\section{\label{sec: Notation}Notation}

The⋅symbol~$\norm{\wc}_{U}$ will stand for the norm on a normed
space~$U$. If⋅$U$ and⋅$V$ are Banach spaces, then $\B\rbr{U,V}$
is the Banach space of bounded linear operators from⋅$U$ to~$V$.
When⋅$U=V$, the space~$\B\rbr U=\B\rbr{U,U}$ becomes a Banach algebra
with the identity map~$\op I$. The⋅norm and the inner product on⋅$\C^{n}$
are denoted by⋅$\abs{\wc}$ and⋅$\abr{\wc,\wc}$. We shall often
identify $\B\rbr{\C^{n},\C^{m}}$ and~$\C^{m\times n}$.

Let⋅$\Sigma$ be a domain in⋅$\R^{d}$ and⋅$U$ a Banach space. Then
$L_{0}\rbr{\Sigma;U}$ is the vector space of all strongly measurable
functions on⋅$\Sigma$ with values in~$U$. The⋅symbol~$L_{p}\rbr{\Sigma;U}$,
$p\in\sbr{1,\infty}$, stands for the⋅$L_{p}$\nobreakdash-space
of  $L_{0}\rbr{\Sigma;U}$\nobreakdash-functions.  For⋅$p<\infty$
and⋅$s\ge0$, we let $W_{p}^{s}\rbr{\Sigma;U}$ denote the usual Sobolev
space or Sobolev–Slobodetskii space of $L_{0}\rbr{\Sigma;U}$\nobreakdash-functions
with norm
\[
\norm u_{W_{p}^{s}\rbr{\Sigma;U}}=\biggrbr{\sum_{\abs{\alpha}=0}^{m}\norm{D^{\alpha}u}_{L_{p}\rbr{\Sigma;U}}^{p}}^{1/p}
\]
if⋅$s=m\in\N_{0}$~and
\[
\norm u_{W_{p}^{s}\rbr{\Sigma;U}}=\biggrbr{\sum_{\abs{\alpha}=m}\Seminorm{D^{\alpha}u}_{W_{p}^{r}\rbr{\Sigma;U}}^{p}+\norm u_{W_{p}^{m}\rbr{\Sigma;U}}^{p}}^{1/p}
\]
if⋅$s=m+r$ with $m\in\N_{0}$ and~$r\in\rbr{0,1}$. Here $D=-i\nabla$~and
\[
\Seminorm u_{W_{p}^{r}\rbr{\Sigma;U}}=\biggrbr{\int_{0}^{\infty}t^{-1-rp}\osc_{p}\rbr{u;t}^{p}\dd t}^{1/p},
\]
where $\osc_{p}\rbr{u;\wc}$ is the⋅$L_{p}$\nobreakdash-modulus
of continuity of⋅$u$, given⋅by
\[
\osc_{p}\rbr{u;t}=\sup_{\abs h\le t}\norm{u\rbr{\wc+h}-u\rbr{\wc}}_{L_{p}\rbr{\Sigma\cap\rbr{-h+\Sigma};U}}.
\]
In⋅the⋅case⋅$U=\C^{n}$, we write⋅$\norm{\wc}_{p,\Sigma}$ and⋅$\norm{\wc}_{s,p,\Sigma}$
for the norms on⋅$L_{p}\rbr{\Sigma}^{n}=L_{p}\rbr{\Sigma;U}$ and⋅$W_{p}^{s}\rbr{\Sigma}^{n}=W_{p}^{s}\rbr{\Sigma;U}$,
respectively, and⋅$\rbr{\wc,\wc}_{\Sigma}$ for the inner product
on~$L_{2}\rbr{\Sigma}^{n}$. The⋅dual space of⋅$W_{p}^{s}\rbr{\Sigma}$
under the pairing~$\rbr{\wc,\wc}_{\Sigma}$ is denoted by⋅$W_{p}^{s}\rbr{\Sigma}^{*}$,
with $\norm{\wc}_{-s,p^{\dual},\Sigma}^{*}$ standing for the norm.
Here $p^{\dual}$ is the exponent conjugate to⋅$p$, that is,⋅$1/p^{\dual}=1-1/p$.
The⋅closure of⋅$C_{c}^{\infty}\rbr{\Sigma}$ in⋅$W_{p}^{s}\rbr{\Sigma}$
is⋅$\mathring{W}_{p}^{s}\rbr{\Sigma}$, and $W_{\smash[t]{\cramped{p^{\dual}}}}^{-s}\rbr{\Sigma}$
is its dual, with norm~$\norm{\wc}_{-s,p^{\dual},\Sigma}$. As⋅usual,
$H^{s}=W_{2}^{s}$, $H^{-s}=W_{2}^{-s}$,~etc.

Let⋅$\cell$ be the closed cube in⋅$\R^{d}$ with center~$0$ and
side length~$1$, sides being parallel to the axes. Then  $\wt W_{p}^{m}\rbr{\cell}$
denotes the completion of⋅$\wt C^{m}\rbr{\cell}$ in the $W_{p}^{m}$\nobreakdash-norm.
Here $\wt C^{m}\rbr{\cell}$ is the class of $m$\nobreakdash-times
continuously differentiable functions  on⋅$\cell$ whose periodic
extension to⋅$\R^{d}$ enjoys the same smoothness. Notice that $\wt L_{p}\rbr{\cell}$
can be identified with the space of all periodic functions in~$L_{p,\text{loc}}\rbr{\R^{d}}$.
In⋅a⋅similar fashion, we define⋅$\wt W_{p}^{m}\rbr{\R^{d}\times\cell}$.
The⋅dual of⋅$\wt W_{p}^{m}$ is denoted by~$\wt W_{\smash[t]{\cramped{p^{\dual}}}}^{-m}$.

For⋅$p\in\sbr{1,\infty}$ and⋅$s=m+r$ with $m\in\N_{0}$ and⋅$r\in(0,1]$,
we also introduce the space~$\Lambda_{p}^{s}\rbr{\Sigma;U}$ of functions~$u\in W_{p}^{m}\rbr{\Sigma;U}$
with finite norm
\[
\abs u_{\Lambda_{p}^{s}\rbr{\Sigma;U}}=\biggrbr{\sum_{\abs{\alpha}=m}\seminorm{D^{\alpha}u}_{\Lambda_{p}^{r}\rbr{\Sigma;U}}^{p}+\norm u_{W_{p}^{m}\rbr{\Sigma;U}}^{p}}^{1/p},
\]
where
\[
\seminorm u_{\Lambda_{p}^{r}\rbr{\Sigma;U}}=\sup_{t\in\R_{+}}\abs t^{-r}\osc_{p}\rbr{u;t}.
\]
We use $\abs{\wc}_{s,p,\Sigma}$ and⋅$\seminorm{\wc}_{s,p,\Sigma}$
as shorthand for the norm and the seminorm on⋅$\Lambda_{p}^{s}\rbr{\Sigma}^{n}$.

Let $B$ be the open unit ball in⋅$\R^{d}$ centered at the origin,
and let $B_{+}$ be the open unit half-ball with~$x_{d}\in\rbr{0,1}$.
We say that a domain~$\Sigma$ in⋅$\R^{d}$ satisfies the uniform
weak Lipschitz condition if it is a $d$\nobreakdash-dimensional
Lipschitz manifold with boundary embedded into~$\R^{d}$. More precisely,
there is a uniformly locally-finite open covering~$\cbr{W_{k}}$
of⋅$\partial\Sigma$ and a sequence of bi-Lipschitz transformations~$\omega_{k}\colon W_{k}\to B$
so that (1)~$\omega_{k}\rbr{W_{k}\cap\Sigma}=B_{+}$ and $\omega_{k}\rbr{W_{k}\cap\partial\Sigma}=\partial B_{+}\!\setminus\partial B$;
(2)~$\sup_{k}\seminorm{\omega_{k}}_{1,\infty,W_{k}}$ and⋅$\sup_{k}\seminorm{\omega_{k}^{-1}}_{1,\infty,B}$
are finite; and (3)~for some⋅$\delta>0$, any open ball~$B_{\delta}\rbr x$
with center~$x\in\partial\Sigma$ and radius~$\delta$ is contained
in some coordinate patch~$W_{k}$. The⋅last
two conditions are automatically satisfied provided that the boundary
of⋅$\Sigma$ is compact. Notice that the domain~$\R^{d}\setminus\wb{\Sigma}$
is uniformly weakly Lipschitz whenever $\Sigma$~is.

For⋅such~$\Sigma$, there exists a $C^{\infty}$\nobreakdash-partition
of unity~$\cbr{\varphi_{k}}$ subordinate to⋅$\cbr{W_{k}}$ with
the property that $\sup_{k}\norm{D^{\alpha}\varphi_{k}}_{\infty,W_{k}}$
is finite for any~$\alpha$, see~\cite[Chapter~6, Section~3]{St:1970}.
Then there is a linear strong $m$\nobreakdash-extension operator~$\op E_{m}$,
which maps $W_{p}^{k}\rbr{\Sigma}$ and⋅$C^{k}\rbr{\wb{\Sigma}}$
into, respectively, $W_{p}^{k}\rbr{\R^{d}}$ and⋅$C^{k}\rbr{\wb{\R}^{d}}$
for every integer~$k\in\sbr{0,m}$ and every⋅$p\in[1,\infty)$, see~\cite{AF:2003}.
It⋅follows that standard density and embedding results which hold
for⋅$\R^{d}$ must also hold for~$\Sigma$. In⋅particular, the
Sobolev  theorem states that $W_{p}^{1}\rbr{\Sigma}$ is continuously
embedded in⋅$L_{q}\rbr{\Sigma}$ for any⋅$q\in\sbr{p,p^{*}}$. Here
$p^{*}$ is the Sobolev conjugate to $p$ given by⋅$1/p^{*}=1/p-1/d$
if⋅$p<d$; $p^{*}$ is any finite number greater than or equal to⋅$p$
if⋅$p=d$; and $p^{*}=\infty$ if~$p>d$. By⋅$p_{*}$ we denote an
exponent such that⋅$W_{p_{*}}^{1}\rbr{\Sigma}$ is embedded in⋅$L_{p}\rbr{\Sigma}$,
that is, $p_{*}=1$ if⋅$p\in[1,d^{\dual})$, $p_{*}$ satisfies⋅$1/p_{*}=1/p+1/d$
if⋅$p\in[d^{\dual},\infty)$ and⋅$p_{*}$ is any number greater than⋅$d$
if~$p=\infty$. We also note that $\Lambda_{p}^{s}\rbr{\Sigma;U}$
coincides with the Sobolev space~$W_{p}^{s}\rbr{\Sigma;U}$ if⋅$s\in\N$
and with the Lipschitz–Besov space~$B_{p,\infty}^{s}\rbr{\Sigma;U}$
if $s\notin\N$ (see, e.g.,~\cite{AF:2003} and⋅\cite{Agranovich:2013}
for the definition); in particular, $\Lambda_{\infty}^{s}\rbr{\Sigma;U}$,
with $s=m+r$ as above, is the space~$C^{m,r}\rbr{\wb{\Sigma};U}$
of H\"{o}lder continuous functions. By⋅interpolation, $\op E_{m}$
is then an extension operator for⋅$\Lambda_{p}^{s}\rbr{\Sigma;U}$
with any real~$s\in\sbr{0,m}$ and any⋅$p\in\sbr{1,\infty}$.

We shall also need the $\BMO\rbr{\R^{d}}$ and⋅$\VMO\rbr{\R^{d}}$
spaces. The⋅former consists of all $u\in L_{1,\text{loc}}\rbr{\R^{d}}$
such that
\[
\norm u_{\BMO}=\sup_{B_{R}}\dashint_{B_{R}}\abs{u\rbr x-m_{B_{R}}\rbr u}\dd x<\infty,
\]
where $B_{R}\subset\R^{d}$ is a ball of radius~$R$ and $m_{B_{R}}\rbr u=\dashint_{B_{R}}u\rbr y\dd y$
is the mean value of $u$ over~$B_{R}$. The⋅latter is the subspace
in⋅$\BMO\rbr{\R^{d}}$ of all functions~$u$ for which the $\VMO$\nobreakdash-modulus,
given~by
\[
\eta\rbr{u;r}=\sup_{B_{R}\colon R<r}\dashint_{B_{R}}\abs{u\rbr x-m_{B_{R}}\rbr u}\dd x,
\]
tends to zero as~$r\to0$. We refer the reader to⋅\cite{Gra:2014-2}
and⋅\cite{Gar:2007} for more on this matter.

For⋅a⋅set~$\Sigma\subset\R^{d}$, we let $\Sigma_{\delta}$ denote
a neighborhood of~$\Sigma$:
\[
\Sigma_{\delta}=\bigcbr{x\in\R^{d}\colon\dist\rbr{x,\Sigma}<\cellradius\delta},
\]
where $2\cellradius=\diam\cell=d^{1/2}$. Thus, $\Sigma+\delta\cell\subset\Sigma_{\delta}$
for any~$\delta\ge0$.

We will often use the notation~$\alpha\lesssim\beta$  (which is
the same as saying that~$\beta\gtrsim\alpha$) to mean that there
is a positive constant~$C$ depending only on some fixed parameters
(listed in Theorem~\ref{thm: Convergence and Approximation with 1st corrector}–Corollary~\ref{cor: Interior convergence for fractional derivative})
such that~$\alpha\le C\beta$. Finally, $\alpha\meet\beta$ and⋅$\alpha\join\beta$
are, respectively, the smaller and the larger of $\alpha$ and~$\beta$.
{"config": "arxiv", "file": "2006.05856/Sections/Section2.tex"}