# minimal $L^{2}$ integrals for the Hardy spaces and the Bergman spaces

Qi’an Guan  Qi’an Guan: School of Mathematical Sciences, Peking University,
Beijing, 100871, China<EMAIL_ADDRESS>and  Zheng Yuan  Zheng Yuan:
School of Mathematical Sciences, Peking University, Beijing, 100871, China.
<EMAIL_ADDRESS>
###### Abstract.

In this article, we consider the minimal $L^{2}$ integrals for the Hardy
spaces and the Bergman spaces, and we present some relations between them,
which can be regarded as the solutions of the finite points versions of
Saitoh’s conjecture for conjugate Hardy kernels. As applications, we give
optimal $L^{2}$ extension theorems for the Hardy spaces, and characterizations
for the holding of the equality in the optimal $L^{2}$ extension theorems.

###### Key words and phrases:

Hardy space, Bergman space, minimal $L^{2}$ integral, product manifold,
concavity property

###### 2020 Mathematics Subject Classification:

30H10 30H20 31C12 30E20

##  1\. Introduction

Let $D$ be a planar regular region with finite boundary components, which are
analytic Jordan curves (see [18, 22]).

######  Definition 1.1 (see [18, 12]).

We call a holomorphic function $f$ on $D$ belongs to Hardy space $H^{2}(D)$,
if $|f(z)|^{2}$ have harmonic majorants $U(z)$, i.e., $|f(z)|^{2}\leq
U(z)\,\,\text{on}\,\,D.$

Each function $f(z)\in H^{2}(D)$ has Fatou’s nontangential boundary value a.e.
on $\partial D$ belonging to $L^{2}(\partial D)$ (see [1]), and we also denote
the nontangential boundary value by $f$ for simplicity. The conjugate Hardy
$H^{2}$ kernel $\hat{K}_{t}(z,\overline{w})$ is defined as follow:

$f(w)=\frac{1}{2\pi}\int_{\partial
D}f(z)\overline{\hat{K}_{t}(z,\overline{w})}\left(\frac{\partial
G_{D}(z,t)}{\partial v_{z}}\right)^{-1}|dz|$

holds for any $f\in H^{2}(D)$, where $G_{D}(z,t)$ is the Green function on
$D$, and $\partial/\partial v_{z}$ denotes the derivative along the outer
normal unit vector $v_{z}$. Fixed $t\in D$, $\frac{\partial
G_{D}(z,t)}{\partial v_{z}}$ is positive and continuous on $\partial D$
because of the analyticity of the boundary (see [18], [5]). When $t=w=z$,
$\hat{K}(z)$ denotes $\hat{K}_{t}(z,\overline{w})$ for simplicity.

In [5], Guan proved the following theorem, which was conjectured by Saitoh
(see [18]):

######  Theorem 1.2 ([5]).

If $D$ is not simple connected, then $\hat{K}(z)>\pi B(z)$, where $B(z)$ is
the Bergman kernel on $D$.

By discussing the weighted kernel functions, we [12] gave a weighted version
of Saitoh’s conjecture and a weighted version of Saitoh’s conjecture for
higher derivatives.

In [13], we considered two classes of weighted Hardy spaces on products of
planar domains. Let us recall their definitions.

Let $D_{j}$ be a planar region bounded by finite analytic Jordan curves for
any $1\leq j\leq n$. Let $M=\prod_{1\leq j\leq n}D_{j}$ be a bounded domain in
$\mathbb{C}^{n}$. Let $M_{j}=\prod_{1\leq l\leq n,l\not=j}D_{l}$, then
$M=D_{j}\times M_{j}$ and $\partial M=\cup_{1\leq j\leq n}\partial
D_{j}\times\overline{M_{j}}$.

Let $\rho$ be a Lebesgue measurable function on $\partial M$ such that
$\inf_{\partial M}\rho>0$. Now, we recall the Hardy space
$H_{\rho}^{2}(M,\partial M)$ (see [13]). Note that $\partial
M=\cup_{j=1}^{n}\partial D_{j}\times\overline{M_{j}}$. Let $d\mu_{j}$ be the
Lebesgue measure on $M_{j}$ for any $1\leq j\leq n$ and $d\mu$ is a measure on
$\partial M$ defined by

$\int_{\partial M}hd\mu=\sum_{1\leq j\leq
n}\frac{1}{2\pi}\int_{M_{j}}\int_{\partial
D_{j}}h(w_{j},\hat{w}_{j})|dw_{j}|d\mu_{j}(\hat{w}_{j})$

for any $h\in L^{1}(\partial M)$, where
$\hat{w}_{j}:=(w_{1},\ldots,w_{j-1},w_{j+1},\ldots,w_{n})\in M_{j}$. For any
$f\in H^{2}(D_{j})$, $\gamma_{j}(f)$ denotes the nontangential boundary value
of $f$ a.e. on $\partial D_{j}$.

######  Definition 1.3 ([13]).

Let $f\in L^{2}(\partial M,\rho d\mu)$. We call $f\in H^{2}_{\rho}(M,\partial
M)$ if there exists $f^{*}\in\mathcal{O}(M)$ such that for any $1\leq j\leq
n$, $f^{*}(\cdot,\hat{w}_{j})\in H^{2}(D_{j})$ for any $\hat{w}_{j}\in M_{j}$
and $f=\gamma_{j}(f^{*})$ a.e. on $\partial D_{j}\times M_{j}$.

$H^{2}_{\rho}(M,\partial M)$ is a Hilbert space (see [13]) equipped with the
norm $\ll\cdot,\cdot\gg_{\partial M,\rho}$, which is defined by

$\ll f,g\gg_{\partial M,\rho}:=\int_{\partial M}f\overline{g}\rho d\mu.$

Denote that $P_{\partial M}(f)=f^{*}$ for any $f\in H^{2}_{\rho}(M,\partial
M)$. $P_{\partial M}$ is a linear injective map from $H^{2}(M,\partial
D_{j}\times M_{j})$ to $\mathcal{O}(M)$ (see [13]). When $n=1$, $P_{\partial
M}=\gamma_{1}^{-1}$, thus $H^{2}_{\rho}(M,\partial M)$ can be seen as a
weighted generalization on product spaces of $H^{2}(D)$.

Denote that $S:=\prod_{1\leq j\leq n}\partial D_{j}$. Let $\lambda$ be a
Lebesgue measurable function on $S$ such that $\inf_{S}\lambda>0$. Let us
recall another class of Hardy space $H^{2}_{\lambda}(M,S)$.

######  Definition 1.4 ([13]).

Let $f\in L^{2}(S,\lambda d\sigma)$, where
$d\sigma:=\frac{1}{(2\pi)^{n}}|dw_{1}|\ldots|dw_{n}|$. We call $f\in
H^{2}_{\lambda}(M,S)$ if there exists $\\{f_{m}\\}_{m\in\mathbb{Z}_{\geq
0}}\subset\mathcal{O}(M)\cap C(\overline{M})\cap L^{2}(S,\lambda d\sigma)$
such that $\lim_{m\rightarrow+\infty}\|f_{m}-f\|_{S,\lambda}^{2}=0$, where
$\|g\|_{S,\lambda}:=\left(\int_{S}|g|^{2}\lambda d\sigma\right)^{\frac{1}{2}}$
for any $g\in L^{2}(S,\lambda d\sigma)$.

Denote that

$\ll
f,g\gg_{S,\lambda}=\frac{1}{(2\pi)^{n}}\int_{S}f\overline{g}\lambda|dw_{1}|\ldots|dw_{n}|$

for any $f,g\in L^{2}(S,\lambda d\sigma),$ then $H_{\lambda}^{2}(M,S)$ is a
Hilbert space equipped with the inner product $\ll\cdot,\cdot\gg_{S,\lambda}$
(see [13]). There exists a linear injective map
$P_{S}:H^{2}_{\lambda}(M,S)\rightarrow\mathcal{O}(M)$ satisfying that
$P_{S}(f)=f$ for any $f\in\mathcal{O}(M)\cap C(\overline{M})\cap
L^{2}(S,\lambda d\sigma)$ (see [13]). When $n=1$, $P_{S}=\gamma_{1}^{-1}$,
thus $H^{2}_{\lambda}(M,S)$ can also be seen as a weighted generalization on
product spaces of $H^{2}(D)$.

In [13], we discussed some properties and kernel functions for the spaces
$H^{2}_{\rho}(M,\partial M)$ and $H^{2}_{\lambda}(M,S)$, and we discussed the
relations between them and the weighted Bergman kernels on $M$, which can be
regarded as the solutions of the product versions of Saitoh’s conjecture.

Note that the above mentioned kernel functions for the Hardy spaces and the
Bergman spaces can be seen as the reciprocal of some minimal $L^{2}$ integrals
related to one point, such as:

$\hat{K}(z)=\frac{1}{\inf\left\\{\frac{1}{2\pi}\int_{\partial
D}|f(z)|^{2}\left(\frac{\partial G_{D}(z,t)}{\partial
v_{z}}\right)^{-1}|dz|:f\in H^{2}(D)\,\&\,f(z)=1\right\\}},$

and

$B(z)=\frac{1}{\inf\left\\{\int_{D}|f|^{2}:f\in\mathcal{O}(D)\,\&\,f(z)=1\right\\}}.$

In this article, we consider more general minimal $L^{2}$ integrals for the
Hardy spaces and the Bergman spaces, and we give some relations between them.
As applications, we give optimal $L^{2}$ extension theorems for the Hardy
spaces, and characterizations for the holding of equality in the optimal
$L^{2}$ extension theorems.

###  1.1. Minimal $L^{2}$ integrals on a planar region

Let $D$ be a planar region bounded by finite analytic Jordan curves, and let
$Z_{0}:=\\{z_{1},\ldots,z_{m}\\}\subset D$, where $m$ is a positive integer.

Let $\psi$ be a Lebesgue measurable function on $\overline{D}$, which
satisfies that $\psi$ is subharmonic on $D$, $\psi\equiv 0$ on $\partial D$
and the Lelong number $v(dd^{c}\psi,z_{j})>0$ for any $z_{j}\in Z_{0}$, where
$d^{c}=\frac{\partial-\bar{\partial}}{2\pi\sqrt{-1}}$. Assume that $\psi\in
C^{1}(U\cap\overline{D})$ and $\frac{\partial\psi}{\partial v_{z}}$ is
positive on $\partial D$, where $U$ is an open neighborhood of $\partial D$
and $\partial/\partial v_{z}$ denotes the derivative along the outer normal
unit vector $v_{z}$.

Let $k_{j}$ be a nonnegative integer for $1\leq j\leq m$. Let $\varphi$ be a
Lebesgue measurable function on $\overline{D}$ satisfying that $\varphi+2\psi$
is subharmonic on $D$, the Lelong number

$v(dd^{c}(\varphi+2\psi),z_{j})\geq 2(k_{j}+1)$

for any $1\leq j\leq n$, and $\varphi$ is continuous at $z$ for any
$z\in\partial D$. Besides, we assume that one of the following two statements
holds:

$(1)$ $(\psi-p_{j}G_{D}(\cdot,z_{j}))(z_{j})>-\infty$, where
$p_{j}=v(dd^{c}(\psi),z_{j})>0$ for any $1\leq j\leq m$;

$(2)$ for any $1\leq j\leq m$, there exists $a_{j}\in[0,1)$ such that
$\varphi+2a_{j}\psi$ is subharmonic near $z_{j}$.

Let $c$ be a positive Lebesgue measurable function on $[0,+\infty)$ satisfying
that $c(t)e^{-t}$ is decreasing on $[0,+\infty)$, $\lim_{t\rightarrow
0+0}c(t)=c(0)=1$ and $\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$. Denote that

$\tilde{\rho}:=e^{-\varphi}c(-2\psi),$

and assume that $\tilde{\rho}$ has a positive lower bound on any compact
subset of $D\backslash Z$, where $Z\subset\\{\psi=-\infty\\}$ is a discrete
subset of $D$. Denote that

$\rho:=e^{-\varphi}\left(\frac{\partial\psi}{\partial v_{z}}\right)^{-1}$

on $\partial D$.

Let us consider the following two minimal integrals. Let
$\mathfrak{a}=(a_{j,l})$ $(1\leq j\leq m,0\leq l\leq k_{j})$, where
$a_{j,l}\in\mathbb{C}$ such that $\sum_{1\leq j\leq m}\sum_{0\leq l\leq
k_{j}}|a_{j,l}|\not=0$. Denote that

$\begin{split}M(Z_{0},\mathfrak{a},\tilde{\rho}):=\inf\bigg{\\{}&\int_{D}|f|^{2}\tilde{\rho}:f\in\mathcal{O}(D)\\\
&\text{ s.t. }f^{(l)}(z_{j})=l!a_{j,l}\text{ for any $0\leq l\leq k_{j}$ and
any $1\leq j\leq m$}\bigg{\\}}.\end{split}$

and

$\begin{split}M_{H}(Z_{0},\mathfrak{a},\rho):=\inf\bigg{\\{}\frac{1}{2\pi}&\int_{\partial
D}|f|^{2}\rho|dz|:f\in H^{2}(D)\\\ &\text{ s.t.
}f^{(l)}(z_{j})=l!a_{j,l}\text{ for any $0\leq l\leq k_{j}$ and any $1\leq
j\leq m$}\bigg{\\}}.\end{split}$

We recall some notations (see [3], see also [14, 9, 6]). Let
$p:\Delta\rightarrow D$ be the universal covering from unit disc $\Delta$ to
$D$. we call the holomorphic function $f$ on $\Delta$ a multiplicative
function, if there is a character $\chi$, which is the representation of the
fundamental group of $D$, such that $g^{\star}f=\chi(g)f$, where $|\chi|=1$
and $g$ is an element of the fundamental group of $D$. It is known that for
any harmonic function $u$ on $D$, there exists a $\chi_{u}$ and a
multiplicative function $f_{u}\in\mathcal{O}^{\chi_{u}}(D)$, such that
$|f_{u}|=p^{\star}\left(e^{u}\right)$. Recall that for the Green function
$G_{D}(z,z_{j})$, there exist a $\chi_{z_{j}}$ and a multiplicative function
$f_{z_{j}}\in\mathcal{O}^{\chi_{z_{j}}}(D)$, such that
$|f_{z_{j}}(z)|=p^{\star}\left(e^{G_{D}(z,z_{j})}\right)$ (see [22, 21]).

We present a relation between $M_{H}(Z_{0},\mathfrak{a},\rho)$ and
$M(Z_{0},\mathfrak{a},\tilde{\rho})$ as follows:

######  Theorem 1.5.

Assume that $M(Z_{0},\mathfrak{a},\tilde{\rho})<+\infty$. Then

(1.1)
$M_{H}(Z_{0},\mathfrak{a},\rho)\leq\frac{M(Z_{0},\mathfrak{a},\tilde{\rho})}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}$

holds, and the equality holds if and only if the following statements hold:

$(1)$ $\varphi+2\psi=2\sum_{1\leq j\leq m}(k_{j}+1)G_{D}(\cdot,z_{j})+2u$,
where $u$ is a harmonic function on $D$;

$(2)$ $\psi=\sum_{1\leq j\leq m}p_{j}G_{D}(\cdot,z_{j})$, where
$p_{j}=v(dd^{c}(\psi),z_{j})>0$;

$(3)$ $\chi_{-u}=\prod_{1\leq j\leq m}\chi_{z_{j}}^{k+1}$, where $\chi_{-u}$
and $\chi_{z_{j}}$ are the characters associated to the functions $-u$ and
$G_{D}(\cdot,z_{j})$ respectively;

$(4)$ For any $1\leq j\leq m$, $\lim_{z\rightarrow
z_{j}}\frac{p_{*}\left(f_{u}\left(\prod_{1\leq j\leq
m}f_{z_{j}}^{k_{j}+1}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{(z-z_{j})^{k_{j}}dz}=c_{0}a_{j,k_{j}}$
and $a_{j,l}=0$ for any $l<k_{j}$, where $c_{0}\not=0$ is a constant
independent of $j$.

When $m=1$, Theorem 1.5 is a solution of the weighted version of Saitoh’s
conjecture for higher derivatives, which can be referred to [12].

######  Remark 1.6.

Assume that the four statements in Theorem 1.5 hold, then we know
$\frac{p_{*}\left(f_{u}\left(\prod_{1\leq j\leq
m}f^{k_{j}+1}_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{c_{0}dz}$ is a (single-
valued) holomorphic function on $D$, and we denote it by $F_{0}$. Then
$F_{0}^{(l)}(z_{j})=l!a_{j,l}$ for any $0\leq l\leq k_{j}$ and any $1\leq
j\leq m$, and there exists $f_{0}\in H^{2}(D)$ such that $f_{0}^{*}=F_{0}$,

$M(Z_{0},\mathfrak{a},\tilde{\rho})=\int_{D}|F_{0}|^{2}\tilde{\rho}\,\,\,\,and\,\,\,\,M_{H}(Z_{0},\mathfrak{a},\rho)=\frac{1}{2\pi}\int_{\partial
D}|f_{0}|^{2}\rho|dz|.$

We prove the remark in Section 3.

Let $Z_{0}:=\\{z_{j}:1\leq k\leq m\\}$ be a subset of $D$. Let $\lambda$ be a
positive continuous function on $\partial D$. By solving Dirichlet problem,
there exists a positive continuous function on $\overline{D}$ denoted also by
$\lambda$, such that $\log\lambda$ is harmonic on $D$. Let $c_{\beta}(z)$ be
the logarithmic capacity (see [19]) on $D$, which is locally defined by

$c_{\beta}(z):=\exp\lim_{\tilde{z}\rightarrow
z}(G_{D}(\tilde{z},z)-\log|\tilde{z}-z|).$

Using Theorem 1.5, we present the following optimal $L^{2}$ extension theorem
for the Hardy space, and give a characterization for the holding of the
equality in this extension theorem.

######  Corollary 1.7.

Let $k_{j}$ be a nonnegative integer, and let $a_{j}\in\mathbb{C}$ for any
$j$. Assume that $\sum_{1\leq j\leq
m}\frac{2|a_{j}|^{2}t_{j}}{(k_{j}+1)c_{\beta}(z_{j})^{2(k_{j}+1)}}\lambda(z_{j})\in(0,+\infty)$.
Then there exists $f\in H^{2}(D)$ such that $f^{(l)}(z_{j})=0$ for $0\leq
l<k_{j}$ and $f^{(k_{j})}(z_{j})=k_{j}!a_{j}$ for any $1\leq j\leq m$, and

$\frac{1}{2\pi}\int_{\partial
D}|f|^{2}\lambda\left(\frac{\partial\psi}{\partial
v_{z}}\right)^{-1}|dz|\leq\sum_{1\leq j\leq
m}\frac{2|a_{j}|^{2}t_{j}}{(k_{j}+1)c_{\beta}(z_{j})^{2(k_{j}+1)}}\lambda(z_{j}),$

where $\psi:=\sum_{1\leq j\leq m}(k_{j}+1)G_{D}(\cdot,z_{j})$ and
$t_{j}:=e^{-2\sum_{1\leq j_{1}\leq
m,j_{1}\not=j}(k_{j}+1)G_{D}(z_{j},z_{j_{1}})}$.

Moreover, denote that $M_{H}:=\inf\\{\frac{1}{2\pi}\int_{\partial
D}|f|^{2}\lambda\left(\frac{\partial\psi}{\partial v_{z}}\right)^{-1}|dz|:f\in
H^{2}(D)$ such that $f^{(l)}(z_{j})=0$ for $0\leq l<k_{j}$ and
$f^{(k_{j})}(z_{j})=k_{j}!a_{j}$ for any $1\leq j\leq m\\}$, then equality

$M_{H}=\sum_{1\leq j\leq
m}\frac{2|a_{j}|^{2}t_{j}}{(k_{j}+1)c_{\beta}(z_{j})^{2(k_{j}+1)}}\rho(z_{j})$

holds if and only if the following statements hold:

$(1)$ $\chi_{\frac{1}{2}\log\lambda}=\prod_{1\leq j\leq
m}\chi_{z_{j}}^{k_{j}+1}$;

$(2)$ For any $1\leq j\leq m$,

$\lim_{z\rightarrow
z_{j}}\frac{p_{*}\left(f_{-\frac{1}{2}\log\lambda}\left(\prod_{1\leq j\leq
m}f_{z_{j}}^{k_{j}+1}\right)\left(\sum_{1\leq j\leq
m}(k_{j}+1)\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{(z-z_{j})^{k_{j}}dz}=c_{0}a_{j},$

where $c_{0}\not=0$ is a constant independent of $j$.

Corollary 1.7 implies the following result.

######  Corollary 1.8.

Let $k$ be a nonnegative integer. Then there is a constant $C$ (depending on
$k$), such that for any $a_{j,l}\in\mathbb{C}$, where $1\leq j\leq m$ and
$0\leq l\leq k$, there exists $f\in H^{2}(D)$ such that
$f^{(l)}(z_{j})=a_{j,l}$ for any $1\leq j\leq m$ and $0\leq l\leq k$, and

$\frac{1}{2\pi}\int_{\partial D}|f|^{2}|dz|\leq C\sum_{1\leq j\leq
m}\sum_{0\leq l\leq k}|a_{j,l}|^{2}.$

###  1.2. Minimal $L^{2}$ integrals for the Hardy space
$H_{\rho}^{2}(M,\partial M)$

Let $D_{j}$ be a planar region bounded by finite analytic Jordan curves for
any $1\leq j\leq n$. Let $M=\prod_{1\leq j\leq n}D_{j}$ be a bounded domain in
$\mathbb{C}^{n}$.

Let $Z_{j}=\\{z_{j,1},z_{j,2},...,z_{j,m_{j}}\\}\subset D_{j}$ for any
$j\in\\{1,2,...,n\\}$, where $m_{j}$ is a positive integer. Denote that

$Z_{0}:=\prod_{1\leq j\leq n}Z_{j}\subset M.$

Let $\psi=\max_{1\leq j\leq n}\\{\sum_{1\leq k\leq
m_{j}}p_{j,k}G_{D_{j}}(\cdot,z_{j,k})\\}$ on $M$. Let $V_{z_{j,k}}\Subset
D_{j}$ be a neighborhood of $z_{j,k}$ satisfying $V_{z_{j,k}}\cap
V_{z_{j,k^{\prime}}}=\emptyset$ for any $j$ and $k\not=k^{\prime}$. Denote
that $I_{1}:=\\{(\beta_{1},\beta_{2},...,\beta_{n}):1\leq\beta_{j}\leq m_{j}$
for any $j\in\\{1,2,...,n\\}\\}$, $V_{\beta}:=\prod_{1\leq j\leq
n}V_{z_{j,\beta_{j}}}$ and
$z_{\beta}:=(z_{1,\beta_{1}},z_{2,\beta_{2}},\ldots,z_{n,\beta_{n}})\in M$ for
any $\beta=(\beta_{1},\beta_{2},...,\beta_{n})\in I_{1}$.

Let $\varphi_{j}$ be a subharmonic function on $D_{j}$, which satisfies that
$\varphi_{j}$ is continuous at $z$ for any $z\in\partial D_{j}$. Denote that

$\varphi(w_{1},\ldots,w_{n}):=\sum_{1\leq j\leq n}\varphi_{j}(w_{j})$

on $M$. Let $f_{0}$ be a holomorphic function $\cup_{\beta\in
I_{1}}V_{\beta}$. For any $\beta\in I_{1}$, let $J_{\beta}$ be an ideal of
$\mathcal{O}_{z_{\beta}}$ satisfying
$\mathcal{I}(\varphi+2\psi)_{z_{\beta}}\subset J_{\beta}$. Note that for any
$\tilde{z}\in D_{j}$, $\frac{\partial G_{D_{j}}(z,\tilde{z})}{\partial v_{z}}$
is a positive continuous function on $\partial D_{j}$ by the analyticity of
the boundary (see [18],[5]), where $\partial/\partial v_{z}$ denotes the
derivative along the outer normal unit vector $v_{z}$. Let $\rho$ be a
Lebesgue measurable function on $\partial M$ such that

$\rho(w_{1},\ldots,w_{n})=\left(\sum_{1\leq k\leq m_{j}}p_{j,k}\frac{\partial
G_{D_{j}}(w_{j},z_{j,k})}{\partial v_{w_{j}}}\right)^{-1}\times\prod_{1\leq
l\leq n}e^{-\varphi_{l}(w_{l})}$

on $\partial D_{j}\times{M_{j}}$. Let $c$ be a positive function on
$[0,+\infty)$, which satisfies that $c(t)e^{-t}$ is decreasing on
$[0,+\infty)$, $\lim_{t\rightarrow 0+0}c(t)=c(0)=1$ and
$\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$. Denote that

$\tilde{\rho}=c(-2\psi)\prod_{1\leq j\leq n}e^{-\varphi_{j}}$

on $M$.

Let us consider the following two minimal integrals. Denote that

$\begin{split}M(Z_{0},J,\tilde{\rho}):=\inf\bigg{\\{}&\int_{M}|f|^{2}\tilde{\rho}:f\in\mathcal{O}(D)\text{
s.t. }(f-f_{0},z_{\beta})\in J_{\beta}\text{ for any $\beta\in
I_{1}$}\bigg{\\}}\end{split}$

and

$\begin{split}M_{H}(Z_{0},J,\rho):=\inf\bigg{\\{}&\|f\|^{2}_{\partial
M,\rho}:f\in H_{\rho}^{2}(M,\partial M)\\\ &\text{ s.t.
}(f^{*}-f_{0},z_{\beta})\in J_{\beta}\text{ for any $\beta\in
I_{1}$}\bigg{\\}}.\end{split}$

Denote that

$\begin{split}G(t):=\inf\bigg{\\{}\int_{\\{2\psi<-t\\}}|f|^{2}\tilde{\rho}:&f\in\mathcal{O}(\\{2\psi<-t\\})\\\
&\text{ s.t. }(f-f_{0},z_{\beta})\in J_{\beta}\text{ for any $\beta\in
I_{1}$}\bigg{\\}}\end{split}$

for any $t\geq 0$. Note that $\tilde{\rho}=c(-2\psi)\prod_{1\leq j\leq
n}e^{-\varphi_{j}}$ and $G(0)=M(Z_{0},J_{\beta},\tilde{\rho})$. As
$\mathcal{I}(\varphi+2\psi)_{z_{\beta}}\subset J_{\beta}$ for any $\beta\in
I_{1}$, it follows from Theorem 2.23 that $G(h^{-1}(r))$ is concave, where
$h(t)=\int_{t}^{+\infty}c(s)e^{-s}ds$.

We present a relation between $M_{H}(Z_{0},J,\rho)$ and
$M(Z_{0},J,\tilde{\rho})$.

######  Theorem 1.9.

Assume that $M(Z_{0},J,\tilde{\rho})<+\infty$. Then

(1.2)
$M_{H}(Z_{0},J,\rho)\leq\frac{M(Z_{0},J,\tilde{\rho})}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}$

holds, and equality holds if and only if $G(h^{-1}(r))$ is linear on
$[0,\int_{0}^{+\infty}c(t)e^{-t}dt]$ and there exists $f\in
H^{2}_{\rho}(M,\partial M)$, such that $(f^{*}-f_{0},z_{\beta})\in J_{\beta}$
for any $\beta\in I_{1},$ $M_{H}(Z_{0},J,\rho)=\|f\|^{2}_{\partial M,\rho}$
and $M(Z_{0},J,\tilde{\rho})=\int_{M}|f^{*}|^{2}\tilde{\rho}$.

######  Remark 1.10.

Let $\hat{\rho}$ be any Lebesgue measurable function on $\overline{M}$, which
satisfies that $\inf_{\overline{M}}\hat{\rho}>0$, $-\log\hat{\rho}$ is
plurisubharmonic on $M$ and
$\hat{\rho}(w_{j},\hat{w}_{j})\leq\liminf_{w\rightarrow
w_{j}}\hat{\rho}(w,\hat{w}_{j})$ for any $(w_{j},\hat{w}_{j})\in\partial
D_{j}\times M_{j}\subset\partial M$ and any $1\leq j\leq n$, where
$M_{j}=\prod_{l\not=j}D_{l}$. Let $\rho(w_{1},\ldots,w_{n})=\left(\sum_{1\leq
k\leq m_{j}}p_{j,k}\frac{\partial G_{D_{j}}(w_{j},z_{j,k})}{\partial
v_{w_{j}}}\right)^{-1}\times\hat{\rho}$ on $\partial D_{j}\times{M_{j}}$, and
let $\tilde{\rho}=c(-2\psi)\hat{\rho}$ on $M$. Inequality (1.2) in Theorem 1.9
also holds for this case (We prove the remark in the Step 1 of the proof of
Theorem 1.9).

Using Theorem 1.5 and Theorem 2.29 (a characterization for the concavity of
$G(h^{-1}(r))$ degenerating to linearity), we obtain the following theorem.

######  Theorem 1.11.

Assume that $J_{\beta}=\mathcal{I}(2\psi)_{z_{\beta}}$ for any $\beta\in
I_{1}$, and $f_{0}=\prod_{1\leq j\leq n}(w_{j}-z_{j,1})^{\tilde{\alpha}_{j}}$
on $V_{\beta^{*}}$, where $\beta^{*}=(1,1,...,1)\in I_{1}$. Then equality

$M_{H}(Z_{0},J,\rho)=\frac{M(Z_{0},J,\tilde{\rho})}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}$

holds if and only if the following statements hold:

$(1)$ $\varphi_{j}=2\log|g_{j}|+2u_{j}$ for any $j\in\\{1,2,...,n\\}$, where
$u_{j}$ is a harmonic function on $D_{j}$ and $g_{j}$ is a holomorphic
function on $\mathbb{C}$ satisfying $g_{j}(z_{j,k})\not=0$ for any
$k\in\\{1,2,...,m_{j}\\}$;

$(2)$ There exists a nonnegative integer $\gamma_{j,k}$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$, which satisfies that
$\Pi_{1\leq k\leq m_{j}}\chi_{j,z_{j,k}}^{\gamma_{j,k}+1}=\chi_{j,-u_{j}}$ and
$\sum_{1\leq j\leq n}\frac{\gamma_{j,\beta_{j}}+1}{p_{j,\beta_{j}}}=1$ for any
$\beta\in I_{1}$, where $\chi_{-u_{j}}$ and $\chi_{z_{j,k}}$ are the
characters associated to the functions $-u_{j}$ and $G_{D_{j}}(\cdot,z_{j,k})$
respectively;

$(3)$ $f_{0}=c_{\beta}\Pi_{1\leq j\leq
n}(w_{j}-z_{j,\beta_{j}})^{\gamma_{j,\beta_{j}}}+g_{\beta}$ on $V_{\beta}$ for
any $\beta\in I_{1}$, where $c_{\beta}$ is a constant and $g_{\beta}$ is a
holomorphic function on $V_{\beta}$ such that
$(g_{\beta},z_{\beta})\in\mathcal{I}(\psi)_{z_{\beta}}$;

$(4)$ $\lim_{z\rightarrow z_{\beta}}\frac{c_{\beta}\Pi_{1\leq j\leq
n}(w_{j}-z_{j,\beta_{j}})^{\gamma_{j,\beta_{j}}}dw_{1}\wedge
dw_{2}\wedge...\wedge dw_{n}}{\wedge_{1\leq j\leq
n}g_{j}(P_{j})_{*}\left(f_{u_{j}}\left(\Pi_{1\leq k\leq
m_{j}}f_{z_{j,k}}^{\gamma_{j,k}+1}\right)\left(\sum_{1\leq k\leq
m_{j}}p_{j,k}\frac{df_{z_{j,k}}}{f_{z_{j,k}}}\right)\right)}=c_{0}$ for any
$\beta\in I_{1}$, where $c_{0}\in\mathbb{C}\backslash\\{0\\}$ is a constant
independent of $\beta$, $P_{j}:\Delta\rightarrow D_{j}$ is the universal
covering, $f_{u_{j}}$ is a holomorphic function $\Delta$ such that
$|f_{u_{j}}|=P_{j}^{*}(e^{u_{j}})$ and $f_{z_{j,k}}$ is a holomorphic function
on $\Delta$ such that
$|f_{z_{j,k}}|=P_{j}^{*}\left(e^{G_{D_{j}}(\cdot,z_{j,k})}\right)$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$.

When $m_{j}=0$ for any $1\leq j\leq n$, the above theorem is a solution of the
product version of Saitoh’s conjecture, which can be referred to [13].

######  Remark 1.12.

Assume that the four statements in Theorem 1.11 hold, then we know
$\frac{c_{0}\wedge_{1\leq j\leq
n}g_{j}(P_{j})_{*}\left(f_{u_{j}}\left(\Pi_{1\leq k\leq
m_{j}}f_{z_{j,k}}^{\gamma_{j,k}+1}\right)\left(\sum_{1\leq k\leq
m_{j}}p_{j,k}\frac{df_{z_{j,k}}}{f_{z_{j,k}}}\right)\right)}{dw_{1}\wedge
dw_{2}\wedge...\wedge dw_{n}}$ is a (single-valued) holomorphic function on
$D_{j}$, and we denote it by $F_{0}$. Then
$(F_{0}-f_{0},z_{\beta})\in\mathcal{I}(2\psi)_{z_{\beta}}$ for any $\beta\in
I_{1}$, and there exists $\tilde{F}_{0}\in H^{2}_{\rho}(M,\partial M)$ such
that $\tilde{F}_{0}^{*}=F_{0}$,

$M(Z_{0},J,\tilde{\rho})=\int_{M}|F_{0}|^{2}\tilde{\rho}\,\,\,\,and\,\,\,\,M_{H}(Z_{0},J,\rho)=\|\tilde{F}_{0}\|^{2}_{\partial
M,\rho}.$

We prove the remark in Section 4.

Denote that
$E_{\beta}:=\left\\{(\alpha_{1},\alpha_{2},...,\alpha_{n}):\sum_{1\leq j\leq
n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}=1\,\&\,\alpha_{j}\in\mathbb{Z}_{\geq
0}\right\\}$ for any $\beta\in I_{1}$, and assume that $f_{0}=\sum_{\alpha\in
E_{\beta}}d_{\beta,\alpha}\prod_{1\leq j\leq
n}(w_{j}-z_{j,\beta_{j}})^{\alpha_{j}}$ on $V_{\beta}$. Denote that

$c_{j,k}:=\exp\lim_{z\rightarrow z_{j,k}}\left(\frac{\sum_{1\leq k_{1}\leq
m_{j}}p_{j,k_{1}}G_{D_{j}}(z,z_{j,k_{1}})}{p_{j,k}}-\log|w_{j,k}(z)|\right)$

for any $j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$.

Using Theorem 1.11, we obtain the following optimal $L^{2}$ extension theorem
for the Hardy space on product spaces, and give a characterization for the
holding of the equality in this extension theorem.

######  Corollary 1.13.

Assume that $\sum_{\beta\in I_{1}}\sum_{\alpha\in
E_{\beta}}\frac{|d_{\beta,\alpha}|^{2}2^{n}\pi^{n-1}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)c_{j,\beta_{j}}^{2\alpha_{j}+2}}\in(0,+\infty)$. Then
there exists $f\in H^{2}_{\rho}(M,\partial M)$, satisfying that
$(f^{*}-f_{0},z_{\beta})\in\mathcal{I}(2\psi)_{z_{\beta}}$ for any $\beta\in
I_{1}$ and

$\|f\|_{\partial M,\rho}^{2}\leq\sum_{\beta\in I_{1}}\sum_{\alpha\in
E_{\beta}}\frac{|d_{\beta,\alpha}|^{2}2^{n}\pi^{n-1}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)c_{j,\beta_{j}}^{2\alpha_{j}+2}}.$

Moreover, assume that $f_{0}=\prod_{1\leq j\leq
n}(w_{j}-z_{1,\beta_{1}})^{\tilde{\alpha}_{j}}$ on $V_{\beta^{*}}$, where
$\beta^{*}=(1,1,...,1)\in I_{1}$, then equality

$M_{H}(Z_{0},\mathcal{I}(2\psi),\rho)=\sum_{\beta\in I_{1}}\sum_{\alpha\in
E_{\beta}}\frac{|d_{\beta,\alpha}|^{2}2^{n}\pi^{n-1}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)c_{j,\beta_{j}}^{2\alpha_{j}+2}}$

holds if and only if the following statements hold:

$(1)$ $\varphi_{j}=2\log|g_{j}|+2u_{j}$ for any $j\in\\{1,2,...,n\\}$, where
$u_{j}$ is a harmonic function on $\mathcal{C}$ and $g_{j}$ is a holomorphic
function on $D_{j}$ satisfying $g_{j}(z_{j,k})\not=0$ for any
$k\in\\{1,2,...,m_{j}\\}$;

$(2)$ there exists a nonnegative integer $\gamma_{j,k}$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$, which satisfies that
$\Pi_{1\leq k\leq m_{j}}\chi_{j,z_{j,k}}^{\gamma_{j,k}+1}=\chi_{j,-u_{j}}$ and
$\sum_{1\leq j\leq n}\frac{\gamma_{j,\beta_{j}}+1}{p_{j,\beta_{j}}}=1$ for any
$\beta\in I_{1}$;

$(3)$ $f_{0}=c_{\beta}\Pi_{1\leq j\leq
n}(w_{j}-z_{j,\beta_{j}})^{\gamma_{j,\beta_{j}}}+g_{\beta}$ on $V_{\beta}$ for
any $\beta\in I_{1}$, where $c_{\beta}$ is a constant and $g_{\beta}$ is a
holomorphic function on $V_{\beta}$ such that
$(g_{\beta},z_{\beta})\in\mathcal{I}(\psi)_{z_{\beta}}$;

$(4)$ $\lim_{z\rightarrow z_{\beta}}\frac{c_{\beta}\Pi_{1\leq j\leq
n}(w_{j}-z_{j,\beta_{j}})^{\gamma_{j,\beta_{j}}}dw_{1}\wedge
dw_{2}\wedge...\wedge dw_{n}}{\wedge_{1\leq j\leq
n}g_{j}(P_{j})_{*}\left(f_{u_{j}}\left(\Pi_{1\leq k\leq
m_{j}}f_{z_{j,k}}^{\gamma_{j,k}+1}\right)\left(\sum_{1\leq k\leq
m_{j}}p_{j,k}\frac{df_{z_{j,k}}}{f_{z_{j,k}}}\right)\right)}=c_{0}$ for any
$\beta\in I_{1}$, where $c_{0}\in\mathbb{C}\backslash\\{0\\}$ is a constant
independent of $\beta$.

Denote that $L_{k}:=\\{\sum_{1\leq j\leq n}\alpha_{j}\leq
k:\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}_{\geq 0}^{n}\\}$.

######  Corollary 1.14.

Let $k$ be a nonnegative integer. Then there is a constant $C$ (depending on
$k$ and $Z_{0}$), such that for any $a_{\beta,\alpha}\in\mathbb{C}$, where
$\beta\in I_{1}$ and $\alpha\in L_{k}$, there exists $f\in
H^{2}_{\rho}(M,\partial M)$ such that
$\partial^{\alpha}f^{*}(z_{\beta})=a_{\beta,\alpha}$ for any $\beta\in I_{1}$
and $\alpha\in L_{k}$, and

$\|f\|_{\partial M,\rho}^{2}\leq C\sum_{\beta\in I_{1},\alpha\in
L_{k}}|a_{\beta,\alpha}|^{2},$

where $\partial^{\alpha}=\left(\frac{\partial}{\partial
w_{1}}\right)^{\alpha_{1}}\dots\left(\frac{\partial}{\partial
w_{n}}\right)^{\alpha_{n}}$.

###  1.3. Minimal $L^{2}$ integrals for the Hardy space $H^{2}_{\lambda}(M,S)$

Let $D_{j}$ be a planar regular region with finite boundary components which
are analytic Jordan curves for any $1\leq j\leq n$. Let $M=\prod_{1\leq j\leq
n}D_{j}$ be a bounded domain in $\mathbb{C}^{n}$. Denote that $S:=\prod_{1\leq
j\leq n}\partial D_{j}$.

Let $Z_{j}=\\{z_{j,1},z_{j,2},...,z_{j,m_{j}}\\}\subset D_{j}$ for any
$j\in\\{1,2,...,n\\}$, where $m_{j}$ is a positive integer. Denote that

$Z_{0}:=\prod_{1\leq j\leq n}Z_{j}\subset M.$

Let $\psi=\max_{1\leq j\leq n}\\{\sum_{1\leq k\leq
m_{j}}2G_{D_{j}}(\cdot,z_{j,k})\\}$. Let $V_{z_{j,k}}\Subset D_{j}$ be a
neighborhood of $z_{j,k}$ satisfying $V_{z_{j,k}}\cap
V_{z_{j,k^{\prime}}}=\emptyset$ for any $j$ and $k\not=k^{\prime}$. Denote
that $I_{1}:=\\{(\beta_{1},\beta_{2},...,\beta_{n}):1\leq\beta_{j}\leq m_{j}$
for any $j\in\\{1,2,...,n\\}\\}$, $V_{\beta}:=\prod_{1\leq j\leq
n}V_{z_{j,\beta_{j}}}$ and
$z_{\beta}:=(z_{1,\beta_{1}},z_{2,\beta_{2}},\ldots,z_{n,\beta_{n}})\in M$ for
any $\beta=(\beta_{1},\beta_{2},...,\beta_{n})\in I_{1}$.

Let $\varphi_{j}$ be a subharmonic function on $D_{j}$, which satisfies that
$\varphi_{j}$ is continuous at $z$ for any $z\in\partial D_{j}$. Denote that

$\varphi(w_{1},\ldots,w_{n}):=\sum_{1\leq j\leq n}\varphi_{j}(w_{j})$

on $M$. Let $f_{0}$ be a holomorphic function $\cup_{\beta\in
I_{1}}V_{\beta}$. Let $\rho$ be a Lebesgue measurable function on $\partial M$
such that

$\rho(w_{1},\ldots,w_{n})=\left(\sum_{1\leq k\leq m_{j}}2\frac{\partial
G_{D_{j}}(w_{j},z_{j,k})}{\partial v_{w_{j}}}\right)^{-1}\times\prod_{1\leq
l\leq n}e^{-\varphi_{l}(w_{l})}$

on $\partial D_{j}\times{M_{j}}$. Let $c$ be a positive function on
$[0,+\infty)$, which satisfies that $c(t)e^{-t}$ is decreasing on
$[0,+\infty)$, $\lim_{t\rightarrow 0+0}c(t)=c(0)=1$ and
$\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$. Let

$\lambda(w_{1},\ldots,w_{n})=\prod_{1\leq j\leq n}\left(\sum_{1\leq k\leq
m_{j}}2\frac{\partial G_{D_{j}}(w_{j},z_{j,k})}{\partial
v_{w_{j}}}\right)^{-1}e^{-\varphi_{j}(w_{j})}$

on $S=\prod_{1\leq j\leq n}\partial D_{j}$. Note that $\lambda$ is continuous
on $S$.

Let $h_{j}$ be a holomorphic function on a neighborhood of $Z_{j}$ for any
$1\leq j\leq n$ satisfying that there exists $k\in\\{1,\ldots,m_{j}\\}$ such
that $h_{j}(z_{j,k})\not=0$. Denote that $h_{0}=\prod_{1\leq j\leq n}h_{j}$.
Let us consider the following minimal integral. Let $J_{\beta}$ be the maximal
ideal of $\mathcal{O}_{z_{\beta}}$ for any $\beta\in I_{1}$. Denote that

$\begin{split}M_{S}(Z_{0},J,\lambda):=\inf\bigg{\\{}&\|f\|^{2}_{S,\lambda}:f\in
H_{\lambda}^{2}(M,S)\\\ &\text{ s.t. }f^{*}(z_{\beta})=h_{0}(z_{\beta})\text{
for any $\beta\in I_{1}$}\bigg{\\}}\end{split}$

and

$\begin{split}M_{H}(Z_{0},J,\rho):=\inf\bigg{\\{}&\|f\|^{2}_{\partial
M,\rho}:f\in H_{\rho}^{2}(M,\partial M)\\\ &\text{ s.t.
}f^{*}(z_{\beta})=h_{0}(z_{\beta})\text{ for any $\beta\in
I_{1}$}\bigg{\\}}.\end{split}$

We present a relation between $M_{S}(Z_{0},J,\lambda)$ and
$M_{H}(Z_{0},J,\rho)$.

######  Theorem 1.15.

Assume that $M_{H}(Z_{0},J,\rho)<+\infty$. Then

(1.3) $M_{S}(Z_{0},J,\lambda)\leq\frac{M_{H}(Z_{0},J,\rho)}{n\pi^{n-1}}$

holds, and equality holds if and only if the following three statements hold

$(1)$ $\varphi_{j}=2u_{j}$ for any $1\leq j\leq n$, where $u_{j}$ is a
harmonic function on $D_{j}$;

$(2)$ $\prod_{1\leq k\leq m_{j}}\chi_{j,z_{j,k}}=\chi_{j,-u_{j}}$ for any
$1\leq j\leq n$;

$(3)$ For any $j$, there exists a constant $c_{j}\not=0$ such that

$\lim_{z\rightarrow z_{j,k}}\frac{P_{j}^{*}\left(f_{u_{j}}\left(\prod_{1\leq
k\leq m_{j}}f_{z_{j,k}}\right)\left(\sum_{1\leq k\leq
m_{j}}\frac{df_{z_{j,k}}}{f_{z_{j,k}}}\right)\right)}{h_{j}dw_{j}}=c_{j}$

holds for any $1\leq k\leq m_{j}$.

##  2\. Preparations

In this section, we do some preparations.

###  2.1. Some results on Hardy space $H^{2}(D)$

Let $D$ be a planar regular region with finite boundary components which are
analytic Jordan curves (see [18], [22]). In this section, we recall some
propoerties related to Hardy space $H^{2}(D)$.

Let $H^{2}(D)$ (see [18]) denote the analytic Hardy class on $D$ defined as
the set of all analytic functions $f(z)$ on $D$ such that the subharmonic
functions $|f(z)|^{2}$ have harmonic majorants $U(z)$:

$|f(z)|^{2}\leq U(z)\,\,\text{on}\,\,D.$

Then each function $f(z)\in H^{2}(D)$ has Fatou’s nontangential boundary value
a.e. on $\partial D$ belonging to $L^{2}(\partial D)$ (see [1]). It is well
know (see [17]) that if a subharmonic function has a harmonic majorant in $D$,
then there exists a least harmonic majorant. Denote the least harmonic
majorant of $|f|^{2}$ by $u_{f}$.

Let $z_{0}\in D$. Let $L^{2}(\partial D,\rho)$ be the space of complex valued
measurable function $h$ on $\partial D$, normed by

$\|h\|_{\partial D,\rho}^{2}=\frac{1}{2\pi}\int_{\partial D}|h|^{2}\rho|dz|,$

where $\rho=\frac{\partial G_{D}(z,z_{0})}{\partial v_{z}}$ is a positive
continuous function on $\partial D$ by the analyticity of $\partial D$,
$G_{D}(z,z_{0})$ is the Green function on $D$, and $\partial/\partial v_{z}$
denotes the derivative along the outer normal unit vector $v_{z}$.

The following lemma gives some properties related to the Hardy space
$H^{2}(D)$.

######  Lemma 2.1 ([17]).

$(a)$ If $f\in H^{2}(D)$, there is a function $f_{*}$ on $\partial D$ such
that $f$ has nontangential boundary value $f_{*}$ almost everywhere on
$\partial D$. The map $\gamma:f\mapsto f_{*}$ is an injective linear map from
$H^{2}(D)$ into $L^{2}(\partial D,\rho)$ and

$\|f_{*}\|_{\partial D,\rho}^{2}=u_{f}(z_{0})$

holds for any $f\in H^{2}(D)$, where $u_{f}$ is the least harmonic majorant of
$|f|^{2}$.

$(b)$ $g\in\gamma(H^{2}(D))$ if and only if

$\int_{\partial D}g(z)\phi(z)dz=0$

holds for any holomorphic function $\phi$ on a neighborhood of $\overline{D}$.

$(c)$ The inverse of $\gamma$ is given by

(2.1) $f(w)=\frac{1}{2\pi\sqrt{-1}}\int_{\partial D}\frac{f_{*}(z)}{z-w}dz$

for any $z\in D$.

Equality (2.1) in Lemma 2.1 implies the following lemma.

######  Lemma 2.2.

If $\lim_{n\rightarrow+\infty}\|\gamma(f_{n})\|_{\partial D,\rho}=0$ for
$f_{n}\in H^{2}(D)$, then $f_{n}$ uniformly converges to $0$ on any compact
subset of $D$.

######  Lemma 2.3.

For any compact set $V\subset D$ and nonnegative integer $k$, there is a
constant $C>0$, such that

$|f^{(k)}(w)|^{2}\leq C\int_{\partial D}|f_{*}|^{2}|dz|$

for any $w\in V$ and $f\in H^{2}(D)$.

###### Proof.

By equality (2.1), we have

$f^{(k)}(w)=\frac{(-1)^{k+1}k!}{2\pi\sqrt{-1}}\int_{\partial
D}\frac{f_{*}(z)}{(z-w)^{k+1}}dz$

for any $w\in D$ and $f\in H^{2}(D)$. Hence, there exists a constant $C>0$,
such that $|f^{(k)}(w)|^{2}\leq C\int_{\partial D}|f_{*}|^{2}|dz|$ for any
$w\in V$. ∎

######  Lemma 2.4 ([17]).

$H^{2}(D)$ is a Hilbert space equipped with the inner product

$\ll f,g\gg_{\partial D,\rho}=\frac{1}{2\pi}\int_{\partial
D}f_{*}\overline{g_{*}}\rho|dz|,$

where $\rho=\frac{\partial G_{D}(z,z_{0})}{\partial v_{z}}$. Moreover,
$\mathcal{O}(D)\cap C(\overline{D})$ is dense in $H^{2}(D)$.

######  Lemma 2.5.

Let $f_{n}\in H^{2}(D)$ for any $n\in\mathbb{Z}_{>0}$. Assume that $f_{n}$
uniformly converges to $0$ on any compact subset of $D$ and there exists $f\in
L^{2}(\partial D,\rho)$ such that
$\lim_{n\rightarrow+\infty}\|\gamma(f_{n})-f\|_{\partial D,\rho}=0$. Then we
have $f=0$.

###### Proof.

It follows from Lemma 2.1 and Lemma 2.4 that there exists $f_{0}\in H^{2}(D)$
such that $\gamma(f_{0})=f$. Using Lemma 2.2, we get that $f_{n}-f_{0}$
uniformly converges to $0$ on any compact subset of $D$, i.e. $f_{0}=0$, which
implies that $f=0$. ∎

Let $\\{D_{k}\\}_{k\in\mathbb{Z}_{>0}}$ be an increasing sequence of domains
with analytic boundaries, such that $z_{0}\in D_{1}$ and
$\cup_{k=1}^{+\infty}D_{k}=D$. Let $G_{D_{k}}(\cdot,z_{0})$ be the Green
function of $D_{k}$.

######  Lemma 2.6 (see [17]).

$\|f_{*}\|^{2}_{\partial
D,\rho}=\lim_{k\rightarrow+\infty}\frac{1}{2\pi}\int_{\partial
D_{k}}|f|^{2}\frac{\partial G_{D_{k}}(z,z_{0})}{\partial v_{z}}|dz|$ holds for
any $f\in H^{2}(D)$.

We recall a well-known property for the Green function $G_{D}(\cdot,z_{j})$ on
$D$.

######  Lemma 2.7 (see [10]).

Let $Z_{0}^{\prime}:=\\{z_{j}:j\in\mathbb{Z}_{\geq 1}\,\&\,j<\gamma\\}$ be a
discrete subset of $D$, where $\gamma\in\mathbb{Z}_{\geq 1}\cup\\{+\infty\\}$.
Let $\psi$ be a negative subharmonic function on $D$ such that
$\frac{1}{2}v(dd^{c}\psi,z_{j})\geq p_{j}>0$ for any $j$, where $p_{j}$ is a
constant. Then $2\sum_{1\leq j<\gamma}p_{j}G_{D}(\cdot,z_{j})$ is a
subharmonic function on $D$ satisfying that $2\sum_{1\leq
j<\gamma}p_{j}G_{D}(\cdot,z_{j})\geq\psi$ and $2\sum_{1\leq
j<\gamma}p_{j}G_{D}(\cdot,z_{j})$ is harmonic on $D\backslash Z_{0}^{\prime}$.

We recall the following basic formula.

######  Lemma 2.8 (see [13]).

$\frac{\partial\psi}{\partial v_{z}}=\left(\left(\frac{\partial\psi}{\partial
x}\right)^{2}+\left(\frac{\partial\psi}{\partial
y}\right)^{2}\right)^{\frac{1}{2}}$ on $\partial D$, where $\partial/\partial
v_{z}$ denotes the derivative along the outer normal unit vector $v_{z}$.

Let $\psi=\sum_{1\leq j\leq m}p_{j}G_{D}(\cdot,z_{j})$, where $p_{j}>0$ and
$\\{z_{j}\\}\subset D$ satisfying $z_{j}\not=z_{k}$ for $j\not=k$. Then there
exist a neighborhood $U$ of $\partial D$ and $r_{0}\in(0,1)$ such that
$\\{z\in D:\psi(z)\geq\log r_{0}\\}\Subset U$ and $dG_{D}(\cdot,z_{j})\not=0$
on $U\cap\overline{D}$ for any $j$.

The following lemma will be used in the proof of Theorem 1.5.

######  Lemma 2.9.

Let $\varphi$ be a positive Lebesgue measurable function on
$U\cap\overline{D}$ satisfying that
$\lim_{z\rightarrow\tilde{z}}\varphi(z)=\varphi(\tilde{z})$ for any
$\tilde{z}\in\partial D$. Then

(2.2) $\int_{\partial D}|f|^{2}\varphi|dz|=\lim_{r\rightarrow
1-0}\int_{\partial D_{r}}|f|^{2}\varphi|dz|$

holds for any $f\in H^{2}(D)$, where $D_{r}=\\{z\in\overline{D}:\psi(z)<\log
r\\}$ for $r\in[r_{0},1]$.

###### Proof.

Following from Lemma 2.7, we know that $\psi-\log r=\sum_{1\leq j\leq
m}p_{j}G_{D_{r}}(\cdot,z_{j})$ on $D_{r}$. Thus, using Lemma 2.6, we get that

(2.3) $\begin{split}\lim_{r\rightarrow 1-0}\int_{\partial
D_{r}}|\tilde{f}|^{2}\frac{\partial\psi}{\partial v_{z}}|dz|&=\sum_{1\leq
j\leq m}p_{j}\lim_{r\rightarrow 1-0}\int_{\partial
D_{r}}|\tilde{f}|^{2}\frac{\partial G_{D_{r}}(z,z_{j})}{\partial v_{z}}|dz|\\\
&=\sum_{1\leq j\leq m}p_{j}\int_{\partial D}|\tilde{f}|^{2}\frac{\partial
G_{D}(z,z_{j})}{\partial v_{z}}|dz|\\\ &=\int_{\partial
D}|\tilde{f}|^{2}\frac{\partial\psi}{\partial v_{z}}|dz|\end{split}$

holds for any $\tilde{f}\in H^{2}(D)$. As
$\lim_{z\rightarrow\tilde{z}}\varphi(z)=\varphi(\tilde{z})$ for any
$\tilde{z}\in\partial D_{1}$ and $dG_{D}(\cdot,z_{j})\not=0$ on
$U\cap\overline{D}$ for any $j$ , there exists a positive number $L_{1}$ such
that

$\frac{1}{L_{1}}<\inf_{\\{z\in\overline{D}:\psi(z)\geq\log
r_{0}\\}}\min\\{|\bigtriangledown\psi|,\varphi\\}\leq\sup_{\\{z\in\overline{D}:\psi(z)\geq\log
r_{0}\\}}\max\\{|\bigtriangledown\psi|,\varphi\\}<L_{1},$

where $|\bigtriangledown\cdot|^{2}=\left(\frac{\partial\cdot}{\partial
x}\right)^{2}+\left(\frac{\partial\cdot}{\partial y}\right)^{2}$.

By Lemma 2.4, there exists
$\\{f_{n}\\}_{n\in\mathbb{Z}_{>0}}\subset\mathcal{O}(D)\cap C(\overline{D})$
such that

(2.4) $\lim_{n\rightarrow+\infty}\int_{\partial D}|f_{n}-f|^{2}\varphi|dz|=0.$

It follows from equality (2.3) and Lemma 2.8 that

(2.5) $\begin{split}\limsup_{r\rightarrow 1-0}\int_{\partial
D_{r}}|f_{n}-f|^{2}\varphi|dz|&\leq L_{1}^{2}\limsup_{r\rightarrow
1-0}\int_{\partial D_{r}}|f_{n}-f|^{2}\frac{\partial\psi}{\partial
v_{z}}|dz|\\\ &\leq L_{1}^{2}\int_{\partial
D}|f_{n}-f|^{2}\frac{\partial\psi}{\partial v_{z}}|dz|\\\ &\leq
L_{1}^{4}\int_{\partial D}|f_{n}-f|^{2}\varphi|dz|.\end{split}$

Using the dominated convergence theorem, we know that

(2.6) $\lim_{r\rightarrow 1-0}\int_{\partial
D_{r}}|f_{n}|^{2}\varphi|dz|=\int_{\partial D}|f_{n}|^{2}\varphi|dz|$

holds for any $n\in\mathbb{Z}_{>0}$. Following from equality (2.4), inequality
(2.5) and equality (2.6), we have

$\begin{split}&\limsup_{r\rightarrow 1-0}\left(\int_{\partial
D_{r}}|f|^{2}\varphi|dz|\right)^{\frac{1}{2}}\\\
\leq&\liminf_{n\rightarrow+\infty}\left(\limsup_{r\rightarrow
1-0}\left(\int_{\partial
D_{r}}|f_{n}|^{2}\varphi|dz|\right)^{\frac{1}{2}}+\limsup_{r\rightarrow
1-0}\left(\int_{\partial
D_{r}}|f_{n}-f|^{2}\varphi|dz|\right)^{\frac{1}{2}}\right)\\\
\leq&\liminf_{n\rightarrow+\infty}\left(\left(\int_{\partial
D}|f_{n}|^{2}\varphi|dz|\right)^{\frac{1}{2}}+L_{1}^{2}\left(\int_{\partial
D}|f_{n}-f|^{2}\varphi|dz|\right)^{\frac{1}{2}}\right)\\\
=&\left(\int_{\partial D}|f|^{2}\varphi|dz|\right)^{\frac{1}{2}}.\end{split}$

By Fatou’s Lemma, we have

$\liminf_{r\rightarrow 1-0}\left(\int_{\partial
D_{r}}|f|^{2}\varphi|dz|\right)^{\frac{1}{2}}\geq\left(\int_{\partial
D}|f|^{2}\varphi|dz|\right)^{\frac{1}{2}}.$

Thus, equality (2.2) holds. ∎

Let $Z_{0}:=\\{z_{j}:1\leq j\leq m\\}$ be a subset of $D$. Let $\rho$ be a
positive continuous function on $\partial D$. Let $\mathfrak{a}=(a_{j,l})$
$(1\leq j\leq m,0\leq l\leq k_{j})$, where $a_{j,l}\in\mathbb{C}$ such that
$\sum_{1\leq j\leq m}\sum_{0\leq l\leq k_{j}}|a_{j,l}|\not=0$. Denote that

$\begin{split}M(Z_{0},\mathfrak{a},\tilde{\rho}):=\inf\bigg{\\{}&\int_{D}|f|^{2}\tilde{\rho}:f\in\mathcal{O}(D)\\\
&\text{ s.t. }f^{(l)}(z_{j})=l!a_{j,l}\text{ for any $0\leq l\leq k_{j}$ and
any $1\leq j\leq m$}\bigg{\\}}.\end{split}$

and

$\begin{split}M_{H}(Z_{0},\mathfrak{a},\rho):=\inf\bigg{\\{}\frac{1}{2\pi}&\int_{\partial
D}|f|^{2}\rho|dz|:f\in H^{2}(D)\\\ &\text{ s.t.
}f^{(l)}(z_{j})=l!a_{j,l}\text{ for any $0\leq l\leq k_{j}$ and any $1\leq
j\leq m$}\bigg{\\}}.\end{split}$

######  Lemma 2.10.

If $M_{H}(Z_{0},\mathfrak{a},\rho)<+\infty$, then there exists a unique $f\in
H^{2}(D)$ such that
$M_{H}(Z_{0},\mathfrak{a},\rho)=\frac{1}{2\pi}\int_{\partial
D}|f|^{2}\rho|dz|$, and $f^{(l)}(z_{j})=l!a_{j,l}$ for any $0\leq l\leq k_{j}$
and any $1\leq j\leq m$.

###### Proof.

Firstly, we prove the existence of $f$. As
$M_{H}(Z_{0},\mathfrak{a},\rho)<+\infty,$ then there is
$\\{f_{s}\\}_{s\in\mathbb{Z}_{>0}}\subset H^{2}(D)$ such that

$\lim_{s\rightarrow+\infty}\frac{1}{2\pi}\int_{\partial
D}|f_{s}|^{2}\rho|dz|=M_{H}(Z_{0},\mathfrak{a},\rho),$

and $f_{s}^{(l)}(z_{j})=l!a_{j,l}$ for any $0\leq l\leq k_{j}$ and any $1\leq
j\leq m$. Thus, there exists a subsequence of
$\\{f_{s}\\}_{s\in\mathbb{Z}_{>0}}$ (denoted also by
$\\{f_{s}\\}_{s\in\mathbb{Z}_{>0}}$), which satisfies that
$\\{f_{s}\\}_{s\in\mathbb{Z}_{>0}}$ weakly converges to a function $g\in
L^{2}(\partial D,\rho)$ in the Hilbert space $L^{2}(\partial D,\rho)$ and
$\\{f_{s}\\}_{s\in\mathbb{Z}_{>0}}$ uniformly converges to a function
$f\in\mathcal{O}(D)$ on any compact subset of $D$. Then we have

(2.7) $\frac{1}{2\pi}\int_{\partial
D}|g|^{2}\rho|dz|\leq\lim_{s\rightarrow+\infty}\frac{1}{2\pi}\int_{\partial
D}|f_{s}|^{2}\rho|dz|=M_{H}(Z_{0},\mathfrak{a},\rho).$

By Lemma 2.1, we have

$\begin{split}\int_{\partial D}g(z)\phi(z)dz&=\int_{\partial
D}g(z)\left(\phi(z)\frac{dz}{|dz|\rho(z)}\right)\rho(z)|dz|\\\
&=\lim_{s\rightarrow+\infty}\int_{\partial
D}f_{s}(z)\left(\phi(z)\frac{dz}{|dz|\rho(z)}\right)\rho(z)|dz|\\\
&=\lim_{s\rightarrow+\infty}\int_{\partial D}f_{s}(z)\phi(z)dz\\\
&=0\end{split}$

for any holomorphic function $\phi$ on a neighborhood of $\overline{D}$, and

$\begin{split}f(w)&=\lim_{s\rightarrow+\infty}f_{s}(w)\\\
&=\lim_{s\rightarrow+\infty}\frac{1}{2\pi\sqrt{-1}}\int_{\partial
D}\frac{f_{s}(z)}{z-w}dz\\\ &=\lim_{s\rightarrow+\infty}\int_{\partial
D}f_{s}(z)\left(\frac{dz}{|dz|(z-w)\rho(z)}\right)\rho(z)|dz|\\\
&=\frac{1}{2\pi\sqrt{-1}}\int_{\partial D}\frac{g(z)}{z-w}dz.\end{split}$

Thus, it follows from Lemma 2.1 that $f\in H^{2}(D)$ and $\gamma(f)=g$. By
inequality (2.7) and the definition of $M_{H}(Z_{0},\mathfrak{a},\rho)$, we
get

$\frac{1}{2\pi}\int_{\partial
D}|f|^{2}\rho|dz|=M_{H}(Z_{0},\mathfrak{a},\rho).$

Thus, we obtain the existence of $f$.

Secondly, we prove the uniqueness of $f$ by contradiction: if not, there exist
two different $g_{1}\in H^{2}(D)$ and $g_{2}\in H^{2}(D)$ satisfying that
$\frac{1}{2\pi}\int_{\partial
D}|g_{s}|^{2}\rho|dz|=M_{H}(Z_{0},\mathfrak{a},\rho)$, and
$g_{s}^{(l)}(z_{j})=l!a_{j,l}$ for any $0\leq l\leq k_{j}$ and any $1\leq
j\leq m$, where $s=1,2$. Note that

$\begin{split}\frac{1}{2\pi}\int_{\partial
D}|\frac{g_{1}+g_{2}}{2}|^{2}\rho|dz|+\frac{1}{2\pi}\int_{\partial
D}|\frac{g_{1}-g_{2}}{2}|^{2}\rho|dz|=M_{H}(Z_{0},\mathfrak{a},\rho),\end{split}$

hence we obtain that

$\frac{1}{2\pi}\int_{\partial
D}|\frac{g_{1}+g_{2}}{2}|^{2}\rho|dz|<M_{H}(Z_{0},\mathfrak{a},\rho),$

which contradicts the definition of $M_{H}(Z_{0},\mathfrak{a},\rho)$.

Thus, Lemma 2.10 has been proved. ∎

In the following, let $\psi$ be as in Theorem 1.5. The following lemma will be
used in the proof of Lemma 2.13.

######  Lemma 2.11 ([12]).

Let $f$ be a holomorphic function on $D$. Assume that

$\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in D:\psi(z)\geq\log
r\\}}|f(z)|^{2}}{1-r}<+\infty,$

then we have $f\in H^{2}(D)$.

We recall the following coarea formula.

######  Lemma 2.12 (see [2]).

Suppose that $\Omega$ is an open set in $\mathbb{R}^{n}$ and $u\in
C^{1}(\Omega)$. Then for any $g\in L^{1}(\Omega)$,

$\int_{\Omega}g(x)|\bigtriangledown
u(x)|dx=\int_{\mathbb{R}}\left(\int_{u^{-1}(t)}g(x)dH_{n-1}(x)\right)dt,$

where $H_{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.

Let $\tilde{\rho}$ be a Lebesgue measurable function on $\overline{D}$, which
satisfies that $\inf_{\overline{D}}\tilde{\rho}>0$ and
$\tilde{\rho}(z)\leq\liminf_{w\rightarrow z}\tilde{\rho}(w)$ for any
$z\in\partial D$. Denote that

$\rho=\left(\frac{\partial\psi}{\partial v_{z}}\right)^{-1}\tilde{\rho}$

on $\partial D$. In the following, we give a sufficient condition for $f\in
H^{2}(D)$ .

######  Lemma 2.13.

Let $f$ be a holomorphic function on $D$. Assume that

$\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in D:\psi(z)\geq\log
r\\}}|f(z)|^{2}\tilde{\rho}}{1-r}<+\infty,$

then we have $f\in H^{2}(D)$ and

(2.8) $\int_{\partial D}|f|^{2}\rho|dz|\leq\liminf_{r\rightarrow
1-0}\frac{\int_{\\{z\in D:\psi(z)\geq\log r\\}}|f(z)|^{2}\tilde{\rho}}{1-r}.$

###### Proof.

Note that $\inf_{\overline{D}}\tilde{\rho}>0$, then Lemma 2.11 tells us that
$f\in H^{2}(D)$. Thus, it suffices to prove inequality (2.8).

Note that $f$ has Fatou’s nontangential boundary value on $\partial D$. By
Lemma 2.8, we have

$\int_{\partial D}|f|^{2}\rho|dz|=\int_{\partial
D}|f|^{2}\left(\frac{\partial\psi}{\partial
v_{z}}\right)^{-1}\tilde{\rho}|dz|=\int_{\partial
D}|f|^{2}\tilde{\rho}\left|\bigtriangledown\psi\right|^{-1}|dz|.$

As $\frac{\partial\psi}{\partial v_{z}}>0$ on $\partial D$, $\psi=0$ on
$\partial D$ and $\tilde{\rho}(z)\leq\liminf_{w\rightarrow z}\tilde{\rho}(w)$
for any $z\in\partial D$, it follows from Fatou’s Lemma and Lemma 2.12 that

$\begin{split}&\int_{\partial
D}|f|^{2}\tilde{\rho}\left|\bigtriangledown\psi\right|^{-1}|dz|\\\
\leq&\liminf_{r\rightarrow 1-0}\frac{\int_{\log r}^{0}\left(\int_{\\{z\in
D:\psi(z)=t\\}}|f|^{2}\rho\left|\bigtriangledown\psi\right|^{-1}|dz|\right)dt}{-\log
r}\\\ =&\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in D:\psi(z)\geq\log
r\\}}|f|^{2}\tilde{\rho}}{1-r}\times\frac{1-r}{-\log r}\\\
=&\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in D:\psi(z)\geq\log
r\\}}|f|^{2}\tilde{\rho}}{1-r}.\end{split}$

Thus, inequality (2.8) holds. ∎

###  2.2. The Hardy space over $\partial M$

Let $D_{j}$ be a planar regular region with finite boundary components which
are analytic Jordan curves for any $1\leq j\leq n$. Let

$M=\prod_{1\leq j\leq n}D_{j}$

be a bounded domain in $\mathbb{C}^{n}$. In this section, we recall and give
some properties on the Hardy space over $\partial M$, which will be used in
the proofs of the main theorems.

####  2.2.1. Some results on $H^{2}_{\rho}(M,\partial D_{j}\times M_{j})$

Let $M_{j}=\prod_{1\leq l\leq n,l\not=j}D_{l}$, then $M=D_{j}\times M_{j}$.
Let $z_{j}\in D_{j}$ for any $1\leq j\leq n$. Recall that $H^{2}(D_{j})$
denotes the Hardy space on $D_{j}$ and there exists a norm-preserving linear
map $\gamma_{j}:H^{2}(D_{j})\rightarrow L^{2}(\partial D_{j},\frac{\partial
G_{D_{j}}(z,z_{j})}{\partial v_{z}})$ (see Section 2.1) satisfying that
$\gamma_{j}(f)$ denotes the nontangential boundary value of $f$ a.e. on
$\partial D_{j}$ for any $f\in H^{2}(D_{j})$, where $G_{D_{j}}(\cdot,z_{j})$
is the Green function on $D_{j}$.

Let $d\mu_{j}$ be the Lebesgue measure on $M_{j}$ for any $1\leq j\leq n$, and
let $d\mu$ be a measure on $\partial M$ defined by

$\int_{\partial M}hd\mu=\sum_{1\leq j\leq
n}\frac{1}{2\pi}\int_{M_{j}}\int_{\partial
D_{j}}h(w_{j},\hat{w}_{j})|dw_{j}|d\mu_{j}(\hat{w}_{j})$

for any $h\in L^{1}(\partial M)$, where
$\hat{w}_{j}:=(w_{1},\ldots,w_{j-1},w_{j+1},\ldots,w_{n})$. For simplicity,
denote $d\mu|_{\partial D_{j}\times M_{j}}$ by $d\mu$. Let us consider a space
over $\partial D_{j}\times M_{j}$. Denote

$\begin{split}\\{f\in L^{2}(\partial D_{j}\times M_{j},d\mu):\exists
f^{*}\in\mathcal{O}(M)\text{, s.t. }&f^{*}(\cdot,\hat{w}_{j})\in
H^{2}(D_{j})\text{ for any $\hat{w}_{j}\in M_{j}$}\\\
&\,\&\,f=\gamma_{j}(f^{*})\text{ a.e. on }\partial D_{j}\times
M_{j}\\}\end{split}$

by $H^{2}(M,\partial D_{j}\times M_{j})$. In [13], we proved that there exists
a unique linear injective map $P_{\partial M,j}$ from $H^{2}(M,\partial
D_{j}\times M_{j})$ to $\mathcal{O}(M)$ such that $P_{\partial M,j}(f)$
(denoted by $f^{*}$ for simplicity) satisfies the following conditions for any
$f\in H^{2}(M,\partial D_{j}\times M_{j})$:

$(1)$ $P_{\partial M,j}(f)(\cdot,\hat{w}_{j})\in H^{2}(D_{j})$ for any
$\hat{w}_{j}\in M_{j}$;

$(2)$ $f=\gamma_{j}(P_{\partial M,j}(f))$ a.e. on $\partial D_{j}\times
M_{j}$.

Let $\rho$ be a Lebesgue measurable function on $\partial M$ such that
$\inf_{\partial M}\rho>0$. Denote that

$\ll f,g\gg_{\partial D_{j}\times
M_{j},\rho}:=\frac{1}{2\pi}\int_{M_{j}}\int_{\partial
D_{j}}f(w_{j},\hat{w}_{j})\overline{g(w_{j},\hat{w}_{j})}\rho|dw_{j}|d\mu_{j}(\hat{w}_{j})$

for any $f,g\in L^{2}(\partial D_{j}\times M_{j},\rho d\mu)\subset
L^{2}(\partial D_{j}\times M_{j},d\mu)$. Denote that

$H^{2}_{\rho}(M,\partial D_{j}\times M_{j}):=\\{f\in H^{2}(M,\partial
D_{j}\times M_{j}):\|f\|_{\partial D_{j}\times M_{j},\rho}<+\infty\\}.$

$H^{2}_{\rho}(M,\partial D_{j}\times M_{j})$ is a Hilbert space equipped with
the inner product $\ll\cdot,\cdot\gg_{\partial D_{j}\times M_{j},\rho}$ (see
[13]).

We recall the following lemma.

######  Lemma 2.14 ([13]).

For any compact subset $K$ of $M$, there exists a positive constant $C_{K}$
such that

$|f^{*}(z)|\leq C_{K}\|f\|_{\partial D_{j}\times M_{j},\rho}$

holds for any $z\in K$ and $f\in H_{\rho}^{2}(M,\partial D_{j}\times M_{j})$.

In the following, assume that $\rho|_{\partial D_{1}\times
M_{1}}=\rho_{1}\times\lambda_{1}$, where $\rho_{1}$ is a positive Lebesgue
measurable function on $\partial D_{1}$, and $\lambda_{1}$ is a positive
Lebesgue measurable function on $M_{1}$ such that
$A^{2}(M_{1},\lambda_{1}):=\\{f\in\mathcal{O}(M_{1}):\int_{M_{1}}|f|^{2}\lambda_{1}<+\infty\\}$
is a Hilbert space with the inner $\ll
f,g\gg_{M_{1},\lambda_{1}}:=\int_{M_{1}}f\overline{g}\lambda_{1}$, i.e.,
$\lambda_{1}$ is an admissible weight on $M_{1}$ (see Section 2.4).

######  Lemma 2.15 ([13]).

Assume that $H^{2}_{\rho}(M,\partial D_{1}\times M_{1})\not=\\{0\\}$. Then we
have $H^{2}_{\rho_{1}}(D_{1},\partial D_{1})\not=\\{0\\}$ and
$A^{2}(M_{1},\lambda_{1})\not=\\{0\\}$. Furthermore,
$\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$ is a complete
orthonormal basis for $H^{2}_{\rho}(M,\partial D_{1}\times M_{1})$, where
$\\{e_{m}\\}_{m\in\mathbb{Z}_{>0}}$ is a complete orthonormal basis for
$H^{2}_{\rho_{1}}(D_{1},\partial D_{1})$, and
$\\{\tilde{e}_{m}\\}_{m\in\mathbb{Z}_{>0}}$ is a complete orthonormal basis
for $A^{2}(M_{1},\lambda_{1})$.

Let $p_{j,k}$ be positive real number for any $1\leq j\leq n$ and $1\leq k\leq
m_{j}$. Let $\psi_{1}=\sum_{1\leq k\leq m_{1}}p_{1,k}G_{D_{1}}(\cdot,z_{1,k})$
on $D_{1}$, and let $\hat{\psi}_{1}=\max_{2\leq j\leq n}\\{\sum_{1\leq k\leq
m_{j}}p_{j,k}G_{D_{j}}(\cdot,z_{j,k})\\}$ on $M_{1}$.

######  Lemma 2.16.

Assume that $\rho_{1}$ is a positive Lebesgue measurable function on
$U\cap\overline{D}_{1}$ satisfying that
$\lim_{z\rightarrow\tilde{z}}\rho_{1}(z)=\rho_{1}(\tilde{z})$ for any
$\tilde{z}\in\partial D_{1}$, where $U$ is a neighborhood of $\partial D_{1}$.
Then

(2.9) $\|f\|_{\partial D_{1}\times
M_{1},\rho}=\frac{1}{2\pi}\lim_{r\rightarrow 1-0}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|f^{*}(w_{1},\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})$

holds for any $f\in H_{\rho}^{2}(M,\partial D_{1}\times M_{1})$, where
$D_{1,r}=\\{z\in\overline{D}_{1}:\psi_{1}(z)<\log r\\}$ and $M_{1,r}=\\{z\in
M_{1}:\hat{\psi}_{1}(z)<\log r\\}$ for $r\in[0,1]$.

###### Proof.

Following from Lemma 2.7, we know that

(2.10) $\psi_{1}-\log r=\sum_{1\leq k\leq
m_{1}}p_{1,k}G_{D_{1,r}}(\cdot,z_{1,k})$

on $D_{1,r}$, where $G_{D_{1,r}}(\cdot,z_{1,k})$ is the Green function on
$D_{1,r}$. For any holomorphic function $h$ on $D_{1}$ and any $\tilde{z}\in
D_{1}$, as $|h|^{2}$ is subharmonic, we know that $\int_{\partial
D_{1,r}}|h(z)|^{2}\frac{G_{D_{1,r}}(z,\tilde{z})}{\partial v_{z}}|dz|$ is
increasing with respect to $r$. Combining Lemma 2.6, we have

$\lim_{r\rightarrow+\infty}\int_{\partial
D_{1,r}}|h(z)|^{2}\frac{G_{D_{1,r}}(z,\tilde{z})}{\partial
v_{z}}|dz|=\int_{\partial
D_{1}}|h(z)|^{2}\frac{G_{D_{1}}(z,\tilde{z})}{\partial v_{z}}|dz|$

for $h\in H^{2}(D_{1})$. Thus, it follows from Levi’s Theorem and equality
(2.10) that

(2.11) $\begin{split}&\lim_{r\rightarrow 1-0}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|\tilde{f}^{*}(w_{1},\hat{w}_{1})|^{2}\frac{\partial\psi_{1}}{\partial
v_{z}}|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\\\
=&\lim_{r\rightarrow 1-0}\sum_{1\leq k\leq
m_{1}}p_{1,k}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|\tilde{f}^{*}(w_{1},\hat{w}_{1})|^{2}\frac{\partial
G_{D_{1,r}}(w_{1},z_{1,k})}{\partial
v_{z}}|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\\\ =&\sum_{1\leq
k\leq m_{1}}p_{1,k}\int_{M_{1}}\int_{\partial
D_{1}}|\tilde{f}(w_{1},\hat{w}_{1})|^{2}\frac{\partial
G_{D_{1}}(w_{1},z_{1,k})}{\partial
v_{z}}|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\\\
=&\int_{M_{1}}\int_{\partial
D_{1}}|\tilde{f}(w_{1},\hat{w}_{1})|^{2}\frac{\partial\psi_{1}}{\partial
v_{z}}\lambda_{1}(\hat{w}_{1})|dw_{1}|d\mu_{1}(\hat{w}_{1})\end{split}$

holds for any $\tilde{f}\in H_{\frac{\partial\psi_{1}}{\partial
v_{z}}\lambda_{1}}^{2}(M,\partial D_{1}\times M_{1})$. There exist positive
numbers $L_{1}$ and $r_{0}\in[0,1]$ such that

$\frac{1}{L_{1}}<\inf_{\\{z\in\overline{D}_{1}:\psi_{1}(z)\geq\log
r_{0}\\}}\min\\{|\bigtriangledown\psi_{1}|,\rho_{1}\\}\leq\sup_{\\{z\in\overline{D}_{1}:\psi_{1}(z)\geq\log
r_{0}\\}}\max\\{|\bigtriangledown\psi_{1}|,\rho_{1}\\}<L_{1}.$

By Lemma 2.15, there exist $\\{f_{l}\\}_{l\in\mathbb{Z}_{>0}}\subset
H_{\rho_{1}}^{2}(D_{1},\partial D_{1})$ and
$\\{g_{l}\\}_{l\in\mathbb{Z}_{>0}}\subset A^{2}(M_{1},\lambda_{1})$ such that

(2.12) $f=\sum_{l=1}^{+\infty}f_{l}g_{l}.$

Denote that $F_{m}:=\sum_{l=m+1}^{+\infty}f_{l}g_{l}\in
H^{2}_{\rho}(M,\partial D_{1}\times M_{1})$. It follows from equality (2.11)
and Lemma 2.8 that

(2.13) $\begin{split}&\limsup_{r\rightarrow 1-0}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|F^{*}_{m}(w_{1},\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\\\
\leq&L_{1}^{2}\limsup_{r\rightarrow 1-0}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|F^{*}_{m}(w_{1},\hat{w}_{1})|^{2}\frac{\partial\psi_{1}}{\partial
v_{z}}|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\\\
\leq&L_{1}^{2}\int_{M_{1}}\int_{\partial
D_{1}}|F_{m}(w_{1},\hat{w}_{1})|^{2}\frac{\partial\psi_{1}}{\partial
v_{z}}\lambda_{1}(\hat{w}_{1})|dw_{1}|d\mu_{1}(\hat{w}_{1})\\\
\leq&L_{1}^{4}\int_{M_{1}}\int_{\partial
D_{1}}|F_{m}(w_{1},\hat{w}_{1})|^{2}\rho|dw_{1}|d\mu_{1}(\hat{w}_{1}).\end{split}$

Using Lemma 2.9, we have

(2.14) $\begin{split}&\limsup_{r\rightarrow
1-0}\left(\int_{M_{1,r}}\int_{\partial D_{1,r}}|\sum_{1\leq l\leq
m}f_{l}(w_{1})g_{l}(\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\right)^{\frac{1}{2}}\\\
\leq&\limsup_{r\rightarrow 1-0}\sum_{1\leq l\leq
m}\left(\int_{M_{1,r}}\int_{\partial
D_{1,r}}|f_{l}(w_{1})g_{l}(\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\right)^{\frac{1}{2}}\\\
=&\limsup_{r\rightarrow 1-0}\sum_{1\leq l\leq
m}\left(\int_{M_{1,r}}|g_{l}(\hat{w}_{1})|^{2}\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\int_{\partial
D_{1,r}}|f_{l}(w_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\right)^{\frac{1}{2}}\\\
=&\sum_{1\leq l\leq
m}\left(\int_{M_{1}}|g_{l}(\hat{w}_{1})|^{2}\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\int_{\partial
D_{1}}|f_{l}(w_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\right)^{\frac{1}{2}}\\\
=&\left(\int_{M_{1}}\int_{\partial D_{1}}|\sum_{1\leq l\leq
m}f_{l}(w_{1})g_{l}(\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\right)^{\frac{1}{2}}\end{split}$

for any $m\in\mathbb{Z}_{>0}$. Following from equality (2.12), inequality
(2.13) and (2.14), we have

$\begin{split}&\limsup_{r\rightarrow 1-0}\left(\int_{M_{1,r}}\int_{\partial
D_{1,r}}|f^{*}(w_{1},\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\right)^{\frac{1}{2}}\\\
\leq&\liminf_{m\rightarrow+\infty}\Bigg{(}\limsup_{r\rightarrow
1-0}\Big{(}\int_{M_{1,r}}\int_{\partial D_{1,r}}|\sum_{1\leq l\leq
m}f_{l}g_{l}|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\Big{)}^{\frac{1}{2}}\\\
&+\limsup_{r\rightarrow 1-0}\Big{(}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|F_{m}^{*}|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\Big{)}^{\frac{1}{2}}\Bigg{)}\\\
\leq&\liminf_{m\rightarrow+\infty}\Bigg{(}\Big{(}\int_{M_{1}}\int_{\partial
D_{1}}|\sum_{1\leq l\leq
m}f_{l}(w_{1})g_{l}(\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1})\Big{)}^{\frac{1}{2}}\\\
&+\Big{(}L_{1}^{4}\int_{M_{1}}\int_{\partial
D_{1}}|F_{m}(w_{1},\hat{w}_{1})|^{2}\rho|dw_{1}|d\mu_{1}(\hat{w}_{1})\Big{)}^{\frac{1}{2}}\Bigg{)}\\\
=&(2\pi)^{\frac{1}{2}}\|f\|_{\partial D_{1}\times M_{1},\rho}.\end{split}$

By Fatou’s Lemma, we have

$2\pi\|f\|^{2}_{\partial D_{1}\times M_{1},\rho}\leq\liminf_{r\rightarrow
1-0}\int_{M_{1,r}}\int_{\partial
D_{1,r}}|f^{*}(w_{1},\hat{w}_{1})|^{2}\rho_{1}(w_{1})|dw_{1}|\lambda_{1}(\hat{w}_{1})d\mu_{1}(\hat{w}_{1}).$

Thus, equality (2.9) holds. ∎

####  2.2.2. Some results on $H^{2}_{\rho}(M,\partial M)$

Let $\rho$ be a Lebesgue measurable function on $\partial M$ such that
$\inf_{\partial M}\rho>0$. Denote that

$\ll f,g\gg_{\partial M,\rho}:=\sum_{1\leq j\leq
n}\frac{1}{2\pi}\int_{M_{j}}\int_{\partial
D_{j}}f(w_{j},\hat{w}_{j})\overline{g(w_{j},\hat{w}_{j})}\rho|dw_{j}|d\mu_{j}(\hat{w}_{j})$

for any $f,g\in L^{2}(\partial M,\rho d\mu)\subset L^{2}(\partial M,d\mu)$.
The weighted Hardy space over $\partial M$ is defined as follows:

_For any $f\in L^{2}(\partial M,\rho d\mu)$, we call $f\in
H^{2}_{\rho}(M,\partial M)$ if $f\in H^{2}_{\rho}(M,\partial D_{j}\times
M_{j})$ for any $1\leq j\leq n$ and $P_{\partial M,j}(f)=P_{\partial M,k}(f)$
for any $j\not=k$._

Denote that $P_{\partial M}(f):=P_{\partial M,j}(f)$ for any $f\in
H^{2}_{\rho}(M,\partial M)$ (denote also by $f^{*}$ for simplicity), and
$P_{\partial M}$ is a linear injective map from $H^{2}_{\rho}(M,\partial M)$
to $\mathcal{O}(M)$. $H^{2}_{\rho}(M,\partial M)$ is a Hilbert space with the
inner product $\ll\cdot,\cdot\gg_{\partial M,\rho}$ (see [13]). Let $Z_{0}$ be
any subset of $M$, and let $J_{z}$ be an ideal of $\mathcal{O}_{z}$ for any
$z\in Z_{0}$. Let $f_{0}$ be a holomorphic function on a neighborhood of
$Z_{0}$. Denote that

$\begin{split}M_{H}(Z_{0},J,\rho):=\inf\bigg{\\{}&\|f\|^{2}_{\partial
M,\rho}:f\in H_{\rho}^{2}(M,\partial M)\\\ &\text{ s.t. }(f^{*}-f_{0},z)\in
J_{z}\text{ for any $z\in Z_{0}$}\bigg{\\}}.\end{split}$

The following Lemma will be used in the proof of Lemma 2.18.

######  Lemma 2.17 (see [4]).

Let $N$ be a submodule of $\mathcal{O}_{\mathbb{C}^{n},o}^{q}$, $1\leq
q<+\infty$, and let $f_{j}\in\mathcal{O}_{\mathbb{C}^{n}}(U)^{q}$ be a
sequence of $q-$tuples holomorphic in an open neighborhood $U$ of the origin
$o$. Assume that the $f_{j}$ converge uniformly in $U$ towards a $q-$tuple
$f\in\mathcal{O}_{\mathbb{C}^{n}}(U)^{q}$, assume furthermore that all germs
$(f_{j},o)$ belong to $N$. Then $(f,o)\in N$.

######  Lemma 2.18.

Assume that $M_{H}(Z_{0},J,\rho)<+\infty$. Then there is a unique $f\in
H^{2}_{\rho}(M,\partial M)$ satisfying that $(f^{*}-f_{0},z)\in J_{z}$ for any
$z\in Z_{0}$ and $M_{H}(Z_{0},J,\rho)=\|f\|_{\partial M,\rho}^{2}$.

###### Proof.

Firstly, we prove the existence of $f$. As $M_{H}(Z_{0},J,\rho)<+\infty,$
there is $\\{f_{j}\\}_{j\in\mathbb{Z}_{>0}}\subset H^{2}_{\rho}(M,\partial M)$
such that $\lim_{j\rightarrow+\infty}\|f_{j}\|^{2}_{\partial
M,\rho}=M_{H}(Z_{0},J,\rho)<+\infty$ and $(f^{*}_{j}-f_{0},z)\in J_{z}$ for
any $z\in Z_{0}$ and any $j$. Then there is a subsequence of
$\\{f_{j}\\}_{j\in\mathbb{Z}_{>0}}$ denoted also by
$\\{f_{j}\\}_{j\in\mathbb{Z}_{>0}}$, which weakly converges to an element
$f\in H_{\rho}^{2}(M,\partial M)$, i.e.,

(2.15) $\lim_{j\rightarrow+\infty}\ll f_{j},g\gg_{\partial M,\rho}=\ll
f,g\gg_{\partial M,\rho}$

holds for any $g\in H^{2}_{\rho}(M,\partial M)$. Hence we have

(2.16) $\|f\|_{\partial
M,\rho}^{2}\leq\lim_{j\rightarrow+\infty}\|f_{j}\|^{2}_{\partial
M,\rho}=M_{H}(Z_{0},J,\rho).$

It follows from Lemma 2.14 that there is a subsequence of
$\\{f_{j}\\}_{j\in\mathbb{Z}_{>0}}$ denoted also by
$\\{f_{j}\\}_{j\in\mathbb{Z}_{>0}}$, which satisfies that $f_{j}^{*}$
uniformly converges to a holomorphic function $g_{0}$ on $M$ on any compact
subset of $M$. Following from Lemma 2.14, for any $z\in M$, there exists
$g_{z}\in H^{2}_{\rho}(M,\partial M)$ such that

(2.17) $\ll g,g_{z}\gg_{\partial M,\rho}=g(z)$

holds for any $g\in H^{2}_{\rho}(M,\partial M)$. By equality (2.15) and
(2.17), we get that

$\lim_{j\rightarrow+\infty}f_{j}^{*}(z)=f^{*}(z)$

for any $z\in M$, hence we know that $f^{*}=g_{0}$ and $f_{j}^{*}$ uniformly
converges to $f^{*}$ on any compact subset of $M$. Following from Lemma 2.17
and $(f^{*}_{j}-f_{0},z)\in J_{z}$ for any $z\in Z_{0}$ and any $j$, we get

$(f^{*}-f_{0},z)\in J_{z}$

for any $z\in Z_{0}$. By definition of $M_{H}(Z_{0},J,\rho)$ and inequality
(2.16), we have

$\|f\|_{\partial M,\rho}^{2}=M_{H}(Z_{0},J,\rho).$

Thus, we obtain the existence of $f$.

Now, we prove the uniqueness of $f$ by contradiction: if not, there exist two
different $g_{1}\in H^{2}_{\rho}(M,\partial M)$ and $g_{2}\in
H^{2}_{\rho}(M,\partial M)$ satisfying that $\|g_{1}\|_{\partial
M,\rho}^{2}=\|g_{1}\|_{\partial M,\rho}^{2}=M_{H}(Z_{0},J,\rho)$,
$(g_{1}^{*}-f_{0},z)\in J_{z}$ and $(g_{2}^{*}-f_{0},z)\in J_{z}$ for any
$z\in Z_{0}$. It is clear that

$(\frac{g^{*}_{1}+g^{*}_{2}}{2}-f_{0},z)\in J_{z}.$

Note that

$\begin{split}\|\frac{g_{1}+g_{2}}{2}\|_{\partial
M,\rho}^{2}+\|\frac{g_{1}-g_{2}}{2}\|_{\partial
M,\rho}^{2}=\frac{\|g_{1}\|_{\partial M,\rho}^{2}+\|g_{2}\|_{\partial
M,\rho}^{2}}{2}=M_{H}(Z_{0},J,\rho),\end{split}$

then we obtain that

$\|\frac{g_{1}+g_{2}}{2}\|_{\partial M,\rho}^{2}<M_{H}(Z_{0},J,\rho),$

which contradicts the definition of $M_{H}(Z_{0},J,\rho)$.

Thus, Lemma 2.18 has been proved. ∎

Let $Z_{j}=\\{z_{j,1},z_{j,2},...,z_{j,m_{j}}\\}\subset D_{j}$ for any
$j\in\\{1,2,...,n\\}$, where $m_{j}$ is a positive integer. Denote that

$Z_{0}:=\prod_{1\leq j\leq n}Z_{j}\subset M.$

Let $\psi=\max_{1\leq j\leq n}\\{\sum_{1\leq k\leq
m_{j}}p_{j,k}G_{D_{j}}(\cdot,z_{j,k})\\}$.

Let $\hat{\rho}$ be a Lebesgue measurable function on $\overline{M}$, which
satisfies that $\inf_{\overline{M}}\hat{\rho}>0$ and
$\hat{\rho}(w_{j},\hat{w}_{j})\leq\liminf_{w\rightarrow
w_{j}}\hat{\rho}(w,\hat{w}_{j})$ for any $(w_{j},\hat{w}_{j})\in\partial
D_{j}\times M_{j}\subset\partial M$ and any $1\leq j\leq n$, where
$M_{j}=\prod_{l\not=j}D_{l}$. Let $\rho$ be a Lebesgue measurable function on
$\partial M$ such that

$\rho(w_{1},\ldots,w_{n}):=\left(\sum_{1\leq k\leq m_{j}}p_{j,k}\frac{\partial
G_{D_{j}}(w_{j},z_{j,k})}{\partial v_{w_{j}}}\right)^{-1}\hat{\rho}$

on $\partial D_{j}\times{M_{j}}$ for any $1\leq j\leq n$, thus we have
$\inf_{\partial M}\rho>0$.

The following proposition gives an sufficient condition for $f\in
H_{\rho}^{2}(M,\partial M)$.

######  Proposition 2.19.

Let $g$ be a holomorphic function on $M$. Assume that

$\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in M:2\psi(z)\geq\log
r\\}}|g(z)|^{2}\hat{\rho}}{1-r}<+\infty,$

then there is $f\in H^{2}_{\rho}(M,\partial M)$ such that $f^{*}=g$ and

$\|f\|_{\partial M,\rho}^{2}\leq\frac{1}{\pi}\liminf_{r\rightarrow
1-0}\frac{\int_{\\{z\in M:2\psi(z)\geq\log r\\}}|g(z)|^{2}\hat{\rho}}{1-r}.$

###### Proof.

If $Z_{0}$ is a single point set, Proposition 2.19 can be referred to [13].
Let $\tilde{\psi}(w_{1},\ldots,w_{n})=\max_{1\leq j\leq
n}\\{2G_{D_{j}}(w_{j},z_{j,1})\\}$ on $M$. There exist $t_{0}>0$ and $C$ such
that

$\\{z\in M:\tilde{\psi}(z)\geq-t\\}\subset\\{z\in M:\psi(z)\geq-Ct\\}$

for any $t\in(0,t_{0})$ (see [12]), which implies that

$\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in M:\tilde{\psi}(z)\geq\log
r\\}}|g(z)|^{2}\hat{\rho}}{1-r}<+\infty.$

As Proposition 2.19 holds when $Z_{0}$ is a single point set, there is $f\in
H^{2}(M,\partial M)$ such that $f^{*}=g$.

In the following part, we will prove that

$\sum_{1\leq j\leq n}\int_{M_{j}}\int_{\partial
D_{j}}|f|^{2}\rho|dw_{j}|d\mu_{j}(\hat{w}_{j})\leq 2\liminf_{r\rightarrow
1-0}\frac{\int_{\\{z\in M:2\psi(z)\geq\log r\\}}|g(z)|^{2}\hat{\rho}}{1-r}.$

Choose any compact subset $K_{j}$ of $M_{j}$ for any $1\leq j\leq n$, and
denote that

$\Omega_{j,r}:=\\{z\in D_{j}:2\sum_{1\leq k\leq
m_{j}}p_{j,k}G_{D_{j}}(z,z_{j,k})\geq\log r\\}\times K_{j}\subset\\{z\in
M:2\psi(z)\geq\log r\\}$

for any $1\leq j\leq n$. There exists $r_{1}\in(0,1)$ such that
$\Omega_{j,r_{1}}\cap\Omega_{j^{\prime},r_{1}}=\emptyset$ for any
$j\not=j^{\prime}$ and $|\bigtriangledown\psi_{j}|\not=0$ on $\\{z\in
D_{j}:2\psi_{j}\geq\log r\\}$, where $\psi_{j}=\sum_{1\leq k\leq
m_{j}}p_{j,k}G_{D_{j}}(\cdot,z_{j,k})$ on $D_{j}$. Note that
$f(\cdot,\hat{w}_{1})=\gamma_{1}(g(\cdot,\hat{w}_{1}))$ denotes the
nontangential boundary value of $g(\cdot,\hat{w}_{1})$ a.e. on $\partial
D_{1}$ for any $\hat{w}_{1}\in M_{1}$. As

$\hat{\rho}(w_{1},\hat{w}_{1})\leq\liminf_{w\rightarrow
w_{1}}\hat{\rho}(w,\hat{w}_{1})$

for any $(w_{1},\hat{w}_{1})\in\partial D_{1}\times M_{1}$, it follows from
Fatou’s lemma, Lemma 2.8 and Lemma 2.12 that

$\begin{split}&\int_{K_{1}}\int_{\partial
D_{1}}|f(w_{1},\hat{w}_{1})|^{2}\rho|dw_{1}|d\mu_{1}(\hat{w}_{1})\\\
=&\int_{K_{1}}\left(\int_{\partial
D_{1}}\frac{|\gamma_{1}(g(\cdot,\hat{w}_{1}))|^{2}}{\sum_{1\leq k\leq
m_{j}}p_{1,k}\frac{\partial G_{D_{1}}(w_{1},z_{1,k})}{\partial
v_{w_{1}}}}\hat{\rho}|dw_{1}|\right)d\mu_{1}(\hat{w}_{1})\\\
\leq&\liminf_{r\rightarrow 1-0}\frac{\int_{\log
r}^{0}\left(\int_{K_{1}}\left(\int_{\\{\psi_{1}=s\\}}\frac{|g|^{2}\hat{\rho}}{|\bigtriangledown\psi_{1}|}|dw_{1}|\right)d\mu_{1}(\hat{w}_{1})\right)ds}{-\log
r}\\\ =&\liminf_{r\rightarrow
1-0}\frac{\int_{\Omega_{1,r^{2}}}|g|^{2}\hat{\rho}}{-\log r}\\\
=&2\liminf_{r\rightarrow 1-0}\frac{\int_{\Omega_{1,r}}|g|^{2}\hat{\rho}}{-\log
r}.\end{split}$

By similar discussion, we have

(2.18) $\int_{K_{j}}\int_{\partial
D_{j}}|f(w_{j},\hat{w}_{j})|^{2}\rho|dw_{j}|d\mu_{j}(\hat{w}_{j})\leq
2\liminf_{r\rightarrow 1-0}\frac{\int_{\Omega_{j,r}}|g|^{2}\hat{\rho}}{-\log
r}$

for any $1\leq j\leq n$. As $\Omega_{j,r}\cap\Omega_{j^{\prime},r}=\emptyset$
for any $j\not=j^{\prime}$ and $r\in(r_{1},1)$, following from the
arbitrariness of $K_{j}$ and inequality (2.18) that

$\begin{split}&\sum_{1\leq j\leq n}\int_{M_{j}}\int_{\partial
D_{j}}|f(w_{j},\hat{w}_{j})|^{2}\rho|dw_{j}|d\mu_{j}(\hat{w}_{j})\\\
\leq&2\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in M:2\psi(z)\geq\log
r\\}}|g|^{2}\hat{\rho}}{-\log r}\\\ =&2\liminf_{r\rightarrow
1-0}\frac{\int_{\\{z\in M:2\psi(z)\geq\log r\\}}|g|^{2}\hat{\rho}}{1-r}\\\
<&+\infty.\end{split}$

Thus, Proposition 2.19 holds. ∎

The following lemma will be used in the proof of Lemma 2.21.

######  Lemma 2.20 ([20]).

Let $u$ is a subharmonic function on $\Omega$. If $v(dd^{c}u,z_{0})<1$, then
$e^{-2u}$ is $L^{1}$ on a neighborhood of $z_{0}$.

Let $\varphi_{j}$ be a subharmonic function on $D_{j}$, which satisfies that
$\varphi_{j}$ is continuous at $z$ for any $z\in\partial D_{j}$. Let
$\rho=\prod_{1\leq j\leq n}e^{-\varphi_{j}}$ on $\overline{M}$.

######  Lemma 2.21.

Assume that $n>1$. Let $f\in H_{\rho}^{2}(M,\partial M)$. Then for any compact
subset $K$ of $M$, we have $\int_{K}|f^{*}|^{2}e^{-\varphi}<+\infty$.

###### Proof.

Since $\varphi_{j}$ is continuous at $z$ for any $z\in\partial D_{j}$, for any
$j$, it follows from Weierstrass theorem (see [3]) and Siu’s Decomposition
Theorem that there exists a holomorphic function $g_{j}$ on $\mathbb{C}$ such
that $\varphi_{j}-2\log|g_{j}|$ is subharmonic on $D_{j}$ and the Lelong
number

$v(dd^{c}(\varphi_{j}-2\log|g_{j}|),z)\in[0,2)$

holds for any $z\in D_{j}$. Lemma 2.20 shows that $\prod_{1\leq j\leq
n}e^{-(\varphi_{j}-2\log|g_{j}|)}$ is locally integrable on $M$. Thus, it
suffices to prove that $\frac{f^{*}}{\prod_{1\leq j\leq n}g_{j}}$ is
holomorphic.

As $f^{*}(\cdot,\hat{w}_{1})\in H^{2}(D_{1})$ for any $\hat{w}_{1}\in M_{1}$
and $\gamma_{1}(f^{*})=f$ a.e. on $\partial D_{1}\times M_{1}$, it follows
from Lemma 2.1 that for any $K_{1}\Subset D_{1}$, there is $C_{K_{1}}>0$ such
that

(2.19) $\sup_{w_{1}\in K_{1}}\left|f^{*}(w_{1},\hat{w}_{1})\right|^{2}\leq
C_{K_{1}}\frac{1}{2\pi}\int_{\partial
D_{1}}\left|f(z_{1},\hat{w}_{1})\right|^{2}|dz_{1}|,$

holds for a.e. $\hat{w}_{1}\in M_{1}$. Note that
$\inf_{M}\frac{\rho}{e^{-2\log|g_{l}|}}>0$. As $f\in H_{\rho}^{2}(M,\partial
M),$ we have

(2.20) $\begin{split}&\int_{M_{j}}\int_{\partial
D_{j}}\left|\frac{f^{*}}{g_{l}}\right|^{2}|dz_{j}|d\mu_{j}(\hat{w}_{j})\\\
\leq&C_{0}\int_{M_{j}}\int_{\partial
D_{j}}\left|f^{*}\right|^{2}e^{-2\log|g_{l}|}\frac{\rho}{e^{-2\log|g_{l}|}}|dz_{j}|d\mu_{j}(\hat{w}_{j})\\\
\leq&C_{0}\|f\|^{2}_{\partial M,\rho}\\\ <&+\infty\end{split}$

for any $1\leq j,l\leq n$. Since $g_{2}\not\equiv 0$, inequality (2.19) and
(2.20) imply that

$\begin{split}&\int_{M_{1}}\int_{K_{1}}\left|\frac{f^{*}}{g_{2}}(w_{1},\hat{w}_{1})\right|^{2}\\\
\leq&C_{1}\int_{M_{1}}\sup_{w_{1}\in
K_{1}}\left|\frac{f^{*}}{g_{2}}(w_{1},\hat{w}_{1})\right|^{2}d\mu_{1}(\hat{w}_{1})\\\
\leq&C_{1}C_{K_{1}}\int_{M_{1}}\left(\frac{1}{2\pi}\int_{\partial
D_{1}}\left|\frac{f^{*}}{g_{2}}(z_{1},\hat{w}_{1})\right|^{2}|dz_{1}|\right)d\mu_{1}(\hat{w}_{1})\\\
<&+\infty\end{split}$

for any $K_{1}$, which implies that $\frac{f^{*}}{g_{2}}$ is holomorphic on
$M$. Thus, $\frac{f^{*}}{\prod_{1\leq j\leq n}g_{j}}\in\mathcal{O}(M)$. ∎

###  2.3. Concavity property of minimal $L^{2}$ integrals

In this section, we recall some results about the concavity property of
minimal $L^{2}$ integrals (see [9, 10, 11]).

Let $M$ be an $n-$dimensional Stein manifold, and let $K_{M}$ be the canonical
(holomorphic) line bundle on $M$. Let $\psi$ be a plurisubharmonic function on
$M$, and let $\varphi$ be a Lebesgue measurable function on $M$, such that
$\varphi+\psi$ is a plurisubharmonic function on $M$. Take
$T=-\sup_{M}\psi>-\infty$.

######  Definition 2.22.

We call a positive measurable function $c$ on $(T,+\infty)$ in class
$\mathcal{P}_{T}$ if the following two statements hold:

$(1)$ $c(t)e^{-t}$ is decreasing with respect to $t$;

$(2)$ there is a closed subset $E$ of $M$ such that $E\subset\\{z\in
Z:\psi(z)=-\infty\\}$ and for any compact subset $K\subseteq M\backslash E$,
$e^{-\varphi}c(-\psi)$ has a positive lower bound on $K$, where $Z$ is some
analytic subset of $M$.

Let $Z_{0}$ be a subset of $\\{\psi=-\infty\\}$ such that $Z_{0}\cap
Supp(\\{\mathcal{O}/\mathcal{I(\varphi+\psi)}\\})\neq\emptyset$. Let
$U\supseteq Z_{0}$ be an open subset of $M$ and let $f$ be a holomorphic
$(n,0)$ form on $U$. Let $\mathcal{F}\supseteq\mathcal{I}(\varphi+\psi)|_{U}$
be an analytic subsheaf of $\mathcal{O}$ on $U$.

Denote

$\begin{split}\inf\Bigg{\\{}\int_{\\{\psi<-t\\}}|\tilde{f}|^{2}e^{-\varphi}c(-\psi):(\tilde{f}-f)\in
H^{0}(Z_{0},&(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})\\\
&\&{\,}\tilde{f}\in
H^{0}(\\{\psi<-t\\},\mathcal{O}(K_{M}))\Bigg{\\}},\end{split}$

by $G(t;c)$ (without misunderstanding, we denote $G(t;c)$ by $G(t)$), where
$t\in[T,+\infty)$, $c\in\mathcal{P}_{T}$ satisfying
$\int_{T}^{+\infty}c(l)e^{-l}dl<+\infty$,
$|f|^{2}:=\sqrt{-1}^{n^{2}}f\wedge\bar{f}$ for any $(n,0)$ form $f$ and
$(\tilde{f}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ means
$(\tilde{f}-f,z_{0})\in(\mathcal{O}(K_{M})\otimes\mathcal{F})_{z_{0}}$ for all
$z_{0}\in Z_{0}$.

We recall some results about the concavity for $G(t)$.

######  Theorem 2.23 ([9]).

Assume that $G(T)<+\infty$. Then $G(h^{-1}(r))$ is concave with respect to
$r\in(0,\int_{T}^{+\infty}c(l)e^{-l}dl)$, $\lim_{t\rightarrow T+0}G(t)=G(T)$
and $\lim_{t\rightarrow+\infty}G(t)=0$, where
$h(t)=\int_{t}^{+\infty}c(l)e^{-l}dl$.

The following corollary gives a necessary condition for the concavity of
$G(h^{-1}(r))$ degenerating to linearity.

######  Corollary 2.24 ([9]).

Assume that $G(T)\in(0,+\infty)$. If $G({h}^{-1}(r))$ is linear with respect
to $r\in(0,\int_{T}^{+\infty}c(l)e^{-l}dl)$, where
${h}(t)=\int_{t}^{+\infty}c(l)e^{-l}dl$, then there is a unique holomorphic
$(n,0)$ form $F$ on $M$ satisfying $(F-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and
$G(t;c)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)$ for any $t\geq T$.
Furthermore,

$\int_{\\{-t_{1}\leq\psi<-t_{2}\\}}|F|^{2}e^{-\varphi}a(-\psi)=\frac{G(T_{1};c)}{\int_{T_{1}}^{+\infty}c(l)e^{-l}dl}\int_{t_{2}}^{t_{1}}a(t)e^{-t}dt$

for any nonnegative measurable function $a$ on $(T,+\infty)$, where
$+\infty\geq t_{1}>t_{2}\geq T$.

We recall the existence and uniqueness of the holomorphic $(n,0)$ form related
to $G(t)$.

######  Lemma 2.25 ([9]).

Assume that $G(t)<+\infty$ for some $t\in[T,+\infty)$. Then there exists a
unique holomorphic $(n,0)$ form $F_{t}$ on $\\{\psi<-t\\}$ satisfying
$(F_{t}-f)\in H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$
and $\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)=G(t)$. Furthermore,
for any holomorphic $(n,0)$ form $\hat{F}$ on $\\{\psi<-t\\}$ satisfying
$(\hat{F}-f)\in H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$
and $\int_{\\{\psi<-t\\}}|\hat{F}|^{2}e^{-\varphi}c(-\psi)<+\infty$, we have
the following equality

$\begin{split}&\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)+\int_{\\{\psi<-t\\}}|\hat{F}-F_{t}|^{2}e^{-\varphi}c(-\psi)\\\
=&\int_{\\{\psi<-t\\}}|\hat{F}|^{2}e^{-\varphi}c(-\psi).\end{split}$

In the following, we recall some characterizations for the concavity of
$G(h^{-1}(r))$ degenerating to linearity.

Assume that $M=\Omega$ is an open Riemann surface, which admitted a nontrivial
Green function $G_{\Omega}$. Let
$Z_{0}=\\{z_{1},z_{2},\ldots,z_{m}\\}\subset\Omega$ be a finite subset of
$\Omega$ satisfying that $z_{j}\not=z_{k}$ for any $j\not=k$.

We recall some notations (see [3], see also [14, 9, 6]). Let
$p:\Delta\rightarrow\Omega$ be the universal covering from unit disc $\Delta$
to $\Omega$. we call the holomorphic function $f$ on $\Delta$ a multiplicative
function, if there is a character $\chi$, which is the representation of the
fundamental group of $\Omega$, such that $g^{\star}f=\chi(g)f$, where
$|\chi|=1$ and $g$ is an element of the fundamental group of $\Omega$. It is
known that for any harmonic function $u$ on $\Omega$, there exists a
$\chi_{u}$ and a multiplicative function
$f_{u}\in\mathcal{O}^{\chi_{u}}(\Omega)$, such that
$|f_{u}|=p^{\star}\left(e^{u}\right)$. Recall that for the Green function
$G_{\Omega}(z,z_{j})$, there exist a $\chi_{z_{j}}$ and a multiplicative
function $f_{z_{j}}\in\mathcal{O}^{\chi_{z_{j}}}(\Omega)$, such that
$|f_{z_{j}}(z)|=p^{\star}\left(e^{G_{\Omega}(z,z_{j})}\right)$ (see [22, 21]).

The following Theorem gives a characterization of the concavity of
$G(h^{-1}(r))$ degenerating to linearity.

######  Theorem 2.26 ([10], see also [7]).

Let $G(0)\in(0,+\infty)$ and $p_{j}=\frac{1}{2}v(dd^{c}(\psi),z_{j})>0$ for
any $j\in\\{1,2,\ldots,m\\}$. For any $j\in\\{1,2,\ldots,m\\}$, assume that
one of the following conditions holds:

$(A)$ $\varphi+a\psi$ is subharmonic near $z_{j}$ for some $a\in[0,1)$;

$(B)$ $(\psi-2p_{j}G_{\Omega}(\cdot,z_{j}))(z_{j})>-\infty$.

Then $G(h^{-1}(r))$ is linear with respect to $r$ if and only if the following
statements hold:

$(1)$ $\psi=2\sum_{1\leq j\leq m}p_{j}G_{\Omega}(\cdot,z_{j})$;

$(2)$ $\varphi+\psi=2\log|g|+2\sum_{1\leq j\leq m}G_{\Omega}(\cdot,z_{j})+2u$
and $\mathcal{F}_{z_{j}}=\mathcal{I}(\varphi+\psi)_{z_{j}}$ for any
$j\in\\{1,2,\ldots,m\\}$, where $g$ is a holomorphic function on $\Omega$ such
that $ord_{z_{j}}(g)=ord_{z_{j}}(f)$ for any $j\in\\{1,2,\ldots,m\\}$ and $u$
is a harmonic function on $\Omega$;

$(3)$ $\prod_{1\leq j\leq m}\chi_{z_{j}}=\chi_{-u}$, where $\chi_{-u}$ and
$\chi_{z_{j}}$ are the characters associated to the functions $-u$ and
$G_{\Omega}(\cdot,z_{j})$ respectively;

$(4)$ $\lim_{z\rightarrow z_{k}}\frac{f}{gp_{*}\left(f_{u}\left(\prod_{1\leq
j\leq m}f_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{d{f_{z_{j}}}}{f_{z_{j}}}\right)\right)}=c_{0}$ for any
$k\in\\{1,2,\ldots,m\\}$, where $c_{0}\in\mathbb{C}\backslash\\{0\\}$ is a
constant independent of $k$, $f_{u}$ is a holomorphic function on $\Delta$
such that $|f_{u}|=p^{*}(e^{u})$ and $f_{z_{j}}$ is a holomorphic function on
$\Delta$ such that $|f_{z_{j}}|=p^{*}\left(e^{G_{\Omega}(\cdot,z_{j})}\right)$

######  Remark 2.27 ([10]).

When the four statements in Theorem 2.26 hold,

$c_{0}gp_{*}(f_{u}(\Pi_{1\leq j\leq m}f_{z_{j}})(\sum_{1\leq j\leq
m}p_{j}\frac{d{f_{z_{j}}}}{f_{z_{j}}}))$

is the unique holomorphic $(1,0)$ form $F$ on $\Omega$ such that
$(F-f,z_{j})\in(\mathcal{O}(K_{\Omega}))_{z_{j}}\otimes\mathcal{F}_{z_{j}}$
for any $j\in\\{1,2,...,m\\}$ and
$G(t)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)$ for any $t\geq 0$.

We recall the following characterization for the holding of the equality in
the optimal $L^{2}$ extension problem.

######  Theorem 2.28 ([10]).

Let $k_{j}$ be a nonnegative integer for any $j\in\\{1,2,...,m\\}$. Let $\psi$
be a negative subharmonic function on $\Omega$ satisfying that
$\frac{1}{2}v(dd^{c}\psi,z_{j})=p_{j}>0$ for any $j\in\\{1,2,...,m\\}$. Let
$\varphi$ be a Lebesgue measurable function on $\Omega$ such that
$\varphi+\psi$ is subharmonic on $\Omega$,
$\frac{1}{2}v(dd^{c}(\varphi+\psi),z_{j})=k_{j}+1$ and
$\alpha_{j}:=(\varphi+\psi-2(k_{j}+1)G_{\Omega}(\cdot,z_{j}))(z_{j})>-\infty$
for any $j$. Let $c(t)$ be a positive measurable function on $(0,+\infty)$
satisfying $c(t)e^{-t}$ is decreasing on $(0,+\infty)$ and
$\int_{0}^{+\infty}c(s)e^{-s}ds<+\infty$. Let $a_{j}$ be a constant for any
$j$.

Let $f$ be a holomorphic $(1,0)$ form on $V_{0}$ satisfying that
$f=a_{j}w_{j}^{k_{j}}dw_{j}$ on $V_{z_{j}}$. Then there exists a holomorphic
$(1,0)$ form $F$ on $\Omega$ such that
$(F-f,z_{j})\in(\mathcal{O}(K_{\Omega})\otimes\mathcal{I}(2(k_{j}+1)G_{\Omega}(\cdot,z_{j})))_{z_{j}}$
and

(2.21)
$\int_{\Omega}|F|^{2}e^{-\varphi}c(-\psi)\leq(\int_{0}^{+\infty}c(s)e^{-s}ds)\sum_{1\leq
j\leq
m}\frac{2\pi|a_{j}|^{2}e^{-\alpha_{j}}}{p_{j}c_{\beta}(z_{j})^{2(k_{j}+1)}}.$

Moreover, equality $(\int_{0}^{+\infty}c(s)e^{-s}ds)\sum_{1\leq j\leq
m}\frac{2\pi|a_{j}|^{2}e^{-\alpha_{j}}}{p_{j}c_{\beta}(z_{j})^{2(k_{j}+1)}}=\inf\\{\int_{\Omega}|\tilde{F}|^{2}e^{-\varphi}c(-\psi):\tilde{F}$
is a holomorphic $(1,0)$ form on $\Omega$ such that
$(\tilde{F}-f,z_{j})\in(\mathcal{O}(K_{\Omega})\otimes\mathcal{I}(2(k_{j}+1)G_{\Omega}(\cdot,z_{j})))_{z_{j}}$
for any $j\\}$ holds if and only if the following statements hold:

$(1)$ $\psi=2\sum_{1\leq j\leq m}p_{j}G_{\Omega}(\cdot,z_{j})$;

$(2)$ $\varphi+\psi=2\log|g|+2\sum_{1\leq j\leq
m}(k_{j}+1)G_{\Omega}(\cdot,z_{j})+2u$, where $g$ is a holomorphic function on
$\Omega$ such that $g(z_{j})\not=0$ for any $j\in\\{1,2,...,m\\}$ and $u$ is a
harmonic function on $\Omega$;

$(3)$ $\Pi_{1\leq j\leq m}\chi_{z_{j}}^{k_{j}+1}=\chi_{-u}$, where $\chi_{-u}$
and $\chi_{z_{j}}$ are the characters associated to the functions $-u$ and
$G_{\Omega}(\cdot,z_{j})$ respectively;

$(4)$ $\lim_{z\rightarrow z_{k}}\frac{f}{gp_{*}(f_{u}(\Pi_{1\leq j\leq
m}f_{z_{j}}^{k_{j}+1})(\sum_{1\leq j\leq
m}p_{j}\frac{d{f_{z_{j}}}}{f_{z_{j}}}))}=c_{0}$ for any $k\in\\{1,2...,m\\}$,
where $c_{0}\in\mathbb{C}\backslash\\{0\\}$ is a constant independent of $k$.

In the following, we consider the case $M$ is a product manifold of open
Riemann surfaces.

Let $\Omega_{j}$ be an open Riemann surface, which admits a nontrivial Green
function $G_{\Omega_{j}}$ for any $1\leq j\leq n$. Let

$M=\prod_{1\leq j\leq n}\Omega_{j}$

be an $n-$dimensional complex manifold, and let $\pi_{j}$ be the natural
projection from $M$ to $\Omega_{j}$. Let $K_{M}$ be the canonical
(holomorphic) line bundle on $M$. Let $\varphi_{j}$ be a subharmonic function
on $\Omega_{j}$, and let

$\varphi=\sum_{1\leq j\leq n}\pi_{j}^{*}(\varphi_{j}).$

Let $Z_{j}=\\{z_{j,1},z_{j,2},...,z_{j,m_{j}}\\}\subset\Omega_{j}$ for any
$j\in\\{1,2,...,n\\}$, where $m_{j}$ is a positive integer. Denote that

$Z_{0}:=\prod_{1\leq j\leq n}Z_{j}\subset M.$

Let $\psi=\max_{1\leq j\leq n}\\{\pi_{j}^{*}(2\sum_{1\leq k\leq
m_{j}}p_{j,k}G_{D_{j}}(\cdot,z_{j,k}))\\}$, where $p_{j,k}>0$ is a constant.
Let $\mathcal{F}_{z}=\mathcal{I}(\psi)_{z}$ for any $z\in Z_{0}$.

Let $w_{j,k}$ be a local coordinate on a neighborhood
$V_{z_{j,k}}\Subset\Omega_{j}$ of $z_{j,k}\in\Omega_{j}$ satisfying
$w_{j,k}(z_{j,k})=0$ for any $j\in\\{1,2,...,n\\}$ and
$k\in\\{1,2,...,m_{j}\\}$, where $V_{z_{j,k}}\cap
V_{z_{j,k^{\prime}}}=\emptyset$ for any $j$ and $k\not=k^{\prime}$. Denote
that $I_{1}:=\\{(\beta_{1},\beta_{2},...,\beta_{n}):1\leq\beta_{j}\leq m_{j}$
for any $j\in\\{1,2,...,n\\}\\}$, $V_{\beta}:=\prod_{1\leq j\leq
n}V_{z_{j,\beta_{j}}}$ for any $\beta=(\beta_{1},\beta_{2},...,\beta_{n})\in
I_{1}$ and $w_{\beta}:=(w_{1,\beta_{1}},w_{2,\beta_{2}},...,w_{n,\beta_{n}})$
is a local coordinate on $V_{\beta}$ of
$z_{\beta}:=(z_{1,\beta_{1}},z_{2,\beta_{2}},...,z_{n,\beta_{n}})\in M$.

Let $f_{0}$ be a holomorphic $(n,0)$ form on $\cup_{\beta\in I_{1}}V_{\beta}$.
Let $c$ be a positive function on $[0,+\infty)$, which satisfies that
$c(t)e^{-t}$ is decreasing on $[0,+\infty)$, $\lim_{t\rightarrow
0+0}c(t)=c(0)=1$ and $\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$.

######  Theorem 2.29 ([11]).

Assume that $G(0)\in(0,+\infty)$ and $\varphi(z_{\beta})>-\infty$ for any
$\beta\in I_{1}$. $G(h^{-1}(r))$ is linear with respect to
$r\in(0,\int_{0}^{+\infty}c(s)e^{-s}ds]$ if and only if the following
statements hold:

$(1)$ $\varphi_{j}=2\log|g_{j}|+2u_{j}$ for any $j\in\\{1,2,...,n\\}$, where
$u_{j}$ is a harmonic function on $\Omega_{j}$ and $g_{j}$ is a holomorphic
function on $\Omega_{j}$ satisfying $g_{j}(z_{j,k})\not=0$ for any
$k\in\\{1,2,...,m_{j}\\}$;

$(2)$ There exists a nonnegative integer $\gamma_{j,k}$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$, which satisfies that
$\Pi_{1\leq k\leq m_{j}}\chi_{j,z_{j,k}}^{\gamma_{j,k}+1}=\chi_{j,-u_{j}}$ and
$\sum_{1\leq j\leq n}\frac{\gamma_{j,\beta_{j}}+1}{p_{j,\beta_{j}}}=1$ for any
$\beta\in I_{1}$, where $\chi_{-u_{j}}$ and $\chi_{z_{j,k}}$ are the
characters associated to the functions $-u_{j}$ and
$G_{\Omega_{j}}(\cdot,z_{j,k})$ respectively;

$(3)$ $f=(c_{\beta}\Pi_{1\leq j\leq
n}w_{j,\beta_{j}}^{\gamma_{j,\beta_{j}}}+g_{\beta})dw_{1,\beta_{1}}\wedge
dw_{2,\beta_{2}}\wedge...\wedge dw_{n,\beta_{n}}$ on $V_{\beta}$ for any
$\beta\in I_{1}$, where $c_{\beta}$ is a constant and $g_{\beta}$ is a
holomorphic function on $V_{\beta}$ such that
$(g_{\beta},z_{\beta})\in\mathcal{I}(\psi)_{z_{\beta}}$;

$(4)$ $\lim_{z\rightarrow z_{\beta}}\frac{c_{\beta}\Pi_{1\leq j\leq
n}w_{j,\beta_{j}}^{\gamma_{j,\beta_{j}}}dw_{1,\beta_{1}}\wedge
dw_{2,\beta_{2}}\wedge...\wedge dw_{n,\beta_{n}}}{\wedge_{1\leq j\leq
n}\pi_{j}^{*}(g_{j}(P_{j})_{*}(f_{u_{j}}(\Pi_{1\leq k\leq
m_{j}}f_{z_{j,k}}^{\gamma_{j,k}+1})(\sum_{1\leq k\leq
m_{j}}p_{j,k}\frac{df_{z_{j,k}}}{f_{z_{j,k}}})))}=c_{0}$ for any $\beta\in
I_{1}$, where $c_{0}\in\mathbb{C}\backslash\\{0\\}$ is a constant independent
of $\beta$, $P_{j}:\Delta\rightarrow\Omega_{j}$ is the universal covering,
$f_{u_{j}}$ is a holomorphic function $\Delta$ such that
$|f_{u_{j}}|=P_{j}^{*}(e^{u_{j}})$ and $f_{z_{j,k}}$ is a holomorphic function
on $\Delta$ such that
$|f_{z_{j,k}}|=P_{j}^{*}(e^{G_{\Omega_{j}}(\cdot,z_{j,k})})$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$.

The following Lemma will be used in the proof of Remark 2.31.

######  Lemma 2.30 (see [11]).

Let $\psi=\max_{1\leq j\leq n}\\{2p_{j}\log|w_{j}|\\}$ be a plurisubharmonic
function on $\mathbb{C}^{n}$, where $p_{j}>0$. Let
$f=\sum_{\alpha\in\mathbb{Z}_{\geq 0}^{n}}b_{\alpha}w^{\alpha}$ (Taylor
expansion) be a holomorphic function on $\\{\psi<-t_{0}\\}$, where $t_{0}>0$.
Then

$\int_{\\{\psi<-t\\}}|f|^{2}d\lambda_{n}=\sum_{\alpha\in\mathbb{Z}_{\geq
0}^{n}}e^{-\sum_{1\leq j\leq
n}\frac{\alpha_{j}+1}{p_{j}}t}\frac{|b_{\alpha}|^{2}\pi^{n}}{\Pi_{1\leq j\leq
n}(\alpha_{j}+1)}$

holds for any $t\geq t_{0}$.

######  Remark 2.31.

The requirement “$\varphi(z_{\beta})>-\infty$ for any $\beta\in I_{1}$” in
Theorem 2.29 can be removed.

###### Proof.

It suffices to prove that the linearity of $G(h^{-1}(r))$ can deduce
$\varphi(z_{\beta})>-\infty$ for any $\beta\in I_{1}$.

Assume that $G(h^{-1}(r))$ is linear with respect to
$r\in(0,\int_{0}^{+\infty}c(t)e^{-t}dt]$. It follows from Corollary 2.24 that
there is a holomorphic $(n,0)$ form $F$ on $M$ such that
$(F-f,z_{\beta})\in\mathcal{I}(\psi)_{z_{\beta}}$ for any $\beta\in I_{1}$,
and

(2.22)
$\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}=\frac{G(0)}{\int_{0}^{+\infty}c(t)e^{-t}dt}e^{-t}$

for any $t\geq 0$.

Firstly, we prove that $(F,z_{\beta})\not\in\mathcal{I}(\psi)_{z_{\beta}}$ for
any $\beta\in I_{1}$. We prove this by contradiction: if not, there exists
$\beta_{0}\in I_{1}$ such that
$(F,z_{\beta_{0}})\in\mathcal{I}(\psi)_{z_{\beta_{0}}}$. Then we have

$(f,z_{\beta_{0}})\in\mathcal{I}(\psi)_{z_{\beta_{0}}}.$

There exists $t>0$ such that $\\{\psi<-t\\}\cap V_{{\beta_{0}}}\Subset
V_{\beta_{0}}.$ Corollary 2.24 tells us that $F$ is the unique “minimal form”
on any sublevel set of $\psi$, thus we have $F\equiv 0$ on $\\{\psi<-t\\}\cap
V_{{\beta_{0}}}$, which implies that

$F\equiv 0$

on $M$. Then we get that $(f,z_{\beta})\in\mathcal{I}(\psi)_{z_{\beta}}$ for
any $\beta\in I_{1}$, which contradict to $G(0)>0$.

Now, we prove $\varphi_{j}(z_{\beta})>-\infty$ for any $\beta\in I_{1}$. Fixed
any $\beta\in I_{1}$, without loss of generality, assume that
$|w_{j}(z)|=e^{\sum_{1\leq k\leq
m_{j}}\frac{p_{j,k}}{p_{j,\beta_{j}}}G_{\Omega_{j}}(z,z_{j,k})}$ on
$V_{z_{j,\beta_{j}}}$, hence $\psi=\max_{1\leq j\leq
n}\\{2p_{j,\beta_{j}}\log|w_{j}|\\}$ on $V_{\beta}$. There is $t_{0}>0$ such
that

$\\{\psi<-t_{0}\\}\cap V_{\beta}\Subset V_{\beta}.$

Denote that

$c_{t}:=\sup_{\\{\psi<-t\\}\cap V_{\beta}}\varphi<+\infty$

for any $t\geq t_{0}$. As $\varphi=\sum_{1\leq j\leq n}\varphi_{j}$ is
plurisubharmonic, we know that

$\lim_{t\rightarrow+\infty}c_{t}=\varphi(z_{\beta})=\sum_{1\leq j\leq
n}\varphi_{j}(z_{j,\beta_{j}}).$

Let $F=\sum_{\alpha\in\mathbb{Z}_{\geq
0}^{n}}d_{\alpha}w^{\alpha}dw_{1}\wedge\ldots\wedge dw_{n}$ near $z_{\beta}$.
Denote that $E_{\beta}=\\{\alpha\in\mathbb{Z}_{\geq 0}^{n}:\sum_{1\leq j\leq
n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}\leq 1\\}$. Since
$(F,z_{\beta})\not\in\mathcal{I}(\psi)_{z_{\beta}}$, we have

$\sum_{\alpha\in E_{\beta}}|d_{\alpha}|^{2}>0.$

Lemma 2.30 tells us that

(2.23) $\begin{split}\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}&\geq
e^{-c_{t}}\int_{\\{\psi<-t\\}\cap V_{\beta}}|F|^{2}\\\
&=e^{-c_{t}}\sum_{\alpha\in\mathbb{Z}_{\geq 0}^{n}}e^{-\sum_{1\leq j\leq
n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}t}\frac{|d_{\alpha}|^{2}(2\pi)^{n}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)}\\\ &\geq e^{-c_{t}}\sum_{\alpha\in
E_{\beta}}e^{-\sum_{1\leq j\leq
n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}t}\frac{|d_{\alpha}|^{2}(2\pi)^{n}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)}\end{split}$

for any $t\geq t_{0}$. It follows from equality (2.22) and inequality (2.23)
that

(2.24)
$\begin{split}\frac{G(0)}{\int_{0}^{+\infty}c(t)e^{-t}dt}&=\lim_{t\rightarrow+\infty}e^{t}\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}\\\
&\geq\lim_{t\rightarrow+\infty}e^{-c_{t}}\sum_{\alpha\in
E_{\beta}}e^{\left(1-\sum_{1\leq j\leq
n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}\right)t}\frac{|d_{\alpha}|^{2}(2\pi)^{n}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)},\end{split}$

Note that $\frac{G(0)}{\int_{0}^{+\infty}c(t)e^{-t}dt}\in(0,+\infty)$ and
$1-\sum_{1\leq j\leq n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}\geq 0$ for any
$\alpha\in E_{\beta}$, inequality (2.24) shows that

$\lim_{t\rightarrow+\infty}c_{t}>-\infty,$

hence we have $\varphi(z_{\beta})>-\infty$. ∎

Denote that

$c_{j,k}:=\exp\lim_{z\rightarrow z_{j,k}}\left(\frac{\sum_{1\leq k_{1}\leq
m_{j}}p_{j,k_{1}}G_{\Omega_{j}}(z,z_{j,k_{1}})}{p_{j,k}}-\log|w_{j,k}(z)|\right)$

for any $j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$.

######  Remark 2.32 ([11]).

When the four statements in Theorem 2.29 hold,

$c_{0}\wedge_{1\leq j\leq n}\pi_{j}^{*}(g_{j}(P_{j})_{*}(f_{u_{j}}(\Pi_{1\leq
k\leq m_{j}}f_{z_{j,k}}^{\gamma_{j,k}+1})(\sum_{1\leq k\leq
m_{j}}p_{j,k}\frac{df_{z_{j,k}}}{f_{z_{j,k}}})))$

is the unique holomorphic $(n,0)$ form $F$ on $M$ such that
$(F-f,z_{\beta})\in(\mathcal{O}(K_{M}))_{z_{\beta}}\otimes\mathcal{I}(\psi)_{z_{\beta}}$
for any $\beta\in I_{1}$ and

$G(t)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)=\left(\int_{t}^{+\infty}c(s)e^{-s}ds\right)\sum_{\beta\in
I_{1}}\frac{|c_{\beta}|^{2}(2\pi)^{n}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq j\leq
n}(\gamma_{j,\beta_{j}}+1)c_{j,\beta_{j}}^{2\gamma_{j,\beta_{j}}+2}}$

for any $t\geq 0$.

Denote that $E_{\beta}:=\\{(\alpha_{1},\alpha_{2},...,\alpha_{n}):\sum_{1\leq
j\leq
n}\frac{\alpha_{j}+1}{p_{j,\beta_{j}}}=1\,\&\,\alpha_{j}\in\mathbb{Z}_{\geq
0}\\}$. Let $f$ be a holomorphic $(n,0)$ form on $\cup_{\beta\in
I_{1}}V_{\beta}$ such that $f=\sum_{\alpha\in
E_{\beta}}d_{\beta,\alpha}w_{\beta}^{\alpha}dw_{1,\beta_{1}}\wedge
dw_{2,\beta_{2}}\wedge...\wedge dw_{n,\beta_{n}}$ on $V_{\beta}$ for any
$\beta\in I_{1}$.

We recall the following characterization for the holding of the equality in
the optimal $L^{2}$ extension problem.

######  Theorem 2.33 ([11]).

If $\sum_{\beta\in I_{1}}\sum_{\alpha\in
E_{\beta}}\frac{|d_{\beta,\alpha}|^{2}(2\pi)^{n}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)c_{j,\beta_{j}}^{2\alpha_{j}+2}}\in(0,+\infty)$, there
exists a holomorphic $(n,0)$ form $F$ on $M$, which satisfies that
$(F-f,z_{\beta})\in(\mathcal{O}(K_{M})\otimes\mathcal{I}(\psi))_{z_{\beta}}$
for any $\beta\in I_{1}$ and

$\int_{M}|F|^{2}e^{-\varphi}c(-\psi)\leq(\int_{0}^{+\infty}c(s)e^{-s}ds)\sum_{\beta\in
I_{1}}\sum_{\alpha\in
E_{\beta}}\frac{|d_{\beta,\alpha}|^{2}(2\pi)^{n}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)c_{j,\beta_{j}}^{2\alpha_{j}+2}}.$

Moreover, assume that $f=w_{\beta^{*}}^{\alpha_{\beta_{*}}}dw_{1,1}\wedge
dw_{2,1}\wedge...\wedge dw_{n,1}$ on $V_{\beta^{*}}$, where
$\beta^{*}=(1,1,...,1)\in I_{1}$, then equality
$\inf\\{\int_{M}|\tilde{F}|^{2}e^{-\varphi}c(-\psi):\tilde{F}\in
H^{0}(M,\mathcal{O}(K_{M}))\,\&\,(\tilde{F}-f,z_{\beta})\in(\mathcal{O}(K_{M})\otimes\mathcal{I}(\max_{1\leq
j\leq n}\\{2\sum_{1\leq k\leq
m_{j}}p_{j,k}\pi_{j}^{*}(G_{\Omega_{j}}(\cdot,z_{j,k}))\\}))_{z_{\beta}}$ for
any $\beta\in I_{1}\\}=(\int_{0}^{+\infty}c(s)e^{-s}ds)\sum_{\beta\in
I_{1}}\sum_{\alpha\in
E_{\beta}}\frac{|d_{\beta,\alpha}|^{2}(2\pi)^{n}e^{-\varphi(z_{\beta})}}{\Pi_{1\leq
j\leq n}(\alpha_{j}+1)c_{j,\beta_{j}}^{2\alpha_{j}+2}}$ holds if and only if
the following statements hold:

$(1)$ $\varphi_{j}=2\log|g_{j}|+2u_{j}$ for any $j\in\\{1,2,...,n\\}$, where
$u_{j}$ is a harmonic function on $\Omega_{j}$ and $g_{j}$ is a holomorphic
function on $\Omega_{j}$ satisfying $g_{j}(z_{j,k})\not=0$ for any
$k\in\\{1,2,...,m_{j}\\}$;

$(2)$ there exists a nonnegative integer $\gamma_{j,k}$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$, which satisfies that
$\Pi_{1\leq k\leq m_{j}}\chi_{j,z_{j,k}}^{\gamma_{j,k}+1}=\chi_{j,-u_{j}}$ and
$\sum_{1\leq j\leq n}\frac{\gamma_{j,\beta_{j}}+1}{p_{j,\beta_{j}}}=1$ for any
$\beta\in I_{1}$;

$(3)$ $f=(c_{\beta}\Pi_{1\leq j\leq
n}w_{j,\beta_{j}}^{\gamma_{j,\beta_{j}}}+g_{\beta})dw_{1,\beta_{1}}\wedge
dw_{2,\beta_{2}}\wedge...\wedge dw_{n,\beta_{n}}$ on $V_{\beta}$ for any
$\beta\in I_{1}$, where $c_{\beta}$ is a constant and $g_{\beta}$ is a
holomorphic function on $V_{\beta}$ such that
$(g_{\beta},z_{\beta})\in\mathcal{I}(\psi)_{z_{\beta}}$;

$(4)$ $\lim_{z\rightarrow z_{\beta}}\frac{c_{\beta}\Pi_{1\leq j\leq
n}w_{j,\beta_{j}}^{\gamma_{j,\beta_{j}}}dw_{1,\beta_{1}}\wedge
dw_{2,\beta_{2}}\wedge...\wedge dw_{n,\beta_{n}}}{\wedge_{1\leq j\leq
n}\pi_{j}^{*}(g_{j}(P_{j})_{*}(f_{u_{j}}(\Pi_{1\leq k\leq
m_{j}}f_{z_{j,k}}^{\gamma_{j,k}+1})(\sum_{1\leq k\leq
m_{j}}p_{j,k}\frac{df_{z_{j,k}}}{f_{z_{j,k}}})))}=c_{0}$ for any $\beta\in
I_{1}$, where $c_{0}\in\mathbb{C}\backslash\\{0\\}$ is a constant independent
of $\beta$, $f_{u_{j}}$ is a holomorphic function $\Delta$ such that
$|f_{u_{j}}|=P_{j}^{*}(e^{u_{j}})$ and $f_{z_{j,k}}$ is a holomorphic function
on $\Delta$ such that
$|f_{z_{j,k}}|=P_{j}^{*}(e^{G_{\Omega_{j}}(\cdot,z_{j,k})})$ for any
$j\in\\{1,2,...,n\\}$ and $k\in\\{1,2,...,m_{j}\\}$.

###  2.4. Some other required results

In this section, we recall and give some lemmas, which will be used in the
proofs of the main theorems.

Let $U\subset\mathbb{C}^{n}$ be an open set. Let us recall the definition of
admissible weight given in [15] and [16].

######  Definition 2.34 (see [15, 16]).

A nonnegative measurable function $\rho$ on $U$ is called an admissible
weight, if for any $z_{0}\in U$ the following condition is satisfied: there
exists a neighborhood $V_{z_{0}}$ in $U$ and a constant $C_{z_{0}}>0$ such
that

$|f(z)|^{2}\leq C_{z_{0}}\int_{U}|f|^{2}\rho$

holds for any $z\in V_{z_{0}}$ and any holomorphic function $f$ on $U$.

Let $\rho$ be an admissible weight on $U$. The weighted Bergman space
$A^{2}(U,\rho)$ is defined as follows:

$A^{2}(U,\rho):=\left\\{f\in\mathcal{O}(U):\int_{U}|f|^{2}\rho<+\infty\right\\}.$

Denote that

$\ll f,g\gg_{U,\rho}:=\int_{U}f\overline{g}\rho$

and $||f||_{U,\rho}:=(\int_{U}|f|^{2}\rho)^{\frac{1}{2}}$ for any $f,g\in
A^{2}(U,\rho)$.

######  Lemma 2.35 (see [15, 16]).

$A^{2}(U,\rho)$ is a separable Hilbert space equipped with the inner product
$\ll\cdot,\cdot\gg_{U,\rho}$.

We recall a sufficient condition for a weight to be an admissible weight.

######  Lemma 2.36 (see [13]).

Let $\rho$ be a nonnegative Lebesgue measurable function on $U$, and let $S$
be an analytic subset of $U$. Assume that for any $K\Subset U\backslash S$,
there is $a>0$ such that $\int_{K}\rho^{-a}<+\infty$. Then $\rho$ is an
admissible weight on $U$.

Let $U\subset\mathbb{C}^{n}$ and $W\subset\mathbb{C}^{m}$ be two open sets.
Let $\rho_{1}$ and $\rho_{2}$ be two nonnegative Lebesgue measurable functions
on $U$ and $W$ respectively. Assume that for any relatively compact set
$U_{1}\Subset U$ ($W_{2}\Subset W$), there exists a real number $a_{1}>0$
($a_{2}>0$) such that $\rho_{1}^{-a_{1}}$ ($\rho_{2}^{-a_{2}}$) is integrable
on $U_{1}$ ($W_{2}$). Let $M:=U\times W$ and $\rho=\rho_{1}\times\rho_{2}$. By
Lemma 2.36, we know that $\rho_{1}$, $\rho_{2}$ and $\rho$ are admissible
weights on $U$, on $W$ and on $M$ respectively.

The following lemma gives a product property of Bergman spaces.

######  Lemma 2.37 (see [8]).

Let $\\{f_{i}(z)\\}_{i\in\mathbb{Z}_{\geq 0}}$ and
$\\{g_{j}(w)\\}_{j\in\mathbb{Z}_{\geq 0}}$ be the complete orthonormal basis
of $A^{2}(U,\rho_{1})$ and $A^{2}(W,\rho_{2})$ respectively. Then
$\\{f_{i}(z)g_{j}(w)\\}_{i,j\in\mathbb{Z}_{\geq 0}}$ is a complete orthonormal
basis of $A^{2}(M,\rho)$.

Let $D_{j}$, $M$, $M_{j}$, $Z_{j}$, $Z_{0}$, $I_{1}$ be as in Section 2.2.
Denote that

$S:=\prod_{1\leq j\leq n}\partial D_{j}.$

Let us recall the definition and some properties of the Hardy space over $S$.

Let $\lambda$ be a Lebesgue measurable function on $S$ such that
$\inf_{S}\lambda>0$. Let $f\in L^{2}(S,\lambda d\sigma)$, where
$d\sigma:=\frac{1}{(2\pi)^{n}}|dw_{1}|\ldots|dw_{n}|$. We call $f\in
H^{2}_{\lambda}(M,S)$ if there exists $\\{f_{m}\\}_{m\in\mathbb{Z}_{\geq
0}}\subset\mathcal{O}(M)\cap C(\overline{M})\cap L^{2}(S,\lambda d\sigma)$
such that $\lim_{m\rightarrow+\infty}\|f_{m}-f\|_{S,\lambda}^{2}=0$, where
$\|g\|_{S,\lambda}:=\left(\int_{S}|g|^{2}\lambda d\sigma\right)^{\frac{1}{2}}$
for any $g\in L^{2}(S,\lambda d\sigma)$.

Denote that

$\ll
f,g\gg_{S,\lambda}=\frac{1}{(2\pi)^{n}}\int_{S}f\overline{g}\lambda|dw_{1}|\ldots|dw_{n}|$

for any $f,g\in L^{2}(S,\lambda d\sigma),$ then $H_{\lambda}^{2}(M,S)$ is a
Hilbert space equipped with the inner product $\ll\cdot,\cdot\gg_{S,\lambda}$
(see [13]). There exists a linear injective map
$P_{S}:H^{2}_{\lambda}(M,S)\rightarrow\mathcal{O}(M)$ satisfying that
$P_{S}(f)=f$ for any $f\in\mathcal{O}(M)\cap C(\overline{M})\cap
L^{2}(S,\lambda d\sigma)$ (see [13]). For simplicity, denote $P_{S}(f)$ by
$f^{*}$.

We recall three lemmas about $H^{2}_{\lambda}(M,S)$, which will be used in the
proof of Lemma 2.42.

######  Lemma 2.38 ([13]).

For any compact subset $K$ of $M$, there exists a positive constant $C_{K}$
such that

$|f^{*}(z)|\leq C_{K}\|f\|_{S,\lambda}$

holds for any $z\in K$ and $f\in H_{\lambda}^{2}(M,S)$.

######  Lemma 2.39 ([13]).

Assume that $M_{S}(Z_{0},J,\lambda)<+\infty$. Then there is a unique
holomorphic function $f\in H^{2}_{\lambda}(M,S)$ such that
$f^{*}(z_{\beta})=h_{0}(z_{\beta})$ for any $\beta\in I_{1}$, and
$M_{S}(Z_{0},J,\lambda)=\|f\|_{S,\lambda}^{2}$.

Let $M_{a}=\prod_{1\leq j\leq n_{a}}D_{j}$ be a bounded domain in
$\mathbb{C}^{n_{a}}$, where $D_{j}$ is planar regular region with finite
boundary components which are analytic Jordan curves for any $1\leq j\leq
n_{a}$. Denote that $S_{a}:=\prod_{1\leq j\leq n_{a}}\partial D_{j}$. Let
$M_{b}=\prod_{1\leq j\leq n_{b}}\tilde{D}_{j}$ be a bounded domain in
$\mathbb{C}^{n_{b}}$, where $\tilde{D}_{j}$ is planar regular region with
finite boundary components which are analytic Jordan curves for any $1\leq
j\leq n_{b}$. Denote that $S_{b}:=\prod_{1\leq j\leq
n_{b}}\partial\tilde{D}_{j}$. Denote that $M:=M_{a}\times
M_{b}\subset\mathbb{C}^{n_{a}+n_{b}}$ ($n=n_{a}+n_{b}$) and $S:=S_{a}\times
S_{b}$.

######  Lemma 2.40 ([13]).

Let $\lambda_{a}$ be a Lebesgue measurable function on $S_{a}$ such that
$\inf_{S_{a}}\lambda_{a}>0$, and let $\lambda_{a}$ be a Lebesgue measurable
function on $S_{b}$ such that $\inf_{S_{b}}\lambda_{b}>0$. Denote that
$\lambda:=\lambda_{a}\lambda_{b}$ on $S$. Assume that
$H^{2}_{\lambda}(M,S)\not=\\{0\\}$. Then we have
$H^{2}_{\lambda_{a}}(M_{a},S_{a})\not=\\{0\\}$ and
$H^{2}_{\lambda_{b}}(M_{b},S_{b})\not=\\{0\\}$. Furthermore,
$\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$ is a complete
orthonormal basis for $H^{2}_{\lambda}(M,S)$, where
$\\{e_{m}\\}_{m\in\mathbb{Z}_{>0}}$ is a complete orthonormal basis for
$H^{2}_{\lambda_{a}}(M_{a},S_{a})$, and
$\\{\tilde{e}_{m}\\}_{m\in\mathbb{Z}_{>0}}$ is a complete orthonormal basis
for $H^{2}_{\lambda_{b}}(M_{b},S_{b})$.

In the following, we give three product properties, which will be used in the
proof of Theorem 1.15.

Let $Z_{j}=\\{z_{j,1},z_{j,2},...,z_{j,m_{j}}\\}\subset D_{j}$ for any
$j\in\\{1,2,...,n\\}$, where $m_{j}$ is a positive integer. Denote that

$Z_{0}:=\prod_{1\leq j\leq n}Z_{j}\subset M.$

Denote that $I_{1}:=\\{(\beta_{1},\beta_{2},...,\beta_{n}):1\leq\beta_{j}\leq
m_{j}$ for any $j\in\\{1,2,...,n\\}\\}$, $V_{\beta}:=\prod_{1\leq j\leq
n}V_{z_{j,\beta_{j}}}$ and
$z_{\beta}:=(z_{1,\beta_{1}},z_{2,\beta_{2}},\ldots,z_{n,\beta_{n}})\in M$ for
any $\beta=(\beta_{1},\beta_{2},...,\beta_{n})\in I_{1}$ . Let $h_{j}$ be a
holomorphic function on a neighborhood of $Z_{j}$ for any $1\leq j\leq n$
satisfying that there exists $k\in\\{1,\ldots,m_{j}\\}$ such that
$h_{j}(z_{j,k})\not=0$. Denote that $h_{0}=\prod_{1\leq j\leq n}h_{j}$.

Let $\rho_{1}$ and $\rho_{2}$ be two Lebesgue measurable functions on
$\partial D_{1}$ and $S_{1}:=\prod_{2\leq j\leq n}\partial D_{j}$
respectively, which satisfy that $\inf_{\partial D_{1}}\rho_{1}>0$ and
$\inf_{\ S_{1}}\rho_{2}>0$. Let $\lambda_{1}$ and $\lambda_{2}$ be two
nonnegative Lebesgue measurable functions on $D_{1}$ and $M_{1}$ respectively,
which satisfy that for any relatively compact subset $R_{1}\Subset D_{1}$
($R_{2}\Subset M_{1}$), there is $a>0$ such that $\lambda_{1}^{-a}$
($\lambda_{2}^{-a}$) is integrable on $R_{1}$ ($R_{2}$). By Lemma 2.36, we
know that $\lambda_{1}$ and $\lambda_{2}$ are admissible weights on $D_{1}$
and on $M_{1}$ respectively.

Let us consider the following minimal integrals. Let $J_{\beta}$ be the
maximal ideal of $\mathcal{O}_{z_{\beta}}$ for any $\beta\in I_{1}$. Denote
that

$\begin{split}M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2}):=\inf\bigg{\\{}&\|f\|^{2}_{\partial
D_{1}\times M_{1},\rho_{1}\lambda_{2}}:f\in H_{\rho}^{2}(M,\partial
D_{1}\times M_{1})\\\ &\text{ s.t. }f^{*}(z_{\beta})=h_{0}(z_{\beta})\text{
for any $\beta\in I_{1}$}\bigg{\\}},\end{split}$ $\begin{split}M_{\partial
D_{1}}:=\inf\Bigg{\\{}\frac{1}{2\pi}\int_{\partial
D_{1}}&|f|^{2}\rho_{1}|dz_{1}|:f\in H^{2}(D_{1})\\\ &\text{ s.t.
}f(z_{j,k})=h_{1}(z_{j,k})\text{ for any $1\leq k\leq
m_{1}$}\Bigg{\\}}\end{split}$

and

$\begin{split}M_{M_{1}}:=\inf\bigg{\\{}&\int_{M_{1}}|f|^{2}\lambda_{2}:f\in\mathcal{O}(M_{1})\\\
&\text{ s.t. }f(z_{\gamma})=\prod_{2\leq l\leq n}h_{l}(z_{l,\gamma_{l}})\text{
for any $\gamma\in I_{1,1}$}\bigg{\\}},\end{split}$

where
$I_{1,1}:=\\{\gamma=(\gamma_{2},\ldots,\gamma_{n})\in\mathbb{Z}^{n-1}:1\leq\gamma_{l}\leq
m_{l}$ for any $2\leq l\leq n\\}$ and
$z_{\gamma}:=(z_{2,\gamma_{2}},\ldots,z_{n,\gamma_{n}})\in M_{j}$ for any
$\gamma\in I_{1,1}$.

######  Lemma 2.41.

$M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})=M_{\partial D_{1}}\times M_{M_{1}}$.

###### Proof.

By definitions of $M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})$, $M_{\partial D_{1}}$
and $M_{M_{1}}$, we have

$M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})\leq M_{\partial D_{1}}\times M_{M_{1}}.$

Thus, it suffices to prove $M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})\geq
M_{\partial D_{1}}\times M_{M_{1}}$.

Without loss of generality, assume that
$M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})<+\infty$. There exists $f_{0}\in
H^{2}_{\rho_{1}\lambda_{2}}(M,\partial D_{1},\times M_{1})$ satisfying
$f_{0}^{*}(z_{\beta})=h_{0}(z_{\beta})$ for any $\beta\in I_{1}$ and

(2.25) $M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})=\|f_{0}\|^{2}_{\partial
D_{1}\times M_{1},\rho_{1}\lambda_{2}}.$

As $H^{2}_{\rho_{1}\lambda_{2}}(M,\partial D_{1}\times M_{1})\not=\emptyset$,
by Lemma 2.15, $H^{2}_{\rho_{1}}(D_{1},\partial D_{1})\not=\\{0\\}$ and
$A^{2}(M_{1},\lambda_{2})\not=\\{0\\}$.

Let $\\{e_{l}\\}_{l\in\mathbb{Z}_{>0}}$ be a complete orthonormal basis for
$H^{2}_{\rho_{1}}(D_{1},\partial D_{1})$, which satisfies that
$e_{l}(z_{1,l})\not=0$ for $1\leq l\leq m_{1}$ and $e_{l}(z_{1,k})=0$ for
$0\leq k<l$. Denote that

$K_{1}:=\\{e_{l}\\}_{l>m_{1}}.$

We call $\gamma<\tilde{\gamma}$ for $\gamma,\tilde{\gamma}\in I_{1,1}$ if
there exists $s\in\\{2,\ldots,n\\}$ such that $\gamma_{l}=\tilde{\gamma}_{l}$
when $l<s$ and $\gamma_{s}>\tilde{\gamma}_{s}$. Let
$\\{\tilde{e}_{m}\\}_{m\in\mathbb{Z}_{>0}}$ be a complete orthonormal basis
for $A^{2}(M_{1},\lambda_{2})$, which satisfies that there exists
$N_{1}\in\mathbb{Z}_{>0}$ such that $\sum_{\gamma\in
I_{1,1}}|\tilde{e}_{m}(z_{\gamma})|=0$ when $m>N_{1}$ and $\sum_{\gamma\in
I_{1,1}}|\tilde{e}_{m}(z_{\gamma})|\not=0$ when $m\leq N_{1}$, and $s_{m}$ is
strictly increasing with respect to $m$ when $m\leq N_{1}$, where
$s_{m}:=\inf\\{\gamma\in I_{1,1}:\tilde{e}_{m}(z_{\gamma})\not=0\\}$. Denote
that

$K_{2}:=\\{\tilde{e}_{m}\\}_{m>N_{1}}.$

Lemma 2.15 shows that $\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$
is a complete orthonormal basis for $H^{2}_{\rho_{1}\lambda_{2}}(M,\partial
D_{1}\times M_{1})$. Then we have

$f_{0}=\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e_{l}\tilde{e}_{m}.$

By Lemma 2.14, we know that

$f_{0}^{*}=\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}\tilde{e}_{m}\,\,\,\,\text{(compactly
uniform convergence)}.$

Since there exists $k\in\\{1,\ldots,m_{1}\\}$ such that
$h_{1}(z_{1,k})\not=0$, without loss of generality, assume that
$h_{z}(z_{1,1})\not=0.$ As $f_{0}^{*}(z_{\beta})=h_{0}(z_{\beta})=\prod_{1\leq
j\leq n}h_{j}(z_{j,\beta_{j}})$ for any $\beta\in I_{1}$, we obtain that

$\frac{\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}(z_{1,1})\tilde{e}_{m}(z_{\gamma})}{h_{1}(z_{1,1})}=\prod_{2\leq
j\leq n}h_{j}(z_{j,\gamma_{j}})$

for any $\gamma\in I_{1,1}$. Note that
$\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}(z_{1,1})\tilde{e}_{m}\in\mathcal{O}(M_{1})$
and

$\begin{split}&\int_{M_{1}}|\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}(z_{1,1})\tilde{e}_{m}|^{2}\lambda_{2}\\\
=&\sum_{l,m\in\mathbb{Z}_{>0}}|a_{l,m}e^{*}_{l}(z_{1,1})|^{2}\\\
=&|e^{*}(z_{1,1})|^{2}\|f_{0}\|^{2}_{\partial D_{1}\times
M_{1},\rho_{1}\lambda_{2}}\\\ <&+\infty.\end{split}$

Thus, we have $M_{M_{1}}<+\infty$. As $H^{2}_{\rho_{1}}(D_{1},\partial
D_{1})\not=\emptyset$ and $D_{1}$ is a planar regular region bounded by finite
analytic Jordan curves, we know that $M_{\partial D_{1}}<+\infty$.

Let $f_{1}\in H^{2}_{\rho_{1}}(D_{1},\partial D_{1})$ satisfy
$f_{1}^{*}(z_{1,k})=h_{1}(z_{1,k})$ for any $1\leq k\leq m_{1}$ and

$M_{\partial D_{1}}=\frac{1}{2\pi}\int_{\partial
D_{1}}|f_{1}|^{2}\rho_{1}|dz_{1}|.$

Let $f_{2}\in\mathcal{O}(M_{1})$ satisfy that $f_{2}(z_{\gamma})=\prod_{2\leq
j\leq n}h_{j}(z_{j,\gamma_{j}})$ for any $\gamma\in I_{1,1}$ and

$M_{M_{1}}=\int_{M_{1}}|f_{2}|^{2}\lambda_{2}.$

Then we know that

(2.26) $\int_{\partial D_{1}}f_{1}\overline{f}\rho_{1}|dz_{1}|=0$

for any $f\in K_{1}$, and

(2.27) $\int_{M_{1}}f_{2}\overline{g}\lambda_{2}=0$

for any $g\in K_{2}$. Denote that

(2.28) $F_{0}:=f_{0}-f_{1}f_{2},$

then we have $F_{0}\in H^{2}_{\rho_{1}\lambda_{2}}(M,\partial D_{1}\times
M_{1})$ and $F^{*}_{0}(z_{\beta})=0$ for any $\beta\in I_{1}$. As
$\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$ is a complete
orthonormal basis for $H^{2}_{\rho_{1}\lambda_{2}}(M,\partial D_{1}\times
M_{1})$, there exists $\\{b_{l,m}\\}_{l,m\in\mathbb{Z}_{>0}}\subset\mathbb{C}$
such that

$F_{0}=\sum_{l,m\in\mathbb{Z}_{>0}}b_{l,m}e_{l}\tilde{e}_{m}=F_{1}+F_{2},$

where $F_{1}:=\sum_{1\leq l\leq m_{1}}\sum_{1\leq m\leq
N_{1}}b_{l,m}e_{l}\tilde{e}_{m}$ and $F_{2}:=\sum_{e_{l}\in K_{1}\text{ or
}\tilde{e}_{m}\in K_{2}}b_{l,m}e_{l}\tilde{e}_{m}$. Note that
$F^{*}_{2}(z_{\beta})=0$ for any $\beta\in I_{1}$, then
$F^{*}_{1}(z_{\beta})=0$ for any $\beta\in I_{1}$. By the construction of
$\\{e_{l}\\}_{1\leq l\leq m_{1}}$ and $\\{\tilde{e}_{m}\\}_{1\leq m\leq
N_{1}}$, we know $b_{l,m}=0$ for $1\leq l\leq m_{1}$ and $1\leq m\leq N_{1}$,
i.e.,

$F_{1}\equiv 0.$

Note that $F_{0}=F_{2}=\sum_{e_{l}\in K_{1}\text{ or }\tilde{e}_{m}\in
K_{2}}b_{l,m}e_{l}\tilde{e}_{m},$ then it follows from equality (2.25),
(2.26), (2.27) and (2.28) that

$\begin{split}M_{H,1}(Z_{0},J,\rho_{1}\lambda_{2})&=\|f_{0}\|^{2}_{\partial
D_{1}\times M_{1},\rho_{1}\lambda_{2}}\\\ &=\|F_{2}\|^{2}_{\partial
D_{1}\times M_{1},\rho_{1}\lambda_{2}}+\|f_{1}f_{2}\|^{2}_{\partial
D_{1}\times M_{1},\rho_{1}\lambda_{2}}\\\ &\geq M_{\partial D_{1}}\times
M_{M_{1}}.\end{split}$

Thus, Lemma 2.41 holds. ∎

Let us consider the following minimal integrals. Denote that

$\begin{split}M_{S}(Z_{0},J,\rho_{1}\rho_{2}):=\inf\bigg{\\{}&\|f\|^{2}_{S,\rho_{1}\rho_{2}}:f\in
H_{\rho_{1}\rho_{2}}^{2}(M,S)\\\ &\text{ s.t.
}f^{*}(z_{\beta})=h_{0}(z_{\beta})\text{ for any $\beta\in
I_{1}$}\bigg{\\}},\end{split}$

and

$\begin{split}M_{S_{1}}:=\inf\bigg{\\{}&\|f\|^{2}_{S_{1},\rho_{2}}:f\in
H_{\rho_{2}}^{2}(M_{1},S_{1})\\\ &\text{ s.t. }f^{*}(z_{\gamma})=\prod_{2\leq
j\leq n}h_{j}(z_{j,\gamma_{j}})\text{ for any $\gamma\in
I_{1,1}$}\bigg{\\}}.\end{split}$

We give a product property as follows:

######  Lemma 2.42.

$M_{S}(Z_{0},J,\rho_{1}\rho_{2})=M_{\partial D_{1}}\times M_{S_{1}}$.

###### Proof.

The proof is similar to the proof of Lemma 2.41. By definitions of
$M_{S}(Z_{0},J,\rho_{1}\rho_{2})$, $M_{\partial D_{1}}$ and $M_{S_{1}}$, we
have

$M_{S}(Z_{0},J,\rho_{1}\rho_{2})\leq M_{\partial D_{1}}\times M_{S_{1}}.$

Thus, it suffices to prove $M_{S}(Z_{0},J,\rho_{1}\rho_{2})\geq M_{\partial
D_{1}}\times M_{S_{1}}$.

Without loss of generality, assume that
$M_{S}(Z_{0},J,\rho_{1}\rho_{2})<+\infty$. By Lemma 2.39, there exists
$f_{0}\in H^{2}_{\rho_{1}\rho_{2}}(M,S)$ satisfying
$f_{0}^{*}(z_{\beta})=h_{0}(z_{\beta})$ for any $\beta\in I_{1}$ and

(2.29) $M_{S}(Z_{0},J,\rho_{1}\rho_{2})=\|f_{0}\|^{2}_{S,\rho_{1}\rho_{2}}.$

As $H^{2}_{\rho_{1}\rho_{2}}(M,S)\not=\emptyset$, by Lemma 2.40,
$H^{2}_{\rho_{1}}(D_{1},\partial D_{1})\not=\\{0\\}$ and
$H^{2}_{\rho_{2}}(M_{1},S_{1})\not=\\{0\\}$.

Let $\\{e_{l}\\}_{l\in\mathbb{Z}_{>0}}$ be a complete orthonormal basis for
$H^{2}_{\rho_{1}}(D_{1},\partial D_{1})$, which satisfies that
$e_{l}(z_{1,l})\not=0$ for $1\leq l\leq m_{1}$ and $e_{l}(z_{1,k})=0$ for
$0\leq k<l$. Denote that

$K_{1}:=\\{e_{l}\\}_{l>m_{1}}.$

We call $\gamma<\tilde{\gamma}$ for $\gamma,\tilde{\gamma}\in I_{1,1}$ if
there exists $s\in\\{2,\ldots,n\\}$ such that $\gamma_{l}=\tilde{\gamma}_{l}$
when $l<s$ and $\gamma_{s}>\tilde{\gamma}_{s}$. Let
$\\{\tilde{e}_{m}\\}_{m\in\mathbb{Z}_{>0}}$ be a complete orthonormal basis
for $H^{2}_{\rho_{2}}(M_{1},S_{1})$, which satisfies that there exists
$N_{1}\in\mathbb{Z}_{>0}$ such that $\sum_{\gamma\in
I_{1,1}}|\tilde{e}_{m}(z_{\gamma})|=0$ when $m>N_{1}$ and $\sum_{\gamma\in
I_{1,1}}|\tilde{e}_{m}(z_{\gamma})|\not=0$ when $m\leq N_{1}$, and $s_{m}$ is
strictly increasing with respect $m$ when $m\leq N_{1}$, where
$s_{m}:=\inf\\{\gamma\in I_{1,1}:\tilde{e}_{m}(z_{\gamma})\not=0\\}$. Denote
that

$K_{2}:=\\{\tilde{e}_{m}\\}_{m>N_{1}}.$

Lemma 2.40 shows that $\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$
is a complete orthonormal basis for $H^{2}_{\rho_{1}\rho_{2}}(M,S)$. Then we
have

$f_{0}=\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e_{l}\tilde{e}_{m}.$

By Lemma 2.38, we know that

$f_{0}^{*}=\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}\tilde{e}^{*}_{m}\,\,\,\,\text{(compactly
uniform convergence)}.$

Since there exists $k\in\\{1,\ldots,m_{1}\\}$ such that
$h_{1}(z_{1,k})\not=0$, without loss of generality, assume that
$h_{z}(z_{1,1})\not=0.$ Note that
$\sum_{l,m\in\mathbb{Z}_{>0}}|a_{l,m}e^{*}_{l}(z_{1,1})|^{2}<+\infty$, then we
have

$\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}(z_{1,1})\tilde{e}_{m}\in
H^{2}_{\rho_{2}}(M_{1},S_{1}).$

As $f^{*}(z_{\beta})=h_{0}(z_{\beta})=\prod_{1\leq j\leq
n}h_{j}(z_{j,\beta_{j}})$ for any $\beta\in I_{1}$, we obtain that

$\frac{\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e^{*}_{l}(z_{1,1})\tilde{e}^{*}_{m}(z_{\gamma})}{h_{1}(z_{1,1})}=\prod_{2\leq
j\leq n}h_{j}(z_{j,\gamma_{j}})$

for any $\gamma\in I_{1,1}$. Thus, we have $M_{S_{1}}<+\infty$. Similarly, we
have $M_{\partial D_{1}}<+\infty$.

Let $f_{1}\in H^{2}_{\rho_{1}}(D_{1},\partial D_{1})$ satisfy
$f_{1}^{*}(z_{1,k})=h_{1}(z_{1,k})$ for any $1\leq k\leq m_{1}$ and

$M_{\partial D_{1}}=\frac{1}{2\pi}\int_{\partial
D_{1}}|f_{1}|^{2}\rho_{1}|dz_{1}|.$

Let $f_{2}\in H^{2}_{\rho_{2}}(M_{1},S_{1})$ satisfy that
$f^{*}_{2}(z_{\gamma})=\prod_{2\leq j\leq n}h_{j}(z_{j,\gamma_{j}})$ for any
$\gamma\in I_{1,1}$ and

$M_{M_{1}}=\|f_{2}\|^{2}_{S_{1},\rho_{2}}.$

Then we know that

(2.30) $\int_{\partial D_{1}}f_{1}\overline{f}\rho_{1}|dz_{1}|=0$

for any $f\in K_{1}$, and

(2.31) $\ll f_{2},g\gg_{S_{1},\rho_{2}}=0$

for any $g\in K_{2}$. Denote that

(2.32) $F_{0}:=f_{0}-f_{1}f_{2},$

then we have $F_{0}\in H^{2}_{\rho_{1}\rho_{2}}(M,S)$ and
$F^{*}_{0}(z_{\beta})=0$ for any $\beta\in I_{1}$. As
$\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$ is a complete
orthonormal basis for $H^{2}_{\rho_{1}\rho_{2}}(M,S)$, there exists
$\\{b_{l,m}\\}_{l,m\in\mathbb{Z}_{>0}}\subset\mathbb{C}$ such that

$F_{0}=\sum_{l,m\in\mathbb{Z}_{>0}}b_{l,m}e_{l}\tilde{e}_{m}=F_{1}+F_{2},$

where $F_{1}:=\sum_{1\leq l\leq m_{1}}\sum_{1\leq m\leq
N_{1}}b_{l,m}e_{l}\tilde{e}_{m}$ and $F_{2}:=\sum_{e_{l}\in K_{1}\text{ or
}\tilde{e}_{m}\in K_{2}}b_{l,m}e_{l}\tilde{e}_{m}$. Note that
$F^{*}_{2}(z_{\beta})=0$ for any $\beta\in I_{1}$, then
$F^{*}_{1}(z_{\beta})=0$ for any $\beta\in I_{1}$. By the construction of
$\\{e_{l}\\}_{1\leq l\leq m_{1}}$ and $\\{\tilde{e}_{m}\\}_{1\leq m\leq
N_{1}}$, we know $b_{l,m}=0$ for $1\leq l\leq m_{1}$ and $1\leq m\leq N_{1}$,
i.e.,

$F_{1}\equiv 0.$

Note that $F_{0}=F_{2}=\sum_{e_{l}\in K_{1}\text{ or }\tilde{e}_{m}\in
K_{2}}b_{l,m}e_{l}\tilde{e}_{m},$ then it follows from equality (2.29),
(2.30), (2.31) and (2.32) that

$M_{S}(Z_{0},J,\rho_{1}\rho_{2})=\|f_{0}\|^{2}_{S,\rho_{1}\rho_{2}}=\|F_{2}\|^{2}_{S,\rho_{1}\rho_{2}}+\|f_{1}f_{2}\|^{2}_{S,\rho_{1}\rho_{2}}\geq
M_{\partial D_{1}}\times M_{S_{1}}.$

Thus, Lemma 2.42 holds. ∎

Let us consider the following minimal integrals. Denote that

$\begin{split}M_{D_{1}}:=\inf\bigg{\\{}&\int_{D_{1}}|f|^{2}\lambda_{1}:f\in\mathcal{O}(D_{1})\\\
&\text{ s.t. }f(z_{1,k})=h_{1}(z_{1,k})\text{ for any $1\leq k\leq
m_{1}$}\bigg{\\}},\end{split}$

and

$\begin{split}M_{M}:=\inf\bigg{\\{}&\int_{M}|f|^{2}\lambda_{1}\lambda_{2}:f\in\mathcal{O}(M)\\\
&\text{ s.t. }f(z_{\beta})=h_{0}(z_{\beta})\text{ for any $\beta\in
I_{1}$}\bigg{\\}}.\end{split}$

######  Lemma 2.43.

$M_{M}=M_{D_{1}}\times M_{M_{1}}.$

###### Proof.

The proof is similar to the proof of Lemma 2.41. By definitions of $M_{M}$,
$M_{D_{1}}$ and $M_{M_{1}}$, we have $M_{M}\leq M_{D_{1}}\times M_{M_{1}}$.
Thus, it suffices to prove $M_{M}\geq M_{D_{1}}\times M_{M_{1}}$.

Without loss of generality, assume that $M_{M}<+\infty$. There exists
$f_{0}\in\mathcal{O}(M)$ satisfying $f_{0}(z_{\beta})=h_{0}(z_{\beta})$ for
any $\beta\in I_{1}$ and

(2.33) $M_{M}=\int_{M}|f_{0}|^{2}\lambda_{1}\lambda_{2}.$

Let $\\{e_{l}\\}_{l\in\mathbb{Z}_{>0}}$ be a complete orthonormal basis for
$A^{2}(D_{1},\lambda_{1})$, which there exists $N_{1}\in\mathbb{Z}_{>0}$ such
that $\sum_{1\leq k\leq m_{1}}|e_{l}(z_{1,k})|=0$ when $l>N_{1}$ and
$\sum_{1\leq k\leq m_{1}}|e_{l}(z_{1,k})|\not=0$ when $l\leq N_{1}$, and
$s_{l}$ is strictly increasing with respect $l$ when $l\leq N_{1}$, where
$s_{l}:=\inf\\{k\in\\{1,\ldots,m_{1}\\}:e_{l}(z_{1,k})\not=0\\}$. Denote that

$K_{1}:=\\{e_{l}\\}_{l>N_{1}}.$

Let $\\{\tilde{e}_{m}\\}_{m\in\mathbb{Z}_{>0}}$ be a complete orthonormal
basis for $A^{2}(M_{1},\lambda_{2})$, which satisfies that there exists
$N_{2}\in\mathbb{Z}_{>0}$ such that $\sum_{\gamma\in
I_{1,1}}|\tilde{e}_{m}(z_{\gamma})|=0$ when $m>N_{2}$ and $\sum_{\gamma\in
I_{1,1}}|\tilde{e}_{m}(z_{\gamma})|\not=0$ when $m\leq N_{2}$, and
$\tilde{s}_{m}$ is strictly increasing with respect $m$ when $m\leq N_{2}$,
where $\tilde{s}_{m}:=\inf\\{\gamma\in
I_{1,1}:\tilde{e}_{m}(z_{\gamma})\not=0\\}$. Denote that

$K_{2}:=\\{\tilde{e}_{m}\\}_{m>N_{2}}.$

Lemma 2.37 shows that $\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$
is a complete orthonormal basis for $A^{2}(M,\lambda_{1}\lambda_{2})$. Then we
have

$f_{0}=\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e_{l}\tilde{e}_{m}.$

Since there exists $k\in\\{1,\ldots,m_{1}\\}$ such that
$h_{1}(z_{1,k})\not=0$, without loss of generality, assume that
$h_{z}(z_{1,1})\not=0.$ As $f_{0}(z_{\beta})=h_{0}(z_{\beta})=\prod_{1\leq
j\leq n}h_{j}(z_{j,\beta_{j}})$ for any $\beta\in I_{1}$, we obtain that

$\frac{\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e_{l}(z_{1,1})\tilde{e}_{m}(z_{\gamma})}{h_{1}(z_{1,1})}=\prod_{2\leq
j\leq n}h_{j}(z_{j,\gamma_{j}})$

for any $\gamma\in I_{1,1}$. Note that
$\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e_{l}(z_{1,1})\tilde{e}_{m}\in\mathcal{O}(M_{1})$
and

$\begin{split}&\int_{M_{1}}|\sum_{l,m\in\mathbb{Z}_{>0}}a_{l,m}e_{l}(z_{1,1})\tilde{e}_{m}|^{2}\lambda_{2}\\\
=&\sum_{l,m\in\mathbb{Z}_{>0}}|a_{l,m}e_{l}(z_{1,1})|^{2}\\\
=&|e(z_{1,1})|^{2}\int_{M}|f_{0}|^{2}\lambda_{1}\lambda_{2}\\\
<&+\infty.\end{split}$

Thus, we have $M_{M_{1}}<+\infty$. Similarly, we have $M_{D_{1}}<+\infty$.

Let $f_{1}\in\mathcal{O}(D_{1})$ satisfy $f_{1}(z_{1,k})=h_{1}(z_{1,k})$ for
any $1\leq k\leq m_{1}$ and

$M_{D_{1}}=\int_{D_{1}}|f_{1}|^{2}\lambda_{1}.$

Let $f_{2}\in\mathcal{O}(M_{1})$ satisfy that $f_{2}(z_{\gamma})=\prod_{2\leq
j\leq n}h_{j}(z_{j,\gamma_{j}})$ for any $\gamma\in I_{1,1}$ and

$M_{M_{1}}=\int_{M_{1}}|f_{2}|^{2}\lambda_{2}.$

Then we know that

(2.34) $\int_{D_{1}}f_{1}\overline{f}\lambda_{1}=0$

for any $f\in K_{1}$, and

(2.35) $\int_{M_{1}}f_{2}\overline{g}\lambda_{2}=0$

for any $g\in K_{2}$. Denote that

(2.36) $F_{0}:=f_{0}-f_{1}f_{2},$

then we have $F_{0}\in A^{2}(M,\lambda_{1}\lambda_{2})$ and
$F_{0}(z_{\beta})=0$ for any $\beta\in I_{1}$. As
$\\{e_{m}(z)\tilde{e}_{l}(w)\\}_{m,l\in\mathbb{Z}_{>0}}$ is a complete
orthonormal basis for $A^{2}(M,\lambda_{1}\lambda_{2})$, there exists
$\\{b_{l,m}\\}_{l,m\in\mathbb{Z}_{>0}}\subset\mathbb{C}$ such that

$F_{0}=\sum_{l,m\in\mathbb{Z}_{>0}}b_{l,m}e_{l}\tilde{e}_{m}=F_{1}+F_{2},$

where $F_{1}:=\sum_{1\leq l\leq N_{1}}\sum_{1\leq m\leq
N_{2}}b_{l,m}e_{l}\tilde{e}_{m}$ and $F_{2}:=\sum_{e_{l}\in K_{1}\text{ or
}\tilde{e}_{m}\in K_{2}}b_{l,m}e_{l}\tilde{e}_{m}$. Note that
$F_{2}(z_{\beta})=0$ for any $\beta\in I_{1}$, then $F_{1}(z_{\beta})=0$ for
any $\beta\in I_{1}$. By the construction of $\\{e_{l}\\}_{1\leq l\leq N_{1}}$
and $\\{\tilde{e}_{m}\\}_{1\leq m\leq N_{2}}$, we know $b_{l,m}=0$ for $1\leq
l\leq N_{1}$ and $1\leq m\leq N_{2}$, i.e.,

$F_{1}\equiv 0.$

Note that $F_{0}=F_{2}=\sum_{e_{l}\in K_{1}\text{ or }\tilde{e}_{m}\in
K_{2}}b_{l,m}e_{l}\tilde{e}_{m},$ then it follows from equality (2.33),
(2.34), (2.35) and (2.36) that

$M_{M}=\int_{M}|f_{0}|^{2}\lambda_{1}\lambda_{2}=\int_{M}|F_{2}|^{2}\lambda_{1}\lambda_{2}+\int_{M}|f_{1}f_{2}|^{2}\lambda_{1}\lambda_{2}\geq
M_{D_{1}}\times M_{M_{1}}.$

Thus, Lemma 2.43 holds. ∎

##  3\. Proofs of Theorem 1.5, Remark 1.6, Corollary 1.7 and Corollary 1.8

In this section, we prove Theorem 1.5, Remark 1.6, Corollary 1.7 and Corollary
1.8.

###  3.1. Proof of Theorem 1.5

We prove Theorem 1.5 in three steps.

_Step 1: proof of inequality ( 1.1)_

Denote

$\begin{split}\inf\bigg{\\{}&\int_{\\{2\psi<-t\\}}|f|^{2}\tilde{\rho}:f\in\mathcal{O}(\\{2\psi<-t\\})\\\
&\text{ s.t. }f^{(l)}(z_{j})=a_{j,l}\text{ for any $0\leq l\leq k_{j}$ and any
$1\leq j\leq m$}\bigg{\\}}\end{split}$

by $G(t)$ for $t\geq 0$. Note that $\tilde{\rho}=e^{-\varphi}c(-2\psi)$ and
$G(0)=M(Z_{0},\mathfrak{a},\tilde{\rho}).$ As
$v(dd^{c}(\varphi+2\psi),z_{j})\geq 2(k_{j}+1)$ for any $1\leq j\leq m$, it
follows from Theorem 2.23 that $G(h^{-1}(r))$ is concave, where
$h(t)=\int_{t}^{+\infty}c(s)e^{-s}ds$.

By Lemma 2.25, there exists a holomorphic function $F_{0}$ on $D$ such that
$f^{(l)}(z_{j})=a_{j,l}$ for any $0\leq l\leq k_{j}$ and any $1\leq j\leq m$,
and

$G(0)=\int_{D}|F_{0}|^{2}\tilde{\rho}.$

By definition of $G(t)$, we have

$G(-\log r)\leq\int_{\\{2\psi<\log r\\}}|F_{0}|^{2}\tilde{\rho}$

for any $r\in(0,1]$, then combining the concavity of $G(h^{-1}(r))$, we obtain
that

(3.1) $\frac{\int_{\\{z\in D:2\psi(z)\geq\log
r\\}}|F_{0}(z)|^{2}\tilde{\rho}}{\int_{0}^{-\log
r}c(t)e^{-t}dt}\leq\frac{G(0)-G(-\log r)}{\int_{0}^{-\log
r}c(t)e^{-t}dt}\leq\frac{G(0)}{\int_{0}^{+\infty}c(t)e^{-t}dt}<+\infty.$

Since $\lim_{t\rightarrow 0+0}c(t)=c(0)=1$ and $\lim_{w\rightarrow
z}\varphi(w)=\varphi(z)$ for any $z\in\partial D$, it follows from Lemma 2.13
and inequality (3.1) that $F_{0}\in H^{2}(D)$ and

(3.2)
$\begin{split}M_{H}(Z_{0},\mathfrak{a},\rho)&\leq\frac{1}{2\pi}\int_{\partial
D}|F_{0}|^{2}\rho|dz|\\\ &\leq\frac{1}{2\pi}\liminf_{r\rightarrow
1-0}\frac{\int_{\\{z\in D:\psi\geq\log r\\}}|F_{0}|^{2}\tilde{\rho}}{1-r}\\\
&=\frac{1}{2\pi}\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in D:2\psi\geq\log
r\\}}|F_{0}|^{2}\tilde{\rho}}{\int_{0}^{-\log
r}c(t)e^{-t}dt}\times\frac{\int_{0}^{-\log
r}c(t)e^{-t}dt}{1-r^{\frac{1}{2}}}\\\
&\leq\frac{1}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}M(Z_{0},\mathfrak{a},\tilde{\rho})\end{split}$

Thus, inequality (1.1) holds.

_Step 2: necessity of the characterization_

Assume that the equality

(3.3)
$M_{H}(Z_{0},\mathfrak{a},\rho)=\frac{M(Z_{0},\mathfrak{a},\tilde{\rho})}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}$

holds. Combining inequality (3.1) and inequality (3.2), we get that

$\liminf_{r\rightarrow 1-0}\frac{\int_{\\{z\in D:2\psi(z)\geq\log
r\\}}|F_{0}(z)|^{2}\tilde{\rho}}{\int_{0}^{-\log
r}c(t)e^{-t}dt}=\liminf_{r\rightarrow 1-0}\frac{G(0)-G(-\log
r)}{\int_{0}^{-\log
r}c(t)e^{-t}dt}=\frac{G(0)}{\int_{0}^{+\infty}c(t)e^{-t}dt}.$

Since $G(h^{-1}(r))$ is concave, we know that $G(h^{-1}(r))$ is linear with
respect to $r\in(0,\int_{0}^{+\infty}c(t)e^{-t}dt)$. By Theorem 2.26, we get
that

$(1)$ $\varphi+2\psi=2\log|g_{1}|+2\sum_{1\leq j\leq
m}G_{D}(\cdot,z_{j})+2u_{1}$, where $g_{1}$ is a holomorphic function on $D$
such that $ord_{z_{j}}(g_{1})=\min\\{l:a_{j,l}\not=0\\}$, and $u_{1}$ is a
harmonic function on $D$;

$(2)$ $\psi=\sum_{1\leq j\leq m}p_{j}G_{D}(\cdot,z_{j})$;

$(3)$ $\chi_{-u_{1}}=\prod_{1\leq j\leq m}\chi_{z_{j}}$;

$(4)$ For any $1\leq j\leq m$,

(3.4) $\lim_{z\rightarrow
z_{j}}\frac{g_{1}p_{*}\left(f_{u_{1}}\left(\prod_{1\leq j\leq
m}f_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{\sum_{0\leq l\leq
k_{j}}a_{j,l}(z-z_{j})^{l}dz}=c_{0}$

holds, where $c_{0}\not=0$ is a constant independent of $j$.

As $v(dd^{c}(\varphi+2\psi),z_{j})\geq 2(k_{j}+1)$, it follows from statements
$(1)$ and $(4)$ above that, for any $1\leq j\leq m$,
$ord_{z_{j}}(g_{1})=k_{j}$ and $a_{j}=0$ for any $l<k_{j}$. By Weierstrass
theorem (see [3]), there exists a holomorphic function $g_{0}$ on $D$ such
that $dg_{0}\not=0$ on $\Omega\backslash Z_{0}$ and $ord_{z_{j}}(g)=k_{j}$ for
any $j$. Denote that

$g_{2}:=\frac{g_{1}}{g_{0}}\quad\text{and}\quad
u_{2}:=u_{1}+\log|g_{0}|-\sum_{1\leq j\leq m}k_{j}G_{D}(\cdot,z_{j})$

on $D$. Note that $u_{2}$ is harmonic on $D$, and $g_{2}$ is harmonic on $D$
satisfying $dg_{2}(z_{j})\not=0$ for any $1\leq j\leq m$. Combining statements
$(1)$ and $(3)$ above, we have

$\varphi+2\psi=2\log|g_{2}|+2\sum_{1\leq j\leq
m}(k_{j}+1)G_{D}(\cdot,z_{j})+2u_{2}$

and

$\chi_{-u_{2}}=\prod_{1\leq j\leq m}\chi_{z_{j}}^{k_{j}+1}.$

In the following, we prove that $g_{2}\not=0$ on $D$. Then
$u:=\log|g_{2}|+u_{2}$ is harmonic on $D$ and $\chi_{-u}=\prod_{1\leq j\leq
m}\chi_{z_{j}}^{k_{j}+1}$. Combining equality (3.4),
$\log|g_{1}|+u_{1}=\log|g_{0}|+\log|g_{2}|+u_{1}=u+\sum_{1\leq j\leq
m}k_{j}G_{D}(\cdot,z_{j})$ and $a_{j,l}=0$ for any $l<k_{j}$, we have

$\lim_{z\rightarrow z_{j}}\frac{p_{*}\left(f_{u}\left(\prod_{1\leq j\leq
m}f^{k_{j}+1}_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{a_{j,k_{j}}(z-z_{j})^{k_{j}}dz}=c_{0},$

thus the necessity of the characterization holds.

Denote that $h:=\varphi+2\psi-2\sum_{1\leq j\leq
m}(k_{j}+1)G_{D}(\cdot,z_{j})$ on $\overline{D}$. Then $h$ is subharmonic on
$D$ and $h$ is continuous at $z$ for any $z\in\partial D$. It suffices to
prove that $h$ is harmonic on $D$. By solving Dirichlet problem, there is a
continuous function $\tilde{h}$ on $\overline{D}$, which satisfies that
$\tilde{h}=h$ on $\partial D$ and $\tilde{h}$ is harmonic on $D$. As $h$ is
subharmonic on $D$, we have

$h\leq\tilde{h}$

on $\overline{D}$. Denote that

$\tilde{\varphi}:=\varphi+\tilde{h}-h.$

Then we have $\tilde{\varphi}|_{\partial D}=\varphi|_{\partial D}$ and
$\tilde{\varphi}+2\psi=2\sum_{1\leq j\leq
m}(k_{j}+1)G_{D}(\cdot,z_{j})+\tilde{h}$. Denote that

$\tilde{\rho}_{1}:=e^{-\tilde{\varphi}}c(-2\psi)$

on $\overline{D}$. Note that $\tilde{\rho}_{1}\leq\tilde{\rho}$. By
definition, we have

$M(Z_{0},\mathfrak{a},\tilde{\rho})\geq
M(Z_{0},\mathfrak{a},\tilde{\rho}_{1}).$

Combining equality (3.3) and inequality (1.1), we have

$M_{H}(Z_{0},\mathfrak{a},\rho)\leq\frac{M(Z_{0},\mathfrak{a},\tilde{\rho}_{1})}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}\leq\frac{M(Z_{0},\mathfrak{a},\tilde{\rho})}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}=M_{H}(Z_{0},\mathfrak{a},\rho),$

which shows that

$M(Z_{0},\mathfrak{a},\tilde{\rho})=M(Z_{0},\mathfrak{a},\tilde{\rho}_{1}).$

As $M(Z_{0},\mathfrak{a},\tilde{\rho})<+\infty$ and $\sum_{1\leq j\leq
m}\sum_{0\leq l\leq k_{j}}|a_{j,l}|\not=0$, we have
$\tilde{\rho}_{1}=\tilde{\rho}$, which implies that $2\log|g_{2}|$ is harmonic
on $D$, i.e. $g_{2}\not=0$ on $D$.

Thus, the necessity of characterization in Theorem 1.5 has been proved.

_Step 3: sufficiency of the characterization_

Assume that the four statements $(1)-(4)$ in Theorem 1.5 hold. By Weierstrass
theorem (see [3]), there exists a holomorphic function $g_{0}$ on $D$ such
that $dg_{0}\not=0$ on $\Omega\backslash Z_{0}$ and $ord_{z_{j}}(g)=k_{j}$ for
any $j$. Denote that $\tilde{u}=u+\sum_{1\leq j\leq
m}k_{j}G_{D}(\cdot,z_{j})-\log|g_{0}|$ is a harmonic on $\Omega$. Thus, we
have $\varphi+2\psi=2\log|g_{0}|+\sum_{1\leq j\leq
m}2G_{D}(\cdot,z_{j})+2\tilde{u}$, $\chi_{-\tilde{u}}=\chi_{-u}\prod_{1\leq
j\leq m}\chi_{z_{j}}^{-k_{j}}=\prod_{1\leq j\leq m}\chi_{z_{j}}$ and

$\lim_{z\rightarrow
z_{j}}\frac{g_{0}p_{*}\left(f_{\tilde{u}}\left(\prod_{1\leq j\leq
m}f_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{a_{j,k_{j}}(z-z_{j})^{k_{j}}dz}=c_{0}$

for any $j$. Then, by Theorem 2.26, we know that $G(h^{-1}(r))$ is linear with
respect to $r\in(0,\int_{0}^{+\infty}c(t)e^{-t}dt)$, where the definition of
$G(t)$ comes from Step 1.

Using Corollary 2.24 and Remark 2.27, we obtain that

(3.5) $G(t)=\int_{\\{2\psi<-t\\}}|F_{0}|^{2}\tilde{\rho}$

for any $t\geq 0$ and

(3.6)
$\begin{split}F_{0}=&\frac{g_{0}p_{*}\left(f_{\tilde{u}}\left(\prod_{1\leq
j\leq m}f_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{c_{0}dz}\\\
=&\frac{p_{*}\left(f_{u}\left(\prod_{1\leq j\leq
m}f^{k_{j}+1}_{z_{j}}\right)\left(\sum_{1\leq j\leq
m}p_{j}\frac{df_{z_{j}}}{f_{z_{j}}}\right)\right)}{c_{0}dz},\end{split}$

where $p$ is the universal covering from unit disc $\Delta$ to $D$,
$f_{\tilde{u}}$ is a holomorphic function on $\Delta$ such that
$|f_{\tilde{u}}|=p^{*}(e^{\tilde{u}})$, $f_{z_{0}}$ is a holomorphic function
on $\Delta$ such that $|f_{z_{j}}|=p^{*}(e^{G_{D}(\cdot,z_{j})})$ for any $j$,
and $f_{u}=\frac{f_{\tilde{u}}g_{0}}{\prod_{1\leq j\leq m}f_{z_{j}}^{k_{j}}}$
satisfies that $|f_{u}|=p^{*}\left(e^{u}\right)$. Note that
$u=\frac{\varphi}{2}+\psi-\sum_{1\leq j\leq m}(k_{j}+1)G_{D}(\cdot,z_{j})$ can
be extended to a continuous function on $\overline{D}$, then we know

$|F_{0}|\in C(\overline{D}).$

Let $f\in H^{2}(D)$ satisfying $f^{(l)}(z_{j})=l!a_{j,l}$ for any $1\leq l\leq
k_{j}$ and any $1\leq j\leq m$. Note that
$(f-F_{0},z_{j})\in\mathcal{I}(\varphi+2\psi)_{z_{j}}$ for any $j$,
$c(t)e^{-t}$ is decreasing and $\\{\psi<-t\\}\Subset D$ for any $t>0$, then it
follows from $\int_{D}|F_{0}|^{2}\tilde{\rho}<+\infty$ that

$\begin{split}\int_{\\{2\psi<-t\\}}|f|^{2}\tilde{\rho}&\leq
2\int_{\\{2\psi<-t\\}}|f-F_{0}|^{2}e^{-\varphi}c(-2\psi)+2\int_{D}|F_{0}|^{2}\tilde{\rho}\\\
&\leq
2C\int_{\\{2\psi<-t\\}}|f-F_{0}|^{2}e^{-\varphi-2\psi}+2\int_{D}|F_{0}|^{2}\tilde{\rho}\\\
&<+\infty\end{split}$

for any $t>0$. Following from Lemma 2.25, we have

$\begin{split}\int_{\\{2\psi<-t\\}}|f|^{2}\tilde{\rho}=&\int_{\\{2\psi<-t\\}}|F_{0}|^{2}\tilde{\rho}+\int_{\\{2\psi<-t\\}}|f-F_{0}|^{2}\tilde{\rho},\end{split}$

which implies that

$\int_{\\{2\psi<-t\\}}F_{0}\overline{F_{0}-f}\tilde{\rho}=0$

for any $t>0$. It follows from Lemma 2.12 and Lemma 2.8 that there exists
$r_{1}>0$ such that

$\int_{\\{z\in
D:\psi(z)=r\\}}F_{0}\overline{F_{0}-f}e^{-\varphi}\left(\frac{\partial\psi}{\partial
v_{z}}\right)^{-1}|dz|=0$

holds for any $r\in(-r_{1},0)$, which implies that

(3.7) $\int_{\\{z\in
D:\psi(z)=r\\}}|f|^{2}e^{-\varphi}\left(\frac{\partial\psi}{\partial
v_{z}}\right)^{-1}|dz|\geq\int_{\\{z\in
D:\psi(z)=r\\}}|F_{0}|^{2}e^{-\varphi}\left(\frac{\partial\psi}{\partial
v_{z}}\right)^{-1}|dz|.$