence \eqref{eq-C-entailed} follows by the transitivity of
$\models_{\text{\rm d}}$.
It remains to prove (i) and (ii).
For the inclusion $\supseteq$ in (i), let $(c_1,\dots,c_n)\in C$.
By Claim~\ref{clm-c-d}, there exists $(d_1,\dots,d_n)\in D$ such that
$(d_1,\dots,d_n)\equiv_\beta(c_1,\dots,c_n)$ and $d_t=c_t$ for all $t\in T$.
Let $b_i=\phi_i(d_i)$ for each $i\in[n]$.
By the definition of $B$, this choice ensures that
$(b_1,\dots,b_n)\in B$.
Furthermore, $c_i\equiv_{\beta_i} d_i$ and $b_i=\phi_i(d_i)$ imply
$(c_i,b_i)\in\beta_i\circ\phi_i$ for every $i\in[n]$, while
$d_t=c_t$ and $b_t=\phi_t(d_t)$ imply
$b_t=\phi_t(c_t)$ for every $t\in T$.
Hence, $(c_1,\dots,c_n,b_1,\dots,b_n)\in X$, so
$(c_1,\dots,c_n)\in\mathrm{proj}_{[n]}(X)$.
The inclusion $\subseteq$ in (i)
and the claim in (ii) will follow if we prove the following statement:
\begin{enumerate}
\item[{}]
for every $(c_1,\dots,c_n)\in\mathrm{proj}_{[n]}(X)$ we have that
\begin{enumerate}
\item[--]
$(c_1,\dots,c_n)\in C$, and
\item[--]
there is a unique tuple $(b_1,\dots,b_n)$
such that $(c_1,\dots,c_n,b_1,\dots,b_n)\in X$.
\end{enumerate}
\end{enumerate}
So, let $(c_1,\dots,c_n)\in\mathrm{proj}_{[n]}(X)$.
Then $(c_1,\dots,c_n,b_1,\dots,b_n)\in X$ for at least one
tuple $(b_1,\dots,b_n)$. By the definition of $X$
it must be the case that $(b_1,\dots,b_n)\in B$,
$(c_i,b_i)\in\beta_i\circ\phi_i$ for all $i\in[n]$, and
$b_t=\phi_t(c_t)$ for all $t\in T$.
For any $t\in T$, define $d_t:=c_t$; so $c_t\equiv_{\beta_t}d_t$ and
$b_t=\phi_t(d_t)$.
For $i\in[n]\setminus T$,
use the definition of $\beta_i\circ\phi_i$ to get
$d_i$ such that $c_i\equiv_{\beta_i} d_i$ and $b_i=\phi_i(d_i)$.
Thus, $(c_1,\dots,c_n)\equiv_{\beta}(d_1,\dots,d_n)$ and
$(d_1,\dots,d_n)\in\phi^{-1}[(b_1,\dots,b_n)]$.
We established in Claim~\ref{clm-Bs-alphas} that
$D=\phi^{-1}[B]$, so
$(d_1,\dots,d_n)\in\phi^{-1}[(b_1,\dots,b_n)]$
implies that $(d_1,\dots,d_n)\in D$.
Since $D\subseteq C$ and $C$ is $\beta$-saturated,
$(c_1,\dots,c_n)\equiv_{\beta}(d_1,\dots,d_n)$ yields that
$(c_1,\dots,c_n)\in C$.
For the uniqueness of $(b_1,\dots,b_n)$ observe that
our argument in the preceding paragraph shows the following:
if $(b_1,\dots,b_n)$ is such that
$(c_1,\dots,c_n,b_1,\dots,b_n)\in X$, then there exists
$(d_1,\dots,d_n)\in D$ with the properties
\[
(c_1,\dots,c_n)\equiv_\beta(d_1,\dots,d_n)\in\phi^{-1}[(b_1,\dots,b_n)]
\quad\text{and}\quad
\text{$c_t=d_t$ for all $t\in T$.}
\]
So, if $(b_1',\dots,b_n')$ is another tuple with
$(c_1,\dots,c_n,b_1',\dots,b_n')\in X$, then there exists
$(d_1',\dots,d_n')\in D$ such that
\[
(c_1,\dots,c_n)\equiv_\beta(d_1',\dots,d_n')\in\phi^{-1}[(b_1',\dots,b_n')]
\quad\text{and}\quad
\text{$c_t=d_t'$ for all $t\in T$.}
\]
It follows from Claim~\ref{clm-c-d}(2) that
$(d_1,\dots,d_n)\equiv_{\kappa_\al D}(d_1',\dots,d_n')$.
Since $\kappa_{\al D}$ is the kernel of $\phi$ we get that
\[
(b_1,\dots,b_n)
=\phi\bigl((d_1,\dots,d_n)\bigr)
=\phi\bigl((d_1',\dots,d_n')\bigr)
=(b_1',\dots,b_n'),
\]
proving the uniqueness of $(b_1,\dots,b_n)$.
$\diamond$
Statement $(*)$ of Theorem~\ref{thm-reduction} was proved in
Claim~\ref{clm-Bs-alphas}. Statement $(**)$ of Theorem~\ref{thm-reduction}
follows from Claim~\ref{clm-C-entailed} and the fact
(established in the proof of Claim~\ref{clm-C-entailed}) that
each one of the relations $A_i,B_i,\phi_i,\beta_i\circ\phi_i$
($i\in[n]$)
is a member of $\rel{R}_m(\al A)$ for some $m\le 1+\mathsf{p}$.
This completes the proof of Theorem~\ref{thm-reduction}.
\end{proof}
\section{Abelian Congruences and Modules}\label{modules}
In \cite[Chapter~9]{freese-mckenzie} Freese and McKenzie
describe how matrix rings of any size can be associated
to a congruence modular variety $\var{V}$, and then
modules over these rings
to any abelian congruence $\alpha$ of an algebra $\al C$
in $\var{V}$.
They prove in \cite[Theorem~9.9]{freese-mckenzie}
that there is a strong connection between
the pair $(\al C,\alpha)$ and the associated modules, namely,
if all $\alpha$-classes are represented in the module, then
the interval $\interval{0}{\alpha}$ in $\Con(\al C)$
is isomorphic to the lattice of submodules of
the associated module.
Our goal in this section is to prove an analogous result for subalgebras
in place of congruences.
We will start by recalling the relevant definitions and basic facts
from \cite[Chapter~9]{freese-mckenzie}, but we will
slightly change the notation.
Throughout this section we will work under the following global assumptions:
\begin{itemize}
\item
$\var{V}$ is a congruence modular variety,
\item
$d$ is a fixed difference term for $\var{V}$, and
\item
$\mathcal{O}$ is a fixed nonempty set of constant symbols not occurring
in the language of $\var{V}$.
\end{itemize}
Recall that every
congruence modular variety has a difference term
(see \cite[Theorem~5.5]{freese-mckenzie}), which is a ternary term $d$
satisfying the following conditions
for every algebra $\al C\in\var{V}$ and congruence $\vartheta$ of $\al C$:
\begin{align}
d^{\al C}(x,x,y) & {}= y
\quad \text{for all $x,y\in\al C$, and}
\label{eq-diffterm1}\\
d^{\al C}(y,x,x) & {}\equiv_{[\vartheta,\vartheta]} y
\quad \text{for all $x,y\in\al C$ such that $x\equiv_\vartheta y$.}
\label{eq-diffterm2}
\end{align}
In addition, $d$ also satisfies
the following condition (see \cite[Proposition~5.7]{freese-mckenzie}): for
every term $t=t(x_1,\dots,x_k)$ and for arbitrary abelian congruence $\alpha$
of an algebra $\al C\in\var{V}$,
\begin{multline}
\label{eq-diffterm3}
d^{\al C}(t^{\al C}(u_1,\dots,u_k),t^{\al C}(v_1,\dots,v_k),
t^{\al C}(w_1,\dots,w_k))\\
=t^{\al C}(d^{\al C}(u_1,v_1,w_1),\dots,d^{\al C}(u_k,v_k,w_k))\\
\text{whenever
$u_i,v_i,w_i\in\al C$ are such that
$u_i\equiv_\alpha v_i\equiv_\alpha w_i$ for all $i$.}
\end{multline}
We will use the set $\mathcal{O}$ of new constant symbols
to expand the language of $\var{V}$.
Terms in the expanded language will
be called \emph{$\mathcal{O}$-terms}, and the algebras obtained
from the members of $\var{V}$ by interpreting all constant symbols $o\in O$
will be called \emph{$\mathcal{O}$-algebras}.
The class of algebras obtained in this way will
be denoted by $\Ovar{V}$. We will restrict the use of the phrases
`subalgebra', `homomorphism', and `product' to the algebras in
$\var{V}$ (i.e., to algebras in the original language),
and use the phrases `$\mathcal{O}$-subalgebra',
`$\mathcal{O}$-homomorphism', and
`$\mathcal{O}$-product' for $\mathcal{O}$-algebras.
There will be no need for using the phrase `$\mathcal{O}$-congruence',
because every $\mathcal{O}$-algebra has the same congruence lattice,
with the same commutator operation, as its reduct to the language of
$\var{V}$.
If we apply Corollary~5.8 (along with Lemma 5.6)
from \cite{freese-mckenzie}
to $\mathcal{O}$-algebras in $\Ovar{V}$, we get the following.
\begin{lm}[From \cite{freese-mckenzie}]
\label{lm-group}
If $\alpha$ is an abelian congruence of
an $\mathcal{O}$-algebra $\al C$ in $\Ovar{V}$, then
for every $o\in\mathcal{O}$, the $\alpha$-class containing $o^{\al C}$
is an abelian group
\[
\Calpha{o}:=(o^{\al C}/\alpha;+_o,-_o,o^{\al C})
\]
with zero element $o^{\al C}$ for the operations
$+_o$ and $-_o$ defined by
\[
u +_o v := d^{\al C}(u,o^{\al C},v)
\quad\text{and}\quad
-_o u := d^{\al C}(o^{\al C},u,o^{\al C})
\quad\text{for all $u,v\in o^{\al C}/\alpha$.}
\]
Furthermore,
\begin{enumerate}
\item[{\rm(i)}]
$d^{\al C}(u,v,w)=u -_o v +_o w$ for all $u,v,w\in o^{\al C}/\alpha$, and
\item[{\rm(ii)}]
the $\mathcal{O}$-term operations of $\al C$
are linear between the $\alpha$-classes; more precisely,
if $r(x_1,\dots,x_k)$ is an $\mathcal{O}$-term and
$o_1,\dots,o_k,o\in\mathcal{O}$ are such that
$r(o_1,\dots,o_k)=o$, then
\[
r^{\al C}{\restriction}_{(o_1^{\al C}/\alpha)\times\dots\times(o_k^{\al C}/\alpha)}\colon
\Calpha{o_1}\times\dots\times\Calpha{o_k}
\to\Calpha{o}
\]
is a group homomorphism.
Consequently, for the unary $\mathcal{O}$-terms
$r_i(x):=r(o_1,\dots,o_{i-1},x,o_{i+1},\dots,o_k)$ $(1\le i\le k)$
the maps
$r_i^{\al C}{\restriction}_{o_i^{\al C}/\alpha}\colon
\Calpha{o_i}\to \Calpha{o}$
are also group homomorphisms, and
\begin{multline}
\label{eq-linDecomp}
\qquad\qquad
r^{\al C}(u_1,\dots,u_k)=r_1^{\al C}(u_1) +_o \dots +_o r_k^{\al C}(u_k)\\
\quad\text{for all $u_i\in o_i^{\al C}/\alpha$ $(i=1,\dots,k)$.}
\qquad
\end{multline}
\end{enumerate}
\end{lm}
In Section~5 the following straightforward consequence of
Lemma~\ref{lm-group}(i) will be useful.
\begin{cor}
\label{cor-dMaltsev}
Under the same assumptions as in Lemma~\ref{lm-group},
if $e$ is an integer that is a multiple of the exponent of
the group $\Calpha{o}$, then
\begin{multline}
\label{eq-dMaltsev}
d^{\al C}(d^{\al C}(\dots d^{\al C}(d^{\al C}(u_1,u_{e+1},u_2),u_{e+1},u_3)
\dots),u_{e+1},u_e)
=u_1 +_o \dots +_o u_{e+1}\\
\quad\text{for all $u_1,\dots,u_{e+1}\in o^{\al C}/\alpha$.}
\end{multline}
\end{cor}
\begin{proof}
By Lemma~\ref{lm-group}(i) the left hand side of \eqref{eq-dMaltsev}
equals $u_1 +_o u_2 +_o u_3 +_o \dots +_o u_e -_o (e-1)u_{e+1}$,
which equals the right hand side of \eqref{eq-dMaltsev}, because
$e$ is a multiple of the exponent of the group $\Calpha{o}$.
\end{proof}
Next we define the matrix ring $\al R(\var{V},\mathcal{O})$
of the variety $\var{V}$ (where the size of the matrices is
$|\mathcal{O}|$). Let $\al F$ be the free algebra in
$\var{V}$ with free generating set $\{x\}\cup\mathcal{O}$.
In other words, $\al F$ is the algebra of all $\mathcal{O}$-terms
of $\Ovar{V}$ in one variable $x$ (modulo the identities of $\Ovar{V}$).
For each $o\in\mathcal{O}$, let $\epsilon_o$ denote the unique
endomorphism of $\al F$ that maps $x$ to $o$ and fixes all elements of
$\mathcal{O}$; that is, $\epsilon_o$ maps every $\mathcal{O}$-term
$r=r(x)$ to $r(o)$.
Furthermore, let $\gamma_o:=[\theta_o,\theta_o]$ where
$\theta_o:=\ker(\epsilon_o)$.
Now for any $o,o'\in\mathcal{O}$, let
\[
\bar H_{o,o'}:=\{r/\gamma_{o}:r=r(x)\in\al F,\ r(o)=o'\}.
\]
Then each $\bar H_{o,o'}$ has a natural abelian group structure with zero element
$o'/\gamma_{o}$ and addition and inversion defined by
\begin{align*}
r/\gamma_{o}+s/\gamma_{o}&{}:=d(r,o',s)/\gamma_{o}
\quad\text{and}\\
-(r/\gamma_{o})&{}:=d(o',r,o')/\gamma_{o}
\quad
\text{for all $r/\gamma_{o},s/\gamma_{o}\in \bar H_{o,o'}$.}
\end{align*}
Moreover, composition of
$\mathcal{O}$-terms yields a multiplication that is defined by
\[
(t/\gamma_{o'})(r/\gamma_{o}):=t(r(x))/\gamma_{o}
\quad
\text{for all $t/\gamma_{o'}\in \bar H_{o',o''}$ and
$r/\gamma_{o}\in \bar H_{o,o'}$.}
\]
It can be also shown that multiplication distributes over addition.
Thus, the
$\mathcal{O}\times\mathcal{O}$
matrices $(m_{o',o})$ with the properties that
\begin{itemize}
\item
the entry $m_{o',o}$ in position $(o',o)$
(i.e., row $o'$, column $o$) satisfies the condition
$m_{o',o}\in \bar H_{o,o'}$
for every $o,o'\in\mathcal{O}$, and
\item
each column contains only finitely
many nonzero entries
\end{itemize}
form a ring for the ordinary matrix operations.
This is the matrix ring $\al R(\var{V},\mathcal{O})$
associated to $\var{V}$.
The proofs of all claims made throughout the definition
above and the lemma below can be found at the beginning of
Chapter~9 (before Theorem~9.4) in \cite{freese-mckenzie}.
\begin{lm}[From \cite{freese-mckenzie}]
\label{lm-module}
If $\alpha$ is an abelian congruence of
an $\mathcal{O}$-algebra $\al C$ in $\Ovar{V}$, then
the direct sum
\begin{align*}
\qquad
\MCalpha
:={}&\bigoplus_{o\in\mathcal{O}}\Calpha{o}\\
={}&\{(a_o)_{o\in\mathcal{O}}\in\prod_{o\in\mathcal{O}}\Calpha{o}:
\text{$a_o=o^{\al C}$ for all but finitely many $o$'s}\}
\end{align*}
of the abelian groups $\Calpha{o}$ $(o\in O)$ is
an $\al R(\var{V},\mathcal{O})$-module
for the ordinary matrix-vector multiplication if
multiplication for the entries is defined as follows:
for $a\in\Calpha{o}$ and $m_{o',o}=r/\gamma_{o}\in\bar H_{o,o'}$
$(\text{with } r=r(x)\in\al F,\ r(o)=o')$,
\begin{equation}
\label{eq-moduleMult}
m_{o',o}a=(r/\gamma_{o}) a:=r^{\al C}(a).
\end{equation}
\end{lm}
Now we are ready to state our first theorem which, for every pair
$(\al C,\alpha)$ consisting of an $\mathcal{O}$-algebra $\al C$ in $\Ovar{V}$
and an abelian congruence $\alpha$ of $\al C$,
relates the subalgebras of $\al C$ to the submodules of the associated
$\al R(\var{V},\mathcal{O})$-module $\MCalpha$.
Recall our convention that the restriction of $\alpha$ to any subalgebra
$\al U$ (subset $U$) of $\al C$ is denoted by $\alpha_{\al U}$
($\alpha_U$, respectively).
\begin{thm}
\label{thm-subalg-submod}
Let $\alpha$ be an abelian congruence of an $\mathcal{O}$-algebra
$\al C$ in $\Ovar{V}$ such that the constants $o^{\al C}$ $(o\in\mathcal{O})$
represent all $\alpha$-classes of $\al C$, and let
$\al E:=\langle\mathcal{O}^{\al C}\rangle$ be the least $\mathcal{O}$-subalgebra
of $\al C$.
\begin{enumerate}
\item[{\rm(1)}]
If $\al U$ is an $\mathcal{O}$-subalgebra of $\al C$, then
$\MUalpha$ is an
$\al R(\var{V},\mathcal{O})$-submodule
of $\MCalpha$ that contains $\MEalpha$ as a submodule.
\item[{\rm(2)}]
The following conditions on any subset $U$ of $\al C$
are equivalent:
\begin{enumerate}
\item[{\rm(a)}]
$U$ is the universe of a subalgebra of $\al C$ that contains $\al E$;
\item[{\rm(b)}]
$U$ is the universe of an $\mathcal{O}$-subalgebra of $\al C$;
\item[{\rm(c)}]
$U$ is closed under all functions
$t^{\al C}
(x_1,\dots,x_k){\restriction}_{(o_1^{\al C}/\alpha)\times\dots\times (o_k^{\al C}/\alpha)}$
where $t$ is an $\mathcal{O}$-term and $o_1,\dots,o_k\in\mathcal{O}$;
\item[{\rm(d)}]
the set
\[
\qquad\quad
\bigoplus_{o\in\mathcal{O}} (o^{\al C}/\alpha_U):=
\{(u_o)_{o\in\mathcal{O}}\in\prod_{o\in\mathcal{O}}(o^{\al C}/\alpha_U):
\text{$u_o=o^{\al C}$ for all but finitely many $o$'s}\}
\]
is the universe of an $\al R(\var{V},\mathcal{O})$-submodule
of $\MCalpha$ that contains \break
$\MEalpha$.
\end{enumerate}
\item[{\rm(3)}]
If $U$ is the universe of an $\mathcal{O}$-subalgebra
$\al U$ of $\al C$, then
the $\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$ described in {\rm(d)}
is $\MUalpha$.
\end{enumerate}
\end{thm}
\begin{proof}{}
For (1),
assume that $\al U$ is an $\mathcal{O}$-subalgebra
of $\al C$; in particular, $o^{\al U}=o^{\al C}$ for all $o\in\mathcal{O}$.
Then $\alpha_{\al U}$
is an abelian congruence of $\al U$, and
the universe of $\MUalpha$ is
a subset of the universe of
$\MCalpha$.
The definition of the modules
$\MUalpha$ and $\MCalpha$
shows that they have the same zero element
$(o^{\al U})_{o\in\mathcal{O}}=(o^{\al C})_{o\in\mathcal{O}}$,
the group operations in the two modules
are determined, in each coordinate, by the
same $\mathcal{O}$-terms $d(x,o,y)$, $d(o,x,o)$
(see Lemma~\ref{lm-group}),
and for every matrix in $\al R(\var{V},\mathcal{O})$,
multiplication for the entries in the two modules are determined by the same
$\mathcal{O}$-terms (see Lemma~\ref{lm-module}).
Since $\al U$ is an $\mathcal{O}$-subalgebra of $\al C$,
this implies that $\MUalpha$ is an
$\al R(\var{V},\mathcal{O})$-submodule of $\MCalpha$.
$\al E$ is an $\mathcal{O}$-subalgebra of $\al U$,
therefore we can repeat the argument in the preceding paragraph
for $\al U$ and $\al E$ (in place of $\al C$ and $\al U$) to
conclude that $\MEalpha$ is an
$\al R(\var{V},\mathcal{O})$-submodule of
$\MUalpha$.
To verify (2) and (3),
we will prove the equivalence of the first three conditions in (2) by showing
that (a) $\Leftrightarrow$ (b) and (b) $\Leftrightarrow$ (c). To establish that
the fourth condition is also equivalent to them we will prove that
(b) $\Rightarrow$ (d) and (d) $\Rightarrow$ (c). Statement (3) will
be verified along with the proof of the implication (b) $\Rightarrow$ (d).
If $U$ is the universe of a subalgebra $\al U$ of $\al C$
containing $\al E=\langle\mathcal{O}^{\al C}\rangle$, then
we can define $o^{\al U}:=o^{\al C}$ for every
$o\in\mathcal{O}$ to make $\al U$ an $\mathcal{O}$-subalgebra of $\al C$.
This proves (a) $\Rightarrow$ (b). The converse is a tautology, so
(a) $\Leftrightarrow$ (b) is proved.
$U$ is the universe of an $\mathcal{O}$-subalgebra of $\al C$ if and only if
$U$ is closed under all functions $t^{\al C}$ induced by $\mathcal{O}$-terms $t$.
Since our assumption is that
the constants $o^{\al C}$ $(o\in\mathcal{O})$
represent all $\alpha$-classes of $\al C$, it follows that for every
$\mathcal{O}$-term $t$, $U$ is closed under the term function $t^{\al C}$
if and only if it is closed under all its restrictions described in (c).
This proves (b) $\Leftrightarrow$ (c).
Now assume that $U$ is the universe of an $\mathcal{O}$-subalgebra $\al U$
of $\al C$;
in particular, $o^{\al U}=o^{\al C}$ for all $o\in\mathcal{O}$.
Then $\alpha_U=\alpha_{\al U}$
is an abelian congruence of $\al U$. The definition of the
$\al R(\var{V},\mathcal{O})$-module
$\MUalpha$
shows that its universe is exactly
$\bigoplus_{o\in\mathcal{O}} (o^{\al C}/\alpha_U)$.
Therefore statement (1) proves that
$\bigoplus_{o\in\mathcal{O}} (o^{\al C}/\alpha_U)$ is indeed
the underlying set of an $\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$ that contains $\MEalpha$ as a submodule, namely
$\MUalpha$.
This finishes the proof of both (b) $\Rightarrow$ (d) and (3).
Finally, we want to argue that
(d) $\Rightarrow$ (c).
We will start by proving a claim which is independent of $U$.
\begin{clm}
\label{clm-moduleops}
Let $\al C$ and $\alpha$ be as in the theorem.
For arbitrary $\mathcal{O}$-term $t=t(x_1,\dots,x_k)$ and constant symbols
$o_1,\dots,o_k\in\mathcal{O}$,
\begin{enumerate}
\item[{\rm(1)}]
$t^{\al C}(o_1^{\al C},\dots,o_k^{\al C})\in o^{\al C}/\alpha$
for some $o\in\mathcal{O}$, and
\item[{\rm(2)}]
for every such $o\in\mathcal{O}$ there exist
$m_i=r_i/\gamma_{o_i}\in\bar H_{o_i,o}$
$($with $r_i=r_i(x)\in\al F$, $r_i(o_i)=o${}$)$ for
$i=1,\dots,k$ such that
\begin{multline}
\label{eq-moduleops}
t^{\al C}(u_1,\dots,u_k)=m_1u_1 +_o \dots +_o m_ku_k +_o
t^{\al C}(o_1^{\al C},\dots,o_k^{\al C})\\
\text{for all $(u_1,\dots,u_k)\in
(o_1^{\al C}/\alpha)\times\dots\times(o_k^{\al C}/\alpha)$.}
\end{multline}
\end{enumerate}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-moduleops}.}
To simplify notation, let $\bar{x}:=(x_1,\dots,x_k)$ and
$\bar{o}:=(o_1,\dots,o_k)$.
(1) holds, because
our assumption that the constants $o^{\al C}$ $(o\in\mathcal{O})$
represent all $\alpha$-classes of $\al C$ implies that the element
$t^{\al C}(\bar{o}^{\al C})$ of $\al C$ is in $o^{\al C}/\alpha$
for some $o\in\mathcal{O}$.
To prove (2)
let us fix such an $o\in\mathcal{O}$ for the rest of the argument.
Since $t=t(\bar{x})$ is an $\mathcal{O}$-term, so is
\begin{equation}
\label{eq-translation}
r(\bar{x}):=d(t(\bar{x}),t(\bar{o}),o).
\end{equation}
As in Lemma~\ref{lm-group}(ii), consider the unary $\mathcal{O}$-terms
$r_i(x):=r(o_1,\dots,o_{i-1},x,o_{i+1},\dots,o_k)$ for each $i$ ($1\le i\le k$).
Since $d$ satisfies the identity $d(x,x,y)=y$ (see \eqref{eq-diffterm1}),
we get that $r(\bar{o})=o$, and hence
$r_i(o_i)=o$ for every $i$ ($1\le i\le k$).
This shows that $r$ satisfies the assumptions of Lemma~\ref{lm-group}(ii),
and $m_i:=r_i/\gamma_{o_i}$ is a member of $\bar H_{o_i,o}$
for every $i$ ($1\le i\le k$).
Thus, first using equality \eqref{eq-linDecomp}
from Lemma~\ref{lm-group}(ii), and then
equality \eqref{eq-moduleMult} from Lemma~\ref{lm-module} we obtain that
\begin{multline}
\label{eq-moduleops-almost}
r^{\al C}(u_1,\dots,u_k)
=r_1^{\al C}(u_1) +_o \dots +_o r_k^{\al C}(u_k)
=m_1u_1 +_o \dots +_o m_ku_k\\
\text{for all $(u_1,\dots,u_k)\in
(o_1^{\al C}/\alpha)\times\dots\times(o_k^{\al C}/\alpha)$.}
\end{multline}
On the other hand, since
$r^{\al C}(u_1,\dots,u_k)\in o^{\al C}/\alpha$
for all $(u_1,\dots,u_k)\in
(o_1^{\al C}/\alpha)\times\dots\times(o_k^{\al C}/\alpha)$
(cf.\ Lemma~\ref{lm-group}(ii)), we can apply
first \eqref{eq-translation}, and then the equality in Lemma~\ref{lm-group}(i)
to get that
\begin{multline*}
r^{\al C}(u_1,\dots,u_k)
=d^{\al C}(t^{\al C}(u_1,\dots,u_k),t^{\al C}(\bar{o}^{\al C}),o^{\al C})
=t^{\al C}(u_1,\dots,u_k) -_o t^{\al C}(\bar{o}^{\al C})\\
\text{for all $(u_1,\dots,u_k)\in
(o_1^{\al C}/\alpha)\times\dots\times(o_k^{\al C}/\alpha)$.}
\end{multline*}
The last displayed equality shows that by adding $t^{\al C}(\bar{o}^{\al C})$
to both sides of \eqref{eq-moduleops-almost}
in the group $\Calpha{o}$ we get the
equality \eqref{eq-moduleops} we wanted to prove.
$\diamond$
To prove the implication (d) $\Rightarrow$ (c), assume that
(d) holds for $U$, that is, the subset
$S_U:=\bigoplus_{o\in\mathcal{O}} ({o}^{\al C}/\alpha_U)$
of $\MCalpha$ is the universe of an
$\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$, and contains
all elements of $\MEalpha$.
In particular, the zero element $(o^{\al C})_{o\in\mathcal{O}}$ of
$\MCalpha$ belongs to $S_U$, which implies that
$\mathcal{O}^{\al C}\subseteq U$.
Let $t=t(x_1,\dots,x_k)$ be an arbitrary $\mathcal{O}$-term,
and let $o_1,\dots,o_k\in\mathcal{O}$.
We want to show that $U$ is closed under the function
$t^{\al C}
(x_1,\dots,x_k){\restriction}_{(o_1^{\al C}/\alpha)\times\dots\times (o_k^{\al C}/\alpha)}$.
By Claim~\ref{clm-moduleops}
there exist $o\in\mathcal{O}$
and $m_i=r_i/\gamma_{o_i}\in\bar H_{o_i,o}$
$($with $r_i=r_i(x)\in\al F$, $r_i(o_i)=o${}$)$ for $i=1,\dots,k$ such that
$t^{\al C}(o_1^{\al C},\dots,o_k^{\al C})\in o^{\al C}/\alpha$ and
\eqref{eq-moduleops} holds.
Let us fix an arbitrary tuple
$(u_1,\dots,u_k)\in
(o_1^{\al C}/\alpha)\times\dots\times (o_k^{\al C}/\alpha)$
that belongs to $U$; that is,
$(u_1,\dots,u_k)\in
(o_1^{\al C}/\alpha_U)\times\dots\times (o_k^{\al C}/\alpha_U)$.
Furthermore, let
$a:=t^{\al C}(o_1^{\al C},\dots,o_k^{\al C})$.
We will be done if we show that the element
\[
b:=t^{\al C}(u_1,\dots,u_k)=m_1u_1 +_o \dots +_o m_ku_k +_o a
\]
belongs to $U$.
Note that $b=t^{\al C}(u_1,\dots,u_k)\equiv_\alpha
t^{\al C}(o_1^{\al C},\dots,o_k^{\al C})=a$, therefore
$a\in o^{\al C}/\alpha$ implies that
$b\in o^{\al C}/\alpha$. Moreover, since
$a=t^{\al C}(o_1^{\al C},\dots,o_k^{\al C})$ is an element of the least
$\mathcal{O}$-subalgebra $\al E$ of $\al C$ (in which
$o^{\al E}=o^{\al C}$), we get from $a\in o^{\al C}/\alpha$ that
$a\in o^{\al E}/\alpha_{\al E}$.
Now, for each $i$ ($1\le i\le k$),
let $\tilde{m}_i$ denote the
$\mathcal{O}\times\mathcal{O}$ matrix which has $m_i$ in position $(o,o_i)$
and zeros (i.e., $o''/\gamma_{o'}$) in all other positions $(o'',o')$,
and let $\tilde{u}_i$ denote the $\mathcal{O}$-tuple
which has $u_i$ in its $o_i$-th position
and zeros (i.e., ${o'}^{\al C}$) in all other positions $o'\in\mathcal{O}$.
Furthermore, let $\tilde{a}$ and $\tilde{b}$ denote the $\mathcal{O}$-tuples
which have
$a$ and $b$, respectively, in their
$o$-th positions, and zeros in all other positions.
Since $m_i\in\bar H_{o_i,o}$ for every $i$ ($1\le i\le k$),
the definition of $\al R(\var{V},\mathcal{O})$ shows that
$\tilde{m}_1,\dots,\tilde{m}_k\in\al R(\var{V},\mathcal{O})$.
Since $u_i\in o_i^{\al C}/\alpha_U$ for every $i$ ($1\le i\le k$) and
$\mathcal{O}^{\al C}\subseteq U$,
we get that
$\tilde{u}_1,\dots,\tilde{u}_k
\in S_U
\subseteq\MCalpha$.
Similarly, $a\in o^{\al E}/\alpha_{\al E}$ implies that
$\tilde{a}
\in \MEalpha$. Therefore, our assumption that $S_U$
contains the elements of $\MEalpha$ yields that
$\tilde{a}\in S_U
\subseteq\MCalpha$.
Finally, we have
$\tilde{b}
\in\MCalpha$, because
$b\in o^{\al C}/\alpha$.
Since $b=m_1u_1 +_o \dots +_o m_ku_k +_o a$,
the construction of the tuples $\tilde{u}_i$, $\tilde{a}$,
$\tilde{b}$ and the matrices $\tilde{m}_i$ makes sure that
the equality
$\tilde{b}
=\tilde{m}_1\tilde{u}_1 + \dots + \tilde{m}_k\tilde{u}_k + \tilde{a}$
holds in the $\al R(\var{V},\mathcal{O})$-module
$\MCalpha$.
Since
$\tilde{u}_1,\dots,\tilde{u}_k,\tilde{a}
\in S_U$
and, by our assumption, $S_U$
is closed under the
$\al R(\var{V},\mathcal{O})$-module operations of
$\MCalpha$, we get that
$\tilde{b}\in S_U$.
Hence, $b\in o^{\al C}/\alpha_U\subseteq U$.
This completes the proof of (d) $\Rightarrow$ (c), and also the proof of
Theorem~\ref{thm-subalg-submod}
\end{proof}
Our second theorem is the analog of Theorem~9.9 in \cite{freese-mckenzie}
mentioned at the beginning of this section.
Given a pair $(\al C,\alpha)$ consisting of an $\mathcal{O}$-algebra in
$\Ovar{V}$ and an abelian congruence $\alpha$ of $\al C$, the
theorem establishes
an isomorphism between the lattice of $\mathcal{O}$-subalgebras of
$\al C$ and an interval
in the submodule lattice of the associated
$\al R(\var{V},\mathcal{O})$-module $\MCalpha$,
provided all $\alpha$-classes are represented in the module.
\begin{thm}
\label{thm-lattice-iso}
Let $\al C$ be an $\mathcal{O}$-algebra
in $\Ovar{V}$, and let $\al E:=\langle\mathcal{O}^{\al C}\rangle$ be
its least $\mathcal{O}$-subalgebra.
If $\alpha$ is an abelian congruence of
$\al C$ such that the constants $o^{\al C}$ $(o\in\mathcal{O})$
represent all $\alpha$-classes of $\al C$, then the mapping
\begin{equation}
\label{eq-lattice-iso}
\al U\mapsto\MUalpha
\end{equation}
is an isomorphism between the
lattice of all $\mathcal{O}$-subalgebras of $\al C$
and the lattice of all $\al R(\var{V},\mathcal{O})$-submodules of
$\MCalpha$ that contain
$\MEalpha$.
\end{thm}
\begin{proof}
It follows from Theorem~\ref{thm-subalg-submod}(1)
that for every $\mathcal{O}$-subalgebra $\al U$ of $\al C$,
$\MUalpha$ is an
$\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$ that contains
$\MEalpha$.
Let $\al U$ and $\al V$ be arbitrary $\mathcal{O}$-subalgebras of
$\al C$.
If $\al U\le\al V$, then
for every $o\in\mathcal{O}$, the congruence class
$o^{\al U}/\alpha_{\al U}=o^{\al C}/\alpha_{\al U}$
in $\al U$ is contained in the corresponding congruence class
$o^{\al V}/\alpha_{\al V}=o^{\al C}/\alpha_{\al V}$
of $\al V$.
This implies that
$\MUalpha
\le\MValpha$,
showing that the map
\eqref{eq-lattice-iso} is order-preserving.
Now suppose that $\al U\not\le\al V$.
The assumption that the interpretations of the
constant symbols $o\in\mathcal{O}$
represent all $\alpha$-classes of $\al C$,
implies the existence of $o^{\al C}=o^{\al U}=o^{\al V}$
such that $o^{\al U}/\alpha_{\al U}\not\subseteq o^{\al V}/\alpha_{\al V}$.
Thus,
$\MUalpha
\not\le\MValpha$.
This shows that the map \eqref{eq-lattice-iso}
preserves and reflects $\le$; hence, in particular, it
is one-to-one.
The surjectivity of the map \eqref{eq-lattice-iso} is proved by
the next claim.
\begin{clm}
\label{clm-allSubmods}
Let $\al C$, $\al E$, and $\alpha$ be as in
Theorem~\ref{thm-lattice-iso}.
If $\al N$ is an $\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$ that contains $\MEalpha$, then
\begin{enumerate}
\item[{\rm(1)}]
as an abelian group, $\al N=\bigoplus_{o\in\mathcal{O}}\al N_o$
where for every $o\in\mathcal{O}$, $\al N_o$ is the subgroup of
$\Calpha{o}$ that
consists of the $o$-components of all tuples in $\al N$;
\item[{\rm(2)}]
$N_o=N_{o'}$ whenever $o^{\al C}/\alpha={o'}^{\al C}/\alpha$
$(o,o'\in\mathcal{O})$;
\item[{\rm(3)}]
the union $U:=\bigcup_{o\in\mathcal{O}}N_o$
of the universes of the groups $\al N_o$
is the universe of an $\mathcal{O}$-subalgebra $\al U$ of $\al C$, and
$\al N=\MUalpha$.
\end{enumerate}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-allSubmods}.}{}
For (1) notice first that
since the group operations in $\MCalpha$
are performed coordinatewise, it follows immediately
from the definition of the $\al N_o$'s
that $\al N_o$ is a subgroup of
$\Calpha{o}$ (in particular, $o^{\al C}\in\al N_o$)
for each $o\in\mathcal{O}$, and
$\al N\subseteq\bigoplus_{o\in\mathcal{O}}\al N_o$.
To prove that equality holds, consider
an arbitrary element
$(a_o)_{o\in\mathcal{O}}\in\bigoplus_{o\in\mathcal{O}}\al N_o
\bigl(\subseteq \MCalpha\bigr)$.
Then (i)~$a_o\in N_o$
for every $o\in\mathcal{O}$, and
(ii)~$a_o=o^{\al C}$ for all but finitely many $o$'s.
Condition~(i) implies that for every $o\in\mathcal{O}$
there exists a tuple $\bar{a}_o\in N$ such that the $o$-th
coordinate of $\bar{a}_o$ is $a_o$. Let $e_o$ be the matrix in
$\al R(\var{V},\mathcal{O})$ which has $x/\gamma_o$ in position
$(o,o)$, and zeros (that is, $o''/\gamma_{o'}$) in all other positions
$(o'',o')$. Since $\al N$ is an $\al R(\var{V},\mathcal{O})$-submodule
of $\MCalpha$,
each tuple $e_o\bar{a}_o$ is an element of $\al N$.
It follows from the construction of $e_o$ that the tuple $e_o\bar{a}_o$
has $a_o$ in the $o$-th coordinate and zeros elsewhere.
By condition~(ii),
only finitely many of the tuples $e_o\bar{a}_o$ ($o\in\mathcal{O}$)
is not the zero element of $\al N$, therefore
their sum in $\MCalpha$, and hence also in $\al N$,
is $(a_o)_{o\in\mathcal{O}}$. Thus $(a_o)_{o\in\mathcal{O}}\in\al N$,
which finishes the proof that $\al N=\bigoplus_{o\in\mathcal{O}}\al N_o$.
To prove (2)
assume that $o,o'\in\mathcal{O}$ are such that
$o^{\al C}/\alpha= {o'}^{\al C}/\alpha$.
Since $o^{\al E}=o^{\al C}$ and ${o'}^{\al E}={o'}^{\al C}$, we also have that
$o^{\al E}/\alpha_{\al E}={o'}^{\al E}/\alpha_{\al E}$.
Therefore
our assumption that $\al N$ is an $\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$ that contains $\MEalpha$ implies that
\[
o^{\al E}/\alpha_{\al E}={o'}^{\al E}/\alpha_{\al E}\subseteq N_o\cap N_{o'}
\subseteq N_o\cup N_{o'}\subseteq o^{\al C}/\alpha= {o'}^{\al C}/\alpha.
\]
In particular, we have that $o^{\al E},{o'}^{\al E}\in N_o\cap N_{o'}$,
or equivalently, $o^{\al C},{o'}^{\al C}\in N_o\cap N_{o'}$.
For any element $a\in N_o$ let $\tilde{a}$ denote the $\mathcal{O}$-tuple
which has $a$ in its $o$-th position, and zeros
in all other positions (i.e., ${o''}^{\al C}$ in the $o''$-th position for
$o''\not=o$).
It follows from statement (1) that $\tilde{a}\in\al N$ for all $a\in N_o$.
Now let $r$ denote the $\mathcal{O}\times\mathcal{O}$
matrix which has $d(x,o,o')/\gamma_o$ in position
$(o',o)$ and zeros (i.e., $o'''/\gamma_{o''}$) in all other positions
$(o''',o'')$. Since $d(o,o,o')=o'$, we have that
$d(x,o,o')/\gamma_o\in\bar{H}_{o,o'}$, and hence
$r\in\al R(\var{V},\mathcal{O})$.
As $\al N$ is an $\al R(\var{V},\mathcal{O})$-submodule of
$\MCalpha$, we get that
$r\tilde{a}\in\al N$ for all $a\in N_o$.
Thus
the $o'$-th component of $r\tilde{a}$ is in $N_{o'}$; that is,
\begin{equation}
\label{eq-transl}
d^{\al C}(a,o^{\al C},{o'}^{\al C})=\bigl(d(x,o,o')/\gamma_o\bigr)a\in N_{o'}
\quad\text{for all $a\in N_o$.}
\end{equation}
By Lemma~\ref{lm-group}(i),
$d^{\al C}(u,o^{\al C},{o'}^{\al C})=u+_o{o'}^{\al C}$ holds in
$\Calpha{o}$ for all $u\in o^{\al C}/\alpha$.
Therefore \eqref{eq-transl} can be restated as follows:
the image of $N_o$ under the translation
$u\mapsto u+_o{o'}^{\al C}$ is contained in $N_{o'}$.
Because of ${o'}^{\al C}\in N_o$ this translation
is a bijection of the abelian group $\al N_o$ onto itself,
so we get that $N_o\subseteq N_{o'}$.
A similar argument, with the roles of $o$ and $o'$ switched, shows that
$N_{o'}\subseteq N_o$ also holds, and hence finishes the proof of
statement (2).
Finally, to establish (3),
recall that $U:=\bigcup_{o\in\mathcal{O}} N_o$.
Since
$(o^{\al C}\in)N_o\subseteq o^{\al C}/\alpha(\subseteq C)$
for every $o\in\mathcal{O}$, we get that $U\subseteq C$ and
that the sets $N_o$ and $N_{o'}$ are disjoint
whenever $o^{\al C}\not\equiv_\alpha {o'}^{\al C}$.
By statement (2),
$N_o=N_{o'}$
whenever $o^{\al C}\equiv_\alpha {o'}^{\al C}$.
Therefore $\{N_o:o\in\mathcal{O}\}$
is the partition of $U$ corresponding to the equivalence relation
$\alpha_U$.
It follows now from statement (1) that the universe of $\al N$ is the set
$\bigoplus_{o\in\mathcal{O}} N_o=\bigoplus_{o\in\mathcal{O}} (o^{\al C}/\alpha_U)$.
Hence, the equivalence of conditions (d) and (b) in
Theorem~\ref{thm-subalg-submod}(2) implies that $U$ is the universe
of an $\mathcal{O}$-subalgebra $\al U$ of $\al C$.
Moreover, by Theorem~\ref{thm-subalg-submod}(3), we have that
$\al N=\MUalpha$.
This finishes the proof of Claim~\ref{clm-allSubmods}.
$\diamond$
The proof of Theorem~\ref{thm-lattice-iso} is complete.
\end{proof}
We conclude this section by an auxiliary result concerning the modules
associated to $\alpha$-saturated subalgebras of direct products
where $\alpha$ is an abelian product congruence.
\begin{lm}
\label{lm-dir-pr}
Let $\al C$ be an $\mathcal{O}$-subalgebra of an $\mathcal{O}$-product
$\prod_{i=1}^n\al B_i$ of $\mathcal{O}$-algebras
$\al B_1,\dots,\al B_n$ in $\Ovar{V}$, and let $\alpha=\prod_{i=1}^n\alpha_i$
be the product congruence of $\prod_{i=1}^n\al B_i$
where for each $i$, $\alpha_i$ is an abelian congruence of $\al B_i$.
If $\al C$ is $\alpha$-saturated in $\prod_{i=1}^n\al B_i$,
then
\begin{enumerate}
\item[{\rm(1)}]
$\alpha_{\al C}$ is an abelian congruence of $\al C$ and
$o^{\al C}/\alpha_{\al C}=
\prod_{i=1}^n(o^{\al B_i}/\alpha_i)$ holds for all
$o\in\mathcal{O}$; moreover,
\item[{\rm(2)}]
the $\al R(\var{V},\mathcal{O})$-modules
$\MCCalpha$ and
$\prod_{i=1}^n\MBialpha$
are naturally isomorphic via the map defined by
\begin{equation}
\label{eq-saturated}
\begin{aligned}
\MCCalpha
& \to
\prod_{i=1}^n\MBialpha \\
(a_o)_{o\in\mathcal{O}}=\bigl((a_{o1},\dots,a_{on})\bigr)_{o\in\mathcal{O}}
& \mapsto
\bigl((a_{o1})_{o\in\mathcal{O}},\dots,(a_{on})_{o\in\mathcal{O}}\bigr)
\end{aligned}
\end{equation}
for all
$a_o=(a_{o1},\dots,a_{on})\in o^{\al C}/\alpha_{\al C}=
\prod_{i=1}^n(o^{\al B_i}/\alpha_i)$ $(o\in\mathcal{O})$.
\end{enumerate}
\end{lm}
\begin{proof}{}
First we prove (1).
Since each $\alpha_i$ is an abelian congruence of $\al B_i$, their
product, $\alpha$, is an abelian congruence of $\prod_{i=1}^n\al B_i$,
and hence $\alpha_{\al C}$ is an abelian congruence of $\al C$.
Therefore the module $\MCCalpha$
exists.
By our assumption, $\prod_{i=1}^n\al B_i$ is an $\mathcal{O}$-product
and $\al C$ is an $\mathcal{O}$-subalgebra of
$\prod_{i=1}^n\al B_i$, so $o^{\al C}=(o^{\al B_1},\dots,o^{\al B_n})$
for every $o\in\mathcal{O}$. This implies that
$o^{\al C}/\alpha_{\al C}\subseteq\prod_{i=1}^n (o^{\al B_i}/\alpha_i)$
for every $o\in\mathcal{O}$.
In fact, $=$ holds here, because $\al C$ is $\alpha$-saturated in
$\prod_{i=1}^n\al B_i$.
Now we prove (2).
It follows from the equalities in (1) that
the elements of $\MCCalpha$
are exactly the tuples $(a_o)_{o\in\mathcal{O}}$ such that
(i)~$a_o\in o^{\al C}/\alpha_{\al C}=\prod_{i=1}^n(o^{\al B_i}/\alpha_i)$, that is,
$a_o=(a_{o1},\dots,a_{on})$ with $a_{oi}\in o^{\al B_i}/\alpha_i$
for every $o$ and $i$, and
(ii)~$a_o=o^{\al C}=(o^{\al B_1},\dots,o^{\al B_n})$ for all but finitely many
$o$'s.
The image of each such tuple $(a_o)_{o\in\mathcal{O}}$ under the map
\eqref{eq-saturated} clearly belongs to the module
$\prod_{i=1}^n\MBialpha$.
The map \eqref{eq-saturated}
just regroups coordinates, so it is one-to-one. To see that it is also onto,
let $\bigl((b_{o1})_{o\in\mathcal{O}},\dots,(b_{on})_{o\in\mathcal{O}}\bigr)\in
\prod_{i=1}^n\MBialpha$.
Then, for every $i$, we have that
(i)$'$~$b_{oi}\in o^{\al B_i}/\alpha_i$ for all $o\in\mathcal{O}$, and
(ii)$'$~$b_{oi}=o^{\al B_i}$ for all but finitely many $o$'s.
Therefore the tuples $b_o:=(b_{o1},\dots,b_{on})$ satisfy the conditions
that
(i)~$b_o\in \prod_{i=1}^n(o^{\al B_i}/\alpha_i)=o^{\al C}/\alpha_{\al C}$
for all $o\in\mathcal{O}$, and
(ii)~$b_o=(o^{\al B_1},\dots,o^{\al B_n})=o^{\al C}$ for all but finitely many $o$'s.
Hence $(b_o)_{o\in\mathcal{O}}\in\MCCalpha$, and
$\bigl((b_{o1})_{o\in\mathcal{O}},\dots,(b_{on})_{o\in\mathcal{O}}\bigr)$
is its image under the map \eqref{eq-saturated}. This proves that
\eqref{eq-saturated} is a bijection.
To see that \eqref{eq-saturated} is a group isomorphism, recall from
Lemma~\ref{lm-module} that
in each one of the modules $\MCCalpha$ and
$\MBialpha$ ($1\le i\le n$), $+$
is defined to be the coordinatewise operation that is $+_o$ in
coordinate $o$ for each $o\in\mathcal{O}$.
Therefore, in the module
$\prod_{i=1}^n\MBialpha$, $+$ is the
coordinatewise operation that is the operation
$+_o$ on $o^{\al B_i}/\alpha_i\,(\subseteq\al B_i)$ in coordinate $(o,i)$.
In the module $\MCCalpha$,
the operation $+_o$ on $o^{\al C}/\alpha_{\al C}=\prod_{i=1}^n(o^{\al B_i}/\alpha_i)$
also acts coordinatewise, because $+_o$ is an $\mathcal{O}$-term
and $\al C$ is an $\mathcal{O}$-subalgebra of $\prod_{i=1}^n\al B_i$.
Therefore, in $\MCCalpha$, too,
$+$ is the
coordinatewise operation that is the operation
$+_o$ on $o^{\al B_i}/\alpha_i\,(\subseteq\al B_i)$ in coordinate $(o,i)$.
This shows that \eqref{eq-saturated} is a group isomorphism.
Finally, we will argue that \eqref{eq-saturated} is
an $\al R(\var{V},\mathcal{O})$-module isomorphism.
Recall again from Lemma~\ref{lm-module} and the discussion
preceding it that
$\al R(\var{V},\mathcal{O})$ is a ring of matrices, and multiplication
by elements of $\al R(\var{V},\mathcal{O})$ in each one of the modules
$\MCCalpha$ and
$\MBialpha$ ($1\le i\le n$) is defined by
matrix multiplication. Furthermore, in the module
$\prod_{i=1}^n\MBialpha$, multiplication by
ring elements is performed coordinatewise.
Therefore we will be done if we show that
\begin{enumerate}
\item[$(*)$]
multiplication by ring elements acts coordinatewise
on the components $a_o=(a_{o1},\dots,a_{on})$ of the elements
$(a_o)_{o\in\mathcal{O}}$ of $\MCCalpha$.
\end{enumerate}
Since \eqref{eq-saturated} is a group isomorphism, it suffices to prove
$(*)$ for the multiplication of entries.
Following the definition of multiplication of entries in Lemma~\ref{lm-module},
let $m_{o',o}=r/\gamma_o\in\bar{H}_{o,o'}$ (where $r=r(x)\in\al F$ and
$r(o)=o'$), and let
$a_o=(a_{o1},\dots,a_{on})\in
o^{\al C}/\alpha_{\al C}=\prod_{i=1}^n(o^{\al B_i}/\alpha_i)$.
By definition, $m_{o',o}a_o=r^{\al C}(a_o)$.
Since $r$ is an $\mathcal{O}$-term and $\al C$ is an $\mathcal{O}$-subalgebra
of $\prod_{i=1}^n\al B_i$, we get that
$r^{\al C}(a_o)=
\bigl(r^{\al B_i}(a_{oi})\bigr)_{i=1}^n
=(m_{o',o}a_{oi})_{i=1}^n$.
This proves that multiplication by $m_{o',o}$ acts coordinatewise
on $a_o$, and hence finishes the proof of the lemma.
\end{proof}
The isomorphism \eqref{eq-saturated} shows that under the assumptions of
Lemma~\ref{lm-dir-pr} the only difference between the modules
$\MCCalpha$ and
$\prod_{i=1}^n\MBialpha$ is how we group their
coordinates. Therefore we may identify
$\MCCalpha$ and
$\prod_{i=1}^n\MBialpha$ via the isomorphism
\eqref{eq-saturated}, and view the submodules of
$\MCCalpha$ as
submodules of the direct product
$\prod_{i=1}^n\MBialpha$.
\section{Proof of Theorem~\ref{thm-main}}\label{mainthm-sec}
Throughout this section $\al A$ will be
a finite algebra with a $k$-parallelogram term, and
$\var{V}$ the variety generated by $\al A$.
Our goal is to combine the results of Sections~\ref{reduc} and \ref{modules}
to prove that if $\al A$ satisfies the
split centralizer condition, then $\al A$ is dualizable.
In view of Theorem~\ref{thm-WZ}, we will be done if we can show
that there exists a constant $\mathsf{c}$,
depending only on $\al A$, such that
\begin{equation}
\label{eq-ToProve}
\rel{R}_\mathsf{c}({\al A})\models_{\text{\rm d}}\rel{R}(\al A).
\end{equation}
Recall that $\rel{R}(\al A)$ stands for
the set of all (finitary) compatible relations of
$\al A$, and for every positive integer $n$,
$\rel{R}_n({\al A})$ denotes the set of all compatible relations
of $\al A$ of arity $\le n$.
The relation $\models_{\text{\rm d}}$ is described in Theorem~\ref{thm-dentails}.
In addition to the natural numbers $\mathsf{a}$, $\mathsf{s}$, $\mathsf{i}$,
and $\mathsf{p}$ introduced in Section~\ref{reduc}, we will need a few other
parameters related to $\al A$, which we introduce now.
The list includes the constant $\mathsf{c}$ that we will use in the proof of
\eqref{eq-ToProve}.
Before defining the new parameters, let us fix a set $\mathcal{O}$ of
constant symbols not occurring in the language of $\var{V}$ such that
$|\mathcal{O}|=\mathsf{i}$.
Furthermore, let $\text{\rm Mod}(\al A,\mathsf{p})$ denote the
set of all $\al R(\var{V},\mathcal{O})$-modules $\MTalpha$
where $\al T$ is a subalgebra of $\al A^p$
for some $1\le p\le\mathsf{p}$ that is isomorphic to a section of $\al A$,
$\alpha$ is a nontrivial
abelian congruence of $\al T$, and $\al T$ is made into an
$\mathcal{O}$-algebra by interpreting
the new constant symbols $o\in\mathcal{O}$
in $\al T$ in such a way that the
elements $o^{\al T}$ ($o\in\mathcal{O}$) represent every $\alpha$-class.
\begin{itemize}
\item
For every
$\al R(\var{V},\mathcal{O})$-module
$\al Q$, let
\[
\mathsf{h}_{\al Q}:=\sum\{|\Hom(\al M,\al Q)|:\al M\in\text{\rm Mod}(\al A,\mathsf{p})\},
\]
and
let $\mathsf{h}$ be the maximum of the numbers $\mathsf{h}_{\al Q}$ as
$\al Q$ runs over all subdirectly irreducible
$\al R(\var{V},\mathcal{O})$-modules.
($\mathsf{h}$ stands for `$\underline{\text{h}}$omomorphisms'.)
\item
Let $\mathsf{e}$ be the least common multiple of the exponents of the additive
groups of all modules $\al M\in\text{\rm Mod}(\al A,\mathsf{p})$.
($\mathsf{e}$ stands for `$\underline{\text{e}}$xponent'.)
\item
Let
$\mathsf{c}_0:=\max(1+\mathsf{p},k-1)$, and let
$\mathsf{c}:=\max(\mathsf{c}_0,\mathsf{p}\mathsf{h}(\mathsf{e}+1))$.
($\mathsf{c}$ stands for `$\underline{\text{c}}$onstant'.)
\end{itemize}
Note that the ring $\al R(\var{V},\mathcal{O})$ is finite, because
it consists of $\mathcal{O}\times\mathcal{O}$ matrices where each entry
is determined by an element of the free algebra $\al F$ in
$\var{V}$ with free generating set $\{x\}\cup\mathcal{O}$,
and $\al F$ is finite, since $\var{V}$ is generated by $\al A$
(a finite algebra).
The set $\text{\rm Mod}(\al A,\mathsf{p})$ is also finite, since $\al A$, $\mathsf{p}$,
and $\mathcal{O}$ are finite.
It follows from the main result of \cite{kearnes-module} that
$|\al Q|\le|\al R(\var{V},\mathcal{O})|$ holds
for every subdirectly irreducible $\al R(\var{V},\mathcal{O})$-module
$\al Q$. Therefore there are only finitely many
subdirectly irreducible $\al R(\var{V},\mathcal{O})$-modules,
up to isomorphism, and all are finite, so $\mathsf{h}$ is a natural number.
The finiteness of $\text{\rm Mod}(\al A,\mathsf{p})$ implies also that $\mathsf{e}$ is a natural number,
and hence so are $\mathsf{c}_0$ and $\mathsf{c}$.
\begin{remrks}
\label{rem-about-rhqc}
(a)
The definition of $\text{\rm Mod}(\al A,\mathsf{p})$ shows that
$\text{\rm Mod}(\al A,\mathsf{p})\not=\emptyset$ if and only if
some power $\al A^p$ of $\al A$ with $1\le p\le\mathsf{p}$ has a subalgebra
that is isomorphic to a section of $\al A$
with a nontrivial abelian congruence.
(b)
If $\text{\rm Mod}(\al A,\mathsf{p})=\emptyset$, then
$\mathsf{p}=1$,
$\mathsf{e}=1$, $\mathsf{h}=0$,
and
$\mathsf{c}=\mathsf{c}_0=\max(2,k-1)$,
while if $\text{\rm Mod}(\al A,\mathsf{p})\not=\emptyset$, then
$\mathsf{p}\ge1$,
$\mathsf{e}\ge2$, and
$\mathsf{h}\ge|\text{\rm Mod}(\al A,\mathsf{p})|\ge 1$, because
the summands in the definition of $\mathsf{h}_{\al Q}$
satisfy $|\Hom(\al M,\al Q)|\ge 1$
for all $\al M\in\text{\rm Mod}(\al A,\mathsf{p})$ and all
$\al R(\var{V},\mathcal{O})$-modules $\al Q$.
(c)
In particular, for the trivial $\al R(\var{V},\mathcal{O})$-module
$\al Q_0$ we have
$\mathsf{h}_{\al Q_0}=|\text{\rm Mod}(\al A,\mathsf{p})|$, therefore the inequality in (b)
implies that $\mathsf{h}\ge\mathsf{h}_{\al Q_0}$.
\end{remrks}
We start the proof of Theorem~\ref{thm-main} by considering
compatible relations $B$ of $\al A$ that are constructed in
Theorem~\ref{thm-reduction}.
We will use the same notation for these relations as in
Theorem~\ref{thm-reduction}, except that the
$\tilde{\phantom{n}}$'s from the notation of
the congruences $\tilde\alpha_i$ will be omitted.
So, the set of relations
we will consider, and will denote by $\rel{R}^*(\al A)$,
consists of all compatible relations $B$ of $\al A$
that satisfy the following condition $(*)$ from Theorem~\ref{thm-reduction},
for some $n$:
\begin{enumerate}
\item[$(*)$]
There exist
\begin{enumerate}
\item[{\rm(I)}]
subalgebras $\al B_i\le\al A^{p_i}$ with $p_i\le\mathsf{p}$
for each $i\in[n]$
such that $\al B_i$ is isomorphic to a section of $\al A$, and
\item[{\rm(II)}]
nontrivial abelian congruences
$\alpha_i\in\Con(\al B_i)$ $(i\in[n])$
\end{enumerate}
such that
\begin{enumerate}
\item[{\rm(III)}]
$B$ is the universe of a
subdirect product $\al B$ of
$\al B_1,\dots,\al B_n$, and
\item[{\rm(IV)}]
the product congruence
$\alpha:=\prod_{i=1}^n \alpha_i$ of $\prod_{i=1}^n\al B_i$
restricts to $\al B$ as a congruence
$\alpha_{\al B}$
of index $\le\mathsf{i}$.
\end{enumerate}
\end{enumerate}
We will refer to $n$ as the \emph{$*$-arity of $B$}.
Since
$\al B\le\prod_{i=1}^n\al B_i\le\prod_{i=1}^n\al A^{p_i}$,
the arity of $B$, as a compatible
relation of $\al A$, is $\sum_{i=1}^n p_i$.
\begin{lm}
\label{lm-main}
Let $B\in\rel{R}^*(\al A)$ be a compatible relation of $\al A$
of $*$-arity $n$, and let $\al B_i$, $\alpha_i$, $\al B$, and $\alpha$
be as in $(*)$.
If
$n>\mathsf{h}(\mathsf{e}+1)$
and $\al B$ is
a $\cap$-irreducible subalgebra
of its $\alpha$-saturation $\al B[\alpha]$ in $\prod_{i=1}^n\al B_i$,
then there exist
\begin{enumerate}
\item[{\rm(1)}]
a compatible relation $B'\in\rel{R}^*(\al A)$ of $\al A$ of $*$-arity
$\le n-\mathsf{e}$
and
\item[{\rm(2)}]
subalgebras $\al D\le\al B_i^{\mathsf{e}+1}$ and
$\al D'\le\al B_i^2$
for some $i\in[n]$
\end{enumerate}
such that
\begin{enumerate}
\item[{\rm(3)}]
$\{B_1,\dots,B_n,B',D,D'\}\models_{\text{\rm d}} B$.
\end{enumerate}
\end{lm}
\begin{proof}
Let $B$ satisfy the hypotheses of the lemma,
including the assumptions that
$n>\mathsf{h}(\mathsf{e}+1)$
and
$\al B$ is a $\cap$-irreducible subalgebra of $\al B[\alpha]$.
Since
$\al A$ has a parallelogram term, we know from Theorem~\ref{thm-parterm-cm}
that the variety $\var{V}$
it generates is congruence modular.
As in Section~\ref{modules}, we
expand the language of $\var{V}$ by $\mathcal{O}$.
Let us fix interpretations $o^{\al B}$ for each $o\in\mathcal{O}$ in $\al B$
in such a way that the elements $o^{\al B}$ ($o\in\mathcal{O}$) represent all
$\alpha_{\al B}$-classes of $\al B$; this is possible, since one of our
assumptions is that $\alpha_{\al B}$ has index $\le\mathsf{i}$.
So, $\al B$ becomes an $\mathcal{O}$-algebra in $\Ovar{V}$.
Now we fix interpretations $o^{\al B_i}$
for every constant symbol $o\in\mathcal{O}$
in each $\al B_i$ ($i\in[n]$) such that
$o^{\al B}=(o^{\al B_1},\dots,o^{\al B_n})$. Thus, each $\al B_i$ will become
an $\mathcal{O}$-algebra in $\Ovar{V}$.
(Note that $\al B_i$ and $\al B_j$ might become different
$\mathcal{O}$-algebras even if $\al B_i$ and $\al B_j$ are the same
as algebras in the original language of $\var{V}$.)
This choice makes sure that $\al B$ is a subdirect
$\mathcal{O}$-subalgebra of the $\mathcal{O}$-product
$\prod_{i=1}^n \al B_i$
of the $\mathcal{O}$-algebras $\al B_1,\dots,\al B_n$.
Since the elements $o^{\al B}$ ($o\in\mathcal{O}$)
represent all $\alpha_{\al B}$-classes of $\al B$, it follows that
for every $i\in[n]$, the elements $o^{\al B_i}$ ($o\in\mathcal{O}$)
represent all $\alpha_i$-classes of $\al B_i$.
Hence $\MBialpha\in\text{\rm Mod}(\al A,\mathsf{p})$ for every $i\in[n]$.
To make $\al B[\alpha]$ into an $\mathcal{O}$-algebra in $\Ovar{V}$,
let $o^{\al B[\alpha]}:=o^{\al B}$ for every $o\in\mathcal{O}$.
Then $\al B[\alpha]$ is also a subdirect
$\mathcal{O}$-subalgebra of the $\mathcal{O}$-product
$\prod_{i=1}^n \al B_i$
of the $\mathcal{O}$-algebras $\al B_1,\dots,\al B_n$,
and $\al B$ is an $\mathcal{O}$-subalgebra of
$\al B[\alpha]$.
Moreover, since
every $\alpha_{\al B[\alpha]}$-class contains an $\alpha_{\al B}$-class,
the elements $o^{\al B[\alpha]}=o^{\al B}$ ($o\in\mathcal{O}$) represent all
$\alpha_{\al B[\alpha]}$-classes of $\al B[\alpha]$.
Now let us consider the $\al R(\var{V},\mathcal{O})$-modules
$\MBalpha$ and $\MBsatalpha$.
The fact that $\al B$ is an $\mathcal{O}$-subalgebra of $\al B[\alpha]$
implies, by Theorem~\ref{thm-lattice-iso},
that $\MBalpha$ is an $\al R(\var{V},\mathcal{O})$-submodule of $\MBsatalpha$.
Our additional assumption that $\al B$ is
a $\cap$-irreducible subalgebra
of $\al B[\alpha]$ implies
that $\al B$ is also $\cap$-irreducible
in the lattice of $\mathcal{O}$-subalgebras of $\al B[\alpha]$,
hence, again by Theorem~\ref{thm-lattice-iso},
$\MBalpha$ is a $\cap$-irreducible $\al R(\var{V},\mathcal{O})$-submodule
of $\MBsatalpha$.
Recall also that by Lemma~\ref{lm-dir-pr} (applied to $\al C=\al B[\alpha]$),
the $\al R(\var{V},\mathcal{O})$-modules
$\MBsatalpha$ and $\prod_{i=1}^n\MBialpha$ are
naturally isomorphic.
Therefore we can identify the modules
$\MBsatalpha$ and $\prod_{i=1}^n\MBialpha$
via this isomorphism, and get that
$\MBalpha$ is a $\cap$-irreducible $\al R(\var{V},\mathcal{O})$-submodule
of $\prod_{i=1}^n\MBialpha$.
Thus, the quotient
\[
\al Q:=\Bigl(\prod_{i=1}^n\MBialpha\Bigr)\Big/\MBalpha
\]
is a subdirectly irreducible
or trivial
$\al R(\var{V},\mathcal{O})$-module
(according to whether $\al B<\al B[\alpha]$ or
$\al B=\al B[\alpha]$).
Let $\phi\colon\prod_{i=1}^n\MBialpha\to\al Q$
be the natural homomorphism.
For each $i\in[n]$,
the mapping $\psi_i\colon\MBialpha\to\al Q$ defined by
$\psi_i(z)=\phi(0,\dots,0,z,0,\dots,0)$ (with $z$ in the $i$-th position)
for all $z\in\MBialpha$
is an $\al R(\var{V},\mathcal{O})$-module homomorphism, and
\[
\phi(z_1,\dots,z_n)=\sum_{i=1}^n\psi_i(z_i)
\quad
\text{for all $(z_1,\dots,z_n)\in\prod_{i=1}^n\MBialpha$.}
\]
Thus, for every element $(z_1,\dots,z_n)$ of $\prod_{i=1}^n\MBialpha$,
\begin{equation}
\label{eq-solset}
(z_1,\dots,z_n)\in M_{\al B}(\alpha_{\al B},\mathcal{O})
\quad\text{if and only if}\quad
\sum_{i=1}^n\psi_i(z_i)=0;
\end{equation}
in other words, $M_{\al B}(\alpha_{\al B},\mathcal{O})$ is the solution set of the equation
$\sum_{i=1}^n\psi_i(z_i)=0$ in $\prod_{i=1}^n\MBialpha$.
Since the $\al R(\var{V},\mathcal{O})$-module
$\al Q$ is subdirectly irreducible
or trivial,
and
the $\al R(\var{V},\mathcal{O})$-modules $\MBialpha$
all belong to $\text{\rm Mod}(\al A,\mathsf{p})$, the definition of $\mathsf{h}$
(combined with Remarks~\ref{rem-about-rhqc}~(c))
makes sure that
there are at most $\mathsf{h}$ distinct homomorphisms among the $\psi_i$'s
($i\in [n]$).
Therefore our assumption
$n>\mathsf{h}(\mathsf{e}+1)$
forces that
at least
$\mathsf{e}+2$
of the $\psi_i$'s are equal.
By permuting coordinates we may assume that
for $i=1,\dots,\mathsf{e}+2$
the $\psi_i$' are equal;
hence for $i=1,\dots,\mathsf{e}+2$
the $\MBialpha$'s are equal as $\al R(\var{V},\mathcal{O})$-modules,
the $\al B_i$'s are equal as $\mathcal{O}$-algebras, and
the $\alpha_i$'s are equal.
Let
\begin{align}
\bar{\psi} &{}:= \psi_1=\dots=\psi_{\mathsf{e}+2},\label{eq-psi-s}\\
\bar{\al B} &{}:=\al B_1=\dots=\al B_{\mathsf{e}+2},\label{eq-B-s}\\
\bar{\alpha} &{}:=\alpha_1=\dots=\alpha_{\mathsf{e}+2},\label{eq-alpha-s}
\end{align}
and
\begin{equation}
\label{eq-o-s}
o^{\bar{\al B}}:=o^{\al B_1}=\dots=o^{\al B_{\mathsf{e}+2}}\quad
\text{for every $o\in\mathcal{O}$}.
\end{equation}
From now on we will
partition the coordinates of $\prod_{i=1}^n\al B_1$
(and its subalgebras $\al B[\alpha]$, $\al B$, etc.)
into two blocks, the first block consisting of the first
$\mathsf{e}+1$ coordinates, and the second from coordinates $\mathsf{e}+2,\dots,n$.
Accordingly, we will
use the notation $\bar{\wec{x}}=(x_1,\dots,x_{\mathsf{e}+1})$
for the elements of
$\bar{\al B}^{\mathsf{e}+1}=\prod_{i=1}^{\mathsf{e}+1}\al B_i$
and the notation $\wec{x}=(x_{\mathsf{e}+2},\dots,x_n)$
for the elements of $\prod_{i=\mathsf{e}+2}^n\al B_i$.
Hence, an element of $\prod_{i=1}^n\al B_i$ (in particular, of
$\al B$ or $\al B[\alpha]$) will be written as $(\bar{\wec{x}},\wec{x})$.
For any tuple $(\bar{\wec{x}},\wec{x})\in\al B[\alpha]$ let
$\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$ denote
the set of all $o\in\mathcal{O}$ such that
$(\bar{\wec{x}},\wec{x})\in o^{\al B[\alpha]}/\alpha_{\al B[\alpha]}$.
\begin{clm}
\label{clm-alpha}
The following hold for every element $(\bar{\wec{x}},\wec{x})\in\al B[\alpha]$:
\begin{enumerate}
\item[{\rm(1)}]
$\mathcal{O}_{(\bar{\wec{x}},\wec{x})}\not=\emptyset$, and for any
$o\in\mathcal{O}$,
\begin{align*}
o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}
\quad &\Leftrightarrow\quad
x_i\in o^{\al B_i}/\alpha_i\ \ \text{for all\ \ $i\in[n]$}\\
\quad &\Leftrightarrow\quad
x_i\in o^{\bar{\al B}}/\bar{\alpha}\ \ \text{for all\ \ $i\in[\mathsf{e}+1]$, and}\\
\quad &\phantom{{}\Leftrightarrow{}}\quad
x_i\in o^{\al B_i}/\alpha_i\ \ \text{for all\ \ $i\in[n]\setminus[\mathsf{e}+1]$.}
\end{align*}
\item[{\rm(2)}]
In particular,
$x_1\equiv_{\bar{\alpha}}x_2\equiv_{\bar{\alpha}}\dots\equiv_{\bar{\alpha}}
x_{\mathsf{e}+1}$.
\end{enumerate}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-alpha}.} Let
$(\bar{\wec{x}},\wec{x})\in\al B[\alpha]$.
To prove (1) observe first that
since the elements $o^{\al B[\alpha]}$ ($o\in\mathcal{O}$) represent all
$\alpha_{\al B[\alpha]}$-classes, there is an $o\in\mathcal{O}$ such that
$o^{\al B[\alpha]}$ is in the $\alpha_{\al B[\alpha]}$-class of
$(\bar{\wec{x}},\wec{x})$.
Hence $o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$, proving that
$\mathcal{O}_{(\bar{\wec{x}},\wec{x})}\not=\emptyset$.
For the second statement in (1), recall that
$o^{\al B[\alpha]}=(o^{\al B_1},\dots,o^{\al B_n})$ for every
$o\in\mathcal{O}$, and that $\alpha_{\al B[\alpha]}$ is the restriction of
the product congruence $\alpha=\prod_{i=1}^n\alpha_i$ to $\al B[\alpha]$.
Thus, $(\bar{\wec{x}},\wec{x})\in o^{\al B[\alpha]}/\alpha_{\al B[\alpha]}$
if and only if
$x_i\in o^{\al B_i}/\alpha_i$ for all $i\in[n]$.
This, together with the definition of $\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$
implies the first $\Leftrightarrow$ in the displayed statement.
The second $\Leftrightarrow$ follows by \eqref{eq-B-s}--\eqref{eq-o-s}.
For (2),
we get from (1) that if $o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$ then
$x_1,\dots,x_{\mathsf{e}+1}\in o^{\bar{\al B}}/\bar{\alpha}$. Since
$\mathcal{O}_{(\bar{\wec{x}},\wec{x})}\not=\emptyset$, this implies that
$x_1,\dots,x_{\mathsf{e}+1}$ are in the same $\bar{\alpha}$-class.
$\diamond$
\begin{clm}
\label{clm-y}
The following conditions on
an element $(\bar{\wec{x}},\wec{x})\in\al B[\alpha]$
are equivalent:
\begin{enumerate}
\item[{\rm(a)}]
$(\bar{\wec{x}},\wec{x})\in\al B$;
\item[{\rm(b)}]
$(y,\dots,y,\wec{x})\in\al B$ (with $\mathsf{e}+1$ occurrences of $y$)
for the element $y$ defined by
\begin{equation}
\label{eq-y}
y=d^{\bar{\al B}}(d^{\bar{\al B}}(\dots
d^{\bar{\al B}}(d^{\bar{\al B}}(x_1,x_{\mathsf{e}+1},x_2),x_{\mathsf{e}+1},x_3)\dots),x_{\mathsf{e}+1},x_\mathsf{e}).
\end{equation}
\end{enumerate}
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-y}.}
Throughout the proof, we will work with a fixed (but arbitrary) tuple
$(\bar{\wec{x}},\wec{x})\in\al B[\alpha]$, and $y$ will denote
the element defined in \eqref{eq-y}.
By Claim~\ref{clm-alpha}(1),
$\mathcal{O}_{(\bar{\wec{x}},\wec{x})}\not=\emptyset$ and
we have $x_i\in o^{\al B_i}/\alpha_i$ whenever
$o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$ and $i\in[n]$.
Consequently, for each $o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$ and $i\in[n]$,
the $\mathcal{O}$-tuple
$\otupl{x_i}$ that has $x_i$
in position $o$ and zeros (i.e., ${o'}^{\al B_i}$)
in all other positions $o'$ is an element of $\MBialpha$.
Similarly,
the $\mathcal{O}$-tuple
$\otupl{(\bar{\wec{x}},\wec{x})}$
that has $(\bar{\wec{x}},\wec{x})=(x_i)_{i\in[n]}$ in position $o$ and
zeros (i.e., ${o'}^{\al B[\alpha]}=({o'}^{\al B_1},\dots,{o'}^{\al B_n})$)
in all other positions $o'$ is
an element of $\MBsatalpha$.
By inspecting the isomorphism that we use to identify
$\MBsatalpha$ with $\prod_{i=1}^n\MBialpha$ it is easy to see that
this identification yields that
$\otupl{(\bar{\wec{x}},\wec{x})}=(\otupl{x_i})_{i\in[n]}$.
Therefore, if $(\bar{\wec{x}},\wec{x})\in\al B$,
then
$(\otupl{x_i})_{i\in[n]}=\otupl{(\bar{\wec{x}},\wec{x})}\inM_{\al B}(\alpha_{\al B},\mathcal{O})$ for all
$o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$, so
using \eqref{eq-solset} we see that
(a) implies the following condition:
\begin{enumerate}
\item[(a)$'$]
$\sum_{i=1}^n \psi_i(\otupl{x_i})=0$
for all $o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$.
\end{enumerate}
Conversely, assume that (a)$'$ holds for $(\bar{\wec{x}},\wec{x})$.
Since $\mathcal{O}_{(\bar{\wec{x}},\wec{x})}\not=\emptyset$,
we can fix an $o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$
and use \eqref{eq-solset} to conclude that
$\otupl{(\bar{\wec{x}},\wec{x})}=(\otupl{x_i})_{i\in[n]}\inM_{\al B}(\alpha_{\al B},\mathcal{O})$.
By the definition of $\MBalpha$, this implies that
$(\bar{\wec{x}},\wec{x})\in\al B$.
Thus, (a) $\Leftrightarrow$ (a)$'$.
Now let us consider the element $y\in\bar{\al B}$.
By Claim~\ref{clm-alpha}(1), for every $o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$
we have that $x_1,\dots,x_{\mathsf{e}+1}\in o^{\bar{\al B}}/\bar{\alpha}$,
so the idempotence of $d$ implies that
$y\in o^{\bar{\al B}}/\bar{\alpha}$.
Since $\mathcal{O}_{(\bar{\wec{x}},\wec{x})}\not=\emptyset$,
it follows that $(\bar{\wec{x}},\wec{x})\equiv_{\alpha}(y,\dots,y,\wec{x})$.
Thus, $(y,\dots,y,\wec{x})\in\al B[\alpha]$ and
the tuples $(\bar{\wec{x}},\wec{x})$ and
$(y,\dots,y,\wec{x})$ belong to the same $\alpha_{\al B[\alpha]}$-classes
$o^{\al B[\alpha]}/\alpha_{\al B[\alpha]}$ ($o\in\mathcal{O}$).
Consequently,
$\mathcal{O}_{(\bar{\wec{x}},\wec{x})}=\mathcal{O}_{(y,\dots,y,\wec{x})}$.
For every
$o\in\mathcal{O}_{(y,\dots,y,\wec{x})}=\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$
the $\mathcal{O}$-tuple
$\otupl{y}$ that has $y$
in position $o$ and zeros (i.e., ${o'}^{\bar{\al B}}$)
in all other positions $o'$ is an
element of $\al{M}_{\bar{\al B}}(\bar{\alpha},\mathcal{O})$, and
the $\mathcal{O}$-tuple
$\otupl{(y,\dots,y,\wec{x})}$
that has $(y,\dots,y,\wec{x})=(y,\dots,y,x_{\mathsf{e}+2},\dots,x_n)$
in position $o$ and
zeros (i.e., ${o'}^{\al B[\alpha]}=({o'}^{\al B_1},\dots,{o'}^{\al B_n})$)
in all other positions $o'$ is an
element of $\MBsatalpha$.
As before, the identification of $\MBsatalpha$
with $\prod_{i=1}^n\MBialpha$ yields that
$\otupl{(y,\dots,y,\wec{x})}=
(\otupl{y},\dots,\otupl{y},\otupl{x_{\mathsf{e}+2}},\dots,\otupl{x_n})$.
Hence, if we apply
the equivalence of conditions (a) and (a)$'$ to the tuple
$(y,\dots,y,\wec{x})\in\al B[\alpha]$, we obtain that
(b) $\Leftrightarrow$ (b)$'$ for the condition
\begin{enumerate}
\item[(b)$'$]
$\sum_{i=1}^{\mathsf{e}+1}\psi_i(\otupl{y})+\sum_{i=\mathsf{e}+2}^n \psi_i(\otupl{x_i})=0$
for all $o\in\mathcal{O}_{(y,\dots,y,\wec{x})}=
\mathcal{O}_{(\bar{\wec{x}},\wec{x})}$.
\end{enumerate}
Thus, our claim (a) $\Leftrightarrow$ (b) will follow if we prove that
(a)$'$ $\Leftrightarrow$ (b)$'$.
The only difference between conditions (a)$'$ and (b)$'$ are in the first
$\mathsf{e}+1$ summands. Therefore we will be done if we show that
\begin{equation}
\label{eq-q-sum}
\sum_{i=1}^{\mathsf{e}+1}\psi(\otupl{x_i})=\sum_{i=1}^{\mathsf{e}+1}\psi_i(\otupl{y})
\quad
\text{for every
$o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}=\mathcal{O}_{(y,\dots,y,\wec{x})}$.}
\end{equation}
So, let
$o\in\mathcal{O}_{(\bar{\wec{x}},\wec{x})}=\mathcal{O}_{(y,\dots,y,\wec{x})}$.
Then, by Claim~\ref{clm-alpha}(1) and by our earlier argument on $y$,
we have that
$x_1,\dots,x_{\mathsf{e}+1},y\in o^{\bar{\al B}}/\bar{\alpha}$.
By combining the definition of $y$ in \eqref{eq-y} with
Corollary~\ref{cor-dMaltsev} we get that
$y=x_1 +_o x_2 +_o \dots +_o x_{\mathsf{e}+1}$
holds in the abelian group $\al M_{\bar{\al B}}(\bar{\alpha},o)$.
Hence, by the definition of $+$ in the module $\al{M}_{\bar{\al B}}(\bar{\alpha},\mathcal{O})$,
\begin{equation}
\label{eq-yo}
\otupl{y}=\otupl{x_1} + \otupl{x_2} + \dots + \otupl{x_{\mathsf{e}+1}}
\end{equation}
holds in $\al{M}_{\bar{\al B}}(\bar{\alpha},\mathcal{O})$.
So, using first \eqref{eq-psi-s}, then the fact that
$\bar{\psi}$ is a module homomorphism $\al{M}_{\bar{\al B}}(\bar{\alpha},\mathcal{O})\to\al Q$,
and finally \eqref{eq-yo},
we get that
\[
\sum_{i=1}^{\mathsf{e}+1}\psi_i(\otupl{x_i})
=\sum_{i=1}^{\mathsf{e}+1}\bar{\psi}(\otupl{x_i})
=\bar{\psi}\Big(\sum_{i=1}^{\mathsf{e}+1}\otupl{x_i}\Bigr)
=\bar{\psi}(\otupl{y}).
\]
Similarly,
\[
\sum_{i=1}^{\mathsf{e}+1}\psi_i(\otupl{y})
=(\mathsf{e}+1)\bar{\psi}(\otupl{y})
=\bar{\psi}((\mathsf{e}+1)\otupl{y})
=\bar{\psi}(\otupl{y}),
\]
where the last equality is true, because the
definition of $\mathsf{e}$ and the fact that
$\otupl{y}\in\al{M}_{\bar{\al B}}(\bar{\alpha},\mathcal{O})\in\text{\rm Mod}(\al A,\mathsf{p})$
ensure that $\mathsf{e}$ is a multiple of the additive order of $\otupl{y}$.
This establishes \eqref{eq-q-sum}, and hence completes the proof of the claim.
$\diamond$
Now we define the relations $B'$ and $D$ that we will use to prove
Lemma~\ref{lm-main}. Let
\begin{gather}
B':=\{(y,\wec{x})\in\bar{B}\times\prod_{i=\mathsf{e}+2}^n B_i:
(y,\dots,y,\wec{x})\in B\},\notag\\
D:=\{(y,\bar{\wec{x}})\in\bar{B}^{\mathsf{e}+2}:
\text{\eqref{eq-y} holds and \ }
x_1\equiv_{\bar{\alpha}}\dots\equiv_{\bar{\alpha}}x_{\mathsf{e}+1}\}.
\notag
\end{gather}
\begin{clm}
\label{clm-B'}
$B'$ has the following properties.
\begin{enumerate}
\item[(1)]
$\bar{\al B}$ has an $\mathcal{O}$-subalgebra
$\bar{\al B}'$ such that
$B'$ is the universe of a
subdirect $\mathcal{O}$-subalgebra $\al B'$ of
the $\mathcal{O}$-product $\bar{\al B}'\times\prod_{i=\mathsf{e}+2}^n\al B_i$
of the $\mathcal{O}$-algebras $\bar{\al B}'$ and $\al B_i$
$(\mathsf{e}+2\le i\le n)$.
\item[(2)]
All the congruences
$\bar{\alpha}':=\bar{\alpha}{\restriction}_{\bar{\al B}'}\in\Con(\bar{\al B}')$
and $\alpha_i\in\Con(\al B_i)$ $(\mathsf{e}+2\le i\le n)$ are abelian, and all
but possibly $\bar{\alpha}'$ are nontrivial.
\item[(3)]
For the product congruence
$\alpha':=\bar{\alpha}'\times\prod_{i=\mathsf{e}+2}^n\alpha_i$
of\/ $\bar{\al B}'\times\prod_{i=\mathsf{e}+2}^n\al B_i$,
the elements $o^{\al B'}$ $(o\in\mathcal{O})$ represent all
$\alpha'_{\al B'}$-classes of $\al B'$, therefore $\alpha'$
restricts to $\al B'$ as a congruence $\alpha'_{\al B'}$ of index $\le\mathsf{i}$.
\end{enumerate}
Consequently, $\al B'$ satisfies all conditions in (I)--(IV),
with the possible exception that the abelian congruence
$\bar{\alpha}'$ in the first factor might be trivial.
Hence, if $\bar{\alpha}'$ is a nontrivial
congruence of $\bar{\al B}'$, then
$B'\in\rel{R}^*(\al A)$ is a compatible relation of $\al A$ of $*$-arity
$n-\mathsf{e}$.
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-B'}}.
Notice
first that
$B'$ is a compatible relation of $\al A$, because
$B'\not=\emptyset$ by Claim~\ref{clm-y}, and
$B'$ is definable
by a primitive positive formula using the compatible relations
$\bar B=B_1$, $B_i$ ($\mathsf{e}+2\le i\le n$), and $B$.
Thus the definition of $B'$ shows that
$B'$ is, in fact, the universe of a subalgebra $\al B'$ of
$\bar{\al B}\times\prod_{i=\mathsf{e}+2}^n \al B_i$.
Moreover, \eqref{eq-o-s} implies that all
tuples $(o^{\bar{\al B}},o^{\al B_{\mathsf{e}+2}},\dots,o^{\al B_n})$ ($o\in\mathcal{O}$)
are in $\al B'$, therefore by defining
$o^{\al B'}:=(o^{\bar{\al B}},o^{\al B_{\mathsf{e}+2}},\dots,o^{\al B_n})$
for every $o\in\mathcal{O}$, $\al B'$ becomes an $\mathcal{O}$-subalgebra of
$\bar{\al B}\times \prod_{i=\mathsf{e}+2}^n \al B_i$.
Claim~\ref{clm-y} implies that $\al B$ and $\al B'$ have the same
projections onto their last $n-(\mathsf{e}+1)$ coordinates, so
letting $\bar{\al B}'$
denote the projection of $\al B'$ onto its
first coordinate, we get an
$\mathcal{O}$-subalgebra
$\bar{\al B}'$ of $\bar{\al B}$
such that $\al B'$ is a subdirect $\mathcal{O}$-subalgebra of
the $\mathcal{O}$-product $\bar{\al B}'\times\prod_{i=\mathsf{e}+2}^n\al B_i$
of the $\mathcal{O}$-algebras $\bar{\al B}'$ and $\al B_i$
($\mathsf{e}+2\le i\le n$).
This proves (1).
Considering $\bar{\al B}'$ as
an algebra in the original language of $\var{V}$, we have
$\bar{\al B}'\le\bar{\al B}=\al B_1\le\al A^{p_1}$
where $p_1\le\mathsf{p}$.
Furthermore, since $\bar{\al B}$ is
isomorphic to a section of $\al A$, so is its subalgebra
$\bar{\al B}'$.
The other algebras $\al B_{\mathsf{e}+2},\dots\al B_n$ are unchanged, therefore
we get that conditions (I) and (III) hold for $B'$.
Condition (II) for $\al B$ implies that
$\bar{\alpha}'$ is an abelian congruence of $\bar{\al B}'$ and
$\alpha_i$ is a nontrivial abelian congruence of $\al B_i$ for every $i$
($\mathsf{e}+2\le i\le n$). This proves (2) and that (II) holds for $\al B'$
with the possible exception that $\bar{\alpha}'$ may be trivial.
Finally, since
the elements
$o^{\al B}=(o^{\bar{\al B}},\dots,o^{\bar{\al B}},o^{\al B_{\mathsf{e}+2}},\dots,o^{\al B_n})$
($o\in\mathcal{O}$)
represent all $\alpha_{\al B}$-classes in $\al B$, the definition of
$\al B'$ implies that the elements
$o^{\al B'}=(o^{\bar{\al B}},o^{\al B_{\mathsf{e}+2}},\dots,o^{\al B_n})$
($o\in\mathcal{O}$)
represent all
$\alpha'_{\al B'}$-classes in $\al B'$.
Hence
$\alpha'_{\al B'}$
has index $\le|\mathcal{O}|=\mathsf{i}$.
This proves
(3) and
that condition (IV) also hold for $B'$.
Thus
we have $B'\in\rel{R}^*(\al A)$
if $\bar{\alpha}'$ is a nontrivial congruence of $\bar{\al B}'$.
It is clear from the construction of $B'$ that its $*$-arity is $n-\mathsf{e}$,
completing the proof of
Claim~\ref{clm-B'}.
$\diamond$
\begin{clm}
\label{clm-D}
$D$ is the universe of a subalgebra $\al D$ of $\bar{\al B}^{\mathsf{e}+1}$.
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-D}}.
The following fact will be useful:
property \eqref{eq-diffterm3} of the difference term $d$
for $\bar{\al B}$ and its abelian congruence $\bar{\alpha}$
is equivalent to saying that
\[
D_1:=\bigl\{\bigl(d^{\bar{\al B}}(u,v,w),u,v,w\bigr)\in\bar{B}^4:
u\equiv_{\bar{\alpha}}v\equiv_{\bar{\alpha}}w\bigr\}
\]
is the universe of a subalgebra of
$\bar{\al B}^4$.
Hence $D_1$ is a compatible relation of $\al A$.
Observing that all four coordinates of the tuples in $D_1$ are
$\bar{\alpha}$-related, one can easily check that
$D$ is definable by a primitive positive formula using $D_1$.
Clearly, $D\not=\emptyset$.
Therefore it follows that $D$ is a compatible relation of $\al A$.
The construction of $D$ shows that $D$ is, in fact,
the universe of a subalgebra
of $\bar{\al B}^{\mathsf{e}+1}=\al B_1^{\mathsf{e}+1}$.
$\diamond$
\begin{clm}
\label{clm-entailB}
$\{B_1,\dots,B_n,B',D\}\models_{\text{\rm d}} B$.
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-entailB}}.
Let
\[
W:=\{(y,\bar{\wec{x}},\wec{x})\in\bar{B}^{\mathsf{e}+2}\times\prod_{i=\mathsf{e}+2}^n B_i:
(y,\bar{\wec{x}})\in D\ \ \text{and}\ \ (y,\wec{x})\in B'\}.
\]
Our goal is to show that
\begin{enumerate}
\item[(i)]
The projection map $W\to \mathrm{proj}_{[n+1]\setminus\{1\}}(W)$ that omits
the first coordinate of $W$
is one-to-one, and
\item[(ii)]
its image is
$B$, that is,
\begin{equation}
\label{eq-ontoB}
\mathrm{proj}_{[n+1]\setminus\{1\}}(W)=B.
\end{equation}
\end{enumerate}
(i)--(ii) will imply the statement of Claim~\ref{clm-entailB}
for the following reason.
By (ii), $W$ is nonempty, and it
is easy to see from the definition of $W$ that
$W$ is definable
by a quantifier-free primitive positive formula, using
$B_1,\dots,B_n$, $B'$ and $D$.
Hence
$\{B_1,\dots,B_n,B',D\}\models_{\text{\rm d}} W$.
By
(i) and (ii),
$B$ is obtained from $W$ by bijective projection, so
$\{W\}\models_{\text{\rm d}} B$.
By the transitivity of $\models_{\text{\rm d}}$ we get that
$\{B_1,\dots,B_n,B',D\}\models_{\text{\rm d}} B$,
as claimed.
To prove (i) observe that the
projection map $W\to \mathrm{proj}_{[n+1]\setminus\{1\}}(W)$ is one-to-one,
because for every element $(y,\bar{\wec{x}},\wec{x})\in W$ we have
$(y,\bar{\wec{x}})\in D$, and hence $y$ is uniquely determined
by $\bar{\wec{x}}$, via \eqref{eq-y}.
To prove the inclusion $\supseteq$ in \eqref{eq-ontoB}, let
$(\bar{\wec{x}},\wec{x})\in B$ and define $y\in\bar{B}$ by \eqref{eq-y}.
Then Claim~\ref{clm-y} implies that $(y,\dots,y,\wec{x})\in B$,
so $(y,\wec{x})\in B'$.
On the other hand, by Claim~\ref{clm-alpha}(2) we have that
$x_1\equiv_{\bar{\alpha}}\dots\equiv_{\bar{\alpha}}x_{\mathsf{e}+1}$,
therefore $(y,\bar{\wec{x}})\in D$.
This shows that $(y,\bar{\wec{x}},\wec{x})\in W$, and hence
$(\bar{\wec{x}},\wec{x})\in \mathrm{proj}_{[n+1]\setminus\{1\}}(W)$.
For the inclusion $\subseteq$ in \eqref{eq-ontoB}, assume that
$(\bar{\wec{x}},\wec{x})\in \mathrm{proj}_{[n+1]\setminus\{1\}}(W)$.
Then there exists $y\in\bar{B}$ such that
$(y,\bar{\wec{x}},\wec{x})\in W$. Let us fix such a $y$.
By the definition of $W$ it follows that
$(y,\bar{\wec{x}})\in D$ and $(y,\wec{x})\in B'$.
The latter implies that $(y,\dots,y,\wec{x})\in\al B$, while the
former implies that
$x_1\equiv_{\bar{\alpha}}\dots\equiv_{\bar{\alpha}}x_{\mathsf{e}+1}$
and the equality in \eqref{eq-y} holds for $y$.
Since $d$ is idempotent, we get that
$x_1\equiv_{\bar{\alpha}}\dots\equiv_{\bar{\alpha}}x_{\mathsf{e}+1}
\equiv_{\bar{\alpha}}y$, so
$(\bar{\wec{x}},\wec{x})\equiv_\alpha(y,\dots,y,\wec{x})$.
Since $(y,\dots,y,\wec{x})\in\al B$, this shows that
$(\bar{\wec{x}},\wec{x})\in B[\alpha]$.
Therefore Claim~\ref{clm-y} applies, and we obtain that
$(\bar{\wec{x}},\wec{x})\in B$.
$\diamond$
Now we are ready to prove the statements (1)--(3) of Lemma~\ref{lm-main}.
We will use the notation and the conclusions of
Claims~\ref{clm-B'}--\ref{clm-entailB}.
There are two cases to consider.
{\sc Case 1:} $\bar{\alpha}'\in\Con(\bar{\al B}')$ is nontrivial.
Then we know from Claim~\ref{clm-B'} that
$B'\in\rel{R}^*(\al A)$ is a compatible relation of $\al A$ of $*$-arity
$n-\mathsf{e}$, so (1) holds. For (2), choose $\al D'$ to be any subalgebra
of $\bar{\al B}^2$, say $\al D'=\bar{\al B}^2$.
Since $\bar{\al B}=\al B_1$, Claim~\ref{clm-D} proves that
(2) also holds. Finally, (3) follows from Claim~\ref{clm-entailB}
(the choice of $D'$ is irrelevant).
{\sc Case 2:} $\bar{\alpha}'\in\Con(\bar{\al B}')$ is trivial.
By Claim~\ref{clm-B'},
$\bar{\al B}'$ is an $\mathcal{O}$-subalgebra of $\bar{\al B}$,
therefore $o^{\bar{\al B}'}=o^{\bar{\al B}}$ for all $o\in\mathcal{O}$.
Since the elements $o^{\bar{\al B}}$ ($o\in\mathcal{O}$)
represent all $\bar{\alpha}$-classes of $\bar{\al B}$ and
$\bar{\alpha}'=\bar{\alpha}{\restriction}_{\bar{\al B}'}$ is a trivial
congruence of $\bar{\al B}'$, we get that the underlying set
of $\bar{\al B}'$ is $\bar{B}'=\mathcal{O}^{\bar{\al B}}\,
(=\mathcal{O}^{\bar{\al B}'})$,
and $o_1^{\bar{\al B}}=o_2^{\bar{\al B}}$
whenever $o_1^{\bar{\al B}}\equiv_{\bar{\alpha}}o_2^{\bar{\al B}}$
in $\bar{\al B}$
($o_1,o_2\in\mathcal{O}$).
It follows that the assignment $o^{\bar{\al B}}\mapsto o^{\bar{\al B}}/\bar{\alpha}$
is an isomorphism $\bar{\al B}'\to\bar{\al B}/\bar{\alpha}$. Hence
the map $\nu\colon\bar{\al B}\to\bar{\al B}'$ that assigns to every
$\bar{b}\in\bar{B}$ the unique element
$o^{\bar{\al B}}\in\mathcal{O}^{\bar{\al B}}=\bar{B}'$
such that $o^{\bar{\al B}}\equiv_{\bar{\alpha}}\bar{b}$
is a well-defined homomorphism with kernel $\bar{\alpha}$.
Now let $\al D'$ denote the subalgebra of $\bar{\al B}^2$ whose universe
\[
D'=\{(x,y)\in \bar{B}^2:y=\nu(x)\}
\]
is the graph of $\nu$, and let
$\al B''=\mathrm{proj}_{[n-\mathsf{e}]\setminus\{1\}}(\al B')$
be the image of $\al B'$ under the
projection homomorphism onto all coordinates except the first.
Since $\bar{\al B}=\al B_1$,
Claim~\ref{clm-D} and the definition of $\al D'$
imply that condition (2) of Lemma~\ref{lm-main} holds
for $\al D$ and $\al D'$.
It follows from the conclusions of Claim~\ref{clm-B'} that
$B''$ is a compatible relation of $\al A$ that belongs to
$\mathcal{R}^*(\al A)$ and has $*$-arity $n-\mathsf{e}-1$.
Thus, condition (1) of Lemma~\ref{lm-main} holds for $B''$ in place of $B'$.
It remains to establish that condition (3) of Lemma~\ref{lm-main}
also holds with $B''$ replacing $B'$, that is,
\begin{equation}
\label{eq-modified-entailB}
\{B_1,\dots,B_n,B'',D,D'\}\models_{\text{\rm d}} B.
\end{equation}
First we prove that
\begin{equation}
\label{eq-partial-dentails}
B'=\{(y,x_{\mathsf{e}+2},\dots,x_n)\in \bar{B}\times B'': (x_{\mathsf{e}+2},y)\in D'\}.
\end{equation}
To verify the inclusion $\subseteq$,
let $(y,\wec{x})=(y,x_{\mathsf{e}+2},\dots,x_n)$
be an arbitrary element of $\al B'$.
Clearly, $(y,\wec{x})\in \bar{B}\times B''$, so we need to
show that $(x_{\mathsf{e}+2},y)\in D'$, that is, $y=\nu(x_{\mathsf{e}+2})$.
By Claim~\ref{clm-B'}, $\al B'$ is a subdirect $\mathcal{O}$-subalgebra
of $\bar{\al B}'\times\al B_{\mathsf{e}+2}\times\dots\times\al B_n$, and the
elements $o^{\al B'}=(o^{\bar{\al B}},o^{\al B_{\mathsf{e}+2}},\dots,o^{\al B_{n}})$
represent all $\alpha'_{\al B'}$-classes of $\al B'$.
Also,
recall from \eqref{eq-B-s}--\eqref{eq-o-s}
that at the beginning of the proof of Lemma~\ref{lm-main}
we arranged that
$\al B_{\mathsf{e}+2}=\bar{\al B}$, $\alpha_{\mathsf{e}+2}=\bar{\alpha}$, and
$o^{\al B_{\mathsf{e}+2}}=o^{\bar{\al B}}$ for all $o\in\mathcal{O}$.
Therefore, for the given element $(y,\wec{x})\in\al B'$,
there exists $o\in\mathcal{O}$
such that $o^{\al B'}\in(y,\wec{x})/\alpha'$, and so
$y\equiv_{\bar{\alpha}'} o^{\bar{\al B}}=o^{\al B_{\mathsf{e}+2}}\equiv_{\alpha_{\mathsf{e}+2}} x_{\mathsf{e}+2}$.
Since $\bar{\alpha}'$ is trivial and
$\al B_{\mathsf{e}+2}=\bar{\al B}$, $\alpha_{\mathsf{e}+2}=\bar{\alpha}$, we get that
$y=o^{\bar{\al B}}\equiv_{\bar{\alpha}} x_{\mathsf{e}+2}\, (\in\bar{B})$.
Thus, $y=\nu(x_{\mathsf{e}+2})$.
This completes the proof of $\subseteq$ in
\eqref{eq-partial-dentails}.
For $\supseteq$, assume that
$(y,x_{\mathsf{e}+2},\dots,x_n)\in \bar{B}\times B''$ is such that
$(x_{\mathsf{e}+2},y)\in D'$, that is, $y=\nu(x_{\mathsf{e}+2})$.
Since $(x_{\mathsf{e}+2},\dots,x_n)\in B''=\mathrm{proj}_{[n-\mathsf{e}]\setminus\{1\}}(B')$,
there exists $y'\in\bar{B}'$ such that
$(y',x_{\mathsf{e}+2},\dots,x_n)\in B'$.
The inclusion $\subseteq$ in \eqref{eq-partial-dentails}
proved in the preceding paragraph implies that
$y'=\nu(x_{\mathsf{e}+2})$.
Thus $y=y'$ and $(y,x_{\mathsf{e}+2},\dots,x_n)=(y',x_{\mathsf{e}+2},\dots,x_n)\in B'$.
This completes the proof of \eqref{eq-partial-dentails}.
By \eqref{eq-partial-dentails}, $B'$ is definable by
a quantifier-free primitive positive formula, using
$\bar{B}=B_1$, $B''$ and $D'$. Hence $\{B_1,B'',D'\}\models_{\text{\rm d}} B'$.
It follows that
\[
\{B_1,\dots,B_n,B'',D,D'\}\models_{\text{\rm d}} \{B_1,\dots,B_n,B',D\}.
\]
Combining this with the result of Claim~\ref{clm-entailB}, and using
the transitivity of $\models_{\text{\rm d}}$, we obtain the desired conclusion
\eqref{eq-modified-entailB}.
The proof of Lemma~\ref{lm-main} is complete.
\end{proof}
\begin{cor}
\label{cor-main}
If some power $\al A^p$ of $\al A$ with $1\le p\le \mathsf{p}$ has a subalgebra
that is isomorphic to a section of $\al A$ with
a nontrivial abelian congruence, then
$\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)\models_{\text{\rm d}}\rel{R}^*(\al A)$.
\end{cor}
\begin{proof}
As we noted in Remarks~\ref{rem-about-rhqc}
(b),
the assumption that some $\al A^p$ with
$1\le p\le \mathsf{p}$ has a subalgebra isomorphic to a section of $\al A$
with a nontrivial abelian congruence implies that
$\mathsf{h}\ge1$ and $\mathsf{e}\ge2$.
To prove the corollary we have to show that
\begin{equation}
\label{eq-ehq-B}
\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)\models_{\text{\rm d}} B
\end{equation}
for every $B\in\rel{R}^*(\al A)$.
We proceed by induction on the $*$-arity $n$ of $B$.
We will use the same notation for the data
$\al B_i$, $\alpha_i$ ($i\in[n]$) and $\alpha$
associated to $B$
as in (I)--(IV).
If $n\le\mathsf{h}(\mathsf{e}+1)$,
then the arity of $B$ (as a compatible relation of $\al A$)
is $\sum_{i=1}^n p_i\le n\mathsf{p}$, therefore $B\in\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)$ and
\eqref{eq-ehq-B} is trivial.
So assume that $n>\mathsf{h}(\mathsf{e}+1)$.
The algebra $\al B$ is the intersection of a family
$\mathcal{I}$ of $\cap$-irreducible
subalgebras of $\al B[\alpha]$.
For every $\hat{\al B}\in\mathcal{I}$ we have
$\al B\le\hat{\al B}\le\al B[\alpha]$, therefore
$\hat{\al B}$ is a subdirect product of $\al B_1,\dots,\al B_n$
and
$\hat{\al B}[\alpha]=\al B[\alpha]$. The latter implies that
the index of $\alpha_{\hat{\al B}}$ in $\hat{\al B}$ is the same as the
index of $\alpha_{\al B}$ in $\al B$, because both are equal to the index of
$\alpha_{\al B[\alpha]}$ in $\al B[\alpha]$.
This shows that for every $\hat{\al B}\in\mathcal{I}$,
$\hat{B}$ is a relation in $\rel{R}^*(\al A)$ with $*$-arity $n$.
Since $\cap$ is an $\models_{\text{\rm d}}$-construct, it suffices
to prove \eqref{eq-ehq-B} for the case when
$\al B$ is a $\cap$-irreducible subalgebra of $\al B[\alpha]$.
Then, by Lemma~\ref{lm-main},
$\{B_1,\dots,B_n,B',D,D'\}\models_{\text{\rm d}} B$
for some $B'\in\rel{R}^*(\al A)$ of $*$-arity
$\le n-\mathsf{e}$ and some compatible
relations $D$ and $D'$
of $\al A$
that are universes
of algebras $\al D\le\al B_i^{\mathsf{e}+1}\le\al A^{p_i(\mathsf{e}+1)}$
and $\al D'\le\al B_i^2\le\al A^{2p_i}$
for some $i$
($i\in[n]$).
Now the induction hypothesis implies that
$\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)\models_{\text{\rm d}} B'$, because $n-\mathsf{e}<n$,
while
the inequalities $\mathsf{h}\ge1$, $\mathsf{e}\ge2$ from
the first paragraph of this
proof imply that
$B_1,\dots,B_n,D,D'\in\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)$, because
$p_i\le 2p_i\le p_i(\mathsf{e}+1)\le\mathsf{p}(\mathsf{e}+1)\le\mathsf{p}\mathsf{h}(\mathsf{e}+1)$.
Hence
\[
\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)\models_{\text{\rm d}} \{B_1,\dots,B_n,B',D,D'\}\models_{\text{\rm d}} B,
\]
so \eqref{eq-ehq-B}
follows by the transitivity of $\models_{\text{\rm d}}$.
\end{proof}
Now we are ready to prove Theorem~\ref{thm-main}.
\begin{proof}[Proof of Theorem~\ref{thm-main}]
Assume that $\al A$ satisfies the hypotheses of Theorem~\ref{thm-main},
that is, in addition to our global assumptions in this section
that $\al A$ is a finite algebra with a $k$-parallelogram term, $\al A$
also satisfies the split centralizer condition. (For the definition
of the split centralizer condition, see the Introduction.)
Our aim is to prove that
\begin{equation}
\label{eq-ToProve2}
\rel{R}_\mathsf{c}(\al A)\models_{\text{\rm d}}\rel{R}(\al A).
\end{equation}
In addition to the notation
$\rel{R}^*(\al A)$ introduced before Lemma~\ref{lm-main}, we will
write $\rel{R}_{{\rm crit}}(\al A)$ for the set of all critical
relations of $\al A$. (Critical relations are defined in Section~\ref{prelim},
subsection~\ref{prelim}.2.)
First we will argue that
\begin{equation}
\label{eq-crit-entails}
\rel{R}_{{\rm crit}}(\al A)\models_{\text{\rm d}}\rel{R}(\al A).
\end{equation}
Every compatible relation of $\al A$ is an intersection of
$\cap$-irreducible compatible relations. Furthermore, if a
$\cap$-irreducible compatible relation of $\al A$ is not critical
(i.e., not directly indecomposable), then up to a permutation
of coordinates, it has the form $\rho\times A^\ell$, so it follows
that $\rho$ is a critical relation of $\al A$;
$A$ itself is clearly a critical relation of $\al A$.
Therefore every compatible relation of $\al A$ can be obtained from
critical relations of $\al A$ by product and intersection.
Since product and intersection are $\models_{\text{\rm d}}$-constructs,
\eqref{eq-crit-entails} follows.
Theorem~\ref{thm-reduction} implies that
\begin{equation}
\label{eq-*-entails}
\rel{R}_{\mathsf{c}_0}(\al A)\cup\rel{R}^*(\al A)\models_{\text{\rm d}}\rel{R}_{{\rm crit}}(\al A),
\end{equation}
as we will show now.
Let $C$ be a critical relation of $\al A$ of arity $n$.
It is clear that
$\rel{R}_{\mathsf{c}_0}(\al A)\cup\rel{R}^*(\al A)\models_{\text{\rm d}} C$
if $n\le \mathsf{c}_0$
(i.e., if $C\in\rel{R}_{\mathsf{c}_0}(\al A)$), so let us assume that
$n>\mathsf{c}_0$. It follows from the definition of $\mathsf{c}_0$ that
$n\ge\max(3,k)$.
Since $\al A$ has a $k$-parallelogram term
and satisfies the split centralizer condition,
all hypotheses of Theorem~\ref{thm-reduction} are satisfied.
Hence the theorem yields the existence of a compatible relation
$B$ of $\al A$ such that $B\in\rel{R}^*(\al A)$ and
$\rel{R}_{1+\mathsf{p}}(\al A)\cup\{B\}\models_{\text{\rm d}} C$.
Since $1+\mathsf{p}\le\mathsf{c}_0$, this
implies that $\rel{R}_{\mathsf{c}_0}(\al A)\cup\rel{R}^*(\al A)\models_{\text{\rm d}} C$,
and
proves \eqref{eq-*-entails}.
If no power $\al A^p$ ($1\le p\le \mathsf{p}$) of $\al A$ has a subalgebra
that is isomorphic to a section of $\al A$ with a nontrivial
abelian congruence, then $\rel{R}^*(\al A)=\emptyset$ and
$\text{\rm Mod}(\al A,\mathsf{p})=\emptyset$. As we saw in Remarks~\ref{rem-about-rhqc}
(b),
the latter implies that $\mathsf{h}=0$ and hence $\mathsf{c}=\mathsf{c}_0$.
Thus, in this case \eqref{eq-crit-entails} and \eqref{eq-*-entails} combine to
show that
\[
\rel{R}_{\mathsf{c}}(\al A)
=\rel{R}_{\mathsf{c}_0}(\al A)
\stackrel{\eqref{eq-*-entails}}{\models_{\text{\rm d}}}\rel{R}_{{\rm crit}}(\al A)
\stackrel{\eqref{eq-crit-entails}}{\models_{\text{\rm d}}}\rel{R}(\al A).
\]
Hence \eqref{eq-ToProve2}
follows by the transitivity of $\models_{\text{\rm d}}$.
In the opposite case, when some $\al A^p$ ($1\le p\le\mathsf{p}$)
has a subalgebra isomorphic to a section of $\al A$ with
a nontrivial abelian congruence,
Corollary~\ref{cor-main} shows that
\begin{equation}
\label{eq-ehq-entails}
\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)\models_{\text{\rm d}}\rel{R}^*(\al A).
\end{equation}
Since $\mathsf{c}=\max(\mathsf{c}_0,\mathsf{p}\mathsf{h}(\mathsf{e}+1))$, we have that
$\rel{R}_{\mathsf{c}_0}(\al A)\cup\rel{R}_{\mathsf{p}\mathsf{h}(\mathsf{e}+1)}(\al A)\subseteq
\rel{R}_{\mathsf{c}}(\al A)$,
so
\eqref{eq-crit-entails}, \eqref{eq-*-entails}, and
\eqref{eq-ehq-entails} together imply that
\[
\rel{R}_{\mathsf{c}}(\al A)
\stackrel{\eqref{eq-ehq-entails}}{\models_{\text{\rm d}}}
\rel{R}_{\mathsf{c}}(\al A)\cup\rel{R}^*(\al A)
\stackrel{\eqref{eq-*-entails}}{\models_{\text{\rm d}}} \rel{R}_{{\rm crit}}(\al A)
\stackrel{\eqref{eq-crit-entails}}{\models_{\text{\rm d}}} \rel{R}(\al A).
\]
By the transitivity of $\models_{\text{\rm d}}$ this shows that
\eqref{eq-ToProve2} holds.
This proves \eqref{eq-ToProve2} in all cases.
In view of Theorem~\ref{thm-WZ}, \eqref{eq-ToProve2}
is sufficient to conclude that $\al A$ is dualizable.
\end{proof}
\section{Applications}\label{appl}
In this section we apply the main
theorem of the paper to establish dualizability within
some well known classes of algebras.
Some of these results were known before.
In the first part of the section we identify
some conditions on a variety $\mathcal V$
which guarantee that every finite member of $\mathcal V$
is dualizable. We end the section by proving that
if $\m a$ is a finite algebra with a parallelogram term
and $\mathsf{S}\mathsf{P}(\m a)$ is a variety, then $\m a$ is dualizable
(although some of the other algebras in
$\mathsf{S}\mathsf{P}(\m a)$ need not be dualizable).
We will often apply Theorem~\ref{thm-main} to prove dualizability
simultaneously for all members of a class $\mathcal{C}$ of finite algebras
with parallelogram terms,
where $\mathcal{C}$ is closed under taking subalgebras.
In such cases, to get the desired conclusion,
it will be enough to check that for every $\al A\in\mathcal{C}$,
each relevant triple $(\delta,\theta,\nu)$ of $\al A$ is split
by a triple $(\alpha,\beta,\kappa)$ relative to $\mathsf{S}\mathsf{P}(\al A)$.
Indeed, if every $\al A\in\mathcal{C}$ has this property, then
for every $\al A\in\mathcal{C}$ and $\al B\le\al A$ we have that
$\al B\in\mathcal{C}$, so every relevant triple of $\al B$ is split
by a triple relative to $\mathsf{S}\mathsf{P}(\al B)$, and therefore relative to
$\mathsf{S}\mathsf{P}(\al A)$ as well.
This shows that every $\al A\in\mathcal{C}$ satisfies the split centralizer
condition, which implies by Theorem~\ref{thm-main} that every
$\al A\in\mathcal{C}$ is dualizable.
We will refer to some commutator
identities by the number assigned to them
in \cite[Chapter~8]{freese-mckenzie}:
\begin{align}
[x\wedge y,y]&= x\wedge[y,y],\tag{C1}\\
[x,y]&= x\wedge y.\tag{C3}\\
[1,x]&= x.\tag{C8}
\end{align}
\noindent
An algebra satisfying a given (Ci), $i\in \{1,3,8\}$,
may be called a
\emph{(Ci)-algebra}. It is known that if
$\mathcal V$ is congruence modular, then the subclass
of (Ci)-algebras in $\mathcal V$
is closed under the formation of finite subdirect products
and quotient algebras. (See
\cite[Chapter~8]{freese-mckenzie}.)
We have met (C1) before: any residually
small congruence modular variety consists
of
(C1)-algebras,
and conversely any
congruence modular variety
generated by a finite algebra whose subalgebras
are all (C1) is residually small.
(C3)-algebras are also called \emph{neutral}.
{}From \cite[Chapter~8]{freese-mckenzie}
we know that an algebra is neutral if and only if
it has no nontrivial abelian congruence intervals.
Therefore, we will call an interval in a congruence lattice
\emph{neutral} if it has
no nontrivial abelian subintervals.
\begin{lm}\label{solvable-or-perfect}
Let $\mathcal V$ be a congruence modular
variety in which every finite
subdirectly irreducible algebra is
either solvable or is a (C8)-algebra.
If $\m a\in {\mathcal V}$ is a finite algebra,
then
\begin{enumerate}
\item[{\rm(1)}]
$\m a$
has a unique pair $(\sigma,\rho)$
of complementary factor congruences such that
$\m a\cong \m a/\sigma\times \m a/\rho$,
$\m a/\sigma$ is solvable and $\m a/\rho$ is a (C8)-algebra; moreover
\item[{\rm(2)}]
every congruence $\chi$ on $\al A$
is a product congruence relative to the factorization
$\m a\cong \m a/\sigma\times \m a/\rho$, meaning that
$\chi = (\chi\vee\sigma)\wedge(\chi\vee\rho)$.
\end{enumerate}
\end{lm}
\begin{proof}{}
To prove (1)
recall that the classes
of solvable algebras and (C8)-algebras in $\mathcal V$
are closed under finite subdirect products and quotients.
Therefore, if
$\m a\in \mathcal V$ is any finite
algebra, then it has a least congruence
$\sigma$ such that $\m a/\sigma$ is solvable
and also a least congruence $\rho$ such that
$\m a/\rho$ is a (C8)-algebra.
The congruence
$\sigma$ is contained in the kernel of any
homomorphism of $\m a$ onto a solvable subdirectly irreducible
algebra and the congruence $\rho$
is contained in the kernel of any
homomorphism of $\m a$ onto a (C8) subdirectly irreducible
algebra. Hence $\sigma\wedge \rho$ is contained in the kernel
of any homomorphism of $\m a$ onto a subdirectly irreducible algebra,
implying that $\sigma\wedge\rho = 0$.
The algebra $\m a/(\sigma\vee\rho)$ is a quotient
of the solvable algebra $\m a/\sigma$ and is also
a quotient of the (C8)-algebra $\m a/\rho$,
so it is both solvable and (C8). This forces it to be trivial,
and therefore $\sigma\vee\rho = 1$.
Congruences $\sigma$ and $\rho$ permute,
since $\sigma$ is cosolvable and cosolvable
congruences permute with all congruences
by \cite[Theorem~6.2]{freese-mckenzie}.
This completes the proof
that $(\sigma,\rho)$ is a pair of complementary factor
congruences, and the argument shows that
$\m a/\sigma$ is solvable and $\m a/\rho$ is (C8).
If $(\sigma',\rho')$ were a second pair
of complementary factor congruences such that
$\m a\cong \m a/\sigma'\times \m a/\rho'$,
$\m a/\sigma'$ is solvable and $\m a/\rho'$ is a (C8)-algebra,
then the solvability of $\m a/\sigma'$ implies that
$\sigma'\supseteq \sigma$,
and a similar argument shows that $\rho'\supseteq\rho$.
But if $(\sigma,\rho)$ and $(\sigma',\rho')$
are pairs of complementary factor congruences where
$\sigma\subseteq \sigma'$ and $\rho\subseteq \rho'$,
then $\sigma=\sigma'$ and $\rho=\rho'$.
For (2) notice that
the set of product congruences on $\m a\cong \m a/\sigma\times \m a/\rho$
is closed under meet, so to prove
claim (2)
it suffices to show that
the meet irreducible congruences on $\m a$ are product
congruences. Each one has been shown to be above $\sigma$ or $\rho$,
so is in fact a product congruence.
\end{proof}
Under the
assumptions of Lemma~\ref{solvable-or-perfect} a congruence
$\chi$ of $\al A$ is central
(i.e., satisfies $[1,\chi]=0$) if and only if
the congruence
$\bar{\chi}:=(\chi\vee\sigma)/\sigma$ of $\al A/\sigma$ is central
and $\chi\le\rho$.
This can be verified as follows.
By Lemma~\ref{solvable-or-perfect}(2),
$\chi$ is a product congruence relative to the factorization
$\al A\cong\al A/\sigma\times\al A/\rho$, therefore it follows that
$\chi$ is a central congruence of $\al A$ if and only if
$\bar{\chi}:=(\chi\vee\sigma)/\sigma$ is a central congruence of $\al A/\sigma$
and $(\chi\vee\rho)/\rho$ is a central congruence of $\al A/\rho$.
But $\al A/\rho$ is a (C8)-algebra, therefore
$(\chi\vee\rho)/\rho$ is central if and only if it is the trivial congruence
of $\al A/\rho$, that is, $\chi\le\rho$.
The statement in the preceding paragraph implies that
if $\zeta$ is the center of
$\al A$, then $\bar{\zeta}=(\zeta\vee\sigma)/\sigma$ is the center of
$\al A/\sigma$ and $\zeta\le\rho$.
Consequently,
if $0=:\zeta_0\le\zeta=:\zeta_1\le\zeta_2\le\dots$ is the ascending
central series of $\al A$, then
$0=\bar{\zeta}_0\le\bar{\zeta}=\bar{\zeta_1}\le\bar{\zeta_2}\le\dots$
is the ascending central series of $\al A/\sigma$ and $\zeta_i\le\rho$
for all $i$.
\begin{cor}
\label{nilpotent-or-perfect}
If $\al A$ is a finite algebra in a congruence modular variety $\var{V}$
such that every finite subdirectly irreducible algebra in $\var{V}$
is either nilpotent or
(C8), then in the factorization
$\al A\cong\al A/\sigma\times\al A/\rho$ in Lemma~\ref{solvable-or-perfect}
the first factor
$\m a/\sigma$ is nilpotent,
and $\sigma$ and $\rho$ are the
final congruences in the descending and
ascending central series of $\al A$, respectively.
\end{cor}
\begin{proof}
By the construction in Lemma~\ref{solvable-or-perfect},
$\sigma$ is the least congruence such that $\al A/\sigma$ is
solvable, so
$\al A/\sigma$ is a subdirect product of solvable subdirectly
irreducible algebras in $\var{V}$.
Since every solvable subdirectly irreducible
algebra in $\var{V}$ is nilpotent,
we get that $\al A/\sigma$ is nilpotent, and
$\sigma$ is the least congruence such that $\al A/\sigma$ is nilpotent.
Hence $\sigma$ is the final congruence in the descending central series
of $\al A$.
To prove our claim on $\rho$, let
$0=\zeta_0\le\zeta=\zeta_1\le\zeta_2\le\dots$ be the ascending
central series of $\al A$. As we saw earlier, this implies that
$0=\bar{\zeta}_0\le\bar{\zeta}=\bar{\zeta_1}\le\bar{\zeta_2}\le\dots$
is the ascending central series of $\al A/\sigma$ and $\zeta_i\le\rho$
for all $i$.
Since $\al A/\sigma$ is nilpotent, $\bar{\zeta_c}=1$ for some $c$,
so $\zeta_c\vee\sigma=1$ and $\zeta_c\le\rho$.
This implies that
$\rho=1\wedge\rho=(\zeta_c\vee\sigma)\wedge\rho
\stackrel{\bf mod}{=}\zeta_c\vee(\sigma\wedge\rho)=\zeta_c\vee 0=\zeta_c$.
Moreover, $\rho=\zeta_c\le\zeta_i\le\rho$ for every $i\ge c$, therefore
$\rho=\zeta_c$ is the final congruence in the ascending central series
of $\al A$.
\end{proof}
We will only use the following special case.
\begin{cor}
\label{abelian-or-perfect}
If $\al A$ is a finite algebra in a congruence modular variety $\var{V}$
such that every finite subdirectly irreducible algebra in $\var{V}$
is either abelian or (C8), then in the factorization
$\al A\cong\al A/\sigma\times\al A/\rho$ in Lemma~\ref{solvable-or-perfect}
the first factor
$\m a/\sigma$ is abelian,
$\sigma=[1,1]$ is the derived congruence,
and $\rho=\zeta$ is the center.
\end{cor}
\begin{proof}
Restricting the argument in the proof of Corollary~\ref{nilpotent-or-perfect}
to the case when all finite nilpotent subdirectly irreducible algebras
are abelian yields that $\al A/\sigma$ is abelian and
$\sigma$ is the least congruence such that $\al A$ is abelian, so
$\sigma=[1,1]$.
Moreover, the center of $\al A/\sigma$ is $1=\bar{\zeta}=\bar{\zeta_1}$,
showing that we can use $c=1$ in the argument.
Therefore $\rho=\zeta_1=\zeta$ is the center of $\al A$.
\end{proof}
We extend the ``neutral'' terminology by calling
a subdirectly irreducible algebra $\m a$ \emph{almost neutral}
if it is nonabelian and fails (C3)
($[x,y]=x\wedge y$) in exactly one way:
$[\mu,\mu]=0$, where $\mu$ is the monolith of $\m a$.
Equivalently, a subdirectly irreducible algebra
$\m a$ with monolith $\mu$ is almost neutral if it is nonabelian
and
\begin{enumerate}
\item[(i)] $(0:\mu)=\mu$, and
\item[(ii)] the congruence interval $\interval{\mu}{1}$ is neutral.
\end{enumerate}
\begin{thm} \label{maincor}
Let $\mathcal V$ be a variety with a parallelogram term.
Assume that every finite subdirectly
irreducible algebra
in ${\mathcal V}$ is abelian, neutral or almost neutral.
Then every finite algebra in $\mathcal V$ is dualizable.
\end{thm}
\begin{proof}
Any neutral or almost neutral subdirectly irreducible
algebra is a (C8)-algebra, hence the finite subdirectly
irreducible algebras in $\mathcal V$
are abelian or (C8).
{}From Corollary~\ref{abelian-or-perfect}
we have that
any finite algebra $\m a\in \mathcal V$ has
the properties that
\begin{enumerate}
\item[(a)]
$[1,1]$ and $\zeta$
are complementary factor congruences,
\item[(b)] $\m a/[1,1]$ is abelian and
$\m a/\zeta$ is a (C8)-algebra, and
\item[(c)] every meet irreducible congruence
on $\m a$ lies above $[1,1]$ or $\zeta$.
\end{enumerate}
Our task is to verify
that for every
relevant triple $(\delta,\theta,\nu)$
of $\al A$, where
$\delta$ is a meet irreducible congruence,
$\delta\prec\theta$,
$\nu = (\delta:\theta)$, and
$\theta/\delta$ is abelian,
there exists a triple $(\alpha,\beta,\kappa)$
which splits $(\delta,\theta,\nu)$ relative to
$\var{Q}:=\mathsf{S}\mathsf{P}(\al A)$.
The conditions (i)--(v) that define what this means
for $(\alpha,\beta,\kappa)$ are listed
in the Introduction.
{\sc Case 1:} $[1,1]\leq \delta$.
We have $[1,\theta]\leq [1,1] \leq \delta$,
proving that $1 \leq (\delta:\theta)$. Hence
$1 = (\delta:\theta)=\nu$.
If we choose $(\alpha,\beta,\kappa)=(\zeta,[1,1],0)$,
then we have
\begin{enumerate}
\item[(i)]
$\kappa=0$ is a $\var{Q}$-congruence,
\item[(ii)]
$\beta=[1,1]\leq \delta$,
\item[(iii)]
$\alpha\wedge\beta = \zeta\wedge[1,1] = 0 = \kappa$,
\item[(iv)]
$\alpha\vee\beta = \zeta\vee[1,1] = 1 =
\nu$, and
\item[(v)]
$[\alpha,\alpha]=[\zeta,\zeta]=0\leq \kappa$.
\end{enumerate}
Hence
the conditions required for $(\alpha,\beta,\kappa)$
are met.
{\sc Case 2:} $\zeta\leq \delta$.
In this case, $\m a/\delta$ is a nonabelian subdirectly
irreducible algebra. Since $(\delta,\theta,\nu)$ is relevant
it must be that the monolith $\mu=\theta/\delta$
of $\m a/\delta$ is abelian.
$\m a/\delta$ must be almost neutral, hence
\[
\theta/\delta\leq \nu/\delta=(0:\mu) = \mu=\theta/\delta,
\]
and therefore $\nu=\theta$.
In this case it is our aim to show that there is a
congruence $\gamma$ covering $\zeta$ such that
$\gamma\not\leq \delta$.
For such a congruence we have a perspectivity
$\interval{\zeta}{\gamma}\nearrow\interval{\delta}{\theta}$.
In this situation $(\alpha,\beta,\kappa)=(\gamma\wedge[1,1],\delta,0)$
is a splitting triple for $(\delta,\theta,\nu)$, since
\begin{enumerate}
\item[(i)]
$\kappa=0$ is a $\var{Q}$-congruence,
\item[(ii)]
$\beta= \delta$ (we need only $\beta\leq \delta$ here),
\item[(iii)]
$\beta\wedge\alpha = \delta\wedge\gamma\wedge[1,1] =
\zeta\wedge[1,1] = 0 = \kappa$,
\item[(iv)]
$\beta\vee\alpha =
\delta\vee(\gamma\wedge[1,1]) =
\delta\vee\zeta\vee(\gamma\wedge[1,1]) \stackrel{\bf mod}{=}
\delta\vee(\gamma\wedge(\zeta\vee[1,1])) =
\delta\vee\gamma =
\theta = \nu$, and
\item[(v)]
$[\alpha,\alpha]\leq\kappa$.
(There are perspectivities
$\interval{\delta}{\theta}\searrow
\interval{\zeta}{\gamma}\searrow
\interval{0}{\alpha}=
\interval{\kappa}{\alpha}$
and the first is abelian, so the last is.)
\end{enumerate}
To reiterate the conclusion just drawn, it suffices
to show that $\m a$ has a congruence $\gamma$ covering
$\zeta$ such that $\gamma\not\leq \delta$.
Since $\delta$ is also above $\zeta$, we may work
modulo $\zeta$ and henceforth assume that
$\m a$ is a (C8)-algebra.
In this situation, every subdirectly irreducible quotient
of $\m a$ is neutral or almost neutral.
So, we will be done of we prove the following claim.
\begin{clm}
\label{clm-atomic}
If $\m a$ is a finite (C8)-algebra in $\mathcal V$,
and $(\delta,\theta,\nu)$ is a relevant triple of $\m a$,
then $\al A$ has
an atomic congruence $\gamma$
such that $\gamma\not\leq \delta$.
\end{clm}
\noindent
{\it Proof of Claim~\ref{clm-atomic}.}
Let $\delta_i$, $i=1,\dots,n$, be the set of meet irreducible
congruences of $\m a$, and for each $i$ let $\theta_i$
be the upper cover of $\delta_i$. For each $i$ choose
$\nabla_i\in \{\delta_i,\theta_i\}$ according to
\[
\nabla_i =
\begin{cases}
\delta_i & \textrm{if $\m a/\delta_i$ is neutral};\\
\theta_i & \textrm{if $\m a/\delta_i$ is almost neutral.}
\end{cases}
\]
Equivalently,
$\nabla_i\in \{\delta_i,\theta_i\}$ is chosen
as small as possible so that
the interval
$\interval{\nabla_i}{1}$
is neutral.
Let $\nabla = \bigcap_{i=1}^n \nabla_i$. Since each $\m a/\nabla_i$
is neutral and the class
of neutral algebras in $\mathcal V$
is closed under finite subdirect products
we get that the interval
$\interval{\nabla}{1}$
is neutral.
$\m a$ itself is not a neutral algebra, since it
has a relevant triple $(\delta,\theta,\nu)$ (and therefore
an abelian congruence interval
$\interval{\delta}{\theta}$),
so $0<\nabla$.
For each $i$, define $\Delta_i = \delta_i\wedge \nabla$.
If $\delta_i=\nabla_i\;(\geq \nabla)$ for some $i$,
then $\Delta_i=\nabla$.
In the alternative case where
$\theta_i=\nabla_i$, the interval
$\interval{\Delta_i}{1}$
contains the abelian prime quotient
$\interval{\delta_i}{\theta_i}$, and $\interval{\nabla}{1}$
contains no such prime quotient, so $\Delta_i\neq\nabla$.
Hence
$\nabla\not\le\delta_i$,
so
there is a perspectivity
$\interval{\delta_i}{\theta_i}\searrow\interval{\Delta_i}{\nabla}$.
Thus, for each $i$, $\Delta_i=\nabla$
if $\al A/\delta_i$ is neutral, and
$\Delta_i$
is a lower cover of $\nabla$ for which the interval
$\interval{\Delta_i}{\nabla}$
is abelian
if $\al A/\delta_i$ is almost neutral.
The intersection $\bigcap_{i=1}^n \Delta_i$ equals
$\bigcap_{i=1}^n \delta_i = 0$. Since the former is a meet
of lower covers of $\nabla$, and the interval
$\interval{0}{\nabla}$
is modular, this interval is complemented.
Thus, for every $i$ where $\Delta_i\prec\nabla$
there is an atom $\gamma_i$ below $\nabla$ such that
$\interval{0}{\gamma_i}\nearrow\interval{\Delta_i}{\nabla}$.
Choose the
value of $i$ where $\delta_i$ is the first congruence
in the relevant triple $(\delta,\theta,\nu)$.
Since
$\theta/\delta$ is abelian,
$\al A/\delta_i=\al A/\delta$ is almost neutral.
Therefore
$\interval{0}{\gamma_i}\nearrow
\interval{\Delta_i}{\nabla}\nearrow
\interval{\delta_i}{\theta_i}=
\interval{\delta}{\theta}$.
For this $i$
we have found an atom $\gamma = \gamma_i$
satisfying $\gamma\not\leq \delta$.
$\diamond$
This completes the proof of the theorem.
\end{proof}
\begin{cor}\label{abelian-or-neutral}
Let $\mathcal V$ be a variety with a parallelogram term.
If every finite subdirectly irreducible algebra in
$\mathcal V$ is abelian or neutral, then
every finite algebra in $\mathcal V$ is dualizable.
\end{cor}
\begin{proof}
This is Theorem~\ref{maincor} restricted
to the situation where $\mathcal V$ has no
finite almost
neutral subdirectly irreducible algebras.
\end{proof}
\begin{cor}[See \cite{davey-werner}]
Any finite algebra with a near unanimity term is dualizable.
\end{cor}
\begin{proof}
A near unanimity term is a parallelogram term,
and any algebra having such a term is neutral.
Hence this corollary is a further
restriction of Theorem~\ref{maincor}.
(In fact, it is exactly the
restriction of
Theorem~\ref{maincor}
to the situation where $\mathcal V$ has no
finite abelian or almost
neutral subdirectly irreducible algebras.)
\end{proof}
\begin{cor}
\label{cor-dirrep}
Any finite algebra in a directly representable variety
is dualizable.
\end{cor}
\begin{proof}
This is a corollary to Corollary~\ref{abelian-or-neutral}.
To see this, recall that (i)~a finite algebra
in a directly representable variety has a Maltsev term
(see \cite[Theorem~5.11]{mckenzie_narrowness}),
(ii)~a Maltsev term is a parallelogram term, and
(iii)~the finite nonabelian subdirectly irreducible algebras
in a directly representable variety are simple, hence neutral
(see \cite[Theorem~5.11]{mckenzie_narrowness}).
\end{proof}
\begin{cor}[Cf.~\cite{davey-quackenbush}]
Any finite algebra in a variety generated by a
paraprimal algebra is dualizable.
\end{cor}
\begin{proof}
This follows from Corollary~\ref{cor-dirrep} and
the fact that a variety generated by a paraprimal algebra
is directly representable
(see \cite[Theorem~1.6]{clark-krauss-paraprimal}).
\end{proof}
\begin{cor}
\label{cor-affine}
Any finite affine algebra is dualizable.
In particular, any finite module is dualizable.
\end{cor}
\begin{proof}
This is another restriction of Theorem~\ref{maincor}.
This time we are restricting to the case where all
finite subdirectly irreducible algebras in $\mathcal V$ are abelian.
\end{proof}
Modules and affine algebras for which dualizability was known before
include finite abelian groups (using the restriction of Pontryagin duality,
see \cite[Chapter~4]{clark-davey}),
finite affine spaces (see \cite{pszczola}), and
finite algebras in a variety generated by a finite simple affine algebra
(see \cite{davey-quackenbush}). All these special cases of
Corollary~\ref{cor-affine}
are covered also by Corollary~\ref{cor-dirrep}.
However, Corollary~\ref{cor-affine} is not a consequence of
Corollary~\ref{cor-dirrep}, because not every finite module lies in
a directly representable variety.
(A finite faithful $R$-module generates a directly representable variety
if and only if $R$ is of finite representation type.)
\begin{cor}
\label{cor-K-alg}
Let $\mathbb K$ be a commutative unital ring.
Let $\mathcal V$ be a residually small variety
of $\mathbb K$-algebras
(commutative or not, unital or not).
Any finite algebra in $\mathcal V$ is dualizable.
\end{cor}
\begin{proof}
It follows from Theorems~3.1 and 3.2 in \cite{mckenzie_kalg} that
for every nonabelian subdirectly irreducible $\mathbb{K}$-algebra
$S$ in
a residually small variety the ring reduct of $S$ is
also subdirectly irreducible and lies in a residually small variety.
The
possible structure of a subdirectly irreducible ring in a
residually small variety is described in Section~7 of
\cite{mckenzie_kalg}.
These results imply that
every nonabelian
subdirectly irreducible algebra $S\in \mathcal V$
is either simple or else (i) has abelian monolith
equal to the radical, $J = \textrm{rad}(S)$, and
(ii) has $S/J$ isomorphic to a field or a product of two fields.
This is enough to guarantee that
every subdirectly irreducible
algebra in $\mathcal V$ is abelian, neutral or
almost neutral.
So, our statement follows from Theorem~\ref{maincor}.
\end{proof}
\begin{cor}
\label{cor-ring}
Any finite ring
(commutative or not, unital or not)
that generates a residually small variety
is dualizable.
\end{cor}
\begin{proof}
This is Corollary~\ref{cor-K-alg} restricted to the case when
$\mathbb K = \mathbb Z$.
\end{proof}
The special case of Corollary~\ref{cor-ring} when the
ring is assumed to be
commutative and unital was proved in \cite{commutative_rings}.
For a commutative ring $R$ with unit, the condition that $R$ generates
a residually small variety is equivalent to the condition that
its radical, $J=\text{\rm rad}(R)$, is abelian (i.e., $J^2=0$).
For a finite group $G$, the condition that
$G$ generates a residually small variety is equivalent
to the condition that the Sylow subgroups of $G$ are abelian.
Now we explain how to derive from Theorem~\ref{thm-main} the
result that any finite group with abelian Sylow subgroups is dualizable.
(This does not follow from Theorem~\ref{maincor}.)
The crucial result is the following.
\begin{thm}[From {\cite[Chapter~2]{nickodemus}}]
\label{matt-nick}
If $G$ is a finite group with abelian Sylow subgroups
and $(\delta,\theta,\nu)$ is a relevant triple of $G$,
then there is an endomorphism $\varepsilon\colon G\to G$
such that
\[
[\nu,\nu]\leq \ker(\varepsilon)\leq \delta.
\]
\end{thm}
\begin{cor}[From {\cite{nickodemus}}]
\label{groups}
Any finite group with abelian Sylow subgroups is dualizable.
\end{cor}
\begin{proof}
Let $G$ be a finite group with abelian Sylow subgroups.
We have to show that every relevant triple $(\delta,\theta,\nu)$
of $G$ is split by a triple $(\alpha,\beta,\kappa)$
relative to ${\mathcal Q}=\mathsf{S}\mathsf{P}(G)$.
Choose $(\alpha,\beta,\kappa) = (\nu,\ker(\varepsilon),\ker(\varepsilon))$
where $\epsilon$ is the endomorphism from Theorem~\ref{matt-nick}.
Then
\begin{enumerate}
\item[(i)]
$\kappa$ is a $\mathcal Q$-congruence, since
$\kappa$ is the kernel of an endomorphism of $G$.
\item[(ii)]
$\beta=\ker(\varepsilon)\leq \delta$,
\item[(iii)]
$\alpha\wedge\beta = \nu\wedge\ker(\varepsilon)=
\ker(\varepsilon) = \kappa$,
\item[(iv)]
$\alpha\vee\beta = \nu\vee\ker(\varepsilon)=\nu$, and
\item[(v)]
$[\alpha,\alpha]=[\nu,\nu]\leq \ker(\varepsilon)=\kappa$.
\end{enumerate}
Hence the
conditions required for $(\alpha,\beta,\kappa)$
are met.
\end{proof}
\begin{exmp}
Pawe{\l} Idziak proved in \cite{idziak}
that the algebra $\m a$ obtained from the
six-element symmetric group $S_3$ by adjoining
all six nullary operations is not a dualizable algebra.
This is in contrast to Corollary~\ref{groups},
which establishes that $S_3$ without the additional
constants \emph{is} dualizable. It is worth pointing
out how Idziak's example fails the conditions
of Theorem~\ref{thm-main}. The algebra $\m a$ has
three congruences, $0 < \delta < 1$, where
$\delta$ is the congruence on $S_3$ corresponding
to the alternating group. The triple $(\delta,\theta,\nu)=(\delta,1,1)$
is relevant. For $\mathcal Q =
\mathsf{S}\mathsf{P}(\m a)$, the only $\mathcal Q$-congruences on $\m a$
are $0$ and $1$, and $1\not\leq \delta$,
so if $(\alpha,\beta,\kappa)$ is a splitting triple
for $(\delta,\theta,\nu)$
then
\begin{enumerate}
\item[(i)]
$\kappa$ must equal $0$.
\end{enumerate}
To establish dualizability our theorem
also requires that
\begin{enumerate}
\item[(ii)]
$\beta\leq \delta$,
\item[(iii)]
$\alpha\wedge\beta = \kappa \;(= 0)$,
\item[(iv)]
$\alpha\vee\beta = \nu \;(= 1)$, and
\item[(v)]
$[\alpha,\alpha]\leq\kappa \;(= 0)$.
\end{enumerate}
But $\delta$ is the largest abelian congruence on $\m a$,
so $\alpha\leq \delta$. This and property
(ii)
yield
that $\alpha\vee\beta\leq\delta<1$, contrary to
(iv).
If we had not added the additional nullary operations,
then the congruence
$\delta$ would be another $\mathcal Q$-congruence and in this situation
$(\alpha,\beta,\kappa)=(1,\delta,\delta)$ would be a splitting
triple for $(\delta,1,1)$.
\end{exmp}
We close with a different kind of application of Theorem~\ref{thm-main}.
\begin{thm}\label{quasivariety}
If $\m a$ is a finite algebra with a parallelogram term
and $\mathsf{S}\mathsf{P}(\m a)$ is a variety, then $\m a$ is dualizable.
\end{thm}
\begin{proof}
If $\mathcal Q:=\mathsf{S}\mathsf{P}(\m a)$ is a variety, then
it must be residually small, hence satisfies (C1)
(by Theorems~\ref{thm-rs} and \ref{thm-parterm-cm}).
Every
congruence on every algebra
in $\var{Q}$
is a $\mathcal Q$-congruence,
so
if
$(\delta,\theta,\nu)$ is relevant, then
it is split by $(\nu,\delta,\delta)$,
because (C1) implies that
$[\nu,\nu]\le\delta$.
\end{proof}
Here it is worth pointing out that if
$\mathcal V$ is a variety with only finitely
many subdirectly irreducible algebras,
each
finite and
having a 1-element subalgebra, then the product
$\m a$ of
all these subdirectly irreducible algebras
is a finite algebra with the property
that $\mathcal V = \mathsf{S}\mathsf{P}(\m a)$.
If a variety $\mathcal V$, like this, has a parallelogram term,
then Theorem~\ref{quasivariety},
combined with the main result of \cite{davey-willard},
proves that each quasivariety generator for $\mathcal V$
is dualizable.
\noindent
{\bf Acknowledgment.}
The research reported in this paper was carried out while visiting
La Trobe University as a guest of Brian Davey.
We gratefully acknowledge
the support and encouragement provided by our host.
We also thank Ross Willard for reading a preliminary draft of the paper and
recommending improvements.
\end{document} | math | 156,147 |
\begin{document}
\title{Torsion exceptional sheaves on weak del Pezzo surfaces of Type A}
\date{\today}
\author{Pu Cao}
\address{
Graduate School of Mathematical Sciences, The University of Tokyo
3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan}
\email{[email protected]}
\author{Chen Jiang}
\address{
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan.}
\email{[email protected]}
\thanks{The first author
was supported by Government Scholarship and the Program for Leading Graduate Schools, MEXT, Japan. The second author was supported by JSPS KAKENHI Grant Number JP16K17558, the Program for Leading Graduate Schools, and World Premier International Research Center Initiative (WPI), MEXT, Japan}
\begin{abstract}
We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most $A_n$-singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a $(-1)$-curve by spherical twists.
\end{abstract}
\subjclass[2010]{13D09, 14J26}
\keywords{derived categories, exceptional objects, weak del Pezzo surfaces}
\maketitle
\pagestyle{myheadings} \markboth{
P. Cao \& C. Jiang
}{
Torsion exceptional sheaves on weak del Pezzo surfaces of Type A
}
\section{Introduction}
\noindent We work over the complex number field ${\mathbb C}$. Let $X$ be a smooth projective variety and $D(X):= D^b(\Coh X)$ the bounded derived category of coherent sheaves on $X$. The category $D(X)$ carries a lot of geometric information on $X$ and has drawn a lot of interest in the study of algebraic varieties. An object $\alpha \in D(X)$ is called \emph{exceptional} if
$$
\Hom (\alpha,\alpha[i])
\cong
\begin{cases} {\mathbb C} & i= 0;\\
0 & i \ne 0.
\end{cases}
$$
Exceptional objects are related to semi-orthogonal decompositions of derived categories and appear in many contexts (see, for example, \cite{Huybrechts}). Hence it is natural to consider the
classification of exceptional objects.
Exceptional objects on {\it del Pezzo surfaces} (i.e., smooth projective surfaces with ample anticanonical bundles) were investigated by Kuleshov and Orlov in \cite{KO} where it is proved that any exceptional object on a del Pezzo surface is isomorphic to a shift of an exceptional vector bundle or a line bundle on a $(-1)$-curve.
As exceptional objects on del Pezzo surfaces are well-understood, it is natural to consider {\it weak del Pezzo surfaces} (i.e., smooth projective surfaces with nef and big anticanonical bundles). In this case something interesting happens since spherical twist functors (see Definition \ref{def spherical}) are involved due to the existence of $(-2)$-curves on weak del Pezzo surfaces. We could not expect that exceptional objects are as simple as those on del Pezzo surfaces (see Section \ref{section example}), but still we expect that they are so after acting by autoequivalences of the derived category.
\begin{conj}[{cf. \cite[Conjecture 1.3]{OU}}]\label{conj}Let $X$ be a weak del Pezzo surface.
For any exceptional object $\EE\in D(X)$, there exists an autoequivalence
$\Phi \in \Auteq(D(X))$ such that $\Phi(\EE)$ is an exceptional vector bundle, or a line bundle on a $(-1)$-curve on $X$.
\end{conj}
Recently, Okawa and Uehara \cite{OU} considered the Hirzebruch surface ${\mathbb F}_2$, the simplest weak del Pezzo surface. They classified exceptional sheaves on ${\mathbb F}_2$ and confirmed Conjecture \ref{conj} for those sheaves.
Note that on ${\mathbb F}_2$, there is no torsion exceptional sheaf due to the absence of $(-1)$-curves. Motivated by Okawa--Uehara's work and this observation, we are interested in torsion exceptional sheaves (and objects) on weak del Pezzo surfaces.
In \cite{C}, the first author treated torsion exceptional sheaves and objects on a weak del Pezzo surface of degree greater than one whose anticanonical model has at most $A_1$-singularities.
On the other hand, one may compare Conjecture \ref{conj} to \cite[Proposition 1.6]{Ishii-Uehara}, where Ishii and Uehara showed that a spherical object on the minimal resolution of an $A_n$-singularity on a surface can be obtained from a line bundle on a $(-2)$-curve by autoequivalence. But the situation for torsion exceptional objects seems to be more complicated since its scheme theoretic support might be non-reduced (see Example \ref{example 2}) while the support of such a spherical object is always reduced (see \cite[Corollary 4.10]{Ishii-Uehara}).
In this article, we give an affirmative answer to Conjecture \ref{conj} for torsion exceptional sheaves on weak del Pezzo surfaces of degree greater than two of {\it Type A} (i.e., those whose anticanonical model has at most $A_n$-singularities). Namely, we prove the following theorem.
\begin{thm}\label{main1}
Let $X$ be a weak del Pezzo surface of degree $d>2$ of Type A, and $\EE$ a torsion exceptional sheaf on $X$. Then there exist a $(-1)$-curve $D$, an integer $d$, and a sequence of spherical twist functors $\Phi_1,\ldots, \Phi_n$ associated to line bundles on chains of $(-2)$-curves such that
$$
\EE\cong \Phi_1 \circ \cdots \circ \Phi_n(\OO_D(d)).
$$
\end{thm}
\begin{remark}
To be more concisely, Theorem \ref{main1} implies that every torsion exceptional sheaf on such $X$ is a line bundle on a $(-1)$-curve up to the action by $\text{Br}(X)$, where $\text{Br}(X)\subset\Auteq(D(X))$ is the group generated by all spherical twists functors. It is worth to mention that every spherical twist in $\text{Br}(X)$ can be decomposed into twists or inverse twists associated to line bundles on $(-2)$-curves by Ishii and Uehara \cite{Ishii-Uehara}. The point of Theorem \ref{main1} is that we show that the torsion exceptional sheaf can be obtained from a line bundle on a $(-1)$-curve by acting spherical twist functors associated to line bundles on chains of $(-2)$-curves successively which can be constructed explicitly according to the proof, and no inverse spherical twist functors are involved here.
\end{remark}
In fact, we can prove the following slightly general theorem on torsion exceptional sheaves.
\begin{thm}\label{main2}
Let $X$ be a smooth projective surface and $\EE$ a torsion exceptional sheaf on $X$.
Assume the following conditions hold:
\begin{enumerate}
\item $\supp(\EE)$ only contains one $(-1)$-curve $D$ and $(-2)$-curves;
\item the restriction of $\EE$ to $D$ is a line bundle (see Definition \ref{def restriction});
\item $(-2)$-curves in $\supp(\EE)$ forms a disjoint union of $A_n$-configurations with $n\leq 6$;
\item the intersection number of $D$ with any chain of $(-2)$-curves in $\supp(\EE)$ is at most one.
\end{enumerate}
Then there exist an integer $d$, and a sequence of spherical twist functors $\Phi_1,\ldots, \Phi_n$ associated to line bundles on chains of $(-2)$-curves such that
$$
\EE\cong \Phi_1 \circ \cdots \circ \Phi_n(\OO_D(d)).
$$
\end{thm}
The idea of the proof is based on the observation that, under some good conditions, we can ``factor" a spherical sheaf out of a torsion exceptional sheaf to get another one (see Lemmas \ref{lem EES} and \ref{lemma main2}), and this step actually corresponds to a spherical twist functor. After this factorization, we get an exceptional sheaf with smaller support. Iterating this process, eventually we get an exceptional sheaf supported on a $(-1)$-curve. To check the conditions that allow us to factor out spherical sheaves, we need a detailed classification of certain torsion rigid sheaves supported on $(-2)$-curves (see Section \ref{section rigid -2}).
We expect that this idea also works for torsion exceptional sheaves on arbitrary weak del Pezzo surfaces. At least, Lemma \ref{lem EES} is very useful for cutting down the support of the torsion exceptional sheaf. However, to find an appropriate way to apply Lemma \ref{lem EES}, our proof is based on the classification of torsion rigid sheaves supported on $(-2)$-curves (see Section \ref{section rigid -2}). If the configuration of the support gets complicated, then the classification becomes tedious.
So we hope that one could replace the latter part by more systematical method which can avoid tedious classifications and works for arbitrary weak del Pezzo surfaces.
\noindent{\bf Notation and Conventions.}
Let $X$ be a smooth projective surface. For a coherent sheaf $\EE$ on $X$, we denote by $\supp(\EE)$
the support of $\EE$ with reduced induced scheme structure.
For $\EE, \FF\in D(X)$, we denote $h^i(\EE, \FF):=\dim \Ext^i(\EE, \FF)=\dim \Hom(\EE, \FF[i]),$
and the Euler characteristic $\chi(\EE, \FF):=\sum_{i}(-1)^ih^i(\EE, \FF).$
A $(-1)$-curve (resp. $(-2)$-curve) is a smooth rational curve on $X$ with self-intersection number $-1$ (resp. $-2$).
We say $Z=C_1\cup\cdots\cup C_n$ is a chain of $(-2)$-curves on $X$ if $C_i$ is a $(-2)$-curve and
$$
C_i\cdot C_j=\begin{cases}
1 &|i-j|=1;\\
0 &|i-j|>1.
\end{cases}
$$
We regard $Z$ as a closed subscheme of $X$ with respect to the reduced induced structure. Sometimes we also regard $Z$ as its fundamental cycle $\sum_iC_i$. For a coherent sheaf $\RR$ on $Z$, we denote by $\deg_{C_l} \RR$ the degree of the restriction $\RR|_{C_l}$ on $C_l\cong \mathbb{P}^1$. We denote by $\RR_0 =\OO_{C_1\cup \cdots \cup C_n}(a_1,\ldots,a_n)$
the line bundle on $Z$ such that $\deg_{C_l} \RR_0 = a_l$ for all $l$. Sometimes we also consider
$
\RR_1 =\OO_{r_1C_1\cup \cdots \cup r_nC_n}(a_1,\ldots,a_n)
$
for $r_l\in \{1,2\}$ for all $l$. Here $\RR_1$ is the line bundle on $r_1C_1\cup \cdots \cup r_nC_n$ such that $\deg_{C_l} \RR_1 = a_l$ for all $l$. In other words, $\RR_1|_{r_lC_l}\cong \OO_{r_lC_l}(a_l)$, where $\OO_{2C}(a)$ is the unique non-trivial extension of $\OO_{C}(a)$ by $\OO_{C}(a+2)$ for a $(-2)$-curve $C$ on $X$.
\section{Preliminaries}
\subsection{Exceptional and spherical objects}
We recall the definition of exceptional and spherical objects.
\begin{definition}
Let
$X$ be a smooth projective variety.
An object $\alpha \in D(X)$ is \emph{exceptional} if
$$
\Hom (\alpha,\alpha[i])
\cong
\begin{cases} {\mathbb C} & i= 0;\\
0 & i \ne 0.
\end{cases}
$$
\end{definition}
\begin{example}
\begin{enumerate}
\item Let $X$ be a smooth projective variety with $H^i(X, \OO_X)=0$ for $i>0$ (e.g. Fano manifolds). Then every line bundle on $X$ is an exceptional object.
\item Let $X$ be a smooth projective surface and $C$ a $(-1)$-curve on $X$. Then any line bundle on $C$ is an exceptional object.
\end{enumerate}
\end{example}
\begin{definition}[\cite{ST}]\label{def spherical}
Let
$X$ be a smooth projective variety.
\begin{enumerate}
\item An object $\alpha \in D(X)$ is \emph{spherical} if
$\alpha \otimes \omega_{X} \cong \alpha$ and
$$
\Hom (\alpha,\alpha[i])
\cong
\begin{cases}
{\mathbb C} & i=0,\dim X;\\
0 & i\ne 0,\dim X.
\end{cases}
$$
\item Let $\alpha\in D(X)$ be a spherical object.
Consider the mapping cone
$$
\mathcal{C}={\rm Cone}(\pi_1^*\alpha^\vee\otimes \pi_2^*\alpha \to \mathcal{O}_{\Delta})
$$
of the natural evaluation
$
\pi_1^*\alpha^\vee\otimes \pi_2^*\alpha \to \mathcal{O}_{\Delta}
$,
where
$
\Delta \subset X \times X
$
is the diagonal and $\pi_i$ is the projection from
$X\times X$ to the $i$-th factor. Then the integral functor
$T_{\alpha}:=\Phi^{\mathcal{C}}_{X\to X}$ defines an autoequivalence of $D(X)$,
called the \emph{twist functor} associated to the spherical object $\alpha$.
By definition, for $\beta\in D(X)$, there is an exact triangle
$$
\RHom(\alpha,\beta)\otimes
\alpha\xrightarrow[]{\textrm{evaluation}}
\beta \to
T_{\alpha}\beta.
$$
\end{enumerate}
\end{definition}
\begin{example}[cf. {\cite[Example 4.7]{Ishii-Uehara}}]
Let $X$ be a smooth projective surface and $Z$ a chain of $(-2)$-curves. Then any line bundle on $Z$ is a spherical object in $D(X)$.
\end{example}
\subsection{Rigid sheaves}In this subsection, we assume that $X$ is a smooth projective surface. All sheaves are considered to be coherent on $X.$
A coherent sheaf $\RR$ is said to be {\it rigid} if $h^1(\RR, \RR) = 0$.
Kuleshov \cite{Kuleshov} systematically investigated rigid sheaves on surfaces with
anticanonical class without base components. We collect some interesting properties for rigid sheaves in this subsection for applications. We will use the following easy lemma without mention.
\begin{lem}\label{lem ext non-trivial} Consider an extension of coherent sheaves
$$0 \to \GG_2 \to \RR \to \GG_1 \to 0$$
such that $\RR$ is rigid. Then $h^1(\GG_1, \GG_2)>0$ if and only if this extension is non-trivial.
\end{lem}
\begin{proof}
The `if' part is trivial. For the `only if' part, assume that $h^1(\GG_1, \GG_2)>0$ and this extension is trivial, then
$\RR\cong \GG_1\oplus \GG_2,$
which implies that
$
h^1(\RR, \RR)\geq h^1(\GG_1, \GG_2)>0,
$
a contradiction.
\end{proof}
We have the following Mukai's lemma for rigid sheaves.
\begin{lem}[Mukai's lemma, {\cite[2.2 Lemma]{KO}}]\label{lem GEG}For each exact sequence
$$0 \to \GG_2 \to \RR \to \GG_1 \to 0$$
of coherent sheaves such that $$h^1(\RR,\RR)=h^0(\GG_2, \GG_1) = h^2 (\GG_1, \GG_2) = 0,$$
the following hold:
\begin{enumerate}
\item $h^1(\GG_1, \GG_1)=h^1(\GG_2, \GG_2)=0$;
\item $h^0(\RR, \RR) = h^0(\GG_1, \GG_1) + h^0(\GG_2, \GG_2) +\chi(\GG_1, \GG_2)$;
\item $h^2(\RR, \RR) = h^2(\GG_1, \GG_1) + h^2(\GG_2, \GG_2) +\chi(\GG_2, \GG_1)$;
\item $h^1(\GG_1, \GG_2)\leq h^0(\GG_1, \GG_1) +h^0(\GG_2, \GG_2)-1$.
\end{enumerate}
\end{lem}
\begin{proof}
(1)--(3) are from \cite[2.2 Lemma]{KO}. We prove (4) here. In the proof of \cite[2.2 Lemma]{KO}, we know that the natural map
$$
\Hom(\GG_1, \GG_1) \oplus \Hom(\GG_2, \GG_2) \xrightarrow{d_1} \Ext^1(\GG_1, \GG_2)
$$
is surjective. Note that the image of $(\id_{\GG_1}, \id_{\GG_2})$ is zero. Hence we get the inequality by comparing the dimensions.
\end{proof}
\begin{lem} \label{lem EES} Consider an exact sequence of coherent sheaves
$$0 \to \EE' \to \EE \to \SSS \to 0,$$
where $\EE$ is rigid, $\SSS$ is spherical, $h^0(\EE', \SSS) =0$, and $\chi(\SSS, \EE')=-1$.
Then $h^i(\EE', \EE')=h^i(\EE, \EE)$ for $i=0,1,2$. In particular, $\EE$ is exceptional if and only if so is $\EE'$, and in this case, $\EE\cong T_\SSS( \EE')$.
\end{lem}
\begin{proof}
Since $\SSS$ is spherical, $h^2(\SSS, \EE') =0$ and $\chi(\EE', \SSS)=-1$ by Serre duality. By Lemma \ref{lem GEG}, $h^i(\EE', \EE')=h^i(\EE, \EE)$ for $i=0,1,2$. In particular, $\EE$ is exceptional if and only if so is $\EE'$.
Suppose that $\EE$ and $\EE'$ are exceptional, then by Lemma \ref{lem GEG}(4),
$$h^1(\SSS,\EE')\leq h^0(\SSS, \SSS) +h^0(\EE', \EE')-1=1.$$
Since $\chi(\SSS,\EE')=-1$, we have $h^1(\SSS,\EE')=1$ and $h^0(\SSS, \EE')=h^2(\SSS, \EE') =0$.
By definition of twist functor, we have a distinguished triangle
$$
\SSS[-1] \to \EE' \to T_{\SSS}(\EE'),
$$
which corresponds to the exact sequence
$$0 \to \EE' \to \EE \to \SSS \to 0.$$
Hence $\EE\cong T_\SSS (\EE')$.
\end{proof}
We can say more about torsion rigid sheaves.
\begin{remark}\label{remark restriction}
Let $\RR$ be a torsion rigid sheaf, then $\RR$ is pure one-dimensional by \cite[Corollary 2.2.3]{Kuleshov}. Suppose that
$\supp(\RR)=Z\cup Z'$ where $Z$ and $Z'$ are unions of curves with no common components. Then there exists an exact sequence
$$
0\to \RR' \to \RR\to \RR_{Z} \to 0
$$
where $\RR'=\HH^0_{Z'}(\RR)$ is the subsheaf with supports (see \cite[II, Ex. 1.20]{H}) in $Z'$ and $\RR_{Z}$ is the quotient sheaf.
Then $\supp(\RR')=Z'$, $\supp(\RR_{Z})=Z$, and $h^0(\RR', \RR_{Z})=h^2(\RR_{Z}, \RR')=0$ by the support condition. By Lemma \ref{lem GEG}, $\RR'$ and $\RR_{Z}$ are torsion rigid (pure one-dimensional) sheaves. Moreover, if we write $Z=\cup_{i}C_i$ and $Z'=\cup_j C'_j$, then we can write the first Chern class of $\RR$ uniquely as $$c_1(\RR)=\sum_ir_iC_i+\sum_j s_jC'_j$$
in the sense that $c_1(\RR_{Z})=\sum_ir_iC_i$ and $c_1(\RR')=\sum_j s_jC'_j$ for some positive integers $r_i$ and $s_j$.
\end{remark}
\begin{definition}[restriction to curves]\label{def restriction}
Let $\RR$ be a torsion rigid sheaf such that
$\supp(\RR)=Z\cup Z'$ where $Z$ and $Z'$ are unions of curves with no common components.
The sheaf $\RR_{Z}$ constructed as in Remark \ref{remark restriction} is called {\it the restriction of $\RR$ to $Z$}.
\end{definition}
Note that for an irreducible curve $C\subset \supp(\RR)$, the restriction $\RR_C$ of $\RR$ to $C$ is sometimes different from the restriction $\RR|_C:=\RR \otimes \OO_C$. For example, if $\RR=\OO_{2C}$, then $\RR_C=\OO_{2C}$ while $\RR|_C=\OO_C$.
Here we remark that for a chain of $(-2)$-curves $Z=C_1\cup\cdots\cup C_n$ and a torsion sheaf $\RR$, $\supp(\RR)\subset Z$ if and only if $c_1({\RR})$ can be written as $\sum_ir_iC_i$ for non-negative integers $r_i$. In this case, $r_i$ is uniquely determined for all $i$.
\begin{lem}\label{lem supp}
Let $\RR$ be a torsion rigid sheaf on $X$, then any irreducible component of $\supp(\RR)$ is a curve with negative self-intersection.
\end{lem}
\begin{proof}
Let $C$ be an irreducible component of $\supp(\RR)$. Take the restriction to $C$, we have an exact sequence
$$0\to \RR'\to \RR\to \RR_C\to 0. $$
By Remark \ref{remark restriction}, $\RR_C$ is rigid and in particular,
$\chi(\RR_C,\RR_C)>0$. On the other hand, by Riemann--Roch formula (see Subsection \ref{subsection RR}), we have
$\chi(\RR_C,\RR_C)=-c_1(\RR_C)^2.$ Hence $c_1(\RR_C)^2<0$, which implies that $C^2<0$.
\end{proof}
\subsection{Weak del Pezzo surfaces}
A smooth projective surface $X$ is a {\it weak del Pezzo surface} if $-K_X$ is nef and big. The {\it degree} of $X$ is the self-intersection number $(-K_X)^2$. A weak del Pezzo surface is of {\it Type A} if its anticanonical model has at most $A_n$-singularities. We collect some basic facts on weak del Pezzo surfaces.
\begin{lem}[{cf. \cite[Theorem 8.3.2]{Dolgachev}}]\label{lem no base}
Let $X$ be a weak del Pezzo surface. Then $|-K_X|$ has no base components.
\end{lem}
\begin{lem}[{cf. \cite[Theorem 8.2.25]{Dolgachev}}]\label{lem no of -2}
Let $X$ be a weak del Pezzo surface of degree $d$. Then the number of $(-2)$-curves on $X$ is at most $9-d$.
\end{lem}
\begin{lem}\label{lem d>1}
Let $X$ be a weak del Pezzo surface of degree $d>1$.
Then the intersection number of a $(-1)$-curve with a chain of $(-2)$-curves is at most one.
\end{lem}
\begin{proof}
Take a chain of $(-2)$-curves $C_1\cup\cdots\cup C_n$ and a $(-1)$-curve $D$. Assume that $\sum_{i=1}^nC_i\cdot D\geq 2$, then
$$\Big(\sum_{i=1}^nC_i+D\Big)^2=\Big(\sum_{i=1}^nC_i\Big)^2+2\sum_{i=1}^nC_i\cdot D+D^2=-2+2\sum_{i=1}^nC_i\cdot D-1\geq 1.$$
By the Hodge index theorem,
$$
(-K_X)^2\cdot\Big(\sum_{i=1}^nC_i+D\Big)^2\leq \Big((-K_X)\cdot\Big(\sum_{i=1}^nC_i+D\Big)\Big)^2=1.
$$
This implies that $(-K_X)^2=1$, which is a contradiction.
\end{proof}
We remark that on weak del Pezzo surfaces of degree one, it is possible that one $(-1)$-curve intersects with a chain of $(-2)$-curves at two points (cf. \cite[Lemma 2.8]{Kosta}).
\subsection{Riemann--Roch formula}\label{subsection RR}
We recall Riemann--Roch formula for torsion sheaves on surfaces (cf. \cite[(1.1)]{KO}).
\begin{lem}[Riemann--Roch formula for torsion sheaves]
For two torsion sheaves $\EE$ and $\FF$ on a smooth projective surface $X$, the Euler characteristic can be calculated by
$
\chi(\EE, \FF)=-c_1(\EE)\cdot c_1(\FF).$
\end{lem}
\subsection{A polynomial inequality}\label{polynomial}
In this subsection, we treat a special polynomial which naturally appears in self-intersection numbers of a union of negative curves.
For positive integers $r_1, r_2, \ldots, r_n$, and $1\leq k\leq n$, define the polynomial
$$f(r_1, r_2, \ldots, r_n; k)=\sum_{i=1}^n r_i^2-\sum_{i=1}^{n-1}r_ir_{i+1}-r_k.$$
\begin{prop}\label{prop polynomial}
For positive integers $r_1, r_2, \ldots, r_n$, and $k$,
$$f(r_1, r_2, \ldots, r_n; k)\geq 0$$
always holds. Moreover, $f(r_1, r_2, \ldots, r_n; k)= 0$ if and only if the following conditions hold:
\begin{enumerate}
\item $r_1=r_n=1$,
\item $0\leq r_{i+1}-r_i\leq 1$ if $i<k$,
\item $0\leq r_{i}-r_{i+1}\leq 1$ if $i\geq k$.
\end{enumerate}
\end{prop}
\begin{proof}
It is easy to see that
\begin{align*}
{}&2f(r_1, r_2, \ldots, r_n; k)\\
={}&r^2_1+\sum_{i=1}^{n-1}(r_i-r_{i+1})^2+r^2_n-2r_k\\
\geq {}&r_1+\sum_{i=1}^{n-1}|r_i-r_{i+1}|+r_n-2r_k\\
\geq {}&\left(r_1+\sum_{i=1}^{k-1}(r_{i+1}-r_i)\right)+\left(\sum_{i=k}^{n-1}(r_i-r_{i+1})+r_n\right)-2r_k=0.
\end{align*}
This proves the inequality $f(r_1, r_2, \ldots, r_n; k)\geq 0$. If the equality holds, then all the above inequalities become equalities.
From the first inequality, we get $r_1=r_n=1$ and $|r_i-r_{i+1}|\leq 1$ for $1\leq i\leq n-1$. From the second, we get $r_{i+1}-r_i\geq 0$ if $i<k$, and $ r_{i}-r_{i+1}\geq 0$ if $i\geq k$.
\end{proof}
\section{Examples}\label{section example}
\noindent In this section, before proving the theorems, we provide several interesting examples of torsion exceptional sheaves on weak del Pezzo surfaces.
The examples illuminate how we may apply Lemma \ref{lem EES} to reduce the torsion exceptional sheaf to a line bundle on a $(-1)$-curve by spherical twists.
\begin{example}
Let $X$ be a weak del Pezzo surface of degree $d>1$ whose anticanonical model has at most $A_1$-singularities. Then by our proof, every torsion exceptional sheaf on $X$ has the form $\OO_{D\cup C_1\cup\cdots \cup C_n}(d, a_1, \ldots, a_n)$, where $C_i$ are disjoint $(-2)$-curves, $D$ is a $(-1)$-curve intersecting with each $C_i$, and $d, a_i$ are integers. Note that $n$ can be $0$ which means there is no $(-2)$-curve in the support. Similar result holds true for weak del Pezzo surfaces of degree $d>1$ whose anticanonical model has at most $A_2$-singularities.
\end{example}
The following example shows that the scheme theoretic support of a torsion exceptional sheaf can be non-reduced.
\begin{example}\label{example 2}Let $X$ be a smooth projective surface, $C_1\cup C_2\cup C_3$ a chain of three $(-2)$-curves, and $D$ a $(-1)$-curve. Assume that $D\cdot C_2=1$ and $D\cdot C_1=D\cdot C_3=0$. Then the structure sheaf $\OO_{D\cup C_1\cup 2C_2\cup C_3}$ is a torsion exceptional sheaf on $X$ with non-reduced support. In fact, by applying Lemma \ref{lem EES} to the following exact sequences
$$
0\to \OO_{D\cup C_1\cup C_2\cup C_3}(-1, -1, 2, -1) \to \OO_{D\cup C_1\cup 2C_2\cup C_3} \to \OO_{C_2} \to 0,
$$
$$
0\to \OO_{D}(-2) \to \OO_{D\cup C_1\cup C_2\cup C_3}(-1, -1, 2, -1) \to \OO_{C_1\cup C_2\cup C_3}( -1, 2, -1) \to 0,
$$
we get
$$
\OO_{D\cup C_1\cup 2C_2\cup C_3}=T_{\OO_{C_2}}\circ T_{\OO_{C_1\cup C_2\cup C_3}( -1, 2, -1)}( \OO_{D}(-2) ).
$$
\end{example}
The following example shows that the support of a torsion exceptional sheaf on a weak del Pezzo surface of degree one can contain loops.
\begin{example}
Let $X$ be a smooth projective surface, $C_1\cup C_2\cup C_3$ a chain of three $(-2)$-curves, and $D$ a $(-1)$-curve. Assume that $D\cdot C_2=0$ and $D\cdot C_1=D\cdot C_3=1$, that is, $C_1, C_2, C_3, D$ form a loop (this might happen, for example, on the minimal resolution of a singular del Pezzo surface of degree one with one $A_3$-singularity, cf. \cite[Lemma 2.8]{Kosta}). Then the unique non-trivial extension $\EE$ of $\OO_{C_2\cup C_3}$ by $\OO_{D\cup C_1\cup C_2}$ is a torsion exceptional sheaf on $X$ whose support is a loop. In fact,
$$
h^1(\OO_{C_2\cup C_3}, \OO_{D\cup C_1\cup C_2})=1
$$
since
\begin{align*}
h^0(\OO_{C_2\cup C_3}, \OO_{D\cup C_1\cup C_2}){}&=0,\\
\chi(\OO_{C_2\cup C_3}, \OO_{D\cup C_1\cup C_2}){}&=-1,\\
h^0(\OO_{D\cup C_1\cup C_2}, \OO_{C_2\cup C_3}){}&=0.
\end{align*}
By applying Lemma \ref{lem EES} twice, we get
$$
\EE=T_{\OO_{C_2\cup C_3}}\circ T_{\OO_{C_1\cup C_2}}( \OO_{D}(-1) ).
$$
\end{example}
From these examples, one can get a rough idea on how we apply Lemma \ref{lem EES} to cut down the support of a torsion exceptional sheaf. In fact, it should be easy to verify Conjecture \ref{conj} for the structure sheaf $\OO_\Gamma$ on arbitrary weak del Pezzo surface where $\Gamma$ is a proper subscheme and $\OO_\Gamma$ is assumed to be a torsion exceptional sheaf. However, in general torsion exceptional sheaf could not have such a good structure.
\section{Factorizations of rigid sheaves}
\noindent In this section, we assume that $X$ is a smooth projective surface. All sheaves are considered to be coherent on $X.$ We will define factorizations of rigid sheaves and give basic properties.
\begin{definition}
A coherent sheaf $\RR$ has a {\it factorization}
$
(\GG_1, \GG_2,\ldots, \GG_n)
$
if there exists a filtration of coherent sheaves
$$
0= \FF_0\subset \FF_1\subset \FF_2 \subset \cdots \subset \FF_n=\RR,
$$
such that
$\FF_i/\FF_{i-1}\cong \GG_i$ for $1\leq i\leq n$
and we write this factorization as
$$\RR\equiv (\GG_1, \ldots, \GG_n).$$
This factorization is said to be {\it perfect} if $h^0(\GG_i, \GG_j)=h^2(\GG_j, \GG_i)=0$ for all $i<j$.
\end{definition}
\begin{remark}For a factorization $\RR\equiv (\GG_1, \ldots, \GG_n)$ and some index $1\leq i_0\leq n$, if $\GG_i\cong \GG_i\otimes \omega_X$ for all $i\neq i_0$ (for example, $\GG_i$ is a torsion sheaf whose support consists of $(-2)$-curves), then $h^0(\GG_i, \GG_j)=h^2(\GG_j, \GG_i)$ for all $i<j$ by duality. Hence to check the perfectness of this factorization, one only need to check that $h^0(\GG_i, \GG_j)=0$ for all $i<j$. Note that in subsequent sections, we are always in such situation when we deal with factorizations.
\end{remark}
\begin{example}[{\cite[Lemma 2.4]{OU}}]\label{example HN filtration}
Let $C$ be a $(-2)$-curve on a smooth projective surface $X$.
Let $\FF$ be a pure one-dimensional sheaf on the scheme $mC$.
Then the subquotients of the Harder--Narasimhan filtration
$$
0=\FF^0\subset\FF^1 \subset\cdots \subset\FF^n =\FF$$
of $\FF$ are of the form
$$
(\FF^1 / \FF^0,
\FF^2 / \FF^1,
\dots,
\FF^n / \FF^{n-1})
=
(\OO_C(a_1)^{\oplus r_1}, \OO_C(a_2)^{\oplus r_2}, \dots, \OO_C(a_n)^{\oplus r_n})
$$
with $a_1>a_2>\cdots>a_n$ and $r_i>0$, which gives a perfect factorization of $\FF$.
\end{example}
The following lemma is a direct consequence of Lemma \ref{lem GEG}.
\begin{lem}\label{lem G rigid}
Let $\RR$ be a rigid sheaf with a perfect factorization
$
(\GG_1,\ldots, \GG_n)
$
and $\FF_0\subset \FF_1\subset \FF_2 \subset \cdots \subset \FF_n
$ the corresponding filtration.
Then $\FF_j/\FF_i$ is rigid for all $i<j$.
\end{lem}
In applications, we usually need to get new factorizations from old ones.
Here we give some lemmas about operations on factorizations.
\begin{lem}\label{lem change}
Let $\RR$ be a coherent sheaf with a factorization
$
(\GG_1,\ldots, \GG_n)
$
and $\FF_0\subset \FF_1\subset \FF_2 \subset \cdots \subset \FF_n
$ the corresponding filtration.
If $\FF_{i+1}/\FF_{i-1}$ has another factorization $(\GG'_{i}, \GG'_{i+1})$ for some $i$,
then $\RR$ has another factorization
$$
(\GG_1,\ldots, \GG_{i-1}, \GG'_i,\GG'_{i+1}, \GG_{i+2}, \ldots, \GG_n).
$$
In particular, if $h^1(\GG_{i+1}, \GG_i)=0$, then we are free to change the order of $\GG_{i+1}$ and $\GG_i$ in a factorization.
\end{lem}
\begin{proof}Easy.
\end{proof}
\begin{lem}
Let $\RR$ be a coherent sheaf with a perfect factorization
$
(\GG_1,\ldots, \GG_n)
$
and $\FF_0\subset \FF_1\subset \FF_2 \subset \cdots \subset \FF_n
$ the corresponding filtration. Then
$
(\FF_{n-1}, \GG_n)
$
is a perfect factorization of $\RR$.
\end{lem}
\begin{proof}Easy.
\end{proof}
\begin{lem}\label{lem extend GS}
Let $\GG$ and $\SSS$ be two coherent sheaves and $r$ a positive integer. Assume that $\SSS$ is simple (i.e., $h^0(\SSS,\SSS)=1$).
\begin{enumerate}
\item Assume that $h^1 (\SSS, \GG)=1$. Denote by $\GG'$ the unique non-trivial extension of $\SSS$ by $\GG$.
Then $h^0(\SSS, \GG)=h^0(\SSS, \GG')$, and any non-trivial extension of $\SSS^{\oplus r}$ by $\GG$ is isomorphic to $\SSS^{\oplus (r-1)}\oplus \GG'$.
\item Assume that $h^1 (\GG, \SSS)=1$. Denote by $\GG''$ the unique non-trivial extension of $\GG$ by $\SSS$.
Then $h^0(\GG, \SSS)=h^0(\GG'', \SSS)$, and any non-trivial extension of $\GG$ by $\SSS^{\oplus r}$ is isomorphic to $\SSS^{\oplus (r-1)}\oplus \GG''$.
\end{enumerate}
\end{lem}
\begin{proof}
(1) Consider the exact sequence
$$
0\to \Hom(\SSS, \GG) \to \Hom(\SSS, \GG') \to \Hom(\SSS, \SSS) \overset{\delta}{\to} \Ext^1(\SSS, \GG)
$$
induced by the extension $$0\to \GG\to \GG'\to \SSS\to 0.$$ Because the extension is non-trivial and $\SSS$ is simple, the map $\delta$ is injective. Hence $h^0(\SSS, \GG)=h^0(\SSS, \GG')$.
Consider a non-trivial extension $\RR$ corresponding to
$$
\eta=(\eta_1, \ldots, \eta_r)\in \Ext^1(\SSS^{\oplus r}, \GG)\cong \CC^r.
$$
As $\SSS$ is simple, $\Aut(\SSS^{\oplus r})=\GL(r, \CC)$. Since $\Aut(\SSS^{\oplus r})$ acts on $\Ext^1(\SSS^{\oplus r}, \GG)\cong \CC^r$
through the natural action of $\GL(r, \CC)$, after taking an automorphism of $\SSS^{\oplus r}$, we may assume that $\eta_i=0$ except for one index $i_0$. Hence $\RR\cong \SSS^{\oplus (r-1)}\oplus \GG'$.
(2) can be proved similarly.
\end{proof}
\begin{lem}\label{lem GSH}
Let $\RR$ be a rigid sheaf with a perfect factorization
$$
(\GG_1,\ldots, \GG_n, \SSS^{\oplus r}, \HH_1,\ldots, \HH_m).
$$
Assume that $\SSS$ is spherical.
\begin{enumerate}
\item Suppose that $h^0 (\SSS, \GG_n)=0$ and $\chi (\SSS, \GG_n)=-1$, then
there is a new perfect factorization
$$
(\GG_1,\ldots, \GG_{n-1}, \SSS^{\oplus (r-1)}, \GG'_n, \HH_1,\ldots, \HH_m).
$$
Here $\GG'_n$ is the (unique) non-trivial extension of $\SSS$ by $\GG_n$.
\item Suppose that $h^0 (\HH_1, \SSS)=0$ and $\chi (\HH_1, \SSS)=-1$, then
there is a new perfect factorization
$$
(\GG_1,\ldots, \GG_n, \HH'_1, \SSS^{\oplus (r-1)}, \HH_2,\ldots, \HH_m).
$$
Here $\HH'_1$ is the (unique) non-trivial extension of $\HH_1$ by $\SSS$.
\end{enumerate}
\end{lem}
\begin{proof}(1)
Since the factorization is perfect, $h^0 (\GG_n, \SSS)=h^2 (\SSS, \GG_n)=0$. Hence $\chi (\SSS, \GG_n)=-1$ implies that $h^1 (\SSS, \GG_n)=1$. The unique non-trivial extension $\GG'_n$ of $\SSS$ by $\GG_n$ is well-defined.
Note that the perfect factorization
$$
(\GG_1,\ldots, \GG_n, \SSS^{\oplus r}, \HH_1,\ldots, \HH_m).
$$
induces another
perfect factorization
$$
(\GG_1,\ldots, \GG_{n-1}, \FF', \HH_1,\ldots, \HH_m).
$$
where $\FF'$ is an extension of $\SSS^{\oplus r}$ by $\GG_n$. By Lemma \ref{lem G rigid}, $\FF'$ is rigid, hence the extension is non-trivial. By Lemma \ref{lem extend GS}(1), $\FF'\cong \SSS^{\oplus (r-1)}\oplus \GG'_{n}$. Hence there exists a factorization
$$
(\GG_1,\ldots, \GG_{n-1}, \SSS^{\oplus r},\GG'_{n}, \HH_1,\ldots, \HH_m).
$$
It is easy to check that this factorization is perfect, since $h^0(\SSS,\GG'_{n})=h^0(\SSS,\GG_{n})=0$ by Lemma \ref{lem extend GS}(1) and $h^2(\GG'_{n},\SSS)=h^0(\SSS,\GG'_{n})=0$ by duality.
(2) can be proved similarly.
\end{proof}
\section{Torsion rigid sheaves supported on $(-2)$-curves}\label{section rigid -2}
\noindent In this section, we assume that $X$ is a smooth projective surface. All sheaves are considered to be coherent on $X.$ We will classify certain torsion rigid sheaves
supported on $(-2)$-curves.
\begin{prop}\label{prop 12}
Let $C_1\cup C_2$ be a chain of two $(-2)$-curves and $\RR$ a torsion rigid sheaf with $c_1(\RR)=C_1+2C_2$. Then $\RR$ has one of the following perfect factorizations:
\begin{enumerate}
\item $(\OO_{C_2}(a_2), \OO_{C_1\cup C_2}(a_1, a_2));$
\item $( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-1));$
\item$ ( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-2)).$
\end{enumerate}
Here $a_1, a_2$ are integers.
\end{prop}
\begin{proof}
Taking the restriction to $C_2$, we have an exact sequence
$$
0\to \RR_1\to \RR \to \RR_2\to 0.
$$
By Remark \ref{remark restriction}, $\RR_1$ and $\RR_2$ are rigid. Note that $c_1(\RR_1)=C_1$ and $c_1(\RR_2)=2C_2$. Hence $\RR_1=\OO_{C_1}(a_1-1)$ is a line bundle on $C_1$ for some integer $a_1$, and $\RR_2$ has a perfect factorization induced by Harder--Narasimhan filtration, which is
{\bf Case 1.} $(\OO_{C_2}(a_2)^{\oplus 2})$ for some integer $a_2$, or
{\bf Case 2.} $(\OO_{C_2}(a_2), \OO_{C_2}(b_2))$, for integers $a_2>b_2$.
In Case 1, $\RR$ has a perfect factorization
$$(\OO_{C_1}(a_1-1), \OO_{C_2}(a_2)^{\oplus 2}),$$
for which we can apply Lemma \ref{lem GSH} to get a new perfect factorization
$$(\OO_{C_2}(a_2), \OO_{C_1\cup C_2}(a_1, a_2)).$$
This gives (1).
In Case 2, since $\RR_2$ is rigid, by Lemma \ref{lem GEG}(4), we have
$$h^1( \OO_{C_2}(b_2),\OO_{C_2}(a_2))\leq 1,$$
which implies that $a_2\leq b_2+2$, that is, $b_2=a_2-1$ or $a_2-2$.
In this case,
$\RR$ has a perfect factorization
$$(\OO_{C_1}(a_1-1), \OO_{C_2}(a_2),\OO_{C_2}(b_2)),$$
for which we can apply Lemma \ref{lem GSH} to get a new perfect factorization
$$( \OO_{C_1\cup C_2}(a_1, a_2),\OO_{C_2}(b_2)).$$
This gives (2) and (3).
\end{proof}
\begin{prop}\label{prop 123}
Let $C_1\cup C_2\cup C_3$ be a chain of three $(-2)$-curves and $\RR$ a torsion rigid sheaf with $c_1(\RR)=C_1+2C_2+3C_3$. Then $\RR$ has one of the following perfect factorizations:
\begin{enumerate}
\item[(1-1)] $(\OO_{C_3}(a_3),\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3));$
\item[(1-2)] $(\OO_{C_3}(a_3),\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3), \OO_{C_2\cup C_3}(a_2-1, a_3));$
\item[(1-3)] $(\OO_{C_3}(a_3),\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3), \OO_{C_2\cup C_3}(a_2-2, a_3));$
\item[(2-1)] $(\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3), \OO_{C_3}(b_3));$
\item[(2-2)] $(\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3), \OO_{C_2\cup C_3}(a_2-1, a_3),\OO_{C_3}(b_3));$
\item[(2-3)] $(\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3), \OO_{C_2\cup C_3}(a_2-2, a_3),\OO_{C_3}(b_3));$
\item[(3-1)] $(\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3),\OO_{C_3}(b_3), \OO_{C_2\cup C_3}(a_2, b_3));$
\item[(3-2)] $(\OO_{C_2\cup C_3}(a_2, a_3),\OO_{C_3}(b_3), \OO_{C_1\cup C_2 \cup C_3}(a_1, a_2+1, b_3));$
\item[(3-3)] $(\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3),\OO_{C_3}(b_3), \OO_{C_2\cup C_3}(a_2-1, b_3));$
\item[(3-4)] $(\OO_{C_1\cup 2C_2 \cup C_3}(a_1, a_2, a_3)\oplus \OO_{C_3}(a_3-1)^{\oplus 2});$
\item[(4-1)] $(\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, a_3), \OO_{C_2\cup C_3}(a_2, b_3),\OO_{C_3}(c_3));$
\item[(4-2)] $(\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_1\cup C_2 \cup C_3}(a_1, a_2+1, b_3), \OO_{C_3}(c_3));$
\item[(4-3)] $(\OO_{C_1\cup C_2 \cup C_3}(a_1, a_2, b_3+1), \OO_{C_2\cup C_3}(a_2-1, b_3),\OO_{C_3}(c_3));$
\item[(4-4)] $(\OO_{C_1\cup 2C_2 \cup C_3}(a_1, a_2, b_3+1)\oplus \OO_{C_3}(b_3), \OO_{C_3}(c_3)).$
\end{enumerate}
Here $a_i, b_i, c_i$ are integers and $a_3>b_3>c_3$.
\end{prop}
\begin{proof}
Taking the restriction to $C_3$, we have an exact sequence
$$
0\to \RR_{12}\to \RR \to \RR_3\to 0.
$$
By Remark \ref{remark restriction}, $\RR_{12}$ and $\RR_{3}$ are rigid. Note that $c_1(\RR_{12})=C_1+2C_2$ and $c_1(\RR_3)=3C_3$. $\RR_3$ has a perfect factorization induced by Harder--Narasimhan filtration, we have 4 cases:
{\bf Case 1.} $(\OO_{C_3}(a_3)^{\oplus 3})$ for some integer $a_3$;
{\bf Case 2.} $(\OO_{C_3}(a_3)^{\oplus 2}, \OO_{C_3}(b_3))$, for integers $a_3>b_3$;
{\bf Case 3.} $(\OO_{C_3}(a_3), \OO_{C_3}(b_3)^{\oplus 2})$, for integers $a_3>b_3$;
{\bf Case 4.} $(\OO_{C_3}(a_3), \OO_{C_3}(b_3),\OO_{C_3}(c_3))$, for integers $a_3>b_3>c_3$.
Each case can be divided in to 3 subcases according to the perfect factorization $\RR_{12}\equiv (\GG_1, \GG_2)$ in Proposition \ref{prop 12}.
In Case 1, applying Lemma \ref{lem GSH} twice with $\SSS=\OO_{C_3}(a_3)$ to the perfect factorization $\RR\equiv (\GG_1, \GG_2, \OO_{C_3}(a_3)^{\oplus 3}),$ we get a new perfect factorization, which gives (1-1), (1-2), or (1-3) by changing $a_2$ appropriately.
In Case 2, applying Lemma \ref{lem GSH} twice with $\SSS=\OO_{C_3}(a_3)$ to the perfect factorization $\RR\equiv (\GG_1, \GG_2, \OO_{C_3}(a_3)^{\oplus 2}, \OO_{C_3}(b_3)),$ we get a new perfect factorization, which gives (2-1), (2-2), or (2-3) by changing $a_2$ appropriately.
In Case 3, we have 3 subcases:
{\bf Subcase 3.1.} $\RR_{12}\equiv (\OO_{C_2}(a_2), \OO_{C_1\cup C_2}(a_1, a_2));$
{\bf Subcase 3.2.} $\RR_{12}\equiv ( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-1));$
{\bf Subcase 3.3.} $\RR_{12}\equiv ( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-2)).$
In Subcase 3.1, $\RR$ has a perfect factorization
$$(\OO_{C_2}(a_2), \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_3}(a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Applying Lemma \ref{lem GSH}, we get a new perfect factorization
$$(\OO_{C_2}(a_2), \OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Note that Hom's and $\chi$ between the first two factors are trivial, we get
$$
h^1( \OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3), \OO_{C_2}(a_2))=0,
$$
and we can exchange the first two factors to get a new perfect factorization
$$(\OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3), \OO_{C_2}(a_2), \OO_{C_3}(b_3)^{\oplus 2}).$$
Applying Lemma \ref{lem GSH}(1), we get a new perfect factorization
$$(\OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3), \OO_{C_3}(b_3), \OO_{C_2\cup C_3}(a_2+1, b_3)).$$
This gives (3-1) by changing $a_2$ appropriately.
In Subcase 3.2, $\RR$ has a perfect factorization
$$( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-1), \OO_{C_3}(a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Applying Lemma \ref{lem GSH}, we get a new perfect factorization
$$( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Note that Hom's and $\chi$ between the first two factors are trivial, we get
$$
h^1(\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_1\cup C_2}(a_1, a_2))=0,
$$
and we can exchange the first two factors to get a new perfect factorization
$$(\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_3}(b_3)^{\oplus 2}).$$
Applying Lemma \ref{lem GSH}(1), we get a new perfect factorization
$$(\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_3}(b_3), \OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, b_3)).$$
This gives (3-2).
In Subcase 3.3, $\RR$ has a perfect factorization
$$( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-2), \OO_{C_3}(a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Applying Lemma \ref{lem GSH}, we get a new perfect factorization
$$( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2\cup C_3}(a_2-1, a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Note that
$$h^1(\OO_{C_2\cup C_3}(a_2-1, a_3), \OO_{C_1\cup C_2}(a_1, a_2))=1$$
since
\begin{align*}
\chi ( \OO_{C_2\cup C_3}(a_2-1, a_3), \OO_{C_1\cup C_2}(a_1, a_2)){}&=0,\\
h^0 ( \OO_{C_2\cup C_3}(a_2-1, a_3), \OO_{C_1\cup C_2}(a_1, a_2)){}&=1,\\
h^0( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2\cup C_3}(a_2-1, a_3)){}&=0,
\end{align*}
and the unique non-trivial extension is $\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3)$, we get a new perfect factorization
$$( \OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3), \OO_{C_3}(b_3)^{\oplus 2}).$$
Note that $\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3)$ can be also viewed as the extension of $\OO_{C_2}(a_2-1)$ by $\OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, a_3-1)$.
Now if $a_3>b_3+1$, then $\RR$ has a new factorization
$$
( \OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, a_3-1), \OO_{C_2}(a_2-1), \OO_{C_3}(b_3)^{\oplus 2}),
$$
which is perfect by checking Hom's. Applying Lemma \ref{lem GSH}, we get a new perfect factorization
$$
( \OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, a_3-1), \OO_{C_3}(b_3), \OO_{C_2\cup C_3}(a_2, b_3)).
$$
This gives (3-3) by changing $a_2, a_3$ appropriately.
If $a_3=b_3+1$, then
$$
h^1( \OO_{C_3}(b_3),\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3))=0
$$
since
\begin{align*}
\chi ( \OO_{C_3}(b_3),\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3)){}&=0,\\
h^0 ( \OO_{C_3}(b_3),\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3)){}&=0,\\
h^0( \OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3), \OO_{C_3}(b_3)){}&=0.
\end{align*}
Hence
$$
\RR\cong \OO_{C_1\cup 2C_2 \cup C_3}(a_1+1, a_2-1, a_3)\oplus \OO_{C_3}(a_3-1)^{\oplus 2},
$$
which gives (3-4) by changing $a_1, a_2$ appropriately.
Finally we consider Case 4. Again we have 3 subcases:
{\bf Subcase 4.1.} $\RR_{12}\equiv (\OO_{C_2}(a_2), \OO_{C_1\cup C_2}(a_1, a_2));$
{\bf Subcase 4.2.} $\RR_{12}\equiv ( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-1));$
{\bf Subcase 4.3.} $\RR_{12}\equiv ( \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-2)).$
In Subcase 4.1, arguing as Subcase 3.1, we have
\begin{align*}
\RR
\equiv {}&(\OO_{C_2}(a_2), \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_3}(a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_2}(a_2), \OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3), \OO_{C_2}(a_2), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_1\cup C_2\cup C_3 }(a_1, a_2+1, a_3),\OO_{C_2\cup C_3}(a_2+1, b_3), \OO_{C_3}(c_3)).
\end{align*}
This gives (4-1) by changing $a_2$ appropriately.
In Subcase 4.2, arguing as Subcase 3.2, we have perfect factorizations
\begin{align*}
\RR
\equiv {}&(\OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-1), \OO_{C_3}(a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_2\cup C_3}(a_2, a_3), \OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_2\cup C_3}(a_2, a_3),\OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, b_3), \OO_{C_3}(c_3)).
\end{align*}
This gives (4-2).
In Subcase 4.3, arguing as Subcase 3.3, we have perfect factorizations
\begin{align*}
\RR
\equiv {}&(\OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2}(a_2-2), \OO_{C_3}(a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_1\cup C_2}(a_1, a_2), \OO_{C_2\cup C_3}(a_2-1, a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&(\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, a_3), \OO_{C_3}(b_3), \OO_{C_3}(c_3)).
\end{align*}
If $a_3>b_3+1$, then $a_3=b_3+2$ in this case since the extension of $\OO_{C_3}(b_3)$ by $\OO_{C_3}(a_3)$ is rigid (see Case 2 of proof of Proposition \ref{prop 12}). Arguing as Subcase 3.3, we have perfect factorizations
\begin{align*}
\RR\equiv {}&( \OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, a_3-1), \OO_{C_2}(a_2-1), \OO_{C_3}(b_3), \OO_{C_3}(c_3))\\
\equiv {}&( \OO_{C_1\cup C_2\cup C_3}(a_1, a_2+1, b_3+1), \OO_{C_2\cup C_3}(a_2, b_3), \OO_{C_3}(c_3)).
\end{align*}
This gives (4-3) by changing $a_2$ appropriately.
If $a_3=b_3+1$, then as Subcase 3.3, we have a perfect factorization
\begin{align*}
\RR\equiv {}&( \OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, b_3+1)\oplus \OO_{C_3}(b_3), \OO_{C_3}(c_3))
\end{align*}
since
$$
h^1( \OO_{C_3}(b_3),\OO_{C_1\cup 2C_2\cup C_3}(a_1+1, a_2-1, b_3+1))=0.
$$
This gives (4-4) by changing $a_1, a_2$ appropriately.
\end{proof}
We get the following corollary directly.
\begin{cor}\label{cor 123}
Let $C_1\cup C_2\cup C_3$ be a chain of three $(-2)$-curves and $\RR$ a torsion rigid sheaf with $c_1(\RR)=C_1+2C_2+3C_3$. Then one of the following holds
\begin{enumerate}
\item $\RR$ has a perfect factorization $(\GG, \LL)$
where $\LL$ is a line bundle supported on the chain $C_i\cup \cdots \cup C_3$ for some $1\leq i \leq 3$, or
\item
$\RR\cong \OO_{C_1\cup 2C_2 \cup C_3}(a_1, a_2, a_3)\oplus \OO_{C_3}(a_3-1)^{\oplus 2}$ for some integers $a_1, a_2, a_3$.
\end{enumerate}
\end{cor}
\begin{cor}\label{cor 12321}
Let $C_1\cup \cdots \cup C_5$ be a chain of five $(-2)$-curves and $\RR$ a torsion rigid sheaf with $c_1(\RR)=C_1+2C_2+3C_3+2C_4+C_5$. Then $\RR$ has a perfect factorization $(\GG, \LL)$
where $\LL$ is a line bundle supported on either the chain $C_i\cup \cdots \cup C_3$ for some $1\leq i \leq 5$, or the chain $C_1\cup \cdots \cup C_5$.
\end{cor}
\begin{proof}
Taking the restriction to $C_1\cup C_2\cup C_3$ and $C_3\cup C_4\cup C_5$, we have exact sequences
\begin{align*}
0\to \RR_{45}\to \RR \to \RR_{123}\to 0,\\
0\to \RR_{12}\to \RR \to \RR_{345}\to 0.
\end{align*}
Note that $c_1(\RR_{123})=C_1+2C_2+3C_3$ and $c_1(\RR_{345})=3C_3+2C_4+C_5$.
If one of $\RR_{123}$ and $\RR_{345}$ satisfies Corollary \ref{cor 123}(1), then we can get the desired perfect factorization.
Suppose that both $\RR_{123}$ and $\RR_{345}$ satisfy Corollary \ref{cor 123}(2), note that their restriction on $C_3$ are the same, for simplicity and without loss of generality, we may write
\begin{align*}
\RR_{123}{}&\cong \OO_{C_1\cup 2C_2 \cup C_3}\oplus \OO_{C_3}(-1)^{\oplus 2},\\
\RR_{345}{}&\cong \OO_{C_3\cup 2C_4 \cup C_5}\oplus \OO_{C_3}(-1)^{\oplus 2}.
\end{align*}
In this case we have an exact sequence
$$
0\to \OO_{C_1\cup 2C_2}(0, -1) \oplus\OO_{2C_4\cup C_5}(-1, 0)\to \RR\to\OO_{C_3}\oplus \OO_{C_3}(-1)^{\oplus 2}\to 0.
$$
This gives perfect factorizations
\begin{align*}
\RR\equiv{}& (\OO_{C_1\cup 2C_2}(0, -1) \oplus\OO_{2C_4\cup C_5}(-1, 0), \OO_{C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
\equiv{}& (\OO_{C_2} \oplus\OO_{C_4}, \OO_{C_1\cup C_2}(0, -1) \oplus\OO_{C_4\cup C_5}(-1, 0), \OO_{C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
\equiv{}& (\OO_{C_2} \oplus\OO_{C_4}, \OO_{C_1\cup C_2}(0, -1), \OO_{C_4\cup C_5}(-1, 0), \OO_{C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
\equiv{}& (\OO_{C_2} \oplus\OO_{C_4}, \OO_{C_1\cup C_2}(0, -1), \OO_{C_3\cup C_4\cup C_5}, \OO_{C_3}(-1)^{\oplus 2})\\
\equiv{}& (\OO_{C_2} \oplus\OO_{C_4}, \OO_{C_1\cup C_2\cup C_3\cup C_4\cup C_5}, \OO_{C_3}(-1)^{\oplus 2}).
\end{align*}
Here we apply Lemma \ref{lem GSH} in the last two steps. Note that
$$
h^1( \OO_{C_3}(-1), \OO_{C_1\cup C_2\cup C_3\cup C_4\cup C_5})=0
$$
by computing Hom's and $\chi$.
Hence we exchange the last two factors and get a perfect factorization
$$
\RR\equiv (\OO_{C_2} \oplus\OO_{C_4}, \OO_{C_3}(-1)^{\oplus 2}, \OO_{C_1\cup C_2\cup C_3\cup C_4\cup C_5}),
$$
and the proof is completed.
\end{proof}
\begin{cor}\label{cor 123321}
Let $C_1\cup \cdots \cup C_6$ be a chain of six $(-2)$-curves and $\RR$ a torsion rigid sheaf with $c_1(\RR)=C_1+2C_2+3C_3+3C_4+2C_5+C_6$. Then $\RR$ has a perfect factorization $(\GG, \LL)$
where $\LL$ is a line bundle supported on one of the following chains:
\begin{enumerate}
\item $C_i\cup \cdots \cup C_3$ for some $1\leq i \leq 3$;
\item $C_2\cup \cdots \cup C_j$ for some $4\leq j \leq 6$;
\item $C_3\cup C_4$.
\end{enumerate}
\end{cor}
\begin{proof}
Taking the restriction to $C_1\cup C_2\cup C_3$, we have an exact sequence
\begin{align*}
0\to \RR_{456}\to \RR \to \RR_{123}\to 0.
\end{align*}
Note that $c_1(\RR_{123})=C_1+2C_2+3C_3$.
If $\RR_{123}$ satisfies Corollary \ref{cor 123}(1), then we get the first case.
Now suppose that $\RR_{123}$ satisfies Corollary \ref{cor 123}(2). For simplicity and without loss of generality, we may assume that
$$
\RR_{123}\cong \OO_{C_1\cup 2C_2 \cup C_3}\oplus \OO_{C_3}(-1)^{\oplus 2},
$$
and we have a perfect factorization
$$
\RR\equiv (\RR_{456}, \OO_{C_1\cup C_2}(-1, 1),\OO_{C_2\cup C_3}, \OO_{C_3}(-1)^{\oplus 2}).
$$
On the other hand, $c_1(\RR_{456})=3C_4+2C_5+C_6$.
Suppose that $\RR_{456}$ has a perfect factorization $(\GG', \LL')$
where $\LL'$ is a line bundle supported on the chain $C_4\cup \cdots \cup C_j$ for some $4\leq j \leq 6$. For simplicity and without loss of generality, we may assume that $\LL'\cong \OO_{C_4\cup \cdots \cup C_j}$. Hence we have perfect factorizations
\begin{align*}
\RR{}& \equiv (\GG', \OO_{C_4\cup \cdots \cup C_j}, \OO_{C_1\cup C_2}(-1, 1),\OO_{C_2\cup C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
{}& \equiv (\GG', \OO_{C_1\cup C_2}(-1, 1),\OO_{C_4\cup \cdots \cup C_j},\OO_{C_2\cup C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
{}& \equiv (\GG', \OO_{C_1\cup C_2}(-1, 1),\OO_{C_2\cup C_3\cup C_4\cup \cdots \cup C_j}(0,0,1,\ldots), \OO_{C_3}(-1)^{\oplus 2})\\
{}& \equiv (\GG', \OO_{C_1\cup C_2}(-1, 1),\OO_{C_3}(-1)^{\oplus 2}, \OO_{C_2\cup C_3\cup C_4\cup \cdots \cup C_j}(0,0,1,\ldots)).
\end{align*}
We apply Lemma \ref{lem GSH} in the second step, and the last step is because
$$
h^1(\OO_{C_3}(-1), \OO_{C_2\cup C_3\cup C_4\cup \cdots \cup C_j}(0,0,1,\ldots))=0
$$
by computing Hom's and $\chi$. This gives the second case of this corollary.
Now suppose that $\RR_{456}$ satisfies Corollary \ref{cor 123}. For simplicity and without loss of generality, we may assume that
$$
\RR_{456}\cong \OO_{C_4\cup 2C_5 \cup C_6}\oplus \OO_{C_4}(-1)^{\oplus 2}.
$$
Then we have perfect factorizations
\begin{align*}
\RR{}& \equiv (\OO_{C_4\cup 2C_5 \cup C_6}, \OO_{C_4}(-1)^{\oplus 2}, \OO_{C_1\cup C_2}(-1, 1),\OO_{C_2\cup C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
{}& \equiv (\OO_{C_4\cup 2C_5 \cup C_6}, \OO_{C_1\cup C_2}(-1, 1),\OO_{C_4}(-1)^{\oplus 2}, \OO_{C_2\cup C_3}, \OO_{C_3}(-1)^{\oplus 2})\\
{}& \equiv (\OO_{C_4\cup 2C_5 \cup C_6}, \OO_{C_1\cup C_2}(-1, 1), \OO_{C_2\cup C_3\cup{C_4}},\OO_{C_4}(-1), \OO_{C_3}(-1)^{\oplus 2})\\
{}& \equiv (\OO_{C_4\cup 2C_5 \cup C_6}, \OO_{C_1\cup C_2}(-1, 1), \OO_{C_2\cup C_3\cup{C_4}},\OO_{C_3}(-1), \OO_{C_3\cup C_4}(-1, 0)).
\end{align*}
Here we apply Lemma \ref{lem GSH} in the last two steps. This gives the third case of this corollary.
\end{proof}
\section{Classification of torsion exceptional sheaves}
\noindent In this section, we prove Theorems \ref{main1} and \ref{main2}.
\begin{lem}\label{lemma main2}
Let $\EE$ be a torsion exceptional sheaf on a smooth projective surface $X$ satisfying conditions in Theorem \ref{main2}. Assume that there exists at least one $(-2)$-curve in $\supp(\EE)$. Then there exists a chain of $(-2)$-curves $Z$ in $\supp(\EE)$, and a line bundle $\LL$ on $Z$, such that $c_1(\EE)\cdot c_1(\LL)=-1$ and there is an exact sequence
$$
0\to \EE'\to \EE\to \LL\to 0
$$
with $h^0(\EE', \LL)=0$.
\end{lem}
\begin{proof}
By assumption,
we may write
$$\supp(\EE)=D\cup\bigcup_{j=1}^m\bigcup_{i=1}^{n_j}C^j_i, $$
where $C^j_1\cup \cdots \cup C^j_{n_j}$ is a chain of $(-2)$-curves for each $j$ and they are disjoint from each other. Since $\EE$ is exceptional, $\supp(\EE)$ is connected. Hence we assume that $D$ intersects with the chain $C^j_1\cup \cdots \cup C^j_{n_j}$ on the curve $C^j_{k_j}$ at one point for each $j$.
We may write
$$c_1(\EE)=D+\sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i C^j_i, $$
in the sense of Remark \ref{remark restriction} since the first Chern class of the restriction to every chain of $(-2)$-curves is uniquely determined.
Since $\EE$ is exceptional, by Riemann--Roch formula,
$c_1(\EE)^2=-\chi(\EE, \EE)=-1.$
On the other hand,
\begin{align*}
c_1(\EE)^2{}&=\Big(D+\sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i C^j_i\Big)^2\\
{}&=D^2+2D\cdot \sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i C^j_i+\Big(\sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i C^j_i\Big)^2\\
{}&=D^2+2D\cdot \sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i C^j_i+\sum_{j=1}^m\Big(\sum_{i=1}^{n_j}r^j_i C^j_i\Big)^2\\
{}&=-1+2\sum_{j=1}^mr^j_{k_j}+\sum_{j=1}^m\Big(-2\sum_{i=1}^{n_j}(r^j_i)^2+2\sum_{i=1}^{n_j-1}r^j_ir^j_{i+1}\Big).
\end{align*}
This implies that
$
\sum_{j=1}^m f(r^j_1, \ldots, r^j_{n_j}; k_j)=0,
$
where $f$ is the polynomial defined in Subsection \ref{polynomial}. By Proposition \ref{prop polynomial}, $f(r^j_1, \ldots, r^j_{n_j}; k_j)=0$ for each $j$ and $\{r^j_1, \ldots, r^j_{n_j}, k_j\}$ satisfies the conditions in Proposition \ref{prop polynomial}.
For convenience, we write $n_1=n$, $r_i^1=r_i$, $C^1_i=C_i$, $k_1=k$.
Note that $n\leq 6$ by assumption, and hence $r_k\leq 3$.
Reversing the order of $\{C_i\}$ if necessary, by the conditions in Proposition \ref{prop polynomial}, we only have the following 6 cases:\begin{enumerate}
\item $k=n=1$, $r_1=1$;
\item $k\geq 2$ and $r_1=r_2=1$;
\item $k=2$ and $r_1=1, r_2=2, r_3=1$;
\item $k\geq 3$ and $r_1=1, r_2=r_3=2$;
\item $k=3$, $n=5$ and $(r_1, \ldots, r_5)=(1,2,3,2,1)$;
\item $k=4$, $n=6$ and $(r_1, \ldots, r_6)=(1,2,3,3,2,1)$.
\end{enumerate}
In Case (1) and (2), taking the restriction to $C_1$,
we have an exact sequence
$$
0\to \EE'\to \EE \to \RR_1\to 0.
$$
Then $c_1(\RR_1)=C_1$ and hence $\RR_1$ is a line bundle on $C_1$. Moreover, $h^0(\EE', \RR_1)=0$ by construction, and
$$
c_1(\EE)\cdot c_1(\RR_1)=
\begin{cases}(C_1+D)\cdot C_1=-1 & \text{Case (1)};\\
(C_1+C_2)\cdot C_1=-1 & \text{Case (2)}.
\end{cases}
$$
Hence we may take $\LL=\RR_1$.
In Case (3) and (4), taking the restriction to $C_1\cup C_2$,
we have an exact sequence
$$
0\to \EE_{12}\to \EE \to \RR_{12}\to 0.
$$
Then $c_1(\RR_{12})=C_1+2C_2$. By Proposition \ref{prop 12},
there exists a line bundle $\LL$ supported on $C_2$ or the chain $C_1 \cup C_2$ with an exact sequence
$$
0\to \GG\to \RR_{12}\to \LL\to 0
$$
such that $h^0(\GG, \LL)=0$. Consider the exact sequence
$$
0\to \EE'\to \EE\to \LL\to 0
$$ given by the surjection $\EE\to \RR_{12}\to\LL$. Then $h^0(\EE', \LL)=0$ since $\EE'$ is an extension of $\GG$ by $\EE_{12}$ and $\supp(\EE_{12})$ does not contain $C_1$ or $C_2$. Note that $c_1(\LL)=C_2$ or $C_1+C_2$, we have
$$
c_1(\EE)\cdot c_1(\LL)=
\begin{cases}(C_1+2C_2+C_3+D)\cdot c_1(\LL)=-1 & \text{Case (3)};\\
(C_1+2C_2+2C_3)\cdot c_1(\LL)=-1 & \text{Case (4)}.
\end{cases}
$$
This $\LL$ satisfies all conditions we require.
In Case (5), taking the restriction to $C_1\cup \cdots \cup C_5$,
we have an exact sequence
$$
0\to \EE_{1}\to \EE \to \RR\to 0.
$$
Then $c_1(\RR)=C_1+2C_2+3C_3+2C_4+C_5$. By Corollary \ref{cor 12321},
$\RR$ has a perfect factorization $(\GG, \LL)$
where $\LL$ is a line bundle supported on either the chain $C_i\cup \cdots \cup C_3$ for some $1\leq i \leq 5$, or the chain $C_1\cup \cdots \cup C_5$.
This induces an exact sequence
$$
0\to \EE'\to \EE\to \LL\to 0
$$
where $\EE'$ is an extension of $\GG$ by $\EE_{1}$. In particular, we have $h^0(\EE', \LL)=0$. By construction, $c_1(\LL)=\sum_{j=i}^3C_j$ for some $1\leq i \leq 5$ or $\sum_{j=1}^5C_j$. Note that $D$ only intersects with $C_3$, it is easy to compute that
$$
c_1(\EE)\cdot c_1(\LL)=(C_1+2C_2+3C_3+2C_4+C_5+D)\cdot c_1(\LL)=-1.
$$
This $\LL$ satisfies all conditions we require.
In Case (6), taking the restriction to $C_1\cup \cdots \cup C_6$,
we have an exact sequence
$$
0\to \EE_{1}\to \EE \to \RR\to 0.
$$
Then $c_1(\RR)=C_1+2C_2+3C_3+3C_4+2C_5+C_6$. By Corollary \ref{cor 123321},
$\RR$ has a perfect factorization $(\GG, \LL)$
where $\LL$ is a line bundle supported on the chain \item $C_i\cup \cdots \cup C_3$ for some $1\leq i \leq 3$, or the chain $C_2\cup \cdots \cup C_j$ for some $4\leq j \leq 6$, or the chain $C_3\cup C_4$.
This induces an exact sequence
$$
0\to \EE'\to \EE\to \LL\to 0
$$
where $\EE'$ is an extension of $\GG$ by $\EE_{1}$. In particular, we have $h^0(\EE', \LL)=0$. By construction, $c_1(\LL)=\sum_{l=i}^3C_l$ for some $1\leq i \leq 3$, or $\sum_{l=2}^jC_l$ for some $4\leq j \leq 6$, or $C_3+C_4$. Note that $D$ only intersects with $C_4$, it is easy to compute that
$$
c_1(\EE)\cdot c_1(\LL)=(C_1+2C_2+3C_3+3C_4+2C_5+C_6+D)\cdot c_1(\LL)=-1.
$$
This $\LL$ satisfies all conditions we require.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main2}]
As in the proof of Lemma \ref{lemma main2}, we may write
$$c_1(\EE)=D+\sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i C^j_i, $$
where $C^j_1\cup \cdots \cup C^j_{n_j}$ is a chain of $(-2)$-curves for each $j$ and they are disjoint from each other.
Assume that there exists at least one $(-2)$-curve in $\supp(\EE)$, then by Lemma \ref{lemma main2} there exists a chain of $(-2)$-curves $Z$ in $\supp(\EE)$, and a line bundle $\LL$ on $Z$, such that $c_1(\EE)\cdot c_1(\LL)=-1$ and there is an exact sequence
$$
0\to \EE'\to \EE\to \LL\to 0
$$
with $h^0(\EE', \LL)=0$.
Note that $\LL$ is a spherical object, and
$$
\chi(\LL, \EE')=\chi(\LL, \EE)-\chi(\LL, \LL)=-c_1(\EE)\cdot c_1(\LL)-2=-1.
$$
By Lemma \ref{lem EES}, $\EE'$ is exceptional and $\EE\cong T_{\LL}\EE'$. Moreover, by the proof of Lemma \ref{lemma main2},
$$c_1(\EE')=c_1(\EE)-c_1(\LL)=D+\sum_{j=1}^m\sum_{i=1}^{n_j}(r^j_i)' C^j_i,, $$
where $(r^j_i)'=\begin{cases}
r^j_i-1 & \text{ if }C^j_i\subset \supp(\LL);\\
r^j_i & \text{ otherwise}.
\end{cases}$
By induction on the number $\sum_{j=1}^m\sum_{i=1}^{n_j}r^j_i$, after finitely many steps, we may assume that $c_1(\EE)=D$. This implies that $\EE$ is a line bundle on $D$ and the proof is completed.
\end{proof}
\begin{lem}\label{lem D in E}
Let $\EE$ be a torsion exceptional sheaf on a weak del Pezzo surface $X$, then there exists exactly one $(-1)$-curve $D$ in $\supp(\EE)$, and the restriction of $\EE$ in $D$ is a line bundle.
\end{lem}
\begin{proof}
Since $|-K_X|$ has no base component by Lemma \ref{lem no base}, choose a general element in $E\in |-K_X|$ which is not contained in $\supp(\EE)$. There is a short exact sequence
$$0\rightarrow \omega _X \rightarrow \OO_X\rightarrow \OO_E \rightarrow 0.$$
Tensoring with $\EE$, since $\EE$ is pure one-dimensional and $E\not \subset\supp(\EE)$, we get an exact sequence
\begin{align*}
0\rightarrow \EE\otimes \omega_X \rightarrow \EE \rightarrow \EE|_E\rightarrow 0.
\end{align*}
Applying $\Hom(\EE,-)$ to this sequence, we get an exact sequence
$$
\Hom(\EE,\EE\otimes \omega_X)\rightarrow \Hom(\EE,\EE)\rightarrow \Hom(\EE,\EE|_E)
\rightarrow \Ext^1(\EE,\EE\otimes \omega_X).
$$
Since $\EE$ is exceptional, $h^0(\EE,\EE)=1$ and $h^0(\EE,\EE\otimes \omega_X)=h^1(\EE,\EE\otimes \omega_X)=0$ by Serre duality. Hence
$
\Hom(\EE,\EE|_E)\cong\mathbb{C}.
$
By Lemma \ref{lem supp}, $\supp(\EE)$ only contains $(-1)$-curves and $(-2)$-curves.
Note that each $(-1)$-curve intersects with $E$ at one point and each $(-2)$-curve does not intersect with $E$,
we conclude that there is only one $(-1)$-curve $D$ in $\supp(\EE)$.
Moreover, taking restriction to $D$, we get an exact sequence
$$
0\to \EE'\to \EE\to\EE_D\to 0,
$$
where the support of $\EE'$ only contains $(-2)$-curves.
Combining with the fact that $
\Hom(\EE,\EE|_E)\cong\mathbb{C}
$, we have
$\Hom(\EE_D,\EE_D|_E)\cong\mathbb{C}.$
Since $\EE_D$ is pure one-dimensional, $\EE_D$ is a line bundle on $D$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main1}]
It suffices to check that any torsion exceptional sheaf $\EE$ on a weak del Pezzo surface $X$ of $d>2$ of Type A satisfies conditions (1)-(4) in Theorem \ref{main2}.
By Lemma \ref{lem supp}, any irreducible component of $\supp(\EE)$ is a curve with negative self-intersection, hence is a $(-1)$-curve or $(-2)$-curve. By Lemma \ref{lem D in E}, conditions (1)--(2) are satisfied.
By the assumption $d>2$ and Lemma \ref{lem no of -2}, there are at most $6$ $(-2)$-curves on $X$, hence condition (3) is satisfied since $X$ is of Type A. Again by the assumption $d>2$ and Lemma \ref{lem d>1}, condition (4) is satisfied.
\end{proof}
\section*{Acknowledgment}
\noindent The first author would like to express his deep gratitude to his supervisor Professor Yujiro Kawamata for discussion and warm encouragement. The authors are grateful to the anonymous refree(s) for valuable comments and suggestions.
\end{document} | math | 58,476 |
\begin{document}
\title{
space{-2.5cm}
\begin{abstract}
Variational weak-coupling perturbation theory yields
converging approximations, uniformly in the coupling strength. This allows us
to calculate directly the coefficients of {\em strong-coupling\/}
expansions. For the anharmonic oscillator we explain the physical origin
of the empirically observed convergence behavior which is exponentially
fast with superimposed oscillations.
\end{abstract}
\section{Introduction}
An important problem of perturbation theory is the calculation of
physically meaningful numbers from expansions which are
usually divergent asymptotic series with coefficients growing $\propto
k!$ in high orders $k$. For small expansion parameters $g$ a direct
evaluation of the series truncated at a finite order $k \approx 1/g$ can
yield a reasonably good approximation, but for larger couplings such
series become completely useless and require some kind of resummation.
Well-known examples are field theoretical $\epsilon$-expansions for
the computation of critical exponents of phase transitions, but also the
standard Stark and Zeeman effects in atomic physics lead to divergent
perturbation expansions.
The paradigm for studying this problem is the quantum
mechanical anharmonic oscillator with a potential
$
V(x) = \frac{1}{2} \omega^2 x^2 + \frac{1}{4} g x^4 \hspace{0.2cm}
( \omega^2,g>0).
$
The Rayleigh-Schr\"{o}dinger perturbation theory yields for the
ground-state energy a power-series expansion
\begin{equation}
E^{(0)}(g) = \omega \sum_{k=0}^{\infty} E^{(0)}_{k} \left(
\frac{g/4}{\omega^3} \right)^k,
\label{eq:E_exp}
\end{equation}
where the $E^{(0)}_k$ are rational numbers $1/2$, $3/4$,
$-21/8$, $333/16$, $-30885/128$, \dots ~, which can easily be
obtained to very high orders from the recursion relations of
Bender and Wu \cite{bewu}. Their large-order behavior is analytically
known to exhibit the typical factorial growth,
\begin{equation}
E^{(0)}_k = -(1/\pi) (6/\pi)^{1/2} (-3)^k k^{-1/2} k! (1 + {\cal O}(1/k)).
\label{eq:Ek_asy}
\end{equation}
Standard resummation methods are Pad\'e or Borel techniques whose accuracy,
however, decreases rapidly in the strong-coupling limit. In this note
we summarize recent work on a new approach based on variational perturbation
theory \cite{syko,PI}. Our results demonstrate that by this means the
divergent series expansion (\ref{eq:E_exp})
can be converted into a sequence of exponentially fast converging
approximations, uniformly in the coupling strength $g$
\cite{jk1,jk2,jk3,conv}. This allows us
to take all expressions directly to the strong-coupling limit, yielding
a simple scheme for calculating the coefficients $\alpha_i$ of the convergent
strong-coupling series expansion,
$E^{(0)}(g)= (g/4)^{1/3}\left[ \alpha _0
+ \alpha _1 (4 \omega^3/g)^{2/3}
+ \alpha _2 (4 \omega^3/g)^{4/3}
+\dots\right]$.
\section{Variational Perturbation Theory}
The origin of variational perturbation theory can be traced back to
a variational principle for the evaluation of quantum partition
functions in the path-integral formulation \cite{PI,variational}.
While in many applications
the accuracy was found to be excellent over a wide range of temperatures,
slight deviations from exact or simulation results at very low temperatures
motivated a systematic study of higher-order corrections \cite{syko,PI}.
In the zero-temperature limit the calculations simplify and lead to a
resummation scheme for the energy eigenvalues which can be summarized as
follows. First, the harmonic term of the potential is
split into a new harmonic term with a trial frequency $\Omega$ and a remainder,
$\omega^2 x^2 = \Omega^2 x^2 + \left(\omega^2-\Omega^2\right)x^2$, and
the potential is rewritten as
$
V(x) = \frac{1}{2} \Omega^2 x^2 + \frac{1}{4} g (-2 \sigma x^2/ \Omega + x^4),
$
where $\sigma = \Omega ( \Omega^2 - \omega^2)/g$.
One then performs a perturbation expansion in powers of $\hat{g} \equiv
g/\Omega^3$ at a fixed $\sigma$,
\begin{equation}
\hat{E}_{N}^{(0)}(\hat g,\sigma) = \sum_{k=0}^{N} \varepsilon^{(0)}_{k}(\sigma)
\left( \hat g/4 \right)^k,
\label{eq:E_reexp}
\end{equation}
where $\hat E_N^{(0)} \equiv E_N^{(0)}/\Omega$ is the dimensionless reduced
energy. The new expansion coefficients
$\varepsilon^{(0)}_{k}$ are easily found by inserting
$\omega = \sqrt{\Omega^2 -g \sigma/\Omega}
= \Omega \sqrt{1 - {\hat g}\sigma}$
in (\ref{eq:E_exp}) and reexpanding in powers of $\hat g$,
\begin{equation}
\varepsilon^{(0)}_{k}( \sigma ) = \sum_{j=0}^{k} E^{(0)}_{j}
\left( \begin{array}{c} (1 - 3 j)/2 \\ k-j \end{array} \right)
(-4 \sigma )^{k-j}.
\label{eq:eps_k}
\end{equation}
The truncated power series
$W_{N}(g,\Omega) \equiv \Omega \hat{E}^{(0)}_{N} \left(\hat{g},\sigma\right)$
is certainly independent of $\Omega$ in the limit $N \rightarrow \infty$.
At any finite order, however, it {\em does} depend on $\Omega$,
the approximation having its fastest speed of convergence where it depends
least on $\Omega$, i.e., at points where $\partial W_N/\partial \Omega = 0$.
If we denote the order-dependent optimal value of $\Omega$ by $\Omega_{N}$,
the quantity $W_{N}(g,\Omega_{N})$ is the new approximation to $E^{(0)}(g)$.
At first sight the extremization condition $\partial W_N/\partial \Omega = 0$
seems to require the determination of the roots of a polynomial in
$\Omega$ of degree $3N$, separately for each value of $g$. In Ref.~\cite{jk1}
we observed, however, that this task can be greatly simplified. While
$W_N$ does depend on both $g$ and $\Omega$ separately, we could prove
that the derivative can be written as $\partial W_N/\partial \Omega=
(\hat g/4)^N P_N(\sigma)$, where
$P_N(\sigma) = -2 d \varepsilon^{(0)}_{N+1}(\sigma)/d \sigma$ is a
polynomial of degree $N$ in $\sigma$. The optimal values
of $\sigma$ were found to be well fitted by
\begin{equation}
\sigma_N = cN \left( 1 + 6.85/N^{2/3}\right),
\label{eq:sigma_opt}
\end{equation}
with $c=0.186\,047\,272\dots$
determined analytically (cp. Sec.~3).
This observation simplifies the calculations
considerably and shows that the optimal solutions $\Omega_N$ depend
only trivially on $g$ through $\sigma_N = \Omega_N(\Omega_N^2 -
\omega^2)/g$. Since the explicit knowledge of $\Omega_N$ is only needed
in the final step when going back from ${\hat E}_N^{(0)}$ to $E_N^{(0)}$,
this suggests that the variational resummation scheme can be
taken directly to the strong-coupling limit.
To this end we introduce the reduced frequency
$\hat{\omega} = \omega/\Omega$, write the
approximation as
$W_N = \left( g/\hat{g} \right)^{1/3} w_N(\hat{g},\hat{\omega}^2)$,
and expand the function $w_N(\hat g,\hat \omega^2)$
in powers of $\hat{\omega}^2 = (\omega^3/g)^{2/3} \hat{g}^{2/3}$.
This gives \cite{jk2}
\begin{equation}
W_N = (g/4)^{1/3} \left[ \alpha_0
+ \alpha_1 \left(4\omega^3/g\right)^{2/3}
+ \alpha_2 \left(4\omega^3/g\right)^{4/3} +\dots \right],
\label{eq:W_N}
\end{equation}
with the coefficients,
\begin{equation}
\alpha_n = (\hat{g}/4)^{(2n-1)/3}
\sum_{k=0}^N (-1)^{k+n} \sum_{j=0}^{k-n} E_{j}^{(0)}
\left( \begin{array}{c} (1 - 3 j)/2 \\ k-j \end{array} \right)
\left( \begin{array}{c} k-j \\ n \end{array} \right)
(-\hat{g}/4)^j.
\label{eq:alpha_n}
\end{equation}
If this is evaluated at ${\hat g} = 1/\sigma_N$ with $\sigma_N$ given in
(\ref{eq:sigma_opt}), we obtain the exponentially fast approach to the
exact limit as shown in Fig.~\ref{fig:contour} for $\alpha_0$. The
exponential falloff is modulated by oscillations. Our result,
$\alpha_0 = 0.667\,986\,259\,155\,777\,108\,270\,96$,
agrees to all 23 digits with the most accurate 62-digit value in the
literature. The computation of the higher-order coefficients
$\alpha_n$ for $n>0$ proceeds similarly and the results up
to $n=22$ are given in
Table~1 of Ref.~\cite{jk2}.
\section{Convergence Behavior}
To explain the convergence behavior \cite{jk3,conv} we recall that
the ground-state energy $E^{(0)}(g)$ satisfies a subtracted dispersion
relation which leads to an integral representation of the original
perturbation coefficients,
\begin{equation}
E^{(0)}_k=\frac{4^k}{2{\pi}i}\int_0^{-\infty}
\frac{d g}{g^{k+1}}
\mbox{disc} E^{(0)}(g),
\label{eq:disc}
\end{equation}
where $ \mbox{disc}\,E^{(0)}(g) = 2i{\rm Im\,}E^{(0)}(g-i \eta )$
denotes the discontinuity across the left-hand cut
in the complex $g$-plane.
For large $k$, only its $g \longrightarrow 0^-$ behavior
is relevant and a semiclassical calculation
yields
$\mbox{disc}\,E^{(0)}(g) \! \approx \! -2i \omega (6/\pi)^{1/2}
(-4\omega^3\!/3g)^{1/2} \exp(4\omega^3\!/3g)$, which in turn implies the
large-order behavior (\ref{eq:Ek_asy}) of $E_k^{(0)}$.
The reexpanded series (\ref{eq:E_reexp}) is obtained
from (\ref{eq:E_exp}) by the replacement of
$\omega \longrightarrow \Omega \sqrt{1- \sigma \hat g}$.
In terms of the coupling constant, the above replacement amounts to
$\bar g \equiv g/ \omega ^3 \longrightarrow \hat g/(1- \sigma \hat
g)^{3/2}$. Using this mapping it is straightforward to show~\cite{jk3}
\begin{figure}\label{fig:contour}
\end{figure}
that ${\hat E}^{(0)} \equiv E^{(0)}/\Omega$ satisfies a dispersion
relation in the complex $\hat g$-plane.
If $C$ denotes the cuts in this plane
and $ \mbox{disc}_C {\hat E}^{(0)}(\hat g)$
is the discontinuity across these cuts,
the dispersion integral for the expansion coefficients
$\varepsilon_k^{(0)}$ reads
\begin{equation}
\varepsilon^{(0)} _k=\frac{4^k}{2{\pi}i}\int_C
\frac{d\hat g}{{\hat g^{k+1}}}
\mbox{disc}_C\hat E^{(0)}(\hat g).
\label{eq:eps_disc}
\end{equation}
In the complex $\hat g$-plane, the cuts $C$ run along the contours
$C_1,C_{\bar 1},C_2,C_{\bar 2}$, and $C_3$, as shown on the r.h.s. of
Fig.~\ref{fig:contour}. The first four cuts are the images of the
left-hand cut in the complex $g$-plane, and the curve $C_3$ is due to the
square root of $1- \sigma \hat g$ in the mapping from $\bar g$ to $\hat g$.
Let us now discuss the contributions of the various cuts to the $k$th
term $S_k$. For the cut $C_1$ and the empirically observed
optimal solutions
$\sigma_N = cN(1+b/N^{2/3})$, a saddle-point approximation
shows \cite{jk3}
that this term gives a convergent contribution,
$S_N(C_1)\propto e^{ -[ -b\log(-\gamma)+(cg)^{-2/3}] N^{1/3} }$,
only if one chooses $c=0.186\,047\,272\dots$ and
\mbox{$\gamma = -0.242\,964\,029\dots$}.
Inserting the fitted value of $b=6.85$ this yields an exponent
of
$-b\log(-\gamma) = 9.7$,
in rough agreement with the convergence seen in Fig.~\ref{fig:contour}.
If this was the only contribution the convergence behavior could be changed
at will by varying the parameter $b$. For $b < 6.85$, a slower convergence
was indeed observed. The convergence cannot be improved, however, by
choosing $b > 6.85$, since the optimal convergence is limited by the
contributions of the other cuts.
The cut $C_{\bar 1}$ is still harmless; it contributes a last term
$S_N(C_{\bar 1})$ of the negligible order $e^{-N\log N}$.
The cuts $C_{2,\bar 2,3}$, however, deserve a careful consideration.
If they would really start
at $\hat g=1/ \sigma $, the
leading behavior would be
$\varepsilon _k^{(0)}(C_{2,\bar 2,3}) \propto \sigma ^{k}$, and therefore
$S_N(C_{2,\bar 2,3}) \propto ( \sigma \hat g)^N$,
which
would be in contradiction to the empirically observed
convergence in the strong-coupling limit. The important point is
that
the cuts in Fig.~\ref{fig:contour}
do not really reach the point
$\sigma \hat g=1$.
There exists a small circle of radius $ \Delta \hat g>0$
in which $\hat E^{(0)}(\hat g)$ has no singularities at all,
a consequence of the fact that the strong-coupling expansion
(\ref{eq:W_N}) converges for $g>g_{\rm s}$.
The complex conjugate pair of singularities
gives a contribution,
\begin{equation}
S_N(C_{2,\bar 2,3})\approx
e^{-N^{1/3}a\cos \theta}\cos(N^{1/3}a\sin \theta),
\label{eq:S_N_2}
\end{equation}
with $a=1/(|\bar{g}_{\rm s}|c)^{2/3}$. By analyzing the convergence
behavior of the strong-coupling series we find
$|\bar g_{\rm s}| \! \approx \! 0.160$
and $\theta \! \approx \! -0.467$, which implies for the envelope an asymptotic
falloff of $e^{-9.23N^{1/3}}$, and furthermore also explains the
oscillations in the data~\cite{jk3}.
\section{Conclusions}
To summarize, we have shown how variational perturbation theory
can be used to convert the divergent weak-coupling perturbation series
of the anharmonic oscillator into a sequence of converging
approximations for the strong-coupling expansion.
By making use of dispersion relations
and identifying the relevant singularities
we are able to explain
the exponentially fast convergence with superimposed oscillations
in the strong-coupling limit.\\[-0.20cm]
W.J. thanks the Deutsche Forschungsgemeinschaft for a Heisenberg
fellowship.
\end{document} | math | 12,819 |
\begin{document}
\title{Birational geometry of defective varieties}
\author{Edoardo Ballico and Claudio Fontanari}
\date{}
\maketitle
\begin{small}
\begin{center}
\textbf{Abstract}
\end{center}
Here we investigate the birational geometry of projective varieties of
arbitrary dimension having defective higher secant varieties.
We apply the classical tool of tangential projections and we determine
natural conditions for uniruledness, rational connectivity, and
rationality.
\noindent AMS Subject Classification: 14N05.
\noindent Keywords: higher secant variety, tangential projection,
uniruledness, rational connectivity, rationality.
\end{small}
\section{Introduction}
We work over an algebraically closed field $\mathbb{K}$ with $\mathrm{char}(\mathbb{K})=0$.
Let $X \subset \mathbb{P}^r$ be an integral nondegenerate $n$-dimensional
variety. Recall that for every integer $k \ge 0$ the
$k$-\emph{secant variety} $S^k(X)$ is defined as the Zariski closure
of the set of the points in $\mathbb{P}^r$ lying in the span of $k+1$
independent points of $X$. An easy parameter count shows that the
expected dimension of $S^k(X)$ is exactly $\min \{r, n(k+1)+k \}$.
However, there are natural examples of projective varieties having
secant varieties of strictly lower dimension: for instance, the
first secant variety of the $2$-Veronese embedding of $\mathbb{P}^2$ in
$\mathbb{P}^5$ has dimension $4$ instead of $5$. More generally, one
defines the $k$-\emph{defect} $\delta_k(X) = \min \{r, n(k+1)+k \}
- \dim S^k(X)$ and says that $X$ is $k$-\emph{defective}
if $\delta_k(X) \ge 1$. It seems reasonable to regard defective
varieties as exceptional and try to classify them.
The first result in this direction, stated by Del Pezzo in 1887
and proved by Severi in 1901 (see \cite{DelPezzo:87}, \cite{Severi:01},
and also \cite{Dale:85} for a modern proof), characterizes the $2$-Veronese
of $\mathbb{P}^2$ as the unique $1$-defective surface which is not a cone. Along
the same lines, subsequent contributions by Palatini (\cite{Palatini:06}
and \cite{Palatini:09}), Scorza (\cite{Scorza:08} and \cite{Scorza:09}),
and Terracini (\cite{Terracini:11} and \cite{Terracini:21}) set up
the classification of $k$-defective surfaces and of $1$-defective
varieties in dimension up to four. This classical work has been
recently reconsidered and generalized by Chiantini and Ciliberto
in \cite{CC1} and \cite{CC2}.
It turns out that one of the main tools for understanding defective
varieties is provided by the technique of tangential projections.
Namely, assume that $X$ is not $(k-1)$-defective and let $p_1, \ldots,
p_k$ be general points on $X \subset \mathbb{P}^r$. The general
$k$-\emph{tangential projection} $\tau_{X,k}$ is the projection
of $X$ from the linear span of its tangent spaces at $p_1, \ldots,
p_k$. By the classical Lemma of Terracini (see \cite{Terracini:11}
and \cite{Dale:84} for a modern version), $X_k := \tau_{X,k}(X)$
is lower dimensional than $X$ if and only if $X$ is $k$-defective.
Therefore, the classification of defective varieties reduces to the
classification of varieties which drop dimension in the general
tangential projection. However, we believe that the only reasonable
goal in arbitrary dimension is to determine some geometrical properties
of defective varieties, and the present paper
is indeed a first attempt in this direction. In order to state our
main results, we recall from \cite{CC1} that the \emph{contact locus}
of a general hyperplane section $H$ tangent at $k+1$ general points
$p_1, \ldots, p_{k+1}$ of $X$ is the union $\Sigma$ of the irreducible
components of the singular locus of $H$ containing $p_1, \ldots, p_{k+1}$.
One defines $\nu_k(X) := \dim \Sigma$ and says that $X$ is $k$-\emph{weakly
defective} whenever $\nu_k(X) \ge 1$. The reason for this terminology
is simply that a $k$-defective variety is always weakly defective
(indeed, we are going to show in Proposition~\ref{defects} that
$\nu_k(X) \ge \delta_k(X)$), but the converse is not true (look at
cones). We point out the following:
\begin{Lemma}\label{birational}
Fix integers $k \ge 1$, $n \ge 2$, $r \ge (k+1)(n+1)$, and let
$X \subset \mathbb{P}^r$ be an integral nondegenerate $n$-dimensional variety.
If $X$ is not $k$-weakly defective, then $\tau_{X,k}$ is birational.
\end{Lemma}
Moreover, for every $k \ge 1$ and $n \ge 2$ we exhibit a projective
$n$-dimensional variety being $k$-weakly defective but not $k$-defective
such that $\tau_{X,k}$ is not birational (see Example~\ref{counter1}).
Next, as an application of Lemma~\ref{birational}, we investigate the
birational geometry of a $k$-defective variety of arbitrary dimension:
\begin{Theorem}\label{uniruled}
Fix integers $k \ge 1$, $n \ge 2$, $r \ge (k+1)(n+1)-1$, and let
$X \subset \mathbb{P}^r$ be an integral nondegenerate $n$-dimensional variety
which is $k$-defective but not $(k-1)$-defective.
Assume that $\nu_k(X) = \delta_k(X)$ and, for $k \ge 2$, that
$X$ is not $(k-1)$-weakly defective. Then $X$ is uniruled.
Assume moreover that the general contact locus of $X$ is irreducible.
Then $X$ is rationally connected and for $\nu_k(X) = \delta_k(X)=1$
it is rational.
\end{Theorem}
We stress that the hypotheses for uniruledness cannot be removed:
in Examples~\ref{counter2} and \ref{counter3} we collect a series
of non-uniruled defective varieties of any dimension which do
not satisfy exactly one of the listed assumptions. An analogous remark
applies to the second part of the statement, where irreducibility
turns out to be essential. For instance, if $X$ is a cone
over a curve of positive genus, then two general points on $X$ cannot be
joined by a rational curve. Indeed, we suspect that a defective variety
with $\nu_k = \delta_k$ and reducible general contact locus should be a
cone (this is certainly the case in dimension up to four by \cite{CC2},
proof of Proposition~4.2, and \cite{Scorza:09}, \S~14).
This research is part of the T.A.S.C.A. project of I.N.d.A.M.,
supported by P.A.T. (Trento) and M.I.U.R. (Italy).
\section{The proofs}
Let $k$ be the minimal integer such that $X$ is $k$-defective.
Then $\tau_{X,k-1}(X)$ is generically finite and by definition
we have
\begin{equation}\label{one}
\nu_k(X)=\nu_1(X_{k-1}).
\end{equation}
Moreover, by applying twice the equality $\delta_h(Y) = \dim Y
- \dim \tau_{Y,h}(Y)$, first with $Y = X$ and $h=k$, then with
$Y = X_{k-1}$ and $h=1$, we deduce
\begin{equation}\label{two}
\delta_k(X)=\delta_1(X_{k-1}).
\end{equation}
\begin{Proposition}~\label{defects}
Fix integers $k \ge 1$, $n \ge 2$, $r \ge (k+1)(n+1)-1$, and let
$X \subset \mathbb{P}^r$ be an integral nondegenerate $n$-dimensional variety.
Assume that $k$ is the minimal integer such that $X$ is $k$-defective.
Then we have $\nu_k(X) \ge \delta_k(X)$.
\end{Proposition}
\proof Assume first $k=1$. If $\delta_1(X)=1$, we are just claiming
that $\nu_1(X)>0$, which is well-known (see for instance \cite{CC1},
Theorem~1.1). If instead $\delta_1(X)\ge 2$, we take a general hyperplane
section $H$. By \cite{CC2}, Lemma~3.6, we have $\nu_1(H)=\nu_1(X)-1$
and $\delta_1(H)=\delta_1(X)-1$, so we conclude by induction.
Assume now $k > 1$. By (\ref{one}) and (\ref{two}), we are reduced to
the previous case, so the proof is over.
\qed
\emph{Proof of Lemma~\ref{birational}.} Recall that $\tau_{X,k}$ is
defined as the projection of $X$ from $<~T_{p_1}(X), \ldots, T_{p_k}(X)~>$,
where $p_i$ is a general
point on $X$ and $T_{p_i}(X)$ denotes the tangent space to $X$ at $p_i$.
Pick a general point $p_{k+1} \in X$ and let $q = \tau_{X,k}(p_{k+1})$.
By Proposition~\ref{defects}, $X$ is not $k$-defective, so $\tau_{X,k}$
is generically finite. Therefore $\tau_{X,k}^{-1}(q)= \{ p_{k+1}, \ldots,
p_{k+d} \}$ and we want to show that $d=1$.
Since $\tau_{X,k}(T_{P_{k+h}}(X)) = T_q(X_k)$
for every $h$ with $1 \le h \le d$, it follows that
$< T_{p_1}(X), \ldots, T_{p_k}(X), T_{P_{k+h}}(X) >$ does not
depend on $h$. On the other hand, by \cite{CC1}, Theorem~1.4,
the general hyperplane which is tangent to $X$ at $p_1, \ldots,
p_{k+1}$ is not tangent to $X$ at any other point, so we have
$d = h =1$ and the proof is over.
\qed
In order to construct nontrivial examples of projective varieties
whose general $k$-tangential projection is not birational, we are
going to apply the following criterion:
\begin{Lemma}~\label{criterion}
Fix integers $k \ge 1$, $n \ge 2$, $r \ge (k+1)(n+1)-1$, and let
$X \subset \mathbb{P}^r$ be an integral nondegenerate $n$-dimensional variety.
Assume that $X$ is $k$-weakly defective, but not $k$-defective.
If $X$ is not uniruled, then $\tau_{X,k}(X)$ is generically finite
but not birational.
\end{Lemma}
\proof Notice that $X$ is a fortiori not $(k-1)$-defective, so we can
apply Proposition~3.6 in \cite{CC1} with $h=k$ to obtain that $X_k$ is
$0$-defective. Hence $X_k$ is a developable scroll (see for instance
\cite{CC1}, Remark~3.1~(ii)), in particular it is uniruled and it
follows that $X_k$ is not birational to $X$.
\qed
\begin{Example}~\label{counter1} Let $k \ge 1$, $n \ge 2$, and
$r \ge (k+1)(n+1)-1$. Take a $(n-1)$-dimensional variety $C$ in
$\mathbb{P}^r$ which is not uniruled, a linear space $V \subset \mathbb{P}^r$
with $\dim V = k$, and a smooth hypersurface $H \subset \mathbb{P}^r$ of degree
$d \ge k+2$ such that $V \not\subset H$. Let $W$ be the cone over
$C$ with vertex $V$ and define $X := H \cap W$. By \cite{CC1},
Example~4.3, $X$ is $k$-weakly defective but not $k$-defective.
We claim that $X$ is not uniruled. Let $\pi: X \to C$ be the
projection and take a general point $p \in X$. If $R \subseteq X$
is a rational curve passing through $p$, then $R$ is contained in
a fiber of $\pi$, otherwise its projection would be a rational
curve through a general point of $C$, in contradiction with the
non-uniruledness of $C$. On the other hand, the general fiber of
$\pi$ is set-theoretically the intersection of $H$ with a
$(k+1)$-dimensional linear space, hence it is a smooth hypersurface
of high degree, which is not covered by rational curves. Hence the
claim is established and from Lemma~\ref{criterion} we deduce that
$\tau_{X,k}(X)$ is generically finite but not birational.
\end{Example}
\emph{Proof of Theorem~\ref{uniruled}.} Assume first $k = 1$.
If $\nu_1(X) = \delta_1(X) = 1$, from the so-called Terracini's
Theorem (\cite{CC1}, Theorem~1.1) it follows that the general contact locus
$\Sigma$ of $X$ imposes only three conditions on hyperplanes containing
it, in particular it is a plane curve. Moreover, if $\Sigma$ had degree
$d > 2$, then the general secant line to $X$ would be a multisecant,
which is a contradiction (the above argument is borrowed from
\cite{CC2}, proof of Proposition~4.2).
Hence $X$ is uniruled and if $\Sigma$ is irreducible then $X$ is
rationally connected. More precisely, if $d = 1$ then two
general points on $X$ would be joined by a straight line, a
contradiction since $X$ is nondegenerate. Therefore if we fix a
general $p \in X$ and for general $q \in X$ we assume that
the corresponding contact locus $\Sigma_{pq}$ is irreducible,
then $\Sigma_{pq}$ is a smooth conic and a natural birational map
from $X$ to its tangent space $T_p(X) \cong \mathbb{P}^n$ is defined by
sending $q$ to the intersection point between the tangent lines
to $\Sigma_{pq}$ at $p$ and $q$ (see \cite{Scorza:09}, \S~15).
If instead $\delta_1(X) \ge 2$, let $H$ be a general hyperplane
section. By \cite{CC2}, Lemma~3.6, we have $\nu_1(H)=\nu_1(X)-1$
and $\delta_1(H)=\delta_1(X)-1$, so uniruledness and rational
connectivity follow by induction. Assume now $k > 1$.
By (\ref{one}) and (\ref{two}), the previous cases apply to
$X_{k-1}$. On the other hand, from Lemma~\ref{birational} we
get a birational map between $X$ and $X_{k-1}$, so the proof is over.
\qed
\begin{Example}\label{counter2} Here we show that the assumption $X$
not $(k-1)$-weakly defective is essential for every $n \ge 2$.
Fix $r \ge 3n+2$ and let $C \subset \mathbb{P}^r$ be a $(n-1)$-dimensional
variety which is not uniruled. Take a line $L \subset \mathbb{P}^r$
and a smooth hypersurface $H \subset \mathbb{P}^r$ of degree $d \ge 3$
such that $L \not\subset H$. Let $W$ be the cone over
$C$ with vertex $V$ and define $X := H \cap W$. By \cite{CC1},
Example~4.3, we have $\delta_2(X) = \nu_2(X) = 1$ and $X$ is also
$1$-weakly defective. Moreover, arguing as in Example~\ref{counter1},
it is easy to check that $X$ is not uniruled.
\end{Example}
Finally we focus on the assumption $\nu_k(X) = \delta_k(X)$.
By Proposition~\ref{defects}, it is always satisfied in the
case of surfaces; however, in higher dimension it is no more
automatic.
\begin{Example}\label{counter3} As in \cite{CC2}, Example~2.2,
we consider a $n$-dimensional variety $X$ contained in a
$(n+1)$-dimensional cone $W$ over a curve $C$ in $\mathbb{P}^r$
with $r \ge 2n+1$. We have $\delta_1(X)=n-2$ and
$\nu_1(X)=n-1$; moreover, if we assume $g(C)\ge 1$ and we
take $X := H \cap W$, where $H$ is a general hypersurface
in $\mathbb{P}^r$ of high degree, then the same argument as in
Example~\ref{counter1} shows that $X$ is not uniruled.
\end{Example}
\noindent
Edoardo Ballico \newline
Universit\`a degli Studi di Trento \newline
Dipartimento di Matematica \newline
Via Sommarive 14 \newline
38050 Povo, Trento, Italy \newline
e-mail: [email protected]
\noindent
Claudio Fontanari \newline
Universit\`a degli Studi di Trento \newline
Dipartimento di Matematica \newline
Via Sommarive 14 \newline
38050 Povo, Trento, Italy \newline
e-mail: [email protected]
\end{document} | math | 13,746 |
\begin{document}
\begin{abstract}
This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$.
We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.
\end{abstract}
\maketitle
\section{Introduction}
\iffalse
The class of dynamical systems whose ergodic theory is best understood is the class of \emph{hyperbolic systems}. This is due to several reasons, one of them is the fact that these systems often have a compact symbolic model whose dynamics is well known \cite{bo, Ruellebook}. For real one-dimensional maps, (uniformly) hyperbolic maps are defined to be the class of maps where all points are either uniformly expanded or map into an attracting basin. This class is large even within families of maps with critical points such as the quadratic family, in which case it is a dense set, see \cite{Lyu, GraSw}. Note that these maps do have a compact symbolic model (see
\cite[Chapter 16]{kh}). In the example of the quadratic family, maps which are not uniformly hyperbolic are nowhere dense, but nevertheless have positive Lebesgue measure, see \cite{Jak, BenCar}. Due to the rich dynamics of these systems, the expansion properties of such systems, can be more delicate. These can be quantified by means of the Lyapunov exponent. For a $C^1$ interval map $f$, the \emph{lower/upper pointwise Lyapunov exponent}
at a point $x$ is defined as $$\underline\lambda_f(x):=\liminf_{n\to\infty} \frac{1}{n} \sum_{j=0}^{n-1} \log|Df(f^j(x))|, \text{ and } \overline\lambda_f(x):=\limsup_{n\to\infty} \frac{1}{n} \sum_{j=0}^{n-1} \log|Df(f^j(x))|$$ respectively. If $\underline\lambda_f(x)=\overline\lambda_f(x)$, then the \emph{Lyapunov exponent} of the map $f$ at $x$ is defined by
$\lambda_f(x) = \lambda(x)=\underline\lambda_f(x)=\overline\lambda_f(x)$.
It measures the exponential rate of divergence of infinitesimally close orbits. The Birkhoff ergodic theorem implies that if $\mu$ is an ergodic $f-$invariant measure, such that $\int \log |Df| \ d \mu$ is finite, then $ \lambda_f(x)$ is constant $\mu$-almost everywhere. We will denote such number by
\[\lambda(\mu):= \int \log |Df| \ d\mu. \]
\fi
The class of dynamical systems whose ergodic theory is best understood is the class of \emph{hyperbolic dynamical systems}, or, more generally, systems where the interesting dynamics behaves in a uniformly hyperbolic way: Axiom A maps. This is due to several reasons, one of them is the fact that these systems often have a compact symbolic model whose dynamics is well known \cite{bo, Ruellebook}. For real one-dimensional maps, Axiom A maps are defined to be the class of maps where all points are either uniformly expanded or map into an attracting basin. This class is large even within families of maps with critical points such as the quadratic family, in which case it is a dense set, see \cite{Lyu, GraSw}. Note that these maps do have a compact symbolic model (see
\cite[Chapter 16]{kh}). In the example of the quadratic family, maps which are not Axiom A are nowhere dense, but nevertheless have positive Lebesgue measure, see \cite{Jak, BenCar}. Due to the rich dynamics of these systems, the expansion properties of such systems, can be very delicate.
In recent years a great deal of attention has been paid to non-Axiom A systems which are expanding on most of the phase space, but not in all of it. The simplest example of these type of maps, namely \emph{non-uniformly hyperbolic} dynamical systems, are interval maps with a parabolic fixed point (e.g. the Manneville-Pomeau map \cite{pm}). The ergodic theory for these maps is fairly well understood \cite{pm, PreSl, Sarphase} and qualitatively different from the one observed in the hyperbolic case.
We will study the ergodic theory of class of maps for which the lack of hyperbolicity can be even stronger: interval maps with critical points. The techniques we develop are different from the ones used to study hyperbolic systems and systems with a parabolic fixed point.
In this paper we will be devoted to study a particular branch of ergodic theory, namely \emph{thermodynamic formalism}. This is a set of ideas and techniques which derive from statistical mechanics \cite{bo, Kellbook, Ruellebook,Waltbook}. It can be thought of as the study of certain procedures for the choice of invariant measures. Let us stress that the dynamical systems we will consider have many invariant measures, hence the problem is to choose relevant ones. The main object in the theory is the topological pressure:
\begin{defi}
Let $f$ be an endomorphism
of a compact metric space and denote by $\mathcal{M}_f$ the set of $f-$invariant probability measures. Let $\varphi: I \to [-\infty, \infty]$ be a \emph{potential}. Assuming that $\mathcal{M}_f\neq {\emptyset}$, the \emph{topological pressure} of $\varphi$ with respect to $f$ is defined, via the Variational Principle, by
\begin{equation*}
P_f(\varphi)=P(\varphi) = \sup \leqslantft\{ h(\mu) + \int \varphi \ d\mu : \mu \in \mathcal{M}_f \textrm{ and } - \int \varphi \ d\mu < \infty\right\},
\end{equation*}
where $ h(\mu)$ denotes the measure theoretic entropy of $f$ with respect to $\mu$. We refer to the quantity in the curly brackets as the \emph{free energy of $\mu$} with respect to $(I, f, \varphi)$. Note that this is sometimes thought of as being minus the free energy; see for example \cite{Kellbook} for a discussion of this terminology.
\end{defi}
Note that we do not specify the regularity properties we require on the potential $\varphi$. If it is a continuous function, then the above definition coincides with classical notions of topological pressure (see \cite[Chapter 9]{Waltbook}). In this paper we will be interested in the geometric potential $x\mapsto -t\log|Df(x)|$ for some parameter $t\in {\mathbb R}$. This function is continuous in the uniformly hyperbolic case, but is not upper/lower semicontinuous for $t$ positive/negative for the class of dynamical systems that we will consider.
A measure $\mu_{\varphi} \in \mathcal{M}_f$ is called an \emph{equilibrium state} for $\varphi$ if it satisfies:
\[ h(\mu_{\varphi}) + \int \varphi \ d\mu_{\varphi}= P(\varphi). \]
In such a way, the topological pressure provides a natural way to pick up measures. Questions about existence, uniqueness and ergodic properties of equilibrium states are at the core of the theory. For instance, if the dynamical system $f$ is transitive, uniformly hyperbolic and the potential $\varphi$ is
H\"older continuous then there exist a unique equilibrium state $\mu_{\varphi}$ for $\varphi$ and it has strong ergodic properties \cite{bo,Ruellebook}. Moreover, the hyperbolicity of the system is reflected on the regularity of the pressure function $t \mapsto P(t \varphi)$. Indeed, the function is real analytic. When the system is no longer hyperbolic, as in the case of the Manneville-Pomeau map, then uniqueness of equilibrium states may break down \cite{PreSl}
and the pressure function might exhibit points where it is not analytic, the so called \emph{phase transitions} \cite{Sarphase}.
As mentioned above, we will consider maps for which the lack of hyperbolicity is strong: not only do the maps have critical points, but the orbit of these points can be dense. We consider a family of real multimodal maps.
To be more precise the class of maps we will consider is defined as follows.
\iffalse Let $\mathcal F$ be the collection of $C^2$ multimodal interval maps $f:I \to I$, where $I=[0,1]$, satisfying:
\newcounter{Lcount}
\begin{list}{\alph{Lcount})}
{\usecounter{Lcount} \itemsep 1.0mm \topsep 0.0mm \leqslantftmargin=7mm}
\item the critical set ${\mathcal Cr} = {\mathcal Cr}(f)$ consists of finitely many critical points $c$ with critical order $1 < \ell_c < \infty$, i.e., $f(x) = f(c) + (g(x-c))^{\ell_c}$ for some diffeomorphisms $g:{\mathbb R} \to {\mathbb R}$ with $g(0) = 0$ and $x$ close to $c$;
\item $f$ has negative Schwarzian derivative, i.e., $1/\sqrt{|Df|}$ is convex; \textbf{GI: don't we need class $C^3$ for the Schwarzian derivative to be defined? maybe we can just ask for the convex part to be true. Maybe I already mentioned this.. MT: this convexity is enough for the distortion properties we require for the inducing schemes}
\item the non-wandering set ${\mathcal O}mega$ (the set of points $x\in I$ such that for arbitrarily small neighbourhoods $U$ of $x$ there exists $n(U)\geqslant 1$ such that $f^n(U)\cap U\neq {\emptyset}$) consists of a single interval;
\item $f^n({\mathcal Cr}) \neq f^m({\mathcal Cr})$ for $m \neq n$.
\end{list}
Conditions c) and d) are for ease of exposition, but not crucial. In particular, Condition c) excludes that $f$ is renormalisable. For multimodal maps satisfying a) and b), the set ${\mathcal O}mega$ consists of finitely many elements ${\mathcal O}mega_k$, on each of which $f$ is topologically transitive, see \cite[Section III.4]{MSbook}. In the case where there is more than one transitive element in ${\mathcal O}mega$, for example the renormalisable case, the analysis presented here can be applied to any one of the transitive elements consisting of a union of intervals. We also note that in this case ${\mathcal O}mega$ contains a (hyperbolic) Cantor set outside the elements of ${\mathcal O}mega$ which consist of intervals. The work of Dobbs \cite{Dobphase} shows that for renormalisable maps these hyperbolic Cantor sets can give rise to phase transitions in the pressure function not accounted for by the behaviour of critical points themselves.
Condition d) excludes that one critical point is mapped onto another. If that happened, it would be possible to consider these critical points as a `block', but to simplify the exposition, we will not do that here. Condition d) also excludes that critical points are preperiodic, a case which is easier to handle (for example by combining \cite[Chapter 16]{kh} and \cite{bo}) and does not require the theory we present here, see Section~\ref{sec:preper}.
\fi
We say that an interval map $f$ is $C^{1+}$ if it is $C^1$ and the derivative $Df$ is $\alpha$-H\"older for some $\alpha>0$. Let $\mathcal F$ be the collection of $C^{1+}$ multimodal interval maps $f:I \to I$, where $I=[0,1]$, satisfying:
\newcounter{Lcount}
\begin{list}{\alph{Lcount})}
{\usecounter{Lcount} \itemsep 1.0mm \topsep 0.0mm \leqslantftmargin=7mm}
\item the critical set ${\mathcal Cr} = {\mathcal Cr}(f)$ consists of finitely many critical points $c$ with critical order $1 < \ell_c < \infty$, i.e., there exists a neighbourhood $U_c$ of $c$ and a diffeomorphism $g_c:U_c \to g_c(U_c)$ with $g_c(c) = 0$
$f(x) = f(c) + g_c(x)^{\ell_c}$;
\item $f$ has negative Schwarzian derivative, i.e., $1/\sqrt{|Df|}$ is convex; \iffalse\textbf{GI: don't we need class $C^3$ for the Schwarzian derivative to be defined? maybe we can just ask for the convex part to be true. Maybe I already mentioned this.. MT: this convexity is enough for the distortion properties we require for the inducing schemes}
\item the non-wandering set ${\mathcal O}mega$ (the set of points $x\in I$ such that for arbitrarily small neighbourhoods $U$ of $x$ there exists $n(U)\geqslant 1$ such that $f^n(U)\cap U\neq {\emptyset}$) consists of a single interval;\fi
\item $f$ is topologically transitive on $I$;
\item $f^n({\mathcal Cr}) \cap f^m({\mathcal Cr})={\emptyset}$ for $m \neq n$.
\end{list}
Conditions c) and d) are for ease of exposition, but not crucial. In particular, Condition c) excludes that $f$ has any attracting cycles, or homtervals (a homterval is an interval $U$ such that $U, f(U), f^2(U), \ldots$ are disjoint and the omega limit set is not a periodic orbit).
Condition d) excludes that one critical point is mapped onto another. If that happened, it would be possible to consider these critical points as a `block', but to simplify the exposition, we will not do that here. Condition d) also excludes that critical points are preperiodic, a case which is easier to handle (for example by combining \cite[Chapter 16]{kh} and \cite{bo}) and does not require the theory we present here, see Section~\ref{sec:preper}. Together c) and d) exclude the renormalisable case.
\begin{rem}
Usually in ergodic theory for one-dimensional dynamics it is assumed that the map is $C^2$. A significant reason is that $C^2$ multimodal maps satisfying a) and b) have no homtervals and the non-wandering set ${\mathcal O}mega$ (the set of points $x\in I$ such that for arbitrarily small neighbourhoods $U$ of $x$ there exists $n(U)\geqslant 1$ such that $f^n(U)\cap U\neq {\emptyset}$) can be broken down into finitely many elements ${\mathcal O}mega_k$, on each of which $f$ is topologically transitive, see \cite[Section III.4]{MSbook}. However, for the maps we consider, assumption c) removes the necessity of a $C^2$ assumption.
We note that in the case where there is more than one transitive element in ${\mathcal O}mega$, for example the renormalisable case, the analysis presented in this paper can be applied to any one of the transitive elements consisting of a union of intervals permuted by $f$.
Now let ${\mathcal O}mega_{int}$ denote the union of all elements of ${\mathcal O}mega$ which consist of intervals permuted by $f$. If, contrary to the assumptions on $\mathcal{F}$ above, ${\mathcal O}mega_{int}$ did not cover $I$ then there would be a (hyperbolic) Cantor set consisting of points which are always outside ${\mathcal O}mega_{int}$. Dobbs \cite{Dobphase} showed that for renormalisable maps these hyperbolic Cantor sets can give rise to phase transitions in the pressure function not accounted for by the behaviour of critical points themselves.
\label{rmk:wandering set}
\end{rem}
\begin{rem}
The smoothness of our maps is important for two further reasons: to allow us to bound distortion on iterates, and to guarantee the existence of `local unstable manifolds'. For the first, the tool we use is the Koebe Lemma, see \cite[Section IV]{MSbook}. The negative Schwarzian condition we impose still allows us to use this for $C^{1+}$ maps. For a detailed explanation of this issue see \cite{Cedthesis}.
Given a measure $\mu\in \mathcal{M}_f$, the existence of local unstable manifolds was used in \cite{BrCMP, BTeqgen} to show the existence of some natural `inducing schemes' (see Section~\ref{sec:ind schemes}). As shown by Ledrappier \cite{Led}, and later generalised by Dobbs \cite{Dobcusp} (see the appendix), we only need a $C^+$ condition on $f$ to guarantee the existence of local unstable manifolds.
\label{rmk:c2 vs c1}
\end{rem}
Note that our class $\mathcal F$ includes transitive Collet-Eckmann maps, that is maps where $|Df^n(f(c))|$ grows exponentially fast. Therefore the set of quadratic maps in $\mathcal F$ has positive Lebesgue measure in the parameter space of quadratic maps (see \cite{Jak, BenCar}). In the appendix we show that our theory can be extended to a slightly more general class of maps, similar to the above, but only piecewise continuous.
As mentioned above, we will be particularly interested in the thermodynamic formalism for the geometric potentials $x\mapsto -t\log|Df|$. The study of these potentials has various motivations, for example the relevant equilibrium states and the pressure function are related to the Lyapunov spectrum, see for example \cite{T}.
Moreover, important geometric features are captured by this potential. Indeed, in several settings, the equilibrium states for this family are associated to conformal measures on the interval. This allows the study of the fractal geometry of dynamically relevant subsets of the space. Moreover, by \cite{Led} any equilibrium state $\mu$ for the potential $x\mapsto -\log|Df|$ is an absolutely continuous invariant probability measure (acip) provided $\lambda(\mu)>0$.
For $\mu\in \mathcal{M}_f$, we define the \emph{Lyapunov exponent of $\mu$} as
\[\lambda(\mu):= \int \log |Df| \ d\mu. \]
We let
$$\lambda_M:=\sup\{\lambda(\mu):\mu\in \mathcal{M}_f\}, \ \lambda_m:=\inf\{\lambda(\mu):\mu\in \mathcal{M}_f\}.$$
\begin{rem}
Our assumptions on $f\in \mathcal{F}$, particularly non-flatness of critical points and a lack of attracting periodic cycles, means that by \cite{Prz}, $\lambda_m \geqslant 0$.
\label{rmk:prz}
\end{rem}
We let
$$p(t):=P(-t\log|Df|)$$
and define
\begin{equation} \label{eq:t plus minus}
t^-:=\inf\{t:p(t)>-\lambda_M t\} \text{ and }
t^+:=\sup\{t:p(t)>-\lambda_m t\}.
\end{equation}
Note that if $t^-\in {\mathbb R}$ (resp $t^+\in {\mathbb R}$) then $p$ is linear for all $t\leqslant t^-$ (resp $t\geqslant t^+$).
We will later prove that for maps in $\mathcal{F}$, $t^-=-\infty$.
We prove in Proposition~\ref{prop:regular} that $t^+>0$.
In some cases $t^+=\infty$. As we will show later, for non-Collet Eckmann maps with quadratic critical point, $\lambda_m=0$ and $t^+=1$. \cite{MakSm} suggests that there should also be Collet-Eckmann maps with $t^+\in (1,\infty)$.
In Proposition \ref{prop:pos press} we prove that under certain assumptions $t^+ \geqslant 1$: we expect that to be true for any map $f \in \mathcal{F}$.
The following is our main theorem.
\begin{maintheorem} \label{thm:eq_exist_unique}
For $f\in \mathcal{F}$ and $t \in (-\infty , t^+)$ there exists a unique equilibrium measure $\mu_t$ for the potential $-t \log |Df|$. Moreover, the measure $\mu_t$ has positive entropy.
\end{maintheorem}
A classical way to show the existence of equilibrium states is to use upper semicontinuity of entropy and the potential $\varphi$ (see \cite[Chapter 4]{Kellbook}), and in particular the upper semicontinuity of $\mu\mapsto \int\varphi~d\mu$.
However, in our setting even though, as noted in \cite{BrKell}, for $f\in \mathcal{F}$ the entropy map is upper semicontinuous, the existence of equilibrium measures in the above theorem is not guaranteed since the potential $-t \log |Df|$ is not upper semicontinuous for $t>0$. So for example, by \cite[Proposition 2.8]{BrKell} for unimodal maps satisfying the Collet-Eckmann condition, $\mu \mapsto -\lambda(\mu)$ is not upper semicontinuous. Theorem~\ref{thm:eq_exist_unique} generalises \cite{BrKell} which applies to unimodal Collet-Eckmann maps for a small range of $t$ near 1; \cite{PeSe} which applies to a subset of Collet-Eckmann maps, but for all $t$ in a neighbourhood of $[0,1]$; and \cite[Theorem 1]{BTeqnat} which applies to a class of non-Collet Eckmann multimodal maps with $t$ in a left-sided neighbourhood of 1.
In order to prove Theorem~\ref{thm:eq_exist_unique} we use the theory of inducing schemes developed in \cite{BrCMP, BTeqgen, BTeqnat, T}. Let us note that the thermodynamic formalism is understood for certain complex rational maps. For example, Przytycki and Rivera-Letelier \cite{PrzRL} proved that if $f: \overline{\mathbb{C}} \to \overline{\mathbb{C}}$ is a rational map of degree at least two, is expanding away from the critical points and has `arbitrarily small nice couples' then the pressure function $p$ is real analytic in a certain interval. These conditions are met for a wide class of rational maps including topological Collet-Eckmann rational maps, any at most finitely renormalisable polynomial with no indifferent periodic orbits, as well as every real quadratic polynomial.
Related to the above are the regularity properties of the pressure function.
\begin{defi} \label{def:kink}
Let $\varphi :[0,1] \to {\mathbb R}$ be a potential. The pressure function has a \emph{first order phase transition} at $t_0 \in {\mathbb R}$ if $p$ is not differentiable at $t=t_0$.
\end{defi}
The pressure function is continuous and convex (see \cite[Theorem 9.7]{Waltbook}), which implies that the left and right derivatives $D^-p(t)$ and $D^+p(t)$ at each $t$ exist. Moreover, the pressure, when finite, can have at most a countable number of points $t_i$ where it is not differentiable (i.e, $Dp^-(t_i)\neq D^+p(t_i)$), hence of first order phase transitions. The regularity of the pressure is related to several dynamical properties of the system. For example, it has deep connections to large deviations \cite{el} and to different modes of recurrence \cite{Sarphase, Sarcrit}. In Section~\ref{sec:smooth and convex} we prove that the pressure function restricted to the interval $(-\infty, t^+)$ not only does not have first order phase transitions, but it is $C^1$.
\begin{maintheorem}
For $f\in \mathcal{F}$, the pressure function $p$ is $C^1$, strictly convex and strictly decreasing in $t \in (-\infty , t^+)$.
\label{thm:smooth}
\end{maintheorem}
First order phase transitions are also related to the existence of absolutely continuous invariant probability measures. If $p(t)=0$ for all $t\geqslant 1$ and there is an acip, then the pressure function is not differentiable at $t=1$. This occurs for example if $f\in \mathcal{F}$ is unimodal and non-Collet Eckmann, but has an acip (see \cite{NoSa}).
The following proposition gives the converse result.
\begin{prop} Let $f\in \mathcal{F}$ be such that $p(1)=0.$
If the pressure function has a first order phase transition at $t=1$ then the map $f$ has an acip. \label{prop:acip}
\end{prop}
We summarise some of the other results we present here for the potential $x\mapsto -t\log|Df(x)|$ in the simpler case of unimodal maps with quadratic critical point in the following proposition.
\begin{prop}
If $f\in \mathcal{F}$ is unimodal, non-Collet Eckmann and $\ell_c=2$ then $p$ is $C^1$, strictly convex and decreasing throughout $(-\infty, 1)$ and $p(t)=0$ for all $t\geqslant 1$. Moreover,
\begin{list}{$\bullet$}{\itemsep 0.2mm \topsep 0.2mm \itemindent -0mm \leqslantftmargin=5mm}
\item[(a)] if $f$ has no acip then $p$ is $C^1$ throughout ${\mathbb R}$;
\item[(b)] if $f$ has an acip then $p$ has a first order phase transition at $t=1$.
\end{list}
\label{prop:collected results}
\end{prop}
The paper is organised as follows. In Section~\ref{sec:CMS} we give an introduction to the theory of thermodynamic formalism for countable Markov shifts, which is primarily due to Sarig. In Section~\ref{sec:ind schemes} we give some preliminary results on inducing schemes, which will allow us to code any of our systems by a countable Markov shift. In Section~\ref{sec:zero press} we show that the inducing schemes in Section~\ref{sec:ind schemes} have some of the properties which will allow us to produce equilibrium states for our systems. In Section~\ref{sec:Gibbs integ} we prove the most technically complex part of our paper which gives us the existence of equilibrium states for our systems. Section~\ref{sec:unique} gives details of the uniqueness of these equilibrium states which then allows us to prove Theorem~\ref{thm:eq_exist_unique} in Section~\ref{sec:main thm}. In Section~\ref{sec:smooth and convex} we prove Theorem~\ref{thm:smooth} and in Section~\ref{sec:kinks acips} we prove Propositions~\ref{prop:acip} and \ref{prop:collected results}. In Section~\ref{sec:remarks} we discuss statistical properties of the measures constructed, the ergodic optimisation problem and the case in which the critical points are preperiodic. Finally in the appendix we show how the results of this paper extend to a class of Lorenz-like maps, of the kind studied by Rovella \cite{Rovella} and Keller and St Pierre \cite{KellStP}.
Note that many of the results we quote in this paper are proved using the theory of Markov extensions introduced by Hofbauer. To prove our main theorems it is not necessary to explain this theory in any detail since it is sufficient to quote results from elsewhere. However, for a short description of this construction, see the appendix.
\emph{Acknowledgements:}
We would like to thank N.\ Dobbs for his comments on earlier versions of this paper which improved both the results and the exposition. We would also like to thank H.\ Bruin and J.\ Rivera-Letelier for their useful remarks. MT would also like to thank the mathematics department at PUC, where some of this work was done, for their hospitality.
\section{Preliminaries: countable Markov shifts}
\label{sec:CMS}
In this section we present the theory of countable Markov shifts: an extension of the finite case, and the relevant model for many non-uniformly hyperbolic systems, including maps in $\mathcal{F}$.
Let $\sigma \colon {\mathcal S}igma \to {\mathcal S}igma$ be a one-sided Markov shift
with a countable alphabet $S$. That is, there exists a matrix
$(t_{ij})_{S \times S}$ of zeros and ones (with no row and no column
made entirely of zeros) such that
\[
{\mathcal S}igma=\{ x\in S^{{\mathbb N}_0} : t_{x_{i} x_{i+1}}=1 \ \text{for every $i
\in {\mathbb N}_0$}\},
\]
and the shift map is defined by $\sigma(x_0x_1 \cdots)=(x_1 x_2
\cdots)$. We say that $({\mathcal S}igma,\sigma)$ is a \emph{countable
Markov shift}.
We equip ${\mathcal S}igma$ with the topology generated by the cylinder sets
\[ C_{i_0 \cdots i_n}= \{x \in
{\mathcal S}igma : x_j=i_j \text{ for } 0 \leqslant j \leqslant n \}.\] Given a function
$\varphi\colon {\mathcal S}igma \to{\mathbb R}$, for each $n \geqslant 1$ we set
\[
V_{n}(\varphi) = \sup \leqslantft\{|\varphi(x)-\varphi(y)| : x,y \in {\mathcal S}igma,\
x_{i}=y_{i} \text{ for } 0 \leqslant i \leqslant n-1 \right\}.
\]
We say that $\varphi$ has \emph{summable variations} if
$\sum_{n=2}^{\infty} V_n(\varphi)<\infty$. We will sometimes refer to $\sum_{n=2}^{\infty} V_n(\varphi)$ as the \emph{distortion bound} for $\varphi$. Clearly, if $\varphi$ has
summable variations then it is continuous. We say that $\varphi$ is \emph{weakly H\"older continuous} if $V_n(\varphi)$ decays exponentially. If this is the case then it has summable variations. In what follows we assume
$({\mathcal S}igma, \sigma)$ to be topologically mixing (see \cite[Section 2]{Sartherm} for a precise definition).
It is a subtle matter to define a notion of topological pressure for countable Markov shifts. Indeed, the classical definition for continuous maps on compact metric spaces is based on the notion of
$(n,\varepsilon)$-separated sets (see \cite[Chapter 9]{Waltbook}). This notion depends upon the metric of the space. In the compact setting, since all metrics generating the same topology are equivalent, the value of the pressure does not depend upon the metric. However, in non-compact settings this is no longer the case. Based on work of Gurevich \cite{Gutopent, Gushiftent}, Sarig \cite{Sartherm} introduced a notion of pressure for countable Markov shifts which does not depend upon the metric of the space and which satisfies a Variational Principle.
Let $({\mathcal S}igma, \sigma)$ be a topologically mixing countable Markov shift, fix a symbol $i_0$ in the alphabet $S$ and
let $\varphi \colon {\mathcal S}igma \to {\mathbb R}$ be a potential of summable variations. We let
\begin{equation}
Z_n(\varphi, C_{i_0}):=\sum_{x:\sigma^{n}x=x} \exp \leqslantft(S_n\varphi(x)\right) \chi_{C_{i_{0}}}(x)
\label{eq:Zn}
\end{equation}
where $\chi_{C_{i_{0}}}$ is the characteristic function of the
cylinder $C_{i_{0}} \subset {\mathcal S}igma$, and
$$S_n\varphi(x):=\varphi(x)+\cdots +\varphi\circ\sigma^{n-1}(x).$$
Moreover, the so-called \emph{Gurevich pressure} of $\varphi$ is defined by
\[
P^G(\varphi) := \lim_{n \to
\infty} \frac{1}{n} \log Z_n(\varphi, C_{i_0}).
\]
Since $\sigma$ is topologically mixing, one can show that $P^G(\varphi)$ does not depend on $i_0$. We define
$$\mathcal{M}_\sigma(\varphi):=\leqslantft\{\mu\in \mathcal{M}_\sigma:-\int\varphi~d\mu<\infty\right\}.$$
If $({\mathcal S}igma, \sigma)$ is the full-shift on a countable alphabet then the Gurevich pressure coincides with the notion of pressure introduced by Mauldin and Urba\'nski \cite{MUifs}.
Furthermore, the following property holds (see \cite[Theorem 3]{Sartherm}):
\begin{prop}[Variational Principle] \label{prop:VarPri}
If $\varphi: {\mathcal S}igma \to \mathbb{R}$ has summable variations and $P^G(\varphi)<\infty$ then
\begin{equation*}
P^G(\varphi)= \sup \leqslantft\{ h_{\mu}(\sigma) +\int_{\mathcal S}igma \varphi \, d
\mu : \mu\in \mathcal{M}_\sigma (\varphi)\right\}.
\end{equation*}
\end{prop}
Let us stress that the right hand side of the above inequality only depends on the Borel structure of the space and not on the metric. Therefore, a notion of pressure which is to satisfy the Variational Principle need not depend upon the metric of the space.
The Gurevich pressure also has the property that it can be approximated by its restriction to compact sets. More precisely
\cite[Corollary 1]{Sartherm}:
\begin{prop}[Approximation property] \label{prop:approx}
If $\varphi: {\mathcal S}igma \to \mathbb{R}$ has summable variations then
\begin{equation*}
P^G( \varphi) = \sup \{ P_{\sigma|K}( \varphi) : K \subset {\mathcal S}igma : K \ne \emptyset \text{ compact and } \sigma\text{-invariant} \},
\end{equation*}
where $P_{\sigma|K}( \varphi)$ is the classical topological pressure on
$K$.
\end{prop}
We consider a special class of invariant measures. We say that $\mu\in \mathcal{M}_\sigma$ is a \emph{Gibbs measure} for the
function $\varphi \colon {\mathcal S}igma \to {\mathbb R}$ if for some constants $P$,
$C>0$ and every $n\in {\mathbb N}$ and $x\in C_{i_0 \cdots i_n}$ we have
\begin{equation*}
\frac{1}C \leqslant \frac{\mu(C_{i_0\cdots i_n})}{\exp \leqslantft(-nP + S_n\varphi(x)\right)} \leqslant C.
\end{equation*}
We refer to any such $C$ as a \emph{distortion constant} for the Gibbs measure.
It was proved by Mauldin and Urba\'nski \cite{muGIBBS} and by Sarig in \cite{SarBIP} that if $({\mathcal S}igma, \sigma)$ is a full-shift and the
function $\varphi$ is of summable variations with finite Gurevich pressure $P^G(\varphi)$ then it has an invariant Gibbs measure. Moreover $P=P^G(\varphi)$, and if $-\int\varphi~d\mu<\infty$ then $\mu$ is an equilibrium state for $\varphi$. Furthermore, this is the unique equilibrium state for $\varphi$ by \cite[Theorem 3.5]{muGIBBS} and \cite{BuSar}.
\section{Inducing schemes}
\label{sec:ind schemes}
In order to prove Theorem~\ref{thm:eq_exist_unique} we will use the machinery of inducing schemes.
We will use the fact that inducing schemes for the system $(I,f)$ can be coded by the full-shift on countably many symbols.
Given $f\in \mathcal{F}$, we say that $(X,F,\tau)$ is an \emph{inducing scheme} for $(I,f)$ if
\begin{list}{$\bullet$}{\itemsep 0.2mm \topsep 0.2mm \itemindent -0mm \leqslantftmargin=5mm}
\item $X$ is an interval containing a finite or countable
collection of disjoint intervals $X_i$ such that $F$ maps each $X_i$
diffeomorphically onto $X$, with bounded distortion on all iterates (i.e. there
exists $K>0$ so that if there exist $i_0, \ldots, i_{n-1}$ and $x,y$ such that $F^j(x), F^j(y)\in X_{i_j}$ for $j=0, 1, \ldots, n-1$ then $1/K\leqslant DF^n(x)/DF^n(y) \leqslant K$);
\item $\tau|_{X_i} = \tau_i$ for some $\tau_i \in {\mathbb N}$ and $F|_{X_i} = f^{\tau_i}$. If $x \notin \cup_iX_i$ then $\tau(x)=\infty$.
\end{list}
The function $\tau:\cup_i X_i \to {\mathbb N}$ is called the {\em inducing time}. It may
happen that $\tau(x)$ is the first return time of $x$ to $X$, but
that is certainly not the general case. For ease of notation, we will frequently write $(X,F)=(X,F,\tau)$. We denote the set of points $x\in I$ for which there exists $k\in {\mathbb N}$ such that $\tau(F^n(f^k(x)))<\infty$ for all $n\in {\mathbb N}$ by $(X,F)^\infty$.
Given an inducing scheme $(X,F, \tau)$, we say that a probability measure $\mu_F$ is a \emph{lift} of $\mu$ if for any $\mu$-measurable subset $A\subset I$,
\begin{equation} \mu(A) = \frac1{\int_X \tau \ d\mu_F} \sum_i \sum_{k = 0}^{\tau_i-1} \mu_F( X_i \cap f^{-k}(A)). \label{eq:lift}
\end{equation}
Conversely, given a measure $\mu_F$ for $(X,F)$, we say that
$\mu_F$ \emph{projects} to $\mu$ if \eqref{eq:lift} holds. Note that if \eqref{eq:lift} holds then $\mu_F$ is $F$-invariant if and only if $\mu$ is $f$-invariant. We call a measure
$\mu$ \emph{compatible with} the inducing scheme $(X,F,\tau)$ if
\begin{list}{$\bullet$}{\itemsep 1.0mm \topsep 0.0mm \leqslantftmargin=5mm}
\item $\mu(X)> 0$ and $\mu\leqslantft(X \setminus (X,F)^\infty\right) = 0$; and
\item there exists a measure $\mu_F$ which projects to $\mu$ by
\eqref{eq:lift}: in particular $\int_X \tau \ d\mu_F <
\infty$ (equivalently $\mu_F\in \mathcal{M}_F(-\tau)$).
\end{list}
\begin{rem}
Given an ergodic measure $\mu\in \mathcal{M}_f$ with positive Lyapunov exponent there exists an inducing scheme $(X, F, \tau)$ with a corresponding $F-$invariant measure $\mu_F$, see for example \cite[Theorem 3]{BTeqnat}.
\end{rem}
\begin{defi}
Let $(X, F, \tau)$ be an inducing scheme for the map $f$. Then for a potential $\varphi:I\to {\mathbb R}$, the induced potential ${\mathcal P}hi$ for $(X,F, \tau)$ is given by $${\mathcal P}hi(x)={\mathcal P}hi^F(x):=S_{\tau(x)}\varphi(x).$$
\end{defi}
Note that in particular for the potential $\log|Df|$, the induced potential for a scheme $(X,F)$ is $\log|DF|$. Moreover, the map $x\mapsto \log|DF(x)|$ has summable variations (see for example \cite[Lemma 8]{BTeqnat}).
\begin{rem} \label{rmk:conj to shift}
Let $(X, F, \tau)$ be some inducing scheme for the map $f$. We suppose that $\partial X\notin (X,F)^\infty$. Then the system $F:(X,F)^\infty \to (X,F)^\infty$ is topologically conjugated to the full-shift on a countable alphabet.
\end{rem}
For an inducing scheme $(X, F,\tau)$ and a potential $\varphi:X \to [-\infty, \infty]$ with summable variations, we can define the Gurevich pressure as in Section~\ref{sec:CMS}, and denote it by
$$P_F^G(\varphi),$$
where we drop the subscript if the dynamics is clear.
In fact the domains for the inducing schemes used above come from the natural cylinder structure of the map $f\in \mathcal{F}$. More precisely, the domains $X$ are $n$-cylinders coming from the so-called \emph{branch partition}: the set ${\mathcal P}_1^f$ consisting of maximal intervals on which $f$ is monotone. So if two domains ${\rm C}_1^i, {\rm C}_1^j\in {\mathcal P}_1^f$ intersect, they do so only at elements of ${\mathcal Cr}$. The set of corresponding $n$-cylinders is denoted ${\mathcal P}_n^f:=\vee_{k=1}^nf^{-k}{\mathcal P}_1$. We let ${\mathcal P}_0^f:=\{I\}$. For an inducing scheme $(X,F)$ we use the same notation for the corresponding $n$-cylinders ${\mathcal P}_n^F$. Note the transitivity assumption on our maps $f$ implies that ${\mathcal P}_1^f$ is a generating partition for any Borel probability measure.
\section{Zero pressure schemes} \label{sec:zero press}
For $t\in {\mathbb R}$, we let $$\psi_t:=-t\log|Df|-p(t).$$ Similarly, for an inducing scheme $(X,F)$ the induced potential is ${\mathcal P}si_t$.
As in \cite{PeSe, BTeqgen, BTeqnat} in order to apply Sarig's theory we need to find an inducing scheme $(X, F, \tau)$ so that $P^G({\mathcal P}si_t)=0$. Then Sarig's theory gives a Gibbs measure for $(X, F, {\mathcal P}si_t)$, which if it projects to a measure in $\mathcal{M}_f$ by \eqref{eq:lift}, must be an equilibrium state by the Abramov formula. The main purpose of this section is to show that there are inducing schemes with $P^G({\mathcal P}si_t)=0$.
We note that a major difficulty when working with inducing schemes is that, in general, no single inducing scheme is compatible with all measures of positive Lyapunov exponent. As a direct consequence of work by Bruin and Todd \cite[Remark 6]{BTeqnat} we obtain in Lemma \ref{lem:bdd ind int} that there exists a finite number of inducing schemes for which any measure of entropy bounded away from zero is compatible with one of them. This will allow us to prove that for each $t\in (t^-,t^+)$ there exists an inducing scheme for which $P({\mathcal P}si_t)=0$ and such that the pressure, $p(t)$, can be approximated with $f$-invariant measures of positive entropy compatible with the inducing scheme.
\begin{prop}
For each $t\in (t^-,t^+)$, there exist an inducing scheme $(X,F)$
and a sequence $(\mu_n)_n\subset \mathcal{M}_f$ all compatible with $(X,F)$ and such that $$h(\mu_n)-t\lambda(\mu_n)\to p(t) \text{ and } \inf_n h(\mu_n)>0.$$ Moreover, $P^G({\mathcal P}si_t)=0$.
\label{prop:zero Gur}
\end{prop}
We need some lemmas and a definition for the proof.
\begin{lema} \label{lem:press less zero}
For each $t \in {\mathbb R}$ and any inducing scheme $(X,F)$, we have $P^G({\mathcal P}si_t)\leqslant 0.$
\end{lema}
\begin{proof}
We let $(X^N,F_N, \tau_N)$ denote the subsystem of $(X,F, \tau)$ where $X^N=\cup_{n=1}^NX_n$ and $F_N, \tau_N$ are the restrictions of $F, \tau$ to $X^N$. Similarly, $P_{F_N}^G({\mathcal P}si_t)$ is defined in the obvious way. By Proposition \ref{prop:approx}, $P_F^G({\mathcal P}si_t)>0$ implies that for large enough $N$, $P_{F_N}^G({\mathcal P}si_t)>0$. Hence there is an equilibrium state $\mu_{F_N}$ for this system so that $\int\tau_N~d\mu_{F_N}<\infty$ and $$h(\mu_{F_N})-t\int \log|DF|~d\mu_{F_N}-p(t) \int \tau_N~d\mu_{F_N}>0.$$ Similarly to the use of the Abramov formula above, the corresponding projected measure $\mu_{f_N}$ as in \eqref{eq:lift} has
\[ h(\mu_{f_N}) -t \int \log |Df|~d\mu_{f_N}> p(t). \]
This contradiction to the Variational Principle proves the lemma.
\end{proof}
\begin{rem}
By \cite[Lemma 8]{BTeqnat}, the potentials ${\mathcal P}si_t$ we consider for the inducing schemes $(X,F)$ in Lemma~\ref{lem:bdd ind int} are weakly H\"older continuous.
\label{rmk:weak Holder}
\end{rem}
\begin{defi}
Given a function $g:[a,b] \to {\mathbb R}$, for $x_0\in {\mathbb R}$, as in \cite[p115]{Royden}, we refer to $s:[a,b] \to {\mathbb R}$ as a \emph{supporting line for $g$ at $x_0$} if $s(x)=g(x_0)+p(x-x_0)$ for some $p\in {\mathbb R}$, and $g(x)\geqslant s(x)$ for all $x\in {\mathbb R}$.
\end{defi}
\begin{lema}
For each $t\in (t^-,t^+)$, there exists $\eta>0$ such that any measure $\mu$ with free energy w.r.t. $\psi_t$ close enough to 0 has $h(\mu)>\eta$. \label{lem:pos ent}
\end{lema}
\begin{proof}
Let $t_0 \in (t^-, t^+)$.
Suppose, by contradiction, that there exists a sequence of invariant measures $\mu_n$ such that $\lim_{n \to \infty} h(\mu_n)=0$ and
$$p(t_0)=\lim_{n \to \infty} \leqslantft( h(\mu_n) -t_0\lambda(\mu_n) \right).$$
Denote by $L(t)$ the line passing through the origin which corresponds to the limit
$$\lim_{n \to \infty} h(\mu_n) -t \lim_{n \to \infty} \lambda(\mu_n).$$
We have that $L(t)$ is a supporting line of the pressure $p(t)$ and $p(t_0)= L(t_0)$.
Since $\lim_{n \to \infty} h(\mu_n)=0$, for every $t \geqslant t_0$ we have $p(t)=L(t)$. But this implies that $t_0= t^+$. This contradiction proves the statement.
\end{proof}
\begin{lema}
For each $\varepsilonilon>0$ there exists $\theta>0$ and a finite number of inducing schemes $\{(X^n, F_n, \tau_n)\}_{n=1}^N$ such that any ergodic measure with $h(\mu)>\varepsilonilon$ is compatible with one of these schemes $(X^n, F_n, \tau_n)$ and $\int\tau_n~d\mu_{F_n}<\theta$.
\label{lem:bdd ind int}
\end{lema}
\begin{proof}
This follows from \cite[Remark 6]{BTeqnat}. We give a brief sketch of the ideas there. That remark gives, for $\varepsilonilon>0$, a set $\{(X^n, F_n, \tau_n)\}_{n=1}^N$ such that for each $\mu\in \mathcal{M}_f$ with $h(\mu)>\varepsilonilon$, $\mu$ must be compatible with some $(X^n, F_n, \tau_n)$. These schemes are constructed from sets $\hat X^n$ on the so-called Hofbauer extension (see the appendix for details). The map $F$ is derived from a first return map $\hat F$ in this tower. Measures $\mu\in \mathcal{M}_f$ with $h(\mu)>0$ can be lifted to the tower, and if they have $h(\mu)>\varepsilonilon$ they must give one of the sets $\hat X_n$ mass greater than some $\eta=\eta(\varepsilonilon)>0$. Since $\hat F$ is a first return map with return time $\hat\tau_n$, we use Kac's lemma to get $$\int\tau_n~d\mu =\int\hat\tau_n~d\hat\mu=\hat\mu(\hat X_n)^{-1}<\eta^{-1},$$
as required.
\end{proof}
As in \cite[Remark 6]{BTeqnat}, we denote this set of inducing schemes by $Cover(\varepsilonilon)$.
\begin{proof}[Proof of Proposition~\ref{prop:zero Gur}]
By Lemmas~\ref{lem:bdd ind int} and \ref{lem:pos ent}, we can take a sequence of ergodic measures $\mu_p$ such that $$h(\mu_p)+\int\psi_t~d\mu_p =\varepsilonilon_p \text{ where } \varepsilonilon_p \to 0 \text{ as } p\to \infty,$$ $h(\mu_p)>\eta$ (some $\eta>0$), all $\mu_p$ are compatible with some inducing scheme $(X,F, \tau)\in Cover(\varepsilonilon)$ and $\int\tau~d\mu_p<\theta$ for all $p\in {\mathbb N}$. This implies that $P^G({\mathcal P}si_t)\geqslant 0$ since we have a sequence of measures $\mu_{F,p}$ such that
$$h(\mu_{F,p})+\int{\mathcal P}si_t~d\mu_{F,p} = \leqslantft(\int\tau~d\mu_{F,p}\right)\leqslantft(h(\mu_{p})+\int\psi_t~d\mu_{p}\right) \geqslant \theta\varepsilonilon_p$$
On the other hand $P^G({\mathcal P}si_t)\leqslant 0$ by Lemma~\ref{lem:press less zero}. So the proposition is proved.
\end{proof}
Since the inducing scheme $(X,F)$ can be coded by the full-shift on countably many symbols we have, as explained in Section~\ref{sec:CMS}, a Gibbs measure $\mu_{{\mathcal P}si_t}$ for ${\mathcal P}si_t$. We need to show that this measure has integrable inducing time and thus that it projects to a measure in $\mathcal{M}_f$.
\section{The Gibbs measure has integrable inducing times}
\label{sec:Gibbs integ}
This section is devoted to proving that the inducing time is integrable with respect to the Gibbs measure constructed in Section \ref{sec:zero press}. In particular, this implies that the measure has finite entropy and that it is an equilibrium state for the induced potential.
It also implies that it can be projected to a measure in $\mathcal{M}_f$.
\begin{prop}
Let $t\in (t^-, t^+)$ and $\psi=\psi_t$. Suppose that we have an inducing scheme $(X,\tilde F)$.
Then there exists $k\in {\mathbb N}$ such that replacing $(X, \tilde F)$ by $(X,F)$, where $F=\tilde F^k$, the following holds. There exist $\gamma_0\in (0,1)$ and, for any cylinder ${\rm C}_n^j\in {\mathcal P}_n^F$ any $n\in {\mathbb N}$, a constant $\delta_{n}^j<0$ such that any measure $\mu_F\in \mathcal{M}_F$ with $$\mu_F({\rm C}_n^j)\leqslant (1-\gamma_0) m_{\mathcal P}si({\rm C}_n^j) \text{ or } \mu_F({\rm C}_n^j) \geqslant \frac{m_{\mathcal P}si({\rm C}_n^j)}{1-\gamma_0},$$ where $m_{\mathcal P}si$ denotes the conformal measure for the system $(X,F,{\mathcal P}si)$, must have $h(\mu_F)+\int{\mathcal P}si~d\mu_F\leqslant \delta_n^j$. \label{prop:conv to Gibbs}
\end{prop}
Note that $\delta_{n}^j \to 0$ as $m_{\mathcal P}si({\rm C}_n^j) \to 0$. Also note that
if $K=\exp\leqslantft(\sum_{k=1}^\infty V_k(\tilde{\mathcal P}si)\right)$ is a distortion constant for the potential $\tilde{\mathcal P}si$ for the inducing scheme $(X, \tilde F)$ then it is also a distortion constant the potential ${\mathcal P}si$ on $(X,F)$.
The following lemma will allow us to choose $k$ in the proof of Proposition~\ref{prop:conv to Gibbs}. It is true for ${\mathcal P}si={\mathcal P}si_t$, but also for more general potentials of summable variation.
\begin{lema}
Suppose that we have an inducing scheme $(X,F)$ and potential ${\mathcal P}si={\mathcal P}si_t$ with distortion constant $K=\exp\leqslantft(\sum_{k=1}^\infty V_k({\mathcal P}si)\right)$ and $P^G({\mathcal P}si)=0$. We let $m_{\mathcal P}si$ denote the conformal measure for the system $(X,F,{\mathcal P}si)$. Then for any ${\rm C}_n\in {\mathcal P}_n^F$ and $n\in {\mathbb N}$, $$m_{\mathcal P}si({\rm C}_n)\leqslant e^{-\lambda n}$$ where $\lambda:=-\log\leqslantft(K\sup_{{\rm C}_1\in {\mathcal P}_1^F} m_{\mathcal P}si({\rm C}_1)\right)$.
\label{lem:shrinking cylinders}
\end{lema}
\begin{proof}
Since $m_{\mathcal P}si$ is a conformal measure, for ${\rm C}_n^i\in {\mathcal P}_n^F$ we have
$$ 1=m_{\mathcal P}si(F^n({\rm C}_n^i))=\int_{{\rm C}_n^i}e^{-S_n{\mathcal P}si}~dm_{\mathcal P}si.$$ So by the Intermediate Value Theorem we can choose $x\in {\rm C}_n^i$ so that $e^{S_n{\mathcal P}si(x)}=m_{\mathcal P}si({\rm C}_n^i)$. For future use we will write $S_n^i{\mathcal P}si:=S_n{\mathcal P}si(x)$. Therefore, $$m_{\mathcal P}si({\rm C}_n^i)=e^{S_n^i{\mathcal P}si}\leqslant e^{n\sup{\mathcal P}si}.$$
By the Gibbs property, $$e^{\sup{\mathcal P}si} \leqslant K \sup_{{\rm C}_1\in {\mathcal P}_1^F}m_{\mathcal P}si({\rm C}_1).$$ Therefore
$$\sup{\mathcal P}si \leqslant \log\leqslantft(K\sup_{{\rm C}_1\in {\mathcal P}_1^F}m_{\mathcal P}si({\rm C}_1)\right).$$ We can choose this as our value for $-\lambda$.
\end{proof}
In the following proof we use the notation $A=\theta^{\pm C}$ to mean $\theta^{-C} \leqslant A \leqslant \theta^C$.
\begin{proof}[Proof of Proposition~\ref{prop:conv to Gibbs}]
Suppose that the distortion of the potential $\tilde{\mathcal P}si$ for the scheme $(X,\tilde F)$ is bounded by $K\geqslant 1$.
We first prove that measures giving cylinders very small mass compared to $m_{\mathcal P}si$ must have low free energy. Note that for any $k\in {\mathbb N}$, the potential ${\mathcal P}si$ for the scheme $(X,F)$ where $F=\tilde F^k$ also has distortion bounded by $K$. We will choose $k$ later so that $\lambda=\lambda(K,\sup_im_{\mathcal P}si(X_i))$ for $(X,F)$ as defined in Lemma~\ref{lem:shrinking cylinders}, is large enough to satisfy the conditions associated to \eqref{eq:flat estimate}, \eqref{eq:sharp estimate} and \eqref{eq:gamma sharp}.
Note that as in \cite[Lemma 3]{Sarphase} we also have $P^G({\mathcal P}si)=0$.
In Lemma~\ref{lem:flat low FE} below, we will use the Variational Principle to bound the free energy of measures for the scheme which, for some $\gamma$, have $\mu({\rm C}_n^i)\leqslant K m_{\mathcal P}si({\rm C}_n^i)(1-\gamma)/(1-m_{\mathcal P}si({\rm C}_n^i))^n$ in terms of the Gurevich pressure.
However, instead of using ${\mathcal P}si$, which, in the computation of Gurevich pressure weights points $x\in {\rm C}_n^i$ by $e^{{\mathcal P}si(x)}$, we use a potential which weights points in ${\rm C}_n^i$ by $(1-\gamma) e^{\mathcal P}si(x)$. That is, we consider $(X,F, {\mathcal P}si^\flat)$ where
\begin{equation*}
{\mathcal P}si^\flat(x)= \begin{cases} {\mathcal P}si(x)+\log(1-\gamma)& \text{if } x\in {\rm C}_n^i,\\
{\mathcal P}si(x) & \text{if } x\in {\rm C}_n^j,\text{ for } j\neq i.
\end{cases}
\end{equation*}
Firstly we will compute $P^G({\mathcal P}si^\flat)$.
\begin{lema}
$P^G({\mathcal P}si^\flat)= \log\leqslantft(1-\gamma m_{\mathcal P}si({\rm C}_n^i)\right).$ \label{lem:pres psi flat}
\end{lema}
\begin{proof}
We prove the lemma assuming that $n=1$ since the general case follows similarly. We will estimate $Z_j({\mathcal P}si^\flat, {\rm C}_1^i)$, where $Z_j$ is defined in \eqref{eq:Zn}. The ideas we use are similar to those in the proof of Claim 2 in the proof of \cite[Proposition 2]{BTeqnat}. As can be seen from the definition,
$$Z_j({\mathcal P}si^\flat, {\rm C}_1^i) = e^{\pm \sum_{k=0}^{j-1}V_k({\mathcal P}si)} \sum_{{\rm C}_j\in {\mathcal P}_j^F\cap {\rm C}_1^i} \ \sum_{\text{any } x\in {\rm C}_j} e^{S_j{\mathcal P}si^\flat(x)}.$$
As in the proof of Lemma~\ref{lem:shrinking cylinders}, the conformality of $m_{\mathcal P}si$ and the Intermediate Value Theorem imply that for each $k$ there exists $x_{{\rm C}_1^k}\in {\rm C}_1^k$ such that $m_{\mathcal P}si({\rm C}_1^k) = e^{{\mathcal P}si(x_{{\rm C}_1^k})}$. For the duration of this proof we write ${\mathcal P}si_k := {\mathcal P}si(x_{{\rm C}_1^k})$. As above, we have $e^{{\mathcal P}si_i^\flat}:=(1-\gamma) e^{{\mathcal P}si_i}$. Therefore, $$\sum_i e^{{\mathcal P}si_i^\flat}= 1-\gamma e^{{\mathcal P}si_i}.$$
For each ${\rm C}_j\in{\mathcal P}_j^F$ and for any $k\in {\mathbb N}$, there exists a unique ${\rm C}_{j+1}\subset {\rm C}_j$ such that $F^j({\rm C}_{j+1})={\rm C}_1^k$. Moreover, there exists $x_{{\rm C}_{j+1}}\in {\rm C}_{j+1}$ such that $F^j(x_{{\rm C}_{j+1}}) = x_{{\rm C}_1^k}$. Then for ${\rm C}_j\subset {\rm C}_1^i$,
\begin{align*}
\sum_{{\rm C}_{j+1}\subset{\rm C}_j} e^{S_{j+1}{\mathcal P}si^\flat(x_{{\rm C}_{j+1}})} &= e^{\pm V_{j+1}({\mathcal P}si)} e^{S_j{\mathcal P}si^\flat(x_{{\rm C}_{j}})} \leqslantft(\sum_i e^{{\mathcal P}si_i^\flat}\right)\\
&= e^{\pm V_{j+1}({\mathcal P}si)} e^{S_j{\mathcal P}si^\flat(x_{{\rm C}_{j}})} (1-\gamma e^{{\mathcal P}si_i}).
\end{align*}
Therefore,
$$Z_{j+1}({\mathcal P}si^\flat, {\rm C}_1^i) = (1-\gamma e^{{\mathcal P}si_i}) e^{\pm\leqslantft(V_{j+1}({\mathcal P}si)+\sum_{k=0}^{j-1}V_k({\mathcal P}si)\right)} Z_j({\mathcal P}si^\flat, {\rm C}_1^i),$$
hence
$$Z_{j+1}({\mathcal P}si^\flat, {\rm C}_1^i) = (1-\gamma e^{{\mathcal P}si_i})^j e^{\pm\sum_{k=0}^{j} (k+1)V_{k}({\mathcal P}si)}.$$
As in Remark~\ref{rmk:weak Holder}, ${\mathcal P}si$ is weakly H\"older, so $\sum_{k=0}^{j} (k+1)V_{k}({\mathcal P}si)< \infty$. Therefore we have $P^G({\mathcal P}si^\flat)= \log(1-\gamma e^{{\mathcal P}si_i})= \log(1-\gamma m_{\mathcal P}si({\rm C}_1^i))$, proving the lemma.
\end{proof}
For the next step in the proof of the upper bound on the free energy of measures giving ${\rm C}_n^i$ small mass, we relate properties of $(X,F, {\mathcal P}si)$ and $(X,F, {\mathcal P}si^\flat)$.
\begin{lema}
$\mathcal{M}_F({\mathcal P}si)=\mathcal{M}_F({\mathcal P}si^\flat)$ and for any ${\rm C}_n^i\in {\mathcal P}_n^F$ we have
\begin{align*}
&\sup\leqslantft\{h_F(\mu) +\int{\mathcal P}si~d\mu:\mu\in \mathcal{M}_F({\mathcal P}si),\ \mu({\rm C}_n^i)<\frac{K(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n} m_{\mathcal P}si({\rm C}_n^i) \right\}\\
&\hspace{5mm} \leqslant \sup\leqslantft\{h_F(\mu) +\int{\mathcal P}si^\flat~d\mu:\mu\in \mathcal{M}_F({\mathcal P}si^\flat),\ \mu({\rm C}_n^i)<\frac{K(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n} m_{\mathcal P}si({\rm C}_n^i) \right\}\\
&\hspace{10mm}-\leqslantft[\frac{K(1-\gamma)\log(1-\gamma)}{ (1-m_{\mathcal P}si({\rm C}_n^i))^n} \right] m_{\mathcal P}si({\rm C}_n^i) \\
&\hspace{5mm} \leqslant P^G({\mathcal P}si^\flat)- \leqslantft[\frac{K(1-\gamma)\log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n} \right] m_{\mathcal P}si({\rm C}_n^i).
\end{align*}
\label{lem:flat low FE}
\end{lema}
Note that we can prove that the final inequality is actually an equality, but since we don't require this here we will not prove it.
\begin{proof}
The fact that $\mathcal{M}_F({\mathcal P}si)=\mathcal{M}_F({\mathcal P}si^\flat)$ is clear from the definition.
Suppose that $\mu\in \mathcal{M}_F({\mathcal P}si)$ and $\mu({\rm C}_n^i)\leqslant m_{\mathcal P}si({\rm C}_n^i) K(1-\gamma)/(1-m_{\mathcal P}si({\rm C}_n^i))^n$.
Then
\begin{align*}
\leqslantft(h_F(\mu)+\int{\mathcal P}si~d\mu\right)- \leqslantft(h_F(\mu)+\int{\mathcal P}si^\flat~d\mu\right)= \int{\mathcal P}si-{\mathcal P}si^\flat~d\mu\\
=\mu({\rm C}_n^i)(-\log(1-\gamma))\leqslant -\leqslantft[\frac{K(1-\gamma)\log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n}\right] m_{\mathcal P}si({\rm C}_n^i),
\end{align*}
proving the first inequality in the Lemma. The final inequality follows from the definition of pressure.
\iffalse
The final equality in the statement of the lemma essentially follows from the fact that for the Gibbs measure $\mu_{{\mathcal P}si^\flat}$, we have $$\mu_{{\mathcal P}si^\flat}({\rm C}_n^i) \leqslant K e^{S_n{\mathcal P}si^\flat(x)}e^{-nP^G({\mathcal P}si^\flat)} = \frac{K(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n}e^{S_n{\mathcal P}si(x)}$$ for all $x\in {\rm C}_n^i$. We replace $S_n{\mathcal P}si(x)$ with $S_n^i{\mathcal P}si$ as defined in the proof of Lemma~\ref{lem:shrinking cylinders} (recall $e^{S_n^i{\mathcal P}si}=m_{\mathcal P}si({\rm C}_n^i)$) which implies that $\mu_{{\mathcal P}si^\flat}({\rm C}_n^i) \leqslant m_{\mathcal P}si({\rm C}_n^i) K(1-\gamma)/(1-m_{\mathcal P}si({\rm C}_n^i))^n$. If we knew that $\mu_{{\mathcal P}si^\flat}$ was an equilibrium state for $(X,F, {\mathcal P}si^\flat)$ we could immediately conclude that the pressure $P^G({\mathcal P}si^\flat)$ is equal to\begin{equation} \label{eq:vpr}
\sup\leqslantft\{h_F(\mu) +\int{\mathcal P}si^\flat~d\mu:\mu\in \mathcal{M}_F({\mathcal P}si^\flat),\ \mu({\rm C}_n^i)<\frac{K(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n} m_{\mathcal P}si({\rm C}_n^i) \right\},
\end{equation}
hence completing the proof of the lemma. However, at this stage, we don't know that $\mu_{{\mathcal P}si^\flat}$ is an equilibrium state for $(X,F, {\mathcal P}si^\flat)$, since $-\int{\mathcal P}si^\flat~d\mu_{{\mathcal P}si^\flat}$ and/or $h_F(\mu_{{\mathcal P}si^\flat})$ could be infinite, so the notion may make no sense. However, by virtue of Proposition \ref{prop:approx} we can choose measures $\nu_p$ which approximate $\mu_{{\mathcal P}si^\flat}$, supported on compact sets with $\nu_p({\rm C}_n^i) \leqslant m_{\mathcal P}si({\rm C}_n^i) K(1-\gamma)/(1-m_{\mathcal P}si({\rm C}_n^i))^n$ \textbf{GI: what is $\mu$ in the previous inequality, do you mean $\nu_p$? MT:changed this} and with $$h_F(\nu_p) +\int{\mathcal P}si^\flat~d\nu_p \to P^G({\mathcal P}si^\flat) \text{ as } p\to \infty,$$
and hence $P^G({\mathcal P}si^\flat)$ is equal to the expression in \eqref{eq:vpr}.\textbf{GI: Do we really use the equality ("the final equality") or it is enough to have an inequality, which is trivial by definition of the pressure? MT:changed this.}
\fi
\end{proof}
Lemmas~\ref{lem:pres psi flat} and \ref{lem:flat low FE} imply that any measure $\mu_F$ with $\mu_F({\rm C}_n^i)<K(1-\gamma)m_{\mathcal P}si({\rm C}_n^i)/(1-m_{\mathcal P}si({\rm C}_n^i))^n$ must have
\iffalse
\begin{align}h(\mu_F)+\int{\mathcal P}si~d\mu_F & \leqslant P^G({\mathcal P}si^\flat)-\leqslantft[\frac{K(1-\gamma) \log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n}\right] m_{\mathcal P}si({\rm C}_n^i) \nonumber\\
&\leqslant \log\leqslantft(1-\gamma m_{\mathcal P}si({\rm C}_n^i)\right) -\leqslantft[\frac{K(1-\gamma) \log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n}\right]m_{\mathcal P}si({\rm C}_n^i)
\end{align}
\fi
\begin{align}
h(\mu_F)+\int{\mathcal P}si~d\mu_F & \leqslant P^G({\mathcal P}si^\flat)-\leqslantft[\frac{K(1-\gamma) \log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n}\right] m_{\mathcal P}si({\rm C}_n^i) \label{eq:inter flat estimate}\\
&\leqslant \log\leqslantft(1-\gamma m_{\mathcal P}si({\rm C}_n^i)\right) -\leqslantft[\frac{K(1-\gamma)\log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n} \right]m_{\mathcal P}si({\rm C}_n^i).
\label{eq:flat estimate}
\end{align}
\iffalse
\begin{equation}
\label{eq:flat estimate}
\begin{split}
h(\mu_F)+\int{\mathcal P}si~d\mu_F & \leqslant P^G({\mathcal P}si^\flat)-\leqslantft[\frac{K(1-\gamma) \log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n}\right] m_{\mathcal P}si({\rm C}_n^i) \nonumber\\
&\leqslant \log\leqslantft(1-\gamma m_{\mathcal P}si({\rm C}_n^i)\right) -\leqslantft[\frac{K(1-\gamma)\log(1-\gamma)}{(1-m_{\mathcal P}si({\rm C}_n^i))^n} \right]m_{\mathcal P}si({\rm C}_n^i)
\end{split}
\end{equation}
\fi
If $m_{\mathcal P}si({\rm C}_n^i)$ is very small then $\log\leqslantft(1-\gamma m_{\mathcal P}si({\rm C}_n^i)\right) \approx -\gamma m_{\mathcal P}si({\rm C}_n^i)$ and so choosing $\gamma\in (0,1)$ close enough to 1 the above is strictly negative. By Lemma~\ref{lem:shrinking cylinders}, $m_{\mathcal P}si({\rm C}_n^i)<e^{-\lambda n}$ so ${\rm C}_n^i$ is small if $\lambda$ large. Hence if $\lambda$ is sufficiently large then we can set $\gamma=\tilde\gamma^{\flat}\in (0,1)$ so that
$$\log\leqslantft(1-\tilde\gamma^\flat e^{-\lambda n}\right)-\leqslantft[\frac{K(1-\tilde\gamma^\flat)\log(1-
\tilde\gamma^\flat)}{(1- e^{-\lambda n})^n}\right]e^{-\lambda n}$$
is strictly negative for all $n\in {\mathbb N}$. This implies that \eqref{eq:flat estimate} with $\gamma=\tilde\gamma^{\flat}$ is strictly negative for any ${\rm C}_n^i\in {\mathcal P}_n^F$ and any $n$, so we set \eqref{eq:flat estimate} to be the value $\delta_{n}^{i,\flat}$.
For the upper bound on the free energy of measures giving ${\rm C}_n^i$ relatively large mass, we follow a similar proof, but with
\begin{equation*}
{\mathcal P}si^\sharp(x)= \begin{cases} {\mathcal P}si(x)-\log(1-\gamma)& \text{if } x\in {\rm C}_n^i,\\
{\mathcal P}si(x) & \text{if } x\in {\rm C}_n^j,\text{ for } j\neq i.
\end{cases}
\end{equation*}
Similarly to above, one can show that
$\mathcal{M}_F({\mathcal P}si)=\mathcal{M}_F({\mathcal P}si^\sharp)$ and
\begin{align*}
&\sup\leqslantft\{h_F(\mu) +\int{\mathcal P}si~d\mu:\mu\in \mathcal{M}_F({\mathcal P}si),\ \mu({\rm C}_n^i)> \frac{m_{\mathcal P}si({\rm C}_n^i)}{K (1-\gamma) \leqslantft[1+m_{\mathcal P}si({\rm C}_n^i)\leqslantft(\frac\gamma{1-\gamma}\right)\right]^n} \right\}\\
&\hspace{2mm} \leqslant \sup\leqslantft\{h_F(\mu) +\int{\mathcal P}si^\sharp~d\mu:\mu\in \mathcal{M}_F({\mathcal P}si^\sharp),\ \mu({\rm C}_n^i)> \frac{m_{\mathcal P}si({\rm C}_n^i)}{K (1-\gamma) \leqslantft[1+m_{\mathcal P}si({\rm C}_n^i)\leqslantft(\frac\gamma{1-\gamma}\right)\right]^n} \right\}\\
&\hspace{10mm}+\frac{\log(1-\gamma)m_{\mathcal P}si({\rm C}_n^i)}{K (1-\gamma) \leqslantft[1+m_{\mathcal P}si({\rm C}_n^i)\leqslantft(\frac\gamma{1-\gamma}\right)\right]^n} \\
&\hspace{2mm}\leqslant P^G({\mathcal P}si^\sharp)+ \frac{\log(1-\gamma)m_{\mathcal P}si({\rm C}_n^i)}{K (1-\gamma) \leqslantft[1+m_{\mathcal P}si({\rm C}_n^i)\leqslantft(\frac\gamma{1-\gamma}\right)\right]^n}.
\end{align*}
Moreover, we can show that
$$P^G({\mathcal P}si^\sharp) = \log\leqslantft(1+ m_{\mathcal P}si({\rm C}_n^i) \leqslantft(\frac{\gamma}{1-\gamma}\right)\right)\leqslant m_{\mathcal P}si({\rm C}_n^i) \leqslantft(\frac{\gamma}{1-\gamma}\right).$$
Therefore, if $\mu({\rm C}_n^i)> \frac{m_{\mathcal P}si({\rm C}_n^i)}{K (1-\gamma) \leqslantft[1+m_{\mathcal P}si({\rm C}_n^i)\leqslantft(\frac\gamma{1-\gamma}\right)\right]^n}$, we have
\begin{equation}h_F(\mu) +\int{\mathcal P}si~d\mu \leqslant m_{\mathcal P}si({\rm C}_n^i) \leqslantft(\frac{\gamma}{1-\gamma}\right)+\frac{\log(1-\gamma) m_{\mathcal P}si({\rm C}_n^i)}{K (1-\gamma) \leqslantft[1+m_{\mathcal P}si({\rm C}_n^i)\leqslantft(\frac\gamma{1-\gamma}\right)\right]^n}.
\label{eq:sharp estimate}
\end{equation}
If $\lambda$ is sufficiently large then we can choose $\gamma=\tilde\gamma^\sharp\in (0,1)$ so that this is strictly negative and can be fixed to be our value $\delta_n^{i, \sharp}$. This can be seen as follows: let and $\gamma=p/(p+1)$ for some $p$ to be chosen later. Then the right hand side of \eqref{eq:sharp estimate} becomes
\begin{equation}
m_{\mathcal P}si({\rm C}_n^i)\leqslantft(p+1\right)\leqslantft[\frac{p}{p+1} -\frac{\log(p+1)}{K(1+p e^{-\lambda n})^n} \right].
\label{eq:p sharp}
\end{equation} If $\lambda$ is sufficiently large, then there exists some large $\lambda'\in (0,\lambda)$ such that $(1+p e^{-\lambda n})^n \leqslant 1+ p e^{-\lambda'n}$ for all $n\in {\mathbb N}$. Hence with this suitable choice of $\lambda$ we can choose $p$ so that the quantity in the square brackets in \eqref{eq:p sharp} is negative for all $n$. So we can choose $\delta_n^{i,\sharp}<0$ to be \eqref{eq:sharp estimate} with $\gamma=\tilde\gamma^\sharp$.
We let
\begin{equation}
\gamma^\sharp=1-(1-\tilde\gamma^\sharp)\leqslantft(1+e^{-\lambda n} \leqslantft(\frac{\tilde\gamma^\sharp}{1- \tilde\gamma^\sharp}\right)\right)^n.
\label{eq:gamma sharp}
\end{equation}
For appropriately chosen $\lambda$ this is in $(0,1)$.
We set $\gamma_0':=\max\{\gamma^\flat, \gamma^\sharp\}$ and for each ${\rm C}_n^i\in {\mathcal P}_n^F$ we let $\delta_n^i:=\max\{\delta_n^{i, \flat}, \delta_n^{i, \sharp}\}$. The proof of the proposition is completed by setting $\gamma_0:=1-K(1-\gamma_0')$, which we may assume is in $(0,1)$.
\end{proof}
\begin{prop}
There exists an inducing scheme $(X,F)$ such that for $t\in (t^-, t^+)$ and $\psi=\psi_t$, any sequence of measures $(\mu_n)_n$ with $h(\mu_n)-\int\psi~d\mu_n \to 0$ as $n\to \infty$ has a limit measure $\mu_\psi$ which is an equilibrium state for $\psi$.
\label{prop:conv to eq}
\end{prop}
Note that $(X,F)$ and $(\mu_n)_n$ can be chosen as in Proposition~\ref{prop:zero Gur}.
\begin{proof}
By Proposition~\ref{prop:zero Gur}, we can find $\theta>0$, an inducing scheme $(X,\tilde F)$ and a sequence of measures $(\mu_n)_n$ with $h(\mu_n)+\int\psi~d\mu_n \to 0$ each compatible with $(X,\tilde F)$ and with $\int\tilde\tau~d\mu_{\tilde F,n}<\theta$. Proposition~\ref{prop:zero Gur} also implies $P^G(\tilde {\mathcal P}si_t)=0$. Taking $F=\tilde F^k$ for $k$ as in Proposition~\ref{prop:conv to Gibbs}, that proposition then implies that there exists $K'>0$ such that for any ${\rm C}_k\in {\mathcal P}_k^F$, for all large enough $n$,
$$\frac1{K'} \leqslant \frac{\mu_{F,n}({\rm C}_k)}{e^{S_k{\mathcal P}si(x)}} \leqslant K'$$ for all $x\in {\rm C}_k$ (note that as in Proposition~\ref{prop:conv to Gibbs}, we can actually take $K'=K/(1-\gamma_0)$ where $K$ is the distortion bound for $\tilde{\mathcal P}si_t$).
\iffalse Note that $(\mu_{F,n})_n$ is tight (see \cite[Section 25]{Bill} for a discussion of this notion) and that any limit of the sequence $\mu_{F,\infty}$ must satisfy the Gibbs property with distortion constant $K'$ and must have $\int\tau~d\mu_{F,\infty}<\theta$. By the uniqueness of Gibbs measures (\cite{BuSar}), $\mu_{F,\infty}=\mu_{\mathcal P}si$. Thus we can project $\mu_{\mathcal P}si$ to $\mu_\psi$.\fi
Note that $(\mu_{F,n})_n$ is tight (see \cite[Section 25]{Bill} for a discussion of this notion) and that any limit of the sequence $\mu_{F,\infty}$ must satisfy the Gibbs property with distortion constant $K'$. By the uniqueness of Gibbs measures (\cite{BuSar}), $\mu_{F,\infty}=\mu_{\mathcal P}si$. We now show that $\int\tau~d\mu_{{\mathcal P}si}<\theta k$. First note that $\int\tau~d\mu_{F,n}= \int\tilde\tau^k~d\mu_{\tilde F,n}<\theta k$. For the purposes of this proof we let $\tau_N:=\min\{\tau,N\}$. By the Monotone Convergence Theorem,
$$\int\tau~d\mu_{{\mathcal P}si}=\lim_{N\to \infty}\int\tau_N~d\mu_{{\mathcal P}si}\leqslant \lim_{N\to \infty}\limsup_{n\to \infty} \int\tau_N~d\mu_{F,n} \leqslant \theta k.$$
Thus we can project $\mu_{\mathcal P}si$ to $\mu_\psi$ by \eqref{eq:lift}.
The fact that $\mu_\psi$ is a weak$^*$ limit of $(\mu_n)_n$ follows as in, for example \cite[Section 6]{FreiTo}. The fact that we have a uniform bound $\mu_{F,n}\leqslantft\{\tau \geqslant j\right\}\leqslant \theta k/j$ for all $n\in {\mathbb N}$ is again crucial in proving this.
The Abramov formula implies that $$\int{\mathcal P}si~d\mu_{\mathcal P}si= \leqslantft(\int\tau~d\mu_{\mathcal P}si\right)\leqslantft(\int\psi~d\mu_\psi\right)= \leqslantft(\int\tau~d\mu_{\mathcal P}si\right)(\lambda(\mu_\psi)-p(t)).$$
Since $\lambda(\mu)\in [\lambda_m, \lambda_M]$ and both $p(t)$ and $\int\tau~d\mu_{\mathcal P}si$ are finite, this implies that $-\int{\mathcal P}si~d\mu_{\mathcal P}si<\infty$ and hence $\mu_{\mathcal P}si$ is an equilibrium state for ${\mathcal P}si$. Using the Abramov formula again we have that $\mu_\psi$ is an equilibrium state for $\psi$.
\end{proof}
\begin{rem}
\iffalse
Here we give an example of a way our setting can be changed so that the arguments in Proposition~\ref{prop:conv to Gibbs} and \ref{prop:conv to eq} fail. In the case of the (appropriately scaled) quadratic Chebyshev polynomial, $t^-\in (-\infty, 0)$. In this case there is a periodic point $p$ such that the Dirac measure $\mu$ on the orbit of $p$ has $\lambda(\mu)=\lambda_M$. Then for each $t<t^-$,
there is an inducing scheme $(X,F)$ such that for any sequence $(\mu_n)_n$ with $h(\mu_n)>0$ for all $n$ and
$$h(\mu_n)-t\lambda(\mu_n) \to p(t)$$ we have $\int\tau~d\mu_{F,n} \to \infty$. Therefore it is important that $t\in (t^-, t^+)$.
\fi
Here we give an example of a way our setting can be changed so that the arguments in Proposition~\ref{prop:conv to Gibbs} and \ref{prop:conv to eq} fail. In the case where $f$ is the (appropriately scaled) quadratic Chebyshev polynomial, $t^-\in (-\infty, 0)$. In this case there is a periodic point $p$ such that the Dirac measure $\delta_p$ on the orbit of $p$ has $\lambda(\delta_p)=\lambda_M$.
The point $p$ is the image of the critical point which means that our class of inducing schemes can not be compatible with $\delta_0$ (indeed the only inducing scheme for $\delta_0$ has only one domain and the only measure compatible to it is $\delta_0$).
However, any measure $\mu\in \mathcal{M}_f$ orthogonal to $\delta_0$ must have $h(\mu)-t\lambda(\mu) \leqslant h(\mu_1)-t\lambda(\mu_1)$ for all $t\in {\mathbb R}$ where $\mu_1$ is the acip. In particular, $h(\mu)-t\lambda(\mu)<p(t)$ for $t<t^-$. If $P^G({\mathcal P}si_t)=0$ then arguments similar to those in the proofs of Lemma~\ref{lem:press less zero} and Proposition~\ref{prop:zero Gur} imply that there are measures with free energy w.r.t. $\psi_t$ is arbitrarily close to zero and positive entropy. This contradiction implies that for $t<t^-$, $P^G({\mathcal P}si_t)<0$ so we cannot begin to apply the arguments above to that case. So it is important that $t\in (t^-, t^+)$.
\end{rem}
\section{Uniqueness of equilibrium states}
\label{sec:unique}
The result in Proposition~\ref{prop:conv to eq} gives the existence of equilibrium states for $-t\log|Df|$ for each $t\in (t^-, t^+)$. In this section we obtain uniqueness. To do this we will use more properties of the inducing schemes described in \cite{BTeqnat}. They were produced in as first return maps to an interval in the so-called Hofbauer tower. This theory was further developed in \cite{BTeqgen} and \cite{T}. The following theorem gives some of their properties.
\begin{teo}\label{thm:schemes}
There exists a countable collection $\{(X^n,F_n)\}_n$ of inducing schemes with $\partial X^n \notin (X^n,F_n)^\infty$ such that:
\newcounter{Mcount}
\begin{list}{\alph{Mcount})}{\usecounter{Mcount} \itemsep 1.0mm \topsep 0.0mm \leqslantftmargin=5mm}
\item any ergodic invariant probability measure $\mu$ with $\lambda(\mu)>0$ is compatible with one of the inducing schemes $(X^n, F_n)$. In particular there exists and ergodic $F_n$-invariant probability measure $\mu_{F_n}$ which projects to $\mu$ as in \eqref{eq:lift};
\item any ergodic equilibrium state for $-t\log|Df|$ where $t\in {\mathbb R}$ with $\lambda(\mu)>0$ is compatible with all inducing schemes $(X^n, F_n)$.
\end{list}
\end{teo}
\begin{rem}
Note that it is crucial in our applications of Theorem~\ref{thm:schemes}, for example in the proofs of Proposition~\ref{prop:unique} and Proposition~\ref{prop:t^-}, that in b) we are able to weaken the condition $h(\mu)>0$ to $\lambda(\mu)>0$ when we wish to lift measures. This is why we need to use a countable number of inducing schemes in Theorem~\ref{thm:schemes} rather than the finite number in \cite[Remark 6]{BTeqnat}.
\label{rmk:LE schemes}
\end{rem}
Before proving Theorem~\ref{thm:schemes}, we prove the following easy lemma.
\begin{lema} \label{lem:press zero}
If $t \in (t^-, t^+)$ and an equilibrium state $\mu_t$ from Proposition~\ref{prop:conv to eq} is compatible with an inducing scheme $(X,F)$, then $P^G({\mathcal P}si_t)=0$. Moreover the lifted measure $\mu_{t,F}$ is a Gibbs measure and an equilibrium state for ${\mathcal P}si_t$.
\end{lema}
\begin{proof}
First note that by Lemma~\ref{lem:press less zero}, $P^G({\mathcal P}si_t)\leqslant 0$.
Denote by $\mu_t$ an equilibrium measure for the potential $-t \log |Df|$ of positive Lyapunov exponent and let $\mu_{t,F}$ be the lifted measure.
Note that by Proposition~\ref{prop:VarPri} and by the Abramov formula, see for example \cite[Theorem 2.3]{PeSe}, we have
\begin{align*}
P^G\leqslantft({\mathcal P}si_t\right) &\geqslant h(\mu_{t,F})+\int{\mathcal P}si_t~d\mu_{t, F} =h(\mu_{t,F}) -t \int \log|DF| \ d \mu_{t,F} -p(t) \int \tau \ d \mu_{t,F} \\
&= \leqslantft(\int \tau \ d \mu_{t,F}\right) \leqslantft( \frac{h(\mu_{t,F})}{\int \tau \ d \mu_{t,F}} -t\leqslantft( \frac{\int \log|DF|~d \mu_{t,F}}{\int \tau~d \mu_{t,F}}\right) -p(t) \right) \\
&=\leqslantft(\int \tau \ d \mu_{t,F}\right) \leqslantft( h(\mu_t) -t \int \log |Df| \ d \mu_t -p(t) \right).\end{align*}
But recall that $\mu_t$ is an equilibrium measure:
\[ p(t) = h(\mu_t) -t \int \log |Df| \ d \mu_t.\]
Therefore $P^G({\mathcal P}si_t) \geqslant 0$.
Since $P^G({\mathcal P}si_t)=0$ there exists a unique Gibbs measure $\mu_F$ corresponding to $(X,F, {\mathcal P}si_t)$. By the Abramov formula,
$$h(\mu_{t,F})+\int {\mathcal P}si_t~d \mu_{t,F}=0,$$ so $\mu_{t, F}$ is an equilibrium state for $(X,F, {\mathcal P}si_t)$. Since, in this setting, equilibrium states are unique (see \cite{BuSar}) we have that $\mu_{t,F}=\mu_F$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:schemes}]
Part (a) of the theorem follows from the proof of \cite[Theorem 3]{BTeqnat}. Part (b) is proved similarly to \cite[Proposition 2]{BTeqnat}, but with added information from our Proposition~\ref{prop:conv to Gibbs}. We sketch some details. Suppose that $\mu$ is compatible to $(X^n, F_n)$. Then Lemma~\ref{lem:press zero} implies that $P^G({\mathcal P}si_n)=0$. Claim 1 of the proof of \cite[Proposition 2]{BTeqnat} implies that for any other inducing scheme $(X^{n'}, F_{n'})$ is `topologically connected' to $(X^n, F_n)$. Proposition~\ref{prop:conv to Gibbs}, which is an improved version of Claim 2 in the proof of \cite[Proposition 2]{BTeqnat}, then can be used as in that proof to give a `metric connection' which means that an equilibrium state compatible with $(X^n,F_n)$ must be compatible with $(X^{n'}, F_{n'})$.
\end{proof}
\begin{prop}
For any $t\in (-\infty,t^+)$ there is at most one equilibrium state for $-t\log|Df|$. Moreover, if $t^+>1$ then for any $t\in {\mathbb R}$ there is at most one equilibrium state for $-t\log|Df|$. \label{prop:unique}
\end{prop}
Clearly the equilibrium states, when unique, must be ergodic.
\begin{proof}
The idea here is first to show that any equilibrium state can be decomposed into a sum of countably many measures, each of which is an equilibrium state and is compatible with an inducing scheme as in Theorem~\ref{thm:schemes}. \cite{BuSar} implies that there is only one equilibrium state per inducing scheme. Lemma~\ref{lem:press zero} then implies that this equilibrium state must be unique.
We suppose that $\mu$ is an equilibrium state for $-t\log|Df|$ for $t\in (-\infty,t^+)$. We first note that $\mu$ may be expressed in terms of its ergodic decomposition, see for example \cite[Section 2.3]{Kellbook}, $\mu(\cdot)=\int\mu_y(\cdot)~d\mu(y)$ where $y\in I$ is a generic point of the ergodic measure $\mu_y\in \mathcal{M}_f$. Clearly, for any set $A\subset I$ such that $\mu(A)>0$, the measure $\mu_A(\cdot):=\frac1{\mu(A)}\int_A\mu_y(\cdot)~d\mu(y)$ must have
$$h(\mu_A)-t\lambda(\mu_A)=p(t),$$
i.e. it must be an equilibrium state itself (otherwise, removing $\mu_A$ from the integral for $\mu$ would increase $h_\mu-t\lambda(\mu)$).
As in the proof of Lemma~\ref{lem:pos ent}, $\lambda(\mu_A)>0$.
Theorem~\ref{thm:schemes}(a) implies that any such $\mu_A$ must decompose into a sum $\mu=\sum_n\alpha_n\mu_n$ where $\mu_n$ is a probability measure compatible with the scheme $(X^n,F_n)$ and $\alpha_n\in (0,1]$. Then there are $F_n$-invariant probability measures $\mu_{F_n}$, each of which projects to $\mu_n$ by \eqref{eq:lift}.
By Lemma~\ref{lem:press zero} and \cite{BuSar}, $\mu_{F_n}$ must be the unique equilibrium state for the scheme $(X^n, F_n, \tau_n)$ with potential $-t\log|DF_n|-p(t)\tau_n$. Therefore, $\mu_n$ is the only equilibrium state for $-t\log|Df|$ which is compatible with $(X^n,F_n)$.
We finish the proof by using Theorem~\ref{thm:schemes} b) which implies that any of these equilibrium states compatible with an inducing scheme $(X^n,F_n)$ as above must be compatible with each of the other inducing schemes $(X_j,F_j)$. Hence $\mu_i=\mu_j$ for every $i,j\in {\mathbb N}$. Since $\mu$ was an arbitrary equilibrium state, this argument implies that $\mu$ is ergodic and is the unique equilibrium state for $-t\log|Df|$, as required.
Suppose that $t^+>1$. Since $\lambda_m\geqslant 0$ this means that $t\mapsto p(t)$ must be strictly decreasing in the interval $(1, t^+)$. Since Bowen's formula implies that $p(t)\leqslant 0$ this means that $p(t)<0$. Ruelle's formula \cite{Ruelleineq} then implies that we must have $\lambda_m>0$. Therefore, if $t^+>1$ then $\lambda(\mu)>0$ for all $\mu\in \mathcal{M}_f$ and so we can apply Theorem~\ref{thm:schemes} to the case $t\geqslant t^+$ also.
\end{proof}
\section{Proof of Theorem~\ref{thm:eq_exist_unique}}
\label{sec:main thm}
The previous sections give most of the information we need to prove
Theorem~\ref{thm:eq_exist_unique}. In this section we prove the remaining part: that the critical parameter $t^-$, defined in equation \eqref{eq:t plus minus}, is not finite. We then put the proof of Theorem~\ref{thm:eq_exist_unique} together.
\begin{lema}
There exists a measure $\mu_M$ such that $\lambda(\mu_M)=\lambda_M$.
\label{lem:max meas}
\end{lema}
\begin{proof}
This follows from the compactness of $\mathcal{M}_f$ and the upper semicontinuity of $x\mapsto \log|Df(x)|$.
\end{proof}
\begin{prop}
$t^-=-\infty$.
\label{prop:t^-}
\end{prop}
\begin{proof}
Suppose, for a contradiction, that $t^->-\infty$. This implies that for $t\leqslant t^-$, the measure $\mu_M$ in Lemma~\ref{lem:max meas} also maximises $h(\mu)-t\lambda(\mu)$ for $\mu\in \mathcal{M}_f$, and must have $h(\mu_M)=0$.
By Theorem~\ref{thm:schemes}, we can choose an inducing scheme $(X,F)$ compatible with $\mu_M$.
\begin{claim}
$P^G({\mathcal P}si_{t})=0$ for all $t\leqslant t^-$.
\end{claim}
\begin{proof}
$P^G({\mathcal P}si_{t})\leqslant 0$ follows by Lemma~\ref{lem:press less zero}. $P^G({\mathcal P}si_{t})\geqslant 0$ follows since $\mu_M$ is compatible with our scheme.
\end{proof}
Since by construction, $\mu_M$ is compatible with $(X,F)$, the induced measure being denoted by $\mu_{F,M}$, and since $h(\mu_M)+\int\psi_t~d\mu_M=0$, we have $$h(\mu_{F,M})+\int{\mathcal P}si_t~\mu_{F,M}=0,$$
and so $\mu_{F,M}$ is an equilibrium state for ${\mathcal P}si_t$. However, by Theorems 1.1 and 1.2 of \cite{BuSar} any equilibrium state of ${\mathcal P}si_t$ must have positive entropy, a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:eq_exist_unique}]
The existence of the equilibrium state for $-t\log|Df|$ and $t\in (t^-, t^+)$ follows from Proposition~\ref{prop:conv to eq}. Uniqueness follows from Proposition~\ref{prop:unique}. Positivity of the entropy of $\mu_t$ comes from Lemma~\ref{lem:pos ent}. Finally the fact that $t^-=-\infty$ comes from Proposition~\ref{prop:t^-}.
\end{proof}
\section{The pressure is of class $C^1$ and strictly convex in $(-\infty, t^+)$} \label{sec:smooth and convex}
As discussed in the introduction, for general systems the pressure function $t \mapsto p(t)$ is convex, therefore it can have at most a countable number of first order phase transitions.
In \cite{Saran} an example is constructed with the property that the set of parameters at which the pressure function is not analytic has positive measure (in this case, there also exist \emph{higher order phase transitions}, see \cite{Sarcrit}). Nevertheless, for multimodal maps it has been shown that in certain intervals the pressure function is indeed real analytic, see \cite{BTeqgen,BTeqnat}. Dobbs \cite[Proposition 9]{Dobphase} proved that in the quadratic family $x\mapsto \gamma x(1-x)$, $\gamma \in (3,4)$ there exists uncountably many parameters for which the pressure function admits infinitely many phase transitions. However, these transitions are caused by the existence of an infinite sequence renormalisations of the map, so for these parameters the corresponding quadratic maps do not have a representative in the class $\mathcal{F}$. He also notes \cite[Proposition 4]{Dobphase} that in the quadratic family there is a always a phase transition for negative $t$ caused by the repelling fixed point at 0. Since, this fixed point is not in the transitive part of the system (which actually must be contained in $[f^2(c), f(c)]$), from our perspective this point is not dynamically relevant, so any representative of such a map in $\mathcal{F}$ would miss this part of the dynamics, and hence not exhibit this transition.
\begin{prop} \label{prop:regular}
Let $f \in \mathcal{F}$, the pressure function $p$ is $C^1$ in the interval $(-\infty, t^+)$.
\end{prop}
\begin{proof}
We first show that $p$ is differentiable. By Theorem~\ref{thm:schemes}, we can choose an inducing scheme $(X,F,\tau)$ which is compatible with $\mu_t$ for each $t\in (-\infty, t^+)$. Then we have the limits
$$\lim_{t'\to t}\int\log|DF|~d\mu_{{\mathcal P}si_{t'}}= \int\log|DF|~d\mu_{{\mathcal P}si_{t}} \text{ and } \lim_{t'\to t}\int\tau~d\mu_{{\mathcal P}si_{t'}}= \int\tau~d\mu_{{\mathcal P}si_{t}}.$$ We emphasise that these limits are the same if $t'$ are taken to the left or to the right of $t$. Hence $\lambda(\mu_{\psi_{t}})$ is continuous in $(t^-, t^+)$. Since the derivative of $p$ is $-\lambda(\mu_{\psi_{t}})$,
the derivative is continuous, proving the lemma.
This standard fact can be seen as follows (see also, for example, \cite[Theorem 4.3.5]{Kellbook}): given $\varepsilon>0$, by the definition of pressure the free energy of $\mu_t$ with respect to $\psi_{t+\varepsilon}$ is no more than $p(t+\varepsilon)$. Similarly the free energy of $\mu_{t+\varepsilon}$ with respect to $\psi_{t}$ is no more than $p(t)$. Hence
$$\frac{(-(t+\varepsilon)+t)\lambda(\mu_{t+\varepsilon})}\varepsilon \geqslant \frac{p(t+\varepsilon)-p(t)}\varepsilon \geqslant \frac{(-(t+\varepsilon)+t)\lambda(\mu_{t})}\varepsilon.$$
So whenever $t\mapsto \lambda(\mu_{t})$ is continuous, $Dp(t) = -\lambda(\mu_{t})$.
\end{proof}
\begin{prop}
\label{prop:strictconv}
For $f\in \mathcal{F}$, $t^+>0$ and the pressure function $p$ is strictly convex in $(-\infty, t^+)$.
\end{prop}
Before proving this proposition, we need two lemmas: the first guarantees that $t^+>0$, while the second will be used to obtain strict convexity of the pressure function (both these facts are in contrast with the quadratic Chebyshev case).
\begin{lema}
For $f\in \mathcal{F}$, $\lambda(\mu_0)>\lambda_m$ where $\mu_0$ is the measure of maximal entropy for $f$.
\label{lem:not affine}
\end{lema}
\begin{proof}
The existence of a (unique) measure of maximal entropy $\mu_0$ is guaranteed by \cite{Hofpw}. Suppose for a contradiction that the lemma is false and hence $\lambda(\mu_0)=\lambda_m$. Since when the derivative of $p$ exists at a point $t$, it is equal to $-\lambda(\mu_t)$ (see \cite{Ruellebook} as well as the computation in the proof of Proposition~\ref{prop:regular}) and by convexity, the pressure function must be affine with constant slope $-\lambda_m$. i.e. $p(t)=h_{top}(f)-t\lambda_m$ for $t\in [0, \infty)$.
This implies that $\mu_0$ must be an equilibrium state for the potential $-t\log|Df|$ for every $t\in {\mathbb R}$. In particular this applies when $t=1$. Moreover, by Ruelle's inequality \cite{Ruelleineq}, we have $\lambda(\mu_0)>0$, so $\mu_0$ must be an acip by \cite{Led}. By \cite[Proposition 3.1]{Dobbsvis}, this implies that $f$ has finite postcritical set, which is a contradiction.
\iffalse Moreover, since by Ruelle's inequality \cite{Ruelleineq}, $\lambda_m=\lambda(\mu_0)>0$, the pressure function $p$ is strictly decreasing. This implies that $\mu_0$ must be an equilibrium state for the potential $-t\log|Df|$ for every $t\in {\mathbb R}$. In particular this applies when $t=1$. By \cite{Led} $\mu_0$ must be an acip. By \cite[Proposition 3.1]{Dobbsvis}, this implies that $f$ has finite postcritical set, which is a contradiction.\fi
\end{proof}
\begin{lema}
For any $\varepsilon>0$ there exists an inducing scheme $(X,F)$ a sequence $i_k \to \infty$ such that the domains $X_{i_k}$ have
$$|X_{i_k}| \geqslant e^{-(\lambda_m+\varepsilon)\tau_{i_k}}.$$
\label{lem:good schemes}
\end{lema}
\begin{proof}
It is standard to show that for any $\varepsilon>0$, there exists a periodic point $p$ with Lyapunov exponent $\leqslant \lambda_m +\varepsilon/3$, see for example \cite[Lemma 19]{Dobphase}. We can choose $(X,F)$ as in Theorem~\ref{thm:schemes} so that the orbit of $p$ is disjoint from $X$. We may further assume that $(X,F)$ has distortion bounded by $e^{\delta}$ for some $\delta>0$, i.e. $$\frac{|DF(x)|}{|DF(y)|} \leqslant e^{\delta}$$ for all $x, y\in X_i$ for any $i\in {\mathbb N}$. In this case, by the transitivity of $(I,f)$, which is reflected in our inducing scheme, there must exist an infinite sequence of domains $X_{n_k}$ of $(X,F)$ which shadow the orbit of $p$ for longer and longer. One can use standard distortion arguments to prove that for all large $k$, $|X_i| \geqslant |X| e^{-\delta} e^{-(\lambda_m+\varepsilon/2)\tau_i}$. Choosing $\delta>0$ appropriately completes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:strictconv}]
For the first part of the proposition, $t^+>0$ is guaranteed by Lemma~\ref{lem:not affine}.
For the second part of the proposition, since $p$ is convex, we only have to rule out $p$ being affine in some interval.
Suppose first that $p$ is affine in an interval $[t_1, t_2] \subset (-\infty, t^*)$ where $t^*:=\inf\{t:Dp(t) = -\lambda_m\}$. I.e. for some $\beta>\lambda_m$, $t\in [t_1, t_2]$ implies $p(t) = p(-t_1)-(t-t_1)\beta$. We let $\varepsilon>0$ be such that $\beta>\lambda_m+\varepsilon$.
By Lemma~\ref{lem:good schemes}, there exists $i_k\to \infty$ such that
$$|X_{i_k}| \geqslant e^{-(\lambda_m+\varepsilon)\tau_{i_k}}.$$
The fact that the pressure function is affine in $[t_1, t_2]$ implies that the equilibrium state is the same $\mu$ for every $t\in [t_1, t_2]$. Denote the induced version of $\mu$ by $\mu_F$.
By the Gibbs property of our inducing schemes, $\mu_F(X_i) \asymp |X_i|^te^{-\tau_ip(t)}$ for all $t\in [t_1, t_2]$. Therefore, $$\frac{|X_i|^{t_1} e^{-\tau_ip(t_1)}}{|X_i|^t e^{-\tau_i(p(t_1) -(t-t_1)\beta)}}\asymp 1,$$
which implies that $$|X_i| \asymp e^{-\tau_i\beta}$$ for all $i$.
Since
$$|X_{i_k}| \geqslant e^{-(\lambda_m+\varepsilon)\tau_{i_k}}$$ for an infinite sequence of domains $X_{i_k}$, and $\beta>\lambda_m+\varepsilon$, this yields a contradiction.
We next want to prove that $t^+ =t^*$. We suppose not in order to get a contradiction. In the first case suppose that $\lambda_m=0$. Then $p(t)\geqslant 0$ for all $t\in {\mathbb R}$. Coupled with Bowen's formula this implies that $p(1)= 0$. So the convexity of $p$ implies $t^+=t^*$, as required. Now suppose that $\lambda_m>0$. Since we assumed $t^+$ the graph of $p(t)$ must be above, and parallel to $t\mapsto -t\lambda_m$ on $[t^*,\infty)$. This implies that $t^+=\infty$ and so Theorem~\ref{thm:eq_exist_unique} gives equilibrium states for all $t\in {\mathbb R}$. Hence we can mimic the argument above, with the inducing scheme as in Theorem~\ref{thm:schemes} compatible with $\mu_{t^*}$, but instead taking $[t_1,t_2] \subset [t^*, \infty)$ and $\beta=\lambda_m$. Noting that the argument of Lemma~\ref{lem:good schemes} ensures that we chose the scheme $(X,F)$ so that there is a sequence of domains $|X_{i_k}| \leqslant e^{-(\lambda_M-\varepsilon)\tau_{i_k}}$, we can complete the argument.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:smooth}]
The convexity of $p$ follows from Proposition~\ref{prop:strictconv}, the smoothness from \ref{prop:regular} and the fact that the pressure is decreasing from \cite{Prz}.
\end{proof}
\section{Phase transitions in the positive spectrum}
\label{sec:kinks acips}
In this section we study the relation between the existence of first order phase transitions at the point $t=1$ and the existence of an acip. The following proposition has Proposition~\ref{prop:acip} as a corollary.
\begin{prop}
Suppose that $f\in \mathcal{F}$ has $\lambda_m=0$. Then $f$ has an acip if and only if $p$ has a first order phase transition at $t=1$. \label{prop:acip kink}
\end{prop}
\begin{rem}
Note that if $\lambda_m>0$ then the situation is quite different. For example if $f\in \mathcal{F}$ satisfies the Collet-Eckmann condition (which by \cite{BrvS} implies $\lambda_m>0$), in which case the map also has an acip, then by \cite[Theorem 3]{BTeqnat}, $p$ is real analytic in a neighbourhood of $t=1$.
\end{rem}
The following lemma will be used to prove Proposition~\ref{prop:acip kink}.
\begin{lema}
Suppose that $t^+\in (0, \infty)$ and there is a first order phase transition at $t^+$. Then there exists an inducing scheme $(X,F)$, an equilibrium state $\mu_{{\mathcal P}si}$ for ${\mathcal P}si={\mathcal P}si_{t^+}$, and an equilibrium state $\mu_\psi$ for $\psi=\psi_{t^+}$ with $h(\mu_\psi)>0$.
\label{lem:meas at kink}
\end{lema}
\begin{proof}
The fact that there is a first order phase transition implies that the left derivative of $p$ at $t^+$ has $Dp^-(t^+)<-\lambda_m$.
The convexity of the pressure function implies that the graph of the pressure lies above the line $t\mapsto D^-p(t^+)t-t^+(\lambda_m+D^-p(t^+))$.
This means that we can take a sequence of equilibrium states $\mu_{\psi_t}$ for $t$ arbitrarily close to, and less than, $t^+$ with free energy converging to $p(t^+)$ with
$$h(\mu_{\psi_t}) \geqslant -t^+(Dp^-(t^+)+\lambda_m)>0.$$ Hence the arguments used to prove Proposition~\ref{prop:conv to eq} give us an equilibrium state for $\mu_{t^+}$ with positive entropy.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:acip kink}]
If there exists an acip $\mu$ then $Dp^-(1)=-\lambda(\mu)<0$. Since $\lambda_m=0$ implies $p(t)=0$ for all $t\geqslant 1$, the existence of an acip implies that there is a first order phase transition at $t=1$.
On the other hand, if there exists a first order phase transition at $t=1$ then Lemma~\ref{lem:meas at kink} implies that there is an equilibrium state $\mu_1$ for $-\log|Df|$, with $h(\mu_1)>0$. By \cite{Led} this must be an acip.
\end{proof}
\begin{rem}
If $\lambda_m>0$ and there is a measure $\mu_m$ such that $\lambda(\mu_m)=\lambda_m$, then by Lemma~\ref{lem:meas at kink} and the arguments in the proof of Proposition~\ref{prop:t^-}, we have that $t^+=\infty$.
\label{rmk:min meas}
\end{rem}
\begin{rem}
There are examples of maps in $\mathcal{F}$ with $\{\mu\in \mathcal{M}_f:\lambda(\mu)=\lambda_m\} = {\emptyset}$, for example \cite[Lemma 5.5]{BrKell}, a quadratic map in $\mathcal{F}$ is defined so that $\lambda_m=0$, but there are no measures with zero Lyapunov exponent. There are also examples of maps $f\in \mathcal{F}$ with $\{\mu\in \mathcal{M}_f:\lambda(\mu)=\lambda_m\} \neq {\emptyset}$, for example in \cite{Brminim} examples of quadratic maps in $\mathcal{F}$ are given for which the omega-limit set of the critical point supports (multiple) ergodic measures with zero Lyapunov exponent. Moreover, Cortez and Rivera-Letelier \cite{CJ} proved that given $\mathcal{E}$ a non-empty, compact, metrisable and totally disconnected topological space then there exists a parameter $\gamma \in (0,4]$ such that set of invariant probability measures of $x\mapsto \gamma x(1-x)$, supported on the omega-limit set of the critical point is homeomorphic to $\mathcal{E}$.
\end{rem}
It is plausible that there are maps $f\in \mathcal{F}$ for which
\begin{equation*}
\inf \leqslantft\{ t \in \mathbb{R} : p(t) \leqslant 0 \right\} <1.
\end{equation*}
However, the following argument shows that this is not true for unimodal maps with quadratic critical point in $\mathcal{F}$.
Given an interval map $f:I \to I$, we say that $A\subset I$ is a \emph{metric attractor} if $B(A):=\{\omega(x)\subset A\}$ has positive Lebesgue measure and there is no proper subset of $A$ with this property. On the other hand $A$ is a \emph{topological attractor} if $B(A)$ is residual and there is no proper subset of $A$ with this property. We say that $f$ has a \emph{wild attractor} if there is a set $A$ which is a metric attractor, but not a topological one.
\begin{prop}
\label{prop:pos press}
If $f\in \mathcal{F}$ is a unimodal map with no wild attractor then for $t<1$, $p(t)>0$.
\end{prop}
\begin{rem} It was shown in \cite{BKNS} that there are unimodal maps with wild attractors in $\mathcal{F}$. However, if $\ell_c=2$ then this is not possible by \cite{Lyuwild}.
\label{rmk:wild}
\end{rem}
\begin{lema}\label{lem:start}
If $f\in \mathcal{F}$ is a unimodal map with no wild attractor then for each $\varepsilon>0$ there exists a measure $\mu\in \mathcal{M}_f$ so that $$\frac{h(\mu)}{\lambda(\mu)}>1-\varepsilon.$$
\end{lema}
\begin{proof}
By \cite[Theorem V.1.4]{MSbook}, originally proved by Martens, there must be an inducing scheme $(X,F)$ such that $Leb(X\setminus \cup_i X_i)=0$. For any $\delta>0$ we can truncate $(X,F)$ to a finite scheme $(X^N, F_N)$ where $X^N =\cup_{i=1}^N X_i$ so that $Leb(\cup_{i=1}^N X_i) > (1-\delta)|X|.$
We therefore have
$$\dim_H\leqslantft\{x:\tau^k(x)<\infty\text{ for all } k\in {\mathbb N}\right\}>1-\delta'$$ where $\delta'$ depends on $\delta$ and the distortion of $F$ (in particular $\to 0$ as $\delta\to 0$).
It follows from the Variational Principle and the Bowen formula (see \cite[Chapter 7]{pe}) that there is an $F$-invariant measure, $\mu_F$, for this system with $$\frac{h(\mu_F)}{\lambda(\mu_F)} >1-\delta'.$$
By the Abramov formula, for $\mu$ the projection of $\mu_F$,
$$\frac{h(\mu)}{\lambda(\mu)}>1 -\delta'$$
also. Choosing $\delta>0$ so small that $\delta'\leqslant\varepsilon$ completes the proof.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:pos press}]
Let $t<1$ and choose $\varepsilon=1-t>0$. Then the measure $\mu$ in Lemma~\ref{lem:start} has $h(\mu)-t\lambda(\mu)>0$. Hence by the definition of pressure, $p(t)>0$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:collected results}]
By Proposition~\ref{prop:pos press} and Remark~\ref{rmk:wild} we can take $t^+=1$. Hence we can conclude that $p$ is $C^1$ strictly convex decreasing in $(-\infty, t^+)$ by Theorem~\ref{thm:smooth}. The fact that $p(t)=0$ for all $t\geqslant 1$ follows from \cite{NoSa}.
Part (a) follows from Proposition~\ref{prop:acip kink} since this implies that both left and right derivatives of $p(t)$ at $t=1$ are zero. Part (b) is the converse of this since the left derivative is strictly negative and the right derivative is zero.
\end{proof}
\section{Remarks on statistical properties and Chebyshev polynomials} \label{sec:remarks}
In this section we collect some further comments on our results.
\subsection{Statistical properties}
Given $f\in \mathcal{F}$ and an equilibrium state $\mu$ as in Theorem~\ref{thm:eq_exist_unique}, one can ask about the statistical properties of the system $(I,f,\mu)$. For general equilibrium states we expect it should be possible to prove exponential decay of correlations along with many other statistical laws. From the theory presented here allied to \cite{BTret}, one can show that the system $(I, f,\mu)$ has `exponential return time statistics' (for definitions see for example \cite{BTret}).
\subsection{Ergodic Optimisation}
Let $f \in \mathcal{F}$ and $\varphi: [0,1] \to \mathbb{R}$ a function. The study of invariant probability measures whose ergodic
$\varphi-$average is as large (or as small) as possible is known as
\emph{ergodic optimisation}. A measure $\mu \in \mathcal{M}_f$ is called $\varphi-$minimising/maximising if
\[\int \varphi \ d\mu = \inf \leqslantft\{ \int \varphi \ d\nu : \nu \in \mathcal{M}_f \right\} \text{ or } \int \varphi \ d\mu = \sup \leqslantft\{ \int \varphi \ d\nu : \nu \in \mathcal{M}_f \right\} \]
respectively.
For a survey on the subject see \cite{OJ}. Let $t\in (-\infty, t^+)$ and denote by $\mu_t$ the unique equilibrium state corresponding to the potential $-t\log|Df|$. A consequence of the results in this paper is that: any accumulation point $\mu$ of a sequence of measures $\mu_{t_n}$, given by Theorem~\ref{thm:eq_exist_unique}, where $t_n \to -\infty$ is a $\log|Df|$-maximising measure. This is because $\log|Df|$ is upper semicontinuous; $Dp(t)=-\lambda(\mu_t)$; and this derivative is asymptotic to $-\lambda_M$. Hence there is a subsequence of these measures $(\mu_{t_{n_k}})_k$ so that $$\lim_{k\to \infty} \lambda(\mu_{t_{n_k}})=\lambda(\mu)=\lambda_M.$$
Note that Lemma~\ref{lem:max meas} guarantees the existence of a $\log |Df|-$maximising measure. (We do not assert anything about the uniqueness of this measure.) Actually, any measure $\mu$, which is an accumulation point of $\mu_{t_n}$ as $t_n \to -\infty$, is a measure maximising entropy among all measures which maximise $\log |Df|$. Then in fact $p(t)$ is asymptotic to the line $h(\mu)-t\lambda_M$ as $t\to -\infty$.
\iffalse
If there is no kink at $t=t^+$ then any accumulation point of a sequence $(\mu_{t_n})_n$ where $t_n\nearrow t^+$ is a $\log |Df|-$minimising measure. As in Remark~\ref{rmk:min meas}, having a kink at $t^+$ with $t^+>1$ is incompatible with having a minimising measure.
\fi
\subsection{The preperiodic critical point case}
\label{sec:preper}
For our class of maps $\mathcal{F}$ we assumed that the orbit of points in ${\mathcal Cr}$ are infinite. Here we comment on an alternative case.
In the case of the quadratic Chebychev polynomial $x\mapsto 4x(1-x)$ on $I$, it is well known that the two relevant measures are the acip $\mu_1$, which has $\lambda(\mu_1)=\log 2=\lambda_m$, and the Dirac measure $\delta_0$ on the fixed point at 0, which has $\lambda(\delta_0)= \log 4=\lambda_M$. So $t^-=-1$ and
\begin{equation*}
p(t)= \begin{cases} (1-t)\log 2& \text{if } t\geqslant -1,\\
-t\log 4& \text{if } t \leqslant -1.
\end{cases}
\end{equation*}
Note that the above piecewise affine form for the pressure function does not conflict with Theorem~\ref{thm:smooth}, which might be expected to apply in the interval $(t^-, t^+)$, since $t^+=t^*=-1$, where $t^*$ is defined in the proof of Proposition~\ref{prop:strictconv}.
\iffalse
\begin{rem}[Return time statistics] Given $f\in \mathcal{F}$ and an equilibrium state $\mu$ as in Theorem~\ref{thm:eq_exist_unique}, one can ask about the statistical properties of the system $(I,f,\mu)$. For general equilibrium states we expect it should be possible to prove exponential decay of correlations along with many other statistical laws. \textbf{MT:tweaked} From the theory presented here allied to \cite{BTret}, one can show that the system $(I, f,\mu)$ has `exponential return time statistics' (for definitions see for example \cite{BTret}).
\end{rem}
\begin{rem}[Ergodic Optimisation]
Let $f \in \mathcal{F}$ and $\varphi: [0,1] \to \mathbb{R}$ a function. The study of invariant probability measures whose ergodic
$\varphi-$average is as large (or as small) as possible is known as
\emph{ergodic optimisation}. A measure $\mu \in \mathcal{M}_f$ is called $\varphi-$minimising if
\[\int \varphi \ d\mu = \inf \leqslantft\{ \int \varphi \ d\nu : \nu \in \mathcal{M}_f \right\}. \]
For a survey on the subject see \cite{OJ}. Let $t\in (-\infty, t^+)$ and denote by $\mu_t$ the unique equilibrium state corresponding to the potential $-t\log|Df|$. A consequence of the results in this paper is that:
If there is no kink at $t=t^+$ then any accumulation point of a sequence $(\mu_{t_n})_n$ where $t_n\nearrow t^+$ is a $\log |Df|-$minimising measure. As in Remark~\ref{rmk:min meas}, having a kink at $t^+$ with $t^+>1$ is incompatible with having a minimising measure.
\end{rem}
\fi
\appendix
\section{Cusp maps}
In this section we outline how to extend the above results to some maps which are not smooth. This class includes the class of contracting Lorenz-like maps, see for example \cite{Rovella}.
\begin{defi}
$f:\cup_jI_j\to I$ is a \emph{cusp map} if there exist constants $C,\alpha>1$ and a set $\{I_j\}_j$ is a finite collection of disjoint open subintervals of $I$ such that
\begin{enumerate}
\item $f_j:=f|_{I_j}$ is $C^{1+\alpha}$ on each $I_j=:(a_j, b_j)$ and $|Df_j|\in (0, \infty)$.
\item $D^+f(a_j), \ D^-f(b_j)$ exist and are equal to 0 or $\pm\infty$.
\item For all $x,y\in \overline{I_j}$ such that $0<|Df_j(x)|, |Df_j(y)|\leqslant 2$ we have $|Df_j(x)-Df_j(y)|<C|x-y|^\alpha$.
\item For all $x,y\in \overline{I_j}$ such that $|Df_j(x)|, |Df_j(y)|\geqslant 2$, we have $|Df_j^{-1}(x)-Df_j^{-1}(y)|<C|x-y|^\alpha$.
\end{enumerate}
We denote the set of points $a_j, b_j$ by ${\mathcal Cr}$.
\end{defi}
\begin{rem}
Notice that if for some $j$, $b_j=a_{j+1}$, i.e. $I_j\cap I_{j+1}$ intersect, then $f$ may not continuously extend to a well defined function at the intersection point $b_j$, since the definition above would then allow $f$ to take either one or two values there. So in the definition above, the value of $f_j(a_j)$ is taken to be $\lim_{x\searrow a_j}f_j(x)$ and $f_j(b_j)=\lim_{x\nearrow b_j}f_j(x)$, so for each $j$, $f_j$ is well defined on $\overline{I_j}$.
\label{rmk:boundaries}
\end{rem}
\begin{rem}
In contrast to the class of smooth maps $\mathcal{F}$ considered previously in this paper, for cusp maps we can have $\lambda_M=\infty$ and/or $\lambda_m=-\infty$. The first possibility follows since we allow singularities (points where the one-sided derivative is $\infty$). The second possibility follows from the presence of critical points (although is avoided for smooth multimodal maps with non-flat critical points by \cite{Prz}). Examples of both of these possibilities can be found in \cite[Section 3.4]{Dobthes}.
\label{rmk:lyap infinity}
\end{rem}
We will ultimately be interested in cusp maps without singular points with negative Schwarzian derivative (in fact the latter rules out the former). Note that since we are only interested in the transitive parts the system, transitive multimodal maps as in the rest of the paper can be considered to fit into this class.
We show below that we can build a Hofbauer extension $(\hat I, \hat f)$.
We note that the possible issue of $f$ not being well defined at the boundaries of $I_j$, discussed in Remark~\ref{rmk:boundaries}, does not change anything in the definition of the Hofbauer tower.
We next define the Hofbauer extension. The setup we present here can be applied to general dynamical systems, since it only uses the structure of dynamically defined cylinders. An alternative way of thinking of the Hofbauer extension specifically for the case of multimodal interval maps, which explicitly makes use of the critical set, is presented in \cite{BrBr}.
We let ${\rm C}_n[x]$ denote the member of ${\mathcal P}_n$, which defined as above, containing $x$. If $x\in \cup_{n\geqslant 0}f^{-n}({\mathcal Cr})$ there may be more than one such interval, but this ambiguity will not cause us any problems here.
The \emph{Hofbauer extension} is defined as $$\hat
I:=\bigsqcup_{k\geqslant 0}\bigsqcup_{{\rm C}_{k}\in {\mathcal P}_{k}}
f^k({\rm C}_{k})/\sim$$ where $f^k({\rm C}_{k})\sim
f^{k'}({\rm C}_{k'})$ as components of the disjoint union $\hat I$ if $f^k({\rm C}_{k})= f^{k'}({\rm C}_{k'})$ as subsets in $I$. Let
${\mathcal D}$ be the collection of domains of $\hat I$ and $\pi:\hat
I \to I$ be the natural inclusion map. A point $\hat x\in \hat I$ can
be represented by $(x,D)$ where $\hat x\in D$ for $D\in {\mathcal D}$ and
$x=\pi(\hat x)$. Given $\hat x\in \hat I$, we can denote the domain $D\in {\mathcal D}$ it belongs to by $D_{\hat x}$.
The map $\hat f:\hat I \to \hat I$ is defined by
$$\hat f(\hat x) = \hat f(x,D) = (f(x), D')$$
if there are cylinder sets ${\rm C}_k \supset {\rm C}_{k+1}$ such that $x \in
f^k({\rm C}_{k+1}) \subset f^k({\rm C}_{k}) = D$ and $D' = f^{k+1}
({\rm C}_{k+1})$.
In this case, we write $D \to D'$, giving $({\mathcal D}, \to)$ the
structure of a directed graph. Therefore, the map $\pi$
acts as a semiconjugacy between $\hat f$ and $f$: $$\pi\circ \hat
f=f\circ \pi.$$
We denote the `base' of $\hat I$, the copy of $I$ in $\hat I$, by $D_0$. For $D\in {\mathcal D}$, we define $\leqslantvel(D)$ to be the length of the shortest path $D_0 \to \dots \to D$ starting at the base $D_0$. For each $R \in {\mathbb N}$, let $\hat I_R$ be the compact
part of the Hofbauer tower defined by
$$
\hat I_R := \sqcup \{ D \in {\mathcal D} : \leqslantvel(D) \leqslant R \}.$$
For maps in $\mathcal{F}$, we can say more about the graph structure of $({\mathcal D}, \to)$ since Lemma 1 of \cite{BTeqnat} implies that if $f\in \mathcal{F}$ then there is a closed primitive subgraph ${\mathcal D}_{{\mathcal T}}$ of ${\mathcal D}$. That is, for any $D,D' \in{\mathcal D}_{{\mathcal T}}$ there is a path $D\to \cdots \to D'$; and for any $D\in {\mathcal D}_{{\mathcal T}}$, if there is a path $D\to D'$ then $D'\in {\mathcal D}_{{\mathcal T}}$ too. We can denote the disjoint union of these domains by $\hat I_{{\mathcal T}}$. The same lemma says that if $f\in \mathcal{F}$ then $\pi(\hat I_{{\mathcal T}})={\mathcal O}mega$, the non-wandering set and $\hat f$ is transitive on $\hat I_{{\mathcal T}}$. Theorem~\ref{thm:cusp facts} gives these properties for transitive cusp maps.
Given an ergodic measure $\mu\in \mathcal{M}_f$, we say that $\mu$ \emph{lifts to $\hat I$} if there exists an ergodic $\hat f$-invariant probability measure $\hat\mu$ on $\hat I$ such that $\hat\mu\circ\pi^{-1}=\mu$. For $f\in \mathcal{F}$, if $\mu\in \mathcal{M}_f$ is ergodic and $\lambda(\mu)>0$ then $\mu$ lifts to $\hat I$, see \cite{Kellift, BrKell}.
Property $(*)$ is that for any $\hat x, \hat y\notin \partial \hat I$ with $\pi(x)=\pi(y)$ there exists $n$ such that $\hat f^n(\hat x)=\hat f^n(\hat y)$. This follows for cusp maps by the construction of $\hat I$ using the branch partition.
We will only use the following result in the context of equilibrium states for cusp maps with no singularities. However, for interest we state the theorem in greater generality.
\begin{teo}
Suppose that $f:I \to I$ is a transitive cusp map with $h_{top}(f)>0$. Then:
\begin{enumerate}
\item there is a transitive part $\hat I_{{\mathcal T}}$ of the tower such that $\pi(\hat I_{{\mathcal T}})=I$;
\item any measure $\mu\in \mathcal{M}_f$ with $0<\lambda(\mu)<\infty$ lifts to $\hat\mu$ with $\mu=\hat\mu\circ \pi^{-1}$;
\item for each $\varepsilon>0$ there exists $\eta>0$ and a compact set $\hat K\subset \hat I_{{\mathcal T}}\setminus \partial \hat I$ such that any measure $\mu\in \mathcal{M}_f$ with $h(\mu)>\varepsilon$ and $0<\lambda(\mu)<\infty$ has $\hat\mu(\hat K)>\eta$.
\end{enumerate}
\label{thm:cusp facts}
\end{teo}
\begin{proof}
\text{Part (1):}
The first part can be shown as in \cite[Lemma 2]{BTeqnat}, but we argue as in \cite{Hofpw} (see also \cite[Theorem 6]{KellStP}). Theorem 11 of that paper gives a decomposition of $\hat I$ into a countable union $\Gamma$ of irreducible (maximal with these properties) closed (if there is a path $D\to D'$ for $D\in {\mathcal E}$ then $D'\in {\mathcal E}$) primitive (there is a path between any $D,D'\in {\mathcal E}$) subgraphs ${\mathcal E}$ along with some sets which carry no entropy. Since $h_{top}(\hat f)=h_{top}(f)$ and we have positive topological entropy, this means that $\Gamma \neq {\emptyset}$. Let ${\mathcal E}\in \Gamma$. Clearly $\pi({\mathcal E})$ is open, so by the transitivity of $f$, there must be a point $x\in \pi({\mathcal E})$ which has a dense orbit in $I$. By definition, $\omega(x)\subset\pi({\mathcal E})$.
By property $(*)$, $\pi({\mathcal E})\cap \pi({\mathcal E}') = {\emptyset}$ for any ${\mathcal E},{\mathcal E}'\in \Gamma$ which implies that $\#\Gamma =1$. That there is a dense orbit in $({\mathcal E},\hat f)$ follows from the Markov property of this subgraph, so we let $\hat I_{{\mathcal T}}=\sqcup_{D\in {\mathcal E}}D$.
\text{Part (2):}
Ledrappier, in \cite[Propostion 3.2]{Led} proved the existence of non-trivial local unstable manifolds for a more general class of maps (so-called PC-maps) with an ergodic measure $\mu\in \mathcal{M}_f$ with $\lambda(\mu)>0$. However, he also required a non-degeneracy condition. For cusp maps, Dobbs \cite[Theorem 13]{Dobcusp} was able to this but without the non-degeneracy requirement.
Keller showed in \cite[Theorem 6]{Kellift} that the existence of such unstable manifolds means that any non-atomic ergodic measure $\mu\in \mathcal{M}_f$ with $\lambda(\mu)>0$ lifts to $\hat\mu$ on $(\hat I, \hat f)$ and that $\mu= \hat\mu\circ\pi^{-1}$. Using Dobbs and assuming that $\mu$ is not supported on $\cup_{n\geqslant 0}f^n({\mathcal Cr})$ we can drop the non-atomic assumption (see also \cite[Theorem 3.6]{BrKell}).
\text{Part (3):}
The third part follows exactly as in \cite[Lemma 4]{BTeqnat}.
\end{proof}
Suppose now that $f$ is a cusp map without singularities (i.e. $|Df|$ is bounded above), with negative Schwarzian and such that the non-wandering set ${\mathcal O}mega$ is an interval. We consider $f:{\mathcal O}mega \to {\mathcal O}mega$.
For each $t\in (t^-, t^+)$, we can find a finite number of inducing schemes as in Proposition~\ref{thm:schemes} with which all measures with large enough free energy w.r.t. $\psi_t$ will be compatible. It is important here that we assume negative Schwarzian derivative since we need bounded distortion for our inducing schemes. This then allows us to prove Theorem~\ref{thm:eq_exist_unique} for this class of maps, but we may have $t^->-\infty$. If we exclude maps with preperiodic critical points then we again have $t^-=-\infty$. Similarly we can prove Theorem~\ref{thm:smooth} for this class of maps, although again we only get $t^-=-\infty$ if we exclude maps with preperiodic critical points. Note also that the fact that $\lambda_m$ can be negative, and may even be $-\infty$, implies that $t^+$, which for the class $\mathcal{F}$ had to lie in $[1,\infty]$, could be any value in the range $[0,\infty]$ for cusp maps.
Note that for the maps considered by Rovella in \cite{Rovella}, the critical values are periodic and so the measure supported on them is not seen by our inducing schemes. This is like the Chebyshev case, so as in that situation, the pressure function could be piecewise affine.
\iffalse
\textbf{MT: I'm beginning to wonder how useful this subsection is. It seems rather obvious, no?}
Let $f:[0,1] \to [0,1]$ be a dynamical system and $\varphi: [0,1] \to \mathbb{R}$ a function. The study of invariant probability measures whose ergodic
$\varphi-$average is as large (or as small) as possible is known as
\emph{ergodic optimisation}. A measure $\mu \in \mathcal{M}_f$ is called $\varphi-$maximising if
\[\int \varphi \ d\mu = \sup \leqslantft\{ \int \varphi \ d\nu : \nu \in \mathcal{M}_f \right\}. \]
The study of such measures is closely related to the study of points $x \in [0,1]$ for which the time average
\[ \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \varphi(f^i x), \]
is a large as possible. For a survey on the subject see \cite{OJ}.
If the potential $\varphi$ is upper-semicontinuous and the space of invariant measures is compact then maximising measures always exist. Some of them can be obtained as accumulation points of equilibrium states. Indeed:
\begin{prop}
Let $f \in \mathcal{F}$ and $\varphi\in \mathcal{P}\cap USC$. Denote by $\mu_t$ the unique equilibrium state corresponding to the potential $t \varphi$, with $t \in (0, t^+)$.
If there is no first order phase transition at $t=t^+$ then any accumulation point of a sequence $(\mu_{t_n})_n$ where $t_n\nearrow t^+$ is a $\varphi-$maximising measure.
\end{prop}
\begin{proof}
Note that the measures $\mu_{t_n}$ do exist in virtue of Theorem \ref{thm:eq cts pots}. Since there is no first order phase transition at $t=t^+$ (or equivalently, if $t^+=\infty$) then the number
\[M= \lim_{n \to \infty} P'(t \varphi) \]
corresponds to the slope of the asymptote of the pressure function
$t \to P(t \varphi)$. Since the pressure function is convex, we have that
the graph $t \to P(t \varphi)$ lies above (or it is equal) to the line
$t \to h(\nu) + t \int \varphi \ d \nu$, for every $\nu \in \mathcal{M}_f$. Therefore
\[ M \geqslant \int \varphi \ d \nu \]
for every invariant measure $\nu$. The result now follows since for any accumulation point $\mu$ of $\{\mu_{t_n} \}$ we have
\[M = \int \varphi \ d \mu.\]
\end{proof}
We stress that, contrary to the standard argument in theory \cite[Lemma 2]{je}, in the above result we are not assuming the pressure to be differentiable in the whole interval $(0, \infty)$.
We can also address the problem of finding maximising measures for the potential $-\log |Df|$ (which is not upper-semicontinuous). Indeed
\begin{prop}
Let $f \in \mathcal{F}$. Denote by $\mu_t$ the unique equilibrium state corresponding to the potential $-t \log |Df|$, with $t \in (0, t^+)$.
If there is no first order phase transition at $t=t^+$ (or if $t^+= \infty$) then any accumulation point of a sequence $(\mu_{t_n})_n$ where $t_n\to t^+$ is a $\varphi-$maximising measure.
\end{prop}
\textbf{MT: seems a bit obvious. No first order phase transition at $t^+$ means no first order phase transition at 1, so presumably if $t^+<\infty$ then any accumulation point of $\mu_n$ must be an infinite measure.}
\begin{proof}
The existence of the measures $\{ \mu_{t_n} \}$ is guaranteed by Theorem
\ref{thm:eq_exist_unique}. The rest of the proof follows as in Proposition \ref{opti-bounded}
\end{proof}
Note that for the potential $-\log |Df|$ it is possible that no maximising measure exists. Consider, for instance, the example of Bruin and Keller \cite[Lemma 5.5]{BrKell}.
Nevertheless, if $t^->-\infty$ then there is a maximising measure for $\log|Df|$, and if $t^+\in (1,\infty)$ there is a maximising measure for $-\log|Df|$.
\textbf{MT: We need to make a comment like: if $t^->-\infty$ then there is a maximising measure for $\log|Df|$, and if $t^+\in (1,\infty)$ there is a maximising measure for $-\log|Df|$.}
\textbf{GI: added the comment. MT: I changed it again since strict convexity rules out both $t^->-\infty$ and $t^+\in (1,\infty)$. This is precisely why I'm a bit dubious about this subsection these days}.
\fi
\end{document} | math | 110,210 |
\begin{document}
\title{ On Borel summation and Stokes phenomena for rank one
nonlinear systems of ODE's }
\author{Ovidiu Costin \thanks {Mathematics Department,
University of Chicago, 5734 University Avenue, Chicago
IL 60637; e-mail: costin\symbol{64}math.uchicago.edu}}
\date{ }
\maketitle
\hyphenation{trans-series}
\section{Introduction}
In this paper we study analytic (linear or) nonlinear systems of
ordinary differential equations, at an irregular singularity of rank
one, under nonresonance conditions. It is shown that the formal
asymptotic exponential series solutions ({\mathrm{e}m transseries}
solutions: countable linear combinations of formal power series
multiplied by small exponentials) are Borel summable in a
generalized sense along any direction in which the exponentials
decay. Conversely, any solution that decreases along some direction
is the Borel sum of a transseries.
The summation procedure introduced is an extension of Borel
summation which is linear, multiplicative, commutes with
differentiation and complex conjugation. The summation algorithm
uses the formal solutions alone (and not the differential
equation that they solve). Along singular (Stokes) directions, the
functions reconstructed by summation are shown to be given by Laplace integrals
along special paths, a subset of \'Ecalle's median paths.
The one-to-one correspondence established between actual solutions
and generalized Borel sums of transseries is constant between Stokes
lines and {changes} if a Stokes line is crossed (local Stokes
phenomenon). We analyze the connection between local and classical
Stokes phenomena.
We study the analytic properties of the Borel (formal inverse
Laplace) transform of the series contained in the transseries of the
transseries and give a systematic description of their
singularities. These Borel transforms satisfy a hierarchy of
convolution equations, for which we give the general solution in a
space of hyperfunctions. In addition, we show that they are {\mathrm{e}m
resurgent functions} in the sense of \'Ecalle.
The summation
procedure is not unique; we classify all proper
extensions of Borel summation to transseries solutions of nonresonant
systems.
We find formulas connecting the
different series contained in the transseries among themselves (resurgence
equations). Resurgence turns out to be closely linked to the local
Stokes phenomenon.
The connection to Berry's hyperasymptotics and applications to the
classification of differential equations are briefly discussed.
\subsection{General setting}
We consider the differential system
\begin{eqnarray}
\label{eqor1}
{\bf y}'=\mathbf{f}(x,{\bf y}) \qquad {\bf y}\mathrm{i}n\mathbb{C}^n
\mathrm{e}nd{eqnarray}
\noindent under the following {\mathrm{e}m assumptions:}
(a1) The function $\mathbf{f}$ is analytic
at $(\mathrm{i}nfty,0)$.
(a2) Nonresonnance: the eigenvalues $\lambda_i$ of the linearization
\begin{eqnarray}
\label{linearized}
\hat{\Lambda}:=-\left(\frac{\partial f_i}{\partial
y_j}(\mathrm{i}nfty,0)\right)_{i,j=1,2,\ldots n}
\mathrm{e}nd{eqnarray}
\noindent are linearly independent over $\mathbb{Z}$ (in particular nonzero) and
such that the Stokes lines are distinct (a somewhat less
restrictive condition is actually used, cf. \S\ref{nonres}).
\noindent {\mathrm{e}m Normalization}. It is convenient
to prepare (\ref{eqor1}) in the following way. Pulling out the inhomogeneous
and the linear terms (relevant to leading order
asymptotics) we get
\begin{eqnarray}\label{eqor}
{\bf y}'={\bf f}_0(x)-\hat\Lambda {\bf y}-
\frac{1}{x}\hat B {\bf y}+{\bf g}(x,{\bf y})
\mathrm{e}nd{eqnarray}
Under the assumptions (a1) and (a2), by means of normal form
calculations it is possible to arrange (\ref{eqor}) so that
(\cite{Wasow}, \cite{To1})
(n1)
$\hat\Lambda=\mbox{diag}(\lambda_i)$ and
(n2) $\hat B=\mbox{diag}(\beta_i)$
\noindent For convenience, we rescale $x$ and
reorder the components of ${\bf y}$ so that
(n3) $\lambda_1=1$, and, with $\phi_i=\arg(\lambda_i)$, we have
$\phi_i<\phi_j$ if $i<j$. To simplify notations, we formulate some of
our results relative to $\lambda_1$; they can be easily adapted to any
other eigenvalue.
To unify the treatment we
make, by taking ${\bf y}={\bf y}_1 x^{-N}$ for some $N>0$,
(n4) $\Re(\beta_j)<0,\ j=1,2,\ldots,n$.
\noindent (there is an asymmetry at this point: the opposite inequality cannot be
achieved, in general, as simply and without violating analyticity at
infinity). Finally, through a transformation of the form
${\bf y}\leftrightarrow{\bf y}-\sum_{k=1}^M{\bf a}_k x^{-k}$ we arrange that
(n5) $ \mathbf{f}_0=O(x^{-M-1})\mbox{ and }\mathbf{g}(x,{\bf y})=
O({\bf y}^2,x^{-M-1}{\bf y}) $. We choose $M>1+\max_i\Re(-\beta_i)$.
{\mathrm{e}m Formal solutions.} In prepared form, given (a1)
and (a2), (\ref{eqor}) admits
an $n$--parameter family of formal exponential series solutions (transseries)
\begin{eqnarray}
\label{eqformgen,n}
\tilde{{\bf y}}=\tilde{{\bf y}}_0+\sum_{\mathbf{k}\ge 0; |\mathbf{k}|>0}C_1^{k_1}\cdots C_n^{k_n}
\mathrm{e}^{-({\bf k}\cdot{\boldsymbol \lambda}) x}x^{{\bf k}\cdot{\bf m}}\tilde{{\bf y}}_{{\bf k}}
\mathrm{e}nd{eqnarray}
\noindent (see \cite{Wasow}, \cite{Cope},\cite{Iwano}, and also \S~\ref{sec:For} below) where
$m_i=1-\lfloor\beta_i\rfloor$,
($\lfloor\cdot\rfloor=$ integer part), ${\bf C}\mathrm{i}n\mathbb{C}^n$ is an arbitrary vector
of constants, and $\tilde{\mathbf{y}}_{\bf k}=x^{-{\bf k}({\boldsymbol \beta}+{\bf m})}
\sum_{l=0}^{\mathrm{i}nfty}{\bf a}_{{\bf k};l} x^{-l}$ are formal power series.
When $x$ is large in some direction $d$ in $\mathbb{C}$, an important role is
played by the subset of transseries which are at the same time {\mathrm{e}m
asymptotic} expressions\footnote{An asymptotic
expansion of a function carries immediate information about behavior
of the
function near the expansion point (in contrast to
antiasymptotic expansions, e.g. a convergent doubly infinite Laurent series)}: When there are infinitely many
exponentials in (\ref{eqformgen,n}) we ask that for all $i$ with
$C_i{\cal N}e 0$ we have $|\mathrm{e}^{-\lambda_i x}|\ll 1$ for large $x$ in
the given direction $d$ in $\mathbb{C}$. Formally, agreeing to omit the
terms with $C_i=0$, with $x$ in $d$,
any {\mathrm{e}m ascending} chain $\Re(-{\bf k}_1\cdot{\boldsymbol \lambda} x)\le
\Re(-{\bf k}_2\cdot{\boldsymbol \lambda} x)\le\ldots$, ${\bf k}_i{\cal N}e{\bf k}_j$, in
(\ref{eqformgen,n}) must be {\mathrm{e}m finite} (the terms of an asymptotic
transseries are well-ordered with respect to $''\ll''$). Thus for $x$
in some direction $d$ we only consider those transseries
that satisfy the condition:
(c1) $\xi+\phi_i:=\arg(x)+\phi_i\mathrm{i}n (-\pi/2,\pi/2)$ for all $i$ such that $C_i{\cal N}e
0$. In other words, $C_i{\cal N}e 0$ implies that $\lambda_i$ lies in a
half-plane centered on $\overline{d}$, the complex conjugate direction
to $d$.
From now on, ${{\boldsymbol \lambda}}
=(\lambda_{i_1},\ldots,\lambda_{i_{n_1}})$,
${\boldsymbol \beta}=(\beta_{i_1},\ldots,\beta_{i_{n_1}})$,
${\bf m}=(m_{i_1},\ldots,m_{i_{n_1}})$ and
${\boldsymbol \beta}'={\boldsymbol \beta}+{\bf m}$ where the indices $i_1,\ldots,i_{n_1}$ satisfy
(c1).
We will henceforth consider that (\ref{eqor}) is presented in prepared
form, and use the designation transseries only for those formal solutions
satisfying (c1).
The series $\tilde{{\bf y}}_0$ is a formal solution of (\ref{eqor})
while, for ${\bf k}{\cal N}e 0$, $\tilde{{\bf y}}_{\bf k}$ satisfy a hierarchy of
linear differential equations \cite{Wasow} (see also \S~\ref{sec:For}
for a brief exposition and notations). Generically all the series
$\tilde{{\bf y}}_k$ are factorially divergent and there is no immediate
way to uniquely associate actual functions to them. Neither can
$\tilde{{\bf y}}$ be viewed as a {\mathrm{e}m classical} asymptotic expansion since
the $\tilde{{\bf y}}_{\bf k}$ are beyond all orders of each
other (e.g., for ${\bf k}{\cal N}e 0$ and all $l\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N$, $\mathrm{e}^{-{\boldsymbol \lambda}\cdot{\bf k} x}x^{{\bf k}\cdot{\bf m}}
\tilde{{\bf y}}_{\bf k} =o(x^{-l})$).
One question is therefore to understand the relation between these
(algorithmically obtained) formal solutions and the actual solutions of
(\ref{eqor}). In the present paper we show that a suitable
generalization of Borel summation provides a one-to-one correspondence
between transseries and actual solutions of (\ref{eqor}):
\begin{eqnarray}
\label{symbol}
{{\bf y}}\rightleftharpoons\tilde{{\bf y}}_0+\sum_{\mathbf{k}\ge 0; |\mathbf{k}|>0
}C_1^{k_1}\cdots C_n^{k_n}
\mathrm{e}^{-({\bf k}\cdot{\boldsymbol \lambda}) x}x^{{\bf k}\cdot{\bf m}}\tilde{{\bf y}}_{{\bf k}}
\ \ \ (x\rightarrow\mathrm{i}nfty, \arg(x)=\xi)
\mathrm{e}nd{eqnarray}
Given ${\bf y}$, the value of $C_i$ can change only when
$\xi+\arg(\lambda_i-{\bf k}\cdot{\boldsymbol \lambda})=0$, $k_i\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N\cup\{0\}$, i.e.
when crossing one of the (finitely many by (c1)) Stokes lines. The
correspondence (\ref{symbol}) defines a summation method, in the sense
that it is an extension of convergent summation which
preserves its basic properties: linearity, multiplicativity,
commutation with differentiation and with complex conjugation. These
properties are essential for obtaining true solutions out of
transseries for nonlinear differential equations. Our procedure is
similar to the medianization proposed by \'Ecalle, but (due to the
structure of (\ref{eqor})) requires substantially fewer analytic
continuation paths. In addition we classify in the context of
(\ref{eqor}) all admissible summation methods (there is a
one-parameter family of them, preserving the properties of usual
summation). Summation recovers from transseries actual solutions of
(\ref{eqor}) without resorting to (\ref{eqor}) in the process. In
addition, the analysis reveals a rich analytic structure and formulas
linking the various $\tilde{{\bf y}}_{\bf k}$ among themselves (resurgence
relations). In \cite{Costin} we studied this problem under further
restrictions on the transseries (decay of the exponentials in a full
half-plane) and on the differential equation. Removing those
restrictions creates difficulties that required a new approach. New
resurgence relations are found and in addition we provide a complete
description, needed in applications, of the singularity structure of
the Borel transforms of $\tilde{{\bf y}}_{\bf k}$.
\subsubsection{Notes on Borel summation}
\label{sec:bsum}
The following is a very brief description; for more details on classical
Borel summation see \cite{Hardy}, \cite{Borel} and for recent
developments see \cite{Balser} and especially \cite{Ecalle}.
If $\tilde{f}=\sum_{k=0}^{\mathrm{i}nfty} a_kx^{-k-r}$ is a formal series with
$\Re(r)>0$, its Borel transform is defined as the (still formal)
series
$\mathcal{B}{\tilde{f}}=\sum_{k=0}^{\mathrm{i}nfty}p^{k-1+r}/\Gamma(k+r+1)$,
the term-by-term inverse Laplace transform of $\tilde{f}$. If
$r\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^+$ and $\tilde{f}$ \mathrm{e}mph{converges} (to $f$), then
$\mathcal{B}\tilde{f}$ converges in $\mathbb{C}$ to an analytic
function which is Laplace (${\cal L}$) transformable and
${\cal L}{\cal B}\tilde{f} =f$. A similar property holds more generally when
$\Re(r)>0$, with now $f$ and ${\cal B}\tilde{f}$ ramified analytic
functions. Even when $\tilde{f}$ is divergent (not faster than
factorially), ${\cal B} \tilde{f}$ may have a nonzero radius of
convergence and define a germ of an analytic function $F(p)$. If
$F(p)$ can be analytically continued along a ray $\arg(p)=\phi$, and
its growth is at most exponential, then
$f={\cal L}_{\phi}F=\mathrm{i}nt_{\mathrm{e}^{\mathrm{i}\phi}\mathbb{R}^+}F(p)\mathrm{e}^{-xp}\mathrm{d}p$
defines a function with the property $f\sim\tilde{f}$ as
$x\rightarrow\mathrm{i}nfty$ with $\Re(x \mathrm{e}^{\mathrm{i}\phi})>0$. In
general now $F(p)$ is singular (not only for $p=0$), and
$\mathcal{L}_\phi F$ (when it exists) will depend on $\phi$; the usual
convention is to choose $\phi$ so that
\begin{equation}\label{convxp}xp\mathrm{i}n\mathbb{R}^+\mathrm{e}nd{equation}
Thus, the Borel sum of $\tilde{f}$ in the
direction $x$, if it exists, is defined as
${\cal L}_{\phi(x)}\mathcal{B}f$ with $-\phi(x)=\xi:=\arg(x)$. However, when
$\tilde{f}$ is a series with real coefficients, it is a common occurrence that
$F(p)$ is singular for $p\mathrm{i}n\mathbb{R}^+$ (because of conjugation symmetry),
and then the {\mathrm{e}m classical} Borel sum of $\tilde{f}$ along the real axis
(the interesting direction in many cases) is undefined.
The difficulty is more serious than it may seem. Summation along paths
that avoid the singularities from above or from below give different
results and thus would lead to an ambiguous (or unnatural) procedure.
More importantly, a ``summation'' procedure using such paths would not
commute with complex-conjugation since the ``sum'' will be, in
general, complex for real $\tilde{f}$ and would thus fail to be a
(proper) summation method. Symmetry considerations suggest a first
step towards {\mathrm{e}m averaging}: summation along the half-sum of the two paths
does commutes with complex conjugation. But this solves a problem only to
create another one. The half-sum process fails to commute with
multiplication (of series) and is thus not a summation method, either.
It turns out that there exist more sophisticated averages which have
all the required properties to define a summation procedure. The
technique of averaging, as well as the fundamental concepts of
analyzable functions and transseries, were discovered and studied by
Ecalle in his constructive approach to the Dulac conjecture (see
\cite{Ecalle-book}, \cite{Ecalle} and \cite{Ecalle2}). The concept of
analyzable function (also discovered by Kruskal in the context of
surreal analysis) is regarded as a very comprehensive generalization
of analyticity/ quasianalyticity. The widely held belief is that
all functions of ``natural origin'' must be analyzable. In particular,
analyzable functions have uniquely associated transseries which are
generalized-Borel summable, after a finite number of transformations
\cite{Ecalle2}. We show that, in the particular case of
(\ref{eqor}), decreasing solutions are analyzable.
There is a wide class of admissible, all-purpose averaging methods
(\cite{Ecalle3}). As yet there is no unique, natural average and the
problem in its full generality is highly nontrivial. We obtain the
{\mathrm{e}m balanced average} directly from the study of the general solution
of the inverse Laplace transform of (\ref{eqor}). Its potential
nonuniqueness is lifted, in our context, by imposing compatibility
with {\mathrm{e}m hyperasymptotics} an important improvement in asymptotic
calculations proposed by M. Berry (\cite{Berry}, \cite{Berry-hyp},
\cite{Berry-Howls}, \cite{Berry-gamma}).
\subsubsection{Nonresonance}\label{nonres} (1) $\lambda_i,\,i=1,...,n_1$ are
assumed $\mathbb{Z}$-linearly independent for any $d$. (2) Let $\theta\mathrm{i}n[0,2\pi)$ and
$\tilde{\boldsymbol{\lambda}}=(\lambda_{i_1},...,\lambda_{i_p})$ where
$\left|\arg\lambda_{i_j}-\theta\right|\mathrm{i}n(-\pi/2,\pi/2)$ (those
eigenvalues contained in the open half-plane $H_\theta$ centered along
$\mathrm{e}^{\mathrm{i}\theta}$). We require that
for any $\theta$ the complex numbers in the set
$\{\tilde{\lambda}_{i}-\mathbf{k}
\cdot\tilde{\boldsymbol{\lambda}}\mathrm{i}n H_\theta:{\bf k}\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^p,\,
i=1,...,p\}$
(note: the set is \mathrm{e}mph{finite}) have
{\mathrm{e}m distinct} directions. These are the Stokes lines $d_{i;{\bf k}}$.
That the set of $\boldsymbol{\lambda}$ which satisfy (1) and (2) has full
measure follows from the fact that (1) and (2) follow from the condition:
\begin{eqnarray}\label{strongnonr}
\Big({\bf m},{\bf m}'\mathrm{i}n\mathbb{Z}^n ,\
\alpha\mathrm{i}n\mathbb{R}\ \mbox{and}\
({\bf m}-\alpha{\bf m}')\cdot\boldsymbol{\lambda}=0
\Big)\ \Rightarrow \Big({\bf m}=\alpha{\bf m}'\Big)
\mathrm{e}nd{eqnarray}
\noindent Indeed, if (\ref{strongnonr}) fails, one of $\Re\lambda_j,
\Im\lambda_j$ is a rational function with rational coefficients of the
other $\Re\lambda_j$ and $\Im\lambda_j$, corresponding to a zero
measure set in $\mathbb{R}^{2n}$.
\subsection{Further notations and conventions}
\label{sec:anset}
If $y_{1}$ and $y_2$ are inverse Laplace transformable functions, then in a
neighborhood of the origin
${\cal L}^{-1}(y_1y_2)=({\cal L}^{-1}y_1)*({\cal L}^{-1}y_2)$, where for $f,
g\mathrm{i}n{L^1}$ convolution is given by
\begin{eqnarray}\label{defconv}
f*g:=p\mapsto\mathrm{i}nt_0^p f(s)g(p-s)\mathrm{d}s
\mathrm{e}nd{eqnarray}
\noindent We use the convention $\hat{B}ox{\it I\hskip -2pt N}N{\cal N}i 0$. Let
\begin{eqnarray}
\label{defW}
\mathcal{W}=\left\{p\mathrm{i}n\mathbb{C}:p{\cal N}e k\lambda_i\,,\forall
k\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N,i=1,2,\ldots,n\right\}
\mathrm{e}nd{eqnarray}
The directions $d_j=\{p:\arg(p)=\phi_j\}, j=1,2,\ldots,n$ (cf. (a2))
are the {\mathrm{e}m Stokes lines} of $\tilde{{\bf y}}_0$ (note: sometimes known as {\mathrm{e}m anti-}Stokes
lines!). We construct over $\mathcal{W}$ a surface $\mathcal{R}$,
consisting of homotopy classes of smooth curves in $\mathcal{W}$
starting at the origin, moving away from it, and crossing at most one
Stokes line, at most once (see Fig. 1):
\begin{eqnarray}\label{defpaths}
{\cal R}:=\Big\{\hat{\Gamma}mma:(0,1)\mapsto \mathcal{W}:\
\hat{\Gamma}mma(0_+)=0;\ \frac{\mathrm{d}}{\mathrm{d}t}|\hat{\Gamma}mma(t)|>0;\
\arg(\hat{\Gamma}mma(t))\ \mbox{monotonic}\Big\}\cr \mbox{modulo homotopies}
\mathrm{e}nd{eqnarray}
\noindent Define $\mathcal{R}_1\subset \mathcal{R}$ by (\ref{defpaths}) with
the supplementary restriction $\arg(\hat{\Gamma}mma)\mathrm{i}n(\psi_n-2\pi,\psi_2)$
where $\psi_n=\max\{-\pi/2,\phi_n-2\pi\}$ and
$\psi_2=\min\{\pi/2,\phi_2\}$. $\mathcal{R}_1$ may be viewed as the
part of the covering $\mathcal{R}$, above a sector containing the real
axis. Similarly we let $\mathcal{R'}_1\subset \mathcal{R}_1$
with the restriction that the curves $\hat{\Gamma}mma$ do not cross
the Stokes directions $d_{i,{\bf k}}$ (cf. \S\ref{nonres}), other than $\mathbb{R}^+$,
and we let $\psi_{\pm}=\pm\max
(\pm\arg \hat{\Gamma}mma)$ with $\hat{\Gamma}mma\mathrm{i}n\mathcal{R'}_1$.
\begin{picture}(0,0)
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\begin{picture}(10890,7989)(4801,-9310)
\mathrm{e}nd{picture}
\centerline{{{Fig 1.} \mathrm{e}mph{The paths
near $\lambda_2$ belong to $\mathcal{R}$. }}}
\centerline{\mathrm{e}m The paths
near $\lambda_1$ relate to the balanced average}
By $AC_\hat{\Gamma}mma(f)$ we denote the analytic
continuation of $f$ along a curve $\hat{\Gamma}mma$.
For the analytic continuations near a Stokes line $d_{i;{\bf k}}$ we use symbols
similar to \'Ecalle's: $f^-$ is the branch of $f$ along a path
$\hat{\Gamma}mma$ with $\arg(\hat{\Gamma}mma)<\phi_i$, while $f^{-j+}$ denotes the
branch along a path that crosses the Stokes line between $j\lambda_i$
and $(j+1)\lambda_i$ (see also \cite{Costin}).
We use the notations $\mathcal{P}f$ for $\mathrm{i}nt_0^p f(s)\mathrm{d}s$ and
$\mathcal{P}_\hat{\Gamma}mma f$ if integration is along the curve $\hat{\Gamma}mma$.
We write ${\bf k}\succeq{\bf k}'$ if $k_i\ge k'_i$ for all $i$ and
${\bf k}\succ{\bf k}'$ if ${\bf k}\succeq{\bf k}'$ and ${\bf k}{\cal N}e{\bf k}'$. The relation
$\succ$ is a well ordering on $\hat{B}ox{\it I\hskip -2pt N}N^{n_1}$. We let $\mathbf{e}_j$ be
the unit vector in the $j^{\rm th}$ direction in $\hat{B}ox{\it I\hskip -2pt N}N^{n_1}$.
Formal expansions are denoted with a tilde, and capital letters
$\bfY,\mathbf{V}\ldots$ will usually denote Borel transforms or other
functions naturally
associated to Borel space. For notational convenience, we
will not however distinguish between the series
$\tilde{\bfY}_k={\cal B}\tilde{{\bf y}}_{\bf k}$, which in our case turn out to be
convergent, and the sums $\bfY_{\bf k}$ of these series as germs
of ramified analytic functions.
By symmetry (renumbering the directions) it suffices to analyze the
singularity structure of $\bfY_0$ in $\mathcal{R}_1$ only. However,
(c1) breaks this symmetry for ${\bf k}{\cal N}e 0$ and the properties of these
$\bfY_{\bf k}$
will be analyzed along some other directions as well.
$\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_A$ will denote the characteristic function of the set $A$.
We write $|{\bf f}|:=\max_{i}\{|f_i|\}$.
We have
\begin{eqnarray}\label{Taylor series}
{\bf g}(x,{{\bf y}})=\sum_{|{\bf l}|\ge 1}{\bf g}_{\bf l}(x) {\bf
y}^{\bf l}=\sum_{s\ge 0;|{\bf l}|\ge 1}{\bf g}_{s,\bf l}x^{-s}
{{\bf y}}^{\bf l} \ \ (|x|>x_0,|{\bf y}|<y_0)
\mathrm{e}nd{eqnarray}
\noindent where ${{\bf y}}^{\bf l}=y_1^{l_1}\cdots y_n^{l_n}$ and
$|{\bf l}|=l_1+\cdots+l_n$. By construction ${\bf g}_{s,{\bf l}}=0$
if $|{\bf l}|=1$ and $s\le M$.
The formal inverse Laplace transform of ${\bf g}(x,{\bf y}(x))$ (formal since ${\bf y}$ is still unrestricted)
is given by:
\begin{eqnarray}\label{lapdef}
{\cal L}^{-1}\left(\sum_{|\bf l|\ge 1}
{\bf y}(x)^{\bf l}\sum_{s\ge 0}{\bf g}_{s,\bf l}x^{-s}\right)
=\sum_{|\bf l|\ge 1}{{\bf G}}_{\bf l}*{\bf Y}^{*\bf l}+
\sum_{|\bf l|\ge 2}{\bf g}_{0,{\bf l}}{\bf Y}^{*\bf l}=:{{{\cal N}b}}(\bf Y)
\cr&&
\mathrm{e}nd{eqnarray}
\noindent
with $ {{\bf G}}_{\bf l}(p)=\sum_{s=1}^{\mathrm{i}nfty}{\bf g}_{s,\bf
l}{p^{s-1}}/{s!}$ and $({{\bf
G_{l}}*\bfY^{*{\bf l}}})_j:= \left({{\bf
G}_{{\bf l}}}\right)_j*Y_1^{*l_1}*..*Y_{n}^{*l_n}$. By
(n5), ${{\bf G}}^{(l)}_{1,{\bf l}}(0)=0\ \mbox{ if
}|{\bf l}|=1$ and $l\le M$. The inverse
Laplace transform of (\ref{eqor}) is the convolution equation:
\begin{eqnarray}\label{eqil}
-p{\bf Y}={\bf F}_0-\hat\Lambda {\bf Y}-
\hat B\mathcal{P}\bfY+{\cal N}({\bf Y})
\mathrm{e}nd{eqnarray}
Let $\mathbf{D}d_{\bf j}(x):=\sum_{ \bf l\ge j} \binom{{\bf l}}{{\bf j} }
\mathbf{g}_{\bf l}(x) \tilde{{\bf y}}_0^{{\bf l} -{\bf j} }$. Straightforward
calculation (see Appendix \S~\ref{sec:For}; cf. also \cite{Costin})
shows that the components $\tilde{{\bf y}}_{\bf k}$ of the transseries
satisfy the hierarchy of differential equations
\begin{eqnarray}
\label{systemformv}
&&{\bf y}'_{\bf k}+ \left(\hat\Lambda+\frac{1}{x}\left(\hat
B+{\bf k}\cdot{\bf m}\right)-{\bf k}\cdot{\boldsymbol \lambda}
\right){\bf y}_k+\sum_{|{\bf j}|=1}\mathbf{D}d_{{\bf j}}(x)({\bf y}_{\bf k})^{{\bf j}}
={\bf t}_{\bf k}
\cr&&
\mathrm{e}nd{eqnarray}
\noindent where ${\bf t}_{\bf k}={\bf t}_{\bf k}\left({\bf y}_0,\left\{{\bf y}_{{\bf k}'}\right\}_{0\prec{\bf k}'\prec{\bf k}}\right)$ is a {\mathrm{e}m polynomial} in
$\left\{{\bf y}_{{\bf k}'}\right\}_{0\prec{\bf k}'\prec{\bf k}}$
and in $\{\mathbf{D}d_{\bf j}\}_{{\bf j}\le{\bf k}}$ (see (\ref{eqmygen})), with
${\bf t}({\bf y}_0,\mathrm{e}mptyset)=0$; ${\bf t}_{\bf k}$ satisfies the homogeneity
relation
\begin{eqnarray}
\label{homogeq2}\label{homogeq}
{\bf t}_{\bf k}\left({\bf y}_0,\left\{C^{{\bf k}'}{\bf y}_{{\bf k}'}\right\}_{0\prec{\bf k}'\prec{\bf k}}\right)
=C^{{\bf k}}{\bf t}_{\bf k}\left({\bf y}_0,\left\{{\bf y}_{{\bf k}'}\right\}_{0\prec{\bf k}'\prec{\bf k}}\right)
\mathrm{e}nd{eqnarray}
\noindent Taking ${\cal L}i$ in (\ref{systemformv}) we get, with $\mathbf{D}_{\bf j}=\sum_{
\bf l\ge j} \binom{{\bf l}}{{\bf j} }\!\left[ {\bf G}_{\bf l}*\bfY_0^{*({\bf l}
-{\bf j}) }+\mathbf{g}_{0,{\bf l}}*\bfY_0^{*({\bf l} -{\bf j})}\right]$,
\begin{eqnarray}
\label{invlapvk}\label{eqMv}\label{invlapyk}
&&\left(-p+\hat{\Lambda}-{\bf k}\cdot{\boldsymbol \lambda}\right)\bfY_{\bf k}
+\left(\hat{B}+{\bf k}\cdot{\bf m}\right)
\mathcal{P}\bfY_{\bf k}+\sum_{|{\bf j}|=1}\mathbf{D}_{\bf j}*\bfY_{\bf k}^{*{\bf j}}
={\bf T}_{\bf k}\cr&&
\mathrm{e}nd{eqnarray}
\noindent where $\mathbf{T}_{\bf k}$ is now a {\mathrm{e}m convolution} polynomial, cf.
(\ref{defT}).
\subsection{Main results}
\label{sec:MR}
(a) {\mathrm{e}m Analytic structure}. \begin{Theorem}\label{AS} (i) $\bfY_0={\cal B}\tilde{{\bf y}}_0$
is analytic in $\mathcal{R}\cup \{0\}$.
The singularities of $\bfY_0$ (which are contained in the set
$\{l\lambda_j:l\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^+,j=1,2,\ldots,n\}$) are described as follows.
For $l\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^+$ and small $z$, using the notations explained
in \S\ref{sec:anset},
\begin{multline}
\label{SY0}
\bfY_0^{\pm}(z+l\lambda_j)=\pm\Big[(\pm S_j)^l(\ln z)^{0,1}
\bfY_{l\mathbf{e}_j}(z)\Big]^{(lm_j)}+{\bf B}_{lj}(z)=\cr
\Big[z^{l\beta_j'-1}(\ln
z)^{0,1}\,{\bf A}_{lj}(z)\Big]^{(lm_j)}+{\bf B}_{lj}(z) \ (l=1,2,\ldots)
\mathrm{e}nd{multline}
\noindent where the power of $\ln z$ is one iff
$l\beta_j\mathrm{i}n\mathbb{Z}$, and ${\bf A}_{lj},{\bf B}_{lj}$ are analytic for small
$z$. The functions $\bfY_{\bf k}$ are, exceptionally, analytic at
$p=l\lambda_j$, $l\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^+$, iff,
\begin{eqnarray}\label{defSj}
S_j=r_j\Gamma(\beta'_j)\left({\bf A}_{1,j}\right)_j(0)=0
\mathrm{e}nd{eqnarray}
\noindent where $r_j=1-\mathrm{e}^{2\pi i(\beta'_j-1)}$
if $l\beta_j{\cal N}otin\mathbb{Z}$ and $r_j=-2\pi i$
otherwise. The $S_j$ are Stokes constants,
see Theorem~\ref{Stokestr}.
(ii) $\bfY_{\bf k}={\cal B}{\tilde{{\bf y}}}_{\bf k}$, $|{\bf k}|>1$, are analytic in
$\mathcal{R}\backslash
\{-{\bf k}'\cdot{{\boldsymbol \lambda}}+\lambda_i:{\bf k}'\le{\bf k},1\le i\le n\}$. For
$l\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N$ and $p$
near $l\lambda_j$, $j=1,2,\ldots,n$ there exist ${\bf A}={\bf A}_{{\bf k} jl}$ and
${\bf B}={\bf B}_{{\bf k} jl}$ analytic at zero so that ($z$ is as above)
\begin{multline}
\label{SYK}
\bfY_{\bf k}^{\pm}(z+l\lambda_j)=
\pm\Big[(\pm S_j)^l\binom{k_j+l}{
l}(\ln z)^{0,1}
\bfY_{{\bf k}+l\mathbf{e}_j}(z)\Big]^{(lm_j)}+{\bf B}_{{\bf k} lj}(z)=\cr
\Big[z^{{\bf k}\cdot{{\boldsymbol \beta}}'+l\beta_j'-1}(\ln z)^{0,1}\,{\bf A}_{{\bf k} l
j}(z)\Big]^{(lm_j)}+{\bf B}_{{\bf k} l j}(z)\ (l=0,1,2,\ldots)
\mathrm{e}nd{multline}
\noindent where the power of $\ln z$ is $0$ iff $l=0$
or
${\bf k}\cdot{{\boldsymbol \beta}}+l\beta_j-1{\cal N}otin\mathbb{Z}$ and ${\bf A}_{{\bf k} 0 j}=\mathbf{e}_j/\Gamma(\beta'_j)$.
Near $p\mathrm{i}n\{\lambda_i-{\bf k}'\cdot{{\boldsymbol \lambda}}:0\prec{\bf k}'\le{\bf k}\}$, (where
$\bfY_0$ is analytic) $\bfY_{\bf k},\,{\bf k}{\cal N}e 0$ have convergent
Puiseux series.
\mathrm{e}nd{Theorem}
{\sc Remark:} The fact that the singular part of
$\bfY_{\bf k}(p+l\lambda_j)$ in (\ref{SY0}) and (\ref{SYK}) is a {\mathrm{e}m
multiple} of $\bfY_{{\bf k}+l\mathbf{e}_j}(p)$ is the effect
of {\mathrm{e}m resurgence} and provides a way of
determining
the $\mathbf{Y}_{\bf k}$ given $\mathbf{Y}_0$
provided the $S_j$ are nonzero. Since, generically,
the $S_j$ are nonzero this is a surprising upshot: given one formal
solution, (generically) an $n$ parameter family of solutions can be
constructed out of it, without using (\ref{eqor}) in the process;
the differential equation itself is then recoverable (\cite{Costin3}).
By Theorem~\ref{AS} the Borel transforms
$\bfY_{\bf k}={{\cal B}}\tilde{{\bf y}}_{\bf k}$ define germs of ramified analytic
functions and are continuable on the surface $\mathcal{R}$. In order
to be able to take Laplace transforms we need to define
${\cal B}\tilde{{\bf y}}_{\bf k}$ along any direction $d$ in $\mathcal{S}$. If
$d{\cal N}e d_{j,{\bf k}}$ then $\bfY_{\bf k}$ are analytically continuable along $d$ and
the continuations turn out to have all the properties that we need.
But along Stokes lines $d_{j,{\bf k}}$ analytic continuation is impossible:
in general the functions $\bfY_{\bf k}$ have an infinite array of
branch points (\ref{SYK}). In addition, while both $\bfY_{\bf k}^+$ and
$\bfY_{\bf k}^-$ turn out to be Laplace transformable (in distributions)
along $d_{j,{\bf k}}$, ${\cal L}\bfY_{\bf k}^+$ and ${\cal L}\bfY_{\bf k}^-$ are generically
{\mathrm{e}m different}. Neither the upper nor the lower continuation would
give rise to a definition of Borel summation which commutes with
complex-conjugation, as discussed in the introduction. Other analytic
continuations along paths $\hat{\Gamma}mma$ that {\mathrm{e}m cross} $d_{j,{\bf k}}$ have even
worse problems, namely that $AC_\hat{\Gamma}mma(\bfY_{\bf k}*\bfY_{\bf k}){\cal N}e
AC_\hat{\Gamma}mma(\bfY_{\bf k})*_\hat{\Gamma}mma AC_\hat{\Gamma}mma(\bfY_{\bf k})$, (see
\cite{Costin}; for notations only, see also \S\ref{sec:twolemmas}). As ${\cal B}$ transforms differential equations into
convolution equations, the implication is that with such a $\hat{\Gamma}mma$,
${\cal L} AC_\hat{\Gamma}mma(\bfY_{\bf k})$ would {\mathrm{e}m not} be, in general, solutions
of their differential equations. Individual analytic continuations are
thus inadequate for solving (\ref{eqor}), but some
{\mathrm{e}m averages} of analytic continuations do satisfy all the
requirements. Let $\alpha=1/2+\mathrm{i}\sigma$ with $\sigma\mathrm{i}n\mathbb{R}$
and ${\cal B}\tilde{{\bf y}}_{\bf k}$ be extended along $d_{j,{\bf k}}$ by
the weighted average of analytic continuations
\begin{eqnarray}
\label{defmed}
{\cal B}_\alpha\tilde{{\bf y}}_{\bf k}=\bfY_{\bf k}^{\alpha}=
\bfY_{\bf k}^++\sum_{j=1}^{\mathrm{i}nfty}\alpha^j\left(\bfY_{\bf k}^{-}
-\bfY_{\bf k}^{-({j-1})+}\right)
\mathrm{e}nd{eqnarray}
\begin{Remark}\label{RF} Relation (\ref{defmed}) gives the most general reality preserving,
linear operator mapping formal power series solutions of
(\ref{eqor}) to solutions of
(\ref{eqil}) in distributions (more precisely
in $\mathcal{D}'_{m,{\cal N}u}$; see \S\ref{sec:NC}).
\mathrm{e}nd{Remark}
\noindent This remark follows easily from Proposition~\ref{SRY0} and
Theorem~\ref{RE} below.
The choice $\alpha=1/2$ has special properties; we call
${\cal B}_{\frac{1}{2}}\tilde{{\bf y}}_{\bf k}=\bfY^{ba}_{\bf k}$ the balanced average
of $\bfY_{\bf k}$. For this choice the expression (\ref{defmed})
coincides with the one in which $+$ and $-$ are interchanged (Proposition~\ref{medianization}), accounting for the
reality-preserving property. Clearly, if $\bfY_{\bf k}$ is analytic along
$d_{j,{\bf k}}$, then the terms in the infinite sum vanish and
$\bfY_{\bf k}^{\alpha}=\bfY_{\bf k}$; we also let $\bfY_{\bf k}^{\alpha}=\bfY_{\bf k}$ if
$d{\cal N}e d_{j,{\bf k}}$, where again $\bfY_{\bf k}$ is analytic. It follows from
(\ref{defmed}) and Theorem~\ref{CEQ} below that the Laplace integral
of $\mathbf{Y}^{\alpha}_{\bf k}$ along $\mathbb{R}^+$ can be deformed into contours as
those depicted in Fig. 1, with weight $-(-\alpha)^{k}$ for a contour
turning around $k\lambda_1$.
In addition to symmetry (the balanced average equals the half sum
of the upper and lower continuations on $(0,2\lambda_j)$,
\cite{Costin3}), an
asymptotic property uniquely picks $C=1/2$. Namely, for $C=1/2$
alone are the ${\cal L}{\cal B}\tilde{{\bf y}}_{\bf k}$ always {\mathrm{e}m summable to the
least term} cf. \S~\ref{sec:aver}.
(b) {\mathrm{e}m Connection with (\ref{eqor}) and (\ref{homogeq})}.
Generalized Borel summation coincides with the usual Borel summation
when the transseries consists of only one term, the first series, when
that series is classically Borel summable. This is clear from
theorem~\ref{CEQ} (ii) below. Furthermore, generalized summation is a
map from a class of formal series to functions which is linear,
multiplicative, commutes with differentiation and complex conjugation
(cf. \S~\ref{sec:aver}, \S~\ref{sec:NC}) so it is a summation
procedure, which furthermore, establishes along every direction a one
to one correspondence between transseries and decaying actual solutions of
(\ref{eqor}) cf. \S~\ref{sec:aver}, Proposition~\ref{medianization}
and Theorem~\ref{CEQ} below.
For
clarity we state the results for $x\mathrm{i}n S_x$, a sector in the
right half plane containing
$\lambda_1=1$ in which (c1) holds and for $p$ in the associated domain
$\mathcal{R'}_1$, but $\lambda_1$ plays no special role as discussed
in the introduction.
\begin{Theorem}\label{CEQ} (i) The branches of $(\bfY_{\bf k})_\hat{\Gamma}mma$ in
$\mathcal{R'}_1$ ($\mathcal{R}_1$ if ${\bf k}=0$)
have limits in a $C^*$-algebra of distributions,
$\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)\subset\mathcal{D}'$ (cf. \S~\ref{sec:NC}).
Their Laplace transforms in $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$
${\cal L}(\bfY_{\bf k})_\hat{\Gamma}mma$ exist simultaneously and with
$x\mathrm{i}n\mathcal{S}_x$ and for any $\delta>0$ there is a constant $K$ and
an $x_1$ large enough, so that for $\Re(x)>x_1$ we have
$\left|{\cal L}(\bfY_{\bf k}\right)_\hat{\Gamma}mma(x)|\le K\delta^{|{\bf k}|}$.
In addition, $\mathbf{Y}_{\bf k}(p\mathrm{e}^{\mathrm{i}\phi})$ are
continuous in $\phi$ with respect to the $\mathcal{D}'_{m,{\cal N}u}$
topology, (separately) on $(\psi_-,0]$ and $[0,\psi_+)$ .
If
$m>\max_i(m_i)$ and $l<\min_i |\lambda_i|$ then
$\mathbf{Y}_0(p\mathrm{e}^{\mathrm{i}\phi})$ is continuous in
$\phi\mathrm{i}n[0,2\pi]\backslash\{\phi_i:i\le n\}$ in the
$\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+,l)$ topology and has (at most) jump
discontinuities for $\phi=\phi_i$. For each ${\bf k}$, $|{\bf k}|\ge 1$ and
any $K$ there is an $l>0$ and an $m$ such that
$\mathbf{Y}_{\bf k}(p\mathrm{e}^{\mathrm{i}\phi})$ are continuous in
$\phi\mathrm{i}n[0,2\pi]\backslash\{\phi_i; -{\bf k}'\cdot{\boldsymbol \lambda}+\lambda_i:i\le n
,{\bf k}'\le{\bf k}\}$ in the $\mathcal{D}'_{m,{\cal N}u}((0,K),l)$ topology and
have (at most) jump discontinuities on the boundary.
(ii) The sum (\ref{defmed}) converges in $\mathcal{D}'_{m,{\cal N}u}$ (and
coincides with the analytic continuation of $\bfY_{\bf k}$ when
$\bfY_{\bf k}$ is analytic along $\mathbb{R}^+$). For any $\delta$ there is a
large enough $x_1$ {\mathrm{e}m independent of ${\bf k}$} so that
$\bfY^{ba}_{\bf k}(p)$ with $p\mathrm{i}n\mathcal{R}'_1$ are Laplace transformable
in $\mathcal{D}'_{m,{\cal N}u}$
for $\Re(xp)>x_1$ and furthermore $|({\cal L}\bfY^{ba}_{\bf k})(x)|\le
\delta^{|{\bf k}|}$. In addition, if $d{\cal N}e\mathbb{R}^+$, then for large ${\cal N}u$,
$\bfY_{\bf k}\mathrm{i}n L^1_{\cal N}u(d)$.
The functions ${\cal L}\bfY_{\bf k}^{ba}$ are analytic for
$\Re(xp)>x_1$. For any ${\bf C}\mathrm{i}n\mathbb{C}^{n_1}$ there is an $x_1({\bf C})$
large enough so that the sum
\begin{eqnarray}
\label{soleqn}
{\bf y}={\cal L}\bfY_0^{ba}+\sum_{|{\bf k}|> 0}{\bf C}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda}
x}x^{-{\bf k}\cdot{\boldsymbol \beta}}{\cal L}\bfY_{\bf k}^{ba}
\mathrm{e}nd{eqnarray}
\noindent converges uniformly for $\Re(xp)>x_1({\bf C})$, and ${\bf y}$ is a solution
of (\ref{eqor}). When the direction
of $p$ is not the real axis then, by definition, $\bfY^{ba}_{\bf k}=\bfY_{\bf k}$,
$\mathcal{L}$ is the usual Laplace transform
and (\ref{soleqn}) becomes
\begin{eqnarray}
\label{soleqnpm}
{\bf y}={\cal L}\bfY_0+\sum_{|{\bf k}|> 0}{\bf C}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda}
x}x^{-{\bf k}\cdot{\boldsymbol \beta}}{\cal L}\bfY_{\bf k}
\mathrm{e}nd{eqnarray}
In addition, ${\cal L}\bfY_{\bf k}^{ba}\sim \tilde{{\bf y}}_{\bf k}$ for large $x$
in the half plane $\Re(xp)>x_1$, for all ${\bf k}$, uniformly.
iii) More generally, for any $\alpha$ and any
solution ${\bf y}$ of (\ref{eqor}) such that ${\bf y}\sim \tilde{{\bf y}}_0$ for
large $x$ along a ray
in $S_x$ there exists a constant
vector ${\bf C}={\bf C}_{\alpha;{\bf y}}$ so that
\begin{eqnarray}
\label{soleqnpa}
{\bf y}={\cal L}{\cal B}_\alpha\tilde{\mathbf{y}}_0+\sum_{|{\bf k}|> 0}{\bf C}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda}
x}x^{-{\bf k}\cdot{\boldsymbol \beta}}{\cal L}{\cal B}_\alpha\tilde{\mathbf{y}}_{\bf k}
\mathrm{e}nd{eqnarray}
Given $\alpha$ the representation (\ref{soleqnpa}) of ${\bf y}$ is unique
(see also \S~\ref{sec:bsum} above for the convention on the direction
of Laplace integration).
\mathrm{e}nd{Theorem}
Of special interest are
the cases $\alpha=1/2$, discussed above, and also $\alpha=0,1$ which give:
\begin{eqnarray}
\label{soleqnp}
{\bf y}={\cal L}\bfY_0^{\pm}+\sum_{|{\bf k}|> 0}{\bf C}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda}
x}x^{-{\bf k}\cdot{\boldsymbol \beta}}{\cal L}\bfY_{\bf k}^{\pm}
\mathrm{e}nd{eqnarray}
{\mathrm{e}m (c) Resurgence properties; local Stokes phenomenon}.
It turns out that the formal series $\tilde{{\bf y}}_{\bf k}$ are connected
among each-other via their Borel transforms. Resurgence
formulas link $\bfY_{\bf k}$ to analytic continuations of
$\bfY_{{\bf k}'}$ with ${\bf k}'\prec{\bf k}$, in a way that, generically,
$\bfY_0$ contains enough information to compute all $\bfY_{\bf k}$.
Various resurgence properties have been observed in different
contexts, and the term resurgence
has been used with slightly different interpretations. In the
hyperasymptotic theory of M. Berry, it was discovered that the first
asymptotic series reappears in various shapes in the process of
computing higher terms of the expansions. J. \'Ecalle, in his
comprehensive theory of analyzable functions, has obtained a general
resurgence principle, the {\mathrm{e}m bridge equation} \cite{Ecalle}. The common
denominator of resurgence is the reappearance of ``earlier'' terms in the
formulas of ``later'' ones. It turns out that,
for our problem, resurgence is fundamentally linked to the {\mathrm{e}m
Stokes phenomenon}. In the following formulas we make the convention
$\bfY_{\bf k}(p-j)=0$ for $p<j$ as an element
of $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$. We again state
the results
is stated for $p\mathrm{i}n\mathcal{R}'_1$ and
$x\mathrm{i}n S_x$
but hold in any sector where (c1) is valid.
\begin{Theorem}\label{RE}
i) For all ${\bf k}$ and $\Re(p)>j,\Im(p)>0$ as well
as in $\mathcal{D}'_{m,{\cal N}u}$ we have
\begin{eqnarray}
\label{mainresur}
\bfY_{{\bf k}}^{\pm j\mp}(p)-\bfY_{{\bf k}}^{\pm (j-1) \mp}(p) = (\pm S_1)^j\binom{k_1+j}{j}
\left(\bfY^\mp_{{\bf k}+j\mathbf{e}_1}(p-j)\right)^{(mj)}
\mathrm{e}nd{eqnarray}
\noindent and also,
\begin{eqnarray}
\label{thirdresu}
\mathbf{Y}_{{\bf k}}^\pm=\bfY_{\bf k}^{\mp}+\sum_{j\ge 1} \binom{j+k_1}{ k_1}(\pm S_1)^{j}(\mathbf{Y}^\mp_{{\bf k}+j\mathbf{e}_1}(p-j))^{(mj)}
\mathrm{e}nd{eqnarray}
ii) {\mathrm{e}m Local Stokes transition.}
\noindent Consider the expression of a fixed solution ${\bf y}$ of (\ref{eqor})
as a Borel summed transseries (\ref{soleqn}). As $\arg(x)$ varies,
(\ref{soleqn}) changes only through ${\bf C}$, and that change occurs
when Stokes lines are crossed (cf. \S\ref{nonres}; the Stokes lines
of $\bfY_0$ are the directions of $\lambda_i$). We have, in the neighborhood
of $\mathbb{R}^+$, with $S_1$
defined in (\ref{defSj}):
\begin{eqnarray}
\label{microsto}
{\bf C}(\xi)=\left\{\begin{array}{ccc} &{\bf C}^-={\bf C}(-0)\qquad\mbox{for
$\xi<0$}\\ &{\bf C}^0={\bf C}(-0)+\frac{1}{2}S_1\mathbf{e}_1\qquad\mbox{for
$\xi=0$}\\ &{\bf C}^+={\bf C}(-0)+S_1\mathbf{e}_1\qquad\mbox{for
$\xi>0$}\\ &\mathrm{e}nd{array}\right.
\mathrm{e}nd{eqnarray}
\mathrm{e}nd{Theorem}
\noindent {\mathrm{e}m (d) Classical Stokes phenomena and local Stokes transitions}.
Again we formulate the result below for $\lambda_1$ but with
straightforward adjustments it holds relative to any other eigenvalue.
Let $\mathbf{C}$ be of the form
$C_1\mathbf{e}_1$. Along the imaginary axis, condition
(c1) fails. The positive and negative imaginary are the {\mathrm{e}m
antistokes} lines corresponding to $\lambda_1=1$ (note: sometimes
called Stokes lines!). If we choose paths in the right half plane
approaching the positive/negative imaginary axis in such a way that
$|x^{-\beta_1-l}\mathrm{e}^{-x}|\rightarrow K{\cal N}e 0$ along them, where
$l+\beta\mathrm{i}n(0,M)$, then ${\bf y}\sim
C^{\pm} x^{-l-\beta_1}\mathrm{e}^{-x}+\tilde{{\bf y}}_0$ for large $x$ and the term
multiplied by $K$ is now the {\mathrm{e}m leading} behavior of ${\bf y}$. The
particular choice of $K$ and $l$ within this range is rather
arbitrary, the main point being that along such special curves, the
constant ${\bf C}$ is definable in terms of {\mathrm{e}m classical} asymptotics.
Within the right half plane, it is only near the imaginary axis that
this happens, since otherwise the exponential term is smaller than all
terms of $\tilde{{\bf y}}_0$. On the other hand Borel summation makes
possible the definition of ${\bf C}$ throughout the right half plane, and we now
address the issue of the relation between classical asymptotics and
exponential asymptotics.
\begin{Theorem}\label{Stokestr} Let $\hat{\Gamma}mma^{\pm}$ be two paths in the right half plane,
near the positive/ negative imaginary axis such that
$|x^{-\beta_1+1}e^{-x\lambda_1}|\rightarrow 1$ as $x\rightarrow\mathrm{i}nfty$
along $\hat{\Gamma}mma^{\pm}$. Consider the solution ${\bf y}$ of (\ref{eqor})
given in (\ref{soleqn}) with $\mathbf{C}=C\mathbf{e}_1$ and where
the path of integration is $p\mathrm{i}n\mathbb{R}^+$. Then
\begin{gather}\label{classicS}
{\bf y}=
(C\pm\frac{1}{2}S_1)\mathbf{e}_1 x^{-\beta_1+1}e^{-x\lambda_1}(1+o(1))
\mathrm{e}nd{gather}
\noindent for large $x$ along $\hat{\Gamma}mma^{\pm}$, where $S_1$ is the same as in
(\ref{defSj}), (\ref{microsto}).
\mathrm{e}nd{Theorem}
Classical asymptotics loses track of the value of ${\bf C}$ along any
ray other than the imaginary directions, as the terms multiplied by
${\bf C}$ will be hidden ``beyond all orders'' of the classically divergent series
$\tilde{{\bf y}}_0$. In contrast to the classical picture, we see that
through generalized Borel summation the constant ${\bf C}$ is precisely
defined throughout the positive half-plane and the question of where
the change in ${\bf C}$ occurs is well defined.
Formula (\ref{microsto}) is the exponential asymptotic expression of
the Stokes phenomenon. It shows that the constant jumps as the Stokes
line is crossed, (\ref{microsto}), as originally predicted by Stokes
himself \cite{Stokes}. Subsequently, the original ideas of Stokes,
based on optimal truncation of series were greatly refined by M.
Berry, leading to his theory of hyperasymptotic expansions and a
description of Stokes transitions for saddle integrals \cite{Berry}.
If more than one component of ${\bf C}$ is nonzero, then in general there
is no direction along which ${\bf C}$ can be defined through {\mathrm{e}m
classical} asymptotics. Part of the difficulty of studying nonlinear
Stokes phenomena using classical tools stems from this fact.
Relation (\ref{microsto}) expresses the evolution of ${\bf C}$ and the
presence of a Stokes phenomenon beyond all orders of Poincar\'e
asymptotics.
\section{Proofs and further results}
\label{sec:proofs}
The layout of the proofs is as follows. We study the formal inverse
Laplace transforms of (\ref{eqor}) and (\ref{systemformv})
in a $C^*$-algebra of distributions that we introduce.
Using a fixed point principle we find the general solution
of these convolution equations, and then study their properties. We
then show that generalized Laplace transforms of these distributions
exist and have all the required properties. The resurgence formulas
are obtained by comparing different expressions for the same solution
of (\ref{eqor}) near a Stokes line.
Since the proofs rely to a large extent on the detailed study of the
convolution equations (\ref{eqil}), (\ref{eqMv}), we start with a few
heuristic remarks. In a convolution equation such as
(\ref{eqil}), the term $(-p+\hat{\Lambda})\bfY$ plays a role similar to that of the highest derivative term
in a differential equation. To illustrate this, assume a solution
$\bfY$ is already given on an interval, say $(0,a)$, and we wish to
extend it to $(0,a+\mathrm{e}psilon)$. We look for such a solution in the form
$\bfY+ \boldsymbol\delta$, where we take $\bfY=0$ on
$(a,a+\mathrm{e}psilon)$ and $\boldsymbol\delta=0$ on $(0,a)$. If $\mathrm{e}psilon$
is small, then $\boldsymbol\delta*\boldsymbol\delta=0$ and the
equation in $\boldsymbol\delta$ is linear inhomogeneous. The terms
that involve integrals of $\boldsymbol\delta$ are of order
$O(\mathrm{e}psilon)\|\boldsymbol\delta\|$ as $\mathrm{e}psilon\rightarrow 0$, so that the
dominant terms are the forcing term, together with
$(-p+\hat{\Lambda})\boldsymbol\delta$ provided the coefficient is invertible. If in addition
the forcing is non-singular then
$\boldsymbol\delta$ can be found, e.g., by a convergent $\mathrm{e}psilon$
expansion; this is the analog of an ordinary
point of a differential equation. $\boldsymbol \delta$ can be singular if either
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_p=\mbox{det\,}(p-\hat{\Lambda})=0$
or the forcing is singular. To
understand
the qualitative behavior near a zero of $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_p$ one has to keep
(at least) one more term, the leading term among those previously
discarded (i.e. the second term in the notation (\ref{eqMv})). In
this approximation, $\boldsymbol\delta$ satisfies a differential
equation.
In our problem, there are $n$ roots of $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_p$ but because of the
nonlinearity and nonlocality of the equations, a singularity generates
(when convolved with itself) a whole array of singularities affecting
$\boldsymbol\delta$ through the forcing term. Through convolution, a
nonintegrable singularity produces further singularities of even lower
regularity. We introduce a distribution space,
$\mathcal{D}'_{m,{\cal N}u,l}$, whose degree of regularity decreases with
the distance from the origin, at a ``convolution-like'' rate; these
distributions form a convolution Banach algebra (cf.
\S~\ref{sec:NC}).
Technically, the proofs rely on suitable fixed-point theorems in
spaces having some of the properties we want to prove, notably in
terms of regularity and behavior at infinity. This is combined with a
local analysis near noninvertibility points of the dominant term,
which is done by treating the convolution equations as a
perturbation of the approximating differential equation mentioned
above, which splits the singularity thus making again possible the use
of fixed point theorems. This analysis is used in order to find the
resurgence properties, which in turn are used to prove (using among
others Lemma~\ref{cuteCauchy} below) the sharper results on global
analyticity and structure of singularities.
We start by introducing some useful functional spaces
and derive specific fixed point theorems.
\subsection{Technical constructions and results}\label{sec:Tec}
\subsubsection{Focusing spaces and algebras}
We say that a family of norms $\|\|_{{\cal N}u}$ depending on a parameter
${\cal N}u\mathrm{i}n\mathbb{R}^+ $ is {\bf focusing} if for any $f$ with $\|f\|_{{\cal N}u_0}<\mathrm{i}nfty$
\begin{eqnarray}
\label{focus-pocus}
\|f\|_{\cal N}u\downarrow 0 \mbox{ as }{\cal N}u\uparrow\mathrm{i}nfty
\mathrm{e}nd{eqnarray}
Let $\mathcal{E}$ be a linear space and $\{\|\|_{\cal N}u\}$ a family of
norms satisfying (\ref{focus-pocus}). For each ${\cal N}u$ we define a
Banach space $\mathcal{B}_{\cal N}u$ as the completion of
$\{f\mathrm{i}n\mathcal{E}:\|f\|_{{\cal N}u}<\mathrm{i}nfty\}$. Enlarging $\mathcal{E}$ if
needed, we may assume that $\mathcal{B}_{\cal N}u\subset\mathcal{E}$. For
$\alpha<\beta$, (\ref{focus-pocus}) shows that the identity is an
embedding of $\mathcal{B}_\alpha$ in $\mathcal{B}_\beta$. Let
$\mathcal{F}\subset\mathcal{E}$ be the projective limit of the
$\mathcal{B}_{\cal N}u$. That is to say
\begin{eqnarray}
\label{focuproj}
\mathcal{F}:=\bigcup_{{\cal N}u>0}\mathcal{B}_{\cal N}u
\mathrm{e}nd{eqnarray}
\noindent is endowed with the topology in which a sequence is convergent if it
converges in {\mathrm{e}m some} $\mathcal{B}_{\cal N}u$. We call $\mathcal{F}$
a {\bf focusing space}.
Consider now the case when
$\left(\mathcal{B}_{{\cal N}u},+,*,\|\|_{\cal N}u\right)$ are commutative Banach
algebras. Then $\mathcal{F}$ inherits a structure of a commutative
algebra, in which $*$ (``convolution'') is continuous. We say that
$\left(\mathcal{F},*,\|\|_{\cal N}u\right)$ is a {\bf focusing algebra}.
\subsubsection{Examples}
\label{sec:NC}
Let $K\mathrm{i}n\mathbb{R}^+$ and
$\mathcal{S}=\mathcal{S}_{K,\alpha_1,\alpha_2}=\{p:\arg(p)\mathrm{i}n[\alpha_1,\alpha_2]\subset
(-\pi/2,\pi/2),|p|\le K\}$ (or a finite union of such sectors)
and $\mathcal{V}$ be a small neighborhood
of the origin. $\overline{\mathcal{V}}$ will be the closure of
$\mathcal{V}$, cut along the negative axis, and together with these
upper and
lower cuts.
{\bf (1)}. ${L^1}_{\cal N}u(\mathcal{K})$. Let
$\mathcal{K}=\mathcal{S}_{K,\phi,\phi}$.
The space
${L^1}_{\cal N}u(\mathcal{K})$ with
convolution (\ref{defconv}) is a commutative
Banach algebra under each of the (equivalent) norms
\begin{eqnarray}
\label{norm00}
\|f\|_{\cal N}u=\mathrm{i}nt_0^K \mathrm{e}^{-{\cal N}u t}|f(t\mathrm{e}xp(\mathrm{i}\phi))|\mathrm{d}t
\mathrm{e}nd{eqnarray}
\noindent Indeed, with $F(s):=f(s\mathrm{e}^{\mathrm{i}\phi})$ and $G(s):=g(s\mathrm{e}^{\mathrm{i}\phi})$ we have:
\begin{multline}\label{prfRem}
\mathrm{i}nt_0^{K}\mathrm{d}t\mathrm{e}^{-{\cal N}u t}\left|\mathrm{i}nt_0^{t}\mathrm{d}sF(s)G(t-s)\right|\le
\mathrm{i}nt_0^{K}\mathrm{d}t\mathrm{e}^{-{\cal N}u t}\mathrm{i}nt_0^{t}\mathrm{d}s|F(s)G(t-s)|\cr
=\mathrm{i}nt_0^{K}\mathrm{i}nt_{0}^{K-v}\mathrm{e}^{-{\cal N}u (u+v)}|F(v)||G(u)|\mathrm{d}u\mathrm{d}v\cr\le
\mathrm{i}nt_0^{K}\mathrm{i}nt_{0}^{K}\mathrm{e}^{-{\cal N}u (u+v)}|F(v)||G(u)|
\mathrm{d}u\mathrm{d}v=\|f\|_{{\cal N}u }\|g\|_{{\cal N}u }
\mathrm{e}nd{multline}
\noindent By dominated convergence
$\|f\|_{\cal N}u\downarrow 0$ as ${\cal N}u\uparrow\mathrm{i}nfty$
and thus ${L^1}(\mathcal{K})$ is a focusing algebra.
{\bf (2)} If $K=\mathrm{i}nfty$ in example $(1)$, then the norms (\ref{norm00})
are not equivalent anymore for different ${\cal N}u$, but convolution is
still continuous in (\ref{norm00}) and the projective limit of
the $L^1_{\cal N}u(\mathbb{R}^+ \mathrm{e}^{\mathrm{i}\phi})$, $\mathcal{F}(\mathbb{R}^+
\mathrm{e}^{\mathrm{i}\phi})\subset L^1_{loc}(\mathbb{R}^+ \mathrm{e}^{\mathrm{i}\phi})$, is a focusing algebra.
{\bf (3a)} $\mathcal{T}_\beta(\mathcal{S}\cup\overline{\mathcal{V}})$.
For $\Re(\beta)> 0$ and $\phi_1{\cal N}e\phi_2$,
this space is given by $\{f:f(p)=p^{\beta}F(p)\}$,
where $F$ is analytic in the interior
of $\mathcal{S}\cup\mathcal{V}$ and continuous in
its closure. We take the family of (equivalent) norms
\begin{eqnarray}
\label{normF1}
\|f\|_{{\cal N}u,\beta}=K\sup_{s\mathrm{i}n
\mathcal{S}\cup\overline{\mathcal{V}}}\left|\mathrm{e}^{-{\cal N}u p}f(p)\right|
\mathrm{e}nd{eqnarray}
\noindent It is clear that convergence of $f$ in $\|\|_{{\cal N}u,\beta}$ implies
uniform convergence of $F$ on compact sets in
$\mathcal{S}\cup\mathcal{V}$ (for $p$ near zero, this follows from
Cauchy's formula). $\mathcal{T}_{\beta}$ are thus Banach spaces and
focusing spaces by (\ref{normF1}). The spaces
$\{\mathcal{T}_{\beta}\}_{\beta}$ are isomorphic to each-other.
Taking $s=pt$ in (\ref{defconv}) we find that
\begin{gather}
\label{defconv31}
p^{-\beta_1-\beta_2-1} (f_1*f_2)(p)=
\mathrm{i}nt_0^1 t^{\beta_1}F_1(pt) (1-t)^{\beta_2}
F_2(p(1-t))\mathrm{d}t=F(p)
\mathrm{e}nd{gather}
\noindent where $F$ is manifestly analytic, and that the application
\begin{eqnarray}
\label{convodom1}
(\cdot *\cdot):\mathcal{T}_{\beta_1}\times
\mathcal{T}_{\beta_2}\mapsto \mathcal{T}_{\beta_1+\beta_2+1}
\mathrm{e}nd{eqnarray}
\noindent is continuous:
\begin{eqnarray}
\label{normconvo1}
&&\|f_1*f_2\|_{{\cal N}u,\beta_1+\beta_2+1}=
K\sup_p\left|\mathrm{e}^{-{\cal N}u
p}\mathrm{i}nt_0^ps^{\beta_1}F_1(s)(p-s)^{\beta_2}F_2(p-s)\mathrm{d}s\right|\cr&&\le
K^{-1}\sup_p\mathrm{i}nt_0^p\left|K\mathrm{e}^{-{\cal N}u s}s^{\beta_1}F_1(s)K\mathrm{e}^{-{\cal N}u (p-s)}
(p-s)^{\beta_2}F_2(p-s)\right|\mathrm{d}|s|\cr&&
\le \|f_1\|_{{\cal N}u,\beta_1}\|f_2\|_{{\cal N}u,\beta_2}
\mathrm{e}nd{eqnarray}
A natural generalization of $\mathcal{T}_\beta$ is obtained taking
$\beta_1,\ldots,\beta_N\mathrm{i}n\mathbb{C}$ with positive real parts, no two of
them differing by an integer. If $f_\beta=\sum_{i=1}^k p^{\beta_i}A_i(p)$
with $A_i$ analytic, then $f_\beta\mathrm{e}quiv 0$ iff $A_i\mathrm{e}quiv 0$ for all $i$
(e.g., by a Puiseux series argument). It is then
natural to identify the space $\mathcal{T}_{\{\beta_1,\ldots,\beta_k\}}$
of functions of the form $f_\beta$ with
$\oplus_{i=1}^k \mathcal{T}_{\beta_i}$. Convolution with analytic
functions is defined on $\mathcal{T}_{\{\beta_1,\ldots,\beta_k\}}$ while
convolution of two functions in $\mathcal{T}_{\{\beta_1,\ldots,\beta_k\}}$
takes values in $\mathcal{T}_{\{\beta_i+\beta_j \,\mbox{mod 1}\} }$.
We write $\mathcal{T}_{\{\cdot\}}$ when the concrete
values of $\beta_1,\ldots,\beta_k$ do not matter.
{\bf (3b)} A particular case of the preceding example is
$\mathcal{A}_{z,l}(\mathcal{S\cup \mathcal{V}})$ consisting of
analytic functions in the interior of $\mathcal{S}\cup\mathcal{V}$,
continuous on its closure, and vanishing at the origin together with
the first $l$ derivatives. $\mathcal{A}_{z,l}$ can be identified with
$\mathcal{T}_l$.
{\bf (4)} $\mathcal{D}'_{m,{\cal N}u}$, the ``staircase distributions''.
{\mathrm{e}m Proofs} of the properties stated in this paragraph and more details are
given in \S~\ref{starca}. Let $\mathcal{D}(0,x)$ be the test
functions on $(0,x)$ and $\mathcal{D}=\mathcal{D}(0,\mathrm{i}nfty)$. Let
$\mathcal{D}'_m\subset\mathcal{D}'$ be the distributions $f$ for which
$f=F_k^{(km)}$ on $\mathcal{D}(0,k+1)$ with $F_k\mathrm{i}n{L^1}[0,k+1]$.
There is a uniquely associated ``staircase decomposition'', a sequence
$\left\{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i(f)\right\}_{i\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N}=\left\{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\right\}_{i\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N}$
such that $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\mathrm{i}n{L^1}(\mathbb{R}^+)$,
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i=\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[i,i+1]}$ and
\begin{eqnarray}
\label{stdec0}
f=\sum_{i=0}^{\mathrm{i}nfty}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i^{(mi)}
\mathrm{e}nd{eqnarray}
\noindent Convolution is well defined on $\mathcal{D}'_m$ by
\begin{eqnarray}
\label{formulaforconv}
\label{stdeccomv}
f*\tilde{f}:=(F_k*\tilde{F}_k)^{(2km)}\ \ \mbox{in } \mathcal{D}'(0,k+1)
\mathrm{e}nd{eqnarray}
\noindent and $(\mathcal{D}'_m,+,*)$ is a commutative algebra. We define, for
$f\mathrm{i}n\mathcal{D}'_{m}$,
\begin{eqnarray}
\label{normdistr00}
\|f\|_{{\cal N}u ,m}:=c_m\sum_{i=0}^{\mathrm{i}nfty}{\cal N}u ^{im}\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\|_{\cal N}u
\mathrm{e}nd{eqnarray}
\noindent where $c_m$ is defined in Lemma 39, and $\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta\|_{\cal N}u$ is
computed from (\ref{norm00}) with $K=\mathrm{i}nfty$. Then
(\ref{normdistr00}) is a norm on $\mathcal{D}'_m$ and
$\mathcal{D}'_{m,{\cal N}u}=\left(\mathcal{D}'_m,+,*,\|\|_{m,{\cal N}u}\right)$ is
a Banach algebra. With respect to the family of norms $\|\|_{m,{\cal N}u}$,
the projective limit of the $\mathcal{D}'_{m,{\cal N}u}$, $\mathcal{F}_{m}$
is a focusing algebra.
For any $f\mathrm{i}n L^1_{{\cal N}u_0}(\mathbb{R}^+)$ there is a constant
$C({\cal N}u,{\cal N}u_0)$ such that $f\mathrm{i}n \mathcal{D}'_{m,{\cal N}u}$ for all
${\cal N}u>{\cal N}u_0$ and
\begin{eqnarray}
\label{majornorm}
\|f\|_{\mathcal{D}'_{m,{\cal N}u}}\le C({\cal N}u_0,{\cal N}u)\|f\|_{ L^1_{{\cal N}u_0}}
\mathrm{e}nd{eqnarray}
\noindent and formula (\ref{formulaforconv}) is equivalent to (\ref{defconv}),
in this case.
The operator $f(p)\mapsto pf(p)$ is continuous from $\mathcal{D}'_{m,{\cal N}u}$
to $\mathcal{D}'_{m,{\cal N}u+\delta}$ for any $\delta>0$.
For $a\mathrm{i}n\mathbb{R}^+$,
$\mathcal{D}'_{m,{\cal N}u}(a,\mathrm{i}nfty)=\{f\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}:
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i(x)=0$ for $x<a\}$ is a closed ideal in $\mathcal{D}'_{m,{\cal N}u}$
(isomorphic to the restriction $\mathcal{D}'_{m,{\cal N}u}(a,\mathrm{i}nfty)$ of
$\mathcal{D}'_{m,{\cal N}u}$ to $\mathcal{D}(a,\mathrm{i}nfty)$). The restrictions
$\mathcal{D}'_{m,{\cal N}u}(a,b)$ of $\mathcal{D}'_{m,{\cal N}u}$
to $\mathcal{D}(a,b)$ are for $0<a<b<\mathrm{i}nfty$ Banach spaces
with respect to the norm
(\ref{normdistr00}) restricted to $(a,b)$.
The functions in
$\mathcal{D}\left(\mathbb{R}^+\backslash\hat{B}ox{\it I\hskip -2pt N}N\right)$
are dense in $\mathcal{D}'_{m,{\cal N}u}$, with respect to the norm
(\ref{normdistr00}) (Lemma~\ref{imbeddi}).
If we choose a different interval length $l>0$ instead of $l=1$ in the
partition associated to (\ref{stdec0}), we then write
$\mathcal{D}'_{m,{\cal N}u}(l)$. Obviously, dilation gives a natural
isomorphism between these structures. If
$d=\{t\mathrm{e}^{\mathrm{i}\phi}:t\mathrm{i}n\mathbb{R}^+\}$ is any ray,
$\mathcal{D}'_{m,{\cal N}u}(d)$ and
$\mathcal{F}_{m;\phi}$ are defined in an analogous way and have
the same properties as their real counterpart.
Laplace transforms are naturally defined in
$\mathcal{D}'_{m,{\cal N}u}$.
\begin{Lemma}\label{existe} Laplace transform extends continuously
from $\mathcal{D}(\mathbb{R}^+\backslash\hat{B}ox{\it I\hskip -2pt N}N)$ to $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$ by the formula
\begin{eqnarray}
\label{laptra}
({\cal L} f)(x):=\sum_{k=0}^{\mathrm{i}nfty}x^{mk}\mathrm{i}nt_0^{\mathrm{i}nfty}\mathrm{e}^{-sx}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k(s)\mathrm{d}s
\mathrm{e}nd{eqnarray}
\noindent In particular, with $f,g, h'\mathrm{i}n\mathcal{D}'_{m,{\cal N}u }$ we have
\begin{eqnarray}
\label{lapcomuta}
&& {\cal L}(f*g)={\cal L}(f){\cal L}(g)\cr &&
{\cal L}(h')=x{\cal L}(h)-h(0)\cr &&
{\cal L}(pf)=-({\cal L}(f))'
\mathrm{e}nd{eqnarray}
For $x\mathrm{i}n S_{\cal N}u= \{x:\Re(x)>{\cal N}u\} $ the sum (\ref{laptra}) converges
absolutely. Laplace transform is, for fixed $x\mathrm{i}n S_{\cal N}u$, a continuous
functional (of norm less than one) on $\mathcal{D}'_{m,{\cal N}u}$.
$({\cal L} f)(x)$ is analytic in $S_{\cal N}u$.
Furthermore, $\mathcal{L}$ is {\mathrm{e}m injective} in $\mathcal{D}'_{m,{\cal N}u}$.
\mathrm{e}nd{Lemma}
\noindent The proof is given in \S~\ref{sec:LT}.
We conclude this section with a few remarks.
\begin{Remark}\label{substiconv}\label{monotprop} Let $\mathcal{U}$ be one
of the spaces considered in the examples and ${\cal N}u$ be large:
i) if $g$ is analytic, (and if $g\mathrm{i}n\mathcal{U}$ in Examples (2) and (4))
then $L_g:=f\mapsto f*g$
is a bounded operator and $\|L_g\|=O({\cal N}u^{-1})$
($\mathcal{P}$ is such an operator, since $\mathcal{P}f=f*1$);
ii) replacing $*$ by $*_\phi$ defined as $f*_\phi g:=\mathrm{e}^{\mathrm{i}\phi}(f*g)$ leads
to an isomorphic structure;
iii) if $g\mathrm{i}n\mathcal{U}$ is a real valued nonnegative function and
$|f|\le g$ ($|\mathcal{P}^{km}f|\le \mathcal{P}^{km}g$ for all $k$, on
$(0,k+1)$ if the space is $\mathcal{D}'_{m,{\cal N}u}$) then
$\|f\|\le\|g\|$.
\mathrm{e}nd{Remark}
(i) In $L^1_{\cal N}u$ and $\mathcal{D}'_{m,{\cal N}u}$ this follows from
the continuity of convolution and (\ref{majornorm}). In
the examples (3), the natural inclusion
$\mathcal{T}_{\beta+\hat{B}ox{\it I\hskip -2pt N}N}\subset\mathcal{T}_{\beta}$ together
with (\ref{convodom1}) and (\ref{normconvo1})
makes convolution with an analytic function
continuous in $\mathcal{T}_\beta$ and the claim follows from
the estimate
$\|f*g\|\le\max|g|\, \|f\|\, \|\mathcal{P}|\mathrm{e}^{{\cal N}u p}|\|$.
(ii) The isomorphism is given by $f\mapsto \mathrm{e}^{-\mathrm{i}\phi}f$.
(iii) Since $\mathcal{P}$ is positivity preserving, writing $|f|\le g
$ as $ -g\le (\Re,\Im) f\le g$ the property is obvious when $f,g$ are
functions, while for $\mathcal{D}'_{m,{\cal N}u}$ it follows from equation
(\ref{defDelI}) below.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\subsubsection{Vectorial convolution and focusing spaces}
We endow $\mathcal{B}_{{\cal N}u}^n$ with a Banach space structure by
identifying it with $\mathcal{B}_{\cal N}u\oplus
\cdots\oplus\mathcal{B}_{\cal N}u$ ($n$ times). The projective limit of the
$\mathcal{B}_{\cal N}u$, $\mathcal{F}^n$ is, clearly, a focusing space. We
define a convolution on
$(\cdot\,*\,\cdot):\mathcal{F}^n\mapsto\mathcal{F}$ ({\mathrm{e}m not}
$\mathcal{F}^n\mapsto \mathcal{F}^n$) by
\begin{eqnarray}
\label{convect2}
\mathbf{V}*\mathbf{W}:=\sum_{i,j=1}^n V_i*W_j
\mathrm{e}nd{eqnarray}
\noindent
We write $\mathbf{V}^{*{\bf l}}:=V_1^{*l_1}*V_2^{*l_2}*\cdots*V_n^{*l_n}$
with the conventions $V^{*1}=V$ and that the factors
with $l_i=0$ are omitted.
\subsubsection{A fixed point property}
\begin{Lemma}\label{gennonsense} Let $\mathcal{F}$ be a focusing space and
$\mathcal{N}$ be a (linear or nonlinear) operator defined on
$\mathcal{F}$.
Equivalently, in view of (\ref{focus-pocus}), let $\mathcal{N}$
be defined on
$\bigcup_{{\cal N}u>{\cal N}u_0}B_{\cal N}u(\delta)$ with
$B_{\cal N}u(\delta)=\{f:\|f\|_{\cal N}u\le\delta\}$ for some $\delta>0$.
Assume $\mathcal{N}$ is
{\bf eventually contractive} in the following sense. There exist
${\cal N}u_0,\,\mathrm{e}psilon>0$, $\alpha<1$, so that if ${\cal N}u\ge{\cal N}u_0$ and
$\|f\|_{\cal N}u + \|g\|_{\cal N}u\le \mathrm{e}psilon$ then
\begin{eqnarray}
\label{condop}
\|\mathcal{N}(f+g)
-\mathcal{N}(g)\|_{\cal N}u\le \alpha\|g\|_{\cal N}u
\mathrm{e}nd{eqnarray}
\noindent Then $\mathcal{N}$ has a unique
fixed point $f_0\mathrm{i}n\mathcal{F}$.
If $\mathcal{N}$ depends continuously (in the strong topology)
on a parameter
$\phi$ for ${\cal N}u>{\cal N}u_0$ and if
the constants ${\cal N}u_0,\alpha$ and $\mathrm{e}psilon$ above
do not depend on $\phi$, then
the fixed point $f_\phi$ is also continuous in $\phi$. Furthermore,
$\lim_{{\cal N}u\rightarrow\mathrm{i}nfty}\sup_{\phi}\|f_\phi\|_{{\cal N}u}=0$.
\mathrm{e}nd{Lemma}
The proof is straightforward. To show existence, take ${\cal N}u>{\cal N}u_0$
large enough so that, by (\ref{focus-pocus})
$\|\mathcal{N}(0)\|_{\cal N}u<(1-\alpha)\mathrm{e}psilon$. Then the closed ball
$B_{\cal N}u(\mathrm{e}psilon)$ is mapped by $\mathcal{N}$ into itself
for any $\phi$ by (\ref{condop}) and $\mathcal{N}$ is contractive
there. The fixed point obtained, for instance, as the limit of the
(convergent, uniformly in $\phi$) iteration
$\phi_{n+1}=\mathcal{N}(\phi_n); \phi_0=0$ is continuous in $\phi$
since $\mathcal{N}$ is. By construction
$\|f_0\|_{\cal N}u\le \mathrm{e}psilon$, for all $\phi$.
For uniqueness, let $f_0$ and $f_1$ be fixed points of
$\mathcal{N}$; by (\ref{focus-pocus}) there is a ${\cal N}u>{\cal N}u_0$ so that
$\|f_{0,1}\|_{\cal N}u<\mathrm{e}psilon$. Then by (\ref{condop}),
$\|f_0-f_1\|_{\cal N}u\le\alpha\|f_0-f_1\|_{\cal N}u$ and thus $f_0=f_1$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
.
Let ${\cal N}u_0>0$ and let $\{\mathbf{M}\}_{{\bf l}}$ be a sequence of linear
operators $\mathbf{M}_{\bf l}:\mathcal{B}_{\cal N}u\mapsto\mathcal{B}_{\cal N}u^n$
for ${\cal N}u\ge{\cal N}u_0$.
Assume that for some $\kappa$ and all ${\bf l}$, $|{\bf l}|\ge 1$,
\begin{eqnarray}
\label{condgG}
&&\|\mathbf{M}_{{\bf l}}\|_{\cal N}u\le C_{\cal N}u\kappa^{|{\bf l}|}\ \mbox{ and }
C_{1,{\cal N}u}:=\lim_{{\cal N}u\rightarrow\mathrm{i}nfty}\max_{|{\bf l}|=1}\|\mathbf{M}_{{\bf l}}\|_{\cal N}u=0
\mathrm{e}nd{eqnarray}
\noindent Let ${\bf F}_0\mathrm{i}n\mathcal{F}^n$ and
$\mathbf{M}:\mathcal{F}^n\mapsto\mathcal{F}^n$ be defined by
\begin{eqnarray}\label{Mdef}
\mathbf{M}(\mathbf{Y}):=\mathbf{F}_0+
\sum_{|\bf l|\ge 1}\mathbf{M}_{\bf l}\left({\bf Y}^{*\bf l}\right)
\mathrm{e}nd{eqnarray}
\begin{Lemma}\label{infico}
$\mathbf{M}$ satisfies the assumptions of Lemma~\ref{gennonsense}.
$\mathbf{M}$ has therefore a unique
fixed point in $\mathcal{F}^n$.
\mathrm{e}nd{Lemma}
\noindent {\mathrm{e}m Proof}. We first need the following estimate.
\begin{Remark}\label{4} Let $\mathbf{V},\mathbf{W}\mathrm{i}n\mathcal{F}^n$. For $|{\bf l}|>0$
and any ${\cal N}u$ we have,
with $\|\|=\|\|_{\cal N}u$,
\begin{eqnarray}\label{estconvn}
\|\mathbf{W}_{\bf l}\|:=\|({\bf V}+{\bf W})^{*{\bf l}}-\mathbf{V}^{*{\bf l}}\|\le |{\bf l}|\left(
\|\mathbf{V}\|+\|\mathbf{W}\|\right)^{|{\bf l}|-1}\|\mathbf{W}\|
\mathrm{e}nd{eqnarray}
\mathrm{e}nd{Remark}
This inequality is obtained by induction on ${\bf l}$, with respect to
$\prec$. For $|{\bf l}|=1$, (\ref{estconvn}) is trivial. Assume
(\ref{estconvn}) holds for all ${\bf l}\prec{\bf l}_1$; without loss of
generality we may consider that ${\bf l}_1 ={\bf l}_0+\mathbf{e}_1$. We have:
\begin{multline*}
\|({\bf V}+{\bf W})^{*{\bf l}_1}-\mathbf{V}^{*{\bf l}_1}\|= \|({\bf
V}+{\bf W})^{*{\bf l}_0}*({\bf V}_1+{\bf
W}_1)-\mathbf{V}^{*{\bf l}_1}\|\cr
=\|(\mathbf{V}^{*{\bf l}_0}+\!
\mathbf{W}_{{\bf l}_0})*(V_1+W_1)-\!\mathbf{V}^{*{\bf l}_1}\|=
\|\mathbf{V}^{*{\bf l}_0}\!*\!W_1+\mathbf{W}_{{\bf l}_0}\!*\!V_1+
\mathbf{W}_{{\bf l}_0}\!*\!W_1\|\cr
\le
\|\mathbf{V}\|^{|{\bf l}_0|}\|\mathbf{W}\|+
\|\mathbf{W}_{{\bf l}_0}\|\|\mathbf{V}\|+
\|\mathbf{W}_{{\bf l}_0}\|\|\mathbf{W}\|\cr\le \|\mathbf{W}\|
\left(\|\mathbf{V}\|^{|{\bf l}_0|}+
|{\bf l}_0|(\|\mathbf{V}\|+\|\mathbf{W}\|)^{|{\bf l}_0|}\right)\le
\|\mathbf{W}\|(|{\bf l}_0|+1)(\|\mathbf{V}\|+\|\mathbf{W}\|)^{|{\bf l}_0|}
\mathrm{e}nd{multline*}
\noindent and (\ref{estconvn}) is proven.
For the sum in $\mathbf{M}$ to converge in $\|\|_{\cal N}u $ it suffices to
choose ${\cal N}u $ such that $\|\mathbf{V}\|_{\cal N}u <\kappa^{-1}$. Let
$\mathrm{e}psilon<\kappa^{-1}$ and $\mathbf{V},\mathbf{W}$ be such that
$\|\mathbf{V}\|+\|\mathbf{W}\|<\mathrm{e}psilon$. We have
\begin{multline}
\label{difN}
\left\|\mathbf{M}(\mathbf{V}+\mathbf{W})-\mathbf{M}(\mathbf{V})\right\|_{\cal N}u
\le \left(nC_{1,{\cal N}u}+C_{\cal N}u\sum_{|\bf l|\ge 2}
|{\bf l}|(\kappa\mathrm{e}psilon)^{|{\bf l}|-1}\right)\|\mathbf{W}\|_{\cal N}u\cr
\le \left(nC_{1,{\cal N}u}+n2^nC_{\cal N}u\kappa\mathrm{e}psilon+nC_{\cal N}u\frac{(2-\kappa\mathrm{e}psilon)(\kappa\mathrm{e}psilon)^{1+2(n-1)}}
{(1-\kappa\mathrm{e}psilon)^{n+1}}\right)\|\mathbf{W}\|_{\cal N}u=K_{\cal N}u\|\mathbf{W}\|_{\cal N}u
\mathrm{e}nd{multline}
\noindent where we separated out the terms with $|{\bf l}|=2, l_i=0$ or $1$ and
for the rest of the terms used the identity $\sum_{l_i\ge
2}|{\bf l}|\mathrm{e}^{-\hat{\Gamma}mma|{\bf l}|}=- \frac{\mathrm{d}}{\mathrm{d}\hat{\Gamma}mma}\left(\sum_{l\ge
2}\mathrm{e}^{-\hat{\Gamma}mma l}\right)^n$. We see that, in fact,
$\lim_{{\cal N}u\rightarrow\mathrm{i}nfty}K_{\cal N}u=0$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\subsubsection{Two lemmas on analytic structure}\label{sec:twolemmas}
\begin{Lemma}\label{cuteCauchy} Let $f$ be analytic in the unit disc cut along the
positive axis and let $0<g(x)\mathrm{i}n C^1[0,1]$. Assume that
$\lim_{\mathrm{e}psilon\downarrow
0}f(x\pm \mathrm{i}\mathrm{e}psilon g(x)) = f^{\pm}(x)$ in ${L^1}[0,1]$ and
\begin{eqnarray}
\label{condipo}
f^{+}(x )-f^{-}(x)=f_\delta(x)=x^rA(x )
\mathrm{e}nd{eqnarray}
\noindent with $\Re(r)>-1$, where $A(\xi)$ extends to an analytic function for
$|\xi |<a\le 1$. Then there exists a function
$B$ analytic in $|\xi|<a$ so that
\begin{eqnarray}
\label{concluelempo}
&&f(\xi )=\frac{1}{1-\mathrm{e}^{2\pi i r}}\xi ^rA(\xi )+B(\xi )\ \ \
(r{\cal N}otin\hat{B}ox{\it I\hskip -2pt N}N)\cr&&
f(\xi )=\frac{i}{2\pi}\ln(\xi )\xi^r A(\xi )+B(\xi ) \ \ \
(r\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N)\cr&&
\mathrm{e}nd{eqnarray}
\noindent If $ f^{+}(x )-f^{-}(x)$ is a linear combination
$\sum_{i=1}^N x^{r_i}A_i(x )$ (under the same assumptions
on $r_i$ and $A_i$), then $f$ is given by
the corresponding superposition of terms of the
form (\ref{concluelempo}).
\mathrm{e}nd{Lemma}
The proof is given in \S~\ref{sec:A0}.
In the following, $\hat{\Gamma}mma:\mathbb{R}^+\mapsto\mathbb{C}$ will denote smooth curves
in $\mathcal{R}'_1$, $\hat{\Gamma}mma_k$ denotes a curve that crosses through
the interval $(k,k+1)$, $\hat{\Gamma}mma_{\mathrm{e}psilon}=\Re(\hat{\Gamma}mma)+i
\mathrm{e}psilon\Im(\hat{\Gamma}mma)$ (cf.\S~\ref{sec:anset}).
Let $a,b\mathrm{i}n(0,\pi/2)$ and
$\mathcal{S}_0=\{p:\arg(p)\mathrm{i}n (\psi_-,0)\cup(0,\psi_+)\}$.
Let $f$ be a function analytic in $\mathcal{R}'_1$ so that
$f\circ\hat{\Gamma}mma_\mathrm{e}psilon\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}$ has limits in $
\mathcal{D}'_{m,{\cal N}u}$ as $\mathrm{e}psilon\downarrow 0$. We denote
the space of such functions by $\mathcal{D}'_{m,{\cal N}u}(\mathcal{R}_1)$.
Let $ f ^{\pm}_0=f^{\pm}$, $F_{j}=\mathcal{P}^{(mj)}f$, and for $j>0$
\begin{gather}
\label{decom0}
f_{j}^+(z-j)=F_j^{-j+}(z)-F_j^{-(j-1)+}(z);\ f_j^-(z-j)=F_j^{+j-}(z)-F_j^{+(j-1)-}(z)
\mathrm{e}nd{gather}
\noindent By construction the $ f_j^+$
are in ${L^1}[0,1-\mathrm{e}psilon)$, analytic in a sectorial neighborhood of
$z=0$ and can be extended analytically for $\Im(z)>0,\Re(z)>0$ (this
last property motivates the choice $+$ for superscript, while the
right shift is chosen in view of our application). Also by
construction $ f_j^{\pm}(z)=0$ for $z<0$; it is convenient to extend
$ f_j^{\pm}$ by zero throughout $\Re(z)<0$. We have the
``telescopic'' decomposition
\begin{eqnarray}
\label{defindecom2}
f^{\mp\,j\,\pm}(z)=\sum_{i=0}^{j}( f_i^{\pm})^{(mi)}(z-i)
\mathrm{e}nd{eqnarray}
Relation (\ref{defindecom2}) holds in
$\mathcal{D}'$ along the real axis, {\mathrm{e}m and} as an equality of analytic
functions for $\Re(z)>j,\Im(z)>0$.
For instance, for
$f=(z-2)^{-1}\mathrm{i}n\mathcal{D}'_{1{\cal N}u}$ we have $ f_1^{+}=0$,
$ f_2^{+}=-2\pi i z$ for $\Re(z)>0$, and $f_2^{+}=0$ otherwise.
Conversely, a decomposition of the form (\ref{defindecom2}) together
with analyticity in $\mathcal{S}_0$ implies analyticity in
$\mathcal{R}'_1$. More precisely, assume that
$f(t\mathrm{e}xp(\mathrm{i}\phi))\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$ for $\phi\mathrm{i}n
(\psi_-,0)\cup(0,\psi_+)$ and that $\lim_{\phi\downarrow
0}f(\cdot\,\mathrm{e}xp(\pm \mathrm{i}\phi)=f^{\pm}$ in $\mathcal{D}'_{m,{\cal N}u}$. Assume in
addition that there exists the decomposition
\begin{eqnarray}
\label{decom}
f^{\pm}=\sum_{k=0}^{\mathrm{i}nfty}
\left[ f ^{\mp}_k(p-k)\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,\mathrm{i}nfty]}\right]^{(mk)}
\mathrm{e}nd{eqnarray}
\noindent where for each
$k$, $ f ^{\mp}_k\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$ (note: the $ f ^{\mp}_k\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}$
are uniquely determined, cf. Remark~\ref{density}). Assume in addition
that $ f ^{\mp}_k$
extend analytically to $\mathcal{S}_0^{\mp}$ in the following sense:
there exist $g_k^{\mp}$ analytic in $\mathcal{S}_0^{\mp}$, with
$g_k^{\mp}(\mp t\mathrm{e}xp(\mathrm{i}\phi))\mathrm{i}n \mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$ and such
that $\lim_{\phi\downarrow 0}g_k^{\mp}(\mp t\mathrm{e}xp(\mathrm{i}\phi))= f_k^{\mp}$ in
$\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$.
\begin{Lemma}\label{STR1} (i) Under the above conditions, $f$ extends analytically
to $\mathcal{R}'_1$.
(ii) If for small argument $ f_k\mathrm{i}n\mathcal{T}_\beta$ then
\begin{eqnarray}
\label{ASTR1}
f^{\pm}(p+k)= \frac{1}{1-\mathrm{e}^{2\pi \mathrm{i}\beta}}(f^{\mp}_k(p))^{(mk)}+a(p)
\mbox{ or }\frac{i}{2\pi}( f ^{\mp}_k(p)\ln p)^{(mk)}+a(p)
\mathrm{e}nd{eqnarray}
\noindent according to whether $\beta{\cal N}otin\mathbb{Z}$ or $\beta\mathrm{i}n\mathbb{Z}$
respectively, where $a$ is analytic at zero. As in Lemma
~\ref{cuteCauchy}, if $ f_k=\sum_{i\le j} f_{ki}$
with $ f_{ki}\mathrm{i}n\mathcal{T}_{\beta_i}$ then
$f^{\pm}(p+k)$ is the corresponding superposition
of terms of the form (\ref{ASTR1}).
\mathrm{e}nd{Lemma}
The proof is given in \S~\ref{sec:A0}.
\subsubsection{Convolutions, analyticity and averaging}
\label{sec:aver}
Define $\mathcal{F}(\mathcal{R}'_1)$
as the functions in $\mathcal{D}'_{m,{\cal N}u}(\mathcal{R}'_1)$
such that,
in the decomposition (\ref{defindecom2}) $\|( f_{j}(p-j))^{(mj)}\|\le
K_f({\cal N}u)^j$ where $\lim_{{\cal N}u\rightarrow\mathrm{i}nfty}K_f({\cal N}u)=0$.
If $\hat{\Gamma}mma$ is a straight line in $\mathcal{R}'_1$
then $AC_\hat{\Gamma}mma(f*g)=AC_\hat{\Gamma}mma(
f)_\hat{\Gamma}mma*AC_\hat{\Gamma}mma(g)$ (if $\hat{\Gamma}mma$
is {\mathrm{e}m not} equivalent to a straight line, this equality
is generally {\mathrm{e}m false} \cite{Costin}).
Convolution does however commute with suitable averages of analytic
continuations, as Proposition~\ref{medianpropo} below shows. In view of symmetry, we only need to look at the
properties of the $+$ decomposition.
For $\alpha\mathrm{i}n\mathbb{C}$, consider the operator
$\mathcal{A}_\alpha:\mathcal{F}(\mathcal{R}'_1)\mapsto \mathcal{F}_m(\mathbb{R}^+)$
given by
\begin{eqnarray}
\label{generalmed}
\mathcal{A}_\alpha(f):=f^{[\alpha]}(p)=\sum_{i=0}^{\mathrm{i}nfty}\alpha^i\left( f ^+_i(p-j)\right)^{(mi)}
\mathrm{e}nd{eqnarray}
\noindent In our assumptions, convergence is ensured in
$\mathcal{D}'_{m,{\cal N}u}$ for large enough ${\cal N}u$. An important case is
the balanced average, $\alpha=1/2$. The operator $\mathcal{A}_{\frac{1}{2}}$ is
similar to \'Ecalle's medianization, and is designed to substitute for
analytic continuation along the singularity line $\mathbb{R}^+$ in a way
compatible with the $*-$algebra structure. As mentioned before, it
can be shown that under our assumptions on (\ref{eqor}), only for the
choice $\alpha=1/2$ is the difference between the
$f={\cal L}\mathcal{A}_\alpha F$ and the optimally truncated asymptotic
series of $f$ always of the order of magnitude of the least term of
the series \cite{CK2}. Borel summability techniques and
hyperasymptotic methods \cite{Berry}, \cite{Berry-hyp} give, whenever
they both apply, the same association between transseries and actual
functions.
\begin{Proposition}\label{medianpropo} i) If $f,g\mathrm{i}n\mathcal{F}(\mathcal{R}'_1)$ then $f*g$ defined
for small argument by (\ref{defconv}) extends
to a function in $\mathcal{F}(\mathcal{R}'_1)$. We have
\begin{eqnarray}
\label{defindecom3}
(f*g)_j^{\pm}=\sum_{s=0}^j f_s^{\pm}*g_{j-s}^{\pm}
\mathrm{e}nd{eqnarray}
\noindent and $K_{f*g}({\cal N}u)\le K_f({\cal N}u)+K_g({\cal N}u)$. If $h$ is analytic and
bounded in the right half plane and $f\mathrm{i}n\mathcal{F}(\mathcal{R}'_1)$
then $hf\mathrm{i}n\mathcal{F}(\mathcal{R}'_1)$.
ii) If $h$ is analytic in $\mathcal{R}'_1\cup\mathbb{R}^+$ and
$f,g\mathrm{i}n\mathcal{F}(\mathcal{R}'_1)$, $a,b\mathrm{i}n\mathbb{C}$, then
\begin{eqnarray}
\label{assertmed}
\mathcal{A}_\alpha(af+bg)&=&a\mathcal{A}_\alpha(f)+b
\mathcal{A}_\alpha(g)
\cr \mathcal{A}_\alpha(hf)&=&h\mathcal{A}_\alpha(f)\cr
\mathcal{A}_\alpha(1)&=&1\cr
\mathcal{A}_\alpha(f*g)&=&\mathcal{A}_\alpha(f)*\mathcal{A}_\alpha(g)
\mathrm{e}nd{eqnarray}
\noindent If $\mathbf{M}$ satisfies the hypothesis of Lemma~\ref{gennonsense} and in
addition
$\mathbf{M}_{{\bf l}}(\mathcal{F}(\mathcal{R}'_1))\subset\mathcal{F}(\mathcal{R}'_1)$ and $\mathcal{A}_\alpha
\mathbf{M}_{{\bf l}}=\mathbf{M}_{{\bf l}}\mathcal{A}_\alpha$
then
\begin{eqnarray}
\label{assertmed3}
\mathcal{A}_\alpha \mathbf{M}=\mathbf{M}\mathcal{A}_\alpha
\mathrm{e}nd{eqnarray}
\noindent In particular, if $\bfY$ is a fixed point of $\mathcal{M}$ then so
is $\mathcal{A}_\alpha\bfY$. An example is the case
$\mathbf{M}_{{\bf l}}(\bfY)= \mathbf{G}_{{\bf l}}*\bfY^{*{\bf l}}$ with
$\mathbf{G}_{{\bf l}}$ analytic in $\mathcal{R}'_1\cup\mathbb{R}^+$ and such that
for some $\kappa$ and all ${\bf l}$ we have $|\mathbf{G}_{{\bf l}}(p)|\le\mathrm{e}xp(\kappa
p)$.
\mathrm{e}nd{Proposition}
\noindent The proof and further details are given in
Appendix~{\ref{sec:A0}}.
Let now
$\mathcal{F}_r(\mathcal{R}'_1)\subset\mathcal{F}(\mathcal{R}'_1)$
consist in functions $f$ whose only singularities are regular, in the
sense that the elements $f_{j}$ (cf. (\ref{decom0})) are of the form
$(\sum_{i=1}^{N_j}p^{a_{i,j}}A_{i,j})^{(mj)}$ where $A_{i,j}$ are
analytic near $p=0$.
\begin{Remark}\label{preservestruct} $\mathcal{F}_r(\mathcal{R}'_1)$ is stable with respect to
convolution.
\mathrm{e}nd{Remark}
\noindent By construction, (Proposition~\ref{medianpropo})
and (\ref{defindecom3}), for small $p$,
$(f_{1}*f_2)_j $ is a sum of terms of the form
$$p^{a_1}A_1*p^{a_2}A_2$$
\noindent and the proof follows without any difficulty from
(\ref{defconv31}).
\subsection{Main proofs}
\begin{Proposition}\label{9}\label{estimd} i)
For any $\kappa>\max\{x_0,y_0^{-1}\}$, cf. (\ref{Taylor
series}), there is a constant $K>0$ such that for all ${\bf l}\succ 0$
\begin{eqnarray}\label{exponesti}
\sup_{p\mathrm{i}n\mathbb{C}}\mathrm{e}^{-\kappa |p|}|{\bf G}_{\bf l}(p)|\le K \kappa^{|\bf l|}
\mathrm{e}nd{eqnarray}
\noindent (cf. (\ref{eqil})).
ii) Let $\mathcal{F}$ be one of the focusing algebras in
\S~\ref{sec:NC} and $\bfY\mathrm{i}n\mathcal{F}$. Let
$$ \mathbf{D}_{\bf j}= \sum_{ \bf l\ge j} \binom{{\bf l}}{{\bf j}
}\left[ {\bf G}_{\bf l}*\bfY^{*({\bf l} -{\bf j}) }+\mathbf{g}_{0,{\bf l}}*\bfY^{*({\bf l}
-{\bf j})}\right]$$
\noindent Then for large ${\cal N}u$ and some $\kappa_1>0$, $\|\mathbf{D}_{\bf j}\|\le
\kappa_1^{|{\bf j}|}$ while for $|\mathbf{j}|=1$,
$\|\mathbf{D}_{\bf j}\|=
o({\cal N}u^{-M})$.
\mathrm{e}nd{Proposition}
{\mathrm{e}m Proof.}
\noindent (i) From the analyticity of $\mathbf{g}$ it follows
that
\begin{eqnarray}\label{unifesti}
|{\bf g}_{m,\bf l}|<\mbox {const. } \kappa^{m+|{\bf l}|}
\mathrm{e}nd{eqnarray}
\noindent where the constant is independent on $m$ and ${\bf l}$.
Then, by (\ref{lapdef}),
\begin{equation}\label{estimmodg}
|{\bf G}_{\bf l}(p)|<\mathrm{const.}\ \kappa^{|{\bf l}|+1}\frac{\mathrm{e}^{\kappa| p|}-1}{\kappa |p|}<
\mathrm{const.}\ \kappa^{|{\bf l}|+1}\mathrm{e}^{\kappa |p|}
\mathrm{e}nd{equation}
\noindent The last claim is a direct consequence of (n5).
(ii)
Note first that $\sum_{|{\bf l}|=l}1\le l^n\le 2^{nl}$ (since
$l_i\le l$). Also, $\binom{l_i}{j_i}\le 2^{l_i}$ so that
$\binom{{\bf l}}{{\bf j}}\le 2^{n|{\bf l}|}$. By (n5) we have $\mathbf{g}_{0,{\bf l}}=0$ if
$|{\bf l}|\le 1$. Choosing $\delta<2^{2nl+1}\kappa$ and ${\cal N}u$ such that
$\|\bfY\|\le\delta$, $\|\mathbf{D}_{\bf j}\|$ is estimated
by
$$\sum_{l\ge j}2^{2nl}\delta^{l-j}\kappa_l^l$$
\noindent where $j=|{\bf j}|,\kappa_l=\|\mathbf{G}_{\bf l}\|+|\mathbf{g}_{0,{\bf l}}|\le 2\kappa$ if $l>1$ and $\kappa_l=o({\cal N}u^{-M})$
for $l=1$, by (i), and the result follows.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
Without loss of generality, we analyze (\ref{eqil}) in
a neighborhood of $d_1$, the Stokes line corresponding
to $\lambda_1=1$. (For the equations (\ref{eqMv})
we will need, in addition, to study a direction where
$p-\lambda_i+\mathbf{k}\cdot \boldsymbol\lambda=0$.)
\noindent Let $\mathrm{e}psilon$
and $c_0$ be small and positive, $\mathcal{V}=\{p:|p|<1\}$,
\begin{equation}
\label{defSC}
\mathcal{S}_c=\{p:\arg(p)\mathrm{i}n [\psi_n-2\pi+c,-c]\cup[c,\psi_2-c]\}
\mathrm{e}nd{equation}
\noindent $\mathcal{S}_0=\cup_{0<c<c_0}\mathcal{S}_c$, $\mathcal{S}_c^{\pm}=
\mathcal{S}_c\cap\{p:\pm\arg(p)>0\}$, and let
$\mathcal{S}'_0,\mathcal{S}'_c,{\mathcal{S}_c^{\pm}}'$ be defined
correspondingly, with $\psi_-$ and $\psi_+$ replacing $\psi_n-2\pi$
and $\psi_2$, respectively.
\begin{Lemma}\label{analyticase} i) For any ray $d\subset \mathcal{S}_0$, there is a
unique solution of (\ref{eqil}) in $\mathcal{D}'_m({d},l)$ for any
$m\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N$, $l\mathrm{i}n(0,1)$. This solution, $\bfY_0$, is in fact
analytic in $\mathcal{S}_0\cup\mathcal{V}$ and, for large enough
${\cal N}u(d)$, $\bfY_0\mathrm{i}n L^1_{\cal N}u(d)$.
ii) The function $\bfY_0(\cdot\,\mathrm{e}^{\mathrm{i}\phi})$ is continuous in
$\phi\mathrm{i}n(\psi_n,0]$ and (separately in)
$\phi\mathrm{i}n[0,\psi_2)$ in the
$\mathcal{D}'_{{\cal N}u,m}(\mathbb{R}^+,l)$ topology and
$\sup_{\phi\mathrm{i}n[\psi_1,\psi_2]}\|\bfY_0(p\mathrm{e}^{\mathrm{i}\phi})\|_{m,{\cal N}u}
\rightarrow 0$ as ${\cal N}u\rightarrow\mathrm{i}nfty$.
iii) The description (\ref{SY0}) holds for $j=1,l=1$.
iv) For $a>1$, there is a one-parameter family
of solutions of (\ref{eqil}) in $\mathcal{D}'_m(0,a)$.
\mathrm{e}nd{Lemma}
\noindent {\mathrm{e}m Note:} the hyperfunctions
$\bfY_0(p\mathrm{e}^{\pm 0i})=\bfY_0^{\pm}$ are different, in general.
\begin{Corollary}\label{prim} For $k\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N\cup\{0\}$, the function
$\mathcal{P}^{mk}\bfY_0(p\mathrm{e}^{\mathrm{i}\phi})$ is continuous in
$S_k^{-}:=\{p:0\le|p|<k+1;\arg(p)\mathrm{i}n (\psi_n,0]\}$ and in
$S_k^{+}:=\{p:0\le|p|<k+1;\arg(p)\mathrm{i}n [0,\psi_2)\}$ (and, of course,
analytic in $S_0$). \mathrm{e}nd{Corollary}
{\mathrm{e}m Proof of Lemma~\ref{analyticase}.}
We write (\ref{eqil}) in the form:
\begin{eqnarray}\label{eqilm}
\bfY=
\left(\hat{\Lambda}-p\right)^{-1}\left({\bf F}_0-
\hat B\mathcal{P}\bfY+{\cal N}({\bf Y})\right)=\mathcal{M}(\bfY)
\mathrm{e}nd{eqnarray}
\noindent Let $d_K$ be an initial segment of $d$ of length $K<\mathrm{i}nfty$. As the matrix
$\hat{\Lambda}-p$ is invertible in $\mathcal{S}_c$, it is easy to see
that the operator $\mathcal{M}$ in (\ref{eqilm}) satisfies the
conditions of Lemma~\ref{infico}, in the spaces $ L^1_{\cal N}u(d)$,
$\mathcal{A}_{z,l}(\mathcal{S})$ ($l\le M$) (cf. Examples (1) through
(4) in \S\ref{sec:NC} and Remark~\ref{9}). The conditions are in
addition also satisfied in $\mathcal{D}'_{m,{\cal N}u}(d)$; due to the
special structure of this space, the proof the boundedness of the
operator $U=(\Lambda-p)^{-1}$ is more delicate, and is given in
Lemma~\ref{cinftycase}.
Thus, $\mathcal{M}$ has a unique
fixed point in each of these spaces. The obvious inclusions between
these spaces complete the proof of part (i). For the rest of
Lemma~\ref{analyticase} we need more results.
\begin{Proposition}\label{IN0}
The properties stated in Lemma~\ref{analyticase}
hold in $\mathcal{S}_0\cap\{p:|p|<1+\mathrm{e}psilon\}$
\mathrm{e}nd{Proposition}
The proof is given in \S~\ref{sec:A1}. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Proposition}\label{contidis}
Let $\mathbf{W}_0$ be a solution of (\ref{eqil}) in $\mathcal{D}'_m(0,a)$
with $a>1$. For $b\ge a$ there exists a unique solution of (\ref{eqil})
in $\mathcal{D}'_m(0,b)$, which
agrees
with $\mathbf{W}_0$ on $\mathcal{D}(0,a)$.
\mathrm{e}nd{Proposition}
\noindent We use the decomposition $\mathcal{D}'_m(0,b)=
\mathcal{D}'_m(0,a)\bigoplus\mathcal{D}'_m(a,b)$, $(a<b\le\mathrm{i}nfty)$ We
identify $\mathbf{W}_0$ with an element of $\mathcal{D}'_m(0,b)$ by
extending it with zero and define $\mathcal{M}_1$ on
$\mathcal{F}_m(a,b)=\cup_{{\cal N}u>{\cal N}u_0}\mathcal{D}'_{m,{\cal N}u}(a,b)$ by
\begin{eqnarray}
\label{decom2}
\mathcal{M}_1(\mathbf{W}_1)=\mathcal{M}(\mathbf{W}_0+\mathbf{W}_1)-\mathcal{M}(\mathbf{W}_0)
\mathrm{e}nd{eqnarray}
\noindent and (\ref{eqilm}) becomes
\begin{eqnarray}
\label{eqilm1}
\mathbf{W}_1=\mathcal{M}_1(\mathbf{W}_1)
\mathrm{e}nd{eqnarray}
\noindent By Lemma~\ref{gennonsense}, $\mathcal{M}$ is eventually contractive
and by (\ref{decom2}), clearly, so is $\mathcal{M}_1$. By
Lemma~\ref{gennonsense} then, $\mathcal{M}_1$ has a unique fixed point
in $\mathcal{F}_{m}(a,b)$ ($b\le\mathrm{i}nfty$). In view of
the inclusions between $\mathcal{F}_{m}(a,\mathrm{i}nfty)$ and
$\mathcal{F}_{m}(a,b)$, the proof is complete. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
.
{\mathrm{e}m Proof of Lemma~\ref{analyticase} (ii)}
Let $\phi\mathrm{i}n[0,\mathrm{e}psilon]$ with $\mathrm{e}psilon$ small. We regard the space
$\mathcal{F}_{m}$ as fixed and $\mathcal{M}_1$ as being dependent on
$\phi$ through $p$ and $*_\phi$ (cf. Remark~\ref{substiconv}).
Firstly, $\mathbf{W}_0(\phi)$ is continuous in $\phi\mathrm{i}n[0,\mathrm{e}psilon]$ and
$\|\mathbf{W}_0(p\mathrm{e}^{\mathrm{i}\phi})\|_{\cal N}u=O(1/{\cal N}u)$ uniformly in $\phi$ as follows
from Proposition~\ref{IN0}. Then the infinite sum in the definition
of $\mathcal{M}_1$ is uniformly convergent in
$\mathcal{D}'_{m,{\cal N}u}(a,\mathrm{i}nfty)$ for ${\cal N}u$ large enough by
(\ref{exponesti}). The operator
$U=(p\mathrm{e}^{\mathrm{i}\phi}-\hat{\Lambda})^{-1}=\mathrm{e}^{-\mathrm{i}\phi}(p-\mathrm{e}^{-\mathrm{i}\phi}\hat{\Lambda})^{-1}$
is strongly continuous in $\phi$ in $\mathcal{F}_{m}(a,\mathrm{i}nfty)$,
$a>1$, cf. Lemma~\ref{cinftycase}. By Remark~\ref{substiconv},
$\mathcal{M}_1$ is $\phi$-continuous and Lemma~\ref{gennonsense}
applies.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\noindent We now study the convolution equations associated to the
higher terms in the transseries, (\ref{eqMv}).
\begin{Lemma}\label{higherterms} i) Given the vector of constants ${\bf C}\mathrm{i}n\mathbb{C}^{n_1}$
and and in addition
given $\bfY_0$ for $d=\mathbb{R}^+$, there is a unique solution of
(\ref{invlapvk}) in $\mathcal{F}_{m}$ with the (singular) initial
condition
\begin{eqnarray}
\label{condinit}
\bfY_{\mathbf{e}_j}(p)= C_j
\Gamma(\beta_j')^{-1}p^{\beta_j'}(\mathbf{e}_j+o(1)) \qquad(p\rightarrow
0,\
j=1,2,\ldots,n_1)
\mathrm{e}nd{eqnarray}
\noindent The general solution
of (\ref{eqsvec0}), (\ref{invlapvk}) is
\begin{eqnarray}
\label{gesovk}
\mathbf{C}^{\bf k}\mathbf{Y}_{\bf k}, \ \mathbf{C}\mathrm{i}n\mathbb{C}^{n_1}
\mathrm{e}nd{eqnarray}
\noindent where $\mathbf{Y}_{\bf k}$ is the solution for
$\mathbf{C}=(1,1,\ldots,1)$.
ii) In a neighborhood
of $p=0$ we have
$$\bfY_{\bf k}(p)=p^{{\bf k}{\boldsymbol \beta}'-1}{\bf A}_{\bf k}(p)$$
\noindent with ${\bf A}_{\bf k}$ analytic near the origin.
iii) The functions $\bfY_{\bf k}$, ${\bf k}\succeq 0$ are analytic in
$\mathcal{R}'_1$ and
$\bfY_{\bf k}(x\mathrm{e}^{\mathrm{i}\phi})$ are continuous in $\phi$ with respect to the
$\mathcal{D}'_{m,{\cal N}u}$ topology for $\phi\mathrm{i}n(\psi_-,0]$
and for $\phi\mathrm{i}n[0,\psi_+)$.
iv) Each $\bfY_{\bf k}$ is in $\mathcal{F}(\mathcal{R}'_1)$
(cf. \S~\ref{sec:aver}).
Furthermore, there is a constant
$K$ and a function $\delta({\cal N}u)$ such that
$\lim_{{\cal N}u\rightarrow\mathrm{i}nfty}\delta({\cal N}u)=0$ and in the decomposition
(\ref{defindecom2}) of $\mathbf{Y}_{\bf k}$ we have
$\mathbf{Y}_{{\bf k};j}\mathrm{i}n\mathcal{T}_{({\bf k}+j\mathbf{e}_1){\boldsymbol \beta}'-1}$ and
\begin{eqnarray}
\label{normVK}
&&\sup_{\phi,{\bf k}}\delta({\cal N}u)^{-|{\bf k}|}\|\mathbf{Y}_{{\bf k}}\|_{\mathcal{D}'_{m,{\cal N}u}(\mathrm{e}^{\mathrm{i}\phi}\mathbb{R}^+)}<K\cr&&
\sup_{\phi,{\bf k},j}\delta({\cal N}u)^{-|{\bf k}|+j}\|\mathbf{Y}_{{\bf k};j}\|_{\mathcal{D}'_{m,{\cal N}u}(\mathrm{e}^{\mathrm{i}\phi}\mathbb{R}^+)}<K
\mathrm{e}nd{eqnarray}
\noindent where $\phi$ runs in $(\psi_-,\psi_+)$.
v) The functions $\mathbf{Y}_{\bf k}(\cdot\,\mathrm{e}^{\mathrm{i}\phi})$,
$\phi\mathrm{i}n(\psi_-,\psi_+)$, are simultaneously Laplace transformable in
$\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$ and their Laplace transforms are
solutions of (\ref{systemformv}). The expression
\begin{eqnarray}
\label{solupperlower}
{\bf y}^{\pm} ={\cal L}\bfY_0^{\pm}+\sum_{{\bf k}\succ 0}x^{{\bf m}\cdot{\bf k}}{\bf C}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda} x}
{\cal L} \bfY^{\pm}_{\bf k}
\mathrm{e}nd{eqnarray}
\noindent
is uniformly convergent for large enough $x(\mathbf{C})$ in some open
sector. In addition, (\ref{solupperlower}) is a solution
of $(\ref{eqor})$ and ${\cal L}\bfY^{\pm}_{\bf k}\sim\tilde{{\bf y}}_{\bf k}$
for large $x$ in the half plane $\Re(xp)>0$.
\mathrm{e}nd{Lemma}
Without loss of generality, we analyze (\ref{invlapvk}) in
$\mathcal{T}_{\{\cdot\}}({\mathcal{S}_c^+}')$, $\mathcal{D}'_{m,{\cal N}u}(d)$,
with $d\mathrm{i}n{\mathcal{S}_c^+}'$, and in $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$. We
denote all the corresponding norms by $\|\|_{\cal N}u$.
\begin{Remark}\label{stableconv} Assume that for ${\bf k}'\prec{\bf k}$ we have
$\bfY_{{\bf k}'}\mathrm{i}n\mathcal{T}_{{\bf k}'{\boldsymbol \beta}'-1}$. Then, in (\ref{defT}),
we have
${\bf T}_{\bf k}(\bfY_0,\{\bfY_{{\bf k}'}\})\mathrm{i}n\mathcal{T}_{{\bf k}{\boldsymbol \beta}'-1}$.
\mathrm{e}nd{Remark}
This follows immediately from Equations (\ref{defconv31}) and
(\ref{convodom1}) and from the homogeneity of $\mathcal{T}$ implicit
in the sum $\sum_{(\mathbf{i}_{mp};{\bf k})}$ (the notation is explained
after (\ref{eqmygen})). {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
For $|{\bf k}|>1$ we take
$\mathbf{W}_{\bf k}:= \bfY_{\bf k}$ and $\mathbf{R}_{\bf k}:=\mathbf{T}_{\bf k}$ and
write (\ref{invlapvk}) as
\begin{equation}\label{eqabstractm}
(1+J_{\bf k})\mathbf{W}_{\bf k}=\hat Q_{\bf k}^{-1} {\bf R}_{\bf k}
\mathrm{e}nd{equation}
\noindent with $\hat Q_{\bf k}:=(-\hat\Lambda+p+{\bf k}\cdot{\boldsymbol \lambda})$ (notice that
for $|{\bf k}|>1$ and $p\mathrm{i}n\mathcal{S}_0'$
we have $\mathrm{det}\,\hat Q_{\bf k}(p){\cal N}e 0 $).
\begin{gather}\label{defjm}
(J_{\bf k}\mathbf{W})(p):=\hat Q_{\bf k}^{-1}\left(\left(\hat
B+{\bf m}\cdot{\bf k}\right)\mathrm{i}nt_0^p\mathbf{W}(s)\mathrm{d}s-\sum_{j=1}^n
\mathrm{i}nt_0^p W_j(s)\mathbf{D}_j(p-s)\mathrm{d}s\right)\cr
\mathrm{e}nd{gather}
The case $|{\bf k}|=1$ is special in that $p=0$ is a singularity. The
corresponding statements of Lemma~\ref{higherterms} for $|p|<\mathrm{e}psilon
$ with $\mathrm{e}psilon$ small are proven in Proposition~\ref{localresult}:
let $\mathbf{W}^0_{\bf k}$ be the functions provided there. For the
analytic part of Lemma~\ref{higherterms} we need to show that
$\mathbf{W}^0_{\bf k}$ extend analytically to solutions of (\ref{eqMv}).
To unify the treatment we derive equations of the form
(\ref{eqabstractm}) for these continuations. Let
$\delta\mathrm{i}n{\mathcal{S}_c^+}',|\delta|<\mathrm{e}psilon$. Using Proposition~\ref{localresult} and standard analyticity
arguments, it suffices to show that $\mathbf{W}^0_{\bf k}$ extend
analytically in any sector ${\subset\mathcal{S}_c^+}'$ centered at $\delta$,
and that the corresponding convolution equation is satisfied
along the ray $d_\delta{\cal N}i\delta$. We let
$\mathbf{a}=\mathbf{W}^0_{\bf k}(\delta)$, and with $\mathbf{W}^1_{\bf k}=\mathbf{W}^0_{\bf k}$
for $|p|<|\delta|$ along $d_\delta$ and zero otherwise, we write
$\bfY_{\bf k}(p)=\mathbf{W}^1_{\bf k}(p)+\mathbf{a}+\mathbf{W}_{\bf k}(p-\delta)$. For
$p\mathrm{i}n d_\delta$ we find that $\mathbf{W}_{\bf k}$ must satisfy
(\ref{eqabstractm}) where $\hat
Q_{\bf k}:=(-\hat\Lambda+p+{\bf k}\cdot{\boldsymbol \lambda}+\delta)$ and ${\bf R}_{\bf k}(p)$
is given by
$$
\left({\bf m}\cdot{\bf k}+\hat B\right)
\left(\mathrm{i}nt_0^{\delta}\mathbf{W}^0_{\bf k}(s)\mathrm{d}s-\mathbf{a}p\right)
+\sum_{j=1}^n
\mathrm{i}nt_0^{\delta}(\mathbf{W}^0_{\bf k})_j(s)\mathbf{D}_{\mathbf{e}_j}(p-s)\mathrm{d}s
-\mathbf{a}\mathbf{D}_{\mathbf{e}_j}(p)
$$
\noindent ${\bf R}_{\bf k}(p)$ is manifestly analytic in ${\mathcal{S}^+}'$.
Since $\mathbf{W}_{\bf k}(p)=\mathbf{W}^0_{\bf k}(p+\delta)-\mathbf{a}$ is already a
solution of (\ref{eqabstractm}) for small $p$, and in this case the
left side of (\ref{eqabstractm}) vanishes for $p=0$, we have ${\bf
R}_{\bf k}\mathrm{i}n\mathcal{T}_1$, for $|{\bf k}|=1$.
Combined with Remark~\ref{stableconv} and induction on ${\bf k}$, the
following result completes the proof of Lemma~\ref{higherterms}, parts
(i) and (ii).
\begin{Proposition}\label{Uniformnorm}
i) For large ${\cal N}u$ and constants $K_1$ and $K_2({\cal N}u)$ independent of
${\bf k}$, with $K_2({\cal N}u)=O({\cal N}u^{-1})$ we have $\|Q_{\bf k}^{-1}\|\le\frac{K_1}{|{\bf k}|}$
and
\begin{eqnarray}
\label{normQk}
\|J_{\bf k}\|\le
K_2({\cal N}u)
\mathrm{e}nd{eqnarray}
ii) For large ${\cal N}u$, the operators $(1+J_{\bf k})$ defined in $\mathcal{D}'_{m,{\cal N}u}$, and
also in
$\mathcal{T}_{{\bf k}{\boldsymbol \beta}'-1}$ for $|{\bf k}|>1$ and
in $\mathcal{T}_1$ for $|{\bf k}|=1$ are simultaneously invertible.
Given $\bfY_0$ and ${\bf C}$, the $\mathbf{W}_{\bf k}, \,|{\bf k}|\ge 1$ are
uniquely determined. For any $\delta>0$ there is a large enough ${\cal N}u$,
so that
\begin{equation}\label{estimunifk}
\|\mathbf{W}_{\bf k}\|\le\delta^{|{\bf k}|},\ k=0,1,..
\mathrm{e}nd{equation}
\noindent (in the $\mathcal{D}'_{m,{\cal N}u}$ topology, (\ref{estimunifk})
hold uniformly in $\phi\mathrm{i}n[\psi_-+\mathrm{e}psilon,0]$ and
$\phi\mathrm{i}n[0,\psi_+-\mathrm{e}psilon]$ for any small $\mathrm{e}psilon>0$).
\mathrm{e}nd{Proposition}
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof.}
(i)
For $\mathcal{T}_{{\bf k}{\boldsymbol \beta}'-1}$, this follows immediately from
Remark~\ref{substiconv}, and the constants will depend on the
parameter $c$ in $\mathcal{S}_c^+\cap\mathcal{S}_{\mathbb{R}}$. Given $\bfY_0$, the estimates
(\ref{normQk}) are also true in $\mathcal{D}'_{m,{\cal N}u;\phi}$ this time
{\mathrm{e}m uniformly} in $\phi$, down to $\phi=0$. The proof of
(\ref{normQk}) in this case is given in Appendix~\ref{sec:A1}, in
Lemma~\ref{cinftycase}, from which the continuity of $J_{\bf k}$ in
$\phi$ also follows.
(ii) From (\ref{eqabstractm}) and (i) we get, for some $K$ and $j\ge
1$ $\|\mathbf{W}_{\bf k}\|\le K\|{\bf R}_{\bf k}\|$. We first show inductively that
the $\mathbf{W}_{\bf k}$ are bounded. Choosing a suitably large ${\cal N}u(\mathrm{e}psilon)$
we can make $\max_{|{\bf k}|\le 1}\|\mathbf{W}_{\bf k}\|_{\cal N}u\le \mathrm{e}psilon$ for any
positive $\mathrm{e}psilon$ (uniformly in $\phi$). We show by induction that
$\|\mathbf{W}_{\bf k}\|_{\cal N}u\le \mathrm{e}psilon$ for all $k$. Using (\ref{estimunifk}),
(\ref{eqMv}), (\ref{combineq}), Proposition~\ref{estimd} and the crude
estimate $\binom{a}{ b}\le 2^a$ we get
\begin{equation}\label{finesti1}\|\mathbf{W}_{\bf k}\|_{\cal N}u\le K\|{\bf R}_{\bf k}\|_{\cal N}u\le
\sum_{{\bf l}\le{\bf k}}\kappa_1^{|{\bf l}|}
\mathrm{e}psilon^{|{\bf k}|}\sum_{({\bf I}i_{mp})}1\le \mathrm{e}psilon^{|{\bf k}|}
\sum_{s=0}^{|{\bf k}|} \kappa_1^s
2^{n_1(|{\bf k}|+s)}2^{s+n_1}\le (C_1 \mathrm{e}psilon)^{|{\bf k}|}
\mathrm{e}nd{equation}
\noindent where $C_1$ does not depend on $\mathrm{e}psilon,{\bf k}$. Choosing $\mathrm{e}psilon$
so that $\mathrm{e}psilon<C_1^{-2}$ we have, for $|{\bf k}|\ge 2$ $(C_1
\mathrm{e}psilon)^{|{\bf k}|}<\mathrm{e}psilon$ completing the induction step . But as we
now know that
$\|\mathbf{W}_{\bf k}\|_{\cal N}u\le \mathrm{e}psilon$, the same
inequalities (\ref{finesti1}) show that in fact
$\|\mathbf{W}_{\bf k}\|_{\cal N}u\le (C_1\mathrm{e}psilon)^{|{\bf k}|}$. Choosing $\mathrm{e}psilon$
small enough, the first part of Proposition~\ref{Uniformnorm}, (ii)
follows.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Proposition}\label{SRY0}
i) Let $\bfY_0$ be given by Lemma~\ref{analyticase}. We
have, in $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$
\begin{eqnarray}
\label{firstdecY}
\mathbf{Y}_0^{\pm}=\mathbf{Y}_0^{\mp}+\sum_{k=1}^{\mathrm{i}nfty}(\pm
S_1)^k\left(\bfY^{\mp}_{k\mathbf{e}_1}(p-k)\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,\mathrm{i}nfty]}\right)
^{(mk)}
\mathrm{e}nd{eqnarray}
\noindent (cf. (\ref{defSj})) and $\mathbf{Y}_0$ is analytic in $\mathcal{R}_1$.
ii) The
general solution of (\ref{eqil}) in $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$
is
\begin{eqnarray}
\label{SGEQIL}
\bfY_0^++\sum_{k=1}^{\mathrm{i}nfty}{C}^k\left(\bfY^+_{k\mathbf{e}_1}(p-k)\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,\mathrm{i}nfty]}\right)
^{(mk)}
\mathrm{e}nd{eqnarray}
\noindent with arbitrary $C$ (a similar statement
holds with $\bfY_0^-$ replacing $\bfY_0^+$).
\mathrm{e}nd{Proposition}
{\mathrm{e}m Proof}.
\noindent We start with (ii). Assuming first (\ref{SGEQIL}) is indeed a
solution of (\ref{eqil}), to see that there are no others, it suffices
by Proposition~\ref{contidis} to check that (\ref{SGEQIL}) is the
general solution on $[0,1+\mathrm{e}psilon)$. The latter part is immediate
from Remark~\ref{coinci} and Proposition~\ref{localresult} below. Now
$\bfY_0^+$ is a solution of (\ref{eqil}), by Lemma~\ref{analyticase}
(ii); the sum (\ref{SGEQIL}) is convergent in $\mathcal{D}'_{m,{\cal N}u}$
by (\ref{finesti1}). Since
$\left(\bfY_{k\mathbf{e}_1}(p-k)\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,\mathrm{i}nfty]}\right)
^{(mk)}\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}(k,\mathrm{i}nfty)$, to show that (\ref{SGEQIL})
is a solution on $\mathbb{R}^+$, we check inductively on $j$ that
${\bf H}}\def\bfY{{\bf Y}_j=\sum_{k=0}^{j}\left(\bfY_{k\mathbf{e}_1}(p-k)\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,\mathrm{i}nfty]}\right)
^{(mk)}$ solves (\ref{eqil}) in $\mathcal{D}'[0,j+1)$. Assuming this
for $j'<j$ and looking for a solution on $[0,j+1)$ in the form
${\bf H}}\def\bfY{{\bf Y}_{j+1}={\bf H}}\def\bfY{{\bf Y}_j+\left(\tilde{\bfY}_{j}(p-j)\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[j,\mathrm{i}nfty]}\right)
^{(mj)}$ we obtain, by a straightforward calculation, using the
induction hypothesis and (\ref{u4})
\begin{multline}
\label{eqykj}
(\hat\Lambda-p) \tilde{\bfY}_j^{(mj)}+
\hat B\mathcal{P}\tilde{\bfY}_j^{(mj)}-\sum_{j=1}^n
(\tilde{\bfY}_j)_j^{(mj)}*\mathbf{D}_{j\mathbf{e}_j}\cr =
\sum_{|{\bf l}|> 1}\mathbf{D}l *
\sum_{\Sigma s=j}*\prod_{i=1}^n*\prod_{j=1}^{l_i}(\tilde{\bfY}_{s_{i,j}})^{(s_{i,j})}_i=:
{\bf R}_j(p)
\mathrm{e}nd{multline}
\noindent which integrated $(mj)$ times is exactly the equation for
$\bfY_{j\mathbf{e}_j}$, cf. also \S~\ref{sec:For}. The claim now
follows from Lemma~\ref{higherterms}, (iii). For (i), we note as
before that $\bfY_0^{\pm}$ are indeed solutions of (\ref{eqil}).
Applying (ii), we only need to identify $C$ for which purpose we
compare the left side with the right side on $(1,1+\mathrm{e}psilon)$, where
all the terms except for $k=1$ vanish and Remark~\ref{identif} below
applies. Lemma~\ref{STR1} completes the proof.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof of Lemma~\ref{higherterms}, (iv)}.
We
let $\bfY_{j}=\bfY_{j\mathbf{e}_j}$. By Lemma~\ref{STR1},
$\mathbf{D}_{j\mathbf{e}_j}\mathrm{i}n\mathcal{F}(\mathcal{R}'_1)$ since
\begin{multline}\label{exprdj}
\mathbf{D}_{\bf j}^+=\sum_{{\bf l}\ge{\bf j}}\binom{{\bf l}}{{\bf j}}{\bf G}_{{\bf l}}*(\bfY_0^{+})^{*({\bf l}-{\bf j})}=
\sum_{{\bf l}\ge{\bf j}}{\bf G}_{{\bf l}}*\Bigg[\bfY_0^{-}\cr+\sum_{s\ge
1} S^s(\bfY_s^{-}(p-s))^{(ms)}\Bigg]^{*({\bf l}-{\bf j})}=
\sum_{{\bf k}\succeq 0}\left(\sum_{s\ge 1}
S^s(\bfY_s^{-})^{(ms)}\right)^{*{\bf k}}{\bf Q}_{{\bf k}{\bf j}}\cr =
\sum_{l=0}^{\mathrm{i}nfty} S^l(\mathbf{D}_{{\bf j}; l}^-)^{(ml)}
\mathrm{e}nd{multline}
\noindent where ${\bf Q}_{{\bf k}{\bf j}}=\sum_{{\bf l}\ge{\bf j}+{\bf k}}
\binom{{\bf l}}{{\bf j}}\binom{{\bf l}-{\bf j}}{{\bf k}}
{\bf G}_{\bf l}(\bfY_0^-)^{*({\bf l}-{\bf k}-{\bf j})}$ and
\begin{eqnarray}\label{definition dj}
&&(\mathbf{D}_{{\bf j}; l}^-)^{(ml)}=
\sum_{\begin{subarray}{\bf k}\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^n\cr|{\bf k}|\mathrm{i}n[0,l]\mathrm{e}nd{subarray}}{\bf Q}_{{\bf k}{\bf j}}*
\sum_{(i_{rs}:l)}
\sideset{^*}{}\prod_{r=1}^n\sideset{^*}{}
\prod_{s=1}^{k_r}\left(\bfY^{-}_{i_{rs}}\right)^{(mi_{rs})}_r\cr&&=
\left(\sum_{\begin{subarray}{\bf k}\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^n\cr|{\bf k}|\mathrm{i}n[0,l]\mathrm{e}nd{subarray}}{\bf Q}_{{\bf k}{\bf j}}*
\sum_{(i_{rs}:l)}
\sideset{^*}{}\prod_{r=1}^n\sideset{^*}{}
\prod_{s=1}^{k_r}\left(\bfY^{-}_{i_{rs}}\right)_r\right)^{(ml)}\cr&&
\mathrm{e}nd{eqnarray}
\noindent and the notation $\sum_{(i_{rs}:l)}$ is explained after Eq.
(\ref{eqmygen})); in particular, (1) there are only finitely many
terms in (\ref{definition dj}) and (2) by homogeneity,
$\mathbf{D}_{{\bf j};l}\mathrm{i}n\mathcal{T}_{l\beta'_1-1}$. In addition, it
follows, as in (\ref{finesti1}), (noting that we only need the
${\bf j}$ with $|{\bf j}|=1$) that $\|\mathbf{Q}_{{\bf k};{\bf j}}\|\le
K(4\kappa)^{|{\bf k}|}$ and $\|\mathbf{D}_{{\bf j};l}\|\le K
(2^{2n+3}\kappa C_1\mathrm{e}psilon)^l$. If we look for $\bfY_{\bf k}^{+}$ in
the form $\mathbf{Y}_{\bf k}^-+\sum_{l=1}^{\mathrm{i}nfty}(\bfY^{-}_{{\bf k};
l}(p-l))^{(ml)}$ then the equation for $\bfY^-_{{\bf k}; l}$, $l\ge
1$, reads
\begin{multline}
\label{eqcomplk}
\left(-p+\hat{\Lambda}-{\bf k}\cdot{\boldsymbol \lambda}-l\right)\bfY^-_{{\bf k};l}+
\left(\hat{B}+{\bf m}\cdot{\bf k}+m_1 l\right)
\mathcal{P}\bfY^-_{{\bf k};l}\cr+\sum_{|{\bf j}|=1}
\mathbf{D}^{-}_{{\bf j};0}*\left(\bfY^-_{{\bf k};l}\right)^{\bf j}
=\left({\bf T}_{\bf k}(\bfY^-_{\cdot;l})\right)-
\sum_{s=1}^{l-1}\sum_{|{\bf j}|=1}
\mathbf{D}_{{\bf j};s}*\left(\bfY^{-}_{{\bf k};l-s}\right)^{{\bf j}}
\mathrm{e}nd{multline}
\noindent By induction, exactly as in (iii), it follows that
$\mathbf{Y}_{{\bf k};l}(\cdot\,\mathrm{e}^{\mathrm{i}\phi})\mathrm{i}n\mathcal{T}_{{\bf k}{\boldsymbol \beta}'+l\beta'_1-1}$,
are $\phi-$ continuous in $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$ and that, with ${\cal N}u$
large enough independent of ${\bf k},l,\phi$,
$\|\mathbf{Y}_{{\bf k};l}\|_{\mathcal{D}'_{m,{\cal N}u}}\le
K\delta^{|{\bf k}|+l}$. Analyticity in $\mathcal{R}'_1$ follows now from
Lemma~\ref{STR1}.
(v) Laplace transformability as well as the fact that ${\bf y}_{\bf k}$
solve (\ref{systemformv}) follow immediately from (\ref{estimunifk})
and Lemma~\ref{existe}. Uniform convergence follows from
(\ref{normVK}) and Lemma~\ref{existe}. Let
${\bf y}_{\bf k}^{\pm}={\cal L}\bfY_{\bf k}^{\pm}$. Now since ${\bf y}_0^{\pm}+\sum
x^{{\bf m}\cdot{\bf k}}\mathbf{C}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda} x} {\bf y}_k^{\pm}$
formally solves (\ref{eqor}) (by the very construction of
(\ref{systemformv}), see \S\ref{sec:For}) and is a uniformly
convergent function series, the conclusion follows together with
the fact that ${\cal L}\bfY_{\bf k}\sim \tilde{{\bf y}}_{\bf k}$ since by (ii),
and (iv) ${\cal L}\bfY_{\bf k}$ have power series
asymptotics which by construction must be formal solutions of
(\ref{systemformv}).
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof of Theorem~\ref{AS}, (ii)} \noindent Modulo relabeling of the
spatial directions and rescaling, the proofs of the properties of the
solution $\mathbf{Y}_0$ of (\ref{eqil}) are valid in a sector
centered around any eigenvalue $\lambda_i$. The above proofs of the
analytic properties work with virtually no change along any direction
so that $p+{\bf k}'\cdot{{\boldsymbol \lambda}}-\lambda_i{\cal N}e 0$ and the same is true
with respect to the $\mathcal{D}'_{m,{\cal N}u}$ properties of
$\mathbf{Y}_{\bf k}$ restricted to {\mathrm{e}m compact sets} in $\mathbb{C}$. The
analysis in $\mathcal{D}'_{m,{\cal N}u}(e^{i\phi}\mathbb{R}^+)$ (i.e., along the
infinite ray) and the associated {\mathrm{e}m norm} estimates are {\mathrm{e}m not}
valid along directions so that $\Re(p-\lambda_i+{\bf k}'\cdot{{\boldsymbol \lambda}})\le 0$
(because of possible presence of small denominators in
$(1+J_{\bf k})^{-1}$, the only difference in this case, but a
significant one). However, we do not take Laplace
transforms along such directions (cf. (c1)) and the associated
infinite ray norms are not
needed for our purposes. For the analytic properties in the
neighborhood of $-{\bf k}'\cdot{{\boldsymbol \lambda}}+\lambda_i$, see
Remark~\ref{wrongdirection}. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\subsubsection{Higher resurgence relations}
\label{sec:str-sing}
\begin{Proposition}\label{asymptrick} i) Let ${\bf y}_1$ and ${\bf y}_2$ be solutions of (\ref{eqor})
so that ${\bf y}_{1,2}\sim\tilde{{\bf y}}_0$ for large $x$ in an
open sector $S$ (or in some direction
$d$);
then ${\bf y}_1-{\bf y}_2=\sum_{j}C_j\mathrm{e}^{-\lambda_{i_j}
x}x^{-\beta_{i_j}}(\mathbf{e}_{i_j}+o(1))$ for some constants $C_j$, where the indices run over
the eigenvalues $\lambda_{i_j}$ with the property $\Re(\lambda_{i_j}
x)>0$ in $S$ (or $d$).
If ${\bf y}_1-{\bf y}_2=o(\mathrm{e}^{-\lambda_{i_j}
x}x^{-\beta_{i_j}})$ for all $j$, then ${\bf y}_1={\bf y}_2$.
ii) Let ${\bf y}_1$ and ${\bf y}_2$ be solutions of
(\ref{eqor}) and assume that ${\bf y}_1-{\bf y}_2$ has differentiable
asymptotics of the form
$\mathbf{K}a\mathrm{e}xp(-ax)x^b(1+o(1))$ with $\Re(ax)>0$ and $\mathbf{K}{\cal N}e 0$, for large $x$.
Then $a=\lambda_i$ for some $i$.
iii) Let $\mathbf{U}_{\bf k}\mathrm{i}n\mathcal{T}_{\{\cdot\}}$ for all ${\bf k}$,
$|{\bf k}|>1$. Assume in addition that for
large ${\cal N}u$ there is a function $\delta({\cal N}u)$ vanishing as
${\cal N}u\rightarrow\mathrm{i}nfty$ such that
\begin{gather}
\label{cd1}
\sup_{{\bf k}}\delta^{-|{\bf k}|}\mathrm{i}nt_{d}\left|\mathbf{U}_{{\bf k}}(p)\mathrm{e}^{-{\cal N}u p}\right|\mathrm{d}|p|<K<\mathrm{i}nfty
\mathrm{e}nd{gather}
\noindent Then, if ${\bf y}_1,{\bf y}_2$ are solutions of (\ref{eqor}) in $S$ where in addition
\begin{eqnarray}
\label{lapcond}
{\bf y}_1-{\bf y}_2=\sum_{|{\bf k}|>1}
\mathrm{e}^{-{\boldsymbol \lambda}\cdot{\bf k} x}x^{{\bf m}\cdot{\bf k}}\mathrm{i}nt_d\mathbf{U}_{\bf k}(p)
\mathrm{e}xp(-xp)\mathrm{d}p
\mathrm{e}nd{eqnarray}
\noindent where ${\boldsymbol \lambda},x$ are as in (c1), then ${\bf y}_1={\bf y}_2$, and
$\mathbf{U}_{\bf k}=0$ for all ${\bf k}$, $|{\bf k}|>1$.
\mathrm{e}nd{Proposition}
{\mathrm{e}m Proof}. (i) is a classical result (see \cite{Iwano}
for the general treatment and \cite{Wasow} for a brief presentation of
special cases and further references). However, what
is actually needed for
our purposes can be reduced to the more familiar {\mathrm{e}m linear}
asymptotic theory in the following way. Let $d$ be a direction in the
complex plane and let ${\bf y}_0$, ${\bf y}_1$ be solutions of (\ref{eqor})
such that ${\bf y}_{0,1}\sim\tilde{{\bf y}}_0$ for large $x$ along $d$.
Then, by (n5), ${\bf y}_{0,1}(x)=O(x^{-M})$ and for any $j$,
${\bf g}^{(\mathbf{e}_j)}(x,{\bf y}_{0,1}(x))=O(x^{-M})$. If
$\boldsymbol\delta={\bf y}_{1}-{\bf y}_{0}$ then by hypothesis
$\boldsymbol\delta(x)=o(x^{-l})$ along $d$, for all $l$. The function
$\boldsymbol\delta$ is locally analytic and satisfies the equation
\begin{multline}
\label{eqdel}
\boldsymbol\delta'=-\hat{\Lambda}\boldsymbol\delta-\frac{1}{x}\hat{B}\boldsymbol\delta
+\sum_{|{\bf k}|=1}{\bf g}^{({\bf k})}(x,{\bf y}_0)\boldsymbol\delta^{{\bf k}}
+\sum_{|{\bf k}|>1}{\bf g}^{({\bf k})}(x,{\bf y}_0)\boldsymbol\delta^{{\bf k}}=\cr
-\hat{\Lambda}\boldsymbol\delta-\frac{1}{x}\hat{B}\boldsymbol\delta
+\frac{1}{x^M}\sum_{j=1}^n(\boldsymbol\delta)_j{\bf h}_{\mathbf{e}_j}(x)
\mathrm{e}nd{multline}
\noindent where ${\bf h}_{\bf k}(x)$ are bounded along $d$. Obviously, because of
the link between $\boldsymbol\delta$ and ${\bf h}_{\bf k}$, the $\boldsymbol\delta$ we started
with might be the only solution of (\ref{eqdel}) which is also a
difference of solutions of (\ref{eqor}). The asymptotic
characterization we need holds nevertheless for {\mathrm{e}m all} decaying
solutions solutions of (\ref{eqdel}): since no two eigenvalues are
equal, there exists by the well-known linear asymptotic theory
\cite{Wasow} a fundamental set $\{\boldsymbol\delta_i\}_{1\le i\le n}$ of
solutions of (\ref{eqdel}) such that $\boldsymbol\delta_i\sim \mathrm{e}^{-\lambda_i x}
x^{-\beta_i}(\mathbf{e}_i+o(1))$. Thus $\boldsymbol\delta=\sum_{i=1}^n
C_i\boldsymbol\delta_i=\sum_{i=1}^n C_i \mathrm{e}^{-\lambda_i x}
x^{-\beta_i}(\mathbf{e}_i+o(1))$. Since $\Re(-\beta_i)>0$ and the
$\lambda_i$ are distinct we must have $C_i=0$ for all $i$ for which
$\Re(-\lambda_i x)\ge 0$, otherwise $|\boldsymbol\delta(x)|$ would be unbounded
for large $x$; the first part of (i) is
proven. If on the other hand $\boldsymbol\delta=o(\mathrm{e}^{-\lambda_{i_j}
x}x^{-\beta_{i_j}})$ for all $j$, again because the $\lambda_i$ are
independent, it follows that $C_i=0$ for all $i=1,2,\ldots,n$, thus
$\boldsymbol\delta=0$.
(ii) is now obvious.
For (iii), note first that by (\ref{cd1}) and (c1) the RHS of
(\ref{lapcond}) converges uniformly for large $x$ in some open sector.
In addition, by an arbitrarily small change in $\xi=\arg(x)$, we can
make the set $\{\Re(x\lambda_i)\}_i$ $\mathbb{Z}$-independent (the existence
of ${\bf k}(\xi){\cal N}e 0$ s.t. $\Re(\mathrm{e}^{\mathrm{i}\xi}{\bf k}\cdot{\boldsymbol \lambda})=0$ for $\xi$ in an
interval of would imply the existence of a {\mathrm{e}m common} ${\bf k}$ for a
set of $\xi$ with an accumulation point, giving ${\bf k}{\boldsymbol \lambda}=0$). We
choose such a $\xi$. Assume now there exist ${\bf k}$ so that
$\mathbf{U}_{\bf k}{\cal N}e 0$; among them let ${{\bf k}_0}$ have the least
$\Re(x{\bf k}\cdot{\boldsymbol \lambda})$. By (\ref{cd1}) for large $x$,
${\bf y}_1-{\bf y}_2\sim \mathrm{e}^{-{\boldsymbol \lambda}\cdot{\bf k}_0 x} x^{{\bf m}\cdot{\bf k}_0}{\cal L}_\phi
\mathbf{U}_{{{\bf k}_0}}(1+o(1))$. Because
$\mathbf{U}_{{{\bf k}_0}}\mathrm{i}n\mathcal{T}_{\{\cdot \}}$, and by (\ref{cd1}),
${\cal L}_\phi \mathbf{U}_{{\bf k}_0}$ has a differentiable power series
asymptotics which is the term-by-term Laplace transform of the Puiseux
series at the origin of $\mathbf{U}_{{{\bf k}_0}}$, and thus non-zero. This
contradicts (i) because with
$|{{\bf k}_0}|>1$ we have ${\boldsymbol \lambda}\cdot{{\bf k}_0} {\cal N}e\lambda_j$ for all $j$
($\mathbb{Z}-$independence).
Thus $\mathbf{U}_{\bf k}=0$ for all ${\bf k}$.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
We let ${\bf C}\mathrm{i}n(\mathbb{C}\backslash\{0\})^{n_1}$ be an arbitrary
constant vector.
\noindent For $x$ large enough, ${\bf y}^+$ defined in
(\ref{solupperlower}) is a solution of
(\ref{eqor}) in an open sector containing $x$. We now use
Lemma~\ref{higherterms} to write (\ref{solupperlower}) in terms of
functions analytic in the lower half plane:
\begin{multline}
\label{solupper-}
{\bf y}^+ ={\cal L}\bfY_0^-+\sum_{j=1}^{\mathrm{i}nfty} x^{mj}\mathrm{e}^{-jx}
{\cal L}\bfY_{0;j}^-+ \sum_{{\bf k}\succ 0,j\ge
0}x^{{\bf m}\cdot{\bf k}+mj}{\bf C}^{{\bf k}}\mathrm{e}^{-({\bf k}\cdot{\boldsymbol \lambda}+j\lambda_1)
x} {\cal L} \bfY^-_{{\bf k};j}\cr ={\cal L}\bfY_0^-+\sum_{{\bf k}\succ
0}x^{{\bf m}{\bf k}}\mathrm{e}^{-({\bf k}\cdot{\boldsymbol \lambda})
x}\sum_{{\bf k}';j:{\bf k}'+j\mathbf{e}_1={\bf k}}{\bf C}^{{\bf k}'}{\cal L}
\bfY^-_{{\bf k}';j}
\mathrm{e}nd{multline}
\noindent where, by (\ref{firstdecY}) we have
$\bfY_{0;j}=S_1^j\bfY^-_{j\mathbf{e}_1}$.
On the other hand, the expression
\begin{eqnarray}
\label{sollower}
{\bf y}^- ={\cal L}\bfY_0^-+\sum_{{\bf k}\succ 0}x^{{\bf m}\cdot{\bf k}}\tilde{{\bf C}}^{{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda} x}
{\cal L} \bfY^-_{\bf k}
\mathrm{e}nd{eqnarray}
\noindent is, for any $\mathbf{\tilde{C}}$,
a solution of (\ref{eqor}) as well. Choosing
$\tilde{C}_1=C_1+S_1;\tilde{C}_i=C_i;\ (i>1)$ all the
terms with $|{\bf k}|\le 1$
in (\ref{sollower}) and (\ref{solupper-}) coincide and thus
\begin{eqnarray}
\label{difey}
{\bf y}^+-{\bf y}^-=\sum_{|{\bf k}|>
1}x^{{\bf m}\cdot{\bf k}}\mathrm{e}^{-{\bf k}\cdot{\boldsymbol \lambda} x}{\cal L}\mathbf{U}_{\bf k}
\cr&&
\mathrm{e}nd{eqnarray}
\noindent where
\begin{eqnarray}\label{defU}
\mathbf{U}_{\bf k}=
\sum_{{\bf k}'+j\mathbf{e}_1={\bf k}}{\bf C}^{{\bf k}'}
\bfY^-_{{\bf k}';j}-\tilde{{\bf C}}^{{\bf k}}\bfY^-_{\bf k}
\mathrm{e}nd{eqnarray}
\noindent so that
applying Proposition~\ref{asymptrick} (ii)
\begin{eqnarray}
\label{u=0}
\mathbf{U}_{\bf k}=0
\mathrm{e}nd{eqnarray}
\noindent Since $C_i{\cal N}e 0$ we have, for any $C_1$,
\begin{eqnarray}
\label{nearfinalresu}
(C_1+S_1)^{k_1}\mathbf{Y}^-_{{\bf k}}=
\sum_{{\bf k}'+j\mathbf{e}_1={\bf k}}C_1^{k'_1}
\bfY^-_{{\bf k}';j}
\mathrm{e}nd{eqnarray}
\noindent with arbitrary $C_1$ so that
\begin{eqnarray}
\label{primaresu}
\bfY^-_{{\bf k};j}= \binom{k_1+j}{j}S_1^{j}\mathbf{Y}^-_{{\bf k}+j\mathbf{e}_1}
\mathrm{e}nd{eqnarray}
\noindent Combined with the definition
of $\bfY^-_{{\bf k};j}$
this gives (\ref{thirdresu}).
Solving for $\mathbf{Y}^-_{{\bf k}+j\mathbf{e}_1}$, (\ref{primaresu})
determines later series in the transseries in terms of earlier ones. The same
arguments work of course with $-/+$ and $+S_1/(-S_1)$ interchanged.
Theorem~\ref{CEQ} part (iii) follows from the following.
\begin{Proposition}\label{classic} Any solution ${\bf y}$ of (\ref{eqor}) so that
${\bf y}\sim\tilde{{\bf y}}_0$ along some direction $d\subset S_x$ is of
the form (\ref{solupperlower}), for a unique ${\bf C}^+$
(a similar statement holds with $+/-$ interchanged). Alternatively,
a solution ${\bf y}$ of (\ref{eqor}) so that ${\bf y}\sim\tilde{{\bf y}}_0$
along some direction $d\subset S_x$ can be represented as
(\ref{soleqn}) or more generally as (\ref{soleqnpa}) where Laplace
integration is along $\mathbb{R}^+$ (in distributions), for a unique
${\bf C}$.
\mathrm{e}nd{Proposition}
{\mathrm{e}m Proof.} Let ${\bf y}$ be an arbitrary solution of (\ref{eqor}) so
that ${\bf y}\sim\tilde{{\bf y}}_0$ along $d\subset S_x$. Then, by
Proposition~\ref{asymptrick},
${\bf y}-{\bf y}^+=\sum_{j}\overline{\mathbf{C}}_j\mathrm{e}^{-\lambda_{i_j}
x}x^{-\beta_{i_j}}(\mathbf{e}_{i_j}+o(1))$ for some constant
$\overline{\mathbf{C}}$. Therefore ${\bf y}_1$ defined as the ``+''
solution in (\ref{solupperlower}) with
$\mathbf{C}={\overline{\mathbf{C}}}$ will have the property
${\bf y}_1-{\bf y}=o(C_j\mathrm{e}^{-\lambda_{i_j} x}x^{-\beta_{i_j}})$ for
all $j$, hence ${\bf y}={\bf y}_1$. Formula (\ref{soleqnpa}). The last part,
as well as the middle formula in (\ref{microsto}), follow through a
straightforward calculation from the first part, (\ref{primaresu})
using (\ref{normVK}) to control convergence.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof of theorem~\ref{Stokestr}}. Let ${\bf y}^{\pm}$ be defined by
(\ref{solupperlower}) with ${\bf C}=(\pm\frac{1}{2} S_1+C)\mathbf{e}_1$,
respectively.
The same arguments leading to (\ref{u=0}) show that
${\bf y}^+={\bf y}^-=:{\bf y}$. All the exponentials in the transseries of
${\bf y}$ are generated by construction by $e^{-\lambda x}$. Choosing
$p$ in the path of integration above/below $\mathbb{R}^+$ and consequently the $+/-$
representation (\ref{solupperlower})
of ${\bf y}$ we have by Lemma~\ref{higherterms} that
${\cal L}\bfY_{\bf k}^{\pm}\sim\tilde{{\bf y}}_{\bf k}$ in (\ref{solupperlower}),
in the half plane $\Re(xp)>0$.
By construction
${\cal L}\bfY_{\mathbf{e}_1}=x^{-\beta'_1}(\mathbf{e}_1+o(1))$
(cf. \ref{eqneuman}) while for $j>1$ we have
${\cal L}\bfY_{j\mathbf{e}_1}\sim x^{-j\beta'_1}$
by Lemma~\ref{higherterms} (ii). The condition
$|x^{-\beta_1+1}e^{-x\lambda_1}|\rightarrow 1$ together with
Lemma~\ref{higherterms} guarantee the uniform convergence of the series
(\ref{solupperlower}). The conclusion is immediate.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{subsubsection}{Local analysis near $p=1$.}
\label{sec:A1}
\noindent
We treat (\ref{eqil}) near $p=1$ as a perturbation of a differential equation
having the same type of singularity. The associated differential
equation splits the singularity, and our convolution equation
is a regular perturbation of it, which is then solved by
fixed point methods.
Let $\bfY_0$ be the unique solution in $\mathcal{A}_{z,l}$ of (\ref{eqil}) and let
$\mathrm{e}psilon>0$ be small. Define
\begin{eqnarray}\label{defbfh}
&&{\bf H}}\def\bfY{{\bf Y}(p):=\left\{\begin{array}{cc}
\bfY_0(p)\ \ \mbox{for $p\mathrm{i}n\mathcal{S}_0$\,,$|p|<1-\mathrm{e}psilon$}\cr
0\ \ \ \mbox{otherwise}
\mathrm{e}nd{array}\right.\cr&&\mbox{and}\ \ \mathbf{W}(1-p):=\bfY_0(p)-{\bf H}}\def\bfY{{\bf Y}(p)
\mathrm{e}nd{eqnarray}
\noindent ($\bfY(p)-{\bf H}}\def\bfY{{\bf Y}(p)=\mathbf{W}(p-1)$ would be more ``natural'',
but would later complicate notations).
In terms of $\mathbf{W}$, for real $z=1-p, z<\mathrm{e}psilon$, (\ref{eqil}) becomes:
\begin{eqnarray}\label{eq002}
-(1-z){\bf W}(z)=\mathbf{F}_1(z)-\hat\Lambda \mathbf{W}(z)+
\hat B\mathrm{i}nt_{\mathrm{e}psilon}^z\mathbf{W}(s)\mathrm{d}s+{\cal N}b({\bf H}+{\bf W})
\mathrm{e}nd{eqnarray}
\noindent where
$${\bf F}_1(1-s):={\bf f}z(s)-{\hat B}\mathrm{i}nt_0^{1-\mathrm{e}psilon}{\bf H}}\def\bfY{{\bf Y}(s)\mathrm{d}s
$$
\begin{Proposition}\label{P8}
i) For small $\mathrm{e}psilon$, ${\bf H}}\def\bfY{{\bf Y}^{*{\bf l}}(1+z)$ extends to an analytic
function in the disk
${\DD_\epsilon}:=\{z:|z|<\mathrm{e}psilon\}$. Furthermore,
for any $\delta$ there is an $\mathrm{e}psilon$ and a
constant $K_1:=K_1(\delta,\mathrm{e}psilon)$ such that for
$z\mathrm{i}n{\DD_\epsilon}$
\begin{eqnarray}\label{estHl2}
|{\bf H}}\def\bfY{{\bf Y}^{*{\bf l}}(1+z)|<K_1\delta^{|{\bf l}|}
\mathrm{e}nd{eqnarray}
ii)
The equation (\ref{eq002}) can be written as
\begin{gather}\label{formN}
-(1-z){\bf W}(z)={\bf F}(z)-\hat\Lambda \mathbf{W}(z)+
\hat B\mathrm{i}nt_{\mathrm{e}psilon}^z\mathbf{W}(s)\mathrm{d}s-\sum_{k=1}^n
\mathrm{i}nt_\mathrm{e}psilon^zh_j(s)\mathbf{D}_j(s-z)\mathrm{d}s
\mathrm{e}nd{gather}
\noindent where
\begin{eqnarray}\label{defD1}
{\bf F}(z):=
{\cal N}b({\bf H}}\def\bfY{{\bf Y})(1-z)+{\bf F}_0(z)
\mathrm{e}nd{eqnarray}
\begin{eqnarray}\label{defderiv}
\mathbf{D}_j=
\sum_{|{\bf l}|\ge 1}l_j{\bf G_{l}}*{\bf H}}\def\bfY{{\bf Y}^{*\bar{\bf l}^j}+
\sum_{|{\bf l}|\ge 2}l_j{\bf g_{\rm 0,\bf l}}{\bf H}}\def\bfY{{\bf Y}^{*\bar{\bf l}^j};\ \bar{\bf l}^j:=(l_1,l_2,..(l_j-1),..l_n)
\mathrm{e}nd{eqnarray}
\noindent extend to analytic functions in ${\DD_\epsilon}$.
Moreover, if ${\bf H}}\def\bfY{{\bf Y}$ is a vector in ${L^1}_{\cal N}u(\mathbb{R}^+)$ then, for large
${\cal N}u$, $\mathbf{D}_j\mathrm{i}n{L^1}_{{\cal N}u}(\mathbb{R}^+)$ and the functions
$ {\bf F}(z)$ and ${\bf D}_j$ extend to analytic functions in
${\DD_\epsilon}$.
Furthermore, ${\bf D}_j\mathrm{i}n\mathcal{A}_{z,M}$.
iii) Near $p=1$ we have (cf. Lemma~\ref{analyticase})
\begin{eqnarray}
\label{Y0p=1}
\mathcal{P}^{m+1}\bfY_0&=&(1-p)^{\beta'}\mathbf{A}+\mathbf{B}\ \ (\beta{\cal N}otin\mathbb{Z})\cr
\mathcal{P}^{m+1}\bfY_0&=&(p-1)\left(\ln(p-1)\mathbf{A}(p)+\mathbf{B}(p)\right)\ \ (\beta\mathrm{i}n\mathbb{Z})
\mathrm{e}nd{eqnarray}
\noindent where $\bf A,B$ analytic at $p=1$.
\mathrm{e}nd{Proposition}
{\mathrm{e}m Proof}.
Parts (i) and (ii), except for the last claim, are proven
in \cite{Costin}, Propositions 18
and 19. To see that $\mathbf{D}_j\mathrm{i}n\mathcal{A}_m$ it is enough to remark
again
that $\mathcal{A}_{z,M}$ is a convolution ideal of $\mathcal{A}_{z,0}$ and that
$\mathbf{G}_{\bf j}\mathrm{i}n\mathcal{A}_{z,M}$ for
$|\mathbf{j}|=1$, by (n5).
For (iii), consider again equation (\ref{formN}).
Let $\hat{\Gamma}=\hat\Lambda-(1-z){\hat{1}}$, where
$\hat{1}$ is the identity matrix. By construction
$\hat{\Gamma}$ and $\hat{B}$ are diagonal, $\hat{\Gamma}_{11}=z$ and $\hat{B}_{11}=\beta_1=:\beta$.
We write this as $\hat{\Gamma}=z\oplus\hat{\Gamma}_c(z)$ and similarly,
$\hat{B}=\beta\oplus\hat{B}_c$, where $\hat{\Gamma}_c$ and $\hat{B}_c$
are $(n-1)\times(n-1)$ diagonal matrices. $\hat{\Gamma}_c(z)$ and
$\hat{\Gamma}_c^{-1}(z)$ are analytic in ${\DD_\epsilon}$.
\noindent Let
\begin{equation}\label{defQ}
{\bf Q}:=\mathcal{P}_\mathrm{e}psilon^{m+1} \mathbf{W}
\mathrm{e}nd{equation}
\noindent with $\mathcal{P}_\mathrm{e}psilon:=
\mathbf{W}\mapsto\left(z\mapsto\mathrm{i}nt_\mathrm{e}psilon^z\mathbf{W}(s)\mathrm{d}s\right)$. By
Lemma~\ref{analyticase}, (i),
${\bf Q}$ is
analytic in ${\DD_\epsilon}\cap(\{z:z+1\mathrm{i}n\mathcal{S}_0\})$. From (\ref{formN}) we obtain
\begin{multline}\label{difeqbfQ}
\noindentpp{\bf Q}^{(m+1)}(z)-(\beta\oplus\bc) {\bf Q}^{(m)}(z)\cr={\bf F}(z)-
\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z\mathbf{D}_j(s-z)Q_j^{(m+1)}(s)\mathrm{d}s
\mathrm{e}nd{multline}
\noindent or, after $m$ integrations by parts in the r.h.s.
of (\ref{difeqbfQ}), by Proposition~\ref{P8} (ii), we get
\begin{multline}\label{difeqbfQ2}
\noindentpp{\bf Q}^{(m+1)}(z)-(\beta\oplus\bc) {\bf Q}^{(m)}(z)\cr={\bf
F}(z)-(-1)^{m}\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z\mathbf{D}_j^{(m)}
(s-z)Q_j'(s)\mathrm{d}s
\mathrm{e}nd{multline}
\noindent so that with $\beta'=\beta'_1, \hat{B}_cp=m_1+\hat{B}_c$,
\begin{multline}\label{difeqbfQ2i}
\noindentpp{\bf Q}'(z)-(\beta'\oplus\hat{B}_cp) {\bf Q}(z)\cr=\mathcal{P}_\mathrm{e}psilon^{m}{\bf
F}(z)-(-\mathcal{P}_\mathrm{e}psilon)^{m}\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z\mathbf{D}_j^{(m)}(s-z)Q_j'(s)\mathrm{d}s\cr
=\mathbf{P}(z)+\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z\mathbf{D}_j'(s-z)Q_j(s)\mathrm{d}s
\mathrm{e}nd{multline}
\noindent where $\mathbf{P}(z)=\mathcal{P}_\mathrm{e}psilon^{m}{\bf F}(z)$.
With the notation
$(Q_1,{\bf Q}_\perp):=(Q_1,Q_2,..,Q_n)$
we write the system in the form
\begin{align}\label{syt1}
(z^{-{\beta'}} Q_1(z))'&=z^{-{\beta'}-1}\left(P_1(z)+
\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z D_{1j}'(s-z)Q_j(s)\mathrm{d}s\right)\cr
(\mathrm{e}^{\hat C(z)}{\bf Q}_\perp)'&=\mathrm{e}^{\hat C(z)
}\hat{\Gamma}_c(z)^{-1}\left(\mathbf{P}_\perp+\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z
{\bf D}_\perp'(s-z)Q_j(s)\mathrm{d}s\right)\cr
\hat C(z)&:=-\mathrm{i}nt_0^z \hat{\Gamma}_c(s)^{-1}\hat{B}_cp(s)\mathrm{d}s
\cr
{\bf Q}(\mathrm{e}psilon)&=0
\mathrm{e}nd{align}
\noindent After integration we get:
\begin{align}\label{eqperp12}
Q_1(z)&=R_1(z)+J_1({\bf Q})\cr
{\bf Q}_\perp(z)&={\bf R}_\perp(z)+J_\perp({\bf Q})
\mathrm{e}nd{align}
\noindent with
\begin{align}\label{eqperp2}
J_1({\bf Q})&=z^{{\beta'}}\mathrm{i}nt_\mathrm{e}psilon^zt^{-{\beta'}-1}\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^tQ_j(s)D_{1j}'(t-s)\mathrm{d}s\mathrm{d}t\cr
J_\perp({\bf Q})(z)&:=\mathrm{e}^{-\hat C(z)}
\mathrm{i}nt_\mathrm{e}psilon^z \mathrm{e}^{\hat C(t)}\hat{\Gamma}_c(t)^{-1}
\left(\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z
{\bf D}_\perp'(s-z)Q_j(s)\mathrm{d}s\right)\mathrm{d}t\cr
{\bf R}_\perp(z)&:=\mathrm{e}^{-\hat C(z)}
\mathrm{i}nt_\mathrm{e}psilon^z \mathrm{e}^{\hat C(t)}\hat{\Gamma}_c(t)^{-1}
{\bf F}_\perp(t)\mathrm{d}t
\cr
R_1(z)&=z^{{\beta'}}\mathrm{i}nt_\mathrm{e}psilon^zt^{-{\beta'}-1}P_1(t)\mathrm{d}t
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ({\beta'}{\cal N}e 1)\cr
R_1(z)&=P_1(0)z\ln z+
z\mathrm{i}nt_{\mathrm{e}psilon}^z\frac{P_1(s)-P_1(0)}{s}\mathrm{d}s
\ \ ({\beta'} = 1)
\mathrm{e}nd{align}
\noindent Consider the space $\mathcal{U}_{\beta'}$ given by
\begin{align}\label{funspace}
{\cal U_{\beta'}}&=
\Big\{{\bf Q}\ \mbox{analytic in }\{z:0<|z|<\mathrm{e}psilon, \arg(z){\cal N}e \pi\}:
{\bf Q}=z^{\beta'}{\bf A}(z)+{\bf B}(z)\Big\}\\* \mathrm{i}ntertext{for $\beta'{\cal N}e 1$ and}
{\cal U}_1&=\Big\{{\bf Q}\ \mbox{analytic in }\{z:0<|z|<\mathrm{e}psilon, \arg(z){\cal N}e \pi\}:
{\bf Q}=z\ln z{\bf A}(z)+z{\bf B}(z)\Big\}{\cal N}otag
\mathrm{e}nd{align}
\noindent where ${\bf A},{\bf B}$ are analytic in ${{\DD_\epsilon}}$.
(The decomposition of $\bf Q$ in (\ref{funspace})
is unambiguous since $z^{\beta'}$ and
$z\ln z$ are not meromorphic in ${\DD_\epsilon}$.)
The norm
\begin{equation}\label{normT}
\|{\bf Q}\|=\sup\left\{
|{\bf A}(z)|,|{\bf B}(z)|:z\mathrm{i}n{\DD_\epsilon}\right\}
\mathrm{e}nd{equation}
\noindent makes $\cal U_{\beta'}$ a Banach space.
\begin{Proposition}\label{eqinte}
The operator $J:=({\bf Q}\mapsto (J_1{\bf Q},J_\perp{\bf Q})$ has
norm
$O(\mathrm{e}psilon)$, for small $\mathrm{e}psilon$,
in ${\cal U_{\beta'}}$ as well as in ${L^1}[-\mathrm{e}psilon,\mathrm{e}psilon]$.
Along any segment $d_\mathrm{e}psilon$ originating at $z=\mathrm{e}psilon$ in the region $|z|<\mathrm{e}psilon, \arg(z){\cal N}e \pi$, Equation
(\ref{difeqbfQ2i})
has a unique solution in $L^1_{\cal N}u(d_\mathrm{e}psilon)$. This solution belongs to
$\mathcal{T}_\beta$.
\mathrm{e}nd{Proposition}
\noindent The proof uses the following
elementary identities:
\begin{eqnarray}\label{convert}
&&\mathrm{i}nt_\mathrm{e}psilon^z A(s)s^r \mathrm{d}s ={\mbox{const.}}+z^{r+1}\mathrm{i}nt
_{0}^{1}\!{ A}(zt)t^{r}{\mathrm{d}t}={\mbox{const.}}+
z^{r+1}Analytic(z)\cr &&\mathrm{i}nt_0^z s^r\ln
s\,A(s)\mathrm{d}s=z^{r+1}\ln z\mathrm{i}nt _{0}^{1}\!{
A}(zt)t^{r}{\mathrm{d}t}+z^{r+1}\mathrm{i}nt _{0}^{1 }\!{ A}(zt)t^{r}\ln
t{\mathrm{d}t}\cr &&
\mathrm{e}nd{eqnarray}
\noindent where the second equality is obtained by differentiating
with respect to $r$ the first equality.
Using
(\ref{convert}) it is straightforward to check that the r.h.s.
of (\ref{eqperp12}) extends to a linear inhomogeneous
operator
on $\cal U_{\beta'}$ with image in $\cal U_{\beta'}$ and that
the norm of $J$ is $O(\mathrm{e}psilon)$ for small $\mathrm{e}psilon$.
For instance, one of the terms in $J$ for $\beta'=1$,
\begin{multline}\label{arr1}
z\mathrm{i}nt_0^z t^{-2}
\mathrm{i}nt_0^t s\ln s\,A(s)D'(t-s)\mathrm{d}s\cr= z^{2}\ln z\mathrm{i}nt _{0}^{1}
\mathrm{i}nt _{0}^{1}\sigma A(z\tau\sigma)D'
(z\tau-z\tau
\sigma){\mathrm{d}\sigma}{\mathrm{d}\tau}
\cr +
z^{2}\mathrm{i}nt _{0}^{1}\mathrm{d}\tau\ln\tau
\mathrm{i}nt _{0}^{1}\mathrm{d}\sigma\,\sigma(1+\ln\sigma)
A(z\tau\sigma)D'(z\tau-z\tau\sigma)
\mathrm{e}nd{multline}
\noindent manifestly in $\cal U_{\beta'}$ if $A$ is
analytic in ${\DD_\epsilon}$. Comparing with (\ref{funspace}), the extra power
of $z$
accounts for a norm $O(\mathrm{e}psilon)$ for this term.
Therefore, in (\ref{syt1}) $(1-J)$ is invertible and the solution $\bf
Q\mathrm{i}n\cal U_{\beta'}$. In view of the the
uniqueness of the solution of (\ref{eqil}) in $\mathcal{S}_0$,
(Lemma~\ref{analyticase}, (i)) the rest of the proof of
Proposition~\ref{P8}, (iii) is immediate.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
A short calculation shows that:
\begin{Remark}\label{wrongdirection} i) The equation for $\mathbf{Y}_{\bf k}$
near $z=0$ where $z=-p-{\bf k}\cdot{\boldsymbol \lambda}+\lambda_i$ can be written
in the form (\ref{formN}), for a different $\mathbf{F}$, and with
$\hat{B}+{\bf k}\cdot{\bf m}$ instead of $\hat{B}$. Thus
$\mathbf{Y}_{\bf k}(z)\mathrm{i}n\mathcal{T}_{\{\cdot\}}$.
ii) The equation for $\mathbf{Y}_{\bf k}$
near $z=0$ where $z=-p-{\bf k}'\cdot{\boldsymbol \lambda}+\lambda_i$ where
${\bf k}'\prec{\bf k}$ can be written as
\begin{eqnarray}
\label{wrongdir}
(1+J_{\bf k})\mathbf{Y}_{\bf k}(z)=\mathbf{R}_{\bf k}(z)
\mathrm{e}nd{eqnarray}
\noindent where
$$J_{\bf k}\bfY=(\hat{B}+{\bf m}\cdot{\bf k})\hat{M}^{-1}\mathcal{P}\bfY+
\hat{M}^{-1}\sum_{|{\bf j}|=1}\overline{\mathbf{D}}_{\bf j}*\bfY^{\bf j},$$
\noindent $\hat{M}=z+\hat{\Lambda}-\lambda_i+({\bf k}'-{\bf k})\cdot{\boldsymbol \lambda}$,
$\mathbf{R}_{\bf k}=\hat{M}^{-1}\mathbf{T}_{\bf k}$ and
$\overline{\mathbf{D}}_{\bf j}$ analytic for small $z$. Thus, arguments
virtually identical to those for (\ref{eqperp12}) imply that
$\mathbf{Y}_{\bf k}\mathrm{i}n\mathcal{T}_{\{\cdot\}}$ near these points.
\mathrm{e}nd{Remark}
\mathrm{e}nd{subsubsection}
\begin{subsubsection}{The solutions of (\ref{difeqbfQ2}) on
$[-\mathrm{e}psilon,\mathrm{e}psilon]$}
\label{sec:-e,e}
Let ${\bf Q}_0$ be the solution given by Proposition \ref{eqinte}, take
$\mathrm{e}psilon$ small enough and denote by $\cal O_\mathrm{e}psilon$ a neighborhood
in $\mathbb{C}$ of width $\mathrm{e}psilon$ of the interval $[0,1+\mathrm{e}psilon]$. We look
for solutions of (\ref{difeqbfQ2i}) in ${L^1}[-\mathrm{e}psilon,\mathrm{e}psilon]$.
The main difference with respect to the previous section is that in
integrating (\ref{difeqbfQ2i}) to the analog of (\ref{eqperp12})for
negative $z$, the constant of integration will now be undetermined leading
to a one-parameter family of solutions. See also Remark~\ref{lincomb}
below.
\begin{Remark}\label{uniformlb}.
As $\phi\rightarrow\pm 0$,
${\bf Q}_0(z\mathrm{e}^{\mathrm{i}\phi})\rightarrow{\bf Q}_0^\pm(z)$ in the
sense of ${L^1}([0,1+\mathrm{e}psilon])$ and also in the sense of
pointwise convergence for
$z{\cal N}e 0$, where
\begin{align}\label{defsolpm}
{\bf Q}_0^\pm(z)&:=\left\{\begin{array}{cc}
{\bf Q}_0(z)
\phantom{\pm0i)^{\beta'}{\bf a}_1(p)+{\bf a}_2(p)}&\ \ \ \ {z>0}\cr
|z|^{\beta'}\mathrm{e}^{\mp \mathrm{i}\pi(\beta')}{\bf a}_1(p)+{\bf a}_2(p)\
\ \ \ \ \ &{z<0}
\cr
\mathrm{e}nd{array}\ \ (\beta{\cal N}e 1)
\right.\cr
{\bf Q}_0^\pm&:=\left\{\begin{array}{cc}
{\bf Q}_0(z)
\phantom{(\ln(|z|)\mp\pi i){\bf a}_1(z)+{\bf a}_2(z)}&{z>0}\cr
z(\ln(|z|)-\pi i){\bf a}_1(z)+z{\bf a}_2(z)&{z<0}
\cr
\mathrm{e}nd{array}\ \ (\beta'= 1)
\right.
\mathrm{e}nd{align}
\noindent Moreover, ${\bf Q}_0^{\pm}$ are ${L}^1_{loc}$
solutions of (\ref{eqinte})
on the interval $[-\mathrm{e}psilon,\mathrm{e}psilon]$.
\mathrm{e}nd{Remark}
The proof is immediate from Propositions
\ref{P8} and \ref{eqinte}.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Remark}\label{lincomb}
For any $\lambda\mathrm{i}n\mathbb{C}$ the combination
${\bf Q}_\lambda=\lambda{\bf Q}_0^++(1-\lambda){\bf Q}_0^-$
is a solution of (\ref{difeqbfQ2}) in ${L^1}[-\mathrm{e}psilon,\mathrm{e}psilon]$.
\mathrm{e}nd{Remark}
{\mathrm{e}m Proof. } Follows from Remark~\ref{uniformlb} as (\ref{difeqbfQ2}) is linear.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\centerline{*}
\noindent Let now ${\bf Q}_0$ be any solution of (\ref{difeqbfQ2}) in
${L^1}[-\mathrm{e}psilon,\mathrm{e}psilon]$. We search for other solutions in
the form ${\bf Q}={\bf Q}_0+{\bf q}$. Since (\ref{difeqbfQ2}) is linear and
${\bf Q}_0$ is already a solution we have
\begin{eqnarray}\label{difeqbfq2}
&&\noindentpp{\bf q}'(z)-(\beta'\oplus\hat{B}_cp) {\bf q}(z)=\sum_{j=1}^n\mathrm{i}nt_\mathrm{e}psilon^z\mathbf{D}_j'(s-z)q_j(s)\mathrm{d}s\cr&&
\mathrm{e}nd{eqnarray}
\noindent and, by the uniqueness of ${\bf Q}_0$ for $z>0$ we have ${\bf q}=0$
for $q<0$ and the equation becomes
\begin{eqnarray}\label{difeqbfq3}
&&\noindentpp{\bf q}'(z)-(\beta'\oplus\hat{B}_cp) {\bf q}(z)=\sum_{j=1}^n\mathrm{i}nt_0^z\mathbf{D}_j'(s-z)q_j(s)\mathrm{d}s\cr&&
\mathrm{e}nd{eqnarray}
\noindent with the initial condition ${\bf q}(0)=0$. Changing variables to
$z=-p$ ($p>0$ now corresponds to going beyond the singularity) and
${\bf q}(z)=\bfY(-p)$ we have
\begin{eqnarray}\label{difeqbfq3+}
&&(p\oplus\gc(-p))\bfY'(p)-(\beta'\oplus\hat{B}_cp) \bfY(p)+\sum_{j=1}^n\mathrm{i}nt_0^p\mathbf{D}_j'(p-t)Y_j(t)\mathrm{d}t=0\cr&&
\mathrm{e}nd{eqnarray}
\noindent We recognize in (\ref{difeqbfq3+}) the equation
for $\mathcal{P}\mathbf{Y}_{\mathbf{e}_1}$:
\begin{Remark}\label{coinci}
Equation (\ref{difeqbfq3+}) is at the same time
the equation for $\mathcal{P}\mathbf{Y}_{\mathbf{e}_1}$ and for
the difference $\mathcal{P}^{m+1}(\mathbf{Y}_0^{[1]}-\mathbf{Y}_0^{[2]})$
where $\mathbf{Y}_0^{[1,2]}$ are any solutions of (\ref{eqil}).
\mathrm{e}nd{Remark}
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Proposition}\label{localresult} Let $\mathrm{e}psilon$ be small. In
$\mathcal{T}_{\beta'}(\{|p|<\mathrm{e}psilon\})$ as well as in
$\mathcal{D}'_{m,{\cal N}u}(0,\mathrm{e}psilon \mathrm{e}^{\mathrm{i}\phi})$ for any $\phi$, there is
a unique solution of (\ref{difeqbfq3+}) $\mathbf{W}_0$ such that, for
small $p$, $\mathbf{W}_0=
\Gamma(\beta')^{-1}p^{\beta'}(\mathbf{e}_1+o(1))$. The general solution of
(\ref{difeqbfq3+}) is $\bfY=C\mathbf{W}_0$, with $C\mathrm{i}n\mathbb{C}$ arbitrary.
\mathrm{e}nd{Proposition}
Notes: (1) Modulo relabeling of the spatial directions,
the statement and proof hold for any of the equations for
$\bfY_{\mathbf{e}_j}$.
(2) The point $p=0$ is singular, and so
is the ``initial condition'' $\mathbf{W}_0\sim
\Gamma(\beta')^{-1}p^{\beta'}\mathbf{e}_1$.
{\mathrm{e}m Proof.} We have
\begin{eqnarray}\label{syt12}
&&(p^{-\beta'} Y_1(z))'=-p^{-\beta'-1}
\sum_{j=1}^n\mathrm{i}nt_0^p D_{1j}'(p-t)Y_j(t)\mathrm{d}t\cr
&&(\mathrm{e}^{\hat{E}(p)}\bfY_\perp)'=-\mathrm{e}^{\hat{E}(p)
}\hat{\Gamma}_c(-p)^{-1}\sum_{j=1}^n\mathrm{i}nt_0^p
{\bf D}_\perp'(p-t)Y_j(t)\mathrm{d}t\cr
&&\hat{E}(p):=-\mathrm{i}nt_0^p \hat{\Gamma}_c(-t)^{-1}\hat{B}_cp(t)\mathrm{d}s
\cr
&&\bfY(0)=0
\mathrm{e}nd{eqnarray}
\noindent After integration we get:
\begin{eqnarray}\label{eqperp122}
(1+J_1)\bfY_1(z)=C R_1(p)\phantom{0}\cr
(1+J_\perp)\bfY_\perp(p)=0\phantom{R_1(z)}
\mathrm{e}nd{eqnarray}
\noindent with $C\mathrm{i}n\mathbb{C}$ arbitrary and
\begin{eqnarray}\label{eqperp22}
&&J_1\,{\bf V}=p^{\beta'}\mathrm{i}nt_0^pt^{-\beta'-1}\sum_{j=1}^n\mathrm{i}nt_0^tY_j(s)D_{1j}'(t-s)\mathrm{d}s\mathrm{d}t\cr
&&
J_\perp\,{\bf V}:=\mathrm{e}^{-\hat{E}(p)}
\mathrm{i}nt_0^p \mathrm{e}^{\hat{E}(t)}\hat{\Gamma}_c(-t)^{-1}
\left(\sum_{j=1}^n\mathrm{i}nt_0^t
{\bf D}_\perp'(t-s)Y_j(s)\mathrm{d}s\right)\mathrm{d}t\cr
&&R_1(p)=p^{\beta'}\cr
\cr
&&
\mathrm{e}nd{eqnarray}
In a region $|p|<\mathrm{e}psilon$, for small $\mathrm{e}psilon$,
the norm of the operator $J$ defined
on $\mathcal{T}_{\beta'}$ is $O(\mathrm{e}psilon)$, exactly as in
Proposition~\ref{eqinte}. Given
$C$ the solution of the system (\ref{eqperp122})
is unique and can be written as
\begin{equation}\label{eqneuman}
\bfY=C \mathbf{W}_0;\ \ \mathbf{W}_0:=\Gamma(\beta')^{-1}(\hat 1+J)^{-1}{\bf R}{\cal N}e 0
\mathrm{e}nd{equation}
\noindent (The prefactor $\Gamma(\beta')^{-1}$ was introduced so that the
coefficient of the leading power in the asymptotic series of
${\cal L}\mathbf{W}_0$ is one).
The proof is essentially the same if we consider (\ref{eqperp122}) in
$L^1_{\cal N}u(0,\mathrm{e}psilon \mathrm{e}^{\mathrm{i}\phi}),$ which coincides with
$\mathcal{D}'_{m,{\cal N}u}(0,\mathrm{e}psilon \mathrm{e}^{\mathrm{i}\phi})$ for small $\mathrm{e}psilon$.
\begin{Remark}\label{identif}
On $(1,1+\mathrm{e}psilon)$, $(\bfY_0^+-\bfY_0^-)(p)=
S_1\mathbf{Y}_{\mathbf{e}_1}^{(m)}(p-1)$.
\mathrm{e}nd{Remark}
The existence of some $S_1$ is obvious from Remark~\ref{coinci}
and Proposition~\ref{localresult}. Its value follows
by comparing (\ref{Y0p=1}), (\ref{primaresu}) and (\ref{eqneuman}).
\mathrm{e}nd{subsubsection}
\begin{Proposition}\label{medianization} i) Let
$\mathbf{Y}_{\bf k}^+,{\bf k}\ge 0$ solve (\ref{eqil}),
(\ref{systemformv}) in $\mathcal{T}_{\{\cdot\}}({\mathcal{S}^+}')$.
Then $\mathbf{Y}_{\bf k}^{ba}$, cf.
(\ref{defmed}) solve (\ref{eqil}), (\ref{systemformv}) in
$\mathcal{D}'_{m,{\cal N}u} (\mathbb{R}^+)$
ii) For any of the functions $\bfY_{\bf k}$, interchanging
$+$ with $-$ in (\ref{defmed}) does not change
the balanced average.
\mathrm{e}nd{Proposition}
{\mathrm{e}m Proof} (i) The fact that $\mathbf{Y}_0^{ba}$ is a solution of
(\ref{eqil}) follows from Proposition~\ref{SRY0}. From
Proposition~\ref{SRY0} and Proposition~\ref{medianpropo} we see that
$\mathbf{D}_{\bf j}^{ba}$ is obtained
by simply replacing $\mathbf{Y}_0^{+}$ by $\mathbf{Y}_0^{ba}$
in (\ref{exprdj}) (notice that on any finite
interval, there are finitely many terms
in the expression of $\mathbf{D}_{\bf j}^{ba}-\mathbf{D}_{\bf j}^{+}$.)
The rest of the proof merely consists in inductively
applying $\mathcal{A}_\alpha$ to the equations (\ref{systemformv}),
noting that each contains finitely many convolutions,
and applying the commutation relation (\ref{assertmed}).
(ii) This is true for $\mathbf{Y}_0$ as an immediate verification
shows that the $+$ and $-$ averages coincide on $(0,2)$ (where they
consist in two terms). Thus by Proposition~\ref{contidis} they have to
coincide on $\mathbb{R}^+$. With this, for the rest of the $\mathbf{Y}_{\bf k}$ the
property follows by an obvious induction from
Proposition~\ref{medianpropo}.
\subsection{Appendix}
\subsubsection{The $C^*$--algebra $\mathcal{D}'_{m,{\cal N}u}$}
\label{starca}
\noindent Let $\mathcal{D}$ be the space of test functions (compactly
supported $C^\mathrm{i}nfty$ functions on $(0,\mathrm{i}nfty)$) and $\mathcal{D}(0,x)$
be the test functions on $(0,x)$.
We say that $f\mathrm{i}n\mathcal{D}'$ is a staircase distribution if for any $k=0,1,2,...$ there is an ${L^1}$ function on
$[0,k+1]$ so that $f=F_k^{(km)}$ (in the sense of distributions) when
restricted to $\mathcal{D}(0,k+1)$ or
\begin{eqnarray}
\label{imposecond0}
F_k:=\mathcal{P}^{mk}f \mathrm{i}n L_1(0,k+1)
\mathrm{e}nd{eqnarray}
\noindent (since ${f}\mathrm{i}n{L}^1_{loc}[0,1-\mathrm{e}psilon]$ and
by Remark~\ref{density}, $\mathcal{P}f$ is well defined).
With this choice we have
\begin{eqnarray}
\label{imposecond}
F_{k+1}=\mathcal{P}^m{F_k}\ \mbox{on }[0,k]
\mbox{ and } F_k^{(j)}(0)=0 \ \mbox{for}\ j\le mk-1
\mathrm{e}nd{eqnarray}
We denote these distributions by $\mathcal{D}'_m$
($\mathcal{D}'_m(0,k)$ respectively, when restricted to
$\mathcal{D}(0,k)$) and observe that
$\bigcup_{m>0}\mathcal{D}'_m\supset S'$, the distributions of slow
growth. The inclusion is strict since any element of $S'$ is of finite
order.
Let $f\mathrm{i}n{L^1}$. Taking $F=\mathcal{P}^j f\mathrm{i}n C^j$ we have, by integration
by parts and noting that the boundary terms vanish,
\begin{eqnarray}
\label{sampleconv}
(F*F)(p)=\mathrm{i}nt_0^pF(s)F(p-s)\mathrm{d}s=\mathrm{i}nt_0^p F^{(j)}(s)\mathcal{P}^jF(p-s)
\mathrm{e}nd{eqnarray}
\noindent so that $F*F\mathrm{i}n C^{2j}$ and
\begin{eqnarray}
\label{sample2}
(F*F)^{(2j)}=f*f
\mathrm{e}nd{eqnarray}
\noindent This motivates the following definition:
for $f,\tilde{f}\mathrm{i}n\mathcal{D}'_m$ let
\begin{eqnarray}
\label{defconvd}
f*\tilde{f}:=(F_k*\tilde{F}_k)^{(2km)}\ \ \mbox{in } \mathcal{D}'(0,k+1)
\mathrm{e}nd{eqnarray}
\noindent
We first check that the definition is consistent in the sense that
$$(F_{k+1}*F_{k+1})^{(2m(k+1))}=(F_{k}*F_{k})^{(2mk)}$$ on
$\mathcal{D}(0,k+1)$. For $p<k+1$ integrating by parts and using
(\ref{imposecond}) we obtain
\begin{eqnarray}
\label{consisit1}
&&\frac{\mathrm{d}^{2m(k+1)}}{\mathrm{d}p^{2m(k+1)}}\mathrm{i}nt_0^p F_{k}(s)\mathcal{P}^{2m}
\tilde{F}_{k}(p-s)\mathrm{d}s=
\frac{\mathrm{d}^{2mk}}{\mathrm{d}p^{2mk}}\mathrm{i}nt_0^p F_{k}(s)
\tilde{F}_{k}(p-s)\mathrm{d}s\cr&&
\mathrm{e}nd{eqnarray}
\noindent The same argument shows that the definition
is compatible with the embedding of $\mathcal{D}'_{m}$
in $\mathcal{D}'_{m'}$ with $m'>m$. Convolution is commutative and
associative: with $f,g,h\mathrm{i}n\mathcal{D}'_m$
and identifying
$(f*g)$ and $h$ by the natural inclusion
with elements in $\mathcal{D}'_{2m}$ we obtain
$(f*g)*h=((F*G)*H)^{(4mk)}=f*(g*h)$.
The following staircase decomposition
exists in $\mathcal{D}'_m$.
\begin{Lemma}\label{Stcase}. For each $f\mathrm{i}n\mathcal{D}'_m$ there is a {\mathrm{e}m unique}
sequence
$\left\{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\right\}_{i=0,1,..}$ such that $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\mathrm{i}n{L^1}(\mathbb{R}^+)$,
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i=\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[i,i+1]}$ and
\begin{eqnarray}
\label{stdec}
f=\sum_{i=0}^{\mathrm{i}nfty}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i^{(mi)}
\mathrm{e}nd{eqnarray}
Also (cf. (\ref{imposecond})),
\begin{eqnarray}
\label{decompik}
F_i=\sum_{j\le i}\mathcal{P}^{m(i-j)}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i \ \mbox{on } [0,i+1)
\mathrm{e}nd{eqnarray}
\mathrm{e}nd{Lemma}
\noindent Note that the infinite sum is $\mathcal{D}'-$convergent since
for a given test function only a finite number of
distributions are nonzero.
{\mathrm{e}m{Proof}}
\noindent We start by showing (\ref{decompik}). For $i=0$ we take
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{0}=F_0\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}[0,1]$ (where $F_0\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}[0,1]$ $:=$ $
\phi\mapsto\mathrm{i}nt_0^1F_0(s)\phi(s)\mathrm{d}s$). Assuming
(\ref{decompik}) holds for $i<n$ we simply note that
\begin{multline}
\label{defDelI}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{n}:=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,n+1]}\left(F_n-\sum_{j\le
n-1}\mathcal{P}^{m(n-j)}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{j}\right)\cr
=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,n+1]}
\left(F_n-\mathcal{P}^m(F_{n-1}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,n]})\right)=
\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,n+1]}
\left(F_n-\mathcal{P}^m(F_{n-1}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,n]})\right)
\mathrm{e}nd{multline}
\noindent (with $\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty]}F_n$ defined in the same way as
$F_0\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}[0,1]$ above) has, by the induction hypothesis and
(\ref{imposecond}) the required properties. Relation (\ref{stdec}) is
immediate. It remains to show uniqueness. Assuming (\ref{stdec})
holds for the sequences $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i,\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_i$ and restricting
$f$ to $\mathcal{D}(0,1)$ we see that $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0=\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_0$.
Assuming $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i=\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_i$ for $i<n$ we then have
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n^{(mn)} =\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_n^{(mn)}$ on $\mathcal{D}(0,n+1)$. It
follows from Remark~\ref{density} that $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n(x)
=\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_n(x)+P(x)$ on $[0,n+1)$ where $P$ is a polynomial (of
degree $<mn$). Since by definition
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n(x)=\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_n(x)=0$ for $x<n$ we have
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n=\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_n(x)$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
The expression (\ref{defconvd}) hints to decrease in regularity, but this
is not the case. In fact, we
check that the regularity of convolution is not worse than that of
its arguments.
\begin{Remark}\label{weldefi}
\begin{eqnarray}\label{welldef}
(\cdot\,*\,\cdot):\mathcal{D}_n\mapsto\mathcal{D}_n
\mathrm{e}nd{eqnarray}
\mathrm{e}nd{Remark}
\noindent Since
\begin{eqnarray}
\label{identi0}
\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[a,b]}*\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[a',b']}=\left(\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[a,b]}*\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[a',b']}\right)
\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[a+a',b+b']}
\mathrm{e}nd{eqnarray}
\noindent we have
\begin{eqnarray}
\label{verifreg}
&&F*\tilde{F}=\sum_{j+k\le \lfloor p\rfloor}\mathcal{P}^{m(i-j)}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{j}*\mathcal{P}^{m(i-k)}\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_{k}
=\sum_{j+k\le \lfloor p\rfloor}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{j}*\mathcal{P}^{m(2i-j-k)}\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}_{k}\cr&&
\mathrm{e}nd{eqnarray}
\noindent which is manifestly in $C^{2mi-m(j+k)}[0,p)\subset C^{2mi-m\lfloor
p\rfloor}[0,p)$.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\subsubsection{Norms on $\mathcal{D}'_m$}
\noindent For $f\mathrm{i}n\mathcal{D}'_m$ define
\begin{eqnarray}
\label{nord1}
\|f\|_{{\cal N}u ,m}:=c_m\sum_{i=0}^{\mathrm{i}nfty}{\cal N}u ^{im}\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\|_{L^1_{\cal N}u }
\mathrm{e}nd{eqnarray}
\noindent (the constant $c_m$, immaterial for the moment, is defined in
(\ref{newcm}). When no confusion is possible we will simply write
$\|f\|_{\cal N}u $ for $\|f\|_{{\cal N}u ,m}$ and $\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta\|_{\cal N}u $ for
$\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\|_{L^1_{\cal N}u }$ (no other norm is used for the $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta$'s).
Let $\mathcal{D'}_{m,{\cal N}u }$ be the distributions in $\mathcal{D}'_m$
such that $\|f||_{\cal N}u <\mathrm{i}nfty$.
\begin{Remark}\label{normisnorm}
$\|\cdot\|_{\cal N}u $ is a norm on $\mathcal{D'}_{m,{\cal N}u }$.
\mathrm{e}nd{Remark}
\noindent If $\|f\|_{\cal N}u =0$ for all $i$, then $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i=0$
whence $f=0$. In view of Lemma~\ref{Stcase} we have
$\|0\|_{\cal N}u =0$. All the other properties are immediate.
\begin{Remark}\label{completen} $\mathcal{D'}_{m,{\cal N}u }$ is a Banach space.
The topology
given by $\|\cdot\|_{\cal N}u $ on $\mathcal{D'}_{m,{\cal N}u }$ is stronger
than the topology inherited from $\mathcal{D}'$.
\mathrm{e}nd{Remark}
{\mathrm{e}m Proof.} If we let $\mathcal{D}'_{m,{\cal N}u}(k,k+1)$ be the subset
of $\mathcal{D'}_{m,{\cal N}u }$ where all $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i=0$ except
for $i=k$, with the norm (\ref{nord1}), we have
\begin{eqnarray}
\label{directsum}
\mathcal{D'}_{m,{\cal N}u }=\bigoplus_{k=0}^{\mathrm{i}nfty}\mathcal{D}'_{m,{\cal N}u}(k,k+1)
\mathrm{e}nd{eqnarray}
\noindent and we only need to check completeness of each
$\mathcal{D}'_{m,{\cal N}u}(k,k+1)$ which is immediate: on ${L^1}[k,k+1]$,
$\|\cdot\|_{\cal N}u $ is equivalent to the usual ${L^1}$ norm and thus if
$f_n\mathrm{i}n \mathcal{D}'_{m,{\cal N}u}(k,k+1)$ is a Cauchy sequence
then
$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k,n}\stackrel{L_{\cal N}u}{\rightarrow}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k$ (whence weak
convergence) and
$f_n\stackrel{\mathcal{D}'_{m,{\cal N}u}(k,k+1)}{\rightarrow} f$
where $f=\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k^{(mk)}$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Lemma}\label{C*}
The space $\mathcal{D'}_{m,{\cal N}u}$ is a $C^*$ algebra with
respect to convolution.
\mathrm{e}nd{Lemma}
{\mathrm{e}m Proof.} Let
$f,\tilde{f}\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}$ with
$$f=\sum_{i=0}^\mathrm{i}nfty \raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i^{(mi)}\ \ ,\ \
\tilde{f}=\sum_{i=0}^\mathrm{i}nfty \tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}
_i^{(mi)}$$
Then
\begin{equation}\label{desf}
f* \tilde{f}=\sum_{i,j=0}^\mathrm{i}nfty\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i^{(mi)}*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}
_j^{(mj)}=\sum_{i,j=0}^\mathrm{i}nfty\left( \raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j
\right)^{m(i+j)}
\mathrm{e}nd{equation}
and the support of $ \raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j$ is in
$[i+j,i+j+2]$ i.e. $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta}
_j=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[i+j,i+j+2]}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j$.
We first evaluate the norm in $\mathcal{D}'_{m,{\cal N}u}$ of the terms
$\left( \raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j\right)^{m(i+j)} $.
\noindent {\bf{I. Decomposition formula.}} Let $f=F^{(mk)}\mathrm{i}n\mathcal{D}'(\mathbb{R}_+)$, where
$F\mathrm{i}n L^1(\mathbb{R}_+)$, and $F$ is supported in $[k,k+2]$ i.e.,
$F=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,k+2]}F $ ($k\geq 0$). Then $f\mathrm{i}n\mathcal{D}'_m$ and the
decomposition of $f$ (cf. (\ref{stdec})) has the terms:
\begin{equation}\label{zerodelta}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0=\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_1=...=\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k-1}=0\ \ ,\ \
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,k+1]}F
\mathrm{e}nd{equation}
and
\begin{equation}\label{decdelta}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+n,k+n+1]}G_n,\ \mbox{ where }G_n=\mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+n,\mathrm{i}nfty)}G_{n-1}\right),\ \ G_0=F
\mathrm{e}nd{equation}
\noindent {\mathrm{e}m{Proof of Decomposition Formula}}. We use first line of (2.98)
of the paper
\begin{equation}\label{deltaform}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_j=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[j,j+1]}\left
( F_j-\sum_{i=0}^{j-1}\mathcal{P}^{m(j-i)}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\right)
\mathrm{e}nd{equation}
where, in our case, $F_k=F,\
F_{k+1}=\mathcal{P}^m F,\, ...,\, F_{k+n}=\mathcal{P}^{mn}F,\, ...$.
The relations (\ref{zerodelta}) follow directly from
(\ref{deltaform}). Formula (\ref{decdelta}) is shown by induction on $n$.
For $n=1$ we have
$$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+1}=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,k+2]}\left( \mathcal{P}^m
\, F-\mathcal{P}^m\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k} \right)$$
$$=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,k+2]}\mathcal{P}^m\left(
\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,\mathrm{i}nfty)}F-\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,k+1]}F
\right)=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,k+2]}\mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,\mathrm{i}nfty)}F
\right)$$
Assume (\ref{decdelta}) holds for $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+j}$, $j\le
n-1$. Using (\ref{deltaform}), with $\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+n,k+n+1]}$ we
have
$$\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}\left(\mathcal{P}^{mn}F-\sum_{i=k}^{n-1}\mathcal{P}^{m(n-i)}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\right)=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}
\mathcal{P}^m\left( G_{n-1}-\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{n-1}\right)$$
$$=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}\mathcal{P}^m\left(
\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+n-1,\mathrm{i}nfty)}G_{n-1}-\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+n-1,k+n]}G_{n-1}\right)=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}\mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+n,\mathrm{i}nfty)}G_{n-1}\right){
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
$$
\noindent {\bf{II. Estimating $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}$.}} For $f$ as in {\bf{I}}, we have
\begin{equation}\label{normkp1}
||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+1}||_{\cal N}u\leq {\cal N}u^{-m}|| F||_{\cal N}u\ \ ,\ \
||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+2}||_{\cal N}u\leq {\cal N}u^{-2m}|| F||_{\cal N}u
\mathrm{e}nd{equation}
and, for $n\geq 3$
\begin{equation}\label{normkpn}
||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}||_{\cal N}u\leq e^{2{\cal N}u-n{\cal N}u}(n-1)^{nm-1} \frac{1}{(nm-1)!}|| F||_{\cal N}u
\mathrm{e}nd{equation}
\noindent {\mathrm{e}m{Proof of estimates of $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}$}}.
\noindent (A) Case $n=1$.
\begin{multline}
\label{fest}
||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+1}||_{\cal N}u\leq \mathrm{i}nt_{k+1}^{k+2} \, dt\, e^{-{\cal N}u t} \mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,\mathrm{i}nfty)}|F|
\right)(t)\\
= \mathrm{i}nt_{k+1}^{k+2} \, dt\, e^{-{\cal N}u t} \mathrm{i}nt_{k+1}^t\, ds_1\,
\mathrm{i}nt_{k+1}^{s_1}\, ds_2\, ...\mathrm{i}nt_{k+1}^{s_{m-1}}\, ds_m |F(s_m)|\\
\le \mathrm{i}nt_{k+1}^{k+2}\, ds_m |F(s_m)|\, \mathrm{i}nt_{s_m}^{\mathrm{i}nfty} \,
ds_{m-1}\, ... \mathrm{i}nt_{s_2}^\mathrm{i}nfty \, ds_1\, \mathrm{i}nt_{s_1}^{\mathrm{i}nfty} \, dt\,
e^{-{\cal N}u t} \\
= \mathrm{i}nt_{k+1}^{k+2}\, ds_m |F(s_m)|e^{-{\cal N}u s_m}{\cal N}u^{-m}
\leq {\cal N}u^{-m}|| F||_{\cal N}u
\mathrm{e}nd{multline}
\noindent (B) Case $n=2$:
$$||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+1}||_{\cal N}u\leq \mathrm{i}nt_{k+2}^{k+3} \, dt\, e^{-{\cal N}u t}\mathcal{P}^m
\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+2,\mathrm{i}nfty)} \mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,\mathrm{i}nfty)}|F|
\right)\right)$$
$$= \mathrm{i}nt_{k+2}^{k+3} \, dt\, e^{-{\cal N}u t} \mathrm{i}nt_{k+2}^t\, dt_1\,
\mathrm{i}nt_{k+2}^{t_1}\, dt_2\, ...\mathrm{i}nt_{k+2}^{t_{m-1}}\, dt_m
\mathrm{i}nt_{k+1}^{t_m}\, ds_1\,
\mathrm{i}nt_{k+1}^{s_1}\, ds_2\, ...\mathrm{i}nt_{k+1}^{s_{m-1}}\, ds_m |F(s_m)|$$
$$\le \mathrm{i}nt_{k+2}^{k+3}\, ds_m |F(s_m)|\, \mathrm{i}nt_{s_m}^{\mathrm{i}nfty}\, ds_{m-1}\,
...\mathrm{i}nt_{s_2}^{\mathrm{i}nfty}\, ds_1 \, \mathrm{i}nt_{\max\{s_1,k+2\}} ^{\mathrm{i}nfty}\,
dt_m\, \mathrm{i}nt_{t_m}^\mathrm{i}nfty \, dt_{m-1}\, ... \mathrm{i}nt_{t_1}^\mathrm{i}nfty \, dt\,
e^{-{\cal N}u t} $$
$$= \mathrm{i}nt_{k+2}^{k+3}\, ds_m |F(s_m)|\, \mathrm{i}nt_{s_m}^{\mathrm{i}nfty}\, ds_{m-1}\,
...\mathrm{i}nt_{s_2}^{\mathrm{i}nfty}\, ds_1
e^{-{\cal N}u \max\{s_1,k+2\}}{\cal N}u^{-m-1}$$
$$ \le\mathrm{i}nt_{k+2}^{k+3}\, ds_m |F(s_m)|\,
\mathrm{i}nt_{s_m}^{\mathrm{i}nfty}\, ds_{m-1}\,...\mathrm{i}nt_{s_3}^\mathrm{i}nfty \, ds_2 e^{-{\cal N}u
s_2}{\cal N}u^{-m-2}=\mathrm{i}nt_{k+2}^{k+3}\, ds_m |F(s_m)|e^{-{\cal N}u s_m}{\cal N}u^{-2m}$$
\noindent (C) Case $n\geq 3$. We first estimate $G_2,...,G_n$:
$$|G_2(t)|\leq \mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+2,\mathrm{i}nfty)}
\mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,\mathrm{i}nfty)}|F| \right)\right) (t)$$
$$= \mathrm{i}nt_{k+2}^t\, dt_1\, \mathrm{i}nt_{k+2}^{t_1}\, dt_2\,
...\mathrm{i}nt_{k+2}^{t_{m-1}}\, dt_m \mathrm{i}nt_{k+1}^{t_m}\, ds_1\,
\mathrm{i}nt_{k+1}^{s_1}\, ds_2\, ...\mathrm{i}nt_{k+1}^{s_{m-1}}\, ds_m |F(s_m)|$$
and using the inequality
$$|F(s_m)|=|F(s_m)|\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k,k+2]}(s_m)\le |F(s_m)|e^{-{\cal N}u
s_m}e^{{\cal N}u(k+2)}$$
we get
$$|G_2(t)|\le e^{{\cal N}u(k+2)} ||F||_{\cal N}u \mathrm{i}nt_{k+1}^t\, dt_1\,
\mathrm{i}nt_{k+1}^{t_1}\, dt_2\, ...\mathrm{i}nt_{k+1}^{t_{m-1}}\, dt_m
\mathrm{i}nt_{k+1}^{t_m}\, ds_1\,
\mathrm{i}nt_{k+1}^{s_1}\, ds_2\, ...\mathrm{i}nt_{k+1}^{s_{m-2}}\,
ds_{m-1}$$
$$=e^{{\cal N}u(k+2)} ||F||_{\cal N}u (t-k-1)^{2m-1}\frac{1}{(2m-1)!}$$
\noindent The estimate of $G_2$ is used for bounding $G_3$:
$$|G_3(t)|\leq \mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+3,\mathrm{i}nfty)}
|G_2|\right)\le \mathcal{P}^m\left( \mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1,\mathrm{i}nfty)}
|G_2|\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le e^{{\cal N}u(k+2)} ||F||_{\cal N}u (t-k-1)^{3m-1}\frac{1}{(3m-1)!}$$
and similarly (by induction)
$$|G_n(t)|\le e^{{\cal N}u(k+2)} ||F||_{\cal N}u (t-k-1)^{nm-1}\frac{1}{(nm-1)!}$$
\noindent Then
$$||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}||_{\cal N}u\leq e^{{\cal N}u(k+2)} ||F||_{\cal N}u \frac{1}{(nm-1)!}
\mathrm{i}nt_{k+n}^{k+n+1}\, dt\, e^{-{\cal N}u t}(t-k-1)^{nm-1} $$
and, for ${\cal N}u\geq m$ the integrand is decreasing, and the inequality
(\ref{normkpn}) follows.
\noindent {\bf{III. Final Estimate}}. Let ${\cal N}u_0>m$ be fixed. For $f$ as in
{\bf{I}}, we have for any ${\cal N}u >{\cal N}u_0$,
\begin{equation}\label{Fk}
||f||\le c_m {\cal N}u^{km}||F||_{\cal N}u
\mathrm{e}nd{equation}
\noindent for some $c_m$, if ${\cal N}u>{\cal N}u_0>m$.
\noindent {\mathrm{e}m{Proof of Final Estimate}}
$$||f||=\sum_{n\geq 0}{\cal N}u^{km+kn}||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_{k+n}||_{\cal N}u \le
{\cal N}u^{km}||F||_{\cal N}u \left[ 3+\sum_{n\geq 3}
{\cal N}u^{nm}e^{2{\cal N}u-n{\cal N}u}\frac{(n-1)^{nm-1}}{(nm-1)!}\right]$$
and,
using $n-1\leq (mn-1)/m$ and a crude Stirling estimate we obtain
\begin{equation}\label{newcm}||f||\le {\cal N}u^{km}||F||_{\cal N}u \left[ 3+ me^{2{\cal N}u-1}\sum_{n\geq 3} \left(
e^{m-{\cal N}u}{\cal N}u^m/m^m\right)^{n} \right]\le c_m {\cal N}u^{km}||F||_{\cal N}u
\mathrm{e}nd{equation}
\noindent Thus (\ref{Fk}) is proven for
${\cal N}u>{\cal N}u_0>m$.
\noindent {\bf{End of the proof}}. From (\ref{desf}) and (\ref{Fk}) we get
$$||f*\tilde{f}||\leq \sum_{i,j=0}^\mathrm{i}nfty||\left(
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j \right)^{m(i+j)}||$$
$$\leq \sum_{i,j=0}^\mathrm{i}nfty c_m^2 {\cal N}u^{m(i+j)}
||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i*\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j ||_{\cal N}u \leq c_m^2 \sum_{i,j=0}^\mathrm{i}nfty
{\cal N}u^{m(i+j)} ||\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i||_{\cal N}u \, ||\tilde{\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta} _j ||_{\cal N}u=c_m^2
||f||\, ||\tilde{f}||$$ ${
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
$
\begin{Remark}\label{conveB} Let $f\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}$ for some ${\cal N}u >{\cal N}u _0$ where
${\cal N}u _0^m=\mathrm{e}^{{\cal N}u _0}$. Then
$f\mathrm{i}n\mathcal{D}'_{m,{\cal N}u'}$ for all ${\cal N}u '>{\cal N}u $ and furthermore,
\begin{eqnarray}
\label{tendzero}
\|f\|_{\cal N}u \downarrow 0\ \mbox{as } {\cal N}u \uparrow\mathrm{i}nfty
\mathrm{e}nd{eqnarray}
\mathrm{e}nd{Remark}
{\mathrm{e}m Proof.} We have
\begin{eqnarray}
\label{difB}
{\cal N}u ^{mk}\mathrm{i}nt_k^{k+1}|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k(s)|\mathrm{e}^{-{\cal N}u s}\mathrm{d}s=
({\cal N}u ^m\mathrm{e}^{-{\cal N}u })^k\mathrm{i}nt_0^1 |\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k(s+k)|\mathrm{e}^{-{\cal N}u s}\mathrm{d}s
\mathrm{e}nd{eqnarray}
\noindent which is decreasing in ${\cal N}u $. The rest follows from
the monotone convergence theorem.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\subsubsection{Embedding of $L^1_{\cal N}u$ in $\mathcal{D}'_m$}
\begin{Lemma}\label{imbeddi}
i) Let $f\mathrm{i}n L^1_{{\cal N}u_0}$ (cf. Remark~\ref{conveB}).
Then $f\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}$ for all ${\cal N}u>{\cal N}u_0$.
ii) $\mathcal{D}(\mathbb{R}^+\backslash\hat{B}ox{\it I\hskip -2pt N}N)\cap L^1_{\cal N}u(\mathbb{R}^+) $ is dense in $\mathcal{D}_{m,{\cal N}u }$ with
respect to the norm $\|\|_{\cal N}u $.
\mathrm{e}nd{Lemma}
{\mathrm{e}m Proof. }
Note that if for some ${\cal N}u_0$ we have $f\mathrm{i}n L^1_{{\cal N}u_0}(\mathbb{R}^+)$
then
\begin{eqnarray}
\label{unifversusint}
\mathrm{i}nt_0^p|f(s)|\mathrm{d}s\le \mathrm{e}^{{\cal N}u_0 p}\mathrm{i}nt_0^p|f(s)|\mathrm{e}^{-{{\cal N}u_0 s}}\mathrm{d}s
\le \mathrm{e}^{{\cal N}u_0 p}\|f\|_{{\cal N}u_0}
\mathrm{e}nd{eqnarray}
\noindent to which, application of $\mathcal{P}^{k-1}$ yields
\begin{eqnarray}
\label{genn}
\mathcal{P}^k|f|\le {\cal N}u_0^{-k+1}\mathrm{e}^{{\cal N}u_0 p} \|f\|_{{\cal N}u_0}
\mathrm{e}nd{eqnarray}
\noindent Also, $\mathcal{P}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty)}\mathrm{e}^{{\cal N}u_0 p}\le
{{\cal N}u_0}^{-1}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty)}\mathrm{e}^{{\cal N}u_0 p}$ so that
\begin{eqnarray}
\label{estimexp5}
\mathcal{P}^m\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty)}\mathrm{e}^{{\cal N}u_0 p}\le
{{\cal N}u_0}^{-m}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty)}\mathrm{e}^{{\cal N}u_0 p}
\mathrm{e}nd{eqnarray}
\noindent so that, by (\ref{defDelI}) (where now $F_n$ and
$\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty]}F_n$ are in ${L}^1_{loc}(0,n+1)$) we have for
$n>1$,
\begin{eqnarray}
\label{festDel}
|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n|\le \|f\|_{{\cal N}u_0}\mathrm{e}^{{\cal N}u_0 p}{{\cal N}u_0}^{1-mn}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,n+1]}
\mathrm{e}nd{eqnarray}
\noindent Let now ${\cal N}u$ be large enough.
We have
\begin{multline}
\label{sumn}
\sum_{n=2}^{\mathrm{i}nfty}{\cal N}u^{mn}\mathrm{i}nt_0^{\mathrm{i}nfty}|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n|\mathrm{e}^{-{\cal N}u p}\mathrm{d}p\le
{\cal N}u_0\|f\|_{{\cal N}u_0}
\sum_{n=2}^{\mathrm{i}nfty}\mathrm{i}nt_n^{n+1}\mathrm{e}^{-({\cal N}u-{\cal N}u_0)
p}\left(\frac{{\cal N}u}{{\cal N}u_0}\right)^p\mathrm{d}p\cr=
\frac{\mathrm{e}^{-2({\cal N}u-{\cal N}u_0-\ln({\cal N}u/{\cal N}u_0))}}{{\cal N}u-{\cal N}u_0-\ln({\cal N}u/{\cal N}u_0)}{\cal N}u_0\|f\|_{{\cal N}u_0}
\mathrm{e}nd{multline}
\noindent For $n=0$ we simply have $\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0\|\le\|f\|$, while
for $n=1$ we write
\begin{eqnarray}
\label{case1}
\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_1\|_{{\cal N}u}\le \|1^{*(m-1)}*|f|\|_{{\cal N}u}\le
{\cal N}u^{m-1}\|f\|_{{\cal N}u}
\mathrm{e}nd{eqnarray}
Combining the estimates above, the proof of (i) is complete.
To show (ii), let $f\mathrm{i}n\mathcal{D}'_{m,{\cal N}u }$ and let
$k_\mathrm{e}psilon$ be such that
$c_m\sum_{i=k_\mathrm{e}psilon}^{\mathrm{i}nfty}{\cal N}u ^{im}\|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\|_{\cal N}u <\mathrm{e}psilon$.
For each $i\le k_\mathrm{e}psilon$ we take a function $\delta_i$ in $\mathcal{D}(i,i+1)$
such that $\|\delta_i-\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_i\|_{\cal N}u <\mathrm{e}psilon 2^{-i}$. Then
$\|f-\sum_{i=0}^{k_\mathrm{e}psilon}\delta_i^{(mi)}\|_{m,{\cal N}u }<2\mathrm{e}psilon$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof} of continuity of $f(p)\mapsto pf(p)$. If $f(p)=
\sum_{k=0}^{\mathrm{i}nfty}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k^{(mk)}$ then
$pf=\sum_{k=0}^{\mathrm{i}nfty}(p\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k)^{(mk)}-
\sum_{k=0}^{\mathrm{i}nfty}mk\mathcal{P}(\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k^{(mk)})$=
$\sum_{k=0}^{\mathrm{i}nfty}(p\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k^{(mk)})-1*\sum_{k=0}^{\mathrm{i}nfty}
(mk\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k)^{(mk)}$. The rest is obvious from continuity of
convolution, the embedding shown above and the definition of the
norms.
\subsubsection{Laplace transforms}
\label{sec:LT}
{\mathrm{e}m Proof of Lemma~\ref{existe} }. Let ${\cal N}u >{\cal N}u _0$ (cf.
Remark~\ref{conveB}). Equation (\ref{lapcomuta}) follows most easily
from the corresponding well property of Laplace transforms on
$\mathcal{D}$, from the continuity of ${\cal L}$ and Lemma~\ref{imbeddi} (ii).
For the second notice that by the definition of $\mathcal{D}'_m,$
$f'\mathrm{i}n\mathcal{D}'_m$ implies $f\mathrm{i}n AC(0,1-\mathrm{e}psilon)$ and the property
follows by density from the $\mathcal{D}$ identity
$\mathcal{L}(\mathcal{P}g)= x^{-1}{\cal L}(g)$. The third equality also
follows by density. The rest of the properties, except
injectivity, follow immediately from the definitions and the topology
used.
To show injectivity assume that $\mathcal{L}d(x)=0$ where
$d\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}$, ${\cal N}u<x_0<x\mathrm{i}n\mathbb{R}^+$. By analyticity,
$\mathcal{L}d(x)=0$ in (say) $S_2:=\{z:|z|>2x_0:|\arg(z)|<\pi/4\}$.
Using dominated convergence, assuming $x_0$ is large enough, we have
$$\left|\sum_{k=1}^{\mathrm{i}nfty}
x^{m(k-1)}\mathrm{e}^{-(k-1)x}\mathrm{i}nt_0^{1}\mathrm{e}^{-sx}|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k(s+k)|\mathrm{d}s\right| \le 1 $$
in $S_2$. Thus $|f(x)|=$
$\left|\mathrm{i}nt_0^1 \mathrm{e}^{px}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0(1-p)\mathrm{d}p\right|$
$\le |x|^m$ in $S_2$. The function $f^{(m)}(x)$ is entire, of
exponential order less than $1+\mathrm{e}psilon$ for any $\mathrm{e}psilon$ and, using
the previous inequality in Cauchy's formula we see that
$|f^{(m)}(x)|<\mbox{const.}$ in $(x_0,\mathrm{i}nfty)$. Since for
$\phi\mathrm{i}n(\pi/2,3\pi/2)$ we obviously also have
$f^{(m)}(r\mathrm{e}^{\mathrm{i}\phi})\rightarrow 0$ as $r\rightarrow\mathrm{i}nfty$, an
elementary instance of the Phragm\'en-Lindel\"of principle
\cite{Holland} implies that $f^{(m)}$ is bounded in $\mathbb{C}$, therefore
constant, so $f$ itself is a polynomial that decays in the left half
plane, thus $f=0$. Therefore $\mathrm{i}nt_0^1
\mathrm{e}^{-px}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0(p)\mathrm{d}p={\cal L}\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0=0$ so that $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_0=0$.
Inductively and in the same way, we see that $\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_k=0$, $k\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N$.
\label{sec:A0}
{\mathrm{e}m Proof of Lemma~\ref{cuteCauchy}}.
Take first $r{\cal N}otin\mathbb{Z}$.
Choose $a_1,a_2$
so that $0<a_1<a_2<a$ and consider the closed contour
$C$ going along the upper cut from $\xi =0$ to $\xi =a_2$, continuing towards the lower cut
anticlockwise along the circle $C(a_2)$ of radius $a_2$ centered at
origin, and finally coming from $\xi =a_2$ back to $\xi =0$
along the lower cut. For $|\xi|<a_1$ we have, by the assumptions
of the lemma,
\begin{eqnarray}
\label{intepathf}
2\pi if(\xi )=\oint_C\frac{f(s)}{s-\xi }\mathrm{d}s=
\oint_{C(a_2)}\frac{f(s)}{s-\xi }\mathrm{d}s+
\mathrm{i}nt_{0}^{a_2}\frac{s^rA(s)}{s-\xi }\mathrm{d}s\cr&&
\mathrm{e}nd{eqnarray}
\noindent On the other hand, defining $z^rA(z)$ in the interior
of $C(a)$ cut along the positive axis (with the usual
convention $\arg(z)=0$ on the upper cut), we have, for
the same contour as above and $\xi \mathrm{i}n\mathcal{V}_{a_1}$
\begin{eqnarray}
\label{intepathfA}
2\pi\,\mathrm{i}\xi ^rA(\xi )=
\oint_{C(a_2)}\frac{f(s)}{s-\xi }\mathrm{d}s+
\left(1-\mathrm{e}^{2\pi i r}\right)\mathrm{i}nt_{0}^{a_2}\frac{s^rA(s)}{s-\xi }\mathrm{d}s\cr&&
\mathrm{e}nd{eqnarray}
\noindent Comparing (\ref{intepathf}) to (\ref{intepathfA}) we get:
\begin{eqnarray}
\label{resultf}
&&f(\xi )=\frac{1}{1-\mathrm{e}^{2\pi i r}} \xi^rA(\xi )\cr&&-\frac{1}
{2\pi i(1-\mathrm{e}^{2\pi i r})}\oint_{C(a_2)}\frac{A(s)}{s-\xi}\mathrm{d}s+
\frac{1}{2\pi i}\oint_{C(a_2)}\frac{f(s)}{s-\xi}\mathrm{d}s\cr&&
\mathrm{e}nd{eqnarray}
\noindent As integrals of analytic functions with respect to complex
absolutely continuous measures ($A(s)\mathrm{d}s$ and $f(s)\mathrm{d}s$), the last two
terms in (\ref{resultf}) are analytic in $\xi$ for $|\xi|<a_1$. Since
$a_1$ can be chosen arbitrarily close to $a$, the case $r{\cal N}otin\mathbb{Z}$ is
proven. For $r\mathrm{i}n\mathbb{Z}$ the argument is essentially the same, in terms
of $A(\xi)\xi^r\ln\xi$ instead of $\xi^r A(\xi)$. The proof
generalizes immediately to linear combinations
of $\xi^r A(\xi)$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof of Lemma~\ref{STR1}}. On the interval $(k,k+1)$ we have
$ f^+=\sum_{i=1}^k (f^-_i)^{(mi)} $ or
\begin{eqnarray}
\label{deco1}
\mathcal{P}^{mk+1}f^+=\sum_{i=1}^k\mathcal{P}^{m(k-i)+1} f^-_i
\mathrm{e}nd{eqnarray}
\noindent Let $\mathrm{e}psilon$ be small and positive. Since $f(t\mathrm{e}^{\mathrm{i}\phi})$ and
$g_i^{-}(t\mathrm{e}^{\mathrm{i}\phi})$ converge as $\phi\rightarrow 0$
in $\mathcal{D}'_{m,{\cal N}u}$ we have
that $\mathcal{P}^{m(k-i)+1}f(t\mathrm{e}^{\mathrm{i}\phi})$ and
$\mathcal{P}^{m(k-i)+1}g_i^{-}(\mathrm{e}^{\mathrm{i}\phi}t)$ converge on
$[0,k+1-\mathrm{e}psilon]$ uniformly to $\mathcal{P}^{mk+1}f^+$ and
$\mathcal{P}^{m(k-i)+1}f_i^{-}$ respectively. The left side of
(\ref{deco1}) is the limit on $I=[k+\mathrm{e}psilon,k+1-\mathrm{e}psilon]$ of a
function analytic in a neighborhood in the {\mathrm{e}m upper} half plane of
$I$ and continuous on $I$ while the right side is the limit of a
function analytic in a neighborhood in the {\mathrm{e}m lower} half plane of
$I$ and continuous on $I$. The equality of their continuous limits on
$I$ implies in particular that $\mathcal{P}^{mk+1}f(t\mathrm{e}^{\mathrm{i}\phi})$
extends analytically through $I$ in the lower half plane, and its
continuation is analytic where $\sum_{i=1}^k\mathcal{P}^{m(k-i)+1}
g_i^{-}(t\mathrm{e}^{\mathrm{i}\phi})$ is. A corresponding
statement is true for the upper plane continuation of
$\mathcal{P}^{mk+1}f(t\mathrm{e}^{-\mathrm{i}\phi})$ and (i) follows. Part (ii) now
follows also, as an immediate application of Lemma~\ref{cuteCauchy}.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
{\mathrm{e}m Proof of Proposition~\ref{medianpropo}}.
The fact that multiplication by a bounded analytic function
is well defined on $\mathcal{F}(\mathcal{R}'_1)$ is immediate.
Since
\begin{gather}\label{polar}
2f*g = (f+g)*(f+g)-f*f-g*g
\mathrm{e}nd{gather}
\noindent we may take $f=g$. With
$h=\mathcal{P}^{mk+1}f\mathrm{i}n\mathcal{F}(\mathcal{R}'_1)$ it suffices to
show for every $k$ that $h*h$ (defined near zero by (\ref{defconv})
and which equals $\mathcal{P}^{2mk+2}(f*f)$ there) extends
analytically to $\mathcal{R}'_1$ for $\Re(x)<k$. Since $f$
is analytic in $\mathcal{R}'_1$ so is $h$.
In particular $h$ can
be analytically continued along any ray $d\subset\mathcal{R}'_1$ other
than the real line, and we have, by analyticity and with $*_d$ meaning
convolution along $d$,
\begin{equation}\label{forac}
AC(h*h)=AC(h)*_d AC (h)
\mathrm{e}nd{equation}
\noindent Also, by (\ref{defindecom2})
\begin{eqnarray}
\label{eqhh}
h^-(p)=h^+(p)+\sum_{j=1}^{\mathrm{i}nfty}(h_j^+(p-j))^{(mj)}
\mathrm{e}nd{eqnarray}
Let $H_0=h^+$ and $H_j(p)=(h_j^+(p))^{(mj)}$. By construction
$H_j'$ have ${L^1}$ boundary values on $[0,k-j+1)$ as
$\Re(z)>0,\Im(z)\downarrow 0$ and so $H_j$ extend {\mathrm{e}m continuously}
to the strip $0<\Re(z)<k-j+1,\Im(z)\ge 0$. We
have, by (\ref{defindecom2}) and continuity
\begin{eqnarray}
\label{defindecom2H}
h^{-}(z)=\sum_{i=0}^{j}H_i(z-j)
\mathrm{e}nd{eqnarray}
\noindent for $\Re(z)\mathrm{i}n[0,j)$ and $\Im(z)\ge 0$ since $H_i(x)=0$ in the left
half plane, by definition. For the same reason we have, with
$p'=p-i-j$ and $i+j\ge 1$,
\begin{eqnarray}
\label{anahi}
&&\mathrm{i}nt_0^p H_i(x-i)H_j(p-x-j)\mathrm{d}x\cr&&=\left\{\begin{array}{cc}
\displaystyle{
\mathrm{i}nt_0^{p'} H_i(x)H_j(p'-x)\mathrm{d}x=J_{i,j}(p')\ }&(\Re(p)'>i+j)\cr
0&(\Re(p')<i+j)\mathrm{e}nd{array}\right.
\mathrm{e}nd{eqnarray}
\noindent As both $H_i$ and $H_j$ are analytic in an open strip $\mathcal{S}$
in the first quadrant and continuous on $[0,k+1-\mathrm{e}psilon]$ we see from
(\ref{anahi}) that $J_{ij}(p')$ are also analytic in $\mathcal{S}$ and
continuous on $[0,k+1-\mathrm{e}psilon]$. For $p\mathrm{i}n (0,l+1)$, $l\le k$ we
have
\begin{eqnarray}
\label{firstequali}
\big(H^{-}*H^{-}\big)(p)=
\sum_{i=0}^{l}\sum_{j=0}^i
J_{ij}(p-i)\cr&&
\mathrm{e}nd{eqnarray}
\noindent Now, by (\ref{forac}) and using the continuity of $H$ and of
convolution, we note that the left side of (\ref{firstequali})
represents the continuous limit along $(l,l+1) $ of $(H*H)^-$, a
function analytic in a domain in the lower half plane while the right
side is the limit of a function analytic in the upper half plane
and $(l,l+1)$ is contained in the common boundary. As in the
proof of Lemma~\ref{STR1} we conclude that $h*h$, thus
$f*f$ extend analytically
in $\mathcal{R}'_1$.
Going back to the definition of $H$ we get on $(0,l+1)$,
\begin{eqnarray}
\label{firstequali2}
\big(f^{-}*f^{-}\big)(p)=\big(f^{+}*f^{+}\big)(p)
+\sum_{i=1}^{l}\sum_{j=0}^i
(f_j^+)^{(mj)}*(f_{i-j}^+)^{m(i-j)}(p-i)\cr&&
\mathrm{e}nd{eqnarray}
\noindent where $f_j*f_{i-j}=(H_j*H_{i-j})^{(2mk+2)}$ is the convolution in
$\mathcal{D}'_{m,{\cal N}u}$ and in our case gives a function analytic in
the open region $\mathcal{S}$. By comparing
with (\ref{defindecom2}) and (\ref{defindecom2H}) we get by induction
$(f*f)_j=\sum_{s=0}^j f_s*f_{j-s}$ or, using (\ref{polar})
we get (\ref{defindecom2}).
Since by assumption $f_s$ and $g_s$ belong to $\mathcal{D}'_{m,{\cal N}u}$
and the sum in (\ref{defindecom3}) only contains a finite number of terms, it
follows that all analytic continuations
of $(f*g)$ also belong to $\mathcal{D}'_{m,{\cal N}u}$. Furthermore,
it follows immediately that $K(f*g,{\cal N}u)\le 2K(f,{\cal N}u)K(g,{\cal N}u)$.
Only the last equality in (\ref{assertmed}) needs a proof; we have
\begin{multline}
\label{verifymedC}
(\mathcal{A}_\alpha(f))^{*2}=\left(\sum_{i=0}^{\mathrm{i}nfty} \alpha^i( f_i(p-i))^{(mi)}\right)^{*2}=
\sum_{k=0}^{\mathrm{i}nfty} \alpha^k\sum_{j=0}^k (f_j*f_{k-j})^{(mk)}(p-k)\cr
=\sum_{k=0}^{\mathrm{i}nfty} \alpha^k((f*f)_k)^{(mk)}=\mathcal{A}_\alpha(f*f)
\mathrm{e}nd{multline}
We have $\|\mathcal{A}_C(f)\|_{m,{\cal N}u}\le
\|f\|_{m,{\cal N}u}\sum_{j=0}^{\mathrm{i}nfty}C^j
K(f,{\cal N}u)^j=(1-KC)^{-1}\|f\|_{m,{\cal N}u}$ so that if
$\bfY\mathrm{i}n\mathcal{F}_r$ then $\|\mathcal{A}_C(\bfY^{*{\bf l}})\|_{m,{\cal N}u}
=\|(\mathcal{A}_C\bfY)^{*{\bf l}}\|_{m,{\cal N}u}\le
(\|\bfY\|_{m,{\cal N}u}/(1-KC))^{|{\bf l}|}$ so that if ${\cal N}u$ is large
enough the sum involved in the expression of $M$
is uniformly convergent in $\mathcal{D}'_{m,{\cal N}u}(\mathbb{R}^+)$
and (\ref{defindecom3}) follows.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Lemma}\label{cinftycase} (i) Let $k_0\ge 0$ and let $\lambda$ be such that
$\Re(\lambda)<\alpha_1<k_0$ and
$\left|\Im(\lambda)\right|<\alpha_2$. Alternatively, let $k_0\ge 0$
and $\lambda$ be such that
$0<\alpha_1<\left|\Im(\lambda)\right|<\alpha_2$. There exists a
constant $C(\alpha_1,\alpha_2)$ independent of $k_0,{\cal N}u$ and
$\lambda$ so that
\begin{eqnarray}
\label{normlemm}
\|U\|_{\mathcal{D}'_{m,{\cal N}u}(k_0,\mathrm{i}nfty)\mapsto
\mathcal{D}'_{m,{\cal N}u}(k_0,\mathrm{i}nfty)}\le
C(\alpha_1,\alpha_2)(1+|\lambda|)^{-1}
\mathrm{e}nd{eqnarray}
(ii) In both cases in (i),
$U$ is
strongly continuous in $\lambda$.
\mathrm{e}nd{Lemma}
{\mathrm{e}m Proof}
The impediments in the proof come on the one hand
from having to estimate
quotients of the form $\mathrm{i}nt|\mathcal{P}^n U f|/\mathrm{i}nt|\mathcal{P}^n f|$
and on the other hand from the nonlocal character
of the action of $U$ in our space.
In view of Eq. (\ref{directsum}) it is enough
to find a $k-$independent upper bound
for the norms of the restrictions of $U$ to $\mathcal{D}'_{m,{\cal N}u}(k,k+1)$,
$U:\mathcal{D}'_{m,{\cal N}u}(k,k+1)\mapsto \mathcal{D}'_{m,{\cal N}u}$. We are interested
in $\Re(\lambda)<1$ in
the cases (a) $b>\left|\Im(\lambda)\right|>a>0$, (b) $\lambda<-a<0$ real,
(c) $\lambda\mathrm{i}n\mathbb{R}^+$ or $\lambda$ complex, $|\Im(\lambda)|<b$
but with $\mbox{supp}(f)\mathrm{i}n (k_1,\mathrm{i}nfty)$
with $k_1>a>\Re(\lambda)$, $k_1\mathrm{i}n(a,a+1)$. We let $k_1=0$ in (a) and (b).
\noindent We have the following identity
\begin{eqnarray}
\label{indentino0}
\frac{f^{(r)}}{p-\lambda}=\left(r(p-\lambda)^{r-1}\mathrm{i}nt_k^p
\frac{f(s)}{(s-\lambda)^{r+1}}\mathrm{d}s+\frac{f(p)}{p-\lambda}\right)^{(r)}
\mathrm{e}nd{eqnarray}
\noindent which is proved by straightforward
differentiation of the r.h.s. or by writing
$f^{(r)}=(p-\lambda)g^{(r)}=(pg-\lambda g)^{(r)}-
rg^{(r-1)}$ so that $f=(p-\lambda)(\mathcal{P}g)'-r\mathcal{P}g$
and solving for $\mathcal{P}g$. We take $k\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N$ with
$k+1>k_1$, a distribution
$f$ with $\mbox{supp}(f)\mathrm{i}n (k_0,k+1)$ where
$k_0=\max\{k,k_1\}$ and we let
\begin{eqnarray}
\label{defc}
c=\mathrm{i}nt_{k_0}^{k+1}
\frac{f(s)}{(s-\lambda)^{r+1}}\mathrm{d}s \ \ \ (r:=mk)
\mathrm{e}nd{eqnarray}
\noindent For $\mathrm{e}psilon$ small (to be made zero
in the end), we write the decomposition
\begin{multline}
\label{decomf}
\frac{f^{(r)}}{p-\lambda}\cr=\left(r(p-\lambda)^{r-1}\mathrm{i}nt_{k_0}^p
\frac{f(s)}{(s-\lambda)^{r+1}}\mathrm{d}s+\frac{f(p)}{p-\lambda}
-cr(p-\lambda)^{r-1}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1-\mathrm{e}psilon,\mathrm{i}nfty]}\right)^{(r)}\cr
+\left( cr(p-\lambda)^{r-1}\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[k+1-\mathrm{e}psilon,\mathrm{i}nfty]}\right)^{(r)}
=f_1+f_2=f_1+c\,r\, g_2^{(r)}
\mathrm{e}nd{multline}
\noindent where by construction
$f_1\mathrm{i}n\mathcal{D}'_{m,{\cal N}u}(k,k+1)$
whence, for $f\mathrm{i}n \mathcal{D}(k_0,k+1)$ we have
\begin{multline}
\label{est11}
\left\|\frac{f^{(r)}}{p-\lambda}\right\|_{m,{\cal N}u}\le
\|f_1\|_{m,{\cal N}u}+\|f_2\|_{m,{\cal N}u}=\|f_1\|_{\mathcal{D}'_{m,{\cal N}u}(k,k+1)}+\|f_2\|_{m,{\cal N}u}
\cr\le
\|f_1-f_2\|_{\mathcal{D}'_{m,{\cal N}u}(k,k+1)}+2\|f_2\|_{m,{\cal N}u}
\mathrm{e}nd{multline}
\noindent and then, for some $C_1$
\begin{multline}
\label{term01}
\|f_1-f_2\|_{\mathcal{D}'_{m,{\cal N}u}(k,k+1)}\cr\le
{\cal N}u^{r}\mathrm{i}nt_{k_0}^{k+1}\left|r(p-\lambda)^{r-1}\mathrm{i}nt_{k_0}^p
\frac{f(s)}{(s-\lambda)^{r+1}}\mathrm{d}s+
\frac{f(p)}{p-\lambda}\right|\mathrm{e}^{-{\cal N}u p} \mathrm{d}p\cr \le
{\sup}_*|p-\lambda|^{-1}\|f\|_{m,{\cal N}u}+{\sup}_*
\left|\frac{(p-\lambda)^{r-1}}{(s-\lambda)^{r+1}}\right|\|f\|_{m,{\cal N}u}\cr
\le C_1{\sup }_*|p-\lambda|^{-1}\|f\|_{m,{\cal N}u}
\mathrm{e}nd{multline}
\noindent where the supremum is taken over
$\{k\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N,p, s\mathrm{i}n[k,k+1]\cap(k_0,\mathrm{i}nfty) \}$.
For the constant $c$ in (\ref{decomf}) we have, for some
$C_2$ depending on $a$ and otherwise independent
of $\lambda,k$ the estimate
\begin{eqnarray}
\label{estimc00}
|c|\le \frac{C_2 \mathrm{e}^{{\cal N}u (k+1)}}{|k_0-\lambda|^{r+1}}\mathrm{i}nt_{k_0}^{k+1}|f(s)|\mathrm{e}^{-{\cal N}u
s}\mathrm{d}s={\cal N}u^{-r}\frac{C_2 \mathrm{e}^{{\cal N}u (k+1)}}{|k_0-\lambda|^{r+1}}\|f\|_{m,{\cal N}u}
\mathrm{e}nd{eqnarray}
\noindent Let $k'=k+1-\mathrm{e}psilon$. For some
$C_3=C_3(a)\le \mathrm{e}xp\left[(k_0+1)|k_0-\lambda|^{-1}\right]$ we have
\begin{multline}
\label{normk}
\|g_2\|_{m,{\cal N}u,k}={\cal N}u^r \mathrm{i}nt_{k'}^{k+1}{\mathrm{e}^{-{\cal N}u
x}}{|x-\lambda|^{r-1}}\mathrm{d}x\le {C_3{\cal N}u^{r-1}}{|k_0-\lambda|^{r-1}}\mathrm{e}^{-{\cal N}u
k'}\cr
\Rightarrow c\,r\,\|g_2\|_{m,{\cal N}u,k}\le
{\cal N}u^{-r}{mk}\frac{\mathrm{e}^{{\cal N}u (k+1)}}{|k_0-\lambda|^{r+1}}\|f\|_{m,{\cal N}u}
{C_2C_3{\cal N}u^{r-1}}{|k_0-\lambda|^{r-1}}\mathrm{e}^{-{\cal N}u
k'}\cr=\frac{C_4 mk
\mathrm{e}^{{\cal N}u\mathrm{e}psilon}}{{\cal N}u|k_0-\lambda|^2}\|f\|_{m,{\cal N}u}
\mathrm{e}nd{multline}
\noindent For $n\ge k+1$ we write (\ref{defDelI}) as
\begin{eqnarray}
\label{deliprel3}
\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta_n(g_2^{(r)})=\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,n+1]}\mathcal{P}^{m}\left(\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty)}\mathcal{P}^{m(n-k-1)}g_2\right)
\mathrm{e}nd{eqnarray}
\noindent (cf. (\ref{defDelI})). For $\lambda$ complex we take $K_1(a)=\sup_{s\ge k_0} (s-\lambda)^{-1}
(s+1+|\lambda|)$. We let
$\tilde{\lambda}=
\lambda$ if $\lambda$ is real and
$\tilde{\lambda}=-1-|\lambda|$ otherwise and write $q=m(n-k-1)$. Noting that $K_1^{mk_0+m-1}\le C_5(a,b)=K$ we have
\begin{eqnarray}
\label{primasconv}
&&\Gamma(q)\mathcal{P}^{q}|g_2|\le K\mathrm{i}nt_{k'}^{x}{(x-s)^{q-1}}(s-\tilde{\lambda})^{r-1}\mathrm{d}s\cr&&
\le K\mathrm{i}nt_0^x{(x-s)^{q-1}}(s-\tilde{\lambda})^{r-1}\mathrm{d}s
=\frac{(x-\tilde{\lambda})^{q+r-1}\Gamma(q)\Gamma(r)}{\Gamma(q+r)}
\mathrm{e}nd{eqnarray}
\noindent The estimate above is true for $\tilde{\lambda}\le 1$ but
is ``optimal'' only when
the maximum of the integrand is inside the region
of integration i.e. when
$\tilde{\lambda}>-(2+m^{-1})k(n-k)^{-1}+m^{-1}$. If this is
not the case
we prefer to
simply estimate the integral in terms of the maximum of the integrand
over the region of integration. So for, say, $\tilde{\lambda}<-3k$
we use the inequality
\begin{eqnarray}
\label{lambdareal6}
&&\Gamma(q)\mathcal{P}^{q}|g_2|\le K(x-k')^{q}(k'-\tilde{\lambda})^{r-1}
\mathrm{e}nd{eqnarray}
\noindent Now, for $\tilde{\lambda}>-3k$, using (\ref{primasconv})
and (\ref{deliprel3})
\begin{eqnarray}
\label{lambdareal3}
&&\mathcal{P}^{m}\!\left(\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[n,\mathrm{i}nfty)}\mathcal{P}^{m(n-k-1)}|g_2|\right)
\le\frac{\Gamma(r)}{\Gamma(q+r)\Gamma(m)} \mathrm{i}nt_n^x
(x-s)^{m-1}(s-\tilde{\lambda})^{q+r-1}\mathrm{d}s\cr&&\le
\frac{\Gamma(r)}{\Gamma(q+r)\Gamma(m)}(x-n)^m(x-\tilde{\lambda})^{q+r-1}
\cr&&
\mathrm{e}nd{eqnarray}
\noindent (as $m$ is fixed we do not
lose too much by this evaluation which has the advantage
of preserving the behavior near $x=n$). Further, we have
\begin{multline}
\label{evalsum0}
\mathrm{i}nt_{n}^{n+1}\mathrm{e}^{-{\cal N}u x}(x-n)^m(x-\tilde{\lambda})^{q+r-1}\mathrm{d}x
\le (n+1-\tilde{\lambda})^{q+r-1}\mathrm{i}nt_{n}^{n+1}
\mathrm{e}^{-{\cal N}u x}(x-n)^m \mathrm{d}x\cr
\le\frac{\Gamma(m)}{{\cal N}u^m} (n+1-\tilde{\lambda})^{q+r-1}
\mathrm{e}nd{multline}
\noindent and
\begin{eqnarray}
\label{evalsum1}
&&\sum_{n=k+1}^{\mathrm{i}nfty}{\cal N}u^{mn} \|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta\|_{m,{\cal N}u}\le
\Gamma(mk)K\sum_{n=k+1}^{\mathrm{i}nfty}\frac{{\cal N}u^{m(n-1)} \mathrm{e}^{-{\cal N}u
n}(n+1-\tilde{\lambda})^{mn-1-m}}{\Gamma(m(n-1))}\cr&&
\mathrm{e}nd{eqnarray}
\noindent The ratio of two successive terms $s_{n+1}/s_n$ of
the infinite series above is estimated by:
\begin{eqnarray}
\label{evalsum2}
{\cal N}u^m
\mathrm{e}^{-{\cal N}u}\mathrm{e}^{\frac{mn-1-m}{n+1-\tilde{\lambda}}}\left(\frac{n+2-\tilde{\lambda}}{mn-m}\right)^m
\le \frac{1}{2}
\mathrm{e}nd{eqnarray}
\noindent when ${\cal N}u>C_1$ for some $C_1$ independent of $k,n,\tilde{\lambda}$ in the
region $k>1,n>k,\tilde{\lambda}\mathrm{i}n(-3k,1)$.
This means that
\begin{eqnarray}
\label{evalsum3}
&&\sum_{n=k+1}^{\mathrm{i}nfty}{\cal N}u^{mn} \|\raisebox{1ex}{\small e}\mathcal{D}'_{m,\nu}elta\|_{m,{\cal N}u}\le
2K{{\cal N}u^{mk} \mathrm{e}^{-{\cal N}u
(k+1)}(k+2-\tilde{\lambda})^{mk-1}}\cr&&
\mathrm{e}nd{eqnarray}
\noindent and combining with (\ref{estimc00}) and (\ref{est11})
we have
\begin{multline}
\label{fiest1}
2\|f_2\|\le
4K{\cal N}u^{-mk}\frac{\mathrm{e}^{{\cal N}u (k+1)}}{(k-\tilde{\lambda})^{mk+1}}\|f\|_{m,{\cal N}u}
{{\cal N}u^{mk} \mathrm{e}^{-{\cal N}u
(k+1)}(k+2-\tilde{\lambda})^{mk-1}}\cr\le
4K\mathrm{e}^{2\frac{(mk+1)}{k-\tilde{\lambda}}}\frac{\|f\|_{m,{\cal N}u}}{(k-\tilde{\lambda})^2}\le
4K\mathrm{e}^{\frac{4m}{k_0\tilde{\lambda}}}\frac{\|f\|_{m,{\cal N}u}}{(k_0-\tilde{\lambda})^2}\le\frac{C_6(a,b)\|f\|_{m,{\cal N}u}}{|k_0-\lambda|^2}
\mathrm{e}nd{multline}
\noindent If we started with (\ref{lambdareal6}) we would have obtained in the same way,
for $\tilde{\lambda}<-3k$,
instead of (\ref{lambdareal3}),
\begin{eqnarray}
\label{lambdareal10}
\frac{1}{\Gamma(q)\Gamma(m)}(x-n)^m(x-k')^{q}(k'-\tilde{\lambda})^{r-1}
\mathrm{e}nd{eqnarray}
\noindent and the calculations
are similar from this point on. Condition (\ref{evalsum2}) is of the same type, with $k'$ replacing
$-\tilde{\lambda}$ and final estimate is
\begin{eqnarray}
\label{fiesta10}
C_7 {\|f\|_{m,{\cal N}u}}
\frac{(k'-\tilde{\lambda})^{r-1}}{(k-\tilde{\lambda})^{r+1}}
\le C_8 \frac{\|f\|_{m,{\cal N}u}}{(k_0-\tilde{\lambda})^2}
\mathrm{e}nd{eqnarray}
\noindent Finally we take the limit $\mathrm{e}psilon\rightarrow 0$
and noting that $K\rightarrow 0$ as $\lambda\rightarrow\mathrm{i}nfty$,
(i) is proved.
For (ii), merely notice that $U(\lambda_2)-U(\lambda_1)
=(\lambda_2-\lambda_1)U(\lambda_1)U(\lambda_2)$.
{
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\begin{Remark}\label{density}
Let $\psi\mathrm{i}n{L^1}[0,1]$ with the property
$\mathrm{i}nt_0^1\psi(t)\phi^{(m)}(t)\mathrm{d}t=0$ for all $\phi\mathrm{i}n\mathcal{D}(0,1)$.
Then $\psi$ is a polynomial of degree at most $m-1$.
\mathrm{e}nd{Remark}
\noindent This is a well-known property. We sketch an elementary proof for
$m=1$ (for general $m$ the proof is similar). Let $x\mathrm{i}n(0,1)$, and
consider a sequence $\chi_n$ in $ \mathcal{D}(0,x)$
${L^1}$--convergent to $\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,x]}$. Then
$\phi_n(t):=\chi_n(t)-\kappa^{-1}\chi_{n}(\kappa(1-t))$ with
$\kappa=x(1-x)^{-1}$ converges to $\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,x]}-\kappa\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[x,1]}$.
Furthermore, since $\mathrm{i}nt_0^1\phi_n(t)\mathrm{d}t=0$ we have
$\Phi_n(t):=\mathrm{i}nt_0^t\phi_n(s)\mathrm{d}s\mathrm{i}n\mathcal{D}(0,1)$. Since
$\Phi_n'+\kappa^{-1}\rightarrow (1+\kappa^{-1})\mbox{\raisebox{.4ex}{\begin{Large}$\chi$\end{Large}}}_{[0,x]}$ it
follows that $\mathrm{i}nt_0^x(\psi-C)=0$, where $C=\mathrm{i}nt_0^1\psi(t)\mathrm{d}t$. Thus
$\psi=C\ a.e.$
\subsection{Derivation of the equations for the transseries.}
\label{sec:For}
Consider first the scalar equation
\begin{eqnarray}
\label{eqscal0}
y'=f_0(x)-\lambda y-{x}^{-1}By+g(x,y)=
-y+x^{-1}By+\sum_{k=1}^{\mathrm{i}nfty}g_k(x)y^k
\mathrm{e}nd{eqnarray}
\noindent For $x\rightarrow+\mathrm{i}nfty$ we take
\begin{eqnarray}\label{tr1}
{y}=\sum_{k=0}^{\mathrm{i}nfty}{y}_k \mathrm{e}^{-kx}
\mathrm{e}nd{eqnarray}
\noindent where ${y}_k$ will be either formal series
$x^{-s_k}\sum_{n=0}^{\mathrm{i}nfty}a_{kn}x^{-n}$, with $a_{k,0}{\cal N}e 0$ or
actual functions with the condition that (\ref{tr1}) converges uniformly. As a
transseries, (\ref{tr1}) can be also understood as a well ordered
double sequence $t_{kn}=x^{p_{kn}}\mathrm{e}^{-kx}$, with $p_{k\,n+1}<p_{kn}$.
(The order relation is $x^{p}\mathrm{e}^{-kx}\gg x^{p'}\mathrm{e}^{-k'x}
$ as $x\rightarrow+\mathrm{i}nfty$ iff $k<k'$ or
$k=k'$ and $p>p'$; thus a strictly {\mathrm{e}m increasing} sequence of
terms of a transseries necessarily terminates.) Power series are a
special case of transseries, with $y_1=y_2=\ldots=0$. Two transseries
$\sum_{k=0}^{\mathrm{i}nfty}y_k\mathrm{e}^{-kx}$ coincide iff all corresponding
component power series $y_k$ coincide. Transseries of this type are
closed under addition, multiplication and infinite sums of the form
involved in (\ref{eqscal0}) (this last aspect will become clear in the
calculation leading to (\ref{eqcompl}) below). Note that well-ordering
plays an important part in defining multiplication of transseries; in
contrast, for the unrestricted formal expansion
$S=\sum_{k=-\mathrm{i}nfty}^{\mathrm{i}nfty}x^k$, no immediate meaning can be given to
$S^2$. Let $y_0$ be the first term in (\ref{tr1}) and $\delta=y-y_0$.
We have
\begin{multline}
\label{part00}
y^k-y_0^k-ky_0^{k-1}\delta=\sum_{j=2}^k\binom{k}{j}y_0^{k-j}\delta^j=
\sum_{j=2}^k\binom{k}{j}y_0^{k-j}\sum_{i_1,\ldots,i_j=1}^{\mathrm{i}nfty}
\prod_{s=1}^j\left(y_{i_s}\mathrm{e}^{-i_s x}\right)\cr
=\sum_{m=1}^{\mathrm{i}nfty}\mathrm{e}^{-mx}\sum_{j=2}^k\binom{k}{j}y_0^{k-j}\sum_{(i_s)}^{(m;j)}
\prod_{s=1}^j y_{i_s}
\mathrm{e}nd{multline}
\noindent where $\sum_{(i_s)}^{(m;j)}$ means the sum over
all positive integers $i_1,i_2,\ldots,i_j$ with the restriction
$i_1+i_2+\cdots+i_j=m$. Let $d_1=\sum_{k\ge 1}k g_k y_0^{k-1}$.
Introducing $y=y_0+\delta$ in (\ref{eqscal0})
and equating the coefficients of $\mathrm{e}^{-lx}$ we get, by separating
the terms containing $y_l$ for $l\ge 1$ and
interchanging the $j,k$ orders of summation,
\begin{multline}
\label{eqcompl}
y_l'+(\lambda(1-l)+x^{-1}B-d_1(x))y_l=\sum_{j=2}^{\mathrm{i}nfty}\sum_{(i_s)}^{(l;j)}
\prod_{s=1}^j y_{i_s}\sum_{k\ge\{2,j\}}\binom{k}{j} g_k
y_0^{k-j}
\cr=\sum_{j=2}^{l}\sum_{(i_s)}^{(l;j)}
\prod_{s=1}^j y_{i_s}\sum_{k\ge\{2,j\}}\binom{k}{ j} g_k
y_0^{k-j}=:\sum_{j=2}^{l}d_j(x)\sum_{(i_s)}^{(l;j)}
\prod_{s=1}^j y_{i_s}
\mathrm{e}nd{multline}
\noindent where for the middle equality we note that the infinite sum
terminates because $i_s\ge 1$ and $\sum_{s=1}^j i_s=l$. The fact
mentioned before that $\sum_{k=1}^{\mathrm{i}nfty}g_k(x)y_k$ is well defined
when $y_k$ are formal series is now visible: collecting the
coefficient of $x^{p}\mathrm{e}^{-kx}$, only \mathrm{e}mph{finite} sums of coefficients
appear.
For a vectorial equation like (\ref{eqor}) we first write
\begin{eqnarray}
\label{eqsvec0}
{\bf y}'={\bf f}_0(x)
-{\hat{\Lambda}}{\bf y}+{x}^{-1}{{\hat{B}}}{\bf y}+\sum_{{\bf k}\succ 0}{\bf g}_{\bf k}(x){\bf y}^{\bf k}
\mathrm{e}nd{eqnarray}
\noindent with ${\bf y}^{{\bf k}}:=\prod_{i=1}^{n_1} ({\bf y})_i^{k_i}$.
The formal operations
and ordering extend naturally to the vectorial
general transseries (\ref{eqformgen,n}), under
the restriction $\Re({\bf k}\cdot{\boldsymbol \lambda} x)>0$
As with (\ref{eqcompl}), we introduce the transseries
(\ref{eqformgen,n}) in (\ref{eqsvec0}) and equate the
coefficients of $\mathrm{e}xp(-{\bf k}\cdot{\boldsymbol \lambda} x)$. Let
$\mathbf{v}_{\bf k}=x^{-{\bf k}\cdot{\bf m}}\mathbf{y}_{\bf k}$ and
\begin{eqnarray}
\label{defD}
\mathbf{D}d_{\bf j}(x)=\sum_{{\bf l}\ge {\bf j}}\binom{{\bf l}}{
{\bf j}}\mathbf{g}_{\bf l}(x){\bf v}_0^{{\bf l}-{\bf j}}
\mathrm{e}nd{eqnarray}
\noindent Noting that,
by assumption, ${\bf k}\cdot{\boldsymbol \lambda}={\bf k}'\cdot{\boldsymbol \lambda}\Leftrightarrow
{\bf k}={\bf k}'$ we obtain, for ${\bf k}\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^{n_1}$, ${\bf k}\succ 0$
\begin{eqnarray}
\label{eqmygen}
&&{\bf v}_{\bf k} '+\left({\hat{\Lambda}}-{\bf k}\cdot{\boldsymbol \lambda} \hat{I}
+x^{-1}{\hat{B}}\right){\bf v}_{\bf k}+\sum_{|{\bf j}|=1}\mathbf{D}d_{{\bf j}}(x)({\bf v}_{\bf k})^{{\bf j}}\cr&& = \sum_{\stackrel{\scriptstyle \bf \phantom{0}j\le
k}{|{\bf j}|\ge 2}}\mathbf{D}d_{\bf j} (x)\sum_{({\bf I}i_{mp}:{\bf k})}
\prod_{m=1}^n\prod_{p=1}^{j_m}
\left( {\bf v}_{{\bf I}i_{mp}}\right)_m=\mathbf{t}_{\bf k}({\bf v})
\mathrm{e}nd{eqnarray}
\noindent where $\binom{\bf l}{j}= \prod_{j=1}^n\binom{l_i}{j_i}$, $({\bf v})_m$
means the component $m$ of ${\bf v}$, and $\sum_{({\bf I}i_{mp}:{\bf k})}$
stands for the sum over all vectors ${\bf I}i_{mp}\mathrm{i}n\hat{B}ox{\it I\hskip -2pt N}N^n$, with $p\le
j_m,m\le n$, such that ${\bf I}i_{mp}\succ 0$ and
$\sum_{m=1}^n\sum_{p=1}^{j_m}{\bf I}i_{mp}={\bf k}$. We use the convention
$\prod_{\mathrm{e}mptyset} =1, \sum_{\mathrm{e}mptyset}=0$. With
$m_i=1-\lfloor\beta_i\rfloor$ we obtain for ${\bf y}_{\bf k}$
\begin{gather}
\label{homogeqv}
{\bf y}_{\bf k} '+\left({\hat{\Lambda}}-{\bf k}\cdot{\boldsymbol \lambda} \hat{I}
+x^{-1}({\hat{B}}+{\bf k}\cdot{\bf m})\right){\bf y}_{\bf k}+\sum_{|{\bf j}|=1}\mathbf{D}d_{{\bf j}}(x)({\bf y}_{\bf k})^{{\bf j}}
= {\bf t}_{\bf k}({\bf y})
\mathrm{e}nd{gather}
There are clearly finitely many
terms in $\mathbf{t}_{\bf k}({\bf y})$. To find a (not too unrealistic) upper
bound for this number of terms, we compare with
$\sum_{({\bf I}i_{mp})'}$ which stands for the same as
$\sum_{({\bf I}i_{mp})}$ except with ${\bf I}i\ge 0$ instead of
${\bf I}i\succ 0$. Noting that $\binom{k+s-1}{ s-1}=\sum_{a_1+\ldots+a_s=k} 1$
is the number of ways $k$ can be written as a sum of $s$ integers,
we have
\begin{gather}
\label{combineq}
\sum_{({\bf I}i_{mp})}1\le \sum_{({\bf I}i_{mp})'} 1
=\prod_{l=1}^{n_1}\sum_{({\bf I}i_{mp})_l}1=
\prod_{l=1}^{n_1}\binom{k_l+|{\bf j}|-1}{|{\bf j}|-1}\le \binom{|{\bf k}|+|{\bf j}|-1}{
|{\bf j}|-1}^{n_1}
\mathrm{e}nd{gather}
\begin{Remark}\label{homogstruct}
Equation (\ref{eqmygen}) can be written in the form (\ref{homogeq})
\mathrm{e}nd{Remark}
{\mathrm{e}m Proof.} The fact that only predecessors of ${\bf k}$ are involved
in ${\bf t}({\bf y}_0,\cdot)$ and the homogeneity property of
${\bf t}({\bf y}_0,\cdot)$ follow immediately by combining the conditions
$\sum {{\bf I}i_{mp}}={\bf k}$ and ${\bf I}i_{mp}\succ 0$. {
\hat{B}ox{\enspace${\mathchoice\sqr54\sqr54\sqr{4.1}3\sqr{3.5}3}$}}
\noindent The formal inverse Laplace transform of (\ref{homogeqv}) is
then
\begin{eqnarray}
\label{invlapvk11}
&&\left(-p+\hat{\Lambda}-{\bf k}\cdot{\boldsymbol \lambda}\right)\bfY_{\bf k}
+\left(\hat{B}+{\bf k}\cdot{\bf m}\right)
\mathcal{P}\bfY_{\bf k}+\sum_{|{\bf j}|=1}\mathbf{D}_{\bf j}*\left(\bfY_{\bf k}\right)^{\bf j}
={\bf T}_{\bf k}(\bfY)\cr&&
\mathrm{e}nd{eqnarray}
\noindent with
\begin{equation}
\label{defT}
{\bf T}_{\bf k}(\bfY)={\bf T}\left(\bfY_0,\{\bfY_{{\bf k}'}\}_{0\prec{\bf k}'\prec{\bf k}}\right)=
\sum_{{\bf j}\le k;\ |{\bf j}|>1}\mathbf{D}_{\bf j}(p)*\sum_{({\bf I}i_{mp};{\bf k})}
\sideset{^*}{}\prod_{m=1}^{n_1}\sideset{^*}{}\prod_{p=1}^{j_m}\left(\bfY_{{\bf I}i_{mp}}\right)_m
\mathrm{e}nd{equation}
\noindent and
\begin{gather}
\label{definitionDj}
\mathbf{D}_\mathbf{j}=\sum_{{\bf l}\ge{\bf m}}\binom{{\bf l}}{{\bf m}}\mathbf{G}_{\bf l}*
\mathbf{Y}_0^{*({\bf l}-{\bf m})}+\sum_{{\bf l}\ge{\bf m};|{\bf l}|\ge
2}
\binom{{\bf l}}{{\bf m}}\mathbf{g}_{0,{\bf l}}\mathbf{Y}_0^{*({\bf l}-{\bf m})}
\mathrm{e}nd{gather}
\begin{subsection}{Useful formulas}
\label{usefulfor}
A straightforward computation shows that
\begin{eqnarray}\label{u1}{\cal B}(\frac{1}{x^n})=\frac{p^{n-1}}{\Gamma(n)}
\ \mbox{or}\ {\cal L}(p^{n})=\frac{\Gamma(n+1)}{x^{n+1}}\mathrm{e}nd{eqnarray}
\begin{eqnarray}\label{u3}p^q*p^r=\frac{\Gamma(q+1)\Gamma(r+1)}{\Gamma(q+r+2)}
p^{q+r+1}\mathrm{e}nd{eqnarray}
Also, with $f_{1,2}(p):=p\mapsto{\cal H}(p-k_{1,2})g_{1,2}(p-k_{1,2})$
we have
\begin{eqnarray}\label{u4}\Big(f_1*f_2\Big)(p)={\cal H}(p-k_1-k_2)\Big(g_1*g_2\Big)(p-k_1-k_2)\mathrm{e}nd{eqnarray}
\mathrm{e}nd{subsection}
\begin{section}{Acknowledgments}
Special thanks are due to Professors Martin Kruskal and Jean
\'Ecalle for their many in-depth comments, and to Professor B. L. J.
Braaksma for carefully reading the manuscript and pointing out an
error in a formula. The author is grateful to Professors Michael
Berry and Percy Deift for very interesting discussions.
\mathrm{e}nd{section}
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\mathrm{e}nd{document} | math | 197,707 |
\begin{document}
\title{Adiabatic geometric phases in hydrogenlike atoms}
\author{Erik Sj\"oqvist$^{1}$\footnote{Electronic address:
[email protected]}, X. X. Yi$^{2}$\footnote{Electronic address:
[email protected]}, and Johan {\AA}berg$^{1}$\footnote{Electronic address:
[email protected]}}
\affiliation{$^{1}$Department of Quantum Chemistry, Uppsala University,
Box 518, Se-751 20 Uppsala, Sweden \\
$^{2}$Department of Physics, Dalian University of Technology,
Dalian 116024, China}
\date{\today}
\begin{abstract}
We examine the effect of spin-orbit coupling on geometric phases in
hydrogenlike atoms exposed to a slowly varying magnetic field. The
marginal geometric phases associated with the orbital angular momentum
and the intrinsic spin fulfill a sum rule that explicitly relates them
to the corresponding geometric phase of the whole system. The marginal
geometric phases in the Zeeman and Paschen-Back limit are analyzed.
We point out the existence of nodal points in the marginal phases
that may be detected by topological means.
\end{abstract}
\pacs{03.65.Vf}
\maketitle
Imagine a quantal spin evolving under influence of a magnetic field so
that the initial and final states of the spin coincide. Cyclic
evolution of this kind results in a phase factor divisible into a
dynamic part and a part that only depends upon the global geometry
associated with the evolution of the spin. The latter is the geometric
phase, first delineated by Berry \cite{berry84} in the adiabatic
case. This adiabatic geometric phase is proportional to the solid
angle enclosed by the direction of a slowly changing magnetic field
and where the proportionality factor is given by the spin projection
quantum number. The geometric phase structure for this system
resembles exactly that of a charged particle in a magnetic monopole
field. A similar result in the special case of spin$-\frac{1}{2}$ was
subsequently found for nonadiabatic evolution
\cite{aharonov87}, in case of which the solid angle is the area
enclosed on the Bloch sphere.
These results opened up the possibility to study magnetic monopole
structures in the laboratory; a fact that has triggered considerable
interest in the geometric phase for quantal systems carrying angular
momentum. Extensions of the spin-monopole problem to systems
consisting of several coupled angular momenta have been theoretically
put forward
\cite{tang95,sjoqvist00a,ekert00,tong03a,bertlmann04,carollo05,kay05}
and experimentally implemented \cite{jones00,du03}. In particular, the
issue concerning the relation between the overall geometric phase and
the geometric phases of the subsystems has been addressed for
adiabatically evolving pairs of uniaxially coupled spin$-\frac{1}{2}$
\cite{yi04a,yi04b}.
Coupling generally leads to entangled multiparticle systems, which
implies that the marginal states of the concomitant subsystems are
mixed. Geometric phases for mixed states take the form of weighted
averages of geometric phase factors, with weight factors given by the
time-independent \cite{sjoqvist00b} or time-dependent \cite{tong04}
eigenvalues of the corresponding marginal states.
In this paper, we address the issue of coupled angular momenta in
terms of hydrogenlike atoms coupled to a slowly varying
magnetic field. For such systems, there is a natural bipartite
decomposition of the total angular momentum into two subsystems
consisting of the orbital part (L) and the intrinsic spin part (S),
exposed to spin-orbit (LS) coupling. We wish to examine the effect of
the LS coupling on the overall geometric phase as well as on those
pertaining to the two subsystems.
Consider a hydrogenlike atom driven by a uniform magnetic field ${\bf
B} = B_0 {\bf n}$ with $B_0$ the nonzero magnetic field strength and
${\bf n} = (\sin \theta \cos \phi , \sin \theta \sin \phi,\cos\theta)$,
$\theta$ and $\phi$ being slowly varying parameters. Let ${\bf L}$ and
${\bf S}$ be the orbital angular momentum and intrinsic spin, respectively.
The spin-orbit Hamiltonian reads \cite{sakurai94}
\begin{eqnarray}
H_{\bf n} & = & g {\bf n} \cdot ({\bf L} + 2{\bf S}) +
2{\bf L} \cdot {\bf S}
\nonumber \\
& = & U_L (\theta,\phi) U_S (\theta,\phi) H_z
U_L^{\dagger} (\theta,\phi) U_S^{\dagger} (\theta,\phi)
\nonumber \\
& = & U_J (\theta,\phi) H_z
U_J^{\dagger} (\theta,\phi) ,
\label{eq:soham}
\end{eqnarray}
where we may choose ($\hbar = 1$ from now on)
\begin{eqnarray}
U_X (\theta,\phi) & = &
e^{-i\phi X_z} e^{-i\theta X_y} e^{i\phi X_z} , \ X=L,S,J ,
\end{eqnarray}
${\bf J} = {\bf L} + {\bf S}$ being the total angular momentum. Here,
$g$ is the Zeeman-LS strength ratio, $H_z$ is the Hamiltonian at the
north pole ${\bf n}=(0,0,1)$ of the parameter sphere, and
$[H_z,J_z]=0$; the latter implying that the eigenvectors of $H_z$
can be labeled by the eigenvalues $\mu$ of $J_z$.
$H_z$ is block-diagonalizable in one- and two-dimensional blocks with
respect to the product basis with elements $\ket{l,m}\ket{\frac{1}{2},
\pm \frac{1}{2}} \equiv \ket{l,m} \ket{\pm}$ being the common
eigenvectors of ${\bf L}^2,L_z,{\bf S}^2,S_z$. Each block may be
labeled by the eigenvalue $\mu=-l-\frac{1}{2},-l+\frac{1}{2},
\ldots, l+\frac{1}{2}$ of $J_z$. The two extremal subspaces characterized
by $|\mu|=l+\frac{1}{2} \equiv \mu_{\textrm{\scriptsize{e}}}$ are
one-dimensional corresponding to the two product vectors
\begin{eqnarray}
\ket{\psi^{(l;\pm \mu_{\textrm{\scriptsize{e}}}})} =
\ket{l,\pm l} \ket{\pm}.
\label{eq:extevect}
\end{eqnarray}
The remaining blocks are two-dimensional, each of which corresponding
to the vectors $\ket{l,m=\mu-\frac{1}{2}} \ket{+}, \ket{l,m+1=
\mu+\frac{1}{2}} \ket{-}$, $|\mu| < l+\frac{1}{2}$. For each
such $\mu$, the corresponding two-dimensional Hamiltonian
suboperator has the form
\begin{eqnarray}
H_z^{(l;\mu)} & = & E^{(l;\mu)} I^{(l;\mu)} + \Delta E^{(l;\mu)}
\left( \sin \alpha^{(l;\mu)} \ \sigma_x^{(l;\mu)} \right.
\nonumber \\
& & \left. + \cos \alpha^{(l;\mu)} \ \sigma_z^{(l;\mu)} \right) ,
\end{eqnarray}
with $I^{(l;\mu)}$, $\sigma_x^{(l;\mu)}$, and $\sigma_z^{(l;\mu)}$ the
standard unit and Pauli operators acting on the relevant subspace.
Furthermore
\begin{eqnarray}
E^{(l;\mu)} & = & g\mu - \frac{1}{2},
\nonumber \\
\Delta E^{(l;\mu)} & = & \frac{1}{2}
\sqrt{g^2 + 4g \mu +
\big( 2l+1 \big)^2} ,
\nonumber \\
\cos \alpha^{(l;\mu)} & = &
\frac{2\mu + g}{\sqrt{g^2 + 4g \mu + \big( 2l+1 \big)^2}} ,
\end{eqnarray}
in terms of which the energy eigenvalues read $E_{\pm}^{(l;\mu)} =
E^{(l;\mu)} \pm \Delta E^{(l;\mu)}$ with corresponding entangled
eigenvectors
\begin{eqnarray}
\ket{\psi_+^{(l;\mu)}} & = &
\cos \big( \frac{1}{2} \alpha^{(l;\mu)} \big) \
\ket{l,\mu-\frac{1}{2}} \ket{+}
\nonumber \\
& & +
\sin \big( \frac{1}{2} \alpha^{(l;\mu)} \big) \
\ket{l,\mu+\frac{1}{2}} \ket{-} ,
\nonumber \\
\ket{\psi_-^{(l;\mu)}} & = &
-\sin \big( \frac{1}{2} \alpha^{(l;\mu)} \big) \
\ket{l,\mu-\frac{1}{2}} \ket{+}
\nonumber \\
& & +
\cos \big( \frac{1}{2} \alpha^{(l;\mu)} \big) \
\ket{l,\mu+\frac{1}{2}} \ket{-} .
\label{eq:evect}
\end{eqnarray}
The $g$-dependence of the eigenvectors is due to the fact that
the Zeeman and LS term do not commute.
With the above choice of rotation operators, we have
$U_X (0,\phi) = I$ and $U_X (\pi,\phi) = e^{-i\pi X_y} e^{2i\phi
X_z}$. This entails that the corresponding energy eigenvectors cannot
be unique simultaneously at the north and south pole. For example, by
choosing the phase of the eigenvectors $\ket{\psi^{(l;\mu)}}$ of $H_z$
to be independent of $\phi$, as in Eqs. (\ref{eq:extevect}) and
(\ref{eq:evect}), the resulting instantaneous eigenvectors $U_X
(\theta,\phi) \ket{\psi^{(l;\mu)}}$ are unique at the north pole but
yields a singular gauge potential at the south pole. For the same
reference eigenvectors, one may move this singularity to the north pole
by instead choosing the rotation operators $\widetilde{U}_X (\theta,\phi)
= e^{-i\phi X_z} e^{-i\theta X_y} e^{-i\phi X_z}$. On the other
hand, the section \cite{wu75} $\{ U_X (\theta,\phi),\theta \in [0,\pi);
\widetilde{U}_X (\theta,\phi),\theta \in (0,\pi]\}$ is globally
well-defined for any choice of eigenvectors of $H_z$. This
single-valued section captures the monopole structure corresponding to
the $2j+1$ fold degeneracy at $g=0$, $j$ being the eigenvalue of ${\bf
J}^2$.
We now compute the adiabatic geometric phases for the atom and its
subsystems L and S under the assumption $g\neq 0$. Let us start with
the extremal states $\mu = \pm
\mu_{\textrm{\scriptsize{e}}}$. We note that
\begin{eqnarray}
\ket{\psi^{(l;\pm \mu_{\textrm{\scriptsize{e}}})}; \theta,\phi} & = &
U_L (\theta,\phi) \ket{l,\pm l} U_S (\theta,\phi)
\ket{\pm}
\end{eqnarray}
are product eigenvectors of $H_{\bf n}$. Assume that the external
magnetic field slowly traverses a loop $\mathcal{C}$ such that the
adiabatic approximation is valid. Then, the adiabatic geometric phase
becomes
\begin{eqnarray}
\Gamma_J^{(l;\pm \mu_{\textrm{\scriptsize{e}}})} [\mathcal{C}] =
\mp \mu_{\textrm{\scriptsize{e}}} \Omega
\label{eq:cs}
\end{eqnarray}
with $\Omega$ the solid angle enclosed by the loop. From the product
form of the extremal states, we obtain the corresponding marginal
geometric phases for L and S as $\mp l\Omega$ and $\mp \frac{1}{2}
\Omega$, respectively, which are $g$-independent and sum up to
$\Gamma_J^{(l;\pm \mu_{\textrm{\scriptsize{e}}})} [\mathcal{C}]$ since
$\mu_{\textrm{\scriptsize{e}}} = l+\frac{1}{2}$.
Next we compute the adiabatic geometric phases for $|\mu| <
l+\frac{1}{2}$. The eigenvectors of the instantaneous Hamiltonian
$H_{{\bf n}}$ take the form
\begin{eqnarray}
\ket{\psi_{\pm}^{(l;\mu)};\theta,\phi} & = &
U_J(\theta,\phi) \ket{\psi_{\pm}^{(l;\mu)}}
\nonumber \\
& = & U_L(\theta,\phi) U_S(\theta,\phi) \ket{\psi_{\pm}^{(l;\mu)}} .
\end{eqnarray}
We obtain the $g$-independent pure state geometric phase as
\begin{eqnarray}
\Gamma_{J,\pm}^{(l;\mu)} [\mathcal{C}] = -\mu \Omega ,
\label{eq:totalgp}
\end{eqnarray}
which follows directly from the fact that the energy eigenvectors
$\ket{\psi_{\pm}^{(l;\mu)}; \theta,\phi}$ are also eigenvectors of
${\bf n} \cdot {\bf J}$ both with the eigenvalue $\mu$. The marginal
states read
\begin{eqnarray}
\rho_{L,\pm}^{(l;\mu)} (\theta,\phi) & = &
\text{Tr}_S \ket{\psi_{\pm}^{(l;\mu)};\theta,\phi}
\bra{\psi_{\pm}^{(l;\mu)};\theta,\phi}
\nonumber \\
& = & U_L(\theta,\phi) \rho_{L,\pm}^{(l;\mu)}
U_L^{\dagger} (\theta,\phi) ,
\nonumber \\
\rho_{S,\pm}^{(l;\mu)} (\theta,\phi) & = &
\text{Tr}_L \ket{\psi_{\pm}^{(l;m)};\theta,\phi}
\bra{\psi_{\pm}^{(l;\mu)};\theta,\phi}
\nonumber \\
& = & U_S(\theta,\phi) \rho_{S,\pm}^{(l;\mu)}
U_S^{\dagger}(\theta,\phi) ,
\end{eqnarray}
with
\begin{eqnarray}
\rho_{L,\pm}^{(l;\mu)} & = &
\text{Tr}_S \ket{\psi_{\pm}^{(l;\mu)}} \bra{\psi_{\pm}^{(l;\mu)}}
\nonumber \\
& = & \frac{1}{2} \left( 1 \pm \cos \alpha^{(l;\mu)} \right)
\ket{l,\mu-\frac{1}{2}} \bra{l,\mu-\frac{1}{2}}
\nonumber \\
& & + \frac{1}{2} \left( 1 \mp \cos \alpha^{(l;\mu)} \right)
\ket{l,\mu+\frac{1}{2}} \bra{l,\mu+\frac{1}{2}} ,
\nonumber \\
\rho_{S,\pm}^{(l;\mu)} & = &
\text{Tr}_L \ket{\psi_{\pm}^{(l;\mu)}} \bra{\psi_{\pm}^{(l;\mu)}}
\nonumber \\
& = &
\frac{1}{2} \left( 1 \pm \cos \alpha^{(l;\mu)} \right)
\ket{+} \bra{+}
\nonumber \\
& & + \frac{1}{2} \left( 1 \mp \cos \alpha^{(l;\mu)} \right)
\ket{-} \bra{-} .
\end{eqnarray}
Since the marginal density operators $\rho_{L,\pm}^{(l;\mu)}$ and
$\rho_{S,\pm}^{(l;\mu)}$ evolve unitarily under $U_L$ and $U_S$,
respectively, it follows that the marginal geometric phases can
be computed using the approach in Ref. \cite{sjoqvist00b}. Explicitly,
for a magnetic field whose direction traces out a loop $\mathcal{C}$,
this yields
\begin{widetext}
\begin{eqnarray}
\exp \left( i\Gamma_{L,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] \right) & = &
\Phi \Big[ \left( 1 \pm \cos \alpha^{(l;\mu)} \right)
e^{-i(\mu-\frac{1}{2})\Omega} +
\left( 1 \mp \cos \alpha^{(l;\mu)} \right)
e^{-i(\mu+\frac{1}{2})\Omega} \Big]
\nonumber \\
& \Rightarrow & \Gamma_{L,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] =
-\mu \Omega \pm \arctan \left( \cos \alpha^{(l;\mu)}
\tan \frac{\Omega}{2} \right) ,
\nonumber \\
\exp \left( i\Gamma_{S,\pm}^{(l;\mu)}
\big[ \mathcal{C};g \big] \right) & = &
\Phi \Big[ \left( 1 \pm \cos \alpha^{(l;\mu)} \right) e^{-i\Omega/2} +
\left( 1 \mp \cos \alpha^{(l;\mu)} \right) e^{i\Omega/2} \Big]
\nonumber \\
& \Rightarrow & \Gamma_{S,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] =
\mp \arctan \left( \cos \alpha^{(l;\mu)}
\tan \frac{\Omega}{2} \right) .
\label{eq:marginalgp}
\end{eqnarray}
\end{widetext}
Here, $\Phi [z]=z/|z|$ for any nonzero complex number $z$. Notice
that the above marginal geometric phases are $g$-dependent through $\cos
\alpha^{(l;\mu)}$. They obey the symmetry $\Gamma_{X,\pm}^{(l;-\mu)}
\big[ \mathcal{C};-g \big] = -\Gamma_{X,\pm}^{(l;\mu)}
\big[ \mathcal{C};g \big]$, $X=L,S$, which is expected since the change
$(\mu,g,\Omega)\rightarrow (-\mu,-g,\Omega)$ is physically equivalent
to reversing the orientation of the loop $\mathcal{C}$. Furthermore, by
comparing Eqs. (\ref{eq:totalgp}) and (\ref{eq:marginalgp}), it
follows that the marginal geometric phases of the L and S subsystems
fulfill the sum rule
\begin{eqnarray}
\Gamma_{L,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] +
\Gamma_{S,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] =
\Gamma_{J,\pm}^{(l;\mu)} [\mathcal{C}]
\end{eqnarray}
that explicitly relates them to the corresponding geometric phases for
the pure entangled states.
Let us now consider the extreme cases $|g| \gg 1$ (Paschen-Back regime
\cite{remark}) and $0<|g| \ll 1$ (Zeeman regime). In the Paschen-Back
limit, we have $\cos \alpha^{(l;\mu)} \approx 1$, which implies
\begin{eqnarray}
\Gamma_{S,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] & \approx &
\mp \frac{1}{2} \Omega ,
\nonumber \\
\Gamma_{L,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] & \approx &
-\left( \mu \mp \frac{1}{2} \right) \Omega .
\end{eqnarray}
These phases are those of the pure vectors $U_S (\theta,\phi)\ket{\pm}$
and $U_L (\theta,\phi)\ket{l,\mu \pm \frac{1}{2}}$, respectively, as
expected as the LS term in $H_{{\bf n}}$ is negligible in this limit.
The Zeeman condition $0<|g|\ll 1$ yields $\cos \alpha^{(l;\mu)} \approx
\mu /(l+\frac{1}{2})$, leading to
\begin{eqnarray}
\Gamma_{S,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] & \approx & \mp
\arctan \left( \frac{\mu}{l+\frac{1}{2}} \tan \frac{\Omega}{2} \right) ,
\nonumber \\
\Gamma_{L,\pm}^{(l;\mu)} \big[ \mathcal{C};g \big] & \approx & -\mu \Omega
\pm \arctan \left( \frac{\mu}{l+\frac{1}{2}} \tan \frac{\Omega}{2} \right) .
\end{eqnarray}
From this we can conclude that the marginal geometric phases may in
general not be small for $0<g \ll 1$, contrary to the cases discussed
in Refs. \cite{yi04a,yi04b}, where all geometric phases were found to be
quenched in this regime due to the uniaxial coupling term. The reason
for this quenching effect in the uniaxial case is that the coupling
term defines a fixed preferred quantization axis that makes the
eigenstates essentially unaffected by a weak magnetic field. In the LS
case, though, no particular direction in space is singled out by the
spherically symmetric coupling term and the quantization axis of the
instantaneous eigenstates still coincides with the direction of the
applied magnetic field. This feature is true no matter how small $g$
is as long as it is nonzero. On the other hand, if $g=0$, then the
magnetic field decouples from the atom and no change in the atomic
eigenstates can take place when the direction of the magnetic field
varies. Thus, $g=0$ is a singular point in the sense that the
geometric phases become independent of the enclosed solid angle
$\Omega$ of the magnetic field. It should be noted that the same
singular behavior is present for the standard case \cite{berry84} of a
single spin in a slowly rotating magnetic field. In physically
realistic scenarios, though, it is reasonable to expect that the
singularity at $g=0$ becomes invisible as the atom is increasingly
exposed to noise and decoherence effects.
\begin{figure}
\caption{\label{fig:1}
\label{fig:1}
\end{figure}
The geometric phase $\Gamma_{S,-}^{(2;-\frac{1}{2})} \big[
\mathcal{C};g \big]$ of the intrinsic spin as a function of
coupling strength $g$ and solid angle $\Omega$ is shown in
Fig. \ref{fig:1}. Notice in particular that this graph confirms the
expected asymptotic behavior in the Paschen-Back limit $|g| \gg 1$.
The marginal density operators are degenerate when $\cos
\alpha^{(l;\mu)}=0$, which happens along the line $g=-2\mu$ in the
space spanned by $(\Omega,g)$. For $\mu = - \frac{1}{2}$ this line
occurs at $g=1$, as indicated in Fig. \ref{fig:1}. While the cyclic
geometric phase of the whole system is well-defined at these lines,
the corresponding marginal geometric phases are undefined there
\cite{sjoqvist00b}. Furthermore, the visibilities of the subsystems,
defined as \cite{sjoqvist00b}
\begin{eqnarray}
\mathcal{V}_{\pm,S}^{(l;\mu)} \big[ \mathcal{C};g \big] & = &
\mathcal{V}_{\pm,L}^{(l;\mu)} \big[ \mathcal{C};g \big]
\nonumber \\
& \equiv & \frac{1}{2} \left| \left( 1 \pm \cos \alpha^{(l;\mu)} \right)
e^{-i\Omega/2} \right.
\nonumber \\
& & + \left. \left( 1 \mp \cos \alpha^{(l;\mu)} \right)
e^{i\Omega/2} \right| ,
\end{eqnarray}
reduces to
\begin{eqnarray}
\mathcal{V}_{\pm,S}^{(l;\mu)} \big[ \mathcal{C};g \big] =
\mathcal{V}_{\pm,L}^{(l;\mu)} \big[ \mathcal{C};g \big] =
\left| \cos \frac{\Omega}{2} \right|
\end{eqnarray}
at the points where the marginal geometric phases are undefined. The
marginal visibilities vanish at their common nodal points $\Omega =
(2p+1) \pi$, $p$ integer, which is manifested as a jump at
$(\Omega,g)=(\pi,1)$ in Fig. \ref{fig:1}. These nodal points
can be detected topologically by considering loops in the space
spanned by $(\Omega,g)$. By continuously monitoring the marginal
phases along a loop, a resulting $2\pi$ phase shift signals the
existence of such a nodal point \cite{bhandari02}. For example,
as indicated in Fig. \ref{fig:1}, by traversing a loop in the
counterclockwise (clockwise) direction such that it encloses
the singular point at $(\Omega,g)=(\pi,1)$ once, we end up at a
phase shift $2\pi$ ($-2\pi$).
In conclusion, we have computed adiabatic geometric phases in
hydrogenlike atoms coupled to a slowly varying magnetic field. We have
shown that while the total geometric phase is independent of the
strength of the spin-orbit coupling, this is not the case for the
corresponding marginal phases. It turns out, though, that the latter
phases sum up to the pure state geometric phase of the whole
system. Further consideration as to the generality of this sum rule
for other systems of coupled angular momenta seems pertinent. We have
examined the marginal geometric phases in the Zeeman and Paschen-Back
limit. Finally, we have pointed out the
existence of nodal points where the marginal geometric phases become
undefined and we have argued that these points may be detected by
topological means.
\vskip 0.3 cm
X.X.Y. acknowledges financial support by NSF of China (Project
No. 10305002).
\end{document} | math | 19,342 |
\begin{document}
\begin{abstract}
We study the Ginzburg-Landau energy of a superconductor with a
variable magnetic field and a pinning term in a bounded smooth two
dimensional domain $\Omega$. Supposing that the Ginzburg-Landau
parameter and the intensity of the magnetic field are large and of the
same order, we determine an accurate asymptotic formula for the
minimizing energy. This asymptotic formula displays the influence of
the pinning term. Also, we discuss the existence of non-trivial
solutions and prove some asymptotics of the third critical field.
\end{abstract}
\maketitle
\section{Introduction}
We consider a bounded, open and simply connected set
$\Omega\subset\mathbb R^2$ with smooth boundary. We suppose that $\Omega$
models an inhomogeneous superconducting sample submitted to an
applied external magnetic field. The energy of the sample is given
by the so called pinned Ginzburg-Landau functional,
\begin{equation}\label{eq-2D-GLf}
\mathcal E_{\kappa,H,a,B_{0}}(\psi,\mathbf A)= \int_\Omega\left( |(\nabla-i\kappa
H\mathbf A)\psi|^2+\frac{\kappa^2}{2}(a(x,\kappa)-|\psi|^2)^2\right)\,dx
+\kappa^2H^2\int_{\Omega}|\curl\mathbf A-B_0|^2\,dx\,.
\end{equation}
Here $\kappa$ and $H$ are two positive parameters such that $\kappa$ describes the properties of the material, and $H$ measures the variation of the intensity of the applied magnetic field. The modulus $|\psi|^{2}$ of the wave function
(order parameter) $\psi\in H^1(\Omega;\mathbb C)$ measures the density of the superconducting electron Cooper pairs. The magnetic potential
$\mathbf A$ belongs to $H^1_{\Div}(\Omega)$ where
\begin{equation}\label{eq-2D-hs} H^1_{\Div}(\Omega)=\{\mathbf A=(\mathbf A_{1},\mathbf A_{2})\in
H^1(\Omega)^{2}~:~\Div \mathbf A=0~{\rm in}~\Omega \,,\,\mathbf A\cdot\nu=0~{\rm
on}\,
\partial\Omega \,\}\,,
\end{equation}
with $\nu$ being the unit interior normal vector of
$\partial\Omega$.\\
The function $\kappa H\curl\mathbf A$ gives the induced magnetic field.
When $\psi\equiv0$ and $(\psi,\mathbf A)$ is a minimizer or a critical point of the functional, we call this pair normal state. In our case it is easy to see normal minimizers (if any) are necessarily in the form $(0,\mathbf A)$ with $\mathbf A$ in $H^1_{\Div} (\Omega)$ such that $\curl\mathbf A=B_{0}$. This solution is unique
and denoted by $\mathbf F$. A natural question will be to determine under which condition this normal solution is a minimizer.
The function $B_{0}\in C^{\infty}(\overline{\Omega})$ is the intensity of the external magnetic field which is variable in our problem. Let
\begin{equation}\label{gamma}
\Gamma=\{x\in\overline{\Omega}: B_{0}(x)=0\}\,.
\end{equation}
We assume that either $\Gamma$ is empty or that $B_{0}$ satisfies :
\begin{equation}\label{B(x)}
\left\{
\begin{array}{ll}
|B_{0}| + |\nabla B_0 | >0&\mbox{ in } \overline{\Omega}\\
\nabla B_{0}\times\vec{n}\neq 0 &\mbox{ on } \Gamma\cap\partial\Omega\,.
\end{array}
\right.
\end{equation}
The assumption in \eqref{B(x)} implies that for any open set $\omega$ relatively compact in $\Omega$, $\Gamma\cap\omega$ is either empty, or consists of a union of smooth curves.
The energy $\mathcal E_{\kappa,H,a,B_{0}}$ considered here is slightly different from the classical Ginzburg-Landau energy in the sense that there is a varying term denoted by $a(x,\kappa)$ penalizing the variations of the order parameter $\psi$ and called the pinning term. This term arises also naturally in the microscopic derivation of the Ginzburg-Landau theory from BCS theory (see \cite{FHSS})
without any a priori assumption on the sign of $a$.\\
In this paper, we will assume that the pining term $a$ satisfies:
\begin{ass}\label{assumption}
The function $a(x,\kappa)$ is real, defined on $\overline{\Omega}\times[\kappa_{0},+\infty)$, and satisfies for some $\kappa_0 >0$ the following assumptions:
\begin{enumerate}
\item[$(A_{1})$] \begin{equation} \label{a1} \forall \kappa\geq \kappa_0\,, a(\cdot,\kappa)\in C^{1}(\overline{\Omega})\,.\end{equation}
\item[$(A_{2})$]
\begin{equation}\label{a2}
\sup_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}| a(x,\kappa)| <+\infty \,.
\end{equation}
\item[$(A_{3})$]
\begin{equation}\label{a3}
\sup_{{x\in\overline{\Omega},\,\kappa \geq \kappa_{0}}} |\nabla_{x}\,a(x,\kappa)|< +\infty \,.
\end{equation}
\item[$(A_{4})$]
There exists a positive constant $C_{1}$, such that,
\begin{equation}\label{a4}
\forall \kappa\geq \kappa_0\,,\qquad \mathcal{L} \left(\partial\{a(x,\kappa)>0\}\right)\leq C_{1}\,\kappa^{\frac{1}{2}}\,,
\end{equation}
where $\mathcal{L}$ is the "length" of $\partial\{a(x,\kappa)>0\}$ in $\Omega$ in a sense that will be explained in \eqref{defA4}.
\end{enumerate}
\end{ass}
Let us introduce for later use,
\begin{equation}\label{def:L}
L(\kappa) = \sup_x |\nabla_{x}\,a(x,\kappa)| \,,
\end{equation}
\begin{equation}\label{def:sup-a}
\overline{a}=\sup_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}a(x,\kappa)
\end{equation}
and
\begin{equation}\label{def:inf-a}
\underline{a}=\inf_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}a(x,\kappa).
\end{equation}
The assumption in ($A_{3}$) gives a uniform control for any $\kappa$ of the oscillation of $a(.,\kappa)$ which will be made precise later by an assumption on $L(\kappa)$.
Notice that the normal state $(0,\mathbf F)$ is a critical point of the functional in \eqref{eq-2D-GLf}.
It is standard, starting from a minimizing sequence, to prove the existence of minimizers in $ H^1(\Omega;\mathbb C)\times
H^1_{\Div}(\Omega)$ of the functional $\mathcal E_{\kappa,H,a,B_{0}}$. A minimizer $(\psi,\mathbf A)$ of \eqref{eq-2D-GLf} is a weak solution of
the Ginzburg-Landau equations,
\begin{equation}\label{eq-2D-GLeq}
\left\{
\begin{array}{llll}
-(\nabla-i\kappa H\mathbf A)^2\psi=\kappa^2\, (a(x,\kappa)-|\psi|^2)\, \psi&{\rm in}&
\Omega&(a)
\\
-\nabla^{\bot}\curl(\mathbf A-\mathbf F)=\displaystyle\frac1{\kappa
H}\IM(\overline{\psi}\,(\nabla-i\kappa
H\mathbf A)\psi) &{\rm in}&\Omega&(b)\\
\nu\cdot(\nabla-i\kappa H\mathbf A)\psi=0&{\rm
on}&\partial\Omega&(c)\\
\curl\mathbf A=\curl\mathbf F&{\rm on}&\partial\Omega&(d)\,.
\end{array}\right.
\end{equation}
Here, $\curl\mathbf A=\partial_{x_1}\mathbf A_{2}-\partial_{x_2}\mathbf A_{1}$ and
$\nabla^{\bot}\curl\mathbf A=(\partial_{x_2}(\curl\mathbf A),
-\partial_{x_1}(\curl\mathbf A)).$
Let us introduce the magnetic Schr\"odinger operator in an open set $\widetilde{\Omega}$ in $\mathbb R^2$:
\begin{equation}\label{def:P}
P_{A,V}^{\widetilde{\Omega}}=-(\nabla-iA)^{2}+V(x)\,,
\end{equation}
where $A\in H^{1}_{\Div}(\widetilde{\Omega})$ and $V$ is a continuous function bounded from below.\\
The form domain of $P_{A,V}^{\widetilde{\Omega}}$ is
$$
\mathcal{V}(\widetilde{\Omega})=\{u\in L^{2}(\widetilde{\Omega})\,,\quad (\nabla-iA)u\in L^{2}(\widetilde{\Omega})\,,\quad(V+C)^{\frac{1}{2}}u\in L^{2}(\widetilde{\Omega})\}\,,
$$
and its operator domain is given by
$$
D(P_{A,V}^{\widetilde{\Omega}}):=\{u\in\mathcal{V}(\widetilde{\Omega})\,,\quad P_{A,V}^{\widetilde{\Omega}}u\in L^{2}(\widetilde{\Omega}),\quad \nu\cdot(\nabla-i A)u=0~{\rm on}~\partial\widetilde{\Omega}\}\,.
$$
Then, $\eqref{eq-2D-GLeq}_{a,c}$ reads $$P_{A,V}^{\Omega}\,\psi=-\kappa^{2}\,|\psi|^{2}\psi\,,$$ with $A=\kappa H \mathbf A$, $\psi \in D(P_{A,V}^{\Omega})$ and $V=-\kappa^{2}\,a\,$.\\
There are many papers on the Ginzburg-Landau functional with a
pinning term, most of them study the influence of the pinning term
on the location of {\it vortices}, i.e. the zeros of the minimizing
order parameter. For the functional without a magnetic field (i.e.
$B_0=0$ in \eqref{eq-2D-GLf}), the influence of the pinning term is
studied in \cite{LM} and more recently in \cite{DSM} and the
references therein. The pinning term (i.e. the function $a$) in
\cite{LM} is a step function independent of $\kappa$; more
complicated $\kappa$-dependent periodic step functions are
considered in \cite{DSM}. The magnetic version of the functional in
\cite{LM} is studied in \cite{A.K, K-cocv}.
In \cite{ASS}, Aftalion, Sandier and Serfaty considered a {\bf
smooth} and $\kappa$-dependent pinning term $a$ satisfying:
\begin{enumerate}
\item[$(H_{1})$]
$L(\kappa) \ll\kappa H.$
\item[$(H_{2})$] There exist a continuous function $a(x)$, a positive constant $a_{0}$ and, for all $\kappa\geq 0$, there exist two functions $
\sigma(\kappa)=\textit{o}\left(\left(\ln\left|\ln\frac{1}{\kappa}\right|\right)^{-\frac{1}{2}}\right)
$
and $ \beta(x,\kappa)\geq 0$ such that,
$$\displaystyle \min_{B(x,\sigma(\kappa))} \beta(x,\kappa)=0\,, \qquad a(x,\kappa)= a(x)+\beta(x,\kappa)\,,\qquad{\rm and}\qquad 0< a_{0}\leq a(x)\leq 1\,.$$
\end{enumerate}
The study contains the case when $a(x,\kappa)=a(x)$ ($\beta=0$) but
also cases with a $\kappa$- control of the $x$-oscillation of
$\beta(\cdot,\kappa)$ which could increase with $\kappa$. In the
scales of this paper, the results in \cite{ASS} are valid
when the parameter
$H$ is of order $\frac{|\ln\kappa|}{\kappa}$ as $\kappa \longrightarrow +\infty$.
Extending the discussion, the
functional in \eqref{eq-2D-GLf} is close to models of Bose-Einstein
condensates (see e.g. \cite{AfAB, AlBr}).
In this paper, we will analyze how the pinning term appears in
the asymptotics of the energy in the presence of a strong external
variable magnetic field (see Theorem~\ref{thm-2D-main} below). Also,
we discuss the influence of the pinning on the asymptotic expression
of the third critical field $H_{C_3}$ (see Theorems~\ref{thm:HC3}
and \ref{thm:HC3-vr}).
We focus on the regime of large values of $\kappa$, $\kappa\rightarrow+\infty$ and we study the ground state energy defined as follows,
\begin{equation}\label{eq-2D-gs}
\E0(\kappa,H,a,B_{0})=\inf\big\{ \mathcal
E_{\kappa,H,a,B_{0}}(\psi,\mathbf A)~:~(\psi,\mathbf A)\in H^1(\Omega;\mathbb C)\times
H^1_{\Div}(\Omega)\big\}\,.
\end{equation}
More precisely, we give an asymptotic estimate which is valid in the simultaneous limit $\kappa\longrightarrow+\infty$ and $H(\kappa)\longrightarrow+\infty$ with the constraint that $\frac{H(\kappa)}{\kappa}$ remains asymptotically of uniform size, that is satisfying
\begin{equation}\label{cond-H}
\lambda_{\min}\leq \frac{H(\kappa)}{\kappa}\leq\lambda_{\max}\qquad(\kappa\geq\kappa_{0})\,,
\end{equation}
where $\lambda_{\min},\,\lambda_{\rm max}$ are positive constants such that $\lambda_{\min}<\lambda_{\max}$.\\
The behavior of $\E0(\kappa,H,a,B_{0})$ involves a function $\hat{f}:[0,+\infty)\longrightarrow[0,\frac{1}{2}]$ introduced in \cite[Theorem~2.1]{KA2}. The function $\hat{f}$ is increasing, continuous and $\hat{f}(b)=\frac{1}{2}$, for all $b\geq 1$.
\begin{thm}\label{thm-2D-main}
Suppose that Assumption~\ref{assumption} and \eqref{cond-H} hold,
and
\begin{equation}
L(\kappa)= \mathcal O(\kappa^{\frac{1}{2}})\qquad{\rm as}~\kappa \rightarrow +\infty\,.
\end{equation}
The ground state energy in \eqref{eq-2D-gs} satisfies
\begin{multline}\label{eq-2D-thm}
\E0(\kappa,H,a,B_{0})=\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,,\qquad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{multline}
\end{thm}
When $\Omega\cap\{a(x,\kappa)>0\}=\varnothing$, we obtain directly from \eqref{eq-2D-gs}
$$
\mathcal E_{\kappa,H,a,B_{0}}(\psi,\mathbf A)\geq\frac{\kappa^{2}}{2}\int_{\Omega}a(x,\kappa)^{2}\,dx = \mathcal E_{\kappa,H,a,B_{0}}(0,\mathbf F)\,.
$$
Hence the minimizer of $\mathcal E_{\kappa,H,a,B_{0}}$ is the normal state. In physical terms, this case corresponds to the case when we are above the critical temperature.
We will describe later cases when the remainder term in \eqref{eq-2D-thm} is indeed small compared with the leading order term (see Section~\ref{examples}).
The assumptions in Theorem~\ref{thm-2D-main} contain the case when
the function $a$ is constant and equals~$1$, which was proved in
\cite{KA} under Assumption~\eqref{cond-H}.
Along the proof of Theorem~\ref{thm-2D-main}, we obtain an estimate
of the `magnetic energy' as follows:
\begin{corol}\label{corol-2D-main}
Under the assumptions of Theorem~\ref{thm-2D-main}, we have
\begin{equation}
(\kappa H)^2\int_{\Omega}|\curl\mathbf A-B_0|^2\,dx=\textit{o}(\kappa^{2})\,,\qquad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{equation}
\end{corol}
If $\mathcal D$ is a domain in $\Omega$,
we introduce the local energy in $\mathcal D$ of $(\psi,\mathbf A) \in H^1(\Omega;\mathbb C)\times H^1_{\Div}(\Omega)$ by:
\begin{equation}\label{eq-GLe0}
\mathcal E_{0}(\psi,\mathbf A;a,\mathcal{D})=\int_{\mathcal{D}}|(\nabla -i\kappa
H\mathbf A)\psi|^2\,dx+\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a(x,\kappa)-|\psi|^{2})^2\,dx\,.
\end{equation}
The next theorem gives an estimate of the local energy $\mathcal E_{0}(\psi,\mathbf A;a,\mathcal{D})$.
\begin{theorem}\label{lc-en}
Under the assumptions of Theorem~\ref{thm-2D-main}, if $(\psi,\mathbf A)$
is a minimizer of \eqref{eq-2D-GLf} and $\mathcal{D}$ is regular set
such that $\mathcal{\overline{D}}\subset\Omega$, then
\begin{multline}\label{eq-lc-en}
\mathcal E_{0}(\psi,\mathbf A;a,\mathcal{D})=\kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,,\qquad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{multline}
\end{theorem}
Theorem~\ref{lc-en} will be useful in the proof of the next theorem which gives the asymptotic behavior of the order parameter $\psi$, when $(\psi,\mathbf A)$ is a global minimizer.
\begin{theorem}\label{est-psi-main}
Under the assumptions of Theorem~\ref{thm-2D-main}, if $(\psi,\mathbf A)$ is a
minimizer of \eqref{eq-2D-GLf} and $\mathcal{D}$ is a regular set such that $\overline{\mathcal{D}}\subset \Omega$, then
\begin{equation}\label{est-psi-D}
\int_{\mathcal{D}}|\psi(x)|^{4}\,dx=-\int_{\mathcal{D} \cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,,\quad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{equation}
\end{theorem}
Formula \eqref{est-psi-D} indicates that $\psi$ is asymptotically localized in the region where $a>0$. When $a(x,\kappa)=1$, Theorem~\ref{est-psi-main} was proved in \cite{KA}.
The techniques that we are going to use here are inspired from those of \cite{KA} and \cite{KA2} (where the case $a=1$ was treated). At a technical level, our proof is slightly different than the proofs in \cite{KA,FK2,SS} since we do not use the uniform elliptic estimates. These important estimates are frequently used in the papers about the Ginzburg-Landau functional (see \cite{FH1}) with a constant pinning term. They appeared first in \cite{LP} and were then extended to the full regime in \cite{FH2}.
Compared with other papers studying the pinned functional, one
novelty here is that the pinning term has no definite sign, another
one being the consideration of a variable (and a potentially
vanishing) applied magnetic field.
The rest of this paper is devoted to the study of third critical field, i.e. the field above which the normal state $(0,\mathbf F)$ is the only critical point of the functional in \eqref{eq-2D-GLf}, in the case when the pining term $a$ is independent of $\kappa$ (i.e. $a(x,\kappa)=a(x)$). We define the set:
\begin{equation}\label{def:Ncp}
\mathcal{N}^{\rm cp}(\kappa)=\{H>0: \mathcal E_{\kappa,H,a,B_{0}}~\text{\rm has a non-normal critical point}\}\,.
\end{equation}\label{def:N}
Notice that the above set is bounded (see Theorem~\ref{thm:GP}). We also introduce the two sets:
\begin{equation}
\mathcal{N}(\kappa)=\{H>0:\mathcal E_{\kappa,H,a,B_{0}}~\text{\rm has a non-normal minimizer}\}\,.
\end{equation}
\begin{equation}\label{def:Nloc}
\mathcal{N}^{\rm loc}(\kappa)=\{H>0:\mu_{1}(\kappa,H)<0\}\,.
\end{equation}
Here, $\mu_{1}(\kappa,H)$ is the ground state energy of the semi-bounded quadratic form
\begin{equation}\label{Quad}
\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\phi)=\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\phi|^{2}-\kappa^{2}\,a(x,\kappa)|\phi|^{2}\right)\,dx\,,
\end{equation}
i.e.
\begin{equation}\label{def:mu1}
\mu_{1}(\kappa,H)=\inf_{\substack{\phi\in H^{1}(\Omega)\\ \phi\neq 0}}\left(\frac{\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\phi)}{\|\phi\|^{2}_{L^{2}(\Omega)}}\right)\,.
\end{equation}
Note that $\mu_{1}(\kappa,H)$ is the lowest eigenvalue of $P_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}$. Here, we refer to \cite{CR,KIH,JPM,XB-KH} for previous contributions.\\
We introduce the following critical fields (cf. e.g.\cite{FH3,LP})\,.
\begin{align}
&\overline{H}_{C_3}^{cp}(\kappa)=\sup\,\mathcal{N}^{cp}(\kappa)\,,\qquad\underline{H}_{C_3}^{cp}(\kappa)=\inf\,(\mathbb R_{+}\setminus\mathcal{N}^{cp}(\kappa))\label{def:HC3-o}\,,\\
&\overline{H}_{C_3}(\kappa)=\sup\,\mathcal{N}(\kappa)\,,\qquad\quad\underline{H}_{C_3}(\kappa)=\inf\,(\mathbb R_{+}\setminus\mathcal{N}(\kappa))\,,\label{def:HC3}\\
&\overline{H}_{C_3}^{loc}(\kappa)=\sup\,\mathcal{N}^{loc}(\kappa)\,,\qquad\underline{H}_{C_3}^{loc}(\kappa)=\inf\,(\mathbb R_{+}\setminus\mathcal{N}^{loc}(\kappa))\label{def:HC3-u}\,.
\end{align}
Below $\underline{H}_{C_3}$, normal states will loose their stability and above $\overline{H}_{C_3}$, the normal state is (up to a gauge transformation) the only critical point of the functional in \eqref{eq-2D-GLf}.\\
Our aim is to determine the asymptotics of all the critical fields as $\kappa\longrightarrow+\infty$. This involves spectral quantities related to three models depending on $\Gamma$ being empty or not. \\
Let us introduce
$$\displaystyle\Theta_{0}=\inf_{\xi\in\mathbb R} \mu(\xi)\,,$$
where $\mu$ is the lowest eigen value of the operator
$$
\mathfrak{h}^{N,\xi}:=-\frac{d^2}{dt^2}+(t+\xi)^2\qquad{\rm in}~L^{2}(\mathbb R_+)\,,
$$
subject to the Neumann boundary condition $u'(0)=0$.
\begin{theorem}\label{thm:HC3}
Suppose that $\Gamma=\{x\in\Omega: B_{0}(x)=0\}=\varnothing$ and that $a\in C^{1}(\overline{\Omega})$ satisfies $\{a>0\}\neq\varnothing$. Then, as $\kappa\longrightarrow+\infty$, all the six critical fields satisfy an asymptotic expansion in the form:
\begin{equation}
H_{C_3}(\kappa)=\max\left(\sup_{x\in\Omega}\frac{a(x)}{|B_{0}(x)|},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}|B_{0}(x)|}\right)\,\kappa+\mathcal{O}(\kappa^{\frac{1}{2}})\,.
\end{equation}
\end{theorem}
We introduce
\begin{equation}\label{lambda0}
\lambda_{0}=\inf_{\tau\in \mathbb R} \lambda(\tau)\,,
\end{equation}
where $\lambda(\tau)$ is the lowest eigenvalue of the selfadjoint realization of the differential operator
\begin{equation}\label{defM}
M(\tau) = -\frac{d^2}{dt^2} +\frac 14 (t^2+2\tau)^2\qquad{\rm in}~L^{2}(\mathbb R)\,.
\end{equation}
We consider, for any $\theta\in(0,\pi)$ the bottom of the spectrum $\lambda(\mathbb R_{+}^{2},\theta)$ of the operator
\begin{equation}\label{def:lambda-theta}
P_{\mathbf A_{\rm app,\theta},0}^{\mathbb R^{2}_{+}}\quad{\rm with}\quad\mathbf A_{\rm app,\theta}=-\left(\frac{x^{2}_{2}}{2}\cos\,\theta,\frac{x^{2}_{1}}{2}\sin\,\theta \right)\,.
\end{equation}
\begin{theorem}\label{thm:HC3-vr}
Suppose that $\Gamma=\{x: B_{0}(x)=0\}\neq\varnothing$, that
\eqref{B(x)} holds and that $a\in C^{1}(\overline{\Omega})$ satisfies $\{a>0\}\neq\varnothing$. As
$\kappa\longrightarrow+\infty$, the six critical fields in
\eqref{def:HC3-o}-\eqref{def:HC3-u} satisfy the asymptotic
expansion:
$$
H_{C_3}(\kappa)=\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+\mathcal{O}\left(\kappa^{\frac{7}{4}}\right)\,.
$$
Here $\theta(x)$ denotes the angle between $\nabla B_{0}(x)$ and the inward normal vector $-\nu(x)$.
\end{theorem}
\subsection*{Organization of the paper} The rest of the paper is split into twelve sections. Section~\ref{2} analyzes the model problem with a constant magnetic field and a constant pinning term. Section~\ref{upperbound} establishes an upper bound on the ground state energy. Section~\ref{section:P.E.} contains useful estimates on minimizers. The estimates in Section~\ref{section:P.E.} are used in Section~\ref{section5} to establish a lower bound of the ground state energy and to finish the proof of Theorem~\ref{thm-2D-main}, Corollary~\ref{corol-2D-main} and Theorem~\ref{lc-en}. In Section~\ref{examples}, we discuss the conclusion in Theorem~\ref{thm-2D-main} by providing various examples of pinning terms obeying Assumption~\ref{assumption}. Section~{7} is devoted to the proof of Theorem~\ref{est-psi-main}. Section~\ref{GP} generalizes a theorem of Giorgi-Phillips concerning the breakdown of superconductivity under a large applied magnetic field. Sections~\ref{Section:4} and \ref{10} are devoted to the proof of Theorem~\ref{thm:HC3}. The proof of Theorem~\ref{thm:HC3-vr} is the purpose of Sections~\ref{Section:Asympt-m1-vanish} and \ref{12}.
\subsection* {Notation.}
{Throughout the paper, we use the following notation:}
\begin{itemize}
\item If $b_{1}(\kappa)$ and $b_{2}(\kappa)$ are two positive functions on $[\kappa_{0},+\infty)$, we write $b_{1}(\kappa)\ll b_{2}(\kappa)$ if $b_{1}(\kappa)/b_{2}(\kappa)\to0$ as $\kappa\to\infty$.
\item If $b_{1}(\kappa)$ and $b_{2}(\kappa)$ are two functions with $b_{2}(\kappa)\not=0$, we write $b_{1}(\kappa)\sim b_{2}(\kappa)$\\
if $b_{1}(\kappa)/b_{2}(\kappa)\to1$ as $\kappa\to\infty$.
\item If $b_{1}(\kappa)$ and $b_{2}(\kappa)$ are two positive functions, we write $b_{1}(\kappa)\approx b_{2}(\kappa)$
if there exist positive constants $c_1$, $c_2$ and $\kappa_0$ such
that $c_1b_{2}(\kappa)\leq b_{1}(\kappa)\leq c_2b_{2}(\kappa)$ for all
$\kappa\geq\kappa_0$.
\item Let $a_{+}(\widetilde{x}_{0},\kappa)=[a(\widetilde{x}_{0},\kappa)]_{+}$ and $a_{-}(\widetilde{x}_{0},\kappa)=[a(\widetilde{x}_{0},\kappa)]_-$ where, for any $x\in\mathbb R$, $[x]_+=\max(x,0)$ and $[x]_{-}=\max(-x,0)$.
\item Given $R>0$ and $x=(x_{1},x_{2})\in\mathbb R^{2}$, $Q_{R}(x)=(-R/2+x_{1},R/2+x_{1})\times(-R/2+x_{2},R/2+x_{2})$ denotes the
square of side length $R$ centered at $ x=(x_1,x_2)$ and we write $Q_{R}=Q_{R}(0)$.
\end{itemize}
\section{A reference problem}\label{2}
The reference problem is obtained by freezing the pinning term and the magnetic field. This approximation will appear to be reasonable in squares avoiding the boundary and the zero set $\Gamma$ of the magnetic field $B_0$.
\subsection{A useful function}\label{uf}
Consider $R>0$, $b>0$, $\zeta\in\{-1,+1\}$ and $\alpha\in\mathbb R\,$. We define the following Ginzburg-Landau energy with constant magnetic field on $H^1(Q_R)$ by
\begin{equation}\label{eq-GL-F}
u\mapsto F^{\zeta,\alpha}_{b,Q_{R}}(u)=\int_{Q_{R}}\left(b|(\nabla-i\zeta\mathbf A_0)u|^2+\frac{1}{2}\left(\alpha-|u|^2\right)^{2}\right)\,dx\,,
\end{equation}
where
\begin{equation}\label{eq-hc2-mpA0}
\mathbf A_{0}(x)=\frac1{2}(-x_2,x_1)\,,\qquad\forall\,x=(x_1,x_2)\in\mathbb R^{2}\,.
\end{equation}
We have two cases according to the sign of $\alpha$\,:~\\
\textbf{Case~1}. $\alpha>0$:~\\
We notice that
\begin{equation}\label{change-F}
F^{\zeta,\alpha}_{b,Q_{R}}(u)=\alpha^{2} F^{\zeta,1}_{\widetilde{b},Q_{R}}(\widetilde{u})\,,
\end{equation}
where
\begin{equation}\label{def-b-u}
\widetilde{b}=\frac{b}{\alpha}\qquad\text{and}\qquad\widetilde{u}=\frac{u}{\sqrt{\alpha}}\,.
\end{equation}
We introduce the two ground state energies
\begin{eqnarray}
e_{N}(b,R,\alpha)=\inf\left\{F^{+1,\alpha}_{b,Q_{R}}(u): u\in H^{1}(Q_{R};\mathbb C)\right\}\label{eN}\\
e_{D}(b,R,\alpha)=\inf\left\{F^{+1,\alpha}_{b,Q_{R}}(u): u\in H^{1}_{0}(Q_{R};\mathbb C)\right\}\label{eD}\,.
\end{eqnarray}
As $F^{+1,\alpha}_{b,Q_{R}}(u)=F^{-1,\alpha}_{b,Q_{R}}(\overline{u})$, it is immediate that,
\begin{equation}\label{F+=F-}
\inf F^{+1,\alpha}_{b,Q_{R}}(u)=\inf F^{-1,\alpha}_{b,Q_{R}}(u)\,.
\end{equation}
Using \eqref{eN} and \eqref{eD}, we get from \eqref{change-F}
\begin{equation}\label{eNa=eN}
e_{N}(b,R,\alpha)=\alpha^{2}\,e_{N}\left(\frac{b}{\alpha},R,1\right)=\alpha^{2}\,e_{N}\left(\frac{b}{\alpha},R\right)\,,
\end{equation}
and
\begin{equation}\label{eDa=eD}
e_{D}(b,R,\alpha)=\alpha^{2}\,e_{D}\left(\frac{b}{\alpha},R,1\right)=\alpha^{2}\,e_{D}\left(\frac{b}{\alpha},R\right)\,.
\end{equation}
As a consequence of \eqref{change-F} and \eqref{def-b-u}, $\widetilde{u}$ is a minimizer of $F^{\zeta,1}_{\widetilde{b},Q_{R}}$ if and only if $u$ is a minimizer of $F^{\zeta,\alpha}_{b,Q_{R}}$. In particular any minimizer of $F^{\zeta,\alpha}_{b,Q_{R}}$ satisfies
\begin{equation}\label{up-u-a}
|u|\leq \sqrt{\alpha}\,.
\end{equation}
Recall from \cite[Theorem~2.1]{FK2} that,
\begin{equation}\label{f(x)}
\displaystyle \hat{f}\left(\mathfrak{b}\right)=\lim_{R\longrightarrow\infty}\frac{e_{D}(\mathfrak{b},R)}{R^{2}}\,.
\end{equation}
The next proposition was proved in \cite[Lemma~2.2, Proposition~2.4]{KA2} in the case $\alpha=1$. It's present form can be deduced immediately from \eqref{eNa=eN}.
\begin{prop}\label{pro-f(b)}
For all $M>0$, there exist universal constants $C_{M}$ and $R_{M}$ such that $\forall R\geq R_{M},\,\forall\,b>0,\,\forall\,\alpha>0$ such that $\displaystyle 0<\frac{b}{\alpha}\leq M$, we have
\begin{equation}\label{eN>eD}
e_{N}(b,R,\alpha)\geq\,e_{D}\left(b,R,\alpha\right)-C_{M}\alpha^{2}R\left(\frac{b}{\alpha}\right)^{\frac{1}{2}}
\end{equation}
\begin{equation}\label{est-f(b)-eD}
\alpha^{2}\hat{f}\left(\frac{b}{\alpha}\right) \leq \frac{e_{D}(b,R,\alpha)}{R^{2}}\leq\alpha^{2}\hat{f}\left(\frac{b}{\alpha}\right)+C_{M}\frac{\alpha^{\frac{3}{2}}\sqrt{b}}{R}.
\end{equation}
\end{prop}
\textbf{Case~2}. $\alpha\leq 0\,$:~\\
When $\alpha\leq 0$, we write $\alpha=-\alpha_{0}$, $\alpha_{0}\geq 0$ and \eqref{eq-GL-F} becomes
\begin{equation}\label{eq-GL-F2}
F^{\zeta,\alpha}_{b,Q_{R}}(u)=\int_{Q_{R}}\left(b|(\nabla-i\zeta\mathbf A_0)u|^2+\frac{1}{2}\left(\alpha_{0}+|u|^2\right)^{2}\right)\,dx\,.
\end{equation}
It is clear that,
$$F^{\zeta,\alpha}_{b,Q_{R}}(u)\geq \frac{1}{2}\alpha_{0}^{2} R^{2}\qquad{\rm and}\qquad F^{\zeta,\alpha}_{b,Q_{R}}(0)=\frac{1}{2}\alpha_{0}^{2} R^{2}\,.$$
As a consequence, we have
$$
\frac{1}{2}\alpha_{0}^{2}R^{2}\leq e_{D}(b,R,\alpha)\leq F^{\zeta,\alpha}_{b,Q_{R}}(0)=\frac{1}{2}\alpha_{0}^{2}R^{2}\,.
$$
When $\alpha=0$, it is easy to show that
$$F^{\zeta,\alpha}_{b,Q_{R}}(u)=0\,.$$
Notice that the only minimizer of $F^{\zeta,\alpha}_{b,Q_{R}}$ is $u=0\,$.
Thus, for any $\alpha\leq 0\,$, we obtain
\begin{equation}\label{F=}
\frac{e_{D}(b,R,\alpha)}{R^{2}}=\frac{1}{2}\alpha^{2}\,.
\end{equation}
\section{Upper bound of the energy}\label{upperbound}
The aim of this section is to give an upper bound of the ground state energy $\E0(\kappa,H,a,B_{0})$ introduced in \eqref{eq-2D-gs} under Assumption~\eqref{cond-H}.
For this we cover $\Omega$ by (the closure of) disjoint open squares $(Q_{\ell}(\gamma))_{\gamma}$ whose centers $\gamma$ belong to a square lattice $\Gamma_\ell= \ell \mathbb Z \times \ell \mathbb Z$.
We will get an upper bound by matching together approximate minimizers, in each square $Q_{\ell}(\gamma)$ contained in $\Omega$, obtained by freezing the pinning term and the magnetic field at a suitable point $\tilde \gamma$. The size $\ell$ of the square will be chosen as a function of $\kappa$. We start with estimates in a given square $Q_\ell (x_0)$ and will take later $x_0=\gamma\,$.\\
{\bf About Assumption $(A_4)$.}\\
We first explain what was meant in Assumption $(A_4)$. By $\mathcal L(\partial\{a >0\}) \leq C_1 \kappa^\frac 12$ we mean the existence of $C_2 >0$ and $\kappa_0$ such that:
\begin{equation}\label{defA4}
\forall \kappa \geq \kappa_0\,,\, \forall \ell \leq C_2 \kappa^{-\frac 12}\,,\, {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{a >0\} \cap \Omega \neq \emptyset\} \leq C_1 \kappa^\frac 12 \ell^{-1}\,.
\end{equation}
~\\
Using Assumption~\eqref{def:L}, for any $\widetilde{x}_{0}\in \overline{Q_{\ell}(x_{0})}$ and $\kappa \geq \kappa_0$, we observe that,
\begin{equation}\label{app-a}
|a(x,\kappa)-a(\widetilde{x}_{0},\kappa)|\leq \left(\sup_x |\nabla_{x}\,a(x,\kappa)|\right)\,|x-x_{0}|\leq \frac{\ell}{\sqrt{2}}\,L (\kappa) \,,\qquad\forall x\in Q_{\ell}(x_{0})\,.
\end{equation}
\begin{definition}[$\rho$-admissible]\label{rho-adm}
Let $\rho\in(0,1)$. We say that triple $(\ell,x_0,\widetilde{x}_{0})$ is $\rho$-admissible if $\overline{Q_{\ell}(x_0)}\subset\{|B_{0}|>\rho\}\cap\Omega$ and $\widetilde{x}_{0}\in\overline{Q_{\ell}(x_{0})}$. In this case, we also say that the pair $(\ell,x_{0})$ is $\rho$-admissible and the corresponding square $Q_{\ell}(x_{0})$ is $\rho$ admissible.
\end{definition}
We recall from \cite[Section~3]{KA2} the definition of the test function,
\begin{equation}\label{def-w2}
\widetilde{w}_{\ell,x_0,\widetilde{x}_{0}}(x)=\begin{cases}
e^{i\kappa H\varphi_{x_{0},\widetilde{x}_{0}}}\widetilde{u}_{R}\left(\frac{R}{\ell}(x-x_{0})\right)&{\rm if}~x\in Q_{\ell}(x_0)\subset\{B_{0}>\rho\}\cap\Omega\\
e^{i\kappa H\varphi_{x_{0},\widetilde{x}_{0}}}\overline{\widetilde{u}}_{R}\left(\frac{R}{\ell}(x-x_{0})\right)&{\rm if}~x\in Q_{\ell}(x_0)\subset\{B_{0}<-\rho\}\cap\Omega\,,
\end{cases}
\end{equation}
where $\widetilde{u}_{R}\in H^{1}_{0}(\Omega)$ is a minimizer of $F^{+1,1}_{b,Q_{R}}$ satisfying by \eqref{up-u-a} $|\widetilde{u}_{R}|\leq 1$ and $\varphi_{x_{0},\widetilde{x}_{0}}$ is the function introduced in \cite[Lemma~A.3]{KA} that satisfies
\begin{equation}\label{F-A}
|\mathbf F(x)-B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_{0})-\nabla\varphi_{x_{0},\widetilde{x}_{0}}(x)|\leq C\, \ell^{2},\,\qquad\,\,\forall x \in Q_{\ell}(x_{0})\,.
\end{equation}
Here $B_{0}=\curl\mathbf F$ and $\mathbf A_{0}$ is the magnetic potential introduced in \eqref{eq-hc2-mpA0}.
Let us introduce the function:
\begin{equation}\label{def-w}
w_{\ell,x_0,\widetilde{x}_{0}}(x)=\sqrt{a_{+}(\widetilde{x}_{0},\kappa)}\,\widetilde{w}_{\ell,x_0,\widetilde{x}_{0}}(x)\,, \qquad \forall x\in Q_{\ell}(\widetilde{x}_{0})\,.
\end{equation}
Using the bound $|\widetilde{w}_{\ell,x_0,\widetilde{x}_{0}}|\leq1$, which is immediately deduced from the bound of $|\widetilde{u}_{R}|$, we get from \eqref{def-w},
\begin{equation}\label{up-w}
|w_{\ell,x_0,\widetilde{x}_{0}}|^{2} \leq a_{+}(\widetilde{x}_{0},\kappa)\,.
\end{equation}
\begin{prop}\label{pp-up-Eg}
Under Assumptions~\eqref{B(x)}-\eqref{a3}, there exist positive constants $C$ and $\kappa_{0}$ such that if $\kappa\geq\kappa_{0}$, $\ell\in(0,1)$, $\delta\in(0,1)$, $\rho>0$, $\ell^{2}\kappa H\rho>1$ and $(\ell,x_{0},\widetilde{x}_{0})$ is a $\rho$-admissible triple, then,
\begin{multline}\label{up-Eg-eq}
\frac{1}{|Q_{\ell}(x_0)|}\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a,Q_{\ell}(x_0))\leq (1+\delta)\kappa^{2}\left[a_{+}(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H\,|B_{0}(\widetilde{x}_{0})|}{\kappa\,a_{+}(\widetilde{x}_{0},\kappa)}\right)+\frac{1}{2}a_{-}(\widetilde{x}_{0},\kappa)^{2}\right]\\
+C\left(\frac{1}{\kappa\ell}+\delta^{-1}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{2}\ell^{4}\right)\kappa^{2}\,.
\end{multline}
\end{prop}
\begin{proof}~\\
Let
\begin{equation}\label{def-Rb}
R=\ell\sqrt{\kappa H |B_{0}(\widetilde{x}_{0})|}\qquad{\rm and}\qquad b=\frac{H\,|B_{0}(\widetilde{x}_{0})|}{\kappa}\,.
\end{equation}
First we estimate $\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}(a(x,\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2})^{2}\,dx$ from above. Using \eqref{app-a}, we get the existence of a constant $C>0$ such that for any $\delta\in(0,1)$ and any $\kappa\geq \kappa_{0}$,
\begin{align}\label{up-a}
\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(x,\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\right)^{2}\,dx&\leq (1+\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\right)^{2}\,dx\nonumber\\
&\qquad\qquad\qquad+(1+\delta^{-1})\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-a(x,\kappa)\right)^{2}\,dx\nonumber\\
&\leq(1+\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\right)^{2}\,dx\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C\delta^{-1}\kappa^{2}\ell^{4}L(\kappa)^{2}\,.
\end{align}
The estimate of $\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H \mathbf F)w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\,dx$ from above is the same as in \cite[Proposition~3.1]{KA2}. We have
\begin{align}\label{up-F}
&\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H \mathbf F)w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\,dx\nonumber\\
&\qquad\qquad\leq (1+\delta)\int_{Q_{\ell}(x_0)}\left|\left(\nabla-i\kappa H (B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}})\right)w_{\ell,x_0,\widetilde{x}_{0}}\right|^{2}\,dx\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C\delta^{-1}\kappa^{4}\ell^{6}|\, w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\,.
\end{align}
From \eqref{def:sup-a}, by collecting \eqref{up-a}, \eqref{up-F} and \eqref{up-w}, we find that,
\begin{multline}\label{1up-loc-en}
\mathcal E_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a,Q_{\ell}(x_0))\leq(1+\delta)\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},\,B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)\\
+C\delta^{-1}(\kappa^{2}\ell^{4}L(\kappa)^{2}+\kappa^{4}\ell^{6}\,a_{+}(\widetilde{x}_{0},\kappa))\,.
\end{multline}
As we did in \cite{KA2}, we use the change of variable $y=\frac{R}{\ell}(x-x_0)$ and obtain
\begin{align*}
&\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},\,B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)\\
&=\int_{Q_{R}}\left[a_{+}(\widetilde{x}_{0},\kappa)\left|\left(\frac{R}{\ell}\nabla-i\frac{R}{\ell}\zeta_{\ell}\,\mathbf A_{0}(y)\right)\widetilde{u}_{R}(y)\right|^{2}+\frac{\kappa^{2}}{2}\left(a(\widetilde{x}_{0},\kappa)-a_{+}(\widetilde{x}_{0},\kappa)\left|\widetilde{u}_{R}(y)\right|^2\right)^2\right]\frac{\ell^2}{R^2}dy.
\end{align*}
Here, we denote by $\zeta_{\ell}$ the sign of $B_{0}(x_{0})$.\\
We distinguish between two cases:\\
\textbf{Case~1:} When $a(\widetilde{x}_{0},\kappa)>0$, we get
$$
\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)=\frac{a(\widetilde{x}_{0},\kappa)^{2}}{b}F^{\zeta_{\ell},1}_{b/a(\widetilde{x}_{0},\kappa),Q_{R}}(\widetilde{u}_{R})\,.
$$
From \eqref{F+=F-} and \eqref{eNa=eN}, we obtain,
\begin{equation}\label{2up-loc-en}
\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)=\frac{1}{b}e_{D}(b,R,a(\widetilde{x}_{0},\kappa))\,.
\end{equation}
As a consequence of the upper bound in \eqref{est-f(b)-eD}, the ground state energy $e_{D}(b,R,a(\widetilde{x}_{0},\kappa))$ in \eqref{2up-loc-en} is bounded for all $b>0$ and $R\geq 1$ by:
\begin{align}\label{F=eD}
e_{D}(b,R,a(\widetilde{x}_{0},\kappa))&\leq a(\widetilde{x}_{0},\kappa)^{2}\,R^{2}\,\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)+C_{M}a(\widetilde{x}_{0},\kappa)^{\frac{3}{2}}\,R\,\sqrt{b}\,.
\end{align}
With the choice of $R$ in \eqref{def-Rb}, we have effectively $R\geq 1$ which follows from the assumption $R\geq \ell\sqrt{\kappa H \rho}>1$.\\
We get from \eqref{2up-loc-en} and \eqref{F=eD} the estimate
\begin{multline}\label{3up-loc-en}
\mathcal E_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\zeta_{\ell}|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq a(\widetilde{x}_{0},\kappa)^{2}\frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)\\
+C_{M}\frac{a(\widetilde{x}_{0},\kappa)^{\frac{3}{2}}\,R}{\sqrt{b}}\,,
\end{multline}
with $(b,R)$ defined in \eqref{def-Rb}.\\
By collecting the estimates in \eqref{1up-loc-en}-\eqref{3up-loc-en} we get,
\begin{multline}\label{up-E0}
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq (1+\delta)\, a(\widetilde{x}_{0},\kappa)^{2}\, \frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)\\
+C_{M}\frac{\overline{a}^{\frac{3}{2}}\,R}{\sqrt{b}}+C\delta^{-1}(\kappa^{2}\ell^{4}L(\kappa)^{2}+\kappa^{4}\ell^{6}\overline{a})\,.
\end{multline}
Here, we have used the fact that $\displaystyle a(\widetilde{x}_{0},\kappa)\leq\sup_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}a(x,\kappa)=\overline{a}\,$.\\
\textbf{Case~2:} When $a(\widetilde{x}_{0},\kappa)\leq 0\,$, we have,
$$
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))=\frac{\kappa^2}{2}\int_{Q_{\ell}(x_0)}a(x,\kappa)^{2}\,dx\,.
$$
From \eqref{app-a}, we get the existence of a constant $C>0$ such that for any $\delta\in(0,1)$,
\begin{equation}\label{up-E0-2}
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq (1+\delta)\, \frac{\kappa^{2}}{2} \, a(\widetilde{x}_{0},\kappa)^{2}\ell^{2}
+C\,\delta^{-1}\,\kappa^{2}\ell^{4}L(\kappa)^{2}\,.
\end{equation}
\textbf{The results of cases 1-2}, we obtain,
\begin{multline}
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq (1+\delta)\kappa^{2}\left[a_{+}(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H\,|B_{0}(\widetilde{x}_{0})|}{\kappa\,a_{+}(\widetilde{x}_{0},\kappa)}\right)+\frac{1}{2}a_{-}(\widetilde{x}_{0},\kappa)^{2}\right]\ell^{2}\\
+C\left(\frac{\kappa}{\ell}\,\overline{a}^{\frac{3}{2}}+\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{4}\ell^{4}\,\overline{a}\right)\ell^{2}\,,
\end{multline}
which finishes the proof of Proposition~\ref{pp-up-Eg}.
\end{proof}
\begin{app}\label{app:1}~\\
We select $\ell,\,\rho,\,\delta$ and the constraint on $L(\kappa)$ as follows:
\begin{equation}\label{choice-ell-rho}
\ell=\kappa^{-\frac{7}{12}}\,,\qquad\rho=\kappa^{-\frac{17}{24}}\,,\qquad L(\kappa)\leq C\,\kappa^{\frac{1}{2}}\,.
\end{equation}
and
\begin{equation}\label{choice-delta}
\delta=\kappa^{-\frac{1}{12}}
\end{equation}
Under Assumption~\eqref{cond-H}, this choice permits to verify the assumptions in Proposition~\ref{pp-up-Eg} and to obtain error terms of order $\textit{o}(\kappa^{2})$. We have indeed as $\kappa\longrightarrow\infty$
$$\frac{\kappa}{\ell}=\kappa^{\frac{19}{12}}\ll\kappa^{2}\,,$$
$$\delta^{-1}\kappa^{2}\ell^{2}\,L(\kappa)^{2}\leq\kappa^{\frac{23}{12}}\ll\kappa^{2}\,,$$
$$\delta^{-1}\kappa^{4}\ell^{4}=\kappa^{\frac{21}{12}}\ll \kappa^{2}\,,$$
$$\ell^{2}\kappa H \rho=\kappa^{\frac{3}{24}}\gg 1\,.$$
\end{app}
\begin{theorem}\label{up-Eg}
Under Assumptions~\eqref{B(x)}-\eqref{a4}, if \eqref{cond-H} holds and $L(\kappa)\leq C\,\kappa^{\frac{1}{2}}$, then, the ground state energy $\E0(\kappa,H,a,B_{0})$ in \eqref{eq-2D-gs} satisfies
\begin{multline}\label{up-Eg-eq1}
\E0(\kappa,H,a,B_{0})\leq \kappa^{2}\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\,\hat{f}\left(\frac{H\,|B_{0}(x)|}{\kappa\,a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx+\textit{o}(\kappa^{2})\,,\quad{\rm as}~\kappa\longrightarrow\infty\,.
\end{multline}
\end{theorem}
\begin{proof}
Let $\ell\in(0,1)$, $\delta$ and $\rho$ be chosen as in \eqref{choice-ell-rho} and \eqref{choice-delta}.
We consider the lattice $\Gamma_{\ell}:=\ell\mathbb Z\times\ell\mathbb Z$ and write, for $\gamma \in\Gamma_{\ell}$, $ Q_{\gamma,\ell}=Q_{\ell}(\gamma)$. In the next decomposition we keep the $\rho$-admissible boxes $Q_\ell(\gamma)$ in $\Omega$ which in addition are either contained in $\{a >0\}$ or in $\{ a \leq 0\}$. Hence we introduce \begin{equation}\label{def-card-squares}
\mathcal I_{\ell, \rho}^{+}=\left\{\gamma; \,\overline{Q_{\gamma, \ell}}\subset \Omega\cap\left\{|B_{0}|>\rho\,; a>0\right\}\right\},\quad \mathcal I_{\ell, \rho}^{-}=\left\{\gamma; \,\overline{Q_{\gamma, \ell}}\subset \Omega\cap\left\{ |B_{0}|>\rho\,; a\leq 0\right\}\right\}\,,
\end{equation}
and
\begin{equation}
N^{+}={\rm card}~ \mathcal I_{\ell, \rho}^{+}\,,\qquad N^{-}={\rm card}~ \mathcal I_{\ell, \rho}^{-}\,.
\end{equation}
Under Assumption \eqref{a4}, we have,
\begin{equation}\label{N}
N^{+}+N^{-}=|\Omega|\ell^{-2}+\mathcal{O}(\kappa^{\frac{1}{2}}\ell^{-1}+\ell^{-1} + \rho\ell^{-2})\,,\qquad{\rm as}~\kappa\rightarrow+\infty\,.
\end{equation}
In \eqref{N}, $\kappa^{\frac{1}{2}}\ell^{-1}$ appears when treating the boundary of the set $\{a(x,\kappa)>0\}$ (using Assumption $(A_4)$ as explained in \eqref{defA4}), $\ell^{-1}$ appears in the treatment of the boundary and $\rho\ell^{-2}$ appears when treating the neighborhood of $\Gamma$.\\
In each $\rho$-admissible $Q_\ell(\gamma)$, we consider some $\widetilde \gamma$ (to be chosen later) such that $(\ell, \gamma,\widetilde{\gamma})$ be a $\rho$-admissible triple. We consider $w_{\ell,\gamma,\widetilde{\gamma}}$ and extend it by $0$ outside of $Q_{\gamma,\ell}$, keeping the same notation for this extension. Then we define
\begin{equation}
s(x)=
\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}} \,w_{\ell,\gamma,\widetilde{\gamma}}(x)\,.
\end{equation}
We compute the Ginzburg-Landau energy of the test configuration $(s,\mathbf F)$ in $\Omega$. Since $\curl \mathbf F=B_{0}\,$, we get,
\begin{align}\label{sumE1}
\mathcal{E}_{\kappa,H,a,B_{0}}(s,\mathbf F,\Omega)=\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}}\mathcal{E}_{0}(w_{\ell,\gamma,\widetilde{\gamma}},\mathbf F;a(\widetilde{\gamma},\kappa),Q_{\gamma, \ell})\,.
\end{align}
Notice that for any $\widetilde{\gamma}\in Q_{\gamma,\ell}\,$, $a(\widetilde{\gamma},\kappa)$ satisfies \eqref{app-a} with $x=\gamma$ and $\widetilde{x}_{0}=\widetilde{\gamma}\,$, and $B_{0}(\widetilde{\gamma})$ satisfies \eqref{F-A}. We recall that $\hat{f}$ is a continuous, non-decreasing function (see \cite[Theorem~2.1]{KA2}) and that $B_{0}$ and $ a(\cdot,\kappa)$ are in $C^{1}$. Then, in each box $Q_{\gamma,\ell}$, we select $\widetilde{\gamma}\in \overline{Q_{\gamma,\ell}}$ such that
$$
|a(\widetilde{\gamma},\kappa)|^{2}\,\hat{f}\left(\frac{H\,B_{0}(\widetilde{\gamma})}{\kappa\,a(\widetilde{\gamma},\kappa)}\right)=\inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} |a(\widehat{\gamma},\kappa)|^{2}\,\hat{f}\left(\frac{H\,B_{0}(\widehat{\gamma})}{\kappa\,a(\widehat{\gamma},\kappa)}\right)\quad(\text{if}~ \gamma\in \mathcal I_{\ell, \rho}^{+})
$$
and
$$
|a(\widetilde{\gamma},\kappa)|^{2}=\inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} |a(\widehat{\gamma},\kappa)|^{2}\quad(\text{if}~ \gamma\in \mathcal I_{\ell, \rho}^{-})\,.
$$
Using Proposition~\ref{pp-up-Eg} and noticing that $|Q_{\gamma,\ell}|=\ell^2$, we get the existence of $C>0$ such that, for any $\delta\in(0,1)$
\begin{multline}\label{sumE2}
\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}}\mathcal{E}_{0}(w_{\ell,\gamma,\widetilde{\gamma}},\mathbf F;a(\widetilde{\gamma},\kappa),Q_{\gamma, \ell})\leq \kappa^{2}(1+\delta)\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}} \inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} [a(\widehat{\gamma},\kappa)]_{+}^{2}\,\hat{f}\left(\frac{H\,B_{0}(\widehat{\gamma})}{\kappa\,a(\widehat{\gamma},\kappa)}\right)\ell^{2}\\
+\kappa^{2}(1+\delta)\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{-}} \inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} \frac{[a(\widehat{\gamma},\kappa)]_{-}^{2}}{2}\ell^{2}+\, C\, \sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}}\left(\frac{\kappa}{\ell}+\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{4}\ell^{4}\right)\,\ell^{2}\,.
\end{multline}
We recognize the lower Riemann sum of the function $x\longmapsto [a(x,\kappa)]_+^{2}\,\hat{f}\left(\frac{H\,B_{0}(x)}{\kappa\,a(x,\kappa)}\right)$ in $(\cup_{\gamma\in\mathcal{I}^{+}_{\ell,\rho}}Q_{\gamma,\ell})$ and the function $x\longmapsto [a(x,\kappa)]_-^{2}$ in $(\cup_{\gamma\in\mathcal{I}^{-}_{\ell,\rho}} Q_{\gamma,\ell})$ . Notice that $\{\cup_{\gamma\in \mathcal{I}_{\ell, \rho}}Q_{\gamma, \ell}\}\subset\Omega$. Thanks to Application~\ref{app:1}, using \eqref{N} and the non negativity of $\hat f$, we get by collecting \eqref{sumE1}-\eqref{sumE2} that,
\begin{equation}\label{final-est}
\mathcal{E}_{\kappa,H,a,B_{0}}(s,\mathbf F,\Omega)\leq\kappa^{2}\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2} \hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+C\,\kappa^{\frac{23}{12}}\,.
\end{equation}
Since $(\psi,\mathbf A)$ is a minimizer of the functional $\mathcal E_{\kappa,H,a,B_{0}}$ in \eqref{eq-2D-GLf}, we get
$$
\E0(\kappa,H,a,B_{0})\leq \mathcal{E}_{\kappa,H,a,B_{0}}(s,\mathbf F,\Omega)\,.
$$
This finishes the proof of Theorem~\ref{up-Eg}.
\end{proof}
\section{A priori estimates of minimizers}\label{section:P.E.}
The aim of this section is to give a priori estimates for the solutions of the Ginzburg-Landau equations \eqref{eq-2D-GLeq}. In the case when $a(x,\kappa)=1$ the starting point is an $L^{\infty}$ estimate of $\psi$. This estimate can be easly extended in the general case considered in this paper when $\eqref{eq-2D-GLeq}_{a}$ and $\eqref{eq-2D-GLeq}_{c}$ hold. Let us introduce:
\begin{equation}
\overline{a}(\kappa)= \sup_{x\in\overline{\Omega}}a(x,\kappa)\,.
\end{equation}
\begin{prop}\label{prop-psi<a}
Let $\kappa>0$; if $(\psi,\mathbf A)$ is a critical point (see \eqref{eq-2D-GLeq}), then,
\begin{equation}\label{eq-psi<supa}
|\psi(x)|^{2}\leq \max\left\{\overline{a}(\kappa),0\right\}\,,\qquad\forall x\in\overline{\Omega}\,.
\end{equation}
\end{prop}
\begin{proof}
We distinguish between two cases:\\
\textbf{Case 1:} \textbf {$\displaystyle\overline{a}(\kappa)\leq 0\,$.}\\
Multiplying the equation for $\psi$ in \eqref{eq-2D-GLeq}$_a$ by $\overline{\psi}$ and integrating over $\Omega$, we get
\begin{equation}\label{eq:A-psi}
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\psi|^{2}\,dx=\kappa^{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})|\psi|^{2}\,dx\,.
\end{equation}
Since $(a(x,\kappa)-|\psi|^{2})\leq -|\psi|^{2}$, we obtain that $ |\psi|^{2}=0\,$ almost everywhere.\\
\textbf{Case 2:} \textbf{ $\displaystyle\overline{a}(\kappa)>0\,$}.\\
We will show that $\psi\in C^{0}(\overline{\Omega})$. In fact, $(\psi,\mathbf A)$ satisfies \eqref{eq-2D-GLeq}$_{a}$, $\psi\in L^{p}(\Omega)$ for all $2\leq p<+\infty$ and $\mathbf A\in H_{\rm div}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$. Thus, $\psi \in W^{2,q}(\Omega)$ for all $q<2$. As a consequence of the continuous Sobolev embedding of $W^{j+m,q}(\Omega)$ into $C^{j}(\overline{\Omega})$ for any $q>\frac{2}{m}$, we obtain that $\psi\in C^{0}(\overline{\Omega})$. Define for any $\kappa>0$ the following open set:
\begin{equation}\label{omega+}
\Omega_{+}=\left\{x\in\Omega:\,|\psi(x)|>\sqrt{\overline{a}(\kappa)}\right\}\,,
\end{equation}
and the following functions on $\Omega_{+}$
$$\phi=\frac{\psi}{|\psi|}\,,\qquad \widehat{\psi}=\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}\phi\,.$$
It is clear that
$$\nabla\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}=1_{\Omega_{+}}\nabla\left(|\psi|-\sqrt{\overline{a}(\kappa)}\right)=1_{\Omega_{+}}\nabla|\psi|\,.$$
Notice that $\psi\in H^{1}(\Omega)$, so applying \cite[Proposition~3.1.2]{FH1}, we get the property that\break $\displaystyle\nabla\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}\in L^{2}(\Omega)$, which implies that $\displaystyle\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}\in H^{1}(\Omega)$.\\
We introduce an increasing cut-off function $\chi\in C^{\infty}(\mathbb R)$ such that,
\begin{equation}\label{chi}
\chi(t)=\left\{
\begin{array}{ll}
0 & \text{for }~t\leq\frac{1}{4}\displaystyle\sqrt{\overline{a}(\kappa)}\\
1& \text{for}~t\geq\frac{3}{4}\displaystyle\sqrt{\overline{a}(\kappa)}\,,
\end{array}
\right.
\end{equation}
and define
\begin{equation}\label{phi1}
\widehat{\phi}=\chi(|\psi|)\frac{\psi}{|\psi|}\,.
\end{equation}
Since $\chi(|\psi|)\frac{\psi}{|\psi|}$ is smooth with bounded derivatives and $\psi\in H^{1}(\Omega)$, the chain rule gives that $\widehat{\phi}\in H^{1}(\Omega)\,.$
Furthermore,
\begin{equation}\label{phi2}
(\nabla-i\kappa H \mathbf A)\widehat{\psi}=1_{\Omega_{+}}\widehat{\phi}\,\nabla|\psi|+\displaystyle\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}(\nabla-i\kappa H\mathbf A)\widehat{\phi}.
\end{equation}
Using \eqref{chi} and \eqref{phi1}, we get
\begin{equation}\label{phi3}
1_{\Omega_{+}}(\nabla-i\kappa H \mathbf A)\psi=1_{\Omega_{+}}(\nabla-i\kappa H\mathbf A)(|\psi|\widehat{\phi})=1_{\Omega_{+}}\{\widehat{\phi}\,\nabla|\psi|+|\psi|(\nabla-i\kappa H\mathbf A)\widehat{\phi}\}\,.
\end{equation}
We have on $\Omega_{+}$ that $|\phi|=|\widehat{\phi}|=1\,$. Therefore
\begin{align*}
\phi\nabla\overline{\phi}+\overline{ \phi\nabla\overline{\phi}}&=\phi\nabla\overline{\phi}+\overline{\phi}\nabla\phi\\
&=\nabla|\phi|^{2}\\
&=0\,.
\end{align*}
So, $\mathbb RE(1_{\Omega_{+}}\phi\nabla\overline{\phi})=0\,$. This implies by using \eqref{phi2} and \eqref{phi3} that
$$
\mathbb RE\left\{\overline{(\nabla-i\kappa H\mathbf A)\widehat{\psi}}\cdot(\nabla-i\kappa H\mathbf A)\psi \right\}=
1_{\Omega_{+}}\left(|\nabla|\psi||^{2}+\left(|\psi|-\sqrt{\overline{a}(\kappa)}\right)|\psi||(\nabla-i\kappa H\mathbf A)\widehat{\phi}|^{2}\right).
$$
Multiplying $\eqref{eq-2D-GLeq}_{a}$ by $\overline{\widehat{\psi}}$ and using $\eqref{eq-2D-GLeq}_{c}$, it results from an integration by parts over $\Omega$ that
\begin{align*}
0&=\mathbb RE\left\{\int_{\Omega} \overline{(\nabla-i\kappa H\mathbf A)\widehat{\psi}}(\nabla-i\kappa H\mathbf A)\psi+\overline{\widehat{\psi}}(|\psi|^{2}-a)\psi\,dx \right\}\\
&\geq \mathbb RE\left\{\int_{\Omega} \overline{(\nabla-i\kappa H\mathbf A)\widehat{\psi}}(\nabla-i\kappa H\mathbf A)\psi+\overline{\widehat{\psi}}\left(|\psi|^{2}-\overline{a}(\kappa)\right)\psi\,dx \right\}\\
&\geq \int_{\Omega_{+}} |\nabla|\psi||^{2}+\left(|\psi|-\overline{a}(\kappa)\right)|\psi||(\nabla-i\kappa H\mathbf A)\widehat{\phi}|^{2}\\
&\qquad\qquad\qquad\qquad+\left(|\psi|+\sqrt{\overline{a}(\kappa)}\right)\left(|\psi|-\sqrt{\overline{a}(\kappa)}\right)^{2}|\psi|\,dx\,.
\end{align*}
Since the integrand is non-negative in $\Omega_{+}$, we easily conclude that $\Omega_{+}$ has measure zero, and consequently, we get that $|\psi|\in L^{\infty}(\Omega)$\,.\\
Since $\Omega_{+}$ has measure zero and $\psi\in C^{0}(\overline{\Omega})$, we get
$$
|\psi(x)|^{2}\leq \overline{a}(\kappa) \,,\qquad\forall x\in\overline{\Omega}\,.
$$
\end{proof}
\begin{corol}
Let $\kappa>0$; If $(\psi,\mathbf A)\in H^1(\Omega;\mathbb C)\times H^1_{\Div}(\Omega)$ is a critical point, we have,
\begin{equation}\label{eq-psi<a}
|\psi(x)|^{2}\leq \max\left\{\overline{a},0\right\}\,,\qquad\forall x\in\overline{\Omega}\,,
\end{equation}
where $ \overline{a}= \sup_{\kappa} \overline{a}(\kappa)$ was introduced in \eqref{def:sup-a}.
\end{corol}
The following estimates play an essential role in controlling the errors resulting from various approximations (see Section~\ref{section5}). These estimates are simpler than the delicate elliptic estimates in \cite{FH2} and \cite{LP}.
\begin{prop}\label{pr-est}
Suppose that \eqref{cond-H} holds. Let $\beta\in(0,1)$. There exist positive constants $\kappa_{0}$ and $C$ such that, if $\kappa\geq\kappa_{0}$ and $(\psi, \mathbf A)$ is a minimizer of \eqref{eq-2D-GLf}, then
\begin{equation}\label{eq-curlAF}
\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq \frac{C}{H}\,.
\end{equation}
\begin{equation}\label{est-A-F}
\|\mathbf A-\mathbf F\|_{H^{2}(\Omega)}\leq \frac{C}{H}\,,
\end{equation}
\begin{equation}\label{est-A-F2}
\|\mathbf A-\mathbf F\|_{C^{0,\beta}(\overline{\Omega})}\leq \frac{C}{H}\,.
\end{equation}
Here we recall that $\mathbf F$ is the magnetic potential defined by
\begin{equation}\label{div-curlF}
\curl \mathbf F = B_0\,,\, \mathbf F \in H^1_{\Div}(\Omega)\,.
\end{equation}
\end{prop}
\begin{proof}
Under Assumption~\eqref{cond-H}, Theorem~\ref{up-Eg} yields
\begin{align}
&\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq\frac{1}{\kappa H}\E0(\kappa,H,a,B_{0})^{\frac{1}{2}}\nonumber\\
&\qquad\qquad\leq\frac{1}{\kappa H}\left(\kappa^{2}\,\int_{\{a(x,\kappa)>0\}} \,a(x,\kappa)^{2} \hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}\,a(x,\kappa)^{2}\,dx\right)^{\frac{1}{2}}\,.
\end{align}
Using \eqref{a2} and the bound $\hat{f}(b)\leq\frac{1}{2}$, we get,
\begin{equation}\label{est-curlAF}
\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq\frac{C}{H}\,.
\end{equation}
As in \cite[Proposition~4.1]{KA2}, we prove that
\begin{equation}\label{est-F2}
\|\mathbf A-\mathbf F\|_{H^{2}(\Omega)}\leq\frac{C}{H}\,.
\end{equation}
Now, the estimate in $C^{0,\beta}$-norm is a consequence of the continuous Sobolev embedding of $H^{2}(\Omega)$ in $C^{0,\beta}(\overline{\Omega})$.
\end{proof}
\section{Lower bounds for the global and local energies} \label{section5}
In this section, we suppose that $\mathcal{D}$ is an open set with smooth boundary such that $\overline{\mathcal{D}}\subset\Omega$ (or $\mathcal{D}=\Omega$). We will give a lower bound of the ground state energy $\E0(\kappa,H,a,B_{0})$ introduced in \eqref{eq-2D-gs}.
\begin{prop}\label{prop-lb}
Under Assumptions~\eqref{B(x)}-\eqref{a3}, there exist for all $\beta\in(0,1)$ positive
constants $C$ and $\kappa_{0}$ such that if $\kappa\geq\kappa_{0}$, $\ell \in (0,\frac 12)$, $\delta\in(0,1)$, $\rho>0$, $\ell^{2}\kappa H \rho>1$, $(\psi,\mathbf A)$ is a minimizer of \eqref{eq-2D-GLf}, $h\in C^{1}(\overline{\Omega})$, $\|h\|_{\infty}\leq 1$ and $(\ell,x_{0},\widetilde{x}_{0})$ is a $\rho$-admissible triple, then,
\begin{multline}\label{lb}
\frac{1}{|Q_{\ell}(x_{0})|}\mathcal E_0(h\psi,\mathbf A;a,Q_{\ell}(x_0))\geq(1-\delta)\kappa^{2}\left\{ a_{+}(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(\widetilde{x}_{0})|}{a_{+}(\widetilde{x}_{0},\kappa)}\right)+\frac{1}{2}a_{-}(\widetilde{x}_{0},\kappa)^{2}\right\}\\
-C\kappa^{2}\left(\delta^{-1}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{2}\ell^{4}+\delta^{-1}\ell^{2\beta}+(\kappa\ell)^{-1}+\ell\,L(\kappa)\right)\,,
\end{multline}
where $L(\kappa)$ is introduced in \eqref{def:L}.
\end{prop}
\begin{proof}
We distinguish between two cases according to the sign of $a(\widetilde{x}_{0},\kappa)$.\\
\textbf{We begin with the case when $a(\widetilde{x}_{0},\kappa)\leq 0\,$.} We have,
\begin{align*}
\mathcal E_0(h\psi,\mathbf A;a,Q_{\ell}(x_0))&=\int_{Q_{\ell}(x_0)}|(\nabla -i\kappa
H\mathbf A)h\psi|^2\,dx+\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)}(a(x,\kappa)-|h\psi|^{2})^2\,dx\\
&\geq\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} a(x,\kappa)^{2}\,dx-\kappa^{2}\int_{Q_{\ell}(x_0)}a(x,\kappa)|h\psi|^{2}\,dx\,.
\end{align*}
Using \eqref{app-a}, \eqref{eq-psi<a} and the assumptions on $h$, the simple decomposition $a(x,\kappa)=a(\widetilde{x}_{0},\kappa)+(a(x,\kappa)-a(\widetilde{x}_{0},\kappa))$ yields for any $\delta\in(0,1)$
\begin{align}\label{up-a2}
\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} a(x,\kappa)^{2}\,dx&\geq (1-\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} a(\widetilde{x}_{0},\kappa)^{2}\,dx\nonumber\\
&\qquad\qquad\qquad+(1-\delta^{-1})\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} (a(x,\kappa)-a(\widetilde{x}_{0},\kappa))^{2}\,dx\nonumber\\
&\geq (1-\delta)\,\frac{\kappa^{2}}{2}\,a(\widetilde{x}_{0},\kappa)^{2}\,|Q_{\ell}(x_0)|-C\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}\,|Q_{\ell}(x_0)|\,,
\end{align}
and
\begin{align}\label{up-apsi}
-\kappa^{2}\int_{Q_{\ell}(x_0)} a(x,\kappa)|h\psi|^{2}\,dx&\geq -\kappa^{2}\int_{Q_{\ell}(x_0)} a(\widetilde{x}_{0},\kappa)|h\psi|^{2}\,dx-C\,\ell\,L(\kappa)\,\kappa^{2}\,|Q_{\ell}(x_0)|\nonumber\\
&\geq-C\,\ell\,L(\kappa)\,\kappa^{2}\,|Q_{\ell}(x_0)|\,.
\end{align}
Collecting \eqref{up-a2} and \eqref{up-apsi}, we get,
\begin{equation}\label{es of A2}
\frac{1}{|Q_{\ell}(x_{0})|}\mathcal E_0(h\psi,\mathbf A;a,Q_{\ell}(x_0))\geq (1-\delta)\,\frac{\kappa^{2}}{2}\,a(\widetilde{x}_{0},\kappa)^{2}-C\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}-C'\,\ell\,L(\kappa)\,\kappa^{2}\,.
\end{equation}
\textbf{Now, we treat the case when $a(\widetilde{x}_{0},\kappa)>0\,$.} Let $\phi_{x_{0}}(x)=(\mathbf A(x_0)-\mathbf F(x_0))\cdot x$, where $\mathbf F$ is the magnetic potential introduced in \eqref{div-curlF}. Using the estimate of $\|\mathbf A-\mathbf F\|_{C^{0,\beta}(\Omega)}$ given in Proposition~\ref{pr-est}, we get for any $\beta\in (0,1)$ the existence of a constant $C$ such that for all $x\in Q_{\ell}(x_0)$,
\begin{equation}\label{alpha}
|\mathbf A(x)-\nabla\phi_{x_0}-\mathbf F(x)|\leq C\,\frac{\ell^{\beta}}{H}\,.
\end{equation}
Let $\widetilde{x}_{0}\in\overline{ Q_{\ell}(x_{0})}$ and $\varphi=\varphi_{x_{0},\widetilde{x}_{0}}+\phi_{x_{0}}$ with $\varphi_{x_{0},\widetilde{x}_{0}}$ satisfying \eqref{F-A}. We define the function in $Q_{\ell}(x_{0})$,
\begin{equation}\label{defu}
u(x)=e^{-i\kappa H\varphi}h\psi(x)\,.
\end{equation}
Similarly to \eqref{up-a}, we have, for any $\delta\in(0,1)$,
\begin{align}\label{lw-a}
\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(x,\kappa)-|h\psi|^{2}\right)^{2}\,dx\geq(1-\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-|h\psi|^{2}\right)^{2}\,dx-C\delta^{-1}\kappa^{2}\ell^{4}L(\kappa)^{2}\,.
\end{align}
Using the same techniques as in \cite[Lemma~4.1]{KA}, we get, for any $\beta\in(0,1)$,
\begin{multline}\label{lw-A}
\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H \mathbf A)h\psi|^{2}\,dx\geq (1-\delta)\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H(\zeta_{\ell}|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_{0})+\nabla\varphi(x)))h\psi|^{2}\,dx\\
-C\delta^{-1}(\kappa H)^{2}\left(\ell^{4}+\frac{\ell^{2\beta}}{H^{2}}\right)\int_{Q_{\ell}(x_0)}|h\psi|^2\,dx\,.
\end{multline}
Thus, by collecting \eqref{lw-a} and \eqref{lw-A}, using \eqref{a3}, \eqref{eq-psi<a} and $\|h\|_{L^{\infty}(\Omega)}\leq 1$, we get
\begin{multline}\label{es of A}
\mathcal E_0(h\psi,\mathbf A;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\geq (1-\delta)\mathcal
E_0(e^{-i\kappa H\varphi}h\psi(x),\zeta_{\ell}|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_{0});a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\\
-C\delta^{-1}\kappa^{2}\ell^{4}L(\kappa)^{2}
-C_{1}\delta^{-1}\kappa^{2}H^{2}\left(\ell^{4}+\frac{\ell^{2\beta}}{H^{2}}\right)\ell^{2}\,.
\end{multline}
Let $R$ and $b$ be as in \eqref{def-Rb}. Let us introduce the function $v_{\ell,x_0,\widetilde{x}_{0}}$ in $Q_{R}$ as follows:
\begin{equation}\label{def-v}
v_{\ell,x_0,\widetilde{x}_{0}}(x)=\begin{cases}
u\left(\frac{\ell}{R} x+x_{0}\right)&{\rm if}~x\in Q_{R}\subset\{B_{0}>\rho\}\cap\Omega\\
\overline{u}\left(\frac{\ell}{R} x+x_{0}\right)&{\rm if}~x\in Q_{R}\subset\{B_{0}<-\rho\}\cap\Omega\,,
\end{cases}
\end{equation}
where $u$ is defined in \eqref{defu}.\\
Similarly to \eqref{2up-loc-en}, we use the change of variable $y=\frac{R}{\ell}(x-x_{0})$ and get
\begin{equation}\label{E0<}
\mathcal E_0(e^{-i\kappa H\varphi}h\psi(x),\zeta_{\ell}\,\kappa H|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_0);a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))=\frac{1}{b} F^{+1,a(\widetilde{x}_{0},\kappa)}_{b,Q_{R}}(v_{\ell,x_{0},\widetilde{x}_{0}})\,,
\end{equation}
where $F^{+1,a(\widetilde{x}_{0},\kappa),}_{b,Q_{R}}$ is introduced in \eqref{eq-GL-F}.\\
Since $v_{\ell,x_{0},\widetilde{x}_{0}}\in H^{1}(Q_{R})$ then, using \eqref{eN>eD} and \eqref{est-f(b)-eD}, we get
\begin{align}\label{F>f}
\frac{1}{b}F^{+1,a(\widetilde{x}_{0},\kappa)}_{b,Q_{R}}(v_{\ell,x_{0},\widetilde{x}_{0}})&\geq \frac{1}{b} e_{N}\left(b,R,a(\widetilde{x}_{0},\kappa)\right)\nonumber\\
&\geq \frac{1}{b}e_{D}\left(b,R,a(\widetilde{x}_{0},\kappa)\right)-C_{M}\,a(\widetilde{x}_{0},\kappa)^{\frac{3}{2}}\frac{R}{\sqrt{b}}\nonumber\\
&\geq a(\widetilde{x}_{0},\kappa)^{2}\frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)-\widehat{C}_{M}\,\frac{R}{\sqrt{b}}\,.
\end{align}
Inserting \eqref{F>f} into \eqref{E0<}, we get
\begin{multline}\label{E0<2}
\mathcal E_0(e^{-i\kappa H\varphi}h\psi(x),\zeta_{\ell}\,\kappa H|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_0);a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\geq a(\widetilde{x}_{0},\kappa)^{2}\frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)\\
-\widehat{C}_{M}\frac{R}{\sqrt{b}}\,.
\end{multline}
Having in mind \eqref{def-Rb} and \eqref{E0<2}, we get from \eqref{es of A},
\begin{multline}\label{E0<4}
\frac{1}{|Q_{\ell}(x_{0})|}\mathcal E_0(h\psi,\mathbf A;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\geq(1-\delta)\kappa^{2}a(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(\widetilde{x}_{0})|}{a(\widetilde{x}_{0},\kappa)}\right)\\
-C\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}
-C_{1}\delta^{-1}\kappa^{2}H^{2}\left(\ell^{4}+\frac{\ell^{2\beta}}{H^{2}}\right)
-C_{2}\frac{\kappa}{\ell}\,.
\end{multline}
The estimates in \eqref{es of A2} and \eqref{E0<4} achieve the proof of Proposition~\ref{prop-lb}.
\end{proof}
\begin{app}
We keep the same choice of $\ell$, $\rho$, $L(\kappa)$ and $\delta$ as in \eqref{choice-ell-rho}, \eqref{choice-delta} and choose:
\begin{equation}\label{choice-delta-alpha}
\beta=\frac{3}{4}\,.
\end{equation}
This choice and Assumption~\eqref{cond-H} permit to have the assumptions in Proposition~\ref{prop-lb} satisfied and make the error terms in its statement
of order $\textit{o}(\kappa^{2})$.
We have as $\kappa\longrightarrow\infty\,$,
$$\delta^{-1}\kappa^{4}\ell^{4}=\kappa^{\frac{21}{12}}\ll \kappa^{2}\,,$$
$$\delta^{-1}\kappa^{2}\ell^{2\beta}=\kappa^{\frac{29}{24}}\ll \kappa^{2}\,,$$
$$\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}=\kappa^{\frac{23}{12}}\ll\kappa^{2}\,,$$
$$\frac{\kappa}{\ell}=\kappa^{\frac{19}{12}}\ll \kappa^{2}\,,$$
$$
\ell\,L(\kappa)\,\kappa^{2}=\kappa^{\frac{23}{12}}\ll\kappa^{2}\,,
$$
$$\ell^{2}\kappa H\rho=\kappa^{\frac{3}{24}}\gg 1\,.$$
\end{app}
The next theorem presents a lower bound of the local energy in a relatively compact smooth domain $\mathcal{D}$ in $\Omega$. We deduce the lower bound of the global energy by replacing $\mathcal{D}$ by $\Omega$.
\begin{theorem}\label{lw-Eg}~\\
Under Assumptions~\eqref{B(x)}-\eqref{a4}, if \eqref{cond-H} holds, $L(\kappa)\leq C\,\kappa^{\frac{1}{2}}$ with $C>0$, $h\in C^{1}(\overline{\Omega})$, $\|h\|_{\infty}\leq 1$, $(\psi,\mathbf A)$ is a minimizer of \eqref{eq-2D-GLf} and $\mathcal D$ an open set in $\Omega$, then as $\kappa\longrightarrow+\infty$,
\begin{multline}\label{fianl-Eg1}
\mathcal{E} (h\psi,\mathbf A;a,B_{0},\mathcal{D})\geq\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D}) \geq \kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
\end{theorem}
\begin{proof}
The proof is similar to the one in Theorem~\ref{up-Eg} and we keep the same notation. Let
$$
\mathcal{D}^{+}_{\ell,\rho}={\rm int}\left(\cup_{\gamma\in \mathcal I_{\ell, \rho}^{+}} \overline{Q_{\gamma,\ell}}\right)\qquad{\rm and}\qquad\mathcal{D}^{-}_{\ell,\rho}={\rm int}\left(\cup_{\gamma\in \mathcal I_{\ell, \rho}^{-}} \overline{Q_{\gamma,\ell}}\right)\,,
$$
where $\gamma\in \mathcal I_{\ell, \rho}^{+}$ and $\gamma\in \mathcal I_{\ell, \rho}^{-}$ are introduced in \eqref{def-card-squares}.\\
Thanks to Proposition~\ref{prop-lb}, we can easily prove the existence of positive constant $C$ such that for any $\delta\in(0,1)$ and $\beta\in(0,1)$,
\begin{multline*}
\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\geq \kappa^{2}(1-\delta)\left\{\int_{\mathcal{D}^{+}_{\ell,\rho}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\right.\\
\left.+\frac{1}{2}\int_{\mathcal{D}^{-}_{\ell,\rho}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}-C\,r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)\,,
\end{multline*}
where
\begin{equation}\label{asympr}
r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)=\kappa^{2}\ell+\kappa^{2}\rho+\frac{\kappa}{\ell}+\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{4}\ell^{4}+\delta^{-1}\kappa^{2}\ell^{2\beta}+\ell\,L(\kappa)\,\kappa^{2}\,.
\end{equation}
Notice that using the regularity of $\partial\mathcal{D}$, \eqref{B(x)} and \eqref{a4} (see \eqref{defA4}), we get the existence of constants $C_{1}>0$ and $C_{2}>0$ such that,
\begin{equation}\label{eq:D/D+-}
\forall \ell\leq C_{2}\,\kappa^{-\frac{1}{2}}\,,\quad \forall \rho\in(0,1)\,,\qquad |\mathcal{D}\setminus\mathcal{\mathcal{D}^{+}_{\ell,\rho}}|+ |\mathcal{D}\setminus\mathcal{\mathcal{D}^{-}_{\ell,\rho}}|\leq C_{1}(\kappa^{\frac{1}{2}}\,\ell+\rho)\,.
\end{equation}
This implies by using \eqref{a3} and the upper bound $\hat{f}\leq \frac{1}{2}$,
\begin{multline}\label{eq:1st-main}
\int_{\mathcal{D}^{+}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\geq\int_{\mathcal{D}^{+}_{\ell,\rho}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
-\frac{1}{2}\,\overline{a}\,|\mathcal{D}\setminus\mathcal{\mathcal{D}_{\ell,\rho}}|
\end{multline}
and
\begin{equation}\label{eq:2nd-main}
\frac{1}{2}\int_{\mathcal{D}^{-}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\geq\frac{1}{2}\int_{\mathcal{D}_{\ell,\rho}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx-\frac{1}{2}\,\overline{a}\,|\mathcal{D}\setminus\mathcal{\mathcal{D}^{-}_{\ell,\rho}}|\,,
\end{equation}
where $\overline{a}$ is introduced in \eqref{def:sup-a}.\\
Collecting \eqref{eq:1st-main} and \eqref{eq:2nd-main}, using Assumptions \eqref{a2} and \eqref{eq:D/D+-}, we find that,
\begin{multline}\label{fianl-E0}
\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\geq \kappa^{2}(1-\delta)\left\{\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\right.\\
\left.+\frac{1}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}-C\,\hat r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)\,,
\end{multline}
where $\hat r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)$ satisfies \eqref{asympr}.\\
Under Assumption~\eqref{cond-H}, the choice of the parameters $\rho$, $\ell$, $L(\kappa)$ in \eqref{choice-ell-rho}, $\delta$ in \eqref{choice-delta} and $\beta$ in \eqref{choice-delta-alpha}, implies that all error terms are of lower order compared to $\kappa^{2}$.\\
As a consequence of \eqref{cond-H}, the inequality \eqref{fianl-E0} becomes as $\kappa\longrightarrow+\infty$
\begin{multline}\label{fianl-E01}
\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\geq \kappa^{2}\left\{\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{1}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}\\
+\textit{o}(\kappa^{2})\,.
\end{multline}
Moreover, we know that
$$\mathcal{E}(h\psi,\mathbf A;a,B_{0},\mathcal{D})\geq \mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\,.$$
This achieves the proof of Theorem~\ref{lw-Eg}.
\end{proof}
As we now show, Theorem~\ref{lw-Eg} permits to achieve the proof of two statements presented in the introduction:
\begin{proof}[\textbf{Proof of Corollary~\ref{corol-2D-main}}]~\\
If $(\psi,\mathbf A)$ is a minimizer of \eqref{eq-2D-GLf}, we have
\begin{equation}\label{eq-glob-en}
\E0(\kappa,H)=\mathcal E_0(\psi,\mathbf A;a,\Omega) + (\kappa H)^2
\int_{\Omega} |\curl\big(\mathbf A - \mathbf F\big)|^2\,dx \,,
\end{equation}
where $\mathcal E_{0}(\psi,\mathbf A;a,\Omega)$ is defined in \eqref{eq-GLe0}.\\
Using \eqref{eq-2D-thm} and \eqref{fianl-E01} (with $\mathcal{D}=\Omega$), then under Assumption~\eqref{cond-H} as $\kappa\longrightarrow+\infty$
\begin{equation}\label{eq-2D}
\mathcal E_{0}(\psi,\mathbf A;a,\Omega)=\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,.
\end{equation}
Putting \eqref{eq-2D} and \eqref{eq-2D-thm} into \eqref{eq-glob-en}, we finish the proof of Corollary~\ref{corol-2D-main}.
\end{proof}
~\\
\begin{proof}[\textbf{Proof of Theorem~\ref{lc-en}}.]~\\
Noticing that \eqref{fianl-E01} is valid when $h=1$ and $\mathcal{D}$ replaced by $\mathcal{\overline{D}}^{c}:=\Omega \setminus \overline{\mathcal D}$ for any open domain $\mathcal{D}\subset\Omega$ with smooth boundary, then we get:
\begin{multline}\label{fianl-E01c}
\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{\overline{D}}^{c})\geq \kappa^{2}\left\{\int_{\mathcal{\overline{D}}^{c}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\right.\\
\left.+\frac{1}{2}\int_{\mathcal{\overline{D}}^{c}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}+\textit{o}(\kappa^{2})\,.
\end{multline}
We can decompose $\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{D})$ as follow:
$$
\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{D})=\mathcal{E}_{0} (\psi,\mathbf A;a,\Omega)-\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{\overline{D}}^{c})\,.
$$
Using \eqref{eq-2D} and \eqref{fianl-E01c}, we get
\begin{multline}\label{fianl-E01-l}
\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{D})\leq \kappa^{2}\left\{\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{1}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}\\
+\textit{o}(\kappa^{2})\,.
\end{multline}
\end{proof}
\section{study of examples}\label{examples}
In this section, we will describe situations where the remainder term in \eqref{eq-2D-thm} is indeed small as $\kappa \rightarrow +\infty$ compared with the leading order term
\begin{equation}\label{def:leading-term}
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0}):=\kappa^{2} \left(\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{1}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx\right)\,,
\end{equation}
where,
\begin{equation}\label{def:sigma}
\sigma=\frac{H}{\kappa}\,.
\end{equation}
Note that $0<\lambda_{\min}\leq\sigma\leq\lambda_{\max}$, so that $\sigma$ will be considered as an independent parameter in $[\lambda_{\min}\,,\,\lambda_{\max}]$.\\
We will also explore, case by case how one can verify Assumption $(A_4)$ as formulated precisely in \eqref{defA4}.
\subsection{The case of a $\kappa$-independent pinning}~\\
\begin{prop}
Suppose \eqref{B(x)} and \eqref{cond-H} hold. Let $a(x,\kappa)=a(x)$ where $a(x)\in C^{1}(\overline{\Omega})$ is a function independent of $\kappa$ and satisfies,
\begin{equation}\label{cond-a1}
\left\{
\begin{array}{lll}
\{x\in\Omega:\,a(x)>0\}\neq\varnothing\,,\\
{\rm or}\\
\{x\in\Omega:\,a(x)<0\}\neq\varnothing\,.
\end{array}
\right.
\end{equation}
There exist positive constants $C$ and $\kappa_{0}$ such that,
$$
\forall\kappa\geq\kappa_{0}\,,\qquad E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})\geq C\,\kappa^{2}\,.
$$
\end{prop}
\begin{proof}
Since $a(x,\kappa)=a(x)$, the energy $E_{\rm g}^{\textbf{L}}$ becomes:
$$
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0}):=\kappa^{2} \left(\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x)}\right)\,dx\\
+\frac{1}{2}\int_{\{a(x)\leq 0\}}a(x)^{2}\,dx\right)\,.
$$
Each term being positive, it is clear that the leading term is positive if $\{x\in\Omega:\,a(x)<0\} \neq\varnothing$.\\
If $ \{x\in\Omega:\,a(x)<0\}=\varnothing$ and $\{x\in\Omega:\,a(x)>0\} \neq\varnothing$, there exist $\rho_{0}>0$, $a_{0}>0$ and a disk $D(x_{0},r_{0})$ such that
$$
D(x_{0},r_{0})\subset\{a(x)>a_{0}\}\cap\{|B_{0}|>\rho_{0}\}\,.
$$
Using the monotonicity of $\hat{f}$ and the bound of $a(x)$ in \eqref{a2}, we may write
\begin{align}\label{est:a-k-ind}
\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x)}\right)\,dx&\geq \int_{D(x_{0},r_{0})} a(x)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x)}\right)\,dx\nonumber\\
&\geq \pi\,r_{0}^{2}\,a_{0}^{2}\,\hat{f}\left(\frac{\rho_{0}}{\overline{a}}\sigma\right)\,,
\end{align}
where $\overline{a}$ is introduced in \eqref{def:sup-a}.\\
In particular, when \eqref{cond-H} is satisfied, there exists $\kappa_{0}>0$ such that
\begin{equation}\label{ex1}
\forall\kappa\geq \kappa_{0}\,,\qquad\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x)}\right)\,dx\geq \pi\,r_{0}^{2}\,a_{0}^{2}\,\hat{f}\left(\frac{\rho_{0}}{\overline{a}}\lambda_{\min}\right)\,.
\end{equation}
\end{proof}
\begin{figure}
\caption{Schematic representation of $\Omega$ with pinning term independent of $\kappa$ and with variable magnetic field.}
\label{example:1}
\end{figure}
\begin{prop}[\textbf{Verification of $(A_{4})$}]
Suppose that the function $a$ satisfies (see Fig.\ref{example:1}),
\begin{equation}\label{cond-a}
\left\{
\begin{array}{ll}
|a| + |\nabla a | >0&\mbox{ in } \overline{\Omega}\,,\\
\nabla a\times\vec{n}\neq 0 &\mbox{on}~ \widetilde{\Gamma}\cap\partial\Omega\,,
\end{array}
\right.
\end{equation}
where $\widetilde{\Gamma}$ defined as follows:
\begin{equation}\label{gamma-tilde}
\widetilde{\Gamma}=\{x\in\overline{\Omega}: a(x)=0\}\,.
\end{equation}
Then Assumption $(A_{4})$ is satisfied.
\end{prop}
\begin{proof}
From \eqref{cond-a}, we observe that,
$$
{\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{a >0\} \neq \emptyset\}={\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \widetilde{\Gamma} \neq \emptyset\}\,.
$$
Let $\epsilon\in(0,1)$, we introduce the domain
$$
D_{\epsilon}=\{x\in\Omega: \dist(x,\widetilde{\Gamma})\leq \epsilon\}\,.
$$
\textbf{Now we give a rough upper bound for the area of $D_\epsilon$.}\\
By assumption $\widetilde\Gamma$ consists of a finite number of connected curves, which are either closed in $\Omega$ or join two points of $\partial\Omega$. Let us consider the first case, we denote by $\widetilde\Gamma^{(1)}$ such a curve. We can parametrize this curve using the standard tubular coordinates $(s,t)$, where $s$ measures the arc-length in $\widetilde\Gamma^{(1)}$ and $t$ measures the distance to $\widetilde\Gamma^{(1)}$ (see \cite[Appendix~F]{FH1} for the detailed construction of these coordinates).
In the neighborhood of $\widetilde{\Gamma}^{(1)}$, we choose one point $\gamma_{0}$ on $\widetilde{\Gamma}^{(1)}$ corresponding to $(0,0)$. Let $N\in\mathbb N$ and $\mathcal{L}$ the length of $\widetilde\Gamma^{(1)}$. We consider for $i=0,...,N$, $s_{i}=\frac{i}{N}\,\mathcal{L}\,\,({\rm modulo}\,\,\mathcal{L}\mathbb Z)$ and $\gamma_{i}=(s_{i},0)$.\\
Notice that, there exists a positive constant $C$ such that,
$$
|\dist(\gamma_{i},\gamma_{i+1})|=(1+\epsilon_{i})|s_{i}-s_{i+1}|\,,\qquad\left(-\frac{C}{N}\leq \epsilon_{i}<0\right)\,.
$$
Thus,
\begin{align*}
\left|\left\{x\in\Omega: \dist\Big(x,\widetilde{\Gamma}^{(1)}\Big)\leq\frac{\mathcal{L}}{N}\right\}\right|&\leq \sum_{i} \left|Q_{\frac{\mathcal{L}}{N}}((s_i,0))\right|\,.
\end{align*}
Coming back to our problem, we select $N=\left[\frac{\mathcal{L}}{\epsilon}\right]$ and we note that $$\frac{\mathcal{L}}{N +1} \leq \epsilon\leq \frac{\mathcal{L}}{N}\,,$$
which implies that,
\begin{align*}
|D_{\epsilon}|&\leq \frac{\mathcal{L}^2}{N}\left(1+\mathcal{O}\left(\frac{1}{N}\right)\right)\\
&\leq \mathcal{L}\,\epsilon\,\left(1+\mathcal{O}\left(\frac{1}{N}\right)\right) = \epsilon \mathcal L (1+ \mathcal O(\epsilon))\,.
\end{align*}
Hence we have shown that,
$$\limsup_{\epsilon\to 0}\frac{|D_{\epsilon}|}{\epsilon}\leq\mathcal{L}\,.$$
In a similar fashion, we prove that
$$\liminf_{\epsilon\to 0}\frac{|D_{\epsilon}|}{\epsilon}\geq\mathcal{L}\,.$$
and, as a consequence, we end up with the following conclusion:
\begin{equation}\label{Area:D}
\lim_{\epsilon\to 0}\frac{|D_{\epsilon}|}{\epsilon}=\mathcal{L}\,.
\end{equation}
Coming back to Assumption $(A_4)$,
we now observe that all the $Q_{\ell}(\gamma)$ touching $\widetilde \Gamma$ are inside $D_{\sqrt{2}\ell}$, hence
we get, by comparison of the area
$$
\ell^2 {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \widetilde{\Gamma} \neq \emptyset\}\leq C\,\ell \,,
$$
and consequently,
there exist positive constants $C_1$, $C_{2}$ and $\kappa_{0}$ such that
$$
\forall \kappa \geq \kappa_0\,,\, \forall \ell \leq C_2 \kappa^{-\frac 12}\,,\, {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{a >0\} \neq \emptyset\}\leq C_{1}\,\ell^{-1}\,,
$$
which is a stronger form of $(A_4)$.
\end{proof}
\subsection{The case with a $\kappa$-dependent oscillation. }
\subsubsection{Preliminaries}
We start with two lemmas which are standard in homogenization theory (see \cite[Section~16-17]{BLP})
\begin{lem}\label{lem:a-}
Let $D\subset\mathbb R^{2}$ be a bounded open set and $\varphi$ be a $\Gamma_{T_{1},T_{2}}$-periodic continuous function in $\mathbb R^{2}$ with $\Gamma_{T_{1},T_{2}}=T_{1}\mathbb Z\times T_{2}\mathbb Z$. There exists a positive constant $M_{0}$ such that if $M\geq M_{0}$, then,
$$
\int_{D}\varphi(M x)\,dx=\frac{|D|}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\varphi(t_{1},t_{2})dt_{1}dt_{2}+\mathcal{O}(M^{-1})\,.
$$
\end{lem}
\begin{lem}\label{lem:a+}
Let $D\subset\mathbb R^{2}$ be a bounded open set
and $\phi:\mathbb R^{2}\times \overline{D} \longrightarrow\mathbb R^{2}$ be a continuous function satisfying:
\begin{equation}\label{1st:cond}
\phi(t+T,x)=\phi(t,x)\,,\qquad\forall T\in T_{1}\mathbb Z\times T_{2}\mathbb Z\,,
\end{equation}
and uniformly Lipschitz, i.e. with the property that there exist constants $C>0$ and $\epsilon_0$, such that,
\begin{equation}\label{2nd:cond}
|\phi(t,x)-\phi(t,\widetilde{x})|\leq C\, |x-\widetilde{x}|\,,\quad \, \forall t \in \mathbb R^2\,, \,\forall x, \widetilde{x}\in\overline{D}, \; {\rm s.t.}\, |x-\widetilde x|<\epsilon_0\,.
\end{equation}
There exists a positive constant $M_{0}$ such that if $M\geq M_{0}$, then,
$$
\int_{D}\phi(M x,x)\,dx=\int_{D}\overline{\phi}(x)\,dx+\mathcal{O}(M^{-1})\,,
$$
where,
\begin{equation}\label{def:phi}
\overline{\phi}(x)=\frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\phi((t_{1},t_{2}),x)\,dt_{1}dt_{2}\,.
\end{equation}
\end{lem}
\subsubsection{First example:}
\begin{prop}\label{prop:1st-ex}
Suppose that \eqref{B(x)} and \eqref{cond-H} hold. Let $
a(x,\kappa)=\alpha(\kappa^{\frac{1}{2}}\,x)
$
where $\alpha(\cdot)\in C^{1}(\overline{\Omega})$ is a $\Gamma_{T_{1},T_{2}}$-periodic function\footnote{see Fig.~\ref{example:3}}. Then the leading order term $E_{\rm g}^{\textbf{L}}$ defined in \eqref{def:leading-term} satisfies,
$$
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})=\kappa^{2} \, \int_{\Omega}\overline{\phi}_+(x)\,dx+\kappa^{2} |\Omega| \, \overline{\phi}_{-} +\textit{o}(\kappa^{2})\,,\quad{\rm as}~\kappa\to+\infty\,.
$$
Here,
$$
\overline{\phi}_{+}(x)=\frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{+}(t_{1},t_{2})^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t_{1},t_{2})}\right)\,dt_{1}dt_{2}\,,
$$
and
$$
\overline{\phi}_{-} = \frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{-}(t_{1},t_{2})^2\,dt_{1}dt_{2}\,.
$$
\end{prop}
\begin{proof}~\\
{\bf We first estimate the second term in \eqref{def:leading-term}.} We apply Lemma~\ref{lem:a-} with $D=\Omega$, $ M=\kappa^\frac{1}{2}$ and $\varphi=\alpha_{-}^2$, we obtain,
$$
\int_{\Omega}a_{-}(x,\kappa)^{2}\,dx=\frac{|\Omega|}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{-}(t_{1},t_{2})^{2}\,dt_{1}dt_{2}+\mathcal{O}(\kappa^{-\frac{1}{2}})\,,
$$
and consequently,
$$
\kappa^{2}\int_{\{a(x)\leq 0\}}a(x,\kappa)^{2}\,dx=\kappa^{2}\,\frac{|\Omega|}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{-}(t_{1},t_{2})^2 \,dt_{1}dt_{2}+\mathcal{O}(\kappa^{\frac{3}{2}})\,.
$$
{\bf Now, we estimate the first term in \eqref{def:leading-term}.}
We first prove that $\hat{f}$ is a Lipschitz function in $[\mathfrak{b}_{0},1]$ with $\mathfrak{b}_{0}\in(0,1)$. We consider this restriction because when $\mathfrak{b}\to 0_{+}$ (see \cite[Theorem~2.1]{KA2}), $\hat{f}$ satisfies,
\begin{equation}\label{ashatf}
\hat{f}(\mathfrak{b})=\frac{\mathfrak{b}}{2}\ln\frac{1}{\mathfrak{b}}(1+\textit{o}(1))\,,
\end{equation}
and $\hat{f}$ is not a Lipschitz function at $0$. We recall the definition of $\hat{f}$
$$
\displaystyle \hat{f}\left(\mathfrak{b}\right)=\lim_{R\longrightarrow\infty}\frac{e_{D}(\mathfrak{b},R)}{R^{2}}\qquad(\forall \mathfrak{b}\in[0,1])\,,
$$
where
$$
e_{D}(\mathfrak{b},R)=\inf_{u}F^{+1,+1}_{\mathfrak{b},Q_{R}}(u):=\inf_{u}\int_{Q_{R}}\left(\mathfrak{b}|(\nabla-i\mathbf A_0)u|^2+\frac{1}{2}\left(1-|u|^2\right)^{2}\right)\,dx\,.
$$
From the definition, we can conclude that $\hat f$ is concave and hence locally Lipschitz in $(0,+\infty)$ (see \cite[Theorem~2.35]{MG}). For completion we write below a proof making explicit the Lipschitz constant. For $\mathfrak{b}' >0$, let $u_{\mathfrak{b}',R}\in H^{1}_{0}(Q_{R})$ be a minimizer of $F^{+1,+1}_{\mathfrak{b}',Q_{R}}$. Then for all $\mathfrak{b}\in (0,1)$, we have,
$$
e_{D}(\mathfrak{b},R)\leq F^{+1,+1}_{\mathfrak{b},Q_{R}} (u_{\mathfrak{b}',R})\leq e_{D}(\mathfrak{b}',R)+\|(\nabla-i\mathbf A_0)u_{\mathfrak{b}',R}\|^{2}_{L^{2}(Q_{R})}|\mathfrak{b}-\mathfrak{b}'|\,.
$$
Now, we estimate $\|(\nabla-i\mathbf A_0)u_{\mathfrak{b}',R}\|^{2}_{L^{2}(Q_{R})}$ from above. Coming back to the definition, we get the existence of a positive constant $C$, such that for any $\mathfrak{b}\in[\mathfrak{b}_0,1]$ and for any $\mathfrak{b}'\in[\mathfrak{b}_0,1]$,
\begin{align}\label{eq:est-from-above2}
\|(\nabla-i\mathbf A_0)u_{\mathfrak{b}',R}\|^{2}_{L^{2}(Q_{R})}&\leq \frac{e_{D}(\mathfrak{b}',R)}{\mathfrak{b}'}\nonumber \,.
\end{align}
This implies that,
$$
e_{D}(\mathfrak{b},R)\leq e_{D}(\mathfrak{b}',R)+ \frac{e_{D}(\mathfrak{b}',R)}{\mathfrak{b}'} |\mathfrak{b}-\mathfrak{b}'|\,.
$$
Dividing by $R^2$ and taking the limit as $R\rightarrow +\infty$, we obtain
$$
\hat f (\mathfrak{b}) \leq \hat f (\mathfrak{b}') + \frac{ | \hat f (\mathfrak{b}') |}{\mathfrak{b}'} |\mathfrak{b}-\mathfrak{b}'|\,.
$$
Using the asymptotic behavior of $\hat f$ in \eqref{ashatf} as $\mathfrak{b}'\rightarrow 0_{+}$, we finally obtain the existence of $C$ such that
$$
\hat f (\mathfrak{b}) \leq \hat f (\mathfrak{b}') + C \left(\log \frac {1}{\mathfrak{b}_0}\right) \, |\mathfrak{b}-\mathfrak{b}'|\,,\, \forall \mathfrak{b}, \mathfrak{b}' \mbox{ with } 1 > \mathfrak{b}>\mathfrak{b}_{0} \mbox{ and } 1 > \mathfrak{b}' >\mathfrak{b}_0\,.
$$
Exchanging $\mathfrak{b}$ and $\mathfrak{b}'$, we have proved the
\begin{lemma} $\hat f$ is locally Lipschitz in $(0,+\infty)$. More precisely, there exists $C$ such that for any $\mathfrak{b}_0 >0$,
\begin{equation}
| \hat f (\mathfrak{b}) - \hat f (\mathfrak{b}')| \leq C \,\left(\log \frac {1}{\mathfrak{b}_0}\right) \, |\mathfrak{b}-\mathfrak{b}'|\,,\, \forall \mathfrak{b}, \mathfrak{b}' \mbox{ with } 1>\mathfrak{b}>\mathfrak{b}_0 \mbox{ and } 1 > \mathfrak{b}' >\mathfrak{b}_0\,.
\end{equation}
In addition, we have
\begin{equation}
| \hat f (\mathfrak{b}) - \hat f (\mathfrak{b}')| \leq 2 \, |\mathfrak{b}-\mathfrak{b}'|\,,\, \forall \mathfrak{b}, \mathfrak{b}' \mbox{ with } \mathfrak{b}>\frac 12 \mbox{ and } \mathfrak{b}' > \frac 12\,.
\end{equation}
\end{lemma}
To continue, we consider
$$ \mathbb R^2 \times \Omega_\rho \ni (t,x) \mapsto \phi(t,x)=\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)\,,$$
where, $\Omega_{\rho}:=\Omega\cap\{|B_0|>\rho\}$.\\
The periodicity condition in \eqref{1st:cond} is clear. Let us verify the Lipschitz property.
Let
$$
\mathfrak{b}_{0}=\frac{\lambda_{\min}}{\alpha_{0}}\,\rho \,,
$$
where, $\lambda_{\min}$ is introduced in \eqref{cond-H} and $\alpha_{0}=\sup \alpha_{+}(t)$.\\
Let $\epsilon>0$, $\mathcal{I}_{+}=\{t\in\mathbb R: \alpha_{+}(t)\geq \epsilon\}$ and $\mathcal{I}_{-}=\{t\in\mathbb R: \alpha_{+}(t)\leq \epsilon\}$, we distinguish between two cases:\\
\textbf{Case 1:} $( \alpha_{+}(t)\geq \epsilon)$. We observe that for $(x,t) \in \Omega_\rho\times \mathcal{I}_{+}$, we have
$$
\mathfrak{b}_0\leq\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\leq \frac{\sigma\,|B_{0}(x)|}{\epsilon} \,.
$$
Thus, for any $t\in \mathcal{I}_{+}$ and for any $x,x'\in\overline{\Omega}_{\rho}$, we get
\begin{align}\label{eq:lipschitz1}
\left|\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)-\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x')|}{\alpha_{+}(t)}\right)\right|&=\alpha_{+}(t)^{2}|\hat{f}\left(\mathfrak{b}\right)-\hat{f}\left(\mathfrak{b}'\right)|\nonumber\\
&\le
C\,\left(\log \frac {1}{\rho}\right)\,\Big||B_{0}(x)|-|B_{0}(x')|\Big|\,.
\end{align}
Therefore, using also the Lipschitz property for $x\mapsto |B_0(x)|$, we get that $\Omega_\rho\ni x \mapsto \phi (t,x)$ is uniformly Lipschitz for $t\in \mathcal{I}_{+}$.\\
\textbf{Case 2:} $( \alpha_{+}(t)\leq \epsilon)$. We observe that for $(x,t) \in \Omega_\rho\times\mathcal{I}_{-}$,
$$
\frac{\sigma\,|B_{0}(x)|}{\alpha_{+}(t)}\geq \frac{\sigma\,|B_{0}(x)|}{\epsilon}\,.
$$
We note that $
\hat{f}(\mathfrak{b})=\frac{1}{2},\,\forall \mathfrak{b}\geq 1$ (see \cite[Theorem~2.1]{FK2}). For this reason we choose
$$
\epsilon=\frac{\lambda_{\min}}{2}\rho \,,
$$
which implies that for $(x,t) \in \Omega_\rho\times \mathcal{I}_{-}$,
$$\frac{\sigma\,|B_{0}(x)|}{\alpha_{+}(t)}\geq 2\qquad{\rm and}\qquad\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)=\frac{1}{2}\,.
$$
Thus, for any $t\in\mathcal{I}_{-}$ and for any $x,x'\in\overline{\Omega}_{\rho}$, we get
\begin{align}\label{eq:lipschitz2}
\left|\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)-\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x')|}{\alpha_{+}(t)}\right)\right|&=\left|\frac{\alpha_{+}(t)^{2}}{2}-\frac{\alpha_{+}(t)^{2}}{2}\right|\nonumber\\
&=0\,.
\end{align}
Hence we get that $\Omega_\rho\ni x \mapsto \phi (t,x)$ is uniformly Lipschitz for $t\in \mathcal{I}_{-}\,$.\\
Now, we apply Lemma~\ref{lem:a+} with $D=\Omega_\rho$ and $M=\kappa^\frac 12$ and we obtain,
\begin{align}\label{eq:1st-ex-2}
\int_{\Omega_{\rho}}a_{+}(x,\kappa)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a_{+}(x,\kappa)}\right)\,dx&=\int_{\Omega_{\rho}}\overline{\phi}(x)\,dx+\mathcal{O}_{\rho}(\kappa^{-\frac{1}{2}})\,,
\end{align}
where $\overline{\phi}$ is introduced in \eqref{def:phi}.\\
Coming back to the integral over $\Omega$, we get, for any $\rho\in(0,\rho_{0})$ and for any $\kappa\geq\kappa_{0}$ with $\rho_{0}$ small enough and $\kappa_{0}$ large enough,
\begin{equation}\label{eq:final}
\int_{\Omega}a_{+}(x,\kappa)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a_{+}(x,\kappa)}\right)\,dx=\int_{\Omega}\overline{\phi}(x)\,dx+\mathcal{O}(\rho)+\mathcal{O}_{\rho}(\kappa^{-\frac{1}{2}})\,.
\end{equation}
Here, we have used the fact that $\overline{\phi}$ is a bounded function in $\Omega$.
Let us show that the remainder term $s(\kappa)$ in the right hand side in \eqref{eq:final} is $\textit{o}(1)$. The remainder term has the form $s_{1}(\kappa)+s_{2}(\kappa)$ with $s_{1}(\kappa)=\mathcal{O}(\rho)$ and $s_{2}(\kappa)= \mathcal{O}_{\rho}(\kappa^{-\frac{1}{2}})$. Let us show that it is $o(1)$.
Given $\varepsilon > 0$, there exists $\rho_{\varepsilon}> 0$ such that $|s_{1}(\kappa)|\leq \frac{\varepsilon}{2}$, for all $\kappa \geq \kappa_0$. Then, $\rho=\rho_{\varepsilon}$ being chosen, we can find $\kappa_{\varepsilon}\geq \kappa_0$ such that, for any $\kappa\geq\kappa_{\epsilon}$, $|s_{2}(\kappa)|\leq \frac{\varepsilon}{2}$.
\begin{figure}
\caption{Schematic representation of a domain with a $\kappa$-dependent oscillation pinning and with vanishing magnetic field along $\Gamma$.}
\label{example:3}
\end{figure}
\end{proof}
\begin{prop}[Verification of $(A_{4})$]
Suppose that the function $\alpha$ defined in Proposition~\ref{prop:1st-ex} satisfies
\begin{equation}\label{cond-alpha}
|\alpha| + |\nabla \alpha | >0\quad\mbox{ in } \mathbb R^2\,.
\end{equation}
Then Assumption $(A_{4})$ is satisfied.
\end{prop}
\begin{proof}
Using \eqref{cond-alpha}, a change of variable $y=\kappa^{\frac{1}{2}}\,x$ and $\gamma'=\kappa^{\frac{1}{2}}\,\gamma$ yields,
\begin{align*}
& {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{x\in\Omega:\,a(x,\kappa)>0\} \neq \emptyset\}\\
&\hspace*{7cm}={\rm card}\,\{ \gamma' \in \Gamma_{\kappa^{\frac{1}{2}}\ell} \cap \kappa^{\frac{1}{2}}\Omega \mbox{ with } Q_{\kappa^{\frac{1}{2}}\ell} (\gamma') \cap \widehat{\Gamma} \neq \emptyset\}\,,
\end{align*}
where,
$$
\widehat{\Gamma}=\{y\in \mathbb R^2 \,|\, \alpha(y)=0\}\,.
$$
Let $\epsilon\in(0,1)$, we introduce the domain
$$
\widehat D_{\epsilon, M}=\{y\in M \,\cdot \,\Omega: \dist(y,\widehat{\Gamma})\leq \epsilon\}\,.
$$
Thanks to \eqref{Area:D} and the periodicity assumption, we get the existence of positive constants $C$, $M_0$ and $\epsilon_0$ such that, for any $\epsilon \in (0,\epsilon_0)$, $M\geq M_0$
$$
|\widehat D_{\epsilon, M}|\leq C\,M\,\epsilon\,.
$$
In the sequel, we choose $M=\kappa^{\frac{1}{2}}$ and $\epsilon=M\,\sqrt{2}\,\ell$. We note that, there exist constants $c > 0$ and $ \kappa_{0}> 0$ such that,
$$
\forall \kappa\geq \kappa_{0}\,,\quad\forall \ell\leq c\,\kappa^{-\frac{1}{2}}\,,\qquad 0< \epsilon\leq \epsilon_0 \,.
$$
We now observe that all the $Q_{\kappa^{\frac{1}{2}}\ell}(\gamma)$ touching $\widehat{\Gamma}$ are inside $\widehat D_{\kappa^{\frac{1}{2}}\,\sqrt{2}\,\ell,\kappa^{\frac{1}{2}}}$, hence
we get, by comparison of the areas
$$
\kappa\,\ell^2 {\rm card}\,\{ \gamma' \in \Gamma_{\kappa^{\frac{1}{2}}\ell} \cap \kappa^{\frac{1}{2}}\Omega \mbox{ with } Q_{\kappa^{\frac{1}{2}}\ell} (\gamma') \cap \widehat{\Gamma}_{\kappa} \neq \emptyset\}\leq C\sqrt{2} \,\kappa\,\ell \,.
$$
There exist positive constants $C_1$ and $C_{2}$, such that,
$$
\forall \kappa \geq \kappa_0\,,\, \forall \ell \leq C_2 \kappa^{-\frac 12}\,,\, {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{x\in\Omega:\,a(x,\kappa)>0\} \neq \emptyset\}\leq C_{1}\,\ell^{-1}\,.
$$
\end{proof}
\subsubsection{Second example.}
This example was considered by Aftalion, Sandier and Serfaty (see $(H_2)$).
\begin{prop}
Suppose that \eqref{B(x)} and \eqref{cond-H} hold. Let $a(x,\kappa)=a(x)+\beta(x,\kappa)$, where $\beta(x,\kappa)$ is a nonnegative function and $\{ a>0\}\cap\Omega\neq\varnothing$, (see Fig.~\ref{example:2}). There exist positive constants $\tau_1$ and $\kappa_{0}$ such that,
$$
\forall\kappa\geq\kappa_{0}\,,\qquad E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})\geq \tau_1 \,\kappa^{2}\,.
$$
\end{prop}
\begin{proof}
We can write,
\begin{align}\label{cp-err}
\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx&\geq \kappa^{2}\int_{\{a(x)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\nonumber\\
&\geq \kappa^{2}\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{\overline{a}}\right)\,dx\,.
\end{align}
Here we have used that $\hat{f}$ is increasing, the nonnegativity of $\beta$ to get $a(x,\kappa)\geq a(x)$, Assumption $(A_{2})$ to estimate $\hat{f}$ from below, and $\{a(x)>0\}\subset\{a(x,\kappa)>0\}$.\\
Proceding like in \eqref{est:a-k-ind}, there exist $\tau_{1}>0$ and $\kappa_{0}>0$ such that,
\begin{equation}\label{lb:a-k2}
\forall\kappa\geq \kappa_{0}\,,\qquad\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\geq\tau_{1}\,\kappa^{2}\,.
\end{equation}
\end{proof}
\begin{figure}
\caption{Schematic representation of some domain with pinning term dependent of $\kappa$ and with vanishing magnetic field along $\Gamma$.}
\label{example:2}
\end{figure}
\subsubsection{Third example:} This example is similar to the previous example, but here we suppose that
$$\beta(x,\kappa)=\alpha(\kappa^{\frac{1}{2}}x)\,,$$
where $\alpha(\cdot)$ is a $\Gamma_{T_{1},T_{2}}$-periodic positive function in $\mathbb R^2$.
\begin{prop}\label{prop:3rd-ex}
Suppose that \eqref{B(x)} and \eqref{cond-H} hold. Let $a(x,\kappa)=a(x)+\alpha(\kappa^{\frac{1}{2}}x)$, where $\alpha(\cdot)$ is a $\Gamma_{T_{1},T_{2}}$-periodic positive bounded function in $\mathbb R^2$, $a(\cdot)\in C^{1}(\overline{\Omega})$ and $\{a<0\}\cap\Omega=\varnothing$. Then the leading order term $E_{\rm g}^{\textbf{L}}$ defined in \eqref{def:leading-term} satisfies,
$$
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})=\kappa^{2} \, \int_{\Omega}\overline{\phi}(x)\,dx+\textit{o}(\kappa^{2})\,,\quad{\rm as}~\kappa\to+\infty\,.
$$
Here,
$$
\overline{\phi}(x)=\frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}(a(x)+\alpha(t_1,t_2))^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x)+\alpha(t_1,t_2)}\right)\,dt_{1}dt_{2}\,.
$$
\end{prop}
The proof of Proposition~\ref{prop:3rd-ex} is similar to that of Proposition~\ref{prop:1st-ex}.
\subsection{Upper bound of the main term.}~\\
It is easy to show that $ E_{\rm g}^{\textbf{L}}$ is less than $C\kappa^{2}$ for some $C>0$. Indeed, using the bound of $a$ in \eqref{a2} and the bound $\hat{f}(b)\leq\frac{1}{2}$, we have,
$$
\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\leq C\kappa^{2}\,,
$$
and
$$
\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx\leq C\kappa^{2}\,.
$$
\section{Proof of Theorem~\ref{est-psi-main}}\label{7}
The technique that will be used in this proof has been introduced by Helffer-Kachmar in \cite{HK} for the case $a(x,\kappa)=\,1$. The proof is decomposed into three steps:\\
\textbf{Step 1: Case $\mathcal{D}=\Omega\,$.}\\
Let $(\psi,\mathbf A)$ be a solution of \eqref{eq-2D-GLeq}. Thanks to \eqref{eq:A-psi}, we have,
\begin{align*}
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\psi|^{2}\,dx&=\kappa^{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})|\psi|^{2}\,dx\\
&=\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)^{2}-|\psi (x)|^{4})\,dx-\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})^{2}\,dx\,.
\end{align*}
Having in mind the definition of $\mathcal E_{0}(\psi,\mathbf A;a,\Omega)$, we get,
\begin{equation}
\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)^{2}-|\psi (x)|^{4})\,dx=\mathcal E_{0}(\psi,\mathbf A;a,\Omega)\,.
\end{equation}
Using \eqref{eq-2D}, we get that as $\kappa\longrightarrow+\infty$
\begin{multline} \label{previousone}
\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)^{2}-|\psi(x)|^{4})\,dx=\kappa^{2}\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
Notice that
$$\int_{\Omega}a(x,\kappa)^{2}\,dx=\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\int_{\{a(x,\kappa)> 0\}}a(x,\kappa)^{2}\,dx\,.$$
Therefore, dividing \eqref{previousone} by $\kappa^2$, we get
\begin{equation}\label{asy-psi-Omega}
\int_{\Omega}|\psi(x)|^{4}\,dx=-\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\textbf{Step 2: Upper bound.} \\
Let $\mathcal{D}\subset\Omega$ be a regular domain and, for $\ell \in (0,1)$,
\begin{equation}
\mathcal{D}_{\ell}=\{x\in\mathcal{D}: {\rm dist}(x,\partial\mathcal{D})\geq\ell\}\,.
\end{equation}
We introduce a cut-off function $\chi_{\ell}\in C^{\infty}_{c}(\mathbb R^{2})$ such that
\begin{equation}\label{chi-l}
0\leq \chi_{\ell} \leq 1~{\rm in}~\mathbb R^{2}\,,\quad {\rm supp}\chi_{\ell}\subset \mathcal{D}\,,\quad \chi_{\ell}=1~{\rm in}~\mathcal{D}_{\ell} \quad{\rm and}\quad|\nabla\chi_{\ell}|\leq\frac{C}{\ell}~{\rm in}~\mathbb R^{2}\,,
\end{equation}
where $C$ is a positive constant. We multiply both sides of $\eqref{eq-2D-GLeq}_{a}$ by $\chi_{\ell}^{2}\psi$. It results from an integration by parts that
\begin{align}\label{eq:1}
\int_{\mathcal{D}} \left(|(\nabla-i\kappa H \mathbf A)\chi_{\ell}\psi|^{2}-\kappa^{2}a\,\chi_{\ell}^{2}|\psi|^{2}+\kappa^{2}\chi_{\ell}^{2}|\psi|^{4}\right)\,dx&=\int_{\mathcal{D}}|\nabla\chi_{\ell}|^{2}\, |\psi|^{2}\,dx\nonumber\\
&=\mathcal{O}(\ell^{-1})\,.
\end{align}
Here, we have used the fact that $|\nabla\,\chi_{\ell}|^{2}=\mathcal{O}(\ell^{-2})$, $|\mathcal{D}_{\ell}|=\mathcal{O}(\ell)$ and the bound of $\psi$ in \eqref{eq-psi<a}.\\
We notice that $\chi_{\ell}^{4}\leq\chi_{\ell}^{2}\leq 1$. We add to both sides the term $\frac{\kappa^{2}}{2}\int_{\mathcal{D}}a^{2}\,dx$ to obtain,
$$
\int_{\mathcal{D}} \left(|(\nabla-i\kappa H \mathbf A)\chi_{\ell}\psi|^{2}+\frac{\kappa^{2}}{2}a^{2}-\kappa^{2}a\,|\chi_{\ell}\,\psi|^{2}+\kappa^{2}|\chi_{\ell}\,\psi|^{4}\right)\,dx\leq\,C\,\ell^{-1}+\frac{\kappa^{2}}{2}\int_{\mathcal{D}}a^{2}\,dx\,.
$$
This implies that
$$
\mathcal E_0(\chi_{\ell}\psi,\mathbf A;a,\mathcal{D})\leq
\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a^{2}-\chi_{\ell}^{4}|\psi|^{4})\,dx+C\,\ell^{-1}\,.
$$
Using \eqref{chi-l}, we get
\begin{align}\label{up-psi42}
\int_{\mathcal{D}}|\psi|^{4}\,dx&= \int_{\mathcal{D}}\chi_{\ell}^{4}|\psi|^{4}\,dx+ \int_{\mathcal{D}}(1-\chi_{\ell}^{4})|\psi|^{4}\,dx\nonumber\\
&\leq \int_{\mathcal{D}}\chi_{\ell}^{4}|\psi|^{4}\,dx+C'\,\ell\,,
\end{align}
and consequently,
\begin{equation}\label{up-psi4}
\mathcal E_0(\chi_{\ell}\psi,\mathbf A;a,\mathcal{D})\leq
\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a^{2}-|\psi|^{4})\,dx+C(\ell^{-1}+\ell)\,.
\end{equation}
Using \eqref{fianl-E01} with $h=\chi_{\ell}$ and taking the choice of $\ell$ defined in \eqref{choice-ell-rho}, we get, as $\kappa\to+\infty$,
\begin{multline}
\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a^{2}-|\psi|^{4})\,dx\geq \kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\\
+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
Notice that,
$$\int_{\mathcal{D}}a(x,\kappa)^{2}\,dx=\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\int_{\mathcal{D}\cap\{a(x,\kappa)> 0\}}a(x,\kappa)^{2}\,dx\,.$$
Therefore,
\begin{multline}
-\frac{\kappa^{2}}{2}\int_{\mathcal{D}}|\psi|^{4}\,dx\geq \kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx-\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)> 0\}} a(x,\kappa)^{2}\,dx\\
+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
Dividing both sides by $-\frac{\kappa^{2}}{2}$, we obtain, as $\kappa\longrightarrow +\infty\,$,
\begin{equation}\label{up-psi45}
\int_{\mathcal{D}}|\psi|^{4}\,dx\leq-\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\begin{rem}\label{rm-up-psi-ng}
We can replace $\mathcal{D}$ by $\mathcal{\overline{D}}^{c}$ such that the estimate in \eqref{up-psi45} is still true. That is:
\begin{equation}\label{up-psi-ng}
\int_{\mathcal{\overline{D}}^{c}}|\psi|^{4}\,dx\leq-\int_{\mathcal{\overline{D}}^{c}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\end{rem}
\textbf{Step 3: Lower bound.} \\
We can decompose $\int_{\mathcal{D}}|\psi|^{4}\,dx$ as follows:
$$
\int_{\mathcal{D}}|\psi|^{4}\,dx=\int_{\Omega}|\psi|^{4}\,dx-\int_{\mathcal{\overline{D}}^{c}}|\psi|^{4}\,dx
$$
Thanks to Remark~\ref{rm-up-psi-ng}, using the asymptotics in \eqref{asy-psi-Omega} obtained in Step~1 when $\mathcal{D}=\Omega$ and the upper bound in Step~2 , we get
\begin{equation}\label{up-psi-D}
\int_{\mathcal{D}}|\psi|^{4}\,dx\leq-\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\section{Extension of the Giorgi-Phillips Theorem}\label{GP}
In this section we extend a result of Giorgi-Phillips \cite{GP}, in the two cases when the external magnetic field $B_{0}$ is variable (i.e. $\Gamma\neq\varnothing$) and when the external magnetic field $B_{0}$ is constant (i.e. $\Gamma=\varnothing$), with a pinning term.
We recall that the normal solution $(0,\mathbf F)$ is a trivial solution of the Ginzburg-Landau system \eqref{eq-2D-GLeq}. We will show that this solution is a global minimizer, when $\kappa$ and $H$ are sufficiently large. We first establish a priori estimates for a critical point $(\psi, \mathbf A)$ of the G-L-functional.
\subsection{Estimates of $\mathbf A$ and of $\|(\nabla-i\kappa H\mathbf F)\psi\|$.}~\\
We need the following estimates on $\mathbf A$ and on $\|(\nabla-i\kappa H\mathbf F)\psi\|$ which are independent of the assumption of $\Gamma$.
\begin{thm}\label{thm-2D-apriori}
There exist positive constants $C_{1}$, $C_{2}$ and $C_{3}$ such that, if \eqref{a2} hold, $\kappa>0$, $H>0$ and $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, then,
\begin{eqnarray}
\|(\nabla-i\kappa H\mathbf A)\psi\|_{L^{2}(\Omega)}\leq C_{1}\,\kappa\, \|\psi\|_{L^{2}(\Omega)}\label{2nd-<}\,,\\
\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq \frac{C_{2}}{H}\, \|\psi\|_{L^{2}(\Omega)}\|\psi\|_{L^{4}(\Omega)}\label{3d-<}\,,\\
\|(\nabla-i\kappa H\mathbf F)\psi\|_{L^{2}(\Omega)}\leq C_{3}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\label{8d-<}\,.
\end{eqnarray}
\end{thm}
\begin{proof}
\textbf{We first prove \eqref{2nd-<}.} In the case when $\overline{a}\leq 0$ with $\overline{a}$ introduced in \eqref{def:sup-a}, we get using \eqref{eq-psi<a} that $\psi=0$ and hence \eqref{2nd-<} is proved.\\
In the case when $\overline{a}>0$, thanks to \eqref{eq-psi<a}, we have,
\begin{equation}\label{eq:a-psi}
0\leq(\overline{a}-|\psi|^{2})\leq \overline{a}\,.
\end{equation}
We recall that if $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq} then, (see \eqref{eq:A-psi})
$$
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\psi|^{2}\,dx=\kappa^{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})|\psi|^{2}\,dx\,.
$$
Using \eqref{a2} and \eqref{eq:a-psi}, we obtain \eqref{2nd-<}.\\
\textbf{Now, we prove \eqref{3d-<}.} We obtain from the equation in \eqref{eq-2D-GLeq}$_{b}$ the following estimate (see \cite[Equation~(11.9b)]{FH1}):
$$
\kappa H\int_{\Omega}|\curl(\mathbf A-\mathbf F)|^{2}\,dx\leq \|(\nabla-i\kappa H\mathbf A)\psi\|_{L^{2}(\Omega)}\, \|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}\,.
$$
Using \eqref{2nd-<} and applying H$\rm\ddot{o}$lder's inequality, we get
$$
\kappa H\int_{\Omega}|\curl(\mathbf A-\mathbf F)|^{2}\,dx\leq C\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|\psi\|_{L^{4}(\Omega)}\, \|\mathbf A-\mathbf F\|_{L^{4}(\Omega)}\,.
$$
We get by regularity of the $\curl$-$\Div$ system (see \cite[A.7]{FH1}),
\begin{equation}\label{eq:curl-div}
\|\mathbf A-\mathbf F\|_{H^{1}(\Omega)}\leq C\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\,,
\end{equation}
where $C$ is a positive constant.\\
By the Sobolev embedding theorem, we get,
\begin{align}\label{sb-emb-L4}
\|\mathbf A-\mathbf F\|_{L^{4}(\Omega)}&\leq C_{\rm Sob}\, \|\mathbf A-\mathbf F\|_{H^{1}(\Omega)}\nonumber\\
&\leq \widehat C \, \|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\,.
\end{align}
Consequently,
$$
\kappa H\, \int_{\Omega}|\curl(\mathbf A-\mathbf F)|^{2}\,dx\leq \kappa\, \|\psi\|_{L^{2}(\Omega)}\|\psi\|_{L^{4}(\Omega)}\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\,,
$$
which leads to \eqref{3d-<}.\\
\textbf{Finally, we prove \eqref{8d-<}.}
Using \eqref{3d-<} and \eqref{sb-emb-L4}, H\"older's inequality gives,
\begin{align}\label{5d-<}
\|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}&\leq \|\mathbf A-\mathbf F\|_{L^{4}(\Omega)}^{2}\|\psi\|_{L^{4}(\Omega)}^{2}\nonumber\\
& \leq \frac{C'}{H^{2}}\|\psi\|_{L^{4}(\Omega)}^{4}\|\psi\|_{L^{2}(\Omega)}^{2}\,,
\end{align}
Using \eqref{2nd-<}, \eqref{5d-<} and the bound of $\psi$ above, Young's inequality gives,
\begin{align}
\|(\nabla-i\kappa H\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}&\leq 2\|(\nabla-i\kappa H\mathbf A)\psi\|_{L^{2}(\Omega)}^{2}+2\,(\kappa H)^{2}\|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}\nonumber\\
&\leq 2\,C''\,\kappa^{2}\|\psi\|^{2}_{L^{2}(\Omega)}\,.
\end{align}
\end{proof}
\subsection{The case $\Gamma=\varnothing$.}~\\
For $\xi\in\mathbb R$, we consider the Neumann realization $\mathfrak{h}^{N,\xi}$ in $L^{2}(\mathbb R_+)$ associated with the operator $-\frac{d^2}{dt^2}+(t+\xi)^2$, i.e.
\begin{equation}
\mathfrak{h}^{N,\xi}:=-\frac{d^2}{dt^2}+(t+\xi)^2\,,\qquad \mathcal{D}(\mathfrak{h}^{N,\xi})=\{u\in B^{2}(\mathbb R_{+}): u'(0)=0\}\,,
\end{equation}
where,
$$
B^2(\mathbb R_+)=\{u\in L^{2}(\mathbb R_+): t^p u^{(q)}\in L^{2}(\mathbb R_+), \forall p,q\in\mathbb N~s.t.~p+q\leq 2\}\,.
$$
M. Dauge and B. Helffer \cite{DH} (see also Fournais-Helffer \cite[Proposition~4.2.2]{FH1}) have proved that the lowest eigenvalue $\mu$ of $\mathfrak{h}^{N,\xi}$ admits a minimum $\Theta_{0}$, which is attained at a unique point $\xi_{0}<0$, and satisfies:
\begin{equation}\label{Theta0}
\Theta_{0}=\inf_{\xi} \mu(\xi)=\mu(\xi_0)<1\,.
\end{equation}
Moreover
\begin{equation}
\Theta_{0}=\xi_{0}^2\,.
\end{equation}
We introduce the notation:
\begin{equation}\label{inf:B0}
\inf_{x\in\overline{\Omega}} |B_{0}(x)|=b_{0}\qquad{\rm and}\qquad \inf_{x\in\partial\Omega} |B_{0}(x)|=b'_{0}\,.
\end{equation}
We denote by $\mu^{N}(\mathcal{B}\mathbf F;\Omega)$ the lowest eigenvalue of the $\rm Schr\ddot{o}dinger$ operator $P_{\mathcal{B}\mathbf F,0}^{\Omega}$ (see \eqref{def:P}) with Neumann condition in $L^{2}(\Omega)$:
\begin{equation}\label{muN(kHF)}
\mu^{N}(\mathcal{B}\mathbf F;\Omega)=\inf_{\substack{\psi\in H^{1}(\Omega)\\ \psi\neq 0}}\frac{\langle P_{\mathcal{B}\mathbf F,0}^{\Omega}\,\psi,\psi\rangle}{\|\psi\|^{2}_{L^{2}(\Omega)}}\,.
\end{equation}
In \cite{FH1}, it is proved that
\begin{theorem}\label{thm:FH}
Suppose that $\Omega\subset\mathbb R^2$ is an open bounded set with smooth boundary and $\Gamma=\varnothing$. Then,
\begin{equation}\label{eq:FH}
\lim_{\mathcal{B}\longrightarrow+\infty}\frac{\mu^{N}(\mathcal{B}\mathbf F,\Omega)}{\mathcal{B}}=\min(b_0,\Theta_{0}\,b'_{0})\,.
\end{equation}
\end{theorem}
In the next theorem, we give a simple proof of the result which says that $(0,\mathbf F)$ is the unique minimizer of the functional when $H$ is sufficiently large and when the magnetic field $B_0$ is constant with pinning term.
\begin{theorem}\label{thm:GP2}
Let $\Omega\subset\mathbb R^2$ be a smooth, bounded, simply-connected open set and $\Gamma=\varnothing$. Then, there exist positive constants $C$ and $\kappa_{0}$, such that, if
$$
H\geq C \kappa\,,\qquad\kappa\geq\kappa_{0}\,,
$$
then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.
\end{theorem}
\begin{proof}
We assume that we have a \textbf{non normal} critical point $(\psi, \mathbf A)$ for $\mathcal E_{\kappa,H,a,B_{0}}$. This means that $(\psi, \mathbf A)\in H^{1}(\Omega)\times H^{1}_{{\rm div}}(\Omega)$ is a solution of \eqref{eq-2D-GLeq} and
\begin{equation}\label{psi>0}
\int_{\Omega}|\psi|^{2}\,dx>0\,.
\end{equation}
Therefore, we get from \eqref{eq-psi<a} that,
$$
|\psi(x)|^{2}\leq \overline{a}\,\qquad\forall x\in\overline{\Omega}\,,
$$
where $\overline{a}$ is introduced in \eqref{def:sup-a}.\\
Let
\begin{equation}\label{eq:B=kH}
\mathcal{B}=\kappa H\,.
\end{equation}
Theorem~\ref{thm-2D-apriori} tells us that,
$$
\|(\nabla-i\mathcal{B}\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}\leq C\,\kappa^{2}\,\|\psi\|^{2}_{L^{2}(\Omega)}\,.
$$
Since $\psi$ satisfies \eqref{psi>0}, this implies by assumption that the lowest Neumann eigenvalue\\
$\mu^{N}(\mathcal{B}\mathbf F;\Omega)$ of $P_{\mathcal{B}\mathbf F,0}^{\Omega}$ in $L^{2}(\Omega)$ satisfies,
\begin{equation}\label{muN<}
\mu^{N}(\mathcal{B}\mathbf F;\Omega)\leq C\,\kappa^{2}\,.
\end{equation}
Thanks to Theorem~\ref{thm:FH}, we get the existence of a constant $C>0$, such that, if $H\geq C\,\kappa$, then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.
\end{proof}
\subsection{The case $\Gamma\neq\varnothing$.}~\\
We recall the definition of $\lambda_{0}$ in \eqref{lambda0}, the definition of $\Gamma$ in \eqref{gamma} and for any $\theta\in(0,\pi)$ we recall that $\lambda(\mathbb R_{+}^{2},\theta)$ is the bottom of the spectrum of the operator
$ P_{\mathbf A_{\rm app,\theta},0}^{\mathbb R^{2}_{+}}$, with
$$\mathbf A_{\rm app,\theta}=-\left(\frac{x^{2}_{2}}{2}\cos\,\theta,\frac{x^{2}_{1}}{2}\sin\,\theta \right)\,.$$
Define
\begin{equation}\label{alpha1}
\alpha_{1}(B_{0})=\min\left\{\lambda_{0}^{\frac{3}{2}}\min_{x\in \Gamma\cap\Omega}|\nabla B_{0}(x)|,\min_{x\in \Gamma\cap\partial\Omega}\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|\right\}\,.
\end{equation}
In \cite{XB-KH}, it is proved that
\begin{thm}\label{thm:BK}
Suppose that \eqref{B(x)} holds and $\Gamma\neq\varnothing$. Then
\begin{equation}\label{eq:pan}
\lim_{\mathcal{B}\longrightarrow+\infty}\frac{\mu^{N}(\mathcal{B}\mathbf F,\Omega)}{\mathcal{B}^{\frac{2}{3}}}=\alpha_{1}(B_{0})^{\frac{2}{3}}\,.
\end{equation}
\end{thm}
In the next theorem, we give a simple proof of the result which says that $(0,\mathbf F)$ is the unique minimizer of the functional when $H$ is sufficiently large and when $B_{0}$ is variable. This result was obtained in \cite{GP} for the case with constant magnetic field and with a constant pinning term.
\begin{theorem}\label{thm:GP}
Let $\Omega\subset\mathbb R^2$ be a smooth, bounded, simply-connected open set, the pinning term $a$ satisfying \eqref{a2}, and the magnetic field $B_{0}$ satisfying \eqref{B(x)}. Then, there exist positive constants $C$ and $\kappa_{0}$, such that, if
$$
H\geq C \kappa^{2}\,,\qquad\kappa\geq\kappa_{0}\,.
$$
Then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.
\end{theorem}
\begin{proof}
Similarly to the proof of Theorem~\ref{thm:GP2}, we assume that we have a \textbf{non normal} critical point $(\psi, \mathbf A)$ for $\mathcal E_{\kappa,H,a,B_{0}}$.\\
Therefore, we get from \eqref{8d-<} that,
$$
\mu^{N}(\mathcal{B}\mathbf F;\Omega)\leq C\,\kappa^{2}\quad(\mathcal B=\kappa H)\,.
$$
Thanks to Theorem~\ref{thm:BK}, we get a contradiction, if $ \displaystyle H\geq C \kappa^{2}$ and $C$ is sufficiently large.
\end{proof}
\section{Asymptotics of $\mu_{1}(\kappa,H)$: the case with non vanishing magnetic field}\label{Section:4}
The aim of this section is to give an estimate for the lowest
eigenvalue $\mu_{1}(\kappa,H)$ of the operator $P_{\kappa
H\mathbf F,-\kappa^{2}a}^{\Omega}$ (see \eqref{def:mu1}) in the case when
$\Gamma=\varnothing$ with a $\kappa$-independent pinning (i.e.
$a(x,\kappa)=a(x)$). Recall that the set $\Gamma$ is introduced in
\eqref{gamma}.
\subsection{Lower bound}~\\
Without loss of generality we suppose that $B_{0}>0 \mbox{ in } \overline{ \Omega}$. Our results will be formulated by introducing:
\begin{equation}\label{Lambda11}
\Lambda_{1}(B_{0},a,\sigma)=\min\left\{\inf_{ x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\},\inf_{ x\in\partial\Omega}\left\{\Theta_{0}\,\sigma\,B_{0}(x)-a(x)\right\}\right\}\,,
\end{equation}
where $\sigma$ is a positive constant.\\
In the case when the pinning term is constant (i.e. $a(x)=a_0$), \eqref{Lambda11} becomes as follows:
$$
\Lambda_{1}(B_{0},a,\sigma)=\sigma\min\left\{\inf_{ x\in\Omega}\left\{B_{0}(x)\right\},\Theta_{0}\,\inf_{ x\in\partial\Omega}\left\{B_{0}(x)\right\}\right\}-a_0\,.
$$
This case was treated by Pan and Kwek \cite{LP1}.\\
Let $\mathcal{Q}_{\mathcal{B}\,\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}$ be the quadratic form of $P_{\mathcal{B}\mathbf F,- \frac{\mathcal{B}}{\sigma}\,a}^{\Omega}$, i.e.
\begin{equation}\label{def:quad}
\mathcal{Q}_{\mathcal{B}\,\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(\psi)=\int_{\Omega}\left(|(\nabla-i\mathcal{B}\mathbf F)\psi|^{2}-\frac{\mathcal{B}}{\sigma} \,a(x)|\psi|^{2}\right)\,dx\,.
\end{equation}
\begin{prop}\label{prop:mu1-cst}
Let $\Omega\subset\mathbb R^2$ be an open bounded set with smooth boundary, $I$ a closed interval in $(0,+\infty)$ and $\Gamma=\varnothing$. There exist positive constant $C$ and $\mathcal{B}_{0}$ such that if $\sigma\in I$, $\mathcal{B}\geq\mathcal{B}_{0}$, $\psi\in H^{1}(\Omega)\setminus \{0\} $ and $a\in C^{1}(\overline{\Omega})$, then,
\begin{equation}\label{eq:lower-bound-constant}
\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(\psi)}{\|\psi\|_{L^{2}(\Omega)}^{2}}\geq \frac{\mathcal B}{\sigma}\,\,\Lambda_{1}(B_{0},a,\sigma)-C\,\mathcal{B}^{\frac{3}{4}}\,.
\end{equation}
\end{prop}
\begin{proof}
The proof is a consequence of the following inequality that we take
from \cite[Prop.~9.2.1]{FH1},
$$
\forall~\psi\in H^1(\Omega)\,,\quad\int_\Omega|(\nabla-i\mathcal{B}\mathbf F)\psi|^2\,dx\geq \int_\Omega \big(U(x)-\bar CB^{3/4}\big)|\psi|^2\,dx\,,
$$
where \begin{equation}\label{eq:U}
U(x)= \left\{
\begin{array}{ll}
\mathcal B\,B_0(x)&{\rm if~}{\rm dist}(x,\partial\Omega)\geq B^{-3/8}\,,\\
\Theta_0\mathcal B\,B_0(x)&{\rm if~}{\rm dist}(x,\partial\Omega)< B^{-3/8}\,,
\end{array}
\right.
\end{equation}
$\mathcal B\geq \bar{\mathcal B_0}$, $\bar{\mathcal B_0}$ and $\bar
C$ are two constants independent of $\mathcal B$.
Clearly, there exist two constants $C'>0$ and $\mathcal B_0>0$ such
that, for all $\sigma\in I$, we have,
$$
U(x)-\frac{\mathcal B}{\sigma}a(x)\geq\frac{\mathcal B}{\sigma}\Lambda_1(B_0,a,\sigma)-C'B^{3/4}\,.
$$
\end{proof}
Coming back to our initial parameters $\kappa$ and $H$, we obtain:
\begin{theorem}\label{thm:mu1-cst}
Let $\Omega\subset\mathbb R^2$ be an open bounded set with smooth boundary and $\Gamma=\varnothing$. Suppose that \eqref{cond-H} holds and $a\in C^1(\overline{\Omega})$, then,
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac H \kappa\right)+\mathcal{O}(\kappa^{\frac{3}{2}})\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
Here, $\Lambda_{1}$ is introduced in \eqref{Lambda11}.
\end{theorem}
\begin{proof}
We apply Proposition~\ref{prop:mu1-cst} with
$$
\mathcal{B}=\kappa H\,,\quad \sigma=\frac{H}{\kappa}\quad{\rm and}\quad I=[\lambda_{\min},\lambda_{\max}]\,.
$$
Let us verify that the conditions of the proposition are satisfied for this choice.\\
It is trivial that $\sigma\in I$. Now, as $\kappa\to+\infty$, we have,
$$
\mathcal{B}=\sigma\,\kappa^{2}\to+\infty\,.
$$
This implies that, as $\kappa\to+\infty$,
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac{H}{\kappa} \right)+\mathcal{O}(\kappa^{\frac{3}{2}})\,.
$$
This finishes the proof of the theorem.
\end{proof}
\subsection{Upper bound}
\begin{prop}[Upper bound in the bulk]\label{prop:up-blk-cst}
Suppose that $\Omega\subset\mathbb R^2$ is an open bounded set with smooth boundary $\partial\Omega$, $\lambda_{\max} >0$ and $\Gamma=\varnothing$. For any $ x_0\in\Omega$, there exist positive constants $C$ and $\mathcal{B}_{0}$ such that, if $\sigma\in (0,\lambda_{\max}]$, $\mathcal{B}\geq\mathcal{B}_{0}$ and $a\in C^{1}(\overline{\Omega})$, then,
\begin{equation}\label{eq:up-blk-cst}
\mu_{\mathcal{B},\sigma}\leq \frac{\mathcal B}{\sigma}\,\left\{\sigma\,B_{0}(x_{0})-a(x_{0})\right\}+C\,\mathcal{B}^{\frac{1}{2}}\,.
\end{equation}
Here,
\begin{equation}\label{def:mu1-cst}
\mu_{\mathcal{B},\sigma}=\inf_{\psi\in H^{1}(\Omega)\setminus \{0\}}\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(\psi)}{\|\psi\|_{L^{2}(\Omega)}^{2}}\,,
\end{equation}
where $\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}$ is introduced in \eqref{def:quad}.
\end{prop}
\begin{proof}
Thanks to \eqref{def:quad}, we have,
$$
\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(u) = \int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)u(x)|^{2}\,dx - \frac{ \mathcal B}{\sigma}\,\int_{\Omega}a(x)|u(x)|^{2}\,dx \,.
$$
The upper bound of the first term in the right hand side above is based on the construction of Gaussian quasi-mode (see \cite[Subsection~2.4.2]{FH1} for the case with constant pinning) centered at $x_0\in\Omega$,
$$
\varphi_{1}(x)=\,e^{i\,\mathcal{B}\,\phi_{0}}\,\chi\left(\mathcal{B}^{+\frac{1}{2}}(x-x_{0})\right)\,u\left(\sqrt{\mathcal{B}B_{0}(x_0)}\,(x-x_{0})\right)\,.
$$
Here, $\chi$ is a cut-off function equal to $1$ in a neighborhood of $0$ such that ${\rm supp}\,\chi\subset D(0,1)$, the function $\phi_{0}$ satisfies \eqref{F-A} and the function $u$ defined as follows:
$$
u(x)=\frac{\pi^{-\frac{1}{4}}}{\sqrt{2}}e^{-\frac{|x|^2}{2}}\,.
$$
We note that ${\rm supp}\,\varphi_{1}\subset \Omega$ for $\mathcal{B}$ large enough. As in \cite[(2.35)]{FH1}, we get the existence of a positive constant $\mathcal{B}_{0}$ such that, for any $\mathcal{B}\geq\mathcal{B}_{0}$,
\begin{equation}\label{up:FA-cst}
\frac{\int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)\varphi_{1}(x)|^{2}\,dx}{\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx}\leq \mathcal{B}\,B_{0}(x_0)+\mathcal{O}(\mathcal{B}^{\frac{1}{2}})\,.
\end{equation}
To derive the upper bound for the second term, we use Taylor's formula for the function $a$ near $x_0$,
\begin{equation}\label{eq:app-a-cst}
|a(x)-a(x_0)|\leq C\,\,\mathcal{B}^{-\frac{1}{2}}\,,\qquad\left(x\in D\left(x_0,\mathcal{B}^{-\frac{1}{2}}\right)\right)\,.
\end{equation}
Using \eqref{eq:app-a-cst}, since ${\rm supp}\,\varphi_{1}\subset D\left(x_0,\mathcal{B}^{-\frac{1}{2}}\right)$, we get,
\begin{equation}\label{upp:a1-cst}
-\int_{\Omega}a(x)|\varphi_{1}(x)|^{2}\,dx\leq -a(x_{0})\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx+C\,\mathcal{B}^{-\frac{1}{2}}\,\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx\,,
\end{equation}
and consequently
\begin{equation}\label{upp:a-cst}
-\frac{\mathcal B}{\sigma}\frac{\int_{\Omega}a(x)|\varphi_{1}(x)|^{2}\,dx}{\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx}\leq -\frac{\mathcal B}{\sigma}\,a(x_{0})+C\,\mathcal{B}^{\frac{1}{2}}\,.
\end{equation}
Collecting \eqref{up:FA-cst} and \eqref{upp:a-cst}, we finish the proof of Proposition~\ref{def:mu1-cst}.
\end{proof}
\begin{rem}\label{rem:first term}
When
$$
\inf_{x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\} < \inf_{ x\in\partial\Omega}\left\{\Theta_{0}\,\sigma\,B_{0}(x)-a(x)\right\}\,,
$$
we notice that, if the infimum of $\sigma\,B_{0}(x)-a(x)$ was attained on $\partial\Omega$, (i.e. there exists $x_0\in\partial\Omega$ such that $\inf_{x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\} =\sigma\,B_{0}(x_0)-a(x_0)$), we would have,
$$
\sigma\,B_{0}(x_0)-a(x_0) < \Theta_{0}\,\sigma\,B_{0}(x_0)-a(x_0)\,,
$$
which is impossible, since $\Theta_{0}<1$. Hence, we can choose $x_0\in\Omega$, such that,
$$
\sigma\,B_{0}(x_0)-a(x_0)=\inf_{x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\}\,,
$$
and we apply Proposition~\ref{prop:up-blk-cst} with
$$
\mathcal{B}=\kappa H\quad{\rm and}\quad\sigma=\frac{H}{\kappa}\,.
$$
Thus, we get the existence of a positive constant $\kappa_0$ such
that, if,
\begin{equation}\label{cond:H-cst}
\kappa\geq\kappa_{0}\quad{\rm and}\quad\kappa_{0}\,\kappa^{-1}<H< \lambda_{\max}\,\kappa\,,
\end{equation}
then,
\begin{equation}\label{eq:up-mu1-cst}
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\inf_{x\in\Omega}\left\{\frac{H}{\kappa}\,B_{0}(x)-a(x)\right\}+ \mathcal{O}(\kappa)\,,\qquad{\rm as}\,\kappa\to+\infty\,.
\end{equation}
\end{rem}
\begin{prop}[Upper bound near the boundary]\label{prop:up-bnd-cst}
Suppose that $\Omega\subset\mathbb R^2$ is an open bounded set with a smooth boundary, $\lambda_{\max} >0$ and $\Gamma=\varnothing$. For any $ x_0\in\partial\Omega$ and for any $\sigma\in (0,\lambda_{\max}]$, we have,
\begin{equation}\label{eq:up-bnd-cst}
\mu_{\mathcal{B},\sigma}\leq \frac{\mathcal B}{\sigma}\big(\sigma\,\Theta_{0}\,B_{0}(x_{0})-a(x_0)\big)+ \mathcal{O}(\mathcal{B}^{\frac{1}{2}})\,,\qquad{\rm as}~\mathcal{B}\to+\infty\,.
\end{equation}
Here, $\Theta_{0}$ is introduced in \eqref{Theta0}.
\end{prop}
\begin{proof}
We recall the definition of $\mu_{\mathcal{B},\sigma}$ as follows:
$$
\mu_{\mathcal{B},\sigma}=\inf_{u\in H^{1}(\Omega)\setminus \{0\}} \left(\frac{\int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)u(x)|^{2}\,dx}{\int_{\Omega}|u(x)|^{2}\,dx}-\frac{\mathcal B}{\sigma}\,\frac{\int_{\Omega}a(x)|u(x)|^{2}\,dx}{\int_{\Omega}|u(x)|^{2}\,dx}\right)\,.
$$
The first term in the right hand side is studied by Helffer-Morame (see \cite[Theorem~9.1]{HM} with $h=\mathcal{B}^{-1}$ and $\mu_{\mathcal{B},\sigma}=\frac{\mu^{(1)}(h)}{h^{2}}$) or Fournais-Helffer (see \cite[Section~9.2.1]{FH1}). They proved for any $x_0\in \partial \Omega$ the existence of $\mathcal B_0$ such that for $\mathcal{B}\geq \mathcal B_0$ one can construct a trial function $\widehat{u}$ such that,
$$
\frac{\int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)\widehat u(x)|^{2}\,dx}{\int_{\Omega}|\widehat u(x)|^{2}\,dx}\leq \mathcal{B}\,\Theta_{0}\,B_{0}(x_{0})+ \mathcal{O}(\mathcal{B}^{\frac{1}{2}})\,,\qquad{\rm as}~\mathcal{B}\to+\infty \,.
$$
The estimates of the second term in the right hand side are just as in \eqref{upp:a-cst} and this achieves the proof of the proposition.
\end{proof}
\begin{rem}\label{rem:second term}
$\partial \Omega$ being compact, we can choose $ x_0\in\partial\Omega$, such that,
$$
\sigma\,\Theta_{0}\,B_{0}(x_0)-a(x_0)=\inf_{x\in\partial\Omega}\left\{\sigma\,\Theta_{0}\,B_{0}(x)-a(x)\right\}\,,
$$
and we apply Proposition~\ref{prop:up-bnd-cst} with
$$
\mathcal{B}=\kappa H\quad{\rm and}\quad \sigma=\frac{H}{\kappa}\,,
$$
which implies under Assumption~\ref{cond:H-cst} that,
\begin{equation}\label{eq:up-mu1-cst2}
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\inf_{ x\in\partial\Omega}\left\{\frac{H}{\kappa}\,\Theta_{0}\,B_{0}(x)-a(x)\right\}+\mathcal{O}(\kappa)\,,\qquad{\rm as}\,\kappa\to+\infty\,.
\end{equation}
\end{rem}
Remarks~\ref{rem:first term} and ~\ref{rem:second term} lead to the conclusion in:
\begin{theorem}\label{thm:mu1-cst-upp}
Let $\Omega\subset\mathbb R^2$ is an open bounded set with a smooth boundary and $\Gamma=\varnothing$. Suppose that \eqref{cond:H-cst} hold and $a\in C^1(\overline{\Omega})$, we have
$$
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac H \kappa\right)+ \mathcal{O}(\kappa)\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
Here, $\Lambda_{1}$ is introduced in \eqref{Lambda11}.
\end{theorem}
Notice that the conclusion in Theorem~\ref{thm:mu1-cst-upp} is valid under the assumption $ \kappa H\geq \mathcal B_0$ with $\mathcal B_0 >0$
sufficiently large. Lemma~\ref{lem-H=kappa} below takes care of the
regime where $\kappa H=\mathcal O(1)$.
\begin{lem}\label{lem-H=kappa}
Let $C_{\max}>0$. Suppose that $\{a>0\}\not=\emptyset$. There exists a constant $\kappa_0>0$ such that, if
$$\kappa\geq \kappa_0\quad{\rm and}\quad 0\leq H\leq C_{\max}\kappa^{-1}\,,$$
then
$$\mu_1(\kappa,H)<0\,.$$
\end{lem}
\begin{rem}\label{rem:h=0}
The conclusion in Lemma~\ref{lem-H=kappa} is valid in both cases
where $\Gamma=\emptyset$ and $\Gamma\not=\emptyset$.
\end{rem}
\begin{proof}[Proof of Lemma~\ref{lem-H=kappa}]~\\
Let $\ell>0$. Choose $x_0\in\Omega$ such that $a(x_0)>0$. We
introduce a cut-off function $\chi_{\ell}\in C_{c}^{\infty}(\mathbb R^{2})$
satisfying:
\begin{equation}\label{def:chil}
0\leq \chi_{\ell} \leq 1~{\rm in}~\mathbb R^{2}\,,\quad {\rm supp}\chi_{\ell}\subset B(x_{0},\ell)\,,\quad \chi_{\ell}=1~{\rm in}~B\left(x_{0},\ell/2\right)\quad{\rm and}\quad |\nabla\chi_{\ell}|\leq \frac{C}{\ell}\,.
\end{equation}
The min-max principle yields,
$$
\mu^{(1)}(\kappa,H)\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq \int_{\Omega}|(\nabla-i\kappa H\mathbf F)\chi_{\ell}|^{2}\,dx-\kappa^{2}\int_{\Omega}a(x)|\chi_{\ell}(x)|^{2}\,dx\,.
$$
Using the assumptions on $\chi_{\ell}$ and the fact that $\mathbf F\in C^{\infty}(\overline{\Omega})$, a trivial estimate is,
\begin{align}\label{up:F-varphi3}
\int_{\Omega}|(\nabla-i\kappa H\mathbf F)\chi_{\ell}|^{2}\,dx&=\int_{B(x_{0},\ell)}|\nabla\chi_{\ell}(x)|^{2}\,dx+\kappa^{2}H^{2}\int_{B(x_{0},\ell)}|\mathbf F\,\chi_{\ell}(x)|^{2}\,dx\nonumber\\
&\leq C\,(1+(\kappa\,H\,\ell)^{2})\,.
\end{align}
We write by Taylor's formula applied to the function $a$ near $x_0$,
\begin{equation}\label{up:a-varphi3}
-\kappa^{2}\int_{\Omega}a(x)|\chi_{\ell}(x)|^{2}\,dx\leq -a(x_0)\,\kappa^{2}\,\ell^{2}+C\,\kappa^{2}\,\ell^{3}\,.
\end{equation}
Collecting \eqref{up:F-varphi3} and \eqref{up:a-varphi3}, we obtain,
$$
\mu^{(1)}(\kappa,H)\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq-a(x_0)\,\kappa^{2}\,\ell^{2}+C(\kappa^{2}\,\ell^{3}+1+(\kappa\,H\,\ell)^{2})\,.
$$
We select $\ell=\kappa^{-\frac{1}{2}}$ and note that $\kappa H<C_{\max}$. We find that,
$$
\mu^{(1)}(\kappa,H)\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq-a(x_0)\,\kappa+C\Big(\kappa^{\frac{1}{2}}+1+C_{\max}^{2}\kappa^{-1}\Big)\,.
$$
Since $\chi_\ell\not=0$ and $a(x_0)>0$, we deduce that, for $\kappa$
sufficiently large,
$$
\mu^{(1)}(\kappa,H)<0\,.
$$
\end{proof}
\section{Proof of Theorem~\ref{thm:HC3}}\label{10}
\subsection{Analysis of $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$.}~\\
In this subsection we give a lower bound of the critical field $\underline{H}_{C_3}^{loc}$ (see \eqref{def:HC3-u}) and we give an upper bound of the critical field $\overline{H}_{C_3}^{loc}$ in the case when the magnetic field $B_0$ is constant with a pining term.
\begin{prop}\label{prop:mu1<0}
Suppose that $\{a>0\}\neq\varnothing$ and $\Gamma=\varnothing$. There exist constants $C>0$ and $\kappa_{0}\geq 0$ such that if
\begin{equation}\label{cond:HC3}
\kappa\geq \kappa_{0}\,,\qquad H\leq \kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\,,
\end{equation}
then,
$$
\mu_{1}(\kappa,H)<0\,.
$$
Moreover,
$$
\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\leq \underline{H}_{C_{3}}^{loc}\,.
$$
\end{prop}
\begin{proof}
To apply the previous results, we take
$$\lambda_{max} = \max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right) +1\,.
$$
We have two cases:\\
\textbf{Case 1.} If
$$\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)}>\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\,.$$
then, there exists $x_0\in\Omega$ (the supremum of $\frac{a(x)}{B_{0}(x)}$ can not be attained on the boundary, since $\frac{a(x)}{\Theta_{0}\,B_{0}(x)}>\frac{a(x)}{B_{0}(x)}$), such that,
$$
\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)} = \frac{a(x_0)}{B_{0}(x_0)} \,.
$$
If \eqref{cond:HC3} is satisfied for some $C>0$, then,
$$
\frac{H}{\kappa}\leq \frac{a(x_0)}{B_{0}(x_0)}-C\,\kappa^{-\frac{1}{2}}\,,
$$
that we can write in the form,
$$
\kappa^{2}\left(\frac{H}{\kappa}B_{0}(x_0)-a(x_0)\right)\leq -C\,M \,\kappa^{\frac{3}{2}}\,,
$$
where $M>0$ is a constant independent of $C$.\\
Suppose that $\kappa H\geq \mathcal B_0$ where $\mathcal B_0$ is selected sufficiently large such that we can apply Remark~\ref{rem:first term}. (Thanks to Lemma~\ref{lem-H=kappa}, $\mu_1(\kappa,H)<0$ when $\kappa H<\mathcal B_0$).\\
Remark~\ref{rem:first term} tells us that there exist positive
constants $C_{1}$ and $\kappa_{0}$ such that, for $\kappa\geq
\kappa_{0}$,
\begin{align}
\mu_{1}(\kappa,H)&\leq \kappa^{2}\inf_{x\in\Omega} \left(\frac{H}{\kappa}B_{0}(x)-a(x)\right)+C_{1}\,\kappa\nonumber\\
&\leq \kappa^{2}\left(\frac{H}{\kappa}B_{0}(x_0)-a(x_0)\right)+C_{1}\,\kappa^{\frac{3}{2}}\\
&\leq (C_1-C\,M) \,\kappa^\frac 32 \,.
\end{align}
By choosing $C$ such that $C\,M> C_{1}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
\textbf{Case 2.} Here, we suppose that
$$\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\geq \sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)}\,.$$
By compactness, there exists $x'_0\in\partial\Omega$, such that,
$$\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\ = \frac{a(x'_0)}{\Theta_{0}\,B_{0}(x'_0)}
$$
If \eqref{cond:HC3} is satisfied for some $C>0$, then,
$$
\kappa^{2}\left(\frac{H}{\kappa}\Theta_{0}\,B_{0}(x'_0)-a(x'_0)\right)\leq -C\,M'\,\kappa^{\frac{3}{2}}\,.
$$
Thanks to Remark~\ref{rem:second term}, we get the existence of positive constants $C_{2}$ and $\kappa_{0}$ such that, for $\kappa\geq \kappa_0$,
\begin{align}
\mu_{1}(\kappa,H)&\leq \kappa^{2}\inf_{x\in\partial\Omega} \left(\frac{H}{\kappa}\,\Theta_{0}\,B_{0}(x)-a(x)\right)+C_{2}\,\kappa\nonumber\\
&\leq \kappa^{2}\left(\frac{H}{\kappa}\,\Theta_{0}\,B_{0}(x'_0)-a(x'_0)\right)+C_{2}\,\kappa^{\frac{3}{2}}\\
&\leq (C_2-C\,M')\,\kappa^\frac 32\,.
\end{align}
By choosing $C$ such that $C\,M'> C_{2}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
This finishes the proof of the proposition.
\end{proof}
\begin{prop}\label{prop:mu>0}
Suppose that $\{a>0\}\neq\varnothing$, $\lambda_{\max} >0$ and $\Gamma=\varnothing$. There exist constants $C>0$ and $\kappa_{0} > 0$ such that if
\begin{equation}\label{cond:HC3-2}
\kappa\geq \kappa_{0}\,,\qquad \lambda_{\max}\,\kappa\geq H>\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,,
\end{equation}
then,
$$
\mu_{1}(\kappa,H)>0\,.
$$
Moreover,
$$
\overline{H}_{C_{3}}^{loc}\leq\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,.
$$
\end{prop}
\begin{proof}
To apply the previous results, we take
$$\lambda_{min} = \frac 12 \max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right) \,.$$
If \eqref{cond:HC3-2} holds for some $C>0$, then, for any $x\in\Omega$, we have,
\begin{equation}\label{eq:bulk}
\frac{H}{\kappa}B_{0}(x)-a(x)\geq C\,\kappa^{-\frac{1}{2}}\,,
\end{equation}
and, for any $x'\in\partial\Omega$, we have,
\begin{equation}\label{eq:bnd}
\frac{H}{\kappa}\Theta_{0}B_{0}(x')-a(x')\geq C\,\kappa^{-\frac{1}{2}}\,.
\end{equation}
Having in mind the definition of $\Lambda_1$ in \eqref{Lambda11}, the estimates in \eqref{eq:bulk} and in \eqref{eq:bnd} give us that for $\kappa$ large enough,
$$
\Lambda_{1}\left(B_0,a,\frac{H}{\kappa}\right)\geq C\,\kappa^{-\frac{1}{2}}\,.
$$
Thanks to Theorem~\ref{thm:mu1-cst}, we get the existence of positive constants $C'$ and $\kappa_0$ such that, for $\kappa\geq\kappa_0$,
\begin{align}
\mu_{1}(\kappa,H)&\geq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac H \kappa\right)-C'\,\kappa^{\frac{3}{2}}\nonumber\\
&\geq (C-C')\,\kappa^{\frac{3}{2}}\,.
\end{align}
To finish this proof, we choose $C>C'$.
\end{proof}
As a consequence, we have proved Theorem~\ref{thm:HC3} for $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$
\subsection{Analysis of $\underline{H}_{C_3}^{cp}$ and $\overline{H}_{C_3}^{cp}$.}~\\
In this subsection we give a lower bound of the critical field $\underline{H}_{C_3}^{cp}$ (see \eqref{def:HC3-o}) and we give an upper bound of the critical field $\overline{H}_{C_3}^{cp}$ in the case when the magnetic field $B_0$ is constant with a pining term. We start with a proposition which measures the effect of the localization at the boundary when $H$ is sufficiently large.
\begin{prop}\label{lem:psi-2<4}
Suppose that $\Gamma=\varnothing$ and \eqref{cond:HC3-2} holds. There exists a positive constant $C$, such that if $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, then the following estimate holds:
\begin{equation}
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac{3}{8}}\|\psi\|^{2}_{L^{4}(\Omega)}\,.
\end{equation}
\end{prop}
\begin{proof}~\\
The techniques that will be used in this proof are similar with the ones in \cite[Lemma~2.6]{FK2}. If $H$ satisfies \eqref{cond:HC3-2} for some $C>0$, then, for any $x\in\Omega$, we have.
\begin{equation}\label{bulk2}
\kappa\,H\,B_{0}(x)-\kappa^{2}\,a(x)\geq C\,\kappa^{\frac{3}{2}}\,.
\end{equation}
First, we let $\chi\in C^{\infty}(\mathbb R)$ be a standard cut-off function such that
\begin{equation}\label{def:chi-cst}
\chi=1\quad{\rm in}~[1,\infty]\qquad{\rm and }\qquad\chi=0\quad{\rm in}~]-\infty,1/2]\,.
\end{equation}
Next, we define $\lambda=\kappa^{-\frac{3}{4}}$, and $\chi_{\kappa}$ as follows:
\begin{equation}
\chi_{\kappa}(x)=\chi\left(\frac{\dist(x,\partial\Omega)}{\lambda}\right)\,,\qquad \forall x\in\Omega\,.
\end{equation}
Referring to \eqref{eq:1}, we have
\begin{equation}\label{eq:2}
\int_{\Omega} \left(|(\nabla-i\kappa H \mathbf A)\chi_{\kappa}\psi|^{2}-|\nabla\chi_{\kappa}|^{2}|\psi|^{2}\right)\,dx=\kappa^{2}\int_{\Omega}|\chi_{\kappa}|^{2}(a(x)-|\psi|^{2})|\psi|^{2}\,dx\,.
\end{equation}
We estimate $\int_\Omega|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx$ from below. As in \cite[Proposition~6.2]{HK}, we can prove that,
$$
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx\geq \kappa\,H\int_\Omega \curl\mathbf F \,|\chi_{\kappa}\psi|^2\,dx-\kappa\,H\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\|\chi_{\kappa}\psi\|_{L^{4}(\Omega)}^{2}\,.
$$
Noticing that $\displaystyle \curl\mathbf F=B_{0}(x)$ and $\displaystyle\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq \frac{c}{H}\|\psi\|_{L^{2}(\Omega)}$, we have,
$$
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx\geq \kappa\,H\int_\Omega\,B_{0}(x)\,|\chi_{\kappa}\psi|^2\,dx-c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|\chi_{\kappa}\psi\|_{L^{4}(\Omega)}^{2}\,.
$$
Implementing a Cauchy-Schwarz inequality, we get
\begin{equation}\label{eq:grd-eng}
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx\geq \kappa\,H\int_\Omega\,B_{0}(x)\,|\chi_{\kappa}\psi|^2\,dx-c^{2}\,\|\psi\|_{L^{2}(\Omega)}^{2}-\kappa^{2}\|\chi_{\kappa}\psi\|_{L^{4}(\Omega)}^{4}\,.
\end{equation}
Inserting \eqref{eq:grd-eng} into \eqref{eq:2}, we obtain,
$$
\int_\Omega\,\left(\kappa\,H\,B_{0}(x)-\kappa^{2}\,a(x)\right)\,|\chi_{\kappa}\psi|^2\,dx\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla\chi_{\kappa}|^{2}|\psi|^{2}\,dx-\kappa^{2}\int_{\Omega}\left(\chi_{\kappa}^{2}-\chi_{\kappa}^{4}\right)|\psi|^{4}\,dx\,.
$$
As a consequence of \eqref{bulk2}, the inequality above becomes,
$$
C\,\kappa^{\frac{3}{2}}\int|\chi_{\kappa}\psi(x)|^{2}\,dx\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla\chi_{\kappa}|^{2}|\psi|^{2}\,dx-\kappa^{2}\int_{\Omega}\left(\chi_{\kappa}^{2}-\chi_{\kappa}^{4}\right)|\psi|^{4}\,dx\,.
$$
Notice that $-\kappa^{2}\int_{\Omega}\left(\chi_{\kappa}^{2}-\chi_{\kappa}^{4}\right)|\psi|^{4}\,dx\leq 0\,$.\\
Decomposing the integral $\displaystyle\int_{\Omega}|\psi|^{2}\,dx=\int_{\Omega}|\chi_{\kappa}\psi|^{2}\,dx+\int_{\Omega}(1-\chi_{\kappa}^{2})|\psi|^{2}\,dx$, using \eqref{bulk2} and choosing $C$ such that $C\geq 2 c^{2}$, we get,
$$
\frac{1}{2}C\,\kappa^{\frac{3}{2}}\int|\chi_{\kappa}\psi(x)|^{2}\,dx\leq \left(c^{2}+\|\chi'\|_{L^{\infty}(\mathbb R)}^{2}\,\lambda^{-2}\right)\int_{\left\{x\in\Omega:\,\dist(x,\Gamma)\leq\lambda\right\}}|\psi|^{2}\,dx\,.
$$
Recall that $\lambda=\kappa^{-\frac{3}{4}}$, we observe that,
$$
\int|\chi_{\kappa}\psi(x)|^{2}\,dx\leq 4\|\chi'\|_{L^{\infty}(\mathbb R)}^{2} \int_{\left\{x\in\Omega:\,\dist(x,\Gamma)\leq\lambda\right\}}|\psi|^{2}\,dx\,,
$$
and consequently, we get,
$$
\int|\psi(x)|^{2}\,dx\leq\left(4\|\chi'\|_{L^{\infty}(\mathbb R)}^{2}+1\right) \int_{\left\{x\in\Omega:\,\dist(x,\Gamma)\leq\lambda\right\}}|\psi|^{2}\,dx\,.
$$
By choosing $C=\max \left(2\,c^{2},4\|\chi'\|_{L^{\infty}(\mathbb R)}^{2}+1\right)$, we obtain,
$$
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac{3}{8}}\|\psi\|^{2}_{L^{4}(\Omega)}\,.
$$
\end{proof}
\begin{theorem}\label{thm:lb-H}
Supose that $\Gamma=\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $H$ satisfies
\begin{equation}\label{cond:HC3-2w}
H>\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,,
\end{equation} then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.\\
Moreover,
$$
\overline{H}_{C_3}^{cp}\leq\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,.
$$
\end{theorem}
\begin{proof}
We first observe that it results from Giorgi-Phillips like Theorem \ref{thm:GP2} that it remains only to prove the theorem under the stronger Assumption \eqref{cond:HC3-2}.
Suppose now that $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq} with $\psi\neq 0$, we observe that,
\begin{equation}\label{l5est}
0<\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}=-\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf A)\psi|^2-\kappa^{2}a(x)|\psi|^{2}\right)\,dx:=\top\,.
\end{equation}
We can write,
\begin{align}\label{main-eq}
-\top&\geq (1-\sqrt{\top}\,\kappa^{-1})\int_{\Omega}|(\nabla-i\kappa H\mathbf F)\psi|^{2}\,dx-\kappa^{2}\,\int_{\Omega}\,a(x)|\psi|^{2}\,dx-\frac{(\kappa H)^{2}}{\sqrt{\top}\kappa^{-1}}\int_{\Omega}|(\mathbf A-\mathbf F)\psi|^{2}\,dx\nonumber\\
&\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-\sqrt{\top}\,\kappa^{-1}\|(\nabla-i\kappa H\mathbf F)\psi\|^{2}_{L^{2}(\Omega)}-\frac{(\kappa H)^{2}}{\sqrt{\top}\kappa^{-1}}\|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}\,.
\end{align}
We reffer to \eqref{8d-<} and \eqref{5d-<}, we have,
\begin{equation}\label{est:top1}
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C\,\sqrt{\top}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}^{2}\,.
\end{equation}
Thanks to Proposition~\ref{lem:psi-2<4}, using \eqref{l5est}, we get,
\begin{equation}\label{est:psi1}
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac{11}{8}}\,\sqrt{\top}\,.
\end{equation}
As a consequence of \eqref{est:psi1}, \eqref{est:top1} becomes,
\begin{equation}\label{est:top2}
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C'\,\kappa^{-\frac{3}{8}}\,\top\,.
\end{equation}
Having in mind that $\psi\neq 0$ and $\top>0$ (see \eqref{l5est}), we deduce for $\kappa$ sufficiently large $\mu_{1}(\kappa,H)<0$, which is in contradiction with Proposition~\ref{prop:mu>0}. Therefore, we conclude that $\psi=0$, which is what we needed to prove.
\end{proof}
\begin{prop}\label{prop:cp}
Supose that $\Gamma=\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $H$ satisfies
\begin{equation}\label{cond:HC3-2w2}
H\leq \kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\,,
\end{equation}
then there exists a solution $(\psi,\mathbf A)$ of \eqref{eq-2D-GLeq} with $\|\psi\|_{L^{2}(\Omega)}\neq 0$.\\
Moreover,
$$
\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\leq \underline{H}_{C_3}^{cp}\,.
$$
\end{prop}
\begin{proof}
We use $(t\psi_{\ast},\mathbf F)$, with $t$ sufficiently small and $\psi_{\ast}$ an eigenfunction associated with $\mu_{1}(\kappa,H)$, as a test configuration for the functional \eqref{eq-2D-GLf}, i.e.
$$
\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\psi_{\ast}|^{2}-\kappa^{2}\,a(x)|\psi_{\ast}|^{2}\right)\,dx=\mu_{1}(\kappa,H)\|\psi_{\ast}\|^{2}_{L^{2}(\Omega)}\,.
$$
Proposition~\ref{prop:mu1<0} tells us that there exists a constant $C$ such that, under Assumption \eqref{cond:HC3-2w2}, $\mu_{1}(\kappa,H)<0\,$.\\
Therefore,
$$
C_{1}(\kappa,H):=\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\psi_{\ast}|^{2}-\kappa^{2}\,a(x)|\psi_{\ast}|^{2}\right)\,dx<0\,.
$$
We can write,
\begin{align*}
\mathcal E_{\kappa,H,a,B_{0}}(t\psi_{\ast},\mathbf F)&=t^{2}\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\psi_{\ast}|^{2}-\kappa^{2}\,a(x)|\psi_{\ast}|^{2}\right)\,dx+t^{4}\,\frac{\kappa^2}{2}\int_{\Omega}|\psi_{\ast}|^{4}\,dx+\frac{\kappa^2}{2}\int_{\Omega}a(x)\,dx\\
&=t^2\left(C_{1}(\kappa,H)+t^{2}\,\frac{\kappa^2}{2}\int_{\Omega}|\psi_{\ast}|^{4}\,dx\right)+ \mathcal E_{\kappa,H,a,B_{0}}(0,\mathbf F)\,.
\end{align*}
We choose $t$ such that,
$$
C_{1}(\kappa,H)+t^{2}\,\frac{\kappa^2}{2}\int_{\Omega}|\psi_{\ast}|^{4}\,dx<0\,.
$$
Thus, we get
$$
\mathcal E_{\kappa,H,a,B_{0}}(t\psi_{\ast},\mathbf F)<\mathcal E_{\kappa,H,a,B_{0}}(0,\mathbf F)\,.
$$
Hence a minimizer, which is a solution of \eqref{eq-2D-GLeq}, will be non-trivial.
\end{proof}
\subsection{End of the proof of Theorem~\ref{thm:HC3}}
First, we will prove the following inclusion,
$$
\mathcal{N}^{\rm loc}(\kappa)\subset \mathcal{N}(\kappa)\,.
$$
We see that if $H\notin \mathcal{N}(\kappa)$, then $(0,\mathbf F)$ is a local minimizer of $\mathcal E_{\kappa,H,a,B_{0}}$. Thus, the Hessian of the functional $\mathcal E_{\kappa,H,a,B_{0}}$ at the normal state $(0,\mathbf F)$ should be positive.\\
For every $(\widetilde{\phi},\widetilde{\mathbf A})\in H^1(\Omega)\times H^1_{\rm div}(\Omega)$ we have,
$$
\mathcal E_{\kappa,H,a,B_{0}}(t\widetilde{\phi},\mathbf F+t\widetilde{\mathbf A})=t^{2}\left[\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\widetilde{\phi})+(\kappa H)^{2}\int_{\Omega}|\curl \widetilde{\mathbf A}|^{2}\,dx\right]+\mathcal{O}(t^3)\,.
$$
This implies that the Hessian of the functional $\mathcal E_{\kappa,H,a,B_{0}}$ at the normal state $(0,\mathbf F)$ can be written as follows:
$$
Hess_{(0,\mathbf F)}[\widetilde{\phi},\widetilde{\mathbf A}]=\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\widetilde{\phi})+(\kappa H)^{2}\int_{\Omega}|\curl \widetilde{\mathbf A}|^{2}\,dx\,.
$$
Since $Hess_{(0,\mathbf F)}[\widetilde{\phi},\widetilde{\mathbf A}]\geq 0$, we get that $\mu_{1}(\kappa H)\geq 0\,,$ and consequently $H\notin \mathcal{N}^{\rm loc}(\kappa)$.
Hence we obtain the above inclusion.\\
On the other hand, if $(\psi,\mathbf A)$ is a minimizer of the functional in \eqref{eq-2D-GLf} with $\psi\neq 0$, then $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, and we have the following inclusion,
$$
\mathcal{N}(\kappa)\subset \mathcal{N}^{\rm cp}(\kappa)\,,
$$
and consequently,
\begin{equation}\label{inclusion-N}
\mathcal{N}^{\rm loc}(\kappa)\subset \mathcal{N}(\kappa)\subset \mathcal{N}^{\rm cp}(\kappa)\,.
\end{equation}
Having in mind the definition of all the critical fields in \eqref{def:HC3-o}, \eqref{def:HC3} and \eqref{def:HC3-u}, we deduce that,
\begin{equation}\label{eq:ov}
\overline{H}_{C_3}^{loc}(\kappa)\leq\overline{H}_{C_3}(\kappa)\leq\overline{H}_{C_3}^{cp}(\kappa)\,,
\end{equation}
Using \eqref{inclusion-N}, we observe that,
$$
\mathbb R^{+}\setminus\mathcal{N}^{\rm cp}(\kappa)\subset\mathbb R^{+}\setminus\mathcal{N}(\kappa)\subset\mathbb R^{+}\setminus\mathcal{N}^{\rm loc}(\kappa)\,.
$$
From the definition of all the critical fields, we conclude that,
\begin{equation}\label{eq:un}
\underline{H}_{C_3}^{loc}(\kappa)\leq\underline{H}_{C_3}(\kappa)\leq\underline{H}_{C_3}^{cp}(\kappa)\,.
\end{equation}
We note that $\underline{H}_{C_{3}}^{loc}\leq\overline{H}_{C_{3}}^{loc}$ and $\underline{H}_{C_{3}}^{cp}\leq\overline{H}_{C_{3}}^{cp}$. Therefore, all the critical fields are contained in the interval $[\underline{H}_{C_{3}}^{loc},\overline{H}_{C_{3}}^{cp}]$.\\
By Proposition~\ref{prop:mu1<0} and Theorem~\ref{thm:lb-H}, we get the existence of positive constants $C$ and $\kappa_0$, such that for $\kappa\geq\kappa_0$,
\begin{multline}
\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\leq \underline{H}_{C_{3}}^{loc}\leq\overline{H}_{C_{3}}^{cp}\\
\leq\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,.
\end{multline}
As a consequence, we have proved Theorem~\ref{thm:HC3} for the six critical fields.
\begin{rem}
As in \cite{FH1}, it would be interesting to show that all the critical fields coincide when $\kappa$ is large enough.
\end{rem}
\section{Asymptotics of $\mu_{1}(\kappa,H)$: the case with vanishing magnetic field}\label{Section:Asympt-m1-vanish}
In this section we give an estimate for the lowest eigenvalue
$\mu_{1}(\kappa,H)$ of the operator $P_{\kappa
H\mathbf F,-\kappa^{2}a}^{\Omega}$ (see \eqref{def:mu1}) in the case when
$\Gamma=\varnothing$ with a $\kappa$-independent pinning, i.e.
$a(\kappa,x)=a(x)$. The results in this section are valid under the
assumption $\Gamma\not=\emptyset$, where the set $\Gamma$ is introduced in
\eqref{gamma}. Let
\begin{equation}
\mathcal{B}=\kappa H\qquad{\rm and}\qquad \widehat{\sigma}=\frac{H}{\kappa^2}\,.
\end{equation}
We observe that,
$$
P_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}=P_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}\,.
$$
We will give an estimate for the lowest eigenvalue
$\mu_{\mathcal{B},\widehat{\sigma}}$ of
$P_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}$.
After a change of notation, we deduce an estimate for
$\mu_{1}(\kappa,H)$.
\subsection{Lower bound}
In the
absence of a pinning term, that is when $a=1$, Pan and Kwek
\cite{XB-KH} gave the lower bound for the lowest eigenvalue
$\mu(\mathcal{B}\mathbf F)$ of $P_{\mathcal{B}\mathbf F,0}^{\Omega}$ when
$\mathcal{B}\to+\infty$.
In this subsection, we determine a lower bound for $\mu_{1}$ when $\kappa\to+\infty$ and the pinning term is present.\\
We first recall the definition of $\lambda_{0}$ in \eqref{lambda0}, the definition of $\Gamma$ in \eqref{gamma} and for any $\theta\in(0,\pi)$ we recall that $\lambda(\mathbb R_{+}^{2},\theta)$ is the bottom of the spectrum of the operator
$ P_{\mathbf A_{\rm app,\theta},0}^{\mathbb R^{2}_{+}}$, with
$$\mathbf A_{\rm app,\theta}=-\left(\frac{x^{2}_{2}}{2}\cos\,\theta,\frac{x^{2}_{1}}{2}\sin\,\theta \right)\,.$$
We then define for any $\widehat{\sigma}>0$,
\begin{multline}\label{Lambda1}
\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})=\min\left\{\inf_{x\in \Gamma\cap\Omega}\left\{\lambda_{0}\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\},\right.\\
\left.\inf_{x\in \Gamma\cap\partial\Omega}\left\{\lambda(\mathbb R^{2}_{+},\theta(x))\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\}\right\}\,.
\end{multline}
Here, for $x\in\Gamma\cap\partial\Omega$, $\theta(x)$ denotes the angle between $\nabla B_{0}(x)$ and the inward normal vector $-\nu(x)$.
We start with a proposition that states a lower bound of $\mu_{1}(\kappa,H)$
in the case when $\Gamma\neq\varnothing$.
\begin{prop}\label{prop:mu1-variable}
Let $I$ be a closed interval in $(0,\infty)$. There exist two
positive constants $\mathcal{B}_{0}>0$ and $C>0$ such that if
$\mathcal{B}\geq\mathcal{B}_{0}$, $\widehat\sigma\in I$, $\psi\in
H^{1}(\Omega)\setminus \{0\} $ and $a\in C^{1}(\overline{\Omega})$,
then,
\begin{equation}\label{eq:lower-bound-variable}
\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}(\psi)}{\|\psi\|^{2}_{L^{2}(\Omega)}}\geq
\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\Big(\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})
-C\mathcal{B}^{-\frac{1}{18}}\Big)\,.
\end{equation}
\end{prop}
\begin{proof}
Let $\ell=B^{-7/29}$. We define the following sets,
\begin{align*}
&D_{1}=\{x\in\Omega: \dist(x,\Gamma)<2\,\ell\}\,,
&D_{2}=\{x\in\Omega: \dist(x,\Gamma)>\ell\}\,.
\end{align*}
Let $h_{j}$ be a partition of unity satisfying
$$
\sum_{j=1}^{2} h_{j}^{2}=1\,,\qquad \sum_{j=1}^{2}|\nabla h_{j}|^{2}\leq C\,\ell^{-2}=C\mathcal B^{14/29}\qquad{\rm and}\qquad \supp h_{j}\subset D_{j}\quad(j\in\{1,2\})\,.
$$
There holds the following decomposition,
\begin{multline}\label{eq:pu}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} a}^{\Omega}(\psi)
=\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(h_{1}\psi)+\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)-\sum_{j=1}^{2}\int_{\Omega}|\nabla h_{j}|^{2}|\psi|^{2}\,dx\\
\geq \mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(h_{1}\psi)+\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)
-C\mathcal B^{14/29}\int_{\Omega}|\psi|^{2}\,dx\,.
\end{multline}
We cover the curve $\Gamma$ by a family of disks
$$D(\omega_{j},\ell)\subset\{x\in\mathbb R^{2}:\dist(x,\Gamma)\leq 2\ell\}\qquad{\rm and}\qquad D_{1}\subset\bigcup_{j} D(\omega_{j},\ell) \qquad\left(\omega_{j}\in \Gamma\right)\,.$$
Consider a partition of unity satisfying
$$
\sum_{j} \chi_{j}^{2}=1\,,\qquad \sum_{j} |\nabla \chi_{j}|^{2}\leq C\,\ell^{-2}\qquad{\rm and}\qquad \supp \chi_{j}\subset D(\omega_{j},\ell)\,.
$$
Moreover, we can add the property that:
$$
{\rm either~supp}\chi_{j}\cap\Gamma\cap\partial\Omega=\varnothing\,,\quad{\rm either}~\omega_{j}\in\Gamma\cap\partial\Omega\,.
$$
We may write,
\begin{equation}\label{eq:main-bulk+bnd}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} a}^{D_{1}}(h_{1}\psi)=\sum_{int}\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)+\sum_{bnd}\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)-\sum_{j}\int_{D_{1}}|\nabla\chi_{j}|^{2}|h_{1}\psi|^{2}\,dx\,,
\end{equation}
where `int' is in reference to the $j$'s such that $\omega_{j}\in\Gamma\cap\Omega$ and `bnd' is in reference to the $j$'s such that $\omega_{j}\in\Gamma\cap\partial\Omega$.\\
For the last term on the right side of \eqref{eq:main-bulk+bnd}, we get using the assumption on $\chi_{j}$:
\begin{equation}\label{eq:est-error}
\int_{D_{1}}|\nabla\chi_{j}|^{2}|h_{1}\psi|^{2}\,dx\leq C\,\ell^{-2}\,\int_{D_{1}}|h_{1}\psi|^{2}\,dx
= C\,\mathcal B^{14/29}\,\int_{D_{1}}|h_{1}\psi|^{2}\,dx\,.
\end{equation}
We have to find a lower bound for
$\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(h_{1}\psi)$
for each $j$ such that $\omega_{j}\in\Gamma\cap\Omega$ and for each
$j$ such that $\omega_{j}\in\Gamma\cap\partial\Omega$. Thanks to
\cite{JPM}, we have,
$$\int_\Omega|(\nabla-i\mathcal B\mathbf F)\chi_jh_1\psi|^2\,dx\geq \mathcal
B^{\frac23}\int_\Omega\Big((\lambda_{0}\,|\nabla B_{0}(\omega_{j})|\Big)^{\frac{2}{3}}-CB^{-1/18}\Big)|\chi_jh_1\psi|^2\,dx\,.
$$
Using Taylor's formula, we can write in every disk $D(w_j,\ell)$,
\begin{equation}\label{eq:aT}
|a(x)-a(w_j)|\leq C\ell=C\mathcal B^{-7/29}\leq C\mathcal B^{-1/18}\,.
\end{equation}
In that way, we get,
\begin{align}\label{eq:lb-bulk-vr}
&\sum_{int}\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)\nonumber\\
&\quad\geq\sum_{int}\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} \left(\lambda_{0}\,\Big(\widehat{\sigma}\,|\nabla B_{0}(\omega_{j})|\Big)^{\frac{2}{3}}-a(\omega_{j})-C\mathcal B^{-1/18}
\right) \int|\chi_{j}h_{1}\psi|^{2}\,dx\nonumber\\
&\quad\geq\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\left(
\inf_{x\in \Gamma\cap\Omega}\left\{\lambda_{0}\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\}-C\mathcal B^{-1/18}\right)
\sum_{int} \int|\chi_{j}h_{1}\psi|^{2}\,dx\,.
\end{align}
In a similar fashion, the analysis in \cite{JPM} and \eqref{eq:aT}
yields,
\begin{align}\label{eq:lb-bnd-vr}
&\sum_{bnd}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)\nonumber\\
&\quad\geq\sum_{bnd}
\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}
\left(\lambda(\mathbb R^{2}_{+},\theta(\omega_{j}))\,\Big(\widehat{\sigma}\,|\nabla B_{0}(\omega_{j})|\Big)^{\frac{2}{3}}-a(\omega_{j})-C\mathcal B^{-1/18}\right)
\int|\chi_{j}h_{1}\psi|^{2}\,dx\nonumber\\
&\quad\geq\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}
\left(\inf_{x\in \Gamma\cap\partial\Omega}\left\{\lambda(\mathbb R^{2}_{+},\theta(x))\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\}-C\mathcal B^{-1/18}\right)
\sum_{bnd} \int|\chi_{j}h_{1}\psi|^{2}\,dx\,.
\end{align}
We insert \eqref{eq:lb-bulk-vr}, \eqref{eq:lb-bnd-vr}
and \eqref{eq:est-error}
into \eqref{eq:main-bulk+bnd} to obtain,
\begin{equation}\label{eq:main-bulk+bnd-D2}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} a}^{D_{1}}(h_{1}\psi)\geq \left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\left(\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})
-C\mathcal B^{-1/18}\right)\,\int|h_{1}\psi|^{2}\,dx\,.
\end{equation}
Now, we will bound $\int_\Omega |(\nabla-i\mathcal B\mathbf F)h_{2}\psi|^2\,dx$ from below. Let $\ell_{1}<\ell$, we cover $D_{2}$ by a family of disks
$$D(\omega_{j}',\ell_{1})\subset\{x\in\mathbb R^{2}:\dist(x,\Gamma)\geq \ell_{1}\}\qquad\left(\omega_{j}'\in \overline \Omega\right)\,.$$
Consider a partition of unity satisfying
$$
\sum_{j} \chi_{j}^{2}=1\,,\qquad \sum_{j} |\nabla \chi_{j}|^{2}\leq C\,\ell_{1}^{-2}\qquad{\rm and}\qquad \supp \chi_{j}\subset D(\omega_{j}',\ell_{1})\,.$$
There holds the decomposition formula,
\begin{align}\label{eq:int-bnd}
\int_\Omega |(\nabla-i\mathcal B\mathbf F)h_{2}\psi|^2\,dx&=\sum_{j}\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx-\sum_{j}\int_{\Omega}|\nabla \chi_{j}|^{2} |h_{2}\psi|^{2}\,dx\nonumber\\
&\geq \sum_{j}\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx-C \ell_{1}^{-2}\int_{\Omega} |h_{2}\psi|^{2}\,dx\,,
\end{align}
We observe that there exists a gauge function $\varphi_{j}$ satisfying (see \cite[Equation~(A.3)]{KA}),
$$
\left|\mathbf F(x)-(B_{0}(\omega_{j}')\mathbf A_{0}(x-\omega_{j}')+\nabla\varphi_{j})\right|\leq C\,\ell_{1}^{2} \quad{\rm in}~D(\omega_{j}',\ell_{1}')\,.
$$
Using Cauchy-Schwarz inequality, we may write,
\begin{multline*}
\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx\geq \frac{1}{2}\int_\Omega |(\nabla-i\mathcal{B}\,B_{0}(\omega_{j}')\mathbf A_{0}(x-\omega_{j}'))e^{-i\mathcal{B}\varphi_{j}}\chi_{j}\,h_{2}\psi|^2\,dx\\
-C\,\mathcal{B}^{2}\,\ell_{1}^{4}\int_{\Omega}|\chi_{j}\,h_{2}\psi|^2\,dx\,.
\end{multline*}
We are reduced to the analysis of the Neumann realization of the Schr\"odinger operator with a constant magnetic field equal to $\mathcal{B}\,B_{0}(\omega_{j}')$ in our case.\\
Notice that by the assumption on $h_{2}$, there exist constants $M>0$ and $\mathcal B_0>0$ such that, for all $j$, $|B_{0}(\omega_{j}')|\geq M\,\ell$ in the support of $h_{2}$. Thus,
$$
\forall j,\quad\mathcal{B}|B_{0}(\omega_{j}')|\geq M\,\mathcal{B}\,\ell=M\mathcal B^{22/29}\gg 1\,.
$$
Moreover, the magnetic potentials $\mathbf A_{0}(x)$ and $\mathbf A_{0}(x-\omega_{j}')$ are gauge equivalent since
$$
\mathbf A_{0}(x-\omega_{j}')=\mathbf A_{0}(x)-\mathbf A_{0}(\omega_{j}')=\mathbf A_{0}(x)-\nabla(\mathbf A_{0}(\omega_{j}')\cdot x)\,.
$$
Thanks to Theorem~\ref{thm:FH}, there exists a constant $\mathcal{B}_{0}$ such that, for any $\mathcal{B}\geq \mathcal{B}_{0}$, we write by the min-max principle,
\begin{align}\label{eq:first-term}
\sum_{j}\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx&\geq \frac{\Theta_0\mathcal{B}\,|B_{0}(\omega_{j}')|}{2}\sum_{int}\int_{\Omega}|\chi_{j}\, h_{2} \psi|^{2}\,dx-C\,\mathcal{B}^{2}\,\ell_{1}^{4}\sum_{int}\int_{\Omega}|\chi_{j}\,h_{2}\psi|^2\,dx\nonumber\\
&\geq \left(\frac{M\Theta_0}2\mathcal{B}\,\ell-C\mathcal{B}^{2}\,\ell_{1}^{4}\right)\sum_{j}\int_{\Omega}|\chi_{j}\,h_{2}\psi|^2\,dx\nonumber\\
&=\left(\frac{M\Theta_0}2\mathcal{B}\,\ell-C\mathcal{B}^{2}\,\ell_{1}^{4}\right)\int_{\Omega}|h_{2}\psi|^2\,dx\,.
\end{align}
Putting \eqref{eq:first-term} into \eqref{eq:int-bnd}, we obtain
\begin{align}\label{eq:D2}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)&=\int_\Omega |(\nabla-i\mathcal B\mathbf F)h_{2}\psi|^2\,dx-\left(\frac{\mathcal B}{\widehat\sigma}\right)^{2/3}\int_{\Omega} a(x)|h_{2}\psi|^2\,dx\nonumber\\
&\geq \left(\frac{M\Theta_0}2\mathcal{B}\,\ell-C\mathcal{B}^{2}\,\ell_{1}^{4}-C\ell_{1}^{-2}\right)\int_{\Omega}|h_{2}\psi|^2\,dx-\left(\frac{\mathcal B}{\widehat\sigma}\right)^{2/3}\int_{\Omega} a(x)|h_{2}\psi|^2\,dx\,.
\end{align}
We choose $\ell_1=B^{-\rho}$ and $\frac{9}{22}<\rho<\frac{11}{29}$. We observe that,
$$\mathcal{B}^{2}\,\ell_{1}^{4}=\mathcal{B}^{2-4\rho}\ll \mathcal B^{22/29}= \mathcal{B}\,\ell\,,\quad
\ell_{1}^{-2}=B^{2\rho}\ll\mathcal B\,\ell\,,\quad \mathcal{B}^{2/3}\ll \mathcal B^{22/29}=\mathcal B\,\ell\,.$$
In this way, we infer from \eqref{eq:D2}, that there exists a constant $c>0$ such that, for $\mathcal{B}$ sufficiently large,
\begin{equation}\label{eq:D2'}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)\geq c\mathcal B^{22/9}\int_{\Omega} |h_{2}\psi|^2\,dx
\geq \left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})\int_{\Omega} |h_{2}\psi|^2\,dx\,.
\end{equation}
Collecting \eqref{eq:pu}, \eqref{eq:main-bulk+bnd-D2} and \eqref{eq:D2'}, we finish the proof of Proposition~\ref{prop:mu1-variable}.
\end{proof}
Theorem~\ref{thm:mu1-up-vr} is valid under the assumption that,
\begin{equation}\label{cond:sigma-hat}
\widehat{\lambda}_{\min}\leq \frac{H}{\kappa^2}\leq \widehat{\lambda}_{\max}\,,
\end{equation}
where $0<\widehat{\lambda}_{\min}<\widehat{\lambda}_{\max}<\infty$ are constants independent of $\kappa$ and $H$.
\begin{theorem}\label{thm:mu1-up-vr}
Let $\Omega\subset\mathbb R^2$ is an open bounded set with a smooth boundary and $\Gamma\neq\varnothing$. Suppose that \eqref{cond:sigma-hat} hold and $a\in C^1(\overline{\Omega})$, we have
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\widehat\Lambda_{1}\left(B_{0},a,\frac {H}{\kappa^{2}}\right)+ \mathcal{O}(\kappa^{\frac{11}{6}})\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
Here, $\widehat\Lambda_{1}$ is introduced in \eqref{Lambda1}.
\end{theorem}
\begin{proof}
We apply Proposition~\ref{prop:mu1-variable} with
$$
\mathcal{B}=\kappa H\,,\quad \widehat{\sigma}=\frac{H}{\kappa^2}\quad{\rm and}\quad I=[\widehat{\lambda}_{\min},\widehat{\lambda}_{\max}]\,.
$$
Let us verify that the conditions of the proposition are satisfied for this choice.
Thanks to \eqref{cond:sigma-hat}, $\widehat{\sigma}\in I$. Now, as $\kappa\to+\infty$, we have,
$$
\mathcal{B}=\widehat{\sigma}\,\kappa^{3}\to+\infty\,.
$$
This implies that, as $\kappa\to+\infty$,
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\widehat\Lambda_{1}\left(B_{0},a,\frac{H}{\kappa^2}\right)+\mathcal{O}(\kappa^{\frac{11}{6}})\,.
$$
This finish the proof of the theorem.
\end{proof}
\subsection{Upper bound}~\\
The next theorem is a generalization of the results in \cite{XB-KH}
and \cite{JPM} valid when the pinning term $a(\kappa,x)=a(x)$ is
independent of $\kappa$ and non-constant.
We denote by $\mu_{\mathcal B,\widehat{\sigma}}$ the lowest eigenvalue of the operator $P_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}$ i.e.
$$\mu_{\mathcal B,\widehat{\sigma}}=\inf_{\psi\in H^{1}(\Omega)\setminus\{0\}}\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}(\psi)}{\|\psi\|^{2}_{L^{2}(\Omega)}}\,.$$
\begin{prop}\label{prop:up-var}
Suppose that $\Gamma\neq\varnothing$ and $\widehat\lambda_{\max}>0$. There exist positive constants $C$ and $B_{0}$ such that, for $\widehat\sigma\in (0,\widehat\lambda_{\max}]$, $a\in C^{1}(\overline{\Omega})$ and $\mathcal{B}\geq\mathcal{B}_0$, we have,
\begin{equation}\label{eq:upper-bound-variable}
\mu_{\mathcal{B},\widehat{\sigma}}\leq
\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\Big(\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})
-C\mathcal{B}^{-\frac{1}{18}}\Big)\,.
\end{equation}
\end{prop}
\begin{proof}
Let $x_0\in\Gamma$. In \cite{XB-KH, JPM}, a quasi-mode $u(\mathcal
B,x_0;x)$ is constructed such that, ${\rm supp}\,u(\mathcal
B,x_0;\cdot)\subset \overline{\Omega}\cap B(0,\mathcal B^{-1/18})$
and,
$$\forall~\mathcal B\geq \mathcal B_0\,,\quad\frac{\displaystyle\int_\Omega|(\nabla-i\mathcal B\mathbf F)u(\mathcal
B,x_0;x)|^2\,dx}{\displaystyle\int_\Omega|u(\mathcal
B,x_0;x)|^2\,dx}\leq \mathcal B^{\frac{2}{3}}\Big(\Lambda(x_0)+C\mathcal
B^{-1/18}\Big)\,,$$ where $\mathcal B_0$ and $C$ are constants independent of the point $x_0$ and the parameter $\mathcal B$, and
$$\Lambda(x_0)=
\left\{
\begin{array}{ll}
\lambda_{0}\,|\nabla B_{0}(x_0)|^{\frac{2}{3}}&{\rm if~}x_0\in\Gamma\cap\Omega\,,\\
\lambda(\mathbb R^{2}_{+},\theta(x_0))\,|\nabla
B_{0}(x_0)|^{\frac{2}{3}}&{\rm if~}x_0\in\Gamma\cap\partial\Omega\,.
\end{array}
\right.
$$
Using the smoothness of the function $a(\cdot)$, we get in the
support of $u(\mathcal B,x_0;\cdot)$,
$$|a(x)-a(x_0)|\leq C\mathcal B^{-1/18}\,.$$
Thus, we deduce that,
$$\frac{\mathcal Q^\Omega_{ \mathcal B\mathbf F,-\left(\frac{\mathcal B}{\widehat\sigma}\right)^{\frac{2}{3}}a }(u(\mathcal B,x_0;\cdot)}{\|u(\mathcal B,x_0;\cdot)\|^2_{L^2(\Omega)}}
\leq \left(\frac{\mathcal
B}{\widehat\sigma}\right)^{\frac{2}{3}}\Big(\widehat\sigma^{\frac{2}{3}}\Lambda(x_0)-a(x_0)+C\mathcal B^{-1/18}\Big)\,.$$
Thanks to the min-max principle, we deduce that,
$$\mu_{\mathcal B,\widehat\sigma}\leq
\left(\frac{\mathcal
B}{\widehat\sigma}\right)^{\frac{2}{3}}\Big(\widehat\sigma^{\frac{2}{3}}\Lambda(x_0)-a(x_0)+C\mathcal B^{-1/18}\Big)\,.$$
Since this is true
for all $x_0\in\Gamma$, we deduce that,
$$\mu_{\mathcal B,\widehat\sigma}\leq
\left(\frac{\mathcal
B}{\widehat\sigma}\right)^{\frac{2}{3}}\Big(\widehat\Lambda_1(B_0,a,\widehat\sigma)+C\mathcal
B^{-1/18}\Big)\,,$$ where $\widehat\Lambda_1(B_0,a,\widehat\sigma)$ is introduced in \eqref{Lambda1}.
\end{proof}
Proposition~\ref{prop:up-var} permits to obtain:
\begin{theorem}\label{thm:mu1-upp-var}
Let $\widehat{\lambda}_{\max}>0$. Suppose that $\Gamma\neq\varnothing$ and $a\in C^1(\overline{\Omega})$. There exist two constants $C_1>0$ and $\kappa_0>0$ such that,
if,
\begin{equation}\label{cond:H-m1<}
\kappa\geq\kappa_0\,,\quad {\rm and}\quad \kappa_0\kappa^{-1}<H<\widehat{\lambda}_{\max}\kappa^2\,
\end{equation}
then
$$
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\widehat\Lambda_{1}\left(B_{0},a,\frac{H}{\kappa^2}\right)+C_1\kappa^{\frac{11}{6}}\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
\end{theorem}
\begin{proof}
To apply the results of Proposition~\ref{prop:up-var}, we take $\mathcal{B}=\kappa H$ and $\widehat{\sigma}=\frac{H}{\kappa^2}$. We see for $\kappa$ sufficiently large that $\widehat{\sigma}\in (0,\widehat{\lambda}_{\max})$ and $\mathcal{B}$ large.
\end{proof}
Theorem~\ref{thm:mu1-upp-var} is valid when $\kappa H\geq \kappa_0$ and $\kappa_0$ is sufficiently large.
\section{Proof of Theorem~\ref{thm:HC3-vr}}\label{12}
\subsection{Analysis of $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$.}~\\
In this subsection we will prove Theorem~\ref{thm:HC3-vr} for $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$.
We first recall some useful results from \cite{XB-KH} about the relation between the eigenvalues $\lambda_0$ and $ \lambda(\mathbb R_{+}^{2},\theta)$, introduced in \eqref{lambda0} and in \eqref{def:lambda-theta}.
\begin{thm}\label{thm:PK-R2+}~
\begin{enumerate}
\item[(i)] $\lambda(\mathbb R^{2}_{+},0)=\lambda_{0}$\,.
\item[(ii)] If $0<\theta<\pi$, then $\lambda(\mathbb R^{2}_{+},\theta)<\lambda_{0}$.
\end{enumerate}
\end{thm}
The next proposition gives the region where $\mu_{1}(\kappa,H)<0$ that allows us to obtain an information about $\underline{H}_{C_3}^{loc}$ (see \eqref{def:HC3-u}) in the case when the magnetic field $B_0$ is constant with a pining term.
\begin{prop}\label{prop:mu1<0-var}
Suppose that $\{a>0\}\neq\varnothing$ and $\Gamma\neq\varnothing$. There exist constants $C>0$ and $\kappa_{0}\geq 0$ such that if
\begin{equation}\label{cond:HC3-var}
\kappa\geq \kappa_{0}\,,\qquad H\leq \max\left(\sup_{x\in\Gamma\cap\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\,,
\end{equation}
then,
$$
\mu_{1}(\kappa,H)<0\,.
$$
Moreover,
$$
\max\left(\sup_{x\in\Gamma\cap\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}- C\,\kappa^{\frac{11}{6}}\leq \underline{H}_{C_{3}}^{loc}\,.
$$
\end{prop}
\begin{proof}
We have two cases:\\
\textbf{Case 1.} Here, we suppose that,
$$\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}>\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\,.$$
Thanks to the assumption in \eqref{B(x)}, we have, for all $x\in\Gamma\cap\partial\Omega$, $0<\theta(x)<\pi$. Theorem~\ref{thm:PK-R2+} then tells us that,
$$
\forall~x\in\Gamma\cap\partial\Omega\,,\quad \frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}>\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}\,.
$$
Thus, there exists $x_0\in\Omega\cap\Gamma$ such that (the supremum of $\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}$ in $\Gamma\cap\overline{\Omega}$ can not be attained on the boundary),
$$
\sup_{x\in\Gamma\cap\overline\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}= \frac{a(x_0)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x_0)|} \,.
$$
If \eqref{cond:HC3-var} is satisfied for some $C>0$, then,
$$
\frac{H}{\kappa^{2}}\leq\frac{a(x_0)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x_0)|} -C\,\kappa^{-\frac{1}{6}}\,,
$$
that we can write in the form,
\begin{equation}\label{eq:appendix}
\kappa^{2}\left(\lambda_0\left(\frac{H}{\kappa^2}|\nabla B_{0}(x_0)|\right)^{\frac{2}{3}}-a(x_0)\right)\leq -C \,M\,\kappa^{\frac{11}{6}}\,,
\end{equation}
where $M>0$ is a constant independent of $C$.
Suppose that $\kappa H\geq \mathcal B_0$ where $\mathcal B_0$ is selected sufficiently large such that we can apply Theorem~\ref{thm:mu1-upp-var}. (Thanks to Lemma~\ref{lem-H=kappa}, $\mu_1(\kappa,H)<0$ when $\kappa H<\mathcal B_0$).
By Theorem~\ref{thm:mu1-upp-var}, there exist positive constants $C_{1}$ and $\kappa_{0}$ such that, for $\kappa\geq \kappa_{0}$,
\begin{align}
\mu_{1}(\kappa,H)&\leq \kappa^{2}\inf_{x\in\Gamma\cap\overline\Omega} \left(\lambda_0\left(\frac{H}{\kappa^2}|\nabla B_{0}(x)|\right)^{\frac{2}{3}}-a(x)\right)+C_{1}\,\kappa^{\frac{11}{6}}\nonumber\\
&\leq \kappa^{2}\left(\lambda_0\left(\frac{H}{\kappa^2}|\nabla B_{0}(x_0)|\right)^{\frac{2}{3}}-a(x_0)\right)+C_{1}\,\kappa^{\frac{11}{6}}\nonumber\\
&\leq (C_1-C\,M) \,\kappa^\frac {11}{6} \,.
\end{align}
By choosing $C$ such that $C\,M> C_{1}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
\textbf{Case 2.} Here, we suppose that
$$\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\geq\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}\,.$$
The assumption in \eqref{cond:HC3-var} and the upper bound in Theorem~\ref{thm:mu1-upp-var} give us, for all $\kappa\geq \kappa_0$, $\kappa H\geq \mathcal B_0$ and $\mathcal B_0$ a sufficiently large constant,
$$
\mu_{1}(\kappa,H)\leq (C_1-C\,\widetilde M)\,\kappa^\frac{11}{6}\,.
$$
where $\widetilde M>0$ is a constant independent of $C$.
By choosing $C$ such that $C\,\widetilde M> C_{1}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
This finishes the proof of the proposition.
\end{proof}
The next proposition gives us a lower bound of $\overline{H}_{C_3}^{loc}$ (see \eqref{def:HC3-u}). This is obtained by localizing the region where $\mu_{1}(\kappa,H)>0$ holds.
\begin{prop}\label{prop:mu>0-var}
Suppose that $\{a>0\}\neq\varnothing$, $\widehat\lambda_{\max} >0$ and $\Gamma=\varnothing$. There exist constants $C>0$ and $\kappa_{0} > 0$ such that if
\begin{equation}\label{cond:HC3-2-var}
\begin{aligned}
\kappa\geq \kappa_{0}\,,\qquad \widehat\lambda_{\max}\,\kappa&\geq H\\
&>\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,,
\end{aligned}
\end{equation}
then,
$$
\mu_{1}(\kappa,H)>0\,.
$$
Moreover,
$$
\overline{H}_{C_{3}}^{loc}\leq\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,.
$$
\end{prop}
\begin{proof}
Having in mind the definition of $\widehat\Lambda_1$ in \eqref{Lambda1}, under the assumption in \eqref{cond:HC3-2-var}, we get for $\kappa$ large enough,
\begin{equation}\label{eq:appendix2}
\widehat\Lambda_{1}\left(B_0,a,\frac{H}{\kappa^2}\right)\geq C\,M\,\kappa^{-\frac{1}{6}}\,,
\end{equation}
where $M>0$ is a constant independent of the constant $C$.
Thanks to Theorem~\ref{thm:mu1-up-vr}, we get the existence of positive constants $C'$ and $\kappa_0$ such that, for $\kappa\geq\kappa_0$,
$$
\mu_{1}(\kappa,H)\geq (C\,M-C')\,\kappa^{\frac{11}{6}}
$$
To finish the proof, we choose $C$ sufficiently large such that $C\,M>C'$.
\end{proof}
\subsection{Analysis of $\underline{H}_{C_3}^{cp}$ and $\overline{H}_{C_3}^{cp}$.}~\\
Proposition~\ref{prop:est-psi-var} below is an adaptation of an analogous result obtained in \cite{HK} for the functional in \eqref{eq-2D-GLf} with a constant pinning term. Proposition~\ref{prop:est-psi-var} is valid when $\Gamma\not=\emptyset$.
Proposition~\ref{prop:est-psi-var} says that, if $(\psi,\mathbf A)$ is a critical point of the functional in \eqref{eq-2D-GLf} and $H$ is of order $\kappa^2$, then $|\psi|$ is concentrated near the set $\Gamma$.
\begin{prop}\label{prop:est-psi-var}
Let $\varepsilon>0$. There exist two positive constants $C$ and $\kappa_0$ such that, if
\begin{equation}\label{cond:H>kappa}
\kappa\geq\kappa_0\,,\quad H\geq\varepsilon\,\kappa^{2}\,,
\end{equation}
and $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, then
\begin{equation}\label{l3est}
\|\psi\|^{2}_{L^2(\Omega)}\leq C\,\kappa^{-\frac{1}{4}}\|\psi\|^{2}_{L^4(\Omega)}\,.
\end{equation}
\end{prop}
\begin{proof}
Let $\lambda=\kappa^{-\frac 12}$ and $\Omega_{\lambda}=\{x\in\Omega:\dist(x,\partial\Omega)>\lambda~\&~\dist(x,\Gamma)>\lambda\}$. We introduce a function $h\in C^{\infty}_{c}(\Omega)$ satisfying
$$
0\leq h \leq 1~{\rm in}~\Omega\,,\quad h=1~{\rm in}~\Omega_{\lambda}\,,\quad {\rm supp}\,h\subset\Omega_{\lambda/2}\,,
$$
and
$$
|\nabla h |\leq \frac{C}{\lambda}\quad{\rm in}~\Omega\,,
$$
where $C$ is a positive constant.\\
Using \eqref{3d-<}, we can prove that (see the detailed proof in
\cite[Eq.~(6.6)]{HK} when $a$ is constant),
$$
\kappa\,H\int_\Omega|B_{0}(x)|\,|h\psi|^2\,dx-c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|h\psi\|_{L^{4}(\Omega)}^{2}\leq \int_{\Omega}|(\nabla-i\kappa H\mathbf A)h\psi|^2\,dx\,.
$$
Now, the Cauchy-Schwarz inequality yields,
$$
c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|h\psi\|_{L^{4}(\Omega)}^{2}\leq c^{2}\|\psi\|_{L^{2}(\Omega)}^{2}+\kappa^2\|h\psi\|_{L^{4}(\Omega)}^{4}\,,
$$
which implies that
\begin{multline*}
\int_\Omega\,\left(\kappa\,H\,|B_{0}(x)|-\kappa^{2}\,a(x)\right)\,|h\psi|^2\,dx\leq\int_{\Omega}|(\nabla-i\kappa H\mathbf A)h\psi|^2\,dx-\kappa^{2}\int_{\Omega}\,a(x)\,|h\psi|^2\,dx\\
+c^{2}\|\psi\|_{L^{2}(\Omega)}^{2}+\kappa^{2}\|h\psi\|_{L^{4}(\Omega)}^{4}\,.
\end{multline*}
We may use a localization formula as the one in \eqref{eq:2} (but with $\chi_\kappa=h$) to write,
\begin{align*}
\int_\Omega\,\left(\kappa\,H\,|B_{0}(x)|-\kappa^{2}\,a(x)\right)\,|h\psi|^2\,dx&\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx+\kappa^{2}\int_{\Omega}(h^{4}-h^{2})|\psi|^{4}\,dx\\
&\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx\,.
\end{align*}
Here, we have used the fact that $h^{4}\leq h^{2}$ since $0\leq h\leq 1$.
By assumption \eqref{B(x)}, $|\nabla B_{0}|$ does not vanish on $\Gamma$, hence
\begin{equation}
|B_0(x)|\geq \frac{1}{M}\,\kappa^{-\frac 12}\qquad{\rm in}\quad\{\dist(x,\Gamma)\geq \lambda\}\,,
\end{equation}
for some constant $M>0$.\\
Thus, by using \eqref{def:sup-a} and \eqref{cond:H>kappa}, we get,
$$
\left(\frac{\varepsilon}{M}\,\kappa^{\frac{5}{2}}-\kappa^{2}\,\overline{a}\right)\int_\Omega|h\psi|^2\,dx\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx\,.
$$
Writing $\displaystyle\int_{\Omega}|\psi|^{2}\,dx=\int_{\Omega}|h\psi|^{2}\,dx+\int_{\Omega}(1-h^{2})|\psi|^{2}\,dx$ and using the assumption on $h$, we have,
$$
\left(\frac{\varepsilon}{M}\,\kappa^{\frac{5}{2}}-\kappa^{2}\,\overline{a}-c^{2}\right)\int_{\Omega}|h\psi(x)|^{2}\,dx\leq (c^{2}+C\,\kappa)\int_{\Omega\setminus\Omega_{\lambda}}|\psi|^{2}\,dx\,.
$$
For $\kappa$ large enough, $\frac{\varepsilon}{M}\,\kappa^{\frac{5}{2}}-\kappa^{2}\,\overline{a}-c^{2}\geq \frac{\varepsilon}{2M}\,\kappa^{\frac{5}{2}}$ and
$$
\int_\Omega| h\psi(x)|^{2}\,dx\leq 2\frac{M}{\varepsilon}C\,\kappa^{-\frac{3}{2}} \int_{\Omega\setminus\Omega_{\lambda}}|\psi|^{2}\,dx\,.
$$
Thanks to the assumption on the support of $h$, we get further,
$$
\int_\Omega|\psi(x)|^{2}\,dx\leq \left(2\frac{M}{\varepsilon}C\,\kappa^{-\frac{3}{2}}+1\right)\int_{\Omega\setminus\Omega_{\lambda}}|\psi|^{2}\,dx\,.
$$
Recall that $\lambda=\kappa^{-\frac{1}{2}}$. The Cauchy Schwarz inequality yields,
$$
\int_{\Omega\setminus\Omega_\lambda}|\psi(x)|^{2}\,dx\leq |\Omega\setminus\Omega_\lambda|^{1/2} \left(\int_{\Omega\setminus\Omega_\lambda} |\psi|^{4}\,dx\right)^{\frac{1}{2}}
\leq C\,\kappa^{-\frac{1}{4}}\left(\int_\Omega |\psi|^{4}\,dx\right)^{\frac{1}{2}}\,.
$$
This finishes the proof of the proposition.
\end{proof}
Now, we can give an upper bound of the critical field $\overline{H}_{C_3}^{cp}$ in the case when $\Gamma\neq\varnothing$ and with a pining term.
\begin{theorem}\label{thm:lb-H-var}
Supose that $\Gamma\neq\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $H$ satisfies
\begin{equation}\label{cond:HC3-2w-var}
H>\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,,
\end{equation} then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.\\
Moreover,
$$
\overline{H}_{C_3}^{cp}\leq\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,.
$$
\end{theorem}
\begin{proof}
In light of the result in Theorem~\ref{thm:GP}, we may assume the extra condition that $H\leq \lambda_{\max}\kappa^2$ for a sufficiently large constant $\lambda_{\max}$.
We take the constant $C$ in \eqref{cond:HC3-2w-var} as in Proposition~\ref{prop:mu>0-var}. In that way, under the assumption in \eqref{cond:HC3-2w-var}, we have
\begin{equation}\label{eq:prop:mu>0-var''}
\mu_1(\kappa,H)<0\,.
\end{equation}
Suppose now that $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq} with $\psi\neq 0$. Similarly, as in the proof of Theorem~\ref{thm:lb-H}, we have,
\begin{equation}\label{est:top1-var}
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C\,\sqrt{\top}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}^{2}\,,
\end{equation}
where $\top=\kappa^2\|\psi\|^4_{L^4(\Omega)}$ is introduced in \eqref{l5est}.
To apply the result of Proposition~\ref{prop:est-psi-var}, we take
$$\varepsilon=\frac{1}{2}\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,,
$$
and get,
\begin{equation}\label{est:psi1-var}
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac14}\|\psi\|^2_{L^4(\Omega)}= C\kappa^{-\frac{5}{4}}\,\sqrt{\top}\,.
\end{equation}
Putting \eqref{est:psi1-var} into \eqref{est:top1-var}, we obtain,
$$
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C'\,\kappa^{-\frac{1}{4}}\,\top\,.
$$
We conclude that, for $\kappa\geq\kappa_0$ and $\kappa_0$ a sufficiently large constant, $\mu_{1}(\kappa,H)<0$, which is in contradiction with \eqref{eq:prop:mu>0-var''}. Therefore, we conclude that $\psi=0$.
\end{proof}
Following the argument given in Proposition~\ref{prop:cp}, we get:
\begin{prop}\label{prop:cp-var}
Supose that $\Gamma\neq\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $\kappa\geq\kappa_0$ and $H$ satisfies
\begin{equation}\label{cond:HC3-2w2-var}
H\leq \max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\,,
\end{equation}
then there exists a solution $(\psi,\mathbf A)$ of \eqref{eq-2D-GLeq} with $\|\psi\|_{L^{2}(\Omega)}\neq 0$.\\
Moreover,
$$
\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\leq \underline{H}_{C_3}^{cp}\,.
$$
\end{prop}
\subsection*{End of the proof of Theorem~\ref{thm:HC3-vr}}
All the critical fields are contained in the interval $[\underline{H}_{C_{3}}^{loc},\overline{H}_{C_{3}}^{cp}]$ (the proof of this statement is exactly as the one given for \eqref{eq:ov} and \eqref{eq:un}).\\
By Proposition~\ref{prop:mu1<0-var} and Theorem~\ref{thm:lb-H-var}, we get the existence of positive constants $C$ and $\kappa_0$, such that for $\kappa\geq\kappa_0$,
\begin{multline}
\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\leq \underline{H}_{C_{3}}^{loc}\leq\overline{H}_{C_{3}}^{cp}\\
\leq\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,.
\end{multline}
As a consequence, we have proved that the asymptotics in Theorem~\ref{thm:HC3-vr} is valid for for the six critical fields in \eqref{def:HC3-o}, \eqref{def:HC3} and \eqref{def:HC3-u}.
\end{document} | math | 179,356 |
\begin{document}
\title[Maxwell--Stefan--Cahn--Hilliard systems]{Existence and weak-strong uniqueness
for Maxwell--Stefan--Cahn--Hilliard systems}
\author[X. Huo]{Xiaokai Huo}
\address{Institute of Analysis and Scientific Computing, Technische Universit\"at
Wien, Wiedner Hauptstra\ss e 8--10, 1040 Wien, Austria}
\email{[email protected]}
\author[A. J\"ungel]{Ansgar J\"ungel}
\address{Institute of Analysis and Scientific Computing, Technische Universit\"at
Wien, Wiedner Hauptstra\ss e 8--10, 1040 Wien, Austria}
\email{[email protected]}
\author[A. Tzavaras]{Athanasios E. Tzavaras}
\address{Computer, Electrical and Mathematical Science and Engineering Division,
King Abdullah University of Science and Technology (KAUST),
Thuwal 23955-6900, Saudi Arabia}
\email{[email protected]}
\date{\today}
\thanks{XH and AJ acknowledge partial support from
the Austrian Science Fund (FWF), grants P33010, W1245, and F65.
AET acknowledges support from the King Abdullah University of Science and Technology (KAUST).
This work has received funding from the European
Research Council (ERC) under the European Union's Horizon 2020 research and
innovation programme, ERC Advanced Grant no.~101018153.}
\begin{abstract}
A Maxwell--Stefan system for fluid mixtures with driving forces depending on
Cahn--Hilliard-type chemical potentials is analyzed. The corresponding parabolic
cross-diffusion equations contain fourth-order derivatives and are considered
in a bounded domain with no-flux boundary conditions. The main difficulty of
the analysis is the degeneracy of the diffusion matrix, which is overcome by
proving the positive definiteness of the matrix on a subspace and using the
Bott--Duffin matrix inverse. The global existence of weak solutions and a
weak-strong uniqueness property are shown by a careful combination of
(relative) energy and entropy estimates, yielding $H^2(\Omega)$ bounds for
the densities, which cannot be obtained from the energy or entropy inequalities alone.
\end{abstract}
\keywords{Cross-diffusion systems, global existence, weak-strong uniqueness,
relative entropy, relative free energy, parabolic fourth-order equations,
Maxwell--Stefan equations, Cahn--Hilliard equations.}
\subjclass[2000]{35A02, 35G20, 35G31, 35K51, 35K55, 35Q35.}
\maketitle
\section{Introduction}
The evolution of fluid mixtures is important in many scientific fields like
biology and nanotechnology to understand the
diffusion-driven transport of the species. The transport can be modeled
by the Maxwell--Stefan equations \cite{Max66,Ste71}, which
consist of the mass balance equations and the relations between the driving forces
and the fluxes. The driving forces involve the chemical potentials of the
species, which in turn are determined by the (free) energy.
When the fluid is immiscible, the energy can be assumed to consist
of the thermodynamic entropy and the phase separation energy,
given by a density gradient \cite{CaHi58}. The gradient energetically penalizes
the formation of an interface and restrains the segregation.
This leads to a system of cross-diffusion equations with fourth-order derivatives.
The aim of this paper is to provide a
global existence and weak-strong uniqueness analysis for the multicomponent
Maxwell--Stefan--Cahn--Hilliard system.
\subsection{Model equations and state of the art}
The equations for the partial densities $c_i$ and partial velocities $u_i$
are given by
\begin{align}
\pa_t c_i + \operatorname{div}(c_iu_i) &= 0, \quad i = 1,\ldots, n, \label{1.eq1} \\
\label{1.eq2}
c_i\na\mu_i - \frac{c_i}{\sum_{k=1}^n c_k}\sum_{j=1}^n c_j\na\mu_j
&= -\sum_{j=1}^n K_{ij}(\bm{c})c_ju_j, \\
\sum_{j=1}^n c_ju_j &= 0, \label{1.eq3}
\end{align}
supplemented by the initial and boundary conditions
\begin{equation}\label{1.bic}
\bm{c}(\cdot,0)=\bm{c}^0\quad\mbox{in }\Omega, \quad
c_iu_i\cdot\nu = \na c_i\cdot\nu = 0\quad\mbox{on }\pa\Omega,\ t>0,\ i=1,\ldots,n,
\end{equation}
where $\Omega\subset{\mathbb R}^d$ ($d=1,2,3$) is a bounded domain, $\nu$ is the
exterior unit normal vector on the boundary $\pa\Omega$, $\bm{c}=(c_1,\ldots,c_n)$
is the density vector, and $K_{ij}(\bm{c})$ are the friction coefficients.
The left-hand side of \eqref{1.eq2} can be interpreted as the driving forces
of the thermodynamic system, and the right-hand side is the sum of the friction forces.
The chemical potentials
\begin{equation}\label{1.mu}
\mu_i=\frac{\delta{\mathcal E}}{\delta c_i} = \log c_i-\Delta c_i, \quad i=1,\ldots,n,
\end{equation}
are the variational derivatives of the (free) energy
\begin{equation}\label{1.HE}
{\mathcal E}(\bm{c}) = {\mathcal H}(\bm{c}) + \frac{1}{2}\sum_{i=1}^n\int_\Omega|\na c_i|^2 dx, \quad
{\mathcal H}(\bm{c}) = \sum_{i=1}^n\int_\Omega\big(c_i(\log c_i-1)+1\big)dx,
\end{equation}
and ${\mathcal H}(\bm{c})$ is the thermodynamic entropy. We assume that
$\sum_{i=1}^n K_{ij}(\bm{c})=0$ for $j=1,\ldots,n$, meaning that the
linear system in $\na\mu_j$ is invertible only on a subspace, and that
$\sum_{i=1}^n c_i^0=1$ in $\Omega$, which implies that $\sum_{i=1}^n c_i(t)=1$
in $\Omega$ for all time $t>0$. This means that the mixture is saturated
and $c_i$ can be interpreted as volume fraction.
For simplicity, we have normalized all physical constants.
Model \eqref{1.eq1}--\eqref{1.mu} has been derived rigorously in \cite{HJT19}
in the high-friction
limit from a multicomponent Euler--Korteweg system for a general convex energy
functional depending on $\bm{c}$ and $\na\bm{c}$. A thermodynamics-based derivation
can be found in \cite{MiSc09}. When the energy equals
${\mathcal E}(\bm{c})={\mathcal H}(\bm{c})$, the model reduces to the classical Maxwell--Stefan
equations, analyzed first in \cite{Bot11,GoMa98,HMPW17} for local-in-time smooth
solutions and later in \cite{JuSt13} for global-in-time weak solutions.
In the single-species case, model \eqref{1.eq1}--\eqref{1.mu} becomes the
fourth-order Cahn--Hilliard equation with potential $\phi(c)=c(\log c-1)$,
which was analyzed in, e.g., \cite{ElGa96,Jin92}. Only few works are concerned
with the multi-species situation, and all of them require additional conditions.
The mobility matrix in \cite{BoLa06,MaZi17} is assumed to be diagonal and that one in
\cite{KRS21} has constant entries, while the
works \cite{EMP21,ElGa97} suppose a particular (but nondiagonal)
structure of the mobility matrix. We also mention the works
\cite{BaEh18,BBEP20} on related models with free energies of the type ${\mathcal H}$.
The proof of the uniqueness of solutions to cross-diffusion or fourth-order
systems is quite delicate due to the lack of a maximum principle and regularity
of the solutions.
The uniqueness of strong solutions to Maxwell--Stefan systems has been shown in
\cite{HMPW17,HuSa18}, and uniqueness results for weak solutions in a very
special case can be found in \cite{ChJu18}. A weak-strong uniqueness result
for Maxwell--Stefan systems was proved in \cite{HJT21}.
Concerning uniqueness results for fourth-order equations,
we refer to \cite{CGPS13} for single-species Cahn--Hilliard equations,
\cite{Joh15} for single-species thin-film equations,
and \cite{F13} for the quantum drift-diffusion equations.
Up to our knowledge, there are no uniqueness results for multicomponent
Cahn--Hilliard systems.
In this paper, we analyze these equations in a general setting for the first time.
\subsection{Key ideas of the analysis}
Before stating the main results, we explain the mathematical ideas needed
to analyze model \eqref{1.eq1}--\eqref{1.mu}. First, we rewrite
\eqref{1.eq2}
by introducing the matrix $D(\bm{c})\in{\mathbb R}^{n\times n}$ with
entries
$$
D_{ij}(\bm{c}) = \frac{1}{\sqrt{c_i}} K_{ij}(\bm{c})\sqrt{c_j}
$$
in the unknowns $(\sqrt{c_1}u_1,\ldots,\sqrt{c_n}u_n)$:
\begin{equation}\label{1.D}
\begin{aligned}
\sqrt{c_i}\na\mu_i - \frac{\sqrt{c_i}}{\sum_{k=1}^n c_k}\sum_{j=1}^n c_j\na\mu_j
&= -\sum_{j=1}^n D_{ij}(\bm{c})\sqrt{c_j}u_j, \\
\sum_{i=1}^n\sqrt{c_i}\big(\sqrt{c_i}u_i\big) &= 0.
\end{aligned}
\end{equation}
We show in Lemma \ref{lem.Dz} that this linear system has a unique solution
in the space $L(\bm{c}):=\{\bm{z}\in{\mathbb R}^n:\sum_{i=1}^n\sqrt{c_i}z_i=0\}$,
and the solution reads as
$$
\sqrt{c_i}u_i = -\sum_{j=1}^n D_{ij}^{BD}(\bm{c})\sqrt{c_j}\na\mu_j,
$$
where $D^{BD}(\bm{c})$ is the so-called Bott--Duffin matrix inverse; see
Lemmas \ref{lem.Dz} and \ref{lem.DB} for the definition and some properties. Then,
defining the matrix $B(\bm{c})\in{\mathbb R}^{n\times n}$ with elements
\begin{equation}\label{1.B}
B_{ij}(\bm{c}) = \sqrt{c_i}D_{ij}^{BD}(\bm{c})\sqrt{c_j}, \quad i,j=1,\ldots,n,
\end{equation}
system \eqref{1.eq1}--\eqref{1.eq2} can be formulated as
(see Section \ref{sec.BD} for details)
$$
\pa_t c_i = \operatorname{div}\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j, \quad i=1,\ldots,n.
$$
The matrix $B(\bm{c})$ is often called Onsager or mobility matrix in the literature.
The major difficulty of the analysis consists in the fact
that the matrix $B(\bm{c})$ is singular
and degenerates when $c_i\to 0$ for some $i\in\{1,\ldots,n\}$.
Computing formally the energy identity
$$
\frac{d{\mathcal E}}{dt}(\bm{c})
+ \sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na\mu_i\cdot\na\mu_j dx = 0,
$$
the degeneracy at $c_i=0$ prevents uniform estimates for $\na\mu_i$ in $L^2(\Omega)$.
In some works, this issue has been compensated. For instance, there exists
an entropy equality for the model of \cite{ElGa97} yielding an $L^2(\Omega)$
bound for $\Delta c_i$, and the decoupled mobilities in \cite{CMN19,MaZi17} allow for
decoupled entropy estimates. In our model, the energy identity does not
provide a gradient estimate for the full vector
$(\na\mu_1,\ldots,\na\mu_n)$ but only for a projection:
$$
\frac{d{\mathcal E}}{dt}(\bm{c}) + C_1\sum_{i=1}^n\int_\Omega
\bigg|\sum_{j=1}^n(\delta_{ij}-\sqrt{c_ic_j})\sqrt{c_j}\na\mu_j\bigg|^2 dx \le 0,
$$
where $\delta_{ij}$
is the Kronecker delta; see Lemma \ref{lem.fei}.
(The constant $C_1>0$ and all constants that follow do not depend on $\bm{c}$.)
To address the degeneracy issue, we compute the time derivative of the entropy:
$$
\frac{d{\mathcal H}}{dt}(\bm{c}) + \sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na\log c_i
\cdot\na\mu_j dx = 0.
$$
This does not provide a uniform estimate for $\Delta c_i$, but we show
(see Lemma \ref{lem.fei}) that
$$
\frac{d{\mathcal H}}{dt}(\bm{c}) + C_2\sum_{i=1}^n\int_\Omega(\Delta c_i)^2 dx
\le C_3\sum_{i=1}^n\int_\Omega\bigg|\sum_{j=1}^n(\delta_{ij}-\sqrt{c_ic_j})
\sqrt{c_j}\na\mu_j\bigg|^2 dx.
$$
Combining the energy and entropy inequalities in a suitable way, the last
integral cancels:
\begin{equation}\label{1.cee}
\frac{d}{dt}\bigg({\mathcal H}(\bm{c})+\frac{C_3}{C_1}{\mathcal E}(\bm{c})\bigg)
+ C_2\sum_{i=1}^n\int_\Omega(\Delta c_i)^2 dx \le 0.
\end{equation}
This provides the desired $H^2(\Omega)$ bound for $c_i$.
Note that the energy or entropy inequality alone does not give estimates
for $c_i$. The combined energy-entropy inequality is the key idea of the paper
for both the existence and weak-strong uniqueness analysis.
\subsection{Main results}
We make the following assumptions:
\begin{itemize}
\item[(A1)] Domain: $\Omega\subset{\mathbb R}^d$ with $d\le 3$ is a bounded domain.
We set $Q_T=\Omega\times(0,T)$ for $T>0$.
\item[(A2)] Initial data: $c_i^0\in H^1(\Omega)$ satisfies $c_i^0\ge 0$ in $\Omega$,
$i=1,\ldots,n$, and $\sum_{i=1}^n c_i^0=1$ in $\Omega$.
\end{itemize}
The assumption $d\le 3$ is made for convenience, it can be relaxed for higher space
dimension, by choosing another regularization in the existence proof;
see \eqref{3.regul1}.
The constraint $\sum_{i=1}^n c_i^0=1$ expresses the saturation of the mixture
and it propagates to the solution.
We introduce the matrix $D_{ij}(\bm{c})=(1/\sqrt{c_i})K_{ij}(\bm{c})\sqrt{c_j}$
for $i,j=1,\ldots,n$ and set
\begin{equation}\label{1.L}
L(\bm{c}) = \{\bm{x}\in{\mathbb R}^n:\sqrt{\bm{c}}\cdot\bm{x}=0\}, \quad
L^\perp(\bm{c}) = \operatorname{span}\{\sqrt{\bm{c}}\},
\end{equation}
where $\sqrt{\bm{c}}=(\sqrt{c_1},\ldots,\sqrt{c_n})$.
The projections $P_L(\bm{c})$, $P_{L^\perp}(\bm{c})\in{\mathbb R}^{n\times n}$
on $L(\bm{c})$, $L(\bm{c})^\perp$, re\-spec\-tive\-ly, are given by
\begin{equation}\label{1.PL}
P_L(\bm{c})_{ij} = \delta_{ij}-\sqrt{c_ic_j}, \quad
P_{L^\perp}(\bm{c})_{ij} = \sqrt{c_ic_j}\quad\mbox{for }i,j=1,\ldots,n.
\end{equation}
We impose for any given $\bm{c}\in[0,1]^n$ the following assumptions on
$D(\bm{c})=(D_{ij}(\bm{c}))\in{\mathbb R}^{n\times n}$:
\begin{itemize}
\item[(B1)]$D(\bm{c})$ is symmetric and $\operatorname{ran} D(\bm{c})=L(\bm{c})$,
$\ker(D(\bm{c})P_L(\bm{c}))=L^\perp(\bm{c})$.
\item[(B2)] For all $i,j=1,\ldots,n$, $D_{ij}\in C^1([0,1]^n)$ is bounded.
\item[(B3)] The matrix $D(\bm{c})$ is positive semidefinite, and there exists
$\rho>0$ such that all eigenvalues $\lambda\neq 0$ of
$D(\bm{c})$ satisfy $\lambda\ge\rho$.
\item[(B4)] For all $i,j=1,\ldots,n$, $K_{ij}(\bm{c})
=\sqrt{c_i}D_{ij}(\bm{c})/\sqrt{c_j}$ is bounded in $[0,1]^n$.
\end{itemize}
Examples of matrices $D(\bm{c})$ satisfying these assumptions
are presented in Section \ref{sec.exam}.
Our first main result is the global existence of weak solutions.
\begin{theorem}[Global existence]\label{thm.ex}
Let Assumptions (A1)--(A2) and (B1)--(B4) hold. Then there exists a weak solution
$\bm{c}$ to \eqref{1.eq1}--\eqref{1.mu} satisfying $0\le c_i\le 1$,
$\sum_{i=1}^n c_i=1$ in $\Omega\times(0,\infty)$,
$$
c_i\in L_{\rm loc}^\infty(0,\infty;H^1(\Omega))\cap
L_{\rm loc}^2(0,\infty;H^2(\Omega)), \quad
\pa_t c_i\in L_{\rm loc}^2(0,\infty;H^1(\Omega)'),
$$
the initial condition in \eqref{1.bic} is satisfied in the sense of
$H^1(\Omega)'$, and for all $\phi_i\in C_0^\infty(\Omega\times(0,\infty))$,
\begin{align}\label{1.weak}
0 &= -\int_0^\infty\int_\Omega c_i\pa_t\phi_i dxdt
+ \sum_{j=1}^n\int_0^\infty\int_\Omega B_{ij}(\bm{c})\na\log c_i\cdot\na\phi_i dxdt \\
&\phantom{xx}{}+ \sum_{j=1}^n\int_0^\infty\int_\Omega\operatorname{div}(B_{ij}(\bm{c})\na\phi_i)
\Delta c_j dxdt, \nonumber
\end{align}
where $B_{ij}(\bm{c})$ is defined in \eqref{1.B}. Furthermore,
\begin{align}\label{1.EH}
{\mathcal H}(\bm{c}(\cdot,T)) &+ C_1{\mathcal E}(\bm{c}(\cdot,T))
+ C_2\int_0^T\int_\Omega(|\na\sqrt{\bm{c}}|^2+|\Delta\bm{c}|^2) dxdt \\
&{}+ C_2\int_0^T\int_\Omega|\bm{\zeta}|^2 dxdt
\le {\mathcal H}(\bm{c}^0) + C_1{\mathcal E}(\bm{c}^0), \nonumber
\end{align}
where $C_1>0$ depends on $\rho$, $n$, $\|D(\bm{c})\|_F$ and $C_2>0$ depends
on $n$, $\|D(\bm{c})\|_F$ ($\|\cdot\|_F$ is the Frobenius matrix norm
and $\rho$ is introduced in Assumption (B3)). Moreover,
$\bm{\zeta}$ is the weak $L^2(\Omega)$ limit of an approximating sequence of
$\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\mu_j$.
\end{theorem}
Some comments are in order. First, by Assumption (B2), the elements of the matrix
$D(\bm{c})$ are bounded for any $\bm{c}\in[0,1]^n$ and therefore, the quantity
$\|D(\bm{c})\|_F$ is bounded uniformly in $\bm{c}$.
Second, the weak formulation \eqref{1.weak} makes sence
since $B_{ij}(\bm{c})\na\log c_i \in L^2(Q_T)$.
Indeed, by the definition of $B(\bm{c})$, we have
$$
B_{ij}(\bm{c})\na\log c_j = \sqrt{c_i}D^{BD}_{ij}(\bm{c})\frac{1}{\sqrt{c_j}}\na c_j,
$$
and the matrix $\sqrt{c_i}D^{BD}_{ij}(\bm{c})/\sqrt{c_j}$ is bounded for all
$\bm{c}\in[0,1]^n$; see Lemma \ref{lem.DB} (iii) below. However, note that
the expression $\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j$ is generally not an element
of $L^2(Q_T)$. In particular, we cannot expect that $\na\Delta c_i\in L^2(Q_T)$.
Third, we have not been able to identify the weak limit $\bm{\zeta}$ because of
low regularity. However, if
$\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\mu_j\in
L^2_{\rm loc}(0,\infty;L^2(\Omega))$ holds for all $i=1,\ldots,n$, then
we can identify $\zeta_i=\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\mu_j$;
see Lemma \ref{lem.ident}.
To prove Theorem \ref{thm.ex}, we first introduce a truncation with parameter
$\delta\in(0,1)$ as in \cite{ElGa97} to avoid the degeneracy.
Then we reduce the cross-diffusion system to $n-1$
equations by replacing $c_n$ by $1-\sum_{i=1}^{n-1}c_i$. The advantage is that
the diffusion matrix of the reduced system is positive definite
(with a lower bound depending on $\delta$). The existence of solutions
$c_i^\delta$ to the truncated,
reduced system is proved by an approximation as in \cite{Jue16}
and the Leray--Schauder fixed-point theorem; see Section \ref{sec.approx}.
An approximate version of the
free energy estimate \eqref{1.EH}
(proved in Lemma \ref{lem.eei} in Section \ref{sec.unif})
provides suitable uniform bounds that allow
us to perform the limit $\delta\to 0$. The approximate densities $c_i^\delta$
may be negative but, by exploiting the entropy bound for $c_i^\delta$,
its limit $c_i$ turns out to be nonnegative.
The limit $\delta\to 0$ is then performed in Section \ref{sec.exproof}, using
the uniform estimates and compactness arguments.
Our second main result is concerned with the weak-strong uniqueness.
For this, we define the relative entropy and free energy
in the spirit of \cite{GLT17} by, respectively,
\begin{align}
{\mathcal H}(\bm{c}|\bar{\bm{c}}) &:= {\mathcal H}(\bm{c}) - {\mathcal H}(\bar{\bm{c}})
- \frac{\pa{\mathcal H}}{\pa\bm{c}}(\bar{\bm{c}})\cdot(\bm{c}-\bar{\bm{c}})
= \sum_{i=1}^n\int_\Omega\bigg(c_i\log\frac{c_i}{\bar{c}_i} - (c_i-\bar{c}_i)
\bigg)dx, \label{1.relH} \\
{\mathcal E}(\bm{c}|\bar{\bm{c}}) &:= {\mathcal E}(\bm{c}) - {\mathcal E}(\bar{\bm{c}})
- \frac{\pa{\mathcal E}}{\pa\bm{c}}(\bar{\bm{c}})\cdot(\bm{c}-\bar{\bm{c}})
= {\mathcal H}(\bm{c}|\bar{\bm{c}}) + \frac12\sum_{i=1}^n\int_\Omega
|\na(c_i-\bar{c}_i)|^2dx. \label{1.relE}
\end{align}
\begin{theorem}[Weak-strong uniqueness]\label{thm.wsu}
Let Assumptions (A1)--(A2), (B1)--(B4) hold, let $\bm{c}$ be a weak solution
to \eqref{1.eq1}--\eqref{1.mu} with initial datum $\bm{c^0}$, and let
$\bar{\bm{c}}$ be a strong solution to \eqref{1.eq1}--\eqref{1.mu} with
initial datum $\bar{\bm{c}}^0$. We assume that the weak solution
$\bm{c}$ satisfies
\begin{equation}\label{1.cregul}
\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\mu_j
\in L^2_{\rm loc}(0,\infty;L^2(\Omega))
\mbox{ for }i,j=1,\ldots,n
\end{equation}
(see \eqref{1.PL} for the definition of $P_L(\bm{c})$)
and for all $T>0$ the energy and entropy inequalities
\begin{align}
{\mathcal E}(\bm{c}(T)) + \sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})
\na\mu_i\cdot\na\mu_j dxdt &\le {\mathcal E}(\bm{c}^0), \label{1.dEdt} \\
{\mathcal H}(\bm{c}(T)) + \sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})
\na\log c_i\cdot\na\mu_j dxdt &\le {\mathcal H}(\bm{c}^0). \label{1.dHdt}
\end{align}
The strong solution $\bar{\bm{c}}$ is supposed to be strictly positive,
i.e., there exists $m>0$ such that $\bar{c}_i\ge m$ in $\Omega$, $t>0$, and
satisfies the regularity
$$
\bar{c}_i\in L_{\rm loc}^\infty(0,\infty;W^{3,\infty}(\Omega)),
\quad \na\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)
\in L^\infty_{\rm loc}(0,\infty;L^\infty(\Omega))
$$
for $i=1,\ldots,n$,
as well as for any $T>0$ the energy and entropy conservation identities
\begin{align}
{\mathcal E}(\bar{\bm{c}}(T)) + \sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bar{\bm{c}})
\na\bar{\mu}_i\cdot\na\bar{\mu}_j dxdt &= {\mathcal E}(\bar{\bm{c}}^0), \label{1.dEdtbar} \\
{\mathcal H}(\bar{\bm{c}}(T)) + \sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bar{\bm{c}})
\na\log \bar{c}_i\cdot\na\bar{\mu}_j dxdt &= {\mathcal H}(\bar{\bm{c}}^0),
\label{1.dHdtbar}
\end{align}
where $\mu_i=\log c_i-\Delta c_i$ and $\bar{\mu}_i=\log\bar{c}_i-\Delta \bar{c}_i$.
Then, for any $T>0$, there exist constants $C_1$, only depending on $\|D(\bm{c})\|_F$,
$n$, $\rho$, and $C_2(T)>0$, only depending on $T$,
$\operatorname{meas}(\Omega)$, $n$, $\rho$, such that
\begin{equation}\label{1.comb}
{\mathcal H}(\bm{c}(T)|\bar{\bm{c}}(T)) + C_1{\mathcal E}(\bm{c}(T)|\bar{\bm{c}}(T))
\le C_2(T)\big({\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0) + C_1{\mathcal E}(\bm{c}^0|\bar{\bm{c}}^0)\big).
\end{equation}
In particular, if $\bm{c}^0=\bar{\bm{c}}^0$ then the weak and strong solutions
coincide.
\end{theorem}
Assumption \eqref{1.cregul} guarantees that the flux
$\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j$ lies in $L^2(Q_T)$. Indeed, we prove in
Lemma \ref{lem.DB} (i) in Section \ref{sec.mobil} that $D_{ij}^{BD}(\bm{c})$
is bounded for $\bm{c}\in[0,1]^n$. Therefore, since
$D^{BD}(\bm{c})=D^{BD}(\bm{c})P_L(\bm{c})$, assumption \eqref{1.cregul}
and $c_i\in L^\infty(Q_T)$ imply that
\begin{equation}\label{1.regflux}
\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j
= \sqrt{c_i}\sum_{j,k=1}^n D_{ik}^{BD}(\bm{c})P_L(\bm{c})_{kj}\sqrt{c_j}\na\mu_j
\in L^2(Q_T).
\end{equation}
By the way, it follows from
$\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\log c_j = 2\na\sqrt{c_i}\in L^2(Q_T)$ that
\begin{equation}\label{1.regc}
\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\Delta c_j
= \sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na(\log c_j-\mu_j) \in L^2(Q_T).
\end{equation}
Since $\na\Delta c_i$ may be not in $L^2(Q_T)$, we interpret \eqref{1.regc}
in the sense of distributions, i.e.\ for all $\Phi\in
C_0^\infty(\Omega;{\mathbb R}^d)$,
$$
\bigg\langle\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\Delta c_j,\Phi
\bigg\operatorname{ran}gle
= -\sum_{j=1}^n\int_\Omega\big(\na(P_L(\bm{c})_{ij}\sqrt{c_j})\cdot\Phi
+ P_L(\bm{c})_{ij}\sqrt{c_j}\operatorname{div}\Phi\big)\Delta c_j dx.
$$
For the proof of Theorem \ref{thm.wsu}, we estimate first the time derivative of
the relative entropy \eqref{1.relH}:
\begin{align*}
\frac{d{\mathcal H}}{dt}&(\bm{c}|\bar{\bm{c}})
+ C_1\sum_{i=1}^n\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}
\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dx
+ C_1\sum_{i=1}^n\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dx \\
&\le C_2\sum_{i=1}^n\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}
\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dx + C_3\int_\Omega
{\mathcal E}(\bm{c}|\bar{\bm{c}})dx,
\end{align*}
where $C_i>0$ are some constants depending only on the data.
The first term on the right-hand side can be handled by estimating the time
derivative of the relative energy \eqref{1.relE}:
\begin{align*}
\frac{d{\mathcal E}}{dt}&(\bm{c}|\bar{\bm{c}}) + C_4\sum_{i=1}^n\int_\Omega
\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dx \\
&\le \theta\sum_{i=1}^n\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}
\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dx
+ \theta\sum_{i=1}^n\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dx \\
&\phantom{xx}{}+ C_5(\theta)\int_\Omega{\mathcal E}(\bm{c}|\bar{\bm{c}})dx,
\end{align*}
where $\theta>0$ can be arbitrarily small. Choosing $\theta=C_1C_4/C_2$,
we can combine both estimates leading to
$$
\frac{d}{dt}\bigg({\mathcal H}(\bm{c}|\bar{\bm{c}}) + \frac{C_2}{C_4}{\mathcal E}(\bm{c}|\bar{\bm{c}})
\bigg)
\le \bigg(C_3+\frac{C_2C_5}{C_4}\bigg){\mathcal E}(\bm{c}|\bar{\bm{c}}),
$$
and the theorem follows after applying Gronwall's lemma. As the computations
are quite involved, we compute first in Section \ref{sec.wsu.formal}
the time derivative of the relative entropy and energy for {\em smooth} solutions.
The rigorous proof of the combined relative entropy-energy inequality
for weak solutions $\bm{c}$ and strong solutions $\bar{\bm{c}}$ is
then performed in Section \ref{sec.wsu.rig}.
The paper is organized as follows. The Bott--Duffin matrix inverse is introduced
in Section \ref{sec.mobil}, some properties of the mobility matrix $B(\bm{c})$ are
proved, and the combined energy-entropy inequality \eqref{1.cee} is derived for
smooth solutions. The global existence of solutions (Theorem \ref{thm.ex}) is
shown in Section \ref{sec.ex}, while Section \ref{sec.wsu} is concerned with the
proof of the weak-strong uniqueness property (Theorem \ref{thm.wsu}). Finally,
we present some examples verifying Assumptions (B1)--(B4) in Section \ref{sec.exam}.
\subsection*{Notation}
Elements of the matrix $A\in{\mathbb R}^{n\times n}$ are denoted by $A_{ij}$,
$i,j=1,\ldots,n$, and the elements of a vector $\bm{c}\in{\mathbb R}^n$ are $c_1,\ldots,c_n$.
We use the notation $f(\bm{c})=(f(c_1),\ldots,f(c_n))$ for $\bm{c}\in{\mathbb R}^n$
and a function $f:{\mathbb R}\to{\mathbb R}$. The expression $|\na f(\bm{c})|^2$ is defined by
$\sum_{i=1}^n|\na f(c_i)|^2$ and $|\cdot|$ is the usual Euclidean norm.
The matrix $R(\bm{c})\in{\mathbb R}^{n\times n}$
is the diagonal matrix with elements $\sqrt{c_1},\ldots,\sqrt{c_n}$, i.e.\
$R_{ij}(\bm{c})=\sqrt{c_i}\delta_{ij}$ for $i,j=1,\ldots,n$, where
$\delta_{ij}$ denotes the Kronecker delta.
We understand by $\na\bm{\mu}$ the matrix with entries $\pa_{x_i}\mu_j$.
Furthermore, $C>0$, $C_i>0$ are generic constants
with values changing from line to line.
\section{Properties of the mobility matrix and a priori estimates}\label{sec.mobil}
We wish to express the fluxes $c_iu_i$ as a linear combination of the gradients
of the chemical potentials. Since $K(\bm{c})$ has a nontrivial kernel,
we need to use a generalized matrix inverse, the Bott--Duffin inverse.
This inverse and its properties are studied in Section \ref{sec.BD}. The properties
allow us to derive in Section \ref{sec.apriori} some a priori estimates
for the Maxwell--Stefan--Cahn--Hilliard system.
\subsection{The Bott--Duffin inverse}\label{sec.BD}
We wish to invert \eqref{1.eq2} or, equivalently, \eqref{1.D}.
We recall definition \eqref{1.PL} of the projection matrices
$P_L(\bm{c})\in{\mathbb R}^{n\times n}$ on $L(\bm{c})$ and
$P_{L^\perp}(\bm{c})\in{\mathbb R}^{n\times n}$ on $L^\perp(\bm{c})$,
where $L(\bm{c})$ and $L^\perp(\bm{c})$ are defined in \eqref{1.L}.
Then \eqref{1.D} is equivalent to the problem:
\begin{equation}\label{2.Dz}
\mbox{Solve}\quad D(\bm{c})\bm{z} = -P_L(\bm{c})R(\bm{c})\na\bm{\mu}
\quad\mbox{in the space }\bm{z}\in L(\bm{c}),
\end{equation}
where $z_i=\sqrt{c_i}u_i$, recalling that
$R(\bm{c})=\operatorname{diag}(\sqrt{\bm{c}})$.
\begin{lemma}[Solution of \eqref{2.Dz}]\label{lem.Dz}
Suppose that $D(\bm{c})$ satisfies Assumption (B1). The Bott--Duffin inverse
$$
D^{BD}(\bm{c}) = P_L(\bm{c})\big(D(\bm{c})P_L(\bm{c})+P_{L^\perp}(\bm{c})\big)^{-1}
$$
is well-defined, symmetric, and satisfies $\ker D^{BD}(\bm{c})=L^\perp(\bm{c})$.
Furthermore, for any $\bm{y}\in L(\bm{c})$, the linear problem
$D(\bm{c})\bm{z}=\bm{y}$ for $\bm{z}\in L(\bm{c})$ has a unique solution
given by $\bm{z}=D^{BD}(\bm{c})\bm{y}$.
\end{lemma}
We refer to \cite[Lemma 17]{HJT21} for the proof. The property for the kernel
follows from $\ker D^{BD}(\bm{c})=\ker P_L(\bm{c})=L^\perp(\bm{c})$.
Since $P_L(\bm{c})R(\bm{c})\na\bm{\mu}\in L(\bm{c})$
(this follows from the definition of $P_L(\bm{c})$ and $\sum_{i=1}^n c_i=1$),
we infer from Lemma \ref{lem.Dz} that \eqref{2.Dz} has the unique solution
$\bm{z}=-D^{BD}(\bm{c})P_L(\bm{c})R(\bm{c})\na\bm{\mu}\in L(\bm{c})$ or,
componentwise,
$$
c_iu_i = \sqrt{c_i}z_i
= -\sum_{j=1}^n \sqrt{c_i}\big(D^{BD}(\bm{c})P_L(\bm{c})\big)_{ij}\sqrt{c_j}\na\mu_j
= -\sum_{j=1}^n \sqrt{c_i}D^{BD}(\bm{c})_{ij}\sqrt{c_j}\na\mu_j
$$
for $i=1,\ldots,n$, where the last equality follows from
$D^{BD}(\bm{c})P_L(\bm{c})=D^{BD}(\bm{c})$; see \cite[(81)]{HJT21}.
Then we can formulate equation \eqref{1.eq1} as
\begin{equation}\label{2.B}
\pa_t c_i = \operatorname{div}\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j, \quad
\mbox{where }B_{ij}(\bm{c})=\sqrt{c_i}D^{BD}_{ij}(\bm{c})\sqrt{c_j},
\quad i,j=1,\ldots,n.
\end{equation}
The boundary conditions $c_iu_i\cdot\nu=0$ on $\pa\Omega$ yield
\begin{equation}\label{2.bc}
\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j\cdot\nu = 0\quad\mbox{on }\pa\Omega,\ t>0,\
i=1,\ldots,n.
\end{equation}
We recall some properties of the Bott--Duffin inverse.
\begin{lemma}[Properties of $D^{BD}(\bm{c})$]\label{lem.DB}
Suppose that $D(\bm{c})\in{\mathbb R}^{n\times n}$ satisfies Assumptions (B1)--(B4). Then:
\begin{itemize}
\item[\rm (i)] The coefficients $D^{BD}_{ij}\in C^1([0,1]^n)$ are bounded for
$i,j=1,\ldots,n$.
\item[\rm (ii)] Let $\lambda(\bm{c})$ be an eigenvalue of
$(D(\bm{c})P_L(\bm{c})+P_{L^\perp}(\bm{c}))^{-1}$. Then
$\lambda_m\le\lambda(\bm{c})\le\lambda_M$, where
$$
\lambda_m = (1+n\|D(\bm{c})\|_F)^{-1}, \quad
\lambda_M = \max\{1,\rho^{-1}\},
$$
$\|\cdot\|_F$ is the Frobenius matrix norm, and
$\rho>0$ is a lower bound for the eigenvalues of $D(\bm{c})$; see Assumption (B3).
\item[\rm (iii)] The functions $\bm{c}\mapsto \sqrt{c_i}D^{BD}_{ij}(\bm{c})/\sqrt{c_j}$
are bounded in $[0,1]^n$ for $i,j=1,\ldots,n$.
\end{itemize}
\end{lemma}
A consequence of (ii) are the inequalities
\begin{equation}\label{2.DBD}
\lambda_m|P_L(\bm{c})\bm{z}|^2 \le \bm{z}^T D^{BD}(\bm{c})\bm{z}
\le \lambda_M|P_L(\bm{c})\bm{z}|^2 \quad\mbox{for }\bm{z}\in{\mathbb R}^n.
\end{equation}
Note that the Frobenius norm of $D(\bm{c})$ is bounded uniformly in $\bm{c}\in[0,1]^n$,
since $D_{ij}$ is bounded by Assumption (B1).
\begin{proof} The points (i) and (ii) are proved in \cite[Lemma 11]{HJT21}
in an interval $[m,1]^n$ for some $m>0$. In fact,
we can conclude (i)--(ii) in the full interval $[0,1]^n$,
since our Assumptions (B2)--(B3) are stronger than those in \cite{HJT21}.
For the proof of (iii),
dropping the argument $\bm{c}$ and observing that $RDR^{-1}=K$, we obtain
\begin{align*}
R D^{BD}R^{-1} &= RP_L(DP_L+P_{L^\perp})^{-1}R^{-1}
= RP_L(R^{-1}R)(DP_L+P_{L^\perp})^{-1}R^{-1} \\
&= RP_LR^{-1}\big(R(DP_L+P_{L^\perp})R^{-1}\big)^{-1} \\
&= RP_LR^{-1}\big(RDR^{-1}RP_LR^{-1}+RP_{L^\perp}R^{-1}\big)^{-1} \\
&= RP_LR^{-1}\big(KRP_LR^{-1}+RP_{L^\perp}R^{-1}\big)^{-1}.
\end{align*}
The determinant of the expression in the brackets equals
$$
\det\big(R(DP_L+P_{L^\perp})R^{-1}\big) = \det(DP_L+P_{L^\perp}).
$$
Therefore, denoting by ``adj'' the adjugate matrix, it follows that
\begin{equation}\label{2.RDR}
R D^{BD}R^{-1} = \frac{RP_LR^{-1}\operatorname{adj}(KRP_LR^{-1}+RP_{L^\perp}R^{-1})}{
\det(DP_L+P_{L^\perp})}.
\end{equation}
By Assumption (B3), the eigenvalues of $D$ are not smaller than $\rho>0$.
The proof of \cite[Lemma 11]{HJT21} shows that the eigenvalues of
$DP_L+P_{L^\perp}$ are not smaller than $\rho>0$, too. This implies that
$\det(DP_L+P_{L^\perp})\ge\rho^{n-1}>0$. The coefficients
$$
(RP_LR^{-1})_{ij} = \delta_{ij}-c_i, \quad (RP_{L^\perp}R^{-1})_{ij} = c_i
$$
are bounded for $\bm{c}\in[0,1]^n$ and, by Assumption (B4), the coefficients of
$K$ are also bounded. Therefore, all elements of
$\operatorname{adj}(KRP_LR^{-1}+RP_{L^\perp}R^{-1})$ are bounded.
We conclude from \eqref{2.RDR} that the entries of $RD^{BD}R^{-1}$ are
bounded in $[0,1]^n$, i.e., point (iii) holds.
\end{proof}
The most important property is the positive definiteness of $D^{BD}(\bm{c})$
on $L(\bm{c})$; see \eqref{2.DBD}. This property implies the a priori estimates
proved in the following subsection.
\subsection{A priori estimates}\label{sec.apriori}
We show an energy inequality for smooth solutions.
\begin{lemma}[Free energy inequality]\label{lem.fei}
Let $\bm{c}\in C^\infty(\Omega\times(0,\infty);{\mathbb R}^n)$ be a positive, bounded, smooth
solution to \eqref{1.eq1}--\eqref{1.mu}. Then, for any
$0<\lambda<\lambda_m$,
\begin{align*}
\frac{d}{dt}\bigg({\mathcal H}(\bm{c}) + \frac{(\lambda_M-\lambda)^2}{\lambda_m\lambda}
{\mathcal E}(\bm{c})\bigg) &+ 2\lambda\int_\Omega|\na\sqrt{\bm{c}}|^2 dx
+ \lambda\int_\Omega|\Delta\bm{c}|^2 dx \\
&{}+ \frac{(\lambda_M-\lambda)^2}{2\lambda}\int_\Omega
|P_L(\bm{c})R(\bm{c})\na\bm{\mu}|^2dx \le 0.
\end{align*}
where the entropy ${\mathcal H}(\bm{c})$ and the free energy ${\mathcal E}(\bm{c})$ are given by
\eqref{1.HE} and $\lambda_m$, $\lambda_M$ are defined in Lemma \ref{lem.DB}.
\end{lemma}
\begin{proof}
We derive first the energy inequality. To this end, we multiply equation
\eqref{2.B} for $c_i$ by $\mu_i=(\pa{\mathcal E}/\pa c_i)(\bm{c})$, integrate over $\Omega$,
integrate by parts (using the boundary conditions \eqref{2.bc}), and take
into account the lower bound \eqref{2.DBD} for $D^{BD}(\bm{c})$:
\begin{align}\label{2.dEdt}
\frac{d{\mathcal E}}{dt}(\bm{c}) &= \sum_{i=1}^n\int_\Omega\frac{\pa{\mathcal E}}{\pa c_i}(\bm{c})
\pa_t c_i dx
= -\sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na\mu_i\cdot\na\mu_j dx \\
&= -\sum_{i,j=1}^n D_{ij}^{BD}(\bm{c})(\sqrt{c_i}\na\mu_i)\cdot
(\sqrt{c_j}\na\mu_j)dx
\le -\lambda_m\int_\Omega|P_L(\bm{c})R(\bm{c})\na\bm{\mu}|^2 dx. \nonumber
\end{align}
The entropy inequality is derived by multiplying \eqref{2.B} by $\log c_i$,
integrating over $\Omega$, and integrating by parts (using the boundary conditions
\eqref{2.bc}):
\begin{equation*}
\frac{d{\mathcal H}}{dt}(\bm{c})
= \sum_{i=1}^n\int_\Omega(\log c_i)\pa_t c_i dx
= -\sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na\log c_i
\cdot\na\mu_j dx.
\end{equation*}
To estimate the right-hand side, we
set $G=RP_LR$ (omitting the argument $\bm{c}$) and $M:=B-\lambda G$ for
$\lambda\in(0,\lambda_m)$. Then
\begin{equation}\label{2.dHdt}
\frac{d{\mathcal H}}{dt}(\bm{c}) = -\sum_{i,j=1}^n\int_\Omega M_{ij}\na\log c_i
\cdot\na\mu_j dx - \lambda\sum_{i,j=1}^n\int_\Omega G_{ij}\na\log c_i
\cdot\na\mu_j dx =: I_1 + I_2.
\end{equation}
Before estimating the integrals $I_1$ and $I_2$, we start with some preparations.
We use Lemma \ref{lem.DB} (ii) and $P_L^TP_L=P_L$ to obtain
$$
\bm{z}^TB\bm{z} = (R\bm{z})^T D^{BD}R\bm{z} \ge \lambda_m|P_LR\bm{z}|^2
= \lambda_m(P_LR\bm{z})^T (P_LR\bm{z}) = \lambda_m\bm{z}^T G\bm{z}\quad\mbox{for }
\bm{z}\in{\mathbb R}^n.
$$
The matrix $M$ is positive semidefinite since for any $\bm{z}\in{\mathbb R}^n$,
\begin{equation}\label{psd.zMz}
\bm{z}^T M\bm{z} = \bm{z}^TB\bm{z} - \lambda\bm{z}^T G\bm{z}
\ge (\lambda_m-\lambda)\bm{z}^T G\bm{z} = (\lambda_m-\lambda)|P_LR\bm{z}|^2.
\end{equation}
Furthermore, by Lemma \ref{lem.DB} (ii) again, we have the upper bound
\begin{equation}\label{3.zMz}
\bm{z}^T M\bm{z} = \bm{z}^T(B-\lambda G)\bm{z}
\le (\lambda_M-\lambda)\bm{z}^TG\bm{z} = (\lambda_M-\lambda)|P_LR\bm{z}|^2.
\end{equation}
We are now in the position to estimate the integral $I_1$,
using Young's inequality for any $\theta>0$:
\begin{align*}
I_1 &\le \frac{\theta}{2}\sum_{i,j=1}^n\int_\Omega M_{ij}\na\log c_i
\cdot\na\log c_j dx + \frac{1}{2\theta}\sum_{i,j=1}^n\int_\Omega
M_{ij}\na\mu_i\cdot\na\mu_j dx \\
&\le \frac{\theta}{2}(\lambda_M-\lambda)\int_\Omega|P_LR\na\log\bm{c}|^2 dx
+ \frac{\lambda_M-\lambda}{2\theta}\int_\Omega|P_LR\na\bm{\mu}|^2 dx \\
&= 2\theta(\lambda_M-\lambda)\int_\Omega|\na\sqrt{\bm{c}}|^2 dx
+ \frac{\lambda_M-\lambda}{2\theta}\int_\Omega|P_LR\na\bm{\mu}|^2 dx,
\end{align*}
where the last step follows from $\sum_{j=1}^n (P_L)_{ij}R_j\na\log c_j
=2\na\sqrt{c_i}$, which is a consequence of $\sum_{j=1}^n\na c_j=0$.
For the integral $I_2$, we use the definitions $G_{ij} = c_i\delta_{ij}-c_ic_j$ and
$\mu_j=\log c_j-\Delta c_j$:
\begin{align*}
I_2 &= -\lambda\sum_{i,j=1}^n\int_\Omega (c_i\delta_{ij}-c_ic_j)
\frac{\na c_i}{c_i}\cdot\na(\log c_j-\Delta c_j) dx \\
&= -\lambda\sum_{i=1}^n\int_\Omega\na c_i\cdot\na(\log c_i-\Delta c_i) dx
+ \lambda\int_\Omega\sum_{i=1}^n \na c_i\cdot\sum_{j=1}^n
c_j\na(\log c_j-\Delta c_j)dx \\
&= -\lambda\sum_{i=1}^n\int_\Omega\na c_i\cdot\na(\log c_i-\Delta c_i) dx
= -\lambda\int_\Omega\big(4|\na\sqrt{\bm{c}}|^2 + |\Delta\bm{c}|^2\big)dx,
\end{align*}
where we integrated by parts in the last step.
Inserting the estimates for $I_1$ and $I_2$ into \eqref{2.dHdt} yields
\begin{align*}
\frac{d{\mathcal H}}{dt}(\bm{c}) &+ 4\lambda\int_\Omega|\na\sqrt{\bm{c}}|^2 dx
+ \lambda\int_\Omega|\Delta\bm{c}|^2 dx \\
&\le 2\theta(\lambda_M-\lambda)\int_\Omega|\na\sqrt{\bm{c}}|^2 dx
+ \frac{\lambda_M-\lambda}{2\theta}\int_\Omega|P_LR\na\bm{\mu}|^2 dx.
\end{align*}
We set $\theta=\lambda/(\lambda_M-\lambda)$ to conclude that
\begin{equation}\label{2.dHdt2}
\frac{d{\mathcal H}}{dt}(\bm{c}) + 2\lambda\int_\Omega|\na\sqrt{\bm{c}}|^2 dx
+ \lambda\int_\Omega|\Delta\bm{c}|^2 dx
\le \frac{(\lambda_M-\lambda)^2}{2\lambda}\int_\Omega|P_LR\na\bm{\mu}|^2 dx.
\end{equation}
The right-hand side can be absorbed by the corresponding term in \eqref{2.dEdt}.
Indeed, adding the previous inequality to \eqref{2.dEdt} times
$(\lambda_M-\lambda)^2/(\lambda_m\lambda)$ finishes the proof.
\end{proof}
Note that the energy inequality \eqref{2.dEdt} or the entropy inequality
\eqref{2.dHdt2} alone are not sufficient to control the derivatives of $\bm{c}$ but
only a suitable linear combination. We will prove these inequalities rigorously
in the following section for weak solutions; see Lemma \ref{lem.eei}.
\section{Proof of Theorem \ref{thm.ex}}\label{sec.ex}
We prove the existence of global weak solutions to \eqref{1.eq1}--\eqref{1.bic}.
For this, we construct an approximate system depending on a parameter $\delta>0$,
similarly as in \cite{ElGa97}, and then pass to the limit $\delta\to 0$.
\subsection{An approximate system}\label{sec.approx}
In order to deal with the degeneracy of the matrix $B(\bm{c})$ when a component
of $\bm{c}$ vanishes, we introduce the cutoff function $\chi_\delta:{\mathbb R}^n\to{\mathbb R}^n$ by
$$
(\chi_\delta\bm{c})_i := \left\{\begin{array}{ll}
\delta &\quad\mbox{for }c_i<\delta, \\
c_i &\quad\mbox{for }\delta\le c_i\le 1-\delta, \\
1-\delta &\quad\mbox{for }c_i>1-\delta,
\end{array}\right.
$$
and define the approximate matrix
\begin{equation}\label{3.Bdelta}
B^\delta(\bm{c}) := R(\chi_\delta\bm{c})D^{BD}(\chi_\delta\bm{c})R(\chi_\delta\bm{c}),
\end{equation}
recalling that $R(\chi_\delta\bm{c})=\operatorname{diag}(\sqrt{\chi_\delta\bm{c}})$.
We wish to solve the approximate problem
\begin{align}
& \pa_t c_i^\delta = \operatorname{div}\sum_{j=1}^n B_{ij}^\delta(\bm{c}^\delta)\na\mu_j^\delta,
\quad \mu_j^\delta = \frac{\pa{\mathcal E}^\delta}{\pa c_j}(\bm{c}^\delta)
\quad\mbox{in }\Omega,\ t>0, \label{3.eq} \\
& c_i^\delta(\cdot,0)=c_i^0 \quad\mbox{in }\Omega, \quad
\sum_{j=1}^n B_{ij}^\delta(\bm{c}^\delta)\na\mu_j^\delta\cdot\nu = 0,\
\na c_i^\delta\cdot\nu=0 \quad\mbox{on }\pa\Omega, \label{3.bic}
\end{align}
where $i=1,\ldots,n$, $\sum_{i=1}^n c^0_i =1$ and the approximate energy is defined by
\begin{align}
& {\mathcal E}^\delta(\bm{c}) := {\mathcal H}^\delta(\bm{c})
+ \frac12\sum_{i=1}^n\int_\Omega|\na c_i|^2 dx,
\quad {\mathcal H}^\delta(\bm{c}) := \sum_{i=1}^n\int_\Omega h_i^\delta(c_i)dx, \nonumber \\
& h_i^\delta(r) = \left\{\begin{array}{ll}
r\log\delta - \delta/2 + r^2/(2\delta) &\quad\mbox{for }r<\delta, \\
r\log r &\quad\mbox{for }\delta\le r\le 1-\delta, \\
r\log(1-\delta) - (1-\delta)/2 + r^2/(2(1-\delta))
&\quad\mbox{for }r>1-\delta.
\end{array}\right. \label{3.hdelta}
\end{align}
Observe that the solutions $c_i^\delta$ may be negative.
We will show below that $c_i^\delta$ converges to a nonnegative function as
$\delta\to 0$. The approximate entropy density is chosen in such a way that
$h_i^\delta\in C^2({\mathbb R})$. Indeed, we obtain
$$
(h_i^\delta)'(c_i) = \left\{\begin{array}{ll}
\log\delta + c_i/\delta &\quad\mbox{for }c_i<\delta, \\
\log c_i + 1 &\quad\mbox{for }\delta< c_i< 1-\delta, \\
\log(1-\delta) + c_i/(1-\delta) &\quad\mbox{for }c_i>1-\delta,
\end{array}\right. \quad
(h_i^\delta)''(c_i) = \frac{1}{(\chi_\delta\bm{c})_i}.
$$
With these definitions, we obtain $\mu_i^\delta = (h_i^\delta)'(c_i^\delta)-\Delta
c_i^\delta$ for $i=1,\ldots,n$.
\begin{theorem}[Existence for the approximate system]\label{thm.approx}\quad
Let Assumptions (A1)--(A2) and (B1)--(B4) hold and let $\delta>0$. Then there
exists a weak solution $(\bm{c}^\delta,\bm{\mu}^\delta)$ to \eqref{3.eq}--\eqref{3.bic}
satisfying $\sum_{i=1}^n c_i^\delta(t)=1$ in $\Omega$, $t>0$,
\begin{align*}
& c_i^\delta\in L_{\rm loc}^\infty(0,\infty;H^1(\Omega))\cap
L_{\rm loc}^2(0,\infty;H^2(\Omega)), \\
& \pa_t c_i\in L_{\rm loc}^2(0,\infty;H^2(\Omega)'), \quad
\mu_i^\delta\in L_{\rm loc}^2(0,\infty;H^1(\Omega)), \quad i=1,\ldots,n,
\end{align*}
and the first equation in \eqref{3.eq} as well as the initial condition
in \eqref{3.bic} are satisfied in the sense of
$L_{\rm loc}^2(0,\infty;H^2(\Omega)')$.
\end{theorem}
Before we prove this theorem, we show some properties of the matrix
$B^\delta(\bm{c})$. We introduce the matrices
$P_L(\chi_\delta\bm{c})$, $P_{L^\perp}(\chi_\delta\bm{c})\in{\mathbb R}^{n\times n}$
with entries
$$
P_L(\chi_\delta\bm{c})_{ij} = \delta_{ij} - \frac{\sqrt{(\chi_\delta\bm{c})_i
(\chi_\delta\bm{c})_j}}{\sum_{k=1}^n(\chi_\delta\bm{c})_k}, \quad
P_{L^\perp}(\chi_\delta\bm{c})_{ij} = \frac{\sqrt{(\chi_\delta\bm{c})_i
(\chi_\delta\bm{c})_j}}{\sum_{k=1}^n(\chi_\delta\bm{c})_k}, \quad i,j=1,\ldots,n.
$$
\begin{lemma}[Properties of $B^\delta(\bm{c})$]\label{lem.Bdelta}\
Suppose that $D(\bm{c})$ satisfies Assumptions (B1)--(B4). Then Lemmas
\ref{lem.Dz} and \ref{lem.DB} hold with $P_L(\bm{c})$, $P_{L^\perp}(\bm{c})$, and
$D^{BD}(\bm{c})$ replaced by $P_L(\chi_\delta\bm{c})$,
$P_{L^\perp}(\chi_\delta\bm{c})$, and $D^{BD}(\chi_\delta\bm{c})$. As a consequence,
the matrix $B^\delta(\bm{c})$, defined in \eqref{3.Bdelta}, satisfies
\begin{equation}\label{3.zBdeltaz}
\bm{z}^TB^\delta(\bm{c})\bm{z}\ge\lambda_m|P_L(\chi_\delta\bm{c})R(\chi_\delta\bm{c})
\bm{z}|^2 \quad\mbox{for any }\bm{z},\bm{c}\in{\mathbb R}^n,
\end{equation}
and the first $(n-1)\times(n-1)$ submatrix $\widetilde{B}^\delta(\bm{c})$ of
$B^\delta(\bm{c})$ is positive definite and satisfies for
$\eta(\delta)=\lambda_m\delta^2/n$,
\begin{equation}\label{3.tilde}
\widetilde{\bm{z}}^T\widetilde{B}^\delta(\bm{c})\widetilde{\bm{z}}
\ge \eta(\delta)|\widetilde{\bm{z}}|^2\quad\mbox{for any }\widetilde{\bm{z}}
\in{\mathbb R}^{n-1}.
\end{equation}
\end{lemma}
\begin{proof}
It can be verified that Assumptions (B1)--(B2) hold for $D(\chi_\delta\bm{c})$,
so Lemmas \ref{lem.Dz} and \ref{lem.DB} still hold for the matrix
$D(\chi_\delta\bm{c})$. Inequality \eqref{3.zBdeltaz} is a direct consequence of
Lemma \ref{lem.DB} (ii). It remains to prove \eqref{3.tilde}.
We define for given $\widetilde{\bm{z}}\in{\mathbb R}^{n-1}$ the vector $\bm{z}\in{\mathbb R}^n$ with
$z_i=\widetilde{z}_i$ for $i=1,\ldots,n-1$ and $z_n=0$. Then \eqref{3.zBdeltaz}
becomes
\begin{equation}\label{3.aux}
\widetilde{\bm{z}}^T\widetilde{B}^\delta(\bm{c})\widetilde{\bm{z}}
\ge \lambda_m\big|\widetilde{P}_L(\chi_\delta\bm{c})\widetilde{R}(\chi_\delta\bm{c})
\widetilde{\bm{z}}\big|^2
= \lambda_m\big(\widetilde{R}(\chi_\delta\bm{c})\widetilde{\bm{z}}\big)^T
\widetilde{P}_L(\chi_\delta\bm{c})
\big(\widetilde{R}(\chi_\delta\bm{c})\widetilde{\bm{z}}\big),
\end{equation}
where $\widetilde{A}$ denotes the first $(n-1)\times(n-1)$ submatrix of a given
matrix $A\in{\mathbb R}^{n\times n}$. It follows from the Cauchy--Schwarz inequality that
for any $\zeta\in{\mathbb R}^{n-1}$,
\begin{align*}
\zeta^T\widetilde{P}_L(\chi_\delta\bm{c})\zeta
&= \sum_{i=1}^{n-1}\zeta_i^2 - \left(\sum_{j=1}^{n-1}
\sqrt{\frac{(\chi_\delta\bm{c})_j}{\sum_{k=1}^n(\chi_\delta\bm{c})_k}}
\zeta_j\right)^2
\ge |\zeta|^2 - \sum_{j=1}^{n-1}\frac{(\chi_\delta\bm{c})_j}{\sum_{k=1}^n
(\chi_\delta\bm{c})_k}|\zeta|^2 \\
&= \frac{(\chi_\delta\bm{c})_n}{\sum_{k=1}^n(\chi_\delta\bm{c})_k}|\zeta|^2
\ge \frac{\delta}{n}|\zeta|^2.
\end{align*}
Therefore, \eqref{3.aux} becomes
$$
\widetilde{\bm{z}}^T\widetilde{B}^\delta(\bm{c})\widetilde{\bm{z}}
\ge \frac{\lambda_m\delta}{n}\sum_{i=1}^{n-1}\big|\sqrt{(\chi_\delta\bm{c})_i}
\widetilde{z}_i\big|^2
= \frac{\lambda_m\delta}{n}
\sum_{i=1}^{n-1}(\chi_\delta\bm{c})_i\big|\widetilde{z}_i\big|^2
\ge \frac{\lambda_m\delta^2}{n}|\widetilde{\bm{z}}|^2,
$$
which proves \eqref{3.tilde}.
\end{proof}
We proceed to the proof of Theorem \ref{thm.approx}. The proof is divided
into four steps. First, we reformulate \eqref{3.eq} using the first $n-1$
components. Second, a time-discretized regularized system, similarly
as in \cite[Chapter 4]{Jue16}, is constructed and the existence of weak solutions
to this system is proved. Third, we derive some uniform estimates from the
energy inequality. Finally, we perform the de-regularization limit.
{\em Step 1: Reformulation in $n-1$ components.}
We reformulate the approximate system in terms of the $n-1$ relative chemical
potentials
$$
w_i^\delta = \mu_i^\delta-\mu_n^\delta, \quad i=1,\ldots,n-1.
$$
It holds that
$$
\sum_{j=1}^n \big(P_L(\chi_\delta\bm{c})R(\chi_\delta\bm{c})\big)_{kj}
= \sum_{j=1}^n\bigg(\delta_{kj} - \frac{\sqrt{(\chi_\delta\bm{c})_k
(\chi_\delta\bm{c})_j}}{\sum_{\ell=1}^n(\chi_\delta\bm{c})_\ell}\bigg)
\sqrt{(\chi_\delta\bm{c})_j} = 0.
$$
Then, using $D^{BD}(\bm{c})=D^{BD}(\bm{c})P_L(\bm{c})$ (which is a general
property of the Bott--Duffin inverse; see \cite[(81)]{HJT21}),
\begin{align*}
\sum_{j=1}^n B_{ij}^\delta(\bm{c}) &= \sum_{j=1}^n\sqrt{(\chi_\delta\bm{c})_i}
D_{ij}^{BD}(\bm{c})\sqrt{(\chi_\delta\bm{c})_j} \\
&= \sum_{j,k=1}^n\sqrt{(\chi_\delta\bm{c})_i}D_{ik}^{BD}(\bm{c})
\big(P_L(\chi_\delta\bm{c})R(\chi_\delta\bm{c})\big)_{kj} = 0.
\end{align*}
This shows that
$$
\sum_{j=1}^n B_{ij}^\delta(\bm{c})\na\mu_j^\delta
= \sum_{j=1}^{n-1}B_{ij}^\delta(\bm{c})\na\mu_j^\delta
+ B_{in}^\delta(\bm{c})\na\mu_n^\delta
= \sum_{j=1}^{n-1}B_{ij}^\delta(\bm{c})\na(\mu_j^\delta-\mu_n^\delta).
$$
Consequently, we can rewrite the first equation in \eqref{3.eq} as
\begin{equation}\label{3.cdelta}
\pa_t c_i^\delta = \operatorname{div}\sum_{j=1}^{n-1}\widetilde{B}_{ij}^\delta(\bm{c}^\delta)
\na w_j^\delta, \quad i=1,\ldots,n-1, \quad
c_n^\delta = 1-\sum_{i=1}^{n-1}c_i^\delta,
\end{equation}
recalling that $\widetilde{B}^\delta$ is the first $(n-1)\times(n-1)$ submatrix
of $B^\delta$.
{\em Step 2: Existence for a regularized system.}
We consider for given $\delta>0$, $T>0$, $N\in{\mathbb N}$, and
$(c_1^{k-1},\ldots,c_{n-1}^{k-1})$ the regularized system
\begin{align}\label{3.regul1}
& \frac{1}{\tau}(c_i^k-c_i^{k-1})
= \operatorname{div}\sum_{j=1}^{n-1}\widetilde{B}_{ij}^\delta( \widetilde{\bm{c}}^k )
\na w_j^k - \eps(\Delta^2 w_i^k + w_i^k)\quad\mbox{in }\Omega, \\
& w_i^k = (h_i^\delta)'(c_i^k) - (h_n^\delta)'(c_n^k)
- \Delta(c_i^k-c_n^k), \quad i=1,\ldots,n-1, \label{3.regul2}
\end{align}
where $\tau=T/N$ and $c_n^k=1-\sum_{i=1}^{n-1}c_i^k$.
Equation \eqref{3.regul1} is understood in the weak sense
$$
\frac{1}{\tau} \int_\Omega(c_i^k-c_i^{k-1})\phi_i dx
+ \sum_{j=1}^{n-1} \int_\Omega \widetilde{B}_{ij}^\delta(\bm{c}^k)
\na\phi_i\cdot\na w_j^k dx
+ \eps \int_\Omega(\Delta w_i^k\Delta\phi_i + w_i^k\phi_i)dx = 0
$$
for test functions $\phi_i\in H^2(\Omega)$.
The $\eps$-regularization ensures that $w_i^k\in H^2(\Omega)\hookrightarrow
L^\infty(\Omega)$ since $d\le 3$. In higher space dimensions, we can replace
$\Delta^2 w_i^k$ by $(-\Delta)^m w_i^k$ with $m>d/2$, which gives
$w_i^k\in H^m(\Omega)\hookrightarrow L^\infty(\Omega)$.
We prove the solvability of \eqref{3.regul1}--\eqref{3.regul2} in two steps.
\begin{lemma}[Solvability of \eqref{3.regul2}]\label{lem.solv}
Let $\bm{w}\in L^2(\Omega;{\mathbb R}^{n-1})$.
Then there exists a unique strong solution $\widetilde{\bm{c}}\in H^2(\Omega;{\mathbb R}^{n-1})$
to
\begin{equation}\label{3.deltac}
w_i = (h_i^\delta)'(c_i) - (h_n^\delta)'(c_n) - \Delta(c_i-c_n)\quad\mbox{in }\Omega,
\quad \na c_i\cdot\nu=0\quad\mbox{on }\pa\Omega
\end{equation}
for $i=1,\ldots,n-1$, where $c_n=1-\sum_{i=1}^{n-1}c_i$. This defines the
operator ${\mathcal L}:L^2(\Omega;{\mathbb R}^{n-1})
\to H^2(\Omega;{\mathbb R}^{n-1})$, ${\mathcal L}(\bm{w})=\widetilde{\bm{c}}$.
\end{lemma}
\begin{proof}
The system of equations can be written as
$$
\operatorname{div}(M\na\widetilde{\bm{c}})_i = (h_i^\delta)'(c_i) - (h_n^\delta)'(c_n) - w_i
\quad\mbox{in }\Omega,
$$
where the entries of the diffusion matrix $M$ are $M_{ii}=2$ and
$M_{ij}=1$ for all $i\neq j$. In particular, $M$ is symmetric and positive definite.
Thus, we can apply
the theory for elliptic systems with sublinear growth function and conclude the
existence of a unique weak solution $\widetilde{\bm{c}}\in H^1(\Omega;{\mathbb R}^{n-1})$.
It remains to verify that this solution lies in $H^2(\Omega;{\mathbb R}^{n-1})$.
Summing \eqref{3.deltac} over $i=1,\ldots,n-1$, we find that
$$
\Delta c_n = -\sum_{i=1}^{n-1}\Delta c_i
= \frac{1}{n}\sum_{i=1}^{n-1}(w_i-(h_i^\delta)'(c_i))
+ \frac{n-1}{n} (h_n^\delta)'(c_n)
\in L^2(\Omega)
$$
with the boundary condition $\na c_n\cdot\nu=0$ on $\pa\Omega$.
We infer from elliptic regularity theory that $c_n\in H^2(\Omega)$.
Consequently, $\Delta c_n\in L^2(\Omega)$ and elliptic regularity again
implies that $c_i\in H^2(\Omega)$.
\end{proof}
It follows from Lemma \ref{lem.solv} that we can write \eqref{3.regul1} as
\begin{equation}\label{3.Bregul}
\frac{1}{\tau}({\mathcal L}(\bm{w})_i - c_i^{k-1})
= \operatorname{div}\sum_{j=1}^{n-1} \widetilde{B}_{ij}^\delta(\widetilde{\bm{c}}^k)
\na w_j^k - \eps(\Delta^2 w_i^k+w_i^k)\quad\mbox{in }\Omega,\ i=1,\ldots,n-1.
\end{equation}
\begin{lemma}[Solvability of \eqref{3.Bregul}]
Let $\widetilde{\bm{c}}^{k-1}\in H^2(\Omega;{\mathbb R}^{n-1})$.
Then there exists a weak solution $\bm{w}^k\in H^2(\Omega;{\mathbb R}^{n-1})$
to \eqref{3.Bregul} such that for all $\phi_i\in L^2(0,T;H^2(\Omega))$,
\begin{align*}
\frac{1}{\tau}\int_\Omega({\mathcal L}(\bm{w})_i - c_i^{k-1})\phi_i dx
&+ \sum_{i,j=1}^{n-1}\int_\Omega\widetilde{B}_{ij}^\delta({\mathcal L}(\bm{w}))
\na\phi_i \cdot\na w_j^k dx \\
&{}+ \eps\sum_{i=1}^{n-1}\int_\Omega(\Delta w_i^k\Delta\phi_i+w_i^k\phi_i)dx = 0.
\end{align*}
\end{lemma}
\begin{proof}
Given $\bar{\bm{w}}\in L^\infty(\Omega;{\mathbb R}^{n-1})$ and $\sigma\in[0,1]$,
we wish to find a solution to the linear problem
\begin{equation}\label{3.LM}
{\mathcal A}(\bm{w},\bm{\phi}) = {\mathcal F}(\bm{\phi})\quad\mbox{for }
\bm{\phi}\in H^2(\Omega;{\mathbb R}^{n-1}),
\end{equation}
where
\begin{align*}
{\mathcal A}(\bm{w},\bm{\phi}) &= \sum_{i,j=1}^{n-1}\int_\Omega
\widetilde{B}_{ij}^\delta({\mathcal L}(\bar{\bm{w}}))\na\phi_i\cdot\na w_j dx
+ \eps\sum_{i=1}^{n-1}\int_\Omega(\Delta w_i\Delta\phi_i + w_i\phi_i)dx, \\
{\mathcal F}(\bm{\phi}) &= -\frac{\sigma}{\tau}\int_\Omega({\mathcal L}(\bar{\bm{w}})
- \widetilde{\bm{c}}^{k-1})\cdot\bm{\phi}dx.
\end{align*}
We infer from the boundedness of $\widetilde{B}_{ij}^\delta
({\mathcal L}(\bar{\bm{w}}))$
that the bilinear form ${\mathcal A}$ is continuous on $H^2(\Omega;{\mathbb R}^{n-1})$.
Furthermore, by the positive definiteness of
$\widetilde{B}_{ij}^\delta({\mathcal L}(\bar{\bm{w}}))$, thanks to \eqref{3.tilde},
${\mathcal A}$ is coercive. Moreover, ${\mathcal F}$ is a continuous linear form
on $H^2(\Omega;{\mathbb R}^{n-1})$. We conclude from the Lax--Milgram theorem that there
exists a unique solution $\bm{w}\in H^2(\Omega;{\mathbb R}^{n-1})$ to \eqref{3.LM}.
Since $d\le 3$ by Assumption (A1), we have $H^2(\Omega)\hookrightarrow
L^\infty(\Omega)$ and therefore $\bm{w}\in L^\infty(\Omega;{\mathbb R}^{n-1})$.
This defines the fixed-point operator $S:L^\infty(\Omega;{\mathbb R}^{n-1})\times[0,1]\to
L^\infty(\Omega;{\mathbb R}^{n-1})$, $S(\bar{\bm{w}},\sigma)=\bm{w}$. The operator $S$
is continuous, and it satisfies $S(\bar{\bm{w}},0)=\bm{0}$ for all $\bar{\bm{w}}
\in L^\infty(\Omega;{\mathbb R}^{n-1})$. In view of the compact embedding
$H^2(\Omega)\hookrightarrow L^\infty(\Omega)$, $S$ is also compact. It remains to
verify that all fixed points of $S(\cdot,\sigma)$ are uniformly bounded.
To this end, let $\bm{w}\in L^\infty(\Omega;{\mathbb R}^{n-1})$ be such a fixed point.
Then $\bm{w}\in H^2(\Omega;{\mathbb R}^{n-1})$ solves \eqref{3.LM} with
$\bar{\bm{w}}=\bm{w}$. We choose the test function $\bm{\phi} = \bm{w}$ in \eqref{3.LM}
to find that
\begin{equation}\label{3.aux2}
\frac{\sigma}{\tau}\int_\Omega(\widetilde{\bm{c}}-\widetilde{\bm{c}}^{k-1})
\cdot\bm{w} dx + \sum_{i,j=1}^{n-1}\int_\Omega \widetilde{B}_{ij}^\delta
(\widetilde{\bm{c}})\na w_i\cdot\na w_j dx
+ \eps\sum_{i=1}^{n-1}\int_\Omega((\Delta w_i)^2+w_i^2)dx = 0,
\end{equation}
where $\widetilde{\bm{c}}={\mathcal L}(\bm{w})=(c_1,\ldots,c_{n-1})$ and $c_i$ solves
\eqref{3.regul2} with $w_i^k$ replaced by $w_i$. Using the test function
$c_i-c_i^{k-1}$ in the weak formulation of \eqref{3.regul2} leads to
\begin{align*}
\sum_{i=1}^{n-1}\int_\Omega(c_i-c_i^{k-1})w_i dx
&= \sum_{i=1}^{n-1}\int_\Omega\big(\na(c_i-c_n)\cdot\na(c_i-c_i^{k-1}) \\
&\phantom{xx}{}+ ((h_i^\delta)'(c_i)-(h_i^\delta)'(c_n))(c_i-c_i^{k-1})\big)dx.
\end{align*}
The convexity of the function $h_i^\delta$ and $\sum_{i=1}^{n-1}c_i=1-c_n$ imply that
\begin{align*}
\sum_{i=1}^{n-1}(c_i-c_i^{k-1})(h_i^\delta)'(c_i)
&\ge \sum_{i=1}^{n-1}\big(h_i^\delta(c_i)-h_i^\delta(c_i^{k-1})\big), \\
-\sum_{i=1}^{n-1}(c_i-c_i^{k-1})(h_n^\delta)'(c_n)
&= (c_n-c_n^{k-1})(h_n^\delta)'(c_n) \ge h_n^\delta(c_n)-h_n^\delta(c_n^{k-1}).
\end{align*}
Moreover, since $\sum_{i=1}^{n-1}\na c_i=-\na c_n$ and
$\sum_{i=1}^{n-1}\na c_i^{k-1}=-\na c_n^{k-1}$,
\begin{align*}
\sum_{i=1}^{n-1}\na(c_i-c_n)\cdot\na (c_i-c_i^{k-1})
&= \sum_{i=1}^{n}|\na c_i|^2 - \sum_{i=1}^{n}\na c_i^{k-1}\cdot\na c_i \\
&\ge \frac12\sum_{i=1}^n|\na c_i|^2 - \frac12\sum_{i=1}^n|\na c_i^{k-1}|^2.
\end{align*}
This yields
\begin{align*}
\sum_{i=1}^{n-1}\int_\Omega(c_i-c_i^{k-1})w_i dx
&\ge \sum_{i=1}^{n}\int_\Omega\big(h_i^\delta(c_i)-h_i^\delta(c_i^{k-1})\big)dx
+ \frac12\sum_{i=1}^{n}\int_\Omega\big(|\na c_i|^2 - |\na c_i^{k-1}|^2\big) dx \\
&\ge \widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}})
- \widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}^{k-1}),
\end{align*}
where
$$
\widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}) := \widetilde{{\mathcal H}}^\delta(\widetilde{\bm{c}})
+ \sum_{i=1}^n\int_\Omega|\na c_i|^2 dx, \quad
\widetilde{{\mathcal H}}^\delta(\widetilde{\bm{c}}) := {\mathcal H}^\delta(\bm{c}).
$$
Inserting this inequality into \eqref{3.aux2} finally gives
\begin{equation}\label{3.rfei}
\sigma\widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}})
+ \tau\sum_{i,j=1}^{n-1}\int_\Omega \widetilde{B}_{ij}^\delta(\widetilde{\bm{c}})
\na w_i\cdot\na w_j dx + \eps\tau\int_\Omega(|\Delta\bm{w}|^2 + |\bm{w}|^2)dx
\le \sigma\widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}^{k-1}).
\end{equation}
By the positive definiteness of $\widetilde{B}^\delta$ (positive semidefiniteness
is sufficient), this gives a uniform $H^2(\Omega)$ bound and
consequently a uniform $L^\infty(\Omega)$ bound for $\bm{w}$.
The Leray--Schauder fixed-point theorem now implies the existence of a solution
to \eqref{3.regul1}--\eqref{3.regul2}.
\end{proof}
{\em Step 3: Uniform estimates.}
We wish to derive estimates uniform in $\eps$ and $\tau$. The starting point is
the regularized energy estimate \eqref{3.rfei} and the positive definiteness
estimate \eqref{3.tilde}. First, we introduce the piecewise constant in time functions
$\bm{w}^{(\tau)}(x,t)=\bm{w}^k(x)$, $\widetilde{\bm{c}}^{(\tau)}(x,t)
= {\mathcal L}(\bm{w}^k(x))$ for $x\in\Omega$ and $t\in((k-1)\tau,k\tau]$,
$k=1,\ldots,N$, and set
$\bm{w}^{(\tau)}(x,0)=(\pa\widetilde{{\mathcal E}}/\pa\widetilde{\bm{c}})(\widetilde{\bm{c}}^0)$
and $\widetilde{\bm{c}}^{(\tau)}(x,0)=\widetilde{\bm{c}}^0$. Introducing the shift
operator $(\sigma_\tau\bm{w}^{(\tau)})(x,t)=\bm{w}^{(\tau)}(x,t-\tau)$ for
$x\in\Omega$ and $t\ge \tau$, we can formulate \eqref{3.regul1}--\eqref{3.regul2} as
\begin{align}
& \frac{1}{\tau}(\widetilde{\bm{c}}^{(\tau)}-\sigma_\tau\widetilde{\bm{c}}^{(\tau)})
= \operatorname{div}(\widetilde{B}^\delta(\widetilde{\bm{c}})\na\bm{w}^{(\tau)})
- \eps(\Delta^2\bm{w}^{(\tau)}+\bm{w}^{(\tau)}), \label{3.tau1} \\
& w_i^{(\tau)} = (h_i^\delta)'(c_i^{(\tau)}) - (h_n^\delta)'(c_n^{(\tau)})
- \Delta(c_i^{(\tau)}-c_n^{(\tau)}), \quad i=1,\ldots,n-1, \label{3.tau2}
\end{align}
recalling that $\widetilde{\bm{c}}^{(\tau)}={\mathcal L}(\bm{w}^{(\tau)})$
is a function of $\bm{w}^{(\tau)}$. Then \eqref{3.rfei} can be
written after summation over $k=1,\ldots,N$ as
\begin{equation*}
\widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}^{(\tau)}(T))
+ \eta(\delta)\int_0^T\int_\Omega|\na\bm{w}^{(\tau)}|^2 dxdt
+ \eps C\int_0^T\|\bm{w}^{(\tau)}\|_{H^2(\Omega)}^2 dt
\le \widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}^0),
\end{equation*}
where we used \eqref{3.tilde} and the generalized Poincar\'e inequality with
constant $C>0$. This implies the estimates
\begin{equation}\label{3.west}
C(\delta)\|\bm{w}^{(\tau)}\|_{L^2(0,T;H^1(\Omega))}
+ \sqrt{\eps}\|\bm{w}^{(\tau)}\|_{L^2(0,T;H^2(\Omega))} \le C,
\end{equation}
where $C>0$ denotes here and in the following a constant independent of
$\eps$ and $\tau$.
To derive a uniform estimate for $\widetilde{\bm{c}}^{(\tau)}$, we multiply
\eqref{3.tau2} by $-\Delta c_i^{(\tau)}$, integrate over $Q_T=\Omega\times(0,T)$,
integrate by parts, and sum over $i=1,\ldots,n-1$:
\begin{align*}
\sum_{i=1}^{n-1}\int_0^T\int_\Omega&\na w_i^{(\tau)}\cdot\na c_i^{(\tau)} dxdt
= \sum_{i=1}^{n-1}\int_0^T\int_\Omega\na\big((h_i^\delta)'(c_i^{(\tau)})
- (h_n^\delta)'(c_n^{(\tau)})\big)\cdot\na c_i^{(\tau)} dxdt \\
&\phantom{xx}{}+ \sum_{i=1}^{n-1}\int_0^T\int_\Omega\big((\Delta c_i^{(\tau)})^2
- \Delta c_i^{(\tau)}\Delta c_n^{(\tau)}\big)dxdt =: I_3+I_4.
\end{align*}
Since $\na(h_i^\delta)'(c_i^{(\tau)})=(h_i^\delta)''(c_i^{(\tau)})\na c_i^{(\tau)}
= \na c_i^{(\tau)}/(\chi_\delta\bm{c}^{(\tau)})_i$ and
$\sum_{i=1}^{n-1}\na c_i^{(\tau)}=-\na c_n^{(\tau)}$, the term $I_3$ can be written as
$$
I_3 = \sum_{i=1}^n\int_0^T\int_\Omega
\frac{|\na c_i^{(\tau)}|^2}{(\chi_\delta\bm{c}^{(\tau)})_i}dxdt.
$$
Using the property $\sum_{i=1}^{n-1}\Delta c_i^{(\tau)}=-\Delta c_n^{(\tau)}$,
the remaining term $I_4$ becomes
$$
I_4 = \sum_{i=1}^n\int_0^T\int_\Omega(\Delta c_i^{(\tau)})^2 dxdt.
$$
Therefore, by Young's inequality,
\begin{align*}
\sum_{i=1}^n&\int_0^T\int_\Omega(\Delta c_i^{(\tau)})^2 dxdt
+ \sum_{i=1}^n\int_0^T\int_\Omega
\frac{|\na c_i^{(\tau)}|^2}{(\chi_\delta\bm{c}^{(\tau)})_i}dxdt
= \sum_{i=1}^{n-1}\int_0^T\int_\Omega\na w_i^{(\tau)}\cdot\na c_i^{(\tau)}dxdt \\
&\le \frac12\sum_{i=1}^{n-1}\int_0^T\int_\Omega\bigg(\frac{|\na c_i^{(\tau)}|^2}{
(\chi_\delta\bm{c}^{(\tau)})_i}
+ (\chi_\delta\bm{c}^{(\tau)})_i|\na w_i^{(\tau)}|^2\bigg) dxdt \\
&\le \frac12\sum_{i=1}^{n-1}\int_0^T\int_\Omega\frac{|\na c_i^{(\tau)}|^2}{
(\chi_\delta\bm{c}^{(\tau)})_i}dxdt
+ \frac12\sum_{i=1}^{n-1}\int_0^T\int_\Omega|\na w_i^{(\tau)}|^2dxdt.
\end{align*}
The first term on the right-hand side is absorbed by the left-hand side. Thus,
we deduce from \eqref{3.west} that
$$
\sum_{i=1}^n\int_0^T\int_\Omega(\Delta c_i^{(\tau)})^2 dxdt
+ \frac12\sum_{i=1}^n\int_0^T\int_\Omega
\frac{|\na c_i^{(\tau)}|^2}{(\chi_\delta\bm{c}^{(\tau)})_i}dxdt
\le \frac12\|\na \bm{w}^{(\tau)}\|^2_{L^2(Q_T)}
\le C.
$$
Since $c_i^{(\tau)}\in L^\infty(Q_T)$, we infer from the
previous estimate that
\begin{equation}\label{3.cH2}
\|c_i^{(\tau)}\|_{L^2(0,T;H^2(\Omega))} \le C, \quad i=1,\ldots,n.
\end{equation}
Finally, we derive an estimate for the discrete time derivative. It follows from
\eqref{3.Bregul} that
\begin{align*}
\frac{1}{\tau}\|c_i^{(\tau)}-\sigma_\tau c_i^{(\tau)}\|_{L^2(0,T;H^2(\Omega)')}
&\le \sum_{j=1}^{n-1}\|\widetilde{B}^\delta_{ij}(\widetilde{\bm{c}}^{(\tau)})
\|_{L^\infty(Q_T)}\|\na w_j^{(\tau)}\|_{L^2(Q_T)} \\
&\phantom{xx}{}+ \eps\|w_i^{(\tau)}\|_{L^2(0,T;H^2(\Omega))}.
\end{align*}
The entries of $\widetilde{B}^\delta(\widetilde{\bm{c}}^{(\tau)})$ are bounded since
$\delta\le(\chi_\delta\bm{c}^{(\tau)})_i\le 1-\delta$. Thus, by \eqref{3.west},
\begin{equation}\label{3.ctime}
\tau^{-1}\|c_i^{(\tau)}-\sigma_\tau c_i^{(\tau)}\|_{L^2(0,T;H^2(\Omega)')}
\le C, \quad i=1,\ldots,n-1.
\end{equation}
{\em Step 4: Limit $(\eps,\tau)\to 0$.}
In view of estimates \eqref{3.cH2} and \eqref{3.ctime}, we can apply the
Aubin--Lions lemma in the version of \cite[Theorem 1]{DrJu12} to conclude the
existence of a subsequence, which is not relabeled, such that
as $(\eps,\tau)\to 0$,
$$
c_i^{(\tau)}\to c_i\quad\mbox{strongly in }L^2(0,T;H^1(\Omega)),\ i=1,\ldots,n-1.
$$
We deduce from \eqref{3.west}--\eqref{3.ctime} that, possibly for another subsequence,
\begin{align*}
c_i^{(\tau)} \rightharpoonup c_i &\quad\mbox{weakly in }L^2(0,T;H^2(\Omega)), \\
\tau^{-1}(c_i^{(\tau)}-\sigma_\tau c_i^{(\tau)}) \rightharpoonup \pa_t c_i
&\quad\mbox{weakly in }L^2(0,T;H^2(\Omega)'), \\
w_i^{(\tau)} \rightharpoonup w_i &\quad\mbox{weakly in }L^2(0,T;H^1(\Omega)), \\
\eps w_i^{(\tau)}\to 0 &\quad\mbox{strongly in }L^2(0,T;H^2(\Omega)),
\quad i=1,\ldots,n-1.
\end{align*}
We define $c_n:=1-\sum_{i=1}^{n-1}c_i$. Then $c_n^{(\tau)}\to c_n$ strongly
in $L^2(0,T;H^1(\Omega))$ and weakly in $L^2(0,T;H^2(\Omega))$.
Furthermore, $(c_i^{(\tau)})$ converges, up to a subsequence, pointwise a.e.,
and its limit satisfies $\delta\le(\chi_\delta\bm{c})_i\le 1-\delta$, $i=1,\ldots,n$.
The matrix $\widetilde{B}_{ij}^\delta(\widetilde{\bm{c}}^{(\tau)})$
is uniformly bounded and
$$
\widetilde{B}_{ij}^\delta(\widetilde{\bm{c}}^{(\tau)})
\to \widetilde{B}_{ij}^\delta(\widetilde{\bm{c}})\quad\mbox{strongly in }
L^q(Q_T)\mbox{ for any } q<\infty,\ i,j=1,\ldots,n.
$$
These convergence results allow us to pass to the limit $(\eps,\tau)\to 0$
in the weak formulation of \eqref{3.tau1}--\eqref{3.tau2} to find that
$\bm{c}$ solves
$$
\pa_t c_i = \operatorname{div}\sum_{j=1}^{n-1}\widetilde{B}_{ij}^\delta(\widetilde{\bm{c}})
\na w_j, \quad w_i = (h_i^\delta)'(c_i)-(h_n^\delta)'(c_n)
- \Delta(c_i-c_n)
$$
for $i=1,\ldots,n-1$. Transforming back to the chemical potential $\bm{\mu}$
via $w_i=\mu_i-\mu_n$ and $c_n=1-\sum_{i=1}^{n-1}c_i$, we see that
$\bm{c}^\delta:=\bm{c}$ solves system \eqref{3.eq}--\eqref{3.bic},
where $\mu_i=(h_i^\delta)'(c_i)-\Delta c_i$.
\subsection{Uniform estimates}\label{sec.unif}
We derive energy and entropy estimates for the solutions to \eqref{3.eq},
being uniform in $\delta$.
\begin{lemma}[Energy and entropy inequalities]\label{lem.eei}
Let $\bm{c}^\delta$ be a weak solution to \eqref{3.eq}--\eqref{3.bic},
constructed in Theorem \ref{thm.approx}. Then the following inequalities hold
for any $T>0$,
\begin{align}
& {\mathcal E}^\delta(\bm{c}^\delta(\cdot,T)) + \sum_{i,j=1}^n\int_0^T\int_\Omega
B_{ij}^\delta(\bm{c}^\delta)\na\mu_i^\delta\cdot\na\mu_j^\delta dxdt
\le {\mathcal E}^\delta(\bm{c}^0), \label{3.Ed} \\
& {\mathcal H}^\delta(\bm{c}^\delta(\cdot,T)) + \sum_{i,j=1}^n\int_0^T\int_\Omega
B_{ij}^\delta(\bm{c}^\delta)\na(h_i^\delta)'(c_i^\delta)\cdot\na\mu_j^\delta dxdt
\le {\mathcal H}^\delta(\bm{c}^0), \label{3.Hd} \\
& {\mathcal H}^\delta(\bm{c}^\delta(\cdot,T)) + \frac{(\lambda_M-\lambda)^2}{2\lambda_m\lambda}
{\mathcal E}^\delta(\bm{c}^\delta(\cdot,T)) + \lambda\sum_{i=1}^n\int_0^T\int_\Omega
\frac{|\na c_i^\delta|^2}{(\chi_\delta\bm{c}^\delta)_i}dxdt \label{3.EHd} \\
&\phantom{xxxx}{}+ \lambda\sum_{i=1}^n\int_0^T\int_\Omega(\Delta c_i^\delta)^2 dxdt
+ \frac{(\lambda_M-\lambda)^2}{2\lambda}\int_0^T\int_\Omega
\big|P_L(\chi_\delta\bm{c}^\delta)R(\chi_\delta\bm{c}^\delta)\na\bm{\mu}^\delta
\big|^2 dxdt \nonumber \\
&\phantom{xx}{}\le {\mathcal H}^\delta(\bm{c}^0) + \frac{(\lambda_M-\lambda)^2}{2\lambda_m\lambda}
{\mathcal E}^\delta(\bm{c}^0), \nonumber
\end{align}
where $0<\lambda<\lambda_m$, $\lambda_m$, $\lambda_M$ are introduced in
Lemma \ref{lem.DB}, and $R(\chi_\delta\bm{c}^\delta)
=\operatorname{diag}(\sqrt{\chi_\delta\bm{c}^\delta})$.
\end{lemma}
\begin{proof}
Summing \eqref{3.rfei} with $\sigma=1$ over $k=1,\ldots,N$, we find that
\begin{align*}
\widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}^{(\tau)}(\cdot,T))
&+ \sum_{i,j=1}^{n-1}\int_0^T\int_\Omega \widetilde{B}_{ij}^\delta
(\widetilde{\bm{c}}^{(\tau)})
\na w_i^{(\tau)}\cdot\na w_j^{(\tau)} dxdt \\
&{}+ \eps\sum_{i=1}^n\int_0^T\int_\Omega\big((\Delta w_i^{(\tau)})^2
+ (w_i^{(\tau)})^2\big)dxdt \le \widetilde{{\mathcal E}}^\delta(\widetilde{\bm{c}}^0).
\end{align*}
We know from \eqref{3.west} and the construction of $\chi_\delta$ that
$(\bm{w}^{(\tau)})$ is bounded in $L^2(0,T;H^1(\Omega))$ and
$(\widetilde{B}_{ij}^\delta(\widetilde{\bm{c}}))$ is bounded in
$L^\infty(Q_T)$ with respect to $(\eps,\tau)$.
Therefore, we can pass to the limit $(\eps,\tau)\to 0$ in the previous
inequality, and weak lower semicontinuity of the integral functionals
leads to \eqref{3.Ed}.
To show \eqref{3.Hd}, we use $(h_i^\delta)'(c_i^\delta)-(h_i^\delta)'(c_n^\delta)$
as a test function in the weak formulation of \eqref{3.cdelta} and sum over
$i=1,\ldots,n-1$:
$$
{\mathcal H}^\delta(\bm{c}(\cdot,T)) + \sum_{i,j=1}^{n-1}\int_0^T\int_\Omega
\widetilde{B}_{ij}^\delta(\widetilde{\bm{c}}^\delta)
\na\big((h_i^\delta)'(c_i^\delta)-(h_i^\delta)'(c_n^\delta)\big)\cdot
\na w_j^\delta dxdt \le {\mathcal H}^\delta(\bm{c}^0).
$$
This inequality can be rewritten as \eqref{3.Hd} using
$w_i^\delta=\mu_i^\delta-\mu_n^\delta$. Finally, we derive \eqref{3.EHd}
by combining \eqref{3.Hd} and \eqref{3.Ed} and proceeding as in the proof
of Lemma \ref{lem.fei}.
\end{proof}
\subsection{Proof of Theorem \ref{thm.ex}}\label{sec.exproof}
We perform the limit $\delta\to 0$ to finish the proof of Theorem \ref{thm.ex}.
It follows from \cite[Lemma 2.1]{ElLu91} that for sufficiently small $\delta>0$,
there exists $C>0$ (independent of $\delta$)
such that for all $r_1,\ldots,r_n\in{\mathbb R}$ satisfying $\sum_{i=1}^n r_i=1$,
\begin{equation}\label{3.lower}
\sum_{i=1}^n h_i^\delta(r_i)\ge -C.
\end{equation}
Therefore, estimate \eqref{3.EHd} implies that
\begin{equation}
\label{eqnhelp1}
\begin{aligned}
\sum_{i=1}^n\int_\Omega|\na c_i^\delta(\cdot,T)|^2 dx
&+ \sum_{i=1}^n\int_0^T\int_\Omega\frac{|\na c_i^\delta|^2}{
(\chi_\delta\bm{c}^\delta)_i}dxdt
+ \sum_{i=1}^n\int_0^T\int_\Omega(\Delta c_i^\delta)^2 dxdt
\\
&{}+ \int_0^T\int_\Omega\big|P_L(\chi_\delta\bm{c}^\delta)
R(\chi_\delta\bm{c}^\delta)\na\bm{\mu}^\delta\big|^2 dxdt \le C,
\end{aligned}
\end{equation}
and the constant $C>0$ depends on $\lambda_m$, $\lambda_M$, and $\bm{c}^0$.
Mass conservation (or using the test function $\phi_i=1$ in the weak
formulation of \eqref{3.eq}) shows that
$\int_\Omega c_i^\delta(\cdot,T)dx=\int_\Omega c_0^\delta dx$ for any $T>0$,
i.e.\ $\|\bm{c}^\delta\|_{L^\infty(0,T;L^1(\Omega))}\le C$. We conclude from the
Poincar\'e--Wirtinger inequality that
\begin{equation}\label{3.cd}
\|\bm{c}^\delta\|_{L^\infty(0,T;H^1(\Omega))}
+ \|\bm{c}^\delta\|_{L^2(0,T;H^2(\Omega))} \le C.
\end{equation}
Next, we estimate $\pa_t c_i^\delta$. Lemma \ref{lem.Bdelta} implies that
the entries of
$$(D(\chi_\delta\bm{c}^\delta)P_L(\chi_\delta\bm{c}^\delta)
+ P_{L^\perp}(\chi_\delta\bm{c}^\delta))^{-1}$$
are uniformly bounded. Thus,
by the definition of $D^{BD}(\chi_\delta\bm{c}^\delta)$ and \eqref{2.DBD},
$$
\int_0^T\int_\Omega\bigg|\sum_{j=1}^n B_{ij}^\delta(\bm{c}^\delta)
\na\mu_j^\delta\bigg|^2 dxdt \le \lambda_M\int_0^T\int_\Omega
\big|P_L(\chi_\delta\bm{c}^\delta)R(\chi_\delta\bm{c}^\delta)
\na\bm{\mu}^\delta\big|^2 dxdt,
$$
and the right-hand side is bounded by \eqref{eqnhelp1}. Setting
$J_i^\delta:=\sum_{j=1}^n B_{ij}^\delta(\bm{c}^\delta)\na\mu_j^\delta$,
this means that $(J_i^\delta)$ is bounded in $L^2(Q_T)$. Therefore, there exists
a subsequence that is not relabeled such that, as $\delta\to 0$,
$$
J_i^\delta\rightharpoonup J_i\quad\mbox{weakly in }L^2(Q_T).
$$
This implies that
\begin{equation}
\label{eqnhelp2}
\|\pa_t c_i^\delta\|_{L^2(0,T;H^1(\Omega)')} \le C.
\end{equation}
We conclude from \eqref{3.cd} and \eqref{eqnhelp2}, using the Aubin--Lions lemma,
that, for a subsequence (if necessary),
\begin{equation}\label{3.cconv}
\begin{aligned}
c_i^\delta\to c_i &\quad\mbox{strongly in }L^2(0,T;H^1(\Omega)), \\
c_i^\delta \stackrel{\star}{\rightharpoonup} c_i
&\quad\mbox{weakly-$\star$\, in }L^\infty(0,T;H^1(\Omega)), \\
c_i^\delta\rightharpoonup c_i &\quad\mbox{weakly in }L^2(0,T;H^2(\Omega)), \\
\pa_t c_i^\delta\rightharpoonup \pa_t c_i &\quad\mbox{weakly in }
L^2(0,T;H^1(\Omega)').
\end{aligned}
\end{equation}
Performing the limit $\delta\to 0$ in \eqref{3.eq}, we see that
$\pa_t c_i=\operatorname{div} J_i$ holds in the sense of $L^2(0,T;H^1(\Omega)')$.
We prove that $c_i\ge 0$ in $Q_T$, $i=1,\ldots,n$,
following \cite{ElLu91}.
By definition \eqref{3.hdelta} and the lower bound \eqref{3.lower}, we have
for $0<\delta<1$,
\begin{align*}
C&\ge \int_\Omega h_i^\delta(c_i^\delta)dx \ge -C
+ \int_{\{c_i^\delta<\delta\}}\bigg(c_i^\delta\log\delta
- \frac{\delta}{2} + \frac{(c_i^\delta)^2}{2\delta}\bigg)dx \\
&\ge -C + \int_{\{c_i^\delta<0\}}c_i^\delta\log\delta dx
+ \int_{\{0<c_i^\delta<\delta\}}c_i^\delta\log\delta dx - C\delta \\
&\ge -C + \log\delta\int_{\{c_i^\delta<0\}}c_i^\delta dx
+ C\delta\log\delta - C\delta.
\end{align*}
Hence, we obtain
$$
\int_\Omega \max\{0,-c_i^\delta\}dx
= \int_{\{c_i^\delta<0\}}|c_i^\delta|dx \le \frac{C}{|\log\delta|}.
$$
The limit $\delta\to 0$ leads to
$$
\int_\Omega\max\{0,-c_i\}dx \le 0,
$$
implying that $c_i\ge 0$ in $Q_T$. The limit $\delta\to 0$ in
$\sum_{i=1}^n c_i^\delta=1$ gives $\sum_{i=1}^n c_i=1$, hence $c_i\le 1$
holds in $Q_T$.
Next, we identify $J_i$ by showing that $J_i=\sum_{j=1}^n
B_{ij}(\bm{c})\na(\log c_j-\Delta c_j)$ in the sense of distributions.
Inserting the definition of $\mu_i^\delta$ and choosing a test function
$\phi_i\in L^\infty(0,T;$ $W^{2,\infty}(\Omega))$ satisfying $\na\phi_i\cdot\nu=0$
on $\pa\Omega$, we find that
\begin{align}
\int_0^T&\int_\Omega J_i^\delta\cdot\na\phi_i dxdt
= \sum_{j=1}^n\int_0^T\int_\Omega B_{ij}^\delta(\bm{c}^\delta)
\na\phi_i\cdot\na\big((h_j^\delta)'(c_j^\delta)-\Delta c_j^\delta\big) dxdt
\nonumber \\
&= \sum_{j=1}^n\int_0^T\int_\Omega B_{ij}^\delta(\bm{c}^\delta)
\na\phi_i\cdot\na(h_j^\delta)'(c_j^\delta)dxdt
+ \sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j^\delta\operatorname{div}(B_{ij}^\delta(\bm{c}^\delta)
\na\phi_i)dxdt \label{3.Jdelta} \\
&=: I_5 + I_6. \nonumber
\end{align}
By definition \eqref{3.Bdelta} of $B_{ij}^\delta(\bm{c}^\delta)$, we have
$$
I_5 = \sum_{j=1}^n\int_0^T\int_\Omega \sqrt{(\chi_\delta\bm{c}^\delta)_i}
D_{ij}^{BD}(\chi_\delta\bm{c}^\delta)\na\phi_i\cdot
\frac{\na c_j^\delta}{\sqrt{(\chi_\delta\bm{c}^\delta)_j}}dxdt.
$$
Lemma \ref{lem.DB} shows that $\sqrt{c_i}D_{ij}^{BD}(\bm{c})/\sqrt{c_j}$ is
bounded in $[0,1]^n$ and in particular when $c_k=0$ for some index $k$.
The strong convergence $\bm{c}^\delta\to \bm{c}$ implies that
$\chi_\delta\bm{c}^\delta\to\bm{c}$ in $L^q(0,T;L^q(\Omega))$ for any $q<\infty$
such that
$$
I_5\to \sum_{j=1}^n\int_0^T\int_\Omega\sqrt{c_i}D_{ij}^{BD}(\bm{c})
\frac{1}{\sqrt{c_j}}\na\phi_i\cdot\na c_j dxdt
= \sum_{j=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})\na\phi_i\cdot\na\log c_j dxdt.
$$
The limit in $I_6$ is more involved. We decompose $I_6=I_{61}+I_{62}$, where
$$
I_{61} = \sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j^\delta
B_{ij}^\delta(\bm{c}^\delta)\Delta\phi_i dxdt, \quad
I_{62} = \sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j^\delta
\na B_{ij}^\delta(\bm{c}^\delta)\cdot\na\phi_i dxdt.
$$
We deduce from the strong convergence of $\bm{c}^\delta$ and the weak
convergence of $\Delta c_j^\delta$ that
$$
I_{61} \to \sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j B_{ij}(\bm{c})\Delta\phi_i dxdt.
$$
To show the convergence of $I_{62}$, we consider
\begin{align*}
\int_0^T&\int_\Omega\big|\na\big(B_{ij}^\delta(\bm{c}^\delta)-B_{ij}(\bm{c})\big)
\big|^2 dxdt \\
&= \int_0^T\int_\Omega\bigg|\sum_{k=1}^n\bigg\{\bigg(
\frac{\pa B_{ij}^\delta}{\pa c_k}(\bm{c}^\delta) - \frac{\pa B_{ij}}{\pa c_k}(\bm{c})
\bigg)\na c_k + \frac{\pa B_{ij}^\delta}{\pa c_k}(\bm{c}^\delta)
\na(c_k^\delta-c_k)\bigg\}\bigg|^2 dxdt.
\end{align*}
By Lemma \ref{lem.DB} (i), $\pa D_{ij}^{BD}/\pa c_k$ exists and is bounded
in $[0,1]^n$. Then, by the definition of $B_{ij}(\bm{c})$, we have
$(\pa B_{ij}^\delta/\pa c_k)(\bm{c}^\delta)\to (\pa B_{ij}/\pa c_k)(\bm{c})$
strongly in $L^2(Q_T)$. It follows from $\na c_k^\delta\to \na c_k$ strongly
in $L^2(Q_T)$ that the right-hand side of the previous identity
converges to zero. We infer that
$$
I_{62}\to \sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j \na B_{ij}(\bm{c})\cdot
\na\phi_i dxdt.
$$
Consequently, we have
\begin{align*}
I_6 \to &\sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j\big(B_{ij}(\bm{c})\Delta\phi_i
+ \na B_{ij}(\bm{c})\cdot\na\phi_i\big) dxdt \\
&= \sum_{j=1}^n\int_0^T\int_\Omega\Delta c_j\operatorname{div}(B_{ij}(\bm{c})\na\phi_i)dxdt.
\end{align*}
We have shown that \eqref{3.Jdelta} becomes in the limit $\delta\to 0$
$$
\int_0^T\int_\Omega J_i\cdot\na\phi dxdt
= \sum_{j=1}^n\int_0^T\int_\Omega\big(B_{ij}(\bm{c})\na\phi_i\cdot\na\log c_j
+ \Delta c_j\operatorname{div}(B_{ij}(\bm{c})\na\phi_i)\big)dxdt
$$
and hence, in the sense of distributions,
$$
J_i = \sum_{j=1}^n B_{ij}(\bm{c})\na(\log c_j-\Delta c_j), \quad i=1,\ldots,n.
$$
{\em Step 2: Energy and entropy inequalities.}
The limit $c_i^\delta\rightharpoonup c_i$ weakly-$\star$ in $L^\infty(0,T;H^1(\Omega))$
(see \eqref{3.cconv}) and the weak lower semicontinuity of the energy and
entropy show that
$$
{\mathcal H}(\bm{c}(\cdot,T))\le \liminf_{\delta\to 0}{\mathcal H}^\delta(\bm{c}^\delta(\cdot,T)), \quad
{\mathcal E}(\bm{c}(\cdot,T))\le \liminf_{\delta\to 0}{\mathcal E}^\delta(\bm{c}^\delta(\cdot,T)).
$$
Moreover, because of the weak convergence of $\Delta c_i^\delta$
in $L^2(Q_T)$ from \eqref{3.cconv},
$$
\sum_{i=1}^n\int_0^T\int_\Omega(\Delta c_i)^2 dxdt
\le \liminf_{\delta\to 0}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta c_i^\delta)^2 dxdt.
$$
The combined energy-entropy inequality \eqref{3.EHd} and the property
$|\na(\chi_\delta\bm{c}^\delta)_i|\le|\na c_i^\delta|$ give
$$
\big\|\na\sqrt{(\chi_\delta\bm{c}^\delta)_i}\big\|_{L^2(Q_T)}
= \frac12\bigg\|\frac{\na c_i^\delta}{\sqrt{(\chi_\delta\bm{c}^\delta)_i}}
\bigg\|_{L^2(Q_T)} \le C,
$$
which, together with $(\chi_\delta\bm{c}^\delta)_i\to c_i$ strongly in
$L^2(Q_T)$ leads to
\begin{equation}\label{3.sqrtc}
\na\sqrt{(\chi_\delta\bm{c}^\delta)_i}\rightharpoonup\na\sqrt{c_i}
\quad\mbox{weakly in }L^2(Q_T).
\end{equation}
We conclude that
$$
\|\na\sqrt{c_i}\|_{L^2(Q_T)}\le \liminf_{\delta\to 0}
\big\|\na\sqrt{(\chi_\delta\bm{c}^\delta)_i}\big\|_{L^2(Q_T)}.
$$
Finally, by \eqref{3.EHd}, we observe that $P_L(\chi_\delta\bm{c}^\delta)
R(\chi_\delta\bm{c}^\delta)\na\bm{\mu}^\delta$ is uniformly bounded
in $L^2(Q_T)$ such that, up to a subsequence,
$$
P_L(\chi_\delta\bm{c}^\delta)R(\chi_\delta\bm{c}^\delta)\na\bm{\mu}^\delta
\rightharpoonup \bm{\zeta}\quad\mbox{weakly in }L^2(Q_T).
$$
Hence, again by weak lower semicontinuity of the norm,
$$
\|\bm{\zeta}\|_{L^2(0,T;L^2(\Omega))}
\le\liminf_{\delta\to 0}\big\|(P_L(\chi_\delta\bm{c}^\delta)
R(\chi_\delta\bm{c}^\delta)\na\bm{\mu}^\delta\big\|_{L^2(0,T;L^2(\Omega))}.
$$
It remains to take the limit inferior $\delta\to 0$ in \eqref{3.EHd}
to conclude that the combined energy-entropy inequality \eqref{1.EH} holds.
\begin{lemma}[Identification of $\bm{\zeta}$]\label{lem.ident}
Let \eqref{1.cregul} hold and let $\bm{\zeta}$ be the weak $L^2(Q_T)$ limit
of $P_L(\chi_\delta\bm{c}^\delta)R(\chi_\delta\bm{c}^\delta)\na\bm{\mu}^\delta$.
Then $\bm{\zeta}=P_L(\bm{c})R(\bm{c})\na\bm{\mu}$.
\end{lemma}
\begin{proof}
Let $\phi_i\in C_0^\infty(Q_T)$ be a test function. Then, inserting the definition
$\mu_j^\delta=(h_j^\delta)'(c_j^\delta)-\Delta c_j^\delta$ and integrating by parts,
\begin{align}
\sum_{j=1}^n&\int_0^T\int_\Omega\Big(P_L(\chi_\delta\bm{c}^\delta)_{ij}
\sqrt{(\chi_\delta\bm{c}^\delta)_j}\na\mu_j^\delta
- P_L(\bm{c})_{ij}\sqrt{c_j}\na\mu_j\Big)\cdot\na\phi_i dxdt \nonumber \\
&= \sum_{j=1}^n\int_0^T\int_\Omega\Big(P_L(\chi_\delta\bm{c}^\delta)_{ij}
\sqrt{(\chi_\delta\bm{c}^\delta)_j}\na (h_j^\delta)'(c_j^\delta)
- P_L(\bm{c})_{ij}\sqrt{c_j}\na\log c_j\Big)\cdot\na\phi_i dxdt \label{4.id} \\
&\phantom{xx}{}+ \sum_{j=1}^n\int_0^T\int_\Omega\operatorname{div}\Big\{
\Big(P_L(\chi_\delta\bm{c}^\delta)_{ij}
\sqrt{(\chi_\delta\bm{c}^\delta)_j}-P_L(\bm{c})_{ij}\sqrt{c_j}\Big)\na\phi_i\Big\}
\Delta c^\delta_j dxdt \nonumber \\
&\phantom{xx}{}+ \sum_{j=1}^n\int_0^T\int_\Omega\operatorname{div}\big(P_L(\bm{c})_{ij}\sqrt{c_j}
\na\phi_i\big)\Delta(c_j^\delta-c_j)dxdt. \nonumber
\end{align}
The bracket in the first integral on the right-hand side can be written as
\begin{align*}
P_L(&\chi_\delta\bm{c}^\delta)_{ij}
\sqrt{(\chi_\delta\bm{c}^\delta)_j}\na (h_j^\delta)'(c_j^\delta)
- P_L(\bm{c})_{ij}\sqrt{c_j}\na\log c_j \\
&= P_L(\chi_\delta\bm{c}^\delta)_{ij}
\frac{\na c_j^\delta}{\sqrt{(\chi_\delta\bm{c}^\delta)_j}}
- P_L(\bm{c})_{ij}\frac{\na c_j}{\sqrt{c_j}}.
\end{align*}
Thanks to the convergences \eqref{3.cconv} and \eqref{3.sqrtc}, we can pass to
the limit $\delta\to 0$ in \eqref{4.id}:
$$
\lim_{\delta\to 0}\sum_{j=1}^n\int_0^T\int_\Omega
\Big(P_L(\chi_\delta\bm{c}^\delta)_{ij}
\sqrt{(\chi_\delta\bm{c}^\delta)_j}\na\mu_j^\delta
- P_L(\bm{c})_{ij}\sqrt{c_j}\na\mu_j\Big)\cdot\na\phi_i dxdt = 0.
$$
By the uniqueness of the limit, the claim
$\bm{\zeta}=P_L(\bm{c})R(\bm{c})\na\bm{\mu}$ follows.
\end{proof}
\section{Proof of Theorem \ref{thm.wsu}}\label{sec.wsu}
In this section, we prove the weak-strong uniqueness property.
First, we compute a combined {\em relative} energy-entropy inequality.
Then we use this inequality to derive a stability estimate, which leads
to the desired weak-strong uniqueness result.
\subsection{Evolution of the relative energy and entropy}\label{sec.wsu.formal}
We start by calculating the time evolution of the relative entropy \eqref{1.relH}
and the relative energy \eqref{1.relE} for {\em smooth} solutions $\bm{c}$ and
$\bar{\bm{c}}$. Inserting \eqref{2.B} and integrating by parts leads to
\begin{align*}
\frac{d{\mathcal H}}{dt}(\bm{c}|\bar{\bm{c}})
&= \sum_{i=1}^n\int_\Omega\bigg(\log\frac{c_i}{\bar{c}_i}\pa_t c_i
- \bigg(\frac{c_i}{\bar{c}_i}-1\bigg)\pa_t\bar{c}_i\bigg)dx \\
&= -\sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})
\na\log\frac{c_i}{\bar{c}_i}\cdot\na\mu_j dx
+ \sum_{i,j=1}^n\int_\Omega B_{ij}(\bar{\bm{c}})\na\bigg(\frac{c_i}{\bar{c}_i}
\bigg)\cdot\na\bar{\mu}_j dx \\
&= -\sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na\log\frac{c_i}{\bar{c}_i}
\cdot\na(\mu_j-\bar{\mu}_j)dx \\
&\phantom{xx}{}
- \sum_{i,j=1}^n\int_\Omega\bigg(B_{ij}(\bm{c}) - \frac{c_i}{\bar{c}_i}
B_{ij}(\bar{\bm{c}})\bigg)\na\log\frac{c_i}{\bar{c}_i}\cdot\na\bar{\mu}_j dx.
\end{align*}
Next, we compute
\begin{align}
\frac{d{\mathcal E}}{dt}(\bm{c}|\bar{\bm{c}})
&= \sum_{i=1}^n\int_\Omega\bigg(\log\frac{c_i}{\bar{c}_i}\pa_t c_i
- \bigg(\frac{c_i}{\bar{c}_i}-1\bigg)\pa_t\bar{c}_i\bigg)dx
+ \sum_{i=1}^n\int_\Omega\na(c_i-\bar{c}_i)\cdot\na\pa_t(c_i-\bar{c}_i)dx \nonumber \\
&= \sum_{i=1}^n\bigg\{\bigg(\log\frac{c_i}{\bar{c}_i} - \Delta(c_i-\bar{c}_i)\bigg)
\pa_t c_i - \bigg(\frac{c_i}{\bar{c}_i}-1-\Delta(c_i-\bar{c}_i)\bigg)\pa_t\bar{c}_i
\bigg\}dx \label{4.aux0} \\
&= -\sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)\cdot\na\mu_j dx
\nonumber \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_\Omega B_{ij}(\bar{\bm{c}})
\na\bigg(\frac{c_i}{\bar{c}_i}-1-\Delta(c_i-\bar{c}_i)\bigg)\cdot
\na\bar{\mu}_j dx. \nonumber
\end{align}
We add and subtract the expression $\sum_{i=1}^n\int_\Omega
B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)\cdot\na\bar{\mu}_j dx$:
\begin{align}
\frac{d{\mathcal E}}{dt}(\bm{c}|\bar{\bm{c}})
&= -\sum_{i=1}^n\int_\Omega B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)
\cdot\na(\mu_j-\bar{\mu}_j) dx \nonumber \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_\Omega\bigg\{B_{ij}(\bar{\bm{c}})
\bigg(\frac{c_i}{\bar{c}_i}\na\log\frac{c_i}{\bar{c}_i} - \na\Delta(c_i-\bar{c}_i)
\bigg) - B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)\bigg\}\cdot\na\bar{\mu}_j dx
\label{4.aux} \\
&= -\sum_{i,j=1}^n\int_\Omega B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)
\cdot\na(\mu_j-\bar{\mu}_j) dx \nonumber \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_\Omega\bigg(B_{ij}(\bm{c})
- \frac{c_i}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\bigg)\na(\mu_i-\bar{\mu}_i)\cdot
\na\bar{\mu}_j dx \nonumber \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_\Omega B_{ij}(\bar{\bm{c}})
\bigg(\frac{c_i}{\bar{c}_i}-1\bigg)\na\Delta(c_i-\bar{c}_i)\cdot\na\bar{\mu}_j dx.
\nonumber
\end{align}
We want to reformulate the expression $\bar{c}_i^{-1}(c_i-\bar{c}_i)
\na\Delta(c_i-\bar{c}_i)$ in the last integral.
For this, we observe that for any smooth function $f$, it holds that
\begin{align*}
f\na\Delta f &= \na(f\Delta f) - \na f\Delta f
= \na\big(\operatorname{div}(f\na f)- |\na f|^2\big)
- \operatorname{div}(\na f\otimes\na f) + \frac12\na|\na f|^2 \\
&= \na\operatorname{div}(f\na f) - \frac12\na|\na f|^2 - \operatorname{div}(\na f\otimes\na f).
\end{align*}
Therefore,
\begin{align*}
(c_i-\bar{c}_i)\na\Delta(c_i-\bar{c}_i)
&= \na\operatorname{div}\big((c_i-\bar{c}_i)\na(c_i-\bar{c}_i)\big)
- \frac12\na|\na(c_i-\bar{c}_i)|^2 \\
&\phantom{xx}{}- \operatorname{div}\big(\na(c_i-\bar{c}_i)\otimes\na(c_i-\bar{c}_i)\big).
\end{align*}
Inserting this expression into the last term of \eqref{4.aux} and integrating
by parts, we find that
\begin{align*}
\frac{d{\mathcal E}}{dt}(\bm{c}|\bar{\bm{c}})
&= -\sum_{i=1}^n\int_\Omega B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)
\cdot\na(\mu_j-\bar{\mu}_j) dx \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_\Omega\bigg(B_{ij}(\bm{c})
- \frac{c_i}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\bigg)\na(\mu_i-\bar{\mu}_i)\cdot
\na\bar{\mu}_j dx \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_\Omega (c_i-\bar{c}_i)\na(c_i-\bar{c}_i)
\cdot\na\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dx \\
&\phantom{xx}{}+ \frac12\sum_{i,j=1}^n\int_\Omega|\na(c_i-\bar{c}_i)|^2
\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dx \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_\Omega
\na(c_i-\bar{c}_i)\otimes\na(c_i-\bar{c}_i)
:\na\otimes\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dx,
\end{align*}
where $\na\otimes(\bar{c}_i^{-1}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j)$ is
a matrix with entries $\pa_{x_k}(\bar{c}_i^{-1}B_{ij}(\bar{\bm{c}})
\pa_{x_\ell}\bar{\mu}_j)$
for $k,\ell=1,\ldots,n$ and ``:'' denotes the Frobenius matrix product.
The following lemma states the rigorous result. Since we suppose that
the weak solution satisfies energy and entropy {\em inequalities}
instead of {\em equalities}, we obtain also inequalities for the relative
energy and entropy.
\begin{lemma}[Relative energy and entropy]
Let $\bm{c}$ and $\bar{\bm{c}}$ be a weak and strong solution to
\eqref{1.eq1}--\eqref{1.mu} with initial data $\bm{c}^0$ and $\bar{\bm{c}}^0$,
respectively. Assume that $\bm{c}$ satisfies the regularity \eqref{1.cregul}
and the energy and entropy inequalites \eqref{1.dEdt}--\eqref{1.dHdt}.
Furthermore, we suppose that $\bar{\bm{c}}$ is strictly positive and
satisfies the regularity
$$
\bar{\mu}_i = \log\bar{c}_i-\Delta\bar{c}_i\in L_{\rm loc}^2(0,\infty;H^2(\Omega)),
\quad \bar{c}_i\in L_{\rm loc}^\infty(0,\infty;W^{3,\infty}(\Omega)),
\quad i=1,\ldots,n.
$$
Then the following relative energy and entropy inequalities hold for any $T>0$:
\begin{align}\label{4.relE}
{\mathcal E}(&\bm{c}(T)|\bar{\bm{c}}(T))
+ \sum_{i=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)
\cdot\na(\mu_j-\bar{\mu}_j)dxdt \\
&\le {\mathcal E}(\bm{c}^0|\bar{\bm{c}}^0)
- \sum_{i,j=1}^n\int_0^T\int_\Omega\bigg(B_{ij}(\bm{c})
- \frac{c_i}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\bigg)\na(\mu_i-\bar{\mu}_i)\cdot
\na\bar{\mu}_j dxdt \nonumber \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_0^T\int_\Omega (c_i-\bar{c}_i)\na(c_i-\bar{c}_i)
\cdot\na\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt
\nonumber \\
&\phantom{xx}{}+ \frac12\sum_{i,j=1}^n\int_0^T\int_\Omega|\na(c_i-\bar{c}_i)|^2
\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt
\nonumber \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_0^T\int_\Omega
\na(c_i-\bar{c}_i)\otimes\na(c_i-\bar{c}_i)
:\na\otimes\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt,
\nonumber \\
{\mathcal H}(&\bm{c}(T)|\bar{\bm{c}}(T)) \le {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0)
- \sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})\na\log\frac{c_i}{\bar{c}_i}
\cdot\na(\mu_j-\bar{\mu}_j)dxdt \label{4.relH} \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_0^T\int_\Omega
\bigg(B_{ij}(\bm{c}) - \frac{c_i}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\bigg)
\na\log\frac{c_i}{\bar{c}_i}\cdot\na\bar{\mu}_j dxdt. \nonumber
\end{align}
\end{lemma}
The integrals in \eqref{4.relE} and \eqref{4.relH} are well defined
because of the regularity
properties for weak solutions $\bm{c}$ and the regularity assumptions
on the strong solution $\bar{\bm{c}}$.
Indeed, we have $B_{ij}(\bm{c})\na\mu_j\in L^2(Q_T)$ (see \eqref{1.regflux}),
$B_{ij}(\bm{c})\na\log c_i=2D_{ij}^{BD}(\bm{c})\sqrt{c_j}\na\sqrt{c_i}
\in L^2(Q_T)$ (see \eqref{1.EH}), and using the definition \eqref{1.B},
the assumption \eqref{1.cregul}, and Lemma \ref{lem.DB} (i), we have
$$
B_{ij}(\bm{c})\na\mu_i\cdot\na\mu_j
= D_{ij}^{BD}(\bm{c})\big(2\na\sqrt{c_i} - \sqrt{c_i}\na\Delta c_i\big)
\cdot\big(2\na\sqrt{c_j} - \sqrt{c_j}\na\Delta c_j\big)\in L^1(Q_T).
$$
\begin{proof}
The relative energy and entropy inequalities are proved from the weak
formulation of \eqref{1.eq1} by choosing suitable test functions. For this,
we observe that, by \eqref{1.weak}, $c_i-\bar{c}_i$ satisfies
\begin{align}
0 &= \int_0^\infty\int_\Omega(c_i-\bar{c}_i)\pa_t\phi_i dxdt
+ \int_\Omega ( c_i^0 (x) - \bar c_i^0 (x) ) \phi_i (x,0) dx \label{4.start} \\
&\phantom{xx}{} - \sum_{j=1}^n\int_0^\infty\int_\Omega\big(B_{ij}(\bm{c})\na\log c_j
- B_{ij}(\bar{\bm{c}})\na\log\bar{c}_j\big)\cdot\na\phi_i dxdt \nonumber \\
&\phantom{xx}{}- \sum_{j=1}^n\int_0^\infty\int_\Omega
\Big(\operatorname{div} \big (B_{ij}(\bm{c})\na\phi_i \big) \Delta c_j
- \operatorname{div} \big ( B_{ij}(\bar{c})\na\phi_i \big )
\Delta\bar{c}_j\Big)dxdt. \nonumber
\end{align}
By density, this formulation also holds for $\phi_i=\bar{\mu}_i\theta_\eps(t)$, where
$$
\theta_\eps(t) = \left\{\begin{array}{ll}
1 &\quad\mbox{for }0\le t\le T, \\
(T-t)/\eps + 1 &\quad\mbox{for }T<t<T+\eps, \\
0 &\quad\mbox{for }t\ge T+\eps.
\end{array}\right.
$$
Then, passing to the limit $\eps\to 0$ and summing over $i=1,\ldots,n$, we arrive at
\begin{align*}
\sum_{i=1}^n&\int_\Omega(c_i-\bar{c}_i)\bar{\mu}_i dx\Bigg|_0^T
= \sum_{i=1}^n\int_0^T\langle \pa_t\bar{\mu}_i,c_i-\bar{c}_i\operatorname{ran}gle dt \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_0^T\int_\Omega\big(B_{ij}(\bm{c})
\na\log c_j\cdot\na\bar{\mu}_i
+ \operatorname{div}(B_{ij}(\bm{c})\na\bar{\mu}_i)\Delta c_j\big)dxdt \\
&\phantom{xx}{}+\sum_{i,j=1}^n\int_0^T\int_\Omega\big(B_{ij}(\bar{\bm{c}})
\na\log\bar{c}_j\cdot\na\bar{\mu}_i
+ \operatorname{div}(B_{ij}(\bar{\bm{c}})\na\bar{\mu}_i)\Delta\bar{c}_j\big)dxdt \\
&=: I_7 + I_8 + I_9,
\end{align*}
where $\langle\cdot,\cdot\operatorname{ran}gle$ is the duality bracket between $H^1(\Omega)'$ and
$H^1(\Omega)$. This product is well defined ,since it holds
in the sense of $H^1(\Omega)'$ that
$$
\pa_t\bar{\mu}_i = \pa_t(\log\bar{c}_i - \Delta\bar{c}_i)
= \sum_{j=1}^n\frac{1}{\bar{c}_i}\operatorname{div}(B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j)
- \sum_{j=1}^n\Delta\operatorname{div}(B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j).
$$
Inserting this expression into $I_7$, the dual
product can be written as an integral:
\begin{align*}
I_7 &= -\sum_{i,j=1}^n\int_0^T\int_\Omega\bigg(B_{ij}(\bar{\bm{c}})
\na\bigg(\frac{c_i}{\bar{c}_i}-1\bigg)\cdot\na\bar{\mu}_j
+ \Delta(c_i-\bar{c}_i)\operatorname{div}(B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j)\bigg)dxdt \\
&= -\sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bar{\bm{c}})
\na\bigg(\frac{c_i}{\bar{c}_i}-1\bigg)\cdot\na\bar{\mu}_j dxdt \\
&\phantom{xx}{} - \sum_{i,j=1}^n\int_0^T\int_\Omega\bar{c}_i\Delta(c_i-\bar{c}_i)
\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_0^T\int_\Omega\frac{1}{\bar{c}_i}
B_{ij}(\bar{\bm{c}})\Delta(c_i-\bar{c}_i)\na\bar{c}_i\cdot\na\bar{\mu}_j dxdt.
\end{align*}
Replacing $\Delta c_j$ by $\log c_j-\mu_j$ in $I_8$ and integrating by parts
in the term involving the divergence, some terms cancel and we find that
\begin{align*}
I_8 &= -\sum_{i,j=1}^n\int_0^T\int_\Omega\big( B_{ij}(\bm{c})
\na\bar{\mu}_i\cdot\na\log c_j + \operatorname{div}(B_{ij}(\bm{c})\na\bar{\mu}_i)
(\log c_j-\mu_j)\big)dxdt \\
&=-\sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})\na\bar{\mu}_i\cdot\na\mu_j dxdt.
\end{align*}
Assumption \eqref{1.cregul} guarantees that the flux
has the regularity $\sum_{j=1}^n B_{ij}(\bm{c})\na\mu_j\in L^2(Q_T)$
such that the last integral is defined.
The remaining term $I_9$ is reformulated in a similar way, leading to
$$
I_9 = \sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bar{\bm{c}})\na\bar{\mu}_i
\cdot\na\bar{\mu}_j dxdt.
$$
It follows from the definition of the relative energy, the inequality \eqref{1.dEdt}
for ${\mathcal E}(\bm{c})$, and the identity \eqref{1.dEdtbar} for ${\mathcal E}(\bar{\bm{c}})$ that
\begin{align*}
{\mathcal E}(&\bm{c}(T)|\bar{\bm{c}}(T)) - {\mathcal E}(\bm{c}^0|\bar{\bm{c}}^0) \\
&= \big({\mathcal E}(\bm{c}(T))-{\mathcal E}(\bm{c}^0)\big)
- \big({\mathcal E}(\bar{\bm{c}}(T))-{\mathcal E}(\bar{\bm{c}}^0)\big)
- \int_\Omega\bar{\bm{\mu}}\cdot(\bm{c}-\bar{\bm{c}})dx \Big|_0^T \\
&\le -\sum_{i,j=1}^n\int_0^T\int_\Omega\big(B_{ij}(\bm{c})\na\mu_i\cdot\na\mu_j
- B_{ij}(\bar{\bm{c}})\na\bar{\mu}_i\cdot\na\bar{\mu}_j\big)dxdt
- (I_7 + I_8 + I_9) \\
&= -\sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)
\cdot\na\mu_j dxdt \\
&\phantom{xx}{}-\sum_{i,j=1}^n\int_0^T\int_\Omega B_{ij}(\bar{\bm{c}})
\na\bigg(\frac{c_i}{\bar{c}_i}-1\bigg)\cdot\na\bar{\mu}_j dxdt \\
&\phantom{xx}{} - \sum_{i,j=1}^n\int_0^T\int_\Omega\bar{c}_i\Delta(c_i-\bar{c}_i)
\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_0^T\int_\Omega\frac{1}{\bar{c}_i}
B_{ij}(\bar{\bm{c}})\Delta(c_i-\bar{c}_i)\na\bar{c}_i\cdot\na\bar{\mu}_j dxdt.
\end{align*}
This inequality is just a reformulation of \eqref{4.aux0}, which leads,
by proceeding as in \eqref{4.aux} and the subsequent calculations, to \eqref{4.relE}.
Next, we verify the relative entropy inequality. Taking the test function
$\phi_i=(\log\bar{c}_i)\theta_\eps (t)$ in \eqref{4.start}, passing to the limit
$\eps\to 0$, and summing over $i=1,\ldots,n$ leads to
\begin{align*}
\sum_{i=1}^n&\int_\Omega(c_i-\bar{c}_i)\log\bar{c}_i dx\bigg|_0^T
= \sum_{i=1}^n\int_0^T \int_\Omega(c_i-\bar{c}_i)\pa_t(\log\bar{c}_i)dxdt \\
&\phantom{xx}{} - \sum_{j=1}^n\int_0^\infty\int_\Omega\big(B_{ij}(\bm{c})\na\log c_j
- B_{ij}(\bar{\bm{c}})\na\log\bar{c}_j\big)\cdot\na \log \bar c_i dxdt \nonumber \\
&\phantom{xx}{}- \sum_{j=1}^n\int_0^\infty\int_\Omega
\Big(\operatorname{div} \big (B_{ij}(\bm{c})\na \log \bar c_i \big) \Delta c_j
- \operatorname{div} \big ( B_{ij}(\bar{c})\na \log \bar c_i \big )
\Delta\bar{c}_j\Big)dxdt. \nonumber
\end{align*}
This yields, together with \eqref{1.dHdt}, \eqref{1.dHdtbar}, an integration by
parts, and regularity assumption \eqref{1.cregul}, that
\begin{align*}
{\mathcal H}(&\bm{c}(T)|\bar{\bm{c}}(T)) - {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0) \\
&= \big({\mathcal H}(\bm{c}(T))-{\mathcal H}(\bm{c}^0)\big) - \big({\mathcal H}(\bar{\bm{c}}(T))-{\mathcal H}(\bar{\bm{c}}^0)
\big) - \int_\Omega(\bm{c}-\bar{\bm{c}})\cdot \log\bar{\bm{c}}\,dx\bigg|_0^T
\\
&\le -\sum_{i,j=1}^n\int_0^T\int_\Omega \Big ( B_{ij}(\bm{c}) \na \log c_i
\cdot \na \mu_j - B_{i j}(\bar{\bm{c}}) \na \log \bar c_i \cdot
\nabla \bar \mu_j \Big ) dx dt \\
&\phantom{xx}{} - \sum_{i=1}^n\int_0^T \int_\Omega(c_i-\bar{c}_i)
\pa_t(\log\bar{c}_i)dxdt \\
&\phantom{xx}{} + \sum_{i,j=1}^n\int_0^\infty\int_\Omega
\Big ( B_{ij}(\bm{c}) \na \mu_j \cdot \na \log \bar c_i
- B_{ij}(\bar{\bm{c}}) \na \bar \mu_j \cdot \na \log \bar c_i \big )
\Big)dxdt. \nonumber \\
&= -\sum_{i,j=1}^n\int_0^T\int_\Omega
\Big ( B_{ij}(\bm{c}) \na\mu_j \cdot \nabla \bigg ( \log \frac{c_i}{\bar c_i} \bigg )
- \nabla \bigg ( \frac{c_i}{\bar c_i} - 1 \bigg ) \cdot B_{i j}(\bar{\bm{c}})
\nabla \bar \mu_j \Big ) dx dt,
\end{align*}
which readily gives \eqref{4.relH}.
\end{proof}
\subsection{Proof of the weak-strong uniqueness property}\label{sec.wsu.rig}
We proceed with the proof of Theorem \ref{thm.wsu}. First, we estimate the
relative entropy inequality \eqref{4.relH} and then the relative energy
inequality \eqref{4.relE}. A combination of both estimates shows \eqref{1.comb},
which proves the weak-strong uniqueness property.
{\em Step 1: Estimating the relative entropy.} As in the proof of Lemma \ref{lem.fei},
we decompose the matrix $B(\bm{c})$ by setting $M(\bm{c}):=B(\bm{c})-\lambda G(\bm{c})$
such that $B(\bm{c}) = M(\bm{c}) + \lambda G(\bm{c})$, where
$G(\bm{c})=R(\bm{c})P_L(\bm{c})R(\bm{c})$ has the entries
$G_{ij}(\bm{c}) = c_i\delta_{ij}-c_ic_j$ and $0<\lambda<\lambda_m$.
In terms of these matrices, we can formulate \eqref{4.relH} as
\begin{align}\label{4.HGM}
{\mathcal H}(&\bm{c}(T)|\bar{\bm{c}}(T)) - {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0)
\le -\sum_{i,j=1}^n\int_0^T\int_\Omega M_{ij}(\bm{c})\na\log\frac{c_i}{\bar{c}_i}\cdot
\na(\mu_j-\bar{\mu}_j)dxdt \\
&{}- \lambda\sum_{i,j=1}^n\int_0^T\int_\Omega G_{ij}(\bm{c})
\na\log\frac{c_i}{\bar{c}_i}\cdot\na(\mu_j-\bar{\mu}_j)dxdt \nonumber \\
&{}- \sum_{i,j=1}^n\int_0^T\int_\Omega\bigg(B_{ij}(\bm{c})
- \frac{c_i}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\bigg)
\na\log\frac{c_i}{\bar{c}_i}\cdot\na\bar{\mu}_j dxdt
=: I_{10} + I_{11} + I_{12}. \nonumber
\end{align}
{\em Step 1a: Estimate of $I_{10}$.}
We know from \eqref{psd.zMz} and \eqref{3.zMz} that $M(\bm{c})$
is positive semidefinite and satisfies
$\bm{z}^T M(\bm{c})\bm{z}\le (\lambda_M-\lambda)|P_L(\bm{c})R(\bm{c})\bm{z}|^2$ for all
$\bm{z}\in{\mathbb R}^n$. Therefore, using Young's inequality with $\theta>0$,
\begin{align}\label{4.I10}
I_{10} &\le \frac{\theta}{4}\sum_{i,j=1}^n\int_0^T\int_\Omega M_{ij}(\bm{c})
\na\log\frac{c_i}{\bar{c}_i}\cdot\na\log\frac{c_j}{\bar{c}_j} dxdt \\
&\phantom{xx}{}+ \frac{1}{\theta}\sum_{i,j=1}^n\int_0^T\int_\Omega M_{ij}(\bm{c})
\na(\mu_i-\bar{\mu}_i)\cdot \na(\mu_j-\bar{\mu}_j)dxdt \nonumber \\
&\le \frac{\theta}{4}(\lambda_M-\lambda)\sum_{i=1}^n\int_0^T\int_\Omega
\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_i}{\bar{c}_i}\bigg|^2
dxdt \nonumber \\
&\phantom{xx}{}+ \frac{1}{\theta}(\lambda_M-\lambda)\sum_{i=1}^n\int_0^T\int_\Omega
\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt.
\nonumber
\end{align}
{\em Step 1b: Estimate of $I_{11}$.}
In the term $I_{11}$, we replace $\mu_j-\bar{\mu}_j$ by $\log(c_j/\bar{c}_j)
-\Delta(c_j-\bar{c}_j)$ and compute both terms in the difference separately.
The definition $G_{ij}(\bm{c})=\sqrt{c_i}P_L(\bm{c})_{ij}\sqrt{c_j}$ and
the property $P_L(\bm{c})^2=P_L(\bm{c})$ lead to
\begin{align}\label{4.G1}
\sum_{i,j=1}^n&\int_0^T\int_\Omega G_{ij}(\bm{c})\na\log\frac{c_i}{\bar{c}_i}
\cdot\na\log\frac{c_j}{\bar{c}_j}dxdt \\
&= \sum_{i,j=1}^n\int_0^T\int_\Omega\sqrt{c_i} P_L(\bm{c})_{ij}\sqrt{c_j}
\na\log\frac{c_i}{\bar{c}_i}\cdot\na\log\frac{c_j}{\bar{c}_j}dxdt \nonumber \\
&= \sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}
\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt. \nonumber
\end{align}
Furthermore, we use $G_{ij}(\bm{c}) = c_i\delta_{ij}-c_ic_j$ and integration by
parts to find that
\begin{align*}
\sum_{i,j=1}^n&\int_0^T\int_\Omega
G_{ij}(\bm{c})\na\log\frac{c_i}{\bar{c}_i}\cdot\na\Delta(c_j-\bar{c}_j)dx dt \\
&= -\sum_{i,j=1}^n\int_0^T\int_\Omega\operatorname{div}\bigg(( c_i\delta_{ij}-c_ic_j)
\na\log\frac{c_i}{\bar{c}_i}\bigg)\Delta(c_j-\bar{c}_j)dxdt \\
&= -\sum_{i=1}^n\int_0^T\int_\Omega\operatorname{div}(\na c_i-c_i\na\log\bar{c}_i)
\Delta(c_i-\bar{c}_i)dxdt \\
&\phantom{xx}{}+ \sum_{i,j=1}^n\int_0^T\int_\Omega\operatorname{div}(c_j\na c_i-c_ic_j
\na\log\bar{c}_i)\Delta(c_j-\bar{c}_j)dxdt \\
&= -\sum_{i,j=1}^n\int_0^T\int_\Omega\operatorname{div}(\na c_i-c_i\na\log\bar{c}_i)
\Delta(c_i-\bar{c}_i)dxdt \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_0^T\int_\Omega\operatorname{div}(c_ic_j
\na\log\bar{c}_i)\Delta(c_j-\bar{c}_j)dxdt,
\end{align*}
where we used $\sum_{i=1}^n c_j\na c_i=0$ in the last step.
We mention that $\sum_{j=1}^n G_{ij}(\bm{c})\na\Delta c_j\in L^2(Q_T)$
because of \eqref{1.regc}, so the first integral in the previous computation
is well defined. It follows from $\Delta c_i\Delta(c_i-\bar{c}_i)
= (\Delta(c_i-\bar{c}_i))^2 + \Delta\bar{c}_i\Delta(c_i-\bar{c}_i)$ that
\begin{align}\label{4.G2}
\sum_{i,j=1}^n&\int_0^T\int_\Omega G_{ij}(\bm{c})\na\log\frac{c_i}{\bar{c}_i}
\cdot\na\Delta(c_i-\bar{c}_i) dxdt
= -\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt \\
&\phantom{xx}{}
- \sum_{i=1}^n\int_0^T\int_\Omega\operatorname{div}(\na\bar{c}_i-c_i\na\log\bar{c}_i)
\Delta(c_i-\bar{c}_i)dxdt \nonumber \\
&\phantom{xx}{}- \sum_{i,j=1}^n\int_0^T\int_\Omega\operatorname{div}(c_ic_j\na\log\bar{c}_i)
\Delta(c_j-\bar{c}_j)dxdt. \nonumber
\end{align}
We multiply \eqref{4.G1} by $-\lambda$ and \eqref{4.G2} by $\lambda$ and
sum both expressions to find that
\begin{align}\label{4.I11}
I_{11} &= -\lambda\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}
\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt
- \lambda\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt \\
&\phantom{xx}{}- \lambda\sum_{i=1}^n\int_0^T\int_\Omega
\operatorname{div}(\na\bar{c}_i-c_i\na\log\bar{c}_i)\Delta(c_i-\bar{c}_i)dxdt \nonumber \\
&\phantom{xx}{}- \lambda\sum_{i,j=1}^n\int_0^T\int_\Omega
\operatorname{div}(c_ic_j\na\log\bar{c}_i)\Delta(c_j-\bar{c}_j)dxdt. \nonumber
\end{align}
We apply Young's inequality to the last two terms. The third term in \eqref{4.I11}
becomes
\begin{align*}
-&\lambda\sum_{i=1}^n\int_0^T\int_\Omega
\operatorname{div}(\na\bar{c}_i-c_i\na\log\bar{c}_i)\Delta(c_i-\bar{c}_i)dxdt \\
&\le \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt
+ \lambda\sum_{i=1}^n\int_0^T\int_\Omega|\operatorname{div}((c_i-\bar{c}_i)\na\log\bar{c}_i)|^2
dxdt \\
&\le \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2
dxdt \\
&\phantom{xx}{}+ \lambda\sum_{i=1}^n\|\na\log\bar{c}_i\|_{L^\infty(Q_T)}
\int_0^T\int_\Omega|\na(c_i-\bar{c}_i)|^2 dxdt \\
&\phantom{xx}{}+ \lambda\sum_{i=1}^n
\|\Delta\log\bar{c}_i\|_{L^\infty(Q_T)}
\int_0^T\int_\Omega(c_i-\bar{c}_i)^2 dxdt \\
&\le \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt
\\
&\phantom{xx}{}+ \lambda C\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2+|\na(c_i-\bar{c}_i)|^2\big)dxdt,
\end{align*}
where the constant $C>0$ depends on the $L^\infty$ norms of $\na\log\bar{\bm{c}}$
and $\Delta\log\bar{\bm{c}}$. Next, for the fourth term in \eqref{4.I11},
\begin{align*}
-\lambda&\sum_{i,j=1}^n\int_0^T\int_\Omega
\operatorname{div}(c_ic_j\na\log\bar{c}_i)\Delta(c_j-\bar{c}_j)dxdt \\
&\le \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt
+ \lambda\sum_{j=1}^n\int_0^T\int_\Omega\bigg|\sum_{i=1}^n\operatorname{div}(c_ic_j
\na\log\bar{c}_i)\bigg|^2 dxdt.
\end{align*}
We estimate the integrand of the last term, taking into account that
$\na\sum_{i=1}^n\bar{c}_i\na\log\bar{c}_i=\sum_{i=1}^n\na\bar{c}_i=0$:
\begin{align*}
\sum_{i=1}^n&\operatorname{div}(c_ic_j\na\log\bar{c}_i) = \sum_{i=1}^n\operatorname{div}
\big((c_i-\bar{c}_i)c_j\na\log\bar{c}_i\big) \\
&= \sum_{i=1}^n c_j\operatorname{div}((c_i-\bar{c}_i)\na\log\bar{c}_i)
+ \sum_{i=1}^n(c_i-\bar{c}_i)\na\log\bar{c}_i\cdot\na c_j \\
&= \sum_{i=1}^n c_j\operatorname{div}((c_i-\bar{c}_i)\na\log\bar{c}_i)
+ \sum_{i=1}^n c_i\na\log\bar{c}_i\cdot\na (c_j-\bar{c}_j)
+ \sum_{i=1}^n(c_i-\bar{c}_i)\na\log\bar{c}_i\cdot\na\bar{c}_j
\\
&\le C\sum_{i=1}^n\big(|c_i-\bar{c}_i| + |\na(c_i-\bar{c}_i)|\big),
\end{align*}
where $C>0$ depends on the $L^\infty$ norms of $\na\log\bar{\bm{c}}$ and
$\Delta\log\bar{\bm{c}}$. This yields
\begin{align*}
-\lambda&\sum_{i,j=1}^n\int_0^T\int_\Omega
\operatorname{div}(c_ic_j\na\log\bar{c}_i)\Delta(c_j-\bar{c}_j)dxdt \\
&\le \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt
+ \lambda C\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt.
\end{align*}
Using these estimates in \eqref{4.I11}, we arrive at
\begin{align}\label{4.I11final}
I_{11} &\le -\lambda\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt
- \frac{\lambda}{2}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2 dxdt \\
&\phantom{xx}{}+ \lambda C\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt. \nonumber
\end{align}
{\em Step 1c: Estimate of $I_{12}$.}
By definition of $B_{ij}(\bm{c})$ and Young's inequality with $\theta'>0$,
\begin{align*}
I_{12} &= -\sum_{i,j=1}^n\int_0^T\int_\Omega\sqrt{c_i}\bigg(
D_{ij}^{BD}(\bm{c})\sqrt{c_j} - \sqrt{\frac{c_i}{\bar{c}_i}}D_{ij}^{BD}(\bar{\bm{c}})
\sqrt{\bar{c}_j}\bigg)\na\log\frac{c_i}{\bar{c}_i}\cdot\na\bar{\mu}_j dxdt \\
&\le \frac{\theta'}{4}\sum_{i=1}^n\int_0^T\int_\Omega c_i\bigg|\na\log
\frac{c_i}{\bar{c}_i}\bigg|^2 dxdt \\
&\phantom{xx}{}+ \frac{n}{\theta'}\sum_{i,j=1}^n\int_0^T\int_\Omega\bigg(
D_{ij}^{BD}(\bm{c})\sqrt{c_j} - \sqrt{\frac{c_i}{\bar{c}_i}}
D_{ij}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}\bigg)^2|\na\bar{\mu}_j|^2 dxdt.
\end{align*}
The bracket of the second term can be estimated according to
\begin{align}
\bigg|D_{ij}^{BD}&(\bm{c})\sqrt{c_j} - \sqrt{\frac{c_i}{\bar{c}_i}}
D_{ij}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}\bigg| \label{4.estD} \\
&= \bigg|D_{ij}^{BD}(\bm{c})\sqrt{c_j} - D_{ij}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}
- \frac{\sqrt{c_i}-\sqrt{\bar{c}_i}}{\sqrt{\bar{c}_i}}D_{ij}^{BD}(\bar{\bm{c}})
\sqrt{\bar{c}_j}\bigg| \nonumber \\
&\le \frac{C}{\sqrt{m}}\sum_{i=1}^n\big(|c_i-\bar{c}_i|
+ |\sqrt{c_i}-\sqrt{\bar{c}_i}|\big)
\le C(m)\sum_{i=1}^n|c_i-\bar{c}_i|, \nonumber
\end{align}
using the assumption $\bar{c}_i\ge m>0$ and the boundedness of $D_{ij}^{BD}$
(see Lemma \ref{lem.DB} (i)). It follows that
\begin{equation}
I_{12} \le \frac{\theta'}{4}\sum_{i=1}^n\int_0^T\int_\Omega c_i\bigg|\na\log
\frac{c_i}{\bar{c}_i}\bigg|^2 dxdt + C(m,\theta')\sum_{i=1}^n
\int_0^T\int_\Omega(c_i-\bar{c}_i)^2 dxdt. \label{4.I12}
\end{equation}
{\em Step 1d: Combining the estimates.}
We deduce from \eqref{4.HGM}, after inserting estimates \eqref{4.I10},
\eqref{4.I11final}, and \eqref{4.I12} for $I_{10}$, $I_{11}$, and $I_{12}$,
respectively, that
\begin{align}
{\mathcal H}(&\bm{c}(T)|\bar{\bm{c}}(T)) \le {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0) \nonumber \\
&{}+ \bigg(\frac{\theta}{4}(\lambda_M-\lambda) - \lambda\bigg)\sum_{i=1}^n\int_0^T
\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}
\bigg|^2 dxdt \label{4.I012} \\
&{}+ \frac{\lambda_M-\lambda}{\theta}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|
\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt
\nonumber \\
&{}- \frac{\lambda}{2}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2dxdt
+ \lambda C\sum_{i=1}^n\int_0^T\int_\Omega\big((c_i-\bar{c}_i)^2
+ |\na(c_i-\bar{c}_i)|^2\big)dxdt \nonumber \\
&{}+ \frac{\theta'}{4}\sum_{i=1}^n\int_0^T\int_\Omega c_i\bigg|\na\log
\frac{c_i}{\bar{c}_i}\bigg|^2 dxdt + C(m,\theta')\sum_{i=1}^n
\int_0^T\int_\Omega(c_i-\bar{c}_i)^2 dxdt. \nonumber
\end{align}
The last but one term on the right-hand side still needs to be estimated.
To this end, we decompose $I=P_L(\bm{c})+P_{L^\perp}(\bm{c})$:
$$
\sum_{i=1}^n c_i\bigg|\na\log\frac{c_i}{\bar{c}_i}\bigg|^2
= \sum_{i=1}^n\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}
\na\log\frac{c_j}{\bar{c}_j}\bigg|^2
+ \sum_{i=1}^n\bigg|\sum_{j=1}^n P_{L^\perp}(\bm{c})_{ij}\sqrt{c_j}
\na\log\frac{c_j}{\bar{c}_j}\bigg|^2.
$$
The first term on the right-hand side can be absorbed for sufficiently
small $\theta'>0$ by the second term of the left-hand side of \eqref{4.I012}.
For the other term, we use the definition $P_{L^\perp}(\bm{c})_{ij}=\sqrt{c_ic_j}$
and $\sum_{j=1}^n\na c_j=\sum_{j=1}^n\na\bar{c}_j=0$:
$$
\sum_{j=1}^n P_{L^\perp}(\bm{c})_{ij}\sqrt{c_j}
\na\log\frac{c_j}{\bar{c}_j} = \sqrt{c_i}\sum_{j=1}^n c_j
\na\log\frac{c_j}{\bar{c}_j}
= \sqrt{c_i}\sum_{j=1}^n (c_j-\bar{c}_j)\na\log\bar{c}_j.
$$
This gives
\begin{align}\label{4.clogcc}
\sum_{i=1}^n\int_0^T\int_\Omega c_i\bigg|\na\log
\frac{c_i}{\bar{c}_i}\bigg|^2 dxdt
&\le \sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}
\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt \\
&\phantom{xx}{}+ \sum_{j=1}^n\|\na\log\bar{c}_j\|_{L^\infty(Q_T)}
\int_0^T\int_\Omega(c_i-\bar{c}_i)^2 dxdt. \nonumber
\end{align}
Hence, choosing $\theta=\lambda/(\lambda_M-\lambda)$ and $\theta'=\lambda$,
we conclude from \eqref{4.I012} that
\begin{align}\label{4.Hfinal}
{\mathcal H}(&\bm{c}(T)|\bar{\bm{c}}(T)) + \frac{\lambda}{2}\sum_{i=1}^n\int_0^T
\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}
\bigg|^2 dxdt \\
&\phantom{xx}{}+ \frac{\lambda}{2}\sum_{i=1}^n\int_0^T\int_\Omega
(\Delta(c_i-\bar{c}_i))^2dxdt \nonumber \\
& \le {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0) + \frac{(\lambda_M-\lambda)^2}{\lambda}
\sum_{i=1}^n\int_0^T\int_\Omega\bigg|
\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt
\nonumber \\
&\phantom{xx}{}+ C\sum_{i=1}^n\int_0^T\int_\Omega\big((c_i-\bar{c}_i)^2
+ |\na(c_i-\bar{c}_i)|^2\big)dxdt. \nonumber
\end{align}
We show in the next step that the second term on the right-hand side can be
estimated by the relative energy inequality.
{\em Step 2: Estimating the relative energy.} We start from the relative energy
inequality \eqref{4.relE}. Observing that due to Lemma \ref{lem.DB} (ii),
\begin{align*}
\sum_{i,j=1}^n B_{ij}(\bm{c})\na(\mu_i-\bar{\mu}_i)\cdot\na(\mu_j-\bar{\mu}_j)
&= \sum_{i,j=1}^n D_{ij}^{BD}(\bm{c})\big(\sqrt{c_i}\na(\mu_i-\bar{\mu}_i)\big)
\cdot\big(\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\big) \\
&\ge \lambda_m\sum_{i=1}^n\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}
\na(\mu_j-\bar{\mu}_j)\bigg|^2,
\end{align*}
inequality \eqref{4.relE} becomes
\begin{align}\label{4.aux2}
& {\mathcal E}(\bm{c}(T)|\bar{\bm{c}}(T))
+ \lambda_m\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}
\na(\mu_j-\bar{\mu}_j)\bigg|^2dxdt \\
&\phantom{xx}{}\le {\mathcal E}(\bm{c}^0|\bar{\bm{c}}^0) + I_{13} + I_{14} + I_{15} + I_{16},
\qquad\mbox{where} \nonumber \\
& I_{13} = -\sum_{i,j=1}^n\int_0^T\int_\Omega\bigg(B_{ij}(\bm{c})
- \frac{c_i}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\bigg)\na(\mu_i-\bar{\mu}_i)\cdot
\na\bar{\mu}_j dxdt, \nonumber \\
& I_{14} = \sum_{i,j=1}^n\int_0^T\int_\Omega(c_i-\bar{c}_i)\na(c_i-\bar{c}_i)
\cdot\na\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt,
\nonumber \\
& I_{15} = \frac12\sum_{i,j=1}^n\int_0^T\int_\Omega|\na(c_i-\bar{c}_i)|^2
\operatorname{div}\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt,
\nonumber \\
& I_{16} = \sum_{i,j=1}^n\int_0^T\int_\Omega
\na(c_i-\bar{c}_i)\otimes\na(c_i-\bar{c}_i)
:\na\bigg(\frac{1}{\bar{c}_i}B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j\bigg)dxdt.
\nonumber
\end{align}
The terms $I_{14}$, $I_{15}$, and $I_{16}$ can be estimated directly by
using the regularity assumption
$\na\operatorname{div}((1/\bar{c}_i)B_{ij}(\bar{\bm{c}})\na\bar{\mu}_j)\in L^\infty(Q_T)$:
\begin{equation}\label{4.I1456}
I_{14}+I_{15}+I_{16} \le C\sum_{i=1}^n\int_0^T\int_\Omega\big(
(c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt.
\end{equation}
The estimate for $I_{13}$ is more involved. First, we use the definition of
$B(\bm{c})$ and decompose $I=P_L(\bm{c})+P_{L^\perp}(\bm{c})$. Then
\begin{align*}
& I_{13} = \sum_{i,j=1}^n\int_0^T\int_\Omega\sqrt{c_i} E_{ij}(\bm{c},\bar{\bm{c}})
\na(\mu_i-\bar{\mu}_i)\cdot\na\bar{\mu}_j dxdt
=: I_{131} + I_{132}, \quad\mbox{where} \\
& E_{ij}(\bm{c},\bar{\bm{c}}) = D_{ij}^{BD}(\bm{c})\sqrt{c_j}
- \sqrt{\frac{c_i}{\bar{c}_i}}D_{ij}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}, \\
& I_{131} = \sum_{i,j,k,\ell=1}^n\int_0^T\int_\Omega
P_L(\bm{c})_{i\ell}E_{\ell j}(\bm{c},\bar{\bm{c}})
P_L(\bm{c})_{ik}\sqrt{c_k}\na(\mu_k-\bar{\mu}_k)\cdot\na\bar{\mu}_j dxdt, \\
& I_{132} = \sum_{i,j,k,\ell=1}^n\int_0^T\int_\Omega
P_{L^\perp}(\bm{c})_{i\ell}E_{\ell j}(\bm{c},\bar{\bm{c}})
P_{L^\perp}(\bm{c})_{ik}\sqrt{c_k}\na(\mu_k-\bar{\mu}_k)\cdot
\na\bar{\mu}_j dxdt.
\end{align*}
For $I_{131}$, it is sufficient to apply Young's inequality and to use
estimate \eqref{4.estD} for $E_{ij}(\bm{c},\bar{\bm{c}})$:
\begin{align}\label{4.I131}
I_{131} &\le \frac{\lambda_m}{2}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt \\
&\phantom{xx}{}+ \frac{n}{2\lambda_m}\sum_{i,j=1}^n\int_0^T\int_\Omega
|E_{ij}(\bm{c},\bar{\bm{c}})|^2|\na\bar{\mu}_j|^2 dxdt \nonumber \\
&\le \frac{\lambda_m}{2}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt \nonumber \\
&\phantom{xx}{}+ C(m)\sum_{i=1}^n\int_0^T\int_\Omega(c_i-\bar{c}_i)^2 dxdt,
\nonumber
\end{align}
where $C(m)>0$ depends on $m$, $n$, $\lambda_m$, and the $L^\infty(Q_T)$ norm
of $\na\bar{\bm{\mu}}$.
For $I_{132}$, we observe that the property $\operatorname{ran} D^{BD}(\bm{c})=L(\bm{c})$,
which follows from Lemma \ref{lem.Dz}, implies that
$P_{L^\perp}(\bm{c})D^{BD}(\bm{c})\bm{z}=\bm{0}$ for all $\bm{z}\in{\mathbb R}^n$.
Hence,
$$
\sum_{\ell=1}^n P_{L^\perp}(\bm{c})_{i\ell}E_{\ell j}(\bm{c},\bar{\bm{c}})
= -\sum_{\ell=1}^n P_{L^\perp}(\bm{c})_{i\ell}\sqrt{\frac{c_\ell}{\bar{c}_\ell}}
D_{\ell j}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}.
$$
We infer from the definitions
$P_{L^\perp}(\bm{c})_{ik}=\sqrt{c_ic_k}$ and
$\mu_k-\bar{\mu}_k=\log(c_k/\bar{c}_k)-\Delta(c_k-\bar{c}_k)$ that
\begin{align}\label{4.I132}
I_{132} &= -\sum_{i,j,k,\ell=1}^n\int_0^T\int_\Omega P_{L^\perp}(\bm{c})_{ik}
\sqrt{c_k}P_{L^\perp}(\bm{c})_{i\ell}\sqrt{\frac{c_\ell}{\bar{c}_\ell}}
D_{\ell j}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}\na(\mu_k-\bar{\mu}_k)
\cdot\na\bar{\mu}_j dxdt \\
&= -\sum_{j,k,\ell=1}^n\int_0^T\int_\Omega\sum_{i=1}^n c_ic_k
\frac{c_\ell}{\sqrt{\bar{c}_\ell}}D_{\ell j}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}
\na(\mu_k-\bar{\mu}_k)\cdot\na\bar{\mu}_j dxdt \nonumber \\
&= -\sum_{j,k,\ell=1}^n\int_0^T\int_\Omega c_k
\frac{c_\ell-\bar{c}_\ell}{\sqrt{\bar{c}_\ell}}D_{\ell j}^{BD}(\bar{\bm{c}})
\sqrt{\bar{c}_j}\na\log\frac{c_k}{\bar{c}_k}\cdot\na\bar{\mu}_j dxdt \nonumber \\
&\phantom{xx}{}- \sum_{j,k,\ell=1}^n\int_0^T\int_\Omega \operatorname{div}\bigg(c_k
\frac{c_\ell-\bar{c}_\ell}{\sqrt{\bar{c}_\ell}}D_{\ell j}^{BD}(\bar{\bm{c}})
\sqrt{\bar{c}_j}\na\bar{\mu}_j\bigg)\Delta(c_k-\bar{c}_k) dxdt \nonumber \\
&=: J_1 + J_2, \nonumber
\end{align}
where we added the expression $-\sum_{\ell=1}^n\sqrt{\bar{c}_\ell}
D_{\ell j}^{BD}(\bar{\bm{c}}) = 0$, which follows from
$\ker D^{BD}(\bar{\bm{c}})=L^\perp(\bar{\bm{c}})=\operatorname{span}
\{\sqrt{\bar{\bm{c}}}\}$ (see Lemma \ref{lem.DB}) and the symmetry of
$D^{BD}(\bar{\bm{c}})$ (see Lemma \ref{lem.Dz}), and we integrated by
parts in the last integral.
To estimate $J_1$, we use Young's inequality with $\theta>0$,
Lemma \ref{lem.DB} (iii), and \eqref{4.clogcc}:
\begin{align*}
J_1 &\le \frac{\theta}{4}\sum_{k=1}^n\int_0^T\int_\Omega c_k\bigg|
\na\log\frac{c_k}{\bar{c}_k}\bigg|^2 dxdt \\
&\phantom{xx}{}+ \frac{n}{\theta}\sum_{j,k,\ell=1}^n\int_0^T\int_\Omega
(c_\ell-\bar{c}_\ell)^2\frac{c_k}{\bar{c}_\ell}D_{\ell j}^{BD}(\bar{\bm{c}})^2
\bar{c}_j|\na\bar{\mu}_j|^2 dxdt \\
&\le \frac{\theta}{4}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt
+ C\theta\sum_{i=1}^n\int_0^T\int_\Omega(c_i-\bar{c}_i)^2 dxdt \\
&\phantom{xx}{}+ \frac{C}{\theta}\sum_{\ell=1}^n\int_0^T\int_\Omega
(c_\ell-\bar{c}_\ell)^2 dxdt,
\end{align*}
where $C>0$ depends on the $L^\infty(Q_T)$ norms of $\na\bar{\bm{c}}$ and
$\na\bar{\bm{\mu}}$.
Next, we use again Young's inequality with
$\theta'>0$:
\begin{equation}\label{4.J2}
J_2 \le \frac{\theta'}{4}\sum_{k=1}^n\int_\Omega(\Delta(c_k-\bar{c}_k))^2dxdt
+ \frac{n}{\theta'}\sum_{k,\ell=1}^n\int_0^T\int_\Omega
\big|\operatorname{div}\big(c_k(c_\ell-\bar{c}_\ell)Q_\ell(\bar{\bm{c}})\big)\big|^2 dxdt,
\nonumber
\end{equation}
where we defined
$$
Q_\ell(\bar{\bm{c}}) := \sum_{j=1}^n\frac{1}{\sqrt{\bar{c}_\ell}}
D_{\ell j}^{BD}(\bar{\bm{c}})\sqrt{\bar{c}_j}\na\bar{\mu}_j.
$$
Estimating
\begin{align*}
\big|\operatorname{div}\big(c_k(c_\ell-\bar{c}_\ell)Q_\ell(\bar{\bm{c}})\big)\big|
&= \big|c_k(c_\ell-\bar{c}_\ell)\operatorname{div} Q_\ell(\bar{\bm{c}})
+ c_k\na(c_\ell-\bar{c}_\ell)\cdot Q_\ell(\bar{\bm{c}}) \\
&\phantom{xx}{}+ (c_\ell-\bar{c}_\ell)\na(c_k-\bar{c}_k)\cdot Q_\ell(\bar{\bm{c}})
+ (c_\ell-\bar{c}_\ell)\na\bar{c}_k\cdot Q_\ell(\bar{\bm{c}})\big| \\
&\le C\big(|c_\ell-\bar{c}_\ell| + |\na(c_\ell-\bar{c}_\ell)|
+ |\na(c_k-\bar{c}_k)|\big),
\end{align*}
where $C>0$ depends on the $L^\infty(Q_T)$ norm of $Q_\ell(\bar{\bm{c}})$,
we deduce from \eqref{4.J2} that
$$
J_2 \le \frac{\theta'}{4}\sum_{k=1}^n\int_\Omega(\Delta(c_k-\bar{c}_k))^2dxdt
+ \frac{C}{\theta'}\sum_{i=1}^n\int_0^T\int_\Omega\big((c_i-\bar{c}_i)^2
+ |\na(c_i-\bar{c}_i)|^2\big)dxdt.
$$
Inserting the estimates for $J_1$ and $J_2$ into \eqref{4.I132} leads to
\begin{align*}
I_{132} &\le \frac{\theta}{4}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt
+ \frac{\theta'}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2dxdt \\
&\phantom{cc}{}+ C(\theta,\theta')\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt.
\end{align*}
Then, together with \eqref{4.I131}, we find that
\begin{align}
I_{13} &\le \frac{\lambda_m}{2}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt \nonumber \\
&\phantom{xx}{}+ \frac{\theta}{4}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt
+ \frac{\theta'}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2dxdt
\label{4.I13} \\
&\phantom{cc}{}+ C(\theta,\theta')\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt. \nonumber
\end{align}
Finally, we insert this estimate and estimate \eqref{4.I1456} for
$I_{14}$, $I_{15}$, and $I_{16}$ into \eqref{4.aux2}, observing that the
first term on the right-hand side of \eqref{4.I13}
is absorbed by the second term on the left-hand side of \eqref{4.aux2}:
\begin{align}\label{4.Efinal}
{\mathcal E}(&\bm{c}(T)|\bar{\bm{c}}(T)) + \frac{\lambda_m}{2}\sum_{i=1}^n
\int_0^T\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}
\na(\mu_j-\bar{\mu}_j)\bigg|^2dxdt \\
&\le {\mathcal E}(\bm{c}^0 |\bar{\bm{c}}^0)
+ \frac{\theta}{4}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt \nonumber \\
&\phantom{xx}{}
+ \frac{\theta'}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2dxdt
\nonumber \\
&\phantom{xx}{}+ C(\theta,\theta')\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt. \nonumber
\end{align}
{\em Step 3: Combining the relative energy and relative entropy inequalities.}
Next, multiply \eqref{4.Efinal} by
$4(\lambda_M-\lambda)^2/(\lambda_m\lambda)$, choose
$\theta'=\lambda_m\lambda^2/(4(\lambda_M-\lambda)^2)$, and add this expression to
\eqref{4.Hfinal} (which estimates ${\mathcal H}(\bm{c}|\bar{\bm{c}})$).
Then some terms on the right-hand
side can be absorbed by the corresponding expressions on the left-hand side,
leading to
\begin{align*}
{\mathcal H}(&\bm{c}(T)|\bar{\bm{c}}(T)) + \frac{4(\lambda_M-\lambda)^2}{\lambda_m\lambda}
{\mathcal E}(\bm{c}(T)|\bar{\bm{c}}(T)) \\
&\phantom{xx}{}+ \frac{(\lambda_M-\lambda)^2}{\lambda}
\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n P_L(\bm{c})_{ij}\sqrt{c_j}
\na(\mu_j-\bar{\mu}_j)\bigg|^2 dxdt \\
&\phantom{xx}{}+ \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega\bigg|\sum_{j=1}^n
P_L(\bm{c})_{ij}\sqrt{c_j}\na\log\frac{c_j}{\bar{c}_j}\bigg|^2 dxdt
+ \frac{\lambda}{4}\sum_{i=1}^n\int_0^T\int_\Omega(\Delta(c_i-\bar{c}_i))^2dxdt \\
&\le {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0) + \frac{4(\lambda_M-\lambda)^2}{\lambda_m\lambda}
{\mathcal E}(\bm{c}^0|\bar{\bm{c}}^0) \\
&\phantom{xx}{}+ C(\theta,\theta')\sum_{i=1}^n\int_0^T\int_\Omega
\big((c_i-\bar{c}_i)^2 + |\na(c_i-\bar{c}_i)|^2\big)dxdt.
\end{align*}
The last term can be bounded in terms of the free energy, since
$c_i\log(c_i/\bar{c}_i)-(c_i-\bar{c}_i)\ge (c_i-\bar{c}_i)^2/2$
\cite[Lemma 18]{HJT21}:
\begin{align*}
{\mathcal H}(\bm{c}(T)|\bar{\bm{c}}(T)) + \frac{4(\lambda_M-\lambda)^2}{\lambda_m\lambda}
{\mathcal E}(\bm{c}(T)|\bar{\bm{c}}(T))
&\le {\mathcal H}(\bm{c}^0|\bar{\bm{c}}^0) + \frac{4(\lambda_M-\lambda)^2}{\lambda_m\lambda}
{\mathcal E}(\bm{c}^0|\bar{\bm{c}}^0) \\
&\phantom{xx}{}+ C\int_0^T{\mathcal E}(\bm{c}(t)|\bar{\bm{c}}(t))dt.
\end{align*}
Then the theorem follows after applying Gronwall's lemma.
\section{Examples}\label{sec.exam}
We present some models which satisfy Assumptions (B1)--(B4).
\subsection{A phase separation model}
Elliott and Garcke have studied in \cite{ElGa97} equations
\eqref{1.eq1}--\eqref{1.mu}, formulated in terms of the mobility matrix
\eqref{1.B}, where
$$
B_{ij}(\bm{c}) = b_i(c_i)\bigg(\delta_{ij} - \frac{b_j(c_j)}{\sum_{k=1}^n b_k(c_k)}
\bigg), \quad i,j=1,\ldots,n.
$$
The functions $b_i\in C^1([0,1])$ are nonnegative and satisfy
$\beta_1 c_i\le b_i(c_i)\le \beta_2 c_i$ for $c_i\in[0,1]$ and some constants
$0<\beta_1\le \beta_2$.
This model describes phase transitions in multicomponent systems;
it has been suggested in \cite{AkTo90} to model the dynamics of polymer mixtures
with $b_i(c_i)=\beta_i c_i$ and $\beta_i>0$.
The subspace $L(\bm{c})$ becomes
$$
L(\bm{c}) = \bigg\{\bm{z}\in{\mathbb R}^n:\sum_{i=1}^n \sqrt{b_i(c_i)}z_i=0\bigg\},
$$
and the matrix $D^{BD}(\bm{c})$ is determined directly from the mobility matrix:
$$
D_{ij}^{BD}(\bm{c}) = \frac{B_{ij}(\bm{c})}{\sqrt{b_i(c_i)b_j(c_j)}}
= \delta_{ij} - \frac{\sqrt{b_i(c_i)b_j(c_j)}}{\sum_{k=1}^n b_k(c_k)}.
$$
Instead of checking Assumptions (B1)--(B4), it is more convenient to verify
the statements of Lemma \ref{lem.DB} directly. This has been done in
\cite[Section 2]{HJT21}. Although the global existence of weak solutions
has been already proved in \cite{ElGa97}, we obtain the weak-strong uniqueness
property as a new result.
\subsection{Classical Maxwell--Stefan system}
In the classical Maxwell--Stefan model, the matrix $K(\bm{c})$ has the entries
$K_{ij}(\bm{c}) = \delta_{ij}\sum_{\ell=1}^n k_{i\ell}c_\ell - k_{ij}c_i$ for
$i,j=1,\ldots,n$. The associated matrix $D^{MS}(\bm{c})$ is given by
$$
D_{ij}^{MS}(\bm{c}) = \frac{1}{\sqrt{c_i}}K_{ij}(\bm{c})\sqrt{c_j}
= \delta_{ij}\sum_{\ell=1}^n k_{i\ell} c_\ell - k_{ij}\sqrt{c_ic_j},
\quad i,j=1,\ldots,n.
$$
It is proved in \cite[Sec.~5.4]{HJT21} that this matrix satisfies
Assumptions (B1)--(B4). Thus, Theorems \ref{thm.ex} and \ref{thm.wsu} hold
for the model
\begin{align*}
& \pa_t c_i + \operatorname{div}(c_iu_i) = 0, \quad \sum_{i=1}^n c_iu_i = 0, \quad
i=1,\ldots,n, \\
& c_i\na\mu_i - \frac{c_i}{\sum_{k=1}^n c_k}\sum_{j=1}^n c_j\na\mu_j
= -\sum_{j=1}^n k_{ij}c_ic_j(u_i-u_j),
\end{align*}
where $\mu_i=\log c_i-\Delta c_i$.
Compared to \cite{HJT21}, the mobility does not only depend on $c_i$ but also
on $\Delta c_i$. This extends the existence and weak-strong uniqueness results
to a more general case.
\subsection{A physical vapor decomposition model for solar cells}
Thin-film crystalline solar cells can be fabricated as thin coatings on a
substrate by the physical vapor decomposition process. The dynamics of the
volume fractions of the process components can be described by model
\eqref{1.eq1}--\eqref{1.bic} with the chemical potentials $\mu_i=\log c_i$
and the mobility matrix
$$
B_{ij}(\bm{c}) = \delta_{ij}\sum_{\ell=1}^n k_{i\ell} \, c_i c_\ell - k_{ij} c_ic_j,
\quad i,j=1,\ldots,n.
$$
In this case, the Bott--Duffin matrix is given by
$D_{ij}^{BD}(\bm{c}) = B_{ij}(\bm{c})/\sqrt{c_ic_j} = D_{ij}^{MS}(\bm{c})$,
where $D^{MS}(\bm{c})$ is the Maxwell--Stefan matrix of the previous subsection.
Thus, Assumptions (B1)--(B4) are verified for this matrix. We infer that
Theorems \ref{thm.ex} and \ref{thm.wsu} hold for the model
$$
\pa_t c_i = \operatorname{div}\sum_{j=1}^n k_{ij}c_ic_j\na(\mu_i-\mu_j), \quad
\mu_i = \log c_i-\Delta c_i, \quad i=1,\ldots,n.
$$
When $\mu_i=\log c_i$ for all $i$, the global existence of weak solutions was proved in
\cite{BaEh18} and the weak-strong uniqueness of solutions was shown in
\cite{HoBu21}. A global existence result was obtained in \cite{EMP21} for
$\mu_1 = \log c_1 - \delta c_1 + \beta(1-2c_1)$ with $\beta>0$
and $\mu_i=\log c_i$ for $i=2,\ldots,n$. Our theorems extend these results
to a more general case.
\end{document} | math | 121,216 |
\betaegin{document}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{algo}[theorem]{Algorithm}
\newtheorem{corol}[theorem]{Corollary}
\newtheorem{prop}[theorem]{Proposition}
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\partialef\noindent{\em Proof. }{\noindent{\varepsilonm Proof. }}
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\betaegin{frontmatter}
\alphauthor[label1]{Istv\'an Bal\'azs}
\varepsilonad{[email protected]}
\alphauthor[label2,label4]{Philipp Getto\corref{cor1}}
\varepsilonad{[email protected]}
\alphauthor[label3,label5]{Gergely R\"ost}
\varepsilonad{[email protected]}
\fracntext[label4]{The research of the author was funded by the
DFG (Deutsche Forschungsgemeinschaft), project number 214819831, and by
the ERC starting grant EPIDELAY (658, No. 259559).}
\fracntext[label5]{The research of the author was funded by the
ERC starting grant EPIDELAY (658, No. 259559), by the Marie Sklodowska-Curie Grant No. 748193, and by NKFIH FK124016.}
\cortext[cor1]{corresponding author}
\tauitle{A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology}
\alphaddress[label1]{Hungarian Academy of Sciences, 1245 Budapest, P.O. Box 1000, Hungary}
\alphaddress[label2]{Center For Dynamics, Technische Universit\"at Dresden, 01062 Dresden, Germany}
\alphaddress[label3]{Bolyai Institute, University of Szeged,
Aradi v\'ertan\'uk tere 1, H-6720 Szeged, Hungary; Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG Oxford, United
Kingdom}
\betaegin{abstract}
We establish variants of existing results on existence, uniqueness and continuous dependence for a class of delay differential equations (DDE). We apply these to continue the analysis of a differential equation from cell biology with state-dependent delay, implicitly defined as the time when the solution of a nonlinear ODE, that depends on the state of the DDE, reaches a threshold. For this application, previous results are restricted to initial histories
belonging to the so-called solution manifold. We here generalize the results
to a set of nonnegative Lipschitz initial histories which is much larger than the
solution manifold and moreover convex.
Additionally, we show that the solutions define a semiflow that is continuous in the state-component in the $C([-h,0],{\mathds R}^2)$ topology, which is a variant of established differentiability of the semiflow in $C^1([-h,0],{\mathds R}^2)$.
For an associated system we show invariance of convex and compact sets under the semiflow for finite time.
\varepsilonnd{abstract}
\betaegin{keyword}
State-dependent delay \sigmaep threshold type delay \sigmaep Well-posedness \sigmaep Continuous dependence \sigmaep Almost locally Lipschitz \sigmaep Stem cell model
\varepsilonnd{keyword}
\varepsilonnd{frontmatter}
\tauableofcontents
\sigmaection{Introduction}
With this paper we would like to contribute to the development of methods to analyze differential equations with state-dependent delay (SD-DDE)
and to continue the analysis of a model from cell population biology, which can be formulated as a SD-DDE. In the cell population equation the delay is implicitly defined as the time when the solution of a nonlinear ordinary differential equation meets a threshold (see (\rhoef{eq11}--\rhoef{eq14}) below). The SD-DDE additionally features continuously distributed delays.
In \cite{Getto}, the authors have elaborated conditions to
guarantee via application of results of \cite{Walther, Walther1} that the solutions of the
cell population equation define a differentiable semiflow on the {\it solution manifold}, for $n=2$ a sub-manifold of ${\cal C}^1:=C^1([-h,0],{\mathds R}^n)$. An advantage of
the approach in \cite{Walther, Walther1} is the associability of a linear variational equation, from which a characteristic equation, which allows to analyze local stability of equilibria, can be deduced.
Motivated by simulations (see the discussion section), a future objective is the proof of existence of periodic solutions for the cell population equation.
One way to do this is to use fixed point arguments for the
Poincar\'e operator, which is done for a general class of SD-DDE in \cite{MN1}.
As in many fixed point arguments, also in \cite{MN1} convexity and compactness of the domain is used, properties the solution manifold in general does not have.
Next, note that differentiability of the semiflow in the ${\cal C}^1$-topology as
established in \cite{Getto} implies continuous dependence on initial values in
${\cal C}^1$, i.e., convergence of sequences of solution segments
in ${\cal C}^1$, if sequences of initial histories converge in ${\cal C}^1$. The
latter however can appear as too strong in applications, see again the discussion section.
We here show how - sometimes slightly modified - existing strategies can be combined to show existence, uniqueness and continuous dependence for a large class of SD-DDE. We apply the results to generalize (global) existence and uniqueness of solutions of the SD-DDE (\rhoef{eq11}--\rhoef{eq14}) for initial histories in the solution manifold to initial histories in a set of nonnegative Lipschitz functions, the latter being a much larger set than the former and moreover convex. Additionally, we show that the solutions define a semiflow that is continuous in the ${\cal C}:=C([-h,0],{\mathds R}^n)$ topology. Compared to the above discussed established continuous dependence with respect to initial data in ${\cal C}^1$, the prerequisite of convergence of initial histories (as well as the conclusion of convergence of solutions) is weaker here - ${\cal C}$ instead of ${\cal C}^1$ - and we refer to the discussion section for possibilities to exploit this.
In \cite{Hale} the existence of noncontinuable and global solutions is established
for systems of delay differential equations defined by functionals that are continuous on
domains that are open in the ${\cal C}$-topology (${\cal C}$-open). Continuous dependence on initial values is shown under the precondition that the solution is unique. Uniqueness
of solutions is shown if the functional is Lipschitz on a ${\cal C}$-open domain.
A known problem is that for SD-DDE the functional is in general not
Lipschitz on a ${\cal C}$-open domain. A hint to see this is that the evaluation operator
(see (\rhoef{eq15}) below) is in general not Lipschitz, if functions in the domain are not.
In \cite{Mallet} the problem is overcome for one-dimensional SD-DDE, where dimension refers to the range space of the functional defining the equation, with the help of the concept of {\it almost local Lipschitzianity}, which roughly means local Lipschitzianity on a domain of Lipschitz functions. It is shown that almost local Lipschitzianity in combination with the discussed results in \cite{Hale} yields existence and uniqueness on a domain of Lipschitz functions.
Functionals derived from applications are typically, and in our case, not defined on the whole space but have a domain restricted to a subset of the space.
In \cite{Mallet} results are first established for an arbitrary functional defined on the whole space $C([-h,0],{\mathds R})$.
Then, to work with restricted domains, a retraction from $C([-h,0],{\mathds R})$ to $C([-h,0],[-B,A])$
is constructed and the results are transferred to the case where the functional
is defined on $C([-h,0],[-B,A])$ only. A negative feedback condition for the functional
ensures that solutions remain in the retracted domain.
We here start with a general functional defined on ${\cal C}$. We argue that almost local Lipschitzianity and its use to conclude uniqueness for Lipschitz initial histories can be generalized from one to $n$ dimensions in a straightforward way, conclude uniqueness, and combine it with results from \cite{Hale} on (global) existence and continuous dependence to get existence, uniqueness and continuous dependence for Lipschitz initial histories for a large class of functionals defined on (all of) ${\cal C}$.
To allow for a domain of the form ${\cal D}=C([-h,0],[-B,\infty)^n)$ of the functional, i.e., in particular, a domain that can be specified to our application, we modify the above discussed construction of retractions
and feedback conditions from \cite{Mallet}. One then can work with a retraction from ${\cal C}$ to ${\cal D}$ and a component-wise feedback condition and transfer the general results on solutions to the case of a functional defined on ${\cal D}$. We conclude that the solutions define a semiflow, in the sense of e.g. \cite{Amann}, that is continuous in the ${\cal C}$-topology on a set of Lipschitz functions and use this continuity to derive some
further properties.
We then establish compactness results employing the following ideas. In \cite{MN1} it is used that by the Arzela-Ascoli theorem a set of functions that share the same finite bound and finite Lipschitz constant is compact in ${\cal C}$. As will be motivated, the approach of \cite{MN1} to show that a time $t$ map leaves such a set invariant for arbitrarily
large $t$ does not work here directly. However, a class of two-dimensional systems that contains (\rhoef{eq11}--\rhoef{eq14}) can be transformed
to a one-dimensional equation. For the latter, invariance of a compact set for finite time can be elaborated. We refer to the discussion section for more details on future implementation of these results.
After having established the general results, we consider the SD-DDE
\betaegin{eqnarray}
w'(t)&=&q(v(t))w(t),
\lambdaabel{eq11}\\
v'(t)&=&
\fracrac{\gamma(v(t-\tau(v_t)))}{g(x_1,v(t-\tau(v_t)))}g(x_2,v(t))w(t-\tau(v_t))e^{\int_0^{\tau(v_t)}
[d-D_1g](y(s,v_t),v(t-s))ds}
\nonumber\\
&&-\mu v(t),
\lambdaabel{eq12}
\varepsilonnd{eqnarray}
where $y=y(\cdot,\psi)$ and $\tau=\tau(\psi)$ are defined as the respective solutions of
\betaegin{eqnarray}
&&y'(s)=g(y(s),\psi(-s)),\;s>0,\;\;y(0)=x_2\;\;{\rhom and}
\lambdaabel{eq13}\\
&&y(\tau,\psi)=x_1,
\lambdaabel{eq14}
\varepsilonnd{eqnarray}
where $x_1<x_2$ are given parameters. As common in delay differential equations (DDE) we use the notation $x_t(s):=x(t+s)$, $s<0$, for functions $x$ defined in
$t+s\in{\mathds R}$.
The system describes the dynamics of a stem cell population ($w$) regulated by the mature cell population ($v$). We refer to \cite{Getto} and references therein, in particular \cite{Getto1}, for biological background of the model.
The SD-DDE can be deduced via integration along the characteristics from a partial
differential equation of transport type which features a progenitor cell maturity density
and maturity structure, see \cite{Getto}.
We apply our general results to (\rhoef{eq11}--\rhoef{eq14}).
To guarantee some of the required conditions, we show that the functional that defines the
system is almost locally Lipschitz. To handle the implicitly defined state-dependent delay
we consider evaluation operators and implicitly defined operators and analyze them on
Lipschitz subsets of continuous functions.
The paper is structured top down: In Section \rhoef{s2} we consider our most general class of equations. Section \rhoef{s3} contains results for an intermediate class and Section \rhoef{s1} an application of these results to the stem cell SD-DDE; in each of these two sections a subsection on main results precedes one on proofs. Finally, Section \rhoef{ss5} contains examples of modelling ingredients and Section \rhoef{s6} a discussion of our results and potential future applications.
\sigmaection{Solving DDE on a state space of Lipschitz functions}\lambdaabel{s2}
\sigmaubsection{Initial value problem}\lambdaabel{ss0}
\betaegin{definition}\rhom
Suppose that $\phi\in{\cal D}\sigmaubset{\cal C}$ and $f:{\cal D}\lambdaongrightarrow{\mathds R}n$. By a {\it solution} of
\betaegin{eqnarray}
x'(t)&=&f(x_t),\;\;\;t\gammae t_0,
\lambdaabel{eq7}
\\
x_{t_0}&=&\phi,
\lambdaabel{eq1}
\varepsilonnd{eqnarray}
or a solution of (\rhoef{eq7}) through $\phi$,
we mean a continuous function $x^\phi:[t_0-h,t_0+\alpha]\lambdaongrightarrow{\mathds R}n$ for some $\alpha>0$,
that is such that on $[t_0,t_0+\alpha]$ one has $x_t^\phi\in{\cal D}$, the function $x^\phi$ is differentiable and satisfies (\rhoef{eq7}--\rhoef{eq1}).
Solutions on half-open intervals
$[t_0-h,t_0+\alpha)$ for $\alpha\in(0,\infty]$ are defined analogously.
\varepsilonnd{definition}
We shall sometimes write $x$ instead of $x^\phi$.
\sigmaubsection{Domain of the functional is ${\cal C}$}\lambdaabel{ss3}
\sigmaubsubsection{Noncontinuable and global solutions}
\betaegin{theorem}\lambdaabel{theo3}
Suppose that $F: {\cal C}\lambdaongrightarrow{\mathds R}n$ is continuous and $\phi\in {\cal C}$. Then
\betaegin{itemize}
\item[(a)] there exists a unique $c=c(\phi)\in(0,\infty]$ such that $x^\phi:[t_0-h,t_0+c)\lambdaongrightarrow{\mathds R}n$ is a non-continuable solution of
\betaegin{eqnarray}
x'(t)=F(x_t),\;t\gammae t_0,\;\;x_{t_0}=\phi.
\lambdaabel{eq6}
\varepsilonnd{eqnarray}
\varepsilonnd{itemize}
If additionally $F(U)$ is bounded whenever $U\sigmaubset {\cal C}$ is closed and bounded then the following hold:
\betaegin{itemize}
\item[(b)] If $c<\infty$ then for any closed and bounded $U\sigmaubset {\cal C}$ there exists some $t_U\in(0,c)$
such that $x_t^\phi\notin U$ for all $t\in [t_0+t_U, t_0+c)$.
\item[(c)] If $\{x_t^\phi: t\in[t_0,t_0+\alpha)\}\sigmaubset {\cal C}$ is bounded, whenever $\alpha<\infty$ and $x^\phi$ is defined on
$[t_0,t_0+\alpha)$, then $c=\infty$, i.e., the solution is global.
\varepsilonnd{itemize}
\varepsilonnd{theorem}
The existence of a solution $x^\phi:[t_0-h,t_0+\alpha]\lambdaongrightarrow{\mathds R}n$ for some $\alpha>0$
follows from \cite[Theorem 2.2.1]{Hale} and the statement in (a) is concluded in \cite[Section 2.3]{Hale} from Zorn's lemma. Next, (b) follows from \cite[Theorem 2.3.2]{Hale}. Then (c) is standard: If $c<\infty$ define
\[
U:={\overline v}erline{\{x_t^\phi:\;t\in[t_0,t_0+c)\}}.
\]
Then by (b) there exists some $t_U\in(0,c)$ such that $x_{t_0+t_U}^\phi\notin U$, which contradicts the definition of $U$.
\betaegin{remark}
Note that the cited results in \cite{Hale} hold for non-autonomous equations. Since
our motivation here is an autonomous system and moreover the uniqueness result that we
will use is also for autonomous systems we have rewritten these results
for the autonomous case.
\varepsilonnd{remark}
\sigmaubsubsection{Uniqueness}
To guarantee uniqueness, the notion of almost local Lipschitzianity for $n=1$ from \cite{Mallet} can
be generalized to arbitrary finite dimensions in a straightforward way.
As common, we define for any $\phi\in{\cal C}$
\[
lip\;\phi:=\sigmaup\lambdaeft\{\fracrac{|\phi(s)-\phi(t)|}{|s-t|}:\;s,t\in[-h,0],\;s\neq t\rhoight\}\in[0,\infty]
\]
and $B_\partiale(x_0):=\{x:\;\|x-x_0\|<\partiale\}$, where $\partiale>0$, $|\cdot|$ denotes norms in ${\mathds R}^n$
with $n$ depending on context, and the choice of norm $\|\cdot\|$ should
also be clear from the context, e.g., the choice of $x_0$. In the following, however, we denote by $\|\cdot\|$ the sup-norm on ${\cal C}$. Then, a function $\phi$ is Lipschitz
with Lipschitz constant $k$ (we will write $k$-{\it Lipschitz}) whenever $\infty>k\gammae lip\;\phi$.
For each $\phi_0\in {\cal C}$, $\partiale>0$, $R>0$ define
\[
V(\phi_0;\partiale,R):=\{\phi\in{{\overline v}erline B}_\partiale(\phi_0):\;lip\;\phi\lambdae R\}.
\]
\betaegin{definition}\lambdaabel{def1}\rhom
A functional $f:{\cal D}\sigmaubset {\cal C}=C([-h,0],{\mathds R}^n)\lambdaongrightarrow{\mathds R}^m$ is called {\it almost locally Lipschitz} if $f$ is continuous and
for all $\phi_0\in{\cal D}$, $R>0$ there exists some
$\partiale=\partiale(\phi_0,R)>0$, $k=k(\phi_0,R,\partiale)\gammae 0$ such that for all $\varphi,\psi\in V(\phi_0;\partiale,R)\cap{\cal D}$
\[
|f(\varphi)-f(\psi)|\lambdae k\|\varphi-\psi\|.
\]
\varepsilonnd{definition}
The following theorem is proven as \cite[Theorem 1.2]{Mallet} for the case $n=1$. The proof
for general $n$ is analogous and we omit it.
For ${\cal D}\sigmaubset {\cal C}$, define $V_{\cal D}:=\{\phi\in{\cal D}:\;lip\;\phi<\infty\}$. Note that if ${\cal D}$ is
convex, so is $V_{\cal D}$.
\betaegin{theorem}\lambdaabel{theo2}
Suppose that $F:{\cal C}\lambdaongrightarrow{\mathds R}n$ is almost locally Lipschitz. Let $\phi\in V_{\cal C}$ and $t_0\in{\mathds R}$. If $\alpha>0$ and $y,z:[t_0-h,t_0+\alpha]\lambdaongrightarrow{\mathds R}n$ are both solutions of (\rhoef{eq6}),
then $y(t)=z(t)$ for all $t\in[t_0,t_0+\alpha]$.
\varepsilonnd{theorem}
\sigmaubsubsection{Continuous dependence on initial values}
The following result follows directly from \cite[Theorem 2.2.2]{Hale} if we use our uniqueness
result.
\betaegin{theorem}\lambdaabel{theo5}
Suppose that $F:{\cal C}\lambdaongrightarrow{\mathds R}^n$ is almost locally Lipschitz, $\phi\in V_{\cal C}$
and let $\alpha>0$ be such that a solution $x^\phi$ through $\phi$ exists on $[t_0-h,t_0+\alpha]$. Let $(\phi^k)\in V_{\cal C}^{\bf N}$ with
$\phi^k\lambdaongrightarrow\phi$. Then $x^\phi$ is unique on $[t_0-h,t_0+\alpha]$, for some $k\gammae k_0$
there exist unique solutions $x^k$ through $\phi^k$ on $[t_0-h,t_0+\alpha]$ for all $k\gammae k_0$ and $x^k\lambdaongrightarrowx^\phi$ uniformly on $[t_0-h,t_0+\alpha]$.
\varepsilonnd{theorem}
\betaegin{remark}
Note that similarly as in \cite[Theorem 2.2.2]{Hale} we could include continuous dependence
on functional and initial time in the above formulation. We did not do this, since, especially
when transferring these results to restricted domains, the exposition would suffer from
further technicalities and moreover we currently see no direct use for these properties.
\varepsilonnd{remark}
\sigmaubsection{Retraction onto a specific domain}\lambdaabel{ss4}
It is remarked in \cite{Mallet} (without proof) that the following result holds in case $n=1$. The proof for general $n$ is analogous and we present it for completeness.
Recall that a {\it retraction} is a continuous
map of a topological space into a subset that on the subset equals the identity.
\betaegin{lemma}\lambdaabel{lem1}
Let ${\cal D}\sigmaubset {\cal C}$, $\rho :{\cal C}\lambdaongrightarrow{\cal D}$ be a locally Lipschitz retraction. Suppose that for all $\phi_0\in {\cal C}$, $\partiale>0$, $R>0$
\[
\sigmaup\{lip\;\rho(\phi):\;\phi\in V(\phi_0;\partiale,R)\}<\infty.
\]
Then, if $f:{\cal D}\lambdaongrightarrow{\mathds R}^n$ is almost locally Lipschitz, so is $F:{\cal C}\lambdaongrightarrow{\mathds R}^n;F:=f\circ\rho$.
\varepsilonnd{lemma}
\noindent{\em Proof. } First, $F$ is continuous as a composition of continuous functions. Next, let $\phi_0\in{\cal C}$, $R>0$. Define $L:=\sigmaup\{lip\;\rho(\phi):\;\phi\in V(\phi_0;1,R)\}<\infty$. Choose $\varepsilon=\varepsilon(\rho(\phi_0),L)$, $k=k(\rho(\phi_0),L)$ such that $f$ is $k$-Lipschitz on $V(\rho(\phi_0);\varepsilon,L)$. Choose $\partiale<1$,
$l\gammae 0$ such that $\rho(B_\partiale(\phi_0))\sigmaubset B_\varepsilon(\rho(\phi_0))$ and $\rho$ is $l$-Lipschitz
on $B_\partiale(\phi_0)$. Then for $\varphi,\psi\in V(\phi_0;\partiale,R)$, one has
\betaegin{eqnarray}
|F(\varphi)-F(\psi)|=|f(\rho(\varphi))-f(\rho(\psi))|\lambdae k|\rho(\varphi)-\rho(\psi)|\lambdae kl\|\varphi-\psi\|.
\nonumber
\varepsilonnd{eqnarray}
Hence, $F$ is $kl$-Lipschitz on $V(\phi_0,\partiale,R)$ and thus almost locally Lipschitz.
$\sigmaqr45$
\sigmaubsubsection{A specific retraction for a specific domain}
For the remainder of the section we will use the following construction (unless specified
otherwise).
\betaegin{remark}
The construction is a modification of the retraction in \cite{Mallet}, the latter of which maps $C([-h,0],{\mathds R})$ onto $C([-h,0],[-B,A])$ with $-\infty<-B<A<\infty$, to a retraction
of $C([-h,0],{\mathds R}^n)$ onto $C([-h,0],[-B,\infty)^n)$ with $-\infty<-B$. With the result we can work with
nonnegative solutions, if $B=0$, of multi-dimensional systems. The construction could probably be generalized to the range
$C([-h,0],\Pi_{i=1}^{n}[-B_i,A_i])$, $-\infty\lambdae -B_i <A_i\lambdae\infty$, $ i=1,...,n$.
\varepsilonnd{remark}
Let $B\in{\mathds R}$ and define
\betaegin{eqnarray}
{\cal D}:=C([-h,0],[-B,\infty)^n).
\lambdaabel{eq2}
\varepsilonnd{eqnarray}
Note that the convexity of ${\cal D}$ implies convexity of $V_{\cal D}$.
We define a map
\betaegin{eqnarray}
r:{\mathds R}\lambdaongrightarrow[-B,\infty),\;
r(u):=
\betaegin{cases}
u,& u\in[-B,\infty),
\\
-B,& u<-B.
\varepsilonnd{cases}
\lambdaabel{eq3}
\varepsilonnd{eqnarray}
Then $r$ is a retraction and Lipschitz with $lip\;r\lambdae 1$. With $r$ we define
another map
\betaegin{eqnarray}
\rho:{\cal C}\lambdaongrightarrow{\cal D}, \rho=(\rho_1,...,\rho_n),
\rho_i(\phi)(t):=r(\phi_i(t)),\;
i=1,...,n.
\lambdaabel{eq4}
\varepsilonnd{eqnarray}
\betaegin{lemma}
$\rho$ is a retraction and maps bounded sets into bounded sets.
\varepsilonnd{lemma}
\noindent{\em Proof. } It is clear that $\rho$ (is onto,) preserves the subset and maps bounded sets into bounded sets. Regarding continuity, suppose
that $\phi^n\lambdaongrightarrow\phi$, and let $\varepsilon>0$. Then
\betaegin{eqnarray}
|[\rho_i(\phi^n)-\rho_i(\phi)](t)|=|r(\phi^n_i(t))-r(\phi_i(t))|.
\nonumber
\varepsilonnd{eqnarray}
Choose $N\in{\bf N}$, $\partiale>0$ such that $\|\phi^n-\phi\|\lambdae\partiale$ for all $n\gammae N$. Then
\[
|\phi^n(t)|\lambdae\|\phi\|+\partiale,\;|\phi(t)|\lambdae\|\phi\|+\partiale,\;\fracorall t\in[-h,0], \;n\gammae N.
\]
Now, continuity follows by uniform continuity of $r$ on compact sets.
$\sigmaqr45$
The following result follows by definition of $\rho$ from Lipschitzianity of $r$ with $lip\;r\lambdae 1$. We omit the straightforward proofs of the two following results.
\betaegin{lemma}\lambdaabel{lem15}
One has $lip\;\rho(\phi)\lambdae lip\;\phi$, hence if $\phi$ is Lipschitz so is $\rho(\phi)$. Moreover, $\rho$ is Lipschitz with $lip\;\rho\lambdae 1$.
\varepsilonnd{lemma}
The result implies that $\sigmaup\{lip\;\rho(\phi):\;\phi\in V(\phi_0;\partiale,R)\}\lambdae R<\infty$ for all $\phi_0\in {\cal C}$, $\partiale>0$, $R>0$.
We can use the latter to directly apply Lemma \rhoef{lem1} to $F:=f\circ\rho$ with $\rho$
being our (locally) Lipschitz retraction:
\betaegin{lemma}\lambdaabel{lem2}
Suppose that $f:{\cal D}\sigmaubset{\cal C}\lambdaongrightarrow{\mathds R}n$ is almost locally Lipschitz. Then so is $F$.
\varepsilonnd{lemma}
\sigmaubsubsection{Noncontinuable and global solutions and uniqueness}
To guarantee that a solution remains within a domain a feedback condition can be used. The proof of the following result is a modification of a
similar result for one dimension \cite[Theorem 1.3]{Mallet}.
\betaegin{lemma}\lambdaabel{lem17}
Suppose that $f:{\cal D}\lambdaongrightarrow{\mathds R}^n$ satisfies
\[
f_i(\phi)\gammae 0,\;{\rhom if}\;\phi_i(0)=-B,\;\fracorall\phi=(\phi_1,...,\phi_n)\in{\cal D},\;i=1,...,n
\tauag{F}.
\]
Now fix $\phi\in{\cal D}$ and assume that $x$ is a solution of $x'(t)=f(\rho(x_t))$ through $\phi$ on some
interval $[t_0-h,t_0+\alpha]$. Then $x_t\in{\cal D}$ and thus $\rho(x_t)=x_t$ for all $t\in[t_0,t_0+\alpha]$ and hence $x$ is a solution of (\rhoef{eq7}--\rhoef{eq1}) on $[t_0,t_0+\alpha]$.
\varepsilonnd{lemma}
\noindent{\em Proof. } The statement would follow if
$x_i(t+\tauh)\gammae-B$ for all $t\gammae t_0$, $\tauh\in[-h,0]$, $i=1,...,n$. First, $\phi\in{\cal D}$ implies
that $\phi_i(\tauh)\gammae-B$ for all $\tauh\in[-h,0]$, $i=1,...,n$. Suppose that for some $i\in\{1,...,n\}$
and $x=x^\phi$ one has $x_i(t_1)<-B$ for some $t_1>t_0$. Then $\tau:=\sigmaup\{t\in[t_0,t_1]:
\;x_i(t)=-B\}\in[t_0,t_1)$. Then $x_i(\tau)=-B$, $x_i(t)<-B$ for all $t\in(\tau,t_1]$. By the mean value
theorem $x_i'(t)<0$ for some $t\in(\tau,t_1)$. Then $\rho_i(x_t)(0)=r(x_i(t))=-B$. Hence by (F)
we have $x_i'(t)=f_i(\rho(x_t))\gammae 0$, which is a contradiction.
$\sigmaqr45$
\betaegin{theorem}\lambdaabel{theo6}
Suppose that $f:{\cal D}\lambdaongrightarrow{\mathds R}n$ is continuous and satisfies (F). Then the following hold.
\betaegin{itemize}
\item[(a)] For every $\phi\in{\cal D}$ there exists a unique $c=c(\phi)\in(0,\infty]$ and a non-continuable solution
$x^\phi$ on $[t_0-h,t_0+c)$ of (\rhoef{eq7}--\rhoef{eq1}).
\varepsilonnd{itemize}
\betaegin{itemize}
\item[(b)] If $f(U)$ is bounded, whenever $U\sigmaubset {\cal D}$ is bounded, and if for some
$\phi\in {\cal D}$ the set $\{x_t^\phi:\;t\in[t_0,t_0+\alpha)\}\sigmaubset {\cal D}$ is bounded, whenever
$\alpha<\infty$ and $x^\phi$ defined on $[t_0,t_0+\alpha)$, then $c=\infty$, i.e., the solution is global.
\item[(c)] If $f$ is almost locally Lipschitz and $\phi\in V_{\cal D}$, then $x^\phi$ is unique.
\varepsilonnd{itemize}
\varepsilonnd{theorem}
\noindent{\em Proof. }
Since $F:=f\circ\rho$ is continuous, by Theorem \rhoef{theo3} (a) there exists a noncontinuable
solution of (\rhoef{eq6}) for this $F$. Next, suppose
that $U\sigmaubset {\cal C}$ is (closed and) bounded. Then, as remarked, $\rho(U)\sigmaubset{\cal D}$ is bounded
and hence by the assumption of (b) $F(U)=f(\rho(U))$ is bounded. Thus by Theorem \rhoef{theo3} (c) we have shown
that if $\{x_t^\phi:t\in[t_0,t_0+\alpha)\}\sigmaubset {\cal C}$ is bounded
whenever $\alpha<\infty$ and $x^\phi$ defined on $[t_0,t_0+\alpha)$, then $c=\infty$.
If $f$ is almost locally Lipschitz, then by Lemma \rhoef{lem2} so is $F$ and thus
by Theorem \rhoef{theo2} we get uniqueness. To complete the proof note that (F) guarantees via Lemma \rhoef{lem17} that $\{x_t^\phi:t\in[t_0,t_0+\alpha)\}\sigmaubset {\cal D}$ and that
$x^\phi$ is a solution of (\rhoef{eq7}--\rhoef{eq1}).
$\sigmaqr45$
\betaegin{remark}
If $f$ would map only the closed and bounded sets on bounded sets, as required in Theorem
\rhoef{theo3}, we could not guarantee that $F(U)=(f\circ\rho)(U)$ is bounded if $U$ is
closed and bounded: the above defined retraction $\rho$ maps bounded on bounded,
but in general does not map closed and bounded on closed sets. To see the latter,
consider e.g. ${\cal C}:=C([0,2],{\mathds R})$, ${\cal D}:=\{x\in{\cal C}:x(t)\gammae0,\;\fracorall\;t\in[-h,0]\}$ and $r$ and $\rho$ defined as above, but for $n=1$, $B=0$ and the modified ${\cal C}$ and ${\cal D}$.
Define $U:=\{x_n:\;n\gammae 2\}\sigmaubset{\cal C}$, where
\betaegin{eqnarray}
x_n(t):=\betaegin{cases}
\fracrac{1}{n}&,t<1-\fracrac{1}{n}
\\
1-t&,1-\fracrac{1}{n}\lambdae t<1
\\
-n(t-1)&,1\lambdae t<1+\fracrac{1}{n}
\\
-1&,1+\fracrac{1}{n}\lambdae t\lambdae 2.
\varepsilonnd{cases}
\nonumber
\varepsilonnd{eqnarray}
Then it is easy to see that $U$ is closed and bounded but
\[
\rho(U)=\{x:\;\varepsilonxists\;n\gammae2\;s.th.\;x(t)=x_n(t)\;\fracorall\;t\in[0,1],\;x(t)=0\;\fracorall\;t\in[1,2]\}.
\]
is not closed.
\varepsilonnd{remark}
\sigmaubsubsection{Continuous dependence on initial values}
The negative feedback condition $(F)$ now ensures that our results on continuous
dependence can be transferred to our case of a specific retraction onto the domain of
the functional.
\betaegin{theorem}\lambdaabel{theo7}
Suppose that $f:{\cal D}\lambdaongrightarrow{\mathds R}^n$ is almost locally Lipschitz and satisfies (F), let $\phi\in V_{\cal D}$ and $\alpha>0$ be such that a solution $x^\phi$ of (\rhoef{eq7}--\rhoef{eq1}) through $\phi$ exists on $[t_0-h,t_0+\alpha]$.
Let $(\phi^k)\in V_{\cal D}^{\bf N}$ with $\phi^k\lambdaongrightarrow\phi$. Then $x^\phi$ is unique on $[t_0-h,t_0+\alpha]$, for some $k\gammae k_0$ there exist unique solutions $x^k$ through
$\phi^k$ on $[t_0-h,t_0+\alpha]$ and
$x^k\lambdaongrightarrowx^\phi$ uniformly.
\varepsilonnd{theorem}
\noindent{\em Proof. } Since $x^\phi$ is a solution of (\rhoef{eq7}--\rhoef{eq1}) we have $x_t^\phi\in{\cal D}$ for all
$t\gammae t_0$. Thus, for $F:=f\circ\rho$, one has $F(x^\phi_t)=f(x^\phi_t)$ and $x^\phi$ is a solution of $x'(t)=F(x_t)$ through $\phi$. Since
$f$ is almost locally Lipschitz, by Lemma \rhoef{lem2} so is $F$ and since $\phi\in V_{\cal D}$ the solution is unique. By Theorem \rhoef{theo5} there exists some $k_0$, such that for all $k\gammae k_0$ there exist unique solutions
$x^k$ of $x'(t)=F(x_t)$ through $\phi^k$ on $[t_0-h,t_0+\alpha]$ and $x^k\lambdaongrightarrowx^\phi$ uniformly. By Lemma
\rhoef{lem17} we have $x_t^k\in{\cal D}$ for all $t\gammae t_0$, hence the $x^k$ solve also (\rhoef{eq7}--
\rhoef{eq1}).
$\sigmaqr45$
\sigmaubsubsection{A continuous semiflow on a state-space of Lipschitz functions}
If $f$ satisfies the assumptions for global existence and uniqueness, we can use the concept of a semiflow, e.g., in the sense of \cite[Section 10]{Amann}.
We start with some definitions:
\betaegin{definition}\rhom
Let $(X,d)$ be a metric space. A map $\Sigma:[0,\infty)\tauimes X\lambdaongrightarrow X$ is called a {\it continuous
semiflow} if
\betaegin{itemize}
\item[(i)] $\Sigma(0,x)=x$ for all $x\in X$,
\item[(ii)] $\Sigma(t,\Sigma(s,x))=\Sigma(t+s,x)$ for all $s,t\in[0,\infty)$, $x\in X$ (``semigroup property''),
\item[(iii)] $\Sigma$ is continuous.
\varepsilonnd{itemize}
A {\it trajectory} of the semiflow $\Sigma$ is a map $\sigma:I\lambdaongrightarrow X$ defined on an interval $I\sigmaubset {\mathds R}$
with positive length, such that for $s$ and $t$ in $I$ with $s\lambdae t$ one has
\[
\sigma(t)=\Sigma(t-s,\sigma(s)).
\]
The $\omegaega$-limit set of a trajectory $\sigma:I\lambdaongrightarrow X$ with $\sigmaup I=\infty$ is defined as
\[
\omegaega(\sigma)=\{x\in X:\;\varepsilonxists\;(t_n)\in I^{\bf N},\;{\rhom s.th.}\;t_n\lambdaongrightarrow\infty,\;\sigma(t_n)\lambdaongrightarrow x\;{\rhom as}\;n\rhoightarrow\infty\}.
\]
\varepsilonnd{definition}
\betaegin{remark}
Note that the definitions in \cite[Section 10]{Amann} and \cite[Definition VII 2.1]{3Diekmann} include also semiflows induced by local solutions. Moreover \cite[Definition VII 2.1]{3Diekmann} additionally requires completeness of the metric space,
which we here cannot expect, since by the Weierstrass approximation theorem
$V_{\cal D}$ is not complete. On the other hand to our understanding this
completeness is not necessary here.
Note also that \cite[Definition VII 2.1]{3Diekmann} merely requires continuity in each of the components, point-wise with respect to the other.
The definitions of trajectories
and $\omegaega$-limit sets are consistent with \cite[Definitions VII 2.3 and 2.4]{3Diekmann}. Note that the reference also contains similar results for $\alpha$-limit sets.
\varepsilonnd{remark}
The following properties of trajectories are proven in \cite[Section VII]{3Diekmann}.
We here merely will use the result on invariance of the $\omegaega$-limit set - for an alternative proof of Corollary \rhoef{corol1} below.
\betaegin{lemma}\lambdaabel{lem20}
Let $\sigma:I\lambdaongrightarrow X$ be a trajectory, then $\sigma$ is continuous. If $\sigmaup I=\infty$, then
\[
\omegaega(\sigma)=\betaigcap_{t\gammae 0} {\overline v}erline{\sigma(I\cap[t,\infty))}.
\]
If additionally ${\overline v}erline{\sigma(I)}$ is compact, then $\omegaega(\sigma)$ is nonempty, compact and connected, $dist(\sigma(t),\omegaega(\sigma))\lambdaongrightarrow0$ as $t\rhoightarrow\infty$ and for $x\in\omegaega(\sigma)$ one has $\Sigma(t,x)\in\omegaega(\sigma)$ for all $t\gammae 0$.
\varepsilonnd{lemma}
We now conclude continuity of the semiflow from continuous
dependence on initial values and the semigroup property from uniqueness.
In the following we assume that $t_0=0$.
\betaegin{theorem}\lambdaabel{theo8}
Suppose that $f:{\cal D}\lambdaongrightarrow{\mathds R}^n$ is almost locally Lipschitz and satisfies (F),
that $f(U)$ is bounded whenever $U\sigmaubset{\cal D}$ is bounded and that $\{x_t^\phi:\;t\in[0,\alpha)\}$ is bounded whenever $\phi\in V_{\cal D}$ and whenever $x^\phi$ is defined on $[0,\alpha)$. Then for any $\phi\in V_{\cal D}$ there exists a unique global solution and for all $t\gammae0$ one has $x_t^\phi\in V_{\cal D}$. Hence, we can define a map
\[
S:[0,\infty)\tauimes V_{\cal D}\lambdaongrightarrowV_{\cal D};\;S(t,\phi):=x_t^\phi.
\]
This map defines a continuous semiflow on $V_{\cal D}$ with respect to the $\sigmaup$-norm.
\varepsilonnd{theorem}
\noindent{\em Proof. } Existence of a unique global solution for all $\phi\in V_{\cal D}$ follows from Theorem \rhoef{theo6}.
Let $\phi\in V_{\cal D}$ and $t>0$. By definition of a solution we have $x_t^\phi\in{\cal D}$. Let $r,s\in[-h,
0]$. Then
\[
|x^\phi_t(r)-x^\phi_t(s)|=|x^\phi(t+s)-x^\phi(t+r)|.
\]
First, $x^\phi$ is Lipschitz on $[-h,0]$, since $\phi$ is Lipschitz. Next, $x^\phi$ is as a solution
differentiable on $[0,t]$ and satisfies (\rhoef{eq7}--\rhoef{eq1}). Hence, $(x^\phi)'$ is continuous. Thus $x^\phi$ is Lipschitz on $[0,t]$ by the mean value theorem. Hence $x^\phi$ is Lipschitz on $[-h,t]$ and thus $x_t^\phi\in V_{\cal D}$. Next, it is clear that $S(0,\phi)=\phi$ for all
$\phi\in V_{\cal D}$.
To see that the semigroup property holds, fix $\phi$ and define for some $t>0$ and $\tau>0$
\betaegin{eqnarray}
y(s)&:=&\betaegin{cases}
\phi(s),&s\in[-h,0]
\\
S(s,\phi)(0),& s\in[0,t]
\\
S(s-t,S(t,\phi))(0),&s\in[t,t+\tau]
\varepsilonnd{cases}
\nonumber\\
z(s)&:=&\betaegin{cases}
\phi(s),&s\in[-h,0]
\\
S(s,\phi)(0),& s\in[0,t+\tau].
\varepsilonnd{cases}
\nonumber
\varepsilonnd{eqnarray}
We have $y=z$ on $[-h,t]$, hence in particular on $[t-h,t]$, thus $y_t=z_t$.
Now suppose that
$s\in(t,t+\tau]$. Let $\tauh\in[-h,0]$. If $s-t+\tauh\gammae0$, then
\betaegin{eqnarray}
x_{s-t}^{S(t,\phi)}(\tauh)=x_{s-t+\tauh}^{S(t,\phi)}(0)=S(s+\tauh-t,S(t,\phi))(0)=y(s+\tauh)=y_s(\tauh).
\nonumber
\varepsilonnd{eqnarray}
If $s-t+\tauh<0$ then
\betaegin{eqnarray}
x_{s-t}^{S(t,\phi)}(\tauh)&=&S(t,\phi)(s-t+\tauh)=x_t^{\phi}(s-t+\tauh)=x^\phi(s+\tauh)
\nonumber\\
&=&\betaegin{cases}
\phi(s+\tauh),&s+\tauh\lambdae 0
\\
S(s+\tauh,\phi)(0),&s+\tauh>0
\varepsilonnd{cases}
=y(s+\tauh)=y_s(\tauh).
\nonumber
\varepsilonnd{eqnarray}
Thus $x_{s-t}^{S(t,\phi)}=y_s$. Hence
\betaegin{eqnarray}
y'(s)=\fracrac{d}{ds}x_{s-t}^{S(t,\phi)}(0)=(x^{S(t,\phi)})'(s-t)=f(x_{s-t}^{S(t,\phi)})=f(y_s).
\nonumber
\varepsilonnd{eqnarray}
Hence with $t$ and $t_0$ replaced by $s$ and $t$ respectively, $y$ is a solution
of (\rhoef{eq7}) through $z_t$ on $[-h,t+\tau]$. One similarly shows that so is $z$. By uniqueness we have $y=z$ on $[-h,t+\tau]$. If we fill $s=t+\tau$ and use the
definitions of $y$ and $z$, we see that this implies the semigroup property.
To see continuity of $S$ note that
\betaegin{eqnarray}
&&|S(t,\phi)(\tauh)-S({{\overline v}erline \tauau}e,{\overline \phi})(\tauh)|
\nonumber\\
&\lambdae&|S(t,\phi)(\tauh)-S(t,{\overline \phi})(\tauh)| +|S(t,{\overline \phi})(\tauh)-S({{\overline v}erline \tauau}e,{\overline \phi})(\tauh)|
\nonumber\\
&=&|x^\phi(t+\tauh)-x^{\overline \phi}(t+\tauh)| +|x^{\overline \phi}(t+\tauh)-x^{\overline \phi}({{\overline v}erline \tauau}e+\tauh)|.
\nonumber
\varepsilonnd{eqnarray}
The first term can be estimated using our result on continuous
dependence (Theorem \rhoef{theo7}), the second using continuity of solutions in time.
$\sigmaqr45$
Continuous dependence and the semigroup property can be combined to prove the following
result:
\betaegin{corol}\lambdaabel{corol1}
Suppose that $f$ satisfies the assumptions of Theorem \rhoef{theo8},
$\phi\in V_{\cal D}$, $x^\phi(t)\lambdaongrightarrowx^*\in{\mathds R}$ as $t\rhoightarrow\infty$. Then $x^*$ is an equilibrium solution.
\varepsilonnd{corol}
\noindent{\em Proof. } Let $(t_k)\in[0,\infty)^{\bf N}$, $t_k\rhoightarrow\infty$, and fix $t>0$. Define a sequence via $\phi^k:=S(t_k,\phi)\in{\cal D}$ and
denote by $\phi^*$ the constant function with value $x^*$ on $[-h,0]$. Then
$\phi^k=x^\phi_{t_k}\lambdaongrightarrow\phi^*$ (uniformly) by our assumption. Similarly $S(t+t_k,\phi)\lambdaongrightarrow\phi^*$.
But also $S(t+t_k,\phi)=S(t,S(t_k,\phi))=S(t,\phi^k)\lambdaongrightarrowS(t,\phi^*)$ by Theorem \rhoef{theo8}. Hence $S(t,\phi^*)=\phi^*$. One can conclude that $x^*$ is an
equilibrium solution.
$\sigmaqr45$
The result can also be concluded from Lemma \rhoef{lem20}:
\betaigskip
\noindent
{\betaf Proof of Corollary \rhoef{corol1} via Lemma \rhoef{lem20}.}
Define $I:=[0,\infty)$, choose any $\phi\in V_{\cal D}$, and define $\sigma(t):=S(t,\phi)$. Then
$\sigma$ is a trajectory. We show that ${\overline v}erline {\sigma(I)}$
is compact, i.e., that $\sigmaigma(I)=S([0,\infty),\phi)$ is relative compact. Let $(t_n)\in
[0,\infty)^{\bf N}$. Case 1: $t_n\in[0,T]$ for all $n\in{\bf N}$ and some $T>0$. Hence, there exists
$(t_{n_j})\sigmaubset (t_n)$, ${{\overline v}erline \tauau}e\in[0,T]$ such that $t_{n_j}\lambdaongrightarrow{{\overline v}erline \tauau}e$ as $j\rhoightarrow\infty$. Then
$(S(t_{n_j},\phi))\sigmaubset (S(t_n,\phi))$ and $S(t_{n_j},\phi)\lambdaongrightarrowS({{\overline v}erline \tauau}e,\phi)$ by continuity.
Case 2: $(t_n)$ is unbounded. Then there exists some $(t_{n_j})\sigmaubset(t_n)$ such that
$t_{n_j}\lambdaongrightarrow\infty$ as $j\rhoightarrow\infty$. Thus $S(t_{n_j},\phi)\lambdaongrightarrow\phi^*$ where $\phi^*\in V_{\cal D}$
is defined as $\phi^*(t)=x^*$ for all $t\in[-h,0]$. Hence, in any case,
$(S(t_n,\phi))$ has a Cauchy subsequence, thus ${\overline v}erline {\sigma(I)}$ is compact. Now
note that for the $\omegaega$-limit set of the trajectory one has $\omegaega(\sigma)=\{\phi^*\}$. Then by Lemma \rhoef{lem20} one has $S(t,\phi^*)\in\omegaega(\sigma)=\{\phi^*\}$,
i.e., $S(t,\phi^*)=\phi^*$ for all $t\gammae 0$.
$\sigmaqr45$
\sigmaection{Invariant compact sets}\lambdaabel{s3}
\sigmaubsection{Assumptions, main results and discussion}
In the setting of Section \rhoef{s2} we now set $n=2$ and $B=0$, such that ${\cal D}=C([-h,0],{\mathds R}_+^2)\sigmaubset {\cal C}=C([-h,0],{\mathds R}^2)$, where ${\mathds R}_+=[0,\infty)$, and consider a functional
$j:{\cal D}\lambdaongrightarrow{\mathds R}_+$, a function $q:{\mathds R}_+\lambdaongrightarrow{\mathds R}$ and a DDE of the form
\betaegin{eqnarray}
w'=q(v)w,\;\;v'(t)=-\mu v(t)+j(w_t,v_t),\;\;t>0,\;\;(w_0,v_0)=(\varphi,\psi)\in{\cal D},
\nonumber\\
\lambdaabel{eq18}
\varepsilonnd{eqnarray}
where $\mu>0$ is a parameter.
Define ${{\overline v}erline q}:=\sigmaup q$ and suppose throughout the section that ${{\overline v}erline q}<\infty$, $q$ is locally Lipschitz, $j$ is almost
locally Lipschitz and that for some $k_j>0$ at least one of the two,
\betaegin{eqnarray}
j(\varphi,\psi)\lambdae k_j\|\varphi\|,\;\;{\rhom or}
\lambdaabel{eq19}\\
j(\varphi,\psi)\lambdae k_j\varphi(-\tau(\psi)),
\lambdaabel{eq20}
\varepsilonnd{eqnarray}
where $\tau:C([-h,0],{\mathds R}_+)\lambdaongrightarrow[{\underline \tau},h)$ for some ${\underline \tau}\in(0,h)$, holds.
Obviously (\rhoef{eq19}) is a weaker requirement. As we will see, however,
(\rhoef{eq20}) may lead to better results while still applicable to our model.
Our proofs in the context of invariant sets of bounded functions rely on an exponential
estimate for the $w$-component that uses the linearity of the $w$-equation. Exponential estimates can be derived for general DDE, see e.g. \cite[Corollary 6.1.1]{Hale}, so our approach possibly works for systems more general than (\rhoef{eq18}) too. In the context of our application, however, we found (\rhoef{eq18}) a good compromise between the wishes to be general and to provide sharp estimates for our model.
Now note that, supposing a solution through $(\varphi,\psi)\in{\cal D}$ exists, one has that
\betaegin{eqnarray}
w(t)&=&\betaegin{cases}
\varphi(t),&t\in[-h,0]
\\
\varphi(0)e^{\int_0^tq(v(s))ds},&t>0,
\varepsilonnd{cases}
\lambdaabel{eq21}
\varepsilonnd{eqnarray}
hence
\betaegin{eqnarray}
w(t)&\lambdae&\|\varphi\|q_e(t), \fracorall\;t\gammae-h,
\;\;{\rhom where}\;\;
q_e(t):=\betaegin{cases}
1,&t\in[-h,0]
\\
e^{{{\overline v}erline q} t},&t>0.
\varepsilonnd{cases}
\lambdaabel{eq22}
\varepsilonnd{eqnarray}
Note that $q_e$ is continuous, nondecreasing, increasing on $[0,\infty)$ and differentiable
on $[-h,0)\cup(0,\infty)$.
An important case is that $q$ is decreasing and has one positive zero, see also Section
\rhoef{ss5} and \cite{Getto1}. Hence, positivity of $q$ is not out-ruled, and
thus, looking at (\rhoef{eq21}), we
cannot expect that a next-state operator $\phi\mapsto S(t,\phi)$ maps
a set of the form $C([-h,0],[0,A]\tauimes[0,B])$, $A,B\in(0,\infty)$ into itself
(to avoid subindices, we here, other than in the previous section, let both $A$ and $B$ denote upper bounds). For similar reasons (see the proof of Theorem \rhoef{theo10}
(b) below) we cannot expect this for a set of $R$-Lipschitz functions either. On the other hand, filling (\rhoef{eq21})
into the second equation of (\rhoef{eq18}) yields a closed system in $v$ (depending on both initial histories).
Motivated by this, next to an initial result for the $w$-component, we will establish an invariant set for the $v$-component. We refer to the discussion section for possible extensions of this research.
Define for any $B>0$ and $R>0$ the set
\betaegin{eqnarray}
C_{B,R}:=\{\chi\in C([-h,0],[0,B]),\;lip\;\chi\lambdae R\}.
\lambdaabel{eq25}
\varepsilonnd{eqnarray}
Note that $C_{B,R}$ is convex and, by the Arzela-Ascoli theorem, compact.
Next, we formulate the main
results of this section and give proofs in the next subsection.
With the cases (\rhoef{eq19}--\rhoef{eq20}), respectively, we associate functions $f_l,f_\tauau:{\mathds R}_+\lambdaongrightarrow{\mathds R}_+;$
\betaegin{eqnarray}
f_l(t)&:=&\fracrac{k_j}{\mu+{{\overline v}erline q}}(e^{{{\overline v}erline q} t}-e^{-\mu t}),
\nonumber\\
f_\tau(t)&:=&
\betaegin{cases}
\fracrac{k_j}{\mu}(1-e^{-\mu t}),&{\rhom if}\;t\lambdae{\underline \tau}
\\
k_j\fracrac{{{\overline v}erline q}(e^{-\mu(t-{\underline \tau})}-e^{-\mu t})+\mu(e^{{{\overline v}erline q} (t-{\underline \tau})}-e^{-\mu t})}{\mu(\mu+q)},&{\rhom if}\;t>{\underline \tau},
\varepsilonnd{cases}
\nonumber
\varepsilonnd{eqnarray}
(where l stands for linear in reference to (\rhoef{eq19})). When writing about
these functions we will assume that the respective case holds, sometimes only implicitly.
\betaegin{theorem}\lambdaabel{theo10}
Under the assumptions of this subsection, the following holds for any $(\varphi,\psi)\in V_{\cal D}$ .
\betaegin{itemize}
\item[(a)] The system (\rhoef{eq18}) has a unique solution $x=(w,v)$ through $(\varphi,\psi)$ on
$[0,\infty)$. The solutions define a continuous semiflow in the sense
of Theorem \rhoef{theo8}.
\item[(b)] Choose $A$, $R$ and $T$ such that ${{\overline v}erline q} A e^{{{\overline v}erline q} T}\lambdae R$. Then, if $\|\varphi\|\lambdae A$ and $lip\;\varphi\lambdae R$ one has $lip\;w_t\lambdae R$ for all $t\in[0,T]$.
\item[(c)] Both, $f_l$ and $f_\tauau$ are
zero in zero, tend to $\infty$ at $\infty$, are increasing and continuous, $f_l$ is differentiable, and $f_\tauau$ is differentiable on $[0,{\underline \tau})\cup({\underline \tau},\infty)$. The functions
\[
t\lambdaongmapsto\fracrac{f_l(t)}{1-e^{-\mu t}}\;{\rhom and}\;
t\lambdaongmapsto\fracrac{f_\tauau(t)}{1-e^{-\mu t}},
\]
respectively, increase from $k_j/\mu$ to infinity on ${\mathds R}_+$, and equal $k_j/\mu$ on $[0,{\underline \tau}]$ and increase to infinity on $[{\underline \tau},\infty)$. Finally, $f_l(t)>f_\tau(t)$ for all $t>0$.
\item[(d)]
Assume that (\rhoef{eq19}) holds and choose $A$, $B$, $R$ and $T$ such that
$\fracrac{Af_l(T)}{1-e^{-\mu T}}\lambdae B$ and $R\gammae \max\{\mu B, k_jAe^{{{\overline v}erline q} T}\}$. Then, if $\|\varphi\|\lambdae A$ and $\psi\in C_{B,R}$ one has $v_t\in C_{B,R}$ for all $t\in[0,T]$.
\varepsilonnd{itemize}
If (\rhoef{eq20}) holds, then the following hold.
\betaegin{itemize}
\item[(e)]
Choose $A$, $B$, $R$ and $T$ such that $\fracrac{Af_\tauau(T)}{1-e^{-\mu T}}\lambdae B$ and
\[
R\gammae \max\{\mu B, k_jAq_e(T-{\underline \tau})\}.
\]
Then, if $\|\varphi\|\lambdae A$ and $\psi\in C_{B,R}$, one has $v_t\in C_{B,R}$ for all $t\in[0,T]$.
\item[(f)]
Choose $A$, $B$ and $R$ such that
$Ak_j< B\mu\lambdae R$ and $\partiale$ such that
$Ak_je^{{{\overline v}erline q}\partiale}=\mu B$.
Then, if $\|\varphi\|\lambdae A$ and $\psi\in C_{B,R}$, one has $v_t\in C_{B,R}$ for all $t\in[0,{\underline \tau}+\partiale]$.
\varepsilonnd{itemize}
\varepsilonnd{theorem}
For further discussion of the theorem we state some technical results.
\betaegin{lemma}\lambdaabel{lem31}
One has $\fracrac{k_je^{{{\overline v}erline q} t}}{\mu}>\fracrac{f_l(t)}{1-e^{-\mu t}}$ for all $t>0$
and $k_je^{{{\overline v}erline q}(t-{\underline \tau})}>\mu\fracrac{f_\tau(t)}{1-e^{-\mu t}}$ for $t>{\underline \tau}$.
\varepsilonnd{lemma}
Now, note that (f) is a simple corollary of (e). To prove this, define $T={\underline \tau}+\partiale$ in (e), and apply the second estimate of the lemma with $t=T$. We omit further details.
By the previous lemma, in Theorem \rhoef{theo10} (d) and (e) it would be sufficient to assume
that
\betaegin{eqnarray}
R\gammae\mu B\gammae
\betaegin{cases}
k_jAe^{{{\overline v}erline q} T},& {\rhom respectively},
\\
k_jAe^{{{\overline v}erline q}(T-{\underline \tau})},&
\nonumber
\varepsilonnd{cases}
\nonumber
\varepsilonnd{eqnarray}
which is stronger but easier to check than the present assumptions.
Note that (e) allows to establish for the solution a lower bound and a lower Lipschitz constant
than (d): Fix $A$ and $T$. Then the lowest bound we can achieve through (d) is $B_d:=\fracrac{Af_l(T)}{1-e^{-\mu T}}$, whereas through (e) we can achieve the bound $B_e:=\fracrac{Af_\tau(T)}{1-e^{-\mu T}}<B_d$. The lowest Lipschitz constant we can achieve through (d) is
$R_d:=\max\{\mu B_d,k_jAe^{{{\overline v}erline q} T}\}>\max\{\mu B_e,k_jAq_e(T-{\underline \tau})\}=:R_e$, where $R_e$ is a
(the lowest) Lipschitz constant we can achieve through (e).
We get invariance for a longer time through (e) than through (d): Fix $A$, $B$ and $R$ such that $\fracrac{Ak_j}{\mu}<B$ and $R\gammae\mu B$. Then the largest time spans which (d) and (e) yield are respectively $t_d:=\min\{t_{d1},t_{d2}\}$ and $t_e=\min\{t_{e1},t_{e2}\}$, where the involved quantities are defined via $\fracrac{Af_l(t_{d1})}{1-e^{-\mu t_{d1}}}=B$, $\fracrac{Af_\tau(t_{e1})}{1-e^{-\mu t_{e1}}}=B$, $R=\max\{\mu B,Ak_je^{{{\overline v}erline q} t_{d2}}\}$
and $R=\max\{\mu B,Ak_jq_e(t_{e2}-{\underline \tau})\}$. One has $t_{dj}<t_{ej}$, $j=1,2$,
hence $t_d<t_e$.
Theorem \rhoef{theo10} (f) shows that, if (\rhoef{eq20}) holds,
there is a lower bound (${\underline \tau}$) for the time for which invariance holds, which is uniform for all $A$, $B$ satisfying $\fracrac{Ak_j}{\mu}<B$. If merely (\rhoef{eq19})
holds we cannot get such a lower bound through (d).
\sigmaubsection{Proofs}
We start with some general facts regarding the computation with almost locally Lipschitz
functions. In the following lemma we let $f$ and $g$ denote arbitrary functions and ${\cal D}\sigmaubset {\cal C}$ an arbitrary domain.
\betaegin{lemma}\lambdaabel{lem16}
(a) Suppose that $f,g:{\cal D}\sigmaubset {\cal C}\lambdaongrightarrow{\mathds R}$ are almost locally Lipschitz. Then so are $fg$, $(f,g)$ and $f+g$.
\\
(b) Let $f:{\cal D}\sigmaubset{\cal C}\lambdaongrightarrow{\mathds R}$ be almost locally Lipschitz and $g:f({\cal D})\sigmaubset{\mathds R}\lambdaongrightarrow{\mathds R}$
locally Lipschitz, then $g\circ f:{\cal D}\lambdaongrightarrow{\mathds R}$ is almost locally Lipschitz.
\varepsilonnd{lemma}
\noindent{\em Proof. } (a) Clearly $fg$ is continuous. Now let $\phi_0\in{\cal D}$, $R>0$. Choose $\partiale_{f}$, $k_{f}$, $\partiale_{g}$, $k_{g}$
in notation similar as in Definition \rhoef{def1} and according to the definition. Define $k:=\max\{
k_{f},k_{g}\}$. By continuity of $f$ and $g$ we can
choose $M$ and $\partiale_1$ such that $f$ and $g$ are bounded by $M$ on ${{\overline v}erline B}_{\partiale_1}(\phi_0)$. Define
$\partiale:=\min\{\partiale_{f},\partiale_{g},\partiale_1\}$. Then $fg$ is $k$-Lipschitz on $V(\phi_0;R,\partiale)$, hence almost locally Lipschitz. The remainder of the proof of (a) is obvious.
\\
(b) First, clearly $g\circ f$ is continuous. Next, let $\phi_0\in{\cal D}$, $R>0$, choose $\varepsilon$, $k_1$ such that $g$ is $k_1$-Lipschitz
on $B_\varepsilon(f(\phi_0))$. Choose, $\partiale$, $k_2$ such that $f$ is $k_2$-Lipschitz on
$V(\phi_0;\partiale,R)$ and $f(B_\partiale(\phi_0))\sigmaubset B_\varepsilon(f(\phi_0))$. Let $\varphi, \psi\in V(\phi_0;\partiale,R)$.
Then the following estimate implies the statement:
\betaegin{eqnarray}
|g(f(\varphi))-g(f(\psi))|\lambdae k_1|f(\varphi)-f(\psi)|\lambdae k_1k_2\|\varphi-\psi\|.
\nonumber
\varepsilonnd{eqnarray}
$\sigmaqr45$
In view of applying the general theory we next would like to show that a functional $f$
associated with (\rhoef{eq18}) is almost locally Lipschitz.
Due to the previous result it is sufficient to show that so are the components of $f$ and we
start with the first component.
\betaegin{lemma}\lambdaabel{lem21} The functional $f_1:{\cal D}\lambdaongrightarrow{\mathds R};\; f_1(\varphi,\psi):=q(\psi(0))\varphi(0)$ is locally Lipschitz, in particular
almost locally Lipschitz.
\varepsilonnd{lemma}
\noindent{\em Proof. } First note that the projection map and the evaluation map
\betaegin{eqnarray}
&&C([-h,0],{\mathds R}_+^2)\lambdaongrightarrowC([-h,0],{\mathds R}_+);\;(\varphi,\psi)\mapsto\varphi,
\;{\rhom and}
\nonumber\\
&&C([-h,0],{\mathds R}_+)\lambdaongrightarrow{\mathds R};\;\varphi\mapsto\varphi(0)
\nonumber
\varepsilonnd{eqnarray}
and analogous maps for the $\psi$-component are locally Lipschitz. Hence, by the
preservation of local Lipschitzianity under composition and the Lipschitz property of $q$
it follows that $(\varphi,\psi)\mapsto q(\psi(0))$ is locally Lipschitz. Moreover $(\varphi,\psi)\mapsto\varphi(0)$ is locally Lipschitz. Thus by the product rule for locally Lipschitz functions so is $f_1$.
$\sigmaqr45$
\noindent
{\betaf Proof of Theorem \rhoef{theo10} (a).} By Lemmas \rhoef{lem16} and \rhoef{lem21} it follows that
\betaegin{eqnarray}
f(\varphi,\psi)=(q(\psi(0))\varphi(0),-\mu\psi(0)+j(\varphi,\psi))^T
\nonumber
\varepsilonnd{eqnarray}
is almost locally Lipschitz. Property $(F)$ is guaranteed by non-negativity of $j$. The
boundedness property of $f$ required in Theorem \rhoef{theo8} is guaranteed by continuity of
$q$
and (\rhoef{eq19}--\rhoef{eq20}). The required boundedness property of the trajectory can be
guaranteed by (\rhoef{eq21}--\rhoef{eq22}) if one integrates the $v$-equation in (\rhoef{eq18}) using
${{\overline v}erline q}<\infty$ and the
variations of constants formula. Application of Theorem \rhoef{theo8} completes the proof.
$\sigmaqr45$
\betaigskip
\noindent
{\betaf Proof of Theorem \rhoef{theo10} (b).} Let $t\in[0,T]$. It is equivalent to show that $lip\;w|_{[t-h,t]}\lambdae R$. Since $lip\;\varphi\lambdae R$ it follows that
$w$ is $R$-Lipschitz on $[t-h,t]\cap[-h,0]$. On $[t-h,t]\cap[0,\infty)$ the function $w$ is
differentiable with
\betaegin{eqnarray}
|w'(t)|\lambdae|q(v(t))|\|\varphi\|q_e(t)\lambdae{{\overline v}erline q} A e^{{{\overline v}erline q} t}\lambdae{{\overline v}erline q} A e^{{{\overline v}erline q} T}\lambdae R.
\nonumber
\varepsilonnd{eqnarray}
$\sigmaqr45$
In subsequent proofs we will sometimes omit bars in ${{\overline v}erline q}$ and ${\underline \tau}$ for the sake of the presentation.
\betaegin{lemma}\lambdaabel{lem22}
One has for any $t>0$
\betaegin{eqnarray}
v(t)&\lambdae&
\betaegin{cases}
e^{-\mu t} \psi(0)+\|\varphi\|f_l(t),&{\rhom if} (\rhoef{eq19})\;{\rhom holds},
\\
e^{-\mu t} \psi(0)+\|\varphi\|f_\tauau(t),&{\rhom if} (\rhoef{eq20})\;{\rhom holds}.
\varepsilonnd{cases}
\lambdaabel{eq23}
\varepsilonnd{eqnarray}
Moreover $f_l(t)>f_\tau(t)$ for all $t>0$.
\varepsilonnd{lemma}
\noindent{\em Proof. } By the variation of constants formula
\betaegin{eqnarray}
v(t)=e^{-\mu t}\psi(0)+e^{-\mu t}\int_0^te^{\mu s}j(w_s,v_s)ds.
\nonumber
\varepsilonnd{eqnarray}
If (\rhoef{eq19}) holds,
\betaegin{eqnarray}
e^{-\mu t}\int_0^te^{\mu s}j(w_s,v_s)ds\lambdae k_je^{-\mu t}\int_0^te^{\mu s}\|w_s\|ds\lambdae
\|\varphi\|k_je^{-\mu t}\int_0^te^{(\mu+q)s}ds,
\nonumber
\varepsilonnd{eqnarray}
which yields the first statement.
If (\rhoef{eq20}) holds, then
\betaegin{eqnarray}
&&e^{-\mu t}\int_0^te^{\mu s}j(w_s,v_s)ds\lambdae e^{-\mu t}k_j\int_0^te^{\mu s}w(s-\tau(v_s))ds
\nonumber\\
&\lambdae&e^{-\mu t}k_j\|\varphi\|\int_0^te^{\mu s}q_e(s-\tau(v_s))ds
\lambdae e^{-\mu t}k_j\|\varphi\|\int_0^te^{\mu s}q_e(s-\tau)ds.
\nonumber
\varepsilonnd{eqnarray}
If $t\lambdae\tau$ the statement is obvious. If $t>\tau$, then
\betaegin{eqnarray}
&&e^{-\mu t}\int_0^te^{\mu s}j(w_s,v_s)ds\lambdae k_j\|\varphi\|
[e^{-\mu t}\int_0^\tau e^{\mu s}ds+e^{-\mu t}\int_\tau^te^{q(s-\tau)+\mu s}ds],
\nonumber
\varepsilonnd{eqnarray}
which also yields the first statement. Now note that by the above estimates
\betaegin{eqnarray}
f_l(t)=k_je^{-\mu t}\int_0^te^{(\mu+q)s}ds, \;\;
f_\tau(t)=k_je^{-\mu t}\int_0^te^{\mu s}q_e(s-\tau)ds.
\nonumber
\varepsilonnd{eqnarray}
Hence $f_l(t)>f_\tau(t)$ for all $t>0$ if $e^{qs}>q_e(s-\tau)$ for all $s>0$, which is the case.
$\sigmaqr45$
\noindent
{\betaf Proof of Theorem \rhoef{theo10} (c).} First note that by the previous lemma
$f_l(t)>f_\tau(t)$ for all $t>0$.
Next, if (\rhoef{eq19}) holds,
\betaegin{eqnarray}
&&{\rhom sgn}\fracrac{d}{dt}\fracrac{f_l(t)}{1-e^{-\mu t}}
={\rhom sgn}[(qe^{qt}+\mu e^{-\mu t})(1-e^{-\mu t})-(e^{qt}-e^{-\mu t})\mu e^{-\mu t}]
\nonumber\\
&=&{\rhom sgn}[qe^{qt}-(q+\mu)e^{(q-\mu)t}+\mu e^{-\mu t}]={\rhom sgn}\;g(t)
\nonumber
\varepsilonnd{eqnarray}
for obviously defined $g$. Then $g(0)=0$ and
\betaegin{eqnarray}
g'(t)&=&q^2e^{qt}+(\mu^2-q^2)e^{(q-\mu)t}-\mu^2e^{-\mu t}
\nonumber\\
&=&q^2e^{qt}(1-e^{-\mu t})+\mu^2e^{-\mu t}(e^{qt}-1)>0.
\nonumber
\varepsilonnd{eqnarray}
Thus $g(t)>0$ for all $t>0$ and hence $t\mapsto f_l(t)/(1-e^{-\mu t})$ is increasing.
If (\rhoef{eq20}) holds, to see that $t\mapsto f_\tauau(t)/(1-e^{-\mu t})$ is increasing, it is sufficient to show that
\betaegin{eqnarray}
g(t):=\fracrac{q(e^{-\mu(t-\tau)}-e^{-\mu t})+\mu(e^{q (t-\tau)}-e^{-\mu t})}{1-e^{-\mu t}}
\nonumber
\varepsilonnd{eqnarray}
is increasing for $t>\tau$. One has
\betaegin{eqnarray}
&&{\rhom sgn}\;g'(t)={\rhom sgn}\;\{[q(\mu e^{-\mu t}-\mu e^{-\mu(t-\tau)})+\mu(qe^{q(t-\tau)}+\mu e^{-\mu t})](1-e^{-\mu t})
\nonumber\\
&&-\mu e^{-\mu t}[q(e^{-\mu(t-\tau)}-e^{-\mu t})+\mu(e^{q(t-\tau)}-e^{-\mu t})]\}
\nonumber\\
&=&{\rhom sgn}\;\{[q(e^{-\mu t}- e^{-\mu(t-\tau)})+qe^{q(t-\tau)}+\mu e^{-\mu t}](1-e^{-\mu t})
\nonumber\\
&&
-e^{-\mu t}[q(e^{-\mu(t-\tau)}-e^{-\mu t})+\mu(e^{q(t-\tau)}-e^{-\mu t})]\}
\nonumber\\
&=&{\rhom sgn}\;\{q(e^{-\mu t}- e^{-\mu(t-\tau)})+qe^{q(t-\tau)}+\mu e^{-\mu t}
+q(e^{-\mu(2t-\tau)}-e^{-2\mu t})
\nonumber\\
&&-qe^{(q-\mu)t-q\tau}-\mu e^{-2\mu t}+q(e^{-2\mu t}-e^{-\mu(2t-\tau)})
+\mu(e^{-2\mu t}-e^{(q-\mu)t-q\tau})\}
\nonumber\\
&=&{\rhom sgn}\;\{q(e^{-\mu t}- e^{-\mu(t-\tau)})+qe^{q(t-\tau)}+\mu e^{-\mu t}
-(q+\mu)e^{(q-\mu)t-q\tau}\}
\nonumber\\
&=&sgn\;h(q)
\nonumber
\varepsilonnd{eqnarray}
for obviously defined $h$. Then $h(0)=0$. Next,
\betaegin{eqnarray}
h'(q)&=&e^{-\mu t}- e^{-\mu(t-\tau)}+e^{q(t-\tau)}+q(t-\tau)e^{q(t-\tau)}
\nonumber\\
&&-[e^{(q-\mu)t-q\tau}+(q+\mu)(t-\tau)e^{(q-\mu)t-q\tau}],
\nonumber\\
h'(0)&=&1-e^{-\mu(t-\tau)}-\mu(t-\tau)e^{-\mu t}=:j(t)
\nonumber
\varepsilonnd{eqnarray}
in obvious notation. Then $j'(t)=\mu e^{-\mu t}[e^{\mu\tau}-1+\mu(t-\tau)]>0$, hence $j(t)> j(\tau)=0$
and thus $h'(0)> 0$. Next,
\betaegin{eqnarray}
h''(q)&=&(t-\tau) e^{q(t-\tau)}\{2+q (t-\tau)-[2+(q+\mu)(t-\tau)]e^{-\mu t}\}
\nonumber\\
&=&(t-\tau)e^{q(t-\tau)}k(q)
\nonumber
\varepsilonnd{eqnarray}
for obviously defined $k$. Then, applying $e^x\gammae 1+x$ to $x=\mu(t-\tau)$,
\betaegin{eqnarray}
k(0)&=&2-[2+\mu (t-\tau)]e^{-\mu t}\gammae 1-e^{-\mu t}+1-e^{-\mu\tau}> 0,
\nonumber\\
k'(q)&=&t-\tau-e^{-\mu t}(t-\tau)>0.
\nonumber
\varepsilonnd{eqnarray}
Hence, $k$ is positive for $q>0$, thus so is $h''$, hence so is $h'$, thus so is $h$,
hence so is $sgn\;g'$. We have shown that $t\mapsto f_\tau(t)(1-e^{-\mu t})$ is
increasing. Monotonicity of $f_l$ follows from monotonicity of
$f_l(t)/(1-e^{-\mu t})$ and the same conclusion holds for $f_\tauau$.
Using that $(1-e^{-\mu t})^{-1}$ is bounded at infinity the remaining statements are
easy to see.
$\sigmaqr45$
\betaegin{lemma}\lambdaabel{lem28}
Assume that (\rhoef{eq19}) holds and that $A$, $B$ and $T$ are such that
$\fracrac{Af_l(T)}{1-e^{-\mu T}}\lambdae B$. Then $\|\varphi\|\lambdae A$ and $\|\psi\|\lambdae B$ imply
that $v(t)\lambdae B$ for all $t\in[-h,T]$.
\varepsilonnd{lemma}
\noindent{\em Proof. } By (\rhoef{eq23}) one has
$
v(t)\lambdae Be^{-\mu t}+Af_l(t)
$
for $t\in(0,T]$.
Hence $v(t)\lambdae B$ if $Af_l(t)/(1-e^{-\mu t})\lambdae B$ and the latter follows by assumption and Theorem \rhoef{theo10} (c).
$\sigmaqr45$
An elaboration of the maximum in the following lemma will be carried out further down.
\betaegin{lemma}\lambdaabel{lem27}
Let $\|\varphi\|\lambdae A$ and $\|\psi\|\lambdae B$. Let $T>0$ and choose
\betaegin{eqnarray}
R&\gammae&
\betaegin{cases}
\max_{t\in[T-h,T]\cap[0,\infty)}\max\{k_jq_e(t)A, \mu(e^{-\mu t}B+Af_l(t))\},&
\\
{\rhom if}\;(\rhoef{eq19})\; {\rhom holds},&
\\
\max_{t\in[T-h,T]\cap[0,\infty)}\max\{k_jq_e(t-{\underline \tau})A, \mu(e^{-\mu t}B+Af_\tauau(t))\},
\\
{\rhom if}\;(\rhoef{eq20})\; {\rhom holds}.
\varepsilonnd{cases}
\nonumber
\varepsilonnd{eqnarray}
Then, if $lip\;\psi\lambdae R$, also $lip\;v_T\lambdae R$.
\varepsilonnd{lemma}
\noindent{\em Proof. } We should show that $lip\;v|_{[T-h,T]}\lambdae R$. First,
\betaegin{eqnarray}
lip\;v|_{[T-h,T]\cap[-h,0]}=lip\;\psi|_{[T-h,T]\cap[-h,0]}
\lambdae R.
\nonumber
\varepsilonnd{eqnarray}
Next, if (\rhoef{eq19}) holds, we get
$
v'(t)\lambdae j(w_t,v_t)\lambdae k_j\|w_t\|\lambdae k_j q_e(t)\|\varphi\|.
$
If (\rhoef{eq20}) holds, then
$
v'(t)\lambdae k_jw(t-\tau(v_t))\lambdae k_j q_e(t-\tau(v_t))\|\varphi\|\lambdae k_j q_e(t-{\underline \tau})\|\varphi\|.
$
Moreover
\betaegin{eqnarray}
v'(t)\gammae-\mu v(t)\gammae \betaegin{cases}
-\mu(e^{-\mu t}|\psi(0)|+\|\varphi\|f_l(t)),&{\rhom if}\;(\rhoef{eq19})\;{\rhom holds}
\\
-\mu(e^{-\mu t}|\psi(0)|+\|\varphi\|f_\tauau(t)),&{\rhom if}\;(\rhoef{eq20})\;{\rhom
holds}.
\varepsilonnd{cases}
\nonumber
\varepsilonnd{eqnarray}
Hence for $t>0$ one has
\betaegin{eqnarray}
|v'(t)|&\lambdae&
\max\{k_jq_e(t)\|\varphi\|,\mu(|\psi(0)|e^{-\mu t}+\|\varphi\|f_l(t))\},
\nonumber\\
&&{\rhom if}\;(\rhoef{eq19})\;{\rhom holds}
\nonumber\\
|v'(t)|&\lambdae&\max\{k_jq_e(t-{\underline \tau})\|\varphi\|,\mu(|\psi(0)|e^{-\mu t}+\|\varphi\|f_\tauau(t))\},
\nonumber\\
&&{\rhom if}\;(\rhoef{eq20})\;{\rhom
holds}.
\nonumber
\varepsilonnd{eqnarray}
Hence $
lip\;v|_{[T-h,T]\cap[0,\infty)}\lambdae\max_{t\in[T-h,T]\cap[0,\infty)}|v'(t)|\lambdae R.
$
$\sigmaqr45$
\betaegin{lemma}\lambdaabel{lem26}
Assume that (\rhoef{eq19}) holds, choose $A$ and $B$ such that $Ak_j/\mu<B$ and define $t_1$ via $A\fracrac{f_l(t_1)}{1-e^{-\mu t_1}}=B$, then if $T\lambdae t_1$
one has
\betaegin{eqnarray}
\max_{t\in[0,T]}\mu[ e^{-\mu t}B+Af_A(t)]=\mu B.
\nonumber
\varepsilonnd{eqnarray}
\varepsilonnd{lemma}
\noindent{\em Proof. } Define $g:[0,t_1]\lambdaongrightarrow{\mathds R}_+;\;g(t):=\mu[ e^{-\mu t}B+Af_l(t)]$. Then,
\betaegin{eqnarray}
g'(t)&=&\mu[\fracrac{Ak_j}{\mu+q}(qe^{qt}+\mu e^{-\mu t})-\mu Be^{-\mu t}], \;
g'(0)=\mu(Ak_j-\mu B)<0,
\nonumber\\
g'(t_1)&=&B\mu[\fracrac{(1-e^{-\mu t_1})(qe^{qt_1}+\mu e^{-\mu t_1})}{e^{q t_1}-e^{-\mu t_1}}
-\mu e^{-\mu t_1}]
\nonumber\\
&=&
\fracrac{B\mu}{e^{q t_1}-e^{-\mu t_1}}
[qe^{qt_1}+\mu e^{-\mu t_1}-(q+\mu )e^{-(\mu-q) t_1}]
=\fracrac{B\mu}{e^{q t_1}-e^{-\mu t_1}}h(t_1)
\nonumber
\varepsilonnd{eqnarray}
for obviously defined $h$. We have seen in the proof of Theorem \rhoef{theo10} (c) that $h(t_1)>0$.
Thus $g'(t_1)>0$. Next
\betaegin{eqnarray}
g''(t)&=&\mu[\fracrac{Ak_j}{\mu+q}(q^2e^{qt}-\mu^2e^{-\mu t})+\mu^2Be^{-\mu t}]
\nonumber\\
&=&\mu[\fracrac{Ak_j}{\mu+q}(q^2e^{qt}-\mu^2e^{-\mu t})+\mu^2e^{-\mu t}\fracrac{Af_l(t_1)}{1-e^{-\mu t_1}}]
\nonumber\\
&>&\mu[\fracrac{Ak_j}{\mu+q}(q^2e^{qt}-\mu^2e^{-\mu t})+\mu^2e^{-\mu t}\fracrac{Af_l(t)}{1-e^{-\mu t}}]
\nonumber\\
&=&\fracrac{\mu Ak_j}{\mu+q}[q^2e^{qt}-\mu^2e^{-\mu t}+\mu^2e^{-\mu t}\fracrac{e^{qt}-e^{-\mu t}}{1-e^{-\mu t}}]
\nonumber\\
&=&\fracrac{\mu Ak_j}{(\mu+q)(1-e^{-\mu t})}[q^2e^{qt}(1-e^{-\mu t})+\mu^2e^{-\mu t}
(e^{qt}-1)]>0.
\nonumber
\varepsilonnd{eqnarray}
Hence, $g'$ increases monotonously from a negative value to a positive value. Thus $g$
decreases monotonously to a minimum, then increases monotonously, hence assumes a
maximum either in zero, or in $t_1$. Since $g(0)=g(t_1)=\mu B$ the statement follows.
$\sigmaqr45$
\noindent
{\betaf Proof of Theorem \rhoef{theo10} (d).}
First note that $T<t_1$ for $t_1$ as in Lemma \rhoef{lem26}. Hence by this lemma and
Lemma \rhoef{lem27} one has $lip\;v_t\lambdae R$ for all $t\in[0,T]$. The boundedness property follows by
Lemma \rhoef{lem28}.
$\sigmaqr45$
\betaigskip
\noindent
{\betaf Proof of Theorem \rhoef{theo10} (e).} The stated boundedness is implied by the monotonicity shown in (c) and Lemma \rhoef{lem22}.
Moreover, since $\mu B\gammae\mu( Be^{-\mu t}+Af_\tauau(t))$ for $t\in[0,T]$, one has
\betaegin{eqnarray}
R&\gammae&\max\{Ak_jq_e(T-\tau),\mu B\}
\nonumber\\
&\gammae&\max_{t\in[0,T]}\max\{Ak_jq_e(t-\tau),\mu( Be^{-\mu t}+Af_\tauau(t))\}.
\nonumber
\varepsilonnd{eqnarray}
Hence the Lipschitz-property follows by Lemma \rhoef{lem27}.
$\sigmaqr45$
\betaigskip
\noindent
{\betaf Proof of Lemma \rhoef{lem31}.} For $t>s\gammae0$ one has
\betaegin{eqnarray}
&&e^{q(t-s)}>\fracrac{q(e^{-\mu(t-s)}-e^{-\mu t})+\mu(e^{q (t-s)}-e^{-\mu t})}{(\mu+q)(1-e^{-\mu t})}
\lambdaabel{eq24}\\
&\Lambdaeftrightarrow&qe^{q(t-s)}+(\mu+q)e^{-\mu t}-(\mu+q)e^{q(t-s)-\mu t}-qe^{-\mu(t-s)}> 0
\nonumber\\
&\Lambdaeftrightarrow& e^{q(t-s)}f(t)>0,\;\;{\rhom where}
\nonumber\\
f(t)&:=&q+(\mu+q)e^{-q(t-s)-\mu t}-(\mu+q)e^{-\mu t}-qe^{-(q+\mu)(t-s)}.
\nonumber
\varepsilonnd{eqnarray}
Then
\betaegin{eqnarray}
f(s)&=&0,
\nonumber\\
f'(t)&=&(\mu+q)e^{-\mu t}[-(\mu+q)e^{-q(t-s)}+\mu+qe^{-q(t-s)+\mu s}]
\nonumber\\
&=&(\mu+q)e^{-\mu t}[qe^{-q(t-s)}(e^{\mu s}-1)+\mu(1-e^{-q(t- s)})]>0.
\nonumber
\varepsilonnd{eqnarray}
Hence $f(t)>0$ for all $t>s$ and (\rhoef{eq24}) holds. Setting $s=\tau$ and
$s=0$ shows the respective statements.
$\sigmaqr45$
\sigmaection{The stem cell model formulated as a SD-DDE}\lambdaabel{s1}
In this section, in regard to Section \rhoef{s3} and the DDE (\rhoef{eq18}), we keep the assumptions on $q$, $\mu$ and ${\cal D}$, but specify $j$ and $\tau$, such that the DDE (\rhoef{eq18})
becomes the SD-DDE (\rhoef{eq11}--\rhoef{eq14}) that describes the stem cell
dynamics. Then we apply the previous results to analyze this SD-DDE.
\sigmaubsection{Assumptions and main results}\lambdaabel{ss1}
Suppose that the function $g$ satisfies the following property, which we denote
by (G):
There exist $x_1, x_2, b, K,\varepsilon \in{\mathds R}$, such that $x_1<x_2$, $0<\varepsilon<K$ and $b>0$,
and $g:{{\overline v}erline B}_b(x_2)\tauimes {\mathds R}_+\lambdaongrightarrow{\mathds R}$
\betaegin{itemize}
\item[($G_1$)] is locally Lipschitz in the second argument, uniformly with respect to the first,
\item[($G_2$)] is partially differentiable with respect to the first argument with $D_1g$
Lipschitz and
\[
\sigmaup_{(y,z)\in {{\overline v}erline B}_b(x_2)\tauimes {\mathds R}_+}|D_1g(y,z)|<\fracrac{K}{b},
\]
\item[($G_3$)] satisfies $\varepsilon\lambdae g(y,z)\lambdae K$ on ${{\overline v}erline B}_b(x_2)\tauimes {\mathds R}_+$ and $x_2-x_1\in(0,\fracrac{b}{K}\varepsilon)$.
\varepsilonnd{itemize}
Note that $(G_3)$ implies that $x_1\in{{\overline v}erline B}_b(x_2)$. We now define
$h:=\fracrac{b}{K}$. The following result is an application of the Picard-Lindel\"of
theorem.
\betaegin{lemma}
Let $\psi\in C([-h,0],{\mathds R}_+)$. Then there exists a unique solution $y=y(\cdot,\psi)$ on $[0,h]$ of
(\rhoef{eq13})
with $y([0,h],\psi)\sigmaubset{{\overline v}erline B}_b(x_2)$. Moreover, there exists a unique
\[
\tau=\tau(\psi)\in[\fracrac{x_2-x_1}{K},\fracrac{x_2-x_1}{\varepsilon}]\sigmaubset(0,h)
\]
solving (\rhoef{eq14}).
\varepsilonnd{lemma}
\noindent{\em Proof. } Define $f_\psi:[0,h]\tauimes{{\overline v}erline B}_b(x_2)\lambdaongrightarrow{\mathds R};\;f_\psi(s,y):=-g(y,\psi(-s))$ and with $f_\psi$
a non-autonomous ODE $y'(s)=f_\psi(s,y(s))$. Then (G) guarantees directly that $f_\psi$
satisfies the conditions of the Picard-Lindel\"of Theorem, e.g. \cite[Theorem II.1.1]{Hartman},
which guarantees that there exists a unique solution $y$ on $[0,h]$, since we defined $h:=\fracrac{b}{K}$. The remaining statements can be shown by integrating the ODE and using (G3).
$\sigmaqr45$
Accordingly, with ${\underline \tau}:=(x_2-x_1)/K$ we can now define a functional
\[
\tau:C([-h,0],{\mathds R}_+)\lambdaongrightarrow[{\underline \tau},h)
\]
to describe the state-dependence of the delay. Moreover, we
suppose that $d:{{\overline v}erline B}_b(x_2)\tauimes {\mathds R}_+\lambdaongrightarrow{\mathds R}$ is bounded and Lipschitz and that
$\gamma:{\mathds R}_+\lambdaongrightarrow{\mathds R}_+$ is bounded and locally Lipschitz and define
\betaegin{eqnarray}
j(\varphi,\psi):=
\fracrac{\gamma(\psi(-\tau(\psi)))}{g(x_1,\psi(-\tau(\psi)))}g(x_2,\psi(0))\varphi(-\tau(\psi))e^{\int_0^{\tau(\psi)}
[d-D_1g](y(s,\psi),\psi(-s))ds}.
\nonumber\\
\lambdaabel{eq17}
\varepsilonnd{eqnarray}
Then clearly the DDE (\rhoef{eq18}) becomes the SD-DDE (\rhoef{eq11}--\rhoef{eq14}) and (\rhoef{eq20}) holds
with
\betaegin{eqnarray}
k_j:=\fracrac{K}{\varepsilon}
e^{(\fracrac{K}{b}+\sigmaup_{(y,z)\in{{\overline v}erline B}_b(x_2)\tauimes{\mathds R}_+}|d(y,z)|)h}\sigmaup_{z\in{\mathds R}_+}\gamma(z)<\infty.
\nonumber
\varepsilonnd{eqnarray}
The following result will be proven in the next subsection.
\betaegin{theorem}\lambdaabel{theo9}
For any $\phi=(\varphi,\psi)\in V_{\cal D}$, under the conditions given in this subsection, the SD-DDE (\rhoef{eq11}--\rhoef{eq14}) has a unique
solution $x^\phi=(w,v)$ on ${\mathds R}_+$ through $\phi$. The solutions
define a continuous semiflow in the
sense of Theorem \rhoef{theo8} and with $f_\tau$ as in Theorem \rhoef{theo10} (c) satisfy the invariance properties Theorem \rhoef{theo10} (e-f).
\varepsilonnd{theorem}
\sigmaubsection{Proofs}\lambdaabel{ss2}
We can apply Theorem \rhoef{theo10} to obtain the statement of Theorem \rhoef{theo9} if we show that $j$ is almost locally Lipschitz. To show this, it is useful to introduce a notation that summarizes
model ingredients with the same type of delay:
Let first $\beta:{\mathds R}_+\lambdaongrightarrow{\mathds R}$, $r:C([-h,0],{\mathds R}_+)\lambdaongrightarrow[0,h]$ and ${\cal G}:C([-h,0],{\mathds R}_+)\lambdaongrightarrow{\mathds R}$ be arbitrary maps.
As a tool to prove several results that follow we define the evaluation operator
\betaegin{eqnarray}
C([-h,0],{\mathds R}_+)\tauimes[-h,0]\lambdaongrightarrow{\mathds R};\;ev(\varphi,s):=\varphi(s).
\lambdaabel{eq15}
\varepsilonnd{eqnarray}
Trivially, $ev$ inherits continuity from the functions in its domain. We will show that $j$ is a special
case of the functional defined in the following lemma.
\betaegin{lemma}\lambdaabel{lem14}
Suppose that $\beta$ is locally Lipschitz and that $r$ and ${\cal G}$ are almost locally Lipschitz, then
the functional ${\cal D}\lambdaongrightarrow{\mathds R}_+;$
\betaegin{eqnarray}
(\varphi,\psi)\lambdaongmapsto\beta(\psi(-r(\psi)))\varphi(-r(\psi)){\cal G}(\psi)
\lambdaabel{eq16}
\varepsilonnd{eqnarray}
is almost locally Lipschitz.
\varepsilonnd{lemma}
\noindent{\em Proof. } By the discussed sum - and product rules and by other rules, which are straightforward, it suffices to show that
the two maps $\psi\mapsto\beta(\psi(-r(\psi)))$ and $(\varphi,\psi)\mapsto\varphi(-r(\psi))$ are almost locally Lipschitz. Now note that the first map can be decomposed as
\[
\psi\mapsto(\psi,-r(\psi)){x_m}apsto{ev}\psi(-r(\psi)){x_m}apsto{\beta}\beta(\psi(-r(\psi))).
\]
Hence it is continuous as a composition by continuity of $r$, $ev$ and $\beta$. Similarly the
second map can be written as
\[
(\varphi,\psi)\mapsto(\varphi,-r(\psi)){x_m}apsto{ev}\varphi(-r(\psi))
\]
and continuity can be concluded. Next, let $\psi_0\in C([-h,0],{\mathds R}_+)$, $R>0$. Choose $\partiale>0$,
$k$ such that $r$ is $k$-Lipschitz on $V(\psi_0;\partiale,R)$. Now note that for $\psi,\chi\in V(\psi_0;\partiale,R)$
\betaegin{eqnarray}
&&|\psi(-r(\psi)-\chi(-r(\chi)))|
\nonumber\\
&\lambdae&|\psi(-r(\psi)-\psi(-r(\chi)))|+|\psi(-r(\chi)-\chi(-r(\chi)))|
\nonumber\\
&\lambdae&R|r(\psi)-r(\chi)|+\|\psi-\chi\|\lambdae(Rk+1)\|\psi-\chi\|.
\nonumber
\varepsilonnd{eqnarray}
Hence, $\psi\mapsto\psi(-r(\psi))$ is almost locally Lipschitz. Since $\beta$ is locally Lipschitz,
$\psi\mapsto\beta(\psi(-r(\psi)))$ is almost local Lipschitz by the discussed composition rule. The stated
Lipschitz property of the second map follows similarly.
$\sigmaqr45$
A Gronwall-Lemma type estimate and use of ($G_1$) and ($G_2$) lead to the following result.
\betaegin{lemma}\lambdaabel{lem8}
The map $Y: C([-h,0],{\mathds R}_+)\lambdaongrightarrowC([0,h],{{\overline v}erline B}_b(x_2));\;Y(\psi)(t):=y(t,\psi)$ is locally Lipschitz.
\varepsilonnd{lemma}
\noindent{\em Proof. } Let $\psi_0,\psi, {{\overline v}erline \psi} \in C([-h,0],{\mathds R}_+)$.
One has
\betaegin{eqnarray}
&&|g(y(s,\psi),\psi(-s))-g(y(s,{{\overline v}erline \psi}),{{\overline v}erline \psi}(-s))|
\nonumber\\
&\lambdae&|g(y(s,\psi),\psi(-s))-g(y(s,{{\overline v}erline \psi}),\psi(-s))|
\nonumber\\
&&+|g(y(s,{{\overline v}erline \psi}),\psi(-s))-g(y(s,{{\overline v}erline \psi}),{{\overline v}erline \psi}(-s))|
=:(I)+(II)
\nonumber
\varepsilonnd{eqnarray}
in obvious notation. By ($G_2$) and the mean value theorem one has
\betaegin{eqnarray}
(I)&\lambdae&L_1|y(s,\psi)-y(s,{{\overline v}erline \psi})|\lambdae L_1\|y(\cdot,\psi)-y(\cdot,{{\overline v}erline \psi})\|= L_1\|Y(\psi)-Y({{\overline v}erline \psi})\|,
\nonumber
\varepsilonnd{eqnarray}
where $L_1:=\sigmaup_{(y,z)\in{{\overline v}erline B}_b(x_2)\tauimes {\mathds R}_+}|D_1g(y,z)|$.
By ($G_1$), one has $(II)\lambdae L_2\|\psi-{{\overline v}erline \psi}\|$ for some $L_2\gammae 0$ and $\psi$ and ${{\overline v}erline \psi}$ in a neighborhood of $\psi_0$. Now combine
\betaegin{eqnarray}
|y(t,\psi)-y(t,{{\overline v}erline \psi})|\lambdae \int_0^t|g(y(s,\psi),\psi(-s))-g(y(s,{{\overline v}erline \psi}),{{\overline v}erline \psi}(-s))ds|
\nonumber
\varepsilonnd{eqnarray}
with the previous estimates and $L_1<\fracrac{1}{h}$, which follows from ($G_2$), to complete the proof.
$\sigmaqr45$
We can use this result to deduce
\betaegin{lemma}\lambdaabel{lem9}
The map $C([-h,0],{\mathds R}_+)\lambdaongrightarrow[0,h];\;\psi\mapsto\tau(\psi)$ is locally Lipschitz.
\varepsilonnd{lemma}
\noindent{\em Proof. } Let ${{\overline v}erline \psi}, \psi\in C([-h,0],{\mathds R}_+)$. By definition of $\tau(\psi)$ and $\tau({{\overline v}erline \psi})$ one has
\betaegin{eqnarray}
y(\tau(\psi),\psi)=y(\tau({{\overline v}erline \psi}),{{\overline v}erline \psi})\;\;(=x_1).
\nonumber
\varepsilonnd{eqnarray}
Hence,
\betaegin{eqnarray}
|y(\tau(\psi),\psi)-y(\tau(\psi),{{\overline v}erline \psi})|=|y(\tau(\psi),{{\overline v}erline \psi})-y(\tau({{\overline v}erline \psi}),{{\overline v}erline \psi})|.
\nonumber
\varepsilonnd{eqnarray}
The left hand side is dominated by $\|Y(\psi)-Y({{\overline v}erline \psi})\|$. There exists some $t\in[0,h]$, such
that the right hand side equals
\betaegin{eqnarray}
&&|D_1y(t,{{\overline v}erline \psi})||\tau(\psi)-\tau({{\overline v}erline \psi})|
\nonumber\\
&=&|g(y(t,{{\overline v}erline \psi}),{{\overline v}erline \psi}(-t))||\tau(\psi)-\tau({{\overline v}erline \psi})|\gammae\varepsilon|\tau(\psi)-\tau({{\overline v}erline \psi})|
\nonumber
\varepsilonnd{eqnarray}
by $(G_3)$. Thus $|\tau(\psi)-\tau({{\overline v}erline \psi})|\lambdae\fracrac{1}{\varepsilon}|Y(\psi)-Y({{\overline v}erline \psi})|$ and the proof
is completed using Lipschitzianity of $Y$.
$\sigmaqr45$
\betaegin{lemma}\lambdaabel{lem10}
Let $G:C([-h,0],{\mathds R}_+)\tauimes C([0,h],{{\overline v}erline B}_b(x_2)) \lambdaongrightarrowC([0,h],{\mathds R})$ be an arbitrary locally Lipschitz operator with
\[
\sigmaup_{(\psi,z)}lip\;G(\psi,z)<\infty.
\]
Define ${\cal G}:C([-h,0],{\mathds R}_+)\lambdaongrightarrow{\mathds R};\;{\cal G}(\psi):=g(x_2,\psi(0))e^{G(\psi,Y(\psi))(\tau(\psi))}$. Then ${\cal G}$
is locally Lipschitz.
\varepsilonnd{lemma}
\noindent{\em Proof. } Choose $\varphi_0\in C([-h,0],{\mathds R}_+)$ and $R:=\sigmaup_{(\psi,z)}lip\;G(\psi,z)$. Choose
$k$ and $\partiale$ such that $G$ is $k$-Lipschitz on ${{\overline v}erline B}_\partiale((\varphi_0,Y(\varphi_0)))$ and $Y$ and $\tau$
are $k$-Lipschitz on ${{\overline v}erline B}_\partiale(\varphi_0)$. Choose $\varepsilon\lambdae\partiale$ such that
\[
|G(\psi,Y(\psi))-G(\varphi_0,Y(\varphi_0))|\lambdae\partiale,\;\;{\rhom if}\;\psi\in{{\overline v}erline B}_\varepsilon(\varphi_0).
\]
Let $\varphi,\psi\in{{\overline v}erline B}_\varepsilon(\varphi_0)$. Now note that $ev$ is $\max\{{\overline v}erline R,1\}$-Lipschitz on $V({\overline v}i;{\overline v}erline\partiale,{\overline v}erline R)\tauimes[-h,0]$ for any ${\overline v}i$, ${\overline v}erline\partiale$, ${\overline v}erline R$. Hence,
for $\psi, \chi\in{{\overline v}erline B}_\varepsilon(\varphi_0)$
\betaegin{eqnarray}
&&|ev(G(\psi,Y(\psi)),\tau(\psi))-ev(G(\chi,Y(\chi)),\tau(\chi))|
\nonumber\\
&\lambdae&\max\{R,1\}\{|G(\psi,Y(\psi))-G(\chi,Y(\chi))|+|\tau(\psi)-\tau(\chi)|\}
\nonumber\\
&\lambdae&\max\{R,1\}\{k[\|\psi-\chi\|+\|Y(\psi)-Y(\chi)\|]+|\tau(\psi)-\tau(\chi)|\}
\nonumber\\
&\lambdae&\max\{R,1\}\max\{k,k^2\}\|\psi-\chi\|.
\nonumber
\varepsilonnd{eqnarray}
Thus $\psi\mapsto ev(G(\psi,Y(\psi)),\tau(\psi))$ is locally Lipschitz. This implies that ${\cal G}$
is locally Lipschitz.
$\sigmaqr45$
\betaegin{lemma}\lambdaabel{lem13}
Let $J\sigmaubset{\mathds R}$, $k:J\tauimes {\mathds R}_+\lambdaongrightarrow{\mathds R}$ be an arbitrary Lipschitz and bounded map. Define
\betaegin{eqnarray}
&&G:C([-h,0],{\mathds R}_+)\tauimes C([0,h],J)
\lambdaongrightarrowC([0,h],{\mathds R});
\nonumber\\
&&G(\psi,z)(t):=\int_0^tk(z(s),\psi(-s))ds.
\nonumber
\varepsilonnd{eqnarray}
Then $G$ is Lipschitz and
\[
\sigmaup_{(\psi,z)}lip\;G(\psi,z)<\infty.
\]
\varepsilonnd{lemma}
\noindent{\em Proof. } The first result follows from the estimates
\betaegin{eqnarray}
&&|G(\psi,z)(t)-G({{\overline v}erline \psi},{\overline z})(t)|\lambdae\int_0^t|k(z(s),\psi(-s))-k({\overline z}(s),{{\overline v}erline \psi}(-s))|ds,
\nonumber\\
&&|k(z(s),\psi(-s))-k({\overline z}(s),{{\overline v}erline \psi}(-s))|\lambdae L[\|z-{\overline z}\|+\|\psi-{{\overline v}erline \psi}\|],
\nonumber
\varepsilonnd{eqnarray}
for some $L\gammae 0$.
Boundedness of $k$ implies the second statement.
$\sigmaqr45$
\betaegin{remark}
In Lemma \rhoef{lem10} and below we merely need local Lipschitzianity of $G$. We presented a
sketch of the rather straightforward proof of the previous lemma to also hint that
mere local Lipschitzianity of $k$ would not yield local Lipschitzianity of G, however.
The point is that continuous functions being close in a point obviously in general does not make them close in the sup-norm. See \cite{Appell} for more details on smoothness
properties of related Nemytskii-operators.
\varepsilonnd{remark}
We now combine our results and prove
\betaegin{prop}\lambdaabel{prop1}
The functional $j$ as defined in (\rhoef{eq17}) is almost locally Lipschitz.
\varepsilonnd{prop}
\noindent{\em Proof. } By Lemma \rhoef{lem14} it is sufficient to show
that $\fracrac{\gamma(\cdot)}{g(x_1,\cdot)}$ is locally Lipschitz and that $\tau$ and $\psi\mapsto g(x_2,\psi(0))\varepsilonxp\{\int_0^{\tau(\psi)}[d-D_1g](y(s,\psi),\psi(-s))ds\}$ are almost locally
Lipschitz. Local Lipschitzianity of the first map follows directly from local Lipschitzianity of
$\gamma$, ($G_1$) and ($G_3$).
(Almost) local Lipschitzianity of $\tau$ is shown in Lemma \rhoef{lem9}. (Almost) local Lipschitzianity
of the third map follows by Lemma \rhoef{lem10}, provided we show local Lipschitzianity
of $G$, defined as $G(\psi,z)(t):=\int_0^t[d-D_1g](z(s),\psi(-s))ds$ and that for this $G$ one
has $\sigmaup_{(\psi,z)}lip\;G(\psi,z)<\infty$. The latter follow
by Lemma \rhoef{lem13} from boundedness and Lipschitzianity of $k:=d-D_1g$ with $J:={{\overline v}erline B}_b(x_2)$. Thus, $j$ is almost locally
Lipschitz.
$\sigmaqr45$
\sigmaection{Examples of model ingredients}\lambdaabel{ss5}
In the previous section we have elaborated conditions on the model ingredients specified as functions $q$, $\gamma$, $g$ and $d$ and the nonnegative parameter $\mu$. The exact nature of the cellular and sub-cellular processes related to
these ingredients is subject to current research \cite{Vivanco}. In \cite{Getto1} a combination of available knowledge with mathematical considerations led to the specification
\betaegin{eqnarray}
&&q(z):=[2s_w(z)-1]d_w(z)-\mu_w,\;\;\gamma(z):=2[1-s_w(z)]d_w(z),\;\;{\rhom where}
\nonumber\\
&& s_w(z):=\fracrac{a_w}{1+k_a z},\;\;d_w(z):=\fracrac{p_w}{1+k_pz}
\nonumber
\varepsilonnd{eqnarray}
with $a_w\in[0,1]$ and $p_w$, $\mu_w$, $k_a$ and $k_p$ nonnegative parameters. It
is obvious that for these examples $q$ and $\gamma$ are Lipschitz, in particular locally
Lipschitz. The function $d$ considered is of the form
\[
d(y,z)=\fracrac{\alpha(y)}{1+k_dz}-\mu_u(y)
\]
for a nonnegative parameter $k_d$ and nonnegative functions $\alpha$ and $\mu_u$. Note that we
here assumed the $y$-component of the domain to be compact (${{\overline v}erline B}_b(x_2)$). Hence, if $\alpha$ and $\mu_u$ are Lipschitz, then $d$ is Lipschitz and bounded.
In \cite{Doumic} based on \cite{Stiehl} the authors consider $g$ of the shape
\betaegin{eqnarray}
g(y,z)=2[1-\fracrac{a(y)}{1+k_gz}]p(y)
\lambdaabel{eq26}
\varepsilonnd{eqnarray}
for nonnegative $k_g$, $a$ and $p$.
Further specifications are considered, which lead to $y$- and $z$-independent $g$ respectively.
We here suppose that $a$ and $p$ are differentiable and that $a'$ and $p'$ are Lipschitz. If we slightly modify (\rhoef{eq26}) such that $g(y,z)\gammae\varepsilon$ on ${{\overline v}erline B}_b(x_2)\tauimes{\mathds R}_+$, and choose the
constants in (G) appropriately, we can guarantee that $g$ satisfies (G).
Note that, though our assumption that $g$ is bounded away from zero
has a mathematical motivation, a nonzero maturation rate also has biological consistency. An example of a $g$ that is decreasing in $z$ could be
\betaegin{eqnarray}
g(y,z):=\varepsilon+e^{-z}\gamma_g(y)
\nonumber
\varepsilonnd{eqnarray}
with $\gamma_g$ differentiable and $\gamma_g'$ Lipschitz.
A choice $g(y,z)\varepsilonquiv 1$ also fulfills the requirements and with this choice $y$
could be interpreted as the age of a progenitor cell.
\sigmaection{Discussion and outlook}\lambdaabel{s6}
Note that in \cite[Theorem 6.8]{Nishi} a large class of SD-DDE is analyzed. An alternative approach to proving well-posedness for (1.1-1.4) could be, to investigate whether the cited result can be modified to include distributed delays and whether the there required smoothness conditions can be guaranteed. Possibly also with that approach the implementation
of retractions could be useful. For results on differentiability of solutions with respect to parameters and initial data, which are related to our results on continuous dependence on
initial values, we refer to the work of Hartung, e.g. \cite{Hartung1, Hartung2}.
For the specifications in Section \rhoef{ss5} and under some additional assumptions, see \cite{Getto1}, the here analyzed model (\rhoef{eq11}--\rhoef{eq14}) has a unique positive equilibrium emerging from the trivial equilibrium in a transcritical bifurcation: the rate $q$ can be assumed to be
decreasing to a negative value, hence the bifurcation parameter should guarantee that $q(0)>0$.
In a manuscript in preparation Ph.G. and G.R.
are using the theory of \cite{Smith} to show that the trivial equilibrium is globally
asymptotically stable in absence of the positive equilibrium, whereas in its
presence, there is uniform strong population persistence. The latter can be
concluded, essentially, if the system is dissipative.
In the manuscript, Ph.G. and G.R.
encounter a situation in which there either is dissipativity or ${\cal C}$-convergence of the solution to a constant, where the constant depends on the initial condition. A priori it is not clear how dissipativity can be concluded from
the second case. By Corollary \rhoef{corol1}, however, it can be concluded that the constant is an equilibrium solution and, as the equilibrium is unique, this implies dissipativity.
Note also that Corollary \rhoef{corol1} follows from continuous dependence of the solution on the initial value in the ${\cal C}$-topology.
Using continuous dependence of the solution on the initial value in the ${\cal C}^1$-topology, as established in \cite{Getto}, one could possibly prove similarly that the limit is an equilibrium, if the convergence of the solution to the constant is in ${\cal C}^1$.
In the manuscript in preparation, the authors, however, are not able to show this convergence in ${\cal C}^1$.
Hence, a ${\cal C}^1$-variant of Corollary \rhoef{corol1} would not be applicable in that manuscript. In summary
the present Corollary \rhoef{corol1} can be expected to be a necessary and sufficient tool to show dissipativity and uniform strong persistence for
(\rhoef{eq11}--\rhoef{eq14}).
In \cite{Getto}, the derivative of the semiflow defined on the solution manifold is
computed, such that a linearization is at hand. General theorems of linearized stability, applicable to our system, are shown in \cite{Walther} (stability) and \cite{Stumpf} (instability).
By the analysis of the characteristic equation derived from this linerization
in a manuscript in preparation by Mats Gyllenberg, Yukihiko Nakata, Francesca Scarabel and Ph. G., the positive equilibrium is stable upon emergence in the neighborhood of a transcritical bifurcation point and destabilizes by a pair of eigenvalues crossing into the right half plane. Based on this analysis and on unpublished numerical simulations with DDE-biftool \cite{Sieber} (by Jan Sieber) and pseudo-spectral methods \cite{Breda} (by F. Scarabel)
there is evidence for a Hopf bifurcation and the emergence of a limit cycle.
This motivates the idea of a future analysis of Hopf bifurcations and periodic solutions.
We refer to \cite{Huwu} for Hopf bifurcation analysis for related equations.
To establish periodicity for a general class of equations, in \cite{MN1} the authors include
the assumption that the initial function should be at equilibrium value at time zero.
If for our model this assumption is included, one can investigate convex and compact sets that are invariant under the original untransformed system (\rhoef{eq11}-\rhoef{eq14}), i.e., sets that are invariant for both components of the state.
Motivated by the fact that periodicity for infinite times often can be concluded from behavior in a finite time interval, we also have some hope that the here
established invariance for finite time may be sufficient.
\betaigskip
\noindent
{\betaf Acknowledgements:} The manuscript was inspired by discussions with
Tibor Krisztin during a postdoctoral stay of Ph.G. at the University
of Szeged. Ph.G. thanks Stefan Siegmund und Reinhard Stahn at Technische Universit\"at Dresden for help with the manuscript.
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\begin{document}
\title{Hyperbolic surfaces with sublinearly many systoles that fill}
\author{Maxime Fortier Bourque}
\address{School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, United Kingdom, G12 8QQ}
\email{[email protected]}
\begin{abstract}
For any ${\varepsilon}>0$, we construct a closed hyperbolic surface of genus $g=g({\varepsilon})$ with a set of at most ${\varepsilon} g$ systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most ${\varepsilon} g$ for the systole function, disproving the lower bound of $2g-1$ posited by Schmutz Schaller.
\end{abstract}
\maketitle
\section{Introduction}
The moduli space ${\mathcal M}_{g,n}$ of Riemann surfaces of genus $g$ with $n$ punctures is an object of great interest to many geometers and topologists. It encodes all the different complex structures, conformal structures, or hyperbolic structures (provided $2g+n > 2$) supported on a surface with given topology. The topology of moduli space is largely encoded in its orbifold fundamental group $\Gamma_{g,n}$, the mapping class group.
All the torsion-free finite index subgroups of $\Gamma_{g,n}$ have the same cohomological dimension, which is called the \emph{virtual cohomological dimension} (vcd) of $\Gamma_{g,n}$. Harer computed the vcd of $\Gamma_{g,n}$ for all $g$ and $n \geq 0$ and found a spine (a deformation retract) for ${\mathcal M}_{g,n}$ with this smal\-lest possible dimension whenever $n>0$ \cite{Harer}. When $n=0$, the vcd of the mapping class group is equal to $4g-5$, but a spine of this dimension has yet to be found \cite[Question 1]{BridsonVogtmann}. The largest codimension attained so far is equal to $2$ \cite{Ji} (the space ${\mathcal M}_{g,0}$ has dimension $6g-6$).
In an unpublished preprint \cite{Thurston}, Thurston claimed that the set ${\calX}_g$ of closed hyperbolic surfaces of genus $g\geq 2$ whose systoles fill forms a spine for ${\mathcal M}_g = {\mathcal M}_{g,0}$. Recall that a \emph{systole} is a closed geodesic of minimal length, and a set of curves \emph{fills} if each component of the complement of their union is simply connected. Thurston's proof that ${\mathcal M}_g$ deformation retracts onto ${\calX}_g$ appears to be difficult to complete \cite{Ji}. Furthermore, the dimension of ${\calX}_g$ is still not known, mostly because we do not understand which filling sets of curves can be systoles. Indeed, Thurston writes: \begin{quote}Unfortunately, we do not have a combinatorial characterization of collections of curves which can be the collection of shortest geodesics on a surface. This seems like a challenging problem, and until more is understood about how to answer it, there are probably not many applications of the current result.\end{quote}
The paper \cite{APP} provides some partial answers to Thurston's question. On a closed hyperbolic surface, systoles do not self-intersect and distinct systoles can intersect at most once. This obvious necessary condition is, however, far from being sufficient. Indeed, a filling set of systoles must contain at least $\sim \pi g / \log g$ curves \cite[Theorem 3]{APP}, but there exist filling collections of $\sim2\sqrt{g}$ geodesics pairwise intersecting at most once \cite[Corollary 2]{APP}. There is a discrepancy in the opposite direction as well: a closed hyperbolic surface can have at most $C g^2 / \log g$ systoles \cite[Corollary 1.4]{ParlierKissing}, but there exist filling collections with more than $g^2$ geodesics pairwise intersecting at most once \cite[Theorem 1.1]{Malestein}.
Our main result here is a construction of closed hyperbolic surfaces with filling sets of systoles containing sublinearly many curves in terms of the genus. Compare this with \cite{SchmutzManySystoles,SchmutzMoreSystoles} where surfaces with superlinearly many systoles are found. Though we are still very far\footnote{The genus $g$ in \thmref{thm:fewsystoles} grows like a tower of exponentials of length roughly $1/{\varepsilon}$.} from the lower bound of $\pi g / \log g$, our examples improve upon the previous record of surfaces with filling sets of $2g$ systoles \cite[Section 5]{APP} \cite{Sanki}.
\begin{thm} \label{thm:fewsystoles}
For every ${\varepsilon}>0$, there exist an integer $g\geq 2$ and a closed hyperbolic surface of genus $g$ with a filling set of at most ${\varepsilon} g$ systoles.
\end{thm}
Near a surface with a filling set of at most ${\varepsilon} g$ systoles, Thurston's set ${\calX}_g$ contains the set of solutions to the same number of equations. This should imply that ${\calX}_g$ has codimension at most ${\varepsilon} g$ in ${\mathcal M}_g$. However, the equations requiring the curves to have equal length can be redundant, preventing us from applying the implicit function theorem. We only manage to prove that ${\calX}_g$ has dimension at least $4g-5$ when $g$ is even, but conjecture the following.
\begin{conj} \label{conj:dimension}
For every ${\varepsilon}>0$, there exists an integer $g\geq 2$ such that ${\calX}_g$ has dimension at least $(6-{\varepsilon})g$.
\end{conj}
On the other hand, we can prove that a closely related spine, the Morse--Smale complex for the systole function, has dimension much larger than the virtual cohomological dimension of the mapping class group.
In a series of papers \cite{SchmutzMaxima,SchmutzMoreExamples,SchmutzSurvey,SchmutzMorse}, Schmutz Schaller initiated the study of the systole function $\sys: {\mathcal M}_{g,n} \to {\mathbb R}_+$, which records the length of any of the shortest closed geodesics on a surface. He proved that the systole function is a topological Morse function on the Teichm\"uller space ${\mathcal T}_{g,n}$ whenever $n>0$ \cite{SchmutzMorse} and Akrout extended this result to $n=0$ (and to a more general class of functions) in \cite{Akrout}.
Schmutz Schaller constructed a critical point of index $2g-1$ for the systole function in every genus $g \geq 2$ and thought it was ``quite possible'' that this was the smallest achievable index \cite[p.439]{SchmutzMorse}. He verified this hypothesis for $g=2$ by finding all the critical points in ${\mathcal M}_2$. If this were true in general, it would imply that the Morse--Smale complex for the systole function has the smallest possible dimension $4g-5 = (6g-6)-(2g-1)$ for a spine of ${\mathcal M}_g$. However, our surfaces show that no such inequality holds.
\begin{thm} \label{thm:critical}
For every ${\varepsilon}>0$, there exist an integer $g\geq 2$ and a critical point of index at most ${\varepsilon} g$ for the systole function on ${\mathcal T}_g$.
\end{thm}
\subsection*{Organization} The surfaces arising in Theorems \ref{thm:fewsystoles} and \ref{thm:critical} are built in two steps, in a similar fashion as the local maxima from \cite{localmax}. First, in \secref{sec:block}, we define a buil\-ding block (depending on some parameters) which is a surface whose systoles are the boundary components. This surface is modelled on a flag-transitive surface map (a generalization of Platonic solids) and can be cut into isome\-tric right-angled polygons along a collection of geodesic arcs. We then glue building blocks together according to the combinatorics of certain graphs of large girth with strong transitivity properties in \secref{sec:gluing}. We do this in such a way that the boundaries of the blocks remain systoles in the larger surface and that the arcs in the blocks connect up to form systoles as well (see \secref{sec:systoles}). In \secref{sec:isometries}, we show that $X / \isom(X)$ is isometric to a triangle or a quadrilateral. This easily implies that $X$ is a critical point of the systole function, which we prove in \secref{sec:critical}. Finally, we discuss our failed attempt to prove \conjref{conj:dimension} in \secref{sec:dimension}.
\section{Building blocks} \label{sec:block}
\subsection*{Graphs}
A \emph{graph} is a $1$-dimensional cell complex, where there can be multiple edges between two vertices and edges from vertices to themselves. The \emph{valence} of a vertex in a graph is the number of half-edges adjacent to it. If every vertex in a graph has the same valence, then this number is called the valence of the graph.
\subsection*{Flag-transitive maps}
A \emph{map} $M$ is a graph embedded on a surface $S$ such that the closure of each complementary component is an embedded closed disk (called a \emph{face} of the map). All maps considered in this paper will be \emph{orientable}, meaning that the surface $S$ is required to be orientable. If all the faces of a map $M$ have the same number $p$ of edges and all the vertices have the same valence $q$, then $M$ is said to have \emph{type} $\{p,q\}$.
A \emph{flag} in a map is a triple consisting of a vertex $v$, an edge $e$ containing $v$, and a face $f$ containing $e$. A \emph{map-automorphism} of $M$ is an automorphism of the underlying graph which can be realized by a homeomorphism of the surface $S$. A map is \emph{flag-transitive}\footnote{These maps are usually called \emph{regular}, but if we stuck to standard terminology, this word would be used with five different meanings throughout the paper.} if its group of map-automorphisms acts transitively on flags.
Any flag-transitive map has type $\{p,q\}$ for some $p\geq 1$ and $q\geq 2$. The five Platonic solids are the only flag-transitive maps on the sphere with $p,q\geq 3$; their types are $\{3,3\}$, $\{4,3\}$, $\{3,4\}$, $\{5,3\}$ and $\{3,5\}$. Beach balls assembled from $q$ spherical bigons are flag-transitive maps of type $\{2,q\}$.
\subsection*{Maps of large girth}
A \emph{cycle} in a graph is a sequence of oriented edges $(e_1,\ldots,e_k)$ such that the endpoint of $e_i$ coincides with the starting point of $e_{i+1}$ for every $i \in {\mathbb Z}_k$. Cycles are considered up to cyclic permutation of their edges and reversal of orientation. The \emph{length} of a cycle is the number of edges that it uses. A cycle is \emph{non-trivial} if it cannot be homotoped to a point by deleting \emph{backtracks}, that is, consecutive edges (modulo $k$) with opposite orientations. The \emph{girth} of a graph is the length of any of its shortest non-trivial cycles. These shortest non-trivial cycles will be called \emph{girth cycles}. A graph of girth at most $2$ is often called a \emph{multigraph}, and a graph of girth larger than $2$ is \emph{simple}.
The girth of a flag-transitive map $M$ of type $\{p,q\}$ is at most $p$ since the faces are non-trivial cycles of length $p$. If $M$ is finite, then one can actually unwrap all the cycles shorter than $p$ by taking a suitable finite normal cover, thereby obtaining a finite flag-transitive map $N$ of girth $p$ \cite[Theorem 11]{Evans}. That such covers exist follows from Mal'cev's theorem on the residual finiteness of finitely generated linear groups \cite{Malcev}.
\begin{thm}[Evans] \label{thm:reg_map}
For any $p,q \geq 2$, there exists a finite flag-transitive map of type $\{p,q\}$ and girth $p$.
\end{thm}
See also \cite{Nedela} and \cite{Siran} for constructive proofs of this result.
\subsection*{Regular polygons}
Let $q\geq 3$. Up to isometry, there exists a unique polygon $P$ in the hyperbolic plane with $2q$ sides of the same length $L$ and all interior angles equal to $\pi/2$. We will call $P$ the \emph{regular right-angled $2q$-gon}. By connecting the center of $P$ to the midpoint of a side and one of its vertices, we obtain a triangle with interior angles $\pi/2$, $\pi/4$ and $\pi/2q$ and a side of length $L/2$ from which we obtain the equation
\begin{equation} \label{eq:trig}
\cosh(L/2)= \cos(\pi/2q) / \sin(\pi/4) = \sqrt{2} \cos(\pi/2q)
\end{equation}
(see \cite[p.454]{Buser}). We color the sides of $P$ red and blue in such a way that adjacent sides have different colors.
\begin{lem} \label{lem:arc_in_poly}
Any arc $\alpha$ between two disjoint sides in the regular right-angled $2q$-gon $P$ has length at least $L$, with equality only if $\alpha$ is a side of $P$.
\end{lem}
\begin{proof}
Let $\alpha$ have minimal length among such arcs. Then $\alpha$ must be geodesic and orthogonal to $\partial P$ at its endpoints. These endpoints are separated by $m$ sides of $P$ in one direction and $n$ sides in the other, where $m+n+2=2q$ and $m\leq n$. First suppose that $m>1$. Let $d$ be a main diagonal of $P$ which is linked with $\alpha$ and has one endpoint at an extremity of one of the two sides of $P$ joined by $\alpha$. Let $z$ be the intersection point between $\alpha$ and $d$, and let $\alpha_\pm$ be the two components of $\alpha \setminus\{ z\}$ labelled in such a way that $\alpha_+$ and $d$ have endpoints in a common side of $P$. If $R_d$ denotes the reflection about $d$, then the arc $\gamma=\alpha_- \cup R_d(\alpha_+)$ has the same length as $\alpha$ and joins two disjoint sides of $P$ (because $m>1$). By minimality, $\gamma$ must be geodesic and orthogonal to $\partial P$, which is absurd. This shows that $m=1$, in which case $\alpha$ is a side of $P$ (the orthogonal segment between two geodesics in the hyperbolic plane is unique when it exists).
\end{proof}
One can also prove this using trigonometry (see \cite[p.91]{APP}).
\subsection*{Gluing regular polygons along maps}
Let $M$ be an oriented map of type $\{p,q\}$ where $q\geq 3$. Let $P$ be the unique right-angled regular hyperbolic $2q$-gon with sides colored red and blue as above. We now define a hyperbolic surface $B$ modelled on $M$. For each vertex $v\in M$, take a copy $P_v$ of $P$. The blue sides of $P_v$ are labelled in counterclockwise order by the edges adjacent to $v$ in $M$, which come with a cyclic ordering from the orientation. For each edge $e=\{u,v\}$ in $M$, we glue the polygons $P_u$ and $P_v$ along their sides labelled $e$ by an orientation-reversing isometry. The resulting surface is denoted $B$ and will be called a \emph{block} in the sequel. The polygons $P_v \subset B$ are its \emph{tiles}.
Topologically, $B$ is the same as the surface $S \supset M$ with a hole cut out in each face. Indeed, if we join the center of each polygon $P_v$ to the midpoints of its blue sides, we obtain an embedded copy of $M$ in $B$. Since each $P_v$ deformation retracts onto the star $M\cap P_v$, the surface $B$ deformation retracts onto $M$. Each boundary component of $B$ is the concatenation of $p$ red sides of polygons $P_v$ coming from the $p$ vertices $v$ around a face of $M$. In particular, each boundary component of $B$ has length $p L$, where $L$ is the positive number implicitly defined by \eqnref{eq:trig}.
\begin{lem} \label{lem:systoles_in_block}
Let $M$ be a map of type $\{p,q\}$ and girth $p$, where $p \geq 2$ and $q \geq 3$. Then the systoles in $B$ are the boundary geodesics, of length $p L$.
\end{lem}
\begin{proof}
Let $\gamma$ be a systole in $B$. As explained above, the map $M$ embeds in $B$ as the dual graph to the decomposition into the $2q$-gons $P_v$. Let $\pi :B\to M$ be the nearest point projection. The image $\pi(\gamma)$ must be non-trivial in $M$ since $\gamma$ is non-trivial in $B$ and $\pi$ is a deformation retraction. It follows that the combinatorial length of $\pi(\gamma)$ in $M$ is at least $p$. In other words, $\gamma$ intersects at least $p$ tiles $P_v$, joining distinct blue sides each time. By \lemref{lem:arc_in_poly}, the length of $\gamma \cap P_v$ is at least $L$ for any tile $P_v$ that $\gamma$ intersects. The total length of $\gamma$ is therefore greater than or equal to $p L$. If equality occurs, then $\gamma$ must be a concatenation of red arcs, that is, a boundary geodesic.
\end{proof}
\begin{cor} \label{cor:arc_boundary_to_self}
Let $M$ be a map of type $\{p,q\}$ and girth $p$, where $p \geq 2$ and $q \geq 3$. Then any arc from a boundary component to itself in $B$ which cannot be homotoped into $\partial B$ has length strictly larger than $p L/2$.
\end{cor}
\begin{proof}
Suppose that $\alpha$ is a non-trivial arc of length at most $p L/2$ from a boundary geodesic $b$ to itself. The arc $\alpha$ followed by the shorter of the two subarcs of $b$ between its endpoints is a non-trivial closed curve $\gamma$ of length at most $p L$ in $B$. The closed geodesic homotopic to $\gamma$ is strictly shorter, contradicting \lemref{lem:systoles_in_block}.
\end{proof}
\begin{lem} \label{lem:arc_boundary_to_boundary}
Let $M$ be a map of type $\{p,q\}$ and girth $p$, where $p \geq 2$ and $q \geq 3$. Then any arc $\alpha$ from $\partial B$ to $\partial B$ which cannot be homotoped into $\partial B$ has length at least $L$, with equality only if $\alpha$ is a blue arc.
\end{lem}
\begin{proof}
Let $\alpha$ be a geodesic arc from $\partial B$ to $\partial B$. By \lemref{lem:systoles_in_block}, we may assume that $\alpha$ joins consecutive sides of any tile $P_v$ it intersects. Since the starting point of $\alpha$ in on a red side, it has to next intersect a blue side, and then a red. This means that $\alpha$ is homotopic to a blue arc in a union $P_u \cup P_v$ of two adjacent tiles. This blue arc is shortest among all arcs in $P_u \cup P_v$ joining the same two sides, as it is orthogonal to the boundary at both endpoints.
\end{proof}
The above results do not require the map $M$ to be finite or flag-transitive, but we will impose these conditions in the next sections.
\section{Gluing graphs} \label{sec:gluing}
In this section, we explain how to glue blocks together along certain graphs of large girth with large automorphism groups in order to get closed hyperbolic surfaces with many symmetries and few systoles.
\subsection*{Strict polygonal graphs}
A \emph{strict polygonal graph} is a gra\-ph $G$ such that any embedded path of length $2$ in $G$ is contained in a unique girth cycle (where cycles are considered up to cyclic reordering and reversal). This notion was introduced by Perkel in his thesis \cite{PerkelThesis}. Examples of strict polygonal graphs include polygons, the tetrahedron, the dodecahedron, and the cube of any dimension. See \cite{Seress} for a short survey on the subject.
Archdeacon and Perkel \cite{ArchPerkel} found a way to double the girth of a strict polygonal graph $G$ (or any graph) by taking an appropriate normal covering space. The girth cycles in this cover $\widetilde G$ are precisely those that wrap twice around a girth cycle in $G$ under the covering map. Repeated applications of their construction yield strict polygonal graphs of arbitrarily large girth and constant valence (equal to the valence of $G$).
Seress and Swartz \cite[Theorem 3.2]{SeressSwartz} proved that any automorphism of the base graph $G$ lifts to an automorphism of the girth-doubling cover $\widetilde G$. They concluded that if $G$ is vertex transitive, edge transitive, arc transitive or 2-arc transitive, then so is $\widetilde G$. We will need an even stronger transitivity property, described in the next paragraph.
\subsection*{Isotropic graphs}
The \emph{star} ${\, \big| \,}arr(v)$ of a vertex $v$ in a graph is the set of half-edges adjacent to $v$. A graph $G$ is \emph{locally symmetric} if for every vertex $v \in V(G)$, any bijection of ${\, \big| \,}arr(v)$ can be extended to an automorphism of $G$ that fixes $v$. We say that a graph is \emph{isotropic} if it is vertex transitive and locally symmetric. To spell it out, $G$ is isotropic if every injection ${\, \big| \,}arr(u) \hookrightarrow {\, \big| \,}arr(v)$ between stars in $G$ extends to an automorphism of $G$.
In an isotropic graph, there is a girth cycle passing through any embedded path of length $2$, but there can be more than one.
\begin{ex}
The Petersen graph $P$ (the quotient of the dodecahedron by the antipodal involution) is an isotropic graph of valence $3$ and girth $5$ on $10$ vertices. However, $P$ is not strict polygonal since every embedded path of length $2$ is contained in two distinct girth cycles in $P$.
\end{ex}
Lubotzky \cite{Lubotzky} constructed infinitely many isotropic Cayley graphs of any valence $d\geq 3$ and any even girth $\geq 6$ (the generators are involutions, allowing the valence to be odd). Since we want better control on the girth cycles of our isotropic graphs, we use the girth-doubling construction of Archdeacon and Perkel instead. The proof that the girth-doubling cover $\widetilde{G}$ of a graph $G$ is isotropic provided that $G$ is isotropic follows immediately from \cite[Theorem 3.2]{SeressSwartz}, which states that any automorphism of $G$ lifts to $\widetilde{G}$, and the fact that the covering $\widetilde G \to G$ is normal, so that its deck group acts transitively on fibers.
The simplest isotropic strict polygonal graph is a pair of vertices joined by $d\geq 2$ edges. Repeated applications of the girth-doubling construction to this graph ${\mathcal T}heta$ yield a sequence of finite, isotropic, strict polygonal graphs of any valence and arbitrarily large girth.
\begin{thm}[Archdeacon--Perkel, Seress--Swartz] \label{thm:poly_graph}
For any $d\geq 2$ and $n\geq 1$, there exists a finite, isotropic, strict polygonal graph $G$ of valence $d$ and girth $2^n$. In fact, $G$ can be chosen to be a covering space of the bipartite graph ${\mathcal T}heta$ of valence $d$ on $2$ vertices, in which case the girth cycles in $G$ project to powers of girth cycles in ${\mathcal T}heta$ under the covering map.
\end{thm}
\subsection*{Gluing}
We now explain how to glue copies of the block $B$ from \secref{sec:block} along a finite isotropic strict polygonal graph $G$ to get a closed hyperbolic surface $X$ with a small set of systoles that fill.
Let $q\geq 3$, let $n\geq 1$, and write $p=2^n$. Let $M$ be a finite flag-transitive map of type $\{p,q\}$ and girth $p$ whose existence is guaranteed by \thmref{thm:reg_map}.
Let $B$ be the block obtained by gluing regular right-angled $2q$-gons along the map $M$ as in \secref{sec:block}. Let $d$ be the number of boundary components of $B$, which is is equal to the number of faces in $M$.
Let $G$ be a finite, isotropic, strict polygonal graph of valence $d$ and girth $p=2^n$ covering the bipartite graph ${\mathcal T}heta$ on two vertices as in \thmref{thm:poly_graph}, and let $\pi : G \to {\mathcal T}heta$ be a covering map. Let $\sigma : V({\mathcal T}heta) \to \{-1,1\}$ and $\chi: E({\mathcal T}heta) \to \{1,\ldots,d\}$ be bijections, where $V({\mathcal T}heta)$ and $E({\mathcal T}heta)$ are the sets of vertices and edges of ${\mathcal T}heta$ respectively. These induce proper colorings $\sigma\circ \pi$ and $\chi\circ \pi$ of the vertices and edges of $G$ respectively.
For each $v \in V(G)$, let $B_v$ be a copy of the block $B$, equipped with its standard orientation if $\sigma(v)=1$ and with the reverse orientation if $\sigma(v)=-1$. Let $b_1, \ldots , b_d$ be the boundary components of $B$ and label the boundary components of any copy $B_v$ in the same way so that the isometric identification $B_v \cong B$ preserves the indices of boundary components.
Here is how we define the closed hyperbolic surface $X$ given the above combinatorial data. For any edge $e=\{u,v\}$ in $G$, glue $B_u$ to $B_v$ by the identity map along their $j$-th boundary component, where $j=\chi\circ \pi(e)$. The surface $X$ is defined as the quotient of $\sqcup_{v \in V(G)} B_v$ by these gluings. Since the gluing maps are orientation-reversing, $X$ is an oriented surface. It has empty boundary since the coloring $\chi\circ \pi$ takes all values in $\{1,\ldots,d\}$ on the edges containing a given vertex $v$, so that all the boundary components of $B_v$ are glued. Lastly, $X$ is compact because $M$ and $G$ are finite.
The main reason for using strict polygonal graphs in this construction is so that the blue arcs in the blocks $B_v$ all close up to curves of the same length in $X$.
\begin{lem} \label{lem:blue}
Any blue arc in a block $B_v \subset X$ is part of a closed geodesic of length $p L$ in $X$.
\end{lem}
\begin{proof}
Any blue arc $\alpha_v$ in $B_v$ connects two boundary geodesics $b_i$ and $b_j$. The block $B_v$ is glued to two other blocks $B_u$ and $B_w$ via these boundary components, and there are blue arcs $\alpha_u \subset B_u$ and $\alpha_w \subset B_w$ corresponding to $\alpha_v$ under the isometric identifications $B_u \cong B_v \cong B_w$. The concatenation $\alpha_u \cup \alpha_v \cup \alpha_w$ is geodesic since $\alpha_v$ is orthogonal to $\partial B_v$.
By our convention, the arc $\alpha_u$ (resp. $\alpha_w$) connects the boundary components of $B_u$ (resp. $B_w$) labelled $b_i$ and $b_j$. By repeating the above reflection process with $\alpha_u$ or $\alpha_w$ instead of $\alpha_v$ (and so on), we obtain a bi-infinite path ${\rm d}elta=(\ldots,u,v,w,\ldots)$ in the graph $G$ whose edges alternate between the colors $i$ and $j$. There is also a bi-infinite geodesic \[\beta = \cdots \cup \alpha_u \cup \alpha_v \cup \alpha_w \cup \cdots\] in $X$ obtained by concatenating the corresponding blue arcs.
Since $G$ is a strict polygonal graph, the path $(u,v,w)$ is contained in a unique non-trivial cycle $\gamma$ of length $p$ (the girth of $G$). Furthermore, \thmref{thm:poly_graph} stipulates that $\gamma$ covers a closed cycle of length $2$ in ${\mathcal T}heta$ under the covering map $\pi:G\to{\mathcal T}heta$. This cycle of length $2$ is necessarily formed by the edges $\chi^{-1}(i)$ and $\chi^{-1}(j)$ since $\pi$ respects the coloring of edges. This means that the edges of $\gamma$ alternate between the colors $i$ and $j$, and hence that the path ${\rm d}elta$ wraps around $\gamma$ periodically in both directions. In other words, ${\rm d}elta$ closes up after $p$ steps. Similarly, the geodesic $\beta$ is closed and its length is equal to $pL$ since each of its $p$ subarcs has length $L$.
\end{proof}
Note that we have not used the hypotheses that $M$ is flag-transitive nor that $G$ is isotropic yet. This will come up in \secref{sec:isometries} where we determine the isometry group of $X$.
\section{Systoles} \label{sec:systoles}
In this section, we determine and count the systoles in the surface $X$ constructed above.
\begin{prop} \label{prop:count}
Let $X$ be the surface constructed in \secref{sec:gluing}. The systoles in $X$ are the red curves and the blue curves. These systoles fill $X$ and there are $\frac{4q}{(q-2) p}(g-1)$ of them, where $q$ is the valence of the map $M$, $p$ is the girth of $M$ and the gluing graph $G$, and $g$ is the genus of $X$.
\end{prop}
\begin{proof}
Let $\gamma$ be a systole in $X$. If $\gamma$ is contained in a single block $B_v \subset X$, then $\gamma$ is a red curve (of length $pL$) by \lemref{lem:systoles_in_block}. Now assume that $\gamma$ is not contained in any block. Then the blocks $B_{v_1}, \ldots, B_{v_k}$ ($k\geq 2$) that it visits define a closed cycle $s=(v_1,\ldots v_k,v_1)$ in the graph $G$. First suppose that $s$ is trivial in $G$. Then $s$ contains at least two backtracks, that is, vertices $v_j$ in the sequence such that $v_{j-1}=v_{j+1}$. If $s$ backtracks at a vertex $u\in G$, this means that a subarc $\omega$ of $\gamma$ enters and leaves the block $B_u$ via the same boundary component. By \corref{cor:arc_boundary_to_self}, $\omega$ has length strictly larger than $pL/2$. Since there are at least two disjoint subarcs like this, $\gamma$ is longer than $pL$. We conclude that $s$ is non-trivial in $G$, so that its length is at least $p$, the girth of $G$. But for each vertex $u$ along $s$, the corresponding subarc of $\gamma$ in $B_v$ has length at least $L$ by \lemref{lem:arc_boundary_to_boundary}. Thus the total length of $\gamma$ is at least $pL$. If equality occurs, then $\gamma$ is a concatenation of blue arcs. Conversely, any concatenation of blue arcs has length $pL$ by \lemref{lem:blue}.
The complementary components of the set of systoles in $X$ are precisely the interiors of the tiles from which the blocks are assembled. In particular, the systoles fill.
The number of systoles in $X$ is equal to the total number of red arcs and blue arcs divided by $p$. This is because the red arcs are joined in groups of $p$ to form systoles, and similarly for the blue arcs. Each such arc $\alpha$ (either red or blue) belongs to exactly two tiles. The rhombus with one vertex in the center of each of these two tiles and diagonal $\alpha$ has area $\pi(q-2)/q$ by the Gauss--Bonnet formula (it has two right angles and two angles $\pi/q$). These rhombi tile $X$, which has area $4\pi(g-1)$. Therefore, the number of systoles is $4\pi(g-1)$ divided by $\pi(q-2)/q$, divided by $p$.
\end{proof}
Recall that in the construction of $X$ we could take any $q\geq 3$ and $p=2^n$ for any $n\geq 1$. Given any ${\varepsilon}>0$, taking $n$ sufficiently large and any $q\geq 3$ gives a surface with a filling set of at most ${\varepsilon} g$ systoles. This proves \thmref{thm:fewsystoles}. At the other extreme, the largest number of systoles is obtained when $q=3$ and $p=2$, which gives $6g-6$ systoles. By \cite[Theorem 2.8]{SchmutzMaxima}, such a surface has too few systoles to be a local maximum of the systole function, but we will see later that it is nevertheless a critical point of lower index.
\begin{ex} \label{example}
For any $g\geq 2$, if we take the map $M$ to be the bipartite graph of valence $g+1$ on two vertices (as a map on the sphere), then the resulting block $B$ has $g+1$ boundary components. Taking the gluing graph $G$ to be equal to $M$, we obtain a surface $X$ which is the double of $B$ across its boundary. The genus of $X$ is then equal to $g$. Since $q=g+1$ and $p=2$, the number of systoles is $2g+2$ according to the formula in \propref{prop:count}. Removing any two intersecting systoles leaves a filling set of $2g$ systoles. This example was previously described in \cite[Theorem 36]{SchmutzMorse} and \cite[Section 5]{APP} and was the starting point of this paper.
\end{ex}
\begin{remark}
We could allow the graphs $G$ and $M$ to have different girths $p$ and $r$ by replacing the polygons $P$ in the blocks to be semi-regular with side lengths $L_\text{blue}$ and $L_\text{red}$ satisfying $p L_\text{blue} = r L_\text{red}$. A version of \propref{prop:count} still holds for this generalization, with the count of systoles coming to \[\frac{2q}{(q-2)}\left(\frac{1}{p}+\frac{1}{r}\right)(g-1).\]
All one has to do is change \lemref{lem:arc_in_poly} to say that the distance between any two blue sides is at least $L_\text{red}$ and the distance between any two red sides is at least $L_\text{blue}$, and modify the other lemmata accordingly.
\end{remark}
\section{Isometries} \label{sec:isometries}
In this section, we determine the isometry group of the surface $X$ up to index $2$. Recall that the blocks $B_v \subset X$ (where $v\in V(G)$) are tiled by regular right-angled $2q$-gons $P_u$ (where $u \in V(M)$). By connecting the center of each polygon $P_u$ to the midpoints of its edges with geodesics, we obtain a tiling ${\mathcal Q}$ of $X$ by $(2,2,2,q)$-quadrilaterals (i.e., quadrilaterals with three right angles and one angle equal to $\pi/q$). Since any isometry of $X$ preserves the set of systoles, it permutes the complementary polygons $P_u$ and therefore the quadrilaterals in ${\mathcal Q}$. In fact, any quadrilateral can be sent to any other by an isometry.
\begin{prop} \label{prop:isom}
The isometry group of $X$ acts transitively on the quadrilaterals in the tiling ${\mathcal Q}$.
\end{prop}
\begin{proof}
The hypothesis that $M$ is flag-transitive implies that the isometry group of $B$ acts transitively on its $(2,2,2,q)$-quadrilaterals. This is because there is a one-to-one correspondence between the flags in $M$ and the quadrilaterals in $B$. The correspondence works as follows. Recall that $M$ naturally embeds in $B$, connecting the centers of polygons $P$ to their blue sides. A flag in $M$ is the same as a half-edge $e$ together with a choice of a face $f$ containing $e$, either on the left or the right. In the tiling of $B$ by quadrilaterals, there are exactly two quadrilaterals that have $e$ as an edge. The side of $e$ on which $f$ lies determines which quadrilateral to pick. Since any map-automorphism of $M$ can be realized as an isometry of $B$ and $M$ is flag-transitive, the claim follows.
Let $v \in V(G)$ and let $\phi: B_v \to B_v$ be an isometry. We claim that $\phi$ extends to an isometry ${\mathbb P}hi$ of $X$. First, the isometry $\phi$ induces a permutation $\tau$ on $\{1,\ldots,d\}$ such that $\phi$ sends the boundary component $b_i$ of $B_v$ to the component $b_{\tau(i)}$ for every $i$. Now the edges adjacent to $v$ in $G$ are colored with the numbers $\{1,\ldots,d\}$ according to the coloring $\chi\circ \pi$. Thus the permutation $\tau$ induces a bijection on the star of $v$. Since $G$ is locally symmetric, this bijection can be extended to an automorphism $\psi$ of $G$. If $x \in B_u \subset X$, then define ${\mathbb P}hi(x)$ to be the point $\phi(x)$ in $B_{\psi(u)}$, where we use the canonical identifications $B_w \cong B_v$ to transport the action of $\phi$ onto any block. This map is well-defined, for if $x\in B_u \cap B_v$ then $x$ belongs to the boundary component labelled $i=\chi\circ \pi(\{u,v\})$ of $B_u$ and $B_v$. By definition, $\phi(x)$ belongs to the $\tau(i)$-th boundary component of $B_v$. Now $B_{\psi(u)}$ and $B_{\psi(v)}$ are glued along their boundary component labelled $\chi\circ\pi(\psi(\{u,v\}))$. This number equals $\tau(i)$ provided that the automorphism $\psi$ is chosen to be a lift of the automorphism of ${\mathcal T}heta$ induced by $\tau$, and this is possible according to \cite[Theorem 3.2]{SeressSwartz}. The map ${\mathbb P}hi$ is an isometry since it is a locally isometry as well as a bijection.
Similarly, any automorphism $\psi$ of $G$ which preserves the coloring $\chi\circ \pi$ defines an isometry ${\mathbb P}si$ of $X$ by sending $x\in B_v$ to the corresponding $x$ in $B_{\psi(v)}$. This simply shuffles the blocks around, acting by the identity map on the blocks. Note that the group of such automorphisms $\psi$ acts transitively on the vertices of $G$.
Combining these two types of isometries gives the desired result. In order to send a quadrilateral $Q\subset B_u$ to another quadrilateral $Q' \subset B_v$, first apply an isometry ${\mathbb P}si$ as in the previous paragraph to send $B_u$ to $B_v$. Then move ${\mathbb P}si(Q)$ to $Q'$ via an isometry ${\mathbb P}hi$ of the first type, preserving the block $B_v$.
\end{proof}
Since there are at most two isometries of $X$ sending one quadrilateral to another, this determines the isometry group of $X$ up to index $2$. We can reformulate this as follows. Subdivide ${\mathcal Q}$ further into a tiling ${\mathcal T}$ by $(2,4,2q)$-triangles by bisecting the quadrilaterals at their smallest angle. Then the isometry group of $X$ may or may not act transitively on the tiles of ${\mathcal T}$ depending on the graphs $M$ and $G$ used to construct $X$.
In \exref{example}, the isometry group of $X$ acts transitively on these triangles, but that is not the case in general. That is, there can be an asymmetry between the red and blue curves in $X$. For example, let $M$ be the flag-transitive map of type $\{4,4\}$ obtained by subdividing the square torus into a $5\times 5$ grid (so that $B$ is a torus with $25$ holes) and let $G$ be the $1$-skeleton of the $25$-dimensional cube. Then each component of $X \setminus \{\text{blue curves}\}$ is a torus with $16$ boundary components corresponding to a $4$-dimensional subcube of $G$, while the complementary components of the red curves are the blocks with $25$ boundary curves each. In this case, no isometry of $X$ can interchange the two families of systoles.
\section{Critical point and index} \label{sec:critical}
A real-valued function $f$ on an $n$-dimensional manifold $M$ is a \emph{topological Morse function} if for every $p\in M$, there is an open neighborhood $U$ of $p$ and an injective continuous map $\phi : U \to {\mathbb R}^n$ with $\phi(p)=0$ such that $f\circ \phi^{-1}-f(p)$ takes either the form
\[(x_1,\ldots,x_n) \mapsto x_1 \]
or
\[(x_1,\ldots,x_n) \mapsto - \sum_{i=1}^j x_i^2 + \sum_{i=j+1}^n x_i^2 \]
for some $j \in \{0,\ldots,n\}$. In the first case, $p$ is an \emph{ordinary point} and in the second case $p$ is \emph{a critical point of index $j$}. Critical points of index $0$ and $n$ are local minima and maxima respectively.
Let $g\geq 2$ and let ${\mathcal T}_g$ be the Teichm\"uller space of marked, connected, oriented, closed, hyperbolic surfaces of genus $g$. This space is a smooth manifold diffeomorphic to ${\mathbb R}^{6g-6}$. The systole $\sys(Y)$ of a surface $Y\in {\mathcal T}_g$ is the length of any of its shortest closed geodesics. As mentionned in the introduction, Akrout \cite{Akrout} proved that $\sys:{\mathcal T}_g \to {\mathbb R}_+$ is a topological Morse function.
Let $Y\in {\mathcal T}_g$ and let ${\mathcal S}$ be the set of (homotopy classes of) systoles in $Y$. For each $\alpha \in {\mathcal S}$ and $Z \in {\mathcal T}_g$, we let $\ell_\alpha(Z)$ be the length of the unique closed geodesic homotopic to $\alpha$ in $Z$. These functions are differentiable on ${\mathcal T}_g$ and we denote their differentials by $d\ell_\alpha$.
\begin{defn}
The point $Y\in {\mathcal T}_g$ is \emph{eutactic} if for every $v \in T_Y {\mathcal T}_g$, the following implication holds: if $d\ell_\alpha(v) \geq 0$ for every $\alpha \in {\mathcal S}$, then $d\ell_\alpha(v)=0$ for every $\alpha \in {\mathcal S}$. The \emph{rank} of a eutactic point $Y$ is the dimension of the image of the linear map ($d\ell_\alpha)_{\alpha\in{\mathcal S}}: T_Y {\mathcal T}_g \to {\mathbb R}^{\mathcal S}$.
\end{defn}
With these definitions, we have the following characterization of the cri\-ti\-cal points of $\sys$ \cite[Theorem 1]{Akrout}.
\begin{thm}[Akrout]
The critical points of index $j$ of the systole function are the eutactic points of rank $j$.
\end{thm}
We can now show that the surface $X$ constructed in \secref{sec:gluing} is a critical point of $\sys$ and give an upper bound for its index.
\begin{prop}
Let $X$ be as in \secref{sec:gluing}. Then $X$ is a critical point of index at most $\frac{4q}{(q-2) p}(g-1)$ for the systole function.
\end{prop}
\begin{proof}
Let ${\mathcal S}$ be the set of systoles of $X$ (the red curves and the blue curves). Suppose that $v\in T_X {\mathcal T}_g$ is such that $d\ell_\alpha(v) \geq 0$ for every $\alpha \in {\mathcal S}$ and let
\[
w = \sum_{f\in \isom(X)} f_*v.
\]
Then
\begin{equation} \label{eq:eutactic}
d\ell_\alpha(w) = \sum_{f\in \isom(X)} d\ell_\alpha(f_* v) = \sum_{f\in \isom(X)} d\ell_{f(\alpha)}(v) \geq d\ell_{\alpha}(v) \geq 0
\end{equation}
for every $\alpha \in {\mathcal S}$. On the other hand, $w$ is the lift to $X$ of a deformation of the quotient orbifold $Q = X /\isom(X)$. By \propref{prop:isom}, $Q$ is either a $(2,4,2q)$-triangle or a $(2,2,2,q)$-quadrilateral. If $Q$ is a triangle, then $w=0$ so that $d\ell_\alpha(w)$ and $d\ell_{\alpha}(v)$ are both zero by \eqnref{eq:eutactic}, for every $\alpha \in {\mathcal S}$. If $Q$ is a quadrilateral, then its deformation space is $1$-dimensional. This is because any $(2,2,2,q)$-quadrilateral is determined by the lengths $a$ and $b$ of the two sides disjoint from the angle $\pi/q$, which satisfy the relation
\[
\sinh a \sinh b = \cos(\pi/q)
\]
(see \cite[p.454]{Buser}). This equation implies that the lengths of the red curves and the blue curves in ${\mathcal S}$ have opposite derivatives in the direction of $w$. Since the derivatives are non-negative, they must all be zero. We conclude that $d\ell_{\alpha}(v) = 0$ for every $\alpha \in {\mathcal S}$ from \eqnref{eq:eutactic}. This shows that $X$ is eutactic. The number of systoles in $X$ is a trivial upper bound for the rank of $X$, and this number is equal to $\frac{4q}{(q-2) p}(g-1)$ by \propref{prop:count}.
\end{proof}
Once again, by taking $p$ sufficiently large we obtain critical points of index at most ${\varepsilon} g$ for any ${\varepsilon}>0$, thereby proving \thmref{thm:critical}. This disproves the possibility envisaged by Schmutz Schaller \cite[p.410]{SchmutzMorse} that the minimal index were $2g-1$.
\section{Deformations preserving the systoles} \label{sec:dimension}
Let ${\calX}_g$ be the subset of ${\mathcal T}_g$ whose systoles fill. We would like to show that ${\calX}_g$ has relatively small codimension in ${\mathcal T}_g$. By \propref{prop:count}, the systoles of any surface $X$ constructed in \secref{sec:gluing} fill (recall that $X$ depends on several parameters). Let ${\mathcal S}$ be the set of systoles in $X$. If we deform $X$ in such a way that the curves in ${\mathcal S}$ remain of equal length, then these curves will still be the systoles for sufficiently small deformations. This is because the second shortest curve on $X$ is longer by a definite amount and length varies continuously.
In other words, the intersection between the inverse image of the diagonal ${\mathbf D}elta \subset {\mathbb R}^{\mathcal S}$ by the map $(\ell_\alpha)_{\alpha\in{\mathcal S}}:{\mathcal T}_g\to{\mathbb R}^{\mathcal S}$ and a small neighborhood of $X$ is contained in ${\calX}_g$. One might be tempted to conclude directly that ${\calX}_g$ has codimension at most $|{\mathcal S}|-1$ in ${\mathcal T}_g$. The subtlety is that the image of $(\ell_\alpha)_{\alpha\in{\mathcal S}}$ is not necessarily transverse to ${\mathbf D}elta$. Indeed, the rank of $X$ can be strictly less than $|{\mathcal S}|-1$. For instance, the surface in \exref{example} has rank $2g-1$ according to \cite[Theorem 36]{SchmutzMorse}, while $|{\mathcal S}|=2g+2$.
To remedy this, one could try to get rid of redundant equations, i.e., to find a filling subset of curves ${\mathcal R} \subset {\mathcal S}$ for which the differential $(d_X \ell_\alpha)_{\alpha\in{\mathcal R}}$ is surjective and apply the implicit function theorem. The problem is that even if the curves in ${\mathcal R}$ stay of equal length, the curves in ${\mathcal S} \setminus {\mathcal R}$ might become shorter and so the systoles might not fill anymore.
Another approach would be to find a nearby surface $X_\theta$ which has the same set of systoles as $X$, and hope that the differential $(d_{X_\theta} \ell_\alpha)_{\alpha\in{\mathcal S}}$ has full rank there. Below we will describe a $1$-dimensional family of deformations of $X$ with the same systoles. This fixes the issue of rank in some (but not all) cases. A similar idea was used in \cite{Sanki} to find a path of surfaces in ${\calX}_g$ with $2g$ systoles.
\subsection*{A $1$-dimensional deformation}
Recall that $X$ is assembled from right-angled regular $2q$-gons $P$ whose sides are colored alternatingly red and blue, where $q \geq 3$. Given any $\theta \in (0,\pi)$, there exists a unique polygon $P_{\theta}$ with $2q$ equal sides and interior angles alternating between $\theta$ and $\pi - \theta$ (start with a triangle with angles $\pi/q$, $\theta /2$ and $(\pi - \theta)/2$ and reflect repeatedly across the two sides at angle $\pi/q$). To fix ideas, let us say that $\theta$ is the counter-clockwise (interior) angle from a red side to a blue side when going clockwise around $P_{\theta}$ and $\pi - \theta$ is the angle from blue to red. Now replace all the polygons $P$ in $X$ by $P_{\theta}$ while keeping the same gluing combinatorics. By construction, the total angle around vertices of the resulting tiling is $2\pi$ so the deformed surface $X_{\theta}$ is still a closed hyperbolic surface. Moreover, the red sides still line up to form closed geodesics and similarly for the blue sides. These closed geodesics all have equal length, namely, $p$ times the side length of $P_{\theta}$. As long as $\theta$ is close enough to $\pi/2$, these curves will remain the systoles.
The goal is then to show that the linear map $(d_{X_\theta} \ell_\alpha)_{\alpha\in{\mathcal S}}$ has full rank when $\theta \neq \pi/2$. We can do this for some small examples (see below), but we do not know how to handle surfaces with complicated gluing graphs of large girth. We present examples with full rank for girth $2$ and $3$ below.
\subsection*{Computing the rank}
To prove that the derivative of lengths has full rank, it suffices to find a set of tangent vectors $\{ v_\beta\}_{\beta \in {\mathcal S}}$ to Teichm\"uller space for which the square matrix $(d_{X_\theta} \ell_\alpha(v_\beta))_{\alpha,\beta\in{\mathcal S}}$ has non-zero determinant. For this, we can choose each vector $v_\beta$ to be the Fenchel-Nielsen twist deformation (i.e., left earthquake) around the curve $\beta$. The cosine formula of Wolpert \cite{WolpertTwist} and Kerckhoff \cite{Kerckhoff} then says that
\[
d\ell_\alpha(v_\beta) = \sum_{p \in \alpha \cap \beta} \cos \angle_ p(\alpha,\beta)
\]
whenever $\alpha$ and $\beta$ are transverse, where $\angle_ p(\alpha,\beta)$ is the counter-clockwise angle from $\alpha$ to $\beta$ at the point $p$. In our case, two distinct curves $\alpha, \beta \in {\mathcal S}$ intersect at most once, with angle $\theta$ from red to blue or $\pi - \theta$ from blue to red. If we split the rows and columns of $D=(d_{X_\theta} \ell_\alpha(v_\beta))_{\alpha,\beta\in{\mathcal S}}$ by color we get a block matrix of the form
\[
D = \cos \theta \begin{pmatrix}
0 & A \\
-A^\intercal & 0
\end{pmatrix}
\]
where $A$ is the matrix of zeros and ones recording which red curves intersect which blue curves. If $\theta \neq \pi / 2$, then $D$ has full rank if and only if the matrix
\[
\widetilde D = \begin{pmatrix}
0 & A \\
A^\intercal & 0
\end{pmatrix}
\]
does. This matrix is the adjacency matrix of some graph $I_{\mathcal S}$, namely, the graph whose vertices are the systoles of $X$ and where two vertices are joined by an edge if and only if the corresponding systoles intersect.
The determinant of the adjacency matrix of a graph counts something combinatorial on the graph. Indeed, according to \cite{Harary} we have
\[
{\rm d}et(\widetilde D) = \sum_{J \subset I_{\mathcal S}} (-1)^{\#\{\text{even components of }J\}} 2^{\#\{\text{cycles in }J\}}
\]
where the sum is over all spanning subgraphs of $J\subset I_{\mathcal S}$ (subgraphs containing all vertices) which are \emph{elementary}, meaning that their components are either edges or embedded cycles. The even components are those with an even number of vertices. In our case, $I_{\mathcal S}$ is bipartite so that all its cycles are even.
We are now ready to give some examples where $\widetilde D$ is invertible.
\subsection*{Examples of girth 2}
The first family of examples comes from \exref{example}. In that example, the red and blue curves form a chain, that is, $I_{\mathcal S}$ is a cycle of length $2g+2$. It follows that $I_{\mathcal S}$ has exactly three elementary spanning subgraphs: $I_{\mathcal S}$ itself, and two subgraphs obtained by deleting every other edge in $I_{\mathcal S}$. If $g = 2m$ is even, then $I_{\mathcal S}$ has $4m+2$ edges and
\begin{equation} \label{eq:det}
{\rm d}et(\widetilde D) = (-1)^1 2^1 + (-1)^{2m+1} 2^0 + (-1)^{2m+1} 2^0 = -4 \neq 0.
\end{equation}
Alternatively, one could compute the determinant of $\widetilde D$ by using the fact that $A$ is a circulant matrix in this case.
The fact that $\widetilde D$ has non-zero determinant implies that the derivative of $\ell=(\ell_\alpha)_{\alpha\in{\mathcal S}}:{\mathcal T}_g\to{\mathbb R}^{\mathcal S}$ has full rank at $X_\theta$ whenever $\theta \neq \pi /2$. By the implicit function theorem, near $X_\theta$ we have that $\ell^{-1}({\mathbf D}elta)$ is a smooth submanifold of codimension \[|{\mathcal S}|-1=(2g+2)-1 = 2g+1,\] hence of dimension $4g-7$. As explained earlier, $\ell^{-1}({\mathbf D}elta)$ intersected with a sufficiently small ball around $X_\theta$ is contained in ${\calX}_g$. We have thus proved that ${\calX}_g$ has dimension at least $4g-7$ when $g$ is even. We can push the proof a little further to obtain the following.
\begin{thm} \label{thm:dimension}
For every even $g \geq 2$, the set ${\calX}_g \subset {\mathcal T}_g$ of closed hyperbolic surfaces of genus $g$ whose systoles fill contains a cell of dimension $4g-5$.
\end{thm}
\begin{proof}
Let $X$ be the surface of genus $g$ from \exref{example} and let \[{\mathcal S} = \{\alpha_1, \ldots, \alpha_{2g+2}\}\] be its set of systoles labelled in such a way that $\alpha_j$ intersects $\alpha_{j-1}$ and $\alpha_{j+1}$ for every $j$, where the indices are taken modulo $2g+2$. Let $X_\theta$ be the deformation of $X$ described above, where $\theta$ is close enough to $\pi/2$ so that its sets of systoles is still equal to ${\mathcal S}$. By Equation \eqref{eq:det}, the map $\ell=(\ell_\alpha)_{\alpha\in{\mathcal S}}:{\mathcal T}_g\to{\mathbb R}^{\mathcal S}$ is a submersion at the point $X_\theta$. In particular, $\ell$ is open in a neighborhood of $X_\theta$. This implies that there exist surfaces $Y$ arbitrarily close to $X_\theta$ such that
\[
\ell_{\alpha_1}(Y) = \cdots = \ell_{\alpha_{2g}}(Y)
\]
and such that these lengths are strictly less than $\ell_{\alpha_{2g+1}}(Y)$ and $\ell_{\alpha_{2g+2}}(Y)$. If $Y$ is close enough to $X_\theta$, then its set of systoles is a subset of ${\mathcal S}$ by continuity of the length functions. Therefore, there is a sequence $Y_n$ converging to $X_\theta$ such that the systoles in $Y_n$ are given by the set ${\mathcal R} = \{ \alpha_1 , \ldots, \alpha_{2g}\}$.
If $n$ is large enough, then the square matrix $(d_{Y_n}\ell_\alpha(v_\beta))_{\alpha,\beta \in {\mathcal R}}$ will have non-zero determinant. Indeed, in the limit the matrix has the form
\[
(d_{X_\theta}\ell_\alpha(v_\beta))_{\alpha,\beta \in {\mathcal R}} = \cos \theta \begin{pmatrix}
0 & B \\
-B^\intercal & 0
\end{pmatrix}
\]
which is invertible because $\begin{pmatrix}
0 & B \\
B^\intercal & 0
\end{pmatrix}$ is the adjacency matrix of a tree $I_{\mathcal R}$ with an even number of vertices. Up to sign, its determinant is the number of perfect matchings (spanning subgraphs whose components are edges) in $I_{\mathcal R}$, which is equal to one. Furthermore, the entries of $(d_{Y}\ell_\alpha(v_\beta))_{\alpha,\beta \in {\mathcal R}}$ depend continuously on the surface $Y$ in the same way that the angles of intersection between geodesics do.
Let $Y=Y_n$ for any such large enough $n$. Then the systoles in $Y$ are given by the set ${\mathcal R}$ and the map $(\ell_\alpha)_{\alpha\in{\mathcal R}}:{\mathcal T}_g\to{\mathbb R}^{\mathcal R}$ is a submersion at $Y$. By the implicit function theorem, the inverse image of the diagonal by this map is a submanifold of codimension $2g-1$ near $Y$. Since the curves in ${\mathcal R}$ fill, a small neighborhood of $Y$ in this submanifold is contained in ${\calX}_g$. The curves in ${\mathcal R}$ fill because the complement of the curves in ${\mathcal S}$ is a union of four polygons which meet at the intersection of $\alpha_{2g+1}$ and $\alpha_{2g+2}$. Adding these two curves fuses the four polygons into a single one.
\end{proof}
When $g$ is odd, the matrix $\widetilde D$ is singular, but this does not ne\-cessarily imply that the image of $(\ell_\alpha)_{\alpha \in {\mathcal S}}$ is not transverse to the diagonal.
\subsection*{An example of girth 3}
Next, we present an example of genus $g=6$ where the underlying graphs $M$ and $G$ for the surface $X$ have girth $3$ and the matrix $\widetilde D$ is non-singular. Let $M$ be the $1$-skeleton of a regular tetrahedron (as a map of type $\{3,3\}$ on the sphere) and let $B$ be the corresponding block. This is a sphere with $4$ holes and tetrahedral symmetry. Although this does not fit in the theory of \secref{sec:gluing}, it is possible to glue five copies of $B$ along the complete graph $K_5$ in such a way that the blue arcs connect up in groups of three to form closed geodesics. To see this, it is convenient to draw $K_5$ in ${\mathbb R}^3$ with a $3$-fold symmetry as in \figref{figure}.
\begin{figure}
\caption{An embbeding of $K_5$ in ${\mathbb R}
\label{figure}
\end{figure}
The tetrahedral pieces are glued as suggested by the figure, in the simplest possible way (without twist). By inspection, the blue arcs connect in groups of three. The proof of \propref{prop:count} applies without change to show that the systoles in $X$ are the red curves and the blue curves. The genus of $X$ is equal to the number of edges in the complement of any spanning tree in $K_5$, which is $10-4=6$. Let us label the red curves from $1$ to $10$ and the blue curves from $a$ to $j$ as in \figref{figure} (the red curves correspond to the edges in $K_5$). Then the intersection matrix $A$ is given by
\[
A = \begin{pmatrix}
1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\
1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{pmatrix}
\]
which has determinant $48 \neq 0$. Therefore, the derivative of lengths $d\ell$ has full rank at $X_\theta$ whenever $\theta \neq \pi /2$. Note that this only gives us that ${\calX}_g$ has codimension at most $19$ in ${\mathcal T}_g$, hence dimension at least $11 = 4g - 13$. The conclusion is weaker than that of \thmref{thm:dimension}, but we wanted to include this example to show that $\widetilde D$ can have full rank for more complicated graphs.
\subsection*{Questions}
We conclude with a few questions related to the strategy we have just outlined.
\begin{question}
Is there a sequence of graphs $M$ and $G$ as in \secref{sec:gluing} with girth going to infinity such that the corresponding intersection matrices $\widetilde D$ have non-zero determinants?
\end{question}
In view of the above reasoning and the counting of \propref{prop:count}, a po\-si\-tive answer would imply \conjref{conj:dimension}. A major difficulty is that $M$ and $G$ are given to us in a non-explicit way from \thmref{thm:reg_map} and \thmref{thm:poly_graph}.
As the proof of \thmref{thm:dimension} shows, one could bypass the determinant issue by fin\-ding a filling subset ${\mathcal R} \subset {\mathcal S}$ of even cardinality such that the corresponding intersection graph $I_{\mathcal R}$ is a tree, and a surface $Y$ near $X_\theta$ whose systoles are exactly the curves in ${\mathcal R}$.
\begin{question}
Given a surface $X$ constructed as in \secref{sec:gluing} with set of systoles ${\mathcal S}$, is there an induced subtree in $I_{\mathcal S}$ with an even number of vertices such that the union of the corresponding curves fill?
\end{question}
\begin{question}
Let $X$ be any hyperbolic surface, let ${\mathcal S}$ be its set of systoles and let ${\mathcal R} \subset {\mathcal S}$ be a non-empty subset. Does there exist, in every neighborhood of $X$, a surface whose set of systoles is equal to ${\mathcal R}$?
\end{question}
Even if these questions have negative answers, they suggest how one should modify the construction of surfaces with sublinearly many systoles that fill in order to show that ${\calX}_g$ has large dimension: the systoles should cut the surface into a single polygon instead of several.
\end{document} | math | 55,427 |
\begin{document}
\title{On De Rham Cohomology of Linear Categories}
\author{Andrei Chite\c{s}, M\u{a}d\u{a}lin Ciungu and Drago\c s \c Stefan}
\address{University of Bucharest, Faculty of Mathematics and Computer
Science, 14 Academiei Street, Bucharest Ro-010014, Romania}
\email{[email protected]}
\email{[email protected]}
\email{[email protected]}
\subjclass[2000]{Primary 18G60; Secondary 13D15}
\begin{abstract}
We define the Chern map from the Grothendieck group of a linear category $
\mathcal{C}$ to the de Rham cohomology of $\mathcal{C}$ with coefficients in
a $DG$-category $\Omegamega^{\ast}$. In order to achieve our goal, we define the
notion of connection on a $\mathcal{C}$-module, and we show that the trace
of the curvature of a connection is a de Rham cocycle, whose cohomology
class does not depend on the choice of the connection.
\end{abstract}
\keywords{linear category; de Rham cohomology; Chern map}
\maketitle
\section*{Introduction}
The classical Chern character is a morphism from the $K_{0}$-theory group of
a manifold to its de Rham cohomology groups. In his work on the foliations
of a manifold, Alain Connes extended the construction of the Chern character
to show that there is a pairing between the Grothendieck group $K_{0}(A)$ of
the Banach algebra $A$ associated to a foliation and the cyclic cohomology $
A $, which replaces de Rham cohomology in Noncommutative Geometry ; see \cite
{Co}. In \cite{Ka}, the definition of the Chern map was extended to $
K_{n}(A),$ the higher $K$-theory groups of $A.$ The method used in loc. cit.
is purely algebraic, working for any finitely generated projective module
over a, not necessarily commutative, algebra $A$. See also \cite[Chapter 8]
{Lo}.
Linear categories were defined and investigated in the seminal paper \cite
{Mi}. Since their appearance, they have been used as a very important tool
not only in algebra, but in many other fields, including algebraic topology,
logic, computer science, etc.
In this paper, proceeding as in \cite{Ka}, we define a Chern map from the
Grothendieck group $K_{0}(\mathcal{C})$ of a linear category $\mathcal{C}$
to the de Rham cohomology of $\mathcal{C}.$ For, we recall the definition of
linear categories and their basic properties in the first section of the
paper. In this part, we also recall some homological properties of the
category of (right) modules over a linear category, and show that the
Hattori-Stallings trace map can be defined for any finitely generated
projective module over a linear category.
In the second section we deal with connections on a finitely generated right
$\mathcal{C}$-module $M$. As an example, we introduce the Levi-Civita
connection on a finitely generated projective module. We are also able to
describe all connections on a finitely generated free $\mathcal{C}$-module.
In the last part of this section, we define the curvature of a connection
and we compute it for the above mentioned examples of connections.
In the third section we define the de Rham cohomology of a linear category $
\mathcal{C}$, with respect to a $DG$-category $\Omegamega ^{\ast }$ such that $
\Omegamega ^{0}=\mathcal{C}.$ We prove that the trace of the curvature of a
connection $\mathbb Nabla$ on a finitely generated projective $\mathcal{C}$-module $
M$ is a de Rham $2q$-cocycle $\mathbb Ch^{q}(M,\mathbb Nabla ).$ We also show that the de
Rham cohomology class of $\mathbb Ch^{q}(M,\mathbb Nabla )$ does not depend on the choice
of the connection $\mathbb Nabla $ on $M.$ We shall denote this cohomology class by
$\mathbb Ch^{q}(M)$. In particular we deduce our main result, stating that $
[M]\mapsto \mathbb Ch^q(M)$ defines a morphism from the Grothendieck group $K_{0}(
\mathcal{C})$ of $\mathcal{C}$ to the de Rham cohomology of $\mathcal{C}$
with coefficients in $\Omegamega ^{\ast }.$
\section{Preliminaries}
In this section we recall the definitions of the notions that we work with.
Throughout, $\Bbbk $ will denote a commutative field. The tensor product
over $\Bbbk $ will be denoted by $\otimesimes .$
\begin{fact}[$\Bbbk $-linear categories.]
A category $\mathcal{C}$\ is called $\Bbbk $-linear if every hom-set in $
\mathcal{C}$\ is a $\Bbbk $-linear space and the composition maps in $
\mathcal{C}$\ are bilinear. We shall denote the space of morphisms from $x$
to $y$ by $_{y}\mathcal{C}_{x}$, so the composition map can be seen as a
linear transformation from $_{x}\mathcal{C}_{y}\otimesimes {}_{y}\mathcal{C}
_{z}\ $to ${}_{x}\mathcal{C}_{z},$ for any objects $x$, $y$ and $z$ in $
\mathcal{C}$. In this paper $\mathcal{C}$ will always denote a small linear
category. Thus $\mathcal{C}_{0},$ the class of objects in $\mathcal{C},$ is
a set.
A functor $F:\mathcal{C}\lambdaongrightarrow \mathcal{D}$ between linear
categories is $\Bbbk $-linear if the corresponding map from ${}_{x}\mathcal{C
}_{y}$ to ${}_{F(x)}\mathcal{D}_{F(y)}$ is $\Bbbk $-linear, for any objects $
x$ and $y$ in $\mathcal{C}_{0}.$
\end{fact}
\begin{fact}[Examples of $\Bbbk $-linear categories.]
Of course the category of $\Bbbk $-linear spaces is the prototype of $\Bbbk $
-linear categories. Other examples are listed below.
\begin{enumerate}
\item Let $\mathcal{C}$\ be a given $\Bbbk $-linear category. The opposite
category of $\mathcal{C}$ is also a linear category. Recall that $\mathcal{C}
$\ and $\mathcal{C}^{op}$\ have the same objects, while $_{x}(\mathcal{C}
^{op})_{y}={}_{y}\mathcal{C}_{x}$. If we denote the composition in $\mathcal{
C}^{op}$ by $\bullet $, then $f\bullet g=g\circ f$\ \ for any $g\in {}_{x}
\mathcal{C}_{y}$\ and $f\in {}_{y}\mathcal{C}_{z}$.
\item Let $\mathcal{C}$\ and $\mathcal{D}$\ be two $\Bbbk $-linear
categories. The tensor product $\mathcal{C\boxtimes D}$\ is the $\Bbbk $
-linear category:
\begin{enumerate}
\item With the set of objects $\mathcal{(C\boxtimes D)}_{0}=\mathcal{C}_{0}
\mathcal{\times D}_{0}$;
\item The space of morphisms $_{(x,y)}\mathcal{(C\boxtimes D)}_{(x^{\prime
},y^{\prime })}={}_{x}\mathcal{C}_{x^{\prime }}\otimesimes {}_{y}\mathcal{D}
_{y^{\prime }}$, for any $(x,y)$ and $(x^{\prime },y^{\prime })$ in $
\mathcal{C}_{0}\times \mathcal{D}_{0}$.
\end{enumerate}
The composition in $\mathcal{C\boxtimes D}$\ is defined by
\begin{equation*}
(f\otimesimes g)\circ \ (f^{\prime }\otimesimes g^{\prime })=(f\circ f^{\prime
})\otimesimes (g\circ g^{\prime }),
\end{equation*}
for any $f\otimesimes g\in {}_{x}\mathcal{C}_{y}\otimesimes {}_{x^{\prime }}\mathcal{
D}_{y^{\prime }},$ and $f^{\prime }\otimesimes g^{\prime }\in {}_{y}\mathcal{C}
_{z}\otimesimes {y^{\prime }}\mathcal{D}_{z^{\prime }}$. In particular, for a
linear category $\mathcal{C}$, one defines its enveloping category by $
\mathcal{C}^{e}:=\mathcal{C\boxtimes C}^{op}$.
\item The category of left $\mathcal{C}$-modules is a $\Bbbk $-linear
category. By definition, a left $\mathcal{C}$-module is a covariant linear
functor from $\mathcal{C}$ to $\Bbbk $-${\mathcal{M}}od$. It is easy to see that a $
\mathcal{C}$-module $M$ is a family $\{{}_{x}M\}_{x\in \mathcal{C}_{0}}$ of
linear spaces, together with maps $\cdotot :{}_{x}\mathcal{C}_{y}\otimesimes
{}_{y}M\lambdaongrightarrow {}_{x}M$ such that
\begin{equation*}
1_{z}\cdotot m=m\qquad \text{and\qquad }f\cdotot (g\cdotot m)=(f\circ g)\cdotot m,
\end{equation*}
for any $m\in {}_{z}M$ and for any $f\in {}_{x}\mathcal{C}_{y}$ and $g\in
{}_{y}\mathcal{C}_{z}$. A morphism of $\mathcal{C}$-modules $u:M\rhoightarrow
N $ is a natural transformation between the corresponding functors.
Equivalently, a morphism $u$ as above is given by a family $\{_{x}u\}_{x\in
\mathcal{C}_{0}}$ of $\Bbbk $-linear maps $_{x}u:{}_{x}M\rhoightarrow {}_{x}N$
such that, for all $f\in {}_{y}\mathcal{C}_{x}$\ and $m\in {}_{x}{M}$ we have
\begin{equation*}
_{y}u(f\cdotot m)=f\cdotot {}_{x}u(m).
\end{equation*}
The category ${\mathcal{M}}od$-$\mathcal{C}$ of right $\mathcal{C}$-modules is defined
similarly.
\item The category $\mathcal{C}$-${\mathcal{M}}od$-$\mathcal{C}$\ of $\mathcal{C}$
-bimodules will play an important role in our paper. By definition, a $
\mathcal{C}$-bimodule is a left $\mathcal{C\boxtimes C}^{op}$-module.
Therefore, a bimodule $M$\ is a family $\{_{x}M_{y}\}_{x,y\in \mathcal{C}
_{0}}$, endowed with maps $\cdotot :{}_{x}\mathcal{C}_{y}\otimesimes
{}_{y}M_{z}\lambdaongrightarrow {}_{x}M_{z}$ and $\cdotot :{}_{x}M_{y}\otimesimes {}_{y}
\mathcal{C}_{z}\lambdaongrightarrow {}_{x}M_{z}.$ By definition, for any $y\in
\mathcal{C}_{0},$ the family $_{\bullet }M_{y}:=\{_{x}M_{y}\}_{x\in \mathcal{
C}_{0}}$ is a left module with respect to the former maps. Similarly, for
any $x\in \mathcal{C}_{0}$ the family $_{x}M_{\bullet }:=\{_{x}M_{y}\}_{y\in
\mathcal{C}_{0}}$ is a right module with respect to the latter maps. In
addition, these module structures are compatible in the sense that, for any $
m\in {}_{y}{M}_{z},$ and any morphisms $f\in {}_{x}\mathcal{C}_{y}$ and $
g\in {}_{z}\mathcal{C}_{t},$ we have
\begin{equation*}
(f\cdotot m)\cdotot g=f\cdotot (m\cdotot g).
\end{equation*}
A morphism of bimodules $u:M\rhoightarrow N$ is given by a family $
\{_{x}u_{y}\}_{x,y\in \mathcal{C}_{0}}$ of $\Bbbk $-linear transformations $
_{x}u_{y}:{}_{x}M_{y}\lambdaongrightarrow {}_{x}N_{y}$, which for $m,$ $f$ and $g$
as above verify the relation
\begin{equation*}
_{x}u_{t}(f\cdotot m\cdotot g)=f\cdotot {}_{y}u_{z}(m)\cdotot g.
\end{equation*}
\end{enumerate}
It is well-known that $\mathcal{C}$-${\mathcal{M}}od$ has enough injective objects, for
any linear category $\mathcal{C}$. In $\mathcal{C}$-${\mathcal{M}}od$ there are enough
projective objects as well. Clearly, ${\mathcal{M}}od$-$\mathcal{C}$ and $\mathcal{C}$-$
{\mathcal{M}}od$-$\mathcal{C}$ also have the above properties, since they can be
regarded as categories of left modules over $\mathcal{C}^{op}$ and $\mathcal{
C\boxtimes C}^{op},$ respectively.
\end{fact}
\begin{fact}[The tensor product of two (bi)modules.]
Let $\mathcal{C}$ be a linear category. We assume that ${M}$ is a right $
\mathcal{C}$-module and that $N$ is a left $\mathcal{C}$-module. The vector
space${\ M}\otimesimes _{\mathcal{C}}N$ is defined as the quotient of $
\opluslus_{z\in \mathcal{C}_{0}}(M_{z}\otimesimes {}_{z}N)$ through the linear
subspace generated by the elements $m\cdotot f\otimesimes n-m\otimesimes f\cdotot n,$
for arbitrary $m\in {}M_{u},$ $f\in {}_{u}\mathcal{C}_{v}$ and $n\in
{}_{v}N. $ The class of $m\otimesimes n\in {}M_{u}\otimesimes {}_{u}N$ in the
quotient linear space ${M}\otimesimes _{\mathcal{C}}N$ will be denoted by $
m\otimesimes _{\mathcal{C}}n.$ These tensor monomials generate ${M}\otimesimes _{
\mathcal{C}}N$ as a vector space.
Now one can define the tensor product of two bimodules $X$ and $Y$ as
follows. For $x$ and $y$ in $\mathcal{C}_{0}$ we know that $_{x}X_{\bullet }$
is a a right module and that $_{\bullet }Y_{y}$ is left module. Thus the
tensor product $_{x}{X}_{\bullet }\otimesimes _{\mathcal{C}}{}_{\bullet }Y_{y}$\
makes sense. By definition $X\otimesimes _{\mathcal{C}}Y$ is the bimodule whose
components are the vector spaces
\begin{equation*}
_{x}({X}\otimesimes _{\mathcal{C}}Y)_{y}:={}{X}_{\bullet }\otimesimes _{\mathcal{C}
}{}_{\bullet }Y_{y}.
\end{equation*}
The bimodule structure on ${X}\otimesimes _{\mathcal{C}}Y$ is induced by the
left action on ${X}$ and the right action on $Y$.
Note that the category $\mathcal{C}$-${\mathcal{M}}od$-$\mathcal{C}$ is a monoidal
category with respect to the tensor product of $\mathcal{C}$-bimodules. Its
unit object is the bimodule $\mathcal{C}$. In $\mathcal{C}$-${\mathcal{M}}od$-$\mathcal{
C}$ there are arbitrary coproducts, and it is easy to see that the tensor
product $\otimesimes _{\mathcal{C}}$ is distributive over coproducts.
\end{fact}
\begin{fact}[Exemples of $\mathcal{C}$-bimodules.]
By definition, $\mathcal{C}\otimesimes \mathcal{C}$ is the $\mathcal{C}$
-bimodule whose components are the vector spaces
\begin{equation*}
_{x}(\mathcal{C}\otimesimes \mathcal{C})_{y}=\textstyle\bigoplus\lambdaimits_{z\in
\mathcal{C}_{0}}(_{x}\mathcal{C}_{z}\otimesimes {}_{z}\mathcal{C}_{y}).
\end{equation*}
The left and right actions are induced by the composition in $\mathcal{C}$.
Thus the bimodule structure is defined by the relation
\begin{equation*}
f^{\prime }\cdotot (g^{\prime \prime \prime })\cdotot f^{\prime \prime
}=(f^{\prime }\circ g^{\prime })\otimesimes (g^{\prime \prime }\circ f^{\prime
\prime }),
\end{equation*}
where $f^{\prime }$, $f^{\prime \prime }$, $g^{\prime }$ and $g^{\prime
\prime }$ are arbitrary morphisms in $\mathcal{C}$ such that $f^{\prime
}\circ g^{\prime }$ and $g^{\prime \prime }\circ f^{\prime \prime }$ make
sense. One can prove easily that $\mathcal{C}\otimesimes \mathcal{C}$ is
projective as a $\mathcal{C}$-bimodule.
Another example of $\mathcal{C}$-bimodule is $\mathcal{C}$ itself. Its
components are the linear spaces $_{x}\mathcal{C}_{y}$\ and the actions are
defined by the composition in $\mathcal{C}$. Therefore, for any object $x$
in $\mathcal{C}$ we can consider the right $\mathcal{C}$-module $_{x}
\mathcal{C}_{\bullet }.$ In view of the remarks from the previous
subsection, for any family $\mathcal{I}:=\{x_{i}\}_{i\in I}$ of objects in $
\mathcal{C}_{0},$ we can consider the coproduct $\mathcal{C}{(}\mathcal{I}{)}
:=\opluslus_{i\in I}{}_{x_{i}}\mathcal{C}_{\bullet }$ in the category of right $
\mathcal{C}$-modules. Of course, a similar coproduct exists in the category
of left $\mathcal{C}$-modules.
\end{fact}
\begin{fact}[Finitely generated projective $\mathcal{C}$-modules and the
trace map.]
\lambdaabel{fa: fgp}We fix, as before, a $\Bbbk $-linear category $\mathcal{C}$.
Let $M$ be a right $\mathcal{C}$-module. By definition, $M$ is finitely
generated if there is a finite family $\mathcal{I}:=\{x_{i}\}_{i\in I}$ of
objects in $\mathcal{C}$ such that $M$ is a quotient of ${}\mathcal{C}{(}
\mathcal{I}{)}$. Equivalently, there is a set $\lambdaeft\{ m_{i}\rhoight\} _{i\in
I}$, with $m_{i}\in M_{x_{i}}$, such that any $m\in M_{x}$ can be written as
a linear combination
\begin{equation}
m=\textstyle\sum\lambdaimits_{i\in I}m_{i}\cdotot f_{i}, \lambdaabel{ec:m}
\end{equation}
for some $f_{i}\in {}_{x_{i}}\mathcal{C}_{x}$. If, in addition, $M$ is
projective then the canonical projection from $\mathcal{C}{(}\mathcal{I}{)}$
to $M$ has a section in ${\mathcal{M}}od$-$\mathcal{C}.$ Therefore $M$ is a direct
summand of $\mathcal{C}{(\mathcal{I})}$. The converse also holds, as $
\mathcal{C}{(}\mathcal{I}{)}$ is a finitely generated projective right $
\mathcal{C}$-module, for any finite family $\mathcal{I}$ of objects in $
\mathcal{C}.$
From the above characterization we deduce that $(M,\cdotot )$ is a finitely
generated projective right $\mathcal{C}$-module if and only if there are
\emph{dual bases} $\{m_{i}\}_{i\in I}$ and $\{\varphi ^{i}\}_{i\in I}$ on $M$
. By definition, $\mathcal{I}:=\{x_{i}\}_{i\in I}$ is a finite family of
objects, $m_{i}\in M_{x_{i}}$ and $\varphi ^{i}:M\rhoightarrow {}_{x_{i}}
\mathcal{C}_{\bullet }$ is a morphism of right $\mathcal{C}$-modules, for
all $i\in I$. We shall denote the components of $\varphi ^{i}$ by $\varphi
_{x}^{i}:M_{x}\rhoightarrow {}_{x_{i}}\mathcal{C}_{x}$. In addition, for any $
m\in M_{x}$, the following relation holds
\begin{equation}
m=\textstyle\sum\lambdaimits_{i\in I}m_{i}\cdotot \varphi _{x}^{i}(m).
\lambdaabel{ec:phi}
\end{equation}
Using the existence of dual bases for finitely generated projective $
\mathcal{C}$-modules, one proves that
\begin{equation}
\Hom_{\mathcal{C}}(M,M)\overset{\cong }{\lambdaongrightarrow }M\otimesimes _{\mathcal{
C}}M^{\ast }, \lambdaabel{ec:izo}
\end{equation}
where $M^{\ast }$ is the dual of $M.$ By definition, $M^{\ast }$ is a left $
\mathcal{C}$-module, and its components are the linear spaces $_{x}M^{\ast
}:=\Hom_{\mathcal{C}}(M,{}_{x}\mathcal{C}_{\bullet })$. If $\varphi
=\{\varphi _{y}\}_{y\in \mathcal{C}_{0}}$ is a morphism of right $\mathcal{C}
$-modules from $M$ to $_{x}\mathcal{C}_{\bullet }$ and $f\in {}_{y}\mathcal{C
}_{x},$ then the component $\lambdaeft( f\cdotot \varphi \rhoight)
_{z}:M_{z}\rhoightarrow {}_{y}\mathcal{C}_{z}$ of $f\cdotot \varphi \in
{}_{y}M^{\ast }$ maps $m\in M_{z}$ to $f\cdotot \varphi _{z}(m).$
As in the case of $\Bbbk $-algebras, we can speak about the commutator $
[f,g] $ of two morphisms $f\in {}_{x}\mathcal{C}_{y}$ and $g\in {}_{y}
\mathcal{C}_{x}.$ By definition, $[f,g]:=f\circ g-g\circ f$ is an element in
$\opluslus _{z\in \mathcal{C}_{0}}{}_{z}\mathcal{C}_{z}.$ Let $\mathcal{C}_{ab}$
denote the quotient vector space of $\opluslus _{z\in \mathcal{C}_{0}}{}_{z}
\mathcal{C}_{z}$ through the subspace $[\mathcal{C},\mathcal{C}]$ spanned by
all commutators in $\mathcal{C}.$ The class of $f\in{}_x\mathcal{C}_x$ in $
\mathcal{C}_{ab}$ will be denoted by $f+[\mathcal{C},\mathcal{C}]$.
If $M$ is a right $\mathcal{C}$-module, then the \textit{evaluation map} $\ev
_{M}:M\otimesimes _{\mathcal{C}}M^{\ast }\rhoightarrow \mathcal{C}_{ab}$ is
uniquely defined by
\begin{equation*}
\ev_{M}(m\otimesimes _{\mathcal{C}}\varphi ):=\varphi _{x}(m)+[\mathcal{C},
\mathcal{C}],
\end{equation*}
for any morphism of right $\mathcal{C}$-modules $\varphi =\{\varphi
_{y}\}_{y\in \mathcal{C}_{0}}$ in $_{x}M^{\ast }$ and $m\in M_{x}$.
We can now define the \emph{Hattori-Stallings trace map} $\Tr_{M}:\Hom_{
\mathcal{C}}(M,M)\rhoightarrow \mathcal{C}_{ab},$ for any finitely generated
projective right $\mathcal{C}$-module $M,$ as being the composition of the
evaluation map $\ev_{M}$ with the isomorphism (\rhoef{ec:izo}). Therefore, if $
\{m_{i}\}_{z\in I}$ and $\{\varphi ^{i}\}_{z\in I}$ are finite dual bases on
$M,$ and $u:=\{u_{x}\}_{x\in \mathcal{C}_{0}}$ is an endomorphism of $M,$
then
\begin{equation}
\Tr_{M}(u)=\textstyle\sum\lambdaimits_{i\in I}\varphi
_{x_{i}}^{i}(u_{x_{i}}(m_{i}))+[\mathcal{C},\mathcal{C}]. \lambdaabel{ec:Tr}
\end{equation}
Note that the formula for $\Tr_{M}(u)$ does not depend on the choice of the
dual bases. Furthermore, if $v:N\rhoightarrow M$ and $w:M\rhoightarrow N$ are
morphisms between two finitely generated projective right $\mathcal{C}$
-modules, then by using \eqref{ec:Tr} one can easily see that $\Tr
_{M}(v\circ w)=\Tr_{N}(w\circ v).$
\end{fact}
\section{Connections. The curvature of a connection.}
In this section we introduce the main tools that we need for the
construction of the Chern map. We start by recalling the definition of $DG$
-categories, which explicitly appear in the definition of connections.
\begin{fact}[$DG$-categories.]
We say that a category $\Omegamega ^{\ast }$ is graded if every hom-space $
_{x}\Omegamega _{y}^{\ast }$ is endowed with a decomposition $_{x}\Omegamega
_{y}^{\ast }=\opluslus _{n\geq 0\,}${}$_{x}\Omegamega _{y}^{n}$ such that $f\circ
g\in {}_{x}\Omegamega _{z}^{n+m},$ for any $f\in {}_{x}\Omegamega _{y}^{n}$ and $
g\in {}_{y}\Omegamega _{z}^{m}.$ For simplicity, the composition of two forms in
a $DG$-category $\Omegamega ^{\ast }$ will be denoted by concatenation, that is $
\omega \circ \mathbb Zeta =\omega \mathbb Zeta .$
A differential $d^{\ast }$ on $\Omegamega ^{\ast }$ is a family of linear maps $
_{x}d_{y}^{n}:{}_{x}\Omegamega _{y}^{n}\rhoightarrow {}_{x}\Omegamega _{y}^{n+1}$ such
that $_{x}d_{y}^{n+1}\circ {}_{x}d_{y}^{n}=0$ and
\begin{equation}
_{x}d_{z}^{n+m}(\omega \mathbb Zeta )={}_{x}d_{y}^{n}(\omega )\mathbb Zeta +(-1)^{n}\omega
\,{}_{y}d_{z}^{m}(\mathbb Zeta ), \lambdaabel{ec:Leibniz2}
\end{equation}
for $\omega \in {}_{x}\Omegamega _{y}^{n}$ and $\mathbb Zeta \in {}_{y}\Omegamega _{z}^{m}$
. We refer to the relation (\rhoef{ec:Leibniz2}) saying that $d^{\ast }$
satisfies the (graded) Leibniz rule.
Recall that a $DG$-category is a couple $(\Omegamega ^{\ast },d^{\ast }),$ with $
\Omegamega ^{\ast }$ a graded category and $d^{\ast }\ $a differential on $
\Omegamega ^{\ast }.$ If $(\Omegamega ^{\ast },d^{\ast })$ is a $DG$-category, then
the vector spaces $\lambdaeft\{ _{x}\Omegamega _{y}^{0}\rhoight\} _{x,y\in \Omegamega
_{0}^{\ast }}$ define a subcategory $\Omegamega ^{0}$ of $\Omegamega ^{\ast }$.
Moreover, for every $n\in \mathbb{N},$ the family $\Omegamega ^{n}:=\lambdaeft\{
_{x}\Omegamega _{y}^{n}\rhoight\} _{x,y\in \Omegamega _{0}^{\ast }}$ has a natural
structure of $\Omegamega ^{0}$-bimodule.
An element $\omega \in {}_{x}\Omegamega _{y}^{n}$ will be called a differential
form in $\Omegamega ^{\ast }$ from $y$ to $x$ of degree $n.$ The degree of $
\omega $ will also be denoted by $\lambdaeft\vert \omega \rhoight\vert $ and, if
there is no danger of confusion, we shall write $d\omega$ instead of $
{}_{x}d_{y}^{n}(\omega ).$
\end{fact}
\begin{fact}[Connections.]
\lambdaabel{fact:conn}Let $M$ be a finitely generated right module over a linear
category $\mathcal{C}$. We fix a $DG$-category $(\Omegamega ^{\ast },d^{\ast })$
such that $\Omegamega ^{0}=\mathcal{C}$. Since $\Omegamega ^{1}$ is a $\mathcal{C}$
-bimodule, the tensor product $M\otimesimes _{\mathcal{C}}{}_{\bullet }\Omegamega
_{x}^{1}$ makes sense for any $x\in\mathcal{C}_0$.
By definition, a \emph{connection} $\mathbb Nabla :M\rhoightarrow M\otimesimes _{\mathcal{
C}}\Omegamega ^{1}$ on $M$ (with respect to $\Omegamega ^{\ast })$ is a family $
\{\mathbb Nabla _{x}\}_{x\in \mathcal{C}_{0}}$ of $\Bbbk $-linear maps $\mathbb Nabla
_{x}:M_{x}\rhoightarrow M\otimesimes _{\mathcal{C}}{}_{\bullet }\Omegamega _{x}^{1}$
such that, for all $m\in M_{x}$ and $f\in {}_{x}\mathcal{C}_{y}$,
\begin{equation}
\mathbb Nabla _{y}(m\cdotot f)=\mathbb Nabla _{x}(m)\cdotot f+m\otimesimes _{\mathcal{C}}df.
\lambdaabel{ec:conn}
\end{equation}
By assumption, there is a finite set $\lambdaeft\{ m_{i}\rhoight\} _{i\in I}$ of
generators for $M,$ with $m_{i}\in M_{x_{i}}$ for some $x_{i}\in \mathcal{C}
_{0}.$ In view of equation \eqref{ec:conn}, the connection $\mathbb Nabla $ is
uniquely determined by the elements $\lambdaambda _{ij}\in {}_{x_{i}}\Omegamega
_{x_{j}}^{1}$ such that
\begin{equation}
\mathbb Nabla _{x_{i}}(m_{i})=\textstyle\sum\lambdaimits_{j\in I}m_{j}\otimesimes _{\mathcal{
C}}\lambdaambda _{ji}. \lambdaabel{ec:lambda}
\end{equation}
Indeed, if $m\in M_{x},$ then there is $\{f_{i}\}_{i\in I}$ in $\mathcal{C}{(
}\mathcal{I}{)}_{x}$ such that the relation (\rhoef{ec:m}) holds. Thus,
\begin{equation}
\mathbb Nabla _{x}(m)=\textstyle\sum\lambdaimits_{i,j\in I}m_{i}\otimesimes _{\mathcal{C}
}\lambdaambda _{ij}f_{j}+\textstyle\sum\lambdaimits_{i\in I}m_{i}\otimesimes _{\mathcal{C}
}df_{i}. \lambdaabel{ec:coord}
\end{equation}
We shall denote the matrix $(\lambdaambda _{ij})_{i,j\in I}$ by $\Lambda $.
\end{fact}
In the following lemma we give two methods for constructing new connections.
\begin{lemma}
\lambdaabel{le:conn}Let $(\Omegamega ^{\ast },d^{\ast })$ be a $DG$-category such
that $\Omegamega ^{0}=\mathcal{C}.$
\begin{enumerate}
\item[(i)] Let $\mathbb Nabla _{p}:M_{p}\rhoightarrow M_{p}\otimesimes _{\mathcal{C}
}\Omegamega ^{1}$ be a connection on $M_{p},$ where $p\in \{1,2\}$. There exists
a unique connection $\mathbb Nabla _{1}\opluslus \mathbb Nabla _{2}$ on $M_{1}\opluslus M_{2}$
such that the diagram below is commutative for any $p\in \{1,2\}$.
\begin{equation*}
\xymatrix{ M_{1}\opluslus M_{2}\ar[r]^-{\mathbb Nabla_1\opluslus\mathbb Nabla_2} & \lambdaeft(
M_{1}\opluslus M_{2}\rhoight) \otimesimes _{\mathcal{C}}\Omegamega ^{1} \\
M_{p}\ar[r]_-{\mathbb Nabla_p}\ar@{^{(}->}[u] & M_{p} \otimesimes _{\mathcal{C}}\Omegamega
^{1}\ar@{^{(}->}[u] }
\end{equation*}
\item[(ii)] A connection $\mathbb Nabla:M\to M\otimesimes_{\mathcal{C}}\Omegamega^1$
induces a connection on every direct summand $N$ of $M$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let us sketch the proof of the first part of the lemma. The details are left
to the reader. Let $m_{p}$ be an element in $\lambdaeft( M_{p}\rhoight) _{x},$
where $p\in \{1,2\}$. Then
\begin{equation*}
\mathbb Nabla _{p}(m_{p})=\textstyle\sum_{j=1}^{n_{p}}m_{pj}\otimesimes _{\mathcal{C}
}\omega _{pj},
\end{equation*}
for some $m_{pj}\in \lambdaeft( M_{p}\rhoight) _{y_{j}}$ and $\omega _{pj}\in
{}_{y_{j}}\Omegamega _{x}^{1}.$ We now define the connection $\mathbb Nabla _{1}\opluslus
\mathbb Nabla _{2}$ by
\begin{equation*}
\lambdaeft( \mathbb Nabla _{1}\opluslus \mathbb Nabla _{2}\rhoight) _{x}(m_{1},m_{2})=\textstyle
\sum_{j=1}^{n_{1}}\lambdaeft( m_{1j},0\rhoight) \otimesimes _{\mathcal{C}}\omega _{1j}+
\textstyle\sum_{j=1}^{n_{2}}\lambdaeft( 0,m_{2j}\rhoight) \otimesimes _{\mathcal{C}
}\omega _{2j}.
\end{equation*}
To prove the second part of the lemma, we choose a morphism $\pi
:M\rhoightarrow N$ in ${\mathcal{M}}od$-$\mathcal{C}$ which has a section $\sigma
:N\rhoightarrow M$ in this category. Then, for an object $x$ in $\mathcal{C},$
we define $\mathbb Nabla _{x}^{\prime }:N_{x}\rhoightarrow N\otimesimes _{\mathcal{C}
}{}_{\bullet }\Omegamega _{x}^{1}$ by
\begin{equation*}
\mathbb Nabla _{x}^{\prime }=(\pi \otimesimes _{\mathcal{C}}\Id_{\Omegamega ^{1}\mathcal{C}
})_{x}\circ \mathbb Nabla _{x}\circ \sigma _{x}.
\end{equation*}
It is easy to see that $\mathbb Nabla ^{\prime }:=\{\mathbb Nabla _{x}^{\prime }\}_{x\in
\mathcal{C}_{0}}$ is a connection on $N$.
\end{proof}
\begin{fact}[Examples of connections.]
\lambdaabel{fa:ExConn}We start by describing all connections on the right module $
\mathcal{C}{(}\mathcal{I}{)}$, where $\mathcal{I}:=\{x_{i}\}_{i\in I}$ is a
finite family of objects in $\mathcal{C}.$ Clearly, $\lambdaeft\{ e_{{i}}\rhoight\}
_{i\in I}$ is a set of generators for $M$, where $e_{i}\in \mathcal{C}{(}
\mathcal{I}{)}_{x_{i}}$ is the family $\{\delta _{i,j}1_{x_{i}}\}_{j\in I}.$
Here $\delta _{i,j}$ denotes the Kronecker symbol. Hence, for an element $
\{f_{i}\}_{i\in I}$ in $\mathcal{C}{(}\mathcal{I}{)}_{x},$ we get
\begin{equation}
\mathbb Nabla _{x}(\{f_{i}\}_{i\in I})=\mathbb Nabla _{x}(\textstyle\sum\lambdaimits_{i\in
I}e_{i}\cdotot f_{i})=\textstyle\sum\lambdaimits_{i,j\in I}e_{i}\otimesimes _{\mathcal{C
}}\lambdaambda _{ij}f_{j}+\textstyle\sum\lambdaimits_{i\in I}e_{i}\otimesimes _{\mathcal{C}
}df_{i}, \lambdaabel{ec:conn_free}
\end{equation}
where $\lambdaambda _{ij}\in {}_{x_{i}}\Omegamega _{x_{j}}^{1}$. Conversely, if $
\Lambda :=(\lambdaambda _{ij})_{i,j\in I}$ is an arbitrary matrix of differential
forms of degree $1$ such that $\lambdaambda _{ij}\in {}_{x_{i}}\Omegamega
_{x_{j}}^{1},$ then \eqref{ec:conn_free} defines a connection $\mathbb Nabla
^{\Lambda }$ on $\mathcal{C}{(}\mathcal{I}{)}.$ In particular, if $\Lambda
=0 $, then
\begin{equation}
\mathbb Nabla _{x}(\{f_{i}\}_{i\in I})=\textstyle\sum\lambdaimits_{i\in I}e_{i}\otimesimes _{
\mathcal{C}}df_{i} \lambdaabel{ec:d2}
\end{equation}
defines a connection on $\mathcal{C}{(}\mathcal{I}{)}.$
Another important example is the Levi-Civita connection, which is defined on
a finitely generated projective module $M$. Let $\{m_{i}\}_{i\in I}$ and $
\{\varphi ^{i}\}_{i\in I}$ be dual bases on $M.$ Recall that $m_{i}\in
M_{x_{i}}$ and $\varphi ^{i}$ is a linear map from $M$ to $_{x_{i}}\mathcal{C
}_{\bullet }$. If $\sigma _{x}:M_{x}\rhoightarrow \mathcal{C}{(}\mathcal{I}{)}
_{x}$ is the application
\begin{equation*}
\sigma _{x}(m)=\lambdaeft\{ \varphi _{x}^{i}(m)\rhoight\} _{i\in I},
\end{equation*}
then $\sigma =\{\sigma _{x}\}_{x\in \mathcal{C}_{0}}$ is a morphism in ${\mathcal{M}}od$
-$\mathcal{C}.$ Moreover, $\sigma $ is a section of the morphism $\pi :
\mathcal{C}{(}\mathcal{I}{)}\rhoightarrow M$ whose component $\pi _{x}$ maps $
\{f_{i}\}_{i\in I}\in \mathcal{C}{(}\mathcal{I}{)}_{x}$ to $\textstyle
\sum\mathbb Nolimits_{i\in I}m_{i}\cdotot f_{i}.$ Thus $M$ can be regarded via $\pi $
and $\sigma $ as a direct summand of $\mathcal{C}{(}\mathcal{I}{)}.$ By the
second part of the preceding lemma, the connection on $\mathcal{C}{(}
\mathcal{I}{)}$ given by the equation (\rhoef{ec:d2}) induces a connection $
\mathbb Nabla ^{LC}$ on $M$. We call $\mathbb Nabla ^{LC}$ the Levi-Civita connection on $
M.$ For every $m\in M_{x}$ we have
\begin{equation}
\mathbb Nabla _{x}^{LC}(m)=\textstyle\sum\lambdaimits_{i\in I}m_{i}\otimesimes _{\mathcal{C}
}d\varphi _{x}^{i}(m). \lambdaabel{ec:LC}
\end{equation}
\end{fact}
\begin{fact}[The curvature of a connection.]
We fix a connection $\mathbb Nabla :M\rhoightarrow M\otimesimes _{\mathcal{C}}\Omegamega ^{1}$
on a finitely generated right $\mathcal{C}$-module $M$. Our goal is to
extend $\mathbb Nabla $ to an endomorphism $R(\mathbb Nabla )$ of degree $2$ of the graded
$\Omegamega ^{\ast }$-module $M\otimesimes _{\mathcal{C}}\Omegamega ^{\ast }.$ First,
for $n\geq 0$ and $x\in \mathcal{C}_{0},$ we define $\mathbb Nabla
_{x}^{n}:(M\otimesimes _{\mathcal{C}}\Omegamega ^{n})_{x}\rhoightarrow (M\otimesimes _{
\mathcal{C}}\Omegamega ^{n+1})_{x}$ by
\begin{equation*}
\mathbb Nabla _{x}^{n}(m\otimesimes _{\mathcal{C}}\omega ):=\mathbb Nabla _{x}(m)\cdotot \omega
+(-1)^{n}m\otimesimes _{\mathcal{C}}d\omega .
\end{equation*}
In the above defining relation $m$ and $\omega $ are elements in $M_{y}$ and
${}_{y}\Omegamega _{x}^{n},$ respectively. By equation (\rhoef{ec:conn}), it
easily follows that $\mathbb Nabla ^{n}$ is well defined, that is
\begin{equation*}
\mathbb Nabla _{x}^{n}(m\cdotot f\otimesimes _{\mathcal{C}}\mathbb Zeta )=\mathbb Nabla
_{x}^{n}(m\otimesimes _{\mathcal{C}}f\mathbb Zeta ),
\end{equation*}
for any $m\otimesimes f\otimesimes \mathbb Zeta \in M_{y}\otimesimes {}_{y}\mathcal{C}
_{z}\otimesimes {}_{z}\Omegamega _{x}^{n}$. Since $d^{\ast }$ satisfies the Leibniz
rule, we get
\begin{equation*}
\mathbb Nabla _{x}^{n+m}(m\otimesimes _{\mathcal{C}}\omega \mathbb Zeta )=\mathbb Nabla
_{z}^{n}(m\otimesimes _{\mathcal{C}}\omega )\cdotot \mathbb Zeta +(-1)^{n}m\otimesimes _{
\mathcal{C}}\omega d\mathbb Zeta ,
\end{equation*}
for all $m\otimesimes \omega \otimesimes \mathbb Zeta \in {}M_{y}\otimesimes {}_{y}\Omegamega
_{z}^{n}\otimesimes {}_{z}\Omegamega _{x}^{m}$. Now it is not difficult to prove
that the family $R(\mathbb Nabla )=\{R(\mathbb Nabla )_{x}\}_{x\in \mathcal{C}_{0}}$
defines a morphism of graded right $\Omegamega ^{\ast }$-modules, where
\begin{equation*}
R^{n}(\mathbb Nabla )_{x}:(M\otimesimes _{\mathcal{C}}\Omegamega ^{n})_{x}\rhoightarrow
(M\otimesimes _{\mathcal{C}}\Omegamega ^{n+2})_{x},\quad R^{n}(\mathbb Nabla )_{x}=\mathbb Nabla
_{x}^{n+1}\circ \mathbb Nabla _{x}^{n}.
\end{equation*}
We shall say that $R(\mathbb Nabla )$ is the \emph{curvature} of $\mathbb Nabla $.
Note that $M_{x}$ can be identified with $(M\otimesimes _{\mathcal{C}}\Omegamega
^{0})_{x}$ via the map $m\mapsto m\otimesimes _{\mathcal{C}}1_{x}.$ On the other
hand, if $\{m_{i}\}_{i\in I}$ is a set of generators of $M,$ such that $
m_{i} $ belongs to a certain $M_{x_{i}},$ then $M\otimesimes _{\mathcal{C}
}\Omegamega ^{\ast }$ is generated by $\{m_{i}\otimesimes _{\mathcal{C}
}1_{x_{i}}\}_{i\in I}. $ It follows that the curvature of $\mathbb Nabla $ in
completely determined by its values$\ $at each $m_{i}.$ Using the matrix $
\Lambda =\lambdaeft( \lambdaambda _{ij}\rhoight) _{i,j\in I}$ that corresponds to $
\mathbb Nabla $ and taking into account (\rhoef{ec:lambda}), we obtain
\begin{equation*}
R(\mathbb Nabla )_{x_{i}}(m_{i})=\textstyle\sum\lambdaimits_{j\in I}\mathbb Nabla (m_{j})\cdotot
\lambdaambda _{ji}+\textstyle\sum\lambdaimits_{j\in I}m_{j}\otimesimes _{\mathcal{C}
}d\lambdaambda _{ji}=\textstyle\sum\lambdaimits_{j,k\in I}m_{j}\otimesimes _{\mathcal{C}
}\lambdaambda _{jk}\lambdaambda _{ki}+\textstyle\sum\lambdaimits_{j\in I}m_{j}\otimesimes _{
\mathcal{C}}d\lambdaambda _{ji}.
\end{equation*}
Hence $R(\mathbb Nabla )$ satisfies the relation
\begin{equation}
R(\mathbb Nabla )_{x_{i}}(m_{i})=\textstyle\sum\lambdaimits_{j\in I}m_{j}\otimesimes _{
\mathcal{C}}\gamma _{ji}, \lambdaabel{ec:R}
\end{equation}
where $\gamma _{ij}=d\lambdaambda _{ij}+\textstyle\sum\mathbb Nolimits_{k\in I}\lambdaambda
_{ik}\lambdaambda _{kj}.$ If $\Gamma =(\gamma _{ij})_{i,j\in I}$, then the
relations that define the elements of $\Gamma $ are equivalent to the matrix
equation
\begin{equation}
\Gamma =d\Lambda +\Lambda ^{2}. \lambdaabel{ec:Gamma}
\end{equation}
For example, let $\mathbb Nabla ^{\Lambda }$ be the connection on $\mathcal{C}(
\mathcal{I})$ that we constructed in \S \rhoef{fa:ExConn} for a matrix $
\Lambda =(\lambdaambda _{ij})_{i,j\in I}$. Since $\{e_{i}\}_{i\in I}$ generates $
\mathcal{C}(\mathcal{I}),$ and the matrix corresponding to $\mathbb Nabla ^{\Lambda
}$ is precisely $\Lambda ,$ the curvature $R(\mathbb Nabla ^{\Lambda })$ verifies
the formula (\rhoef{ec:R}), in which one substitutes $m_{i}$ by $e_{i}$, and $
\Gamma =(\gamma _{ij})_{i,j\in I}$ is given by (\rhoef{ec:Gamma}).
In a similar way one computes the curvature of the Levi-Civita connection on
a finitely generated projective $\mathcal{C}$-module $M$. If $
\{m_{i}\}_{i\in I}$ and $\{\varphi ^{i}\}_{i\in I}$ are dual bases, then by
equation (\rhoef{ec:LC}) we have
\begin{equation*}
\mathbb Nabla _{x_{i}}^{LC}(m_{i})=\textstyle\sum\lambdaimits_{j\in I}m_{j}\otimesimes _{
\mathcal{C}}d\varphi _{x_{i}}^{j}(m_{i}).
\end{equation*}
Hence, for the curvature of $\mathbb Nabla ^{LC},$ the elements of the matrix $
\Lambda $ are $\lambdaambda _{ij}:=d\varphi _{x_{i}}^{j}(m_{i}).$ Since $d\Lambda
=0$, we have $\Gamma =\Lambda ^{2}.$ Thus, the curvature of the Levi-Civita
connection verifies the equations
\begin{equation*}
R(\mathbb Nabla )_{x_{i}}(m_{i})=\textstyle\sum\lambdaimits_{j,k\in I}m_{j}\otimesimes _{
\mathcal{C}}d\varphi _{x_{k}}^{j}(m_{k})d\varphi _{x_{i}}^{k}(m_{i}).
\end{equation*}
\end{fact}
\begin{fact}[The powers of $R(\protect\mathbb Nabla )$.]
Let $\mathbb Nabla :M\lambdaongrightarrow M\otimesimes _{\mathcal{C}}\Omegamega ^{1}$ be a
connection on $M$. We have seen that $R(\mathbb Nabla )$ is an endomorphism of $
M\otimesimes _{\mathcal{C}}\Omegamega ^{\ast }$ of degree $2$. Thus $R(\mathbb Nabla )^{q},$
the $q^{\text{th}}$ power of $R(\mathbb Nabla ),$ is an endomorphism of degree $2q$
. Let $\Lambda :=\lambdaeft( \lambdaambda _{ij}\rhoight) _{i,j\in I}$ be the matrix
associated to $\mathbb Nabla $ with respect to a set of generators $\{m_{i}\}_{i\in
I}.$ By induction on $q$, it follows that the elements $R(\mathbb Nabla
)_{x_{i}}^{q}(m_{i})\in (M\otimesimes _{\mathcal{C}}\Omegamega ^{2q})_{x_{i}}$ are
given by the relation
\begin{equation}
R(\mathbb Nabla )_{x_{i}}^{q}(m_{i})=\textstyle\sum\lambdaimits_{i_{1},\dots ,i_{q}\in
I}m_{i_{1}}\otimesimes _{\mathcal{C}}\gamma _{i_{1}i_{2}}\gamma
_{i_{2}i_{3}}\cdotots \gamma _{i_{q-1}i_{q}}\gamma _{i_{q}i},
\lambdaabel{ec:R_nabla}
\end{equation}
where $\Gamma =(\gamma _{ij})_{i,j\in I}$ is defined by the formula (\rhoef
{ec:Gamma}).
For the connection $\mathbb Nabla ^{\Lambda }$ associated to the matrix $\Lambda
=(\lambdaambda _{ij})_{i,j\in I}$ we have seen that $\Gamma =(\gamma
_{ij})_{i,j\in I}$ is given by $\Gamma =d\Lambda +\Lambda ^{2},$ so
\begin{equation*}
R(\mathbb Nabla ^{\Lambda })_{x_{i}}^{q}(e_{i})=\textstyle\sum\lambdaimits_{i_{1},\dots
,i_{q}\in I}e_{i_{1}}\otimesimes _{\mathcal{C}}\gamma _{i_{1}i_{2}}\gamma
_{i_{2}i_{3}}\cdotots \gamma _{i_{q-1}i_{q}}\gamma _{i_{q}i}.
\end{equation*}
Let $\{m_{i}\}_{i\in I}$ and $\{\varphi ^{i}\}_{i\in I}$ be dual bases on a
finitely generated projective module $M.$ For the Levi-Civita connection on $
M,$ we get
\begin{equation*}
R(\mathbb Nabla ^{LC})_{x_{i}}^{q}(m_{i})=\textstyle\sum\lambdaimits_{i_{1},\dots
,i_{2q}\in I}m_{i_{1}}\otimesimes _{\mathcal{C}}d\varphi
_{x_{i_{2}}}^{i_{1}}(m_{i_{2}})\cdotots d\varphi
_{x_{i_{_{2q}}}}^{i_{2q-1}}(m_{i_{2q}})d\varphi _{x_{i}}^{i_{2q}}(m_{i}).
\end{equation*}
\end{fact}
\section{de Rham cohomology and the Chern map.}
In this section, which is the main part of our paper, we define the de Rham
cohomology of a linear category $\mathcal{C}$ with coefficients in a $DG$
-category $\Omegamega ^{\ast }$. Our goal is to associate to every finitely
generated projective $\mathcal{C}$-module $M$ some de Rham cohomology
classes, that will lead us to the construction of the Chern map.
\begin{fact}[de Rham cohomology $H^{\ast }(\mathcal{C},\Omegamega ^{\ast }).$]
We fix a $\Bbbk $-linear category $\mathcal{C}$, and we suppose that $
(\Omegamega ^{\ast },d^{\ast })$ is a $DG$-category such that $\Omegamega ^{0}=
\mathcal{C}.$ For $\omega \in {}_{x}\Omegamega _{y}^{p}$ and $\mathbb Zeta \in
{}_{y}\Omegamega _{x}^{n-p}$ we define the graded commutator
\begin{equation}
\lambdabrack \omega ,\mathbb Zeta ]:=\omega \mathbb Zeta -(-1)^{p(n-p)}\mathbb Zeta \omega ,
\lambdaabel{ec:grcomm}
\end{equation}
which is an element in $\opluslus _{x\in \Omegamega _{0}^{\ast }\,}{}_{x}\Omegamega
_{x}^{n}$. The subspace spanned by all graded commutators in $\opluslus _{x\in
\Omegamega _{0}^{\ast }}{}_{x}\Omegamega _{x}^{n}$ will be denoted by $[\Omegamega
^{\ast },\Omegamega ^{\ast }]^{n}.$ Note the usual commutator $\omega \mathbb Zeta
-\mathbb Zeta \omega $ makes sense in $\Omegamega ^{\ast }$ as well. However, in a
graded linear category, we shall always use the notation $[\omega ,\mathbb Zeta ]$
for the graded commutator of $\omega $ and $\mathbb Zeta .$
For a $DG$-category $(\Omegamega ^{\ast },d^{\ast })$ as above we now define
\begin{equation*}
\Omegamega _{ab}^{n}:=(\textstyle\bigoplus_{x\in \mathcal{C}_{0}}{}_{x}\Omegamega
_{x}^{n})/\lambdaeft[ \Omegamega ^{\ast },\Omegamega ^{\ast }\rhoight] ^{n}.
\end{equation*}
For example, $[\Omegamega ^{\ast },\Omegamega ^{\ast }]^{0}$ coincides with the
subspace $[\mathcal{C},\mathcal{C}],$ used in \S \rhoef{fa: fgp} to define $
\mathcal{C}_{ab},$ the target of the Hattori-Stallings trace map. Hence, $
\Omegamega _{ab}^{0}=\mathcal{C}_{ab}.$
As a consequence of the Leibniz rule, we immediately deduce that $d^{n}$
maps commutators to commutators. Hence $d^{n}$ factorizes through a linear
map $d_{ab}^{n}:\Omegamega _{ab}^{n}\rhoightarrow \Omegamega _{ab}^{n+1}.$ Obviously,
the sequence
\begin{equation*}
\xymatrix{ 0\ar[r] & \Omegamega^0_{ab}\ar[r]^-{d^0_{ab}} &
\Omegamega^1_{ab}\ar[r]^-{d^1_{ab}} & \dotsb \ar[r] &
\Omegamega^n_{ab}\ar[r]^-{d^n_{ab}} &\Omegamega^{n+1}_{ab}\ar[r] &\dotsb }
\end{equation*}
is a cochain complex $(\Omegamega _{ab}^{\ast },d_{ab}^{\ast }),$ that will be
called the de Rham complex. By definition, the de Rham cohomology $
H_{DR}^{\ast }(\mathcal{C},\Omegamega ^{\ast })$ of $\mathcal{C}$ with respect
to the $DG$-category $(\Omegamega ^{\ast },d^{\ast })$ is the cohomology of $
(\Omegamega _{ab}^{\ast },d_{ab}^{\ast })$.
\end{fact}
\begin{fact}[The cocycles $\mathbb Ch^{\ast}(M,\protect\mathbb Nabla)$.]
\lambdaabel{fa:Chern} Our aim now is to associate certain cohomology classes in $
H_{DR}^{\ast }(\mathcal{C},\Omegamega ^{\ast })$ to any finitely generated
projective $\mathcal{C}$-module $M$ which is endowed with a connection $
\mathbb Nabla :M\lambdaongrightarrow M\otimesimes _{\mathcal{C}}\Omegamega ^{1}.$
First, we remark that the subspaces $\opluslus {}_{n\in \mathbb{N\,}
}{}_{x}\Omegamega _{y}^{2n}$ define a linear subcategory $\Omegamega ^{2\ast }$ of $
\Omegamega ^{\ast },$ which contains $\mathcal{C}.$ The vector space $\lambdaeft(
\Omegamega ^{2\ast }\rhoight) _{ab},$ that corresponds to the subcategory of even
forms, is the quotient of the coproduct of the family $\{_{x}\Omegamega
_{x}^{2q}\}_{(q,x)\in \mathbb{N}\times \mathcal{C}_{0}}$ through the
subspace spanned by the commutators of two even forms. On the other hand, $
\Omegamega _{ab}^{2q}$ is obtained by killing all commutators of degree $2q$ in $
\opluslus _{x\in \mathcal{C}_{0}\,x}\Omegamega _{x}^{2q},$ including those that
corresponds to two odd forms. Hence, $\lambdaeft( \Omegamega ^{2\ast }\rhoight) _{ab}$
and $\opluslus _{q\geq 0}\Omegamega _{ab}^{2q}$ are not identical, but there is a
canonical linear transformation from the former vector space to the latter
one, that respects the canonical $\mathbb{N}$-gradings, in the sense that
the equivalence class of a form $\omega \in {}_{x}\Omegamega _{x}^{2q}$ is
mapped to its equivalence class in $\Omegamega _{ab}^{2q}.$ In conclusion, for
any finitely generated projective right $\Omegamega ^{2\ast }$-module $N,$ we
can compose the Hattori-Stallings trace map $\Tr_{N}:\End_{\Omegamega ^{2\ast
}}(N)\rhoightarrow $ $\lambdaeft( \Omegamega ^{2\ast }\rhoight) _{ab}$ with the above
canonical map. We still denote the resulting map by $\Tr_{N}$. Throughout
the remaining part of this paper we shall work only with this new trace map,
whose codomain is $\opluslus _{q\geq 0}\Omegamega _{ab}^{2q}.$
Recall that, by assumption, $M$ is a finitely generated projective $\mathcal{
C}$-module. Thus, the right $\Omegamega ^{\ast }$-module $M\otimesimes _{\mathcal{C}
}\Omegamega ^{2\ast }$ has the same properties. Therefore, it makes sense to
speak about the trace of its endomorphisms, which are elements of $\opluslus
_{q\geq 0}\Omegamega _{ab}^{2q},\ $in view of the foregoing remarks.
We have seen that $R(\mathbb Nabla )^{q}$ is an endomorphism of $M\otimesimes _{
\mathcal{C}}\Omegamega ^{\ast }$ of degree $2q.$ Thus, it induces a homogeneous
endomorphism of $M\otimesimes _{\mathcal{C}}\Omegamega ^{2\ast }\ $of the same
degree, so we can compute its trace
\begin{equation*}
\mathbb Ch^{q}(M,\mathbb Nabla ):=\Tr_{M\otimesimes _{\mathcal{C}}\Omegamega ^{2\ast }}\lambdaeft(
R(\mathbb Nabla )^{q}\rhoight) .
\end{equation*}
We claim that $\mathbb Ch^{q}(M,\mathbb Nabla )$ is a $2q$-cochain in the de Rham complex $
(\Omegamega _{ab}^{\ast },d_{ab}^{\ast }).$ Let $\{m_{i}\}_{i\in I}$ and $
\{\varphi ^{i}\}_{i\in I}$ be dual bases on $M$ such that each $m_{i}$
belongs to a certain component $M_{x_{i}}.$ Then $\{m_{i}\otimesimes _{\mathcal{C
}}1_{x_{i}}\}_{i\in I}$ and $\{\varphi ^{i}\otimesimes _{\mathcal{C}}\Id_{\Omegamega
^{2\ast }}\}_{i\in I}$ are dual bases on $M\otimesimes _{\mathcal{C}}\Omegamega
^{2\ast }.$ Therefore, by the definition of the trace and the relation (\rhoef
{ec:R_nabla}) we get that
\begin{align*}
\mathbb Ch^{q}(M,\mathbb Nabla )& =\textstyle\sum_{i_{0}\in I}\lambdaeft( \varphi
^{i_{0}}\otimesimes _{\mathcal{C}}\Id_{\Omegamega ^{2\ast }}\rhoight) \lambdaeft( R(\mathbb Nabla
)^{q}(m_{i_{0}})\rhoight) +\lambdaeft[ \Omegamega ^{\ast },\Omegamega ^{\ast }\rhoight] ^{2q}
\\
& =\textstyle\sum\lambdaimits_{i_{0},i_{1},\dots ,i_{q}=1}\varphi
_{x_{i_{1}}}^{i_{0}}(m_{i_{1}})\gamma _{i_{1}i_{2}}\lambdadots \gamma
_{i_{q-1}i_{q}}\gamma _{i_{q}i_{0}}+\lambdaeft[ \Omegamega ^{\ast },\Omegamega ^{\ast }
\rhoight] ^{2q}.
\end{align*}
In particular, our claim has been proved, as $\mathbb Ch^{q}(M,\mathbb Nabla )$ is the
class of $\omega ^{q}(M,\mathbb Nabla )$ in $\Omegamega _{ab}^{2q}$, where
\begin{equation}
\omega ^{q}(M,\mathbb Nabla )=\textstyle\sum\lambdaimits_{i_{0},i_{1},\dots
,i_{q}=1}\varphi _{x_{i_{1}}}^{i_{0}}(m_{i_{1}})\gamma _{i_{1}i_{2}}\lambdadots
\gamma _{i_{q-1}i_{q}}\gamma _{i_{q}i_{0}}. \lambdaabel{ec:wq}
\end{equation}
Note that, since the trace of an endomorphism does not depend on the choice
of the dual bases on $M$, the class of $\omega ^{q}(M,\mathbb Nabla )$ in $\Omegamega
_{ab}^{\ast }$ is also independent of such a choice.
On the projective and finitely generated module $\mathcal{C}(\mathcal{I})$
we take the dual bases $\{e_{i}\}_{z\in I}$ and $\{\varphi ^{i}\}_{i\in I}.$
The components of the morphism $\varphi ^{i}$ are the canonical projections $
\varphi _{x}^{i}:\,_{x_{i}}\mathcal{C}(\mathcal{I})_{x}\rhoightarrow \,_{x_{i}}
\mathcal{C}_{x}$ that maps $\{f_{i}\}_{i\in I}$ to $f_{i}.$ On $\mathcal{C}(
\mathcal{I})$ we take, as usual, the connection $\mathbb Nabla ^{\Lambda }$
associated to a matrix $\Lambda =(\lambdaambda _{ij})_{i,j\in I},$ with $\lambdaambda
_{ij}\in \,_{x_{i}}\Omegamega _{x_{j}}^{1}.$ Since the matrix that corresponds
to $\mathbb Nabla ^{\Lambda }$ with respect to the generating set $\{e_{i}\}_{i\in
I}$ is $\Lambda ,$ and $\varphi _{x_{i_{1}}}^{i_{0}}(e_{i_{1}})=\delta
_{i_{0},i_{1}}1_{x_{i_{1}}},$ the equation (\rhoef{ec:wq}) is equivalent in
this setting to
\begin{equation}
\omega ^{q}(M,\mathbb Nabla ^{\Lambda })=\textstyle\sum\lambdaimits_{i_{1},\lambdadots
,i_{q}\in I}\gamma _{i_{1}i_{2}}\lambdadots \gamma _{i_{q-1}i_{q}}\gamma
_{i_{q}i_{1}}, \lambdaabel{ec:wLambda}
\end{equation}
where $\Gamma =(\gamma _{ij})_{i,j\in I}$ is given $\Gamma =d\Lambda
+\Lambda ^{2}.$
With respect to the dual bases $\{m_{i}\}_{i\in I}$ and $\{\varphi
^{i}\}_{i\in I},$ the Levi-Civita connection $\mathbb Nabla ^{LC}$ on a finitely
generated projective module $M$ has the matrix $\Lambda :=(d\varphi
_{x_{j}}^{i}(m_{j}))_{i,j\in I}.$ It follows that $\omega ^{q}(M,\mathbb Nabla
^{LC})$ satisfies the following equation
\begin{equation}
\omega ^{q}(M,\mathbb Nabla ^{LC})=\textstyle\sum\lambdaimits_{i_{0},i_{1},\dots
,i_{2q}}\varphi _{x_{i_{1}}}^{i_{0}}(m_{{i_{1}}})d\varphi
_{x_{i_{2}}}^{i_{1}}(m_{{i_{2}}})\cdotots d\varphi _{x_{i_{2q}}}^{i_{2q-1}}(m_{
{i_{2q}}})d\varphi _{x_{i_{0}}}^{i_{2q}}(m_{{i_{0}}}). \lambdaabel{ec:wLC}
\end{equation}
\end{fact}
\begin{theorem}
Let $\Bbbk $ be a field of characteristic different of $2.$ If $M$ is a
finitely generated projective $\mathcal{C}$-module and $\mathbb Nabla :M\rhoightarrow
M\otimesimes _{\mathcal{C}}\Omegamega ^{1}$ is a connection on $M,$ then $\mathbb Ch
^{q}(M,\mathbb Nabla )$ is a $2q$-cocycle in the de Rham complex $(\Omegamega
_{ab}^{\ast },d_{ab}^{\ast }).$
\end{theorem}
\begin{proof}
We have to prove that $d\omega ^{q}(M,\mathbb Nabla )$ is a sum of graded
commutators. Since $M$ is projective and finitely generated, there is a
right $\mathcal{C}$-module $N$ such that $M\opluslus N=\mathcal{C}(\mathcal{I}),
$ for a certain finite subset $I$ of $\mathcal{C}_{0}$. Since $N$ is a
direct summand of $\mathcal{C}(\mathcal{I})$ we can choose dual bases $
\{n_{i}\}_{i\in I}$ and $\{\psi ^{i}\}_{i\in I}$ on $N.$ By Lemma \rhoef
{le:conn}(i), $\mathbb Nabla \opluslus \mathbb Nabla ^{LC}$ is a connection on $M\opluslus N,$
where $\mathbb Nabla ^{LC}$ denotes the Levi-Civita connection on $N.$ Obviously,
\begin{equation*}
\omega ^{q}(M,\mathbb Nabla )+\omega ^{q}\lambdaeft( N,\mathbb Nabla ^{LC}\rhoight) =\omega
^{q}\lambdaeft( \mathcal{C}(\mathcal{I}),\mathbb Nabla \opluslus \mathbb Nabla ^{LC}\rhoight) .
\end{equation*}
In conclusion, it is enough to show that $d\omega ^{q}(N,\mathbb Nabla ^{LC})$ and $
d\omega ^{q}(\mathcal{C}(\mathcal{I}),\mathbb Nabla \opluslus \mathbb Nabla ^{LC})$ can be
written as sums of commutators.
We consider first the case of the Levi-Civita connection on $N.$ For a
matrix $X=(\omega _{ij})_{i,j\in I},$ with $\omega _{ij}\in {}_{x_{i}}\Omegamega
_{x_{j}}^{n}$ we define $\Tr(X)=\textstyle\sum\mathbb Nolimits_{i\in I}\omega _{ii}.
$ If $Y=(\mathbb Zeta _{ij})_{i,j\in I}$ is another matrix, with $\mathbb Zeta _{ij}\in
{}_{x_{i}}\Omegamega _{x_{j}}^{m},$ then
\begin{equation*}
\Tr(YX)=(-1)^{nm}\Tr(XY)+[\Omegamega ^{\ast },\Omegamega ^{\ast }]^{n+m}.
\end{equation*}
By the equation (\rhoef{ec:wLC}) and the definition of the trace of a matrix
we get
\begin{equation*}
d\omega ^{q}(N,\mathbb Nabla ^{LC})=\textstyle\sum\lambdaimits_{i_{0},i_{1},\dots
,i_{2q}\in I}d\psi _{x_{i_{1}}}^{i_{0}}(m_{{i_{1}}})d\psi
_{x_{i_{2}}}^{i_{1}}(m_{{i_{2}}})\cdotots d\psi _{x_{i_{2q}}}^{i_{2q-1}}(m_{{
i_{2q}}})d\psi _{x_{i_{0}}}^{i_{2q}}(m_{{i_{0}}})=\Tr(\lambdaeft( d\Psi \rhoight)
^{2q+1}),
\end{equation*}
where $\Psi :=(\psi _{x_{j}}^{i}(n_{j}))_{i,j\in I}.$ On the other hand,
since $\{n_{i}\}_{i\in I}$ and $\{\psi ^{i}\}_{i\in I}$ are dual bases we
have $n_{i}=\textstyle\sum\mathbb Nolimits_{j\in I}n_{j}\cdotot \psi _{xi}^{j}(n_{i}),
$ for all $i\in I.$ Therefore, for any $k\in I$, we get
\begin{equation*}
\psi _{i}^{k}(n_{i})=\textstyle\sum\mathbb Nolimits_{j\in I}\psi
_{j}^{k}(n_{j})\circ \psi _{x_{i}}^{j}(n_{i}).
\end{equation*}
These identities are equivalent to the matrix equation $\Psi =\Psi ^{2}.$ We
can now proceed as in the proof \cite[Theorem 1.19]{Ka}. Namely, let $\Pi
=2\Psi -1$. Hence $\Pi ^{2}=1$ and $\Pi (d\Psi )=-(d\Psi )\Pi .$ Thus,
\begin{equation*}
(d\Psi )^{2q+1}=\Pi ^{2}(d\Psi )^{2q+1}=-\Pi (d\Psi )^{2q+1}\Pi .
\end{equation*}
By the foregoing computations we deduce that
\begin{equation*}
d\omega ^{q}(N,\mathbb Nabla ^{LC})=\Tr\lambdaeft( (d\Psi )^{2q+1}\rhoight) =-\Tr\lambdaeft(
\Pi (d\Psi )^{2q+1}\Pi \rhoight) =-\Tr\lambdaeft( (d\Psi )^{2q+1}\Pi ^{2}\rhoight)
+[\Omegamega ^{\ast },\Omegamega ^{\ast }]^{2q+1}.
\end{equation*}
Since $2$ is invertible in $\Bbbk $ we conclude that $d\omega ^{q}(N,\mathbb Nabla
^{LC})$ is a commutator.
It remains to show that $d\omega ^{q}(\mathcal{C}(\mathcal{I}),\mathbb Nabla \opluslus
\mathbb Nabla ^{LC})$ is a commutator as well. Since $\mathbb Nabla \opluslus \mathbb Nabla ^{LC}$
is a connection on $\mathcal{C}(\mathcal{I})$ there exists a matrix $\Lambda
=(\lambdaambda _{ij})_{i,j\in I},$ with $\lambdaambda _{ij}\in {}_{x_{i}}\Omegamega
_{x_{j}},$ such that $\mathbb Nabla \opluslus \mathbb Nabla ^{LC}=\mathbb Nabla ^{\Lambda }.$ Let $
\Gamma :=(\gamma _{ij})_{i,j\in I}$ be the matrix $\Gamma =d\Lambda +\Lambda
^{2}.$ By (\rhoef{ec:wLambda}) we have
\begin{equation*}
\omega ^{q}\lambdaeft( \mathcal{C}(\mathcal{I}),\mathbb Nabla ^{\Lambda }\rhoight) =
\textstyle\sum\lambdaimits_{i_{1},\lambdadots ,i_{q}\in I}\gamma _{i_{1}i_{2}}\lambdadots
\gamma _{i_{q-1}i_{q}}\gamma _{i_{q}i_{1}}=\Tr(\Gamma ^{q}).
\end{equation*}
On the other hand, by induction on $q,$ one shows that $d\lambdaeft( \Gamma
^{q}\rhoight) =\Gamma ^{q}\Lambda -\Lambda \Gamma ^{q}$. As the trace map and $
d^{\ast }$ commute we have
\begin{equation*}
d\omega ^{2q}(\mathcal{C}(\mathcal{I}),\mathbb Nabla ^{\Lambda })=\Tr\lambdaeft(
d(\Gamma ^{q})\rhoight) =\Tr\lambdaeft( \Gamma ^{q}\Lambda \rhoight) -\Tr(\Lambda
\Gamma ^{q}).
\end{equation*}
We conclude the proof by remarking that the elements of $\Gamma $ are all of
even degree, so $\Tr\lambdaeft( \Gamma ^{q}\Lambda \rhoight) -\Tr(\Lambda \Gamma
^{q})$ is a commutator.
\end{proof}
Our next aim is to prove that the cohomology class of $\mathbb Ch^{q}(M,\mathbb Nabla )$
in $H_{DR}^{2q}(\mathcal{C},\Omegamega ^{\ast })$ does not depend on the
connection $\mathbb Nabla $. We start by proving some preliminary results. First of
all, we shall associate to a $DG$-category $(\Omegamega ^{\ast },d^{\ast })$ two
new $DG$-categories $\Omegamega ^{\ast }[t]$ and $\widetilde{\Omegamega }^{\ast }.$
Recall that $\Omegamega ^{0}=\mathcal{C}.$
\begin{fact}[{The $DG$-category $\Omegamega ^{\ast }[t]$.}]
By definition, the objects of $\Omegamega ^{\ast }[t]$ are the elements of $
\mathcal{C}_{0}$, and we set$\ {}_{x}\Omegamega ^{n}[t]_{y}:={}_{x}\Omegamega
_{y}^{n}\otimesimes \Bbbk \lambdabrack t]$. Therefore, a morphism $\omega $ in$\
{}_{x}\Omegamega ^{n}[t]_{y}$ can be uniquely written as a polynomial $\omega =
\textstyle\sum\mathbb Nolimits_{i=0}^{p}\omega _{i}t^{i}$ with coefficients in $
{}_{x}\Omegamega _{y}^{n}$. The composition in $\Omegamega ^{\ast }[t]$ is given by
the relation
\begin{equation*}
(\textstyle\sum\mathbb Nolimits_{i=0}^{p}\omega _{i}t^{i})\circ (\textstyle
\sum\mathbb Nolimits_{j=0}^{q}\mathbb Zeta _{j}t^{j})=\textstyle\sum\mathbb Nolimits_{k=0}^{p+q}(
\textstyle\sum\mathbb Nolimits_{r=0}^{k}\omega _{r}\mathbb Zeta _{k-r})t^{k},
\end{equation*}
while the identity morphism of $x$ in $\Omegamega ^{\ast }[t]$ is $1_{x}\in
\,_{x}\mathcal{C}_{x}$, regarded as a constant polynomial in ${}_{x}\Omegamega
^{0}[t]_{x}.$ The differential of $\Omegamega ^{\ast }[t]$ satisfies, for any
polynomial in $_{x}\Omegamega \lambdabrack t]_{y}^{n},$ the following relation
\begin{equation*}
d(\textstyle\sum\mathbb Nolimits_{i=0}^{p}\omega _{i}t^{i})=\textstyle
\sum\mathbb Nolimits_{i=0}^{p}\lambdaeft( d\omega _{i}\rhoight) t^{i}.
\end{equation*}
\end{fact}
\begin{fact}[The $DG$-category $\widetilde{\Omegamega }^{\ast }$.]
\lambdaabel{fa:Omega_tilde} The set of objects in $\widetilde{\Omegamega }^{\ast }$
is $\mathcal{C}_{0},$ while $_{x}\widetilde{\Omegamega }_{y}^{n}$ is defined by
\begin{equation*}
_{x}\widetilde{\Omegamega }_{y}^{n}={}_{x}\Omegamega ^{n}[t]_{y}\opluslus {}_{x}\Omegamega
^{n-1}[t]_{y}.
\end{equation*}
Note that $_{x}\widetilde{\Omegamega }_{y}^{0}=$\thinspace $_{x}\Omegamega
^{0}[t]_{y}{}.$ It is convenient to write an element $\omega $ in $_{x}
\widetilde{\Omegamega }_{y}^{n}$ as formal sum $\omega =\omega _{0}+\omega
_{1}\varepsilon ,$ where $\omega _{0}\in {}_{x}\Omegamega ^{n}[t]_{y}$ and $
\omega _{1}\in {}_{x}\Omegamega ^{n-1}[t]_{y}.$ In $\widetilde{\Omegamega }^{\ast }$
the composition of morphisms is defined by the formula
\begin{equation*}
(\omega _{0}+\omega _{1}\varepsilon )\circ (\mathbb Zeta _{0}+\mathbb Zeta _{1}\varepsilon
)=(\omega _{0}\mathbb Zeta _{0})+(\omega _{0}\mathbb Zeta _{1}+(-1)^{\lambdaeft\vert \mathbb Zeta
_{0}\rhoight\vert }\omega _{1}\mathbb Zeta _{0}),
\end{equation*}
and we take the identity of $x$ in ${}_{x}\widetilde{\Omegamega }_{x}^{\ast }$
to be $1_{x}+0\varepsilon .$ For $\omega _{0}+\omega _{1}\varepsilon \in
{}_{x}\widetilde{\Omegamega }_{y}^{n}$ we set
\begin{equation*}
\partial (\omega _{0}+\omega _{1}\varepsilon )=d\omega _{0}+\lambdaeft( d\omega
_{1}+(-1)^{n}\dot{\omega}_{0}\rhoight) \varepsilon ,
\end{equation*}
where $\dot{\omega}_{0}$ is the derivative of $\omega _{0}$ with respect to $
t.$ Hence, for a polynomial $\mathbb Zeta =\textstyle\sum\mathbb Nolimits_{i=1}^{p}\mathbb Zeta
_{i}t^{i}$ with coefficients in $_{x}\Omegamega _{y}^{n-1}$, we have $\dot{\mathbb Zeta}
=\textstyle\sum\mathbb Nolimits_{i=1}^{p}i\mathbb Zeta _{i}t^{i-1}$. It is not difficult
to check that $(\widetilde{\Omegamega }^{\ast },\partial ^{\ast })$ is a $DG$
-category.
In the following lemma we describe the de Rham complex associated to $(
\widetilde{\Omegamega }^{\ast },\partial ^{\ast }).$ To simplify the notation,
we shall write $\lambdaeft\lambdaangle \omega \rhoight\rhoangle $ for the class of $\omega
\in \Omegamega ^{n}[t]$ in $\Omegamega ^{n}[t]_{ab}.$
\end{fact}
\begin{lemma}
\lambdaabel{le:DeRham} The de Rham complex associated to $\widetilde{\Omegamega }
^{\ast }$ is isomorphic to the complex $(\mathrm{C}^{\ast },\delta ^{\ast })
$, where $\mathrm{C}^{n}=\Omegamega ^{n}[t]_{ab}\opluslus \Omegamega ^{n-1}[t]_{ab}$,
and
\begin{equation*}
\delta ^{n}(\lambdaeft\lambdaangle \omega _{0}\rhoight\rhoangle +\lambdaeft\lambdaangle \omega
_{1}\rhoight\rhoangle \varepsilon )=\lambdaeft\lambdaangle d\omega _{0} \rhoight\rhoangle
+\lambdaeft\lambdaangle d \omega _{1} +(-1)^{n}\dot{\omega}_{1}\rhoight\rhoangle
\varepsilon .
\end{equation*}
\end{lemma}
\begin{proof}
It is enough to prove the following identity
\begin{equation*}
\lambdaeft[ \widetilde{\Omegamega }^{\ast },\widetilde{\Omegamega }^{\ast }\rhoight] ^{n}=
\lambdaeft[ \Omegamega ^{\ast }[t],\Omegamega ^{\ast }[t]\rhoight] ^{n}\textstyle\bigoplus
\lambdaeft[ \Omegamega ^{\ast }[t],\Omegamega ^{\ast }[t]\rhoight] ^{n-1}\varepsilon .
\end{equation*}
The inclusion $\subseteq $ is a simple consequence of the following relation
\begin{equation*}
\lambdabrack \omega _{0}+\omega _{1}\varepsilon ,\theta _{0}+\theta
_{1}\varepsilon ]=[\omega _{0},\theta _{0}]+\lambdaeft( [\omega _{0},\theta
_{1}]+(-1)^{\lambdaeft\vert \theta _{0}\rhoight\vert }[\omega _{1},\theta
_{0}]\rhoight) \varepsilon .
\end{equation*}
To prove the other inclusion we notice that the following relations hold
\begin{equation*}
\lambdabrack \omega ,\theta ]=[\omega +0\varepsilon ,\theta +0\varepsilon ]\quad
\text{and\quad }[\mathbb Zeta ,\xi ]\varepsilon =[\mathbb Zeta +0\varepsilon ,0+\xi
\varepsilon ],
\end{equation*}
for any commutators $[\omega ,\theta ]\in \Omegamega ^{n}[t]$ and $[\mathbb Zeta ,\xi
]\in \Omegamega ^{n-1}[t].$ Therefore, $[\omega ,\theta ]+[\mathbb Zeta ,\xi
]\varepsilon $ is a sum of commutators in $\widetilde{\Omegamega }^{n}$.
\end{proof}
\begin{fact}[The evaluation map.]
\lambdaabel{fa:ev_map}Let $\Omegamega ^{\ast }$ be a $DG$-category. For every
polynomial $\omega _{0}=\textstyle\sum\mathbb Nolimits_{i=0}^{p}\omega _{i}t^{i}$
with coefficients in $_{x}\Omegamega _{y}^{n}$ and $a\in \Bbbk ,$ let $\omega
_{0}(a):=\textstyle\sum\mathbb Nolimits_{i=0}^{p}\omega _{i}a^{i}.$ Furthermore,
for a couple of objects $x$ and $y$ in $\mathcal{C}_{0},$ we define the
linear map $_{x}(ev_{a}^{n})_{y}$ from ${}_{x}\Omegamega ^{n}[t]_{y}\opluslus
\Omegamega ^{n-1}[t]_{ab}$ to ${}_{x}\Omegamega _{y}^{n},$ by
\begin{equation*}
\quad _{x}(ev_{a}^{n})_{y}(\omega _{0}+\lambdaeft\lambdaangle \omega _{1}\rhoight\rhoangle
\varepsilon )=\omega _{0}(a).
\end{equation*}
Since $_{x}(ev_{a}^{n})_{x}$ maps a commutator $[\omega _{0},\theta _{0}]$
in ${}_{x}\Omegamega ^{n}[t]_{x}$ to the commutator $[\omega _{0}(a),\theta
_{0}(a)]$ in ${}_{x}\Omegamega _{x}^{n}$, the family $\{_{x}(ev_{a}^{n})_{x}
\}_{x\in \mathcal{C}_{0}}$ induces a linear transformation
\begin{equation*}
ev_{a}^{n}:\Omegamega ^{n}[t]_{ab}\opluslus \Omegamega ^{n-1}[t]_{ab}\lambdaongrightarrow
\Omegamega _{ab}^{n},\quad ev_{a}^{n}\lambdaeft( \lambdaeft\lambdaangle \omega
_{0}\rhoight\rhoangle +\lambdaeft\lambdaangle \omega _{1}\rhoight\rhoangle \varepsilon \rhoight)
=\lambdaeft\lambdaangle \omega _{0}(a)\rhoight\rhoangle .
\end{equation*}
\end{fact}
\begin{lemma}
The family $\{ev_{a}^{n}\}_{n\in
\mathbb{N}
}$ is a morphism of complexes between $(\widetilde{\Omegamega }_{ab}^{\ast
},\partial_{ab}^{\ast })$ and $(\Omegamega _{ab}^{\ast },d_{ab}^{\ast }).$
\begin{proof}
We identify $(\widetilde{\Omegamega }_{ab}^{\ast },\partial _{ab}^{\ast })$ with
the cochain complex from Lemma \rhoef{le:DeRham}. Therefore, if $\lambdaeft\lambdaangle
\omega _{0}\rhoight\rhoangle +\lambdaeft\lambdaangle \omega _{1}\rhoight\rhoangle \varepsilon $
is a cochain of degree $n,$ then
\begin{equation*}
\lambdaeft( ev_{a}^{n+1}\circ \partial ^{n}\rhoight) (\lambdaeft\lambdaangle \omega
_{0}\rhoight\rhoangle +\lambdaeft\lambdaangle \omega _{1}\rhoight\rhoangle \varepsilon
)=ev_{a}^{n+1}\lambdaeft( \lambdaeft\lambdaangle d\omega _{0}\rhoight\rhoangle +\lambdaeft\lambdaangle
d\omega _{1}+(-1)^{n}\dot{\omega}_{0}\rhoight\rhoangle \varepsilon \rhoight)
=(d\omega _{0})(a).
\end{equation*}
We conclude the proof by remarking that $\lambdaeft( d\omega _{0}\rhoight)
(a)=d(\omega _{0}(a))=\lambdaeft( d^{n}\circ ev_{a}^{n}\rhoight) (\lambdaeft\lambdaangle
\omega _{0}\rhoight\rhoangle +\lambdaeft\lambdaangle \omega _{1}\rhoight\rhoangle \varepsilon
) $.
\end{proof}
\end{lemma}
In order to prove that $\{ev_{0}^{n}\}_{n\in
\mathbb{N}
}$ and $\{ev_{1}^{n}\}_{n\in
\mathbb{N}
}$ induce the same maps in cohomology, we are going to construct a homotopy
map between them.
\begin{fact}[The homotopy operator.]
We keep the notation and the assumptions from \S \rhoef{fa:ev_map}. In
addition, we assume that the characteristic of $\Bbbk $ is zero. For any $
\omega :=\textstyle\sum\mathbb Nolimits_{i=0}^{p}\omega _{i}t^{i}$ with
coefficients in ${}_{x}\Omegamega _{y}^{n-1}$ we define
\begin{equation*}
\textstyle\int\mathbb Nolimits_{0}^{1}\omega dt:=\textstyle\sum
\mathbb Nolimits_{i=0}^{p}(i+1)^{-1}\omega _{i}.
\end{equation*}
It is easy to see that the map $k^{n}:\Omegamega ^{n}[t]_{ab}\opluslus {}\Omegamega
^{n-1}[t]_{ab}\varepsilon \lambdaongrightarrow {}\Omegamega _{ab}^{n-1}$ given by
\begin{equation*}
\quad _{x}k_{y}^{n}(\lambdaeft\lambdaangle \omega \rhoight\rhoangle _{0}+\lambdaeft\lambdaangle
\omega _{1}\rhoight\rhoangle \varepsilon )=(-1)^{n}\lambdaangle \textstyle
\int\mathbb Nolimits_{0}^{1}\omega _{1}dt\rhoangle
\end{equation*}
is well defined, as $\textstyle\int\mathbb Nolimits_{0}^{1}[\mathbb Zeta ,\xi ]dt$ is a
sum of commutators in $\Omegamega ^{\ast },$ for any $\mathbb Zeta \in \,_{x}\Omegamega
^{p}[t]_{y} $ and $\xi \in \,_{y}\Omegamega ^{n-p-1}[t]_{x}.$
\end{fact}
\begin{lemma}
The operators $(-1)^{\ast }k^{\ast }$ define a homotopy between $
ev_{1}^{\ast }$ and $ev_{0}^{\ast }$.
\end{lemma}
\begin{proof}
Let $\varpi :=\lambdaangle \omega _{0}\rhoangle +\lambdaangle \omega _{1}\rhoangle
\varepsilon $ be a cochain of degree $n$ in $(\widetilde{\Omegamega }_{ab}^{\ast
},\partial _{ab}^{\ast }).$ Thus
\begin{align*}
\lambdaeft( k^{n+1}\circ \partial ^{n}\rhoight) (\varpi )& =(-1)^{n+1}\lambdaangle
\textstyle\int\mathbb Nolimits_{0}^{1}(d^{n-1}\omega _{1})dt\rhoangle +\lambdaangle
\textstyle\int\mathbb Nolimits_{0}^{1}\dot{\omega}_{0}dt\rhoangle \\
& =(-1)^{n+1}\lambdaangle d^{n-1}(\textstyle\int\mathbb Nolimits_{0}^{1}\omega
_{1}dt)\rhoangle +\lambdaangle \omega _{0}(1)\rhoangle -\lambdaangle \omega _{0}(0)\rhoangle
\\
& =-\lambdaeft( d^{n-1}\circ k^{n}\rhoight) (\varpi )+\lambdaeft(
ev_{1}^{n}-ev_{0}^{n}\rhoight) (\varpi ).
\end{align*}
This computation shows us that $(-1)^{\ast }k^{\ast }$ is a homotopy between
$ev_{1}^{\ast }$ and $ev_{0}^{\ast }$.
\end{proof}
\begin{corollary}
\lambdaabel{co:cobord} If $\varpi $ is a cocycle of degree $n$ in $(\widetilde{
\Omegamega }_{ab}^{\ast },\partial _{ab}^{\ast })$ then $ev_{1}^{n}(\varpi
)-ev_{0}^{n}(\varpi )$ is a coboundary.
\end{corollary}
Now we can prove that the cohomology class of $\mathbb Ch^{2q}(M,\mathbb Nabla)$ does not
depend on $\mathbb Nabla.$ Let $B^{n}(\mathcal{C},\Omegamega^{\ast })$ denote the space
of $n$-coboundaries in the de Rham complex. The cohomology class of an $n$
-cocycle $\omega$ will be denoted by $\omega+B^{n}(\mathcal{C},\Omegamega^{\ast
})$.
\begin{theorem}
Let $\mathcal{C}$ be a $\Bbbk $-linear category. We assume that $\Bbbk $ is
a field of characteristic zero and that $\Omegamega ^{\ast }$ is a $DG$-category
such that $\Omegamega ^{0}=\mathcal{C}.$ If $M$ is a finitely generated
projective right $\mathcal{C}$-module and $\mathbb Nabla _{1},\mathbb Nabla _{2}:M\to
M\otimesimes _{\Omegamega ^{0}}\Omegamega ^{1}$ are connections on $M$, then
\begin{equation*}
\mathbb Ch^{q}(M,\mathbb Nabla _{1})+B^{2q}(\mathcal{C},\Omegamega ^{\ast })=\mathbb Ch^{q}(M,\mathbb Nabla
_{2})+B^{2q}(\mathcal{C},\Omegamega ^{\ast }).
\end{equation*}
\end{theorem}
\begin{proof}
Let $N$ be a right $\mathcal{C}$-module such that $M\opluslus N=\mathcal{C}(
\mathcal{I})$. The Levi-Civita connection $\mathbb Nabla ^{LC}$ on $N$ exists, as $
N $ is projective and finitely generated, being a direct summand of $
\mathcal{C}(\mathcal{I})$. For $i\in \{1,2\}$ we have
\begin{equation*}
\mathbb Ch^{q}(M\opluslus N,\mathbb Nabla _{i}\opluslus \mathbb Nabla ^{LC})=\mathbb Ch^{q}(M,\mathbb Nabla _{i})+\mathbb Ch
^{q}(N,\mathbb Nabla ^{LC}).
\end{equation*}
Therefore, by subtracting these two equations, we get
\begin{equation*}
\mathbb Ch^{q}(M,\mathbb Nabla _{1})=\mathbb Ch^{q}(M,\mathbb Nabla _{2})+\mathbb Ch^{q}(M\opluslus N,\mathbb Nabla
_{1}\opluslus \mathbb Nabla ^{LC})-\mathbb Ch^{q}(M\opluslus N,\mathbb Nabla _{2}\opluslus \mathbb Nabla ^{LC}).
\end{equation*}
In conclusion, it is enough to prove that $\mathbb Ch^{q}(\mathcal{C}(\mathcal{I}
),\mathbb Nabla )$ is a coboundary for any connection $\mathbb Nabla $ on $\mathcal{C}(
\mathcal{I}).$ We may assume that $\mathbb Nabla =\mathbb Nabla ^{\Lambda },$ where $
\Lambda =(\lambdaambda _{ij})_{i,j\in I}$ is a matrix with $\lambdaambda _{ij}\in
{}_{x_{i}}\Omegamega _{x_{j}}^{1}$.
Let $\widetilde{\Omegamega }^{\ast }$ be the $DG$-category that we constructed
in the subsection \rhoef{fa:Omega_tilde}. Recall that, by construction, $
\widetilde{\Omegamega }^{0}$ is the $\Bbbk $-linear category $\mathcal{C}[t]$.
It has the same objects as $\mathcal{C}$, and its morphism from $y$ to $x$
are polynomials with coefficients in $_{x}\mathcal{C}_{y}.$ On the right $
\mathcal{C}[t]$-module $\widetilde{M}:=\mathcal{C}[t](\mathcal{I})$, we
consider the connection $\widetilde{\mathbb Nabla }:\widetilde{M}\to \widetilde{M}
\otimesimes _{\mathcal{C}[t]}\widetilde{\Omegamega }^{1}$ associated to the matrix $
\widetilde{\Lambda }:=(\lambdaambda _{ij}t+0\varepsilon )_{i,j\in I}=\Lambda
t+0\varepsilon .$ Note that $\lambdaambda _{ij}t+0\varepsilon \in {}_{x_{i}}
\widetilde{\Omegamega }_{x_{j}}^{1}.$
We have already proved that $\varpi :=\mathbb Ch^{q}(\widetilde{M},\widetilde{
\mathbb Nabla })$ is a $2q$-cocycle in $\widetilde{\Omegamega }_{ab}^{\ast }$, so by
Corollary \rhoef{co:cobord} it follows that $ev_{1}(\varpi )-ev_{0}(\varpi )$
is a coboundary in $(\Omegamega _{ab}^{\ast },d_{ab}^{\ast })$. On the other
hand, by the computation that we performed in \S \rhoef{fa:Chern}, the cocycle
$\varpi $ equals the class in $\widetilde{\Omegamega }_{ab}^{2q}$ of the trace
of $\widetilde{\Gamma }=\partial \widetilde{\Lambda }+\widetilde{\Lambda }
^{2}$. Since $\partial \widetilde{\Lambda }=(d\Lambda )t+\Lambda \varepsilon
$ and $\widetilde{\Lambda }^{2}=t^{2}\Lambda ^{2}+0\varepsilon ,$ we have
\begin{equation*}
\widetilde{\Gamma }^{q}=(\partial \widetilde{\Lambda }+\widetilde{\Lambda }
^{2})^{q}=\lambdaeft( (td\Lambda +t^{2}\Lambda ^{2})+\Lambda \varepsilon \rhoight)
^{q}=(td\Lambda +t^{2}\Lambda ^{2})^{q}+\Lambda ^{\prime }\varepsilon ,
\end{equation*}
where $\Lambda ^{\prime }:=\lambdaeft( \lambdaambda _{ij}^{\prime }\rhoight) _{i,j\in I}$
is a certain matrix with $\lambdaambda _{ij}^{\prime }\in {}_{x_{i}}\Omegamega
_{x_{j}}^{2q-1}.$ Hence $\varpi =\Tr\lambdaeft( (td\Lambda +t^{2}\Lambda
^{2})^{q}\rhoight) +\Tr(\Lambda ^{\prime })\varepsilon .$ By the definition of
the evaluation map, $ev_{a}(\varpi )=\Tr\lambdaeft( ad^{1}\Lambda +a^{2}\Lambda
^{2})^{q}\rhoight) .$ Thus,
\begin{equation*}
\mathbb Ch^{q}(\mathcal{C}(\mathcal{I}),\mathbb Nabla )=\Tr\lambdaeft( (d^{1}\Lambda +\Lambda
^{2})^{q}\rhoight) =ev_{1}\lambdaeft( \varpi \rhoight) -ev_{0}\lambdaeft( \varpi \rhoight) .
\end{equation*}
We conclude that $\mathbb Ch^{q}(\mathcal{C}\lambdaeft( \mathcal{I}\rhoight) ,\mathbb Nabla )$ is
a coboundary in $\Omegamega _{ab}^{2q}$, so the theorem is proved.
\end{proof}
\begin{fact}[The Chern classes.]
Let $M$ be a finitely generated projective $\mathcal{C}$-module, where $
\mathcal{C}$ is a linear category over a field of characteristic zero. We
assume that $\Omegamega^{\ast}$ is a $DG$-category such that its homogeneous
component of degree zero equals $\mathcal{C}$. On $M$ we consider a
connection $\mathbb Nabla$ on $M.$ Note that such a connection always exists, as $M$
is finitely generated and projective, so on $M$ we can take for instance the
Levi-Civita connection. We have just proved that the de Rham cohomology
class of $\mathbb Ch^{q}(M,\mathbb Nabla )$ in $H^{2q}_{DR}(\mathcal{C},\Omegamega^{\ast})$
does not depend on $\mathbb Nabla.$ We shall call this cohomology class the $q^{
\text{th}}$ \emph{Chern class} of $M$, and we shall denote it by $\mathbb Ch
^{q}(M,\Omegamega^{\ast})$.
\end{fact}
\begin{fact}[The Grothendieck group of a linear category.]
We keep the notation and the assumptions from the previous subsection. Let
us denote the isomorphism class of $M$ by $\lambdaeft[ M\rhoight] .$ The
Grothendieck group of $\mathcal{C}$ is, by definition, the quotient of the
free abelian group generated by the set $\{\lambdaeft[ M\rhoight] :M$ is finitely
generated projective$\}$ through the subgroup generated by the elements $
\lambdaeft[ M^{\prime }\rhoight] +\lambdaeft[ M^{\prime \prime }\rhoight] -\lambdaeft[
M^{\prime }\opluslus M^{\prime \prime }\rhoight] ,$ where $M^{\prime }$ and $
M^{\prime \prime }$ are arbitrary finitely generated projective modules. We
shall denote the Grothendieck group of $\mathcal{C}$ by $K_{0}(\mathcal{C}).$
We can now prove the main result of this paper.
\end{fact}
\begin{theorem}
Let $\mathcal{C}$ be a $\Bbbk $-linear category. We assume that $\Bbbk $ is
a field of characteristic zero and that $\Omegamega ^{\ast }$ is a $DG$-category
such that $\Omegamega ^{0}=\mathcal{C}.$ The mapping $\lambdaeft[ M\rhoight] \mapsto \mathbb Ch
^{2q}(M)$ induces a morphism of groups from $K_{0}(\mathcal{C})$ to $
H_{DR}^{\ast }(\mathcal{C},\Omegamega ^{\ast }).$
\end{theorem}
\begin{proof}
Let $M^{\prime }$ and $M^{\prime \prime }$ be two finitely generated
projective $\mathcal{C}$-module. Let $\mathbb Nabla ^{\prime }$ and $\mathbb Nabla
^{\prime \prime }$ be connections on $M^{\prime }$ and $M^{\prime \prime },$
respectively. We have seen that there is a unique connection $\mathbb Nabla
^{\prime }\opluslus \mathbb Nabla ^{\prime \prime }$ on $M^{\prime }\opluslus M^{\prime
\prime },$ such that its restrictions to $M^{\prime }$ and $M^{\prime \prime
}$ coincide with $\mathbb Nabla ^{\prime }$ and $\mathbb Nabla ^{\prime \prime },$
respectively. Since the definition of the Chern class $\mathbb Ch^{q}(M^{\prime
}\opluslus M^{\prime \prime })$ does not depend on the connection, and
\begin{equation*}
\mathbb Ch^{q}(M^{\prime }\opluslus M^{\prime \prime },\mathbb Nabla^{\prime }\opluslus
\mathbb Nabla^{\prime \prime }=\mathbb Ch^{q}(M^{\prime },\mathbb Nabla^{\prime })+\mathbb Ch
^{q}(M^{\prime \prime },\mathbb Nabla^{\prime \prime }),
\end{equation*}
it follows that $\mathbb Ch^{q}(M^{\prime }\opluslus M^{\prime \prime })=\mathbb Ch
^{q}(M^{\prime })+\mathbb Ch^{q}(M^{\prime \prime }).$ Hence the theorem is proved.
\end{proof}
\mathbb Noindent\textbf{Acknowledgement.} The authors of the paper were financially
supported by UEFISCDI, Contract 560/2009 (CNCSIS code ID\_69).
\end{document} | math | 69,029 |
\begin{document}
\twocolumn[\hsize\textwidth\columnwidth\hsize\csname
@twocolumnfalse\endcsname
\draft
\title{The Quantum-Classical Transition in Nonlinear Dynamical Systems}
\author{Salman Habib$^1$, Kurt Jacobs$^1$, Hideo Mabuchi$^2$, Robert
Ryne$^3$, Kosuke Shizume$^4$, and Bala Sundaram$^5$}
\address{$^1$ T-8, Theoretical Division, MS B285, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545}
\address{$^2$ Mail Code 12-33, California Institute of Technology,
Pasadena, CA 91125}
\address{$^3$ LANSCE-1, LANSCE Division, MS H817, Los Alamos National
Laboratory, Los Alamos, NM 87545}
\address{$^4$University of Library and Information Science, 1-2
Kasuga, Tsukuba, Ibaraki 305, Japan}
\address{$^5$Department of Mathematics \& Graduate Faculty in Physics,
CSI-CUNY, Staten Island, NY 10314}
\date{\today}
\maketitle
\begin{abstract}
Viewed as approximations to quantum mechanics, classical evolutions
can violate the positive-semidefiniteness of the density matrix. The
nature of this violation suggests a classification of dynamical
systems based on classical-quantum correspondence; we show that this
can be used to identify when environmental interaction (decoherence)
will be unsuccessful in inducing the quantum-classical transition. In
particular, the late-time Wigner function can become positive without
any corresponding approach to classical dynamics. In the light of
these results, we emphasize key issues relevant for experiments
studying the quantum-classical transition.
\end{abstract}
\pacs{{PACS Numbers: 03.65.Yz, 05.45.-a}
LAUR-00-4046}
\vskip1pc]
In recent years much effort has been expended, both theoretically and
experimentally, to explore the transition from quantum to classical
behavior in a controlled way. In this context, the interaction of
trapped, cold atoms with optical potentials, both time-dependent and
independent, has become a topic of considerable interest and activity
\cite{exp,latt}. The experimental ability to systematically study
dissipative quantum dynamics of nonlinear systems is an exciting new
area where the frontier between classical and quantum mechanics may be
carefully examined.
In this Letter we wish to explore some of the key qualitative features
of the quantum-classical transition. We establish that with $\hbar$
fixed at a finite value, and classical dynamical evolution equations
for phase space distribution functions viewed as approximations to the
underlying quantum equations, classical Liouville and Master equations
violate the quantum constraint of positive-semidefiniteness of the
density matrix: We refer to this property of the density matrix as
`rho-positivity'. There is thus a global obstruction to the classical
limit arising directly from quantum evolutions. We argue below that
rho-positivity violation can (1) serve as a guide in classifying
dynamical systems with regard to classical-quantum correspondence:
weak violation as Type I, strong violation as Type II, and (2) explain
robustness to decoherence in the sense of avoidance of the classical
limit as exemplified by dynamical localization in the (open-system)
quantum delta kicked rotor (QDKR). Our results impact directly on the
interpretation and design of experiments to test various aspects of
the quantum-classical transition.
As described in more detail below, certain nonclassical aspects of the
dynamics of the QDKR turn out to be stable to decohering effects of
external noise and decoherence due to spontaneous emission. A
dynamical description in terms of the Wigner function leads to two
alternatives to explain this stability: (1) diffusion in the quantum
Master equation is simply not efficient at suppressing quantum
interference terms present in the Wigner function, or (2) the much
more intriguing possibility that the diffusion is successful at
decohering the Wigner function, i.e., interference terms are
suppressed and the Wigner function is (almost) everywhere positive,
yet the late-time distribution is not the solution of the
corresponding classical Fokker-Planck equation. We find that the
second possibility is the one actually realized, and show how it
arises as a consequence of the fact that the classical Fokker-Planck
equation violates rho-positivity, while the quantum Master equation
does not.
The singular nature of the classical limit $\hbar\rightarrow 0$ in
quantum mechanics has been appreciated for a long time. However, what
has not been stressed sufficiently is the reason for this singular
behavior, that classical dynamics violates unitarity and
rho-positivity, and thus $\hbar=0$ cannot be connected smoothly to
$\hbar\rightarrow 0$. A simple example suffices to make this point
clear. Let us consider as initial condition a pure Gaussian state. Let
us suppose that we evolve the corresponding (positive) Wigner function
classically in some nonlinear potential (for linear systems classical
and quantum dynamics are identical \cite{ah}), then the distribution
becomes no longer Gaussian, but is still positive-definite. Three
possibilities now present themselves: the evolved object can be
interpreted as (1) a pure quantum state (unitarity is preserved), (2)
a mixed quantum state (rho-positivity is preserved), and (3) cannot be
interpreted as a quantum state (rho-positivity is violated). The first
possibility can be dismissed using Hudson's theorem: the {\em only}
pure state with positive Wigner function is a Gaussian state with a
(necessarily) Gaussian Wigner function \cite{rh}. But our distribution
is non-Gaussian. As to the second, we first note that the phase space
integral of any function of the phase space distribution is preserved
under a Liouville flow. In particular $\int f^2(x,p)dxdp$ remains
constant. For Wigner functions this quantity is proportional to
$Tr\rho^2$ which is a direct measure of whether a state is mixed or
not -- since this measure cannot change, the evolved object is not
interpretable as a mixed state. Thus we are forced to the third
alternative, that the evolved object cannot be interpreted as a
quantum state at all, i.e., the Weyl transform of the evolved
classical distribution yields a `classical density matrix' which is
non-rho-positive \cite{shcl}.
The above analysis makes it clear that the classical Liouville
equation can never arise as a formal limit of quantum theory. However,
all real experiments deal with open systems, i.e., systems interacting
with their environment, of which the particular case of a measuring
apparatus (necessary to deduce classical behavior) is an important
example. Quantum decoherence and conditioned evolution arising as a
consequence of such system-environment couplings and the act of
measurement and observation provide a natural pathway to the classical
limit as has been quantitatively demonstrated, {\em e.g.}, in
Ref. \cite{bhj}. Thus, it is important to inquire into the role of
rho-positivity violation in this context: When is it important, and
when not?
Conditions have been previously derived that apply to the extraction
of (noisy) classical trajectories via continuous quantum measurement
\cite{bhj}. Once these (strict) conditions are satisfied (typically in
the macroscopic limit $\hbar\ll S$, where $S$ is the system action),
measurement induces classical behavior, and in this regime all systems
are therefore Type I. However, when these conditions are not
satisfied, as is the case in most current experiments, the differences
are indeed important. The key point is that, under continuous
measurement, Type I systems can violate the classicality conditions in
the sense that individual classical trajectories cannot be extracted,
yet expectation values are close to the classical results, whereas in
Type II systems, violation of the classicality condition also implies
a violation of correspondence at the level of expectation values. We
will demonstrate this for the QDKR below.
A quantum Master equation representing a weakly coupled, high
temperature environment often utilized in studies of decoherence is
\begin{eqnarray}
{\partial \over \partial t}f_W&=&L_{cl}f_W+ L_q
f_W + D{\partial^2 \over \partial p^2}f_W;
\label{qme}\\
L_{cl}&\equiv&-{p\over m}{\partial\over \partial
x}+ {\partial V\over\partial x}{\partial \over \partial
p}, \label{lcl}\\
L_q &\equiv& \sum_{\lambda~odd}{1\over\lambda!}\left({\hbar\over
2i}\right)^{\lambda-1}{\partial^{\lambda}V(x)\over\partial
x^{\lambda}}{\partial^{\lambda}\over\partial p^{\lambda}}.
\label{lq}
\end{eqnarray}
An unraveling of the weakly-coupled, high temperature environment,
Master equation (\ref{qme}) is provided by a continuous measurement of
position. This process is described by a stochastic Master equation
for the density matrix $\rho (t)$, conditioned on the measurement
record $\langle X \rangle + \xi(t)$ with $\xi(t) \equiv (8\eta
k)^{-1/2}dW/dt$, \cite{measx}
\begin{eqnarray}
\rho(t+dt) &=& \rho - (\frac{i}{\hbar} [H,\rho] - k [X,[X,\rho]])
\mathbin{} dt
\nonumber\\
&& {} + \sqrt{2\eta k} \mathbin{} ( [X,\rho]_+ -
2 \rho \mathop{\rm Tr} \rho X ) \mathbin{} dW \,,
\label{sme}
\end{eqnarray}
where $k$ is a constant specifying the strength of the measurement,
$\eta$ is the measurement efficiency and is a number between 0 and 1,
and $dW$ is a Wiener process, satisfying $(dW)^2=dt$. When $\eta=1$,
the evolution preserves the purity of the state and can be rewritten
in a way which allows it to be understood as a series of diffuse
projection measurements~\cite{cqm2}. Averages over the resulting
Schr\"odinger trajectories reproduce expectation values computed using
the reduced density matrix $\rho$ or the corresponding Wigner function
$f_W(x,p)$ obtained from solving the Master equation (\ref{qme}). The
strength of the measurement is related to the diffusion coefficient of
Eq. (\ref{qme}) by $D=\hbar^2 k$.
When the diffusion constant $D=0$, Eq. (\ref{qme}) is just the quantum
Liouville equation for the closed system. Note that the linearity of
the quantum Liouville equation implies that in order for the evolution
to be unitary, $L_q$ cannot be unitary since $L_{cl}$ is not (the sum
$L_{cl}+L_q$ is unitary but not the operators separately). The
familiar heuristic argument for obtaining the classical limit from the
quantum Master equation is that the diffusion term smooths out the
interference effects generated by $L_q$ in such a way that quantum
corrections to the classical dynamics are much reduced. It has also
been argued, that at finite $\hbar$, the appropriate limiting case of
the quantum Master equation is in fact the classical Fokker-Planck
equation [setting $L_q=0$ in Eqn. (\ref{qme})] rather than the
classical Liouville equation \cite{hsz}. In any case, one immediately
appreciates that if {\em either of the classical equations are
strongly rho-positivity-violating, i.e., are Type II} then this
implies the existence of compensatory `large' quantum corrections in
the quantum Master equation, and hence the above heuristic argument
must fail: $L_q$ is responsible for more than just the generation of
interference fringes in the Wigner evolution.
Previous work has already suggested the possibility that closed
dynamical systems may be roughly divided into two types depending on
the (dynamical) classical-quantum correspondence as follows: (1) Type
I systems in which quantum expectation values and classical averages
track each other relatively closely as a function of time
\cite{hsz,close}, {\em e.g.}, the driven Duffing oscillator with
Hamiltonian,
\begin{equation}
H_{\hbox{duff}}=p^2/2m + B x^4 - A x^2 + \Lambda x \cos(\omega t),
\label{hduff}\\
\end{equation}
and (2) Type II systems in which the quantum and classical averages
diverge sharply after some finite time, e.g., dynamical localization
in the QDKR \cite{dloc}. The QDKR Hamiltonian is
\begin{equation}
H_{\hbox{dkr}}={1\over 2}p^2+\kappa \cos q \sum_n\delta(t-n).
\label{hdkr}
\end{equation}
We solved the classical and quantum Master equations corresponding to
Eqs. (\ref{hduff}) and (\ref{hdkr}) using a high-resolution spectral
solver implemented on parallel supercomputers. The solver explicitly
respects rho-positivity conservation.
We verified that in both the QDKR and the Duffing oscillator numerical
examples discussed below the localization condition \cite{bhj}
necessary to obtain classical trajectories was violated. The relevant
condition is $8\eta k\gg (\partial^2_x F /F)\sqrt{\partial_x F/2m}$
where the force $F$ and its derivatives are evaluated at typical
points in phase space. For both cases we have in fact, $8\eta k\sim
(\partial^2_x F /F)\sqrt{\partial_x F/2m}$ thus localization does not
occur (direct numerical solution of the corresponding stochastic
Schr\"odinger equation confirms this result) and, as discussed
earlier, a meaningful distinction between Types I and II is
possible. As the value of $\hbar$ is reduced (with $D$ fixed and
non-zero) one does expect an approach to the classical limit
\cite{bhj}, though the trajectory in the space of $D$ and $\hbar$ need
not be simple \cite{bhjs}.
\begin{figure}
\caption{Eigenvalues of the quantum density matrix (solid) and the
classical approximation (long-dashed) computed from the quantum and
classical Master equation evolutions for the QDKR at $t=6$. Also shown
(short-dashed) is the classical result for the Duffing oscillator at
$t=10$.}
\label{fig1}
\end{figure}
Our numerical code returns us the classical distribution function, the
quantum density matrix and the Wigner function as functions of
time. We then numerically solve for the eigenvalues of the quantum
density matrix and the eigenvalues of the Weyl transform of the
classical phase space distribution (the `classical density
matrix'). Results of one such computation are displayed in Fig. 1 for
the QDKR (Type II) and Duffing system (Type I). For the QDKR, initial
conditions are pure Gaussian Wigner functions characterized by the
position width $\Delta x=2.5$, momentum width $\Delta p=1$, centered
on the point $(x,p)=(0,0)$, and with $\hbar=5$ and $\kappa=10$. The
diffusion coefficient $D=0.1$, corresponding to $k=0.004$. The
horizontal axis refers to the index $i$ corresponding to the
eigenvalues $\lambda_i$, which are themselves plotted on the vertical
axis. The solid line is a result from a numerical solution of the
quantum Master equation, as expected all eigenvalues are positive (the
pure initial state has one eigenvalue equaling unity, the rest being
zero). The dashed line is the corresponding result from the classical
Fokker-Planck equation, which is characterized by a strong
contribution from negative eigenvalues. It is thus clear that the true
quantum density matrix and that provided by the classical
approximation are in fact quite different. In contrast, results from
classical Duffing calculations show a very small contribution from
negative eigenvalues [Parameter values in the particular case shown in
Fig. 1 were $m=1$, $A=10$, $B=0.5$, $\Lambda=10$, $\omega=6.07$,
$\Delta x=0.05$, $\Delta p=1$, $(x,p)=(-3,8)$, $\hbar=0.1$, $D=0.02$.]
These results show how rho-positivity violation may be used to
distinguish the two types of dynamical systems. An important point to
emphasize is that it is sufficient to only carry out the
classical dynamical calculation in order to classify a dynamical
system as being Type I or II. (The initial condition must of course be
a Wigner function.) Also, it should be clear that non-violation of
rho-positivity is a necessary but not sufficient condition for
quantum-classical correspondence in terms of agreement of expectation
values.
\begin{figure}
\caption{The Wigner function negativity measure $\Gamma$ as a function
of time for $D=0$ and $D=0.1$ for the QDKR.}
\label{fig2}
\end{figure}
It is well-known that dynamical localization in the QDKR can be
destroyed (in the sense that $\langle p^2(t)\rangle$ no longer
saturates at late times) by coupling to external noise or to
dissipative channels (e.g., spontaneous emission)
\cite{locdiff}. However, what is important to note is that even in the
presence of quite strong coupling to these decohering channels, the
evolution does not go over to the classical one, and in this sense the
QDKR is quite different from the Duffing system investigated in
Ref. \cite{hsz}. In addition to numerically solving the Master
equation (\ref{qme}) we have investigated in detail the effects of
including amplitude and phase noise, timing jitter in the kicked
system, and carried out more realistic simulations taking into account
the effects of spontaneous emission. Stability to decoherence in the
sense above was manifest in all of these cases. Since we have
established that the QDKR is a Type II system (Fig. 1), this behavior
is essentially forced: as long as the classical evolution strongly
violates rho-positivity, it is impossible for the full evolution to
ever become close to the classical limit as the quantum corrections
must always be concomitantly large. The question remains whether the
resulting Wigner function at least has a classical interpretation. In
order to investigate this we computed as a function of time, the
quantity $\Gamma=\int dxdp(|f_W|-f_W)$, which provides a global
measure of negativity of the Wigner function. With $D=0$, one sees
that $\Gamma$ increases monotonically as the Wigner function develops
the expected oscillatory structure as a consequence of quantum
interference in phase space. When $D\neq 0$, diffusion in phase space
wipes out the interference and produces an essentially positive
distribution which one may interpret classically. However, because
rho-positivity must be maintained, classical evolution cannot
connect two such positive distributions. Thus, in Type II systems
decoherence can be successful in rendering the Wigner distribution
positive, but yet not lead to a classical limit. We note that in NMR
systems there is an interesting question regarding when classical
evolution of spins can reproduce quantum evolutions connecting spin
states that have no entanglement and thus may be interpreted
classically \cite{caves}. We have shown that a similar situation can
arise even in single-particle evolution where entanglement is not an
issue.
Recent experiments have attempted to directly address the issue of
environment-induced decoherence in the QDKR, in the context of cold
atom optics \cite{exp}. Despite some complications stemming from
non-ideal realizations, the results indicate that classical and
quantum evolutions agree only at inordinately large noise levels. Note
that in these experiments, parametric noise or spontaneous emission
was used as the decohering mechanism. (The non-selective Master
equation for atomic motion in far-detuned laser light has a similar
form to that of a particle subjected to continuous position
measurement. However, arguments can be made that only the weak
decoherence regime can be accessed in this manner.) The parameter
values in our numerical work are close to those actually used in the
experiments thus, as with our simulations, the experiments are not
carried out in a classical regime in the sense of Ref. \cite{bhj}. And
since we have demonstrated the strongly Type II nature of the QDKR, it
follows immediately that to observe true classical behavior, either
the current experiments have to switch to a Type I system or employ lower
values of $\hbar$. Simply increasing the measurement constant $k$, or
equivalently $D$, while it produces localization, adds noise into the
trajectory which must be kept small in order to achieve the classical
limit. This final condition requires a reduction in $\hbar$ as $k$ is
increased \cite{bhj}.
The authors acknowledge helpful discussions with Tanmoy Bhattacharya,
Doron Cohen, and Andrew Doherty. The work of BS was supported by the
National Science Foundation and a grant from the City University of
New York PSC-CUNY Research Award Program. Large-scale numerical
simulations were carried out at the Advanced Computing Laboratory
(ACL), LANL and at the National Energy Research Scientific Computing
Center (NERSC), LBNL.
\end{document} | math | 20,177 |
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