# Uniform estimates of the Cauchy-Riemann equation on product domains

Yuan Yuan  Department of Mathematics, Syracuse University, Syracuse, NY 13244,
USA<EMAIL_ADDRESS>
###### Abstract.

We observe that the continuity assumption on $f$ for the uniform estimates of
the canonical solution to $\bar{\partial}u=f$ on products of $C^{2}$ bounded
planar domains in [9] can be reduced to the boundedness assumption. This
completely answers the original question raised by Kerzman in 1971. Moreover,
the $L^{p}$ estimates of $\bar{\partial}$ is obtained for all
$p\in[1,\infty]$.

##  1\. Introduction

The estimates of the $\bar{\partial}$-equation in various function spaces are
one of the most important problems in several complex variables and partial
differential equations and have invaluable applications in differential
geometry, algebraic geometry and other subjects (cf. [3, 7, 13, 22, 27, 29,
33]). In this paper, we aim to answer the following fundamental question of
the uniform estimates of the $\bar{\partial}$-equation raised by Kerzman in
1971 (cf. [24] pp. 311-312). This question recently was reminisced in [5] (cf.
lines 22-24 on page 409) and attracted substantial attentions.

Remark in [24] (cf. pp.311-312): We do not know whether Grauert-Lieb’s and
Henkin’s theorem holds in polydiscs, i.e., whether there exists a bounded
solution $u$ to $\bar{\partial}u=f$ on $\Delta^{n}$ whenever $f$ is bounded in
$\Delta^{n}$, $\bar{\partial}f=0$.

On bounded strictly pseudoconvex domains in $\mathbb{C}^{n}$, the uniform
estimate for $\bar{\partial}$-equation is obtained by Grauert-Lieb [17] and
Henkin [20] in 1970. Sibony later constructed a smooth bounded weakly
pseudoconvex domain in $\mathbb{C}^{3}$ and a $\bar{\partial}$-closed
$(0,1)$-form $f$, continuous on the closure of the domain, such that every
solution of $\bar{\partial}u=f$ is unbounded [32] (cf. [2, 16] for examples in
$\mathbb{C}^{2}$). In 1986 Fornæss proved uniform estimates for a class of
pseudoconvex domains in $\mathbb{C}^{2}$, which include the Kohn-Nirenberg
example [14]. The uniform estimates for finite type domains in
$\mathbb{C}^{2}$ and convex, finite type domains in $\mathbb{C}^{n}$ are
solved by Fefferman-Kohn [12] (cf. [29] as well) and Diederich-Fischer-Fornæss
[8]. More recently, Grundmeier-Simon-Stensønes proved the uniform estimates
for a wide class of finite type pseudoconvex domains in $\mathbb{C}^{n}$,
including the bounded, pseudoconvex domains with real-analytic boundary [18].
The uniform estimates for $\bar{\partial}$-equation has been an attractive
problem for many authors and we refer the interested readers to [15, 19, 24,
30, 31] and references therein for detailed account of the subject and related
problems.

On the other hand, when the domain is not smooth, in particular, a product
domain, the problem becomes somehow different. In 1971, Henkin obtained the
uniform estimates for $\bar{\partial}u=f$ on the bidisk provided that $f$ is
$C^{1}$ up to the boundary [21]. Landucci in 1975 proved the uniform estimates
for the canonical solution on the bidisc provided that $f$ is $C^{2}$ up to
the boundary [26]. More recently, a very useful new solution integral operator
was used by Chen-McNeal to prove many interesting results on product domains,
including the $L^{p}$ estimates for $\bar{\partial}$-equation [5, 6], and also
by Fassina-Pan to prove the uniform estimates for $\bar{\partial}$-equation on
the high dimensional product of planar domains [11]. Dong-Pan-Zhang later also
apply the integral operator to further obtain the uniform estimates for the
canonical solution to $\bar{\partial}u=f$ on the product of planar domains by
assuming $f$ is merely continuous up to the boundary [9]. This is not only a
deep result but also the proof contains fascinating ideas by combining the
above-mentioned new integral operator and Kerzman’s celebrated estimates of
the Green function [25] as observed in [1]. For more related studies of the
$\bar{\partial}$-equation on product domains, the interested readers may refer
to [4, 23, 28, 34, 35] and references therein.

Heavily relying on the ideas developed in [9], we are able to answer Kerzman’s
original question. The key difference is that in [9], the canonical solution
is re-written using the integral formula against $f$, which contains certain
boundary integrals. This requires $f$ to be at least continuous to make sense
of the boundary integral. However, we observe that the boundary integrals
actually do not appear because the integral kernels vanish on the boundary.
This already provides $L^{p}$ estimates when $f$ is sufficiently smooth,
combining the deep estimates of the Green function by Kerzman. When $f$ is
merely $L^{p}$, the estimates is achieved by using approximation as in [9].

The main theorem of the paper is the following $L^{p}$ estimates of
$\bar{\partial}$.

######  Theorem 1.1.

Let $\Omega=D_{1}\times\cdots\times D_{n}$, where, for each $1\leq j\leq n$,
$D_{j}$ is a $C^{2}$-smooth bounded planar domain. For any $p\in[1,\infty]$,
assume $f\in L^{p}_{(0,1)}(\Omega)$. Then there exists a constant
$C_{\Omega}>0$ (independent of $p$) such that the canonical solution to
$\bar{\partial}u=f$ satisfies $\|{\bf T}f\|_{L^{p}}\leq
C_{\Omega}\|f\|_{L^{p}}$.

In particular, when $\Omega=\Delta^{n}$ is the polydisc, this answers
Kerzman’s question.

######  Corollary 1.2.

There exists a constant $C>0$ such that for any $f\in
L^{\infty}_{(0,1)}(\Delta^{n})$ with $\bar{\partial}f=0$, the canonical
solution to $\bar{\partial}u=f$ satisfies $\|u\|_{\infty}\leq
C\|f\|_{\infty}$.

##  2\. Proof of the Theorem

The majority of the proof was already carried out by Dong-Pan-Zhang in [9]. We
try to make the argument here as self-contained as possible.

###  2.1. One dimensional case

We follow the argument by Barletta-Landucci [1] and Dong-Pan-Zhang in [9]. Let
$D\subset\mathbb{C}$ be a bounded planar with $C^{2}$ boundary and
$H(w,z)=\frac{1}{2\pi i(z-w)}$ be the Cauchy kernel on $D$. The following
kernel is defined by Barletta-Landucci [1],

$S(w,z)=L(w.z)-H(w,z),$

where for any $w\in D$, $L(w,z)$ solves the Dirichlet problem

$\left\\{\begin{aligned} \Delta L(w,z)&=0,\quad z\in D;\\\
L(w,z)&=H(w,z),\quad z\in\partial D.\end{aligned}\right.$

It is known that for fixed $w\in D$, $L(w,z)\in C^{1,\alpha}(\overline{D})$
for any $\alpha\in(0,1)$ (cf. [9]). Define ${\bf
T}f(w)=\int_{D}S(w,z)f(z)d\bar{z}\wedge dz$. Then ${\bf T}$ is the canonical
solution operator for $\bar{\partial}u=fd\bar{z}$ on $D$ (cf. Theorem in [1]
or Proposition 2.3 in [9]).

Here is the key observation: $S(w,z)=0$ for any $w\in D,z\in\partial D$.

######  Example 1.

Let $D=\Delta$ be the unit disc in $\mathbb{C}$. Then it is very easy to
verify that $L(w,z)=\frac{\bar{z}}{2\pi i(1-w\bar{z})}$ and thus
$S(w,z)=\frac{1-|z|^{2}}{2\pi i(1-w\bar{z})(w-z)}$ vanishes for
$z\in\partial\Delta,w\not=z$.

###  2.2. High dimensional case when $f$ is sufficiently smooth

Let $\Omega=D_{1}\times\cdots\times D_{n}\subset\mathbb{C}^{n}$ be a bounded
domain, where each $D_{j}$ is a bounded planar domain with $C^{2}$ boundary,
$1\leq j\leq n$. We first handle the case when $f$ is sufficiently smooth.
Assume $f=\sum_{j=1}^{n}f_{j}d\bar{z}_{j}$ to be a $\bar{\partial}$-closed
$(0,1)$-form on $\Omega$ with $f_{j}\in C^{n-1}(\overline{\Omega})$, $1\leq
j\leq n$, and define

(2.1) ${\bf T}f=\sum_{s=1}^{n}(-1)^{s-1}\sum_{1\leq i_{1}\leq\cdots\leq
i_{s}\leq n}{\bf T}_{i_{1}}\cdots{\bf
T}_{i_{s}}\left(\frac{\partial^{s-1}f_{i_{s}}}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{s-1}}}\right),$

where ${\bf T}_{j}$ is the canonical solution operator on $D_{j}$. Let

$B_{1,2,\cdots,k}(w,z)=\sum_{j=1}^{k}\prod_{m\not=j,m=1}^{k}|w_{m}-z_{m}|^{2},$
$e^{1,2,\cdots,k}_{j}(w,z)=\left(\prod_{l=1}^{k}S_{l}(w_{l},z_{l})\right)\frac{\prod_{m\not=j,m=1}^{k}|w_{m}-z_{m}|^{2}}{B_{1,2,\cdots,k}(w,z)}.$

Then $\sum_{j=1}^{k}e^{1,2,\cdots,k}_{j}=\prod_{l=1}^{k}S_{l}(w_{l},z_{l})$.
Here is the key lemma.

######  Lemma 2.1.

Then

(2.2) $\int_{D_{1}\times\cdots\times
D_{k}}e^{1,2,\cdots,k}_{k}(w,z)\frac{\partial^{k-1}f_{k}(z)}{\partial\bar{z}_{1}\cdots\partial\bar{z}_{k-1}}dV(z)=\int_{D_{1}\times\cdots\times
D_{k}}(-1)^{k}f_{k}(z)\frac{\partial^{k-1}e^{1,2,\cdots,k}_{k}(w,z)}{\partial\bar{z}_{1}\cdots\partial\bar{z}_{k-1}}dV(z).$

###### Proof.

Fix $w\in D_{1}\times\cdots\times D_{k}$. Let
$A_{j}^{\epsilon}=D_{j}\setminus\Delta_{\epsilon}(w_{j})$ for $0<\epsilon<<1$.
Note that for any $0\leq m\leq k-2,1\leq i_{1}<\cdots<i_{m}\leq k-2$ and
$l\not\in\\{i_{1},\cdots,i_{m},k\\}$,
$\frac{\partial^{m}e^{1,2,\cdots,k}_{k}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{m}}}=0$
on $D_{1}\times\cdots\times\partial D_{l}\times\cdots\times D_{k}$ and
$\frac{\partial^{m}e^{1,2,\cdots,k}_{k}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{m}}}\rightarrow
0$ uniformly for $z\in
D_{1}\times\cdots\times\partial\Delta_{\epsilon}(w_{l})\times\cdots\times
D_{k}$ as $\epsilon\rightarrow 0$.

Applying the Stokes formula to

$e^{1,2,\cdots,k}_{k}(w,z)\frac{\partial^{k-2}f_{k}(z)}{\partial\bar{z}_{2}\cdots\partial\bar{z}_{k-1}}dz_{1}\wedge
d\bar{z}_{2}\wedge dz_{2}\wedge\cdots\wedge d\bar{z}_{k}\wedge dz_{k}$

on $A_{1}^{\epsilon}\times D_{2}\times\cdots\times D_{k}$, we have

$\begin{split}&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\int_{A_{1}^{\epsilon}\times\cdots\times
D_{k}}\left(e^{1,2,\cdots,k}_{k}(w,z)\frac{\partial^{k-1}f_{k}(z)}{\partial\bar{z}_{1}\cdots\partial\bar{z}_{k-1}}+\frac{\partial
e^{1,2,\cdots,k}_{k}(w,z)}{\partial\bar{z}_{1}}\frac{\partial^{k-2}f_{k}(z)}{\partial\bar{z}_{2}\cdots\partial\bar{z}_{k-1}}\right)d\bar{z}_{1}\wedge
dz_{1}\wedge\cdots\wedge d\bar{z}_{k}\wedge dz_{k}\\\ &=\left(\int_{\partial
D_{1}\times D_{2}\times\cdots\times
D_{k}}-\int_{\partial\Delta_{\epsilon}(w_{j})\times D_{2}\times\cdots\times
D_{k}}\right)e^{1,2,\cdots,k}_{k}(w,z)\frac{\partial^{k-2}f_{k}(z)}{\partial\bar{z}_{2}\cdots\partial\bar{z}_{k-1}}dz_{1}\wedge
d\bar{z}_{2}\wedge dz_{2}\wedge\cdots\wedge d\bar{z}_{k}\wedge dz_{k}\\\
&=-\int_{\partial\Delta_{\epsilon}(w_{j})\times D_{2}\times\cdots\times
D_{k}}e^{1,2,\cdots,k}_{k}(w,z)\frac{\partial^{k-2}f_{k}(z)}{\partial\bar{z}_{2}\cdots\partial\bar{z}_{k-1}}dz_{1}\wedge
d\bar{z}_{2}\wedge dz_{2}\wedge\cdots\wedge d\bar{z}_{k}\wedge
dz_{k},\end{split}$

as $e^{1,2,\cdots,k}_{k}(w,z)=0$ for $z_{1}\in\partial D_{1}$. As
$\epsilon\rightarrow 0$, $e^{1,2,\cdots,k}_{k}(w,z)\rightarrow 0$ uniformly
for $z\in\Delta_{\epsilon}(w_{j})\times D_{2}\times\cdots\times D_{k}$. It
follows that

$\int_{D_{1}\times\cdots\times
D_{k}}e^{1,2,\cdots,k}_{k}(w,z)\frac{\partial^{k-1}f_{k}(z)}{\partial\bar{z}_{1}\cdots\partial\bar{z}_{k-1}}dV(z)=-\int_{D_{1}\times\cdots\times
D_{k}}\frac{\partial
e^{1,2,\cdots,k}_{k}(w,z)}{\partial\bar{z}_{1}}\frac{\partial^{k-2}f_{k}(z)}{\partial\bar{z}_{2}\cdots\partial\bar{z}_{k-1}}dV(z).$

The lemma then follows by applying the argument repeatedly in all other
variables $z_{2},\cdots,z_{k-1}$. ∎

The next result provides $L^{p}$-estimates for $\bar{\partial}u=f$ when $f$ is
smooth, where the key is an important application of the Kerzman’s estimates
of the Green function proved in [9] (cf. Proposition 4.5). Since the integrals
here do not involve the boundary terms, the corresponding estimates become
simpler.

######  Proposition 2.2.

For $f\in C^{n-1}_{(0,1)}(\overline{\Omega})$ with $\bar{\partial}f=0$, let

${\bf K}f(w)=\sum_{s=1}^{n}\sum_{1\leq i_{1}\leq\cdots\leq i_{s}\leq
n}\sum_{j=1}^{s}\int_{D_{i_{1}}\times\cdots\times
D_{i_{s}}}f_{i_{j}}(z)\frac{\partial^{s-1}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}d\bar{z}_{i_{1}}\wedge\cdots
dz_{i_{s}}.$

Then ${\bf T}f={\bf K}f.$ Moreover, for any $p\in[1,\infty]$, there exists a
universal constant $C_{\Omega}>0$ (independent of $p$) such that $u={\bf T}f$
is the canonical solution to $\bar{\partial}u=f$ and satisfies $\|{\bf
T}f\|_{L^{p}}\leq C_{\Omega}\|f\|_{L^{p}}$.

###### Proof.

Since $f$ is $\bar{\partial}$-closed,
$\frac{\partial^{s-1}f_{i_{1}}}{\partial\bar{z}_{i_{2}}\cdots\partial\bar{z}_{i_{s}}}=\frac{\partial^{s-1}f_{i_{2}}}{\partial\bar{z}_{i_{1}}\partial\bar{z}_{i_{3}}\cdots\partial\bar{z}_{i_{s}}}=\cdots=\frac{\partial^{s-1}f_{i_{s}}}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{s-1}}}$.
It follows that

$\left(\frac{\partial^{s-1}f_{i_{s}}(z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{s-1}}}\right)\prod_{j=i_{1}}^{i_{s}}S_{j}(w_{j},z_{j})=\sum_{j=1}^{s}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)\frac{\partial^{s-1}f_{i_{j}}(z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}$

and thus by Lemma 2.1,

${\bf T}_{i_{1}}\cdots{\bf
T}_{i_{s}}\left(\frac{\partial^{s-1}f_{i_{s}}}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{s-1}}}\right)=(-1)^{s-1}\sum_{j=1}^{s}\int_{D_{i_{1}}\times\cdots\times
D_{i_{s}}}f_{i_{j}}(z)\frac{\partial^{s-1}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}d\bar{z}_{i_{1}}\wedge\cdots
dz_{i_{s}}.$

Moreover, we have

${\bf T}f=\sum_{s=1}^{n}\sum_{1\leq i_{1}\leq\cdots\leq i_{s}\leq
n}\sum_{j=1}^{s}\int_{D_{i_{1}}\times\cdots\times
D_{i_{s}}}f_{i_{j}}(z)\frac{\partial^{s-1}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}d\bar{z}_{i_{1}}\wedge\cdots
dz_{i_{s}}.$

By Proposition 4.5 in [9],
$\frac{\partial^{s-1}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}\in
L^{1}(D_{i_{1}}\times\cdots\times D_{i_{s}})$ and the $L^{1}$ norm is
independent of $w$. Therefore, $L^{p}$ estimates $\|{\bf T}f\|_{L^{p}}\leq
C_{\Omega}\|f\|_{L^{p}}$ follows from Young’s convolution inequality.
Furthermore, It follows from Theorem 4.3 in [9] that ${\bf T}f$ is the
canonical solution. ∎

###  2.3. High dimensional case when $f$ is $L^{p}$

In this section, we will use approximation to handle the case when $f\in
L^{p}_{(0,1)}(\Omega)$ for $p\in[1,\infty]$. In fact, we will show the
following general $L^{p}$ estimates.

######  Theorem 2.3.

For any $p\in[1,\infty]$, assume $f\in L^{p}_{(0,1)}(\Omega)$. Then $u={\bf
K}f$ is the canonical solution to $\bar{\partial}u=f$ and satisfies $\|{\bf
K}f\|_{L^{p}}\leq C_{\Omega}\|f\|_{L^{p}}$.

###### Proof.

We note by the same estimates that ${\bf K}$ is bounded from $L^{p}$ to
$L^{p}$. It suffices to show that $u={\bf K}f$ is the canonical solution. This
is proved in Proposition 5.1 and Theorem 1.1 in [9]. For completeness, we
sketch the proof here. First, let $\\{\Omega^{(l)}\\}$ be an increasing
sequence of relatively compact subdomains of $\Omega$ with each $\Omega^{(l)}$
being the product of $C^{2}$ planar domains such that
$\cup_{l}\Omega^{(l)}=\Omega$. Let $e^{(l)},{\bf K}^{(l)}$ be defined
accordingly on $\Omega^{(l)}$. Then it was showed in [9] (cf. equation (5.8))
using Kerzman’s deep estimates of the Green function that

(2.3)
$\frac{\partial^{s-1}\left(e^{(l)}\right)^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}\rightarrow\frac{\partial^{s-1}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}$

as $l\rightarrow\infty$. Moreover, write
$\Omega_{\epsilon}=\\{z\in\Omega:\text{dist}(z,\partial\Omega)>\epsilon\\}$.
Since $f\in L^{p}_{(0,1)}(\Omega)$ for any $p\geq 1$, by the standard
mollification argument, there exists a sequence of $\bar{\partial}$-closed
smooth $(0,1)$-form $f^{\epsilon}$ in $\Omega_{\epsilon}$, such that
$f^{\epsilon}\rightarrow f$ in $L^{p}_{(0,1)}(\Omega)$ for all $p>1$. Second,
let $\chi$ be a $(n,n-1)$-form in $\Omega$ with compact support. Choosing $l$
sufficiently large and $\epsilon$ sufficiently small, such that
$supp(\chi)\subset\Omega^{(l)}\subset\overline{\Omega^{(l)}}\subset\Omega_{\epsilon}$,
then $f^{\epsilon}\in C^{\infty}_{(0,1)}(\overline{\Omega^{(l)}})$ and thus
$u={\bf K}^{(l)}f^{\epsilon}$ is the canonical solution to
$\bar{\partial}u=f^{\epsilon}$ on $\Omega^{(l)}$ by Proposition 2.2. It
follows that

$\begin{split}\int_{\Omega}\bar{\partial}{\bf
K}f\wedge\chi&=-\int_{\Omega^{(l)}}{\bf
K}f\wedge\bar{\partial}\chi=-\lim_{l\rightarrow\infty}\lim_{\epsilon\rightarrow
0}\int_{\Omega^{(l)}}{\bf K}^{(l)}f^{\epsilon}\wedge\bar{\partial}\chi\\\
&=\lim_{l\rightarrow\infty}\lim_{\epsilon\rightarrow
0}\int_{\Omega^{(l)}}f^{\epsilon}\wedge\chi=\lim_{\epsilon\rightarrow
0}\int_{\Omega}f^{\epsilon}\wedge\chi=\int_{\Omega}f\wedge\chi,\end{split}$

where the second inequality follows from (2.3). This implies that
$\bar{\partial}{\bf K}f=f$ weakly. Lastly, assume $g$ to be a $L^{2}$
holomorphic function in $\Omega$. Then similarly

$\int_{\Omega}{\bf
K}f(z)\overline{g(z)}dV(z)=\lim_{l\rightarrow\infty}\lim_{\epsilon\rightarrow
0}\int_{\Omega^{(l)}}{\bf
K}^{(l)}f^{\epsilon}(z)\overline{g(z)}dV(z)=\lim_{l\rightarrow\infty}\lim_{\epsilon\rightarrow
0}0=0.$

This means that $u={\bf K}f$ is the canonical solution to $\bar{\partial}u=f$.
∎

######  Remark 2.4.

When $\Omega$ is the product of star-shaped planar domains, for instance,
$\Omega=\Delta^{n}$, there is no need to approximate $\Omega$ by
$\Omega^{(l)}$ as above. By the standard mollification argument, any $L^{p}$
integrable $\bar{\partial}$-closed $(0,1)$-form $f$ on $\Omega$ can be
approximated in $L^{p}_{(0,1)}(\Omega)$ by a sequence of
$\bar{\partial}$-closed, smooth $(0,1)$-forms $\\{f^{\epsilon}\\}$ on
$\Omega$. Therefore that ${\bf K}f=\lim_{\epsilon\rightarrow 0}{\bf
K}f^{\epsilon}$ is the canonical solution is the direct consequence of the
boundedness of ${\bf K}$ from $L^{p}$ to $L^{p}$. On the other hand, in the
case of $\Omega=\Delta^{n}$, it can be verified directly without using
Kerzman’s estimates that
$\frac{\partial^{s-1}e^{i_{1},\cdots,i_{s}}_{i_{j}}(w,z)}{\partial\bar{z}_{i_{1}}\cdots\partial\bar{z}_{i_{j-1}}\partial\bar{z}_{i_{j+1}}\cdots\partial\bar{z}_{i_{s-1}}}\in
L^{1}(\Delta_{i_{1}}\times\cdots\times\Delta_{i_{s}})$ and the $L^{1}$ norm is
also independent of $w$, and therefore ${\bf K}$ is bounded from $L^{p}$ to
$L^{p}$ on $\Delta^{n}$.

Acknowledgement: The author would like to thank Zhenghui Huo for helpful
discussions. The work was done when the author was visiting BICMR in Spring
2022. He thanks the center for providing him the wonderful research
environment.

## References

  * [1] Barletta, E.; Landucci, M. Optimal $L^{\infty}$ estimates for the canonical solution of the CR-equation in the nonsmooth case, Complex Variables Theory Appl. 16 (1991), no. 2-3, 93-106.
  * [2] Berndtsson, B. A smooth pseudoconvex domain in $C^{2}$ for which $L^{\infty}$-estimates for $\bar{\partial}$ do not hold, Ark. Mat. 31 (1993), 209-218.
  * [3] Boas, H. P.; Straube, E. J. Global regularity of the $\bar{\partial}$-Neumann problem: a survey of the $L^{2}$-Sobolev theory, Several complex variables (Berkeley, CA, 1995-1996), 79-111, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999.
  * [4] Chakrabarti, D.; Shaw, M.-C. The Cauchy-Riemann equations on product domains, Math. Ann. 349 (2011), no. 4, 977-998.
  * [5] Chen, L.; McNeal, J. D. A solution operator for $\bar{\partial}$ on the Hartogs triangle and $L^{p}$ estimates, Math. Ann. 376 (2020), no. 1-2, 407-430.
  * [6] Chen, L.; McNeal, J. D. Product domains, Multi-Cauchy transforms, and the $\bar{\partial}$ equation, Adv. Math. 360 (2020), 106930, 42 pp.
  * [7] Chen, S.-C.; Shaw, M.-C. Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, 19. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. xii+380 pp. ISBN: 0-8218-1062-6
  * [8] Diederich, K., Fischer, B., Fornæss, J.E. Hölder estimates on convex domains of finite type, Math. Z. 232(1), 43-61 (1999)
  * [9] Dong, X.; Pan, Y.; Zhang, Y. Uniform estimates for the canonical solution to the $\bar{\partial}$-equation on product domains, arXiv:2006.14484 (with an appendix by Martino Fassina).
  * [10] Evans, L. C. Partial differential equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp. ISBN: 978-0-8218-4974-3
  * [11] Fassina, M.; Pan, Y. Supnorm estimates for $\bar{\partial}$ on product domains in $C^{n}$, arXiv:1903.10475v3
  * [12] Fefferman, C. L. and Kohn, J. J. Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds, Adv. in Math. 69 (1988), no. 2, 223-303.
  * [13] Folland, G. B.; Kohn, J. J. The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. viii+146 pp.
  * [14] Fornæss, J. E. Sup-norm estimates for $\bar{\partial}$ in $C^{2}$, Ann. of Math. (2) 123 (1986), no. 2, 335-345.
  * [15] Fornæss, J. E.; Lee, L.; Zhang, Y. On supnorm estimates for $\bar{\partial}$ on infinite type convex domains in $C^{2}$, J. Geom. Anal. 21 (2011), no. 3, 495-512.
  * [16] Fornæss, J. E.; Sibony, N. Smooth pseudoconvex domains in $C^{2}$ for which the corona theorem and $L^{p}$ estimates for $\bar{\partial}$ fail, Complex analysis and geometry, 209-222, Univ. Ser. Math., Plenum, New York, 1993.
  * [17] Grauert, H., Lieb, I. Das Ramirezsche Integral und die Lösung der Gleichung $\bar{\partial}f=\alpha$ im Bereich der beschränkten Formen, Rice Univ. Stud. 56(2), 29-50 (1970), (1971).
  * [18] Grundmeier, D.; Simon, L.; Stensønes, B. Sup-norm Estimates for $\bar{\partial}$ in $C^{3}$, Pure Appl. Math. Q. 18 (2022), no. 2, 531-571.
  * [19] Henkin, G.M. Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications, Mat. Sb. 78(120), 611-632 (1969)
  * [20] Henkin, G.M. Integral representation of functions in strictly pseudoconvex domains and applications to the $\bar{\partial}$-problem, Mat. Sb. 124(2), 300-308 (1970)
  * [21] Henkin, G.M. A uniform estimate for the solution of the $\bar{\partial}$-problem in a Weil region, Usp. Mat. Nauk 26(3(159)), 211-212 (1971).
  * [22] Hörmander, L. An introduction to complex analysis in several variables, Third edition. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990.
  * [23] Jin, M.; Yuan, Y. On the canonical solution of $\bar{\partial}$ on polydisks, C. R. Math. Acad. Sci. Paris 358 (2020), no. 5, 523-528.
  * [24] Kerzman, N. Hölder and $L^{p}$ estimates for solutions of $\bar{\partial}u=f$ in strongly pseudoconvex domains, Comm. Pure Appl. Math. 24 (1971), 301-379.
  * [25] Kerzman, N. Topics in complex analysis: two estimates for Green’s function, unpublished notes, MIT, 1975/1976.
  * [26] Landucci, M. On the projection of $L^{2}(D)$ into $H(D)$, Duke Math. J. 42 (1975), 231-237.
  * [27] Ohsawa, T. $L^{2}$ approaches in several complex variables. Towards the Oka-Cartan theory with precise bounds, Second edition. Springer Monographs in Mathematics. Springer, Tokyo, 2018. xi+258 pp. ISBN: 978-4-431-56851-3; 978-4-431-56852-0
  * [28] Pan, Y.; Zhang, Y. Hölder estimates for the $\bar{\partial}$ problem for $(p,q)$ forms on product domains, Internat. J. Math. 32 (2021), no. 3, 2150014, 20 pp.
  * [29] Range, R.M. Integral kernels and Hölder estimates for $\bar{\partial}$ on pseudoconvex domains of finite type in $C^{2}$, Math. Ann. 288(1), 63-74 (1990).
  * [30] Range, R. M. Holomorphic functions and integral representations in several complex variables, Grad. Texts in Math., 108, Springer-Verlag, New York, 1986. 2nd corrected printing 1998.
  * [31] Range, R. M.; Siu, Y.-T. Uniform estimates for the $\bar{\partial}$-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 206 (1973), 325-354.
  * [32] Sibony, N. Un exemple de domaine pseudoconvexe regulier où l’équation $\bar{\partial}u=f$ n’admet pas de solution bornée pour f bornée, Invent. Math. 62 (1980), 235-242.
  * [33] Straube, E. J. Lectures on the $L^{2}$-Sobolev theory of the $\bar{\partial}$-Neumann problem, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2010. viii+206 pp. ISBN: 978-3-03719-076-0
  * [34] Yuan, Y.; Zhang, X.; Zhang, Y. Weighted Sobolev estimates of $\bar{\partial}$ on domains covered by polydiscs, submitted
  * [35] Zhang, Y. Optimal Hölder regularity for the $\bar{\partial}$ problem on product domains in $C^{2}$, Proc. Amer. Math. Soc. 150 (2022), no. 7, 2937-2943.