ON $L^{\infty}$ ESTIMATES FOR COMPLEX MONGE-AMPÈRE EQUATIONS 111Work supported
in part by the National Science Foundation under grant DMS-1855947.

Bin Guo, Duong H. Phong, and Freid Tong

###### Abstract

A PDE proof is provided for the sharp $L^{\infty}$ estimates for the complex
Monge-Ampère equation which had required pluripotential theory before. The
proof covers both cases of fixed background as well as degenerating background
metrics. It extends to more general fully non-linear equations satisfying a
structural condition, and it also gives estimates of Trudinger type.

##  1 Introduction

The main goal of this paper is to answer a long-standing question in the
theory of complex Monge-Ampère equations and its applications to complex
geometry, namely whether sharp $L^{\infty}$ estimates can be established by
PDE methods and without pluripotential theory. We shall see that the answer is
affirmative and, as may have been anticipated, the PDE proof also gives new
estimates as well as extensions to many other fully non-linear equations.

$L^{\infty}$ estimates have a uniquely storied history in the theory of
complex Monge-Ampère equations. Early on, they were recognized as the defining
difficulty in the problem of finding Kähler-Einstein metrics. Yau’s
introduction in 1978 of Moser iteration was the key step in his solution of
the Calabi conjecture, and it ushered in a new era for complex Monge-Ampère
equations [25]. Yau’s Moser iteration method works for equations whose right
hand sides are in $L^{q}$ for $q>n$, where $n$ is the complex dimension of the
underlying space. The next major advance was the 1998 result of Kolodziej
[14], which established $L^{\infty}$ estimates for the solution when the right
hand side is in $L^{q}$, for $q>1$. This improvement in the range of $q$ is no
mere technicality, and it has a profound geometric significance: that
$L^{\infty}$ estimates fail for $q=1$ is indicative of the necessity of
stability conditions in the Kähler-Ricci flow [16], while the cases $1<q\leq
n$ are needed in a wide range of problems, including singular Kähler-Einstein
metrics on manifolds of general type [12, 9], the analytic minimal model
program [18], and degenerating Calabi-Yau metrics [21]. Kolodziej’s method of
proof relied heavily on the pluripotential theory developed in the late 1970’s
by Bedford and Taylor [3, 4]. Actually, for many applications to geometry, an
extension of Kolodziej’s results to the more general case of degenerating
background metrics is necessary. Such an extension was developed in 2007
independently by Demailly and Pali [9] and Eyssidieux, Guedj, and Zeriahi
[12], and pluripotential theory continued to be essential.

An immediate question for the theory of geometric partial differential
equations is whether the above $L^{\infty}$ estimates can be derived by PDE
methods, instead of pluripotential theory which is specific to Monge-Ampère
equations. This question gained considerable attention over the years, as more
and more fully non-linear equations without corresponding pluripotential
theory emerged in complex differential geometry. A completely different proof
for right hand sides in $L^{q}$ with $q>2$ was found in 2011 by Blocki [5],
using the Alexandrov-Bakelman-Pucci (ABP) maximum principle.111The possibility
of applying ABP maximum principle to the complex Monge-Ampère equation were
suggested a while ago by S.Y. Cheng and S.T. Yau. His methods turn out to be
remarkably powerful, and have since been applied successfully to many
problems, including subsolutions [19, 17], equations with gradient terms [23],
and the constant scalar curvature problem [7]. Even so, extensions to
degenerating backgrounds as well as the full range $q>1$ remained out of
reach. In a different direction, a PDE proof of $L^{\infty}$ estimates for the
complex Monge-Ampère equation was obtained by J.X. Wang, X.J. Wang, and B.
Zhou [24] for domains in ${\bf C}^{n}$. However, their methods do not appear
adaptable to the compact manifold case, even in the simplest case when the
background Kähler metric is fixed. Thus a fully effective approach to
$L^{\infty}$ estimates remained an open question.

The PDE approach to $L^{\infty}$ estimates which we present in this paper
combines the methods of Wang, Wang, and Zhou [24] with a fundamental new idea
of Chen and Cheng [7] in their recent work on constant scalar curvature Kähler
metrics, namely to compare the given equation with an auxiliary complex Monge-
Ampère equation. A key novelty in our paper resides in the choice of auxiliary
complex Monge-Ampère equation, as well as of the test function $\Phi$ for the
comparison. We now formulate our main results.

Let $X$ be a compact Kähler manifold without boundary of dimension $n$, and
$\omega_{X}$ its Kähler form. If $\varphi$ is a real smooth function on $X$,
we let $\omega_{\varphi}=\omega_{X}+i\partial\bar{\partial}\varphi$, and let
$h_{\varphi}$ be the corresponding endomorphism relative to the metric
$\omega_{X}$. Explicitly, if we write $\omega_{X}=ig_{\bar{k}j}dz^{j}\wedge
d\bar{z}^{k}$ in local holomorphic coordinates, then
$(h_{\varphi})^{j}{}_{k}=g^{j\bar{m}}(\omega_{\varphi})_{\bar{m}k}$. Let
$\lambda[h_{\varphi}]$ be the vector of eigenvalues of $h_{\varphi}$, and
consider the non-linear partial differential equation

$\displaystyle f(\lambda[h_{\varphi}])=\,e^{F},\quad{\rm
sup}_{X}\varphi=0,\quad\lambda[h_{\varphi}]\in\Gamma,$ (1.1)

for a given function $f(\lambda)$ and real function $F$ normalized such that
$\int_{X}e^{nF}\omega_{X}^{n}=V=\int_{X}\omega_{X}^{n}$. Here the function
$f(\lambda)$ is assumed to be invariant under permutations of the components
of $\lambda$, and defined on an open cone
$\Gamma\subset\\{\lambda:\lambda_{1}+\ldots+\lambda_{n}>0\\}$ with vertex at
the origin and containing the first octant
$\\{\lambda:\lambda_{1}>0,\ldots,\lambda_{n}>0\\}$. We assume throughout $f$
is elliptic in the sense that $\frac{\partial
f(\lambda)}{\partial\lambda_{j}}>0$ for any $\lambda\in\Gamma$, and that

$\displaystyle f(r\lambda)=r\,f(\lambda),\quad r>0,\ \lambda\in\Gamma,$ (1.2)

which can be viewed as a normalization, as the same homogeneous equation can
be expressed with many different functions $f(\lambda)$. We shall need the
$L^{1}(\,{\rm log}\,L)^{p}$ norm of $e^{nF}$ with respect to the measure
$\omega_{X}^{n}$, which can be recognized as a generalized entropy ${\rm
Ent}_{p}(F)$,

$\displaystyle{\rm Ent}_{p}(F)={1\over
V}\int_{X}e^{nF}|F|^{p}\omega_{X}^{n}={1\over n^{p}V}\|e^{nF}\|_{L^{1}(\,{\rm
log}\,L)^{p}}.$ (1.3)

###### Theorem 1

Consider the equation (1.1), and assume that the function $f(\lambda)$
satisfies the following structural condition, namely that there exists a
constant $\gamma>0$ so that

$\displaystyle\ {\rm det}\big{(}\frac{\partial f(\lambda[h])}{\partial
h_{ij}}\big{)}\geq\gamma,\quad\mbox{for all }\lambda\in\Gamma.$ (1.4)

Fix $p>n$. Then for any solution $\varphi\in C^{2}(X)$, we have the estimate

$\displaystyle{\rm sup}_{X}|\varphi|\leq C$ (1.5)

where the constant $C$ depends only on $n,p,\gamma$, $\omega_{X}$, and the
entropy ${\rm Ent}_{p}(F)$.

We observe that many equations satisfy the structural condition (1.4). They
include the Monge-Ampère equation
$f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$, more generally Hessian
equations $f(\lambda)=\sigma_{k}(\lambda)^{1/k}$, $k=1,\ldots,n$, where
$\sigma_{k}(\lambda)$ is the $k$-th symmetric function, and quotient Hessian
equations such as
$f(\lambda)=\big{(}{\sigma_{k}\over\sigma_{l}}\big{)}^{1/(k-l)}+c\sigma_{m}^{1/m}$
for some $c>0$. Applying Theorem 1 to the Monge-Ampère equation, we obtain
immediately Kolodziej’s [14] sharp $L^{\infty}$ bounds. Applying it to Hessian
equations, we obtain the $L^{\infty}$ bounds of Dinew and Kolodziej [11]. In
fact, our result is stronger, as the bound in [11] requires that $\,{\rm
exp}\,(nF)$ be in $L^{q}$ for $q>1$, while we only need that $\,{\rm
exp}\,(nF)$ be in $L^{1}(\,{\rm log}\,L)^{p}$ for $p>n$. Beyond these cases,
the $L^{\infty}$ estimates in Theorem 1 are new, and appear to be the first
obtained in any generality for fully non-linear equations from Kähler
geometry.

We discuss now estimates, particularly important for many geometric
applications, where the background metric is allowed to degenerate. It is
convenient to set up the equation as follows. Let $(X,\omega_{X})$ be a
compact Kähler manifold of dimension $n$ as before, and let $\chi$ be a fixed
closed and non-negative $(1,1)$-form. For $t\in(0,1]$, set

$\displaystyle\omega_{t}=\chi+t\omega_{X}$ (1.6)

and for each $\varphi\in C^{2}(X)$, let
$\omega_{t,\varphi}=\omega_{t}+i\partial\bar{\partial}\varphi$,
$h_{t,\varphi}$ be the corresponding endomorphism relative to the metric
$\omega_{X}$, and $\lambda[h_{t,\varphi}]$ be the vector of eigenvalues of
$h_{t,\varphi}$. Consider the family of non-linear partial differential
equations

$\displaystyle f(\lambda[h_{t,\varphi_{t}}])=c_{t}\,e^{F_{t}},\quad{\rm
sup}_{X}\varphi_{t}=0,\quad\lambda[h_{t,\varphi_{t}}]\in\Gamma,\quad
t\in(0,1]$ (1.7)

for given function $f(\lambda)$, real functions $F_{t}$, and positive constant
coefficients $c_{t}$. Here the functions $F_{t}$ are normalized by
$\int_{X}e^{nF_{t}}\omega_{X}^{n}=\int_{X}\omega_{X}^{n}$, so that the
constants $c_{t}$ are determined. Denote by $V_{t}$ the volume of the metric
$\omega_{t}$, $V_{t}=\int_{X}\omega_{t}^{n}$, and define the energy
$E_{t}(\varphi)$ by

$\displaystyle
E_{t}(\varphi)=\frac{1}{V_{t}}\int_{X}(-\varphi)f(\lambda([h_{t,\varphi}])^{n}\omega_{X}^{n}.$
(1.8)

For functions $\varphi_{t}$ solving the equation (1.7), we obviously have

$\displaystyle
E_{t}(\varphi_{t})=\frac{c^{n}_{t}}{V_{t}}\int_{X}(-\varphi_{t}){\rm
exp}(nF_{t})\omega_{X}^{n}.$ (1.9)

###### Theorem 2

Consider the family of equations (1.7), and assume that $f(\lambda)$ satisfies
the structural condition (1.4). Let $\varphi_{t}$ be $C^{2}$ functions on $X$
satisfying (1.7). Fix $p>n$. Then for any $t\in(0,1]$, we have

$\displaystyle\sup_{X}|\varphi_{t}|\leq C$ (1.10)

where $C$ is a constant depending only on $\omega_{X},\chi,p,n,\gamma$, and
upper bounds for the following three quantities

$\displaystyle{c_{t}^{n}\over V_{t}},\quad E_{t}(\varphi_{t}),\quad{\rm
Ent}_{p}(F_{t}).$ (1.11)

All three quantities in (1.11) have attractive interpretations. We have
already noted from (1.3) that ${\rm Ent}_{p}(F_{t})$ is a generalized entropy.
The quantity $c_{t}^{n}\over V_{t}$ can be viewed as a relative volume, and a
substitute for a cohomological constraint when dealing with general equations
$f(\lambda)$. The quantity $E_{t}(\varphi_{t})$ is clearly an energy
functional, as it reduces to the Dirichlet integral in the case of the
Laplacian on surfaces.

As a special case, Theorem 2 applied to the Monge-Ampère equation gives back
immediately the $L^{\infty}$ estimates of Eyssidieux, Guedj, Zeriahi [12], and
Demailly, Pali [9], in the full generality of degenerating background metrics.
The point here is that the relative volumes $c_{t}^{n}/V_{t}$ must be $1/V$
because of a cohomological constraint, and it follows easily from Jensen’s
inequality that the energies $E_{t}$ are uniformly bounded (see the fuller
discussion in Theorem 4 in §4 below). Thus Theorem 2 gives the pure PDE proof
of these $L^{\infty}$ estimates that we sought. In fact, it is particularly
simple as the ABP maximum principle is not even needed.

More generally, as long as the admissible cone $\Gamma$ is the one
corresponding to PSH functions, we can obtain uniform bounds for the energies
$E_{t}$. For example, this applies to the equations
$f(\lambda)=\big{(}\frac{\sigma_{k}}{\sigma_{l}}\big{)}^{1/(k-l)}+c\;\sigma_{n}^{1/n}$.

For Hessian equations $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$, $1\leq
k<n$, nothing was known in the case of degenerating background metrics, and
Theorem 2 is now the only available result. As we shall see in Theorem 5 in §5
below, if $\|e^{nF_{t}}\|_{L^{p}}$ is uniformly bounded for $p>{n\over k}$, we
can bound the energy $E_{t}$ by a multiple of ${c_{t}^{n}V_{t}^{-1}}$. In
particular, for big cohomology classes $[\omega_{t}]$, the volumes $V_{t}$ do
not tend to $0$, the coefficients $c_{t}^{n}$ are bounded by the entropy, and
we obtain then uniform $L^{\infty}$ bounds. Such bounds are completely new for
fully non-linear equations, and in particular Theorem 6 is new for Hessian
equations.

This paper is organized as follows. In §2 and §3, we give the proofs of
Theorem 1 and Theorem 2. We actually begin with the proof of Theorem 2 in §2,
as Theorem 1 would follow from Theorem 2 upon control of the energy term
$E_{t}$. As in [7], we use a comparison of the given equation to an auxiliary
complex Monge-Ampère equation. However, for Theorem 2, it is very important to
choose the auxiliary equation so as to avoid having to use the ABP maximum
principle, as this maximum principle would be an impediment in the case of
degenerating background metrics. The additional step of estimating the energy
terms to get Theorem 1 from Theorem 2 is provided by Theorem 3, which is the
one requiring the ABP maximum principle. In §4 and §5, we show how our results
apply to Monge-Ampère and Hessian equations respectively, and how they improve
on many recent results in the literature. Finally, we have discussed
exclusively so far $L^{\infty}$ estimates. But not surprisingly, the same
methods apply to $L^{p}$ and exponential estimates as well. We illustrate this
in §6 by some applications to Trudinger exponential inequalities for fully
non-linear equations, which are either new or independent proofs of known
sharp estimates. We also observe that the assumption (1.2) that $f(\lambda)$
was homomegenous of degree $1$ was only for simplicity. The results of this
paper still hold with this assumption replaced by the weaker Euler inequality
with some fixed positive constant $\Lambda$,

$\displaystyle\sum_{j=1}^{n}{\partial
f\over\partial\lambda_{j}}\lambda_{j}\leq\Lambda\,f(\lambda),\quad\lambda\in\Gamma.$

Furthermore, our methods can be adapted to the setting of families of Kähler
manifolds $(X_{j},\omega_{j})$ of the same dimension, as long as the
$\alpha$-invariant estimates hold uniformly.

##  2 Proof of Theorem 2

Let $\varphi_{t}$ solve the equation (1.7). Fix $p>n$, and any upper bound
${\overline{E}}_{t}>0$ for $E_{t}(\varphi_{t})$. We shall actually show that

$\sup_{X}|\varphi_{t}|\leq C_{0}\Big{\\{}\frac{c_{t}^{n}}{V_{t}}\big{(}{\rm
Ent}_{p}+1+{\rm
exp}(C_{1}{\overline{E}}_{t})\big{)}\Big{\\}}^{\frac{n}{p-n}}{\overline{E}}_{t}+C(n,p),$

for constants $C_{0},C_{1}$ depending only on $n,\omega_{X},\chi,\gamma$, and
$C(n,p)$ depending only on $n,p$. Throughout the proof we will fix
$t\in(0,1]$, but the constants will be independent of $t$, unless stated
explicitly otherwise.

For any $s>0$, we let $\Omega_{s}:=\\{\varphi_{t}\leq-s\\}$ be the sub-level
set of $\varphi_{t}$.

###### Lemma 1

There are constants $C=C(n,\omega_{X},\chi,\gamma)>0$ and
$\beta_{0}=\beta_{0}(n,\omega_{X},\chi,\gamma)>0$ such that for any $s>0$

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}\beta_{0}\big{(}\frac{-(\varphi_{t}+s)}{A_{s}^{1/(n+1)}}\big{)}^{\frac{n+1}{n}}\Big{\\}}\omega_{X}^{n}\leq
C\,{\rm exp}\,(C{\overline{E}}_{t}),$

where
$A_{s}:=\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t}-s)e^{nF_{t}}\omega_{X}^{n}$
is the energy of $(\varphi_{t}+s)_{-}$.

Proof. We choose a sequence of smooth positive functions $\tau_{k}:{\bf
R}\to{{\bf R}}_{+}$ such that

$\tau_{k}(x)=x+\frac{1}{k},\quad\mbox{ when }x\geq 0,$ (2.1)

and

$\tau_{k}(x)=\frac{1}{2k},\quad\mbox{ when }x\leq-\frac{1}{k},$

and $\tau_{k}(x)$ lies between $1/2k$ and $1/k$ for $x\in[-1/k,0]$. Clearly
$\tau_{k}$ converge pointwise to $\tau_{\infty}(x)=x\cdot\chi_{{\bf
R}_{+}}(x)$ as $k\to\infty$, where $\chi_{{\bf R}_{+}}$ denotes the
characteristic function of ${\bf R}_{+}$.

We solve an auxiliary complex Monge-Ampère equation on $X$

$(\omega_{t}+i\partial\bar{\partial}\psi_{t,k})^{n}=\frac{\tau_{k}(-\varphi_{t}-s)}{A_{s,k}}f(\lambda[h_{\varphi_{t}}])^{n}\omega_{X}^{n}=\frac{\tau_{k}(-\varphi_{t}-s)}{A_{s,k}}c_{t}^{n}e^{nF_{t}}\omega_{X}^{n},$
(2.2)

with $\sup\psi_{t,k}=0$ where
$A_{s,k}:=\frac{c_{t}^{n}}{V_{t}}\int_{X}\tau_{k}(-\varphi_{t}-s)e^{nF_{t}}\omega_{X}^{n}$
is chosen so that the integrals of both sides of (2.2) are equal. Note that
(2.2) admits a unique smooth solution by Yau’s theorem [25]. We also observe
that as $k\to\infty$

$A_{s,k}\to
A_{s}=\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t}-s)e^{nF_{t}}\omega_{X}^{n},$
(2.3)

which follows from Lebesgue’s dominated convergence theorem. The limit $A_{s}$
satisfies $A_{s}\leq{\overline{E}}_{t}$, the assumed upper bound of
$E_{t}(\varphi_{t})$.

Denote $\Phi$ to be the smooth function

$\displaystyle\Phi:=-\varepsilon(-\psi_{t,k}+\Lambda)^{\frac{n}{n+1}}-(\varphi_{t}+s)$
(2.4)

where

$0<\varepsilon:=(\frac{n+1}{n^{2}})^{\frac{n}{n+1}}A_{s,k}^{\frac{1}{n+1}}\gamma^{{-\frac{1}{n+1}}},\quad
0<\Lambda:=\frac{1}{(n+1)n^{n-1}}\frac{A_{s,k}}{\gamma}$ (2.5)

where $\gamma>0$ is the constant in the structure condition (1.4) of $f$.

Since $X$ is compact without boundary, the maximum of $\Phi$ must be attained
at some point, say, $x_{0}\in X$. If $x_{0}\in X\backslash\Omega_{s}^{\circ}$,
then

$\sup_{X}\Phi=\Phi(x_{0})=-\varepsilon(-\psi_{t,k}(x_{0})+\Lambda)^{\frac{n}{n+1}}-(\varphi_{t}(x_{0})+s)<-\varphi_{t}(x_{0})-s\leq
0.$

Otherwise $x_{0}\in\Omega_{s}^{\circ}$. Then at $x_{0}$,
$i\partial\bar{\partial}\Phi(x_{0})\leq 0$, and on the right hand side of
(2.2) we have

$\tau_{k}(-\varphi_{t}-s)(x_{0})=-(\varphi_{t}(x_{0})+s)+1/k>0$

by the definition of $\tau_{k}$ in (2.1).

We denote by $G^{i\bar{j}}=\frac{\partial\,{\rm log}\,f(\lambda[h])}{\partial
h_{ij}}=\frac{1}{f}\frac{\partial f(\lambda[h])}{\partial h_{ij}}$ the
coefficients of the linearization of the operator $\,{\rm log}\,f(\lambda[h])$
with $h=\omega_{X}^{-1}\cdot\omega_{t,\varphi_{t}}$. By the ellipticity
assumption of $f(\lambda[\cdot])$, $(G^{i\bar{j}})$ is positive definite.
Moreover, by the structure condition (1.4) on $f$, we have

${\rm det}\,G^{i\bar{j}}=f^{-n}{\rm det}\big{(}\frac{\partial
f(\lambda[h])}{\partial h_{ij}}\big{)}\geq\frac{\gamma}{f(\lambda)^{n}}.$

Recall that the eigenvalues of $h$ are by definition
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$. Working in a basis where $h$ is
diagonal and $\omega_{X}$ the identity, we find using the definition of
$G^{i\bar{j}}$ that

$\sum_{i,j}G^{i\bar{j}}(\omega_{t,\varphi_{t}})_{\bar{j}i}=\frac{1}{f(\lambda)}\sum_{j}\frac{\partial
f(\lambda)}{\partial\lambda_{j}}\lambda_{j}=1$

where we have used the assumption that $f$ is homogeneous of degree one, so
that $\sum_{i}\lambda_{i}\frac{\partial
f(\lambda)}{\partial\lambda_{i}}=f(\lambda)$. At the maximum point $x_{0}$ of
$\Phi$, we can write

$\displaystyle 0$ $\displaystyle\geq$ $\displaystyle
G^{i\bar{j}}\Phi_{\bar{j}i}(x_{0})$ $\displaystyle=$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}G^{i\bar{j}}(\psi_{t,k})_{\bar{j}i}+\frac{n\varepsilon}{(n+1)^{2}}(-\psi_{t,k}+\Lambda)^{-\frac{n+2}{n+1}}G^{i\bar{j}}(\psi_{t,k})_{\bar{j}}(\psi_{t,k})_{i}-G^{i\bar{j}}(\varphi_{t})_{\bar{j}i}$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}G^{i\bar{j}}(\omega_{t,\psi_{t,k}})_{\bar{j}i}-G^{i\bar{j}}(\omega_{t,\varphi_{t}})_{\bar{j}i}+\big{(}1-\frac{\varepsilon
n}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}\big{)}G^{i\bar{j}}(\omega_{t})_{\bar{j}i}$
$\displaystyle\geq$ $\displaystyle\frac{\varepsilon
n}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}n\Big{(}{\rm
det}G^{i\bar{j}}\cdot{\rm
det}(\omega_{t,\psi_{t,k}})_{\bar{j}i}\Big{)}^{1/n}-1+\big{(}1-\frac{\varepsilon
n}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}\big{)}G^{i\bar{j}}(\omega_{t})_{\bar{j}i}$
$\displaystyle\geq$ $\displaystyle\frac{\varepsilon
n^{2}}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}\gamma^{1/n}\Big{(}\frac{\tau_{k}(-\varphi_{t}-s)}{A_{s,k}}\Big{)}^{1/n}-1+\big{(}1-\frac{\varepsilon
n}{n+1}\Lambda^{-\frac{1}{n+1}}\big{)}G^{i\bar{j}}(\omega_{t})_{\bar{j}i}$
$\displaystyle\geq$ $\displaystyle\frac{\varepsilon
n^{2}\gamma^{1/n}}{n+1}(-\psi_{t,k}+\Lambda)^{-\frac{1}{n+1}}\Big{(}\frac{-\varphi_{t}-s+1/k}{A_{s,k}}\Big{)}^{1/n}-1$

where in the third inequality we used the arithmetic-geometric inequality and
in the last one we used the choice of $\varepsilon$ and $\Lambda$ in (2.5).
Thus at $x_{0}$ we have

$-(\varphi_{t}+s)(x_{0})<A_{s,k}\Big{(}\frac{n+1}{n^{2}\varepsilon\gamma^{1/n}}\Big{)}^{n}(-\psi_{t,k}(x_{0})+\Lambda)^{n/(n+1)}=\varepsilon(-\psi_{t,k}(x_{0})+\Lambda)^{n/(n+1)}$

which implies that $\Phi(x_{0})\leq 0$. Hence we can conclude that
$\sup_{X}\Phi\leq 0$, that is, on $X$

$\frac{-(\varphi_{t}+s)}{A_{s,k}^{1/(n+1)}}\leq(\frac{n+1}{n^{2}})^{\frac{n}{n+1}}\gamma^{-\frac{1}{n+1}}\big{(}-\psi_{t,k}+\frac{1}{(n+1)n^{n-1}}\frac{A_{s,k}}{\gamma}\big{)}^{\frac{n}{n+1}}\leq
C_{n}\big{(}-\psi_{t,k}+\frac{A_{s,k}}{\gamma}\big{)}^{\frac{n}{n+1}},$ (2.6)

for some constant $C_{n}$ depending only on $n$ and $\gamma$. Taking the
$\big{(}\frac{n+1}{n}\big{)}$-th power of both sides of the previous equation,
multiplying it by some small $\beta_{0}>0$, taking the exponential on both
sides and then integrating the resulting inequality over $\Omega_{s}$, we
obtain

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}\beta_{0}\big{(}\frac{-(\varphi_{t}+s)}{A_{s,k}^{1/(n+1)}}\big{)}^{\frac{n+1}{n}}\Big{\\}}\omega_{X}^{n}\leq\,{\rm
exp}\,(C_{n}\beta_{0}A_{s,k})\int_{\Omega_{s}}\,{\rm
exp}\,(-C_{n}\beta_{0}\psi_{t,k})\omega_{X}^{n}.$ (2.7)

Recall that $\omega_{t}+i\partial\bar{\partial}\psi_{t,k}>0$ and
$\omega_{t}=\chi+t\omega_{X}$. We may assume $\chi\leq(a_{0}-1)\omega_{X}$ for
some $a_{0}=a_{0}(\chi,\omega)>1$, so $\psi_{t,k}$ is also
$(a_{0}\omega_{X})$-plurisubharmonic. Now it is a basic fact in Kähler
geometry that, for any Kähler class $\hat{\chi}$ on $X$, there is a constant
$\alpha=\alpha(X,\hat{\chi})$ so that

$\displaystyle\int_{X}e^{-\alpha_{0}\psi}\omega_{X}^{n}\leq
C(\alpha_{0},n,\hat{\chi},\omega_{X})$ (2.8)

for any $\alpha_{0}<\alpha$ and any $\hat{\chi}$-plurisubharmonic function
$\psi$ with $\sup_{X}\psi=0$. The local version of this statement is in [13],
and the above global version in [20]. We apply this statement with
$\hat{\chi}=a_{0}\omega_{X}$, and fix $\alpha_{0}$ with
$0<\alpha_{0}<\alpha(X,\hat{\chi})$. Then we choose
$\beta_{0}=\beta_{0}(n,\omega_{X},\chi,\gamma)>0$ in (2.7) such that
$\beta_{0}C_{n}=\alpha_{0}$, and from (2.7) we can then deduce that

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}{\beta_{0}\big{(}\frac{-(\varphi_{t}+s)}{A_{s,k}^{1/(n+1)}}\big{)}^{\frac{n+1}{n}}}\Big{\\}}\omega_{X}^{n}\leq
Ce^{CA_{s,k}},$ (2.9)

for some constant $C=C(n,\omega_{X},\chi,\gamma)>0$. Letting $k\to\infty$ in
(2.9) we obtain from (2.3)

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}{\beta_{0}\big{(}\frac{-(\varphi_{t}+s)}{A_{s}^{1/(n+1)}}\big{)}^{\frac{n+1}{n}}}\Big{\\}}\omega_{X}^{n}\leq
Ce^{CA_{s}}\leq Ce^{C{\overline{E}}_{t}},$ (2.10)

for some constant $C=C(n,\omega_{X},\chi,\gamma)>0$. The proof of Lemma 1 is
complete.

We come now to the proof of Theorem 2 proper. Fix $p>n$, and define $\eta:{\bf
R}_{+}\to{\bf R}_{+}$ by $\eta(x)=(\,{\rm log}\,(1+x))^{p}$. Note that $\eta$
is a strictly increasing function with $\eta(0)=0$, and let $\eta^{-1}$ be its
inverse function. If we let

$\displaystyle
v:=\frac{\beta_{0}}{2}\big{(}\frac{-\varphi_{t}-s}{A_{s}^{1/(n+1)}}\big{)}^{(n+1)/n}$
(2.11)

then we have for any $z\in\Omega_{s}$, by the generalized Young’s inequality
with respect to $\eta$,

$\displaystyle v(z)^{p}e^{nF_{t}(z)}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\,{\rm
exp}\,({nF_{t}(z)})}\eta(x)dx+\int_{0}^{v(z)^{p}}\eta^{-1}(y)dy$
$\displaystyle\leq$ $\displaystyle\,{\rm exp}\,({nF_{t}(z)})(\,{\rm
log}\,(1+\,{\rm exp}\,({nF_{t}(z)})))^{p}+\int_{0}^{\,{\rm
exp}\,({v(z)-1})}x\eta^{\prime}(x)dx$ $\displaystyle\leq$ $\displaystyle\,{\rm
exp}\,({nF_{t}(z)})(1+n|F_{t}(z)|)^{p}+v(z)^{p}\,{\rm exp}\,({v(z)})$
$\displaystyle\leq$ $\displaystyle\,{\rm
exp}\,({nF_{t}(z)})(1+n|F_{t}(z)|)^{p}+C(p)\,{\rm exp}\,({2v(z)})$

We integrate both sides in the inequality above over $z\in\Omega_{s}$, and get
by Lemma 1 that

$\displaystyle\int_{\Omega_{s}}v(z)^{p}e^{nF_{t}(z)}\omega_{X}^{n}$
$\displaystyle\leq$
$\displaystyle\int_{\Omega_{s}}e^{nF_{t}}(1+n|F_{t}(z)|)^{p}\omega_{X}^{n}+\int_{\Omega_{s}}e^{2v(z)}\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle\|e^{nF_{t}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{\overline{E}}_{t}},$

where the constant $C>0$ depends only on $n,\omega_{X},\chi,\gamma,p$. In view
of the definition of $v$, this implies

$\int_{\Omega_{s}}(-\varphi_{t}-s)^{\frac{(n+1)p}{n}}e^{nF_{t}(z)}\omega_{X}^{n}\leq
2^{p}\beta_{0}^{-p}A_{s}^{\frac{p}{n}}\big{(}\|e^{nF_{t}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{\overline{E}}_{t}}\big{)}.$ (2.12)

From the definition of $A_{s}$ in (2.3), it follows from Hölder inequality
that

$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t}-s)e^{nF_{t}}\omega_{X}^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t}-s)^{\frac{(n+1)p}{n}}e^{nF_{t}}\omega^{n}_{X}\Big{)}^{\frac{n}{(n+1)p}}\cdot\Big{(}\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}e^{nF_{t}}\omega_{X}^{n}\Big{)}^{1/q}$
$\displaystyle\leq$ $\displaystyle
A_{s}^{\frac{1}{n+1}}\Big{(}\frac{c_{t}^{n}}{V_{t}}2^{p}\beta_{0}^{-p}\big{(}\|e^{nF_{t}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{\overline{E}}_{t}}\big{)}\Big{)}^{\frac{n}{(n+1)p}}\cdot\Big{(}\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}e^{nF_{t}}\omega_{X}^{n}\Big{)}^{1/q}$

where $q>1$ satisfies $\frac{n}{p(n+1)}+\frac{1}{q}=1$, i.e.
$q=\frac{p(n+1)}{p(n+1)-n}$. The inequality above yields

$A_{s}\leq\Big{(}\frac{c_{t}^{n}}{V_{t}}2^{p}\beta_{0}^{-p}\big{(}\|e^{nF_{t}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{\overline{E}}_{t}}\big{)}\Big{)}^{1/p}\cdot\Big{(}\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}e^{nF_{t}}\omega_{X}^{n}\Big{)}^{\frac{1+n}{qn}}.$
(2.13)

Observe that the exponent of the integral on the right hand of (2.13)
satisfies

$\frac{1+n}{qn}=\frac{pn+p-n}{pn}=1+\delta_{0}>1,$

for $\delta_{0}:=\frac{p-n}{pn}>0$. For notation convenience, set

$B_{0}:=\Big{(}\frac{c_{t}^{n}}{V_{t}}2^{p}\beta_{0}^{-p}\big{(}\|e^{nF_{t}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{\overline{E}}_{t}}\big{)}\Big{)}^{1/p}.$ (2.14)

From (2.13) we then get

$A_{s}\leq
B_{0}\Big{(}\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}e^{nF_{t}}\omega_{X}^{n}\Big{)}^{1+\delta_{0}}.$
(2.15)

For any $r\in[0,1]$, we note that $-\varphi_{t}-s\geq r$ on
$\Omega_{s+r}=\\{\varphi_{t}\leq-s-r\\}$. Thus

$A_{s}=\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t}-s)e^{nF_{t}}\omega_{X}^{n}\geq
r\cdot\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s+r}}e^{nF_{t}}\omega_{X}^{n}.$
(2.16)

If we define $\phi:{\bf R}_{+}\to{\bf R}_{+}$ by

$\phi(s):=\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}e^{nF_{t}}\omega_{X}^{n}$

then (2.15) and (2.16) imply that

$r\phi(s+r)\leq B_{0}\phi(s)^{1+\delta_{0}},\quad\forall r\in[0,1]\mbox{ and
}s\geq 0.$ (2.17)

$\phi$ is clearly nonincreasing and continuous, so the lemma below applies to
$\phi$. It is a classic lemma due to De Giorgi, which was also used in [14,
12]. We include a sketch of the proof for the readers’ convenience, and to
exhibit the dependence of $\|\varphi_{t}\|_{L^{\infty}}$ on the given data.

###### Lemma 2

Let $\phi:{\bf R}_{+}\to{\bf R}_{+}$ be a decreasing right-continuous function
with $\lim_{s\to\infty}\phi(s)=0$. Assume that $r\phi(s+r)\leq
B_{0}\phi(s)^{1+\delta_{0}}$ for some constant $B_{0}>0$ and all $s>0$ and
$r\in[0,1]$. Then there exists some
$S_{\infty}=S_{\infty}(\delta_{0},B_{0},\phi)>0$ such that $\phi(s)=0$ for all
$s\geq S_{\infty}$.

Proof. Fix an $s_{0}>0$ such that $\phi(s_{0})^{\delta_{0}}<\frac{1}{2B_{0}}$.
This $s_{0}$ exists since $\phi(s)\to 0$ as $s\to\infty$. Define an increasing
sequence $(s_{j})$ of positive real numbers inductively by

$s_{j+1}:=\sup\\{s>s_{j}|~{}\phi(s)>\frac{1}{2}\phi(s_{j})\\}.$

If at some stage $\phi(s_{j})=0$, we stop there. By the right-continuity of
$\phi$, it follows that $\phi(s_{j+1})\leq\frac{\phi(s_{j})}{2}$ and
$s_{j+1}\leq 1+s_{j}$ since $\phi(1+s_{j})\leq\frac{1}{2}\phi(s_{j})$. It
follows from the assumptions on $\phi$ that $s_{j+1}-s_{j}\leq
2^{-j\delta_{0}}$, which implies that the sequence $(s_{j})$ converges to

$S_{\infty}=s_{0}+\sum_{j\geq 0}(s_{j+1}-s_{j})\leq
s_{0}+\frac{1}{1-2^{-\delta_{0}}}.$

It is then clear that $\phi(s)=0$ for any $s>S_{\infty}$. The lemma is proved.

We return now to the proof of Theorem 2. By Chebyshev’s inequality, we have

$\phi(s)=\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}e^{nF_{t}}\omega_{X}^{n}\leq\frac{1}{s}\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t})e^{nF_{t}}\omega_{X}^{n}\leq\frac{{\overline{E}}_{t}}{s}\to
0\mbox{ as }s\to\infty.$

Thus we may choose $s_{0}=(2B_{0})^{1/\delta_{0}}{\overline{E}}_{t}$ in the
proof of Lemma 2. By (2.17) and Lemma 2, we deduce that

$\Omega_{S_{\infty}}=\\{\varphi_{t}\leq-S_{\infty}\\}=\emptyset,$

so hence

$\inf_{X}\varphi_{t}\geq-
S_{\infty}=-(2B_{0})^{1/\delta_{0}}{\overline{E}}_{t}-\frac{1}{1-2^{-\delta_{0}}},$
(2.18)

where $B_{0}$ is the constant in (2.14) and $\delta_{0}:=\frac{p-n}{pn}>0$
depends only on $n$ and $p$. The proof of Theorem 2 is complete.

##  3 Proof of Theorem 1

Let $\varphi$ solve the equation (1.1), where $F$ is a given smooth function
and $f(\lambda[\cdot])$ is the nonlinear operator as introduced in Section 1.
The setting of Theorem 1 with a fixed background $\omega_{X}$ can be viewed as
a special case of the setting of Theorem 2 with $\chi=0$, and $t$ taken to be
$1$, $\omega_{1}=\omega_{X}$, and in the notations (1.7) and (1.1) for the two
settings,

$\displaystyle e^{F}=c_{1}e^{F_{1}},\quad V=V_{1}.$ (3.1)

We observe that, in view of the normalization ${1\over
V}\int_{X}e^{nF_{1}}\omega_{X}^{n}=1$ for $F_{1}$,

$\displaystyle c_{1}^{n}={1\over V}\int_{X}e^{nF}\omega_{X}^{n}\leq{\rm
Ent}_{p}(F)+e^{n}$ (3.2)

for any $p\geq 1$. Thus, applying Theorem 2 and assuming that ${\rm Ent}_{p}$
is bounded, we find that $L^{\infty}$ bounds for $\varphi$ would follow if we
can control the energy $E=E_{t=1}$. However, an easy application of Hölder’s
inequality gives

$E(\varphi)=\frac{1}{V}\int_{X}(-\varphi)e^{nF}\omega_{X}^{n}\leq\Big{(}\frac{1}{V}\int_{X}(-\varphi)^{\frac{n}{n-1}}e^{nF}\omega_{X}^{n}\Big{)}^{(n-1)/n}\leq
C$ (3.3)

so it suffices to control the right hand side. This is done in the following
theorem, part (c), which completes the proof of Theorem 1:

###### Theorem 3

If $f(\lambda[\cdot])$ satisfies the structure condition (1.4), then the
following holds:

(a) Assume that $p\in(0,n)$. Then there exist constants $c_{p}$, $C_{p}>0$
depending only on $\omega_{X}$, $n$, $p$, $\gamma$ and the generalized entropy
${\mathrm{Ent}}_{p}(F)$ such that

$\displaystyle\int_{X}{\rm exp}\big{\\{}c_{p}(-\varphi)^{n\over
n-p}\big{\\}}\omega_{X}^{n}\leq C_{p}.$ (3.4)

(b) Assume that $p=n$. Then for any $N>0$, there exists constants $c_{N}>0$,
$C_{N}>0$ depending on $n,\omega_{X}$, $N$, $\gamma$, and the generalized
entropy ${\mathrm{Ent}}_{n}(F)$ so that

$\displaystyle\int_{X}{\rm
exp}\big{\\{}c_{N}(-\varphi)^{N}\big{\\}}\omega_{X}^{n}\leq C_{N}.$ (3.5)

(c) We have the energy estimate:

$\displaystyle\int_{X}(-\varphi)^{N}e^{nF}\omega_{X}^{n}\leq C$ (3.6)

for $N={n\over n-p}$ if $p\in[1,n)$, and for any $N>0$ if $p=n$, where the
constant $C$ on the right hand side of (3.6) depends on
$n,\omega_{X},\gamma,N$ and the entropy ${\mathrm{Ent}}_{p}(F)$.

We observe that, in the special case of the Monge-Ampère equation and when
$p=1$, these estimates have been established in [10], using pluripotential
theory. Further if $p>n$, then Theorem 1 implies the solutions are
$L^{\infty}$ bounded. Theorem 3 gives a more complete integral estimate of
such solutions for the full range of $p\in(0,\infty)$, using pure PDE methods.

For the proof of Theorem 1, we only need part (c) of Theorem 3, but we give
the proofs of the other parts as well, as they are of independent interest. As
mentioned in the Introduction, for this we need the ABP method developed in
[5] and [7] in order to get better integral bounds for $\varphi$ as stated in
(3.4) and (3.5). In particular, we have fix a background metric $\omega_{X}$,
as this method is not effective for handling degenerating families.

Suppose $p\in(0,n]$ and write

$\Psi_{p}:=\frac{1}{V}\int_{X}(F^{2}+1)^{p/2}e^{nF}\omega_{X}^{n}\sim\frac{1}{V}\int_{X}|F|^{p}e^{nF}\omega_{X}^{n}={\mathrm{Ent}}_{p}(F).$

We can solve the complex Monge-Ampère equation

$(\omega_{X}+i\partial\bar{\partial}\psi)^{n}=\frac{(F^{2}+1)^{p/2}}{\Psi_{p}}e^{nF}\omega_{X}^{n},\quad\sup_{X}\psi=0.$
(3.7)

###### Lemma 3

Let $\varphi,\psi$ be the solutions to the equations (1.1), (3.7),
respectively. There exist constants $\Lambda,\lambda,C>0$ depending on
$n,\omega_{X},\gamma,p,$ and $\Psi_{p}$ such that

$\sup_{X}(-(-\psi+\Lambda)^{\beta}-\lambda\varphi)\leq C$

where $\beta=\frac{n-p}{n}\in(0,1)$ if $p<n$, and if $p=n$, we choose an
arbitrary $\beta=N^{-1}\in(0,1)$ and the constants $\lambda,\Lambda,C$ depend
additionally on $N$.

In the proof below, for a smooth function $u$ on $X$, we denote

$\Box u:=G^{i\bar{j}}u_{\bar{j}i},\quad|\nabla
u|_{G}^{2}:=G^{i\bar{j}}\nabla_{\bar{j}}u\nabla_{i}u$ (3.8)

and

${\rm tr}_{G}\alpha:=G^{i\bar{j}}\alpha_{\bar{j}i},\quad\mbox{ for a smooth
$(1,1)$-form $\alpha=\alpha_{\bar{j}i}\sqrt{-1}dz^{i}\wedge d\bar{z}^{j}$},$

where as in Lemma 1, $G^{i\bar{j}}=\frac{\partial\,{\rm
log}\,f(\lambda[h])}{\partial h_{ij}}$ is the coefficient matrix of the
linearized operator of $\,{\rm log}\,f(\lambda[\cdot])$ at
$h=\omega_{X}^{-1}\cdot\omega_{\varphi}$, and by the structure condition (1.4)
on $f$, we have ${\rm det}(G^{i\bar{j}})\geq\gamma f^{-n}$ and $G^{i\bar{j}}$
is positive definite.

Proof of Lemma 3. We choose constants as follows:

$\Lambda=\Big{(}\frac{4^{n}}{n^{n}\gamma}\frac{2^{p}}{\alpha_{0}^{p}}\Psi_{p}\Big{)}^{1/n(1-\beta)},\quad\lambda=4\beta\Lambda^{-(1-\beta)}$
(3.9)

where as usual $\alpha_{0}=\alpha_{0}(X,\omega_{X})$ is a fixed constant
smaller than the $\alpha$-invariant of $(X,\omega_{X})$. We denote

$\rho:=-(-\psi+\Lambda)^{\beta}-\lambda\varphi.$ (3.10)

For notation convenience, we set $\phi_{\delta}(t)=t+\sqrt{t^{2}+\delta}>0$,
which is a smoothing of $2\max(t,0)$ and converges to it as $\delta\to 0$.
Here we will first fix a $\delta>0$ small and later on $\delta$ will be sent
to zero. All constants appearing in the proof are independent of $\delta$,
unless stated otherwise. From now on we will consider $\phi_{\delta}(\rho)$
which is monotone decreasing and converges to $2\rho_{+}$ as $\delta\to 0$. We
will omit the $\delta$ in $\phi_{\delta}(\rho)$ and simply write $\phi(\rho)$.

We define a smooth function

$H=\phi(\rho)^{b},$

where $b=1+\frac{1}{4n}>1$ is constant. Since $X$ is compact, $H$ must achieve
its maximum at some point in $X$, say, $x_{0}$, and we denote $\sup_{X}H=:M>0$
(if $M=0$ there is nothing to prove). Let $r=\min\\{1,r(X,\omega_{X})\\}$
where $r(X,\omega_{X})>0$ is the injectivity radius of $(X,\omega_{X})$ viewed
as a compact Riemannian manifold. So we can identify the geodesic ball
$B_{r}(x_{0})$ as an open smooth domain in ${\bf R}^{2n}$ with Euclidean
diameter bounded by $3r$, say. Let $\theta\in(0,1)$ be a small constant
defined by

$\theta:=\min\\{\frac{r^{2}\beta\Lambda^{-(1-\beta)}}{100M^{1/b}},\frac{r^{2}}{100n}\\}<\frac{1}{10}.$
(3.11)

As in [7], we choose an auxiliary smooth function $\eta$ defined on
$B_{r}(x_{0})$ so that $\eta\equiv 1$ on $B_{r/2}(x_{0})$ and $\eta\equiv
1-\theta$ on $X\backslash B_{3r/4}(x_{0})$, and $\eta$ lies between $1$ and
$1-\theta$ in the annulus $B_{3r/4}(x_{0})\backslash B_{r/2}(x_{0})$. Moreover
$\eta$ can be chosen to satisfy

$\displaystyle|\nabla\eta|_{g}^{2}\leq\frac{10\theta^{2}}{r^{2}},\quad|\nabla^{2}\eta|_{g}\leq\frac{10\theta}{r^{2}},$
(3.12)

where we identify $\omega_{X}$ with its associated Riemannian metric $g$.

We can now calculate,

$\Box(H\eta)=\eta\Box
H+H\Box\eta+2Re\big{(}G^{i\bar{j}}\nabla_{\bar{j}}H\nabla_{i}\eta\big{)}.$
(3.13)

We observe that the middle term in (3.13) satisfies

$H\Box\eta=H{\rm
tr}_{G}i\partial\bar{\partial}\eta\geq-H\frac{10\theta}{r^{2}}{\rm
tr}_{G}\omega_{X}.$

The last term in (3.13) satisfies

$\displaystyle 2Re\big{(}G^{i\bar{j}}\nabla_{\bar{j}}H\nabla_{i}\eta\big{)}$
$\displaystyle=$ $\displaystyle
2b\phi(\rho)^{b-1}Re\big{(}G^{i\bar{j}}\nabla_{\bar{j}}\phi(\rho)\nabla_{i}\eta\big{)}$
$\displaystyle\geq$
$\displaystyle-\frac{b(b-1)}{2}\phi(\rho)^{b-2}|\nabla\phi(\rho)|^{2}_{G}-\frac{2b}{b-1}\phi(\rho)^{b}|\nabla\eta|^{2}_{G}$
$\displaystyle\geq$
$\displaystyle-\frac{b(b-1)}{2}\phi(\rho)^{b-2}|\nabla\phi(\rho)|^{2}_{G}-\frac{2b}{b-1}\phi(\rho)^{b}\frac{10\theta^{2}}{r^{2}}{\rm
tr}_{G}\omega_{X}$

where in the first inequality we applied the Cauchy-Schwarz inequality. The
first term in (3.13) is

$\displaystyle\eta\Box H$ $\displaystyle=$ $\displaystyle
b\eta\phi(\rho)^{b-1}\Box\phi(\rho)+b(b-1)\eta\phi(\rho)^{b-2}|\nabla\phi(\rho)|_{G}^{2}$
$\displaystyle=$ $\displaystyle
b\eta\phi(\rho)^{b-1}\phi^{\prime}(\rho)\Box\rho+b\eta\phi(\rho)^{b-1}\phi^{\prime\prime}(\rho)|\nabla\rho|^{2}_{G}+b(b-1)\eta\phi(\rho)^{b-2}|\nabla\phi(\rho)|_{G}^{2}.$

We note that the middle term in (3) is nonnegative due to the fact that

$\phi^{\prime\prime}(t)=\frac{1}{\sqrt{t^{2}+\delta}}-\frac{t^{2}}{(\sqrt{t^{2}+\delta})^{3}}=\frac{\delta}{(t^{2}+\delta)^{3/2}}>0.$

To deal with the first term in (3) we note by the homogeneity of degree one
assumption on $f$ that

$\Box\varphi={\rm tr}_{G}i\partial\bar{\partial}\varphi={\rm
tr}_{G}\omega_{\varphi}-{\rm tr}_{G}\omega_{X}=1-{\rm tr}_{G}\omega_{X}.$

Then we calculate

$\displaystyle\Box\rho$ $\displaystyle=$
$\displaystyle\Box(-(-\psi+\Lambda)^{\beta}-\lambda\varphi)$ (3.15)
$\displaystyle=$
$\displaystyle\beta(-\psi+\Lambda)^{\beta-1}\Box\psi+\beta(1-\beta)(-\psi+\Lambda)^{\alpha-2}|\nabla\psi|_{G}^{2}-\lambda\Box\varphi$
$\displaystyle\geq$ $\displaystyle\beta(-\psi+\Lambda)^{\beta-1}{\rm
tr}_{G}\omega_{\psi}-\beta(-\psi+\Lambda)^{\beta-1}{\rm
tr}_{G}\omega_{X}-\lambda+\lambda{\rm tr}_{G}\omega_{X}$ $\displaystyle\geq$
$\displaystyle
n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{(F^{2}+1)^{p/2}}{\Psi_{p}}\big{)}^{1/n}+(\lambda-\beta(-\psi+\Lambda)^{\beta-1}){\rm
tr}_{G}\omega_{X}-\lambda$ $\displaystyle\geq$ $\displaystyle
n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}+(\lambda-\beta\Lambda^{\beta-1}){\rm
tr}_{G}\omega_{X}-\lambda,$

where in the second inequality we used the arithmetic-geometric inequality and
the equations for $\varphi$ and $\psi$. Plugging these inequalities into
(3.13), we obtain

$\displaystyle\Box(H\eta)$ $\displaystyle\geq$
$\displaystyle-\frac{10\theta}{r^{2}}\phi(\rho)^{b}{\rm
tr}_{G}\omega_{X}-\frac{2b}{b-1}\phi(\rho)^{b}\frac{10\theta^{2}}{r^{2}}{\rm
tr}_{G}\omega_{X}$
$\displaystyle+b\eta\phi(\rho)^{b-1}\phi^{\prime}(\rho)\Big{(}n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}+(\lambda-\beta\Lambda^{\beta-1}){\rm
tr}_{G}\omega_{X}-\lambda\Big{)}$ $\displaystyle\geq$ $\displaystyle
b\phi(\rho)^{b-1}\Big{\\{}\frac{9}{10}(\lambda-\beta\Lambda^{\beta-1})\phi^{\prime}(\rho){\rm
tr}_{G}\omega_{X}-\frac{20\theta}{r^{2}b}\phi(\rho){\rm tr}_{G}\omega_{X}$
$\displaystyle-\frac{20\theta^{2}}{(b-1)r^{2}}\phi(\rho){\rm
tr}_{G}\omega_{X}+n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}-\lambda\phi^{\prime}(\rho)\Big{\\}},$

where we ignored $\eta$ in the last inequality since $\eta>9/10$. To deal with
the right hand side in the equation (3), we note that on the set $\\{\rho\leq
0\\}$

$0\leq\phi(\rho)=\rho+\sqrt{\rho^{2}+\delta}=\frac{\delta}{\sqrt{\rho^{2}+\delta}-\rho}\leq\sqrt{\delta},$

and on this same set the function $\phi^{\prime}(\cdot)$ satisfies

$1\geq\phi^{\prime}(\rho)=1+\frac{\rho}{\sqrt{\rho^{2}+\delta}}=\frac{\phi(\rho)}{\sqrt{\rho^{2}+\delta}}\geq
0.$

So on the set $\\{\rho\leq 0\\}$ the right hand side of (3) is greater or
equal to

$\displaystyle
b\phi(\rho)^{b-1}\Big{(}-\frac{20\theta}{r^{2}b}\sqrt{\delta}{\rm
tr}_{G}\omega_{X}-\frac{20\theta^{2}}{(b-1)r^{2}}\sqrt{\delta}{\rm
tr}_{G}\omega_{X}-\lambda\Big{)}$

On the other hand, on the set $\\{\rho>0\\}$, we know
$2\geq\phi^{\prime}(\rho)>1$, so the first three terms on the right hand side
of (3) are positive due to the choice of $\theta$ in (3.11), and
$\Lambda,\lambda$ in (3.9) and the fact that $\phi(\rho)\leq M^{1/b}$. So on
the set $\\{\rho>0\\}$ the right hand side of (3) is greater or equal to

$b\phi(\rho)^{b-1}\Big{(}n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}-\lambda\Big{)}.$

Combining the above two cases, we obtain

$\displaystyle\Box(H\eta)$ $\displaystyle\geq$ $\displaystyle
b\phi(\rho)^{b-1}\Big{(}n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}-\lambda\Big{)}\cdot\chi_{\\{\rho>0\\}}$
$\displaystyle-b\phi(\rho)^{b-1}\Big{(}\frac{20\theta}{r^{2}b}\sqrt{\delta}{\rm
tr}_{G}\omega_{X}+\frac{20\theta^{2}}{(b-1)r^{2}}\sqrt{\delta}{\rm
tr}_{G}\omega_{X}+\lambda\Big{)}\cdot\chi_{\\{\rho\leq 0\\}},$

where $\chi_{E}$ denotes the characteristic function of a set $E$.

We now apply the ABP maximum principle to the function $H\eta$ on the domain
$B_{r}(x_{0})=:B_{0}$ in ${\bf R}^{2n}$. It follows that

$\displaystyle\sup_{B_{0}}(H\eta)$ $\displaystyle\leq$
$\displaystyle\sup_{\partial
B_{0}}(H\eta)+C(n)r\Big{\\{}\int_{B_{0}\cap\\{\rho>0\\}}\frac{\phi(\rho)^{2n(b-1)}(n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}-\lambda\big{)}_{-}^{2n}}{({\rm
det}G^{i\bar{j}})^{2}}\omega_{X}^{n}$ (3.17)
$\displaystyle+\int_{B_{0}\cap\\{\rho\leq
0\\}}\frac{\phi(\rho)^{2n(b-1)}(\frac{20\theta}{r^{2}b}\sqrt{\delta}{\rm
tr}_{G}\omega_{X}+\frac{20\theta^{2}}{(b-1)r^{2}}\sqrt{\delta}{\rm
tr}_{G}\omega_{X}+\lambda)^{2n}}{({\rm
det}G^{i\bar{j}})^{2}}\omega_{X}^{n}\Big{\\}}^{1/2n}$ $\displaystyle\leq$
$\displaystyle\sup_{\partial
B_{0}}(H\eta)+C(n)\Big{\\{}\int_{B_{0}\cap\\{\rho>0\\}}\frac{\sqrt{\phi(\rho)}\big{(}n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}-\lambda\big{)}_{-}^{2n}}{e^{-2nF}}\omega_{X}^{n}$
$\displaystyle+C^{\prime}\delta^{n(b-1)}\Big{\\}}^{1/2n}$

where the constant $C^{\prime}=C^{\prime}(n,F,G,\omega_{X})$ in the last term
may not be uniformly bounded, but this is not a concern, since later on we
will let $\delta\to 0$. We observe that the integral involved in the last
inequality is in fact integrated over the set where
$n\gamma^{1/n}\beta(-\psi+\Lambda)^{\beta-1}\big{(}\frac{|F|^{p}}{\Psi_{p}}\big{)}^{1/n}-\lambda<0$
and $\rho>0$, and over this set we have from the choice of constants in (3.9)

$|F|\leq(\Psi_{p})^{1/p}\Big{(}\frac{\lambda}{n\gamma^{1/n}\beta}\Big{)}^{n/p}(-\psi+\Lambda)^{(1-\beta)n/p}=\frac{\alpha_{0}}{2}(-\psi+\Lambda)^{(1-\beta)n/p}.$

At the same time, on the same set, we have $0<\rho\leq-\lambda\varphi$ and
$\phi(\rho)\leq 2\rho+\sqrt{\delta}$. Therefore, we obtain from (3.17) that

$\displaystyle M$ $\displaystyle\leq$ $\displaystyle(1-\theta)\sup_{\partial
B_{0}}H+C\Big{(}\int_{B_{0}}(-\varphi+\sqrt{\delta})^{1/2}\,{\rm
exp}\,\Big{(}\frac{\alpha_{0}}{2}(-\psi+\Lambda)^{(1-\beta)n/p}\Big{)}\omega_{X}^{n}+C^{\prime}\delta^{n(b-1)}\Big{)}^{1/2n}$
(3.18) $\displaystyle\leq$
$\displaystyle(1-\theta)M+C\Big{(}\int_{B_{0}}\left(-\varphi+\,{\rm
exp}\,\Big{(}{\alpha_{0}}(-\psi+\Lambda)^{(1-\beta)n/p}\Big{)}\right)\omega_{X}^{n}+C^{\prime}\delta^{n(b-1)}\Big{)}^{1/2n}$
$\displaystyle\leq$
$\displaystyle(1-\theta)M+C_{0}+C^{\prime}\delta^{(b-1)/2}$

where $C_{0}=C_{0}(n,\omega_{X},\gamma,\Psi_{p},p)>0$ is independent of
$\delta$ and in the last inequality we have used the following inequalities:

(1) $\int_{X}(-\varphi)\omega_{X}^{n}\leq C(n,\omega_{X})$ which follows from
the Green’s formula, $n+\Delta_{\omega_{X}}\varphi>0$ (since
$\lambda[h_{\varphi}]\in\Gamma\subset\\{\lambda_{1}+\cdots+\lambda_{n}>0\\}$)
and the normalization condition $\sup_{X}\varphi=0$.

(2) $\int_{X}\,{\rm
exp}\,\Big{(}{\alpha_{0}}(-\psi+\Lambda)^{(1-\beta)n/p}\Big{)}\omega_{X}^{n}\leq
C(n,\omega_{X},\Psi_{p},p)$. By the choice of $\beta$, if $p<n$,
$(1-\beta)n/p=1$; and if $p=n$, then $(1-\beta)n/p=1-N^{-1}<1$, Young’s
inequality gives the desired estimate.

Hence we conclude that with the choice of $\theta$ in (3.11)

$\min(M^{1-\frac{1}{b}},M)\leq
C_{0}+C^{\prime}\delta^{(b-1)/2}\,\,\Rightarrow\,M\leq
C_{0}+C^{\prime}\delta^{2b},$

which implies that

$\sup_{X}2\rho_{+}\leq\sup_{X}\phi(\rho)=M\leq C_{0}+C^{\prime}\delta^{2b}.$

Finally letting $\delta\to 0$ yields the desired estimate
$\sup_{X}\rho_{+}\leq C_{0}$ for some positive constant
$C_{0}=C_{0}(n,\omega_{X},\gamma,\Psi_{p})$ (which may be different from the
$C_{0}$ in (3.18)). The proof of Lemma 3 is complete.

Proof of Theorem 3. (a) and (b) follow easily from Lemma 3 and the fact the
$\omega_{X}$-plurisubharmonic function $\psi$ satisfies
$\int_{X}e^{-\alpha_{0}\psi}\omega_{X}^{n}\leq C(n,\omega_{X})$.

The inequality (3.6) in (c) is an immediate consequence of the estimates in
(a) and (b), and Jensen’s inequality. More precisely, we have

$\frac{1}{V}\int_{X}e^{-nF+c_{0}(-\varphi)^{N}}e^{nF}\omega_{X}^{n}=\frac{1}{V}\int_{X}e^{c_{0}(-\varphi)^{N}}\omega_{X}^{n}\leq
C(n,\omega_{X},\gamma,\Psi_{p},N).$

Taking the logarithms of both sides and applying Jensen’s inequality yields

$\frac{1}{V}\int_{X}\big{(}c_{0}(-\varphi)^{N}-nF\big{)}e^{nF}\omega_{X}^{n}\leq
C(n,\omega_{X},\gamma,\Psi_{p},N),$

from which (3.6) follows after noting that $\Psi_{p}$ is equivalent to the
entropy ${\mathrm{Ent}}_{p}$ and if $p\geq 1$

$\frac{1}{V}\int_{X}Fe^{nF}\omega_{X}^{n}\leq{\mathrm{Ent}}_{p}.$

The proof of Theorem 3 is complete.

##  4 Monge-Ampère equations

In this and the next section, we apply Theorems 1 and 2 to the specific cases
of the Monge-Ampère and Hessian equations on a compact Kähler manifold
$(X,\omega_{X})$.

We begin by noting that the structural condition (1.4) holds for many
equations and is usually easy to check:

###### Lemma 4

Assume that $f:{\bf R}^{n}\to{\bf R}_{+}$ is a concave and homogeneous
function of degree one, which satisfies $\frac{\partial
f(\lambda)}{\partial\lambda_{j}}>0$ for any $\lambda$ in an admissible cone
$\Gamma\subset{\bf R}^{n}$. Assume that there is a $\gamma>0$ such that

$f(\mu)\geq n\gamma^{1/n}(\prod_{j}\mu_{j})^{1/n},\quad\mbox{for all
}\mu\in\Gamma_{n}:=\\{\lambda\in{\bf
R}^{n}:\lambda_{1}>0,\ldots,\lambda_{n}>0\\}.$ (4.1)

Then $f$ satisfies the structural condition (1.4).

Proof. By the concavity of $f$ on $\Gamma$, for any $\lambda,\mu\in\Gamma$ we
have

$f(\mu)\leq f(\lambda)+\sum_{j=1}^{n}(-\lambda_{j}+\mu_{j})\frac{\partial
f(\lambda)}{\partial\lambda_{j}}=\sum_{j=1}^{n}\mu_{j}\frac{\partial
f(\lambda)}{\partial\lambda_{j}},$ (4.2)

where we have used the homogeneity of degree one assumption on $f$, which
implies that $\sum_{j}\lambda_{j}\frac{\partial
f(\lambda)}{\partial\lambda_{j}}=f(\lambda)$. Taking the infimum of the right
hand side of (4.2) over all $\mu\in\Gamma_{n}$ with
$\prod_{j=1}^{n}\mu_{j}=1$, by the arithmetic-geometric inequality and the
assumption (4.1) on $f$, we get

$\prod_{j=1}^{n}\frac{\partial f(\lambda)}{\partial\lambda_{j}}\geq
n^{-n}\big{\\{}\inf_{\mu\in\Gamma_{n},\prod_{j}\mu_{j}=1}f(\mu)\big{\\}}^{n}\geq\gamma>0.$

The desired inequality (1.4) follows from this inequality upon diagonalizing
the matrix $h$. The proof of Lemma 4 is complete.

It follows immediately from this lemma that the functions
$f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$,
$f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$, and
$f(\lambda)=({\sigma_{k}(\lambda)\over\sigma_{\ell}(\lambda)})^{1\over
k-\ell}+c\sigma_{p}(\lambda)^{\frac{1}{p}}$ for $c>0$, $n\geq k\geq\ell\geq
1$, $n\geq p\geq 1$, corresponding respectively to the Monge-Ampère, the
Hessian, and the quotient Hessian equations, all satisfy the structural
condition (1.4), and the constant $\gamma$ depends only on the given numbers
$n,k,\ell,p,c>0$, and the admissible cone $\Gamma=\Gamma_{k}=\\{\lambda\in{\bf
R}^{n}:\sigma_{1}(\lambda)>0,\ldots,\sigma_{k}(\lambda)>0\\}$. The last
equation appeared recently in [8].

In this section, we focus on the Monge-Ampère equation. To discuss the
underlying geometry, it is convenient to rewrite it in the more usual form

$\displaystyle(\omega_{t}+i\partial\bar{\partial}\varphi_{t})^{n}=c_{t}^{n}e^{nF_{t}}\omega_{X}^{n},\quad\lambda[h_{t,\varphi_{t}}]\in\Gamma,\quad\sup_{X}\varphi_{t}=0,$
(4.3)

where $\omega_{t}=t\omega_{X}+\chi$ is the family of degenerating background
metrics. To apply Theorem 2, we need to control the ratio $c_{t}^{n}/V_{t}$
and the energy $E_{t}$. This can be readily done using the following two easy
lemmas:

###### Lemma 5

Let $V=\int_{X}\omega_{X}^{n}$ be the volume of $(X,\omega_{X})$, then

$V^{-1}=\frac{c_{t}^{n}}{V_{t}},\quad\forall t\in(0,1].$

Proof. Integrating both sides of (4.3), we get

$c_{t}^{n}V=c_{t}^{n}\int_{X}e^{nF_{t}}\omega_{X}^{n}=\int_{X}(\omega_{t}+i\partial\bar{\partial}\varphi_{t})^{n}=\int_{X}\omega_{t}^{n}=V_{t}.$

###### Lemma 6

There is a uniform constant $C>0$ depending only on
$n,\|e^{nF_{t}}\|_{L^{1}(\,{\rm log}\,L)^{1}(\omega_{X}^{n})}$, $\omega_{X}$,
$\chi$ such that for all $t\in(0,1]$

$\displaystyle E_{t}(\varphi_{t})\leq C.$ (4.4)

Proof. Recall that

$E_{t}(\varphi_{t})=\frac{c_{t}^{n}}{V_{t}}\int_{X}(-\varphi_{t})e^{nF_{t}}\omega_{X}^{n}=\frac{1}{V_{t}}\int_{X}(-\varphi_{t})\omega_{\varphi_{t}}^{n},$

so that it suffices to show that
$\frac{1}{V_{t}}\int_{X}(-\varphi_{t})\omega_{\varphi_{t}}^{n}\leq C$,
$\forall t\in(0,1].$ We may assume without loss of generality that
$\chi\leq\omega_{X}$, so the $\omega_{t}$-plurisubharmonic function
$\varphi_{t}$ is also $2\omega_{X}$-plurisubharmonic and by the
$\alpha$-invariant estimate, there is an
$\alpha_{0}=\alpha_{0}(X,\omega_{X})>0$ such that

$\frac{1}{V_{t}}\int_{X}\,{\rm exp}\,\Big{(}-\,{\rm
log}\,\frac{\omega_{\varphi_{t}}^{n}}{\omega_{X}^{n}}-\alpha_{0}\varphi_{t}\Big{)}\omega_{\varphi_{t}}^{n}=\frac{1}{V_{t}}\int_{X}e^{-\alpha_{0}\varphi_{t}}\omega_{X}^{n}\leq\frac{C(n,\omega_{X})}{V_{t}}$

By Jensen’s inequality it follows that

$\frac{1}{V_{t}}\int_{X}\Big{(}-\,{\rm
log}\,\frac{\omega_{\varphi_{t}}^{n}}{\omega_{X}^{n}}-\alpha_{0}\varphi_{t}\Big{)}\omega_{\varphi_{t}}^{n}\leq\,{\rm
log}\,C-\,{\rm log}\,V_{t}$

which implies that

$\displaystyle\frac{1}{V_{t}}\int_{X}(-\alpha_{0}\varphi_{t})\omega_{\varphi_{t}}^{n}$
$\displaystyle\leq$ $\displaystyle\frac{1}{V_{t}}\int_{X}\,{\rm
log}\,(e^{nF_{t}}c_{t}^{n})\omega_{\varphi_{t}}^{n}+\,{\rm log}\,C-\,{\rm
log}\,V_{t}$ $\displaystyle=$
$\displaystyle\int_{X}(nF_{t})e^{nF_{t}}\frac{c_{t}^{n}}{V_{t}}\omega_{X}^{n}+\,{\rm
log}\,C+\,{\rm log}\,\frac{c^{n}_{t}}{V_{t}}$ $\displaystyle\leq$
$\displaystyle C\|e^{nF_{t}}\|_{L^{1}(\,{\rm log}\,L)^{1}(\omega_{X})}+C,$

from which the estimate follows since $c^{n}_{t}$ and $V_{t}$ are uniformly
equivalent by Lemma 5.

Theorem 2 together with the preceding lemmas implies at once the following
basic estimates of Kolodziej [14], Eyssidieux, Guedj, and Zeriahi [12], and
Demailly and Pali [9]:

###### Theorem 4

Consider the above family (4.3) of complex Monge-Ampère equations, with
respect to the degenerating background metrics $\omega_{t}$, $t\in(0,1]$. Fix
any $q>n$. If $\varphi_{t}$ is a family of $C^{2}$ solution, normalized by
$\sup_{X}\varphi_{t}=0$, and if $\|e^{nF_{t}}\|_{L^{1}(\,{\rm
log}\,L)^{q}(\omega_{X}^{n})}$ is uniformly bounded in $t$, then
$\|\varphi_{t}\|_{L^{\infty}(X)}$ is uniformly bounded in $t$ as well.

In many applications, the $(1,1)$-form $\chi$ is chosen to be
$\chi=\pi^{*}\omega_{Y}$, where $\pi:X\to Y$ is a holomorphic map between
Kähler manifolds and $\omega_{Y}$ is a Kähler metric on $Y$.

##  5 Fully non-linear and Hessian equations

In this section, we consider applications of Theorem 2 to fully non-linear
equations besides Monge-Ampère equations. For these applications, we need to
consider the relative volumes $c_{t}^{n}V_{t}^{-1}$ and the energies
$E_{t}(\varphi_{t})$. We begin with a simple estimate for
$E_{t}(\varphi_{t})$, which generalizes the simple considerations which
applied earlier to Monge-Ampère equations and is a straightforward application
of the Hölder inequality,

###### Lemma 7

Consider the energy $E_{t}(\varphi_{t})$ in the formalism for degenerating
background metrics as in the set-up (1.7). Then we have for any $p,q>1$ and
${1\over p}+{1\over q}=1$,

$\displaystyle E_{t}(\varphi_{t})\leq{c_{t}^{n}\over
V_{t}}\,\|e^{nF_{t}}\|_{L^{q}}\,\|\varphi_{t}\|_{L^{p}}.$ (5.1)

Next we note the following uniform $L^{p}$ estimate for general functions $u$,
whose Hessian is in a cone $\Gamma_{k}=\\{\lambda;\sigma_{\ell}>0,\
1\leq\ell\leq k\\}$, which is an analogue of the $\alpha$-invariant estimate
for plurisubharmonic functions. It is well-known to experts, but we supply a
statement and proof without pluripotential theory, as we could not find a
convenient reference:

###### Lemma 8

For any $p\in(0,\frac{n}{n-k})$, there is a uniform constant
$C=C(n,p,\omega_{X})>0$ such that

$\|u\|_{L^{p}(\omega_{X}^{n})}\leq C,$

with $\sup_{X}u=0$ and $\lambda[\omega_{u}]\in\Gamma_{k}$,
$\omega_{u}=\omega_{X}+i\partial\bar{\partial}u$.

Proof. We use an idea in [11]. Without loss of generality we may assume
${\mathrm{Vol}}(X,\omega_{X})=1$.

Fix an $s>0$ and a small $\epsilon>0$. Let $K=\\{u\leq-s\\}\subset X$ be
compact sub-level set of $u$. We choose a sequence of smooth positive
functions $\eta_{j}$ with $\int_{X}\eta_{j}\omega_{X}^{n}=1$ which converge to
$\eta_{\infty}:=aV_{K}^{2\epsilon-1}\chi_{K}+a\cdot\chi_{X\backslash K}$ in
$L^{1+\epsilon}(\omega_{X}^{n})$ and also pointwise, where
$V_{K}=\int_{K}\omega_{X}^{n}$ is the volume of the set $K$ and $a>0$ is a
constant such that

$\int_{X}\eta_{\infty}\omega_{X}^{n}=\int_{X}(aV_{K}^{2\epsilon-1}\chi_{K}+a\cdot\chi_{X\backslash
K})\omega_{X}^{n}=aV_{K}^{2\epsilon}+a{\mathrm{Vol}(X\backslash K)}=1.$

It is not hard to see that $1/2\leq a\leq\max(2,2^{2\epsilon})=2$. Hence

$\int_{X}\eta_{\infty}^{1+\epsilon}\omega_{X}^{n}=a^{1+\epsilon}V_{K}^{\epsilon+2\epsilon^{2}}+a^{1+\epsilon}{\mathrm{Vol}(X\backslash
K)}\leq 4.$

Thus we may assume $\|\eta_{j}\|_{L^{1+\epsilon}(\omega_{X})}\leq 5$ for $j$
large enough. We solve the complex Monge-Ampère equations

$(\omega_{X}+i\partial\bar{\partial}v_{j})^{n}=\eta_{j}\omega_{X}^{n},\quad\sup_{X}v_{j}=0.$

By Theorem 1 (or [14]), it holds that $\|v_{j}\|_{L^{\infty}}\leq C_{0}$ for a
uniform $C_{0}=C_{0}(n,\omega_{X},\epsilon)$.

By integration by parts we have

$\displaystyle\int_{X}(-u)(\omega_{X}+i\partial\bar{\partial}v_{j})^{k}\wedge\omega_{X}^{n-k}$
$\displaystyle=$
$\displaystyle\int_{X}(-u)\omega_{X}\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}+(-v_{j})(\omega_{u}-\omega_{X})\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}$
$\displaystyle\leq$
$\displaystyle\int_{X}(-u)\omega_{X}\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}+C_{0}\int_{X}\omega_{u}\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}$
$\displaystyle=$
$\displaystyle\int_{X}(-u)\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k+1}+C_{0}.$

Applying (5) inductively we get

$\int_{X}(-u)(\omega_{X}+i\partial\bar{\partial}v_{j})^{k}\wedge\omega_{X}^{n-k}\leq\int_{X}(-u)\omega_{X}^{n}+kC_{0}\leq
C,$ (5.3)

for some uniform constant $C=C(n,k,\omega_{X},\epsilon)>0$. On the other hand,
by Newton-Maclaurin inequality that
$\big{(}\frac{\omega_{v_{j}}^{k}\wedge\omega_{X}^{n-k}}{\omega_{X}^{n}}\big{)}^{1/k}\geq
c(n,k)\big{(}\frac{\omega_{v_{j}}^{n}}{\omega_{X}^{n}}\big{)}^{1/n}$ we derive
from (5.3) that

$\int_{X}(-u)(\eta_{j})^{k/n}\omega_{X}^{n}\leq C(n,k,\omega_{X},\epsilon).$

Letting $j\to\infty$ and applying Fatou’s lemma we get

$\int_{K}(-u)V_{K}^{(2\epsilon-1)k/n}\omega_{X}^{n}\leq
C(n,k,\omega_{X},\epsilon)$

from which we obtain that

$V_{K}={\mathrm{Vol}}(\\{u\leq-s\\})\leq
C(n,k,\omega_{X},\epsilon)s^{-\frac{n}{(2\epsilon-1)k+n}}.$

For any $p<\frac{n}{n-k}$, we have

$\int_{X}(-u)^{p}\omega_{X}^{n}\leq
1+pC(n,k,\omega_{X},\epsilon)\int_{1}^{\infty}s^{p-1-\frac{n}{(2\epsilon-1)k+n}}ds\leq
C(n,k,\omega_{X},p)$

if $\epsilon=\epsilon(p)>0$ is chosen small enough so that the integral above
is integrable. The proof of Lemma 8 is complete.

Returning to the applications of Theorem 2, we observe that the condition that
$p<{n\over n-k}$ is equivalent to the dual exponent $q$ satisfying $q>{n\over
k}$. Thus Theorem 2 combined with Lemmas 7 and 8 imply at once:

###### Theorem 5

Consider the family of fully non-linear equations (1.7) with respect to the
degenerating background metrics $\omega_{t}$, $t\in(0,1]$. Assume that we have
solutions $\varphi_{t}\in C^{2}$, normalized by $\sup_{X}\varphi_{t}=0$.
Assume that $\lambda[h_{t,\varphi_{t}}]\in\Gamma_{k}$, for some fixed $k$,
$1\leq k\leq n$. Fix $q>n/k$. Then $\|\varphi_{t}\|_{L^{\infty}}$ is uniformly
bounded by a constant $C$ depending only on $n,k,q$ $\omega_{X},\chi$
$\|e^{nF_{t}}\|_{L^{q}(\omega_{X}^{n})}$ and $\frac{c_{t}^{n}}{V_{t}}$.

We illustrate this theorem by specializing now to the case of Hessian
equations, where $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$ for some $1\leq
k\leq n$. The more familiar form of this equation is

$(\omega_{t}+i\partial\bar{\partial}\varphi_{t})^{k}\wedge\omega_{X}^{n-k}=c_{t}^{k}e^{kF_{t}}\omega_{X}^{n},\quad\sup_{X}\varphi_{t}=0,$
(5.4)

and the condition $\lambda[\omega_{t,\varphi_{t}}]\in\Gamma_{k}$ is part of
the equation222We remark that usually in the equation (5.4), one normalizes
the function $F_{t}$ such that $\int_{X}e^{kF_{t}}\omega_{X}^{n}=V$. However,
our normalization is that $\int_{X}e^{nF_{t}}\omega_{X}^{n}=V$.. Thus Theorem
5 applies and, assuming uniform bounds for $\|e^{nF}\|_{L^{q}}$ for some
$q>n/k$, it reduces the uniform estimates for $\varphi_{t}$ to a uniform
estimate for the relative volumes $c_{t}^{n}/V_{t}$. An important geometric
case when the relative volumes can be controlled is when the classe $\chi$ is
big, in the sense that its volume $[\chi^{n}]=\int_{X}\chi^{n}$ is strictly
positive. In this case, we obtain

###### Theorem 6

Fix $1\leq k<n$, and consider the family (5.4) of Hessian equations with
respect to the degenerating background metrics. Assume that $\chi$ is big.
Then for any $q>{n\over k}$, $\|\varphi\|_{L^{\infty}}$ is bounded uniformly
by a constant $C$ depending only on $n,k,q$, $\omega_{X},\chi$ and an upper
bound for $\|e^{nF_{t}}\|_{L^{q}(\omega_{X}^{n})}$.

Proof of Theorem 6. In view of Theorem 5, it suffices to show that
$c_{t}^{n}V_{t}^{-1}$ is uniformly bounded. Since $V_{t}\geq[\chi^{n}]$ for
any $t$, this reduces to showing that $c_{t}$ are themselves uniformly
bounded.

The factors $c_{t}$ are determined by

$c_{t}^{k}\int_{X}e^{kF_{t}}\omega_{X}^{n}=\int_{X}\omega_{t}^{k}\wedge\omega_{X}^{n-k}=O(1).$
(5.5)

To estimate $c_{t}$, we still need a uniform lower bound of
$\int_{X}e^{kF_{t}}\omega_{X}^{n}$. We use Hölder’s inequality as before. Thus
we write

$\int_{X}e^{kF_{t}}\omega_{X}^{n}\geq\int_{\\{F_{t}\leq
0\\}}e^{nF_{t}}\omega_{X}^{n}+\int_{\\{F_{t}>0\\}}\omega_{X}^{n}.$ (5.6)

Recall that by our normalization on $F_{t}$

$V=\int_{X}e^{nF_{t}}\omega_{X}^{n}=\int_{\\{F_{t}\leq
0\\}}e^{nF_{t}}\omega_{X}^{n}+\int_{\\{F_{t}>0\\}}e^{nF_{t}}\omega_{X}^{n}.$
(5.7)

If the first term on the right hand side of (5.7) is greater than $V/2$, then
(5.6) shows that $\int_{X}e^{kF_{t}}\omega_{X}^{n}\geq V/2$; otherwise the
second term in (5.7) is greater than $V/2$. Then we have by the Hölder
inequality

$\int_{\\{F_{t}>0\\}}e^{nF_{t}}\omega_{X}^{n}\leq\Big{(}\int_{X}e^{qnF_{t}}\Big{)}^{1/q}\Big{(}\int_{\\{F_{t}>0\\}}\omega_{X}^{n}\Big{)}^{\frac{q-1}{q}}$

which yields a uniform lower bound of the second term in (5.6) depending
additionally on the assumed $\|e^{nF_{t}}\|_{L^{q}}$. Therefore we conclude
from (5.5) that $c_{t}\leq C$ for a uniform constant $C>0$. The proof of
Theorem 6 is complete.

##  6 Trudinger Inequalities

In this section, we illustrate the versatility of our approach by establishing
also inequalities of Trudinger type for general non-linear energies. Let
$f(\lambda)$ be a fully nonlinear operator satisfying the same structural
conditions as in Theorem 1, and define for each $p>0$,

$E_{p}(\varphi)=\frac{1}{V}\int_{X}(-\varphi)^{p}f^{n}(\lambda[h_{\varphi}])\omega_{X}^{n}$
(6.1)

(this notation is slightly different from the notation $E_{t}$ used earlier
for degenerating background metrics, but there should be no confusion, as the
background metric is here fixed, and the index $t$ can be dropped). Recall
that we had established before integral estimates for $\varphi$ in terms of
the entropy ${\rm Ent}_{p}$. Trudinger estimates are also exponential
estimates, but in terms of the energy $E_{p}$. We have

###### Theorem 7

Let $\varphi$ be a plurisubharmonic function such that $\sup\varphi=-1$. Then

$\int_{X}{\rm
exp}\Big{\\{}c(n,p,\gamma)\alpha\Big{(}\frac{-\varphi}{E_{p}(\varphi)^{\frac{1}{n+p}}}\Big{)}^{\frac{n+p}{n}}\Big{\\}}\omega_{X}^{n}\leq
2C_{\alpha}$ (6.2)

where $\alpha$ and $C_{\alpha}$ are the constants coming from the
$\alpha$-invariant estimate of $(X,\omega_{X})$.

We note that when specialized to the case when $f(\lambda)$ is the Monge-
Ampère operator $f(\lambda)=(\prod_{k=1}^{n}\lambda_{k})^{\frac{1}{n}}$, our
theorem recovers an inequality proved in [2, 10]. Moreover, our estimate has
the major advantage that all constants there depend only on the
$\alpha$-invariant of the underlying manifold, hence is uniform over
degenerating families with uniform $\alpha$-invariants.

Proof of Theorem 7: Let us solve the following auxiliary Monge-Ampère equation
with $\sup\psi=0$, which is solvable due to Yau’s theorem [25].

$(\omega_{X}+i\partial\bar{\partial}\psi)^{n}=\frac{(-\varphi)^{p}e^{nF}}{E_{p}(\varphi)}\omega_{X}^{n}$
(6.3)

then by the same maximum argument as in Lemma 1, we obtain the inequality

${c(n,p,\gamma)\Big{(}\frac{-\varphi}{E_{p}(\varphi)^{1/(n+p)}}\Big{)}^{\frac{n+p}{n}}}\leq-\psi+C(n,p,\gamma)E_{p}(\varphi)^{\frac{1}{p}}$
(6.4)

Now we pick $\kappa=2^{\frac{n}{n+p}}C^{\frac{n}{n+p}}c^{-\frac{n}{n+p}}$,
where $C$ and $c$ are constants in estimate above, which only depends on $n,p$
and $\gamma$. Now set $U_{\kappa}=\\{-\varphi\leq\kappa
E_{p}(\varphi)^{\frac{1}{p}}\\}$. Then on $X\setminus U_{\kappa}$, we have by
our choice of $\kappa$,

$\frac{1}{2}c(n,p,\gamma)\Big{(}\frac{-\varphi}{E_{p}(\varphi)^{1/(n+p)}}\Big{)}^{\frac{n+p}{n}}\leq-\psi$
(6.5)

and on $U_{\kappa}$, we have

$c(n,p,\gamma)\Big{(}\frac{-\varphi}{E_{p}(\varphi)^{1/(n+p)}}\Big{)}^{\frac{n+p}{n}}\leq-c(n,p,\gamma)\kappa^{\frac{p}{n}}\varphi$
(6.6)

now multiplying by $\min(c^{-1}\kappa^{-\frac{p}{n}},1/2)\alpha$ and
integrating, we get

$\int_{X}{\rm
exp}\Bigg{\\{}c^{\prime}(n,p,\gamma)\alpha\Big{(}\frac{-\varphi}{E_{p}(\varphi)^{1/(n+p)}}\Big{)}^{\frac{n+p}{n}}\Bigg{\\}}\omega_{X}^{n}\leq\int_{U_{\kappa}}e^{-\alpha\psi}\omega_{X}^{n}+\int_{X\setminus
U_{\kappa}}e^{-\alpha\varphi}\omega_{X}^{n}\leq 2C_{\alpha}$ (6.7)

which is the desired result.

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Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
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Department of Mathematics, Columbia University, New York, NY 10027 USA

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