# SOBOLEV INEQUALITIES ON KÄHLER SPACES

Bin Guo∗, Duong H. Phong†, Jian Song‡ and Jacob Sturm†† ∗ Department of
Mathematics & Computer Science, Rutgers University, Newark, NJ 07102
<EMAIL_ADDRESS>† Department of Mathematics, Columbia University, New York,
NY 10027<EMAIL_ADDRESS>‡ Department of Mathematics, Rutgers
University, Piscataway, NJ 08854<EMAIL_ADDRESS>†† Department of
Mathematics & Computer Science, Rutgers University, Newark, NJ 07102
<EMAIL_ADDRESS>
###### Abstract.

We establish a uniform Sobolev inequality for Kähler metrics, which only
require an entropy bound and no lower bound on the Ricci curvature. We further
extend our Sobolev inequality to singular Kähler metrics on Kähler spaces with
normal singularities. This allows us to build a general theory of global
geometric analysis on singular Kähler spaces including the spectral theorem,
heat kernel estimates, eigenvalue estimates and diameter estimates. Such
estimates were only known previously in very special cases such as Bergman
metrics. As a consequence, we derive various geometric estimates, such as the
diameter estimate and the Sobolev inequality, for Kähler-Einstein currents on
projective varieties with definite or vanishing first Chern class.

Work supported in part by the National Science Foundation under grants
DMS-22-03273, DMS-22-03607 and DMS-23-03508, and the collaboration grant
946730 from Simons Foundation.

###### Contents

  1. 1 Introduction
  2. 2 Uniform bounds for smooth Kähler manifolds
  3. 3 Applications to singular Kähler spaces
    1. 3.1 Sobolev inequality, heat kernel and diameter estimates
    2. 3.2 Singular Kähler-Einstein metrics
    3. 3.3 Finite time singularities of the Kähler-Ricci flow
  4. 4 The Sobolev-type inequality
  5. 5 Upper bound of the heat kernel
    1. 5.1 Off diagonal estimates
    2. 5.2 On diagonal estimates
  6. 6 Local Sobolev inequality and Poincaré inequality
  7. 7 Smooth approximation for $\mathcal{AK}$-currents
    1. 7.1 Kähler varieties
  8. 8 Sobolev inequalities for $\mathcal{AK}$-currents
  9. 9 Noncollapsing and diameter bounds for $\mathcal{AK}$-currents
  10. 10 Heat kernel estimates for $\mathcal{AK}$-currents
  11. 11 A spectral theorem for normal Kähler spaces
  12. 12 Kähler-Einstein spaces
  13. 13 Finite time solutions of the Kähler-Ricci flow

##  1\. Introduction

The Sobolev inequality is a fundamental tool in global geometric analysis. The
optimal Sobolev constant reflects the geometric structure of the underlying
Riemannian manifold, and geometric curvature bounds will in turn control the
Sobolev constant. The classical works of Yau and his collaborators have built
a wide range of geometric estimates for compact or complete Riemannian
manifolds assuming a lower bound for the Ricci curvature (c.f. [30, 5, 6]). In
[17], we were able to establish uniform Green’s formula and diameter estimates
for a large family of Kähler metrics on a Kähler manifold without such a Ricci
curvature lower bound. Rather, these estimates only depended on an entropy
bound. One of the first main results in the present paper is to obtain a
uniform Sobolev inequality for Kähler metrics, again assuming only an entropy
bound and no Ricci curvature lower bound. As a consequence, under similar
assumptions, we also obtain a wide range of geometric estimates such as heat
kernel estimate and eigenvalue estimates.

In his celebrated work [40] on the Calabi conjecture, Yau initiated the theory
of complex Monge-Ampère equations with singular data, in the hope of
understanding canonical Kähler metrics with singularities and anticipating
future applications to algebraic geometry. The analytic theory was
subsequently developed by Kołodziej [22] using pluripotential theory, and
later further extended to complex Monge-Ampère equations on singular Kähler
varieties [12, 9]. However, very little geometric information can be extracted
by pluripotential theory for singular Kähler metrics, and particularly for
Kähler-Einstein metrics on projective varieties with log terminal
singularities. Very recently, an alternative proof of Kołodziej’s estimates
has been found by Guo, Phong, and Tong in [19], using PDE methods. These PDE
methods have had many important consequences [16, 15, 17]. Their key features
are their flexibility and uniformity, which allow many geometric estimates to
be passed on to Kähler spaces with normal singularities.

The next main goal of the present paper is to develop a geometric theory of
complex Monge-Ampère equations with singularities. A theory with sharp
geometric estimates had been a well-known challenge in presence of
singularities. This is because geometric invariants such as the diameter and
the Sobolev inequality are defined in terms of the underlying metric, whose
very definition requires two derivatives of the potential. Bounds for such
high derivatives of the potential are usually not available in presence of
singularities. Another basic and subtle issue right at the start, is to
identify suitable Kähler spaces with singularities which would contain Kähler
spaces of interest, such as those carrying canonical metrics such as Kähler-
Einstein metrics. In this context, we note that the standard notion on smooth
Kähler metrics on normal varieties, defined as restrictions of smooth Kähler
metrics on the ambiant space, is inadequate, since such notions would exclude
Kähler-Einstein metrics.

Nevertheless, building in part on the methods of [19, 18, 17] and especially
the uniform Sobolev estimates on Kähler manifolds obtained in the first part
of the paper, we shall identify a natural class of singular Kähler spaces
(c.f. Definition 3.1) and establish for them a whole package of estimates such
as Sobolev inequalities, the spectral theorem, heat kernel estimates, diameter
estimates and volume non-collapsing estimates. These results provide both
conceptual and technical platforms for a geometric regularity theory on
singular Kähler spaces, which had long been an objective in both analytic and
algebraic geometry. As an application, we derive a unique diameter bound for
singular Kähler-Einstein spaces. Applying our methods to the Kähler-Ricci
flow, we can extend the Sobolev inequality to the metric completion of the
singular Kähler current and get the diameter/noncollapsing bound from the
Sobolev inequality without any curvature assumptions. We note that diameter
bounds for the metric completion $(\hat{X},d)$ of normal Kähler spaces have
been an open problem for a long time except for special cases when
$(X,\omega)$ is a limit of a sequence of Kähler manifolds with suitable Ricci
curvature bounds (cf. [28, 32]). On the analytic side, our Sobolev inequality
allows classical PDE techniques such as the Moser’s iteration to work on
singular Kähler spaces. Regarding the general problem of geometric analysis
with singularities, we would like to mention the recent beautiful work of Cao
et al. [4] on Hermitian-Einstein metrics on spaces with normal singularities
in the sense of Grauert, which also relied in an essential manner on the
estimates of [19, 18].

The paper is organized as follows. Sections 2 to 5 are devoted to a precise
formulation of our results, with the case of smooth Kähler manifolds described
in Section 2, and the case of Kähler spaces with singularities described in
Section 3, which includes applications to singular Kähler-Einstein metrics and
finite-time limits of the Kähler-Ricci flow. Throughout Sections 4 to 6, we
will assume that $X$ is a smooth Kähler manifold. In Section 4, we will
present the proof of Theorem 2.1, while Theorem 2.2 will be proven in Section
5. Additionally, Section 6 will contain statements of local versions of
Sobolev inequalities. In the remaining sections, we will assume that $X$ is a
normal Kähler variety. We will consider then the applications of Theorems 2.1
and 2.2, and in particular provide the proof of the results formulated in
Section 3.

##  2\. Uniform bounds for smooth Kähler manifolds

We first state our results for smooth Kähler manifolds. Let $(X,\omega_{X})$
be a compact Kähler manifold without boundary with $\omega_{X}$ a fixed Kähler
metric. Without loss of generality we normalize $\omega_{X}$ such that
$\int_{X}\omega_{X}^{n}=1$. Suppose that the complex dimension of $X$ is $n$.
Given a Kähler metric $\omega$ on $X$, we denote its volume by

$V_{\omega}=[\omega]^{n}=\int_{X}\omega^{n}$

and define the relative volume function by

$e^{F_{\omega}}=\frac{1}{V_{\omega}}\frac{\omega^{n}}{\omega_{X}^{n}}.$

Given $p\geq 1$ we define the $p$-th Nash-Yau entropy of $\omega$ by

${\mathcal{N}}_{p}(\omega)=\frac{1}{V_{\omega}}\int_{X}\Big{|}\log\frac{1}{V_{\omega}}\frac{\omega^{n}}{\omega_{X}^{n}}\Big{|}^{p}\;{\omega^{n}}=\int_{X}|F_{\omega}|^{p}e^{F_{\omega}}\omega_{X}^{n}.$

For a given nonnegative continuous function $\gamma\in C^{0}(X)$, and given
parameters $0<A\leq+\infty,K>0$, as in [17], we consider the following subset
of the space of Kähler metrics on $X$:

(2.1)
${\mathcal{W}}:={\mathcal{W}}(n,p,A,K,\gamma):=\Big{\\{}\omega:\;[\omega]\cdot[\omega_{X}]^{n-1}<A,\,{\mathcal{N}}_{p}(\omega)\leq
K,\,e^{F_{\omega}}\geq\gamma\Big{\\}}.$

It has been shown in [17] that if $p>n$ and the vanishing locus of $\gamma$ is
relatively small (c.f. (2.2)), then the Kähler metrics
$\omega\in{\mathcal{W}}(n,p,A,K,\gamma)$ have uniformly bounded diameter and
satisfy local noncollapsing. In this paper, we aim to study the heat kernels
of the Laplacian $\Delta_{\omega}$ of Kähler metrics $\omega$ in this subset,
and as applications, we give lower bounds of eigenvalues of
$-\Delta_{\omega}$, by using the approach of Cheng-Li [5]. The main tool is
the following Sobolev-type inequality. Throughout the paper, as in [17] we
always assume $\gamma\in C^{0}(X)$ satisfies

(2.2) $\dim_{{\mathcal{H}}}\\{\gamma=0\\}<2n-1,\quad\gamma\geq 0,$

where ${\mathrm{dim}}_{{\mathcal{H}}}E$ denotes the Hausdorff dimension of a
closed subset $E\subset X$. In our earlier work [17], uniform estimates are
established for the Green’s function and the diameter associated to Kähler
metrics in $\mathcal{W}(n,p,A,K,\gamma)$ . We denote

(2.3) $I_{\omega}=[\omega]\cdot[\omega_{X}]^{n-1}$

to be the intersection of the Kähler classes $[\omega]$ and $[\omega_{X}]$. To
state results for possibly unbounded Kähler classes, we will write the
admissible set of Kähler metrics with $A=\infty$ in (2.1) as
${\mathcal{W}}(n,p,\infty,K,\gamma)$.

######  Theorem 2.1.

Given $p>n$, $K>0$ and $\gamma\in C^{0}(X)$ satisfying (2.2), there exist
$q=q(n,p)>1$ and $C=C(n,p,K,\gamma,q)>0$ such that for any Kähler metric
$\omega\in{\mathcal{W}}(n,p,\infty,K,\gamma)$ and any $u\in W^{1,2}(X)$, we
have the following Sobolev-type inequality

(2.4)
$\Big{(}\frac{1}{V_{\omega}}\int_{X}|u-\overline{u}|^{2q}\omega^{n}\Big{)}^{1/q}\leq
C\frac{I_{\omega}}{V_{\omega}}\int_{X}|\nabla u|_{\omega}^{2}\omega^{n},$

where $\overline{u}=\frac{1}{V_{\omega}}\int_{X}u\omega^{n}$ is the average of
$u$ over $(X,\omega)$.

We remark that the Sobolev constant in Theorem 2.1 does not depend on the
curvature of the Kähler metric $\omega$. The Sobolev inequality (2.4) is
indeed scale invariant, although the exponent $q$ may be smaller than
$\frac{n}{n-1}$. We also stress that if we impose the Kołodziej-type condition
on $e^{F_{\omega}}$, i.e. its $L^{1+\epsilon^{\prime}}(X,\omega_{X}^{n})$-norm
is uniformly bounded for some $\epsilon^{\prime}>0$, then the constant $q>1$
can be chosen as close as possible to the exponent $\frac{n}{n-1}$ as in the
Euclidean case, with the constant $C_{q}$ possibly blowing up as $q$
approaches $\frac{n}{n-1}$. This fact will be clear from the proof of Theorem
2.1 by combining the almost sharp estimates on the Green’s functions
established in [18].111In [18] the metrics are assumed to be uniformly big,
however, the same proof remains applicable to the degenerating metrics by
slight modifications. However, for the geometric applications in this paper,
having $q>1$ suffices.

We note that the Sobolev inequality (2.4) can also be used to prove the
diameter estimate of $(X,\omega)$, which has been established in our previous
work [17] using the Green’s formula. We are informed by Guedj, Guenancia and
Zeriahi that they very recently obtained diameter estimates under hypotheses
different from ours (c.f. [14]).

With the Sobolev inequality in Theorem 2.1, we can now state the heat kernel
upper bounds and eigenvalue lower bounds for Kähler metrics in
$\mathcal{W}(n,p,\infty,K,\gamma)$. Recall that the heat kernel of the
Laplacian $\Delta_{\omega}$ satisfies the equation

(2.5) $\frac{\partial}{\partial
t}H(x,y,t)=\Delta_{\omega,y}H(x,y,t)\quad\mbox{and }\lim_{t\to
0^{+}}H(x,y,t)=\delta_{x}(y).$

It is well-known that $H(x,y,t)$ is a positive smooth function on $X\times
X\times(0,\infty)$, and it is symmetric in $x$ and $y$.

######  Theorem 2.2.

Let $q=q(n,p)>1$ be the constant in Theorem 2.1. There exists
$C=C(n,p,K,\gamma)>0$ such that for any $x,y\in X$

(2.6)
$H(x,y,t)\leq\left\\{\begin{array}[]{ll}\frac{C}{V_{\omega}}\big{(}\frac{I_{\omega}}{t}\big{)}^{\frac{q}{q-1}}\exp\big{(}{-\frac{d_{\omega}(x,y)^{2}}{10t}}\big{)},&\quad\mbox{if
}t\in(0,I_{\omega}]\\\
\frac{C}{V_{\omega}}\exp\big{(}{-\frac{d_{\omega}(x,y)^{2}}{10t}}\big{)},&\quad\mbox{if
}t\in(I_{\omega},\infty).\end{array}\right.$

Here $d_{\omega}(x,y)$ is the geodesic distance between $x$ and $y$ with
repsect to the metric $\omega$. When $x=y$, a stronger upper bound of the heat
kernel on the diagonal of $X\times X$ holds as below,

(2.7)
$H(x,x,t)\leq\frac{1}{V_{\omega}}+\frac{C}{V_{\omega}}I_{\omega}^{\frac{q}{q-1}}t^{-\frac{q}{q-1}}.$

Moreover, if $t\geq I_{\omega}$, we have for some positive constant $c>0$

(2.8) $\sup_{x,y\in
X}\big{|}H(x,y,t)-\frac{1}{V_{\omega}}\big{|}\leq\frac{C}{V_{\omega}}e^{-cI_{\omega}^{-1}t}.$

The classical heat kernel upper bound is achieved in [6]. Theorem 2.2 does not
require any assumption on the Ricci curvature and it will also give uniform
lower bounds for the eigenvalues associated to metrics in
$\mathcal{W}(n,p,\infty,K,\gamma)$. Let
$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\ldots$ (counting multiplicity)
be the increasing sequence of eigenvalues of the Laplacian $-\Delta_{\omega}$.
We have the following lower bound on these eigenvalues.

######  Corollary 2.1.

There exists $c=c(n,p,K,\gamma)>0$ such that for any $k\in{\mathbb{N}}$,

$\lambda_{k}\geq c\,I_{\omega}^{-1}k^{\frac{q-1}{q}}.$

We observe that the uniform positive lower bound of $\lambda_{1}$ was proved
in [17]. The known estimates on higher order eigenvalues required lower bounds
on the Ricci curvature [5].

##  3\. Applications to singular Kähler spaces

In this section, we will apply the Sobolev inequality and the heat kernel
estimates to normal Kähler varieties. We will focus on the family of closed
positive $(1,1)$-currents whose log volume measure has only log type analytic
singularities (c.f. Definition 7.2).

######  Definition 3.1.

Let $X$ be an $n$-dimensional compact normal Kähler space. Let
$\pi:Y\rightarrow X$ be a log resolution of singularities and let $\theta_{Y}$
be a smooth Kähler metric on the nonsingular model $Y$. We define the set of
admissible semi-Kähler currents

${\mathcal{AK}}(X,\theta_{Y},n,p,A,K,\gamma),$

to be the set of any semi-Kähler current $\omega$ on $X$ satisfying the
following conditions.

  1. (1)

$[\omega]$ is a Kähler class on $X$ and $\omega$ has bounded local potentials.

  2. (2)

$[\pi^{*}\omega]\cdot[\theta_{Y}]^{n-1}\leq A$ and $[\omega]^{n}\geq A^{-1}$.

  3. (3)

The $p$-th Nash-Yau entropy is bounded for some $p>n$, i.e.

${\mathcal{N}}_{p}(\omega)=\frac{1}{V_{\omega}}\int_{Y}\left|\log\frac{1}{V_{\omega}}\frac{(\pi^{*}\omega)^{n}}{(\theta_{Y})^{n}}\right|^{p}(\pi^{*}\omega)^{n}\leq
K,$

where $V_{\omega}=[\omega]^{n}$.

  4. (4)

The log volume measure ratio

$\log\left(\frac{(\pi^{*}\omega)^{n}}{(\theta_{Y})^{n}}\right)$

has log type analytic singularities (c.f. Definition 7.2). There also exists a
non-negative $\gamma\in C^{0}(Y)$ such that $\\{\gamma=0\\}$ is contained in a
proper analytic subvariety of $Y$ and

$\frac{(\pi^{*}\omega)^{n}}{(\theta_{Y})^{n}}\geq\gamma.$

The nonsingular model $Y$ is indeed Kähler (cf. Lemma 2.2 [7]) in Definition
3.1. The $\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$ space is the extension of
$\mathcal{W}(n,p,A,K,\gamma)$ with two key additional conditions: (1) uniform
lower bound of the volume for $[\omega]$ to prevent collapsing; (2) the volume
measure ratio has log type analytic singularities to guarantee smoothness of
$\omega$ in an open Zariski subset of $X$.

Smooth Kähler metrics on a normal variety $X$ are defined locally as the
restriction of smooth Kähler metrics in the ambient space (cf. Definition
7.1). In general, canonical metrics on $X$ such as Kähler-Einstein metrics are
not smooth Kähler metrics in this sense. For example, if $X$ is a projective
variety with orbifold singularities, the smooth orbifold Kähler metrics are
never smooth Kähler metrics on $X$. However, the Kähler-Einstein metrics on
projective varieties belong to the $\mathcal{AK}$-space of Definition 3.1 and
they are not smooth near the algebraic singularities. This is one of the
reasons we consider the $\mathcal{AK}$-spaces.

######  Remark 3.1.

The space $\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$ in Definition (3.1) can
be easily generalized for positive currents $\omega$ with minimal
singularities in a big and nef class. Most of our results still hold for such
positive currents $\omega$. We choose to state our results for
$\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$ with the Kähler assumptions for
convenience and simplicity. In fact, the definition for $\mathcal{AK}$-space
of currents can be further extended without the nef assumption.

For any $\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$, we let
$\mathcal{S}_{X,\omega}$ be the union of the singular set of $X$ and the
singular set of
$\pi_{*}\left(\log\frac{(\pi^{*}\omega)^{n}}{(\theta_{Y})^{n}}\right)$. By
definition, $\mathcal{S}_{X,\omega}$ is an analytic subvariety of $X$. The
current $\omega$ is a smooth Kähler metric on
$X\setminus\mathcal{S}_{X,\omega}$ (c.f. Proposition 7.1).

######  Definition 3.2.

Let $X$ be an $n$-dimensional a compact normal Kähler variety and

$\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma).$

We define

$(\hat{X},d)=\overline{(X\setminus\mathcal{S}_{X,\omega},\omega|_{X\setminus\mathcal{S}_{X,\omega}})}$

to be the metric completion of
$\left(X\setminus\mathcal{S}_{X,\omega},\omega|_{X\setminus\mathcal{S}_{X,\omega}}\right)$.
We also denote the unique metric measure space associated with $(X,\omega)$ by

$(\hat{X},d,\omega^{n}).$

We remark that $\omega^{n}$ extends uniquely to a volume measure on $\hat{X}$
because both $\omega$ and $\omega^{n}$ does not carry mass on
$\mathcal{S}_{X,\omega}$.

Our goal is to extend Theorem 2.1 and Theorem 2.2 to the metric measure space
$(\hat{X},d,\omega^{n})$ associated to $(X,\omega)$. We then apply the Sobolev
inequality to establish the volume non-collapsing and diameter bounds for
$(\hat{X},d,\omega^{n})$ in Section 9. A spectral theorem and heat kernel
estimates will also be proved in Section 10 and Section 11. Finally, we will
apply our estimates to singular Kähler-Einstein spaces in Section 12 and
limiting spaces of non-collapsing finite time solutions of the Kähler-Ricci
flow in Section 13.

###  3.1. Sobolev inequality, heat kernel and diameter estimates

Let $X$ be an $n$-dimensional normal Kähler variety. For any $p>n$ and any
$\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$, we let
$(\hat{X},d,\omega^{n})$ the metric measure space associated to $(X,\omega)$.
We will consider the Sobolev space $W^{1,2}(\hat{X},d,\omega^{n})$ as in
Definition 8.1 on the metric measure space $(\hat{X},d,\omega^{n})$ associated
to $\omega$.

######  Theorem 3.1.

Let $X$ be an $n$-dimensional compact normal Kähler space. For any
$\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$, the metric measure space
$(\hat{X},d,\omega^{n})$ associated to $(X,\omega)$ satisfies the following
properties.

  1. (1)

There exists $C=C(X,\theta_{Y},n,p,A,K,\gamma)>0$ such that

${\textnormal{diam}}(\hat{X},d)\leq C.$

In particular, $(\hat{X},d)$ is a compact metric space.

  2. (2)

There exist $q>1$ and $C_{S}=C_{S}(X,\theta_{Y},n,p,A,K,\gamma,q)>0$ such that

$\Big{(}\int_{\hat{X}}|u|^{2q}\omega^{n}\Big{)}^{1/q}\leq
C_{S}\left(\int_{\hat{X}}|\nabla
u|^{2}~{}\omega^{n}+\int_{\hat{X}}u^{2}\omega^{n}\right).$

for all $u\in W^{1,2}(\hat{X},d,\omega^{n})$.

  3. (3)

There exists $C=C(X,\theta_{Y},n,p,A,K,\gamma)>0$ such that the following
trace formula holds for the heat kernel of $(\hat{X},d,\omega^{n})$

$H(x,x,t)\leq\frac{1}{V_{\omega}}+\frac{C}{V_{\omega}}t^{-\frac{q}{q-1}}.$

  4. (4)

Let $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq...$ be the increasing
sequence of eigenvalues of the Laplacian $-\Delta_{\omega}$ on
$(\hat{X},d,\omega^{n})$. Then there exists
$c=c(X,\theta_{Y},n,p,A,K,\gamma)>0$ such that

$\lambda_{k}\geq ck^{\frac{q-1}{q}}.$

To the best of our knowledge, Theorem 3.1 is the first to provide general
formulas for the Sobolev inequality, the heat kernel, and eigenvalue estimates
on complex spaces with singularities. Such estimates were only known
previously to hold in special cases such as the following:

  1. (1)

$X$ is projective and $\omega$ is a Bergman metric via projective embedding
(cf. [24, 3]).

  2. (2)

$(X,\omega)$ is a sequential Gromov-Hausdorff limit of Kähler-Einstein
manifolds (cf. [11]).

Geometric estimates are traditionally studied for smooth Kähler metrics on
singular Kähler spaces. The most immediate concern is that, as we pointed out
earlier, canonical Kähler metrics near algebraic singularities cannot be in
general the restriction of smooth metrics via local or global embeddings. In
addition, most algebraic singularities are not smoothable and so most singular
Kähler metrics cannot be approximated by smooth Kähler metrics with Ricci
curvature uniformly bounded from below.

Thus the traditional geometric estimates cannot be directly applied to the
study of canonical metrics. Nor do the traditional methods of proof, which
usually require that the metrics are smooth Kähler metrics on normal Kähler
varieties. For example, the heat kernel trace formula and Sobolev inequalities
are obtained in [27, 24] for Bergman metrics on projective varieties. Bergman
metrics are restrictions of Fubini-Study metrics via global projective
embeddings and so they are smooth Kähler metrics on normal Kähler varieties.

Canonical metrics, such as Kähler-Einstein metrics, require rather a different
and much larger class of singular Kähler spaces, such as the classes
$\mathcal{AK}$-spaces as in Definition 3.1. These lie beyond the reach of
traditional methods, but they can be treated by our Sobolev inequality and
trace formula for the heat kernel. We also observe that our Sobolev inequality
can be applied to the study of Hermitian-Einstein metrics on stable sheaves
over singular normal Kähler spaces as developed by [2, 4].

As we had stressed in the Introduction, diameter bounds for
$(X\setminus\mathcal{S}_{X,\omega})$ or $(\hat{X},d)$ have been an open
problem for a long time. The uniform diameter bound in Theorem 3.1 is
particularly striking, in the sense that it mostly relies on the entropy bound
rather than the topological and complex structures of singularities.

###  3.2. Singular Kähler-Einstein metrics

We will apply the diameter estimates and Sobolev estimates of Theorem 3.1 to
Kähler-Einstein spaces.

Let $X$ be a projective Calabi-Yau variety of $\dim X=n$, i.e. $X$ is normal
and $c_{1}(X)=0$. It is proved in [12], for any Kähler class $\alpha$ on $X$,
there exists a unique positive closed $(1,1)$-current $\omega_{CY}\in\alpha$
with bounded local potential and volume measure if $X$ has at worst log
terminal singularities. In particular, $\omega_{CY}$ is a smooth Kähler metric
on $X^{\circ}$, the regular part of $X$. We will apply the Sobolev inequality
(3.1) to obtain the following geometric estimates.

######  Theorem 3.2.

Let $X$ be a projective Calabi-Yau variety with log terminal singularities.
Then for any Kähler class $\alpha$, there exists a unique Ricci-flat current
$\omega_{CY}\in\alpha$ satisfying the following.

  1. (1)

Let $(\hat{X},d_{CY})$ be the metric completion of $(X^{\circ},\omega_{CY})$.
Then $(\hat{X},d_{CY},(\omega_{CY})^{n})$ is a compact metric measure space.

  2. (2)

For any $q\in(1,\frac{n}{n-1})$, there exists $C_{S}=C_{S}(X,n,\alpha,q)>0$
such that

(3.1) $\Big{(}\int_{\hat{X}}|u|^{2q}(\omega_{CY})^{n}\Big{)}^{1/q}\leq
C_{S}\left(\int_{\hat{X}}|\nabla
u|^{2}_{\omega_{CY}}~{}(\omega_{CY})^{n}+\int_{\hat{X}}u^{2}(\omega_{CY})^{n}\right).$

for all $u\in W^{1,2}(\hat{X},d_{CY},(\omega_{CY})^{n})$.

  3. (3)

For any $\mu>n$, there exists $\kappa=\kappa(X,n,\alpha,\mu)>0$ such that for
any $p\in\hat{X}$ and $r\in(0,\textnormal{diam}(\hat{X},d_{CY})]$,

$\frac{{\mathrm{Vol}}(B(p,r))}{r^{2\mu}}\geq\kappa,$

where $B(p,r)$ denotes the ball in $(\hat{X},d)$ with center $p$ and radius
$r$.

We note that Theorem 3.2 remains valid when $X$ is only a _Kähler_ Calabi-Yau
variety. The proof remains unchanged.

We will also derive geometric estimates for Kähler-Einstein currents on
canonical models. Let $X$ be an $n$-dimensional projective normal variety with
ample $K_{X}$ and log terminal singularities. There exists a unique Kähler-
Einstein current $\omega_{KE}\in[K_{X}]$ constructed in [12] such that
$\omega_{KE}$ has bounded local potential and volume measure satisfying

${\mathrm{Ric}}(\omega_{KE})=-\omega_{KE}$

smoothly on $X^{\circ}$, the regular part of $X$.

######  Theorem 3.3.

Let $X$ be an $n$-dimensional projective normal variety with ample $K_{X}$ and
log terminal singularities. Then the unique Kähler-Einstein current
$\omega_{KE}\in[K_{X}]$ satisfies the following properties:

  1. (1)

Let $(\hat{X},d_{KE})$ be the metric completion of $(X^{\circ},\omega_{KE})$.
Then $(\hat{X},d_{KE},(\omega_{KE})^{n})$ is a compact metric measure space.

  2. (2)

For any $q\in(1,\frac{n}{n-1})$, there exists $C_{S}=C_{S}(X,n,q)>0$ such that

$\Big{(}\int_{\hat{X}}|u|^{2q}(\omega_{KE})^{n}\Big{)}^{1/q}\leq
C_{S}\left(\int_{\hat{X}}|\nabla
u|^{2}_{\omega_{KE}}(\omega_{KE})^{n}+\int_{\hat{X}}u^{2}(\omega_{KE})^{n}\right).$

for all $u\in W^{1,2}(\hat{X},d_{KE},(\omega_{KE})^{n})$.

  3. (3)

For any $\mu>n$, there exists $\kappa=\kappa(X,n,\mu)>0$ such that for any
$p\in\hat{X}$ and $r\in(0,\textnormal{diam}(\hat{X},d_{KE})]$,

$\frac{{\mathrm{Vol}}(B(p,r))}{r^{2\mu}}\geq\kappa.$

The existence of Kähler-Einstein currents on a K-stable $\mathbb{Q}$-Fano
variety is established in [23, 25] as a singular version of the Yau-Tian-
Donaldson conjecture. The analogue of Theorem 3.3 also holds for such singular
Kähler-Einstein metrics with positive scalar curvature.

######  Theorem 3.4.

Let $X$ be an $n$-dimensional $\mathbb{Q}$-Fano variety with log terminal
singularities. If there exists a Kähler-Einstein current $\omega\in-[K_{X}]$
with bounded local potentials, then $\omega\in C^{\infty}(X^{\circ})$ and
$(\hat{X},d_{KE},(\omega_{KE})^{n})$ associated with $(X,\omega_{KE})$ is a
compact metric measure space satisfying estimates in (2) and (3) of Theorem
3.3.

We stress that the above diameter and non-collapsing estimates go considerably
farther than those obtained in [17]. Indeed, the diameter and non-collapsing
estimates in [17] are for smooth Kähler manifolds, and even though they were
uniform in many key parameters, they still do not apply to the present setting
of singular spaces $(X,\omega)$ by taking limits. This is because the
convergence is only uniform on any compact subset of the regular part of $X$,
which is too weak and not sufficient to obtain the convergence of geodesics or
geodesic balls. Furthermore, the arguments in [17] cannot be adapted to the
singular setting because of the lack of estimates of the Green’s functions on
the singular space. Instead, we have to rely on the Sobolev inequality to
prove these estimates.

The Sobolev inequality has the advantage that it can be passed down to the
limit or singular space, thanks to the uniform Sobolev inequality we establish
on the smooth resolution. However, this is not obvious for the Green’s
function because the gradient of Green’s function is not in $L^{2}$. This
latter is the technique we used in [17].

It is also the Sobolev inequality which enables us to estimate the heat kernel
of the Laplacian, and its eigenvalues, hence obtain a spectral theorem on
singular spaces (more precisely on the metric completion of such spaces).

###  3.3. Finite time singularities of the Kähler-Ricci flow

We consider the following unnormalized Kähler-Ricci flow on a Kähler manifold
$X$ with an initial Kähler metric $g_{0}$

(3.2) $\left\\{\begin{array}[]{l}{\displaystyle\frac{\partial g}{\partial
t}=-\textnormal{Ric}(g),}\\\ \\\ g|_{t=0}=g_{0}\in H^{1,1}(X,\mathbb{R})\cap
H^{2}(X,\mathbb{\mathbb{Q}}).\end{array}\right.$

Let

(3.3) $T=\sup\\{t>0~{}|~{}[g_{0}]+t[K_{X}]>0\\}\in\mathbb{R}\cup\\{\infty\\}.$

The Kähler-Ricci flow has a maximal solution $g(t)$ for $t\in[0,T)$. If
$T<\infty$, the flow (3.2) must develop singularities at $t=T$ and the
curvature tensors must blow up as $t\rightarrow T^{-}$. In this case,
$\alpha_{T}=[g_{0}]+T[K_{X}]$ is a nef class on $X$. Since $[g_{0}]\in
H^{2}(X,\mathbb{Q})$, $\alpha_{T}$ is semi-ample by Kawamata’s base point free
theorem and the numerical dimension of $\alpha_{T}$ coincides with its Kodaira
dimension. In particular, $\alpha_{T}$ induces a surjective holomorphic map

(3.4) $\Phi:X\rightarrow Y.$

We consider the case when $\alpha_{T}$ is big, which is equivalent to either
of the following conditions

  1. (1)

$(\alpha_{T})^{n}=\lim_{t\rightarrow T}{\rm Vol}_{g(t)}(X)>0.$

  2. (2)

$\dim Y=n$.

We let $S$ be the exceptional locus of $\Phi$, i.e., where $\Phi$ is not
biholomorphic. $S$ is then an analytic subvariety of $X$.

######  Theorem 3.5.

Let $(X,g_{0})$ be a Kähler manifold equipped with a Kähler metric $g_{0}$. If
$g(t)$ is the maximal solution of the Kähler-Ricci flow (3.2) for $t\in[0,T)$
for some $T\in\mathbb{R}^{+}$ and if the limiting class $\alpha_{T}$ is big,
then the following hold.

  1. (1)

$g(t)$ converges smoothly a Kähler metric $g(T)$ on $X\setminus S$. There
exists $D=D(X,g_{0},n)>0$ such that for all $t\in[0,T)$,

$diam(X,g(t))\leq D.$

For any $\mu>n$, there exists $\kappa=\kappa(X,g_{0},n,\mu)>0$ such that for
all $t\in[0,T)$ and $r\in(0,D]$,

$\frac{{\mathrm{Vol}}_{g(t)}(B_{g(t)}(x,r))}{r^{2\mu}}\geq\kappa.$

  2. (2)

Let $\omega(T)$ be the unique positive current extended by $g(T)$ and
$(\hat{X},d_{T})$ be the metric completion of $(X\setminus S,g(T))$ . Then the
metric measure space $(X_{T},d_{T},\omega(T)^{n})$ is a compact metric space.
For any $\mu>n$, there exists $\kappa=\kappa(\mu)>0$ such that for any
$r\in(0,\textnormal{diam}(X_{T},d_{T}))$, we have

$\frac{{\mathrm{Vol}}_{\omega(T)^{n}}(B_{d_{T}}(x,r))}{r^{2\mu}}\geq\kappa$

for any $x\in X_{T}$.

As in the previous section, it is worth pointing out that these results go
beyond the corresponding ones in [17], as they apply directly to the limiting
Kähler current at the singular time $T$. In [17], only uniform bounds for the
diameter and noncollapsing for the smooth metrics along the KR flow were
obtained, but it is not possible to take limits because the convergence on
compact subsets of the smooth parts is too weak. Again, it is the Sobolev
inequality extended to the metric completion of the singular Kähler current
that can apply to get the present diameter and noncollapsing estimates.

The first statement of Theorem 3.5 implies that after passing to a sequence
$t_{j}\rightarrow T^{-}$, $(X,g(t_{j}))$ always converges in Gromov-Hausdorff
sense to a compact metric space $(\hat{X}_{T},\hat{d}_{T})$ which satisfies
the same volume non-collapsing condition. In fact, the identity map between
$X\setminus S$ and itself extends to a surjective Lipschitz map from
$(X_{T},d_{T})$ to $(\hat{X}_{T},\hat{d}_{T})$. We conjecture that
$(\hat{X}_{T},\hat{d}_{T})$ and $(X_{T},d_{T})$ are isomorphic and they are
both homeomorphic to the original variety $Y$. This conjecture holds for
Kähler surfaces by the work of [34, 35, 36, 31]. We would also like to remark
that Theorem 3.5 still holds if $[g_{0}]$ is a general Kähler class as a
consequence of Remark 3.1, since the limiting class $\alpha$ is big and nef.
Theorem 3.5 can also be generalized to the unique analytic solution $g(t)$ to
the Kähler-Ricci flow on projective varieties with log terminal singularities
whenever $t>0$, because the semi-Kähler current $g(t)$ lies in
$\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$ for suitable choices of a log
resolution $\pi:Y\rightarrow X$, $\theta_{Y}$, $A$, $p>n$ and $K>0$ whenever
$t$ is uniformly bounded away from $0$.

##  4\. The Sobolev-type inequality

In this section we will prove the Sobolev inequality in Theorem 2.1. We first
let $A>0$ be a finite number and consider the parameters $p>n,K>0$ and
$\gamma\in C^{0}(X)$ satisfying (2.2). We will fix a Kähler metric
$\omega\in{\mathcal{W}}(n,p,A,K,\gamma)$. We recall the Green’s function
$G_{\omega}(x,y)$ of the Laplacian $\Delta_{\omega}$ on $X$ is defined by

$\Delta_{\omega}G_{\omega}(x,\cdot)=\frac{1}{V_{\omega}}-\delta_{x}(\cdot),\quad\int_{X}G_{\omega}(x,\cdot)\omega^{n}=0,$

where $\delta_{x}(\cdot)$ is the Dirac function at the base point $x\in X$. It
is well-known that $G_{\omega}$ always exists and is smooth in $X\times X$ off
the diagonal. Moreover, $G_{\omega}$ is symmetric in the sense that
$G_{\omega}(x,y)=G_{\omega}(y,x)$ for any $x\neq y$. We first recall the lower
bound of $G_{\omega}$ from [17].

######  Lemma 4.1 ([17]).

There exists a constant $C_{0}=C_{0}(n,p,A,K,\gamma)>0$ such that

$\inf_{x,y\in X}G_{\omega}(x,y)\geq-C_{0}/V_{\omega}.$

Given this lemma, we denote the positive function

${\mathcal{G}}_{\omega}=G_{\omega}+\frac{C_{0}+1}{V_{\omega}}\geq\frac{1}{V_{\omega}},$

which still satisfies the Green’s formula. We also need the following uniform
integral estimates of ${\mathcal{G}}_{\omega}$ and
$\nabla{\mathcal{G}}_{\omega}$:

######  Lemma 4.2 ([17, 18]).

(i). There exist constants $C_{1}=C_{1}(n,p,A,K,\gamma)>0$ and
$\varepsilon_{0}=\varepsilon_{0}(n,p)>0$ such that

(4.1) $\sup_{x\in
X}\int_{X}{\mathcal{G}}_{\omega}(x,\cdot)^{1+\varepsilon_{0}}\omega^{n}\leq\frac{C_{1}}{V_{\omega}^{\varepsilon_{0}}}.$

(ii). For any $\beta>0$ we have

(4.2) $\sup_{x\in
X}\int_{X}\frac{|\nabla{\mathcal{G}}_{\omega}(x,\cdot)|^{2}_{\omega}}{{\mathcal{G}}_{\omega}(x,\cdot)^{1+\beta}}\omega^{n}\leq\frac{V_{\omega}^{\beta}}{\beta}.$

With Lemma 4.2, we now give the proof of Theorem 2.1.

Proof of Theorem 2.1: By approximation, we may assume the function $u\in
C^{1}(X)$. We apply the Green’s formula associated to the metric $\omega$ to
obtain

$u(x)=\frac{1}{V_{\omega}}\int_{X}u\omega^{n}+\int_{X}\langle\nabla{\mathcal{G}}_{\omega}(x,\cdot),\nabla
u\rangle_{\omega}\omega^{n}$

for any $x\in X$. Denote
${\overline{u}}=\frac{1}{V_{\omega}}\int_{X}u\omega^{n}$ to be the average of
$u$ over $(X,\omega^{n})$. By the triangle inequality this gives

$\displaystyle|u(x)-\overline{u}|$ $\displaystyle\leq$
$\displaystyle\int_{X}|\nabla{\mathcal{G}}_{\omega}(x,\cdot)|_{\omega}|\nabla
u|_{\omega}\omega^{n}$ $\displaystyle=$
$\displaystyle\int_{X}\frac{|\nabla{\mathcal{G}}_{\omega}(x,\cdot)|_{\omega}}{{\mathcal{G}}_{\omega}(x,\cdot)^{(1+\beta)/2}}\cdot{\mathcal{G}}_{\omega}(x,\cdot)^{(1+\beta)/2}|\nabla
u|_{\omega}\omega^{n}$ (by Hölder’s ineq.) $\displaystyle\leq$
$\displaystyle\Big{(}\int_{X}\frac{|\nabla{\mathcal{G}}_{\omega}(x,\cdot)|^{2}_{\omega}}{{\mathcal{G}}_{\omega}(x,\cdot)^{1+\beta}}\omega^{n}\Big{)}^{1/2}\Big{(}\int_{X}{\mathcal{G}}_{\omega}(x,\cdot)^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}\Big{)}^{1/2}$ (4.3) by (4.2) $\displaystyle\leq$
$\displaystyle\Big{(}\frac{V_{\omega}^{\beta}}{\beta}\Big{)}^{1/2}\Big{(}\int_{X}{\mathcal{G}}_{\omega}(x,\cdot)^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}\Big{)}^{1/2}.$

Here we fix the constant $\beta=\varepsilon_{0}/2$, where $\varepsilon_{0}>0$
is as in (4.1), and

(4.4)
$q=\frac{1+\varepsilon_{0}}{1+\beta}=\frac{1+\varepsilon_{0}}{1+(\varepsilon_{0}/2)}>1.$

Taking $(2q)$-th power on both sides of (4.3), we obtain

$|u(x)-\overline{u}|^{2q}\leq\frac{V_{\omega}^{\beta
q}}{\beta^{q}}\Big{(}\int_{X}{\mathcal{G}}_{\omega}(x,y)^{1+\beta}|\nabla
u(y)|^{2}_{\omega}\omega^{n}(y)\Big{)}^{q}.$

We first integrate this inequality over $X$ against the measure $\omega^{n}$,
and then take the $q$-th root of the resulted integrals, which reads

(4.5)
$\Big{(}\int_{X}|u(x)-\overline{u}|^{2q}\omega^{n}\Big{)}^{1/q}\leq\frac{V_{\omega}^{\beta}}{\beta}\Big{\\{}\int_{x\in
X}\Big{(}\int_{X}{\mathcal{G}}_{\omega}(x,y)^{1+\beta}|\nabla
u(y)|^{2}_{\omega}\omega^{n}(y)\Big{)}^{q}\omega^{n}(x)\Big{\\}}^{1/q}.$

Applying the generalized Minkowski inequality to the right-hand side of (4.5),
we get

(4.6) $\displaystyle\Big{\\{}\int_{x\in
X}\Big{(}\int_{X}{\mathcal{G}}_{\omega}(x,y)^{1+\beta}|\nabla
u(y)|^{2}_{\omega}\omega^{n}(y)\Big{)}^{q}\omega^{n}(x)\Big{\\}}^{1/q}$
$\displaystyle\leq$ $\displaystyle\int_{X}|\nabla
u(y)|^{2}_{\omega}\Big{(}\int_{x\in
X}{\mathcal{G}}_{\omega}(x,y)^{(1+\beta)q}\omega^{n}(x)\Big{)}^{1/q}\omega^{n}(y).$

By the symmetry of ${\mathcal{G}}_{\omega}$, the integral factor in (4.6)
satisfies

$\displaystyle\Big{(}\int_{x\in
X}{\mathcal{G}}_{\omega}(x,y)^{(1+\beta)q}\omega^{n}(x)\Big{)}^{1/q}$
$\displaystyle=$
$\displaystyle\Big{(}\int_{X}{\mathcal{G}}_{\omega}(y,x)^{1+\varepsilon_{0}}\omega^{n}(x)\Big{)}^{1/q}$
$\displaystyle\leq$ $\displaystyle\Big{(}\sup_{y\in
X}\int_{X}{\mathcal{G}}_{\omega}(y,x)^{1+\varepsilon_{0}}\omega^{n}(x)\Big{)}^{1/q}$
by (4.1) $\displaystyle\leq$
$\displaystyle\Big{(}\frac{C_{1}}{V_{\omega}^{\varepsilon_{0}}}\Big{)}^{1/q}.$

Combining this with (4.5) and (4.6), we get

(4.7)
$\displaystyle\noindent\Big{(}\int_{X}|u(x)-\overline{u}|^{2q}\omega^{n}\Big{)}^{1/q}\leq\frac{V_{\omega}^{\beta}}{\beta}\frac{C_{1}^{1/q}}{V_{\omega}^{\varepsilon_{0}/q}}\int_{X}|\nabla
u|_{\omega}^{2}\omega^{n}=\frac{C_{2}}{V_{\omega}^{1-\frac{1}{q}}}\int_{X}|\nabla
u|_{\omega}^{2}\omega^{n},$

where $C_{2}=C_{1}^{1/q}/\beta>0$ depends only on $n,p,A,K,\gamma$. From
(4.7), it is clear that the Sobolev inequality holds, i.e.

(4.8)
$\displaystyle\Big{(}\frac{1}{V_{\omega}}\int_{X}|u(x)-\overline{u}|^{2q}\omega^{n}\Big{)}^{1/q}\leq\frac{C_{2}}{V_{\omega}}\int_{X}|\nabla
u|_{\omega}^{2}\omega^{n}.$

To see the scale-invariant inequality (2.4), suppose
$\omega\in{\mathcal{K}}(X)$ satisfies $F_{\omega}\geq\gamma$ and
$\|e^{F_{\omega}}\|_{L^{1}(\log L)^{p}(\omega_{X}^{n})}\leq K$. Set
$\hat{\omega}=I_{\omega}^{-1}\omega$. It is elementary to see that
$\hat{\omega}\in{\mathcal{W}}(n,p,A,K,\gamma)$ with $A=1$. So the inequality
(4.8) holds for the metric $\hat{\omega}$ with a constant $C_{2}^{\prime}$
which is independent of $A$ as stated, i.e.,

(4.9)
$\displaystyle\Big{(}\frac{1}{V_{\hat{\omega}}}\int_{X}|u(x)-\overline{u}|^{2q}\hat{\omega}^{n}\Big{)}^{1/q}\leq\frac{C_{2}^{\prime}}{V_{\hat{\omega}}}\int_{X}|\nabla
u|_{\hat{\omega}}^{2}\hat{\omega}^{n}.$

Then the inequality (2.4) follows by observing that $|\nabla
u|_{\hat{\omega}}^{2}=I_{\omega}|\nabla u|_{\omega}^{2}$ and
$\frac{\hat{\omega}^{n}}{V_{\hat{\omega}}}=\frac{\omega^{n}}{V_{\omega}}$. The
proof of Theorem 2.1 is completed. ∎

By combining (4.8) with the Minkowski inequality, we obtain the following form
of the Sobolev inequality, which is better suited for the Moser’s iteration
scheme:

(4.10)
$\Big{(}\frac{1}{V_{\omega}}\int_{X}|u|^{2q}\omega^{n}\Big{)}^{1/q}\leq\frac{C_{3}}{V_{\omega}}\int_{X}(u^{2}+|\nabla
u|_{\omega}^{2})\omega^{n},$

where $C_{3}=\max(2C_{2}^{2},1)$.

In case the Kähler metric $\omega$ satisfies the conditions

$e^{F_{\omega}}\geq\gamma,\quad\|e^{F_{\omega}}\|_{L^{1+\varepsilon^{\prime}}}\leq
K,\quad[\omega]\cdot[\omega_{X}]^{n-1}\leq A,$

where $\gamma$ satisfies the condition (2.2), and $\varepsilon^{\prime}>0$ is
a positive constant, we have the following improved estimates on $(X,\omega)$
which may be of independent interest.

######  Theorem 4.1.

Let the assumptions be as above. For any $1\leq p^{\prime}<n$ and
$q^{\prime}\in(p^{\prime},\frac{np^{\prime}}{n-p^{\prime}})$, there exists a
constant $C>0$ which depends on $n,K,A,\gamma,\varepsilon^{\prime}$ and
$p^{\prime},q^{\prime}$ such that

(4.11)
$\Big{(}\frac{1}{V_{\omega}}\int_{X}|u-{\overline{u}}|^{2q^{\prime}}\omega^{n}\Big{)}^{1/q^{\prime}}\leq
C\Big{(}\frac{1}{V_{\omega}}\int_{X}|\nabla
u|_{\omega}^{2p^{\prime}}\omega^{n}\Big{)}^{1/p^{\prime}}$

for all $u\in C^{1}(X)$. In particular, we have the imbedding
$W^{1,2p^{\prime}}(X,\omega)\hookrightarrow L^{2q^{\prime}}(X,\omega)$.

###### Proof.

We know from [18, 17] that the Green’s function satisfies

$\int_{X}{\mathcal{G}}_{\omega}(x,y)^{s}\omega^{n}(y)\leq\frac{C(s)}{V_{\omega}^{s-1}}$

for all $s\in[1,\frac{n}{n-1})$, for some constant $C(s)>0$ depending
additionally on $s$ besides the given parameters. Given the constants
$p^{\prime},q^{\prime}$ as in the statement of the theorem, we define

$r=({1+\frac{1}{q^{\prime}}-\frac{1}{p^{\prime}}})^{-1}\in(1,\frac{n}{n-1}).$

We choose the constant $\beta\in(0,1)$ small enough in (4.3) such that
$(1+\beta)r<\frac{n}{n-1}$. We define a linear operator as follows: (below
$r^{*}=\frac{r}{r-1}$)

${\mathbf{T}}:L^{r^{*}}\cap L^{1}\to L^{\infty}\cap
L^{r},\quad{\mathbf{T}}f(x)=\int{\mathcal{G}}_{\omega}(x,y)^{1+\beta}f(y)\omega^{n}(y).$

We can verify that ${\mathbf{T}}$ is of strong types $(1,r)$ and
$(r^{*},+\infty)$. Indeed, we calculate as follows.

$\displaystyle\|{\mathbf{T}}f\|_{L^{r}(\omega^{n})}$ $\displaystyle=$
$\displaystyle\Big{(}\int_{X}|{\mathbf{T}}f(x)|^{r}\omega^{n}(x)\Big{)}^{\frac{1}{r}}\leq\int_{X}\Big{(}\int_{x\in
X}{\mathcal{G}}_{\omega}(x,y)^{r(1+\beta)}\omega^{n}(x)\Big{)}^{\frac{1}{r}}|f(y)|\omega^{n}(y)$
$\displaystyle\leq$ $\displaystyle
C_{1}V_{\omega}^{-(1+\beta)+1/r}\|f\|_{L^{1}(\omega^{n})},$

where in the first inequality we have applied the generalized Minkowski
inequality. For any $x\in X$, by Hölder’s inequality we have

$|{\mathbf{T}}f(x)|\leq\Big{(}\int{\mathcal{G}}_{\omega}(x,y)^{r(1+\beta)}\omega^{n}(y)\Big{)}^{1/r}\|f\|_{L^{r^{*}}(\omega^{n})}\leq
C_{1}V_{\omega}^{-(1+\beta)+1/r}\|f\|_{L^{r^{*}}(\omega^{n})},$

which implies that $\|{\mathbf{T}}f\|_{L^{\infty}}\leq
C_{1}V_{\omega}^{-(1+\beta)+1/r}\|f\|_{L^{r^{*}}(\omega^{n})}$. Then we apply
the Riesz-Thorin Interpolation Theorem to conclude that for any $t\in(0,1)$,
and the pair of numbers $p_{t},q_{t}$ given by

$\frac{1}{p_{t}}=\frac{1-t}{r^{*}}+{t},\quad\frac{1}{q_{t}}=\frac{t}{r},\,\mbox{respectively,
}$

${\mathbf{T}}$ is of strong type $(p_{t},q_{t})$. More precisely, the
following holds,

(4.12) $\|{\mathbf{T}}f\|_{L^{q_{t}}(\omega^{n})}\leq
C_{1}V_{\omega}^{-(1+\beta)+1/r}\|f\|_{L^{p_{t}}(\omega^{n})}.$

In particular, taking $t={r}/{q^{\prime}}\in(0,1)$, we see that
$p_{t}=p^{\prime}$ and $q_{t}=q^{\prime}$. To finish the proof we recall that
from (4.3), we have the following

$|u(x)-{\overline{u}}|^{2}\leq\frac{V_{\omega}^{\beta}}{\beta}{\mathbf{T}}(|\nabla
u|_{\omega}^{2})(x)$

for all $x\in X$. We can now apply the inequality (4.12) with $f:=|\nabla
u|^{2}_{\omega}$ to conclude that

(4.13)
$\||u-{\overline{u}}|^{2}\|_{L^{q^{\prime}}(\omega^{n})}\leq\frac{C_{1}}{\beta}V_{\omega}^{\frac{1}{q^{\prime}}-\frac{1}{p^{\prime}}}\||\nabla
u|^{2}_{\omega}\|_{L^{p^{\prime}}(\omega^{n})}.$

Clearly here the $\beta$ can be chosen depending only on $p^{\prime}$ and
$q^{\prime}$. The desired inequality (4.11) follows easily from some
rearrangement of (4.13). ∎

We next discuss a few examples when the Sobolev inequalities (4.8) and (4.10)
hold.

Example 4.1 Let $\pi:(X,\omega_{X})\to(Y,\omega_{Y})$ be a surjective
holomorphic map between compact Kähler manifolds $X$ and $Y$. Assume that
${\mathrm{dim}}(X)=n\geq{\mathrm{dim}}(Y)=m$. Consider the continuous family
of Kähler metrics $\omega=\omega_{t}:=t\omega_{X}+\pi^{*}\omega_{Y}$ for
$t\in(0,1]$. It is easy to see that $V_{\omega}\sim t^{n-m}$, and

$e^{F_{\omega}}=\frac{1}{V_{\omega}}\frac{\omega^{n}}{\omega_{X}^{n}}\sim\frac{1}{V_{\omega}}\sum_{k=0}^{m}t^{n-k}\frac{(\pi^{*}\omega_{Y})^{k}\wedge\omega_{X}^{n-k}}{\omega_{X}^{n}}\leq
C$

for some constant $C>0$ depending only on $n,m,\omega_{X},\pi^{*}\omega_{Y}$
but independent of $t$. The intersection number
$[\omega]\cdot[\omega_{X}]^{n-1}$ is clearly bounded. The expected lower bound
$\gamma$ of $e^{F_{\omega}}$ can be taken as suitable multiple of
$\frac{(\pi^{*}\omega_{Y})^{m}\wedge\omega_{X}^{n-m}}{\omega_{X}^{n}}$, which
is equal to the squared Jacobian of the holomorphic map
$\pi:(X,\omega_{X})\to(Y,\omega_{Y})$. The Jacobian of $\pi$ consists of
nontrivial (local) holomorphic functions hence its vanishing locus
$\\{\gamma=0\\}$ is a proper analytic subvariety in $X$. Hence the assumption
(2.2) on $\gamma$ is also satisfied. Given these, we find that the Sobolev
inequalities (4.8) and (4.10) hold for this continuous family of Kähler
metrics, with the constants depending on $n,\pi,\omega_{X},\omega_{Y}$, but
not $t\in(0,1]$.

We remark that when $\pi:X\to Y$ is the blown-up of a smooth center in $Y$,
the uniform Sobolev inequality (4.10) is known by the work of Bando-Siu in [2]
with $q=\frac{n}{n-1}$. However, for the case of general surjective
holomorphic maps, as far as we know, the uniform Sobolev inequalities (4.8)
and (4.10) are new.

Example 4.2 Consider the finite-time Kähler-Ricci flow $\omega=\omega_{t}$:
$\frac{\partial\omega}{\partial t}=-\textnormal{Ric}(\omega)$ with $t\in[0,T)$
and $T<\infty$. If the limit current $[\omega_{T}]$ is big, in the sense that
$[\omega_{T}]^{n}>0$, then by [17], the metrics $\omega_{t}$ lie in
${\mathcal{W}}(n,p,A,K,\gamma)$ for suitable parameters depending only on
$n,\omega_{t}|_{t=0}$ and $[\omega_{T}]$. Hence the uniform Sobolev
inequalities (4.8) and (4.10) hold for these metrics $\omega_{t}$ with
$t\in[0,T)$.

Example 4.3 Let $X$ be a compact Kähler manifold with nef canonical line
bundle $K_{X}$ and nonnegative Kodaira dimension. Suppose $\omega=\omega_{t}$
satisfies the normalized Kähler-Ricci flow $\tfrac{\partial\omega}{\partial
t}=-\textnormal{Ric}(\omega)-\omega$. Again by [17], the metrics $\omega_{t}$
lie in ${\mathcal{W}}(n,p,A,K,\gamma)$ for suitable parameters depending only
on $n,\omega_{t}|_{t=0}$ and $K_{X}$. Hence the uniform Sobolev inequalities
(4.8) and (4.10) hold for these metrics $\omega_{t}$ with $t\in[0,\infty)$.

##  5\. Upper bound of the heat kernel

Let $\omega\in{\mathcal{W}}(n,p,A,K,\gamma)$ be a fixed Kähler metric, and
$H(x,y,t)$ be the heat kernel of $\Delta_{\omega}$ as defined in the equation
(2.5).

###  5.1. Off diagonal estimates

Given the Sobolev inequality (4.10), we will make use of the ideas of Davies
[8] (see also [41]) and Saloff-Coste [29] to prove a Gaussian-type upper bound
of the hear kernel $H(x,y,t)$.

Let $u_{0}\in C^{\infty}(X)$ be an arbitrary positive smooth function on $X$.
For a given $\alpha\in{\mathbb{R}}$ and an arbitrary Lipschitz function $\phi$
with $\|\phi\|_{L^{\infty}(\omega)}\leq 1$, we consider the function $u(x,t)$
defined by

(5.1)
$u(x,t)=e^{-\alpha\phi(x)}\int_{X}H(x,y,t)e^{\alpha\phi(y)}u_{0}(y)\omega^{n}(y),$

which is positive and satisfies the equation

(5.2) $\frac{\partial u}{\partial
t}=e^{-\alpha\phi}\Delta_{\omega}(ue^{\alpha\phi}),\quad u(x,0)=u_{0}(x)$

for all $x\in X$. Fix any time $T>0$ and a constant $\beta\in(0,1)$ (for our
purpose, it suffices to take $\beta=1/2$). We define an increasing, piecewise
smooth, and continuous function $r(t)$ of $t\in(0,T)$ by

$r=r(t)=\left\\{\begin{array}[]{ll}\frac{2^{\beta}-1}{(T/2)^{\beta}}t^{\beta}+1,&\quad\mbox{if
}t\in[0,T/2),\\\ \frac{T^{\beta}}{(T-t)^{\beta}},&\quad\mbox{if
}t\in[T/2,T),\end{array}\right.$

which satisfies $r(0)=1$ and $r(T^{-})=\infty$. Denote $\|u\|_{r}$ to be the
$L^{r}(X,\omega^{n})$-norm of $u$. By straightforward calculations, we have

(5.4) $\displaystyle\frac{\partial}{\partial t}\log\|u\|_{r}$ $\displaystyle=$
$\displaystyle-\frac{r^{\prime}}{r}\log\|u\|_{r}+\frac{1}{\|u\|_{r}^{r}}\int_{X}u^{r-1}e^{-\alpha\phi}\Delta_{\omega}(ue^{\alpha\phi})\omega^{n}$
$\displaystyle\quad+\frac{r^{\prime}}{r}\frac{1}{\|u\|_{r}^{r}}\int_{X}u^{r}(\log
u)\,\omega^{n}.$

By integration by parts, the last term in (5.4) is equal to

(5.5)
$\displaystyle\frac{1}{\|u\|_{r}^{r}}\int_{X}u^{r-1}e^{-\alpha\phi}\Delta_{\omega}(ue^{\alpha\phi})\omega^{n}$
$\displaystyle=$
$\displaystyle-\frac{1}{\|u\|_{r}^{r}}\int_{X}\langle\nabla(u^{r-1}e^{-\alpha\phi}),\nabla(ue^{\alpha\phi})\rangle_{\omega}\omega^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{\|u\|_{r}^{r}}\int_{X}-(r-1)u^{r-2}|\nabla
u|^{2}-\alpha(r-2)u^{r-1}\langle\nabla
u,\nabla\phi\rangle_{\omega}+\alpha^{2}u^{r}|\nabla\phi|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\|u\|_{r}^{r}}\int_{X}-(r-1)(1-\delta)u^{r-2}|\nabla
u|^{2}+\frac{\alpha^{2}(r-2)^{2}}{4(r-1)\delta}u^{r}|\nabla\phi|^{2}+\alpha^{2}u^{r}|\nabla\phi|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\|u\|_{r}^{r}}\int_{X}-\frac{4(r-1)(1-\delta)}{r^{2}}|\nabla
u^{r/2}|^{2}\omega^{n}+\frac{\alpha^{2}(r-2)^{2}}{4(r-1)\delta}+\alpha^{2},$

where in the first inequality we apply the Cauchy-Schwarz inequality with
$\delta\in(0,1)$ a constant to be chosen later, and in the last line we use
the condition that $\|\nabla\phi\|_{\infty}\leq 1$.

For notational convenience we set $v=\frac{u^{r/2}}{\|u^{r/2}\|_{2}}$, which
satisfies $\|v\|_{2}=1$. Note that $\|u^{r/2}\|_{2}^{2}=\|u\|_{r}^{r}.$ The
term in (5.4) becomes

$\frac{r^{\prime}}{r}\frac{1}{\|u\|_{r}^{r}}\int_{X}u^{r}\log
u\,\omega^{n}=\frac{r^{\prime}}{r^{2}}\int_{X}v^{2}\log
v^{2}\,\omega^{n}+\frac{r^{\prime}}{r}\log\|u\|_{r}.$

Therefore, we have from (5.4) and (5.5) that

$\displaystyle\frac{\partial}{\partial t}\log\|u\|_{r}$ $\displaystyle\leq$
$\displaystyle-\frac{4(r-1)(1-\delta)}{r^{2}}\int_{X}|\nabla
v|_{\omega}^{2}\omega^{n}+\frac{r^{\prime}}{r^{2}}\int_{X}v^{2}\log
v^{2}\,\omega^{n}$
$\displaystyle\quad+\frac{\alpha^{2}(r-2)^{2}}{4(r-1)\delta}+\alpha^{2}.$

We next apply the Sobolev inequality (4.10) to deal with the last term in
(5.1).

$\displaystyle\int_{X}v^{2}\log v^{2}\omega^{n}$ $\displaystyle=$
$\displaystyle\frac{1}{q-1}\int_{X}v^{2}\log v^{2(q-1)}\omega^{n}$ (by
Jensen’s inequality) $\displaystyle\leq$
$\displaystyle\frac{1}{q-1}\log\frac{1}{V_{\omega}}\int_{X}v^{2q}\omega^{n}+\frac{\log
V_{\omega}}{q-1}$ $\displaystyle\leq$
$\displaystyle\frac{q}{q-1}\log\Big{(}\frac{C}{V_{\omega}}\int_{X}(v^{2}+|\nabla
v|^{2})\omega^{n}\Big{)}+\frac{\log V_{\omega}}{q-1}$ $\displaystyle=$
$\displaystyle\frac{q}{q-1}\log\Big{(}\int_{X}(v^{2}+|\nabla
v|^{2})\omega^{n}\Big{)}-\log V_{\omega}+\frac{q}{q-1}\log C.$

Combining this with (5.1), we obtain

(5.7) $\displaystyle\frac{\partial}{\partial t}\log\|u\|_{r}$
$\displaystyle\leq$
$\displaystyle-\frac{4(r-1)(1-\delta)}{r^{2}}\int_{X}|\nabla
v|_{\omega}^{2}\omega^{n}+\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}\log\Big{(}1+\int_{X}|\nabla
v|^{2}\omega^{n}\Big{)}$
$\displaystyle\quad+\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}\log{C}-\frac{r^{\prime}}{r^{2}}\log
V_{\omega}+\frac{\alpha^{2}(r-2)^{2}}{4(r-1)\delta}+\alpha^{2}.$

Claim: The following inequality holds:

$\displaystyle-\frac{4(r-1)(1-\delta)}{r^{2}}\int_{X}|\nabla
v|_{\omega}^{2}\omega^{n}+\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}\log\Big{(}1+\int_{X}|\nabla
v|^{2}\omega^{n}\Big{)}$ $\displaystyle\leq$
$\displaystyle\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}\log\Big{(}\frac{r^{\prime}q}{4(r-1)(1-\delta)(q-1)}\Big{)}+\frac{4(r-1)(1-\delta)}{r^{2}}-\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}.$

This claim follows easily from the calculus inequality that the concave
function

$f(x):=-\frac{4(r-1)(1-\delta)}{r^{2}}x+\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}\log(1+x)$

achieves the maximum at $x:=\frac{r^{\prime}q}{4(r-1)(1-\delta)(q-1)}-1$.

This claim together with the inequality (5.7) implies that

$\displaystyle\frac{\partial}{\partial t}\log\|u\|_{r}$ $\displaystyle\leq$
$\displaystyle\frac{r^{\prime}}{r^{2}}\frac{q}{q-1}\log\Big{(}\frac{r^{\prime}}{r-1}\Big{)}+\frac{4(r-1)(1-\delta)}{r^{2}}+\frac{r^{\prime}}{r^{2}}\Psi$
$\displaystyle\quad+\frac{\alpha^{2}(r-2)^{2}}{4(r-1)\delta}+\alpha^{2},$

where the constant $\Psi$ is given by

$\Psi:=\frac{q}{q-1}\log\frac{q}{4(q-1)(1-\delta)}-\frac{q}{q-1}+\frac{q}{q-1}\log{C}-\log
V_{\omega}.$

We need the following elementary identities:

$\displaystyle\int_{0}^{T}\frac{r^{\prime}}{r^{2}}dt=-\frac{1}{r(t)}\Big{|}_{t=0}^{T}=1.$
$\displaystyle\int_{0}^{T}\frac{r^{\prime}}{r^{2}}\log\Big{(}\frac{r^{\prime}}{r-1}\Big{)}dt$
$\displaystyle=$
$\displaystyle\int_{0}^{T/2}\frac{r^{\prime}}{r^{2}}\log\frac{\beta}{t}dt+\int_{T/2}^{T}\frac{r^{\prime}}{r^{2}}\log\Big{(}\frac{\beta
T^{\beta}}{(T-t)(T^{\beta}-(T-t)^{\beta})}\Big{)}dt$ $\displaystyle=$
$\displaystyle\int_{0}^{1/2}\frac{r_{s}^{\prime}}{r^{2}}\log\frac{\beta}{Ts}ds+\int_{1/2}^{1}\frac{r_{s}^{\prime}}{r^{2}}\log\Big{(}\frac{\beta}{T(1-s)(1-(1-s)^{\beta})}\Big{)}ds$
$\displaystyle=$
$\displaystyle\log\frac{1}{T}\int_{0}^{1}\frac{r_{s}^{\prime}}{r^{2}}ds+\int_{0}^{1/2}\frac{\beta(4^{\beta}-2^{\beta})s^{\beta-1}}{((4^{\beta}-2^{\beta})s^{\beta}+1)^{2}}\log\frac{\beta}{s}ds$
$\displaystyle\quad+\int_{1/2}^{1}\frac{\beta}{(1-s)^{1-\beta}}\log\Big{(}\frac{\beta}{(1-s)(1-(1-s)^{\beta})}\Big{)}ds$
$\displaystyle=$ $\displaystyle-\log T+A_{\beta},$

where we denote $A_{\beta}\in{\mathbb{R}}$ to be the sum of the two integrals
which are convergent by the choice of $\beta\in(0,1)$, and we note that
$A_{\beta}\approx 2.455$ if $\beta=1/2$.

$\displaystyle\int_{0}^{T}\frac{(r-2)^{2}}{r-1}dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T/2}\frac{[(4^{\beta}-2^{\beta})(t/T)^{\beta}-1]^{2}}{(4^{\beta}-2^{\beta})(t/T)^{\beta}}dt+\int_{T/2}^{T}\frac{[T^{\beta}-2(T-t)^{\beta}]^{2}}{(T-t)^{\beta}[T^{\beta}-(T-t)^{\beta}]}dt$
$\displaystyle=$ $\displaystyle
T\Big{(}\int_{0}^{1/2}\frac{[(4^{\beta}-2^{\beta})s^{\beta}-1]^{2}}{(4^{\beta}-2^{\beta})s^{\beta}}ds+\int_{1/2}^{1}\frac{[1-2(1-s)^{\beta}]^{2}}{(1-s)^{\beta}[1-(1-s)^{\beta}]}ds\Big{)}$
$\displaystyle=$ $\displaystyle TB_{\beta},$

where $B_{\beta}>0$ is the sum of the two integrals. (If $\beta=1/2$, then
$B_{\beta}\approx 2.008$).

$\int_{0}^{T}\frac{r-1}{r^{2}}dt=T\Big{(}\int_{0}^{1/2}\frac{(4^{\beta}-2^{\beta})s^{\beta}}{[(4^{\beta}-2^{\beta})s^{\beta}+1]^{2}}ds+\int_{1/2}^{1}(1-(1-s)^{\beta})(1-s)^{\beta}ds\Big{)}=TC_{\beta},$

where $C_{\beta}\approx 0.192$ if $\beta=1/2$.

Together with these identities, integrating (5.1) over $t\in(0,T)$ yields that

$\log\frac{\|u_{T}\|_{\infty}}{\|u_{0}\|_{1}}\leq\frac{q}{q-1}(A_{\beta}-\log
T)+4(1-\delta)C_{\beta}T+\Psi+\frac{B_{\beta}}{4\delta}\alpha^{2}T+\alpha^{2}T$

where $u_{T}(x)=u(x,T)$. This implies that

(5.9) $\displaystyle\|u_{T}\|_{\infty}$ $\displaystyle\leq$ $\displaystyle
T^{-\frac{q}{q-1}}\exp\Big{(}4(1-\delta)C_{\beta}T+\frac{A_{\beta}q}{q-1}+\Psi+\frac{B_{\beta}}{4\delta}\alpha^{2}T+\alpha^{2}T\Big{)}\|u_{0}\|_{1}$
$\displaystyle\leq$ $\displaystyle
C_{0}T^{-\frac{q}{q-1}}V_{\omega}^{-1}e^{4(1-\delta)C_{\beta}T+\frac{B_{\beta}}{4\delta}\alpha^{2}T+\alpha^{2}T}\|u_{0}\|_{1},$

where the constant $C_{0}>0$ depends on $\beta$ and $\delta\in(0,1)$. Since
(5.9) holds for any positive function $u_{0}$, by duality and the
representation formula (5.1), the inequality (5.9) implies the pointwise
estimate for $H(x,y,t)$, for any two points $x,y\in X$, that

(5.10) $H(x,y,T)\leq
C_{0}T^{-\frac{q}{q-1}}V_{\omega}^{-1}e^{4(1-\delta)C_{\beta}T+\frac{B_{\beta}}{4\delta}\alpha^{2}T+\alpha^{2}T}e^{\alpha(\phi(x)-\phi(y))}.$

For two fixed points $x,y\in X$, if we take
$\alpha=-\frac{\phi(x)-\phi(y)}{(2+B_{\beta}/2\delta)T}$, then from (5.10) we
obtain

(5.11) $H(x,y,T)\leq
C_{0}T^{-\frac{q}{q-1}}V_{\omega}^{-1}e^{4(1-\delta)C_{\beta}T}e^{-\frac{(\phi(x)-\phi(y))^{2}}{(4+B_{\beta}/\delta)T}}.$

Since (5.11) holds for any smooth function $\phi$ with
$\|\nabla\phi\|_{\infty}\leq 1$, we can take a sequence of such functions
which converge uniformly to $z\mapsto d_{\omega}(x,z)$, where
$d_{\omega}(x,z)$ is the function of distance of $z\in X$ to the fixed point
$x$. Therefore, from (5.11), we conclude that for any $T>0$, it holds that

(5.12) $H(x,y,T)\leq
C_{0}T^{-\frac{q}{q-1}}V_{\omega}^{-1}e^{4(1-\delta)C_{\beta}T}e^{-\frac{d_{\omega}(x,y)^{2}}{(4+B_{\beta}/\delta)T}}.$

We next claim that the factor $e^{4(1-\delta)C_{\beta}T}$ on the right-hand
side of (5.12) can be replaced by a uniform constant independent of $T>0$. We
have shown that (5.12) holds for any metric
$\omega\in{\mathcal{W}}(n,p,A,K,\gamma)$. If we take $A=1$, the constant
$C_{0}$ in (5.12) can be chosen to be independent of the parameter $A$, which
we will assume next.

Recall that we denote $I_{\omega}=[\omega]\cdot[\omega_{X}]^{n-1}>0$ to be the
intersection number of the Kähler classes $[\omega]$ and $[\omega_{X}]$. Given
a metric $\omega\in{\mathcal{W}}(n,p,\infty,K,\gamma)$, we consider the
rescaled metric $\tilde{\omega}=I_{\omega}^{-1}\omega$, which meets the
equations $F_{\tilde{\omega}}=F_{\omega}$ and
$I_{\tilde{\omega}}=I_{\omega}^{-1}[\omega]\cdot[\omega_{X}]^{n-1}=1$. Hence
we have $\tilde{\omega}\in{\mathcal{W}}(n,p,1,K,\gamma)$. Applying (5.12) to
the metric $\tilde{\omega}$ we get

(5.13) $H_{\tilde{\omega}}(x,y,T)\leq
C_{0}T^{-\frac{q}{q-1}}V_{\tilde{\omega}}^{-1}e^{4(1-\delta)C_{\beta}T}e^{-\frac{d_{\tilde{\omega}}(x,y)^{2}}{(4+B_{\beta}/\delta)T}},$

and we stress that the constant $C_{0}$ depends only on $n,p,K$, $\gamma$ and
$\delta\in(0,1)$. It is a standard fact that the heat kernels of $\omega$ and
$\tilde{\omega}$ are related by

(5.14)
$H_{\omega}(x,y,T)=I_{\omega}^{-n}H_{\tilde{\omega}}(x,y,T/I_{\omega}).$

Combining (5.13) and (5.14) gives

(5.15) $\displaystyle H_{\omega}(x,y,T)$ $\displaystyle\leq$ $\displaystyle
C_{0}I_{\omega}^{-n}(T/I_{\omega})^{-\frac{q}{q-1}}V_{\tilde{\omega}}^{-1}e^{4(1-\delta)C_{\beta}T/I_{\omega}}e^{-\frac{I_{\omega}d_{\tilde{\omega}}(x,y)^{2}}{(4+B_{\beta}/\delta)T}}$
$\displaystyle=$ $\displaystyle
C_{0}I_{\omega}^{\frac{q}{q-1}}T^{-\frac{q}{q-1}}V_{\omega}^{-1}e^{4(1-\delta)C_{\beta}T/I_{\omega}}e^{-\frac{d_{\omega}(x,y)^{2}}{(4+B_{\beta}/\delta)T}}.$

The inequality (5.15) yields an upper bound of $H_{\omega}(x,y,T)$ when $T\leq
I_{\omega}$ by choosing $\delta=1/2$, i.e.

(5.16) $H_{\omega}(x,y,T)\leq
C_{0}I_{\omega}^{\frac{q}{q-1}}T^{-\frac{q}{q-1}}V_{\omega}^{-1}e^{2C_{\beta}}e^{-\frac{d_{\omega}(x,y)^{2}}{(4+2B_{\beta})T}}.$

If $T\geq I_{\omega}$, we choose $1-\delta=\frac{I_{\omega}}{4T}\leq 1/4$,
i.e. $1>\delta\geq 3/4$. Note that the constant $C_{0}\sim
e^{\Psi}\sim(1-\delta)^{-\frac{q}{q-1}}$, hence, the inequality (5.15) implies
that if $T\geq I_{\omega}$

(5.17) $H_{\omega}(x,y,T)\leq
C_{1}V_{\omega}^{-1}e^{C_{\beta}}e^{-\frac{d_{\omega}(x,y)^{2}}{(4+2B_{\beta})T}},$

for some constant $C_{1}>0$ which depends only on $n,p,K$ and $\gamma$. The
upper bound of $H_{\omega}(x,y,t)$ in (2.6) follows from (5.16) and (5.17). ∎

###  5.2. On diagonal estimates

In this subsection, we will derive a more precise upper bound of the heat
kernel on the diagonal. We follow the classical arguments of Cheng-Li in [5].
We define

$\tilde{H}(x,y,t)=H(x,y,t)-\frac{1}{V_{\omega}},$

which satisfies $\int_{X}\tilde{H}(x,y,t)\omega^{n}(y)=0$ for any $t>0$.

######  Lemma 5.1.

$\tilde{H}$ satisfies the following semi-group property

(5.18)
$\tilde{H}(x,z,t)=\int_{X}\tilde{H}(x,y,s)\tilde{H}(y,z,t-s)\omega^{n}(y)$

for all $s\in(0,t)$.

This lemma is well-known, but for completeness, we provide a sketched proof.

Proof of Lemma 5.1. Denote
$f(s)=\int_{X}\tilde{H}(x,y,s)\tilde{H}(y,z,t-s)\omega^{n}(y)$. Taking
derivative on $f(s)$, we get

$\displaystyle f^{\prime}(s)$ $\displaystyle=$
$\displaystyle\int_{X}[\frac{\partial}{\partial
s}\tilde{H}(x,y,s)\tilde{H}(y,z,t-s)-\tilde{H}(x,y,s)\frac{\partial}{\partial
t}\tilde{H}(y,z,t-s)]\omega^{n}(y)$ $\displaystyle=$
$\displaystyle\int_{X}[\Delta_{\omega,y}\tilde{H}(x,y,s)\tilde{H}(y,z,t-s)-\tilde{H}(x,y,s)\Delta_{\omega,y}\tilde{H}(y,z,t-s)]\omega^{n}(y)$
$\displaystyle=$ $\displaystyle 0,$

by integration by parts. Hence $f(s)$ is independent of $s\in(0,t)$. It is not
hard to see $f(s)\to\tilde{H}(x,z,t)$ as $s\to 0^{+}$. Hence the lemma
follows. ∎

In particular, setting $s=t/2$ and $x=z$ in Lemma 5.1 we get the equation

$\tilde{H}(x,x,t)=\int_{X}\tilde{H}(x,y,t/2)^{2}\omega^{n}(y).$

Taking derivative with respect to $t$, this gives

(5.19) $\displaystyle\frac{\partial}{\partial t}\tilde{H}(x,x,t)$
$\displaystyle=$
$\displaystyle\int_{X}\tilde{H}(x,y,t/2)\frac{\partial}{\partial
s}\Big{|}_{s=\frac{t}{2}}\tilde{H}(x,y,s)\omega^{n}(y)$ $\displaystyle=$
$\displaystyle\int_{X}\tilde{H}(x,y,t/2)\Delta_{\omega}\tilde{H}(x,y,t/2)\omega^{n}(y)$
$\displaystyle=$
$\displaystyle-\int_{X}|\nabla_{y}\tilde{H}(x,y,t/2)|^{2}_{\omega}\omega^{n}(y).$

For a fixed $x\in X$ and $t>0$, we consider the function
$u(y):=\tilde{H}(x,y,t/2)$. By the definition of $\tilde{H}(x,y,t/2)$, we have
$\overline{u}=\frac{1}{V_{\omega}}\int_{X}u\omega^{n}=0$, hence the Sobolev
inequality (2.4) implies that

(5.20)
$\Big{(}\frac{1}{V_{\omega}}\int_{X}|u|^{2q}\Big{)}^{1/q}\leq\frac{C}{V_{\omega}}\int_{X}|\nabla
u|_{\omega}^{2}\omega^{n}.$

By Hölder’s inequality, we have

$\displaystyle\frac{1}{V_{\omega}}\int_{X}u^{2}\omega^{n}$ $\displaystyle\leq$
$\displaystyle\Big{(}\frac{1}{V_{\omega}}\int_{X}|u|^{2q}\omega^{n}\Big{)}^{\frac{1}{2q-1}}\Big{(}\frac{1}{V_{\omega}}\int_{X}|u|\omega^{n}\Big{)}^{\frac{2q-2}{2q-1}}$
by (5.20) $\displaystyle\leq$
$\displaystyle\Big{(}\frac{C}{V_{\omega}}\int_{X}|\nabla
u|_{\omega}^{2}\omega^{n}\Big{)}^{\frac{q}{2q-1}}\Big{(}\frac{2}{V_{\omega}}\Big{)}^{\frac{2q-2}{2q-1}}.$

Thus it holds that

$\displaystyle
0<\tilde{H}(x,x,t)=\int_{X}u^{2}\omega^{n}\leq{C}{V_{\omega}^{-\frac{q-1}{2q-1}}}\Big{(}\int_{X}|\nabla
u|_{\omega}^{2}\omega^{n}\Big{)}^{\frac{q}{2q-1}}.$

Combining this with (5.19), we obtain the differential inequality:

$\frac{\partial}{\partial
t}\tilde{H}(x,x,t)+CV_{\omega}^{\frac{q-1}{q}}\tilde{H}(x,x,t)^{\frac{2q-1}{q}}\leq
0,$

which implies that for a possible different constant $C>0$,
$\frac{\partial}{\partial t}\tilde{H}(x,x,t)^{-\frac{q-1}{q}}\geq
C(1-\frac{1}{q})V_{\omega}^{1-\frac{1}{q}}$. Note that
$\tilde{H}(x,x,t)\to+\infty$ as $t\to 0^{+}$. Integrating the inequality above
over $t\in(0,T]$ for any arbitrary $T>0$ yields that

$\tilde{H}(x,x,T)\leq\frac{C}{V_{\omega}}\frac{1}{T^{\frac{q}{q-1}}},$

which combined with $\tilde{H}=H-\frac{1}{V_{\omega}}$ implies the upper bound
of $H(x,x,t)$ (we change $T$ to $t$),

$H(x,x,t)\leq\frac{1}{V_{\omega}}+\frac{C}{V_{\omega}}\frac{1}{t^{\frac{q}{q-1}}}.$

A simple scaling argument using (5.14) shows that the constant $C$ can be
chosen to be independent of $A>0$ in the following form:

$H(x,x,t)\leq\frac{1}{V_{\omega}}+C\frac{I_{\omega}^{\frac{q}{q-1}}}{V_{\omega}}\frac{1}{t^{\frac{q}{q-1}}}.$

To see (2.8), we observe that from the expansion formula (5.21) we have when
$t\geq I_{\omega}$,

$\displaystyle|\tilde{H}(x,y,t)|$ $\displaystyle\leq$
$\displaystyle\sum_{k=1}^{\infty}e^{-\lambda_{k}t}|\phi_{k}(x)||\phi_{k}(y)|\leq
e^{-\lambda_{1}t/2}\sum_{k=1}^{\infty}e^{-\lambda_{k}t/2}|\phi_{k}(x)||\phi_{k}(y)|$
$\displaystyle\leq$ $\displaystyle
e^{-\lambda_{1}t/2}\tilde{H}(x,x,t)^{1/2}\tilde{H}(y,y,t)^{1/2}$
$\displaystyle\leq$
$\displaystyle\frac{C}{V_{\omega}}e^{-\lambda_{1}t/2}I_{\omega}^{\frac{q}{q-1}}t^{-\frac{q}{q-1}}\leq\frac{C}{V_{\omega}}e^{-cI^{-1}_{\omega}t},$

where we have used the estimate $\lambda_{1}\geq 2cI_{\omega}^{-1}$ for some
constant $c>0$. In particular, the heat kernel $H(x,y,t)$ converges to the
constant function $1/V_{\omega}$ exponentially as $t\to\infty$. ∎

Proof of Corollary 2.1. We turn to the applications of the upper bound of heat
kernel (2.7). Let $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\ldots$
(counting multiplicity) be the increasing sequence of eigenvalues of the
Laplacian $-\Delta_{\omega}$. We choose eigenfunctions $\phi_{k}$ with
eigenvalue $\lambda_{k}$ such that $\\{\phi_{k}\\}$ are orthonormal in
$L^{2}(X,\omega^{n})$. It is a classical fact that $\\{\phi_{k}\\}$ is an
orthonormal basis of $L^{2}(X,\omega^{n})$, $\phi_{0}=1/\sqrt{V_{\omega}}$,
and the heat kernel $H(x,y,t)$ can be expanded as

(5.21)
$H(x,y,t)=\sum_{j=0}^{\infty}e^{-\lambda_{j}t}\phi_{j}(x)\phi_{j}(y)=\frac{1}{V_{\omega}}+\sum_{j=1}^{\infty}e^{-\lambda_{j}t}\phi_{j}(x)\phi_{j}(y).$

The upper bound (2.7) implies that

(5.22)
$\sum_{j=1}^{\infty}e^{-\lambda_{j}t}\phi_{j}(x)^{2}\leq\frac{CI_{\omega}^{q/(q-1)}}{V_{\omega}}\frac{1}{t^{\frac{q}{q-1}}}.$

Integrating (5.22) over $X$ gives

(5.23)
$\sum_{j=1}^{\infty}e^{-\lambda_{j}t}\leq\frac{CI^{q/(q-1)}_{\omega}}{t^{\frac{q}{q-1}}}.$

For any given $k\in{\mathbb{N}}$, setting $t=1/\lambda_{k}$, the equation
(5.23) implies

$CI_{\omega}^{q/(q-1)}\lambda_{k}^{\frac{q}{q-1}}\geq\sum_{j=1}^{\infty}e^{-\lambda_{j}/\lambda_{k}}\geq\sum_{j=1}^{k}e^{-\lambda_{j}/\lambda_{k}}\geq
e^{-1}k.$

Hence we conclude that there is a positive constant
$c=1/(Ce)^{\frac{q-1}{q}}>0$ such that

$\lambda_{k}\geq cI_{\omega}^{-1}\,k^{\frac{q-1}{q}}.$

This finishes the proof of Corollary 2.1. ∎

We also have the following estimate on the time-derivative of the heat kernel,
which will be useful in section 10. In the following we assume
$\omega\in{\mathcal{W}}(n,p,A,K,\gamma)$ as usual, and $I_{\omega}\leq A$ by
definition.

######  Lemma 5.2.

The following inequalities hold: for any $t>0$

(5.24) $\int_{X}\big{(}\dot{H}(x,y,t)\big{)}^{2}\omega^{n}(y)\leq
t^{-2}\tilde{H}(x,x,t)\leq\frac{C}{V_{\omega}}t^{-2-\frac{q}{q-1}},$

and

(5.25) $\sup_{x,y\in
X}|\dot{H}(x,y,t)|\leq\frac{C}{V_{\omega}}t^{-1-\frac{q}{q-1}},$

where we denote $\dot{H}(x,y,t)=\frac{\partial}{\partial t}H(x,y,t)$.

###### Proof.

By the expansion formula (5.21) of the heat kernel $H(x,y,t)$, we have

$\dot{H}(x,y,t)=-\sum_{j=1}^{\infty}e^{-\lambda_{j}t}\lambda_{j}\phi_{j}(x)\phi_{j}(y).$

Then we integrate $\dot{H}(x,y,t)^{2}$ and obtain

$\int_{X}\big{(}\dot{H}(x,y,t)\big{)}^{2}\omega^{n}(y)=\sum_{j=1}^{\infty}e^{-2\lambda_{j}t}\lambda_{j}^{2}\phi_{j}^{2}(x)\leq\frac{1}{t^{2}}\sum_{j=1}^{\infty}e^{-\lambda_{j}t}\phi_{j}^{2}(x)=\frac{1}{t^{2}}\tilde{H}(x,x,t)\leq\frac{C}{V_{\omega}}t^{-2-\frac{q}{q-1}},$

where we have used the elementary inequality
$0<e^{-2\lambda_{j}t}\lambda_{j}^{2}=t^{-2}e^{-\lambda_{j}t}\cdot(\lambda_{j}t)^{2}e^{-\lambda_{j}t}<t^{-2}e^{-\lambda_{j}t}$
since the maximum of the function $x^{2}e^{-x}|_{x>0}$ is $(2/e)^{2}$ which is
$<1$.

To see (5.25), we observe that the function $\dot{H}(x,y,t)$ as a function of
$(y,t)\in X\times(0,\infty)$ satisfies the heat equation, hence by the
representation formula for heat equations, we have for a fixed $t_{0}>0$ and
any $t>t_{0}$

(5.26) $\displaystyle|\dot{H}(x,y,t)|$ $\displaystyle=$
$\displaystyle|\int_{X}H(z,y,t-t_{0})\dot{H}(x,z,t_{0})\omega^{n}(z)|=|\int_{X}\tilde{H}(z,y,t-t_{0})\dot{H}(x,z,t_{0})\omega^{n}(z)|$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{X}\tilde{H}(z,y,t-t_{0})^{2}\omega^{n}(z)\Big{)}^{1/2}\Big{(}\int_{X}\dot{H}(x,z,t_{0})^{2}\omega^{n}(z)\Big{)}^{1/2}$
(5.27) $\displaystyle=$
$\displaystyle\tilde{H}(y,y,2t-2t_{0})^{1/2}\Big{(}\int_{X}\dot{H}(x,z,t_{0})^{2}\omega^{n}(z)\Big{)}^{1/2},$

where (5.26) follows from Hölder’s inequality and in (5.27) we have applied
the semi-group property (5.18) of $\tilde{H}$. For a fixed $t>0$ we take
$t_{0}=t/2$. We then get (5.25) by applying the estimates (5.24) and (2.7).
Alternatively, the inequality (5.25) also follows from the expansion of
$\dot{H}(x,y,t)$ and Cauchy-Schwarz inequality. ∎

We remark that $L^{\infty}$ estimates of the eigenfunctions $\phi_{k}$ also
follow from (5.22). In fact, setting $t=1/\lambda_{k}$ in (5.22) yields that

$\phi_{k}(x)^{2}\leq\sum_{j=1}^{\infty}e^{-\lambda_{j}/\lambda_{k}}\phi_{j}^{2}(x)\leq\frac{CI^{q/(q-1)}_{\omega}}{V_{\omega}}\lambda_{k}^{\frac{q}{q-1}}.$

Choose $x\in X$ such that $\phi_{k}(x)^{2}=\max_{X}\phi_{k}^{2}$. We then get

$\max_{X}|\phi_{k}|^{2}\leq\frac{C}{V_{\omega}}(I_{\omega}\lambda_{k})^{\frac{q}{q-1}}.$

##  6\. Local Sobolev inequality and Poincaré inequality

In this section, we show that the Sobolev inequality and Poincaré inequality
continue to hold on domains in $X$. Let $\Omega^{\prime}\subset X$ be an open
domain which for simplicity is assumed connected, and
$\Omega\subset\Omega^{\prime}$ be a relatively compact subdomain of
$\Omega^{\prime}$. We fix a compactly supported function $\eta\in
C^{\infty}_{0}(\Omega^{\prime})$ such that $\eta\equiv 1$ in $\Omega$. The
main result is

######  Lemma 6.1.

There is a uniform constant $C=C(n,p,A,K,\gamma)>0$ such that for any $u\in
W^{1,2}(\Omega^{\prime})$, we have:

(1). The local Sobolev inequality:

(6.1)
$\Big{(}\frac{1}{V_{\omega}}\int_{\Omega}|u-\bar{u}_{\Omega}|^{2q}\omega^{n}\Big{)}^{1/q}\leq\frac{C}{V_{\omega}}\int_{\Omega^{\prime}}|\nabla(\eta
u)|^{2}_{\omega}\omega^{n},$

and (2). The local Poincaré inequality:

(6.2)
$\int_{\Omega}|u-\bar{u}_{\Omega}|^{2}\omega^{n}{}\leq{C}\int_{\Omega^{\prime}}|\nabla(\eta
u)|^{2}_{\omega}\omega^{n},$

where $\bar{u}_{\Omega}=\frac{1}{V_{\omega}(\Omega)}\int_{\Omega}u\omega^{n}$
is the average of $u$ over $\Omega$ under the metric $\omega$ and
$V_{\omega}(\Omega)=\int_{\Omega}\omega^{n}$ is the $\omega$-volume of
$\Omega$.

###### Proof.

By approximation, we may assume $u\in C^{1}(\Omega^{\prime})$ and
$\int_{\Omega^{\prime}}(u^{2}+|\nabla u|^{2}_{\omega})\omega^{n}<\infty$. For
simplicity we denote $\tilde{u}=\eta u$ which can be extended trivially to $X$
as a $C^{1}$ function. By the Green’s formula, we have for the function
$\tilde{u}$,

(6.3) $\tilde{u}(x)=\frac{1}{V_{\omega}}\int_{X}\tilde{u}\omega^{n}+\int_{y\in
X}\langle\nabla{\mathcal{G}}(x,y),\nabla\tilde{u}(y)\rangle_{\omega}\omega^{n}.$

Applying (6.3) to different points $x,z\in\Omega$ and taking the difference
result in the equation

(6.4) $\tilde{u}(x)-\tilde{u}(z)=\int_{y\in
X}\langle\nabla{\mathcal{G}}(x,y),\nabla\tilde{u}(y)\rangle_{\omega}\omega^{n}-\int_{y\in
X}\langle\nabla{\mathcal{G}}(z,y),\nabla\tilde{u}(y)\rangle_{\omega}\omega^{n}.$

Integrating (6.4) over $z\in\Omega$ against $\omega^{n}$ and dividing the
obtained equation by $V_{\omega}(\Omega)$, we get

(6.5) $\tilde{u}(x)-\bar{\tilde{u}}_{\Omega}=\int_{y\in
X}\langle\nabla{\mathcal{G}}(x,y),\nabla\tilde{u}(y)\rangle_{\omega}\omega^{n}-\frac{1}{V_{\omega}(\Omega)}\int_{z\in\Omega}\omega^{n}\int_{y\in
X}\langle\nabla{\mathcal{G}}(z,y),\nabla\tilde{u}(y)\rangle_{\omega}\omega^{n},$

which implies by triangle inequality that

$\displaystyle|\tilde{u}(x)-\bar{\tilde{u}}_{\Omega}|$ $\displaystyle\leq$
$\displaystyle\int_{X}|\nabla{\mathcal{G}}|(x,y)|\nabla\tilde{u}(y)|_{\omega}\omega^{n}+\frac{1}{V_{\omega}(\Omega)}\int_{z\in\Omega}\omega^{n}\int_{y\in
X}|\nabla{\mathcal{G}}|(z,y)|\nabla\tilde{u}(y)|_{\omega}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{X}\frac{|\nabla{\mathcal{G}}(x,y)|^{2}}{{\mathcal{G}}(x,y)^{1+\beta}}\omega^{n}\Big{)}^{1/2}\Big{(}\int_{X}{\mathcal{G}}(x,y)^{1+\beta}|\nabla\tilde{u}|^{2}\omega^{n}\Big{)}^{1/2}$
$\displaystyle~{}+\frac{1}{V_{\omega}(\Omega)}\int_{z\in\Omega}\omega^{n}\Big{(}\int_{X}\frac{|\nabla{\mathcal{G}}(z,y)|^{2}}{{\mathcal{G}}(z,y)^{1+\beta}}\omega^{n}\Big{)}^{1/2}\Big{(}\int_{X}{\mathcal{G}}(z,y)^{1+\beta}|\nabla\tilde{u}|^{2}\omega^{n}\Big{)}^{1/2}$
(6.6) by (4.2) $\displaystyle\leq$
$\displaystyle\frac{V_{\omega}^{\beta/2}}{\beta^{1/2}}\Big{[}\Big{(}\int_{X}{\mathcal{G}}(x,\cdot)^{1+\beta}|\nabla\tilde{u}|^{2}\Big{)}^{1/2}+\frac{1}{V_{\omega}(\Omega)}\int_{\Omega}\Big{(}\int_{X}{\mathcal{G}}(z,\cdot)^{1+\beta}|\nabla\tilde{u}|^{2}\omega^{n}\Big{)}^{1/2}\omega^{n}\Big{]}$
$\displaystyle\leq$
$\displaystyle\frac{V_{\omega}^{\beta/2}}{\beta^{1/2}}\Big{[}\Big{(}\int_{X}{\mathcal{G}}(x,\cdot)^{1+\beta}|\nabla\tilde{u}|^{2}\Big{)}^{1/2}+\Big{(}\frac{1}{V_{\omega}(\Omega)}\int_{z\in\Omega}\int_{X}{\mathcal{G}}(z,\cdot)^{1+\beta}|\nabla\tilde{u}|^{2}\Big{)}^{1/2}\Big{]},$

where the last inequality follows from Hölder’s inequality. To get the local
Poincaré inequality (6.2), we take square on both sides of (6.6) and integrate
the inequality over $x\in\Omega$, then we obtain

$\displaystyle\int_{\Omega}|\tilde{u}-\bar{\tilde{u}}_{\Omega}|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{2V_{\omega}^{\beta}}{\beta}\Big{[}\int_{x\in\Omega}\int_{X}{\mathcal{G}}(x,\cdot)^{1+\beta}|\nabla\tilde{u}|^{2}+\int_{z\in\Omega}\int_{X}{\mathcal{G}}(z,\cdot)^{1+\beta}|\nabla\tilde{u}|^{2}\Big{]},$
$\displaystyle\leq$
$\displaystyle\frac{C}{\beta}\int_{X}|\nabla\tilde{u}|_{\omega}^{2}\omega^{n},$

where we apply the inequality (4.1) in the last line by choosing
$\beta=\varepsilon_{0}$. The inequality (6.2) follows since $\tilde{u}=\eta
u=u$ in $\Omega$ and $\tilde{u}$ is supported in $\Omega^{\prime}$.

To prove the local Sobolev inequality (6.1), we choose
$\beta=\varepsilon_{0}/2$ and $q>1$ as in (4.4). Taking $(2q)$-th power of
both sides of (6.6) and arguing as that in the proof of Theorem 2.1, the
inequality (6.1) follows. ∎

We denote $W^{1,2}_{0}(\Omega)$ the $W^{1,2}$-functions with compact support
in $\Omega$. For these functions, we have the following local Sobolev and
Poincaré inequalities:

######  Lemma 6.2.

There is a uniform constant $C=C(n,p,A,K,\gamma)>0$ such that for any $u\in
W_{0}^{1,2}(\Omega)$, we have:

(1). The local Soblev inequality:

(6.7)
$\Big{(}\frac{1}{V_{\omega}}\int_{\Omega}|u|^{2q}\omega^{n}\Big{)}^{1/q}\leq
C\Big{(}1+\frac{V_{\omega}(\Omega)}{V_{\omega}(\Omega^{c})}\Big{)}\frac{1}{V_{\omega}}\int_{\Omega}|\nabla
u|^{2}_{\omega}\omega^{n},$

and (2). The local Poincaré inequality:

(6.8)
$\int_{\Omega}|u|^{2}\omega^{n}{}\leq{C}\Big{(}1+\frac{V_{\omega}(\Omega)}{V_{\omega}(\Omega^{c})}\Big{)}\int_{\Omega}|\nabla
u|^{2}_{\omega}\omega^{n},$

where $\Omega^{c}=X\backslash\Omega$ is the complement of $\Omega$ in $X$.

###### Proof.

The proof is similar to that of the above lemma. So we just outline the main
steps. We may assume $u\in C^{1}_{0}(\Omega)$ and $u$ can be viewed as a
$C^{1}$ function in $X$. By the Green’s formula, we have for any $x\in X$

(6.9) $u(x)=\frac{1}{V_{\omega}}\int_{X}u\omega^{n}+\int_{y\in
X}\langle\nabla{\mathcal{G}}(x,y),\nabla u(y)\rangle_{\omega}\omega^{n}.$

Applying (6.9) to $x\in\Omega^{c}=X\backslash\Omega$ at which $u=0$ and
integrating the equation over $\Omega^{c}$, we obtain

(6.10)
$\frac{1}{V_{\omega}}\int_{X}u\omega^{n}=-\frac{1}{V_{\omega}(\Omega^{c})}\int_{\Omega^{c}}\int_{y\in
X}\langle\nabla{\mathcal{G}}(x,y),\nabla u(y)\rangle_{\omega}\omega^{n}.$

The equations (6.10) and (6.9) together yield that for any $x\in\Omega$

(6.11) $u(x)=-\frac{1}{V_{\omega}(\Omega^{c})}\int_{z\in\Omega^{c}}\int_{y\in
X}\langle\nabla{\mathcal{G}}(z,y),\nabla
u(y)\rangle_{\omega}\omega^{n}+\int_{y\in
X}\langle\nabla{\mathcal{G}}(x,y),\nabla u(y)\rangle_{\omega}\omega^{n}.$

Taking square on both sides of (6.11) gives that for any $x\in\Omega$

(6.12) $|u(x)|^{2}\leq\frac{C}{V_{\omega}(\Omega^{c})}\int_{X}|\nabla
u|^{2}_{\omega}\omega^{n}+\frac{V_{\omega}^{\beta}}{\beta}\int_{X}{\mathcal{G}}(x,y)^{1+\beta}|\nabla
u|_{\omega}^{2}\omega^{n}.$

Integrating (6.12) over $\Omega$ yields that

$\int_{\Omega}|u|^{2}\omega^{n}\leq
C\frac{V_{\omega}(\Omega)}{V_{\omega}(\Omega^{c})}\int_{X}|\nabla
u|^{2}_{\omega}\omega^{n}+C\int_{X}|\nabla u|_{\omega}^{2}\omega^{n},$

which is the inequality (6.8). The local Sobolev inequality (6.7) can be
proved by taking $q$-th power of both sides of (6.12) and arguing similarly to
that in Lemma 6.1. ∎

We next consider a mean value inequality for sub-solutions of the Laplacian
$\Delta_{\omega}$ on geodesic balls in $X$. We fix a point $x\in X$ and denote
the geodesic ball $B_{\omega}(x,R)$ by $B_{R}$ for any $R>0$. Suppose $u\in
W^{1,2}(B_{R})$ satisfies

(6.13) $\Delta_{\omega}u\geq f,\quad\mbox{in }B_{R}.$

By the classical Moser’s iteration argument (e.g. [20]), we can derive the
following geometric mean value inequality.

######  Lemma 6.3.

Given a constant $N>\frac{q}{q-1}$, there exists a constant
$C=C(n,p,A,K,\gamma,N)>0$ such that for any $s>0$

(6.14)
$\sup_{B_{r}}u^{+}\leq\frac{C}{(R-r)^{\frac{1}{q-1}}}\big{(}\frac{1}{V_{\omega}}\int_{B_{R}}|f|^{N}\big{)}^{\frac{1}{N}}+\frac{C}{(R-r)^{\frac{2}{s(q-1)}}}\big{(}\frac{1}{V_{\omega}}\int_{B_{R}}(u^{+})^{s}\big{)}^{1/s},$

for any $0<r<R$.

###### Proof.

We denote
$k=(R-r)^{\beta}\|f\|_{L^{N}(B_{R},\omega^{n})}=(R-r)^{\beta}(\frac{1}{V_{\omega}}\int_{B_{R}}|f|^{N}\omega^{n})^{1/N}$
if $|f|\not\equiv 0$, and $k>0$ is an positive number if $f\equiv 0$, and here
$\beta=\frac{2(q-1)N-2q}{(q-1)N}>0$. Replacing $u$ by $u^{+}+k\geq k$, we may
assume $u\geq k$. Fix any constants $0<r<R_{2}<R_{1}<R$. Choose a cut-off
function $\eta$ that depends on $d_{\omega}(x,\cdot)$, the distance function
to the center $x\in B_{R}$, such that $\eta\equiv 1$ in $B_{R_{2}}$,
supp$(\eta)\subset B_{R_{1}}$, and

$\sup|\nabla\eta|_{\omega}\leq\frac{4}{R_{1}-R_{2}}.$

Multiplying both sides of (6.13) by $\eta^{2}u^{m}$ for some $1\leq m$, and
applying integration by parts, we obtain (we omit the integral domain $B_{R}$
and the measure $\omega^{n}$ in the following calculations.)

$\displaystyle\int m\eta^{2}u^{m-1}|\nabla u|^{2}_{\omega}$
$\displaystyle\leq$ $\displaystyle\int 2\eta u^{m}|\nabla\eta||\nabla u|+\int
f\eta^{2}u^{m}$ $\displaystyle\leq$ $\displaystyle\frac{1}{2}\int
m\eta^{2}u^{m-1}|\nabla u|^{2}+\frac{2}{m}\int u^{m+1}|\nabla\eta|^{2}+\int
f\eta^{2}u^{m},$

hence

$\frac{2m}{(m+1)^{2}}\int\eta^{2}|\nabla
u^{\frac{m+1}{2}}|^{2}\leq\frac{2}{m}\int u^{m+1}|\nabla\eta|^{2}+\int
f\eta^{2}u^{m},$

which is equivalent to

$\int|\nabla(\eta u^{\frac{m+1}{2}})|^{2}\leq 8\int
u^{m+1}|\nabla\eta|^{2}+2m\int f\eta^{2}u^{m}.$

By the Sobolev inequality (2.4), this implies

(6.15)
$\Big{(}\frac{1}{V_{\omega}}\int\eta^{2q}u^{(m+1)q}\Big{)}^{1/q}\leq\frac{C}{V_{\omega}}\Big{(}\int(\eta^{2}u^{m+1}+u^{m+1}|\nabla\eta|^{2})+m\int
f\eta^{2}u^{m}\Big{)}.$

We observe that the last term in (6.15) satisfies

(6.16) $\displaystyle\frac{m}{V_{\omega}}\int f\eta^{2}u^{m}$
$\displaystyle\leq$
$\displaystyle\frac{m}{V_{\omega}}\int\frac{|f|}{k}\eta^{2}u^{m+1}\leq\frac{m}{k}\|f\|_{L^{N}}\|\eta^{2}u^{m+1}\|_{L^{N^{*}}}$
$\displaystyle\leq$
$\displaystyle{m}(R-r)^{-\beta}\|\eta^{2}u^{m+1}\|_{L^{N^{*}}}.$

Applying Hölder’s inequality to the right-hand side of (6.16), we obtain

(6.17)
$\displaystyle{m}(R-r)^{-\beta}\|\eta^{2}u^{m+1}\|_{L^{N^{*}}}\leq{m}(R-r)^{-\beta}\|\eta^{2}u^{m+1}\|_{L^{q}}^{\frac{q}{(q-1)N}}\cdot\|\eta^{2}u^{m+1}\|_{L^{1}}^{\frac{(q-1)N-q}{N(q-1)}}$
$\displaystyle\leq$ $\displaystyle
m(R-r)^{-\beta}\epsilon\|\eta^{2}u^{m+1}\|_{L^{q}}+Cm(R-r)^{-\beta}\epsilon^{-\frac{q}{(q-1)N-q}}\|\eta^{2}u^{m+1}\|_{L^{1}},$

where in (6.17) we make use of Young’s inequality. We choose
$\epsilon=\frac{1}{2Cm(R-r)^{-\beta}}>0$ for $C>0$ the constant on the right-
hand side of (6.15). Then combining (6.17) and (6.15), we get

(6.18)
$\Big{(}\frac{1}{V_{\omega}}\int\eta^{2q}u^{(m+1)q}\Big{)}^{1/q}\leq\frac{C}{V_{\omega}}\Big{(}\int(\frac{m^{\alpha}\eta^{2}u^{m+1}}{(R-r)^{2}}+u^{m+1}|\nabla\eta|^{2})\Big{)}$

where we denote $\alpha=\frac{(q-1)N}{(q-1)N-q}>0$. By the choice of the cut-
off function $\eta$ and the inequality
$\frac{1}{R-r}\leq\frac{1}{R_{1}-R_{2}}$, this yields

(6.19)
$\Big{(}\frac{1}{V_{\omega}}\int_{B_{R_{2}}}u^{(m+1)q}\omega^{n}\Big{)}^{1/q(m+1)}\leq
C^{\frac{1}{m+1}}\frac{(m+1)^{\alpha/(m+1)}}{(R_{2}-R_{1})^{2/(m+1)}}\Big{(}\frac{1}{V_{\omega}}\int_{B_{R_{1}}}u^{m+1}\omega^{n}\Big{)}^{\frac{1}{m+1}}.$

We will iteratively apply (6.19) to a sequence of numbers $m_{j}=2q^{j}$ and
radii

$R_{j}=r+\frac{R-r}{2^{j}},\quad\mbox{for }j=0,1,2,\ldots$

It is clear that $R_{0}=R$, $R_{\infty}=r$, $R_{j}-R_{j+1}={2^{-j-1}}(R-r)$.
Iterating the ineuquality (6.19) with $R_{1}=R_{j}$ and $R_{2}=R_{j+1}$ for
$j=0,\ldots,\ell$ gives

(6.20)
$\Big{(}\frac{1}{V_{\omega}}\int_{B_{R_{\ell+1}}}u^{m_{\ell+1}}\omega^{n}\Big{)}^{\frac{1}{m_{\ell+1}}}\leq\frac{(2^{\alpha}C)^{\sum_{j}\frac{1}{2q^{j}}}q^{\alpha\sum_{j}jq^{-j}}2^{\sum_{j}q^{-j}}}{(R-r)^{\sum_{j=0}^{\ell}q^{-j}}}\Big{(}\frac{1}{V_{\omega}}\int_{B_{R}}u^{2}\omega^{n}\Big{)}^{\frac{1}{2}}.$

Letting $\ell\to\infty$, we get from (6.20) that

(6.21)
$\sup_{B_{r}}u\leq\frac{C}{(R-r)^{\frac{q}{q-1}}}\Big{(}\frac{1}{V_{\omega}}\int_{B_{R}}u^{2}\omega^{n}\Big{)}^{\frac{1}{2}},$

which implies by our choice of $u:=u^{+}+\|f\|_{L^{N}}$ that

(6.22)
$\sup_{B_{r}}u^{+}\leq\frac{C}{(R-r)^{\frac{q}{q-1}}}\Big{(}\|u^{+}\|_{L^{2}(B_{R})}+\frac{V_{\omega}(B_{R})^{1/2}}{V_{\omega}^{1/2}}(R-r)^{\beta}\|f\|_{L^{N}(B_{R})}\Big{)}.$

This proves (6.14) with $s=2$, and hence for all $s\geq 2$ by Hölder’s
inequality. Next let $s\in(0,2)$ be a given number. Note that (6.22) tells
that for any $0<r<R^{\prime}\leq R$

(6.23) $\displaystyle\|u^{+}\|_{L^{\infty}(B_{r})}$ $\displaystyle\leq$
$\displaystyle\frac{C}{(R^{\prime}-r)^{\frac{1}{q-1}}}\Big{(}\|u^{+}\|_{L^{2}(B_{R^{\prime}})}+\|f\|_{L^{N}(B_{R^{\prime}})}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{C}{(R^{\prime}-r)^{\frac{1}{q-1}}}\Big{(}\|u^{+}\|_{L^{\infty}(B_{R^{\prime}})}^{1-\frac{s}{2}}\|u^{+}\|_{L^{s}(B_{R^{\prime}})}^{s/2}+\|f\|_{L^{N}(B_{R^{\prime}})}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|u^{+}\|_{L^{\infty}(B_{R^{\prime}})}+\frac{C}{(R^{\prime}-r)^{\frac{2}{s(q-1)}}}\|u^{+}\|_{L^{s}(B_{R^{\prime}})}+\frac{C}{(R^{\prime}-r)^{\frac{1}{q-1}}}\|f\|_{L^{N}(B_{R^{\prime}})}.$

If we denote $\phi(r):=\|u^{+}\|_{L^{\infty}(B_{r})}$ which is monotonically
increasing and by (6.23) it satisfies the inequality

$\phi(r)\leq\frac{1}{2}\phi(R^{\prime})+\frac{C}{(R^{\prime}-r)^{\frac{2}{s(q-1)}}}A+\frac{C}{(R^{\prime}-r)^{\frac{1}{q-1}}}B$

with $A:=\|u^{+}\|_{L^{s}(B_{R})}$ and $B:=\|f\|_{L^{N}(B_{R})}$. We can now
apply the standard iteration lemma (see e.g. Lemma 4.3, Chap. 4, in [20]) to
conclude that

$\phi(r)\leq\frac{C}{(R-r)^{\frac{2}{s(q-1)}}}\|u^{+}\|_{L^{s}(B_{R})}+\frac{C}{(R-r)^{\frac{1}{q-1}}}\|f\|_{L^{N}(B_{R})},$

which is (6.14). ∎

As a corollary of Lemma 6.3, we have the following pointwise upper bound of
the Green’s function $G_{\omega}(x,y)$, which we do not expect to be sharp.

######  Corollary 6.1.

The Green’s function $G_{\omega}(x,y)$ satisfies the upper bound: for any
$x\neq y$

$G_{\omega}(x,y)\leq\frac{C}{V_{\omega}}d_{\omega}(x,y)^{-\frac{2}{q-1}}.$

###### Proof.

For any two different points $x,y\in X$, we let $0<R=d_{\omega}(x,y)/2$, and
consider the function $u(x)=G_{\omega}(x,y)$ as a smooth function on
$B_{R}:=B_{\omega}(y,R)$, which satisfies the differential equation

$\Delta_{\omega}u=\frac{1}{V_{\omega}},\quad\mbox{in }B_{R}.$

We apply Lemma 6.3 with $r=R/2$ and $s=1$ to conclude that

$\sup_{B_{R/2}}u^{+}\leq
C\Big{(}V_{\omega}^{-1}R^{-\frac{1}{q-1}}+V_{\omega}^{-1}R^{-\frac{2}{q-1}}\Big{)}\leq
CV_{\omega}^{-1}R^{-\frac{2}{q-1}}.$

This completes the proof. ∎

##  7\. Smooth approximation for $\mathcal{AK}$-currents

###  7.1. Kähler varieties

We first recall the definition of a smooth Kähler metric on a Kähler space.

######  Definition 7.1.

Let $X$ be a complex space of dimension $n$. We say $\theta$ is a smooth
Kähler metric on $X$, if for any point $x\in X$ there is an open subset $U$ of
$X$ together with a local holomorphic embedding
$\iota_{x}:U\hookrightarrow{\mathbb{C}}^{N_{x}}$ such that $\theta|_{U}$ is
the pullback of some smooth Kähler metric on ${\mathbb{C}}^{N_{x}}$ under
$\iota_{x}$.

If $\theta$ is a smooth Kähler metric in the sense of Definition 7.1, then
$\theta|_{X^{reg}}$ is a genuine Kähler metric on the smooth manifold
$X^{\circ}$, the regular part of $X$. $X$ is said to be a Kähler space or
variety if there exists a smooth Kähler metric on $X$. If $X$ is a compact
normal Kähler space, then there exists a resolution of singularities $\pi:Y\to
X$ such that the smooth model $Y$ of $X$ is also Kähler by [21, 7]. It is
straightforward to check that $[\pi^{*}\theta]$ is big and nef on $Y$ since
one can construct a Kähler current on $Y$ by [10, Theorem 2.12]. Then there
exists an effective divisor $E$ on $Y$ such that $[\pi^{*}\theta]-\delta[E]$
is a Kähler class for any sufficiently small $\delta>0$ by Demailly’s
regularization techniques (cf. [10, 7]).

Starting from this section, we will assume $X$ is an $n$-dimensional compact
normal Kähler variety. Let $\pi:Y\rightarrow X$ be a log resolution of $X$
with a fixed smooth Kähler metric $\theta_{Y}$ on $Y$.

######  Definition 7.2.

A nonnegative function $f$ on $Y$ is said to have log type analytic
singularities if the following hold.

  1. (1)

There exist holomorphic effective smooth divisors $D_{j}$ on $Y$ with simple
normal crossings, for $j=1,....,N$. Let $\sigma_{j}$ be the defining section
of $D_{j}$ and $h_{j}$ be a smooth hermitian metric associated to $D_{j}$.

  2. (2)

$f=\sum_{k=1}^{K}a_{k}(-\log)^{k}\Big{(}\prod_{j=1}^{N}e^{f_{k,j}}|\sigma_{j}|^{2b_{k,j}}_{h_{j}}\Big{)},$

where $(-\log)^{k}$ is the $k^{th}$ composition of $(-\log)$,
$a_{k}\in\mathbb{R}$, $b_{k,j}\geq 0$, $f_{k,j}\in C^{\infty}(Y)$ and
$K\in\mathbb{Z}^{+}$.

One can easily expand Definition 7.2 to a class of more general functions.
However, Definition 7.2 is sufficient to study most canonical metrics such as
singular Kähler-Einstein metrics.

For any semi-Kähler current
$\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$, we choose a smooth
closed $(1,1)$-form $\omega_{0}\in[\omega]$. Then there exists a unique
$\varphi\in PSH(X,\omega_{0})\cap C^{0}(X)$ such that

$\omega=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi,~{}\sup_{X}\varphi=0.$

Let

$F=\log\Big{(}\frac{1}{V_{\omega}}\frac{(\pi^{*}\omega)^{n}}{(\theta_{Y})^{n}}\Big{)}.$

Then $F$ has log type analytic singularities and

$\mathcal{N}_{p}(\omega)=\frac{1}{V_{\omega}}\int_{Y}|F|^{p}\omega^{n}=\frac{1}{V_{\omega}}\int_{Y}|F|^{p}e^{F}(\theta_{Y})^{n}\leq
K.$

As in the definition, let

(7.1) $\pi:Y\rightarrow X$

be a log resolution of $X$ and $\theta_{Y}$ be a Kähler metric on $Y$. We can
find a sequence of $\\{F_{j}\\}_{j=1}^{\infty}$ that approximate $F$
satisfying the following.

  1. (1)

$F_{j}\in C^{\infty}(X)$ and $e^{F_{j}}\geq 0.5\pi^{*}\gamma$, for all
sufficiently large $j$.

  2. (2)

There exists $K^{\prime}>0$ such that for all $j\geq 1$

(7.2) $\int_{Y}|F_{j}|^{p}e^{F_{j}}(\theta_{Y})^{n}\leq K^{\prime}.$

In the case when
$\|e^{F}\|_{L^{1+\epsilon^{\prime}}(Y,(\theta_{Y})^{n})}<\infty$ for some
$\epsilon^{\prime}>0$, there exists $K^{\prime\prime}>0$ such that

(7.3) $\|e^{F_{j}}\|_{L^{1+\epsilon^{\prime}}(Y,(\theta_{Y})^{n})}\leq
K^{\prime\prime}.$

  3. (3)

Let $D$ be an effective divisor of $Y$ such that the support of $D$ contains
the exceptional locus of $\pi$ and the singular locus of $F$. We can assume
that

$\pi^{*}[\omega]-\delta[D]$

is a Kähler class for some sufficiently small $\delta>0$. Let $\sigma_{D}$ be
the defining section of $D$ and $h_{D}$ be a smooth hermitian metric
associated to $D$. Then there exists $N>0$ and $C>0$ such that

$\sup_{Y}|\sqrt{-1}\partial\overline{\partial}F_{j}|_{\theta_{Y}}\leq
C|\sigma_{D}|^{-2N}_{h_{{\color[rgb]{0,0,0}D}}}.$

The simplest choice for $F_{j}$ will be

$F_{j}=\sum_{k=1}^{K}a_{k}(-\log)^{k}\Big{(}\prod_{{\color[rgb]{0,0,0}i}=1}^{N}e^{f_{k,{\color[rgb]{0,0,0}i}}}\left(|\sigma_{{\color[rgb]{0,0,0}i}}|^{2}_{h_{{\color[rgb]{0,0,0}i}}}+j^{-1}\right)^{b_{k,{\color[rgb]{0,0,0}i}}}\Big{)},$

We now consider the following complex Monge-Ampère equation

(7.4)
$(\pi^{*}\omega_{0}+\epsilon_{j}\theta_{Y}+\sqrt{-1}\partial\overline{\partial}\varphi_{j})^{n}=e^{F_{j}+c_{j}}(\theta_{Y})^{n},,~{}~{}\sup_{X}\varphi_{j}=0,$

where $c_{j}$ is the normalizing constant satisfying

$\int_{Y}e^{F_{j}+c_{j}}(\theta_{Y})^{n}=\left(\pi^{*}[\omega_{0}]+\epsilon_{{\color[rgb]{0,0,0}j}}[\theta_{Y}]\right)^{n}.$

We define

(7.5)
$\omega_{j}=\pi^{*}\omega_{0}+\epsilon_{j}\theta_{Y}+\sqrt{-1}\partial\overline{\partial}\varphi_{j}.$

######  Lemma 7.1.

There exist $N^{\prime}>0$, $C>0$ such that

$\|\varphi_{j}\|_{L^{\infty}(Y)}\leq C,~{}\Delta_{\theta_{Y}}\varphi_{j}\leq
C|\sigma_{D}|_{h_{D}}^{-2N^{\prime}}.$

###### Proof.

The $L^{\infty}$-estimate directly follows from [22, 12, 19] by the uniform
bound (7.2). For the second order estimate for $\varphi_{j}$, we let
$\omega_{j}=\pi^{*}\omega_{0}+\epsilon_{j}\theta_{Y}+\sqrt{-1}\partial\overline{\partial}\theta_{j}$
and let $h_{D}$ be a smooth hermtian metric associated to $D$ such that

$\pi^{*}\omega_{0}-\delta{\mathrm{Ric}}(h_{D})>0.$

We recall that $\omega_{j}$ satisfies the complex Monge-Ampère equation (7.4).
It follows from standard calculations [40] that

(7.6) $\Delta_{\omega_{j}}{\rm
tr}_{\theta_{Y}}\omega_{j}\geq\Delta_{\theta_{Y}}F_{j}-C_{1}({\rm
tr}_{\omega_{j}}\theta_{Y})\,({\rm
tr}_{\theta_{Y}}\omega_{j})+\frac{|\nabla{\rm
tr}_{\theta_{Y}}\omega_{j}|^{2}_{\omega_{j}}}{{\rm
tr}_{\theta_{Y}}\omega_{j}},$

where $-C_{1}$ is a lower bound of the bisectional curvature of
$(Y,\theta_{Y})$. Let
$\hat{\varphi}_{j}=\varphi_{j}-\delta\log|\sigma_{D}|^{2}_{h_{D}}$. For some
$\mu>0$ to be determined, we define $Q=e^{-\mu\hat{\varphi}_{j}}{\rm
tr}_{\theta_{Y}}\omega_{j}$. We will choose $\mu>0$ sufficiently large so that
$Q$ is $C^{2}$ in $Y$. Using (7.6), we calculate the Laplacian of $Q$ as
follows:

(7.7) $\displaystyle\Delta_{\omega_{j}}Q$ $\displaystyle\geq$ $\displaystyle
e^{-\mu\hat{\varphi}_{j}}\big{(}\Delta_{\theta_{Y}}F_{j}-C_{1}({\rm
tr}_{\omega_{j}}\theta_{Y})\,({\rm
tr}_{\theta_{Y}}\omega_{j})+\frac{|\nabla{\rm
tr}_{\theta_{Y}}\omega_{j}|^{2}_{\omega_{j}}}{{\rm
tr}_{\theta_{Y}}\omega_{j}}\big{)}+\mu^{2}e^{-\mu\hat{\varphi}_{j}}{\rm
tr}_{\theta_{Y}}\omega_{j}|\nabla\hat{\varphi}_{j}|^{2}_{\omega_{j}}$
$\displaystyle+\mu Q\big{(}-n+\epsilon_{j}{\rm tr}_{\omega_{j}}\theta_{Y}+{\rm
tr}_{\omega_{j}}(\omega_{0}-\delta\textnormal{Ric}(h_{D}))\big{)}-2\mu
e^{-\mu\hat{\varphi}_{j}}{\mathrm{Re}}\langle\nabla\hat{\varphi}_{j},\bar{\nabla}{\rm
tr}_{\theta_{Y}}\omega_{j}\rangle_{\omega_{j}}.$

By Cauchy-Schwarz inequality, the last term in (7.7) satisfies

$-2\mu
e^{-\mu\hat{\varphi}_{j}}{\mathrm{Re}}\langle\nabla\hat{\varphi}_{j},\bar{\nabla}{\rm
tr}_{\theta_{Y}}\omega_{j}\rangle_{\omega_{j}}\geq-e^{-\mu\hat{\varphi}_{j}}\frac{|\nabla{\rm
tr}_{\theta_{Y}}\omega_{j}|^{2}_{\omega_{j}}}{{\rm
tr}_{\theta_{Y}}\omega_{j}}-\mu^{2}e^{-\mu\hat{\varphi}_{j}}{\rm
tr}_{\theta_{Y}}\omega_{j}|\nabla\hat{\varphi}_{j}|^{2}_{\omega_{j}}.$

We fix a constant $\mu_{1}>0$ such that

$\mu_{1}(\omega_{0}-\delta\textnormal{Ric}(h_{D}))\geq 2\theta_{Y}.$

The constant $\mu$ is chosen so that $\mu\geq\mu_{1}$. Then (7.7) yields that

(7.8) $\Delta_{\omega_{j}}Q\geq
e^{-\mu\hat{\varphi}_{j}}\Delta_{\theta_{Y}}F_{j}-n\mu Q.$

Note that $|\sqrt{-1}\partial\overline{\partial}F_{j}|_{\theta_{Y}}\leq
C|\sigma_{D}|_{h_{D}}^{-2N}$. If additionally we take $\mu>0$ such that
$\mu\delta\geq N$, from (7.8) we obtain

(7.9) $\Delta_{\omega_{j}}Q\geq-C|\sigma_{D}|_{h_{D}}^{2(\mu\delta-N)}-n\mu
Q\geq-n\mu Q-C,$

where we have assumed $|\sigma_{D}|_{h_{D}}^{2}\leq 1$. Recall that the
Green’s function $G_{j}$ associated to $\omega_{j}$ satisfies the uniform
integral estimates as in [18, 17]. Applying Green’s formula at a point $x\in
Y$ where $Q$ achieves its maximum $\bar{Q}$, we have

(7.10) $\displaystyle\bar{Q}$ $\displaystyle\leq$
$\displaystyle\frac{1}{V_{\omega_{j}}}\int_{Y}Q\omega_{j}^{n}+\int_{Y}{\mathcal{G}}_{j}(x,\cdot)(n\mu
Q+C)\omega_{j}^{n}$ $\displaystyle\leq$
$\displaystyle\frac{\bar{Q}^{1-\epsilon}}{V_{\omega_{j}}}\int_{Y}Q^{\epsilon}\omega_{j}^{n}+C+\bar{Q}^{1-\frac{\epsilon}{2n}}\big{(}\int_{Y}Q^{\epsilon}\omega_{j}^{n}\big{)}^{1/2n},$

where the inequality (7.10) follows from the Hölder’s inequality and
$\epsilon=\frac{\epsilon^{\prime}}{1+\epsilon^{\prime}}>0$ is a constant
chosen so that the following holds

$\int_{Y}Q^{\epsilon}\omega_{j}^{n}\leq C\int_{Y}({\rm
tr}_{\theta_{Y}}\omega_{j})^{\epsilon}e^{F_{j}}\theta_{Y}^{n}\leq
C\big{(}\int_{Y}({\rm
tr}_{\theta_{Y}}\omega_{j})\theta_{Y}^{n}\big{)}^{\frac{\epsilon^{\prime}}{1+\epsilon^{\prime}}}\leq
C$

where the constant $C>0$ depends on $\|e^{F_{j}}\|_{L^{1+\epsilon^{\prime}}}$
which is uniformly bounded. Then (7.10) gives that

$\bar{Q}\leq C\bar{Q}^{1-\epsilon}+C\bar{Q}^{1-\frac{\epsilon}{2n}}+C,$

which implies the upper bound of $\bar{Q}$, hence that of $Q$. ∎

Immediately, we can obtain local higher order estimate for $\varphi_{j}$
uniformly away from $Y\setminus D$. Let $\mathcal{S}_{X,\omega}$ be the union
of the singular set of $X$ and the singular set of $\pi_{*}F$. Since
$\pi^{-1}(\mathcal{S}_{X,\omega})$ is contained in the ample locus of
$[\pi^{*}\omega]$, for every point $p\in X\setminus\mathcal{S}_{X,\omega}$, we
can choose $D$ such that $\pi^{-1}(p)\in Y\setminus D$. The standard Schauder
estimates give the following regularity result for
$\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$.

######  Proposition 7.1.

The smooth Kähler metrics $\omega_{j}$ on $Y$ satisfy the following
conditions. Then for $\mathcal{K}\subset\subset
X\setminus\mathcal{S}_{X,\omega}$ and $k>0$, we have

$\lim_{j\rightarrow\infty}\|\varphi_{j}-\pi^{*}\varphi\|_{L^{\infty}(\mathcal{K})}=0,$
$\lim_{j\rightarrow\infty}\|\omega_{j}-\omega\|_{C^{k}(\mathcal{K})}=0.$

Here we identify $\omega_{j}$ and $\pi_{*}\omega_{j}$ on
$X\setminus\mathcal{S}_{X,\omega}$. In particular, $\omega$ is a smooth Kähler
metric on $X\setminus\mathcal{S}_{X,\omega}$.

##  8\. Sobolev inequalities for $\mathcal{AK}$-currents

Let $\omega\in\mathcal{AK}(X,\theta_{Y},n,p,A,K,\gamma)$ as in Definition 3.1
and $\mathcal{S}_{X,\omega}$ be the union of the singular set of $X$ and the
image of the singular set of $\log\frac{\pi^{*}\omega^{n}}{(\theta_{Y})^{n}}$
under $\pi$. We define the distance function $d$ on
$(X\backslash\mathcal{S}_{X,\omega},\omega)$ as follows: for any $x,y\in
X\backslash\mathcal{S}_{X,\omega}$

(8.1) $d(x,y)=\inf\\{L_{\omega}(c):~{}c:[0,1]\to
X\backslash\mathcal{S}_{X,\omega},\,c(0)=x,\,c(1)=y\\},$

where $c$ is a piecewise differentiable curve in
$X\backslash\mathcal{S}_{X,\omega}$, and
$L_{\omega}(c)=\int_{0}^{1}|\dot{c}(t)|_{\omega}dt$ is the length of $c$ under
the smooth metric $\omega$. It is not hard to see $d$ is a distance function
on $X\backslash\mathcal{S}_{X,\omega}$. Let $(\hat{X},\hat{d})$ be the metric
completion of $(X\backslash\mathcal{S}_{X,\omega},d)$. For notational
simplicity we will write $\hat{d}$ as $d$.

We will first set up the Sobolev spaces on the metric measure space
$(\hat{X},d,\omega^{n})$ associated to $(X,\omega)$. It is straightforward to
define $L^{q}(\hat{X},\omega^{n})$ by identifying it with
$L^{q}(X,\omega^{n})$ because the singular set $\mathcal{S}_{X,\omega}$ is a
proper analytic subvariety with real codimension at least $2$.

######  Definition 8.1.

The Sobolev space $W^{1,2}(\hat{X},d,\omega^{n})$ is defined by for all
$u:X\rightarrow\mathbb{R}$ such that $u|_{X^{\circ}}\in
W^{1,2}_{loc}(X^{\circ},\omega)$ and

$\sup_{{\mathcal{K}}\subset\subset
X\setminus\mathcal{S}_{X,\omega}}\|u\|_{W^{1,2}({\mathcal{K}},\omega)}{=\sup_{{\mathcal{K}}\subset\subset
X\setminus\mathcal{S}_{X,\omega}}\Big{(}\int_{{\mathcal{K}}}(u^{2}+|\nabla
u|^{2}_{\omega})\omega^{n}\Big{)}^{1/2}}<\infty.$

We then define the $W^{1,2}$-norm of $u$ by

$\|u\|_{W^{1,2}(\hat{X},d,\omega^{n})}=\sup_{{\mathcal{K}}\subset\subset
X\setminus\mathcal{S}_{X,\omega}}\|u\|_{W^{1,2}({\mathcal{K}},\omega)}.$

We can also identify $W^{1,2}(X,\omega)$ with $W^{1,2}(\hat{X},d,\omega^{n})$
as the integrals are calculated on $X\setminus\mathcal{S}_{X,\omega}$. We will
first prove a Sobolev inequality for $u\in
W^{1,2}(X,\omega)=W^{1,2}(\hat{X},d,\omega^{n})$ with compact support in
$X\backslash\mathcal{S}_{X,\omega}$.

######  Lemma 8.1.

There exists $C=C(X,\theta_{Y},n,p,A,K,\gamma)>0$ such that

$\Big{(}\int_{X}|u|^{2q}\omega^{n}\Big{)}^{\frac{1}{q}}\leq
C\Big{(}\int_{X}|\nabla
u|^{2}_{\omega}\omega^{n}+\int_{X}u^{2}\omega^{n}\Big{)}$

for all $u\in W^{1,2}(\hat{X},d,\omega^{n})$ with compact support in
$X\setminus\mathcal{S}_{X,\omega}$, where $q>1$ is the Sobolev exponent in
Theorem 2.1.

###### Proof.

We will approximate $\omega$ by $\omega_{j}$ as in Proposition 7.1 on $Y$, the
blow-up of $X$. By construction, there exist $A^{\prime},K^{\prime}>0$ and a
nonnegative $\gamma^{\prime}=\pi^{*}\gamma\in C^{0}(Y)$ such that for all
$j>0$,

$\omega_{j}\in\mathcal{W}(n,p,\gamma^{\prime},A^{\prime},K^{\prime}).$

By Theorem 2.1, there exists $C>0$ such that for all $f\in
W^{1,2}(Y,\omega_{j})$, we have

$\Big{(}\int_{Y}|f|^{2q}(\omega_{j})^{n}\Big{)}^{\frac{1}{q}}\leq
C\Big{(}\int_{Y}|\nabla
f|_{\omega_{j}}^{2}(\omega_{j})^{n}+\int_{Y}f^{2}(\omega_{j})^{n}\Big{)}.$

For any $u\in W^{1,2}(X,\omega)$ with compact support in
$X\setminus\mathcal{S}_{X,\omega}$, since $\omega_{j}$ converges to $\omega$
locally smoothly on $X\setminus\mathcal{S}_{X,\omega}$, we have $\pi^{*}u\in
W^{1,2}(Y,\omega_{j})$ and

$\Big{(}\int_{Y}|\pi^{*}u|^{2q}(\omega_{j})^{n}\Big{)}^{\frac{1}{q}}\leq
C\Big{(}\int_{Y}|\nabla\pi^{*}u|_{\omega_{j}}^{2}(\omega_{j})^{n}+\int_{Y}(\pi^{*}u)^{2}(\omega_{j})^{n}\Big{)}.$

The lemma is proved by letting $j\rightarrow\infty$. ∎

The following cut-off functions are constructed in [37] (c.f. Lemma 3.7 in
[32]).

######  Lemma 8.2.

Let $X$ be an $n$-dimensional Kähler variety and $E$ is a subvariety of $X$.
Let $\omega$ be a semi-Kähler current on $X$ with bounded local potentials.
Then for any $\epsilon>0$ and $\mathcal{K}\subset\subset X\setminus E$, there
exists a cut-off function $\eta\in C^{\infty}(X)$ satisfying the following.

  1. (1)

$0\leq\eta\leq 1$, $\eta|_{\mathcal{K}}=1$ and $\eta=0$ on an open
neighborhood of $E$.

  2. (2)

$\int_{X}|\nabla\eta|_{{\color[rgb]{0,0,0}\omega}}^{2}\omega^{n}={\color[rgb]{0,0,0}n}\int_{X}\sqrt{-1}\partial\eta\wedge\overline{\partial}\eta\wedge\omega^{n-{\color[rgb]{0,0,0}1}}<\epsilon.$

######  Lemma 8.3.

There exists $C=C(X,\theta_{Y},n,A,K,\gamma)>0$ such that

$\left(\int_{\hat{X}}|u|^{2q}\omega^{n}\right)^{\frac{1}{q}}\leq
C\left(\int_{\hat{X}}|\nabla
u|^{2}_{\omega}\omega^{n}+\int_{\hat{X}}u^{2}\omega^{n}\right)$

for all $u\in W^{1,2}(\hat{X},d,\omega^{n})\cap L^{\infty}(\hat{X})$. Here all
the integrals are calculated on $X\setminus\mathcal{S}_{X,\omega}$.

###### Proof.

For any $\epsilon>0$, we choose a cut-off function $\eta_{\epsilon}$ as in
Lemma 8.2 with support $\mathcal{K}_{\epsilon}\subset\subset
X\setminus\mathcal{S}_{X,\omega}$. We can assume that
$\mathcal{K}_{\epsilon}\rightarrow X\setminus\mathcal{S}_{X,\omega}$
increasingly as $\epsilon\rightarrow 0$. Then

(8.2) $\displaystyle\int_{X}|\nabla(\eta_{\epsilon}u)|^{2}_{\omega}\omega^{n}$
$\displaystyle=$
$\displaystyle\int_{X}|\nabla\eta_{\epsilon}|^{2}_{\omega}u^{2}\omega^{n}+\int_{X}(\eta_{\epsilon})^{2}|\nabla
u|^{2}_{\omega}\omega^{n}+2Re\int_{X}\eta_{\epsilon}u{\color[rgb]{0,0,0}\langle}\nabla\eta_{\epsilon},\overline{\nabla}u{\color[rgb]{0,0,0}\rangle_{\omega}}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\epsilon\|u\|_{L^{\infty}}^{2}+\int_{X}|\nabla
u|^{2}_{\omega}\omega^{n}+2\epsilon^{1/2}\|u\|_{L^{\infty}(X)}\Big{(}\int_{X}|\nabla
u|^{2}_{\omega}\omega^{n}\Big{)}^{1/2}$ $\displaystyle\rightarrow$
$\displaystyle\int_{X}|\nabla u|^{2}_{\omega}\omega^{n},$

as $\epsilon\rightarrow 0$ since $\|u\|_{L^{\infty}(X)}<\infty$. Applying
Lemma 8.1, we have

$\displaystyle\left(\int_{X}|\eta_{\epsilon}u|^{2q}\omega^{n}\right)^{\frac{1}{q}}\leq
C\left(\int_{X}|\nabla(\eta_{\epsilon}u)|^{2}_{\omega}\omega^{n}+\int_{X}(\eta_{\epsilon}u)^{2}\omega^{n}\right)$

for some uniform $C>0$. The lemma is proved by letting $\epsilon\rightarrow
0$. ∎

######  Theorem 8.1.

There exists $C=C(X,\theta_{Y},n,A,K,\gamma)>0$ such that

(8.3) $\left(\int_{\hat{X}}|u|^{2q}\omega^{n}\right)^{\frac{1}{q}}\leq
C\left(\int_{\hat{X}}|\nabla
u|^{2}_{\omega}\omega^{n}+\int_{\hat{X}}u^{2}\omega^{n}\right)$

for all $u\in W^{1,2}(\hat{X},d,\omega^{n})$, where $q>1$ is the Sobolev
exponent in Theorem 2.1.

###### Proof.

We first assume that $u\geq 0$. We define

$u_{A}=\min(u,A)=\frac{u+A-|u-A|}{2}$

for a constant $A\geq 0$. Obviously, by monotone convergence theorem we have

$\lim_{A\rightarrow\infty}\|u_{A}\|_{L^{2q}(X,\omega^{n})}=\|u\|_{L^{2q}(X,\omega^{n})},~{}\lim_{A\rightarrow\infty}\|u_{A}\|_{L^{2}(X,\omega^{n})}=\|u\|_{L^{2}(X,\omega^{n})}.$

Also for any $f\in W^{1,2}(X,\omega)$, we have

(8.4)
$\int_{\hat{X}}|\nabla|f||^{2}_{\omega}\omega^{n}\leq\int_{\hat{X}}|\nabla
f|^{2}_{\omega}\omega^{n}.$

Therefore

$\int_{\hat{X}}|\nabla u_{A}|^{2}\omega^{n}\leq\int_{\hat{X}}\left(|\nabla
u|^{2}_{\omega}+|\nabla|u-A\|^{2}_{\omega}\right)\omega^{n}\leq
2\int_{\hat{X}}|\nabla u|^{2}_{\omega}\omega^{n}$

by Kato’s inequality (8.4). Applying Lemma 8.3 to the bounded function
$u_{A}$, we have

$\Big{(}\int_{\hat{X}}|u_{A}|^{2q}\omega^{n}\Big{)}^{\frac{1}{q}}\leq
C\Big{(}\int_{\hat{X}}|\nabla
u_{A}|^{2}_{\omega}\omega^{n}+\int_{\hat{X}}u_{A}^{2}\omega^{n}\Big{)}\leq
2C\Big{(}\int_{\hat{X}}|\nabla
u|^{2}_{\omega}\omega^{n}+\int_{\hat{X}}u_{A}^{2}\omega^{n}\Big{)}.$

The Sobolev inequality is then proved by letting $A\rightarrow\infty$.  For
general $u\in W^{1,2}(\hat{X},\omega)$, we apply the Sobolev inequality just
proved to $u_{+}$ and $u_{-}$, the positive and negative parts of $u$,
respectively. Therefore, we have

$\Big{(}\int_{\hat{X}}|u|^{2q}\omega^{n}\Big{)}^{\frac{1}{q}}\leq
2C\Big{(}\int_{\hat{X}}|\nabla
u|^{2}_{\omega}\omega^{n}+\int_{\hat{X}}u^{2}\omega^{n}\Big{)}$

for all $u\in W^{1,2}(\hat{X},d,\omega^{n})$ and the theorem is proved. ∎

Remark. With the same proof as that in Theorem 8.1, using the Sobolev
inequality (2.4), we also have the following version of Sobolev inequality for
some constant $C>0$ that

(8.5)
$\Big{(}\frac{1}{V_{\omega}}\int_{\hat{X}}|u-\bar{u}|^{2q}\Big{)}^{\frac{1}{q}}\leq\frac{C}{V_{\omega}}\int_{\hat{X}}|\nabla
u|^{2}_{\omega}\omega^{n}$

for all $u\in W^{1,2}(\hat{X},d,\omega^{n})$, where
$\bar{u}=\frac{1}{V_{\omega}}\int_{\hat{X}}u\,\omega^{n}$ is the average of
$u$.

######  Lemma 8.4.

The following local Sobolev inequality holds on $(X,\omega)$. For any open
subset $\Omega\subset X$, there is a constant $C>0$ such that

(8.6)
$\Big{(}\frac{1}{V_{\omega}}\int_{\Omega}|u|^{2q}\omega^{n}\Big{)}^{1/q}\leq
C\Big{(}1+\frac{V_{\omega}(\Omega)}{V_{\omega}(\Omega^{c})}\Big{)}\frac{1}{V_{\omega}}\int_{\Omega}|\nabla
u|^{2}_{\omega}\omega^{n},$

for all $u\in W^{1,2}_{0}(\Omega)$.

###### Proof.

The argument is similar to that in Theorem 8.1 by making use of the inequality
(6.7), so we only outline the proof. Taking $u_{\epsilon}=u\eta_{\epsilon}$ if
necessary, we may assume $u\in
C^{1}_{0}(\Omega\backslash\mathcal{S}_{X,\omega})$, and identify it with
$\pi^{*}u\in C^{1}_{0}(\pi^{-1}(\Omega))$. We can then apply the local Sobolev
inequality (6.7) to $u$ over $\pi^{-1}(\Omega)\subset Y$ with the
approximating Kähler metrics $\omega_{j}$. Letting $j\to\infty$ gives (8.6). ∎

As a corollary of Theorem 8.1, we have the following Sobolev embedding
theorem.

######  Corollary 8.1.

For any $1\leq\hat{q}<q$, the continuous embedding
$W^{1,2}(X,\omega)\hookrightarrow L^{2\hat{q}}(X,\omega)$ is compact.

###### Proof.

Take a sequence of cut-off functions $\eta_{j}$ such that for each $j$,
$\eta_{j}\leq\eta_{j+1}$, ${\mathcal{K}_{j}}\subset{\mathcal{K}_{j+1}}$,
$\cup_{j}{\mathcal{K}_{j}}=X\backslash\mathcal{S}_{X,\omega}$, where
${\mathcal{K}}_{j}={\mathrm{supp}}(\eta_{j})\subset
X\backslash\mathcal{S}_{X,\omega}$ is the compact support of $\eta_{j}$.
Moreover,

$\int_{X}(1-\eta_{j})\omega^{n}\leq\frac{1}{j}\to 0.$

Such cut-off functions always exist because the measure $\omega^{n}$ does not
charge mass on $\mathcal{S}_{X,\omega}$. To prove Corollary 8.1, we need to
show that for any sequence of functions $\\{f_{a}\\}_{a=1}^{\infty}\subset
W^{1,2}(X,\omega)$ with $\sup_{a}\|f_{a}\|_{W^{1,2}(X,\omega)}<\infty$, there
exists a convergent subsequence in $L^{\hat{q}}(X,\omega)$. We observe first
that by the Sobolev inequality in Theorem 8.1,

(8.7) $M:=\sup_{a}\|f_{a}\|_{L^{2q}(X,\omega)}\leq
C\sup_{a}\|f_{a}\|_{W^{1,2}(X,\omega)}<\infty.$

We next consider the sequence of functions $\\{f_{a}\eta_{1}\\}_{a}$ which are
compactly supported in the smooth manifold
$({\mathcal{K}}_{2}^{\circ},\omega)$ and uniformly bounded222The
$W^{1,2}({\mathcal{K}}_{2}^{\circ},\omega)$-bound of $\\{f_{a}\eta_{1}\\}_{a}$
may depend on $|\nabla\eta_{1}|$, but it is independent of $a$. in
$W^{1,2}({\mathcal{K}}_{2}^{\circ},\omega)$. By the standard Sobolev
precompact embedding theorem on smooth manifolds, we see that there is a
subsequence $\\{f_{a_{1}}\eta_{1}\\}_{a_{1}\in{\mathcal{A}}_{1}}$ which
converges in $L^{2\hat{q}}(X,\omega)$. We then argue similarly on the sequence
of functions $\\{f_{a_{1}}\eta_{2}\\}_{a_{1}\in{\mathcal{A}}_{2}}$ which are
uniformly bounded in $W^{1,2}({\mathcal{K}}_{3}^{\circ},\omega)$, hence there
is a subsequence of functions
$\\{f_{a_{2}}\eta_{2}\\}_{a_{2}\in{\mathcal{A}}_{2}}$ which converges in
$L^{2\hat{q}}(X,\omega)$. Continuing this way, for any $k$ we can find a
sequence of functions $\\{f_{a_{k}}\eta_{k}\\}_{a_{k}\in{\mathcal{A}}_{k}}$
which converges in $L^{2\hat{q}}(X,\omega)$, and the index sets fulfill
${\mathcal{A}}_{1}\supset{\mathcal{A}}_{2}\supset\cdots\supset{\mathcal{A}}_{k}$.
By the classical diagonal argument, we can pick a sequence of functions
$\\{f_{a_{k}}\\}_{k=1}^{\infty}$ such that for any fixed $\ell$, the sequence
$\\{f_{a_{k}}\eta_{\ell}\\}_{k=1}^{\infty}$ is convergent in
$L^{2\hat{q}}(X,\omega)$. We claim that the sequence $\\{f_{a_{k}}\\}_{k}$ is
also convergent in $L^{2\hat{q}}(X,\omega)$. It suffices to show
$\\{f_{a_{k}}\\}$ is a Cauchy sequence in $L^{2\hat{q}}(X,\omega)$. To see
this, for any $\epsilon>0$ we take
$\ell=\big{[}(4M\epsilon^{-1})^{2q\hat{q}/(q-\hat{q})}\big{]}+1$. The sequence
$\\{f_{a_{k}}\eta_{\ell}\\}_{k=1}^{\infty}$ is Cauchy in
$L^{2\hat{q}}(X,\omega)$. Hence there is a $K=K(\epsilon,\ell)>0$ such that
for any $k^{\prime},k\geq K$,
$\|f_{a_{k}}\eta_{\ell}-f_{a_{k^{\prime}}}\eta_{\ell}\|_{\scriptscriptstyle
L^{2\hat{q}}}\leq\frac{\epsilon}{4}$. Then we have

$\displaystyle\|f_{a_{k}}-f_{a_{k^{\prime}}}\|_{\scriptscriptstyle
L^{2\hat{q}}}$ $\displaystyle\leq$
$\displaystyle\|f_{a_{k}}-f_{a_{k}}\eta_{\ell}\|_{\scriptscriptstyle
L^{2\hat{q}}}+\|f_{a_{k}}\eta_{\ell}-f_{a_{k^{\prime}}}\eta_{\ell}\|_{\scriptscriptstyle
L^{2\hat{q}}}+\|f_{a_{k^{\prime}}}-f_{a_{k^{\prime}}}\eta_{\ell}\|_{\scriptscriptstyle
L^{2\hat{q}}}$ $\displaystyle\leq$
$\displaystyle\|f_{a_{k}}\|_{\scriptscriptstyle
L^{2q}}\|1-\eta_{\ell}\|_{\scriptscriptstyle
L^{\frac{q}{q-\hat{q}}}}^{1/2\hat{q}}+\|f_{a_{k^{\prime}}}\|_{\scriptscriptstyle
L^{2q}}\|1-\eta_{\ell}\|_{\scriptscriptstyle
L^{\frac{q}{q-\hat{q}}}}^{1/2\hat{q}}+\frac{\epsilon}{4}$ $\displaystyle\leq$
$\displaystyle\frac{\epsilon}{4}+\frac{\epsilon}{4}+\frac{\epsilon}{4}<\epsilon.$

This proves the claim and hence the Corollary. ∎

######  Proposition 8.1.

The space $C^{\infty}_{0}(X\setminus\mathcal{S}_{X,\omega})$ is dense in
$W^{1,2}(\hat{X},d,\omega^{n})$. In particular, $C^{\infty}(X)\cap W^{1,2}$ is
dense in $W^{1,2}(\hat{X},d,\omega^{n})$ if we identify every $u\in
C^{\infty}(X)$ as a smooth function on $\hat{X}$ by trivial extension of
$u|_{X\setminus\mathcal{S}_{X,\omega}}$ .

###### Proof.

Let $u\in W^{1,2}(\hat{X},d,\omega^{n})$. For any $\epsilon>0$, we choose a
cut-off function $\eta_{\epsilon}$ as in Lemma 8.2 with support
$\mathcal{K}_{\epsilon}\subset\subset X\setminus\mathcal{S}_{X,\omega}$. We
can assume that $\mathcal{K}_{\epsilon}\rightarrow
X\setminus\mathcal{S}_{X,\omega}$ increasingly as $\epsilon\rightarrow 0$.

We will first assume $u\in L^{\infty}(\hat{X})$. Let
$u_{\epsilon}=\eta_{\epsilon}u$. Since $\omega$ is smooth on
$X\setminus\mathcal{S}_{X,\omega}$, there exists $v_{\epsilon}\in
C_{0}^{\infty}(X\setminus\mathcal{S}_{X,\omega})$ such that

$\|v_{\epsilon}-u_{\epsilon}\|_{W^{1,2}(\hat{X},d,\omega^{n})}<\epsilon.$

Obviously,

$\lim_{\epsilon\rightarrow 0}\|u_{\epsilon}-u\|_{L^{2}(X,\omega^{n})}=0.$

By the calculations in (8.2), we have

$\displaystyle\lim_{\epsilon\rightarrow
0}\int_{X}|\nabla(u_{\epsilon}-u)|^{2}_{\omega}\omega^{n}\leq\lim_{\epsilon\to
0}2\int_{X}(1-\eta_{\epsilon})^{2}|\nabla
u|^{2}_{\omega}\omega^{n}+2\|u\|_{L^{\infty}}\int_{X}|\nabla\eta_{\epsilon}|^{2}_{\omega}\omega^{n}=0$

Hence, we have

${\lim_{\epsilon\rightarrow
0}\int_{X}(|\nabla(u_{\epsilon}-u)|^{2}+(u-u_{\epsilon})^{2})\omega^{n}=0.}$

Immediately, we have $\lim_{\epsilon\rightarrow
0}\|v_{\epsilon}-u\|_{L^{2}(X,\omega^{n})}=0$ by the triangle inequality.

In general, we can always replace $u$ by
$\tilde{u}_{A}=(u_{+})_{A}-(u_{-})_{A}$, where $f_{A}:=\min(u,f)$ for a
function $f$ and constant $A\geq 0$. Then

(8.8) $\int_{X}\chi_{\scriptscriptstyle\\{|u|\geq
A\\}}\omega^{n}=\int_{X\cap\\{|u|\geq
A\\}}\omega^{n}\leq\frac{1}{A^{2}}\int_{X}u^{2}\omega^{n}\rightarrow
0,\quad\mbox{as }A\to\infty.$

Hence $\chi_{\scriptscriptstyle\\{|u|\geq A\\}}\to 0$ in the a.e. sense on
$(X,\omega^{n})$. It then follows from dominated convergence theorem that for
$u\in W^{1,2}(X,\omega)$

$\int_{X}|u-\tilde{u}_{A}|^{2}\omega^{n}=\int_{X}|u|^{2}\chi_{\scriptscriptstyle\\{|u|\geq
A\\}}\omega^{n}\to 0,$

and

$\int_{X}|\nabla(u-\tilde{u}_{A})|^{2}_{\omega}\omega^{n}\leq\int_{X}|\nabla
u|^{2}_{\omega}\chi_{\scriptscriptstyle\\{|u|\geq A\\}}\omega^{n}\to 0,$

as $A\to\infty$. We then can apply the same argument earlier for
$\tilde{u}_{A}\in L^{\infty}(\hat{X})$. This completes the proof of the
proposition. ∎

By applying similar arguments, we can also show that $W^{1,2}\cap
C^{\infty}(X\setminus\mathcal{S}_{X,\omega})\cap L^{\infty}(\hat{X})$ is also
dense in $W^{1,2}(\hat{X},d,\omega^{n})$. If we denote $W^{1,2}_{0}(X,\omega)$
the Hilbert space obtained by the completion of
$C^{\infty}_{0}(X\backslash\mathcal{S}_{X,\omega})$ under the norm
$\|\cdot\|_{W^{1,2}(X,\omega)}$, then Proposition 8.1 implies that

$W^{1,2}(X,\omega)=W^{1,2}_{0}(X,\omega).$

##  9\. Noncollapsing and diameter bounds for $\mathcal{AK}$-currents

Let $(X,\theta_{Y})$ be an $n$-dimensional projective normal variety equipped
with a smooth Kähler metric $\theta_{Y}$. Let $\mathcal{S}_{X,\omega}$ be the
union of the singular set of $X$ and
$\log\frac{(\pi^{*}\omega)^{n}}{(\theta_{Y})^{n}}$. We let $(\hat{X},d)$ be
the metric completion of $(X\setminus\mathcal{S}_{X,\omega},\omega)$ since
$\omega$ is a smooth Kähler metric on $X\setminus\mathcal{S}_{X,\omega}$. The
natural question is whether $(\hat{X},d)$ is a compact metric space.

We first verify that the distance function on $\hat{X}$ with a fixed base
point is Lipschitz.

######  Lemma 9.1.

For any fixed point $p\in\hat{X}$, the function

$f(x)=d(p,x)$

is a $1$-Lipschitz function on $(\hat{X},d)$.

###### Proof.

We can assume $p\in X\setminus\mathcal{S}_{X,\omega}$ since any point of
$\hat{X}$ can always be approximated by a sequence of points in
$X\setminus\mathcal{S}_{X,\omega}$. For any $x,y\in X\setminus\mathcal{S}$, we
have by the triangle inequality

$-d(x,y)\leq f(x)-f(y)\leq d(x,y).$

Then lemma immediately follows. ∎

The volume measure $\omega^{n}=e^{F}(\theta_{Y})^{n}$ can be uniquely extended
to a volume measure to $\hat{X}$ by natural extension. Hence we will consider
the metric measure space

$(\hat{X},d,\omega^{n}).$

We also define the Sobolev space

$W^{1,2}(\hat{X},d,\omega^{n})=W^{1,2}(X,\omega).$

by extending $f\in W^{1,2}(X,\omega)$ from $X\setminus\mathcal{S}_{X,\omega}$
to $\hat{X}$ trivially. By Lemma 9.1, for any $p\in\hat{X}$, the distance
function
$d(p,\cdot)\in{\color[rgb]{0,0,0}W^{1,\infty}_{loc}(\hat{X},d,\omega^{n})}$.
The Sobolev inequality in Theorem 8.1 also holds for
${\color[rgb]{0,0,0}W^{1,2}}(\hat{X},d,\omega^{n})$ with the integration
calculated on $X\setminus\mathcal{S}_{X,\omega}$.

We will prove the following non-collapsing result for
$(\hat{X},d,\omega^{n})$.

######  Proposition 9.1.

There exists $\kappa=\kappa(n,A,K,\gamma)>0$ such that for any point
$p\in\hat{X}$ and $r\in(0,1)$, we have

(9.1)
$\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{V_{\omega}}\geq\min\\{\frac{1}{2},\kappa{r^{\frac{2q}{q-1}}}\\},$

where $q>1$ is the Sobolev exponent in Theorem 3.1 and $B_{d}(p,r)$ is the
geodesic ball of radius $r$ centered at $p$ with respect to $d$ and
$\omega^{n}$.

We remark that if $q=\frac{n}{n-1}$, then (9.1) would be the expected

$\frac{\mbox{Vol}_{\omega}(B_{d}(p,r))}{V_{\omega}}\geq\kappa{r^{2n}}.$

###### Proof.

First we assume $p\in X\setminus\mathcal{S}_{X,\omega}$. For a given radius
$r>0$, either $\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{V_{\omega}}\geq 1/2$
or $\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{V_{\omega}}\leq 1/2$. We assume
the latter inequality holds, so
$\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{{\mathrm{Vol}}_{\omega}(X\backslash
B_{d}(p,r))}\leq 1$. So by (8.6) the following local Sobolev inequality holds

(9.2)
$\Big{(}\frac{1}{V_{\omega}}\int_{B_{d}(p,r)}|u|^{2q}\omega^{n}\Big{)}^{1/q}\leq\frac{C}{V_{\omega}}\int_{B_{d}(p,r)}|\nabla
u|^{2}_{\omega}\omega^{n}$

for all $u\in W^{1,2}_{0}(B_{d}(p,r))$. We let
$\rho:\mathbb{R}\rightarrow[0,\infty)$ be the standard cut-off function with
$\rho=1$ on $(-\infty,1/2)$ and $\rho=0$ on $[1,\infty)$. We let

$u(x)=\rho(d(p,x)/r).$

Then $u$ is a Lipschitz function with compact support in $B_{d}(p,r)$, and we
also have $\sup_{\hat{X}}|\nabla u|\leq 4r^{-1}.$ Applying (9.2) to this
function $u$, we obtain

$\Big{(}\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r/2))}{V_{\omega}}\Big{)}^{1/q}\leq
C\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{r^{2}V_{\omega}},$

for some uniform constant $C>0$ independent of $r$. This is equivalent to

(9.3)
$c_{0}\Big{(}\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r/2))}{(r/2)^{\frac{2q}{q-1}}V_{\omega}}\Big{)}^{1/q}\leq\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{r^{\frac{2q}{q-1}}V_{\omega}},$

for some uniform constant $c_{0}>0$. We now apply (9.3) to the radii
$r_{m}=2^{-m}r$ with $m=0,1,2,\cdots,\ell$ and iterate the resulted
inequalities to obtain

(9.4)
$c_{0}^{\sum_{m=0}^{\ell}q^{-m}}\Big{(}\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,2^{-\ell-1}r))}{(2^{-\ell-1}r)^{\frac{2q}{q-1}}V_{\omega}}\Big{)}^{1/q^{\ell+1}}\leq\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,r))}{r^{\frac{2q}{q-1}}V_{\omega}}.$

We observe that the left-hand side of (9.4) satisfies
$\lim_{\ell\to\infty}c_{0}^{\sum_{m=0}^{\ell}q^{-m}}=c_{0}^{\frac{q}{q-1}}$,
and

$\displaystyle\lim_{\ell\to\infty}\Big{(}\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,2^{-\ell-1}r))}{(2^{-\ell-1}r)^{\frac{2q}{q-1}}V_{\omega}}\Big{)}^{\frac{1}{q^{\ell+1}}}$
$\displaystyle=$
$\displaystyle\lim_{\ell\to\infty}\Big{(}\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,2^{-\ell-1}r))}{(2^{-\ell-1}r)^{2n}V_{\omega}}\Big{)}^{\frac{1}{q^{1+\ell}}}r^{(2n-\frac{2q}{q-1})/q^{\ell+1}}2^{-(\ell+1)(2n-\frac{2q}{q-1})/q^{\ell+1}}=1,$

since $p$ is chosen to be a smooth point where $\omega$ is locally smooth,
hence when $\ell$ is sufficiently large the metric ball
$B_{d}(p,2^{-\ell+1}r)$ is approximately Euclidean, so

$\lim_{\ell\to\infty}\frac{{\mathrm{Vol}}_{\omega}(B_{d}(p,2^{-\ell-1}r))}{(2^{-\ell-1}r)^{2n}V_{\omega}}=\frac{c_{n}}{V_{\omega}},$

where $c_{n}$ is the volume of unit ball in ${\mathbb{C}}^{n}$.

In general, if $p\in\hat{X}$, for any $r>0$ and $0<\epsilon<<r$, there exists
$p_{\epsilon}\in X\setminus\mathcal{S}_{X,\omega}$ such that

$d(p,p_{\epsilon})<\epsilon$

and

$\frac{Vol(B(p,r))}{r^{2q/(q-1)}}\geq\frac{Vol(B(p_{\epsilon},r-2\epsilon))}{r^{2q/(q-1)}}\geq\kappa\Big{(}\frac{r-2\epsilon}{r}\Big{)}^{2q/(q-1)}\rightarrow\kappa$

as we take $\epsilon\rightarrow 0$. We have completed the proof of the
proposition. ∎

The non-collapsing estimate in Proposition 9.1 immediately implies the
following diameter estimate.

######  Corollary 9.1.

There exists $C=C(n,A,p,K,\gamma)>0$ such that if
$\omega\in\mathcal{AK}(n,A,p,K,\gamma)$,

$diam(\hat{X},d)\leq C.$

In particular, $(\hat{X},d,\omega^{n})$ is a compact metric space.

##  10\. Heat kernel estimates for $\mathcal{AK}$-currents

Let $X$ be an $n$-dimensional normal Kähler variety. Let
$\omega\in\mathcal{AK}(X,\theta_{Y},n,A,p,K,\gamma)$. Since $\omega\in
C^{\infty}(X\setminus\mathcal{S}_{X,\omega})$, we would like to define the
heat kernel of the Laplacian $\Delta_{\omega}$ by the following parabolic
equation

$\frac{\partial}{\partial
t}H(x,y,t)=\Delta_{\omega,y}H(x,y,t),~{}\lim_{t\rightarrow
0^{+}}H(x,y,t)=\delta_{x}(y)$

for $x,y\in Y^{\circ}=\pi^{-1}(X\setminus\mathcal{S}_{X,\omega})$. However, we
have to show the existence and uniqueness of $H(x,y,t)$ and how it can be
extended naturally to $\hat{X}\times\hat{X}\times[0,\infty)$.

We will apply Proposition 7.1 with $\\{\omega_{j}\\}_{j=1}^{\infty}$ as the
approximating Kähler metrics for $\pi^{*}\omega$ on $Y$, the suitable blow-up
of $X$ as in (7.1). We will then consider the heat kernel $H_{j}(x,y,t)$
associated to $\omega_{j}$ on $Y\times Y\times[0,\infty)$. Theorem 2.2
immediately implies the following lemma.

######  Lemma 10.1.

There exists $C>0$ such that

$\sup_{x,y\in
Y}H_{j}(x,y,t)\leq\frac{1}{V_{\omega_{j}}}+\frac{C}{V_{\omega_{j}}}t^{-\frac{q}{q-1}}.$

###### Proof.

This lemma follows from the diagonal estimates for heat kernel in Theorem 2.2
and the elementary inequality:

$H_{j}(x,y,t)\leq\frac{1}{2}H_{j}(x,x,t)+\frac{1}{2}H_{j}(y,y,t).$

∎

In particular, for any $\epsilon>0$, there exists $C=C(\epsilon)>0$ such that
for all $j\geq 1$,

$\sup_{Y\times Y\times[\epsilon,\infty)}H_{j}(x,y,t)\leq C.$

The following lemma immediately follows by the standard interior estimates for
linear parabolic equations as $\omega_{j}$ converges smoothly to $\omega$ on
$Y^{\circ}$.

######  Lemma 10.2.

For any $\epsilon>0$, $k>0$ and ${\mathcal{K}}\subset\subset Y^{\circ}$, there
exists $C>0$ such that for all $j\geq 1$,

$\|H_{j}(x,y,t)\|_{C^{k}({\mathcal{K}}\times{\mathcal{K}}\times[\epsilon,\infty))}\leq
C,$

where the $C^{k}$ norm of $H_{j}$ is with respect to the fixed metric
$\pi^{*}\omega|_{\mathcal{K}}$.

######  Lemma 10.3.

For any $x\in Y^{\circ}$, any open neighborhood $U_{x}\subset Y^{\circ}$ of
$x$, there exist $\delta>0$ and $C>0$ such that for all $j\geq 1$,

(10.1) $\sup_{y\in Y\backslash
U_{x}}H_{j}(x,y,t)<Ct^{-\frac{q}{q-1}}e^{-\frac{\delta}{t}}\chi_{\scriptscriptstyle(0,A]}(t)+Ce^{-\frac{\delta}{t}}\chi_{\scriptscriptstyle(A,\infty)}(t).$

###### Proof.

Since $x\in Y^{\circ}$, there exists $d_{0}>0$ such that for all $j\geq 1$ and
$y\in Y\setminus U_{x}$,

$d_{\omega_{j}}(x,y)>d_{0},$

by the smooth convergence of $\omega_{j}|_{U_{x}}$ to $\pi^{*}\omega|_{U_{x}}$
if $U_{x}\subset Y^{\circ}$ is chosen sufficiently small. The lemma then
follows from the uniform estimates in Theorem 2.2. ∎

By taking a subsequence, we can assume that $H_{j}(x,y,t)$ converges locally
smoothly on

$\left(Y^{\circ}\times Y^{\circ}\right)\times(0,\infty).$

We let $H_{\infty}(x,y,t)$ be the limit of $H_{j}(x,y,t)$ defined on
$(Y^{\circ}\times Y^{\circ})\times{\color[rgb]{0,0,0}(}0,\infty)$, after
further taking a subsequence. $H_{\infty}$ can be naturally extended to a
locally (in time $t$) bounded function on $Y\times Y\times(0,\infty)$.

Straightforward arguments will give the following lemma by the construction of
$H_{\infty}$ and the uniform estimates of $H_{j}$.

######  Lemma 10.4.

Let $H_{\infty}(x,y,t)$ be a sub-sequential limit of $H_{j}(x,y,t)$. Then the
following hold:

  1. (1)

$H_{\infty}\in C^{\infty}((Y^{\circ}\times Y^{\circ})\times(0,\infty)).$

  2. (2)

$H_{\infty}$ satisfies the same upper bounds as $H_{j}$ in Lemma 10.1, i.e.

(10.2) $\sup_{x,y\in
Y^{\circ}}H_{\infty}(x,y,t)\leq\frac{1}{V_{\omega}}+\frac{C}{V_{\omega}}t^{-\frac{q}{q-1}}.$

  3. (3)

For all $t>0$ and $x\in Y^{\circ}$,

$\int_{Y}H_{\infty}(x,y,t)\omega^{n}(y)=1.$

  4. (4)

For any $x,y\in Y^{\circ}$,

$\lim_{t\rightarrow 0^{+}}H_{\infty}(x,y,t)=\delta_{x}(y).$

  5. (5)

for any $(x,y,t)\in(Y^{\circ}\times Y^{\circ})\times(0,\infty)$,

$\frac{\partial}{\partial
t}H_{\infty}(x,y,t)=\Delta_{\omega,y}H_{\infty}(x,y,t).$

  6. (6)

For a fixed $x\in Y^{\circ}$ and $t>0$, the function $y\mapsto
H_{\infty}(x,y,t)\in W^{1,2}(Y^{\circ},\omega)$ and the following hold:

(10.3) $\int_{Y^{\circ}}|\nabla
H_{\infty}(x,y,t)|_{\omega}^{2}\omega^{n}\leq\frac{C}{V_{\omega}}t^{-1-\frac{q}{(q-1)}},$
(10.4) $\sup_{x,y\in
Y^{\circ}}|\dot{H}_{\infty}(x,y,t)|\leq\frac{C}{V_{\omega}}t^{-1-\frac{q}{q-1}}.$

Furthermore, $H_{\infty}$ satisfies the following semi-group property

(10.5)
$\int_{Y^{\circ}}H_{\infty}(x,z,t)H_{\infty}(y,z,s)\omega^{n}(z)=H_{\infty}(x,y,t+s),$

for all $x$,$y\in Y^{\circ}$ and for all $s$, $t>0$.

###### Proof.

Items (1), (2) and (5) are immediate consequences of locally smooth
convergence of $H_{j}$ to $H_{\infty}$ over $(Y^{\circ}\times
Y^{\circ})\times(0,\infty)$. To prove item (3), for any $\epsilon>0$ we take a
small open neighborhood $U$ of $Y\backslash Y^{\circ}$ such that
$\sup_{j}\int_{U}\omega_{j}^{n}<\epsilon.$ By the uniform convergence of
$H_{j}(x,y,t)$ over $y\in Y^{\circ}\backslash U$, we have

$\displaystyle 1\geq\int_{Y^{\circ}\backslash
U}H_{\infty}(x,y,t)\omega^{n}(y)$ $\displaystyle=$
$\displaystyle\lim_{j\to\infty}\int_{Y^{\circ}\backslash
U}H_{j}(x,y,t)\omega_{j}^{n}(y)$ $\displaystyle=$ $\displaystyle
1-\lim_{j\to\infty}\int_{U}H_{j}(x,y,t)\omega_{j}^{n}(y)$ $\displaystyle\geq$
$\displaystyle
1-\big{(}\frac{1}{V_{\omega}}+\frac{C}{V_{\omega}}t^{-\frac{q}{q-1}}\big{)}\epsilon\to
1,$

if we let $U$ shrink to $Y\backslash Y^{\circ}$ and then $\epsilon\to 0$. This
proves item (3). For item (4), it suffices to show that for any test function
$\rho\in C^{\infty}_{0}(Y^{\circ})$,

(10.6) $\lim_{t\to
0^{+}}\int_{Y^{\circ}}H_{\infty}(x,y,t)\rho(y)\omega^{n}(y)=\rho(x),$

for all $x\in Y^{\circ}$. Note that the function
$u_{j}(x,t)=\int_{Y}H_{j}(x,y,t)\rho(y)\omega_{j}^{n}(y)$ satisfies

$\Big{|}\frac{\partial u_{j}}{\partial
t}\Big{|}=\Big{|}\int_{Y}H_{j}(x,y,t)\Delta_{\omega_{j}}\rho(y)\omega_{j}^{n}(y)\Big{|}\leq
C,$

where we have used $|\Delta_{\omega_{j}}\rho|\leq C$ since $\rho$ is compactly
supported in $Y^{\circ}$. Integrating this inequality gives
$|u_{j}(x,t)-\rho(x)|\leq Ct$. It is clear that
$u_{j}(x,t)\to\int_{Y^{\circ}}H_{\infty}(x,y,t)\rho(y)\omega^{n}(y)$ as
$j\to\infty$. The equation in (10.6) follows easily. To see the last item (6),
multiplying both sides of the heat equation
$\partial_{t}H_{j}(x,y,t)=\Delta_{\omega_{j}}H_{j}(x,y,t)$ by $H_{j}(x,y,t)$
and applying integration by parts, we get

(10.7) $\displaystyle\int_{Y}|\nabla
H_{j}(x,y,t)|^{2}_{\omega_{j}}\omega_{j}^{2}$ $\displaystyle=$
$\displaystyle-\int_{Y}\dot{H}_{j}(x,y,t)\tilde{H}_{j}(x,y,t)\omega^{n}$
$\displaystyle\leq$
$\displaystyle\|\dot{H}_{j}(x,y,t)\|_{L^{2}(\omega_{j})}\tilde{H}_{j}(x,x,2t)^{1/2}$
$\displaystyle\leq$ $\displaystyle\frac{C}{V_{\omega}}t^{-1-\frac{q}{(q-1)}}.$

The inequality (10.3) follows by letting $j\to\infty$ in (10.7) and using
Fatou’s lemma. The semi-group property of $H_{\infty}$ in (10.5) follows by
similar argument using the uniform upper bound of $H_{j}$. The estimate (10.4)
follows from (5.25) and locally smooth convergence of $\dot{H}_{j}$ to
$\dot{H}_{\infty}$. ∎

We remark that (10.6) also holds for any $\rho\in C(Y^{\circ})$. To see this,
we first consider $\rho_{\epsilon}=\rho\eta_{\epsilon}\in C_{0}(Y^{\circ})$
such that $x\in{\mathrm{supp}(\eta_{\epsilon})}$ and $\eta_{\epsilon}(x)=1$.
Then we take a sequence of smooth functions $u_{j}\in
C^{\infty}_{0}(Y^{\circ})$ such that
$\|u_{j}-\rho_{\epsilon}\|_{L^{\infty}}\leq j^{-1}$. It then follows that

$u_{j}(x)=\lim_{t\to
0^{+}}\int_{Y^{\circ}}H_{\infty}(x,y,t)u_{j}(y)\omega^{n}(y)=\lim_{t\to
0^{+}}\int_{Y^{\circ}}H_{\infty}(x,y,t)\rho_{\epsilon}(y)\omega^{n}(y)+O(j^{-1}).$

Letting $j\to\infty$ and then $\epsilon\to 0$ gives the equation (10.6) for
$\rho\in C(Y^{\circ})$.

######  Corollary 10.1.

Suppose there exists another heat kernel $\hat{H}(x,y,t)$ satisfying the
conditions in Lemma 10.4. Then we have

$\hat{H}(x,y,t)=H_{\infty}(x,y,t)$

for all $x,y\in Y^{\circ}$. Hence the heat kernel $H_{\infty}$ is unique in
this sense.

###### Proof.

Take cut-off functions $\eta_{\epsilon}$ as in Lemma 8.2. Fix a time $T>0$. We
claim that the function

$t\in(0,T)\mapsto\phi(t):=\int_{Y^{\circ}}H_{\infty}(x,z,t)\hat{H}(z,y,T-t)\omega^{n}(z)$

is independent of $t$. Denote
$\phi_{\epsilon}(t):=\int_{Y^{\circ}}\eta_{\epsilon}(z)H_{\infty}(x,z,t)\hat{H}(y,z,T-t)\omega^{n}(z)$.
By integration by parts and using the heat equation, we have for any small
$\delta>0$ and $t\in[\delta,T-\delta]$, there is a constant $C>0$ depending
additionally on $\delta>0$ such that

$|\phi_{\epsilon}^{\prime}(t)|=\Big{|}\int_{Y^{\circ}}(-\langle\nabla\eta_{\epsilon},\nabla
H_{\infty}\rangle\hat{H}+\langle\nabla\eta_{\epsilon},\nabla\hat{H}\rangle
H_{\infty})\omega^{n}\Big{|}\leq C\epsilon^{1/2},$

by using the Hölder’s inequality together with the upper bounds of $\|\nabla
H_{\infty}\|_{L^{2}}$ (and $\|\nabla\hat{H}\|_{L^{2}}$) and $H_{\infty}$ (and
$\hat{H}$) in (10.3) and (10.2), respectively. Integrating this over
$[t_{1},t_{2}]\subset[\delta,T-\delta]$ gives

$|\phi_{\epsilon}(t_{1})-\phi_{\epsilon}(t_{2})|\leq C\epsilon^{1/2}.$

Letting $\epsilon\to 0$ yields $\phi(t_{1})=\phi(t_{2})$. Since $\delta>0$ is
an arbitrary constant, this easily implies that $\phi(t)$ remains a constant
for $t\in(0,T)$. By the Dirac property (10.6) of $H_{\infty}$ and $\hat{H}$,
we see that $\phi(t)\to\hat{H}(x,y,T)$ as $t\to 0^{+}$ and $\phi(t)\to
H_{\infty}(x,y,T)$ as $t\to T^{-}$. This finishes the proof of the uniqueness
of the heat kernel. ∎

##  11\. A spectral theorem for normal Kähler spaces

Let $(X,\theta_{Y})$ be a normal Kähler variety of $\dim X=n$, equipped with a
smooth Kähler metric $\theta_{Y}$. Let
$\omega\in\mathcal{AK}(X,\theta_{Y},n,A,p,K,\gamma)$. We consider the Laplace
operator $\Delta=\Delta_{\omega}$ with respect to $\omega$ on
$X^{\circ}:=X\setminus\mathcal{S}_{X,\omega}$. The following lemma is a simple
consequence of the integration by parts.

######  Lemma 11.1.

The linear operator $-\Delta:C^{\infty}_{0}(X^{\circ})\to
C^{\infty}_{0}(X^{\circ})$ is self-adjoint, i.e. for any $u,v\in
C^{\infty}_{0}(X^{\circ})$, $\int_{X}(-\Delta u)v\omega^{n}=\int_{X}(-\Delta
v)u\omega^{n}$.

Note $-\Delta:L^{2}(X,\omega)\to L^{2}(X,\omega)$ is a linear and densely
defined operator. The associated bilinear quadratic form of $-\Delta$ is given
by $Q(u,u)=\int_{X}|\nabla u|_{\omega}^{2}\omega^{n}$, for any $u\in
C^{\infty}_{0}(X^{\circ})$. Let ${\mathcal{D}}(Q)$ be the completion of
$C^{\infty}_{0}(X^{\circ})$ in $L^{2}(X,\omega)$ under the norm
$\|u\|_{L^{2}}+Q(u,u)^{1/2}$, which is equivalent to the
$W^{1,2}(X,\omega)$-norm of $u$. Hence by Proposition 8.1,
${\mathcal{D}}(Q)=W_{0}^{1,2}(X,\omega)=W^{1,2}(X,\omega)$. We then let
$(-\Delta)^{F}:L^{2}(X,\omega)\to L^{2}(X,\omega)$ be the Friedrich extension
of $-\Delta$, which is a self-adjoint, densely defined and nonnegative
operator. We note that the domain of $(-\Delta)^{F}$ is a subspace of
${\mathcal{D}}(Q)=W^{1,2}(X,\omega)$ (c.f. [26]). To ease notations, we will
omit the superscript $F$ and simply write $(-\Delta)^{F}$ as $-\Delta$.

######  Theorem 11.1.

There exists a countable set $\Sigma\subset{\mathbb{R}}_{\geq 0}$ such that
the equation

$-\Delta u=\lambda u,\quad\mbox{with}\quad u\in W^{1,2}(X,\omega)$

admits a unique solution iff $\lambda\not\in\Sigma$. Furthermore, we can write
$\Sigma=\\{\lambda_{k}\\}$ with

(11.1) $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots.$

###### Proof.

We consider the linear operator $Lu=-\Delta u+u$. The associated quadratic