# Big Hankel operators on Hardy spaces of strongly pseudoconvex domains

Bo-Yong Chen  Bo-Yong Chen
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
<EMAIL_ADDRESS>and  Liangying Jiang  Liangying Jiang
Department of Statistics and Mathematics, Shanghai Lixin University of
Accounting and Finance, Shanghai 201209, China<EMAIL_ADDRESS>
###### Abstract.

In this article, we investigate the (big) Hankel operators $H_{f}$ on Hardy
spaces of strongly pseudoconvex domains with smooth boundaries in
$\mathbb{C}^{n}$. We also give a necessary and sufficient condition for
boundedness of the Hankel operator $H_{f}$ on the Hardy space of the unit
disc, which is new in the setting of one variable.

###### Key words and phrases:

Hankel operator, Hardy space, strongly pseudoconvex domain

##  1\. Introduction

Let $\Omega=\\{z\in\mathbb{C}^{n}:\rho(z)<0\\}$ be a strongly pseudoconvex
domain with smooth boundary, where $\rho(z)\in C^{\infty}(\mathbb{C}^{n})$ is
a strictly plurisubharmonic defining function of $\Omega$, with $d\rho\neq 0$
in some neighborhood of $\partial\Omega$.

Let $d\sigma$ denote the surface measure on $\partial\Omega$ induced by the
Lebesgue measure of $\mathbb{C}^{n}$. For $0<p<\infty$, we denote
$L^{p}(\partial\Omega)$ the usual $L^{p}$ space on $\partial\Omega$ with
respect to the measure $d\sigma$. Let $H^{p}(\Omega)$ denote the Hardy space
of holomorphic functions on $\Omega$, whose norm is given by

$\|f\|^{p}_{H^{p}}:=\sup\limits_{\varepsilon>0}\int_{\partial\Omega_{\varepsilon}}|f|^{p}d\sigma_{\varepsilon},$

where $\Omega_{\varepsilon}=\\{z\in\mathbb{C}^{n}:\rho(z)<-\varepsilon\\}$ and
$d\sigma_{\varepsilon}$ is the surface measure on
$\partial\Omega_{\varepsilon}$. It is well-know that any $f\in H^{p}(\Omega)$
has a radial limit at almost all points on $\partial\Omega$ (cf. [42]). Thus
one can identify $H^{p}(\Omega)$ as a closed subspace of
$L^{p}(\partial\Omega)$. Let $H^{\infty}(\Omega)$ denote the space of bounded
holomorphic functions in $\Omega$.

Let $P:L^{2}(\partial\Omega)\rightarrow H^{2}(\Omega)$ be the orthogonal
projection via the Szegö kernel $S(z,w)$, that is

$Pg(z)=\int_{\partial\Omega}S(z,\zeta)g(\zeta)d\sigma(\zeta)$

for $g\in L^{2}(\partial\Omega)$. We may regard this operator as a singular
integral operator on $\partial\Omega$, for $S(z,w)$ is $C^{\infty}$ on
$\partial\Omega\times\partial\Omega\setminus\Delta$ (where $\Delta$ is the
diagonal of $\partial\Omega\times\partial\Omega$). We also define the Poisson-
Szegö kernel by $P(z,w)=S(z,z)^{-1}|S(z,w)|^{2}$ for $z\in\Omega$ and
$w\in\partial\Omega$ (see [42]).

For $f\in L^{2}(\partial\Omega)$, the (big) Hankel operator $H_{f}$ with
symbol $f$ is the operator from $H^{2}(\Omega)$ into $(H^{2}(\Omega))^{\bot}$
defined by

$H_{f}g=(I-P)(fg),\ \ g\in H^{2}(\Omega).$

We may identify $H_{f}$ with the operator $(I-P)M_{f}P$ on the Hilbert space
$L^{2}(\partial\Omega)$, where $M_{f}$ is the multiplication operator given by
$(M_{f}g)(z)=f(z)g(z)$, $z\in\overline{\Omega}$. If $f$ is bounded on
$\partial\Omega$, then clearly $H_{f}$ is bounded with
$\|H_{f}\|\leq\|f\|_{\infty}$. In general, $H_{f}$ is only densely defined,
whose domain contains $H^{\infty}(\Omega)$.

This operator is a generalization of the Hankel operator in the case of the
unit disc $D$, which is different from the (small) Hankel operator $h_{f}$
defined by
$h_{f}g=\overline{P(fg)}:H^{2}(\Omega)\rightarrow\overline{H_{0}^{2}(\Omega)}$,
where $\overline{H_{0}^{2}(\Omega)}$ is the space of complex conjugate of
functions in $H^{2}(\Omega)$ which are zero at the origin of $\mathbb{C}^{n}$.
The theory of Hankel operators of the unit disc is classical. There are
several books dealing with Hankel operators on the Hardy space $H^{2}(D)$, see
[32], [35], [36] and [48] for details.

For the unit disc, the characterization of bounded Hankel operators on the
Hardy space is due to Nehari [30]. The compactness is given by Hartman [20].
Schatten class Hankel operators is characterized by Peller [33], [34] and
Semmes [40].

The extension of these results to higher dimensional domains for (big) Hankel
operators is proved to be resistant in the case of the unit ball $B_{n}$. A
fundamental result due to Coifman, Rochberg and Weiss [13] asserts that
$[M_{f},P]=M_{f}P-PM_{f}$ is bounded on $L^{2}(\partial B_{n})$ for $f\in
BMO$, and $[M_{f},P]$ is compact if $f\in VMO$. The commutator result implies
that $H_{f}=[M_{f},P]P$ is bounded on $L^{2}(\partial B_{n})$ for every $f\in
BMO$ and $H_{f}$ is compact if $f\in VMO$. In fact, by virtue of the relation

$[M_{f},P]=H_{f}-H_{\overline{f}}^{*},$

the study of the commutator $[M_{f},P]$ is equivalent to the problem
concerning both $H_{f}$ and $H_{\overline{f}}$.

The study of $H_{f}$ alone is usually more difficult than corresponding
problems concerning $H_{f}$ and $H_{\overline{f}}$ simultaneously. Since
$H_{f}=H_{f-Pf}$, properties of the Hankel operator $H_{f}$ for general $f\in
L^{2}(\partial\Omega)$ are recovered by properties of $f-Pf$.

For $f\in L^{2}(\partial B_{n})$, it was showed in [46] that $H_{f}$ is
bounded on the Hardy space $H^{2}(B_{n})$ if and only if $f-Pf\in BMO$, and
$H_{f}$ is compact if and only if $f-Pf\in VMO$. Moreover, in [15], the
authors proved that $H_{f}$ belongs to the Schatten class $S_{p}$,
$2n<p<\infty$, if and only if $f$ lies in the Besov space $B_{p}$. Other
characterizations for the Hankel operator $H_{f}$ on the Hardy space
$H^{2}(B_{n})$ were given in [16], [45] and [47].

In particular, Amar [2] established links between the Hankel operator $H_{f}$
on the Hardy space $H^{2}(B_{n})$ and the $\overline{\partial}$ of a Stokes
extension of $f$ in $B_{n}$, which is also the motivation of our work. In
addition, we mention the work of Li and Luecking (see [20-22] and [28]), where
the theory of $\overline{\partial}$-operator is used to give complete
characterizations of boundedness, compactness and Schatten ideals of Hankel
operators acting on Bergman spaces of strongly pseudoconvex domains.

In this article, we try to give some characterizations for boundedness of
Hankel operators on the Hardy space $H^{2}(\Omega)$, based on the theory of
$BMO$, $VMO$, and $\overline{\partial}$-operator, when $\Omega$ is a bounded
strongly pseudoconvex domain in $\mathbb{C}^{n}$. Our main result is the
following.

######  Theorem 1.1.

Let $\Omega$ be a bounded strongly pseudoconvex domain with smooth boundary
and $f\in L^{2}(\partial\Omega)$. If $f\in BMO(\partial\Omega)$, then the
Hankel operator $H_{f}$ is bounded from $H^{2}(\Omega)$ into
$L^{2}(\partial\Omega)$.

The idea for proving this result comes from Amar’s
$\overline{\partial}$-method and the characterization of $BMO$ given by
Varopoulos [44]. This method is different from the proofs of corresponding
results in the setting of the unit disc and the unit ball. Particularly, in
the proof of Theorem 1.1, we get a solution to Gleason’s problem on the Hardy
space $H^{p}(\Omega)$ for $1\leq p\leq\infty$.

In fact, the proof of Theorem 1.1 shows that $H_{f}$ is bounded on
$H^{2}(\Omega)$ if $f$ admits a decomposition $f=f_{1}+f_{2}$ with $f_{1}\in
L^{\infty}(\partial\Omega)$ and $f_{2}$ having an extension
$\widetilde{f_{2}}$ in $\Omega$, such that
$|\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$
are Carleson measures in $\Omega$, where $dV$ is the Lebesgue measure on
$\Omega$. We conjecture that this condition is also necessary.

Based on this observation, we give the following
$\overline{\partial}$-characterization for Hankel operators on the Hardy space
$H^{2}(D)$ of the unit disc.

######  Theorem 1.2.

If $f\in L^{2}(\partial D)$, then the following properties are equivalent.
(1) $H_{f}$ is bounded from $H^{2}(D)$ into $L^{2}(\partial D)$.
(2) The function

$F(z)=\inf\limits_{h\in
H^{2}(D)}\frac{1}{2\pi}\int|f-h|^{2}P_{z}(\theta)d\theta$ $None$

is bounded.
(3) The measure $|\frac{\partial
f}{\partial\overline{z}}|^{2}\log\frac{1}{|z|}dxdy$ is a Carleson measure,
where $f(z)=\int f(\theta)P_{z}(\theta)d\theta$

is the Poisson integral of $f$.

Moreover, applying Berndtsson’s $L^{2}$-estimations for solutions of
$\overline{\partial}_{b}$-equation, we shall answer the question about the
boundedness of Hankel operators on the Hardy space $H^{2}(\Omega)$, when
$\Omega$ is a general pseudoconvex domain. We shall also use Donnelly-
Fefferman’s $L^{2}$-estimates to give a new proof for the
$\overline{\partial}$-characterization of $H_{f}$ on the Bergman space
$A^{2}(\Omega)$. This result has been proved in Li [25], by using the integral
representation of a solution to $\overline{\partial}$-equation.

This paper is divided into six sections. After some necessary preliminaries in
Section 2, we solve the Gleason’s problem on the Hardy space $H^{p}(\Omega)$.
In Section 3, we give the proof of Theorem 1.1 and generalize this result to
Hardy spaces $H^{p}(\Omega)$ for $p>1$. The compactness of these Hankel
operators shall be discussed in this section. In Section 4, we try to
investigate the boundedness of Hankel operators when acting on Hardy spaces of
general pseudoconvex domains. Similarly, we use a different method to prove
the corresponding result on the Bergman space of strongly pseudoconvex domain.
Section 5 devotes to give a new characterization for Hankel operator on the
Hardy space in the case of the unit disc. In last section, we show that both
$H_{f}$ and $H_{\overline{f}}$ are bounded on the Hardy space $H^{2}(\Omega)$
if and only if $f\in BMO(\partial\Omega)$, when $\Omega$ is a bounded strongly
pseudoconvex domain, and $1/S(\cdot,z)$ is holomorphic and bounded in $\Omega$
for fixed $z\in\Omega$.

##  2\. Geometry of strongly pseudoconvex domains

The geometry of strongly pseudoconvex domains is well understood (see [23] and
[37]). In this section, we collect some notations and results needed later.

Let $d(p,q)$ be the Koranyi distance between points on $\partial\Omega$ and
$B(p,r)=\\{\zeta\in\partial\Omega:d(\zeta,p)<r\\}$ the corresponding balls.
The tent $Q(p,r)$ over $B(p,r)$ is

$Q(p,r)=\\{z\in\Omega:d(z^{\prime},p)+d(z)<r\\},$

where $d(z)=d(z,\partial\Omega)$ is the Euclidean distance and $z^{\prime}$ is
the projection of $z$ on $\partial\Omega$. A positive measure $d\mu$ on
$\Omega$ is a Carleson measure if

$\|\mu\|=\sup\limits_{p\in\partial\Omega,r>0}\frac{1}{\sigma(B(p,r))}\int_{Q(p,r)}d\mu=C$

is finite. It is well-known [22] that $d\mu$ is a Carleson measure if and only
if

$\int_{\Omega}|\phi|^{p}d\mu\leq C\|\phi\|^{p}_{H^{p}},\ \ \ \ \phi\in
H^{p}(\Omega).$ $None$

Also, we call $d\mu$ a vanishing Carleson measure if

$\lim\limits_{j\to\infty}\int_{\Omega}|\phi_{j}|^{p}d\mu=0$

for any sequence $\\{\phi_{j}\\}$, which is bounded in $H^{p}(\Omega)$ and
converges to $0$ uniformly on any compact subset of $\Omega$ as $j\to\infty$.

For $0\leq\alpha\leq 1$, we denote by $W^{\alpha}$ the interpolation spaces
between the space $W^{1}$ of Carleson measures and the space $W^{0}$ of finite
measures in $\Omega$. If $0<\alpha<1$, then the measure $d\mu$ is in
$W^{\alpha}$ if and only if $d\mu=\phi d\tau$, where $d\tau\in W^{1}$,
$\phi\in L^{p}(d\tau)$ and $1/p=1-\alpha$ (see [3]). In view of (2.1), if
$\phi\in H^{p}(\Omega)$ and $d\tau\in W^{1}$, then $\phi d\tau\in W^{\alpha}$
and

$\|\phi d\tau\|_{W^{\alpha}}\lesssim\|\phi\|_{H^{p}}\|d\tau\|_{W^{1}}.$ $None$

Let $\omega$ be a $\overline{\partial}$-closed $(0,1)$ form in $\Omega$. If
$u$ is a function defined on $\partial\Omega$ that can be obtained as the
boundary value of a solution to $\overline{\partial}u=\omega$, then we write

$\overline{\partial}_{b}u=\omega$

(see [41], [3] or [8]). Following Skoda [41], this equation can be given by

$\int_{\partial\Omega}uh=\int_{\Omega}\omega\wedge h.$ $None$

for any smooth $\overline{\partial}$-closed form $h$ of bidegree $(n,n-1)$.

In [3], Amar and Bonami gave the following on the smoothly bounded strongly
pseudoconvex domains, which is very important for proving our main result.
Theorem A. Suppose that $\omega$ is a smooth $\overline{\partial}$-closed
$(0,1)$ form on $\overline{\Omega}$ and $p>1$. If the coefficients of $\omega$
and $\frac{\overline{\partial}\rho\wedge\omega}{\sqrt{-\rho}}$ belong to the
space $W^{\alpha}$ with $1/p=1-\alpha$, then there is a solution of
$\overline{\partial}_{b}u=\omega$ continuous on $\overline{\Omega}$ such that

$\|u\|_{L^{p}(\partial\Omega)}\leq
C\biggl{\|}\biggl{(}|\omega|+\frac{|\omega\wedge\overline{\partial}\rho|}{\sqrt{-\rho}}\biggr{)}dV\biggr{\|}_{W^{\alpha}}$

for a constant $C>0$.

The space $BMO(\partial\Omega)$ is defined in terms of the Koranyi distance as
the space of functions such that

$\|f\|_{BMO}:=\sup\limits_{p\in\partial\Omega,r>0}\frac{1}{\sigma(B(p,r))}\int_{B(p,r)}|f(\zeta)-f_{B(p,r)}|d\sigma(\zeta)<\infty,$

where $f_{B(p,r)}$ is the average of $f$ over the ball $B(p,r)$. Moreover,
$VMO(\partial\Omega)$ is defined as the subspace of $BMO$ which is the closure
of the continuous functions in the $BMO$ topology. We also put
$BMOA=BMO\cap\mathcal{O}(\Omega)$ and $VMOA=VMO\cap\mathcal{O}(\Omega)$

When $\Omega$ is strongly pseudoconvex, the reproducing property of the
Poisson-Szegö kernel gives

$f(z)=\int_{\partial\Omega}f(\zeta)P(z,\zeta)d\sigma(\zeta)\ \ \ \ \
\mbox{for}\ f\in H^{2}(\Omega)$

(see [42]). It follows from [27] that $f\in BMOA(\Omega)$ if and only if

$\|f\|^{2}_{BMOA}=\sup\biggl{\\{}\int_{\partial\Omega}|f(\zeta)-f(z)|^{2}P(z,\zeta)d\sigma(\zeta);z\in\Omega\biggr{\\}}<\infty,$

by using similar arguments as in the book of Garnett [18] and the John-
Nireberg Lemma. In fact, as in [18], one may show that these characterizations
also hold for $BMO(\partial\Omega)$ and $VMO(\partial\Omega)$, the only
difference is that $f(z)$ should be replaced by

$\widetilde{f}(z)=\int_{\partial\Omega}f(\zeta)P(z,\zeta)d\sigma(\zeta),$

the Poisson-Szegö integral of $f$.

By [29] there is a smooth function $\Phi(\zeta,z)$ on
$\overline{\Omega}\times\overline{\Omega}$ which is holomorphic in $z$ for
fixed $\zeta\in\overline{\Omega}$ and satisfies

$2\mbox{Re}\Phi\geq-\rho(\zeta)-\rho(z)+\delta|\zeta-z|^{2}$

and
$d_{\zeta}\overline{\Phi}|_{\zeta=z}=-d_{z}\Phi|_{\zeta=z}=-\partial\rho(\zeta)$.
It is well-known that $|\Phi(\zeta,z)|$ is compatible with the Koranyi
distance $d(\zeta,z)$. Moreover, there is a form
$Q(\zeta,z)=\sum\limits_{j=1}^{n}Q_{j}(\zeta,z)d\zeta_{j}$, holomorphic in
$z$, such that
$\Phi(\zeta,z)=\sum\limits_{j=1}^{n}Q_{j}(\zeta,z)(z_{j}-\zeta_{j})$ and the
following Cauchy-Fantappiè formula holds

$Fu(z)=c\int_{\partial\Omega}u(\zeta)\frac{Q\wedge(\overline{\partial}Q)^{n-1}}{\Phi(\zeta,z)^{n}}d\sigma(\zeta),\
\ \ z\in\Omega.$

Clearly, $Fu$ is holomorphic in $\Omega$ if $u\in L^{p}(\partial\Omega)$,
$1\leq p\leq\infty$ and $Fu=u$ if $u$ is the boundary value of a holomorphic
function. In fact, the kernel in the Cauchy-Fantappiè formula is also called
the Henkin-Ramirez reproducing kernel. Thus, $Fu$ has admissible boundary
values a.e. if $u\in L^{p}(\partial\Omega)$, $p>1$, and this operator maps
$L^{p}(\partial\Omega)$ into $H^{p}(\Omega)$, $BMO$ into $BMOA$, we refer to
[21] or [37] for details. We also notice that the Cauchy-Fantappiè formula for
a holomorphic function $u$ can be written as follows

$u(z)=\int_{\partial\Omega}\frac{A(\zeta,z)u(\zeta)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta),$

where $A(\zeta,z)$ is smooth and holomorphic in $z$.

To conclude this section, we will solve the following Gleason’s problem in the
setting of bounded strongly pseudoconvex domains. Let $X$ be some class of
holomorphic functions in a region $\Omega\subset\mathbb{C}^{n}$. Gleason’s
problem for $X$ is the following:

If $a\in\Omega$ and $f\in X$, do there exist functions $g_{1},\ldots,g_{n}\in
X$ such that

$f(z)-f(a)=\sum\limits_{j=1}^{n}(z_{j}-a_{j})g_{j}(z)$

for all $z\in\Omega$?

Gleason [19] originally asked this problem for $\Omega=B_{n}$, $a=0$ and
$X=A(B_{n})$, the ball algebra, which was solved by Leibenson. If $X$ is one
of the spaces $H^{p}(B_{n})$ ($1\leq p\leq\infty$), $A(B_{n})$ and
$\mbox{Lip}_{\alpha}$ ($0<\alpha\leq 1$), then the problem was also solved
(see [38]). The simplest solution is probably the one found by Ahern and
Schneider [1], which also works for strongly pseudoconvex domains when
$X=\mbox{Lip}_{\alpha}$ or $A(\Omega)$. Here, we shall solve the problem for
$X=H^{p}(\Omega)$ when $\Omega$ is strongly pseudoconvex.

First of all, we recall the following two results due to Stout [43].
Theorem B. If $f\in H^{p}(\Omega)$, $1\leq p\leq\infty$ and if $\gamma$ is a
Lipschitz continuous function on $\mathbb{C}^{n}$, then the function

$F(z)=\int_{\partial\Omega}f(\zeta)\gamma(\zeta)\frac{Q\wedge(\overline{\partial}Q)^{n-1}}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)$

belongs to $H^{p}(\Omega)$.

Note that

$F(z)=\int_{\partial\Omega}f(\zeta)[\gamma(\zeta)-\gamma(z)]\frac{Q\wedge(\overline{\partial}Q)^{n-1}}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)+\gamma(z)f(z).$

Thus, in order to proving Theorem B, the author considered the operator
$T=T_{\gamma}$ defined by

$Tg(z)=\int_{\partial\Omega}g(\zeta)[\gamma(\zeta)-\gamma(z)]\frac{Q\wedge(\overline{\partial}Q)^{n-1}}{\Phi(\zeta,z)^{n}}d\sigma(\zeta),$

and obtained the following boundedness properties of the operator $T$.
Theorem C. (1) If $g\in L^{q}(\partial\Omega)$, $q\geq 2n$, then $Tg$ is bound
on $\Omega$. (2) If $g\in L^{p}(\partial\Omega)$, $1\leq p<\infty$, then

$\sup\limits_{r<0}\int_{\rho=r}|Tg(\zeta)|^{p}d\sigma(\zeta)<\infty.$

Now we may solve Gleason’s problem for $H^{p}(\Omega)$.

######  Proposition 2.1.

Let $\Omega$ be a bounded strongly pseudoconvex domain with smooth boundary.
If $f\in H^{p}(\Omega)$, $1\leq p\leq\infty$, and $a\in\Omega$, then there
exist functions $g_{1},\ldots,g_{n}\in H^{p}(\Omega)$ such that

$f(z)-f(a)=\sum\limits_{j=1}^{n}(z_{j}-a_{j})g_{j}(z).$

###### Proof.

We follow the arguments used in solving Gleason’s problem for the case
$X=A(\Omega)$. As in Theorem 4.2 of [37], the reproducing property of the
Henkin-Ramirez kernel gives

$\displaystyle f(z)-f(a)$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}f(\zeta)\biggl{[}\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}-\frac{A(\zeta,a)}{\Phi(\zeta,a)^{n}}\biggr{]}d\sigma(\zeta)$
$\displaystyle=$
$\displaystyle\int_{\partial\Omega}f(\zeta)\biggl{[}\frac{A(\zeta,z)\Phi(\zeta,a)^{n}-A(\zeta,a)\Phi(\zeta,z)^{n}}{\Phi(\zeta,z)^{n}\Phi(\zeta,a)^{n}}\biggr{]}d\sigma(\zeta)$

for $f\in H^{p}(\Omega)$ and $z,a\in\Omega$.

By [31], p. 148, there are functions $L_{j},j=1,\ldots,n$, which are
$C^{\infty}$ on
$(\partial\Omega)\times\overline{\Omega}\times\overline{\Omega}$ and
holomorphic in $z$ and $a$, such that

$A(\zeta,z)\Phi(\zeta,a)^{n}-A(\zeta,a)\Phi(\zeta,z)^{n}=\sum\limits_{j=1}^{n}(z_{j}-a_{j})L_{j}(\zeta,z,a).$

It follows that

$f(z)-f(a)=\sum\limits_{j=1}^{n}(z_{j}-a_{j})\int_{\partial\Omega}f(\zeta)\frac{L_{j}(\zeta,z,a)}{\Phi(\zeta,z)^{n}\Phi(\zeta,a)^{n}}d\sigma(\zeta).$
$None$

Define

$T_{j}f(z,a)=\int_{\partial\Omega}f(\zeta)\frac{L_{j}(\zeta,z,a)}{\Phi(\zeta,z)^{n}\Phi(\zeta,a)^{n}}d\sigma(\zeta).$

Clearly, $T_{j}f$ is holomorphic on $\Omega\times\Omega$. If we show that
$T_{j}f(z,a)\in H^{p}(\Omega)$ for fixed $a$, we were done.

Note that Corollary 3.10 in [37] gives $A(\zeta,z)=h(\zeta)+O(|\zeta-z|)$,
where $h(\zeta)\neq 0$ for all $\zeta\in\partial\Omega$. Set

$\chi(\zeta,z,a)=\frac{L_{j}(\zeta,z,a)}{h(\zeta)\Phi(\zeta,a)^{n}}.$

We have $\chi(\zeta,z,a)=\chi(\zeta,\zeta,a)+O(|\zeta-z|)$. A calculation
shows that

$\displaystyle\frac{L_{j}(\zeta,z,a)}{\Phi(\zeta,z)^{n}\Phi(\zeta,a)^{n}}=\frac{L_{j}(\zeta,z,a)}{h(\zeta)\Phi(\zeta,a)^{n}}\cdot\frac{h(\zeta)}{\Phi(\zeta,z)^{n}}$
$\displaystyle=\chi(\zeta,z,a)\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}+\chi(\zeta,z,a)\frac{O(|\zeta-z|)}{\Phi(\zeta,z)^{n}}$
$\displaystyle=\chi(\zeta,\zeta,a)\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}+O(|\zeta-z|)\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}+\chi(\zeta,z,a)\frac{O(|\zeta-z|)}{\Phi(\zeta,z)^{n}}.$

This implies

$\displaystyle T_{j}f(z,a)$
$\displaystyle=\int_{\partial\Omega}f(\zeta)\chi(\zeta,\zeta,a)\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)$
$\displaystyle+\int_{\partial\Omega}f(\zeta)O(|\zeta-z|)\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)+\int_{\partial\Omega}f\chi\frac{O(|\zeta-z|)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta).\
\ \ \ (2.5)$

Next, we will need a technique of Stout [43] to estimate (2.5), which is
different from that in the proof of Theorem 4.2 in [37]. Since
$\chi(\zeta,\zeta,a)\in C^{\infty}$, it follows from Theorem B that

$F(z)=\int_{\partial\Omega}f(\zeta)\chi(\zeta,\zeta,a)\frac{A(\zeta,z)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)$

belongs to $H^{p}(\Omega)$, $1\leq p\leq\infty$.

Since the functions $A$ and $\chi$ are smooth, we may use similar ideas to
estimate the second term in (2.5). It remains to consider the last term. Set

$Tf(z)=\int_{\partial\Omega}f\chi\frac{O(|\zeta-z|)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)$

and $T^{(r)}f=Tf|_{\\{\rho=r\\}}$ when $r$ is close to $0$. Once we prove the
following claim

$\|T^{(r)}f\|_{L^{\infty}(\\{\rho=r\\})}\leq
C\|f\|_{L^{\infty}(\partial\Omega)}\ \ \mbox{and}\ \
\|T^{(r)}f\|_{L^{1}(\\{\rho=r\\})}\leq C\|f\|_{L^{1}(\partial\Omega)}$

for a constant $C>0$, the Riesz-Thorin theorem implies that if $f\in
L^{p}(\partial\Omega)$, $1<p<\infty$, then

$\|T^{(r)}f\|_{L^{p}(\\{\rho=r\\})}\leq C\|f\|_{L^{p}(\partial\Omega)}.$

Since the constant $C$ is independent of $r$, we obtain that $Tf\in
L^{p}(\partial\Omega)$ for $1<p<\infty$.

To verify the claim, we first assume $f\in L^{\infty}(\partial\Omega)$. Since
$\chi$ is smooth, we have

$|Tf(z)|\lesssim\|f\|_{\infty}\biggl{(}\int_{\partial\Omega}\frac{O(|\zeta-z|)}{|\Phi(\zeta,z)|^{n}}d\sigma(\zeta)\biggr{)}$

for any $z\in\Omega$. In the proof of Theorem C, the author has showed that
the above integral is finite. The proof of Theorem C also yields that

$\int_{\rho=r}\frac{O(|\zeta-z|)}{|\Phi(\zeta,z)|^{n}}d\sigma(z)$

are bounded uniformly in $\zeta,r$, where $\zeta\in\partial\Omega$ and $r$ is
close to $0$. Thus, if $f\in L^{1}(\partial\Omega)$, we get that

$\int_{\rho=r}|Tf(z)|d\sigma(z)$

are bounded uniformly in $r$ near $0$. Thus the claim is verified.

Combining the above arguments with (2.5), we find that $T_{j}f(\cdot,a)\in
H^{p}(\Omega)$ for $1\leq p\leq\infty$. Taking $g_{j}(z)=T_{j}f(z,a)$, we
obtain the desired result from (2.4). ∎

Proposition 2.1 gives a decomposition for $H^{2}(\Omega)$. Similar as the
proof of Lemma 2.5 in [2], we may easily deduce the following lemma.

######  Lemma 2.2.

The space $H^{2}(\Omega)^{\bot}$ can be identified with the space of
$\overline{\partial}_{b}$-closed forms of bidegree $(n,n-1)$ satisfying

$h\in H^{2}(\Omega)^{\bot}\rightarrow\mathcal{H}\in
L^{2}_{(n,n-1)}(\partial\Omega),\ \ \ \ \overline{\partial}_{b}\mathcal{H}=0$

and

$\int_{\partial\Omega}\phi\cdot\overline{h}d\sigma=\int_{\partial\Omega}\phi\cdot\mathcal{H},\
\ \ \ \forall\,\phi\in L^{2}(\partial\Omega).$

As an easy consequence of this lemma and Stokes’ theorem, we obtain the
following result which generalizes a result of Amar [2] in the unit ball.

######  Proposition 2.3.

Suppose that $\Omega$ is a bounded strongly pseudoconvex domain with smooth
boundary. Let $f\in L^{2}(\partial\Omega)$ and $\widetilde{f}$ be any Stokes’
extension of $f$ in $\Omega$. Then

$\overline{\partial}_{b}(H_{f}g)=g\cdot\overline{\partial}\widetilde{f},\ \ \
\ \ \forall\,g\in H^{\infty}(\Omega).$

###### Proof.

The argument is similar to Proposition 2.4 in [2]. For convenience, we give a
simple proof here.

Let $g\in H^{\infty}(\Omega)$ and $u=H_{f}g=fg-P(fg)$. Applying Lemma 2.2 and
Stokes’ theorem, for any $h\in H^{2}(\Omega)^{\bot}$, we get

$\displaystyle\int_{\partial\Omega}u\cdot\overline{h}d\sigma$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}[fg-P(fg)]\cdot\overline{h}d\sigma=\int_{\partial\Omega}fg\cdot\overline{h}d\sigma$
$\displaystyle=$
$\displaystyle\int_{\partial\Omega}fg\cdot\mathcal{H}=\int_{\Omega}g\cdot\overline{\partial}\widetilde{f}\wedge\mathcal{H}.$

This combined with (2.3) yields
$\overline{\partial}_{b}u=g\cdot\overline{\partial}\widetilde{f}$, which
completes the proof. ∎

Remark. In fact, if $g\in H^{\infty}(\Omega)$ and $f\in
L^{2}(\partial\Omega)$, then $P(fg)\in H^{2}(\Omega)$. Now, applying the
definition of $\overline{\partial}_{b}$ in Page 166 of [12], we easily deduce
that $H_{f}g=fg-P(fg)$ is a solution of
$\overline{\partial}_{b}u=g\cdot\overline{\partial}\widetilde{f}$ for the
extension $\widetilde{f}$ of $f$.

##  3\. Hankel operators on Hardy spaces of strongly pseudoconvex domains with
smooth boundarys

Before proving Theorem 1.1, we shall introduce one more notion. Let $f\in
C^{1}(\Omega)$, we define the nonisotropic gradiant near the boundary of
$\Omega$ as

$|Df|=|\nu_{0}(f)|+|\mu_{0}(f)|+|\rho|^{-1/2}\sum\limits_{j=1}^{2n-2}|\mu_{j}(f)|$

(see [44]), where $\nu_{0}$ is a normalized vector field in some neighborhood
of $\partial\Omega$ that is normal and directed inwards to $\partial\Omega$ at
every point $\zeta\in\partial\Omega$, $\mu_{0}$ is a normalized field that is
orthogonal to the holomorphic tangent space at every point
$\zeta\in\partial\Omega$, and fields $\mu_{1},\ldots,\mu_{2n-2}$ are
constructed to form an orthonormal basis at $\zeta$.

In [13], the authors used the theory of singular integral operators to show
that $H_{f}$ is bounded on the Hardy space $H^{2}(B_{n})$ if $f\in
BMO(\partial B_{n})$. Amar [2] applied the theory of $\overline{\partial}$ to
prove that $H_{f}$ is bounded on $H^{2}(B_{n})$ if $f$ admits a Stokes’
extension $\widetilde{f}$ to $B_{n}$ such that both
$|\overline{\partial}\widetilde{f}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f}\wedge\overline{\partial}\rho|dV$
are Carleson measures. Motivated by this result and the work of Varopoulos
[44] about the $BMO$ theory, we may generalize the result of [13] to the
bounded strongly pseudoconvex domains.

###### Proof of Theorem 1.1.

Let $f\in BMO(\partial\Omega)$. By Theorem 2.1.1 of [44], $f$ admits a
decomposition $f=f_{1}+f_{2}$, such that $f_{1}\in L^{\infty}(\partial\Omega)$
and $f_{2}$ has an extension $\widetilde{f_{2}}$ in $\Omega$, which satisfies
that $\lim\limits_{\lambda\rightarrow
0}\widetilde{f_{2}}(\zeta+\lambda\nu_{0}(\zeta))=\widetilde{f_{2}}(\zeta)$,
$\forall\,\zeta\in\partial\Omega$, and $|D\widetilde{f_{2}}|dV$ is a Carleson
measure.

Since $f_{1}\in L^{\infty}(\partial\Omega)$, it is easy to see that the
multiplication operator $M_{f_{1}}$ is bounded from $H^{2}(\Omega)$ into
$L^{2}(\partial\Omega)$, which implies that $H_{f_{1}}=(I-P)M_{f_{1}}$ is
bounded. Thus, it suffices to prove the boundedness of $H_{f_{2}}$.

Now, $f\in L^{2}(\partial\Omega)$ and $f_{1}\in L^{\infty}(\partial\Omega)$
give that $f_{2}\in L^{2}(\partial\Omega)$. Since $|D\widetilde{f_{2}}|dV$ is
a Carleson measure, we infer from Lemma 2.5.1 in [44] that both
$|\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$
are Carleson measures in $\Omega$. Let $g\in H^{2}(\Omega)$. Applying (2.2) to
$|g\cdot\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|g\cdot\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$,
we deduce that the coefficients of
$g\cdot\overline{\partial}\widetilde{f_{2}}$ and
$g\cdot\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho/\sqrt{-\rho}$
belong to the space $W^{1/2}$. Thus there is, according to Theorem A, a
solution of the equation
$\overline{\partial}_{b}u=g\cdot\overline{\partial}\widetilde{f_{2}}$ which
satisfies $u\in L^{2}(\partial\Omega)$. Moreover, using Theorem A and (2.2),
we obtain

$\displaystyle\|u\|_{L^{2}(\partial\Omega)}$ $\displaystyle\lesssim$
$\displaystyle\||g\cdot\overline{\partial}\widetilde{f_{2}}|dV\|_{W^{1/2}}+\biggl{\|}\frac{1}{\sqrt{-\rho}}|g\cdot\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV\biggr{\|}_{W^{1/2}}$
$\displaystyle\lesssim$
$\displaystyle\|g\|_{H^{2}}\biggl{(}\||\overline{\partial}\widetilde{f_{2}}|dV\|_{W^{1}}+\biggl{\|}\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV\biggr{\|}_{W^{1}}\biggr{)}.$

In view of Proposition 2.3, we have
$\overline{\partial}_{b}(H_{f_{2}}g)=g\cdot\overline{\partial}\widetilde{f_{2}}$
when $g\in H^{\infty}(\partial\Omega)$. Using the uniqueness of the solution
orthogonal to $H^{2}(\Omega)$, we get

$\displaystyle\|H_{f_{2}}g\|_{L^{2}(\partial\Omega)}$ $\displaystyle\leq$
$\displaystyle\|u\|_{L^{2}(\partial\Omega)}$ $\displaystyle\lesssim$
$\displaystyle\|g\|_{H^{2}}\biggl{(}\||\overline{\partial}\widetilde{f_{2}}|dV\|_{W^{1}}+\biggl{\|}\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV\biggr{\|}_{W^{1}}\biggr{)}.$

Since $H^{\infty}(\partial\Omega)$ is dense in $H^{2}(\Omega)$, we see that
$H_{f_{2}}$ is bounded on $H^{2}(\Omega)$ and

$\|H_{f_{2}}\|\lesssim\biggl{(}\||\overline{\partial}\widetilde{f_{2}}|dV\|_{W^{1}}+\biggl{\|}\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV\biggr{\|}_{W^{1}}\biggr{)}.$

It follows that $H_{f}=H_{f_{1}}+H_{f_{2}}$ is bounded from $H^{2}(\Omega)$
into $L^{2}(\partial\Omega)$. ∎

From the proof of Theorem 1.1, we may also give the following characterization
for the compactness of $H_{f}$ on the Hardy space $H^{2}(\Omega)$.

######  Theorem 3.1.

Let $f\in L^{2}(\partial\Omega)$. If $f$ admits a decomposition
$f=f_{1}+f_{2}$ with $f_{1}\in C(\partial\Omega)$ and $\widetilde{f_{2}}$
being an extension of $f_{2}$ in $\Omega$, such that
$|\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$
are vanishing Carleson measures in $\Omega$, then the Hankel operator $H_{f}$
is compact from $H^{2}(\Omega)$ into $L^{2}(\partial\Omega)$.

###### Proof.

For $f_{1}\in C(\partial\Omega)\subset L^{\infty}(\partial\Omega)$, we know
that $H_{f_{1}}$ is bounded on $H^{2}(\Omega)$. Using a consequence of the
Kohn’s solution of $\overline{\partial}_{b}$-Neumann problem (cf. [17]), we
see that $H_{\phi}$ is compact for $\phi\in C^{\infty}(\partial\Omega)$. Since
any $\phi\in C(\partial\Omega)$ can be approximated uniformly by smooth ones,
it follows from Lemma 1 in [39] that

$H_{\phi}=(I-P)M_{\phi}:H^{2}(\partial\Omega)\rightarrow
L^{2}(\partial\Omega)$

is compact for $\phi\in C(\partial\Omega)$, where $H^{2}(\partial\Omega)$
denotes the closure of the $C^{\infty}$-functions on $\partial\Omega$ which
can be extended holomorphically to $\Omega$. This means that $H_{f_{1}}$ is
also compact on $H^{2}(\Omega)$.

Now consider the function $f_{2}$. In the proof of Theorem 1.1, we have shown
that there exists a positive constant $C$ such that

$\|H_{f_{2}}g\|_{L^{2}(\partial\Omega)}\leq
C\biggl{(}\||g\cdot\overline{\partial}\widetilde{f_{2}}|dV\|_{W^{1/2}}+\biggl{\|}\frac{1}{\sqrt{-\rho}}|g\cdot\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV\biggr{\|}_{W^{1/2}}\biggr{)}$

for any $g\in H^{2}(\Omega)$. Let $\\{g_{j}\\}$ be a bounded sequence in
$H^{2}(\Omega)$ and converge to $0$ uniformly on any compact subset of
$\Omega$ as $j\to\infty$. Since $|\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$
are vanishing Carleson measures, we infer from (2.1) and (2.2) that

$\||g_{j}\cdot\overline{\partial}\widetilde{f_{2}}|dV\|_{W^{1/2}}\rightarrow
0$

and

$\biggl{\|}\frac{1}{\sqrt{-\rho}}|g_{j}\cdot\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV\biggr{\|}_{W^{1/2}}\rightarrow
0$

as $j\rightarrow\infty$. This yields that
$\lim\limits_{j\to\infty}\|H_{f_{2}}g_{j}\|_{L^{2}(\partial\Omega)}=0$. Thus
$H_{f_{2}}$ is compact on $H^{2}(\Omega)$. So we complete the proof. ∎

In [44], Varopoulos found that $f\in BMO(\partial\Omega)$ admits a
decomposition $f=f_{1}+f_{2}$, such that $f_{1}\in L^{\infty}(\partial\Omega)$
and $f_{2}$ has an extension $\widetilde{f_{2}}$ in $\Omega$, which satisfying
that $\lim\limits_{\lambda\rightarrow
0}\widetilde{f_{2}}(\zeta+\lambda\nu_{0}(\zeta))=\widetilde{f_{2}}(\zeta)$ for
any $\zeta\in\partial\Omega$ and $|D\widetilde{f_{2}}|dV$ is a Carleson
measure. This follows from a well-known result of Carleson, i.e. if $f\in
BMO(\partial\Omega)$ then there exists some Carleson measure $d\mu$ in
$\Omega$ such that

$f(\zeta)-\int_{\Omega}P(z,\zeta)d\mu(z)\in L^{\infty}(\partial\Omega).$

For $f\in VMO(\partial\Omega)$ it is natural to make the following conjecture:
Conjecture 1. If $f\in VMO(\partial\Omega)$, then there exists some vanishing
Carleson measure $d\mu$ in $\Omega$ such that

$f(\zeta)-\int_{\Omega}P(z,\zeta)d\mu(z)\in C(\partial\Omega).$

Remark. Once this conjecture is true, one may use similar arguments as Theorem
2.1.1 and Lemma 2.5.1 in [44] to obtain the following decomposition: if $f\in
VMO(\partial\Omega)$, then $f=f_{1}+f_{2}$ with $f_{1}\in C(\partial\Omega)$
and $\widetilde{f_{2}}$ being an extension of $f_{2}$, such that the measures
$|\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$
are vanishing Carleson measures in $\Omega$. As a consequence, Theorem 3.1
implies that $H_{f}$ is compact from $H^{2}(\Omega)$ into
$L^{2}(\partial\Omega)$ for $f\in VMO(\partial\Omega)$.

Now, for $0<p<\infty$ and $s>-1$, we recall that the weighted Bergman spaces
$A^{p}_{s}(\Omega)$ consist of all holomorphic functions $f$ on $\Omega$
satisfying

$\|f\|_{p,s}=\biggl{(}\int_{\Omega}|f|^{p}(-\rho)^{s}dV\biggr{)}^{1/p}<\infty.$

For $s=-1$, we define $A^{p}_{-1}(\Omega)$ to be the Hardy space
$H^{p}(\Omega)$. Let $\Omega$ be a strongly pseudoconvex domain in
$\mathbb{C}^{n}$, Theorem 5.4 in [6] gives that
$\mathcal{O}(\overline{\Omega})$ is dense in $A^{p}_{s}(\Omega)$ for
$0<p<\infty$ and $s\geq-1$. This implies that $H^{\infty}(\Omega)$ is dense in
$H^{p}(\Omega)$ for $0<p<\infty$. Since the projection $P$ via the Szegö
kernel $S(z,w)$ is bounded from $L^{p}(\partial\Omega)$ into $H^{p}(\Omega)$
if $p>1$, we may define Hankel operators $H_{f}$ on Hardy spaces
$H^{p}(\Omega)$. On the other hand, Theorem A and (2.2) hold for all $p>1$.
Therefore, we might use the same proof to obtain the following.

######  Theorem 3.2.

Let $\Omega$ be a bounded strongly pseudoconvex domain with smooth boundary in
$\mathbb{C}^{n}$. Assume $f\in L^{p}(\partial\Omega)$ and $p>1$. If $f\in
BMO(\partial\Omega)$, then the Hankel operator $H_{f}$ is bounded from
$H^{p}(\Omega)$ into $L^{p}(\partial\Omega)$.

Finally, we try to give a necessary condition for Hankel operator $H_{f}$ to
be bounded on the Hardy space $H^{2}(\Omega)$, which shall build a bridge to
solve the boundedness conjecture pointed out in Section 5.

######  Theorem 3.3.

Let $\Omega$ be a smoothly bounded strongly pseudoconvex domain. If the Hankel
operator $H_{f}$ is bounded from $H^{2}(\Omega)$ into $L^{2}(\partial\Omega)$,
then the following inequality

$\inf\limits_{h\in
H(B(w,r))}\frac{1}{\sigma(B(w,r))}\int_{B(w,r)}|f(\zeta)-h(\zeta)|^{2}d\sigma(\zeta)<\infty$

holds for any $w\in\partial\Omega$ and $r>0$.

###### Proof.

Let $\Phi(z,w)$ be the function introduced in Section 2 and

$v(z,w)=\Phi(z,w)-\rho(z).$

It follows that $|v(z,w)|\thicksim d(z)+d(w)+d(z^{\prime},w^{\prime})$. Set

$g_{z}(\zeta)=\frac{d(z)^{n/2}}{v(z,\zeta)^{n}},\ \ \ \ \ z\in\Omega,\
\zeta\in\overline{\Omega}.$

Since $v(z,\zeta)$ is holomorphic in $\zeta$ for fixed $z$, and the well-known
estimate

$\int_{\partial\Omega}\frac{d\sigma(\zeta)}{|v(z,\zeta)|^{n+\alpha}}\lesssim\frac{1}{d(z)^{\alpha}}$

for $\alpha>0$, we have $\sup\limits_{z\in\Omega}\|g_{z}\|_{H^{2}}\leq C$ for
some constant $C>0$. If $H_{f}$ is bounded, then
$\sup\limits_{z\in\Omega}\|H_{f}(g_{z})\|_{L^{2}(\partial\Omega)}\leq
C\|H_{f}\|$. It follows that

$\displaystyle\infty$ $\displaystyle>$
$\displaystyle\sup\limits_{z\in\Omega}\int_{\partial\Omega}|fg_{z}-P(fg_{z})|^{2}d\sigma$
$\displaystyle=$
$\displaystyle\sup\limits_{z\in\Omega}\int_{\partial\Omega}\biggl{|}f(\zeta)-\frac{1}{g_{z}(\zeta)}P(fg_{z})(\zeta)\biggr{|}^{2}|g_{z}(\zeta)|^{2}d\sigma(\zeta).$

Let $w\in\partial\Omega$ and $B(w,r)$ be a ball on $\partial\Omega$ in the
Koranyi distance. Then for $\zeta\in B(w,r)$ and $\widetilde{w}=w-rv^{(1)}$,
where $\\{v^{(1)},\cdots,v^{(n)}\\}$ is the orthonormal basis at $w$ with
$v^{(1)}$ transversal to the boundary, we see that

$|g_{\widetilde{w}}(\zeta)|^{2}=\frac{d(\widetilde{w})^{n}}{|v(\widetilde{w},\zeta)|^{2n}}\gtrsim\frac{1}{\sigma(B(w,r))}.$

Since $\frac{1}{g_{\widetilde{w}}(\zeta)}P(fg_{\widetilde{w}})(\zeta)$ is
holomorphic on $\partial\Omega$, we may choose a holomorphic function $h$ on
$B(w,r)$ satisfying

$\frac{1}{\sigma(B(w,r))}\int_{B(w,r)}|f(\zeta)-h(\zeta)|^{2}d\sigma(\zeta)$
$\lesssim\int_{\partial\Omega}\biggl{|}f(\zeta)-\frac{1}{g_{\widetilde{w}}(\zeta)}P(fg_{\widetilde{w}})(\zeta)\biggr{|}^{2}|g_{\widetilde{w}}(\zeta)|^{2}d\sigma(\zeta).$

As a consequence, we obtain the desired inequality. ∎

##  4\. Hankel operators on Hardy spaces of general pseudoconvex domains

Let $\Omega$ be a bounded domain with $C^{2}$ boundary in $\mathbb{C}^{n}$. A
positive measure $d\mu$ on $\Omega$ is called a Carleson measure if there
exists a constant $C>0$ such that

$\int_{\Omega}|g|^{2}d\mu\leq C\int_{\partial\Omega}|g|^{2}d\sigma,\ \ \
\forall\,g\in H^{2}(\Omega).$

The starting point is the following result due to Berndtsson:
Theorem D [cf. [9], see also [11]]. Let $\Omega$ be a bounded pseudoconvex
domain with $C^{2}$ boundary in $\mathbb{C}^{n}$ and let $\rho$ be a negative
$C^{2}$ function on $\Omega$. Suppose there exists a $C^{2}$ psh function
$\psi$ on $\Omega$ which is continuous on $\overline{\Omega}$ and satisfies

$\Theta:=(-\rho)\partial\bar{\partial}\psi+\partial\bar{\partial}\rho>0.$
(4.1)

Let $u_{0}$ be the $L^{2}_{\psi}(\Omega)-$minimal solution of the equation
$\bar{\partial}u=v$. Then

$\int_{\partial\Omega}|u_{0}|^{2}e^{-\psi}d\sigma/|\nabla\rho|\leq\int_{\Omega}|v|^{2}_{\Theta}e^{-\psi}dV.$

Applying this result, we shall give the following characterization for the
boundedness of Hankel operators.

######  Theorem 4.1.

Let $\Omega$ be a bounded pseudoconvex domain with $C^{2}$ boundary in
$\mathbb{C}^{n}$ and let $f\in L^{2}(\partial\Omega)$. Suppose the following
conditions hold:

  1. (1)

there exist a negative $C^{2}$ function $\rho$ on $\Omega$ and a $C^{2}$ psh
function $\psi$ on $\Omega$ which is continuous on $\overline{\Omega}$ such
that $(\ref{eq:curvature})$ holds.

  2. (2)

there is a decomposition $f=f_{1}+f_{2}$, where $f_{1}\in
L^{\infty}(\partial\Omega)$ and $f_{2}$ admits a Stokes extension
$\tilde{f}_{2}$ to $\Omega$ such that
$|\bar{\partial}\tilde{f}_{2}|^{2}_{\Theta}dV$ is a Carleson measure on
$\Omega$.

Then $H_{f}:H^{2}(\Omega)\rightarrow L^{2}(\partial\Omega)$ is bounded.

###### Proof.

Clearly $H_{f_{1}}:H^{2}(\Omega)\rightarrow L^{2}(\partial\Omega)$ is bounded.
For given $g\in H^{2}(\Omega)$, $H_{f_{2}}g$ is nothing but the
$L^{2}(\partial\Omega)-$minimal solution of the equation

$\bar{\partial}_{b}u=\bar{\partial}(\tilde{f}_{2}g)=g\cdot\bar{\partial}\tilde{f}_{2}.$

It follows from Theorem D that

$\displaystyle\int_{\partial\Omega}|H_{f}g|^{2}d\sigma/|\nabla\rho|$
$\displaystyle\leq$
$\displaystyle\int_{\partial\Omega}|u_{0}|^{2}d\sigma/|\nabla\rho|=\lim_{r\rightarrow
1-}(1-r)\int_{\Omega}|u_{0}|^{2}(-\rho)^{-r}dV$ $\displaystyle\lesssim$
$\displaystyle\lim_{r\rightarrow
1-}(1-r)\int_{\Omega}|u_{0}|^{2}(-\rho)^{-r}e^{-\psi}dV$
$\displaystyle\lesssim$
$\displaystyle\int_{\partial\Omega}|u_{0}|^{2}e^{-\psi}d\sigma/|\nabla\rho|$
$\displaystyle\lesssim$
$\displaystyle\int_{\Omega}|g|^{2}|\bar{\partial}\tilde{f}_{2}|^{2}_{\Theta}e^{-\psi}dV$
$\displaystyle\lesssim$
$\displaystyle\int_{\Omega}|g|^{2}|\bar{\partial}\tilde{f}_{2}|^{2}_{\Theta}dV\lesssim\int_{\partial\Omega}|g|^{2}d\sigma,$

since $|\bar{\partial}\tilde{f}_{2}|^{2}_{\Theta}dV$ is a Carleson measure.
The proof is complete. ∎

Let $\Omega$ be a bounded strongly pseudoconvex domain in $\mathbb{C}^{n}$ and
let $\rho\geq-1/e^{2}$ be a definition function which is strictly psh on
$\overline{\Omega}$. Given $\varepsilon>0$, let

$\psi=-[-\log(-\rho)]^{-\varepsilon}.$

A straightforward calculation shows that if $\varepsilon\leq 1/2$ then

$\displaystyle\partial\bar{\partial}\psi$ $\displaystyle=$
$\displaystyle\varepsilon[-\log(-\rho)]^{-\varepsilon-1}\partial\bar{\partial}[-\log(-\rho)]$
$\displaystyle-\varepsilon(1+\varepsilon)[-\log(-\rho)]^{-\varepsilon-2}\partial\log(-\rho)\wedge\bar{\partial}\log(-\rho)$
$\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{4}[-\log(-\rho)]^{-\varepsilon-1}\partial\bar{\partial}[-\log(-\rho)]$
$\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{4}[-\log(-\rho)]^{-\varepsilon-1}\rho^{-2}\partial\rho\wedge\bar{\partial}\rho,$

so that

$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle(-\rho)\partial\bar{\partial}\psi+\partial\bar{\partial}\rho$
$\displaystyle\geq$ $\displaystyle
C\left[\partial\bar{\partial}\rho+\frac{\partial\rho\wedge\bar{\partial}\rho}{(-\rho)[-\log(-\rho)]^{1+\varepsilon}}\right].$

Set
$N=|\bar{\partial}\rho|^{-1}\sum_{j}\frac{\partial\rho}{\partial\bar{z}_{j}}\cdot\frac{\partial}{\partial\bar{z}_{j}}$.
For any $(0,1)-$form $\omega$ on $\Omega$, we set

$\omega_{N}=\langle\omega,N\rangle\cdot\frac{\bar{\partial}\rho}{|\bar{\partial}\rho|}\
\ \ \text{and}\ \ \ \omega_{T}=\omega-\omega_{N}.$

It follows that $|\bar{\partial}\tilde{f}_{2}|^{2}_{\Theta}dV$ is a Carleson
measure provided that $|(\bar{\partial}\tilde{f}_{2})_{T}|^{2}dV$ and

$(-\rho)[-\log(-\rho)]^{1+\varepsilon}|(\bar{\partial}\tilde{f}_{2})_{N}|^{2}dV$

are Carleson measures. Consequently, we obtain

######  Corollary 4.2.

Let $\Omega$ be a bounded strongly pseudoconvex domain in $\mathbb{C}^{n}$ and
$\rho$ a strictly psh defining function on $\Omega$. Let $f\in
L^{2}(\partial\Omega)$. Suppose $f=f_{1}+f_{2}$, where $f_{1}\in
L^{\infty}(\partial\Omega)$ and $f_{2}$ admits a Stokes extension
$\tilde{f}_{2}$ such that $|(\bar{\partial}\tilde{f}_{2})_{T}|^{2}dV$ and

$(-\rho)[-\log(-\rho)]^{1+\varepsilon}|(\bar{\partial}\tilde{f}_{2})_{N}|^{2}dV$

are Carleson measures for some $\varepsilon>0$. Then
$H_{f}:H^{2}(\Omega)\rightarrow L^{2}(\partial\Omega)$ is bounded.

Similar ideas work for more general domains, e.g. pseudoconvex domains of
finite D’Angelo type (compare [11], Proposition 5.2).

In the setting of strongly pseudoconvex domains, the theory of the
$\overline{\partial}$-operator has been used to investigate Hankel operators
$H_{f}$ on the Bergman space $A^{2}(\Omega)$. The boundedness, compactness,
essential norms and Schatten ideals of $H_{f}$ have been completely
characterized (see [20-22] and [5]). In the proofs of these results, an
essential tool is the integral representation of a solution to the
$\overline{\partial}$-equation.

To conclude this section, we shall apply the $L^{2}$-estimates of
$\overline{\partial}$-equation due to Donnelly-Fefferman [14] to give a new
proof for the boundedness of $H_{f}$ on $A^{2}(\Omega)$.

Let $\Omega$ be a bounded pseudoconvex domian and let $\psi$ be a $C^{2}$
plurisubharmonic function on $\Omega$. For any $(0,1)$ form $f$, let

$|f|^{2}_{\partial\overline{\partial}\psi}:=\sum\limits_{i,j=1}^{n}\psi^{i\,\overline{j}}f_{i}\overline{f_{j}},$

where $(\psi^{i\,\overline{j}})=(\psi_{i\,\overline{j}})^{-1}$. According to
an estimate of Donnelly and Fefferman, one can solve $\overline{\partial}u=f$
with the following estimate

$\int_{\Omega}|u|^{2}e^{-\varphi}dV\leq
C_{0}\int_{\Omega}|f|^{2}_{\partial\overline{\partial}\psi}e^{-\varphi}dV$

for some numerical constant $C_{0}$, whenever $\psi$ satisfies
$\partial\overline{\partial}\psi\geq\partial\psi\wedge\overline{\partial}\psi$
and $\varphi$ is plurisubharmonic.

We first consider the case of the unit ball.

######  Theorem 4.3.

Let $f\in L^{2}(B_{n})\cap C^{1}(B_{n})$ and $\rho=|z|^{2}-1$. If

$|\rho\cdot\overline{\partial}f|+|\rho|^{1/2}|\overline{\partial}f\wedge\overline{\partial}\rho|\leq
C$

for a constant $C>0$. Then the Hankel operator $H_{f}=(I-P_{B})M_{f}$ is
bounded from $A^{2}(B_{n})$ into $L^{2}(B_{n})$, here $P_{B}$ is the Bergman
projection.

###### Proof.

Take $\psi=-\log(-\rho)$. A calculation shows that

$\partial\overline{\partial}\psi=-\frac{\partial\overline{\partial}\rho}{\rho}+\frac{\partial\rho\wedge\overline{\partial}\rho}{\rho^{2}}\geq\partial\psi\wedge\bar{\partial}\psi$

and

$\partial\overline{\partial}\psi=\frac{\frac{1}{|z|^{2}}\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}dz_{i}\wedge
d\overline{z_{j}}}{(1-|z|^{2})^{2}}+\frac{\sum\limits_{i=1}^{n}dz_{i}\wedge
d\overline{z_{j}}-\frac{1}{|z|^{2}}\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}dz_{i}\wedge
d\overline{z_{j}}}{1-|z|^{2}}.$

Write $\partial\overline{\partial}\psi=\sum\psi_{i\,\overline{j}}dz_{i}\wedge
d\overline{z_{j}}$. Let $A=\frac{1}{|z|^{2}}(\overline{z_{i}}z_{j})$ and
$B=I-A$, where $I$ is the identity matrix. It is easy to check $A^{2}=A$ and
$AB=0$. Thus we get

$\displaystyle\sum\psi^{i\,\overline{j}}dz_{i}\wedge d\overline{z_{j}}$
$\displaystyle=$
$\displaystyle(1-|z|^{2})^{2}\frac{1}{|z|^{2}}\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}dz_{i}\wedge
d\overline{z_{j}}$
$\displaystyle+(1-|z|^{2})\biggl{(}\sum\limits_{i=1}^{n}dz_{i}\wedge
d\overline{z_{j}}-\frac{1}{|z|^{2}}\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}dz_{i}\wedge
d\overline{z_{j}}\biggr{)}.$

Now, for $\overline{\partial}u=\sum\limits_{i=1}^{n}\frac{\partial
u}{\partial\overline{z_{i}}}d\overline{z_{i}}$, we compute that

$\displaystyle|\overline{\partial}u|^{2}_{\partial\overline{\partial}\psi}$
$\displaystyle=$
$\displaystyle(1-|z|^{2})^{2}\frac{1}{|z|^{2}}\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}\frac{\partial
u}{\partial\overline{z_{i}}}\overline{\frac{\partial
u}{\partial\overline{z_{j}}}}$
$\displaystyle+(1-|z|^{2})\biggl{(}\sum\limits_{i=1}^{n}\biggl{|}\frac{\partial
u}{\partial\overline{z_{i}}}\biggr{|}^{2}-\frac{1}{|z|^{2}}\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}\frac{\partial
u}{\partial\overline{z_{i}}}\overline{\frac{\partial
u}{\partial\overline{z_{j}}}}\biggr{)}$ $\displaystyle=$
$\displaystyle-(1-|z|^{2})\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}\frac{\partial
u}{\partial\overline{z_{i}}}\overline{\frac{\partial
u}{\partial\overline{z_{j}}}}+(1-|z|^{2})|z|^{2}\sum\limits_{i=1}^{n}\biggl{|}\frac{\partial
u}{\partial\overline{z_{i}}}\biggr{|}^{2}$
$\displaystyle+(1-|z|^{2})^{2}\sum\limits_{i=1}^{n}\biggl{|}\frac{\partial
u}{\partial\overline{z_{i}}}\biggr{|}^{2}$ $\displaystyle=$
$\displaystyle(1-|z|^{2})^{2}\sum\limits_{i=1}^{n}\biggl{|}\frac{\partial
u}{\partial\overline{z_{i}}}\biggr{|}^{2}+(1-|z|^{2})\biggl{(}|z|^{2}\sum\limits_{i=1}^{n}\biggl{|}\frac{\partial
u}{\partial\overline{z_{i}}}\biggr{|}^{2}-\sum\limits_{i,j=1}^{n}\overline{z_{i}}z_{j}\frac{\partial
u}{\partial\overline{z_{i}}}\overline{\frac{\partial
u}{\partial\overline{z_{j}}}}\biggr{)}$ $\displaystyle=$
$\displaystyle(1-|z|^{2})^{2}|\overline{\partial}u|^{2}+(1-|z|^{2})\biggl{|}\overline{\partial}u\wedge\overline{\partial}|z|^{2}\biggr{|}^{2}$
$\displaystyle=$
$\displaystyle\rho^{2}|\overline{\partial}u|^{2}+|\rho|\cdot|\overline{\partial}u\wedge\overline{\partial}\rho|^{2}.$

Hence, for any solution $u$ orthogonal to $A^{2}(B_{n})$, the estimate of
Donnelly and Fefferman yields that

$\displaystyle\int_{B_{n}}|u|^{2}dV$ $\displaystyle\leq$ $\displaystyle
C_{0}\int_{B_{n}}|\overline{\partial}u|^{2}_{\partial\overline{\partial}\psi}dV$
$\displaystyle=$ $\displaystyle
C_{0}\int_{B_{n}}\biggl{(}\rho^{2}|\overline{\partial}u|^{2}+|\rho|\cdot|\overline{\partial}u\wedge\overline{\partial}\rho|^{2}\biggr{)}dV.$

Next, for any $g\in A^{2}(B_{n})$, consider the equation

$\overline{\partial}u=\overline{\partial}(fg)=g\cdot\overline{\partial}f.$

It is obvious that $g\cdot\overline{\partial}f$ is
$\overline{\partial}$-closed and $u=(I-P)(fg)=H_{f}g$ is the minimal solution
to this equation. Thus, the above discussion gives that

$\displaystyle\int_{B_{n}}|H_{f}g|^{2}dV$ $\displaystyle\leq$ $\displaystyle
C_{0}\int_{B_{n}}|g\cdot\overline{\partial}f|^{2}_{\partial\overline{\partial}\psi}dV$
$\displaystyle=$ $\displaystyle
C_{0}\int_{B_{n}}|g|^{2}\biggl{(}|\rho\cdot\overline{\partial}f|^{2}+|\rho|\cdot|\overline{\partial}f\wedge\overline{\partial}\rho|^{2}\biggr{)}dV$
$\displaystyle\leq$ $\displaystyle C_{0}C^{2}\int_{B_{n}}|g|^{2}dV,$

since
$|\rho\cdot\overline{\partial}f|+|\rho|^{1/2}|\overline{\partial}f\wedge\overline{\partial}\rho|\leq
C$. It follows that $H_{f}$ is bounded from $A^{2}(B_{n})$ into
$L^{2}(B_{n})$. Moreover, we obtain the following norm estimate

$\|H_{f}\|\leq C_{0}\sup\limits_{z\in
B_{n}}\biggl{(}|\rho\cdot\overline{\partial}f|^{2}+|\rho|\cdot|\overline{\partial}f\wedge\overline{\partial}\rho|^{2}\biggr{)}^{1/2}.$

So the proof is complete. ∎

As pointed out on page 89 of [4], if $f$ is a $(0,1)$ form defined in the
strongly pseudoconvex domain $\Omega=\\{\rho<0\\}$, we also have

$|f|^{2}_{\partial\overline{\partial}\log(1/-\rho)}=\frac{1}{B}\biggl{(}\rho^{2}|f|^{2}_{\beta}+|\rho|\cdot|f\wedge\overline{\partial}\rho|^{2}_{\beta}\biggr{)},$

where $B=-\rho+|\partial\rho|_{\beta}\thicksim 1$ and $|\cdot|_{\beta}$
denotes the norm induced by the metric form
$\beta=\frac{i}{2}\partial\overline{\partial}\rho$, which is equivalent to the
Euclidean metric, since $\rho$ is strictly plurisubharmonic. Therefore, using
similar ideas as Theorem 4.3, we get the following result.

######  Theorem 4.4.

Let $\Omega$ be a strongly pseudoconvex domain with smooth boundary and $f\in
L^{2}(\Omega)\cap C^{1}(\Omega)$. If

$|\rho\cdot\overline{\partial}f|+|\rho|^{1/2}|\overline{\partial}f\wedge\overline{\partial}\rho|\leq
C$

for a constant $C>0$. Then the Hankel operator $H_{f}=(I-P_{B})M_{f}$ is
bounded from $A^{2}(\Omega)$ into $L^{2}(\Omega)$.

This result is just Theorem 3.5 of [25], where the authors also showed that
this condition is necessary. Now, we once again find the relationship between
Hankel operators and $\overline{\partial}$-theory. Moreover, using the
inequality

$\|H_{f}g\|^{2}_{A^{2}(\Omega)}\lesssim\int_{\Omega}|g|^{2}\biggl{(}|\rho\cdot\overline{\partial}f|^{2}+|\rho|\cdot|\overline{\partial}f\wedge\overline{\partial}\rho|^{2}\biggr{)}dV,$

we may easily deduce the compactness of $H_{f}$ on $A^{2}(\Omega)$ as Theorem
3.6 in [25].

##  5\. Proof of Theorem 1.2

$(1)\Rightarrow(2)$. For any $z\in D$, the normalized reproducing kernel
$k_{z}\in H^{2}(D)$ and $\|k_{z}\|_{H^{2}(D)}=1$. So we have

$\frac{1}{2\pi}\int|fk_{z}-P(fk_{z})|^{2}d\theta=\|H_{f}(k_{z})\|^{2}_{H^{2}(D)}\leq
C_{1}$

for a positive constant $C_{1}$. It is easy to see that
$\frac{1}{k_{z}}P(fk_{z})\in H^{2}(D)$ and

$P_{z}(\theta)=\frac{1-|z|^{2}}{|e^{i\theta}-z|^{2}}=|k_{z}|^{2}.$

Thus we get

$\displaystyle\inf\limits_{h\in
H^{2}}\frac{1}{2\pi}\int|f-h|^{2}P_{z}(\theta)d\theta\leq\frac{1}{2\pi}\int|f-\frac{1}{k_{z}}P(fk_{z})|^{2}|k_{z}|^{2}d\theta$
$\displaystyle=\frac{1}{2\pi}\int|fk_{z}-P(fk_{z})|^{2}d\theta\leq C_{1},$

which implies (2).

$(2)\Rightarrow(3)$. There exists a constant $M>0$, such that $F(z)\leq M$ for
any $z\in D$. For $z=0$, we have

$F(0)=\inf\limits_{h\in H^{2}}\frac{1}{2\pi}\int|f-h|^{2}d\theta\leq M.$

Choose $h_{0}\in H^{2}(D)$ which satisfies

$\frac{1}{2\pi}\int|f-h_{0}|^{2}d\theta=\inf\limits_{h\in
H^{2}}\frac{1}{2\pi}\int|f-h|^{2}d\theta.$

It is clear that $h_{0}$ is the orthogonal projection of $f$ onto $H^{2}(D)$
and $f-h_{0}$ is conjugate analytic. Moreover, we have $f(0)-h_{0}(0)=0$.
Hence,

$|\nabla(f-h_{0})|^{2}=2|\partial(f-h_{0})/\partial\overline{z}|^{2}=2|\partial
f/\partial\overline{z}|^{2}.$

Combining this with the Littlewood-Paley indetity (see, p. 230), we obtain

$\frac{1}{2\pi}\int|f-h_{0}|^{2}d\theta=\frac{1}{\pi}\iint\limits_{D}|\nabla(f-h_{0})|^{2}\log\frac{1}{|z|}dxdy$
$=\frac{2}{\pi}\iint\limits_{D}\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy\leq M.$ $None$

For fixed $z_{0}\in D$, let $h_{1}$ be chosen to attain the infimum in (1.1)
with respect $P_{z_{0}}(\theta)d\theta$. Applying similar arguments as in the
proof of Theorem 3.5 on page 372 of [18], we see that $f-h_{1}$ is conjugate
analytic and $f(z_{0})-h_{1}(z_{0})=0$, because in the Hilbert space
$L^{2}(P_{z_{0}}(\theta)d\theta)$, the function $h_{1}$ is the orthogonal
projection of $f$ onto $H^{2}(D)$. This means

$|\nabla(f-h_{1})|^{2}=2|\partial(f-h_{1})/\partial\overline{z}|^{2}=2|\partial
f/\partial\overline{z}|^{2}.$

Thus the Littlewood-Paley indetity gives

$\frac{1}{2\pi}\int|f-h_{1}|^{2}P_{z_{0}}(\theta)d\theta=\frac{2}{\pi}\iint\limits_{D}\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\biggl{|}\frac{1-\overline{z_{0}}z}{z-z_{0}}\biggr{|}dxdy\leq
M.$ $None$

Notice that

$(1-|z|^{2})\frac{1-|z_{0}|^{2}}{|1-\overline{z_{0}}z|^{2}}=1-\biggl{|}\frac{z-z_{0}}{1-\overline{z_{0}}z}\biggr{|}^{2}\leq
2\log\biggl{|}\frac{1-\overline{z_{0}}z}{z-z_{0}}\biggr{|}.$

Combining this inequality with (5.2), we have a constant $C_{2}>0$ such that

$\displaystyle\iint\limits_{|z|>1/4}\frac{1-|z_{0}|^{2}}{|1-\overline{z_{0}}z|^{2}}\cdot\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy$ $\displaystyle\leq
C_{2}\iint\limits_{|z|>1/4}\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}(1-|z|^{2})\frac{1-|z_{0}|^{2}}{|1-\overline{z_{0}}z|^{2}}dxdy$
$\displaystyle\leq 2C_{2}\iint\limits_{D}\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\biggl{|}\frac{1-\overline{z_{0}}z}{z-z_{0}}\biggr{|}dxdy\leq
C_{2}M\pi.$

On the other hand, we infer from (5.1) that

$\displaystyle\iint\limits_{|z|\leq
1/4}\frac{1-|z_{0}|^{2}}{|1-\overline{z_{0}}z|^{2}}\cdot\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy$ $\displaystyle\leq
C_{3}\iint\limits_{|z|\leq 1/4}\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy\leq C_{3}M\pi.$

for a constant $C_{3}>0$. All these arguments yield that

$\sup\limits_{z_{0}\in
D}\iint\limits_{D}\frac{1-|z_{0}|^{2}}{|1-\overline{z_{0}}z|^{2}}\cdot\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy\leq C$

for a positive constant $C$. In view of Lemma 3.3 in Chapter VI of [18], we
see that $|\frac{\partial f}{\partial\overline{z}}|^{2}\log\frac{1}{|z|}dxdy$
is a Carleson measure.

$(3)\Rightarrow(1)$. For any $g\in H^{2}(D)$, we see that $fg-P(fg)$ is
conjugate analytic and $f(0)g(0)-P(fg)(0)=0$, because
$(H^{2}(D))^{\perp}=\overline{H^{2}(D)}$. If $|\frac{\partial
f}{\partial\overline{z}}|^{2}\log\frac{1}{|z|}dxdy$ is a Carleson measure,
then the Littlewood-Paley identity implies

$\displaystyle\|H_{f}(g)\|^{2}_{H^{2}(D)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int|fg-P(fg)|^{2}d\theta$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\iint\limits_{D}\biggl{|}\nabla(fg-P(fg))\biggr{|}^{2}\log\frac{1}{|z|}dxdy$
$\displaystyle=$
$\displaystyle\frac{2}{\pi}\iint\limits_{D}|g|^{2}\cdot\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy\lesssim\|g\|^{2}_{H^{2}(D)}.$

Thus, $H_{f}$ is bounded on $H^{2}(D)$. This completes the proof of Theorem
1.2. $\Box$

Remark. Recall that $f\in BMO(\partial D)$ if and only if

$d\lambda_{f}=|\nabla f(z)|^{2}\log\frac{1}{|z|}dxdy$

is a Carleson measure by Theorem 3.4 of [18]. Let

$d\nu_{f}=\biggl{|}\frac{\partial
f}{\partial\overline{z}}\biggr{|}^{2}\log\frac{1}{|z|}dxdy.$

Since $|\nabla f(z)|^{2}=2(|\partial f/\partial\overline{z}|^{2}+|\partial
f/\partial z|^{2})$ and $\overline{\partial f/\partial
z}=\partial\overline{f}/\partial\overline{z}$, we have
$\lambda_{f}=2(\nu_{f}+\nu_{\overline{f}})$. In particular, when $f$ is real-
valued, $\lambda_{f}$ and $\nu_{f}$ are equivalent. Thus, Theorem 1.2 implies
a well-known result, that is, $H_{f}$ is bounded on $H^{2}(D)$ if and only if
$f\in BMO(\partial D)$ for real-valued $f$.

On the other hand, observing Theorem 1.2 and the corresponding result on the
Bergman space $A^{2}(\Omega)$, we have the following conjecture for the
boundedness of Hankel operators on $H^{2}(\Omega)$.
Conjecture 2. Suppose that $\Omega$ is a bounded strongly pseudoconvex domain
with smooth boundary and $f\in L^{2}(\partial\Omega)$. Then the Hankel
operator $H_{f}$ is bounded from $H^{2}(\Omega)$ into $L^{2}(\partial\Omega)$,
if and only if $f$ admits a decomposition $f=f_{1}+f_{2}$ with $f_{1}\in
L^{\infty}(\partial\Omega)$ and $f_{2}$ having an extension
$\widetilde{f_{2}}$ in $\Omega$, which satisfies that
$|\overline{\partial}\widetilde{f_{2}}|dV$ and
$\frac{1}{\sqrt{-\rho}}|\overline{\partial}\widetilde{f_{2}}\wedge\overline{\partial}\rho|dV$
are Carleson measures in $\Omega$.

##  6\. $H_{f}$ and $H_{\overline{f}}$ on Hardy spaces of strongly
pseudoconvex domains

For the unit disc $D$, it is well known that for $f\in L^{2}(\partial D)$,
both $H_{f}$ and $H_{\overline{f}}$ are bounded on $H^{2}(D)$ if and only if
$f\in BMO(\partial D)$. This result can be generalized to the unit ball [45]
and the key tool is the following inequality:

$\|H_{f}k_{z}\|^{2}_{2}+\|H_{\overline{f}}k_{z}\|^{2}_{2}\geq\|fk_{z}\|^{2}_{2}+|\langle
fk_{z},k_{z}\rangle|^{2},$

where $k_{z}$ is the normalized reproducing kernel, which can be proved by
using automorphisms of the ball. In this section, we denote the norm
$\|\cdot\|_{2}:=\|\cdot\|_{H^{2}}$.

Let us consider the corresponding result on the strongly pseudoconvex domain
$\Omega$. Since there might be no nontrivial holomorphic automorphisms for
general strongly pseudoconvex domains in $\mathbb{C}^{n}$, we borrow some
ideas from [7] to obtain the following.

######  Theorem 6.1.

Let $\Omega$ be a strongly pseudoconvex domain with smooth boundary and $f\in
L^{2}(\partial\Omega)$. Suppose that for each fixed $z\in\Omega$,
$1/S(\cdot,z)$ is holomorphic and bounded in $\Omega$. Then the Hankel
operators both $H_{f}$ and $H_{\overline{f}}$ are bounded from $H^{2}(\Omega)$
into $L^{2}(\partial\Omega)$ if and only if $f$ lies in $BMO(\partial\Omega)$.

###### Proof.

Let $S(\zeta,z)$ be the Szegö kernel and
$k_{z}(\zeta)=\frac{S(\zeta,z)}{\sqrt{S(z,z)}}$ be the normalized kernel. For
fixed $\zeta\in\partial\Omega$, the function $S(z,\zeta)$ is holomorphic in
$\Omega$. Let $b_{z}(\zeta)=\frac{S(\zeta,z)}{S(z,\zeta)}$. Then for any
$z\in\Omega$, we have

$\langle H_{f}(k_{z}),b_{z}\overline{H_{\overline{f}}(k_{z})}\rangle=\langle
H_{f}(k_{z})H_{\overline{f}}(k_{z}),b_{z}\rangle.$

Note that

$\displaystyle H_{f}(k_{z})H_{\overline{f}}(k_{z})$ $\displaystyle=$
$\displaystyle(I-P)(fk_{z})(I-P)(\overline{f}k_{z})$ $\displaystyle=$
$\displaystyle|f|^{2}k^{2}_{z}-fk_{z}P(\overline{f}k_{z})-\overline{f}k_{z}P(fk_{z})+P(fk_{z})P(\overline{f}k_{z}).$

Since $k_{z}^{2}\,\overline{b_{z}}=|k_{z}|^{2}$, we obtain

$\langle|f|^{2}k_{z}^{2},b_{z}\rangle=\|fk_{z}\|^{2}_{2}.$

From $\overline{k_{z}}\,b_{z}=k_{z}$, we see that

$\langle fk_{z}P(\overline{f}k_{z}),b_{z}\rangle=\langle
P(\overline{f}k_{z}),\overline{fk_{z}}\,b_{z}\rangle=\langle
P(\overline{f}k_{z}),\overline{f}k_{z}\rangle=\|P(\overline{f}k_{z})\|^{2}_{2}$

and similarly

$\langle\overline{f}k_{z}P(fk_{z}),b_{z}\rangle=\|P(fk_{z})\|^{2}_{2}.$

Since $1/S(\cdot,z)$ is holomorphic and bounded in $\Omega$ for each fixed
$z\in\Omega$, the function $k_{z}^{-1}$ is well defined. Using the reproducing
property of the Szegö kernel, we have

$\displaystyle\langle P(fk_{z})P(\overline{f}k_{z}),b_{z}\rangle$
$\displaystyle=$ $\displaystyle\langle
P(fk_{z})P(\overline{f}k_{z})k_{z}^{-1},k_{z}\rangle$ $\displaystyle=$
$\displaystyle P(fk_{z})(z)P(\overline{f}k_{z})(z)S(z,z)^{-1}.$

Now, $P(z,\zeta)=\frac{S(\zeta,z)S(z,\zeta)}{S(z,z)}$ is the Poisson-Szegö
kernel. Thus, we get

$\displaystyle P(fk_{z})$ $\displaystyle=$
$\displaystyle\sqrt{S(z,z)}\int_{\partial\Omega}f(\zeta)\frac{S(\zeta,z)S(z,\zeta)}{S(z,z)}d\sigma(\zeta)$
$\displaystyle=$
$\displaystyle\sqrt{S(z,z)}\int_{\partial\Omega}f(\zeta)P(z,\zeta)d\sigma(\zeta)=\sqrt{S(z,z)}\widetilde{f}(z),$

It follows that

$\langle
P(fk_{z})P(\overline{f}k_{z}),b_{z}\rangle=\widetilde{f}(z)\widetilde{\overline{f}}(z)=|\widetilde{f}(z)|^{2}$

because of $\widetilde{\overline{f}}=\overline{\widetilde{f}}$. Therefore, the
above discussions show that

$\langle
H_{f}(k_{z}),b_{z}\overline{H_{\overline{f}}(k_{z})}\rangle=\|fk_{z}\|^{2}_{2}-\|P(fk_{z})\|^{2}_{2}-\|P(\overline{f}k_{z})\|^{2}_{2}+|\widetilde{f}(z)|^{2}.$

Recall that
$\|H_{f}(k_{z})\|^{2}_{2}=\|fk_{z}\|^{2}_{H^{2}}-\|P(fk_{z})\|^{2}_{2}$ and
$\|H_{\overline{f}}(k_{z})\|^{2}_{2}=\|\overline{f}k_{z}\|^{2}_{2}-\|P(\overline{f}k_{z})\|^{2}_{2}$.
This means

$\langle
H_{f}(k_{z}),b_{z}\overline{H_{\overline{f}}(k_{z})}\rangle=\|H_{f}(k_{z})\|^{2}_{2}+\|H_{\overline{f}}(k_{z})\|^{2}_{2}-\|\overline{f}k_{z}\|^{2}_{2}+|\widetilde{f}(z)|^{2}.$

Applying this identity and the Schwarz inequality, we obtain

$\|\overline{f}k_{z}\|^{2}_{2}-|\widetilde{f}(z)|^{2}\leq\frac{3}{2}\biggl{(}\|H_{f}(k_{z})\|^{2}_{2}+\|H_{\overline{f}}(k_{z})\|^{2}_{2}\biggr{)}.$
$None$

On the other hand, in Section 2, we have talked about an equivalent norm for
$BMO$, i.e. $f\in BMO(\partial\Omega)$ if and only if

$\sup\limits_{z\in\Omega}\int_{\partial\Omega}|f(\zeta)-\widetilde{f}(z)|^{2}P(z,\zeta)d\sigma(\zeta)<\infty.$

In fact, $P(z,\zeta)=|k_{z}(\zeta)|^{2}$, so we get

$\displaystyle\int_{\partial\Omega}|f(\zeta)-\widetilde{f}(z)|^{2}P(z,\zeta)d\sigma(\zeta)$
$\displaystyle=$
$\displaystyle\int_{\partial\Omega}|f(\zeta)|^{2}P(z,\zeta)d\sigma(\zeta)-\overline{\widetilde{f}(z)}\int_{\partial\Omega}f(\zeta)P(z,\zeta)d\sigma(\zeta)$
$\displaystyle-\widetilde{f}(z)\int_{\partial\Omega}\overline{f(\zeta)}P(z,\zeta)d\sigma(\zeta)+|\widetilde{f}(z)|^{2}\int_{\partial\Omega}P(z,\zeta)d\sigma(\zeta)$
$\displaystyle=$
$\displaystyle\|\overline{f}k_{z}\|^{2}_{2}-|\widetilde{f}(z)|^{2}.$

As a consequence, if both $H_{f}$ and $H_{\overline{f}}$ are bounded, then
(6.1) gives

$\int_{\partial\Omega}|f(\zeta)-\widetilde{f}(z)|^{2}P(z,\zeta)d\sigma(\zeta)\leq\frac{3}{2}\biggl{(}\|H_{f}(k_{z})\|^{2}_{2}+\|H_{\overline{f}}(k_{z})\|^{2}_{2}\biggr{)}<\infty.$

This implies that $f\in BMO(\partial\Omega)$.

For another direction, Theorem 1.1 gives the desired result and the proof is
complete. ∎

## References

  * [1] P. Ahern and R. Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math., 101 (1979), 543–565.
  * [2] E. Amar, Big Hankel operator and $\overline{\partial}_{b}$-equation, J. Oper. Theory, 33 (1995), 223–233.
  * [3] E. Amar and A. Bonami, Measures de Carleson d’ordre $\alpha$ et solutions au bord de l’équation $\overline{\partial}$, Bull. Soc. Math. France, 107 (1979), 23–48.
  * [4] M. Andersson and H. Carlsson, Estimates of solutions of the $H^{p}$ and BMOA corona problem, Math. Ann., 316 (2000), 83–102.
  * [5] S. Asserda. The essential norm of Hankel operators on the Bergman spaces of strongly pseudoconvex domain, Integr. equ. oper. theory, 36 (2000), 379–395.
  * [6] F. Beatrous, $L^{p}$ estimates for extensions of holomorphic functions, Michigan Math. J., 32(1985), 361–380.
  * [7] F. Beatrous and S. Li, Trace ideal criteria for operators of Hankel type, Illinois J. Math., 39 (1995), 723–754.
  * [8] B. Berndtsson, $\overline{\partial}_{b}$ and Carleson inequalities, Complex Analysis. II, (Univ. of Maryland, 1985-86), Lecture Notes in Math., vol. 1276, Springer, 1987, pp. 42–54.
  * [9] B. Berndtsson, Weighted estimates for the $\bar{\partial}$-equation, Complex Analysis and Complex Geometry (J. D. McNeal eds.), de Gruyter, pp. 43–57, 2001.
  * [10] P. Charpentier and Y. Dupain, Weighted and boundary $L^{p}$ estimates for solutions of the $\partial$-equation on lineally convex domains of finite type and applications, Math. Z., 290 (2018), 195–220.
  * [11] B.-Y. Chen and S. Fu, Comparison of the Bergman and Szegö kernels, Adv. Math., 228 (2011), 2366–2384.
  * [12] S. Chen and M. Shaw, Partial Differential Equations in Several Complex Variables, AMS/IP, 2000.
  * [13] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math., 103 (1976), 611–635.
  * [14] H. Donnelly and C. Fefferman, $L^{2}$-cohomology and index theorem for the Bergman metric, Ann. Math., 118 (1983), 593–618.
  * [15] Q. Fang and J. Xia, Schatten class membership of Hankel operators on the unit sphere, J. Funct. Anal., 257 (2009), 3082–3134.
  * [16] M. Feldman and R. Rochberg, Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group, in: Analysis and Partial Differential Equations, in: Lect. Notes Pure Appl. Math., vol. 122, Dekker, New York, 1990, 121–159.
  * [17] G. Folland and J. Kohn, The Neumann Problem for Cauchy-Riemann Complex, Annals of math. studies, 75 (1972).
  * [18] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
  * [19] A. Gleason, Finitely generated ideals in Banach algebras, J. Math. Mech., 13 (1964), 125–132.
  * [20] P. Hartman, On completely continuous Hankel metrices, Proc. Amer. Math. Soc., 9 (1958), 862–866.
  * [21] G. Henkin, Integral representation of a function in a strictly pseudo-convex domain and applications to the $\overline{\partial}$-problem (Russian), Mat. Sb., 82 (1970), 300–308.
  * [22] L. Hörmander, $L^{p}$-estimates for plurisubharmonic functions, Math. Scand., 20 (1967), 65–78.
  * [23] L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, New York, 1973.
  * [24] H. Li, Schatten class Hankel operators on strongly pseudoconvex domains, Proc. Amer. Math. Soc. 199 (1993), 1211–1221.
  * [25] H. Li, Hankel operators on the Bergman spaces of strongly pseudoconvex domains, Integral Equ. Oper. Theory, 19 (1994), 458–476.
  * [26] H. Li and D. Luecking, BMO on strongly pseudoconvex domains: Hankel operators, duality and $\overline{\partial}$-estimates, Trans. Amer. Math. Soc., 346(1994), 661–691.
  * [27] S. Li and W. Luo, Analysis on Besov spaces II: embedding and duality theorems, J. Math. Anal. Appl., 333 (2007), 1189–1202.
  * [28] D. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disc, J. Funct. Anal., 110 (1992), 247–271.
  * [29] J. Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math., 98 (1976), 529–569.
  * [30] Z. Nehari, On bounded linear forms, Ann Math., 65 (1957), 153–162.
  * [31] N. Øvrelid, Integral representation formulas and $L^{p}$ estimates for a equation, Math. Scand, 29 (1971), 137–160.
  * [32] J. Partington, An Introduction to Hankel Operators, London Math. Soc. Student Texts 13, Cambridge University Press, Cambridge, 1988.
  * [33] V. Peller, Hankel operators of class $C_{p}$ and their applications (rational approximation, Gaussian processes, the problem of majarizing operators), Math. USSR-Sbornik, 41 (1982), 443–479.
  * [34] V. Peller, A description of Hankel operators of class $S_{p}$ for $p>0$, an inverstigation of the rate of rational approximation, and other application, Math. USSR Sbornik,- 50 (1985), 465-494.
  * [35] V. Peller, Hankel Operators and Their Applications, Springer-Verlag, New York, 2003.
  * [36] S. Power, Hankel Operators on Hilbert Spaces, Pitman Research Notes in Math. 64, Pitman, London, 1982.
  * [37] R. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, New York, 1986.
  * [38] W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^{n}$, Springer, New York, 1980.
  * [39] H. Sato and K. Yabuta, Toeplitz operators on strongly pseudoconvex domains in stein spaces, Tohoku Math. Journ., 30 (1978), 153–162.
  * [40] S. Semmes, Trace ideal criteria for Hankel operators, and applications to Besov spaces, Integral Equ. Oper. Theory, 7 (1984), 241–281.
  * [41] H. Skoda, Valeurs au bord pour les solutions de opérateur $d^{\prime\prime}$ et caractérisation de zéros des fonctions de la classe de Nevanlinna, Bull. Soc. Math. France, 104 (1976), 225–299.
  * [42] E. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972.
  * [43] E. Stout, $H^{p}$-Functions on strictly pseudoconvex domains, Amer. J. Math., 98 (1976), 821–852.
  * [44] N. Varopoulos, BMO functions and the $\overline{\partial}$-equation, Pacific J. Math., 71 (1977), 221–273.
  * [45] J. Xia, Bounded functions of vanishing mean oscillation on compact metric spaces, J. Funct. Anal., 209 (2004), 444–467.
  * [46] J. Xia, Boundedness and compactness of Hankel operators on the sphere, J. Funct. Anal., 255 (2008), 25–45.
  * [47] D. Zheng, Toeplitz operators and Hankel operators on the Hardy space of the unit sphere, J. Funct. Anal., 149 (1997), 1–24.
  * [48] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Mathematical Surveys and Monographs, vol.138, American Mathematical Society, Providence, RI, 2007.