# The Laplace and Leray transforms on some (weakly) convex domains in
$\mathbb{C}^{2}$

Agniva Chatterjee  Department of Mathematics, Indian Institute of Science,
Bangalore 560012, India<EMAIL_ADDRESS>
###### Abstract.

The space of Laplace transforms of holomorphic Hardy-space functions have been
characterized as weighted Bergman spaces of entire functions in two cases:
that of planar convex domains (Lutsenko–Yumulmukhametov, 1991), and that of
strongly convex domains in higher dimensions (Lindholm, 2002). In this paper,
we establish such a Paley–Weiner result for a class of (weakly) convex
Reinhardt domains in $\mathbb{C}^{2}$ that are well-modelled by the so-called
egg domains. We consider Hardy spaces on these domains with respect to a
canonical choice of boundary Monge–Ampère measure. This class of domains was
introduced by Barrett–Lanzani (2009) to study the $L^{2}$-boundedness of the
Leray transform in the absence of either strongly convexity or
$\mathcal{C}^{2}$-regularity. The boundedness of the Leray transform plays a
crucial role in understanding the image of the Laplace transform. As a
supplementary result, we expand the known class of convex Reinhardt domains
for which the Leray transform is $L^{2}$-bounded (with respect to the
aforementioned choice of boundary measure). Finally, we also produce an
example to show that the Lutsenko–Yumulmukhametov result cannot be expected to
generalize to all convex domains in higher dimensions.

##  1\. Introduction

Let $\mathcal{X}$ be a topological vector space of functions on a subset of
${\mathbb{C}^{n}}$ such that all exponential functions of the form
$\zeta\mapsto e^{\langle z,\zeta\rangle}$, $z\in{\mathbb{C}^{n}}$, belong to
$\mathcal{X}$. It is sometimes useful to identify $\mathcal{X}^{\prime}$, the
dual space of $\mathcal{X}$, with a space of functions on ${\mathbb{C}^{n}}$
via the Laplace transform

$\mathcal{L}(f)(z)=f(e^{\langle z,\cdot\rangle}),\quad
f\in\mathcal{X}^{\prime}.$

Some classical examples of $\mathcal{X}^{\prime}$, for which this
identification has been used to great effect, include the space of
$L^{2}$-functions with compact convex support in $\mathbb{R}^{n}$ (the
Paley–Wiener theorem), the space of distributions with compact convex support
in $\mathbb{R}^{n}$ (the Paley-Wiener-Schwartz theorem), the space of
hyperfunctions with convex compact support in $\mathbb{R}^{n}$ (the Paley-
Weiner-Martineau theorem), and the space of analytic functionals carried by
compact convex sets in $\mathbb{C}^{n}$ (the Polya-Martineau-Ehrenpreis
theorem). More recently, results in this spirit have appeared in the context
of interpolation and sampling, [31, 8], and in the proofs of certain geometric
inequalities for convex bodies, [28, 9].

###  1.1. Laplace transforms of holomorphic Hardy functions

In this paper, we consider the case when $\mathcal{X}=\mathcal{X}^{\prime}$ is
a Hardy space of holomorphic functions on a convex Reinhardt domain of the
type considered by Barrett–Lanzani [4]. Our motivation is as follows. Given a
bounded convex domain $\Omega\subset{\mathbb{C}^{n}}$ and a Radon measure
$\mu$ on $b\Omega$, let $L^{2}(b\Omega,\mu)$ denote the space of complex-
valued functions on $b\Omega$ that are square-integrable with respect to
$\mu$, and $\mathcal{A}(\Omega)$ denote the space of continuous
$\mathbb{C}$-valued functions on $\overline{\Omega}$ that are holomorphic on
$\Omega$. Consider the Laplace transform $\mathcal{L}$ on $L^{2}(b\Omega,\mu)$
given by

$\mathcal{L}:f\mapsto F(z)=\int_{bD}\overline{f(\zeta)}\,e^{\langle
z,\zeta\rangle}\,d\mu(\zeta),\quad\forall f\in L^{2}(b\Omega,\mu).$

Since $\mathcal{L}$ annihilates the orthogonal complement of

$\mathcal{H}^{2}(\Omega,\mu)=\text{the $L^{2}(\mu)$-closure of
$\mathcal{A}(\Omega)|_{b\Omega}$ in }L^{2}(b\Omega,\mu),$

we might as well consider the restriction of $\mathcal{L}$ to
$\mathcal{H}^{2}(\Omega,\mu)$ (also denoted by $\mathcal{L}$).

The Polya-Martineau-Ehrenpreis theorem gives a necessary condition for an
entire function to be the Laplace transform of some
$f\in\mathcal{H}^{2}(\Omega,\mu)$. In some cases, it is possible to completely
characterize the range of $\mathcal{L}$ as a weighted Bergman space of the
form

$A^{2}({\mathbb{C}^{n}},\omega)=\left\\{F\in\mathcal{O}({\mathbb{C}^{n}}):\|F\|_{\omega}^{2}=\int_{\mathbb{C}^{n}}|F|^{2}d\omega<\infty\right\\}\text{
with norm }\|\cdot\|_{\omega}.$

One advantage of shifting the analysis to a weighted Bergman space is the
availability of Hörmander’s $L^{2}$-theory to construct functions with desired
properties. The following rather comprehensive result completely tackles the
case of planar convex domains when $\mu$ is the arc-length measure; see [26].

######  Theorem 1.1 (Lutsenko–Yumulmukhametov).

Let $\Omega\subset\mathbb{C}$ be a bounded convex domain, and
$\sigma_{\Omega}$ be the arc-length measure on $b\Omega$. Let $H_{\Omega}$
denote the support function of $\Omega$ (see (2.6)). Then, $\mathcal{L}$ is a
normed space isomorphism between $\mathcal{H}^{2}(\Omega,\sigma_{\Omega})$ and
$A^{2}(\mathbb{C},\widetilde{\omega}_{\Omega})$, where

$\widetilde{\omega}_{\Omega}(z)=\|e^{\langle
z,\cdot\rangle}\|^{-2}_{L^{2}(\sigma_{\Omega})}\Delta H_{\Omega}(z),\quad
z\in\mathbb{C},$ (1.1)

and $\Delta H_{\Omega}$ is the Riesz measure of the convex function
$H_{\Omega}$.

Special cases of this theorem were previously proved by Levin (polygons),
Likht (discs), Katsnel’son and Lyubarskii (strongly convex domains); see [23,
Appendix I], [24], [19] and [27], respectively. The case of strongly convex
domains in ${\mathbb{C}^{n}}$ is also well-understood.

######  Theorem 1.2 (Lindholm).

Let $\Omega\subset{\mathbb{C}^{n}}$ be a bounded strongly convex domain, and
$\sigma_{\Omega}$ be the surface area measure on $b\Omega$. Let $H_{\Omega}$
denote the support function of $\Omega$. Then, $\mathcal{L}$ is a normed space
isomorphism between $\mathcal{H}^{2}(\Omega,\sigma_{\Omega})$ and
$A^{2}({\mathbb{C}^{n}},\widetilde{\omega}_{\Omega})$, where

$\widetilde{\omega}_{\Omega}(z)=\|e^{\langle
z,\cdot\rangle}\|^{-2}_{L^{2}(\sigma_{\Omega})}\left(dd^{c}H_{\Omega}\right)^{n}(z),\quad
z\in{\mathbb{C}^{n}},$ (1.2)

and $(dd^{c}H_{\Omega})^{n}$ is the Monge–Ampère measure of $H_{\Omega}$.

The measure $\widetilde{\omega}_{\Omega}$ in (1.2) is different from the one
appearing in Lindholm’s original text [25], but we show in Proposition 7.2
below that $\widetilde{\omega}_{\Omega}$ works equally well. Theorem 1.2
raises the

######  Question.

Are the Laplace transforms of holormorphic Hardy functions on weakly convex
domains in ${\mathbb{C}^{n}}$ characterized by a weighted Bergman condition?

A natural class of model domains in this context are the so-called egg
domains:

$\left\\{(z_{1},z_{2})\in\mathbb{C}^{2}:a_{1}|z_{1}|^{p_{1}}+a_{2}|z_{2}|^{p_{2}}<1\right\\},\quad
a_{1},a_{2}\in(0,\infty),p_{1},p_{2}\in(1,\infty).$ (1.3)

These are also sometimes called complex ellipsoids. We show that, instead of
the surface area measure $\sigma_{\Omega}$, if one considers the boundary
Monge–Ampère measure $\mu_{\Omega}$ associated to the Minkowski functional of
$\Omega$, see (2.11), then Theorem 1.2 generalizes to a class of Reinhardt
domains in $\mathbb{C}^{2}$ that are modelled by egg domains. Note that the
measures $\mu_{\Omega}$ and $\sigma_{\Omega}$ are equivalent for $\Omega$ as
in Theorems 1.1 and 1.2, but not for the domains considered in this paper. For
some remarks on this particular choice of measure, see Section 1.5.

######  Theorem 1.3.

Let $\Omega\subset\mathbb{C}^{2}$ be a bounded convex Reinhardt domain that is
$\mathcal{C}^{2}$-smooth and strongly convex away from
$\mathcal{Z}=\left\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}z_{2}=0\right\\}$.
Given $\zeta=(\zeta_{1},\zeta_{2})\in b\Omega\setminus\mathcal{Z}$, let
$\kappa_{1}(\zeta)$, $\kappa_{2}(\zeta)$, and $\kappa_{3}(\zeta)$ be the
principal curvatures of $b\Omega$ at $\zeta$, where the first two correspond
to the directions $\left(i\zeta_{1},0\right)$ and $\left(0,i\zeta_{2}\right)$,
respectively. Let

$\omega_{\Omega}(z)=\|e^{\langle
z,\cdot\rangle}\|^{-2}_{L^{2}(\mu_{\Omega})}\left(dd^{c}H_{\Omega}\right)^{2}(z),\quad
z\in\mathbb{C}^{2}.$ (1.4)

If the function $\kappa_{\Omega}=(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}$ is
uniformly continuous and bounded away from zero on
$b\Omega\setminus\mathcal{Z}$, then $\Omega$ has $\mathcal{C}^{1}$-smooth
boundary, and $\mathcal{L}$ is a normed space isomorphism between
$\mathcal{H}^{2}(\Omega,\mu_{\Omega})$ and
$A^{2}(\mathbb{C}^{2},\omega_{\Omega})$.

See Section 1.6 for bounds on $\|\mathcal{L}\|_{op}$ and
$\|\mathcal{L}^{-1}\|_{op}$ for $\Omega$ as above. On a strictly convex
Reinhardt domain, the function
$\kappa_{\Omega}=(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}$ is well-defined on
$b\Omega\setminus\mathcal{Z}$. The additional hypothesis on $\kappa_{\Omega}$
in Theorem 1.3 forces the domain to be in the class $\mathcal{\widetilde{R}}$
defined in (1.2). All domains of finite type in $\mathcal{\widetilde{R}}$,
including the egg domains, satisfy the hypothesis of Theorem 1.3; see
Proposition 2.2. Although the conditions on $\Omega$ may seem very
restrictive, we show that even within the class of convex Reinhardt domains,
the conclusion of Theorem 1.3 fails to hold if $\kappa_{3}$ is allowed to
vanish; see Theorem 1.6.

###  1.2. Boundedness of the Leray transform

A key ingredient in the proof of Theorem 1.3 is the $L^{2}$-boundedness of the
so-called Leray transform. Since the Leray transform is of independent
interest, we state the relevant result separately. Classically, the Leray
integral of a function $f$ on the boundary of a $\mathcal{C}^{2}$-smooth
convex domain $\Omega\subset{\mathbb{C}^{n}}$ is defined as

$\mathbb{L}(f)(z)=\int_{b\Omega}\frac{f(\zeta)}{\langle\partial\rho(\zeta),\zeta-z\rangle^{n}}\lambda_{\rho}(\zeta),\quad
z\in\Omega,\quad$ (1.5)

where $\rho$ is a $\mathcal{C}^{2}$-smooth defining function of $\Omega$,
$\partial\rho$ is shorthand for
$(\partial\rho/{\partial\zeta_{1}},...,\partial\rho/{\partial\zeta_{n}})$, and
$\lambda_{\rho}$ is the Leray–Levi measure given by the pull-back of the form
$d^{c}\rho\wedge\left(dd^{c}\rho\right)^{n-1}$ under the inclusion of
$b\Omega$ into $\mathbb{C}^{n}$. Due to the convexity of $\Omega$,
$\mathbb{L}$ is a well-defined operator from $L^{2}(b\Omega,\lambda_{\rho})$
to $\mathcal{O}(\Omega)$. It reproduces holomorphic functions that are
continuous up to the boundary; see [30, Theorem 3.14]. If there is a dense
subspace $\mathcal{D}\subset L^{2}(b\Omega,\lambda_{\rho})$ such that
holomorphic functions in $\mathbb{L}(\mathcal{D})$ admit boundary values in
$L^{2}(b\Omega,\lambda_{\rho})$, then one obtains a densely-defined operator
$\mathbb{L}_{b}:\mathcal{D}\rightarrow L^{2}(b\Omega,\lambda_{\rho})$ given by

$\displaystyle\mathbb{L}_{b}:f$ $\displaystyle\mapsto$
$\displaystyle\text{b.v. }\mathbb{L}(f).$

This is the Leray transform on $\Omega$, and one can ask the

######  Question.

Does $\mathbb{L}_{b}$ extend to a bounded operator from
$L^{2}(b\Omega,\lambda_{\rho})$ onto $\mathcal{H}^{2}(\Omega,\lambda_{\rho})$?

If yes, then the reproducing property of $\mathbb{L}$ allows for functions in
$\mathcal{H}^{2}(\Omega,\lambda_{\rho})$ to be decomposed into rational
functions. In the plane, $\mathbb{L}_{b}$ is the Cauchy transform, and
$\lambda_{\rho}$ is equivalent to the arc-length measure $\sigma_{\Omega}$ on
$b\Omega$. The $L^{p}(\sigma)$-regularity, $1<p<\infty$, of the Cauchy
transform, was established in the general setting of Lipschitz domains in the
works of Calderón [10], Coifman-McIntosh-Meyer [12], and David [13]. For
$\mathcal{C}^{2}$-smooth domains in higher dimensions, the
$L^{2}(\lambda_{\rho})$-regularity of $\mathbb{L}_{b}$ was established for egg
domains by Hansson [17], and for strongly convex bounded domains by
Kerzman–Stein [20] and Lanzani–Stein [21]. Lanzani–Stein [22] also gave a
counterexample to $L^{2}(\lambda_{\rho})$-regularity of $\mathbb{L}_{b}$ when
strong convexity is dropped. Some unbounded hypersurfaces were considered by
Barrett–Edholm [3].

The Leray integral can be defined for any $\mathcal{C}^{1}$-smooth convex
domain $\Omega\subset{\mathbb{C}^{n}}$ by replacing $\rho$ and
$\lambda_{\rho}$ in (1.5) by the Minkowski functional $m_{\Omega}$ of
$\Omega$, and the boundary Monge–Ampère measure $\mu_{\Omega}$ associated to
$m_{\Omega}$, respectively. When $\Omega$ is $\mathcal{C}^{2}$-smooth, this
coincides with the original definition of $\mathbb{L}$. For
$\mathcal{C}^{1,1}$ domains, the $L^{p}(\mu_{\Omega})$-boundedness of
$\mathbb{L}_{b}$, $1<p<\infty$, was established for strongly
$\mathbb{C}$-linearly convex domains by Lanzani–Stein [21]. The class

$\displaystyle\widetilde{\mathcal{R}}=\\{\Omega\subset\mathbb{C}^{2}:\Omega\
\text{is a bounded convex $\mathcal{C}^{1}$-smooth Reinhardt domain}\quad$
$\displaystyle\qquad\text{that is $\mathcal{C}^{2}$-smooth and strongly convex
away from $\mathcal{Z}$}\\}.$ (1.6)

was studied by Barrett–Lanzani as a generalization of egg domains. They showed
that for each $\Omega\in\widetilde{\mathcal{R}}$, $\mathbb{L}_{b}$ is a
densely-defined operator on $L^{2}(b\Omega,\mu)$, where $\mu$ belongs to a
class of rotation-invariant finite measures on $b\Omega$, which always
includes $\mu_{\Omega}$. They further described a subclass of domains
$\mathcal{R}\subset\widetilde{\mathcal{R}}$ for which $\mathbb{L}_{b}$ extends
as a bounded projection operator from $L^{2}(b\Omega,\mu)$ onto
$\mathcal{H}^{2}(\Omega,\mu)$. The class $\mathcal{R}$ has a technical
definition, but essentially consists of domains in $\widetilde{\mathcal{R}}$
that are osculated by weighted $L^{p}$-balls even at the axes. For
$\mu=\mu_{\Omega}$, we describe a larger class of domains in
$\widetilde{\mathcal{R}}$ for which $\mathbb{L}_{b}$ is
$L^{2}(\mu_{\Omega})$-bounded. Membership in this class imposes a weaker
condition on the limiting behavior of $\kappa_{1},\kappa_{2}$ and $\kappa_{3}$
than in $\mathcal{R}$; see Example 3.1.

######  Theorem 1.4.

Let $\Omega\in\mathcal{\widetilde{R}}$, and $\mathbb{L}_{b}$ denote the
(densely-defined) Leray transform on $\Omega$.

  1. $(i)$

If $(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}$ is uniformly continuous and bounded
away from zero on $b\Omega\setminus\mathcal{Z}$, then $\mathbb{L}_{b}$ extends
to a bounded projection operator from $L^{2}(b\Omega,\mu_{\Omega})$ onto
$\mathcal{H}^{2}(\Omega,\mu_{\Omega})$.

  2. $(ii)$

If $(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}$ is either not bounded above or not
bounded away from zero, then $\mathbb{L}_{b}$ does not extend as a bounded
operator to $L^{2}(b\Omega,\mu_{\Omega})$.

The case where $(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}$ is bounded above and
away from zero, but does not extend continuously to $b\Omega$, appears to be
quite subtle.

###  1.3. Relating the Leray and the Laplace transform

Lindholm’s original version of Theorem 1.2 states that on strongly convex
domains,
$\mathcal{L}(\mathcal{H}^{2}(\Omega,\sigma))=A^{2}({\mathbb{C}^{n}},\nu_{\Omega})$,
where

$\nu_{\Omega}(z)=e^{-2H_{\Omega}(z)}\|z\|^{n-\frac{1}{2}}\left(dd^{c}H_{\Omega}\right)^{n}(z),\quad
z\in{\mathbb{C}^{n}}.$ (1.7)

We show that this choice of measure works for precisely those domains in
$\mathcal{\widetilde{R}}$ whose Leray transform is $L^{2}$-bounded. In order
to recover Theorem 1.3 from this result, we must compare
$A^{2}({\mathbb{C}^{n}},\nu_{\Omega})$ and
$A^{2}({\mathbb{C}^{n}},\omega_{\Omega})$. While these spaces are the same
(with comparable norms) for strongly convex domains, see Proposition 7.2, we
are only able to provide a sufficient condition for this equality to hold for
a domain in $\mathcal{\widetilde{R}}$.

######  Theorem 1.5.

Let $\Omega\in\widetilde{\mathcal{R}}$, and $\mathbb{L}_{b}$ be the Leray
transform on $\Omega$. Let $\kappa_{1},\kappa_{2}$ and $\kappa_{3}$ be as in
Theorem 1.3. Then, the following hold.

  1. $(i)$

The Leray transform $\mathbb{L}_{b}$ extends to a bounded operator on
$L^{2}(b\Omega,\mu_{\Omega})$ if and only if $\mathcal{L}$ is a normed space
isomorphism between $\mathcal{H}^{2}(\Omega,\mu_{\Omega})$ and
$A^{2}(\mathbb{C}^{2},\nu_{\Omega})$.

  2. $(ii)$

If the function $(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}$ is bounded above and
away from zero on $b\Omega\setminus\mathcal{Z}$, then the identity map is a
normed space isomorphism between $A^{2}(\mathbb{C}^{2},\omega_{\Omega})$ and
$A^{2}(\mathbb{C}^{2},\nu_{\Omega})$.

See Section 1.6 for bounds on $\|\mathcal{L}\|_{op}$ and
$\|\mathcal{L}^{-1}\|_{op}$ in the case when $\mathbb{L}_{b}$ is
$L^{2}(\mu_{\Omega})$-bounded.

We elaborate briefly on the role that the Leray transform plays in studying
the range of $\mathcal{L}$. On strongly convex domains, the Laplace and Leray
transforms are related via the following commutative diagram:

${\mathcal{H}^{2}(\Omega,\mu_{\Omega})}$${\mathcal{L}^{2}(b\Omega^{*},\mu_{\Omega^{*}})}$${\mathcal{H}^{2}(\Omega^{*},\mu_{\Omega^{*}})}$${A^{2}({\mathbb{C}^{n}},\omega_{\Omega})=A^{2}({\mathbb{C}^{n}},\nu_{\Omega})}$$\scriptstyle{\mathcal{L}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathbb{L}_{b}}$$\scriptstyle{\mathcal{B}}$

Here, $\Omega^{*}$ denotes the dual complement of $\Omega$, see (2.5). There
is a natural bijection $s:b\Omega\rightarrow b\Omega^{*}$, which induces the
map $\mathcal{S}$. The map $\mathcal{F}_{b}=\mathbb{L}_{b}\circ\mathcal{S}$ is
the restriction to $\mathcal{H}^{2}(\Omega,\mu_{\Omega})$ of the well-known
Fantappié transform, which is an isomorphism betweel
$\mathcal{O}^{\prime}(\overline{\Omega})$ and $\mathcal{O}(\text{int
\\!}\Omega^{*})$. The map $\mathcal{B}$ is the restriction to
$A^{2}({\mathbb{C}^{n}},\nu_{\Omega})$ of the so-called Borel transform, see
[25, Definition 14]. In order to establish that $\mathcal{L}$ is an
isomorphism, Lindholm shows that $\mathcal{F}$ is an isomorphism between
$\mathcal{H}^{2}(\Omega,\mu_{\Omega})$ and
$\mathcal{H}^{2}(\Omega^{*},\mu_{\Omega^{*}})$, and $\mathcal{B}$ is an
isomorphism between $A^{2}({\mathbb{C}^{n}},\nu_{\Omega})$ and
$\mathcal{H}^{2}(\Omega^{*},\mu_{\Omega^{*}})$. In our case, it can be shown
that

  1. (a)

$\mathcal{B}:A^{2}(\mathbb{C}^{2},\nu_{\Omega})\rightarrow\mathcal{H}^{2}(\Omega^{*},\mu_{\Omega^{*}})$
is a well-defined isomorphism for all $\Omega\in\widetilde{\mathcal{R}}$,

  2. (b)

$\mathcal{F}:\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)\rightarrow\mathcal{H}^{2}\left(\Omega^{*},\mu_{\Omega^{*}}\right)$
is a well-defined isomorphism if and only if the Leray transform
$\mathbb{L}_{b}$ is a bounded projection operator from
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$ onto
$\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$.

However, the symmetries of the domains allow us to take a more direct approach
via series expansions.

###  1.4. A negative result

The following example shows that the measure $\omega_{\Omega}$ will not yield
as comprehensive a result in higher dimensions as Theorem 1.1.

######  Theorem 1.6.

Let $\Omega=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|+|z_{2}|<1\\}$. Then,

$\mathcal{L}\left(\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)\right)\subsetneq
A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)\subsetneq
A^{2}\left(\mathbb{C}^{2},\omega_{\Omega}\right).$

Here, unlike in the case of $\mathcal{\widetilde{R}}$, the measure
$(dd^{c}H_{\Omega})^{2}$ is supported on a three-dimensional surface in
$\mathbb{C}^{2}$. This is analogous to the case of planar polygons, where the
Riesz measure $\triangle H_{\Omega}$ is supported on a union of real lines.
However, for planar polygons, we have that
$A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)\subsetneq
A^{2}\left(\mathbb{C}^{2},\omega_{\Omega}\right)=\mathcal{L}\left(\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)\right)$.
Currently, we do not know of any $\mathcal{C}^{2}$-smooth domains in
$\mathbb{C}^{2}$ for which
$\mathcal{L}\left(\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)\right)\neq
A^{2}\left(\mathbb{C}^{2},\omega_{\Omega}\right)$.

###  1.5. Remarks on the choice of boundary measure

The suitability of $\mu_{\Omega}$ for studying $\mathcal{L}$ was already
indicated by Lindholm [25]. We collect some observations to support this.

  1. (a)

Since Leray–Levi measures appear explicitly in the Leray kernel, boundedness
of the Leray operator has often been studied with respect to these measures.
Hansson makes a strong case for this measure in [17]. However, Leray–Levi
measures are not independent of the choice of defining functions. For convex
domains, the Minkowski functional allows for a canonical choice of defining
function. Furthermore, this choice behaves well under the duality operation:
$s^{*}(\mu_{\Omega^{*}})=\mu_{\Omega}$, where $s:b\Omega\rightarrow
b\Omega^{*}$ is the natural bijection mentioned above.

  2. (b)

Up to a constant factor, $\mu_{\Omega}$ coincides with the special measure
$\mu_{0}$ of order $0$ considered in [4, §8]; see (3.13). The $L^{2}$-adjoint
of $\mathbb{L}_{b}$ with respect to this measure is somewhat special, as
demonstrated in [4, Prop. 49].

  3. (c)

When tackling the Borel transform $\mathcal{B}$, Lindholm uses a coarea-type
formula over the level sets of $\mu_{\Omega}$; see Proposition 2.5 for a
version for $\mathcal{\widetilde{R}}$. Such a formula is needed to transfer
data from $b\Omega$ to ${\mathbb{C}^{n}}$. It is in this formula that the
measures $\omega_{\Omega}$ and $\mu_{\Omega}$ appear naturally as the
analogues of $\omega_{std}$ and $\sigma_{\Omega}$ in the classical coarea
formula.

###  1.6. Bounds on the operator norms

In applications, the classical Paley–Wiener theorem is used in conjunction
with the Plancherel theorem. In all the results above, $\mathcal{L}$ is only
shown to be a normed space isomorphism (as opposed to an isometric
isomorphism). By tracking the constants in our proofs, we obtain the following
bounds.

  1. (a)

If $\Omega\in\mathcal{\widetilde{R}}$ and $\mathbb{L}_{b}$ extends as a
bounded operator on $L^{2}(b\Omega,\mu_{\Omega})$, then

$\frac{\sqrt{\pi}}{e^{2}}c_{\Omega}^{\frac{3}{2}}\|f\|_{\mu_{\Omega}}^{2}\leq\|\mathcal{L}f\|_{\nu_{\Omega}}^{2}\leq\frac{5^{3}\sqrt{e}}{2^{15/2}\pi}C_{\Omega}^{\frac{3}{2}}\|\mathbb{L}_{b}\|_{op}^{2}\|f\|_{\mu_{\Omega}}^{2},$
(1.8)

for all $f\in\mathcal{H}^{2}(\Omega,\mu_{\Omega})$, where
$c_{\Omega}^{-1}=\sup_{\|z\|=1}H_{\Omega}(z)$ and
$C_{\Omega}^{-1}=\inf_{\|z\|=1}H_{\Omega}(z).$

  2. (b)

If $\nu_{\Omega}$ is replaced by the equivalent measure

$\widetilde{\nu}_{\Omega}=e^{-2H_{\Omega}(z)}H_{\Omega}(z)^{\frac{3}{2}}\left(dd^{c}H_{\Omega}\right)^{2}(z),\quad
z\in{\mathbb{C}^{n}},$

above, then $c_{\Omega}^{3/2}$ and $C_{\Omega}^{3/2}$ drop out of the above
inequalities.

  3. (c)

If $\Omega\in\mathcal{\widetilde{R}}$ satisfies the hypothesis of Theorem 1.5
$(ii)$, $p_{\ell}=\inf\\{p(\zeta):\zeta\in b\Omega\\}$, and
$p_{g}=\sup\\{p(\zeta):\zeta\in b\Omega\\}$, where $p$ is defined as in
Section 2.2. Then

$k_{1}(\alpha_{\Omega}C_{\Omega}^{-1})^{\frac{3}{2}}\|F\|_{\nu_{\Omega}}^{2}\leq\|F\|_{\omega_{\Omega}}^{2}\leq
k_{2}(\beta_{\Omega}c_{\Omega}^{-1})^{\frac{3}{2}}\|F\|_{\nu_{\Omega}}^{2},\quad
F\in A^{2}(\mathbb{C}^{2},\omega_{\Omega}),$ (1.9)

where $\alpha_{\Omega}=p_{g}^{-1}$ and
$\beta_{\Omega}={p_{g}}^{-1}{(p_{g}-1)}$ when $p_{\ell}\geq 2$,
$\alpha_{\Omega}={p_{g}^{-1}}{(p_{\ell}-1)}$ and
$\beta_{\Omega}={p_{g}}^{-1}{(p_{l}-1)^{-1}}{(p_{g}-1)}$ when $1<p_{\ell}\leq
2$, $c_{\Omega}$ and $C_{\Omega}$ are as in (1.8), and $k_{1},k_{2}$ are two
positive constants that are independent of $\Omega$.

###  1.7. Organization of the paper

In Section 2, we elaborate on the domains, measures and function spaces
appearing in the main results. In Section 3, we collect some technical tools
that are used extensively in the rest of the paper. Of particular note are the
following two ingredients from [4]:

  * (i)

a special parametrization of any domain in $\mathcal{\widetilde{R}}$; see
(3.1), and

  * (ii)

a decomposition of the Leray transform on any domain in
$\mathcal{\widetilde{R}}$ into bounded rank-one projection operators; see
Lemma 3.5.

The proof of Theorem 1.4 is presented in Section 4. It is a variation of the
proof of Theorem 45 in [4], wherein one performs an asymptotic analysis of the
norms of the rank-one operators obtained in (ii). The main distinction in our
case is the weaker regularity of the parametrizing functions appearing in
(3.3) and (3.4). The proof of Theorem 1.5 is presented in Section 5. The first
part of the proof heavily relies on the series representations obtained in
Section 3.2. The second half of the proof involves a direct comparison of the
two measures in question. The proofs of Theorem 1.3 and Theorem 1.6 are placed
in Section 6. Finally, in Section 7, we establish the equality of the two
weighted Bergman spaces $A^{2}\left(\mathbb{C}^{n},\omega_{\Omega}\right)$ and
$A^{2}\left(\mathbb{C}^{n},\nu_{\Omega}\right)$ for strongly convex domains,
thus unifying Lindholm’s higher-dimensional result with the planar one.
Acknowledgements. I am grateful to my thesis advisor, Purvi Gupta, for
suggesting this problem and sharing many invaluable insights, as well as
engaging in fruitful discussions throughout the course of this project. I
would also like to thank her for helping me with the writing of this paper. I
am also thankful to Koushik Ramachandran for directing me towards literature
on regularly varying functions. This work is supported by a scholarship from
the Indian Institute of Science, and the DST-FIST programme (grant no. DST
FIST-2021 [TPN-700661]).

##  2\. Preliminaries

We collect here some preliminary results and observations regarding our main
objects of study.

###  2.1. Notation

The following notation will be used throughout the paper.

  * (1)

$\mathbb{B}^{n}(R)$ denotes the Euclidean ball in $\mathbb{C}^{n}$ centered at
the origin and of radius $R>0$.

  * (2)

$\mathbb{B}^{n}=\mathbb{B}^{n}(1)$ denotes the unit Euclidean ball in
$\mathbb{C}^{n}$.

  * (3)

$\mathcal{Z}$ denotes the set
$\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}z_{2}=0\\}$.

  * (4)

$\langle\zeta,z\rangle$ denotes the pairing
$\zeta_{1}z_{1}+\zeta_{2}z_{2}\cdots+\zeta_{n}z_{n}$,
$\zeta=(\zeta_{1},\cdots,\zeta_{n}),z=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$.

  * (5)

$\omega_{std}$ denotes the Lebesgue volume measure on $\mathbb{C}^{n}$.

  * (6)

Given a bounded convex domain $\Omega\subset{\mathbb{C}^{n}}$,

    * –

$b\Omega_{+}=b\Omega\setminus\mathcal{Z}$,

    * –

$m_{\Omega}$ denotes the Minkowski functional of $\Omega$,

    * –

$H_{\Omega}$ denotes the support function of $\Omega$,

    * –

$\mu_{\Omega}$ denotes the boundary Monge-Ampère measure of $\Omega$,

    * –

$\sigma_{\Omega}$ denotes the Euclidean surface area measure on $b\Omega$,

    * –

$\omega_{\Omega}$ denotes the measure on ${\mathbb{C}^{n}}$ defined in (1.4),

    * –

$\nu_{\Omega}$ denotes the measure on ${\mathbb{C}^{n}}$ defined in (1.7).

  * (7)

$\|.\|_{\nu}$ denotes the $L^{2}(\nu)$-norm for a given measure $\nu$.

  * (8)

$\|.\|_{C(K)}$ denotes the supremum norm on the space of continuous functions
on the compact set $K$.

  * (9)

$d=\partial+\overline{\partial}$ denotes the standard exterior derivative.

  * (10)

$d^{c}=\frac{i}{4\pi}\left({\overline{\partial}-\partial}\right).$

  * (11)

$\mathds{1}_{A}$ denotes the indicator function of a set
$A\subset\mathbb{C}^{n}$.

  * (12)

Given two $\mathbb{R}$-valued functions $f$ and $g$ on set a $X$, $f\approx g$
on $X$ denotes the existence of $C_{1},C_{2}>0$ such that $C_{1}g(x)\leq
f(x)\leq C_{2}g(x)$ for all $x\in X$.

  * (13)

Given two positive measures $\mu$ and $\nu$ on a set $X$,

    * –

$\mu\ll\nu$ denotes that $\mu$ is absolutely continuous with respect to $\nu.$

    * –

$\mu\approx\nu$ denotes that $\mu$ and $\nu$ are mutually absolutely
continuous, and $d\mu/d\nu\approx 1$ as functions on $X$.

###  2.2. The class $\mathbf{\mathcal{\widetilde{R}}}$

Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^{2}$ that is
strongly convex away from $\mathcal{Z}$. Then $\Omega$ is modelled by weighted
$L^{p}$-balls in the following sense. Given $\zeta=(\zeta_{1},\zeta_{2})\in
b\Omega_{+}=b\Omega\setminus\mathcal{Z}$, there exists a unique triplet
$(p(\zeta),a_{1}(\zeta),a_{2}(\zeta))\in(1,\infty)\times(0,\infty)\times(0,\infty)$
such that the domain

$\left\\{(z_{1},z_{2})\in\mathbb{C}^{2}:a_{1}(\zeta)|z_{1}|^{p(\zeta)}+a_{2}(\zeta)|z_{2}|^{p(\zeta)}<1\right\\}$

osculates $\Omega$ at the point $\zeta$ up to order $2$; see [4, Prop. 8]
whose proof does not depend on the regularity of $b\Omega$ at points in
$\mathcal{Z}$. The functions $a_{1},a_{2}$ and $p$ are continuous on
$b\Omega_{+}$ and rotation-invariant in both the coordinates. The exponent
function $p$ is expressible in terms of the principal curvatures of
$b\Omega_{+}$, as shown in Lemma 2.1 below.

Due to its rotational symmetry, much of the geometry of $\Omega$ is captured
by its Reinhardt shadow. Let
$\gamma_{\Omega}=\left\\{(r_{1},r_{2})\in\mathbb{R}^{2}_{\geq
0}:(r_{1},r_{2})\in b\Omega\right\\}.$ Then,

$\gamma_{\Omega}=\\{(r_{1},r_{2}):r_{2}=\phi(r_{1}),\ 0\leq r_{1}\leq
b_{1}\\}$ (2.1)

for some $b_{1}>0$ and continuous function
$\phi:[0,b_{1}]\rightarrow[0,\infty)$ satsifying

  * $(i)$

$\phi>0$ on $[0,b_{1})$ and $\phi(b_{1})=0$,

  * $(ii)$

$\phi$ is $\mathcal{C}^{2}$-smooth and strongly concave on $(0,b_{1})$.

As a consequence of $(ii)$, one also obtains that

  * $(iii^{\prime})$

$\lim_{t\rightarrow 0^{+}}\phi^{\prime}(t)\in(-\infty,0]$ and
$\lim_{t\rightarrow b_{1}^{-}}\phi^{\prime}(t)\in[-\infty,0)$.

Conversely, any $b_{1}>0$ and continuous $\phi:[0,b_{1}]\rightarrow[0,\infty)$
satisfying $(i)$ and $(ii)$ uniquely determine a bounded convex Reinhardt
domain in $\mathbb{C}^{2}$ that is strongly convex off of $\mathcal{Z}$. In
this paper, for a given $\Omega,$ $b_{1}$ and $\phi$ denote the unique
positive number and continuous function granted by (2.1). The intercept of
$\gamma_{\Omega}$ with the $y$-axis, $\phi(0)$, is denoted by $b_{2}$.

######  Lemma 2.1.

Let $\Omega$ be as above and $\rho$ be a $\mathcal{C}^{1}$-smooth defining
function of $\Omega$ near $\zeta\in b\Omega_{+}$. Then,

$\displaystyle p(\zeta)$ $\displaystyle=$ $\displaystyle
1+\left(\frac{\kappa_{3}}{\kappa_{1}\kappa_{2}}\left(\zeta\right)\times\frac{\|\nabla\rho(\zeta)\|}{2\operatorname{Re}\left<\partial{\rho}(\zeta),\zeta\right>}\right)$
(2.2) $\displaystyle=$ $\displaystyle
1+\frac{\left|\zeta_{1}\right|\phi^{\prime\prime}(\left|\zeta_{1}\right|)\phi(\left|\zeta_{1}\right|)}{\phi^{\prime}(\left|\zeta_{1}\right|)\left(\phi(\left|\zeta_{1}\right|)-\left|\zeta_{1}\right|\phi^{\prime}(\left|\zeta_{1}\right|)\right)}.$
(2.3)

Equation (2.2) is proved in Section 3, while (2.3) was already proved in [4,
Prop. 8].

Recall that $\mathcal{\widetilde{R}}$ is the class of bounded
$\mathcal{C}^{1}$-smooth convex Reinhardt domains in $\mathbb{C}^{2}$ that are
strongly convex away from $\mathcal{Z}$. We see in Section 3.1 that membership
in $\mathcal{\widetilde{R}}$ imposes a condition on the limiting behavior of
$p$ at the axes. In terms of the graphing function $\phi$ of its shadow,
$\Omega\in\mathcal{\widetilde{R}}$ if and only if, in addition to $(i)$ and
$(ii)$, it also satisfies

  * $(iii)$

$\lim_{t\rightarrow 0^{+}}\phi^{\prime}(t)=0$ and $\lim_{t\rightarrow
b_{1}^{-}}\phi^{\prime}(t)=-\infty$.

It’s clear from (2.3) that $(iii)$ does not guarantee the continuous extension
of $p$ to $b\Omega$. Let

$\displaystyle\mathcal{R}^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{\Omega\in\widetilde{\mathcal{R}}:p\ \text{extends as a
continuous function from $b\Omega$ to }(1,\infty)\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\Omega\in\widetilde{\mathcal{R}}:(\kappa_{1}\kappa_{2})^{-1}\kappa_{3}\
\text{extends as a continuous function from $b\Omega$ to
}(0,\infty)\right\\}.$

Several of the results in [4] are proved for domains in the class
$\mathcal{R}$; see [4, Definition 16]. It can be shown that domains in
$\mathcal{R}$ are precisely those $\Omega\in\mathcal{\widetilde{R}}$ for which
$(p,a_{1},a_{2})$ extends continuously as a function from $b\Omega$ to
$(1,\infty)\times(0,\infty)\times(0,\infty)$. It is clear the
$\mathcal{R}\subseteq\mathcal{R}^{\prime}$. See Example 3.1 for a concrete
example in $\mathcal{R}^{\prime}\setminus\mathcal{R}.$

For domains in $\mathcal{\widetilde{R}}$, the limiting behaviour of $p$
captures some information about the regularity and contact type of $\Omega$ at
$\mathcal{Z}$. Note the following result, for instance. Since it has no
bearing on the proofs of our main results, the reader is directed to [11] for
its proof.

######  Proposition 2.2.

Let $\Omega\in\mathcal{\widetilde{R}}$ and $w\in b\Omega\cap\mathcal{Z}$.

  1. $(a)$

If $\lim_{\zeta\rightarrow w}p\left(\zeta\right)$ exists and is greater than
$2$, then $b\Omega$ is $\mathcal{C}^{2}$-smooth at $w$.

  2. $(b)$

If $b\Omega$ is $\mathcal{C}^{2}$-smooth at $w$, then
$\limsup_{\zeta\rightarrow w}p\left(\zeta\right)\geq 2$.

Suppose $b\Omega$ is $\mathcal{C}^{\infty}$-smooth at $w$. Let
$m\in\mathbb{Z}_{+}$.

  1. $(c)$

Then $\Omega$ is of finite type $2m$ at $w$ if and only if
$\lim_{\zeta\rightarrow w}p\left(\zeta\right)$ exists and is equal to $2m$.

  2. $(d)$

Then $\Omega$ is of infinite type at $w$ if and only if
$\limsup_{\zeta\rightarrow w}p(\zeta)=\infty$.

We note one final feature of the class $\mathcal{\widetilde{R}}$, which has a
significant bearing on the proofs of the main results. Given a set
$E\subset\mathbb{C}^{n}$, recall that its dual complement is the set given by

$E^{*}=\\{z\in\mathbb{C}^{n}:\langle\zeta,z\rangle\neq 1,\,\forall\zeta\in
E\\}.$ (2.5)

For a detailed discussion on the significance of the dual complement of a set,
see [2, Chapter 2]. We recall from [4, §7] that $\mathcal{\widetilde{R}}$ is
closed under the action of taking dual complements, i.e.,
$\Omega\in\mathcal{\widetilde{R}}$ if and only if (the interior of)
$\Omega^{*}\in\mathcal{\widetilde{R}}$. The class $\mathcal{R}^{\prime}$ is
also closed under the action of taking dual complements.

###  2.3. Monge–Ampère measures associated to domains in
$\mathcal{\widetilde{R}}$

Recall that, for a bounded convex domain $\Omega\subset\mathbb{C}^{n}$, the
support function of $\Omega$ is the function
$H_{\Omega}:{\mathbb{C}^{n}}\rightarrow\mathbb{R}$ given by

$H_{\Omega}(z)=\sup_{\zeta\in\Omega}\operatorname{Re}\langle\zeta,z\rangle,\qquad
z\in\mathbb{C}^{n}.$ (2.6)

Since $H_{\Omega}$ is a convex function, one can make sense of
$(dd^{c}H_{\Omega})^{n}$ as a Radon measure on $\mathbb{C}^{2}$; see [6, §2].
In particular, if $n=2$, $\nabla H_{\Omega}\in
L^{\infty}_{loc}(\mathbb{C}^{2})$, and

$\int_{\mathbb{C}^{2}}\psi(dd^{c}H_{\Omega})^{2}=-\int_{\mathbb{C}^{2}}dH_{\Omega}\wedge
d^{c}H_{\Omega}\wedge
dd^{c}\psi,\qquad\forall\psi\in\mathcal{C}_{c}^{\infty}(\mathbb{C}^{2}).$
(2.7)

In general, $(dd^{c}H_{\Omega})^{n}$ may be mutually singular with respect to
$\omega_{std}$, the Lebesgue measure on $\mathbb{C}^{n}$. However, when
$\Omega$ is strongly convex,
$H_{\Omega}\in\mathcal{C}^{2}(\mathbb{C}^{n}\setminus\\{0\\})$ (see [32,
§2.5]) and consequently $(dd^{c}H_{\Omega})^{n}\approx\|z\|^{-n}\omega_{std}$.
When $\Omega\in\mathcal{\widetilde{R}}$, we have that
$(dd^{c}H_{\Omega})^{2}\ll\omega_{std}$ on $\mathbb{C}^{2}$.

######  Proposition 2.3.

Let $\Omega\in\mathcal{\widetilde{R}}$. Then,
$(dd^{c}H_{\Omega})^{2}=\mathfrak{H}_{\Omega}\omega_{std}$ as measures, where

$\displaystyle\mathfrak{H}_{\Omega}(z)=\begin{cases}\frac{2}{\pi^{2}}\left(\dfrac{\partial^{2}H_{\Omega}}{\partial
z_{1}\partial{\overline{z}_{1}}}\dfrac{\partial^{2}H_{\Omega}}{\partial
z_{2}\partial{\overline{z}_{2}}}-\dfrac{\partial^{2}H_{\Omega}}{\partial
z_{1}\partial\overline{z}_{2}}\dfrac{\partial^{2}H_{\Omega}}{\partial
z_{2}\partial{\overline{z}_{1}}}\right)(z),&\text{if }z\notin\mathcal{Z},\\\
0,&\text{if }z\in\mathcal{Z}.\end{cases}$ (2.8)

###### Proof.

Let $\mu=(dd^{c}H_{\Omega})^{2}\Big{|}_{\mathbb{C}^{2}\setminus\mathcal{Z}}$
and
$\nu=\mathfrak{H}_{\Omega}\omega_{std}\Big{|}_{\mathbb{C}^{2}\setminus\mathcal{Z}}$.
By [6, Corollary 3.5], $(dd^{c}H_{\Omega})^{2}$ does not charge $\mathcal{Z}$.
Thus, it suffices to show that, $\mu=\nu$ on
$\mathbb{C}^{2}\setminus\mathcal{Z}$. Since $\mu$ and $\nu$ are Radon measures
on $\mathbb{C}^{2}\setminus\mathcal{Z}$, by the Riesz representation theorem
for positive functionals, we must show that

$\int_{\mathbb{C}^{2}\setminus\mathcal{Z}}\varphi
d\mu=\int_{\mathbb{C}^{2}\setminus\mathcal{Z}}\varphi
d\nu,\quad\forall\varphi\in C_{c}(\mathbb{C}^{2}\setminus\mathcal{Z}).$ (2.9)

Since $C_{c}^{\infty}(\mathbb{C}^{2}\setminus\mathcal{Z})$ is dense in
$C_{c}(\mathbb{C}^{2}\setminus\mathcal{Z})$ in the sup-norm, and $\mu$ and
$\nu$ act as continuous linear functionals on
$C_{c}(\mathbb{C}^{2}\setminus\mathcal{Z})$ via integration, it suffices to
establish (2.9) for $\varphi\in
C_{c}^{\infty}(\mathbb{C}^{2}\setminus\mathcal{Z})$. But this follows from
(2.7) and Stokes’ theorem as $H_{\Omega}$ is $\mathcal{C}^{2}$-smooth on
$\mathbb{C}^{2}\setminus\mathcal{Z}$. ∎

Following Demailly [14], one can also consider Monge–Ampère boundary measures
on the boundary of a bounded convex domain $\Omega\subset\mathbb{C}^{n}$. Let
$\rho:\overline{\Omega}\rightarrow(-\infty,0]$ be a continuous convex
exhaustion function of $\Omega$, i.e., satisfying
$\\{z\in\overline{\Omega}:\rho(z)<-c\\}\Subset\Omega$ for all $c>0$, such that
$\int_{\Omega}(dd^{c}\rho)^{n}<\infty$. Then, by [14, Theorem 3.1],
$\\{(dd^{c}\max\\{\rho,r\\})^{2}:r<0\\}$ is a family of Radon measures on
$\overline{\Omega}$ that converge weakly as $r\rightarrow 0$, and

$\mu_{\rho}=\lim_{r\rightarrow 0}(dd^{c}\max\\{\rho,r\\})^{2}$ (2.10)

is a Radon measure supported on $b\Omega$. In particular, let

$\mu_{{}_{\Omega}}=\lim_{r\rightarrow 0}(dd^{c}\max\\{m_{\Omega}-1,r\\})^{2},$
(2.11)

where $m_{\Omega}$ is the Minkowski functional of $\Omega$ given by
$m_{\Omega}(z)=\inf\\{t>0:\frac{z}{t}\in\Omega\\}$. When the exhaustion
function $\rho$ is $\mathcal{C}^{2}$-smooth on a neighborhood of $w\in
b\Omega$ and $\nabla\rho(w)\neq 0$, then $\mu_{\rho}(w)$ coincides with the
so-called Leray–Levi measure associated to $\rho$ at $w$, i.e.,

$\mu_{\rho}(w)=j^{*}\left(d^{c}\rho\wedge(dd^{c}\rho)^{n-1}\right)(w)={\mathcal{M}[\rho](w)\sigma_{{}_{\\!\Omega}}(w)},$

where $j:b\Omega\rightarrow\mathbb{C}^{2}$ is the inclusion map,
$\sigma_{{}_{\\!\Omega}}$ is the surface area measure on $b\Omega$, and

$\mathcal{M}[\rho]=\frac{{-}\operatorname{det}\begin{pmatrix}0&{\partial\rho}/{\partial
z_{j}}\\\ {\partial\rho}/{\partial\bar{z}_{j}}&{\partial^{2}\rho}/{\partial
z_{j}\partial\bar{z}_{k}}\end{pmatrix}_{1\leq j,k\leq
n}}{\pi^{n}\|\nabla\rho\|}.$

Thus, if $\rho$ extends as a $\mathcal{C}^{2}$-smooth defining function of
$\Omega$ on a neighborhood of $\overline{\Omega}$, then
$\mu_{\rho}\ll\sigma_{{}_{\\!\Omega}}$, and further, if $\Omega$ is strongly
pseudoconvex, then $\mu_{\rho}\approx\sigma_{{}_{\\!\Omega}}$.

When $\Omega\in\mathcal{\widetilde{R}}$, we use the fact that $m_{\Omega}$ is
$\mathcal{C}^{2}$-smooth on $b\Omega_{+}$ and the Monge–Ampère measures
appearing on the R.H.S. of (2.11) do not charge $\mathcal{Z}$ to conclude the
following.

######  Proposition 2.4.

Let $\Omega\in\mathcal{\widetilde{R}}$, and $\rho$ be a
$\mathcal{C}^{1}$-smooth defining function of $\Omega$ that is
$\mathcal{C}^{2}$-smooth on $b\Omega_{+}$. Then,
$\mu_{{}_{\Omega}}=\left(\mathds{1}_{b\Omega_{+}}\mathcal{M}[{m_{\Omega}}]\right)\sigma_{{}_{\\!\Omega}}\approx\mathds{1}_{b\Omega_{+}}j^{*}\left(d^{c}\rho\wedge(dd^{c}\rho)\right).$

We end this section with a connection between the two measures appearing
above. This is an analogue of integration using polar coordinates and was
proved by Lindholm for strongly convex domains; see [25, Lemma 3.8].

######  Proposition 2.5.

Let $\Omega\in\mathcal{\widetilde{R}}$. The map $T_{\Omega}:(0,\infty)\times
b\Omega\rightarrow\mathbb{C}^{2}\setminus\\{0\\}$ given by

$T_{\Omega}(r,\zeta)=2r\left(\dfrac{\partial{m_{\Omega}}}{\partial{z_{1}}}(\zeta),\dfrac{\partial{m_{\Omega}}}{\partial{z_{2}}}(\zeta)\right)$
(2.12)

is a homeomorphism from $(0,\infty)\times b\Omega$ onto
$\mathbb{C}^{2}\setminus\\{0\\}$ that restricts to a
$\mathcal{C}^{1}$-diffeomorphism from $(0,\infty)\times b\Omega_{+}$ onto
$\mathbb{C}^{2}\setminus\mathcal{Z}$. Moreover, if $\psi$ is a
$(dd^{c}H_{\Omega})^{2}$-integrable function, then

$\int_{\mathbb{C}^{2}}\psi(z)\left(dd^{c}H_{\Omega}\right)^{2}(z)=\int_{0}^{\infty}\int_{b\Omega}\psi\left(T_{\Omega}\left(r,\zeta\right)\right)r\,d\mu_{\Omega}(\zeta)dr.$
(2.13)

###### Proof.

Let $T^{1}_{\Omega}(\cdot)=T_{\Omega}(1,\cdot)$. Since $\Omega$ is Reinhardt
and $m_{\Omega}$ is $1$-homogenous, we have that
$\left<T^{1}_{\Omega}(\zeta),\zeta\right>=1$, for all $\zeta\in b\Omega$.
Thus, $T^{1}_{\Omega}$ maps $b\Omega$ into $b\Omega^{*}$. Similarly,
$T^{1}_{\Omega^{*}}$ maps $b\Omega^{*}$ into $b\Omega$. So, for any $w\in
b\Omega^{*}$, $\langle
T_{\Omega}^{1}\left(T^{1}_{\Omega^{*}}(w)\right),T^{1}_{\Omega^{*}}(w)\rangle=1=\langle
T^{1}_{\Omega^{*}}(w),w\rangle.$ By [25, Proposition 18], the Minkowski
functional of $\Omega^{*}$ is $H_{\Omega}$. Thus, abusing notation, we have
that $T_{\Omega^{*}}(r,w)=2r\partial H_{\Omega}(w)$. Now, the complex line
$\ell_{w}=\\{z\in\mathbb{C}^{2}:\langle 2\partial H_{\Omega}(w),z\rangle=1\\}$
is tangential to $b\Omega^{*}$ at $w$. By the strict convexity of
$b\Omega^{*}$, $\ell_{w}$ cannot intersect $b\Omega^{*}$ at any point other
than $w$. Thus, it must be that
$T^{1}_{\Omega}\left(T^{1}_{\Omega^{*}}(w)\right)=w$ for all $w\in
b\Omega^{*}$. Similarly,
$T^{1}_{\Omega^{*}}\left(T^{1}_{\Omega}(\zeta)\right)=\zeta$ for all $\zeta\in
b\Omega$. Thus, $T_{\Omega}^{1}$ is a continuous bijective map from $b\Omega$
onto $b\Omega^{*}$. Moreover, the unique complex tangent space at any
$\zeta\in b\Omega$ is parallel to the axes if and only if
$\zeta\in\mathcal{Z}$. Since $m_{\Omega}$ is $\mathcal{C}^{2}$-smooth off of
$\mathcal{Z}$, $T_{\Omega}^{1}$ maps $b\Omega_{+}$ $\mathcal{C}^{1}$-smoothly
onto $b\Omega^{*}_{+}$. This yields the first claim.

To prove (2.13), let $T_{+}$ denote the restriction of $T_{\Omega}$ to
$(0,\infty)\times b\Omega_{+}$. Then, following the computation presented in
[7, §8], where we set $\rho=m_{\Omega}$ and $\varphi=H_{\Omega}$, we have that

$T_{+}^{*}\left(\mathfrak{H}_{\Omega}d\omega_{std}\right)=rdr\wedge
j^{*}\left(d^{c}m_{\Omega}\wedge dd^{c}m_{\Omega}\right),$

where $\mathfrak{H}_{\Omega}$ is as in Proposition 2.3. The claim now follows
from Propositions 2.3 and 2.4. ∎

###  2.4.  Hardy spaces on domains in $\mathcal{\widetilde{R}}$

Recall that $\mathcal{H}^{2}(\Omega,\mu)$ is defined as the
$L^{2}(b\Omega,\mu)$-closure of
$\mathcal{A}(\Omega)=\mathcal{C}(\overline{\Omega})\cap\mathcal{O}(\Omega)$.
Since convex domains have the Mergelyan property, one can also define
$\mathcal{H}^{2}(\Omega,\mu)$ to be the $L^{2}(b\Omega,\mu)$-closure of
holomorphic polynomials, as done in [4]. We comment briefly on what it means
for this space to be a reproducing kernel Hilbert space of holomorphic
functions on $\Omega$.

Suppose $\mathcal{A}(\Omega)$ is strongly admissible in the sense of [16,
Definition 3.4], then the map $F|_{b\Omega}\mapsto F$ extends from
$\mathcal{A}(\Omega)$ to an isometric isomorphism
$\mathcal{I}:\mathcal{H}^{2}(\Omega,\mu)\rightarrow\mathfrak{X}(\Omega,\mu)$,
where $\mathfrak{X}(\Omega,\mu)$ is a reproducing kernel Hilbert space of
holomorphic functions on $\Omega$ containing $\mathcal{A}(\Omega)$. This gives
a unique function $s:\Omega\times\text{supp}\>\mu\rightarrow\mathbb{C}$, with
$s(\cdot,w)\in\mathfrak{X}(\Omega,\nu)$ for all $w\in\text{supp}\>\mu$ and
$s(z,\cdot)\in\mathcal{H}^{2}(\Omega,\mu)$ for all $z\in\Omega$, such that

$(\mathcal{I}f)(z)=\int_{b\Omega}f(w)s(z,w)d\mu(w),\quad
z\in\Omega,f\in\mathcal{H}^{2}(\Omega,\mu).$

This extends to a well-defined bounded transformation
$\mathbb{S}:L^{2}(b\Omega,\mu)\rightarrow\mathfrak{X}(\Omega,\mu)$ that
“produces” and “reproduces” holomorphic functions:

$\mathbb{S}f(z)=\int_{b\Omega}f(w)s(z,w)d\mu(w),\quad z\in\Omega.$

Further, a projection operator
$\mathbb{S}_{b}:L^{2}(b\Omega,\mu)\rightarrow\mathcal{H}^{2}(\Omega,\mu)$ can
be defined as follows.

$\mathbb{S}_{b}f=\mathcal{I}^{-1}(\mathbb{S}f),\quad f\in L^{2}(b\Omega,\mu).$

In the case when $\Omega$ is $\mathcal{C}^{2}$-smooth and convex, and
$\mu=\sigma$ is the surface area measure, $\mathcal{A}$ is indeed strongly
admissible, $\mathbb{S}$ and $\mathbb{S}_{b}$ are bounded operators on
$L^{2}(b\Omega,\sigma)$, $\mathbb{S}_{b}f(\zeta)=\lim_{r\rightarrow
1^{-}}\mathbb{S}f(r\zeta)$ for $\sigma_{\Omega}$-a.e. $\zeta\in b\Omega$ and
$f\in L^{2}(b\Omega,\sigma)$, and $\mathbb{S}_{b}$ is in fact the (orthogonal)
Szegő projection from $L^{2}(b\Omega,\sigma)$ onto
$\mathcal{H}^{2}(\Omega,\sigma)$; see [1]. For a general convex domain
$\Omega\subset{\mathbb{C}^{n}}$ and $\mu=\mu_{\rho}$ as in (2.10), the obvious
candidate for $\mathfrak{X}(\Omega,\mu)$ is the Poletsky–Stessin Hardy space
on $\Omega$ corresponding to $\rho$; see [29]. However, the strong
admissibility of $\mathcal{A}$ or the existence of boundary values for
functions in $\mathfrak{X}(\Omega,\mu)$ is not known in this general setting.
For $\Omega\in\mathcal{\widetilde{R}}$, we give an explicit description of
$\mathfrak{X}(\Omega,\mu)$ using series expansions in Section 3.2.

##  3\. Some technical tools

Our computations are facilitated by a special parametrization of domains in
$\mathcal{\widetilde{R}}$. We describe the relevant function spaces and
operators in terms of this parametrization in this section.

###  3.1. An alternate parametrization

Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^{2}$ that is
strongly convex away from $\mathcal{Z}$. Let $\phi,b_{1}$ and $b_{2}$ be as
granted by (2.1). Since $\Omega$ is Reinhardt, any parameterization of the
curve $\gamma_{\Omega}$ induces a parameterization of $b\Omega$. Thus, the
parameterization of $\gamma_{\Omega}$ given in (2.1) induces a
parameterization of $b\Omega$. However, it is fruitful to consider a different
parameterization of $\gamma_{\Omega}$ that was introduced in [4, §2]. Let

$\displaystyle
s(r)=\begin{cases}\frac{-r\phi^{\prime}(r)}{\phi(r)-r\phi^{\prime}(r)},&r\in(0,b_{1}),\\\
0,&r=0,\\\ 1,&r=b_{1}.\end{cases}$ (3.1)

Then, due to properties $(i)$, $(ii)$ and $(iii^{\prime})$ of $\phi$,
$s:[0,b_{1}]\rightarrow[0,1]$ is a continuous monotone function that is
$\mathcal{C}^{1}$-smooth on $\left(0,b_{1}\right)$; see the discussion after
Prop. 8 in [4]. Denote $s^{-1}$ by $r_{1}$. Then, we have that

$b\Omega=\left\\{\left(r_{1}(s)e^{i\theta_{1}},r_{2}(s)e^{i\theta_{2}}\right):0\leq
s\leq 1,(\theta_{1},\theta_{2})\in[0,2\pi)\times[0,2\pi)\right\\},$ (3.2)

where $r_{2}=\phi\circ r_{1}$. The functions, $r_{1}$ and $r_{2}$ are related
to the exponent function $p$ from (2.2) as follows. Let
$\check{p}:(0,1)\rightarrow(1,\infty)$ be given by
$\check{p}(s)=p\left(r_{1}(s),r_{2}(s)\right)$. Then, for $s\in(0,1)$,

$\displaystyle
r_{1}(s)=b_{1}\exp\left(-\int_{s}^{1}\frac{dt}{t{\check{p}}(t)}\right),$ (3.3)
$\displaystyle
r_{2}(s)=b_{2}\exp\left(-\int_{0}^{s}\frac{dt}{(1-t)\check{p}(t)}\right).$
(3.4)

Moreover,

$\displaystyle\int_{0}^{1}\frac{dt}{t{\check{p}}(t)}=\int_{0}^{1}\frac{dt}{(1-t){\check{p}}(t)}=\infty.$
(3.5)

If $\Omega\in\mathcal{\widetilde{R}}$, then in addition to (3.5),

$\displaystyle\int_{0}^{1}\left(1-\frac{1}{\check{p}(t)}\right)\frac{dt}{t}=\int_{0}^{1}\left(1-\frac{1}{\check{p}(t)}\right)\frac{dt}{(1-t)}=\infty.$
(3.6)

Conversely, by [4, Theorem 9], each tuple
$\left(\check{p},b_{1},b_{2}\right)$, where $b_{1},b_{2}>0$ and
$\check{p}:(0,1)\rightarrow(1,\infty)$ is a continuous function satisfying
(3.5) and (3.6), determines a unique $\Omega\in\mathcal{\widetilde{R}}$.

######  Example 3.1.

Domains in $\mathcal{R}^{\prime}$ (see (2.2)) are characterized by tuples of
the form $(\check{p},b_{1},b_{2})$, where $b_{1},b_{2}$ are positive constants
and $\check{p}:[0,1]\rightarrow(1,\infty)$ is a continuous function which
satisfies (3.5) and (3.6). By [4, Theorem 19], the additional condition

$\int_{0}^{1}\left(\frac{1}{\check{p}(s)}-\frac{1}{\check{p}(0)}\right)\frac{ds}{s}<\infty\quad\text{and}\quad\int_{0}^{1}\left(\frac{1}{\check{p}(s)}-\frac{1}{\check{p}(1)}\right)\frac{ds}{1-s}<\infty$
(3.7)

characterizes domains in $\mathcal{R}$. Thus,
$\mathcal{R}\subseteq\mathcal{R}^{\prime}$. The tuple $(\check{p},1,1)$, where

$\check{p}(s)=2+\frac{1}{\log\left(\frac{10}{s}\right)},\quad s\in[0,1],$

satisfies (3.5) and (3.6), but does not satisfy the integral conditions in
(3.7). Hence the corresponding domain is in $\mathcal{R}^{\prime}$, but not in
$\mathcal{R}$.

The parametrization discussed above also yields a quick proof of Lemma (2.1).

###### Proof of Lemma 2.1.

Let $\zeta\in b\Omega_{+}$. By (3.2), we have that
$\zeta=\left(r_{1}(s)e^{i\theta_{1}},r_{2}(s)e^{i\theta_{2}}\right)$ for some
$\left(s,\theta_{1},\theta_{2}\right)\in(0,1)\times[0,2\pi)^{2}$. From [4,
(4.5)], the three principal curvatures at $\zeta$ are given by

$\displaystyle\kappa_{1}\left(\zeta\right)$
$\displaystyle=\frac{s}{{r_{1}(s)}^{2}\sqrt{\left(\frac{s}{r_{1}(s)}\right)^{2}+\left(\frac{1-s}{r_{2}(s)}\right)^{2}}},$
$\displaystyle\kappa_{2}\left(\zeta\right)$
$\displaystyle=\frac{1-s}{{r_{2}(s)}^{2}\sqrt{\left(\frac{s}{r_{1}(s)}\right)^{2}+\left(\frac{1-s}{r_{2}(s)}\right)^{2}}},$
$\displaystyle\kappa_{3}\left(\zeta\right)$
$\displaystyle=\left(\check{p}(s)-1\right)\frac{s(1-s)}{{r_{1}(s)}^{2}{r_{2}(s)}^{2}}\left(\left(\frac{s}{r_{1}(s)}\right)^{2}+\left(\frac{1-s}{r_{2}(s)}\right)^{2}\right)^{-\frac{3}{2}}.$

Since $p$ is rotation-invariant in each coordinate,
$p\left(\zeta\right)=\check{p}(s)$. Thus,

$p(\zeta)=1+\left(\frac{\kappa_{3}}{\kappa_{1}\kappa_{2}}\left(\zeta\right)\times\sqrt{\left(\frac{s}{r_{1}(s)}\right)^{2}+\left(\frac{1-s}{r_{2}(s)}\right)^{2}}\right).$
(3.8)

Let $\phi$ be as in (2.1), and

$\widetilde{\rho}(z_{1},z_{2})=|z_{2}|-\phi(|z_{1}|),\qquad(z_{1},z_{2})\in\overline{\mathbb{B}^{1}(b_{1})}\times\mathbb{C}.$
(3.9)

Note that $\widetilde{\rho}$ is a $\mathcal{C}^{2}$-smooth defining function
of $\Omega$ near any $\zeta\in b\Omega_{+}$. By [4, (2.23)],
$\phi^{{}^{\prime}}\left({r_{1}(s)}\right)=-\frac{sr_{2}(s)}{(1-s)r_{1}(s)}$.
Thus, for this specific choice of defining function, we obtain that

$\frac{\|\nabla\widetilde{\rho}(\zeta)\|}{2\operatorname{Re}\left<\partial{\widetilde{\rho}}(\zeta),\zeta\right>}=\frac{\sqrt{1+\phi^{{}^{\prime}}\left({r_{1}(s)}\right)^{2}}}{r_{2}(s)-r_{1}(s)\phi^{{}^{\prime}}\left({r_{1}(s)}\right)}=\sqrt{\left(\frac{s}{r_{1}(s)}\right)^{2}+\left(\frac{1-s}{r_{2}(s)}\right)^{2}}.$

Combining this with (3.8), we obtain (3.9) for this specific choice of
defining. However,
$\frac{\|\nabla\rho(\zeta)\|}{2\operatorname{Re}\left<\partial{\rho}(\zeta),\zeta\right>}$
is independent of the choice of $\mathcal{C}^{1}$-defining function near
$\zeta$. Our proof is complete.

∎

The above parameterization transforms particularly well under the action of
taking the dual complement of $\Omega$; see [4, §7]. Let
$\left(\check{p}^{*},b_{1}^{*},b_{2}^{*}\right)$ be the tuple corresponding to
$\Omega^{*}$. Then

$\displaystyle\frac{1}{\check{p}}+\frac{1}{\check{p}^{*}}\equiv
1\,\,\text{and}\,\,b_{1}^{*}b_{1}=b_{2}^{*}b_{2}=1.$ (3.10)

Consequently,
$b\Omega^{*}=\left\\{\left(r_{1}^{*}(s)e^{i\theta_{1}},r_{2}^{*}(s)e^{i\theta_{2}}\right):0\leq
s\leq 1,(\theta_{1},\theta_{2})\in[0,2\pi)\times[0,2\pi)\right\\}$, where

$\displaystyle
r_{1}^{*}(s)=\frac{s}{r_{1}(s)}\,\,\text{and}\,\,r_{2}^{*}(s)=\frac{(1-s)}{r_{2}(s)},\quad\text{for
all }s\in[0,1].$ (3.11)

To compute the map $T_{\Omega}:(0,\infty)\times
b\Omega\rightarrow\mathbb{C}^{2}$ as in (2.12), observe that as $\Omega$ is
Reinhardt and $m_{\Omega}$ is $1$-homogeneous $2\left<\partial
m_{\Omega}(\zeta),\zeta\right>=1,\,\forall\zeta\in b\Omega$; see [25,
Proposition 18]. Thus, for any $\zeta\in b\Omega_{+}$,

$2\dfrac{\partial{m_{\Omega}}}{\partial{z_{l}}}(\zeta)=\frac{\dfrac{\partial{m_{\Omega}}}{\partial{z_{l}}}(\zeta)}{\left<\partial
m_{\Omega}(\zeta),\zeta\right>}=\frac{\dfrac{\partial{\widetilde{\rho}}}{\partial{z_{l}}}(\zeta)}{\left<\partial\widetilde{\rho}(\zeta),\zeta\right>},\quad
l=1,2,$

where $\widetilde{\rho}$ is as in (3.9), and the last equality holds since
both $m_{\Omega}-1$ and $\tilde{\rho}$ are local defining functions of
$b\Omega_{+}$. Thus, for
$\zeta=\left(r_{1}(s)e^{i\theta_{1}},r_{2}(s)e^{i\theta_{2}}\right)$,

$T_{\Omega}(\zeta)=r\left(\frac{s}{r_{1}(s)}e^{-i\theta_{1}},\frac{(1-s)}{r_{2}(s)}e^{-i\theta_{2}}\right)=r\left(r_{1}^{*}(s)e^{-i\theta_{1}},r_{2}^{*}(s)e^{-i\theta_{2}}\right).$
(3.12)

###  3.2. Series expansions for Hardy and Bergman-space functions

It is well-known that the functions in the Hardy space of the unit disc are
precisely those whose Fourier expansions have no negative Fourier
coefficients. Due to the toric symmetry and convexity of domains in
$\mathcal{\widetilde{R}}$, an analogous statement holds for functions in Hardy
spaces on such domains. To see this, let
$\vartheta:(0,1)\times[0,2\pi)^{2}\rightarrow b\Omega_{+}$ denote the
parametrization given by
$(s,\theta_{1},\theta_{2})\mapsto\left(r_{1}(s)e^{i\theta_{1}},r_{2}(s)e^{i\theta_{2}}\right)$.
Then, by Proposition 2.4 and [4, (4.2)],

$\left(\vartheta^{*}\mu_{\Omega}\right)(s,\theta_{1},\theta_{2})=\frac{1}{16\pi^{2}}\>ds\,d\theta_{1}d\theta_{2}.$
(3.13)

Thus, there is a normed space isomorphism between
$L^{2}(b\Omega,\mu_{\Omega})$ and
$L^{2}((0,1)\times[0,2\pi)^{2},ds\>d\theta_{1}d\theta_{2})$. In view of this,
for any $f\in L^{2}(b\Omega,\mu_{\Omega})$, we abbreviate
$f(r_{1}(s)e^{i\theta_{1}},r_{2}(s)e^{i\theta_{2}})$ to
$f(s,\theta_{1},\theta_{2})$, and identify the two spaces without further
comment.

######  Lemma 3.2.

Let $\Omega\in\mathcal{\widetilde{R}}$. Then,
$f\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$ if and only if

$f(s,\theta_{1},\theta_{2})=\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}a_{m_{1},m_{2}}{r_{1}(s)}^{m_{1}}{r_{2}(s)}^{m_{2}}e^{i({\theta_{1}{m_{1}}+\theta_{2}{m_{2}}})}\quad\left(\text{in
}L^{2}(b\Omega,\mu_{\Omega})\right)$ (3.14)

for some sequence of complex numbers $(a_{m_{1},m_{2}})_{\mathbb{N}^{2}}$
satisfying

$\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\right)<\infty.$
(3.15)

Moreover, if $f$ and $(a_{m_{1},m_{2}})_{\mathbb{N}^{2}}$ are related as in
(3.14), then

$\|f\|^{2}_{\mathcal{H}^{2}(\Omega,\mu_{\Omega})}\approx\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\right).$
(3.16)

###### Proof.

Let $\Omega\in\mathcal{\widetilde{R}}$, and
$f\in\mathcal{H}^{2}(\Omega,\mu_{\Omega})$. For each $s\in(0,1)$, let
$f_{s}:[0,2\pi)\times[0,2\pi)\rightarrow\mathbb{C}$ be given by

$f_{s}\left(\theta_{1},\theta_{2}\right)=f\left(s,\theta_{1},\theta_{2}\right).$

Then, $f_{s}\in L^{2}\left(d\theta_{1}d\theta_{2}\right)$ for almost every
$s\in(0,1).$ Thus, for almost every $s\in(0,1)$,

$f_{s}(\theta_{1},\theta_{2})=\sum_{(m_{1},m_{2})\in\mathbb{Z}^{2}}\widehat{f_{s}}(m_{1},m_{2})e^{i(\theta_{1}m_{1}+\theta_{2}m_{2})}\quad\text{in
}L^{2}\left(d\theta_{1}d\theta_{2}\right),$ (3.17)

where $\widehat{f_{s}}(m_{1},m_{2})$, $(m_{1},m_{2})\in\mathbb{Z}^{2}$, are
the Fourier coefficients of $f_{s}$.

Since holomorphic polynomial are dense in
$\mathcal{H}^{2}(\Omega,\mu_{\Omega})$, there is a sequence of polynomials

$p_{k}\left(z_{1},z_{2}\right)=\sum_{m_{1}=0}^{j_{1}(k)}\sum_{m_{2}=0}^{j_{2}(k)}a_{m_{1},m_{2}}^{(k)}z_{1}^{m_{1}}z_{2}^{m_{2}},\quad
k\in\mathbb{N},$

that converges to $f$ in $L^{2}\left(b\Omega,\mu_{\Omega}\right)$. For each
$s\in(0,1)$, we have that

$\displaystyle\widehat{\left(p_{k}\right)_{s}}(m_{1},m_{2})=\begin{cases}a_{m_{1},m_{2}}^{(k)}r_{1}(s)^{m_{1}}r_{2}(s)^{m_{2}},\quad&0\leq
m_{1}\leq j_{1}(k),0\leq m_{2}\leq j_{2}(k),\\\
0,&\text{otherwise}.\end{cases}$

By Parseval’s formula and Fubini’s theorem, we obtain that

$\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{2\pi}|p_{k}-f|^{2}d\theta_{1}d\theta_{2}\,ds\approx\sum_{(m_{1},m_{2})\in\mathbb{Z}^{2}}\int_{0}^{1}\left|\widehat{\left(p_{k}\right)_{s}}(m_{1},m_{2})-\widehat{f_{s}}(m_{1},m_{2})\right|^{2}ds.$

Since the sequence on the left-hand side converges to $0$ as
$k\rightarrow\infty$,

$\displaystyle\lim_{k\rightarrow\infty}\int_{0}^{1}\left|\widehat{\left(p_{k}\right)_{s}}(m_{1},m_{2})-\widehat{f_{s}}(m_{1},m_{2})\right|^{2}ds=0,\quad\forall(m_{1},m_{2})\in\mathbb{Z}^{2}.$
(3.18)

In particular, for each $(m_{1},m_{2})\in\mathbb{Z}^{2}$, there exists some
$a_{m_{1},m_{2}}\in\mathbb{C}$ such that

$\displaystyle\lim_{k\rightarrow\infty}\int_{0}^{1}\widehat{\left(p_{k}\right)_{s}}(m_{1},m_{2})\>ds$
$\displaystyle=\left(\int_{0}^{1}r_{1}(s)^{m_{1}}r_{2}(s)^{m_{2}}ds\right)\lim_{k\rightarrow\infty}a_{m_{1},m_{2}}^{(k)}$
$\displaystyle=\left(\int_{0}^{1}r_{1}(s)^{m_{1}}r_{2}(s)^{m_{2}}ds\right)a_{m_{1},m_{2}}.$

Since $\left(a_{m_{1},m_{2}}^{(k)}\right)_{k\in\mathbb{N}}$ is the constant
zero sequence whenever $\min\\{m_{1},m_{2}\\}<0$, we have that
$a_{m_{1},m_{2}}=0$ whenever $\min\\{m_{1},m_{2}\\}=0$. Now, by the fact that
$L^{2}$-convergence implies a.e. pointwise convergence of a subsequence, we
obtain that, for a.e. $s\in(0,1)$,

$\displaystyle\widehat{f_{s}}(m_{1},m_{2})=\begin{cases}a_{m_{1},m_{2}}r_{1}(s)^{m_{1}}r_{2}(s)^{m_{2}},\quad&\min\\{m_{1},m_{2}\\}\geq
0,\\\ 0,&\min\\{m_{1},m_{2}\\}<0.\end{cases}$

This yields that (3.14), (3.15), and (3.16) hold for any
$f\in\mathcal{H}^{2}(\Omega,\mu_{\Omega})$.

Conversely, if $(a_{m_{1},m_{2}})_{\mathbb{N}^{2}}$ is a sequence of complex
numbers satisfying (3.15), then the function $f$ defined as in (3.14) is in
$L^{2}(b\Omega,\mu_{\Omega})$, and is approximable therein by holomorphic
polynomials of the form
$p_{k}(z_{1},z_{2})=\sum_{m_{1},m_{2}=0}^{k}a_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}$,
$k\in\mathbb{N}$. Thus, $f\in\mathcal{H}^{2}(\Omega,\mu_{\Omega})$. ∎

It is now possible to write the power series expansion of the Laplace
transform of a Hardy-space function.

######  Lemma 3.3.

Let $\Omega\in\mathcal{\widetilde{R}}$, and
$f\in\mathcal{H}^{2}(\Omega,\mu_{\Omega})$ admit the expansion (3.14) in
$L^{2}(b\Omega,\mu_{\Omega})$. Then, the Laplace transform of $f$ is an entire
function on $\mathbb{C}^{2}$ with power series expansion

$\mathcal{L}\left(f\right)(z_{1},z_{2})=\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}t_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}},$
(3.19)

where the coefficients $t_{{m_{1}},{m_{2}}}$ are given by

$t_{m_{1},m_{2}}=\frac{1}{4}\frac{\overline{a_{m_{1},m_{2}}}}{m_{1}!m_{2}!}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\right).$
(3.20)

###### Proof.

By the Cauchy–Schwarz inequality, for any compact set
$K\subset\mathbb{C}^{2}$, there is a $C_{K}>0$ such that

$\|\mathcal{L}(f)\|^{2}_{\mathcal{C}(K)}\leq
C_{K}\|f\|^{2}_{\mu_{\Omega}},\quad f\in\mathcal{H}^{2}(\Omega,\mu_{\Omega}).$

Thus, it suffices to establish the claim for polynomials of the form
$p_{k}\left(z_{1},z_{2}\right)=\sum\limits_{m_{1},m_{2}=0}^{k}a_{m_{1},m_{2}}z_{1}^{m_{1}}\,z_{2}^{m_{2}}.$
Expanding $e^{\langle.,z\rangle}$ in terms of its power series and using the
parameterization of $b\Omega$ discussed in Section 3.1, we have that

$\displaystyle\mathcal{L}\left(p_{k}\right)\left(z_{1},z_{2}\right)$
$\displaystyle=\int_{b\Omega}\overline{{p_{k}}(\zeta)}e^{\langle\zeta,z\rangle}d\mu_{\Omega}(\zeta)$
$\displaystyle=\frac{1}{16\pi^{2}}\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{2\pi}\left({\sum_{m_{1},m_{2}=0}^{k}\overline{a_{m_{1},m_{2}}}{r_{1}(s)}^{m_{1}}{r_{2}(s)}^{m_{2}}e^{-i({\theta_{1}m_{1}+\theta_{2}m_{2}})}}\right)$
$\displaystyle\hskip
113.81102pt\times\left(\sum_{(k_{1},k_{2})\in\mathbb{N}^{2}}\frac{r_{1}(s)^{k_{1}}r_{2}(s)^{k_{2}}e^{i(k_{1}\theta_{1}+k_{2}\theta_{2})}}{k_{1}!k_{2}!}z_{1}^{k_{1}}z_{2}^{k_{2}}\right)d\theta_{1}d\theta_{2}ds$
$\displaystyle=\frac{1}{4}{\sum_{m_{1},m_{2}=0}^{k}}\frac{\overline{a_{m_{1},m_{2}}}}{m_{1}!m_{2}!}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}\,ds\right)z_{1}^{m_{1}}z_{2}^{m_{2}},$

where the last equality is obtained by integrating over the $\theta_{1}$ and
$\theta_{2}$ variables. ∎

######  Lemma 3.4.

Let $\Omega\in\mathcal{\widetilde{R}}$. Let $F$ be an entire function on
$\mathbb{C}^{2}$ with power series expansion

$F(z_{1},z_{2})=\sum_{\left(m_{1},m_{2}\right)\in\mathbb{N}^{2}}\beta_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}},$
(3.21)

for some sequence of complex numbers $(\beta_{m_{1},m_{2}})_{\mathbb{N}^{2}}$.
Then, $F\in\mathcal{A}^{2}(\mathbb{C}^{2},\nu_{\Omega})$ if and only if

$\sum_{\left(m_{1},m_{2}\right)\in\mathbb{N}^{2}}|\beta_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}(r_{1}^{*}(s))^{2m_{1}}(r_{2}^{*}(s))^{2m_{2}}ds\right)<\infty.$
(3.22)

Moreover, if $F\in\mathcal{A}^{2}(\mathbb{C}^{2},\nu_{\Omega})$, then

$\|F\|^{2}_{\nu_{\Omega}}\approx\sum_{\left(m_{1},m_{2}\right)\in\mathbb{N}^{2}}|\beta_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}(r_{1}^{*}(s))^{2m_{1}}(r_{2}^{*}(s))^{2m_{2}}ds\right).$
(3.23)

###### Proof.

First, since $B\left(0,r_{1}\right)\Subset\Omega\Subset
B\left(0,r_{2}\right)$, for some $r_{1},r_{2}>0$, we have that $r_{1}\|z\|\leq
H_{\Omega}(z)\leq r_{2}\|z\|$, i.e., $H_{\Omega}(z)\approx\|z\|$. Now, we
apply (2.13) to $\psi=|F|^{2}e^{-2H_{\Omega}}H_{\Omega}^{3/2}$, noting that
$H_{\Omega}(T_{\Omega}(r,\zeta))=r$. Then, parametrizing $b\Omega$ as in
Section 3.1, expressing $T_{\Omega}$ and $\mu_{\Omega}$ in terms of this
parametrization as in (3.12) and (3.13), we obtain that

$\displaystyle\|F\|_{\nu_{\Omega}}^{2}$
$\displaystyle=\int_{\mathbb{C}^{2}}\left|F(z)\right|^{2}e^{-2H_{\Omega}(z)}\|z\|^{\frac{3}{2}}\left(dd^{c}H_{\Omega}\right)^{2}(z)$
$\displaystyle\approx\int_{\mathbb{C}^{2}}\left|F(z)\right|^{2}e^{-2H_{\Omega}(z)}H_{\Omega}(z)^{\frac{3}{2}}\left(dd^{c}H_{\Omega}\right)^{2}(z)$
$\displaystyle\approx\int_{0}^{\infty}\int_{b\Omega}\left|F\left(T_{\Omega}\left(r,\zeta\right)\right)\right|^{2}e^{-2r}r^{\frac{5}{2}}d\mu_{\Omega}(\zeta)dr$
$\displaystyle\approx\int_{0}^{\infty}\int_{0}^{1}\,\int_{0}^{2\pi}\,\int_{0}^{2\pi}\left|F(rr_{1}^{*}(s)e^{-i\theta_{1}},rr_{2}^{*}(s)e^{-i\theta_{2}})\right|^{2}e^{-2r}r^{\frac{5}{2}}d\theta_{1}d\theta_{2}dsdr$
$\displaystyle=\int_{0}^{\infty}\left(4\pi^{2}\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}r^{2(m_{1}+m_{2})}|\beta_{m_{1},m_{2}}|^{2}\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}ds\right)e^{-2r}r^{\frac{5}{2}}dr$
$\displaystyle\approx\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|\beta_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}r_{1}^{*}(s)^{2m_{1}}r_{2}^{*}(s)^{2m_{2}}\,ds\right),$

where, in the final step, we use the following consequence of Stirling’s
approximation:

$\left(\int_{0}^{\infty}r^{2k+1}e^{-2r}r^{-\frac{1}{2}}\,dr\right)\approx(k!)^{2}.$
(3.24)

From this comparison, the lemma immediately follows. ∎

###  3.3. The Leray transform.

We briefly recall the construction and properties of the Leray transform as
given in [4, §3]. Given $\Omega\in\mathcal{\widetilde{R}}$,

$\mathbb{L}(f)(z)=\int_{b\Omega}f(\zeta)L(\zeta,z),\quad z\in\Omega,$

where, the (Leray) kernel is given by

$L(\zeta,z)=\frac{d\mu_{\Omega}(\zeta)}{\langle\partial
m_{\Omega}(\zeta),\zeta-z\rangle^{2}}=\mathds{1}_{b\Omega_{+}}(\zeta)\frac{j^{*}\left(d^{c}m_{\Omega}\wedge
dd^{c}m_{\Omega}\right)(\zeta)}{\langle\partial
m_{\Omega}(\zeta),\zeta-z\rangle^{2}},$

by Proposition 2.4. Since the right-hand side is independent of the choice of
$\mathcal{C}^{2}$-smooth defining function of $b\Omega_{+}$, we replace
$m_{\Omega}$ by $\rho$ from (3.9), and use the parametrization given in (3.2).
Thus, for
$\zeta=\left(r_{1}(s)e^{i\theta_{1}},r_{2}(s)e^{i\theta_{2}}\right)\in
b\Omega$ and $z=(z_{1},z_{2})\in\Omega$,

$\displaystyle
L(\zeta,z)=\frac{ds\,d\theta_{1}\,d\theta_{2}}{4\pi^{2}}\sum\limits_{m_{1},m_{2}=0}^{\infty}\frac{(m_{1}+m_{2}+1)!}{m_{1}!m_{2}!}r_{1}^{*}(s)^{m_{1}}r_{2}^{*}(s)^{m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}e^{-i\left(m_{1}\theta_{1}+m_{2}\theta_{2}\right)},$

where the above series converges uniformly in $z$ on compact subsets of
$\Omega$.

Following [4], a function of the form
$g(s)e^{i\left(m_{1}\theta_{1}+m_{2}\theta_{2}\right)}$ is called an
$(m_{1},m_{2})$-monomial, and the space of all $(m_{1},m_{2})$-monomials that
are in $L^{2}(b\Omega,\mu_{\Omega})$ is denoted by
$L^{2}_{m_{1},m_{2}}(b\Omega,\mu_{\Omega})$. The spaces
$L^{2}_{m_{1},m_{2}}\left(b\Omega,\mu_{\Omega}\right)$,
$(m_{1},m_{2})\in\mathbb{Z}\times\mathbb{Z}$, are mutually orthogonal, and

$L^{2}\left(b\Omega,\mu_{\Omega}\right)=\underset{(m_{1},m_{2})\in\mathbb{Z}^{2}}{\bigoplus}L^{2}_{m_{1},m_{2}}\left(b\Omega,\mu_{\Omega}\right).$
(3.25)

By direct computation, it is shown in [4, Corollary 23] that $\mathbb{L}$ and
$\mathbb{L}_{b}$ are well-defined on each
$L^{2}_{m_{1},m_{2}}(b\Omega,\mu_{\Omega})$, and therefore densely-defined on
$L^{2}(b\Omega,\mu_{\Omega})$. Furthermore, $\mathbb{L}_{b}$ reproduces (the
restrictions to $b\Omega$ of) holomorphic polynomials, and annihilates any
$(m_{1},m_{2})$-monomial with $\min\\{m_{1},m_{2}\\}<0$. Finally, since the
measure $\mu_{\Omega}$ is admissible in the sense of [4, Definition 28], we
have the following result as a combination of Proposition 26, Theorem 30 and
Proposition 32 in [4].

######  Lemma 3.5.

Let $\Omega\in\mathcal{\widetilde{R}}$. Given
$(m_{1},m_{2})\in\mathbb{Z}\times\mathbb{Z}$, let $\mathbb{L}_{m_{1},m_{2}}$
denote the restriction of $\mathbb{L}_{b}$ to
$L^{2}_{m_{1},m_{2}}\left(b\Omega,\mu_{\Omega}\right)$.

  1. (1)

If $\min\\{m_{1},m_{2}\\}\geq 0$, then $\mathbb{L}_{m_{1},m_{2}}$ is a bounded
rank-one projection operator on
$L^{2}_{m_{1},m_{2}}\left(b\Omega,\mu_{\Omega}\right)$, with

$\displaystyle\|\mathbb{L}_{m_{1},m_{2}}\|^{2}_{\mu_{\Omega}}=\gamma_{m_{1},m_{2}}^{2}\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}{ds},$
(3.26)

where $\gamma_{m_{1},m_{2}}=\frac{\left(m_{1}+m_{2}+1\right)!}{m_{1}!m_{2}!}$.

  2. (2)

The densely-defined operator $\mathbb{L}_{b}$ extends as a bounded operator on
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$ if and only if the quantities
$\|\mathbb{L}_{m_{1},m_{2}}\|^{2}_{\mu_{\Omega}}$ is uniformly bounded for
$(m_{1},m_{2})\in\mathbb{N}\times\mathbb{N}$.

  3. (3)

If $\mathbb{L}_{b}$ extends as a bounded operator on
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$, it extends as a bounded projection
operator from $L^{2}\left(b\Omega,\mu_{\Omega}\right)$ onto
$\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$.

##  4\. Proof Of Theorem 1.4

For the first claim, we essentially mimic the proof of Theorem 1 in [4]. The
main technical tool used therein is Theorem 45, which gives the asymptotic
values of the operator-norms of $\mathbb{L}_{m_{1},m_{2}}$ along special
sequences in $\mathbb{N}\times\mathbb{N}$. Since we assume a weaker condition
on the exponent function $p$ (but our measure is more special), we require a
modification of Theorem 45 in [4]. This modification is stated as Lemma 4.1
below. For the second claim, we produce a sequence in
$\mathbb{N}\times\mathbb{N}$ for which the corresponding operator-norms of
$\mathbb{L}_{m_{1},m_{2}}$ are unbounded, and then invoke Lemma 3.5.

###  4.1. Proof of Theorem 1.4 (i)

Recall, from (2.2), that the exponent function ${p}$ associated to $\Omega$
extends continuously from $b\Omega$ to $(1,\infty)$. Since $s:[0,1]\rightarrow
b\Omega$ is a continuous function that assumes the values $0$ and $1$ at the
axes, $\check{p}=p\circ s$ extends as a continuous function from $[0,1]$ to
$(1,\infty)$.

######  Lemma 4.1.

Let $\Omega\in\mathcal{R}^{{}^{\prime}}$. Let
$({m_{1}}_{j},{m_{2}}_{j})_{j\in\mathbb{N}}\subset\mathbb{N}\times\mathbb{N}$
be a sequence such that
$\lim_{j\rightarrow\infty}\max\\{{m_{1}}_{j},{m_{2}}_{j}\\}=\infty$. Then,

$\lim_{j\rightarrow\infty}\|\mathbb{L}_{{m_{1}}_{j},{m_{2}}_{j}}\|^{2}_{\mu_{\Omega}}\
\text{exists}$

in each of the following cases:

  1. (a)

if $\lim_{j\rightarrow\infty}min\\{{m_{1}}_{j},{m_{2}}_{j}\\}=\infty$, and
there is an $x\in[0,\infty]$ such that
$\lim_{j\rightarrow\infty}\frac{{m_{1}}_{j}}{{m_{2}}_{j}}=x$;

  2. (b)

if ${m_{1}}_{j}$ is independent of $j$, and
$\lim_{j\rightarrow\infty}{m_{2}}_{j}=\infty$;

  3. (c)

if ${m_{2}}_{j}$ is independent of $j$, and
$\lim_{j\rightarrow\infty}{m_{1}}_{j}=\infty$.

Assuming the above lemma, we complete the proof of Theorem 1.4 (i). By Lemma
3.5, it suffices to show that $\|\mathbb{L}_{m_{1},m_{2}}\|_{\mu_{\Omega}}$ is
uniformly bounded for $(m_{1},m_{2})\in\mathbb{N}\times\mathbb{N}$. If the
latter doesn’t hold, there exists a sequence
$({m_{1}}_{j},{m_{2}}_{j})_{j\in\mathbb{N}}\subset\mathbb{N}\times\mathbb{N}$
with $\lim_{j\rightarrow\infty}\max\\{{m_{1}}_{j},{m_{2}}_{j}\\}=\infty$ such
that $\|\mathbb{L}_{{m_{1}}_{j},{m_{2}}_{j}}\|_{\mu_{\Omega}}$ is unbounded
along any subsequence of $(m_{1j},m_{2j})_{j\in\mathbb{N}}$. Such a sequence
always admits a subsequence, which we still denote by
$({m_{1}}_{j},{m_{2}}_{j})_{j\in\mathbb{N}}$, that satisfies one of the
conditions $(a)$, $(b)$ or $(c)$ in Lemma 4.1. But, by Lemma 4.1,
$\|\mathbb{L}_{{m_{1}}_{j},{m_{2}}_{j}}\|_{\mu_{\Omega}}$ converges to a
positive constant, which is a contradiction. Thus, the quantities
$\|\mathbb{L}_{{m_{1}},{m_{2}}}\|_{\mu_{\Omega}}$ are uniformly bounded, and
$\mathbb{L}_{b}$ extends as a bounded projection operator from
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$ onto
$\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$. But for the proof of Lemma
4.1, we are done.

Proof of Lemma 4.1. In the case of $(a)$, the proof is exactly the same as
that of [4, Theorem 45, part (a)] with $h_{k}=\omega^{k}$ set as $1$; see[4,
Page 2807]. The proof therein only depends on the continuous extendability of
$\check{p}$ to $[0,1]$. See Section 4.2 for a summary of this proof.

We prove the result in the case of $(b)$. By switching the roles of $r_{1}$
and $r_{2}$ in the following argument, we also obtain the result in the case
of $(c)$. Suppose $m_{1j}=m_{0}\in\mathbb{N}$ for all $j\in\mathbb{N}$. Let
$\check{p}_{0}=\lim_{s\rightarrow 0}\check{p}(s)$. By (3.3),

$r_{1}(s)=s^{1/\check{p}_{0}}\tau(s),\quad s\in(0,1),$ (4.1)

where
$\tau(s)=b_{1}\exp\left({\int_{s}^{1}\left({\frac{1}{\check{p}_{0}}-\frac{1}{\check{p}(t)}}\right)\frac{dt}{t}}\right).$
Then, for any $x>0$,

$\lim_{s\rightarrow 0}\frac{r_{1}(xs)}{r_{1}(s)}=x^{1/\check{p}_{0}}.$ (4.2)

This implies that $r_{1}$ is a regularly varying function at $0$, with
$1/\check{p}_{0}$ as the index of regular variation at $0$ (see [18] for the
definition). This allows us to use known estimates on regularly varying
functions to tackle the integrals appearing in the following expression from
(3.26):

$\displaystyle\|\mathbb{L}_{m_{1},m_{2}}\|^{2}_{\mu_{\Omega}}=\left(\frac{\left(m_{1}+m_{2}+1\right)!}{m_{1}!m_{2}!}\right)^{2}\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}{ds}.$
(4.3)

Specifically, we will show that, as $j\rightarrow\infty$,

$\displaystyle E_{j}$ $\displaystyle=$
$\displaystyle{\left({{m_{2}}_{j}}\right)^{1+\frac{2m_{0}}{\check{p}_{0}}}}{r_{2}(0)}^{-2{m_{2}}_{j}}{\tau\left(\frac{1}{{m_{2}}_{j}}\right)}^{-2m_{0}}\left(\int_{0}^{1}{r_{1}(s)}^{2m_{0}}{r_{2}(s)}^{2{m_{2}}_{j}}ds\right)\rightarrow
C_{m_{0}},$ (4.4) $\displaystyle E_{j}^{*}$ $\displaystyle=$
$\displaystyle{\left({{m_{2}}_{j}}\right)^{1+\frac{2m_{0}}{\check{p}_{0}^{*}}}}{r_{2}(0)}^{2{m_{2}}_{j}}{\tau\left(\frac{1}{{m_{2}}_{j}}\right)}^{2m_{0}}\left(\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{0}}{r_{2}^{*}(s)}^{2{m_{2}}_{j}}{ds}\right)\rightarrow
C^{*}_{m_{0}},$ (4.5)

where $C_{m_{0}},C^{*}_{m_{0}}>0$ are constants depending only on $m_{0}$, and
$\check{p}_{0}^{*}$ is the conjugate of $\check{p}_{0}$.

Assume that (4.4) holds for any $\Omega\in\mathcal{R}^{\prime}$. Then it also
holds for $\Omega^{*}\in\mathcal{R}^{\prime}$, i.e.,

$\lim_{j\rightarrow\infty}{\left({{m_{2}}_{j}}\right)^{1+\frac{2m_{0}}{\check{p}_{0}^{*}}}}{{r_{2}^{*}(0)}^{-2{m_{2}}_{j}}{\tau^{*}\left(\frac{1}{{m_{2}}_{j}}\right)}^{-2m_{0}}}\left(\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{0}}{r_{2}^{*}(s)}^{2{m_{2}}_{j}}ds\right)=C^{\prime}_{m_{0}},$

for some positive constant $C^{\prime}_{m_{0}}$. However, it follows from
(3.11) that $r_{2}^{*}(0)=1/r_{2}(0)$ and
$\tau^{*}\left(\frac{1}{{m_{2}}_{j}}\right)=\tau\left(\frac{1}{{m_{2}}_{j}}\right)^{-1}$.
Thus, (4.5) holds if (4.4) holds. Further, by (3.26),

$\displaystyle\|\mathbb{L}_{m_{0},m_{2j}}\|^{2}_{\mu_{\Omega}}=\left({\frac{\left(m_{0}+{m_{2}}_{j}+1\right)!}{m_{0}!{m_{2}}_{j}!}}\right)^{2}{m_{2}}_{j}^{-2(1+m_{0})}E_{j}E_{j}^{*}.$

Thus, by (4.4), (4.5), and Stirling’s approximation, we obtain that
$\lim_{j\rightarrow\infty}\|\mathbb{L}_{m_{0},m_{2j}}\|^{2}_{\mu_{\Omega}}$
exists, and we are done in this case.

It remains to prove (4.4). For convenience, we denote ${m_{2}}_{j}$ by
$m_{j}$. Assume $j$ is large enough so that $m_{j}>2$. Splitting the integral
appearing in $E_{j}$ over $[0,1/2]$ and $(1/2,1]$, we write
$E_{j}=E_{j}^{1}+E_{j}^{2}$. We first show that

$\lim_{j\rightarrow\infty}E_{j}^{2}=\lim_{j\rightarrow\infty}{\left({m_{j}}\right)^{1+\frac{2m_{0}}{\check{p}_{0}}}}{{r_{2}(0)}^{-2m_{j}}{\tau\left(\frac{1}{m_{j}}\right)}^{-2m_{0}}}\left(\int_{\frac{1}{2}}^{1}{r_{1}(s)}^{2m_{0}}{r_{2}(s)}^{2m_{j}}ds\right)=0.$
(4.6)

Since $r_{2}(s)$ is a strictly decreasing function on $[0,1]$,
$r_{2}(1)=0<r_{2}(s)<r_{2}\left(\frac{1}{2}\right)$, for any
$s\in\left(\frac{1}{2},1\right)$. From the definition of $\tau$,

$\tau\left(\frac{1}{m_{j}}\right)>b_{1}\exp\left(-\int_{\frac{1}{m_{j}}}^{1}\frac{dt}{t}\right)=\frac{b_{1}}{m_{j}}.$

Thus, there is a positive constant $D_{m_{0}}$ depending only on $m_{0}$ such
that

$E_{j}^{3}\leq
D_{m_{0}}{m_{j}^{1+2m_{0}+\frac{2m_{0}}{\check{p}_{0}}}}{r_{2}(0)}^{-2m_{j}}{r_{2}\left(\frac{1}{2}\right)}^{2m_{j}},$

Since $r_{2}(\frac{1}{2})<r_{2}(0)$, the right-hand side tends to $0$ as
$j\rightarrow\infty$. Thus, (4.6) holds.

The limit of $E_{j}^{1}$ is handled using the dominated convergence theorem,
after applying an appropriate change of variables. Let $s=w/m_{j}$. Then,
using (4.1), we have that

$E_{j}^{1}=\int_{0}^{\infty}\mathbbm{1}_{[0,m_{j}/2]}r_{2}\left(\frac{w}{m_{j}}\right)^{2m_{j}}{r_{2}(0)}^{-2m_{j}}w^{\frac{2m_{0}}{\check{p}_{0}}}\tau\left(\frac{w}{m_{j}}\right)^{2m_{0}}\tau\left(\frac{1}{m_{j}}\right)^{-2m_{0}}dw,$

where $\mathbbm{1}_{A}$ denotes the indicator function of
$A\subset\mathbb{R}$. From $\eqref{eq:r_2 in s}$, it follows that $r_{2}$ is a
$\mathcal{C}^{1}$-smooth, decreasing function on $(0,1)$. Since $\check{p}$
extends as a continuous function on $[0,1]$, $r_{2}^{\prime}$ extends
continuously at $0$, with $\lim\limits_{s\rightarrow
0}r_{2}^{\prime}(s)=-1/({r_{2}(0)\check{p}_{0}})$. Thus, for each
$w\in[0,\infty)$,

$\displaystyle\lim_{j\rightarrow\infty}{r_{2}\left(\frac{w}{m_{j}}\right)}^{2m_{j}}{r_{2}(0)}^{-2m_{j}}$
$\displaystyle=\lim_{j\rightarrow\infty}\exp\left(m_{j}\left(\log
r_{2}\left(\frac{w}{m_{j}}\right)-\log r_{2}(0)\right)\right)$
$\displaystyle=\exp\left(w\left(\log{r_{2}}\right)^{\prime}(0)\right)=\exp\left(-{2w}{\check{p}_{0}}^{-1}\right).$
(4.7)

Also, from the expression of $\tau$ it follows that for each $x>0$,
$\lim\limits_{s\rightarrow 0}\frac{\tau(xs)}{\tau(s)}=1$. Hence,

$\displaystyle\lim_{j\rightarrow\infty}r_{2}\left(\frac{w}{m_{j}}\right)^{2m_{j}}{r_{2}(0)}^{-2m_{j}}w^{\frac{2m_{0}}{\check{p}_{0}}}\tau\left(\frac{w}{m_{j}}\right)^{2m_{0}}$
$\displaystyle\tau\left(\frac{1}{m_{j}}\right)^{-2m_{0}}=\exp\left(-\frac{2}{\check{p}_{0}}w\right)w^{2m_{0}/\check{p}_{0}}$

for each fixed $w\in[0,\infty)$. We now bound the integrand of $E_{j}^{1}$ by
a function in $L^{1}(0,\infty).$

From the discussion about $r_{2}$ above, we have that
${w^{-1}}{\log\left({r_{2}(w)}{r_{2}(0)}^{-1}\right)}$ is a negative
continuous function on $[0,\frac{1}{2}]$. Thus, there is a $\beta>0$ such that
for each $m_{j}$,

$\displaystyle{r_{2}\left(\frac{w}{m_{j}}\right)}^{2m_{j}}{r_{2}(0)}^{-2m_{j}}\leq\exp\left(-\beta
w\right)\quad\forall w\in\left[0,\frac{m_{j}}{2}\right].$ (4.8)

By [18, Theorem 2] on regularly varying functions, the convergence in (4.2) is
uniform on $x\in[0,1]$. In particular, the functions

$w^{1/{\check{p}_{0}}}{\tau\left(\frac{w}{m_{j}}\right)}{\tau\left(\frac{1}{m_{j}}\right)}^{-1}\
\text{ are uniformly bounded on $[0,1]$.}$ (4.9)

Next, assume $w\in(1,\frac{m_{j}}{2}]$. From the definition of $\tau$,

${\tau\left(\frac{w}{m_{j}}\right)}{\tau\left(\frac{1}{m_{j}}\right)}^{-1}=\exp\int_{\frac{1}{m_{j}}}^{\frac{w}{m_{j}}}\left(\frac{1}{\check{p}_{0}}-\frac{1}{\check{p}(t)}\right)\frac{dt}{t}<\exp\left(\int_{\frac{1}{m_{j}}}^{\frac{w}{m_{j}}}\frac{1}{t}dt\right)\,={w}.$
(4.10)

Combining (4.9) and (4.10) with (4.8), we obtain a $D_{m_{0}}^{\prime}>0$ such
that

$\mathbbm{1}_{[0,m_{j}/2]}r_{2}\left(\frac{w}{m_{j}}\right)^{2m_{j}}{r_{2}(0)}^{-2m_{j}}w^{2m_{0}/\check{p}_{0}}\tau\left(\frac{w}{m_{j}}\right)^{2m_{0}}\tau\left(\frac{1}{m_{j}}\right)^{-2m_{0}}\leq
D^{\prime}_{m_{0}}f(w),$

where

$f(w)=\begin{cases}\exp\left(-\beta w\right),&\ w\in[0,1],\\\ \exp\left(-\beta
w\right)w^{2m_{0}\left(\frac{1}{\check{p}_{0}}+1\right)},&w\in(1,\infty).\end{cases}$

Finally, since $f\in L^{1}\left(0,\infty\right)$, by the dominated convergence
theorem,

$\displaystyle\lim_{j\rightarrow\infty}E_{j}^{1}=\int_{0}^{\infty}\exp\left(-\frac{2}{\check{p}_{0}}w\right)w^{2m_{0}/{\check{p}_{0}}}dw.$
(4.11)

Finally, combining (4.11) and (4.6), we obtain (4.4).

∎

###  4.2. Proof of Theorem 1.4 (ii)

The hypothesis on $\kappa_{\Omega}$ implies that either the exponent
$\check{p}$ or its conjugate $\check{p}^{*}$ is not bounded above. In other
words, there exists a sequence $(x_{l})_{l\in\mathbb{N}}\subset(0,\infty)$,
such that

$\lim_{l\rightarrow\infty}{\sqrt{\check{p}\left(\frac{x_{l}}{1+x_{l}}\right)\check{p}^{*}\left(\frac{x_{l}}{1+x_{l}}\right)}}=\infty.$
(4.12)

For each $l\in\mathbb{N}$, let
$({m}_{l,j},{n}_{l,j})_{j\in\mathbb{N}}\subset\mathbb{N}\times\mathbb{N}$ be a
sequence such that
$\lim_{j\rightarrow\infty}\min\\{{m}_{l,j},{n}_{l,j}\\}=\infty$ and
$\lim_{j\rightarrow\infty}{m_{l,j}}/{n_{l,j}}=x_{l}$. We claim that for each
$l\in\mathbb{N}$,

$\lim_{j\rightarrow\infty}\|\mathbb{L}_{m_{l,j},n_{l,j}}\|^{2}_{\mu_{\Omega}}=\frac{1}{2}\sqrt{\check{p}\left(\frac{x_{l}}{1+x_{l}}\right)\check{p}^{*}\left(\frac{x_{l}}{1+x_{l}}\right)}.$
(4.13)

Assuming (4.13), we obtain from (4.12) that
$\|\mathbb{L}_{m,n}\|^{2}_{\mu_{\Omega}}$ is not uniformly bounded for
$(m,n)\in\mathbb{N}\times\mathbb{N}$. Thus, by Lemma 3.5 $(2)$,
$\mathbb{L}_{b}$ does not extend as a bounded operator on
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$.

The limit in (4.13) can be established by making minor modifications to the
proof of [4, Theorem 45 (a)], which itself is a variation of Laplace’s method
for approximating integrals. Since the proof therein is rather long, we only
provide a brief summary, indicating the modifications required in our case.

Fix an $l\in\mathbb{N}$. For convenience, we denote $(m_{l,j},n_{l,j})$ by
$(m_{j},n_{j})$. Now, note that

$\displaystyle\frac{2\|\mathbb{L}_{m,n}\|^{2}_{\mu_{\Omega}}}{\sqrt{\check{p}\left({s_{m,n}}\right)\check{p}^{*}\left({s_{m,n}}\right)}}=\left(\frac{\alpha_{m,n,0}A_{m,n,0}}{I_{m,n,0}}\right)^{2}\cdot\frac{I_{m,n,1}}{\alpha_{m,n,1}A_{m,n,1}}\cdot\frac{I_{m,n,-1}}{\alpha_{m,n,-1}A_{m,n,-1}},$
(4.14)

where $s_{m,n}=\frac{m}{m+n}$,
$\alpha_{m,n,k}=g_{1,k}(s_{m,n})^{m}g_{2,k}(s_{m,n})^{n}$,

$A_{m,n,k}=\sqrt{\frac{2mn}{\left(\frac{2k}{\check{p}\left(s_{m,n}\right)}+1-k\right)(m+n)^{3}}},\quad
I_{m,n,k}=\int_{0}^{1}g_{1,k}(s)^{m}g_{2,k}(s)^{n}ds,\quad$

$g_{1,k}(s)=r_{1}(s)^{2k}s^{1-k}$ and $g_{2,k}(s)=r_{2}(s)^{2k}(1-s)^{1-k}$,
for $m,n\in\mathbb{N}$ and $k\in\\{-1,0,1\\}$. Thus, to prove (4.13), it
suffices to prove that

$\lim_{j\rightarrow\infty}\frac{I_{m_{j},n_{j},k}}{\alpha_{m_{j},n_{j},k}A_{m_{j},n_{j},k}}=c$
(4.15)

for some constant $c>0$. For a fixed $m,n,k$, one sets $s=s_{m,n}+tA_{m,n,k}$
to obtain

$\frac{I_{m,n,k}}{A_{m,n,k}}=\alpha_{m,n,k}\int_{-\infty}^{\infty}\mathbbm{1}_{\left(-\frac{s_{m,n}}{A_{m,n,k}},\frac{1-s_{m,n}}{A_{m,n,k}}\right)}g_{m,n,k}(t)dt,$

where
$g_{m,n,k}(t)=\dfrac{g_{1,k}(s_{m,n}+tA_{m,n,k})^{m}g_{2,k}(s_{m,n}+tA_{m,n,k})^{n}}{\alpha_{m,n,k}}$.
The rest of the proof relies on the fact that for all $k\in\\{-1,0,1\\}$,

$\displaystyle\frac{2k}{\check{p}(s_{m_{j},n_{j}})}+1-k$ $\displaystyle\leq$
$\displaystyle 2$ $\displaystyle
C_{k,l}=\inf_{j\in\mathbb{N}}\left(\frac{2k}{\check{p}\left(s_{{m_{j}},{n_{j}}}\right)}+1-k\right)$
$\displaystyle>$ $\displaystyle 0.$

These are the necessary modifications of [4, (5.4)] and [4, (5.5)] needed in
our case. As in the case of [4, (5.13), (5.17), (5,19)], it follows that, for
all $k\in\\{-1,0,1\\},$

  1. (1)

$\lim_{j\rightarrow\infty}{A_{m_{j},n_{j},k}}/{s_{m_{j},n_{j}}}=\lim_{j\rightarrow\infty}{A_{m_{j},n_{j},k}}/{1-s_{m_{j},n_{j}}}=0$,

  2. (2)

$\lim_{j\rightarrow\infty}g_{m_{j},n_{j},k}(t)=e^{-t^{2}}$ for all
$t\in\mathbb{R}$, and

  3. (3)

$g_{m_{j},n_{j},k}(t)\leq e^{C_{k,l}(1-2^{-|k|/2}|t|)}$ for all
$t\in\mathbb{R}$, when $j$ is large enough.

By the dominated convergence theorem, (4.15) holds with $c=\sqrt{\pi}$.
Combining this with (4.14), we obtain (4.13).

##  5\. Proof of Theorem 1.5

For the first claim, we use the series expansions obtained in Section 3.2 to
write the $A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$-norm of
$\mathcal{L}(f)$ of any $f\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$
in terms of the $\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$-norm of $f$
and the operator norms of $\mathbb{L}_{m_{1},m_{2}}$,
$m_{1},m_{2}\in\mathbb{N}$. This allows us to use the characterization given
by Lemma 3.5. The bulk of the proof of the second claim is devoted to showing
that the two measures $\nu_{\Omega}$ and $\omega_{\Omega}$ are comparable
outside a compact set. For balls (in all dimensions), this is done in Section
7. For other domains satisfying the hypothesis of the claim, this is done by
comparing the Radon–Nikodym derivative of $\nu_{\Omega}$ with respect to
$\omega_{\Omega}$ with that of the unit ball.

###  5.1. Proof of Theorem 1.5 $\mathbf{(i)}$

First, assume that $\mathbb{L}_{b}$ extends as a bounded operator on
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$. Let
$f\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$. Then, by Lemma 3.2,

$f(s,\theta_{1},\theta_{2})=\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}a_{m_{1},m_{2}}{r_{1}(s)}^{m_{1}}{r_{2}(s)}^{m_{2}}e^{i({\theta_{1}m_{1}+\theta_{2}m_{2}})}\quad\left(\text{in
}L^{2}(b\Omega,\mu_{\Omega})\right).$

with $(a_{m_{1},m_{2}})_{\mathbb{N}^{2}}$ satisfying (3.15). By Lemma 3.4,
$\mathcal{L}(f)\in A^{2}(\mathbb{C}^{2},\nu_{\Omega})$ if

$\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|t_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}r_{1}^{*}(s)^{2m_{1}}r_{2}^{*}(s)^{2m_{2}}\,ds\right)<\infty,$

where $t_{m_{1},m_{2}}$ is related to $a_{m_{1},m_{2}}$ as in (3.20). From
(3.20) and (3.26), the uniform boundedness of
$\|\mathbb{L}_{m_{1},m_{2}}\|_{\mu_{\Omega}}^{2}$ from Lemma 3.5, and (3.16),
we have that

$\displaystyle\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|t_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}r_{1}^{*}(s)^{2m_{1}}r_{2}^{*}(s)^{2m_{2}}\,ds\right)$
$\displaystyle\approx$
$\displaystyle\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\|\mathbb{L}_{m_{1},m_{2}}\|_{\mu_{\Omega}}^{2}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\right)$
$\displaystyle\approx$
$\displaystyle\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\right)\approx\|f\|^{2}_{\mu_{\Omega}}<\infty.$

Thus, $\mathcal{L}$ is an operator from
$\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$ into
$A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$. Moreover, by Lemma 3.4,

$\|\mathcal{L}(f)\|^{2}_{\nu_{\Omega}}\approx\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|t_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}r_{1}^{*}(s)^{2m_{1}}r_{2}^{*}(s)^{2m_{2}}\,ds\right)\approx\|f\|^{2}_{\mu_{\Omega}}<\infty.$

Thus, $\mathcal{L}$ is injective.

We now show that $\mathcal{L}$ is in fact onto
$A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$. Let $F\in
A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$ be given by

$F(z_{1},z_{2})=\sum_{\left(m_{1},m_{2}\right)\in\mathbb{N}^{2}}\beta_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}.$

By Lemma 3.4,

$\|F\|_{\mu_{\Omega}}^{2}\approx\sum_{\left(m_{1},m_{2}\right)\in\mathbb{N}^{2}}|\beta_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}(r_{1}^{*}(s))^{2m_{1}}(r_{2}^{*}(s))^{2m_{2}}ds\right).$
(5.1)

Define $g$ on $b\Omega$ as
$g(s,\theta_{1},\theta_{2})=\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}\alpha_{m_{1},m_{2}}r_{1}(s)^{m_{1}}r_{2}(s)^{m_{2}}e^{i(\theta_{1}m_{1}+\theta_{2}m_{2})}$,
where

$\alpha_{m_{1},m_{2}}={4}{\overline{\beta_{m_{1},m_{2}}}}{\left(m_{1}!m_{2}!\right)}{\left(\int_{0}^{1}{r_{1}(s)}^{2m_{1}}{r_{2}(s)}^{2m_{2}}ds\right)^{-1}}.$
(5.2)

Then, by (3.26) and the uniform boundedness of
$\|\mathbb{L}_{m_{1}.m_{2}}\|_{\mu_{\Omega}}^{2}$, we have that

$\displaystyle\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|\alpha_{m_{1},m_{2}}|^{2}\left(\int_{0}^{1}r_{1}(s)^{2m_{1}}r_{2}(s)^{2m_{2}}ds\right)$
$\displaystyle\approx$
$\displaystyle\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|\beta_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\frac{1}{\|\mathbb{L}_{m_{1}.m_{2}}\|_{\mu_{\Omega}}^{2}}\left(\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}ds\right)$
$\displaystyle\approx$
$\displaystyle\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|\beta_{m_{1},m_{2}}|^{2}(m_{1}+m_{2}+1)!^{2}\left(\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}ds\right)<\infty.$

Thus, by Lemma 3.2, $g\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$, and
by Lemma 3.3, $\mathcal{L}(g)=F$. This concludes the proof of the claim that
$\mathcal{L}$ is a normed space isomorphism between
$\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$ and
$A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$.

Now, we prove the converse claim by contraposition. Suppose $\mathbb{L}_{b}$
does not extend as a bounded operator on
$L^{2}\left(b\Omega,\mu_{\Omega}\right)$. By Lemma 3.5,
$\sup_{(m_{1},m_{2})\in\mathbb{N}^{2}}\|\mathbb{L}_{m_{1}.m_{2}}\|_{\mu_{\Omega}}^{2}=\infty$.
Thus, there exists a sequence
$\left(m_{1_{k}},m_{2_{k}}\right)_{k\in\mathbb{Z}_{+}}\subset\mathbb{N}\times\mathbb{N}$
such that $\|\mathbb{L}_{m_{1_{k}},m_{2_{k}}}\|_{\mu_{\Omega}}^{2}\geq k$,
$\forall k\in\mathbb{Z}_{+}$. Consider the following function on $b\Omega$

$\widetilde{g}\left(s,\theta_{1},\theta_{2}\right)=\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}\widetilde{\alpha}_{m_{1},m_{2}}r_{1}(s)^{m_{1}}r_{2}(s)^{m_{2}}e^{i(\theta_{1}m_{1}+\theta_{2}m_{2})},$

where

$\widetilde{\alpha}_{m_{1},m_{2}}=\begin{cases}\frac{1}{k}\left(\int_{0}^{1}{r_{1}(s)}^{2m_{1_{k}}}{r_{2}(s)}^{2m_{2_{k}}}ds\right)^{-\frac{1}{2}},\quad&\text{when}\,m_{1}=m_{1_{k}},m_{2}=m_{2_{k}},\\\
0,\quad&\text{otherwise}.\end{cases}$

Then, by Lemma 3.2,
$\widetilde{g}\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$, but

$\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|\widetilde{\alpha}_{m_{1},m_{2}}|^{2}\|\mathbb{L}_{m_{1},m_{2}}\|_{\mu_{\Omega}}^{2}\left(\int_{0}^{1}(r_{1}(s))^{2m_{1}}(r_{2}(s))^{2m_{2}}ds\right)\geq\sum_{k=1}^{\infty}\frac{1}{k}=\infty.$

Thus, by Lemma 3.4, $\mathcal{L}(\widetilde{g})\notin
A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$. This completes the proof of
Theorem 1.5 $(i)$.

###  5.2. Proof of Theorem 1.5 $\mathbf{(ii)}$ assuming Lemma 5.1

Let $\Omega\in\mathcal{\widetilde{R}}$ be a domain that satisfies the
hypothesis of Theorem 1.5 (ii). Recall that $\Omega^{*}$ is the dual
complement of $\Omega$. It suffices to prove that, for some $k>0$,

  * (i)

for all $F\in\mathcal{O}\left(\mathbb{C}^{2}\right)$,

$\int_{k\Omega^{*}}\left|F(z)\right|^{2}d\omega_{\Omega}(z)\approx\int_{k\Omega^{*}}\left|F(z)\right|^{2}d\nu_{\Omega}(z),$
(5.3)

  * (ii)

for $z\in\mathbb{C}^{2}\setminus k\Omega^{*}$,

$\|e^{\langle z,\cdot\rangle}\|^{2}_{\mu_{\Omega}}\approx
e^{2H_{\Omega}(z)}\|z\|^{-\frac{3}{2}}.$ (5.4)

.

For the proof of (i), fix a $k>0$. Note that since both $e^{-2H_{\Omega}(z)}$
and $\|e^{\langle z,\cdot\rangle}\|^{-2}_{\mu_{\Omega}}$ are positive
continuous functions on $\mathbb{C}^{2}$ and $\Omega^{*}$ is a bounded set, it
suffices to show that for all $F\in\mathcal{O}(\mathbb{C}^{2})$,

$\int_{k\Omega^{*}}|F(z)|^{2}\|z\|^{\frac{3}{2}}\left(dd^{c}H_{\Omega}\right)^{2}(z)\approx\int_{k\Omega^{*}}|F(z)|^{2}\left(dd^{c}H_{\Omega}\right)^{2}(z).$

By the density of holomorphic polynomials in $\mathcal{O}(\mathbb{C}^{2})$, it
further suffices to consider $F$ to be a holomorphic polynomial, i.e., of the
form
$F(z)=\sum_{m_{1}=0}^{k_{1}}\sum_{m_{2}=0}^{k_{2}}t_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}.$
Recall, from the proof of Proposition 2.5 we have $T_{\Omega}\left((0,k)\times
b\Omega\right)=k\Omega^{*}.$ Thus, applying (2.13) to
$\psi(z)=\mathds{1}_{k\Omega^{*}}(z)|F(z)|^{2}\|z\|^{3/2}$ and
$\psi(z)=\mathds{1}_{k\Omega^{*}}(z)|F(z)|^{2}$, and mimicking the computation
in Lemma 3.4, we obtain that

$\displaystyle\int_{k\Omega^{*}}|F(z)|^{2}\|z\|^{\frac{3}{2}}\left(dd^{c}H_{\Omega}\right)^{2}(z)$
$\displaystyle\approx$
$\displaystyle\sum_{m_{1}=0}^{k_{1}}\sum_{m_{2}=0}^{k_{2}}\frac{\left|t_{m_{1},m_{2}}\right|^{2}}{\left(2m_{1}+2m_{2}+\frac{7}{2}\right)}\left(\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}ds\right),$
$\displaystyle\int_{k\Omega^{*}}|F(z)|^{2}\left(dd^{c}H_{\Omega}\right)^{2}(z)$
$\displaystyle\approx$
$\displaystyle\sum_{m_{1}=0}^{k_{1}}\sum_{m_{2}=0}^{k_{2}}\frac{\left|t_{m_{1},m_{2}}\right|^{2}}{2\left(m_{1}+m_{2}+1\right)}\left(\int_{0}^{1}{r_{1}^{*}(s)}^{2m_{1}}{r_{2}^{*}(s)}^{2m_{2}}ds\right).$

Finally, as $\left(2m_{1}+2m_{2}+\frac{7}{2}\right)\approx
2\left(m_{1}+m_{2}+1\right)$, the proof of (5.3) is complete.

We now establish (ii) for a large enough $k>0$ (to be chosen later). We first
reduce (5.4) to the integral estimate (5.8) below. Since the function
${e^{-2H_{\Omega}(z)}}\|z\|^{\frac{3}{2}}\|e^{\langle
z,\cdot\rangle}\|^{2}_{\mu_{\Omega}}$ is continuous on $\mathbb{C}^{2}$, it
suffices to prove (5.4) for $z\in\mathbb{C}^{2}\setminus(\mathcal{Z}\cup
k\Omega^{*})$. For such a $z$, by Proposition 2.5, there exists
$\left(r,w\right)\in[k,\infty)\times b\Omega_{+}$ such that
$z=T_{\Omega}(r,w)=rT_{\Omega}(1,w)$. Since $T_{\Omega}(1,w)\in
b\Omega^{*}_{+}$, there exists
$\left(t,\phi_{1},\phi_{2}\right)\in(0,1)\times[0,2\pi)^{2}$ such that
$z=r\left(r_{1}^{*}(t)e^{i\phi_{1}},r_{2}^{*}(t)e^{i\phi_{2}}\right).$ Writing
${e^{-2H_{\Omega}(z)}}\|z\|^{\frac{3}{2}}\|e^{\langle
z,\cdot\rangle}\|^{2}_{\mu_{\Omega}}$ in terms of this parameterization, and
using that $H_{\Omega}\left(T_{\Omega}(r,w)\right)=r$, we have that

$\displaystyle\frac{\|z\|^{\frac{3}{2}}}{{e^{2H_{\Omega}(z)}}}\|e^{\langle
z,\cdot\rangle}\|^{2}_{\mu_{\Omega}}$ $\displaystyle=$
$\displaystyle\|z\|^{\frac{3}{2}}\int_{b\Omega}e^{2\operatorname{Re}\langle\zeta,{z}\rangle-2H_{\Omega}(z)}d\mu_{\Omega}(\zeta)$
(5.5) $\displaystyle\approx$ $\displaystyle
r^{\frac{3}{2}}\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{2\pi}e^{2r\left(r_{1}(s)r_{1}^{*}(t)\cos(\theta_{1}-\phi_{1})+r_{2}(s)r_{2}^{*}(t)\cos(\theta_{2}-\phi_{2})-1\right)}d\theta_{1}d\theta_{2}ds$
$\displaystyle\approx$ $\displaystyle
r^{\frac{3}{2}}\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{\pi}e^{2r\left(r_{1}(s)r_{1}^{*}(t)\cos(\theta_{1})+r_{2}(s)r_{2}^{*}(t)\cos(\theta_{2})-1\right)}d\theta_{1}d\theta_{2}ds.$

Now, let $F_{\Omega}:[0,1]^{2}\times[0,2\pi)^{2}\rightarrow\mathbb{R}$,

$F_{\Omega}(x,y,\psi_{1},\psi_{2})=r_{1}(x)r_{1}^{*}(y)\cos(\psi_{1})+r_{2}(x)r_{2}^{*}(y)\cos(\psi_{2}).$
(5.6)

Since $\left(r_{1}(x)e^{i\psi_{1}},r_{2}(x)e^{i\psi_{2}}\right)\in b\Omega$
and $\left(r_{1}^{*}(y),r_{2}^{*}(y)\right)\in b\Omega^{*}$, we have that
$F_{\Omega}(x,y,\psi_{1},\psi_{2})\leq 1$ for all
$\left(x,y,\psi_{1},\psi_{2}\right)\in[0,1]^{2}\times[0,2\pi)^{2}$. Moreover,
since both ${\Omega}$ and $\Omega^{*}$ are strictly convex, for each $\zeta\in
b\Omega$, there exists a unique $\eta\in b\Omega^{*}$ such that
$\operatorname{Re}\langle\zeta,\eta\rangle=1$. Thus,
$F_{\Omega}(x,y,\psi_{1},\psi_{2})=1$ if and only if $x=y$ and
$\psi_{1}=\psi_{2}=0$. Considering this, and the fact that $F_{\Omega}$ is
continuous on $[0,1]^{2}\times[0,2\pi)^{2}$, we obtain a $C>0$ such that

$\displaystyle
F_{\Omega}(x,y,\psi_{1},\psi_{2})-1<-C,\quad\forall\left(x,y,\psi_{1},\psi_{2}\right)\in[0,1]^{2}\times\left[\frac{\pi}{2},\pi\right]^{2}.$
(5.7)

From (5.5) and (5.7), it follows that proving (5.4) is equivalent to proving

$r^{\frac{3}{2}}\int_{0}^{1}\int_{0}^{\pi/2}\int_{0}^{\pi/2}e^{2r\left(F_{\Omega}(s,t,\theta_{1},\theta_{2})-1\right)}d\theta_{1}d\theta_{2}ds\approx
1,\quad r>k.$ (5.8)

To establish (5.8) for any $\Omega$ satisfying the hypothesis of Theorem 1.5
$(ii)$, we use the following result which allows us to reduce the problem to
the case of balls.

######  Lemma 5.1.

Let $\Omega\in\mathcal{\widetilde{R}}$ satisfy the hypothesis of Theorem 1.5
$(ii)$. Let $F_{\Omega}$ be as in (5.6). There exist $C_{1},C_{2}>0$ such that

$C_{1}(1-F_{\mathbb{B}^{2}})\leq 1-F_{\Omega}\leq
C_{2}(1-F_{\mathbb{B}^{2}}),\quad\text{on
}(0,1)^{2}\times\left(0,\frac{\pi}{2}\right)^{2}.$ (5.9)

Deferring the proof of Lemma 5.1 for now, we complete the proof of (ii). By
Lemma 7.1, (5.4), and therefore (5.8), holds for $\Omega=\mathbb{B}^{2}$ and
$k=1$. In particular, for any $C>0$

$r^{\frac{3}{2}}\int_{0}^{1}\int_{0}^{\pi/2}\int_{0}^{\pi/2}e^{2rC\left(F_{\mathbb{B}^{2}}(s,t,\theta_{1},\theta_{2})-1\right)}d\theta_{1}d\theta_{2}ds\approx
1,\quad r>\frac{1}{C}.$ (5.10)

The estimate (5.8), and therefore (5.4), now follows from (5.9) and (5.10) for
$k>1/C_{1}$.

###  5.3. Proof of Lemma 5.1

Using elementary calculus, we optimize the functions
$1-F_{\mathbb{B}^{2}}-C_{2}(1-F_{\Omega})$ and
$1-F_{\Omega}-C_{1}(1-F_{\mathbb{B}^{2}})$ on
$(0,1)^{2}\times\left(0,{\pi}/{2}\right)^{2}$, for appropriate choices of
$C_{1}$ and $C_{2}$. It is simpler to do this for egg domains first, and then,
for a general $\Omega$.

Upper bound for egg domains. Let $p>1$ and
$\Omega_{p}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|^{p}+|z_{2}|^{p}<1\\}$.
We claim that, for all
$(s,t,\theta_{1},\theta_{2})\in(0,1)^{2}\times\left(0,\pi/2\right)^{2}$,

$\displaystyle{}1-F_{\mathbb{B}^{2}}(s,t,\theta_{1},\theta_{2})\leq
C_{p}\left(1-F_{\Omega_{p}}\left(s,t,\theta_{1},\theta_{2}\right)\right),$
(5.11)

where $C_{p}=p/2$, when $p\geq 2$ and $C_{p}=p/(2p-2)$, when $1<p<2.$

Assume $p\geq 2$. Fix
$(t,\theta_{1},\theta_{2})\in(0,1)\times\left(0,{\pi}/{2}\right)^{2}$ and
consider $G_{1}:(0,1)\rightarrow\mathbb{R}$ given by

$\displaystyle G_{1}(s)$ $\displaystyle=$ $\displaystyle
1-F_{\mathbb{B}^{2}}(s,t,\theta_{1},\theta_{2})-\frac{p}{2}\left(1-F_{\Omega_{p}}\left(s,t,\theta_{1},\theta_{2}\right)\right)$
$\displaystyle=$ $\displaystyle
1-\frac{p}{2}-\left(s^{\frac{1}{2}}t^{\frac{1}{2}}-\frac{p}{2}s^{\frac{1}{p}}t^{\frac{p-1}{p}}\right)\cos\theta_{1}-\left((1-s)^{\frac{1}{2}}(1-t)^{\frac{1}{2}}-\frac{p}{2}(1-s)^{\frac{1}{p}}(1-t)^{\frac{p-1}{p}}\right)\cos\theta_{2},$

for $s\in(0,1)$. Then, for all $s\in(0,1)$,

$\displaystyle
G_{1}^{\prime}(s)=\frac{1}{2}\left(\left(\frac{t}{s}\right)^{\frac{p-1}{p}}-\left(\frac{t}{s}\right)^{\frac{1}{2}}\right)\cos\theta_{1}+\frac{1}{2}\left(\left(\frac{1-t}{1-s}\right)^{\frac{1}{2}}-\left(\frac{1-t}{1-s}\right)^{\frac{p-1}{p}}\right)\cos\theta_{2}.$

Now using the fact that $a^{\frac{p-1}{p}}>a^{\frac{1}{2}}$ whenever $a>1$, we
have that $G^{\prime}(s)>0$ if $s\in(0,t)$ and $G^{\prime}(s)<0$ if
$s\in(t,1)$. Thus, $G_{1}$ has a global maximum on $(0,1)$ at $s=t$. However,

$G_{1}(t)=\left(1-\frac{p}{2}\right)\left(1-t\cos\theta_{1}-(1-t)\cos\theta_{2}\right)\leq
0,$

which proves the claim in the case when $p\geq 2$.

The proof of (5.11) when $1<p\leq 2$ follows once we observe that $p^{*}\geq
2$,
$F_{\Omega_{p}}(s,t,\theta_{1},\theta_{2})=F_{\Omega_{p}^{*}}(t,s,\theta_{1},\theta_{2})$
and $p^{*}/{2}=p/(2p-2)$.

Lower bound for egg domains. We claim that for all
$(s,t,\theta_{1},\theta_{2})\in(0,1)^{2}\times(0,{\pi}/{2})^{2}$,

$\displaystyle
1-F_{\Omega_{p}}\left(s,t,\theta_{1},\theta_{2})\right)\leq\frac{2}{(2C_{p})^{*}}\left(1-F_{\mathbb{B}^{2}}(s,t,\theta_{1},\theta_{2})\right).$

The same method of proof as in the previous case is applied to the function

$G_{2}(t)=1-F_{\Omega_{p}}(s,t,\theta_{1},\theta_{2})-\frac{2}{(2C_{p})^{*}}\left(1-F_{\mathbb{B}^{2}}\left(s,t,\theta_{1},\theta_{2}\right)\right),\quad
t\in(0,1),$

for a fixed $(s,\theta_{1},\theta_{2})\in(0,1)\times(0,{\pi}/{2})^{2}$.

Upper bound in the general case. Let $\Omega\in\mathcal{\widetilde{R}}$
satisfy the hypothesis of Theorem 1.5 $(ii)$. Then,
$p_{\ell}=\inf\\{\check{p}(u):{u\in(0,1)}\\}>1$ and
$p_{g}=\sup\\{\check{p}(u):{u\in(0,1)}\\}<\infty$. We claim that, for all
$\left(s,t,\theta_{1},\theta_{2}\right)\in\left(0,1\right)^{2}\times\left(0,{\pi}/{2}\right)^{2}$,

$\displaystyle
1-F_{\Omega_{p_{\ell}}}\left(s,t,\theta_{1},\theta_{2}\right)\leq\frac{p_{g}}{p_{\ell}}\left(1-F_{\Omega}\left(s,t,\theta_{1},\theta_{2}\right)\right).$
(5.12)

From (3.3) and (3.4), it follows that that

$\displaystyle{}r_{1}(s)$ $\displaystyle=$ $\displaystyle
s^{\frac{1}{p_{\ell}}}h_{1}(s),\,\text{where}\,h_{1}(s)=b_{1}\exp\left(\int_{s}^{1}\left(\frac{1}{p_{\ell}}-\frac{1}{\check{p}(u)}\right)\frac{du}{u}\right),$
(5.13) $\displaystyle r_{2}(s)$ $\displaystyle=$
$\displaystyle(1-s)^{\frac{1}{p_{\ell}}}h_{2}(s),\,\text{where}\,h_{2}(s)=b_{2}\exp\left(\int_{0}^{s}\left(\frac{1}{p_{\ell}}-\frac{1}{\check{p}(u)}\right)\frac{du}{1-u}\right).$
(5.14)

Note that $h_{1}$ is a decreasing function and $h_{2}$ is an increasing
function in $s$.

Now, fix $s,t\in(0,1)$ and consider the function
$G:\left[0,{\pi}/{2}\right]^{2}\rightarrow\mathbb{R}$ given by

$\displaystyle G(\theta_{1},\theta_{2})$ $\displaystyle=$ $\displaystyle
1-F_{\Omega_{p_{\ell}}}\left(s,t,\theta_{1},\theta_{2}\right)-\frac{p_{g}}{p_{\ell}}\left(1-F_{\Omega}\left(s,t,\theta_{1},\theta_{2}\right)\right)$
(5.16) $\displaystyle=$ $\displaystyle
1-\frac{p_{g}}{p_{\ell}}-s^{\frac{1}{p_{\ell}}}t^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{1}(s)}{h_{1}(t)}\right)\cos\theta_{1}$
$\displaystyle\qquad\qquad-(1-s)^{\frac{1}{p_{\ell}}}(1-t)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{2}(s)}{h_{2}(t)}\right)\cos\theta_{2},$

for $(\theta_{1},\theta_{2})\in(0,\pi/2)^{2}.$ Since $G$ is continuous on the
rectangle $\left[0,{\pi}/{2}\right]^{2}$, $G$ attains its maximum on
$\left[0,{\pi}/{2}\right]^{2}$. In fact, we claim that the maximum is attained
at one of the corners. Suppose the maximum is attained at a point
$\left(\theta_{1}^{{}^{\prime}},\theta_{2}^{{}^{\prime}}\right)\in\left(0,{\pi}/{2}\right)^{2}$.
Then $0=\nabla
G\left(\theta_{1}^{{}^{\prime}},\theta_{2}^{{}^{\prime}}\right)$, which, is
only possible if
$0=1-\frac{p_{g}}{p_{\ell}}\frac{h_{1}(s)}{h_{1}(t)}=1-\frac{p_{g}}{p_{\ell}}\frac{h_{2}(s)}{h_{2}(t)}$,
since $\sin(\theta_{j}^{{}^{\prime}})\neq 0$ for $j=\\{1,2\\}$. In this case,
$\nabla G\equiv 0$, and $G$ is a constant function.

Alternatively, suppose the maximum is attained at
$\left(\theta_{1}^{{}^{\prime}},0\right)$ for some
$\theta_{1}^{{}^{\prime}}\in\left(0,\pi/2\right)$. Then,

$0=\dfrac{\partial{G}}{\partial{\theta_{1}}}\left(\theta_{1}^{\prime},0\right)=-s^{\frac{1}{p_{\ell}}}t^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{1}(s)}{h_{1}(t)}\right)\sin\theta_{1}^{\prime}.$

This implies that
$\frac{\partial{G}}{\partial{\theta_{1}}}\left(\cdot,0\right)\equiv 0$, and
$G(\cdot,0)$ is a constant function. The interior of the other three edges can
be tackled in a similar fashion. Thus, to show that $G\leq 0$ on
$[0,\pi/2]^{2}$, we show that the value of $G$ is nonpositive at the four
corners.

  * (1)

Since $p_{\ell}\leq p_{g}$,
$G\left(\frac{\pi}{2},\frac{\pi}{2}\right)=1-p_{g}/{p_{\ell}}\leq 0$.

  * (2)

Note that

$\displaystyle
G\left(\frac{\pi}{2},0\right)=1-\frac{p_{g}}{p_{\ell}}-(1-s)^{\frac{1}{p_{\ell}}}(1-t)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{2}(s)}{h_{2}(t)}\right).$

For a fixed $t\in(0,1)$, we consider the right-hand side to be a function of
$s$, i.e., let

$L(s)=1-\frac{p_{g}}{p_{\ell}}-(1-s)^{\frac{1}{p_{\ell}}}(1-t)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{2}(s)}{h_{2}(t)}\right).$

As $r_{1}$ and $r_{2}$ extend continuously to $[0,1]$, so does $L$. We show
that $L$ is nonpositive at $s=0,1$ and at any $s\in(0,1)$ such that
$L^{\prime}(s)=0$. Thus, $L(s)\leq 0$ on $[0,1]$.

    * (a)

Since $p_{\ell}\leq p_{g}$, $h_{2}$ is an increasing function and
$h_{2}(0)=b_{2}$,

$\displaystyle
L(0)=\left(1-\frac{p_{g}}{p_{\ell}}\right)+(1-t)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(\frac{p_{g}}{p_{\ell}}\frac{b_{2}}{h_{2}(t)}-1\right)\leq
0.$

    * (b)

Since $p_{\ell}\leq p_{g}$, $L(1)=1-{p_{g}}/{p_{\ell}}\leq 0$.

    * (c)

Suppose $L^{\prime}(s_{t})=0$ at some $s_{t}\in(0,1)$. Then, since
$h_{2}^{\prime}(s)=\frac{h_{2}(s)}{1-s}\left(\frac{1}{p_{\ell}}-\frac{1}{\check{p}(s)}\right)$,

$\displaystyle
0=L{{}^{\prime}}(s_{t})=\left(\frac{1-t}{1-s_{t}}\right)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(\frac{1}{p_{\ell}}-\frac{p_{g}}{p_{\ell}}\frac{h_{2}(s_{t})}{\check{p}(s_{t})h_{2}(t)}\right).$

Thus, $\check{p}(s_{t})=p_{g}{h_{2}(s_{t})}/{h_{2}(t)}$. Plugging this into
$L(s_{t})$, we have that

$L(s_{t})=\left(1-\frac{p_{g}}{p_{\ell}}\right)+(1-s_{t})^{\frac{1}{p_{\ell}}}(1-t)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(\frac{\check{p}(s_{t})}{p_{\ell}}-1\right)\leq
0.$

From (a), (b) and (c), we obtain that $G\left(\frac{\pi}{2},0\right)=L(s)\leq
0$.

  * (3)

The same method of proof as in (2) works for $G(0,\pi/2)$ to yield that
$G\left(0,{\pi}/{2}\right)\leq 0$.

  * (4)

Note that $G(0,0)$ is

$\displaystyle
1-\frac{p_{g}}{p_{\ell}}-s^{\frac{1}{p_{\ell}}}t^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{1}(s)}{h_{1}(t)}\right)-(1-s)^{\frac{1}{p_{\ell}}}(1-t)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(1-\frac{p_{g}}{p_{\ell}}\frac{h_{2}(s)}{h_{2}(t)}\right).$

For a fixed $t\in(0,1)$, let $L(s)$ denote the above expression. Then,

$\displaystyle L^{{}^{\prime}}(s)$
$\displaystyle=\left(\frac{t}{s}\right)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(\frac{p_{g}}{p_{\ell}\check{p}(s)}\frac{h_{1}(s)}{h_{1}(t)}-\frac{1}{p_{\ell}}\right)+\left(\frac{1-t}{1-s}\right)^{\frac{p_{\ell}-1}{p_{\ell}}}\left(\frac{1}{p_{\ell}}-\frac{p_{g}}{p_{\ell}\check{p}(s)}\frac{h_{2}(s)}{h_{2}(t)}\right).$

Now, using that $h_{1}$ is increasing, $h_{2}$ is decreasing,
$p_{g}\geq\check{p}(s)$ for all $s\in(0,1)$, and $p_{\ell}-1>0$, we obtain
that ${L}^{{}^{\prime}}(s)\geq 0$ for all $s\in(0,t]$. Similarly,
${L}^{{}^{\prime}}(s)\leq 0$ for all $s\in[t,1)$. Thus, $L$ attains its
maximum at $s=t$, which is $L(t)=0$. Consequently, $G(0,0)\leq 0$.

Since $G$ must attain it maximum at one of the corners, we obtain (5.12) from
(1)-(4).

Lower bound for a general $\Omega$. Using the same method of proof as in the
case of (5.12), but switching the roles of $s$ and $t$, we obtain that for any
$\left(s,t,\theta_{1},\theta_{2}\right)\in(0,1)^{2}\times\left[0,{\pi}/{2}\right]^{2}$,

$\displaystyle
1-F_{\Omega}\left(s,t,\theta_{1},\theta_{2}\right)\leq\frac{p^{*}_{g}}{p^{*}_{\ell}}\left(1-F_{\Omega_{p_{\ell}}}\left(s,t,\theta_{1},\theta_{2}\right)\right),$
(5.17)

where $p^{*}_{g}=\sup\\{\check{p}^{*}(u):u\in(0,1)\\}$,
$p^{*}_{\ell}=\inf\\{\check{p}^{*}(u):u\in(0,1)\\}$.

##  6\. Proofs of Theorem 1.3 and Theorem 1.6

The proofs of Theorem 1.3 and 1.6 are now relatively straightforward.

###  6.1. Proof of Theorem 1.3

By (2.3) and (2.2), we obtain that for
$\zeta=\left(\zeta_{1},\zeta_{2}\right)\in b\Omega_{+}$,

$\kappa_{\Omega}(\zeta)=\frac{\left|\zeta_{1}\right|\phi^{\prime\prime}(\left|\zeta_{1}\right|)\phi(\left|\zeta_{1}\right|)}{\phi^{\prime}(\left|\zeta_{1}\right|)\left(1+\phi^{\prime}(\left|\zeta_{1}\right|)^{2}\right)^{\frac{1}{2}}}.$

The hypothesis on $\kappa_{\Omega}$ implies that $\lim_{|\zeta_{1}|\rightarrow
0}\kappa_{\Omega}(\zeta)$ and $\lim_{|\zeta_{1}|\rightarrow
b_{2}}\kappa_{\Omega}(\zeta)$ are finite and positive. Due to the strong
convexity of $b\Omega_{+}$, $\phi^{\prime}$ is a strictly decreasing negative
function on $(0,b_{1})$. Hence, $\lim_{\left|\zeta_{1}\right|\rightarrow
0}\phi^{\prime}(\left|\zeta_{1}\right|)=l\in(-\infty,0]$. However, if
$l\in(-\infty,0)$, then from the above expression it follows that
$\lim_{\left|\zeta_{1}\right|\rightarrow
0}\left|\zeta_{1}\right|\phi^{\prime\prime}(\left|\zeta_{1}\right|)<0.$ This
implies that $\lim_{\left|\zeta_{1}\right|\rightarrow
0}\phi^{\prime}(\left|\zeta_{1}\right|)=\infty$, which is a contradiction.
Thus, $\lim_{\left|\zeta_{1}\right|\rightarrow
0}\phi^{\prime}(\left|\zeta_{1}\right|)=0$. Similarly, we can show that
$\lim_{\left|\zeta_{1}\right|\rightarrow
b_{1}}\phi^{\prime}(\left|\zeta_{1}\right|)=-\infty$. As discussed in Section
2.2, this implies that $\Omega$ has a $\mathcal{C}^{1}$-smooth boundary, and
thus, $\Omega\in\mathcal{\widetilde{R}}$. Furthermore, since $\kappa_{\Omega}$
is bounded away from zero on $b\Omega_{+}$ and extends continuously to the
axes, it follows that $\Omega\in\mathcal{R}^{\prime}$. Theorem 1.3 now follows
from Theorem 1.4 and Theorem 1.5.

###  6.2. Proof of Theorem 1.6

Note that

$\Omega=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|+|z_{2}|<1\\}$

is a convex Reinhardt domain that is $\mathcal{C}^{2}$-smooth but not strongly
convex away from the axes. Thus, the preliminaries established in Sections 2
and 3 do not directly apply to $\Omega$. However, several of the features of
domains in $\mathcal{\widetilde{R}}$ can be recovered for $\Omega$ using the
same arguments. We note the relevant features here, but leave the details to
the reader.

  * (i)

The boundary of $\Omega$ admits the following parameterization

$\displaystyle\vartheta:[0,1]\times[0,2\pi)^{2}$ $\displaystyle\rightarrow$
$\displaystyle b\Omega,$ $\displaystyle(s,\theta_{1},\theta_{2})$
$\displaystyle\mapsto$
$\displaystyle\left(se^{i\theta_{1}},(1-s)e^{i\theta_{2}}\right).$ (6.1)

  * (ii)

The Minskwoski functional and the support function of $\Omega$ are given by
$m_{\Omega}(z_{1},z_{2})=|z_{1}|+|z_{2}|$ and
$H_{\Omega}(z_{1},z_{2})=\max\\{|z_{1}|,|z_{2}|\\}$, respectively.

  * (iii)

The boundary Monge–Ampère measure of $\Omega$ with respect to $m_{\Omega}$ (in
terms of the given parametrization) is

$\left(\vartheta^{*}\mu_{\Omega}\right)(s,\theta_{1},\theta_{2})=\frac{1}{16\pi^{2}}\>ds\,d\theta_{1}d\theta_{2}.$

  * (iv)

The Monge–Ampère measure of $H_{\Omega}$ is a Radon measure on
$\mathbb{C}^{2}$ that is supported on the real hypersurface
$M=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|=|z_{2}|\\}$; see [5, Theorem 1].
In fact, following the computations in [5, Theorem 1] for
$E=\mathbb{C}^{2}\setminus\mathcal{Z}$ and $u=H_{\Omega}$, we have that for
$z=(z_{1},z_{2})\in\mathbb{C}^{2}$,

$\displaystyle(dd^{c}H_{\Omega})^{2}(z)$
$\displaystyle\approx\mathds{1}_{M}(z)\frac{1}{|z_{1}||z_{2}|}\sigma_{M}(z)$
$\displaystyle\approx
j^{*}\left(\frac{\overline{z_{1}}}{|z_{1}|}\;dz_{1}dz_{2}d\overline{z_{2}}-\frac{\overline{z_{2}}}{|z_{2}|}\;dz_{1}dz_{2}d\overline{z_{1}}\right).$

Parametrizing $M$ via
$\vartheta_{M}\left(r,\psi_{1},\psi_{2}\right)=\left(re^{i\psi_{1}},re^{i\psi_{2}}\right)$,,
we obtain that

$\vartheta_{M}^{*}(dd^{c}H_{\Omega})^{2}\left(r,\psi_{1},\psi_{2}\right)\approx
drd\psi_{1}d\psi_{2},\quad(r,\psi_{1},\psi_{2})\in(0,\infty)\times[0,2\pi)^{2}.$
(6.2)

  * (v)

Analogous to Lemma 3.2, any
$f\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$ if and only if

$f(s,\theta_{1},\theta_{2})=\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}a_{m_{1},m_{2}}{s}^{m_{1}}{(1-s)}^{m_{2}}e^{i({\theta_{1}{m_{1}}+\theta_{2}{m_{2}}})}\quad\left(\text{in
}L^{2}(b\Omega,\mu_{\Omega})\right)$ (6.3)

for some sequence $(a_{m_{1},m_{2}})_{\mathbb{N}^{2}}\subset\mathbb{C}$
satisfying

$\displaystyle\sum_{\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\left(\int_{0}^{1}s^{2m_{1}}(1-s)^{2m_{2}}ds\right)=\sum_{\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\dfrac{2m_{1}!2m_{2}!}{(2m_{1}+2m_{2}+1)!}<\infty.$
(6.4)

Moreover,

$\|f\|^{2}_{\mu_{\Omega}}\approx\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}|a_{m_{1},m_{2}}|^{2}\dfrac{2m_{1}!2m_{2}!}{(2m_{1}+2m_{2}+1)!}.$
(6.5)

  * (vi)

Analogous to Lemma 3.3, if $f$ is as in (6.3), we have that $\mathcal{L}(f)$
is entire on $\mathbb{C}^{2}$ and admits the following power series expansion:

$\mathcal{L}(f)(z_{1},z_{2})=\frac{1}{4}\sum_{(m_{1},m_{2})\in\mathbb{N}^{2}}\dfrac{\overline{a_{m_{1},m_{2}}}}{m_{1}!m_{2}!}\dfrac{2m_{1}!2m_{2}!}{(2m_{1}+2m_{2}+1)!}z_{1}^{m_{1}}z_{2}^{m_{2}},$
(6.6)

where the above series converges uniformly on compact subsets of
$\mathbb{C}^{2}$.

  * (vii)

Analogous to Lemma 3.4, if $F\in\mathcal{O}(\mathbb{C}^{2})$ has expansion
$\sum_{\mathbb{N}^{2}}\beta_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}$, then

$\displaystyle\|F\|^{2}_{\nu_{\Omega}}$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{2\pi}\left|F\left(re^{i\psi_{1}},re^{i\psi_{2}}\right)\right|^{2}e^{-2r}r^{\frac{3}{2}}d\psi_{1}d\psi_{2}dr$
$\displaystyle\approx\sum\limits_{(m_{1},m_{2})\in\mathbb{N}^{2}}\frac{|\beta_{m_{1},m_{2}}|^{2}}{2^{2m_{1}+2m_{2}}}\Gamma\left(2m_{1}+2m_{2}+5/2\right).$
(6.7)

It remains to characterize the power series expansions of entire functions in
$A^{2}(\Omega,\omega_{\Omega})$. For this, we analyze the function
$\|e^{\langle z,.\rangle}\|^{2}_{\mu_{\Omega}}$ on $M$.

######  Lemma 6.1.

Let $\Omega,\mu_{\Omega}$ be as above. Then

$\|e^{\langle
z,.\rangle}\|^{2}_{\mu_{\Omega}}\approx{e^{2H_{\Omega}(z)}}{\|z\|^{-1}},\quad
z\in M\cap\\{z:\mathbb{C}^{2}:H_{\Omega}(z)>1\\}.$ (6.8)

###### Proof.

Parametrizing $M\cap\\{z\in\mathbb{C}^{2}:H_{\Omega}(z)>1\\}$ as
$\\{\left(re^{i\psi_{1}},re^{i\psi_{2}}\right):r>0,\psi_{1},\psi_{2}\in[0,2\pi)\\}$,
we have that

$\displaystyle{e^{-2H_{\Omega}(z)}}\|z\|\|e^{\langle
z,.\rangle}\|^{2}_{\mu_{\Omega}}$ $\displaystyle\approx
r\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{2\pi}e^{2r\left(s\cos(\theta_{1}-\psi_{1})+(1-s)\cos(\theta_{2}-\psi_{2})-1\right)}d\theta_{1}d\theta_{2}ds$
$\displaystyle\approx
r\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{\pi}e^{2r\left(s\cos(\theta_{1})+(1-s)\cos(\theta_{2})-1\right)}d\theta_{1}d\theta_{2}ds$
$\displaystyle=r\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{\pi}e^{-4r\left(s\sin^{2}(\theta_{1}/2)+(1-s)\sin^{2}(\theta_{2}/2)\right)}d\theta_{1}d\theta_{2}ds.$

Now, using that, $\sin^{2}\frac{\theta}{2}\approx\theta^{2}$ on $[0,\pi]$, and
that

$r\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{\pi}e^{-2rC\left(s\theta_{1}^{2}+(1-s)\theta_{2}^{2}\right)}d\theta_{1}d\theta_{2}ds\approx\frac{1}{C}\int_{0}^{1}\frac{1}{s^{\frac{1}{2}}(1-s)^{\frac{1}{2}}}ds,$

we obtain the desired result. ∎

Combining the above lemma with (6.2), we conclude that if
$F\in\mathcal{O}(\mathbb{C}^{2})$ has expansion
$\sum_{\mathbb{N}^{2}}\beta_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}$, then

$\displaystyle\|F\|^{2}_{\omega_{\Omega}}$ $\displaystyle\approx$
$\displaystyle\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{2\pi}\left|F\left(re^{i\psi_{1}},re^{i\psi_{2}}\right)\right|^{2}e^{-2r}rd\psi_{1}d\psi_{2}dr$
$\displaystyle\approx$
$\displaystyle\sum\limits_{(m_{1},m_{2})\in\mathbb{N}^{2}}\frac{|\beta_{m_{1},m_{2}}|^{2}}{2^{2m_{1}+2m_{2}}}\left(2m_{1}+2m_{2}+1\right)!.$

To see that $A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)\subsetneq
A^{2}\left(\mathbb{C}^{2},\omega_{\Omega}\right),$ consider
$G(z_{1},z_{2})=t_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}$, where

$\displaystyle
t_{m_{1},m_{2}}=\begin{cases}\frac{2^{2k}}{k^{\frac{3}{4}}{\left(4k+1\right)!}^{\frac{1}{2}}},&\text{when
}m_{1}=m_{2}=k>1,\\\ 0,&\text{otherwise}.\end{cases}$

Then $G\in\mathcal{O}\left(\mathbb{C}^{2}\right)$, and by (6.2),
$\|G\|^{2}_{\omega_{\Omega}}\approx\sum\limits_{k=1}^{\infty}\frac{1}{k^{\frac{3}{2}}}<\infty$.
Thus $G\in A^{2}\left(\mathbb{C}^{2},\omega_{\Omega}\right)$. However, using
Stirling’s approximation and (6.7),
$\|G\|^{2}_{\nu_{\Omega}}\approx\sum\limits_{k=1}^{\infty}\frac{1}{k}=\infty$.
Thus $G\notin A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$.

To see that $\mathcal{L}(\mathcal{H}^{2}(\Omega,\mu_{\Omega}))\subsetneq
A^{2}(\mathbb{C}^{2},\nu_{\Omega})$, consider
$F(z_{1},z_{2})=\sum\limits_{m_{1},m_{2}=0}^{\infty}b_{m_{1},m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}},$
where

$\displaystyle
b_{m_{1},m_{2}}=\begin{cases}\frac{2^{2k}}{k^{\frac{3}{4}}{\Gamma\left(4k+\frac{5}{2}\right)}^{\frac{1}{2}}},&\text{when
}m_{1}=m_{2}=k>1,\\\ 0,&\text{otherwise}.\end{cases}$

Then, $F\in\mathcal{O}\left(\mathbb{C}^{2}\right)$, and, by (6.7)
$\|F\|^{2}_{\nu_{\Omega}}\approx\sum\limits_{k=1}^{\infty}\frac{1}{k^{\frac{3}{2}}}<\infty.$
Thus, $F\in A^{2}\left(\mathbb{C}^{2},\nu_{\Omega}\right)$. Now, suppose there
is an $f\in\mathcal{H}^{2}\left(\Omega,\mu_{\Omega}\right)$ such that
$\mathcal{L}(f)=F$. Then, $f$ admits an expansion as in (6.3) for some
sequence $(a_{m_{1},m_{2}})_{\mathbb{N}^{2}}$, satisfying (6.4). Then, by
(6.6),

$\displaystyle a_{m_{1},m_{2}}$
$\displaystyle=4\frac{\overline{b_{m_{1},m_{2}}}}{2m_{1}!2m_{2}!}{\left(2m_{1}+2m_{2}+1\right)!}{\left(m_{1}!m_{2}!\right)},$
$\displaystyle=\begin{cases}\frac{2^{2k+2}k!^{2}}{k^{\frac{3}{4}}(2k!)^{2}}{\Gamma\left(4k+\frac{5}{2}\right)}^{-\frac{1}{2}}(4k+1)!,&\text{when
}m_{1}=m_{2}=k>1,\\\ 0,&\text{otherwise}.\end{cases}$

Thus, by (6.4),

$\displaystyle\|f\|_{\mu_{\Omega}}^{2}\approx\sum_{k=1}^{\infty}\frac{2^{4k}(k!)^{4}}{k^{\frac{3}{2}}(2k!)^{2}}\frac{(4k+1)!}{\Gamma\left(4k+\frac{5}{2}\right)}=\infty,$

where we have used the uniform boundedness of

$\frac{k^{\frac{1}{2}}(2k!)^{2}}{2^{4k}(k!)^{4}}{\frac{\Gamma\left(4k+\frac{5}{2}\right)}{(4k+1)!}}$

by Stirling’s approximation. This is a contradiction, and our proof is
complete.

##  7\. Comparison of two weighted Bergman spaces

For a strongly convex domain $\Omega$ in the plane, it is known that

$\omega_{\Omega}(z)\approx
e^{-2H_{\Omega}(z)}|z|^{\frac{1}{2}}dd^{c}H_{\Omega}(z)=\nu_{\Omega}(z),\quad\text{for
}|z|>1.$ (7.1)

However, unlike $\omega_{\Omega}$, $\nu_{\Omega}$ does not capture the growth
rate of the Laplace transforms of Hardy space functions on other convex planar
domains, such as polygons. Thus, $\omega_{\Omega}$ has the advantage of
providing a unified expression for the Paley–Wiener spaces of Hardy spaces on
convex planar domains. Although tacitly implied in [25], it is never shown
that the weights $\omega_{\Omega}$ and $\nu_{\Omega}$ are similarly comparable
for higher-dimensional strongly convex domains. For the sake of completeness,
we provide a proof here.

######  Lemma 7.1.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded strongly convex domain. For
$z\in\mathbb{C}^{n}\setminus\mathbb{B}^{n}$,

$\|e^{\langle
z,\cdot\rangle}\|^{2}_{\sigma_{\Omega}}\approx\frac{e^{2H_{\Omega}(z)}}{\|z\|^{n-\frac{1}{2}}}.$
(7.2)

###### Proof.

First, we specialize to the case of balls. Let $R>0$. We show that, for
$\|z\|>1$,

$\left\lVert e^{\langle
z,\cdot\rangle}\right\rVert^{2}_{\sigma_{\mathbb{B}^{n}(R)}}\approx\frac{e^{2R{\|z\|}}}{\|z\|^{n-\frac{1}{2}}}.$
(7.3)

Fix $z$ such that $\|z\|>1$. Then,

$e^{-2R{\|z\|}}\|z\|^{n-\frac{1}{2}}\|e^{\langle
z,\cdot\rangle}\|^{2}_{\sigma_{\mathbb{B}^{n}(R)}}=\|z\|^{n-\frac{1}{2}}\int_{b\mathbb{B}^{n}(R)}e^{2R\|z\|\left(\operatorname{Re}\langle\lambda,\frac{z}{R\|z\|}\rangle-1\right)}d\sigma_{\mathbb{B}^{n}(R)}(\lambda).$

After applying the change of variable $\lambda=R\zeta$, we obtain that

$\|z\|^{n-\frac{1}{2}}\int_{b\mathbb{B}^{n}(R)}e^{2R\|z\|\left(\operatorname{Re}\langle\lambda,\frac{z}{R\|z\|}\rangle-1\right)}d\sigma_{\mathbb{B}^{n}(R)}(\lambda)\approx\|z\|^{n-\frac{1}{2}}\int_{b\mathbb{B}^{n}}e^{2R\|z\|\left(\operatorname{Re}\langle\zeta,\frac{z}{\|z\|}\rangle-1\right)}d\sigma_{\mathbb{B}^{n}}(\zeta).$

Next, consider the unitary transformation
$U_{z}:{\mathbb{C}^{n}}\rightarrow{\mathbb{C}^{n}}$ such that
$U_{z}\left(\frac{\overline{z}}{\|z\|}\right)=\left(1,0,\cdots,0\right)$.
After applying the change of variable $\zeta=U_{z}^{*}\left(\eta\right)$,
combined with the fact that $\sigma_{b\mathbb{B}^{n}}$ is invariant under
unitary transformations, we obtain that

$\displaystyle\int_{b\mathbb{B}^{n}}e^{2R\|z\|\left(\operatorname{Re}\langle\zeta,\frac{z}{\|z\|}\rangle-1\right)}d\sigma_{\mathbb{B}^{n}}(\zeta)$
$\displaystyle=\int_{b\mathbb{B}^{n}}e^{-2R\|z\|\left(1-\operatorname{Re}\left\langle\eta,\left(1,0,\cdots,0\right)\right\rangle\right)}d\sigma_{\mathbb{B}^{n}}(\eta)$
$\displaystyle=\int_{b\mathbb{B}^{n}}e^{-2R\|z\|\left(1-\operatorname{Re}\eta_{1}\right)}d\sigma_{\mathbb{B}^{n}}(\eta),$

where $\eta=\left(\eta_{1},\eta_{2},\cdots,\eta_{n}\right)$. Using spherical
co-ordinates on $b\mathbb{b}\mathbb{B}^{n}$, we obtain that

$\int_{b\mathbb{B}^{n}}e^{-2R\|z\|\left(1-\operatorname{Re}\eta_{1}\right)}d\sigma_{b\mathbb{B}^{n}}(\eta)\approx\int_{0}^{\pi}e^{-2R\|z\|\left(1-\cos\theta\right)}\left(\sin\theta\right)^{2n-2}d\theta.$

When $\theta\in\left(\frac{\pi}{2},\pi\right)$,
$1<\left(1-\cos(\theta_{1})\right)<2$. Thus,

$0\leq\|z\|^{n-\frac{1}{2}}\int_{\frac{\pi}{2}}^{\pi}e^{-2R\|z\|\left(1-\cos\theta\right)}\left(\sin\theta\right)^{2n-2}d\theta\lesssim
1.$ (7.4)

For $\theta\in(0,\frac{\pi}{2})$, $\sin\theta\approx\theta.$ Thus, for all
$\theta\in(0,\frac{\pi}{2})$, we have that

$\int_{0}^{\frac{\pi}{2}}e^{-\alpha\theta^{2}}\theta^{2n-2}d\theta\lesssim\int_{0}^{\frac{\pi}{2}}e^{-2R\|z\|\left(1-\cos\theta\right)}\left(\sin\theta\right)^{2n-2}d\theta\lesssim\int_{0}^{\frac{\pi}{2}}e^{-\beta\theta^{2}}\theta^{2n-2}d\theta,$

where $\alpha=C_{1}\|z\|$ and $\beta=C_{2}\|z\|$ for some constants
$C_{1},C_{2}>0$ that are independent of $z$.
Fix a $k>0$, then applying the change of variable $x=\sqrt{\gamma}\theta$, we
have that, for $\gamma\geq k$,

$\int_{0}^{\frac{\pi}{2}}e^{-\gamma\theta^{2}}\theta^{2n-2}d\theta=\gamma^{\frac{1}{2}-n}\int_{0}^{\frac{\sqrt{\gamma}\pi}{2}}e^{-x^{2}}x^{2n-2}dx\approx\gamma^{\frac{1}{2}-n}.$

Hence,

$\|z\|^{n-\frac{1}{2}}\int_{0}^{\frac{\pi}{2}}e^{-2R\|z\|\left(1-\cos\theta\right)}\left(\sin\theta\right)^{2n-2}d\theta\approx
1.$ (7.5)

Finally, combining (7.4) and (7.5), we conclude the proof in the special case
of balls.

Let $\Omega\subset{\mathbb{C}^{n}}$ be a bounded strongly convex domain.
Without loss of generality, assume $0\in\Omega$. We collect some facts about
the support function $H_{\Omega}$ of $\Omega$; see [32, §2.5]. Note that the
definition of support function $h_{\Omega}$ considered in [32] is different
from $H_{\Omega}$. However, they are related by
$H_{\Omega}(z)=h_{\Omega}(\bar{z})$, $z\in\mathbb{C}^{n}$.

  1. (1)

$H_{\Omega}$ is $\mathcal{C}^{2}$ smooth on $\mathbb{C}^{2}\setminus\\{0\\}$.

  2. (2)

The map $\lambda\rightarrow 2\partial H_{\Omega}(\overline{\lambda})$ is a
$\mathcal{C}^{1}$-difeomorphsim from $b\mathbb{B}^{n}$ onto $b\Omega$.

  3. (3)

Owing to the $1$-homegeneity of $H_{\Omega}$, we may write (abusing notation):

$H_{\Omega}(z)=2\operatorname{Re}\langle\partial H_{\Omega}(z),z\rangle,\quad
z\in{\mathbb{C}^{n}}\setminus\\{0\\}.$ (7.6)

  4. (4)

For any real-valued function $f$ that is measurable with respect to
$\sigma_{\Omega}$,

$\int_{b\Omega}f(\zeta)d\sigma_{\Omega}(\zeta)\approx\int_{b\mathbb{B}^{n}}f(2\partial
H_{\Omega}(\overline{\lambda}))d\sigma_{\mathbb{B}^{n}}(\lambda).$ (7.7)

  5. (5)

The polar domain $\Omega^{\circ}$ of $\Omega$ given by
$\Omega^{\circ}=\left\\{z:H_{\Omega}(\overline{z})<1\right\\}$ is also
strongly convex.

By (7.7) and (7.6), we have that

$\frac{1}{e^{2H_{\Omega}(\overline{z})}}\int_{b\Omega}e^{2\operatorname{Re}\langle\zeta,\overline{z}\rangle}d\sigma_{\Omega}(\zeta)\approx\int_{b\mathbb{B}^{n}}e^{4\|z\|\left(\operatorname{Re}\left\langle\partial
H_{\Omega}(\overline{\lambda})-\partial
H_{\Omega}(\overline{z}),\frac{\overline{z}}{\|z\|}\right\rangle\right)}d\sigma_{\mathbb{B}^{n}}(\lambda).$
(7.8)

Note that, using (7.6) twice, we obtain that for $\left(\lambda,z\right)\in
b\mathbb{B}^{n}\times\mathbb{C}^{n}\setminus\\{0\\}$,

$\displaystyle\operatorname{Re}\left\langle\partial
H_{\Omega}(\overline{z})-\partial
H_{\Omega}(\overline{\lambda}),\frac{\overline{z}}{\|z\|}\right\rangle$
$\displaystyle\approx$ $\displaystyle\operatorname{Re}\left\langle\partial
H_{\Omega}(\overline{z})-\partial
H_{\Omega}(\overline{\lambda}),\frac{\overline{z}}{H_{\Omega}(\overline{z})}\right\rangle$
(7.9) $\displaystyle=$
$\displaystyle\frac{1}{2}-\operatorname{Re}\left\langle\partial
H_{\Omega}(\overline{\lambda}),\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}-\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}+\frac{\overline{z}}{H_{\Omega}(\overline{z})}\right\rangle$
$\displaystyle=$ $\displaystyle\operatorname{Re}\left\langle\partial
H_{\Omega}(\overline{\lambda}),\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}-\frac{\overline{z}}{H_{\Omega}(\overline{z})}\right\rangle$
$\displaystyle=$ $\displaystyle\operatorname{Re}\left\langle\partial
H_{\Omega}\left(\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}\right),\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}-\frac{\overline{z}}{H_{\Omega}(\overline{z})}\right\rangle.$

Now, by the definition of $\Omega^{\circ}$, it is clear that for any $w\neq
0$, $\frac{w}{H_{\Omega}(\overline{w})}\in b\Omega^{\circ}$. Considering
$H_{\Omega}(\bar{z})-1$ to be the defining function of $\Omega^{\circ}$, we
get from [15, Lemma 5.1] that, for all $(\lambda,z)\in
b\mathbb{B}^{n}\times\mathbb{C}^{n}\setminus\\{0\\}$,

$\operatorname{Re}\left\langle\partial
H_{\Omega}\left(\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}\right),\frac{\overline{\lambda}}{H_{\Omega}(\overline{\lambda})}-\frac{\overline{z}}{H_{\Omega}(\overline{z})}\right\rangle\approx\left\|\frac{z}{H_{\Omega}(\overline{z})}-\frac{\lambda}{H_{\Omega}(\overline{\lambda})}\right\|^{2}.$
(7.10)

Moreover, for all $(\lambda,z)\in
b\mathbb{B}^{n}\times\mathbb{C}^{n}\setminus\\{0\\}$,

$\left\|\frac{z}{H_{\Omega}(\overline{z})}-\frac{\lambda}{H_{\Omega}(\overline{\lambda})}\right\|^{2}\approx\left\|\frac{z}{\|z\|}-\lambda\right\|^{2}=2\operatorname{Re}\left\langle\frac{z}{\|z\|}-\lambda,\frac{\overline{z}}{\|z\|}\right\rangle.$
(7.11)

Combining (7.9), (7.10), (7.11), and (7.8), we note that it suffices to
estimate

$\int_{b\mathbb{B}^{n}}e^{2C\|z\|\left(\operatorname{Re}\left\langle\lambda,\frac{\bar{z}}{\|z\|}\right\rangle-1\right)}d\sigma_{\mathbb{B}^{n}}(\lambda)=C^{1-2n}\|e^{\langle\overline{z},\cdot\rangle}\|^{2}_{\sigma_{\mathbb{B}^{n}(C)}}e^{-2C\|z\|},$

where $C>0$ is some constant. Now, applying (7.3) to the right-hand side, we
conclude the proof. ∎

######  Proposition 7.2.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded strongly convex domain. Then,
the identity map is a normed space isomorphism between
$A^{2}(\mathbb{C}^{n},\omega_{\Omega})$ and
$A^{2}(\mathbb{C}^{n},\nu_{\Omega})$.

###### Proof.

We must show that

$\int_{\mathbb{B}^{n}}|F(z)|^{2}d\omega_{\Omega}(z)\approx\int_{\mathbb{B}^{n}}|F(z)|^{2}d\nu_{\Omega}(z),$

uniformly for all $F\in\mathcal{O}\left(\mathbb{C}^{2}\right).$ Due to Lemma
7.1, it suffices to prove that

$\int_{\mathbb{B}^{n}}|F(z)|^{2}\left(dd^{c}H_{\Omega}\right)^{n}(z)\approx\int_{\mathbb{B}^{n}}|F(z)|^{2}\|z\|^{n-\frac{1}{2}}\left(dd^{c}H_{\Omega}\right)^{n}(z),$

where $F$ is a holomorphic polynomial. This can be verified easily using
spherical coordinates and the fact that
$\|z\|^{n}\left(dd^{c}H_{\Omega}\right)^{n}\approx\omega_{std}$. ∎

## References

  * [1] L. Aizenberg, V. Gotlib, and A. Vidras.  Duality for Hardy spaces in Domains of $\mathbb{C}^{n}$ and Some Applications.  Complex Anal. Oper. Theory, 8:1341–1366, 2014.
  * [2] M. Andersson, M. Passare, and R. Sigurdsson.  Complex Convexity and Analytic Functionals, volume 225.  Springer Science & Business Media, 2004.
  * [3] D. E. Barrett and L. D. Edholm.  High frequency behavior of the Leray transform: model hypersurfaces and projective duality.  arXiv preprint arXiv:2111.13954, 2021.
  * [4] D. E. Barrett and L. Lanzani.  The spectrum of the Leray transform for convex Reinhardt domains in $\mathbb{C}^{2}$.  J. Funct. Anal., 257(9):2780–2819, 2009.
  * [5] E. Bedford and S. Ma’u.  Complex Monge-Ampére of a Maximum.  Proc. Amer. Math. Soc., 136(1):95–101, 2008.
  * [6] E. Bedford and B. A. Taylor.  The Dirichlet Problem for a Complex Monge-Ampére Equation.  Invent. Math., 37:1–44, 1976.
  * [7] B. Berndtsson.  Weighted Integral Formulas.  In Several Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987–1988, volume 38, pages 160–187. Princeton University Press, 1993.
  * [8] B. Berndtsson.  An Inequality for Fourier–Laplace transforms of Entire functions, and the existence of Exponential Frames in Fock space.  J. Funct. Anal., 149(1):83–101, 1997.
  * [9] B. Berndtsson.  Bergman kernels for Paley–Wiener spaces and Nazarov’s proof of the Bourgain–Milman theorem.  Pure Appl. Math. Q, 18(2):395–409, 2022.
  * [10] A. P. Calderón.  Cauchy integrals on Lipschitz curves and related operators.  Proc. Natl. Acad. Sci. USA, 74(4):1324–1327, 1977.
  * [11] A. Chatterjee.  Cauchy-type and other boundary operators on convex domains in $\mathbb{C}^{2}$.  PhD thesis, in preparation.
  * [12] R. R. Coifman, A. McIntosh, and Y. Meyer.  L’intégrale de Cauchy définit un opérateur borné sur $\operatorname{L}^{2}$ pour les courbes lipschitziennes.  Ann. of Math. (2), pages 361–387, 1982.
  * [13] G. David.  Une minoration de la norme de l’opérateur de Cauchy sur les graphes lipschitziens.  Trans. Amer. Math. Soc, 302:741–750, 1987.
  * [14] J.-P. Demailly.  In Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, number 19, pages 1–124. Société mathématique de France, 1985.
  * [15] F. Forstnerič.  Embedding Strictly Pseudoconvex Domains Into Balls.  Trans. Amer. Math. Soc, 295(1):347–368, 1986.
  * [16] A.-K. Gallagher, P. Gupta, L. Lanzani, and L. Vivas.  Hardy spaces for a class of singular domains.  Math. Z., 299(3-4):2171–2197, 2021.
  * [17] T. Hansson.  On Hardy spaces in complex ellipsoids.  Ann. Inst. Fourier (Grenoble), 49(5):1477–1501, 1999.
  * [18] Y. Huang, J. Ibbotson, and Z. Zielezny.  An application of Regularly varying functions.  Rocky Mountain J. Math., pages 505–515, 1998.
  * [19] V. Katsnelson.  A generalization of the Paley–Wiener theorem on a representation for entire functions of exponential type.  Teor. Funktsii. Funktsional. Anal. i Prilozhen., 1:99–110, 1965 (in Russian).
  * [20] N. Kerzman and E. M. Stein.  The Szegö kernel in terms of Cauchy-Fantappiè kernels.  Duke Math. J., 45:197–224, 1978.
  * [21] L. Lanzani and E. M. Stein.  The Cauchy integral in $\mathbb{C}^{n}$ for domains with minimal smoothness.  Adv. Math., 264:776–830, 2014.
  * [22] L. Lanzani and E. M. Stein.  The Cauchy-Leray Integral: Counter-examples to the $\operatorname{L}^{p}$-theory.  Indiana Univ. Math. J., 68(5):1609–1621, 2017.
  * [23] B. Ja. Levin.  Distribution of Zeros of Entire Functions.  Translations of Mathematical Monographs, American Mathematical Society, 1964.
  * [24] M. K. Likht.  A remark about the Paley–Wiener theorem on entire functions of exponential type.  Uspekhi Mat. Nauk, 19 1(115):169–171, 1964 (in Russian).
  * [25] N. Lindholm.  A Paley–Wiener theorem for convex sets in $\mathbb{C}^{n}$.  Bull. Sci. Math., 126(4):289–314, 2002.
  * [26] V. I. Lutsenko and R. S. Yulmukhametov.  A generalization of Wiener–Paley theorem to functionals in Smirnov spaces.  Proc. Steklov Inst. Math., 200:245–254, 1991.