# Quasi-plurisubharmonic envelopes 1: uniform estimates on Kähler manifolds

Vincent Guedj & Chinh H. Lu  Institut de Mathématiques de Toulouse
Université de Toulouse
118 route de Narbonne
31400 Toulouse, France
<EMAIL_ADDRESS>https://www.math.univ-toulouse.fr/ guedj/
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405,
Orsay, France<EMAIL_ADDRESS>https://www.imo.universite-paris-saclay.fr/ lu/

###### Abstract.

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate
complex Monge-Ampère equations on complex manifolds. It only relies on
compactness and envelopes properties of quasi-plurisubharmonic functions. Our
method allows one to obtain new and efficient proofs of several fundamental
results in Kähler geometry as we explain in this article.

In a sequel we shall explain how this approach also applies to the hermitian
setting producing new relative a priori bounds, as well as existence results.

###### Key words and phrases:

Monge-Ampère equation, a priori estimates

###### 2010 Mathematics Subject Classification:

32W20, 32U05, 32Q15, 35A23

###### Contents

  1. 1 Quasi-plurisubharmonic envelopes
    1. 1.1 Monge-Ampère operators
      1. 1.1.1 Quasi-plurisubharmonic functions
      2. 1.1.2 Monge-Ampère measure
    2. 1.2 Envelopes
      1. 1.2.1 Basic properties
      2. 1.2.2 A key lemma
  2. 2 Global $L^{\infty}$ bounds
    1. 2.1 Measures which integrate quasi-plurisubharmonic functions
    2. 2.2 Absolutely continuous measures
    3. 2.3 Stability estimate
  3. 3 Refinements and extensions
    1. 3.1 Big cohomology classes
    2. 3.2 Degenerating families
    3. 3.3 Relative a priori $L^{\infty}$-bounds
  4. 4 The local context
    1. 4.1 Cegrell classes
    2. 4.2 Dirichlet problem

## Introduction

Complex Monge-Ampère equations have been one of the most powerful tools in
Kähler geometry since Yau’s solution to the Calabi conjecture [Yau78]. A
notable application is the construction of Kähler-Einstein metrics: given
$(X,\omega)$ a compact Kähler manifold of complex dimension $n$ and $\mu$ an
appropriate volume form normalized by $\mu(X)=\int_{X}\omega^{n}$, one seeks
for a solution $\varphi:X\rightarrow\mathbb{R}$ to

$(\omega+dd^{c}\varphi)^{n}=e^{-\lambda\varphi}\mu,$

where $d=\partial+\overline{\partial}$,
$d^{c}=i(\partial-\overline{\partial})$ and $\lambda\in\mathbb{R}$ is a
constant whose sign depends on that of $c_{1}(X)$. The metric
$\omega_{\varphi}:=\omega+dd^{c}\varphi$ is then Kähler-Einstein as ${\rm
Ric}(\omega_{\varphi})=\lambda\omega_{\varphi}$.

When $\lambda\leq 0$, Yau [Yau78] (see also [Aub78] when $\lambda<0$) showed
the existence of a unique (normalized) solution $\varphi$ by establishing a
priori estimates along a continuity method, the most delicate one being the
uniform a priori estimate that he established by using Moser iteration
process.

In recent years degenerate complex Monge-Ampère equations have been
intensively studied by many authors. In relation to the Minimal Model Program,
they led to the construction of singular Kähler-Einstein metrics (see [EGZ09,
GZ, BBEGZ] and the references therein). The main analytical input came here
from pluripotential theory which allowed Kolodziej [Kol98] to establish
uniform a priori estimates when $\mu=fdV_{X}$ has density in $L^{p}$ for some
$p>1$.

Using different methods (Gromov-Hausdorff techniques), the case $\lambda>0$
(Yau-Tian-Donaldson conjecture) has been settled by Chen-Donaldson-Sun [CDS15,
Don18]. Again establishing a uniform a priori estimate in this context turned
out to be the most delicate issue, a key step being obtained by Donaldson-Sun
[DS14] through a refinement of Hörmander $L^{2}$-techniques. An alternative
pluripotential variational approach has been developed by Berman-Boucksom-
Jonsson in [BBJ21], based on finite energy classes studied in [GZ07] and
variational tools obtained in [BBGZ13]. This approach has been pushed one step
further by Li-Tian-Wang who have settled the case of singular Fano varieties
[LTW20].

The main goal of this article is to provide yet another approach for
establishing such uniform a priori estimates. While the pluripotential
approach consists in measuring the Monge-Ampère capacity of sublevel sets
$(\varphi<-t)$, we directly measure the volume of the latter, avoiding
delicate integration by parts. Our approach thus extends with minor
modifications to the hermitian (non Kähler) setting, providing several new
results that will be discussed in a companion paper [GL21]: the hermitian
setting introduces several technicalities and new challenges that might affect
the clarity of exposition and could scare the Kähler reader away.

In the whole article we let thus $X$ denote a compact Kähler manifold of
complex dimension $n$. We fix $\omega$ a closed semi-positive $(1,1)$-form
which is big, i.e.

$V:=\int_{X}\omega^{n}>0.$

We let ${\rm PSH}(X,\omega)$ denote the set of $\omega$-plurisubharmonic
functions: these are functions $u:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$
which are locally given as the sum of a smooth and a plurisubharmonic
function, and such that $\omega+dd^{c}u\geq 0$ is a positive current.

Our first main result is a brand new proof of the following a priori estimate
:

######  Theorem A.

Let $\omega$ be semi-positive and big. Let $\mu$ be a probability measure such
that ${\rm PSH}(X,\omega)\subset L^{m}(\mu)$ for some $m>n$. Any bounded
solution $\varphi\in{\rm PSH}(X,\omega)$ to
$V^{-1}(\omega+dd^{c}\varphi)^{n}=\mu$ satisfies a uniform a priori bound

${\rm Osc}_{X}(\varphi)\leq T_{\mu}$

for some uniform constant $T_{\mu}=T(A_{m}(\mu))$ which depends on an upper
bound on

$A_{m}({\mu}):=\sup\left\\{\int_{X}(-\psi)^{m}d\mu,\;\psi\in{\rm
PSH}(X,\omega)\text{ with }\sup_{X}\psi=0\right\\}.$

Hölder inequality shows that this result covers the case when $\mu=fdV_{X}$ is
absolutely continuous with respect to Lebesgue measure, with density $f$
belonging to $L^{p}$, $p>1$, or to an appropriate Orlicz class, as we explain
in Section 2.2.

A crucial particular case of this estimate is due to Kolodziej [Kol98]. Other
important special cases have been previously obtained in [EGZ09, EGZ08, DP10].
Our new method covers all these settings at once, it also permits to recover
the main estimates of [BEGZ10] (big cohomology classes) and [DnGG20]
(collapsing families) as we explain in Sections 3.1 and 3.2. A slight
refinement of our technique allows one to establish an important stability
estimate (see Theorem 2.4).

There are several geometric situations when one can not expect the Monge-
Ampère potential $\varphi$ to be globally bounded. We next consider the
equation

$V^{-1}(\omega+dd^{c}\varphi)^{n}=fdV_{X},$

where the density $f\in L^{1}(X)$ does not belong to any good Orlicz class.
Since the measure $\mu=fdV_{X}$ is non pluripolar, there exists a unique
finite energy solution $\varphi$ (see [GZ], [Din09]). It is crucial to
understand its locally bounded locus.

As $\omega$ is a semi-positive and big $(1,1)$ form, we can find $\rho$ an
$\omega$-psh function with analytic singularities such that
$\omega+dd^{c}\rho\geq\delta\omega_{X}$ is a Kähler current (see [DP04,
Theorem 0.5]). For $\psi$ quasi-psh and $c>0$, we set

$E_{c}(\psi):=\\{x\in X,\;\nu(\psi,x)\geq c\\},$

where $\nu(\psi,x)$ denotes the Lelong number of $\psi$ at $x$. A celebrated
theorem of Siu ensures that for any $c>0$, the set $E_{c}(\psi)$ is a closed
analytic subset of $X$.

Our second main result provides the following a priori estimate, which extends
a result of DiNezza-Lu [DnL17]:

######  Theorem B.

Assume $f=ge^{-\psi}$, where $0\leq g\in L^{p}(dV_{X})$, $p>1$, and $\psi$ is
a quasi-psh function. Then there exists a unique
$\varphi\in\mathcal{E}(X,\omega)$ such that

  * •

$\alpha(\psi+\rho)-\beta\leq\varphi\leq 0$ with $\sup_{X}\varphi=0$;

  * •

$\varphi$ is locally bounded in the open set
$\Omega:=X\setminus\\{\rho=-\infty\\}\cup E_{\frac{1}{q}}(\psi)$;

  * •

$V^{-1}(\omega+dd^{c}\varphi)^{n}=fdV_{X}\;\;\text{ in }\;\;\Omega$,

where $\alpha,\beta>0$ depend on an upper bound for $||g||_{L^{p}}$ and
$\frac{1}{p}+\frac{1}{q}=1$.

Again the proof we provide is direct, and can be extended to the hermitian
setting (see [GL21]). We finally show in Section 4 how the same arguments can
be applied to efficiently solve the Dirichlet problem in pseudoconvex domains.

Comparison with other works. Yau’s proof of his famous $L^{\infty}$-a priori
estimate [Yau78] goes through a Moser iteration process. Although Yau could
deal with some singularities, the method does not apply when the right hand
side is too degenerate (see however [Cao85, Tos10] for further applications of
Yau’s method).

An important generalization of Yau’s estimate has been provided by Kolodziej
[Kol98] using pluripotential techniques. These have been further generalized
in [EGZ09, EGZ08, DP10, BEGZ10] in order to deal with less positive or
collapsing families of cohomology classes on Kähler manifolds. As this
approach relies on delicate integration by parts, it is difficult to extend to
the hermitian setting.

Blocki has provided a different approach in [Blo05] based on the Alexandroff-
Bakelman-Pucci maximum principle and a local stability estimate due to Cheng-
Yau ($L^{2}$-case) and Kolodziej ($L^{p}$-case). This has been pushed further
by Szekelehydi in [Szek18]. It requires the reference form $\omega$ to be
strictly positive.

A PDE proof of the $L^{\infty}$-estimate has been very recently provided by
Guo-Phong-Tong [GPT21] using an auxiliary Monge-Ampère equation, inspired by
the recent breakthrough by Chen-Cheng on constant scalar curvature metrics
[CC21].

Our approach consists in showing that the sublevel set $(\varphi<-t)$ becomes
the empty set in finite time by directly measuring its $\mu$-size. It only
uses weak compactness of normalized $\omega$-plurisubharmonic functions and
basic properties of quasi-psh envelopes, allowing us to deal with semi-
positive forms.

######  Acknowledgements.

This work has benefited from State aid managed by the ANR projects PARAPLUI
and ANR-11-LABX-0040. We thank Ahmed Zeriahi for carefully reading the paper
and giving numerous useful comments.

##  1\. Quasi-plurisubharmonic envelopes

In the whole article we let $X$ denote a compact Kähler manifold of complex
dimension $n\geq 1$. We fix $\omega$ a smooth closed real $(1,1)$-form on $X$.

###  1.1. Monge-Ampère operators

####  1.1.1. Quasi-plurisubharmonic functions

A function is quasi-plurisubharmonic if it is locally given as the sum of a
smooth and a psh function. Quasi-psh functions
$\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ satisfying
$\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$ in the weak sense of currents
are called $\omega$-plurisubharmonic ($\omega$-psh for short).

######  Definition 1.1.

We let ${\rm PSH}(X,\omega)$ denote the set of all $\omega$-plurisubharmonic
functions which are not identically $-\infty$.

Constant functions are $\omega$-psh functions if (and only if) $\omega$ is
semi-positive. A ${\mathcal{C}}^{2}$-smooth function $u$ has bounded Hessian,
hence $\varepsilon u$ is $\omega$-psh if $0<\varepsilon$ is small enough and
$\omega$ is positive. It is useful to consider as well the case when $\omega$
is not necessarily positive, in order to study big cohomology classes (see
section 3.1).

######  Definition 1.2.

A semi-positive closed $(1,1)$-form $\omega$ is big if
$V_{\omega}:=\int_{X}\omega^{n}>0$.

The set ${\rm PSH}(X,\omega)$ is a closed subset of $L^{1}(X)$, for the
$L^{1}$-topology. Subsets of $\omega$-psh functions enjoy strong compactness
and integrability properties, we mention notably the following: for any fixed
$r\geq 1$,

  * •

${\rm PSH}(X,\omega)\subset L^{r}(X)$; the induced $L^{r}$-topologies are
equivalent;

  * •

${\rm PSH}_{A}(X,\omega):=\\{u\in{\rm PSH}(X,\omega),\,-A\leq\sup_{X}u\leq
0\\}$ is compact in $L^{r}$.

We refer the reader to [Dem, GZ] for further basic properties of $\omega$-psh
functions.

####  1.1.2. Monge-Ampère measure

The complex Monge-Ampère measure

$(\omega+dd^{c}u)^{n}=\omega_{u}^{n}$

is well-defined for any $\omega$-psh function $u$ which is bounded, as follows
from Bedford-Taylor theory (see [BT82] for the local theory, and [GZ] for the
compact Kähler context). It also makes sense in the ample locus of a big
cohomology class [BEGZ10], as we shall briefly discuss in section 3.1.

The mixed Monge-Ampère measures
$(\omega+dd^{c}u)^{j}\wedge(\omega+dd^{c}v)^{n-j}$ are also well defined for
any $0\leq j\leq n$, and any bounded $\omega$-psh functions $u,v$. We note for
later use the following classical inequality:

######  Lemma 1.3.

Let $\varphi,\psi$ be bounded $\omega$-psh functions such that
$\varphi\leq\psi$, then

$1_{\\{\psi=\varphi\\}}(\omega+dd^{c}\varphi)^{j}\wedge(\omega+dd^{c}\psi)^{n-j}\leq
1_{\\{\psi=\varphi\\}}(\omega+dd^{c}\psi)^{n},$

for all $1\leq j\leq n$.

###### Proof.

To simplify notations we just treat the case $j=n$. It follows from Bedford-
Taylor theory [BT82] that for any bounded $\omega$-psh functions
$\varphi,\psi$,

$1_{\\{\psi\leq\varphi\\}}\omega_{\varphi}^{n}+1_{\\{\psi>\varphi\\}}\omega_{\psi}^{n}\leq(\omega+dd^{c}\max(\varphi,\psi))^{n}$

When $\varphi\leq\psi$ we infer
$1_{\\{\psi=\varphi\\}}\omega_{\varphi}^{n}\leq
1_{\\{\psi=\varphi\\}}\omega_{\psi}^{n}.$ ∎

We shall also need the following (see [GZ, Proposition 10.11]):

######  Proposition 1.4.

[Domination principle] If $u,v$ are bounded $\omega$-psh functions such that
$u\geq v$ a.e. with respect to $\omega_{u}^{n}$. Then $u\geq v$.

###  1.2. Envelopes

Upper envelopes of (pluri)subharmonic functions are classical objects in
Potential Theory. They were considered by Bedford and Taylor to solve the
Dirichlet problem for the complex Monge-Ampère equation in strictly pseudo-
convex domains [BT76]. We consider here envelopes of $\omega$-psh functions.

####  1.2.1. Basic properties

######  Definition 1.5.

Given a Lebesgue measurable function $h:X\rightarrow\mathbb{R}$, we define the
$\omega$-psh envelope of $h$ by

$P_{\omega}(h):=\left(\sup\\{u\in{\rm PSH}(X,\omega);u\leq
h\,\,\text{in}\,\,X\\}\right)^{*},$

where the star means that we take the upper semi-continuous regularization.

The following is a combination of [GLZ19, Propositions 2.2 and 2.5, Lemma
2.3]:

######  Proposition 1.6.

If $h$ is bounded from below and quasi-continuous, then

  * •

$P_{\omega}(h)$ is a bounded $\omega$-plurisubharmonic function;

  * •

$P_{\omega}(h)\leq h$ in $X\setminus P$, where $P$ is pluripolar;

  * •

$(\omega+dd^{c}P_{\omega}(h))^{n}$ is concentrated on the contact
$\\{P_{\omega}(h)=h\\}$.

Recall that a function $h$ is quasi-continuous if for any $\varepsilon>0$,
there exists an open set $G$ of capacity smaller than $\varepsilon$ such that
$h$ is continuous in $X\setminus G$. Quasi-psh functions are quasi-continuous
(see [BT82]), as well as differences of the latter: we shall use this fact
during the proof of Theorem 3.3.

When $h$ is ${\mathcal{C}}^{1,1}$-smooth, so is $P_{\omega}(h)$ [Ber19, CZ19]
and one can further has

(1.1)
$(\omega+dd^{c}P_{\omega}(h))^{n}=1_{\\{P_{\omega}(h)=h\\}}(\omega+dd^{c}h)^{n}.$

####  1.2.2. A key lemma

The following is a key technical tool to our new approach:

######  Lemma 1.7.

Fix $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ a concave increasing
function such that $\chi^{\prime}(0)\geq 1$. Let $\varphi,\phi$ be bounded
$\omega$-psh functions with $\varphi\leq\phi$. If
$\psi=\phi+\chi\circ(\varphi-\phi)$ then

$(\omega+dd^{c}P_{\omega}(\psi))^{n}\leq
1_{\\{P_{\omega}(\psi)=\psi\\}}(\chi^{\prime}\circ(\varphi-\phi))^{n}(\omega+dd^{c}\varphi)^{n}.$

###### Proof.

Using that $\chi^{\prime\prime}\leq 0$ and $\chi^{\prime}\geq 1$, we observe
that

$\displaystyle\omega+dd^{c}\psi$ $\displaystyle=$
$\displaystyle\omega_{\phi}+\chi^{\prime}\circ(\varphi-\phi)(\omega_{\varphi}-\omega_{\phi})+\chi^{\prime\prime}\circ(\varphi-\phi)d(\varphi-\phi)\wedge
d^{c}(\varphi-\phi)$ $\displaystyle\leq$
$\displaystyle\chi^{\prime}\circ(\varphi-\phi)\omega_{\varphi}+[1-\chi^{\prime}\circ(\varphi-\phi)]\omega_{\phi}\leq\chi^{\prime}\circ(\varphi-\phi)\omega_{\varphi}.$

When $\varphi,\phi$ and $\chi$ are ${\mathcal{C}}^{1,1}$-smooth, we can invoke
(1.1) to conclude that

$(\omega+dd^{c}P_{\omega}(\psi))^{n}=1_{\\{P_{\omega}(\psi)=\psi\\}}\omega_{\psi}^{n}\leq
1_{\\{P_{\omega}(\psi)=\psi\\}}(\chi^{\prime}\circ(\varphi-\phi))^{n}\omega_{\varphi}^{n}.$

The last inequality follows from
$\omega+dd^{c}\psi\leq\chi^{\prime}\circ(\varphi-\phi)\omega_{\varphi}$ and
the fact that $\psi$ is $\omega$-psh on $\\{P_{\omega}(\psi)=\psi\\}$, where
these inequalities can be interpreted pointwise.

When these functions are less regular we take a different route. We set
$\tau=\chi^{-1}:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$. This is a convex
increasing function such that $\tau^{\prime}=(\chi^{\prime}\circ\tau)^{-1}\leq
1$. Set $\rho=P_{\omega}(\psi)-\phi$. The function
$v=\phi+\tau\circ(P_{\omega}(\psi)-\phi)$ is $\omega$-psh with

$\displaystyle\omega+dd^{c}v$ $\displaystyle=$
$\displaystyle\omega_{\phi}+\tau^{\prime\prime}\circ\rho\,d\rho\wedge
d^{c}\rho+\tau^{\prime}\circ\rho\,dd^{c}(P_{\omega}(\psi)-\phi)$
$\displaystyle\geq$
$\displaystyle[1-\tau^{\prime}\circ\rho]\omega_{\phi}+\tau^{\prime}\circ\rho\,(\omega+dd^{c}P_{\omega}(\psi))$
$\displaystyle\geq$
$\displaystyle\tau^{\prime}\circ\rho\,(\omega+dd^{c}P_{\omega}(\psi)).$

Thus $\omega_{P_{\omega}(\psi)}^{n}\leq
1_{\\{P_{\omega}(\psi)=\psi\\}}(\tau^{\prime}\circ(P_{\omega}(\psi)-\phi))^{-n}\omega_{v}^{n}.$
On $\\{P_{\omega}(\psi)=\psi\\}$ we get

$\tau^{\prime}\circ(P_{\omega}(\psi)-\phi)=\tau^{\prime}\circ(\psi-\phi)=[\chi^{\prime}\circ(\varphi-\phi)]^{-1}.$

Now $v\leq\phi+\tau\circ(\psi-\phi)=\varphi$ on $X$, with equality on the
contact set $\\{P_{\omega}(\psi)=\psi\\}$. It follows therefore from Lemma 1.3
that $\omega_{v}^{n}\leq\omega_{\varphi}^{n}$ on
$\\{P_{\omega}(\psi)=\psi\\}$. ∎

##  2\. Global $L^{\infty}$ bounds

In this section we prove Theorem A, as well as a stability estimate.

###  2.1. Measures which integrate quasi-plurisubharmonic functions

######  Theorem 2.1.

Let $\omega$ be semi-positive and big. Let $\mu$ be a probability measure such
that ${\rm PSH}(X,\omega)\subset L^{m}(\mu)$ for some $m>n$. Any solution
$\varphi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ to
$V^{-1}(\omega+dd^{c}\varphi)^{n}=\mu$ satisfies

${\rm Osc}_{X}(\varphi)\leq T_{\mu}$

for some uniform constant $T_{\mu}=T(A_{m}(\mu))$ which depends on an upper
bound on

$A_{m}({\mu}):=\sup\left\\{\left(\int_{X}(-\psi)^{m}d\mu\right)^{\frac{1}{m}},\;\psi\in{\rm
PSH}(X,\omega)\text{ with }\sup_{X}\psi=0\right\\}.$

Let us stress that this result is not new: it can be derived from the
celebrated a priori estimate of Kolodziej [Kol98], together with its
extensions [EGZ09, EGZ08, DP10]. We provide here an elementary proof that does
not use the theory of Monge-Ampère capacities, and merely relies on the
compactness properties of sup-normalized $\omega$-psh functions and Lemma 1.7.

###### Proof.

Shifting by an additive constant, we normalize $\varphi$ by
$\sup_{X}\varphi=0$. Set

$T_{\max}:=\sup\\{t>0\;:\;\mu(\varphi<-t)>0\\}.$

Our goal is to establish a precise bound on $T_{max}$. By definition,
$-T_{max}\leq\varphi$ almost everywhere with respect to $\mu$, hence
everywhere by the domination principle (Proposition 1.4), providing the
desired a priori bound ${\rm Osc}_{X}(\varphi)\leq T_{max}$.

We let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ denote a concave
increasing function such that $\chi(0)=0$ and $\chi^{\prime}(0)=1$. We set
$\psi=\chi\circ\varphi$, $u=P_{\omega}(\psi)$ and observe that

$\omega+dd^{c}\psi=\chi^{\prime}\circ\varphi\,\omega_{\varphi}+[1-\chi^{\prime}\circ\varphi]\,\omega+\chi^{\prime\prime}\circ\varphi\,d\varphi\wedge
d^{c}\varphi\leq\chi^{\prime}\circ\varphi\,\omega_{\varphi}.$

It follows from Lemma 1.7 that

$MA(u):=\frac{1}{V}(\omega+dd^{c}u)^{n}\leq
1_{\\{u=\psi\\}}(\chi^{\prime}\circ\varphi)^{n}\mu.$

Controlling the energy of $u$. We fix $\varepsilon>0$ so that
$n<n+3\varepsilon=m.$ The concavity of $\chi$ and the normalization
$\chi(0)=0$ yields $|\chi(t)|\leq|t|\chi^{\prime}(t)$. Since
$u=\chi\circ\varphi$ on the contact set $\\{P_{\omega}(\psi)=\psi\\}$, Hölder
inequality yields

$\displaystyle\int_{X}(-u)^{\varepsilon}MA(u)$ $\displaystyle\leq$
$\displaystyle\int_{X}(-\chi\circ\varphi)^{\varepsilon}(\chi^{\prime}\circ\varphi)^{n}d\mu\leq\int_{X}(-\varphi)^{\varepsilon}(\chi^{\prime}\circ\varphi)^{n+\varepsilon}d\mu$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}(-\varphi)^{n+2\varepsilon}d\mu\right)^{\frac{\varepsilon}{n+2\varepsilon}}\left(\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu\right)^{\frac{n+\varepsilon}{n+2\varepsilon}}$
$\displaystyle=$ $\displaystyle
A_{m}({\mu})^{\varepsilon}\left(\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu\right)^{\frac{n+\varepsilon}{n+2\varepsilon}}$

using that $\varphi$ belongs to the set of $\omega$-psh functions $v$
normalized by $\sup_{X}v=0$ which is compact in $L^{n+2\varepsilon}(\mu)$, and
observing that $A_{n+2\varepsilon}({\mu})\leq A_{m}({\mu})$.

Controlling the norms $||u||_{L^{m}}$. We are going to choose below the weight
$\chi$ in such a way that
$\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu=B\leq 2$ is a finite
constant under control. This provides a uniform lower bound on $\sup_{X}u$ as
we now explain: indeed

$0\leq(-\sup_{X}u)^{\varepsilon}=(-\sup_{X}u)^{\varepsilon}\int_{X}MA(u)\leq\int_{X}(-u)^{\varepsilon}MA(u)\leq
2A_{m}({\mu})^{\varepsilon}$

yields $-2^{\frac{1}{\varepsilon}}A_{m}({\mu})\leq\sup_{X}u\leq 0.$ We infer
that $u$ belongs to a compact set of $\omega$-psh functions, hence its norm
$||u||_{L^{m}(\mu)}$ is under control with

$||u||_{L^{m}(\mu)}\leq
A_{m}(\mu)+2^{\frac{1}{\varepsilon}}A_{m}({\mu})\leq[1+2^{\frac{1}{\varepsilon}}]A_{m}({\mu}).$

Since $u\leq\chi\circ\varphi\leq 0$ we infer
$||\chi\circ\varphi||_{L^{m}}\leq||u||_{L^{m}}$. Chebyshev inequality thus
yields

(2.1) ${\mu}(\varphi<-t)\leq\frac{\tilde{A}}{|\chi|^{m}(-t)},\;\;\text{ where
}\;\;\tilde{A}=[1+2^{\frac{1}{\varepsilon}}]A_{m}({\mu}).$

Choice of $\chi$. Lebesgue’s formula ensures that if
$g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is an increasing function such
that $g(0)=1$, then

$\int_{X}g\circ(-\varphi)d\mu=\mu(X)+\int_{0}^{T_{\max}}g^{\prime}(t){\mu}(\varphi<-t)dt.$

Fix $0<T_{0}<T_{\max}$. Setting $g(t)=[\chi^{\prime}(-t)]^{n+2\varepsilon}$ we
define $\chi$ by imposing $\chi(0)=0$, $\chi^{\prime}(0)=1$, and

$g^{\prime}(t)=\begin{cases}\dfrac{1}{(1+t)^{2}{\mu}(\varphi<-t)},\;\text{if}\;t\leq
T_{0}\\\ \;\\\ \frac{1}{(1+t)^{2}}\;\;\;\;\text{ if}\;t>T_{0}\end{cases}.$

This choice guarantees that $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ is
concave increasing with $\chi^{\prime}\geq 1$, and

$\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu\leq\mu(X)+\int_{0}^{+\infty}\frac{dt}{(1+t)^{2}}=2.$

Conclusion. We set $h(t)=-\chi(-t)$ and work with the positive counterpart of
$\chi$. Note that $h(0)=0$ and
$h^{\prime}(t)=[g(t)]^{\frac{1}{n+2\varepsilon}}$ is positive increasing,
hence $h$ is convex. Observe also that $g(t)\geq g(0)=1$ hence
$h^{\prime}(t)=[g(t)]^{\frac{1}{n+2\varepsilon}}\geq 1$ yields

(2.2) $h(1)=\int_{0}^{1}h^{\prime}(s)ds\geq 1.$

Together with (2.1) our choice of $\chi$ yields, for all $t\in[0,T_{0}]$,

$\frac{1}{(1+t)^{2}g^{\prime}(t)}={\mu}(\varphi<-t)\leq\frac{\tilde{A}}{h^{m}(t)}.$

For $t\in[0,T_{0}]$, this reads

$h^{m}(t)\leq\tilde{A}(1+t)^{2}g^{\prime}(t)=(n+2\varepsilon)\tilde{A}(1+t)^{2}h^{\prime\prime}(t)(h^{\prime})^{n+2\varepsilon-1}(t).$

Multiplying by $h^{\prime}$, integrating between $0$ and $t$, we infer that
for all $t\in[0,T_{0}]$,

$\displaystyle\frac{h^{m+1}(t)}{m+1}$ $\displaystyle\leq$
$\displaystyle(n+2\varepsilon)\tilde{A}\int_{0}^{t}(1+s)^{2}h^{\prime\prime}(s)(h^{\prime})^{n+2\varepsilon}(s)$
$\displaystyle\leq$
$\displaystyle\frac{(n+2\varepsilon)\tilde{A}(t+1)^{2}}{n+2\varepsilon+1}\left((h^{\prime})^{n+2\varepsilon+1}(t)-1\right)$
$\displaystyle\leq$
$\displaystyle\tilde{A}(1+t)^{2}(h^{\prime})^{n+2\varepsilon+1}(t).$

Recall that $m=n+3\varepsilon$ so that
$\alpha:=m+1>\beta:=n+2\varepsilon+1>2.$ The previous inequality then reads

$(1+t)^{-\frac{2}{\beta}}\leq C{h^{\prime}(t)}{h(t)^{-\frac{\alpha}{\beta}}},$

for some uniform constant $C$ depending on $n,m,\tilde{A}$. Since
$\alpha>\beta>2$ and $h(1)\geq 1$, integrating the above inequality between
$1$ and $T_{0}$ we obtain $T_{0}\leq C^{\prime},$ for some uniform constant
$C^{\prime}$ depending on $C,\alpha,\beta$. Since $T_{0}$ was chosen
arbitrarily in $(0,T_{\max})$ the result follows. ∎

###  2.2. Absolutely continuous measures

Assume $\mu=fdV_{X}$ is absolutely continuous with respect to a volume form
$dV_{X}$, with density $0\leq f\in L^{p}(dV_{X})$ for some $p>1$. Since ${\rm
PSH}(X,\omega)\subset L^{r}(dV_{X})$ for any $1\leq r<+\infty$, we obtain

$\int_{X}|u|^{m}d\mu\leq||f||_{L^{p}(dV_{X})}\cdot\left(\int_{X}|u|^{qm}dV_{X}\right)^{1/q},$

for all $u\in{\rm PSH}(X,\omega)$, where $1/p+1/q=1$, so that ${\rm
PSH}(X,\omega)\subset L^{m}(d\mu)$ for all $m\geq 1$. Thus Theorem 2.1 applies
to this type of measures, providing a new proof of the celebrated a priori
estimate of Kolodziej [Kol98] (see also [EGZ09]).

As in [Kol98] our technique also covers the case of more general densities as
we briefly indicate. Let $w:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a
convex increasing weight. A measurable function $f$ belongs to the Orlicz
class $L^{w}(dV_{X})$ if there exists $\alpha>0$ such that

$\int_{X}w(\alpha|f|)dV_{X}<+\infty.$

The Luxembourg norm of $f$ is defined as

$||f||_{w}:=\inf\\{r>0,\;\int_{X}w(|f|/r)dV_{X}\leq 1\\};$

it turns $L^{w}(dV_{X})$ into a Banach space.

If $w^{*}$ denotes the conjugate convex weight of $w$ (its Legendre
transform), Hölder-Young inequality ensures that for all measurable functions
$f,g$,

$\int_{X}|fg|dV_{X}\leq 2||f||_{w}||g||_{w^{*}}.$

We refer the reader to [RR] for more information on Orlicz classes.

Theorem 2.1 thus allows to reprove [Kol98, Theorem 2.5.2]:

######  Corollary 2.2.

Let $\mu=fdV_{X}$ be a probability measure. Let
$w:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a convex increasing weight that
grows at infinity at least like $t(\log t)^{m}$ with $m>n$. If $f$ belongs to
the Orlicz class $L^{w}$ then any solution $\varphi\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ to $V^{-1}(\omega+dd^{c}\varphi)^{n}=\mu$ satisfies

${\rm Osc}_{X}(\varphi)\leq T_{\mu}$

for some uniform constant $T_{\mu}\in\mathbb{R}^{+}$.

###### Proof.

While this was not required for the case of $L^{p}$ densities, we need here to
invoke Skoda’s uniform integrability result (see [GZ, Theorem 8.11]): there
exists $\alpha>0$ and $C=C(\alpha,M)>0$ such that

$\sup\left\\{\int_{X}e^{2\alpha|u|}dV_{X},\;u\in{\rm PSH}(X,\omega)\text{ and
}-M\leq\sup_{X}u\leq 0\right\\}\leq C.$

The reader will check that, as $s\rightarrow+\infty$, the conjugate weight
$w^{*}(s)$ grows like

$w^{*}(s)\sim
s^{1-\frac{1}{m}}\exp(s^{\frac{1}{m}})\leq\exp(2s^{\frac{1}{m}}).$

It follows therefore from Young inequality that any $\omega$-psh function $u$
satisfies

$\alpha^{m}\int_{X}|u|^{m}d\mu\leq\int_{X}w\circ
fdV_{X}+\int_{X}\exp(2\alpha|u|)dV_{X}<+\infty.$

Thus ${\rm PSH}(X,\omega)\subset L^{m}(\mu)$ and the conclusion follows from
Theorem 2.1. ∎

One can slightly improve the assumption on the density as in [Kol98, Theorem
2.5.2], we leave the technical details to the interested reader.

######  Remark 2.3.

It follows from the Chern-Levine-Nirenberg inequality that if
$\mu=(\omega+dd^{c}\varphi)^{n}$ is the Monge-Ampère measure of a bounded
$\omega$-psh function, then ${\rm PSH}(X,\omega)\subset L^{1}(\mu)$. If $n=1$
this condition is equivalent to $\mu$ having bounded potential (see [DnGL20,
Lemma 3.2]). Note however that when $n\geq 2$,

  * •

the condition ${\rm PSH}(X,\omega)\subset L^{n}(\mu)$,
$\mu=(\omega+dd^{c}\varphi)^{n}$, is not sufficient to guarantee that the
$\omega$-psh function $\varphi$ is bounded ;

  * •

one cannot improve the C-L-N inequality: there are examples of Monge-Ampère
measures with bounded potential and ${\rm PSH}(X,\omega)\not\subset
L^{1+\varepsilon}(\mu)$.

###  2.3. Stability estimate

We now establish the following stability estimate, which can be seen as a
refinement of [GZ12, Proposition 5.2].

######  Theorem 2.4.

Let $\omega,\mu$ be as in Theorem 2.1. Let $\varphi\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ be such that $\sup_{X}\varphi=0$ and
$V^{-1}(\omega+dd^{c}\varphi)^{n}=\mu$. Then

$\sup_{X}(\phi-\varphi)_{+}\leq
T\left(\int_{X}(\phi-\varphi)_{+}d\mu\right)^{\tau},$

for any $\phi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$, where
$\tau=\tau(n,m)>0$ and

$T=T(\mu,\|\phi\|_{L^{\infty}})$

is a uniform constant which depends on an upper bound on
$||\phi||_{L^{\infty}}$ and

$A_{m}({\mu}):=\sup\left\\{\left(\int_{X}(-\psi)^{m}d\mu\right)^{\frac{1}{m}},\;\psi\in{\rm
PSH}(X,\omega)\text{ with }\sup_{X}\psi=0\right\\}.$

###### Proof.

Replacing $\phi$ by $\max(\varphi,\phi)$, we can assume that
$\varphi\leq\phi$. Define

$T_{\max}:=\sup\\{t>0\;:\;\mu(\varphi<\phi-t)>0\\}.$

It follows from Theorem 2.1 that $T_{\max}$ is uniformly controlled by $\mu$
and $||\phi||_{L^{\infty}}$.

We let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ denote a concave
increasing function such that $\chi(0)=0$ and $\chi^{\prime}(0)=1$. We set
$\psi=\phi+\chi\circ(\varphi-\phi)$, $u=P(\psi)$ and observe that

$\displaystyle\omega+dd^{c}\psi$ $\displaystyle=$
$\displaystyle\omega_{\phi}+\chi^{\prime}\circ(\varphi-\phi)(\omega_{\varphi}-\omega_{\phi})+\chi^{\prime\prime}\circ(\varphi-\phi)d(\varphi-\phi)\wedge
d^{c}(\varphi-\phi)$ $\displaystyle\leq$
$\displaystyle\chi^{\prime}\circ(\varphi-\phi)\omega_{\varphi}.$

It follows from Lemma 1.7 that

$MA(u):=\frac{V_{\omega}}{(\omega+dd^{c}u)^{n}}\leq
1_{\\{u=\psi\\}}(\chi^{\prime}\circ(\varphi-\phi))^{n}\mu.$

Controlling the energy of $u$. We fix $0<a<b<c<2c<\varepsilon$ so small that

$q:=\frac{(\varepsilon-a)(n+b)}{b-a}<m=n+\varepsilon.$

The concavity of $\chi$ and the normalization $\chi(0)=0$ yields
$|\chi(t)|\leq|t|\chi^{\prime}(t)$. Since $u=\phi+\chi\circ(\varphi-\phi)$ on
the support of $(\omega+dd^{c}u)^{n}$ and ${\rm PSH}(X,\omega)\subset
L^{n+2c}(\mu)$, Hölder inequality yields

$\displaystyle 0\leq\int_{X}(-u+\phi)^{c}$ $\displaystyle
MA(u)\leq\int_{X}(-\chi\circ(\varphi-\phi))^{c}(\chi^{\prime}\circ(\varphi-\phi))^{n}d\mu$
$\displaystyle\leq\int_{X}(-\varphi+\phi)^{c}(\chi^{\prime}\circ(\varphi-\phi))^{n+c}d\mu$
$\displaystyle\leq\left(\int_{X}(-\varphi+\phi)^{n+2c}d\mu\right)^{\frac{c}{n+2c}}\left(\int_{X}(\chi^{\prime}\circ(\varphi-\phi))^{n+2c}d\mu\right)^{\frac{n+c}{n+2c}}$
$\displaystyle\leq
A_{m}(\mu)^{c}\left(\int_{X}(\chi^{\prime}\circ(\varphi-\phi))^{n+2c}d\mu\right)^{\frac{n+c}{n+2c}}.$

Controlling the norms $||u||_{L^{m}}$. We choose $\chi$ below s.t.
$\int_{X}(\chi^{\prime}\circ(\varphi-\phi))^{n+2c}d\mu\leq B$ is under
control. This provides a uniform lower bound on $\sup_{X}u$. Indeed our
normalizations yield $\chi(t)\leq t$ hence
$u\leq\phi+\chi(\varphi-\phi)\leq\varphi\leq 0,$ while

$0\leq(-\sup_{X}(u-\phi))^{c}\leq\int_{X}(-u+\phi)^{c}MA(u)\leq
A_{m}(\mu)^{c}B^{\frac{n+c}{n+2c}}$

yields a lower bound on $\sup_{X}(u-\phi)$. Now $u=u-\phi+\phi\geq
u-\phi+\inf_{X}\phi$, so $\sup_{X}u\geq\sup_{X}(u-\phi)+\inf_{X}\phi\geq-
A_{m}(\mu)B^{\frac{n+c}{c(n+2c)}}+\inf_{X}\phi$.

Thus $u$ belongs to a compact set of $\omega$-psh functions: its norm
$||u||_{L^{q}(\mu)}$ is under control for any $q\leq m$. Since
$u-\phi\leq\chi\circ(\varphi-\phi)\leq 0$, Hölder inequality yields

$\displaystyle\int_{X}$
$\displaystyle|\chi\circ(\varphi-\phi)|^{m}d\mu\leq\int_{X}|\chi\circ(\varphi-\phi)|^{n+a}(\phi-u)^{\varepsilon-a}d\mu$
$\displaystyle\leq\left(\int_{X}|\chi\circ(\varphi-\phi)|^{n+b}d\mu\right)^{\frac{n+a}{n+b}}\left(\int_{X}(\phi-u)^{q}d\mu\right)^{\frac{b-a}{n+b}}$
$\displaystyle\leq
C_{\mu}^{\prime}\left(\int_{X}|(\phi-\varphi)\chi^{\prime}\circ(\varphi-\phi)|^{n+b}d\mu\right)^{\frac{n+a}{n+b}}$
$\displaystyle\leq
C_{\mu}^{\prime}\left(\int_{X}(\phi-\varphi)^{\frac{(n+c)(n+b)}{c-b}}d\mu\right)^{\frac{(c-b)(n+a)}{(n+c)(n+b)}}\left(\int_{X}|\chi^{\prime}\circ(\varphi-\phi)|^{n+c}d\mu\right)^{\frac{n+a}{n+c}}$
(2.3) $\displaystyle\leq
C_{1}B^{\frac{n+a}{n+c}}\left(\int_{X}(\phi-\varphi)d\mu\right)^{\gamma}=:\tilde{A},$

where $\gamma=\dfrac{(c-b)(n+a)}{(n+c)(n+b)}$, and $C_{1}$ depends on
$C_{\mu}$, $||\varphi||_{L^{\infty}}$ and $||\phi||_{L^{\infty}}$.

It follows therefore from Chebyshev inequality that

(2.4) ${\mu}(\varphi<\phi-t)\leq\frac{\tilde{A}}{|\chi|^{m}(-t)}.$

Choice of $\chi$. Fix $T_{0}\in(0,T_{\max})$. We set
$g(t)=[\chi^{\prime}(-t)]^{n+2c}$ and define $\chi$ by imposing $\chi(0)=0$,
$\chi^{\prime}(0)=1$, and

$g^{\prime}(t)=\begin{cases}\dfrac{1}{{\mu}(\varphi<\phi-t)},\;\text{if}\;t\leq
T_{0}\\\ \;\\\ 1\;\;\;\;\text{ if}\;t>T_{0}\end{cases}.$

This choice guarantees that

$\int_{X}(\chi^{\prime}\circ(\varphi-\phi))^{n+2c}d\mu\leq\mu(X)+\int_{0}^{T_{\max}}dt=1+T_{\max}.$

It follows from Theorem 2.1 that $T_{\max}\leq T_{\mu}$ is uniformly bounded
from above, hence $B:=1+T_{\mu}$ is under control. Together with (2.3) and
(2.4) we thus obtain

(2.5) ${\mu}(\varphi<\phi-t)\leq\frac{C_{2}\delta}{|\chi|^{m}(-t)},$

where $\delta:=\left(\int_{X}(\phi-\varphi)d\mu\right)^{\gamma}$.

Conclusion. Set $h(t)=-\chi(-t)$. It follows from (2.5) that for all
$t\in[0,T_{0}]$,

$\frac{1}{g^{\prime}(t)}={\mu}(\varphi<\phi-t)\leq\frac{C_{2}\delta}{h^{m}(t)},$

hence

$h^{m}(t)\leq C_{2}\delta g^{\prime}(t)=(n+2c)C_{2}\delta
h^{\prime\prime}(t)(h^{\prime})^{n+2c-1}(t).$

Multiplying by $h^{\prime}$, integrating between $0$ and $t$, we infer that
for all $t\in[0,T_{0}]$,

$\displaystyle h^{m+1}(t)$ $\displaystyle\leq$
$\displaystyle(m+1)(n+2c)C_{2}\delta\int_{0}^{t}h^{\prime\prime}(s)(h^{\prime})^{n+2c}(s)ds$
$\displaystyle\leq$ $\displaystyle
C_{3}\delta\left((h^{\prime})^{n+2c+1}(t)-1\right),$

which yields

(2.6)
$1\leq\frac{C_{3}\delta(h^{\prime})^{n+2c+1}(t)}{h^{m+1}(t)+C_{3}\delta}.$

Recall that we have set $m=n+\varepsilon$ so that

$\alpha:=m+1=n+\varepsilon+1>\beta:=n+2c+1.$

Raising both sides of (2.6) to power $1/\beta$ we obtain

$1\leq\frac{C_{4}\delta^{\frac{1}{\beta}}h^{\prime}(t)}{(h(t)^{\alpha}+C_{3}\delta)^{1/\beta}}.$

We integrate between $0$ and $T_{0}$ and make the change of variables
$x=h(t)\delta^{-1/\alpha}$ to conclude $T_{0}\leq C_{5}\delta^{1/\alpha}\leq
C_{5}\left(\int_{X}(\phi-\varphi)_{+}d\mu\right)^{\tau}$, with
$\tau=\gamma/\alpha$. Letting $T_{0}\to T_{\max}$ we obtain the desired
estimate. ∎

##  3\. Refinements and extensions

We explain now how minor modifications of the proof of Theorem 2.1 provide
other important uniform estimates in various contexts of Kähler geometry.

###  3.1. Big cohomology classes

Let $\theta$ be a smooth closed $(1,1)$-form that represents a big cohomology
class $\alpha$. We set

$V_{\theta}(x):=\sup\\{v(x),\;v\in{\rm PSH}(X,\theta)\text{ with }v\leq 0\\}.$

The latter is a $\theta$-psh function with minimal singularities, i.e. any
other $\theta$-psh function $\varphi$ satisfies $\varphi\leq V_{\theta}+C$ for
some constant $C$. It is locally bounded in the ample locus ${\rm
Amp}(\alpha)$, a Zariski open subset of $X$ where the cohomology class
$\alpha$ behaves like a Kähler class.

The Monge-Ampère measure $(\theta+dd^{c}\varphi)^{n}$ of a $\theta$-psh
function with minimal singularities is well defined in ${\rm Amp}(\alpha)$,
and one can show that it has finite mass independent from $\varphi$ and equal
to

$V_{\alpha}={\rm Vol}(\alpha)=\int_{{\rm
Amp}(\alpha)}(\theta+dd^{c}V_{\theta})^{n}>0,$

the volume of the class $\alpha$.

We refer the reader to [BEGZ10] for more details on these notions and focus
here on slightly extending [BEGZ10, Theorem B] by our new approach:

######  Theorem 3.1.

Let $\mu$ be a probability measure on $X$. If ${\rm PSH}(X,\theta)\subset
L^{m}(\mu)$ for some $m>n$, then there exists a unique $\varphi\in{\rm
PSH}(X,\theta)$ with minimal singularities such that
$V_{\alpha}^{-1}(\theta+dd^{c}\varphi)^{n}=\mu$ and $\sup_{X}\varphi=0$.
Moreover

$||\varphi-V_{\theta}||_{L^{\infty}(X)}\leq T_{\mu}$

for some uniform constant $T_{\mu}$.

###### Proof.

It follows from [BEGZ10, Theorem A] that there exists a unique finite energy
solution $\varphi$. The key point for us here is to establish the a priori
estimate. Note that $\varphi\leq V_{\theta}$ since $\sup_{X}\varphi=0$ Our
goal is to show that $V_{\theta}-T_{max}\leq\varphi$, obtaining a uniform
upper bound on $T_{max}$.

A difficulty lies in the fact that $\theta$ is not a positive form. We
consider the positive current $\omega=\theta+dd^{c}V_{\theta}$ and set
$\tilde{\varphi}=\varphi-V_{\theta}\leq 0$. Observe that

$\theta_{\varphi}:=\theta+dd^{c}\varphi=\omega+dd^{c}\tilde{\varphi}=:\omega_{\tilde{\varphi}}\geq
0.$

Our plan is thus to show that the ”$\omega$-psh” function $\tilde{\varphi}$ is
bounded.

As in the proof of Theorem 2.1 we let
$\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ denote a concave increasing
function such that $\chi(0)=0$ and $\chi^{\prime}(0)=1$. We set
$\psi=V_{\theta}+\chi\circ\tilde{\varphi}$ and consider

$u=P_{\theta}(\psi)=P_{\theta}(V_{\theta}+\chi\circ(\varphi-V_{\theta})).$

Observe that

$\displaystyle\theta+dd^{c}\psi$ $\displaystyle=$
$\displaystyle\chi^{\prime}\circ\tilde{\varphi}\,\theta_{{\varphi}}+[1-\chi^{\prime}\circ\tilde{\varphi}]\omega+\chi^{\prime\prime}\circ\tilde{\varphi}\,d\tilde{\varphi}\wedge
d^{c}\tilde{\varphi}$ $\displaystyle\leq$
$\displaystyle\chi^{\prime}\circ(\varphi-V_{\theta})\,(\theta+dd^{c}\varphi).$

The envelopes in the context of big cohomology classes enjoy similar
properties as those reviewed in Section 1.2. In particular the complex Monge-
Ampère measure $(\theta+dd^{c}P_{\theta}(\psi))^{n}$ is concentrated on the
contact set $\\{P_{\theta}(\psi)=\psi\\}$ (see [GLZ19, Theorem 2.7]) and the
big-version of Lemma 1.7 holds, showing that

$V_{\alpha}^{-1}(\theta+dd^{c}u)^{n}\leq
1_{\\{P_{\theta}(\psi)=\psi\\}}(\chi^{\prime}\circ(\varphi-
V_{\theta}))^{n}\mu.$

The rest of the proof is identical to that of Theorem 2.1. ∎

###  3.2. Degenerating families

Families of Kähler-Einstein varieties have been intensively studied in the
past decade, requiring one to analyze the associated family of complex Monge-
Ampère equations. We refer the reader to [Tos09, Tos10, ST12, GTZ13, DnGG20,
Li20] for detailed examples and geometrical motivations.

The most delicate situation is when the volume of the fiber collapses. Theorem
2.1 yields a uniform bound in this case, providing an alternative proof and an
extension of the main results of [EGZ08, DP10]:

######  Corollary 3.2.

Let $\omega_{t}$ be a family of semi-positive and big forms on $X$, and assume
there is a fixed form $\Theta$ such that $0\leq\omega_{t}\leq\Theta$. We let
$V_{t}:=\int_{X}\omega_{t}^{n}>0$ denote the volume of $(X,\omega_{t})$. Let
$\mu$ be a probability measure. If ${\rm PSH}(X,\Theta)\subset L^{m}(\mu)$ for
some $m>n$, then any solution $\varphi_{t}\in{\rm PSH}(X,\omega_{t})\cap
L^{\infty}(X)$ to

$\frac{1}{V_{t}}(\omega_{t}+dd^{c}\varphi_{t})^{n}=\mu$

satisfies ${\rm Osc}_{X}(\varphi_{t})\leq T_{\mu}$ for some uniform constant
$T_{\mu}$.

The point here is that the estimate is uniform in $t$ although the volumes
$V_{t}$ may degenerate to zero (volume collapsing).

###### Proof.

Theorem 2.1 provides a uniform bound ${\rm Osc}_{X}(\varphi_{t})\leq
T(A_{m}(\omega_{t},\mu))$, where

$A_{m}(\omega_{t},{\mu}):=\sup\left\\{\int_{X}(-\psi)^{m}d\mu,\;\psi\in{\rm
PSH}(X,\omega_{t})\text{ with }\sup_{X}\psi=0\right\\}.$

Since ${\rm PSH}(X,\omega_{t})\subset{\rm PSH}(X,\Theta)$ and ${\rm
PSH}(X,\Theta)\subset L^{m}(\mu)$, we obtain that $A_{m}(\omega_{t},{\mu})\leq
A_{m}(\Theta,{\mu})<+\infty$. The uniform upper bound follows. ∎

This uniform estimate shows in particular that in many geometrical contexts, a
uniform control on the $L^{n+\varepsilon}$-norm of the Monge-Ampère potentials
$\varphi_{t}$ suffices to obtain a $L^{\infty}$-control of the latter.

One can obtain similarly uniform estimates when the underlying complex
structure is also changing: let ${\mathcal{X}}$ be an irreducible and reduced
complex Kähler space, and let $\pi:{\mathcal{X}}\rightarrow\mathbb{D}$ denote
a proper, surjective holomorphic map such that each fiber $X_{t}=\pi^{-1}(t)$
is an $n$-dimensional, reduced, irreducible, compact Kähler space, for any
$t\in\mathbb{D}$. Given $\omega$ a Kähler form on ${\mathcal{X}}$ and
$\omega_{t}:=\omega_{|X_{t}}$, one can consider the complex Monge-Ampère
equations

$\frac{1}{V}(\omega_{t}+dd^{c}\varphi_{t})^{n}=\mu_{t},$

where

  * •

the volume $V=\int_{X_{t}}\omega_{t}^{n}$ turns out to be independent of $t$,
and

  * •

$\mu_{t}$ is a family of probability measures on each fiber $X_{t}$ (e.g. the
normalized Calabi-Yau measures of a degenerating family of Calabi-Yau
manifolds).

In many concrete geometrical situations (see e.g. [GTZ13, DnGG20, Li20]), one
can check that $A_{m}(\omega_{t},\mu_{t})\leq A$ is uniformly bounded from
above for some $m>n$ (often any $m>1$). If one can further uniformly compare
$\sup_{X_{t}}\varphi_{t}$ and
$\int_{X_{t}}\varphi_{t}\frac{\omega_{t}^{n}}{V}$, then Theorem 2.1 then
applies and provides a uniform $L^{\infty}$-estimate. It is thus sometimes not
necessary to establish a uniform Skoda integrability theorem in families
(compare with [DnGG20, Li20b]).

###  3.3. Relative a priori $L^{\infty}$-bounds

Fix $\omega$ a semi-positive and big $(1,1)$ form, and $\rho$ an $\omega$-psh
function with analytic singularities such that
$\omega+dd^{c}\rho\geq\delta\omega_{X}$ is a Kähler current which is smooth in
the ample locus ${\rm Amp}(\omega)$. We normalize $\rho$ so that
$\sup_{X}\rho=0$ and set $V=\int_{X}\omega^{n}>0$.

We consider in this section the degenerate complex Monge-Ampère equation

(3.1) $V^{-1}(\omega+dd^{c}\varphi)^{n}=\mu=fdV_{X},$

where $\mu$ is a probability measure whose density $f\in L^{1}(X)$ does not
belong to any good Orlicz class (see section 2.2). Since $\mu$ does not charge
pluripolar sets, there exists a unique ”finite energy solution”
$\varphi\in{\mathcal{E}}(X,\omega)$ (see [GZ]), but one cannot expect
anylonger that $\varphi$ is globally bounded on $X$.

Given $\psi$ a quasi-plurisubharmonic function on $X$ and $c>0$, we set

$E_{c}(\psi):=\\{x\in X,\;\nu(\psi,x)\geq c\\},$

where $\nu(\psi,x)$ denotes the Lelong number of $\psi$ at $x$. A celebrated
theorem of Siu ensures that for any $c>0$, the set $E_{c}(\psi)$ is a closed
analytic subset of $X$.

######  Theorem 3.3.

Assume $f=ge^{-\psi}$, where $0\leq g\in L^{p}(dV_{X})$, $p>1$, and $\psi$ is
a quasi-psh function. Then there exists a unique
$\varphi\in\mathcal{E}(X,\omega)$ such that

  * •

$\alpha(\psi+\rho)-\beta\leq\varphi\leq 0$ with $\sup_{X}\varphi=0$;

  * •

$\varphi$ is locally bounded in the Zariski open set $\Omega:={\rm
Amp}(\omega)\setminus E_{\frac{1}{q}}(\psi)$;

  * •

$V^{-1}(\omega+dd^{c}\varphi)^{n}=fdV_{X}\;\;\text{ in }\;\;\Omega$,

where $\alpha,\beta>0$ depend on an upper bound for $||g||_{L^{p}}$ and
$\frac{1}{p}+\frac{1}{q}=1$.

When $f\leq e^{-\psi}$ for some quasi-psh function $\psi$, it has been shown
by DiNezza-Lu [DnL17, Theorem 2] that the normalized solution $\varphi$ to
(3.1) is locally bounded in the complement of the set $\\{\psi=-\infty\\}$.
The proof of DiNezza-Lu is a generalization of the method of Kołodziej [Kol98]
that makes use of a theory of generalized Monge-Ampère capacities further
developed in [DnL15]. We slightly extend this result here and propose a brand
new proof using envelopes and Theorem 2.1.

###### Proof.

Reduction to analytic singularities. We let $q$ denote the conjugate exponent
of $p$, set $r=\frac{2p}{p+1}$, and note that $1<r<p$. If the Lelong numbers
of $\psi$ are all less than $\frac{1}{q}$, it follows from Hölder inequality
that $f\in L^{r}(dV_{X})$, since

$\int_{X}f^{r}dV_{X}=\int_{X}g^{r}e^{-r\psi}dV_{X}\leq\left(\int_{X}g^{p}dV_{X}\right)^{\frac{r}{p}}\cdot\left(\int_{X}e^{-\frac{pr}{p-r}\psi}dV_{X}\right)^{\frac{p-r}{p}},$

where the last integral is finite by Skoda’s integrability theorem [GZ,
Theorem 8.11] if $\frac{pr}{p-r}\nu(\psi,x)<2$ for all $x\in X$, which is
equivalent to $\nu(\psi,x)<\frac{1}{q}$.

It is thus natural to expect that the solution $\varphi$ will be locally
bounded in the complement of the closed analytic set $E_{q^{-1}}(\psi)$. It
follows from Demailly’s equisingular approximation technique (see [Dem15])
that there exists a sequence $(\psi_{m})$ of quasi-psh functions on $X$ such
that

  * •

$\psi_{m}\geq\psi$ and $\psi_{m}\rightarrow\psi$ (pointwise and in $L^{1}$);

  * •

$\psi_{m}$ has analytic singularities concentrated along $E_{m^{-1}}(\psi)$;

  * •

$dd^{c}\psi_{m}\geq-K\omega_{X}$, for some uniform constant $K>0$;

  * •

$\int_{X}e^{2m(\psi_{m}-\psi)}dV_{X}<+\infty$ for all $m$.

We choose $m=[q]$, set $g_{m}:=ge^{\psi_{m}-\psi}$, and observe that

$\displaystyle\int_{X}g_{m}^{r}$ $\displaystyle\leq$
$\displaystyle\left(\int_{X}e^{2m(\psi_{m}-\psi)}dV_{X}\right)^{\frac{1}{2m}}\cdot\left(\int_{X}g_{m}^{\frac{2mr}{2m-r}}dV_{X}\right)^{\frac{2m-r}{2m}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}e^{2m(\psi_{m}-\psi)}dV_{X}\right)^{\frac{1}{2m}}\cdot\left(\int_{X}g_{m}^{p}dV_{X}\right)^{\frac{2m-r}{2m}}<+\infty$

if we choose $r^{-1}=p^{-1}+(2m)^{-1}<1$ so that $\frac{2mr}{2m-r}=p$. By
replacing $\psi$ by $\psi_{[q]}\geq\psi$ and $g$ by $g_{m}\in L^{r}$ in the
sequel, we can thus assume that

  * •

$\psi$ has analytic singularities and is smooth in $X\setminus
E_{q^{-1}}(\psi)$;

  * •

the functions $\tilde{\psi}:=a\psi+\rho$ is $\omega$-psh, with $a:=\delta/K$.

Uniform integrability of $\varphi$. It is a standard measure theoretic fact
that the density $f$ belongs to an Orlicz class $L^{w}$ for some convex
increasing weight $w:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
$w(t)/t\rightarrow+\infty$ as $t\rightarrow+\infty$. Set
$\chi_{1}(t):=-(w^{*})^{-1}(-t)$, where $w^{*}$ denotes the Legendre transform
of $w$. Thus $\chi_{1}:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ is a convex
increasing weight such that $\chi_{1}(-\infty)=-\infty$ and

$\int_{X}(-\chi_{1}\circ\varphi)(\omega+dd^{c}\varphi)^{n}\leq\int_{X}w\circ
fdV_{X}+\int_{X}(-\varphi)dV_{X}\leq C_{0},$

as follows from the additive version of Hölder-Young inequality and the
compactness of $\sup$-normalized $\omega$-psh functions.

It follows that $\varphi$ belongs to a compact subset of the finite energy
class ${\mathcal{E}}_{\chi_{1}}(X,\omega)$, hence for all
$\lambda\in\mathbb{R}$,

(3.2) $\int_{X}\exp(-\lambda\varphi)dV_{X}\leq C_{\lambda},$

for some $C_{\lambda}$ independent of $\varphi$ (see [GZ07, GZ] for more
information).

The envelope construction. Let $u=P(2\varphi-\tilde{\psi})$ denote the
greatest $\omega$-psh function that lies below $2\varphi-\tilde{\psi}$. Since
$h=2\varphi-\tilde{\psi}$ is bounded from below and quasi-continuous, it
follows from Proposition 1.6 that the measure $(\omega+dd^{c}u)^{n}$ is
supported on the contact set ${\mathcal{C}}=\\{u=2\varphi-\tilde{\psi}\\}$.
Thus

$(\omega+dd^{c}u)^{n}\leq
1_{{\mathcal{C}}}(\omega+dd^{c}(2\varphi-\tilde{\psi}))^{n}\leq
1_{{\mathcal{C}}}(2\omega+dd^{c}(2\varphi))^{n}.$

Since $v\leq w$ on $X$, it follows from Lemma 1.3 that

(3.3) $1_{\\{v=w\\}}(2\omega+dd^{c}v)^{n}\leq
1_{\\{v=w\\}}(2\omega+dd^{c}w)^{n},$

where

  * •

$v=u+\tilde{\psi}$ is $2\omega$-psh and $u+\tilde{\psi}\leq 2\varphi=w$ on
$X$;

  * •

$\\{u+\tilde{\psi}=2\varphi\\}$ coincides with the contact set
${\mathcal{C}}$.

Therefore, it follows from (3.3) that

$\displaystyle 1_{{\mathcal{C}}}(2\omega+dd^{c}(u+\tilde{\psi}))^{n}$
$\displaystyle\leq$ $\displaystyle
1_{{\mathcal{C}}}(2\omega+dd^{c}(2\varphi))^{n}$ $\displaystyle\leq$
$\displaystyle 1_{{\mathcal{C}}}2^{n}cge^{-\psi}dV_{X}$ $\displaystyle\leq$
$\displaystyle 1_{{\mathcal{C}}}2^{n}cge^{u/a}e^{-2\varphi/a}dV_{X}\leq
c_{1}ge^{-2\varphi/a}dV_{X},$

using that $\sup_{X}u\leq c_{2}$ is uniformly bounded from above, as we
explain below.

It follows from Hölder inequality and (3.2) that the measure
$ge^{-2\varphi/a}dV_{X}$ satisfies the assumption of Theorem 2.1. We infer
that $u\geq-M$ is uniformly bounded below, hence

$2\varphi=(2\varphi-\tilde{\psi})+\tilde{\psi}\geq
u+\tilde{\psi}\geq\frac{\delta}{K}\psi+\rho-M.$

The desired a priori estimate follows with $\beta=M/2$ and
$\alpha=\max(1,\delta/2K)$.

Bounding $\sup_{X}u$ from above. We can assume without loss of generality that
$\sup_{X}\tilde{\psi}=0$. Consider $G=\\{\tilde{\psi}>-1\\}$; this is a non
empty plurifine open set. Observe that for all $x\in G$,
$u(x)\leq(2\varphi-\tilde{\psi})(x)\leq 1$, hence

$u(x)-1\leq V_{G,\omega}(x):=\sup\\{w(x),\,w\in{\rm PSH}(X,\omega)\text{ with
}w\leq 0\text{ on }G\\}.$

It follows from [GZ, Theorem 9.17.1] that $\sup_{X}V_{G,\omega}=C$ is finite
since $G$ is non-pluripolar, thus $\sup_{X}u\leq
c_{2}=1+\sup_{X}V_{G,\omega}=1+C$. ∎

##  4\. The local context

###  4.1. Cegrell classes

We fix $\Omega\subset\mathbb{C}^{n}$ a bounded hyperconvex domain, i.e. there
exists a continuous plurisubharmonic function $\rho:\Omega\rightarrow[-1,0)$
whose sublevel sets $\\{\rho<-c\\}\Subset\Omega$ are relatively compact for
all $c>0$.

Let ${\mathcal{T}}(\Omega)$ denote the set of bounded plurisubharmonic
functions $u$ in $\Omega$ such that $\lim_{z\to\zeta}u(z)=0,$ for every
$\zeta\in\partial\Omega,$ and $\int_{\Omega}(dd^{c}u)^{n}<+\infty$. Cegrell
[Ceg98, Ceg04] has studied the complex Monge-Ampère operator
$(dd^{c}\cdot)^{n}$ and introduced different classes of plurisubharmonic
functions on which the latter is well defined:

  * •

${\rm DMA}(\Omega)$ is the set of psh functions $u$ such that for all
$z_{0}\in\Omega$, there exists a neighborhood $V_{z_{0}}$ of $z_{0}$ and
$u_{j}\in{\mathcal{T}}(\Omega)$ a decreasing sequence which converges to $u$
in $V_{z_{0}}$ and satisfies $\sup_{j}\int_{\Omega}(dd^{c}u_{j})^{n}<+\infty$.

  * •

a function $u$ belongs to ${\mathcal{F}}(\Omega)$ iff there exists
$u_{j}\in{\mathcal{T}}(\Omega)$ a sequence decreasing towards $u$ in all of
$\Omega$, which satisfies $\sup_{j}\int_{\Omega}(dd^{c}u_{j})^{n}<+\infty$;

  * •

a function $u$ belongs to ${\mathcal{E}}^{p}(\Omega)$ if there exists a
sequence of functions $u_{j}\in{\mathcal{T}}(\Omega)$ decreasing towards $u$
in $\Omega$ with $\sup_{j}\int_{\Omega}(-u_{j})^{p}(dd^{c}u_{j})^{n}<+\infty$.

  * •

a function $u$ belongs to ${\mathcal{F}}^{p}(\Omega)$ if there exists a
sequence of functions $u_{j}\in{\mathcal{T}}(\Omega)$ decreasing towards $u$
in $\Omega$ with
$\sup_{j}\int_{\Omega}[1+(-u_{j})^{p}](dd^{c}u_{j})^{n}<+\infty$.

Given $u\in\mathcal{E}^{p}(\Omega)$ we define the weighted energy of $u$ by

$E_{p}(u):=\int_{\Omega}(-u)^{p}(dd^{c}u)^{n}<+\infty.$

The operator $(dd^{c}\cdot)^{n}$ is well defined on these sets, and continuous
under decreasing limits. If $u\in\mathcal{E}^{p}(\Omega)$ for some $p>0$ then
$(dd^{c}u)^{n}$ vanishes on all pluripolar sets [BGZ09, Theorem 2.1]. If
$u\in\mathcal{E}^{p}(\Omega)$ and $\int_{\Omega}(dd^{c}u)^{n}<+\infty$ then
$u\in\mathcal{F}^{p}(\Omega)$. Also, note that

${\mathcal{T}}(\Omega)\subset{\mathcal{F}}^{p}(\Omega)\subset{\mathcal{F}}(\Omega)\subset{\rm
DMA}(\Omega)\;\;\text{ and
}\;\;{\mathcal{T}}(\Omega)\subset{\mathcal{E}}^{p}(\Omega)\subset{\rm
DMA}(\Omega).$

Cegrell has characterized the range of the complex Monge-Ampère operator
acting on the classes ${\mathcal{E}}^{p}(\Omega)$:

######  Theorem 4.1.

[Ceg98, Theorem 5.1] Let $\mu$ be a probability measure in $\Omega$. There
exists a function $u\in{\mathcal{F}}^{p}(\Omega)$ such that
$(dd^{c}u)^{n}=\mu$ if and only if ${\mathcal{F}}^{p}(\Omega)\subset
L^{p}(\Omega)$.

A simplified variational proof of this result has been provided in [ACC12].

###  4.2. Dirichlet problem

We have the following local analogue of Theorem 2.1:

######  Theorem 4.2.

Assume $\mu$ is a probability measure in $\Omega$ and
$\mathcal{F}(\Omega)\subset L^{m}(\mu)$, for some $m>n$. Then there exists a
unique bounded function $u\in\mathcal{F}(\Omega)$ such that
$(dd^{c}u)^{n}=\mu$. The upper bound on $\sup_{\Omega}|u|$ only depends on
$A_{m}(\mu),m,n$, where

$A_{m}(\mu):=\sup\left\\{\int_{\Omega}(-u)^{m}d\mu\;:\;u\in\mathcal{T}(\Omega)\text{
with }\int_{\Omega}(dd^{c}u)^{n}\leq 1\right\\}.$

###### Proof.

We first explain why the integrability condition $\mathcal{F}(\Omega)\subset
L^{m}(\Omega,\mu)$ is equivalent to the finiteness of $A_{m}$. Indeed, if
$A_{m}$ is not finite then there exists a sequence $(u_{j})$ in
$\mathcal{T}(\Omega)$ such that $\int_{\Omega}(dd^{c}u_{j})^{n}\leq 1$ but
$\int_{\Omega}|u_{j}|^{m}d\mu\geq 4^{jm}$. Let
$u:=\sum_{j=1}^{+\infty}2^{-j}u_{j}$. Then, by [Ceg04, Corollary 5.6], we have
$u\in\mathcal{F}(\Omega)$, but

$\int_{\Omega}(-u)^{m}d\mu\geq 2^{-jm}\int_{\Omega}(-u_{j})^{m}d\mu\geq
2^{jm}\to+\infty.$

It follows from Theorem 4.1 that there exists $\varphi\in\mathcal{F}(\Omega)$
such that $(dd^{c}\varphi)^{n}=\mu$. We assume for the moment that
$u\in\mathcal{T}$ is bounded and we establish a uniform bound for $\varphi$.
Set

$T_{\max}:=\sup\\{t>0\;:\;\mu(\varphi<-t)>0\\}.$

Our goal is to establish a precise bound on $T_{max}$. By definition,
$-T_{max}\leq\varphi$ almost everywhere with respect to $\mu$, hence
$(dd^{c}\max(\varphi,-T_{\max}))^{n}\geq(dd^{c}\varphi)^{n}$ and the
domination principle, [GZ, Corollary 3.31], gives $\varphi\geq-T_{\max}$,
providing the desired a priori bound $|\varphi|\leq T_{max}$.

We let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ denote a concave
increasing function such that $\chi(0)=0$ and $\chi^{\prime}(0)=1$. We set
$\psi=\chi\circ\varphi$, $u=P(\psi)\in\mathcal{T}(\Omega)$ the largest psh
function in $\Omega$ which lies below $\psi$, and observe that

$\displaystyle dd^{c}\psi$ $\displaystyle=$
$\displaystyle\chi^{\prime}\circ\varphi\omega_{\varphi}+\chi^{\prime\prime}\circ\varphi
d\varphi\wedge d^{c}\varphi\leq\chi^{\prime}\circ\varphi dd^{c}\varphi.$

Since $\psi\geq\chi^{\prime}(-T_{\max})\varphi$ and the latter is in
$\mathcal{T}(\Omega)$ we deduce that $u\geq\chi^{\prime}(-T_{\max})\varphi$
and $u\in\mathcal{T}(\Omega)$.

Although the function $\psi$ is not psh, this provides a bound from above on
the positivity of $dd^{c}\psi$ which allows to control the Monge-Ampère of its
envelope, see [DnGL20, Lemma 4.1 and Lemma 4.2],

$(dd^{c}u)^{n}\leq{\bf
1}_{\\{u=\psi\\}}(dd^{c}\psi)^{n}\leq(\chi^{\prime}\circ\varphi)^{n}\mu.$

The above inequalities hold for smooth functions and the general case of
bounded psh functions can be obtained as in the proof of Lemma 1.7.

We thus get a uniform control on the Monge-Ampère mass of $u$:

$\displaystyle\int_{\Omega}(dd^{c}u)^{n}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}(\chi^{\prime}\circ\varphi)^{n}d\mu.$

We are going to choose below the weight $\chi$ in such a way that
$\int_{\Omega}(\chi^{\prime}\circ\varphi)^{n}d\mu=B\leq 2$ is a finite
constant under control. This provides a uniform upper bound on
$||u||_{L^{m}(\mu)}$. Using Chebyshev inequality we thus obtain

(4.1)
${\mu}(\varphi<-t)\leq\frac{\int_{\Omega}|\chi(\varphi)|^{m}d\mu}{|\chi|^{m}(-t)}\leq\frac{\int_{\Omega}|u|^{m}d\mu}{|\chi|^{m}(-t)}\leq\frac{A_{m}}{|\chi|^{m}(-t)},$

where $A_{m}\geq 1$ is an upper bound for $\int_{\Omega}|u|^{m}d\mu$.

Choice of $\chi$. We use again Lebesgue’s formula: if
$g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is increasing and normalized by
$g(0)=1$ then

$\int_{\Omega}g\circ(-\varphi)d\mu=\mu(\Omega)+\int_{0}^{T_{\max}}g^{\prime}(t){\mu}(\varphi<-t)dt.$

Setting $g(t)=[\chi^{\prime}(-t)]^{n}$ we define $\chi$ by imposing
$\chi(0)=0$, $\chi^{\prime}(0)=1$, and

$g^{\prime}(t)=\begin{cases}\dfrac{1}{(1+t)^{2}{\mu}(\varphi<-t)},\;\text{if}\;t\in[0,T_{0}],\\\
\;\\\ \dfrac{1}{t^{2}+1},\;\text{if}\;t>T_{0}.\end{cases}$

This choice guarantees that

$\int_{\Omega}(\chi^{\prime}\circ\varphi)^{n}d\mu\leq\mu(\Omega)+\int_{0}^{+\infty}\frac{dt}{(1+t)^{2}}=2.$

Conclusion. We set $h(t)=-\chi(-t)$ and work with the positive counterpart of
$\chi$. Note that $h(0)=0$ and $h^{\prime}(t)=[g(t)]^{\frac{1}{n}}$ is
positive increasing, hence $h$ is convex increasing (so $\chi$ is concave
increasing and negative).

Together with (4.1) our choice of $\chi$ yields, for all $t\in[0,T_{0}]$,

$\frac{1}{(1+t)^{2}g^{\prime}(t)}={\mu}(\varphi<-t)\leq\frac{A_{m}}{h^{m}(t)}.$

This reads

$h^{m}(t)\leq
A_{m}(1+t)^{2}g^{\prime}(t)=nA_{m}(1+t)^{2}h^{\prime\prime}(t)(h^{\prime})^{n-1}(t).$

We integrate this inequality as in the proof of Theorem 2.1 and obtain

$T_{0}\leq C^{\prime},$

for some uniform constant $C^{\prime}$ depending on $n,m,A_{m}$.

To finish the proof we write $\mu=f(dd^{c}\phi)^{n}$, where $0\leq f\in
L^{1}(\Omega,(dd^{c}\phi)^{n})$ and $\phi\in\mathcal{T}(\Omega)$. This is
known as Cegrell’s decomposition theorem [Ceg98, Theorem 6.3]. We next solve
$(dd^{c}\varphi_{j})^{n}=\min(f,j)(dd^{c}\phi)^{n}$ with
$\varphi_{j}\in\mathcal{T}(\Omega)$. Since $(dd^{c}\varphi_{j})^{n}\leq\mu$,
our estimate above shows that $|\varphi_{j}|\leq C$ for a uniform constant
$C$. The comparison principle also gives that $\varphi_{j}$ is decreasing and
$\varphi\leq\varphi_{j}$, thus $u:=\lim_{j}\varphi_{j}\in\mathcal{F}(\Omega)$
is bounded and $(dd^{c}u)^{n}=\mu$. It then follows from [Ceg04, Theorem 5.15]
that $u=\varphi$, finishing the proof. ∎

## References

  * [ACC12] P. Åhag, U. Cegrell, R. Czyż, _On Dirichlet’s principle and problem._ Math. Scand. 110 (2012), no. 2, 235–250.
  * [Aub78] T. Aubin, _Equation de type Monge-Ampère sur les variétés kählériennes compactes._ Bull. Sci. Math. 102 (1978), 63–95.
  * [BT76] E. Bedford, B.A. Taylor, _The Dirichlet problem for a complex Monge-Ampère equation_. Invent. Math. 37 (1976), no. 1, 1–44.
  * [BT82] E. Bedford, B. A. Taylor, _A new capacity for plurisubharmonic functions_. Acta Math. 149 (1982), no. 1-2, 1–40.
  * [BGZ09] S. Benelkourchi, V. Guedj, A. Zeriahi, _Plurisubharmonic functions with weak singularities_ , Complex analysis and digital geometry, Acta Univ. Upsaliensis, Skr. Uppsala Univ. C Organ. Hist., vol. 86, Uppsala Universitet, Uppsala, 2009, 57–74.
  * [Ber19] R. J. Berman, _From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit_ , Math. Z. 291 (2019), no. 1-2, 365–394.
  * [BBEGZ] R.J.Berman, S.Boucksom, P.Eyssidieux, V.Guedj, A.Zeriahi, _Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties_ , J. Reine Ang. Math. 751 (2019), 27–89.
  * [BBGZ13] R. J. Berman, S. Boucksom, V. Guedj, and A. Zeriahi, _A variational approach to complex Monge-Ampère equations_ , Pub.Math. I.H.E.S. 117 (2013), 179–245.
  * [BBJ21] R. J. Berman, S.Boucksom, M.Jonsson, _A variational approach to the Yau-Tian-Donaldson conjecture_ , J. Amer. Math. Soc. (2021).
  * [Blo05] Z. Blocki, _On uniform estimate in Calabi-Yau theorem._ Sci. China Ser. A 48 (2005), suppl., 244–247.
  * [BEGZ10] S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, _Monge-Ampère equations in big cohomology classes_ , Acta Math. 205 (2010), no. 2, 199–262.
  * [Cao85] H.D.Cao, _Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds_ , Invent. Math. 81 (1985), no. 2, 359–372.
  * [Ceg98] U. Cegrell, _Pluricomplex energy_ , Acta Math. 180 (1998), no. 2, 187–217.
  * [Ceg04] U. Cegrell, _The general definition of the complex Monge-Ampère operator_ , Ann. Inst. Fourier (Grenoble) 54 (2004), no. 1, 159–179.
  * [CC21] X.X. Chen, J.Cheng _On the constant scalar curvature Kähler metrics._ J. Amer. Math. Soc. (2021).
  * [CDS15] X.X. Chen, S. Donaldson, S. Sun, _Kähler-Einstein metrics on Fano manifolds, I, II & III _, J. Amer. Math. Soc. 28 (2015), 183-197, 199-234 & 235-278.
  * [CZ19] J. Chu, B. Zhou, _Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds_ , Sci. China Math. 62 (2019), no. 2, 371–380.
  * [Dem15] J.P. Demailly, On the cohomology of pseudoeffective line bundles. Complex geometry and dynamics, Abel Symp., vol. 10, Springer, Cham, 2015, pp. 51-99.
  * [Dem] J.P. Demailly, _Analytic methods in algebraic geometry_ , Surveys of Modern Mathematics, 1. International Press; Higher Education Press, Beijing, 2012. viii+231 pp.
  * [DP10] J.P. Demailly, N. Pali, _Degenerate complex Monge-Ampère equations over compact Kähler manifolds._ Internat. J. Math. 21 (2010), no. 3, 357–405.
  * [DP04] J.P. Demailly, M. Păun, _Numerical characterization of the Kähler cone of a compact Kähler manifold._ Ann. of Math. (2) 159 (2004), no. 3, 1247-1274.
  * [Din09] S. Dinew, _Uniqueness in $\mathcal{E}(X,\omega)$_, J. Funct. Anal. 256 (2009), no. 7, 2113–2122.
  * [DnGG20] E. Di Nezza, V. Guedj, H. Guenancia, _Families of singular Kähler-Einstein metrics._ Preprint arXiv:2003.08178. To appear in Journal of the E.M.S.
  * [DnGL20] E. Di Nezza, V. Guedj, C. H. Lu, Finite entropy vs finite energy. Preprint arXiv:2006.07061. To appear in Commentarii Math. Helvetici.
  * [DnL15] E. Di Nezza, C. H. Lu, _Generalized Monge-Ampère capacities_. Int. Math. Res. Not. IMRN 2015, no. 16, 7287–7322.
  * [DnL17] E. Di Nezza, C. H. Lu, _Complex Monge-Ampère equations on quasi-projective varieties_ , J. Reine Angew. Math. 727 (2017), 145–167.
  * [Don18] S. Donaldson, _Some recent developments in Kähler geometry and exceptional holonomy_ , Proc. Int. Congress of Mathematicians, Rio de Janeiro 2018. Vol. I. Plenary lectures, 425–451, World Sci. Publ., Hackensack, NJ, 2018.
  * [DS14] S. Donaldson, S. Sun, _Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry_ , Acta Math. 213 (2014), no. 1, 63-106.
  * [EGZ08] P. Eyssidieux, V. Guedj, A. Zeriahi, _A priori $L^{\infty}$-estimates for degenerate complex Monge-Ampère equations_, I.M.R.N., Vol. 2008, Article ID rnn070, 8 pages.
  * [EGZ09] P. Eyssidieux, V. Guedj, A. Zeriahi, _Singular Kähler-Einstein metrics_ , J. Amer. Math. Soc. 22 (2009), no. 3, 607–639.
  * [GTZ13] M. Gross, V. Tosatti, Y. Zhang, _Collapsing of abelian fibered Calabi-Yau manifolds._ Duke Math. J. 162 (2013), no. 3, 517-551.
  * [GL21] V. Guedj, C. H. Lu, _Quasi-plurisubharmonic envelopes 2 and 3_ , Preprints (2021).
  * [GLZ19] V. Guedj, C. H. Lu, A. Zeriahi, _Plurisubharmonic envelopes and supersolutions_ , J. Differential Geom. 113 (2019), no. 2, 273–313.
  * [GZ07] V. Guedj, A. Zeriahi, _The weighted Monge-Ampère energy of quasiplurisubharmonic functions_ , J. Funct. Anal. 250 (2007), no. 2, 442–482.
  * [GZ12] V. Guedj, A. Zeriahi, _Stability of solutions to complex Monge-Ampère equations in big cohomology classes._ Math. Res. Lett. 19 (2012), no. 5, 1025-1042.
  * [GZ] V. Guedj, A. Zeriahi, _Degenerate complex Monge-Ampère equations_ , EMS Tracts in Mathematics, vol. 26, European Mathematical Society (EMS), Zürich, 2017.
  * [GPT21] B. Guo, D.H. Phong, F. Tong, _On $L^{\infty}$-estimates for complex Monge-Ampère equations._ Preprint arXiv:2106.02224.
  * [Kol98] S. Kołodziej, _The complex Monge-Ampère equation_ , Acta Math. 180 (1998), 69–117.
  * [LTW20] C. Li, G. Tian, F. Wang, On Yau-Tian-Donaldson conjecture for singular Fano varieties, Comm. Pure Appl. Math. (2020).
  * [Li20] Yang Li, Metric SYZ conjecture and non-archimedean geometry. arXiv:2007.01384
  * [Li20b] Yang Li, Uniform Skoda integrability and Calabi-Yau degeneration. arXiv:2006.16961
  * [RR] M.M. Rao, Z.D. Ren, _Theory of Orlicz Spaces,_ Monogr. Textbooks Pure Appl. Math., vol. 146, Dekker, New York,1991, xii+449 pp
  * [ST12] J. Song, G. Tian, _Canonical measures and Kähler-Ricci flow._ J. Amer. Math. Soc. 25 (2012), no. 2, 303-353.
  * [Szek18] G. Szekelyhidi, _Fully non-linear elliptic equations on compact Hermitian manifolds._ J. Differential Geom. 109 (2018), no. 2, 337–378.
  * [Tos09] V. Tosatti, _Limits of Calabi-Yau metrics when the Kähler class degenerates._ J. Eur. Math. Soc. 11 (2009), no. 4, 755-776.
  * [Tos10] V. Tosatti, _Adiabatic limits of Ricci-flat Kähler metrics._ J. Differential Geom. 84 (2010), no. 2, 427-453.
  * [Yau78] S. T. Yau, _On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I._ Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411.