$\int_{U}|F_{1}-F_{2}|^{2}e^{-\varphi}c(-\psi)<+\infty$ for any
$F_{1}\in\mathcal{H}^{2}(c,\tilde{\varphi},t)$ and
$F_{2}\in\mathcal{H}^{2}(c,\varphi,t)$, where $U\subset\subset\\{\psi<-t\\}$.

###### Proof.

We prove the lemma by comparing $G(t;\varphi)$ and $G(t;\tilde{\varphi})$. In
the following, let us assume that there exists a Lebesgue measurable function
$\tilde{\varphi}$ satisfying these properties in Proposition 6.1 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 $(1,0)$ form $F$ on $\Omega$
such that $(F-f,z_{0})\in(\mathcal{O}(K_{\Omega})\otimes\mathcal{F})_{z_{0}}$
and $\forall t\geq 0$ equality

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

holds. As $\tilde{\varphi}+\psi$ is subharmonic and there exists a 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
$\mathcal{I}(\tilde{\varphi}+\psi)=\mathcal{I}(\varphi+\psi)$, 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 subharmonic
functions on $\Omega$, then there exists a subset $V$ of $\Omega$ such that
$e^{-\tilde{\varphi}}<e^{-\varphi}$ on a subset $V$ and $\mu(V)>0$, where
$\mu$ is Lebesgue measure on $\Omega$. As $F\not\equiv 0$, inequality

$\frac{G(T_{0};\tilde{\varphi})}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}\leq\frac{\int_{\\{\psi<-T_{0}\\}}|F|^{2}e^{-\tilde{\varphi}}c(-\psi)}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}<\frac{G(T_{0};\varphi)}{\int_{T_{0}}^{+\infty}c(s)e^{-s}ds}$
(6.1)

holds for some $T_{0}>0$. For $t>0$, there exists a holomorphic $(1,0)$ form
$F_{t}$ on $\\{\psi<-t\\}$ such that
$(F_{t}-f)_{z_{0}}\in(\mathcal{O}(K_{\Omega})\otimes\mathcal{F})_{z_{0}}$ and

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

As there exists a subset $U\subset\subset\Omega$ such that
$\sup_{\Omega\backslash U}(\tilde{\varphi}-\varphi)<+\infty$, we get that

$\begin{split}\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)=&\int_{\\{\psi<-t\\}\cap
U}|F_{t}|^{2}e^{-\varphi}c(-\psi)+\int_{\\{\psi<-t\\}\backslash
U}|F_{t}|^{2}e^{-\varphi}c(-\psi)\\\ \leq&2\int_{\\{\psi<-t\\}\cap
U}|F|^{2}e^{-\varphi}c(-\psi)+2\int_{\\{\psi<-t\\}\cap
U}|F_{t}-F|^{2}e^{-\varphi}c(-\psi)\\\ &+e^{\sup_{\Omega\backslash
U}(\tilde{\varphi}-\varphi)}\int_{\\{\psi<-t\\}\backslash
U}|F_{t}|^{2}e^{-\tilde{\varphi}}c(-\psi)\\\ <&+\infty\end{split}$ (6.2)

holds for small enough $t>0$. It follows from Lemma 2.6 that

$\begin{split}G(t_{1};\tilde{\varphi})-G(t_{2};\tilde{\varphi})&\geq\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F_{t_{1}}|^{2}e^{-\tilde{\varphi}}c(-\psi)\\\
&\geq\left(\inf_{z\in\\{-t_{2}\leq\psi\\}}e^{\varphi-\tilde{\varphi}}\right)\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F_{t_{1}}|^{2}e^{-\varphi}c(-\psi)\\\
&\geq\left(\inf_{z\in\\{-t_{2}\leq\psi\\}}e^{\varphi-\tilde{\varphi}}\right)\int_{\\{-t_{2}\leq\psi<-t_{1}\\}}|F|^{2}e^{-\varphi}c(-\psi)\end{split}$
(6.3)

holds for small enough $t_{1}$ and $t_{2}$ such that $0<t_{1}<t_{2}<+\infty$.
As $\lim_{t\rightarrow
0+0}\sup_{\\{\psi\geq-t\\}}(\tilde{\varphi}-\varphi)=0$, it follows from
inequality (6.1) and (6.3) that

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

which contradicts the concavity of $G(\hat{h}^{-1}(r);\tilde{\varphi})$. Thus
the assumption doesn’t hold, and we complete the proof of Proposition 6.1. ∎

###  6.2. Proof of Theorem 1.16

Firstly, we prove the sufficiency by using Theorem 2.15. The following remark
shows that it suffices to prove the sufficiency for the case
$\psi=2G_{\Omega}(z,z_{0})$.

######  Remark 6.2.

Let $\tilde{\varphi}=\varphi+a\psi$,
$\tilde{c}(t)=c(\frac{t}{1-a})e^{-\frac{at}{1-a}}$ and
$\tilde{\psi}=(1-a)\psi$ for some $a\in(-\infty,1)$. It is clear that
$e^{-\tilde{\varphi}}\tilde{c}(-\tilde{\psi})=e^{-\varphi}c(-\psi)$,
$(1-a)\int_{t}^{+\infty}c(l)e^{-l}dl=\int_{(1-a)t}^{+\infty}\tilde{c}(l)e^{-l}dl$
and $G(t;\varphi,\psi,c)=G((1-a)t;\tilde{\varphi},\tilde{\psi},\tilde{c})$.

Let $\tilde{c}\equiv 1$ on $(0,+\infty)$. Set $\hat{f}=\frac{f}{g}$,
$\hat{\varphi}=\varphi-2\log|g|=2u$, and
$\hat{\mathcal{F}}_{z_{0}}=\mathcal{I}(\hat{\varphi}+\psi)_{z_{0}}=\mathcal{I}(2G_{\Omega}(z,z_{0}))_{z_{0}}$.
Denote

$\inf\\{\int_{\\{\psi<-t\\}}|\tilde{f}|^{2}e^{-\hat{\varphi}}:(\tilde{f}-\hat{f})_{z_{0}}\in(\mathcal{O}(K_{\Omega})\otimes\hat{\mathcal{F}})_{z_{0}}\,\&\,\tilde{f}\in
H^{0}(\\{\psi<-t\\},\mathcal{O}(K_{\Omega}))\\}$

by $\hat{G}(t;\tilde{c})$. Without loss of generality, we can assume that
$\hat{f}(z_{0})=dw$, where $w$ is a local coordinate on a neighborhood
$V_{z_{0}}$ of $z_{0}$ satisfying $w(z_{0})=0$. By definition of
$G(t;\tilde{c})$ and $B_{\Omega,e^{-2u}}(z_{0})$, it is clear that
$G(t;\tilde{c})=\hat{G}(t;\tilde{c})$ and
$\hat{G}(0;\tilde{c})=\frac{2}{B_{\Omega,e^{-2u}}(z_{0})}=\inf\\{\int_{\Omega}|\tilde{f}|^{2}e^{-2u}:\tilde{f}$
is a holomorphic extension of $\hat{f}$ from $z_{0}$ to $\Omega$$\\}$. Theorem
2.15 shows that
$G(0;\tilde{c})=\hat{G}(0;\tilde{c})=2\pi\frac{e^{-2u(z_{0})}}{c_{\beta}^{2}(z_{0})}$.
Note that
$\|\hat{f}\|_{z_{0}}=\pi\int_{z_{0}}\frac{|\hat{f}|^{2}}{dV_{M}}e^{-\hat{\varphi}}dV_{\Omega}[2G_{\Omega}(z,z_{0})]=2\pi\frac{e^{-2u(z_{0})}}{c_{\beta}^{2}(z_{0})}$,
therefore Theorem 1.10 tells us that $G(-\log r;\tilde{c})$ and $\hat{G}(-\log
r;\tilde{c})$ is linear with respect to $r$.

As $\psi=2G_{\Omega}(z,z_{0})$, Lemma 2.12 shows that, for any $t_{0}\geq 0,$
there exists $t>t_{0}$ such that $\\{G_{\Omega}(z,z_{0}<-t)\\}$ is a
relatively compact subset of $\Omega$ and $g$ has no zero point in
$\\{G_{\Omega}(z,z_{0}<-t)\\}\backslash{\\{z_{0}\\}}$. Combining Corollary
1.7, Remark 1.8 and $G(-\log r;\tilde{c})$ is linear with respect to $r$, we
obtain that $G(\hat{h}^{-1}(r))$ is linear with respect to $r$, where
$\hat{h}(t)=\int_{t}^{+\infty}c(l)e^{-l}dl$.

In the following part, we prove the necessity in three steps.

By Remark 6.2, without loss of generality, we can assume that $\varphi$ is
subharmonic near $z_{0}$. As $\varphi+\psi$ is a subharmonic function on
$\Omega$, it follows from Weierstrass Theorem on open Riemann surfaces (see
[15]) and Siu’s Decomposition Theorem that

$\varphi+\psi=2\log|g|+2G_{\Omega}(z,z_{0})+2u,$ (6.4)

where $g$ is a holomorphic function on $\Omega$, and $u$ is a subharmonic
function on $\Omega$ such that $v(dd^{c}u,z)\in[0,1)$ for any $z\in\Omega$.

_Step 1: $\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$,
$ord_{z_{0}}(g)=org_{z_{0}}(f_{1})$ and $v(dd^{c}\psi,z_{0})>0$._

As
$\mathcal{I}(\varphi+\psi)_{z_{0}}=\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))_{z_{0}}\subset\mathcal{F}_{z_{0}}$
and $G(0)\not=0$, we have $ord_{z_{0}}(g)+1>org_{z_{0}}(f_{1})$. Corollary 1.7
tells us there exists a holomorphic $(1,0)$ form on $\Omega$ such that
$(F-f,z_{0})\in(\mathcal{O}(K_{\Omega})\otimes\mathcal{F})_{z_{0}}$ and
$G(t)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)$ for $t\geq 0$. Denote
that $\tilde{c}(t)=\max{\\{c(t),e^{rt}\\}}$ on $(0,+\infty)$, where
$r\in(0,1)$. Set $F=\tilde{F}dw$ on $V_{z_{0}}$, and it follows from Corollary
1.7 and Remark 1.8 that $|\tilde{F}|^{2}e^{-\varphi-r\psi}$ is locally
integrable near $z_{0}$ for any $r\in(0,1)$, which implies that
$ord_{z_{0}}(\tilde{F})\geq ord_{z_{0}}(g)$.

we prove $\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$ by
contradiction: if not, then
$\mathcal{F}_{z_{0}}\subsetneqq\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))_{z_{0}}$.
Since $ord_{z_{0}}(\tilde{F})\geq ord_{z_{0}}(g)$, we have
$(\tilde{F},z_{0})\in\mathcal{F}_{z_{0}}$, which contradicts to $G(0)\not=0$.
Thus $\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$.

As $ord_{z_{0}}(\tilde{F})\geq ord_{z_{0}}(g)$,
$ord_{z_{0}}(g)+1>org_{z_{0}}(f_{1})$ and
$(\tilde{F}-f_{1},z_{0})\in\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))_{z_{0}}$,
we have $ord_{z_{0}}(g)=org_{z_{0}}(f_{1})$.

we prove $v(dd^{c}\psi,z_{0})>0$ by contradiction: if not,
$v(dd^{c}\psi,z_{0})=0$ shows that
$\mathcal{I}(\varphi+\psi)_{z_{0}}=\mathcal{I}(\varphi)_{z_{0}}$. Without loss
of generality, we can assume that $c(t)>1$ for large enough $t$, then
$|\tilde{F}|^{2}e^{-\varphi}$ is locally integrable near $z_{0}$, which
contradicts to $(\tilde{F},z_{0})\not\in\mathcal{F}_{z_{0}}$. Thus
$v(dd^{c}\psi,z_{0})>0$.

_Step 2: $\psi=2pG_{\Omega}(z,z_{0})$ for some $p>0$._

As $\psi$ is subharmonic function on $\Omega$, it follows from Siu’s
Decomposition Theorem that $\psi=2pG_{\Omega}(z,z_{0})+\psi_{1}$ such that
$v(dd^{c}\psi_{1},z_{0})=0$.

Firstly, we prove $\psi_{1}$ is harmonic near $z_{0}$ by contradiction : if
not, there exists a closed positive $(1,1)$ current $T\not\equiv 0$, such that
$suppT\subset\subset V_{z_{0}}$,
$T\leq\frac{1}{2}i\partial\bar{\partial}\psi_{1}$ on $V_{z_{0}}$, where
$V_{z_{0}}$ is an open neighborhood of $z_{0}$, satisfying that $g$ has not
zero point on $\overline{V_{z_{0}}}\backslash\\{z_{0}\\}$, $\varphi$ is
subharmonic on a neighborhood of $\overline{V_{z_{0}}}$ and
$V_{z_{0}}\subset\subset\Omega$. Note that
$\\{z\in\overline{V_{z_{0}}}:\mathcal{I}(\varphi+\psi)_{z}\not=\mathcal{O}_{z}\\}=\\{z_{0}\\}$.

Using Lemma 2.14, there exists a subharmonic function $\Phi<0$ on $\Omega$,
which satisfies the following properties: $i\partial\bar{\partial}\Phi\leq T$
and $i\partial\bar{\partial}\Phi\not\equiv 0$; $\lim_{t\rightarrow
0+0}(\inf_{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\Phi(z))=0$;
$supp(i\partial\bar{\partial}\Phi)\subset V_{z_{0}}$ and
$\inf_{\Omega\backslash V_{z_{0}}}\Phi>-\infty$. It following from Lemma 2.11,
$v(dd^{c}\psi,z_{0})>0$ and $\psi<0$ on $\Omega$, that $\lim_{t\rightarrow
0+0}(\inf_{\\{\psi\geq-t\\}}\Phi(z))=0$.

Set $\tilde{\varphi}=\varphi-\Phi$, then
$\tilde{\varphi}+\psi=\varphi+2pG_{\Omega}(z,z_{0})+\psi_{1}-\Phi$ on
$V_{z_{0}}$, where $\psi_{1}-\Phi$ is subharmonic on $V_{z_{0}}$. It is clear
that $\tilde{\varphi}\geq\varphi$ and $\tilde{\varphi}\not=\varphi$.
$suppT\subset\subset V_{z_{0}}$ and $i\partial\bar{\partial}\Phi\leq T\leq
i\partial\bar{\partial}\varphi_{1}$ on $V_{z_{0}}$ show that
$\tilde{\varphi}+\psi$ is subharmonic on $\Omega$,
$\mathcal{I}(\tilde{\varphi}+\psi)=\mathcal{I}(\varphi+\psi)=\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))$.

Without loss of generality, we can assume that $c(t)>e^{\frac{t}{2}}$ for any
$t>0$. $T\leq\frac{1}{2}i\partial\bar{\partial}\psi_{1}$ on $V_{z_{0}}$ and
$i\partial\bar{\partial}\Phi\subset\subset V_{z_{0}}$ show that
$\frac{1}{2}\psi-\Phi$ is subharmonic on $\Omega$, which implies that
$e^{-\tilde{\varphi}}c(-\psi)\geq e^{-\varphi}e^{\Phi-\frac{1}{2}\psi}$ has a
positive lower bound on $V_{z_{0}}$. Notice that $\inf_{\Omega\backslash
V_{z_{0}}}(\varphi-\tilde{\varphi})=\inf_{\Omega\backslash
V_{z_{0}}}\Phi>-\infty$ and
$\int_{V_{z_{0}}}|F_{1}-F_{2}|^{2}e^{-\varphi}c(-\psi)\leq
C\int_{V_{z_{0}}}|F_{1}-F_{2}|^{2}e^{-\varphi-\psi}<+\infty$ for any
$F_{1}\in\mathcal{H}^{2}(c,\tilde{\varphi},t)$ and
$F_{2}\in\mathcal{H}^{2}(c,\varphi,t)$, where
$V_{z_{0}}\subset\subset\\{\psi<-t\\}$, then $\tilde{\varphi}$ satisfies the
conditions in Proposition 6.1, which contradicts to the result of Proposition
6.1. Thus $\psi_{1}$ is harmonic near $z_{0}$.

Then, we prove $\psi=2pG_{\Omega}(z,z_{0})$. Using Remark 6.2, it suffices to
consider the case $p=1$, where $p=\frac{1}{2}v(dd^{c}\psi,z_{0})$. By Siu’s
Decomposition Theorem and Lemma 2.11, there exists a subharmonic function
$\psi_{2}\leq 0$ on $\Omega$ such that $\psi=2G_{\Omega}(z,z_{0})+\psi_{2}$.
Note that $\psi_{2}(z_{0})>-\infty$.

As $\Omega$ is an open Riemann surface, there exists a holomorphic function
$f_{2}$ on $\Omega$, such that $ord_{z_{0}}(f_{2})=ord_{z_{0}}(f_{1})$ and
$\\{z\in\Omega:f_{2}(z)=0\\}=\\{z_{0}\\}$. Set $\tilde{f}=\frac{f}{f_{2}}$,
$\tilde{\varphi}=\varphi-2\log|f_{2}|$, and
$\tilde{\mathcal{F}}_{z_{0}}=\mathcal{I}(\tilde{\varphi}+\psi)_{z_{0}}=\mathcal{I}(2G_{\Omega}(z,z_{0}))_{z_{0}}$.
Denote

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

by $\tilde{G}(t)$. By the definition of $G(t)$ and $\tilde{G}(t)$, we know
$G(t)=\tilde{G}(t)$ for any $t\geq 0$, therefore $\tilde{G}(\hat{h}^{-1}(r))$
is linear with respect to $r$. Note that $(\tilde{\varphi},\psi)\in W$,
$(\tilde{\varphi}+\psi-2G_{\Omega}(z,z_{0}),2G_{\Omega}(z,z_{0}))\in W$,
$\psi_{2}(z_{0})>-\infty$ and $\psi_{2}\leq 0$, then Theorem 1.15 shows that
$\psi=2G_{\Omega}(z,z_{0})$.

_Step 3. $u$ is harmonic on $\Omega$ and $\chi_{-u}=\chi_{z_{0}}.$_

Without loss of generality, we can assume that $\psi=2G_{\Omega}(z,z_{0})$.
Lemma 2.12 shows that, for any $t_{0}\geq 0,$ there exists $t>t_{0}$ such that
$\\{G_{\Omega}(z,z_{0})<-t\\}$ is a relatively compact subset of $\Omega$ and
$g$ has no zero point in
$\\{G_{\Omega}(z,z_{0})<-t\\}\backslash{\\{z_{0}\\}}$. Combining Corollary
1.7, Remark 1.8 and $G(\hat{h}^{-1}(r);c)$ is linear with respect to $r$, we
obtain that $G(-\log r;\tilde{c}\equiv 1)$ is linear with respect to $r$ and
$G(0;\tilde{c})\in(0,+\infty)$.

Now, we assume that $u$ is not harmonic to get a contradiction. There exists a
closed positive $(1,1)$ current $T\not\equiv 0$, such that
$suppT\subset\subset\Omega$ and $T\leq i\partial\bar{\partial}u$. There exists
an open subset $U\subset\subset\Omega$, such that $suppT\subset U$.

Using Lemma 2.14, there exists a subharmonic function $\Phi<0$ on $\Omega$,
which satisfies the following properties: $i\partial\bar{\partial}\Phi\leq T$
and $i\partial\bar{\partial}\Phi\not\equiv 0$; $\lim_{t\rightarrow
0+0}(\inf_{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\Phi(z))=0$;
$supp(i\partial\bar{\partial}\Phi)\subset U$ and $\inf_{\Omega\backslash
U}\Phi>-\infty$.

Set $\tilde{\varphi}=\varphi-\Phi$, then $\tilde{\varphi}=2\log|g|+2u-\Phi$ is
subharmonic on $\Omega$. It is clear that $\tilde{\varphi}\geq\varphi$,
$\tilde{\varphi}\not=\varphi$ and $\tilde{\varphi}+\psi$ is subharmonic on
$\Omega$,
$\mathcal{I}(\tilde{\varphi}+\psi)=\mathcal{I}(\varphi+\psi)=\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))$.

As $\tilde{\varphi}$ is subharmonic on $\Omega$, we have
$e^{-\tilde{\varphi}}$ has a positive lower bound on $U$. Note that
$\mathcal{I}(\varphi)=\mathcal{I}(\tilde{\varphi})$, then

$\int_{U}|F_{1}-F_{2}|^{2}e^{-\varphi}\leq
2\int_{U}|F_{1}|^{2}e^{-\varphi}+2\int_{U}|F_{2}|^{2}e^{-\varphi}<+\infty$

for any $F_{1}\in\mathcal{H}^{2}(\tilde{c},\tilde{\varphi},t)$ and
$F_{2}\in\mathcal{H}^{2}(\tilde{c},\varphi,t)$, where
$U\subset\subset\\{\psi<-t\\}$ and $\tilde{c}\equiv 1$. Since
$\inf_{\Omega\backslash U}(\varphi-\tilde{\varphi})=\inf_{\Omega\backslash
U}\Phi>-\infty$, then $\tilde{\varphi}$ satisfies the conditions in
Proposition 6.1, which contradicts to the result of Proposition 6.1. Thus $u$
is harmonic on $\Omega$.

Finally, we prove $\chi_{-u}=\chi_{z_{0}}$ by using Theorem 2.15.

Recall some notations in the proof of sufficiency. Set $\hat{f}=\frac{f}{g}$,
$\hat{\varphi}=\varphi-2\log|g|=2u$, and
$\hat{\mathcal{F}}_{z_{0}}=\mathcal{I}(\hat{\varphi}+\psi)_{z_{0}}=\mathcal{I}(2G_{\Omega}(z,z_{0}))_{z_{0}}$.
Denote

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

by $\hat{G}(t;\tilde{c})$. Without loss of generality, we can assume that
$\hat{f}(z_{0})=dw$, where $w$ is a local coordinate on a neighborhood
$V_{z_{0}}$ of $z_{0}$ satisfying $w(z_{0})=0$. By definition of
$G(t;\tilde{c})$ and $B_{\Omega,e^{-2u}}(z_{0})$, it is clear that $G(-\log
r;\tilde{c})=\hat{G}(-\log r;\tilde{c})$ is linear with respect to $r$ and
$\hat{G}(0;\tilde{c})=\frac{2}{B_{\Omega,e^{-2u}}(z_{0})}=\inf\\{\int_{\Omega}|\tilde{f}|^{2}e^{-2u}:\tilde{f}$
is a holomorphic extension of $\hat{f}$ from $z_{0}$ to $\Omega$$\\}$.

Note that
$\|\hat{f}\|_{z_{0}}=2\pi\frac{e^{-2u(z_{0})}}{c_{\beta}^{2}(z_{0})}$ , then
Theorem 1.15 shows that

$\hat{G}(0,\tilde{c})=2\pi\frac{e^{-2u(z_{0})}}{c_{\beta}^{2}(z_{0})},$

i.e., $c_{\beta}^{2}(z_{0})=\pi e^{-2u(z_{0})}B_{\Omega,e^{-2u}}(z_{0})$.
Therefore Theorem 2.15 shows that $\chi_{-u}=\chi_{z_{0}}$.

Thus, Theorem 1.16 holds.

###  6.3. Proof of Theorem 1.17

Theorem 1.16 implies the sufficiency. Thus we just need to prove the
necessity.

As $\varphi+\psi$ is a subharmonic function on $\Omega$, it follows from
Weierstrass Theorem on open Riemann surfaces (see [15]) and Siu’s
Decomposition Theorem that

$\varphi+\psi=2\log|g|+2G_{\Omega}(z,z_{0})+2u,$ (6.5)

where $g$ is a holomorphic function on $\Omega$, and $u$ is a subharmonic
function on $\Omega$ such that $v(dd^{c}u,z)\in[0,1)$ for any $z\in\Omega$.

As
$\mathcal{I}(\varphi+\psi)_{z_{0}}=\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))_{z_{0}}\subset\mathcal{F}_{z_{0}}$
and $G(0)\not=0$, we have $ord_{z_{0}}(g)+1>org_{z_{0}}(f_{1})$. Corollary 1.7
tells us there exists a holomorphic $(1,0)$ form on $\Omega$ such that
$(F-f,z_{0})\in(\mathcal{O}(K_{\Omega})\otimes\mathcal{F})_{z_{0}}$ and
$G(t)=\int_{\\{\psi<-t\\}}|F|^{2}e^{-\varphi}c(-\psi)$ for $t\geq 0$. Let
$\tilde{c}(t)=\max{\\{c(t),e^{rt}\\}}$ defined on $(0,+\infty)$, where
$r\in(0,1)$. Set $F=\tilde{F}dw$ on $V_{z_{0}}$, and it follows from Corollary
1.7 and Remark 1.8 that $|\tilde{F}|^{2}e^{-\varphi-r\psi}$ is locally
integrable near $z_{0}$ for any $r\in(0,1)$. Note that

$\int_{U}|\tilde{F}|^{2}e^{-\frac{\varphi+\psi}{p}}\leq\left(\int_{U}|\tilde{F}|^{2p}e^{-\varphi-\psi+ps\psi}\right)^{\frac{1}{p}}\left(\int_{U}e^{-qs\psi}\right)^{\frac{1}{q}}$

holds for any $p>1$, $\frac{1}{p}+\frac{1}{q}=1$, $U$ is a small open
neighborhood of $z_{0}$, and $s\in(0,1)$. For any $p\in(1,+\infty)$, we can
choose small enough $U$ and small enough $s\in(0,1)$ such that
$\int_{U}|\tilde{F}|^{2p}e^{-\varphi-\psi+ps\psi}<+\infty$ and
$\int_{U}e^{-qs\psi}<+\infty$, which implies that
$(\tilde{F},z_{0})\in\mathcal{I}(\frac{\varphi+\psi}{p})_{z_{0}}\subset\mathcal{I}(\frac{2\log|g|+2G_{\Omega}(z,z_{0})}{p})_{z_{0}}$.
Therefore, we have $ord_{z_{0}}(\tilde{F})\geq ord_{z_{0}}(g)$.

we prove $\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$ by
contradiction: if not, then
$\mathcal{F}_{z_{0}}\subsetneqq\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))_{z_{0}}$.
Since $ord_{z_{0}}(\tilde{F})\geq ord_{z_{0}}(g)$, we have
$(\tilde{F},z_{0})\in\mathcal{F}_{z_{0}}$, which contradicts to $G(0)\not=0$.
Thus $\mathcal{F}_{z_{0}}=\mathcal{I}(\varphi+\psi)_{z_{0}}$.

As $ord_{z_{0}}(\tilde{F})\geq ord_{z_{0}}(g)$,
$ord_{z_{0}}(g)+1>org_{z_{0}}(f_{1})$ and
$(\tilde{F}-f_{1},z_{0})\in\mathcal{I}(2\log|g|+2G_{\Omega}(z,z_{0}))_{z_{0}}$,
we have $ord_{z_{0}}(g)=org_{z_{0}}(f_{1})$.

we prove $v(dd^{c}\psi,z_{0})>0$ by contradiction: if not, as
$|\tilde{F}|^{2}e^{-\varphi-r\psi}$ is locally integrable near $z_{0}$ for any
$r\in(0,1)$ and $ord_{z_{0}}(g)=org_{z_{0}}(\tilde{F})$, we have
$e^{-2G_{\Omega}(z,z_{0})+(1-r)\psi}$ is locally integrable near $z_{0}$.
Therefore there exists $s>0$ such that

$\begin{split}\int_{\Delta_{s}}\frac{e^{(1-r)\psi}}{|w|^{2}}<+\infty,\end{split}$

where $w$ is a local coordinate near $z_{0}$ such that $w(z_{0})=0$.
$e^{(1-r)\psi}$ is subharmonic shows that

$2\pi
e^{(1-r)\psi(z_{0})}\int_{0}^{s}\frac{1}{t}dt=\int_{\Delta_{s}}\frac{e^{(1-r)\psi}}{|w|^{2}}<+\infty,$

which contradicts to $\psi(z_{0})>-\infty$. Thus $v(dd^{c}\psi,z_{0})>0$
holds.

Using Remark 6.2, it suffices to consider the case $p=1$, where
$p=\frac{1}{2}v(dd^{c}\psi,z_{0})$. By Siu’s Decomposition Theorem and Lemma
2.11, there exists a subharmonic function $\psi_{2}\leq 0$ on $\Omega$ such
that $\psi=2G_{\Omega}(z,z_{0})+\psi_{2}$. Following the assumption in Theorem
1.17, we know $\psi_{2}(z_{0})>-\infty$.

As $\Omega$ is an open Riemann surface, there exists a holomorphic function
$f_{2}$ on $\Omega$, such that $ord_{z_{0}}(f_{2})=ord_{z_{0}}(f_{1})$ and
$\\{z\in\Omega:f_{2}=0\\}=\\{z_{0}\\}$. Set $\tilde{f}=\frac{f}{f_{2}}$,
$\tilde{\varphi}=\varphi-2\log|f_{2}|$, and
$\tilde{\mathcal{F}}_{z_{0}}=\mathcal{I}(\tilde{\varphi}+\psi)_{z_{0}}=\mathcal{I}(2G_{\Omega}(z,z_{0}))_{z_{0}}$.
Denote

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

by $\tilde{G}(t)$. By the definition of $G(t)$ and $\tilde{G}(t)$, we know
$G(t)=\tilde{G}(t)$ for any $t\geq 0$, therefore
$\tilde{G}(\tilde{h}^{-1}(r))$ is linear with respect to $r$. Note that
$(\tilde{\varphi},\psi)\in W$,
$(\tilde{\varphi}+\psi-2G_{\Omega}(z,z_{0}),2G_{\Omega}(z,z_{0}))\in W$,
$\psi_{2}(z_{0})>-\infty$ and $\psi_{2}\leq 0$, then Theorem 1.15 shows that
$\psi=2G_{\Omega}(z,z_{0})$.

As $\varphi+\psi$ is subharmonic on $\Omega$ and $\psi=2G_{\Omega}(z,z_{0})$,
we have $\varphi$ is subharmonic on $\Omega$. Then Theorem 1.16 implies that
$u$ is harmonic on $\Omega$ and $\chi_{-u}=\chi_{z_{0}}$.

Thus, Theorem 1.17 holds.

###  6.4. Proof of Corollary 1.18

The following remark shows that it suffices to prove the existence of
holomorphic extension satisfying inequality (1.6) for the case $c(t)$ has a
positive lower bound and upper bound on $(t^{\prime},+\infty)$ for any
$t^{\prime}>0$.

######  Remark 6.3.

Take $c_{j}$ is a positive measurable function on $(0,+\infty)$, such that
$c_{j}(t)=c(t)$ when $t<j$, $c_{j}(t)=\min\\{c(j),\frac{1}{j}\\}$ when $t\geq
j$. It is clear that $c_{j}(t)e^{-t}$ is decreasing with respect to $t$, and
$\int_{0}^{+\infty}c_{j}(t)e^{-t}<+\infty$. As

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

we have

$\lim_{j\rightarrow+\infty}\int_{0}^{+\infty}c_{j}(t)e^{-t}=\int_{0}^{+\infty}c(t)e^{-t}.$

If the existence of holomorphic extension satisfying inequality (1.6) holds in
this case, then there exists a holomorphic $(1,0)$ form $F_{j}$ on $\Omega$
such that $F_{j}(z_{0})=f(z_{0})$ and

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

Note that $\psi$ has locally lower bound on
$\Omega\backslash\psi^{-1}(-\infty)$ and $\psi^{-1}(-\infty)$ is a closed
subset of an analytic subset $Z$ of $\Omega$. For any compact subset $K$ of
$\Omega\backslash Z$, there exists $s_{K}>0$ such that
$\int_{K}e^{-s_{K}\psi}dV_{\Omega}<+\infty$, where $dV_{\Omega}$ is a
continuous volume form on $\Omega$. Then we have

$\int_{K}\left(\frac{e^{\varphi}}{c_{j}(-\psi)}\right)^{s_{K}}dV_{\Omega}=\int_{K}\left(\frac{e^{\varphi+\psi}}{c_{j}(-\psi)}\right)^{s_{K}}e^{-s_{K}\psi}dV_{\Omega}\leq
C\int_{K}e^{-s_{K}\psi}dV_{\Omega}<+\infty,$

where $C$ is a constant independent of $j$. It follows from Lemma 2.4
($g_{j}=e^{-\varphi}c_{j}(-\psi)$) that there exists a subsequence of
$\\{F_{j}\\}$, denoted still by $\\{F_{j}\\}$, which is uniformly convergent
to a holomorphic $(1,0)$ form $F$ on any compact subset of $\Omega$ and

$\begin{split}\int_{\Omega}|F|^{2}e^{-\varphi}c(-\psi)&\leq\lim_{j\rightarrow+\infty}\left(\int_{0}^{+\infty}c_{j}(t)e^{-t}dt\right)\|f\|_{z_{0}}\\\
&=\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}.\end{split}$

Since $F_{j}(z_{0})=f(z_{0})$ for any $j$, we have $F(z_{0})=f(z_{0})$.

As $\psi\in A(z_{0})$ and $e^{-\varphi-\psi}$ is not $L^{1}$ on any
neighborhood of $z_{0}$, it follows from Siu’s Decomposition Theorem and the
following Lemma that $\psi(z)-2G_{\Omega}(z,z_{0})$ and
$\varphi(z)+\psi(z)-2G_{\Omega}(z,z_{0})$ is subharmonic on $\Omega$ with
respect to $z$. Denote that $\psi_{2}(z)=\psi(z)-2G_{\Omega}(z,z_{0})$.

######  Lemma 6.4 ([50]).

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

As $\Omega$ is a Stein manifold and $\varphi+\psi_{2}$ is subharmonic on
$\Omega$, there exist smooth subharmonic functions $\Phi_{l}$ on $\Omega$,
which are decreasingly convergent to $\varphi+\psi_{2}$. We can find a
sequence of open Riemann surfaces $\\{D_{m}\\}_{m=1}^{+\infty}$ satisfying
$z_{0}\in D_{m}\subset\subset D_{m+1}$ for any $m$ and
$\cup_{m=1}^{+\infty}D_{m}=\Omega$, and there is a holomorphic $(n,0)$ form
$\tilde{F}$ on $\Omega$ such that $\tilde{F}(z_{0})=f(z_{0})$.

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

$\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\Phi_{l}-2G_{\Omega}(\cdot,z_{0})}\leq
e^{t_{0}+1}\int_{D_{m}}|\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}<+\infty$

for any $m$, $l$ and $t_{0}>T$. Using Lemma 2.1
($\varphi\sim\Phi_{l}+2G_{\Omega}(\cdot,z_{0})$), for any $D_{m}$,
$l\in\mathbb{N}^{+}$ and $t_{0}>T$, there exists a holomorphic $(1,0)$ form
$F_{l,m,t_{0}}$ on $D_{m}$, such that

$\begin{split}&\int_{D_{m}}|F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}-2G_{\Omega}(\cdot,z_{0})+v_{t_{0},1}(\psi)}c(-v_{t_{0},1}(\psi))\\\
\leq&\left(\int_{0}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\Phi_{l}-2G_{\Omega}(\cdot,z_{0})}\end{split}$
(6.6)

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
$e^{-2G_{\Omega}(\cdot,z_{0})}$ is not $L^{1}$ on any neighborhood of $z_{0}$,
and $b_{t_{0},1}(t)=0$ for large enough $t$, then
$(F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F})(z_{0})=0$, and therefore
$F_{l,m,t_{0}}(z_{0})=f(z_{0})$.

Note that $v_{t_{0},1}(\psi)\geq\psi$ and $c(t)e^{-t}$ is decreasing, then the
inequality (6.6) becomes

$\begin{split}&\int_{D_{m}}|F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)\\\
\leq&\left(\int_{0}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\Phi_{l}-2G_{\Omega}(\cdot,z_{0})}.\end{split}$
(6.7)

There exist smooth subharmonic functions $\Psi_{k}$ on $\Omega$, which are
decreasingly convergent to $\psi_{2}$. By definition of $dV_{\Omega}[\psi]$,
we have

$\begin{split}&\limsup_{t_{0}\rightarrow+\infty}\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\Phi_{l}+\Psi_{k}-\psi}\\\
\leq&\pi\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\Phi_{l}+\Psi_{k}}dV_{\Omega}[\psi]\\\
<&+\infty.\end{split}$ (6.8)

Combining inequality (6.7) and (6.8), let $t_{0}\rightarrow+\infty$, we have

$\begin{split}&\limsup_{t_{0}\rightarrow+\infty}\int_{D_{m}}|F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)\\\
\leq&\limsup_{t_{0}\rightarrow+\infty}\left(\int_{0}^{t_{0}+1}c(t)e^{-t}dt\right)\int_{D_{m}}\mathbb{I}_{\\{-t_{0}-1<\psi<-t_{0}\\}}|\tilde{F}|^{2}e^{-\Phi_{l}+\Psi_{k}-\psi}\\\
\leq&\pi\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\Phi_{l}+\Psi_{k}}dV_{\Omega}[\psi].\end{split}$
(6.9)

Let $k\rightarrow+\infty$, inequality (6.9) implies that

$\begin{split}&\limsup_{t_{0}\rightarrow+\infty}\int_{D_{m}}|F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)\\\
\leq&\pi\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\Phi_{l}+\psi_{2}}dV_{\Omega}[\psi].\end{split}$
(6.10)

Note that

$\limsup_{t_{0}\rightarrow+\infty}\int_{D_{m}}|(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)<+\infty,$

then we have

$\limsup_{t_{0}\rightarrow+\infty}\int_{D_{m}}|F_{l,m,t_{0}}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)<+\infty.$

Using Lemma 2.4, we obtain that there exists a subsequence of
$\\{F_{l,m,t_{0}}\\}_{t_{0}\rightarrow+\infty}$ (also denoted by
$\\{F_{l,m,t_{0}}\\}_{t_{0}\rightarrow+\infty}$) compactly convergent to a
holomorphic (1,0) form on $D_{m}$ denoted by $F_{l,m}$. Then it follows from
inequality (6.10) and Fatou’s Lemma that

$\begin{split}\int_{D_{m}}|F_{l,m}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)=&\int_{D_{m}}\liminf_{t_{0}\rightarrow+\infty}|F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)\\\
\leq&\liminf_{t_{0}\rightarrow+\infty}\int_{D_{m}}|F_{l,m,t_{0}}-(1-b_{t_{0},1}(\psi))\tilde{F}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)\\\
\leq&\pi\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\Phi_{l}+\psi_{2}}dV_{\Omega}[\psi].\end{split}$
(6.11)

As
$\|f\|_{z_{0}}=\pi\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\varphi}dV_{\Omega}[\psi]<+\infty$
and $\Phi_{l}$ are decreasingly convergent to $\varphi+\psi_{2}$, we have

$\lim_{l\rightarrow+\infty}\pi\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\Phi_{l}+\psi_{2}}dV_{\Omega}[\psi]=\|f\|_{z_{0}}<+\infty.$
(6.12)

It follows from inequality (6.11) and (6.12) that

$\limsup_{l\rightarrow+\infty}\int_{D_{m}}|F_{l,m}|^{2}e^{-\Phi_{l}+\psi_{2}}c(-\psi)\leq\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}<+\infty.$
(6.13)

Using Lemma 2.4 ($g_{l}=e^{-\Phi_{l}+\psi_{2}}c(-\psi)$), we obtain that there
exists a subsequence of $\\{F_{l,m}\\}_{l\rightarrow+\infty}$ (also denoted by
$\\{F_{l,m}\\}_{l\rightarrow+\infty}$) compactly convergent to a holomorphic
(1,0) form on $D_{m}$ denoted by $F_{m}$ and

$\int_{D_{m}}|F_{m}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}.$
(6.14)

Inequality (6.14) implies that

$\int_{D_{m}}|F_{m^{\prime}}|^{2}e^{-\varphi}c(-\psi)\leq\pi\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}$

holds for any $m^{\prime}\geq m$. Note that $\varphi+\psi$ and $\psi$ are
subharmonic on $\Omega$ and $\varphi=(\varphi+\psi)-\psi$. Using Lemma 2.4,
the diagonal method and Levi’s Theorem, we obtain a subsequence of
$\\{F_{m}\\}$, denoted also by $\\{F_{m}\\}$, which is uniformly convergent to
a holomorphic $(1,0)$ form $F$ on $\Omega$ satisfying that $F(z_{0})=f(z_{0})$
and

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

Thus the existence of holomorphic extension satisfying inequality (1.6) holds.

In the following part, we prove the characterization for
$\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\|f\|_{z_{0}}=\inf\\{\|\tilde{F}\|_{\Omega}:\tilde{F}$
is a holomorphic extension of $f$ from $z_{0}$ to $\Omega$$\\}$.

Firstly, we prove the necessity. If $\|f\|_{z_{0}}=0$, then $F\equiv 0$, which
contradicts to $F(z_{0})=f(z_{0})\not=0$. Thus, we only consider the case
$\|f\|_{z_{0}}\in(0,+\infty)$.

As $\\{\psi<-t\\}$ is an open Riemann surface. Note that
$dV_{\Omega}[\psi+t]=e^{-t}dV_{\Omega}[\psi]$. By the above discussion
($\psi\sim\psi+t$, $c(\cdot)\sim c(\cdot+t)$ and $\Omega\sim\\{\psi<-t\\}$),
for any $t>0$, there exists a holomorphic $(n,0)$ form $F_{t}$ on
$\\{\psi<-t\\}$ such that $F_{t}(z_{0})=f(z_{0})$ and

$\int_{\\{\psi<-t\\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\leq\left(\int_{t}^{+\infty}c(l)e^{-l}dl\right)\|f\|_{z_{0}}.$

Let $\mathcal{F}|_{Z_{0}}=\mathcal{I}(\psi)_{z_{0}}$, by the definition of
$G(t)$, we obtain that inequality

$\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}\leq\frac{G(0)}{\int_{0}^{+\infty}c(t)e^{-t}dt}$
(6.15)

holds for any $t\geq 0$. Theorem 1.3 tells us $G(\hat{h}^{-1}(r))$ is concave
with respect to $r$. Combining inequality (6.15) and Corollary 1.5, we obtain
that $G(\hat{h}^{-1}(r))$ is linear with respect to $r$. As
$\psi-2G_{\Omega}(z,z_{0})$ is bounded near $z_{0}$ and
$G(0)=(\int_{0}^{+\infty}c(t)e^{-t}dt)\|f\|_{z_{0}}\in(0,+\infty)$, Theorem
1.17 shows that statements $(1)$, $(2)$ and $(3)$ hold.

Now, we prove the sufficiency. Let
$\mathcal{F}|_{Z_{0}}=\mathcal{I}(2G_{\Omega}(z,z_{0}))_{z_{0}}$, then Theorem
1.16 shows that $G(\hat{h}^{-1}(r))$ is linear with respect to $r$. It follows
from Lemma 2.10 and Corollary 1.7 that there exists
$\tilde{c}\in\mathcal{P}_{0}$ such that $\tilde{c}(t)$ is increasing on
$(a,+\infty)$ for some $a>0$, and $G(\hat{h}_{\tilde{c}}^{-1}(r);\tilde{c})$
is linear with respect to $r$, where
$\hat{h}_{\tilde{c}}(t)=\int_{t}^{+\infty}\tilde{c}(l)e^{-l}dl$. Using
Proposition 5.1, we have
$G(0;\tilde{c})=\|f\|_{z_{0}}(\int_{0}^{+\infty}\tilde{c}(l)e^{-l}dl)$.
Following from Corollary 1.7 and Remark 1.8, we obtain that
$G(0;c)=\|f\|_{z_{0}}(\int_{0}^{+\infty}c(l)e^{-l}dl)$, which implies that
$\|f\|_{z_{0}}(\int_{0}^{+\infty}c(l)e^{-l}dl)=\\{\|\tilde{F}\|_{\Omega}:\tilde{F}$
is a holomorphic extension of $f$ from $z_{0}$ to $\Omega$$\\}$.

Thus, Corollary 1.18 holds.

###  6.5. Proof of Corollary 1.19

Note that $2G_{\Omega}(z,z_{0})\in A^{\prime}(z_{0})$ and $\psi_{1}\in
A^{\prime}(z_{0})$, we have

$\|f\|^{*}_{z_{0}}=\pi\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\varphi-\psi_{2}}dV_{\Omega}[\psi_{1}]=\pi\int_{z_{0}}\frac{|f|^{2}}{dV_{\Omega}}e^{-\varphi-\psi+2G_{\Omega}(z,z_{0})}dV_{\Omega}[2G_{\Omega}(z,z_{0})].$
(6.16)

Corollary 1.18 implies the sufficiency. Thus, it suffices to prove the
necessity.

Let $\mathcal{F}|_{Z_{0}}=\mathcal{I}(\psi_{1})_{z_{0}}$. It follows from
Lemma 2.6 that there exists a unique holomorphic extension from $z_{0}$ to
$\Omega$, such that
$\|F\|_{\Omega}\leq\|f\|_{z_{0}}^{*}(\int_{0}^{+\infty}c(l)e^{-l}dl)$. Using
Corollary 1.12, we know that $G(\hat{h}^{-1}(r))$ is linear with respect to
$r$, therefore

$\frac{G(t)}{\int_{t}^{+\infty}c(l)e^{-l}dl}=\|f\|_{z_{0}}^{*}$ (6.17)

holds for any $t\geq 0$.

Let $\tilde{\psi}=2G_{\Omega}(z,z_{0})$. Lemma 2.11 tells us
$\psi-\tilde{\psi}\leq 0$ on $\Omega$. Let
$\tilde{\varphi}=\varphi+\psi-\tilde{\psi}$, then we compare
$G(t;\varphi,\psi)$ and $G(t;\tilde{\varphi},\tilde{\psi})$ to prove
$\psi-\tilde{\psi}\equiv 0$. As $\|f\|_{z_{0}}^{*}<+\infty$ and
$e^{-\tilde{\varphi}}c(-\tilde{\psi})=e^{-\varphi-\psi}e^{\tilde{\psi}}c(-\tilde{\psi})\geq
e^{-\varphi}c(-\psi)$, it follows from Corollary 1.11 and equality (6.16) that
$G(0;\tilde{\varphi},\tilde{\psi})\leq\|f\|^{*}_{z_{0}}\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)$.
Without loss of generality, we can assume that $c(t)e^{-t}$ is strictly
decreasing on $(0,+\infty)$. We prove $\psi-\tilde{\psi}\equiv 0$ by
contradiction: if not, $c(t)e^{-t}$ is strictly decreasing on $(0,+\infty)$
implies that $G(0;\tilde{\varphi},\tilde{\psi})>G(0;\varphi,\psi)$, which
contradicts to
$G(0;\tilde{\varphi},\tilde{\psi})\leq\|f\|^{*}_{z_{0}}\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)=G(0;\varphi,\psi)$.
Thus, we have $\psi=2G_{\Omega}(z,z_{0})$. Combining the linearity of
$G(\hat{h}(r);\varphi,\psi)$,
$G(0;\varphi,\psi)=\|f\|^{*}_{z_{0}}\left(\int_{0}^{+\infty}c(t)e^{-t}dt\right)\in(0,+\infty)$
and Theorem 1.16, we obtain that the other two statements in Corollary 1.19
hold.

Thus, Corollary 1.19 holds.

##  7\. Appendix

###  7.1. Proof of Lemma 2.1

In this section, we prove Lemma 2.1.

It follows from Lemma 7.4 that there exist smooth strongly plurisubharmonic
functions $\psi_{m}$ and $\varphi_{m}$ on $M$ decreasingly convergent to
$\psi$ and $\varphi$ respectively.

The following remark shows that it suffices to consider Lemma 2.1 for the case
that $M$ is a relatively compact open Stein submanifold of a Stein manifold,
and $F$ is a holomorphic $(n,0)$ form on $\\{\psi<-t_{0}\\}$ such that
$\int_{\\{\psi<-t_{0}\\}}|F|^{2}<+\infty$, which implies that
$\sup_{m}\sup_{M}\psi_{m}<-T$ and $\sup_{m}\sup_{M}\varphi_{m}<+\infty$ on
$M$.

######  Remark 7.1.

It is well-known that there exist open Stein submanifolds
$D_{1}\subset\subset\cdots\subset\subset D_{j}\subset\subset
D_{j+1}\subset\subset\cdots$ such that $\cup_{j=1}^{+\infty}D_{j}=M$.

If inequality (2.3) holds on any $D_{j}$ and inequality (2.2) holds on $M$,
then for any $B>0$, we obtain a sequence of holomorphic (n,0) forms
$\tilde{F}_{j}$ on $D_{j}$ such that

$\begin{split}&\int_{D_{j}}|\tilde{F}_{j}-(1-b_{t_{0},B}(\psi))F|^{2}e^{-\varphi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))\\\
\leq&\int_{T}^{t_{0}+B}c(t)e^{-t}dt\int_{D_{j}}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}\leq
C\int_{T}^{t_{0}+B}c(t)e^{-t}dt\end{split}$ (7.1)

is bounded with respect to $j$. Note that for any given $j$,
$e^{-\varphi+v_{t_{0},B}(\psi)}c(-v_{t_{0},B}(\psi))$ has a positive lower
bound, then it follows that for any any given $j$,
$\int_{D_{j}}|\tilde{F}_{j^{\prime}}-(1-b_{t_{0},B}(\psi))F|^{2}$ is bounded
with respect to $j^{\prime}\geq j$. Combining with

$\int_{D_{j}}|(1-b_{t_{0},B}(\psi))F|^{2}\leq\int_{D_{j}\cap\\{\psi<-t_{0}\\}}|F|^{2}<+\infty$
(7.2)

and inequality (2.3), one can obtain that
$\int_{D_{j}}|\tilde{F}_{j^{\prime}}|^{2}$ is bounded with respect to
$j^{\prime}\geq j$.

By the diagonal method, there exists a subsequence $F_{j^{\prime\prime}}$
uniformly convergent on any $D_{j}$ to a holomorphic $(n,0)$ form on $M$
denoted by $\tilde{F}$. Then it follows from inequality (7.1) and Fatou’s
Lemma that

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

then one can obtain Lemma 2.1 when $j$ goes to $+\infty$.

Next, we recall some lemmas on $L^{2}$ estimates for some $\bar{\partial}$
equations.

######  Lemma 7.2.

(see [5, 7]) Let $X$ be a complete Kähler manifold equipped with a (non
necessarily complete) Kähler metric $\omega$, and let $E$ be a Hermitian
vector bundle over $X$. Assume that there are smooth and bounded functions
$\eta$, $g>0$ on $X$ such that the (Hermitian) curvature operator

$\textbf{B}:=[\eta\sqrt{-1}\Theta_{E}-\sqrt{-1}\partial\bar{\partial}\eta-\sqrt{-1}g\partial\eta\wedge\bar{\partial}\eta,\Lambda_{\omega}]$

is positive definite everywhere on $\Lambda^{n,q}T^{*}_{X}\otimes E$, for some
$q\geq 1$. Then for every form $\lambda\in
L^{2}(X,\Lambda^{n,q}T^{*}_{X}\otimes E)$ such that
$D^{\prime\prime}\lambda=0$ and
$\int_{X}\langle\textbf{B}^{-1}\lambda,\lambda\rangle dV_{M}<\infty$, there
exists $u\in L^{2}(X,\Lambda^{n,q-1}T^{*}_{X}\otimes E)$ such that
$D^{\prime\prime}u=\lambda$ and

$\int_{X}(\eta+g^{-1})^{-1}|u|^{2}dV_{M}\leq\int_{X}\langle\textbf{B}^{-1}\lambda,\lambda\rangle
dV_{M}.$

######  Lemma 7.3.

(see [25])Let $X$ and $E$ be as in the above lemma and $\theta$ be a
continuous $(1,0)$ form on $X$. Then we have

$[\sqrt{-1}\theta\wedge\bar{\theta},\Lambda_{\omega}]\alpha=\bar{\theta}\wedge(\alpha\llcorner(\bar{\theta})^{\sharp}\big{)},$

for any $(n,1)$ form $\alpha$ with value in $E$. Moreover, for any positive
$(1,1)$ form $\beta$, we have $[\beta,\Lambda_{\omega}]$ is semipositive.

The following Lemma belongs to Fornaess and Narasimhan on approximation
property of plurisubharmonic functions of Stein manifolds.

######  Lemma 7.4.

[14] Let $X$ be a Stein manifold and $\varphi\in PSH(X)$. Then there exists a
sequence $\\{\varphi_{n}\\}_{n=1,2,\cdots}$ of smooth strongly
plurisubharmonic functions such that $\varphi_{n}\downarrow\varphi$.

For the sake of completeness, let’s recall some steps in the proof in [18]
(see also [23, 25, 27]) with some slight modifications in order to prove Lemma
2.1.

It follows from Remark 7.1 that it suffices to consider that $M$ is a Stein
manifold, and $F$ is holomorphic $(n,0)$ form on $\\{\psi<-t_{0}\\}$ and

$\int_{\\{\psi<-t_{0}\\}}|F|^{2}<+\infty,$ (7.3)

and there exist smooth plurisubharmonic functions $\psi_{m}$ and $\varphi_{m}$
on $M$ decreasingly convergent to $\psi$ and $\varphi$ respectively,
satisfying $\sup_{m}\sup_{M}\psi_{m}<-T$ and
$\sup_{m}\sup_{M}\varphi_{m}<+\infty$.

_Step 1: construct some functions_

Let $\varepsilon\in(0,\frac{1}{8}B)$. Let
$\\{v_{\varepsilon}\\}_{\varepsilon\in(0,\frac{1}{8}B)}$ be a family of smooth
increasing convex functions on $\mathbb{R}$, which are continuous functions on
$\mathbb{R}\cup\\{-\infty\\}$, such that:

$(1)$ $v_{\varepsilon}(t)=t$ for $t\geq-t_{0}-\varepsilon$,
$v_{\varepsilon}(t)=constant$ for $t<-t_{0}-B+\varepsilon$ and are pointwise
convergent to $v_{t_{0},B}(t)$;

$(2)$ $v^{\prime\prime}_{\varepsilon}(t)$ are pointwise convergent to
$\frac{1}{B}\mathbb{I}_{(-t_{0}-B,-t_{0})}$, when $\varepsilon\to 0$, and
$0\leq
v^{\prime\prime}_{\varepsilon}(t)\leq\frac{2}{B}\mathbb{I}_{(-t_{0}-B+\varepsilon,-t_{0}-\varepsilon)}$
for any $t\in\mathbb{R}$;

$(3)$ $v^{\prime}_{\varepsilon}(t)$ are pointwise convergent to
$b_{t_{0},B}(t)$ which is a continuous function on $\mathbb{R}$, when
$\varepsilon\to 0$, and $0\leq v^{\prime}_{\varepsilon}(t)\leq 1$ for any
$t\in\mathbb{R}$.

One can construct the family
$\\{v_{\varepsilon}\\}_{\varepsilon\in(0,\frac{1}{8}B)}$ by the setting

$\begin{split}v_{\varepsilon}(t):=&\int_{-\infty}^{t}\left(\int_{-\infty}^{t_{1}}\left(\frac{1}{B-4\varepsilon}\mathbb{I}_{(-t_{0}-B+2\varepsilon,-t_{0}-2\varepsilon)}*\rho_{\frac{1}{4}\varepsilon}\right)(s)ds\right)dt_{1}\\\
&-\int_{-\infty}^{-t_{0}}\left(\int_{-\infty}^{t_{1}}\left(\frac{1}{B-4\varepsilon}\mathbb{I}_{(-t_{0}-B+2\varepsilon,-t_{0}-2\varepsilon)}*\rho_{\frac{1}{4}\varepsilon}\right)(s)ds\right)dt_{1}-t_{0},\end{split}$
(7.4)

where $\rho_{\frac{1}{4}\varepsilon}$ is the kernel of convolution satisfying
$supp(\rho_{\frac{1}{4}\varepsilon})\subset(-\frac{1}{4}\varepsilon,\frac{1}{4}\varepsilon)$.
Then it follows that

$v^{\prime\prime}_{\varepsilon}(t)=\frac{1}{B-4\varepsilon}\mathbb{I}_{(-t_{0}-B+2\varepsilon,-t_{0}-2\varepsilon)}*\rho_{\frac{1}{4}\varepsilon}(t),$

and

$v^{\prime}_{\varepsilon}(t)=\int_{-\infty}^{t}\left(\frac{1}{B-4\varepsilon}\mathbb{I}_{(-t_{0}-B+2\varepsilon,-t_{0}-2\varepsilon)}*\rho_{\frac{1}{4}\varepsilon}\right)(s)ds.$

Let $\eta=s(-v_{\varepsilon}(\psi_{m}))$ and
$\phi=u(-v_{\varepsilon}(\psi_{m}))$, where $s\in C^{\infty}((S,+\infty))$
satisfies $s>0$ and $s^{\prime}>0$, and $u\in C^{\infty}((S,+\infty))$, such
that $u^{\prime\prime}s-s^{\prime\prime}>0$, and $s^{\prime}-u^{\prime}s=1$.
It follows from $\sup_{m}\sup_{M}\psi_{m}<-S$ and $\max{\\{t,-t_{0}-B\\}}\leq
v_{\epsilon}(t)\leq\max{\\{t,t_{0}\\}}$ that
$\phi=u(-v_{\varepsilon}(\psi_{m}))$ are uniformly bounded on $M$ with respect
to $m$ and $\varepsilon$, and $u(-v_{\varepsilon}(\psi))$ are uniformly
bounded on $M$ with respect to $\varepsilon$. Let
$\Phi=\phi+\varphi_{m^{\prime}}$, and let $\tilde{h}=e^{-\Phi}$.

Let $f(x)=2\mathbb{I}_{(-\frac{1}{2},\frac{1}{2})}*\rho(x)$ be a smooth
function on $\mathbb{R}$, where $\rho$ is is the kernel of convolution
satisfying $supp(\rho)\subset(-\frac{1}{3},\frac{1}{3})$ and $\rho\geq 0$.

Let $g_{l}(x)=\left\\{\begin{array}[]{lcl}lf(lx)&\mbox{if}&x\leq 0\\\
lf(l^{2}x)&\mbox{if}&x>0\end{array}\right.$, then
$\\{g_{l}\\}_{l\in\mathbb{N}^{+}}$ be a family of smooth functions on
$\mathbb{R}$ satisfying that:

$(1)$ $supp(g_{l})\subset[-\frac{1}{l},\frac{1}{l}],$ $g_{l}(x)\geq 0$ for any
$x\in\mathbb{R}$;

$(2)$ $\int_{-\frac{1}{l}}^{0}g_{l}(x)dx=1$,
$\int_{0}^{\frac{1}{l}}g_{l}(x)dx\leq\frac{1}{l}$ for any
$l\in\mathbb{N}^{+}$.

Set $c_{l}(t)=e^{t}\int_{\mathbb{R}}h(e^{y}(t-S)+S)g_{l}(y)dy$, where
$h(t)=e^{-t}c(t)$ and $c\in\tilde{\mathcal{P}}_{S}$. It is easy to get

$c_{l}(t)-c(t)\geq
e^{t}\int_{-\frac{1}{l}}^{0}(h(e^{y}(t-S)+S)-h(t))g_{l}(y)dy\geq 0.$

Set $\tilde{h}(t)=h(e^{t}+S)$ and $\tilde{g}_{l}(t)=g_{l}(-t)$, then
$c_{l}(t)=e^{t}\tilde{h}*\tilde{g}_{l}(\ln(t-S))\in C^{\infty}(S,+\infty)$.
Because $h(t)$ is decreasing with respect to $t$, so is $c_{l}(t)e^{-t}$. And

$\begin{split}\int_{S}^{s}c_{l}(t)e^{-t}dt&=\int_{S}^{s}\int_{\mathbb{R}}h(e^{y}(t-S)+S)g_{l}(y)dydt\\\
&=\int_{\mathbb{R}}e^{-y}g_{l}(y)\int_{S}^{e^{y}(s-S)+S}h(t)dtdy\\\
&\leq\int_{\mathbb{R}}e^{-y}g_{l}(y)dy\int_{S}^{e(s-S)+S}h(t)dt\\\
&<+\infty,\end{split}$

then $c_{l}(t)\in\tilde{\mathcal{P}}_{S}$ for any $l\in\mathbb{N}^{+}$.

As $h(t)$ is decreasing with respect to $t$, then set
$h^{-}(t)=\lim_{s\rightarrow t-0}h(s)\geq h(t)$ and
$c^{-}(t)=\lim_{s\rightarrow t-0}c(s)\geq c(t)$, then we claim that
$\lim_{l\rightarrow+\infty}c_{l}(t)=c^{-}(t)$. In fact, we have

$\begin{split}|c_{l}(t)-c^{-}(t)|\leq&e^{t}\int_{-\frac{1}{l}}^{0}|h(e^{y}(t-S)+S)-h^{-}(t)|g_{l}(y)dy\\\
&+e^{t}\int_{0}^{\frac{1}{l}}h(e^{y}(t-S)+S)g_{l}(y)dy.\end{split}$ (7.5)

$\forall\varepsilon>0$, $\exists\delta>0$ and
$|h(t-\delta)-h^{-}(t)|<\varepsilon$. Then $\exists N>0$, $\forall l>N$, such
that $e^{y}(t-S)+S>t-\delta$ for all $y\in[-\frac{1}{l},0)$ and
$\frac{1}{l}<\varepsilon$. It following from (7.5) that

$|c_{l}(t)-c^{-}(t)|\leq\varepsilon e^{t}+\varepsilon h(t)e^{t},$

hence $\lim_{l\rightarrow+\infty}c_{l}(t)=c^{-}(t)$ for any $t>S$.

_Step 2: Solving $\bar{\partial}-$equation with smooth polar function and
smooth weight_

Now let $\alpha\in\Lambda_{x}^{n,1}T_{M}^{*}$, for any $x\in M$. Using
inequality $s>0$ and the fact that $\varphi_{m}$ is plurisubharmonic on $M$,
we get

$\begin{split}\langle\textbf{B}\alpha,\alpha\rangle_{\tilde{h}}=&\langle[\eta\sqrt{-1}\Theta_{\tilde{h}}-\sqrt{-1}\partial\bar{\partial}\eta-\sqrt{-1}g\partial\eta\wedge\bar{\partial}\eta,\Lambda_{\omega}]\alpha,\alpha\rangle_{\tilde{h}}\\\
\geq&\langle[\eta\sqrt{-1}\partial\bar{\partial}\phi-\sqrt{-1}\partial\bar{\partial}\eta-\sqrt{-1}g\partial\eta\wedge\bar{\partial}\eta,\Lambda_{\omega}]\alpha,\alpha\rangle_{\tilde{h}}.\end{split}$
(7.6)

where $g>0$ is a smooth and bounded function on $M$. We need the following
calculations to determine $g$.

$\begin{split}&\partial\bar{\partial}\eta=-s^{\prime}(-v_{\varepsilon}(\psi_{m}))\partial\bar{\partial}(v_{\varepsilon}(\psi_{m}))+s^{\prime\prime}(-v_{\varepsilon}(\psi_{m}))\partial
v_{\varepsilon}(\psi_{m})\wedge\bar{\partial}v_{\varepsilon}(\psi_{m}),\end{split}$
(7.7)

and

$\begin{split}&\partial\bar{\partial}\phi=-u^{\prime}(-v_{\varepsilon}(\psi_{m}))\partial\bar{\partial}v_{\varepsilon}(\psi_{m})+u^{\prime\prime}(-v_{\varepsilon}(\psi_{m}))\partial
v_{\varepsilon}(\psi_{m})\wedge\bar{\partial}v_{\varepsilon}(\psi_{m}).\end{split}$
(7.8)

Then we have

$\begin{split}&-\partial\bar{\partial}\eta+\eta\partial\bar{\partial}\phi-g(\partial\eta)\wedge\bar{\partial}\eta\\\
=&(s^{\prime}-su^{\prime})\partial\bar{\partial}v_{\varepsilon}(\psi_{m})+((u^{\prime\prime}s-s^{\prime\prime})-gs^{\prime
2})\partial(-v_{\varepsilon}(\psi_{m}))\bar{\partial}(-v_{\varepsilon}(\psi_{m}))\\\
=&(s^{\prime}-su^{\prime})(v^{\prime}_{\varepsilon}(\psi_{m})\partial\bar{\partial}\psi_{m}+v^{\prime\prime}_{\varepsilon}(\psi_{m})\partial(\psi_{m})\wedge\bar{\partial}(\psi_{m}))\\\
&+((u^{\prime\prime}s-s^{\prime\prime})-gs^{\prime
2})\partial(-v_{\varepsilon}(\psi_{m}))\wedge\bar{\partial}(-v_{\varepsilon}(\psi_{m})).\end{split}$
(7.9)

We omit composite item $-v_{\varepsilon}(\psi_{m})$ after
$s^{\prime}-su^{\prime}$ and $(u^{\prime\prime}s-s^{\prime\prime})-gs^{\prime
2}$ in the above equalities. Let
$g=\frac{u^{\prime\prime}s-s^{\prime\prime}}{s^{\prime
2}}(-v_{\varepsilon}(\psi_{m}))$. It follows that
$\eta+g^{-1}=(s+\frac{s^{\prime
2}}{u^{\prime\prime}s-s^{\prime\prime}})(-v_{\varepsilon}(\psi_{m}))$.

As $v^{\prime}_{\varepsilon}\geq 0$ and $s^{\prime}-su^{\prime}=1$, using
Lemma 7.3, equality (7.9) and inequality (7.6), we obtain

$\begin{split}\langle\textbf{B}\alpha,\alpha\rangle_{\tilde{h}}=&\langle[\eta\sqrt{-1}\Theta_{\tilde{h}}-\sqrt{-1}\partial\bar{\partial}\eta-\sqrt{-1}g\partial\eta\wedge\bar{\partial}\eta,\Lambda_{\omega}]\alpha,\alpha\rangle_{\tilde{h}}\\\
\geq&\langle[(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\sqrt{-1}\partial\psi_{m}\wedge\bar{\partial}\psi_{m},\Lambda_{\omega}]\alpha,\alpha\rangle_{\tilde{h}}\\\
=&\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\bar{\partial}\psi_{m}\wedge(\alpha\llcorner(\bar{\partial}\psi_{m})^{\sharp}\big{)},\alpha\rangle_{\tilde{h}}.\end{split}$
(7.10)

Using the definition of contraction, Cauchy-Schwarz inequality and the
inequality (7.10), we have

$\begin{split}|\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\bar{\partial}\psi_{m}\wedge\gamma,\tilde{\alpha}\rangle_{\tilde{h}}|^{2}=&|\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\gamma,\tilde{\alpha}\llcorner(\bar{\partial}\psi_{m})^{\sharp}\big{\rangle}_{\tilde{h}}|^{2}\\\
\leq&\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\gamma,\gamma\rangle_{\tilde{h}}(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})|\tilde{\alpha}\llcorner(\bar{\partial}\psi_{m})^{\sharp}\big{|}_{\tilde{h}}^{2}\\\
=&\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\gamma,\gamma\rangle_{\tilde{h}}\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\bar{\partial}\psi_{m}\wedge(\tilde{\alpha}\llcorner(\bar{\partial}\psi_{m})^{\sharp}\big{)},\alpha\rangle_{\tilde{h}}\\\
\leq&\langle(v^{\prime\prime}_{\varepsilon}\circ\psi_{m})\gamma,\gamma\rangle_{\tilde{h}}\langle\textbf{B}\tilde{\alpha},\tilde{\alpha}\rangle_{\tilde{h}},\end{split}$
(7.11)

for any $(n,0)$ form $\gamma$.

It follows from $s>0$ and $\varphi_{m^{\prime}}$ is strongly plurisubharmonic
that B is positive definite everywhere on $\Lambda^{n,1}T_{M}^{*}$. As $F$ is
holomorphic on $\\{\psi<-t_{0}\\}$ and
$Supp(v^{\prime\prime}_{\varepsilon}(\psi_{m}))\subset\\{\psi<-t_{0}\\}$, then
$\lambda:=\bar{\partial}[(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}]$ is well-
defined and smooth on $M$.

Taking $\gamma=F$, and $\tilde{\alpha}=\textbf{B}^{-1}\lambda$, note that
$\tilde{h}=e^{-\Phi}$, using inequality (7.11), we have

$\langle\textbf{B}^{-1}\lambda,\lambda\rangle_{\tilde{h}}\leq
v^{\prime\prime}_{t_{0},\varepsilon}(\psi_{m})|\tilde{F}|^{2}e^{-\Phi}.$

Then it follows that

$\int_{M}\langle\textbf{B}^{-1}\lambda,\lambda\rangle_{\tilde{h}}\leq\int_{M}v^{\prime\prime}_{t_{0},\varepsilon}(\psi_{m})|\tilde{F}|^{2}e^{-\Phi}.$

Assume that we can choose $\eta$ and $\phi$ such that
$e^{v_{\varepsilon}\circ\psi_{m}}e^{\phi}c_{l}(-v_{\varepsilon}\circ\psi_{m})=(\eta+g^{-1})^{-1}$.
Using Lemma 7.2, we have locally $L^{1}$ function
$u_{m,m^{\prime},\varepsilon,l}$ on $M$ such that
$\bar{\partial}u_{m,m^{\prime},\varepsilon,l}=\lambda$, and

$\begin{split}&\int_{M}|u_{m,m^{\prime},\varepsilon,l}|^{2}e^{v_{\varepsilon}(\psi_{m})-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi_{m})\\\
=&\int_{M}|u_{m,m^{\prime},\varepsilon,l}|^{2}(\eta+g^{-1})^{-1}e^{-\Phi}\\\
\leq&\int_{M}\langle\textbf{B}^{-1}\lambda,\lambda\rangle_{\tilde{h}}\\\
\leq&\int_{M}v^{\prime\prime}_{\varepsilon}(\psi_{m})|F|^{2}e^{-\Phi}\\\
=&\int_{M}v^{\prime\prime}_{\varepsilon}(\psi_{m})|F|^{2}e^{-\phi-\varphi_{m^{\prime}}}.\end{split}$
(7.12)

Let
$F_{m,m^{\prime},\varepsilon,l}:=-u_{m,m^{\prime},\varepsilon,l}+(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}$.
Then inequality (7.12) becomes

$\begin{split}&\int_{M}|F_{m,m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}|^{2}e^{v_{\varepsilon}(\psi_{m})-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi_{m})\\\
&\leq\int_{M}(v^{\prime\prime}_{\varepsilon}(\psi_{m}))|F|^{2}e^{-\phi-\varphi_{m^{\prime}}}.\end{split}$
(7.13)

_Step 3: Singular polar function and smooth weight_

As
$\sup_{m,\varepsilon}\sup_{M}|\phi|=\sup_{m,\varepsilon}|u(-v_{\varepsilon}(\psi_{m}))|<+\infty$
and $\sup_{M}\varphi_{m^{\prime}}<+\infty$, note that

$v^{\prime\prime}_{\varepsilon}(\psi_{m})|F|^{2}e^{-\phi-\varphi_{m^{\prime}}}\leq\frac{2}{B}\mathbb{I}_{\\{\psi<-t_{0}\\}}|F|^{2}\sup_{m,\varepsilon}e^{-\phi-\varphi_{m^{\prime}}}$

on $M$, then it follows from inequality (7.3) and the dominated convergence
theorem that

$\lim_{m\to+\infty}\int_{M}v^{\prime\prime}_{\varepsilon}(\psi_{m})|F|^{2}e^{-\phi-\varphi_{m^{\prime}}}=\int_{M}v^{\prime\prime}_{\varepsilon}(\psi)|F|^{2}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}}$
(7.14)

Note that
$\inf_{m}\inf_{M}e^{v_{\varepsilon}(\psi_{m})-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi_{m})>0$,
then it follows from inequality (7.13) and (7.14) that
$\sup_{m}\int_{M}|F_{m,m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}|^{2}<+\infty$.
Note that

$|(1-v^{\prime}_{\varepsilon}(\psi_{m}))F|\leq|\mathbb{I}_{\\{\psi<-t_{0}\\}}F|,$
(7.15)

then it follows from inequality (7.3) that
$\sup_{m}\int_{M}|F_{m,m^{\prime},\varepsilon,l}|^{2}<+\infty$, which implies
that there exists a subsequence of $\\{F_{m,m^{\prime},\varepsilon,l}\\}_{m}$
(also denoted by $\\{F_{m,m^{\prime},\varepsilon,l}\\}_{m}$) compactly
convergent to a holomorphic $F_{m^{\prime},\varepsilon,l}$ on $M$.

Note that
$e^{v_{\varepsilon}(\psi_{m})-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi_{m})$
are uniformly bounded on $M$ with respect to $m$, then it follows from
$|F_{m,m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}|^{2}\leq
2(|F_{m,m^{\prime},\varepsilon,l}|^{2}+|(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}|^{2})\leq
2(|F_{m,m^{\prime},\varepsilon,l}|^{2}+|\mathbb{I}_{\\{\psi<-t_{0}\\}}F^{2}|)$
and the dominated convergence theorem that

$\begin{split}\lim_{m\to+\infty}&\int_{K}|F_{m,m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi_{m})){F}|^{2}e^{v_{\varepsilon}(\psi_{m})-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi_{m})\\\
=&\int_{K}|F_{m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}e^{v_{\varepsilon}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi)\end{split}$
(7.16)

holds for any compact subset $K$ on $M$. Combining with inequality (7.13) and
(7.14), one can obtain that

$\begin{split}&\int_{K}|F_{m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}e^{v_{\varepsilon}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi)\\\
&\leq\int_{M}v^{\prime\prime}_{\varepsilon}(\psi)|F|^{2}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}},\end{split}$
(7.17)

which implies

$\begin{split}&\int_{M}|F_{m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}e^{v_{\varepsilon}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi)\\\
&\leq\int_{M}v^{\prime\prime}_{\varepsilon}(\psi)|F|^{2}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}}.\end{split}$
(7.18)

_Step 4: Nonsmooth cut-off function_

Note that
$\sup_{\varepsilon}\sup_{M}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}}<+\infty,$
and

$v^{\prime\prime}_{\varepsilon}(\psi)|F|^{2}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}}\leq\frac{2}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}\sup_{\varepsilon}\sup_{M}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}},$

then it follows from inequality (7.3) and the dominated convergence theorem
that

$\begin{split}&\lim_{\varepsilon\to
0}\int_{M}v^{\prime\prime}_{\varepsilon}(\psi)|F|^{2}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}}\\\
=&\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-u(-v_{t_{0},B}(\psi))-\varphi_{m^{\prime}}}\\\
\leq&(\sup_{M}e^{-u(-v_{t_{0},B}(\psi))})\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi_{m^{\prime}}}<+\infty.\end{split}$
(7.19)

Note that
$\inf_{\varepsilon}\inf_{M}e^{v_{\varepsilon}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi)>0$,
then it follows from inequality (7.18) and (7.19) that
$\sup_{\varepsilon}\int_{M}|F_{m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}<+\infty.$
Combining with

$\sup_{\varepsilon}\int_{M}|(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}\leq\int_{M}\mathbb{I}_{\\{\psi<-t_{0}\\}}|F|^{2}<+\infty,$
(7.20)

one can obtain that
$\sup_{\varepsilon}\int_{M}|F_{m^{\prime},\varepsilon,l}|^{2}<+\infty$, which
implies that there exists a subsequence of
$\\{F_{m^{\prime},\varepsilon,l}\\}_{\varepsilon\to 0}$ (also denoted by
$\\{F_{m^{\prime},\varepsilon,l}\\}_{\varepsilon\to 0}$) compactly convergent
to a holomorphic (n,0) form on $M$ denoted by $F_{m^{\prime},l}$. Then it
follows from inequality (7.18), inequality (7.19) and Fatou’s Lemma that

$\begin{split}&\int_{M}|F_{m^{\prime},l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v\circ\psi)\\\
=&\int_{M}\liminf_{\varepsilon\rightarrow
0}|F_{m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}e^{v_{\varepsilon}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi)\\\
\leq&\liminf_{\varepsilon\rightarrow
0}\int_{M}|F_{m^{\prime},\varepsilon,l}-(1-v^{\prime}_{\varepsilon}(\psi)){F}|^{2}e^{v_{\varepsilon}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{\varepsilon}\circ\psi)\\\
\leq&\liminf_{\varepsilon\rightarrow
0}\int_{M}v^{\prime\prime}_{\varepsilon}(\psi)|F|^{2}e^{-u(-v_{\varepsilon}(\psi))-\varphi_{m^{\prime}}}\\\
\leq&(\sup_{M}e^{-u(-v_{t_{0},B}(\psi))})\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi_{m^{\prime}}}.\end{split}$
(7.21)

_Step 5: Singular weight_

Note that

$\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi_{m^{\prime}}}\leq\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}<+\infty,$
(7.22)

and $\sup_{M}e^{-u(-v_{t_{0},B}(\psi))}<+\infty$, then it from (7.21) that

$\sup_{m^{\prime}}\int_{M}|F_{m^{\prime},l}-(1-b(\psi)){F}|^{2}e^{v(\psi)-\varphi_{m^{\prime}}}c_{l}(-v\circ\psi)<+\infty.$

Combining with
$\inf_{m^{\prime}}\inf_{M}e^{v(\psi)-\varphi_{m^{\prime}}}c_{l}(-v(\psi))>0$,
one can obtain that

$\sup_{m^{\prime}}\int_{M}|F_{m^{\prime},l}-(1-b(\psi)){F}|^{2}<+\infty.$

Note that

$\int_{M}|(1-b(\psi)){F}|^{2}\leq\int_{M}|\mathbb{I}_{\\{\psi<-t_{0}\\}}F|^{2}<+\infty.$
(7.23)

Then $\sup_{m^{\prime}}\int_{M}|F_{m^{\prime},l}|^{2}<+\infty$, which implies
that there exists a compactly convergent subsequence of
$\\{F_{m^{\prime},l}\\}_{m^{\prime}}$ (also denoted by
$\\{F_{m^{\prime},l}\\}_{m^{\prime}}$), which converges to a holomorphic (n,0)
form $\tilde{F_{l}}$ on $M$. Then it follows from inequality (7.21),
inequality (7.22) and Fatou’s Lemma that

$\begin{split}&\int_{M}|F_{l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi}c_{l}(-v_{t_{0},B}\circ\psi)\\\
=&\int_{M}\liminf_{m^{\prime}\rightarrow+\infty}|F_{m^{\prime},l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{t_{0},B}\circ\psi)\\\
\leq&\liminf_{m^{\prime}\rightarrow+\infty}\int_{M}|F_{m^{\prime},l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi_{m^{\prime}}}c_{l}(-v_{t_{0},B}\circ\psi)\\\
\leq&\liminf_{m^{\prime}\rightarrow+\infty}(\sup_{M}e^{-u(-v_{t_{0},B}(\psi))})\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi_{m^{\prime}}}\\\
\leq&(\sup_{M}e^{-u(-v_{t_{0},B}(\psi))})\int_{M}\frac{1}{B}\mathbb{I}_{\\{-t_{0}-B<\psi<-t_{0}\\}}|F|^{2}e^{-\varphi}.\end{split}$
(7.24)

_Step 6: ODE system_

we need to find $\eta$ and $\phi$ such that
$(\eta+g^{-1})=e^{-\psi_{m}}e^{-\phi}\frac{1}{c_{l}(-v_{\varepsilon}(\psi_{m}))}$
on $M$ and $s^{\prime}-u^{\prime}s=1$. As $\eta=s(-v_{\varepsilon}(\psi_{m}))$
and $\phi=u(-v_{\varepsilon}(\psi_{m}))$, we have
$(\eta+g^{-1})e^{v_{\varepsilon}(\psi_{m})}e^{\phi}=(s+\frac{s^{\prime
2}}{u^{\prime\prime}s-s^{\prime\prime}})e^{-t}e^{u}\circ(-v_{\varepsilon}(\psi_{m}))$.

Summarizing the above discussion about $s$ and $u$, we are naturally led to a
system of ODEs (see [22, 23, 25, 27]):

$\begin{split}&1).\,\,\left(s+\frac{s^{\prime
2}}{u^{\prime\prime}s-s^{\prime\prime}}\right)e^{u-t}=\frac{1}{c_{l}(t)},\\\
&2).\,\,s^{\prime}-su^{\prime}=1,\end{split}$ (7.25)

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

It is not hard to solve the ODE system (7.25) and get
$u(t)=-\log(\int_{S}^{t}c_{l}(t_{1})e^{-t_{1}}dt_{1})$ and
$s(t)=\frac{\int_{S}^{t}(\int_{S}^{t_{2}}c_{l}(t_{1})e^{-t_{1}}dt_{1})dt_{2}}{\int_{S}^{t}c_{l}(t_{1})e^{-t_{1}}dt_{1}}$
(see [25]). It follows that $s\in C^{\infty}((S,+\infty))$ satisfies $s>0$ and
$s^{\prime}>0$, $u\in C^{\infty}((S,+\infty))$ satisfies
$u^{\prime\prime}s-s^{\prime\prime}>0$.

As $u(t)=-\log(\int_{S}^{t}c_{l}(t_{1})e^{-t_{1}}dt_{1})$ is decreasing with
respect to $t$, then it follows from $-S\geq
v(t)\geq\max\\{t,-t_{0}-B_{0}\\}\geq-t_{0}-B_{0}$ for any $t\leq 0$ that

$\begin{split}\sup_{M}e^{-u(-v(\psi))}\leq\sup_{t\in(S,t_{0}+B]}e^{-u(t)}=\int_{S}^{t_{0}+B}c_{l}(t_{1})e^{-t_{1}}dt_{1},\end{split}$
(7.26)

then it follows from inequality (2.2) and inequality (7.24) that

$\int_{M}|F_{l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi}c_{l}(-v_{t_{0},B}(\psi))\leq
C\int_{S}^{t_{0}+B}c_{l}(t_{1})e^{-t_{1}}dt_{1}.$ (7.27)

_Step 7: non-smooth function $c$_

By the construction of $c_{l}$ in Step $1$, we have

$\begin{split}&\int_{S}^{t_{0}+B}c_{l}(t_{1})e^{-t_{1}}dt_{1}\\\
=&\int_{S}^{t_{0}+B}\int_{\mathbb{R}}h((t_{1}-S)e^{y}+S)g_{l}(y)dydt_{1}\\\
=&\int_{\mathbb{R}}e^{-y}g_{l}(y)\int_{S}^{(t_{0}+B-S)e^{y}+S}h(s)dsdy\\\
=&\int_{\mathbb{R}}e^{-y}g_{l}(y)dy\int_{S}^{t_{0}+B}h(s)ds+\int_{\mathbb{R}}e^{-y}g_{l}(y)\int_{t_{0}+B}^{(t_{0}+B-S)e^{y}+S}h(s)dsdy.\end{split}$
(7.28)

As

$\begin{split}&\lim_{l\rightarrow+\infty}\left|\int_{\mathbb{R}}e^{-y}g_{l}(y)dy-1\right|\\\
\leq&\lim_{l\rightarrow+\infty}\left|\int_{-\frac{1}{l}}^{0}(e^{-y}-1)g_{l}(y)dy\right|+\lim_{l\rightarrow+\infty}\left|\int_{0}^{\frac{1}{l}}e^{-y}g_{l}(y)dy\right|\\\
=&0\end{split}$

and

$\begin{split}&\left|\int_{\mathbb{R}}e^{-y}g_{l}(y)\int_{t_{0}+B}^{(t_{0}+B-S)e^{y}+S}h(s)dsdy\right|\\\
\leq&e^{\frac{1}{l}}\left(1+\frac{1}{l}\right)h((t_{0}+B-S)e^{-1}+S)(t_{0}+B-S)(e^{\frac{1}{l}}-e^{-\frac{1}{l}}),\end{split}$

then it follows from inequality (7.28) that

$\lim_{l\rightarrow+\infty}\int_{S}^{t_{0}+B}c_{l}(t_{1})e^{-t_{1}}dt_{1}=\int_{S}^{t_{0}+B}c(t_{1})e^{-t_{1}}dt_{1}.$
(7.29)

Combining with
$\inf_{l}\inf_{M}e^{v_{t_{0},B}(\psi)-\varphi}c_{l}(-v(\psi))\geq\inf_{M}e^{v_{t_{0},B}(\psi)-\varphi}c(-v(\psi))>0$,
we obtain that

$\sup_{l}\int_{M}|F_{l}-(1-b_{t_{0},B}(\psi)){F}|^{2}<+\infty.$

Note that

$\int_{M}|(1-b_{t_{0},B}(\psi)){F}|^{2}\leq\int_{M}|\mathbb{I}_{\\{\psi<-t_{0}\\}}F|^{2}<+\infty,$
(7.30)

then $\sup_{l}\int_{M}|F_{l}|^{2}<+\infty$, which implies that there exists a
compactly convergent subsequence of $\\{F_{l}\\}$ (also denoted by
$\\{F_{l}\\}$), which converges to a holomorphic (n,0) form $\tilde{F}$ on
$M$. Then it follows from inequality (7.27) and the Fatou’s Lemma that

$\begin{split}&\int_{M}|\tilde{F}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi}c(-v_{t_{0},B}(\psi))\\\
\leq&\int_{M}|\tilde{F}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi}c^{-}(-v_{t_{0},B}(\psi))\\\
=&\int_{M}\liminf_{l\rightarrow+\infty}|F_{l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi}c_{l}(-v_{t_{0},B}(\psi))\\\
\leq&\liminf_{l\rightarrow+\infty}\int_{M}|F_{l}-(1-b_{t_{0},B}(\psi)){F}|^{2}e^{v_{t_{0},B}(\psi)-\varphi}c_{l}(-v_{t_{0},B}(\psi))\\\
\leq&C\liminf_{l\rightarrow+\infty}\int_{S}^{t_{0}+B}c_{l}(t_{1})e^{-t_{1}}dt_{1}\\\
=&C\int_{S}^{t_{0}+B}c(t_{1})e^{-t_{1}}dt_{1}.\end{split}$

Thus we prove Lemma 2.1.

###  7.2. Proof of Lemma 2.14

The proof is from [20] with a few minor modifications.

Choose $p\in suppT\cap U$. By Lemma 2.12, there exist a real number $t>0$ and
a coordinate $(V,w)$, such that $w(p)=0$, $w(V)\cong B(0,1)$ and
$V\subset\subset\\{G_{\Omega}(z,p)<-t\\}\subset\subset U$. There exists a cut-
off function $\theta$ on $\Omega$, such that $\theta\equiv 1$ on
$w^{-1}(B(0,\frac{1}{4}))$ and $supp\theta\subset\subset
w^{-1}(B(0,\frac{1}{2}))$.

Let $\tilde{T}=\theta T$, then $\tilde{T}$ is a closed positive $(1,1)$
current on $\Omega$ with $supp\tilde{T}\subset\subset
w^{-1}(B(0,\frac{1}{2}))$ and $\tilde{T}\not\equiv 0$. Now, we prove that
exists a subharmonic function $\Phi<0$ on $\Omega$, which satisfies the
following properties: $i\partial\bar{\partial}\Phi=\tilde{T}$;
$\lim_{t\rightarrow 0+0}(\inf_{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\Phi(z))=0$;
$\inf_{\Omega\backslash U}\Phi>-\infty$. Then $\Phi$ satisfies the
requirements in Lemma 2.14.

_Step 1: Construct $\Phi$._

Let $\rho\in C^{\infty}(\mathbb{C})$ be a function with $supp\rho\subset
B(0,\frac{1}{2})$ and $\rho(z)$ depends only on $|z|$, $\rho\geq 0$ and
$\int_{\mathbb{C}}\rho(z)d\lambda_{z}=1$. Let $\rho_{n}(z)=n\rho(nz)$,
$\rho_{n}$ is a family of smoothing kernels.

As $w(V)\cong B(0,1)$, without misunderstanding we see $(V,z_{1})$ and
$(B(0,1),w)$ the same. As $supp\tilde{T}\subset\subset
w^{-1}(B(0,\frac{1}{2}))$ and $supp\rho\subset B(0,\frac{1}{2})$, denote that
$T_{n}=\tilde{T}\ast\rho_{n}$ be the convolution of $\tilde{T}$. In fact, for
any test function $h\in C_{c}^{\infty}(\Omega)$, $((h\circ
w^{-1})\ast\rho_{n})(w)$ ($h\ast\rho_{n}(w)$ for short) is well defined on
$w^{-1}(B(0,\frac{1}{2}))$, and $\langle
T_{n}(z_{1}),h(z_{1})\rangle=<\tilde{T}(w),h\ast\rho_{n}(w)>$. Then $T_{n}$ is
a smooth closed positive $(1,1)$ current on $\Omega$ with
$suppT_{n}\subset\subset w^{-1}(B(0,\frac{1}{2}+\frac{1}{2n}))$.

Let $u_{n}(z)=\langle T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle$.
$G_{\Omega}(z,z_{1})$ is locally integrable with respect $z_{1}\in\Omega$ for
any fixed $z\in\Omega$ implies that $u_{n}(z)>-\infty$ for any $z\in\Omega$.
For fixed $z$ and fixed $n$, we will prove $\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle=\langle\tilde{T}(w),(\frac{1}{\pi}G_{\Omega}(z,\cdot)\ast\rho_{n})(w)\rangle$.
For fixed $z$, $G_{\Omega}(z,z_{1})$ is a subharmonic function on $\Omega$.
There exists a sequence of smooth subharmonic functions $G_{m}(z_{1})$
decreasingly converge to $G_{\Omega}(z,z_{1})$ with respect to $m$. As
$G_{m}(z_{1})$ is smooth, we have

$\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{m}(z_{1})\rangle=<\tilde{T}(w),\frac{1}{\pi}G_{m}\ast\rho_{n}(w)>.$
(7.31)

As $\tilde{T}$ and $T_{n}$ are closed positive $(1,1)$ current on $\Omega$
with $suppT_{n}\subset\subset V$ and $supp\tilde{T}\subset\subset V$, and
$G_{m}(z_{1})$ decreasingly converge to $G_{\Omega}(z,z_{1})<0$ with respect
to $m$ on $\Omega$, it follows from Levi’s Theorem and equality (7.31) that

$\begin{split}\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle=&\lim_{m\to+\infty}\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{m}(z_{!})\rangle\\\
=&\lim_{m\to+\infty}<\tilde{T}(w),\frac{1}{\pi}G_{m}\ast\rho_{n}(w)>\\\
=&<\tilde{T}(w),\frac{1}{\pi}G_{\Omega}(z,\cdot)\ast\rho_{n}(w)>.\end{split}$

Fixed $z\in\Omega$, as $\frac{1}{\pi}G_{\Omega}(z,z_{1})$ is subharmonic, then
$\frac{1}{\pi}G_{\Omega}(z,\cdot)\ast\rho_{n}$ is decreasingly convergent to
$\frac{1}{\pi}G_{\Omega}(z,z_{1})$ with respect to $n$. Note that $\tilde{T}$
is a positive $(1,1)$ current on $\Omega$, then $u_{n}(z)$ is decreasing with
respect to $n$. Let $\Phi(z)=\lim_{n\to+\infty}u_{n}(z)$.
$G_{\Omega}(z,z_{1})<0$ on $\Omega\times\Omega$ shows that $u_{n}(z)<0$ and
$\Phi(z)<0$ on $\Omega$.

_Step 2: $i\partial\bar{\partial}\Phi=\tilde{T}$._

Firstly, we show that both $\\{u_{n}\\}$ and $\Phi$ is $L^{1}_{loc}$ function
on $\Omega$. As $u_{n}\leq 0$ on $\Omega$ and $u_{n}$ is decreasingly
convergent to $\Phi$ with respect to $n$ on $\Omega$, it suffices to prove
that for any $q\in\Omega$ there exists an open subset $K\subset\subset\Omega$,
such that $q\in K$ and $\int_{K}|u_{n}|dV_{\Omega}\leq C$, where $dV_{\Omega}$
is some continuous volume form and $C$ is a constant which independent of $n$.

It is clear that there exists a compact subset $D$ of $V$ such that
$supp\tilde{T}\subset D$ and $suppT_{n}\subset D$ for any $n$. When $q\not\in
V$, where exists a coordinate $w_{1}$ on a neighborhood $V^{\prime}$ of $q$,
such that $w_{1}(q)=0$, $V^{\prime}\subset\subset\Omega$,
$w_{1}(V^{\prime})\cong B(3,1)$ and $\overline{V^{\prime}}\cap D=\emptyset$.
Note that for any $(z,z_{1})\in V^{\prime}\times D$, $G(z,z_{1})<0$ on
$\Omega\times\Omega$, $G(z,z_{1})$ is harmonic with respect to $z$ or $z_{1}$
when fixed another one and $\sum_{z_{1}\in V}|G(q,z_{1})|<+\infty$. Without
loss of generality, we see $(V^{\prime},z)$ and $(B(3,1),w_{1})$ the same and
assume that $dV_{\Omega}=d\lambda_{z}$ on $V^{\prime}$, where $d\lambda_{z}$
is the Lebesgue measure on $\mathbb{C}$. Then we have

$\begin{split}\int_{V^{\prime}}|u_{n}|dV_{\Omega}&=\frac{1}{\pi}\int_{z\in
V^{\prime}}\int_{z_{1}\in V}|G_{\Omega}(z,z_{1})|T_{n}(z_{1})d\lambda_{z}\\\
&=\frac{1}{\pi}\int_{z_{1}\in V}\int_{z\in
V^{\prime}}|G_{\Omega}(z,z_{1})|d\lambda_{z}T_{n}(z_{1})\\\
&=\frac{1}{\pi}\int_{z_{1}\in V}\pi|G_{\Omega}(q,z_{1})|T_{n}(z_{1})\\\
&\leq\|T_{n}\|\sup_{z_{1}\in V}|G_{\Omega}(q,z_{1})|\\\
&=\|\tilde{T}\|\sup_{z_{1}\in V}|G_{\Omega}(q,z_{1})|.\end{split}$ (7.32)

When $q\in V$, $G_{\Omega}(w,\tilde{w})=\log|w-\tilde{w}|+v(w,\tilde{w})$ on
$V\times V$, where $v(w,\tilde{w})$ is harmonic with respect to $w$ or
$\tilde{w}$ when fixed another one. Without loss of generality, we see $(V,z)$
and $(B(0,1),w)$ the same and assume that $dV_{\Omega}=d\lambda_{w}$ on $V$,
where $d\lambda_{w}$ is the Lebesgue measure on $\mathbb{C}$. Then we have

$\begin{split}\int_{V}u_{n}dV_{\Omega}&=\frac{1}{\pi}\int_{w_{\in}V}\int_{\tilde{w}\in
V}G_{\Omega}(w,\tilde{w})T_{n}(\tilde{w})d\lambda_{w}\\\
&=\frac{1}{\pi}\int_{\tilde{w}\in V}\int_{w\in
V}G_{\Omega}(w,\tilde{w})d\lambda_{w}T_{n}(\tilde{w})\\\
&=\frac{1}{\pi}\int_{\tilde{w}\in V}\int_{w\in
V}\log|w-\tilde{w}|d\lambda_{w}T_{n}(\tilde{w})+\frac{1}{\pi}\int_{\tilde{w}\in
V}\int_{w\in V}v(w,\tilde{w})d\lambda_{w}T_{n}(\tilde{w}).\end{split}$ (7.33)

Note that

$\int_{w\in V}\log|w-\tilde{w}|d\lambda_{w}\geq-\int_{w\in
B(0,2)}|\log|w||d\lambda_{w}>-\infty$

holds for any $\tilde{w}\in V$,

$\int_{w\in V}v(w,\tilde{w})d\lambda_{w}=\pi v(q,\tilde{w})$

holds for any $\tilde{w}\in V$ and $\inf_{\tilde{w}\in
V}v(q,\tilde{w})>-\infty$, then equality (7.33) implies that there exists a
constant $N>0$ such that

$\int_{V}u_{n}dV_{\Omega}\geq N\|T_{n}\|.$ (7.34)

By the definition of $T_{n}$, we know $\|T_{n}\|=\|\tilde{T}\|<+\infty$. As
$u_{n}\leq 0$, combining inequality (7.32) and (7.34), we obtain that any
$q\in\Omega$ there exists an open subset $K\subset\subset\Omega$, such that
$q\in K$ and $\int_{K}|u_{n}|dV_{\Omega}\leq C$, where $dV_{\Omega}$ is some
continuous volume form and $C$ is a constant which independent of $n$. Hence,
we know $\\{u_{n}\\}\in L^{1}_{loc}(\Omega)$ and $\Phi\in
L^{1}_{loc}(\Omega)$.

Now we consider $i\partial\bar{\partial}\Phi$. Let $g\in C^{\infty}_{c}(X)$ be
a test function. It follows from $\Phi\in L_{loc}^{1}(\Omega)$ and the
dominated convergence Theorem that

$\begin{split}\langle
i\partial\bar{\partial}\Phi,g\rangle&=\langle\Phi(z),i\partial\bar{\partial}g(z)\rangle\\\
&=\lim_{n\to+\infty}\langle
u_{n}(z),i\partial\bar{\partial}g(z)\rangle.\end{split}$ (7.35)

As $u_{n}(z)=\langle T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle$,
using Fubini’s Theorem, equality (7.35) becomes

$\begin{split}\langle
i\partial\bar{\partial}\Phi,g\rangle&=\lim_{n\to+\infty}\langle\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle,i\partial\bar{\partial}g(z)\rangle\\\
&=\lim_{n\to+\infty}\langle
T_{n}(z_{1}),\langle\frac{1}{\pi}G_{\Omega}(z,z_{1}),i\partial\bar{\partial}g(z)\rangle\rangle.\end{split}$
(7.36)

Since $T_{n}$ is positive $(1,1)$ current on $\Omega$, $T_{n}$ converge weakly
to $\tilde{T}$ and
$\frac{i}{\pi}\partial_{z}\bar{\partial}_{z}G_{\Omega}(z,z_{1})=\delta_{z_{1}}$,
it follows from equality (7.36) that

$\begin{split}\langle
i\partial\bar{\partial}\Phi,g\rangle&=\lim_{n\to+\infty}\langle
T_{n}(z_{1}),\langle\frac{1}{\pi}G_{\Omega}(z,z_{1}),i\partial\bar{\partial}g(z)\rangle\rangle\\\
&=\lim_{n\to+\infty}\langle T_{n}(z_{1}),g(z_{1})\rangle\\\
&=\langle\tilde{T},g\rangle,\\\ \end{split}$ (7.37)

which implies that $i\partial\bar{\partial}\Phi=\tilde{T}$.

_Step 3: $\lim_{t\rightarrow
0+0}(\inf_{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\Phi(z))=0$ and
$\inf_{\Omega\backslash U}\Phi>-\infty$_

Let $W\subset\subset\Omega$ be an open set of $\Omega$ which satisfies
$\overline{V}\cup\\{z_{0}\\}\subset W$ and $\overline{W}\cap\\{-t\leq
G_{\Omega}(z,z_{0})\\}=\emptyset$, where $t$ is a small enough positive
number. Then for every fixed $z\in\\{-t\leq G_{\Omega}(z,z_{0})\\}$,
$G_{\Omega}(z,z_{1})$ is harmonic function on a neighborhood of $\overline{W}$
with respect to $z_{1}$. By the Harnack inequality of harmonic function, there
exists a $M>0$ such that

$\sup_{z_{1}\in\overline{W}}(-G_{\Omega}(z,z_{1}))\leq
M\inf_{z_{1}\in\overline{W}}(-G_{\Omega}(z,z_{1})).$

As $z\in\\{-t\leq G_{\Omega}(z,z_{0})\\}$, we have

$Mt>-MG_{\Omega}(z,z_{0})\geq
M\inf_{z_{1}\in\overline{W}}(-G_{\Omega}(z,z_{1})\geq\sup_{z_{1}\in\overline{W}}(-G_{\Omega}(z,z_{1})\geq
0,$

which means that $\lim_{t\rightarrow
0+0}(\inf_{{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\times\overline{W}}G_{\Omega}(z,z_{1}))=0$.

Note that $0\geq u_{n}(z)=\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle\geq\frac{1}{\pi}\inf_{{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\times\overline{W}}G_{\Omega}(z,z_{1})\|T_{n}\|$
holds for any $n$ and $z\in\\{-t\leq G_{\Omega}(z,z_{0})\\}$, as
$\|T_{n}\|=\|\tilde{T}\|<+\infty$ and $u_{n}$ is decreasingly convergent to
$\Phi$, then we have

$\lim_{t\rightarrow
0+0}(\inf_{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\Phi(z))\geq\lim_{t\rightarrow
0+0}\frac{1}{\pi}\inf_{{\\{G_{\Omega}(z,z_{0})\geq-t\\}}\times\overline{W}}G_{\Omega}(z,z_{1})\|\tilde{T}\|=0.$

Next, we prove $\inf_{\Omega\backslash U}\Phi>-\infty$. Note that $p\in
V\subset\subset\\{G_{\Omega}(z,p)<-t\\}\subset\subset U\subset\subset\Omega$,
it follows from Lemma 2.13 that there exists a constant $N>0$, such that

$G_{\Omega}(z,z_{1})\geq NG_{\Omega}(z,p)\geq-Nt$ (7.38)

holds for any $(z,z_{1})\in(\Omega\backslash U,V)$. As $u_{n}(z)=\langle
T_{n}(z_{1}),\frac{1}{\pi}G_{\Omega}(z,z_{1})\rangle$ and
$suppT_{n}\subset\subset V$ for any $n$, then we have
$u_{n}(z)\geq-\frac{Nt}{\pi}\|T_{n}\|$ hold on $z\in\Omega\backslash U$. Note
that $\|T_{n}\|=\|\tilde{T}\|$ and $u_{n}$ is decreasingly convergent to
$\Phi$, then we have $\inf_{\Omega\backslash U}\Phi>-\infty$.

Thus, Lemma 2.14 holds.

Acknowledgements. The authors would like to thank Dr. Shijie Bao and Dr.
Zhitong Mi for checking the manuscript and pointing out some typos. The first
named author was supported by National Key R&D Program of China
2021YFA1003103, NSFC-11825101, NSFC-11522101 and NSFC-11431013.

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