\begin{document}
\title{\vspace{-1.2cm} \bf Uniform estimates for the canonical solution to the $\bar{\partial}$-equation on product domains  \rm}
\author{Robert Xin Dong\footnote{supported by AMS-Simons travel grant}, \, \, \quad Yifei Pan,  \quad \, \,Yuan Zhang\footnote{partially supported by NSF DMS-1501024}}
\date{\fontsize{13}{9}\selectfont \it with an appendix by Martino Fassina}
\maketitle

\begin{abstract}
We obtain uniform estimates for the canonical solution to $\bar \partial u=f$ on the Cartesian product of smoothly bounded planar domains, when $f$ is continuous up to the boundary.
This generalizes Landucci's result for the bidisc toward higher dimensional product domains.
\end{abstract}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnotetext{\hspace*{-7mm}
\begin{tabular}{@{}r@{}p{16.5cm}@{}}
& 2010 Mathematics Subject Classification. Primary 32A25; Secondary 32A26, 32W05\\
& Key words and phrases.
$\bar{\partial}$-equation, Bergman kernel, canonical solution, Green's function, $L^2$ minimal solution
\end{tabular}}

\section{Introduction}

The purpose of the paper is to study uniform estimates for the canonical solution to the $\bar \partial$-equation on the Cartesian product of  bounded planar domains with $C^2$ boundaries. Under the continuity assumption on the data, we prove the following theorem.

\begin{theorem}\label{main}
Let $\Omega:=D_1\times \cdots  \times D_n \subset \mathbb C^n$, $n\geq 2$, where each $D_j$ is a bounded planar domain with $C^2$ boundary. Then there exists a positive constant $C$ depending only on $\Omega$ such that for any $\bar\partial$-closed $(0, 1)$ form $ f$ continuous up to $\bar\Omega$,  the canonical solution  to  $\bar\partial u = f$ (in the sense of distributions) is continuous on $\Omega$ and satisfies 
$$
\|  u   \|_{ L^\infty(\Omega) }\le C\|  f\|_{ L^\infty(\Omega) }.
$$
\end{theorem}
  
The canonical solution is the unique one that is perpendicular to the kernel of $\bar \partial$, and is sometimes referred to as the $L^2$-minimal solution because it has minimal $L^2$-norm among others. 
The canonical solution exists on any bounded pseudoconvex domain in $\mathbb C^n$ whenever the datum is in $L^2$; 
however, uniform estimates for the solutions
do not exist in general.
Indeed, Berndtsson \cite{Be93} and Sibony \cite{Sib80} constructed examples of $\bar \partial$-closed $(0, 1)$-forms $f$ continuous on the closures
of weakly pseudoconvex domains in $\mathbb{C}^2$ and $\mathbb{C}^3$, respectively, which have no bounded solution to $\bar \partial u = f$. 
More strikingly, Forn\ae ss and Sibony \cite{FS93} constructed a smoothly bounded pseudoconvex domain in $\mathbb{C}^2$ whose boundary is strictly pseudoconvex except at one point, but uniform estimates for the solutions fail to hold.
 

\vspace{0.2cm}

On the other hand, on smoothly bounded strictly pseudoconvex domains  in $\mathbb{C}^n$, Henkin \cite{Hen70} and Grauert and Lieb \cite{G-L70} constructed integral solutions that satisfy the uniform estimates $\|u\|_{{\infty}} \leq C\|f\|_{\infty}$ for $C^{\infty}$ forms $f$.
Kerzman  \cite{Ker71} and later Henkin and Romanov  \cite{HR} extended their results and obtained the $\nicefrac{1} {2}$-H\"{o}lder estimates. See also the works of Lieb and Range \cite{LR}, Range \cite{R78, R86, R90, R20}, Range and Siu \cite{R-S72, R-S73}, as well as \cite{Ker73, K76, LTL93, Be97, Be01, CS11, SV14, GSS19}, and the references therein.
\vspace{0.2cm}

The unit polydisc $\mathbb D^n:=\mathbb D(0,1)^n$ is pseudoconvex with non-smooth boundary. Henkin in \cite{Hen71} (see also \cite{F-L-Zh11}) showed that there exist integral solutions with uniform estimates when the $(0,1)$ data are $ C^1$ up to $\overline{\mathbb D^2}$; Landucci in \cite{L75} extended Henkin's result  and obtained  bounded canonical solutions on the bidisc $\mathbb D^2$.
(See Appendix by Fassina at the end of the paper for an alternative proof of uniform estimates on polydiscs.)
Recently, Chen and McNeal \cite{CM18, CM19} and Fassina and the second author \cite{FP19} generalized Henkin's integral solutions \cite{Hen71} to higher dimensional product domains. In particular, in \cite{FP19} uniform estimates were proved when the data are smooth up to the boundaries. As remarked in \cite{CM18}, the integral solution there (and so in \cite{CM19, FP19}) is not canonical.

\vspace{0.2cm}
   
Theorem \ref{main} generalizes Landucci's result \cite{L75} from the bidisc toward higher dimensional product of general bounded planar domains with $C^2$ boundaries. The estimate in Theorem \ref{main} is optimal in the sense that  the supnorm on the left hand side cannot be replaced by any H\"older norm. In \cite[p. 310-311]{Ker71}, Kerzman and Stein constructed an example of a $\bar\partial$-closed $(0, 1)$ form $f$ that is $L^\infty$ on the bidisc but the equation $\bar\partial u =  f$ admits no H\"older solutions. 

\vspace{0.2cm}

 
For differentiable data, our canonical solution operator involves the derivatives of $f$ up to order $n-1$ and takes similar forms as in the inspiring works \cite{Hen71, L75,  CM18, CM19, FP19}.  To achieve uniform estimates, one needs to eliminate  those derivatives making use of the $\bar\partial$-closeness of the data. The crucial ingredients in our proof are the results of Barletta and Landucci \cite{BL91} as well as Kerzman \cite{Ker76} regarding planar domains, in addition to a refined decomposition of the canonical solution motivated by \cite{L75, FP19}. For 
continuous data, we weaken the regularity assumption by introducing a new canonical integral formula which involves no derivatives of $f$. Such type of formula was initially proposed in \cite{FP19} as a non-canonical solution candidate, and we secure such a proposal in Section 5. By a stability result on the canonical solution kernel,  we show for continuous data that our  formula gives a bounded solution that is  also $L^2$-minimal.

\vspace{0.2cm}

The organization of this paper is as follows.  In Section 2, we give a proof of the  uniform estimates for the canonical solutions on planar domains. In  Section 3, we investigate upper bounds for the derivatives of one dimensional canonical solution kernels.
In Section 4, we prove uniform estimates for the canonical solutions on product domains of dimensions two and higher, when the data are differentiable. In Section 5,  the regularity assumption is weakened to continuity, which proves the main theorem completely, and yields Corollary \ref{cor} on uniform estimates for the Bergman projection. 
Lastly in Section 6, we give proofs of upper bounds for Green's function and its derivatives based on \cite{Ker76}.

 


{
\paragraph{Acknowledgements.} 

\setlength{\parskip}{-1.5ex}
  
\small
 
We are  grateful to Emil Straube for sharing the unpublished note of Kerzman.}  
 
\section{One dimensional canonical solution to the $\bar{\partial}$-equation}
For a bounded domain $D$ in $ \mathbb C$ with $C^{2}$  boundary,  let $SH(D)$ be the set of subharmonic functions on $D$. The (negative) Green's function $g(z, w)$ with a pole at $w \in D$ is defined as
$$
g(z, w) = \sup \, \{ u(z):  u < 0,\, u \in SH(D),\, \limsup_{\zeta \to w} \, (u(\zeta) - \log |\zeta - w|) < \infty \}.
$$
Denote by $ G(z, w):\equiv - (2\pi)^{-1}  {g(z, w)}$  the positive Green's function. Let
$$
H(w, z):= \frac{1}{2\pi i (z-w)}
$$
stand for the universal Cauchy kernel on $D$. 

For any fixed $w \in D$ consider the Dirichlet problem: 
\begin{equation}\label{Dirichlet}
\begin{cases}
 \Delta L(z) = 0, & z \in D; \\
L(z)=H (w, z), & z \in bD.
\end{cases}
\end{equation}
Then by the Poisson formula one solves \eqref{Dirichlet} uniquely with a solution expressed as 
 \begin{equation} \label{L def}
L(w, z) \equiv L_w(z):= \frac{1}{\pi}\int_{ b D} \frac{1}{\zeta-w} \frac{\partial G(z, \zeta)}{\partial \zeta}d\zeta \left(= \frac{1}{-2\pi i}\int_{ b D} \frac{1}{\zeta-w} \frac{\partial G(z, \zeta)}{\partial \vec{n}_\zeta }ds_\zeta\right),
\end{equation}
which belongs to $ C^\infty( D)\cap  C^{1, \alpha}(\bar D)$ as a function of  $ z $ for any  $\alpha \in (0, 1)$ (see the proof of Proposition \ref{DD}). Here $\vec{n}$ is the unit outer normal vector and $ds$ is the arc length on $b D$.
Regarded as a function on $D\times D$, $L(w, z)$ is holomorphic with respect to all $w$ in $D$.

\vspace{0.15cm}

In \cite{BL91}, Barletta and Landucci rewrote the function $L$ by using Green's Theorem as 

\begin{equation} \label{L=H+I}
L(w,z) = H(w, z)  - \frac{1}{\epsilon\pi i}\int_0^{2\pi}G(w+\epsilon e^{it}, z)e^{-it}dt, \quad z\neq w,
\end{equation}
where $\epsilon$ can be chosen as any small number such that 
$$
0< \epsilon \le\epsilon (w, z):=2^{-1}{\min} \left \{ {|z-w|} , {\delta(w)}  \right \}.
$$

For any (0, 1)-form $ f(z) d\bar z$ such that $f \in   L^1 ( D)$, define the operator $\mathbf T$ as
\begin{equation}\label{c1}
\mathbf Tf(w):=\int_D S(w, z) { f} (z)  d\bar z\wedge dz,
\end{equation}
where for any fixed $w\in D$, 
\begin{equation}\label{sl}
    S(w, z): = L(w, z) -H(w, z). 
\end{equation} 
The classical Cauchy integral theory
and the holomorphy of $L$ with respect to $w$ imply that 
$$
\frac{\partial  }{\partial  \bar w } (\mathbf T f(w)) =\frac{\partial  }{\partial  \bar w } \left (\frac{1}{2\pi i}\int_{ D}\frac{ { f} (z)}{z-w} dz \wedge d\bar z \right) = { f}(w)
$$
weakly, so $\mathbf T$ is also a solution operator. 
 
 \vspace{0.15cm}
 
In his unpublished note \cite{Ker76}, Kerzman obtained estimates for Green's function on general $C^2$-smooth bounded domains in $\mathbb R^n, n\geq 2$. 

\begin{lem} [Kerzman \cite{Ker76}] \label{Kerzman-Green}

Let $D$ be a  bounded domain with $C^2$ boundary in $\mathbb C$, whose diameter is $d$. Let $\delta(\cdot)$ denote the distance to the boundary $bD$.
Then there exists a positive constant $C$ depending only on $D$ such that for $(w, z) \in \bar D \times \bar D \setminus \{z=w\}$,\\
a) \begin{equation}\label{G}
  G(w, z)\le   \frac{C \delta(w)}{|z-w|}\log\frac{d}{|z- w|},  \quad G(z, w)\le \frac{C \delta(z)}{|z-w|}\log \frac{d}{|z-w|};
\end{equation}
b)\begin{equation}\label{g2}
      G(w, z) \le   \frac{C \delta(z)\delta(w)}{|z-w|^2}\log \frac{d}{|z-w|}.
 \end{equation}
\end{lem}

\vspace{0.15cm}

The proofs of Kerzman's estimates for real dimensions $n\geq 3$ were presented by Krantz in \cite[Propositions 8.2.2, 8.2.6]{K01} with the permission of Kerzman. For the sake of completeness, in Section 6 we shall provide a proof to  Lemma \ref{Kerzman-Green}  on planar domains,  by imitating \cite{K01}. 
In \cite{BL91} Barletta and Landucci   estimated the canonical solution kernel $S$ by using Kerzman's estimates on Green's function. Their idea gives rise to the following result on  planar domains with $C^2$ boundaries.

 

\begin{pro}\label{|L|}

Let $D$ be a  bounded domain with $C^2$ boundary in $\mathbb C$, whose diameter is d. Then there exists a positive constant $C$ depending only on $D$ such that for $(w, z) \in   D \times \bar D \setminus \{z=w\}$,
\begin{equation}\label{S}
|S(w, z)|\le \frac{C }{|z-w|} \log \frac{2d}{|z-w|} ;   \quad \quad
|S(w, z)|  \le    \frac{C \delta(z)}{|z-w|^2}\log \frac{2d}{|z-w|}.
\end{equation}

\end{pro}

\begin{proof}
It suffices to estimate the last term in \eqref{L=H+I}. We  first note that  from the definition of $\epsilon$, 
\begin{equation}\label{comp}
| w+\epsilon  e^{it}- z  |\ge | w-z| -\epsilon \ge  \frac{|z-w|}{2}, \ \ \ \text{and}\ \ \ \delta( w+\epsilon  e^{it}  )\le \delta( w  )+\epsilon\le \frac{3\delta( w  )}{2}.\end{equation}

When $|z-w|\ge \delta(w)$, we have $\epsilon =  2^{-1}{\delta(w)} $. Thus from (\ref{G}) and (\ref{comp}),  
$$
\frac{1}{ \pi\epsilon}\left | \int_0^{2\pi}G(w+\epsilon e^{it}, z)dt   \right | \le \frac{6 C }{|z-w|}  \log \frac{2d}{|z-w|}.
$$
When $|z-w|\le \delta(w)$, $\epsilon =  2^{-1} {|z-w|} $. Since  $|G(w, z)| \le  \frac{1}{2\pi}\log\frac{d}{|z-w|}$  (see \eqref{G<} for instance), we have
$$
\frac{1}{\pi \epsilon} \left | \int_0^{2\pi}G(w+\epsilon e^{it}, z)dt  \right | \le \frac{2\pi ^{-1}}{|z-w|}  \log \frac{2d}{|z-w|}.
$$
Therefore, we get 
$$
|S(w, z)|\le \frac{ 2\max \{\pi ^{-1}, 3 C \} }{|z-w|}   \log \frac{ 2d}{|z-w|}.
$$  
For the second inequality, it suffices to make similar argument replacing  (\ref{G}) and \eqref{G<} by (\ref{g2}) and (\ref{G}), respectively. 

\end{proof}
 
 
 \vspace{0.15cm}
 
Let $\Omega$ be a bounded domain in $\mathbb C^n$. $A^2(\Omega) := L^2(\Omega) \cap \hbox{ker}(\bar \partial)$ represents the space of all square integrable holomorphic functions on  $\Omega$.  The Bergman projection  on $\Omega$, denoted by  $\mathbf P$, is the orthogonal projection of $L^2(\Omega)$ onto its closed subspace $A^2(\Omega)$. Let $K$ be the Bergman kernel such that for all $h \in L^2(\Omega)$ and $w \in \Omega$,
\begin{equation} \label{reproducing}
\mathbf P h(w) = \int_{\Omega} K(w, z) h(z) {d \nu_z},
\end{equation}
where $d\nu$ is the Lebesgue $\mathbb{R}^{2n}$ measure.
A  solution $u \in L^2(\Omega)$ to the $\bar\partial$-equation is  canonical  if $\mathbf P u =0$. 

On a planar domain $D$ with $C^2$ boundary, recall (see \cite{BL91, Bell}) that for $(z, w) \in D \times D $ off the diagonal, $K$ is related to Green's function $G$ below. 
\begin{equation*}
K(z, w)= {-4} \frac{\partial^2 G(z, w)} {\partial z \partial \bar w}.
\end{equation*}
Consequently, it holds that
\begin{equation}\label{LtoK}
 K(w, z) = \frac{ - i}{2\pi} \int_{\zeta\in b D}  \frac{K(\zeta, z) }{\zeta-w} d\zeta= \frac{2 i}{\pi} \int_{\zeta\in b D} \frac{\partial^2  G(z, \zeta)}{\partial \zeta \partial \bar z} \frac{1 }{\zeta-w} d\zeta= {2 i} \frac{\partial L(w, z)}{\partial \bar z},
 \end{equation}
 where $L(w, z)$ is expressed in \eqref {L=H+I}. 

\medskip
 
 
 Barletta and Landucci observed that $\mathbf T$ defined in \eqref{c1} gives the canonical solution to the $\bar\partial$-equation, and the uniform estimate of the solution was written as a remark in \cite[p. 103] {BL91} without proof. For data in the Lebesgue function spaces,  the result below is a consequence of Proposition \ref{|L|}. For abbreviation of  notations, we write $\|\cdot\|_p: = \|\cdot\|_{L^p(\Omega)}$, $1\le p\le \infty$ here and henceforth, when there is no confusion of domains.
 

 
\begin{pro} \label{dim 1} Let $D$ be a  bounded domain with $C^2$ boundary in $\mathbb C$. Then for any $f \in L^p(D),$ $p > 1$, it holds that $\mathbf Tf\in L^2(D)$ and $\mathbf T$ is the canonical solution operator for $\bar\partial u =fd\bar z$. Moreover,  
\begin{enumerate}[label=(\roman*)]

\item for any $ p\in [1,  2]$ and  $q \in (0,  \frac{2p}{2-p})$, there exists a positive constant $C=C(D, p, q)$ such that for any $f\in L^p(D) $, 
\begin{equation}\label{pq}
    \|\mathbf Tf\|_{q}\le C\|f\|_{p };
    \end{equation}
    
\item for any $p \in (2, \infty]$, there exists a positive constant $C=C(D, p)$ such that for any $f \in L^p (D)$,   \begin{equation}\label{pq2}
    \|\mathbf Tf\|_{\infty}\le C\|f\|_{p}.
    \end{equation}
    
\end{enumerate}
\end{pro}

\begin{proof}
By the first inequality in (\ref{S}), the kernel $S$ is bounded by a convolution type function in $ L^r(D)$ for any $r \in [1, 2)$. Young's convolution inequality then implies the boundedness of $\mathbf T$ from $L^p(D)$ into $L^q(D)$ whenever $p^{-1}+ r^{-1} = q^{-1}+1$ with $  p, q, r \in [1, \infty]$. (\ref{pq}) and (\ref{pq2}) thus follow.

It remains to show that the solution operator $\mathbf T$ is canonical.
For any $f \in L^p(D),$ $p > 1$, notice by (i) that  $\mathbf Tf\in L^2(D)$ so $\mathbf P\mathbf T f $ is well defined. By the Cauchy--Pompeiu formula, \eqref{Dirichlet} and \eqref{LtoK},
\begin{align*}
    \mathbf P \mathbf T f  (\zeta) 
        =&\int_D { f} (z) \left(\int_D K(\zeta, w)L(w, z) d\nu_w + \frac{1}{2\pi i}\int_D \frac{K(\zeta, w)}{w-z}d\nu_w\right) d\bar z\wedge dz\\
    =&\int_D {  f} (z) \left(L(\zeta, z) -\frac{1}{2\pi i}\int_D \frac{\partial L(\zeta, w)}{\partial \bar w} \frac{1}{w-z} {dw\wedge d \bar w}  \right )  d\bar z\wedge dz\\
    =&\int_D {  f} (z)\left ( \frac{1}{2\pi i} \int_{bD} \frac{ L(\zeta, w)}{w-z} dw \right)  d\bar z\wedge dz\\
    =& \int_D {  f} (z)  \frac{1}{(2\pi i)^2} \frac{1}{z- \zeta}  \left ( \int_{bD}  \frac{ dw}{w-z } - \int_{bD}  \frac{ dw}{w-\zeta}  \right)  d\bar z\wedge dz =0,
\end{align*}
which means that $(\mathbf T f  ) \perp A^2(D)$ and the proof is complete.

\end{proof} 
  
  
As a side product of Proposition \ref{dim 1}, the corollary below on the Bergman projection follows immediately from the fact that for any $u \in L^2(D) \cap  \text{Dom}(\bar\partial)$,
\begin{equation*}
  \mathbf Pu=u- \mathbf T \bar\partial u.
   \end{equation*}
 
\begin{cor} \label{cor=dim1}
Let $\mathbf P$ be the Bergman projection over a  bounded domain $D$ with $C^2$ boundary in $\mathbb C$. Then there exists a positive constant $C$ depending only on $D$ such that for any $u\in W^{1, \infty}(D)$ 
it holds that
$$
    \|u- \mathbf Pu\|_{ \infty  }\le  C  \|   \bar\partial u  \|_{ \infty }  ;   \quad \quad
  \|\mathbf Pu\|_{ \infty  }\le   \|u\|_{  \infty  }  + C\|   \bar\partial u  \|_{ \infty }.
$$

\end{cor}
 

 \section{Derivatives of one dimensional canonical solution kernel}  

  

\vspace{0.15cm}

Let $D \subset \mathbb C$ be a bounded domain with $C^{2}$  boundary. The goal of the section is to estimate the gradient of the canonical solution kernel $S$ on $D$. In view of \eqref{L=H+I} and \eqref{sl}, we shall need bounds for the derivatives of Green's function as follows. Here  $\nabla G$ represents the first order derivatives of $G$ with respect to either $z$ or $w$ variable, while $\nabla_z G$ represents the first order derivatives of $G$ with respect to $z$ variable.  


\begin{lem}  \label{Kerzman-Green2} Under the same assumptions as in Lemma \ref{Kerzman-Green}, it holds that for all $(w, z) \in  D \times \bar D \setminus \{z=w\}$,
 \begin{equation}\label{Gd}
     |\nabla G( w, z)|\le \frac{C}{|z-w|}\log\frac{d}{|z-w|}; \quad \quad   |\nabla_z G(w, z)| \le \frac{C\delta(w)}{|z-w|^2}\log \frac{d}{|z-w|}.
    \end{equation}
\end{lem}

\medskip


 Lemma \ref{Kerzman-Green2} is essentially an application of the mean value theorem for harmonic functions to Lemma \ref{Kerzman-Green}, and we will give its proof in Section 6.  Such  estimates for real dimensions bigger than two can be found in the comment after Theorem 1.2.8 in \cite{Kenig} without proof. Making use of a similar approach as in  Proposition \ref {|L|}, we obtain the following estimate for the derivatives  of the canonical solution kernel. 


\begin{pro}\label{kkd}
Under the same assumptions as in Proposition \ref{|L|}, it holds that 
\begin{equation*}
|\nabla_z S(w, z) |\le \frac{C }{|z-w|^{2}}  \log \frac{2d}{|z-w|}.
\end{equation*}
\end{pro}


\begin{proof}
Fix $w\in D$ and let $z_0  \in \bar D \setminus \{w\}$. By \eqref{L=H+I}, 
 

\begin{equation} \label{K-est}
  \nabla_z S(w, z_0 )   =   \left. \nabla_z  \left(\frac{i}{\epsilon\pi    }\int_0^{2\pi}G(w+\epsilon e^{it}, z)e^{-it}dt \right) \right |_{z=z_0},
  \end{equation}
where $\epsilon$ can be any positive number no bigger than $
\epsilon (w, z):=2^{-1}{\min} \left \{ {|z-w|} , {\delta(w)}  \right \}>0$.  
Set $\epsilon_0: = 2^{-1}\epsilon (w, z_0) = 4^{-1}{\min} \left \{ {|z_0-w|} , {\delta(w)}  \right \}$. 
For any point $z$ sufficiently close to $z_0$, the continuity of $\epsilon( w, z)$ in $z$ implies that $\epsilon_0\le \epsilon (w, z) $. Then on the right hand side of \eqref{K-est}, $\epsilon$   can be replaced by $\epsilon_0$, as $z$ approaches $z_0$. So
\begin{equation*}
  \nabla_z S(w, z_0 ) =  \frac{i}{\epsilon_0 \pi }  \int_0^{2\pi}  \left. \nabla_z     G(w+ \epsilon_0 e^{it}, z)\right |_{z=z_0} e^{-it}dt.
\end{equation*}
When $\epsilon_0  =   4^{-1} \delta(w) $, by the second part of \eqref{Gd} and (\ref{comp}),  we have
$$ 
|\nabla_z S(w, z_0 )|  \le   \frac{ 4}{ \delta(w) \pi  } \left| \int_0^{2\pi}   \left.  \nabla_z     G(w+\epsilon_0  e^{it}, z)\right |_{z=z_0}  e^{-it}dt \right| \le     \frac{ 48C  }{|z_0-w|^2}\log \frac{2d}{|z_0-w|};
$$
when $\epsilon_0  = 4^{-1} {|z_0-w|}$, the first part of \eqref{Gd} gives the desired estimate
$$
 |\nabla_z S(w, z_0 )|  \le  \frac{32C }{   {|z_0-w|^2}    }\log \frac{2d}{|z_0-w|}.
$$
Since $z_0$ is arbitrary, the proof is complete.

\end{proof}

Recall that the Bergman kernel $K$ is  associated with a derivative of the canonical solution kernel $S$ in terms of (\ref{LtoK}).  As a consequence of this and Proposition \ref{kkd}, we have an estimate of $K$ as follows.  
\begin{cor}
Under the same assumptions as in Proposition \ref{|L|}, it holds that 
$$
|K (w, z)|\le \frac{C }{|z-w|^{2}}  \log \frac{d}{|z-w|}.
$$
\end{cor}
 
\medskip

 The stability for the  Bergman kernel  is already known in literature. See \cite{Boas, CZ} for instance.
Below we study the stability of the canonical solution kernel $S$ and its gradient under the mild variation of the underlying domains. The following locally uniform convergence of $S$ is proved using PDE theory and plays a key role in reducing the regularity of the data to continuity in Section 5. 

     
 
\begin{pro}\label{DD}
Let $D$ be a bounded domain in $\mathbb C$ with $C^2$ boundary. Let $\{D^l\}_{l=1}^\infty$ be an exhausting family of open subsets of $D$ with $C^2$ boundaries such that \\
a) $\bar D^l\subset D^{l+1}$;\\
 b) $h^l: \bar D\rightarrow \bar D^l$ is a $C^2$ diffeomorphism with  $\lim \limits_{l\rightarrow \infty}\|h^l-id\|_{C^2(D)}= 0$.\\
Let  $S$ and $S^l$ be the kernels of the canonical solution operators for the $\bar\partial$-equations on $D$ and $D^l$, respectively.
Then for each compact subset $\kappa \Subset   D$, 
$S^l(w, h^l(z))$ and $\nabla S^l(w, h^l(z)) $ converge uniformly on $\kappa\times \bar D \setminus  \{z=w\}$ to $S(w, z)$ and $\nabla S(w, z) $, respectively.
\end{pro}

\begin{proof}
Assume $\kappa\Subset D^l$ for all $l$, and we will first prove that 
\begin{equation}\label{stable}
 \lim \limits_{l\rightarrow \infty}\sup \limits_{(w, z)\in \kappa\times \bar D} \left  |S^l(w, h^l(z)) - S(w, z) \right | = \, 0.
\end{equation} 
Let $L_w$ and $ L^l_w $ be the solutions to \eqref{Dirichlet} on $D$ and $D^l$, respectively. 
Since both $L_w$ and $L^l_w$ are harmonic on $D^l$, by the Maximum Principle we know that for any $w\in \kappa$,
\begin{equation}\label{DD6}
\sup_{z\in \bar D}|L^l_w(h^l(z))-L_w(h^l(z))| = \sup_{\zeta\in \bar D^l}|L^l_w(\zeta)-L_w(\zeta)|\le \sup_{\zeta\in bD^l}|{H(w, \zeta)-L_w(\zeta)}|. \end{equation}
Since $w$ belongs to $\kappa$,  the universal Cauchy kernel $H (w, \cdot)\in C^\infty(bD)$ and has a uniform $C^2(bD)$ norm that is independent of $w$. By the $L^{p}$ theory in PDE, $L_w\in W^{2, p}(D)$ for all $p>1$ with $\|L_w\|_{W^{2, p}(D)}\le C$ independent of $w$. Thus $L_w\in C^{1, \alpha}(\bar D)$ for all $0<\alpha<1$ by the Sobolev embedding theorem with 
\begin{equation}\label{lll}
    \|L_w\|_{C^{1, \alpha}(D)}\le C
\end{equation}  independent of $w$.  Next, for each   $\zeta\in bD^l$, write $\zeta = h^l(z)$  for some $z\in bD$.  In view of the boundary condition on $L_w$, we know that 
 $$
    |H(w,\zeta )-L_w(\zeta)|   
        \le    |H(w, h^l(z))-H(w, z)|+  |L_w(z)-L_w(h^l(z))|  \le   (C +\|L_w\|_{C^{\alpha}(D)})| h^l(z)-z|^\alpha. 
    $$
By (\ref{lll}) and the assumption of $h^l$, one infers that
\begin{equation*}
    \sup_{(w, \zeta)\in \in \kappa\times bD^l}  |H(w, \zeta)-L_w(\zeta)|\rightarrow 0,
\end{equation*}
as $l\rightarrow \infty$. Furthermore, by (\ref{DD6}) it follows that
$$
    \sup_{(w,z)\in \kappa\times \bar D}|L^l_w(h^l(z))-L_w(h^l(z))| \rightarrow 0,
 $$
which yields \eqref{stable} from  (\ref{sl}).

Adapt the previous argument for $H$, $L_w$ and $ L_w^l$ to $\nabla H$, $\nabla L_w$ and $\nabla L_w^l$, respectively. One proves an identity similar to \eqref{stable} for $\nabla S$ by virtue of the fact that $\nabla L_w\in C^\alpha(D)$ and thus completes the proof.
 
\end{proof}

 

  
\section{Proof of the main theorem under differentiability}
 
In this section we prove the main Theorem \ref{main} on uniform estimates for the canonical solution to $\bar \partial u =f$, under the assumption that $f$ is $C^{n-1}$ up to the boundary of product domains. More precisely, we shall prove 


  
 \begin{theorem} \label{C1}
Let $\Omega:=D_1\times \cdots \times D_n$, $n\geq 2$, where each $D_j$ is a bounded planar domain with $C^2$ boundary. Then there exists a positive constant $C$ depending only on $\Omega$ such that for any $\bar\partial$-closed form $ f \in C^{n-1}_{(0,1)} (\bar\Omega)$,  the canonical solution to  $\bar\partial u = f$ satisfies 
\begin{equation*}
\|  u   \|_{ \infty }\le C\|  f\|_{ \infty },
\end{equation*}
where $\| \cdot  \|_{\infty}$ is the essential supnorm on $\Omega$.
\end{theorem}

 
 
 
 
 In this first subsection, we study the canonical solution to the $\bar \partial$ equation on the Cartesian product of planar domains, and construct the canonical solution operator, inspired by the recent works \cite{CM18, CM19, FP19}. One dimensional canonical solution operator is given together with uniform estimates in Section 2. Since the case of $n=2$ carries the precise idea yet without involving too much technical computations,  we prove Theorem \ref{C1} for this case in the second subsection. After that, we deal with the general case of arbitrary $n$ in the third subsection.  
 
\vspace{0.15cm}

The crucial idea in the proof is to eliminate the derivative terms in (\ref{formula1}) by the $\bar\partial$-closeness of the data, making use of a similar  idea as in \cite{FP19, L75}. In the last two subsections, after conducting  a refined decomposition of the solution kernel and eliminating the derivatives of the data in (\ref{formula1}), we shall verify each decomposed portion still maintains the integrability. The following elementary  geometric mean inequality  serves the role well. 

\begin{lem} \label{ele}
Let $x_j, 1\le j\le n$ be $n$ nonnegative numbers. Let $  \alpha_j \in [0,  1]$, $1\le j\le n$. Then it holds that
$$
  \prod \limits_{j=1}^n x_j^{\alpha_j}   \le  
\begin{cases}
\quad   \sum \limits _{j=1}^n x_j &  \text{whenever \,}   \sum\limits_{j=1}^n\alpha_j =1; \\ 
 \quad \sum \limits_{j=1}^n\prod \limits_{(j \ne)m=1}^n x_m &  \text{whenever \,}   \sum\limits_{j=1}^n\alpha_j =n-1. 
\end{cases}
 $$
\end{lem}

Due to the special geometry of product domains, we are not sure whether the above method work for general pseudoconvex domains in $\mathbb C^n$. For the rest of the paper, $C$ will represent a constant depending only on each domain. It may be different at various contexts.

 

\subsection{Canonical solution formula on the product of planar domains}


A (non-canonical) solution operator, which consists of compositions of the solid Cauchy integral along each slice, first appeared in \cite{CM19, FP19}. It is our observation that, to construct the canonical solution operator on $\Omega$, one only needs to replace each of slice-wise Cauchy operators by the corresponding  canonical solution operator.

  
   

\begin{theorem} \label{diff}
Let $\mathbf f=\sum \limits_{j=1}^nf_jd\bar z_j$ be a $\bar\partial$-closed $(0,1)$ form on $\Omega$ such that
$f_j  \in C^{n-1} (\bar\Omega)$.
 Then the canonical solution $\mathbf T\mathbf f$ to  $\bar\partial u =\mathbf f$ is represented as
\begin{equation}\label{formula1}
  \mathbf T\mathbf f: =  \sum \limits_{s=1}^{n} (-1)^{s-1}\sum \limits_{1\le i_1<\cdots< i_s\le n} \mathbf T_{i_1}\cdots \mathbf T_{i_s}(\frac{\partial^{s-1} f_{i_s}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}}), 
\end{equation}
where $\mathbf  T_j$ is the canonical solution operator over $D_j$. 
\end{theorem}

\begin{proof} Firstly, the operator $\mathbf T$ in (\ref{formula1}) is well defined over $C^{n-1} (\bar\Omega)$, with its supnorm bounded by the $C^{n-1} (\bar\Omega)$ norm of the data in view of the estimate in Proposition \ref{dim 1}.
     
     \vspace{0.2cm}
     
Secondly, denote by $\mathbf P_j$ and $\mathbf  P$  the Bergman projection operator over $D_j$ and  $\Omega$, respectively. Then
  $ \mathbf  P= \mathbf P_1 \cdots \mathbf P_n $  and each term in \eqref{formula1} contains some $\mathbf T_j$ while $\mathbf P_j\mathbf T_j=0$ by the proof of Proposition \ref{dim 1}. Repeated application of Fubini's theorem implies that $\mathbf P\mathbf T\mathbf f=0$, i.e., $$(\mathbf T f) \perp A^2(\Omega).$$


Thirdly, we show that $\bar\partial \mathbf T\mathbf f = \mathbf f$ using a direct computation, similar to \cite{FP19}. The $\bar\partial$-closeness of $\mathbf f$ implies that (\ref{formula1}) is symmetric with respect to the roles of $z_j$ and $z_n$, so we only need to prove $\bar\partial_n \mathbf T\mathbf f = f_n$. Isolate terms containing $\mathbf T_n$ in (\ref{formula1}) and rewrite
\begin{equation*}
 \begin{split} \mathbf T\mathbf f= & \sum_{s=1}^{n-1} (-1)^{s-1}\sum_{1\le i_1<\cdots< i_s\le n-1} \mathbf T_{i_1}\cdots \mathbf T_{i_s}(\frac{\partial^{s-1} f_{i_s}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}}) \, +\\ &+ \mathbf T_n f_n +  \sum_{s=2}^{n} (-1)^{s-1}\sum_{1\le i_1<\cdots< i_{s-1}\le n-1, i_s=n} \mathbf T_{i_1}\cdots \mathbf T_{i_{s-1}}\mathbf T_{n}(\frac{\partial^{s-1} f_{n}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}}).
 \end{split}
\end{equation*}
Applying $\bar\partial_n$ to each term in the above expression, we obtain by $\bar\partial_n \mathbf T_n =id$ that
\begin{equation*}
    \begin{split}
  \bar\partial_n \mathbf T\mathbf f =&    \sum_{s=1}^{n-1} (-1)^{s-1}\sum_{1\le i_1<\cdots< i_s\le n-1} \mathbf T_{i_1}\cdots \mathbf T_{i_s}(\frac{\partial^{s} f_{i_s}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}\partial\bar z_n}) \\ &+ f_n +  \sum_{s=2}^{n} (-1)^{s-1}\sum_{1\le i_1<\cdots< i_{s-1}\le n-1} \mathbf T_{i_1}\cdots \mathbf T_{i_{s-1}}(\frac{\partial^{s-1} f_{n}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}}).
    \end{split}
\end{equation*}

Lastly, since  $\mathbf f$ is $\bar\partial$-closed and
$$
\frac{\partial^{s} f_{i_s}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}\partial\bar z_n} = \frac{\partial^{s} f_{n}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}\partial\bar z_s},
$$
 we replace $s$   by $s-1$ in the last summation above to get 
\begin{align*}
  \bar\partial_n \mathbf T\mathbf f = &    \sum_{s=1}^{n-1} (-1)^{s-1}\sum_{1\le i_1<\cdots< i_s\le n-1} \mathbf T_{i_1}\cdots \mathbf T_{i_s}(\frac{\partial^{s} f_{n}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}\partial\bar z_s})\\ 
  &+ f_n+  \sum_{s=1}^{n-1} (-1)^{s}\sum_{1\le i_1<\cdots< i_{s}\le n-1} \mathbf T_{i_1}\cdots \mathbf T_{i_{s-1}}(\frac{\partial^{s} f_{n}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s}}}) =  f_n.
\end{align*}

\end{proof}


 

\subsection{Uniform estimate for $n=2$}
 
When $n=2$, $\mathbf T \mathbf f = \mathbf T_1f_1+\mathbf T_2f_2-\mathbf T_1\mathbf T_2(\mathcal Df)$, where $\mathcal Df = \frac{\partial f_1}{\partial \bar z_2} = \frac{\partial f_2}{\partial \bar z_1}$. By Proposition \ref{dim 1}, we only need to estimate $\|\mathbf T_1\mathbf T_2(\mathcal Df)\|_{\infty}$. Let $|{w}-z|^2: = |{w}_1-z_1|^2+|{w}_2-z_2|^2$  and $ dV_z:=d\bar z_1\wedge dz_1\wedge d\bar  z_2\wedge dz_2$. Let $K_j$ and $S_j$ be the Bergman kernel and canonical solution kernel, respectively, on $D_j$ for $j=1, 2$.

\begin{proof} [Proof of Theorem \ref{C1} ($n=2$)]

Making use of  the trivial identity
\begin{equation*}
    S_1 S_2  =  \frac{S_1 S_2 |{w}_1-z_1|^2}{|{w}- z|^2} +\frac{S_1 S_2 |{w}_2-z_2|^2}{|{w}- z|^2},
    \end{equation*}
    we rewrite $ \mathbf T_1\mathbf T_2(\mathcal Df)$ as 
\begin{equation*}
\begin{split}
 \mathbf T_1\mathbf T_2(\mathcal Df)({w})=
&\int_{D_1 \times D_2}{\mathcal D (f)(z_1, z_2)}{S_1({w}_1, z_1)S_2({w}_2,z_2)}dV_z\\
=&\int_{D_1 \times D_2}{\frac{\partial f_2(z)}{\partial \bar z_1}}\frac{S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_1-z_1|^2}{|{w}- z|^2}dV_z\\
&+\int_{D_1 \times D_2}{\frac{\partial f_1(z)}{\partial \bar z_2}}\frac{S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_2-z_2|^2}{|{w}- z|^2}dV_z =: I_1+I_2.
\end{split}
\end{equation*}
We shall only estimate  $\|I_2\|_{\infty}$, since $\|I_1\|_{\infty}$ is handled similarly due to symmetry. 

     
     \vspace{0.2cm}
     
We assert that
\begin{equation}\label{stok1}
          \begin{split}
          I_2 =&\int_{{  D}_1} \int_{b{  D}_2} \frac{f_1(z) S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_2-z_2|^2}{|{w} - z|^2} dz_2\wedge d\bar z_1\wedge dz_1  -\int_{D_1\times   D_2} f_1( z)S_1({w}_1, z_1)  \\
          &
 \cdot \Big( \frac{1}{2i}K_2({w}_2, z_2) \frac{|{w}_2- z_2|^2}{|{w}-z|^2} - \frac{ S_2({w}_2, z_2)({w}_2- z_2)|{w}_1-z_1|^2}{|{w}- z|^4}\Big)dV_z.\end{split}
   \end{equation}
 
     
To see this, we write $  R^\epsilon_2: =   D_2 \setminus B({w}_2, \epsilon)$, where $B({w}_2, \epsilon)$ is a disc centered at ${w}_2$ with radius $\epsilon<<1$. Then by Stokes' Theorem,
\begin{equation} \label{stok}
    \begin{split}
   &\int_{D_1\times   R^\epsilon_2}\frac{\partial f_1( z)}{\partial \bar z_2}  \frac{S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_2-z_2|^2}{|{w}- z|^2} dV_z \\
   =&  \int_{{  D}_1} \int_{b{  D}_2}  \frac{f_1(z) S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_2-z_2|^2}{|{w} - z|^2}dz_2\wedge d\bar z_1\wedge dz_1 \\
   &- \int_{{  D}_1}\int_{b{B({w}_2, \epsilon)}}  \frac{f_1(z) S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_2-z_2|^2}{|{w} - z|^2} dz_2\wedge d\bar z_1\wedge dz_1\\
&-\int_{{  D}_1\times R^\epsilon_2} f_1( z)S_1({w}_1, z_1) \Big( {\frac{1}{2i} K_2({w}_2, z_2)}\frac{|{w}_2- z_2|^2}{|{w}-z|^2} - \frac{ S_2({w}_2, z_2)({w}_2- z_2)|{w}_1-z_1|^2}{|{w}- z|^4}\Big)dV_z.    \end{split}
\end{equation}
Notice that by (\ref{S}),  for any $\sigma>0$, for $ (z_1, z_2)\in (D_1\setminus \{ z_1 = {w}_1\}) \times ( D_2\setminus \{ z_2 = {w}_2\})$,
$$ \frac{|S_1({w}_1, z_1) S_2({w}_2,z_2)||{w}_2-z_2|^2}{|{w}- z|^2}\le \frac{C|{w}_2-z_2|^{1-\sigma}}{|{w}_1-z_1|^{1+\sigma}|{w}-z|^{2}}\le  \frac{C}{|{w}_1-z_1|^{1+\sigma}|{w}_2-z_2|^{1+\sigma}}, $$
which belongs to $ L^1(D_1\times D_2)$ by choosing   $\sigma$ less than 1. 
Because $f_1\in C^{1}(\bar\Omega)$,  the left hand side of (\ref{stok}) approaches $ I_2  $ as $\epsilon\rightarrow 0$ by Lebesgue's Dominated Convergence Theorem.

 On the other hand, in Lemma \ref{ele} choosing $\alpha_1=1/4$, $\alpha_2=3/4$ and $x_j=|{w}_j-z_j|$, for $j=1,2$, we know that 
 $|{w}-z|^2\ge C |{w}_1-z_1|^{\frac{1}{2}}|{w}_2-z_2|^{\frac{3}{2}}$. So for any $\sigma>0$, 
 \begin{equation} \label{n1}\frac{|S_1({w}_1, z_1) S_2({w}_2,z_2)||{w}_2-z_2|^2}{|{w}- z|^2}\le    \frac{C}{|{w}_1-z_1|^{\frac{3}{2}+\sigma}|{w}_2-z_2|^{\frac{1}{2}+\sigma}}\in L^1_{loc}(\mathbb C_{z_1}\times \mathbb R_{z_2})  \end{equation}
when choosing $\sigma$ small (say, less than $\frac{1}{2}$). The second term on the right hand side of (\ref{stok}) satisfies
 \begin{equation*}
 \begin{split}
     \bigg|\int_{{  D}_1}  \int_{b{B({w}_2, \epsilon)}}  \frac{f_1(z) S_1({w}_1, z_1)S_2({w}_2,z_2)|{w}_2-z_2|^2}{|{w} - z|^2} dz_2\wedge d\bar z_1\wedge dz_1\bigg|
\le \int_{b{B({w}_2, \epsilon)}} \frac{C|dz_2|}{|{w}_2-z_2|^{\frac{1}{2}+\sigma}} \rightarrow 0,
\end{split}
 \end{equation*} 
 as $\epsilon\rightarrow 0$.
For the third term on the right hand side of (\ref{stok}), by the fact that $|{w}-z|^2\ge C|{w}_1-z_1|^{\frac{1}{2}}|{w}_2-z_2|^{\frac{3}{2}}$,  
we know from (\ref{S}) and Proposition \ref{kkd} that there exists a small $\sigma$ such that
 \begin{equation}\label{n2} \begin{split}
|S_1({w}_1, z_1)   {K_2({w}_2, z_2)}| \frac{|{w}_2- z_2|^2}{|{w}-z|^2}     & \le  \frac{C}{|{w}_1-z_1|^{1+\sigma}|{w}_2-z_2|^{\sigma}|{w}-z|^{2}}\\
 &  \le  \frac{C}{|({w}_1-z_1) ({w}_2-z_2)|^{\frac{3}{2}+\sigma}}\in L^1(D_1\times D_2). 
 \end{split}
 \end{equation}
 Similarly, using $|{w}-z|^4\ge C |{w}_1-z_1|^{\frac{5}{2}}|{w}_2-z_2|^\frac{3}{2}$ and letting $\sigma$ less than $\frac{1}{2}$, we get
 \begin{equation}\label{n3}\begin{split}\frac{ |S_1({w}_1, z_1) S_2({w}_2, z_2) ({w}_2- z_2)| |{w}_1-z_1|^2}{|{w}- z|^4}
 \le& \frac{C|{w}_1-z_1|^{1-\sigma}}{|{w}_2-z_2|^{\sigma}|{w}-z|^4}\\
 \le & \frac{C}{|({w}_1 - z_1) ( {w}_2 - z_2)|^{\frac{3}{2}+\sigma}}\in L^1(D_1\times D_2).\end{split} \end{equation}
As $\epsilon\rightarrow 0$, the last line on the right hand side of (\ref{stok}) approaches $$\int_  {D_1\times D_2} f_1( z)S_1({w}_1, z_1)\Big( { \frac{K_2({w}_2, z_2)}{2i} } \frac{|{w}_2- z_2|^2}{|{w}-z|^2} - \frac{ S_2({w}_2, z_2)({w}_2- z_2)|{w}_1-z_1|^2}{|{w}- z|^4}\Big)dV_z,$$
from which the assertion (\ref{stok1}) follows immediately. Furthermore, by the proof of the assertion, in particular by (\ref{n1}-\ref{n3}), we obtain
 \begin{equation*}
     \|I_2\|_{\infty}\le C \|\mathbf f\|_{\infty},
 \end{equation*}
so the proof of Theorem \ref{C1} is complete for $n=2$, when  the data are $C^1$ up to the boundary.  

\end{proof} 

 

\subsection{Uniform estimate for arbitrary $n$}

For the rest of the paper, for convenience, we will suppress the corresponding measure element from a given integral. We prove by induction on $n$ that for all $1\le i_1<\cdots< i_s\le n$,
\begin{equation} \label{induction}
    \big\|\mathbf T_{i_1}\cdots \mathbf T_{i_s} \left (\frac{\partial^{s-1} f_{i_s}}{\partial \bar z_{i_1}\cdots \partial\bar z_{i_{s-1}}} \right )\big\|_{\infty}\le C \|\mathbf f\|_{\infty}.
\end{equation}
 Assuming \eqref{induction} is true for $n=k-1, k\ge 3$, we  shall verify that \begin{equation}\label{indu1}
  \|\mathbf T_1\cdots \mathbf T_k (\mathcal D^{k-1} \mathbf f)\|_{\infty} =  \left \| \int  _{  D_1 \times\cdots\times   D_k}\mathcal D^{k-1} (\mathbf f)(z)\prod_{j=1}^k S_j({w}_j, z_j) \right \|_{\infty}\le C \|\mathbf f\|_{\infty},
\end{equation}
where $\mathcal D^{k-1} \mathbf f := \frac{\partial^{k-1} f_1}{\partial \bar z_2\cdots  \partial \bar z_k} = \frac{\partial^{k-1} f_2}{\partial \bar z_1\partial \bar z_3=\cdots  \partial \bar z_k} \cdots = \frac{\partial^{k-1} f_k}{\partial \bar z_1\cdots  \partial \bar z_{k-1}}$. The remaining cases are done by symmetry.
 Letting 
 \begin{equation} \label{H}
H : =  \sum_{j=1}^k \prod_{m=1, m\ne j}^{k} |{w}_m- z_m|^2,
 \end{equation}  
 we write
\begin{equation*}
   \mathcal D^{k-1} (\mathbf f)({w})\prod_{j=1}^k S_j({w}_j, z_j) = \sum_{j=1}^k H^{-1}{\prod_{l=1}^k S_l({w}_l, z_l)\prod_{m=1, m\ne j}^{k} |{w}_m- z_m|^2} \frac{\partial^{k-1} f_j(z)}{\partial \bar z_1\cdots \partial \bar z_{j-1}\partial \bar z_{j+1} \cdots\partial \bar z_{k}}.
\end{equation*}
Due to the symmetry again, to prove (\ref{indu1}), it suffices to show
\begin{equation}\label{indu}
     \left\| \int_{  D_1 \times\cdots\times   D_k}  e({w}, z)\frac{\partial^{k-1} f_k({w})}{\partial \bar z_1\cdots \partial \bar z_{k-1}}  \right\|_{\infty}\le C \|\mathbf f\|_{\infty},
\end{equation}
where $$
    e({w}, z): = H^{-1}   {\prod_{l=1}^k S_l({w}_l, z_l)\prod_{m=1}^{k-1} |{w}_m- z_m|^2}.
$$

Before establishing estimates of the derivatives for the function $e({w}, z)$ defined above, the following lemma on $H$ defined in \eqref{H} will be needed. This part is parallel to (5.10) in \cite{FP19}.
 

\begin{lem}\label{ap}
For $ (z_1, \cdots, z_k)\in (D_1\setminus \{z_1={w}_1\}) \times\cdots\times   (D_k\setminus \{ z_k = {w}_k\})$, it holds that
$$
\frac{\partial^m }{\partial \bar z_1\cdots\partial \bar z_m}  \left( \frac{ \prod_{j=1}^{k-1} |{w}_j- z_j|^2 }{H}\right) = \frac{m! |{w}_k-z_k|^{2m}}{H^{m+1}}  \prod_{j=1}^m({w}_j-z_j)|{w}_j-z_j|^{2(m-1)}  \prod_{j=m+1}^{k-1}|{w}_j-z_j|^{2(m+1)}.
$$
\end{lem}

\begin{proof}
We prove it by induction on $m$. Since $m=1$ is trivial, assuming $m=l$ is true, by a straightforward computation  we know that when $m=l+1$,
 $$
 \frac{\partial^{l+1} }{\partial \bar z_1\cdots\partial \bar z_l\partial \bar z_{l+1}} \left( \frac{ \prod_{j=1}^{k-1} |{w}_j- z_j|^2 }{H}\right) ={\frac{(l+1)!}{ H ^{l+2}} \prod_{j=1}^{l+1}({w}_j-z_j)|{w}_j-z_j|^{2l} \prod_{j=l+2}^{k-1}|{w}_j-z_j|^{2(l+2)} } {|{w}_k-z_k|^{2(l+1)}}. 
 $$


\end{proof}


 

We now estimate the derivatives of  $e(w, z)$ point-wisely, which is the key to the proof of (\ref{indu}).

\begin{pro}\label{ee} Let $\{i_1, \cdots, i_m, j_1, \cdots, j_{k-1-m}\}$ be a permutation of $\{1, \cdots, k-1\}$. Then for any $\sigma>0$, for $ (z_1, \cdots, z_k)\in (D_1\setminus \{z_1={w}_1\}) \times\cdots\times   (D_k\setminus \{ z_k = {w}_k\})$,
$$
\left|\frac{\partial^m  e({w}, z)}{\partial \bar  z_{i_1}\cdots\partial\bar z_{i_m}}\right|
     \le  {C} |{w}_k- z_k|^{-\frac{3}{2}- \sigma} { \prod_{l=1}^m |{w}_{i_l}- z_{i_l}|^{-2 +\frac{1}{2(k-1)}-\sigma}\prod_{l=1}^{k-1-m} |{w}_{j_l}-z_{j_l}|^{-1 + \frac{1}{2(k-1)}-\sigma} }.
$$
\end{pro}
 

\begin{proof}
Due to symmetry, we shall only estimate $\frac{\partial^m  e({w}, z)}{\partial \bar  z_{1}\cdots\partial\bar z_{m}}$, for $m\le k-1$. Namely, we prove
\begin{equation}\label{hh}
    \left|\frac{\partial^m  e({w}, z)}{\partial \bar  z_{1}\cdots\partial\bar z_{m}}\right|\le   C  |{w}_k- z_k|^{-\frac{3}{2}-\sigma}  {(\prod_{l=1}^m |{w}_{l}- z_{l}|^{-2 +\frac{1}{2(k-1)}-\sigma})(\prod_{l=m+1}^{k-1-m} |{w}_{l}-z_{l}|^{-1 + \frac{1}{2(k-1)}-\sigma}) }.
\end{equation}
Making use of Lemma \ref{ap}, (\ref{S}) and Proposition \ref{kkd}, one sees for any $\sigma>0$ that
\begin{equation}\label{hh1}\begin{split}
    \bigg|\frac{\partial^m  e({w}, z)}{\partial \bar  z_{1}\cdots\partial\bar z_{m}}\bigg| 
    \le &  \, C  \sum_{1\le k_1, \ldots, k_m\le m}  \bigg|\frac{\partial^{m-t} \prod\limits_{l=1}^k S_l({w}_l, z_l)}{\partial \bar  z_{k_{t+1}}\cdots\partial\bar z_{k_m}} \cdot \frac{\partial^{t} }{\partial \bar  z_{k_{1}}\cdots\partial\bar z_{k_t}}  \frac{\prod\limits_{m=1}^{k-1} |{w}_m- z_m|^2}{ H  } \bigg|\\
\le&  \, C \sum_{1\le k_1, \ldots, k_m\le m}\frac{  \prod\limits  _{l=1}^t |{w}_{k_l}-z_{k_l}|^{2t-2-\sigma} \prod \limits  _{l=t+1}^m|{w}_{k_l}-z_{k_l}|^{2t-\sigma} \prod\limits _{l=m+1}^{ k-1}|{w}_l-z_l|^{2t+1-\sigma}}{ H ^{ t+1}  |{w}_k-z_k|^{-2t+1+\sigma}},
    \end{split}
\end{equation}
where the sum is over all permutations of $(1, \ldots, m)$.
On the other hand, 
 we know from Lemma \ref{ele} that
$$
    H ^{ t+1} 
    \ge C \prod_{l=1}^t |{w}_{k_l}- z_{k_l}|^{2t -\frac{1}{2(k-1)}} \prod_{l=t+1}^m |{w}_{k_l}- z_{k_l}|^{2t +2-\frac{1}{2(k-1)}} \prod_{l=m+1}^{k-1} |{w}_{l} - z_{l}|^{2t+2 - \frac{1}{2(k-1)}}|{w}_k- z_k|^{2t+\frac{1}{2}},
$$
since both sides have the same total degree $2(k-1)(t+1)$.
 Combining the above inequality with (\ref{hh1}), we have proved (\ref{hh}) and thus the proposition.
 
\end{proof}
 

We are now ready to prove (\ref{indu}) and therefore give a complete proof of Theorem \ref{C1} for the arbitrary dimensional case,  when the data are $C^{n-1}$ up to the boundary.  


\begin{proof} [Proof of Theorem \ref{C1} (arbitrary $n$)]

Similar to the case of $n=2$, denote by $R_j^\epsilon: =D_j\setminus B({w}_j, \epsilon)$ with $\epsilon$ small. Repeatedly applying  Stokes' Theorem, one obtains
\begin{equation}\label{st}
    \begin{split}
      &\int _{  R_1^\epsilon \times\cdots\times  R_{k-1}^\epsilon\times   D_k}  e({w}, z)\frac{\partial^{k-1} f_k( z)}{\partial \bar  z_1\cdots \partial \bar  z_{k-1}}  \\
    = & \int_{b   D_1\times   R_2^\epsilon \times\cdots\times  R_{k-1}^\epsilon\times   D_k } e({w}, z)\frac{\partial^{k-2} f_k( z)}{\partial \bar  z_2\cdots \partial \bar  z_{k-1}}  -  \int_{  R_1^\epsilon\times \cdots\times  R_{k-1}^\epsilon\times  D_k} \frac{\partial  e({w}, z)}{\partial \bar  z_1}\frac{\partial^{k-2} f_k( z)}{\partial \bar  z_2\cdots \partial \bar  z_{k-1}} \\
    & - \int_{ b B({w}_1, \epsilon)\times   R_2^\epsilon \times\cdots\times  R_{k-1}^\epsilon\times   D_k} e({w}, z)\frac{\partial^{k-2} f_k( z)}{\partial \bar  z_2\cdots \partial \bar  z_{k-1}}  \quad 
     = \quad \cdots\\
    =&\sum_{m=0}^{k-1} (-1)^m\sum_{1\le i_1<\cdots <i_j\le k-1} \int  _{  R^\epsilon_{i_1}\times \cdots  R^\epsilon_{i_m}\times b  D_{j_1}\times \cdots \times b  D_{j_{k-1-m}}\times   D_k}\frac{\partial^m  e({w}, z)}{\partial \bar  z_{i_1}\cdots\partial\bar z_{i_m}}f_k( z) \\
    &-\sum_{m=0}^{k-2} (-1)^m\sum_{1\le i_1<\cdots <i_j\le k-1} \sum_{l=1}^{k-1-m}\int \limits_{R_{\mathbf i\mathbf j, \epsilon}} \frac{\partial^m e({w}, z)}{\partial \bar  z_{i_1}\cdots\partial\bar z_{i_m}}f_k( z) ,
    \end{split}
\end{equation}
where $R_{\mathbf i\mathbf j, \epsilon}:= R^\epsilon_{i_1}\times \cdots  R^\epsilon_{i_m}\times b B({w}_{j_1}, \epsilon)\times \cdots \times b B({w}_{j_{l}}, \epsilon)\times b   D_{j_{l+1}}\times\cdots\times b   D_{j_{k-1-m}}\times   D_k$,
and $\{1, \cdots, k-1\}$ is a permutation of $\{i_1, \cdots, i_m, j_1, \cdots, j_{k-1-m}\}$. 
Letting $\epsilon\rightarrow 0$, we claim that (\ref{st}) reduces to
\begin{equation}\label{form}
      \int \limits_{  D_1 \times\cdots\times     D_k} e({w}, z)\frac{\partial^{k-1} f_k( z)}{\partial \bar  z_1\cdots \partial \bar  z_{k-1}} 
       =\sum \limits_{m=0}^{k-1} (-1)^m\sum _{1\le i_1<\cdots <i_j\le k-1} \int \limits   _{  D_{i_1}\times \cdots \times D_{i_m}\times b  D_{j_1}\times \cdots \times b  D_{j_{k-1-m}}\times  D_k}\frac{\partial^m  e({w}, z)}{\partial \bar  z_{i_1}\cdots\partial\bar z_{i_m}}f_k( z) .
\end{equation}
To get (\ref{form}),   choose $\sigma = \frac{1}{4(k-1)}(<\frac{1}{4} )$ in Proposition \ref{ee}. Then with  $m=0$, one has
$$|e({w}, z)|\le C |{w}_k- z_k|^{-\frac{3}{2}-\frac{1}{4(k-1)}} \prod_{l=1}^{k-1} |{w}_{l}-z_{l}|^{-1 + \frac{1}{4(k-1)}} \in L^1( D_1 \times\cdots\times     D_k).$$
 So
the left hand side of (\ref{st}) as $\epsilon\rightarrow 0$ satisfies
\begin{equation*}
    \int_{  R_1^\epsilon \times\cdots\times  R_{k-1}^\epsilon\times   D_k}  e({w}, z)\frac{\partial^{k-1} f_k( z)}{\partial \bar  z_1\cdots \partial \bar  z_{k-1}} \rightarrow \int_{  D_1 \times\cdots\times   D_k} e({w}, z)\frac{\partial^{k-1} f_k( z)}{\partial \bar  z_1\cdots \partial \bar  z_{k-1}}.  
\end{equation*}
Similarly, for the last line of (\ref{st}),  by Proposition \ref{ee}, it holds that
 \begin{equation}\label{en}
 \begin{split}
    \left|\frac{\partial^m  e({w}, z)}{\partial \bar  z_{i_1}\cdots\partial\bar z_{i_m}}\right|&\le {C}|{w}_k- z_k|^{-\frac{3}{2}- \frac{1}{4(k-1)}} { \prod_{l=1}^m |{w}_{i_l}- z_{i_l}|^{-2 +\frac{1}{4(k-1)}}\prod_{l=1}^{k-1-m} |{w}_{j_l}-z_{j_l}|^{-1 + \frac{1}{4(k-1)}} }\\
    &\in L^1_{loc}(\mathbb C_{ z_{i_l}}\times\cdots \times\mathbb C_{ z_{i_m}} \times\mathbb R_{ z_{j_1}}\times \cdots\times \mathbb R_{ z_{j_{k-1-m}}}\times \mathbb C_{ z_k}). 
    \end{split}
 \end{equation}
As $\epsilon\rightarrow 0$,  the last line of (\ref{st}) vanishes similarly as the case of $n=2$. The formula (\ref{form}) thus holds. Since (\ref{indu}) follows from (\ref{form}) and (\ref{en}), we have finished the proof of Theorem \ref{C1} completely.

\end{proof} 
 

In \cite{FP19}, a bounded (no-canonical) solution is achieved for smooth data on product domains whose factors have $C^{1, \alpha}$ boundaries, $0<\alpha<1$.    In comparison with \cite{FP19},  our method seems not to work on such domains in general,  since our proof requires a uniform exterior ball condition to estimate Green's function on each portion.

\section{Proof of the main theorem under continuity}

Let $\Omega:= D_1\times\cdots\times D_n$, $n\ge 2$, where each $D_j$ is a bounded  domain with $C^2$ boundary in $\mathbb C$. 
We shall prove in this section the main Theorem \ref{main} by weakening the $C^{n-1}_{(0, 1)}(\bar\Omega)$ assumption in Theorem \ref{C1} to continuity.
In \cite{FP19}, an operator in terms of Cauchy integrals was proposed as a potential candidate to solve the $\bar\partial$-equation with continuous data. The following operator $\tilde {\mathbf T}$ is an analogue  but in the context of canonical solutions. For a given $\mathbf f \in C_{(0,1)}(\bar\Omega)$, we define
$$
\tilde {\mathbf T} \mathbf f: = \sum_{s=1}^n (-1)^{s-1}\sum_{1\le i_1<\cdots<i_s\le n} {\mathbf T}^{[i_1, \cdots, i_s]} {\mathbf f}. 
$$
Here for each $(i_1, \cdots, i_s)$ with $1\le i_1<\cdots<i_s\le n$, let $(z', w'')$ represents the  point whose $j$-th component is $z_j$ if $j\in \{i_1, \ldots, i_s\}$, and is $w_j$ otherwise. Then  for $w\in \Omega$,
\begin{equation*}\label{ns}
     {\mathbf T}^{[i_1, \cdots, i_s]}\mathbf f(w):= \sum_{k=1}^s\sum_{m=0}^{s-1}(-1)^m
\sum_{1\le j_1<\cdots<j_m\le i_s}\int \limits_{D_{j_1}\times \cdots\times D_{j_m}\times bD_{t_1}\times \cdots \times bD_{t_{s-m-1}}\times D_{i_k}} f_{i_k}(z', w'')\frac{\partial^m e_w^{k, i_1, \ldots, i_s}(z)}{\partial \bar{z}_{j_1}\cdots\partial \bar{z}_{j_m}},
\end{equation*}
where the third sum is over all $1\le j_1<\cdots<j_m\le i_s $  such that
$\{j_1, \ldots, j_m\}\cup\{t_1, \ldots, t_{s-m-1}\} = \{i_1, \ldots, \hat i_k, \ldots, i_s\},$
 and
\begin{equation}\label{ee1}
    e_w^{k, i_1, \ldots, i_s}(z): = \left(\sum_{k=1}^s\prod_{l=1,l\neq k}^s |{w}_{i_l}-z_{i_l}|^2\right)^{-1}   {\prod_{l=1}^s S_{i_l}({w}_{i_l}, z_{i_l})\prod_{l=1, l\ne k}^{s} |{w}_{i_m}- z_{i_m}|^2}.  
\end{equation} 
For convenience, we suppress the corresponding measure elements  from integrals. 

In Proposition \ref{ee}, we have shown   that there exist some constants  $\alpha_r \in [0, 2)$ when $1\le r\le m+1$, and $\beta_r \in [0,1) $ when $   1\le r\le s-m-1$, such that 
\begin{equation}\label{G11} 
 \big |\frac{\partial^m e_w^{k, i_1,\ldots, i_s}}{\partial \bar{z}_{j_1}\cdots\partial \bar{z}_{j_m}}\big |\le   {C}{\prod_{r=1}^m|{z}_{j_r}-w_{j_r}|^{-\alpha_r}\prod_{r=1}^{s-m-1}|{z}_{t_r}-w_{t_r}|^{-\beta_r}|{z}_{{i_k}}-w_{i_k}|^{-\alpha_{m+1}}}
\in  L^1(R),
\end{equation}
where $R:={D_{j_1}\times \cdots\times D_{j_m}\times bD_{t_1}\times \cdots bD_{t_{s-m-1}}\times D_{i_k}}$. Thus $\tilde {\mathbf T} \mathbf f$ is well defined over $C_{(0,1)}(\bar\Omega)$.

 
\vspace{0.2cm}


Our goal is to prove that $\tilde {\mathbf T} \mathbf f$ is the canonical solution to  $\bar\partial u = \mathbf f$ in the sense of distributions for all continuous $\bar\partial$-closed $(0,1)$ forms $\mathbf f$  up to $\bar\Omega$. Note that if the datum $\mathbf f\in C^{n-1}_{(0, 1)}(\bar\Omega)$, the proof of  Theorem \ref{C1} already implies that  $\tilde {\mathbf T} \mathbf f$ is equal to ${\mathbf T}\mathbf f$ and thus is the canonical solution. 
 The following proposition proves that $\tilde {\mathbf T} \mathbf f$  solves the $\bar\partial$-equation for continuous data in the sense of distributions, making use of a similar approximation argument as in \cite{PZ}.
 

\begin{pro}\label{main11}
There exists a positive constant $C$ depending only on $\Omega$ such that for any $\bar\partial$-closed $(0, 1)$ form $\mathbf f$ continuous up to $\bar\Omega$,  
  $\tilde {\mathbf T}\mathbf f $  is a continuous solution (in the sense of distributions) to  $\bar\partial u =\mathbf f$,  and satisfies
    $$
    \|\tilde {\mathbf T} \mathbf f\|_{ \infty }\le C \|\mathbf f\|_{ \infty }.  
    $$
\end{pro}

\begin{proof}
The fact that 
$\tilde {\mathbf T}\mathbf f \in C(\Omega)$ follows from (\ref{G11}) and the Dominated Convergence Theorem. Moreover, $\|\tilde {\mathbf T} \mathbf f\|_{\infty}\le C \|\mathbf f\|_{\infty}$ by (\ref{G11}). It remains to show that $\tilde {\mathbf T}\mathbf f$ solves $\bar\partial u =\mathbf f$ in $\Omega$ in the sense of distributions. 
  
  \vspace{0.2cm}
  
 For each $j\in \{1, \ldots, n\}$, let $ \{D^{l}_j\}_{l=1}^\infty$   be a family of strictly increasing open subsets  of $D_j$ such that\\
 a) for  $l (\in \mathbb N) \ge N_0 $, $bD^{l}_j$ is  $C^{2}$ and $(l+1)^{-1}< \dist(D^{l}_j, b D_j)<l^{-1}$;\\
 b) $h_j^{l}: \bar D_j\rightarrow \bar D_j^{l}$ is a $C^2$ diffeomorphism  with $\lim \limits_{l\rightarrow \infty} \|h_j^{l}- id\|_{C^2(D_j)}=0$.\\
 Denote by $\Omega^{l}:= D^{l}_1\times\cdots\times D^{l}_n$  the product of these planar domains, and $h^l:=(h_1^l, \ldots, h_n^l)$ the diffeomorphism between $\bar\Omega$ and $\bar\Omega_j$.  Let ${e^{(l)}}$,  ${\mathbf T}^{(l)}$,  $\tilde {\mathbf T}^{(l)}$ and $( {\mathbf T}^{(l)})^{[i_1, \cdots, i_s]}$ stand for the corresponding  operators on $\Omega^{l}$ instead. Then $\tilde {\mathbf T}^{(l)} \mathbf f\in L^\infty(\Omega^{l})$. 
 
   \vspace{0.2cm}
   
 For each  $l$, we adopt the mollifier argument to $\mathbf f\in C(\bar\Omega)$ and obtain $\mathbf f^\epsilon\in C^{\infty}( {\Omega^{l}}) \cap L^{\infty}(\Omega^{l})$  such that  $\|\mathbf f^\epsilon - \mathbf f\|_{L^\infty(\Omega^{l})}\rightarrow 0$  as $\epsilon\rightarrow 0$.  Since $\bar\partial \mathbf f^\epsilon =0$ on $\Omega^{l}$,   $\bar\partial \tilde {\mathbf T}^{(l)} \mathbf f^\epsilon =\bar\partial {\mathbf T}^{(l)} \mathbf f^\epsilon=\mathbf f^\epsilon$ in $ \Omega^{l}$ with  $ \tilde {\mathbf T}^{(l)} \mathbf f^\epsilon\in L^\infty(\Omega^{l})$ when $\epsilon$ is small. Furthermore, 
 \begin{equation*} \label{DD10}
     \|\tilde {\mathbf T}^{(l)} \mathbf f^\epsilon - \tilde {\mathbf T}^{(l)} \mathbf f\|_{L^\infty(\Omega^{l})} \le C_l \|\mathbf f^\epsilon - \mathbf f\|_{L^\infty(\Omega^{l})}\rightarrow 0,
 \end{equation*} as $\epsilon\rightarrow 0$. We thus  have  $\lim \limits_{\epsilon\rightarrow 0}\tilde {\mathbf T}^{(l)} \mathbf f^\epsilon$ exists almost everywhere on $\Omega^{l}$
and is equal to $\tilde {\mathbf T}^{(l)} \mathbf f \in L^\infty(\Omega^{l})$.


Given a test function $\phi\in C^\infty_0(\Omega)$ with a compact support $\kappa$, let $l_0\ge N_0$ be such that $\kappa \subset \Omega^{l_0-2}$. 
Denote by $\langle  \cdot, \cdot \rangle_{\Omega}$ and $\langle  \cdot, \cdot \rangle_{\Omega^{l_0}} $ the inner products in $L^2(\Omega)$ and $L^2(\Omega^{l_0})$, respectively. Let $\bar\partial^*:=-\partial$ be the formal adjoint of $\bar\partial$. 
For $l\ge l_0$,
\begin{equation}\label{22}
\langle \tilde {\mathbf T}^{(l)}\mathbf f, \bar\partial^*\phi\rangle_{\Omega^{l_0}} =\lim_{\epsilon \rightarrow 0} \langle \tilde {\mathbf T}^{(l)}\mathbf f^\epsilon, \bar\partial^*\phi \rangle_{\Omega^{l_0}}= \lim_{\epsilon \rightarrow 0}\langle \bar\partial {\mathbf T}^{(l)}\mathbf f^\epsilon, \phi \rangle_{\Omega^{l_0}} = \lim_{\epsilon \rightarrow 0} \langle\mathbf f^\epsilon, \phi \rangle_{\Omega^{l_0}} = \langle \mathbf f, \phi \rangle_{\Omega}.
\end{equation}

We further claim that \begin{equation}\label{11}
\langle \tilde {\mathbf T}\mathbf f, \bar\partial^*\phi \rangle_{\Omega}=\lim_{l\rightarrow \infty}  \langle  \tilde  {\mathbf T}^{(l)}\mathbf f, \bar\partial^*\phi  \rangle_{\Omega^{l_0}}.
\end{equation}
To get \eqref{11}, it suffices to show that 
\begin{equation}\label{dd7}
\langle   {\mathbf T}^{[i_1, \cdots, i_s]}\mathbf f, \bar\partial^*\phi \rangle_{\Omega^{l_0}}=\lim_{l\rightarrow \infty}\langle (   {\mathbf T}^{(l)})^{[i_1, \cdots, i_s]}\mathbf f, \bar\partial^*\phi \rangle_{\Omega^{l_0}}. \end{equation}
Without loss of generality, assume $(i_1, \ldots, i_s)=(1, \ldots, s)$, and  $(j_1, \ldots, j_m)=(1, \ldots, m) $ with $i_k= m+1$. The remaining cases can be treated similarly. Write $z=(z', z'')\in \mathbb C^{s}\times \mathbb C^{n-s}$, so (\ref{dd7}) is equivalent to
\begin{equation}\label{do}
         \lim _{l\rightarrow \infty}\int \limits_{w\in \kappa} \int \limits_{z'\in\Gamma^{l}}f_{{m+1}}(z', w'')\bar\partial^*\phi(w)D^m e^{(l)}(w', z')
         =  \int \limits_{w\in \kappa}\int \limits_{z'\in\Gamma} f_{{m+1}}(z', w'')\bar\partial^*\phi(w)D^m e(w', z'),
\end{equation}
where $\Gamma=\Gamma_1\times \Gamma_2: = (D_{1}\times \cdots\times D_{m}\times D_{m+1}) \times (bD_{m+2}\times \cdots \times b D_{s})$, $D^m e(w', z'): = \frac{\partial^m }{\partial \bar z_{1}\cdots\partial \bar z_{m}} e_w^{k, i_1,\ldots, i_s}(z)$.
and $\Gamma^{l}, D^m e^{(l)}$ are defined similarly with respect to each $l$.
Indeed, by change of variables, we have
\begin{align*}
   &\int \limits _{w\in \kappa} \int \limits_{z'\in \Gamma^{l}}f_{{m+1}}(z', w'')\bar\partial^*\phi(w)D^m e^{(l)}(w', z')\\
         =&  \int \limits_{(w, z')\in \kappa \times \Gamma} f_{{m+1}}(h^{l}(z'), w''))\bar\partial^*\phi(w)D^m e^{(l)}(w', h^{l}(z'))\mathcal {\tilde D} h^{l}(z')=: \int \limits_{(w, z')\in  \kappa \times \Gamma} F^{(l)}(w, z').
\end{align*}
 Here 
 $h^{l}(z' )$ represents the first $s$ components of $h^{l}(z)$, and $\mathcal {\tilde D} h^{l}\in C^1(\bar\Omega)$ is an  expression that involves only the first order derivatives of $  h^{l}$. Moreover,  $\mathcal {\tilde D} h^{l}\rightarrow 1$ as $l\rightarrow \infty$.
  
   
   \vspace{0.2cm}
   
Notice that if  $w\in \kappa(\subset \Omega^{l_0-2})$ and $z_k\in b D_k^{l}, k \in [m+2, s], l \in [ l_0, \infty)$, then 
  $$
  |z_k-w_k|\ge \dist((\Omega^{l})^c, \Omega^{l_0-2})\ge \dist((\Omega^{l_0})^c, \Omega^{l_0-2})>  {l_0^{-2}} =:\delta_0.
  $$
Hence  for  all  $(w, z')\in \kappa \times \Gamma$,  there exist $ \alpha_j \in [0, 2)$, $\beta_0\ge 0$, and a positive constant $C$ independent of $l$ by  (\ref{G11}) such that 
 \begin{equation}\label{dd3}
  \left|  F^{(l)}(w, z') \right| \le  {C \delta_0^{-\beta_0}\prod_{j=1}^{m+1} |z_j-w_j|^{-\alpha_j}}\in     L^1( \kappa \times \Gamma).  
 \end{equation}


On the other hand, we will show that 
 for  all $(w, z')\in \kappa  \times \Gamma\setminus \cup_{j=1}^{m+1}\{z_j=w_j\} $, 
\begin{equation}\label{hhh}
    \lim_{l\rightarrow \infty} D^m e^{(l)}( w', h^{l}(z')) = D^m e( w', z').
    \end{equation}
Note that from the definition (\ref{ee1}) for $e$,  besides some explicit continuous functions, $D^m e^{(l)}$ off the diagonal involves only products of  the canonical solution kernels and the Bergman kernels along the first $s$ slices. 
In view of this, (\ref{hhh})  follows from the fact that the canonical solution kernel for the $\bar\partial$-equation and the Bergman kernel converge locally uniformly on planar domains (see Proposition \ref{DD} on the stability).
By the continuity of $\mathbf f$ and  the construction of $\Omega^{l}$,
$$
        \lim_{l\rightarrow \infty} F^{(l)}(w, z') =  f_{{m+1}}(z', w'')\bar\partial^*\phi(w)D^m e( w', z')
 $$
point-wisely on $ \kappa \times \Gamma\setminus \cup_{j=1}^{m+1}\{z_j=w_j\} $. Therefore, (\ref{do})  follows from (\ref{dd3}), \eqref{hhh} and the Dominated Convergence Theorem.

 Finally, combining  (\ref{22}) with (\ref{11}),  we complete the proof by 
 deducing that
$$
\langle  \tilde {\mathbf T}\mathbf f, \bar\partial^*\phi  \rangle_{\Omega}=\lim_{l\rightarrow \infty}  \langle \tilde  {\mathbf T}^{(l)}\mathbf f, \bar\partial^*\phi  \rangle_{\Omega^{l_0}}= \langle \mathbf f, \phi \rangle_\Omega.
 $$


\end{proof}


 \begin{proof}[Proof of Theorem \ref{main}]  In view of Proposition \ref{main11}, we only need to show that for any given $\epsilon>0$, $| \langle \tilde {\mathbf T}\mathbf f, h \rangle |\le \epsilon$ for any $h\in A^2(\Omega)$ with $\|h\|_{L^2(\Omega)}=1$. Without loss of generality, we may further assume that  $\|\mathbf f\|_{L^2(\Omega)} =1$.

 
From (\ref{dd3}) we know for all $w\in \kappa$ that
 \begin{equation*}
  \int_{z'\in \Gamma^{l}}f_{{m+1}}(z', w'')D^m e^{(l)}(w', z')\le C
 \end{equation*}
 and
 $$
        \lim_{l\rightarrow \infty}  \int_{z'\in \Gamma^{l}}f_{{m+1}}(z', w'')D^m e^{(l)}(w', z')         = \int_{z'\in\Gamma} f_{{m+1}}(z', w'')D^m e(w', z').
$$
 So the Dominated Convergence Theorem guarantees that for any $ U \Subset \Omega$, as $l\rightarrow \infty$,
 \begin{equation}\label{DD8}
    \| \tilde {\mathbf T}\mathbf f -\tilde {\mathbf T}^{(l)}\mathbf f\|_{L^2(U)}\rightarrow 0.
 \end{equation}
 
 
 According to   the proof of Proposition \ref{main11}, there exist a family of increasing exhausting subsets $\{\Omega^{l}\}_{l=1}^{\infty}$ of $\Omega$ with component-wise $C^2$ boundaries such that the followings  hold for any given $\epsilon>0$.
  
  \vspace{0.2cm}
  
  a) By the absolute continuity of Lebesgue's integral, there exists an $l_0 \in\mathbb N$ with $\text{Vol}(\Omega \setminus \Omega^{l_0})\le 1$ and 
 \begin{equation}\label{dd2}
  \|h\|_{L^2(\Omega\setminus \Omega^{l_0})}\le   {6^{-1} \epsilon}{ (M    +C_0)^{-1}}.
    \end{equation}
Here $M:=\|\tilde {\mathbf T}\mathbf f\|_{\infty}<\infty$, and  $C_0$ is H\"ormander's $L^2$ bound for the canonical solution operator on $\Omega$.   Since the $L^2$ bound depends only on the diameter of the domain, $C_0$
can be chosen as the $L^2$ bound for the canonical solution operator on $\Omega^{l}$ for each $l\ge l_0$ as well. 

\vspace{0.2cm}


b) From  (\ref{DD8}) there exists an $l_1(\in\mathbb N)\ge l_0$ such that 
\begin{equation*}   
 \|\tilde {\mathbf T}\mathbf f - \tilde {\mathbf T}^{(l_1 )}\mathbf f\|_{L^2(\Omega^{l_0})} \le 6^{-1}{\epsilon}.
\end{equation*}

\vspace{0.2cm}

c) There exist an $l (\in\mathbb N)\ge l_1$ and a $\bar\partial$-closed form $\mathbf f^{(l )} \in C^{n-1}(\Omega^{l_1})$ such that $    \mathbf {\tilde T}^{(l_1)} \mathbf f^{(l )}\perp A^2(\Omega^{l_1})$ and 
$$
    \|\tilde {\bf T}^{(l_1)}\mathbf f - \tilde {\bf T}^{(l_1)}\mathbf f^{(l )}\|_{L^2(\Omega^{l_1})}\le  C_0 \| \mathbf f -  \mathbf f^{(l )}\|_{L^2(\Omega^{l_1})} \le \sqrt {\text{Vol}(\Omega)} C_0 \| \mathbf f -  \mathbf f^{(l )}\|_{L^ \infty(\Omega^{l_1})}  \le 6^{-1}{\epsilon} .$$
 

      
In particular, as  consequences of c),  we obtain
\begin{equation}\label{DD11}
\langle  \tilde {\bf T}^{(l_1)}\mathbf f^{(l)}, h \rangle_{\Omega^{l_1}} =0, \quad \text {and} \quad \| \tilde {\mathbf T}^{(l_1)}\mathbf f \|_{L^2(\Omega^{l_1})}\le C_0.
\end{equation}
 Here the first identity is due to the fact that $h|_{\Omega^{l_1}}\in A^2(\Omega^{l_1})$. Since $\tilde {\mathbf T}^{(l_1)}$ is the  canonical solution operator for the  $\bar\partial$-equation on $\Omega^{l_1}$, the second inequality follows from H\"ormader's $L^2$ estimate on $\Omega^{l_1}$ for the  $\bar\partial$-closed form $\mathbf f\in C_{(0,1)}(\bar \Omega)\subset L^2_{(0,1)}(\Omega^{l_1})$.
Combining (\ref{DD11}) with b) and c), one has 
\begin{align*} 
 | \langle \tilde {\bf T}\mathbf f, h \rangle_{\Omega^{l_1}}|  
 \le &| \langle \tilde {\bf T}\mathbf f - \tilde {\bf T}^{(l_1)}\mathbf f, h \rangle _{\Omega^{l_1}}| +| \langle \tilde {\bf T}^{(l_1)}\mathbf f - \tilde {\bf T}^{(l_1)}\mathbf f^{(l)}, h \rangle_{\Omega^{l_1}}| \\
  \le &\big  \|\tilde {\bf T}\mathbf f-\tilde {\mathbf T}^{(l_1)}\mathbf f  \big  \|_ {L^2 \left (\Omega^{l_0} \right )}+ \big   \|\tilde {\bf T}\mathbf f-\tilde {\mathbf T}^{(l_1)}\mathbf f  \big  \|_{L^2\left (\Omega^{l_1}\setminus \Omega^{l_0} \right)}  \|h\|_{L^2 \left (\Omega^{l_1}\setminus \Omega^{l_0} \right )}+  \big  \|\tilde {\bf T}^{(l_1)}\mathbf f - \tilde {\bf T}^{(l_1)}\mathbf f^{(l)}  \big \|_{L^2 \left (\Omega^{l_1} \right )} \\
   \le & 6^{-1}{\epsilon}+ {6^{-1} \epsilon}{ (M    +C_0)^{-1}}\left(  \|\tilde{\bf T}  \mathbf f\|_{ \infty } + \|\tilde {\mathbf T}^{(l_1)}\mathbf f\|_{L^2 \left(\Omega^{l_1}\right)}  \right) + 6^{-1}{\epsilon}
   \, =   \, 2^{-1}{\epsilon}. 
\end{align*}
Together with (\ref{dd2}), we finally obtain
\begin{equation*}
    | \langle \tilde {\bf T}\mathbf f, h \rangle_\Omega|\le | \langle \tilde {\bf T}\mathbf f, h \rangle_{\Omega^{(l_1)}}|+  | \langle \tilde {\bf T}\mathbf f, h \rangle_{\Omega\setminus \Omega^{(l_1)}}|\le 2^{-1}{\epsilon}+(2M)^{-1}  \epsilon\|\tilde {\bf T}\mathbf f\|_{\infty}\le \epsilon
\end{equation*}
and thus have proved the main theorem completely. 

\end{proof}


One  could compare Theorem \ref{main} with \cite[Theorem 4.1]{DLT}, where uniform estimates for the canonical solution to $\bar\partial u = f$ on polydiscs were obtained without the continuity condition but under a stronger assumption than $f$ being $L^\infty$. 

Lastly, an immediate consequence of Theorem \ref{main} is the following estimate concerning the Bergman projection, which generalizes Corollary \ref{cor=dim1} to product domains.

 

\begin{cor} \label{cor} Under the same assumptions as in Theorem \ref{main}, 
let $\mathbf P$ be the Bergman projection on $\Omega$. Then there exists a positive constant $C$ depending only on $\Omega$ such that for any $
u\in C  (\bar\Omega)  \cap W^{1, \infty}(\Omega)$,
$$
\|\mathbf P u\|_{ \infty }\le \|u \|_{ \infty } + C \|\bar\partial u\|_{ \infty }.
$$
\end{cor}


 \section{Estimates for Green's function}
For the convenience of the reader, we complete the paper by providing proofs to Lemma \ref{Kerzman-Green}  based on \cite{Ker76} and Lemma \ref{Kerzman-Green2}. Kerzman's result  on Green's function serves as a foundational estimate, from  which upper bounds for  the canonical solution kernel of $\bar\partial$ and its derivatives are obtained in Section 2 and 3, respectively.

On a bounded domain $D \subset \mathbb{C}$ with $C^{2}$ boundary, Green's function $G$ satisfies the following properties:

\vspace{0.15cm}

i) $ G(z,w)=G(w, z) $ are positive $C^2$ functions for $(z, w) \in D \times D \setminus \{z=w\}$. 

\vspace{0.15cm}

ii) For each fixed $w\in D$,  $ G(z, w)$ is harmonic in $z \in D \setminus \{w\}$, and $G(z, w) = 0$ when $ z \in bD$.
\vspace{0.15cm}

iii) For each fixed $w\in D$,  $ G(z,w)+(2\pi)^{-1}\log|z-w|$ is harmonic in $z\in D$.
 
\vspace{0.15cm}

Fix $w\in D$ and consider $\rho(z, w):= G(z, w) + (2\pi)^{-1}  \log |z-w|$.  
When $z \in bD$, $\rho(z, w)= (2\pi)^{-1}  \log |z-w| \le  (2\pi)^{-1}  \log d$, where $d$ is the diameter of $D$. By the Maximum Principle, we know that for $(z, w) \in D\times D$ off the diagonal,
$$
G(z, w) +\frac{1}{2\pi}\log |z-w|\le \frac{\log d}{2\pi},
$$
which yields that
\begin{equation}\label{G<}
0<G(z, w) \le  \frac{1}{2\pi}\log\frac{d}{|z-w|}.
\end{equation}
\medskip


\begin{proof}[Proof of Lemma \ref{Kerzman-Green}]

Since $G$ is symmetric in $z$ and $w$ and $G(z, \cdot)=0$ on $bD$, we shall prove the first part of (\ref{G}) only, for any fixed pole $z\in D$. By rotation and translation if necessary, $w$ is on the negative $y$-axis and $0\in bD$ such that $\delta(w) = |w|$. Since $D$ has $C^2$ boundary, it satisfies the exterior ball condition with a uniform radius $r\leq d$.
If $\delta(w) \ge (8d)^{-1}r|z-w|$, then (\ref{G}) follows from \eqref{G<}. So we assume that $\delta(w) \le (8d)^{-1}r|z-w|$.   
Let $\alpha := (4d)^{-1}r|z-w|< 4^{-1}r $ and define the region $R:= B(0, \alpha)\setminus B(\alpha i, \alpha)$. Consider the harmonic function $E(\zeta): =\Real    \{ (\zeta -\alpha i)^{-1}\zeta \} $ on $R$. Then 
$E\ge {c_0}$ on $bB(0, \alpha)\setminus \overline{B(\alpha i, \alpha)}$ for some universal constant
 $$
 {c_0} :=\inf_{\zeta\in bB(0, 1)\setminus \overline{B( i, 1)}}\Real    \frac{\zeta}{\zeta -i}>0,
 $$
 which is particularly independent of $\alpha$. Moreover, at $\zeta=w$,
\begin{equation}\label{E2}
    E(w)\le \frac{|w|}{\alpha}= \frac{ 4d\delta(w)}{  r|z-w|}.
\end{equation}
Consider the sub-region $\tilde R: = R\cap D$, where the function  $E(\zeta)\log\frac{d}{|z-w|}$ is harmonic in $\zeta$.

\vspace{0.15cm}

We assert that
\begin{equation}\label{claim}
E(\zeta) \log \frac{d}{|z-w|}  \ge  2\pi {c_0}G(z, \zeta),
\end{equation}
for $\zeta\in b\tilde R\equiv   (R\cap bD) \cup (bR\cap D)$.

 
\vspace{0.15cm}

If $\zeta\in R\cap bD$, then $E(\zeta)\log\frac{d}{|z-w|}\ge 0=2\pi {c_0}G(z, \zeta)$.

\vspace{0.15cm}

If $\zeta\in bR\cap D$,
$$
|z-\zeta|\ge |z-w|-|w-\zeta|\ge |z-w|-2\alpha\ge \frac{|z-w|}{2},
$$
and thus by \eqref{G<} again,
$$
G(z, \zeta) \leq  \frac{1}{2\pi}\log \frac{d}{|z-w|}.
$$
On the other hand, since $bB(\alpha i, \alpha)\cap  D =\emptyset$ by the choice of $\alpha$, we have $\zeta \in bB(0, \alpha)\cap D\subset bB(0, \alpha)\setminus \overline{B(\alpha i, \alpha)}$ where $E\ge C$. Therefore, for $\zeta \in  bR\cap D$, we get \eqref{claim} and thus have verified the above assertion.


By the Maximum Principle, \eqref{claim} holds true for all $\zeta \in \tilde R$. In particular at $\zeta=w$, one has
\begin{equation}\label{E}
    E(w)\log\frac{d}{|z- w|}\ge 2\pi {c_0}G(z, w).  \end{equation}
Now, the first part of \eqref{G} follows from (\ref{E}) and (\ref{E2}), if we take $C:= 2d  (\pi {c_0}r)^{-1} $.

\vspace{0.15cm}
 
 For \eqref{g2}, it follows directly from \eqref{G} when $\delta(z)\ge (8d)^{-1}r|z-w|$. Otherwise, we argue in the same way as we prove \eqref{G}, and use \eqref{G} in the place of \eqref{G<} to show that the assertion
$$  
C  \frac{E(\zeta)\delta(z) }{|z-w|}\log\frac{d}{|z-w|}\ge {c_0} G(z, \zeta)
$$
holds true for $\zeta\in b\tilde R$. By the Maximum Principle we further know that the above inequality holds true on all $\tilde R$, and particularly at $\zeta=w$. Using (\ref{E2}) again, we get \eqref{g2}.

\end{proof}


\begin{proof}[Proof of Lemma \ref{Kerzman-Green2}]
The symmetry of $G$ implies that the gradient in the first part of \eqref{Gd} could be taken with respect to either $z$ or $w$. Fix  $w\in D$, and let $\epsilon = \min\{\frac{\delta(z_0)}{2}, \frac{|z_0-w|}{2}\}>0$ for  $z_0 \in D \setminus \{w\}$. Then $$|z-w|\ge |z_0-w|-|z_0-z|\ge \frac{|z_0-w|}{2}, \quad \text{and} \quad \delta(z)\le \delta(z_0)+|z_0-z| \le 2\delta(z_0)$$
   for all $z\in B(z_0, \epsilon)$. Since $G(\cdot, w)$ is harmonic on $B(z_0, \epsilon)$, by the Mean Value theorem (cf. \cite[(2.31) on p. 22]{GT}),  it follows that
  \begin{equation*}
     |\nabla_z G(w, z_0)|\le \frac{2}{\epsilon}\sup_{z\in B(z_0, \epsilon)}|G(z, w)|.
 \end{equation*}
 When $\epsilon = 2^{-1} {\delta(z_0)} $, we use \eqref {G} to get
$$
|\nabla_z G(w, z_0)|\le \frac{4}{\delta(z_0)} \frac{4C \delta(z_0)}{|z_0-w|}\log \frac{d}{|z_0-w|};
$$
when $\epsilon = 2^{-1} {|z_0-w|} $, we use \eqref{G<} to get
 $$
 |\nabla_z G(w, z_0)|\le \frac{4}{|z_0-w|} \frac{1}{2\pi}\log \frac{d}{|z-w|}.
 $$
Then, part a) for $z\in D\setminus\{w\}$ follows if the constant is chosen as the maximum of $8C$ and $ {2}{\pi^{-1}}$; part a) for $z\in bD$ follows from the second part of \eqref{G} and the fact that $G( w, \cdot)$  vanishes at $bD$.
Similarly, to obtain the second part of \eqref {Gd}, we  use \eqref{g2} and \eqref{G} in the places of \eqref {G} and \eqref{G<}, respectively.

\end{proof}

  
 
 

 
\begin{appendices} 
  
  \section{Boundedness of the canonical solution on polydiscs\\ (\large Appendix by Martino Fassina\protect\footnote[2]{Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, IL 61801, USA \newline \emph{email}: fassina2@illinois.edu})}
 
 
 
 

 

Let $D$ be a polydisc in $\C^2$. Boundedness of the canonical solution of the $\bar\partial$-problem in $D$ was established by Landucci in \cite{L75}. The goal of this appendix is to prove Landucci's result using the techniques developed in \cite{FP19} in a way that suggests a natural (but technically involved) generalization to higher dimensions. Even though such generalization is only a special case of Theorem \ref{main}, we think that the arguments here presented are of independent interest and might inspire new applications.

\subsection{A representation formula in terms of the operator $ \mathcal{T}$}
 
 In \cite{FP19} the authors defined an integral operator {\bf $\mathcal{T}$} on the product of one-dimensional domains with $C^{1,\alpha}$ boundary.\footnote{This operator is actually called {\bf T} in \cite{FP19}. Here we changed the notation to ${\bf \mathcal{T}}$ to avoid confusion, since the symbol {\bf T} is used in the main paper to denote the canonical solution operator, as in equation \eqref{formula1}.} For a $\bar\partial$-closed $(0,1)$ form $f$ on on such a product domain, the function {\bf $\mathcal{T}$}$f$ satisfies $\bar\partial {\bf \mathcal{T}}f=f$ and $\norm{{\bf \mathcal{T}}}_{\infty}\leq C\norm{f}_{\infty}$, where $C$ is a constant that does not depend on $f$. If $u$ is another solution to $\bar\partial u=f$, then $u-{\bf \mathcal{T}}f$ is of course a holomorphic function. 
 
 Let us consider the polydisc case. As a first step towards proving boundedness of the canonical solution, we give an integral representation in terms of $u$ for the holomorphic function $u-{\bf \mathcal{T}}f$.
\begin{pro}\label{t1}
Let $D=D_1\times D_2$ be the unit polydisc in $\C^2$ and $u\in C^2(\overline{D})$. Let $f=\bar\partial u$. Then 
\begin{equation*}
u(z_1,z_2)-{\bf \mathcal{T}}f(z_1,z_2)=\frac{1}{(2\pi i)^2}\int_{\partial D_1\times \partial D_2}\frac{u(\zeta_1,\zeta_2)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\zeta_1\wedge d\zeta_2
\end{equation*}
for every $(z_1,z_2)\in D$.
\end{pro}
\begin{proof}
Recall the integral representation formula 
\begin{equation}\label{BM}
u(z)=\int_{\partial D}u(\zeta)B(\zeta,z)-\int_{D} \bar\partial u \wedge B(\zeta,z),\quad z\in D,
\end{equation}
where $B(\zeta,z)$ is the Bochner-Martinelli kernel in $\C^2$, given by
\begin{equation*}
B(\zeta,z)=\frac{1}{(2\pi i)^2}\frac{1}{|\zeta-z|^4}\Big{[}(\overline{\zeta_1-z_1})\, d\zeta_1\wedge d\bar{\zeta}_2\wedge d\zeta_2+(\overline{\zeta_2-z_2})\, d\zeta_2\wedge d\bar{\zeta}_1\wedge d\zeta_1 \Big{]}.
\end{equation*}
Here $|\zeta-z|^2=|\zeta_1-z_1|^2+|\zeta_2-z_2|^2$. Fix $z=(z_1,z_2)\in D$, and let $f_j=\partial_{\bar{\zeta}_j}u$ for $j=1,2$. By \eqref{BM}
\begin{equation}\label{f1}
\begin{split}
(2\pi i)^2u(z)=&\int_{\partial D_1\times D_2}\frac{\overline{\zeta_1-z_1}}{|\zeta-z|^4}\, u(\zeta)\, d\zeta_1\wedge d\bar{\zeta}_2\wedge d\zeta_2 + \int_{ D_1\times \partial D_2}\frac{\overline{\zeta_2-z_2}}{|\zeta-z|^4}\, u(\zeta)\, d\zeta_2\wedge d\bar{\zeta}_1\wedge d\zeta_1\\&-\int_{D_1\times D_2}\frac{\overline{\zeta_1 - z_1}}{|\zeta - z|^4}\, f_1(\zeta)\,  dV_\zeta -\int_{D_1\times D_2}\frac{\overline{\zeta_2 - z_2}}{|\zeta - z|^4}\, f_2(\zeta)\,  dV_\zeta.
\end{split}
\end{equation}
where $dV_\zeta=d\bar{\zeta}_1\wedge d\zeta_1\wedge d\bar{\zeta}_2\wedge d\zeta_2.$
Note that
\begin{equation*}
\frac{\partial}{\partial \bar{\zeta}_2}\bigg{[}\frac{\overline{\zeta_2-z_2}}{|\zeta -z|^2}\bigg{]}=\frac{|\zeta_1-z_1|^2}{|\zeta-z|^4}.
\end{equation*}
Using Stokes' theorem we can therefore rewrite the first term in \eqref{f1} as
\begin{equation*}
\begin{split}
\int_{\partial D_1\times D_2}\frac{\overline{\zeta_1-z_1}}{|\zeta-z|^4}\, u(\zeta)\, d\zeta_1\wedge d\bar{\zeta}_2\wedge d\zeta_2 =&\int_{\partial D_1\times \partial D_2}\frac{\overline{\zeta_2-z_2}}{(\zeta_1-z_1)|\zeta-z|^2}\, u(\zeta)\, d\zeta_1\wedge d\zeta_2\\&-\int_{\partial D_1\times D_2}\frac{\overline{\zeta_2-z_2}}{(\zeta_1-z_1)|\zeta-z|^2}\, f_2(\zeta)\, d\zeta_1\wedge d\bar{\zeta}_2\wedge d\zeta_2.\\
\end{split}
\end{equation*}
Arguing in the same way for the other terms of \eqref{f1}, letting $Df=\partial_{\bar{\zeta}_1}f_2=\partial_{\bar{\zeta}_2}f_1$, and using the identity  
\begin{equation}\label{identity}
\frac{1}{(\zeta_1-z_1)(\zeta_2-z_2)}=\frac{\overline{\zeta_1-z_1}}{(\zeta_2-z_2)|\zeta-z|^2}+\frac{\overline{\zeta_2-z_2}}{(\zeta_1-z_1)|\zeta-z|^2},
\end{equation}
we obtain
\begin{equation*}
\begin{split}
(2\pi i)^2u(z)=&\int_{\partial D_1\times \partial D_2}\frac{u(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\zeta_1\wedge d\zeta_2-\int_{D_1\times \partial D_2}\frac{f_1(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\bar{\zeta}_1\wedge d\zeta_1\wedge d\zeta_2\\&-\int_{\partial D_1\times  D_2}\frac{f_2(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\bar{\zeta}_2\wedge d\zeta_2\wedge d\zeta_1+\int_{D_1\times D_2}\frac{Df(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, dV_\zeta.
\end{split}
\end{equation*}
The Cauchy-Pompeiu formula gives
$$
\int_{D_1\times \partial D_2}\frac{f_1(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\bar{\zeta}_1\wedge d\zeta_1\wedge d\zeta_2=\int_{D_1\times D_2}\frac{Df(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, dV_\zeta +2\pi i \int_{D_1}\frac{f_1(\zeta_1,z_2)}{(\zeta_1-z_1)}\, d\bar{\zeta}_1\wedge d\zeta_1,
$$
and an analogous identity holds when the roles of the two variables are reversed. By the formula for ${\bf \mathcal{T}}f$ given in \cite[Section 3]{FP19}, 
\begin{equation*}
\begin{split}
u(z)&=\frac{1}{(2\pi i)^2}\int_{\partial D_1\times\partial D_2}\frac{u(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\zeta_1\wedge d\zeta_2-\frac{1}{2\pi i}\int_{D_1}\frac{f_1(\zeta_1, z_2)}{\zeta_1-z_1}\, d\bar{\zeta}_1\wedge d\zeta_1\\&\phantom{34}-\frac{1}{2\pi i}\int_{D_2}\frac{f_2(z_1, \zeta_2)}{\zeta_2-z_2}\, d\bar{\zeta}_2\wedge d\zeta_2-\frac{1}{(2\pi i)^2}\int_{D_1\times D_2}\frac{Df(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, dV_\zeta\\&=\frac{1}{(2\pi i)^2}\int_{\partial D_1\times\partial D_2}\frac{u(\zeta)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\zeta_1\wedge d\zeta_2+{\bf \mathcal{T}}f(z),
\end{split}
\end{equation*}
which concludes the proof.
\end{proof}

\subsection{Boundedness of the Bergman projection}

As in Landucci \cite{L75}, we consider functions $u$ on the bidisc that satisfy a certain condition, namely the vanishing of an integral. We show that the Bergman projection of $u$ enjoys a supnorm estimate in terms of the supnorm of $\bar\partial u$. (Compare with Corollaries \ref {cor=dim1} and \ref{cor} of the main paper, where the supnorm of the Bergman projection of a general smooth function is estimated by two terms). In the proof we employ techniques and results from \cite{FP19}, some of which are recalled in the remark below. 

\begin{pro} [{\cite[Proposition 1]{L75}}] \label{t2} 
Let $D=D_1\times D_2$ be the unit polydisc in $\C^2$ and $u\in C^2(\overline{D})$. Assume 
\begin{equation*}
\int_{\partial D_1\times \partial D_2}\frac{u(\zeta_1,\zeta_2)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\zeta_1\wedge d\zeta_2=0.
\end{equation*}
Then the Bergman projection $Pu$ satisfies a supnorm estimate \begin{equation*}
\norm{Pu}_{L^{\infty}(D)}\leq C \norm{\bar\partial u}_{L^{\infty}(D)}
\end{equation*}
for some constant $C$ depending only on $D$.
\end{pro}


\begin{remark}\label{r}
By \cite[Lemma 2.3]{FP19}, if $\alpha<1$, then there exists a constant $C$ such that 
\begin{equation*}
\int_{|\zeta|=1}\frac{|d\zeta|}{|1-\bar{\zeta}z|^{\alpha}}\, d\zeta=\int_{|\zeta|=1}\frac{|d\zeta|}{|\zeta-z|^{\alpha}}\, d\zeta\leq C
\end{equation*}
for every $|z|<1$. Moreover, let
\begin{equation*}
g(\zeta,z)=\frac{\zeta-z}{1-\bar{\zeta}z},\quad |\zeta|<1, |z|<1.
\end{equation*}
Then $g$ is holomorphic in $z$, and by the maximum principle $|g|\leq 1$ in $D$. By \cite[Lemma 2.2]{FP19}, for $\alpha<2$, there exists a constant $C$ such that
\begin{equation*}
\int_{|\zeta|<1}\frac{|d\bar{\zeta}\wedge d\zeta|}{|1-\bar{\zeta}z|^{\alpha}}\leq C
\end{equation*}
for every $|z|<1$.
\end{remark}


\begin{proof}[Proof of Proposition \ref{t2}]
The Bergman kernel for the unit polydisc $D$ in $\C^2$ is given by 
\begin{equation*}
K(z,\zeta)=\frac{1}{\pi^2(1-z_1\bar{\zeta}_1)^2(1-z_2\bar{\zeta}_2)^2},
\end{equation*}
and the Bergman projection is \begin{equation*}
Pu(z)=\int_D K(z,\zeta)u(\zeta) .
\end{equation*}
 Hence, to prove the proposition, it is enough to estimate the integral 
 \begin{equation*}
 I=\int_{D_1\times D_2}\frac{u(\zeta)}{(1-\bar{\zeta}_1z_1)^2(1-\bar{\zeta_2}z_2)^2}\, dV_\zeta.
 \end{equation*}
 Note that \begin{equation*}
 \frac{\partial}{\partial\bar{\zeta}_2}\bigg{[}\frac{\bar{\zeta}_2}{(1-\bar{\zeta}_2z_2)}\bigg{]}=\frac{1}{(1-\bar{\zeta}_2z_2)^2}.
 \end{equation*}
Applying Stokes' theorem twice and letting $Df=\partial_{\bar{\zeta}_1}f_2=\partial_{\bar{\zeta}_2}f_1$, we get
 \begin{equation}\label{h}
 \begin{split}
 I=&\int_{\partial D_1\times \partial D_2}\frac{\bar{\zeta}_1\bar{\zeta}_2\,u(\zeta)}{(1-\bar{\zeta}_1z_1)(1-\bar{\zeta}_2z_2)}\, d\zeta_1\wedge d\zeta_2-\int_{ D_1\times \partial D_2}\frac{\bar{\zeta}_1\bar{\zeta}_2\,f_1(\zeta)}{(1-\bar{\zeta}_1z_1)(1-\bar{\zeta}_2z_2)}\, d\bar{\zeta}_1\wedge d\zeta_1\wedge d\zeta_2\\&-\int_{ \partial D_1\times D_2}\frac{\bar{\zeta}_1\bar{\zeta}_2\,f_2(\zeta)}{(1-\bar{\zeta}_1z_1)(1-\bar{\zeta}_2z_2)}\, d\bar{\zeta}_2\wedge d\zeta_2\wedge d\zeta_1+\int_{ D_1\times D_2}\frac{\bar{\zeta}_1\bar{\zeta}_2\,Df(\zeta)}{(1-\bar{\zeta}_1z_1)(1-\bar{\zeta}_2z_2)}\, dV_\zeta.
 \end{split}
 \end{equation}
 By hypothesis, the first term in \eqref{h} is equal to zero. By the Cauchy-Pompeiu formula, we obtain 
 \begin{equation}\label{ll}
 \begin{split}
 I=&-2\pi i\bigg(\int_{D_1}f_1(\zeta_1,z_2)\frac{\bar{\zeta}_1}{(1-\bar{\zeta}_1z_1)}\, d\bar{\zeta}_1\wedge d\zeta_1+\int_{D_2}f_2(z_1,\zeta_2)\frac{\bar{\zeta}_2}{(1-\bar{\zeta}_2z_2)}\, d\bar{\zeta}_2\wedge d\zeta_2\bigg)\\&-\int_{D_1\times D_2}Df(\zeta)\frac{\bar{\zeta}_1\bar{\zeta}_2}{(1-\bar{\zeta}_1z_1)(1-\bar{\zeta}_2z_2)}\, dV_\zeta.
 \end{split}
 \end{equation}
The first two terms in \eqref{ll} can be estimated using Remark \ref{r}. By \cite[equation (3.5)]{FP19}, 
\begin{equation}\label{dec}
\begin{split}
\int_{D_1\times D_2}Df(\zeta)\frac{\bar{\zeta}_1\bar{\zeta}_2}{(1-\bar{\zeta}_1z_1)(1-\bar{\zeta}_2z_2)}\, dV_\zeta=&\int_{D_1\times D_2}\frac{\partial f_1}{\partial \bar{\zeta}_2}(\zeta)\frac{(1-\zeta_2\bar{z}_2)\bar{\zeta}_1\bar{\zeta}_2}{(1-\bar{\zeta}_1z_1)(|1-\bar{\zeta}_1z_1|^2+|1-\bar{\zeta}_2z_2|^2)}\, dV_\zeta\\&+\int_{D_1\times D_2}\frac{\partial f_2}{\partial \bar{\zeta}_1}(\zeta)\frac{(1-\zeta_1\bar{z}_1)\bar{\zeta}_1\bar{\zeta}_2}{(1-\bar{\zeta}_2z_2)(|1-\bar{\zeta}_1z_1|^2+|1-\bar{\zeta}_2z_2|^2)}\, dV_\zeta.\\
\end{split}
\end{equation} 
 By symmetry, it is enough to estimate the first of the two integrals on the right side of \eqref{dec}. We use Stokes' theorem to move the derivative away from $f_1$. Hence 
\begin{equation}\label{last}
\begin{split}
 &\int_{D_1\times D_2}\frac{\partial f_1}{\partial \bar{\zeta}_2}(\zeta)\frac{(1-\zeta_2\bar{z}_2)\bar{\zeta}_1\bar{\zeta}_2}{(1-\bar{\zeta}_1z_1)(|1-\bar{\zeta}_1z_1|^2+|1-\bar{\zeta}_2z_2|^2)}\, dV_\zeta\\
 =& \int_{D_1\times \partial D_2} f_1(\zeta)\frac{(1-\zeta_2\bar{z}_2)\bar{\zeta}_1\bar{\zeta}_2}{(1-\bar{\zeta}_1z_1)(|1-\bar{\zeta}_1z_1|^2+|1-\bar{\zeta}_2z_2|^2)}\, d\bar{\zeta}_1\wedge d\zeta_1\wedge d\zeta_2\\
 &-\int_{D_1\times D_2} f_1(\zeta)\frac{\bar{\zeta}_1(1-\zeta_2\bar{z}_2)}{1-\bar{\zeta}_1z_1}\,\,\frac{\bar{\zeta}_2z_2(1-\zeta_2\bar{z}_2)+|1-\bar{\zeta}_1z_1|^2+|1-\bar{\zeta}_2z_2|^2}{(|1-\bar{\zeta}_1z_1|^2+|1-\bar{\zeta}_2z_2|^2)^2}\, dV_\zeta.
\end{split}
\end{equation} 
The two integrals on the right side of \eqref{last} can be estimated combining Remark \ref{r} with the arguments in \cite[Page 6-7]{FP19}.
\end{proof}

\begin{remark}
Propositions \ref{t1} and \ref{t2} can be generalized to polydiscs in $\C^n$ by considering the general formula for ${\bf \mathcal{T}}$ in \cite[equation 5.1]{FP19}, the higher-dimensional analogue of \eqref{identity} given in \cite[Lemma 5.4]{FP19}, and the techniques of estimation of \cite[Section 5]{FP19}.
\end{remark}

\subsection{Boundedness of the canonical solution on polydiscs: smooth data}

Let us consider again the unit polydisc $D$ in $\C^2$. The argument will be analogous for higher dimensional polydiscs, with the appropriate changes according to the generalizations of Propositions \ref{t1} and \ref{t2}. Let $f=f_1d\bar{z}_1+f_2 d\bar{z}_2$ be a $\bar\partial$-closed $(0,1)$ form on $D$, with components $f_j\in C^{\infty}(\overline{D})$. We consider  
\begin{equation}\label{dibarproblem}
\bar\partial u=f\quad\text{in }D.
\end{equation}
In \cite{PZ}, Pan and Zhang constructed solutions to \eqref{dibarproblem} (on more general product domains) satisfying H\"older estimates. They also proved that the solution ${\bf \mathcal{T}}f$ is in $C^{\infty}(\overline{D})$ whenever $f$ is \cite[Lemma 3.5] {PZ}.
Hence, letting $u={\bf \mathcal{T}}f$, we can apply Proposition \ref{t1} to obtain
\begin{equation*}
\int_{\partial D_1\times \partial D_2}\frac{u(\zeta_1,\zeta_2)}{(\zeta_1-z_1)(\zeta_2-z_2)}\, d\zeta_1\wedge d\zeta_2=0.
\end{equation*}
By Proposition \ref{t2},
\begin{equation*}
\norm{Pu}_{L^{\infty}(D)}\leq C\norm{f}_{L^{\infty}(D)}.
\end{equation*}
Since the canonical solution $\tilde{u}$ of \eqref{dibarproblem} can be written as 
\begin{equation*}
\tilde{u}=u-Pu,
\end{equation*}
and both $u$ and $Pu$ satisfy supnorm estimates, then $\tilde{u}$ does as well. 
\begin{remark}
In \cite{L75} the datum $f$ is just assumed in $C^1(D)$. Under these assumptions, the solution ${\bf \mathcal{T}}f$ is in $C^2(D)$ by \cite{Hen71}. The argument above can still be applied, but a scaling argument is necessary (see \cite[pages 236--237]{L75}).
\end{remark}

\end{appendices}


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