Green’s functions and complex Monge-Ampère equations 111Work supported in part
by the National Science Foundation under grant DMS-1855947.

Bin Guo, Duong H. Phong, and Jacob Sturm

###### Abstract

Uniform $L^{1}$ and lower bounds are obtained for the Green’s function on
compact Kähler manifolds. Unlike in the classic theorem of Cheng-Li for
Riemannian manifolds, the lower bounds do not depend directly on the Ricci
curvature, but only on integral bounds for the volume form and certain of its
derivatives. In particular, a uniform lower bound for the Green’s function on
Kähler manifolds is obtained which depends only on a lower bound for the
scalar curvature and on an $L^{q}$ norm for the volume form for some $q>1$.
The proof relies on auxiliary Monge-Ampère equations, and is fundamentally
non-linear. The lower bounds for the Green’s function imply in turn $C^{1}$
and $C^{2}$ estimates for complex Monge-Ampère equations with a sharper
dependence on the function on the right hand side.

##  1 Introduction

A fundamental theorem in Riemannian geometry is the inequality of Cheng-Li
[3], which provides conditions for a uniform lower bound for the Green’s
function. More precisely, let $(X,g)$ be a compact Riemannian manifold, and
define the Green’s function $G(x,y)$ as the unique function $X\times X\to{\bf
R}\cup\\{\infty\\}$ satisfying the conditions

(1.1)
$\Delta_{g,y}G(x,y)=-\delta_{x}(y)+\frac{1}{{\mathrm{Vol}}_{g}(X)},\quad\int_{X}G(x,y)dV_{g}(y)=0,$

where $\delta_{x}(y)$ denotes the Dirac function at the point $x$. It is well-
known that $G(x,y)=G(y,x)$ and $\Delta_{g,x}G(x,y)=\Delta_{g,y}G(x,y)$ for any
$x\neq y$ . Assume that the Ricci curvature of $(X,g)$ satisfies ${\rm
Ric}(g)\geq-\kappa$ for some constant $\kappa$. Then Cheng-Li [3] prove that
there is a constant $C>0$ depending only on the dimension of $X$ and $\kappa$
such that

(1.2)
$G(x,y)\geq-C\frac{{\mathrm{diam}_{g}}(X)^{2}}{{\mathrm{Vol}}_{g}(X)},\quad\forall
x,y\in X.$

Here ${\mathrm{diam}}_{g}(X)$ is the diameter of $(X,g)$ and
${\mathrm{Vol}}_{g}(X)$ is its volume. The dependence of this inequality on a
lower bound $\kappa$ for the Ricci curvature is crucial, and it does not seem
possible in Riemannian geometry to lessen this dependence in any significant
way.

A first primary goal of the present paper is to show that, in the Kähler
setting, lower bounds for the Green’s function can actually be established
without assumptions on lower bounds for the Ricci curvature. Rather, we assume
integral bounds on the volume form and some specific derivatives. As we shall
explain below, for our purposes, integral bounds are far superior to pointwise
bounds. It turns out that several distinct sets of assumptions can guarantee
lower bounds for the Green’s function, and we shall describe them in detail
later in §2 (Theorem 2). While these sets of assumptions may be difficult to
assess at a glance, an easy comparison with the theorem of Cheng-Li can be
obtained by observing that Theorem 2 implies in particular that the Green’s
function is bounded from below if the scalar curvature is bounded from below,
and the $L^{q}$ norm of the volume form is bounded for some $q>1$ (see
Corollary 2). It may also be worth stressing that, while our goal of
establishing lower bounds for the Green’s function is the same as Cheng-Li’s,
our method of proof is completely different. It builds repeatedly on the key
idea in [9] of comparison with an auxiliary Monge-Ampère equation, and even
though the Green’s function is the solution of a linear partial differential
equation, our method is fundamentally non-linear.

We come now to the second primary goal of this paper, which is also a major
motivation for the first, and which is sharp a priori estimates for general
complex Monge-Ampère equations. The first estimates were obtained in 1976 by
S.T. Yau in his seminal paper [25] solving the Calabi conjecture. However, a
new generation of problems in complex geometry is leading to more complicated
Monge-Ampère equations, which can be degenerate or singular in many different
senses. Thus ever sharper versions of a priori estimates are needed, as each
improvement usually has significant geometric consequences.

The first sharp form of $C^{0}$ estimates for the complex Monge-Ampère
equation was obtained by Kolodziej [16], using pluripotential theory.
Kolodziej’s estimates were extended to the important case of degenerating
background Kähler metrics by Eyssidieux, Guedj, and Zeriahi [5] and Demailly
and Pali [4]. More general $C^{0}$ estimates using the theory of envelopes can
be found in [7, 8]. Another approach to $C^{0}$ estimates, using PDE methods,
was introduced very recently in [9]. This method can also apply to nef classes
[12], and lead to many sharp estimates, including stability estimates [11],
diameter estimates [13], and non-collapse estimates [14]. We have just seen it
applied to lower bounds for Green’s functions in the first part of this paper.
On the other hand, while $C^{1}$ and $C^{2}$ estimates were extended to the
case allowing a divisor, building on ideas of Tsuji [23], Blocki [1], and [20,
19], they still require very restrictive conditions on the right hand side of
the equation, such as pointwise lower or upper bounds, and in the case of
$C^{2}$ estimates, also a bound on its Laplacian. A lower or upper bound
assumption is a particularly severe constraint, as it may rule out equations
which are degenerate or have singularities.

Thus our second primary goal in this paper is to establish sharp $C^{1}$ and
$C^{2}$ estimates for the complex Monge-Ampère equation which depend
essentially only on integral bounds for the right hand side. In order to do
so, we cannot apply the standard maximum principle to the elliptic
differential inequalities satisfied by the derivatives of the solution of the
Monge-Ampère equation. Rather, we apply instead the new lower bounds for the
Green’s function obtained in the first part of the paper.

We now state precisely our main results. In view of many applications in
complex geometry, it is important to obtain estimates which remain uniform as
the Kähler class may degenerate, so we consider the following set-up, which
includes both cases of fixed and degenerating Kähler classes as special cases.
Let $(X,\omega_{X})$ be a compact Kähler manifold with dimension $n$. Suppose
$\chi$ is a $d$-closed $(1,1)$-form on $X$ such that its cohomology class
$[\chi]\in H^{1,1}(X,{\bf C})$ is nef and big, i.e. $[\chi]$ lies in the
closure of the Kähler cone of $X$ and $\int_{X}\chi^{n}>0$. So for each $t>0$,
$[\chi+t\omega_{X}]$ is a Kähler class. For any fixed $t\in(0,1]$ and any
Kähler metric $\omega_{t}\in[\chi+t\omega_{X}]$, we define the function
$F_{\omega_{t}}$ to be the logarithmic of the relative volume form of
$\omega_{t}$ with respect to the fixed volume form $\omega_{X}^{n}$, that is

(1.3) $F_{\omega_{t}}=\,{\rm
log}\,\Big{(}\frac{\omega_{t}^{n}/V_{t}}{\omega_{X}^{n}/V}\Big{)}$

where $V_{t}=\int_{X}\omega_{t}^{n}=\int_{X}(\chi+t\omega_{X})^{n}>0$ is the
volume of the Kähler class $[\chi+t\omega_{X}]$, and
$V=\int_{X}\omega_{X}^{n}$ is the volume of the fixed metric $\omega_{X}$.
Note that $V_{0}=\int_{X}\chi^{n}>0$ under our assumptions.

Fix $p>n$. We denote the $p$-th entropy of the Kähler metric $\omega_{t}$ by

${\mathrm{Ent}}_{p}(\omega_{t})=\frac{1}{V_{t}}\int_{X}|F_{\omega_{t}}|^{p}{\omega_{t}^{n}}=\frac{1}{V}\int_{X}|F_{\omega_{t}}|^{p}e^{F_{\omega_{t}}}\omega_{X}^{n}.$

For any $N>0$, we define a subset of the space of Kähler metrics in
$[\chi+t\omega_{X}]$ by

${\mathcal{M}}_{t}(N,p)=\\{\omega_{t}\in[\chi+t\omega_{X}]|~{}{\mathrm{Ent}}_{p}(\omega_{t})\leq
N\\}.$

Our first main theorem shows that the lower bound of $G_{t}$ is equivalent to
the $L^{1}(X,\omega_{t}^{n})$-norm of $G_{t}$ for
$\omega_{t}\in{\mathcal{M}}_{t}(N,p)$.

###### Theorem 1

Given $p>n$ and $N>0$, there is a constant $C>0$ depending only on
$n,p,\chi,\omega_{X}$ and $N$, such that for any $t\in(0,1]$ and any Kähler
metric $\omega_{t}\in{\mathcal{M}}_{t}(N,p)$, the Green’s function $G_{t}$
associated to $\omega_{t}$ satisfies

$\frac{1}{2V_{t}}\|G_{t}(x,\cdot)\|_{L^{1}(X,\omega_{t}^{n})}\leq-\inf_{X}G_{t}(x,\cdot)\leq
C(1+\|G_{t}(x,\cdot)\|_{L^{1}(X,\omega_{t}^{n})}),$

for any $x\in X$.

We remark that $G_{t}(x,y)$ satisfies the asymptotic behavior [21]

(1.4) $G_{t}(x,y)\sim d_{\omega_{t}}(x,y)^{-2n+2},\mbox{ if }n\geq 2$

and $G_{t}(x,y)\sim-\,{\rm log}\,d_{\omega_{t}}(x,y)$ if $n=1$, when $x$ is
close to $y$. Here $d_{\omega_{t}}(x,y)$ denotes the geodesic distance of $x$
and $y$ under $\omega_{t}$. Thus an upper bound for $G_{t}(x,y)$ cannot be
expected to hold.

For a smaller class of Kähler metrics than ${\mathcal{M}}_{t}(N,p)$, we will
show that the $L^{1}$-norms of the Green’s function are uniformly bounded,
hence by Theorem 1, we also have a pointwise lower bound on the Green’s
functions.

Henceforth we require that $\chi$ is nonnegative and
$V_{0}=\int_{X}\chi^{n}>0$. Let $F_{\omega_{t}}$ be associated with a Kähler
metric $\omega_{t}\in[\chi+t\omega_{X}]$ as in (1.3). For $\epsilon>0$, $N>0$
and $\gamma\geq 1$, we denote for each $t\in(0,1]$

(1.5)
${\mathcal{M}}^{\prime}_{t}(N,\epsilon,\gamma)=\Big{\\{}\omega_{t}\in[\chi+t\omega_{X}]|~{}\frac{1}{V}\int_{X}e^{(1+\epsilon)F_{\omega_{t}}}\omega_{X}^{n}\leq
N\mbox{ and }\sup_{X}e^{-F_{\omega_{t}}}\leq\gamma\Big{\\}}.$

A second class of metrics for $t\in(0,1]$ is given by

$\displaystyle{\mathcal{M}}_{t}^{\prime\prime}(N,\epsilon,\gamma)$
$\displaystyle=$
$\displaystyle\Big{\\{}\omega_{t}\in[\chi+t\omega_{X}]|~{}\frac{1}{V}\int_{X}e^{(1+\epsilon)F_{\omega_{t}}}\omega_{X}^{n}\leq
N$ $\displaystyle\qquad\mbox{ and
}\int_{X}(e^{-F_{\omega_{t}}}+|\Delta_{\omega_{X}}e^{-F_{\omega_{t}}}|)\omega_{X}^{n}\leq\gamma\Big{\\}}.$

The case of Kähler metrics in a fixed Kähler class $\omega\in[\omega_{X}]$ can
be considered as a special case of the above more general set-up by taking
$\chi=\omega_{X}/2$ and $t=1/2$. We can also consider the class:

$\displaystyle\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$
$\displaystyle=$
$\displaystyle\Big{\\{}\omega\in[\omega_{X}]|~{}\frac{1}{V}\int_{X}e^{(1+\epsilon)F_{\omega}}\omega_{X}^{n}\leq
N$ $\displaystyle\qquad\mbox{ and
}\int_{X}(e^{-F_{\omega}}+|\nabla_{\omega_{X}}e^{-F_{\omega}}|_{\omega_{X}}^{2})\omega_{X}^{n}\leq\gamma\Big{\\}}.$

Abusing notations, when we write
$\omega_{t}\in\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$, we mean
that $t=1/2$ and $\chi=\omega_{X}/2$, which corresponds to the case of complex
Monge-Ampère equations with the fixed background metric $\omega_{X}$.

It follows easily from calculus inequalities that the sets of metrics in
(1.5), (1), (1) are contained in ${\mathcal{M}}_{t}(\tilde{N},p)$ for suitable
$\tilde{N}>N$ and $p>n$. Hence Theorem 1 applies to the metrics in these sets.
Our second theorem is:

###### Theorem 2

Given any $\epsilon>0$, $N>0$ and $\gamma\in(0,1)$, for each $t\in(0,1]$ and
each Kähler metric
$\omega_{t}\in{\mathcal{M}}_{t}^{\prime}(N,\epsilon,\gamma)\cup{\mathcal{M}}_{t}^{\prime\prime}(N,\epsilon,\gamma)\cup\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$,
let $G_{t}$ be the Green’s function associated with $(X,\omega_{t})$. Then the
following hold:

(i) There is a constant $C>0$ which depends on $n,\epsilon,\omega_{X},\chi$
and $N,\gamma$ such that for any $x\in X$

(1.8)
$\|G_{t}(x,\cdot)\|_{L^{1}(X,\omega_{t}^{n})}=\int_{X}|G_{t}(x,\cdot)|\omega_{t}^{n}\leq
C,$

and

(1.9) $\inf_{y\in X}G_{t}(x,y)\geq-C.$

(ii) For any given $\delta\in(0,\frac{2n}{2n-1})$, there is a constant
$C_{\delta}>0$ depending additionally on $\delta$ such that $G_{t}$ satisfies

$\int_{X}|G_{t}(x,\cdot)|^{\frac{n}{n-1}-\delta}\omega_{t}^{n}+\int_{X}|\nabla
G_{t}(x,\cdot)|_{{}_{\omega_{t}}}^{\frac{2n}{2n-1}-\delta}\omega_{t}^{n}\leq
C_{\delta},$

for any fixed $x\in X$.

Moreover, when $n=1$, (i) and (ii) hold for any $\omega_{t}$ with
$\|e^{F_{\omega_{t}}}\|_{L^{1+\epsilon}}\leq N$, and no extra conditions on
$e^{-F_{\omega_{t}}}$ as in (1.5), (1) and (1) are needed.

We now turn to the application of the estimates of Green’s functions in
Theorem 2. Again we assume $\chi$ is nonnegative and the class $[\chi]$ is
big. Then by Kodaira’s lemma, there is an effective divisor $D$ on $X$ such
that

$\chi-\varepsilon_{0}{\rm Ric}(h_{D})\geq\delta_{0}\omega_{X}$

for suitable positive constants $\varepsilon_{0}$ and $\delta_{0}$ which we
will fix throughout the paper, where $h_{D}$ is a Hermitian metric on the line
bundle $[D]$ associated with $D$. Let $s_{D}\in{\mathcal{O}}_{X}(D)$ be a
holomorphic section defining $D$ such that

${\rm sup}_{X}|s_{D}|_{h_{D}}^{2}=1.$

Let $\varphi_{t}$ be the Kähler potential of the Kähler metric
$\omega_{t}\in[\chi+t\omega_{X}]$, i.e.
$\omega_{t}=\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t}$. We first
derive an estimate on the gradient of $\varphi_{t}$ with respect to the fixed
metric $\omega_{X}$, for $\omega_{t}$ in the sets of Kähler metrics in Theorem
2.

###### Theorem 3

Given $N,\epsilon,\gamma\in(0,1)$, and $p>n$, for each $t\in(0,1]$ and any
$\omega_{t}\in{\mathcal{M}}_{t}^{\prime}(N,\epsilon,\gamma)\cup{\mathcal{M}}_{t}^{\prime\prime}(N,\epsilon,\gamma)\cup\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$,
the following estimate on $|\nabla\varphi_{t}|_{\omega_{X}}^{2}$ holds. There
is a constant $C>0$ depending on $n,\epsilon,\chi,\omega_{X},N,\gamma$, $p$
and $\int_{X}|\nabla
F_{\omega_{t}}|_{\omega_{X}}^{p}e^{F_{\omega_{t}}}\omega_{X}^{n}$ such that
away from $D$

$|\nabla\varphi_{t}|_{\omega_{X}}^{2}\leq\frac{C}{|s_{D}|_{h_{D}}^{2A}},$

where $A>0$ depends only on $n,\epsilon,\chi,\omega_{X},N$ and $\gamma$.

We observe that gradient estimates for solutions to certain complex Monge-
Ampère equations had been obtained in [1, 20], but they required pointwise
bounds on $|\nabla F_{\omega_{t}}|_{\omega_{X}}$. In [2, 10], the gradient is
shown to depend on certain $L^{p}$ bound of $|\nabla
F_{\omega_{t}}|_{\omega_{X}}$ for $p\geq 2n$. Theorem 3 implies that the
gradient estimate depends only on integral bound of $e^{F_{\omega_{t}}}$ and
$L^{p}$-bound of $|\nabla F_{\omega_{t}}|_{\omega_{X}}$ for smaller $p$ which
turns out to be sharp. In particular, the gradient estimate continues to hold
in some situations even when $e^{F_{\omega_{t}}}$ has zeros or is unbounded.
Theorem 3 also provides immediately a gradient estimate for solutions to
complex Monge-Ampère equations with a fixed background metric (i.e. when
$\omega\in\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$, cf.
Corollary 3 in §4 below), in which case we can take $s_{D}\equiv 1$,
$h_{D}\equiv 1$ and $D$ to be trivial. Example 3.1 shows that the gradient
estimate may fail if $p<n$, so the assumption on $p>n$ in Theorem 3 is sharp.

With regard to the second-order derivatives, we have the following counterpart
of Yau’s $C^{2}$ estimate [25].

###### Theorem 4

Under the same setup as in Theorem 3 with $p>2n$, there is a constant $C>0$
depending on $n,\epsilon,\chi,\omega_{X},N,\gamma$, $p$ and $\int_{X}|\nabla
F_{\omega_{t}}|_{\omega_{X}}^{p}e^{F_{\omega_{t}}}\omega_{X}^{n}$ such that
away from $D$ the following holds

$|i\partial\bar{\partial}\varphi_{t}|_{\omega_{X}}^{2}\leq\frac{C}{|s_{D}|_{h_{D}}^{2B}},$

where $B>0$ depends only on $n,\epsilon,\chi,\omega_{X},N$ and $\gamma$.

We stress that the above $C^{2}$-estimate of $\varphi_{t}$ is independent of
the pointwise estimates on the second order derivatives of $F_{\omega_{t}}$,
and it only depends on certain $L^{p}$-bound of $|\nabla
F_{\omega_{t}}|_{\omega_{X}}$. Theorem 4 improves in particular on the known
estimates [25] for complex Monge-Ampère equations with a fixed background
metric (cf. Corollary 4). We remark that the exponent $p>2n$ in Theorem 4 is
also sharp, as Example 3.2 shows that the estimates may fail if $p<2n$.

By utilizing the estimates of the Green’s functions, we can also improve the
$C^{3}$ estimates of complex Monge-Ampère equations (cf. Theorem 5), which
generalize the known ones in [25, 18] by a weaker dependence of the function
$e^{F}$ on the right-hand side.

##  2 Proof of Theorem 1

Given any $t\in(0,1]$, we fix an arbitrary Kähler metric
$\omega_{t}\in{\mathcal{M}}_{t}(N,p)$. It is clear from the
$i\partial\bar{\partial}$-lemma that (1.3) is equivalent to the following
complex Monge-Ampère equation with
$\omega_{t}=\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t}$

(2.1)
$(\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t})^{n}=c_{t}e^{F_{\omega_{t}}}\omega_{X}^{n},\quad\sup_{X}\varphi_{t}=0,$

where we have normalized $\varphi_{t}$ to make it unique, and we denote
$c_{t}=V_{t}/V$. The case of a fixed Kähler class $\omega_{X}$ corresponds for
example to the special case $\chi={1\over 2}\omega_{X}$ and $t={1\over 2}$.
For simplicity of notations, we will write $F_{\omega_{t}}$ as $F$.

Since $\hat{\omega}_{t}:=\chi+t\omega_{X}$ may not necessarily be positive, we
introduce the following $\hat{\omega}_{t}$-plurisubharmonic (PSH) function.

###### Definition 1

For any $t\in(0,1]$, we denote the envelope associated to the $(1,1)$-form
$\hat{\omega}_{t}$

${\mathcal{V}}_{t}=\sup\\{v\in PSH(X,\hat{\omega}_{t})|~{}v\leq 0\\}.$

Note that if $\chi\geq 0$ is a nonnegative $(1,1)$-form,
${\mathcal{V}}_{t}\equiv 0$ for any $t\in(0,1]$.

The following $L^{\infty}$ estimate for the family of solutions $\varphi_{t}$
to the equation (2.1) is proved in [12] (see also [16, 6]).

######  Lemma 1 ([12])

There is a uniform constant
${C}_{0}={C}_{0}(n,p,\chi,\omega_{X},{\mathrm{Ent}}_{p}(\omega_{t}))>0$ such
that

$\sup_{X}|\varphi_{t}-{\mathcal{V}}_{t}|\leq{C_{0}},\quad\forall t\in(0,1].$

To prove Theorem 1, we need the following mean-value type inequality for
functions satisfying certain linear differential inequality.

###### Lemma 2

Suppose $v\in L^{1}(X,\omega_{t}^{n})$ is a function that satisfies
$\int_{X}v\omega_{t}^{n}=0$ and

(2.2) $v\in C^{2}(\Omega_{-1}),\quad\Delta_{\omega_{t}}v\geq-a\mbox{ in
}\Omega_{0}$

for some $a>0$ and $\Omega_{s}=\\{v>s\\}$ is the super-level set of $v$. Then
there is a constant $C>0$ depending only on
$n,p,\chi,\omega_{X},{\mathrm{Ent}}_{p}(\omega_{t})$ and $a$ such that

$\sup_{X}v\leq C(1+\|v\|_{L^{1}(X,\omega_{t}^{n})}).$

We emphasize that the constant $C>0$ in the lemma above is independent of
$t\in(0,1]$. The proof of Lemma 2 uses the techniques similar to that of the
$L^{\infty}$-estimate for fully nonlinear partial differential equations in
[9]. The key idea to introduce an auxiliary equation to compare with $v$. The
lemma is trivial if $v\equiv 0$, so we assume $v\not\equiv 0$.

Proof.  We break the proof into four steps. Since the proof is uniform in
$t\in(0,1]$, we fix a $t\in(0,1]$. We may assume
$\|v\|_{L^{1}(X,\omega_{t}^{n})}\leq V_{0}$, otherwise, replace $v$ by
$\hat{v}:=V_{0}\cdot v/\|v\|_{L^{1}(X,\omega_{t}^{n})}$ which still satisfies
(2.2) with the same $a>0$. It suffices to show $\sup_{X}v\leq C$ for some
$C>0$ with the dependence as stated in the lemma.

Step 1. We fix a sequence of positive smooth functions $\eta_{k}:{\bf
R}\to{\bf R}_{+}=(0,\infty)$ such that $\eta_{k}(x)$ converges uniformly and
monotonically decreasingly to the function $x\cdot\chi_{\bf R_{+}}(x)$, as
$k\to\infty$. We may choose $\eta_{k}(x)\equiv 1/k$ for any $x\leq-1/2$. For
$s\geq 0$ and large $k$, we consider the following auxiliary complex Monge-
Ampère equations

(2.3)
$(\hat{\omega}_{t}+i\partial\bar{\partial}\psi_{t,k})^{n}=c_{t}\frac{\eta_{k}(v-s)}{A_{s,k}}e^{F}\omega_{X}^{n},\quad\sup_{X}\psi_{t,k}=0,$

where

(2.4)
$A_{s,k}=\frac{c_{t}}{V_{t}}\int_{X}{\eta_{k}(v-s)}e^{F}\omega_{X}^{n}\to\frac{1}{V}\int_{\Omega_{s}}(v-s)e^{F}\omega_{X}^{n}=:A_{s}\mbox{
as }k\to\infty.$

We remark that the right-hand side of (2.3) is smooth and positive, and
$[\hat{\omega}_{t}]$ is a Kähler class, so by Yau’s theorem [25] this equation
admits a unique smooth solution $\psi_{t,k}$. We have also assumed that the
open set $\Omega_{s}\neq\emptyset$ so $A_{s}>0$, otherwise the lemma is
already proved. The assumption that $\|v\|_{L^{1}(X,\omega_{t}^{n})}\leq
V_{0}$ implies that $A_{s}\leq 1$, hence $A_{s,k}\leq 2$ for large $k$.

Step 2. We denote $\Lambda=C_{0}+1$ where $C_{0}$ is the constant in Lemma 1.
Consider the function

$\Phi:=-\varepsilon(-\psi_{t,k}+\varphi_{t}+\Lambda)^{\frac{n}{n+1}}+(v-s),$

where $\varepsilon>0$ is chosen such that

(2.5) $\varepsilon^{n+1}=\Big{(}\frac{n+1}{n^{2}}\Big{)}^{n}(a+\varepsilon
n)^{n}A_{s,k}.$

It follows easily from $A_{s,k}\leq 2$ and equation (2.5) that

(2.6) $\varepsilon\leq C(n,a)A_{s,k}^{1/(n+1)},$

for some $C(n,a)>0$ depending only on $n\mbox{ and }a$. $\Phi$ is a $C^{2}$
function on $\Omega_{-1}$ since $v-s$ is so and

(2.7)
$-\psi_{t,k}+\varphi_{t}+\Lambda=({\mathcal{V}}_{t}-\psi_{t,k})+(\varphi_{t}-{\mathcal{V}}_{t}+C_{0})+1\geq
1.$

We claim that $\Phi\leq 0$ on $X$. Observe that by the definition of
$\Omega_{s}$ it is clear that $\Phi|_{X\backslash\Omega_{s}}<0$, so if
$\max_{\Omega_{s}}\Phi\leq\sup_{X\backslash\Omega_{s}}\Phi<0$, we are done.
Otherwise, $\max_{\Omega_{s}}\Phi>\sup_{X\backslash\Omega_{s}}\Phi$ and $\Phi$
achieves its maximum at some point $x_{0}\in\Omega_{s}$. By maximum principle,
$\Delta_{\omega_{t}}\Phi(x_{0})\leq 0$. Therefore, we calculate (below we
denote
$\omega_{t,\psi_{t,k}}=\hat{\omega}_{t}+i\partial\bar{\partial}\psi_{t,k}$)

$\displaystyle 0$ $\displaystyle\geq$
$\displaystyle\Delta_{\omega_{t}}\Phi(x_{0})$ $\displaystyle\geq$
$\displaystyle\frac{\varepsilon
n}{n+1}(-\psi_{t,k}+\varphi_{t}+\Lambda)^{-\frac{1}{n+1}}({\rm
tr}_{\omega_{t}}\omega_{t,\psi_{t,k}}-{\rm
tr}_{\omega_{t}}\omega_{t})+\Delta_{\omega_{t}}v$ $\displaystyle\geq$
$\displaystyle\frac{\varepsilon
n^{2}}{n+1}(-\psi_{t,k}+\varphi_{t}+\Lambda)^{-\frac{1}{n+1}}\Big{(}\frac{\omega^{n}_{t,\psi_{t,k}}}{\omega_{t}^{n}}\Big{)}^{1/n}-\frac{\varepsilon
n^{2}}{n+1}-a$ $\displaystyle\geq$ $\displaystyle\frac{\varepsilon
n^{2}}{n+1}(-\psi_{t,k}+\varphi_{t}+\Lambda)^{-\frac{1}{n+1}}\frac{(v-s)^{1/n}}{A_{s,k}^{1/n}}-a-\varepsilon
n,$

where in the third line we applied the arithmetic-geometric inequality and in
the last line we use the equation (2.3). By the choice of $\varepsilon$ in
(2.5), it follows by a straightforward calculation that $\Phi(x_{0})\leq 0$,
as claimed.

Step 3. From $\Phi\leq 0$ and (2.6) in the previous step , we have
$(v-s)A_{s,k}^{-1/(n+1)}\leq C_{1}(-\psi_{t,k}+\varphi_{t}+\Lambda)^{n/(n+1)}$
on $X$, for some $C_{1}>0$ depending only on $n$ and $a$. In particular on
$\Omega_{s}=\\{v-s>0\\}$ we have by taking $(n+1)/n$-th power

$\frac{(v-s)^{(n+1)/n}}{A_{s,k}^{1/n}}\leq
C_{1}^{(n+1)/n}(-\psi_{t,k}+\varphi_{t}+\Lambda)\leq
C_{1}^{(n+1)/n}(-\psi_{t,k}+\Lambda)$

where the second inequality follows from the normalization
$\sup_{X}\varphi_{t}=0$. Multiply both sides of above by suitable
$0<\alpha=\alpha(\chi,\omega_{X})>0$ such that $C_{1}^{(n+1)/n}\alpha$ is less
than the alpha invariant of the Kähler manifold $(X,(C_{2}+1)\omega_{X})$
where $\chi\leq C_{2}\omega_{X}$ for some $C_{2}>0$, and integrate the
resulted inequality over $\Omega_{s}$. We thus obtain

(2.8) $\int_{\Omega_{s}}\,{\rm
exp}\,\Big{(}\alpha\frac{(v-s)^{(n+1)/n}}{A_{s,k}^{1/n}}\Big{)}\omega_{X}^{n}\leq
C\int_{X}\,{\rm
exp}\,\Big{(}-C_{1}^{(n+1)/n}\alpha\psi_{t,k}\Big{)}\omega_{X}^{n}\leq C_{3},$

for some uniform constant $C_{3}>0$ independent of $t$. In this last
inequality we apply the $\alpha$-invariant estimate for quasi-PSH functions on
compact Kähler manifolds [15, 22]. By Young’s inequality, (2.8) implies that
for some $C_{4}>0$ depending additionally on $p>n$ and
${\mathrm{Ent}}_{p}(\omega_{t})$ (cf. [9])

(2.9) $\int_{\Omega_{s}}(v-s)^{(n+1)p/n}e^{F}\omega_{X}^{n}\leq
C_{4}A_{s,k}^{p/n}\to C_{4}A_{s}^{p/n},$

as $k\to\infty$, noting that the left-hand side of (2.9) is independent of
$k$. On the other hand, by Hölder inequality and (2.9) we have

$A_{s}\leq\frac{1}{V}\Big{(}\int_{\Omega_{s}}(v-s)^{\frac{p(n+1)}{n}}e^{F}\omega_{X}^{n}\Big{)}^{\frac{n}{p(n+1)}}\Big{(}\int_{\Omega_{s}}e^{F}\omega_{X}^{n}\Big{)}^{\frac{1}{p^{\prime}}}\leq
C_{5}A_{s}^{\frac{1}{n+1}}\Big{(}\int_{\Omega_{s}}e^{F}\omega_{X}^{n}\Big{)}^{\frac{1}{p^{\prime}}},$

where $p^{\prime}>1$ satisfies $\frac{n}{p(n+1)}+\frac{1}{p^{\prime}}=1$. This
implies that $A_{s}\leq
C_{6}\Big{(}\int_{\Omega_{s}}e^{F}\omega_{X}^{n}\Big{)}^{\frac{1+n}{np^{\prime}}}$,
i.e.

(2.10) $\int_{\Omega_{s}}(v-s)e^{F}\omega_{X}^{n}\leq
C_{6}\Big{(}\int_{\Omega_{s}}e^{F}\omega_{X}^{n}\Big{)}^{1+\delta_{0}},$

for $1+\delta_{0}=\frac{1+n}{np^{\prime}}$ with $\delta_{0}=\frac{p-n}{np}>0$.
We denote $\phi(s)=\int_{\Omega_{s}}e^{F}\omega_{X}^{n}$. Then (2.10) yields
easily that

(2.11) $r\phi(s+r)\leq C_{6}\phi(s)^{1+\delta_{0}},\quad\forall s\geq 0,\,r>0$

Step 4. By the assumption $\|v\|_{L^{1}(X,\omega_{t}^{n})}\leq V_{0}$ we have

$\int_{\Omega_{0}}ve^{F}\omega_{X}^{n}\leq\frac{1}{c_{t}}\int_{X}|v|\omega_{t}^{n}\leq\frac{V_{0}V}{V_{t}}\leq
V,$

and this implies that for any $s>0$

(2.12)
$\phi(s)=\int_{\Omega_{s}}e^{F}\omega_{X}^{n}\leq\frac{1}{s}\int_{\Omega_{0}}ve^{F}\omega_{X}^{n}\leq\frac{V}{s}.$

So we can pick $s_{0}=(2C_{6})^{1/\delta_{0}}V$ to guarantee that
$\phi(s_{0})^{\delta_{0}}<1/2C_{6}$. Given (2.11), we can apply the De Giorgi
type iteration argument of Kolodziej [16] to conclude that $\phi(s)=0$ for any
$s>S_{\infty}$ with

$S_{\infty}=s_{0}+\frac{1}{1-2^{-\delta_{0}}}=(2C_{6})^{1/\delta_{0}}V+\frac{1}{1-2^{-\delta_{0}}}.$

This means that $v\leq S_{\infty}$ and we finish the proof of the lemma.

Proof of Theorem 1.  Fix a point $x\in X$. We let $v(y)=-G_{t}(x,y)$ be the
Green’s function of $\omega_{t}$ centered at $x$. This $v$ satisfies the
assumptions in Lemma 2, i.e. $v\in L^{1}(X,\omega_{t}^{n})$, $v$ is smooth on
$X\backslash\\{x\\}$ and

$\int_{X}v(y)\omega_{t}^{n}(y)=0,\quad\Delta_{\omega_{t}}v(y)=-\frac{1}{V_{t}}\geq-\frac{1}{V_{0}}\mbox{
for }y\in\\{v\geq 0\\}$

Lemma 2 gives a constant $C>0$ depending only on $n,p,\omega_{X},\chi,N$ such
that

$v\leq C(1+\|v\|_{L^{1}(X,\omega_{t}^{n})})$

which implies that $\inf_{y\in
X}G_{t}(x,y)\geq-C(1+\|G_{t}(x,\cdot)\|_{L^{1}(X,\omega_{t}^{n})})$.

On the other hand, since $-C_{l}=\inf_{y\in X}G_{t}(x,y)$ is a lower bound of
$G_{t}(x,\cdot)$, we have

$\|G_{t}(x,\cdot)\|_{L^{1}(X,\omega_{t}^{n})}\leq\int_{X}|G_{t}(x,\cdot)+C_{l}|\omega_{t}^{n}+C_{l}V_{t}\leq
2C_{l}V_{t}.$

##  3 Proof of Theorem 2

Given the parameters $\epsilon>0$, $N>0$ and $\gamma\in(0,1)$, we fix a Kähler
metric
$\omega_{t}\in{\mathcal{M}}_{t}^{\prime}(N,\epsilon,\gamma)\cup{\mathcal{M}}_{t}^{\prime\prime}(N,\epsilon,\gamma)\cup\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$.
We will denote $G_{t}$ the associated Green’s function of $(X,\omega_{t})$. As
in the last section, we let
$\omega_{t}=\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t}$ be the
solution to the following complex Monge-Ampère equation

(3.1)
$(\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t})^{n}=c_{t}e^{F_{\omega_{t}}}\omega_{X}^{n},\quad\sup_{X}\varphi_{t}=0.$

In this section, we say a constant $C>0$ is uniform if it depends only on
$n,\chi,\omega_{X}$ and the given parameters $\epsilon,N,\gamma$.

Since $\chi$ is assumed to be nonnegative in Theorem 2, by Lemma 1, there is a
uniform constant $C_{0}>0$ such that for each $t\in(0,1]$ and $\varphi_{t}$
satisfying (3.1)

(3.2) $\sup_{X}|\varphi_{t}|\leq C_{0}$

We observe the following estimate on $L^{2}$-norm of $\nabla\varphi$.

###### Lemma 3

Suppose $\omega=\omega_{X}+i\partial\bar{\partial}\varphi$ is a metric such
that $e^{F}={\omega^{n}}/{\omega_{X}^{n}}\in
L^{1+\epsilon}(X,\omega_{X}^{n})$, then

(3.3) $\int_{X}|\nabla\varphi|_{\omega_{X}}^{2}\omega_{X}^{n}\leq C,$

for some $C>0$ depending only on $n,\epsilon,\omega_{X}$ and
$\|e^{F}\|_{L^{1+\epsilon}}$.

Proof.  If we normalize $\varphi$ such that $\sup_{X}\varphi=0$, by the
$L^{\infty}$-estimates of Kolodziej [16, 9], we have $\sup_{X}|\varphi|\leq C$
for some $C$ depending on $n,\omega_{X}$ and $\|e^{F}\|_{L^{1+\epsilon}}$.
Then we calculate

$\displaystyle\int_{X}(-\varphi)(e^{F}-1)\omega_{X}^{n}$ $\displaystyle=$
$\displaystyle\int_{X}(-\varphi)(\omega^{n}-\omega_{X}^{n})$ $\displaystyle=$
$\displaystyle\int_{X}i\partial\varphi\wedge\bar{\partial}\varphi\wedge(\omega^{n-1}+\cdots+\omega_{X}^{n-1})$
$\displaystyle\geq$
$\displaystyle\frac{1}{n}\int_{X}|\nabla\varphi|^{2}_{\omega_{X}}\omega_{X}^{n}.$

The lemma follows straightforwardly from this.

From Lemma 2, we easily get that

###### Lemma 4

Suppose $v\in C^{2}(X)$ satisfies

(3.4) $|\Delta_{\omega_{t}}v|\leq 1,\quad\mbox{ and
}\int_{X}v\omega_{t}^{n}=0,$

then there is a uniform constant $C>0$ such that

$\sup_{X}|v|\leq C(1+\|v\|_{L^{1}(X,\omega_{t}^{n})}).$

We further have the following lemma which asserts that the function $v$
satisfying (3.4) is in fact bounded uniformly in $L^{\infty}$-norm.

###### Lemma 5

Under the same assumptions as in Lemma 4, we have
$\|v\|_{L^{1}(X,\omega_{t}^{n})}\leq C$ for some uniform constant $C>0$, in
particular by Lemma 4 this implies $\sup_{X}|v|\leq C$.

Assuming Lemma 5, we see how it yields the $L^{1}$-bound on the Green’s
function $G_{t}$ of $(X,\omega_{t})$. For any fixed point $x\in X$, we view
the Green’s function $G_{t}(x,\cdot)=G_{t}(x,y)$ as a function of $y$.
Consider the equation

(3.5) $\Delta_{\omega_{t}}v=-\chi_{\\{G_{t}\leq
0\\}}+\frac{1}{V_{t}}\int_{\\{G_{t}\leq 0\\}}\omega_{t}^{n},\quad\mbox{and
}\int_{X}v\omega_{t}^{n}=0.$

Take a sequence of smooth and bounded functions $f_{k}$ that converge
pointwise (in fact uniformly) to the bounded function $-\chi_{\\{G_{t}\leq
0\\}}+\frac{1}{V_{t}}\int_{\\{G_{t}\leq 0\\}}\omega_{t}^{n}$ and satisfies
$\int_{X}f_{k}\omega_{t}^{n}=0$. Let $v_{k}$ be the smooth solution to
$\Delta_{\omega_{t}}v_{k}=f_{k}$ with $\int_{X}v_{k}\omega_{t}^{n}=0$. It
follows from standard elliptic theory that for fixed $t>0$, $v_{k}$ converges
uniformly to $v$, which is the solution to (3.5). By Green’s formula, we have

$v_{k}(x)=\frac{1}{V_{t}}\int_{X}v_{k}\omega_{t}^{n}+\int_{X}G_{t}(x,\cdot)(-\Delta_{\omega_{t}}v_{k})\omega_{t}^{n}=\int_{X}G_{t}(x,\cdot)(-f_{k})\omega_{t}^{n}.$

From Lemma 5 we have $|v_{k}(x)|\leq C$, for each $k$, since $|f_{k}|\leq 2$,
say. Letting $k\to\infty$ and we get by the choice of $f_{k}$ and the
normalization $\int_{X}G_{t}(x,\cdot)\omega_{t}^{n}=0$ that

$\Big{|}\int_{\\{G_{t}(x,\cdot)\leq
0\\}}G_{t}(x,\cdot)\omega_{t}^{n}\Big{|}\leq C.$

Since $|\int_{\\{G_{t}(x,\cdot)\leq
0\\}}G_{t}(x,\cdot)\omega_{t}^{n}|=\int_{\\{G_{t}(x,\cdot)\geq
0\\}}G_{t}(x,\cdot)\omega_{t}^{n}$, this easily gives the
$L^{1}(X,\omega_{t}^{n})$-bound of $G_{t}(x,\cdot)$. This finishes the proof
of (i) in Theorem 2, assuming Lemma 5.

Now we turn to the proof of Lemma 5. We argue by contradiction. Suppose there
is a sequence of Kähler metrics
$\omega_{j}=\omega_{t_{j}}\in{\mathcal{M}}_{t}^{\prime}(N,\epsilon,\gamma)\cup{\mathcal{M}}_{t}^{\prime\prime}(N,\epsilon,\gamma)\cup\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$
and $F_{j}=F_{\omega_{j}}$ as defined in (1.3), and a sequence of $C^{2}(X)$
functions $v_{j}$ satisfying

$\Delta_{\omega_{j}}v_{j}=h_{j},\quad\mbox{and
}\int_{X}v_{j}\omega_{j}^{n}=0,$

for some function $h_{j}$ with $\sup_{X}|h_{j}|\leq 1$, for which Lemma 5
fails, i.e. they satisfy

(3.6)
$\|v_{j}\|_{L^{1}(X,\omega_{j}^{n})}=\int_{X}|v_{j}|\omega_{j}^{n}\to\infty\mbox{
~{}~{}as }j\to\infty.$

We normalize each $v_{j}$ by

$\hat{v}_{j}=\frac{v_{j}}{\|v_{j}\|_{L^{1}(X,\omega_{j}^{n})}},\quad\mbox{so
}\|\hat{v}_{j}\|_{L^{1}(X,\omega_{j}^{n})}=1.$

It is clear that $\hat{v}_{j}\in C^{2}(X)$ satisfies

(3.7)
$\Delta_{\omega_{j}}\hat{v}_{j}=\frac{h_{j}}{\|v_{j}\|_{L^{1}(X,\omega_{j}^{n})}},\quad\mbox{and
}\int_{X}\hat{v}_{j}\omega_{j}^{n}=0.$

We can apply Lemma 4 to conclude that

(3.8) $\sup_{X}|\hat{v}_{j}|\leq
C(1+\|\hat{v}_{j}\|_{L^{1}(X,\omega_{j}^{n})})\leq C,$

for some uniform constant $C>0$ independent of $j$. Multiplying both sides of
(3.7) by $\hat{v}_{j}$ and applying integration by parts, we get

(3.9)
$\int_{X}|\nabla\hat{v}_{j}|^{2}_{\omega_{j}}\omega_{j}^{n}=\int_{X}-\frac{h_{j}\hat{v}_{j}}{\|v_{j}\|_{L^{1}(X,\omega_{j}^{n})}}\omega_{j}^{n}\leq\frac{1}{\|v_{j}\|_{L^{1}(X,\omega_{j}^{n})}}\to
0,$

as $j\to\infty$ by the hypothesis (3.6). By the Hölder inequality we have

(3.10) $\displaystyle\int_{X}|\nabla\hat{v}_{j}|_{\omega_{X}}\omega_{X}^{n}$
$\displaystyle\leq$
$\displaystyle\int_{X}(|\nabla\hat{v}_{j}|_{\omega_{j}}^{2}{\rm
tr}_{\omega_{X}}\omega_{j})^{1/2}\omega_{X}^{n}$ $\displaystyle\leq$
$\displaystyle\Big{(}\int_{X}|\nabla\hat{v}_{j}|_{\omega_{j}}^{2}e^{F_{j}}\omega_{X}^{n}\Big{)}^{1/2}\Big{(}\int_{X}({\rm
tr}_{\omega_{X}}\omega_{j})e^{-F_{j}}\omega_{X}^{n}\Big{)}^{1/2}.$

Note that the first factor in (3.10) satisfies

$\int_{X}|\nabla\hat{v}_{j}|_{\omega_{j}}^{2}e^{F_{j}}\omega_{X}^{n}=c_{t_{j}}^{-1}\int_{X}|\nabla\hat{v}_{j}|_{\omega_{j}}^{2}\omega_{j}^{n}\to
0$

by (3.9). For the second factor in (3.10), we consider different cases of
$\omega_{j}$.

(a) If $\omega_{j}\in{\mathcal{M}}^{\prime}_{t_{j}}(N,\epsilon,\gamma)$, then
it holds that

$\int_{X}({\rm
tr}_{\omega_{X}}\omega_{j})e^{-F_{j}}\omega_{X}^{n}\leq\frac{1}{n\gamma}\int_{X}\omega_{j}\wedge\omega_{X}^{n-1}\leq
C.$

(b) If $\omega_{j}\in{\mathcal{M}}^{\prime\prime}_{t_{j}}(N,\epsilon,\gamma)$,
then we have

$\displaystyle\int_{X}({\rm
tr}_{\omega_{X}}\omega_{j})e^{-F_{j}}\omega_{X}^{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\int_{X}e^{-F_{j}}(\chi+t_{j}\omega_{X}+i\partial\bar{\partial}\varphi_{t_{j}})\wedge\omega_{X}^{n-1}$
$\displaystyle\leq$ $\displaystyle
C\int_{X}e^{-F_{j}}\omega_{X}^{n}+\int_{X}|\varphi_{t_{j}}||\Delta_{\omega_{X}}e^{-F_{j}}|\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle C,$

by the definition of ${\mathcal{M}}^{\prime\prime}_{t_{j}}(N,\epsilon,\gamma)$
and (3.2).

(c) If $\omega_{j}\in\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$,
then by a similar calculation we have

$\displaystyle\int_{X}({\rm
tr}_{\omega_{X}}\omega_{j})e^{-F_{j}}\omega_{X}^{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\int_{X}e^{-F_{j}}(\omega_{X}+i\partial\bar{\partial}\varphi_{j})\wedge\omega_{X}^{n-1}$
$\displaystyle\leq$ $\displaystyle
C\int_{X}e^{-F_{j}}\omega_{X}^{n}+\int_{X}|\nabla\varphi_{j}|_{\omega_{X}}|\nabla
e^{-F_{j}}|_{\omega_{X}}\omega_{X}^{n}$ $\displaystyle\leq$ $\displaystyle
C\int_{X}e^{-F_{j}}\omega_{X}^{n}+\Big{(}\int_{X}|\nabla\varphi_{j}|_{\omega_{X}}^{2}\omega_{X}^{n}\Big{)}^{1/2}\Big{(}\int_{X}|\nabla
e^{-F_{j}}|^{2}_{\omega_{X}}\omega_{X}^{n}\Big{)}^{1/2}$ $\displaystyle\leq$
$\displaystyle C,$

by Lemma 3 and the definition of the set
$\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$.

(d) If $n=1$, we observe that from (3.9), as $j\to\infty$

(3.11)
$\int_{X}|\nabla\hat{v}_{j}|^{2}_{\omega_{X}}\omega_{X}=\int_{X}i\partial\hat{v}_{j}\wedge\bar{\partial}\hat{v}_{j}=\int_{X}|\nabla\hat{v}_{j}|^{2}_{\omega_{j}}\omega_{j}\to
0.$

Combining all cases discussed above, (3.10) or (3.11) entail that

(3.12) $\int_{X}|\nabla\hat{v}_{j}|_{\omega_{X}}\omega_{X}^{n}\to 0.$

From (3.8) and (3.12) we see the the sequence of functions $\\{\hat{v}_{j}\\}$
is uniformly bounded in the Sobolev space $W^{1,1}(X,\omega_{X})$ (under the
fixed metric $\omega_{X}$). By the Sobolev embedding theorem, there is an
embedding

$W^{1,1}(X,\omega_{X})\hookrightarrow L^{q}(X,\omega_{X}^{n})$

which is compact for any $1\leq q<\frac{2n}{2n-1}$. Therefore, up to a
subsequence we have $\hat{v}_{j}\to\hat{v}_{\infty}$ in
$L^{q}(X,\omega_{X}^{n})$. In particular $\hat{v}_{j}$ also converge to
$\hat{v}_{\infty}$ in $L^{1}(X,\omega_{X}^{n})$ and in the a.e. sense up to a
further subsequence if necessary.

We now claim that $\hat{v}_{\infty}$ is constant in the a.e. sense. Indeed,
for any fixed $C^{2}$ function $\rho$ on $X$, we have

$\Big{|}\int_{X}\hat{v}_{j}\Delta_{\omega_{X}}\rho\omega_{X}^{n}\Big{|}=\Big{|}\int_{X}\langle\nabla\hat{v}_{j},\nabla\rho\rangle_{\omega_{X}}\omega_{X}^{n}\Big{|}\leq\|\nabla\rho\|_{\infty}\int_{X}|\nabla\hat{v}_{j}|_{\omega_{X}}\omega_{X}^{n}\to
0$

as $j\to\infty$. By the dominated convergence theorem and
$\hat{v}_{j}\to\hat{v}_{\infty}$ a.e. we conclude that

$\int_{X}\hat{v}_{\infty}\Delta_{\omega_{X}}\rho\omega_{X}^{n}=0$

which holds for any $\rho\in C^{2}(X)$. By Weyl’s lemma, this implies that
$\hat{v}_{\infty}$ is $\Delta_{\omega_{X}}$-harmonic, hence
$\hat{v}_{\infty}=\alpha_{0}$ in the a.e. sense for some constant
$\alpha_{0}\in\bf R$. We next claim that $\alpha_{0}\neq 0$. Indeed, from the
normalization $\|\hat{v}_{j}\|_{L^{1}(X,\omega_{j}^{n})}=1$ and (3.8), we get

(3.13) $\displaystyle 1$ $\displaystyle=$ $\displaystyle
c_{t_{j}}\int_{X}|\hat{v}_{j}|e^{F_{j}}\omega_{X}^{n}\leq
C\int_{X}|\hat{v}_{j}|^{\eta}e^{F_{j}}\omega_{X}^{n}$ $\displaystyle\leq$
$\displaystyle
C\Big{(}\int_{X}|\hat{v}_{j}|^{\eta\frac{1+\epsilon}{\epsilon}}\omega_{X}^{n}\Big{)}^{\epsilon/(1+\epsilon)}\Big{(}\int_{X}e^{(1+\epsilon)F_{j}}\omega_{X}^{n}\Big{)}^{1/(1+\epsilon)}$

and here we take $\eta=\epsilon/(1+\epsilon)$. From (3.13) we obtain

(3.14) $\int_{X}|\hat{v}_{j}|\omega_{X}^{n}\geq c_{0}>0$

for some uniform constant $c_{0}>0$. Taking limit and applying the dominated
convergence theorem again, we get
$\int_{X}|\hat{v}_{\infty}|\omega_{X}^{n}\geq c_{0}>0$, and this implies that
$|\alpha_{0}|>0$.

However, this will contradict the second equation in (3.7). To see this, we
assert that $\lim_{j\to\infty}\int_{X}\hat{v}_{\infty}\omega_{j}^{n}=0$. In
fact, by (3.7) we have

$\displaystyle|\int_{X}\hat{v}_{\infty}\omega_{j}^{n}|$ $\displaystyle=$
$\displaystyle|\int_{X}(\hat{v}_{j}-\hat{v}_{\infty})\omega_{j}^{n}|$
$\displaystyle\leq$ $\displaystyle
c_{t_{j}}\int_{X}|\hat{v}_{j}-\hat{v}_{\infty}|e^{F_{j}}\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle
C\int_{X}|\hat{v}_{j}-\hat{v}_{\infty}|^{\eta}e^{F_{j}}\omega_{X}^{n}\quad\mbox{
here }\eta=\frac{\epsilon}{1+\epsilon}$ $\displaystyle\leq$ $\displaystyle
C\Big{(}\int_{X}|\hat{v}_{j}-\hat{v}_{\infty}|\omega_{X}^{n}\Big{)}^{\epsilon/(1+\epsilon)}\Big{(}\int_{X}e^{(1+\epsilon)F_{j}}\omega_{X}^{n}\Big{)}^{1/(1+\epsilon)}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\int_{X}|\hat{v}_{j}-\hat{v}_{\infty}|\omega_{X}^{n}\Big{)}^{\epsilon/(1+\epsilon)}\to
0$

since $\hat{v}_{j}\to\hat{v}_{\infty}$ in $L^{1}(X,\omega_{X}^{n})$. But this
is absurd since $\int_{X}\hat{v}_{\infty}\omega_{j}^{n}=\alpha_{0}V_{t_{j}}$
which is strictly away from zero. This finishes the proof of Lemma 5.

Once the $L^{1}(X,\omega_{t}^{n})$-norm of $G_{t}$ is achieved, Theorem 1
provides a lower bound of $G_{t}$. (1.9) is thus proved. Let $C_{l}>0$ be the
constant in (1.9), i.e. $G_{t}\geq-C_{l}$. For notational simplicity, we will
denote the positive Green’s function

(3.15) ${\mathcal{G}}_{t}(x,\cdot)=G_{t}(x,\cdot)+C_{l}+1\geq 1.$

We are ready to prove (ii) in Theorem 2. To begin with, we show that the
$L^{q}(X,\omega_{t}^{n})$-norm of $G_{t}$ is uniformly bounded, for any
$q<\frac{2n}{2n-2}$, which is optimal in view of the asymptotic behavior of
$G_{t}$ in (1.4).

###### Lemma 6

For any $q\in(1,\frac{2n}{2n-2})$, there is a uniform constant $C>0$ depending
on $q$ such that $G_{t}$ satisfies

(3.16) $\int_{X}|G_{t}(x,y)|^{q}\omega_{t}^{n}(y)\leq C,\quad\forall~{}x\in
X.$

Proof.  We break the proof into two steps. We fix a point $x\in X$ and
consider the Green’s function $G_{t}(x,y)$ as a function of $y$. It suffices
to show (3.16) for ${\mathcal{G}}_{t}(x,\cdot)$, since $C_{l}>0$ in (3.15) is
uniform. The first step is to show the $L^{q}(X,\omega_{t}^{n})$ bound of
${\mathcal{G}}_{t}(x,\cdot)$ for any $q<1+1/n$, then we can apply an iteration
argument similar to the Moser iteration process to improve the exponent $q$.

Step 1. We will show the $L^{1+\frac{1}{r_{0}}}(X,\omega_{t}^{n})$-norm of
${\mathcal{G}}_{t}(x,\cdot)$ is uniformly bounded, for any $r_{0}>n$. The
argument is based on the $L^{1}$-bound of ${\mathcal{G}}_{t}(x,\cdot)$ in (i)
of Theorem 2.

Fix a large $k\gg 1$ and consider the function
$H_{k}(y)=\min\\{{\mathcal{G}}_{t}(x,y),k\\}$. By smoothing $H_{k}$ if
necessary we may assume it is a smooth function and $H_{k}(y)$ converges
monotonically increasingly to ${\mathcal{G}}_{t}(x,y)$ as $k\to\infty$. We
solve the following equation

(3.17)
$\left\\{\begin{array}[]{ll}&\Delta_{\omega_{t}}u_{k}=-H_{k}^{1/r_{0}}+\frac{1}{V_{t}}\int_{X}H_{k}^{1/r_{0}}\omega_{t}^{n},\\\
&\frac{1}{V_{t}}\int_{X}u_{k}\omega_{t}^{n}=0.\end{array}\right.$

Equation (3.17) admits a unique smooth solution since the smooth function on
the right-hand side of the first equation has integral $0$. To deal with the
unbounded term $-H_{k}^{1/r_{0}}$ on the right-hand side of (3.17) and non-
uniform ellipticity of the linear operator $\Delta_{\omega_{t}}$, we consider
again an auxiliary complex Monge-Ampère equation

(3.18)
$(\chi+t\omega_{X}+i\partial\bar{\partial}\psi_{k})^{n}=\frac{H_{k}^{n/r_{0}}+1}{V_{t}^{-1}\int_{X}(H_{k}^{n/r_{0}}+1)\omega_{t}^{n}}\omega_{t}^{n}=c_{t}\frac{H_{k}^{n/r_{0}}+1}{B_{k}}e^{F_{\omega_{t}}}\omega_{X}^{n},$

with $\sup_{X}\psi_{k}=0$ and
$B_{k}=\int_{X}(H_{k}^{n/r_{0}}+1)e^{F_{\omega_{t}}}\omega_{X}^{n}$. We stress
that this auxiliary Monge-Ampère equation plays a very different role from the
auxiliary Monge-Ampère equation introduced in the proof of Lemma 2. We note
that

(3.19) $V\leq B_{k}\leq
V+\Big{(}\int_{X}H_{k}e^{F_{\omega_{t}}}\omega_{X}^{n}\Big{)}^{n/r_{0}}\Big{(}\int_{X}e^{F_{\omega_{t}}}\omega_{X}^{n}\Big{)}^{(r_{0}-n)/r_{0}}\leq
C(V),$

and the upper bound holds because of $0<H_{k}\leq{\mathcal{G}}_{t}$ for any
$k$ and the integral bound of ${\mathcal{G}}_{t}$ in (i) of Theorem 2. We note
that the $p$-th (for some $p>n$) entropy of the function on the right-hand
side of (3.18) satisfies

$\displaystyle\frac{1}{B_{k}}\int_{X}(H_{k}^{n/r_{0}}+1)\Big{|}-\,{\rm
log}\,B_{k}+F_{\omega_{t}}+\,{\rm
log}\,(1+H_{k}^{n/r_{0}})\Big{|}^{p}e^{F_{\omega_{t}}}\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle\frac{|\,{\rm
log}\,B_{k}|^{p}}{B_{k}}\int_{X}(H_{k}^{n/r_{0}}+1)e^{F_{\omega_{t}}}\omega_{X}^{n}+\frac{1}{B_{k}}\int_{X}(H_{k}^{n/r_{0}}+1)[\,{\rm
log}\,(H_{k}^{n/r_{0}}+1)]^{p}e^{F_{\omega_{t}}}\omega_{X}^{n}$
$\displaystyle\quad+\frac{1}{B_{k}}\int_{X}(H_{k}^{n/r_{0}}+1)|F_{\omega_{t}}|^{p}e^{F_{\omega_{t}}}\omega_{X}^{n}\leq
C$

for some uniform constant $C>0$ depending on $n,p,\epsilon,\chi,\omega_{X}$
and $\|e^{F_{\omega_{t}}}\|_{L^{1+\epsilon}(\omega_{X}^{n})}$. Here the first
term in (3) is bounded due to the estimate of the constant $B_{k}$ in (3.19)
and Hölder inequality along with the uniform $L^{1}(X,\omega_{t}^{n})$-bound
of $H_{k}$; the second term is bounded because of $\,{\rm log}\,(1+x)\leq
C_{\delta}x^{\delta}$ for any $\delta>0$ and the
$L^{1}(X,\omega_{t}^{n})$-bound of $H_{k}$; and the last term is bounded again
by Hölder inequality. We can now apply Lemma 1 to conclude that

$\sup_{X}|\psi_{k}|\leq C,$

We know $\varphi_{t}$ also satisfies (3.2), so it follows that

(3.21) $\sup_{X}|\psi_{k}-\varphi_{t}|\leq C,$

for some uniform constant $C>0$, which is independent of $k$. We now consider
the function

(3.22)
$v:=(\psi_{k}-\varphi_{t})-\frac{1}{V_{t}}\int_{X}(\psi_{k}-\varphi_{t})\omega_{t}^{n}+\varepsilon^{\prime}u_{k},$

where $\varepsilon^{\prime}>0$ is a suitable constant to be chosen later. By
definition it follows $\frac{1}{V_{t}}\int_{X}v\omega_{t}^{n}=0$ and $v$ is a
smooth function. We then calculate the Laplacian of $v$ in (3.22)

$\displaystyle\Delta_{\omega_{t}}v$ $\displaystyle=$ $\displaystyle{\rm
tr}_{\omega_{t}}\omega_{t,\psi_{k}}-n+\varepsilon^{\prime}\Delta_{\omega_{t}}u_{k}$
$\displaystyle\geq$ $\displaystyle
n\Big{(}\frac{\omega_{t,\psi_{k}}^{n}}{\omega_{t}^{n}}\Big{)}^{1/n}-n-\varepsilon^{\prime}H_{k}^{1/r_{0}}+\frac{\varepsilon^{\prime}}{V_{t}}\int_{X}H_{k}^{1/r_{0}}\omega_{t}^{n}$
$\displaystyle=$ $\displaystyle
nB_{k}^{-1/n}(H_{k}^{n/r_{0}}+1)^{1/n}-n-\varepsilon^{\prime}H_{k}^{1/r_{0}}+\frac{\varepsilon^{\prime}}{V_{t}}\int_{X}H_{k}^{1/r_{0}}\omega_{t}^{n}$
$\displaystyle\geq$ $\displaystyle
nB_{k}^{-1/n}H_{k}^{1/r_{0}}-n-\varepsilon^{\prime}H_{k}^{1/r_{0}}\geq-n,$

if we choose $\varepsilon^{\prime}=nC(V)^{-1/n}$, where $C(V)$ is the upper
bound of $B_{k}$ in (3.19). We apply the Green’s formula to the function $v$
at $x$

$\displaystyle v(x)$ $\displaystyle=$
$\displaystyle\frac{1}{V_{t}}\int_{X}v\omega_{t}^{n}+\int_{X}G_{t}(x,\cdot)(-\Delta_{\omega_{t}}v)\omega_{t}^{n}=\int_{X}{\mathcal{G}}_{t}(x,\cdot)(-\Delta_{\omega_{t}}v)\omega_{t}^{n}$
$\displaystyle\leq$
$\displaystyle\int_{X}{\mathcal{G}}_{t}(x,\cdot)n\omega_{t}^{n}\leq C,$

where the last inequality follows from the uniform
$L^{1}(X,\omega_{t}^{n})$-bound of ${\mathcal{G}}_{t}(x,\cdot)$, as proved in
(i) of Theorem 2. It then follows from (3.21) and (3.22) that $u_{k}(x)\leq C$
for a uniform constant $C>0$.

We apply again the Green’s formula to the function $u_{k}$ at $x$ to get

$\displaystyle C\geq u_{k}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{V_{t}}\int_{X}u_{k}\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x,\cdot)(-\Delta_{\omega_{t}}u_{k})\omega_{t}^{n}$
$\displaystyle=$
$\displaystyle\int_{X}{\mathcal{G}}_{t}(x,y)\Big{(}H_{k}(y)^{1/r_{0}}-\frac{1}{V_{t}}\int_{X}H_{k}^{1/r_{0}}\omega_{t}^{n}\Big{)}\omega_{t}^{n}(y),$

which yields that

(3.23) $\int_{X}{\mathcal{G}}_{t}(x,y)H_{k}(y)^{1/r_{0}}\omega_{t}^{n}(y)\leq
C+C\int_{X}H_{k}^{1/r_{0}}\omega_{t}^{n}\leq C$

where we apply the Hölder inequality and the $L^{1}(X,\omega_{t}^{n})$-bound
of ${\mathcal{G}}_{t}(x,\cdot)$ and $H_{k}$. Finally letting $k\to\infty$ we
conclude from monotone convergence theorem and (3.23) that

(3.24)
$\int_{X}{\mathcal{G}}_{t}(x,y)^{1+\frac{1}{r_{0}}}\omega_{t}^{n}(y)\leq
C(r_{0}).$

Here we write the uniform constant as $C(r_{0})$ to emphasize its addintional
dependence on $r_{0}>n$.

Step 2. We now iterate the arguments in Step 1, given the
$L^{1+1/r_{0}}(X,\omega_{t}^{n})$-bound (3.24) of ${\mathcal{G}}_{t}$ for any
$r_{0}>n$. We take an arbitrary $r_{1}\in(0,r_{0})$ such that
$\frac{n}{r_{1}}<1+\frac{1}{r_{0}}$, and replace $r_{0}$ by $r_{1}$ in the
equations (3.17) and (3.18). Using the better
$L^{1+1/r_{0}}(X,\omega_{t}^{n})$-bound (3.24) of ${\mathcal{G}}_{t}$ instead
of just the $L^{1}(X,\omega_{t}^{n})$-bound as in Step 1, we can repeat the
arguments above to obtain the $L^{1+1/r_{1}}(X,\omega_{t}^{n})$-bound of
${\mathcal{G}}_{t}$. Iterating this process we eventually get for any
$r_{l}\in(0,r_{l-1})$ with $\frac{n}{r_{l}}<1+\frac{1}{r_{l-1}}$, the
$L^{1+1/r_{l}}(X,\omega_{t}^{n})$-bound of ${\mathcal{G}}_{t}$ is achieved,
which depends in addition on $l$. A simple combinatorial argument shows that
one can take any $r_{l}>0$ such that

$\frac{1}{r_{l}}<\frac{1}{n-1}-\frac{1}{n^{l}}\frac{1}{n(n-1)}.$

For $l$ large enough, this implies the bound on $L^{q}(X,\omega_{t}^{n})$-norm
of ${\mathcal{G}}_{t}(x,\cdot)$ for any $q<1+\frac{1}{n-1}=\frac{n}{n-1}$. As
we mentioned at the beginning, the $L^{q}(X,\omega_{t}^{n})$-bound of
$G_{t}(x,\cdot)$ follows from the definition of ${\mathcal{G}}_{t}(x,\cdot)$
in (3.15).

Now we derive the $L^{q}(X,\omega_{t}^{n})$ bound on the gradient of
${\mathcal{G}}_{t}(x,\cdot)$ for any $q\in(1,\frac{2n}{2n-1})$. First we
observe the following elementary estimate which follows easily from the
Green’s formula.

###### Lemma 7

For any $\beta>0$ we have

(3.25)
$\int_{X}\frac{|\nabla_{y}{\mathcal{G}}_{t}(x,y)|_{\omega_{t}(y)}^{2}}{{\mathcal{G}}_{t}(x,y)^{1+\beta}}\omega_{t}^{n}(y)\leq
1/\beta.$

Proof.  The function $u(y):={\mathcal{G}}_{t}(x,y)^{-\beta}$ is a continuous
function with $u(x)=0$ and $u\in C^{\infty}(X\backslash\\{x\\})$. By the
Green’s formula we have

$\displaystyle 0=u(x)$ $\displaystyle=$
$\displaystyle\frac{1}{V_{t}}\int_{X}u\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x,\cdot)(-\Delta_{\omega_{t}}u)\omega_{t}^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{V_{t}}\int_{X}u\omega_{t}^{n}-\beta\int_{X}\frac{|\nabla{\mathcal{G}}_{t}(x,\cdot)|^{2}_{\omega_{t}}}{{\mathcal{G}}_{t}(x,\cdot)^{1+\beta}}\omega_{t}^{n}.$

Here we have applied the integration by parts, which can be justified by the
asymptotic behavior of ${\mathcal{G}}_{t}(x,y)$ as in (1.4). The lemma then
follows easily from the fact that $0\leq u\leq 1$.

###### Lemma 8

For any $\delta\in(0,2/n)$ and $\beta>0$, there is a uniform constant $C>0$
depending on $\delta$ and $\beta$ such that

(3.26)
$\int_{X}{\mathcal{G}}_{t}(x,y)^{1-\frac{(1+\beta)(2-n\delta)}{2n}}|\nabla
G_{t}(x,y)|^{\frac{2}{n}-\delta}_{\omega_{t}(y)}\omega_{t}(y)\leq C.$

Proof.  Given Lemma 7, the proof of Lemma 8 is essentially the same as that in
Step 1 of Lemma 6. We use similar notations as in Lemma 6. Let

$H_{k}(y)=\widetilde{\min}\Big{\\{}\frac{|\nabla_{y}{\mathcal{G}}_{t}(x,y)|_{\omega_{t}(y)}^{2}}{{\mathcal{G}}_{t}(x,y)^{(1+\beta)}},k\Big{\\}}$

where $\widetilde{\min}$ denotes a suitable smoothing of the $\min$ function.
We can solve equations (3.17) and (3.18) with this $H_{k}$ and any $r_{0}>n$.
With the estimate (3.25) in Lemma 7, the same argument gives

$\int_{X}{\mathcal{G}}_{t}(x,y)H_{k}(y)^{1/r_{0}}\omega_{t}^{n}(y)\leq C.$

Letting $k\to\infty$, this yields

$\int_{X}{\mathcal{G}}_{t}(x,y)^{1-(1+\beta)/r_{0}}|\nabla_{y}{\mathcal{G}}_{t}(x,y)|_{\omega_{t}(y)}^{2/r_{0}}\omega_{t}^{n}(y)\leq
C.$

The lemma follows by setting $\delta=\frac{2}{n}-\frac{2}{r_{0}}$.

With Lemmas 7 and 8, we are ready to derive the
$L^{s}(X,\omega_{t}^{n})$-bound of $\nabla G_{t}(x,\cdot)$ for
$s\in[1,\frac{2n}{2n-1})$.

###### Lemma 9

For any given $s\in[1,\frac{2n}{2n-1})$, there is a uniform constant $C>0$
depending on $s$ such that

(3.27) $\int_{X}|\nabla G_{t}(x,y)|^{s}_{\omega_{t}(y)}\omega_{t}^{n}(y)\leq
C.$

Proof.  We fix a constant $1\leq s<\frac{2n}{2n-1}$ and a point $x\in X$. The
Green’s function $G_{t}(x,y)$ is viewed as a function of $y$ and all the
integrals below are integrated over $y\in X$. Then we calculate

$\displaystyle\int_{X}|\nabla
G_{t}(x,y)|^{s}_{\omega_{t}(y)}\omega_{t}^{n}(y)=\int_{X}(|\nabla
G_{t}|_{\omega_{t}}^{s\delta}{\mathcal{G}}_{t}^{a})\frac{|\nabla
G_{t}|_{\omega_{t}}^{(1-\delta)s}}{{\mathcal{G}}_{t}^{b}}{\mathcal{G}}_{t}^{b-a}\omega_{t}^{n}$
(3.28) $\displaystyle\leq$ $\displaystyle\Big{(}\int_{X}|\nabla
G_{t}|_{\omega_{t}}^{s\delta
p}{\mathcal{G}}_{t}^{ap}\omega_{t}^{n}\Big{)}^{1/p}\Big{(}\int_{X}\frac{|\nabla
G_{t}|_{\omega_{t}}^{(1-\delta)sq}}{{\mathcal{G}}_{t}^{bq}}\omega_{t}^{n}\Big{)}^{1/q}\Big{(}\int_{X}{\mathcal{G}}_{t}^{(b-a)r}\omega_{t}^{n}\Big{)}^{1/r}$

where the inequality follows from the generalized Hölder inequality and the
numbers involved are chosen as follows.

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1,\,0<\delta\ll
1,\,0<a<b,\,(b-a)r<\frac{n}{n-1}.$

Furthermore for a small $\beta>0$ we can pick these numbers as

(3.29) $(1-\delta)sq=2,\,\,bq=1+\beta,\,\,s\delta
p=\frac{2}{n}-\beta,\,\,ap=1-\frac{(1+\beta)(2-n\beta)}{2n}.$

With these choices of parameters, Lemma 7 and Lemma 8 imply the first two
factors in (3.28) are bounded. It only remains to verify the last factor in
(3.28) is also bounded, and by Lemma 6, it suffices to make sure that
$(b-a)r<\frac{2n}{2n-2}=\frac{n}{n-1}$. From (3.29), we derive

$q=\frac{2}{(1-\delta)s},\,p=\frac{{2}-n\beta}{ns\delta},\,b=\frac{(1+\beta)(1-\delta)s}{2},\,a=\frac{ns\delta}{{2}-n\beta}\Big{(}1-\frac{(1+\beta)(2-n\beta)}{2n}\Big{)}.$

Therefore, we have

$\displaystyle(b-a)r=:\Psi(\delta)$ $\displaystyle=$
$\displaystyle\Big{(}\frac{(1+\beta)(1-\delta)s}{2}-\frac{ns\delta}{{2}-n\beta}\big{(}1-\frac{(1+\beta)(2-n\beta)}{2n}\big{)}\Big{)}\frac{1}{1-\frac{(1-\delta)s}{2}-\frac{ns\delta}{2-n\beta}}.$

We note that as $s<\frac{2n}{2n-1}$

$\Psi(0)=\frac{(1+\beta)s}{2}\frac{1}{1-\frac{s}{2}}=(1+\beta)\frac{s}{2-s}<\frac{2n}{2n-2}$

if $\beta>0$ is chosen small enough. Since $\Psi(\delta)$ is continuous in
$\delta\geq 0$, $\Psi(\delta)<\frac{2n}{2n-2}$ if $\delta>0$ is sufficiently
small. This verifies the desired inequality $(b-a)r<\frac{n}{n-1}$. The proof
of Lemma 9 is completed.

As a corollary of Lemma 9, we have the following Sobolev-Morrey type
inequality for the metric $\omega_{t}$.

###### Corollary 1

For any $p>2n$ there is a uniform constant $C>0$ depending on $p$ such that

$\sup_{X}\Big{|}u-\frac{1}{V_{t}}\int_{X}u\omega_{t}^{n}\Big{|}\leq
C\Big{(}\int_{X}|\nabla
u|_{\omega_{t}}^{p}\omega_{t}^{n}\Big{)}^{1/p},\,\,\forall u\in C^{1}(X).$

Proof.  This follows immediately from the Green’s formula below, Lemma 9 and
Hölder inequality

$u(x)-\frac{1}{V_{t}}\int_{X}u\omega_{t}^{n}=\int_{X}\langle\nabla_{y}G_{t}(x,y),\nabla
u(y)\rangle_{\omega_{t}(y)}\omega_{t}^{n}(y),$

noting that the conjugate exponent $p^{*}=\frac{p}{p-1}<\frac{2n}{2n-1}$.

Proof of (ii) in Theorem 2.  This follows from Lemma 6 and Lemma 9.

We conclude this section by comparing the lower bound of the Green’s function
obtained in Theorem 2 with the classical one in Cheng-Li [3]. Let
$\omega=\omega_{X}+i\partial\bar{\partial}\varphi\in[\omega_{X}]$ be a Kähler
metric with $\|e^{F_{\omega}}\|_{L^{1+\epsilon}(X,\omega_{X}^{n})}\leq N$.
Suppose ${\rm Ric}(\omega)\geq-\kappa^{\prime}\omega$ for some
$\kappa^{\prime}\geq 0$, then from [13, 6] we know
${\mathrm{diam}}(X,\omega)\leq C(n,\omega_{X},N)$. Then Cheng-Li’s estimate
(1.2) implies the Green’s function associated with $\omega$ is bounded below.

We show now that, in the Kähler setting, under a Kolodziej type [16] condition
on the volume form, Theorem 2 implies the lower bound of the Green’s function
under the less restrictive assumption of a lower bound of the scalar
curvature:

###### Corollary 2

For any Kähler metric $\omega\in[\omega_{X}]$, if its relative volume form
$\|e^{F_{\omega}}\|_{L^{1+\epsilon}(X,\omega_{X}^{n})}\leq N$ for some
$\epsilon>0$, $N>0$, and its scalar curvature $R(\omega)\geq-\kappa$ for some
$\kappa\geq 0$, the Green’s function $G$ of $(X,\omega)$ satisfies

$\inf_{y\in X}G(x,y)\geq-C,\quad\forall x\in X,$

for some constant $C>0$ depending on $n,\omega_{X},\epsilon,N$ and $\kappa$.

Proof.  We claim that under the assumption of a scalar curvature lower bound,
the relative volume form satisfies $\inf_{X}e^{F_{\omega}}\geq\delta^{\prime}$
for some $\delta^{\prime}>0$. Hence it follows that
$\omega\in{\mathcal{M}}^{\prime}(N,\epsilon,\gamma)$ with
$\gamma=1/\delta^{\prime}$ and (i) in Theorem 2 implies a lower bound on the
Green’s function $G$.

To see the claim, note that by definition of the Ricci curvature, ${\rm
Ric}(\omega)={\rm Ric}(\omega_{X})+i\partial\bar{\partial}(-F_{\omega})$, so
the scalar curvature of $\omega$ satisfies $R(\omega)={\rm tr}_{\omega}({\rm
Ric}(\omega_{X}))+\Delta_{\omega}(-F_{\omega})$. The assumption that
$R(\omega)\geq-\kappa$ implies $\Delta_{\omega}(-F_{\omega})\geq-\kappa-{\rm
tr}_{\omega}({\rm Ric}(\omega_{X}))$. By Kolodziej’s $L^{\infty}$ estimate
[16] (see also [9]), the Kähler potential $\varphi$ of
$\omega=\omega_{X}+i\partial\bar{\partial}\varphi$ is bounded, i.e.
$\|\varphi\|_{L^{\infty}}\leq C$ for some $C=C(\epsilon,N)>0$, if we normalize
$\sup_{X}\varphi=0$. For a constant $A>0$ to be determined, we calculate

$\displaystyle\Delta_{\omega}(-F_{\omega}-A\varphi)$ $\displaystyle\geq$
$\displaystyle-\kappa-{\rm tr}_{\omega}({\rm Ric}(\omega_{X}))+A{\rm
tr}_{\omega}\omega_{X}-An$ $\displaystyle\geq$ $\displaystyle{\rm
tr}_{\omega}\omega_{X}-\kappa-An$ $\displaystyle\geq$ $\displaystyle
n\Big{(}\frac{\omega_{X}^{n}}{\omega^{n}}\Big{)}^{1/n}-\kappa-
An=ne^{-F_{\omega}/n}-\kappa-An$

where we take $A=C^{\prime}+1$ and $C^{\prime}>0$ is an upper bound of the
Ricci curvature ${\rm Ric}(\omega_{X})$. Applying maximum principle, we get at
the maximum point of $-F_{\omega}-A\varphi$,
$e^{-F_{\omega}}\leq(A+\kappa/n)^{n}$. Combined with the $L^{\infty}$ bound of
$\varphi$, this easily shows the upper bound of $-F_{\omega}$.

We observe that typically, the assumption of lower bound for the scalar
curvature is much more difficult to work with than the assumption of lower
bound for the Ricci curvature. There seem to be far fewer results under this
assumption. One of which, in a very different direction, is the recent result
of Munteanu-Wang [17] on the decay of the Green’s function on a real three-
dimensional complete manifold with scalar curvature bounded from below.

##  4 Applications

In this section we discuss some applications of the estimates on the Green’s
functions in Theorem 2, and provide the proof of Theorems 3 and 4. Given the
parameters $\epsilon,N,$ and $\gamma$, for each $t\in(0,1]$, we fix a Kähler
metric
$\omega_{t}\in{\mathcal{M}}_{t}^{\prime}(N,\epsilon,\gamma)\cup{\mathcal{M}}_{t}^{\prime\prime}(N,\epsilon,\gamma)\cup\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$.

Recall that $\chi$ a closed and nonnegative $(1,1)$-form such that its class
$[\chi]$ is big. By the definition of $F_{\omega_{t}}$ in (1.3), the metric
$\omega_{t}=\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t}$ satisfies the
following complex Monge-Ampère equation

(4.1)
$(\chi+t\omega_{X}+i\partial\bar{\partial}\varphi_{t})^{n}=c_{t}e^{F_{\omega_{t}}}\omega_{X}^{n},\quad\mbox{and
}\sup_{X}\varphi_{t}=0,$

where as before $c_{t}=V_{t}/V$ is a normalizing constant. By the assumptions
on $\omega_{t}$, the Green’s function $G_{t}$ associated with $\omega_{t}$
satisfies the estimates stated in Theorem 2.

Since $[\chi]$ is assumed to be big, Kodaira’s lemma implies that there is an
effective divisor $D$ such that

(4.2) $\chi-\varepsilon_{0}{\rm Ric}(h_{D})\geq\delta_{0}\omega_{X},$

where $h_{D}$ is a Hermtian metric on the line bundle $[D]$ associated to $D$
and $\varepsilon_{0}>0$ and $\delta_{0}>0$ are fixed constants depending only
on $\chi,\omega_{X}$. Let $s_{D}\in{\mathcal{O}}_{X}(D)$ be a holomorphic
section of $[D]$ defining $D$ and by rescaling $h_{D}$ if necessary we assume
$\sup_{X}|s_{D}|^{2}_{h_{D}}\leq 1$.

To ease the notations, throughout this section we will denote by $\tilde{g}$
and $g$ (omitting the subscript $t$ in $g_{t}$) the associated metrics of
$\omega_{X}$ and $\omega_{t}$, respectively.

We will omit the subscript $t$ in $\varphi_{t}$ which solves (4.1) and simply
write it as $\varphi$. The function $F_{\omega_{t}}$ will be simply written as
$F$, since $\omega_{t}$ is a fixed metric.

Recall that we denote ${\mathcal{G}}_{t}$ the positive Green’s function in
(3.15), which differs from $G_{t}$ by a uniform constant.

###  4.1 Gradient estimates

We will prove Theorem 3 in this subsection. Fix a constant $p>n$.

The lemma below follows from straightforward calculations, so we omit the
proof.

###### Lemma 10

Suppose $\varphi$ satisfies (4.1). We have

(4.3) $\Delta_{g}|\nabla\varphi|_{\tilde{g}}^{2}\geq 2Re\langle\nabla
F,\bar{\nabla}\varphi\rangle_{\tilde{g}}+g^{i\bar{j}}\tilde{g}^{k\bar{l}}(\varphi_{ki}\varphi_{\bar{j}\bar{l}}+\varphi_{k\bar{j}}\varphi_{i\bar{l}})-2K{\rm
tr}_{g}\tilde{g}|\nabla\varphi|^{2}_{\tilde{g}},$

where $-K$ is a lower bound of the bisectional curvature of the fixed metric
$\omega_{X}$ and $\varphi_{ki}$ denote the second-order covariant derivatives
of $\varphi$ with respect to $\omega_{X}$.

###### Lemma 11

The following inequality holds on $X$:

(4.4) $\Delta_{g}H\geq 2e^{-\lambda\hat{\varphi}}Re\langle\nabla
F,\bar{\nabla}\varphi\rangle_{\tilde{g}}+H{\rm tr}_{g}\tilde{g}-(2+n)\lambda
H-CH^{1/2}-C_{s}H^{1/2}{\rm tr}_{g}\tilde{g}.$

where $\hat{\varphi}=\varphi-\varepsilon_{0}\,{\rm log}\,|s_{D}|_{h_{D}}^{2}$,
and $H=e^{-\lambda\hat{\varphi}}|\nabla\varphi|^{2}_{\tilde{g}}$ for suitable
$\lambda>0$ depending only on $\varepsilon_{0},\delta_{0}$ and $\omega_{X}$.
The constants $C>0$ and $C_{s}>0$ in (4.4) are both uniform.

Proof.  As above, we denote by
$H=e^{-\lambda\hat{\varphi}}|\nabla\varphi|^{2}_{\tilde{g}}$ for some
$\lambda>0$ to be determined later. We calculate on $X\backslash D$

$\displaystyle\Delta_{g}H$ $\displaystyle=$ $\displaystyle
e^{-\lambda\hat{\varphi}}\Delta_{g}|\nabla\varphi|_{\tilde{g}}^{2}+2e^{-\lambda\hat{\varphi}}Re\langle-\lambda\nabla\hat{\varphi},\bar{\nabla}|\nabla\varphi|_{\tilde{g}}^{2}\rangle_{g}+H(-\lambda\Delta_{g}\hat{\varphi}+\lambda^{2}|\nabla\hat{\varphi}|_{g})$
$\displaystyle\geq$ $\displaystyle
e^{-\lambda\hat{\varphi}}\Big{(}2Re\langle\nabla
F,\bar{\nabla}\varphi\rangle_{\tilde{g}}+g^{i\bar{j}}\tilde{g}^{k\bar{l}}(\varphi_{ki}\varphi_{\bar{j}\bar{l}}+\varphi_{k\bar{j}}\varphi_{i\bar{l}})-2K{\rm
tr}_{g}\tilde{g}|\nabla\varphi|^{2}_{\tilde{g}}\Big{)}$
$\displaystyle+2e^{-\lambda\hat{\varphi}}Re\langle-\lambda\nabla\hat{\varphi},\bar{\nabla}|\nabla\varphi|_{\tilde{g}}^{2}\rangle_{g}+H(-\lambda\Delta_{g}\hat{\varphi}+\lambda^{2}|\nabla\hat{\varphi}|_{g}).$

We now perform the calculation at a fixed point $x_{0}\in X\backslash D$ and
choose normal coordinates at $x_{0}$ relative to $\tilde{g}$, such that
$\tilde{g}_{i\bar{j}}(x_{0})=\delta_{ij}$, $d\tilde{g}_{i\bar{j}}(x_{0})=0$
and $g_{i\bar{j}}(x_{0})=g_{i\bar{i}}\delta_{ij}$ is diagonal. Then at $x_{0}$
we have

$\displaystyle
2e^{-\lambda\hat{\varphi}}Re\langle-\lambda\nabla\hat{\varphi},\bar{\nabla}|\nabla\varphi|_{\tilde{g}}^{2}\rangle_{g}=-2\lambda
e^{-\lambda\hat{\varphi}}Re(g^{i\bar{i}}(\hat{\varphi}_{i}\varphi_{k\bar{i}}\varphi_{\bar{k}}+\hat{\varphi}_{i}\varphi_{\bar{k}\bar{i}}\varphi_{k}))$
$\displaystyle\geq$ $\displaystyle-2\lambda
e^{-\lambda\hat{\varphi}}Re(g^{i\bar{i}}\hat{\varphi}_{i}\varphi_{i\bar{i}}\varphi_{\bar{i}})-\lambda^{2}e^{-\lambda\hat{\varphi}}|\nabla\hat{\varphi}|_{g}^{2}|\nabla\varphi|_{\tilde{g}}^{2}-e^{-\lambda\hat{\varphi}}g^{i\bar{i}}\varphi_{\bar{k}\bar{i}}\varphi_{ki}$
$\displaystyle=$ $\displaystyle-2\lambda
e^{-\lambda\hat{\varphi}}Re(g^{i\bar{i}}\hat{\varphi}_{i}(g_{i\bar{i}}-1)\varphi_{\bar{i}})-\lambda^{2}H|\nabla\hat{\varphi}|_{g}^{2}-e^{-\lambda\hat{\varphi}}g^{i\bar{i}}\varphi_{\bar{k}\bar{i}}\varphi_{ki}$
$\displaystyle=$ $\displaystyle-2\lambda
e^{-\lambda\hat{\varphi}}\langle\nabla\hat{\varphi},\nabla\varphi\rangle_{\tilde{g}}+2\lambda
e^{-\lambda\hat{\varphi}}\langle\nabla\hat{\varphi},\nabla\varphi\rangle_{g}-\lambda^{2}H|\nabla\hat{\varphi}|_{g}^{2}-e^{-\lambda\hat{\varphi}}g^{i\bar{i}}\varphi_{\bar{k}\bar{i}}\varphi_{ki}.$

And by (4.2) we have

$-\Delta_{g}\hat{\varphi}={\rm
tr}_{g}(\chi+t\omega_{X}-\omega_{t}-\varepsilon_{0}{\rm
Ric}(h_{D}))\geq\delta_{0}{\rm tr}_{g}\tilde{g}-n$

Substituting the above two (in)equalities to (4.1), we obtain that at $x_{0}$

(4.6) $\displaystyle\Delta_{g}H$ $\displaystyle\geq$ $\displaystyle
e^{-\lambda\hat{\varphi}}\Big{(}2Re\langle\nabla
F,\bar{\nabla}\varphi\rangle_{\tilde{g}}+g^{i\bar{j}}\tilde{g}^{k\bar{l}}\varphi_{k\bar{j}}\varphi_{i\bar{l}}-2K{\rm
tr}_{g}\tilde{g}|\nabla\varphi|^{2}_{\tilde{g}}\Big{)}+\lambda
H(\delta_{0}{\rm tr}_{g}\tilde{g}-n)$ $\displaystyle-2\lambda
e^{-\lambda\hat{\varphi}}\langle\nabla\hat{\varphi},\nabla\varphi\rangle_{\tilde{g}}+2\lambda
e^{-\lambda\hat{\varphi}}\langle\nabla\hat{\varphi},\nabla\varphi\rangle_{g}$
$\displaystyle\geq$ $\displaystyle 2e^{-\lambda\hat{\varphi}}Re\langle\nabla
F,\bar{\nabla}\varphi\rangle_{\tilde{g}}+(\lambda\delta_{0}-2K)({\rm
tr}_{g}\tilde{g})H-\lambda nH-2\lambda
e^{-\lambda\hat{\varphi}}|\nabla\varphi|_{\tilde{g}}^{2}+2\lambda
e^{-\lambda\hat{\varphi}}|\nabla\varphi|_{g}^{2}$
$\displaystyle-2\varepsilon_{0}\lambda
e^{-\lambda\varphi}|s_{D}|_{h_{D}}^{2\lambda\varepsilon_{0}-1}|\nabla_{h_{D}}s_{D}|_{\tilde{g},h_{D}}|\nabla\varphi|_{\tilde{g}}-2\varepsilon_{0}\lambda
e^{-\lambda\varphi}|s_{D}|_{h_{D}}^{2\lambda\varepsilon_{0}-1}|\nabla_{h_{D}}s_{D}|_{g,h_{D}}|\nabla\varphi|_{g},$

where we denote by $\nabla_{h_{D}}$ the Chern connection of the hermitian
metric $h_{D}$ on the line bundle $[D]$. We choose $\lambda>1$ large enough so
that $\lambda\delta_{0}>2K+10$ and $2\lambda\varepsilon_{0}>10$. Then the
first term in (4.6) satisfies

(4.7) $-2\varepsilon_{0}\lambda
e^{-\lambda\varphi}|s_{D}|_{h_{D}}^{2\lambda\varepsilon_{0}-1}|\nabla_{h_{D}}s_{D}|_{\tilde{g},h_{D}}|\nabla\varphi|_{\tilde{g}}\geq-
Ce^{-\lambda\varphi/2}|s_{D}|_{h_{D}}^{\lambda\varepsilon_{0}-1}H^{1/2}\geq-
CH^{1/2},$

because $|\nabla_{h_{D}}s_{D}|_{\tilde{g},h_{D}}\leq C$ and
$\|\varphi\|_{L^{\infty}}\leq C$. Similarly the last term in (4.6) satisfies

(4.8) $-2\varepsilon_{0}\lambda
e^{-\lambda\varphi}|s_{D}|_{h_{D}}^{2\lambda\varepsilon_{0}-1}|\nabla_{h_{D}}s_{D}|_{g,h_{D}}|\nabla\varphi|_{g}\geq-
C_{s}H^{1/2}{\rm tr}_{g}\tilde{g},$

for some uniform constant $C_{s}>0$. Plugging (4.7) and (4.8) into (4.6), we
obtain that at $x_{0}\in X\backslash D$

(4.9) $\Delta_{g}H\geq 2e^{-\lambda\hat{\varphi}}Re\langle\nabla
F,\bar{\nabla}\varphi\rangle_{\tilde{g}}+H{\rm tr}_{g}\tilde{g}-(2+n)\lambda
H-CH^{1/2}-C_{s}H^{1/2}{\rm tr}_{g}\tilde{g}.$

Since $x_{0}\in X\backslash D$ is arbitrary, $X\backslash D\subset X$ is
clearly dense and both sides of (4.9) are smooth, we see that (4.9) holds
globally on $X$.

Proof of Theorem 3. Let $C_{s}>0$ be the constant in Lemma 11 and denote
$\Lambda=C_{s}^{2}+1$. From Lemma 11, we have for some uniform constant $C>0$

(4.10) $\Delta_{g}H\geq-C|\nabla
F|_{\tilde{g}}H^{1/2}+(H^{1/2}-C_{s})H^{1/2}{\rm tr}_{g}\tilde{g}-CH-C$

We consider the convex and monotonically increasing function
$\Phi_{\delta}(x)=\frac{1}{2}(\sqrt{x^{2}+\delta}+x)+\Lambda$ for $\delta>0$.
It is clear that

$\Phi_{\delta}(H-\Lambda)\to\max(H,\Lambda),\mbox{ as }\delta\to 0.$

We also have $0\leq\Phi_{\delta}^{\prime}(x)\leq 1$,
$\Phi_{\delta}^{\prime\prime}(x)\geq 0$ for all $x\in{\bf R}$ and

(4.11)
$0\leq\Phi_{\delta}^{\prime}(x)=\frac{1}{2}\frac{\delta}{\sqrt{x^{2}+\delta}(\sqrt{x^{2}+\delta}-x)}\leq\frac{\delta}{2},\mbox{
when }x\leq-1.$

We denote $\hat{H}=H-\Lambda$ and calculate

$\displaystyle\Delta_{g}\Phi_{\delta}(\hat{H})$ $\displaystyle=$
$\displaystyle\,\Phi_{\delta}^{\prime}\Delta_{g}H+\Phi_{\delta}^{\prime\prime}|\nabla
H|_{g}^{2}$ $\displaystyle\geq$
$\displaystyle\,\Phi_{\delta}^{\prime}(\hat{H})\Big{(}-C|\nabla
F|_{\tilde{g}}H^{1/2}+(H^{1/2}-C_{s})H^{1/2}{\rm
tr}_{g}\tilde{g}-CH-C\Big{)}\chi_{\\{\hat{H}\geq-1\\}}$
$\displaystyle\,+\Delta_{g}H\cdot\Phi_{\delta}^{\prime}(\hat{H})\chi_{\\{\hat{H}<-1\\}}$
$\displaystyle\geq$
$\displaystyle\,\Phi_{\delta}^{\prime}(\hat{H})\Big{(}-C|\nabla
F|_{\tilde{g}}H^{1/2}-CH-C\Big{)}\chi_{\\{\hat{H}\geq-1\\}}-\delta|\Delta_{g}H|\chi_{\\{\hat{H}<-1\\}}$

where the second inequality follows since on $\\{\hat{H}\geq-1\\}$,
$H\geq\Lambda-1\geq C_{s}^{2}$, and on $\\{\hat{H}<-1\\}$, we have
$\Phi_{\delta}^{\prime}(\hat{H})\leq\delta$ by (4.11). Applying Green’s
formula to the smooth function $\Phi_{\delta}(\hat{H})$ we obtain for any
$x\in X$

$\displaystyle\Phi_{\delta}(\hat{H})(x)$ $\displaystyle=$
$\displaystyle\,\frac{1}{V_{t}}\int_{X}\Phi_{\delta}(\hat{H})\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x,\cdot)(-\Delta_{g}\Phi_{\delta}(\hat{H}))\omega_{t}^{n}$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{V_{t}}\int_{X}\Phi_{\delta}(\hat{H})\omega_{t}^{n}+\delta\int_{\\{\hat{H}<-1\\}}{\mathcal{G}}_{t}(x,\cdot)|\Delta_{g}H|\omega_{t}^{n}$
$\displaystyle\,+\int_{\\{\hat{H}\geq-1\\}}{\mathcal{G}}_{t}(x,\cdot)\Phi_{\delta}^{\prime}(\hat{H})\Big{(}C|\nabla
F|_{\tilde{g}}H^{1/2}+CH+C\Big{)}\omega_{t}^{n}.$

Letting $\delta\to 0$ in (4.1), we get (denoting
$H_{\Lambda}=\max\\{H,\Lambda\\}$)

(4.13) $\displaystyle
H_{\Lambda}(x)\leq\frac{1}{V_{t}}\int_{X}H_{\Lambda}\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x,\cdot)\Big{(}C|\nabla
F|_{\tilde{g}}H^{1/2}+CH+C\Big{)}\omega_{t}^{n}.$

Let $x_{1}\in X$ be a maximum point of $H_{\Lambda}$. Then we obtain from
(4.13) that

$\displaystyle H_{\Lambda,\max}$ $\displaystyle=$
$\displaystyle\,H_{\Lambda}(x_{1})\leq\frac{c_{t}H_{\Lambda,\max}^{1-\eta}}{V_{t}}\int_{X}H_{\Lambda}^{\eta}e^{F}\omega_{X}^{n}$
$\displaystyle\,+CH_{\Lambda,\max}^{1/2}\int_{X}{\mathcal{G}}_{t}(x_{1},\cdot)|\nabla
F|_{\tilde{g}}\omega_{t}^{n}+CH_{\Lambda,\max}^{1-\eta}\int_{X}{\mathcal{G}}_{t}(x_{1},\cdot)H^{\eta}\omega_{t}^{n}+C$
$\displaystyle\leq$
$\displaystyle\,\frac{c_{t}H_{\Lambda,\max}^{1-\eta}}{V_{t}}\Big{(}\int_{X}e^{(1+\epsilon)F}\omega_{X}^{n}\Big{)}^{\frac{1}{1+\epsilon}}\Big{(}\int_{X}H_{\Lambda}^{\eta(1+\epsilon)/\epsilon}\omega_{X}^{n}\Big{)}^{\frac{\epsilon}{1+\epsilon}}+C$
$\displaystyle\,+CH_{\Lambda,\max}^{1/2}\Big{(}\int_{X}{\mathcal{G}}_{t}(x_{1},\cdot)^{p^{*}}\omega_{t}^{n}\Big{)}^{{1/}{p^{*}}}\Big{(}\int_{X}|\nabla
F|_{\tilde{g}}^{p}c_{t}e^{F}\omega_{X}^{n}\Big{)}^{{1/}{p}}$
$\displaystyle\,+CH_{\Lambda,\max}^{1-\eta}\Big{(}\int_{X}{\mathcal{G}}_{t}(x_{1},\cdot)^{p_{0}}\omega_{t}^{n}\Big{)}^{1/p_{0}}\Big{(}\int_{X}H^{\eta
p_{0}^{*}}\omega_{t}^{n}\Big{)}^{1/p_{0}^{*}},$

where we fix a number $p_{0}\in(1,\frac{n}{n-1})$. By assumption $p>n$ so we
have $p^{*}<\frac{n}{n-1}$. If furthermore we choose $\eta>0$ small (depending
on only $\epsilon$ and $p_{0}$) such that $\eta(1+\epsilon)/\epsilon\leq 1$
and $\eta p_{0}^{*}\leq\epsilon/(1+\epsilon)$, then by Theorem 2, and Lemma 12
below it follows that all the integrals involved in (4.1) are bounded
uniformly from above. It then follows that

$H_{\Lambda,\max}\leq CH_{\Lambda,\max}^{1-\eta}+CH_{\Lambda,\max}^{1/2}+C.$

By Young’s inequality we immediately derive the uniform upper bound of
$H_{\Lambda,\max}$. Hence the upper bound of
$H=e^{-\lambda\varphi}|s_{D}|^{2\varepsilon_{0}\lambda}_{h_{D}}|\nabla\varphi|^{2}_{\omega_{X}}$.
The proof of Theorem 3 is completed.

It only remains to show the integrals of $H$ are bounded.

###### Lemma 12

There is a uniform constant $C>0$ such that

$\int_{X}H\omega_{X}^{n}=\int_{X}e^{-\lambda\varphi}|s_{D}|^{2\lambda\varepsilon_{0}}_{h_{D}}|\nabla\varphi|^{2}_{\omega_{X}}\omega_{X}^{n}\leq
C,$

and when integrated against $\omega_{t}^{n}$ we have

$\int_{X}H^{\frac{\epsilon}{1+\epsilon}}\omega_{t}^{n}\leq C.$

Proof. Recall we write $\hat{\varphi}=\varphi-\varepsilon_{0}\,{\rm
log}\,|s_{D}|^{2}_{h_{D}}$. For any small $\delta>0$, we denote the super-
level set of $|s_{D}|_{h_{D}}$, $E_{\delta}=\\{|s_{D}|_{h_{D}}\geq\delta\\}$.
We write $\hat{\omega}=\chi-\varepsilon_{0}{\rm
Ric}(h_{D})\geq\delta_{0}\omega_{X}$ as in (4.2). Observe that on $E_{\delta}$
the following equation holds

$\displaystyle\omega_{t}^{n}-(\hat{\omega}+t\omega_{X})^{n}$ $\displaystyle=$
$\displaystyle(\hat{\omega}+t\omega_{X}+i\partial\bar{\partial}\hat{\varphi})^{n}-(\hat{\omega}+t\omega_{X})^{n}$
$\displaystyle=$
$\displaystyle\,i\partial\bar{\partial}\hat{\varphi}\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1}).$

Multiplying both sides by $e^{-\lambda\hat{\varphi}}$ and integrating over
$E_{\delta}$, we obtain by integration by parts

$\displaystyle\,\frac{\lambda}{n}\int_{E_{\delta}}e^{-\lambda\hat{\varphi}}|\nabla\hat{\varphi}|^{2}_{\hat{\omega}}\hat{\omega}^{n}\leq{\lambda}\int_{E_{\delta}}e^{-\lambda\hat{\varphi}}i\partial\hat{\varphi}\wedge\bar{\partial}\hat{\varphi}\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1})$
$\displaystyle=$ $\displaystyle\,\int_{\partial
E_{\delta}}e^{-\lambda\hat{\varphi}}i\bar{\partial}\hat{\varphi}\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1})+\int_{E_{\delta}}e^{-\lambda\hat{\varphi}}(\omega_{t}^{n}-(\hat{\omega}+t\omega_{X})^{n})$
$\displaystyle\leq$ $\displaystyle\,\int_{\partial
E_{\delta}}e^{-\lambda\hat{\varphi}}i\bar{\partial}\hat{\varphi}\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1})+C,$

since the function
$e^{-\lambda\hat{\varphi}}=e^{-\lambda\varphi}|s_{D}|^{2\lambda\varepsilon_{0}}_{h_{D}}$
is uniformly bounded. We rewrite the boundary integral as follows.

$\displaystyle\,\int_{\partial
E_{\delta}}e^{-\lambda\hat{\varphi}}i\bar{\partial}(\varphi-\varepsilon_{0}\,{\rm
log}\,|s_{D}|^{2}_{h_{D}})\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1})$
$\displaystyle=$ $\displaystyle\ \int_{\partial
E_{\delta}}e^{-\lambda\varphi}|s_{D}|^{2\lambda\varepsilon_{0}-2}_{h_{D}}\,\eta\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1})$
$\displaystyle=$
$\displaystyle\,\delta^{2\lambda\epsilon_{0}-2}\int_{E_{\delta}}\,d(e^{-\lambda\varphi}\,\eta)\wedge(\omega_{t}^{n-1}+\cdots+(\hat{\omega}+t\omega_{X})^{n-1})$

Here
$\eta=i|s_{D}|^{2}_{{}_{h_{D}}}\bar{\partial}(\varphi-\varepsilon_{0}\,{\rm
log}\,|s_{D}|^{2}_{h_{D}})$ is a smooth $1$-form on $X$ and in the last line
we have integrated by parts again. Now the last integrand is a smooth $(n,n)$
form on $X$ which is independent of $\delta$ and thus the integral remains
bounded as $\delta\rightarrow 0$. On the other hand,
$2\lambda\epsilon_{0}-2\geq 10-2>0$ so the last line tends to zero as
$\delta\rightarrow 0$. Letting $\delta\to 0$ we get from the equivalence of
the fixed metrics $\omega_{X}$ and $\hat{\omega}$ that

$\int_{X}e^{-\lambda\hat{\varphi}}|\nabla\hat{\varphi}|_{\omega_{X}}^{2}\omega_{X}^{n}\leq
C.$

The first inequality in the lemma then follows from the following triangle
inequality

$\int_{X}e^{-\lambda\hat{\varphi}}|\nabla\varphi|_{\omega_{X}}^{2}\omega_{X}^{n}\leq
2\int_{X}e^{-\lambda\hat{\varphi}}|\nabla\hat{\varphi}|_{\omega_{X}}^{2}\omega_{X}^{n}+2\varepsilon_{0}^{2}\int
e^{-\lambda\varphi}|s_{D}|^{2\varepsilon_{0}\lambda}_{h_{D}}|\nabla\,{\rm
log}\,|s_{D}|_{h_{D}}^{2}|^{2}_{\omega_{X}}\omega_{X}^{n}\leq C.$

The second inequality follows from Hölder inequality

$\int_{X}H^{\frac{\epsilon}{1+\epsilon}}\omega_{t}^{n}\leq
c_{t}\Big{(}\int_{X}e^{(1+\epsilon)F}\omega_{X}^{n}\Big{)}^{1/(1+\epsilon)}\Big{(}\int_{X}H\omega_{X}^{n}\Big{)}^{\epsilon/(1+\epsilon)}\leq
C.$

The proof of Lemma 12 is completed.

We recall the gradient estimate proved in Theorem 3 for complex Monge-Ampère
equations with a fixed background metric.

(4.15)
$(\omega_{X}+i\partial\bar{\partial}\varphi)^{n}=e^{F}\omega_{X}^{n},\quad\sup_{X}\varphi=0.$

The conditions on the sets ${\mathcal{M}}^{\prime}(N,\epsilon,\gamma)$,
${\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$ or
$\tilde{\mathcal{M}}^{\prime\prime}(N,\epsilon,\gamma)$ state that

(4.16) $\|e^{F}\|_{L^{1+\epsilon}(X,\omega_{X}^{n})}\leq N,\quad\mbox{and}$
(4.17) $\sup e^{-F}\leq\gamma,\,\mbox{or
}\int_{X}(e^{-F}+|\Delta_{\omega_{X}}e^{-F}|)\omega_{X}^{n}\leq\gamma,\,\mbox{or
}\int_{X}(e^{-F}+|\nabla e^{-F}|_{\omega_{X}}^{2})\omega_{X}^{n}\leq\gamma$

A corollary of Theorem 3 on equation (4.15) states that

###### Corollary 3

Fix a constant $p>n$. Let $\varphi$ solve the equation (4.15). Suppose $F$
satisfies (4.16) and (4.17), then the following gradient estimate of $\varphi$
w.r.t. $\omega_{X}$ holds

(4.18) $\sup_{X}|\nabla\varphi|^{2}_{\omega_{X}}\leq C,$

where $C>0$ depends on $n,p,\omega_{X},N,\epsilon,\gamma$, and additionally
$\int_{X}|\nabla F|_{\omega_{X}}^{p}e^{F}\omega_{X}^{n}$.

Corollary 3 follows from Theorem 3 as a particular case if we choose
$\chi=\omega_{X}/2$, $t=1/2$, $D={\mathbf{0}}$ (the trivial divisor),
$s_{D}\equiv 1$ and $h_{D}\equiv 1$.

We remark that besides the integral bounds on $e^{\pm F}$ and $|\nabla F|$,
the gradient estimate of $\varphi$ can be made to be independent of the
pointwise bounds $\sup_{X}e^{F}$ and $\inf_{X}e^{F}$. The exponent $p>n$ in
the $L^{p}(X,e^{F}\omega_{X}^{n})$-bound of $|\nabla F|_{\omega_{X}}$ is also
sharp in the sense that the gradient estimate may fail if $p<n$ and all other
conditions on $F$ are still valid, as the following example shows. We do not
know whether the statement holds or not when $p=n$.

Example 3.1. Let $0\in{\bf C}^{n}\subset{\bf{CP}}^{n}$ and
$z=(z_{1},\ldots,z_{n})$ be the natural coordinates on ${\bf C}^{n}$. Fix a
number $a\in(0,1)$ and a small $\delta\in(0,1/100)$. We consider the function

$\varphi_{\delta}=\left\\{\begin{array}[]{lll}&(|z|^{2}+\delta)^{a},&\mbox{ if
}|z|<\zeta\\\ &\widetilde{\max}\Big{\\{}(|z|^{2}+\delta)^{a},2\,{\rm
log}\,(1+|z|^{2})\Big{\\}},&\mbox{ if }\zeta\leq|z|\leq 1\\\ &2\,{\rm
log}\,(1+|z|^{2}),&\mbox{ if }|z|\geq 1,\end{array}\right.$

where $\zeta>0$ is a constant (independent of $\delta$) such that
$(|z|^{2}+\delta)^{a}=2\,{\rm log}\,(1+|z|^{2})$ for some $|z|\in(\zeta,1)$.
The metric $\omega_{\delta}=i\partial\bar{\partial}\varphi_{\delta}$ on ${\bf
C}^{n}$ can be naturally extended to a smooth Kähler metric on
$X={\bf{CP}}^{n}$. We express the metric $\omega_{\delta}$ locally near $0$

$\omega_{\delta}=\,\sum_{i,j}a\frac{\delta_{ij}+(a-1){\bar{z}_{i}z_{j}/}{(|z|^{2}+\delta)}}{(|z|^{2}+\delta)^{1-a}}\sqrt{-1}dz_{i}\wedge
d\bar{z}_{j}$

so near $0$ we have

$\omega_{\delta}^{n}=a^{n}\frac{(a|z|^{2}+\delta)/(|z|^{2}+\delta)}{(|z|^{2}+\delta)^{n-an}}\omega_{\bf
C^{n}}^{n}=:e^{F}\omega_{\bf C^{n}}^{n}.$

If $a<1/2$, we can choose $0<p<2an<n$. By straightforward calculations, we see
that near $0$

(4.19) $|\nabla F|^{p}_{\omega_{{\bf C}^{n}}}e^{F}\leq
C\frac{|z|^{p}}{|z|^{2p+2n-2an}}=C|z|^{-p+2an-2n}$

and the function on the right-hand side is integrable near $z=0$. Since
$\omega_{{\bf C}^{n}}$ is equivalent to $\omega_{FS}$ (the Fubini-Study metric
on ${\bf{CP}}^{n}$) near $0$, it follows easily that $\int_{X}|\nabla
F|_{\omega_{FS}}^{p}e^{F}\omega_{FS}^{n}$ is uniformly bounded above
(independent of $\delta>0$). Moreover, the other conditions (4.16) and (4.17)
of $F$ are satisfied for certain $\epsilon,N$ and $\gamma$. Here we have
viewed $F$ as a smooth function on the whole manifold ${\bf{CP}}^{n}$.
However, near $0$ (e.g. at $|z|^{2}=\delta$)

$|\nabla\varphi_{\delta}|_{\omega_{\bf
C^{n}}}\sim\frac{|z|}{(|z|^{2}+\delta)^{1-a}}\sim\delta^{a-\frac{1}{2}}$

blows up as $\delta\to 0$. Therefore when $p<n$, the integral $\int_{X}|\nabla
F|_{\omega_{X}}^{p}e^{F}\omega_{X}^{n}$ is not enough to conclude the gradient
estimate of $\varphi$ which satisfies (4.15).

###  4.2 $C^{2}$ estimate

We consider the $C^{2}$ estimates in this subsection and give the proof of
Theorem 4. We continue to use the same notations as in the previous
subsection. Fix a number $p>2n$.

Let $\varphi_{t}$ be the solution to the equation (4.1). We again omit the
subscript $t$ in $\varphi_{t}$.

###### Lemma 13

The following holds for any $t\in(0,1]$:

$\displaystyle\Delta_{g}{\rm tr}_{\tilde{g}}g$ $\displaystyle\geq$
$\displaystyle\,\Delta_{\tilde{g}}F-K{\rm tr}_{g}\tilde{g}\cdot{\rm
tr}_{\tilde{g}}g+\tilde{g}^{k\bar{l}}\nabla^{\tilde{g}}_{k}g_{i\bar{q}}\bar{\nabla}^{\tilde{g}}_{\bar{l}}g_{p\bar{j}}g^{p\bar{q}}g^{i\bar{j}}$
$\displaystyle\geq$ $\displaystyle\,\Delta_{\tilde{g}}F-K{\rm
tr}_{g}\tilde{g}\cdot{\rm tr}_{\tilde{g}}g+\frac{|\nabla{\rm
tr}_{\tilde{g}}g|^{2}_{g}}{{\rm tr}_{\tilde{g}}g},$

where as before $-K$ is a lower bound of the bisectional curvature of
$\tilde{g}=\omega_{X}$, and $\nabla^{\tilde{g}}$ denotes the covariant
derivatives with respect to $\tilde{g}$.

Lemma 13 follows from standard calculations in [25], so we omit the proof.

###### Lemma 14

For some $\mu>0$ depending only on $\chi$, and $\omega_{X}$, we have

$\Delta_{g}Q\geq e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F-n\mu Q,$

where we have written $Q=e^{-\mu\hat{\varphi}}{\rm tr}_{\tilde{g}}g$ and
$\hat{\varphi}=\varphi-\varepsilon_{0}\,{\rm log}\,|s_{D}|^{2}_{h_{D}}$.

Proof.  We fix a constant $\mu>0$ to be determined later. We calculate using
Lemma 13

$\displaystyle\Delta_{g}Q$ $\displaystyle=$
$\displaystyle\,e^{-\mu\hat{\varphi}}\Delta_{g}{\rm tr}_{\tilde{g}}g-2\mu
e^{-\mu\hat{\varphi}}Re\langle\nabla\hat{\varphi},\bar{\nabla}{\rm
tr}_{\tilde{g}}g\rangle_{g}+Q(-\mu\Delta_{g}\hat{\varphi}+\mu^{2}|\nabla\hat{\varphi}|_{g}^{2})$
$\displaystyle\geq$
$\displaystyle\,e^{-\mu\hat{\varphi}}\big{(}\Delta_{\tilde{g}}F-K{\rm
tr}_{g}\tilde{g}\cdot{\rm tr}_{\tilde{g}}g+\frac{|\nabla{\rm
tr}_{\tilde{g}}g|^{2}_{g}}{{\rm tr}_{\tilde{g}}g}\big{)}-2\mu
e^{-\mu\hat{\varphi}}Re\langle\nabla\hat{\varphi},\bar{\nabla}{\rm
tr}_{\tilde{g}}g\rangle_{g}$ $\displaystyle\,\,+\mu Q\Big{[}-n+t\cdot{\rm
tr}_{g}\tilde{g}+{\rm tr}_{g}(\chi-\varepsilon_{0}{\rm
Ric}(h_{D}))\Big{]}+\mu^{2}e^{-\mu\hat{\varphi}}{\rm
tr}_{\tilde{g}}g\cdot|\nabla\hat{\varphi}|_{g}^{2}.$

Applying Cauchy-Schwarz inequality the following holds

$-2\mu e^{-\mu\hat{\varphi}}Re\langle\nabla\hat{\varphi},\bar{\nabla}{\rm
tr}_{\tilde{g}}g\rangle_{g}\geq-\mu^{2}e^{-\mu\hat{\varphi}}{\rm
tr}_{\tilde{g}}g|\nabla\hat{\varphi}|_{g}^{2}-e^{-\mu\hat{\varphi}}\frac{|\nabla{\rm
tr}_{\tilde{g}}g|^{2}_{g}}{{\rm tr}_{\tilde{g}}g},$

plugging this into (4.2) and applying (4.2) we obtain

$\displaystyle\Delta_{g}Q$ $\displaystyle\geq$
$\displaystyle\,e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F-n\mu
Q+(\mu\delta_{0}-K)e^{-\mu\hat{\varphi}}{\rm tr}_{g}\tilde{g}\cdot{\rm
tr}_{\tilde{g}}g$ $\displaystyle\geq$
$\displaystyle\,e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F-n\mu Q,$

if we choose $\mu>0$ such that $\mu\delta_{0}-K\geq 1$.

Proof of Theorem 4. By Lemma 14 and Green’s formula we have for any $x\in X$

(4.22) $\displaystyle Q(x)$ $\displaystyle=$
$\displaystyle\;\frac{1}{V_{t}}\int_{X}Q\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x,\cdot)(-\Delta_{g}Q)\omega_{t}^{n}$
$\displaystyle\leq$
$\displaystyle\;\frac{1}{V_{t}}\int_{X}Q\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x,\cdot)(-e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F+n\mu
Q)\omega_{t}^{n}.$

Let $x_{0}$ be a maximum point of $Q$. We apply (4.22) at the point $x_{0}$,
then

(4.23) $\displaystyle Q_{\max}$ $\displaystyle=$
$\displaystyle\;Q(x_{0})\leq\frac{1}{V_{t}}\int_{X}Q\omega_{t}^{n}+\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)(-e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F+n\mu
Q)\omega_{t}^{n}$ $\displaystyle\leq$
$\displaystyle\;\frac{Q_{\max}^{1-\eta}}{V_{t}}\int_{X}Q^{\eta}\omega_{t}^{n}+n\mu
Q_{\max}^{1-\eta}\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)Q^{\eta}\omega_{t}^{n}+c_{t}\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)(-e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F)e^{F}\omega_{X}^{n}.$

We observe that by Hölder inequality

(4.24)
$\int_{X}Q^{\frac{\epsilon}{1+\epsilon}}\omega_{t}^{n}=c_{t}\int_{X}Q^{\frac{\epsilon}{1+\epsilon}}e^{F}\omega_{X}^{n}\leq
c_{t}\|e^{F}\|_{L^{1+\epsilon}}\Big{(}\int_{X}e^{-\mu\hat{\varphi}}({\rm
tr}_{\omega_{X}}\omega_{t})\omega_{X}^{n}\Big{)}^{(1+\epsilon)/\epsilon}\leq
C,$

since
$e^{-\mu\hat{\varphi}}=e^{-\mu\varphi}|s_{D}|_{h_{D}}^{2\mu\varepsilon_{0}}\leq
C$. By Hölder inequality and (4.24), the integral in the first term in (4.23)
is bounded if $0<\eta\leq\frac{\epsilon}{1+\epsilon}$. The integral in the
second term in (4.23) satisfies

$\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)Q^{\eta}\omega_{t}^{n}\leq\Big{(}\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)^{p_{0}}\omega_{t}^{n}\Big{)}^{1/p_{0}}\Big{(}\int_{X}Q^{p_{0}^{*}\eta}\omega_{t}^{n}\Big{)}^{1/p_{0}^{*}}\leq
C$

if we choose $p_{0}\in(1,\frac{n}{n-1})$ and $\eta>0$ small so that
$p_{0}^{*}\eta\leq\frac{\epsilon}{1+\epsilon}$. The inequality above then
follows from (4.24) and Theorem 2.

To deal with the last integral in (4.23), we apply integration by parts to
obtain

(4.25)
$\displaystyle\;c_{t}\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)(-e^{-\mu\hat{\varphi}}\Delta_{\tilde{g}}F)e^{F}\omega_{X}^{n}$
$\displaystyle=$
$\displaystyle\;c_{t}\int_{X}\Big{(}{\mathcal{G}}_{t}(x_{0},\cdot)e^{-\mu\hat{\varphi}}|\nabla
F|_{\tilde{g}}^{2}+{\mathcal{G}}_{t}(x_{0},\cdot)\langle\nabla
F,\bar{\nabla}e^{-\mu\hat{\varphi}}\rangle_{\tilde{g}}+e^{-\mu\hat{\varphi}}\langle\nabla{\mathcal{G}}_{t}(x_{0},\cdot),\nabla
F\rangle_{\tilde{g}}\Big{)}e^{F}\omega_{X}^{n}$ $\displaystyle\leq$
$\displaystyle\;c_{t}\int_{X}\Big{(}2{\mathcal{G}}_{t}(x_{0},\cdot)e^{-\mu\hat{\varphi}}|\nabla
F|^{2}_{\tilde{g}}+\mu^{2}{\mathcal{G}}_{t}(x_{0},\cdot)e^{-\mu\hat{\varphi}}|\nabla\hat{\varphi}|_{\tilde{g}}^{2}+e^{-\mu\hat{\varphi}}|\nabla
F|_{\tilde{g}}|\nabla{\mathcal{G}}_{t}(x_{0},\cdot)|_{\tilde{g}}\Big{)}e^{F}.$

The first integral in (4.25) satisfies (recall $p>2n$)

$2c_{t}\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)e^{-\mu\hat{\varphi}}|\nabla
F|^{2}_{\tilde{g}}e^{F}\leq
C\Big{(}\int_{X}{\mathcal{G}}_{t}(x_{0},\cdot)^{(p/2)^{*}}\omega_{t}^{n}\Big{)}^{1/(p/2)^{*}}\cdot\Big{(}\int_{X}|\nabla
F|_{\tilde{g}}^{p}e^{F}\omega_{X}^{n}\Big{)}^{2/p}$

which is bounded above uniformly since
$(p/2)^{*}=\frac{p/2}{p/2-1}<\frac{n}{n-1}$ we can apply Theorem 2 and the
assumption on $F$ in Theorem 4. The second integral in (4.25) is also bounded
because of Theorem 2 and
$e^{-\mu\hat{\varphi}}|\nabla\hat{\varphi}|_{\tilde{g}}^{2}\leq C$ which
follows from Theorem 3 (we may assume $\mu>A$, where $A>0$ is the constant in
Theorem 3). We now deal with the last integral in (4.25). We calculate

$\displaystyle\;\int_{X}e^{-\mu\hat{\varphi}}|\nabla
F|_{\tilde{g}}|\nabla{\mathcal{G}}_{t}(x_{0},\cdot)|_{\tilde{g}}e^{F}\omega_{X}^{n}\leq\int_{X}e^{-\mu\hat{\varphi}}|\nabla
F|_{\tilde{g}}({\rm
tr}_{\tilde{g}}g)^{\frac{1}{2}}|\nabla{\mathcal{G}}_{t}(x_{0},\cdot)|_{g}e^{F}\omega_{X}^{n}$
$\displaystyle=$ $\displaystyle\;\int_{X}e^{-\mu\hat{\varphi}/2}|\nabla
F|_{\tilde{g}}Q^{\frac{1}{2}}|\nabla{\mathcal{G}}_{t}(x_{0},\cdot)|_{g}e^{F}\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle\;CQ_{\max}^{1/2}\Big{(}\int_{X}|\nabla
F|_{\tilde{g}}^{p}e^{F}\omega_{X}^{n}\Big{)}^{1/p}\cdot\Big{(}\int_{X}|\nabla{\mathcal{G}}_{t}(x_{0},\cdot)|_{g}^{p^{*}}\omega_{t}^{n}\Big{)}^{1/p^{*}}$
$\displaystyle\leq$ $\displaystyle\;CQ_{\max}^{1/2},$

where we have used $p^{*}=\frac{p}{p-1}<\frac{2n}{2n-1}$, so the integral on
$L^{p^{*}}(X,\omega_{t}^{n})$ of $|\nabla{\mathcal{G}}_{t}(x_{0},\cdot)|_{g}$
is bounded uniformly by Theorem 2. Plugging these into (4.23), we get

$\displaystyle Q_{\max}\leq CQ_{\max}^{1-\eta}+CQ_{\max}^{1/2}+C,$

from which we conclude $Q_{\max}\leq C$ by Young’s inequality, that is

$\sup_{X}(e^{-\mu\varphi}|s_{D}|^{2\mu\varepsilon_{0}}_{h_{D}}{\rm
tr}_{\omega_{X}}\omega_{t})\leq C.$

The proof of Theorem 4 is complete by noting that $e^{-\mu\varphi}$ is
uniformly bounded.

We note that Theorem 4 also applies to the complex Monge-Ampère equations with
a fixed background metric.

###### Corollary 4

Let $\varphi$ be the solution to (4.15). Suppose $F$ satisfies the conditions
(4.16) and (4.17). Given $p>2n$, the following holds

$\sup_{X}|i\partial\bar{\partial}\varphi|_{\omega_{X}}^{2}\leq C,$

for some constant $C>0$ depending only $n,p,\omega_{X},\epsilon,N,\gamma$ and
$\int_{X}|\nabla F|^{p}_{\omega_{X}}e^{F}\omega_{X}^{n}$.

The example below shows that exponent $p>2n$ in Corollary 4 is also sharp
since the estimate may not hold if $p<2n$, even when other conditions are
valid for $F$. The case when $p=2n$ is not clear to us.

Example 3.2. We take the same metrics and notations as in Example 3.1. Fix an
$a>0$ close but smaller than $1$. We can pick $1<p<2an<2n$, and near $0\in{\bf
C}^{n}$, (4.19) tells that

$|\nabla F|^{p}_{\omega_{{\bf C}^{n}}}e^{F}\leq C|z|^{-p+2an-2n},$

which shows that the $L^{1}({\bf{CP}}^{n},\omega_{FS}^{n})$-norm of $(|\nabla
F|^{p}_{\omega_{FS}}e^{F})$ is uniformly bounded (i.e. independent of
$\delta>0$), while conditions (4.16) and (4.17) on $F$ hold for some
$\epsilon,N,$ and $\gamma$ which are independent of $\delta$. However at the
points $z$ with $|z|^{2}=\delta$

${\rm tr}_{\omega_{{\bf
C}^{n}}}(i\partial\bar{\partial}\varphi_{\delta})\sim\delta^{a-1}\to\infty\quad\mbox{as
}\delta\to 0.$

Finally we mention an application of Corollary 4 to the regularity of complex
Monge-Ampère equations when $e^{F}$ satisfies weaker regularity than being
$C^{2}$. For this we need a theorem from [24]. Corollary 5 below may be known
to experts, but we cannot find a reference in the literature, so we include
the statement and a sketched proof.

###### Corollary 5

Let $\varphi$ be the solution to (4.15) with $F$ a Lipschitz function (i.e.
$|F(x)-F(y)|\leq Ld_{\omega_{X}}(x,y)$ for some $L>0$). Then there is an
$\alpha\in(0,1)$ depending on $n,\omega_{X}$ such that

$\|\varphi\|_{C^{2,\alpha}(X,\omega_{X})}\leq C,$

for some constant $C>0$ depending only $n,\omega_{X}$, and $L$.

Proof of Corollary 5.  By smoothing out $F$ and taking limits if necessary, we
may assume $F$ is a smooth function with Lipschitz constant $\leq L$. Since
$\int_{X}e^{F}\omega_{X}^{n}=V$, and $F$ is Lipschitz, it follows that $\sup
e^{F}$ and $\inf e^{F}>0$ are both bounded depending on $\omega_{X}$ and $L$,
and so is the $L^{p}(e^{F}\omega_{X}^{n})$-norm of $|\nabla F|_{\omega_{X}}$.
It then follows from the $C^{2}$ estimate in Corollary 4 and the equation
(4.15) that $\omega_{X}+i\partial\bar{\partial}\varphi$ is equivalent to
$\omega_{X}$. We can now invoke the main theorem in [24] to conclude the proof
of Corollary 5.

###  4.3 $C^{3}$ estimates

To keep the notations lighter, we only consider the $C^{3}$ estimates for the
equation (4.15) with a fixed background metric. We follow closely the approach
in [18]. We continue to denote $\tilde{g}$ and $g$ the associated metrics of
$\omega_{X}$ and $\omega=\omega_{X}+i\partial\bar{\partial}\varphi$,
respectively. Furthermore we assume there is a $\theta>1$ such that

(4.26) $\theta^{-1}g\leq\tilde{g}\leq\theta g.$

By the $C^{2}$ estimates in Corollary 4, (4.26) holds with $\theta$ depending
additionally on $\inf_{X}e^{F}$. We remark that Theorem 5 below has been known
with the constant $C>0$ depending on the $C^{3}$-norm of $F$ (cf. [18, 25]).

###### Theorem 5

Fix $p>2n$. The following estimate holds:

$\sup_{X}|\nabla_{\tilde{g}}i\partial\bar{\partial}\varphi|_{\tilde{g}}^{2}\leq
C$

for some $C>0$ depending on $n,p,\theta,\omega_{X}$, $\int_{X}|\nabla
F|^{p}_{\omega_{X}}e^{F}\omega^{n}_{X}$ and
$\int_{X}|i\partial\bar{\partial}F|^{p}_{\omega_{X}}e^{F}\omega^{n}_{X}$.

Proof of Theorem 5. As in [18], we denote

$S^{i}_{jk}=\Gamma^{i}_{jk}-\tilde{\Gamma}^{i}_{jk}$

to be the difference of the Christoffel symbols of $g$ and$\tilde{g}$. Note
that $S^{i}_{jk}$ is indeed a tensor. We write

$|S|_{g}^{2}=S^{i}_{jk}\overline{S^{l}_{pq}}g_{\bar{l}i}g^{j\bar{k}}g^{k\bar{q}}.$

Under the assumption (4.26), it is easy to see $|S|_{g}^{2}$ is equivalent to
$|\nabla_{\tilde{g}}i\partial\bar{\partial}\varphi|^{2}_{\tilde{g}}$, so it
suffices to estimate $|S|_{g}^{2}$. By the calculations in [18], we have

$\displaystyle\Delta_{g}|S|_{g}^{2}$ $\displaystyle=$ $\displaystyle\,|\nabla
S|_{g}^{2}+|\bar{\nabla}S|_{g}^{2}-2Re(\overline{S^{i}_{jk}}R_{\bar{i}p,q}g^{p\bar{j}}g^{q\bar{k}})$
$\displaystyle\,+S*S*{\rm Ric}(g)+S*S*Rm(\tilde{g})+S*\nabla^{\tilde{g}}{\rm
Ric}(\tilde{g}),$

where $T*S$ means certain linear combinations of the tensors $T$ and $S$
contracted by $g$ or $\tilde{g}$, and $R_{\bar{i}p,q}$ denotes the covariant
derivative of ${\rm Ric}(g)_{\bar{i}p}$ with respect to $g$. From the equation
(4.15), we have ${\rm Ric}(g)={\rm Ric}(\tilde{g})-i\partial\bar{\partial}F$.
Let $G$ be the Green’s function of $g$ and as before ${\mathcal{G}}$ be the
positive Green’s function defined in (3.15). By Green’s formula, for any $x\in
X$, we have

(4.28)
$\displaystyle\,|S|_{g}^{2}(x)-\frac{1}{V}\int_{X}|S|_{g}^{2}\omega^{n}=\int_{X}{\mathcal{G}}(x,\cdot)\Big{(}-|\nabla
S|_{g}^{2}-|\bar{\nabla}S|_{g}^{2}+2Re(\overline{S^{i}_{jk}}R_{\bar{i}p,q}g^{p\bar{j}}g^{q\bar{k}})$
$\displaystyle\qquad\qquad+S*S*{\rm
Ric}(g)+S*S*Rm(\tilde{g})+S*\nabla^{\tilde{g}}{\rm
Ric}(\tilde{g})\Big{)}\omega^{n}$ $\displaystyle\leq$
$\displaystyle\,\int_{X}{\mathcal{G}}(x,\cdot)\Big{(}-|\nabla
S|_{g}^{2}-|\bar{\nabla}S|_{g}^{2}+C|S|_{g}^{2}+C|S|_{g}^{2}|i\partial\bar{\partial}F|_{\tilde{g}}+C|S|_{g}$
$\displaystyle\,\qquad+|\nabla S|_{g}|{\rm Ric}|_{g}+|\bar{\nabla}S|_{g}|{\rm
Ric}|_{g}\Big{)}\omega^{n}+\int_{X}2|\nabla{\mathcal{G}}|_{g}|S|_{g}|{\rm
Ric}|_{g}\omega^{n}$ $\displaystyle\leq$
$\displaystyle\,C\int_{X}{\mathcal{G}}(x,\cdot)\Big{(}|S|_{g}^{2}+|S|_{g}^{2}|i\partial\bar{\partial}F|_{\tilde{g}}+|i\partial\bar{\partial}F|_{\tilde{g}}^{2}+1\Big{)}\omega^{n}+\int_{X}2|\nabla{\mathcal{G}}|_{g}|S|_{g}|{\rm
Ric}|_{g}\omega^{n},$

where in the last line we apply Cauchy-Schwarz inequality. Recall we assume a
bound on $\int_{X}|i\partial\bar{\partial}F|_{\tilde{g}}^{p}\omega^{n}$ for
$p>2n$. We integrate (13) against $\omega^{n}$ over $X$. We obtain by Hölder
inequality

(4.29) $\int_{X}|S|_{g}^{2}\omega^{n}\leq
C+\int_{X}|\Delta_{\tilde{g}}F|\omega^{n}\leq
C+\int_{X}|i\partial\bar{\partial}F|_{\tilde{g}}\omega^{n}\leq C.$

Let $x_{0}\in X$ be a maximum point of the function $|S|_{g}^{2}$ and denote
$M=|S|_{g}^{2}(x_{0})$. We apply (4.28) at $x_{0}$. It follows that (denote
$q=p/2>n$)

$\displaystyle M$ $\displaystyle\leq$
$\displaystyle\,C+CM^{1-\eta}\big{(}\int{\mathcal{G}}^{q^{*}}\big{)}^{1/q^{*}}\big{(}\int|S|_{g}^{2\eta
q}\big{)}^{1/q}+C\big{(}\int{\mathcal{G}}^{q^{*}}\big{)}^{1/q^{*}}\big{(}\int|i\partial\bar{\partial}F|_{\tilde{g}}^{p}\big{)}^{1/q}+CM^{1/2}$
$\displaystyle\,+CM^{1/2}\big{(}\int|\nabla{\mathcal{G}}|_{g}^{p^{*}}\big{)}^{1/p^{*}}\big{(}\int|i\partial\bar{\partial}F|_{\tilde{g}}^{p}\big{)}^{1/p}+CM^{1-\eta}\big{(}\int{\mathcal{G}}^{\frac{2n}{2n-1}}\big{)}^{\frac{2n-1}{2n}}\big{(}\int_{X}|i\partial\bar{\partial}F|_{\tilde{g}}^{p}\big{)}^{\frac{1}{p}}\big{(}\int|S|_{g}^{2\eta
b}\big{)}^{\frac{1}{b}}$

where $b=(p-2n)/2np$. We can choose $\eta>0$ such that $\max(2\eta q,2\eta
b)=2$. All the integrals above are bounded, due to (4.29) and Theorem 2. Hence
we have

$M\leq C+CM^{1-\eta}+CM^{1/2},$

from which we conclude $M\leq C$. This finishes the proof of Theorem 5.

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Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
07102

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Department of Mathematics, Columbia University, New York, NY 10027

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Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
07102

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