# Solutions to degenerate complex Hessian equations

Lu Hoang Chinh

###### Abstract.

Let $(X,\omega)$ be an $n$-dimensional compact Kähler manifold. We study
degenerate complex Hessian equations of the form
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n}.$ Under
some natural conditions on $F$, this equation has a unique continuous
solution. When $(X,\omega)$ is rational homogeneous we further show that the
solution is Hölder continuous.

##  1\. Introduction

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$. Fix an
integer $m$ between $1$ and $n$, and let $d,d^{c}$ denote the usual real
differential operators
$d:=\partial+\bar{\partial},d^{c}=\frac{\sqrt{-1}}{2\pi}(\bar{\partial}-\partial)$
so that $dd^{c}=\frac{i}{\pi}\partial\bar{\partial}.$ We are studying
degenerate complex Hessian equations of the form

(1.1) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n},$

where the density $F:X\times\mathbb{R}\rightarrow\mathbb{R}^{+}$ satisfies
some natural integrability conditions (see Theorem A below).

The case $m=1$ corresponds to the Laplace equation and the case $m=n$
corresponds to degenerate complex Monge-Ampère equations which have been
studied intensively in recent years (see [Bl03, Bl05, Bl12, BGZ08, BK07,
EGZ09, GKZ08, GZ05, GZ07, Kol98, Kol02, Kol03, Kol05]). So, equation (1.1) is
a generalization of both Laplace and Monge-Ampère equations.

The non degenerate complex Hessian equation on compact Kähler manifold, where
$F(x,\varphi)=f(x)$, with $0<f\in\mathcal{C}^{\infty}(X)$, has been studied
recently in [H09, HMW10, Jb10, DK12]. In [H09] and [Jb10], the authors
independently solved this equation with a strong additional hypothesis,
assuming $(X,\omega)$ has non negative holomorphic bisectional curvature.
Later on, in [HMW10] an a priori $\mathcal{C}^{2}$ estimate was obtained
without curvature assumption. Recently, using this estimate and a blowing up
analysis suggested in [HMW10], Dinew and Kolodziej solved the equation in full
generality.

Following Blocki [Bl05] we develop a potential theory for the complex Hessian
equation on compact Kähler manifold. We define the class of
$(\omega,m)$-subharmonic functions which is a generalization of the class of
$\omega$-plurisubharmonic functions when $m=n.$ The definition of the complex
Hessian operator on bounded $(\omega,m)$-subharmonic functions is delicate due
to difficulties in regularization process.

To go around this difficulty, we introduce a capacity and use it to define the
concept of quasi-uniform convergence. This allows us to define a suitable
class of bounded and quasi-continuous $(\omega,m)$-subharmonic functions on
which the complex Hessian operator is well defined and continuous under quasi-
uniform convergence. We show that this definition coincides with the
definition in the spirit of Bedford and Taylor method for the complex Monge-
Ampère operator. A comparison principle and convergence results for this
operator are also established.

With these potential tools in hand, we then consider the degenerate complex
Hessian equation (1.1). The first main result of this paper is the following:

Theorem A. Let $(X,\omega)$ be a $n$-dimensional compact Kähler manifold. Fix
$1\leq m\leq n$. Let $F:X\times\mathbb{R}\rightarrow[0,+\infty)$ be a function
satisfying the following conditions:

(F1) for all $x\in X$, $t\mapsto F(x,t)$ is non-decreasing and continuous,

(F2) for any fixed $t\in\mathbb{R}$, there exists $p>n/m$ such that the
function $x\mapsto F(x,t)$ belongs to $L^{p}(X)$,

(F3) there exists $t_{0}\in\mathbb{R}$ such that
$\int_{X}F(.,t_{0})\omega^{n}=\int_{X}\omega^{n}.$

Then there exists a function
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ , unique up to an
additive constant, such that

$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n}.$

Moreover if $\forall x\in X,\ t\mapsto F(x,t)$ is increasing, then the
solution is unique.

Note that the condition (F3) is automatically satisfied if $F(.,-\infty)=0$
and $F(.,+\infty)=+\infty.$ An important particular case is the exponential
function $F(x,t)=f(x)e^{t}.$

A particular case of this result has been obtained in [DK11]. The key point in
their proof is a domination between volume and capacity. Our main result is
proved using this technique and the recent result in the smooth case [DK12].

When $(X,\omega)$ is rational homogeneous with $\omega$ being invariant under
the Lie group action, we can easily regularize $(\omega,m)$-subharmonic.
Adapting the techniques in [EGZ09] we obtain Hölder continuity of the
solution:

Theorem B. Under the same assumption as in Theorem A, assume further that
$(X,\omega)$ is rational homogeneous and $\omega$ is invariant under the Lie
group action. Then the unique solution is Hölder continuous with exponent
$0<\gamma<\frac{2(mp-n)}{mnp+2mp-2n}$.

When $m=n$ we get the same exponent $\gamma$ as in [EGZ09].

Acknowledgement. The paper is part of my Ph.D Thesis. I would like to express
my deep gratitude to my advisor, Professor Ahmed Zeriahi, for sacrificing his
very valuable time for me. I wish to express my sincere gratitude to Professor
Vincent Guedj for his very useful suggestions and discussions to improve the
paper. I also wish to say a special word of thanks to Professor Sébastien
Boucksom for his kind invitation to IMJ and useful discussions. This paper
owes much to their help and constant encouragement.

##  2\. Preliminaries

In this section we introduce the notion of $(\omega,m)$-subharmonic functions
following Blocki’s ideas [Bl05] (see also [DK11]). Using classical techniques
for plurisubharmonic functions we obtain similar results.

###  2.1. Elementary symmetric functions

First, we recall some basic properties of elementary symmetric functions (see
[Bl05], [CW01], [Ga59]). We use the notations in [Bl05]. Let $S_{k}$,
$k=1,...,n$ be the $k$-elementary symmetric function, that is, for
$\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}$,

$S_{k}(\lambda)=\sum_{1\leq i_{1}<i_{2}<...<i_{k}\leq
n}\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{k}}.$

Let $\Gamma_{k}$ denote the closure of the connected component of
$\\{S_{k}(\lambda)>0\\}$ containing $(1,...,1).$ It is easy to show that

$\Gamma_{k}=\big{\\{}\lambda\in\mathbb{R}^{n}\ \mid\
S_{k}(\lambda_{1}+t,...,\lambda_{n}+t)\geq 0,\ \forall t\geq 0\big{\\}}.$

and hence

$\Gamma_{k}=\big{\\{}\lambda\in\mathbb{R}^{n}\ \mid\ S_{j}(\lambda)\geq 0,\ \
\forall 1\leq j\leq k\big{\\}}.$

We have an obvious inclusion $\Gamma_{n}\subset...\subset\Gamma_{1}.$
By Gårding [Ga59] the set $\Gamma_{k}$ is a convex cone in $\mathbb{R}^{n}$
and $S_{k}^{1/k}$ is concave on $\Gamma_{k}.$ Let $\mathcal{H}$ denote the
vector space (over $\mathbb{R}$) of complex hermitian $n\times n$ matrices.
For $A\in\mathcal{H}$ we set

$\widetilde{S}_{k}(A)=S_{k}(\lambda(A)),$

where $\lambda(A)\in\mathbb{R}^{n}$ are is the vector of eigenvalues of $A.$
The function $\widetilde{S}_{k}$ can also be defined as the sum of all
principal minors of order $k$,

$\widetilde{S}_{k}(A)=\sum_{|I|=k}A_{II}.$

From the latter we see that $\widetilde{S}_{k}$ is a homogeneous polynomial of
order $k$ on $\mathcal{H}$ which is hyperbolic with respect to the identity
matrix $I$ (that is for every $A\in\widetilde{S}$ the equation
$\widetilde{S}_{k}(A+tI)=0$ has $n$ real roots; see [Ga59]). As in [Ga59] (see
also [Bl05]), the cone

$\widetilde{\Gamma}_{k}:=\big{\\{}A\in\mathcal{H}\ \mid\
\widetilde{S}_{k}(A+tI)\geq 0,\forall t\geq 0\big{\\}}=\\{A\in\mathcal{H}\
\mid\ \lambda(A)\in\Gamma_{k}\\}$

is convex and the function $\widetilde{S}_{k}^{1/k}$ is concave on
$\widetilde{\Gamma}_{k}.$

###  2.2. $\omega$-subharmonic functions

In this section, we consider $\Omega\subset X$ an open subset contained in a
local chart.

######  Definition 2.1.

A function $u\in L^{1}(\Omega)$ is called weakly $\omega$-subharmonic if

$dd^{c}u\wedge\omega^{n-1}\geq 0,$

in the weak sense of currents.

Thanks to Littman [Lit63] we have the following approximation properties.

######  Proposition 2.2.

Let $u$ be a weakly $\omega$-subharmonic function in $\Omega$. Then there
exists a one parameter family of functions $u_{h}$ with the following
properties: for every compact subset $\Omega^{\prime}\subset\Omega$

a) $u_{h}$ is smooth in $\Omega^{\prime}$ for $h$ sufficiently large,

b) $dd^{c}u_{h}\wedge\omega^{n-1}\geq 0$ in $\Omega^{\prime},$

c) $u_{h}$ is non increasing with increasing $h,$ and
$\lim_{h\to\infty}u_{h}(x)=u(x)$ almost every where in $\Omega^{\prime},$

d) $u_{h}$ is given explicitly as

$u_{h}(y)=\int_{\Omega}K_{h}(x,y)u(x)dx,$

where $K_{h}$ is a smooth non negative function and

$\int_{\Omega}K_{h}(x,y)dy\to 1,$

uniformly in $x\in\Omega^{\prime}.$

######  Definition 2.3.

A function $u$ is called $\omega$-subharmonic if it is weakly
$\omega$-subharmonic and for every $\Omega^{\prime}\Subset\Omega$,
$\lim_{h\to\infty}u_{h}(x)=u(x),\forall x\in\Omega^{\prime},$ where $u_{h}$ is
constructed as in Proposition 2.2.

######  Remark 2.4.

Any continuous weakly $\omega$-subharmonic function is $\omega$-subharmonic.
If $(u_{j})$ is a sequence of continuous $\omega$-subharmonic functions
decreasing to $u$ and if $u\neq-\infty$ then $u$ is $\omega$-subharmonic.
If $u$ is weakly $\omega$-subharmonic then the pointwise limit of $(u_{h})$ is
a $\omega$-subharmonic function.
Let $(u_{j})$ be a sequence of $\omega$-subharmonic functions and $(u_{j})$ is
uniformly bounded from above. Then $u:=(\limsup_{j}u_{j})^{\star}$ is
$\omega$-subharmonic. Where for a function $v$, $v^{\star}$ denotes the upper
semicontinuous regularization of $v.$

The following Hartogs lemma can be proved in the same way as in the case of
subharmonic functions.

######  Lemma 2.5.

Let $u_{t}(x),t>0$ be a family of non positive $\omega$-subharmonic functions
in $\Omega$ and $u_{t}$ is uniformly bounded in $L^{1}_{loc}(\Omega)$. Suppose
that for compact subset $K$ in $\Omega$ there exists a constant $C$ such that
$v(x)=[\limsup_{t\to+\infty}u_{t}(x)]^{\star}\leq C$ on $K.$ Then for every
$\epsilon>0,$ there exists $T_{\epsilon}$ such that $u_{t}(x)\leq C+\epsilon$
for $t\geq T_{\epsilon}$ and $x\in K.$

###  2.3. $(\omega,m)$-subharmonic functions

We associate real (1,1)-forms $\alpha$ in $\mathbb{C}^{n}$ with hermitian
matrices $[a_{j\bar{k}}]$ by

$\alpha=\frac{i}{\pi}\sum_{j,k}a_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}}.$

Then the canonical Kähler form $\beta$ is associated with the identity matrix
$I.$ It is easy to see that

$\binom{n}{k}\alpha^{k}\wedge\beta^{n-k}=\widetilde{S}_{k}(A)\beta^{n}.$

######  Definition 2.6.

Let $\alpha$ be a real $(1,1)$-form on $X$. We say that $\alpha$ is
$(\omega,m)$-positive at a given point $P\in X$ if at this point we have

$\alpha^{k}\wedge\omega^{n-k}\geq 0,\ \ \forall k=1,...,m.$

We say that $\alpha$ is $(\omega,m)$-positive if it is $(\omega,m)$-positive
at any point of $X.$

######  Remark 2.7.

Locally at $P\in X$ with local coordinates $z_{1},...,z_{n}$, we have

$\alpha=\frac{i}{\pi}\sum_{j,k}\alpha_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}},$

and

$\omega=\frac{i}{\pi}\sum_{j,k}g_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}}.$

Then $\alpha$ is $(\omega,m)$-positive at $P$ if and only if the eigenvalues
$\lambda(g^{-1}\alpha)=(\lambda_{1},...,\lambda_{n})$ of the matrix
$\alpha_{j\bar{k}}(P)$ with respect to the matrix $g_{j\bar{k}}(P)$ is in
$\Gamma_{m}$. These eigenvalues are independent of any choice of local
coordinates.

We can show easily the following result:

######  Proposition 2.8.

Let $\alpha\in\Lambda^{1,1}(X)$ be a real (1,1)-form on $X.$ Then $\alpha$ is
$(\omega,m)$-positive if and only if

$\alpha\wedge\beta_{1}\wedge...\wedge\beta_{m-1}\wedge\omega^{n-m}\geq 0,$

for all $(\omega,m)$-positive forms $\beta_{1},...,\beta_{m-1}.$

######  Definition 2.9.

A current $T$ of bidegree $(p,p)$ is said to be $(\omega,m)$-positive if

$\alpha_{1}\wedge...\wedge\alpha_{n-p}\wedge T\geq 0,$

for all smooth $(\omega,m)$-positive (1,1)-forms $\alpha_{i}.$

Following Blocki [Bl05] we can define $(\omega,m)$-subharmonicity for non-
smooth functions.

######  Definition 2.10.

A function $\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called
$(\omega,m)$-subharmonic if the following conditions hold
(i) In any local chart $\Omega,$ given $\rho$ the local potential of $\omega$
and set $u:=\rho+\varphi$, then $u$ is $\omega$-subharmonic,
(ii) for every smooth $(\omega,m)$-positive forms $\beta_{1},...,\beta_{m-1}$
we have, in the weak sense of distributions,

$(\omega+dd^{c}\varphi)\wedge\beta_{1}\wedge...\wedge\beta_{m-1}\wedge\omega^{n-m}\geq
0.$

Let $SH_{m}(X,\omega)$ be the set of all $(\omega,m)$-subharmonic functions on
$X.$ Observe that, by definition, any $\varphi\in SH_{m}(X,\omega)$ is upper
semicontinuous.

The following properties of $(\omega,m)$-subharmonic functions are easy to
show.

######  Proposition 2.11.

(i) A $\mathcal{C}^{2}$ function $\varphi$ is $(\omega,m)$-subharmonic if and
only if the form $(\omega+dd^{c}\varphi)$ is $(\omega,m)$-positive or
equivalently

$(\omega+dd^{c}\varphi)\wedge(\omega+dd^{c}u_{1})\wedge...\wedge(\omega+dd^{c}u_{m-1})\wedge\omega^{n-m}\geq
0,$

for all $\mathcal{C}^{2}$ $(\omega,m)$-subharmonic functions
$u_{1},...,u_{m-1}.$

(ii) If $\varphi,\psi\in SH_{m}(X,\omega)$ then $\max(\varphi,\psi)\in
SH_{m}(X,\omega).$

(ii) If $\varphi,\psi\in SH_{m}(X,\omega)$ and $\lambda\in[0,1]$ then
$\lambda\varphi+(1-\lambda)\psi\in SH_{m}(X,\omega).$

(iii) If $(\varphi_{j})\subset SH_{m}(X,\omega)$ is uniformly bounded from
above then $(\limsup_{j}\varphi_{j})^{\star}\in SH_{m}(X,\omega).$

Thanks to Hartogs Lemma 2.5, the following proposition can be proved in the
same way as in the case of $\omega$-plurisubharmonic function (see [GZ05]).

######  Proposition 2.12.

Let $(\varphi_{j})$ be a sequence of functions in $SH_{m}(X,\omega)$.

(i) If $(\varphi_{j})$ is uniformly bounded from above on $X$, then either
$\varphi_{j}$ converges uniformly to $-\infty$ or the sequence $(\varphi_{j})$
is relatively compact in $L^{1}(X).$

(ii) If $\varphi_{j}\rightarrow\varphi$ in $L^{1}(X)$ then

$sup_{X}\varphi=\lim_{j}sup_{X}\varphi_{j}.$

The compactness result can be deduced easily from Proposition 2.12.

######  Lemma 2.13.

There exists a constant $C_{0}>0$ such that for all $\varphi\in
SH_{m}(X,\omega)$ satisfying $\sup_{X}\varphi=0$ we have

$\int_{X}\varphi\omega^{n}\geq-C_{0}.$

It then follows that

$\mathcal{C}:=\\{\varphi\in SH_{m}(X,\omega)\ \mid\ \sup_{X}\varphi\leq
0;\int_{X}\varphi\omega^{n}\geq-C_{0}\\}$

is a convex compact subset of $L^{1}(X).$

###  2.4. Non degenerate complex Hessian equations

We summarize here some recent results on the non degenerate complex Hessian
equation on compact Kähler manifolds,

(2.1) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=f\omega^{n},$

where $0<f$ is smooth such that

(2.2) $\int_{X}f\omega^{n}=\int_{X}\omega^{n}.$

The following existence result was solved by Dinew and Kolodziej:

######  Theorem 2.14.

[DK12] If $(X,\omega)$ is a compact Kähler manifold and
$0<f\in\mathcal{C}^{\infty}(X)$ satisfies (2.2) then equation (2.1) has a
unique (up to an additive constant) smooth solution.

This result was known to hold when $(X,\omega)$ has non negative holomorphic
bisectional curvature [H09, Jb10].

The complex Hessian equation in domains of $\mathbb{C}^{n}$, i.e. equations of
the form

$(dd^{c}u)^{m}\wedge\beta^{n-m}=f\beta^{n},$

where $\beta$ is the canonical Kähler form in $\mathbb{C}^{n},$ was considered
by Li [Li04] and Blocki [Bl05]. Existence and uniqueness of smooth solution to
the Dirichlet problem in smoothly bounded domains with $(m-1)$\- pseudoconvex
boundary was obtained in [Li04]. In [Bl05], a potential theory for
$m$-subharmonic functions was developed and the corresponding degenerate
Dirichlet problem was solved. Recently, Sadullaev and Abdullaev studied
capacities and polar sets for $m$-subharmonic functions [SA12]. Note that the
corresponding problem when $\beta$ is not the euclidean Kähler form is fully
open.

It is important to mention that the study of real Hessian equations is a
classical subject which has been developed previously in many papers, for
example [CNS85, CW01, ITW04, Kr95, La02, Tr95, TW99, Ur01, W09].

##  3\. Complex Hessian operators.

One of the key points in pluripotential theory is the smooth approximation
which holds for quasi plurisubharmonic functions ([BK07], [De92]). Such a
result for $(\omega,m)$-subharmonic functions seems to be very difficult. To
overcome this difficulty we work in an (a priori) restrictive class which is
defined by means of uniform convergence with respect to capacity.

###  3.1. Capacity

######  Definition 3.1.

Let $E\in X$ be a Borel subset. We define the inner $(\omega,m)$-capacity of
$E$ by

$\text{cap}_{\omega,m}(E):=\sup\Big{\\{}\int_{E}\omega_{\varphi}^{m}\wedge\omega^{n-m}\
\mid\ \varphi\in SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X),0\leq\varphi\leq
1\Big{\\}}.$

The outer $(\omega,m)$-capacity of $E$ is defined to be

$\text{Cap}_{\omega,m}(E):=\inf\Big{\\{}\text{cap}_{\omega,m}(U)\ \mid\
E\subset U,\ \ U\ \ \text{is an open subset of X}\Big{\\}}.$

It follows directly from the definition that $\text{Cap}_{\omega,m}$ is
monotone and $\sigma$-sub-additive.
Observe that if $\varphi\in SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$,
$0\leq\varphi\leq M$ then, for any Borel subset $E\subset X,$

(3.1) $\int_{E}\omega_{\varphi}^{m}\wedge\omega^{n-m}\leq
M^{m}\text{cap}_{\omega,m}(E).$

######  Definition 3.2.

A sequence $(\varphi_{j})$ converges in $\text{cap}_{\omega,m}$ to $\varphi$
if for any $\delta>0$ we have

$\lim_{j\to\infty}\text{cap}_{\omega,m}(|\varphi_{j}-\varphi|>\delta)=0.$

######  Definition 3.3.

A sequence of functions $(\varphi_{j})$ converges quasi-uniformly to $\varphi$
on $X$ (w.r.t $\text{Cap}_{\omega,m}$) if for every $\epsilon>0$ there exists
an open subset $U\subset X$ such that $\text{Cap}_{\omega,m}(U)\leq\epsilon$
and $\varphi_{j}$ converges uniformly to $\varphi$ in $X\setminus U.$

This convergence is almost equivalent to the convergence in capacity as the
following result shows

######  Proposition 3.4.

(i) If $\varphi_{j}$ converges quasi-uniformly to $\varphi$, then for each
$\delta>0$,

$\lim_{j\to\infty}\text{Cap}_{\omega,m}(|\varphi_{j}-\varphi|>\delta)=0.$

(ii) Conversely, assume that $(\varphi_{j})$ is a sequence of functions and
$\varphi$ is a function such that, for every $\delta>0,$

$\lim_{j\to\infty}\text{Cap}_{\omega,m}(|\varphi_{j}-\varphi|>\delta)=0.$

Then there exists a subsequence $(\varphi_{j_{k}})$ converging quasi-uniformly
to $\varphi$.

###### Proof.

The first part is obvious, so we only prove the second part. We can find a
subsequence (and for convenience we still denote it by $(\varphi_{j})$) such
that

$\text{Cap}_{\omega,m}(|\varphi_{j}-\varphi|>1/j)\leq 2^{-j},\ \forall j.$

For each $j$, let $U_{j}$ be an open subset of $X$ such that
$(|\varphi_{j}-\varphi|>1/j)\subset U_{j}$ and
$\text{cap}_{\omega,m}(U_{j})\leq 2^{-j+1}.$ Then for each $\epsilon>0$, we
can find $k\in\mathbb{N}$ such that $\cup_{j\geq k}U_{j}$ is the open subset
of $\text{Cap}_{\omega,m}$ less than $\epsilon$ and $\varphi_{j}$ converges
uniformly to $\varphi$ on its complement. ∎

######  Definition 3.5.

We denote $\mathcal{P}_{m}(X,\omega)$ the set of all functions $\varphi\in
SH_{m}(X,\omega)$ such that there exists a sequence of $\mathcal{C}^{2}$,
$(\omega,m)$-subharmonic functions $(\varphi_{j})$ converging quasi-uniformly
to $\varphi$ on $X$. Equivalently, we can replace quasi-uniform convergence by
convergence in Capacity thanks to Proposition 3.4.

######  Proposition 3.6.

(i) Any $\varphi\in\mathcal{P}_{m}(X,\omega)$ is quasi continuous, that means,
for any $\epsilon>0$ there exists an open subset $U\subset X$ of
$\text{Cap}_{\omega,m}$ less than $\epsilon$ such that $\varphi$ is continuous
on $X\setminus U$.
(ii) If $\varphi_{j}\downarrow\varphi$ in $\mathcal{P}_{m}(X,\omega)$ then
$(\varphi_{j})$ converges quasi-uniformly to $\varphi.$

###### Proof.

The first statement follows directly from the definition. From (i) , for each
$\epsilon>0$, there exists an open subset $U$ of $\text{cap}_{\omega,m}$ less
than $\epsilon$ such that $\varphi_{j},\varphi$ are continuous on $X\setminus
U$ which is compact. By Dini’s Theorem, $\varphi_{j}$ converges uniformly to
$\varphi$ on $X\setminus U.$ ∎

We have obvious inclusions

$SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X)\subset\mathcal{P}_{m}(X,\omega)\subset
SH_{m}(X,\omega),$

and

$PSH(X,\omega)\subset\mathcal{P}_{m}(X,\omega).$

######  Remark 3.7.

Quasi-uniform convergence implies convergence point wise outside a subset of
$\text{Cap}_{\omega,m}$ zero. Moreover, if $\varphi_{j}$ is uniformly bounded
and converges quasi-uniformly to $\varphi$, then we have convergence in
$L^{p}$ for every $p>0$. Indeed, for any $\epsilon>0$ and an open subset $U$
as in definition 3.3, we have

$\displaystyle\int_{X}|\varphi_{j}-\varphi|^{p}\omega^{n}$ $\displaystyle\leq$
$\displaystyle\int_{X\setminus
U}|\varphi_{j}-\varphi|^{p}\omega^{n}+\int_{U}|\varphi_{j}-\varphi|^{p}\omega^{n}$
$\displaystyle\leq$ $\displaystyle\int_{X\setminus
U}|\varphi_{j}-\varphi|^{p}\omega^{n}+\sup_{X,j}|\varphi_{j}-\varphi|^{p}.\text{cap}_{\omega,m}(U)$
$\displaystyle\leq$ $\displaystyle\int_{X\setminus
U}|\varphi_{j}-\varphi|^{p}\omega^{n}+C\epsilon.$

Taking the limsup over $j$ and then letting $\epsilon\to 0$ we obtain

$\limsup_{j}\|\varphi_{j}-\varphi\|_{p}=0.$

######  Lemma 3.8.

If $\varphi,\psi$ belong to the class $\mathcal{P}_{m}(X,\omega)$ then so does
$\max(\varphi,\psi).$

###### Proof.

Let $(\varphi_{j}),(\psi_{j})$ be uniformly bounded sequences of functions in
$SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$ converging quasi-uniformly to
$\varphi,\psi$ respectively. Set

$u:=\max(\varphi,\psi);\ u_{j}:=\max(\varphi_{j},\psi_{j});\
v_{j}:=\frac{1}{j}\log(e^{j\varphi_{j}}+e^{j\psi_{j}}).$

For each $\epsilon>0$ there exists an open subset $U$ of
$\text{cap}_{\omega,m}$ less than $\epsilon$ and $\varphi_{j},\psi_{j}$
converges uniformly on $X\setminus U$ to $\varphi,\psi$ respectively. Since
$u_{j}\leq v_{j}\leq\log(2)/j+u_{j}$ and $u_{j}$ converges uniformly to $u$ on
$X\setminus U$ we deduce that $v_{j}$ converges uniformly to $u$ on
$X\setminus U.$ ∎

###  3.2. Hessian measure

In this section we define complex Hessian measure for functions in
$SH_{m}(X,\omega)$ which can be approximated in $\text{Cap}_{\omega,m}$ by
$\mathcal{C}^{2}$-functions in $SH_{m}(X,\omega)$. In particular, for
functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ this notion of
Hessian measure can be defined by Bedford-Taylor’s method.

######  Theorem 3.9.

Let $\varphi\in SH_{m}(X,\omega)$ such that there exists a uniformly bounded
sequence $(\varphi_{j})$ of $\mathcal{C}^{2}$ $(\omega,m)$-subharmonic
functions converging in $\text{cap}_{\omega,m}$ to $\varphi.$ Then the
sequence of measures

$H_{m}(\varphi_{j}):=(\omega+dd^{c}\varphi_{j})^{m}\wedge\omega^{n-m}$

converges (weakly in the sense of measures) to a positive Radon measure $\mu.$
Moreover, the measure $\mu$ does not depend on the choice of the approximating
sequence $(\varphi_{j}).$ We define the Hessian measure of $\varphi$ to be
$H_{m}(\varphi):=\mu.$

###### Proof.

Since all the measures $H_{m}(\varphi_{j})$ have uniformly bounded mass (which
is $\int_{X}\omega^{n}$), they stay in a weakly compact subset. It suffices to
show that all accumulation points of this sequence are just the same. To do
this it is enough to show that for every test function
$\chi\in\mathcal{C}^{\infty}(X),$

$\lim_{j,k\to\infty}\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]=0.$

By integration by part formula we have

$\displaystyle\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]$
$\displaystyle=$ $\displaystyle\int_{X}\chi
dd^{c}(\varphi_{j}-\varphi_{k})\wedge T$ $\displaystyle=$
$\displaystyle\int_{X}(\varphi_{j}-\varphi_{k})dd^{c}\chi\wedge T,$

where

$T=\sum_{l=0}^{m-1}(\omega+dd^{c}\varphi_{j})^{l}\wedge(\omega+dd^{c}\varphi_{k})^{m-1-l}\wedge\omega^{n-m}.$

Fix $\epsilon>0$, and set
$U=U(j,k,\epsilon)=\\{|\varphi_{j}-\varphi_{k}|\geq\epsilon\\}.$ By $C$ we
will denote a constant that does not depend on $j,k,\epsilon.$ Then by (3.2)
and (3.1) there exists $C>0$ such that

$\displaystyle\Big{|}\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]\Big{|}$
$\displaystyle\leq$
$\displaystyle\int_{U}|\varphi_{j}-\varphi_{k}|C\omega\wedge
T+\int_{X\setminus U}|\varphi_{j}-\varphi_{k}|C\omega\wedge T$
$\displaystyle\leq$ $\displaystyle
C\text{cap}_{\omega,m}(U)+C\epsilon\sup_{X\setminus
U}|\varphi_{j}-\varphi_{k}|\int_{X\setminus U}\omega\wedge T.$

Now, it follows that

$\limsup_{j,k\to\infty}\Big{|}\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]\Big{|}\leq
C\epsilon.$

The result follows by letting $\epsilon\downarrow 0.$ For the independence in
the choice of the sequence it is enough to repeat the above arguments. ∎

######  Lemma 3.10.

Let $U\subset X$ be an open subset and $\varphi$ be a bounded function in
$\mathcal{P}_{m}(X,\omega).$ Then

$\int_{U}H_{m}(\varphi)\leq 2(\sup_{X}|\varphi|+1)\text{Cap}_{\omega,m}(U).$

###### Proof.

Let $\varphi_{j}$ be a sequence of $\mathcal{C}^{2}$ functions in
$SH_{m}(X,\omega)$ converging quasi uniformly to $\varphi.$ We can assume that

$-\sup_{X}|\varphi|-1\leq\varphi_{j}\leq\sup_{X}|\varphi|+1,\ \forall j.$

Then

$\int_{U}H_{m}(\varphi)\leq\liminf_{j\to+\infty}\int_{U}H_{m}(\varphi_{j})\leq
2(\sup_{X}|\varphi|+1)\text{Cap}_{\omega,m}(U).$

∎

For functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ we can also
define the Hessian measure in a weak sense following Bedford-Taylor method.

######  Lemma 3.11.

Let $\varphi_{1},\varphi_{2}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$.
Then the current
$\omega_{\varphi_{1}}\wedge\omega_{\varphi_{2}}\wedge\omega^{n-m}$ is well
defined in the weak sense (Bedford-Taylor), symmetric and
$(\omega,m)$-positive. Then we can define inductively the Hessian measure of
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$,

$H_{m}(\varphi):=(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}.$

Moreover, this definition coincides with the one in Theorem 3.9.

###### Proof.

It follows from definition of $(\omega,m)$-subharmonic functions that
$T_{1}=(\omega+dd^{c}\varphi_{1})\wedge\omega^{n-m}$ is a
$(\omega,m)$-positive current. If $\varphi_{2}\in\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$ then $dd^{c}\varphi_{2}\wedge T_{1}$ is the current defined by

$dd^{c}\varphi_{2}\wedge T_{1}=dd^{c}(\varphi_{2}T_{1}).$

We denote by
$T_{2}=\omega_{\varphi_{1}}\wedge\omega_{\varphi_{2}}\wedge\omega^{n-m}.$
Since $\varphi_{1},\varphi_{2}$ are in $\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$, there exist uniformly bounded sequences
$(\varphi_{1}^{j}),(\varphi_{2}^{j})$ in
$\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$ converging quasi-uniformly
to $\varphi_{1},\varphi_{2}$ respectively. The sequence of currents
$T_{2}^{j}=\omega_{\varphi_{1}^{j}}\wedge\omega_{\varphi_{2}^{j}}\wedge\omega^{n-m}$
converges to $T_{2}$ and hence $T_{2}$ is $(\omega,m)$\- positive and

$\omega_{\varphi_{1}}\wedge\omega_{\varphi_{2}}\wedge\omega^{n-m}=\omega_{\varphi_{2}}\wedge\omega_{\varphi_{1}}\wedge\omega^{n-m}.$

To prove that $T_{2}^{j}$ converges to $T_{2}$, let us choose some test form
$\chi$ and prove the following convergence

(3.3) $\lim_{j\to\infty}\int_{X}\chi\wedge
dd^{c}(\varphi_{2}^{j}-\varphi_{2})\wedge T_{1}=0.$

We have

$\displaystyle\Big{|}\int_{X}\chi\wedge
dd^{c}(\varphi_{2}^{j}-\varphi_{2})\wedge T_{1}\Big{|}$ $\displaystyle=$
$\displaystyle\Big{|}\int_{X}(\varphi_{2}^{j}-\varphi_{2})dd^{c}\chi\wedge
T_{1}\Big{|}$ $\displaystyle\leq$ $\displaystyle
C\int_{X}|\varphi_{2}^{j}-\varphi_{2}|\omega_{\varphi_{1}}\wedge\omega^{n-1},$

where the constant $C$ depends only on $\chi,\omega.$ Now (3.3) follows from
the last inequality in view of

$\int_{U}\omega_{\varphi_{1}}\wedge\omega^{n-1}\leq
C\text{cap}_{\omega,m}(U),$

for every open subset $U\subset X.$ ∎

We can prove inductively that the current

$T_{k}=\omega_{\varphi_{1}}\wedge...\wedge\omega_{\varphi_{k}}\wedge\omega^{n-m}$

is well-defined, symmetric, $(\omega,m)$-positive, for each $k\leq m$ and
$\varphi_{i}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$. The Hessian
measure of $\varphi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ is defined
in this way

$H_{m}(\varphi)=\omega_{\varphi}\wedge...\wedge\omega_{\varphi}\wedge\omega^{n-m}.$

Now, given $\varphi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$, it is easy
to see that the Hessian measure of $\varphi$ defined by the above construction
coincides with the Hessian measure $H_{m}(\varphi)$ defined in Theorem 3.9.

###  3.3. Some Convergence results

In this section we state some convergence results and the comparison principle
for functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$. The proofs are
nearly the same as for the Monge-Ampère operator and hence are omitted.

######  Proposition 3.12.

Let $(\varphi^{1}_{j}),...,(\varphi^{m}_{j})$ be uniformly bounded sequence of
functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ converging quasi-
uniformly to $\varphi^{1},...,\varphi^{m}$ respectively. Assume that $(f_{j})$
is a uniformly bounded sequence of quasi continuous functions converging quasi
uniformly to $\varphi.$ Then we have the weak convergence of measures

$f_{j}\omega_{\varphi_{j}^{1}}\wedge...\wedge\omega_{\varphi_{j}^{m}}\wedge\omega^{n-m}\rightharpoonup
f\omega_{\varphi^{1}}\wedge...\wedge\omega_{\varphi^{m}}\wedge\omega^{n-m}.$

###### Proof.

Thanks to Lemma 3.10 we can follow the lines in [Kol05]. ∎

The integration by parts formula is valid for $\mathcal{C}^{2}$ functions (by
Stokes). By Proposition 3.12 we see that it is also valid for functions in
$\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X).$

######  Theorem 3.13 (Integration by parts).

Let $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ and $T$ be a
current of the form

$T=\omega_{\varphi_{1}}\wedge...\wedge\omega_{\varphi_{m-1}}\wedge\omega^{n-m},$

with $\varphi_{i}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X).$ Then

$\int_{X}\varphi dd^{c}\psi\wedge T=\int_{X}\psi dd^{c}\varphi\wedge T.$

The maximum principle for functions in $\mathcal{P}_{m}(X,\omega)$ can be
proved by the same way as in the classical case.

######  Theorem 3.14 (Maximum principle).

If $\varphi,\psi$ be two functions in $\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$ then

$1{\hskip-2.5pt}\hbox{{I}}_{\\{\varphi>\psi\\}}H_{m}(\max(\varphi,\psi))=1{\hskip-2.5pt}\hbox{{I}}_{\\{\varphi>\psi\\}}H_{m}(\varphi).$

From Theorem 3.14 we easily get

######  Corollary 3.15 (Comparison principle).

If $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ then

$\int_{(\varphi>\psi)}H_{m}(\varphi)\leq\int_{(\varphi>\psi)}H_{m}(\psi).$

######  Lemma 3.16.

Let $\varphi,\psi$ be two non positive functions in
$\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X).$ If $s>0$ and $0<t<1$ then we
have

(3.4)
$t^{m}\text{Cap}_{\omega,m}(\varphi-\psi<-t-s)\leq(1+M)^{m}\int_{(\varphi-\psi<-s)}H_{m}(\varphi),$

where $M=\|\psi\|_{L^{\infty}(X)}.$

###### Proof.

We can assume that $\psi$ is continuous on $X$. For the general case we can
approximate $\psi$ quasi-uniformly by sequence of $\mathcal{C}^{2}$ functions
in $SH_{m}(X,\omega)$. In (3.4) We can replace $\text{Cap}_{\omega,m}$ by
$\text{cap}_{\omega,m}$ since they coincide on open sets. Now, it suffices to
repeat the arguments in [EGZ09].

∎

######  Proposition 3.17 (Chern-Levine-Nirenberg inequality).

Let $T$ be any current of the form
$T=\omega_{u_{1}}\wedge...\wedge\omega_{u_{m-1}}\wedge\omega^{n-m}$ with
$u_{1},...,u_{m-1}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X),$ and
$\varphi,\psi$ be two functions in $\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$. Then

(3.5) $\int_{X}|\psi|\omega_{\varphi}\wedge
T\leq\int_{X}|\psi|T\wedge\omega+\Big{(}2|\sup_{X}\psi|+\sup_{X}\varphi-\inf_{X}\varphi\Big{)}\int_{X}\omega^{n}.$

###### Proof.

The proof is nearly the same as in [GZ05] and is omitted. ∎

Applying (3.5) for $T_{i}=\omega_{\varphi}^{i}\wedge\omega^{n-m+i}$ for
$i=m-1,...,0$ we obtain

######  Corollary 3.18.

Let $\varphi,\psi$ be two functions in $\mathcal{P}_{m}(X,\omega)$ such that
$0\leq\varphi\leq 1$. Then

$\int_{X}|\psi|H_{m}(\varphi)\leq\int_{X}|\psi|\omega^{n}+m\Big{(}2|\sup_{X}\psi|+1\Big{)}\int_{X}\omega^{n}.$

Applying Corollary 3.18 we obtain:

######  Corollary 3.19.

There exists a constant $C>0$ such that for all
$\psi\in\mathcal{P}_{m}(X,\omega)$ satisfying $\sup_{X}\psi=-1$ and for every
$t>0$ we have

$\text{Cap}_{\omega,m}(\psi<-t)\leq C/t.$

We end this section by showing that the class $\mathcal{P}_{m}(X,\omega)$ is
stable under decreasing sequences.

######  Proposition 3.20.

Let $(\varphi_{j})$ be a decreasing sequence of functions in
$\mathcal{P}_{m}(X,\omega)$ converging to $\varphi\not\equiv-\infty.$ Then
$\varphi_{j}$ converges to $\varphi$ quasi-uniformly. In particular,
$\varphi\in\mathcal{P}_{m}(X,\omega).$

###### Proof.

It is easy to see that $\varphi$ is $(\omega,m)$-subharmonic. It suffices to
show that there exists a subsequence of $(\varphi_{j})$ converging quasi-
uniformly to $\varphi.$
In view of Corollary 3.19 we can assume that $\varphi$ is bounded.
Fix $k\in\mathbb{N}.$ For each $j>k\in\mathbb{N},$ by applying Lemma 3.16 with
$\varphi=\varphi_{j},\psi=\varphi_{k},s=t$ we obtain

$\displaystyle t^{m}\text{Cap}_{\omega,m}(\varphi_{j}-\varphi_{k}<-2t)$
$\displaystyle\leq$
$\displaystyle(1+M)^{m}\int_{(\varphi_{j}-\varphi_{k}<-t)}H_{m}(\varphi_{j})$
$\displaystyle\leq$
$\displaystyle\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi_{j})H_{m}(\varphi_{j}).$

After extracting a subsequence if necessary we can assume that
$H_{m}(\varphi_{j})\rightharpoonup\mu$ in the weak sense of measures.We apply
Lemma 3.21 below to get

$\lim_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})=\int_{X}\varphi
d\mu.$

From the quasi-continuity of the functions $\varphi_{j},j\in\mathbb{N}$ and
the $\sigma$-subadditivity of $\text{Cap}_{\omega,m}$ we deduce that for each
fixed $\epsilon>0$ there exists an open subset $U$ such that
$\text{Cap}_{\omega,m}(U)<\epsilon$ and there exists a subsequence
$(\tilde{\varphi}_{j})$ of continuous functions on $X$ such that for any $j,$
$\varphi_{j}=\tilde{\varphi}_{j}$ on $X\setminus U.$
From basic properties of $\text{Cap}_{\omega,m}$ we have

$\displaystyle
t^{m}\text{Cap}_{\omega,m}(\varphi_{j}-\tilde{\varphi}_{k}<-2t)$
$\displaystyle\leq$ $\displaystyle
t^{m}\text{Cap}_{\omega,m}(\varphi_{j}-\varphi_{k}<-2t)+t^{m}\epsilon$
$\displaystyle\leq$
$\displaystyle\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi_{j})H_{m}(\varphi_{j})+t^{m}\epsilon.$

Recall that $\text{cap}_{\omega,m}$ is continuous under increasing sequence.
Note also that $\text{Cap}_{\omega,m}$ and $\text{cap}_{\omega,m}$ coincide on
open sets. By taking the limit when $j\to+\infty$ in (3.3), we obtain

$t^{m}\text{Cap}_{\omega,m}(\varphi-\tilde{\varphi}_{k}<-2t)\leq\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi)d\mu+t^{m}\epsilon.$

It follows that

$t^{m}\text{Cap}_{\omega,m}(\varphi-\varphi_{k}<-2t)\leq\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi)d\mu+2t^{m}\epsilon,$

and hence,

$\lim_{k\to+\infty}\text{Cap}_{\omega,m}(\varphi-\varphi_{k}<-2t)=0.$

Now, by Proposition 3.4 there exists a subsequence of $(\varphi_{j})$
converging quasi-uniformly to $\varphi$. To complete the proof it remains to
prove the following lemma. ∎

######  Lemma 3.21.

Assume that $(\varphi_{j})$ is a sequence in $\mathcal{P}_{m}(X,\omega)$
decreasing to $\varphi\in L^{\infty}(X).$ If $H_{m}(\varphi_{j})$ converges
weakly to $\mu$ in the sense of measures then

$\lim_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})=\int_{X}\varphi
d\mu.$

###### Proof.

We prove this lemma by induction. It obviously holds when $m=1.$ Remark also
that

$\limsup_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})\leq\int_{X}\varphi
d\mu.$

Thus, it suffices to prove that

$\liminf_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})\geq\int_{X}\varphi
d\mu.$

Fix $k\in\mathbb{N}$. For each $j>k$, By integration by parts we get

$\displaystyle\int_{X}\varphi_{k}[H_{m}(\varphi_{k})-H_{m}(\varphi_{j})]$
$\displaystyle=$
$\displaystyle\int_{X}\varphi_{k}dd^{c}(\varphi_{k}-\varphi_{j})\wedge
T\wedge\omega^{n-m}$ $\displaystyle=$
$\displaystyle\int_{X}(\varphi_{k}-\varphi_{j})dd^{c}\varphi_{k}\wedge
T\wedge\omega^{n-m}$ $\displaystyle\geq$
$\displaystyle-\int_{X}(\varphi_{k}-\varphi_{j})T\wedge\omega^{n-m+1},$

where

$T=\sum_{p=0}^{m-1}(\omega+dd^{c}\varphi_{k})^{p}\wedge(\omega+dd^{c}\varphi_{j})^{m-1-p}.$

By setting $\psi_{j}=\frac{\varphi_{j}+\varphi_{k}}{2},$ we get

$T\wedge\omega^{n-m+1}\leq
2^{m-1}(\omega+dd^{c}\psi_{j})^{m-1}\wedge\omega^{n-m+1}.$

As a consequence, (3.3) yields

(3.9)
$\displaystyle\int_{X}\varphi_{k}[H_{m}(\varphi_{k})-H_{m}(\varphi_{j})]$
$\displaystyle\geq$
$\displaystyle-2^{m}\int_{X}(\varphi_{k}-\psi_{j})H_{m-1}(\psi_{j})$

After extracting a subsequence if necessary, we can assume that
$H_{m-1}(\psi_{j})\rightharpoonup\nu$ in the weak sense of measures. By
letting $j\to+\infty$ in (3.9), the induction hypothesis gives us

$\int_{X}\varphi_{k}[H_{m}(\varphi_{k})-\mu]\geq-2^{m-1}\int_{X}(\varphi_{k}-\varphi)d\nu.$

We then infer that

$\liminf_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})\geq\int_{X}\varphi
d\mu,$

and the result follows. ∎

##  4\. Stability results

In this section we use the volume-capacity estimate in [DK11] and mimic the
arguments in [EGZ09] to prove stability results for the complex Hessian
equation.

Using Blocki’s technique [Bl03] we obtain the following stability results.

######  Theorem 4.1.

Let $\varphi,\psi\in SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X,\omega)$, $r\geq
2$, and set $\rho=\varphi-\psi$. Then

$\int_{X}|\rho|^{r-2}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}\leq
C\Big{(}\int_{X}|\rho|^{r-2}\rho(H_{m}(\psi)-H_{m}(\varphi))\Big{)}^{2^{1-m}},$

where $C$ is a positive constant depending only on $n,m,r$, and upper bounds
of $\|\varphi\|_{L^{\infty}(X)}$, $\|\psi\|_{L^{\infty}(X)}$, and
$\int_{X}\omega^{n}$.

From Theorem 4.1 and Corollary 3.2 we thus get

######  Corollary 4.2.

Let $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$, and set
$\rho=\varphi-\psi$. Then

$\int_{X}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}\leq
C\Big{(}\int_{X}\rho(H_{m}(\psi)-H_{m}(\varphi))\Big{)}^{2^{1-m}},$

where $C$ is a positive constant depending only on $n,m$, and upper bounds of
$\|\varphi\|_{L^{\infty}(X)}$, $\|\psi\|_{L^{\infty}(X)}$, and
$\int_{X}\omega^{n}$.

Corollary 4.2 is useful to prove uniqueness results as we will see in the
proof of Theorem A.

######  Definition 4.3.

Let $\alpha>0,A>0.$ A Borel measure $\mu$ on $X$ satisfies condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$ if for all Borel subsets $K$ of $X,$

$\mu(K)\leq A\text{Cap}_{\omega,m}(K)^{1+\alpha}.$

######  Proposition 4.4.

Let $\mu$ be a Borel measure satisfying condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$. Suppose that
$\varphi\in\mathcal{P}_{m}(X,\omega)$ solves $H_{m}(\varphi)=\mu$, and
$\sup_{X}\varphi=-1.$ Then there exists a constant $C=C(\alpha,A,\omega,n,m)$
such that

$\sup_{X}|\varphi|\leq C.$

Sketch of proof. Set

$f(s):=[\text{Cap}_{\omega,m}(\varphi<-s)]^{1/m}.$

Observe that $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is right continuous,
decreasing with $\lim_{+\infty}f=0.$ Since $\mu$ satisfies condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$, it follows from Lemma 3.16 applied to the
function $\psi\equiv 0$ that $f$ satisfies the condition in Lemma 2.4 in
[EGZ09]. Moreover it follows from Corollary 3.19 that

$f(s)\leq C_{1}s^{-1/m},$

for some constant $C_{1}$ which only depends on $\omega.$ Thus, by following
the lines in [EGZ09], page 615, we have the desired uniform estimate.

######  Theorem 4.5.

Suppose that $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$
satisfy

$\sup_{X}\varphi=\sup_{X}\psi=-1.$

Assume that $H_{m}(\varphi)$, $H_{m}(\psi)$ satisfy condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$ for some $\alpha,A>0.$ Then there exists
$C=C(\alpha,A,\omega,\|\varphi\|_{L^{\infty}(X)},\|\psi\|_{L^{\infty}(X)})>0$
such that, for any $\epsilon>0,$

$\sup_{X}(\psi-\varphi)\leq\epsilon+C[\text{Cap}_{\omega,m}(\varphi-\psi<-\epsilon)]^{\alpha/m}.$

###### Proof.

The same as in [EGZ09], Proposition 2.6. ∎

The following Proposition is due to Kolodziej and Dinew [DK11, Propsition
2.1]. We include here a slightly different proof.

######  Proposition 4.6.

[DK11] Let $1<p<\frac{n}{n-m}.$ There exists a constant $C=C(p,\omega)$ such
that for every Borel subset $K$ of $X$, we have

$V(K)\leq C\text{Cap}_{\omega,m}(K)^{p},$

where $V(K):=\int_{K}\omega^{n}.$

###### Proof.

Fix an open subset $U$ such that $K\subset U.$ Solve the complex Monge-Ampère
equation to find $u\geq 0$ such that $\omega_{u}^{n}=f\omega^{n}$ on $X$, with
$f=V(U)^{-1}\chi_{U}.$ From [BGZ08], Corollary 3.2, the solution $u$ is
continuous and moreover, for each $r>1$,

$\sup_{X}u\leq C\|f\|_{r}^{1/n},$

where the constant $C=C(r,\omega)$ does not depend on $K.$ The inequality
between mixed complex Monge-Ampère measures [Di09] tells us that

$\omega_{u}^{m}\wedge\omega^{n-m}\geq f^{m/n}\omega^{n}.$

Thus since $u\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$, we obtain

$\text{Cap}_{\omega,m}(U)\geq(\sup_{X}u)^{-m}\int_{U}H_{m}(u)\geq(\sup_{X}u)^{-m}\int_{U}f^{m/n}\omega^{n}\geq
C^{-m}V(U)^{1-\frac{m}{rn}}.$

Thus, for every $r>1$, there exists a constant $C$ not depending on $K$ such
that $V(K)\leq C.\text{Cap}_{\omega,m}(K)^{\frac{nr}{nr-m}}$. The proof is
complete. ∎

As a consequence of Proposition 4.6 we have some examples of measures
satisfying condition $\mathcal{Q}_{m}(\alpha,A,\omega).$

######  Lemma 4.7.

Assume $\mu=f\omega^{n}$ is a Borel measure with $0\leq f\in L^{p}(X)$ for
some $p>n/m$. Then for any $0<\alpha<\frac{mp-n}{(n-m)p},$ there exists
$A_{\alpha}>0$ such that $\mu$ satisfies
$\mathcal{Q}_{m}(\alpha,A_{\alpha},\omega).$

The following stability theorem was established in [EGZ09] for the Monge-
Ampère equation.

######  Theorem 4.8.

Assume $H_{m}(\varphi)=f\omega^{n},\ H_{m}(\psi)=g\omega^{n},$ where
$\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ and $f,g\in
L^{p}(X)$ with $p>n/m.$ Fix $r>0$. Then if $\gamma$ small enough such that
$\frac{\gamma mq}{r-\gamma(r+mq)}<\frac{mp-n}{(n-m)p}$, we have

$\|\varphi-\psi\|_{L^{\infty}(X)}\leq C\|\varphi-\psi\|_{L^{r}(X)}^{\gamma},$

where $q=\frac{p}{p-1}$ denotes the conjugate exponent of $p,$ and the
constant $C$ depends only on $n,m,p,r$ and upper bounds of $\|f\|_{p},\
\|g\|_{p}$.

###### Proof.

Fix $\epsilon>0,$ and $\alpha>0$ to be chosen later. It follows from Theorem
4.5 and Proposition 4.4 that

$\|\varphi-\psi\|_{L^{\infty}(X)}\leq\epsilon+C_{1}[\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)]^{\alpha/m}.$

Applying Lemma 3.16 we see that

$\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)\leq\frac{C_{2}}{\epsilon^{m+r/q}}\int_{X}|\varphi-\psi|^{r/q}(f+g)\omega^{n}.$

It follows thus from Hölder’s inequality that

$\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)\leq\frac{C_{3}\|f+g\|_{p}}{\epsilon^{m+r/q}}\|\varphi-\psi\|_{L^{r}}^{r/q}.$

Choose $\epsilon:=\|\varphi-\psi\|_{L^{r}}^{\gamma}.$ Then

$\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)\leq
C_{4}[\|\varphi-\psi\|_{L^{r}}]^{r/q-\gamma(m+r/q)}.$

We infer that

$\|\varphi-\psi\|_{L^{\infty}(X)}\leq\|\varphi-\psi\|_{L^{r}(X)}^{\gamma}+C_{5}\|\varphi-\psi\|_{L^{r}(X)}^{\gamma^{\prime}},$

where $\gamma^{\prime}=\frac{\alpha}{m}[r/q-\gamma(m+r/q)].$ We finally choose
$\alpha$ so that $\gamma=\gamma^{\prime}$: this yields the desired estimate. ∎

##  5\. Proof of the main results

###  5.1. Proof of Theorem A

We first prove the uniqueness. Suppose that $\varphi$ and $\psi$ are two
continuous solutions of (1.1). Set $\rho:=\varphi-\psi.$ It follows from
Corollary 4.2 that

$\int_{X}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}\leq
C.\Big{(}\int_{X}\rho(H_{m}(\psi)-H_{m}(\varphi))\Big{)}^{2^{1-m}},$

where $C$ is a positive constant. Since $F$ is non decreasing in the second
variable, it follows from Stokes formula that

$0\leq\int_{X}\rho(H_{m}(\psi)-H_{m}(\varphi))=\int_{X}(\varphi-\psi)(F(.,\psi)-F(.,\varphi))\omega^{n}\leq
0.$

Thus,

$\int_{X}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}=0,$

which implies that $\rho$ is constant. If moreover $t\mapsto F(x,t)$ is
increasing for every $x\in X$, it is easy to see that $\rho=0.$

To prove the existence, we consider three cases.

Case 1: $F$ does not depend on the second variable, $F(x,t)=f(x),\forall x,t$.
Take a sequence of smooth strictly positive functions $(f_{j})$ converging to
$f$ in $L^{p}(X).$ We can assume that
$\int_{X}f_{j}\omega^{n}=\int_{X}\omega^{n}$, for every $j$. We use the
existence result in Theorem 2.14 to produce a sequence of smooth solutions
$(\varphi_{j})$ normalized by $\sup_{X}\varphi_{j}=0,\forall j.$ By passing to
a subsequence we can assume that $(\varphi_{j})$ converges in $L^{1}(X).$
Since $\|f_{j}\|_{p}$ is uniformly bounded, by Lemma 4.7 we can find
$\alpha,A$ which do not depend on $j$ such that all the measures
$f_{j}\omega^{n}$ satisfy condition $\mathcal{Q}_{m}(\alpha,A,\omega).$ By
Proposition 4.4, the sequence $(\varphi_{j})$ is uniformly bounded. Now it
follows from Theorem 4.8 that $\varphi_{j}$ converges uniformly to a
continuous function $\varphi\in\mathcal{P}_{m}(X,\omega)$ which solves
equation $H_{m}(\varphi)=f\omega^{n}$.

In the next two cases we will use the Schauder fixed point Theorem.

Case 2: There exists $t_{1}\in\mathbb{R}$ such that
$\int_{X}F(x,t_{1})\omega^{n}>\int_{X}F(x,t_{0})\omega^{n}$.
We set

$\mathcal{C}:=\\{\varphi\in SH_{m}(X,\omega)\ \mid\
\int_{X}\varphi\omega^{n}\geq-C_{0};\ \sup_{X}\varphi\leq 0\\},$

where $C_{0}$ is the constant introduced in Lemma 2.13. It follows that
$\mathcal{C}$ is a compact convex subset of $L^{1}(X).$

Take $\psi\in\mathcal{C},$ we use the result in case 1 to find
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ such that
$\sup_{X}\varphi=0$ and

$H_{m}(\varphi)=F(.,\psi+c_{\psi})\omega^{n},$

where $c_{\psi}\geq t_{0}$ is a constant such that

(5.1) $\int_{X}F(.,\psi+c_{\psi})\omega^{n}=\int_{X}\omega^{n}.$

This can be done because $F$ satisfies conditions (F2) and (F3). Indeed, by
Fatou’s Lemma we have

$\liminf_{t\to+\infty}\int_{X}F(.,\psi+t)\omega^{n}\geq\int_{X}F(.,t_{1})\omega^{n}>\int_{X}\omega^{n}.$

Moreover
$\int_{X}F(.,\psi+t_{0})\omega^{n}\leq\int_{X}F(.,t_{0})=\int_{X}\omega^{n}$.
Thus by continuity of $t\mapsto\int_{X}F(.,\psi+t)\omega^{n}$ we can find
$c_{\psi}$ satisfying (5.1). Observe that $\varphi$ is well-defined and does
not depend on $c_{\psi}.$ Indeed, assume that $c_{1},c_{2}$ are two constants
such that

$\int_{X}F(.,\psi+c_{1})\omega^{n}=\int_{X}F(.,\psi+c_{2})\omega^{n}=\int_{X}\omega^{n},$

and $\varphi_{1},\varphi_{2}$ are two continuous functions in
$\mathcal{P}_{m}(X,\omega)$ such that

$H_{m}(\varphi_{1})=F(.,\psi+c_{1}),\ \ H_{m}(\varphi_{2})=F(.,\psi+c_{2}).$

Since $t\mapsto F(x,t)$ is non decreasing for every $x\in X,$ we have
$F(.,\psi+c_{1})=F(.,\psi+c_{2})$ almost every where on $X$. Thus by the
uniqueness result above, $\varphi_{1}=\varphi_{2}+c$ for some constant $c$
which must be $0$ by the normalization. Then we define the map
$\Phi:\mathcal{C}\rightarrow\mathcal{C},\ \psi\mapsto\varphi.$

Now we prove that $\Phi$ is continuous on $\mathcal{C}.$ Suppose that
$(\psi_{j})$ is a sequence in $\mathcal{C}$ converging to $\psi\in\mathcal{C}$
in $L^{1}(X)$ and let $\varphi_{j}=\Phi(\psi_{j})$. We set
$c_{j}:=c_{\psi_{j}}$ and prove that $(c_{j})$ is uniformly bounded. Suppose
in the contrary that $c_{j}\uparrow+\infty.$ By subtracting a subsequence if
necessary we can assume that $\psi_{j}\to\psi$ almost everywhere in $X$. Then
by Fatou’s lemma we have

$\int_{X}\omega^{n}=\lim_{j\to+\infty}\int_{X}F(.,\psi_{j}+c_{j})\omega^{n}\geq\int_{X}F(.,t_{1})\omega^{n},$

which is impossible. Therefore the sequence $(c_{j})$ is bounded. This implies
that the sequence $(F(.,\psi_{j}+c_{j}))_{j}$ is bounded in $L^{p}(X),$ for
some $p>n/m$ which does not depend on $j.$ To prove the continuity of $\Phi$
it suffices to show that any cluster point of $(\varphi_{j})$ satisfies
$\Phi(\psi)=\varphi.$ Suppose that $\varphi_{j}\to\varphi$ in $L^{1}(X)$. It
follows from Theorem 4.8 that the sequence $(\varphi_{j})$ is Cauchy in
$\mathcal{C}^{0}(X)$. Thus $(\varphi_{j})$ converges to $\varphi$ in
$\mathcal{C}^{0}(X)$ and
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$. By subtracting a
subsequence if necessary we can assume that $\psi_{j}\to\psi$ almost
everywhere on $X$ and $c_{j}\to c.$ Since $t\mapsto F(x,t)$ is continuous we
see that $F(.,\psi_{j}+c_{j})\to F(.,\psi+c)$ almost everywhere. Thus
$H_{m}(\varphi)=F(.,\psi+c)$ which means $\Phi(\psi)=\varphi$ and hence $\Phi$
is continuous on $\mathcal{C}.$

By the Schauder fixed point Theorem, it follows that $\Phi$ has a fixed point
in $\mathcal{C}$, say $\varphi$. By definition of $\Phi$, the function
$\varphi$ must be in the class
$\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ and we have

$H_{m}(\varphi)=F(.,\varphi+c_{\varphi})\omega^{n}.$

The function $\varphi+c_{\varphi}$ is the required solution.

Case 3: $\int_{X}F(.,t)\omega^{n}=\int_{X}F(.,t_{0})\omega^{n},\forall t\geq
t_{0}.$ In this case we have $F(x,t)=F(x,t_{0})$ for all $t\geq t_{0}$ and for
almost $x\in X.$ Thus, for every $t\geq t_{0},$

$\|F(.,t_{0})\|_{L^{p}(X)}=\|F(.,t)\|_{L^{p}(X)}.$

From Proposition 4.4 we can find a positive constant $C_{1}$ such that for any
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ satisfying
$\sup_{X}\varphi=0$ and

$H_{m}(\varphi)=f\omega^{n},$

with $\|f\|_{p}\leq\|F(.,t_{0})\|_{p}$ then

$\varphi\geq-C_{1}.$

We set

$\mathcal{C}^{\prime}:=\\{\varphi\in SH_{m}(X,\omega)\ \mid\
-C_{1}\leq\varphi\leq 0\\}.$

Then $\mathcal{C}^{\prime}$ is a compact convex subset of $L^{1}(X).$

Take $\psi\in\mathcal{C}^{\prime},$ we use the result in case 1 to find
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ such that
$\sup_{X}\varphi=0$ and

$H_{m}(\varphi)=F(.,\psi+c_{\psi})\omega^{n},$

where $t_{0}\leq c_{\psi}\leq t_{0}+C_{1}$ is a constant such that

$\int_{X}F(.,\psi+c_{\psi})\omega^{n}=\int_{X}\omega^{n}.$

This can be done because $F$ satisfies the condition (F2) and (F3) . Indeed,

$\int_{X}F(.,\psi+t_{0})\omega^{n}\leq\int_{X}\omega^{n}\leq\int_{X}F(.,\psi+t_{0}+C_{1})\omega^{n}.$

Thus by continuity we can find $c_{\psi}$ as above. As in case 2, $\varphi$ is
well-defined and does not depend on the choice of $c_{\psi}$. By the choice of
$C_{1}$, we see that $\varphi\in\mathcal{C}^{\prime}.$ So, we can define a map
$\Phi:\mathcal{C}^{\prime}\rightarrow\mathcal{C}^{\prime}$ by setting
$\Phi(\psi)=\varphi.$

Now we prove that $\Phi$ is continuous on $\mathcal{C}^{\prime}.$ Suppose that
$(\psi_{j})$ is a sequence in $\mathcal{C}^{\prime}$ converging to
$\psi\in\mathcal{C}^{\prime}$ in $L^{1}(X)$ and let
$\varphi_{j}=\Phi(\psi_{j})$. We set $c_{j}:=c_{\psi_{j}}$. For each
$j\in\mathbb{N}$,

$\int_{X}[F(.,\psi_{j}+c_{j})]^{p}\omega^{n}\leq\int_{X}[F(.,c_{j})]^{p}\omega^{n}=\int_{X}[F(.,t_{0})]^{p}\omega^{n}.$

Therefore, the sequence $(F(.,\psi_{j}+c_{j}))_{j}$ is bounded in $L^{p}(X).$
As in case 2, we can assume that $\varphi_{j}\to\varphi$ in $L^{1}(X)$. It
follows from Theorem 4.8 that the sequence $(\varphi_{j})$ is Cauchy in
$\mathcal{C}^{0}(X)$. Thus $(\varphi_{j})$ converges to $\varphi$ in
$\mathcal{C}^{0}(X)$ and
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$. By subtracting a
subsequence if necessary we can assume that $\psi_{j}\to\psi$ in $L^{1}(X)$
and $c_{j}\to c.$ Then $H_{m}(\varphi)=F(.,\psi+c)$ and $\Phi(\psi)=\varphi$
which implies that $\Psi$ is continuous on $\mathcal{C}^{\prime}.$

By the Schauder fixed point Theorem, it follows that $\Phi$ has a fixed point
in $\mathcal{C}^{\prime}$, say $\varphi$. By definition of $\Phi$, the
function $\varphi$ must be in the class
$\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ and we have

$H_{m}(\varphi)=F(.,\varphi+c_{\varphi})\omega^{n}.$

The function $\varphi+c_{\varphi}$ is the required solution.

###  5.2. Proof of Theorem B

In this section we consider a special class of compact Kähler manifolds. We
assume that $(X,\omega)$ is a rational homogeneous manifold. That means
$X=G/H$, where $G$ is a complex semi-simple algebraic group and $H$ is a
parabolic subgroup. Let $K$ be a maximal compact subgroup of $G.$ Then $K$
acts transitively on $X.$ We assume moreover that $\omega$ is fixed by action
of $K.$ In this case we can regularize singular $(\omega,m)$-subharmonic
functions by using the group action which preserves the metric.

Let $\varphi$ be a continuous $(\omega,m)$-subharmonic function on $X.$ We
consider the following regularizing sequence

$\varphi_{\epsilon}(x):=\int_{K}\varphi(g^{-1}.x)\chi_{\epsilon}(g)dg,$

where $dg$ is the Haar measure on $K$ and $\chi_{\epsilon}$ are cut-off
functions whose supports decreases to $\\{e\\}$ (the identity of $K$), and
$\int_{K}\chi_{\epsilon}(g)dg=1,\forall\epsilon>0.$
It follows from [G99], [Hu94] that $\varphi_{\epsilon}$ is smooth for every
$\epsilon>0.$

######  Theorem 5.1.

Let $\varphi$ be a continuous $(\omega,m)$-subharmonic function on $X.$ Then
for each $\epsilon>0$, $\varphi_{\epsilon}$ is smooth $(\omega,m)$-subharmonic
and

$\lim_{\epsilon\to 0}\varphi_{\epsilon}=\varphi$

uniformly on $X.$

###### Proof.

The uniform convergence always holds for continuous functions. Let us show the
second assertion. Let $\alpha_{1},...,\alpha_{m-1}$ be $(\omega,m)$-positive
closed (1,1)-forms on $X$, and denote (for short)
$\alpha=\alpha_{1}\wedge...\wedge\alpha_{m-1}.$ Let $\mathcal{L}_{g}$ denote
the left action of $g\in K$, i.e.

$\mathcal{L}_{g}(x)=g.x,\ \ x\in X.$

Then $\mathcal{L}^{*}_{g}\alpha_{j}$ is also $(\omega,m)$-positive for every
$j,$ since $\mathcal{L}^{*}_{g^{-1}}\omega=\omega,$ and

$\mathcal{L}^{*}_{g^{-1}}(\mathcal{L}^{*}_{g}\alpha^{k}\wedge\omega^{n-k})=\alpha^{k}\wedge\omega^{n-k}.$

Fix a positive test function $\psi.$ We have

$\displaystyle\int_{X}\psi(\omega+dd^{c}\varphi_{\epsilon})\wedge\alpha\wedge\omega^{n-m}=\int_{X}\psi\alpha\wedge\omega^{n-m+1}+\int_{X}\varphi_{\epsilon}dd^{c}\psi\wedge\alpha\wedge\omega^{n-m}$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{X}\psi\alpha\wedge\omega^{n-m+1}+\int_{X}\Big{(}\int_{K}\mathcal{L}^{*}_{g}\varphi\chi_{\epsilon}(g)dg\Big{)}dd^{c}\psi\wedge\alpha\wedge\omega^{n-m}$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{X}\psi\alpha\wedge\omega^{n-m+1}+\int_{K}\Big{(}\int_{X}\mathcal{L}^{*}_{g}\varphi
dd^{c}\psi\wedge\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{K}\Big{(}\int_{X}\psi(\omega+dd^{c}\mathcal{L}^{*}_{g}\varphi)\wedge\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{K}\Big{(}\int_{X}\psi(\omega+\mathcal{L}^{*}_{g}dd^{c}\varphi)\wedge\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{K}\Big{(}\int_{X}\psi\mathcal{L}^{*}_{g}\big{[}(\omega+dd^{c}\varphi)\wedge\mathcal{L}^{*}_{g^{-1}}\alpha\wedge\omega^{n-m}\big{]}\Big{)}\chi_{\epsilon}(g)dg\geq
0.$

∎

######  Remark 5.2.

Thanks to Theorem 5.1, every continuous $(\omega,m)$-subharmonic function
belongs to $\mathcal{P}_{m}(X,\omega).$

Proof of Theorem B. Let $\varphi$ be the unique continuous solution to (1.1).
For $h\in K$, let $\varphi_{h}(x):=\varphi(h.x),\ x\in X.$ If $u$ is smooth
then

$\|u_{h}-u\|^{2}_{L^{2}}\leq
Cdist^{2}(h,e)\int_{X}(-u)dd^{c}u\wedge\omega^{n-1},$

where $C$ is some universal constant. Then, it follows from the approximation
theorem (Theorem 5.1) that

$\|\varphi_{h}-\varphi\|_{L^{2}(X)}\leq Cdist(h,e).$

For fixed $h\in K$, observe that $\varphi_{h}$ is $(\omega,m)$-subharmonic and
satisfies

$H_{m}(\varphi_{h})=F(h.x,\varphi(h.x))\omega^{n}.$

Thus, by applying Theorem 4.8 with $r=2$ we obtain

$\|\varphi_{h}-\varphi\|_{L^{\infty}}\leq
C^{\prime}.\|\varphi_{h}-\varphi\|_{L^{2}(X)}^{\gamma},$

where $0<\gamma<\frac{2(mp-n)}{mnp+2mp-2n}$ is a given constant and
$C^{\prime}>0$ is another constant which does not depend on $h.$ We thus get

$\|\varphi_{h}-\varphi\|_{L^{\infty}(X)}\leq C.C^{\prime}dist(h,e)^{\gamma},\
\forall h\in K.$

This yields the $\gamma$-Hölder continuity of $\varphi$ (see [EGZ09]).

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Lu Hoang Chinh

Laboratoire Emile Picard

UMR 5580, Université Paul Sabatier

118 route de Narbonne

31062 TOULOUSE Cedex 04 (FRANCE)

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