# concavity property of minimal $L^{2}$ integrals with Lebesgue measurable
gain

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 present a concavity property of the minimal $L^{2}$
integrals related to multiplier ideal sheaves with Lebesgue measurable gain.
As applications, we give necessary conditions for our concavity degenerating
to linearity, characterizations for 1-dimensional case, and a characterization
for the holding of the equality in optimal $L^{2}$ extension problem on open
Riemann surfaces with weights may not be subharmonic.

###### Key words and phrases:

concavity, minimal $L^{2}$ integral, multiplier ideal sheaf, plurisubharmonic
function, sublevel set

###### 2010 Mathematics Subject Classification:

32D15 32E10 32L10 32U05 32W05

##  1\. Introduction

The multiplier ideal sheaves related to plurisubharmonic functions plays an
important role in complex geometry and algebraic geometry (see e.g. [52, 37,
46, 9, 10, 6, 11, 34, 48, 49, 7]). Recall the definition of the multiplier
ideal sheaves as follows (see [7]).

_The multiplier ideal sheaf $\mathcal{I}(\varphi)$ was defined as the sheaf of
germs of holomorphic functions $f$ such that $|f|^{2}e^{-\varphi}$ is locally
integrable, where $\varphi$ is a plurisubharmonic function on a complex
manifold $M$._

The strong openness conjecture is
$\mathcal{I}(\varphi)=\mathcal{I}_{+}(\varphi):=\cup_{\varepsilon>0}\mathcal{I}((1+\varepsilon)\varphi)$,
which was posed by Demailly (see [6]) and be proved by Guan-Zhou [26] (the
dimension two case was proved by Jonsson-Mustata [31]). When
$\mathcal{I}(\varphi)=\mathcal{O}$, this conjecture is called the openness
conjecture, which was posed by Demailly-Kollár [10], and was proved by
Berndtsson [3] (the dimension two case was proved by Favre-Jonsson [13]) by
establishing an effectiveness result of the openness conjecture.

Simulated by Berndtsson’s effectiveness result, continuing the solution of the
strong openness conjecture [26], Guan-Zhou [27] established a non-sharp
effectiveness result of the strong openness conjecture. Recall that for the
first time Guan-Zhou [27] considered the minimal $L^{2}$ integral related to
multiplier ideals on the sublevel set $\\{\varphi<0\\}$ i.e. the pseudoconvex
domain $D$.

In [18], by considering all the minimal $L^{2}$ integrals on the sublevels of
the weights $\varphi$, Guan presented a sharp version of the effectiveness
result of the strong openness conjecture, and obtained a concavity property of
the minimal $L^{2}$ integrals without gain. In [19], Guan generalized the
concavity property in [18] to minimal $L^{2}$ integrals with smooth gain.

In [20], Guan-Mi gave some applications of the concavity property in [19]: a
necessary condition for the concavity degenerating to linearity, a
characterization for 1-dimensional case, and a characterization for the
holding of the equality in optimal $L^{2}$ extension problem on open Riemann
surfaces with subharmonic weights. Recall that if the subharmonic weights
degenerate to $0$, the characterization for the holding of the equality in
optimal $L^{2}$ extension problem on open Riemann surfaces is the solution of
the equality part of the Suita conjecture in [25]; if the subharmonic weights
degenerate to harmonic, the characterization for the holding of the equality
in optimal $L^{2}$ extension problem on open Riemann surfaces is the solution
of the equality part of the extended Suita conjecture in [25].

In the present article, we point out that the smooth gain of the general
concavity property in [19] (see also [20]) can be replaced by Lebesgue
measurable gain (Definition 1.1 and Theorem 1.3). As applications, we give
necessary conditions for our concavity degenerating to linearity (Section
1.2.2), characterizations for 1-dimensional case (Section 1.2.3), and a
characterization for the holding of the equality in optimal $L^{2}$ extension
problem on open Riemann surfaces with weights may not be subharmonic (Section
1.2.4).

###  1.1. Concavity property of minimal $L^{2}$ integrals with Lebesgue
measurable gain

Let $M$ be a complex manifold. We call $M$ that satisfies condition $(a)$, if
there exists a closed subset $X\subset M$ satisfying the following two
statements:

$(a1)$ $X$ is locally negligible with respect to $L^{2}$ holomorphic
functions; i.e., for any local coordinate neighborhood $U\subset M$ and for
any $L^{2}$ holomorphic function $f$ on $U\backslash X$, there exists an
$L^{2}$ holomorphic function $\tilde{f}$ on $U$ such that
$\tilde{f}|_{U\backslash X}=f$ with the same $L^{2}$ norm;

$(a2)$ $M\backslash X$ is a Stein manifold.

Let $M$ be an $n-$dimensional complex manifold satisfying condition $(a)$, 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 $T=-\sup_{M}\psi$ ($T$ maybe $-\infty$).

######  Definition 1.1.

We call a positive measurable function $c$ (so-called ”gain”) 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 $M$.

######  Remark 1.2.

We recall a class $\mathcal{P}^{\prime}_{T}$ of positive smooth functions in
[19]. A positive smooth function $c$ on $(T,+\infty)$ in class
$\mathcal{P}^{\prime}_{T}$ if the following three statements hold:

$(1)^{\prime}$ $\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$;

$(2)^{\prime}$ $c(t)e^{-t}$ is decreasing with respect to $t$;

$(3)^{\prime}$ for any compact subset $K\subseteq M$, $e^{-\varphi}c(-\psi)$
has a positive lower bound on $K$.

We compare these two classes of functions $\mathcal{P}_{T}$ and
$\mathcal{P}^{\prime}_{T}$. When $c\in\mathcal{P}_{T}$, $c$ maybe non-smooth
on $(T,+\infty)$ and $\int_{T}^{+\infty}c(t)e^{-t}dt$ maybe $+\infty$. When
$\varphi$ is continuous on $M$, condition $(3)^{\prime}$ is equivalent to
$\liminf_{t\rightarrow+\infty}c(t)>0$. When $\varphi$ is continuous on $M$ and
$\psi\in A(S)$ (see Section 1.2.1), the decreasing property of $c(t)e^{-t}$
implies that $c\in\mathcal{P}_{T}$ and $\liminf_{t\rightarrow+\infty}c(t)$ may
be equal to $0$.

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 a 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 $G(t;\varphi,\psi,c)$ (so-called ”minimal $L^{2}$ integrals related to
multiplier ideal sheaves”), where $t\in[T,+\infty)$, $c$ is a nonnegative
function on $(T,+\infty)$, $|f|^{2}:=\sqrt{-1}^{n^{2}}f\wedge\bar{f}$ 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
$z_{0}\in Z_{0}$. If there is no holomorphic holomorphic $(n,0)$ form
$\tilde{f}$ on $\\{\psi<-t\\}$ satisfying $(\tilde{f}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$, we set
$G(t;\varphi,\psi,c)=+\infty$. Without misunderstanding, we denote
$G(t;\varphi,\psi,c)$ by $G(t)$, and when we focus on different $\varphi$,
$\psi$ or $c$, we denote it by $G(t;\varphi)$, $G(t;\psi)$ or $G(t;c)$,
respectively.

In the present article, we obtain the following concavity for $G(t)$.

######  Theorem 1.3.

Let $c\in\mathcal{P}_{T}$. If there exists $t\in[T,+\infty)$ satisfying that
$G(t)<+\infty$, then $G(h^{-1}(r))$ is concave with respect to
$r\in(\int_{T_{1}}^{T}c(t)e^{-t}dt,\int_{T_{1}}^{+\infty}c(t)e^{-t}dt)$,
$\lim_{t\rightarrow T+0}G(t)=G(T)$ and $\lim_{t\rightarrow+\infty}G(t)=0$,
where $h(t)=\int_{T_{1}}^{t}c(t_{1})e^{-t_{1}}dt_{1}$ and
$T_{1}\in(T,+\infty)$.

When $c(t)\in\mathcal{P}^{\prime}_{T}$ and $M$ is a Stein manifold, Theorem
1.3 is the concavity property in [19] (see also [20]).

Theorem 1.3 implies the following corollary.

######  Corollary 1.4.

If $\int_{T_{1}}^{+\infty}c(t)e^{-t}dt=+\infty$, where $c\in\mathcal{P}_{T}$,
and $f\notin H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$,
then $G(t)=+\infty$ for any $t\geq T$, i.e., there is no holomorphic
holomorphic $(n,0)$ form $\tilde{f}$ on $\\{\psi<-t\\}$ satisfying
$(\tilde{f}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and
$\int_{\\{\psi<-t\\}}|\tilde{f}|^{2}e^{-\varphi}c(-\psi)<+\infty$.

In the following, we give two corollaries of Theorem 1.3 when concavity
degenerates to linearity.

######  Corollary 1.5.

Let $c\in\mathcal{P}_{T}$, and let $G(t)\in(0,+\infty)$ for some $t\geq T$,
then $G(h^{-1}(r))$ is concave with respect to
$r\in(\int_{T_{1}}^{T}c(t)e^{-t}dt,\int_{T_{1}}^{+\infty}c(t)e^{-t}dt]$ and
the following three statements are equivalent:

$(1)$
$G(t)=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}\int_{t}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}$
holds for any $t\in[T,+\infty)$, i.e., $G(\hat{h}^{-1}(r))$ is linear with
respect to $r\in[0,\int_{T}^{+\infty}c(s)e^{-s}ds)$, where
$\hat{h}(t)=\int_{t}^{+\infty}c(s)e^{-s}ds$;

$(2)$
$\frac{G(h^{-1}(r_{0}))}{\int_{T_{1}}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}-r_{0}}\leq\lim_{t\rightarrow
T+0}\frac{G(t)}{\int_{t}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}$ holds for some
$r_{0}\in(\int_{T_{1}}^{T}c(t)e^{-t}dt,\int_{T_{1}}^{+\infty}c(t)e^{-t}dt)$,
i.e.,

$\frac{G(t_{0})}{\int_{t_{0}}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}\leq\lim_{t\rightarrow
T+0}\frac{G(t)}{\int_{t}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}$

holds for some $t_{0}\in(T,+\infty)$;

$(3)$
$\lim_{r\to{\int_{T_{1}}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}-0}\frac{G(h^{-1}(r))}{\int_{T_{1}}^{+\infty}c(t)e^{-t}dt-r}\leq\lim_{t\rightarrow
T+0}\frac{G(t)}{\int_{t}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}$ holds, i.e.,

$\lim_{t\to+\infty}\frac{G(t)}{\int_{t}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}\leq\lim_{t\rightarrow
T+0}\frac{G(t)}{\int_{t}^{+\infty}c(t_{1})e^{-t_{1}}dt_{1}}$

holds.

######  Remark 1.6.

Let $M=\Delta\subset\mathbb{C}$ and let $\psi=\psi+\varphi=2\log|z|$. Let
$c(t)\equiv 1$ and let $\mathcal{F}=\mathcal{I}(\varphi+\psi)$. Let $f\equiv
dz$ and $Z_{0}=o$ the origin of $\mathbb{C}$. It is clear that
$G(\hat{h}^{-1}(r))=2\pi r$ is linear with respect to $r$, where
$\hat{h}(t)=\int_{t}^{+\infty}c(l)e^{-l}dl$.

Let $c(t)$ be a nonnegative measurable function on $(T,+\infty)$. Set

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

where $t\in[T,+\infty)$.

######  Corollary 1.7.

Let $c\in\mathcal{P}_{T}$, if $G(t)\in(0,+\infty)$ for some $t\geq T$ and
$G(\hat{h}^{-1}(r))$ is linear with respect to
$r\in[0,\int_{T}^{+\infty}c(s)e^{-s}ds)$, where
$\hat{h}(t)=\int_{t}^{+\infty}c(l)e^{-l}dl$, 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 $t\geq T$.
Equality

$\int_{\\{-t_{1}\leq\psi<-t_{2}\\}}|F|^{2}e^{-\varphi}a(-\psi)=\frac{G(T_{1};c)}{\int_{T_{1}}^{+\infty}c(t)e^{-t}dt}\int_{t_{2}}^{t_{1}}a(t)e^{-t}dt$
(1.1)

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

Furthermore, if
$\mathcal{H}^{2}(\tilde{c},t_{0})\subset\mathcal{H}^{2}(c,t_{0})$ for some
$t_{0}\geq T$, where $\tilde{c}$ is a nonnegative measurable function on
$(T,+\infty)$, we have

$G(t_{0};\tilde{c})=\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\tilde{c}(-\psi)=\frac{G(T_{1};c)}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{0}}^{+\infty}\tilde{c}(s)e^{-s}ds.$
(1.2)

When $c(t)\in\mathcal{P}^{\prime}_{T}$ and $M$ is a Stein manifold, Corollary
1.5 and Corollary 1.7 can be referred to [20] (when $c\equiv 1$, $M$ is a
Stein manifold, $\varphi$ is a smooth plurisubharmonic function on $M$ and
$\\{\psi=-\infty\\}$ is a closed subset of $M$, Xu-Zhou [54] also get the
existence of $F$ in Corollary 1.7 independently).

######  Remark 1.8.

Let $\tilde{c}\in\mathcal{P}_{T}$, if
$\mathcal{H}^{2}(\tilde{c},t_{1})\subset\mathcal{H}^{2}(c,t_{1})$, then
$\mathcal{H}^{2}(\tilde{c},t_{2})\subset\mathcal{H}^{2}(c,t_{2})$, where
$t_{1}>t_{2}>T$. In the following, we give some sufficient conditions of
$\mathcal{H}^{2}(\tilde{c},t_{0})\subset\mathcal{H}^{2}(c,t_{0})$ for
$t_{0}>T$:

$(1)$ $\tilde{c}\in\mathcal{P}_{T}$ and
$\lim_{t\rightarrow+\infty}\frac{\tilde{c}(t)}{c(t)}>0$. Especially,
$\tilde{c}\in\mathcal{P}_{T}$, $c$ and $\tilde{c}$ are smooth on $(T,+\infty)$
and $\frac{d}{dt}(\log(\tilde{c}(t))\geq\frac{d}{dt}(\log c(t))$;

$(2)$ $\tilde{c}\in\mathcal{P}_{T}$, $\mathcal{H}^{2}(c,t_{0})\not=\emptyset$
and there exists $t>t_{0}$, such that
$\\{\psi<-t\\}\subset\subset\\{\psi<-t_{0}\\}$,
$\\{z\in\overline{\\{\psi<-t\\}}:\mathcal{I}(\varphi+\psi)_{z}\not=\mathcal{O}_{z}\\}\subset
Z_{0}$ and
$\mathcal{F}|_{\overline{\\{\psi<-t\\}}}=\mathcal{I}(\varphi+\psi)|_{\overline{\\{\psi<-t\\}}}$.

The sufficiency of condition $(1)$ is clear. For condition (2), assume that
$\mathcal{H}^{2}(\tilde{c},t_{0})\not=\emptyset$, then the following
inequality gives the sufficiency of condition $(2)$:

$\begin{split}&\int_{\\{\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&2\int_{\\{\psi<-t\\}}|\tilde{F}-F|^{2}e^{-\varphi}c(-\psi)+2\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)\\\
&+\int_{\\{-t\leq\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&2C\int_{\\{\psi<-t\\}}|\tilde{F}-F|^{2}e^{-\varphi-\psi}+2\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)\\\
&+\frac{\sup_{s\in(t_{0},t]}c(s)}{\inf_{s\in(t_{0},t]}\tilde{c}(s)}\int_{\\{\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi}\tilde{c}(-\psi)\\\
<&+\infty,\end{split}$

where $\tilde{F}\in\mathcal{H}^{2}(\tilde{c},t_{0})$ and
$F\in\mathcal{H}^{2}(c,t_{0})$.

###  1.2. Applications

In this section, we give some applications of our concavity property.

####  1.2.1. Applications in optimal $L^{2}$ extension theorem

Let $M$ be an $n-$dimensional complex manifold, and let $S$ be an analytic
subset of $M$. Let $dV_{M}$ be a continuous volume form on $M$.

We consider a class of plurisubharmonic functions $\Phi$ from $M$ to
$[-\infty,+\infty)$, such that

$(1)$ $S\subset\Phi^{-1}(-\infty)$, and $\Phi^{-1}(-\infty)$ is a closed
subset of some analytic subset of $M$ satisfying that $\Phi$ has locally lower
bound on $M\backslash\Phi^{-1}(-\infty)$.

$(2)$ if $S$ is $l-$dimensional around a point $x\in S_{reg}$, there is a
local coordinate $(z_{1},...,z_{n})$ on a neighborhood $U$ of $x$ such that
$z_{l+1}=...=z_{n}=0$ on $S\cap U$ and

$\sup_{U-S}|\Phi(z)-(n-l)\log\sum_{l+1}^{n}|z_{j}|^{2}|<+\infty.$

The set of such polar functions $\Phi$ will be denoted by $A(S)$. We call
$\Phi$ is in class $A^{\prime}(S)$, if the condition $(2)$ is replaced by
$(2)^{\prime}$:

$(2)^{\prime}$ if $S$ is $l-$dimensional around a point $x\in S_{reg}$, there
is a local coordinate $(z_{1},...,z_{n})$ on a neighborhood $U$ of $x$ such
that $z_{l+1}=...=z_{n}=0$ on $S\cap U$ and
$\Phi(z)-(n-l)\log\sum_{l+1}^{n}|z_{j}|^{2}$ is continuous on $U$.

Let $\psi\in A(S)$. Following [40] (see also [25]), one can define a positive
measure $dV_{M}[\psi]$ on $S_{reg}$ as the minimum element of the partially
ordered set of positive measures $d\mu$ satisfying

$\int_{S_{l}}fd\mu\geq\limsup_{t\rightarrow+\infty}\frac{2(n-l)}{\sigma_{2n-2l-1}}\int_{M}\mathbb{I}_{\\{-t-1<\psi<-t\\}}fe^{-\psi}dV_{M}$

for any nonnegative continuous function $f$ with $suppf\subset\subset M$. Here
denote by $\sigma_{m}$ the volume of the unit sphere in $\mathbb{R}^{m+1}$. If
$\psi\in A^{\prime}(S)$, then $dV_{M}[\psi]|_{S_{l}}$ is a continuous volume
form on $S_{l}$ and $dV_{M}[\psi+h]|_{S_{l}}=e^{-h}dV_{M}[\psi]|_{S_{l}}$ (see
[25]), where $h$ is a continuous function on $M$.

Let us recall a class of complex manifolds (see [25]). Let $M$ be a complex
manifold with the volume form $dV_{M}$, and let $S$ be an analytic subset of
$M$. We say $(M,S)$ satisfies condition $(ab)$ if there exists a closed subset
$X\subset M$ satisfying the following statements:

$(a)$ $X$ is locally negligible with respect to $L^{2}$ holomorphic functions;

$(b)$ $M\backslash X$ is a Stein manifold which intersects with every
component of $S$, such that $S_{sing}\subset X$.

We give the following $L^{2}$ extension theorem with an optimal estimate. When
$c(t)$ is continuous, the theorem can be referred to [25].

######  Theorem 1.9.

Let $(M,S)$ satisfy condition $(ab)$. Let $\psi\in A(S)$ satisfying $\psi<-T$
on $M$. Let $\varphi$ be a continuous function on $M$, such that
$\varphi+\psi$ is plurisubharmonic on $M$. Let $c(t)$ be a positive function
on $(T,+\infty)$ such that $c(t)e^{-t}$ is decreasing and
$\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$. Then for any holomorphic section $f$
of $K_{M}|_{S}$ on $S$, such that

$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]<+\infty,$

there exists a holomorphic $(n,0)$ form $F$ on $M$ such that $F|_{S}=f$ and

$\int_{M}|F|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi].$

By the definition of $dV_{M}[\psi]$, we know
$\frac{|f|^{2}}{dV_{M}}dV_{M}[\psi]$ is independent of the choice of $dV_{M}$
(see [25]).

Denote that
$\|f\|_{S}:=\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]$
and $\|F\|_{M}:=\int_{M}|F|^{2}e^{-\varphi}c(-\psi)$. Let
$\mathcal{F}|_{Z_{0}}=\mathcal{I}(\psi)|_{S_{reg}}$ and choose the $f$ in the
definition of $G(t)$ by any holomorphic extension of the $f$ in Theorem 1.9.
Then $G(T)=\inf\\{\|F\|_{M}:F$ is a holomorphic extension of $f$ from $S$ to
$M$$\\}$, and Theorem 1.9 tells us that

$G(T)\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{S}$ (1.3)

(when $G(T)<+\infty$, Lemma 2.6 shows that there exists a holomorphic
extension $F$ of $f$ such that $G(T)=\|F\|_{M}$).

Using Corollary 1.7 and Theorem 1.9, we obtain a necessary condition of
inequality (1.3) becomes an equality.

######  Theorem 1.10.

Let $(M,S)$ satisfy condition $(ab)$. Let $\psi\in A(S)$, and let $\psi<-T$.
Let $\varphi$ be a continuous function on $M$, such that $\varphi+\psi$ is
plurisubharmonic on $M$. Let $c(t)$ be a positive function on $(T,+\infty)$
such that $c(t)e^{-t}$ is decreasing and
$\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$. Let $f$ be a holomorphic section of
$K_{M}|_{S}$ on $S$, such that

$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]<+\infty.$

If $G(T)=\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{S},$ then
$G(\hat{h}^{-1}(r))$ is linear with respect to $r$ and there exists a unique
holomorphic $(n,0)$ form $F$ on $M$ such that $F|_{S}=f$ and $G(T)=\|F\|_{M}$.

For any $t\geq T$, there exists a unique holomorphic $(n,0)$ form $F_{t}$ on
$\\{\psi<-t\\}$ such that $F_{t}|_{S}=f$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{t}^{+\infty}c(l)e^{-l}dl\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi].$

In fact, $F_{t}=F$ on $\\{\psi<-t\\}$.

If $\mathcal{H}^{2}(\tilde{c},t)\subset\mathcal{H}^{2}(c,t)$ for some $t\geq
T$, where $\tilde{c}$ is a nonnegative measurable function on $(T,+\infty)$,
then there exists a unique holomorphic $(n,0)$ form $F_{t}$ on $\\{\psi<-t\\}$
such that $F_{t}|_{S}=f$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}\tilde{c}(-\psi)\leq\left(\int_{t}^{+\infty}\tilde{c}(l)e^{-l}dl\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi].$

In fact, $F_{t}=F$ on $\\{\psi<-t\\}$.

When $c(t)\in\mathcal{P}^{\prime}_{T}$ and $M$ is a Stein manifold, Theorem
1.10 was obtained by Guan-Mi in [20].

Using Theorem 1.9, we obtain the following optimal $L^{2}$ extension theorem.

######  Corollary 1.11.

Let $M$ be an $n-$dimensional Stein manifold, and let $S$ be an analytic
subset of $M$. Let $\psi_{1}\in A(S)$, and let $\psi_{2}$ be a
plurisubharmonic function on $M$ such that $\psi=\psi_{1}+\psi_{2}<-T$ on $M$.
Let $\varphi$ be a Lebesgue measurable function on $M$ such that
$\varphi+\psi_{2}$ is plurisubharmonic on $M$. Let $c(t)$ be a positive
function on $(T,+\infty)$, such that $c(t)e^{-t}$ is decreasing,
$\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$ and $e^{-\varphi}c(-\psi)$ has
locally a positive lower bound on $M\backslash Z$, where $Z$ is some analytic
subset of $M$. For any holomorphic section $f$ of $K_{M}|_{S_{reg}}$ on
$S_{reg}$ satisfying

$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]<+\infty,$

there exists a holomorphic $(n,0)$ form $F$ on $M$ such that $F|_{S_{reg}}=f$
and

$\int_{M}|F|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].$

Especially, when $c\equiv 1$ and $\psi_{1}=2\log|w|$, where $w$ is a
holomorphic function on $M$, such that $dw$ does not vanish identically on any
branch of $w^{-1}(0)$ and $S_{reg}=\\{z\in M:w(z)=0\,\&\,dw(z)\not=0\\}$,
Corollary 1.11 can be referred to [22] (see also [25]).

Denote that
$\|f\|_{S}^{*}:=\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]$.
Let $\mathcal{F}|_{Z_{0}}=\mathcal{I}(\psi_{1})|_{S_{reg}}$ and choose the $f$
in the definition of $G(t)$ by any holomorphic extension of the $f$ in
Corollary 1.11. Then $G(T)=\inf\\{\|F\|_{M}:F$ is a holomorphic extension of
$f$ from $S$ to $M$$\\}$, and Corollary 1.11 tells us that

$G(T)\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{S}^{*}.$ (1.4)

Similarly to Theorem 1.10, we give a necessary condition of inequality (1.4)
becomes an equality.

######  Corollary 1.12.

Let $M$ be an $n-$dimensional Stein manifold, and let $S$ be an analytic
subset of $M$. Let $\psi_{1}\in A(S)$, and let $\psi_{2}$ be a
plurisubharmonic function on $M$ such that $\psi=\psi_{1}+\psi_{2}<-T$ on $M$.
Let $\varphi$ be a Lebesgue measurable function on $M$ such that
$\varphi+\psi_{2}$ is plurisubharmonic on $M$. Let $c(t)\in\mathcal{P}_{T}$
such that $\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$. Let $f$ be a holomorphic
section of $K_{M}|_{S_{reg}}$ on $S_{reg}$ satisfying

$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]<+\infty.$

If $G(T)=\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{S}^{*},$ then
$G(\hat{h}^{-1}(r))$ is linear with respect to $r$ and there exists a unique
holomorphic $(n,0)$ form $F$ on $M$ such that $F|_{S}=f$ and $G(T)=\|F\|_{M}$.

For any $t\geq T$, there exists a unique holomorphic $(n,0)$ form $F_{t}$ on
$\\{\psi<-t\\}$ such that $F_{t}|_{S}=f$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{t}^{+\infty}c(l)e^{-l}dl\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].$

In fact, $F_{t}=F$ on $\\{\psi<-t\\}$.

If $\mathcal{H}^{2}(\tilde{c},t)\subset\mathcal{H}^{2}(c,t)$ for some $t\geq
T$, where $\tilde{c}$ is a nonnegative measurable function on $(T,+\infty)$,
then there exists a unique holomorphic $(n,0)$ form $F_{t}$ on $\\{\psi<-t\\}$
such that $F_{t}|_{S}=f$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}\tilde{c}(-\psi)\leq\left(\int_{t}^{+\infty}\tilde{c}(l)e^{-l}dl\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].$

In fact, $F_{t}=F$ on $\\{\psi<-t\\}$.

####  1.2.2. Necessary conditions of $G(\hat{h}^{-1}(r))$ is linear

In this section, we give some necessary conditions of $G(\hat{h}^{-1}(r))$ is
linear.

######  Theorem 1.13.

Let $M$ be an $n-$dimensional complex manifold satisfying condition $(a)$. Let
$c\in\mathcal{P}_{T}$, and assume that there exists $t\geq T$ such that
$G(t)\in(0,+\infty)$. If there exists a Lebesgue measurable function
$\tilde{\varphi}\geq\varphi$ such that $\tilde{\varphi}+\psi$ is
plurisubharmonic function on $M$ and satisfies that:

$(1)$ $\tilde{\varphi}\not=\varphi$;

$(2)$ $\lim_{t\rightarrow
T+0}\sup_{\\{\psi\geq-t\\}}(\tilde{\varphi}-\varphi)=0$;

$(3)$ $\tilde{\varphi}-\varphi$ is bounded on $M$.

Then $G(\hat{h}^{-1}(r))$ is not linear with respect to
$r\in(0,\int_{T}^{+\infty}c(s)e^{-s}ds)$. Especially, if $\varphi+\psi$ is
strictly plurisubharmonic at $z_{1}\in M$, $G(\hat{h}^{-1}(r))$ is not linear
with respect to $r\in(0,\int_{T}^{+\infty}c(s)e^{-s}ds)$.

In the following, we give two necessary conditions for $\psi$ when
$G(\hat{h}^{-1}(r))$ is linear.

######  Theorem 1.14.

Let $M$ be an $n-$dimensional complex manifold satisfying condition $(a)$. Let
$c\in\mathcal{P}_{T}$, and assume that $G(T)\in(0,+\infty)$. If there exists a
plurisubharmonic function $\tilde{\psi}\geq\psi$ on $M$ satisfying that:

$(1)$ $\tilde{\psi}<-T$ on $M$;

$(2)$ $\tilde{\psi}\not=\psi$;

$(3)$ $\lim_{t\rightarrow+\infty}\sup_{\\{\psi<-t\\}}(\tilde{\psi}-\psi)=0$.

Then $G(\hat{h}^{-1}(r))$ is not linear with respect to
$r\in(0,\int_{T}^{+\infty}c(s)e^{-s}ds)$. Especially, if $\psi$ is strictly
plurisubharmonic at $z_{1}\in M\backslash(\cap_{t}\overline{\\{\psi<-t\\}})$,
$G(\hat{h}^{-1}(r))$ is not linear with respect to
$r\in(0,\int_{T}^{+\infty}c(s)e^{-s}ds)$.

Let $M$ be an $n-$dimensional Stein manifold, and let $S$ be an analytic
subset of $M$. Let $\psi$ be a plurisubharmonic function on $M$, and let
$\varphi$ be a Lebesgue measurable function on $M$ such that $\varphi+\psi$ is
plurisubharmonic on $M$.

We call $(\varphi,\psi)$ in class $W$ if there exist two plurisubharmonic
functions $\psi_{1}\in A^{\prime}(S)$ and $\psi_{2}$, such that
$\varphi+\psi_{2}$ is plurisubharmonic function on $M$ and
$\psi=\psi_{1}+\psi_{2}$.

The following theorem gives a necessary condition of $G(\hat{h}^{-1}(r))$ is
linear when $(\varphi,\psi)\in W$.

######  Theorem 1.15.

Let $c\in P_{T}$, and let $(\varphi,\psi)\in W$. Let
$\mathcal{F}|_{Z_{0}}=\mathcal{I}(\psi_{1})|_{S_{reg}}$. Assume that
$G(T)\in(0,+\infty)$ and $\psi_{2}(z)>-\infty$ for almost every $z\in
S_{reg}$. If $G(\hat{h}^{-1}(r))$ is linear with respect to
$r\in(0,\int_{T}^{+\infty}c(s)e^{-s}ds)$, then we have

$\frac{G(T)}{\int_{T}^{+\infty}c(t)e^{-t}dt}=\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}],$
(1.5)

and there is no plurisubharmonic function $\tilde{\psi}\geq\psi$ on $M$
satisfying that:

$(1)$ $\tilde{\psi}<-T$;

$(2)$ $\tilde{\psi}\not=\psi$;

$(3)$ $(\varphi+\psi-\tilde{\psi},\tilde{\psi})\in W$.

####  1.2.3. Characterizations for the linearity of $G(\hat{h}(r))$ in
$1-$dimensional case

In this section, we consider the $1-$dimensional case. Let $M=\Omega$ be an
open Riemann surface admitted a nontrivial Green function $G_{\Omega}$, we
give characterizations of the linearity.

We recall some notations (see [25]). Let $p:\Delta\rightarrow\Omega$ be the
universal covering from unit disc $\Delta$ to $\Omega$. we call the
holomorphic function $f$ (resp. holomorphic $(1,0)$ form $F$) on $\Delta$ a
multiplicative function (resp. multiplicative differential (Prym
differential)), if there is a character $\chi$, which is the representation of
the fundamental group of $\Omega$, such that $g^{*}f=\chi(g)f$ (resp.
$g^{*}(F)=\chi(g)F$), where $|\chi|=1$ and $g$ is an element of the
fundamental group of $\Omega$. Denote the set of such kinds of $f$ (resp. $F$)
by $\mathcal{O}^{\chi}(\Omega)$ (resp. $\Gamma^{\chi}(\Omega)$).

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
$|f_{u}|=p^{*}\left(e^{u}\right)$. If $u_{1}-u_{2}=\log|f|$, where $u_{1}$ and
$u_{2}$ are harmonic function on $\Omega$ and $f$ is holomorphic function on
$\Omega$, then $\chi_{u_{1}}=\chi_{u_{2}}$.

For the Green function $G_{\Omega}(z,z_{0})$, one can also find a
$\chi_{z_{0}}$ and a multiplicative function
$f_{z_{0}}\in\mathcal{O}^{\chi_{z_{0}}}(\Omega)$, such that
$|f_{z_{0}}|=p^{*}\left(e^{G_{\Omega}(z,z_{0})}\right)$.

Let $M=\Omega$ be an open Riemann surface admitted a nontrivial Green function
$G_{\Omega}$. Let $\psi$ be a subharmonic function on $\Omega$ satisfying
$T=-\sup_{\Omega}\psi=0$, and let $\varphi$ be a Lebesgue measurable function
on $\Omega$, such that $\varphi+\psi$ is subharmonic on $\Omega$. Let
$Z_{0}=z_{0}$ be a point in $\Omega$.

Let $w$ be a local coordinate on a neighborhood $V_{z_{0}}$ of
$z_{0}\in\Omega$ satisfying $w(z_{0})=0$. Set $f=f_{1}(w)dw$ on $V_{z_{0}}$,
where $f$ is the holomorphic $(1,0)$ form in the definition of $G(t)$ (see
Section 1.1) and $f_{1}$ is a holomorphic function on $V_{z_{0}}$.

The following two theorems give characterizations of $G(\hat{h}^{-1}(r))$ is
linear with respect to $r\in(0,\int_{0}^{+\infty}c(l)e^{-l}dl)$ for some kinds
of $(\varphi,\psi)$. Set $d^{c}=\frac{1}{2\pi i}(\partial-\bar{\partial})$.

######  Theorem 1.16.

Let $c\in\mathcal{P}_{0}$. Assume that $\varphi+a\psi$ is a subharmonic
function on a neighborhood of $z_{0}$ for some $a\in[0,1)$, and
$G(0)\in(0,+\infty)$. Then $G(\hat{h}^{-1}(r))$ is linear with respect to $r$
if and only if the following statements hold:

$(1)$ $\varphi+\psi=2\log|g|+2G_{\Omega}(z,z_{0})+2u$,
$ord_{z_{0}}(g)=ord_{z_{0}}(f_{1})$ and
$\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$, where $g$ is a
holomorphic function on $\Omega$ and $u$ is a harmonic function on $\Omega$;

$(2)$ $\psi=2pG_{\Omega}(z,z_{0})$ on $\Omega$ for some $p>0$;

$(3)$ $\chi_{-u}=\chi_{z_{0}}$, where $\chi_{-u}$ and $\chi_{z_{0}}$ are the
characters associated to the functions $-u$ and $G_{\Omega}(z,z_{0})$
respectively.

When $\psi=2G_{\Omega}(z,z_{0})$,
$\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$ and
$c(t)\in\mathcal{P}^{\prime}_{0}$, Theorems 1.16 can be referred to [20].

######  Theorem 1.17.

Let $c\in\mathcal{P}_{0}$, and let $Z_{0}=z_{0}$ be a point in $\Omega$.
Assume that $(\psi-2pG_{\Omega}(z,z_{0}))(z_{0})>-\infty$, where
$p=\frac{1}{2}v(dd^{c}\psi,z_{0})$, and $G(0)\in(0,+\infty)$. Then
$G(\hat{h}^{-1}(r))$ is linear with respect to $r$ if and only if the
following statements hold:

$(1)$ $\varphi+\psi=2\log|g|+2G_{\Omega}(z,z_{0})+2u$,
$ord_{z_{0}}(g)=ord_{z_{0}}(f_{1})$ and
$\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$, where $g$ is a
holomorphic function on $\Omega$ and $u$ is a harmonic function on $\Omega$;

$(2)$ $p>0$ and $\psi=2pG_{\Omega}(z,z_{0})$ on $\Omega$;

$(3)$ $\chi_{-u}=\chi_{z_{0}}$, where $\chi_{-u}$ and $\chi_{z_{0}}$ are the
characters associated to the functions $-u$ and $G_{\Omega}(z,z_{0})$
respectively.

####  1.2.4. Characterizations for the holding of the equality in optimal
$L^{2}$ extension problem on open Riemann surfaces with weights may not be
subharmonic.

Let $M=\Omega$ be an open Riemann surface admitted a nontrivial Green function
$G_{\Omega}$. Let $\psi$ be a subharmonic function on $\Omega$ satisfying
$T=-\sup_{\Omega}\psi=0$, and let $\varphi$ be a Lebesgue measurable function
on $\Omega$, such that $\varphi+\psi$ is subharmonic on $\Omega$. Let
$Z_{0}=z_{0}$ be a point in $\Omega$.

Let $w$ be a local coordinate on a neighborhood $V_{z_{0}}$ of
$z_{0}\in\Omega$ satisfying $w(z_{0})=0$. Let $f\equiv dw$ be a holomorphic
$(1,0)$ form on $V_{z_{0}}$. Following the notations in Section 1.2.1. Now, we
give characterizations for the holding of the equality in optimal $L^{2}$
extension problem on open Riemann surfaces with weights may not be
subharmonic.

######  Corollary 1.18.

Let $M=\Omega$, $S=z_{0}$, and $T=0$. Let $\varphi(z_{0})>-\infty$. Assume
that $\psi\in A(z_{0})$, $e^{-\varphi-\psi}$ is not $L^{1}$ on any
neighborhood of $z_{0}$ and $c(t)\in\mathcal{P}_{0}$ satisfying
$\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$.

Then there exists a holomorphic $(1,0)$ form $F$ on $\Omega$ such that
$F(z_{0})=f(z_{0})$ and

$\int_{\Omega}|F|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}.$
(1.6)

Moreover, equality
$\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}=\inf\\{\|\tilde{F}\|_{\Omega}$:
$\tilde{F}$ is a holomorphic extension of $f$ from $z_{0}$ to $\Omega$$\\}$
holds if and only if the following statements hold:

$(1)$ $\varphi=2\log|g|+2u$, where $u$ is a harmonic function on $\Omega$ and
$g$ is a holomorphic function on $\Omega$ such that $g(z_{0})\not=0$;

$(2)$ $\psi=2G_{\Omega}(z,z_{0})$ on $\Omega$;

$(3)$ $\chi_{-u}=\chi_{z_{0}}$, where $\chi_{-u}$ and $\chi_{z_{0}}$ are the
characters associated to the functions $-u$ and $G_{\Omega}(z,z_{0})$
respectively.

When $\psi=2G_{\Omega}(z,z_{0})$ and $c(t)\in\mathcal{P}^{\prime}_{0}$,
Corollary 1.18 can be referred to [20].

######  Corollary 1.19.

Let $M=\Omega$, $S=z_{0}$, and $T=0$. Let $(\varphi,\psi)\in W$ and let
$\|f\|_{z_{0}}^{*}\in(0,+\infty)$. Let $c(t)\in\mathcal{P}_{0}$ such that
$\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$. Then equality
$\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}^{*}=\inf\\{\|F\|_{\Omega}:F$
is a holomorphic extension of $f$ from $z_{0}$ to $\Omega$$\\}$ holds if and
only if the following statements hold:

$(1)$ $\varphi=2\log|g|+2u$, where $u$ is a harmonic function on $\Omega$ and
$g$ is a holomorphic function on $\Omega$ such that $g(z_{0})\not=0$;

$(2)$ $\psi=2G_{\Omega}(z,z_{0})$ on $\Omega$;

$(3)$ $\chi_{-u}=\chi_{z_{0}}$, where $\chi_{-u}$ and $\chi_{z_{0}}$ are the
characters associated to the functions $-u$ and $G_{\Omega}(z,z_{0})$
respectively.

##  2\. Preparation

###  2.1. $L^{2}$ methods

We call a positive measurable function $c$ on $(S,+\infty)$ in class
$\tilde{\mathcal{P}}_{S}$ if $\int_{S}^{s}c(l)e^{-l}dl<+\infty$ for some $s>S$
and $c(t)e^{-t}$ is decreasing with respect to $t$. Note that
$\mathcal{P}_{T}\subset\tilde{\mathcal{P}}_{S}$ when $S>T$.

In this section, we present the following Lemma (proof can be referred to
Section 7.1), whose various forms already appear in [23, 25, 18, 20] etc.:

######  Lemma 2.1.

Let $B\in(0,+\infty)$ and $t_{0}\geq S$ be arbitrarily given. Let $M$ be an
$n-$dimensional Stein manifold. Let $\psi<-S$ be a plurisubharmonic function
on $M$. Let $\varphi$ be a plurisubharmonic function on $M$. Let $F$ be a
holomorphic $(n,0)$ form on $\\{\psi<-t_{0}\\}$, such that

$\int_{K\cap\\{\psi<-t_{0}\\}}|F|^{2}<+\infty$ (2.1)

for any compact subset $K$ of $M$, and

$\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\leq
C<+\infty.$ (2.2)

Then there exists a holomorphic $(n,0)$ form $\tilde{F}$ on $M$, such that

$\begin{split}\int_{M}&|\tilde{F}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\leq
C\int_{S}^{t_{0}+B}c(t)e^{-t}dt\end{split}$ (2.3)

where
$b_{t_{0},B}(t)=\int_{-\infty}^{t}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<s<-t_{0}\\}}ds$,
$v_{t_{0},B}(t)=\int_{-t_{0}}^{t}b_{t_{0},B}(s)ds-t_{0}$, and
$c(t)\in\tilde{\mathcal{P}}_{S}$.

We give the proof of Lemma 2.1 in Section 7.1. It is clear that
$\mathbb{I}_{(-t_{0},+\infty)}(t)\leq
b_{t_{0},B}(t)\leq\mathbb{I}_{(-t_{0}-B,+\infty)}(t)$ and
$\max\\{t,-t_{0}-B\\}\leq v_{t_{0},B}(t)\leq\max\\{t,-t_{0}\\}$.

Lemma 2.1 implies the following lemma, which will be used in the proof of
Theorem 1.3.

######  Lemma 2.2.

Let $M$ be an $n-$dimensional complex manifold satisfying condition $(a)$, and
let $c(t)\in\mathcal{P}_{T}$. Let $B\in(0,+\infty)$ and $t_{0}>t_{1}>T$ be
arbitrarily given. Let $\psi<-T$ be a plurisubharmonic function on $M$. Let
$\varphi$ be a Lebesgue measurable function on $M$, such that $\varphi+\psi$
is plurisubharmonic on $M$. Let $F$ be a holomorphic $(n,0)$ form on
$\\{\psi<-t_{0}\\}$, such that

$\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}c(-\psi)<+\infty.$ (2.4)

Then there exists a holomorphic $(n,0)$ form $\tilde{F}$ on
$\\{\psi<-t_{1}\\}$, such that

$\begin{split}\int_{\\{\psi<-t_{1}\\}}&|\tilde{F}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\leq
C\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt\end{split}$ (2.5)

where
$C=\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi-\psi}<+\infty$,
$b_{t_{0},B}(t)=\int_{-\infty}^{t}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<s<-t_{0}\\}}ds$
and $v_{t_{0},B}(t)=\int_{0}^{t}b_{t_{0},B}(s)ds$.

###### Proof.

As $M$ is an $n-$dimensional complex manifold satisfying condition $(a)$ and
$c(t)\in\mathcal{P}_{T}$, there exist a closed subset $X\subset M$ and a
closed subset $E\subset X\cap\\{\psi=-\infty\\}$ satisfying that $X$ is
locally negligible with respect to $L^{2}$ holomorphic functions, $M\backslash
X$ is a Stein manifold, $e^{-\varphi}c(-\psi)$ has locally a positive lower
bound on $M\backslash E$ and there exists an analytic subset $Z$ of $M$ such
that $E\subset Z$.

Combining inequality (2.4) and $e^{-\varphi}c(-\psi)$ has locally a positive
lower bound on $M\backslash E$, we obtain that

$\int_{K\cap\\{\psi<-t_{0}\\}}|F|^{2}<+\infty$

holds for any compact subset $K$ of $M\backslash X$. Then Lemma 2.1 shows that
there exists a holomorphic $(n,0)$ form $\tilde{F}_{X}$ on
$\\{\psi<-t_{1}\\}\backslash X$, such that

$\begin{split}\int_{\\{\psi<-t_{1}\\}\backslash
X}|\tilde{F}_{X}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\leq
C\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt.\end{split}$ (2.6)

For any $z\in\\{\psi<-t_{1}\\}\cap(X\backslash E)$, there exists an open
neighborhood $V_{z}$ of $z$, such that
$V_{z}\subset\subset\\{\psi<-t_{1}\\}\backslash E$. Note that $c(t)e^{-t}$ is
decreasing on $(T,+\infty)$ and $v_{t_{0},B}(\psi)\geq\psi$, then we have

$\begin{split}&\int_{V_{z}\backslash
X}|\tilde{F}_{X}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\int_{V_{z}\backslash
X}|\tilde{F}_{X}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\\\
<&+\infty\end{split}$ (2.7)

Note that there exists a positive number $C_{1}>0$ such that
$e^{-\varphi}c(-\psi)>C_{1}$ on $V_{z}$ and $\int_{V_{z}\backslash
X}|(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi}c(-\psi)\leq\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}c(-\psi)<+\infty$,
then we have

$\begin{split}&\int_{V_{z}\backslash X}|\tilde{F}_{X}|^{2}\\\
\leq&C_{1}\int_{V_{z}\backslash X}|\tilde{F}_{X}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&2C_{1}\left(\int_{V_{z}\backslash
X}|(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi}c(-\psi)+\int_{V_{z}\backslash
X}|\tilde{F}_{X}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi}c(-\psi)\right)\\\
<&+\infty.\end{split}$ (2.8)

As $X$ is locally negligible with respect to $L^{2}$ holomorphic functions, we
can find a holomorphic extension $\tilde{F}_{E}$ of $\tilde{F}_{X}$ from
$\\{\psi<-t_{1}\\}\backslash X$ to $\\{\psi<-t_{1}\\}\backslash E$ such that

$\int_{\\{\psi<-t_{1}\\}\backslash
E}|\tilde{F}_{E}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\leq
C\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt.$ (2.9)

Note that $E\subset\\{\psi<-t_{0}\\}\subset\\{\psi<-t_{1}\\}$, for any $z\in
E$, there exists an open neighborhood $U_{z}$ of $z$, such that
$U_{z}\subset\subset\\{\psi<-t_{0}\\}$. As $\varphi+\psi$ is plurisubharmonic
on $M$ and $e^{v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))$ has a positive lower
bound on $\\{\psi<-t_{1}\\}$, then we have

$\begin{split}&\int_{U_{z}\backslash
E}|\tilde{F}_{E}-(1-b_{t_{0},B}(\psi))F|^{2}\\\
\leq&C_{2}\int_{\\{\psi<-t_{1}\\}\backslash
E}|\tilde{F}_{E}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\\\
<&+\infty,\end{split}$ (2.10)

where $C_{2}$ is some positive number. As
$U_{z}\subset\subset\\{\psi<-t_{0}\\}$, we have

$\int_{U_{z}}|(1-b_{t_{0},B}(\psi))F|^{2}\leq\int_{U_{z}}|F|^{2}<+\infty.$
(2.11)

Combining inequality (2.10) and (2.11), we obtain that $\int_{U_{z}\backslash
E}|\tilde{F}_{E}|^{2}<+\infty$.

As $E$ is contained in some analytic subset of $M$, we can find a holomorphic
extension $\tilde{F}$ of $\tilde{F}_{E}$ from $\\{\psi<-t_{1}\\}\backslash E$
to $\\{\psi<-t_{1}\\}$ such that

$\int_{\\{\psi<-t_{1}\\}}|\tilde{F}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\leq
C\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt.$ (2.12)

This proves Lemma 2.2. ∎

###  2.2. Some properties of $G(t)$

We present some properties related to $G(t)$ in the this section.

######  Lemma 2.3.

(see [16]) Let $N$ be a submodule of $\mathcal{O}_{\mathbb{C}^{n},o}^{q}$,
$1\leq q<+\infty$, 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 $q-$tuples
$f\in\mathcal{O}_{\mathbb{C}^{n}}(U)^{q}$, assume furthermore that all germs
$(f_{j},o)$ belong to $N$. Then $(f,o)\in N$.

The closedness of submodules will be used in the following discussion.

######  Lemma 2.4.

Let $M$ be a complex manifold. Let $S$ be an analytic subset of $M$. Let
$\\{g_{j}\\}_{j=1,2,...}$ be a sequence of nonnegative Lebesgue measurable
functions on $M$, which satisfies that $g_{j}$ are almost everywhere
convergent to $g$ on $M$ when $j\rightarrow+\infty$, where $g$ is a
nonnegative Lebesgue measurable function on $M$. Assume that for any compact
subset $K$ of $M\backslash S$, there exist $s_{K}\in(0,+\infty)$ and
$C_{K}\in(0,+\infty)$ such that

$\int_{K}{g_{j}}^{-s_{K}}dV_{M}\leq C_{K}$

for any $j$, where $dV_{M}$ is a continuous volume form on $M$.

Let $\\{F_{j}\\}_{j=1,2,...}$ be a sequence of holomorphic $(n,0)$ form on
$M$. Assume that $\liminf_{j\rightarrow+\infty}\int_{M}|F_{j}|^{2}g_{j}\leq
C$, where $C$ is a positive constant. Then there exists a subsequence
$\\{F_{j_{l}}\\}_{l=1,2,...}$, which satisfies that $\\{F_{j_{l}}\\}$ is
uniformly convergent to a holomorphic $(n,0)$ form $F$ on $M$ on any compact
subset of $M$ when $l\rightarrow+\infty$, such that

$\int_{M}|F|^{2}g\leq C.$

###### Proof.

As $S$ is a analytic subset of $M$, by Local Parametrization Theorem (see [8])
and Maximum Principle, for any compact set $K\subset\subset M$ there exists
$K_{1}\subset\subset M\backslash S$ satisfying

$\sup_{z\in K}\frac{|F_{j}(z)|^{2}}{dV_{M}}\leq C_{1}\sup_{z\in
K_{1}}\frac{|F_{j}(z)|^{2}}{dV_{M}}$ (2.13)

for any $j$, where $C_{1}$ is a constant. Then there exists a compact set
$K_{2}\subset\subset M\backslash S$ satisfying $K_{1}\subset K_{2}$ and

$\begin{split}\left(\frac{|F_{j}(z)|^{2}}{dV_{M}}\right)^{r}&\leq
C_{2}\int_{K_{2}}\left(\frac{|F_{j}(z)|^{2}}{dV_{M}}\right)^{r}\\\ &\leq
C_{2}\left(\int_{K_{2}}|F_{j}|^{2}g_{j}\right)^{r}\left(\int_{K_{2}}g_{j}^{-\frac{r}{1-r}}\right)^{1-r}\end{split}$
(2.14)

for any $j$ and any $z\in K_{1}$, where $r\in(0,1)$ and $C_{2}$ is a constant.
Let $\frac{r}{1-r}=s_{K_{2}}$, inequality (2.14) implies

$\sup_{z\in K_{1}}\frac{|F_{j}(z)|^{2}}{dV_{M}}\leq
C_{3}\int_{K_{2}}|F_{j}|^{2}g_{j},$ (2.15)

where $C_{3}$ is a constant. As
$\liminf_{j\rightarrow+\infty}\int_{M}|F_{j}|^{2}g_{j}<C$, combining
inequality (2.13), (2.15) and the diagonal method, we obtain a subsequence of
$\\{F_{j}\\}$, denoted still by $\\{F_{j}\\}$, which is uniformly convergent
to a holomorphic $(n,0)$ form $F$ on $M$ on any compact subset of $M$.

It follows from the Fatou’s Lemma and
$\lim_{j\rightarrow+\infty}\int_{M}|F_{j}|^{2}g_{j}\leq C$ that

$\begin{split}\int_{M}|F|^{2}g=&\int_{M}\lim_{j\rightarrow+\infty}|F_{j}|^{2}g_{j}\\\
\leq&\liminf_{j\rightarrow+\infty}\int_{M}|F_{j}|^{2}g_{j}\\\
\leq&C.\end{split}$

Thus Lemma 2.4 holds. ∎

Let $M$ be an $n-$dimensional complex manifold satisfying condition $(a)$. 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 $M$. Let $c\in\mathcal{P}_{T}$. The following lemma is a
characterization of $G(t)=0$ for any $t\geq T$, where $T=-\sup_{M}{\psi}$ and
the meaning of $G(t)$ can be referred to Section 1.1.

######  Lemma 2.5.

$f\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})\Leftrightarrow
G(t)=0$.

###### Proof.

It is clear that $f\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})\Rightarrow
G(t)=0$.

In the following part, we prove that $G(t)=0\Rightarrow f\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. As $G(t)=0$,
then there exists holomorphic $(n,0)$ forms
$\\{\tilde{f}_{j}\\}_{j\in\mathbb{N}^{+}}$ on $\\{\psi<-t\\}$ such that
$\lim_{j\to+\infty}\int_{\\{\psi<-t\\}}|\tilde{f}_{j}|^{2}e^{-\varphi}c(-\psi)=0$
and $(f_{j}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ for any $j$. As
$e^{-\varphi}c(-\psi)$ has a positive lower bound on any compact subset of
$M\backslash Z$, where $Z$ is some analytic subset of $M$, it follows from
Lemma 2.4 that there exists a subsequence of
$\\{\tilde{f}_{j}\\}_{j\in\mathbb{N}^{+}}$ denoted by
$\\{\tilde{f}_{j_{k}}\\}_{k\in\mathbb{N}^{+}}$ that compactly convergent to
$0$. It is clear that $\tilde{f}_{j_{k}}-f$ is compactly convergent to $0-f=f$
on $U\cap\\{\psi<-t\\}$. It follows from Lemma 2.3 that $f\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. This proves
Lemma 2.5. ∎

The following lemma shows the existence and uniqueness of the holomorphic
$(n,0)$ form related to $G(t)$.

######  Lemma 2.6.

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

$\begin{split}&\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)+\int_{\\{\psi<-t\\}}|\hat{F}-F_{t}|^{2}e^{-\varphi}c(-\psi)\\\
=&\int_{\\{\psi<-t\\}}|\hat{F}|^{2}e^{-\varphi}c(-\psi).\end{split}$ (2.16)

###### Proof.

Firstly, we prove the existence of $F_{t}$. As $G(t)<+\infty$ then there
exists holomorphic (n,0) forms $\\{f_{j}\\}_{j\in\mathbb{N}^{+}}$ on
$\\{\psi<-t\\}$ such that
$\lim_{j\rightarrow+\infty}\int_{\\{\psi<-t\\}}|f_{j}|^{2}e^{-\varphi}c(-\psi)=G(t)$,
and $(f_{j}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. As
$e^{-\varphi}c(-\psi)$ has a positive lower bound on any compact subset of
$M\backslash Z$, where $Z$ is some analytic subset of $M$, it follows from
Lemma 2.4 that there exists a subsequence of $\\{f_{j}\\}$ compact convergence
to a holomorphic $(n,0)$ form $F$ on $\\{\psi<-t\\}$ satisfying
$\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)\leq G(t)$. It follows from
Lemma 2.3 that $(F-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. Then we obtain
the existence of $F_{t}(=F)$.

Secondly, we prove the uniqueness of $F_{t}$ by contradiction: if not, there
exist two different holomorphic (n,0) forms $f_{1}$ and $f_{2}$ on on
$\\{\psi<-t\\}$ satisfying
$\int_{\\{\psi<-t\\}}|f_{1}|^{2}e^{-\varphi}c(-\psi)=\int_{\\{\psi<-t\\}}|f_{2}|^{2}=G(t)$,
$(f_{1}-f)\in H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$
and $(f_{2}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. Note that

$\begin{split}&\int_{\\{\psi<-t\\}}\left|\frac{f_{1}+f_{2}}{2}\right|^{2}e^{-\varphi}c(-\psi)+\int_{\\{\psi<-t\\}}\left|\frac{f_{1}-f_{2}}{2}\right|^{2}e^{-\varphi}c(-\psi)\\\
=&\frac{\int_{\\{\psi<-t\\}}|f_{1}|^{2}e^{-\varphi}c(-\psi)+\int_{\\{\psi<-t\\}}|f_{2}|^{2}e^{-\varphi}c(-\psi)}{2}=G(t),\end{split}$
(2.17)

then we obtain that

$\int_{\\{\psi<-t\\}}\left|\frac{f_{1}+f_{2}}{2}\right|^{2}e^{-\varphi}c(-\psi)<G(t),$

and $(\frac{f_{1}+f_{2}}{2}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$, which
contradicts the definition of $G(t)$.

Finally, we prove equality (2.16). For any holomorphic $h$ on $\\{\psi<-t\\}$
satisfying $\int_{\\{\psi<-t\\}}|h|^{2}e^{-\varphi}c(-\psi)<+\infty$ and $h\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$, it is clear
that for any complex number $\alpha$, $F_{t}+\alpha h$ satisfying
$((F_{t}+\alpha h)-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$, and
$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\int_{\\{\psi<-t\\}}|F_{t}+\alpha
h|^{2}e^{-\varphi}c(-\psi)<+\infty$. Note that

$\int_{\\{\psi<-t\\}}|F_{t}+\alpha
h|^{2}e^{-\varphi}c(-\psi)-\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\geq
0$

implies

$\Re\int_{\\{\psi<-t\\}}F_{t}\bar{h}e^{-\varphi}c(-\psi)=0$

then

$\int_{\\{\psi<-t\\}}|F_{t}+h|^{2}e^{-\varphi}c(-\psi)=\int_{\\{\psi<-t\\}}(|F_{t}|^{2}+|h|^{2})e^{-\varphi}c(-\psi).$

Choosing $h=\hat{F}-F_{t}$, we obtain equality (2.16). ∎

The following Lemma shows the monotonicity and lower semicontinuity property
of $G(t)$.

######  Lemma 2.7.

$G(t)$ is decreasing with respect to $t\in[T,+\infty)$, such that $\lim_{t\to
t_{0}+0}G(t)=G(t_{0})$ for any $t_{0}\in[T,+\infty)$, and if $G(t)<+\infty$
for some $t\geq T$, then $\lim_{t\to+\infty}G(t)=0$. Especially $G(t)$ is
lower semicontinuous on $[T,+\infty)$.

###### Proof.

By the definition of $G(t)$, it is clear that $G(t)$ is decreasing on
$[T,+\infty)$. And it follows from the dominated convergence theorem that if
$G(t)<+\infty$ for some $t\geq T$, then $\lim_{t\to+\infty}G(t)=0$. Then it
suffices to prove $\lim_{t\to t_{0}+0}G(t)=G(t_{0}).$ We prove it by
contradiction: if not, then $\lim_{t\to t_{0}+0}G(t)<G(t_{0})$.

By Lemma 2.6, 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
$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)=G(t)$. Note that $G(t)$
is decreasing implies that
$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\lim_{t\to
t_{0}+0}G(t)$ for any $t>t_{0}$. If $\lim_{t\to t_{0}+0}G(t)=+\infty$, the
equality $\lim_{t\to t_{0}+0}G(t)=G(t_{0})$ is clear, thus it suffices to
prove the case $\lim_{t\to t_{0}+0}G(t)<+\infty$. As $e^{-\varphi}c(-\psi)$
has a positive lower bound on any compact subset of $M\backslash Z$, where $Z$
is some analytic subset of $M$, and
$\int_{\\{\psi<-t_{1}\\}}|F_{t}|e^{-\varphi}c(-\psi)\leq\lim_{t\to
t_{0}+0}G(t)<+\infty$ holds for any $t\in(t_{0},t_{1}]$, where $t_{1}>t_{0}$
is a fixed number, it follows from Lemma 2.4 that there exists
$\\{F_{t_{j}}\\}$ $(t_{j}\to t_{0}+0,$ as $j\to+\infty)$ uniformly convergent
on any compact subset of $\\{\psi<-t_{1}\\}$. Using the diagonal method, we
obtain a subsequence of $\\{F_{t}\\}$ (also denoted by $\\{F_{t_{j}}\\}$),
which is convergent on any compact subset of $\\{\psi<-t_{0}\\}$.

Let $\hat{F}_{t_{0}}=\lim_{j\to+\infty}F_{t_{j}}$, which is a holomorphic
$(n,0)$ form on $\\{\psi<-t_{0}\\}$. Then it follows from the decreasing
property of $G(t)$ that

$\int_{K}|\hat{F}_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq\lim_{j\to+\infty}\int_{K}|F_{t_{j}}|^{2}e^{-\varphi}c(-\psi)\leq\lim_{j\to+\infty}G(t_{j})\leq\lim_{t\to
t_{0}+0}G(t)$

for any compact set $K\subset\\{\psi<-t_{0}\\}$. It follows from Levi’s
Theorem that

$\int_{\\{\psi<-t_{0}\\}}|\hat{F}_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq\lim_{t\to
t_{0}+0}G(t).$

It follows from Lemma 2.3 that $\hat{F}_{t_{0}}\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. Then we obtain
that
$G(t_{0})\leq\int_{\\{\psi<-t_{0}\\}}|\hat{F}_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq\lim_{t\to
t_{0}+0}G(t)$, which contradicts $\lim_{t\to t_{0}+0}G(t)<G(t_{0})$. ∎

We consider the derivatives of $G(t)$ in the following lemma.

######  Lemma 2.8.

Assume that $G(t_{1})<\infty$, where $t_{1}\in(T,+\infty)$, then for any
$t_{0}>t_{1}$ we have

$\frac{G(t_{1})-G(t_{0})}{\int_{t_{1}}^{t_{0}}c(t)e^{-t}dt}\leq\liminf_{B\to
0+0}\frac{G(t_{0})-G(t_{0}+B)}{\int_{t_{0}}^{t_{0}+B}c(t)e^{-t}dt},$

i.e.

$\frac{G(t_{0})-G(t_{1})}{\int_{T_{1}}^{t_{0}}c(t)e^{-t}dt-\int_{T_{1}}^{t_{1}}c(t)e^{-t}dt}\geq\limsup_{B\to
0+0}\frac{G(t_{0}+B)-G(t_{0})}{\int_{T_{1}}^{t_{0}+B}c(t)e^{-t}dt-\int_{T_{1}}^{t_{0}+B}c(t)e^{-t}dt}.$

###### Proof.

It follows from Lemma 2.7 that $G(t)<+\infty$ for any $t\geq t_{1}$. By Lemma
2.6, there exists a holomorphic $(n,0)$ form $F_{t_{0}}$ on
$\\{\psi<-t_{0}\\}$, such that $(F_{t_{0}}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and
$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)=G(t_{0})$.

It suffices to consider that $\liminf_{B\to
0+0}\frac{G(t_{0})-G(t_{0}+B)}{\int_{t_{0}}^{t_{0}+B}c(t)e^{-t}dt}\in[0.+\infty)$
because of the decreasing property of $G(t)$. Then there exists $B_{j}\to 0+0$
$(j\to+\infty)$ such that

$\lim_{j\to+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{\int_{t_{0}}^{t_{0}+B_{j}}c(t)e^{-t}dt}=\liminf_{B\to
0+0}\frac{G(t_{0})-G(t_{0}+B)}{\int_{t_{0}}^{t_{0}+B}c(t)e^{-t}dt}$ (2.18)

and
$\\{\frac{G(t_{0})-G(t_{0}+B_{j})}{\int_{t_{0}}^{t_{0}+B_{j}}c(t)e^{-t}dt}\\}_{j\in\mathbb{N}^{+}}$
is bounded. As $c(t)e^{-t}$ is decreasing and positive on $(T,+\infty)$, then

$\begin{split}\lim_{j\to+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{\int_{t_{0}}^{t_{0}+B_{j}}c(t)e^{-t}dt}=&\left(\lim_{j\to+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}\right)\left(\frac{1}{\lim_{t\rightarrow
t_{0}+0}c(t)e^{-t}}\right)\\\
=&\left(\lim_{j\to+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}\right)\left(\frac{e^{t_{0}}}{\lim_{t\rightarrow
t_{0}+0}c(t)}\right).\end{split}$ (2.19)

Hence
$\left\\{\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}\right\\}_{j\in\mathbb{N}^{+}}$
is bounded with respect to $j$.

As $t\leq v_{t_{0},B_{j}}(t)$, the decreasing property of $c(t)e^{-t}$ shows
that

$e^{-\psi+v_{t_{0},B_{j}}(\psi)}c(-v_{t_{0},B_{j}}(\psi))\geq c(-\psi).$

Lemma 2.2 shows that for any $B_{j}$, there exists holomorphic $(n,0)$ form
$\tilde{F}_{j}$ on $\\{\psi<-t_{1}\\}$, such that
$(\tilde{F}_{j}-F_{t_{0}})\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{I}(\varphi+\psi))|_{Z_{0}})\subseteq
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and

$\begin{split}&\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi+v_{t_{0},B_{j}}(\psi)}c(-v_{t_{0},B_{j}}(\psi))\\\
\leq&\int_{t_{1}}^{t_{0}+B_{j}}c(t)e^{-t}dt\int_{\\{\psi<-t_{1}\\}}\frac{1}{B_{j}}(\mathbb{I}_{\\{-t_{0}-B_{j}<\psi<-t_{0}\\}})|F_{t_{0}}|^{2}e^{-\varphi-\psi}\\\
\leq&\frac{e^{t_{0}+B_{j}}\int_{t_{1}}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)}\int_{\\{\psi<-t_{1}\\}}\frac{1}{B_{j}}(\mathbb{I}_{\\{-t_{0}-B_{j}<\psi<-t_{0}\\}})|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\frac{e^{t_{0}+B_{j}}\int_{t_{1}}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)}\times\Bigg{(}\int_{\\{\psi<-t_{1}\\}}\frac{1}{B_{j}}\mathbb{I}_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\\
&-\int_{\\{\psi<-t_{1}\\}}\frac{1}{B_{j}}\mathbb{I}_{\\{\psi<-t_{0}-B_{j}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\Bigg{)}\\\
\leq&\frac{e^{t_{0}+B_{j}}\int_{t_{1}}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)}\times\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}.\end{split}$
(2.20)

Firstly, we will prove that
$\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)$ is bounded
with respect to $j$.

Note that

$\begin{split}&\left(\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}\\\
\geq&\left(\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}-\left(\int_{\\{\psi<-t_{1}\\}}|(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}},\end{split}$
(2.21)

then it follows from inequality (2.20) that

$\begin{split}&\left(\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}\\\
\leq&\left(\frac{e^{t_{0}+B_{j}}\int_{t_{1}}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)}\right)^{\frac{1}{2}}\left(\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}\right)^{\frac{1}{2}}\\\
&+\left(\int_{\\{\psi<-t_{1}\\}}|(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}.\end{split}$
(2.22)

Since
$\left\\{\frac{G(t_{0}+B_{j})-G(t_{0})}{B_{j}}\right\\}_{j\in\mathbb{N}^{+}}$
is bounded,
$\lim_{j\rightarrow+\infty}\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)=\lim_{t\rightarrow
t_{0}+0}c(t)\in(0,+\infty)$ and

$\int_{\\{\psi<-t_{1}\\}}|(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq\int_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)<+\infty,$

then $\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)$ is
bounded with respect to $j$.

Secondly, we will prove the main result.

It follows from
$\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)$ is bounded
with respect to $j$ and Lemma 2.4 that there exists a subsequence of
$\\{\tilde{F}_{j}\\}$, denoted by
$\\{\tilde{F}_{j_{k}}\\}_{k\in\mathbb{N}^{+}}$, which is uniformly convergent
to a holomorphic $(n,0)$ form $F_{1}$ on $\\{\psi<-t_{1}\\}$ on any compact
subset of $\\{\psi<-t_{1}\\}$ when $k\rightarrow+\infty$, such that

$\int_{\\{\psi<-t_{1}\\}}|F_{1}|^{2}e^{-\varphi}c(-\psi)\leq\liminf_{j\rightarrow+\infty}\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)<+\infty.$

As $(\tilde{F}_{j}-F_{t_{0}})\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ for any $j$, we
have $(F_{1}-F_{t_{0}})\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. Note that

$\lim_{j\rightarrow+\infty}b_{t_{0},B_{j}}(t)=\lim_{j\rightarrow+\infty}\int_{-\infty}^{t}\frac{1}{B_{j}}\mathbb{I}_{\\{-t_{0}-B_{j}<s<-t_{0}\\}}ds=\left\\{\begin{array}[]{lcl}0&\mbox{if}&x\in(-\infty,-t_{0})\\\
1&\mbox{if}&x\in[-t_{0},+\infty)\end{array}\right.$

and

$\lim_{j\rightarrow+\infty}v_{t_{0},B_{j}}(t)=\lim_{j\rightarrow+\infty}\int_{-t_{0}}^{t}b_{t_{0},B_{j}}ds-
t_{0}=\left\\{\begin{array}[]{lcl}-t_{0}&\mbox{if}&x\in(-\infty,-t_{0})\\\
t&\mbox{if}&x\in[-t_{0},+\infty)\end{array}.\right.$

Following from equality (2.19), inequality (2.20) and the Fatou’s Lemma, we
have

$\begin{split}&\int_{\\{\psi<-t_{0}\\}}|F_{1}-F_{t_{0}}|e^{-\varphi-\psi-
t_{0}}c(t_{0})+\int_{\\{-t_{0}\leq\psi<-t_{1}\\}}|F_{1}|^{2}e^{-\varphi}c(-\psi)\\\
=&\int_{\\{\psi<-t_{1}\\}}\lim_{k\rightarrow+\infty}|\tilde{F}_{j_{k}}-(1-b_{t_{0},B_{j_{k}}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi+v_{t_{0},B_{j_{k}}}(\psi)}c(-v_{t_{0},B_{j_{k}}}(\psi))\\\
\leq&\liminf_{k\rightarrow+\infty}\int_{\\{\psi<-t_{1}\\}}|\tilde{F}_{j_{k}}-(1-b_{t_{0},B_{j_{k}}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi+v_{t_{0},B_{j_{k}}}(\psi)}c(-v_{t_{0},B_{j_{k}}}(\psi))\\\
\leq&\liminf_{k\rightarrow+\infty}\left(\frac{e^{t_{0}+B_{j_{k}}}\int_{t_{1}}^{t_{0}+B_{j_{k}}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j_{k}})}c(t)}\times\frac{G(t_{0})-G(t_{0}+B_{j_{k}})}{B_{j_{k}}}\right)\\\
=&\frac{e^{t_{0}}\int_{t_{1}}^{t_{0}}c(t)e^{-t}dt}{\lim_{t\rightarrow
t_{0}+0}c(t)}\lim_{j\rightarrow+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}\\\
=&\int_{t_{1}}^{t_{0}}c(t)e^{-t}dt\lim_{j\rightarrow+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{\int_{t_{0}}^{t_{0}+B_{j}}c(t)e^{-t}dt}.\end{split}$
(2.23)

As $e^{\psi}c(-\psi)\leq e^{-t_{0}}c(t_{0})$ on $\\{\psi<-t_{0}\\}$, it
follows Lemma 2.6, equality (2.18) and inequality (2.23) that

$\begin{split}&\int_{t_{1}}^{t_{0}}c(t)e^{-t}dt\liminf_{B\rightarrow
0+0}\frac{G(t_{0})-G(t_{0}+B)}{\int_{t_{0}}^{t_{0}+B}c(t)e^{-t}dt}\\\
=&\int_{t_{1}}^{t_{0}}c(t)e^{-t}dt\lim_{j\rightarrow+\infty}\frac{G(t_{0})-G(t_{0}+B_{j})}{\int_{t_{0}}^{t_{0}+B_{j}}c(t)e^{-t}dt}\\\
\geq&\int_{\\{\psi<-t_{0}\\}}|F_{1}-F_{t_{0}}|e^{-\varphi-\psi-
t_{0}}c(t_{0})+\int_{\\{-t_{0}\leq\psi<-t_{1}\\}}|F_{1}|^{2}e^{-\varphi}c(-\psi)\\\
\geq&\int_{\\{\psi<-t_{0}\\}}|F_{1}-F_{t_{0}}|e^{-\varphi}c(-\psi)+\int_{\\{-t_{0}\leq\psi<-t_{1}\\}}|F_{1}|^{2}e^{-\varphi}c(-\psi)\\\
=&\int_{\\{\psi<-t_{1}\\}}|F_{1}|^{2}e^{-\varphi}c(-\psi)-\int_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\\
\geq&G(t_{1})-G(t_{0}).\end{split}$ (2.24)

This proves Lemma 2.8. ∎

The following well-known property of concave functions will be used in the
proof of Theorem 1.3.

######  Lemma 2.9.

Let $a(r)$ be a lower semicontinuous function on $(A,B)$ ( $-\infty\leq
A<B\leq+\infty$). Then $a(r)$ is concave if and only if

$\frac{a(r_{2})-a(r_{1})}{r_{2}-r_{1}}\geq\limsup_{r\to
r_{2}+0}\frac{a(r)-a(r_{2})}{r-r_{2}},$ (2.25)

holds for any $A<r_{1}<r_{2}<B$.

###### Proof.

For the convenience of the reader, we recall the proof.

It suffices to prove that inequality (2.25) implies the concavity of $a(r)$.
We prove by contradiction: if not, there exists $A<r_{3}<r_{4}<r_{5}<B$ such
that

$\frac{a(r_{4})-a(r_{3})}{r_{4}-r_{3}}<\frac{a(r_{5})-a(r_{3})}{r_{5}-r_{3}}<\frac{a(r_{5})-a(r_{4})}{r_{5}-r_{4}}.$
(2.26)

Consider
$\tilde{a}(r)=a(r)-a(r_{5})-\frac{a(r_{5})-a(r_{3})}{r_{5}-r_{3}}(r-r_{5})$ on
$(A,B)$. As $a(r)$ is lower semicontinuous on $(A,B)$, then $\tilde{a}(r)$ is
lower semicontinuous on $(A,B)$. Note that
$\tilde{a}(r_{3})=\tilde{a}(r_{5})=0$ and $\tilde{a}(r_{4})<0$, then it
follows from the lower semicontinuity of $\tilde{a}(r)$ that there exists
$r_{6}\in(r_{3},r_{5})$ such that
$\tilde{a}(r_{6})=inf_{r\in[r_{3},r_{5}]}\tilde{a}(r)<0$. It clear that
$\frac{\tilde{a}(r_{6})-\tilde{a}(r_{3})}{r_{6}-r_{3}}<0$ and
$\limsup_{r\rightarrow
r_{6}+0}\frac{\tilde{a}(r)-\tilde{a}(r_{6})}{r-r_{6}}\geq 0$. Then we obtain
that
$\frac{{a}(r_{6})-{a}(r_{3})}{r_{6}-r_{3}}<\frac{a(r_{5})-a(r_{3})}{r_{5}-r_{3}}\leq\limsup_{r\rightarrow
r_{6}+0}\frac{{a}(r)-{a}(r_{6})}{r-r_{6}}$, which contradict inequality
(2.25). ∎

###  2.3. Some results used in the proofs of applications

In this section, we give some results which will be used in the proofs of
applications in Section 1.2.

######  Lemma 2.10.

If $c(t)$ is a positive measurable function on $(T,+\infty)$ such that
$c(t)e^{-t}$ is decreasing on $(T,+\infty)$ and
$\int_{T_{1}}^{+\infty}c(t)e^{-t}dt<+\infty$ for some $T_{1}>T$, then there
exists a positive measurable function $\tilde{c}$ on $(T,+\infty)$, satisfying
the following statements:

$(1)$ $\tilde{c}\geq c$ on $(T,+\infty)$;

$(2)$ $\tilde{c}(t)e^{-t}$ is strictly decreasing on $(T,+\infty)$ and
$\tilde{c}$ is increasing on $(a,+\infty)$, where $a>T$ is a real number;

$(3)$ $\int_{T_{1}}^{+\infty}\tilde{c}(t)e^{-t}dt<+\infty$.

Moreover, if $\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$ and
$c\in\mathcal{P}_{T}$, we can choose $\tilde{c}$ satisfying the above
conditions, $\int_{T}^{+\infty}\tilde{c}(t)e^{-t}dt<+\infty$ and
$\tilde{c}\in\mathcal{P}_{T}$.

###### Proof.

Without loss of generality, we can assume that $T<0$. Let $a_{n}=c(n)e^{-n}$,
where $n\in\mathbb{N}^{+}$. Take $b_{1}=a_{1}$, and we can define
$b_{n}=\max{\\{\frac{b_{n-1}}{e},a_{n}\\}}$ for $n>1$, inductively. Since
$a(n)$ is decreasing with respect to $n$, we have $b_{n}\geq
b_{n+1}\geq\frac{b_{n}}{e}$ and $b_{n}\geq a_{n}$ for any
$n\in\mathbb{N}^{+}$.

Let

$\tilde{a}(t)=\left\\{\begin{array}[]{lcl}eb_{n}(\frac{b_{n+1}}{b_{n}})^{t-n}&\mbox{if}&t\in[n,n+1),\\\
c(t)e^{-t+1}&\mbox{if}&t\in(T,1].\end{array}\right.$

It clear that $\tilde{a}(t)\geq c(t)e^{-t}$, $\tilde{a}(t)$ is decreasing and
continuous on $(T,+\infty)$. Let $\tilde{c}(t)=\tilde{a}(t)e^{t}$. When
$t\in[n,n+1)$, as $eb_{n+1}\geq b_{n}$, we have $\tilde{c}(t)$ is increasing
on $[n,n+1)$, which implies that $\tilde{c}(t)$ is increasing on
$(1,+\infty)$.

As $\int_{0}^{+\infty}c(t)e^{-t}dt<+\infty$, then
$\sum_{n=1}^{+\infty}a_{n}<+\infty$. In the following, we will prove
$\int_{0}^{+\infty}\tilde{c}(t)e^{-t}<+\infty$. By the definition of
$\tilde{c}(t)$, we have

$\int_{0}^{+\infty}\tilde{c}(t)e^{-t}=\int_{0}^{1}\tilde{a}(t)dt+\sum_{n=1}^{+\infty}\int_{n}^{n+1}\tilde{a}(t)dt\leq
c(0)e+e\sum_{n=1}^{+\infty}b_{n}.$ (2.27)

Take $I=\\{n_{i}:n_{i}$ is the $i-$th positive integer such that
$a_{n_{i}}=b_{n_{i}}$$\\}\in\mathbb{N}^{+}$. Note that if
$a_{n+1}\not=b_{n+1}$, then $b_{n+1}=\frac{b_{n}}{e}$, thus we have

$\begin{split}\sum_{n=1}^{+\infty}b_{n}=&\sum_{i=1}\sum_{j=0}^{n_{i+1}-n_{i}-1}b_{n_{i}+j}\\\
=&\sum_{i=1}\sum_{j=0}^{n_{i+1}-n_{i}-1}b_{n_{i}}e^{-j}\\\
\leq&\sum_{i=1}a_{n_{i}}\frac{e}{e-1}\\\ <&+\infty,\end{split}$ (2.28)

where if $n_{i}$ is the largest integer such that $a_{n_{i}}=b_{n_{i}}$, take
$n_{i+1}=+\infty$. Combining inequality (2.27) and (2.28), we obtain
$\int_{0}^{+\infty}\tilde{c}(t)e^{-t}dt<+\infty$. Let
$\tilde{c}(t)=\tilde{c}(t)+1$, we have $\tilde{c}\geq c$, $\tilde{c}$ is
increasing on $(1,+\infty)$, $c(t)e^{-t}$ is strictly decreasing on
$(T,+\infty)$ and $\int_{0}^{+\infty}\tilde{c}(t)e^{-t}dt<+\infty$.

Moreover, if $\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$ and
$c\in\mathcal{P}_{T}$, as $\tilde{c}(t)\geq c(t)$ on $(T,+\infty)$ and
$\tilde{c}(t)=ec(t)+1$ on $(T,1)$, we have
$\int_{T}^{+\infty}\tilde{c}(t)e^{-t}dt<+\infty$ and $c\in\mathcal{P}_{T}$.
Thus, Lemma 2.10 holds. ∎

Let $\Omega$ be an open Riemann surface admitted a nontrivial Green function
$G_{\Omega}$. Let $w$ be a local coordinate on a neighborhood $V_{z_{0}}$ of
$z_{0}\in\Omega$ satisfying $w(z_{0})=0$.

######  Lemma 2.11 (see [42], see also [53]).

$G_{\Omega}(z,z_{0})=\sup_{v\in\Delta_{0}(z_{0})}v(z)$, where
$\Delta_{0}(z_{0})$ is the set of negative subharmonic functions on $\Omega$
satisfying that $v-\log|w|$ has a locally finite upper bound near $z_{0}$.

######  Lemma 2.12.

For any open neighborhood $U$ of $z_{0}$, there exists $t>0$ such that
$\\{G_{\Omega}(z,z_{0})<-t\\}$ is a relatively compact subset of $U$.

###### Proof.

Let $w$ be a coordinate on a neighborhood $V_{z_{0}}\subset\subset U$ of
$z_{0}$, such that $w(z_{0})=0$ and $G_{\Omega}(z,z_{0})=\log|w(z)|+v(w(z))$,
where $v$ is a harmonic function on $V_{z_{0}}$ and
$\sup_{V_{z_{0}}}|v(w(z))|<+\infty$. Then there exists $t>0$ such that
$\\{z\in V_{z_{0}}:\log|w(z)|+v(w(z))<-t\\}\subset\subset V_{z_{0}}$.

We claim that $\\{z\in\Omega:G_{\Omega}(z,z_{0})<-t\\}\subset\subset
V_{z_{0}}$, therefore Lemma 2.12 holds. In fact, set

$\tilde{G}(z)=\left\\{\begin{array}[]{lcl}G_{\Omega}(z,z_{0})&\mbox{if}&z\in
V_{z_{0}},\\\ \max\\{G_{\Omega}(z,z_{0}),-t\\}&\mbox{if}&z\in\Omega\backslash
V_{z_{0}}.\end{array}\right.$

As $\\{z\in V_{z_{0}}:\log|w(z)|+v(w(z))<-t\\}\subset\subset V_{z_{0}}$, we
know $\tilde{G}(z)$ is subharmonic on $\Omega$. Lemma 2.11 tells us
$\tilde{G}(z)\leq G_{\Omega}(z,z_{0})$, therefore
$\\{z\in\Omega:G_{\Omega}(z,z_{0})<-t\\}=\\{z\in
V_{z_{1}}:G_{\Omega}(z,z_{0})<-t\\}\subset\subset V_{z_{0}}$. ∎

######  Lemma 2.13.

For any $z_{0}\in\Omega$ and open subsets $V_{1}$ and $U_{1}$ of $\Omega$
satisfying $z_{0}\in V_{1}\subset\subset U_{1}\subset\subset\Omega$, there
exists a constant $N>0$ such that

$G_{\Omega}(z,z_{1})\geq NG_{\Omega}(z,z_{0})$

holds for any $(z,z_{1})\in(\Omega\backslash U_{1})\times V_{1}$.

###### Proof.

As $V_{1}\subset\subset U_{1}\subset\subset\Omega$, fixed
$z\in\Omega\backslash U_{1}$, $G_{\Omega}(z,z_{1})$ is harmonic with respect
to $z_{1}$ on a open neighborhood of $\overline{V_{1}}$. The Harnack
inequality shows that there exists a constant $N>0$ such that

$\sup_{z_{1}\in\overline{V_{1}}}(-G_{\Omega}(z,z_{1}))\leq
N\inf_{z_{1}\in\overline{V_{1}}}(-G_{\Omega}(z,z_{1}))$ (2.29)

holds of any $z\in\Omega\backslash U_{1}$. As $z_{0}\in V_{1}$, it follows
from inequality (2.29) that

$G_{\Omega}(z,z_{1})\geq NG(z,z_{0})$

holds for any $(z,z_{1})\in(\Omega\backslash U_{1})\times V_{1}$. ∎

The following lemma (proof can be referred to Section 7.2) will be used in the
proof of Theorem 1.16.

######  Lemma 2.14.

Let $T$ be a closed positive $(1,1)$ current on $\Omega$. For any open set
$U\subset\subset\Omega$ satisfying $U\cap suppT\not=\emptyset$, there exists a
subharmonic function $\Phi<0$ on $\Omega$, which satisfies the following
properties:

$(1)$ $i\partial\bar{\partial}\Phi\leq T$ and
$i\partial\bar{\partial}\Phi\not\equiv 0$;

$(2)$ $\lim_{t\rightarrow
0+0}(\inf_{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\Phi(z))=0$;

$(3)$ $supp(i\partial\bar{\partial}\Phi)\subset U$ and $\inf_{\Omega\backslash
U}\Phi>-\infty$.

Now, we recall some notations. Let $c_{\beta}(z)$ be the logarithmic capacity
which is locally defined by

$c_{\beta}(z_{0}):=\exp\lim_{z\rightarrow
z_{0}}(G_{\Omega}(z,z_{0})-\log|w(z)|)$

on $\Omega$ (see [42]). The weighted Bergman kernel $\kappa_{\Omega,\rho}$
with weight $\rho$ of holomorphic $(1,0)$ form on $\Omega$ is defined by
$\kappa_{\Omega,\rho}:=\sum_{i}e_{i}\otimes\bar{e}_{i}$, where
$\\{e_{i}\\}_{i=1,2,...}$ are holomorphic $(1,0)$ forms on $\Omega$ and
satisfy
$\sqrt{-1}\int_{\Omega}\rho\frac{e_{i}}{\sqrt{2}}\wedge\frac{\bar{e}_{j}}{\sqrt{2}}=\delta_{i}^{j}$.
Let $B_{\Omega,\rho}(z):=\frac{\kappa_{\Omega,\rho}(z)}{|dw|^{2}}$ on
$V_{z_{0}}$.

######  Theorem 2.15 ([25]).

(A solution of the extended Suita Conjecture) Let $u$ be a harmonic function
on $\Omega$. $c_{\beta}^{2}(z_{0})\leq\pi
e^{-2u(z_{0})}B_{\Omega,e^{-2u}}(z_{0})$ holds, and the equality holds if and
only if $\chi_{-u}=\chi_{z_{0}}.$

##  3\. proofs of Theorem 1.3, Corollary 1.4, Corollary 1.5 and Corollary 1.7

In this section, we prove Theorem 1.3, Corollary 1.4, Corollary 1.5 and
Corollary 1.7.

###  3.1. Proof of Theorem 1.3

Firstly, we prove that if $G(t_{0})<+\infty$ for some $t_{0}>T$, then
$G(t_{1})<+\infty$ for any $t_{1}\in(T,t_{0})$. It follows from Lemma 2.6 that
there exists a holomorphic $(n,0)$ form $F_{t_{0}}$ on $\\{\psi<-t_{0}\\}$
satisfying $(F_{t_{0}}-f)\in
H^{0}(Z_{0},(\mathcal{O}(F_{M})\otimes\mathcal{F})|_{Z_{0}})$ and
$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)=G(t_{0})<+\infty$.
Using Lemma 2.2 we get a holomorphic $(n,0)$ form $\tilde{F}$ on
$\\{\psi<-t_{1}\\}$, such that

$(\tilde{F}-F_{t_{0}})\in
H^{0}(Z_{0},(\mathcal{O}(F_{M})\otimes\mathcal{I}(\varphi+\psi))|_{Z_{0}})\subset
H^{0}(Z_{0},(\mathcal{O}(F_{M})\otimes\mathcal{F})|_{Z_{0}})$

and

$\begin{split}&\int_{\\{\psi<-t_{1}\\}}|\tilde{F}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\int_{\\{\psi<-t_{1}\\}}|\tilde{F}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi-\psi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\\\
\leq&\left(\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt\right)\int_{\\{\psi<-t_{1}\\}}\frac{1}{B}\mathbb{I}_{\\{-t_{0}+B<\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi-\psi}.\end{split}$
(3.1)

Note that

$\begin{split}&\left(\int_{\\{\psi<-t_{1}\\}}|\tilde{F}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}-\left(\int_{\\{\psi<-t_{1}\\}}|(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}\\\
\leq&\left(\int_{\\{\psi<-t_{1}\\}}|\tilde{F}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}},\end{split}$

combining with inequality (3.1), we obtain

$\begin{split}&\left(\int_{\\{\psi<-t_{1}\\}}|\tilde{F}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}\\\
\leq&\left(\left(\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt\right)\int_{\\{\psi<-t_{1}\\}}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi-\psi}\right)^{\frac{1}{2}}\\\
&+\left(\int_{\\{\psi<-t_{1}\\}}|(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}.\end{split}$
(3.2)

As $b_{t_{0},B}(\psi)=1$ on $\\{\psi\geq t_{0}\\}$, $0\leq
b_{t_{0},B}(\psi)\leq 1$,
$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)<+\infty$, and
$c(t)$ has a positive lower bound on any compact subset of $(T,+\infty)$, then

$\left(\int_{\\{\psi<-t_{1}\\}}|(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}<+\infty$

and

$\begin{split}&\left(\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt\right)\int_{\\{\psi<-t_{1}\\}}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi-\psi}\\\
\leq&\frac{e^{t_{0}+B}\int_{t_{1}}^{t_{0}+B}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B)}c(t)}\int_{\\{\psi<-t_{1}\\}}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\\
<&+\infty,\end{split}$

which implies that

$\int_{\\{\psi<-t_{1}\\}}|\widetilde{F}|^{2}e^{-\varphi}c(-\psi)<+\infty.$

Then we obtain
$G(t_{1})\leq\int_{\\{\psi<-t_{1}\\}}|\widetilde{F}|^{2}e^{-\varphi}c(-\psi)<+\infty$.

Now, assume that $G(t_{0})<+\infty$ for some $t_{0}\geq T$ (otherwise it is
clear that $G(t)\equiv+\infty$). As $G(h^{-1}(r))$ is lower semicontinuous
(Lemma 2.7), then Lemma 2.8 and Lemma 2.9 imply the concavity of
$G(h^{-1}(r))$. It follows from Lemma 2.7 that $\lim_{t\rightarrow
T+0}G(t)=G(T)$ and $\lim_{t\rightarrow+\infty}G(t)=0$, hence we prove Theorem
1.3.

###  3.2. Proof of Corollary 1.4

Note that if there exists a positive decreasing concave function $g(t)$ on
$(a,b)\subset\mathbb{R}$ and $g(t)$ is not a constant function, then
$b<+\infty$. We prove Corollary 1.4 by contradiction: if $G(t)<+\infty$ for
some $t\geq T$, as $f\not\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$, Lemma 2.5 shows
that $G(t)\in(0,+\infty)$. Following from Theorem 1.3 we know $G({h}^{-1}(r))$
is concave with respect to
$r\in(\int_{T_{1}}^{T}c(t)e^{-t}dt,\int_{T_{1}}^{+\infty}c(t)e^{-t}dt)$ and
$G({h}^{-1}(r))$ is not a constant function, therefore we obtain
$\int_{T_{1}}^{+\infty}c(t)e^{-t}dt<+\infty$, which contradicts to
$\int_{T_{1}}^{+\infty}c(t)e^{-t}dt=+\infty$. Thus Corollary 1.4 holds.

###  3.3. Proof of Corollary 1.5

If $G(t)\in(0,+\infty)$ for some $t\geq T$, Corollary 1.4 and Lemma 2.5 show
that $\int_{T_{1}}^{+\infty}c(t)e^{-t}dt<+\infty$. As
$\lim_{t\rightarrow+\infty}G(t)=0$, then $G(h^{-1}(r))$ is concave on
$(\int_{T_{1}}^{T}c(t)e^{-t}dt,\int_{T_{1}}^{+\infty}c(t)e^{-t}dt]$ by
defining $G(+\infty)=0$. Then the concavity of $G(h^{-1}(r))$ implies that the
three statements are equivalent.

###  3.4. Proof of Corollary 1.7

It follows from Corollary 1.5 that
$G(t)=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t}^{+\infty}c(s)e^{-s}ds$
for any $t\in[T,+\infty)$.

Firstly, we prove the existence and uniqueness of $F$.

Following the notations in Lemma 2.8, as
$G(t)=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t}^{+\infty}c(s)e^{-s}ds\in(0,+\infty)$
for any $t\in(T,+\infty)$, by choosing $t_{1}\in(T,+\infty)$ and
$t_{0}>t_{1}$, we know that the inequality (2.24) must be equality, which
implies that

$\int_{\\{\psi<-t_{0}\\}}|F_{1}-F_{t_{0}}|^{2}e^{-\varphi}(e^{-\psi-
t_{0}}c(t_{0})-c(-\psi))=0,$ (3.3)

where $F_{1}$ is a holomorphic $(n,0)$ form on $\\{\psi<-t_{1}\\}$ such that
$(F_{1}-f)\in H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$
and $F_{t_{0}}$ is a holomorphic $(n,0)$ form on $\\{\psi<-t_{0}\\}$ such that
$(F_{t_{0}}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$. As
$\int_{T_{1}}^{+\infty}c(t)e^{-t}<+\infty$ and $c(t)e^{-t}$ is decreasing,
then there exists $t_{2}>t_{0}$ such that
$c(t)e^{-t}<c(t_{0})e^{-t_{0}}-\delta$ for any $t\geq t_{2}$, where $\delta$
is a positive constant. Then equality (3.3) implies that

$\begin{split}&\delta\int_{\\{\psi<-t_{2}\\}}|F_{1}-F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi}\\\
\leq&\int_{\\{\psi<-t_{2}\\}}|F_{1}-F_{t_{0}}|^{2}e^{-\varphi}(e^{-\psi-
t_{0}}c(t_{0})-c(-\psi))\\\
\leq&\int_{\\{\psi<-t_{0}\\}}|F_{1}-F_{t_{0}}|^{2}e^{-\varphi}(e^{-\psi-
t_{0}}c(t_{0})-c(-\psi))\\\ =&0\end{split}$

It follows from $\varphi+\psi$ is plurisubharmonic function and $F_{1}$ and
$F_{t_{0}}$ are holomorphic $(n,0)$ forms that $F_{1}=F_{t_{0}}$ on
$\\{\psi<-t_{0}\\}$. As
$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)=G(t_{0})$ and the
inequality (2.24) becomes equality, we have

$\int_{\\{\psi<-t_{1}\\}}|F_{1}|^{2}e^{-\varphi}c(-\psi)=G(t_{1}).$

Following from Lemma 2.6, 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
$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)=G(t)$ for any $t>T$. By
discussion in the above, we know $F_{t}=F_{t^{\prime}}$ on
$\\{\psi<-\max{\\{t,t^{\prime}\\}}\\}$ for any $t\in(T,+\infty)$ and
$t^{\prime}\in(T,+\infty)$. Hence combining $\lim_{t\rightarrow
T+0}G(t)=G(T)$, we obtain that there exists 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
$\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)=G(t)$ for any $t\geq T$.

Secondly, we prove equality (1.1). As $a(t)$ is nonnegative measurable
function on $(T,+\infty)$, then there exists a sequence of functions
$\\{\sum_{j=1}^{n_{i}}a_{ij}\mathbb{I}_{E_{ij}}\\}_{i\in\mathbb{N}^{+}}$
($n_{i}<+\infty$ for any $i\in\mathbb{N}^{+}$) satisfying
$\sum_{j=1}^{n_{i}}a_{ij}\mathbb{I}_{E_{ij}}$ is increasing with respect to
$i$ and
$\lim_{i\rightarrow+\infty}\sum_{j=1}^{n_{i}}a_{ij}\mathbb{I}_{E_{ij}}(t)=a(t)$
for any $t\in(T,+\infty)$, where $E_{ij}$ is a Lebesgue measurable subset of
$(T,+\infty)$ and $a_{ij}\geq 0$ is a constant. It follows from Levi’s Theorem
that it suffices to prove the case that $a(t)=\mathbb{I}_{E}(t)$, where
$E\subset\subset(T,+\infty)$ is a Lebesgue measurable set .

Note that
$G(t)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t}^{+\infty}c(s)e^{-s}ds$,
then

$\int_{\\{-t_{1}\leq\psi<-t_{2}\\}}|F|^{2}e^{-\varphi}c(-\psi)=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}}^{t_{1}}c(s)e^{-s}ds$
(3.4)

holds for any $T\leq t_{2}<t_{1}<+\infty$. It follows from the dominated
convergence theorem and inequality (3.4) that

$\int_{\\{z\in M:-\psi(z)\in N\\}}|F|^{2}e^{-\varphi}=0$ (3.5)

holds for any $N\subset\subset(T,+\infty)$ such that $\mu(N)=0$, where $\mu$
is Lebesgue measure.

As $c(t)e^{-t}$ is decreasing on $(T,+\infty)$, there are at most countable
points denoted by $\\{s_{j}\\}_{j\in\mathbb{N}^{+}}$ such that $c(t)$ is not
continuous at $s_{j}$. Then there is a decreasing sequence open sets
$\\{U_{k}\\}$, such that $\\{s_{j}\\}_{j\in\mathbb{N}^{+}}\subset
U_{k}\subset(T,+\infty)$ for any $j$, and
$\lim_{k\rightarrow+\infty}\mu(U_{k})=0$. Choosing any closed interval
$[t_{2}^{\prime},t_{1}^{\prime}]\subset(T,+\infty)$. Then we have

$\begin{split}&\int_{\\{-t_{1}^{\prime}\leq\psi<-t_{2}^{\prime}\\}}|F|^{2}e^{-\varphi}\\\
=&\int_{\\{z\in M:-\psi(z)\in(t_{2}^{\prime},t_{1}^{\prime}]\backslash
U_{k}\\}}|F|^{2}e^{-\varphi}+\int_{\\{z\in
M:-\psi(z)\in[t_{2}^{\prime},t_{1}^{\prime}]\cap
U_{k}\\}}|F|^{2}e^{-\varphi}\\\
=&\lim_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\int_{\\{z\in M:-\psi(z)\in
I_{n,i}\backslash U_{k}\\}}|F|^{2}e^{-\varphi}+\int_{\\{z\in
M:-\psi(z)\in[t_{2}^{\prime},t_{1}^{\prime}]\cap
U_{k}\\}}|F|^{2}e^{-\varphi},\end{split}$ (3.6)

where $I_{n,i}=(t_{1}^{\prime}-(i+1)\alpha_{n},t_{1}^{\prime}-i\alpha_{n}]$
and $\alpha_{n}=\frac{t_{1}^{\prime}-t_{2}^{\prime}}{n}$. Note that

$\begin{split}&\lim_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\int_{\\{z\in
M:-\psi(z)\in I_{n,i}\backslash U_{k}\\}}|F|^{2}e^{-\varphi}\\\
\leq&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{n,i}\backslash
U_{k}}c(t)}\int_{\\{z\in M:-\psi(z)\in I_{n,i}\backslash
U_{k}\\}}|F|^{2}e^{-\varphi}c(-\psi).\end{split}$ (3.7)

It follows from equality (3.4) that inequality (3.7) becomes

$\begin{split}&\lim_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\int_{\\{z\in
M:-\psi(z)\in I_{n,i}\backslash U_{k}\\}}|F|^{2}e^{-\varphi}\\\
\leq&\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{n,i}\backslash
U_{k}}c(t)}\int_{I_{i}\backslash U_{k}}c(s)e^{-s}ds.\end{split}$ (3.8)

It is clear that $c(t)$ is uniformly continuous and has a positive lower bound
and upper bound on $[t_{2}^{\prime},t_{1}^{\prime}]\backslash U_{k}$. Then we
have

$\begin{split}&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{n,i}\backslash
U_{k}}c(t)}\int_{I_{n,i}\backslash U_{k}}c(s)e^{-s}ds\\\
\leq&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{\sup_{I_{n,i}\backslash
U_{k}}c(t)}{\inf_{I_{n,i}\backslash U_{k}}c(t)}\int_{I_{n,i}\backslash
U_{k}}e^{-s}ds\\\ =&\int_{(t_{2}^{\prime},t_{1}^{\prime}]\backslash
U_{k}}e^{-s}ds.\end{split}$ (3.9)

Combining inequality (3.6), (3.8) and (3.9), we have

$\begin{split}&\int_{\\{-t_{1}^{\prime}\leq\psi<-t_{2}^{\prime}\\}}|F|^{2}e^{-\varphi}\\\
=&\int_{\\{z\in M:-\psi(z)\in(t_{2}^{\prime},t_{1}^{\prime}]\backslash
U_{k}\\}}|F|^{2}e^{-\varphi}+\int_{\\{z\in
M:-\psi(z)\in[t_{2}^{\prime},t_{1}^{\prime}]\cap
U_{k}\\}}|F|^{2}e^{-\varphi}\\\
\leq&\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{(t_{2}^{\prime},t_{1}^{\prime}]\backslash
U_{k}}e^{-s}ds+\int_{\\{z\in M:-\psi(z)\in[t_{2}^{\prime},t_{1}^{\prime}]\cap
U_{k}\\}}|F|^{2}e^{-\varphi}.\end{split}$ (3.10)

Let $k\rightarrow+\infty$, following from equality (3.5) and inequality
(3.10), we obtain that

$\int_{\\{-t_{1}^{\prime}\leq\psi<-t_{2}^{\prime}\\}}|F|^{2}e^{-\varphi}\leq\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}^{\prime}}^{t_{1}^{\prime}}e^{-s}ds.$
(3.11)

Following from a similar discussion, we obtain

$\int_{\\{-t_{1}^{\prime}\leq\psi<-t_{2}^{\prime}\\}}|F|^{2}e^{-\varphi}\geq\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}^{\prime}}^{t_{1}^{\prime}}e^{-s}ds,$

then combining inequality (3.11), we know that

$\int_{\\{-t_{1}^{\prime}\leq\psi<-t_{2}^{\prime}\\}}|F|^{2}e^{-\varphi}=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}^{\prime}}^{t_{1}^{\prime}}e^{-s}ds.$
(3.12)

Then it is clear that for any open set $U\subset(T,+\infty)$ and compact set
$V\subset(T,+\infty)$

$\int_{\\{z\in M:-\psi(z)\in
U\\}}|F|^{2}e^{-\varphi}=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{U}e^{-s}ds$

and

$\int_{\\{z\in M:-\psi(z)\in
V\\}}|F|^{2}e^{-\varphi}=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{V}e^{-s}ds.$

As $E\subset\subset(T,+\infty)$, then $E\cap(t_{2},t_{1}]$ is Lebesgue
measurable subset of $(T+\frac{1}{n},n)$ for some large $n$, where $T\leq
t_{2}<t_{1}\leq+\infty$. Then there exist a sequence of compact sets
$\\{V_{j}\\}$ and a sequence of open sets $\\{V^{\prime}_{j}\\}$ satisfying
$V_{1}\subset...\subset V_{j}\subset V_{j+1}\subset...\subset
E\cap(t_{2},t_{1}]\subset...\subset V^{\prime}_{j+1}\subset
V^{\prime}_{j}\subset...\subset V^{\prime}_{1}\subset\subset(T,+\infty)$ and
$\lim_{j\rightarrow+\infty}\mu(V^{\prime}_{j}-V_{j})=0$, where $\mu$ is
Lebesgue measure. Then we have

$\begin{split}\int_{\\{-t_{1}\leq\psi<-t_{2}\\}}|F|^{2}e^{-\varphi}\mathbb{I}_{E}(-\psi)=&\int_{\\{z\in
M:-\psi(z)\in E\cap(t_{2},t_{1}]\\}}|F|^{2}e^{-\varphi}\\\
\leq&\liminf_{j\rightarrow+\infty}\int_{\\{z\in M:-\psi(z)\in
V^{\prime}_{j}\\}}|F|^{2}e^{-\varphi}\\\
\leq&\liminf_{j\rightarrow+\infty}\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{V^{\prime}_{j}}e^{-s}\\\
=&\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{E\cap(t_{2},t_{1}]}e^{-s}ds\\\
=&\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}}^{t_{1}}e^{-s}\mathbb{I}_{E}(s)ds\end{split}$

and

$\begin{split}\int_{\\{-t_{1}\leq\psi<-t_{2}\\}}|F|^{2}e^{-\varphi}\mathbb{I}_{E}(-\psi)\geq&\liminf_{j\rightarrow+\infty}\int_{\\{z\in
M:-\psi(z)\in V_{j}\\}}|F|^{2}e^{-\varphi}\\\
\geq&\liminf_{j\rightarrow+\infty}\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{V_{j}}e^{-s}\\\
=&\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}}^{t_{1}}e^{-s}\mathbb{I}_{E}(s)ds,\end{split}$

which implies that
$\int_{\\{-t_{1}\leq\psi<-t_{2}\\}}|F|^{2}e^{-\varphi}\mathbb{I}_{E}(-\psi)=\frac{G(T_{1})}{\int_{T_{1}}^{+\infty}c(s)e^{-s}ds}\int_{t_{2}}^{t_{1}}e^{-s}\mathbb{I}_{E}(s)ds$.
Hence we obtain that equality (1.1) holds.

Finally, we prove equality (1.2).

By the definition of $G(t_{0};\tilde{c})$, we have
$G(t_{0};\tilde{c})\leq\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\tilde{c}(-\psi)$,
then we only consider the case $G(t_{0};\tilde{c})<+\infty$.

By the definition of $G(t_{0};\tilde{c})$, we can choose a holomorphic $(n,0)$
form $F_{t_{0},\tilde{c}}$ on $\\{\psi<-t_{0}\\}$ satisfying
$(F_{t_{0},\tilde{c}}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and
$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\tilde{c}(-\psi)<+\infty$.
As $\mathcal{H}^{2}(\tilde{c},t_{0})\subset\mathcal{H}^{2}(c,t_{0})$, we have
$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}c(-\psi)<+\infty$.
By using Lemma 2.6, we obtain that

$\begin{split}\int_{\\{\psi<-t\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}c(-\psi)=&\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)\\\
&+\int_{\\{\psi<-t\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}c(-\psi)\end{split}$

for any $t\geq t_{0}$, then

$\begin{split}\int_{\\{-t_{3}\leq\psi<-t_{4}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}c(-\psi)=&\int_{\\{-t_{3}\leq\psi<-t_{4}\\}}|F|^{2}e^{-\varphi}c(-\psi)\\\
&+\int_{\\{-t_{3}\leq\psi<-t_{4}\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}c(-\psi)\end{split}$
(3.13)

holds for any $t_{3}>t_{4}\geq t_{0}$. It follows from the dominant
convergence theorem, equality (3.13), equality (3.5) and $c(t)>0$ for any
$t>T$, that

$\int_{\\{z\in
M:-\psi(z)=t\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}=\int_{\\{z\in
M:-\psi(z)=t\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}$ (3.14)

holds for any $t>t_{0}$.

Choosing any closed interval
$[t_{4}^{\prime},t_{3}^{\prime}]\subset(t_{0},+\infty)\subset(T,+\infty)$.
Note that $c(t)$ is uniformly continuous and have positive lower bound and
upper bound on $[t_{4}^{\prime},t_{3}^{\prime}]\backslash U_{k}$, where
$\\{U_{k}\\}_{k}$ is a decreasing sequence of open subsets of $(T,+\infty)$,
such that $c$ is continuous on $(T,+\infty)\backslash U_{k}$ and
$\lim_{k\rightarrow+\infty}\mu(U_{k})=0$. Take $N=\cap_{k=1}^{+\infty}U_{k}$.
Note that

$\begin{split}&\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\\\
=&\lim_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\int_{\\{z\in M:-\psi(z)\in
I_{n,i}\backslash U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\cap
U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\\\
\leq&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{n,i}\backslash
U_{k}}c(t)}\int_{\\{z\in M:-\psi(z)\in I_{n,i}\backslash
U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}c(-\psi)\\\ &+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\cap
U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi},\end{split}$ (3.15)

where $I_{n,i}=(t_{4}^{\prime}-(i+1)\alpha_{n},t_{3}^{\prime}-i\alpha_{n}]$
and $\alpha_{n}=\frac{t_{3}^{\prime}-t_{4}^{\prime}}{n}$. It following from
equality (3.13), (3.14), (3.5), and the dominated theorem that

$\begin{split}&\int_{\\{z\in M:-\psi(z)\in I_{n,i}\backslash
U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}c(-\psi)\\\ =&\int_{\\{z\in
M:-\psi(z)\in I_{n,i}\backslash
U_{k})\\}}|F|^{2}e^{-\varphi}c(-\psi)+\int_{\\{z\in M:-\psi(z)\in
I_{n,i}\backslash
U_{k})\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}c(-\psi).\end{split}$ (3.16)

As $c(t)$ is uniformly continuous and have positive lower bound and upper
bound on $[t_{4}^{\prime},t_{3}^{\prime}]\backslash U_{k}$, combining equality
(3.16), we have

$\begin{split}&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{n,i}\backslash
U_{k}}c(t)}\int_{\\{z\in M:-\psi(z)\in I_{n,i}\backslash
U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}c(-\psi)\\\
=&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{n,i}\backslash
U_{k}}c(t)}(\int_{\\{z\in M:-\psi(z)\in I_{n,i}\backslash
U_{k})\\}}|F|^{2}e^{-\varphi}c(-\psi)\\\ &+\int_{\\{z\in M:-\psi(z)\in
I_{n,i}\backslash
U_{k})\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}c(-\psi))\\\
\leq&\limsup_{n\rightarrow+\infty}\sum_{i=0}^{n-1}\frac{\sup_{I_{n,i}\backslash
U_{k}}c(t)}{\inf_{I_{n,i}\backslash U_{k}}c(t)}(\int_{\\{z\in M:-\psi(z)\in
I_{n,i}\backslash U_{k}\\}}|F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in
M:-\psi(z)\in I_{n,i}\backslash
U_{k}\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi})\\\ =&\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
U_{k}\\}}|F|^{2}e^{-\varphi}+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
U_{k}\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}.\end{split}$ (3.17)

It follows from inequality (3.15) and (3.17), we obtain that

$\begin{split}&\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\\\
\leq&\int_{\\{z\in M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
U_{k}\\}}|F|^{2}e^{-\varphi}+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
U_{k}\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\cap
U_{k}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}.\end{split}$ (3.18)

It follows from $F_{t_{0},\tilde{c}}\in\mathcal{H}^{2}(c,t_{0})$ that
$\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}<+\infty$.
Let $k\rightarrow+\infty$, following from equality (3.5), inequality (3.18)
and the dominated theorem, we have

$\begin{split}\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\leq&\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F|^{2}e^{-\varphi}\\\
&+\int_{\\{z\in M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
N\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\cap
N\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi},\end{split}$ (3.19)

Following from a similar discussion, we can obtain that

$\begin{split}\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\geq&\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F|^{2}e^{-\varphi}\\\
&+\int_{\\{z\in M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
N\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\cap
N\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi},\end{split}$

then combining inequality (3.19) we have

$\begin{split}\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}=&\int_{\\{-t_{3}^{\prime}\leq\psi<-t_{4}^{\prime}\\}}|F|^{2}e^{-\varphi}\\\
&+\int_{\\{z\in M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\backslash
N\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in
M:-\psi(z)\in(t_{4}^{\prime},t_{3}^{\prime}]\cap
N\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}.\end{split}$ (3.20)

Using equality (3.5), (3.14), (3.20) and Levi’s Theorem, we have

$\begin{split}\int_{\\{z\in M:-\psi(z)\in
U\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}=&\int_{\\{z\in M:-\psi(z)\in
U\\}}|F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in M:-\psi(z)\in U\backslash
N\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in M:-\psi(z)\in
U\cap N\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\end{split}$ (3.21)

holds for any open set $U\subset\subset(t_{0},+\infty)$, and

$\begin{split}\int_{\\{z\in M:-\psi(z)\in
V\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}=&\int_{\\{z\in M:-\psi(z)\in
V\\}}|F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in M:-\psi(z)\in V\backslash
N\\}}|F_{t_{0},\tilde{c}}-F|^{2}e^{-\varphi}\\\ &+\int_{\\{z\in M:-\psi(z)\in
V\cap N\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\end{split}$ (3.22)

holds for any compact set $V\subset(t_{0},+\infty)$. For any measurable set
$E\subset\subset(t_{0},+\infty)$, there exists a sequence of compact sets
$\\{V_{l}\\}$, such that $V_{l}\subset V_{l+1}\subset E$ for any $l$ and
$\lim_{l\rightarrow}\mu(V_{l})=\mu(E)$, hence

$\begin{split}\int_{\\{\psi<-t_{0}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\mathbb{I}_{E}(-\psi)\geq&\lim_{l\rightarrow+\infty}\int_{\\{\psi<-t_{0}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\mathbb{I}_{V_{j}}(-\psi)\\\
\geq&\lim_{j\rightarrow+\infty}\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\mathbb{I}_{V_{j}}(-\psi)\\\
=&\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\mathbb{I}_{E}(-\psi).\end{split}$
(3.23)

It is clear that for any $t>t_{0}$, there exists a sequence of functions
$\\{\sum_{j=1}^{n_{i}}\mathbb{I}_{E_{ij}}\\}_{i=1}^{+\infty}$ defined on
$(t,+\infty)$, satisfying $E_{ij}\subset\subset(t,+\infty)$,
$\sum_{j=1}^{n_{i+1}}\mathbb{I}_{E_{i+1j}}(s)\geq\sum_{j=1}^{n_{i}}\mathbb{I}_{E_{ij}}(s)$,
and
$\lim_{i\rightarrow+\infty}\sum_{j=1}^{n_{i}}\mathbb{I}_{E_{ij}}(s)=\tilde{c}(s)$
for any $s>t$. Combining Levi’s Theorem and inequality (3.23), we have

$\int_{\\{\psi<-t_{0}\\}}|F_{t_{0},\tilde{c}}|^{2}e^{-\varphi}\tilde{c}(-\psi)\geq\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\tilde{c}(-\psi).$
(3.24)

By the definition of $G(t_{0},\tilde{c})$, we have
$G(t_{0},\tilde{c})=\int_{\\{\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\tilde{c}(-\psi)$.
Then equality (1.2) holds.

##  4\. Proofs of Theorem 1.9, Theorem 1.10, Corollary 1.11 and Corollary 1.12

In this section, we prove Theorem 1.9, Theorem 1.10, Corollary 1.11 and
Corollary 1.12.

###  4.1. Proof of Theorem 1.9

The following remark shows that it suffices to consider Theorem 1.9 for the
case $c(t)$ has a positive lower bound and upper bound on
$(t^{\prime},+\infty)$ for any $t^{\prime}>T$.

######  Remark 4.1.

Take $c_{j}$ is a positive measurable function on $(T,+\infty)$, such that
$c_{j}(t)=c(t)$ when $t<T+j$, $c_{j}(t)=\min\\{c(T+j),\frac{1}{j}\\}$ when
$t\geq T+j$. It is clear that $c_{j}(t)e^{-t}$ is decreasing with respect to
$t$, and $\int_{T}^{+\infty}c_{j}(t)e^{-t}<+\infty$. As

$\lim_{j\rightarrow+\infty}\int_{T+j}^{+\infty}c_{n}(t)e^{-t}=0,$

we have

$\lim_{j\rightarrow+\infty}\int_{T}^{+\infty}c_{j}(t)e^{-t}=\int_{T}^{+\infty}c(t)e^{-t}.$

If Theorem 1.9 holds in this case, then there exists a holomorphic $(n,0)$
form $F_{j}$ on $M$ such that $F_{j}|_{S}=f$ and

$\int_{M}|F_{j}|^{2}e^{-\varphi}c_{j}(-\psi)\leq\left(\int_{T}^{+\infty}c_{j}(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}|f|^{2}e^{-\varphi}dV_{M}[\psi].$

Note that $\psi$ has locally lower bound on $M\backslash\psi^{-1}(-\infty)$
and $\psi^{-1}(-\infty)$ is a closed subset of some analytic subset of $M$, it
follows from Lemma 2.4 that there exists a subsequence of $\\{F_{j}\\}$,
denoted still by $\\{F_{j}\\}$, which is uniformly convergent to a holomorphic
$(n,0)$ form $F$ on any compact subset of $M$ and

$\begin{split}\int_{M}|F|^{2}e^{-\varphi}c(-\psi)&\leq\lim_{j\rightarrow+\infty}\left(\int_{T}^{+\infty}c_{j}(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}|f|^{2}e^{-\varphi}dV_{M}[\psi]\\\
&=\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}|f|^{2}e^{-\varphi}dV_{M}[\psi].\end{split}$

Since $F_{j}|_{S}=f$ for any $j$, we have $F|_{S}=f$.

By the definition of condition $(ab)$, $\liminf_{t\rightarrow+\infty}c(t)>0$,
it suffices to prove the case that $M$ is Stein manifold and $S_{reg}=S$.
Without loss of generality, we can assume that
$supp(\mathcal{O}_{M}/\mathcal{I}(\psi))=S_{reg}$ (if
$supp(\mathcal{O}_{M}/\mathcal{I}(\psi))\not=S_{reg}$, there exists a analytic
subset $X$ of $M$ such that $(M,X)$ satisfies condition $(ab)$ and
$supp(\mathcal{O}_{M}/\mathcal{I}(\psi))\backslash S_{reg}\in X$).

Since $M$ is Stein, we can find a sequence of Stein manifolds
$\\{D_{m}\\}_{m=1}^{+\infty}$ satisfying $D_{m}\subset\subset D_{m+1}$ for any
$m$ and $\cup_{m=1}^{+\infty}D_{m}=M$, and there is a holomorphic $(n,0)$ form
$\tilde{F}$ on $M$ such that $\tilde{F}|_{S}=f$.

Note that $\int_{D_{m}}|\tilde{F}|^{2}<+\infty$ for any $m$ and

$\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi-\psi}<+\infty$

for any $m$ and $t_{0}>T$. Using Lemma 2.1, for any $D_{m}$ and $t_{0}>T$,
there exists a holomorphic $(n,0)$ form $F_{m,t_{0}}$ on $D_{m}$, such that

$\begin{split}&\int_{D_{m}}|F_{m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\varphi-\psi+v_{t_{0},1}(\psi)}c(-v_{t_{0},1}(\psi))\\\
\leq&\left(\int_{T}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi-\psi}\end{split}$
(4.1)

where
$b_{t_{0},1}(t)=\int_{-\infty}^{t}\mathbb{I}_{\\{-t_{0}-1<s<-t_{0}\\}}ds$,
$v_{t_{0},1}(t)=\int_{0}^{t}b_{t_{0},1}(s)ds$. Note that $e^{-\psi}$ is not
locally integrable along $S$, and $b_{t_{0},1}(t)=0$ for large enough $t$,
then $(F_{m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F})|_{D_{m}\cap S}=0$, and
therefore $F_{m,t_{0}}|_{D_{m}\cap S}=f$.

Note that $v_{t_{0},1}(\psi)\geq\psi$ and $c(t)e^{-t}$ is decreasing, then the
inequality (4.1) becomes

$\begin{split}&\int_{D_{m}}|F_{m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\left(\int_{T}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi-\psi}.\end{split}$
(4.2)

As
$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]<+\infty$,
by definition of $dV_{M}[\psi]$ and
$supp(\mathcal{O}_{M}/\mathcal{I}(\psi))=S_{reg}$, we have

$\begin{split}&\limsup_{t_{0}\rightarrow+\infty}\left(\int_{T}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi-\psi}\\\
\leq&\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}\cap
D_{m}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]\\\ <&+\infty.\end{split}$
(4.3)

Note that $e^{-\varphi}c(-\psi)$ has a positive lower bound on $D_{m}$, then
it follows from inequality (4.2) and (4.3) that
$\sup_{t_{0}}\int_{D_{m}}|F_{m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}<+\infty.$

Combining with

$\sup_{t_{0}}\int_{D_{m}}|(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}\leq\sup_{t_{0}}\int_{D_{m}}\mathbb{I}_{\\{\psi<-t_{0}\\}}|\tilde{F}|^{2}<+\infty,$
(4.4)

one can obtain that $\sup_{t_{0}}\int_{D_{m}}|F_{m,t_{0}}|^{2}<+\infty$, which
implies that there exists a subsequence of
$\\{F_{m,t_{0}}\\}_{t_{0}\rightarrow+\infty}$ (also denoted by
$\\{F_{m,t_{0}}\\}_{t_{0}\rightarrow+\infty}$) compactly convergent to a
holomorphic (n,0) form on $D_{m}$ denoted by $F_{m}$. Then it follows from
inequality (4.2), inequality (4.3) and the Fatou’s Lemma that

$\begin{split}\int_{D_{m}}|F_{m}|^{2}e^{-\varphi}c(-\psi)=&\int_{D_{m}}\liminf_{t_{0}\rightarrow+\infty}|F_{m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\liminf_{t_{0}\rightarrow+\infty}\int_{D_{m}}|F_{m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\varphi}c(-\psi)\\\
\leq&\limsup_{t_{0}\rightarrow+\infty}\left(\int_{T}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\varphi-\psi}\\\
\leq&\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}\cap
D_{m}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]\\\
\leq&\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi],\end{split}$
(4.5)

and $F_{m}|_{D_{m}\cap S}=f$. Inequality (4.5) implies that

$\int_{D_{m}}|F_{m^{\prime}}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]$

holds for any $m^{\prime}\geq m$. As $e^{-\varphi}c(-\psi)$ has a positive
lower bound on any $D_{m}$, by the diagonal method, we obtain a subsequence of
$\\{F_{m}\\}$, denoted also by $\\{F_{m}\\}$, which is uniformly convergent to
a holomorphic $(n,0)$ form $F$ on $M$ on any compact subset of $M$ satisfying
that $F|_{S}=f$ and

$\int_{M}|F|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi].$

Thus Theorem 1.9 holds.

###  4.2. Proof of Theorem 1.10

If
$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]=0$,
it is clear that $F\equiv 0$ satisfying all requirements in Theorem 1.10. In
the following part, we consider the case
$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi]\in(0,+\infty)$.

Using Theorem 1.9, for any $t>T$, there exists a holomorphic $(n,0)$ form
$F_{t}$ on $\\{\psi<-t\\}$ such that $F_{t}|_{S}=f$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{t}^{+\infty}c(l)e^{-l}dl\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi}dV_{M}[\psi].$

Then we have inequality

$\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\leq\frac{G(T)}{\int_{T}^{+\infty}c(t)e^{-t}dt}$
(4.6)

holds for any $t>T$. As $(M,S)$ satisfies condition $(ab)$, and $\psi\in
A(S)$, Theorem 1.3 tells us $G(\hat{h}^{-1}(r))$ is concave with respect to
$r$. Combining inequality (4.6) and Corollary 1.5, we obtain that
$G(\hat{h}^{-1}(r))$ is linear with respect to $r$. Note that
$\frac{G(T)}{\int_{T}^{+\infty}c(t)e^{-t}dt}=\|f\|_{M}$, Corollary 1.7 shows
that the rest results of Theorem 1.10 hold.

###  4.3. Proof of Corollary 1.11

In this section, we prove Corollary 1.11 by using Theorem 1.9.

Since $M$ is Stein, we can find a sequence of Stein manifolds
$\\{D_{l}\\}_{l=1}^{+\infty}$ satisfying $D_{l}\subset\subset D_{l+1}$ for any
$l$ and $\cup_{l=1}^{+\infty}D_{l}=M$. Since $\psi_{2}$ and $\psi_{2}+\varphi$
are plurisubharmonic functions on $M$, there exist smooth plurisubharmonic
functions $\Psi_{m}$ and $\Phi_{m^{\prime}}$, which are decreasingly
convergent to $\psi_{2}$ and $\psi_{2}+\varphi$, respectively.

Fixed $D_{l}$, we can choose large enough $m$ such that $\Psi_{m}+\psi_{1}<-T$
on $D_{l}$. Note that
$dV_{M}[\Psi_{m}+\psi_{1}]=e^{-\Psi_{m}}dV_{M}[\psi_{1}]$ and

$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\Phi_{m^{\prime}}}dV_{M}[\psi_{1}]\leq\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]<+\infty,$

Using Theorem 1.9, for any $D_{l}$, there exists a holomorphic $(n,0)$ form
$F_{l,m^{\prime}}$ on $D_{l}$, satisfying $F_{l,m^{\prime}}|_{S}=f$ and

$\int_{D_{l}}|F_{l,m^{\prime}}|^{2}e^{-\Phi_{m^{\prime}}+\Psi_{m}}c(-\Psi_{m}-\psi_{1})\leq\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\Phi_{m^{\prime}}}dV_{M}[\psi_{1}].$
(4.7)

As $e^{-\Phi_{m^{\prime}}+\Psi_{m}}c(-\Psi_{m}-\psi_{1})$ has locally
uniformly positive lower bound for any $m^{\prime}$ on $D_{m}\backslash Z$,
where $Z$ is some analytic subset of $M$, it follows from Lemma 2.4 that there
exists a subsequence of
$\\{F_{l,m^{\prime}}\\}_{m^{\prime}\rightarrow+\infty}$, also denoted by
$\\{F_{l,m^{\prime}}\\}_{m^{\prime}\rightarrow+\infty}$, which satisfies that
$\\{F_{l,m^{\prime}}\\}_{m^{\prime}\rightarrow+\infty}$ is uniformly
convergent to a holomorphic $(n,0)$ form $F_{l}$ on any compact subset of
$D_{l}$. Following from inequality (4.7), Fatou’s Lemma and $c(t)e^{-t}$ is
decreasing, we have

$\begin{split}\int_{D_{l}}|F_{l}|^{2}e^{-\varphi}c(-\psi_{2}-\psi_{1})\leq&\int_{D_{l}}|F_{l}|^{2}e^{-\varphi-\psi_{2}+\Psi_{m}}c(-\Psi_{m}-\psi_{1})\\\
=&\int_{D_{l}}\lim_{m^{\prime}\rightarrow+\infty}|F_{l}|^{2}e^{-\Phi_{m^{\prime}}+\Psi_{m}}c(-\Psi_{m}-\psi_{1})\\\
\leq&\liminf_{m^{\prime}\rightarrow+\infty}\int_{D_{l}}|F_{l}|^{2}e^{-\Phi_{m^{\prime}}+\Psi_{m}}c(-\Psi_{m}-\psi_{1})\\\
\leq&\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].\end{split}$
(4.8)

Note that $e^{-\varphi}c(-\psi_{2}-\psi_{1})$ has locally a positive lower
bound on $M\backslash Z$, where $Z$ is some analytic subset of $M$, by using
Lemma 2.4 and the diagonal method, we obtain that there exists a subsequence
of $\\{F_{l}\\}$, also denoted by $\\{F_{l}\\}$, which satisfies that
$\\{F_{l}\\}$ is uniformly convergent to a holomorphic $(n,0)$ form $F$ on $M$
on any compact subset of $M$. Following from inequality (4.8) and Fatou’s
Lemma, we have

$\begin{split}\int_{M}|F|^{2}e^{-\varphi}c(-\psi_{2}-\psi_{1})=&\int_{M}\lim_{l\rightarrow+\infty}\mathbb{I}_{D_{l}}|F_{l}|^{2}e^{-\varphi}c(-\psi_{2}-\psi_{1})\\\
\leq&\liminf_{l\rightarrow+\infty}\int_{D_{l}}|F_{l}|^{2}e^{-\varphi}c(-\psi_{2}-\psi_{1})\\\
\leq&\left(\int_{T}^{+\infty}c(t)e^{-t}dt\right)\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].\end{split}$
(4.9)

This proves Corollary 1.11.

###  4.4. Proof of Corollary 1.12

If $\|f\|_{S}^{*}=0$, it is clear that $F\equiv 0$ satisfying all requirements
in Corollary 1.12. In the following part, we consider the case
$\|f\|_{S}^{*}\in(0,+\infty)$.

Using Corollary 1.11, for any $t>T$, there exists a holomorphic $(n,0)$ form
$F_{t}$ on $\\{\psi<-t\\}$ such that $F_{t}|_{S}=f$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{t}^{+\infty}c(l)e^{-l}dl\right)\|f\|_{S}^{*}.$

Then we have inequality

$\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\leq\frac{G(T)}{\int_{T}^{+\infty}c(t)e^{-t}dt}$
(4.10)

holds for any $t>T$. Theorem 1.3 tells us $G(\hat{h}^{-1}(r))$ is concave with
respect to $r$. Combining inequality (4.10) and Corollary 1.5, we obtain that
$G(\hat{h}^{-1}(r))$ is linear with respect to $r$. Note that
$\frac{G(T)}{\int_{T}^{+\infty}c(t)e^{-t}dt}=\|f\|^{*}_{M}$, Corollary 1.7
shows that the rest results of Theorem 1.12 hold.

##  5\. Proofs of Theorem 1.13, Theorem 1.14 and Theorem 1.15

In this section, we prove Theorem 1.13, Theorem 1.14 and Theorem 1.15.

###  5.1. Proof of Theorem 1.13

We prove the theorem by comparing $G(t;\varphi)$ and $G(t;\tilde{\varphi})$.
Let us assume that $G(\hat{h}^{-1}(r);\varphi)$ is linear with respect to $r$
to get a contradiction.

As $G(\hat{h}^{-1}(r);\varphi)$ is linear with respect to $r$, it follows from
Corollary 1.7 that there exists a holomorphic $(n,0)$ form $F$ on $M$ such
that $(F-f)\in H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$
and $\forall t\geq T$ equality

$G(t;\varphi)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)$

holds. As $\tilde{\varphi}+\psi$ is plurisubharmonic and
$\tilde{\varphi}-\varphi$ is bounded on $M$, it follows from Theorem 1.3 that
$G(\hat{h}^{-1}(r);\tilde{\varphi})$ is concave with respect to $r$.

As $\tilde{\varphi}+\psi\geq\varphi+\psi$,
$\tilde{\varphi}+\psi\not=\varphi+\psi$ and both of them are plurisubharmonic
functions on $M$, then there exists a subset $U$ of $M$ such that
$e^{-\tilde{\varphi}}<e^{-\varphi}$ on a subset $U$ and $\mu(U)>0$, where
$\mu$ is Lebesgue measure on $M$. As $F\not\equiv 0$, inequality

$\frac{G(T_{0};\tilde{\varphi})}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}\leq\frac{\int_{\\{\psi<-T_{0}\\}}|F|^{2}e^{-\tilde{\varphi}}c(-\psi)}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}<\frac{G(T_{0};\varphi)}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}$
(5.1)

holds for some $T_{0}>T$. For $t>T$, there exists a holomorphic $(n,0)$ form
$F_{t}$ on $\\{\psi<-t\\}$ such that $(F_{t}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and

$G(t;\tilde{\varphi})=\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\tilde{\varphi}}c(-\psi)<+\infty.$

As $\tilde{\varphi}-\varphi$ is bounded on $M$, we have
$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)<+\infty.$ It follows from
Lemma 2.6 that

$\begin{split}G(t_{1};\tilde{\varphi})-G(t_{2};\tilde{\varphi})&\geq\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F_{t_{1}}|^{2}e^{-\tilde{\varphi}}c(-\psi)\\\
&\geq\left(\inf_{\\{-t_{2}\leq\psi\\}}e^{\varphi-\tilde{\varphi}}\right)\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F_{t_{1}}|^{2}e^{-\varphi}c(-\psi)\\\
&\geq\left(\inf_{\\{-t_{2}\leq\psi\\}}e^{\varphi-\tilde{\varphi}}\right)\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F|^{2}e^{-\varphi}c(-\psi)\end{split}$
(5.2)

holds for $T<t_{1}<t_{2}<+\infty$. As $\lim_{t\rightarrow
T+0}\sup_{z\in\\{\psi\geq-t\\}}((\tilde{\varphi}-\varphi)(z))=0$, it follows
from inequality (5.1) and (5.2) that

$\begin{split}\liminf_{t_{2}\rightarrow
T+0}\frac{G(t_{1};\tilde{\varphi})-G(t_{2};\tilde{\varphi})}{\int_{t_{1}}^{t_{2}}c(s)e^{-s}ds}\geq&\liminf_{t_{2}\rightarrow
T+0}\left(\inf_{z\in\\{-t_{2}\leq\psi\\}}e^{\varphi-\tilde{\varphi}}\right)\frac{\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F|^{2}e^{-\varphi}c(-\psi)}{\int_{t_{1}}^{t_{2}}c(s)e^{-s}ds}\\\
=&\frac{G(T_{0};\varphi)}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}\\\
>&\frac{G(T_{0};\tilde{\varphi})}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds},\end{split}$

which contradicts the concavity of $G(\hat{h}^{-1}(r);\tilde{\varphi})$. Thus
the assumption doesn’t hold, i.e., $G(\hat{h}^{-1}(r);\varphi)$ is not linear
with respect to $r$.

Especially, if $\varphi+\psi$ is strictly plurisubharmonic at $z_{1}\in M$, we
can construct a $\tilde{\varphi}\geq\varphi$ satisfying the three statements
in Theorem 1.13, which implies $G(\hat{h}^{-1}(r);\varphi)$ is not linear with
respect to $r$. In fact, there is a small open neighborhood $(U,w)$ of $z_{1}$
and $w=(w_{1},...,w_{n})$ is the local coordinate on $U$ such that
$i\partial\bar{\partial}(\varphi+\psi)>\epsilon\omega$ for some $\epsilon>0$,
where $\omega=i\sum_{j=1}^{n}dw_{j}\wedge d\bar{w}_{j}$ on $U$. Let $\rho$ be
a smooth nonnegative function on $M$ satisfying $\rho\not\equiv 0$ and
$supp\rho\subset\subset U$. It is clear that there exists a positive number
$\delta$ such that

$i\partial\bar{\partial}(\varphi+\psi+\delta\rho)>0$

holds on $U$. Let $\tilde{\varphi}=\varphi+\delta\rho$, it is clear that
$\tilde{\varphi}$ satisfies the three statements in Theorem 1.13. Thus we
complete the proof of Theorem 1.13.

###  5.2. Proof of Theorem 1.14

Let $\tilde{\varphi}=\varphi+\psi-\tilde{\psi}$, then
$\tilde{\varphi}+\tilde{\psi}=\varphi+\psi$ is a plurisubharmonic function on
$M$. We prove the theorem by comparing $G(t;\varphi,\psi)$ and
$G(t;\tilde{\varphi},\tilde{\psi})$. Let us assume that
$G(\hat{h}^{-1}(r);\varphi,\psi)$ is linear with respect to $r$ to get a
contradiction.

Since $G(T;\varphi,\psi)\in(0,+\infty)$, $G(\hat{h}^{-1}(r))$ is linear and
Corollary 1.7, we have $\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$. As
$G(\hat{h}^{-1}(r);\varphi,\psi)$ is linear with respect to $r$, it follows
from Corollary 1.7, Remark 1.8 and Lemma 2.10 that we can assume $c(t)e^{-t}$
is strictly decreasing on $(T,+\infty)$ and $c(t)$ is increasing on
$(a,+\infty)$ for some $a>T$.

Using Corollary 1.7, there exists a holomorphic $(n,0)$ form $F$ on $M$, such
that $(F-f)\in H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$
and $\forall t\geq T$ equality

$G(t;\varphi,\psi)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)$

holds.

Since $\lim_{t\rightarrow+\infty}\sup_{\\{\psi<-t\\}}(\tilde{\psi}-\psi)=0$,
we have $Z_{0}\subset\\{\psi=-\infty\\}=\\{\tilde{\psi}=-\infty\\}$. As
$c(t)e^{-t}$ is decreasing and $\tilde{\psi}\geq\psi$, we have
$e^{-\varphi}c(-\psi)=e^{-\varphi-\psi}e^{\psi}c(-\psi)\leq
e^{-\tilde{\varphi}-\tilde{\psi}}e^{\tilde{\psi}}c(-\tilde{\psi})=e^{-\tilde{\varphi}}c(-\tilde{\psi})$.
It follows from Theorem 1.3 that
$G(\hat{h}^{-1}(r);\tilde{\varphi},\tilde{\psi})$ is concave with respect to
$r$.

We claim that

$\lim_{t\rightarrow
T+0}\frac{G(t;\tilde{\varphi},\tilde{\psi})}{\int_{t}^{+\infty}c(s)e^{-s}ds}>\frac{G(T;\varphi,\psi)}{\int_{T}^{+\infty}c(s)e^{-s}ds}.$
(5.3)

In fact, we just need to prove the inequality for the case
$G(T;\tilde{\varphi},\tilde{\psi})<+\infty$. It follows from Lemma 2.6 that
there exists a holomorphic $(n,0)$ form $F_{T}$ on $M$ such that $(F_{T}-f)\in
H^{0}(Z_{0},(\mathcal{O}(K_{M})\otimes\mathcal{F})|_{Z_{0}})$ and

$G(T;\tilde{\varphi},\tilde{\psi})=\int_{M}|F_{T}|^{2}e^{-\varphi}c(-\tilde{\psi})\in(0,+\infty),$

where $G(T;\tilde{\varphi},\tilde{\psi})>0$ follows from
$G(T;\varphi,\psi)>0$. As $\tilde{\psi}\geq\psi$, $\tilde{\psi}\not=\psi$ and
both of them are plurisubharmonic functions on $M$, then there exists a subset
$U$ of $M$ such that $\tilde{\psi}>\psi$ on a subset $U$ and $\mu(U)>0$, where
$\mu$ is Lebesgue measure on $M$. As $F_{T}\not\equiv 0$ and $c(t)e^{-t}$ is
strictly decreasing on $(T,+\infty)$, we have

$\begin{split}\frac{G(T;\tilde{\varphi},\tilde{\psi})}{\int_{T}^{+\infty}c(s)e^{-s}ds}=&\frac{\int_{M}|F_{T}|^{2}e^{-\tilde{\varphi}}c(-\tilde{\psi})}{\int_{T}^{+\infty}c(s)e^{-s}ds}\\\
>&\frac{\int_{M}|F_{T}|^{2}e^{-\varphi}c(-\psi)}{\int_{T}^{+\infty}c(s)e^{-s}ds}\\\
\geq&\frac{G(T;\varphi,\psi)}{\int_{T}^{+\infty}c(s)e^{-s}ds}.\end{split}$

Then the claim holds.

As $c(t)$ is increasing on $(a,+\infty)$ and
$\lim_{t\rightarrow+\infty}\sup_{\\{\psi<-t\\}}(\tilde{\psi}-\psi)=0$, we
obtain that

$\begin{split}\lim_{t\rightarrow+\infty}\frac{G(t;\tilde{\varphi},\tilde{\psi})}{\int_{t}^{+\infty}c(s)e^{-s}ds}\leq&\lim_{t\rightarrow+\infty}\frac{\int_{\\{\tilde{\psi}<-t\\}}|F|^{2}e^{-\tilde{\varphi}}c(-\tilde{\psi})}{\int_{t}^{+\infty}c(s)e^{-s}ds}\\\
\leq&\lim_{t\rightarrow+\infty}\frac{\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi-\psi}e^{\tilde{\psi}}c(-\psi)}{\int_{t}^{+\infty}c(s)e^{-s}ds}\\\
\leq&\lim_{t\rightarrow+\infty}\left(\sup_{\\{\psi<-t\\}}e^{\tilde{\psi}-\psi}\right)\frac{\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)}{\int_{t}^{+\infty}c(s)e^{-s}ds}\\\
=&\frac{\int_{\\{\psi<-T\\}}|F|^{2}e^{-\varphi}c(-\psi)}{\int_{T}^{+\infty}c(s)e^{-s}ds}.\end{split}$
(5.4)

Combining inequality (5.3) and (5.4), we have

$\lim_{t\rightarrow+\infty}\frac{G(t;\tilde{\varphi},\tilde{\psi})}{\int_{t}^{+\infty}c(s)e^{-s}ds}<\lim_{t\rightarrow
T+0}\frac{G(t;\tilde{\varphi},\tilde{\psi})}{\int_{t}^{+\infty}c(s)e^{-s}ds},$

which contradicts the concavity of
$G(\hat{h}^{-1}(r);\tilde{\varphi},\tilde{\psi})$. Thus the assumption doesn’t
hold, i.e., $G(\hat{h}^{-1}(r);\varphi,\psi)$ is not linear with respect to
$r$.

Especially, if $\psi$ is strictly plurisubharmonic at $z_{1}\in
M\backslash(\cap_{t}\overline{\\{\psi<-t\\}})$, we can construct a
$\tilde{\psi}\geq\psi$ satisfying the three statements in Theorem 1.14, which
implies $G(\hat{h}^{-1}(r);\varphi,\psi)$ is not linear with respect to $r$.
In fact, there is a small open neighborhood $(U,w)$ of $z_{1}$ and
$w=(w_{1},...,w_{n})$ is the local coordinate on $U$ such that
$U\cap(\cap_{t}\overline{\\{\psi<-t\\}})=\emptyset$ and
$i\partial\bar{\partial}\psi>\epsilon\omega$ for some $\epsilon>0$, where
$\omega=i\sum_{j=1}^{n}dw_{j}\wedge d\bar{w}_{j}$ on $U$. Let $\rho$ be a
smooth nonnegative function on $M$ satisfying $\rho\not\equiv 0$ and
$supp\rho\subset\subset U$. It is clear that there exists a positive number
$\delta$ such that

$i\partial\bar{\partial}(\psi+\delta\rho)>0$

holds on $U$ and $\psi+\delta\rho<-T$ on $M$. Let
$\tilde{\psi}=\psi+\delta\rho$, it is clear that $\tilde{\psi}$ satisfies the
three statements in Theorem 1.14. Thus we complete the proof of Theorem 1.14.

###  5.3. A limiting property of $G(t)$

The following Proposition gives a limiting property of $G(t)$, which will be
used in the proof of Theorem 1.15 and Corollary 1.18.

######  Proposition 5.1.

Let $M$ be an $n-$dimensional Stein manifold, and let $S$ be an analytic
subset of $M$. Let $c\in P_{T}$, and let $(\varphi,\psi)\in W$. Let
$\mathcal{F}|_{Z_{0}}=\mathcal{I}(\psi_{1})|_{S_{reg}}$. Assume that
$G(T)\in(0,+\infty)$ and $\psi_{2}(z)>-\infty$ for almost every $z\in
S_{reg}$.

Assume that $c(t)$ is increasing on $(a,+\infty)$ for some $a>T$. Then we have

$\lim_{t\rightarrow+\infty}\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}=\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].$
(5.5)

###### Proof.

$\lim_{t\rightarrow+\infty}\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\leq\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]$
can be obtained by using Corollary 1.11. Thus, we just need to prove that

$\lim_{t\rightarrow+\infty}\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\geq\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].$

For any $t\geq T$, there exists a holomorphic $(n,0)$ form $F_{t}$ on
$\\{\psi<-t\\}$, such that $F_{t}|_{S}=f$ and
$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)=G(t)$.

Let $\\{U^{\alpha}\\}_{\alpha\in\mathbb{N}}$ be a coordinate patches of
$M\backslash S_{sing}$, biholomorphic to polydiscs, and admite the following
property: if $U^{\alpha}\cap S_{reg}\not=\emptyset$, and we denote the
corresponding coordinates by
$(z^{\alpha},w^{\alpha})\in\Delta^{l}\times\Delta^{n-l}$, where
$z^{\alpha}=(z_{1}^{\alpha},...,z_{l}^{\alpha})$ and
$w^{\alpha}=(w_{1}^{\alpha},...,w_{n-l}^{\alpha})$ for some
$l\in\\{0,1,2...,n-1\\}$, then $U^{\alpha}\cap S=U^{\alpha}\cap
S_{l}=\\{w^{\alpha}=0\\}$. Let $\\{v^{\alpha}\\}$ be a partition of unity
subbordinate to $\\{U^{\alpha}\\}$.

As $\varphi+\psi_{2}$ is plurisubharmonic, then there exist smooth
plurisubharmonic functions $\Phi_{n}$ on $M$ decreasingly convergent to
$\varphi+\psi_{2}$. Thus, we have

$\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\varphi}c(-\psi)\geq\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)$
(5.6)

for any $n\in\mathbb{N}$.

Firstly, we consider
$\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)$,
where $U^{\alpha}\cap S_{l}\not=\emptyset$.

Note that $\psi=\psi_{1}+\psi_{2}$ and $\psi_{1}\in A^{\prime}(S)$, then for
small enough $s>0$, $\psi_{1}=(n-l)\log(|w^{\alpha}|^{2})+h_{1}$ on
$\Delta^{l}\times\\{|w^{\alpha}|<s\\}$ and $h_{1}$ is continuos on
$\Delta^{l}\times\\{|w^{\alpha}|<s\\}$. For any $\epsilon>0$, there exists
$s>0$ such that
$v^{\alpha}(z^{\alpha},w^{\alpha})\geq\max{\\{v^{\alpha}(z^{\alpha},0)-\epsilon,0\\}}$,
$\Phi_{n}(z^{\alpha},w^{\alpha})\leq\Phi_{n}(z^{\alpha},0)+\epsilon$, and
$h_{1}(z^{\alpha},w^{\alpha})\leq h_{1}(z^{\alpha},0)+\epsilon$ on
$\Delta^{l}\times\\{|w^{\alpha}|<s\\}$. Let
$\psi_{s}(z^{\alpha})=\sup_{|w^{\alpha}|<s}\psi_{2}(z^{\alpha},w^{\alpha})$.
As $\psi_{2}(z)>-\infty$ for almost every $z\in S_{reg}$, we know
$\psi_{s}(z^{\alpha})>-\infty$ for almost every $z^{\alpha}\in\Delta^{l}$. Let
$v^{\alpha}_{\epsilon}:=\max{\\{v^{\alpha}(z^{\alpha},0)-\epsilon,0\\}}$. As
$c(t)$ is increasing for $t>a$, then we have

$\begin{split}&\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)\\\
\geq&\int_{\\{\psi_{2}+h_{1}+(n-l)\log(|w^{\alpha}|^{2})<-t\\}\cap\\{|w^{\alpha}|<s\\}}v^{\alpha}_{\epsilon}|F_{t}|^{2}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon+\psi_{2}}c(-\psi)\\\
\geq&\int_{\\{\psi_{s}+h_{1}(z^{\alpha},0)+\epsilon+(n-l)\log(|w^{\alpha}|^{2})<-t\\}\cap\\{|w^{\alpha}|<s\\}}v^{\alpha}_{\epsilon}|F_{t}|^{2}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon+\psi_{2}}\\\
&\times
c(-\psi_{s}-h_{1}(z^{\alpha},0)-\epsilon-(n-l)\log(|w^{\alpha}|^{2}))\end{split}$
(5.7)

for $t>a$.

Without loss of generality, assume that
$dV_{M}=(\wedge_{k=1}^{l}idz_{k}^{\alpha}\wedge
d\bar{z}_{k}^{\alpha})\wedge(\wedge_{k=1}^{n-l}idw_{k}^{\alpha}\wedge
d\bar{w}_{k}^{\alpha})$, $dV_{\alpha}=\wedge_{k=1}^{l}idz_{k}^{\alpha}\wedge
d\bar{z}_{k}^{\alpha}$ on $U^{\alpha}$ and
$dV^{\prime}_{\alpha}=\wedge_{k=1}^{n-l}idw_{k}^{\alpha}\wedge
d\bar{w}_{k}^{\alpha}$. Let
$h_{2}(z^{\alpha}):=\psi_{s}(z^{\alpha})+h_{1}(z^{\alpha},0)+\epsilon$. As
$\frac{|F_{t}|^{2}}{dV_{M}}e^{\psi_{2}}$ is plurisubharmonic on
$\Delta^{l}\times\\{|w^{\alpha}|<s\\}$, then we obtain that inequality

$\begin{split}&\int_{\\{h_{2}+(n-l)\log(|w^{\alpha}|^{2})<-t\\}\cap\\{|w^{\alpha}|<s\\}}\frac{|F_{t}|^{2}}{dV_{M}}e^{\psi_{2}}c(-h_{2}-(n-l)\log(|w^{\alpha}|^{2}))dV^{\prime}_{\alpha}\\\
\geq&\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}e^{\psi_{2}(z^{\alpha},0)}\int_{\\{h_{2}+(n-l)\log(|w^{\alpha}|^{2})<-t\\}\cap\\{|w^{\alpha}|<s\\}}c(-h_{2}-(n-l)\log(|w^{\alpha}|^{2}))dV^{\prime}_{\alpha}\\\
=&2^{n-l}\frac{\sigma_{2n-2l-1}}{2(n-l)}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}e^{\psi_{2}(z^{\alpha},0)}\int_{\max{\\{t,-h_{2}(z^{\alpha})-2(n-l)\log(s)\\}}}^{+\infty}c(l)e^{-l}dl\end{split}$
(5.8)

holds for any $z^{\alpha}\in\Delta^{l}$. It follows from inequality (5.7) and
(5.8) that

$\begin{split}&\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)\\\
\geq&\int_{\Delta^{l}}v^{\alpha}_{\epsilon}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon}\\\
&\times\int_{\\{h_{2}+(n-l)\log(|w^{\alpha}|^{2})<-t\\}\cap\\{|w^{\alpha}|<s\\}}\frac{|F_{t}|^{2}}{dV_{M}}e^{\psi_{2}}c(-h_{2}-(n-l)\log(|w^{\alpha}|^{2}))dV_{M}\\\
\geq&2^{n-l}\frac{\sigma_{2n-2l-1}}{2(n-l)}\int_{\Delta^{l}}v^{\alpha}_{\epsilon}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}e^{\psi_{2}(z^{\alpha},0)}e^{-h_{2}}\\\
&\times\left(\int_{\max{\\{t,-h_{2}(z^{\alpha})-2(n-l)\log(s)\\}}}^{+\infty}c(l)e^{-l}dl\right)dV_{\alpha}\end{split}$
(5.9)

for $t>a$.

Next, we prove that

$\liminf_{t\rightarrow+\infty}\frac{\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\varphi}c(-\psi)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\geq\frac{\pi^{n-l}}{(n-l)!}\int_{S_{l}}v^{\alpha}e^{-\varphi-\psi_{2}}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}dV_{M}[\psi_{1}].$
(5.10)

It follows from $\psi_{s}(z^{\alpha})>-\infty$ for almost every
$z^{\alpha}\in\Delta^{l}$ that $h_{2}(z^{\alpha})>-\infty$ for almost every
$z^{\alpha}\in\Delta^{l}$. Thus we have

$\liminf_{t\rightarrow+\infty}\frac{\int_{\max{\\{t,-h_{2}(z^{\alpha})-2(n-l)\log(s)\\}}}^{+\infty}c(l)e^{-l}dl}{\int_{t}^{+\infty}c(l)e^{-l}dl}=1$
(5.11)

for almost every $z^{\alpha}\in\Delta^{l}$. Combining inequality (5.9),
equality (5.11) and Fatou’s Lemma, we have

$\begin{split}&\liminf_{t\rightarrow+\infty}\frac{\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\\\
\geq&2^{n-l}\frac{\sigma_{2n-2l-1}}{2(n-l)}\int_{\Delta^{l}}v^{\alpha}_{\epsilon}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}e^{\psi_{2}(z^{\alpha},0)}e^{-h_{2}}\\\
&\times\liminf_{t\rightarrow+\infty}\frac{\int_{\max{\\{t,-h_{2}(z^{\alpha})-2(n-l)\log(s)\\}}}^{+\infty}c(l)e^{-l}dl}{\int_{t}^{+\infty}c(l)e^{-l}dl}dV_{\alpha}\\\
=&2^{n-l}\frac{\sigma_{2n-2l-1}}{2(n-l)}\int_{\Delta^{l}}v^{\alpha}_{\epsilon}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}e^{\psi_{2}(z^{\alpha},0)}e^{-h_{2}}dV_{\alpha}\end{split}$
(5.12)

As $dV_{M}=dV_{\alpha}\wedge(\wedge_{k=1}^{n-l}idw_{k}^{\alpha}\wedge
d\bar{w}_{k}^{\alpha})$ and $\psi_{1}=(n-l)\log(|w^{\alpha}|^{2})+h_{1}$, by
definition of $dV_{M}[\psi_{1}]$, we have
$dV_{M}[\psi_{1}]=2^{n-l}e^{-h_{1}}dV_{\alpha}$ on $\Delta^{l}\subset S_{l}$.
Then inequality (5.12) becomes

$\begin{split}&\liminf_{t\rightarrow+\infty}\frac{\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\\\
\geq&\frac{\sigma_{2n-2l-1}}{2(n-l)}\int_{S_{l}}v^{\alpha}_{\epsilon}e^{-\Phi_{n}(z^{\alpha},0)-\epsilon}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}e^{\psi_{2}(z^{\alpha},0)-\psi_{s}(z^{\alpha})-\epsilon}dV_{M}[\psi_{1}].\end{split}$
(5.13)

When $s\rightarrow 0$, $\psi_{s}(z^{\alpha})$ is decreasing to
$\psi_{2}(z^{\alpha},0)$ for any $z^{\alpha}\in\Delta^{l}$. As
$\psi_{2}(z^{\alpha},0)>-\infty$ for almost every $z^{\alpha}\in\Delta^{l}$,
let $s\rightarrow 0$ and $\epsilon\rightarrow 0$, then inequality (5.13)
implies that

$\begin{split}&\liminf_{t\rightarrow+\infty}\frac{\int_{\\{\psi<-t\\}}v^{\alpha}|F_{t}|^{2}e^{-\Phi_{n}+\psi_{2}}c(-\psi)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\\\
\geq&\frac{\pi^{n-l}}{(n-l)!}\int_{S_{l}}v^{\alpha}e^{-\Phi_{n}(z^{\alpha},0)}\frac{|f(z^{\alpha},0)|^{2}}{dV_{M}}dV_{M}[\psi_{1}].\end{split}$
(5.14)

Note that $\Phi_{n}$ decreasing to $\varphi+\psi_{2}$, then inequality (5.14)
implies that inequality (5.10) holds.

Following from inequality (5.10) and the concavity of $G(t)$, we have

$\lim_{t\rightarrow+\infty}\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\geq\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|f|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].$

Thus, Proposition 5.1 holds. ∎

###  5.4. Proof of Theorem 1.15

Assume that $G(\hat{h}^{-1}(r))$ is linear with respect to $r$. As
$G(T)\in(0,+\infty)$, we have $\int_{T}^{+\infty}c(t)e^{-t}dt<+\infty$. It
follows from Corollary 1.7, Remark 1.8 and Lemma 2.10 that we can assume
$c(t)e^{-t}$ is strictly decreasing on $(T,+\infty)$ and $c(t)$ is increasing
on $(a,+\infty)$ for some $a>T$. Thus, Proposition 5.1 shows that equality
(1.5) holds.

In the following part, assume that there exists $\tilde{\psi}$ satisfying the
three statements in Theorem 1.15 to get a contradiction. We prove it by
comparing $G(t;\varphi,\psi)$ and $G(t;\tilde{\varphi},\tilde{\psi})$, where
$\tilde{\varphi}=\varphi+\psi-\tilde{\psi}$. It follows from Proposition 5.1
and the linearity of $G(\hat{h}^{-1}(r);\varphi,\psi)$ that
$\sum_{k=1}^{n}\int_{S_{n-k}}\frac{\pi^{k}}{k!}\frac{|F|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]<+\infty$
and equality

$\frac{G(t;\varphi,\psi)}{\int_{t}^{+\infty}c(l)e^{-l}dl}=\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|F|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]$
(5.15)

holds for any $t\geq T$.

Let $\tilde{\varphi}=\varphi+\psi-\tilde{\psi}$ As
$(\tilde{\varphi},\tilde{\psi})\in W$, there exist plurisubharmonic functions
$\tilde{\psi}_{1}$ and $\tilde{\psi}_{2}$ such that
$\tilde{\psi}=\tilde{\psi}_{1}+\tilde{\psi}_{2}$, $\tilde{\psi}_{1}\in
A^{\prime}(S)$ and $\tilde{\varphi}+\tilde{\psi}_{2}$ is plurisubharmonic on
$M$. $dV_{M}[\psi_{1}]=e^{-\psi_{1}+\tilde{\psi}_{1}}dV_{M}[\tilde{\psi}_{1}]$
implies that
$\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|F|^{2}}{dV_{M}}e^{-\tilde{\varphi}-\tilde{\psi}_{2}}dV_{M}[\tilde{\psi}_{1}]=\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}\frac{|F|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]<+\infty.$
It follows from Corollary 1.11 that

$\begin{split}\frac{G(T;\tilde{\varphi},\tilde{\psi})}{\int_{T}^{+\infty}c(l)e^{-l}dl}\leq&\sum_{k=1}^{n}\int_{S_{n-k}}\frac{\pi^{k}}{k!}\frac{|F|^{2}}{dV_{M}}e^{-\tilde{\varphi}-\tilde{\psi}_{2}}dV_{M}[\tilde{\psi}_{1}]\\\
=&\sum_{k=1}^{n}\int_{S_{n-k}}\frac{\pi^{k}}{k!}\frac{|F|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}].\end{split}$
(5.16)

Since $\tilde{\psi}\geq\psi$, $\tilde{\psi}\not=\psi$, there exists a subset
$U$ of $M$ such that $\mu(U)>0$ and $\tilde{\psi}>\psi$ on $U$, where $\mu$ is
Lebesgue measure on $M$. As $c(t)e^{-t}$ is strictly decreasing on
$(T,+\infty)$, we have $G(T;\tilde{\varphi},\tilde{\psi})>G(T;\varphi,\psi)$.
Then inequality (5.16) implies that

$\sum_{k=1}^{n}\int_{S_{n-k}}\frac{\pi^{k}}{k!}\frac{|F|^{2}}{dV_{M}}e^{-\varphi-\psi_{2}}dV_{M}[\psi_{1}]\geq\frac{G(T;\tilde{\varphi},\tilde{\psi})}{\int_{T}^{+\infty}c(l)e^{-l}dl}>\frac{G(T;\varphi,\psi)}{\int_{T}^{+\infty}c(l)e^{-l}dl},$

which contradicts equality (5.15). Thus Theorem 1.15 holds.

##  6\. Proofs of Theorem 1.16, Theorem 1.17, Corollary 1.18 and Corollary
1.19

In this section, we prove Theorem 1.16, Theorem 1.17, Corollary 1.18 and
Corollary 1.19.

###  6.1. A necessary condition of linearity

The following Proposition give a necessary condition of $G(\hat{h}^{-1}(r))$
is linear, and will be used in the proof of Theorem 1.16.

######  Proposition 6.1.

Let $\Omega$ be an open Riemann surface. Let $c\in\mathcal{P}_{0}$, and assume
that there exists $t\geq 0$ such that $G(t)\in(0,+\infty)$. If
$G(\hat{h}^{-1}(r))$ is linear with respect to $r$, then there is no Lebesgue
measurable function $\tilde{\varphi}\geq\varphi$ such that
$\tilde{\varphi}+\psi$ is subharmonic function on $M$ and satisfies:

$(1)$ $\tilde{\varphi}\not=\varphi$ and
$\mathcal{I}(\tilde{\varphi}+\psi)=\mathcal{I}(\varphi+\psi)$;

$(2)$ $\lim_{t\rightarrow
0+0}\sup_{\\{\psi\geq-t\\}}(\tilde{\varphi}-\varphi)=0$;

$(3)$ there exists an open subset $U\subset\subset\Omega$ such that
$\sup_{\Omega\backslash U}(\tilde{\varphi}-\varphi)<+\infty$,
$e^{-\tilde{\varphi}}c(-\psi)$ has a positive lower bound on $U$ and